Applications of Markov Chains in Chemical Engineering
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Transcript of Applications of Markov Chains in Chemical Engineering
PREFACE
Markov chains enable one to predict the future state of a system from its
present state ignoring its past history.
Surprisingly, despite the widespread use of Markov chains in many areas of
science and technology such as: Polymers, Biology, Physics, Astronomy,
Astrophysics, Chemistry, Operations Research, Economics, Communications,
Computer Networks etc., their applications in Chemical Engineering has been
relatively meager.
A possible reason for this phenomenon might be that books containing
material on this subject have been written in such a way that the simplicity of
Markov chains has been shadowed by the tedious mathematical derivations. This
caused one to abandon the book, thus loosing a potential tool for handling his
problems.
There are many advantages, detailed in Chapter 1, of using the discrete
Markov-chain model in Chemical Engineering. Probably, the most important
advantage is that physical models can be presented in a unified description via state
vector and a one-step transition probability matrix. Consequently, a process is
demonstrated solely by the probability of a system to occupy a state or not to
occupy it. William Shakespeare profoundly stated this in the following way: " to
be (in a state) or not to be (in a state), that is the question".
I believe that Markov chains have not yet acquired their appropriate status in
the Chemical Engineering textbooks although the method has proven very effective
and simple for solving complex processes. Thus, the major objective of writing
this book has been to try to change this situation. The book has been written in an
easy and understandable form where complex mathematical derivations are
abandoned. The demonstration of the fundamentals of Markov chains in Chapter 2
has been done with examples from the bible, art and real life problems. The
majority of the book contains an extremely wide collection of examples viz..
VI
reactions, reactors, reactions and reactors as well as combined processes, including
their solution and a graphical presentation of it. All this, to my opinion,
demonstrates the usefulness of applying Markov chains in Chemical Engineering.
Bearing all the above in mind, leads me also to suggest this book as a useful
textbook for a new course entitled Applications of Markov chains in Chemical
Engineering,
Abraham Tamir
Beer Sheva, Israel
May 1, 1998
ACKNOWLEDGMENTS
A few persons have contributed either directly or indirectly to this book; I
would like to mention them by name.
Professor Arie Dubi, a great teacher and scientist, deserves special thanks.
He was the one who skillfully polished my knowledge in Markov chains to such a
level which made it possible for me to write this book.
Mr. Moshe Golden, a personal friend and a talented progranuner, deserves
many thanks. He assisted me in all technical problems which developed in
producing the book in a camera ready copy form.
Professor T.Z.Fahidy, of Waterloo University, was extremely influential in
the creation of this book, in reviewing part of it; I deeply thank him.
To Ms. Stella Zak, an extremely talented artist, many thanks for helping
design the book cover.
The most significant impact, however, has been that of my graduate students
who participated in my course related to Markov chains. Their proclivity to ask
'why'? has forced me to rethink, recognize and rewrite many parts of the book
again and again. In particular, many thanks are due to my student Adi Wolfson,
who reviewed Chapter 2.
Also thanks are due to Ben Gurion University, which provided generous
assistance and a pleasant atmosphere in which to write this book.
Finally, since I have no co-authors, I must accept responsibility for all errors
in this book.
Abraham Tamir
Table of Contents
Ch. 0 Biblical Origins and Artistic Demonstrations of Markov Chains - a
Humorous Introduction 1
Ch. 1 Why Write this Book? 6
Ch. 2 Fundamentals of Markov Chains 11
2.1 Markov Chains Discrete in Time and Space 11
2.2 Markov Chains Discrete in Space and Continuous in Time 132
2.3 Markov Chains Continuous in Space and Time 170
2.4 Concluding Remarks 180
2.5 Artistic Ending of the Chapter 180
Ch. 3 Applications of Markov Chains in Chemical Reactions 186
3.1 Modeling the Probabilities in Chemical Reactions 187
3.2 Application and Verification of the Modeling 193
3.3 Major Conclusions and General Guidelines for Applying the
Modeling 204
3.4 Application of Kinetic Models to Artistic Paintings 204
3.5 Introduction to Modeling of Chemical Reactions 210
3.6 Single Step Irreversible Reaction 213
3.7 Single Step Reversible Reactions 219
3.8 Consecutive Irreversible Reactions 228
3.9 Consecutive Reversible Reactions 237
3.10 Parallel Reactions Single and Consecutive Irreversible Reaction
Steps 250
3.11 Parallel Reactions Single and Consecutive Reversible Reaction
Steps 287
3.12 Chain Reactions 301
3.13 Oscillating Reactions [55-69] 305
3.14 Non-Existing Reactions with a Beautiful Progression Route 323
Ch. 4 Applications of Markov Chains in Chemical Reactors 334
4.1 Modeling The Probabilities in Flow Systems 335
4.2 Application of the Modeling and General Guidelines 349
4.3 Perfectly Mixed Reactor Systems 353
4.4 Plug Flow-Perfectly Mixed Reactor Systems 406
4.5 Impinging-Stream Systems 462
Ch. 5 Applications of Markov Chains in Chemical Processes 498
5.1 Modeling of the Probabilities 498
5.2 Application of the Modeling and General Guidelines 521
Nomenclature 590
References 599
Chapter 0
BIBLICAL ORIGINS AND ARTISTIC DEMONSTRATIONS OF MARKOV CHAINS
A HUMOROUS INTRODUCTION
The origin of Markov chains, a probabihstic model for predicting the future
given atv^ present of a process and ignoring iXspast, goes back to biblical times, i.e.
to the Book of Books. This we know thanks to what has been said in Exodus 28,
verse 13-14, "Make gold rosettes and two CHAINS of pure gold worked into a
form of ropes, and fix them on the rosettes". A thorough investigation of this verse
led to the conclusion that the word CHAINS is an abbreviation of MARKOV
CHAINS. Thus, it turns out that Markov chains is a very old subject and, as said
in Ecclesiastes 1 verse 9,"... And there is nothing new under the sun".
It is also surprising that available books [2-8, 15-18] related to the subject
matter do not refer at all to biblical Markov processes. Such a process, for
example, can be generated on the basis of Genesis 1 and is related to the order of
the days of the week in the Creation. According to verse 27, man was created on
Friday. The Bible describes this event very nicely as follows: " And God created
man in His image, in the image of God he created him ... And there was evening
and there was morning, the sixth day." Independent of the past history, i.e.
Sunday to Friday, the probability that man will occupy a Saturday on the next day
is 100%. In other words, since the present state is known, namely, Friday, and the
probabihty of moving to the next state is also known, 100%, it is possible to predict
Saturday as the future state of man with respect to the days of the week. The above
example, elaborated later in example 2.11, demonstrates for the first time the
essence of Markov chains proposed by Markov only in 1906 [1], much later than
Biblical times.
An additional example of a Markov process is related to the states day and
night in Genesis 1 verse 4-5. The creation of these complicated states is described
simply as: " God saw that Ught was good, and God separated the light from the
darkness. God called the light day, and the darkness He called night. And there
was evening and there was morning, a first day." The occurrence of the state night
(or day^ depends only on the previous state unless something unexpected happens
in the universe.
The last Biblical example of a Markov process is concerned with the famous
trial of king Solomon [1 Kings 3]. The story develops as follows (verse 16-22):
"Then came there two women, that were harlots, unto the king, and stood before
him. And the one women said: "Oh, my lord, I and this woman dwell in one
house; and I was delivered of a child with her in the house. And it came to pass the
third day after I was delivered, that this woman was delivered also; and we were
together; there was no stranger with us in the house, save we two in the house.
And this woman's child died in the night; because she overlay it. And she arose at
midnight, and took my son from beside me, while thy handmaid slept, and laid her
dead child in my bosom. And when I arose in the morning to give my child suck,
behold, it was dead; but when I had looked well at it in the morning, behold, it was
not my son, whom I did bear". And the other woman said: "Nay; but the living is
my son, and the dead is thy son". And this said: "No; but the dead is thy son, and
the living is my son. Thus they spoke before the king."
King Solomon was faced with an extremely hard human problem of how to
find out to whom does the living child belong? In order to resolve the problem,
king Solomon made a wise decision described in verse 24-25 as: And the king said:
"Fetch me a sword." And they brought a sword before the king. And the king said:
"Divide the living child in two, and give half to the one, and half to the other." The
above example generates the following Markov process. There are two states here,
namely, that of a living child and that of a divided child. By his brave decision,
king Solomon fixed the probability of moving from the first state to the final one to
be 100%. Consequently, if his verdict would have been materialized, an ultimate
state known in Markov chains as dead state would have been reached. Fortunately,
according to verse 26-27 the woman the child belonged to said, for her heart
yearned upon her son: ... "Oh, my lord, give her the living child, and in no wise
slay it" while the other woman said: ... "It shall be neither mine nor thine; divide it." Then the king answered and said: "Give her the living child, and in no wise slay it: she is the mother thereof." In this way, the terrible result predicted by the Markov process was avoided.
The following example of ^ifrog in a lily pond was mentioned by Howard in 1960 in the opening of his book [3, p.3] as a graphic example of a Markov process.
Fig.0-1. Escher-Howard Markov process (M.C.Escher "Frog" © 1998 Cordon Art B.V. - Baarn - Holland. All rights reserved)
Surprisingly, M.C.Escher, the greatest graphic artist (1898-1972), probably unfamiliar with Markov processes, has already demonstrated in 1931 the same situation in his woodcut Frog [10, p.231]. This is reproduced in Fig.0-1. As time goes by, the frog, system, jumps from one lily pad, state, to another according to
his whim of the moment. The latter assures a true Markov process since the effect
of the past history is assumed to be negUgible. The state of the system is the
number of the pad currently occupied by the frog; the state transition is, of course,
his leap. If the number of lily pad is finite, then we have a finite-state process.
In 1955,Escher prepared the lithograph Convex and Concave [10, p.308]
which is reproduced in Fig.0-2. It is interpreted below as a Markov process and
elaborated in Chapter 2 in example 2.14.
Fig.0-2. Demonstration of Escher*s Markov process
("Convex and Concave", © 1955 M.C.Escher Foundation ® - Baam, the Netherlands. AH rights
reserved)
It is interesting, first of all, to explore some interesting phenomena in the
lithograph, which is a visual shock. The columns appearing in the picture can be
seen as either concave or convex. On the right-hand side, the solid floor underfoot
can become the ceiling overhead, and that one may once climb the selfsame
staircase safely, and after some time, while climbing, suddenly be falling down
because the stairs seem upside down. Similarly is the situation with the woman
with the basket walking down the stairs. The upper floor in the middle of the
lithograph, with the flute player, may be seen as convex or concave. Thus, when
it looks concave, the flute player, when climbing out of the window, stands safely
on the vaulting. However, if the appearance looks convex, and the flute player
does not pay attention, he might land far down when leaving the window. In
addition, the element on the floor above the two lizards, may be observed as shell-
shaped ceiling or shell-shaped basin. All above behaviors are phenomena related to
the cognition of vision by the brain.The reader can see in Fig.0-2 six locations,
designated 1 to 6, selected as possible states that a person, system, can occupy. In
principle, the possible occupations depend on the original location of the person,
namely, the initial state vector, and on the probabilities of moving from one state to
the other, i.e. the single-step probability matrix. The above concepts are elaborated
in Chapter 2.1-3. As demonstrated later, a person trying to walk along Escher's
Convex Concave structure, will end up walking up and down along the staircase
connecting states 2 and 3. This result solely depends on Eq.(2-42) which may be
looked upon as the policy-making matrix of the person. The matrix depends on his
mood, but for the sake of simplicity it has been assumed to remain unchanged.
Such a Markov process, i.e. ending walking endlessly between states 2 and 3, is
known as periodic chain. However, more interesting is the fact that the final
situation the person has been trapped in, is independent of the initial state. This is
known as without memory or ergodic process. In conclusion, the aforementioned
examples indicate that the origin of Markov chains goes back to very ancient days
and many wonderful examples can be found in the Book of Books to demonstrate
this process. In addition, some interesting relationships may also arise between the
subject matter and art, which are demonstrated in 2.1-3 of Chapter 2. However,
from Chapter 3 on, applications of Markov chains in Chemical Engineering are
demonstrated.
Chapter 1
WHY WRITE THIS BOOK?
Markov chains or processes are named after the Russian mathematician
A.A.Markov (1852-1922) who introduced the concept of chain dependence and did
basic pioneering work on this class of processes [1]. A Markov process is a
mathematical probabilistic model that is very useful in the study of complex
systems. The essence of the model is that if the initial state of a system is known,
i.e. its present state, and the probabilities to move forward to other states are also
given, then it is possible to predict the future state of the system ignoring its past
history. In other words, past history is immaterial for predicting the future; this is
the key-element in Markov chains. Distinction is made between Markov processes
discrete in time and space, processes discrete in space and continuous in time and
processes continuous in space and time. This book is mainly concerned with
processes discrete in time and space.
Surprisingly, despite the widespread use of Markov chains in many areas of
science and technology such as: Polymers, Biology, Physics, Astronomy,
Astrophysics, Chemistry, Operations Research, Economics, Communications,
Computer Networks etc., their applications in Chemical Engineering has been
relatively meager.
A possible reason for this phenomenon might be that books containing
material on this subject have been written in such a way that the simplicity of
discrete Markov chains has been shadowed by the tedious mathematical derivations.
This caused one to abandon the book, thus loosing a potential tool for handling his
problems. In a humorous way, this situation might be demonstrated as follows.
Suppose that a Chemical Engineer wishes to study Markov processes and has been
suggested several books on this subject. Since the mathematics is rather complex
or looks complicated, the probability of moving to the next book is decreasing and
diminishes towards the last books because the Chemical Engineer remembers
always the difficulties he has encountered in studying the previous books. In other
words, his long-range memory of the past has a significant and accumulative effect
on the probability of moving to the next book. However, had he known Markov
chains, he should have made efforts to forget the past or to remember only the
effect of the last book which might be better than the previous ones. In this way,
his chances of becoming famiUar with Markov chains would have been significantly
increased. M.C.Escher demonstrated the above situation very accurately in his
woodcut Still Life and Street [10, p.271], which is reproduced in Fig.1-1 .
Fig.1-1. Abandoned books on Markov chains according to Escher
(M.C.Escher "Still Life and Street" © 1998 Cordon Art B.V. - Baam - Holland. All rights
reserved)
8
The reader can observe on the right- and left-hand sides of the desk a total of
twelve books on Markov processes among which are, probably, refs.[2-8, 15-18,
84]. Some support to the fact that the books are abandoned is the prominent fact
that the inmiediate continuation of the desk is the street... and that the books are
leaning on the buildings.
There are many advantages of using the discrete Markov-chain model in
Chemical Engineering, as follows.
a) Physical models can be presented in a unified description via state vector
and a one-step transition probability matrix. Consequently, a process is
demonstrated solely by the probability of a system to occupy a state or not to
occupy it. William Shakespeare profoundly stated this in the following way: " to
be (in a state) or not to be (in a state), that is the question". It is shown later
that this presentation coincides with the finite difference equations of the process
obtained from the differential equations. In some cases the process is also of
probabilistic nature, such as a chemical reaction, where the Markov-chain model
presentation seems natural.
b) Markov-chain equations provide a solution to complicated problems. The
increase in the complexity of the problem increases the size of the one-step
transition probability matrix on the one hand, however, it barely increases the
difficulty in solving it, on the other.
c) In some cases, the governing equations of the process are non-linear
differential equations for which an analytical solution is extremely difficult or
impossible. In order to solve the equations, simplifications, e.g. linearization of
expressions and assumptions must be carried out. For these situations the discrete
Markov-chain model may be advantageous.
d) The application of an exact solution is sometimes more complicated in
comparison to Markov-chain finite difference equations. For example, an analytical
solution with one unknown where the equation has no explicit solution, or an
equation with two unknowns where there is no analytical expression of one
unknown versus the other. Both cases may be encountered in problems with
chemical reactions where the solutions involve iterative means.
e) It is extremely easy to obtain all distributions of the state vector versus time
from the Markov-chain solution. However, it is not always easy or convenient to
9
obtain these distributions from the analytical solution.
f) Elements of consecutive state vectors yield the transient response of the
system, undergoing some process, to a pulse input. Thus, RTD of the fluid
elements or particles are obtained, which gives an insight into the mixing properties
of a single-or multiple- reactor systems. Such a solution, given by Eq.(2-24), is
the product of the state vector by the one-step transition probability matrix.
g) One can model various processes in Chemical Engineering via combination
of flows, recycle streams, plug-flow and perfectly-mixed reactors. The processes
may be also associated with heat and mass transfer as well as chemical reactions.
The author believes that Markov Chains have not yet acquired their
appropriate status in the Chemical Engineering textbooks and applications although
the method has proven very effective and simple for solving complex processes.
Surprisingly, correspondence of the author with eminent Professors in Chemical
Engineering revealed they hardly have heard about Markov chains. Thus, the
major objective of the proposed book is to try to change this situation. Additional
objectives are:
a) Present a comprehensive collection of various applications of Markov
chains in Chemical Engineering, viz., reactions, reactors, reactions and reactors as
well as other processes. This is materiaUzed as from Chapter 3.
b) Provide the university Professor with a textbook for a possible course on
"Applications of Markov chains in Chemical Engineering". Alternatively, the book
can be used as reference book in other courses such as Reactor Design where
examples presented in the present book may be very useful.
c) Provide the practical engineer with numerous models and their solutions
in terms of the state vector and the one-step transition probability matrix, which
might be useful in his work. In addition, to convince the engineer about the
simplicity of applying Markov chains in solving complicated problems.
d) Stimulate application of Markov chains so as to become a common tool in
Chemical Engineering.
e) Last, but not least, to demonstrate the application of Markov chains in art
and biblical problems.
The organization of the book is as follows. The fundamentals of Markov chains
will be presented in Chapter 2 in an easy and understandable form where complex
10
mathematical derivations are abandoned and numerous examples are presented including their solution. The chapter contains processes discrete in time and space, processes discrete in space and continuous in time and processes continuous in space and time. In Chapter 3, modeling of chemical reactions is presented as well as demonstrations of their transient behavior. In Chapter 4, modeling of chemical reactors is presented and their dynamic behavior with respect to a pulse input. The latter are important parameters for describing the RTD behavior of a system where graphical presentations follow the modeling. Chapter 5 presents modeling of a few processes encountered in Chemical Engineering where effects of heat and mass transfer as well as chemical reaction are also accounted for. A general presentation of the model and its solution by Markov chains is also provided.
11
Chapter 2
FUNDAMENTALS OF MARKOV CHAINS
Markov chains have extensively been dealt with in refs.[2-8, 15-18, 84],
mainly by mathematicians. Based on the material of these articles and books, a
coherent and a short "distillate" is presented in the following. The detailed
mathematics is avoided and numerous examples are presented, demonstrating the
potential of the Markov-chain method. Distinction has been made with respect to
processes discrete in time and space, processes discrete in space and continuous in
time as well as processes continuous in space and time.
Demonstration of the fundamentals has been performed also on the basis of
examples generated from unusual sources, art and the Bible. Surprisingly, biblical
stories and paintings can be nicely analyzed by applying Markov chains discrete in
time and space.
For each example, a solution was obtained by applying Eq.(2-23) and the
EXCEL software. The solution is presented graphically, which demonstrates the
dynamical behavior of the system in occupying the various states under
consideration. Such an information was missing in the above refs.[2-8, 15-18,
84]. The latter contained only the one-step transition matrix, termed also as policy
making matrix.
2.1 IMARKOV CHAINS DISCRETE IN TIME AND SPACE 2.1-1 The conditional probability
The conception of conditional probability plays a fundamental role in Markov
chains and is presented firstly.
12
In observing the outcomes of random experiments, one is often interested how the outcome of one event S^ is influenced by that of a previous event Sj. For example, in one extreme case the relation between Sk and Sj may be such that Sk always occurs if Sj does, while in the other extreme case, Sk never occurs if Sj does. The first extreme may be demonstrated by the amazing Uthograph Waterfall
by Escher [10, p.323] depicted in Fig.2-0.
Fig.2-0. Conditional probability demonstrated by Esther's Waterfall (M.CEscher "V^aterfall" © 1998 Cordon Art B.V. - Baarn - Holland. All rights reserved)
13
If we follow the various parts of the construction in the figure one by one,
we are unable to discover any mistake in it. Yet the combination is impossible as
one may reveal. The basis for this phenomeon is a particular triangle, "the
impossible triangle", 3-4-5 in Fig.2-0 with sum of angles of 270^, first introduced
by Oscar Reutesverd in 1934. Two sides of such a triangle can exist in reality,
however, the overall element can never be constructed but can easily be perceived
by the brain. Therein lies the ingenuity of the artist who can give free play to the
imagination, relying on optical illusions to make the fanciful look real. As
observed, Fig.2-0 is based on three such triangles, i.e., 1-4-5, 3-4-5 and 2-3-4.
Despite this fact, the conception of conditional probability may be clearly
demonstrated in the following way. Assume that Si, S2, ..., S5 are five events
occurring along the trajectory of the moving water, i.e. that water pass through
points 1, 2,..., 5 in Fig.2-0 along their perpetual motion uphill. According to the
above conception, the passing of water at point 2 is completely dependent on their
previous passing at point 1. In other words, given that water pass at point 1,
ensures also their passing at point 2 with 100% probability. Similar are the
relationships between events S3 and S2 and the other consecutive events. Finally,
it should be noted that Fig.2-0 has also been widely used in Chapter 2.1-5 to
demonstrate the application of the equations developed there in classifying the
various states of Markov chains.
To characterize the relation between events Sk and Sj, the conditional
probability o/Sk occurring under the condition that Sj is known to have occurred,
is introduced. This quantity is defined [5, p.25] by the Bayes' rule which reads:
^ prob{SkS;} (2-1)
For the Escher's example, Eq.(2-1) reads, prob{S2 I Si} = prob{S2Si}/prob{Si}.
probjSk I Sj} is the probability of observing an event Sk under the condition that
event Sj has already been observed or occurred; prob{Sj} is the probability of
observing event Sj. SkSj is the intersection of events Sk and Sj, i.e., that both Sk
and Sj may be observed but not simultaneously. prob{SkSj} is the probability for
the intersection of events Sk and Sj or the probability of observing, not
14
simultaneously, both S^ and Sj. Simultaneous observation of events or,
alternatively, simultaneous occupation of states which is impossible according to
the aforementioned, is nicely demonstrated in Fig.2-1, an oil on canvas painting
[11, p.92]. The Empire of Lights (1955) of Magritte, the greatest surrealist
philosopher.
Fig.2-1. The coexistence of two states, Day and Night, according to
Magritte ("The Empire of Lights", 1954, © R.Magritte, 1998 c/o Beeldrecht Amstelveen)
15
The painting shows a house at night surrounded by trees. The only
bewildering element about this peacefully idyllic scene is the surprising fact that it
has been placed under the light blue clouds of a dayhght sky. This is Magritte's
amazing skill in combining seemingly disparate elements by simultaneously
showing two states that are mutually exclusive in time. Thus, in mathematical
terms, Magritte's painting depicted in Fig.2-1 contradicts the fact that Day and
Night can not coexist, i.e. prob{SkSj} = prob{Day Night} = 0.
Possible explanations for this contradiction by Magritte are the following
ones. The first is that Magritte was not familiar with Markov chains or
probabilistic rules. The second one is based on Genesis 1, which Magritte was,
probably, familiar with. In verse 1-2 is said: "When God began to create the
heaven and earth-the earth being unformed and void...". Thus, recalling the
philosophical character of Magritte, it may be assumed that in The Empire of Lights
Magritte has described the last second before the Creation, i.e. when "...the earth
being unformed..." and Day and Night could live in harmony together. Additional
support is provided by Rashi, the greatest Bible commentator. In Genesis 1 verse
4 it has been said: "God saw that light was good...". According to Rashi: "God
saw that it is good, and that it is not appropriate that darkness and light should be
mixed; he fixed the light for the day and the darkness for the night". In other
words before God's action. Day and Night were mixed and the prob{Day Night}
= 1. Following the above example, Eq.(2-1) gives:
and the question presented by the equation is: what is the probability that Day will
come, knowing that Night has already occurred? Although the answer is trivial,
i.e. prob{Day/Night} = 1, it will be demonstrated on the basis of the above
equation as follows. For a time interval of 24 hours probjDay Night} = 1, i.e. it is
certain that both events will occur within that time interval. In addition,
probfNight} = 1, thus it follows from Eq.(2-la) that, indeed, prob{Day/Night} =
1. On the other hand, because prob{Day Day} = 0 it follows that:
16
proMDay/Day>=P^<,Xyr'°T°° <^-"'>
Let us now return to Eq.(2-1) for further derivations. If the two events Sk
and Sj are independent of one another, then the probability of observing Sk and Sj
not simultaneously is given by the product of the probabihty of observing Sk and
the probability of observing SJ; that is:
prob{SkSj} = prob{Sk}prob{Sj} (2-2)
It follows then from Eq.(2-1) that prob{Sk I Sj} = prob{Sk}. If prob{SkSj} = 0,
Eq.(2-2) yields that prob{Sk} = 0- In general, the events Si, S2,..., Sz are said to
be mutually independent if:
prob{SiS2 ... Sz} = prob{Si}prob{S2} ... prob{Sz} (2-2a)
The following examples demonstrate the above concepts, a) Assume a single
coin is tossed one time where the two sides of it are designated as event Sk and
event Sj. For a single toss prob{Sk} = prob{Sj} = 1/2. However, prob{SkSj} =
0, since in a single toss either event Sk or event Sj may occur. In other words the
two events are dependent according to prob{Sk} + prob{Sj} = 1. b) Assume now
that the coin is tossed twice. Again prob{Sk} = prob{Sj} = 1/2. However, the
two tosses are independent of each other because in the second toss one can obtain
either Sk or Sj. Therefore, according to Eq.(2-2) one obtains that prob{SkSj} =
1/4. If the number of tosses is three, probjSkSkSk} = prob{SkSjSk} = ... =
(l/2)(l/2)(l/2) = 1/8. c) Let Sk be the event that a card picked from a full deck (52
cards) at random is a spade (13 cards in a deck), and Sj the event that it is a queen
(4 cards in a deck). Considering the above information, we may write that
prob{Sk} = 13/52; prob{Sj} =4/52; probfSkSj} = 1/52
where the last equality designates that there is only one card on which both a spade
and queen are marked. It follows from the above probabilities that Eq.(2-2) is
satisfied, namely, the events Sk and Sj are independent. In other words, if it is
known that a spade is withdrawn from a deck, no information is obtained regarding
to the next withdrawn.
17
In calculating the probability of observing an event Sk, the conditional
probability is applied in the following way. Suppose Si, S2, S3,... is a. full set of
mutually exclusive events independent of each other. By a full set is meant that it
includes all possible events and that the events Si, S2, S3, ... may always be
observed, but not simultaneously. If Sk is known to be dependent on Si, S2, S3,
..., then we can find probjSk} by using the total probability formula [5, p.27]
where Z is the total number of events; it reads:
z
prob{Sk} = ^ prob{Sj} prob{Sk I Sj} (2-3) j=i
Alternative conceptions to event and observe, more suitable in Chemical
Engineering, are, respectively, state and occupy. Thus, the prob{Sk} designates
the probability of occupying state Sk at step n+1. The prob{Sk I Sj} designates the
probability of occupying state Sk at step n+1 under the condition that state Sj has
been occupied at step n. The prob{Sj} is the probability that state Sj has been
occupied at step n where Z is the total number of states.
The application of Eq.(2-3) has been demonstrated on the basis of the
painting The Lost Jockey [12, p. 18] of Magritte (1898-1967) in Fig.2-2 which was
slightly modified. The surreal element in the picture has been achieved by the trees
which appear like sketched leaves, or nerve tracts. As seen the rider, defined as
system, leaves point O towards the trees designates as states. Eight such states are
observed in the figure. Si, S2,..., S7 and Sk. From each state the system can also
occupy other states, i.e., riding to other directions from each tree. The question is
what is the probability of the rider to arrive at Sk noting that, at first, he must pass
through one of the trees-states Si, S2, ..., S7? The solution is as follows. The
trees Si, S2v., S7 are equiprobable, since, by hypothesis, the rider initially makes
a completely random choice of one of these when leaving O. Therefore:
prob{Sj} = 1/7, j = l , . . . ,7
Once having arrived at Si, the rider can proceed to Sk only by making the proper
choice of one of the five equiprobable roads demonstrated by the five arrows in
18
Fig.5-2. Hence, the conditional probability of arriving at Sk starting from S\ is
1/5. The latter may be designated by:
Fig.2-2. Application of Eq.(2-3) to Magritte's "Lost Jockey"
("The Lost Jockey", 1948, © R.Magritte, 1998 c/o Beeldrecht Amstelveen)
prob{Sk I Si} = 1/5. Similarly, for the other states S],.. . , S7:
prob{Sk I S2} = 1/3, prob{Sk I S3} = 1/4, prob{Sk I S4} = 1/4,
prob{Sk I S5} = 1/3, prob{Sk I S6} = 1/4, prob{Sk I S7} = 1/3
Similarly, for the other states Si,..., S7:
prob{Sk I S2} = 1/3, prob{Sk I S3} = 1/4, prob{Sk I S4} = 1/4,
prob{Sk I S5} = 1/3, prob{Sk I S6} = 1/4, prob{Sk I S7} = 1/3
Thus, it follows from Eq.(2-3) that the probabihty of arriving at point Sk is:
prob{Sk} = (l/7)(l/5 + 1/3 + 1/4 + 1/4 + 1/3 + 1/4 + 1/3) = 34/190 = 27.9%
19
2.1-2 What are Markov chains? Introduction
A Markov chain is a probabilistic model applying to systems that exhibit a
special type of dependence, that is, where the state of the system on the n+1
observation depends only on the state of the system on the nth observation. In
other words, once this type of a system is in a given state, future changes in the
system depend only on this state and not on the manner the system arrived at this
particular state. This emphasizes the fact that the past history is immaterial and is
completely ignored for predicting the future.
The basic concepts of Markov chains are: system, the state space, i.e., the set
of all possible states a system can occupy and the state transition, namely, the
transfer of the system from one state to the other. Alternative synonyms are event
as well as observation of an event. It should be emphasized that the concepts
system and state Bit of a wide meaning and must be specified for each case under
consideration. This will be elaborated in the numerous examples demonstrated in
the following.
A state must be real, something that can be occupied by the system. Fig.2-3,
probably the most famous of Magritte's pictures, showing a painting in front of a
window, can nicely demonstrate the above. The painting is representing exactly
that portion of the landscape covered by the painting. Assume the tree to be the
state. Thus, the tree in the picture is an unreal state, hiding the tree behind it
outside the room, which is the real state. The latter can be occupied by a system,
for example, the Lost Jockey in Fig.2-2.
Another example of the above concepts is presented by a drunkard, the
system, living in a small town with many bars, the state space. As time goes by,
the system undergoes a transition from one state to another according to the mood
of the system at the moment. The drunkard is also staying in the bar for some time
to drink beer; in other words, the system is occupying the state for some time. If
the system transitions are governed by some probabilistic parameters, then we have
a stochastic process.
20
Fig.2-3. The real and unreal state of Magritte ("The human condition", 1933, © R.Magritte, 1998 c/o Beeldrecht Amstelveen)
Another example is that of particles suspended in a fluid, and moving under the rapid, successive, random impacts of neighboring particles. This physical phenomenon is known as Brownian motion, after the Botanist Robert Brown who first noticed it in 1927. For this case the particle is the system, the position of the particle at a given time is its state, the movement of the particle from one position to the other is its transition and its staying in a certain position is the occupying of the state; all states comprise the state space. The difference between the above
21
examples is that the first one may be considered as discrete with respect to the states
(bars) and the second one is continuous with respect to the states (position of the
particle in the fluid).
To summarize we may say the following about the applications of Markov
chains. Markov chains provide a solution for the dynamical behavior of a system
in occupying various states it can occupy, i.e., the variation of the probability of the
system versus time (number of steps) in occupying the different states. Thus,
possible applications of Markov chains in Chemical Engineering, where the
transient behavior is of interest, might be in the study of chemical reactions, RTD
of reactors and complex processes employing reactors.
Markov chains aim, mainly, at answering the following questions:
1) What is the unconditional probability that at step n the system is occupying some
state where the first occupation of this state occurred at n = 0? The answer is given
by Eqs.(2"23)-(2-25).
2) What is the probabiUty of going from state j to state k in n steps? The answer is
given by Eqs.(2-30)-(2-32).
3) Is there a steady state behavior for a Markov chain?
4) If a Markov chain terminates when it reaches a state k, defined later as absorbing
state or dead state, then what is the expected mean time to reach k (hence, terminate
the chain) given that the chain has started in some particular state j?
A few more examples
In order to examine the characteristics of Markov chains and the application
of the basic conceptions system, state, occupation of state as well as to elaborate the
idea of the irrelevance of the past history on predicting the future, on the one hand,
and the relevance of the present, on the other, the following examples are
considered.
Example 2.1 is the following irreversible first order consecutive reaction
kj ^2 3 4
Ai -> A2 ~> A3 -»A4--> A5 (2-4)
where a molecule is considered as system . The type Aj of a molecule is regarded
as the state of the system where the reaction from state Ai to state Aj is the
22
transition between the states. A molecule is occupying state i if it is in state Aj.
The major characteristic of the above reaction is that the transition to the next state
depends solely on the state a molecule occupies, and on the transition probability of
moving to the next one. How the system arrived at the occupied state, i.e. the past
history, is immaterial. For example, state A4 is govemed by the following equation
^ = k3C3-k4C4 (2-5)
where the finite difference equation between steps n and n+1 (time t and t+At)
reads
C4(n+1) = C3(n)[k3At] + C4(n)[l - k4At] (2-6)
The quantities [k3At] and [1 -k4At] may be looked upon, respectively, as the
probabilities to transit from state A3 to state A4 and the probability to remain in state
A4. Eq.(2-6) indicates that the condition of state A4 at step n+1 depends solely on
the conditions of this state prevailing at step n where the past history of the reaction
prior to step n is irrelevant.
Example 2.2 is also a Markov chain. It deals with a pulse input of some
dye introduced into a perfectly-mixed continuous flow reactor. Here the system is
a fluid element containing some of the dye-pulse. The state of the system is the
concentration of the dye-pulse in the reactor, which is a continuous function of
time. The change of system's concentration with time is the state transition given
by
C(tO/Co = exp(-tVt^) (2-7)
CQ is the initial concentration of the pulse in the reactor, C is the concentration at
each instant t' and t ^ is the mean residence time of the fluid inside the reactor. It
may be concluded that once the state of the system is known at some step, the
prediction of the state at the next step is independent of the past history.
23
Example 2.3 where the outcome of each step is independent of the past
history, is that of tossing repeatedly a fair coin designated as the system. The
possible states the system can occupy are heads or tails. In this case, the
information that the first three tosses were tails on observing a head on the fourth
toss is irrelevant. The probability of the latter is always 1/2, independent of the
past history. Moreover, thtfUture is also independent on the present, and from this
point of view the above chain of tosses is a non-Markov one.
Example 2.4 demonstrates a non-Markov process where the past history
must be taken into account for prediction of the future. We consider the state of
Israel as the system which has undergone many wars during the last fifty years.
This situation is demostrated schematically as follows:
Independence —> Sinai -^ Six day -> Attrition -» Yom kipur -» Lebanon war 1947 war 1956 war 1967 war 1968 war 1973 war 1982
-» The Gulf -> Intifadah uprising war 1991 1987-91/92
It is assumed that the system may occupy the following three states: war, no-war
and peace. In 1995 the system was in a state of war with Lebanon and in a state of
nO'War with Syria. It may be concluded that the prediction of the future state of the
system, if possible at all, depends not only on the present state, but also on the
outcome of preceding wars. In other words such a situation is not without memory
to the past and is affected by it.
Example 2.5 concerns the tossing of a die, the system, numbered 1 to 6,
the states. The probability of obtaining a 6 upward at the 6th toss, conditioned that
previous tosses differed from 6, is 1/6. This is because the probability of obtaining
any number at any toss is 1/6, since the outcome of any toss is independent of the
outcome of a previous toss. Similarly, the probability of obtaining a 6 at the 3rd
toss, conditioned that previous tosses differed from 6, is again 1/6. Thus, the
future is independent neither on the past and on the present, and the tossing chain is
a non-Markov one.
24
Mathematical formulation
The formulation for the discrete processes may be presented as follows. Let
the possible states that a system can occupy be a finite or countably infinite number.
The states are denoted by Si, S2, S3, S4, S5, S6, ..., Si, ... where S stands for
state. The subscript i designates the number of the state and if we write Si =, it
means that after the equality sign must come a short description about the meaning
of the state.
A discrete random variable X(t) is defined, which describes the states of the
system with respect to time. The quantity t designates generally time where in a
discrete process it designates the number of steps from time zero, t is finite or
countably infinite. X(t) designates the fact that the system has occupied some state
at step t. X(t) can be assigned any of the values corresponding to the states Si, S2,
S3, S4,... However, at a certain occupation (observation) of a state by the system,
only one value can be assigned. When the following equality is applied , i.e.,
X(t) = Si (2-8)
this indicates that state i was occupied by the system on step t, or that the random
variable X(t) has been realized by acquiring the value Si. Thus, Si may be looked
upon as the realization of the random variable.
In example 2.5, the number of states is six where the observations are: Si =
1, S2 = 2, S3 = 3, S4 = 4, S5 = 5, S6 = 6. The figures corresponding to the states
are those appearing on the die. If the die was tossed 5 times, a possible outcome
while considering the upward observation in each toss as a result, may be presented
as:
[X(l), X(2), X(3), X(4), X(5)] = [Si, Si, S3, S2, S6]
Thus, for example, on the fourth toss state S2 was obtained, where 2 was observed
upward. In general, after n observations or steps of the system one has a sample
[X(l), ..., X(n)],
In the above examples, the observations are obtained sequentially. An
important question that can be asked is: Does our knowledge of the past history of
the system affect our statements for the probability of the future events? For
example, does knowledge of the outcome on the first k-1 observations affect our
25
statements of the probability of observing some particular state, say Si, on the Alh
observation ? In example 2.4 above, the answer was yes, in examples 2.1 and 2.2
the answer was no, whereas examples 2.3 and 2.5 are non-Markov chains, because
neither the past and the present are relevant for predicting the future.
Therefore, the general question may be presented in the following form: What
is the probability of occupying state Si by the system on step k, knowing the
particular states that were occupied at each of the k-1 steps? This probability may
be expressed as:
prob{X(k) = Si I X(l) = Si, X(2) = S2, ..., X(k-l) = Sk-i} (2-9)
where Sn, for n = 1, 2,..., k-1, is a symbol for the state that was observed on the
nth observation or for the state that was occupied on the nth step. As shown later,
the answer to this question will be given by Eqs.(2-23)-(2-26). Eq.(2-9) contains
the conception of conditional probability previously elaborated, designated as
prob[Sk I Sj] which reads ''the probability of observing Sjc given that Sj was
observed". The Escher's Waterfall in Fig.2-0 is a nice demonstration of Eq.(2-9),
i.e.,
prob{X(5) = S5 I X(l) = Si, X(2) = S2, X(3) = S3, X(4) = S4}
In other words the probability of the system, a water element, to occupy S5, i.e.
point 5, is conditioned of previous occupation by the system of Si to S4.
If the outcomes of the observations of the states in a system are independent
of one another, it can be shown [6, p. 15] that the conditional probability in Eq.(2-
9) is equal to the unconditional probability prob{X(k) = Si}; that is:
prob{X(k) = Si I X(l) = Si,..., X(k-l) = Sk-i} = prob{X(k) = Si) (2-10)
Example 2.6 where observations are independent of one another, is that of
tossing repeatedly a fair coin, the system. The possible states are Si = head and S2
= tail. The probability of observing a head on the fourth toss of the coin, given the
information that the first three tosses were tails, is still simply the probability of
observing a head on the fourth toss, that is, 1/2. Thus, Eq.(2-10) becomes for this
example:
26
prob{X(4) = Si I X(l) = S2, X(2) = S2, X(3) = 82}= prob{X(4) = Si} = 1/2
In general, however, physical systems show dependence, and the state that
occurs on the ^ h observation is conditioned by the particular states through which
the system has passed before reaching the kth state. For a probabilistic system this
fact may be stated mathematically by saying that the probability of being in a
particular state on the kth observation does depend on some or all of the k-1 states
which were observed. This has been demonstrated before in example 2.4.
A Markov chain is a probabilistic model that applies to processes that exhibit
a special type of dependence, that is, where the state of the system on the itth
observation depends only on the state of the system on the (k-l)si observation. In
other words, the processes are conditionally independent of their past provided that
their present values are known. Thus, for a Markov chain:
. prob{X(k) = Si I X(l) = Si, ..., X(k-l) = Sk-i} =
prob{X(k) = Si I X(k-l) = Sk-i} (2-11)
or alternatively:
prob{X(k+n) = Si I X(l) = Si, ..., X(k) = Sk} =
prob{X(k+n) = Si I X(k) = Sk} (2-1 la)
that is, the state of the system on the (k+n)ih observation depends only on the state
of the system on the kxh observation. We, therefore, have a sequence of discrete
random variables X(l), X(2),... having the property that given the value of X(k)
for any time instant k, then for any later time instant k+n the probability distribution
of X(k+n) is completely determined and the values of X(k-l), X(k-2),... at times
earlier than k are irrelevant to its determination. In other words, if the present state
of the system is known, we can determine the probability of any future state
without reference to the past, or on the manner in which the system arrived at this
particular present state. The theory of Markov chains is most highly developed for
homogeneous chains and we shall mostly be concerned with these. A Markov
chain is said to be time-homogeneous or to posses a stationary transition
mechanism when the probability in Eq.(2-1 la) depends on the time interval n but
not on the time k.
27
Example 2.7 is concerned with the application of the above formulation
for two jars, one red and one black. The red jar contains 10 red balls and 10 black
balls. The black jar contains 3 red balls and 9 black balls. A ball is considered as
system and there are two states viz. Si = red ball and S2 = black ball. The process
begins with the red jar where a ball is drawn, its color is noted, and it is then
replaced. If the ball drawn was red, the second drawn is from the red jar; if the ball
drawn was black, the second draw is from the black jar. This process is repeated
with the jar chosen for a draw determined by the color of the ball on the previous
draw. It is also assumed that when drawing from an urn, each ball in that urn has
the same probability of being drawn. The probability that on the fifth drawing one
obtains a red ball, given that the outcomes of the previous drawings were (black,
black, red, black) = (S2, S2, Si, S2), is simply the probability of a red ball on the
fifth draw, given that the fourth draw produced a black ball and that the first draw
was from the red jar. That is:
prob{X(5) = Si I X(l) = S2, X(2) = S2, X(3) = Si, X(4) = S2} =
prob{X(5) = Si I X(4) = S2 } = 3/12 = 1/4
Note that:
prob{X(5) = Si I X(l) = Si, X(2) = Si, X(3) = Si, X(4) = S2} =
prob{X(5) = Si I X(4) = S2} = 3/12 = 1/4
while:
prob{X(5) = Si I X(l) = Si, X(2) = S2, X(3) = Si, X(4) = Si} =
Prob{X(5) = Si I X(4) = Si} = 10/20 = 1/2
2.1-3 Construction elements of Markov chains The basic elements of Markov-chain theory are: the state space, the one-step
transition probability matrix or the policy-making matrix and the initial state vector
termed also the initial probability function In order to develop in the following a
portion of the theory of Markov chains, some definitions are made and basic
probability concepts are mentioned.
28
The state space
Definition. The state space 55 of a Markov chain is the set of all states a
system can occupy. It is designated by:
SS =[Si ,S2 ,S3 , ..., Sz] (2-12)
In Si, S designates state where the subscript stands for the number of the state.
The states are exclusive of one another, that is, no two states can occur or be
occupied simultaneously. This point is clearly elaborated in example 2.9 in the
following and its opposite in Fig.2-1 with the associated explanations. Markov
chains are applicable only to systems where the number of states Z is finite or
countably infinite. In the latter case, an infinite number of states can be arranged in
a simple sequence Si, S2, S3, .... For the preceding example 2.1, the state space
is SS = [Si, S2, S3, S4, S5] = [Ai, A2, A3, A4, A5]. For example 2.3, SS = [Si,
S2] = [heads, tails]. For example 2.5, SS = [Si, S2, S3, S4, S5, S6] = [1, 2, 3, 4,
5, 6] and for example 2.7 in the following, SS = [Si, S2] = [red ball, black ball].
Some properties of the state space are [6, p. 18]:
1 > prob{Si} > 0 i = 1, 2,..., Z (2-12a)
where prob{Si} reads the probability of occupying state Sj. An alternative
expression for Eq.(2-12a) for all Sj in the state space, in terms of the conditional
probability defined in Eq.(2-1), reads:
1 > prob{Sj I Sj} > 0 (2-12b)
Note that Si must be occupied before occupying each of the others. It should be
noted that Eqs.(2-12a) and (2-12b) satisfy:
probJ ^ S i l = l or probJ ^ Sj I sA = 1 (2-12c) I alii in SS J [ a l l j i n S S J
An additional property of the state space is:
29
prob" i=l
X i = S P ^^ i ^ P H S J ' i = S P ^ ^J ' i ^"^^ L j = l j=l
The summation on the left-hand side designates a state (or event) comprised of Z
fundamental states. The prob{sunmiation} means the probability of occupying at
least one of the states [Si, S2, S3,..., Sz].
Definition. A Markov chain is said to be a finite Markov chain if the state
space is finite.
The one-step transition probability matrix
Deflnition. The one-step transition probability function pjk for a Markov
chain is a function that gives the probability of going from state j to state k in one
step (one time interval) for each j and k. It will be denoted by:
Pjk = prob{Sk I Sj} = prob{k I j} for all j and k (2-13)
Note that the concept of conditional probability is imbedded in the definition of pjk-
Considering Eq.(2-1 la), we may write also:
Pjk = prob{X(m+n) = Si I X(m) = Sj} (2-13a)
pjk is time-homogeneous or stationary transition probability function if it satisfies:
Pjk = function(time interval between j and k) (2-14)
Considering Eq.(2-13a), Eq.(2-14) is expressed by:
Pjk = function(n)
Pjk 5 function(m) as well as pjk 9 function(m+n) (2-14a)
Thus, for a time-homogeneous chain, the probability of a transition in a unit time or
in a single step from one given state to another, depends only on the two states and
not on the time.
In general, the one-step transition probability function is given by:
30
Pjk(n, n+1) = prob{X(n+l) = Sk I X(n) = Sj} (2-15)
which gives the probabiUty of occupying state k at time n+1 given that the system
occupied state j at time n. This function is time-dependent, while the function given
by Eq.(2-13) is independent of time, or homogeneous in time. Since the system
must move to some state from any state j , Eq.(2-18) below is satisfied for all j
The one-step transition probabilities can be arranged in a matrix form as
follows:
P = (Pjk) =
Pll P12
P21 P22
PlZ
P2Z
PZl PZ2 Pzz
(2-16)
where pjk denotes the probability of a transition from state] (row suffix) to state k
(column suffix) in one step. The matrix is time-homogeneous or stationary if the
pjk's satisfy Eq.(2-14a). A finite Markov chain is one whose state space consists
of a finite number of states, i.e., the matrix P will be a ZxZ square matrix. In
general the state space may be finite or countably infinite. If the state space is
countably infinite then the matrix P has an infinite number of columns and rows.
Definition. The square matrix P is a stochastic matrix if it satisfies the
following conditions:
1) Its elements obay:
0 < p j k < l (2-17)
otherwise the transition matrix loses meaning.
2) For every row j :
z
SPjk = 1 (2-18)
k=l
31
where Z is the number of states which can be finite or countable infinite. However, z
one may notice that /]Pjk ^ 1, a fact that apparently violates the standard theory of
Markov chains, and encountered, for example, in non-linear chemical reactions.
Other characteristics of the square matrix P are:
3) The elements pjj on the diagonal designate probabilities of remaining in same
state j .
4) The elements pjk above the diagonal designate probabilities of entering state k by
the system, from state j .
5) The elements pjk under the diagonal designate probabilities of leaving state j by
the system, to state k.
6) The sum of the products of the elements pjk (over a certain column k) by the
elements of the state vector (defined below) has the significance of the conditional
probability defined by Eq.(2-3) and is also identical with Eq.(2-23). The latter
equation is of utmost significance, giving the probability of occupying a certain
state at step (n+1) knowing that this state was influenced by other states at step n.
The transition matrix P is, thus, a complete description of the Markov
process. Any homogeneous Markov chain has a stochastic matrix of transition
probabilities and any stochastic matrix defines a homogeneous Markov chain.
In the non-homogeneous case the transition probability:
prob{X(r) = k I X(n) = j} (r > n)
will depend on both n and r. In this case we write:
Pjk(n,r) = prob{X(r) = k I X(n) = j} (2-19)
In particular the one-step transition probabilities pjk(n,n+l) will depend on the time
n and we will have a sequence of the following stochastic matrices corresponding
to Eq.(2-16) for n = 0, 1, 2, 3,...:
32
P(n) = (pjk(n,n+l)) =
pll(n,n+l) pi2(n,n+l)
P2i(n,n+1) p22(n,n+l)
(2-20)
The initial and the n-step state vectors
Definition. The initial state vector, termed also as the initial probability
function, is a function that gives the probabihty that the system is initially (at time
zero or n = 0) in state i, for each i. The initial state vector will be denoted by:
Si(0) = prob{X(0) = Si} i = 1, 2,..., Z (2-21)
designating the probability of the system to occupy state i at time zero. The above
quantities may be arrayed in a row vector form of the initial state vector, i.e.:
S(0) = [si(0), S2(0), S3(0), ..., Sz(0)] (2-22)
designating the initial occupation probability of the states [Si, S2, S3, ..., Sz] by
the system. Similarly:
S(n) = [si(n), S2(n), S3(n), ..., Sz(n)] (2-22a)
is the state vector of the system at time n (step n). Si(n) is the occupation
probability by the system of state i at time n, where:
Si(n) = prob{X(n) = Si} i = 1, 2,..., Z (2-21a)
The relationship between the one-step probability matrix and the
state vector
This relationship takes advantage of the definition of the product of a row
vector by a matrix [7, p. 19]. The product of S(n) defined in Eq.(2-22a), by the
square matrix P defined in Eq.(2-16), yields the new row vector S(n+1). The
Sk(n+1) component of this vector, i.e., the (unconditional) probability of occupying
Sk at the (n + 1) step, reads:
33
Sk(n+1) =X^j(n)pjk (2-23) j=i
It should be noted [15, p.384] that intuitively it is felt that the influence of the initial
state Sj(0) should gradually wear off so that for large n the distribution in Eq.(2-23)
should be nearly independent of the initial state vector S(0), i.e., the state
occupation is without memory to its initial history.
Let us now consider the significance of Eq.(2-23) with respect to Eq.(2-3)
given below, i.e.:
prob{Sk} = ^ prob{Sj} prob{Sk I Sj} j=i
where prob{Sj} is the probability that state Sj has been occupied at step n. Noting
Eq.(2-21a), it may be concluded that prob{Sj} is identical to Sj(n) in Eq.(2-23),
where Sj(n) is the occupation probability by the system of state j at time n. prob{Sk
I Sj} is the probability of occupying state Sk at step n+1 under the condition that
state Sj has been occupied at step n. Thus, it is identical to pjk in Eq.(2-23) which
designates the one-step transition probability from state j to k. Finally, probjSk} ^
Eq.(2-3) designates the probability of occupying state Sk at step n+l, which is
identiced to Sk(n+1). Eq.(2-23), which is a recurrence relation, may be expressed
in matrix notation as:
S(n+1) = S(n)P (2-24)
and on iteration we obtain:
S(n+1) = S(0)P'^+l (n = 0, 1, 2,. . .) (2-25)
where P and S(0) are given by Eqs.(2-16) and (2-22). Alternatively, if S(n)
denotes a column vector, then:
S(n+1) = P'^+1S(0) (2-25a)
since the choise of S(n) as a row vector is arbitrary.
34
Eqs.(2-23)-(2-25) are the fundamental expressions of Markov chains because
of the following reasons:
a) The equations give an answer to question 1 (in the introduction of section
2.1-2), i.e., what is the unconditional probability that at time n+1 (n+1 steps after
the first occupation) the system occupies state kl As can be seen, a Markov chain
is completely described when the state space, initial state vector and the one-step
transition probability matrix are given. For a physical system that is to be
represented by a Markov chain, this means that first, the set of possible states of the
system, 55, must be determined or defined. Second, the initial (at time zero)
probabilities of occupying each of these states, Si(0), must be calculated or
estimated. Finally, the probability of going from state j to state k in one time
interval (one step), pjk, must be determined or estimated for all possible j and k .
Thus, the probabilities of future state of a system, namely, at step n+1, can be
predicted from its present state at step n, and the transition probabilities in one step;
the past has no influence at all in the predictions.
b) On tha basis of Chapters 3 and 4, it may be concluded that Eqs.(2-23)-(2-
25), or the state vector S(n) given by Eq.(2-22a) and the matrix P yielding the
above equations, is an elegant way of writing the Euler integration algorithm for the
differential equations which describe the mechanism of the process.
The discrete Chapman-Kolmogorov equation
In deriving this equation, the following question is considered: W/zar is the
probability of transition of a system from state Sj to state Sjc in exactly n steps i.e.:
Pjk(n) = prob{X(n+t) = Sk I X(t) = Sj} (2-26)
In other words, Pjk(n), the n-step transition probability function, is the conditional
probability of occupying Sk at the nth step, given that the system initially occupied
Sj. pjk(n), termed also higher transition probability, extends the one-step transition
probability pjk(l) = Pjk and gives an answer to question 2 in 2.1-2. Note also that
the function given by Eq.(2-26) is independent of t, since we are concerned in
homogeneous transition probabilities.
In answering the above question, we refer again to The Lost Jockey depicted
in Fig.2-2 defined as system. We designate now point O, the initial state of the
35
system, by Sj. The other states are the trees Si to Sz where the final state is S^. It
may be seen that the system can move from state Sj to Sk by a number of different
paths. For example, if a system has Z possible states, then in two steps it may go
fromSj to Skby:
Sj -^ Si -^ \
Sj ^ S 2 ^ S k
(2-27) S j - > S i - > S k ^ ^
Sj - > S z ^ S k
where Z = 7 in Fig.2-2, excluding Sj and Sk. In order to compute the probability
of the transition S; -^ Sj -> Sj , assuming the states are independent of one
another, one applies the concept of independence given by Eqs.(2-2) and (2-2a). Thus, for a Markov chain, the transition S: -^ Sj in one step is independent of the
transition Sj -^ Sj , yielding:
prob{Sj -> Si and Sj -> S^} = prob{Sj ^ Sj}prob{Si -> Sj } = pjiPi (2-28)
Noting that prob{Sj ^ Sj ^ S^} = prob{Sj -4 Sj and Sj -^ S^}, we may now
have expressions for computing the probabilities of the transitions to the states
listed in Eq.(2-27). Since the transitions to and from the Z states in Eq.(2-27) are
mutually exclusive (that is, no pair of them can be occupied simultaneously), the
probability of the transition from state j to state k in two steps, i.e. pjk(2), is equal
to the sum of the probabilities over the Z different paths; it is given by:
Pjk(2) = Pjipik + Pj2P2k + ... + PjzPzk (2-29)
It should be noted that the above result follows also from the concept of conditional
probability given by Eq.(2-3).
Assume now that the Jockey becomes tired and wants to rest along the paths i to k. Thus, the latter trajectories are performed in the two steps Sj -^ Sj ~> \ ,
i.e., he is resting at state Si for one time interval and then riding towards the state
36
Sk. The corresponding probability for this step is pik(2), i = 1, 2,..., Z, where the
total probability for Si to Sk, i.e., Pik(3), is given recursively by:
Pik(3) = PilPlk(2) + Pi2P2k(2) + ... + PizPzk(2) (2-29a)
The general case is where the Jockey is moving from Sj to Si in n steps and from Si
to state Sk in m steps. Based on the above considerations, one can show that
Pjk(n + m) = 2 ^ pji(n)pik(m) (2-30) 1=1
which is the discrete Chapman-Kolmogorov equation. In order to have Eq.(2-30)
true for all n > 0 we define pjk(O) by pjj(O) = 1 and pjk(O) = 0 for j 9 k as is natural
[15, p.383]. pji(n) and pik(ni) are the n and m-step transition probabilities,
respectively. The latter quantities are arrayed in matrix form denoted by P(n) in the
same way as pjk form the matrix P in Eq.(2-16), i.e.:
P(n) = (pjk(n)) =
Pll(n) pi2(n) Pi3(n) ... piz(n)
P2l(n) P22(n) P23(n) ... P2z(n)
P3l(n) P32(n) P33(n) P3z(n)
Pzl(n) Pz2(n) Pz3(n) ... Pzz(n)
(2-31)
The calculation of the components Pjk(n) is as follows. In general:
P(n) = Pn (2-31a)
where P' is the one-step transition probability matrix multiplied by itself n times.
In a matrix form notation, we may write Eq.(2-30) as:
P(n + m) = P(n)P(m) = P(m)P(n)
37
(2-32)
where also:
P(n+ 1) =P(n)P = PP(n) (2-32a)
The above equations require the multiplication of a matrix by a matrix yielding a
new matrix. According to [7, p. 19], we define the product:
Q = (qjk) = AB
to have the components:
z
r=l
j , k = l , 2 , . . . , Z
where A and B are both a ZxZ square matrices given by:
(2-33)
(2-33a)
A = (ajk) =
an ai2
a2l a22
azi a22
aiz
a2z
azz
B = (bjk) =
bii bi2
b21 b22
bzi bzi
biz
b2z
bzz
(2-34)
Eq.(2-33a) states that to obtain the component qjk of Q in Eq.(2-33), we have to
multiply the elements of theyth row of A by the corresponding components of the
kxh column of B and add all products. This operation is called row-into-column
multiplication of the matrices A and B.
As an example, we consider the case 81-^82 corresponding to the following
2x2 matrix:
38
P = Si
S2
Si
Pll
1 P21
S2
P12
P22
(2-35)
From Eq.(2-33), noting that A = B = P where Z = 2, follows from P2 that:
Pll(2) = PIipi 1 + P12P21 for the paths
Si - » S i - ^ S , s,-^s. Pl2(2) = P11P12 + P12P22 for the paths
S| —> Sj —> S2 Sj — S2 — S2
where Sj-^Sj and 82-^82 indicate "resting" steps at states Si and S2.
P2l(2) = P21P11 + P22P21 for the paths
0 2 —^ >Ji —^ ^\ ^2 —^ * 2 —^ ^\
P22(2) = P21P12 + P22P22 for the paths
0 2 —^ »J| —^ > 2 ^ 2 —^ 2 —^ 2
Similarly, from P^ = PP^ it is obtained that:
Pll(3) = p i ip i ip i i +P11P12P21 +P11P12P21 +P12P22P21 for the paths
Si -^Si-^Si-^Si Si->Si->S2->Si
S i -^S i^S2-^S i Si-^S2->S2-^Si
Pl2(3) = PI ipi 1P12 + Pi 1P12P22 + P12P21P12 + P12P22P22 for the paths
Si-^Si-»Si-^S2 Si ->Si^S2-^S2
vS|—^^2—^^\—^^^2 1—^ 2—^ 2— * 2
P2l(3) = P21P11P11 +P22P21P11 +P21P12P21 +P22P22P21 for the paths
S2->Si-»Si->Si S2-^S2-^Si-^S|
02—^^\—^^2—^ 1 2— * 2—^^ 2—^ 1
P22(3) = P21P11P12 + P22P21P12 + P21pl2p22 + P22P22P22 for the paths
39
02—^^i—^^\—^^^2 2—^*^2—^^1—^^^2
09—^*^l—^*^2—^*^2 2— " 2— * 2—^ 2
2.1-4 Examples In the following examples, the application of Eqs.(2-23)-(2-25) is
demonstrated to describe the dynamics of the occupation of the states by the
system. The basic conceptions system and state are defined in each example
selected from the Bible, art as well as real life problem.
Biblical examples Example 2.8 combines bible and art and refers to the oil on canvas painting
Still Life with Open Bible (1885) by van Gogh [13, p. 15], one of the greatest
expressionists. It should also be noted that the Bible was a symbol of van
Gogh's parents' home. To demonstrate Markov chains on this painting, the
original painting was slightly modified by placing two identical books on the Bible
located to the left-hand side of the candles as depicted in Fig.2-4. As a matter of
fact, the two books are copies of the original one on the right-hand side to the
Bible.
40
Fig.2-4. The modified *StiIl Life with Open Bible*
("still Life with Open Bible", 1885, V. van Gogh )
The following Markov chains is generated. It is clear that if one wants to
study the Bible, the Bible has to rest on the top of the pile. For three books,
designated as system, which are placed one on the top of the other, three states are
possible according to the order of the books: Si = Bible, book, book; S2 = book,
Bible, book; S3 = book, book, Bible where the book on the left-hand side is placed
on the top of the pile. A one-step transition from one state to the other is conducted
by taking a book from the bottom of the pile and placing it on the top of it. Thus,
the three states may be expressed by a 3x3 one-step transition matrix given by:
P =
Si
S2
S3
Si
1
0
1
S2 0
0
0
S3 0
1
0
(2-36)
41
The matrix was applied for the following cases:
1) Assume the system is initially at state S]. In this case pn = 1, namely,
the state is a trapping or a dead state which is impossible to escape from. Thus,
one can study safely the Bible because the Book of books is always on the top of
the other books.
2) The system is initially at state S2, i.e., S2(0) = 1 where si(0) = S3(0) = 0.
The question is how many steps are needed to move to Si where the number of
States Z = 3? The initial state vector reads:
S(0) = [ 0, 1, 0 ]
Applying Eqs.(2-23), (2-24), i.e., multiplying the state vector by the matrix (2-
36), yields:
si(n), S2(n), S3(n)
S(1) = [0, 0, 1 ]
S(2) = [ l , 0, 0 ]
S(3) = [ l , 0, 0 ]
S(4) = [ l , 0, 0 ]
The above calculations indicate that at the second step, as expected, the system will
be at Si. Once it reaches this state, it will remain there forever.
3) If the system is initially at S3, the initial state vector reads:
S(0) = [0, 0, 1 ]
Similar calculations yield:
si(n), S2(n), S3(n)
S(l) = [ l , 0, 0 ]
S(2) = [ l , 0, 0 ]
S(3) = [ l , 0, 0 ]
i.e. a steady state at Si will be reached after one step. It should be noted that Si is
always reached independent of S(0), namely, this state is without memory to the
initial state.
42
Example 2.9 is related to the creation of the two states already mentioned in
Chapter 0, vis., Si = Day and S2 = Night in Genesis 1 verse 4-5. The system
may be a man or a population who may occupy the above states. The two states are
expressed by the following 2x2 matrix corresponding to 24 hours:
P =
Si
S2
Si
0
1
S2
1
0
(2-37)
The significance of the above probabilities is the following. The probability of
remaining in Si is zero because after day comes always night, i.e., p n = 0. The
transition probability from Si to S2 is, of course, pi2 = 1. Similarly, p2i = 1, i.e.,
the transition probability from S2 to Si, as well as probability of remaining in S2,
P22 = 0. It should be noted that above probabilities are independent of time. If the
initial state vector reads:
S(0) = [ 1, 0 ]
application of Eqs.(2-23), (2-24) yields:
si(n), S2(n)
S(l) = [ 0, 1 ]
S(2) = [ 1, 0 ]
S(3) = [ 0, 1 ]
S(4) = [ 1, 0 ]
The behavior of the vector si(n) indicates that if initially there was Si, the
probability of remaining at this state after one step is si(l) = 0. After two steps (24
hours), si(2) = 1, i.e., there will be again a state of Day, as expected. According
to section 2.1-5 Table 2-2, the above chain is defined as a periodic chain.
Si
0
0
S2
1
1
43
Example 2.10 is a Markov chain representation of king Solomon's famous
trial [1 Kings 3] discussed in Chapter 0. The child was selected here as system.
The two states are: Si = Living child and S2 = Divided child. These states are
presented by the following 2x2 matrix where the probability of the system to
remain in state Si is, unfortunately, pn =0 . This is because king Solomon said:
Divide the living child in two. The transition probability from Si to S2 is,
therefore, 100%, i.e., pi2 = 1. Similarly, p2i = 0, which is the transition
probability from S2 to Si. Finally, the probability of remaining in state S2, P22 = 1
and the matrix reads:
Si
P = I I (2-38)
S2
It is interesting to note that the matrix is characterized by the so-called dead state,
i.e., once the system reaches this state, it will remain there for ever. The
application of Eq.(2-24), assuming that initially si(0) = 1, i.e., the system is at Si
= Living child and that:
S(0) = [ 1, 0 ]
yields:
si(n) S2(n)
S(l) = [ 0, 1 ]
S(2) = [ 0, 1 ]
The behavior of the vector si(n) indicates that if initially there was a Living child,
the probability of remaining at this state after one step is, unfortunately, si(l) = 0.
Thus, the Markovian dead state will be also a real description of the actual situation.
Example 2.11 considers the order of the days of the week in the Creation
mentioned in Chapter 0. The states are defined as Si = Sunday, S2 = Monday,...,
and S7 = Saturday. The probability matrix reads:
44
P =
Si S2 S3 S4 S5 S6 S7
Si
S2
S3
S4
S5
S6
S7
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
(2-39)
The system may be a man or the universe, which may occupy the states at some
time according to matrix (2-39).
Assuming that:
S(0) = [ 1, 0, 0, 0, 0, 0, 0 ]
i.e., that the system initially occupying Si, yields the following behavior:
si(n) S2(n) S3(n) S4(n) S5(n) S6(n) S7(n)
S(1) = [ 0 , 1, 0, 0, 0, 0, 0 ]
(2-39a)
S(2) = [ 0,
S(3) = [ 0,
S(4) = [ 0,
S(5) = [ 0,
S(6)= [ 0,
S(7) = [ 1,
0,
0,
0,
0,
0,
0,
1,
0,
0,
0,
0,
0,
0,
1,
0,
0,
0,
0,
0,
0,
1,
0,
0,
0,
0,
0,
0,
1,
0,
0,
0 ]
0 ]
0 ]
0 ]
1 ]
0 ]
The above behavior reveals that all states are periodic of a period of seven days.
Example 2.12 generates A Markov chain based on the biblical story about
the Division of the Promised Land among the twelve tribes. In Book of Books
[Joshua 13, verses 1 and 7] we read: "Now Joshua was old and advanced in years;
and the Lord said to him: You are old and advanced in years, and there remains yet
very much land to be possessed... Now therefore divide this land..." In [25,
p.52], a map depicted below, shows the results of this division, i.e. the boundaries
of the inheritances of the tribes in the 12th century BC. corresponding to the above
verses.
45
The Big Sea / Sj-Asher
1 Acre L
f\ 1 ( 1 / ^^S3=Zebulun ^
1 1 S = Issachar
/ S Jaffa i
1 I S ^ Ephraim /S^= Dan
1 / S = Benjamin
/
1 /Oaza S=Judah
1/ 1 S.y Simeon
8^= Naphtali 1
^ 1 Kinereth 1
. j Sea 1
.=/Manasseh 1 > 1 1
ISg=Gad
1 Jordan 1 V River 1
•*SS = Reuben 1
Deadl SeaJ
A visitor, designated as system, wishes to visit the tribes. His transition
between the states assumes that the probabilities of remaining in a state or moving
to the other states is the same and that the following one-step transition matrix
holds:
46
P =
Si
S2
S3
S4
S5
S6
S7
Sg
S9
siol S l l
Si S2 S3 S4 S5 S6 S7 Sg S9 Sio S u S12
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
1/4 0 0 0 0
1/3 1/3 1/3
1/3 1/3 1/3
1/4 1/4 1/4 0 0 0
0
0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
sd 0 0
0
0
0
6 0
0
0
0
0
1/3 1/3 1/3 0 0 0 0 0 0
1/3 1/3 1/3 0 0 0 0 0
0 0 0 1/4 1/4 1/4 1/4 0 0
0 0 0 1/4 1/4 0 1/4 1/4 0
0 0 1/5 1/5 0 1/5 1/5 0 1/5
0 0 0 1/6 1/6 1/6 1/6 1/6 1/6
0 0 0 0 0 1/4 1/4 1/4 1/4
0 0 0 0 1/4 1/4 1/4 1/4 0
0 0 0 0 0 0 1 / 2 0 1 / 2
(2-40)
The application of Eq.(2-24) yields Fig.2-5(a,b) showing the probability
distribution of visiting the twelve tribes, i.e., Si(n), against the number of steps n.
A Step is equivalent to some time interval the visitor stays in a tribe after which he is
leaving to the next one. Two cases were considered: a) The visitor begins at state
Si = Asher, where the initial state vector S(0) is given at the top of Fig.2-5(a). b)
The visitor begins at state S12 = Simeon, where the initial state vector S(0) is given
atthetopofFig.2-5(b).
47
0.8
(0~
1—I—I—I— \—I—I— \—I—I—I 1—I—T"
S(0) = [1,0,0,0,0,0,0.0,0,0,0,0]
I I I
(a)
s^(n), Sg(n)
.s (n)
0.8 h
0.6 U
0.4
0.2
1 — I — I — \ — I — I — I — I — \ — I — I — I — I — I — I — I — I — I — r
S(0) = [0,0,0,0.0,0.0,0,0.0,0,1]
-s (n)
3^(n) to s^(n)
(b)H
\ y^o(") Sii(");V") ^ " ^f") %(")
f]/ \^\ I I I I I I I I 1 1 I I I \ i* I 10 n
15 20
Fig.2-5(a, b). Probability of visiting the twelve tribes
As observed, in both cases the visiting probabihty distribution of the tribes,
which may be used to decide the policy of visiting of the tribes, reaches a steady
state after several steps. In other words, he will start at the tribe of the highest
probability and then move to the tribe of a lower probability, and so on. According
48
to the results, in case (a) he starts at Si and moves according to the following order
of states where he reaches the last tribe after 12 steps:
Si -^82^83-^84-^85 ->86->87->89->88-^89->8io->8ii-^8i2
It should be noted that the results indicate that S2(l) = S3(l) = 0.333, S5(6) = S6(6)
-= 0.044. Bearing this in mind this fact, the visitor has decided to move according
to 82->S3 and 85 ->86. In case (b), he starts at 812 and his transition was found
to be according to the following trajectory:
812 -^8io-^89->8i 1-^88 -^87-^86
where it has been observed that S9(2) = sii(2) = 0.125. However, the interesting
result is that the visitor terminates his visits after 7 steps and will never reach states
81 to 85 because the values of the probabilities Si(n) = 0, i = 1,..., 5.
Artistic examples Example 2.13 is a Markov-chain model for Magritte's painting The Castle
in the Pyrenees' [14, p.l l6] depicted in Fig.2-6. In 1958-1959 particularly,
Magritte was obsessed by the volume and weight of enormous rocks, but he altered
the laws of gravity and disregarded the weight of matter; for instance, he had a rock
sink or rise. 8imilarly, in Fig.2-6, he visioned a castle on a rock floating above the
sea. Considering the floating effect, two states may be visualized, i.e., 81 = the
rock is floating at some height above sea level, as seen in Fig.2-6; 82 = the rock is
floating at very small distance above sea level. The rock was chosen as system.
Thus, one may assume the following one-step probability matrix given by Eq.(2-
41). The reason for assuming pn and p22 to be unity, i.e., the system remains in
its state, is that since the rock is in a floating state, once it is "located" somewhere,
it will remain there.
49
Fig.2-6. Magritte's gravityless world ("The Castle in the Pyrenees", 1959, © R.Magritte, 1998 c/o Beeldrecht Amstelveen)
50
Si
p =
S2
Si S2
1 0 1
0 1 1
If we assume the following matrix:
Si S2
Si
P =
S2
1 0
1 0
(2-41)
(2-41a)
where p2i = 1» this suggests that if the rock was placed very near the sea level at
S2, it will move up to Si, a dead state, and remains there.
Example 2.14 demonstrates the situation depicted in Fig.0-2 by Escher as
a Markov process in the following way. States Si ( i = 1, 2, ..., 6) are various
locations which the system, a moving man, is occupying along its trajectory. The
one-step transition of the system is according to the following matrix:
P =
Si S2 S3 S4 S5 S6
Si
S2
S3
S4
S5
S6
0
0
0
0
0
0
0
0
1
0
1
0
0
1
0
0
0
0
1
0
0
0
0
1
0
0
0
1
0
0
0
0
0
0
0
0
(2-42)
Some explanations are needed regarding to the underlying assumptions in the
matrix. The probabilities pii on the diagonal are all zero, indicating that the system,
never remain in these states. P45 has been assumed 1, namely, that climbing along
the staircase from state 4 to 5 is possible. This is applicable if the reader covers
with his palm the right half of the staircase. He then sees that the staircase is in the
51
upward direction. However, when he unveils the staircase, the latter seems to turn
over. Therefore, it has been assumed that P55 = 0, i.e., it is impossible to remain
at state 5. Since gravitational forces are effective, the man will fall to state 2.
Praying that he remains alive, it is assumed that P52 = 1. Assume the following
initial state vector, namely, the starting position is at state Si:
S(0) = [ 1, 0, 0, 0, 0, 0 ]
and applying Eq. (2-24) yield that:
si(n) S2(n) S3(n) S4(n) S5(n) S6(n)
S(1) = [ 0 ,
S(2) = [ 0,
S(3)= [ 0,
S(4) = [ 0,
S(5)= [ 0,
S(6) = [ 0,
S(7)=[ 0,
S(8) = [ 0,
0,
0,
1,
0,
1,
0,
1,
0,
0,
0,
0,
1,
0,
1,
0,
1,
1,
0,
0,
0,
0,
0,
0,
0,
0,
1,
0,
0,
0,
0,
0,
0,
0 ]
0 ]
0 ]
0 ]
0 ]
0 ]
0 ]
0 ]
(2-42a)
Inspection of the above state vectors behavior reveals that the system, at
steady state from n = 4, will end up walking up and down along the staircase
connecting states 2 and 3. The latter behavior was also independent of its initial
state vector, i.e., this is an ergodic Markov chain which is without memory to the
initial step.
Example 2.15, demonstrating the common situation of "dead state" (pii =
1) in Markov chains, is based on Escher's lithograph 'Reptiles' [10, p.284]
depicted in Fig.2-7. It demonstrates the life cycle of a little alligator. Amid all
kinds of objects, a drawing book lies open. The drawing on view, is a mosaic of
reptilian figures in three contrasting shades. Evidently one of them has tired of
lying flat and rigid among his fellows, probably in a "dead state", so he puts one
plastic-looking leg over the edge of the book, wrenches himself free and launches
into real life. He climbs up the back of a book on zoology and works his laborious
52
way up a slippery slope of a set square to the highest point of his existence. At
states 5 and 7 he might slip and fall on the book joining again the "dead state"
situation. If this does not happen at this stage, after a quick snort, tired but
fulfilled, he goes downhill again, via an ashtray, to the level surface to that flat
drawing paper, and meekly rejoins his previous friends, taking up once more his
function as an element of surface division, i.e, the "dead state" situation.
Fig.2-7. Escher's reptiles demonstrating life cycle and "dead state"
(M.C.Escher "Reptiles" © 1998 Cordon Art B.V. - Baam - Holland. All rights reserved)
The above description can be framed by a Markov process in the following
way. A reptile was defined as ^y r m and the follo'wmg states shown in the figure
were selected, i.e. Si = reptile at position i, i = 1, 2,..., 11. On the basis of above
states, the following matrix may be established. Some assumptions made were:
53
from S5, the reptile can move to states 85 and S7 with equal probabilities, i.e., 1/2.
Similarly, from S7 it can move to Sg and S9 with equal probabilities. Other
transitions of the reptile are governed by the one-step probability matrix given by
Eq.(2-43) which is the policy-making matrix of the reptile.
Si S2 S3 S4 S5 S6 S7 Sg S9 Sio Sii
P =
Si
S2
S3
S4
S5
S6
s? Sg
S9
Sio
Sii
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.5 0.5 0
1
0
0
0
0
0
0
0
0
0
0
0
0
0.5
1
0
0
0
0
0
0
0
0
0
0.5
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
1
1
(2-43)
To demonstrate the behavior of a reptile along his life cycle we assume that
S(0) = [1, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0]
i.e., the system (reptile) in Fig.2-7 initially at state Si,yields the following
dynamical behavior:
si(n) S2(n) S3(n) S4(n) S5(n) S6(n) S7(n) S8(n) S9(n) sio(n) sii(n)
S(l) = [0,
S(2) = [0,
S(3) = [0,
S(4) = [0,
S(5) = [0,
S(6) = [0,
S(7) = [0,
S(8) = [0,
1,
0,
0,
0,
0.
0,
0,
0,
0,
1,
0,
0,
0,
0,
0,
0,
0,
0,
1,
0,
0,
0,
0,
0,
0,
0,
0,
1,
0,
0,
0,
0,
0,
0,
0,
0,
0.5,
0.5,
0.5,
0.5,
0,
0,
0,
0,
0.5,
0,
0,
0,
0,
0,
0,
0,
0,
0.25,
0.25,
0.25,
0,
0,
0,
0,
0,
0.25,
0,
0,
0 ,
0 ,
0 ,
0 ,
0 ,
0 ,
0.25 ,
0 , 0
0]
0]
0]
0]
0]
0]
0]
.25]
54
S(9) = [0, 0, 0, 0, 0, 0.5, 0, 0 .25,0, 0 , 0.25]
As seen, the system attains a steady state after seven steps where the reptile
has a probability of 50% to occupy state 85 and 25% probability to occupy Sg and
Sii . Note, however, that in all these states the reptile is in a "dead state", pn = 1,
as also demonstrated Escher's Fig.2-7.
If the reptile is initially at S9, i.e.
S(0) = [0, 0, 0, 0, 0, 0, 0, 0, 1, 0 , 0]
it is obtained that:
si(n) S2(n) S3(n) S4(n) S5(n) S6(n) S7(n) sgCn) S9(n) sio(n) sii(n)
S(l) = [0, 0, 0, 0, 0, 0, 0, 0, 0, 1 , 0]
S(2) = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 1]
S(3) = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 1]
i.e., the reptile is in a steady "dead state" at Sn already after two steps.
Example 2.16 models the movement of fish based on Escher's painting
'Fish' [10, p.311] depicted in Fig.2-8. The selected states Si to S12, are various
locations of the fish as shown in Fig.2-8. The system is a fish. The underlying
assumptions on the transition of the system are: a) The probability of remaining in
some state pa = 0. b) The probability of occupying the state of an adjacent fish
moving in counter current flow is also zero, c) A fish can not jump above another
fish, d) There are equal probabilities to occupy two adjacent fish states. Bearing in
mind the above assumptions yields the matrix below.
55
Fig.2-8. Movement of fish according to Escher (M.CEscher "Fish" © 1998 Cordon Art B.V. - Baarn - Holland. All rights reserved)
P =
Si S2 S3 S4 S5 S6 S7 Sg S9 Sio Sii S12
Si
S2
S3
S4
S5
Se S7
Sg
S9
Sio Sii
S12
0 0
0
0
0
0.5
0.5 0
0
0
0
0
0
0
0
0.5
0
0.5
0
0
0
0
0
0
0.5
0.5
0
0
0
0
0
0
0
0
0
0
0.5
0.5
0
0
0
0
0
0
0
0
0
0
0 • 0
0.5
0
0
0.5
0
0
0
0
0
0
0 0 0
0 0 0
0 0 0
0 0 0
0.5 0.5 0
0 0 0
0 0 0.5
0.5 0.5 0
0 0 0
0 0 0.5
0 0 0
0 0 0
0
0
0
0
0
0
0.5
0
0
0
0
0
0
0
0
0
0
0.5
0.5 0
0
0
0.5
0
0
0
0
0.5
0
0
0
0
0
0
0
0.5
0
0
0
0
0
0
0
0
0.5
0
0.5
0
(2-44)
56
Results of the dynamical behavior of the system are presented in Fig.2-9 for
the initial state vector S(0) = [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], i.e., the fish is
initially occupying state 1 in Fig.2-8. As seen, there are two groups of states
attaining an identical steady state after ten steps where one group lags one step
behind the other. In addition, the occupation probability distribution of the states at
steady state, which equals 0.167, is periodic. One group lags by one step behind
the other where the system has to decide at each step what state to occupy among
six possibilities of an equal probability. It has also been observed that the steady
state is independent of the initial state vectors S(0) which classifies this case as an
ergodic Markov chain.
0.8
0.6
c C O '
0.4
0.2
s^ (n) = 82(0) = s^(n) = Sg(n) = Sg(n) = s^ (n)
SgCn) = s^(n) = Sg{n) = s^(n) = 5^ (11) = s^2(n)
8 10 0 2 4 6
n Fig.2-9. Probability versus time of occupying the states by the fish
Examples 2.17-2.22 (and 2.41, 2.42) relate to what is normally called
random walk [7, p.26; 4, p.89]. In principle, we imagine a particle moving in a
straight line in unit steps. Each step is one unit to the right with probability p or
one unit to the left with probability q, where p + q = 1. The particle moves until it
reaches one of the two extreme points called boundary points. The possibilities for
its behavior at these points determine several different kinds of Markov chains
57
demonstrated in the following. An artistic demonstration of the different cases is
based on Escher's painting Sun and Moon [10, p.295] depicted in Fig.2-10.
Fig.2-10. Random walk demonstrations according to Escher
(M.C.Escher "Sun and Moon" © 1998 Cordon Art B.V. - Baarn - Holland. All rights reserved)
We select nine states Si, S2,..., S9, which are various locations in the bird's
state field that can be occupied by the system - the moving bird. The states are
designated by 1,2, ..., 9 in Fig.2-10. Although the movement is not in a straight
line, the transition from one state to the other defined by the transition probability
matrices below, ensures the random walk model. States Si and S9 are the
boundary states and S2, S3,... , Sg the interior states.
58
Example 2.17 is a simple random walk between two absorbing barriers. It
is characterized by the behavior of the moving bird, the system, that when it
occupies states S\ and S9, it remains there from that time on (pn = P99 = 1). In
this case the transition matrix is given by:
Si S2 S3 S4 S5 S6 S7 Sg S9
Si
S2
S3
S4
p = S5
S6
S7
S8
S9
where p + q = 1.
1
q 0
0
0
0
0
0
0
0
0
q 0
0
0
0
0
0
0
p 0
q 0
0
0
0
0
0
0
p 0
q 0
0
0
0
0
0
0
p 0
q 0
0
0
0
0
0
0
p 0
q 0
0
0
0
0
0
0
p 0
q 0
0
0
0
0
0
0
p 0
0
0
0
0
0
0
0
0
p 1
(2-45)
Fig.2-11 presents results for q = 0.5 and S(0) = [0, 0, 0, 0, 1, 0, 0, 0, 0],
i.e., the bird is initially occupying S5 in Fig.2-10. It is observed that as time goes
by, the probabilities of the bird to occupy states 1 and 9 is increasing as well as are
identical. After 90 steps a steady state is achieved where S(90) = [0.5, 0, 0, 0, 0,
0, 0, 0, 0, 0.5]. In other words only the boundary states may be occupied with
identical probabilities. The above behavior is evident recalling that S\ and S9 are
absorbing boundaries. It should be noted that the behavior of the other states is
similar to S5 reported in the figure, i.e., their occupation probability diminishes
versus time. For q = 0.2, the steady state state vector reads S(30) = [0.004, 0, 0,
0, 0, 0, 0, 0, 0.996], i.e., the probability of occupying S9 is much higher than that
of state 1.
59
CO"
1
0.8
0.6
0.4
0.2
n
— 1
r—
r s,(n)
T ;": ;.
i ''/'\ V VIV V ';'.
i 1
s,(n) = s (n) _
'. ''• '"' '\ .'
- 1
H
H
H
10 20 n
30 40
Fig.2-11. Probability distribution of occupying various states for the
two absorbing barriers Si and S9
Example 2.18 is a simple random walk with reflecting barriers. Whenever
the bird reaches states S\ and S9, it returns to the point from which it came, i.e. pn
= P99 = 0. The transition matrix reads
Si S2 S3 S4 S5 85 S7 Sg S9
Si
S2
S3 S4
p = S5
S6
S7
Sg
S9
wherep + q = 1.
0
q 0
0
0
0
0
0
0
1
0
q 0
0
0
0
0
0
0
p 0
q 0
0
0
0
0
0
0
p 0
q 0
0
0
0
0
0
0
p 0
q 0
0
0
0
0
0
0
p 0
q 0
0
0
0
0
0
0
p 0
q 0
0
0
0
0
0
0
p 0
1
0
0
0
0
0
0
0
p 0
(2-46)
60
Fig.2-12 presents results for q = 0.5 and S(0) = [0, 0, 0, 0, 1, 0, 0, 0, 0].
The general trend observed is that after 20 steps, the probabihties of occupying the
various states attain the following steady states corresponding to the groups
designated clearly in Fig.2-12:
S(20) = [0.125, 0, 0.25, 0, 0.25, 0, 0.25, 0, 0.125]
S(21) = [0, 0.25, 0, 0.25, 0, 0.25, 0, 0.25, 0]
S(22) = [0.125, 0, 0.25, 0, 0.25, 0, 0.25, 0, 0.125]
S(23) = [0, 0.25, 0, 0.25, 0, 0.25, 0, 0.25, 0]
This behavior is plausible recalling that the boundaries are of reflecting barrier type.
In other words, the moving bird will never be at rest. A final remark is that the
limiting behavior is independent of S(0), characterizing an ergodic Markov chain.
1
0.8
0.6
0.4
0.2
S2(n) = s^(n) = Sg(n) = Sg(n)
Sg(n) = Sg(n) = s^(n)
s^(n) = Sg(n)
10 15
Fig.2-12. Probability distribution of occupying various states for
the two reflecting barriers Si and S9
Example 2.19 belongs also to the random walk model with reflecting
barrier. However, it is assumed that whenever the moving bird, system, hits the
61
boundary states Si or S9, it goes directly to the central state S5. The corresponding
transition matrix is given by Eq.(2-47) where p + q = 1. Fig.2-13 presents results
for q = 0.5 and S(0) = [1,0,0,0,0,0,0,0,0], i.e., the system is initially at the Si
reflecting barrier shown in Fig.2-10.
P =
Si S2 S3 S4 S5 S6 S7 Sg S9
Si
S2
S3
S4
S5
S6
S7
S8
S9
0
q 0
0
0
0
0
0
0
0
0
q 0
0
0
0
0
0
0
p 0
q
0
0
0
0
0
0
0
p 0
q 0
0
0
0
1
0
0
p 0
q
0
0
1
0
0
0
0
p 0
q 0
0
0
0
0
0
0
p
0
q 0
0
0
0
0
0
0
p 0
0
0
0
0
0
0
0
0
p 0
(2-47)
As seen, the elements of the state vector oscillate towards a steady state
distribution attained for n = 31, i.e., S(31) = [0.029, 0.059, 0.117, 0.177, 0.235,
0.177, 0.117, 0.059, 0.029] where the maximum probability corresponds to S5.
As in previous cases, the steady state distribution is independent of S(0) and if the
system has to occupy some state, it will be, probably, S5, of the highest
probability.
62
Fig.2-13. Probability distribution of occupying the states for the two
reflecting barriers Si and S9 sending the bird directly to S5
Example 2.20 is a random walk with retaining barriers (partially
reflecting). It has been assumed that the occupation probability by the system of
the boundary state, or moving to the other boundary state is 0.5. Thus, the one-
step transition probability matrix reads:
P =
Si
S2
S3
S4
S5
S6 S7
Sg
S9
Si
0.5
q 0
0
0
0
0
0
1 0.5
S2 0
0
q 0
0
0
0
0
0
S3 0
p 0
q 0
0
0
0
0
S4 0
0
p 0
q 0
0
0
0
S5 0
0
0
p 0
q 0
0
0
S6 0
0
0
0
p 0
q 0
0
S7 0
0
0
0
0
p 0
q 0
S8 0
0
0 0
0
0
p 0
0
S9
0.51 0
0
0 0
0
0
p 0.5
(2-48)
63
where p + q = 1. Fig.2-14 presents results for q = 0.5 and S(0) = [0, 0, 0, 0,1, 0,
0, 0, 0], i.e., the system is initially occupying S5. As expected, the following
steady state is obtained at n = 90, i.e., S(90) = [0.500, 0, 0, 0, 0, 0, 0, 0, 0.500].
Under these conditions, the bird, has a 50% probability of occupying the boundary
states Si and S9 in Fig.2-10. It should also be noted that the limiting behavior for
large n, is independent of the initial state, i.e. a situation without memory with
respect to the far past.
0.8
0.6
0.4
0.2
0 10 15 20 25 30 35 40
n
Fig.2-14. Occupation probability distribution of some states for the
partially reflecting barriers Si and S9
Example 2.21 assumes that when the bird reaches one of the boundary
states Si or S9, it moves directly to the other, like in SLping-pong game (pi9 = p9i
= 1). Thus, the transition matrix reads:
64
P =
Si S2 S3 S4 S5 Se S7 Sg S9
Si
S2
S3
S4
S5
S6
S7
Sg
S9
0
q 0
0
0
0
0
0
1
0
0
q 0
0
0
0
0
0
0
p 0
q 0
0
0
0
0
0
0
p 0
q 0
0
0
0
0
0
0
p 0
q
0
0
0
0
0
0
0
p 0
q 0
0
0
0
0
0
0
p
0
q 0
0
0
0
0
0
0
p 0
0
1
0
0
0
0
0
0
p 0
(2-49)
where p + q = 1. Fig.2-15 presents the variation against time of si(n), S3(n) and
S9(n) for q = 0.5 and S(0) = [0,0,1,0,0,0,0,0,0], i.e., the system is initially
occupying S3 as shown in Fig.2-10.
0)"
1
0.8
0.6
0.4
0.2
i:: ;; S •: II :• i i ;; j l - i n :• n ; : !:• i : : M - - M : : M : -
i n a , : : i , : ; i ; : | . : ,h;!h:l i : :
- i - v - v . ' - I - ' -
40 0 5 10 15 20 25 30 35 n
Fig.2-15. Ting-pong' type probability distribution of the boundary
states Si and S9
65
As seen, S3(n) is oscillating and approaches zero at steady state. A similar behavior
was observed also for the other Si(n)'s excluding si(n) and S9(n) which correspond
to the boundary states S\ and S9. These oscillating quantities attain the following
limiting behavior of the state vectors for S(0) = [0,0,1,0,0,0,0,0,0]:
S(86) = [0.75, 0, 0, 0, 0, 0, 0, 0, 0.25]
S(87) = [0.25, 0, 0, 0, 0, 0, 0, 0, 0.75]
S(88) = [0.75, 0, 0, 0, 0, 0, 0, 0, 0.25]
S(89) = [0.25, 0, 0, 0, 0, 0, 0, 0, 0.75]
This behavior is plausible recalling the 'ping-pong' type behavior of the
boundaries. It should be noted that the limiting values of Si and S9 depend on
S(0).
For example:
a) S(0) = [1, 0, 0, 0, 0, 0, 0,0, 0] yields
S(l) = [0, 0, 0, 0, 0, 0, 0, 0, 1]
S(2) = [1, 0, 0, 0, 0, 0, 0, 0, 0]
where for S(0) = [0, 0, 0, 0, 0, 0, 0, 0, 1] the values of S(2) replace these of
S( l ) .
b) S(0) = [0, 1, 0, 0, 0, 0, 0, 0, 0] yields
S(79) = [0.875, 0, 0, 0, 0, 0, 0, 0, 0.125]
S(80) = [0.125, 0, 0, 0, 0, 0, 0, 0, 0.875]
where for S(0) = [0, 0, 0, 0, 0, 0, 0, 1, 0] the values of S(80) replace these of S(79).
c) S(0) = [0, 0, 0, 1, 0, 0, 0, 0, 0] yields
S(89) = [0.625, 0, 0, 0, 0, 0, 0, 0, 0.375]
S(90) = [0.375, 0, 0, 0, 0, 0, 0, 0, 0.625]
66
where for S(0) = [0, 0, 0, 0, 0, 1, 0, 0, 0] the values of S(90) replace these of S(89).
d) S(0) = [0, 0, 0, 0, 1, 0, 0, 0, 0] yields
S(90) = [0.500, 0, 0, 0, 0, 0, 0, 0, 0.500]
S(91) = [0.500, 0, 0, 0, 0, 0, 0, 0, 0.500]
Example 2.22 is a modified version of the random walk. If the system
(bird) occupies one of the seven interior states S2 to S7, it has equal probability of
moving to the right, moving to the left, or occupying its present state. This
probability is 1/8. If the system occupies the boundaries Si and S9, it can not
remain there, but has equal probability of moving to any of the other seven states.
The one-step transition probability matrix, taking into account the above
considerations, is given by:
P =
Si S2
S3 S4
S5
S6
S7
Sg
S9
Si 0
1/3
0
0
0
0
0
0 1/8
S2 1/8
1/3
1/3 0
0
0
0
0 1/8
S3 1/8
1/3
1/3 1/3
0
0
0
0 1/8
S4 1/8
0
1/3 1/3
1/3
0
0
0 1/8
S5 1/8
0
0 1/3
1/3
1/3
0
0 1/8
S6 1/8
0
0 0
1/3
1/3 1/3
0 1/8
S7 Sg
1/8 1/8
0 0
0 0
0 0
0 0
1/3 0
1/3 1/3
1/3 1/3
1/8 1/8
S9 1/8
0
0
0 0
0
0
1/3
0
(2-50)
Fig.2-16 presents results for S(0) = [0,0,0,0,1,0,0,0,0] indicating that the
probability distribution of approaching a steady state for n = 13 reads S(13) =
[0.03, 0.078, 0.134, 0.168, 0.179, 0.168, 0.134, 0.078, 0.03]. It should be
noted that, generally, si(n) = S9(n), S2(n) = sgCn), S3(n) = S7(n), S4(n) = S6(n), i.e.,
the curves are symmetrical. In addition, it was found that the limiting distribution
is independent of S(0), i.e., the resulting Markov process generates an ergodic
chain.
67
0 ) "
15 20 25 30
n Fig.2-16. Probability distribution of occupying various states
Example 2.23 is the last one on artistic examples. In Fig.2-1, Magritte has
demonstrated an impossible situation on the coexistence of Day and Night, two
states which can not coexist. It is also interesting to show how Escher
demonstrated the above situation in his woodcut Day and Night [10, p.273] and to
model this behavior. The situation is depicted in Fig.2-17. One sees gray
rectangular fields develop upwards into silhouettes of white and black birds. The
black ones are flying towards the left and the white ones towards the right, in two
opposing formations. To the left of the picture, the white birds flow together and
merge to form a daylight sky and landscape. To the right, the black birds melt
together into night. The day and night landscapes are mirror images of each other,
united by means of the gray fields out of which, once again, the birds merge. The
difference between the two demonstrations of day and night is therefore the
following. In Magritte's picture, 'half of the picture, i.e., clouds and sky, are at
daylight. The other 'half, house surrounded by trees, are at night. In Escher's
woodcut, however, the right 'half is at night, where the other one, on the left
(mirror image of the right), is at day light.
68
Fig.2-17. The "coexistence" of Day and Night according to Escher
(M.C.Escher "Day and Night" © 1998 Cordon Art B.V. - Baam - Holland. All rights reserved)
In Fig.2-17 sixteen states S\ to Si6, i.e., sixteen possible locations of birds
in the sky along the flying route, are shown; the states are designated by 1, 2,...,
16. The system is a bird. The underlying assumptions for the system are: a) A
bird occupying states Sn to Si6 moves only to the right, b) A bird occupying S5
to Si moves only to the left, c) A bird occupying 85 to S10 can move to the left
and to the right, d) A bird occupying Si and S16 remains there, i.e., p n and
Pi6,16 = 1- Other assumptions can be concluded from the 16x16 one-step
transition probability matrix given by Eq.(2-51).
69
Si 1 S2 !
S3
S4
Ss S6
P = S7
S8
S9
Sio Sii
Sl2
Sl3 Si4
Sl5
Sl6
Si 1
1/2
1/2
0
0
0
0
0
0
0
0
0
0
0
0
0
S2 S3 S4 S5 85 S7 Sg S9 Sio 0
0
0
1/2
1/2 0
1/2
0
0
0
0
0
0
0
0
0
0
0
0
0
1/2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
1 / 2 0 0 0 0 0
1/2 0
1/3
0
0
0
0
0
0
0
0
0
0
0
0 0 0 0 0
0 0 1/3 1/3 0
1/3 0 0 0 1/3 1/3
0
0
0
0
0
0
0
0
0
1/3 0 0 1/3 0
1/5 1/5 1/5 0 1/5
0 1/3 0 1/3 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
Sii 0
0
0
0
0
0
0
1/3
0
0
0
1/3
0
0
0
0
S12 Si3 Si4
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0
0
0
0
0
0
0
0
0
Sl5 S16 0 0
1/5 0
0 1/3 0
1/2 0 1/2
0 1/3 0
1 0 0
0 0 0
0 0 1/2
0 0 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0 0
0 0
1/3 0
0 0
1/2 1/2
0 1/2
0 1
(2-51)
Fig.2-18 presents results for S(0) = [0,0,0,0,0,1/3,1/3,0,1/3,0,0,0,0,0,0,0]
indicating equal initial probabilities, 1/3, of occupying 85, 87 and 89 (6, 7, 9 in
Fig.2-17). 8ince this initial condition enables the bird to fly to the left or to the
right, it is observed in Fig.2-18 that at steady state the bird has a probability of
47.2% to occupy 81 (left) and 52.8% to occupy Si6(right). This behavior is
explained by the fact that 81 and S16 are absorbing (dead) states, thus, the
occupation probability of other states must diminish versus time.
70
(0
0.6
0.5
0.4 h
0.3
0.2
»AaiiUUamM»
15 20
Fig.2-18. Occupation probability of states by the bird
Table 2-1 shows limiting occupation probabilities (for n = ©o) for the case where
the initial occupation of the bird is of a certain state, i.e., Si(0) = 1, i = 1, 2, ....,
16.
Table 2-1. Limiting occupation probabilities of states
Si and Si6 for Si(0) = 1
i 1 1-5
6
7
8
9
10
1 11-16
Si(oo)
1
0.542
0.542
0.292
2/3
0.292
0
Sl6M
0 1 0.458
0.458
0.708
1/3
0.708
1 _
71
Generally, the results comply with the assumptions spelled out above. For
example, if the bird is initially at states Si to S5, it will eventually occupy state Si.
If it is initially at states Si 1 to Si5, it will finally occupy state Si5. If it is initially at
states S6 to Sio, it will at the end occupy state Si or S16 depending on the
magnitude of the relative probability.
Real life examples Examples 2.24 and 2.25 were inspired by the assassination of the Prime
Minister of the state of Israel Itzhak Rabin (1922-1995) on Saturday, November
4th, 1995. During his life, Rabin was a soldier, the Chief of Staff, participating in
all Israeli wars and the greatest motivating force for peace in the Middle East.
Thus, the examples deal with Life and Death as well as with Peace, War and No
peace-No war situations.
Example 2.24 assumes two states a man, designated as system, can
occupy, viz.. Si = Life, S2 = Death. The two states may be expressed by the
following 2x2 matrix:
P = Si
S2
Si
q
0
S2
l-q
1
(2-51)
where q is a parameter depending, among others, upon the age of the man, his
profession (soldier, university professor, worker), health, etc. For q = 0, the
matrix is of the "dead-state" type, i.e., once the system occupies S2, it will remain
there for ever. In other words, if the man is initially alive, after one step he will
die. Fig.2-19 presents results for S(0) = [1, 0] and q = 0.5, i.e., the man is
initially alive and his probability to occupy this state is 50%. His probability to die
is also 50%. It is observed that after ten years the system will occupy state S2,
namely, "dead state". It should be noted that si(n) = 1/2" where S2 is occupied
only as n-^00. Of course si(lO) = 1/2 ^ is nearly zero for all practical purposes.
72
c^ (/)"
0 2 4 6 8 10
n
Fig.2-19. Probability of remaining alive
Example 2.25 presents a three-state model for which S\ = War-No war,
S2 = War and S3 = Peace. The system is at some state. The matrix for this case
is the following one:
P =
Si
S2
S3
Si
q r
u
S2
P t
V
S3 1-q-p
l-r-t
1-u-v
(2-52)
Eq.(2-52) is a multi-parameter matrix which depends on time and other factors not
easy to evaluate. This is because we deal with complicated states. Fig.2-20
presents results for S(0) = [0, 1, 0], i.e., the system (some country) is initially at a
state of war. The following values were also assumed for the transition
probabilities: q = p = 0.1,r = 0.5, t = 0, u = 0.2 and v = 0.1. It is observed that
after five steps the system approaches a steady state for which S(5) = [0.206,
0.091, 0.704]. The state vector indicates that the chances for peace are quite high,
70.4%, promising a bright future.
73
0.8
0.6
0.4
0.2
Fig.2-20. Dynamical probability of Si = War-No war, S2 = War and
S3 = Peace
A very interesting behavior may be obtained by varying the initial state vector S(0).
It is observed that the steady state behavior is independent of S(0), where such a
Markov chain, later discussed, is defined as ergodic.
Example 2.26 is a simulation of a tennis game. The system is a tennis
ball for which the following states are defined and schematically depicted in Fig.2-
21a. Si = the ball is down on the ground on the right-hand side, briefly
designated, DR (down right); S2 = the ball is up in the air on the right-hand side,
UR (up right); S3 = the ball is up in the air on the left-hand side, UL (up left); S4 =
the ball is down on the ground on the left-hand side, DL (down left).
74
S3=UL=UpLeft
m
S4=DL= Down Left
® - tennis ball
S2 = UR = Up Right
o
Fig.2-21a. Scheme of the states in a tennis game
The corresponding matrix reads:
S i=DR S2 = UR S3 = UL 84 = DL
P =
Si
S2
S3
S4
1
P21
P31
0
0
0
P32
0
0
P23
0
0
0
P24
P34
1
(2-53)
where the one-step transition probabilities are:
P21 = UR -^ DR, P23 = UR -^ UL, p24 = UR -^ DL,
P31 = UL -» DR, P32 = UL -> UR, p34 = UL -^ DL.
pi 1 = P44 = 1 means that once the ball is on the ground, it will remain there until the
game starts again. P22 = P33 = 0 indicates that if the ball is in the air the game must
go on. It should also be noted that the various probabilities in the matrix depend on
the characteristics of the players which depend on time and their talent. In the
following demonstrations the probabilities remain unchanged. Assume the
following values:
P2i= 0.01, p23 = 0.99, p24 = 0,
P31=0, P32 = 0.8, p34 = 0.2
75
The above data indicate that the tennis player on the right-hand side of the tennis
court is a better one. This is because p2i is significantly lower than P34, i.e., the
probability to hit the ball to his court is lower. On the other hand, both players will
always hit the ball up to the air to the other court, i.e., p3i = p24 = 0. Fig.2-21b
presents results for the dynamics of the tennis game corresponding to S(0) = [0, 1,
0, 0] indicating that the ball is initially at the right-hand side court. It is observed
that as time goes by, the probability for state S4 to be occupied is increasing, and
the player on the right-hand side is going to win the game. The steady state value
of the state vector reads S(68) = [0.048, 0,0, 0.952], indicating the above trend.
c 0)"
0.8
0.6
0.4
0.2
n
1 ] 1 1
— - "" r 1 ^ '
• . : : : : , ! " sjn), UR
: . 1 A'\*
• ; • ^' .' I i: J/; : ;:
_X \\ »[ 'fT
u / • 1 ' l 1 • 1 • 1 %
1 • 1 ) I * ' ,'•
'''{'-' • ' . - y •••• • • • r « *\
» %»
1
NFV
1 1
s <n), DL 4 1
H
J
s,(n), DR J
/ 1 '^ i\x ' v ' J - . .
10 15 20 n
25 30 35 40
Fig.2-21b. Dynamics of a tennis game (better player on the
right-hand side)
Assume now the following values for the one-step transition probabilities:
P21 = 0.01, P23 = 0.99, P24 = 0,
P31 = 0.12, p32 = 0.8, P34 = 0.08
Considering the values of p2i, P23 and P24, indicates that the performance of the
tennis player on the right-hand side is unchanged. However, the player on the left-
hand side improved his performance; there are good chances he can now hit the ball
from his court down to the ground on the right-hand side, i.e., p3i = 0.12 instead
of nil before. This situation is reflected in Fig.2-22 where the probability for
occupying Si is increased, which is opposite to the behavior in Fig.2-21b. At
76
steady state S(68) = [0.619, 0, 0, 0.381], indicating the above trend. It is
interesting to note that if p3i is increased, i.e., p3i = 0.2 ( p32 = 0.8, P34 = 0),
S(68) = [1, 0, 0, 0], i.e., the left-hand player has excellent chances to win the
game at the end.
1
0.8 h
0.6 h-
0.4
0.2
p 1 p 1 t *
s s H
I ' l "J • 1 ^ 1 , J '
1 , J '
/::'"
• « ' 1 ' 1 '
1
'! ^ •1 • I',
1
^ X " "
' '• •"'• 1 ' •"• .'
UR
1
s/n). DR
s,(n). DL
'\lx V .,r,.**
• =d
H
10 20 n
30 40
Fig.2-22. Tennis game dynamics of the better player
(on left-hand side)
Example 2,27 is a simulation of an ideal pendulum, taken as system,
where drag force is ignored. The four states assumed for the system, demonstrated
in Fig.2-23a, are: Si = maximum height of the pendulum on left-hand side; S24eft
= minimum height of the pendulum while reaching this point from left-hand side;
S2,right = minimum height of the pendulum while reaching this point from right-
hand side; S3 = maximum height of the pendulum on the right-hand side. The
above states are schematically depicted in Fig.2-23a
77
Fig.2-23a. Scheme of the states of an ideal pendulum
The governing one-step transition matrix is given by:
P =
Si
S2,left
S2,right
S3
Si
0
0
1
0
S2,left
1
0
0
0
S2,right
0
0
0
1
S3
0
1
0
0
(2-54)
If the following initial state vector is assumed S(0) = [1, 0, 0, 0], i.e., the
pendulum is initially at Si, the behavior depicted in Fig.2-23b is obtained. As
seen, S\ is occupied after four steps whereas states S2,right and S2,ieft» which are
practically the same point, are occupied after two steps, once from left-hand side
and once from right-hand side. As observed also, the behavior of the system is
un-damped oscillating. The more complicated case of a damped oscillations may be
obtained by varying the transition matrix versus the number of steps.
78
I' V \ V 2 4 6
n
Fig*2-23b. The behavior of an ideal pendulum
Example 2.28, a student progress scheme at a university, is a slightly
modified version of the example appearing in [7, p.30]. In the faculty of
Engineering at Ben-Gurion university of the Negev in Israel, a student, the system,
is studying for four years. The following states are assumed: Si-first year; S2-
second year; Ss-third year and S4-fourth year. Additional states are: Ss-the student
has flunked out; S6- the student has graduated. Let r be the probability of flunking
out, p the probabiUty of repeating a year and q the probability of passing on to next
year where r + p + q = 1. The six states are governed by the following matrix:
P =
Si
S2
S3
S4
S5
S6
Si
P 0
0
0
0
0
S2
q
p 0
0
0
0
S3 0
q
p 0
0
0
S4 0
0
q
p 0
0
S5 r
r
r
q 1
0
S6
0 1 0
0
r
0
1
(2-55)
The first situation considered is a student having some probability of
repeating a year, p = 0.1, and good chances for passing to next year, i.e., q = 0.9;
79
thus, his flunk out probabiUty r = 0. In addition, the student is a first year student,
thus S(0) = [1, 0, 0, 0, 0, 0]. Fig.2-24 demonstrates the progress of the student at
the university, where after eight steps (years) the probability of graduating is
100%, S6(8) = 1. The reason for not graduating after four years are his slight
chances, 10%, for repeating a year. If p = 0, the student will graduate exactly after
four years.
0.8 h
0.6
0.4
0.2 h
4 n
Fig.2-24. Student's progress dynamics at the university in the
absence of flunk out probability (r = 0)
Fig.2-25 demonstrates the progress of the student with good chances to pass
to next year, however, there are chances of 16% to flunk out. The prominent result
is that at his fourth year, he has only 50% chances for graduating, i.e., he has equal
probabilities to occupy S5 or 85. The probability of occupying the other states has
diminished.
80
CO"
1 \ \
p = 0, q = 0.86, r = 0.16
8
Fig.2-25. Student's progress dynamics at the university in the
presence of flunk out probability (r = 0.16)
Examples 2.29-2.31 relate to weather forecast, whereas example 2.31
demonstrates a way to improve the forecast by considering previous days
information.
Example 2.29 [4, p.78] predicts the weather in Tel Aviv (Israel) by a two-
state Markov chain during the rainy period December, January and February. The
system is the city of Tel Aviv where the states it can occupy are: Si = D, Dry day
and S2 = W, Wet day. Using relative frequencies from data over 27 years it was
found that during this season the probability of wet day following a dry day is
0.250 (= pi2) and the probability of a dry day following a wet day is 0.338 (=
p2i). These data resulted in the following transition matrix:
P =
Si
S2
Si = D
0.750
0.338
S2 = W
0.250
0.662
(2-56)
81
Results of the calculation of the state vectors S(n), are depicted in Fig.2-26
for different initial state vectors S(0) spelled out at the top of the figure. Thus for
example, given that January 1st is a dry day (right-hand side of Fig.2-26), yields a
probability of 0.580 that January 6th will be a dry. If January 1st is a wet day
(middle of Fig.2-26), then the probability that January 6th is a dry day is 0.586.
However, after ten days, the equilibrium conditions has for all practical purposes
been reached. Thus, for example, if we call December 31st day 0 and January 10th
day 10, then whatever distribution S[0] we take for day 0, we find that S(10) =
[0.575, 0.425]. Such a Markov chain is defined as ergodic and is without memory
to the initial state.
1
0.8 ^ 0.6 ^-0 .4
0.2 0
S(0) = 1 1
I s n)
1 1
[0.5. 0.5] 1 1
1 1
S(0) = [0,1] — \ — \ — r
J I \ L
S(0) = \ ' ' X?iln)
/ 1 1
= [1.0] 1 1
1 1
-\
—=i
0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10
n n n Fig.2-26. Weather in Tel Aviv for different initial weather conditions
Example 2.30 is related to the weather in the land of Oz [7, p.29] where
they never have two nice days one after the other. If they have a nice day, they are
just as likely to have snow as rain the next day. If they have snow (or rain), they
have an even chance of having the same the next day. If there is a change from
snow or rain, only half of the time is this a change to a nice day. The system is the
land of Oz whereas the states it can occupy are: S] = Rain; S2 = Nice; S3 = Snow.
The transition matrix reads:
P =
Si
S2
S3
Si
1/2
1/2
1/4
S2 1/4
0
1/4
S3 1/4
1/2
1/2
(2-57)
Three cases, differing by the initial state vector S(0) given above each graph, were
studied. The results of the computations are depicted in Fig.2-27, indicating that
82
after two steps (days) the weather will always be snowy; it will continue like this
independent of the weather two days before, i.e., on S(0). The limiting behavior is
similar to that obtained in example 2.29, indicating an ergodic Markov chain.
S(0) = [1.0,0] S(0) = [0,1,0]
vsjn) / ^a^^H
S(0) = [0.0,1]
<t(") 1 L
k
r~
\
1
v /
. s/n)
1,
1 J
=%wj i 1 ;
3 0
Fig.2-27. Weather forecast
1 2 n
Example 2.31. In examples 2.29 and 2.30 the Markov property clearly
held, i.e., the new step solely depends on the previous step. Thus, the forecast of
the weather could only be regarded as an approximation since the knowledge of the
weather of the last two days, for example, might lead us to different predictions
than knowing the weather only on the previous day. One way of improving this
approximation is to take as states the weather of two successive days.
This approach is known as expanding a Markov chain [7, p.30, 140], which
may be summarized as follows. Consider a Markov chain with states Si, S2, ...»
Sz. The states correspond to the land of Oz defined as system. We form a new
Markov chain, called the expanded process where a state is a pair of states (Si, Sj)
in the original chain, for which pij > 0. We denote these states by Sij. Assume
now that in the original chain the transition from Si to Sj and from Sj to S^ occurs
on two successive steps. We shall interpret this as a single step in the expanded
process from the state Sij to the Sjk- With this convention, transition from state Sij
to state Ski in the expanded process is possible only if j = k. Transition
probabilities are given by
P(ij)(jl) = Rl; P(ij)(kl) = 0 for j 9i k
Consider now example 2.30 applying the expanded process approach.
Making the following designations, S = Snowy day, N = Nice day and R = Rainy
day, yields the following states: Si = NR, S2 = NS, S3 = RN, S4 = RR, S5 = RS,
S6 = SN, S7 = SR, Sg = SS. Note that NN is not a state, since PNN = 0 in the
83
original process in example 2.30. The transition matrix for the expanded process
is:
RR RN RS NR NS SR SN SS
P =
RR
RN
RS
NR
NS
SR SN
SS
1/2 0
0
1/2
0
1/2
0
0
1/4 0
0
1/4
0
1/4
0
0
1/4
0
0
1/4
0
1/4
0
0
0
1/2
0
0
0
0
1/2
0
0
1/2
0
0
0
0
1/2
0
0
0
1/4
0
1/4
0
0
1/4
0
0
1/4
0
1/4
0
0
1/4
0
0
1/2
0
1/2
0
0
1/2
(2-58)
The results of the computations are depicted in Fig.2-27a for two initial state
vectors given at the top of the graphs. The one on the left-hand side gives 50%
chances for the two previous days to be either Rainy-Nice or Snowy-Nice. The
one on the right-hand side gives 100% chances for the two previous days to be
Snowy-Rainy. The prominent observation is that the probability distribution of the
states becomes unchanged from the 7th day on and is independent of S(0). It is
given by S(7) = [0.2, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.2], indicating that the weather
on the 7th day has 20% chances to be either Rainy or Snowy. If it was Snowy on
the 7th it will remain like this; if it was Rainy, it will continue Rainy. Recalling the
previous example, where only the information on the previous day was taken into
account, it was obtained that the weather in the winter season will be 100% Snowy
from the 3rd day on.
S(0) = [0,0.5,0,0,0,0,0.5,0] S(0) = [0,0,0,0,0,1,0,0] T — \ 1 — I 1 — r
S2(n) = s (n) = ...=s^(n)
h s (n)
1—I—I—\ 1 — r
8 (11) = s^(n)=... = s (n)
(n) = s^(n)H
Fig.2-27a. Weather forecast by the expanded Markov process
84
Examples 2.32-2.33 below treat the behavior of a drunkard. In the first
example the new step solely depends on the previous step. In example 2.33, the
behavior depends on the last two steps.
Example 2.32. A drunkard, the system, is living in a small town with
four bars, states Si = i, i = 1, ..., 4. As time goes by, the drunkard jumps from
one bar to the other according to the following one-step transition matrix. As seen,
the probabilities of moving from one bar to the other are equal; also, the drunkard
eventually leaves the bar to the next one, i.e. pii = 0.
P =
Si
S2
S3
S3
Si S2 S3 S4
0 1/3 1/3 1/3
1/3 0 1/3 1/3
1/3 1/3 0 1/3
1/3 1/3 1/3 0
(2-59)
Fig.2-28 shows the behavior of the drunkard corresponding to two initial
state vectors S(0) shown above the graphs. It is observed that after six steps the
behavior of the drunkard reaches a steady state where his chances to visit any of the
bars on the next step are 25%. The steady state is independent of S(0), thus the
chain is an ergodic one.
1
0.8
^ 0.6
0)" 0.4
0.2
0 (
S(0) = [1,0,0,0] . 1 1 1 1 1
J}f s (n) = S3(n) = s (n) _
AA^-D 1 2 3 4 5
n 5 (
S(0) = [0.5,0,0,0.5] 1 1 i 1 1 1
- s {n) = s (n) J
/ I 1 1 1 1 1 D 1 2 3 4 5 6
n
Fig.2-28. The drunkard behavior
Example 2.33 applies the expanded process approach (elaborated in
example 2.31) for the drunkard, the system, in example 2.32. The states are
designated in the present example by Sy indicating that before moving to state Sjk
at step n+1, the drunkard spent in state Si some time at step n-1, and at step n he
85
visited state Sj. In this way, the effect of the last two steps, rather than one step, on his next visits can be studied. Considering the assumptions made in example 2.32, yields the following matrix where q = 1/3:
Si2 Si3 Si4 S21 S23 S24S31 S32 S34 S41 S42 S43
O O O q q q O O O O O o l
P =
S12
Sl3
Si4
S2I
S23
S24
S31
S32
S34
S41
S42
S43
0
0
q 0 0
q 0 0
q
0
0
0
0
q 0 0
q 0 0
q 0
0
0
0
q 0 0
q 0 0
q 0
0
0
0
0
0
0
0
q 0 0
q 0
0
0
0
0
0
0
q 0 0
q 0
0
0
0
0
0
0
q 0 0
q 0
q
0
0
q
0
0
0
0
0
0
q
q 0
0
q
0
0
0
0
0
0
q
q 0 0
q
0
0
0
0
0
0
q
0
q 0 0
q 0 0
q 0
0
0
0
q 0
0
q
0
0
q
0
0
0
(2-60)
The results of the computation for two initial state vectors given at the top of the graphs, each comprising of twelve states, are depicted in Fig.2-29. They indicate that after 7 steps, the drunkard is always at the same situation, i.e., his chances to occupy the next state, depending on his past deeds of two steps before, are 8.3%. This situation is independent of his initial step, which is plausible recalling that we deal with a drunkard.
S(0) = [0.5,0 0,0.5]
0.2
0
" 1 — I — I — I — i — I — r s,(n) = s (n)
kl
s^(n) = . . . = Sg(n)
\ s (n) =... = s (n) \ / 9 12
l^' \ L_J I L 1 2 3 4 5 6 7 8
n
S(0) = [1,0 0] I 1 1 1 1 1 1
- \ s (n) =... = s (n)
\ ' J *i" '7" '""i "f^
1 ]
"r 2 3 4 5 6 7 8
n
Fig.2-29. The drunkard*s behavior according to the expanded Markov process
86
Example 2.34 deals with actuarial considerations needed for premium
calculations. The problem may be presented by assuming the following states that
a system, the customer of an insurance company, may occupy: Si = Healthy
customer, S2 = Handicapped customer, and S3 = Dead customer. The following
matrix clarifies the interaction between the states
P =
Si
S2
S3
Si S2 S3
Healthy Handicapped Dead
p q 1-p-q 0 r 1-r
0 0 1
(2-61)
For example, the matrix indicates that a healthy man can remain healthy, can
become handicapped or die, i.e., p n = p, pi2 = q and pi3 = 1- p - q. If he is
handicapped, he can never become healthy again, thus P2i = 0, and if he is dead,
he is in the so-called dead state. It should be noted that the parameters p, q and r
depend strongly on age but are taken here as constants.
Fig.2-30 demonstrates the situation of a young man and an old man after five
steps (years); both are initially healthy, i.e., S(0) = [1, 0, 0]. It has been assumed
that the young man has 95% chances to remain healthy (p = pn = 0.95) whereas
the old man has 50% chances to remain in Si (p = p n = 0.5); other quantities
shown on the graphs, r and q, are identical. It is observed in the figure that the
probability of the young man to remain healthy after five years are quite good, si(5)
= 0.774; for the old man si(5) = 0.0313, quite small to remain healthy. An
interesting observation is related to the effect of r, i.e. to remain handicapped, on
the state vector S(n); for example:
Young man: S(5) = [ 0.774, 0.0613, 0.210] r = 0
S(5) = [ 0.774, 0.0905, 0.136] r = 1
Old man: S(5) = [ 0.0313, 0.0380, 0.930] r = 0
S(5) = [ 0.0313, 0.0013, 0.968] r = 1
It may be observed that si(5) is not affected by r where the other quantities are
influenced by varying r.
87
1
0.8
0.6 c^ vT 0.4
0.2
0
Y(
-
- p = 0.95 q = 0.02
- r = 0.97
-
3ung man i 1
-^ji!!L
-
-
sjn) s (n) -
2 3 n
Old man
Fig.2-30. The future dynamics of a young man and an old man
Example 2.35 is the Ehrenfest diffusion model [6, p.21] for a simple
random walk with reflecting barrier presented in example 2.18. The model
assumes two containers A and B containing Z molecules. The containers are
separated by a permeable membrane so that the molecules may move freely back
and forth between the containers. It is assumed that at each instant of time t, one
of the Z molecules chosen at random, is moving from one container to the other.
The system are molecules in container A and the state Sj of the system is the
number of molecules in container A which equals j - 1. Thus, the following states
are assumed: Si = 0, S2 = 1, S3 = 2, S4 = 3, ..., Sz+i = Z molecules. In the
Ehrenfest model, if A has j molecules, i.e., it is in state Sj+i, it can on the next step
move to Sj or to Sj+2 with probabilities
Pj+lj = j/Z; Pj+i,j+2 = (Z - j)/Z; j = 1,..., Z (2-62)
pj+ij+i = 0; j = 0 ,1 , . . . , z
The transition probability matrix is then given by:
88
P =
Si
S2
S3
S4
Si S2 S3 S4 85
0 1 0 0 0
1/Z 0 (Z-l)/Z 0 0
0 2/Z 0 (Z-2)/Z 0
0 0 3/Z 0 (Z-3)/Z
• Sz
. 0
. 0
. 0
. 0
Sz+1 0
0
0
0
Sz
Sz+1
0
0
0
0
0
0
0
0
0
0
... 0
... 1
1/Z
0
(2-63)
Fig.2-31 presents results for total number of molecules Z = 3 and S(0) = [1,
0,0,0]. The initial state vector indicates that the system (molecules in container A)
is initially at Si, i.e., does not contain any molecule. In other words the three
molecules are in vessel B. It is observed in the figure that the system attains the
following two sets of constant values:
S(9) = [1/4, 0, 3/4, 0], S(10) = [0, 3/4, 0, 1/4]
S(l l ) = [1/4, 0, 3/4, 0], S(12) = [0, 3/4, 0, 1/4]
The results indicate that states S2 and S3 have the same occupation
probability which oscillate against time. Note that state S2 corresponds to one
molecule in container A where in state S3 two molecules should occupy container
A. Thus, if at step n = 10 there is one molecule in A, two molecules will occupy
container B. If at step n = 11 there are two molecules in A, then one molecule will
occupy B and this process repeats its self ad infinitum.
89
CO"
0.8
0.6
0.4
0.2 h"
Fig.2-31. The approach towards equilibrium for a total number of
molecules in the containers Z = 3
Fig.2-32 presents results for an even number of molecules in the containers,
i.e., Z = 8 and an identical initial state vector as before, S(0) = [1, 0, 0, 0]. It is
observed that the system attains the following behavior after about 20 steps, which
is similar to that before, i.e.:
S(25) = [0, 0.063, 0, 0.438, 0, 0.437, 0, 0.062, 0]
S(26) = [0.008, 0, 0.219, 0, 0.547, 0, 0.218, 0, 0.008]
S(27) = [0, 0.063, 0, 0.438, 0, 0.437, 0, 0.062, 0]
S(28) = [0.008, 0, 0.219, 0, 0.547, 0, 0.218, 0, 0.008]
The results indicate that S4(25) = S6(25) and that states S4, S5 and 85 have the
highest occupation probability which oscillate against time; other state probabilities
are lower. Note that S4 corresponds to three molecules in container A, 85 to five
molecules where in 85 four molecules should occupy container A. Thus, if at step
n = 25 there are three or five molecules in A, because the states are of equal
probability, the mean value is four. Thus, since the total number of molecules is m
= 8, four molecules will occupy container B. If at step n = 26 there are four
molecules in A corresponding to 85 with the highest probability, then also four
molecules will occupy container B. Therefore, at steady state the eight molecules
will be equally distributed between the two containers.
90
CO"
Fig.2-32. The approach towards equilibrium for a total number of
molecules in the containers Z = 8
The general conclusion drawn is that in both cases, i.e., with odd and even
number of molecules in the containers, the tendency of the system, molecules in
container A, is to shift towards an equilibrium state of half molecules in each
container. This trend is also expected on physical grounds.
Example 2.36 is the Bernoulli-Laplace model of diffusion [15, p.378],
similar to the one suggested by Ehrenfest. It is a probabilistic analog to the flow of
two incompressible liquids between two containers A and B. This time we have a
total of 2Z particles among which Z are black and Z white. Since these particles are
supposed to represent incompressible liquids, the densities must not change, and so
the number Z of particles in each container remains constant. The system are
particles in container A of a certain color and the state Si of the system, is the
number of these particles in container A where
Si = i - l ( i = l , 2 , ...,Z+1) (2-64)
If we say that the system is in state Sk = k - 1, i.e., the container contains k - 1
black particles, this implies that it contains Z - (k -1) white particles. The transition
probabilities for the system are:
91
pj+l j = 0/Z)2; pj+i,j+i = 2j(Z - j)/Z2; pj+i.j+2 = ((Z - j)/Z)2 ; j = 1,..., Z
Pj+l,k+l = 0 whenever i j - k I >1; j = 0, 1 ,..., Z (2-65)
The transition probability matrix is then given by:
P =
Si
S2
S3
S4
Sz
Sz+1
Si 0
P21 0
0
S2 1
P22
P32 0
Ss 0
P23
P33 P43
S4 0
0
P34
P44
Ss .. 0 ..
0 ..
0 ..
P45 ••
. Sz
0
0
0 0
Sz+1 0
0
0 0
0
0
0
0
0
0
0
0
0
0 Pzz Pz,z+1
1 0
(2-66)
where
P21 = (1/Z)2; P22 = 2(Z - 1)/Z2; p23 = ((Z - 1)/Z)2
P32 = (2/Z)2; p33 = 4(Z - 2)/Z2; P34 = ((Z - 2)/Z)2
P43 = (3/Z)2; P44 = 6(Z - 3)/Z2; P45 = ((Z - 3)/Z)2
Pzz = 2(Z-l)/Z2; p2,2+i = l/Z2
Fig.2-33 presents results for 2Z = 8, i.e., 4 black and 4 white particles. Two
initial state vectors S(0) were considered, given at the top of the figure. On the left-
hand side, it is assumed that initially container A contains 2 black particles. On the
right-hand side, there are 50% chances that container A will contain one black
particle and 50% to contain four black particles. The results indicate that after ten
steps, the system attains a steady state, independent of the initial conditions, where
S(10) = [0.014, 0.229, 0.514, 0.229, 0.014]. Such a process is known as
92
ergodic Markov chain later discussed. As seen, S3(10) = 0.514 corresponding to
two black particles in each container at steady state, is the highest probability with
respect to the other states (Si = 0, S2 = 1, S3 = 2, S4 = 3 and S5 = 4). In other
words, the state of the highest probability will exist at steady state; this is also the
expected physical result. An interesting behavior observed in Fig.2-33 on the left-
hand side is the following. In S(0), it was assumed that S3 = 2 has 100%
probability. As seen, the probability of this state, i.e., S3(n), remains always the
highest along the path towards equilibrium until it reaches its ultimate value. On the
other hand, on the right-hand side, the state S3 = 2 had initially a probability of
0%. As seen, along its approach towards equilibrium, S3(n) is continuously
increasing and attains the value of S3(10) = 0.514, which remains constant from
thereon.
1
0.8
^ 0.6
^' 0.4
0.2
0
S(0) = [0,0,1,0,0] T T
\,>_>fe.''^'___3 . s (n) = s (n)
s (n) = s (n)
S(0) = [0,0.5.0,0,0.5]
4 6 n
10
Fig.2-33. The dynamics of approach towards equilibrium for a total
number of particles in the containers 2Z = 8 (4 black and 4 white)
Example 2.37 considers the random placement of balls [15, p.379] where
a sequence of independent trials, each consisting in placing a ball at random in one
of Z given cells, is performed. The system are balls and the state of the system is
the number of cells occupied by the balls. The state Si of the system is given by:
Si = i - 1 (i = 1, 2,..., Z+1) (2-67)
Thus, if we say that the system is in state Sk, this implies that k-1 cells are occupied
and Z - (k -1) cells are still free. For the placing process of the balls in the cells, the
following transition probabilities apply:
Pj+lo+l=j/Z; pj+i,j+2 = (Z-j)/Z; j = 0, 1,...,Z
yielding the following matrix:
Si
S2
S3 S4
Si 0
0
0
0
S2 S3 1 0
P22 P23
0 P33 0 0
S4 0
0
P34 P44
S5 .. 0
0 ..
0 ..
P45 ••
. s 0
0
0 0
Sz+1 0
0
0 0
Sz
Sz+1
0 0
0 0
0
0
0
0
0
0
Pzz Pz,z+1
0 1
where
93
(2-68)
(2-69)
P22 = 1/Z; P23 = (Z - 1)/Z; P33 = 2/Z; p34 = (Z - 2)/Z
P44 = 3/Z; P45 = (Z - 3)/Z; Pzz = (Z - 1)/Z; pz,z+i = (Z - 1)/Z
Fig.2-34 presents results for three cells, i.e., Z = 3. Thus, the four states
are: Si = 0, S2 = 1, S3 = 2 and S4 = 3 cells occupied. S2 = 1 indicates that one cell
has already been occupied by the ball. It may be observed that in both cases
depicted in the figure, differing by the initial state vectors S(0), the ultimate
situation is identical, i.e., all four cells are occupied, as expected. This occurs after
15 steps. The maximum in the Si(n) curves, i = 2, 3 is interesting and clear. For
example, S2(n) corresponding to S2 = 1 is attaining a maximum after one step, n =
1, because after Si = 0, S2 must come.
94
S(0) = [1.0,0,0] S(0) =
_s,(n) /
[0.25,0.25,0.25,0.25]
^ wS (n)
0 5 10 15 n n
Fig.2-34. The dynamics of cell occupation by balls for Z = 3
Example 2.38 is concerned with cell genetics [15, p.379] where a Markov
chain occurs in a biological problem which may be described roughly as follows.
Each cell of a certain organism contains N particles, some of which are of type A,
others of type B. The system is a cell and the state of the cell is the number of
particles of type A it contains; there are N + 1 states. A cell is said to be in state i if
it contains exactly i -1 particles of type A, where
Si = i - l ( i = l , 2 , . . . , N + l ) (2-70)
The cell is undergoing the following process. Daughter cell are formed by
cell division, but prior to its division each particle replicates itself. The daughter
cell inherits N particles chosen at random from 2i particles of type A and 2N - 2i
particles of type B present in the parental cell. The probability that a daughter cell
occupies state k is given by the following hypergeometric distribution
(2j)! (2N-2J)! (Nir Pj+i.k+1 j.,(2j _ k)! (N - ]f)!(N - 2j + Ic)! (2N)!
(2-71)
j , k = 0, 1,2, ...,N
Note that pi i and PN+I,N+I = 1; Pj+i,k+i = 0 if the expressions in the parenthesis in
the denominator < 0.
95
The behavior predicted by the model is demonstrated for N = 4, yielding the
following states from Eq.(2-70): S] = 0, S2 = 1, S3 = 2, S4 = 3 and S5 = 4
particles of type A. The corresponding matrix obtained from Eq.(2-71) reads:
Si
S2
= S3
S4
S5
Si
1
0.2143
0.0143
0
0
S2 0
0.5714
0.2286
0
0
S3 0
0.2143
0.5143
0.2143
0
S4 0
0
0.2286
0.5714
0
S5 0
0
0.0143
0.2143
1
(2-72)
Results for the formation dynamics of particles of type A from their cells is
demonstrated in Fig.2-35. Three initial state vectors, given at the top of each
figure, were considered. The calculation indicate that a steady state is reached after
thirty steps (generations), always at Si and S5, which are dead or absorbing states.
For the case on the left-hand side, si(30) = 0.75 and S5(30) = 0.25 where for that
on the right-hand side si(30) = 0.25 and S5(30) = 0.75. The state with the highest
probability is always the one nearest to the state of the highest probability in the
initial state vector. The case in the middle is symmetrical, hence, si(50) = S5(50) =
0.50. It is interesting to mention that the results after sufficiently many
generations, n -^00, comply with the following theoretical predictions. The entire
population will be (and remain) in one of the pure states Si and SN+I ; the
probability of these two contingencies at steady state are
Sj(oo) = 1 - i/N, SN+I(«>) = i/N where i is the number of particles of A at the initial
state. For example, i = S3(0) = 2 and N = 4, for the case on the middle in Fig.2.35. Thus, Si(oo) = S5(oo) = 0.5.
S(0) = [0,1,0.0,0] S(0) = [0,0,1,0,0]
: 1
s,n 4"
_\s,(n) =
\/r
\
s 1
1 1
-
(n) = s(n) 5 —
S(0) = 1
\ \~\
\
\ /C^^<^
[0.0.0,1.0] 1
.^(")
-*f- *-
1
_s_^(n)J
JnT
20 0 10 n
15 20 0 10 n
15 20
Fig.2-35. The formation dynamics of A-type particles for N = 4
96
Example 2.39 is taken from population genetics [15, p.380]. Consider the
successive generation of a population which is kept constant in size by the selection
of N individuals in each generation. A particular gene, assuming the forms A and
a, has 2N representatives. If in the nth generation A occurs i times, then a occurs
2N - i times. The system is the population (such as plants in a com field). The
state of the system is the the number of times that the A-gene occurs after some
generations. A population is said to be in state i if A occurs 2i times, i.e.,
Si = i - 1 ( i = l , 2 , . . . , 2N+ 1) (2-73)
where the number of states is 2N + 1. Assuming random mating, the composition
of the following generation is determined by 2N Bernoulli trials in which the
probability of the occurrence of the A-gene is i/2N. We have, therefore, a Markov
chain with
(2N)! M Yfi M" -k)!V2N>' V 2 N ;
2N-k
Pj+iMi - i,,(2N
j , k = 0 , 1, ...,2N
(2-74)
Note that pn and P2N+i,2N+i = 1; Pj+i,k+i = 0 if the expressions (2N - k) in the
denominator < 0. The above indicates that at states Si and S2N+1» called
homozygous, all genes are of the same type, and no exit from these states is
possible.
The behavior predicted by the model is demonstrated for N = 2, yielding the
following states from Eq.(2-73): Si = 0, S2 = 1, S3 = 2, S4 = 3 and S5 = 4 A-gene
occurrence. The corresponding matrix reads:
P =
Si
S2
S3
S4
S5
Si
1
0.3164
0.0625
0.0039
0
S2 0
0.4219
0.2500
0.0469
0
S3 0
0.2109
0.3750
0.2109
0
S4
0
0.0469
0.2500
0.4219
0
S5 0
0.0039
0.0625
0.3164
1
(2-75)
97
Fig 2.35 demonstrates the dynamical behavior of the A-gene occurrence for
two initial state vectors S(0). The behavior is, in general, similar to example 2.38
where the process is terminated at one of the dead states for which pii = 1. In other
words, generally Si and S2N+1 designate the homozygous states at one of which
the ultimate population will be fixed, depending on the magnitude of the
corresponding probabilities. It should be emphasized that the ultimate results are
dependent only on the initial step where the steady state probabilities are given by
Sj(oo) = 1 - i/(2N), S2N+i(° ) = i/(2N). I is the number of A-genes at the initial
step. For example, for the case on the right-hand side in, I = si(0) = 1 and N = 2. Thus, si(oo) = 0.75 and S5(oo) = 0.25.
1
0.8
^ 0 . 6
" "0.4
0.2
0
S(0) = [0,1,0.0.0] I 1 ' 1
^s^(n) ^ ^ ^
i • y"""^^
\l -Tit 3 - : • '
I
— 1 —
^(n) H
"H
s,(n) 1
10 15
S(0) = [0,1/3,1/3,1/3,0]
^s (n) = Sg(n)
n • n Fig.2-35a. The dynamics of A-gene occurrence for N = 2
15
Example 2.40 is a breeding problem [15, p.380] where in the so-called
brother-sister mating, two individuals are mated. Among their direct descendants,
two individuals of opposite sex are selected at random. These are again mated, and
the process continues indefinitely. With three genotypes AA, Aa and aa for each
parent, we have to distinguish six combinations of parents, designated as states,
which we label as follows: Si = AA*AA, S2 = AA*Aa, S3 = Aa*Aa, S4 = Aa*aa,
S5 = aa*aa, S6 = AA*aa. The system is two individuals of opposite sex which are
mating. Based on above reference, the matrix of transition probabilities reads:
98
P =
Si S2 S3 S4 S5 S6
Si
S2
S3
S4
S5
S6
1 0
1/4 1/2
1/16 1/4
0 0
0 0
0 0
0
1/4
1/4
1/4
0
1
0
0
1/4
1/2
0
0
0 0
0 0
1/16 1/8
1/4 0
1 0
0 0
(2-76)
Fig.2-36 demonstrates the results of brother-sister mating against time. As
expected from the matrix given by Eq.(2-76), containing two dead states S\ and S5
(pil and P55 = 1), the system will eventually occupy one of these states, depending
on the magnitude of the relative probabilities si(n) and S5(n). The latter depends on
S(0) given in Fig.2-36 at the top of the graphs.
S(0) = [0,1,0,0,0,0] S(0)
U h^^(")
W^T R u i^ r i 7^-i i/^ 1
= [0,0,0,0,0,1]
^("L-T-^"^"^
• " ^
'
s^(n) = i
. . • • • L ~ ' . " * '
—
's^"jJ —
-
•.... —• . H
n ' n Fig.2-36. Breeding dynamics
10 15
Examples 2.41-2.42 are examples of random walk type on real life
problems, in addition to examples 2.17-2.22.
Example 2.41 demonstrates a game [6, p.21] related to a simple random
walk between two absorbing barriers, considered also in example 2.17. Assume
that Jacob and Moses have five shekels (Israeli currency) divided between them.
On one side there is a symbol of a lectf and on the other side appears the number
one. The shekels are assumed fair ones, so that the probability of a leaf on a toss
equals the probability of a one on a toss, equals 1/2. Jacob tosses a coin first and
records the outcome, lector one. Then Moses tosses a coin. If Moses matches
99
Jacob (obtains the same outcome as Jacob), then Moses wins the shekel, otherwise
Jacob wins. Note that Moses or Jacob win with probability 1/2. Let the states Si =
i - 1 (i = 1, 2, ..., 6) represent the number of shekels that Moses, the system, has
won. The game ends when Moses has 0 or 5 shekels, i.e., states Si and S6 are
then dead or an absorbing states. Given that Moses is in state k (has k -1 shekels),
he goes to state k + 1 (wins) with probability 1/2 or goes to state k - 1 (loses) with
probability 1/2. The above rules of the game may be summarized in the following
matrix:
P =
Si S2 S3 S4 S5 S6
Si
S2
S3
S4
S5
S6
1
1/2
0
0
0
0
0
0
1/2
0
0
0
0
1/2
0
1/2
0
0
0
0
1/2
0
1/2
0
0
0
0
1/2
0
0
0
0
0
0
1/2
1
(2-77)
where, for example, P45 = 1/2 is the probability that Moses wins a fourth coin
given that he already has three coins.
Fig.2-36 demonstrates the dynamics of the game for two cases. On the left-
hand side is observed that initially Moses is in S2, he has one shekel. As time goes
by, his chances to win are decreasing where after 20 steps his situation is given by
S(20) = [0.8, 0, 0, 0, 0, 0.2]. This means that there are 80% chances he will be
left without money. It is also observed that after one step, there are 50% chances
he will be left without money at all (be in Si) or win one shekel (be in S3) and from
thereon his chances to loose are increasing. So if he is smart, on the one hand, and
knows Markov chains, on the other, he should better stop the game after one step.
On the right-hand side, Moses begins with three shekels, he is in S4, and as time
goes by, his chances to win are increasing where S(25) = [0.4, 0, 0, 0, 0, 0.6].
100
S(0) = [0,1,0,0,0,0]
L s
/ \
%
- ;
S(0) = [0,0,0,1,0,0] 1
,(n)
> s(") . -
'^if-i^^'\\\S^
1
^ — -
. \ l X .
sJn) J
s,(n) _
X ^ - > 15 0 10
n n Fig.2-36. Dynamics of the matching shekels game
15
Example 2.42 deals with a political issue of establishing a coalition in
Israel in the eighties. Generally, the treatment of the problem is based on
the
random walk model incorporating the reflecting-absorbing barrier effects, i.e., p n
= t and P99 =1, respectively, assuming that the maximum size of the coalition
comprise nine parties.
In Fig.2-37, the Israeli caricaturist Moshik [19] demonstrates the efforts
made by Mr. Menahem Begin, the Prime Minister these days, to establish a
coalition. He is seen trying to attract to the coalition additional parties in order to
establish a stable government. As observed in the figure, so far five parties have
joined the coalition.
The system is the coalition headed by Mr. Begin, where a state is the number
of parties in the coalition, i.e.. Si = 1, 2,..., 9. The underlying assumptions of the
"political game" are: a) The probability to increase the coalition from one state to the
next one is p, independent of Si. b) There is a probability, r, that the coalition will
remain unchanged in its size, c) Similarly, there is a probability, q, that the size of
the coalition will decrease due to unsuccessful negotiations. Note that q + p + r =
1.
101
Fig.2-37. Establishment of a coalition in Israel
The following matrix summarizes the above considerations:
P =
Si 1 S2
S3 S4
S5
S6 S7
S8
S9
Si t
q 0 0
0
0
0
0
0
S2 i-t
r
q 0
0
0
0
0
0
S3
0
p r
q 0
0
0
0
0
S4
0
0
p r
q 0
0
0
0
S5 0
0
0
p r
q 0
0
0
S6 0
0
0
0
p r
q 0
0
S7
0
0
0
0
0
p r
q 0
Sg 0
0
0
0
0
0
p r
0
S9
0
0
0
0 0
0
0
p 1
(2-78)
102
Fig.2-38, containing the input data and S(0) above the graphs, demonstrates
results of the calculations with respect to the following points. Note that S(0)
corresponds in all cases to a coalition already with five parties, i.e. S5, as seen in
Fig.2-37. In general, depending on time, a stable coalition consisting of nine
parties will be established because S9 = 9 will, eventually, acquire the highest
probability. In case c, the approach towards a stable coalition is very fast because
the 'negotiation-success factor' p = 0.8 is a relatively high value, in comparison to
p = 0.25 in cases a and b. It is also observed in case c that at each step, after
intensive rounds of talks, the highest probability corresponds to the state of a
higher number of parties, i.e., a larger coalition.
In cases a and b, S5 remains of the highest probability until n = 15; from
thereon the probability of S9 becomes the highest. However, this behavior
depends on the reflecting barrier effect governed by the factor t in the matrix given
by Eq.(2-78). If t = 0 (case a), i.e., an ideal reflector, S9(n) > si(n). If t = 1 (case
b). Si becomes an absorbing or dead state like S9, S9(n) = si(n) and the chances to
establish a stable coalition or to fail are the same. It should be noted that for high
p's, the effect of t is negligible, as observed in case c.
S(0) = [0,0,0.0,1.0,0,0,0.0] S(0) = [0.0.0.0.1,0.0.0.0] \ 1
q = 0.25, p = 0.25, r=: 0.5 J t=1
.(n) s,(n) = Sg(n)
^ - ^ ' X ^ 1 - - - - 1 10 n
15 20
(a) (b)
103
1
0.8
0.6
S(0) = [0,0,0,0,1,0,0,0,0]
s.(n) ' s(n) '
u p ® q = 0.1,p = 0.8, r = 0.1 ttfW^^^ , , t=o-i
V ^ ' - ' " ' .,(n) , _ ^ 10 n
(c)
15 20
Fig.2-38. The dynamics of establishing the coalition
Examples 2.43-2.45 treat a few models of periodic chains, also referred
to as recurrent events. Generally speaking, a system undergoes some process as a
result of which it occupies the states sequentially. The latter repeats its self ad
infinitum or attains some steady state.
Example 2.43, recurrent events 1 [15, p.381], obeys the following
transition probabilities matrix:
P =
Si
S2
S3
S4
S5
Si
Pll 1
0
0
0
S2
P12
0
1
0
0
S3
P13
0
0
1
0
S4 . . . P14 . . .
0 . . .
0 . . .
0 . . .
1 . . .
(2-79)
To visualize the process which generates from above matrix, suppose that
initially the system occupies Si. If the first step leads to Sk-i, the system is
bound to pass successively through states Sk-2, Sk-3, ..., and at the k th step it
104
returns to Si, whence the process starts from scratch passing, in principle, the
above steps again and again.
Practical examples conforming with the above model are the following ones.
A drunkard, the system, treated also in examples 2.32 and 2.33, is acting now
according to different rules dictated by the transition matrix, Eq.(2-79). The states
Si to S4 are four bars in the small town the drunkard is occupying. Another
example is concerned with a dancer, the system, acting as follows. There are four
nice dancers, states S\ to S4, standing on a circle at equal distances. The dancer is
moving from one nice dancer to the other, performing with her a dance, and
moving to the next one. His occupation of states are according to Eq.(2-79). The
interesting question is what happens versus time with the system! Fig.2-39a,b
demonstrate the behavior of the system as a function of the Pij's in the matrix,
Eq.(2-79), and S(0). The characteristic behaviors observed in the figure are: a) In
each state the system oscillates until a steady state is achieved, b) The magnitude
of the steady state is independent of S(0), i.e., the chain is ergodic. c) The
magnitude of the steady state depends on the policy-making matrix, Eq.(2-79),
i.e., on the py's. Regarding to the behavior of the system, the drunkard or the
dancer, it is observed in Fig.2-39a,b that at steady state, it will remain in Si, the
state of the highest probability. The ultimate values of the state vectors are:
Fig.2-39a, S(24) = [ 0.333, 0.300, 0.234, 0.133]
Fig.2-39b, S(10) = [ 0.500, 0.300, 0.150, 0.050]
S(0) = [1,0,0.0] S(0) = [0,0,0,1]
20 0 5 10 15
0.2, P13 = 0.3, P14 = 0.4
105
1
0.8
0.6
S(0) = [1,0,0.0] S(0) = [0,0,0,1] « • " I ^
/^s/") s (n) 83(0) s (n)
b) p u = 0.4, P12 = 0.3, P13 = 0.2, P14 = 0.1 Fig.2-39 a,b. The dynamical behavior of the system
Example 2.44, recurrent events 2 [15, p.382], obeys the following transition-probability matrix:
P =
Si
S2
S3
S4
S5
Si S2 S3 S4 S5
qi Pi 0 0 0
qi 0 P2 0 0
q3 0 0 P3 0
q4 0 0 0 p4 (2-80)
The matrix indicates that the system moves from one state to the other, and upon reaching the new state it has always some probability of returning to the initial state.
Fig.2-40 demonstrates results of the calculations for the case of four states, i.e.. Si,..., S4. The corresponding matrix reads:
P =
Si
S2
S3 S4
Si
qi
q2
qa 1
S2
pi 0
0
0
S3 0
P2
0
0
S4 0
0
P3 0
(2-80a)
106
The input data are reported on the figure. On the left-hand side the values of the
qi's are relatively high in comparison to the right-hand side, thus, the probability of
remaining in or returning to Si is high, as a result of which the approach towards a
steady state is after one step. By increasing qi, the occupation of the states by the
system is of a recurrent type, where eventually a steady state is achieved of the
state-probability distribution. In both cases, the Markov chain is ergodic.
0.8 P
0.6
0.4
0.2
0
S(0) = [1,0,0,0]
_V. s(n) q=0.8, p=0.2;i = 1,2
[_ 2 ") ^ (n) s^n)
-_.-"Iy-.-i.--1
S(0) = [1,0,0.0]
s(n)
.s (n) ^ q=0.2,p. = 0.8;i = 1,2
5 10 15 20 0 5 n
Fig.2-40. The dynamical behavior of the system
10 15 20 n
Example 2.45, recurrent event 5, demonstrates 3. periodic chain [4, p. 102]
by considering the behavior of a Saudi sheilc. Fig.2-41 shows the Saudi sheik,
defined as system, opening the door for an amazing beauty symbolizing the West at
the entrance of his harem. The Israeli caricaturist Moshik [19] demonstrates in the
figure the approaching process of the West towards Saudi Arabia at the beginning
of the eighties. The following model is developed for the caricature, which is
applicable for cases 1 and 2 below. The states are seven Saudi beauties, Si =
Saudi beauty , i = 1, ..., 7 and Sg = Western beauty. The caricature may be
understood in several ways, dictating the construction of the 8x8 one-step transition
probability matrix.
107
P =
Si S2 S3 S4 S5 Se 87 Sg Si
S2
S3
S4
S5
S6 s? Sg
0 0
0
0
0
0
1/2
q
0 0
0
0
0
0
1/2
q
1
1
0
0
0
0
0
0
0
0
1/3
0
0
0
0
0
0
0
1/3
0
0
0
0
0
0
0
1/3
0
0
0
0
0
0 0
0 0
0 0
1/2 1/2
1/2 1/2
1/2 1/2
0 0
0 p
(2-81)
Fig.2-41. The harem of the Saudi sheik
The matrix demonstrates several interesting characteristics, for example: from Si the sheik never goes to S2; similarly, from S2 he never goes to Si; however, from these states he always moves to S3. From S3 the sheik has to decide where to go, because P34 = P35 = P35, As seen, by analyzing the various pij's, which are
108
assumed to remain constant, a complete understanding of the sheik's behavior can be obtained.
Case 1. It has been assumed that in the matrix p = 0 and q = 1/2, indicating that Sg has no preference over the other states. Sg has been located on the third floor of the harem, right to S7 as seen in Fig.-41. It has also been assumed that the sheik is initially occupying Si, i.e., S(0) = [1, 0, 0, 0, 0, 0, 0, 0]. The occupation dynamics of the sheik is obtained by applying Eq.(2-24), i.e., multiplying the state vector by the matrix, Eq.(2-81), yielding Fig.2-42.
c^ ^
0.8
0.6
0.4
0.2
0 2 4 6 8 10 12 14
n Fig.2-42. States occupation dynamics of the Saudi sheik
The prominent observation in Fig.2-42 is the periodic behavior of the system
(sheik), where each state is reoccupied after four steps. Hov ever, in general, the s\\<^\]fi is occupying at each step another state, thus acting very very hard, sooner or later might affect his health. The only way to change the dangerous results predicted by the model, is by modifying the matrix, Eq.(2-81); the latter depends on the habits of the sheik. Thus, he should be advised by his doctor accordingly.
Additional interesting observations are: a) Whenever the sheik is moving to 87, he should consider to occupy instead Sg because the occupation probability of
109
these states is 50%. He has the same problem with Si and S2. b) The occupation
problem of S4, S5 and 85 is more complex since he has to decide among three
beauties, c) No problem with S3; the occupation probability of this state, whenever
reached, is 100%.
Case 2. It has been assumed here that in matrix, Eq.(2-81), p = 1, q = 0
and, as in previous case, S(0) = [1, 0, 0, 0, 0, 0, 0, 0]. The result in Fig.2-43
indicate clearly a significant preference for Sg over the other states. After 43 steps
it is obtained that S(43) = [0, 0, 0, 0, 0, 0, 0, 1], i.e., the sheik will be 'absorbed'
at Sg. Noting the amazing beauty in Fig.2-41, his behavior is not surprising at all.
if)'
zyz-r.
S3(n)
J s^(n) = 85(11) = Sg(n)
A s (n)
,;!i\/!\AA.>:/^v --?<N ^''T** ^**t "^ .l-»i t*«ifc
10 15
n
20 25 30
Fig.2-43. States occupation dynamics of tlie Saudi slieili
Case 3. In cases 1 and 2 the Saudi sheik was the system and the beauties
were the states. Now we look at the caricature in Fig.2-41 from a different point of
view. The system is the Western beauty where the states S j , i = 1, ..., 8, are the
eight rooms occupied by the Saudi beauties. Note that number 8 in the figure,
representing the western beauty in the above cases, designates now room number 8
located on the third floor of the harem, right to S7. The figure shows also that
initially the West is invited by the sheik to join the harem, hoping it will occupy
no only one of the rooms, i.e. number 8 which is the only available one. The behavior
of the system may be deduced from the following single-step transition matrix:
P =
Si S2 S3 S4 S5 S6 S7 Sg
Si
S2
S3
S4
S5
S6 S7
S8
1/5
1/4
1/5
1/8
1/6
0
0
0
1/5
1/4
0
1/8
1/6
0
0
0
1/5
0
1/5
1/8
0
1/4
1/6
0
1/5
1/4
1/5
1/8
1/6
1/4
1/6
1/4
1/5
1/4
0
1/8
1/6
0
1/6
1/4
0
0
1/5
1/8
0
1/4
1/6
0
0 0
0 0
1/5 0
1/8 1/8
1/6 1/6
1/4 0
1/6 1/6
1/4 1/4
(2-82)
Two cases were explored for the dynamical behavior of the West. In the first
case, the West occupies initially Si; this is expressed by the initial state vector S(0)
given on the top of Fig.2-44, left-hand side. On the right-hand side, the state
vector corresponds to the case where the West has equal probabilities to occupy
states S2, S4 and S6, i.e., Si(0) = 1/3, i = 2, 4, 6. The results depicted in the
figure are very interesting, indicating that after ten steps the system has a certain
probability to be found in every state, i.e., the domineering process of Saudi Arabia
by the West is very effective. Moreover, this process after some time, becomes
independent of the initial step, i.e., an ergodic Markov chain which is without
memory to the past. The probability distribution at steady state after ten steps is
given by S(10) = [0.119, 0.095, 0.119, 0.190, 0.143, 0.095, 0.143, 0.095]
noting that the probabilities are not too different from each other.
I l l
S(0) = [1,0,0,0,0.0,0,0] S(0) = [0,1/3,0,1/3,0,1/3,0,0]
s,(n)=s^(n) s^(n)=s^(n)
sjn)= sJn) = Sg(n)
0 5 ^ 10 15 0 5 _ 10 15
Fig.2-44. The domineering dynamics of Saudi Arabia by the West
Example 2.46 is of fundamental importance and completes our real life
examples. It considers the imbedded Markov chain of a single-server queuing
process [4, pp.8, 89]. Queuing is encountered in every comer of our life such as
queues at servers, telephone trunk Unes, traffic in public transportation - bus, trains
as well as airports, queues at surgery rooms in hospitals and governmental offices,
department stores, supermarkets and in a variety of industrial and service systems.
One of the simplest models of queuing is the following one. Let customers
arrive at a service point in a Poisson process [see 2.2-3] of rate X [customers
arriving per unit time]. Suppose that customers can be served only one at a time
and that customers arriving to find the server busy queue up in the order of arrival
until their turn for service comes. Such a queuing policy is called First In First Out
(FIFO). Further, suppose that the length of time taken to serve a customer is a
random variable with the exponential p.d.f. = (probability density function) given
by
f(t) = pe"' ^ ( t>0) (2-83)
f(t) [1/time] is the probability density that the length of time it takes to complete the
service is exactly t, f(t)dt is the probability that the length of time to complete the
service is between t and t + dt where
prob{x > t} = f f(t)dt = f pe'P^dt: Jo Jo
1 - e -PT (2-83a)
112
is the probability to complete the service between 0 - x. For x -> ©o the probability
equals unity, i.e. the probability to complete the service at infinite time is unity.
The constant 1/(3 [time/customer] is the expected mean service time per customer
where p is the mean number of customers being serviced per unit time. The
exponential distribution is employed to describe the service probability distribution
function in many queuing systems. It should also be noted that for any other
distribution the problem becomes intractable. It can also be shown that if the
service of a customer is in progress at time t, and the p.d.f. of the service time is
given by Eq.(2-83), then the probability that the service time is completed in the
time (t, t + At)is:
PAt + 0(At2) (2-84)
where O(At^) denotes a function tending to zero at the same rate as At .
The above queuing process, and more precisely between consecutive times of
two customers who completed to receive their service, can be described also by the
so-called discrete imbedded Markov chain of a single server. In the queuing
literature this process is noted by M/M/1. The first M denotes a Poisson arrival
time, the second M is an identical independent exponential probability distribution
service time and the 1 represents a single server. We define the system as the
queue and the state of the system is the number of customers waiting in the queue.
The state space is the possible number of states, i.e., S] = 0, S2 = 1, S3 = 2,... If
the state space is finite. Si = 0, S2 = 1, S3 = 2, ..., Si = i - 1 customers. It should
be noted that the state of the system is evaluated at the moment the nth customer
has completed to receive her/his service. By observing the state of the system at
these points (completion of service time) the queuing process of the system is
described by the Markov property of absence of memory, i.e., ignoring the past
history of the customers which have already been served. Accordingly, let Xn
denote the number of customers in the queue immediately after the nth customer has
completed his service. Thus, Xn includes the customer, if any, whose service is
just commencing. Then we can write down the following equations:
113
Xn-1+Yn+i (Xn>l) (2-85a)
Xn+1 = I Yn+1 (Xn = 0) (2-85b)
where Yn+i is the number of customers arriving during the service time of the
(n+1 )ih customer. Eq.(2-85a) expresses the fact that if the nth customer does not
leave an empty queue behind him (Xn > 1), then during the service time of the
(n+1 )ih customer, Yn+i new customers arrive and his own departure diminishes
the queue size by 1. If the nth customer leaves an empty queue, then the (n+7)th
customer arrives and depart after the completion of his service during which Yn+i
new customers arrive. We can distinguish by a simple argument between two
types of behavior of the system. The rate of arrival of customers is X and within a
long time to the average number of customers arriving is Xto- As indicated before,
the mean service time per customer is 1/(3. If the service of the customers were to
go on continuously, the average number of customers served during time to would
be pto. Hence, if ^ > P, we can expect the queue of unserved customers to
increase indefinitely; in a practical application when this occurs, customers would
be deterred from joining the queue and X will therefore decrease. If, however, X, <
P the server needs to work only for a total time of about Xto/P in order to serve the
customers: that is the server will be idle, i.e., the process will be in state Si = 0, for
about a proportion 1- X/p of the time.
In the following presentation we assume that customers arrive in a Poisson
process of rate X and that their service times are independently distributed. The
distribution function B(t) (0 < t < <») is the probability that the service time ts
satisfies
B(t) = prob{ts<t} (2-86)
where
dB(t) = B'(t)dt (2-87)
114
needed later is the probability that the service time lies between t and t + dt. B'(t) is
the p.d.f. defined in Eq.(2-83).
A useful quantity based on the above concepts, needed for evaluating the
probabilities in the one-step transition matrix, is the following one
bi = prob{Yn = i} (i = 0,1,2,...) (2-88)
i.e., bi is the probability that the number of customers arriving during the service of
the nth customer is equal to i. The above quantity may be calculated in the
following way. For the Poisson process
Pi(t) = prob{N(t) = i} = t-^KUy/il (i = 0, 1, 2,...) (2-89)
where Pi(t) is the probability that the number N(t) of events occurred (customers
arrived) is equal to i, given that the service time is t. From the above definitions,
Eqs.(2-86) and (2-89), it follows that the product Pi(t)dB(t) designates the
probability that i additional customers will be added to the queue (see Eq.2-85a) if
the service time is between t and t + dt. Since the service time ts varies as 0 < tg <
oo, the probability that the number of customers arriving during the service time is
equal to i, is given by [4, pp.88]
Jo i(t)dB(t) (i = 0,1,2,. . .) (2-90)
From Eqs.(2-85a) and (2-85b) the elements of the transition matrix of the process
Xn are given by the following matrix:
P =
Si
S2
S3
S4
Ss
Si
bo bo 0
0
0
S2
bi
bi
bo 0
0
S3
b2
b2
bl
bo 0
S4
b3
b3
b2
bl
bo
Ss . . . b4 . . .
b4 . . .
b3 . . .
b2 . . .
bl . . .
(2-91)
115
It should be emphasized that the transition matrix, Eq.(2-91), applies to the time
interval between two consecutive service completion where the process between the
two completions is of a Markov-chain type discrete in time. The transition matrix
is of a random walk type, since apart from the first row, the elements on any one
diagonal are the same. The matrix indicates also that there is no restriction on the
size of the queue which leads to a denumerable infinite chain. If, however, the size
of the queue is limited, say N - 1 customers (including the one being served), in
such a way that arriving customers who find the queue full are turned away, then
the resulting Markov chain is finite with N states. Immediately after a service
completion there can be at most N -1 customers in the queue, so that the imbedded
Markov chain has the state space SS = [0, 1, 2, ..., N - 1 customers] and the
transition matrix:
P =
Si
S2
S3
S4
S5
SN-1
SN
SI
bo
bo 0
0
0
0
0
S2
bi
bi
bo 0
0
0
0
S3
b2
b2
bl
bo 0
0
0
S4
b3
b3
b2
bl
bo
0
0
Ss .
b4 .
b4 .
b3 .
b2 •
bl .
0 . .
0 . .
. SN-1
• • bN.2
• • bN-2
• • bN-B
• • bN-4
• • bN-S
• bl
. bo
SN
dN-i
dN-i
dN-2
dN-3
dN-4
d2
di
(2-92)
where dN-i = 1 - (bo + bi + ... + bN-2)-
An example demonstrating the above concepts was calculated for N = 4
states; thus, SS = [0, 1,2, 3 customers]. The corresponding matrix is the
following one where the probabilities were selected arbitrarily.
P =
Si
S2
S3
S4
Si S2 S3 S4
0.5 0.2 0.1 0.2
0.5 0.2 0.1 0.2
0 0.5 0.2 0.3
0 0 0.5 0.5
(2-93)
116
Fig.45 shows results for two initial state vectors S(0). On the left hand-side
the system is initially at Si, i.e., there are no customer in the queue. On the right
hand-side the system is initially at S4, i.e., there are three customers in the queue.
The propagation of the probability distribution indicates that the system (queue)
reaches a steady state independent of S(0), i.e., an ergodic Markov chain. The
calculation indicates that the steady state is achieved after nine steps and the
appropriate state vector reads S(9) = [0.211, 0.212, 0.254, 0.322], i.e., S4 with
three customers is the state of the highest probability. Certainly, if the possible
number of states N (with maximum number of customers N -1) is changed, a new
steady state would have been achieved, which depends also on the matrix, Eq.(2-
93).
S(0) = [1,0,0,0] S(0) = [0,0,0,1] 1 1
\
\ _ \ \
L ^'^ '--- -.
r / " h/ spy
1 1 -]
1 2 3 4 5 0 1 2 3 4 n n
Fig.2-45. The dynamics of a queuing process
2.1-5 Classification of states and their behavior The examples presented in section 2.1-4 indicate that the states of a Markov
chain fall into distinct types according to their limiting behavior. In addition, it is
also possible to identify from the transition matrix some characteristic behavior of
the states. Suppose that the system is initially at a given state. If the ultimate
occupation to this state is certain, the state is called recurrent, in this case the time of
first return will be called the recurrence time. If the ultimate return to the state has
probability less than unity, the state is called transient. On the basis of the
examples presented above, the following classification of states may be suggested
[4, p.91; 6, p.28; 15, p.387].
117
Ephemeral state. A state j is called ephemeral if pij = 0 for every i.
Thus, an ephemeral state can never be reached from any other state. For example,
Si in the matrix given by Eq.(2-43) is such a state since pn = 0 for every i.
Accessible state. A state k is said to be accessible from state j if there
exists a positive integer n such that pjk(n) > 0 where pjk(n) is defined in Eq.(2-26).
For example, according to Eq.(2-49), S4 is accessible from S2 since it may be
obtained from Eqs.(2-32) to (2-34) that p24(2) = p^ > 0. However, in the same
matrix S2 is not accessible from Si since pi2(n) = 0. In Eq.(2-51), S2,..., S16
are not accessible from Si since it is an absorbing state.
Inter-communicating states. Two states j and k are said to
communicate if k is accessible from j in a finite number of transitions, i.e., if there
is an integer n such that pjk(n) > 0. We define Ujk to be the smallest integer n for
which this is true. If j is accessible from k and k is accessible from j , then j and k
are said to be inter-communicating.
As indicated before, in Eq.(2-49), state S4 is accessible from S2 since P24(2)
= p2 > 0. In fact, S2 is also accessible from S4 since it may be shown that P42(2)
= q^ > 0. Thus, S2 and S4 are inter-communicating. It may also be shown that
other states in Eq.(2-49) demonstrate the same behavior.
Irreducible chain. Perhaps the most important class of Markov chains is
the class of irreducible chains. An irreducible chain is one in which all pairs of
states communicate, i.e., pjk(n) > 0 for some integer n where j , k = 1,2,..., Z; the
latter is the number of states. In other words, every state can be reached from
every other state. The following matrix, corresponding to example 2.32
P =
Si
S2
S3
S3
Si 0
1/3
1/3
1/3
S2 1/3
0
1/3
1/3
S3 1/3
1/3
0
1/3
S4 1/3
1/3
1/3
0
satisfies the above condition since it follows from Eqs.(2-32) to (2-34) that pii(2),
P22(2), P33(2), P44(2) > 0 as well as the other pjk's. Pjj(n), defined in Eq.(2-26),
118
is the probability of occupying Sj after n steps (or at time n) while initially
occupying also this state.
Absorbing state. State j is said to be an absorbing (dead or a trapping)
state if the occupation probability of the state by the system satisfies pjj = 1. In
other words, once a system occupies this state, it remains there forever. Examples
of such a state are: 2.8, 2.10, 2.13, 2.15, 2.17, 2.23, 2.24, 2.28, 2.34, 2.37,
2.38, 2.39, 2.40, 2.41, 2.42.
Prior to further definition of states, additional notation is established in the
following! 15, p.387]. Throughout the following chapter we will use fjj(n) as the
probability that in a process starting from Sj, the next occupation of Sj occurs
exactly at the nth step. In other words, we may say that conditional on Sj being
occupied initially, fjj(n) is the probability that Sj is avoided at steps (times) 1, 2,
..., n - 1 and re-occupied at step n. fjj(l) = pjj and for n = 2, 3,...,
fjj(n) = prob{X(r) ^ j , r = 1,..., n - 1; X(n) = j I X(0) = j} (2-94)
It is interesting to demonstrate the above concepts considering Fig.2-0 by Escher.
Assume that X(r) designates locations along the water trajectory after r steps.
Thus, X(l) = 1 is location of the system at point 1, the top of the waterfall, after
one step. Similarly, X(5) = 2 designates location at point 2 after 5 steps. On the
basis of Eq.(2-94), the following probabilities may be established, i.e.:
fll(5) = prob{X(r) 5 1, r = 1,..., 4; X(5) = 1 I X(0) = 1} = 1
fll(3) = prob{X(r) t 1, r = 1, 2; X(3) = 1 I X(0) = 1} = 0
On the basis of fjj(n) the following quantities, later applied, are defined:
Probability fy that, starting from Sj, the system will ever pass through Sj,
reads:
fjj = £fj j(n) (2-95) n=l
In other words, ^j designates the probability that Sj is eventually re-entered. The
application of Eq.(2-95) can be demonstrated on the basis of Fig.2-0 in the
following way. Instead of the perpetual motion of the water, the loop is opened at
119
S3, point 3. A certain amount of water is then poured at Si which flows through
S2 and is leaving away at point 3, state S3. For a water element, the system,
Eq.(2-95) yields that fn = f i i( l) + fii(2) + ... = 0 + 0 + ... = 0, i.e., the
probability of the water element to re-enter at point 1 is nil. Similarly, fil = 33 =
0.
Mean recurrence time is given by
^^J=S^¥^) (2-96) n=l
where for the above example, it gives ILII = ^2 = |Lt3 = 0.
Probability fjk(n) that in a process starting from Sj, the first occupation of
Sk (or entry to Sk) occurs at the wth step, is defined in Eq.(2-97). In other words,
fjk(n) indicates that Sk is avoided at steps (times) 1,..., n - 1 and occupied exactly
at step n, given that state Sj is occupied initially. Thus, fjk(l) = Pjk and for n = 2,
3, ..., fjk(n) is give n by:
fjk(n) = prob{X(r) ; k, r = 1,..., n - 1; X(n) = k I X(0) = j} (2-97)
The application of Eq.(2-97) to Fig.2-0 yields that:
fl5(4) = prob{X(r) t 5, r = 1,..., 3; X(4) = 5 I X(0) = 1} = 1
fl5(7) = prob{X(r) ?t 5, r = 1, ..., 6; X(7) = 5 I X(0) = 1} = 0
whereas:
fl3(7) = prob{X(r) ; 3, r = 1, ..., 6; X(7) = 3 I X(0) = 1} = 1
The calculation of fjk(n) is detailed in [15, p.388]. We put fjk(O) = 0 and on the
basis of fjk(n) we define
120
fjk=£fjk(n) (2-98) n=l
fjk is the probability that, starting from Sj, the system will ever pass through S^.
Thus, fjk ^ 1. When fjk = 1, the {fjk(n), n = 1, 2, ...} is a proper probability
distribution and we will refer to it as the first-passage distribution for Sk. In
particular, {fjj(n), n = 1, 2, ...} represents the distribution of the recurrence time
for Sj. It should be noted that the definition in Eq.(2-96) is meaningful only when
fjj = 1, that is, when a return to Sj is certain. In this case ^j < <» is the mean
recurrence time for Sj.
Considering again the case of an open loop between points 1 and 3, states S2
and S3, in Fig.2-0, and applying Eq.(2-98) for a fluid element initially at point 1,
yields:
fl2 = fl2(l) + fl2(2) + fi2(3) + ... = 1 + 0 + 0 + ... = 1
fl3 = fl3(l) + fl3(2) + fl3(3) + ... = 0 + 1 + 0 + ... = 1
i.e., the probability of the water element to pass S2, after one step and S3 after two
steps is 100%.
Transient or non-recurrent state. State j is said to be transient if the
conditional probability of occupying (or returning to) Sj, given that the system
initially occupies Sj, is less than one. Thus, the eventual return to the state is
uncertain. According to [15, pp.389], Sj is transient if, and only if:
Xpjj(n)<oo (2-99)
n=l
In this case:
Xpjk(n)<oo (2-100) n=l
for all j in the state space. On the basis of the quantity defined in Eq.(2-95), Sj is
transient if:
121
fjj < 1 (2-101)
For the open loop between states Si and S3 corresponding to points 1 and 3
in Fig.2-0, where a certain amount of water is poured at Si, Eqs.(2-99) and (2-
100) yield
P22(l) + P22(2) + P22(3) + ... = 0 + 0 + 0 + ... < 00
P13(1) + P13(2) + P13(3) + ... = 0 + 1 + 0 + ... < 00
i.e., S2 and S3 are transient states. By applying Eqs.(2-95), it is obtained that
f22 (= 0) < 1 and f33 (= 0) < 1
results which are in agreement with Eq.(2-101) for characterizing transient states.
Recurrent or persistent state. State j is said to be recurrent if the
conditional probability of occupying Sj or returning to it, given that the system
initially occupied Sj or started in it, is one. Thus, the eventual return to the state is
certain. Taking into account Eq.(2-95), it follows that a state Sj is recurrent if:
fjj=l (2-102)
Considering Eq.(2-96), the mean recurrence time |lj = 00 2ind Sj is called null-
recurrent state or null state. If |ij is finite, Sj is defined as positive-recurrent.
The application of Eq.(2-102) can be demonstrated in the following way on
the basis of Fig.2-0, now for a closed water loop in its perpetual motion uphill.
Considering a fluid element, and applying Eq.(2-95), yields:
fll = f u d ) + fll(2) + ... + fii(5) + ... = 0 + 0 + ... + 1 + ... = 1
hi = f22(l) + f22(2) + ... + f22(5) + ... = 0 + 0 + ... + 1 + ... = 1
and similarly:
122
f33 = UA = f55 = 1
Thus, on the basis of Eq.(2-102), the states Si (points i, i = 1, 2, ..., 5) are
recurrent states. From Eq.(2-96) it follows that:
m = ^2 = ... = ^5 = 5
i.e., each state is reoccupied after 5 steps, where the states are also positive-
recurrent considering the definition following Eq.(2-102).
Periodic state. Suppose that a chain starts in state Sj. Subsequent
occupations of Sj can only occur at steps (times) Iv, 2v, 3v, 4v ... where v is an
integer.
I f v > l (2-103)
and the chain is finite, Sj is periodic. The period of Sj is the greatest common
divisor of the set Iv, 2v, 3v, 4v ... for which Pjj(n) > 0 where n is an integral
multiple of v. In the absence of the latter, pjj(n) = 0.
If v = l (2-104)
and the chain is finite, Sj is called an aperiodic Markov chain. The above concepts
are demonstrated on the basis of Fig.2-0 by considering point 2, state S2.
Occupation of this state occurs at steps 5, 10, 15, 20, ..., hence, Iv = 5, 2v = 10,
3v = 15, 4v = 20,..., yielding that v = 5. From Eq.(2-103) S2 is periodic with a
period of 5. It should be noted that the same results applies also for the other states
Si, S3, S4 and S5.
An additional example is 2.9 for which the transition probability reads:
P =
Si
S2
0
1
1
0
From Eq.(2-33a), it follows that:
123
P2 =
Si
S2
1
0
^2 0
1
= p4 = p6 = p8 = plO
Hence, Iv = 2, 2v = 4, 3v = 6, 4v = 8, 5v = 10, thus v = 2, i.e.. Si and S2 have a
period of 2.
Other examples showing periodic behavior are: Example 2.11 with a 7x7
matrix showing a period of v = 7 in Eq.(2-39a); Example 2.14 with a 6x6 matrix
and a period of v = 2 for states S2 and S3 in Eq.(2-42a) where the probability of
the other states vanishes; Example 2.16 with a 12x12 matrix and a period of v = 2
for all states demonstrated in Fig.2-9; Example 2.18 with a 9x9 matrix and a period
of V = 2 for all states depicted in Fig.2-12; Example 2.21 with a 9x9 matrix and a
period of v = 2 shown in Fig.2-15; Example 2.27 with a 4x4 matrix and a period
of V = 2 for all states demonstrated in Fig.2-23; Example 2.35 with a 4x4 matrix
and a period of v = 2 for all states depicted in Figs.2-31, 32; Example 2.45 case 1
with a 8x8 matrix and a period of v = 4 as demonstrated in Fig.2-42.
The following may be shown:
a) A Markov chain is aperiodic if for a state Sj there exist pjj = Pjj(l). Thus, if any
of the diagonal elements in P is non zero, the chain is aperiodic.
b) A Markov chain is aperiodic if there exists an integer n such that pjk(n) > 0 for
allj andk.
The following examples demonstrate the above behaviors:
If
P =
Si
S2
S3
Si
0
0
1
S2 S3
1 0
1/2 1/2
0 0
the chain is aperiodic, since P22 = 1/2 > 0.
124
If
P =
Si
S2
S3
Si S2 S3
0 1/2 1/2
1/2 0 1/2
1/2 1/2 0
the chain is aperiodic, since
P2 = Si
S2
S3
Si
1/2
1/4
1/4
S2 1/4
1/2
1/4
S3 1/4
1/4
1/2
and Pjk(2) > 0 for all j and k. Other examples exhibiting aperiodic behavior may be
found among examples 2.8-2.46.
Ergodic state. A finite Markov chain is ergodic if there exist probabiUties
TCk such that [6, p.41; 4, p.lOl]:
lim Pjk(n) = % for all j and k (2-105)
n—>oo
These limiting probabilities TCk are the probabilities of being in a state after
equilibrium has been achieved. As can be seen from Eq.(2-105), the TCkare
independent of the initial state j , i.e., they are without memory to the past history.
Similarly, from Eq.(2-25), we may write that:
,n+l l imS(n-hl) = limS(0)P"^' = 7C n-»oo n^oo
where n is the stationary distribution of the limiting state vector. Thus:
TCP = 7C (2-105a)
Hence, if the system starts with the distribution n over states, the distribution over
states for all subsequent times is n. This is the defining property of a stationary
125
distribution. Ergodic systems retain the property of having unique equilibrium
distribution and these are also unique stationary distributions.
The limiting probabilities TCk may be found by solving the following system
of equations:
z fork= 1,2, ...,Z (2-106)
subject to the conditions:
n^>0 for all k
z
X^k= (2-107)
1 k=l
As indicated before, the probability distribution {TCk} defined in Eqs.(2-106)
and (2-107) is called a stationary distribution. If a Markov chain is ergodic, it can
be shown [17, pp.247-255] that it possesses a unique stationary distribution; that
is, there exist 7Ck that satisfy Eqs.(2-105), (2-106) and (2-107). There are Markov
chains, however, that possess distributions that satisfy Eqs.(2-106) and (2-107),
i.e., they have stationary distributions, which are not ergodic. For example, if the
probability transition matrix is given by:
P =
Si
S2
Si
0
1
S2
1
0
then:
Pll (n)={ 1 if n is even
0 if n is odd
126
so the chain is not ergodic. However, one may solve Eqs.(2-106) and (2-107) to
obtain the stationary probabilities Tii and 7t2 = 1/2. Recall that this P is the
transition matrix for an irreducible periodic Markov chain.
Some sufficient conditions for a finite Markov chain to be ergodic are based
on the following theorems, given without proof [17, pp.247-255]. The first one
states that: A finite irreducible aperiodic Markov chain is ergodic. Let:
P =
Si
S2
S3
Si
1/4
0
3/4
S2 1/4
2/3
1/4
S3 1/2
1/3
0
This chain is irreducible (all pairs of states communicate) since pjk(2) > 0 for all j ,
k. It is aperiodic since p n = 1/4 > 0. Hence by above theorem the chain is
ergodic. To find the limiting probabilities, solve Eqs.(2-106) for Z = 3 to obtain
the following equations:
TCi = (l/4)7Ci + (3/4)7C3; 7C2 = (l/4)7Ci + (2/3)7l2 + (1/4)7^3; 7C3 = (l/2)7Ci + (1/3)7C2
The solution of these equations is TC] = 2/7, 7i2 = 3/7, 71:3 = 2/7. Thus, the
asymptotic probability of being in state Si is 2/7, in S2 is 3/7 and in S3 is 2/7.
The following Table summarizes part of the classification of the above states:
Table 2-2. Classification of states (4, p.93)
Type of state
Periodic
Aperiodic
Recurrent
Transient
Positive-recurrent
Null-recurrent
1 Ergodic
Definition of state
(assuming it is initially occupied)
Return to state is possible only at times Iv, 2v, 3v,
..., where the period v > 1
Not periodic. Essentially it has a period of v = 1
Eventual return to state is certain
Eventual retum to state is uncertain
Recurrent, finite mean recurrence time
Recurrent, infinite mean recurrence time, |iij = 00
I Aperiodic, positive-recurrent, |ij < 00
127
Doubly stochastic matrix. A transition probability matrix is said to be
doubly stochastic if each column sums to 1, that is, if
7,Pi], = 1 for each k (2-108)
On the basis of the following theorem, i.e., if the transition matrix P for a
finite irreducible aperiodic Markov chain with Z states is doubly stochastic^ then the
stationary probabilities are given by
Ttk = 1/Z fork= 12, ..., Z
it follows that the transition matrix
Si
S2
S3
S3
Si
1/8
1/2
3/8
0
S2 1/4
1/4
1/4
1/4
S3 1/8
1/8
1/4
1/2
S4
1/2 1 1/8
1/8
1/4
p =
is irreducible, aperiodic, and doubly stochastic. Hence Tlk = 1/4 for k = 1, 2, 3, 4.
Definition. Closed sets of states. A non empty set C of states is called a
closed set if each state in C communicates only with other states in C. In other
words, no state outside C can be reached from any state in C. A single state Sk
forming a closed set is an absorbing state. Once a closed set is entered it is never
vacated. If j belongs to a closed set C then pjk = 0 for all k outside C. Hence, if all
rows and columns of P corresponding to states outside C are deleted from P, we
are still left with a stochastic matrix obeying Eqs.(2-17) and (2-18). A Markov
chain is irreducible if there exists no closed set other than the set of all states.
According to [17, p.210], a closed communicating class C of states
essentially constitutes a Markov chain which can be extracted and studied
independently. If one writes the transition probability matrix P of a Markov chain
so that the states in C are written first, and P can be written as:
128
P = (2-109)
where * denotes a possibly non-zero matrix entry and p^ is the sub matrix of P
giving the transition probabilities for the states in C, then:
pn =
(Pc) 0
(2-109a)
Thus, if a Markov chain consists of one or more closed sets, then these sets are
sub-Markov chains andean be studied independently. The following example
demonstrates the above ideas, i.e.:
P =
Si $2 S3 S4 S5 S6 S7 Sg S9
Si
S2
S3
S4
S5
S6 S7
Sg S9
0
0
0
1
0
0 0
0 0
0
Y 0
0
0
1 (0
0 0
0
5
0
0
0
0 0
1 0
a 0
0
0
0
0 0
0 V
0
e 0
0
1
0 0
0 0
0
0
0
0
0
0
I 0 0
0
0
0
0
0
0 X
0 0
0
0
1
0
0
0 0
0 0
p X
0
0
0
0 0
0
^
(2-110)
In order to find the closed sets, it suffices to know which pjk vanishes and
which are positive. In the fifth row in matrix (2-110) P55 = 1, thus S5 is
absorbing. S3 and Sg form a closed set since p38 = pgs = 1. From Si, passages
are possible into S4 and S9, and from there only to Si, S4, S9. Accordingly the
latter states form another closed set. The appearance of the matrix and the
determination of the closed sets can be simplified by renumbering the states in the
order S5 S3 Sg Si S4 S9 S2 S6 S7 so that the modified matrix reads:
129
P =
S5 S3 Sg Si S4 S9 S2 S6 S7
S5
S3
Sg
Si
S4
S9
S2
S6 S7
1
0
0
0
0
0
e 0 0
0
0
1
0
0
1
6 0 0
0 1
0
0 0
0
0
0
0
0
0
0
0 0
0 0 0 0
0 0
0
a 1
V
0
0 0
0 0
0
p 0
H X
0
0
0 0
0
0
0
0
Y 1
(0
0 0
0
0
0
0
0
0
^
0
0
0
0
0
0
0
0 X
(2-110a)
The closed sets then contain only adjacent states and a glance at the new matrix
reveals the grouping of the states. In addition, the sub matrices whose states form
closed sets may be studied separately and on the basis of Eqs.(2-109) and (2-
109a), matrix (2-110a) reads:
(p)" =
S5
S3
Sg
Si
S4
S9
S2
S6 S7
S5 1
0
0
0
0
0
e 0 0
S3 0
0
1
0
0
1
5
0 0
Sg Si 0 0
1
0
" 0
0
0
0
0
0
0
0
0 0
0 0 0 0
S4 0
0
0
a 1
V
0
0 0
S9 0
0
0
p 0
1 X
0 0
S2 0
0
0
" 0
0
0
Y 1 CO
S6 0
0
0
0
0
0
0
0
^
S7 0
0
0
0
0
0
0
0 X
(2-110b)
Examples
To conclude section 2.1-5 on classification of states, we consider the
following matrix corresponding to five states which belongs to the random walk
model considered in examples 2.17-2.22, 2.41 and 2.42.
130
P =
Si
S2
S3
S4
Ss
Si
P
q 0
0
0
S2 1-p
r
q 0
0
S3 0
p r
q 0
S4 0
0
p r
q
S5 0
0
0
p 1-q
By changing the parameters p, q and r, many behaviors are revealed depicted in
Fig.2-46. It should be noted that examination of the various behaviors has been
done with respect to the state vector S(n) rather than exploration of the quantities
Pij(n) referred to in part of the above definitions. Eqs.(2-23) and (2-25) were used
in the calculation of S(n) and an initial state vector S(0) = [0, 0, 1,0, 0] was
assumed, i.e., S3(0) = 1.
Case a in Fig.2-46, for which q = 0, p = 1 and r = 0, demonstrates that S5
is an absorbing state which is occupied after two steps. Thus, the probability of
occupying the other states vanishes. Si, which is also an absorbing state, can
never be occupied unless it is initially occupied.
Case b where q = 1, p = 0 and r = 0, has a reflecting barrier at the origin.
Fig.2-46 demonstrates 2i periodic behavior of states S] and S2. For the other states
si(n) = S2(n) = 0, n = 1, 2,...; S3(n > 1) = 0. The period equals v = 2.
Case c for which p > q (q = 0.3, p = 0.7 and r = 0) reveals a transient
behavior of the states as well as their being ergodic, i.e., independent of S(0). The
stationary distribution reads S(25) = [0.045, 0.045, 0.104, 0.242, 0.564].
Case rf, also for p > q (q = 0.1, p = 0.7 and r = 0.2), reveals a similar
behavior as in the previous case. However, the approach towards equilibrium is
less oscillatory by comparison to case c due to the damping parameter r > 0, i.e.,
the probability of remaining in the state. The limiting distribution reads S(24) =
[0.001, 0.003, 0.018, 0.122, 0.856].
Case ^, for p = q = 0.5 is transient with damped oscillations towards an
equilibrium of S(32) = [0.2, 0.2, 0.2, 0.2, 0.2], i.e., the occupation probability of
all states is the same. The states are also ergodic.
131
Casef with q = 0.2, p = 0.2 and r = 0.6 is transient, free of oscillations as
well as ergodic. The equilibrium state vector reads S(46) = [0.058, 0.236, 0.236,
0.235, 0.235].
Cases g and h with q = 0.8, p = 0.2, r = 0 and q = 0.4, p = 0.2, r = 0.4,
respectively, behave similar to cases e and f, i.e., they are transient and ergodic.
Note that p < q. The ultimate distributions are: S(65) = [0.430, 0.430, 0.107,
0.027, 0.006] for case g and S(20) = [0.210, 0.420, 0.211, 0.106, 0.053] for
case h.
1
0.8
0.4
0.2
0
q = 0,p = 1,r = 0
^.3(") >
•/f.V-'"' s.(n) = s(n)
q=1,p = 0,r = 0
U M") / s,(n) / W i ) /
\ /
^ ^iTi'i 5 0 1
b) 2 3
n q =
~\ ^ s (n)
\L\ • J ^
0.1,p = 0.7, r = 0.2 ' s (n) ' 1
s,(n) 1
15 0 5 10 d) n
15
132
1
0.8
0.2
0
q = 0.5, p = 0.5, r = 0
s (n)
I/' \^' 2(n) = s (n)
\:J im
s (n) = s (n)
0 5 10 e) n
q = 0.8, p = 0.2, r = 0
q = 0.2, p = \ 1
^ s (n)
0 . 2 , r = 0 . 6
1 1
V , - - - - ' • ! 1 15 0 5 10
f) n q = 0.4, p = 0.2, r = 0.4
15
Fig.2-46. Demonstration of states behavior for S(0) = [0, 0, 1, 0, 0]
2.2 MARKOV CHAINS DISCRETE IN SPACE AND CONTINUOUS IN TIME 2.2-1 Introduction
In the preceding chapter Markov chains has been dealt with as processes
discrete in space and time. These processes involve countably many states Si, S2,
... and depend on a discrete time parameter, that is, changes occur at fixed steps n
= 0, 1, 2, ....
Although this book is mainly concerned with the above mode of Markov
chains, a rather concise presentation will be given in the following on Markov
chains discrete in state space and continuous in time [2, p.57; 4, p. 146; 5, p. 102;
15, p.444]. This is because many processes are associated with this mode, such as
telephone calls, radioactive disintegration and chromosome breakages where
changes, discrete in nature, may occur at any time. In other words, we shall be
concerned with stochastic processes involving only countably many states but
133
depending on a continuous time parameter. Such processes are also referred to as
discontinuous processes. Unfortunately, application of the basic models presented
in 2.2-3 to chemical reactions, behavior of chemical reactors with respect to RTD,
as well as to chemical processes, is very limited or even inapplicable.
While the distinction between discrete time and continuous time is
mathematically clear-cut, we may in applied work use discrete time approximation
to a continuous time phenomena and vice versa. However, discrete time models
are usually easier for numerical analysis, whereas simple analytical solutions are
more likely to emerge in continuous time.
In the following, we derive the Kolmogorov differential equation on the basis
of a simple model and report its various versions. In principle, this equation gives
the rate at which a certain state is occupied by the system at a certain time. This
equation is of a fundamental importance to obtain models discrete in space and
continuous in time. The models, later discussed, are: the Poisson Process, the
Pure Birth Process, the Polya Process, the Simple Death Process and the Birth-
and-Death Process. In section 2.1-3 this equation, i.e. Eq.2-30, has been derived
for Markov chains discrete in space and time.
2.2-2 The Kolmogorov differential equation The following model represents a simplified version of the Kolmogorov
equation. A big state of a total population NQ consists of many cities. The cities
are arranged in two circles, internal and external. The external circle consists of J
cities, j = 1, 2,..., J, where each city is a state designated as Sj = j =jth city in the
external circle. The internal circle consists of K cities, k = 1, 2,..., K. A state in
this circle is designated by Sk = k = kth city in the internal circle. Inhabitants,
designated as system, are moving from cities in the external circle to cities in the
internal circle, i.e., along the trajectory Sj —> S| -> out of kth city, where from each
external city, inhabitants can move only to each internal city. In other words
inhabitants can not move from one external city to another extemal city.
The movement of inhabitants through an internal city was convincingly
demonstrated by Magritte [20] in his painting Golconda depicted in Fig.2-47.
134
4|( M i l M l I I I It I l '
l i l t I l t i ^ 4 / It | | | | i I ^M I i I I flBl ^ 1 I H I i
in 1 I ( f V M r t ( ^
Fig.2-47. Motion of inhabitants through an internal city k according to
Magri t te ("Golconda ", 1953, © R.Magritte, 1998 c/o Beeldrecht Amstelveen)
The figure has slightly been modified by adding a circle on it, demonstrating
a city, as well as arrows indicating 'in' and 'out' from city k. The painting is one
of Magritte's most scrupulous displays of reordering where he allows his figures
no occupation, no purpose, and despite of their rigid formation, no fixed point.
They could be moving up or down or not moving at all.
In setting up the model, we designate by Nj(t) the number of inhabitants
occupying state j (an extemal city j) at time t. Similarly, Nk(t) corresponds to the
number of inhabitants occupying state k (an internal city k) at time t. The change in
the number of inhabitants in state Sk during time interval At is given by:
135
ANk(t) = number of inhabitants entering the city in the time interval (t, t + At)
- number of inhabitants leaving the city in the interval (t, t + At)
(2-111)
The quantities on the right-hand side may be expressed by the following
probabilities [2, p.59]:
qj(t) - a rate (1/time) or an intensity function indicating the rate at which
inhabitants leave state Sj (external city j). This function has the following
interpretation, i.e.,
qj(t)At - stands for the probability of the system (inhabitants) to leave state Sj
in the time interval (t, t + At). This quantity is the probability for a change to occur
at Sj not indicating the 'direction' of the change. The following quantity gives the
'direction', i.e., the probability that inhabitants leaving Sj will occupy exactly S^,
namely, they will occupy city k in the internal circle. Thus,
Qjk(0 - gives the direction of the change at the interval (t, t + At), i.e., the
transition probability of the inhabitants from Sj to occupy S^ at time t [15, p.473].
Note that 0<Qjk(t), Qkk(t)< 1.
Other quantities pertaining to the cities in the intemal circle are:
qk(t) - the rate (1/time) at which inhabitants {system) leave state Sk (intemal
cityk). Thus,
qk(t)At - is the probability of the inhabitants to leave Sk at the interval (t, t +
At).
Accounting for the above quantities and applying Eq.(2-111) yields:
J
ANk(t) = X Nj(t)qj(t)AtQjk(t) - Nk(t)qk(t)At (2-11 la)
Defining the following state probabilities, i.e.,
Pk(0 - probability of the system to occupy state Sk at time t,
Pj(t) - probability of the system to occupy Sj at time t
and assuming that the total number of inhabitants NQ in the big state remains
constant, gives
Nj(t) = Pj(t)No; Nk(t) = Pk(t)No (2-112)
136
Substitution of Eq.(2-112) into Eq.(2-llla) and approaching At to zero, yields a
simplified version of the forward Kolmogorov differential equation for the
transition S; -> S|j~> out of kih city. This equation is continuous in time and
discrete in space; it reads:
dPk(t) ^ - ^ = XPj ^^^J^^^QjkW - Pk(t)qk(t) k = 1, 2, ..., K (2-113)
j=i
It should be noted that Eq.(2-113) may be looked upon as an "unsteady state
probability balance" on city k in the internal circle; J is the total number of cities in
the external circle and K in the internal circle. In other words, the equation gives
the rate of change of the probability of occupying state k at a certain time. Note that
the origin of the equation was an unsteady state "mass balance" on the transition of
inhabitants through city k.
Another version of the Kolmogorov equation is obtained by considering the transition S; -» Sj -» \ . Eq.(2-30) gives the Chapman-Kolmogorov equation,
discrete in time and space. For a continuous time and discrete space, this equation
is generally written as [2, p.61]:
PjkCT 't) = ^Pji(T,s)Pik(s,t) for t > T > 0 (2-114)
and is valid for x < s < t. This relation expresses the fact that a transition from state
Sj at time x to Sk at time t occurs via some state Sj at the intermediate time s, and for
a Markov processes the probability Pik(s,t) of the transition from Si to Sk is
independent of the previous state Sj. Pjk('C,t) is the transition probability of a
system to occupy state k at time t subjected to the fact that the system occupied state
j at time T. Another way of expressing the above is that we shall write Pjk(x,t) for
the conditional probability of finding the system at time t in state Sk, given that at a
previous time T the system occupied Sj. As indicated above, the symbol Pjk('C,t) is
meaningless unless x < t. Differentiation of Eq.(2-114) [15, p.472] yields the
forward Kolmogorov differential equation for the transition S: -> S, —> Sj . It is
continuous in time and discrete in space and reads:
137
dt = 2 PjiC- OqiCOQikCt) - Pik(T,t)qk(t) (2-115)
Here j and T are fixed so that we have, despite of the formal appearance of the
partial derivative, a system of ordinary differential equations for the function
Pjk(T,t). The parameters] and T appear only in the initial conditions, i.e.:
1 forj = k Pjk(T,T)= (2-115a)
0 otherwise
Thus, the system of ordinary differential equations reads:
^ ^ = X Pi(t)qi(t)Qik(t) - Pk(t)qk(t) (2-116) i
The above equation becomes identical to Eq.(2-113) by replacing the notation of the
state transitions, i.e. i with j .
As Pjk('Tjt) stands in Eq.(2-115), it is not time-homogeneous, since it
depends explicitly on t and x. However, if a restriction is made to the time-
homogeneous or stationary case, then:
Pjk(t,t) = Pjk(t-T) (2-117)
i.e., the transition probability Pjk(T,t) depends only on the duration of the time
interval (t - x) and not on the initial time x. Hence, in the time-homogeneous case,
Eq.(2-114) reads:
PjkC + t^^XV^^Pik^^^ fort>T>0 (2-118) i
The significance of Eq.(2-118) is as follows. Given that at time = 0 a system is at
state Sj = j . If we ask about its state at time (x + t), then the probability that the
system will occupy state k at time (T +1), i.e., Pjk(T + t), may be computed from
the above sunmiation based on the fact that the system will occupy an intermediate
138
state Si = i before a transition to occupy Sk takes place. Pji(T) is the probability of
the system to occupy Si at time x and Pik(t) is the probability of occupying Sk at
time (T +1).
2.2-3 Some discontinuous models The following models for the transition S; -^ S^ are considered because they
find applications in many comers of our life, later elaborated. It should be noted
that the above transition indicates the basic property of Markov chains, i.e. that
occupation of Sk is conditioned on a prior occupation of Sj and that the past history
is irrelevant. Unfortunately, the application of the models in Chemical Engineering
is very limited, however, some applications are mentioned. For each case we
derive the difference equation describing the probability law of the process and
report the final solution of the differential equation accompanied by its graphical
presentation. In some cases we derive the differential equation from the
Kolmogorov equation.
The Poisson Process. This process is the simplest of the discontinuous
processes which occupies a unique position in the theory of probability and has
found many applications in biology, physics, and telephone engineering. In
physics, the random emission of electrons from the filament of a vacuum tube, or
from a photosensitive substance under influence of light, and the spontaneous
decomposition of radioactive atomic nuclei, lead to phenomena obeying the Poisson
probability law. This law arises frequently in the field of operations research and
management science, since demands for service, whether upon the cashiers or
salesmen of a department store, the stock clerk of a factory, the runways of an
airport, the cargo-handling facilities of a port, the maintenance man of a machine
shop, and the trunk lines of a telephone exchange, and also the rate at which service
is rendered, often lead to random phenomena either exactly or approximately
obeying the Poisson probability law. Part of the above examples are elaborated in
the light of the governing equation describing the law. Finally, it should also be
noted that the Poisson model obeys the Markov chains fundamental property, i.e.,
that future development depends only on the present state, but not on the past
history of the process or the manner in which the present state has been reached.
139
In deriving the Poisson model, a rather general description arising from the
above applications can be established. We assume that events of a given kind occur
randomly in the course of time. For example, we can think on "service calls" as
(requests for service) arriving randomly at some "server" (service facility) as
events, like inquiries at an information desk, arriving of motorists at a gas station,
telephone calls at an exchange, or emission of electrons from the filament of a
vacuum tube.
Let X(t) be a random variable designating the number of events occurring
during the time interval (0, t). An interesting question regarding to the random
variable X(t) may be presented as what is the probability that the number of events
occurring during the time interval (0, t) = t will be equal to some prescribed value
X. Mathematically it is presented by:
Px(t) = prob{X(t) = X}; X = 0, 1, 2,... (2-119)
where the exact relationship is derived in the following. The above equation
indicates also the realization of the random variable X(t) by acquiring the value x. An expression for Px(t) for the transition S; -^ \ , where the two discrete
states are Sj = X - 1 and Sk = x, may be derived bearing in mind the following
assumptions:
a) The events are independent of one another; more exactly, the random
variables X(ti), X(t2),... are independent of each other if the intervals ti, t2,... are
non-overlapping. In other words, if for example t] = t2 then X(ti) = X(t2). X(ti)
designates the number of events occurring during the time interval (0, ti).
b) The flow of events is stationary, i.e., the distribution of the random
variable X(t) depends only on the length of the interval t and not on the time of its
occurrence.
c) The probability of a change in the time interval (t, t + At), or of a transition
from Sj to Sk in the time interval (t, t + At), or the probability that at least one event
occurs in a small time interval At, is given by:
prob{X(t, t + At) = 1} = U t + o(At) = pjk (2-120)
140
where X,(events/time) is a positive parameter characterizing the rate (or density or
intensity) of occurrence of the events. A possible interpretation of the above
definition, later elaborated, makes use of the conception birth, i.e., the occurrence
of an event in the time interval (t, t + At) may be looked upon as a single birth.
Thus, the parameter X is the birth rate. If we approach At to zero, no change
occurs. Here o(At) is an infinitesimal of a higher order than At, i.e.:
lim—r— = 0 At
At~>0
o(At) emerges from the expansion pjk(At) = X,o + X.At + Xi^X^ + — The second
term on the right-hand side is responsible for the probability of the occurrence of at
least one event where the third term accounts for more than one event to occur
during At, i.e., "twin birth". XQ should be omitted in the expansion noting that the
probability of the number of events per unit time, i.e., pjk(At)/At approaches
infinity as At is approaching to zero.
d) The probability of no change in (t, t + At), or of remaining in Sj, or that no
events occur during At, is given by:
prob{X(t, t + At) = 0} = 1 - >.At + o(At) = pjj (2-121)
e) The probability of more than one change in the interval (t, t + At) is o(At),
thus it is negligible as At is approaching zero. In other words, this assumption
excludes the possibility of a "twin birth". It should be noted that the above
probabilities are independent of the state of the system.
Having established the one-step transition probabilities pjk and pjj, the
differential equation for Px(t) will be derived by setting up an appropriate
expression for Px(t + At). If the system occupies state Sj = x -1 at time t, then the
probability of occupying Sk = x, i.e., making the transition Sj to Sk, is equal to the
product A,At Px-i(t). If the system already occupies state Sk = x at time t, then the
probability of remaining in this state at (t, t + At) is equal to (1 - AAt )Px(t). Thus,
since the above transition probabilities are independent of each other, and following
Eq.(2-3), we may write that:
141
Px(t + At) = (1 - >-At)Px(t) + >^t Px-i(t) + o(At) (2-122)
If we transpose the term Px(t) on the right-hand side, divide by At, and approach At
to zero, we obtain the following differential equation:
dPJx) - i - = -?iP,(t) + ?iP,.i(t) X > 1 (2-123)
When X = 0, Px-i(t) = 0; hence:
dPo(t)
dt = -?iPo(t) (2-124)
Eqs.(2-122) and (2-123) characterize the Poisson process and are to be solved with
the initial conditions:
Po(0)=l
Px(0) = 0 forx= 1,2,... (2-125)
Having obtained Po(t), and using Eq.(2-123), it is possible to obtain by induction
[2, p.74] that:
P (t) = Mle-^^; x = 0,1,2,... (2-126)
which is called the Poisson distribution. Some interpretations of Px(t) are: a) It
gives the probability that at time interval (0, t) = t (> 0) the system occupies state x
(x = 0, 1,2, ...). b) It is also the probability of exactly x changes or events
occurring during the time interval of length t. c) Px(t) indicates the probability to
remain at a prescribed state x during the time interval t.
Let us analyze in more details some examples, assuming they obey the
Poisson model:
1) Requests for service arriving randomly at some single service facility,
arrive at a mean rate X calls per time. If the calls have not yet been answered by the
142
service facility, Eq.(2-126) gives an answer to the following possible questions
which might be of some interest: a) What is the probability that during a prescribed
time interval (0, t), the number of the arriving calls will be equal to a certain value
X? or, b) How long will it take until a call is answered if the probability of
remaining at a certain number x of "waiting calls" is prescribed?
2) A new cemetery of a known size of x graves has been opened at some
town. The mean death rate X in this town is known and the following questions
may arise: a) What is the probability that during the next ten years, i.e., time
interval (0, t) = 10, the cemetery will be full? b) If the probability of occupying
50% of the cemetery is known, how long will it take to reach this state? c) If the
probability to reach a certain time in the future is known, Eq.(2~126) gives an
answer to the occupation state of the cemetery.
3) It has been announced that some urban area became polluted and
consequently men might become infertile. If the process of becoming infertile is
random, thus, Poissonic, a mean parameter X may be defined as the rate of
infertility per day (say!), and Eq.(2-126) becomes applicable yielding the following
information: a) What is the probabiUty that all men in town become infertile during
a week, a month, a year etc.? b) If we prescribe the probability of remaining at
some state x (= number of men which be came infertile), then we may calculate the
time interval of remaining in this state. Certainly, the most important data needed is
X.
4) Electrons are emitted randomly from the filament of a vacuum tube. If the
emission rate, X electron/unit time, is known, then Eq.(2-126) gives the probability
that within the time interval (0, t) the total number of electrons emitted is a
prescribed value x.
5) A mailbox, designated as system, has a finite capacity for letters. The
number of letters is the state of the system x = 0, 1, .... The rate of filling of the
mailbox is X letters per day and Eq.(2-126) gives an answer to several questions of
the type demonstrated above.
Some characteristic properties of the Poisson equation are:
143
a) Fig.2-48 demonstrates the Px(Xt) - Xi, Px(> t) - x relationships computed from
Eq.(2-126).
0 2
Fig.2-48. The Poisson model
Characteristic behavior observed is:
Px = o(0) = 1, Px>o(0) = 0 and Px^o(oo) = 0 (2-126a)
b) 2 ) Px(t) = 1 (2-127)
x-O
c) The mean number of events, m(t), occurring in the time interval of length t is
given by:
m(t).2)^Px(t) = ?t (2-128) x«0
It should be noted that for the Poisson distribution, the variance is equal to the
mean and as'ki^' co the distribution tends to normality.
d) The mean time until the occurrence of the first event, < t >, reads:
< t > = l / X (2-129)
where K is the mean rate of occurence of the events.
144
It is interesting to demonstrate the derivation of Eq.(2-123) from the
Kolmogorov Eq.(2-113). In this equation J = K = 1 as well as:
k = Sk = Sk=i=x
j = Sj = Sj=i = x-1
qj(t) = qj=i(t) = X, Pj(t) = Px.i(t) and Qjk(t) = 1
qk(t) = qk=l(t) = >. and Pk(t) = Px(t)
Substitution of the above quantities into Eq.(2-113) gives Eq.(2-123).
Few applications of the Poisson Distribution in Chemical Engineering are:
For X = 0, Eq.(2-126) reduces to:
Po(t) = e-> t (2-130)
where Po(t) stands for the probability of remaining at the state x = 0, i.e., no
change (or event) is expected to occur in the system during the time interval of
length t. The above equation is applicable in the following cases:
a) 1st order chemical reaction, A -> B, for which -dCA/dt = ICCA occurring
in a batch reactor. The solution for the concentration distribution reads:
7 ^ - = l f - - = e-k (2-131)
where CA(t) and CAQ are, respectively, the concentration of A at time t and t = 0. k
is the reaction rate constant. Similarly, NA(t) and NAG are the number of moles of
A at time t and t = 0. The system in this case is a fluid element containing the
species A and B. The states of the system in Eq.(2-126) are: x = 0 designating
species A and x = 1 for species B. As observed, Eqs.(2-130) and (2-131) are
similar, i.e.:
Po(t) = e-kt (2-132)
therefore, the ratio NA(t)/NAO = Po(t) may be looked upon as the probability that
the system remains at x = 0 until time t.
If t = 0, it is obtained that Po(0) = 1, indicating that the probability of
remaining at the initial state at t = 0 is 100%.
145
If k = 0, Eq.(2-132) yields Po(t) = 1, namely, in the absence of a chemical
reaction, the probability that the system remains in its initial state x = 0 for all t is
100%. If k ^ oo, it is obtained from Eq.(2-132) that Po(t) = 0, namely, the
probability of remaining at the initial state for all t is zero. In other words, in the
presence of an intensive reaction, the initial number of moles will immediately
diminish, i.e., the system will immediately occupy the state x = 1, i.e. in the state
of species B.
If Po(t) = 0.5, this means that the probability of remaining in x = 0 until time t
is 50%. In other words, 50% of A will decompose to B because the probabiUty of
remaining in the initial state until time t is 50%.
b) An additional example is concerned with the introduction of a pulse of an
initial concentration CAO into a single continuous perfectly-mixed reactor, the
system. The states of the system are x = 0, designating the initial concentration
CAO» and X = 1, the concentration of species A at time t, i.e. CACO. The
relationship CACO is given by:
- j ^ = e-^t„ = Po(t) (2-133) ^AO
where tm is the mean residence time of the fluid in the reactor. Thus, If tm -> <», Eq.(2-133) yields Po(t) = 1. This means that the probability of
the pulse to remain at the initial state x = 0 along the time interval t is 100% because
the residence time of the fluid in the reactor is infinity. If tm -» 0, Po(t) = 0. This means that the probability of remaining at x = 0 at
t > tm is zero because of the extremely short mean residence time of the fluid.
c) The following example is concerned with a closed recirculation system
consisting of N perfectly-mixed reactors of identical volumes. If we introduce a
pulse input into the first reactor, then the output signal at the Mh reactors is given
by [21, p.294]:
C = e - ^ ^ ^ J X - ^ ^ J ^ n=l ,2 , . . . ,N (2-134) n=l
146
where C is a dimensionless concentration at the Mh reactor, tm is the mean
residence time of the fluid in a reactor which is equal for all reactors, n is the
number of passes of the fluid through the system. For short times where only one
term in the series expansion is taken into account, n = 1, Eq.(2-134) is reduced to:
^ (^W^^ ' (t/tj ^ 0^^-Xt (2-i34a) (N~l)! x! "
X = N - 1, X = l/tm where the above equation is exactly the Poisson distribution for
X = N - 1. The description about the behavior of the pulse in the previous example,
is also applicable here.
d) The final example is related to a step change in concentration at the inlet to
a single continuous perfectly-mixed reactor from CA,inlet = 0 to CA,inlet = CAQ.
The response at the exit of the reactor, designated as system, is given by:
CA(t) - p — = l - P o ( t ) (2-135)
where Po(t) is given by Eq.(2-133). CA(t) is the concentration of A inside the
reactor. The states of the system are: x = 0, i.e. CA(0) = 0 and x = 1 for CA(t) >
0. The above result can be interpreted as the probability of not remaining at the
initial state along the time interval t. For example: If tjn -^ 0, Po(t) = 0 and Eq.(2-135) gives CA(t)/CAO = 1, indicating that the
probability of not remaining at the initial state x = 0 along t is 100%. Indeed, this is
plausible because the residence time of the fluid in the reactor is extremely short and
the reactor will acquire instantaneously the final concentration CAQ, i.e, the inlet
concentration. If tm-^ oo, Eq.(2-135) gives CA(t)/CAO = 0, i.e, the probability of remaining
at the initial state x = 0 along t is 100% because of the extremely long residence
time.
The Pure Birth Process. The simplest generalization of the Poisson
process is obtained by permitting the transition probabilities to depend on the actual
state of the system. Thus, if at time t the system occupies state Sj = x (x = 0, 1,2,
147
...), and at time (t, t + At) the system occupies state Sk = x + 1 (a single birth), then
the following probabilities may be defined:
a) The probability of the transition Sj to Sk is:
Pjk = M t + o(At) (2-136)
b) The probability of no change reads:
Pjj = 1 - >-xAt + o(At) (2-137)
where A is the mean occurrence rate of the events which is a function of the actual
state X. The dependence of X, on x avoids, in the context of birth, the phenomenon
of infertility unless X^ is a constant.
c) The probability of a transition from x to a state different from x + 1 is o(At), i.e.,
twin or multiple birth is impossible.
In view of the above assumptions and following Eq.(2-3), we may write that:
Px(t -h At) = (1 - >.xAt)Px(t) + Ax_iPx_i(t)At + o(At) (2-138)
The reason that x - 1 occurs in the coefficient of Px-i(t) is that the probability of an
event has to be taken conditional on X(t) = x - 1. If we transpose the term Px(t) on
the right-hand side, divide by At, and approach At to zero, we obtain the following
system of differential equations:
dP^(t) = -Xy?^{i) + >.x-iPx-i(t) X > 1 (2-139) dt
dPo(t) = ->.oPo(t) (2-139a)
dt
subjected to the following initial conditions:
Px(0) = 1 for X = xo
P,(0)=:Oforx>xo (^-1^9^)
148
Depending on the A. - x relationship, two cases will be considered, viz, the linear
birth process and the consecutive-irreversible z-states process.
The linear birth process. This process, sometimes known as the Yule-Furry
process, assumes [2, p.77] that for a constant X:
X^ = Xx x > l , X , > 0 (2-140)
In this case Eq.(2-138) becomes:
Px(t + At) = (1 - ^xAt)Px(t) + X(x - l)At Px_,(t) + o(At) (2-141)
where in terms of the conception of birth, it may be interpreted as
prob{X(t + At) = x} = prob{X(t) = x and no birth occurs in (t, t + At)} +
prob{X(t) = X - 1 and one birth occurs in (t, t + At)}
The following definitions are applicable:
Px(t) = prob{X(t) = x}; 1 - UAt = prob{X(t, t + At) = 0}
Px-l(t) = prob{X(t) = X - 1}; (x - l)XM = prob{X(t, t + At) = 1}
where
X(t) is a random variable designating the population size at time t;
X or X - 1 designate the actual population size
The corresponding physical picture which may be visualized in the light of the
above, is the following one [4, p. 156; 15, p.450]. Consider a population of
members which can, by splitting or otherwise, give birth to new members but can
not die. Assume that during any short time interval of length At, each member has
probability A.At + o(At) to create a new one; the constant X births/(timexmember)
determines the rate of increase of the population. If their is no interactions among
the members and at time t the population size is x, then the probability that an
increase takes place at some time between t and t + At equals X.xAt + o(At).
It is interesting at this stage to compare the "birth characteristics" of the above
process with the Poisson one. In the Pure Birth Process, each member at each time
interval is capable of giving birth. Also each new bom member continues to give
birth and this process repeats its self ad infinitum. Thus, the birth rate is A,x. In
the Poisson process, once a member gave birth, he is becoming impotent but
remains alive, and only the new born member is giving birth and then becoming
149
also impotent. In this case, the birth rate is X, Table 2-3 in the following is a
numerical comparison between the two process for X = Ibirth per unit time and an
initial population size of x = 1.
Table 2-3. Comparison between Pure Birth and Poisson Processes Pure Birth Process Poisson Process
X
population
at timet
1
2
3
4
5
6
7
8
9
1 10
BR = X,x
births per
unit time
1
2
3
4
5
6
7
8
9
10
At*
/
1
1/2
1/3
1/4
1/5
1/6
1/7
1/8
1/9
t*
0
1
1.50
1.83
2.08
2.28
2.45
2.59
2.72
2.83
BR = X At* t*^
births per
unit time
; /
1 ]
1 ] T 1 1
1 1 I ]
1 ]
1 ]
1
1
o | [ 1
I 2
3
[ 4
5
[ 6
[ 7
1 8
1 9J BR = birth rate; At = Ax ^t+At ^t 1
birthrate (BR)t (BR)t
* number of time units
The major conclusion drawn from Table 2-3 is that the increase of the population in the Pure Birth Process is significantly greater because the members don't become infertile after their first birth giving. For example, in the Pure Birth Process the population size becomes 10 after 2.83 time units where in the Poisson Process it takes 9 time units.
The results in the Table can also be demonstrated in Fig.2-49 by Escher's painting Metamorphose [10, p.326] modified by the author of this book. On the left-hand side of the original upper picture hexagons can be seen which make one think of the cells in a honeycomb, and so in every cell there appears a bee larva. The fully grown larvae tum into bees which fly off into space. But they are not
150
vouchsafed a long life of freedom, for soon their black silhouettes join together to
form a background for white fish seen on the right-hand side. The modified
painting below the original one demonstrates the Poisson Process of rate ^ = 1
birth-metamorphosis per unit time corresponding to x = 1, 2, 3, 4. An interesting
question is concerned with the birth mechanism in the figure, which is beyond the
scope of this book.
Fig.2-49. Poisson process by the modiHed painting **Metamorphose**
(M.CEscher "Metamorphose" © 1998 Cordon Art B.V. - Baarn - Holland. All rights reserved)
Returning to Eq.(2-141), the following basic system of differential equations
are obtained:
dP^(t)
dt
dPo(t)
dt
= ->.xP^(t) + ^(x - 1 )Px i (t) X > 1
= 0
(2-142)
(2-142a)
151
where the solution reads [2, p.78]:
P^(t) = e-^Kl - e-^')''"^ for x = 1, 2,... (2-143)
= 0 for X = 0
If the initial population size is denoted by XQ and the initial conditions are PXQ(O) = 1, Px(0) = 0 for X > xo, the solution reads [16, p.450]:
^"^'^ = (x-!fp)Kxo-l)!^"^' '°^' " ^"^'^''"" X > Xo > 0 (2-143a)
The above type of process was first studied by Yule [15, p.450] in
connection with the mathematical theory of evolution. The population consists of
the species within a genus, and the creation of a new element is due to mutations.
The assumption that each species has the same probability of throwing out a new
species neglects the difference in species sizes. Since we have also neglected the
possibility that species may die out, Eq.(2-143) can be expected to give only a
crude approximation for a population with initial size of XQ = 1. Thus, if the
mutation rate X is known, the above equation gives the probability that within the
time interval (0, t) the population will remain at some prescribed state x > XQ.
Some characteristic properties of the above distribution are:
a) Fig.2-50 demonstrates the Px(^t) - Xt, Px(^t) - x relationships computed from
Eq.(2-143) with Xi as parameter. It is observed that for x > 1 and for a constant X,t,
the probability of remaining in a certain state decreases by increasing x. For x > 1,
and at a constant Xi, the probability of remaining in the state decreases as time
increases, where in general, Eq.(2-126a) is satisfied.
152
1—I I — \ — \ — I — r
- A , t = 1
Xt = 0 , x > 0 -
j - . : i":: i " i^ 4 6
X 10
Fig.2-50. The Pure Birth model
b) Px(t) obeys Eq.(2-127). There is, however, some problem with the summation
in the equation. Eq.(2-139) indicates that the solution for Px(t) depends generally
on the value of X . Hence, it is possible for a rapid increase in the A. to lead to the
condition where:
oo
S Px(t) < 1 x=0
i.e., is a dishonest process. However, in order to comply with Eq.(2-127) for all t,
it is necessary that [2, p.81]:
i^ (2-144)
x=0
c) The mean number of events, m(t), occurring in the time interval of length t, is
defined by Eq.(2-128) where for Px(t) given in Eq.(2-143) it reads:
m(t) = ]£xP^(t) = e ^ (2-145)
x=0
The only application of Eq.(2-143) in Chemical Engineering is for x = 1, i.e.,
that during the time interval t only one event occurs with the probability Pi(t) given
by:
153
Pi(t) = e- ^
This is the same equation obtained by the Poisson model, i.e., Eq.(2-130).
The consecutive-irreversible z-state process. In this process, encountered
in chemical engineering reactions, the system undergoes the following succession
of transitions:
So -> Si -> S2 -> S3 -> ... -> Sj -» ... ^ S,_i ->S^ (2-146)
In this context, the system is defined as a fluid element containing chemical species.
The above scheme may be looked upon as a birth process where the bom member
gives birth to only one new member. Once the new member gives birth, it becomes
infertile for any reason. At a certain time interval, it may happen that all members
are alive at different ages. However, at the end of the birth process, only one
member remains and all previous bom ones disappeared. The present birth scheme
is similar to the Poisson process, however, no disappearance of members versus
time occurs in the latter. In addition, the magnitude of the birth rate Xi in Eq.(2-
146) depends on the state Si, where in the linear birth process, according to which
mankind growth is conducted (in the absence of death), the birth rate is
proportional to the state size x according to Eq.(2-140). In view of the above
assumptions, we may conclude that Eqs.(2-138) to (2-139a) are applicable.
Let us now apply the above birth model to a well-known process, i.e., a
consecutive-irreversible z-stage first order chemical reaction, with a single initial
substance, the "first member of the family". The various states are Sj s Aj (i = 0,
1,..., Z) where A designates concentration of a chemical species i acquiring some
chemical formula; from Px(t) it follows that i = x. Considering Eq.(2-146), the
system occupies the various states at different times, i.e. a fluid element contains
different species along its transitions among the states. Px(t) is the probability of
occupying state x, i.e., occupying the chemical state of species i = x. x = 0 is the
initial chemical substance, x = 1 is the second chemical species, etc., where x = Z
is the "last bom member" of the family which remains alive for ever, according to
Eq.(2-146). Another interpretation of Px(t) is the probability that at time interval
154
(0, t), the system will still occupy the state of chemical species x of concentration
p^(t) = Ax/(t)Ao(0) where Ao(0) is the initial concentration of species x = 0. On the
basis of Eqs.(2-138) to (2-139a) we may write for the consecutive-irreversible
reactions the following equations, designating X-i = ki where the latter are the
chemical reaction rate constants (1/time):
(2-147a)
for
for
for
dPo(t)
dt "
species 0.
dPi(t)
dt "
species 1.
dPjd)
dt
specie 2.
-'A^
-koPo(t)
-k,P,(t) ^ -koPo(t)
-k2P2(t) + kiP,(t)
= -k .P_ ,m + k Af " -Z- l ' Z-l^"-' ' "-Z-2^ Z-Z^"-'
for specie z-i. For species z the probability or mass balance reads
Px(0) = 1 for X = 0
Px(0) = 0 for X > 0
(2-147b)
(2-147c)
(2-147d)
dP,(t) - ^ = k,_,P,_i(t) (2-147e)
The following are the initial conditions:
(2-147f)
A general solution for the above set is available [22, p.l 1]. For Z = 2 considered
in the following, it reads:
PQ(t) = e-M (2-148a)
155
Pl(t) = -^-!-r[e-V-e-KV] Iv — 1
(2-148b)
P2(t) = l -Po( t ) - Pi(t) (2-148C)
where K = ki/ko- Some characteristic properties of the above distributions are
demonstrated in Fig.2-51 where the relationship Px(kot) - kot is plotted for K =
0.25.
K = k/k =0.25 1 0
20
Fig.2-51. The Pure Birth model for consecutive reactions
Generally, it is observed that the system, eventually, occupies state 3, i.e.,
only species 3 is present; the other "members of the family" have died during the
time. On the basis of curves in the figure, we may ask also the following questions
which are of some interest. If the occupation probability of some state is, say,
0.368, then what is this state, what is the situation of the other states and during
what time interval are the above occupation probabilities valid? According to
Eq.(2-148c) the summation of all probabilities at each time must be unity.
Therefore, for the above probability, the following is the probability distribution of
the system among the states: Po(l) = 0.368, Pi(l) = 0.548 and P2(l) = 0.084. In
other words, at time kot = 1, 36.8% of the system still occupies the state x = 0 (So
= Ao), 54.8% underwent a transition to state x = 1 (Si = Ai) and 8.4% to state x =
156
2 (S2 = A2). However, at time kot = 20, the whole system, practically, occupies
state X = 2 (S2 = A2).
To conclude birth models, it is interesting to present in Fig.2-52 the painting
Development II by Escher [10, p.276] which is originally a woodcut in three
colors. The painting demonstrates the development of reptiles and at first glance it
seems that their number is increasing along the radius. Although their birth origin
is not so clear from the figure, it was possible, by counting their number along a
certain circumference, to find out that it contains exactly eight reptiles of the same
size. This number is independent of the distance from the center. Thus, it may be
concluded according to Eq.(2-142) that dPx(t)/dt = 0, namely, Px=8(t) = 1 . In other
words, the probability of remaining at the state x = 8 reptiles during time interval
(0, t) is 100% since no birth takes place along the radius.
^ # . • • •_ • ^p*
« *
*
Fig.2-52. Escher*s demonstration for reptiles' birth rate of X = 0
(M.C.Escher "Development 11" © 1998 Cordon Art B.V. - Baarn - Holland. All rights reserved)
157
The Polya Process. In the above models the rate of occurrence of the
events X was independent of time, i.e., homogeneous. In the present process, it
has been assumed that the probability of a transition from Sj to Sk is time
dependent. Two cases are considered in the following.
The inversely proportional time dependence. It has been assumed [2,
p.82] that in the time interval (t, t + At) the transition probability is given by:
1 + OCX
prob{X(t, t + At) = 1} = '^YT^Hkt^^ " ^^^^^" Pjk (2-149)
where a and X, are non negative constants. The above probability designates an
occurrence of one event or one occupation during the time interval At. By
proceeding as before, we obtain the following differential equations:
dPo(t) 1 - i r = -^TTEalPo( t ) (2-150a)
subjected to the following initial conditions:
Px(0) = 1 for X = 0
P x ( 0 ) = O f o r x > 0 ^2-^^^*')
Eq.(2-150a) yields that:
Po(t) = (1 + aXt)- ' '" (2-151)
and the solution of Eq.(2-150) reads:
Px(t) = - ^ ( 1 + aXt r -^^ '^^J^d + ai) X = 1, 2,.. . (2-152) i = l
158
For a = 0 the equation reduces to Poisson model. For a = 1 and by introducing
the new time parameter (1 + Xt) = exp(XT), one obtains the Yule-Furry process
given by Eq.(2-143a).
Some characteristic properties of the above distribution are:
a) Fig.2-53 demonstrates the Px(t) - Xt, Px(t) - x relationships computed from
Eq.(2-152) with Xi as parameter and for a = 1, 10. It is observed that generally
Eq.(2-126a) is satisfied.
1
0.8
^ 0.6
QL 0.4
K h x = 0
1—I—I—I—r
x=1 5 2 1 2 • a -/1 1 1 10 1.0
l^n—I—I—I— \— \—r
L xt = o,x = o
Fig.2-53. The Polya model for inversly proportional time
dependence
b) Px(t) obeys Eq.(2-127).
c) The mean number of events, m(t), occurring in the time interval of length t, is
defmed by Eq.(2-128) where for Px(t) given by Eq.(2-152)
m(t) = 2)^Px(t)»?^t x-O
(2-153)
The exponential time dependence. Another dependence on time of the
transition probability from Sj to S^, leading to a closed form solution of the
differential equation, assumes that the process rate parameter X. is given by:
A. = AQC (2-154)
where KQ and k are adjustable parameters. Let us relate X. to the following
humorous example noting that numerous cases fall in this category. A girls'
159
dormitory accommodates N girls in every academic year. The mean rate of a girl
acquiring a boy-friend throughout the year varies with time according to Eq.(2-154)
where t denotes the number of months lapsed from the beginning of the academic
year. Assume that the same girls stay in the dormitory, there is no departure or
arrival of new girls during the academic year, a girl can have only one boy-friend
throughout the year, and that there are no break-ups during the year. The decrease
of X with time is plausible because as time goes by, the girls become more engaged
in their studies and don't have too much time for other activities. If the system is
the N girls in the dormitory and x is the state of the system, i.e., the number of
girls already acquired a boy-friend, where Px(t) is the probability that at time
interval (0, t) x girls are already occupied, then the set of Eqs.(2-122)-(2-124) is
applicable under the initial conditions Po(0) = 1, Px(0) = 0 for x = 1,2, ..., N.
The solution of Eq.(2-124) with X given by Eq.(2-154) is:
Po(t) = c^^M^M-^^)-i^ (2-155)
The rest of the solution is obtained as follows [23]:
dP,(t) ^ dP,(t) d?i ^ . kt^Px(t)_ , dP,(t) dt " d^ dt —^^0^ dX - ^^ dX
Thus, Eq.(2-123) is expressed in terms of the variable X instead of t by:
dP.(>.) 1
- ^ = ^ [ p , a ) + p , . i a ) ]
where Px(Ao) = 0 for x > 1. If the following solution is assumed:
P^iX) = Po(>.)F,(X) = c^^-'^'^F^iX)
it is possible to obtain for Fx(X,) that:
dF^ (X) 1.
dX = - k F x - i a )
160
where Fx(X.o) = 0. Applying the solutions for Fx(X.) and the above relationships,
yields the following expression for Px(t) which is a modified Poisson model:
Px( t )« -^{[?^o /kIexp( -k t ) - l ]}VVa^^^^ X - 0 , 1 , 2 , . . . (2-156)
If k = 0, the original Poisson model is obtained, i.e., Eq.(2-126).
Some characteristic properties of the probabiUty distributions computed from
Eq.(2-156) are demonstrated in Fig.2-54. The figure on the left-hand side indicates
that the distributions versus time attain a steady state depending on XQ and k. For
example, for x = 1 and Kolk = 1, namely, one girl has already been occupied, it is
observed that the probability of remaining at this state during time interval (0, oo) is
0.368 where the probability of remaining in state x = 0 is zero at t > 0. Thus, the
probabiUty of occupying the other states in the above time interval is 1 - 0.368 =
0.632 , i.e., each girl will acquire a boy-friend throughout the year with some
probabiUty. The figure on the right-hand side indicates that increasing the number
X of the occupied girls, other parameters remain constant, reduces the probability of
remaining in this state. A possible explanation for this behavior is that an increased
number of occupied girls by a boy-friend, indicates that this is a good situation.
Thus, more giris would like to acquire boy-friends which is nicely predicted by the
model.
K 0.8
^ 06 U
Q- 0.4 1-.
0.2
0
"1—I—I—I— \—r
kt=0,x = 0
x=1 2 2 1 X^=1 1 5 5
1—r
^ - i — I * - ! - - ! —I -H H 4 6
kt 10
*^g^ I I 1 1 1 1 1 L ^ kt = 0.x = 0
r kt=i r 0 * =1 i K ^ h - \ j / 1 ^ 4 h \ 5 5 L V ^ ^ j/ L -^XSr'""^' -V- f' r^4»» r i^'i^^r -
r n
-H - n
1 H
1
J
±z2 4 6
X 10
Fig.2-54. The Polya model for exponential time dependence
161
The Simple Death Process. In the above birth processes, the transition
of the system is always from state Sj = x - 1 to Sk = x where x > 0. This is clearly
reflected in Eqs.(2-122) and (2-138, 2-141). In the death process, the transition of
the system is in the opposite direction, i.e., from Sj = x + 1 to S^ = x, and the
derivation of the governing equations follows the assumptions:
a) At time zero the system is in a state x = XQ, i.e., the random variable
X(0) = Xo>l .
b) If at time t the system is in state x (x = 1, 2, ...), then the probability of the transition x -» x -1 at the interval (t, t + At) is:
Pjk = |ixAt + o(At) (2-156)
c) The probability of remaining in state x reads:
Pjj = 1 - iLlx t + o(At) (2-157)
where [X is the mean occurrence rate of the events which is a function of the actual
state. d) The probability of the transition x -^ x - i where i > 1 is o(At), i.e., twin or
multiple death is impossible. In view of the above assumptions and following
Eq.(2-3), we may write that:
Px(t -h At) = (1 - HxAt)Px(t) + |Lix+iPx+i WAt + o(At) (2-158)
The reason that x + 1 occurs in the coefficient of Px+i(t) is that the probability of an
event has to be taken conditional on X(t) = x + 1. The differential equation which
follows reads:
dP.(t) — ^ = - H,P,(t) -h ix+iPx+iCt) (2-159)
where |LIX is the death rate which is a function of the state the system occupies.
If |Lix = |Li = constant, we have a constant rate death process. The case of fx =
1 death per unit time may be demonstrated in Fig.2-49 as follows. If one is
observing the figure, while beginning from the middle and continuing to left or to
162
the right, he can see that the number of fish or beens is reducing up to one, due to
their death for some reason.
If it is assumed that the death rate follows a linear relationship, i.e., jix = Mx
where |i > 0, we obtain the so-called simple death process or linear death process
satisfying:
dP.(t) — ^ = - |ixP,(t) + ^(x + l)Px+,(t) (2-160)
subjected to the following initial conditions:
Px(0)=l for x = xo> 1
Px(0) = 0 for X ;i xo
The solution of the differential equation is [2, p.85]:
(2-161)
Px (t) = , / " ' „e-' o tKe^ t_i)Xo-x o < x < x o (2-162) x!(x — XQ)!
with the following characteristics: a) Fig.2-55 demonstrates the Px(A.t) - x, Px(?tt) - Xi relationships computed from
Eq.(2-162) with \ii as parameter for two initial values of XQ = 5, 10. A typical
behavior revealed in the present model indicates that:
Px = xo(0) = Px = o M = l (2-163)
which is due to XQ > x. On the other hand, in the previous models:
Px = o(0) = 1 and Px>o(oo) = 0 (2-164)
163
1 — I — I — I — I — I — ! — r 1—I—I—I—I—r
x^ = 5
I >i ^ 1 ^ r-->j \ I I L 2 4 6 8 10 0 2 4 6 8 10
X X
1—\—r—r
x^ = 5
J L-L 2 4 6 8 10 0 2 4 6 8 10
Fig.2-55. The Simple Death model
b) Since XQ > x and depending on the magnitude of XQ, the probabiUty of remaining
in a certain state increases versus time.
c) Px(t) obeys Eq.(2-127).
d) The mean number of events, m(t), occurring in the time interval of length t, is
defined by Eq.(2-128) where for Px(t) given in Eq.(2-162):
m(t) = 2 x P ^ ( t ) = Xoe-»* x-O
(2-165)
The Birth-and-Death Process. An example of such a process are cells
undergoing division, i.e., birth, w^here simultaneously they also undergo
apoptosis, i.e., programmed cell death. The latter might be due to signals coming
from exterior of the cells. The existence of human being may also be looked upon,
approximately, as a simultaneous birth-death process although population in the
164
world is generally increasing. One should imagine what would have happened if
death phenomenon would not have existed. Finally, it should be noted that birth-
and'death processes are of considerable theoretical interest and are also encountered
in many fields of application.
The derivation of the governing equations is based on the following
assumptions:
a) If at time t the system is in state x (x = 1, 2,...), the probability of the transition X -> X + 1, i.e. Sj to Sk, in the time interval (t, t + At) is given by:
Pjk = >oc t + o(At) (2-166)
where Xx is the mean birth rate which is an arbitrary functions of x.
b) If at time t the system is in the state x (x = 1,2, ...), the probability of the transition x -^ x - 1, i.e. Sj to Sk, in the time interval (t, t + At) is given by:
Pjk = M t + o(At) (2-167)
where |LIX is the death rate.
c) The probability of a transition to a state other than the neighboring state is o(At).
d) If at time t the system is in the state x (x = 1, 2,...), the probability of remaining
in the state is given by:
pjj = 1 - {\ + ixJAt + o(At) (2-168)
e) The state x = 0 is an absorbing or dead state, i.e., poo =1.
In view of the above assumptions and following Eq.(2-3), we may write that:
P^(t + At) = ;ix-iPx-i WAt + |ix+iPx+i(t)At + [1 - (Xx + ^x)^t)]Px(t) + o(At)
(2-169)
leading to the following differential equation:
dP.(t) - ^ = ?^x-iPx-i(t) - ^ x + ^x)Px(t) + Hx+iPx^i(t) (2-170)
165
In Eq.(2-169), the first two terms on the right-hand side describe the transitions
from states x-1 and x+1 to state x; the third term designates the probability of
remaining in state x. Eq.(2-170) holds for x = 1, 2, ... where for x = 0, noting
assumption e above, we have:
- ^ = ^iPi(t) (2-171)
since X^i = XQ = [XQ = 0. If at t = 0 the system is in the state x = XQ, 0 < XQ < <»,
the initial conditions are:
Px(0)=l forx = xo
Px(0) = 0 forx^txo (2-172)
Depending on the X^-x and |ix " ^ relationships, two cases will be considered,
viz., the linear birth-death process and the z-stage reversible-consecutive process.
The linear birth-death process. It is assumed here that (i^ = fix and A. = X,x
where \i and X are constant values. The solution obtained for XQ = 1[2, p.88],
which satisfies Eqs.(2-172), reads:
Px(t) = [1 - a(t)][l - p(t)][p(t)]^-i X = 1, 2,... (2~173a)
Po(t) = a(t) (2-173b)
where
M[e(^-fi)t - 11 M
m = rK-T-^ (2-174b)
166
Some characteristics of the solution are:
a) Fig.2-56 demonstrates the Px(t) -1 relationship computed by Eqs.(2-173a,b) for
different values of \i and X designated on the graph.
40 50
Fig.2-56. The Birth-Death model
The following trends, also demonstrated in the figure, are observed regarding to the
extinction probability Po(t), i.e., the probability that the population will eventually
die out by time t The trends are obtained from Eq.(2-173b) by letting t -> oo.
Hence:
lim Po(t) -= 1 for X < |ui t-»oo
« Y f or X > ji (2-175)
From the above results we can see that the population dies out with a probability of
unity when the death rate is greater than the birth rate, but when the birth rate is
greater than the death rate, the probability of eventual extinction is equal to the ratio
167
of the rates. Additional computations indicate that if X > |Li (left-hand side graphs),
increasing X at constant i decreases Px(t). For example:
for ^1= 1 and A, = 2, Po(t) = 1 i f t > 3
for|Li=land^=10,Po(t) = 0.1 i f t > l
for |i = 1 and X = 50, Po(t) = 0.02 if t > 1.
If |l > A., increasing |Li causes Po(t) to approach unity and the others to faster
approach zero.
b) Px(t) obeys Eq.(2-127).
c) The mean number of events, m(t), occurring in the time interval of length t, is
defined by Eq.(2-128), where for Px(t) given in Eq.(2-173)
m(t) = e( - )t (2-175)
The following asymptotic behavior is obtained:
limm(t) = 0 for X<\i
= 1 for A = |i
= oo for >.>^ (2-176)
From the above we can see that when A = |i, the expected rate of growth is zero
and the mean population size is stationary.
The Z'State reversible-consecutive process. In this process, widely
encountered in chemical engineering reactions, the system, defined below,
undergoes the following succession of simultaneous birth-death transitions:
XQ X J X,2 X^ X J _ | X- X^_2 A,2_j
So -^ Si -» S2 ^ S3 -^ ...-^ Si -» ... -^ S,_i ^ S, (2-177)
^1 1^2 \^3 1^4 \^i \^M ^^z-1 \^z
where the magnitude of the birth rate Ai and the death rate fij depend on the state Sj.
Let us now apply the above simultaneous birth-death model to a well-known
process, i.e., a z-stage irreversible-consecutive first order chemical reaction, with a single initial substance. The various states are Sj = Aj (i = 0, 1, ..., Z) where Ai
designates concentration of a chemical species i acquiring some chemical formula;
168
considering the quantity Px(t), it follows that i = x. In this context, the system is
defined as a fluid element containing chemical species which undergo a chemical
reaction or not. Considering Eq.(2-177), the system occupies the various states at
different times. Px(t) is the probabiUty of occupying state x, i.e., occupying the
chemical state of species i = x. x = 0 is the initial chemical substance, x = 1 is the
second chemical species, etc., where x = Z is the last born member of the family
which remains alive for ever. Another interpretation of Px(t) is the probabiUty that
at time interval (0, t), the system still occupies the state of chemical species x of
concentration Px(t) = Ax(t)/Ai(0); Ai(0) is the initial concentration of species x = i.
On the basis of Eq.(2-170) we may write the following equations where %{and m
are the chemical reaction rate constant (1/time):
^ ^ = HiPi(t)-?ioPo(t) (2-178a)
for species 0.
^ ^ = |i2P2(t) + K^oii) - Pi(t)[^i + X,] (2-178b)
for species 1.
dPn(t) — ^ = ^3P3(t) + ?LiPi(t) - P2(t)[|Li2 + ^2] (2-178C)
for specie 2.
^ ^ = n,P,(t) + V2P,_2(t) - P.-i(t)[|i.-, + K-{\ (2-178d)
for specie z-1 where for specie z
dp,(t) dt = ^z-iPz-i(t) - HzPz(t) (2-178e)
169
For Z = 2, a solution is available [22, p.42] if we take species 0 with initial
concentration Ao(0) to be the starting substance. Then, Px(t) = Ax(t)/Ao(0) and the
solution reads:
Po(t) = ^^rr + ; e ^^' + Y1Y2 Y I ( Y I - Y 2 )
YVY2(^ll+^2+^l) + ll i2
Y2(Y2-YI)
-Y2t (2-179a)
P,(t) = >.o LY1Y2 YI(Y,-Y2) Y2(Y2-YI)
(2-179b)
P2(t)=^? i [ ^ 7 7 7 : + - r — : « " ' • ' + - r ^ H LY1Y2 Y I (Y I -Y2 ) Y2(Y2-YI) J
(2-179C)
where YJ and Y2 are roots of the following quadratic equation taken with the reverse
signs:
Y + Y(^ + X,i + M-i + 1I2) + ^^1 + ^|A2 + RIH2 = 0 (2-180)
Some characteristic properties of the above probability distributions, or
dimensionless concentration of the species, are demonstrated in Fig.2-57 where the
relationship Px(t) -1 is plotted.
170
1
0.8
^ 0.6 X
^ 0.4
0.2
0
"\ x = 0
- \
-
L 1 IZ- , 1 .
1
1
' 1 ' 1 ' J
0 1 J ^1^ = 1 ^^2=1 1
_J J — —-„ : =" * !
1 1 1 1 1
x = 0
^^ 2- ^^ = 1 X^=0.5
, \ ^ 1 = 0.2 t =0.1
2 3 t
10
Fig.2-57. The Birth-Death model for reversible-consecutive reactions
Generally, it is observed that the system, i.e., a fluid element containing
some chemical species, attains a steady state distribution of its occupation
probability among the various states. The distribution depends on the rate
constants designated in the figure and are characteristic of reversible reactions.
Considering the figure on the left-hand side, it may be concluded that the
probability of remaining in states 0, 1 and 2 at time interval (0, ©o) is 1/3 for each
state, i.e., in a form of the initial component, component 1 and component 2. The
above probabilities indicate also that in the above time interval, exactly three
changes have occurred, i.e., the formation of states 2 and 3 due to the reaction and
also the remaining in state 0. Certainly, the quantities Px(oo) = Ax(c»)/Ao(0), x =
0, 1, 2, may be looked upon as dimensionless concentrations encountered in
reaction engineering. On the basis of the figure on the right-hand side, it may be
concluded that if the rate constants |LII and |i2 will approach zero, i.e., the effect of
the irreversibility is diminished, the probability of remaining at states 0 and 1 will
also diminish at time interval (0, ©o), and the only state to be occupied by the
system is 3, i.e., the state of solely component 3.
2.3 MARKOV CHAINS CONTINUOUS IN SPACE AND TIME 2.3-1 Introduction
The above topic is presented in the following because the equations
developed and the basic underlying mechanism, i.e. diffusion, are of fundamental
171
importance in Chemical Engineering. However, the major problem of applying the
equations is that not always an analytical solution is available.
Let us consider a dancing hall where people are arriving at and leaving from
after they have enjoyed the evening. If the hall is defined as system and the number
X of people present in the hall at some time during the evening is the state of the
system, x = 1, 2, 3,. . . , then this situation may be looked upon as a Markov chain
continuous in time and discrete in space, elaborated in the preceding section 2.2.
More specifically the situation may be categorized as a birth-death model where the
entering people designate birth and the leaving ones symbolize death. The mean
rates of birth and death can easily be determined by counting the people arriving
and leaving the hall. In the above example, one may be interested in the probability
that at a certain time of the evening, the number of dancers in the hall is some
prescribed value. Certainly, early in the evening, x = 0 and Px(0) = 1. The main
characteristic of processes similar to the above one, as well as the ones discussed in
section 2.2, is that in a small time interval there is either no change of state or a
radical change of state. Therefore in a finite interval there is either no change of
state or a finite or possibly denumerably infinite number of discontinuous changes.
Assume now that some time in the evening the dancing hall becomes
extremely congested and the doors are suddenly shut. Under these conditions, the
movement of the dancers in the hall may be governed by random impacts of
neighboring dancers. A typical physical situation, similar to the above one, is that
of particles suspended in a fluid, and moving under the rapid, successive, random
impacts of neighboring particles. If for such a particle the displacement in a given
direction was plotted against time, we would expect to obtain a continuous or
somewhat erratic graph which would, in fact, be a realization of a stochastic
process in continuous time with continuous state space. The characteristic of such
processes is that in a small time interval, displacement or change of state are small.
The physical phenomenon related to the above behavior is known as Brownian
motion, first noticed by the botanist Robert Brown in 1827. The modeling of such
motions is based on the theory of diffusion and kinetic theory of matter. Markov
processes associated with above motions, in which only continuous changes of
state occur versus time, are therefore called diffusion processes. The above points
will be elaborated in the following.
172
2.3-2 Principles of the modeling Brownian motion of particles is the governing phenomenon associated with
transitions between states in the above examples as well as in the mathematical derivations in the following [4, p.203]. If we consider a particle as system and the states are various locations in the fluid which the particle occupies versus time, then the transition from one state to the other is treated by the well-known random walk
model. In the latter, the particle is moving one step up or down (or, alternatively, right and left) in each time interval. Such an approach gives considerable insight into the continuous process and in many cases we can obtain a complete probabilistic description of the continuous process.
The essence of the above model is demonstrated for the simple random walk
in the following. Let X(n) designate the position at time or step n of the moving particle (n = 0, 1,2,...). Initially the particle is at the origin X(0) = 0. At n = 1, there is a jump of one step, upward to position 1 with probability 1/2, and downwards to position -1 with probability 1/2. At n = 2 there is a further jump, again of one step, upwards or downwards with equal probability. Note that the jumps at times n = 1, 2, 3, ... are independent of each other. The results of this fundamental behavior are demonstrated in Fig.2-58 where two trajectories 1 and 2 for a single particle, out of many possible ones, are shown.
T \ \ I I I I r
J I I \ \ L 0 1 2 3 4 5 6 7 8 9 10 11 12
Time, n
Fig.2-58. Two independent realizations of a simple random walk
173
In general, the trajectory of the particle is given by:
X(n) = X(n - 1) + Z(n) (2-181)
where Z(n), the jump at the nth step, is such that the random variables Z(l), Z(2),
... are mutually mdependent and all have the same distribution which reads;
prob{Z(n) = 1} = prob{Z(n) = - 1} = 1/2 (n = 1, 2,...) (2-182)
Finally, it should be noted that Eq.(2-181), taken with the initial condition X(0) =
0, is equivalent to:
X(n) = Z(l) + Z(2) + Z(3) + ... + Z(n) (2-183)
The mathematical elements used in the following derivations are:
X(t) - a random variable designating the position of a particle, system, in the fluid
at time t.
X(t) = X, indicates realization of the random variable, i.e., that at time t the random
variable acquired a value x, or, that the system occupied state x.
p(y, x; X, t) - a probability density function, i.e. probability per unit length, where,
p(y, T; X, t)dx - is the probability of finding a Brownian particle at time t in the
interval (x, x + dx) when it is known that the particle occupied state y at an earlier
timex.
It follows from above definitions that:
prob{a < X(t) < b | X(T) = y} = j p(y, x; x, t)dx (t > x) (2-184)
i.e., the probability of finding a particle at time t between states a and b, subjected
to the fact that the particle occupied state y earlier at time x, is given by the integral
on the right-hand side. Assume now that the lower limit is a = 0, then we define a
new ftmction P(y, x; x, t), the transition probability function by:
174
P(y,T; b, t) = p(y, x; x, t)dx = prob{ X(t) < b | X(T) = y} (t > x) (2-185) Jo
where
p(y,T;x.t) = ^ ^ i i ^ 4 i i ^ (2-186a)
P(y, x; X, t), the probability of finding a particle at time t in the interval (0, b) while
recalUng it occupied state y at time x, also satisfies the following conditions:
lim P(y, x; x, t) = 0
x ^ O 1- T./ A (2-186b) hm P(y, x; X, t) = 1
2.3-3 Some continuous models In the following, two basic models are presented [2, p. 129; 4, p.203]; the
Wiener process and the Kolmogorov equation. Applications of the resulting
equations in Chemical Engineering are also elaborated. The common to all models
concerned is that they are one-dimensional and, certainly, obey the fundamental
Markov concept - thai past is not relevant and thai future may be predicted from the
present and the transition probabilities to the future.
The Wiener process. We consider a particle governed by the transition
probabilities of the simple random walk. The steps of the particle are Z(l), Z(2),
... each having for n = 1, 2,.. . the distribution:
prob{Z(n) = 1} = p, prob{Z(n) = - 1} = q = 1 - p (2-187)
where p and q are constant one-step transition probabilities. Assume that a particle
is in the time coordinate n - 1 . It undergoes a transition to a state coordinate k from
the state coordinates k - 1 and k + 1. Thus, we may write the following forward
equation for a fixed j :
175
Pjk(n) = Pj,k-l(n - l)p + Pj,k+l(n - l)q (2-188)
where pjk(n) is the probability that a particle occupies state k at time n, if it occupied
state j at n = 0. Considering the definition of p(y, T; X, t)dx above, let:
p(yo, 0; X, t)Ax = p(xo, 0; x, t)Ax = p(xo; x, t)Ax (2-189)
be the conditional probability that the particle is at x at time t, given that it started at
yo = xo at time T = 0; x is a continuous state coordinate. We have also XQ = j Ax,
X = kAx and t = nAt, thus, Eq.(2-188) in terms of the new scale reads:
p(xo; X, t) = p(xo; x - Ax, t - At)p + p(xo; x + Ax, t - At)q (2-190)
the factor Ax canceling throughout. Suppose that p(xo; x, t) can be expanded in a
Taylor series and the first and second derivatives exist, we obtain that:
p(xo; X + Ax, t - At) = p(xo; x, t) - A t | ^ + A x - ^ + 0.5(Ax)^|^ (2-191)
If we expand Eq.(2-190) according to Eq.(2-191) and apply the following
expressions for p and q [4, p.206]:
p = 0.5[l + i ! f ^ ] , q = 0 . 5 [ l - i ! f i ] (2-192)
where |i is the mean (length/time) and a^ is the variance (lengths/time) of the
random variable X(t) designating the position of the particle at time t. If At -> 0,
we obtain the forward equation:
^P(xo; X, t) ^^2^^p{xo,x,i) apCxp; X, t) 5t = ^-^^ ; 2 ^ ^ (2-193)
176
which is a partial differential equation of the second order with respect to x and is
first order in t. It is an equation in the state variable x at time t for a given initial
state XQ. The solution of the above equation conditional on X(0) = XQ reads:
with the two parameters i and a. Eq.(2-194) satisfies Eqs.(2-185)-(2-186b).
Eq.(2-193) is familiar in Chemical Engineering in the following cases [24]:
a) Diffusive-convective heat transfer. In this case, the equation of energy for
constant properties, ignoring viscous dissipation, taking into account axial heat
diffusion only in the x-direction which is superimposed on the molecular diffusion
of heat, reads:
9T(x, t) ^,3^T(x,t) ,,aT(x,t) . .Q. ^ — 5 t — = ^ 9^2 " ^—^T- (2-195a)
Comparing the above equation with Eq.(2-193) demonstrates the similarity between
the two for XQ = 0. The parameters appearing in Eq.(2-195a), i.e., the heat thermal
diffusivity a which accounts for the molecular diffusion, and the constant axial
velocity U accounting for the axial diffusion, are characteristic to the heat transfer
process. The underlying model of particle's motion associated with the derivation
of Eq.(2-193) may be applied also to heat transfer. In this case, a fluid element (or
a molecule) at some temperature moves due to both, diffusion as well as axial
diffusion generated by the fluid velocity, transferring heat to the other fluid
elements. The motion of the fluid elements may be looked upon also from
probabilistic arguments which led to Eq.(2-193). Finally, it should be mentioned
that in particular problems, the solution of the equations will depend on initial and
boundary conditions.
b) Diffusive-convective mass transfer. The equation of continuity for species
A in a binary solution of A+B, assuming constant density of the fluid mixture as
well as the diffusion coefficient DAB. taking into account axial mass diffusion only
in the x-direction which is superimposed on the molecular diffusion of mass, reads:
177
dCAx, t) 9^CA(X, t) 9CA(X, t) - V - = D A B — ^ ; 2 U - ^ ^ (2-195b)
CA is the local concentration of species A in the mixture. Other remarks made in
(a) are also applicable here.
c) Heat or mass transfer into semi-infinite bodies by pure conduction or
diffusion. The governing equations may be obtained from the above equations by
ignoring the axial component; thus:
3T(x, t) , .8^(x, t )
^£^ = j£0dl (2-1950
The above equations are similar to Eq.(2-193) for |LI = 0 which assumes that a fluid
element of some concentration, or at some temperature, is moving by pure
molecular diffusion, thus generating mass or heat transfer in the system.
An additional equation of a similar form, belongs to momentum transfer. For
a semi-infinite body of liquid with constant density and viscosity, bounded on one
side by a flat surface which is suddenly set to motion, the equation of motion reads
[24, p. 125]:
^ ^ = v ^ ^ (2-195d)
Ux is the velocity of the liquid in the x-direction, and v is the kinematics viscosity of
the liquid governing the momentum transfer by pure molecular diffusion.
The Kolmogorov equation. In the preceding model, the major
assumption made was that the one-step transition probabilities p and q remain
constant. In the following it is assumed that these probabiUties are state depending,
namely, designated as pk and q . The following forward equation for the present
model reads:
178
Pjk(n) = Pj,k-i(n - l)pk-i + Pj,k+l(n - l)qk+l + Pj,k(n - 1)(1- Pk - Qk) (2-196)
The equation may be interpreted in the following way: the probability of being at
state k after n steps = the probability of being at state k-1 after n-1 steps times the
one-step transition probability (which depends on state k-1) plus the probability of
being at state k+1 after n-1 steps times the one-step transition probability of moving
from state k+1 to k (which depends on state k+l) plus the probability of being at
state k after n-1 steps times the probabihty of remaining in this state, (1- Pk - Qk); all
transitions take place in the time interval n and n+1. Again we consider the process
as a particle taking small steps -Ax, 0, or +Ax in the small time interval of length
At. From Eq.(2-189) the probability density p(xo; x, t)Ax may be looked upon as
the conditional probabihty that a particle is at location x at time t, given that initially
it occupied XQ. Thus, Eq.(2-196) becomes:
p(xo; X, t) = p(xo; x - Ax, t - At)p(x. AX) + p(xo; x + Ax, t - At)q(x + AX)
+ p(xo; X, t - At)(l - qx - px) (2-197)
the factor Ax canceling throughout. It has been shown elsewhere [4, p.214] that:
l i m { p ^ - q j - ^ = p(x)
Ax,At-^0
lim{p^ + qx - (Px - q x ) ^ } - ^ = a(x) (2-198)
Ax,At-^0
where P(x) and a(x) are the instantaneous mean (length/time) and variance
(lengths/time), respectively. Applying Eqs.(2-197) and (2-198), resulted in the
following forward equation:
ap(xo; X, t) a^{a(x)p(xo; x, t)} 3{(3(x)p(xo; x, t)} — 5 ^ - = 0-5 ^;;2 T. (2-199)
179
which is a parabolic partial differential equation. A more general equation can be
obtained by allowing the transition mechanism to depend not only on the state
variable x but also on time t. In this case we are led to define p(x, t) and a(x, t),
both depending on x and t, i.e., the instantaneous mean (length/time) and variance
(lengths/time), respectively. Thus, if XQ denotes the state variable at time to and x
that at a later time t, then the transition probability density p(xo, to; x, t) satisfies the
following/<7rwar<i Kolmogorov equation also called the Fokker-Plank equation:
ap(xo, to ; x, t) aS{a(x, t)p(xo, to*, x, t)} 3{p(x, t)p(xo, to; x, t)} 3t = ^'^ ^;;2 ^ (2-200)
where it has been assumed in the derivations that the above partial derivatives exist.
The functions P(x, t) and a(x, t) are sometimes called the infinitesimal mean and
variance of the process. If these functions are assumed as constant values, the
above equation reduces to Eq.(2-193) where a(x, t) = a^ and P(x, t) = |LI.
Eq.(2-200) is familiar in Chemical Engineering in turbulent flow. For
example, the energy equation for one-dimensional flow [24, p.377] for a fluid of
constant properties, in the absence of viscous dissipation effects and for xo = to =
0, reads:
aT(x,t) ,.3^T(x,t) a{u,(x,t)T(x,t)} — 5 r - = ^ -^ ;^2 ^ (2-201)
where T = T + T'; the thermal diffusivity a has been assumed constant. Note that
in turbulent flow the temperature T is a widely oscillating function of time,
fluctuating about the time-smoothed value of the temperature T, where T' is the
temperature fluctuations.
The major problem of Markov chains continuous in time and space is that the
availability of analytical solutions of the governing equations, which depend also
on the boundary conditions, is limited to simplified situations and for more
compUcated cases, numerical solutions are called for.
180
2.4 CONCLUDING REMARKS In the present chapter, Markov processes discrete in time and space,
processes discrete in space and continuous in time as well as processes continuous
in space and time, have been presented. The major aim of the presentation has been
to provide the reader with a concise summary of the above topics which should
give the reaaer an overview of the subject.
The fundamentals of Markov chains have been presented in an easy and
understandable form where complex mathematical derivations are abandoned on the
one hand, and numerous examples are presented on the other. Despite of the
simplifications made, the author believes that the needed tools have been provided
to the reader so that he can solve complicated problems in reactors, reactions,
reactor plus reactions and other processes encountered in Chemical Engineering,
where Markov chains may provide a useful tool.
The models discrete in space and continuous in time as well as those
continuous in space and time, led many times to non-linear differential equations
for which an analytical solution is extremely difficult or impossible. In order to
solve the equations, simplifications, e.g. linearization of expressions and
assumptions must be carried out. However, if this is not sufficient, one must
apply numerical solutions. This led the author to a major conclusion that there are
many advantages of using Markov chains which are discrete in time and space.
The major reason is that physical models can be presented in a unified description
via state vector and a one-step transition probability matrix. Additional reasons are
detailed in Chapter 1. It will be shown later that this presentation coincides also
with the fact that it yields the finite difference equations of the process under
consideration on the basis of which the differential equations have been derived.
2.5 ARTISTIC ENDING OF THE CHAPTER The present chapter begun with Fig.2-0, the painting Waterfall by Escher,
which made it possible to present in various places of this chapter the basic concept
of conditional probability. We end this chapter by 'Markovization', discrete in time
and space, of the amazing oil on canvas painting by Magritte Carte blanche-
Signature in blank [12, p.45] depicted in Fig.2-59. This demonstrates again a way
of entertaining the combination of art and science. A few words about this
181
painting. Magritte, in an extremely subtle and deceptive way, demonstrated the
simultaneous movement in two planes of the horsewoman. Between two trunks
the normal backdrop of foliage is visible, and this conceals a portion of the horse
and rein, the horse appearing to be passing between the same two trunks.
Spatially, the rider and the woods become an absurdity, due to this section of the
intruding background, the position of one of the horse's hind legs, and another tree
trunk in the background, part of which passes in front of the horse and rider.
Magritte expressed his thoughts on his painting as follows: "Visible things can be
invisible. If somebody rides a horse through a wood, at first one sees them, and
then not, yet one knows that they are there. In Carte blanche, the rider is hiding the
trees, and the trees are hiding her. However, our powers of thoughts grasp both
the visible and the invisible."
182
Fig.2-59. The impossible state S5 ("Carte blanche", 1965, © R.Magritte, 1998 c/o Beeldrecht Amstelveen)
Let us now apply Markov chains to investigate the trajectory of the system,
the horsewoman, riding in the forest. The possible states that the system can
occupy on the basis of Fig.2-59 are defined as follows: Si-the passage between
trees 1 and 2 through which the system can move, S2-the passage between trees 4
and 3, S3-the passage between trees 4 and 5, S4-the passage between trees 7 and 6,
183
and Ss-the impossible situation depicted in Fig.2-59. It is assumed that once the
horsewoman abandons S5, she never retums to it and continues to ride in the forest
according to her mood at the moment. Thus, S5 is an emepheral state. The
policy-making matrix distinguishes among all cases analyzed below where the
conmion to all of them is that S(0) = [0, 0, 0, 0, 1], i.e., the rider is initially at S5.
The results of the computations are depicted in Fig.2-60; on the top of each
figure, the corresponding one-step transition probability matrix is presented.
Generally, in cases a to c the states attain a steady state whereas the stationary
distribution in each case obeys Eq.(2-105a), i.e. the states reveal Ergodic
characteristics. In cases d to f the states exhibit periodic behavior whereas S5 is
eventually abandoned in all cases. In cases a, c, d and f it is abandoned after the
first step whereas in cases b and e, abandoning occurs after a few steps. This is
because in the corresponding matrix, there is also a probability of remaining in the
state, pii ^ 0.
In the following we consider each case separately. In case a the steady state
occupation probability is different for each state; the corresponding state vector
reads, S(l 1) = [0.300, 0.400, 0.200, 0.100, 0.000]. As seen, S2 is of the highest
probability, i.e. S2(l 1) = 0.400. Case b differs from a only by the values of p5i;
thus, causing vanishing of S5(n) to occur after 6 steps rather than after 1 step.
From thereon, the values of S(n) are the same as in a. Case c causes a problem to
the horsewoman because S(2) = [0.25, 0.25, 0.25, 0.25, 0.00]; thus she has a
problem what state to occupy because all states acquire an identical probability.
Case d reveals periodic characteristics of the states with period of v = 2 as follows:
S(21) = [0, 1/3, 0, 2/3, 0]
S(22) = [2/3, 0, 1/3, 0, 0]
S(23) = [0, 1/3, 0, 2/3, 0]
S(24) = [2/3, 0, 1/3, 0, 0]
In other words, in each step two states, S2 and S4 or Si and S3, acquire a certain
occupation probability. Case e demonstrates the following periodic results at
steady state
184
S(12) = [0.133, 0.267, 0.533, 0.067, 0]
S(13) = [0.067, 0.133, 0.267, 0.533, 0]
S(14) = [0.533, 0.067, 0.133, 0.267, 0]
S(15) = [0.267, 0.533, 0.067, 0.133, 0]
where the period of each state is v = 4. Case f is similar to e whereas the
occupation probability of each state at each step is unity. In conclusion we may say
that the major problem of the horsewoman, once abandoning the strange state S5,
is as William Shakespeare profoundly stated:" to be (in a state) or not to be (in a
state), that is the question''.
Si
S2
P = S3
S4
S5
Case a
Si S2 S3 S4 S5
0 1 0 0 0
1/2 0 1/2 0 0
0 1/2 0 1/2 0
1 0 0 0 0
1/2 0 0 1/2 0
Si
S2
P = S3
S4
S5
Caseb
Si S2 S3 S4 S5
0 1 0 0 0
1/2 0 1/2 0 0
0 1/2 0 1/2 0
1 0 0 0 0
1/3 0 0 1/3 1/3
1
0.8
0.6
I \ 1 r n r (a)
s/n) spx) s (n) s^(n).
n I I I r
/^<")
"1—r~ (b)
s,(n) s(n) s(n) s (n)
8 10 0 8 10
185
p =
Si
S2
S3
S4
S5
Si
0
1/2
0
1/2
1/2
Casec S2 S3
1/2 0
0 1/2
1/2 0
0 1/2
0 0
S4
1/2
0
1/2
0
1/2
S5
0
0
0
0
0
Si
S2
P = S3
S4
S5
Si
0
0
0
1
0
S2
1/2
0
0
0
0
S3
0
1
0
0
0
S4
1/2
0
1
0
1
S5
0
0
0
0
0
" 1 — I — \ ! — ! — \ 1 I r~ (c)
s.(n), i = 1,...,4
M . I I I I I I I i I I i I
10 0
Si S2 S3 S4 S5
Si
S2
S3
S4
S5
1
0
0
0
0
0
1
0
0
0
0
0
1
0
1
0
0
0
0
0
'\ /.
8 10 0
., ,; \/ 1 ; *)
y nv ''v•• 2 4 6 8 10
n Fig.2-60. Policy-making matrices and state occupation probabilities
of the horsewoman
186
Chapter 3
APPLICATION OF MARKOV CHAINS IN CHEMICAL REACTIONS
Chemical reactions occur due to collisions between similar or dissimilar
species. The collision process requires, first of all, the coming together of
molecules, which is of probabilistic or stochastic nature. Thus, chemical reactions
may be looked upon as probabilistic or stochastic processes and more specifically
as a Markov chain. According to this model, concentration of the species at time
(n+1) depends only on their concentration at time n and is independent on times
prior to n. Therefore, the governing equations for treating chemical reactions, as
elaborated later, are Eqs.(2-23) and (2-24) below, derived in Chapter 2:
z
Sk(n+1) = X^j(n)pjk (2-23) j=i
S(n+1) = S(n)P) (2-24)
The application of the equations to chemical reactions requires the proper
definition of the above quantities as well as correctly defining the transition
probabilities pjj and pjk*, this is established in the following. It should also be noted
that the models derived below for numerous chemical reactions, are applicable to
chemical reaction occurring in a perfectly-mixed batch reactor or in a single
continuous plug-flow reactor. Other flow systems accompanied with a chemical
reaction will be considered in next chapters.
187
The attractiveness of Markov chains in chemical reactions is particularly due
to the following reasons: a) Simplicity, elegance and the didactic value,
b) Demonstrating the power of probability theory in handling a pric^n deterministic
problems, c) Applying fundamentals of linear algebra to chemical reaction
problems of practical importance, d) The attractiveness of the method increases
with the number of reacting components, if the reactions are higher-order or if they
are non-isothermal. In such cases the governing differential equations are
nonlinear, and an analytical solution is extremely difficult, if not impossible;
finally, e) The solution of Eq.(2-24) yields the transient response of the reaction
towards attainment of a steady state, which is important in practice. The above
points are demonstrated by the numerous examples presented in this book.
Throughout this chapter it has been decided to apply Markov chains which
are discrete in time and space. By this approach, reactions can be presented in a
unified description via state vector and a one-step transition probability matrix.
Consequently, a process is demonstrated solely by the probability of a system to
occupy or not to occupy a state. In addition, complicated cases for which analytical
solutions are impossible are avoided.
3.1 MODELING THE PROBABILITIES IN CHEMICAL REACTIONS
Definitions. The basic elements of Markov chains associated with Eq.(2-
24) are: the system, the state space, the initial state vector and the one-step
transition probability matrix. Considering refs.[26-30], each of the elements will
be defined in the following with special emphasize to chemical reactions occurring
in a batch perfectly-mixed reactor or in a single continuous plug-flow reactor. In
the latter case, which may simulated by perfectly-mixed reactors in series, all
species reside in the reactor the same time.
The system is simply a molecule. The state of the system is the specific
chemical formula of the molecule, or what kind of species is the molecule. The
state space is the set of all states that a molecule can occupy, where a molecule is
occupying a state if it is in the state. For example, in the following irreversible
consecutive reaction:
188
2 3 4
A| -^ A2 —> A3 -> A4 -> A5
type Ai = i of a molecule is regarded as the state of the system, i.e. a specific
chemical formula.
The state space SS, which is the set of all states a system can occupy, is
designated by:
55 = [Ai, A2, A3, A4, A5] = [1, 2, 3, 4, 5]
Finally, the reaction from state Ai to state Aj is the transition between the states.
The initial state vector is given by Eq.(2-22), i.e.:
S(0) = [si(0), S2(0), S3(0), ..., Sz(0)] (2-22)
Si(0) designates the probability of the system to occupy state A s i at time zero,
whereas S(0) designates the initial occupation probability of the states [Ai, A2, A3,
..., Az] by the system. Z designates the number of states, i.e. the number of
chemical species involved in the chemical reaction. In the context of chemical
reactions, as shown later, the probabilities Si(0) may be replaced by the initial
concentration of the molecular species and S(0) will contain the initial concentration
of all species. The one-step transition probability matrix is generally given by
Eqs.(2-16) and (2-20) whereas pjk represent the probability that a molecule Aj will
change into a molecule Ak in one step, pjj represent the probability that a molecule
Aj will remain unchanged within one step. In the following, general expressions
are derived for the determination of pjk and pjj for the model below.
The reacting system. Consider a chemically reacting system containing
the species Ai, A2, A3,...., Az. A chemical reaction among the species induces the
change in the state of the mixture. It is also assumed that a certain species Aj can
react simultaneously in several reactions, designated in the following by
superscripts (i), i = 1,2,..., R where R is the total number of reactions in which
Aj is involved. The following scheme of irreversible reactions by which Aj is
converted to products is assumed, where each set of reactions involving reversible
reactions can always be written according to the scheme below in order to apply the
following derivations.
189
i s t leaction:
2ndieactiQii:
.4%.
. ^ ^ \ .
(1)
(2)
G)
products
products
ith reaction: a^ Ai + ... + ^ Aj + ... -^ ... + a^ Aj^ + ... (R)
Rih reaction: ....f>^... products (3-1)
. ( i ) ; . where a-* is the stoichiometric coefficients of species j in the ith reaction. The rate
of conversion of species j in the ith reaction based on volume of fluid V, i.e. rj*, is
defined by:
(i^ 1 dN: (^^ (i) G) A) J V dt J 1 ^ ' " y •••
(3-2)
where Nj are the number of moles of species j and Cj is its concentration, moles of .0) j/m . kj , in consistent units, is the reaction rate constant with respect to the
(i) conversion of species j in the rth reaction. In the case of a plus sign before kj , this
means moles of j formed/(s m ). The discrete form of Eq.(3-2) reads:
rf(n) = -kS^^C^(n)C^(n)... C[i(n)... (3-2a)
where the reaction rate and the concentrations Cj(n) refer to step n. In addition, the
conservation of the molar rates for all reacting species in the rth reaction in Eq.(3-1)
is governed by:
'1 „(i) Ji)
^k
(3-3)
Eq.(3-3), corresponding to the fth reaction, makes it possible to compute the
reaction rates of all species on the basis of TJ given by Eq.(3-2).
190
Definition of pjj. If species j is converted simultaneously in R reactions,
the overall rate of conversion of species j is the following sum:
dCi V (i) -dt' = rj = 2^r}'^ (3-4)
i
where the summation is over all reactions in which species j is involved, i.e. i = 1,
2,..., R. The integration of the equation between t and t + At, taking into account
that rj ^ is negative according to Eq.(3-2), yields that:
Cj(n) - Cj (n+1) = ^ r f \n)At (3-5)
where Cj(n) and Cj(n+1) are, respectively, the concentration of species j in the
mixture at time interval t and t + At or at step n and n+1. rj (n) is the reaction rate
corresponding to step n, i.e. the concentrations in Eq.(3-2a) correspond to this
step. Here it is assumed that the variation of rj (n) between step n and n+1 is not
significant. If we consider the following quantity:
VCj(n+l) ^ amount of species j present at time t = (n+1) At . rri < \ VCj(n) " amount of species j present at time t = nAt "" v " ^)
where V is the volume of fluid, we may look on it as the probability pjj that a
molecule Aj will remain unchanged within one step. Thus, from Eq.(3-5) it
follows that:
q(n+l) Pjj~ C3(n)
= 1 - I £(n)l CjCn)
At (3-6)
The summation is over all reactions in which species j is converted to products, i.e.
i = 1, 2,..., R. Eq.(3-6) indicates that if At -^ 0, pjj = 1. Indeed Aj will remain
unchanged under such conditions. However, if At is large enough, the probability
that Aj will remain unchanged approaches zero, as expected. Thus, the above expression may, indeed, serve as a probability term, provided that 0 < pjj < 1 •
191
Definition of pjk. In deriving an equation for the probability pjk
corresponding to the transition Aj -> Aj , we consider the ith reaction in Eq.(3-1).
In this case, dCj /dt = r j \ and an integration of it yields that:
Cf \n -H) -Cf (n ) rf(n)At
Cj(n) Cj(n)
amount of species j converted in the time interval At in reaction i amount of j available at time t = nAt (3-7)
Since the formation of the products in the ith reaction is associated with reaction
among all reactants, and each species contributes according to its stoichiometric
coefficient, we account for this effect in the determination of pjk by introducing the
ratio between the stoichoimetric coefficients. In addition, the transition probability
for Aj -^ Ay. depends also on the stoichiometric coefficient of Ak. Thus, on the
basis of Eq.(3-7), the probability of the transition Aj -> Aj for the ith reaction in
Eq.(3-1), i.e. Pjk, reads:
Pjk - % r . « A
^ Ji)
-J At Cj(n)
(3-8)
1=1
where N( ) is the number of the reacting species in reaction i. Indeed, the properties
of the above expression are appropriate for describing a transition probability,
namely, as At = 0, pjk = 0 while for relatively large values of At, py^ is increasing
with a limiting value not exceeding unity.
If the ith reaction in Eq.(3-1) is reduced to:
G)
(0. + a^A
then in order to comply with the results obtained by direct integration of the rate
equations, the transition probability for A; —> Aj must read:
192
a)_ (0 /1 J r f \ n ) | •At (3-8a)
If species j is involved in several reactions, the total of which is R as in
Eq.(3-1), the following expression may be obtained on the basis of Eq.(3-5) for the
overall transition probability of Aj —> Ajj, i.e.:
Pjk=2 r a«) ^
N
3 (i)
Ir;»(n)|
Cj(n) At
1=1
(3-9)
The summation is over all reactions in which species j is involved, i.e. i = 1,2,...,
R. The justification for the above expressions for pjj and pjk is confirmed later by
the agreement with determinations made on the basis of integration of the rate
equations.
If all R reactions are of the type:
Ji) (0, af%-» ... -fa^% +
the transition probability is given by:
r?(n)| V «)/ 1 ^ I £ L ^ At (3-9a)
Finally, if species j undergoes a change solely by a single irreversible reaction
of the type:
... + ajAj + + a Ak + ...
we may ignore, for the sake of simplicity, some of the subscripts and superscripts
in Eq.(3-8), which is reduced to:
/ Pjk = \ \ N
193
-^—!-At (3-10) Cj(n)
N is the number of the reacting species on the left-hand side of the above reaction.
If the above reaction is reduced to:
ajAj -^ ... + a Ak + ...
the transition probability for Aj —> Aj will read:
. 1X ri(n)
3.2 APPLICATION AND VERIFICATION OF THE MODELING
The modeling in section 3.1 will be applied to non-linear and linear reactions.
In section 3.2-1, all the stoichiometric coefficients of the species equal unity, i.e. aj
= 1. In section 3.2-2 part of the stoichiometric coefficients are different from unity,
aj 9t 1. In section 3.2-3 linear reactions are dealt with whereas in section 3.2-4,
linear-non linear reactions with aj t 1 are demonstrated in detail by case 3.13-6.
The validity of the results for Cj(n+1) computed by Eq.(3-20) on the basis of
Pjj and pjk predicted by Eqs.(3-6), (3-8) to (3-10), will be compared to those
obtained by direct integration of the kinetic equations.
3.2-1 Non-linear reversible reactions with all aj = 1 A procedure for determination of the transition probabilities is demonstrated
by a detailed treatment of the following non-linear irreversible reactions:
194
Ai+A2 "^ A3 + A4 (3-1 la)
A2 + A3"^ A4 (3-1 lb)
k-2
The following rate equations are assumed satisfying Eq.(3-3):
n = dCi/dt = - kiCiC2 + k.iC3C4 (3-12a)
12 = dC2/dt = - ki C1C2 - k2C2C3 + k. 1C3C4 + k.2C4 (3-12b)
r3 = dCs/dt = - k2C2C3 - k.iC3C4 + kiCiC2 + k.2C4 (3-12c)
14 = dC4/dt = - k-2C4 - k.iC3C4 + kiCiC2 + k2C2C3 (3-12d)
1st step: Transformation of Eqs.(3-11) into the following set of irreversible
reactions which best demonstrate the transition between the states:
(3-13a)
(3-13b)
(3-13c)
(3-13d)
2nd step: Determination of the reaction rates for Eqs.(3-13) on the basis of
Eqs.(3-12). The definition in Eq.(3-2) yields for species j = 1 reacting in the first
reaction, i= 1, that:
T\^\D) = - kS*^Ci(n)C2(n) = - kiCi(n)C2(n) (3-14a)
where k] = kj according Eq.(3-13) for i = 1. Similarly, for species j = 2 reacting
in the first and third reactions, i = 1, 3:
i = l :
i = 2:
i = 3:
i = 4:
"1
Ai + A2 - A3 + A4
A3 + A4 - Ai + A2
A2 + A3 -> A4
k_2
A4 A2 + A3
195
i[%) = 4^^Ci(n)C2(n) = - kiCi(n)C2(n)
4^\n) = 4^^C2(n)C3(n) = - k2C2(n)C3(n) (3-14b)
wherek2 =kj accordingEq.(3-13)fori= 1 andk2 = k2 for i = 3. For species j
= 3 reacting in the second and third reactions, i = 2, 3:
r^^^n) = - k f C3(n)C4(n) = - k_iC3(n)C4(n)
r'i\n) = - 4^^C2(n)C3(n) = - k2C2(n)C3(n) (3-14c)
For species j = 4 reacting in the fourth reaction, i = 4:
r f (n) = - ki^^C3(n)C4(n) = - k_iC3(n)C4(n)
rj*\n) = - kJ*^C4(n) = - k_2C4(n) (3- 14d)
3rd step: Determination of the probabihties pjj. Applying Eqs.(3-6) and (3-
14a) yields for species j = 1 where i = 1 that:
pil = l-kiC2(n)At (3-15a)
For species j = 2, converted according to Eqs.(3-13), i = 1,3, Eqs.(3-6) and (3-
14b) yield:
P22 = 1 - [kiCi(n) + k2C3(n)]At (3-15b)
For species j = 3, converted according to Eqs.(3-13), i = 2, 3, Eqs.(3-6) and (3-
14c) yield:
P33 = 1 - [k.iC4(n) + k2C2(n)]At (3-15c)
For species j = 4, converted according to Eqs.(3-13), i = 2, 4, Eqs.(3-6) and (3-
14d) yield:
P44 = 1 - [k-iC3(n) + k.2]At (3-15d)
196
4th step: Determination of the probabilities pjk. As observed in the
reactions given by Eqs.(3-13):
pl2 = 0 and P21 = 0 because Ai is not converted to A2 and vice versa.
Applying Eqs.(3-9) and (3-14a) to species j = 1, converted to species j = 3
and j = 4 according to Eqs.(3-13), i = 1, noting that NW = 2, yields:
P13 = (l/2)kiC2(n)At pi4 = (l/2)kiC2(n)At (3-16a)
Applying Eqs.(3-9) and (3-14b) to species j = 2 which is converted to species
j = 3 according to Eq.(3-13), i = 1, and to species j = 4 according to Eq.(3-13), i =
1, 3, noting that N(l) = N(^) = 2, yields:
P23 = (l/2)kiCi(n)At P24 = (l/2)[kiCi(n) + k2C3(n)]At (3-16b)
Applying Eqs.(3-9) and (3-14c) to species j = 3 which is converted to species
j = 1 according to Eq.(3-13), i = 2, to species j = 2 according to Eq.(3-13), i = 2,
and to species j = 4 according to Eq.(3-13), i = 3, noting that N(2) = N(^) = 2,
yields:
P31 = (l/2)k.iC4(n)At P32 = (l/2)k.iC4(n)At
P34 = (l/2)k2C2(n)At (3-16C)
Finally, applying Eqs.(3-9) and (3-14d) to species j = 4 which is converted to
species j = 1 according to Eq.(3-13), i = 2, to species j = 2 according to Eq.(3-13),
i = 2, 4, and to species j = 3 according to Eq.(3-13), i = 4, noting that N(2) = 2 and
N W = 1 , yields:
P41 = (l/2)k.iC3(n)At P42 = [(l/2)k.iC3(n) + L2]At
P43 = k.2At (3-16d)
The above probabilities may be grouped in the matrix given by Eq.(3-17). It
should be noted that Eq.(2-18) is not satisfied along each row because the one-step
transition probabilities pjk depend on time n. This is known as the non-
homogeneous case defined in Eqs.(2-19) and (2-20), due to non-linear rate
equations, i.e. Eqs.(3-12).
197
P =
1
2
3
4
Ai = l
1- kiC2(n)At
0
(l/2)k.iC4(n)At
(l/2)k.iC3(n)At
A2 = 2
0
l - [kiCi(n) +
k2C3(n)]At
(l/2)k.iC4(n)At
[(l/2)k.iC3(n) +
k.2]At
A3 = 3
(l/2)kiC2(n)At
(l/2)kiCi(n)At
l-[k.iC4(n) +
k2C2(n)]At
k.2At
A4 = 4
(l/2)kiC2(n)At
(l/2)[kiCi(n) +
k2C3(n)]
(l/2)k2C2(n)At
1- [k.iC3(n) +
k-2]At
(3-17)
Verification of tlie model. Several assumptions were made in section
3.1 which led to Eqs.(3-6), (3-8) to (3-10) for the determination of pjjj and pj^. For
the reactions given by Eqs.(3-13) the results are summarized in the matrix given by
Eq.(3-17). The validity of the results will be tested by writing the Euler integration
algorithm for the differential equations, Eqs.(3-12), which describe the reaction
mechanisms.
Integration of Eq.(3-12a) yields after a few manipulations that:
Ci(n+1) = Ci(n)pii + C2(n)p2i + C3(n)p3i + C4(n)p4i (3-18a)
where p2i = 0. The other pij's, as well as those for the results below, are given in
Eqs.(3-15a to d) and (3-16a to d) which are summarized in the matrix given by
Eqs.(3-17). Integration of Eq.(3-12b) yields:
C2(n+1) = Ci(n)pi2 + C2(n)p22 + C3(n)p32 + C4(n)p42
where pi2 = 0. Integration of Eqs.(3-12c) and (3-12d) yields:
C3(n+1) = Ci(n)pi3 + C2(n)p23 + C3(n)p33 + C4(n)p43
C4(n+1) = Ci(n)pi4 + C2(n)p24 + C3(n)p34 + C4(n)p44
Equations (3-18) reveals the following characteristics:
(3-18b)
(3-18c)
(3-18d)
198
a) The equations are a function of the transition probabilities pij and pik
detailed in the matrix given by Eq.(3-17).
b) Each of the Eqs.(3-18a to d) obey Eq.(2-23) for a number of states Z = 4
as well as Eq.(2-24). Thus, the following equalities may be obtained:
Sj(n) = Cj(n); Sj(n+1) = C/n+l)
S(n) = C(n) = [Ci(n), CjW, €3(0),..., € > ) ] (3-19)
where C(n) may be looked upon as the state vector of the system at time nAt (step
n). In addition, the initial state vector reads:
S(0) = C(0) = [Ci(0), €2(0), €3(0),..., Cz(0)] (3-19a)
c) Eqs.(3-18a to d) indicate that each Cj(n+1) is a result of the product of the
row vector C(n), defined in Eq.(3-19), by the square matrix P defined in Eq.(3-
17), i.e.:
z
Cj(n+l) = 2^Cj(n)pjk
j=l
C(n+1) = C(n)P (3-20)
where Z is the number of states.
3.2-2 Non-linear reversible and irreversible reactions with a j ^ l
Example a: Aj -^ 2A2 (3-21)
for which:
ri = dCi/dt = - kiCi (3-22a)
It follows from Eq.(3-3) that:
r2 = dC2/dt = 2kiCi (3-22b)
Integration of the above equations yields:
Ci(n+1) = Ci(n)[l - kiAt] (3-23a)
C2(n+1) = Ci(n)[2kiAt] + C2(n) (3-23b)
199
which can be arranged on the basis of Eq.(3-20) in the following matrix form:
Ai = 1 A2 = 2
P =
1 - kiAt 2kiAt
0 1
Applying Eqs.(3-6) and (3-10) yields identical probabilities.
k,
Example b: Ai "^ 2A2
for which:
ri = dCj/dt = - kjCi + kzCj
It follows from Eq.(3-3) that:
T2 = dCz/dt = 2kiCi - 2k2C2
Integration of the above equations gives:
Ci(n+1) = Ci(n)[l - kiAt] + C2(n)[k2C2(n)At]
C2(n+1) = Ci(n)[2kiAt] + C2(n)[l - 2k2C2(n)At]
yielding the following matrix:
P =
Ai = l 1 - kiAt
A2 = 2 2kiAt
k2C2(n)At l-2k2C2(n)At
(3-24)
(3-25)
(3-26a)
(3-26b)
(3-27a)
(3-27b)
(3-28)
Applying Eqs.(3-6), (3-10) and for p2i Eq.(3-10a), yields identical
probabilities while Eq.(3-25) was expressed as the following two irreversible
reactions:
Aj -> 2A2 and 2A2 -> Aj
200
Example c: 2Ai -*• A2
for which:
ri = dCi /d t - -k iCf
gives from Eq.(3-3) that:
r2 = dC2/dt- O.SkjCi
Thus, by integration it is obtained that
Ci(n+l) = Ci(n)[l-kiCi(n)At]
C2(n+1) = Ci(n)[0.5kiCi(n)At] + C2(n)
hence:
Ai = l A2 = 2
1 - kiCi(n)At 0.5kiCi(n)At 1
P =
0 1
Identical pnababihties are obtained from Eqs.(3-6) and (3-10a) for pu-
Example d: 2Ai "* A2
for which:
ri = dCj/dt - - kiCi + kjCj
r2 = dCj/dt - O.SkJCi - O.SkjCj
yields by integration that:
Ci(n+1) = Ci(n)[l - kiCi(n)At] + C2(n)[k2At]
C2(n+1) = Ci(n)[0.5kiCi(n)At] + C2(n)[l - 0.5k2At]
Ai = 1 A2 = 2
1 - kiCi(n)At 0.5kiCi(n)At
P =
k2At 1 - 0.5k2At
(3-29)
(3-30a)
(3-30b)
(3-3 la)
(3-3 lb)
(3-32)
(3-33)
(3-34a)
(3-34b)
(3-35a)
(3-35b)
(3-36)
201
The above probabilities are identical to those computed from Eqs.(3-6), (3-
10) and Eq.(3-10a) for pi2 considering Eq.(3-33) as the irreversible set kj kj
2Ai -> A2 and A2 -» lAj .
3.2-3 Linear reactions Consider the following reactions:
Ai A2 "* A3 -» A4 (3-37)
The kinetics of the reactions satisfying Eq.(3-3) is given by the following
expressions:
ri = dCi/dt = - kiCi + k.iC2
r2 = dC2/dt = - k2C2 - k.iC2 + kiCi + k.2C3
rs = dCa/dt = - ksCs - k.2C3 + k2C2
T4 = dC^dt = k3C3
(3-38a)
(3-38b)
(3-38c)
(3-38d)
1st step: Transformation of Eqs.(3-21) into a set of irreversible reactions as
follows: ki
1 = 1: A i ^ A 2 (3-39a)
= 2: A2->Ai
= 3: A2 — A3
k-2 = 4: A3-J.A2
(3-39b)
(3-39c)
(3-39d)
= 5: A3 —A4 (3-39e)
2nd step: Determination of the reaction rates for Eqs.(3-39) on the basis of
Eqs.(3-38). Following the definition in Eq.(3-2) yields:
202
T\^\n) = - kS^^Ci(n) = - kiCi(n) (3-40a)
Similarly:
T^i\n) = - k^^^C2(n) = - k_iC2(n) (3-40b)
r^^\n) = - k^^^C2(n) = - k2C2(n) (3-40c)
r ' ^n) = - k^%3(n) = - k_2C3(n) (3-40d)
r^^^n) = ~ kfc^in) = - k3C3(n) (3-40e)
3rd step: Determination of the probabilities pjj. Applying Eqs.(3-6) and (3-
40a) yield for species j = 1 where i = 1 that:
pil = l -k iAt (3-41a)
For species j = 2 which is converted according to Eqs.(3-39b, c), i = 2, 3, Eqs.(3-
6) and (3-40b, c) yield:
P22=l-[k-i+k2]At (3-41b)
For species j = 3 which is converted according to Eqs.(3-39d, e), i = 4, 5, Eqs.(3-
6) and (3-40d, e) yield:
P33=l-[k-2 + k-3]At (3-41C)
For species j = 4, formed according to Eq.(3-39e), i = 5, and for remaining at this
state:
P44 = 1 (3-41d)
4th step: Determination of the probabilities pj^. Applying Eqs.(3-9) or (3-
10) and (3-40a) to species, j = 1 which is converted to species j = 2 according to
Eq.(3-39a), i = 1, noting that N(l) = 1 and that A] is not converted to A3 and A4 in
one step, yields:
pi2 = kiAt pi3 = pi4 = 0
203
(3-42a)
Applying Eqs.(3-9) or (3-10) and (3-40b) to species j = 2 which is converted to
species 1 according to Eq.(3-39b), i = 2, to species 3 according to Eq.(3-39c), i =
3, noting that N(2) = 1 and N(3) = 1, yields:
P21 = k.iAt P23 = k2At P24 = 0 (3-42b)
Applying Eqs.(3-9) or (3-10) and (3-40c) to species j = 3 which is converted
to species j = 2 according to Eq.(3-39d), i = 4, to species j = 4 according to Eq.(3-
39e), i = 5, noting that N(4) = N(5) = 1, yields:
P31 = 0 P32 = k.2At p34 = k3At (3-42c)
Species j = 4, formed according to Eq.(3-39e), i = 5, remains in its state,
thus:
P41 = 0 P42 = 0 P43 = 0 (3-42d)
The above probabilities may be grouped in the matrix given by Eq.(3-43). It
should be noted that Eq.(2-18) is satisfied along each row because the one-step
transition probabilities pjk are independent of the time n. This is known as the
time-homogeneous case defined in Eqs.(2-14a) and (2-16), due to the fact that the
rate equations (3-23) are linear.
1
2
3
4
A i = l
1 -kiAt
k-i At
0
0
A2 = 2 kiAt
l - [k - i+k2]At
k-2At
0
A3 = 3
0
k2At
l - [k-2 + k3]At
0
A4 = 4
0
0
ksAt
1
p =
(3-43)
Verification of the model. Integration of Eq.(3-38a to d) yields after a
few manipulations Eqs.(3-18a to d). The pij's are given by Eqs.(3-41a to d) and
(3-42a to d), which are summarized in the matrix governed by Eq.(3-43).
204
3.2-4 Linear-non linear reactions with aj ^ l Example 3.13-6 (chapter 3.13) is presented in details demonstrating also the
derivation of the kinetic equations satisfying Eq.(3-3). This example should be
studied thoroughly since it contains important aspects of applying the equations for
calculating pjj and pj] .
3.3 MAJOR CONCLUSIONS AND GENERAL GUIDELINES FOR APPLYING THE MODELING
The major conclusions drawn from treating the above reactions, and many
others reported in the following without detailed derivations, are:
a) The results obtained by the Euler integration are in complete agreement
with the results obtained by the model presented in section 3.1 yielding Eqs.(3-6)
and (3-8) to (3-10a) for predicting the probabilities pjj and pijk. Thus, one may
apply each of the methods, depending on his conveniece. However, by gaining
enough experience, one starts to Teel' that the method based on the Markov chains
is easier and becomes 'automatic' to apply. In addition, chemical reactions are
presented in unified description via state vector and a one-step transition probability
matrix.
b) Reversible reactions should be transformed into a set of irreversible
reactions.
c) The above reactions, treated in detail, provide the reader with a good
introduction for applying the probabilities pjj and pjk to chemical reactions.
3.4 APPLICATION OF KINETIC MODELS TO ARTISTIC PAINTINGS
Prior to modeling of chemical reactions in the next sections, it is interesting to
demonstrate how simple kinetic models can also be applied to artistic paintings.
No reaction
The first example ^plies to Fig.2-52. The painting in the figure.
Development II by Escher [10, p.276], demonstrates the development of reptiles,
and at first glance it seems that their number is increasing along the radius.
205
Although their birth origin is not so clear from the figure, it was possible, by counting their number along a certain circumference, to conclude that it contains exactly eight reptiles of the same size. This number is independent of the distance from the center. Thus, if we designate by Ci the number of reptiles at some distance from the origin, it follows that
dCj Ti = —^ = 0 or alternatively in a discrete form Ci(n+1) = Ci(n)
This result indicates the absence of a chemical reaction. Although the reptiles become fatter versus the number of steps (time), their number is unchanged.
zero order reaction The second example refers to Fig.3-la showing various kinds of "winged
creatures" in a drawing of a ceiling decoration designed by Escher in 1951 [10, p.79] for the Philips company in Eindhoven. If the number of the "winged creatures", designated as the "concentration" Ci, is counted along the lines corresponding to steps (time) n = 0, 1, 2, ... shown schematically in Fig.3-lb for cases 1 to 3, the results summarized in Table 3-1 are obtained. The general trend observed is a decrease versus time of the "concentration".
206
iTi ^ 1 ^ C i ^ ^ W ^ ^ ^^^ r^ 1
!ic4 ^.-:
r : |r #" f ?l | f -€ 1^ fT, -*
Fig.3-la. "Winged creatures'* demonstrating zero order reaction (M.C.Escher "Ceiling Decoration for Philips" © 1998 Cordon Art B.V. - Baam - Holland. All
rights reserved)
207
0 1 t 9 A 4
n = 9 10 11
U - ^ > — D
-e—©
n = o o—a s—n o
1 1
1
1 A ^ f
: a-1 {
«—«—^-•-' e
T 1
1
• ±
Casel Case 2 Case 3
Fig.3-lb. Three configurations for determination of the "winged
creatures concentration" Ci
Table 3-1. "Concentration" Ci of "winged creatures" versus time n
time n
case 1*, Ci(n)
case 2*, Ci(n)
[case 3*, Ci(n)
0 1 2 3 4 5
12 11 10 9 8 7
25 23 21 19 17 15
48 40 32 24 16 8
6 7
6 5
13 11
8
4
9
9
3
7
10
2
5
111 1
3
* see Fig.3-lb
It should be noted that in case 1 of Fig.3-lb, the hnes for n = 0 and 11 correspond
to twelve butterflies and a single fly, respectively, in Fig.3-la. In case 2 these lines
correspond to twenty five and three "winged creatures", respectively, whereas in
case 3 the lines correspond to forty eight and eight "winged creatures",
respectively.
In order to fit the concentration data Ci (number of "winged creatures" along
a line) versus the time n reported in Table 3-1, the common approach of fitting
experimental data to a kinetic model is applied. Thus, the simplest model of a zero k
order reaction is tested which corresponds Ai -> A2. The rate equation reads
208
dCi
Eqs.(3-6) and (3-8) are applied for determination of the probabilities which
yield the following matrix:
Ai = l A2 = 2
l"
P =
2
Thus,
1 - [k/Ci(n)]At [k/Ci(n)]At
0 1 (3-45)
Ci(n+l) = Ci(n)-kAt
C2(n+1) = kAt + C2(n) (3-46)
Fitting the data in Table 3-1 by Eq.(3-46) for At = 1, yields the following
equations corresponding to Fig.3-lb:
Case 1: Ci(n+1) = Ci(n) - 1; C2(n+1) = 1 + C2(n) where Ci(0) = 12; C2(0) = 0
Case 2: Ci(n+1) = Ci(n) - 2; C2(n+1) = 2 + C2(n) where Ci(0) = 25; C2(0) = 0
Case 3: Ci(n+1) = Ci(n) - 8; C2(n+1) = 8+ C2(n) where Ci(0) = 48; C2(0) = 0
The excellent fit to Eqs.(3-30) of the data given in Table 3-1, indicates that the
concentration-time dependence of the "winged creatures" in Fig.3-la according to
the configurations depicted in Fig.3-lb, obeys a model of zero order reaction. The
significance of the quantities C2(n) is as follows. Since Ci(n) is decreasing versus
time, i.e. the number of the "winged creatures", the conservation of mass requires
that they are found in state A2 according to the reaction A] -^ A2.
mth order reaction
The third example refers to Fig.3-2 which is a woodcut by Escher [10,
p. 118, 325] showing moving fish of changing size. Here Escher demonstrated an
infinite number by a gradual reduction in size of the figures, until reaching the Umit
of infinite smallness on the straight side of the square. If the number of fish along
the square perimeter, designated as "concentration" Ci, is counted, the obtained
results are summarized in Table 3-2. Fig.3-lb, case 3, shows schematically the
209
fish orientation along a squre which was counted, where each circle symbolizes a
fish. Also, along a certain square, each fish is located exactly behind (or before)
the other, and all are of the same size. The case of n = 0 corresponds to the the
square located almost at the sides of the square where n = 6 corresponds to the
most inner square comprising of four fish.
, ^>k<!k> • • * . * * . ••>>*««>*»
4 4:
Fig.3-2. Fish orientation for demonstrating an mih order reaction
(M.C.Escher "Square limit" © 1998 Cordon Art B.V. - Baarn - Holland. All rights reserved)
Table 3-2. "Concentration" Ci versus time of moving fish along the square perimeter
1 timen
Ci(n)
1 Ci.calc(n)*
0
760
1
376
399
2
184
185
3
88
84
4
40
37
5
16
15
6 1 4
5 * rounded values computed by Eq.(3-49)
210
In order to fit the "concentration" data Ci (number of moving fish along a
square) versus the time n reported in Table 3-2, an mth order reaction is tested k
corresponding to Ai -». A2. The rate equation reads:
dCi dt kC^ (3-47)
where the apphcation of Eqs.(3-6) and (3-8) yields the following transition matrix:
Al = 1 A2 = 2
1 - kCim-l(n)At kCim-l(n)At
P =
0
Thus,
1 (3-48)
Ci(n+1) = Ci(n)[l - kAtCr^n)]
C2(n+1) - kACf(n) + CjCn) (3-49)
Fitting the data in Table 3-2 by Eq.(3-49) for At = 1, which was modified to the
following equation [Ci(n) - Ci(n+l)] - kC7(n), yields m = 0.904 and k = 0.896
with a mean deviation of 8.3% between calculated data with respect to coimted
values in Fig.3-2. The above examples indicate that the application of kinetic
models to artistic paintings has been successful.
3.5 INTRODUCTION TO MODELING OF CHEMICAL REACTIONS
In the following, a solution generated by the discrete Markov chains is
presented gr^hically for a large number of chemical reactions of various types.
The solution demonstrates the transient response Cj-nAt and emphasizes some
characteristic behavior of the reaction. The solution is based on the transition
probability matrix P obtained on the basis of the reaction kinetics by applying
211
Eqs.(3-6), (3-8) to (3-lOa) for computing the probabilities pjj and pjk- It should be
emphasized that the rate equations for the kinetics were tested to satisfy Eq.(3-3).
In order to obtain the transient response, Eq.(3-20) is applied where the initial
state vector S(0) is given by Eq.(3-19a). In each case, the magnitude of S(0) are
the quantities on the Ci axis of the response curve corresponding to t = 0. An
important parameter in the computations is the magnitude of the interval At. This
parameter has been chosen recalling that pjj and pjk should satisfy
0 < Pjj and pj] < 1 on the one hand, and that Cj versus nAt should remain
unchanged under a certain magnitude of At, on the other. In addition, a comparison
with the exact solution has been conducted in many cases, which made it possible
to evaluate the accuracy of the solution obtained by Markov chains. The quantities
reported in the comparison are the maximum deviation, Dmax* and the mean
deviation, Dmean- Oi the basis of these comparisons, a representative value of At =
0.01 is reconmiended, which is the parameter of Markov chains solution. Finally,
it should be emphasized that by equating the reaction rate constant (one or a few) to
zero in a certain case, it is possible to generate numerous interesting situations.
The reactions are presented according to the following categories:
1) Single step irreversible reaction.
2) Single step reversible reaction.
3) Consecutive-irreversible reactions.
4) Consecutive-reversible reactions.
5) Parallel reactions: single and consecutive-irreversible reaction steps.
6) Parallel reactions: single and consecutive-reversible reaction steps.
7) Chain reactions.
8) Oscillating reactions.
The following definitions are applicable [31, 32]:
Consecutive chemical reactions are those in which the initial substance and all
the intermediates products can react in one direction only, i.e.:
Parallel chemical reactions are those in which the initial substance reacts to
produce two different substances simultaneously, i.e.:
212
- . < ^3
Reversible reactions are those in which two substances entering a single
simple consecutive chain reaction interact in both forward and backward directions,
i.e.:
A^ ^ A^ A^ A^
Conjugated reactions are two simultaneous reactions in which only one
substance Ai is common to both, i.e.:
\ + ^2 ^ ^ 4
Aj + A3 • A5
All three substances Ai, A2 and A3 must be present in the reaction mass in order
for both reactions to take place concurrently.
Consecutive-reversible reactions are those in which two or more reactions,
each of different type, occur simultaneously, for example:
5
Parallel-consecutive reactions belong to the mixed type which have the
characteristics of both parallel and consecutive reactions. The following example
comprises two parallel chains, each composed of three simple reactions:
<
A parallel-consecutive reaction becomes complex when species that belong to
different chains interact as shown below:
213
Chain reactions. If the initial substance and each intermediate reaction
product interact simultaneously with different substances and in different
directions, such processes are known as chain reactions. For example, the
following scheme is a chain reaction in two stages. Other types of chain reactions
are described in [22].
'^
3.6 SINGLE STEP IRREVERSIBLE REACTION
3.6-1 where
r, = -kCV
(3.6-1)
(3.6-la)
By applying Eqs.(3-6) and (3-10), the following one step transition probability
matrix is obtained:
1 2
1 - kCim-i(n)At kCi'n-i(n)At
P =
0 1 (3.6-lb)
214
where 1, 2 stand for states (chemical species) A] and A2, respectively. From Eqs.(3-19a), (3-20), one obtains that:
Ci(n+1) = Ci(n)[l - kAtCr^n)]
C2(n+1) = kAtCf (n) + C2(n) (3.6-lc)
The variation of Ci against t = nAt for the initial state vector C(0) = [Ci(0), C2(0)] = [1, 0] is depicted in Figs.3.6-1 (a to d) for different values of the parameter m = 0, 0.5, 1 and 3 where also the effect of the reaction rate constant k is demonstrated.
u
Fig.3.6-la. Ci versus t demonstrating the effect of k for m = 0 in Eq.(3.6-la)
1
0.8
0.6
0.4
0.2
0
U
h-
u
1
i = l \
ly
1
1 1 k = 0.5
1 1
H
H
0.5 1.5 2 0
Fig.3.6-lb. Ci versus t demonstrating the effect of k for m = 0.5 in Eq.(3.6-la)
215
u
1
0.8
0.6
0.4
0.2
0
-
r
h-
1 1
iX
1 1
1 y
y ^ k = 0.5-
^"""^-cr
1
1 /
l \ /
7
1
1 1
k = l -
—
1 1 0.5 1.5 2 0 0.5 1.5
Fig.3.6-lc. Ci versus t demonstrating the effect of k for m = 1 in
Eq.(3.6-la)
u
1
0.8
0.6
0.4
0.2
0
u
H
1
i=^^^^^^
1
1 I k = 0.5
1 1
—
-
-•
Y
1
iV
'y
k = 1
1
—
-
0.5 1.5 2 0 0.5 1.5
Fig.3.6-ld. Ci versus t demonstrating the effect of k for m = 3 in
Eq.(3.6- la)
It should be noted that exact solutions for the above models are available in
refs.[32, vol.1, p.361; 34, pp.4-5, 4-6]. For At = 0.01, the agreement between
the Markov chain solution and the exact solution is D^ax = 0.4% and Dmean =
0.2%.
3>6-2 ajAj + ^2^1 ~^ ^3^3
where
ri = ~ kCjC^' r2 = - rkcjc^' r = a2/ai
(3.6-2)
(3.6-2a)
The following one step transition probabiUty matrix is obtained:
216
1
P = 2
3
l-kc'f'(n)C5'(n)At 0 kRcV'(n)C5'(n)At
l-rkC\(n)C^"'(n)At ARC',(n)C^'(n)At
(3.6-2b)
0
0 0 1
where R = 1/(1+ r) and r = aa/ai. 1, 2, 3 stand for states Ai, A2 and A3, respectively.
By applying Eqs.(3-19a), (3-20), one obtains that:
Ci(n+1) = Ci(n)[l-kCl"kn)C^(n)At]
C2(n+1) = C2(n)[l-rkcl(n)C^"'(n)At]
(3.6-2C)
-,1-1 C3(n+1) = Ci(n)[kRCr (n)C^(n)At] + C2(n)[r'kRC;(n)Cr (n)At] + C3(n)
The following cases were explored:
3.6-2a For the initial state vector C(0) = [Ci(0), C2(0), €3(0)] = [1,0.5, 0]:
u
1
0.8
0.6
0.4
0.2
0
r ^
1
J = l
3 , ^ - ^
1
1 r = l
1
1 -H
H
-
1 0.5 1.5
1/ V 2 I
2 0 0.5 1.5
Fig.3.6-2a. Ci versus t demonstrating the effect of r for I = 0, m = 3/2 and k = 5 in Eq.(3.6-2a)
For At = 0.04, the agreement between the Markov chain solution and the
exact solution [32, vol.l, p.361] is Dmax = 1-9% and Dmean = 0.3%.
217
3>6-2b For an initial state vector C(0) = [Ci(0), C2(0), €3(0)] = [2, 3,0]:
3
2.5
2
u" 1.5
1
0.5
0
_ \ 2
\ i = l
r y 3
- / V
y' y ^
1
1
^ - "
-..., ^ -^^^-^7-^-—-^
0.5 1 1.5 0 t t
Fig.3.6-2b. Ci versus t according to Eq.(3.6-2a).
ri = - kCi l /2c2 (left), ri = - kCiCil'^ (right) for k = 1, r = 1
For At = 0.005, the agreement between the Markov chain solution and the
exact solution [32, vol.1, p.361] is Dmax = 1-2% and Dmean = 0.6%.
3.6-3 where
Aj + A2 + A3 -^ A4
rj - r2 - r3 - - r4 - - kC]C2C3
(3.6-3)
(3.6-3a)
yields the following transition probability matrix
P =
2 0
3 0
4 ^kC2(n)C3(n)At
1 1 -
kC2(n)C3(n)At
0 ^~ 0 |kC,(n)C3(n)At kC,(n)C3(n)At ^
0
0
0 . ^ / L . ^ . T' C,(n)C2{n)At kC,(n)C,(n)At 3 '•
0 0 (3.6-3b)
218
Ci(n+1) is obtained by applying Eq.(3-20) where the effect of the initial state
vectors C(0) = [Ci(0), C2(0), €3(0), C4(0)] = [1, 2, 3, 0] and [3, 2, 1, 0] is
demonstrated in Fig.3.6-3a.
3
2.5
2
0 - 1.5
1
0.5
0
\ 1
^3
^ \ ^ ^ v
. \ -Lv< 1/ 1
1
4x
/
\
i"" "
1
y
1
>.--I=iz=:^
^ 1
A
-
^ —-0.2 0.4 0.6 0.8
t
X J
""'•---
p\jz 1
k ,
1 1 i
—
^
— z j
-
-L ' ' ' 1 1 1 1
1 0 0.2 0.4 0.6 0.8 t
Fig.3.6-3. Ci versus t demonstrating the effect of Ci(0) for k = 1
For At = 0.01, the agreement between the Markov chain solution and the
exact solution [31, p.20] is Dmax = 2.3% and Dmean = 1-5%. For At = 0.005,
Dmax = 1.4% and Dmean = !%•
3.6-4 where
rj = - kjCj/Cl + kjCi)
yields the following transition probability matrix:
(3.6-4)
(3.6-4a)
P =
1 2 1 - [kj/(l + k2Ci(n)]At [k,/(l + k2Ci(n)]At
0 1 (3.6-4b)
Fig.3.6-4 demonstrates the effect of the reaction rate constants ki and k2 on
the species concentration distribution for the initial state vector C(0) = [Ci(0),
C2(0)] = [1, 0].
219
u
1
0.8
0.6
0.4
0.2
0 (
1
0.8
0.6
0.4
0.2
0
\ r~
ly^
/ ^ 1 ) 0.5
V 1
i = l \
\y V , 0 0.5
1 k = l , k = 1
1 2
^ "
1 1 t
i k =5,k =5^
1 ^x
/
1 1 t
"~1
1 1.5
^'^
1.5
-
<
2
^
' y / , 0 0.5
2 (
\ y--V 7\ r V ) 0.5
\ = l ,k =0.1
2
1 1 t
= 5,k =0.5 2
1 t
~T
1 1.5
1
1 1.5
- : :
-
H
A
/ •
Fig.3.6-4. Ci versus t demonstrating the effect of ki and k2
3.7 SINGLE STEP REVERSIBLE REACTIONS
3.7-1 Ai ^ A 2 (3.7-1)
where
Tj — — r2 — — ( k j C j — )£.'^'2) (3.7-la)
By applying the approach detailed in section 3.2-1, i.e., treating the
reversible reactions as two irreversible ones demonstrated in Eqs.(3-10a) and (3-
10b), the following transition probability matrix is obtained:
P =
1 2 1 - kjAt kjAt
koAt 1 - k^At (3.7-lb)
220
Thus, from Eqs.(3-19a), (3-20), one obtains that:
Ci(n+1) = Ci(n)[l - kjAt] + C2(n)[k2At]
C2(n+1) = Ci(n)[kiAt] + C2(n)[l - k2At] (3.7-lc)
yielding the following curves for the initial state vector C(0) = [Ci(0), C2(0)] = [1, 0]:
u
1
0.8
0.6
0.4
0.2
0
u
u
V
1
i = r\^
2 /
1 k = 1
2
^ '
1
1
1
V^ ' f
-
-
> , T
1 1 k =5
2
1
-J
-J
A
—
0 0.5 1 1.5 2 0 0.5 1 1.5 2 t t
Fig.3.7-1. Ci versus t demonstrating the effect of k2 for ki = 1
At steady state, the results verified the relationship which follows from
Eq.(3.7-la), i.e., (C2/Ci)eq.= ki/k2. For At = 0.01, the agreement between the
Markov chain solution and the exact solution [34, p.4-7; 48, p.20; 49, p.85] is
Dmax = 0.3% and Dmean = 0.1%.
3.7-2 Ai "^ 2A2 (3.7-2)
k2
The transition matrix based on Eqs.(3-26a,b) was developed before and is
given by Eq.(3-28). The transient response of Ci and C2 for the initial state vector
C(0) = [Ci(0), C2(0)] = [1, 0] is demonstrated in Fig.3.7-2. At equilibrium, the
results verified the relationship which follows from Eqs.(3-26a,b), i.e.,
(C2/Ci2)eq = ki/k2 where no analytical solution is available for comparison.
o
1
0.8
0.6
0.4
0.2
0 0.5 1.5 2 0 0.5 1.5
221
\ 1
i=^'">^
- ;
/ :
1 1 k =1
2
^ ^
1 1
V 1
1
- 1^ -
1 ,
1 1 k =5
2
1 1
Fig.3.7-2. Ci versus t demonstrating the effect of k2 for ki = 1
according to Eqs.(3-26a,b)
3.7-2.1 Ai ^ A2 + A3 (3.7-2. la)
where ri - -12 - - r3 - - kjCj + k2C2C (3.7-2. lb)
yields the following transition probability matrix:
P =
1 2 3 l-kjAt kjAt k,At
^k2C3(n)At l-k2C3(n)At 0
•5-k2C2(n)At 0 l-k2C2(n)At (3.7-2. Ic)
The transient response of Ci, C2 and C3 for the initial state vector C(0) =
[Ci(0), C2(0), C3(0)] = [1, 0, 0] is demonstrated in Fig.3.7-2.1.
222
1
0,8
0.6
0.4
0.2
0
U
h- ^
L_
1/'
1
2
L.
C 3
1 1 k =1
2
. . . . . — • ' - " ' ' "
1 i
H
-H
\ k =5
I ^ ^ I
- c = c U 2 3 - J
0 0.5 1 1.5 2 0 0.5 1 1.5 2 t t
Fig.3,7-2.1. Ci versus t demonstrating the effect of k i f o r k i = 1
For At = 0.01, the agreement between the Markov chain solution and the
exact solution [32, p.79; 44; 48, p.20; 49, p.85] is Dbax = 0.36% and Dmean =
0.23%.
3,7-3 2Ai ^ A2 (3.7-3)
The transition matrix based on Eqs.(3-34a,b) was developed before and is
given by Eq.(3-36). The transient response of Ci and C2 for the initial state vector
C(0) = [Ci(0), C2(0)] = [1,0] is demonstrated in Fig.3.7-3. The remarks made in
3.7-2 are also applicable here.
u
11
0.8
0.6
0.4
0.2
0 (
i = r
2
3
^
1 0.5
1 k =1
2
1 1 t
"T
1 1.5 2 (
- r"
2
3
1
1 0.5
1 k =5
2
1 1 t
1
1 1.5
H
H
-H
Fig.3,7-3. Ci versus t demonstrating the effect of kz
for ki = 1 according to Eqs.(3-34a,b)
3.7-4
where
2Ai ~^ A2 + A3
Ti — - kCj + k2C2C3
Tj = rg = 0.5kCi - O.SkjCjCj
yields the following transition probability matrix:
223
(3.7-4)
(3.7-4a)
(3.7-4b)
P =
l-kiCi(n)At
•jk2C3(n)At
^k2C2(n)At
^kiCi(n)At
l-jk2C3(n)At
0
^k,C,(n)At
l-^k2C2(n)At (3.7-4C)
where the computation of pi2 = pi3 was made by Eq.(3-10a).
The transient response of Ci, C2 and C3 for the initial state vectors C(0) =
[Ci(0), C2(0), C3(0)] = [1, 0.1, 0] and [1, 0, 0] is demonstrated in Fig.3.7-4.
u
1
0.8
0.6
0.4
0.2
0
r-
^
[^
A = l
> -
1
1
k = 1
— -"
1 '
.
1
^
H
-
-
0 0.5 1 1.5 0 t t
Fig.3.7-4. Ci versus t demonstrating the effect of ki for k2 = 1
For At = 0.005, the agreement between the Markov chain solution and the
exact solution [32, p.35; 49, p.86] is Dmax = 3.1% and Dmean = 0.7%. For At =
0.01, Dmax = 7.2% and Dmean = 2.6%.
224
3.7-5
where
2A1 + A2 ^ 2 A 3
rj = - rg = - IkiCjCj + 2k2Cl
2 ^ — '^1^1^2 " 2 3
(3.7-5)
(3.7-5a)
(3.7-5b)
yields the following transition probability matrix:
P =
l-2k,Ci(n)C2(n)At
2k2C3(n)At
2
0
-kiC?(n)At
k2C3(n)At
3
3-kiC,(n)C2(n)At
|kiC^(n)At
l-2k2C3(n)At (3.7-5C)
The transient response of Ci, C2 and C3 for the initial state vector C(0) =
[Ci(0), C2(0), CBCO)] = [1, 1, 0] is demonstrated in Fig.3.7-5.
1
0.8
0.6
0.4
0.2
0
- \
h
r/ V'
1
1
1
/
^ -— -
k = 2
. - - - - I "
_\;/(c^^)_
= 1 1
-
-\
-\
0.5 1.5 2 0 0.5
\ ^ 2 .
- \l^ v Y \ >" ~
1 1 1
—j
-J c;/(c;9 \
k =5 2
1 1 1.5
Fig.3.7-5. Ciand the ratio C3/(CiC2) versus t demonstrating the
effect of k2 for ki = 1
No exact solution is available for this reaction [32, vol.2, p.76]. However, it
should be noted that the ratio C3/(C|C2) approaches at steady state the ratio ki/k2
as predicted from Eqs.(3.7-5a,b).
225
3.7-6 A1 + A2 ^ A 3
where
Tj — T2 — — 1*2 — — K2Cx|V-'2 " ^ 2 3
(3.7-6)
(3.7-6a)
yields the following transition probability matrix:
P =
1
2
3
1 l-kiC2(n)At
0
kjAt
2
0
l-kiCi(n)At
kjAt
3
•^k,C2(n)At
^k,C,(n)At
l-kjAt (3.7-6b)
The transient response of Ci, C2, C3 and C3/(CiC2) for the initial state vector
C(0) = [Ci(0), C2(0), C3(0)] = [1, 0.5, 0] is demonstrated in Fig.3.7-6.
u
1
0.8
0.6
0.4
0.2
0
2 ^ S/^^1^2)
/ 3 ^ —
k = 1 2
\£1-
^-^ 1 1 1 1 1
K_. 2 • ~
C /(C C ) / 3 ^ 1 2 ^
J
k =5-] 2
—J
/ " 3 k "' 1 1 1
0.5 1.5 2 0 0.5 1.5
Fig.3.7-6. Ci and the ratio C3/(CiC2) versus t demonstrating the effect of ki for ki = 1
For At = 0.01, the agreement between the Markov chain solution and the exact solution [33, p.43; 48, p.20] is Dmax = 6.5% and Dmean = 1-5%. In addition, the ratio C3/(CiC2) approaches at steady state the ratio ki/k2 as predicted from Eqs.(3.7-6a,b).
226
3,7-7 Ai + A2 "^A3 + A4 (3.7-7)
where rj = r2 = " 13 = - r4 = - kiCiC2 + k2C^C^ (3.7-7a)
yields the following transition probability matrix:
P =
1 2 3 4
l-kiC2(n)At 0 lkiC2(n)At ^kiC2(n)At
0 l-kiCi(n)At |kiCi(n)At ^kjC^WAt
^k2C4(n)At ^k2C4(n)At l-k2C4(n)At 0
^k2C3(n)At ^k2C3(n)At 0 l-k2C3(n)At (3.7-7b)
k = l , k =1 1 2
C C /(C C ) . 3 4 ^ 1 2^
U
i = l
— ~3 '
k"
" k =l',k =4 I 2
The transient response of Ci to C4 and C3C4/(CiC2) for the initial state
vector C(0) = [Ci(0), CiCO), CsCO), €4(0)] = [1, 0.5, 0.25, 0] is demonstrated in
Fig.3.7-7.
1
0.8
0.6
0.4
0.2
0
C C /(C C ) 3 4 ^ 1 2
K
0.5 1.5 2 0 0.5 1.5
Fig.3.7-7. Ci and the ratio C3C4/(CiC2) versus t demonstrating the
effect of 1 2 for ki = 1
For At = 0.01, the agreement between the Markov chain solution and the
exact solution, not existing for ki = k2, [31, p. 187; 44; 48, p.20; 49, p.86] IS l^max
227
= 8.1% and Dmean = 5.3%. The ratio C3C4/(CiC2) approaches at steady state the
ratio ki/k2 as predicted from Eqs.(3.7-7a).
3,7-8 Ai + A2 + A3 "^ A4
where fl - r2 - "3 ~ ~ ^4 ~ ~ kjCjC2C4 + k2C4
(3.7-8)
(3.7-8a)
yields the following transition probability matrix:
P =
1 1 -
kiC2(n)C3(n)At
0
0
k2At
2 3 4 0 0 jkiC2(n)C3(n)At
1 ^ / N " ^ . X. 0 |k iC i (n )C3(n)At kiCi(n)C3(n)At 3 ^ * ^
0 . r.}^r. ^ ^. |k,Ci(n)C2(n)At kiCi(n)C2(n)At 3 ^ ^ ^
k2At k2At l -k2At (3.7-8b)
The transient response of Ci to C4 and C4/(CiC2C3) for the initial state vector C(0) = [Ci(0), C2(0), €3(0), €4(0)] = [1, 0.9, 0.8, 0] is demonstrated in Fig.3.7-8.
u
0.8
0.6
0.4
0.2
0 "(
V 1 1 ^ \ ^ i
"v \ 2^^^^^""^^" - —
' ^ C /(C C C ) 4 ^ 1 2 3^
- • ^ - " "
1 ^1 } 0.5 1
t
.1 . - - '
1 1.5 2 0 0.5
V I I I 1
^ " 2 ^ _ _ ^ 3 ^ ....
C /(C C C ) 1 ' 4 ^ 1 2 3
— —
-
l- 1 1 1 1.5
Fig.3.7-8. Ci and the ratio C4/(CiC2C3) versus t demonstrating the effect of k2 for ki = 1
228
The present reaction appears in [35, p. 148] with no exact solution. As seen
in Fig.3.7-8, the ratio C4/(CiC2C3) approaches at steady state the ratio k;i/k2 as
predicted from Eqs.(3.7-8a).
3.8 CONSECUTIVE IRREVERSIBLE REACTIONS
3.8-1 aiAi _> A2
a2A2 -^ A3 (3.8-1)
where
ri = - kjCii T2 = - (l/ai)dCi/dt - kzCj rg = - (l/a2)dC2/dt (3.8-la)
yields the following transition probability matrix:
P =
1 2 3
l-k,C^-'(n)At l.kiC?'-'(n)At 0
0
0
l-kjC^^ kn)At J-k2Cr'(n)At a2
0 1 (3.8-lb)
where pi2 and p23 were computed by Eq.(3-10a). The transient response of Ci,
C2 and C3 for the initial state vector C(0) = [Ci(0), C2(0), €3(0)] = [1, 0, 0] is
demonstrated in Figs.3.8-la to d for various combinations of ai and a2 in Eq.(3.8-
1).
229
3,8-la Ai -> A2 -» A3
u
1
0.8
0.6
0.4
0.2
0
y 1 1 1 \ i = 1
\ y^
\ ^
L2 /XT \ . v 1 1 1^-—•
u - \
k =1 2
;;;v^~— 2 3
t
Fig.3.8-la. Ci versus t for ai = a2 = 1 in Eq.(3.8-1) demonstrating the effect of k2 for ki = 1
For At = 0.015, the agreement between the Markov chain solution and the
exact solution [31, p.l66; 49, p.90; 51] is Dmax = 2.2% and Dmean = 1.1%.
3,8-lb 2Ai -.> A2 _> A3
u
1
0.8
0.6
0.4
0.2
0
V 1 1 1 \ i = l 1^2=1
f V ^
1
^ — -
— • - . - -
3.^ L ^ \ / l ^ 2 - , _ _
1 1 1 1 k =5
2
--X J J 1
1 2 3 t
5 0 1 2 3 4 t
Fig.3.8-lb. Ci versus t for ai = 2, a2 = 1 in Eq.(3.8-1) demonstrating the effect of k2 for ki = 1
For At = 0.01, the agreement between the Markov chain solution and the
exact solution [33, p.95], which is very complicated, is Dmax = 0-4% and Dmean =
230
0.3%. It should be noted that for large values of t, C3 should approach 1 whereas for t = 150, C3 = 0.497.
3.8-lc 2Ai ^ A2 2A2 -4 A3
Fig.3.8-lc. Ci versus t for ai = 2, ai = 2 in Eq.(3.8-1) demonstrating the effect of k2 and C2(0) for ki = 1
For At = 0.02, the agreement between the Markov chain solution and the
exact solution [36; 51] is Dmax = 0.7% and Dmean = 0-5%.
231
3.8-ld Ai _> A2 2A2 -> A3
5%
\
_ 3 . "
/ j
\
1
_^
^ '"" '
l_
JHZ
_ I - - 4 - -1
J
C (0) = 0.5 1
2 3 t
Fig.3.8-ld. Ci versus t for ai = 1, ai = 2 in Eq.(3.8-1)
demonstrating the effect of Ci(0) for ki = 1 and ki = 5
For At = 0.01, the agreement between the Markov chain solution and the
exact solution [36; 32, vol.2, p.51] is Dmax = 2.4% and Dmean = 4.9%.
3.8-2
kK A2
(3.8-2)
where
ri = - k iCi + k3C3 r2 = ~ k2C2 + kjCj 13 = - k3C3 + k2C2 (3.8-2a)
yields the following transition probability matrix:
p = 1
2
3
1
l-kjAt
0
kjAt
2
kjAt
l-kjAt
0
3
0
k2At
l-kjAt (3.8-2b)
232
The transient response of Ci, C2 and C3 for the initial state vector C(0) =
[Ci(0), C2(0), C3(0)] = [1, 0, 0] is demonstrated in Fig.3.8-2 for various
combinations of the reaction rate constants. No exact solution is available for
comparison.
h-3/ V_
k 1
1 1 1 = 3,k =2
2 J
--
I [y 1 1 1 4 0
Fig.3.8-2. Ci versus t demonstrating the effect of ki and k2
for k3 = 1
3.8-3
A2 + A3 -> A4
where
Fj = —kjCj r2 = -k2C2C3+kiCi r3 = - r 4 =-k2C2C3
yields the following transition probability matrix:
(3.8-3)
(3.8-3a)
233
1
2
3
4
1 1-kiAt
0
0
0
2 kjAt
l-k2C3(n)At
0
0
3
0
0
l-k2C2(n)At
0
4
0
^k2C3(n)At
•2-k2C2(n)At
1
p =
(3.8-3b)
The transient response of C\ to C4 for the initial state vectors C(0) = [Ci(0),
C2(0), C3(0), C4(0) ] = [1, 0, 1, 0] and [1, 0, 0.25, 0] is demonstrated in Fig.3.8-
3 where the effect of €3(0) is demonstrated.
u
1
0.8
0.6
0.4
0.2
0
r 1 \ \ Cj(0)=l J \
— \
-A L-
1 C (0) = 0.5
3
^ 2 . ' ^ " "
—
-J 1
" \
Fig.3.8-3. Ci versus t demonstrating the effect of €3(0)
for ki = k2 = 5
For At = 0.005, the agreement between the Markov chain solution and the
exact solution [36] is Dmax = 3.1% and Dmean = 1-5%.
234
3.8-4 2 A i ^ A2
A2 + A3 - • A4 (3.8-4)
where
r i - - k i C i rj = - kjCjCg + O.SkjCi rj - - r4 = - kjCjCj (3.8-4a)
yields the following transition probability matrix:
1
2
3
4
1 2 l-kiCi(n)At ikiCi(n)At
0 l-kjCjWAt
0 0
0 0
3
0
0
l-k2C2(n)At
0
4
0
^k2C3(n)At
•jkzCjWAt
1
p =
(3.8-4b)
where P12 was computed by Eq.(3-10a). The transient response of Ci to C4 for the initial state vectors C(0) = [Ci(0), CiCO), €3(0), €4(0)] = [1, 0, 1, 0] and [1, 0,0.5, 0] is demonstrated in Fig.3.8-4 where the effect of €3(0) is demonstrated.
i 1
^ 1
PH \
1 C3(0)
1
= 0.5
^4 .
"T"
1
-
• ^ , ^
2 0 0.5 1.5
Fig.3.8-4. Ci versus t demonstrating the effect of €3(0) for ki = kz = S
For At = 0.01, the agreement between the Markov chain solution and the
exact solution [36] is Dmax = 0.6% and Dmean = 0.3%.
3>8-5 where
Ai + A2 -> A3 -^ A4
Tj — r2 — — kjL-|C-2 3 — — ''2^3 " kjv jt 2 ^4 ~ ^2^3
yields the following transition probability matrix:
235
(3.8-5)
(3.8-5a)
P =
1
2
3
4
1 l-k,C2(n)At
0
0
0
2
0
l-kiCi(n)At
0
0
3
^k,C2(n)At
^kiC,(n)At
l-k2At
0
4
0
0
kjAt
1 (3.8-5b)
The transient response of Ci to C4 for the initial state vector C(0) = [Ci(0),
C2(0), C3(0), C4(0)] = [1, 0, 1, 0] is depicted in Fig.3.8-5 where the effect of ki is
demonstrated.
1 ' ' k = 5 ' V '
i i'-'"' \^K.-[V "-
H
5 0 2 3 t
Fig.3.8-5. Ci versus t demonstrating the effect of ki for ki = 1
3,8-6 where
Ai --» A2 -^ A3 -» A4 (3.8-6)
r — — kjCj 2 ~"" ^2^2 + k|Cj r3 — — k3C3 + k2C2 r — k3C3 (3.8-6a)
236
yields the following transition probability matrix:
1
P = 2
3
4
1 2 3 4
l-kjAt k,At 0 0
0 l-k2At k2At 0
0 0 l-ksAt k3At
0 0 0 1 (3.8-6b)
The transient response of Ci to C4 for the initial state vector C(0) = [Ci(0),
C2(0), C3(0), C4(0)] = [1, 0, 0, 0] is depicted in Fig.3.8-6 where the effect of ks is
demonstrated.
i 1
\ + 1 \
r\ / "• \ L\ / [•''V-^2 ." ' K'-'-/ . - ' I - - - -
1 1 1 k 3 = l
^ 4. - "
^ ^
^ ^ / \ J
\ 3 1
-==
-::;J.,..- 1 2 0 0.5 1.5
Fig.3.8-6. Ci versus t demonstrating the effect of ka
for ki = k2 = 5
A very complicated exact solution is available [33, p.46].
Kj K2 * 3 ^Z-1
3>8-7 Ai -> A2 -> A3 -- A4 ... Az-i _-> Az
where
Tj = kjCj 2 = - k2C2 + kjCj ^3 ~ •" ' S^B " ^^2^2
V l = "" kz- lCz-1 + ^Z-l^Z-1 n = kz_iCz_i
(3.8-7)
(3.8-7a)
yields the following transition probability matrix:
237
1
2
3
1 l-kjAt
0
0
2 kjAt
l-kjAt
0
3
0 k2At
i-M* ...
Z-1
0
0
0
Z
0
0
0
p =
z-1 z
0
0
0
0
0
0
l-kz_iAt k^iAt
0 1 (3.8-7b)
where from Eqs.(3-19a), (3-20), one obtains that:
Ci(n+l)-Ci(n)[l-kiAt] C2(n+1) - Ci(n)[kiAt] + C2(n)[l - kjM]
CjCn+l) - C2(n)[k2At] + C3(n)[l - kg At]
Cz_i(n+1) - C^2(n)[kz^2At] + Cz_i(n)[l - kz_iAt] Cz(n+1) = Cz_i(n)[kz.iAt] + Cz(n) (3.8-7c)
An exact solution for the present case appears in [22, p.9; 33, p.52].
Particular solutions by Markov chains appear above in cases 3.8-la (n = 3) and
3.8-6 (n = 4).
3.9 CONSECUTIVE REVERSIBLE REACTIONS
3.9-1 Ai -». A2 A3
k-2
where
Tj " — kjV^j ^2 " ~ k2V-'2 " —2^^ " 1 1 ^3 " "" —2^3 "^ k2v^2
yields the following transition probability matrix:
(3.9-1)
(3.9-la)
238
1
= 2
3
1 l-kjAt
0
0
2 kjAt
l-ksAt
k_2At
3
0 kjAt
l-k_2At (3.9-lb)
The transient response of Ci, C2, C3 and the ratio C3/C2 for the initial state vectors
C(0) = [Ci(0), C2(0), C3(0)] = [1, 0, 0] and [1, 0, 1] is depicted in Fig.3.9-1
where the effect of €3(0) is demonstrated.
1
0.8
0.6
0.4
0.2
0
U
| \ :
u h-
k'
= 1
^f ^i f
y
T 1 r
•
V'" 1 1 ^ ^ ^
w* " "
~"
TT^^^
C/O): = 0
H
•~j
/ \
I 1 1 ^"":r-^^
-1 - — 1
C^(0) = ll
H
0 0.2 0.4 0.6 t
0.8 1 0 0.2 0.4 0.6 t
0.8
Fig.3.9-1. Ci and the ratio C3/C2 versus t demonstrating tlie effect
of C3(0) for Iti = 1 2 = k.2 = 5
For At = 0.005, the agreement between the Markov chain solution and the
exact solution [32, vol.2, p.26] is Dmax = 6.2% and Dmean = 3.1%. As seen also
in Fig.3.9-1, the ratio C3/C2 approaches at steady state the value k2/k-2 as predicted
from Eqs.(3.9-la).
3.9-2 Ai "^ A2 - • A3 (3.9-2)
where
ri - - kjCi + k.iC2 r2 - - (k.i + ^.^^Q^ + ^\^\ 3 - ^2^2 (3.9-2a)
yields the following transition probability matrix:
239
P =
1
2
3
1
1-kiAt
k.iAt
0
2
kjAt
l-(k_i+ k2)At
0
3
0
kjAt
1 (3.9-2b)
The transient response of Ci, C2 and C3 for the initial state vector C(0) =
[Ci(0), C2(0), C3(0)] = [1, 0, 0] is demonstrated in Fig.3.9-2.
u
1
0.8
0.6
0.4
0.2
0
^ i = l
V \
I r
3
1 / 1
r k =0-i
2
-
H
1
i 1
L \ /
1
JHEL
^
":>-*->-
k =5 H 2
J J
0.5 1.5 2 0 0.5 1.5
Fig.3.9-2. Ci versus t demonstrating the efTect of 1 2
for \L\ = k.1 = 5
For At = 0.01, the agreement between the Markov chain solution and the
exact solution [22, p.24; 51] is Dmax = 3-6% and Dmean= O.J
3.9-3 kj kj
Ai ^ A2 ] ^ A3
k, k .
(3.9-3)
where
rj = — KjC| + k_2(J2 r2 = "~ vK_j + K2/C2 + kjCj + k_2C3
r3 - kjCj - k_2C3 (3.9-3a)
yields the following transition probability matrix:
240
P =
1-kiAt
k_iAt
2 3 kjAt 0
l-(k_i+ k2)At kjAt
0 k_2At l-k_2At I (3.9-3b)
The transient response of Ci, C2, C3 and the ratios C2/C1, C3/C2 for the
initial state vector C(0) = [Ci(0), C2(0), €3(0)] = [1, 0, 0] is demonstrated in
Fig.3.9-3.
Fig.3.9-3. Q , C2/C1 and C3/C2 versus t demonstrating the effect of
ki and k.ifor ki = 5
For At = 0.01, the agreement between the Markov chain solution and the
exact solution [22, p.42; 31, p. 175; 42] is Dmax = 1-4% and Dmean = 0.5%. As
observed in Fig.3.9-3, the ratios C2/C1 and C3/C2 approach at steady state the
limits ki/k-i and k2/k.2, respectively.
3.9-4 (3.9-4)
241
where r = - (kj + k_3)Ci + k_iC2 + k3C3
r2 = - (k i + k2)C2 + kiCi + k_2C3
r3 = - (k3 + k.2)C3 + k_3Ci + k2C2 (3.9-4a)
From Eq.(3.9-4a), the steady state conditions for the system follows from ri = r2 =
rs = 0, yielding
r ^2 1 kik_2 + kik3 -f k_2k,3 Lr^Jeq. " *:—; ;—;; TT" ""^i
1 k_2k_2 " k_j k3 + k2k3
'-C2-'^'J-
C31 kik2 + k_2k,3 + k2k_3
kik„2 + kik3 + k.2k_3
K_iK!_2 + —1* 3 " * 2 3
•=K,
L^aJeq kik2 + k_ik_3 + KiK 1^2
From these conditions it follows that
k-ik-2k-3 = kik2k3
Eqs.(3.9-4a) yield the following transition probability matrix:
(3.9-4b)
(3.9-4C)
P =
1 2 3 l-(ki+ k_3)At kjAt k_3At
k iAt l-(k_i+ k2)At k2At
k3At k_2At l-(k.2+ ^3^^^ (3.9-4d)
The transient response of Ci, C2, C3 and the ratios C1/C3, C2/C1 and C3/C2
for the initial state vector C(0) = [Ci(0), C2(0), €3(0)] = [1, 0, 0] are demonstrated
in Fig.3.9-4.
242
Fig.3.9-4. Ci and Ci/Cj versus t demonstrating the effect of k.i
for ki = k2 = k3 = k4 = 5
For At = 0.01, the agreement between the Markov chain solution and the
exact solution [32, vol.2, p.31; 35, p.92] is Dmax = 2.0% and Dmean = 0.3%. As
observed in Fig.3.9-4, the ratios Ci/Cj approach at steady state values predicted by
Eq.(3.9-4b).
3.9-5 -> Ai A2 -^ A3 --> A4 (3.9-5)
where r| = - kjCi + k_iC2 r2 = - (k_i + k2)C2 + k^Ci
r3 = - k3C3 + k2C2 u = k3C3 (3.9-5a)
yields the following transition probability matrix:
243
1
P = 2
3
4
1 2 3 4 l-kjAt kjAt 0 0
k_iAt l-(k_i+k2)At k2At o
0 0 l-k3At k3At
0 0 0 1 (3.9-5b)
The transient response of Ci to C4 for the initial state vector C(0) = [Ci(0),
C2(0), C3(0), C4(0)] = [1,0,0,0] is depicted in Fig.3.9-5 where the effect of ks is
demonstrated.
Fig.3.9-5. Ci versus t demonstrating the effect of ks for
111 = k.i = I12 = 5 and At = 0.01
A complicated exact solution is available [22, p.27].
3.9-6 1 1 1 2 1 3
Ai A2 A3 A4 «- ^ <-
(3.9-6)
^-2 •^-3
where
rj = - kjCi + k_iC2 12 = - (k^i + k2)C2 + kjCi + k^2C3
13 = k2C2 + k_3C4 - (k_2 + k3)C3 14 = k3C3 - k_3C4 (3.9-6a)
yields the following transition probability matrix:
244
1
P = 2
3
4
1 l-kiAt
k.jAt
0
0
2 kjAt
l-(k_i+ k2)At
k_2At
0
3
0
k2At
l-(k_2+ k3)At
k_3At
4
0
0 kgAt
l-k_3At (3.9-6b)
The transient response of Ci to C4 for an initial state vector C(0) = [Ci(0),
C2(0), C3(0), C4(0)] = [1.0,0,0] is depicted in Rg.3.9-6 where the effect of ki is
demonstrated.
Sv 1 1 1 V k =k =k =1
r--—-- '
L 4
H
— _~~]
0.5 1 1.5 0 0.5 1 t t
Fig.3.9-6, Ci versus t demonstrating the effect of kt for k.1 = k.2 = k-3 = 5 and At = 0.01
1.5
A complicated exact solution is available [22, p.44].
3.9-7 Ai A2 A3 ^ A4 -»• A5 (3.9-7)
where
Fj » - kjCj + k_iC2 r2 =" - (k_i + ^ ^ 2 + k iCi + k_2C3
3 " ' ^ 2 2 "" ' ^-2 "'' ' 3 ^3 ^4 "* ^^3^3 "" k4C4 V^ = K4C4 (3.9-7a)
yields the following transition probability matrix:
245
1
2
= 3
4
5
1 l-kjAt
k_iAt
0
0
0
2 kjAt
l-(k_i+ k2)At
k_2At
0
0
3
0 kjAt
l-(k_2+ k3)At
0
0
4
0
0 k3At
l-k4At
0
5
0
0
0 k4At
1 (3.9-7b)
The transient response of Ci to C5 for the initial state vector C(0) = [Ci(0), C2(0), C3(0), C4(0), C5(0)] = [1, 0, 0, 0, 0] and [1, 0, 0.5, 0] is depicted in Fig.3.9-7 where the effect of k4 is demonstrated.
u
11
0.8
0.6
0.4
0.2
0
« k = l ' 1 4
i-i = l
\ ' k = 5 . - « - -
\ ' • '
\ 2
- - - - -|
- J
H
* "* *1l M 1 1
1 0.5 1 l.f 0 0.5
t t
Fig.3.9-7. Ci versus t demonstrating the effect of k4 for ki = k2 = ka = 10, hL 1 = k.2 = 1 and At = 0.01
A complicated exact solution is available [22, p.29].
1.5
3.9-8
where
A i + A2 ^ A3 ^ A4
k, k .
(3.9-8)
r ** 2 *" "" *^i^i^2 " —13
r3 = - (k^i + k2)C3 + k iCiC2 + ^^2^4
1*4 "" ~k_2C4 + k2C3 (3.9-8a)
246
yields the following transition probability matrix:
P =
1
2
3
4
1 l-kiC2(n)At
0
k_iAt
0
2
0
l-kiCi(n)At
k_,At
0
3
^kiC2(n)At
^kiCi(n)At
l-(k_,+ k2)At
k_2At
4
0
0
kjAt
l-k_2At (3.9-8b)
The transient response of Ci to C4 for the initial state vector C(0) = [Ci(0),
C2(0), C3(0), C4(0)] = [1, 0.9, 0, 0] is depicted in Fig.3.9-8 where the effect of
ki, k-i is demonstrated.
1
0.8
0.6
0.4
0.2
0
h
[ k:
-i Lj "•2 - - ->_
3 - ^ ^ _ _ 4- - - > - -
1 1
k =k =1 1 2 J
k =k =5 1 -1 -2
i-"V
/
r -' I'
1
4 , - -
1
1 1
k = k = 5
k =k = 1 ~
1 0 0.5 1 1.5 0 0.5 1 1.5
t t
Fig.3.9-8. Ci versus t demonstrating the effect of kf and k.i for
At = 0.01
The present reaction is considered in [35, p. 149] without an analytical
solution.
3>9-9 Ai + A2 "^ A3 A4 ^ A5 + Ai (3.9-9)
where
247
ri = - kjCiCj + k_iC3 - lesCiCj + k3C4 rj = - kjCjCj + k_iC3
r3 = - (k_i + k2)C3 + kiCjC2 + k_2C4
r4 = - (k_2 + k3)C4 + k2C3 + k_3CiC5 rs = - k_3CiC5 + k3C4 (3.9-9a)
yields the following transition probability matrix:
1
2
P = 3
4
5
1 1-
2 3 4 5
0 •i-k,C2(n)At ^k_3C5(n)At o [k,C2(n)+k_3C5(n)]At
k_iAt k.jAt
k3At
1-(k_i+k2)At
k_2At
0
k2At
1-(k_2+k3)At
lk_3Ci(n)At
0
k3At
1-k_3Ci(n )At
(3.9-9b)
The transient response of Ci to C5 for the initial state vector C(0) = [Ci(0),
C2(0), C3(0), C4(0), C5(0)] = [1, 0, 0, 0, 1] is depicted in Fig.3.9-9 where the
effect of ki is demonstrated.
u
1
0.8
0.6
0.4
0.2
0
\ i = l
[/
1 I 1 k =k =k =10
1 2 3
5
4
T - 2 ^ , -
\1 1 i
k =k =k =1 1 2 3
2
/
1 " 1 ' ' • ~ ^ • • " ^ - -
-
0 0.5 1.5 2 0 0.5 1.5
Fig.3.9-9. Ci versus t demonstrating the effect of ki = k i = k3
for k.i = k.2 = k.3 = 5 and At = 0.01
248
The present reaction is considered in [35, p. 170] without an analytical
solution.
3.9-10
For the above reaction of Z states, which simulates signal transmition in a T-
cell [37], the following equations are applicable:
rj = Tj = - kjCiCz + k_i(C3 + C4 + C4+
r3 = - (k_ i+k2)C3 + kiC,C2
r4 = - (k_i + k2)C4 + k2C3
rj = - (k_i + k2)C 5 + k2C4
+ Cz_i + Cz)
Tz-l - ~ (k_i + k2)C 2;_i + k2Cz_2 tz = - k-j C z + k2Cz_] (3.9-lOa)
yielding the foUowing transition probability matrix:
P =
1
2
3
4
5
Z-1
Z
1
Pll k_iAt k_iAt
k.jAt
k_iAt
k_,At
k_iAt
2 0
P22 k_iAt
k_iAt
k_iAt
k_iAt
k_iAt
3
Pl3
P23 P33
0
0
0
0
4
0
0
k2At
P44 0
0
0
5 0
0
0
k2At
P55
0
0
z-1 0
0
0
0
0
Pz-i,z-i P7.7..1
z 0
0
0
0
0
k2/5
Pz (3.9-lOb)
249
where
Pn = 1 - kiC2(n)At; p,3 = 0.5k,C2(n)At; P22 = 1 " kiC,(n)At; P23 = 0.5kjC,(n)At
P33 = P44 = P55 = ••• = Pz-i,z-i = 1 - (2k_, + k2)At; p , , = 1 - 2k_,At; (3.9-lOc)
From Eqs.(3-19a), (3-20), one obtains that:
Ci(n+1) = Ci(n)[l - kiC2(n)At] + {CjCn) + CgCn) + - + Cz(n)}[k_iAt]
C2(n+1) = C2(n)[l - kiCi(n)At] + {CjCn) + €4^) + ••• + Cz(n)}[k_iAt]
Cjin+l) = Ci(n)[0.5kiC2(n)At] + C2(n)[0.5kiC,(n)At] + C3(n)[l-(2k_, + k2)At]
C4(n+1) = C3(n)[k2At] + C4(n)[l - (2k_i + k2)At]
Cz_i(n+1) = Cz_2(n)[k2At] + Cz_i(n)[l - (2k_, + k2)At]
Cz(n+1) = Cz_i(n)[k2At] + Cz(n)[l - 2k_iAt] (3.9-lOd)
For Z = 5 the transient response of Ci to C5 for the initial state vector C(0) =
[Ci(0), C2(0), C3(0), C4(0), C5(0) = [1, 1, 0, 0, 0] is depicted in Fig.3.9-10
where the effect of ki is demonstrated.
\ \h2
" V L / \
U—:
1
— K
4
">-— 5 -
1
1 1 1 k = 7
-- J
-
-
1 1 1 1 0 0.2 0.4 0.6 0.8
t
Fig.3.9-10. Ci versus t demonstrating the effect of ki for It-i = 1,
k2 = 5 and At = 0.003
250
3.10 PARALLEL REACTIONS: SINGLE AND CONSECUTIVE IRREVERSIBLE REACTION STEPS
3.10-1 Ai ^ A 2 (3.10-1)
where [38, chap., problem C48]
Ti = - (ki + k2)Ci r2 = -k3C2 + kiCi r3 = k2Ci + k3C2 (3.10-la)
yields the following transition probability matrix:
1
P = 2
3
1 1 - (ki + k2)At
0
0
2 kjAt
1 - k3At
0
3 k2At
k3At
1 (3.10-lb)
The transient response of Ci, C2 and C3 for the initial state vector C(0) =
[Ci(0), C2(0), C3(0)] = [1, 0, 0] is depicted in Fig.3.10-1 where the effect of ks is
demonstrated.
1
0.8
0.6
0.4
0.2
0
u
-\i = l
V
1 1
X /
/
1 i"~"~^—
1 ^ J __ -—
k3 = 5]
.
\ , '
^z^' \
I ' r^^
1 1 1 1
-] k =0 3 J
3 -J
0.2 0.4 0.6 0.8 t
1 0 0.2 0.4 0.6 0.8 t
Fig.3.10-1. Ci versus t demonstrating the effect of ka
for ki = 5, k2 = 1 and At = 0.005
3.10-2
251
(3.10-2)
where [38, chap., problem C55]
r i = - k i C i r2 = - (kj + k3)C2 + k,Ci r3 = k2C2 r4 = k3C2 (3.10-2a)
yields the following transition probability matrix:
1
P = 2
3
4
1 l-kjAt
0
0
0
2 kjAt
l-(k2 + k3)At
0
0
3
0 k2At
1
0
4
0 k3At
0
1 (3.10-2b)
The transient response of Ci to C4 for the initial state vector C(0) = [Ci(0),
C2(0), C3(0), C4(0)] = [1, 0, 0, 0] is depicted in Fig.3.10-2 where the effect of ks
is demonstrated.
1
0.8
0.6
0.4
0.2
0
u
4 i= 1
\ 2 - -V
L <•'
1 1 k =1
3
^ ^ 4 .
r^-—•— 1 '^^
-
\
\j '"^ '>--^^^^^i:i_.j=^
-J
__ —
0.5 1 1.5 0 0.5 1 t t
Fig.3.10-2. Ci versus t demonstrating the effect of ka
for ki = 5, k2 = 2 and At = 0.01
1.5
252
3.10-3
1
1 ^
A 2 - : i l ^ A 4 - ^ I - ^ A 6
A, — - . ^ - As . >-A7 k6 MO
where
ri = - (ki + k2)Ci X2= - (kj + k4)C2 + kjCj
rj = - (ks + k6)C3 + kjCi u = - (ky + k8)C4 + kjCj + kjCs
rj = - (k9 + kio)C 5 + k4C2 + k^Cj
ig = k7C4 + k9C5 r-j = k8C4 + kjoCs
yields the following transition probability matrix:
P =
1
2
3 4
5
6
7
1
Pll 0
0
0
0
0
0
2 kjAt
P22 0
0
0
0
0
3 k2At
0
P33 0
0
0
0
4
0 kjAt
ksAt
P44 0
0
0
5
0 k4At
k At
0
P55 0
0
6
0
0
0 kyAt
kjAt
1
0
7
0
0
0 kgAt
k,oAt
0
1
where
(3.10-3)
(3.10-3a)
(3.10-3b)
Pll = 1 - (ki + k2)At P22 = 1 - ( 3 + k4)At P33 = 1 - {^^ + k6)At P44 = 1 - (k7 + k8)At P55 = 1 - (k9 + kio)At (3.10-3d)
It should be noted that by equating to zero one (or more) of the rate constants ki in
Eq.(3.10-3b), many interesting reactions can be generated.
The transient response of Ci to C7 for the initial state vector C(0) = [Ci(0),
C2(0), C3(0), ..., C7(0)] = [1, 0, 0, ..., 0] is depicted in Fig.3.10-3 where the
effect of k7 is demonstrated.
253
u
0.8
0.6
0.4
0.2
0
1' 4 -r""'
A / " ^ "• -
I k =1
7
c =c 6 7
^ ^ - . c =c " " - 1 ^
1 " ^ ' '
\
tc \ 1
1
/L^
' k = 5 ' 7
c ,.. 6
c 7
. c
-
—
_
-
0 0.5 1 1.5 0 0.5 1 1.5 t t
Fig.3.10-3. Ci versus t demonstrating the effect of k? for
ki = k2 = k4 = ks = 5, k3 = k6 = 10, kg = kp = kio = 1 and At = 0.005
An extremely complicated exact solution is available [22, p.55].
3.10-4 3 ^ A 5
ly< Ai + A s ' ^ |k5
1 ^ A4 • >• Ae k4
where [38, chap.6, problem B37]
(3.10-4)
rj = r2 = - (kj + k2)CiC2
r3 = - k3C3 + kjCiC2 r4 = - k4C4 + k2CjC2
r5 = - ksCs + k3C3 r = )^^C^ + ^^C^ (3.10-4a)
yields the following transition probability matrix:
254
P =
1
2
3
4
5
6
1
Pll
0
0
0
0
0
2
0
P22
0
0
0
0
3 lkiC2(n)At
ikiCi(n)At
l-kgAt
0
0
0
4 Yk2C2(n)At
lk2Ci(n)At
0
l-k4At
0
0
5
0
0
k3At
0
l-k5At
0
6
0
0
0
k4At
k^M
1
where Pll = 1 - (kj + k2)C2(n)At; P22 = 1 - (^i + k2)Ci(n)At
(3.10-4b)
(3.10-4C)
The transient response of Ci to C^ for the initial state vector C(0) = [Ci(0), C2(0),
C3(0), ..., C6(0)] = [1, 0, 0, ..., 0] is depicted in Fig.3.10-4 where the effect of
C2(0) is demonstrated.
1
0.8
0.6 r 0.4
0.2
0
L \ ^
' 0^(0) = 0.5 ' 1
• c 6
" c =c • ^ 3 4
r-"-:^-^-^i . .
I LA
k r - ^ ^ > -
' c^(0)
c =c / 3 4
" • ^ - i ^
- T~~--
= 0 . 6 '
-\
c J
c -:-^k.,-1 2 3 0 1 2 3
t t
Fig.3.10-4. Ci versus t demonstrating tlie ej^ect of C2(0)
for Ici = k2 = 5, k3 = li4 = ks = 2 and At = 0.015
3.10-5
ajAi + a j A j — ^ ^ • A 4
(3.10-5)
255
where [32, vol.2, p.77]
ii = - aikC?iC22 12 = - a2kCi>C22
k = ki + k2 + k3
(3.10-5a)
If the initial conditions are CsCO) = €4(0) = €5(0) = 0, it follows that
ri/r3 = k2/ki or C4/C3 = k2/ki
r5r3 = k3/ki or C5/C3 = k3/ki (3.10-5b)
Thus, the ratio of the amounts of the products is constant during the reaction and
independent of its order. Eqs.(3.10-5a) yield the following transition probability
matrix:
1 2
P = 3 4 5
1
Pll
0
0
0
0
2 0
P2
0
0
0
where
3 4 5
0 NikiC i"' (n)C 2(n)At Nik2Cii"kn)C^2(n)At N^k^Cl^^HnK^MM
P22 N2kiCji(n)C^"Vn)At ^jh^^]'^^^^!^'^^^^^^ N2k3CjKn)C^"Hn)At
1
0
0
0
1
0
Pll = 1 - aikC i""Hn)C22(n)At; P22 = l-a2kCii(n)C22"Hn)At
^1 XT ^ k = k, + ko + ko; Ni = • ; N , = .
(3.10-5C)
(3.10-5d)
For ai = a2 =1, the transient response of Ci to C5 is depicted in Fig.3.10-5
where the effect of C2(0) is demonstrated. It should be noted also that Eqs.(3.10-
5b) are verified by the numerical results.
256
0 0.5 1 1.5 0 0.5 1 1.5 t t
Fig.3.10-5. Ci versus t demonstrating the effect of CiCO) for
Ci(0) = 2, ki = 3, k2 = 2, k3 = 1 and At = 0.005
3 . 1 0 - 6 i = 1: Ai_^A2
i = 2: 2Ai-^A3 (3.10-6)
The derivation of the kinetic equations, based on Eqs.(3-2), (3-3), is:
fi -~^2 - *^iCi
1 J 2 ) _ . ( 2 ) _ _ , , p 2 2" ti - - r3 - - K2i-.i
where from Eq.(3-4) follows that
ri = rV^ -I- xf = - (k,Ci -i- 2k2Ci)
, Ji) u r f - r(2) _ u r2 ^2-h - *'i*-i ^3-r3 -is.2^1 (3.10-6a)
yielding the following transition probability matrix:
1
P = 2
3
1 2 3 l-[ki-I-2k2Ci(n)]At kjAt k2Ci(n)At
0 1 0
0 0 1 (3.10-6b)
where pi4 is computed by Eq.(3-10a).
257
The transient response of Ci, C2 and C3 for the initial state vector C(0) =
[Ci(0), C2(0), C3(0)] = [1, 0, 0] is depicted in Fig.3.10-6 where the effect of k2 is
demonstrated.
Fig.3.10-6. Ci versus t demonstrating the effect of kz for Iti = 5
For At = 0.0025, the agreement between the Markov chain solution and the
exact solution [33, p.35; 39, p.32] is Dmax = 4.8% and Dmean = 4.0%.
3.10-7
where
Ai .^ A2 -> A3
2Ai —> A4
Tj = - kjCj - 2k3Ci
Ta = k2C2 r4 = k3Cj
yields the following transition probability matrix:
T2 — — k2i-^2 " ki^--!
2
1
P = 2
3
4
1 2 3 4 l-[ki + 2k3Ci(n)]At k,At 0 k3C,(n)At
0 l-k2At k2At 0
0 0 1 0
0 0 0 1
(3.10-7)
(3.10-7a)
(3.10-7b)
258
The transient response of Ci to C4 and ZCi for the initial state vector C(0) =
[Ci(0), C2(0), C3(0), C4(0)] = [1, 0, 0, 0] is depicted in Fig.3.10-7 where the
effect of ks is demonstrated.
u
3.5
3
2.5
^ 2
1.5
1 0.5
0 \^'^ ^ ^ r- — r 0.5 1 1.5 0 0.5
t t Fig.3.10-7. Ci versus t demonstrating the effect of ka
for ki = k2 = 1 and At = 0.015
No exact solution is available. However, it should be noted in the above
figure that XCi approaches the limits (2 for k3 = 1 and 3 for ks = 0) according the
stoichiometry in Eq.(3.10-7).
3.10-7,1 Ai -» A2 A3
Ai + A2 -^ A4
where
Ai + A3 -> A4 (3.10-7.1)
ri = - kiCi - k3CiC2 - k4CiC3 x^ = ^\^\ " k2C2 - k3CiC2
1*3 — k2C2 — k4C],C3 r4 = k3CiC2 + k4CiC3 (3.10-7. la)
yields the following transition probabiUty matrix:
259
P =
1
2
3
4
1
l - [ki+ k3C2(n)+k4C3(n)]At
0
0
0
2
kjAt
l-[k2+k3Ci(n)]At
0
0
3
0
k2At
l-k4Ci(n)]At
0
4
•i-[k3C2(n)
+k4C3(n)]At
i-k3Ci(n)At
lk4Ci(n)]At
1
(3.10-7. lb)
The transient response of Ci to C4 for the initial state vector C(0) = [Ci(0), C2(0), C3(0), C4(0)] = [1, 0, 0, 0] is depicted in Fig.3.10-7.1 where the effect of ki is demonstrated.
0 2 4 6 0 2 4 6 t t
Fig.3.10-7.1. Ci versus t demonstrating the effect of ki for k2 = k3 = k4 = 1 and At = 0.01
3.10-7.2
where
k[ kj
Ai + A2 -4 A3 -> A4
^ 3
Ai + A2 -» A5
ri = r2 = ~ (ki + k3)CiC2
3 " l^lCiC2 - k2C3
r4 = k2C3 r5 = k3CiC2
(3.10-7.2)
(3.10-7.2a)
260
yields the following transition probability matrix:
P =
1
2
3
4
5
1 l~(ki+ k3)C2(n)At
0
0
0
0
2
0
l - (ki+ k3)Ci(n)At
0
0
0
3 lkiC2(n)At
ikjCiWAt
l-k2At
0
0
4
0
0
k2At
1
0
5 yk3C2(n)At
lk3Ci(n)At
0
0
1
(3.10-7.2b) The transient response of Ci to C5 for the initial state vector C(0) = [Ci(0),
C2(0), C3(0), C4(0), C5(0)] = [1, 0, 0, 0, 0] is depicted in Fig.3.10-7.2 where the effect of ki is demonstrated.
1 1 1 k =5
1
\ ^ |1C =C . - ' 1 2
Q_ ^ M \c
\V Pv ^ ' L V ^ B '^^^ "V^ . . . .
\' ^\ """ r
c 4
c 5
—
—
6 0
Fig.3.10-7.2. Ci versus t demonstrating the effect of ki for k2 = ii3 = 1 and At = 0.01
3.10-8
where
A i ^ A 2
Ai + A2 -> A3
T] — — vk|Cj + k2C|C2) r2 — k]Cj ~ k20jC-2 '"3 — k2C jC 2
yields the following transition probability matrix:
(3.10-8)
(3.10-8a)
261
1
P =
1, l-[ki + k2C2(n)]At k,At ^k2C2(n)At
0 l-k2Ci(n)At lk2C,(n)At
0 0 1 (3.10-8b)
The transient response of Ci, C2 and C3 for the initial state vector C(0) =
[Ci(0), C2(0), C3(0)] = [1, 0,0] is depicted in Fig.3.10-8 where the effect of Ci(0)
is demonstrated.
4
3 k
U
-
C (0) = 1
J = l
- 3-
-
-
\ - \
\ \ \ \
/ . ' ;
C (0) = 4
2
H
3— H
0.5 1.5 0 0.5 1.5
Fig.3,10-8. Ci versus t demonstrating the effect of Ci(0) for ki = k2 = 2
For At = 0.01, the agreement between the Markov chain solution and the
exact solution [33, p.95; 44; 49, p.91] is Dmax = 4.1% and Dmean = 2.4%.
3.10-9
where
Ai-^A4
Ai + A2-> A3
ri = - kiCj - k2CiC2 r2 = - r3 = - k2CiC2 i^ = k2Ci
yields the following transition probability matrix:
(3.10-9)
(3.10-9a)
262
1 2 3 4
0 4-k2C2(n)At kjAt
1
l-[k, + k2C2(n)]At
0 l-k2Cj(n)At i.k2C,(n)At 0
0 0 1 0
0 0 0 1 (3.10-9b)
The transient response of Ci to C4 for the initial state vector C(0) = [Ci(0),
C2(0), C3(0), C4(0)] = [1, 1, 0, 0] is depicted in Fig.3.10-9 where the effect of ki
is demonstrated.
- \ -"^ -'-
\' 1
k 1
— 4 -
1 = 2
1
3
— 1 -
-
H
0.5 1.5
Fig.3.10-9. Ci versus t demonstrating the effect of ki
for li2 = 2
For At = 0.01, the agreement between the Markov chain solution and the
exact solution [32, vol.2, p.45] is Dmax = 10% and Dmean = 0.5%. It should be
noted that in [27], the transition probability matrix is incorrect.
3,10-9,1
where [53, p.201]
2Ai->A4
Ai + A2 -> A3 (3.10-9.1)
ri = - 2kiCi2 - k2CiC2 r2 = - r3 = - k2C|C2 i^ = k^Cf (3.10-9.la)
263
yields the following transition probability matrix:
1
P =
2 0 ^k2C2(n)At kiCi(n)At l-[2kiCi(n) + k2C2(n)]At
0 l-k2Ci(n)At |k2Ci(n)At 0
0 0 1 0
0 0 0 1
where pi4 is computed by Eq.(3-10a). (3.10-9. lb)
The transient response of Ci to C4 for an initial state vector C(0) = [Ci(0),
C2(0), C3(0), C4(0)] = [1, 1, 0, 0] is depicted in Fig.3.10-9.1 where the effect of
ki is demonstrated.
1
0.8
0.6
0.4
0.2
0
u
r nyi=i
Vr ^ \
\
^2
3^
Ar
1
1 1 k =5
1
• - "
"1 h-
1 1
^
-
[-
0.2 0,4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 t t
Fig.3.10-9.1 Ci versus t demonstrating tlie effect of l i
for 1 2 = 5 and At = 0.02
3,10-10
where
Ai -> A3 + A4
Ai + A2 -» A3 + A5 (3.10-10)
Tj — — 13 — — kjv.'i "~ 2^1^2 2 "" ~ ^2^1^2
14 = kjCj 15 = k2CiC2
yields the following transition probability matrix:
(3.10-lOa)
264
P =
2
3
4
5
1 1 -
[kj + k2C2(n)]At
2 3 4 5 0 ''1^^+ k,At ik2C2(n)At
ik2C2(n)At
0 k2c'^)At i'^^^'^"^^^ 0 T'^^'^"^^'
0 0 1 0 0
0 0 0 1 0
0 0 0 0 1 (3.10-lOb)
The transient response of Ci to C5 for the initial state vector C(0) = [Ci(0),
C2(0), C3(0), C4(0), C5(0)] = [1, 1, 0, 0, 0] is depicted in Fig.3.10-10 where the
effect of k2 is demonstrated.
1
0.8
0.6
0.4
0.2
0
i\\ . ' 3
r \^ ^ 1 ^ ^
. 4 - - -
• - • -
-'-"r 1
~^ 1 = 5 2
—
— 1, 0 0.2 0.4
t 0.6 0.8 0
Fig.3.10-10. Ci versus t demonstrating the effect of k2
for ki = 5 and At = 0.015
3>10-11 Ai + Ai -»A2
Ai + A3 -^ A4
Ai + A2-^A3
Ai + Az-i -> Az Ai + Az-^ Az+i (3.10-1 la)
where
Ti = klCi + 2j^m^l^n m=2
12 = O.SkiCf - k2CiCi 13 = k2CiC2 - k^C^C^
265
yields the following transition probability matrix:
P =
1
2
3
4
5
:
Z
Z+1
1
Pll 0
0
0
0
0
1 0
2
Pl2
22 0
0
0
0
0
0
3
Pl3
P23
P33
0
0
0
0
0
4
Pl4 0
P34
P44
0
0
0
0
5
Pl5 0
0
P45
P55 ••
0
0
0
z Piz 0
0
0
0
0
Pzz 0
z+1
Pl,Z+l
0
0
0
0
0
Pz,z+i 1
where
Pii = l - 2kiCi(n) + ^k„C„(n) m=l
At P12 = kjCi(n)At
Pi.m+1 = 0.5k^C Jn)At m = 2,..., Z
and for the deteraunation of pn see section 3.3.
P22 = 1 - k2Ci(n)At P23 = 0.5k2Ci(n)At
P33 = 1 - k3Ci(n)At P34 = 0.5k3Ci(n)At
P44 = 1 - k4Ci(n)At P45 = 0.5k4Ci (n)At
(3.110-llb)
(3.10-1 Ic)
266
P55 = 1 - k5Ci(n)At p56 = O.SksCi(n)At
pzz = 1 - kzCi(n)At pz,z+i = 0.5kzCi(n)At
k,
(3.10-1 Id)
For Z = 2:
where
2Ai_»A2 kj
Ai + A2 -> A3 (3.10-1 le)
ri = -2k iCi -k2CiC2 r2 = -k2C,C2 + k,C^ r3 = k2CiC2 (3.10-1 If)
yields the following transition probability matrix:
p =
1
2
3
l-[2kiCi(n) + k2C2(n)]At kiCi(n)At ^k2C2(n)At 1
0 l-k2Cj(n)At ik2Ci(n)At
0 0 1 (3.10-1 Ig)
where pi2 is computed by Eq.(3-10a).
The transient response of Ci, C2 and C3 for the initial state vector C(0) =
[Ci(0), C2(0), C3(0)] = [1, 0, 0] is depicted in Fig.3.10-11 where the effect of k2
is demonstrated.
Fig.3.10-11. Ci versus t demonstrating the effect of k2
for ki = 1 and At = 0.03
267
It should be noted that refs.[32, vol.2, p.69; 39, p.47] predict oscillations,
not observed here.
3>10-12 2Ai -^ A2 + A4
Ai + A2 -> A3 + A4
where [32, vol.2, p.70]
(3.10-12)
r — — kiC i — k2v-'iv 2 2 — 1 1 — 12 1 2
3 "" k2CiC2 r4 = kiC? + k2CiC2 (3.10-12a)
yields the following transition probability matrix:
1 1 -
[2kiCi(n)4-k2C2(n)]At
0
0
0
2 3 4 kiCi(n)At ik2C2(n)At [|k2C2(n)+kiCi(n)]At
l-kiCi(n)At ik2Ci(n)At
0 1
0 0
ik2Ci(n)At
0
1
(3.10-12b)
where pi2 was computed by Eq.(3-10a). The transient response of Ci to C4 for
the initial state vectors C(0) = [Ci(0), C2(0), €3(0), €4(0), €5(0)] = [1, 1, 0, 0, 0]
and [0.5, 1, 0, 0, 0] is depicted in Fig.3.10-12 where the effect of Ci(0) is
demonstrated.
268
1
0.8
0.6
0.4
0.2
0
l \ 1 1 1 ^ C(0)=1
V 4 - - - " " 1 \ ' ' / X r ^ - -
1 " " ~ ~ " ~ i — • — H -
-
—
0.5 1.5 2 0
Fig.3.10-12. Ci versus t demonstrating the effect of Ci(0) for ki = k2 = 2 and At = 0.03
3,10-13
where
Ai + 2A2 -4 A3
Ai + A2 -> A4
Tj — - k iCiC2 - k2CiC2 1*2 - - 2kiCiC2 - k2C|C2
yields the following transition probability matrix:
(3.10-13)
(3.10-13a)
P =
1 1 -
[kiC2(n)+k2C2(n)]At
0
0
0
2
0
3 4 lkiC2(n)At ik2C2(n)At
l-[2kiCi(n)C2(n) |k,Ci(n)C2(n)At ik2Ci(n)At
+k2Ci(n)]At
0
0
(3.10-13b)
The transient response of Ci to C4 for the initial state vector C(0) = [Ci(0),
C2(0), C3(0), C4(0)] = [1, 1, 0, 0] is depicted in Fig.3.10-13 where the effect of
k2 is demonstrated.
269
u
1
0.8
0.6
0.4
0.2
0
r 2 \
~ l r ' ' ^ If
1 1 1 k = 5
2
^ J>- - -^
.. ^ ^ 4
1 1 - 4 -
-
0.5 1.5 2 0
Fig.3.10-13. Ci versus t demonstrating the effect of k i for ki = 5 and At = 0.01
3 , 1 0 - 1 4 A i - ^ A 3
Ai + A2 -^ A4
2 A i ^ A 5
where
13 = kjCj r4 = k2CiC2 15 = k3Cj
yields the following transition probability matrix:
1
P =
1 l-[ki+k2C2(n)+
2k3Ci(ii)]At
0
2
0 kjAt ik2C2(n)At k3Ci(n)At
(3.10-14)
(3.10-14a)
l-k2Ci(n)]At 0
3 0 0 1
4 I 0 0 0
5 I 0 0 0 0 1 I (3.10-14b)
where pi5 was calculated by Eq.(3-10a). The transient response of Ci to C5 for the
initial state vector C(0) = [Ci(0), C2(0), €3(0), €4(0), €5(0)] = [1, 1, 0, 0, 0] is
depicted in Fig.3.10-14 where the effect of ki is demonstrated.
i.k2Ci(n)At
0
1
0
0
0
0
1
Fig.3.10-14. Ci versus t demonstrating the effect of ki for k2 = 2 and k3 = 5
For At = 0.005, the agreement between the Markov chain solution and the
exact solution [32, vol.2, p.48] is Dmax = 3.3% and Dmean = 0.6%. For ks = 0, an
exact solution is available [51].
3>10-15 2Ai-^A3
Ai + A2 -> A4
2A2->A5
where [33, p.75; 38, chap.6, problem D82]
(3.10-15)
rj — — ZkjC j — k2v_ j>-'2 r2 - ~ 2k3C2 - k2CiC2
3 ~ '^i^i ^4"" k2CiC2 r5 - k3C2 (3.10-15a)
yields the following transition probability matrix:
271
P =
1
2
3
4
5
1 2
0 kiCi(n)At lk2C2(n)At o l-[2kiCi(n)
+k2C2(n)]At
Q HkjCiCn) Q lk2Ci(n)At k3C2(n)At +k3C2(n)]At
0
0
0
0
0
0
0
0
1 (3.10-15b)
The transient response of C\ to C5 for the initial state vector C(0) = [Ci(0),
C2(0), C3(0), C4(0), C5(0)] = [1, 1, 0, 0, 0] is depicted in Fig.3.10-15 where the
effect of k3 is demonstrated.
u
1
0.8
0.6
0.4
0.2
0
1-
\ ^
r
1
5 - •
_. __ -.
1
' ' \
-
- 3
- r 1.
0.5
[
't \\ \ .
L ^ -!>^
1
- 3 -
""1"^
1 1 1 k =1 3
H
__ 1 ' 5
1.5 1 1.5 2 0 0.5 1 t t
Fig.3.10-15 Ci versus t demonstrating the effect of k^ for ki = k2 = 5 and At = 0.01
3.10-15.1
where
r i = - k i C ,
A i ^ A 2
2A2 -> A3
A 2 ^ A 4
2 ~ '^iCl - k2C2 - k3C2
Tj = O.SkjCj u = kgCj
(3.10-15.1)
(3.10-15.1a)
272
yields the following transition probability matrix:
P =
1
2
3
4
1
l-kjAt
0
0
0
2
kjAt
l-[k2C2(n)+k3]At
0
0
3
0
"J^l^i
1
0
4
0
k3At
0
1 (3.10-15.1b)
where p23 was calculated by Eq.(3-10a). The transient response of Ci to C4 for
the initial state vector C(0) = [Ci(0), C2(0), €3(0), C4(0)] = [1,0,0,0] is depicted
in Fig.3.10-15.1 where the effect of ks is demonstrated.
Fig.3.10-15.1 Ci versus t demonstrating the effect of ks for ki = k2 = 5 and At = 0.01
A rather compUcated exact solution is available [50].
3,10-16 Ai -> A4 + A5
A5 + A2-^A3
Ai+A2->A3 + A4 (3.10-16)
273
where [33, p.86]
fj = - kjCi - k3CiC2 12 = - 13 = - k2C2C5 - k3CiC2
14 = kjCj + k3CiC2 15 = kjCj - k2C2C5
yields the following transition probability matrix:
(3.10-16a)
P =
1 1-
[ki+k3C2(n)]At
0
2 3 4
0 ik3C2(n)]At [ki+lk3C2(n)]At
l-[k2C5(n) [ |k2C5(n) ik3C i(n)At +k3Ci(n)]At :
+i.k3Ci(n)]At
kjAt
0
0
0
0
0
^ i.k2C2(n)]At
0
1
0
0
0
l-k2C2(n)]At
(3.10-16b)
The transient response of C\ to C5 for the initial state vectors C(0) = [Ci(0),
C2(0), C3(0), C4(0), C5(0)] = [1, 1, 0, 0, 0] and [0.5, 1, 0, 0, 01 is depicted in
Fig.3.10-16 where the effect of Ci(0) is demonstrated.
1
0.8
0.6
0.4
0.2
0 0 0.5 1 1.5 0
t
Fig.3.10-16. Ci versus t demonstrating the effect of Ci(0) for ki = k2 = k3 = 5 and At = 0.005
I -' X/
p \=, "" —-
1
C ( 0 ) = 1
-
1
274
3 .10 -17
where
Ai + A2 -^ A3
A2 + A3 ^ A4 (3.10-17)
i i — - k j C i C 2 r 2 - - k i C ] C 2 - k 2 C 2 C 3
T^ ^ lCjV jv 2 ~ ^2^-2^3 4 ^ K2C2^3
yields the following transition probability matrix:
(3.10-17a)
P =
1 2 3 4 l-kiC2(n)]At 0 Yk,C2(n)At 0
0 l-[klCi(n) ikiC,(n)At lk2C3(n)At +k2C3(n)]At
0 l-k2C2(n)]At lk2C2(n)At 0
0 0 (3.10-17b)
The transient response of Ci to C4 for the initial state vectors C(0) = [Ci(0), C2(0), C3(0), C4(0)] = [1, 1, 0, 0], [1, 0.5, 0, 0] and [0.5, 1, 0, 0] is depicted in Fig.3.10-17 where the effect of C2(0) is demonstrated.
y 1
_\1
t K
1 1 C (0) = 0.5
2
4
^.L i
H
2 0 0.5 1.5
275
1.5
U^ 1
0.5
n
[ I I I 1 \ C (0) = 2
\ - 2
v\>--^ v~^^Cj^:_-~^ - - ^ ~~^ _
0.5 1.5
Fig.3.10-17. Ci versus t demonstrating the effect of C2(0) for ki = k2 = 5
For At = 0.01, the agreement between the Markov chain solution and the
exact solution [33, p. 100; 43; 51] is Dmax = 2.5% and Dmean = 1.9%.
3>10-18
where
Ai + A2 -» A3 + A4
Ai + A3 -> A5 + A4
rJ = — JC|C'2C^2 ~ k2v-'iC-'3
3 - ^\^\^2 "" '^2CiC3
4 ^ k.2C|C2 " " 2 1 3
yields the following transition probability matrix:
(3.10-18)
r 2 - - k i C | C 2
^5 = ^\^\^3 (3.10-18a)
P =
1 l-[kiC2(n)
+k2C3(n)]At
0
0
0
0
2 3 4 5
0 ikiC2(n)]At ^[kiC2(n) ik2C3(n)]At
+k2C3(n)]At
l-kiCi(n)]At ikiC,(n)]At i.kjCi(n)]At Q
0 l-k2Ci(n)]At ^k^c,(n)At lk2Ci(n)At
0
0
1
0
0
1
(3.10-18b)
276
The transient response of Ci to C5 for the initial state vectors C(0) = [Ci(0),
C2(0), C3(0), C4(0), C5(0)] = [1, 1, 0, 0, 0] and [1, 0.5, 0, 0, 0] is depicted in
Fig.3.10-18 where the effect of C2(0) is demonstrated.
Fig.3.10-18. Ci versus t demonstrating the effect of C2(0) for ki = k2 = 5
For At = 0.01, the agreement between the Markov chain solution and the
exact solution [32, vol.2, p.61; 47] is Dmax = 0.7% and Dmean = 0.3%. It should
be noted that an exact solution is available only for Ci(0) = 2C2(0) in the first
reference where in the other one it is a comphcated solution.
3,10-19
where
Ai + A2 ^ A3 + A5
Ai + A3 _> A4 + A6 (3J0-19)
Ti - •" \CiC2 - k2C|C3 r2 - - kiC|C2
3 = '^iCiC2 - k2CiC3 r4 = r = k2CiC3
r5 = kiCiC2 (3.10-19a)
yields the following transition probability matrix:
p =
277
1 2 3 4 5 6
P n 0 •jkiC2(n)At lk2C3(n)]At lkiC2(n)At lk2C3(n)]At
0 l-kiCi(n)]At ikiCi(n)At ^kiCi(n)At
l-k2Ci(n)]At ik2Ci(n)At o ~-k2Ci(n)At
0
0
0
0
0
0
0 1 0 0
0 0 1 0
0 0 0 1
(3.10-1%) where Pn = l-[kiC2(n) + k2C3(n)]At
The transient response of Ci to C^ is depicted in Fig.3.10-19 where the
effect of C2(0) is demonstrated for ki = 2 and k2 = 1.
Fig.3.10-19. Ci versus t demonstrating the effect of CiCO)
For At = 0.01, the agreement between the Markov chain solution and the
exact solution [32, vol.2, p.65] is Dmax = 10% and Dmean = 0.7%.
3.10-20 Ai + A2 -» A3 + A6
Ai + A3 -^ A4 + A6
Ai + A4 -> A5 + A6 (3.10-20)
where
278
rj - - kiCiC2 - k2CiC3 - k3CiC4 ^2-^ k]C|C2
3 ~ 1^1^102 - k2C|C3 14 -- k2C2C3 — k3C|C4
1 15 = k3CiC4 16 = 2-(kiCiC2 + k2CiC3 + k3CiC4)
yields the following transition probability matrix:
(3.10-20a)
P =
1
2
3
4
5
6
1 2 3 4 5 6
Pll 0 YkiC2(n)At ik2C3(n)]At ik3C4(n)]At p^^
0 l-kiCi(n)]At ikiCi(n)At 0 0 ^kjCiWAt
0 0 l-k2Ci(n)]At ik2C,(n)At 0 |k2Ci(n)At
l-k3Ci(n)]At ik3Ci(n)At lk3Ci(n)At 0
0
0
0
0
0
0
0
where Pll = l-[kiC2(n) + k2C3(n) + k3C4(n)]At
P16 = y[kiC2(n) + k2C3(n) + k3C4(n)]At
(3.10-20b)
(3.10-20C)
The transient response of Ci to C6 for an initial state vector C(0) = [Ci(0),
C2(0), C3(0), C4(0), C5(0), C6(0)] = [3, 1, 0, 0, 0, 0] is depicted in Fig.3.10-20
where the effect of C2(0) is demonstrated.
0.6 0
Fig.3.10-20. Ci versus t demonstrating the effect of k2 for ki = k3 = 10
279
For At = 0.0025, the agreement between the Markov chain solution and the
exact solution [32, voL2, p.66; 45; 46] is Dmax = 3.6% and Dmean = 2.6%. It
should be noted that an exact solution is available only for Ci(0) = 3C2(0).
3>10-21 Ai + A2 -^ A3
A3 + A2 -> A4
A4 + A5 -^ A6
where [32, vol.2, p.66]
r^ = - kiCiC2 1*2 ~ ~ kiCiC2 ~ k2C2C3
^3 == k |CiC2 - ^^2^1^^
yields the following transition probability matrix:
(3.10-21)
(3.10-21a)
P =
1
2
3
4
5
6
1 l"kiC2(n)]At
0
0
0
0
0
2
0
P22
0
0
0
0
3 •ikiC2(n)At
ikjCiWAt
l-k2C2(n)]At
0
0
0
4
0
ik2C3(n)]At
ik2C2(n)At
l-k3C5(n)At
0
0
5
0
0
0
0
l-k3C4(n)]At
0
6
0
0
0
k3C5(n)At
k3C4(n)lAt
1
(3.10-21b)
where
P22 = l-[kiCi(n) + k2C3(n)]At (3.10-21C)
The transient response of Ci to C6 for an initial state vector C(0) = [Ci(0),
C2(0), C3(0), C4(0), C5(0), C6(0)] = [1, 1, 0, 0, 1, 0] is depicted in Fig.3.10-21
where the effect of k2 is demonstrated.
280
u
1
0.8
U.6
04
0.2
0
I - 1 1 \ \ \ N
\ . - - ^ 3
1
_ 5
6
-j
k =10 2 J
L.
\
^
[--.-
- - - i^_^ 1
V, 2 •---^.__
^ ^ ^ ^ - 1 ^ . - -^ ••- ^ u ~
1 ^5
3
_''~~rrt
-J
k =1 2
— _ 0.5 1.5 2 0 0.5 1.5
Fig.3.10-21. Ci versus t demonstrating the effect of k2 for ki = k3 = 10 and At = 0.01
3,10-22
where
Ai-^A2 + A3 k2
A2 + A3 -4 A6
A3 + A4 ^ A5 + A6
A2 + A5 -^ A6
Tj = - k^Cj 12 = k |Ci - k2C2C3 -- k4C2C5
13 = k |Ci - k2C2C3 - k3C3C4 14= - k3C3C4
^5 "= '^3C3C4 - k4C2C5
^6 "= ^':^'^z + ^30304 + k4C2C5
yields the following transition probability matrix:
(3.10-22)
(3.10-22a)
281
P =
1
2
3
4
5
6
1 l-kjAt
0
0
0
0
0
2 kjAt
P22
0
0
0
0
3 kjAt
0
P33
0
0
0
4
0
0
0
l-k3C3(n)At
0
0
5
0
0
lk3C4(n)At
yk3C3(n)At
l-k4C2(n)At
0
6
0
|[k2C3(n)
+k4C5(n)]At
^[k2C2(n)
+k3C4(n)]At
ik3C3(n)At
•i.k4C2(n)At
1 (3.10-22b)
where P22 = 1 - [k2C3(n) + k4C5(n)]At
P33 = 1 - [k2C2(n) + k3C4(n)]At (3.10-22c)
The transient response of Ci to C6 for the initial state vectors C(0) = [Ci(0),
C2(0), C3(0), C4(0), C5(0), C6(0)] = [1, 0, 0, 1, 0, 0] and [1, 0, 0, 1, 1, 0] is
depicted in Fig.3.10-22 where the effect of C5(0) is demonstrated.
2
1.5
0.5
0
i = 6-
^ \ l \ '
C (0) = 0 '
- '
^ - - - ~ - ^
—
0 1 3 0 t t
Fig.3.10-22. Ci versus t demonstrating the effect of CsCO)
for ki = 5 (i = 1, ..., 5) and At = 0.02
An exact solution is available [31, p. 80] only for extreme conditions.
282
3.10-23 Ai + A2 -^ A3+ A6
Ai + A3 ^ A4+ A7 k3
Ai + A2 -4 A5 + A7
k4 Ai + A5 -^ A4 + A6
where [32, vol.2, p.67; 40]
tj = - (kj + k3)C2 - kjCj - k4C5 T2= - (ki + k3)C2
13 = kjC2 - k2C3 r4 = k2C3 + k4C5
^5 ~ '^3^2" ^ 4 5 Tg = kiC2 + k4C5 r-j = k2C3 + k3C2
yields the following transition probability matrix:
(3.10-23)
(3.10-23a)
P =
where
1
2
3
4
5
6
7
1
Pii
0
0
0
0
0
0
(ki
2
0
1 -+ k3)At
0
0
0
0
0
3
Pl3
| k , A t
l-kjAt
0
0
0
0
C,(n)
4
Pl4
0
ik^At
1
Ik^At
0
0
C.(n)
5
Pl5
ik3At
0
0
l-k4At
0
0
, Cc(n)-
6
PI6
lk ,At
0
0
^ M t
1
0
7
Pl7
ik3At
l k , A t
0
0
0
1
(3.10-23b)
K C '2(n) .
ir C2(n) CsCii)-] ![- C,(n) C2(n)l» .0 1m^^^ P'^ = IL* ! C > ) ^ •' CM^ P'^ == iL'^^ C > ) ^ "3 c l ^ j A t (3.10-23d)
283
The transient response of Ci to C7 for the initial state vector C(0) = [Ci(0),
C2(0), ..., C7(0)] = [1, 0.5, 0, 0, 0, 0, 0] is depicted in Fig.3.10-23 where the
effect of ki is demonstrated.
2.5 0
Fig.3.10-23. Ci versus t demonstrating the effect of ki
for ii2 = k4 = land ka = 2
For At = 0.0025, the agreement between the Markov chain solution and the
exact solution [32, vol.2, p.67; 40] is Dmax = 4.9% and Dmean = 2.4%. It should
be noted that in the exact solution, x should be replaced by t.
3,10-24 ^ A . > l .Ag
^-Aio >^A 12" M3
(3.10-24)
where
284
ri = - ki2Ci r2 = - (k23 + k24 + k25)C2 + ki2Ci
13 = - (k36 + k37 + k3g)C3 + k23C2
14 = — (k45 + k47 + k4g)C4 + k24C2
5 ~ ~ ( 56 + ^ 57 " 1^58)^5 + k25C2
6 = " ( 69 + ^^,10 " ^511)05 + k35C3 + k46C4 + k56C5
Ty = - (k79 + k-jiQ + k7j i )C7 + k37C3 + k47C4 + k57C5
ig = - (kgQ + kg 10 + kg i i)Cg + k3gC3 + k4gC4 + k5gC5
TQ = - k9 12C9 + k59C5 + k79C7 + kg9Cg
^ 10 = - ^10,12^10 "•• 6,10^6 + ^7 10C7 + kgjoCg
^11="" ^11,12^11 + ^M^6 "•• k7,iiC7 + kg i iCg
^12 = "" 12,13^12"*" ^9,12^9"*" ^10,12^10 + k ^ 12^11
^ 13 = ^12,13^12
yields the following transition probability matrix:
P =
(3.10-24a)
1 2 3 4 5 6 7 8 9 10 11 12 13
1 P i i
0 0 0 0 0 0 0 0 0 0 0 0
2 P12
P22
0 0 0 0 0 0 0 0 0 0 0
3 0
P23
P33
0 0 0 0 0 0 0 0 0 0
4 0
P24
0 P44
0 0 0 0 0 0 0 0 0
5 0
P25
0 0
P55
0 0 0 0 0 0 0 0
6 0 0
P36
P46
P56
P66
0 0 0 0 0 0 0
7 0 0
P37
P47
P57
0 P77
0 0 0 0 0 0
8 0 0
P38
P48
P58
0 0
Pss 0 0 0 0 0
9 0 0 0 0 0
P69
P79
P89
P99
0 0 0 0
10 0 0 0 0 0
P6,10
P7.10
P8,10
0 PlO.lO
0 0 0
11 0 0 0 0 0
P6,ll
P7,ll
P8,ll
6 0
P u . i i
0 0
12 0 0 0 0 0 0 0 0
P9,12
PlO.lO
P11.12
P12.12
0
13 0 0 0 0 0 0 0 0 0 0 0
Pl2,13
1
where
pll = l-ki2At; pi2 = ki2At
P22 = 1 - (k23+k24 + k25)At p23 = k23At p24 = k24At P25
P33 = 1 - (k36 + k37 + k38)At p36 = k36At P37 = ksTAt P38
I
(3.10-24b)
= k25At
= k38At
285
P44 = 1 - (k46 + k47 + k48)At p46 = k46At P47 = k47At p48 = k48At
P55 = 1 - (k56 + k57 + k58)At p56 = ksgAt P57 = ks-jAt psg = ksgAt
P66 = 1 - (k69 + k6,10 + k6,l l)At P69 = k69At p6,10 = ke.loAt
P6,ll='f6,llAt
P77 = 1 - (k79 + k7,io + k7,i i)At P79 = k79At P7jo = k7,10At
P7, i i=k7, i lAt
P88 = 1 - (k89 + k8,lO + k8,ll)At p89 = k89At p8,io = k8,ioAt
P8, l l=k8, l lAt
P99 = 1 - k9,i2At p9,i2 = k9,i2At; pio,10 = 1 - kio,12At pio,12 = kio,12At
Pll.ll = 1 -kii,i2At Pii,i2 = kii,i2At
P12,12 = 1 - ki2,i3At pi2,i3 = ki2,l3At (3.10-24c)
The transient response of Ci to C13 C(0) = [Ci(0), C2(0),..., Ci3(0)] = [1,
0, 0, ..., 0] is depicted in Fig.3.10-24.
0.8
0.6 H
0.4 h
0.2
T
|1
\ \
f K / \ ^
V 3^V <^v/
1
y ^
/ /
/ l'^
/
/
. ' 6 / /
/ / -,.5
-. 4
' J~^
1:C 1
2:C^ 3:C =C =C
3 4 5
4:C =C =C 6 7 8
5:C =C =C 9 10 11 6:C
12 7:C
13
- — i
—
—
-
0.5 1.5
Fig.3.10-24. Ci versus t for k n = 15, ky = 5 and At = 0.01
A complicated exact solution is available [22, p.59].
286
3.10-25 A + B _>AB k2
AB + B _> AB2 k3
AB2 + B -> AB3
ABz-i+B ^ A B z
where [41]
B = ~ I^JCACB - k2CABCB ~ ^30^1^5, 3 - kvC Z'^ABZ_,CB
(3.10-25)
l AB - • 'ICA^B - k2CABCB rAB2 " '^2CABCB " k3CAB2CB
^ABj = k3CAB2CB - k4CAB3CB
I'AB^,, - ^Z-ICAB^.I^B - kzCAB2_,^B ABz " kzCAB2_,CB (3- 10-25a)
yields the following transition probability matrix for Z = 4:
A B AB
P =
A
B
AB
AB2
AB3
AB4
AB2
k,c'(„)At « i' ' B " ^ «
AB3
0
AB4
0
0 PBB |k,CA(n)At ik2CAB(n)At IkjCAB/n)^ ik4CAB/n)At
0 0
0 0
^ lk2CB(n)At 0 k2CB(n)At
0 0 0 l-k3CB(n)]At i.k3CB(n)]At
0 l-k4CB(n)]At l.k4CB(n)]At
(3.10-25b)
287
where PBB = l-tkiC^di) + k2CAB(n) + 1^30^3/") + k4CAB/n)]At (3.10-25c)
The transient response of A to AB4 is depicted in Fig.3.10-25 where the effect of CA(0) is demonstrated.
1.5
u"" 1
0.5
0
1 ' C (0)=1 ' \
\ _ B
\ - ^ V " '' <:: - ^ A A B •~'~ -- _ V ^ . - - ^ ' - ' - - A B ^
^ ' ' . . . . . ,3 -L ' ^ ^ - ~ - ~ - * - . - 1 - - ,- -
;
AB / 4
1 "~~ '"^i'-^i^
.. 1 • ' 1
\ - ' - ~
1
r BV A
C (0) = 2 ' A
AB
AB AB AB
^ : • : : : ' : , : ' : •
-
-
0 0.5 1 1.5 0 0.5 1 t t
Fig.3.10-25. Ci versus t demonstrating the effect of CA(0) for ki = 10, ki = 5 (i = 1, 2, 3) and At = 0.005
A complicated exact solution is available [41].
3.11 PARALLEL REACTIONS: SINGLE AND CONSECUTIVE REVERSIBLE REACTION STEPS
1.5
3.11-1
> ^ 1
(3.11-1)
where
rj = - (kj + k2)Ci + k_iC2 r2 = kjC] - k_iC2
yields the following transition probability matrix:
(3.11-la)
288
P =
1
2
3
1 - (kj + k2)At
k_iAt
0
kjAt
1 - k_iAt
0
k2At
0
1 (3.11-lb)
The transient response of Ci, C2 and C3 for the initial state vector C(0) =
[Ci(0), C2(0), C3(0)] = [1,0,0] is depicted in Fig.3.11-1 where the effect of k.i is
demonstrated.
1
0.8
0.6
0.4
0.2
0
u
\i =
r
:1
/ /
1
" - 2
I
1 k =
-1
1
= 4 -
1 •
:i_jr-—-___
0 0.5 1.5 2 0
Fig.3.11-1. Ci versus t demonstrating the effect of k-i
for ki = 5 and k2 = 3
For At = 0.005, the agreement between the Markov chain solution and the
exact solution [33, p.73] is Dmax = 1-5% and Dmean = 0.9%.
3.11-2 k, ^ A ,
Ai
(3.11-2)
where
ri = 12 = - (k, + k2)CiC2 + k_iC3 rg = kiCjCj - k_iC3
r4 = k2CiC2 (3.11-2a)
289
yields the following transition probability matrix:
1
2
3
4
1 1-
(ki+k2)C2(n)At
0
k_iAt
0
2
0
1-(ki+k2)Ci(n)At
k.jAt
0
3 4 ikiC2(n)At lk2C2(n)At
ikiCi(n)At •i-k2Ci(n)At
l - k . j A t 0
0 1 (3.11-2b)
The transient response of Ci to C4 for the initial state vector C(0) = [Ci(0),
C2(0), C3(0), C4(0)] = [1, 1, 0, 0] is depicted in Fig.3.11-2 where the effect of
C2(0) is demonstrated.
u
1
0.8
0.6
0.4
0.2
0
' k =4 -1
fc =C - " '
rt-K y \ ^
/ S .
F e- r ' "'" "—•— 3 ^ - ^
1
4 -\
_
-
u
t 1
0.8
0.6
0.4
0.2
0
6 0
' k =1 ' -1
ic =c
1 1 1
- ' "c 4
• - « = = : — _ _ _ _ _ _
\c =c
\K'" f\^_^
' k =0 ' 1 -1
c 3 . . . _ i
c 4 —
T ~ • — 1
Fig.3.11-2. Ci versus t demonstrating tlie effect of k-i for ki = 5, k2 = 3 and At = 0.005
290
3.11-3
where
Jf" -2
kl
ri = - (kj + k2)C, + k_iC2 + k_2C3
r2 = kjCi - k-iC2 r3 = k2Ci-k_2C3
yields the following transition probability matrix:
(3.11-3)
(3.11-3a)
p =
1
2
3
1 l-(ki+ k2)At
k_iAt
k_2At
2 kiAt
l-k_jAt
0
3 k2At
0
l-k_2At (3.11-3b)
The transient response of Ci to C3 for the initial state vector C(0) = [Ci(0),
C2(0), C3(0)] = [1, 0, 0] is depicted in Fig.3.11-3 where the effect of ki and k-i is
demonstrated.
u
1
0.8
0.6
0.4
0.2
0
L A 1 - 1 I
I \ I
0 0.1 0.2 0.3 t
\ ' -\
\ 1 [- ^
r c >
Ks_
\.^
1 1 1 k =l,k =5 1 -1
'^~z:zr-:=~rz=~
~\ 1 1
-H
H
"1
— -
0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 t
291
u
1
0.8
0.6
0.4
0.2
«.
\ 1
_ \
1 3
— 1 —
1 0.2
1 1 i 1 k = 5 , k = 1
1 -1 1
^ ^ ^ 2
c — ^
1 1 1 0.4 0.6 0.8 ]
t
Fig.3.11-3. Ci versus t demonstrating the effect of ki and k-i
for k2 = k-2 = 5
For At = 0.01, the agreement between the Markov chain solution and the
exact solution [54, p. 140] is Dmax = 3.8% and Dmean = 1.3%.
3.11-4
A1+A2. '
-2
where [35, p. 149] rj = rj = - (ki + k2)CiC2 + k_,C3 + k_2C4
r3 = kiCjC2 - k_jC3 r4 = k2CjC2 - k_2C4
(3.11-4)
(3.11-4a)
yields the following transition probability matrix:
P =
1
2
3
4
1 1-
(ki+k2)C2(n)At
0
k_iAt
k_2At
2
0
1-(ki+k2)Ci(n)At
k_iAt
k_2At
3 lkiC2(n)At
lkiCi(n)At
1 - k_iAt
0
4 lk2C2(n)At
ik2Ci(n)At
0
1 - k_2At (3.11-4b)
292
The transient response of Ci to C4 for the initial state vector C(0) = [Ci(0),
C2(0), C3(0), C4(0)] = [1, 1, 0, 0] is depicted in Fig.3.11-4 where the effect of ki
and k-i is demonstrated.
1
0.8
0.6
0.4
0.2
0
u
y 1
h- X. ^ ^^
V 1
1 1 1 1 k =k =5
1 -1
C =C H
-
C =C
3 4
i 1 1 0 0.1
\ 1
L " - -k
1 /
U - r
1 = l ,k =
-1
c =c 1 2
c 4
c 3 1
1 5
1
~^_
0.4 0.5
Fig.3.11-4. Ci versus t demonstrating the effect of ki and k-i
for k2 = k-2 = 5 and At = 0.01
3.11-5
(3.11-5)
293
where [35, p.lOl]
rj = - (kj + kj + k3)Ci + k_,C2 + k_2C3 + k_3C4
rj = kjCi - k_jC2 r3 = k2Ci-k_2C3 14 = k3Ci - k_3C4 (3.11-5a)
yields the following transition probability matrix:
p =
1
2
3
4
1 1-
(ki+k2+k3)At
k_iAt
k^2^t
k_3At
2 kjAt
1 - k.jAt
0
0
3 k2At
0
1 ~ k_2At
0
4 k3At
0
0
1 - k_3At (3.11-5b)
The transient response of Ci to C4 for the initial state vector C(0) = [Ci(0),
C2(0), C3(0), C4(0)] = [1,0, 0, 0] is depicted in Fig.3.11-5 where the effect of k-i
is demonstrated.
0.5 1 1.5 0 0.5 t t
Fig.3.11-5. Ci versus t demonstrating the effect of k-i
for ki = k-3 = 5, k2 = k-2 = ks = 1 and At = 0.01
1.5
3.11-6 Ai-^A2
-^ 2 A i - ' A 3 (3.11-6)
294
where:
Ti = - (kjCi + 2k2Cf) + k^jCj r2 = k,Ci
r3 — 2k2Ci - k_2C3
yielding the following transition probability matrix:
(3.11-6a)
1
P = 2
3
1 2 3 l-[ki + 2k2Ci(n)]At kjAt 2k2C,(n)At
0 1 0 k_2At 0 l-k_2At (3.11-6b)
where pi3 was computed by Eq.(3-10a). The transient response of Ci, C2 and C3
for the initial state vector C(0) = [Ci(0), C2(0), €3(0)] = [1, 0, 0] is depicted in
Fig.3.11-6 where the effect of k-2 is demonstrated.
Fig.3.11-6. Ci versus t demonstrating the effect of k-2
for ki = k2 = 5 and At = 0.01
3.11-7
where [35, p.3]
A l ^ A 2
k., k2
Ai + A2 ^ A3
K (3.11-7)
295
rj = - (kjCi + kjCjCj) + k_,C2 + k_2C3
rj = - (kjCjCj + k_,C2) + k]Ci + k_2C3
yields the following transition probability matrix:
(3.11-7a)
1
1
P = 2
3
1, l-[ki+k2C2(n)]At kiAt ^k2C2(n)At
k_iAt l-[k_i+k2Ci(n)]At lk2Ci(n)At
k_2At k_2At l-k_2At (3.11-7b)
The transient response of Ci, C2 and C3 for the initial state vector C(0) =
[Ci(0), C2(0), C3(0)] = [1,0, 0] is depicted in Fig.3.11-7 where the effect of k-2 is
demonstrated for ki = k-i = k2 = 5 and At = 0.01.
- \ , \
_ 2
(^ 1
1 1 k =0.5
-2
1 1
1
1 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8
t t Fig.3.11-7. Ci versus t demonstrating the effect of k-2
3,11-8
where [28]
Ai + A2 A3 A3 + A2 -^ A4 (3.11-8)
296
12 = - (kjCiCz + kjCzCj) + k_,C3
rj = kiCjCj - (k_iC3 + kjCjCj) 14 = kjCjCj
yields the following transition probability matrix:
(3.11-8a)
P =
1 l-kiC2(n)At
0
k_,At
2
0 lk,C2(n)At
4
0
[k,C,(n)+k,C,(n)]At 2 ' ' '^ '^"^^ '
k_iAt
0
1 -
LK_IH~K2V_'2'
^k2C3(n)At
, M. |k3C2(n)At (n)]At 2 ^ ^
0 1 (3.11-8b)
The transient response of Ci to C4 for the initial state vector C(0) = [Ci(0), C2(0),
C3(0), C4(0)] = [1, 1, 0, 0] is depicted in Fig.3.11-8 where the effect of k-i is
demonstrated.
1
0.8
0.6
0.4
0.2
^0 0.5 1 1.5 0 0.5 1 t t
Fig.3.11-8. Ci versus t demonstrating the effect of k-i
for ki = k2 = 5 and At = 0.01
u
[ 1 1
2 • —
1 J-^ - - " " "
-\
-
297
3.11-9 Ai + A2 A3 + A4
Ai + A5 " A3 + A6
where Tj — — (kjC|C2 + '^2^1^5-' ''" '^-1^3^4 ''" *^2^3^6
r2 = — kjC-jC-2 " k_jC3C4
r3 = (k_iC3C4+ k_2C3C6) + k , C , C 2 + kjCjCs
14 = K|C|C2 ~ k_jC3C4
15 = k2C|C5 - k_2C3C5
yields the following transition probability matrix:
1
P =
1-
[kjCjCn) +
k2C5(n)]At
0 l-kiCi(n)At ikiCi(n)At i-k,Ci(n)At 0
(3.11-9)
(3.11-9a)
2 3 4 5 6 0 ^{^iC^in) ykiC2(n)At 0 Y^2C5(n)At
+k2C5(n)]At
P31 ik_iC4(n)At l-[k_iC4(n) ^ ik_2C6(n)At Q +k_2C6(n)]At
P41 P41 0 l-k_iC3(n)At 0 0
0 0 |k2Ci(n)At 0 l-k2Ci(n)At ik2Ci(n)At
0 0 ~k^2C3(n)At l-k_2C3(n)At
(3.11-9b)
U-k_2C3(n)At 0
where P31 = y[k_iC4(n) + k_2C6(n)]At; P41 = ik_iC3(n)At (3.11-9C)
298
The transient response of Ci to Ce for the initial state vector C(0) = [Ci(0),
C2(0), C3(0), C4(0), C5(0), C6(0)] = [1, 1, 0, 0, 1, 0] is depicted in Fig.3.11-9
where the effect of ki is demonstrated.
c\ r ' \ L c x r 3
/
r/ ; . - '
1 1 c c • ...
c 6
' 'c 4
1 1
1 1 k =1
1
-rrr-r-:. - ^
1 1 0 0.2 0.4 0.6 0.8
t Fig.3.11-9. Cj versus t demonstrating the effect of ki
for k-i = k2 = k-2 = 2 and At = 0.005
An extremely complicated exact solution is available [32, vol. 2, p.49].
3.11-10 Ai + A 2 ^ A 3
where
Ai + A 4 ^ A 5
A3 + A4 ^ A 6
ri = - (kjCiCj + k2CiC4) + k_]C3
^2 ^ — '^1^1^2 " — 1 3
r3 = - (k_iC3 + k3C3C4) + k,C,C2
14 = — vk2CjC4 + k3C3C4)
15 = k2CiC4 r = k3C3C4
yields the following transition probability matrix:
(3.11-10)
(3.11-lOa)
299
P =
1 2 3 4 5 6 l-[kiC2(n)+ Q lkiC2(n)At o ^k2C4(n)At o |k2C4(n)]At
0 l-kiCi(n)At ikiCi(n)At
k_iAt k_iAt l-[k_i+ k3C4(n)]At
0 •i.k3C4(n)At
0
0
0
0
0 l-[k2C,(n)+ lk2Ci(n)At lk3C3(n)At k3C3(n)]At
0 0
0 0
(3.11-lOb)
The transient response of Ci to C^ for the initial state vectors C(0) = [Ci(0),
C2(0), C3(0), C4(0), C5(0), C6(0)] = [1, 1, 0, 1, 0, 0] and [1, 1, 0, 0, 0, 0] is
depicted in Fig.3.11-10 where the effect of €4(0) is demonstrated.
1
0.8
0.6
0.4
0.2
^0 0.5 1 1.5 0 0.5 1 t t
Fig.3.11-10. Ci versus t demonstrating the effect of €4(0)
for ki = k.i = k2 = k3 = 5 and At = 0.01
u
U I \
l \ ' \ l ' -V \ '
— \ ^ '\ '*'
k^.> L^
1
5 .
v 4
1
C(0) = 4
, - -
• ' • • ^ ~
-
1 '
" - • •
6 ^ ^ - -
~~ l~"
^
'
—
— -r=r-~::::r-.-~
An exact solution is available [32, vol. 2, p.75] only for limiting cases.
300
3,11-11
where
rJ = — (k|2 + k|3 + k |4)C| + k2iC2 + ^3^03 H- K41C4
12 = - (k2i + k23 + k24)C2 + ki2Ci + k32C3 + k42C4
13 = - {k^i + k32 + k34)C3 + ki3Ci H- k23C2 + k43C4
14 = - (k4i H- k42 H- k43)C4 + ^i^Ci + k24C2 + k34C3
yields the following transition probability matrix:
(3.11-11)
(3.11-lla)
1 1-
[ki2+ki3+ki4]At
k2iAt
k3iAt
k4iAt
2 ki2At
1-[k2i+k23+k24]At
k32At
k42At
3 ki3At
k23At
1-[k3i+k32+k34]At
k43At [k4
4 ki4At
k24At
k34At
1-l+k42+k43]At
1
2 P =
3
I LK4i+K42+K43ja
(3.11-llb)
The transient response of Ci to C4 for the initial state vector C(0) = [Ci(0),
C2(0), C3(0), C4(0)] = [1, 0, 0, 0] is depicted in Fig.3.11-11 where the effect of
ki2 is demonstrated.
301
u
1
0.8
0.6
0.4
0.2
0 "(
K ^ 1 \ k =1 \ ^
z^" """C^C =C y^ 2 3 4
X 1 1 3 0.5 1 t
r
-4-
A 1.5 0
1 1 k =10
12
--^c^
c =c 3 4
1 i 0.5 1
t
-
1
Fig.3.11-11. Ci versus t demonstrating the effect of ki2 for kij = 1 (U 5t 12) and At = 0.01
An exact solution is available [31, p. 172; 52].
3.12 CHAIN REACTIONS
3.12-1 ks.
where
"kH^ 3
\K,
ri = - (ki + k2)Ci
rj = - (k3 + k4)C2+ kiCj rj = - (kj + k6)C3+ kjCj
14 = k3C2 Tj = k4C2 Tg = k5C3 Tj = k6C3
(3.12-1)
(3.12-la)
yields the following transition probability matrix:
302
p =
1
2
3
4
5
6
7
1 l-[ki+k2]At
0
0
0
0
0
0
2 kjAt
l-[k3+k4]At
0
0
0
0
0
3 k2At
0
l-[k5+k6]At
0
0
0
0
4
0
k At
0
1
0
0
0
5
0
k4At
0
0
1
0
0
6
0
0
ksAt
0
0
1
0
7
0
0
k^At
0
0
0
1
(3.12-lb) The transient response of Ci to C7 for the initial state vector C(0) = [Ci(0),
C2(0),..., C7(0)] = [1, 0,..., 0] is depicted in Fig.3.12-1 where the effect of ki is demonstrated.
i 1
V 1
k =10 1
7 6
5
4
H
. , -1
0.5 1 1.5 0 0.5 1 1.5 t t
Fig.3.12-1. Ci versus t demonstrating the effect of ki for k2 = 2, k3 = 3, k4 = 4, ks = 6 and At = 0.005
An exact solution is available [22, p.52].
303
3.12-2 ^ ^' }
f vL vL WW W W W
g A9 Alo All A12 Ai3 Ai4 Ai5 (3.12-2)
where
r i = - ( k i 2 + ki3)Ci
2 = - (k24 + k25)C2 + ki2Ci
14 = — (k4g + k49)C4 + k24C2
6 = "" (k^ 12 + k i3)C6 + k36C3
8 k4gC4 Tg = k49C4
42 - k6,12C6 "13 = k< 6,13^6
13 - - (k36 + k37)C3 + ki3Ci
5 = ~ (k5 10 + k5 ii)C5 H- k25C2
17 = - (k7 14 + k7 i5)C7 + k37C3
^io = k5 10C5 rii = k5 11C5
ri4 = k7,i4C7 ri5 = k7,i5C7 (3.12.2a)
yields the following transition probability matrix:
P =
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 Pii 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2 P12
P22 0 0 0 0 0 0 0 0 0 0 0 0 0
3 Pl3 0 P33 0 0 0 0 0 0 0 0 0 0 0 0
4 0 P24 0 P44 0 0 0 0 0 0 0 0 0 0 0
5 0 P25 0 0 P55 0 0 0 0 0 0 0 0 0 0
6 0 0 P36 0 0 P66 0 0 0 0 0 0 0 0 0
7 0 0 P37 0 0 0 P77 0 0 0 0 0 0 0 0
8 0 0 0 P48 0 0 0 1
0 0 0 0 0 0 0
9 0 0 0 P49 0 0 0 0 1
0 0 0 0 0 0
10 0 0 0 0
P5,10
0 0 0 0 1
0 0 0 0 0
11 0 0 0 0
P5,ll 0 0 0 0 0 1
0 0 0 0
12 0 0 0 0 0
P6,12
0 0 0 0 0 1
0 0 0
13 0 0 0 0 0
P6,13
0 0 0 0 0 0 1
0 0
14 0 0 0 0 0 0
P7,14
0 0 0 0 0 0 1
0
15 0 0 0 0 0 0
P7,15
0 0 0 0 0 0 0 1
(3.12-2b)
where
Pll = l - ( k l 2 + ki3)At pi2 = ki2At pi3 = ki3At
P22 = 1 - (k24 + k25)At P24 = k24At P25 = k25At
P33 = 1 - (k36 + k37)At P36 = k36At P37 = ks-jAt
304
P44 = 1 - (k48 + k49)At P48 = k^gAt P49 = k49At
P55 = 1 - (k5,io + k5ji)At p5,io = k5,ioAt p5,ii = ks^nAt
P66 = 1 - (k6,12 + k6,13)At P6,12 = k6,12At P6,13 = k6,13At
P77 = 1 - (k7,i4 + k7j5)At P7,i4 = k7,i4At P7,i5 = k7,i5At (3.12-2C)
The transient response of Ci to C15 for the initial state vector C(0) = [Ci(0),
C2(0),..., Ci5(0)] = [1, 0, ..., 0] is depicted in Fig.3.12-2 where the effect of ki3
is demonstrated.
0.8 k \ l
0.6
U
0.4
0.2
\r'
6:C =C =C =C 9 10 11
7:C =C =C =C 12 13 14 V.
0.5 1.5
305
0.8 h \ l
0.6
U
0.4
0.2
^13 = 2
1:C 1
2:C =C 2 3
3:C =C =C =C 4 5 6 7
4:C =C =C =C = 8 9 10 11
c =c =c =c 12 13 14 15
0.5 1.5
Fig.3.12-2. Ci versus t demonstrating the effect of ki3 for ki2 = 2,
kij = l(ij ^ 12,13) and At = 0.01
An exact solution is available [22, p.70].
3.13 OSCILLATING REACTIONS [55-69]
3 .13-1 (Ai)gas -> A i ki
Ai-->A2
A2-^A3 (3.13-1)
where [55] (Ai)gas denotes a saturated vapor of gas A] in equilibrium with its
condensed phase containing species Ai, A2 and A3. It is assumed that equilibrium
between the phases is established immediately and that the condensed phase is
perfectly mixed so that diffusion effects are negligible. H is the rate of supply of Ai
in moles/sec from the vapor phase (Ai)gas into the condensed phase. The
governing equations are:
306
rj = H - kC, = H - kjCjC,
2 ^ ^^1 ~" ^^2^2 ~ '^1^2^1 "~ ^2^2
13 = k2C2 (3.13-la)
The fact that k = kiC2 indicates species A2 influence autocatalytically its own rate of
formation. The above equations yield the following transition probability matrix:
P =
1 2 3
1 - [kiC2(n)]At + [H/Ci(n)]At kiC2(n)At 0
0 1 - k2At k2At
0 0 1 (3.13-lb)
The term [H/Ci(n)]At, where H is a constant supply rate of Ai, must be added in
pil in order to comply with the integrated form of ri in Eq.(3.13-la).
The transient response of Ci, Ci exact and C2 is depicted in Fig.3.11-3 where
the effect of ki and H is demonstrated. The initial state vector is C(0) = [Ci(0),
C2(0)] = [0.5, 1].
50 100 t
200 400 600 800 1000 lOOOOt
307
1.5
U
0.5 k
A , i.'
c 1,exact
K /^\
1
C 2
1
I 1
k =2, H = 0.5 1
(c)
1 1 10 20
t 30 40
Fig.3.13-1. Ci versus t demonstrating the effect of ki and H
for k2 = 3
An exact solution is available [55]. However, it is restricted to relatively
large values of t. Indeed, as observed in case (a), the exact solution does provide
reasonable results for small t, i.e. Ci < 0. In addition, the agreement between the
Markov chain solution and the exact solution depends on the parameters ki, k2 and
H as shown in Fig.3.13-la,b,c for At = 0.08, 0.00001 and 0.02, respectively. It
should be noted that oscillations occur when H < 4k2^/ki as observed in Fig.3.13-
la,c. A non-oscillatory behavior occurs when H > 4k2^/ki as shown in Fig.3.13-
Ib. This has been obtained by varying ki from 0.1 to 400.
3 > 1 3 - 2 Ai + A2 -> A3 + A4
A3 + A2 -^ 2A4
Ai + A3->2A3 + A5
2A3 -^ Ai + A6
A5^fA2 (3.13-2)
where for the above reaction, known as the Belousov-Zhabotinski reaction [59],
308
T2 = - kjCiCj - k2C2C3 + fkjCs
r3 = kjCiCz - k2C2C3 + k3CiC3 - 2k4C3
r5 = k3CiC3-k5C5 r6 = k4C? (3.13-2a)
For a detailed derivation, the attention of the reader is addressed to case 5.2-1(1).
The following transition probability matrix is obtained:
1 l-[kiC2(n)+
Lc3(n)]At
0
k4C3(n)At
0
0
0
2
0
l-[kiCi(n)+
k2C3(n)]At
0
0
fkjAt
0
3 [i-kiC2(n)+
2k3C3(n)]At
ikiCi(n)At
l-[k2C2(n)+
k3Ci(n)+
2k4C3(n)]At
0
0
0
4 ikiC2(n)At
•i-[kiCi(n)+
2k2C3(n)]At
k2C2(n)At
1
0
0
5 lk3C3(n)At
0
ik3Ci(n)At
0
l-ksAt
0
6
0
0
k4C3(
0
0
1
p =
4
5
6 I
(3.13-2b)
where p3i = p36 was computed by Eq.(3-10a). The transient response of C3 and
C5 is depicted in Fig.3.13-1 where the effect of f in Eq.(3.13-2) is demonstrated
for At = 5x10-5, ki = 0.1, k2 = 6xl08, ks = L5xl05, k4 = 4xl08 and ks = 5x10^
and the initial state vector C(0) = [0.05, 10-6, 10-12,0, 10-12, Q].
309
t — t
oo w-> O
so «n
u- 2
o
cs - ;
, \
c 5
1 1
,* ' ;' '. ;
' .
c 3
V "—^
1
—._^ T
1
.-' '•--• - J
-f = 0.530
-
1
-
i ,.„/,.
.• 1 1
C ' 5
. c
1 1
-
f = 0.535 1
1 1 0 0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 0.04 0.05
t t
Fig.3.13-2. Ci versus t demonstrating the effect of f
The differential Eqs.(3.13-2) were solved numerically [59] for the case of a
continuous perfectly mixed reactor.
3,13-3 Ai + A2 "^ A3
2 A2 -^ A4
A3 ->A2
A3 + A4 -> 3A2
where [65]
A 4 - ^ A5
rj = - k |CiC2 + k_iC3
^2^^ — k l ^ l ^ 2 " — 1 3 " 2 3 — '^'^3^2 "'" •^J'^4^3^4
r3 = kiC|C2 - k_|C3 - k2C3 - k4C3C4
^4 ~ ^302 - k4C3C4 - k5C4
^5 = ^^5^4
(3.13-3)
(3.13-3a)
For a detailed derivation, the attention of the reader is addressed to case 5.2-1(4).
The following transition probabiUty matrix is obtained:
310
P =
1 2 3 4 l-kiC2(n)At 0 ^kiC2(n)At o
0
5
0
l-[kiCi(n)+ lkjCi(n)At k3C2(n)At 2k3C2(n)]At
k_iAt f V l-[k_i+k2+ Q
|k4C4(n)]At k4C4(n)]At
Q l~[k4C3(n)+
kslAt
0 0
0
0
|k4C3(n)At ksAt
(3.13-3b)
where p24 was computed by Eq.(3-10a). The transient response of Ci to C5 for an
initial state vector C(0) = [Ci(0), C2(0), €3(0), €4(0), C5(0)] = [1, 1, 0, 0, 0] is
depicted in Fig.3.13-3 where the effect of ks in Eq.(3.13-3) is demonstrated for At
= 0.01, ki = 2.5, k.1 = 0.1, k2 = 1, k4 = 10 and ks = 1.
k =0.2 3
10 15 0 t t
Fig.3.13-3. Ci versus t demonstrating the effect of ka
An analytical solution is available only for a simplified case [65].
311
3>13-4 (Ai)gas-^Ai
2A3 + A2 -^ 3A3
A3-^A4
Ai ->A3
A3->A2
(3.13-4)
where (Ai)gas denotes a saturated vapor of gas Ai in equilibrium with its condensed
phase containing species Ai, A2, A3 and A4. It is assumed that equilibrium
between the phases is established immediately and that the condensed phase is
perfectly mixed so that diffusion effects are negligible. H is the rate of supply of Ai
in moles/sec from the vapor phase (Ai)gas into the condensed phase. The following
equations, a detailed derivation of which without H appears in case 5.2-1(1), are
known as the Brusselator model [60]:
ri = H - k i C i
1*2 - ~ ^20203 + k3C3
r3 = kiC,
'•4 = ^403
fj = kjCi - 2k2C2C3 + SkjCjCg - (kg + k4)C3
The above equations yield the following transition probability matrix:
(3.13-4a)
P =
1
2
3
4
1 2 l-kiAt+[H/Ci(n)]At Q
0 l-k2Cf(n)At
0 kjAt
0 0
3 k,At
k2C|(n)At
l-(k3+k4) At
0
4
0
0
k4At
1 (3.13-4b)
The term [H/Ci(n)]At, where H is a constant supply rate of Ai, must be added in
pil in order to comply with the integrated form of ri in Eq.(3.13-4a).
The calculation of P33 requires some clarification since according to Eq.(3.13-
4), A3 is simultaneously consumed and formed. This fact must be taken into
312
account in calculating P33 in order to satisfy the result obtained by integration of 13
given by Eq.(3.13-4a). Thus, considering the latter equation, yields
P33 = 1 - [2k2C2C3 + (k3 + k4)]At + (2/3)[3k2C2C32/C3]At
= l - [k3 + k4]At
where the term 1 - [2k2C2C3 + (k3 + k4)]At stands for the probability to remain in
state A3 and (2/3)[3k2C2C32/C3]At is the transition probabiUty from 2A3 to 3A3
for the reaction 2A3 + A2 -^ 3 A3 according to Eq.(3-8).
The transient response of Ci, C2 and C3 for the initial state vector C(0) =
[Ci(0), C2(0), C3(0), C4(0)] = [1, 0, 0, 0] is depicted in Fig.3.13-4 where the
effect of k3 and H in Eq.(3.13-4a) is demonstrated for At = 0.03, ki = 10, k2 = 1,
k 4 = l , H = l a n d 0 .
k = 2 , H = 1 3
40 0 20 t
40
313
r
-\ ^ ^
/ \ 2y'
f X 1 . , - , „ . 1
,K '
1
,
— J
1
• ' " * " "1
J
k = 2 . 5 , H = 0 3 _
1
20 30 40 0 0.5 t t
Fig.3.13-4. Ci versus t demonstrating the effect of ka and H
An analytical solution is available [60].
3 > 1 3 - 5 (Ai)gas -> Ai
i = 1: Ai + A2 "^ A3 + A4 i = 2: A3 + A2"^2A4
k.
i = 3: Ai + A3 "^ 2A5 i = 4:A6 + A5 '/' A3 + A7
i = 5: 2A3 ^ Ai + A4 i = 6: A7-4gA2 + A6 (3.13-5)
where (Ai)gas denotes a saturated vapor of gas Ai in equilibrium with its condensed
phase containing species Ai to A7. It is assumed that equilibrium between the
phases is established immediately and that the condensed phase is perfectly mixed
so that diffusion effects are negligible. H is the rate of supply of Ai in moles/sec
from the vapor phase (Ai)gas into the condensed phase. The last six equations
describe the modified Oregonator mechanism [57] consisting of six steps. The rate
equations, a detailed derivation of which appears in case 5.2-1(3), are:
314
ri = H - kiCjCj - le,C3C4 - kgCjCj + k_^cl + kjCj - k_5C,C4
rj = - kiCjC2 + k_iC3C4 - kjCjCj + k_2C^ + gk6C7
Tj = KjL-]C 2 ~ k_|C^3v-4 — k2C 2 3 "'" k_2^4 — k3C ]C 3 + k_3C5 + k4C5Cg
- k_4C3C7 - 2k5C3 + 2k_5CiC4
T^ = kjCjC-2 ~ k_|C3C4 + 2k2C2C3 — 2k_2C4 + k5C3 — k_5CjC4
fj = 2k3(^jv_3 ~ 2k_3L-5 — k/^\^^\^^ + k_4C 3C,'7
r = - k4C5C6 + k_4C3C7 + k6C7
Tj = k4C5C6 - k_4C3C7 - k6C7
which yield the following transition probability matrix:
(3.13-5a)
P =
1
2
3
4
5
6
7
1 2 3 4 5 6 7
Pll 0 Pi3 Pi4 Pi5 0 0
0 P22 P23 P24 0 0 0
P31 P32 P33 P34 P35 P36 0
P41 P42 P43 P44 0 0 0
P51 0 P53 0 P55 0 P57
0 0 P63 0 0 P66 P67
0 P72 0 0 P75 P76 P77 (3.13-5b)
where
315
Pll = 1 - [kiCzCn) + kgCjdi) + k_5C4(n)]At + [H/Ci(n)]At
Pi3 = [O.SkiCjCn) + k_5C4(n)]At
Pi4 = 0.5kiC2(n)At pi5 = k3C3(n)A
P22 = 1 - [kiCi(n) + k2C3(n)]At P23 = 0.5k,Ci(n)At
P24 = [0.5kiCi(n) + k2C3(n)]At
P31 = [0.5k_iC4(n) + k5C3(n)]At P32 = 0.5k_iC4(n)At
P33 = 1 - [k_iC4(n) + k2C2(n) + k3Ci(n) + k_4C7(n) + 2k5C3(n)]At
P34 = [k2C2(n) + k5C3(n)]At P35 = [k3C,(n) + 0.5k_4C7(n)]At
P36 = 0.5k_4C7(n)At P41 = 0.5k_iC3(n)At
P42 = [0.5k_iC3(n) + k_2C4(n)]At P43 = [k_2C4(n) + k_5Ci (n)]At
P44 = 1 - [k_iC3(n) + 2k^2C4(n) + k_5C,(n)]At
P51 = k.3C5(n)At P53 = [k_3C5(n) + 0.5 k4C6(n)] At
P55 = 1 - [2k_3C5(n) + k4C6(n)]At P57 = 0.5k4C6(n)At
P63 = 0.5k4C5(n)At P66 = 1 - k4C5(n)At P67 = 0.5k4C5(n)At
P72 = gkgAt P75 = 0.5k_4C3(n)At P76 = [0.5k_4C3(n) + k lAt
P77 = 1 - [kg + k_4C3(n)]At (3.13-5c)
The term [H/Ci(n)]At, where H is a constant supply rate of Ai, must be added in
Pll in order to comply with the integrated form of ri in Eq.(3.13-5a).
The transient response of Ci, C2, €5 and C7 for the initial state vector C(0) =
[Ci(0), C2(0), C3(0), C4(0), C5(0), C6(0), CTCO)] = [1, 1, 0. 0, 0, 0, 1] is depicted
in Fig.3.13-5 where the effect of €7(0) = 3 and 1 is demonstrated. The parameters
of the results are [57]: Ci(0) = C2(0) = 1, ki = 1.5, k2 = ks = k4 = ks = 1, k-i = k-
3 = k-4 = k-5 = 0.005, k-2 = 0, kfi = 2, g = 3, H = 1 and At = 0.1.
10 20 ^0 10 20 t t
Fig.3.13-5. Ci versus t demonstrating the effect of Cj
40
In [57] the equations were integrated numerically for the case where the
reactions take place in a continuous perfectly mixed reactor.
3»13-6 The derivations of this example are detailed and serve as a completion to
chapter 3.2 and in particular to section 3.2-4. Of special importance is the
calculation of the probabilities P33 and P34 elaborated below. The following
reactions are considered showing at some conditions a complicated mixed-mode
behavior [69] .
i = 1: Ai + A2 + A3 -> 2A3
i = 2:
i = 3:
i = 4:
i = 5:
i = 6:
i = 7:
Ai
2A3 -^ 2A4
+ A2 + A4 ^ 2A3 k4
A3-»A5
A 4 ^ A 6
A3,0 -> A3
k'7
Ai,o ^ Ai
k'» i = 8: A2,0 -^ A2 (3.13-6)
317
where Ai,o indicate that the initial concentration Ci(0) of species Ai (i = 1,2, 3)
remains unchanged. The following kinetic equations may be derived by
considering the above reactions and the basic relationship given by Eq.(3-3):
i = 1: rS"(n) = ~ kS^^Ci(n)C2(n)C3(n) = - kiCi(n)C2(n)C3(n)
= 12 (Xi) = r3 (nj = - -J r3 Kn)
i = 2: I xf\n) = - k?^C (n) = - k2C3 (n) = - ^ ri'\n)
i = 3: if in) = - k f Ci(n)C2(n)C4(n) = - k3Ci(n)C2(n)C4(n)
= r2 (n) = r4 (n) = - ^ r3 (nj
i = 4: r^\n) = - ^^^€3^) = - k4C3(n) = - r?\n)
i = 5: r f (n) = - ki ^C4(n) = ~ k5C4(n) = - r \n)
i = 6: r f (n) = k ^ C3(0) = k'6C3(0) = k6
i = 7: /7\n) = kf^Ci(O) - k_7Ci(n) = k'vCiCO) - k_7Ci(n)
= k7~k_7Ci(n)
i = 8: r ^ n) = k ^ C2(0) = k'8C2(0) = kg (3.13-6a)
From Eq.(3-4) one obtains the following kinetic equations:
r ^ — K|C^|\^2^3 — k3C^ 2^-2^4 — " —7 1 " ^7
r2 = - kiCiC2C3 - k3CiC2C4 + kg
r3 ^ — KjC 10^2^3 " 2 K | C ' | C ^ 2 ^ 3 — 2K2V^3 — ^40-3 H~ 2k3C^|C^2^4 " k^
r4 = — k3C^|CJ2^4 — ^^5^4 "^ 2k2Cx3
5 = ^^4^3
r6 = k5C4 (3.13.6b)
The first term on the right-hand side in r3 indicates consumption of A3 whereas the
second term stands for the formation of A3 according to Eq.(3.13-6) for i = 1. This
presentation is important in the determination of P33, later elaborated. The
following transition probability matrix is obtained on the basis of Eqs.(3.13-6, 6b):
318
P =
1
2
3
4
5
6
1
Pll
0
0
0
0
0
2
0
P22
0
0
0
0
3
Pl3
P23
P33
P43
0
0
4
0
0
P34
P44
0
0
5
0
0
P35
0
1
0
6
0
0
0
P4
0
1 (3.13-6C)
The calculation of the probabilities was made by applying Eqs.(3-6), (3-9)
and (3-10) as follows:
pl l , the probability of remaining in state Ai, applies Eq.(3-6) for i = 1, 3, 7
inEq.(3.13-6). It is obtained that
PI 1 = 1 - [kiC2(n)C3(n) + k3C2(n)C4(n) + leyjAt + [k7/Ci(n)]At (3.13-6d)
The term [k7/Ci(n)]At, where k-j = k'7Ci(0) is a constant supply rate of Ai, must
be added to p n in order to comply with the integrated form of Eq.(3.13-6b) for ri.
pl3, the transition probability from Ai to A3, is calculated by Eq.(3-9) for i =
1, 3 in Eq.(3.13-6). It is obtained that
P13 = [2(l/3)kiC2(n)C3(n) + 2(l/3)k3C2(n)C4(n)]At (3.13-6e)
P22, the probability of remaining in state A2, applies Eq.(3-6) for i = 1, 3, 8
in Eq.(3.13-6). It is obtained that
P22 = 1 - [kiCi(n)C3(n) + k3Ci(n)C4(n)]At + [k8/C2(n)]At (3.13-6f)
The term [k8/C2(n)]At, where kg = k'8C2(0) is a constant supply rate of A2, must
be added to p22 in order to comply with the integrated form of Eq.(3.13-6b) for r2.
P23, the transition probability from A2 to A3, is calculated by Eq.(3-9) for i =
1, 3 in Eq.(3.13-6). It is obtained that
P23 = [2(l/3)kiCi(n)C3(n) + 2(l/3)k3Ci(n)C4(n)]At (3.13-6g)
319
P33, the probability of remaining in state A3, applies Eq.(3-6) for i = 1, 2, 4,
6 in Eq.(3.13-6). However, the application of this equation needs some
clarification because of the following situation. According to Eq.(3.13-6) for i = 1,
A3 is consumed on the one hand, but it is also formed on the other. This fact must
be introduced in computing the probabilities and the only place is in P33 in a way
which complies with the result obtained by integration of r3 in Eq.(3.13-6b). Thus,
P33 = 1 - [kiCi(n)C2(n) + 2k2C3(n) + k4]At + 2(l/3)kiCi(n)C2(n)At
+ [k6/C3(n)]At (3.13-6h)
where the term 1 - [kiCi(n)C2(n) + 2k2C3(n) + k4]At designates the probability of
remaining in state A3; the term 2(l/3)kiCi(n)C2(n)At designates the transition
probability from A3 to 2A3 in Eq.(3.13-6) for i = 1. The latter was computed by
Eq.(3-10). The term [k6/C3(n)]At, where k6 = k'6C3(0) is a constant supply rate of
A3, must be added to P33 in order to comply with the integrated form of Eq.(3.13-
6b) for r3.
P34, the transition probability from A3 to A4, is calculated by Eq.(3-10a) for i
= 2inEq.(3.13-6). It is obtained that
P34 = 2k2C3(n)At (3.13-6i)
P35, the transition probabiUty from A3 to A5, is calculated by Eq.(3-10) for i
= 4inEq.(3.13-6). It is obtained that
P35 = k4At (3.13.6J)
P43, the transition probability from A4 to A3, is calculated by Eq.(3-10) for i
= 3 in Eq.(3.13-6). It is obtained that
P43 = 2(l/3)k3Ci(n)C2(n) (3.13-6k)
P44, the probability of remaining in state A4, applies Eq.(3-6) for i = 3, 5 in
Eq.(3.13-6). It is obtained that
P44 = 1 - [k3Ci(n)C2(n) + kslAt (3.13-61)
Finally, p46, the transition probability from A4 to A6, is calculated by Eq.(3-
10) for i = 5 in Eq.(3.13-6). It is obtained that
320
P46 = ksAt (3.13-6m)
Transient response curves for Ci to Ce for the initial state vector C(0) = [Ci(0), C2(0), C3(0), C4(0), C5(0), C6(0)] = [3, 20, 0.01435, 0, 0, 0] , where the effect of ki = 0.02, 0.16, 0.2, 0.25 and 1 is demonstrated, are depicted in Fig.3.13-6 for the following data [69]: k2 = 1250, ks = 0.04688, k4 = 20, ks = 1.104, k6 = 0.001, k7 = 0.89, k.7 = 0.1175, kg = 0.5 and At = 0.01.
30
20 U
u
10 \-
k ,= l
h
1
p. 4
/ • .
1
/
6
. ' 5
- --. 2
i = l
3
: • 1
( a ) |
-J
H
50 100 150
Fig.3.13-6a. Cf versus t for ki = 1
321
30
20 p,
u
10
kj = 0.25
H-\
r
1
5.'
6
. . . 2
i = l
3
1 (b)
H
-
50 100 150
Fig.3.13-6b- Ci versus t for ki = 0.25
30
20 Y-
u
10 U
50 100 150
Fig.3.13-6c. Ci versus t for ki = 0.22
322
30
20
u
10 h
kj = 0.2
r~ '.
4^,
1
• '•'• - • '
. . - ^ . 1 - - .
5 / '
6
1
, • 1- • '
^ 1 ' > . ' •
(d)|
-j
A/ 50 100 150
Fig.3.13-6d. Ci versus t for ki = 0.2
Fig.3.13-6e. Ci versus t for ki = 0.16
323
60
40
u
20 K
0 50 100 150
t
Fig.3.13-6f- Ci versus t for ki = 0.02
Figs.3.13-6, a to f, demonstrate the effect of ki on the transient behavior of
Ci to C6. Of particular interest is Figs.3.13-6d which is not actually chaotic but is
a complicated mixed-mode state[69].
3.14 NON-EXISTING REACTIONS WITH A BEAUTIFUL PROGRESSION ROUTE
It has been said that the essence of beauty emerges from the shape. Thus, to
conclude this chapter, two reactions are demonstrated whose routes form beautiful
shapes.
3.14-1 The Shield of David progression-route reaction The Shield of David is a Jewish national and religious symbol whose origin
goes back to the 12th century. It is comprised of two intergrated opposite triangles.
On May 24th 1949, the Parliament of the State of Israel declared that the Shield of
David will appear on the national flag and will be the identification symbol for
every Jew where ever he is. It is demonstrated below as the progression route of
the following reaction.
324
(3.14-1)
For the above reaction scheme, the following kinetic equations are applicable.
It is assumed that a saturated vapor of gas Ai is in equilibrium with its condensed
phase containing species Ai to A12. Other assumptions are as in case 3.13-5
above, where H is the rate of supply of A1 in moles/sec from the vapor phase into
the condensed phase.
rj = H - (ki2 + kj 12)^1 2 ~ ~ vk23 + k24)C2 + ^n^i + '^i,12^12
r7 = - k7gC7 + k^-jC^
Tg = - k9 10C9 + kg9Cg
r4 = — (k45 + k45)C4 + 1 3403 + 1 2402
rg = - (k 39 + kg |o)Cg + k7gC7 + k^gC^
r o = - (k 10,11 + 1 10,12) 10 + ^,10^8 + ^9,10^9
11 - "" 1 11,12 11 "•• 1 10,11 10
ri2 - - k 12,2^12 + ki 12C1 + kji 12C11 + kio,i2Cio (3 .14- la)
The above equations yield the following transition probability matrix:
325
P =
1
2
3
4
5
6
7
8
9
10
11
12
1 P i i
0
0
0
0
0
0
0
0
0
0
0
2 P l 2
P22
0
0
0
0
0
0
0
0
0
Pl2,2
3 0
P23
P33
0
0
0
0
0
0
0
0
0
4 0
P24
P34
P44
0
0
0
0
0
0
0
0
5 0
0
0
P45
P55
0
0
0
0
0
0
0
6 0
0
0
P46
P56
P66
0
0
0
0
0
0
7 0
0
0
0
0
P67
P77
0
0
0
0
0
8 0
0
0
0
0
P68
P78
P88
0
0
0
0
9 0
0
0
0
0
0
0
P89
P99
0
0
0
10 0
0
0
0
0
0
0
P8,10
P9,10
PlO.lO
0
0
11 0
0
0
0
0
0
0
0
0
PlO.l
P l l . l
0
12 Pi,12
0
0
0
0
0
0
0
0
lPlO,12
l P l l , 1 2
Pl2,12
(3.14-lb)
where
Pii = 1 - [ki2 + ki,i2]At + [H/Ci(n)]]At
P l2= ki2At Pi,i2 = ki,i2^t
P22 = 1 - [k23 + k24]At P23 = k23At P24 = k24At
P33= l -k34At P34 = k34At
P44 = 1 - [k45 + k46]At P45 = k45At P46 = k46At
P55= l - k s e A t p56 = k56At
P66 = 1 - [ 67 + k68]^t P67 = k67At Pgg = kggAt
P77= l - k 7 8 A t P78 = k78At
P88 ~ ^ ~ '•' 89 ••• .loJ^t P89 - ''89At Pgjo ~ 1 8,10 *
P99 = 1 - k9 joAt P9 10 = k9 joAt
Pio.io = 1 ~ tl^io.ii + ^ 10,12]' ^ pjo,ii = k]o,iiAt Pio,i2 = kjo,i2At
Pi i .u = 1 ~ kj] 12 At Pii . i2= J'li.n^t
Pl2,12 = 1 - ki2,2 ^t Pi2,2 = ki2,2At (3.14-lc)
326
The term [H/Ci(n)]At, where H is a constant supply rate of Ai, must be added in
pil in order to comply with the integrated form of ri in Eq.(3.14-la).
The transient response of Ci to C12 is depicted in Fig.3.14-1 where the
effect of H = 0, 2 and 3 is demonstrated. Other parameters are ky = 1, Ci(0) = 1,
Ci(0) = 0 (i = 2, 3,... , 12) and At = 0.04. 0.3
0.2 \-
0.1
1
; \
^ / F / ;
/ / ' 1/ .:
1 H = 0
r^'^\ \ ' • • • - . ^ x r • • : . - . . .
\ ^y-. :-<^^::..^ \/ (^y '']y''"i' / \ y ly
il-^r:.::^^^^ 1
1:C 1
3:C =C 3 4
5:C =C 7 8
7:C 11
V • - - z : j-^"-T::-r
2:C 2
4:C =C 5 6
6:C =C 9 10
8:C 12 - j
^S^5?"^^;S*.«S5E^- "w*w
10
Fig.3.14-la. Ci versus t for H = 0
Fig.3.14-lb. Ci versus t for H = 2
4 h
3 \-
U
1 F
1 1:C 2:C
1 2 3:C =C 4:C =C
— 3 4 5 6 5:C =C 6:C =C
7 8 9 10 7: C 8: C
11 12
r --''' ' '•'"''' '^ '
K '•-•"/^•- ,.•<--• y . .--^.--''..•••
v. - ;;-<.-- •• I ^ ..::...::-:•:..-- - i
H = 3
..-' 'J.
- ^ " .-" . .-"'
,.'''" '
1
8
; ' . - ^
..-'
1
^'"^'
^..•-•jj
•"'' - " 1 - ' , * '' -- "i
-
10
Fig.3.14.1c. Ci versus t for H = 3
327
15
3.14-2 The Benzene molecule progression-route reaction The route of this reaction is shown below
(3.14-2)
The kinetic reactions are:
r i = - ( k i 2 + ki6)Ci + k6iC6 l2 = 1 23 2 + kj2Cj + k32C3
rj = - (k34 + k32)C3 + k23C2 r4 = - k45C4 + k34C3 + k54C5
rj = - (k56 + k54)C5 + k45C4 r = - k6,C6 + kigCj + kseCs (3.14-2a)
328
yielding the following transition probability matrix:
P =
1 l -[ki2+ki^
0
0
0
0
k^iAt
]At
2 ki2At
l-k23At
k32At
0
0
0
3
0
k23At
l-[k32+k34]At
0
0
0
4
0
0
k34At
l-k45At
k54At
0
5
0
0
0
k45At
l-[1^54+k565^t
0
6 ki^At
0
0
0
k56At
l-k^iAt
1
2
3
4
5
6
(3.14-2b) The transient response of Ci to Ce is depicted in Fig.3.14-2 where the effect
of ki2 is demonstrated for At = 0.04, ky = 1 (ij ?i 12), C](0) = C5(0) = 1, C2(0) = C4(0) = 2, C3(0) = 3 and CeCO) = 0.
u
1 H
' . . = '
y^~"*""--s ^''
. - - ^ ---
,'''
'* - - - ' ' ' i = l
l 1
- • — - -- - - - — — '
10
Fig.3.14-2a. Cf versus t for ki2 = 1
329
Fig.3.14-2b. Ci versus t for ki2 = 0 As observed in the above Figs.3.14-1 and 2, the transient
curves look very similar to the numerous ones generated before. The question that arises then is why should not the reactions corresponding to the Shield of David and Benzene progress shape routes exist ?
330
3>14-3 The Lorenz system, partially demonstrating chemical reactions, for creating esthetic patterns
The system of equations that Lorenz proposed in 1963 [85, p.697] are: 10
ri = IOC2 - lOCi
12 = 28Ci - C2 - C1C3
r3 = CiC2-(8/3)C3
for which Ai A2 may be written.
10
for which no reaction may be realized. 8/3
for which A3 Ai +A2 may be written.
1
As seen, two equations, may demonstrate reversible reactions. However, the
Lorenz system is in fact a model of thermal convection, which includes not only a
description of the motion of some viscous fluid or atmosphere, but also the
information about distribution of heat, the driving force of thermal convection. The
above set can be described by the following matrix, which, however, does not have
any probabiUstic significance:
1
P =
thus,
Ci(n+1) = Ci(n)[l - lOAt] + C2(n)[10At]
C2(n+1) = Ci(n)[28 - C3 (n)]At + C2(n)[l - At]
C3(n+1) = Ci(n)C2(n)At + C3 (n)[l - (8/3)At]
The transient response of Ci, C2 and C3 is depicted in Fig.3.14-3 for At =
0.01 and C(0) = [0.01, 0.01, 0.01]. The behavior in the figures clearly
demonstrates chaos characterized by:
a) The physical situation is described by non-linear differential equations.
1
2
3
1 - lOAt
lOAt
0
[28-C3(n)]At 0.5C2(n)]At
1-At 0.5Ci(n)]At
0 1 - (8/3)At
331
b) The transient response is characterized by high sensitivity to extremely small
changes in the initial conditions as demonstrated in Fig.3.14-3 case a.
c) Numbers generated by the solution are random as shown in Fig.3.14-3 case b.
d) Order, demonstrated by esthetic patterns in Fig.3.14-4 below, can be generated
from chaos by appropriate representation of the chaotic data in Fig.3.14-3.
30 Initial conditions: 1: CCO) = € (0) = C (0) = 0.01
2: Cj(0) = CCO) = C (0) = 0.01001
(a)
Fig.3.14-3. Ci versus t for the Lorenz system demonstrating extreme
sensitivity to initial conditions in case a and chaos in case b
332
60
50 h
40 h
U
30
20 h
10
0 -30
I ^^^^Hi
^ 1 ^ ^ 1 H |
KpDii'^Dlp^iya D D D Bq^
ir J o 1 a riT ih l l i 1 iTinih « if^ jtinn|nii;rji=-' --V'-
i l infl l 'Jj^^iJj*" i LIT if:V^j\^lj'!l(IJ-^!s^jPQiB!S
1 i ^^mmmatt^^^^^^^^.m^^m
-20 -10 0 10 20 30
Fig.3.14-4a,b. C3-C1 and C3-C2: creation of Order from Chaos
333
30 h
20 h
10 h
V
-10
•20
^0
Fig.3.14-4c. The C2-C1 Lorenz attractor demonstrating the creation
of Order from Chaos
As observed in Figs.3.14-4, all patterns generated by the C3-C1, C3-C2 and
C2~Ci representations, remind, in one way or another, a butterfly. The latter
stands for a basic phenomenon in the chaos model known as the butterfly effect,
after the title of a paper by Edward N.Lorenz 'Can the flap of a butterfly's wing stir
up a tornado in Texas?' An additional point may be summarized as follows, i.e.,
How come that relatively simple mathematical models create very complicated
dynamic behaviors, on the one hand, and how Order, followed by esthetics
patterns, may be created by the specific representation of the transient behavior, on
the other ?
334
Chapter 4
APPLICATION OF MARKOV CHAINS IN CHEMICAL REACTORS
The major aim of the present chapter is to demonstrate how Markov chains
can be appUed to determine the behavior of a compUcated system with respect to the
residence time of fluid elements flowing through it. In other words, to obtain the
response of the system to some tracer input, usually in the form of a pulse.
It is well-known that fluid elements entering simultaneously a continuous
reactor, do not, in general, leave together, owing to the complex flow pattern inside
the reactor. Because of this reason, there is a spread of the residence time of the
flowing elements, i.e. the time each element resides in the reactor; the latter can be
represented by the so-called age distribution function. Thus, if a chemical reaction
is carried out in a reactor, the resulting products depend on the length of time each
element of the reactants spends within the reactor, which affects the overall
conversion.
Several models have been suggested to simulate the behavior inside a reactor
[53, 71, 72]. Accordingly, homogeneous flow models, which are the subject of
this chapter, may be classified into: (1) velocity profile model, for a reactor whose
velocity profile is rather simple and describable by some mathematical expression,
(2) dispersion model, which draws analogy between mixing and diffusion
processes, and (3) compartmental model, which consists of a series of perfectly-
mixed reactors, plug-flow reactors, dead water elements as well as recycle streams,
by pass and cross flow etc., in order to describe a non-ideal flow reactor.
In the following, approach (3) above was adopted. Stimulating the resulting
flow configuration by some input, yields the residence time distribution of fluid
elements in the flow system. In the present chapter, the flow system was treated
335
by Markov chains yielding the transition probability matrix. Each expression in the
matrix is either the probability to remain in a state (reactor) or to leave it to the next
state (reactor). Complicating the flow behavior is usually manifested by additional
terms in the matrix, which, however, does not create proportional difficulties in the
solution. Applying Eqs.(3-20), based on Eqs.(2-23) and (2-24), may yield
important characteristics of the flow behavior, viz., response to a step change or to
a pulse input. Both responses provide necessary information about the residence
time distribution of fluid elements in the flow system.
Many flow systems of interest in Chemical Engineering are presented in the
following, viz., a diagram of the flow system, the transition probability matrix and
the transient response to some input signal. Attention has been paid also to
systems employing impinging streams [73] which is an effective technique for
intensifying technological processes.
4.1 MODELING THE PROBABILITIES IN FLOW SYSTEMS
Definitions. The basic elements of Markov chains in flow systems are: the
system, the state space, the initial state vector and the one-step transition probability
matrix. The system is a fluid element. The state of the system is the concentration
of the fluid element in the reactor, assumed perfectly mixed. The state space is the
set of all states that a fluid element can occupy, where a fluid element is occupying
a state if it is in the state, i.e. at some concentration. For the flow system depicted
in Figs.(4-1) and (4-la), the state space SS, which is the set of all states a system
can occupy, is designated by
55 = [C\, C2,... ,Cz, C^]
Finally, the movement of a fluid element from state Cj to state Ck is the transition
between the states.
The initial state vector given by Eq.(2-22), corresponding to Figs.(4-1) and
(4-la), reads:
S(0) = [si(0), S2(0), S3(0), ..., Si(0), ..., Sz(0), s^(0)]
336
where Si(0) is the probabihty of the system to occupy state i at time zero. S(0) is the initial occupation probabihty of the states [Ci, C2,..., Cz, C ] by the system.
Z+1 designates the number of states, i.e. Z perfectly mixed reactors in the flow
system as well as the tracer collector designated by ^. As shown later, the
probabilities Si(0) may be replaced by the initial concentration of the fluid elements
in each state, i.e. Ci(0) and S(0) will contain all initial concentrations of the fluid
elements. The one-step transition probability matrix is given by Eqs.(2-16) and (2-
20) whereas pjk represent the probability that a fluid element at Cj will change into
Ck in one step, pjj represent the probability that a fluid element will remain
unchanged in concentration within one step.
In the following, general expressions are derived for the transition
probabilities corresponding to two general flow configurations. The latter can be
reduced to numerous systems encountered in Chemical Engineering elaborated
below.
4.1-1. Probabilities in an interacting conflguration
Basic configurations. The models demonstrated in Figs.4-1 and 4-la
comprise of perfectly-mixed reactors, generally, of not the same volume. The
central reactor in Fig.4-1 is designated by j and also termed as reactor junction (j).
If the volume of this reactor is zero, this location designated by j in Fig.4-1 a, is
termed as point junction (j). The above situations are of practical importance and
generate a slightly different transition probability matrix. The mean residence time
of the fluid in the reactor is V/Q whereas in the case of a point it is equal to zero.
The peripheral reactors are designated by a, b,..., Z. The final reactor is the tracer
collector designated by ^. It should be noted that the letters are assigned numerical
values when a specific case is considered where ^ is assigned the highest number,
i.e. ^ = Z + 1. When the central reactor j is considered, usually, j = 1 and a = 2, b
= 3, etc. When the central reactor is not considered, usually, a = 1, b = 2, etc.
The total number of reactors is the number of states. If reactor j is considered, the
total number of states equals ^ -h 1 = Z + 2.
As observed, there are various interacting flows in Figs.4-1 and 4-la as
follows: a) Flows between the central reactor or point j , and the peripheral ones,
and vice versa, Qji and Qy, respectively, as well as between the central reactor, or
point, and collector ^, i.e. qj^. b) Flows between every peripheral reactor and each
337
of the others, Qik, where i, k = a, b, ..., Z, k ?t i. It should be noted that among
these stresims are also the so-called recycle streams, c) Flows between every
peripheral reactor and the collector ^, q^ , where k = a, b,..., Z. d) External flows
into the reactors, Qr, r = j , a, b, ..., Z. For example, in Fig.4-1 we consider
reactor p. Qp indicates an external flow into reactor P. Qkp refers to the flows
from all reactors into reactor p, where k = j , a, b,..., Z, k t p. Qpj demonstrates
all flows from reactor p to all other reactors k, i.e. k = j , a, b, ..., Z, k 9fc p.
Finally, qp^ indicates the flow from reactor p to the collector ^.
Stimulating the system by a tracer input introduced into reactor j or reactors i,
causes a change of the concentration in the entire flow system due to interaction
between the reactors. The following situations my be possible with regard to the
tracer. If its concentration C' = 0 at the exit of reactor ^, the tracer is completely
accumulated in reactor ^; thus, the reactor is considered as "total collector" or "dead
state" or "absorbing state" for the tracer. In other words p^^ = 1, and once the
tracer enters this state, it stays there for ever. If, however, the concentration C^ of
the tracer in the reactor is equal to its concentration at the exit, i.e. C^ = C' , there
is no accumulation of the tracer in reactor ^, If 0 < C' < C , the tracer is partially
accumulated in reactor ^, which is considered as "partial collector". Generally, the
fluid is always at steady state flow. In the case of a closed circulating system, i.e.
Qj. = 0, r = j , a, b,. . . , Z, the tracer is eventually distributed uniformly between all
reactors.
Finally, it should be noted that the schemes depicted in Figs.4-1 and 4-la
cover numerous flow configurations encountered in Chemical Engineering where a
specific configuration is determined by appropriate selection of the interacting
flows and the number of reactors. Numerous examples of high significance will be
treated in the following.
338
a) Overall scheme QkaQak^p
b) Tracer collector
k k
Fig.4-1. A scheme for an interacting flow system with a reactor junction (j)
339
a) Overall scheme
b) Tracer collector
Fig.4-la. An interacting flow system with a point junction (j)
340
Derivation of the probabilities from mass balances
Reactor j . A mass balance on the tracer in the reactor junction (j) in Fig.4-
1 for k = a, b,... , Z where k^] and ^ reads:
dC V j - d f ^ ^ Q k A - Uj Cj + XQjkCj (4-1)
It should be noted that for a specific configuration, j , k and ^ are assigned
numerical values where ^, the collector, should be assigned the highest value. Cj
and Ck are, respectively, the concentrations of the tracer in reactor j and reactors k;
Qkj and Qjk are the interacting flows between reactor k and j and vice versa,
whereas flow qj^ is from the central reactor to the collector reactor ^. Vj is the
volume of the fluid in reactor j , assumed to remain unchanged.
In the case of a point junction (j) in Fig.4-la, Eq.(4-1) for Vj = 0 yields that
the concentration at this point is:
^ Q k j ^ k ^ ^ k j ^ k
Cj = -^ = -^ (4-la)
k k
The following quantities are defined for the flows in Fig.4-1 where k = a, b,
..., Z, k:^i,% gives:
««=^ %=t fe=f («) Qi
Hj = # (4-3)
where |LIJ (1/sec) is a measure for the transition rate of the system (fluid element)
between consecutive states (reactors). Qj is the flow rate into reactor j where Qkj
and Qjk are the flows from reactor k to j and j to k, respectively. In the case of a
closed recirculation configuration, i.e., Qj = Qk = 0 (k = a, b,. . . , Z), one of the
341
internal streams should be selected as reference flow instead of Qj in Eqs.(4-2) and
(4-3) in order to perform the above non-dimensionalization.
Integration of Eq.(4-1) between the times t and t+At, or step n to n+1, while
considering the above definitions, yields for k = a, b,..., Z where k T j , 4. that
Cj(n+l) = Cj(n) 1 - h + X"jk h^^ + X^k(n)[akjtijAt] (4-4)
In terms of probabilities, the above equation reads:
Cj(n+1) = Cj(n)pjj + 2,Ck(n)Pkj (4-4a)
Pjj is the probability to remain in reactor j and pkj are the transition probabilities
from reactors k to j . The definition of the probabilities is obtained from Eq.(4-4).
Reactor i. A mass balance on the tracer in reactor i in Figs.4-1 and 4-la
for i, k = a, b,. . . , Z where k ?i i, j , ^ gives:
dCj ^ Vi-3^ = QjiCj + 2 ^ Q k i C , - qi + Qij + Z Q i k R
k / (4-5)
In the case of the point junction (j) in Fig.4-la, the concentration Cj is given
by Eq.(4-la). Defining the following quantities with respect to the reactor junction
(;)inFig.4-l:
aki = Qki
Qj
Qik Qji Qij ^ i ^ ttik--^ " j i - - ^ " ' j " " ^ ^ ' ^ " " ^
Qj
(4-6)
(4-7)
and expressing Eq.(4-5) in a finite difference form, yields for i, k = a, b, ..., Z
where k9 t i , j , ^ that:
342
Ci(n+1) = Cj(n)[aji|LiiAt] + QCii)
+ XCk(n)[aki^iAt] (4-8) k
An alternative form of the above equation in terms of transition probabilities,
reads:
Ci(n+1) = Cj(n)pji + Ci(n)pii +X^k(n)Pki (4-8a) k
where Cj(n) is the concentration in reactor j . pji is the transition probability from
state j to i, pa is the probability to remain in reactor i and pki are the transition
probabilities from reactors k to i. The definition of the probabilities is obtained
from Eq.(4-8). In the case of the point junction (j) in Fig.4-la, the concentration
Cj is given by Eq.(4-la). In the case of a closed recirculation configuration, i.e.,
Qj = Qk = 0 (k = a, b,.. . , Z), or if Qj = 0 and Qk ^ 0, one of the internal streams
or one of the Qk's should be selected as reference flow instead of Qj in order to
perform the above non-dimensionalization.
Reactor ^. A mass balance on the tracer in reactor ^ in Fig.4-1 for k = a,
b, . . . , Z where k^],^ reads:
^ • ^ = 2<q>c Ck + qj^q (4-9)
yielding:
C^(n+1) = Cj(n)[pj^^^At] +X^k(n)[p^^^At] + C^(n) (4-10) k
In terms of transition probabilities the above equation reads:
C^(n+1) = Cj(n)pj^ +X^k(n)Pk^ + C^(n)p^^ (4-lOa)
343
Pjt is the transition probability from reactor j to ^, p^t are the transition
probabilities from reactors k to ^ whereas p^t = 1 is the probability to remain in
reactor ^ since this reactor is considered as a collector for the tracer. The definition
of the probabilities is obtained from Eq.(4-10). For the case depicted in Fig.4-la,
i.e. iht point junction (j) scheme, Cj(n) is given by Eq.(4-la), where:
H = T^ h = ^ >•=•''•' (4-11)
Finally, the following relationships resulting from mass balances on the
flows, must be satisfied simultaneously. They serve to determine the quantities
otk, cxij, ocji, akj, ttjk, aik, aki as well as pj^ and pk^.
An overall mass balance on the flow system in Figs.4-1 and 4-la, i.e. on
reactors j and k , k = a, b,.. . , Z where k^} and , gives:
Qj + X^k = Qj + X^k^ y''^^^^ 1 + S^k = Pj + XPk^ (4-12a)
ttk is given by Eq.(4-12d) and pj^, pk^ by Eq.(4-11).
A mass balance on the flows of reactor j or point j in Figs.4-1 and 4-la, respectively, for k = a, b,.. . , Z where k ;t j and , yields:
Qj + XQkj = qj^+XQjk yi^i^s i + X^kj = Pj^+X"jk (4-i2b) k k k k
where ttkj and ajk is given by Eq.(4-6). If Qj = 0, one of the in flows Qi where i
= a, b, ..., Z or one of the internal flows, should be taken as a reference flow in
Eqs.(4-2) and (4-6) instead of Qj. In addition, Eq.(4-12a) should be ignored and
in Eq.(4-12b), the figure 1 should be omitted.
A mass balance on reactors i in Figs.4-1 and 4-la for i , k = a, b, ..., Z
where k ?t i, j and ^, reads:
344
Qi + Qji + X ^ w = qi + Qij + X ^ i k yields k k
tti + ttji + ^ a k i = pi + ay + Y^o-m (4-12C)
where aji, ttki, ttik and Pi are defined in Eq.(4-6). In the case of a closed
system, i.e., Qj = Qk = 0 (k = a, b,.... Z), one of the internal streams should be
selected as reference flow in order to perform the non-dimensionalization of the
quantities designated by a. The coefficient ai is defined by:
Qi (4-12d)
where Qj is the flow rate into reactor j in Figs.4-1 or to junction j in 4-la.
For the reactor junction j scheme in Fig.4-1, Eqs.(4-4), (4-4a), (4-8), (4-8a)
(4-10) and (4-lOa) may be expressed by the following matrix:
P =
j
a
i-l
i i+1
Z
^
C j -j
Pjj Paj
Pi-l,j
Pij
Pi+l.j
Pzj
1 0
Ca = a
Pja
Paa
Pi-l,a
Pia
Pi+l,a
Pza
0
i-l
Pj,i-1
Pa,i-1
i
Pji Pai
Q+i = i+1
Pj,i+1
Pa,i+1
Pi-l,i-l Pi-l,i Pi-l,i+l
Pi,i-1 Pii Pi,i+1
Pi+l,i-l Pi+l,i Pi+l,i+l
Pz,i-1 Pzi Pz,i+1 0 0 0 0
Pjz Paz
m Pa4
Pi-l,z Pi-14
Piz Pi
Pi+l,z Pi+1,
Pzz Pz
0 1 (4-13)
The calculation of the new concentration vector C(n+1) is performed by
Eq.(3-20), i.e.:
C(n+1) = C(n)P
where
C(n) = [Cj(n), Ca(n), Cb(n),..., Cz(n), q(n)]
345
(4-13a)
For the point junction j scheme depicted in Fig.4-la, Eq.(4-la) for Cj(n), and
Eqs.(4-8), (4-8a), (4-10) and (4-lOa) for Ci(n) and C^(n), may be expressed by
the following matrix:
P =
j a
i-l
i
i+l
Z
^
j
0
Paj
Pi-l,J
P'ij •*• P i + l j
P'zj
0
a
Pja
Paa
Pi-l,a
Pia
Pi+l,a
Pza
0
i-l
•• Pj,i-l
Pa,i-1
•• Pi-l,i-l
Pi,i-1
•. Pi+l,i-l
Pz,i-1
0 0
i
Pji
Pai
Pi-l,i
Pii
Pi+l,i
Pzi
0
Q+1 = i+l
Pj,i+1
Pa,i+1
Pi-l,i+l •
Pi,i+1
Pi+l,i+l •
Pz,i+1 0
z Pjz
Paz
Pi-l,z
Piz
Pi+l,z
Pzz
0
Pj^
Pa
Pi-1,^
Pi^
Pi+1,^
Pz^
1
(4-13b)
where
Pkj OCkj
pj^+X«jk
k = a, b, ..., Z where k T j , ^ (4.13c)
It should be noted that the p jks are not exactly probabilities but merely
quantities which enable one to present the junction concentration Cj(n), due to
mixing of various streams at this point, given by Eq.(4-la) in the above matrix and
to compute it from C(n+1) = C(n)P. The above matrix differs from Eq.(4-13a) by
the following: pjj = 0 as well as some of the expressions for computing the
probabilities are different, as detailed below.
Summary of probabilities
These quantities resuh in from the above mass balances which yield
equations (4-4), (4-4a), (4-8), (4-8a) (4-10) and (4-lOa).
346
The probability pjj of remaining in state j (reactor j) in a single
step (single time interval At) for k = a, b, ..., Z where k 9 j , ^,
reads:
P j j = l - pj +X%Vj ^ ^^'"^^"^
Pjj stems from Eq.(4-4) and is applicable for the reactor junction j scheme in Fig.4-
1. Pjj = 0 for the case depicted in Fig.4-la.
The transition probability pji (i = a, b, •••, Z) in a single step
(single time interval At), from state j (reactor j) or junction j , to state
i (reactor i) where i 9 j , ^, reads:
Pji = ajiiLiiAt (4-15)
pji stems from Eq.(4-8) and is applicable for the scheme depicted in Figs.4-1 and
4-la. For the latter case Cj is obtained from Eq.4-la.
The transition probability pj^ in a single step (single time
interval At), from state j (reactor j) or junction j , to state ^ (reactor ^)
where j ^ ^, reads:
Pj^ = Pj^^^At (4-15a)
which stems from Eq.(4-10) corresponding to Figs.4-1 and 4-la.
The probability pii of remaining in state i (reactor i) in a single
step (single time interval At) for k, i = a, ..., Z where k ? i, j and ^,
reads:
r Pii = 1 - Pi + oCij + X^ikh^^ (4-16)
V
pii stems from Eq.(4-8) and corresponds to Figs.4-1 and 4-la.
The transition probabilities py and pi in a single step (single
time interval At) from state i (reactor i) to state j or ^ (reactor j or ^)
for i = a, b, ..., Z where i 9t j and I,, reads:
347
Pij = ttij^jAt (4-17a)
which stems from Eq.(4-4) and corresponds to Fig.4-1. From Eq.(4-10) follows
that
Pi^=Pi^^^At (4-17b)
corresponding to Figs.4-1 and 4-la.
The transition probabilities between the peripheral reactors, corresponding to
Figs.4-1 and 4-la, are:
The transition probability pki in a single step (single time
interval At) from state k (reactor k) to state i (reactor i) for k = a, b,
••*, Z where k ? i, j and ^, reads:
Pki = akiHiAt (4-18)
which stems from Eq.(4-8). It follows, for example, from Eq.(4-18) that:
Pi,i+i = ai,i+iM'i+iAt (4-18a)
Pi+l,i = ai+l,iWAt (4-18b)
The transition probability pk^ in a single step (single time interval At) from state k to state ^ (reactor ^) for k = a, ..., Z where k ? i, j and , reads:
Pk^=Pk^|Li^At (4-19)
stemming from Eq.(4-10).
Inspection the probabilities defined in Eqs.(4-14) to (4-19) and the transition
probability matrix given by Eqs.(4-13) and (4-13a), leads to the following
conclusions:
a) In general, for every row:
2jPik '^ 1 1 = a,..., Z (4-20a)
348
b) However, if all reactors are of an identical volume, i.e. |Lii = |ij = |LI according to
Eqs.(4-3) and (4-7), then for each row:
XPjk = 1 (4-20b)
and the matrix given by Eq.(4-13) is time-homogeneous or stationary.
Additional expressions of transition probabilities, without a mathematical
proof, have been suggested [75], viz.:
Pjj ~ ^ Pjk "• ^ ^ (4-21)
These expressions reduce to the above ones if the first term in the Taylor series
expansion is taken, which is justified for short At. At is the time a molecule in
vessel j can either remain where it is or move on to vessel k.
4.1-2. 'Dead state' (absorbing) element. Such an element depicted in Fig.4-2
"cbi rebi
Fig.4-2.'Dead state' element
is characterized by the following transition probabilities:
Pii = 1 Pij = 0 (4-22)
4.1-3. Plug flow element. Such an element depicted in Fig.4-3 is
characterized by the following transition probabilities for a pulse input introduced at
t = 0: pjj = l pji = 0
0 < t < t p : pjj = 0 Pii = 0
t = tp: Pjj = 0 Pii = 1
t>tp: pjj = 0 pii = 0 (4-23)
349
tp = V/Q is residence time in the reactor which is identical for all fluid elements.
Fig.4-3.Plug-flow element
4.2 APPLICATION OF THE MODELING AND GENERAL GUIDELINES
The above modeling is applied to numerous flow configurations which have
appeared in various Chemical Engineering textbooks as well as additional ones of
particular interest, i.e. impinging-stream reactors [73]. In general, any flow
configurations under consideration will consist of a series of perfectly-mixed
reactors, plug-flow reactors, 'dead water' elements as well as recycle streams, by
pass and cross flow etc., or part of the above.
Our major concern is to study the transient behavior of the configuration by
introducing an ideal pulse input at the inlet. The resulting response curve provides
the RTD of fluid elements leaving the system. As shown below, a certain element
in Eq.(3-20), which is the Markov chain key equation, yields the response to the
ideal pulse input, while another element gives the response to a step change input.
According to [38] the RTD is designated by E where:
Edt (4-24)
represents the fraction of the exit stream of age (the time spent by the element in the
reactor) between t and t+dt. Thus:
fEdt=l Jo
whereas
J" Jo
Edt
350
is the fraction of the exit stream younger than age ti. If C(t) designates the
response of a state (reactor) in its exit to a deha function or impulse introduced into
the reactor, then:
E(t)= ^^^^ (4-25)
f C(t)dt Jo
where the integral represents the area under the response curve. An important
quantity frequently applied in the following, which stems from the response curve,
is the mean residence time of fluid elements in the reactor, tm. This quantity is
defined by:
oo
fc(t)tdt
f C(t)dt Jo
t„,=-^ (4-26)
In treating a certain configuration, the first step is to define the states that the
system can occupy. By a state is meant, the concentration Ci in a perfectly mixed
reactor i or at the inlet or the exit of a plug-flow reactor, that the system (fluid
element) can occupy. The states will be designated by Ci, C2,... whereas the state
space SS, will read:
SS = [Ci, C2,... , Cz, q ] = [1, 2, ..., Z, ^]
The next step is to 'break' the complicated flow configuration into basic
elements which were described above in sections 4.1-1 to 4.1-3; thus, the
probabilities of remaining in a state, pjj, or of moving to a new state, pjk, can be
deduced. This yields the transition probabiUty matrix P.
A further step is to specify the initial concentration of each state, i.e. the
initial state vector S(0). This is defined by:
S(0) = C(0) = [Ci(0), C2(0),..., Cz(0), q (0) ]
351
In the present case, the initial concentration is that of the pulse input introduced into
the reactor. The concentration at some time t = nAt or step n, in the various
reactors, i.e., the states of the system, is defined by the following state vector:
S(n) = C(n) = [Ci(n), C2(n),..., Cz(n), q(n)]
whereas the relation between the above quantities, which are connected by the
transition probability matrix P, is given by:
Ck(n+1) = ]^Cj(n)pjk j = 1, 2,..., Z, ^ j
C(n+1) = C(n)P (3-20)
The above equations are based on Eqs.(2-23) and (2-24). The justification
for applying the above equations to flow systems under consideration lies in the
complete agreement obtained by the Euler integration of the linear equations, Eq.(4-
1), (4-5) and (4-9), and expressing their difference presentation, i.e., Eq.(4-4), (4-
4a) (4-8), (4-8a) (4-10) and (4-lOa) in the form of Eq.(3-20) above. Thus, flow
system can easily be treated by Markov chains where the matrix P becomes
'automatic' to construct, once gaining enough experience. In addition, flow
systems are presented in unified description via state vector and a one-step
transition probability matrix.
Detailed demonstrations of the above guidehnes will be made in typical cases
out of numerous ones presented below. The presentation of the examples is made
according to the following categories:
1) Perfectly-mixed reactor systems (chapter 4.3).
2) Plug flow-perfectly mixed reactor systems (chapter 4.4).
3) Impinging-stream systems (chapter 4.5).
In each case are presented a diagram of the flow configuration, the transition
probability matrix and the transient response to a pulse input. It should be
emphasized that the nomenclature in the text is specific to each case. The
magnitude of C(0) are the quantities on the Ci axis of the response curve
corresponding to t = 0. An important parameter in the computations is the
magnitude of the interval At. This parameter has been chosen recalling that pjj and
352
Pjk should satisfy 0 < Pjj and Pjj < 1 on the one hand, and that Cj versus nAt
should remain unchanged under a certain magnitude of At, on the other. In
addition, a comparison with the exact solution has been conducted in many cases,
which made it possible to evaluate the accuracy of the solution obtained by Markov
chains. The quantities reported in the comparison are the maximum deviation,
Dniax» ai d the mean deviation, Dmean- On the basis of these comparisons, a
representative value of At = 0.01 is recommended, which is the parameter of
Markov chains solution.
353
4.3 PERFECTLY MIXED REACTOR SYSTEMS Perfectly mixed reactors are the key element for conducting chemical
processes and in simulation of complex flow systems. Other synonyms are mixed
reactor, back mix reactor, an ideal stirred reactor and the CFSTR (constant flow
stirred tank reactor). As the name implies, it is a reactor in which the contents are
well stirred and uniform throughout. Thus, the exit stream from this reactor has the
same composition as the fluid within the reactor.
In the following a variety of configurations will be treated, which find
importance in practice and in simulation, as well as elaborate the application of the
model in chapter 4.1. In some cases the derivations are also detailed.
4,3-1 The flow configuration comprising of two perfectly-mixed reactors of
volumes Vi and V2 is demonstrated in Fig.4.3-1. A tracer in a form of a pulse
input is introduced into reactor 1 and is transferred by the flow Qi into reactor 2
where it is assumed to accumulate. Thus, this reactor is a "dead" or "absorbing"
state for the tracer, i.e. C' = 0 in Fig.4-1.
Fig.4.3-1. Two perfectly-mixed reactors
There are two states for which the state space is:
SS =[Ci ,C2] = [ l ,2] (4.3.1a)
where
C(n)= [Ci(n), C2(n)] (4.3-lb)
and according to Eq.(3-20)
C(n+1) = C(n)P
The initial state vector reads:
C(0) = [Ci(0), C2(0)] = [1, 0] (4.3-lc)
i.e. it has been assumed that the initial concentration of the tracer in reactor 1 is
unity.
354
Referring to Fig.4-1, yields for the configuration in Fig.4.3-1 that: Qj = Qi (i
= 2, 3, ...) = Qij = Qji = 0 or ttj = ai (i = 2, 3, ...) = ay = aji = 0, i.e., vessel j is
not considered, thus, a = 1 and ^ = 2. From Eq.(4-12a) follows that ai = P12 = 1
while taking Qi as reference flow. Applying Eqs.(4-16) and (4-17b), considering
the above information, yields pn = 1 - jiiAt pi2 = m^t where jiii = QiA^i, |Ll2 =
QlA^2 according to Eq.(4-7). p22 = 1 according to Eq.(4-22), since the tracer does
not leave reactor 2, thus, being a "dead state" with respect to the tracer. P2i = 0
since the system, i.e. a fluid element of the tracer, can not return from reactor 2 to
1. The above probabilities yield the following transition matrix:
P =
C i = l 1 - fXiAt
0
C2 = 2 ^2At
1 (4.3-ld)
Thus,
Ci(n+1) = Ci(n)[l - mAt ] C2(n+1) = Ci(n)|Li2At + C2(n)
In the numerical solution it was assumed that the reactors are of an identical
volume, thus, |ii = |i2 = M- The transient response of Ci and C2 is depicted in
Fig.4.3-la where the effect of |l = 1/tm is demonstrated. As seen, increasing |X (or
decreasing the mean residence time in the reactor) brings the reactors faster to
steady state.
355
2
1.5
"1 r
H = 2 , n ^ = l -
1 I 1 r
Fig.4.3-la. Ci versus t demonstrating the effect of |LII and ^2
For At = 0.005, the agreement between the Markov chain solution and the
exact solution for Ci = Ci(0)exp(-|Liit) is Dmax = 2.5% and Dmean = 1-2% with
respect to |Xi = 2. The agreement is better for C2.
4.3-2 The flow configuration comprising of two perfectly-mixed reactors of
volumes Vi and V2 is demonstrated in Fig.4.3-2. A tracer in a form of a pulse
input is introduced into reactor 1 and is transferred by the flow Qi into reactor 2;
Q - Ql is the by-pass stream. The tracer is assumed to accumulate in reactor 2,
while flow Qi is leaving the reactor. This configuration simulates a situation
demonstrated in ref.[81], designated as "short-circuit". It should be noted that the
behavior in reactor 1 is independent of what happens in reactor 2.
Q,, *
Q-Qi
Fig.4.3-2. Perfectly-mixed reactors with a by-pass
Eqs.(4.3-la) to (4.3-lc) in the aforementioned case are applicable also in the
present case. The following definitions were made establishing the three
parameters of the system, v, \i and q:
356
v = v - ^ l = v - q = - ^ ( l < q < o o )
Thus, Qi ^
H,= V 7 = q ^ 2 = -q
(4.3-2a)
(4.3-2b)
The transition probability matrix is identical to Eq.(4.3-ld) yielding: Ci(n+l) = Ci(n)[l-mAt] C2(n+1) = Ci(n)^2At + C2(n) (4.3-2c) where |i.i and |l2 are given be Eq.(4.3-2b). The pulse input is introduced into
reactor 1 bringing its concentration to unity. The transient response of Ci and C2
is depicted in Fig.4.3-2a where the effect of q is demonstrated. As seen, increasing
q, i.e. decreasing the flow rate Qi into reactor 1, reduces changes in the initial
concentration of the tracer in the reactor, whereas the concentration of the tracer in
reactor 2 remains almost unchanged, i.e. zero.
1.5
U^l
0.5
n
„_
1 1
/ l
1 1
q = 0.5 -1
J
0 0.5
u
1
0.8
0.6
0.4
0.2
0
1.5 2 0
~
I
h-
1 1
1 1
.--'•""" ^ = J
1 0 0.5 1.5 2 0
Fig.4.3-2a. Cf versus t demonstrating the effect of the by-pass ratio q for |i = 1, V = 2 and At = 0.002
357
4>3-3
Fig.4.3-3 demonstrates two perfectly-mixed reactors. Reactor 1 contains a
"dead water" element of volume Vd [21, p.296] where the other part of volume Vi
is perfectly mixed. The total volume of the reactor is Vi+Vd and the volume of the
second reactor is V2. A tracer in a form of a pulse input is introduced into reactor 1
and is transferred by the flow Qi into reactor 2 where it is accumulating.
Fig.4.3-3. Perfectly-mixed reactors with a "dead water" element in
reactor 1
Eqs.(4.3-la) to (4.3-lc) in case 4.3-1 are applicable also in the present case
as well as the transition matrix given by Eq.(4.3-lf). The following definitions
were made: Vi + Vd ,^ ^ ^ Q,
V =
^ll = -rT- = V|I
( l < V < o o ) |X =
Qi
v,+v,
The parameters of the present configuration are: v, |LI and |X2- The transient
response of Ci and C2, computed by Eq.(4.3-2c), and E(t) given by Eq.(4-25), is
presented in Fig.4.3-3a for a pulse input into reactor 1. The effect of increasing v,
the magnitude of the "dead water" element, is clearly demonstrated for values of
0.5,1 and 2. The integration to obtain E(t) was performed numerically.
358
2
1.5
U" 1
0.5
0
[-/'
I 1 _ . J
/ C 2
\ C andE(t)
1 r ——f
V = 0.5-i
-
-~ 1_
-| r
v = H
2 3 t
3k
U 1\-
0
\ 1
\ ~\
^E(t) \
1 \ L \ fe \
\
1
_:!- 1
1
C 2
^ ... 1
1
v = 2-J
-
1 0 0.5 2.5 1 1.5 2
t Fig.4.3-3a. Ci and E(t) versus t demonstrating the effect of v
f or |X = |i2 = 2
For At = 0.0025, the agreement between the Markov chain solution and the
exact solution for E(t) = |Llvexp(-^vt) [21, p.296] is Dmax = 4.4% and Dmean =
2.5% with respect to v = 2. The agreement is better for the smaller v.
Cases 4.3-4 to 4.3-7 in the following, demonstrate examples modeling long
time scale behavior of real stirred reactors [21, p.269].
4.3-4 Increasing the residence time in the system may be achieved by adding reactor
2 in parallel to reactor 1. The pulse input is introduced into reactor 1 and is finally
accumulated in reactor 3.
359
Fig.4.3-4. Configuration demonstrating long time-scale behavior
There are three states here, i.e.:
5S=[Ci ,C2 , C3]
where
C(n)= [Ci(n), C2(n), C3(n)]
and
(4.3-4a)
(4.3.4b)
(3-20) C(n+1) = C(n)P
The initial state vector reads:
C(0) = [Ci(0), C2(0), C3(0)] = [1.25, 0, 0]
j = l, a = 2, ^ = 3 where Q2 = a2 = 0. Eq.(4-12a) gives P13 = 1, i.e. Qi = qn .
Eq.(4-12b) yield that 1 + a2i = a n + a ^ ; thus a2i = ai2, i.e. Q12 = Q21. Eq.(4-
12c) gives for i = a = 2 that ai2 = 0C2b noting that a2 = 0, that the sum on the left-
hand side does not exist and that P23 = 0- ^^ addition:
Vi=fV V2 = ( l-f)V where V = Vi+V2 and 0 < f < l
Vi and V2 are the volumes of reactors 1 and 2 in Fig.4.3-4. From Eq.(4-7):
|Lii = QiA^l = ji/f |Li2 = QlA^2 = H/(l-f) ^3 = Ql/V3 where ^ = QiA^
From Eq.(4-14) for j = 1, i = a = 2, ^ = 3 and the above information:
Pll = 1 - (Pl3 + 0Li2)\i\At = 1 - (1+ ai2)(Myf)At
From Eq.(4-16) for i = 2, considering the above information:
P22 = 1 - cx2iM'2At = 1 - ai2|ii2At = 1 - a\2(\xJ(l - f))At
From Eq.(4-15) for j = 1 and i = 2:
P12 = oci2^2At = ai2(M/(l - f))At
FromEq.(4-15b)forj = l and ^ = 3:
P13 = CXlB^At = l|l3At
From Eq.(4-17a) for i = 2 and j = 1:
P21 = 0C2imAt = ai2(n/f)At
From Eq.(4-17b) for i = 2 and ^ = 3:
360
P23 = p23^3At = 0|l3At = 0
The above probabilities are summarized in the following transition probability matrix with four parameters f = Vi A , b = ai2, |LI and ILI3:
P =
C i = l 1 - (l+b)(M/f)At
b(|i/f)At
0
C2 = 2 b(n/(l-f))At
1 - b(n/(l-f))At
0
C2 = 3 UsAt
0
1
From Eq.(3-20) above it follows that:
Ci(n+1) = Ci(n)[l - (1 + b)(|i/f)At] + C2(n)[b(n/f)At]
C2(n+1) = Ci(n)[b(My(l - f))At] + C2(n)[l - b(^/(l - f))At]
C3(n+1) = Ci(n)[^3At] + C3(n) (4.3-4c)
For 13 = |Lli and a pulse input of C\{0) = 1.25 introduced into reactor 1, the
transient response of Ci, C2 and C3 computed by Eq.(4.3-4c), is presented in Fig.4.3-4a demonstrating the effect of f = ViA^ and b = ai2.
361
u
1.2
1
0.8
0.6
0.4
0.2
0
f X y W
rf\ r " 1
^ ^ —
1 3_
2
J _. — - - j
- 1
f = 0.2, b = 0.25
-
-
0.05 t
0.1 0
f = 0.2,b = 5
1 3 .-'
^
0.05 t
0.1
Fig.4.3-4a. Ci versus t demonstrating the effect of f and b for |LI = 20
For At = 0.0002, the agreement between the Markov chain solution and the
exact solution for Ci [21, p.303] is Dmax = 0.78% and Dmean = 0.48%. The exact
solution is restricted to Ci(0) = 1/f.
4>3-5 The configuration depicted in Fig.4.3-5 comprises of two interacting reactors
forming a closed recirculation system. The pulse introduced into reactor 1 is
distributed at steady state between reactors 1 and 2.
*
i ^ 2 1 = Q l 2
Fig.4.3-5. A closed recirculation system
There are two states, i.e.:
362
SS= [Ci,C2]
where
C(n)= [Ci(n),C2(n)]
and
C(n+1) = C(n)P
The initial state vector reads:
C(0) = [Ci(0), C2(0)] = [1.25, 0]
Referring to Fig.4-1, yields for the configuration in Fig.4.3-5 that a (= i) = 1
and b = 2. As seen, this is a closed recirculation system, for which Eq.(4-12c)
gives that Q12 = Q21 or au = a2l- If Q12 is selected as a reference flow, Eq.(4-6)
gives a i2 = a2i = 1 and from Eq.(4-7) m = Q12/V1 = |LL/f, |i2 = Q12/V2 = M (l - 0
where 0 < f < 1 and |i = Q^A^ noting that V = Vi + V2 and V1/V2 = f/(l-f) =
|i2/|iil. pii and P22 are computed from Eq.(4-16), pi2 and p2i from Eq.(4-18),
yielding the following transition matrix:
P =
Thus,
C i = l
1 - iLiiAt
jilAt
C2 = 2
^2At
1 - |Li2At
Ci(n+1) = Ci(n)[l - mAt ] + C2(n)[mAt ]
C2(n+1) = Ci(n)[H2At] + C2(n)[l - ^2At ] (4.3-5a)
with two parameters |X and f = ViA^. The transient response of Ci and C2
computed by Eq.(4.3-5a), is presented in Fig.4.3-5a demonstrating the effect of f.
u
1.2
1
0.8
0.6
0.4
0.2
0
h
1 1
1
2..-' '"
1
1 1 f=0.8
1 1
1 J
-H
1
r ' --"''
V 1
1 1 f=0.5
- ""*
1 1
1 J
-
1 0.01 0.02 0.03 0.04 0.05 0
t 0.01 0.02 0.03 0.04 0.05
t
363
0.02 0.03 t
0.05
Fig.4.3-5a. Ci versus t demonstrating the effect of f for |X = 20
For At = 0.0002, the agreement between the Markov chain solution and the exact solution for C2 [21, p.303] is Dmax = 1-3% and Dmean = 0.6%. The exact solution is restricted to Ci(0) = 1/f.
4>3-6 This case is an extension of the previous case for an open system. Reactor 3
was added as a collector for the tracer introduced into reactor 1.
*
•fcct" ' ^ 2 3 '
Fig.4.3-6. An open recirculation system
There are three states in this case for which Eqs.(4.3-4a) and (4.3-4b) are relevant. The initial state vector reads:
C(0) = [Ci(0), C2(0), C3(0)] = [1.25, 0, 0] The system depicted in Fig.4.3-6 is deduced from the general scheme in
Fig.4-1 in the following way. Designating j = 1, a = 2 and ^ = 3 while noting that reactor 2 is the only one belonging to the peripheral reactors i in Fig.4-1. In addition, P13 = a2 = 0. Eq.(4-12a) gives that P23 = 1. Eqs.(4-12b) and (4-12c), noting the above results, give both that a i 2 = l + a 2 i . Vi=fV, V2 = ( l - f)V
364
where V = Vi + V2, yield m = QiA^i = | f, 112 = Q1/V2 = ^i/(l - f). ^3 = QlA^3
where 0 < f < 1, l = QiA^ and ViA^2 = f/(l-f) = m'm-
On the basis of the above results, the probabilities are obtained from the
following equations: pn from Eq.(4-14), pi2 from Eq.(4-I5), p2i from Eq.(4-
17a), p22 from Eq.(4-16) and P23 from Eq.(4-17b), yielding the following
transition matrix with four parameters f =V\/W, b = ai2, |l and 1x3:
P =
1
2
3
Ci = l 1 - b(nyf)At
(b-l)(n/f)At
0
C2 = 2 b(n/(l-f))At
l-b(n/(l-f))At
0
C3 = 3 0 1
^3At
1
Thus,
Ci(n+1) = Ci(n)[l - h(\i/f)m + C2(n)[(b-l)(|Li/f)At]
C2(n+1) = Ci(n)[b(^/(1 - f))At] + C2(n)[l - b(|Li/(l - f))At]
C3(n+1) = C2(n)[|ii3At] + €3(11) (4.3-6a)
Taking 1X3 = ^1, yields the transient response of Ci, C2 and C3 computed by
Eq.(4.3-6a), which is presented in Fig.4.3-6a demonstrating the effect of f and b.
As seen, by increasing b = ai2, and after some time, reactors 1 and 2 acquire an
identical concentration of the tracer, which is eventually vanishing and
accumulating in reactor 3.
Z \ L 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1
t t
365
1
1 r 11
i/ 1/ 1
1 1 1 _
?"'" f = 0.2,b = 5i
0 0.02 0.04 0.06 0.08 0.1 t
0.02 0.04 0.06 0.08 0.1 0 t
Fig.4.3-6a. Cf versus t demonstrating the effect of f and b for |Li = 20
For At = 0.0002, the agreement between the Markov chain solution and the
exact solution for C2 [21, p.303] is Dmax = 6.4% and Dmean = 3.4%. The exact
solution is restricted to Ci(0) = 1/f.
4,3-7 The fluid flow is divided into flows Qi and Q2 as shown in Fig.4.3-7. The
tracer, in a form of a pulse input, is introduced into reactor 1 whereas reactor 3 is a
collector for the tracer.
*
'13
\ 2 \ ^23
Q1+Q2
Fig.4.3-7. An open recirculation system with divided flow
There are three states in this case for which Eqs.(4.3-4a) and (4.3-4b) are
relevant, where C(0) = [Ci(0), C2(0), CsCO)] = [1, 0, 0]
The system depicted in Fig.4.3-7 can be deduced from the general scheme in
Fig.4-1 in the following way. Designate a = 1, b = 2 and ^ = 3 where reactor j is
not considered here. Taking Qi + Q2 as a reference flow rate and defining:
366
a i = (4.3-7a) Q1+Q2
resulted in by applying Eq.(4-12a), while noting that Oj = 0, that:
a, + a2 = 1 = Pi3 + P23 (4.3-7b)
Eq.(4-12c) for i =1, 2, respectively, yields:
aj + a2i = aj2 + P13 and a2 + ai2 = 0,21 + P23 (4.3-7c)
It should be noted that the expression on the RHS is obtained from the one on the
LHS by applying Eq.(4.3-7b).
Designating V = Vi + V2, f = VjA^ where f - 1 = V2A/ , yields by considering
Eqs.(4-5) and (4-7), that ^ = (Qi + Q2)A i = fi/f, ^2 = (Ql + Q2)A 2 = H/(l - 0,
^3 = (Ql + Q2)A 3 0 < f < 1 where ^ = (Qi + Q2)A .
On the basis of the above results, probabilities are obtained by applying the followin;
eters f, b = ai2, q = P13, a i , \i and fi3. Thus, from Eq.(4.3-7c) a2i = b + q - a i ,
P23 = 1- q- Note that for a i = q, a2i = ai2 = b.
P =
Thus,
C i = l l-(q+b)(n/f)At
C2 = 2 b(M/(l-f))At
C2 = 3
qR3At
a2l(H/f)At l-[a2i+(l-q)](M/(l-f))At (l-q)|l3At
0 0 1
Ci(n+l) = Ci(n)pii+C2(n)p2i
C2(n+1) = Ci(n)pi2+ C2(n)p22
C3(n+1) = Ci(n)pi3 + C2(n)p23 + C3(n) (4.3-7d)
Taking 13 = -1. yields the transient response of Ci, C2 and C3 computed by
Eq.(4.3-7d), which is presented in Fig.4.3-7a. demonstrating the effect of q = tti
given by Eq.(4.3-7a).
367
1
L .^ \1 3/
1 / - \ /
pK \ ^\---—
. — -r — ""
H
q = 0.5
-
1 0.1 0.2 0.3 0 0.1 0.2
t t
Fig.4.3-7a. Cf versus t demonstrating the effect of q for |i = 20, b = 1, f = 0.5 and At = 0.0005
0.3
4.3-8 The closed recirculation system shown below comprises of Z perfectly-mixed
reactors of not the same volume. If a tracer is introduced in a form of a pulse input
into the first reactor, the recorder will measure the tracer as it flows the first time,
the second time, and so on. In fact, it measures a tracer which passed through Z
reactors, 2Z reactors and so on, i.e. the superposition of all these signals.
[ 1
Q 2 \ i-1
dbn Q r ^ i+i
o^ z h
Fig.4.3-8. A closed recirculation system
Referring to Fig.4-1 gives the following information for the system in
Fig.4.3-8, i.e. a = 1, b = 2, etc. In addition, since the system closed and reactor j
is not included, ai = pi^ = ay = Oji = 0. Therefore, only Eq.(4-12c) is applicable,
yielding, by considering Eq.(4-8), that ai-i^i = ai^i+i = 1 for i = 2,..., Z-1. From
Eqs.(4-16) and (4-19a), respectively, one obtains that pjj = 1 - jUjAt and Pi,i+i =
|Lli+lAt, i = 1, ..., Z-1; pz,i = |LliAt. The parameter JLLI is given by Eq.(4-7), i = 1,
2,..., Z where Qj is replaced by Q in Fig.4.3-8. The above information yields the
following transition matrix:
368
1
2
3
= i-1
i
Z-1
Z
1 l-^ijAt
0
0
0
0
0
^lAt
2 ^,2^1
l-|i2At
0
0
0
0
0
3
0
l3At
1-^3 At
0
0
0
0
. . .
0
0
i-1
0
0
0
1-h-iAt
0
0
0
i
0
0
0
ijAt
1-^At
0
0
. . .
0
0
z-1 0
0
0
0
0
1-l^z-iAt
0
z 0
0
0
0
0
^zAt
l-^zAt
Thus,
Ci(n+1) = Ci(n)[l - mAt] + Cz(n)[|iiiAt]
Ci(n+1) = Ci.i(n)[mAt] + Ci(n)[l - mAt] i = 2, ..., Z-1
Cz(n+1) = Cz-i(n)[^zAt] + Cz(n)[l - M t ]
A particular solution is obtained for Z = 5 and a constant JLII, i.e. all reactors
have the same volume. For the pulse input C(0) = [Ci(0), C2(0), C3(0), C4(0),
^5(0)] = [1, 0, 0, 0,0] the response curve for Ci to C5 is depicted in Fig.4.3-8a.
Fig.4.3-8a. Ci versus t for |LI = 20
For At = 0.0005, the agreement between the Markov chain solution and the
exact solution for C5 [21, p.295] is as follows: for C5,exact = 0.001, D = 8.9%;
369
C5,exact = 0.01, D = 3.3%; for Cs.exact = 0.1, D = 0.074% and for Cs.exact = 0.2,
D = 0.012%.
4.3-9 The following scheme is an open recirculating system with a recycle of
magnitude Qzi-
^1+^21 Qi+Qzi Qi+Qzi
|q5L_»Jro| ^ Jro]—Jrol—Jrol V « z i
^
-1.i ^l,i+1
^ I zrnz+i I
^ Z 1
Fig.4.3-9. An open recirculation system with recycle
The above system can be deduced from Fig.4-1 as follows: j = 1, a = 2, etc.
where ^ = Z+1; in addition Q2 = Q3 =,..., Qz = 0 or a2 = as = ,..., oCz = 0.
Eq.(4-12a) gives 1 = Pz,z+l-
Eq.(4-12b) gives 1+ ttzi = OL12. If we define recycle by R = Qzi/Qi and
consider Eq.(4-2) for kj = zl , it follows that azi = R and that also ai2 = 1 + R.
Eq.(4-12c) gives for i = a = 2that ai2 = cx23,
for i = 3, 4. ..., Z-1, it gives that ai.i j = ai,i+] = 1 + R,
and for i = Z it is obtained that az-i,z = Pz,z+l(=l) + ^z\ = R + 1.
The following probabilities were obtained:
Eq.(4-14) gives pn = 1- ai2|iiAt = 1- (1 + R)fiiAt
Eq.(4-16) gives pii = 1 - ai,i+i|XiAt = 1- (1 + R)mAt i = 1,..., Z-1
where Pzz = 1 - (Pz,z+1 + cXzOMt = 1 - (1 + R)|LLzAt .
Eq.(4-15) gives pi2 = ai2H2At = (1 + R)|i2At
Eq.(4-18a) gives pi,i+i = ai,i+i|ii+iAt = (1 + R)|ii+iAt i = 1,..., Z - 1
Eq.(4-17b) gives pz,z+l = Pz,z+mz+lAt = l|iz+lAt i = Z
Eq.(4-17a) gives pzi = oCzlHlAt = R|LIIAt
For Z = 5, while assuming that ^1 = 11 and considering the above probabilities,
gives that:
370
P =
1
2
3
4
5
6
1
1- (l+R)M.At
0
0
0
RuAt
0
2
(l+R)jiAt
1- (l+R)^At
0
0
0
0
3 4
0 0
(l+R)nAt 0
1- (l+R)nAt (l+R)(iAt
0 1- (l+R)|iAt
0 0
0 0
5
0
0
0
(l+R)nAt
1- (l+R)nAt
0
6
0
0
0
0
|iAt
1
For the pulse input C(0) = [1, 0, 0, 0,0, 0], the response curve for Ci to C6, computed from C(n-i-l) = C(n)P, is depicted in Fig.4.3-9a.
0.5 1 1.5 0 0.5 1 t t
Fig.4.3-9a. Cf versus t demonstrating the effect of R for ^ -10 and At = 0.01
4,3-10
1 W^
2 — ^ '
3 — ^
4 - ^
^
5
^56 ^- 6
Fig.4.3-10. Perfectly mixed reactors with baclc flow
The equations for the above configuration [21, p.298] can be obtained from Fig.4-1 by designating j = 1, a = 2, b = 3,..., Z = 5 and | = 6. In addition, Qji = Qij = 0 or aij = aji = 0,i = 3,...,Z; Qi = ai = 0,i = 2,...,Z.
371
Eq.(4-12a) gives 1 = ^55; Eq.(4-12b) gives 1 + aii = an', Eq.(4-12c) for i = a = 2 gives ai2 + OC32 = 0 21 + 0 23, hence 1 + a32 = CX23; for i = 3, a23 + a43 = a32 + a34, hence 1 + a43 = a34; for i = 4, a34 + a54 = a43 + a45, hence, 1 + OC54 = a45; for i = 5, a45 = p56 + ^54.
The following probabilities were obtained considering the above results: pll = 1 - ai2|iiAt from Eq.(4-14). Pii = 1 - (ttij + ai,i.i + ai,i+i)^iAt from Eq.(4-16), i = 2, ..., Z - 1, i ^ j , ^.
For i > 2, ay = 0. In addition, Pzz = 1 - (Pz + 0Cz,z.i)|XzAt. P12 = 1 - 0Ci2|Lt2At from Eq.(4-15). Pi,i+1 = 0 i,i+lW+lAt from Eq.(4-18a), i = 2, ..., Z - 1 where from Eq.(4-
17b) pz = Pz l At. Finally, Pi+l,i = ai+i,i|liAt from Eq.(4-18b), i = 1,..., Z - 1. Assuming that ^i = \i, a2i = a32 = a43 = a54 = R, thus, ai2 = a23 = a34 =
OC45 = 1 + R where the rest values of a are zero. Considering the above, yields the following transition probability matrix which can be easily extended to a higher number of states:
P =
1
2
3
4
5
6
1
1- (l+R)^lAt
R^At 1
0
0
0
0
2
(l+R)|iAt
- (l+2R) lAt
R^At
0
0
0
3
0
(l+R)^At
1- (l+2R)MAt
RuAt
0
0
4
0
0
(l+R)MAt
1- (l+2R)nAt
R lAt
0
5
0
0
0
(l+R)^At
1- (l+R)nAt
0
6
0
0
0
0
\iAt
1
For the pulse input C(0) = [1, 0, 0, 0, 0, 0] the response curve for Ci to €5, computed from C(n+1) = C(n)P according to above matrix, is depicted in Fig.4.3-10a. It may be observed that for large value of the back flow R, the concentration of the tracer becomes uniform after some time, i.e. all reactors act as a single reactor.
372
u
1
0.8
0.6
0.4
0.2
^0 0.5 1 1.5 0 0.5 1 1.5 t t
Fig.4.3-10a. Ci versus t demonstrating the effect of R for |LI = 10 and
At = 0.0005
r-
i = l
1
6,
'T' ^——
1
R=10
\
4,3-11 The following configuration simulates a flow pattern in a tubular reactor [21,
p.334, case d] in the presence of side-leaving streams.
^ Q i
'15
^23 d o ! 3 4 1 ^
25 *35
45 doi_2.
u Fig.4.3-11. Perfectly mixed reactors with side-leaving streams
The equations for this configuration can be obtained from Fig.4-1 as
follows, designating j = l , a = 2, b = 3,...,Z = 4 and ^ = 5 while noting that ai =
0, Qii = 0 for i = 2,..., 4.
Eq.(4-12a) gives: 1 = P15 + p25 + P35 + P45. Eq.(4-12b) gives: 1 = P15 +
ai2. Eq.(4-12c) gives: for i = a = 2, an = P25 + OC23; for i = 3, a23 = p35 + a34;
for i = 4, a34 = P45. Thus, if three values of p are given, the rest of the
coefficients are known.
The following probabilities were obtained considering the above results:
Pll = 1 - (Pl5 + cxi2)mAt from Eq.(4-14)
Pii = 1 - (Pi5 + 0Ci,i+i)|iiAt from Eq.(4-16), i = 2, 3 where
P44 = 1 - P45M'4At
In addition:
P12 = 0Ci2|Li2At from Eq.(4-15),
373
Pi,i+1 = «i, i+lR+l^t from Eq.(4-18a), i = 2, 3
Pl5 = PiaM-S' t from Eq.(4-15b), and
Pi5 = Pis^sAt from Eq.(4-19), i = 2, 3, 4
Defining |Xi = QiA^i, = QA^ where V = Vi + V2 +V3 +V4 and Vi = mV, V2 =
nV, V3 = pV, V4 = (1 - m - n - p)V for 0 < m + n + p < 1, while considering the
above results, gives the following transition probability matrix:
P =
1
2
3
4
5
1
Pll
0
0
0
0
2
P12
P22
0
0
0
3
0
P23
P33
0
0
4
0
0
P34
P44
0
5
P15
P25
P35
P45
1
A specific example was obtained for V5 = Vi, i.e. (15 = |ii = \Um. Other
parameters were: P15 = 0.06 and 0.6, P25 = 0-2, P35 = 0.1; m = 0.5, n = 0.3, p =
0.1, and |X = 2. For the pulse input C(0) = [1, 0, 0, 0, 0], the response curve for
Ci to C5, computed from C(n+1) = C(n)P, is depicted in Fig.4.3-lla for At =
0.005.
1
0.8
0.6 r 0.4
0.2
0
\ I 1, 1
\ i = 1
1
0 = 0.06 IS
1
\ 1 1 ,J^ \ 5
V - \ •
A?
1 1 H
a =0.6 15 J
" • - .
0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 t t
Fig.4.3-lla. Ci versus t demonstrating the effect of a i 5
2.5
374
4,3-12 The following configuration is in some aspects similar to the previous one. It
simulates a flow pattern in a tubular reactor [21, p.334, case b] in the presence of
side feedings.
h 12
m ''23
QN JO^ Ji^ 34 45
Fig,4.3-12. Perfectly mixed reactors with side feeding
The equations for this configuration are obtained from Fig.4-1 by
designating a = 1, b = 2,..., Z = 4 and ^ = 5 where reactor j is not active and the
reference flow is Q = Qi + Q2 + Q3 + Q4. As seen, the state space comprises of
five reactors. Eq.(4-12a) gives: a i + a2 + as + a4 = 1 = P45; Eqs.(4-12c) gives:
a i = a i2 for i = 1, a2 + an = a23 for i = 2, as + a23 = a34 for i = 3 and a4 +
a34 = p45. From the above equations it may be concluded that if a i , a2 and as
are specified, the rest of the coefficients are known. Thus, the following
probabilities are obtained from Eq.(4-16):
pii = 1 - ai2mAt P22 = 1 - a2sH2At ps3 = 1 - as4|i3At
P44 = 1 - p45R4At
The following probabilities are obtained from Eq.(4-18a):
P12 = ai2^2At P23 = a23|X3At P34 = as4H4At and from Eq.(4-19)
P45 = P45l 2At
From Eq.(4-7) |ii = QA i where \i = QA^ and Q = Qi + Q2 + Qs + Q4. In addition,
V = Vi + V2 +V3 +V4 where Vi = mV, V2 = nV, Vs = pV and V4 = (1 - m - n -
p)V for 0 < m + n + p < 1. Thus, |Xi = p,/m, |i2 = |Li/n, ILL3 = M-Zp and 114 = |i/(l-
m-n-p). Considering the above results, gives the following transition probability
matrix:
375
P =
1
2
3
4
5
1
Pll
0
0
0
0
2
P12
P22
0
0
0
3 0
P23
P33
0
0
4 0
0
P34
P44
0
5 0
0
0
P45
1
A specific example was obtained for V5 = Vi, i.e. 115 = p,i = |i/m. Other
parameters were: a i = 0.2 and 0.6, a2 = 0.2 and as = 0.1; ai is defined by
Eq.(4-12d) where Qj = Qi + Q2 + Q3 + Q4. m = 0.5, n = 0.3, p = 0.1, and |Li = 2.
For the pulse input C(0) = [1, 0, 0, 0, 0], the response curve for Ci to C5,
computed from C(n+1) = C(n)P, is depicted in Fig.4.3-12a for At = 0.005.
tti = 0.2 ttj = 0.6
I v
- A . .•V-rS
r" r
1 . • - 1 - 1
5 /
-1
\ '* --»~.
4 0
Fig.4.3-12a. Cf versus t demonstrating the effect of a^
4.3-13 The following configuration demonstrates an interacting system whereby it is
possible to increase the mean residence time by applying operating different
streams in the flow system. Reactor 5 is the tracer collector.
376
^ 1 f ^
'12
^
13
Q
24
35 ^
4 A
'43
oloL,242jd& A 4
''45
25
Fig.4.3-13. A model for demonstrating the increasing of the
mean residence time
The above configuration is obtained from the general scheme in Fig.4-1 by
designating j = l ,a = 2, b = 3,... . Z = 4 and ^ = 5 which is the 5th state.
Eq.(4-12a) gives: 1 = p25 + p35 + p45
Eq.(4-12b) gives: 1+ aj i + 0.31 = an + ai3
Eq.(4-12c) gives: an + 042 = P25 + «2l + OC24 for i = 2,
a i3 + a43 = P35 + a3i + a34 for i = 3 and a24 + 034 = P45 + 042 + 043
fori = 4. (4.3-13a)
The above five equations contain eleven coefficients; thus, six of which must be
prescribed in order to solve for the others. The following probabilities of
remaining in the state are obtained from Eqs.(4-14) and (4-16): pii = l - ( a i 2 + oi3)|XiAt
P22 = 1 - (P25 + 021 + a24)^2At
P33 = 1 - (P35 + 031 + 034)^3At
p44 = 1 - (P45 + 042 + a43)|X4At (4.3-13b)
The following probabilities of leaving the state are obtained fi-om Eqs.(4-15):
P12 = ai2|i2At pi3 = ai3|X3At
From Eq.(4-18b):
P21 = 02imAt
FromEq.(4-18a):
P24 = 024mAt
From Eq.(4-17b):
P25 = p25^5At
P31 = 03i|XiAt
p34 = 034|X4At
P42 = a42fl2At P43 = a43li3At
P35 = p35^5At P45 = p45R5At (4.3-13C)
The above probabiUties are arranged in the following matrix:
377
P =
1 2 3 4 5
Pll P12 P13 0 0
P21 P22 0 P24 P25
P31 0 P33 P34 P35
0 P42 P43 P44 P45
0 0 0 0 1 (4.3-13d)
The above results are applied for the two cases below, 4.3-13(1) and 4.3-13(2).
4.3-13(1) The following assumption were made in the present case referring to
Fig.4.3-13: Q12 = Q21 = Q34 = Q43, Ql3 = q35 where Q31 = Q24 = Q42 = q25 =
q45 = 0. Thus, ai3 = P35 = 1, ai2 = Oiil = OC34 = a43 = a where the rest of the
coefficients are zero. The above scheme reduces to the following on:
*
12
'13 ^
Q2I ^34
^
»35 *
MB
^
Also, all reactors are of the same volume, i.e. Vj = Vj = V, thus, it follows from
Eqs.(4-3) and (4-7) that ^j = p,i = ^ = QiA . The matrix given by Eq.(4.3-13d) is
reduced to the following one where Eqs.(4.3-13a) to (4.3-13c) above, are
applicable, considering the coefficients which are zero.
P =
1
2
3
4
5
1 l-(l+a)^At
a|iAt
0
0
0
2 a|jAt
l-a^At
0
0
0
3 HAt
0
l-(l+a)|iAt
anAt
0
4
0
0
a|xAt
l-a|xAt
0
5
0
0
^At
0
1
378
For pulse inputs C(0) = [1, 0, 0, 0, 0] and [0, 1, 0, 0, 0], the response
curves for Ci to C5 were computed from C(n+1) = C(n)P. A specific example
was obtained for p, = 1 where the effects of a = 0.5 and 5 as well as Ci(0) and
C2(0) are demonstrated in Fig.4.3-13(l) for At = 0.005. As seen in cases (a) and
(b) for the same Ci(0) = 1, by increasing a, i.e. the fluid exchange rate between
the reactors, the difference in the concentration between reactors 1-2 and 3-4,
diminishes faster. A similar behavior is observed in cases (c) and (d) for the same
C2(0) = 1. The effect of Ci(0), i.e. the initial location of the pulse, is demonstrated
in cases (a) and (c) for a = 0.5 as well as (b) and (d) for a = 5. As seen, for a =
0.5, the concentration profiles are different, whereas for a = 5 they are identical
with respect to reactors 3 and 4 where rector 1 replaces 2 because of the initial
location of the pulse.
The mean residence time tm in the flow configuration demonstrated above
was computed by Eq.(4-26) for the response in reactor 3. Theeffect of a = ai2
= a2i = a34 = 043 on tm is as follows. For a = 0, only reactors 1 and 3 are active
and tm = 2; for a > 1, i.e. all four reactors are active and tm = 4, demonstrating the
interaction between the vessels on tm-
1
0.8
0.6
0.4
0.2
0
, ' ' ' 1 Ua)
r\'"^ •' ^ 3 A •
t — 1 — • ( • • " 1
C^(0)=l,a=0.5
-J 1 r-
(b)
10 0
n 1 r-
C ( 0 ) = l , a = 5
10
379
u
1
0.8
0.6
0.4
0.2
0
r 1 1
\(C)
2*-.
1 1 .•' ''
1 1 1 1
^ C (0 = 1, a = 0.5 2
'' ''''' "r - ^
n 1 r (d)
•.2 C (0) = 1, a = 5
10 0 10
Fig.4.3-13(l). Ci versus t demonstrating the effect of Ci(0), C2(0) and a for |LI = 1
4>3-13(2) Referring to Fig.4.3-13, the following assumption were made in the present
case: Q12 = Q34 = q25 = q45. Thus, an = OC34 = P25 = P45 = a where the rest of the coefficients are zero. The above scheme reduces to the following one:
[4 2
13 rdoi — \ ^ 1
4 r
i
^45
5J I r
Also, all reactors are of the same volume, i.e. Vj = Vi = V, thus, it follows from Eqs.(4-3) and (4-7) that |Xj = ^i = ji = QiA . The matrix given by Eq.(4.3-13d) in case 4.3-13 above reduces to the following one where also Eqs.(4.3-13a) to (4.3-13c) there are applicable, considering the coefficients which are zero.
380
p =
1
2
3
4
5
1 1-HAt
0
0
0
0
2 a|iAt
l-a|iAt
0
0
0
3 (l-a)iiAt
0
l-(l-a)|jAt
0
0
4 5
0 0
0 a|iAt
ajiAt (l-2a)|iAt
l-ajiAt apAt
0 1
For the pulse input C(0) = [1, 0, 0, 0, 0], the response curve for Ci to C5 is
computed from C(n+1) = C(n)P. A specific example was obtained for |x = 1
where the effects of a = 0, 0.4 and 1 is demonstrated in Fig.4.3-13(2) for At =
0.02. As seen in cases a and c, a = 0 and 1, the concentration profiles are
identical whereas rector 2 replaces 3.
The mean residence time tm in the reactors' configuration demonstrated above
was computed by Eq.(4-26) for the mean concentration of streams q25, qss and
q45. The effect of a = an = OC34 = p25 = p45 on tm is as follows. For a = 0,
only reactors 1 and 3 are active and tm = 2; for 0 < a < 1, all four reactors are
active and tm = 4 whereas for a = 1, again two reactors are active, 1 and 2,
demonstrating the flow interaction effect between the vessels on tm-
a = 0
1.5 0
a = 0.4
381
Fig.4.3-13(2). Ci versus t demonstrating the effect a for |i = 1
4,3-14 The following configuration is a generalized scheme for demonstrating non
ideal mixing vessel [77].
Qa'^^a^Q/Qs
Fig.4.3-14. A generalized model for demonstrating non ideal mixing vessel
The above configuration is obtained from the general scheme in Fig.4-1 by designating j = l ,a = 2, b = 3, ..., Z = 5 and the collector ^ = 6 which is the 6th state. Eq.(4-12a) gives: a2 + as +, 04 + as = p56 Eq.(4-12b) gives: a2i + asi + a4i + asi = an + a n + ai4 + OL\S
382
Eq.(4-12c) gives: tti + ai2 + 0.32 + 0152 = 01,21 + 01.23 + 0025 for i = 2,
as + ai3 + a23 + 043 = 0131 + 032 + a34 for i = 3,
04 + ai4 + a34 + a54 = 041 + a43 + 045 for i = 4,
as + ai5 + a25 + "45 = asi + a52 + 054 + P56 for i = 5
(4.3-14a)
The six equations contain twenty one coefficients; thus, fifteen of which must be
prescribed in order to solve for the others.
The following probabilities of remaining in the state are obtained from
Eqs.(4-14) and (4-16):
Pii = 1 - (ai2 + ai3 + ai4 + ai5)mAt
P22 = 1 - (a2i + a23 + a25)|A2At
P33 = 1 - (asi + a32 + a34)^3At
P44 = 1 - (041 + a43 + a45)M4At
P55 = 1 - (asi + a52 + a54 + p56)^5At (4.3- 14b)
The following probabiUties of leaving the state are obtained:
FromEqs.(4-15): P12 = ai2^2At P13 = ai3|X3At P14 = ai4|l4At P15 = ciis^sAt
(4.3-14c)
From Eqs.(4-19a, 19b):
P21 = a2imAt
P31 = asmiAt
P41 = a4imAt
P5i = a5imAt
P56 = p56^6At
P23 = a23|A3 At
P32 = a32H2At
P43 = a43|X3At
P52 = a52^2At
P25 = a25 t5At
P34 = a34|X4At
P45 = a45^5At
P54 = a54|X4At and from Eqs.(4-20)
(4.3-14d)
The above probabilities are arranged in the following matrix:
P =
1
2
3
4
5
6
1
Pll
P21
P31
P41
P51
0
2
P12
P22
P32
0
P52
0
3
P13
P23
P33
P43
0
0
4
P14
0
P34
P44
P54
0
5
P15
P25
0
P45
P55
0
6
0
0
0
0
P56
1 (4.3-14e)
383
A simplified case appears in ref.[77], treated below in 4.3-14(1).
4,3-14(1) The following assumption were made in the present case while referring to
Fig.4.3-14: Q23 = Q12 = Qsi = Q34 = Q43 = Q54 = Q25 = Q52 = 0, or ais = an =
asi = a34 = a43 = a54 = ais = OC52 = 0- Also, Q2 = Q3 = Q5 = 0, or a2 = a3 =
as = 0 as well as pi6 = P26 = p36 = P46 = 0- However, a4 = 1 since Q4 is the
reference flow rate noting that Qj = Qi = 0. Additional assumptions made were:
ai4 = a4i = a; a is = asi = y; a n = a32 = a2i = p. From Eq.(4.3-14a) for i = 4
and 5, it follows that a45 = P56 = 1. AH reactors are of the same volume, i.e. Vj =
Vi = V, thus, it follows from Eqs.(4-3) and (4-7) that |LIJ = p,i = |i = Q4/V.
Consequently, the above scheme reduces to the following one.
The matrix given by Eq.(4.3-14e) in case 4.3-14 above is reduced to the
following one where also Eqs.(4.3-14a) to (4.3-13d) there are applicable,
considering the coefficients which are zero.
P =
1 l-(a+p-py)|xAt
PnAt
0
a|iAt
7|iAt
1 ^
2 0
1-pHAt
PjlAt
0
0
0
3 p^At
0 l-P|aAt
0
0
0
4 a^At
0
0
l-(l+a)^At
0
0
5 6 THAt 0
0 0
0 0
lAt 0
l-(l+Y)HAt ^At
0 1
384
For a pulse input creating a unit concentration in the reactor, which is introduced into reactors 1 or 3 or 4 or 5, the response curves given below for Ci to C6 is computed from C(n+1) = C(n)P.
1
u
n 1 1 r- 1 I r~
0.8 p. i = 1
0.61
r 0.4
0.2
0
C j ( 0 ) = l
6
- : ' l " • ^ '
V 1 1 1 1 1 1
C,(0)=1
^ \
- - 3 '^
1 6^ r -" """ 1 , -' ^ 1 ^... K. . i . . - -1-; - -. I -r :-'r-''-'r-"-'"i-" -
'
5 ^
' 1
2
\---^/-10 0 10
^4(^1= 1
1 1 1 r
10 0
Fig.4.3-14(l). Cf versus t demonstrating the effect of the introduction location of the tracer
Fig.4.3-14(l) demonstrates the effect of the introduction location of the pulse on the response curves for a = ai4 = a4i = 0.05, P = a n = a32 = OL21 = 0.1, y = 0C15 = asi, i = 1 and At = 0.004.
4.3-15 The following configuration is a generalization the example in [78, p.74]
which aimed at describing various conditions of mixing.
Fig.4.3-15. Parallel reactors model
The above configuration is obtained from the general scheme in Fig.4-1 by
designating a = 1, b = 2,..., Z and the collector ^ which is the Z + 1 state,
Eq.(4-12a) gives:
a i + a2 = 1 = pz-l,^ + Pz, (4.3-15a)
where the reference flow is Qi + Q2 and
ai = Qi/(Ql + Q2) i = l , 2 (4.3.15b)
Eq.(4-12c) gives:
cxi + OL2\ + as i = a i2 + a n for i = 1
0 2 + OC12 + a32 + a42 = ^21 + ^23 + ^24 for i = 2
a i3 + a23 + OC43 + OC53 = a3i + a32 + a34 + OC35 for i = 3
0 24 + 0 34 + a54 + a64 = a42 + a43 + a45 + a46 for i = 4
CX35 + OC45 + a65 + OC75 = CX53 + CX54 + OC56 + OC57 for i = 5
0 46 + OC56 + ^76 + OC86 = a64 + CX65 + a67+ OC68 for i = 6
ai-2,i + ai-i,i + ai+i,i + ai+2,i = ai,i.2 + ai,i.i + ai,i+i + ai,i+2 3 < i < Z-2
az-4,z-2 + 0Cz-l,z.2 + CXz,z-2 = az.2,z-4 + CXz-2,z-l + 0Cz-2,z for i = Z-2
0Cz-3,z-l + CXz-2,z-l + az,z-l = 0Cz-l,z-3 + az-l,z-2 + az-l,z + Pz-l,^ for i = Z-l
0Cz-2,z + az-i,z = az,z-2 + cxz,z-i + Pz, for i = Z
(4.3-15C)
where Z = 5, 7, 9,... and ^ = 6, 8, 9,..., respectively.
The following probabilities of remaining in the state are obtained from Eq.(4-16):
pil = l - ( a i 2 + ai3)mAt
P22 = 1 - (a21 + OC23 + a24)|Ll2At
P33 = 1 - (a3i + a32 + a34 + a35)|i3At
P44 = 1 - (CX42 + CX43 + a45 + a46)l^4At
386
Pii = 1 - (ai,i-2 + ai,i.i + ai,i+i + ai,i+2)mAt 3 i < Z-2
Pz-l,z-l = 1 - (az-l,z-3 + az-l,z-2 + Otz-Lz + Pz-l, )^z-lAt
Pzz = 1 - (az,z-2 + CCz,z-l + Pz,4)Mt
The following probabilities of leaving the state are obtained:
From Eqs.(4-17b):
Pz-14 = Pz-l, fi At pz% = Pz i At
From Eqs.(4-18) for k, i = 1, 2,..., Z-2; k vt i
Pki = cCkiRAt
The above probabilities are arranged in the following matrix:
P =
1
2
3
4
5
Z-2
Z-1
z
1 2 3 4 5 6
Pll P12 P13 0 0 0
P21 P22 P23 P24 0 0
P31 P32 P33 P34 P35 0
0 P42 P43 P44 P45 P46
0 0 P53 P54 P55 P56
(4.3-15d)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
owing
i ...
0 ...
0 ...
0 ...
0 ...
0 ..
0 ..
Pii •• 0 ..
0 ..
0 ..
0 ..
0 ..
matrix:
z-2 0
0
0
0
0
0
0
0
Z-1
0
0
0
0
0
0
0
0
Pz-2,z-2 Pz-2,z-l
• Pz-l,z-2 Pz-l,z-l
• Pz,z-2 . 0
Pz,z-1 0
(4.3-15e)
(4.3-15f)
Z
0
0
0
0
0
0
0
0
Pz-2,z
Pz-l,z
Pzz 0
^ 0
0
0
0
0
0
0
0
0
Pz-1,4
Pz4 1
4.3-15(1) The above model was demonstrated graphically for the following
configuration consisting of six states, i.e. Z = 5 and = 6.
387
Q.+Qo
56
Considering Eqs.(4.3-15a) and (4.3-15c) above gives:
1 = P46+P56 oci + a2i + asi = ai2 + ai3 for i = 1
0C2 + Otl2 + ^32 + a42 = OC21 + a23 + a24 for i = 2
CX13 + CX23 + CX43 + a53 = 0631 + OC32 + a34 + a35 for i = 3
CX24 + CX34 + a54 = a42 + a43 + a45 + P46 for i = 4
a35 + a45 = OC53 + a54 + P56 for i = 5
As observed, there are six equations and eighteen unknowns. It has been
assumed that:
CX12 = a2i = b a23 = a32 = d a43 = a34 = e a45 = a54 = f
a42 = a53 = a3i = R (4.3-15(l)a)
Designating ai and a2, defined in Eq.(4.3-15b), by:
ai = a a 2 = l - a (4.3-15(l)b)
where from the above equations it follows that:
a i = p56 CL2 = P46
a24 = R + 1 - a ai3 = a35 = R + a
Thus, the solution is reduced to six parameter: a, b, d, e, f, R and m = \i. The
following probabilities are obtained from Eqs.(4.3-15d, e, f) and the above
parameters:
pi 1 = 1 - (a + b + R)|iAt P12 = b|LiAt P13 = (a + R)|iAt
P 2 2 = l - ( b + d + R + l - a)^At p2i = bfiAt p23 = d|LlAt
P24 = (R + 1 - a )|LiAt
P33 = 1 - (a + d + e + 2R)^At p3i = R|iAt P32 = d|LiAt P34 = e|iAt
P35 = (a + R)|iAt
P 4 4 = l - ( l - a + e + f + R)|iAt p42 = R|iAt P43 = epAt P45 = f|iAt
P46 = (l-a)^At
P55 = 1 - (a + f + R)|iAt P53 = R|iAt P54 = f|iAt p56 = a|iAt
The above probabilities are presented in the following transition matrix:
388
P =
1
2
3
4
5
6
1 2 3 4 5 6
Pll P12 P13 0 0 0
P21 P22 P23 P24 0 0
P31 P32 P33 P34 P35 0
0 P42 P43 P44 P45 P46
0 0 P53 P54 P55 P56
0 0 0 0 0 1
For the pulse input C(0) = [0, 1, 0, 0, 0, 0], the response curve for Ci to C6
was obtained from C(n+1) = C(n)P. A specific example was computed for |ii = 5,
a = 0.5, b = d = e = f=0.1,At = 0.001 where the effect of the recycle R = 0, 5 is
demonstrated in Fig.4.3-15(l). As observed, by increasing the recycle, reactors 2,
4 and 1, 3, 5 acquire sooner the same concentration.
1
0.8
0.6
0.4
0.2
0
{ 1
h\i = 2
[_ \
1 V. "^
R = 0
y^
> '
. 4 5
,,''•
: • • . . • « _ . . . - - .
1
-\
—
0 0.5
Fig.4.3-15(l). Ci versus t demonstrating the effect of the recycle R
The following configurations, 4.3-16 and 4.3-16(1), are multiloop circulation
models [74-76] for fitting experimental residence time distribution data of
continuous stirred vessels.
4.3-16 The following scheme is a three loop model consisting of six perfectly mixer
reactors which simulates a mixer [75].
389
Fig.4.3-16. Three loop model with inflow to impeller
The above configuration is obtained from the general scheme in Fig.4-la by
designating a = l , b = 2, ...,Z = 6 and ^ = 7 which is the 7th state; the junction is
j . The concentration at this point, Cj, is given by Eq.(4-la).
Eq.(4-12a) gives:
1 = P57 where the reference flow is Qj in Fig.4.3-16.
Eq.(4-12b) gives:
1 + a2j + a4j + a6j = aji + aj3 + ajs (4.3-16a)
Eq.(4-12c) gives:
ocji = 0 12 for i = 1; ai2 = a2j for i = 2; OJB = a34 for i = 3;
a34 = a4j for i = 4; Ojs = P57 + a56 = 1 + a56 for i = 5;
0 56 = 0C6j for i = 6 (4.3-16b)
The following probabilities of remaining in the state are obtained from Eq.(4-
16) where m = QjA i:
Pii = 1 - ai2mAt P22 = 1 - a2j^2At
P33 = 1 - a34|X3At P44 = 1 - a4j|l4At
P55 = 1 - (P57 + CX56)|Li5At P66 = 1 " a6j^6At (4.3-16c)
The following probabiUties of leaving the state are obtained:
From Eq.(4-13c): pkj = ockj/ZkOCjk k = 2,4, 6
From Eq.(4-15): pji = OjiiLiiAt k = 1, 3, 5
From Eq.(4-17b): P57 = p57|i7At
where pi2, P34 and p56 are obtained From Eq.(4-18). The above probabilities are
arranged in a transition matrix given below which is of the kind demonstrated by
Eq.(4-13) with pjj = 0, noting those probabilities among pki and pkj which are zero
according to the scheme in Fig.4.3-16.
390
P =
j
1
2
3
4
5
6
7
J 0
0
P'2j
0
P'4j
0
P'6j
0
1
pjl
Pll
0
0
0
0
0
0
2
0
P12
P22
0
0
0
0
0
3
Pj3
0
0
P33
0
0
0
0
4
0
0
0
P34
P44
0
0
0
5 6 7
Pj5 0 0
0 0 0
0 0 0
0 0 0
0 0 0
P55 P56 P57
0 P66 0
0 0 1 (4.3-16d)
Eqs.(4.3-16a,b) is a set of seven equations with nine unknowns, two of
which must be specified. Thus, it has been assumed that:
Ql2 = Q34 = r» or ai2 = a34 = r/Qj = R, i.e. the circulatory flow in the loop
1-2-j. Therefore, Eqs.(4.3-16a,b) yield:
otji = o 2j = 0Cj3 = a4j = R and a6j = a56 and aj5 = 1 + a6j
where it is further assumed that:
0C6j = OC56 = R thus, aj5 = 1 + R
If all reactors have the same volume V, i.e. ILLJ = |LI = QjA , the probabilities in
Eqs.(4.3-16c,d) satisfying the above matrix, read: Pii = 1 - R^At i = 1, 2, 3, 4, 6 P55 = 1 - (1 + R)jiAt
P12 = P34 = P56 = R^At
Pjl = Pj3 = [(R/(3R + l)] iAt pj5 = [(R + 1)/(3R + l)]jiAt
P'2j = P'4j = P'6j = 1/3 As seen, the solution is a function of the parameters |Li, R and At. For the
pulse input given by Eq.(4-13a), i.e. C(0) = [Cj(0), 1, 0, 0, 0, 0, 0, 0], where
Cj(0) is given by Eq.(4-la), the response curve for Ci to C7 was obtained from
C(n+1) = C(n)P. A specific example was computed for |i = 1, At = 0.005 where
the effect of the circulation R = 0.2, 1 and 5 is demonstrated in Fig.4.3-16a. As
observed, by increasing R, reactors 1-6 acquire sooner an identical concentration
and behave as a single reactor.
391
R=l
1, '
4 0
;• v . .
Fig.4.3-16a. Ct versus t demonstrating the effect of the recycle R
4 . 3 - 1 6 ( 1 )
The following scheme is a simplified version of case 4.3-16 for Qj = 0, i.e. a
closed three loop system.
Assuming all reactors are of the same volume, that the reference flow is one
of the internal flows, and that all flows in the loops are identical, thus all aij = 1,
yields the following transition matrix:
392
P =
j
1
2
3
4
5
6
J 0
0
1/3
0
1/3
0
1/3
1 |j.At
l-|iAt
0
0
0
0
0
2
0
|lAt
l-|iAt
0
0
0
0
3 MAt
0
0
l-\lAt
0
0
0
4
0
0
0
pAt
l-pAt
0
0
5 loAt
0
0
0
0
1-MAt
0
6 0
0
0
0
0
^At
l-HAt
For the pulse input C(0) = [Cj(0), 1, 0, 0, 0, 0, 0], where Cj(0) is given by Eq.(4-la), the response curve for Ci to €5 was obtained from C(n+1) = C(n)P. A specific example was computed for At = 0.01 where the effect of |i = 0.1, 1 is demonstrated in Fig.4.3-16(l). As observed, by increasing (i, reactors 1-6 acquire sooner an identical concentration which is equal to 1/6.
2 4 6 8 1 0 0 2 4 6 8 10 t t
Fig.4.3-16(l). Ci versus t demonstrating the effect of |X
4>3-17 This is another configuration simulating a continuous mixer with an equally
divided flow between the upper two circulation loops [75].
'^^^LA
393
Fig.4.3-17. Three loop model with inflow divided
The derivation of the probabiUties is quite similar to the former case. For the
transition matrix given by Eq.(4.3-16d), assuming that all reactors are of the same
volume and that the reference flow is Q in Fig.4.3-17, the following expressions
were obtained:
Pii = 1 - (1/2 + R)|iiAt i = 1, 2, 3, 4
P66 = 1 - R^At
pjl = pj3 = R|lAt
P2j = P4j = (l/2 + R)/(l+3R)
Pl2 = P34 = (l/2 + R)|llAt
where the rest of the probabilities are zero. The solution is a function of the
parameters \i, R and At, and for the pulse input C(0) = [Cj(0), 1, 0, 0, 0, 0, 0, 0],
where Cj(n) is given by Eq.(4-la), the response curve for Ci to C7 was obtained by
C(n+1) = C(n)P. A specific example was computed for p, = 2, At = 0.01 where
the effect of the circulation R = 0, 1 and 5 is demonstrated in Fig.4.3-17a. As
observed, by increasing R, reactors 1-6 acquire sooner an identical concentration
and behave as a single reactor.
P55=l- ( l+R)^At
P77= 1 Pj5=(l+R)HAt
p'6j = ( l + R ) / ( l + 3 R )
P56 = RflAt P57 = |lAt
u
1
0.8
0.6
0.4
0.2
n
y 1
\ i = 1 - \
" Ao /j<'.
1
1 ^ " '""T——
1
J
A -
, ^
2 3 t
5 0
394
o
1
0.8
0.6
0.4
0.2
0
ll
[\\2
r
1 1 R = 5
7
1 ' n^
1
1
H H
-
2 3 t
Fig.4.3-17a. Ci versus t demonstrating the effect of the recycle R
4,3-18 Another configuration simulating a continuous mixer with non divided inflow
[74] is depicted below.
1^
Fig.4.3-18. Three loop model with non divided inflow
Assuming that all reactors are of the same volume, that the reference flow in
Fig.4.3-18 is Q2, yields the following results by applying Eqs.(4-12a) to (4-12c):
Oji = Ojs = asj = asj = a i2 = R where R = Q5J/Q2 = Qjk/Q2 k = 2, 3
a2j = ocj4 = a45 = R+ 1.
The transition matrix reads:
395
P =
j 1
2
3
4
5
6
J 0
0
P'2j
P'3j
0
P'5J 0
1
pjl
Pll
0
0
0
0
0
2 0
P12
P22
0
0
0
0
3
Pj3
0
0
P33
0
0
0
4
Pj4
0
0
0
P44
0
0
5 0
0
0
0
P45
P55
0
6 0
0
0
0
0
P56
1
where the probabilities obtained from Eqs.(4-14) to (4-19) are: pll = P33 = l-R^At pii = 1 - (1+R)nAt i = 2,4,5
Pj 1= pj3 = R^At Pj4 = (1 +R)^At
p'2j = (1 + R)/(l + 3R) p'3j = p'5j = R/(l + 3R) P12 = R lAt P45 = (1 + R)|ilAt P56 = M.At
Cases a and b in Fig.4.3-18a, demonstrate the effect of the recycle R for jl =
1 and At = 0.004. Cases b, c and d demonstrate the effect of the introduction
location of the tracer, reactors 1, 3 or 5, on the response curves for p, = 1 and R =
5.
u
0.8
0.6
0.4
0.2
n
-
z.
1
i = l
3--^ -5
' ' (a)
R= 1, Tracer in 1 H
H
^ - - ^ . _ ^ -
pj^-^T"" 0.5 1.5
396
1
0.8
0.6
•"0.4
0.2
0
U.
1
r \ 1 \
s 1 m I * ' III! — — I I
[^tl.
1
" * " l - ^ 1
. _ _ _ J
' (0
R = 5. Tracer in3-^
A -^^"1
I 1 0.5 1.5
R = 5, Tracer in 5—1
2 0
hl .3 ,4>:
0.5 JL J .
1.5
Fig.4.3-18a. Q versus t demonstrating the effect of the recycle R
and the introduction location of the tracer in reactors 1, 3 and 5
4,3-18fl) The following configuration is a simplified version of case 4.3-18 where Qj
0, i.e. a closed three loop model.
Assuming that all reactors are of the same volume, that the reference flow is
one of the internal flows, and that all flows in the loops are identical, thus all ay =
1, yields the following transition matrix:
397
P =
j
1
2
3
4
5
J 0
0
1/3
1/3
0
1/3
1
M,At
l-fiAt
0
0
0
0
2
0
M.At
l -^At
0
0
0
3
(iAt
0
0
l -nAt
0
0
4
MAt
0
0
0
l-M,At
0
5
0
0
0
0
^At
1-nAt
Cases a and b in F ig .4 .3-18( l ) , demonstrate the effect of fx « 0.1 and 1 for
At = 0 .01 . Cases b , c and d demonstrate the effect of the introduction location of
the tracer, reactors 1, 3 o r 5 , on the response curves for (i = 1. A s observed, by
increasing \i, reactors 1-5 acquire sooner an identical concentration which is equal
to 1/5.
u
u
1
0.8
0.6
0.4
0.2
0
1
0.8
0.6
0.4
0.2
0
L ^ ' - L - 1 . ^ ^ —I — — r—V.I
1 1 r (b)]
\i - 1,Tracer at IH
10 0 10
(d) |i • 1, Tracer at 5 -
1 . 3 ^ ' ' - l , . - •
C^^ I \ L 4 6
t 10
Fig.4.3-18(l). Ci versus t demonstrating the effect of \JL
and the introduction location of the tracer in reactors 1, 3 and 5
398
4 .3 -19 An extension of case 4.3-18 where the central loop contains a varying
number of reactors designated by 3, 4,..., Z-2, Z > 5, is depicted in Fig.4.3-19.
iHi*"^^-dii
Fig.4.3-19. Three loop system with the central loop of variable number of reactors
From Eq.(4-12a) it is obtained that Pz = 1 where the reference flow is Q2.
From Eq.(4-12b) it follows that:
a2j + ocz-2,j + oczj = ocji + aj3 + aj,z-i where from Eq.(4-12c) it is obtained that:
For i = 1: Oji = ai2; i = 2: 1 + ai2 = aij; i = 3: Ojs = a34; i = 4: a34 = a45; ... i = i: ai-i,i = ai,i+i; ... i = Z-2: az.3,z-2 = otz-2,j; i = Z-1: aj,z-i = az-i,z; i = Z: az-i,z = azj + 1
The above set yields Z (Z > 5) independent equations and Z+3 unknowns; thus three unknowns must be fixed. For example, if Z = 6 and ^ = 7, the process is as follows: Fix an = R, thus, oji = R and a2j = 1 + R;
Fix aj3 = R, thus, a34 = a4j = R; Fix a56 = R + 1, thus, aj5 = R + 1 and a6j = R;
For the general case, the following transition matrix is applicable:
399
j
1
2
3
4
P = i-1
i
i+1
Zr2
Z-1
Z
^
J 0
0
P'2j
0
0
0
0
0
P'z-2,j
0
P'zj
0
1
Pjl
Pll
0
0
0
0
0
0
0
0
0
0
2
0
P12
P22
0
0
0
0
0
0
0
0
0
3
Pj3
0
0
P33
0
0
0
0
0
0
0
0
4 ...
0 ...
0 ...
0 ...
P34 ...
P44 ..
0 ..
0 ..
0 ..
0 ..
0 ..
0 ..
0 ..
i-1
0
0
0
0
0
Pi-l,i
0
0
0
0
0
0
i
0
0
0
0
0
-1 Pi-1,
Pii
0
0
0
0
0
i+1 .
0
0
0
0
0
i 0
Pi,i+1 .
Pi+l,i+l .
0
0
0
0
Z-2
0
0
0
0
0
0
0
0
.. Pz-2,z-
0
0
0
Z-l
Pj,z-1
0
0
0
0
0
0
0
21 0
Pz-l.z-
0
0
z 0
0
0
0
0
0
0
0
0
lPz-l,z
Pzz
0
^
0
0
0
0
0
0
0
0
0
0
Pz^
1
specific expressions for the probabilities, obtained from Eqs.(4-15) to (4-19) are:
Pll = 1 - ai2M.lAt = 1- RmAt P22 = 1 - a2jH2At = 1- (l+R)|j,2At
P33 = 1 - a34|a3At = 1- R|X3At P44 = 1 - a45|i4At = 1- R^At
Pii = 1 - ai,i+miAt = 1- RmAt pz.2,z-2 = 1 - az-2,jHz-2At = 1- (l/3)^z.2At
Pz-l,z-l = 1 - az-i,zM z-lAt = 1- (l+R)|J,z-lAt
Pzz= 1 - (l+azj)^zAt = 1- (l+R)^lzAt
Pjl = ajimAt = RmAt
Pj,z-1 = aj,z-mz-lAt = (l+R)^iz.iAt
P12 = ai2^2At = R|X2At
P45 = a45^5At = R|X5At
Pz-l,z = Oz-Lz^z-lAt = (l+R)^z.lAt
p'2j = a2j/(cxji+aj3+aj,z-i) = (1+R)/(1+3R)
p'z-2o = az-2,j /(aji+aj3+aj,z-i) = R/(1+3R)
p'zj = ttzj /(aji+aj3+aj,z.i) = R/(1+3R)
Particular solutions were obtained for a total number of states of 7,9 and 13, in the
following. The increase in the number of states was in the central loop.
For a total number of states of 7, where the central loop contains the two
states 3 and 4, Fig 4.3-19 is reduced to:
Pj3 = aj3^3At = R^3At
P34 = a34|X4At = R|X4At
Pi,i +1 = ai,i+i|li+iAt = R|Xi+iAt
Pz^=Pz^Mt = |At
400
The following matrix is obtained for reactors of an identical volume V, i.e. (ii =|i. =
Q2A :
j
1
2
= 3
4
5
6
7
J 0
0 1+R 1+3R
0 R
l+BR
0 R
1+3R
0
1
RM,At
1-R^At
0 1
0
0
0
0
0
2
0
R iAt
- (l+R)^At
0
0
0
0
0
3
R^At
0
0
1-R^At
0
0
0
0
4
0
0
0
R lAt
1-R^At
0
0
0
5
(l+R)^At
0
0
0
0
1- (l+R)^At
0
0
6
0
0
0
0
0
(l+R)M.At
1- (l+R)^At
0
7
0
0
0
0
0
0
M.At
1
For a total number of states of 9, where the central loop contains the four
states 3 to 6, Fig 4.3-19 is reduced to:
The following matrix is obtained:
fej^
401
j
1
2
3
4
P = 5
6
7
8
9
J 0
0 l+R 1+3R
0
0
0 R
1+3R
0
1+3R
0
1
R iAt
1-R^At
0
0
0
0
0
0
0
0
2
0
RM-At
1-l+R)^lAt
0
0
0
0
0
0
0
3
R^lAt
0
0
l-R^iAt
0
0
0
0
0
0
4
0
0
0
RM,At
l-R|iAt
0
0
0
0
0
5
0
0
0
0
R^At
l-RjiAt
0
0
0
0
6
0
0
0
0
0
R^iAt
l-R^iAt
0
0
0
7
(l+R)^At
0
0
0
0
0
0
1-l+R)^iAt
0
0
8
0
0
0
0
0
0
0
(l+R)^At
1-l+R)^iAt
0
9
0
0
0
0
0
0
0
n
1
For a total number of states of 13, where the central loop contains the eight
states 3 to 10, Fig 4.3-19 is reduced to
irni >^[db|;^|db| ^idbl >JoDl
The following matrix is obtained while designating p = RjiAt, q = 1 - R|LiAt,
r = (l+R)M,At and s = 1 - (l+R)^At.
402
j 1
2
3
4
5
P = 6
7
8
9
10
11
12
13
J 0
0 l+R 1+3R
0
0
0
0
0
0
0 R
1+3R
0 R
1+3R
0
1
P
q 0
0
0
0
0
0
0
0
0
0
0
0
2
0
P s
0
0
0
0
0
0
0
0
0
0
0
3
P 0
0
q 0
0
0
0
0
0
0
0
0
0
4
0
0
0
p
q 0
0
0
0
0
0
0
0
0
5
0
0
0
0
p
q 0
0
0
0
0
0
0
0
6
0
0
0
0
0
p
q 0
0
0
0
0
0
0
7
0
0
0
0
0
0
p
q 0
0
0
0
0
0
8
0
0
0
0
0
0
0
p
q 0
0
0
0
0
9
0
0
0
0
0
0
0
0
p
q 0
0
0
0
10
0
0
0
0
0
0
0
0
0
p
q
0
0
0
11
r
0
0
0
0
0
0
0
0
0
0
s
0
0
12
0
0
0
0
0
0
0
0
0
0
0
r
s
0
13
0
0
0
0
0
0
0
0
0
0
0
0 HAt
1
Fig.4-3.19 demonstrates response curves of various states (reactors) for a
pulse introduced in state 1 raising its concentration to unity. The parameters of the
graphs are the recycle rate R (= ai2 = Oji = Ojs = a34 = a4j), 0.5 and 5, and the
number of states in the central loop, 2,4 and 8 corresponding to a total number of
states 7, 9 and 13, respectively. Common data were |i = 1 and At = 0.01. The
general trends observed were that the approach towards equilibrium becomes
slower by increasing the number of states and that the streams attain faster a
uniform concentration by increasing R. The effect of the number of states is
reflected by curves 4 (case a), 6 (case c) and 10 (case e) corresponding to the exit
reactors of the central loop, as well as the overall effect reflected in curves 7 (case
b), 9 (case d) and 13 (case f) corresponding to the final collector of the tracer. The
effect of R is demonstrated in cases a and b, c and d as well as e and f.
403
u
0
u
1
0.8
0.6
0.4
0.2
0
1
0.8
0.6
0.4
0.2
0
1
0.8
0.6
0.4
0.2
%
Fig.4
\
p-
k
k
1 1 1 1 Number of states = 2, R = 0.5
^ ,
^ \ . 7 2 .
(a)J
-]
r-
, 1 1 1 1 1 Number of states = 2, R = 5
i t\ - - ' ' ' ' h, - '::. . ^
m^
-
-
—
\j • ' \ \ 1 1 2 3
t 5 0 1 2 3
t
1 1 1 1 1 \ Number of states = 4, R = 0.5 (c)
; ^ "
\-
L 2
V . r t ' - ' '
-
" --. ?' ~ . _ _ _ , . ' " " " " • - . . , -
^,1 _ -1 I ^ ~
'
0 2 3 t
n 1 i \ Number of states = 4, R = 5 (d)
/V
5 0 1 I I 1 I
2 3 t
V i l l i \ Number of states = 8, R = 0,5
r ^ M
1 " "--- 13 1 2 > ^ '
[/ , - t - ' - 1 ^10 1 1
(e)
J
A _
—
1 \ 1 \ Number of states = 8, R = 5 (f)
12>
13.
10 / .. ' , : . j " • - 7
I -•-' I 1 2 3 4 5 0 1 2 3 4 5
t t
3-19. Ci versus t demonstrating the effect of the number of states in the central loop and the recycle R
404
4.3-19(1) The following configuration is a simplified version of case 4.3-19 where Q2
= 0, i.e. a closed three loop model.
lilHi*-ii*-^
Assuming that all reactors are of the same volume, that the reference flow is
one of the internal flows, and that all flows in the loops are identical, i.e. all aij =
1, yields the following transition matrix for p = pAt and q = 1 - nAt:
j
1
2
3
4
P = i-1
i
i+1
7/1
Zrl
z
j
0
0
1/3
0
0
0
0
0
1/3
0
1 1/3
1
P q 0
0
0
0
0
0
0
0
0
2
0
P q 0
0
0
0
0
0
0
0
3
P 0
0
q 0
0
0
0
0
0
0
4 ..
0 ..
0 ..
0 ..
p ..
q ..
0 ..
0 .
0 .
0 .
0 .
0 .
. i-1
0
0
0
0
0
. q 0
0
0
0
.. 0
i
0
0
0
0
0
P q 0
0
0
0
i+1 ..
0 ..
0 ..
0 ..
0 ..
0 ..
0 ..
P ..
q .
0 .
0 .
0 .
Z-2
0
0
0
0
0
0
0
0
q 0
0
Z-1
p 0
0
0
0
0
0
0
0
q 0
z 0
0
0
0
0
0
0
0
0
p q
Fig.4-3.19(1) demonstrates response curves of various states (reactors) in the
closed system, computed from C(n+1) = C(n)P for a pulse introduced in state 1
raising its concentration to unity. The parameter of the graphs is the number of
405
states in the central loop, 2, 4 and 8 corresponding to a total number of states 6, 8 and 12, respectively. Common data were \i = 1 and At = 0.01. The general trend observed is that the approach towards equilibrium, i.e. concentration of l/(number of states) becomes slower by increasing the number of states. The effect is reflected by curves 4 (case a), 6 (case b) and 10 (case c) corresponding to the exit reactors of the central loop. In addition, the approach versus time towards equilibrium in case c is slower than in case a where case b is intermediate.
u
1
0.8
0.6
0.4
0.2
0
u "1 1 1 Number of states = 2 (a)| Number of states = 4 (b)
10 0 10
U
1
0.8
0.6
0.4
0.2
0
1
\ — I
A - X^ il' c.
-^127
1 1 1 Number of states = 8
^"""--^ 10
1..- \--^"\'^
{0)1
4
-
-
4 6 t
10
Fig.4.3-19(l). Ci versus t demonstrating the effect of the number of states in the central loop
406
4.4 PLUG FLOW-PERFECTLY MIXED REACTOR SYSTEMS
Plug flow reactors are widely used in industry. In simulation of chemical
processes, such a reactor represents, many times, a certain time delay element in the
process between two stages. A plug flow reactor is characterized by the fact that
the flow of fluid through the reactor is orderly with no element of fluid overtaking
or mixing with any other element ahead or behind. The necessary and sufficient
condition for plug flow is for the residence time in the reactor to be the same for all
elements of fluid. In the following, several flow configurations comprising of plug
flow reactors will be treated.
The treatment of the plug flow-perfectly mixed reactor systems, generally,
comprises the following steps: a) Defining the stages for the transfer process of the
tracer between the states (reactors) according to the residence times, tp, of the plug
flow reactors in the flow system, b) Establishing the transition probability matrix
for each stage, c) Determination of the distribution coefficients aij, Pij from mass
balances given by Eqs.(4-12a), (4-12b) and (4-12c). d) Determination of transition
probabilities by Eqs.(4-14) to (4-19). e) Computation of the response curves
according to C(n+1) = C(n)P. At each step ^ C j = 1 must hold for all states for a
unit concentration pulse.
In the following, a variety of configurations will be treated, which find
importance in practice and in simulation, as well as elaborate the application of the
model in chapter 4.1.
4>4-l The basic scheme of a single plug flow reactor with recycle is shown below
in Fig.4.4-1. Such a reactor may be described also as a combination of perfectly
mixed reactors. Other simplified configurations are described in the following
cases, 4.4-1(1) and 4.4-1(2). The states are the concentration of the tracer in the
perfectly mixed reactors, i.e. SS = [Ci, C2, ..., C^ ]. The residence time in the
plug flow reactor is tp = V/Qi where V is the volume of the reactor. The recycle
stream is Qz2-
407
ifrsiitzitti I z-ll z L 3 VZ2
Fig.4.4-1. Basic scheme of a plug flow reactor composed of perfectly mixed reactors with recycle
Referring to Fig.4-1, yields for the above configuration that: Oj = ai (i = 2,
3,...) = ay = ttji = 0, i.e., reactor j is not considered; hence, a = 1, b = 2,... and ^
= Z+1.
From Eq.(4-12a) follows that pz^ = 1
From Eq.(4-12c) follows: for i = 1 ai = an = 1
fori = 2 ai2 + az2 = 0C23 or alternatively
1 + R = a23 where az2 = Qz2/Ql = R for i = 3 a23 = a34 = 1 + R for i = i ai-i,i = ai,i+i = 1 + R
fori = Z az.i,z = Pz + azi= 1+R
The corresponding probabilities are obtained from Eqs.(4-16), (4-18), (4-18a) and
(4-19) yielding the following probability matrix:
1
2
3
P = i
i+1
Z-1
Z
%
where
P22 =
Pii =
1 l-^ijAt
0
0
0
0
0
0
1 0
2 ijAt
P22
0
0
0
0
R ljAt
0
1 - (l+R)^2At
1 - (l+R)mAt
3
0
P23 P33
0
0
0
0
0
. . .
. . .
. . .
0
0
. . .
P23
i
0
0
0
Pii
0
0
0
0
i+1
0
0
0
0
Pi,i+1
Pi+l,i+l
0
0
0
0
= (l+R)|Ll3At
Pi,i+1 = (1+R)W+
. . .
. . .
. . .
lAt
z-1 0
0
0
0
0
Pz-l,z-l
0
0
z 0
0
0
0
0
Pz-l.z
Pzz
0
^
0
0
0
0
0
0
Pz
1
408
Pz-i,z-i = 1 - (l+R)^z-iAt pz-1,2 = (l+R)Mt Pzz = 1 - (l+R)|LizAt pz = iLi At
The parameters of the solution are: i = QiA i (i = 1,.., Z+1) and R. If all reactors
have the same volume, |ii = (Z-l)|i = (Z-l)/tp where [i = QiA^p; Vp is the volume
of the plug flow reactor and tp is the mean residence time of the fluid in the reactor.
In the numerical solution, the plug flow reactor was divided into ten perfectly
mixed reactors of identical volumes, i.e. Z = 11, ^ = 12 and i = \i. The transient
response of Q (i = 1, 2, 5, 8, 11 and 12) for C(0) = [1, ..., 0] is depicted in
Fig.4.4-1, cases a to d. For cases e and f, C(0) = [0, 1,..., 0], i.e. the tracer was
introduced into reactor 2. The effect of the following parameters was explored for
cases a to d: R = 0 and 10, ^ = 10,50 and 100 where At = 0.0005 for the large m
and 0.004 for the smaller ones. The quantities R = 0, 5, |Lii = 500 and At =
0.00004 were applied in cases e and f.
The effect of an identical value of m = 10 and 50 is demonstrated in cases a
and b. As seen the curves attain faster the steady state for |LLi = 50. The effect of
the recycle R is demonstrated in cases c and d, noting that for R = 10 all transient
responses for reactors i = 2,..., 11 are almost identical. This is because the plug
flow reactor behaves as a single perfectly mixed reactor. Note that in the above
cases the tracer was introduced into reactor 1. In cases e and f the tracer was
introduced into reactor 2, i.e. directly into the entrance of the plug flow reactor.
The response at the exit of the reactor is that of reactor 11, as shown. For the case
of R = 0, a maximum in the response is observed for reactor 11 after about 0.02
time units. Indeed, this figure is the mean residence time of the plug flow reactor
noting that tp = (number of perfectly mixed reactors)(l/p,i) = 10(1/500) = 0.02.
When the recycle is R = 5, as seen in case f, the behavior of the plug flow reactor
becomes perfectly mixed.
409 1
1 - 1
I
8
1 1
1 2 / '
- - -f 1 (b)
^.= 50 a = i....,i2) R = 0 H
' - - I 2 0 0.1 0.2 0.3
t 0.4 0.5
U
1
0.8
0.6
0.4
0.2
0 I
1
0.8
0.6
*0.4
0.2
0
\ . 1
f\2
t K\\
b ; / ' •
1
11
• •
- 1 V:
1
/""
1 12.
^ i
(i = R =
>. ' * ^ - ^
. . . I . . .
= ^^12=
= 100 = 2, ..., :0
* * - L : : ^
m
10 -
1 1 ) -
-
—•»>'»-..:„m
\ . '
t • I 1
1
^ - 1 1
1
1 1 12. (d)
\i. = 100 (i = 2, ..., l l ) -R=10
0.1 0.2 0.3 t
0.4 0.5 0
[
f V -r 1 1
(e)
1.= 500 ~i
(i = 2, ..., 11)J R = 0
1 0.01 0.02
t
0 0.1
L
h
1 1
0.2 0.3 0.4 0 t
(Ol
\^r 5 0 1 0 = 2, ...,11) J R = 5
1 1 0.03 0.04 0 0.01 0.02
t 0.03 0.04
Fig.4.4-1. Ci versus t demonstrating the effect of the recycle R and
4>4-l(l) A simplified approach for treating a plug flow reactor by comparison to case
4.4-1 above, which avoids the division of the reactor into perfectly mixed reactors, is demonstrated in Fig.4.4-l(l). The states are the concentration of the tracer in the two perfectly mixed reactors 1 and 2, at the inlet and exit of the plug flow reactor. The residence time in the plug flow reactor is tp = Vp/Qi where Vp is the volume of
410
the reactor. This case simulates the situation demonstrated in ref.[81], designated
as 'partial mixing and piston flow'.
3i^ 12 f o b ^ = 2|
Fig.4.4-l(l). A simplified scheme of a plug flow reactor
In establishing the probability matrix, distinguish is made between two stages
in the transfer process of the tracer from one state to the other, i.e., 0 < t < tp and
t > tp. For 0 < t < tp, Ci = Ci(0) and C2 = C2(0) whereas for t > tp, the following
matrix is apphcable:
P = 1 2
1 2
Pll P12
0 1
The probabilities are determined as follows ignoring the presence of the
plug flow reactor. Referring to Fig.4-1, yields for the configuration in Fig.4.4-
1(1) that: Qj = Qi (i = 2, 3,...) = Qy = Qji = Oor aj = ai (i = 2, 3,...) = ay = aji
= 0, i.e., vessel j is not considered, hence, a = 1 and ^ = 2. From Eq.(4-12a)
follows that a i = Qi/Qi = P12 = qi2/Ql = 1- Applying Eqs.(4-16) and (4-17b),
noting the above results, yields pn = 1 - [XiAt pi2 = |X2At where m = QiA^i, |Li2
= Ql A^2 according to Eq.(4-7). The parameters of the solution are m , |L12 and tp.
In the numerical solution it has been assumed that the reactors are of the same
volume, thus, m = |i2 = |Li. The transient response of Ci and C2, for C(0) = [1,0]
while C(n+1) = C(n)P is depicted in Fig.4.4-l(la) where the effect of i = QiA^i =
10,20 and 200 is demonstrated for tp = 0.1 and At = 0.001. As seen, increasing
|i (or decreasing the mean residence time in the reactor) brings the reactors faster to
a steady state. The results for \i = 200 are interesting showing an immediate
response of the reactors. For constant Qi, a large |LI indicates that the volimie of the
reactor is very small, i.e. the mean residence time of the fluid in it is extremely
short. Thus, the case for |Li = 200 simulates a situation where reactors 1 and 2 are
411 reduced to 'points', usually for sampling, or the location of some measurement
device.
u
1
0.8
0.6
0.4
0.2
0
-
-
-
1 i = l
2 _ 1
1
/
1
1 1 ^l= loH
-
V "^
-
1
0.05 0.1 t
0.15 0.2 0
U
1
0.8
0.6
0.4
0.2
0
1 1 i = l !
-
-
-2 1
/ /
i 1 I \
I
1 J
-J J
i = 200
-
1
0.05 0.1 t
0.15 0.2
Fig«4.4-l(la). Ci versus t demonstrating the effect of |Li
4 . 4 - 1 ( 2 )
An extension of case 4.4-1(1) is demonstrated in Fig.4.4-1(2). The scheme
comprises of a plug flow reactor, a feeding reactor 1 and a collector 4, as before.
In addition, measurement points of the concentration, simulated by 'small'
perfectly mixed reactors 2 and 3, were added. The residence time of the fluid in the
plug flow reactor is tp. These reactors make it possible to include the recycle
stream Q32, impossible to add in case 4.4-1(1). Reactors 2 and 3 simulate also
reactors 2 and 11 in example 4.4-1.
412
*
32
Fig.4.4-1(2). Plug flow reactor with concentration measurement 'points* 2 and 3
The two stages in the transfer process of a tracer between the states are
expressed in the following transition matrices:
0 < t <tp: t > t n
P =
P =
1
2
3
4
1
Pll 0
0
0
2
P12
P22
P32 0
3
0
P23
P33 0
1 2 1 2 3 4
1 I Pll P12 1 Pll P12 0 0 2 1 0 1 1 2 I 0 P22 P23 0
P34
1
(4.4-1(2))
Referring to Fig.4-1, yields for the above configuration that: Oj = ai (i = 2,
3,...) = ay = ttji = 0, i.e., reactor j is not considered; hence, a = 1, b = 2,... and ^
= 4.
From Eq.(4-12a) follows that P34 = 1
From Eq.(4-12c) follows:
for i = 1 a i = ai2 = 1
fori = 2 ai2 + (X32 = a23 or altematively
1 + R = a23 where a32 = Q32/Q1 = R is the recycle
for i = 3 a23 = P34 + a32 = 1 + R
The probabilities corresponding to the matrix given by Eq.(4.4-1(2)) are
obtamed from Eqs.(4-16), (4-17b), and (4-18a); they read:
pil = l - | I lAt pi2 = |Ll2At
P22 = 1 - (l+R)|Ll2At P23 = (l+R)|l3At
P33 = 1 - (l+R)|l3At P32 = R|l2At P34 = |l4At
413
|Lii = Qi/Vi, i.e. the reactors are of different volumes. The parameters of the
solution are: |Xi (i = 1,..., 4), the recycle R and the residence time tp in the plug
flow reactor the effect of which is depicted in Fig4.4-1(2) for C(0) = [1,0, 0, 0]
and At = 0.004.
Cases a and b demonstrate the effect of the residence time tp in the plug
flow reactor, 0.2 and 1. As seen, in case a tp was too short for the pulse
introduced in reactor 1 to reach its maximum value of unity. The effect of the
recycle, R = 0 and 10 is demonstrated in cases a and c. As observed, the
concentration of the tracer in reactors 2 and 3 becomes identical at tp = 0.2; the
concentration are different in the absence of a recycle. The effect of |Lli is
demonstrated in cases a, d and e. In case a and d all |ii = Qi A i = are equal; in case
a |li = 10 (i = 1, ..., 4) and in d |ii = 100 (i = 1, ..., 4). Note that a large |ii
indicates a smaller volume of reactor for a constant Qi. Thus, it is clearly
demonstrated that the system attains faster its steady state values for m = 100.
Case e demonstrates the effect of large ^i =100 for reactors 2 and 3, by
comparison to the behavior of reactors 1 and 4 for which m = 10. It is observed
that the response of reactors 2, 3 is faster than the response of reactors 1 and 4. It
should be noted that if the tracer is introduced into reactor 2, the behavior is
similar, however the concentration in reactor 2 remains constant and begins to
change only after tp time units have passed.
1.2,
1
0.8 I
0.4
0.2
0
IA u I' 1'/--
4 , ' /
/ /
/ /
— 1 ^ r ^
a = 1,... R = 0 t =0.2
p
{•-::=:;....-T4 .
(a)|
.4)-j
—
(b)|
\ ^
I \ --^1
\ / (i=l,...,4)J
\ ; R=o J A/\ t =1 '
3 A\ P
/, ^ s , 0.2 0.4 0.6
t 0.8 0.5 1.5
414
Q / f
i
1 v^ 1
I 4 ^ 1 / 1 1 \ 1 1 . 1 / 11
i li V
(d)l
^.= 100 "i
(i = l 4)-J
R = 0 J t =0.2 1 p
- 1 0.2 0.4 0.6 O.J
t 10,
1 0 0.1 0.2 t
0.3 0.4
(e)
^ = 1= 10 ^^=^^=100-^ R = 0
Fig.4
0.1 0.2 0.3 0.4 t
4-1(2) . Ci versus t demonstrating the effect of R, tp and m
4,4-1(3) A simplification of case 4.4-1(2) is demonstrated in Fig.4.4-1(3) simulating
the configuration in ref.[80, p.761]. Reactor 2 was added to take into account the
recycle stream Q12. The response of this reactor can be controlled by the
magnitude of ^2- If reactor 2 is a measurement 'point' of the concentration, i.e. a
very small reactor, |LL2 should be assigned a relatively large value.
Fig.4.4-1(3). A plug flow reactor with a closed loop
415 The following transition matrices are applicable:
0 < t <tp:
P =
1
Pll 0
3
P13
1
t > t „
P = 1
2
3
1
PU
P21 0
2
Pl2
P22 0
3
P13 0
1
Applying Eqs.(4-12a) to (4-12c) gives that an = (X2h Pl3 = 1- Eqs.(4-14)
to (4-17a) yield the following probabilities:
For 0 < t < tp: pn = 1 - |LliAt pis = iiisAt
For t > tp: Pll = 1 - (l+ai2)|XiAt pi2 = ai2^2At pi3 = isAt
P22 = 1 - ai2H2At P21 = ai2mAt
The parameters of the solution are: |ii (i = 1,..., 3), the recycle R = ai2 and
the residence time tp in the plug flow reactor. The effect of R = 0 and 5 is depicted
in Fig4.4-l(3a), cases a and b, for C(0) = [1, 0, 0, 0], tp = 0.5 and At = 0.005. m
= 2 has been assumed for all reactors. The effect of tp = 0.5 and 2, is demonstrated
in cases b and c. It is observed that the concentration in reactor 1 diminishes before
the tracer has reached reactor 2 due to the long residence time in the plug flow
reactor.
1
0.8
0.6
0.4
0.2
0
- • . i = l
' ^
, '3
.
(a)
R = 0,t =0.5 p J
1 i i 1 0.5
-M
^
1 '^ /•—..
1
' 3
1
- - - • • ' * ' (b)^
R = 5,t =0.5 p J
-
1 1 1 1.5
t 2.5 0 0.5 1 1.5
t 2.5
416
u
1
0.8
0.6
0.4
0.2
0
- M
h -' ''
2
^ ' 3
' .,
ic)\
R = 5,t = 2 p J
1 1 i 1 1 0 0.5 1 1.5 2 2.5
t Fig.4.4-l(3a). Cf versus t demonstrating the effect of R and tp
4,4-2 Fig.4.4-2 demonstrates a plug flow reactor containing a "dead water" element
of volume Vd where the active part is of volume Vp. The system contains also two
perfectly mixed reactors 1 and 2. A tracer in a form of a pulse is introduced into
reactor 1 and is transferred by the flow Qi into reactor 2 where it accumulates.
SuJ* ^^
iliii:-'::!-. ^ .: —n ^ r
Fig.4.4-2. Plug flow reactor with a **dead water" element
The following definitions were made:
V = — % (l<V<c<.) 11 = Vp + Vd
yielding that
V Qi ~ Jiv , = ^ = ^
This case is similar to case 4.4-1(1). For 0 < t < tp, Ci = Ci(0) and C2
C2(0) whereas for t > tp, the following matrix holds:
417
P =
1 1 - |liAt
0
2
1
where |Xi = QiA^i, i = 1,2. The parameters of the solution are: |ii, \i and v where
C(0) = [1, 0].
Fig.4.4-2a demonstrates results for v = 1 and 10 corresponding to tp = 1
and 0.1, receptively, while |X = 1. Other data are: (li = fi2 = 10 and At = 0.01 for v
= 1 and 0.001 for v = 10. It should be noted that by increasing v, the effective
volume of the reactor is decreased for a constant value of Qi, hence tp is decreased.
This is clearly reflected in the figure below.
u
1
0.8
0.6
0.4
0.2
0
_
-
i = l
/ 1 1
x"""" 2 / ^
\l J
k " •,
1
-
-
\ /
i 1
1 1
2 ^^
"--..,
1
^- ^
v= 10
-
1 0.4 0.5 0 1 2 0 0.1 0.2 0.3
t t Fig4.4-2a. Cf versus t demonstrating the effect of v
4,4-3 The flow configuration in Fig.4.4-3 comprises of two perfectly-mixed
reactors of volumes Vi and V2 and a plug flow reactor of volume Vp. The tracer is introduced into reactor 1 and is transferred by the flow Qi into reactor 2; Q - Qi is the by-pass stream. The tracer is accumulated in reactor 2, while flow Qi is leaving the reactor.
^Hi Q-Qi
Fig.4.4-3. Plug flow reactor reactors with a by-pass
418
The following definitions were made:
q = - ^ ( l<q<oo) 1^ = —
yielding that
This case is similar to case 4.4-1(1) yielding the following matrix for t > tp:
2
1 2 1 - |LiiAt |i2At
0 1
where for 0 < t < tp, Ci = Ci(0) and C2 = C2(0). |ii = Qi/Vi, i = 1, 2. The
parameters of the solution are: m, \i and q where C(0) = [1,0].
Cases a and b in Fig.4.4-3a demonstrate the effect of q. By increasing q,
i.e. the by-pass stream, the mean residence time in the tubular reactor tp is
increased from 0.1 to 0.5 time units. Case a and c demonstrate the effect of \i\ =
|X2. By increasing this quantity from 10 to 500, the mean residence time in the
perfectly mixed reactor is decreased, and the response becomes instantaneous. In
the computations, At = 0.002 for q = 1 and 0.01 for q = 5.
1
0.8
0.6
0.4
0.2
0
1
-
0
i = l \
'\
/
/ / 1
0.1
^
'\
^'.^
1 1 0.2 0.3
t
2 _^ M (a)!
n = n=ioH
q=l H
-
! 0.4 0 0.5 0
k
H
1 /^
'. /
f •
/ \ /
^ ' (b)]
u = u= 10 H ^ = 10 q = 5 i
1 1 1 0.5 1.5
419
u
1
0.8
0.6
0.4
0.2
0
i = l :
-
-
2 J
|X = H = 500 H
' ^ = 1 0
-
1 1 1 1 0.1 0.2 0.3 0.4 0.5
t
Fig4.4-3a. Cf versus t demonstrating the effect of q, |LII and ^2
4,4-4 The following system comprises two plug flow reactors, perfectly mixed
reactors and recycle streams Q45, Q53 and Q52. Reactors 2, 4 and 5 may be
considered also as measurement points of the concentration, and in this case their
|ii's are assigned a large value, say, 500.
ribuj^ 'PI *
Q, '52
'P2 lobuJ^ %=i
h %
Fig.4.4-4. Two plug flow reactors in series with recycle
Noting that the residence time in each plug flow reactor is tpi and tp2 yields the
following matrices:
0 < t <tpi:
P =
1 2
Pll P12 0 1
tpl<t < tp2: 1
1
P = 2
3
Pll 0
0
2
P12
P22 0
3
0
P23 1
420
tp2^t:
P =
1
2
3
4
5
6
1
Pll 0
0
0
0
0
2
P12
P22 0
0
P52 0
3
0
P23
P33 0
P53 0
4
0
0
P34
P44 0
0
5
0
0
0
P45
P55 0
6
0
0
0
P46 0
1
Referring to Fig.4-1, yields for the configuration in Fig.4.4-4 the following
probabilities by considering reactor j (hence j = 1, a = 2,... and ^ = 6) as well as
Eqs.(4-14) to (4-19) and taking Qi as reference flow:
P12 = ai2H2At
P23 = a23H3At
P34 = a34|X4At
P45 = a45^5At
P52 = a52 l2At
pii = l-ai2|i.iAt
P22 = 1 - a23^2At
P33 = 1 - a34^3At
P44 = 1 - (p46 + a45)|X4At
P55 = 1 - (a52 + a53)|X5At
The balances given by Eqs.(4-12a) to (4-12c) yield: ai2 = p46 = 1 a52 + 1 = a23 for i = 2 a23 + 0153 = 0034 for i = 3 a34 = 045 + 1 for i = 4 045 = 052 + a53 for i = 5
Case a: a53 = 0, yields the following matrices designating 0145 = a^i = R:
P46= P46^6At
P53 = a53^3At
0 < t <tpi: tpi<t <tp2:
P =
1 2 l-JljAt 2^t
0 1 P =
1
2
1 2 3 l-H,At H2^t 0
0 1. (l+R)H3At
(l+R)H2At
1 (4.4-4)
421
tp2^t:
P =
1
2
3
4
6
1 l"^iAt
0
0
0
1 0
2 3 M 2 t 0
1. (l+R)^i3At
(l+R)Mt
0 1-(l+R)|ii3At
R^ljAt Q
0 0
4
0
0
(l+R)|LL4At
1-(l+R)Mt
0
6
0
0
0
^ 6
1
The parameters of the solution are: |ii, |L12» 3» 4* l 6i tpi and tp2, R and the
introduction location of the pulse. In cases a, b, c the pulse is introduced into
reactor 1; in cases d and e it is introduced into reactor 4. The effect of the above
parameters is demonstrated in Fig.4.4-4a. Cases a and b as well as d and e
demonstrate the effect of the recycle R = 0 and 10. As observed, increasing of R
causes the overall system to behave as a single perfectly mixed reactor. In case d,
where the pulse is introduced in reactor 4, reactors 2 and 3 are initially inactive;
however, in the presence of the recycle they become active. The effect of |i is
depicted in case c where |ii = 1x5 = 1 and |Li2 = ^3 = ^4 = 25. The effect of tp, i.e.
tpi = 1, for which At = 0.005, and tp2 = 2, for which At = 0.01, is demonstrated
in all cases by the beginning and termination of each response curve.
C(0) = [1,0,0,0,0] C(0) = [1,0,0,0,0]
422
25
20 U
C(0) = [1.0.0.0.0]
(c) _
3.
M ' I '
1.2,4,6 , ^,
^ - |Ll « 1
li « li » u - 25 ^2 'a U R = 0
1-4
U
1
0.8
0.6
0.4
0.2
0
i = 4
2.3
C(0) =
1
\ • \ .'
' 1
L
[0.0.0.1.0]
(d)
* i . i^
6
R =
- iH = 0 J
A
i - - J - 1 — 1 1 — 1 1 1 • 1 1
r ^ h-
p
h-
2,3 1 1
(
2,3
i 1
::(0) = [0,0,0.1.0]
(e)
6. - '
, , '
1 •^^*'"*-' ...^.23^
I 1 - i 1 —
H
-\
^ - i H R=loJ
^ 1 5 t
10 0 5 t
10
Fig4.4-4a. Ci versus t demonstrating the effect of R, i and the
introduction location of the pulse for tpi = 1 and tp2 = 2
Case b: 052 = R, yields the following matrix while the matrices given by
Eq.(4.4^) are applicable also here:
423
tpi^t:
P =
1
2
3
4
5
6
1 1-jiiAt
0
0
0
0
0
2 Ji2At
1-(l+R)JA2^t
0
0
Rpi2^t
0
3
0 (l+R)|i3At
1-
0
RjAsAt
0
4
0
0
(l+2R)M'4At
1-(l+2R)|i4At
0
0
5
0
0
0
2R^5At
1- 2R^5At
0
6
0
0
0
*6
0
1
where the parameters of the solution were spelled above.
The effect of the parameters is demonstrated in Fig.4.4-4b for C(0) =
[0,0,0,0,1,0], i.e., the pulse was introduced into reactor 5. tpi = 1, for which At
= 0.005, and tp2 = 4, for which At = 0.02. Cases a, b and c demonstrate the
effect of the recycle R = 0,0.5 and 10. In case a, all reactors are inactive since R =
0. However, increasing of R causes the entire system to ^proach the behavior of
a single perfectly mixed reactor. The effect of \i is demonstrated in case d, which
can be compared to case b, for m = |J16 = 1 and pi2 = ^3 = M4 = M5 = 10- As
observed, the approach of the system towards equilibrium is much faster.
u
1
0.8
0,6
0.4
0.2
0
^ _ .(a) 1 i = 5
1,23,4,6 1 "'" "l
~J
R = oJ
-J
——^~\
10 15 0
424
1
0.8 I
0.6
'0.4
0.2
0
u.
r 5
U
k
h" 1 1.2,3^4,6 _ 1
(c)
•'*'''**,,^2^.4,5
1
^ "1
^-l1 R=10^
-j "^ - -=\
r 5
k
L [ 1,2,3.4,6
1
1 1
1
1 1
^ ( 4 ) , 6 1
j i . - 1 0 1 (1=2.3.4.5) J
R = 0.5 1
1,2.3.4,5 L ... 1 .,., J 1
0 5 10 15 0 5 10 15 t t
Fig4.4-4b. Q versus t demonstrating the effect of R and (li for
tpi = 1 and tp2 = 4
425
4.4-5 The following system comprises two plug flow reactors and two perfectly
mixed reactors designated by 1 and N+1. The latter may be considered also as measurement points of the concentration. This system has been used to simulate an airlift reactor [79].
TITITI |ob|cb|cb| N-11 N I
fe - ^ N+1
cb|cb|cb| Z |cb|cb|cb| Z irhirhi^ I z I Z-llz-2l i i+li t I i-1 I lN+3N+2r^
Fig.4.4-5. Two plug flow reactors in a closed loop
The need for conservation of mass of the tracer at each step in the calculation of the closed loop, prohibited the use of the simplified approach applied in cases 4.4-1(1) to 4.4-4, i.e. distinguishing between the states according to the residence time in the plug flow reactors, tpi and tp2. The latter approach is useful in open systems. Thus, each of the plug flow reactors was divided to perfectly mixed reactors of equal or unequal volumes. The scheme in Fig.4.4-5 can be deduced from the general model demonstrated in Fig.4-1 by ignoring reactor j , noting that ai (i = 2, 3, ..., Z) = 0 where ay = aji = 0 and ai^i+i = 1 while taking Q as reference flow. Thus, the designation of the reactors in Fig.4.4-5 is a = 1, b = 2, etc. and the following matrix is applicable for the process:
426
1 2
i
i+1
N
N+1
P = N+2
N+3
J
J+1
Z-1
z
1 Pll
0
0
0
0
0
0
0
0
0
0
Pzl
2 3 Pl2 0 P22 P23
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
i+1 0 0
ii Pi, i+1
0 Pi+l, i+1
N 0 0
0
0
0 PNN
0
0
0
0
0
0 0
0
N+1 N+2 N+3 .. 0 0
0
0
0 PN, N+1
P N + 1 , N+1
0
0
0
0
0 0
0
0 0
0
0
0 0
P N + 1 N+2
P N + 2 N+2
0
0
0
0 0
0
0 .. 0 ..
0 ..
0 .
0 . 0 .
0 .
P N + 2 , . N+3
P N + 3 , . N+3
0 .
0 .
0 . 0 .
0 .
• j . 0 . 0
. 0
. 0
.. 0 . 0
. 0
.. 0
.. 0
:: pjj
. . 0
.. 0
. . 0
.. 0
j+1 . 0 . 0 .
0 .
0 .
0 . 0 .
0 .
0 .
0 .
j+1 pj+1 , . j+1
0 . 0 .
0 .
.. z-1
. . 0
.. 0
.. 0
. . 0
.. 0 .. 0
.. 0
.. 0
.. 0
.. 0
.. 0
', ..Pz-1
z-1 .. 0
z 0 0
0
0
0 0
0
0
0
0
0
Pz-1 z
Pzz
0 0
0 0
0 0 0 0
0 0
The corresponding probabilities are obtained from Eqs.(4-16) and (4-17a)
which read:
Pi i=l -mAt ( i = l , 2 , ...,Z) Pi,i+i = li+iAt (i = 1, 2, ..., Z-1)
where |ii = QA i. For prescribed values of tpi, tp2 and n, the number of the
perfectly mixed reactors comprising the plug flow reactor. At = tpi/n.
A numerical solution was obtained by dividing each plug flow reactor into ten
perfectly mixed reactors, i.e. Z = 22 in Fig.4.4-5. It was also assumed that tpi =
tp2 = 0.01, |ii = |i = 80 and 800 for all reactors where At = tpi/10 = 0.001. Thus,
the above probabilities are reduced to:
Pii = 1 - ^At i = 1, 22 pi,i+i = lAt (i = 1,..., 21) p22a = ^At
Fig.4.4-5a demonstrates the effect of |LL on the transient response of C2 and Cn,
i.e. the first and the last perfectly mixed reactors in the upper plug flow reactor in
Fig.4.4-5. The tracer was introduced into reactor 2. It is obsereved that by
increasing |X from 80 to 800, the number of oscillations for reaching the steady state
concentration 1/22 in the system is increased. It should also be noted that for i =
800, the distance between two successive peaks corresponding to C2 and Cn is
427
approximately equal to tp = 0.01. It will approach tp by increasing the number of reactors comprising the plug flow reactor.
u
1
0.8
0.6
0.4
0.2
0
-
i ii = 2
h- \
'-^^'-^ 1
11
1 1
1 = 80
n - ' '
1 1
A
-
-
> - ^ * - ™ .
0.1 0.2 0.3 0.4 0.5
Fig4.4-5a. Ci versus t demonstrating the effect of \i
Fig.4.4-5b shows the relationship between C2 and Cn at identical times. The graph demonstrates the approach towards the ultimate concentration 1/22 in all reactors which is clearly observed for |LI = 800.
428
0.12 I-
0.08
U 0.04 k
u
0.15
:: 0.1
0.05
0
1 = 800
^
•f- / - ' " • ' • ' - .
^•-•••^^S^^te^^..
H
-J
H
1 1 1 0.05 0.1 0.15
Fig4.4-5b. Cii versus C2 demonstrating the effect of |x
4,4-5(1) A simpler solution for the configuration in Fig.4.4-5, avoiding the division of
the plug flow reactor to perfectly mixed reactors, is given in the following for the
scheme in Fig.4.4-5(1). Q
^
pi
Tb ip2
Fig.4.4-5(1). Two plug flow reactors in a closed loop
429
Three cases will be treated below, depending on the magnitudes of tpi and tp2.
Case a: tpi = tp2 = tp
The stages in the transfer process of the tracer between the states, reactors 1
and 2, are expressed by the following transition matrices for which pii = 1 - flAt i
= 1, 2; pi2 = P21 = M-At where M- = Hi = (12:
0 < t <tp:
P =
1
1
0
2
0
1
tp<t <2tp:
P =
1 2
Pll P12 0 1
2tp<t <3tp:
P =
1 2
1 0
P21 P22
3tp<t <4tp:
1
P = Pll P12 0 1
4tp<t <5tp:
1
P =
2
1 0
P21 P22
5tp<t <6tp:
1
P = Pll P12 0 1
6tp<t <7tp:
1
P = Pll 0
P12 1
7tp<t <8tp:
1 2
1 0
P21 P22
The parameters of the solution are: tp, (ii and Ji2- A numerical solution was
obtained for |X = QA' = m = |i2. iO, 20, 100 and 1000, i.e. for reactors of an
identical volume V. The residence time in the reactor tp = 0.01, 0.1 and 1
corresponding to At = 0.0001, 0.001 and 0.01, respectively. The tracer was
introduced into reactor 1. Fig.4.4-5(la) demonstrates in cases a to d the effect of H
and in cases c and d the effect of tp. Note in cases c and d that an identical behavior
was obtained for different combinations of |i and tp.
430
1
0.8
0.6
0.4
0.2
0
-
-
—
sj = l
'''i' 1
^ = 10 t = 0.01 p
, • - •'" '
1 1
•(a)]
-H
\ - - ^
. • --^
-
-
0.05 0.1 t
0.15 0.2 0
^1=100 t =0.01 i = 1000 t = 0.01
10 1
u U
n
•.:i;
n
y
'•.n
y
m
mm iU 0.2 0 0.05 0.1
t 0.15 0.2
Fig4.4-5(la). Ci versus t demonstrating the effect of jii and tp
Case a: tpi > tp2
The following transition matrices apply for which ph = 1 - |iAt i = 1, 2; pi2
= p2i = |xAt where |X = |ii = |X2-
0 < t <tpi:
P =
1 2
1 0
0 1
tp l<t < tpl+tpi: 1 2
1
1 ^= 2 Pll P12 0 1
tpi+tp2<t <2tpi+tp2:
1 2
1
P = 2
1 0
P21 P22 1
2tpi+tp2<t <2tpi+2tp2: 2tpi+2tp2<t <2tpi+3tp2: 2tpi+3tp2<t <3tpi+3tp2:
1 2 1 2 1 2
1 P =
Pll P12 0 1
1 P =
1 0
P21 P22
1 P =
Pll P12 0 1
431
3tpi+3tp2 , < t < 3tpi+4tp2:
1 2
1
P = 2 Pll P12 0 1
3tpi+4tp2 < t <4tpi+4tp2
1 2
1
P = 2
1 0
P21 P22
The parameters of the solution are: tpi, tp2, |Xi and |Li2- A numerical solution
was obtained for |x = m = |i2, 1, 10 and 10, where |Li = QA , i.e. for reactors of an
identical volume V. In addition |Xi = 1, 2, 100 and |X2 = 1, 2, 5,10. The residence
time in the reactors was tpi = 1 and tp2 = 0, 0.05, 0.5 and 1 corresponding to At =
0.0005, 0.005 and 0.01, respectively. The tracer was introduced into reactor 1.
The following behaviors are demonstrated in Fig.4.4-5(lb):the effect of tp2 is
demonstrated in cases a to d; the effect of |LI in cases b, e and f; the effect of |Li2 in
cases c, g and h; the effect of |ii in cases c, i, j and k.
t = 1 t = 0 ^1=1 _El P2.
t = 1 t = 0.05 II = 1 pi P2_
8 10
432
1
0.8
0.6
0.4
0.2
0
i = l
t = pi
1 t I
= 0.05 >2
l = 10
2. _ _. .. . , ^ M
1
I L I I,, } »
1 1 I ,
n 1 V
V
1
-
t pi
= 1 t = P2
D.05 H = = 100
2
1
(f)
i 1 1 1
433
u
1
0.8
0.6
0.4
0.2
0
i =
_2
t = pi
{
J 1
1 t = p2
0.5 H
/
/ ..--.-_.-.J-.-
= IOC 1 h- 1
(kj
H
H
/
J 1
/
1 d
0 2 4 6 8 10 t
Fig4.4-5(lb). Ci versus t demonstrating the effect of |Xi and tpi
4.4-6 The fluid flow Qi is divided into flows Q12 and Q13 as shown in Fig.4.4-6.
The tracer, in a form of a pulse input, is introduced into reactor 1; reactor 4 is the
collector of it.
Fig.4.4-6. An open system with divided flow between two plug flow reactors
Noting the residence times in each plug flow reactor, i.e. tpi and tp2, and assuming that tpi > tp2, yields the following matrices:
0< t <tp2: tp2<t <tpi:
p =
1
2
3
1
Pll 0
0
2
P12
1
0
3
P13
0
1
1
2
P = 3
4
1
Pll 0
0
0
2
P12
P22
0
0
3
P13
0
1
0
4
0
P24
0
1
tpi < t:
1
2
P= 3
4
1
Pll 0
0
0
2
P12
P22
0
0
3
P13 0
P33
0
4
0
P24
P34
1
Referring to Fig.4-1, yields for the configuration in Fig.4.4-6 the following
probabilities considering reactor j (hence j = 1, a = 2,... and ^ = 4) as well as
Eqs.(4-12a) to (4-12c) and (4-14) to (4-17b) and Qi as reference flow:
PI 1 = 1 - m At P12 = ai2|i2At P13 = ais^isAt
where an + a n = 1 P24 = oci2 p34 = OL\3
P22 = 1 - ai2^2At P24 = ai2|^4At
P33 = 1 - aiB^lsAt P34 = (1 - ai2)|l4At
As seen, the parameters of the solutions are: |ii = QiA^i (i = 1,..., 4), ai2 =
Ql2/Qb tpi and tp2. In the numerical solution it has been was assumed that tpi =
0.1, tp2 = 0.05, At = 0.001 as well as C(0) = [1, 0, 0, 0], i.e. the pulse was
introduced into reactor 1 in Fig.4.4-6. The effect of an = Qi2/Ql» 0 and 0.5, is
demonstrated in cases a and c in Fig.4.4-6a, as well as in cases a to d and e, f, for
different values of |ii. The effect of m is demonstrated in cases a to c, c to e and d
to f. In case d, the sudden change of C4 at tp2 = 0.05 and tpi = 0.1, is clearly
observed. In cases e and f, C2 and C3 are greater than unity because the volume of
reactors 2 and 3 are reduced by increasing ^i (for a constant flow rate), i.e. p,2 = ^3
= 100 by comparison to |ii = 14 = 1.
435
u
1
0.8
0.6
0.4
0.2
0
\,^^ 1
1 ^
r " f
.^
1 1 1 |i = 1 a = 0
i 12
^ .,4, - ' '
' - - ' ' ' 2
1 i 1
(a)
-""-'
0 0.2 0.4 0.6 t
0.8 1 0
U
1
0.8
0.6
0.4
0.2
0
-
-
= 1
1 ' \ "
/
(\
1 \
\ / '
l /^ /
1' \ 1
1
• \ 1,2,4 1 1,2 ^
1
(c)l
~1
^1.= 100--
a = 0 12 —
1-3 1
1 ! 1
— i ^
i 1 /
~ 1 3 '
/ 4
V
_m
^ . = 1 0 0 -
a = 0.5 12 —
^ 1-3 1 1 1" 1 1
0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2
10 i = 3/
: / :
- ;
- ^
fi 1
K
— ^
K 1 ^ = ^ = 100 (e^ 2 3
a = 0 12
-
4
1 1
-13
l\
" / ^ 1 1 1
K - \
1
= 1 i = M = 100 (ri 2 3
a =0.5 1 12
- , _ _ _ 4 -
1 1 0 0.2 0.4 0.6
t 0.8 1 0 0.2 1 0.4 0.6 0.8 1
Fig4.4-6a. Ci versus t demonstrating the effect of |Xi and a i2
4.4-6(1) A simplification of case 4.4-6 is demonstrated in Fig.4.4-6(1) simulating the
configuration in ref.[78, p.88].
436
Fig.4.4-6(1). An open system with divided flow
The following transition matries are applicable:
0 < t <tp2:
P =
tp2^t :
1
2
3
4
1
PU 0
0
0
2
P12 1
0
0
3
P13 0
P33 0
4
0
0
P34 1
1
2
P= 3
4
1
Pll 0
0
0
2
P12
P22
0
0
3
P13 0
P33 0
4
0 1 P24
P34 1
Eqs.(4-12a) to (4.12-c) give that P24 = ^12. p34 = CX13 where a i2+a i3 =1.
Eqs.(4-14) to (4.17b) yield the following probabilities:
Pll = 1 - mAt P12 = ai2|li2At P13 = ai3|l3At
P22 = 1 for 0 < t < tp2 and P22 = 1 - ai2^2At for t > tp2 P24 = oci2^4At
P33 = 1 - ai3^3At P34 = ai3^4At
As seen, the parameters of the solutions are: [i[ = Qi/Vi (i = 1, ..., 4), a i2 =
Q12/Q1, tpi and tp2. In the numerical solution it was assumed that tpi = 2, tp2 = 1,
At = 0.01 as well as C(0) = [1, 0,0,0], i.e. the pulse was introduced into reactor 1
in Fig.4.4-6(1). The effect of a i2 = Qi2/Ql» 0» 0-5 and 1, is demonstrated in
Fig.4.4-6(la); cases a to c correspond to |Xi = 10 and cases d to e for |ii = 100.
The effect of |ii, 10 and 100, is demonstrated in cases a and d, b and e and c and f
for various values of ai2.
437
1
0.8
0.6
0.4
0.2
0
V
/ .
LL 1
1
/
— 2 — = : ~ ^ 1 1
- —
4
, .1.
(a^^
a = 0 H 12
1
i _ i
1
4 (b)
/ I /
/ / a = 0.5 "1
^.= 10 ^
\ -
1 1 1 1 1 0 0.2 0.4 0.6 0.8 1 0
t 1
0.8
0.6
0.4
0.2
0
0.5
4 J
\ : -\
\i a = 1 -
, n,= io -
1-3 1 i i 1 1
0.5 1.5
1
0.8
0.6
0.4
0.2
0
\ i = l '
A / \ /
" / \
-/ \
L _ 2 . \
\^ / " "
V / \
/ \ \
1 1
4
a = 12
^ i =
1-3 1 1
(d)
0 -j 100 J
-
0 0.01 0.02 0.03 0.04 0.05 0 t
1
0.8
0.6
r 0.4
0.2
0
-
\-
LJ
2
1.3,4
1
2,4
4 J
(f)^
a = I H 12
i = 100-^
1-3 1
1.5
0.5 1.5
Fig4.4-6(la). Ci versus t demonstrating the effect of ^i and ai2
438
4,4-6(2) This case simulates the situation demonstrated in ref.[81], designated as
'partial mixing with piston flow and short-circuit' in continuous flow systems. It is
an extension of case 4.4-6(1) which is demonstrated in Fig.4.4-6(2) below.
•p2
^
Fig.4.4-6(2). An open system with plug flow and short-circuit
The following transition matrices are applicable:
0 < t <tp2: tp2^t :
p =
1 2
3
4
5
Pll 0
0
0
0
P12 1
0
0
0
P13 0
P33 0
0
P14 0
0
P44 0
0 0
P35
P45 1
P =
1
2
3
4
5
1
Pll 0
0
0
0
2
P12
P22
0
0
0
3
P13 0
P33 0
0
4
P14 0
0
P44 0
5
0
P25
P35
P45 1
Similar to case 4.4-6(1), the following probabilities are obtained:
Pll = 1 - ILliAt P12 = ai2^2At P13 = ai3|l3At P14 =
ai4|i4At
P22 = 1 for 0 < t < tp2 and P22 = 1 - CXi2|i2At for t > tp2 p25 = CXi2|l5At
P33 = 1 - OCislIsAt P35 = ai3|l5At
P44 = 1 - ai4|X4At P45 = ai4|X5At
where a u = 1- (ai2 + a n ) . The parameters of the solution are: tp, m, a i2 and
a i3 and the pulse was introduced into reactor 1 in Fig.4.4-6(2), i.e. C(0) = [1,0,
0, 0, 0]. In the numerical solution it was assumed that |ii = ji2 = 1x4 = 115 = 30,
i.e. the response of the reactors is relatively fast by comparison to the response of
439
reactor 3 for which ^3 = 2. These conditions, approximately, simulate the model in
ref.[81] where reactors 1, 2, 4 and 5 may be considered as measurement 'points'
(very small reactors) of the concentration of the pulse. Cases a, b and c in
Fig.4.4-6(2a) demonstrate the effect of tp2 = 0.03, 0.3 and 3 for a 12 = a 13 = a 14
= 1/3. Cases c and d demonstrate the effect of ai2 and 013. In the later case ai2 =
0.1 and ai3 = 0.9 for tp2 = 3, i.e. Q14 = 0. In the solution. At = 0.0003, 0.003
and 0.03 for tp2 = 0.03, 0.3 and 3, respectively.
u
1
0.8
0.6
0.4
0.2
0
-Aj = i
> " ^ :
' t =0.03 ^ J p2 n
a - a - a - 1/3 (a) 12 13 14 J
H
_5. . -" ' H
1 3 1 H I 0.05
1 \ r t =0.3 p2
a - a - a - 1/3 12 13 14 . . . -
(b)
J I I L 0.1 0.15
U
1
0.8
0.6
0.4
0.2
0
. ^ -
1,
Lv—i^
t
« t = 3 « ' P2 , .
a - a - a - 1/3 12 13 14
. = 1 . . . , ' 5
- " ' 2 ———
> \
"(c^
-A J
H
i . 1 1 „.„. 1
0 0.2
1 1
-
L 1 ' '
u^
0 4
1
t
- 2
0.6
1
t = p2
a -0 .1 12
0.8 1
1 J
(d)J 3
a - 0.9 13 1
-
1 1 1 1 l-
Fig4.4-6(2a). Ci versus t demonstrating the effect of tp2andaij
4.4-7 This example extends the previous one and simulates a contacting pattern of
fluid elements of different ages [21, p.334]. The fluid flow Qi is divided into
flows Q12, Qi3 and Q14 as shown in Fig.4.4-7. The tracer, in a form of a pulse
input, is introduced into reactor 1 whereas reactor 5 is the collector of the tracer.
440
Fig.4*4-7. An open system with divided flow between three plug flow reactors
Noting that the residence time in each plug flow reactor is tpi, tp2 and tp3 and
assuming that tpi > tp2 > tp3, yields the following matrices:
0 < t <tp3:
1
1 1 2
**= 3
4
PU 0
0
0
tp2<t <tpi:
1
1
2
P = 3
4
5
Pll 0
0
0
1 0
2
P12
1
0
0
2
P12
P22
0
0
0
3
P13
0
1
0
3
P13
0
P33
0
0
4
Pl4
0
0
1
4
P14
0
0
1
0
5
0
P25
P35
0
1
tp3<t < tp2:
1
1
2
P = 3
4
5
Pll 0
0
0
0
tpi < t:
1
1
2
P = 3
4
5
Pll 0
0
0
1 0
2
P12
P22
0
0
0
2
P12
P22
0
0
0
3
P13
0
1
0
0
3
P13
0
P33
0
0
4
P14
0
0
1
0
4
P14
0
0
P44
0
5 1 0
P25
0
0
1 1
5 1 0
P25
P35
P45
1
Referring to Fig.4-1, yields for the configuration in Fig.4.4-7 the following probabilities by considering reactor j (hence j = 1, a = 2,... and ^ = 5) and taking Qi as reference flow. From Eqs.(4-12a) to (4-12c) one obtains that ai2 = P25» OC13 = P35 and ai4 = P45 where ajk and pi are defined in Eq.(4-2). In addition ai2 + a\3 +ai4 = 1 where Eqs.(4-14) to (4-17b) yield:
441
pil = 1 - (ai2 + ai3 +ai4)mAt pn = ai2Ji2At pu = aisjisAt
pi4 = (l-ai2-ai3)M4At
P22 = 1 - ai2M'2At P25 = ct^wAt
P33 = 1 - aisfisAt P35 = aisjisAt
P44= l - ( l -a i2-a i3)wAt P45= (1-ai2 - ai3)^5At
As seen, the parameters of the solutions are: Mi = Ql/Vi (i = 1, ..., 5), a 12, a 13,
tpi, tp2 and tp3. In the numerical solution it was assumed that tpi = 0.4, tp2 = 0.2
and tp3 = 0.1; m = 200, At = 0.001 as well as C(0) = [1, 0, 0, 0, 0], i.e. the pulse
was introduced into reactor 1 in Fig.4.4-7. The effect of a 12 = Q12/Q1 = 1.
ai3 = Q13/Q1 = 1» ai4 = QlVQl = 1 and ai2 = 0.2 ai3 = 0.3 and ai4 = 0.5, is
demonstrated in Fig.4.4-7a.
u
1 h
0.8 h
0.6
0.4 [-
0.2 h
0
-0.1
u
-0.1
i = l / 2 : •
L 3-5
0.1 0.2 t
0.1 0.2 t
1-4
0.3
a - 1 12 «J
J I L 0.4 0.5
1
0.8
0.6
0.4
0.2
0
-
i = l (
1
\ 1 1
3
2,4,5 1
1; 1: I 1 ,1 •1
.1 1
5
1-4 1
- j J
a -1 13 J
J
[ , „.
0.3 0.4 0.5
442
u
1
0.8
0.6
0.4
0.2
0
—
—
-
-
i = 11
i 1
1
i
:i \ .
4
1,2,3.5 i 1 i
[; 5 1
1 CL m l \
1 ^ - J
•» H ' V l - 4 1
-0.1
u
1
0.8
0.6 h
0.4
0.2
0
-0.1
i = l
0.1 0.2 0.3 0.4 0.5 t
a - 0.2 a - 0.3 a - 0.5 12 13 14
4,5
'5 -J
JLS \ 1-4 J I I I L 0 0.1 0.2 0.3 0.4 0.5
t
Fig4.4-7a. Q versus t demonstrating the effect of aij
4.4-8 The following configuration of two interacting plug flow reactors was applied
elsewhere [21, p.298] for describing deviation from plug flow and long tails. Due to the interaction between the reactors, it is necessary to divide the reactor into perfectly mixed reactors. In the following example, the reactor was divided to five reactors. Generally, the number of perfectly mixed reactors needed, must be determined by comparing the response curve of the divided system to that of the plug flow reactor.
443
l i o k J opJ op "op BL^4
Yb a\C
w a a
w \a ou
m
" ^ "
a a
^ ^ K 50,
Fig.4.4-8. Two plug flow interacting reactors
Referring to Fig.4-1, yields for the configuration in Fig.4.4-8 the following
probabilities by considering reactor j and taking Qi as reference flow. From
Eqs.(4-12a) to (4-12c) one obtains a set of equations for the determination of aik-
Assuming that:
a = a67 = OC85 = a58 = a94 = a49 = aio,3 = OC3,10 = 0Cii,2
P = OC23 = CX34 = a45 = a 5 6
Y=a76 = a2,ii
5 = a87 = CX98 = cxio,9 = ocnjo (4.4-8a)
as well as
p + 5 = 1 (4.4-8b)
it follows that a + 6 = Y (4.4-8c)
The probability matrix for the above configuration reads:
444
P =
1
2
3
4
5
6
7
8
9
10
11
12
1
Pii
0
0
0
0
0
0
0
0
0
0
0
2
Pl2
P22
0
0
0
0
0
0
0
0
Pll,2
0
3
0
P23
P33
0
0
0
0
0
0
PlO,3
0
0
4
0
0
P34
P44
0
0
0
0
P94
Pl0,9
0
0
5
0
0
0
P45
P55
0
0
P85
0
0
0
0
6
0
0
0
0
P56
P66
P76
0
0
0
0
0
7
0
0
0
0
0
P67
P77
P87
0
0
0
0
8
0
0
0
0
P58
0
0
Pss
P98
0
0
0
9
0
0
0
P49
0
0
0
0
P99
PlO,9
0
0
10
0
0 f
P3,10
0
0
0
0
0
0
PlO,10
Pii,ioP
0
11
0
2,11
0
0
0
0
0
0
0
0
11,11
0
12
0
0
0
0
0
P6,12
0
0
0
0
0
1
Assuming that all m's are equal, i.e. |ii = \i, the following probabilities are
obtained from Eqs.(4-14) to (4-19):
pil = 1 - iLlAt P22 = 1 - (P + Y)| At P33 = P44 = P55 = 1 - (a + P)|liAt
P66 = 1 - (1+ a)|LiAt
P77 = P88 = P99 = P10,10 = P11,11 = 1 - TI At
Pl2 = P6,12 = At
P23 = P34 = P45 = P56 = PjAAt
P87 = P98 = P10,9 = PI 1,10 = 8|XAt P2,ll=P76 = mAt
P3,10 = P49 = P58 = P67 = Pi 1,2 = PlO,3 = P94 = P85 = a|XAt
The parameters of the solution are a and p related by Eqs.(4.4-8b) and (4.4-8c)
as well as \i, Fig.4.4-8a demonstrates the effect of the circulation intensity a
between the reactors on the transient response in various reactors of the tracer,
introduced in reactor 1. In case a, a = 0.05, P = 0.5, 5 = 0.5 and y = 0.55; in case
b, a = 5, P = 0.5, 5 = 0.5, y = 5.5 and At = 0.001 where in both cases \i = 50.
As observed, by increasing a, the C-t curves for reactors 4 and 9 (see Fig.4.4-8
above), reactors 6 and 7 and reactors 2 and 11 become identical.
445
0.8 h-
0.6 U
u 0.4
0.2
[
• i = l
p.
I- / A . 2
\l "••V 11 ,4 / 9
f ''-\ - •'•-'/ [.i? ' - •• ilr.-'-rr*^^-——_
1 1 1
y' y
, 1 2
/
- - ^ ^^''7
• • - • " • - = - ^ "" ' 7 - • • - . . . .
1 1
(a)
-j
-j a = 0.05
0 F^-
0 0.1 0.2 0.3
t
0.4 0.5
U
0.8
0.6
0.4
0.2
0
•i = l
i , 2 , l l
4,9
1
^.
/ •
, ' 12 /
/ /
- . ,6,7
1 1
• • ' " "
- — ' •• ' 1
(b)
a = 0.5 - j
-
1
0 0.1 0.2 0.3 0.4 0.5 t
Fig4.4-8a. Ci versus t demonstrating the effect of the circulation a
The variation of the mean residence time tm in reactors 1 to 11, computed by Eq.4-26, was obtained from the response curve of reactor 6. The resuhs are sununarized in the following Table for the various operating parameters a to 5 listed in the Table and defined by Eq.4.4-8a. The following trends are observed: a) Increasing |i (i.e. the flow rate), decreases tm. b) Taking a = 0 in case c, i.e. decreasing the number of effective reactors in Fig.4.4-8, decreases tm-
446
^ 10
50
100
1 n
(a) a = 0.05
P = 0.5
Y = 0.55
5 = 0.5
tm 1.096
0.219
0.109
1 11
(b) a = 0.05
p = o Y= 1.05
6 = 1
tm 1.080
0.219
0.108
11
(c) 1 a = 0 p = o Y=l
6 = 1
tm 1 0.797
0.159
0.079
8
n - number of effective reactors
4>4-9 The following configuration is a simulation (see also case 4.4-6(2)) of
"partial mixing with piston flow and short-circuit" in continuous flow systems
treated in ref. [81].
%4
Q i o ^ ^ 1 2 1 ^ 4 = 4
^32
t,
ns
yfe
Fig.4.4-9. Mixing with plug flow and short-circuit
The following transition matrices are applicable:
< t < tp: 1
1
»= 2
4
Pll 0
0
2
P12
P22
0
4
P14
P24
1
tp<t:
1
2
P = 3
4
1
Pll 0
0
0
2 3
P12 P13
P22 0
P32 P33 0 0
4
P14
P24
0
1
447
Referring to Fig.4-1, yields for the above configuration the following
information by considering reactor j (hence j = 1, a = 2,... and ^ = 4) as well as
Eqs.(4-12a) to (4-12c) and (4-14) to (4-17b) and Qi as reference flow.
Pi4 + p24 = 1 Pi4 + ai2 + ai3 =1 For 0 < t < tp: ai2 = p24 and a32 = 0; for tp < t: an + OL32 = p24 and
0C13 = a32. The following probabiUties were obtained:
Pll = 1 - (Pl4 + ^12)^1 At for 0 < t < tp, and
Pii = 1 - (Pi4 + ai2 + oci3)mAt for tpi < t.
P12 = ai2M'2At P13 = aiBiiisAt pu = P u M t
P22 = 1 - p24^2At P24 = p24|ll4At
P33 = 1 - ai3M'3At P32 = ai3^2At
The parameters of the solutions are: m = QiA i (i = 1,..., 4), an = Qii/Qi (i
= 2, 3) pi4 = qi4/Qi (i = 1, 2) and tp. In the numerical solution C(0) = [1, 0, 0, 0],
i.e. the pulse was introduced into reactor 1 in Fig.4.4-9. Common quantities in all
cases were: m = p,4 = 10, 13 = 50, an = 0.1, a n = 0.8 and P14 = 0.1. The
effect of tp, 0.1 and 1, is demonstrated in cases a and c in Fig.4.4-9a. The effect
of |I2, 50, 100 and 300, is depicted in cases b, c and d. For tp = 0.1, At = 0.001
and for tp = 1, At = 0.01. An interesting observation is the sudden increase of the
concentration in reactor 2 at t = tp in cases c and d. This is caused by the sudden
supply of the solute at this time due to the plug flow reactor.
u
1
0.8
0.6
0.4
0.2
0
1 1 \ i = 1 4^
" A / - \ /
~ , •' \^ 2
'^ ^ " " ~ - ^ : - i ^ ^ . , . .
1 1 J
(a)|
t = o.i H p
^1=50 - 2
t i l l
I 1
i \ X
b ^ \"--r ^ "^^^^ y ^ - ^
J
t =1 H p
2
-
1 1 1 1 0 0.2 0.4 0.6 0.8 1 0
t 0.5 1.5
448
u
1
0.8
0.6
0.4
0.2
0
1 1 1 \ i = l 4-- " "~
~ \ / - A /
7 /\ y ^-^•='^-^^.
1
(c)l
t = 0.1 -1 p
^1= 100^ 2
1 1 1 1
1 1
,'
L
\ /
f i \ '
r " ^ L. ^ < i i
1 1
1 1 1
—
t = o.H p
l = 300 H 2
-
1 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
t t Fig4.4-9a. Ci versus t demonstrating the effect of tp and |i2
4,4-9(1) The following configuration is another simulation (see also cases 4.4-6(2)
and 4.4-9) of "partial mixing with piston flow and short-circuit" in continuous flow systems treated in ref.[81].
1 ,4
• .J 12 r ^ fefel9i ^ = 4
23 32
fe
Fig.4.4-9(1). Mixing with plug flow and short-circuit
The following transition matrices are applicable:
0 < t <t j )•
1
1
p = 2
4
Pll 0
0
2
Pl2
P22
0
4
P14
P24
1
t p < t :
1
p = 2
3
4
1
Pll 0
0
0
2 3
P12 0
P22 P23
P32 P33
0 0
4
P14
P24
0
1
Similarly to case 4.4-9, the following relations are obtained:
449
Pl4 + p24 = 1 Pl4 + ai2 = 1 yielding that an = p24 and that a23 = OL32
pi 1 = 1 - mAt P12 = ai2|i2At P14 = Pi4^4At
For 0 < t < tp P22 = 1 - ai2^2At; for tp < t P22 = 1 - (OC23 + ai2)|X2At
In addition p24 = ai2|X4At p23 = a23|X3At P33 = 1 - a23M'3At and p32 =
OC23 2At.
The parameters of the solution are: tp, ai2, OC23 = R and |Xi (i = 1, ..., 4).
In the numerical solution C(0) = [1, 0, 0, 0], i.e. the pulse was introduced into
reactor 1 in Fig.4.4-9(1). Common quantities in all cases were: ^i = |i2 = ILL3 = ILI4
= 1 and ai2 = 0.5. The effect of R, 0,0.2 and 2, is depicted in cases a, b and c in
Fig.4.4-9(la). The effect of tp, 2 and 10, is demonstrated in cases c and d. For tp
= 2, At = 0.02 and for tp = 10, At = 0.1. Interesting observations are: a) For R >
2, the transient response of Ci remains unchanged; and b) for tp = 10, the effect of
R is negligible because the response was transferred straight into reactor 4, by
passing reactors 3 and the plug flow.
u
1
0.8
0.6
0.4
0.2
0
U = i "T" ^
\ / _ y
7 ^
4^ (a)
A t = 2 R = 0 p J
-
1 1 1 1
8 10 0 8 10
— \\
A
A
I-
/ /
_-3_. . 1
/
" 2 ,
^ 4 '
1 _
(d)
1
t =10 p
1
-J
R = 2
~
8 10 0 4 6 t
8 10
Fig4.4-9(la). Cf versus t demonstrating the effect of tp and R
450
4,4-10 The following configuration demonstrates a general cell model of a
continuous flow system described in ref.[77]. There are two possibilities to arrive
at state 4, i.e. directly and via the upper plug flow reactor. However, in order to
materialize these possibilities it was necessary to add state 3-a perfectly mixed
reactor 3. The residence time in this reactor is controlled by the quantity 113.
Qi
Fig.4.4-10. A cell model
Three cases will be treated below, depending on the magnitudes of tpi and
tp2. Case a: tpi = tp2
The following transition matrices are applicable:
0 < t <tpi:
P =
1
t p i < t :
1
2
4
Pll
P21
0
P12
P22
0
P14 0
1
1
2
P = 3
4
5
Pll
P21
0
0
0
P12
P22
0
0
0
P13 0
P33 0
0
P14 P15
0 P25
P34 0
P44 P45
0 1
Referring to Fig.4-1, yields for the configuration in Fig.4.4-10 the following
probabilities by considering reactor j (hence j = 1, a = 2,... and ^ = 5) and taking
Ql as reference flow. From Eqs.(4-12a) to (4-12c) one obtains that:
Pl5 + P25 + P45 = 1 (4.4.10a)
1 + OC21 = Pi5 + OL12 + ai3 + ai4 (4.4-lOb)
451
ai2 = P25 + «2l for i = 2; ai3 = a34 for i = 3; ai4+ a34 = P45 for i = 4 (4.4-10c)
The following probabilities are obtained from Eqs.(4-14) to (4-19) for
0 < t <tpi: pii = l - ( a i 2 + ai4)mAt pi2 = ai2^2At pi4 = ai4^4At (4.4-lOd)
P22 = 1 - a2m2At
Fortpi<t: pii = 1 - (ai2 + ai3 + ai4 + Pi5)mAt
P21 = a2imAt
P13 = ai3^3At
where pi2and p 14 are given above.
P22 = 1 - (a21 + p25) 2At P25 = p25^5At
where p2i is given in Eq.(4.4-10e).
P33 = 1 - a34^3At P34 = a34M,4At P44 = 1 " P45M-5At
(4.4-lOe)
P15 = PlSM-sAt
(4.4-lOf)
(4.4-lOg)
P45 = p45^5At
(4.4-lOh)
The parameters of the solutions are: |ii = QiA i (i = 1,..., 5), tpi, tp2, ai2,
cti3, ai4 and a2i = R- The rest of the parameters can be obtained from Eqs.(4.4-
10a) to (4.4-lOc). Fig.4.4-10a demonstrates the effect of tpi (i = 1, 2), 0.01, 0.1
and 1 (corresponding to At = 0.0001, 0.001 and 0.01, respectively) and jlj = 10
and 100 (i = 1,..., 5) for C(0) = [1, 0, 0, 0, 0], i.e. the pulse was introduced into
reactor 1 in Fig.4.4-10. Other unchanged parameters were: ai2 = ai3 = ai4 =
0.25 and R = 0.1. The effect of tpj is demonstrated in cases a, b, c and the effect of
|i.i in cases c, d. In the latter case the response becomes faster by increasing |li. In
addition, increasing tpi causes reactor 3 to become ineffective.
u
1
0.8
0.6
0.4
0.2
0
^=^ t = 0.01 ^ = 10 ^ -i ^ " " \ ^ pi i
- ^.-^.-^-J " 1 1 LJ 1 I
-V - \
- \
t = 0.1 |X.= 10 ^ pi »
5 . - • - ' • J
^•^<^4=--.— J 1 1 1 1 "
0 0.02 0.04 0.06 0.08 0.1 0 0.2 0.4 0.6 0.8 1 t t
452
1
0.8
0.6
0.4
0.2
0
ti=l 5-' A 4 ' ' /
f ^ v _
- • • ' ^
A
t =1 H pi
1 1 0 1 3 0
t t Fig4.4-10a. Ci versus t demonstrating the effect of tpi and [if
Case b: tpi > tp2 The following transition matrices are applicable:
0 < t < tp2:
P =
tpi<t:
tp2^t<tpi:
1
1
2
4
Pll P12
P21 P22
0 0
P14 0
1
1
P = 2
4
5
Pll
P21 0
0
P12 P14 P15
P22 0 P25
0 P44 P45
0 0 1
p =
1 2
3
4
5
Pll
P21
0
0
0
P12
P22
0
0
0
P13 0
P33 0
0
P14 0
P34
P44 0
P15
P25
0
P45 1
Eqs.(4.4-10a) to (4.4-lOc) are applicable also in this case. For 0 < t < tp2, the probabilities pn, pi2, pi4 are given by Eq.(4.4-10d); p22
andp2iby Eq.(4.4-10e). Fortp2<t<tpi:
pii = l - (a i2 + a i4+Pi5)Mt where pi2 and pu are given in Eq.(4.4-10d), pis in Eq.(4.4-10f), p22 in Eq.(4.4-lOg), P21 in Eq.(4.4-10e), p44 and P45 in Eq.(4.4-10h).
453
For tpi < t: PI 1 = 1 - (ai2 + ai3 + ai4 + Pi5)mAt
pl3 and P15 are given in Eq.(4.4-10f), pi2 and pi4 in Eq.(4.4-10d), p22 and p25 by Eq.(4.4-10g) and p2i in Eq.(4.4-10e). Finally, P33, P34, P44 and P45 are give by Eq.(4.4-10h).
The parameters of the solution are: |ii = Qi/Vi (i = 1, ..., 5), tpi, tp2, ai2, ttl3> 0C14 and a2i = R. The rest of the parameters can be obtained from Eqs.(4.4-10a) to (4.4-lOc). Fig.4.4-10b demonstrates the effect of the residence time in the plug flow reactors for tpi = 2tp2 where tp2 = 0.01 and 0.1 (corresponding to At = 0.00005 and 0.0005, respectively) and m = 25 and 250 (i = 1, ..., 5) for C(0) = [1, 0, 0, 0, 0], i.e. the pulse was introduced into reactor 1 in Fig.4.4-10. Other unchanged parameters were: au = ai4 = 0.05, a\3 = 0.9 and R = a2i = 0.025. The effect of tpi, tp2 is demonstrated in cases a, b and c, d, respectively. The effect of |ii is demonstrated in cases a, c and b, d. Note the similarity between cases b and c although the time scale is different by a factor of ten.
t =0.02 t =0.01 ^^ ^pl p2 ^
1 = 25
t =0.02 t =0.01 (c)J pi p2 ^
|i.= 250
0 0.01 0.02 0.03 0.04 0.05 0 0.1 0.2 0.3 0.4 0.5 t t
Fig4.4-10b. Ci versus t demonstrating the effect of tpi and m
454
Case c: tp2 > tp i
The following transition matrices are applicable:
0 < t <tpi:
P =
tp2^t:
tpi<t<tp2:
P =
1
2
4
1 2 4
Pll P12 P14
P21 P22 0
0 0 1
1 2 3
1
2
= 3
4
5
Pll P12 P13
P21 P22 0
0 0 P33
0 0 0
0 0 ( )
P =
4
P14 0
P34
P44 0
1
2
3
4
1
Pll
P21
0
0
5
P15
P25
0
P45 1
2
P12
P22
0
0
3
P13 0
P33
0
4
P14 0
P34
1
Eqs.(4.4-10a) to (4.4-10c) are applicable also in this case.
For 0 < t < tpi, the probabilities pi i, pi2, pi4 are given by Eq.(4.4-10d); p22
andp2iby Eq.(4.4-10e).
Fortpi<t<tp2:
Pll = 1 - (ai2 + ai3 + ai4)|LiiAt
where pi2 and pi4 are given in Eq.(4.4-10d), p n in Eq.(4.4-10f), p22 in Eq.(4.4-
10e)» P21 in Eq.(4.4-10e), P33 and P34 in Eq.(4.4-10h).
Fortp2<t:
Pii = l - ( a i 2 + ai3 + ai4 + Pi5)mAt
P13 and pi5 are given in Eq.(4.4-10f), pi2 and pi4 in Eq.(4.4-10d), p22 and P25 by
Eq.(4.4-10g) and p2i in Eq.(4.4-10e). Finally, P33, P34, P44 and P45 are give by
Eq.(4.4-10h).
The parameters of the solution are: |Lii = Qi/Vi (i = 1, ..., 5), tpi, tp2, a i2,
ai3, a i4 and a2i = R. The rest of the parameters can be obtained from Eqs.(4.4-
10a) to (4.4-10c). Fig.4.4-10c demonstrates the effect of the residence time in the
plug flow reactors for tp2 = 2tpi where tpi = 0.01 and 0.1 (corresponding to At =
0.00005 and 0.0005, respectively) and \i{ = 25 and 250 (i = 1, ..., 5) for C(0) =
455
[1, 0, 0, 0, 0], i.e. the pulse was introduced into reactor 1 in Fig.4.4-10. Other unchanged parameters were: ai2 = ai4 = 0.05, ai3 = 0.9 and R = a2i = 0.025.
The effect of tpi, tp2 is demonstrated in cases a, b and c, d, respectively. The effect of m is demonstrated in cases a, c and b, d. Note the similarity, as in previous example, between cases b and c although the time scale is different by a factor of ten.
u
1
0.8
0.6
0.4
0.2
0
Tr^
^
t =0.01 t =0.02 % pi p2 1
s^ ^.-25 J
^ ' ^ ' ' H . , , ^ ^ ^ ^ - J " ^ ^ .v -J
3_ - - " T ]
— 1 ,1 „.1„..,..2,.„J
psl
^
1 1
t = pi
4
*
1
O l t = p2
**i"
1
\ ' S
^. ^ ' ^ • * - ' — i * ^
1
0 2
25
-"
ao^Ii;.
5
- 2 * _ _
1
~~M . . - -
u
(
1
0.8
0.6
0.4
0.2
0
3
^ \ i •
0.01
= 1
- . ™ J . -
0.02 0.03 t
t =0.01 t = 0.01 pi P2
f i . - 250
r 1 .
- 1 1 . ,
0 . 0 4 O.i
(ci
• ' 5 " " J
-^
H
— 2 .
" r"^ 1
0.1 0.2 0.3 0.4 0.5
0.01 0.02 0.03 t
0.04 0.05 0
Fig4.4-10c. Ci versus t demonstrating the effect of tpi and yu^
4 .4 -11 The configurations in Rg.4.4-11 demonstrate a model for small deviations
from plug flow and long tails in continuous flow system described in ref.[21, p.298]. The deviation from plug flow, as well as adding time delay, is achieved by introducing plug flow reactors in parallel, as shown below. In case a the upper reactor is of plug flow type. In case b the plug flow reactor was divided into perfectly mixed reactors in order to obtain a solution for the following cases, 4.4-11(1, 2, 3). In general, the perfectly mixed reactors are not of equal size, depending on the number of reactors in parallel and on their location.
456
a)
I m in Q,
I
b)
Q,
Time delay at random location
/ ^
I UTT
D
do I CD I do I ... I do I do I do |_»;
I I I
Fig.4.4-11. Schemes for describing deviations from a plug flow reactor
4,4-11(1) The simplest scheme corresponding to Fig.4.4-11 is demonstrated in case
4.4-1(3). An additional scheme is shown in Fig.4.4-11(1) below. Reactors 2 and 4 were added to account for the recycle streams Q12 and Q34. The response of these reactors can be controlled by the magnitudes of |i2 and ^4. If reactor 2, for example, is a measurement "point" of the concentration, i.e. a very small reactor, |i2 should be assigned a relatively large value.
457
Qi 1 I n_j_j ^4=5
Q.
I •^4 i
V 'P2
Fig.4.4-l l ( l ) . Simplifled scheme for describing deviations from
plug flow reactor
Three cases will be treated below, depending on the magnitudes of tpi and
tp2. Case a: tpi = tp2
The following transition matrices are applicable:
0 < t < t p i : t p i< t :
1
P = 3
5
From Eqs.(4-12a) to (4-12c) one obtains that:
P35 = a i3 = 1, "21 = ai2 and 034 = 043 (4.4-1 l(la))
The following probabilities are obtained from Eqs.(4-14) to (4-19) for
0 < t <tpi:
pil = l - m A t pi3 = ^3At p33=l-JA3At p35 = ^5At (4.4-1 l(lb))
Fo r tp i< t :
pil = l - ( l+a i2 )mAt pi2 = ai2H2At pi3 = ^3At (4.4-1 l(lc))
p22 = 1 - a2m2At P21 = a2iHiAt (4.4-1 l(ld))
P33 = 1 - (1+ a34)|X3At P34 = a34^4At p35 = l^sAt (4.4-1 l ( le))
1
pll 0
1 0
3
P13
P33
0
5
0
P35
1
1
2
P= 3
4
5
1
Pll
P21
0
0
0
2
P12
P22
0
0
0
3
P13
0
P33
P43 0
4
0
0
P34
P44
0
5
0
0
P35
0
1
458
P44 = 1 - a43^4At P43 = a43^3At (4.4-1 l(lf))
The parameters of the solutions are: |ii = QiA^i (i = 1,..., 5), tpi = tp2 ,
0 < Ri = ai2 < «> and 0 < R2 = a34 < 00. Fig.4.4-1 l(la) demonstrates the effect
of the residence time in the plug flow reactors for tpi = tp2, 0.1 and 1
(corresponding to At = 0.0005 and 0.005, respectively), the effect of |ii =10 and
100 (i = 1,..., 5) as well as the effect of the recycle Ri = R2,0 and 10, for C(0) =
[1, 0, 0, 0, 0], i.e. the pulse was introduced into reactor 1 in Fig.4.4-11(1). The
effect of Ri = R2 is demonstrated in cases a and b, the effect of tpi = tp2 is
demonstrated in cases b and c and the effect of |LLi is demonstrated in cases c and d.
The following observation should be noted: a) The similarity between cases a and c
although the time scale is different by a factor of ten. b) The effect of Ri = R2 in
case c is negUgible due to the relatively large value of tpi = tp2 = 1. In other words,
no response is observed in reactors 2 and 4.
R =R =0 t =t =0.1 1 2 pi p2
(a5 R =R =10 t =t =0.1 (^ 1 2 pl p2
U
1
0.8
0.6
0.4
0.2
0
^ R =R =0,10 t =t =1 r 1 2 pl p2 \ i = 10 A i =1 • 5 . '
- \
/ - N" ->. 3 - / , ' \ ^ - . ^
1 1 1 1
(cjl
H
-\
-j J:-_ •-
0.1 0.2 0.3 t
0.4 0.5 0 0.01 0.02 0.03 0.04 0.05 t
Fig.4.4-ll(la). Ci versus t demonstrating the effect of tpi, m and
Ri
459
Case b : tpi > tp2
The foUowing transition matrices are applicable:
0 < t <tp2:
P =
tpi<t:
tp2<t<tpi:
1
3
5
1
Pll 0
0
3
P13
P33 0
5
0
P35 1
p =
1 3 4 5
Pll Pl3 0 0
0 P33 P34 P35
0 P43 P44 0
0 0 0 1
P =
1 2 3
Pll P12 P13
P21 P22 0
0 0
0 0
0 0
4
0
0
5
0
0
P33 P34 P35 P43 P44 0
0 0 1
Eq.(4.4-1 l(la)) is applicable also in this case.
For 0 < t < tp2 the probabilities are given by Eq.(4.4-1 l(lb)).
For tp2 t < tpi the probabilities pn and pi3 are given by Eq.(4.4-1 l(lb)).
P33. P34 and P35 are given in Eq.(4.4-1 l(le)); P44 and P43 are given by
Eq.(4.4-ll(lf)).
For tpi < t the probabilities p n , pi2 and pi3 are given in Eq.(4.4-ll(lc));
P33' P34 and P35 are given by Eq.(4.4-ll(le)); P44 and P43 are given by Eq.(4.4-
ll(lf)).
The parameters of the solutions are: m = QiA^i (i = 1,.... 5), tpi, tp2,
0 < Ri = ai2 < °° and 0 < R2 = a34 < 00. A numerical solution was obtained for
tpi = 0.2 and tp2 = 0.1 (corresponding to At = 0.0005), |J.i = 10 and 2 (i = 1, ...,
5), Ri = R2, 0, 10 and 100 for C(0) = [1, 0, 0, 0, 0], i.e. the pulse was introduced
into reactor 1 in Fig.4.4-11(1). The effect of Ri = R2 is demonstrated in cases a, b
and c in Fig.4.4-1 l(lb) and the effect of fii in cases c and d.
460
u
1
0.8
0.6
0.4
0.2
0
-
J.i = l
- / '
(ail
5 - ' ' J
R =R =oH 1 2
1 1 1 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5
t t
U
1
0.8
0.6
0.4
0.2
0
(cj
\ i = l . -•• -J
- \ 5. ' ' R =R =100H \ ' ' 1 2
- \ . ' l= loJ 3/- K -
1 I ' ' ^ 1 1
- p -v .
h
r ^ r r
(dJI
1 R =R =100H 1 1 2 ^ = 2 J
: _M73I' I I~I^^
2 1 1 1
0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 t t
Fig.4.4-ll(lb). Ci versus t demonstrating the effect of jiii and Ri
Case c: tp2 > tpi The following transition matrices are applicable:
0 < t < tp i :
P =
tp2^t:
tpl<t<tp2:
1
3
5
1
Pll 0
0
3
P13
P33
0
5
0
P35
1
p =
p =
1 2 3
Pll P12 P13
P21 P22 0
0 0
0 0
1
2
3
4
5
1 2 3 4 5
Pll P12 P13 0 0 P21 P22 0 0 0
0 0 P33 P34 P35
0 0 P43 P44 0
0 0 0 0 1
5
0
0
P33 P35 0 1
461
Eq.(4.4-11(1 a)) is applicable also in this case. For 0 < t < tpi the probabilities are given by Eq.(4.4-1 l(lb)). For tpi < t < tp2 the probabilities pu, pi2 and pn are given by Eq.(4.4-
ll(lc)), P21 andp22by Eq.(4.4-ll(ld)), P33 andpssby Eq.(4.4-ll(lb)). For tp2 ^ t the probabilities pn, pi2 and pn are given by Eq.(4.4-ll(lc));
P21 and p22by Eq.(4.4-ll(ld)); P33, P34 and P35 are given by Eq.(4.4-ll(le)); P44 and P43 are given by Eq.(4.4-1 l(lf)).
The parameters of the solutions are: m = QiA i (i = 1,..., 5), tpi, tp2, 0 < Ri = ai2 < «> and 0 < R2 = a34 < <». A particular solution was obtained for tpi = 0.1 and tp2 = 0.2 (corresponding to At = 0.0005), fxi = 10 and 2 (i = 1, ..., 5), Rl = R2, 0, 10 and 100 for C(0) = [1, 0, 0, 0, 0], i.e. the pulse was introduced into reactor 1 in Fig.4.4-11(1). The effect of Ri = R2 is demonstrated in cases a, b and c in Fig.4.4-1 l(lc) and the effect of jLii in cases c and d. One should compare the behavior in the above figure corresponding to tp2 > tpi with that in Fig.4.4-
1 l(lb) above for tpi > tp2.
u
1
0.8
0.6
0.4
0.2
0
• \ i = l
- \
3/-- / ^ '
1A_ \ '
(ai
5, • ' ' -j
R =R = 0 -1 2
ki.= 1 0 -
1 1 1 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5
U
1
0.8
0.6
0.4
0.2
0
r-
- \ i = l
Y \
r 1
. - '
1
(cjl
y . • ' \
R =R = 1 0 0 -1 2
^1.= 1 0 -
_ 3 , 4
t
1 ',' ^
\ 1 4 1
(<a|
R =R =10oH 1 2
H = 2 ^
_—__—--r-^.—^~
5 _ 1 1
0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 t t
Fig.4.4-ll(lc). Ci versus t demonstrating the effect of |ii and Ri
462
4.5 IMPINGING-STREAM SYSTEMS Impinging streams [73] is a unique and multipurpose configuration of a two-
phase suspension for intensifying heat and mass transfer processes in the following heterogeneous systems: gas-solid, gas-liquid, liquid-liquid and solid-liquid.
The essence of the method demonstrated in Fig.4.5, lies in bringing two streams of a suspension, flowing along the same axis in opposite directions, into collision. As a result of collisions between the opposed streams, a relatively narrow zone of high turbulent intensity is generated, which offers excellent conditions for intensifying the heat and the mass transfer rates. In this zone, too, the concentration of the particles is the highest [73], and continuously decreases towards the injection point of the system. The opposed flow configuration of the suspensions encourages multiple inter particle collisions as well as mutual penetration and multiple circulation of the particles from one stream into the other. The penetration of particles arises due to their inertia, whereas deceleration takes place due to the opposite flow of the gas stream as a result of drag forces. At the end of the deceleration path, the particle is accelerated and once again regains its original stream. After performing several such oscillatory motions, the particle velocity eventually vanishes until it is withdrawn from the system. The latter might occur even earlier due to inter particle coUisions.
GAS +
PARTICLES
uzzzzzzzzzzzzzn^^ * ^ • ^ • ^ •
EZZZZZZZZZ2ZZZZZ3 • *
• • •-GAS +
• • • : • • • • • zzzzzzzzzzzzzzzn
PARTICLES
IMPINGEMENT PLANE
Fig.4.5. The principle of impinging streams
The intensification of the transfer processes is due to the following effects: a) An increase of the relative velocity between the penetrating particles and
the opposed gas stream. Under extreme conditions, where the particle attains the gas velocity at the point of entering the opposite stream, the relative velocity may
463
reach twice that of the gas velocity; the increase is thus significant. Consequently,
the external resistance will decrease.
b) An increase of the mean residence time of the particles in the system or
their holdup due to their penetration into the opposed stream, consequently
undergoing multiple circulation followed by damped oscillations. However, in a
dense system of particles, inter particle collisions might reduce the effect of the
increase of the mean residence time. Finally, an increase of the mean residence
time also allows a decrease of the geometrical size of the system.
c) In gas-liquid and liquid-liquid systems, shear forces exerted between the
phases or inter droplet colUsions, can result in a breakup of the droplets, leading to
an increase in their surface area, their rejuvenation, and therefore an increase of the
mass transfer rates.
d) Collision of the continuous phase of the opposed streams, namely, jet-jet
impingement, induces pressure pulsations or generates intense radial and axial
velocity components in turbulent flow. Consequently, good mixing is created in
the impingement zone of the streams. The mixing is also enhanced by the multiple
circulation of the particles in this zone.
As indicated previously, the key phenomenon in impinging-stream systems is
the penetration of particles into opposite streams, causing an increase of their
holdup in the reactor, of the relative velocity between the particles and the carrying
stream as well as a longer mean residence time of the particles. Consequently, the
mixing properties of the reactor may be significantly improved. A quantitative
analysis of this phenomenon and others related to impinging streams is presented in
the following examples, where the major aim is to determine the behavior of the
particles by introducing a pulse of these into one of the reactors.
In section 4.5, the following designations are made: Qi, Qy and qy are the
mass flow rates of the particles stream (kg particles/sec) and Ci is the concentration
of the tracer particles (kg tracer particles/kg particles). In such systems, the tracer
particles are, usually, those of the original ones. They are, however, made
radioactive or are painted, in order to distinguish them from the original particles.
The latter makes it possible to determine their concentration versus time in the RTD
experiments [73, p. 176], thus their mean residence time tm in the system, tm =
464
Vi/Qi in a perfectly mixed reactor. An additional key quantity is the holdup of the
particles in the reactor, Vi (kg particles); instead of the volume of the reactor.
Thus, in consistent units, Eqs.(4-1) to (4-11) hold. An important quantity used in
this section for comparing various effects is the mean residence time tm of the
particles in the system.
4,5-1 The simplest model [73, p. 180] of a two impinging-stream reactor is shown
schematically in Fig.4.5-1. On the LHS is demonstrated the actual configuration of
the reactor and on the RHS the Markov-chain model. The latter employs the
following considerations and assumptions:
a) There are three main zones in which mixing of particles takes place: zones
1, 2 and 3. These zones are considered to behave like perfectly mixed vessels, i.e.
states.
b) The lower part of the reactor behaves like a plug flow reactor in which no
back mixing takes place. This is designated as tp in Fig.4.5-1.
c) The time needed for a particle to pass from one vessel to the other is equal
to zero. This is true since the actual borders of one vessel overlap with the
neighboring one. On the other hand, a particle might stay in one of the above zones
during a finite time.
d) The holdups, Vi, of the particles in each model vessel are the same.
The resulting model is shown on the RHS of Fig.4.5-1. Clearly each vessel
represents a state in a Markov process; vessel 4 is the collector of the particles from
which only the carrying stream is leaving at flow rate 2Qi. Q12, Q2i» Q23 ^^ Q 32
are recycle streams. This is due to the penetration of particles from one stream into
the other where the impingement zone is designated by 2 in Fig.4.5-1 and is
simulated as a perfectly mixed reactor. The effect of penetration is emphasized by
the fact that movement of particles to vessel 4 is only possible from vessels 1 and
3. Whenever a particle reaches vessel 4, it remains there, i.e. the vessel is a
trapping state.
465
Qi ^o i f^ ( =q
'* 21 %
i71£~sl
I Fig.4.5-1. A model for a single stage two impinging-stream reactor
Noting that the residence time in the plug flow reactor is tp yields the
following matrices:
0 < t <ti
1
P = 2
3
Considering reactor j in Fig.4-1, hence j = l , a = 2, b = 3 and ^ = 4, as well
as Eqs.(4-12a) to (4-12c), and taking Qi as a reference flow, yields:
1 + CX3 = Pi4 + P34,1 + ^21 = ai2 + Pi4, for i = 2: an + OL32 = a2i + a23,
i = 3: as + a23 = a32 + p34
The following assumption were made:
OC12 = CX21 = a23 = OL32 = R, as = a i = 1; this yields that P14 = P34 = 1
The holdups of particles in all reactors are the same, i.e. m = ^l, i = 1,..., 4
From Eqs.(4-14) to (4-17b) the following probabilities were obtained:
For 0 < t < tp: p n = P33 = 1 - R|iiAt pi2 = P21 = P23 = P32 = RM'At
p-1
Pll
P21
0
2
P12
P22
P32
3
0
P23
P33
tp<t:
1
P= 2
3
4
1
Pll
P21
0
0
2
P12
P22
P32
0
3
0
P23
P33
0
4
P14
0
P34
1
466
P22 = 1 - 2R^At
For tp< t : pii = P33 = 1 - (1 + R)|iAt
P14 = P34 = ILiAt
where the rest of the probabiUties are as for 0 < t < tp.
As seen, the parameters of the solutions are: |LI, R and tp. In the numerical
solution it was assumed that tp = 1 corresponding to At = 0.005, |LI = 10 and that
C(0) = [1, 0, 0, 0], i.e. the pulse was introduced into reactor 1 in Fig.4.5-1, RHS.
The effect of the recycle R, 0, 0.1, 1 and 10 is demonstrated in Fig.Fig.4.5-la. It
is observed that by increasing R, the central reactor 2 becomes active with respect
to the response of the pulse input in reactor 1 and the system of reactors 1, 2 and 3
behaves as a single perfectly-mixed reactor for R = 10.
u
1
0.8
0.6
0.4
0.2
0
_
-
-
i = l
2,3,4
1 • " 1 / \ 1
1/ 1/
»\
1 \
; V_
4 1
_J R = 0
-\
1,2,3 t i l l
u
0.5
0 0.5 1 1.5 t
\ l
L 2 -""^
1 1
4 ^ ' ' ' " '
/ / 1
" 1 / R = 0.1
- ^ _ J 1 1 1
2.5 0 0.5
1
0.8
0.6
0.4
0.2
0
r.^
_ \
h V 1
1
1
y
/ 4'
\
~ \ • V _ 1 1
R = l I
_
1 1 1.5
t 2.5 0 0.5 1 1.5
t
Fig4.5-la. Ci versus t demonstrating the effect of R
467
4,5-2 The reactor [82; 73, p. 188] depicted schematically in Fig.4.5-2 is a modified
form of the original two impinging-stream reactor described in Fig.4.5-1 on the LHS with two additional air streams located below the upper streams where particles are introduced. A brief description of the vessels-flows model in Fig.4.5-2, is as follows. The inlet pipes to the reactor are simulated by two plug flow reactors 1 and 5. The entrance of the particles to the reactor are followed by three zones 2, 3 and 4, simulated by perfectly-mixed vessels. Particles leaving vessels 2 and 4 enter another mixing zone, designated as a perfectly mixed vessel 6, formed by the secondary air stream. The particles leave this reactor through a tubular reactor, where the time needed for a particle to pass from one vessel to the other is zero but it is finite for staying in the reactors. The recycle streams between vessels 2,3,4 simulate the harmonic motion of the particles.
unruji
Fig.4.5-2. An impinging-stream reactor with two pairs of tangential air feeds
468
Case a: tpi > 0
Noting that the residence time in the plug flow reactors is tp and tpi, yields
the following matrices:
0 < t <tp >•
1
1
P= si 1
0
tp + tpi<t:
1
1
2
3
P = 4
5
6
7
Pll 0
0
0
0
0
0
5
0
1
2
P12
P22
P32
0
0
0
0
3
0
P23
P33
P43 0
0
0
4
0
0
P34
P44
P54
0
0
tp
p
5
0
0
0
0
P55
0
0
< t < t p
1
2
= 3
4
5
6
6
0
P26
0
P46 0
P66
0
+ tpi
1
Pll 0
0
0
0
0
7
0
0
0
0
0
P67
1
i:
2
P12
P22
P32
0
0
0
3
0
P23
P33
P43
0
0
4
0
0
P34
P44
P54
0
5
0
0
0
0
P55
0
6
0
P26
0
P46
0
1
Considering reactor j in Fig.4-1, hence j = l ,a = 2, b = 3, c = 4, d = 5, e =
6 and ^ = 7, as well as Eqs.(4-12a) to (4-12c), and taking Qi as reference flow,
yields:
1 + as = p67,0C12 = 1, for i = 2: an + a32 = OL23 + a26,
i = 3: a23 + 0643 = a32 + a34, i = 4: a54 + a34 = a43 + a46,
i = 5: as = as4, i = 6: a26 + OC46 = P67
The following assumption were made:
as = ai = 1, thus, as4 = 1 and P67 = 2
^23 = ^32 = a43 = a34 = R, thus, a26 = a46 = 1
yielding the following probabilities:
469
Pll = l
P22= 1
P33= ]
P44= 1
P55= 1
P66= 1
[ - mAt [ - ( 1 +R)|Ll2At
[ - 2R|Ll3At
[-(l+R)|X4At
L - JlsAt
I - M'6At
P12 = ^2At
P23 = R^At P26 = ^6At
P32 = R|lt2At P34 = R|Ll4At
P43 = R|Li3At P46 = MeAt
P54 = |l4At
P67 = |l7At
In the numerical solution it has been assumed that [i[ = |X, yielding the
following parameters: \i, R, tp and tpi. In addition, tp = 0.2, tpi = 0.4
corresponding to At = 0.001 and that C(0) = [1, 0, 0, 0, 0, 0, 0], i.e. the pulse
was introduced into reactor 1 in Fig.4.5-2. The effect of the recycle R, 0 and 10 is
demonstrated in cases a, b and c, d for p, = 1 and 50, respectively, in Fig.4.5-2a.
The effect of i = 1, 50 is demonstrated in cases a, c and b, d, respectively.
u
1
0.8
0.6
0.4
0.2
0
^
p
L 1 1 1
R = 0 i = H
"^-- .^1
6 - 7
" r 3-5 "i 0 0.2 0.4 0.6 0.8
t
(b)|
k
\ R=10 "-\^^ ^i=H
^^ ' '--. i 1
2 3 J
1.2 0 0.2 0.4 0.6 0.8 t
1.2
U
I
0.8
0.6
0.4
0.2
0
—
-
i = l
....
~1 1 / 1 / 1 ^
:/Y.2
'l ~ 1
6 ;
\
1
(cJJ
R = 0 H 1 = 50
-
1
p
L
1
1
1 1
1 fwl 1 5 1
6
\
1,.„
. (411
R = l o H 1 = 50J
—
1 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6
t t Fig.4.5-2a. Ci versus t demonstrating the effect of R and |X
470
Case b : tpi = 0
In this case, the plug flow reactor before the exit from the system in Fig.4.5-
2, is absent. The results for identical parameters corresponding to case a are
depicted in Fig.4.5-2b. It was observed that for |j, = 1, the results for the present
case coincide with the results in case a above in the investigated range of t. The
results in cases a and b below for i = 6, 7 differ from the results in cases c and d
above due to the effect of tpi.
1
0.8
0.6
0.4
0.2
0
-
-
-
i = l
1
1 1 1 A,
2;/!'
1
>6 VLi-.
1
(a)J 7 1
-J R = 0 1 ^ = 5CM
-
1 1
-
-
p
1
1
I
i 1 ; 1
A 6 2-4
1 5 1 1
(bjl 7 1
R = 1 0 ^ ^ = 50 j
-
1 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6
t t Fig«4.5-2b. Ci versus t demonstrating the effect of R and |X
As indicated before, a possible increase in the mean residence time tm of the
particles might be anticipated due to penetration of particles into the opposed
stream, consequently undergoing multiple circulation followed by damped
oscillations. This behavior is controlled in the above model by the quantity R, i.e.
the recycle stream, and is demonstrated as follows for extreme cases.
R = 0:
In this case the effective reactors are 1, 2 and 6. Excluding reactor 7, yields
the following equation for the total mean residence time in the reactors for a pulse
introduced into reactor 1:
V i V. V^ ^ - n l + ^ + W " m6 "*• tpi - "Q" " tp " Q " " 2 Q " V
Assuming that the holdup V of the particles in the reactors is the same, thus \i =
Ql A , yields that:
471
tm = tp + tpi + | (4.5.2a)
R-»oo:
In this case all reactors are effective and reactors 2, 3 and 4 behave as a
single reactor of holdup V2 + V3 + V4 , thus
V^ V2 + V3 + V4 _V6_
*™ Qi """^"^ 2Qi •*" 2Q, • • Pi
or alternatively
t™ = tp + tp, + | (4.5-2b)
Eqs.(4.5-2a) and (4.5-2b) indicate that the relative increase in tm, due to the recycle
R, with respect to its value for R = 0 is 16.67%.
4>5-3 As indicated at the beginning of this section, there is an increase of the
holdup of the particles at the impingement zone, resulting an increase in the mean
residence time of the particles in the system. However, inter particle collisions
might decrease the above effect as well as back flow of the particles. It is the aim
of this case, depicted in Fig.4.5-3, to investigate these effects. Reactors 1 to 5,
demonstrate the impingement zone and reactor 3 with the highest concentration,
demonstrates the impingement plane. Reactor 6 is the exit of the particles from the
impingement zone and reactor 7 is the collector of the particles.
472
'Miisjte^rftyfefi-^ I X I
% H 6
^ = 7 Q1+Q5
Fig.4.5-3. A scheme for demonstrating characteristics of impinging streams
The following matrix is applicable:
P =
1
2
3
4
5
6
7
1
Pll
P21
0
0
0
P61
0
2
P12
P22
P32 0
0
0
0
3
0
P23
P33
P43 0
0
0
4
0
0
P34
P44
P54
0
0
5
0
0
0
P45
P55
P65
0
6
P16
0
P36 0
P56
P66
0
7
0
0
0
0
0
P67
1
Considering reactor j in Fig.4-1, hence j = l,a = 2, b = 3, c = 4, d = 5, e = 6 and ^ = 7, as well as Eqs.(4-12a) to (4-12c), and taking Qi as reference flow, yields:
1 + CX5 = P67 1 + a6i + a2i = ai2 + ai6
for i = 2: an + a32 = a2i + a23, i = 3: a23 + OL43 = a32 + a34 + a36,
i = 4: a34 + a54 = a43 + a45, i = 5: as + a45 + a65 = ^54 + a56,
i = 6: ai6 + ase + a56 = 0C6i + a65 + p67 Assuming that:
as = a i = 1, a32 = a34 = a2i = a45, ai2 = a54 = a23 = OC43,
473
ai6 = 0156, Ot61 = a65
and taking the following quantities as known, designated by
a2i = R. a36 = a, ai6 = Y
yields that:
«i2 = CC54 = a23 = CC43 = R + a/2
061 = a65 = Y+a /2 -1
a56 = Y
a32 = 034 = 045 = R
P67 = 2
Considering the above coefficients and Eqs.(4-14) to (4-19), yields the following
probabilities:
P12 = (R + a/2)mAt
P21 = R iiAt
P32 = RM.2At
P16 = Y|A6At
P23 = (R + a/2)H3At
P34 = R^4At
P43 = (R + a/2)H3At
P54 = (R + a/2)|A4At P45 = R|A5At
P56 = Y 6At
P6i = (Y+a/2-l)|XiAt i = l , 5
pil = 1 - (R + a/2 + Y)|iiAt
P22=l- (2R + a/2) i2At
P33 = 1 - (2R + a)^3At
P36 = aM.6At
P44 = 1 - (2R + a/2)|i4At
P55=l - (R + a/2 + Y)|X5At
P66= l-(2Y+a)|X6At
P67 = 2 i7At
Because of symmetry, |li = fis = |4., |i2 = 1 4 = \i', yielding the following
parameters: |a,i (i = 3, 6, 7), |i, |i', R, a and Y- The mean residence time in the
system (Fig.4.5-3), for a pulse introduced into reactor 1 and considering reactors 1
to 6, is determined below for R = 0 and R —> <».
R = 0:
Ignoring recycles Qgi and Qes, assuming that the holdups Vi = Vg, thus |i =
Ql/Vi, yields
•" Q, 2Q, 2n
474
R-»«>:
^Yi V2+V3+V4 _ ^ ^ 2 . ± j _ ^ Qi"" 2Qi ^ 2 Q i " 2^1'" III'^2^3
where |LI' = QiA^2 = Q1/V4 and 113 = QiA^s. The above equation indicates that if
the holdup of the particles V3 in the impingement zone is increased, 113 is decreased
and hence tm is increased. The latter is an important property of the impinging-
stream configuration [73, p.3 and 138]. It has also been observed in the
calculations that increasing a = a36 and 7= ai6 = (X56 has a negligible effect on
tm. Finally, it should be noted that values of tm calculated by the above equations,
coincide with numerical values obtained by numerical integration according to
Eq.(4-26) for a pulse introduced into reactor 1 and collected in reactor 6.
Fig.4.5-3a demonstrates the effect of R = a32 = a34 = a45 = 0, 5, 25, 100
on the concentration profile for 7= a56 = 1 and constant jii = 1 (i = 1,..., 7) for a
unit pulse introduced into reactor 1.
475
1
0.8
0.6
0.4
0.2
0
-
-
-
i = l
^ , •
1
R = 0
, . - • • ' • . . . . . . . ^ ' ^
2-5 1
7 '
1
-\
- i
J -|
___-
0 0.5 1 1.5 2 0 0.5 1 t t
1
0.8
0.6
0.4
0.2
0
-
ll-A 2
1 4,5 ^^6 " 1
R = 25
1
7
1
J
:
-
— -
0 0.5 1 1.5 2 0 0.5 1 1.5 t t
Fig.4.5-3a. Ci versus t demonstrating the effect of R
4,5-4 The major characteristic of impinging streams is the penetration of particles
from one stream into the opposite one through the impingement plane (Fig.4.5),
thus, increasing their meem residence time in the reactor as well as their relative
velocity with respect to the air. The scheme in Fig.4.5-4 demonstrates this effect in
the following way. Reactors 1 to 10 simulate the impingement zone of the particles
and in each reactor particles reside for some time. A pulse of particles introduced
in reactor 1 may be divided into three streams. One stream occupies reactors 3 to 5,
the other, reactors 6 to 8, and the third one will occupy reactors 9 and 10.
Eventually, the pulse accumulates in reactor 12 while passing reactor 11.
476
^ OD ^ Ri ^
5 U - Q s
R,>| 6 U-|—I 7 ka I '-j 8 TRi ^lobi ^ o b i '
R3
11
I R3
Q,+Q5
Fig.4.5-4. A scheme for demonstrating the effect of different penetration distances in impinging streams
The following matrix is applicable for the configuration in Fig.4.5-4:
P =
1 1 2
3
4
5
6
7
8
9
10
11
12
1
Pll
P21
0
0
0
P61
0
0
P91
0
0
0
2
P12
P22
P32 0
0
0
0
0
0
0
0
0
3
0
P23
P33
P43
0
0
0
0
0
0
0
0
4
0
0
P34
P44
P54
0
0
0
0
0
0
0
5
0
0
0
P45
P55 0
0
P85
0
P10,5
0
0
6
P16
0
0
0
0
P66
P76
0
0
0
0
0
7
0
0
0
0
0
P67
P77
0
0
0
0
0
8
0
0
0
0
P58 0
P78
P87
0
0
0
0
9
P19
0
0
0
0
0
0
P88
P99
10
0
0
0
0
P5,10 0
0
0
P9,10
P10,9 PlO.lO
0
0
0
0
11
Pi.11 0
0
0
P5,ll 0
0
0
0
0
12
0
0
0
0
0
0
0
0
0
0
Pl l . l l Pll,12
0 1
The following simplifying assumptions were made: |Lii = |Li (i = 1,..., 12). For reactors 1, 2, 3,4, 5 all interactions are equal, i.e., Ri = ay. For reactors 1, 6,7, 8, 5 all interactions are equal, designated as R2. For reactors 1,9,10, 5 all interactions are equal, designated as R3.
477
Considering Eqs.(4-12a) to (4-12c) and Eqs.(4-14) to (4-19), yields the
following probabilities:
Pll = P55 = 1 - (1 + Rl + R2 + Rs^At
P12 = P21 = P23 = P32 = P34 = P43 = P45 = P54 = Rl|lAt
P16 = P61 = P58 = P85 = P67 = P76 = P78 = P87 = R2^At
P19 = P91 = P5,10 = P10,5 = P9,10 = P10,9 = R3^At
Pl,ll = P5,ll = ^At P22 = P33 = P44 = 1 - 2RmAt
P99 = PlO,10=l-2R3^At pii,ii = l-2^At pii,i2 = 2^At
As seen the parameters of the solution are: |X and Ri, R2 and R3.
The mean residence time in the system (Fig.4.5-4), for a pulse introduced
into reactor 1 and considering reactors 1 to 11, is determined below for Ri = 0 and Ri^oo .
Ri = 0:
In this case only reactors 1 and 11 are effective for a unit pulse introduced into
reactor 1. Assuming that the holdups Vi = Vn, thus |LL = QiA^i, yields
^ Q i ^ 2 Q i 2ji
Ri-">oo:
In this case reactors 1 to 11 are effective. Assuming the same holdup of particles in
all reactors, Vi = V, where |LI = QiA , yields
^ Vi + ...-f Vio Vii _ I I V ^ 11 ^ " 2Qi " 2Qi " 2Qi "" 2^
The above equations indicate that the relative increase of tm due to the recycles Ri,
with respect to tm for Ri = 0, amounts to 266.7%.
Fig.4.5-4a demonstrates typical response curves to a unit pulse input
introduced into reactor 1 for Ri = 10, R2 = 5, R3 = 1, M, = 10 and At = 0.00005.
478
0.01 0.02 0.03 0.04 0.05
Fig.4.5-4a. Ci versus t
4,5-5 The following schemes demonstrate configurations comprising of 2, 3 and 4
impinging streams. The effect of the number of impinging streams on the mean residence time of the particles in the system will be investigated below.
479
case a: 2-impinging streams case b: 3-impinging streams
%,T±\. .ri^iJi ^ ^ R R
Mil ^
I [33
^ = 5
case c: 4-impinging streams
case d: 2-impinging streams case e: 2-impinging streams
apt! R *^ Ir* ±
*lS
'?(2Q| 5-4
'p.l
s 'SfiS
^ = 4
Fig.4.5-5. The effect of the number of impinging streams
480
where R = ay = Qi/Qi
Case a: 2-impinging streams
The following matrix applicable for case a in Fig.4.5-5 is:
P =
1
2
3
4
5
1 2 3 4 5
Pll P12 0 P14 0
P21 P22 P23 0 0
0 P32 P33 P34 0 0 0 0 P44 P45
0 0 0 0 1
Considering Eqs.(4-12a) to (4-12c) and assuming that |ii = fi (i = 1,..., 5),
a i2 = a2i = a23 = a32 = R and that au = a34, yields from Eqs.(4-14) to (4-19),
the following probabilities:
Pll = P33 = 1 -(1 + R)^At P22 = 1 -2R^At
P14 = P34 = M t P21 = P12 = P23 = P32 = R^At
P44 = 1 - 2nAt P45 = 2nAt
Fig.4.5-5a demonstrates the effect of the recycle R for |i = 1 and At = 0.001
for a unit pulse introduced into reactor 1. It is observed that for R = 50, reactors 1,
2 and 3 behave as a single reactor due to the relatively high recycle.
R = 50 , . - - i
• • ' • • H 1 5 . - " 1
• •
• •
• 1 1»2,3 / J \ \ 1 m 1 il / •
l i \ / m 1 \ ^
1 '/ 1 T^--f--^H-^ 5 -1
Fig.4.5-5a. Ci versus t demonstrating the effect of R
481
Case b: 3-impinging streams
The following matrix is applicable for case b depicted in Fig.4.5-5:
P =
1
2
3
4
5
6
1
Pll
P21
0
0
0
0
2
P12
P22
P32
P42 0
0
3 0
P23
P33 0
0
0
4 0
P24
0
P44 0
0
5
P15 0
P35
P45
P55 0
6 0
0
0
0
P56 1
Considering Eqs.(4-12a) to (4-12c) and assuming that (Xj = [i (i = 1, ..., 6),
«12 = a21 = 0123 = "32 = ^24 = 042 = R, a i s = 035 = 045, yields from Eqs.(4-
14) to (4-19), the following probabilities:
Pll = P33 = P 4 4 = l - ( l + R)fAAt p22=l-3RfiAt
P12 = P21 = P23 = P32 = P24 = P42 = R^At pi5 = P35 = P45 = ^At
P55 = 1 - SfiAt P56 = 3|iiAt
Fig.4.5-5b demonstrates the effect of the recycle R for fi « 1 and At = 0.005
for a unit pulse introduced into reactor 1. It is observed that for R = 50, reactors 1
to 4 behave as a single reactor due to the relatively high recycle.
Fig.4.5-5b. Ci versus t demonstrating the effect of R
482
Case c: 4-impinging streams
The following matrix is applicable for case c depicted in Fig.4.5-5:
P =
1
2
3
4
5
6
7
1
Pll
P21
0
0
0
0
0
2
P12
P22
P32
P42
P52 0
0
3
0
P23
P33 0
0
0
0
4
0
P24
0
P44 0
0
0
5
0
P25
0
0
P55 0
0
6
P16 0
P36
P46
P56
P66 0
7
0
0
0
0
0
P67 1
As before, the following probabilities are obtained:
Pll = P33 = P44 = P55 = 1 - (1 + R)fiAt P22 = 1 - 4R^At
P12 = P21 = P23 = P32 = P24 = P42 = P25 = P52 = R| At
P16 = P36 = P46 = P56 = I At
P66 = 1 - 4jlAt P67 = 4|xAt
where R = ai2 = a2i = a23 = a32 = CX24 = 0C42 = OC25 = OL52
Fig.4.5-5c demonstrates the effect of the recycle R for |i = 1 and At = 0.005
for a unit pulse introduced into reactor 1. It is observed that for R = 50, reactors 1
to 5 behave as a single perfectly-mixed reactor due to the relatively high recycle.
1
0.8
0.6
0.4
0.2
0
\ R = l
L. \ y \i=l 7 '
\ 3,4,5
H
-.-.u. -1
[-
h-
R = 50
1 7.
/
1-5 / '
-j
-
1
5 -1
Fig.4.5-5c. Ci versus t demonstrating the effect of R
483
Case d: 2-impinging streams
This is another scheme of 2-impinging streams depicted in Fig.4.5-5 case d
where part of the impingement zone is a plug flow reactor. Noting that the
residence time in the plug flow reactor is tp yields the following matrices:
0 < t < t
1
P = 3
4
2tp<t :
1
2
P = 3
4
P-1
Pll 0
0
1
Pll
1 P21
0
0
3
P13
P33 0
2
P12
P22
0
0
4
0
P34 1
3
P13
P23
P33 0
4
0
0
P34 1
tp < t < 2 tp:
1
1
2
P = 3
4
Pll 0
0
0
2
P12
P22 0
0
3
P13
P23
P33 0
4
0
0
P34 1
Considering Eqs.(4-12a) to (4-12c) and assuming that jii = |X (i = 1,..., 4),
o^l2 = o t2 i=R, oci3 = a23, yields from Eqs.(4-14) to (4-19) the following
probabilities:
ForO<t <tp:
pil = l - ^At Pl3 = ^At
P33 = 1 - 2|iAt P34 = 2|ilAt
For tp < t:
P l l = P 2 2 = l - ( l + R ¥ A t
P12 = P21 = R^At P13 = P23 = |XAt
P33 = 1 - 2 lAt P34 = 2p,At
Fig.4.5-5d demonstrates the effect of R, |LI and tp on the distributions Ci -1
for At = 0.005. Cases a and b show the effect of R; if R > 0 reactor 2 becomes
active after some time. Cases b and c depict the effect of \i and cases c, d
484
demonstrate the effect of tp. In the latter case it is interested to note that reactor 2 is
inactive all the time due to the relatively high value of tp.
0
1
0.8
0.6
0.4
0.2
0
\ i = l
-/7
0.1 0.2 0.3 0.4 t
^1= 10 R = 5 t =0.1
•
4 /
9 ^i^^:^^--:^:^,.^.^ . -_
1 1 1 1
0.
(cjl
H
H
0.2 0.3 t
0.5
\ r \ L \
/
r '
^1= 10 R = 5 t =1
/ /
4/ /
3 " " " = ^ - - ^ 2 ^ —
"(dji
H
-J
•J
1 1 1 1 1 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5
t t
Fig.4.5-5d. Ci versus t demonstrating the effect of R, JLI and tp
Case e: 2-impinging streams In this scheme depicted in Fig.4.5-5 case e, an additional tubular reactor is
added, simulating the plug flow at the lower part of the reactor (Fig.4.5-1, LHS) in which no back mixing takes place. The residence time in the plug flow reactor simulating the impingement zone is tp and of the lower part it is tpi. In the following we consider the interesting case where tpi > tp; a unit pulse introduced into reactor 1, yielding the following information:
For 0 < t <tp, and because tpi > tp, no change in concentration takes place in the system.
For tp < t < 2tp (< tpi), there is a transfer of the tracer from reactor 1 to 2 only through the tubular reactor. As a result of this situation Qi = 0 and there are only
485
circulation flows, i.e. Q12 = Q21. However, when t > tpi, Qi > 0. We define R
with respect to this flow, i.e. R = Q12/Q1 = OC12 = a2l- Under this condition, the
following matrix holds:
P =
1
Pll 0
P12 1
where pn = 1 - R|LiAt pi2 = R|xAt
For 2tp < t < 3tp (< tpi), the following matrix holds:
P =
1 2
Pll P12
P21 P22
where pi 1 = p22 = 1 - R|LiAt pi2 = p2i = R^At
For tpi < t, the following matrix holds:
1
2
3
4
1
Pll
P21 0
0
2
P12
P22 0
0
3
P13
P23
P33 0
4
0
0
P34 1
p =
where P l l = P 2 2 = l - ( l + R ¥ A t
P12 = P21 = R^At P13 = P23 = |iAt
P33 = 1 - 2|i,At P34 = 2|LlAt
As seen, the parameters of the solution are: R, \i, tp and tpi. Fig.4.5-5e
demonstrates the effect of the above parameters on the distributions Ci -1 for At =
0.00005 and 0.005. The following quantities were assigned for the parameters: tpi
= 5tp; tp = 0.01 and 1, R = 0, 1 and 10 whereas |i = 50, 100 and 500. Cases a, b
486
and b, e show the effect of |LI; cases a, c depict the effect of tp and tpi; cases d, e, f
demonstrate the effect of R.
u
1
0.8
0.6
0.4
0.2
0
i = l
h-
h
X t =
1/^
/
\ \
= 0.01
:1
1
t Pl
= 0.05
50
(a
>^ /
/ , ' 4
1
t =0.01 1 \ '
F '^ /
I I I
t : pl
= 0.05
100
1 T
(bj
•
/ /
1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.070 0.01 0.02 0.03 0.04 0.05 0.06 0.07
u
1
0.8
0.6
0.4
0.2
0
i = l T""
\
t =1 t =5 p pl
R = l ^ = 50
1,2
3,4
4 1
•H
—
\ 1-3 1 1 1 1 1 1
0 1 3 4 t
1
0.8
0.6
0.4
0.2
0
1 1
i = l t :
- P
R =
-
2-4
= 0.01
= 0
t = pl
1 1
= 0.05 1 /
500 i ;
1; 1
'•iV
' (d)| 4 1
H
-\
1 1 1 1 1 2 1
1 i = 11
1 1 r i
1
1
\- ;
1 1
1 :'1\
• 1 \ \
t = p
R =
\
/
V
1
:0.01
:1
1
1 t = pl
3,4__ 1
1 = 0.05 .
500 '
1'
1
1 (e)l " "4 1
—j
1-3 1
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 t t
487
o
1
0.8
0.6
0.4
0.2
0
~i"=T
-
\
• 2
1 1 I t=0.01 t =0.05 •
p pi R=10 ^ = 500 .
V
3,4 ^'V 1 1 1
4
-
1-3 1
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 t
Fig.4.5-5e, Ci versus t demonstrating the effect of R, \i, tp and tp i
Mean residence time tm of the impinging-stream systems in Fig 4.5-5
The comparison will be made for extreme cases, i.e. R = 0 (part of the
reactors are not interacting) and R —> ©<> (appropriate reactors are interacting and
considered as a reactor of volume equal to the total volume of the single reactors).
Note that R = Q12/Q1 = Q2l/Ql- The following examples elaborate the above for a
unit pulse introduced into reactor 1 and assuming that the holdups of the particles in
each reactor are the same, i.e. Vi = V, thus, |ii = |i = QiA^
Case c
R = 0: tm = tml + tm6 = Vj/Qi 4- V6/(4Qi) = l/^l + 1/(4^) = 5/(4^)
R -^ ^: tm = (Vi + V2 + V3 + V4 )/(4Qi) + V6/(4Qi) = 6/(4^)
Case d R = 0: tm = tml + tm3 = Vi/Qi + V3/(2Qi) = 3/(2^1)
R -^ «^: tm = tml + tp + tm2 + tm3
= Vi/(Qi + Q21) + tp + V2/(Q2 + Q12) + V3/(Ql + Q2)
for Q12 = Q21 and Qi = Q2 it is obtained that
tm = (5 + R)/(2^(l+R)) + tp
where tp = Vp/(2Qi2) = l/(2R^p) and ^p = QiA^p.
488
Case e (tpi > tp)
R = 0: Fort<tpi tm = tpi
For t > tpi tm = tmi + tpi + tm3 = Vi/Qi + tpi + V3/(2Qi) =
3/(2^1)+ tpi
R-^oo: For t^ tpi tm = tpi
For t> tpi
tm = tml + tp + tm2 + tp + tm3 = (5 + R)/(2n(l+ R)) + tp + tpi
where tp is given above in case d, and tpi = Vpi/(2Qi) = l/(2|Xpi) and \ip\ =
QlA^pl Similarly, the following Table may be obtained for a pulse input introduced
into reactor 1:
Table 4.5-5. Expressions for the mean residence time tm in the reactor-system in Fig.4.5-5 (excluding the collector reactor %)
n-number of
impinging streams
2 (case a)
3 (case b)
4(case c)
1 n
2 (case d)
1 2 (case e)
t>tpi
R = 0 1
tin = (3/2)/^
(4/3)7^1 '
(5/4)/^
[(n+l)/n]/^l
(3/2)7^1
(3/2)/^ + tp
R ^ o o
tn, = (4/2)/jl
(5/3)/^
(6/4)/|X
[(n+2)/n]/n R > 0 tm =
(5+R)/(2|x(l+R)) + t„
1 R>0 1 tn,=
(5+R)/(2n(l+R)) 1 +tp + tDl 1
tp and tpj are given in cases d and e above.
It should be emphasized that the value of tm calculated by the above
equations, coincides with numerical values obtained by numerical integration
according to Eq.(4-26) for a pulse introduced into reactor 1 and collected in reactor
489
The following ratios may be obtained from Table 4.5-5:
,R->oo n + 2 a i =
tm,R=0 ^ + 1
ai demonstrates the effect of the recycle R on increasing tm and it approaches 1 for
large n. Maximum ai = 4/3 for a 2-impinging-stream reactor (n = 2).
2 Vn (n+1) ^ ^ r^
tto = z = "7 :^ for R = 0 tn, n+i n(n + 2)
tm,n (n-H)(n + 2) . ^ aa = 7 = 7 -T— for R -» oo
^ Vn+i n(n + 3)
a2 and as show the effect of the number of streams n on the mean residence time
ratio for R = 0 and R -> <». It may be concluded that increasing n causes the ratios
to approach unity. The maximum ratio is obtained for n = 2, yielding a2 = 9/8 =
1.125 and as = 12/10 =1.2
4>5-6 Fig.4.5-6 shows a single stage four impinging-stream reactor [73, p. 186].
On the RHS is schematic of the reactor and on the LHS is the model of reactors and
flows. It is assumed to comprise eight perfectly-mixed reactors designated as 1,
..., 8. All reactors are assumed to have equal holdups. The model assumes also
tangential feed of the streams and that more mixing zones are generated by the
direct impingement of the streams. Recycle occurs through intermediate perfectly-
mixed reactors 2,4, 6, 8 where impingement is expected.
490
is ?L^r^^<^^M^ 5 rv^
y
Fig.4.5-6. A model for a single stage four impinging-stream reactor
For the configuration in Fig.4.5-6, the following matrix holds:
1
2
3
4
5
6
7
8
9
10
1
Pll
P21
0
0
0
0
0
P81
0
0
2
P12
P22
P32
0
0
0
0
0
0
0
3
0
P23
P33
P43
0
0
0
0
0
0
4
0
0
P34
P44
P54 0
0
0
0
0
5
0
0
0
P45
P55
P65
0
0
0
0
6
0
0
0
0
P56
P66
P76
0
0
0
7
0
0
0
0
0
P67
P77
P87
0
Os
8
P18
0
0
0
0
0
P78
P88
0
0
9 10
P19 0
0 0
P39 0
0 0
P59 0
0 0
P79 0
0 0
P99 P9,I0
0 1
Considering reactor j in Fig.4-1 and making the following designations: j = 9, a = 1, b = 2,..., Z = 8 and ^ = 10. Applying Eqs.(4-12a) to (4-12c), taking Qi as reference flow and making the following assumptions, i.e.
ay = a5 = as = ai = 1
491
R = ai2 = a2i = a23 = OC32 = OC34 = a43 = a45 = OC54 = a56 = a65 = a67 =
a76 = OC78 = ag? = CX18 = agi
yields that
0C19 = a39 = a59 = a79 as well as p9,io = 4
Assuming also that all reactors have the same holdup of particles, i.e. Vi = V (i =
1, ..., 10) or |ii = ^1= QiA^, and applying Eqs.(4-14) to (4-19), yields the
following probabiUties:
Pll = P33 = P55 = P77 = 1 - (1 + 2R)|LlAt
P22 = P44 = P66 = P88 = 1 - 2R|XAt
P12 = P21 = P23 = P32 = P34 = P43 = P45 = P54 = P56 = P65 = P67 = P76 = P78
= P87 = P18 = P81 = R^At
P19 = P39 = P59 = P79 = |LiAt p99 = 1 - 4^At P9J0 = 4|lAt
As seen the parameters of the solution are |Li and R.
The mean residence time in the system (Fig.4.5-6), for a pulse introduced
into reactor 1 and considering reactors 1 to 9, is determined below for R = 0 and R - ^ 0 0 .
R = 0: Assuming that the holdups Vi = V9, thus |X = QiA^i, yields
tm = Vi/Qi+V9/(4Qi) = (5/4)/^
R->oo:
tm = (Vi + ... + V8)/(4Qi) + V9/(4Qi) = (9/4)/^
The above equations indicate that the relative increase in the mean residence time
with respect to R = 0, due to the recycles, is 80%.
In the numerical solution the effect of the recycle R, 0, 2 and 50 and of \i, 1
and 0.1, is demonstrated in Fig.4.5-6a for At = 0.005 and a unit pulse introduced
into reactor 1 in Fig.4.5-6. Note that due to symmetry C2 = Cg, C3 = C7 and C4 =
C6. The effect of R is demonstrated in cases a, b and c; the effect of \i in cases c
and d.
492
1
0.8
0.6
0.4
0.2
0
1 \"i=l
- \ /
- X ~ J .9 j^_2;8_
1
1 1 1 0 , . • • • - - " "
1 1
' (a)l
-
| X = 1 - 1 R = 0 J
-
T" ""
1
0.8
0.6
0.4
0.2
0
1
-
1
i = l
1
1
iQ-
1
1 (c^
^i=lJ R = 50
A
1
10. . . - --
^=i i R = 2
1 2 3 4 5 0 1 t
-0.5 0 0.5 1 1.5 t
2 - 1 0 1
Fig.4.5-6a. Ct versus t demonstrating the effect of R and p.
4,5-7 An extension of the single-stage two impinging-stream reactor to a 3-stage
reactor [73, p.l95], and similarly to a multi-stage reactor, is demonstrated in Fig.4.5-7. The reactor is composed of sections separated by plates a, with appropriate openings c, d etc. Two successive plates and their top views A-A' and B-B' are shown. Between two plates there is a partition designated by b. The partition is dividing the gas-particle stream so that the impinging-stream effect is maintained at each of the reactor's stages. In order to operate the multi-stage reactor, two gas-solid streams are fed to the top of the reactor. The streams collide in the impingement zone 2 (Fig.4.5-7, top view). The combined gas-solid stream formed after the impingement enters through opening c and flows downwards where it is divided into equal horizontal streams by the partition b. The streams impinge above opening d and enter the next stage where they are divided again by
493
partition b. Eventually, the solid particles leave at point e and the gas exits through pipe f located at the reactor's exit.
The vessel-flow arrangement which is modeled, is shown on the RHS of Fig.4.5-7. The n stages may be envisioned as n identical reactor segments, shown schematically in Fig.4.5-1, LHS. Each segment consists of four mixed vessels where the multi-stage reactor is terminated by a plug flow reactor. The inlet pipes are also simulated by plug flow reactor. It is assumed that the transition time from one stage to the other is negligible as compared to the time spent in the mixed vessel.
top view
single-stage reactor zone
Fig.4.5-7. Structure and model of a 3-stage two impinging-stream reactor
494
The transition probability matrix for the 3-stage reactor is given in the
following and can easily be extended to a multi-stage reactor. For tp < t < tpi the
matrix is of 12 by 12 since reactor 13 is yet inactive whereas for tpi < t, it reads:
P =
1
2
3
4
5
6
7
8
9
10
11
12
13
1
Pii
P21
0
0
0
0
0
0
0
0
0
0
1 0
2
Pl2
P22
P32
0
0
0
0
0
0
0
0
0
0
3
0
P23
P33
0
0
0
0
0
0
0
0
0
0
4
Pl4
0
P34
P44
0
0
0
0
0
0
0
0
0
5
0
0
0
P45
P55
P65
0
0
0
0
0
0
0
6
0
0
0
0
P56
P66
P76
0
0
0
0
0
0
7
0
0
0
P47
0
P67
8
0
0
0
0
P58
0
P77 P78
0
0
0
0
0
0
P88
0
0
0
0
0
9
0
0
0
0
0
0
0
P89
P99
Pl0,9
0
0
0
10
0
0
0
0
0
0
0
0
P9,10
PlO,10
Pll,10
0
0
11
0
0
0
0
0
0
0
P8,ll
0
PlO,ll
Pll,ll
0
0
12
0
0
0
0
0
0
0
0
P9,12
0
Pll,12
Pl2,12
0
13
0
0
0
0
0
0
0
0
0
0
0
Pl2,13
1
The derivation of the probabilities is similar to case 4.5-1. The following
probabilities were obtained assuming symmetry in the flows with respect to
reactors 2, 4, 6, 8, 10 and 12, and that all reactors have the same holdup of
particles, thus, |ii = |i, i = 1,..., 13:
Pll = P33 = P55 = P77 = P99 = Pll,ll = 1 - (1 + R)| At
P22 = P66 = P10,10 = 1 - 2R|lAt
P44 = P88 = 1 - 2|lAt
P12 = P21 = P23 = P32 = P56 = P65 = P67 = P76 = P9,10 = Pl0,9 = PlO,ll = Pll,10 = R| At
P45 = P47 = P89 = P8ai = ^At
P14 = P34 = P58 = P78 = P9,12 = Pl 1,12 = I At
whereas for tp < t < tpi: pi2,i2 = 1 and for tpi < t: pi2,i2 = 1 - 2|iAt pi2,i3 = 2|LiAt
495
As seen, the parameters of the solutions are: |i, R, tp and tpi. In the
numerical solution it was assumed that tp = 0, tpi = 1,5 corresponding to At =
0.001 and 0.005, respectively. Other parameters are: R = 0, 1, 20, i = 1, 5 and
the unit pulse was introduced into reactor 1 in Fig.4.5-7, RHS. Fig.4.5-7a
demonstrates the relationship Q-t in which the effect of R is demonstrated in cases
a, b and c, the effect of |X is depicted in cases c and d whereas the effect of tpi in
cases dande. Note that due to synmietry Ci =€3, C5 =C7andC9 =Cii.
u
1
0.8
0.6
0.4
0.2
0
1
\ i = l - \
4 ^
/ 5 -
1
1 1 1 1 ^= 1 R = 0 t =5
12
, . • • - • • " \ , /
^ < S ^ '' 8
T2:ro:i3, , 7 ^ ' ^
1
1 3 . •
' ^ -..
1
(aj
'' 1
A
-|
1 : : V
496
1
0.8
0.6
1 ] 1 r
i = l
1 r ^1= 1 R = 20 t = 5
(c)
12 13.''
U
0.6
0.4
0.2
0
. ' Li = 5 R = 20 t =5 i = l 12 ' ^ pi
2\r / / / 5 , 6 , 8
llJ^^. J.
9,10
J I L
0 1
497
u
1
0.8
0.6
0.4
0.2
0
1
i = l
1 •A 5,6
2: V /
«
1 1 1 |i = 5 R = 20 t =1
pi , . . - - •
12 ,'13
/ ' ^ /
;". 9,10
1 1 1
' (e l
-
-
-
1 -0.5 0 0.5 2.5 1 1.5 2
t
Fig.4.5-7a. Ci versus t demonstrating the effect of R, |i and tpi
The mean residence time in the system (Fig.4.5-7), for a pulse introduced
into reactor 1 and considering reactors 1 to 12, is determined below for R= 0 and R->oo.
R = 0:
In this case reactors 1, 4, 5, 7, 8, 9, 11, and 12 are effective for the pulse
introduced into reactor 1. Assuming that the holdups of the reactors are identical,
i.e. Vi = V, thus |Li = QiA^, yields:
tm = tp + tml + tni4 + tm5 + tm7 + tm8 + tni9 + tmll + tml2 + tpi
= tp +tpi+(13/(2^1)
R ^ o o :
In this case reactors 1 to 12 are effective, and reactors 1-2-3, 5-6-7 and 9-10-11
may be considered as single reactors due to the recycle R. The above assumption is
valid also here, thus it follows that:
tm = tp + tpi + (15/(2^)
The above equations indicate that the relative increase of tm due to the recycles R,
with respect to tm for R = 0, and ignoring tp and tpi is 15.4%.
498
Chapter 5
APPLICATIONS OF MARKOV CHAINS IN CHEMICAL PROCESSES
The major objective of this chapter is to demonstrate how Markov chains can
be appUed to determine the transient behavior of comphcated open systems
undergoing simultaneously heat and mass transfer processes as well as chemical
reactions. It is an extension of chapter 4 in which the RTD of a complicated system
was investigated.
5.1 MODELING OF THE PROBABILITIES The model. A multi-component and multi-reactor system, arranged
according to the general model depicted in Fig.5-1, is considered, which extends
the scheme in Fig.4-1. It covers numerous flow arrangements and processes
encountered in Chemical Engineering.
The general scheme consists of a central reactor designated by j and
peripheral reactors a, b,..., Z. A process may terminate at the exit of the peripheral
reactors, but the model makes it possible also to collect in reactor | the streams
leaving the peripheral reactors. In reactor % no chemical or physical processes take
place.
The flow system comprise the following flows: Qj, Qa, Qb> ••» Qz - flows
from feed vessels to reactors j , a, b, ..., Z. Qj, Q'a, Qb, •••» Qz - flows leaving
reactors j , a, b, ..., Z, outside. There are also interacting flows between the
reactors, i.e. each reactor is feeding all the others. Finally, flows qi^ are from each
reactor to the collector ^.
Each of the streams and the reactors may contain species f where f = 1,2,...,
F; the total number of species is F. The concentrations of the species at the exit of
the reactors, i.e., Cg, Cfi (i = a, ..., z) and C'f are not, in general, equal to the
concentrations inside the reactors, Cg, Cfi (i = a, ..., z) and Cf . If the
499
concentrations C'fj, Cfi (i = a, ..., z) are equal to zero, such a flow configuration
simulates, for example, a concentration process. If the concentration of the species
at the exit of reactor ^ equals zero, C'f = 0, the species are completely accumulated
in reactor ^ which is considered as "total collector" or "dead state" for the species.
If C'f = Cf , the species are not accumulated in reactor ^. If 0 < C'f < Cf , the
species are partially accumulated in reactor ^ which is considered as a "partial
collector" of the species.
The basic element in the flow system is the perfectly-mixed reactor. In the
multi-reactor system heat and mass transfer operation (absorption, desorption,
dissolution of solids, heat generation or absorption as well as heat interaction
between the reactor and the surroundings etc.) as well as chemical reactions may
occur simultaneously, or not. The processes are governed by Eqs.(5-8), (5-12),
(5-16), (5-19), (5-23) and (5-25) in the following, on the basis of which transition
probabilities are derived as well as the single step transition matrix.
A specific configuration is determined by appropriate selection of the
interacting flows and the number of reactors as well as the operating conditions.
When the central reactor j is considered, usually, j = 1 and a = 2, b = 3, etc. As
indicated, the collector is designated by ^. When the central reactor is not
considered, usually, a = 1, b = 2, etc. Each reactor designates a state, thus, the
total number of reactors is the number of states.
500
a) Overall scheme
b) Collector
Fig.5-1. Scheme of a continuous flow system
501
Definitions. A^ = fk designates a system with respect to its chemical formula f
and location (reactor) k in the flow system. The system is a fluid element
containing F chemical species for which f = l , 2, 3, ...,F where each figure
designates a certain species with its chemical formula. In addition, k = a, b,..., Z,
where each letter designates a reactor in the flow system composed of Z + 1
perfectly mixed reactors, including reactor .
The states of the system are the following concentrations and enthalpies (or temperatures) of the different species inside the various reactors: a) The concentrations Cf nj of species f at the peripheral feed vessels to reactor j in Fig.5-1 as well as the specific enthalpy hinj (or Tinj) of the fluid in the feed vessel to j ; b) The concentrations Cf,in,k of species f at the feed vessels to reactors k and the specific enthalpy hin,k (or Tin,k) of the fluid in the feed vessels to k; c) The concentrations Cfk of species f in reactors k as well as the specific enthalpies hk, h (or Tk, T^), respectively, of the fluid in reactors k and in collector .
Thus, the state space SS for the system will read:
SS = [Ci^inj, C2,in,J5 C3^in,j, ..., Cp^nJ?
Cl,in,a»
Cl,in,b»
^l,in,z»
Clj,
Cla,
Clb,
Clz,
Ci^,
hinj?
hj.
C2,in,a>
C2,in,b»
C2,in,z»
C2j,
C2a>
C2b.
C2z,
C2^,
hin,a»
ha,
C3,in,a» •••»
C'3,in,b» •••>
C'3,in,z» •••»
C3j , ...,
C3a , ...,
C3b , ...,
C3z , ...,
C3^ , ...,
hin,b > •••»
hb , . . . ,
CF,in,a»
CF,in,b»
C'F,in,z>
Cpj,
Cpa,
Cpb*
Cpz,
Cp^,
hin,z»
hz, h ] (5-1)
where the total number of states, only if all above elements do exist, is (F+l)(2Z+3). Z is the number of perfectly-mixed reactors excluding the collector and F is the total number of species.
A compact form of the above equation for f = 1, 2, 3,..., F reads:
502
SS = [Cf^inJ, Cfjn,a» Cf in,b» •••» C!f in,z>
Cfj, Cfa, Cfb , ..., Cfz, Cf ,
hinj» hin,a> hin,b » ••• hin,z?
hj, ha, hb ,..., hz, h ] (5-la)
If the system occupies a state, it means that the concentrations of species f in
all reactors are prescribed as well as the enthalpies of the fluid in the reactors.
Finally, the movement of a fluid element from a state to another is the transition
between the states. The initial state vector is given by Eq.(2-22), i.e.,
S(0) = [si(0), S2(0), S3(0), ..., sz(0)]
For the case under consideration it reads:
S(0) = [Ci,in,j(0), C2,inj(0), C3,in,j(0) , ..., CpanjCO),
Cl,in,a(0)» C2,in,a(0)» C3,in,a(0) , ..., CF,in,a(0).
Cl,in,b(0), C2,in,b(0), C3,in,b(0) , ..., CF,in,b(0)»
Cl,in,z(0), C2,in,z(0), C3,in,z(0) , ..., CF,in,z(0),
Cij(O),
Cia(O),
Cib(O),
Ciz(O),
Ci^(O),
hind(O),
hi(0),
C2j(0),
C2a(0),
C2b(0),
C2z(0),
C2^(0),
hin,a(0),
ha(0),
C3j(0) , ...
C3a(0) ,...
C3b(0) ,...
C3z(0) , ...
C3^(0) , ...
hin,b(0) ,...
hb(0) ,...
, CFJ(O),
, CFa(O),
, CFb(O),
, CFZ(O),
, CF^CO),
, hi„,z(0).
, hz(0).
The above equation is expressed compactly for f = 1,2, 3 , . . . , F by:
S(0) = [Cf.i„,j(0), Cf,in,l(0), Cf,in.2(0), ..., Cf,in,z(0),
Cfj(O), Cfa(O), Cfb(O) ,..., Cfz(O), Cf^(O),
hin,j(0), hi„,a(0), hi„,b(0) , ..., hin.z(0),
hj(0), ha(0), hb(0) ,..., hz(0), h4(0)] (5-2a)
503
The state vector at step n, or time t is, similarly, given by:
Cl,inj(n),
Cl,in,a(n),
Ci,in,b(n),
Ci,in,z(n),
Cij(n),
Cia(n),
Cib(n),
Ciz(n),
Ci^(n),
hin,j(n),
hj(n),
C2,in,j(n),
C2,in,a(n),
C2,in,b(n),
C2,in,z(n),
C2j(n),
C2a(n),
C2b(n),
C2z(n),
C2^(n),
hin,a(n),
ha(n),
C3,in,j(n) , .
C3,in,a(n) , ..
C3,in,b(^) . ••
C3,in,z(n) , ..
C3j(n) , .
C3a(n) , .
C3b(n) , .
C3z(n) ,
C3^(n) ,
hin,b(n) ,
hb(n)
CF,inj(n),
CF,in,a(n),
CF,in,b(n),
CF,in,z(n),
CFj(n),
CFa(n),
CFb(n),
CFz(n),
CF^(n),
hin,z(n),
.., hz(n), h^(n)] (5-3)
A compact representation for f = 1, 2, 3,..., F is given by:
S(n) = [Cf4n,j(n), Cf,in,i(n), Cf.in,2(n),..., Cf,in,z(n),
Cfj(n), Cfa(n), Cfb(n) ,..., Cfz(n), Cf^(n),
hin,j(n), hin,a(n), hin,b(n) , ..., hin,z(n),
hj(n), ha(n), hb(n) ,..., hz(n), ^(n)] (5-3a)
The transition from step n to n+1 is carried out according to:
S(n+1) = S(n)P + L(n) (5-4)
where P is the one-step transition probability matrix given by Eq.(5-27). S(n) and
S(n+1) are state vectors at times t and t+At, or step n to n+1, given by Eq.(5-3).
L(n) is an expression corresponding to time t or step n which is spelled out in the
appropriate equations below. Specific elements of the state vector, corresponding
to Fig.5-1 are derived in the following and are given by Eqs.(5-9a), (5-13a), (5-
17a), (5-21a), (5-24a) and (5-26a).
504
Scheme of the chemical reactions. In the derivations below, chemical
reactions are involved. To remain consistent with the nomenclature in chapter 3.1
and the modified nomenclature in this chapter, we repeat part of the material as
follows. Consider a chemically reacting system containing species Ai, A2, A3, ....,
Ap. It is assumed that a certain species Af can react simultaneously in several
reactions, designated in the following by superscripts (m), m = 1, 2, ..., R where
R is the total number of reactions in which Af is involved. The following scheme
of irreversible reactions by which Af is converted to products is assumed, where
each set of reactions involving reversible reactions can always be written according
to the scheme below in order to apply the following derivations.
7st reaction:
2nd reaction:
mth reaction:
Rih reaction:
Jl)
J2)
(1)
... + a . A^+ ... -> products (2)
... + a A^ + ... -» products (m)
dL Aj.+ ... + a. Aj. + . . .
JR)
- ^ " \ -(R)
... + a A^ + ... -* products (5-5)
M): a is the stoichiometric coefficients of species f in the mth reaction. The rate of
conversion of species f in the mth reaction based on volume of the fluid in the
reactor V, i.e. r , is defined by:
(m) _ J [ f V dt
d N ^ / V (m) (m) (m) (m) I V m ; ^ a , ^ a
= - k p q i q i ...c;f . . .c^ (5-6)
where Nf are the number of moles of species f and Cf is its concentration in moles
of f/m . k "*, in consistent units, is the reaction rate constant with respect to the
conversion of species f in reaction m. In the case of a plus sign before k , this
means (moles of f formed)/(s m^) in reaction m. The discrete form of Eq.(5-9)
reads:
/ x / \ (Msi (ad (in) On)
r ; % ) = -k;"'^c;i (n)q2 (n)... C f (n)... C^ (n) (5-6a)
505
where the reaction rate and the concentrations Cf(n) refer to step n. In addition, the
conservation of the molar rates for all reacting species in reaction m in Eq.(5-5) is
given by:
(m)
Jm)
(m)
Jm)
- r , .(m)
(m) \
(5-7)
This makes it possible to compute the reaction rates of all species on the basis of
rf^ given by Eq.(5-6).
Derivation of probabilities from mass balances
Reactor j . A mass balance on species f undergoing various mass transfer
processes as well as chemical reactions due to changes in the operating conditions
in the central reactor j in Fig.5-1, gives:
d[V.C^]
dt ^ j f,inj Z-rf^kj fk
^Skfpo^j^S.i +
k
' ft J (5-8)
k = a, b, ..., Z where k t j , ^ f = 1, 2,..., F p = 1, 2, ..., P. The last
summation with respect to m, is on all reactions in Eq.(5-6) that species f is
involved with, i.e. m = 1,..., R.
Cf,in,j is the concentration of species f at the inlet stream Qj flowing from a
feed vessel into reactor j ; Cfk and Cg are, respectively, the concentrations of species
f in reactors k and j . C'fj is the concentration of f in the stream Q j leaving reactor
j -kfpjapjACfp j = ^fp jACfp jVj is the mass transfer rate of species f into (or from)
reactor j by some transfer mechanism designated by p (such as absorption,
dissolution, etc., or simultaneously by several processes). kfpj(m/s) is the mass
transfer coefficient for process p with respect to species f corresponding to
conditions in reactor j ; apj is the mass transfer area for process p corresponding to
conditions prevailing in reactor j . If the mass transfer area apj is not known, the
506
volumetric mass transfer coefficient |LI .(1/S) defined in Eq.(5-10) is used. AC^ •
is the driving force for the transfer process p with respect to species f at conditions
prevailing in reactor j . It should be noted that a positive sign before kfpj indicates
mass transfer into the reactor whereas a negative sign indicates mass transfer from
the reactor outside, t^ is the reaction rate of species f by reaction m per unit
volume of reactor (or fluid in reactor) j corresponding to the conditions in this
reactor. A plus sign before k " in Eq.(5-7) means (moles of f formed in reaction
m)/(s m ).
Integration of Eq.(5-8) between the times t and t+At, or step n to n+1,
assuming that Vj remains constant, yields:
Cg(n+1) = q . , .[ ij At] + C /n)]" 1 - C^V h + «:{Cq(n)/C^/n)} jiij At
+XCfk( )f«kj ijAt] + [X^^fp/Cfp/n)At +X4r^(n) •- p m
At (5-9)
k = a, b, ..., Z where k ; t j , ^ f= 1,2,..., F p= 1,2,..., P m = 1, ..., R
An alternative form of Eq.(5-9) in terms of transition probabilities reads
Cfj(n+1) = Cf i jPinj + Cfj(n)pjj +X^fk(n)Pkj + Lfj(n) (5-9a) k
where detailed expressions for the probabilities are summarized in Eq.(5-28a).
CQ(n+l) is the concentration at time t+At, or step n+1, of species f in reactor
j ; Cf^nj is the concentration of species f at the inlet stream Qj flowing from a feed
vessel into reactor j ; Cg(n) and Cfk(n) are, respectively, the concentrations at time t
or step n of species f in reactors j and k. C'fj(n) is the concentration of f in the (m)
Stream Q j leaving reactor j . AC^ .(n) and r . (n) are, respectively, at time t or step n, the driving force for the transfer process p and the reaction rate per unit volume
of species f by reaction m, corresponding to the conditions in reactor j . pinj is the
single step transition probability from the state of the feed reactor to the state of
reactor j ; Pjj is the probability of remaining in the state of reactor j and pkj is the
transition probability from the state of reactor k to the state of reactor j . The
complete expressions for the probabilities in Eq.(5-9a) are given, respectively, in
507
Eq.(5-9). Lfj(n) is the last term on the RHS of the equation corresponding to
species f (= 1, 2, 3,..., F) in reactor j and at time t. Other definitions are:
. fPdfw M = S i (5-10) i J
where Hfpj, the volumetric mass transfer coefficient (1/s), indicates that this
quantity corresponds to species f for process p in reactor j . In addition,
%=-i %=-i h=t <^-ii> ^ J ^ J ^ J
Reactors i. A mass balance on species f undergoing various mass transfer
processes as well as chemical reactions due to changes in the operating conditions
in reactor i in Fig.5-1, gives:
d[V. C ]
dt '' QiCf,in,i + QjiCfj + X ^ k i C J " ZQikC, H- Q x , + qi^c,+ Q;C, k J L k
p m
i, k = a, b, ..., Z where k ^ t i j , 4 f = l , 2 , ..., F p = l , 2 , ..., P m = l , ..., R.
Cf,in,i is the concentration of species f in the stream Qi flowing from the feed
vessel into reactor i, Cfj is the concentration of species f in reactor j , Cfk is the
concentration of species f in reactor k and Cfi is the concentration of species f in
reactor i. Cfi is the concentration of species f in stream Q'l leaving reactor i.
kfp^apiAC^ . = IX .ACf jV is the mass transfer rate of species f into (or from)
reactor i by some transfer mechanism designated by p (such as absorption,
dissolution, etc., or simultaneously by several processes). kfpj(m/s) is the mass
transfer coefficient with respect to species f for the process p corresponding to
conditions in reactor i; api is the mass transfer area for process p corresponding to
conditions in reactor i. If the mass transfer area apj is not known, the volumetric
mass transfer coefficient |ui .(1/s) defined in Eq.(5-14) is used. AC^ j is the
driving force with respect to species f for the transfer process p at conditions of
reactor i. It should be noted that a positive sign before kfp j indicates mass transfer
+ C ,(n)| 1~
508
into the reactor whereas a negative sign indicates mass transfer from the reactor
outside. T^ is the reaction rate of species f by reaction m per unit volume of
reactor i corresponding to conditions in this reactor. In the case of a plus sign
before k "" in Eq.(5-6), this means (moles of f formed in reaction m)/(s m^).
Integration of Eq.(5-12) between the times t and t+At, or step n to n+1,
assuming that Vj remains constant, yields:
C^/n+1) = Cf i„,i[ai^iAt] + Cfj(n)[ajiJiiAt]
Pi + ocy +X^ik"^ a;{c;.,(n)/q(n)} V A t l V k ^ -I
k L p m J
i, k = a, b , . . . , Z where k ; t i , j , ^ f = l , 2 , . . . , F p = l , 2, . . . , P m = l , . . . , R.
An alternative form of Eq.(5-13) in terms of transition probabilities reads
Cfi(n+1) = Cf i,,iPi„,i + Cg(n)pj. + C,^(n)p, +^C^in)p^, + ^ ( n ) (5-13a) k
where detailed expressions are summarized in Eq.(5-29a).
Pin,i is the single step transition probability from the state of the feed reactor
to the state of reactor i; pji is the transition probability from the state of reactor j to
the state of reactor i; py is the probability of remaining in the state of reactor i and
pki is the transition probability from the state of reactor k to the state of reactor i .
The complete expressions for the probabilities in Eq.(5-13a) are given,
respectively, in Eq.(5-13) where Lfi(n) is the last term on the RHS of Eq.(5-13)
corresponding to species f (= 1, 2, 3 , . . . , F) in reactors i (= a, b,. . . , Z) and at time
t or step n. Other definitions appear after Eq.(5-12) whereas the (n) or (n+1) in
Eq.(5-13a) stands for t and t+At, or step n to n+1. In addition, the following
definitions are appUcable:
_ fp,i pi „ - J l (t -— (5-14) i 1 rJ
509
Wp,i» the volumetric mass transfer coefficient (1/s), indicates that this quantity
corresponds to species f for process p (for example: absorption, p = 1; desorption,
p = 2; dissolution, p = 3; etc.) in reactors i or j . In addition,
^ J ^ J ^ J ^ J ^ J
Reactor ^. A mass balance on species f in reactor ^ in Fig.5-1 for k = a, b,
..., Z where k 9t j , ^ reads:
\-^ = %^n -"^^^ff^ - k +Xqk^k (5-16) k ^ k ^
It is assumed that the volume of the fluid in the reactor remains unchanged and that no chemical reactions take place in the reactor as well as other mass transfer processes. Cf is the concentration of species f in reactor ^, C'f is the concentration of species leaving reactor ^ (not necessarily equal to Cf ), Cfk is the concentration of species f in reactor k and Cg is the concentration of species f in reactor j . If the concentration C'f = 0, the species are completely accumulated in reactor ^ which is considered as "total collector" or "dead state" for the species. If C'f = Cf , the species are not accumulated in reactor ^. If 0 < C'f < Cf , the species are partially accumulated in reactor ^ which is considered as a "partial collector" of the species. Integration of Eq.(5-16) between the times t and t+At, or step n to n+1, yields:
C^(n+1) = Cg(n)[Pj ji At] ^X fk^^^^Pk^J^^^^^ k
+ Cf Cn)!" 1 ~ U.^ + ^p^Ac;^(n)/Cf^(n)}^^Atl (5-17)
An alternative form of Eq.(5-17) in terms of transition probabilities reads:
Cf^(n+1) = Cfj(n)pj +X* fk^ )Pk^ + Cf ( )P^ (5-17a)
510
Detailed expressions for the probabilities are summarized in Eq.(5-28a). Pjt is the
transition probability from the state of reactor j to the state of reactor ^ and pj t is
the transition probability from the state of reactor k to the state of reactor ^. The
complete expressions for the probabilities in Eq.(5-9a) are given, respectively, in
Eq.(5-9). In addition, the following definition are applicable:
Qk . O k . Qi « k = Q - « k = Q - ^rq: (5-18)
Finally, it should be noted that the determination of the parameters of the type
ttij is carried out by the following mass balances which extend Eqs.(4-12a) to (4-
12c). An overall mass balance on the flow system in Fig.5-1, i.e. on reactors j and
k (k = a, b,..., Z where k j) as well as ^, gives:
Qj+XQk = %+21%^+XOk yields 1 + Z«k = Pj +ZPk^+ Z«k k k k k k k
(5-18a)
ttk. ctk'. Pj^, Pk are given by Eq.(5-18). A mass balance on the flows of reactor j , fork = a, b,..., Z where k;tj and ^, yield
Qj-^XQkj = qj^+XQjk+Qi yields i + X«kj = Pj^+E«jk+«i k k k k
(5-18b)
where ttkj, ttjk, ocj' and Pj are given by Eqs.(4-6) and Eq.(5-18). If Qj is not
considered, one of the in flows Qi (i = a, b, ..., Z) or one of the internal flows,
should be taken as a reference flow in Eq.(5-18).
A mass balance on reactors i in Fig.5-1 (k = a, b,..., Z where k ;t i, j and ^),
reads:
511
Qi + Qji + S Q W = %+Qij + S Q i k + Q i yields k k
"i + "ji + S « k i = h + "ij + X«ik + «i (5-18c)
where Oji, aki, ccik and Pi are defined in Eq.(5-15) and ai, a'j in the following:
(5-18d)
Derivation of probabilities from energy balances
Reactor j . The energy balance [83, p.39] on reactor j referring to Fig.5-1,
ignoring kinetic and potential energies as well as shaft work, reads:
dU. f V W V • -1 "dt = inJ J ^2.Mkj - hj j Z h n ^ + hjnj + \^\f\^ - T )
-XX4fVj [f (5-19) f m
k = a, b, ..., Z where k ^ t j , ^ f= 1,2,..., F m = 1,..., R.
The following definitions are applicable: Uj is the internal energy of the content of reactor j (kcal/kg or kg-mole). h-, h^-, h-t and hjj are, respectively, the
total flow rates (moles or mass) into reactor j from the feed vessel, from reactor k to
reactor j , from j to ^ and from j to k. hinj and hj are, respectively, the specific
enthalpies (per unit mole or mass of mixture) of the inlet stream flowing from a
feed vessel into reactor j and the stream leaving reactor j . n'j is the stream leaving
reactor j outside with enthalpy hj. khj(kcal/(s m^ K)) is the heat transfer coefficient
corresponding to the conditions in reactor j and ahj is the heat transfer area to
reactor j . If the heat transfer area ahj is not known, it is accustomed to use the
volumetric heat transfer coefficient |i (1/s) defined in Eq.(5-22). TQJ is the
temperature from which heat is transferred into reactor j and Tj is the temperature in
reactor j . r . is the reaction rate of species f by reaction m per unit volume of the
512
reactor (or fluid in the reactor) j corresponding to the conditions in this reactor. As
Eq.(5-19) stands, the minus sign before the double summation indicates that
species f is consumed by the chemical reaction; thus, a plus sign will appear in
Eq.(5-6) before kf. AH J^ is the heat of reaction m at conditions in reactor j . A
minus sign before the heat of reaction indicates generation of heat.
An altemative expression can be obtained by assuming that U = H - PV = H,
that a mean value is taken for the densities in each reactor, and making the
following transformations:
Hj(n) = VjPjhj(n) ATj(n) = \ . - T/n)
j = QjPind kj = QkjPk ^jk = QjkPj ^j^ = Qj Pj ^r^iPi (^"2°)
Integration of Eq.(5-19) between the times t and t+At, or step n to n+1, while
considering the above definitions and making some rearrangements, yields that:
hj(n+l)= hi„j[Pi„jprVjAt]
+ hj(n)|
+Xh>)KjPkPj"VjAt] k
f m
(5-21)
k = a, b, ..., Z where k ^ j , ^ f = 1, 2,..., F m= 1, ..., R.
An altemative form of Eq.(5-21) in terms of transition probabilities reads:
hj(n+l) = hj jPi j i + hj(n)pjj j +X''k(n)Pkj,h + LQ,h( ) (5-21a)
where detailed expressions are sunmiarized in Eq.(5-28b). hj(n+l) is the enthalpy
per unit mass (or mole) of the fluid in reactors j at time t+At, or step n to n+1; hj(n)
and hk(n) are the enthalpies per unit mass (or mole) of the fluid in reactors j and k at
513
time t, respectively, hj(n) is the enthalpy of the stream leaving reactor j outside.
p|j^: 1 is the transition probability with respect to enthalpy (or temperature) from the
state of the inlet reactor (to reactor j) to the state of reactor j ; p- ^ is the probability
to remain in the state of reactor j with respect to enthalpy (or temperature); pj : j^ is
the transition probability with respect to enthalpy (or temperature) from the state of
reactor k to the state of reactor j . The complete expressions for the probabilities in
Eq.(5-21a) are given, respectively, in Eq.(5-21) where L : j (n) is the last term on
the RHS of Eq.(5-21) corresponding to species f (= 1, 2, 3,. . . , F) in reactor j and
at time t. pinj is the density of the fluid entering reactor j from a feed vessel, pj
and pk are, respectively, the densities of the fluid in reactors j and k and pj is the
density of the stream leaving reactor j ; r|." (n) is the reaction rate at time t, or step n,
of species f by reaction m per unit volume of reactor j corresponding to conditions in this reactor, whereas Ix jCl/s) is defined by:
J pon
Cp j is the specific heat of the fluid mixture in reactor j .
Reactors i. The energy balance on reactor i, ignoring kinetic and potential
energies as well as shaft work, reads [83, p.391:
"dT " in,i i + hjAji + X M k i ~ Vi^ +Z^Vik + ^Ai + hjii; ^ k ^ ^ k ^
^ khiahi(To,i - T.) - X X 4 r ^ ^ i ^ " r f (5-23) f m
i, k = a, b,..., Z where k9ii,j, ^ f=l ,2 , . . . , F m=l, . . . , R.
The definitions of the various quantities appearing in the above equation and
those below are similar to those which follow Eq.(5-19) and (5-21a) whereas j is
replaced by i. Introducing similar transformations to those in Eq.(5-20), as well as
nj = Qjp'j and hj = QjPin i» where Q'i is the volumetric flow rate leaving reactor i
and Q is the flow rate of the fluid from a feed vessel into reactor i; pi is the density
of the fluid leaving reactor i andpjn,i is the density of the fluid from the feed vessel.
514
Thus, the following equation is obtained by integrating Eq.(5-23) between t and
t+At, or step n to n+1: hjCn-f 1) = hi„. KPi^iprViAt] 4- hj(n)[ajiPjp-ViAt]
+ h-(xi)\ 1 - J Pi^+ tty + 2 l ^ i k + ^iPiPr^t K(^)/ hi(n)]l |i jAtl
+ ^hi^(n)[ai^iPi^prViAt]
|i,iCp .ATi(n) - p r ^ X Z ^ 4 f ^^n)AH(f)} f m
At (5-24)
i, k = a, b, ..., Z where k ^ ^ i j , ^ f = l , 2 , . . . , F m = l , ..., R.
An alternative form of Eq.(5-24) in terms of transition probabilities reads:
hi(n+l) = hi^ .pi„. j + h/n)pji 1 + hi(n)pii j +Shk(n)Pki,h + k.h^^) (5-24a) k
where detailed expressions are summarized in Eq.(5-29b). p. ^ j is the transition
probability with respect to enthalpy (or temperature) from the state of the inlet reactor (to reactor i) to the state of reactor i; p. ^ is the transition probability with
respect to enthalpy (or temperature) from state of reactor j to state of reactor i; pj. j
is the probability to remain in the state of reactor i with respect to enthalpy (or
temperature); pj j is the transition probability with respect to enthalpy (or
temperature) from the state of reactor k to the state of reactor i. [i^^^ is given by
Eq.(5-22) while j is replaced by i. The complete expressions for the probabilities in
Eq.(5-24a) are given, respectively, in Eq.(5-24) where Lfi,h(n) is the last term on
the RHS of Eq.(5-24) corresponding to species f (= 1, 2, 3,. . . , F) in reactor i (=
a, b, . . . , Z) and at time t.
Reactor ^. The energy balance on reactor ^, ignoring kinetic and potential
energies as well as shaft work, chemical reactions and mass transfer processes,
reads [83, p.39]:
515
ST = ( j' i +XMk%] - h[^i^ + X"k^ ] (5-25)
k = a, b, ..., Z where k^j,^.
U^ is the internal energy (kcal) of the content of reactor ^. HJ and hj ^ are,
respectively, the total flow rates (moles or mass) into reactor ^ from reactors j and
k. hj and hk are, respectively, the specific enthalpies (kcal/kg or kg-mole) of
reactors j and k, respectively. Applying part of the quantities in Eq.(5-20), substituting U^ = H^ =
(Vtpt)ht, and integrating Eq.(5-25) between t and t+At or step n to n+1, yields:
h^(n+l) = hj(n)[pj PjP^V^At] + X^k(n)[Pk^PkP^ V^At] k
+ h^in)\ 1 - f PjP^^Pj + SpkP^'Pk^^^^At] (5-26)
k = a, b, ..., Z where k 9= j , ^.
ht(n+l) is the specific enthalpy at time t+At of the content in reactor ^; ht(n),
hj (n) and hj(n) are, respectively at time t, the specific enthalpies of the content in
reactors ^, k and j . pt is the density of the fluid in reactor ^, |it and P^E ^^
defined in Eq.(5-18). An alternative form of Eq.(5-26) in terms of transition
probabilities reads
h^(nH-l) = hj(n)pj^ h ^X^^k^^^Pk^h "*" ^ ^ P ^ h (5-26a) k
k = a, b,..., Z where k 9t j , ^
where detailed expressions are summarized in Eq.(5-30b). Pjt h is the transition probability with respect to enthalpy (or temperature)
from the state of reactor j to the state of reactor ^; pj t is the transition probability
with respect to enthalpy (or temperature) from the state of reactor k to the state of reactor ^ and pt t ^ is the probability with respect to the enthalpy (or temperature)
516
to remain in the state of reactor . The complete expressions for the probabilities in
Eq.(5-26a) are given, respectively, in Eq.(5-26).
The single-step probability matrix and summary of the probabilities
The single-step transition probabilities appearing in Eqs.(5-9a), (5-13a), (5-17a), (5-21a), (5-24a), (5-26a), which are defined in Eqs.(5-9), (5-13), (5-17), (5-21), (5-24), (5-26), can be arranged in the following transition matrix P given by Eq.(5-27). The general representation of the above equations is given by Eq.(5-4). For the convenience of the reader the probabilities and the appropriate equations are also summarized below in Eqs.(5-28a,b) to (5-30a,b).
It should be emphasized that the matrix representation becomes possible due to the Euler integration of the differential equations, yielding appropriate difference equations. Thus, flow systems incorporating heat and mass transfer processes as well as chemical reactions can easily be treated by Markov chains where the matrix P becomes "automatic" to construct, once gaining enough experience. In addition, flow systems are presented in unified description via state vector and a one-step transition probability matrix.
517
cl f,in,j|
f.in.a
c1 f.in.b
"c1 f,in,z|
cl fj
fa
c fb
fz
h in.j
h in.a
h in.b
in.z
h j
IT a h b
z
in
c* f.in.j 1
0
0
0
0
0
0
1 ^ 0
0
0
0
0
p p
0
p p
c f.in.a 0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
c f.in,b 0 •
0 •
1 •
0 •
0 •
0 •
0 •
0 •
0 •
0 •
0 •
0 •
0 •
0 •
0 •
0 •
0 •
0 •
c f.in.z
• 0
' 0
• 0
• 1
• 0
• 0
• 0
• 0
• 0
• 0
• 0
• 0
• 0
• 0
• 0
• 0
• 0
• 0
c p
in.j
0
0
0
p jj
p aj
p bj
p zj
0
0
0
0
0
0
0
0
0
0
c fa 0
p in.a
0
0
p ja
p aa
p ba
p za
0
0
0
0
0
0
0
0
0
0
c fb 0 •
0 •
p in.b
0 •
p • jb
p ab
p bb
P zb
0 •
0 •
0 •
0 •
0 •
0 •
0 •
0 •
0 •
0 •
c fz
• 0
• 0
• 0
.. p in.z
• p jz
• p az
• P bz
• P zz
• 0
• 0
• 0
• 0
• 0
• 0
• 0
• 0
• 0
• 0
c 0
0
0
0
',
^^
^b^
'^ p
0
0
0
0
0
0
0
0
0
h in.j
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
h in.a
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
h •• in.b
0 •
0 ••
0 ••
0 •
0 •
0 •
0 •
0 •
0 •
0 •
0 •
1 •
0 •
0 •
0 •
0 •
0 •
0 •
. h in.z
• 0
• 0
• 0
• 0
• 0
• 0
• 0
• 0
• 0
• 0
• 0
• 0
• 1
• 0
0
• 0
• 0
• 0
h* j
0
0
0
0
0
0
0
0
0
p in.j.^
0
0
0
p jj.h
p aj.h
p bj.h
p zj.h
0
h a
0
0
0
0
0
0
0
0
0
0
p in.a.t
0
0
p ja.h
p aa.h
p ba.h
p za.h
0
h .. b
0 •
0 ••
0 ••
0 ••
0 ••
0 ••
0 ••
0 ••
0 •
0 •
0 •• 1
p in.b.h
0 ••
p jb.h
p abji
p bb.h
p zbji
0 •
. h h
• 0 0 1
0 0 1 0 0 1
• 0 0 1
• 0 0 j
• 0 0 1 0 0 1
0 0 1
• 0 0 1 • 0 0 1
• 0 0 j
• 0 0
P 0 1 in,z.h 1
p p 1 jz.h j ,h 1
• p p 1 az.h a Ji 1
• p p 1 bz.h b^h 1
• p p zz.h z ,h 1
• 0 p«
C — Cf inj f,in,j
h* = hi
(5-27)
The probabilities in the above matrix are spelled out in the following. The
various quantities appearing in the equations are defined as follows: Ojk and akj in
Eq.(5-ll); aik, aki, aij and aji in Eq.(5-15); ai in Eq.(5-14). p.. in Eq.(5-15),
Pl ^ in Eq.(5-18) and pj in Eq.(5-ll). ij in Eq.(5-14), |ij in Eq.(5-10) and l in
Eq.(5-18). iilj j in Eq.(5-22) and, similarly, ij j where i replaces j . \if - in Eq.(5-
10) and |Lif • in Eq.(5-14). AC^ • is the driving force with respect to species f for
the transfer process p at conditions of reactor i, ATj(n) = To,i - T-(n), and similarly
518
for j which replaces i. p-j^: and p-^ j are the densities of the inlet streams into
reactors j and i from the feed vessel in Fig.5-1. p-, p: and pt are the densities of
the fluid in reactors i, j and ^, respectively.
Finally, attention should be paid to the remarks corresponding to the
following equations: a) In these equations, i, k = a, b, ..., Z where k ^ i, j , ^ f
= 1, 2, ..., F p = 1, 2,..., P m = 1, 2,..., R. b) The determination of the above
quantities, i.e. aik, ttki, ocij, ocji, oci, Pjt, Pj t and Pjt, is based on Eqs.(4-12a) to
(4-12c) following Eq.(5-18) above.
Reactor j :
For mass transfer
Cfj(n-M) = q .„ .pi, j + C,j(n)pj- +XCfk(n)Pkj + L,/n) (5-28a) k
where
Pinj = i^' Pjj = 1 - f Pj^ + S « j k + ai{Cg(n)/Cg(n)} V At ^ k ^
Pkj = afcj jAt Pjk = aj^^^At Pj = aj^^^At
Lfj(n) =
=
%/") = '
S^^fpJACfp.(n) '- p
X^^fp.J^Cfp/n) ^ P
£4f(n)
+ X4f (n)' m
At +rg(n)At
At
519
For heat transfer
hj(n+l) = hi„ jPi„ j h + hj(n)pjj J, +X^k(n)Pkj,h + \h^^^ k
where
Pind.h = PinjPj"Vj At Pkj.h = ttyPk pj-Vj At
Pjj.h = 1 - f Pj^+ S « j k + a:{h:(n)(n)h/n)}yj At
(5-28b)
Lfi.hW = H,jCp/Tj(n) - p r i ^ X { 4 f (n)AH(f} f m
At
= {^,jCp.AT/n)}At-pri{rg(n)AH,j}At
5(n)AH^ = 5^X^r(f(n)AH(f} f m
If p. . = p., it follows from above equations that p j j = Pin j h ^ ^^^^ ^^^^ Pkj "
Pkj,h'' ^ ^ J ~ ^' PJJ»h = Pjj. As the expression for Lfj,h(n) stands, the minus sign
before the double summation indicates that species f is consumed by the chemical
reaction; thus, a plus sign will appear in Eq.(5-6) before kf. AH^^^ is the heat of
reaction m at conditions in reactor j . A minus sign before the heat of reaction indicates generation of heat.
520
Reactors i = a, b, ..., Z:
For mass transfer
Cf,(n+1) = q i „ iPi„ i + Cq(n)pji + C,,(n)p, +^C^in\i + Lf,(n) (5-29a)
where
Pin,i = "ih^t Pii = 1 - [ pi + tty + X « i k + ai{Cfi(n)/Cfi(n)}yiAt
Pji = «jiMt Pki = aki^iAt Pi = aikRkAt Pi = ai l At
L,(n) = S^^fp.i^Cfp/n) + Xr^^(n) "- p
At
• P
At + rj.,(n)At
For heat transfer
hj(n+l) = hi„ iPi„ i J, + hj(n)pji, + hi(n)pii^ +Xhk(n)Pki,h + ^.h^^) (5-29b) k
where
Pin.i,h = aiPin,iPr^^i i^t p-i = a^iPjprV jAt pj^i, = a^iPkPr'jiiAt
Pii.h = 1 - fPi + «ij +X«ik+ a;{h;(n)/hj(n))| V A I
Lfi.h(n) = [jthiCp,iATj(n) - prl^;£{4?')(n)AH(f} L f m
= { lhiCpiATi(n)}At - {prirf.,(n)AH j A t
rfl(n)AHri = XZ^4i"^^n)AH(f >}
At
f m
521
If p. . = p., it follows from above equations that p^^ i = Pin i h ^^ ^^^^ ^ ^^^ Pki ~
Pki,h'if«'i = 0,pii,h = Pii.
Reactor ^:
For mass transfer
Cf^(n+1) = Cfj(n)pj +X^fk^^^Pk^ •*• Cf^(^)P^^ (5-30a) k
where
Pj^=p.^^^At Pk = Pk^^^At
p^^ = 1 - ( pj + XPk^]^C;^(n)/Cf^(n)}H^At
For heat transfer
h^(n+l) = hj(n)pj^ j^ +X^k(«)Pk^.h + h (n)P^ ,h (5~30b) k
where
Pj^h = Pj^PjP^V^At p^^ 1, = p^^p^p^i^i^At
P th = 1 - f PjP 'Pj + XpkP 'Pk^ W If p. = pt, it follows from above equations that p:. j^ = Pjt as well as that pj^ j^ =
Pj t. If, in addition, C'f = Cf , then p. . jj = ptt•
5.2 APPLICATION OF THE MODELING AND GENERAL GUIDELINES
The above modeling is applied to numerous flow configurations which have
appeared in Chemical Engineering textbooks. Additional ones of particular interest
have also been included. Generally, any flow configurations consists of a series of
perfectly-mixed and plug-flow reactors, as well as recycle streams, by pass and
522
cross flow etc., or part of the above. The guidelines appearing in Chapter 4.2 are also applicable here.
In treating a certain configuration, the first step is to reduce the general configuration in Fig.5-1 to the case under consideration. The second step is to define the state space generally given by Eq.(5-1); thus, the transition probability matrix P given by QE.(5-27). The probabilities appearing in the matrix are given in Eqs.(5-28a) to (5-30b) and the coefficients appearing in the equations are determined by Eqs.(5-18a) to (5-18c). A further step is to specify the initial state vector S(0) given by Eqs.(5-2). The transition from step n to n+1, namely, the determination of S(n+1) given by Eqs.(5-3) is carried out according to Eq.(5-4). Detailed demonstrations of the above guidelines will be made in typical cases presented in the following.
5.2-1 REACTING SYSTEMS
5.2-1(1) The flow system is shown below. It comprises of two reactors where only in
the first one chemical reaction takes place. If the second reactor is assumed as a "total collector" ("dead state") of the reactants and products, p^^ = p22 = 1. If this reactor is not a "dead state", it follows from Eq.(5-30a), assuming C'f = Cf , that:
p ri-fpj - XPk ^ ^ ^ (5.2-l(la))
The carrying fluid in which the reacting species are dissolved, enters the first reactor and leaves the second one at rate Qi.
Fig.5.2-l(l). The flow system
The configuration in Fig.5-1 is reduced to that in Fig.5.2-l(l) by choosing reactors j and ^ designated as j = 1 and ^ = 2, respectively. Considering
523
Eq.(5-18a) for a'k = 0, yields that a i = Qi/Qi = P12 = qi2/Ql = 1 while taking Qi
as reference flow.
From Eq.(5-la) the state space reads:
55 = [Cf,in,l,Cfi,Cf2] f = l , . . . , 4 (5.2-l(lb))
noting that no heat and mass transfer take place in the case under consideration, i.e.
I fpo = Mhj = 0- The latter are given by Eqs.(5-10) and (5-22). From Eq.(5-3a) the
state vector reads:
S(n) = [Cf,in,l(n), Cfi(n), Cf2(n)] (5.2-l(lc))
The probability matrix given by Eq.(5-27) is reduced to the following one:
Cf.in.l
P = Cfi
Cf2
Cf,in,l
1
0
0
Cfi
Pin,l
Pll
0
Cf2
0
P12
P22 5.2.1(ld))
From Eqs.(5-28a), for j = 1, and (5-30a) for ^ = 2, noting that P12 = 1, it
follows:
Pin,l = m At Pll = 1 - |Lli At P12 = |Ll2At P22= 1 - ^2 At (5.2-l(le))
where p22 is obtained from Eq.(5.2-l(la)). Other equations are:
For reactor 1
Cfi(n+1) = Cf,in,lPin,l + Cfi(n)pii + Lfi(n) f = 1,..., 4 (5.2-l(lf))
where
Lfi(^) =X4?^(n)^t = r ]£r("^)(n) At = r/n)At (5.2-l(lg))
Subscript f 1 designates species f in reactor 1. Since chemical reaction takes place
only in reactor 1, subscript f replaces fl.
For reactor 2
Cf2(n+1) = Cfi(n)pi2 + Cf2(n)p22 (5.2.1(lh))
524
The following reactions, similar to case 3.13-4, known as the Brusselator
model [60], are assumed:
i = 1: Ai -^ A3 i = 2: 2A3 + A2 -^ 3A3 i = 3: A3 -^ A2
i = 4: A3 —> A4
thus, m = 1, ..., 4 in the equation for Lfi(n) above. Considering Eq.(5-7), yields
the following relationships:
For i= l : - r l i ) = ra) for i = 2: - ^ = -r^2)= ^
Fori = 3:-r^3) = r 3) Fori = 4: -r ^^ = r ^
Thus,
1 M ~ ^ 1 1 ^ 1 " ^1^1
where kn indicates the rate constant for reaction i = 1 in reactor 1. However, since
chemical reaction takes place only in reactor 1, ki should replace kn and similarly
with the other k's, i.e. one of the subscripts will be omitted for the sake of
simplicity. In addition:
^2 ~ ^2 •*" 2 ~ ~ '^2^2^3 •*" ^ ^ 3
Considering Eqs.(5.2-l(le)) to (5.2-l(lg)) yields for reactor 1 the following
equations where f = 1,..., 4 :
Cii(n+1) = Ci,i„,iPi„j + Cii(n)[pii - kiAt]
C2i(n+1) = C2.in,iPin.i + C2i(n)[pii - kjCf ,(n)At ] + k3C3i(n)At
C3i(n+1) = C3,i„,iPi„ i + C3i(n)[pii - (k3 + k4)At] + kiCii(n)At
+ k2C|i(n)C2i(n)At
525
C4i(n+1) = C4,in,lPin,l + QlWpi l + k4C3i(n)At
where pin,i, pn and pi2 are given by Eq.(5.2-l(ld)) above. In addition, for
reactor 2:
Ci2(n+1) = Cii(n)pi2 + Ci2(n)p22 C22(n+1) = C2i(n)pi2 + C22(n)p22
C32(n+1) = C3i(n)pi2 + C32(n)p22 C42(n+1) = C4i(n)pi2 + C42(n)p22
In the numerical solution it was assumed that the reactors are of an identical
volume, thus, |ii = QiA^i = |i2 = QiA^2 = V^- The transient response of Cn (case
a), C21 and C31, i.e. the concentrations of species 2 in reactor 1 in Fig.5.2-l(l), is
depicted in Fig.5.2-l(la) where the effect of i is demonstrated. The initial state
vector C(0) = [Cf,in,i(0), Cfi(O), Cf2(0)] reads: [100, 1, 0] for f = 1 as well as [0,
0, 0] for the species f = 2, 3, 4. Other parameters are: ki = 10, k2 = 0.1, k3 = 2,
k4 = 1 and At = 0.05. As seen, increasing \i, i.e. the flow rate into reactor 1,
brings the system to oscillate at |i = 0.01 which diminishes at |i = 0.06.
526
u
1
0.8
0.6
0.4
0.2
0
n = o . I I I
T > ^ = ' !|'^
fy \ . - • • " " IS \-'' n '-''^
(a)
-
H
0.5
\i = 0.02
1.5
25
20
15
10
5
2 %
[i = 0.01
-
h
1 _
2 /
A--.-
1
.•••'
1/ '' 3
1 i 1/
~~(b)|
/ H
—
50 100
\x = 0.03
50 100 t
150
150
7
6
5
^4
^ 3
2
1
0
/^;
1 / '"'
Vl
\i = 0.06 1 1
3
V f=2
1 1
(e)| — H
-|
-
-
1 = 0.1
5 10 t
15 %
Fig.5.2-l(la). Cii, C21, C31 versus t demonstrating the effect of |X
5>2-l(2) This is an extension of case 3.13-2 for an open system. It comprises here of
two reactors and only in the first one chemical reaction takes place. The flow
scheme is shown in Fig.5.2-l(l). The configuration in Fig.5-1 is reduced to the
527
present case by choosing reactors j and ^ designated as j = 1 and ^ = 2,
respectively. Considering Eq.(5-18a) for a'k = 0, yields that ai = Qi/Qi = P12 =
qi2/Ql = 1 while taking Qi as reference flow. Eqs.(5.2-l(lb)) to (5.2-l(lh)) of
case 5.2-1(1) are applicable also in the present case for f = 1,..., 6.
The following reactions of case 3.13-2, known as the Belousov-Zhabotinski
reaction [59], were also applied here: k, kj
i= 1: Ai+A2-> A3 + A4 i = 2: A3 + A2-^2A4 kj k4
i = 3: Ai + A3 ^ 2A3 + A5 i = 4: 2A3 Ai + Ae
i = 5:A5^gA2 (5.2-1(2))
thus, m = 1,..., 5 in Eq.(5.2-l(lf)) for Lfi(n). Considering Eq.(5-7), yields the
following relationships:
1.(2)
Fori=l:-r( i> = -r(') = r(i) = r(i) For i = 2: - r ) = - r ) = - 1 -fO) 1.(4)
For i = 3: - r(3> = - r ) = ^ = r ) For i = 4: - - | - = r ) = r ) 1.(5)
For i = 5: - r 5) = —
Thus, 1-1 -— x-i 11-1 • *• 1 ~~ """ ^1 ^ 1 ^ 0 """ ^ "^^1 ^ ' d. '
T^ = r^ +13 +13 +r3 = kjCjC2 - k2C2C3 - k^C^C^ + 2k3C|C3 - 2k4C3
^4~^4 "*' 4 = k|C|C2 + 2k2C2C3 r5 = r3 +T^ = k3CjC3-k^C^
Considering Eqs.(5.2-l(le)) to (5.2-l(lg)) yields for f = 1, ..., 6 the
following equations for reactor 1:
528
Cii(n+l) = Ci,in,lpin,l +Cii(n)[pii - {kiC2i(n)+ k3C3](n)}At]
+ k4C3i2(n)At
C2i(n+1) = C2 ,in,lpin,l + C2i(n)[pii - {kiCii(n) + k2C3i(n)}At]
+ fk5C5i(n)At
C3i(n+1) = C3,in,lPin,l + C3i(n)[pii - {k2C2i(n) + 2k4C3i(n)}At]
+ kiCii(n)C2i(n)At + k3Cii(n)C3i(n)At
C4i(n+1) = C4,in,lPin,l + C4i(n)pii + kiCii(n)C2i(n)At
+ 2k2C2i(n)C3i(n)At
C5i(n+1) = C5,in,lPin,l + k3Cii(n)C3i(n)}At +C5i(n)[pii _ k5Cii(n)At]
C6i(n+1) = C6,in,lPin,l + C6i(n)pii + k4C3i2(n)At
where pin,i, pii and pi2 are given by by Eq.(5.2-l(ld)). In addition, for reactor 2:
Ci2(n+1) = Cii(n)pi2 + Ci2(n)p22 C22(n+1) = C2i(n)pi2 + C22(n)p22 C32(n+1) = C3i(n)pi2 + C32(n)p22 C42(n+1) = C4i(n)pi2 + C42(n)p22
C52(n+1) = C5i(n)pi2 + C52(n)p22 C62(n+1) = C6i(n)pi2 + C62(n)p22
In the numerical solution it was assumed that the reactors are of an identical
volume, i.e. ^ii = Qi/Vi = (X2 = Q1/V2 = |Li. In addition, the second reactor has
been assumed as "total collector" of the reactants and products, i.e. p22 = 1. If this
reactor is not a total collector, i.e. a "dead state", Eq.(5.2-l(a)) is applicable and
P22 = 1 - M 2At. The transient response of C3i,C5i in reactor 1, €32,052 in reactor
2, corresponding to Fig.5.2-l(l), is depicted in Fig.5.2-1(2) where the effect of |X
and g in Eq.(5.2-1(2)), is demonstrated. The initial state vector C(0) = [Cf,in,l(0),
Cfi(O), Cf2(0)] reads: [0.015, 0, 0] for f = 1, [0.004, 0, 0] for f = 2 as well as [0,
0, 0] for f = 3,..., 6. Other parameters were: ki = 0.05, k2 = 100, k3 = 10^, k4 =
10, ks = 5, g = 1, 5, 10 and At = 0.0005. As seen, increasing \i, i.e. the flow rate
into reactor 1 brings the system to oscillate at |LI = 0.5 and g = 1 as seen in cases c
and d. The oscillations diminish at large values of t and also by increasing g as
demonstrated in cases e and f .
529
1.2 lO"
1 lo' U
8 1 0 >
^ 6 lO-'L
^"4 lO-'L
2 lO'^l -
0 10
\i « 0.005, g = 1
u
u
» » (^
f 1 = s i / ^ ' l
/ """'"•? 1 J 12 = 32, 52
1 - 1 1
L h
u
^ i - 0,05, g = l
* ' ^J 3 1 / " ^
/ ' ' ^ 51 J / £=^2,i2j _ ._...j J. _._i
0.5 1.5 0 0.5 1.5
6 10'
5 lO'^k
4 10
^ 3 10"
u"2 lO' U
1 lO' L
0 lOV — — -1 10"
^ - 0 . 5 , g = l
h-
H
h
-
1
fl=3J
1
't-
' (cj
f2 = 32,l
J
> i - l , g = l
0.5 1 t
1.5 0
5 10"
4 10"
3 10' G
^ 2 10-
^"iio-"
0 10°
-1 10'
^ - l , g = 5
JA 1= 1 ' / r >•
/ . " 3 2 / 51 . ^
^ - - - • " 52 ^ - •
1 1
^''H j ^
, - . j - ' # - - -
H
H
Ji- 1, g=10
0.5 1 t
1.5 0
Fig.5.2-1(2). C31, C51, C32andC52 versus t demonstrating the
effect of fiand g
5.2-1(3) This is an extension of case 3.13-5 for an open system comprising of two
reactors where only in the first one chemical reactions take place. The scheme is
530
shown in Fig.5.2-l(l). The configuration in Fig.5-1 is reduced to the present one by choosing reactors j and ^ designated as j = 1 and ^ = 2, respectively. Considering Eq.(5-18a) for a'k = 0, yields that ai = Qi/Qi = P12 = qi2/Ql = 1 while taking Qi as reference flow. Eqs.(5.2-l(la)) to (5.2-l(lg)) in case 5.2-1(1) are applicable also in the present case for f = 1,..., 7.
The following model [57] appearing in case 3.13-5, which simulates the Belousov-Zhabotinski reaction, was appUed also for the open system:
kj k2
i = 1: Ai + A2 " A3 + A4 i = 2: A3 + A2 ~ 2A4
i = 3: Ai + A3 " 2A5 i = 4: A6 + A5 ^ A3 + A7
k_3 k_4
ks h i = 5: 2A3 Ai + A4 i = 6: A7 4 gA2 + Ae
thus, m = 1,..., 6 in Eq.(5.2-l(lf)) for Lfi(n). Considering Eq.(5-7), yields the following relationships:
1.(2)
Fori = l:-r(i) = -ry) = r») = rV) Fori = 2: - f ) = -42) = - i -
r; .(3) 5 Fori = 3:-r|3) = -r(3> = - | - For i = 4: - r ) = - r 4) = r ) = r )
r(5) 1.(6)
Fori = 5 : - 4 - =r^^^ = r'i^ For i = 6: - r ) = ^ = r 6)
Thus,
rj = rj +rj +rj =--kjCjC2+k_|C3C4-k3CjC3 + k_3C5 + k5C3
r ^ ^ ^ 9 ' ^ 9 2 ^ — 1 1 9 —1 ' ^^4 9 9 ' —2 4 S 6 7
r ^ ro 4"ro iro ' " " "" i i 9 —1 " 4. 9 9 ' ' —9 4
" % ^ i ^ 3 ••" ^- -3 5 4 5 6 -4 3^7 5 3 - i ^ v jv 4
531
1 = 1^ +1*4 "^^4 ~'^iCjC2 "" K.jC^C^ + 2k2C2C2~ 2K_2C4 + k^C^
" ^ - 5 ^ 1 ^ 4
^5~^5 "*" 5 ~ 2k3CjC 3 - 2k_3C5 - k^C^C^ - k_^C3C ^
r^ = r^ + r ^ = - ^ € 5 0 ^ + k_4C3C j + k^C 7
r = /"*) + r( ^ = k r r - k T C - k C 7 7 7 4 5 6 *'-4^3^ 7 *6 7
Considering Eqs.(5.2-l(le)) to (5.2-l(lg)) yields for f = 1, ..., 7 the following
equations for reactor 1 where the different rf(n) are given above:
Cfi(n+1) = Cf,in,lPin,l + Cfi(n)pii + rf(n)At
Pin,l> Pll above and pi2 below are given by by Eq.(5.2-l(ld)). In addition, for
reactor 2:
Cf2(n+1) = Cfi(n)pi2 + Cf2(n)p22
where P22 = 1 - M'2At is obtained from Eq.(5-30a) noting that C'f2 = Cf2.
In the numerical solution it was assumed that the reactors are of an identical
volume, i.e. |Lii = Qi/Vi = |i2 = Q1/V2 = |ii. The initial state vector C(0) =
[Cf,in,l(0), Cfi(O), Cf2(0)] reads: [1, 0, 0] for f = 1, [0.2, 0, 0] for f = 2, 6 as well
as [0, 0, 0] for f = 3, 4, 5, 7. Additional parameters were: ki = 1, k-i = 10^, k2 =
40, k.2 = 0, k3 = 100, k.3 = 20, k4 = 1000, k.4 = 100, ks = 1, k.5 = 0, k6 = 1 and
At = 0.015.
The transient response of C n , C21, C31 and C71, i.e. the concentrations in
reactor 1 where reactions take place, is depicted in Fig.5.2-1(3). The effect of |Ll =
0.00001, 0.0001, 0.0005 as well as g = 0 and 0.5 is demonstrated.
532
0.0005
0.00041
0.0003
u 0.00021
0.0001
o|
-0.0001
0.02|
0.015
0.01
^0.005
01
k
L_
\i = 0.00001,g = 0 1 1 1 1
n = \y^
71 t i l l
/H -j
-J
- -
10 20 30 40 50 t
-0.005.
\i = 0.0005, g = 0
k
\-
r^ • *•'•'
1 1
fl = l l
21. .
1 1
1, 71;' '.
: . • • * '
.v''3r\'
1
' (c)|
-j
y^ H
1 0 10 20 30 40
t 50
\i = 0.0002, g = 0
i = 0.0005, g = 0.5
-0.005
Fig.5.2-1(3). Ci i , C21, C31 and C71 versus t demonstrating the effect of III and g
5.2-1(4^ This is an extension of case 3.13-3 for an open system comprising of two
reactors; in the first one chemical reaction takes place. The flow scheme is shown
in Fig.5.2-l(l). The configuration in Fig.5-1 is reduced to the present one by
choosing reactors j and ^ designated as j = 1 and ^ = 2, respectively. Considering
Eq.(5-18a) for a'k = 0 = 0, yields that a i = Qi/Qi = P12 = qi2/Ql = 1 while taking
Ql as reference flow. Eqs.(5.2-l(lb)) to (5.2-l(lh)) in case 5.2-1(1) are
applicable also in the present case where f = 1,..., 7.
The following model [65], appearing in case 3.13-3, was applied also for the
open system:
533 ki k, k3
i = 1: Ai + A2 ~* A3 i = 2: A3 - • A2 i = 3: 2A2 - • A4
i = 4: A3 + A4 -* 3A2 i = 5: A4 - • A5
thus, m = 1,.... 6 in Eq.(5.2-l(10) for Lfi(n). Considering Eq.(5-7), yields the
following relationships: r(3)
Fori- l:-r<i)--r^»>-r(i> Fori -2: - r f - 42) For i - 3 : - - | - - r f
r(4) Fori - 4: - r > - - i ^ --j- Fori - 5: - i ) = i s)
Thus,
rj - r - - kjCjCj + k.jCg
= - kjCjC^ + k.iC3 + k2C3 - 2k3C 2 + ^^4^^^^
^3 *" ^3 •*" ^3 •*• ^3 " ^ 1 ^ 2 " - 1 ^ 3 " ' ^ ^ 3 ~ k 4 C 3 C 4
r - r^f + r(f + r(|) - kgC^ - k^CgC - kjC 4
^ 5 - 4 ' ^ - k 5 C ,
Considering Eqs.(5.2-l(le)) to (5.2-l(lg)) yields for f = 1, ..., 5 the following
equations for reactor 1 where the different rf(n)'s are given above:
Cfi(n+1) = Cf4n,iPin,l + Cfi(n)pii + rf(n)At
Pin,l> Pll above and pi2 below are given by by Eq.(5.2-l(ld)). In addition, for
reactor 2
Cf2(n+1) = Cfi(n)pi2 + Cf2(n)p22
where p22 = 1 - \^2^^ is obtained from Eq.(5-30a) assuming that C*f2 = Cf2. In the
numerical solution it was assumed that the reactors are of an identical volume, i.e.
m = QiA/ i = ji2 = Q1/V2 = Ji. The initial state vector C(0) = [Cf4n,i(0), Cfi(O),
534
Cf2(0)] reads: [20, 1, 1] for f = 1, 2 and [0, 0, 0] for f = 3, 4 and 5. Additional parameters were: ki = 2.5, k.i = 0.1, k2 = 1, ks = 0.03, k4 = 1, ks = 1 and At = 0.005. The transient response of Cn, C21, C31 and C41 in reactor 1 shown in Fig.5.2-l(l), is depicted in Fig.5.2-1(4) where the effect of \i is demonstrated.
2
1.5
ii = 0
U 1 U
0.5
0
1
k
1
fl =
r
I
1
11
1 ""
1 1
21
3 1
^" 1 " I
1
--
1
\i = OA
10
i = 0.01
1.5
1
0.5
0
14
12
10
8
6
4
2
0
-2
1 1 1 1
21 . --"" 11
1 ^ vL ^- --H i l l
1 1
H
-
1 4 6
t i = 5
10
-
h
h
1 1
/
1 1
IK r 1 1 1
1 1 31
21
11
1 1
1 1
- H
H
1
10
Fig.5.2-1(4). Cii, C21 and C31 versus t demonstrating the effect of |Li
5>2-lf5) This is an extension of case 5.2-1(1) for an open system comprising of three
reactors; in the first two ones chemical reaction takes place, and the third reactor is a "total collector" of the reactants and products. If this reactor is not a total collector, QE.(5.2-l(a)) is applicable, i.e. in the following matrix P33 = 1 - ILI3 At. Note that in previous cases, 5.2-1(1) to 5.2-1(4), chemical reaction took place only in one reactor. The flow scheme is shown in Fig.5.2-1(5), which is slightly different than the scheme in case 4.3-4.
535
Qi+Qs
!S ' Fig.5.2-1(5). The flow system
The general configuration in Fig.5-1 is reduced to the present one by choosing reactors j , a and ^ designated as j = 1, a = 2 and ^ = 3, respectively. Considering Eqs.(5-18a) to (5-18c) for a'k = 0, yields that P13 = 1 + a2, 1 + OL21
= CX12 + Pi3 and a2 + an = a2i, where a2 = Q2/Q1, Pi3 = qis/Ql, OC12 = Q12/Q1 and a2i = Q2l/Ql- A numerical solution was obtained in the following for a2 = 1; thus Pi3 = 2 and a2i = ai2 + 1. From Eq.(5-la) the state space in the present case reads:
SS = [Cf,in,l, Cf,in,2, Cfi, Cf2, Cf3] f = 1, ..., 4 (5.2-l(5a))
noting that no heat and mass transfer take place, i.e. |ifpj = |ihj = 0 which are given
by Eqs.(5-10) and (5-22). From Eq.(5-3a) the state vector reads:
S(n) = [Cf,in,i(n), Cf,in,2(n), Cfi(n), Cf2(n), Cf3(n)] (5.2.1(5b))
The probability matrix given by Eq.( 5-27) is reduced to:
Cf,in,l
Cf,in,2
P = Cfi
Cf2
Co
Cf,in.l 1
0
0
0
0
Cf,in,2 0
1
0
0
0
Cfi
Pin,l
0
Pll
P21
0
Cf2
0
Pin,2
P12
P22
0
Cf3
0
0
P13
0
1 5.2-l(5c))
From Eqs.(5-28a) for j = 1, (5-29a) for a = 2 and (5-30a) for ^ = 3, it
follows, noting that P13 = 2 and a2i = O-u + 1, for reactor 1 that:
Pin,i=HiAt pii = l - (2 + ai2)mAt pi2 = ai2|X2At pi3 = 2M.3At
536
for reactor 2:
Pin,2 = H2At p 2 2 = l - (l+ai2)^l2At p2i = (1 + ai2)^2At P23 = 0
for reactor 3:
P33 = 1 (5.2-l(ld))
In addition, for all species, i.e. f = 1,..., 4:
for reactor 1: Cfi(n+1) = Cf,in,iPin,l + Cfi(n)pii + Cf2(n)p2i + Lfi(n)
for reactor 2: Cf2(n+1) = Cf,in,2Pin,2 + Cfi(n)pi2 + Cf2(n)p22 + Lf2(n)
for reactor 3: Co(n+l) = Cfi(n)pi3 + Cf3(n) (5.2-l(le))
where
Lfi(n)=^r(7)(n)At = rf,(n)At L jCn) =^r[™Hn)At = r jWAt (5.2-l(lf)) m m
Subscripts fl, f2, f3 designate species fin reactors 1, 2, 3, respectively.
The following reactions, as in cases 5.2-1(1) and 3.13-4, known as the
Brusselator model [60], are assumed:
i = 1: Ai —> A3 i = 2: 2A3 + A2 --> 3A3 i = 3: A3 -> A2
i = 4: A3 -> A4
thus, m = 1, ..., 4 in the equations for Lfi(n) and Lf2(n) above. Considering the
derivations in case 5.2-1(1), the following equations are obtained which enables
one to compute Lfi(n) and Lf2(n):
537
f = l :
for reactor 1: rjj(n) = - k^Cjidi) for reactor 2: rj2(n) = - kjCj2(n)
f=2 :
for reactor 1: r2|(n) = - k2C2i(n)C^j(n) + k3C3j(n)
for reactor 2: r22(n) = - k2C22(n)C^2(^^ "*" ^3^32(11)
f = 3 :
for reactor 1: r3j(n) = kjCjj(n) + k2C2iC^i(n) - (k3+ k^)C^^(xi)
for reactor 2: r32(n) = kjCj2(n) + k2C22C32(n) - (k3+ k4)C32(n)
f = 4:
for reactor 1: r jCn) = k4C3j(n) for reactor 2: r42(n) = k4C32(n)
Thus, the above equations as well as Eqs.(5.2-l(le)) to (5.2-l(lg)), makes it
possible to calculate the concentration distributions in reactors 1,2 and 3 of species
f = 1, ..., 4. In the numerical solution it was assumed that the reactors are of an
identical volume, thus, |ii = QiA^i = |i2 = Q1/V2 = II3 = QiA' s = |ii. The initial
state vector C(0) = [Cf,in,i(0), Cf,in,2(0), Cfi(O), Cf2(0), CoCO)] reads: [100, 10,
1, 1, 0] for f = 1 as well as [0, 0, 0, 0, 0] for f = 2, 3, 4. Other parameters were:
ki = 10, k2 = 0.1, k3 = 2, k4 = 1, an = 0, 1, 5, 10 and 50, |ii = 0, 0.02, 0.03 and
0.05 and At = 0.05.
The transient response of C21 and C31 versus t, i.e. the concentrations in
reactor 1 in Fig.5.2-l(l) and of €22* the concentrations in reactor 2, as well as the
attractor C31 against C21, are depicted in Fig.5.2-l(5a) where the effect of ai2 is
demonstrated for [i = 0.02. In Fig.5.2-l(5b) the effect of |Li is shown for ai2 = 5.
538
6
5
4
U3
2
1
n
1
—
-
^ 1
«,2=1 t i l l
1 1 1 1
1 1
-H
^ v H
, (d)
0 2 4 6 p 8 S i
10 12 14
^ 1 2 " ^^
Fig.5.2-l(5a). C21, C22 versus t and C31 versus C21 demonstrating the effect of ai2
539
n = o n = o 1 1 1 1 1
i 1 1 1 1
1 1
1 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
[ f 1
i = 0.03
1 1 1 1
1 1
1 -2 0 4 6 8 10 12
0.06 1
1
i = 0.06 1 1 1 1 1
1 1 1 1 1
1 1
1 -1 0 1 2 3 ^ 4 5
S i 6 7
Fig.5.2-l(5b). C21, C31 versus t and C31 versus C21 demonstrating the effect of |i
540
5,2-1(6) IMPOSSIBLE PRODUCTS' BEHAVIOR IN BELOUSOV-ZHABOTINSKI MODEL CREATING A
HUMOROUS PATTERN This case terminates the examples of chapter 5.2-1 on reacting systems. We
apply here the Belousov-Zhabotinski model [57], presented in case 5.2-1(3), for
operating conditions which generate non-realistic results, on the one hand, but a
humorous unexpected behavior, on the other, due to a certain presentation of the
results. The equations are those appearing in case 5.2-1(3) for the following data.
The initial state vector C(0) = [Cf,in,i(0), Cfi(O), Cf2(0)] reads: [5.0 10-3, 10-5, Q]
for f = 1, [3.5 10-5, 0, 0] for f = 2, [0, 0, 0] for f = 3, 4, 6, 7 and for f = 5 it reads
[1.24 10- , 10-11,0]. Additional parameters were: ki = 0.084, k.i = 1.0 10 ' k2 =
4.0 108, k.2 = 0, k3 = 2000, k.3 = 2.0 10 , k4 = 1.3 10 , k.4 = 2.4 10 , ks = 4.0
106, k.5 = 4.0 10-11, k6 = 1.0 106» g = 1, ^ = 0.03 and At = 0.01. These data are
slightly similar to those appearing in [57].
The transient response of C21 and C31, i.e. the concentrations in reactor 1
where reactions take place, is depicted in Fig.5.2-l(6a) as well as the plot of C21
against C31 is demonstrated. As observed, the C2i-t and C3i-t behavior is
unrealistic after 25 time units since negative concentrations are obtained and
somehow chaotic behavior. However, the plot of C21 against C31 creates a
combined eyes and nose which were complimented by the author to a face.
541
5 10"
4 10
3 10" h
2 10"' h
u l l O ' h
- l l O ' h
-2 10"
Fig.5.2-l(6a). C21 and C31 versus t
-2 10-' -1 10
Fig.5.2.1(6b). C21 versus C31
542
5.2-2 ABSORPTION SYSTEMS In the following cases absorption processes with and without chemical
reaction will be demonstrated.
5.2-2(1) The flow system shown below comprises of three reactors. In the first two
reactors absorption of a single component takes place, whereas the third reactor is
assumed as "total collector" for the absorbed gas. If this is not the case, Eq.(5.2-
1(c)) is applicable and in the following matrix P33 =1-113 t. The fluid in which
the species are absorbed, enters the first reactor and leaves the third one at a rate
Qi.
%, [oh 1
•
V 2
^23 do 3
^13
Fig.5.2-2(1). The flow system
The configuration in Fig.5-1 is reduced to that in Fig.5.2-2(1) by choosing
reactors j , a and ^ designated as j = 1, a = 2 and ^ = 3, respectively. Considering
Eqs.(5-18a) to (5-18c) for a'k = 0, yields that P13 + P23 = 1, 0C2l + p23 = a i2
where Ojk and Pj^ are given by Eq.5-11.
From Eq.(5-la), the state space for species f = 1 reads:
SS = [Ci,in,i, Cii , C12, C13]
where from Eq.(5-3a) the state vector reads:
S(n) = [Ci,in,i(n), Cii(n), Ci2(n), Ci3(n)]
The probability matrix given by Eq.( 5-27) is reduced to:
(5.2-2(la))
(5.2-2(lb))
543
P =
'l,in,l
Ci i
Cl2
Cl3
Cl,in,l
1
0
0
0
Cu Pin.l
Pll
P21
0
Cl2
0
P12
P22
0
Cl3
0
P13
P23
1 5.2-2(lc))
From Eqs.(5-28a) for j = 1, (5-29a) for a = 2 and (5-30a) for ^ = 3, noting
that there is a single mass transfer process, i.e. absorption and hence p = 1, it
follows that:
Pin,l = mAt pii = l-(ai2+Pl3)HlAt p2l=a2imAt
P12 = ai2 M'2At P22 = 1 - (a21 + p23)|X2 At
Pl3 = Pl3H3At P23 = p23^3At (5.2-2(ld))
For reactor 1:
Cii(n+1) = Ci,in,lPin,l + Cii(n)pii + Ci2(n)p2i + Lii(n) (5.2-2(le))
where from Eq.(5-28a)
Lii(n) = ^ii,iACn,iAt (5.2-2(lf))
andfromEq.(5-14)
C* J is the equilibrium concentration of the species 1 absorbed on the surface of the
liquid in reactor 1 corresponding to its partial pressure in the gas phase above the
liquid.
For reactor 2:
Ci2(n+1) = Cii(n)pi2 + Ci2(n)p22 + Li2(n) (5.2-2(lg))
where
544
Li2(n) = ^ii,2ACn,2^t (5.2-2(lh))
and 11 2^12 *
M'11,2 ~ V ^^11,2 ~ ^12 ~ C^2(n)
C*2 is the equilibrium concentration of species 1 absorbed on the surface of the
liquid in reactor 2 corresponding to its partial pressure in the gas phase above the
liquid.
For reactor 3:
Ci3(n+1) = Cii(n)pi3 + Ci2(n)p23 + Ci3(n)p33 (5.2-2(li))
In the numerical solution it was assumed that the reactors are of an identical
volume, thus, |ii = Qi/Vi = |X2 = Q1/V2 = II3 = Q1/V3 = |LI, and that the third
reactor behaves as "total collector" for the reactants and products, namely, P33 =
l.The transient response of Cn , C12 and C13, i.e. the concentrations of species 1
in reactors 1, 2 and 3 is depicted in Fig.5.2-2(la) where the effect of |l = 1, 10
(cases a, b in the figure) and the mass transfer coefficient for absorption (for which
p = 1 in Eq.5-28a) of species lin reactor 2, i.e. |LIII,2 = 1, 50 (cases d, e) as well
as C*2 = 3-10-6' 6-10-6 (cases c, d) is demonstrated. The initial state vector C(0) =
[Cf,in,i(0), Cfi(O), Cf2(0), Cf3(0)] = [0, 0, 0, 0] for f = 1. Other parameters were: OC12 = 1, Pl3 = 0, mi,l = 50, C*i = 310-6 and At = 0.0001.
545
3 10
2.5 10" h
.- 2 10- h
1.5 10" h
1 10
5 10
j-6
)-' ,-6
,-6
,-6
H = l
y^ i w
- / -
- /
_/
f =i - " " "
^l= 10
^0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 0.2 0.25 0.3 t t
1.5 10
1 10" k
u 5 10"T
C =3 10 , i = 50 12 11.2
_ ~
p
c
1
^^
1
• 1
1
1
3 ^
1
2
1
*-
J (d)-
^ J
1 0 0.05 0.1 0.15 0.2 0.25 0.3 Q O.05 0.1 0.15 0.2 0.25 0.3
t t
1.5 10 •5 C = 3 10 , i = 1 12 11.2
1 10" r5r
u 5 10'
-
h
' ' ' ' ' (e)J
3 ^ ^ i =1 ^ ^ ^
-_--x: - r 1 1 1 •
6h
0 0.05 0.1 0.15 0.2 0.25 0.3 t
Fig.5.2-2(la). Cn, C12 and C13 versus t demonstrating the effect of
A simpler case which can be treated by applying the above model is the absorption of one component, f = 1, into a single reactor, i.e. reactor 1 where reactor
546
3 is the "total collector". Under these conditions, ai2 = 0 and P13 = 1 in Fig.5.2-
2(1). In the numerical solution it was assumed that the reactors are of an identical
volume, i.e. a constant |i. Other parameters kept unchanged were: C*^ = 3-10"^ and
C*2 = 0. The initial state vector C(0) = [Ci,in,i(0), Cii(O), Ci3(0)] = [0, 0, 0] and
At = 0.0001. The transient response of Cn and C13, i.e. the concentrations of
species 1 in reactors 1 and 3 is depicted in Fig.5.2-2(lb) where the effect of |x = 0.1,
10 (cases b, c in the figure) and |Liiij = 10, 50 (cases a, c) is demonstrated.
3 10-
2.5 10"
2 10-
^1.5 10-
1 10-
5 10"
0
-
- /
— /
^^=°-^'V.= 1 1
l i = 1 1 ^ - " " ^
13 1 1 r
10
1 • " " ^
-\
-\
-J -J
\i = OA,\i =50 11,1
t - 1 1 r 1 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 0.2 0.25 0.3
t t
3 10'
2.5 10'
.- 2 10"
^L5 10-
1 10"
5 10"''
0
\i= 10,
1 1 /' i l = 11 •'
~ / ^ "^• '
4 / I / k 1 1
i = 50 11,1
1 1
1 1
1 1 (c)
A
-\
-
1 0 0.05 0.1 0.15 0.2 0.25 0.3
t Fig.5.2-2(lb). Ci i and C13 versus t demonstrating the effect of \i and
^ 11,1
547
5>2-2(2) The flow system is shown in Fig.5.2-l(l). It comprises of two reactors and
only in the first one the chemical reaction given below takes place, i.e.:
i = l : A i + A2->A3 (5.2-2(2a))
The carrying fluid of reactant 1 (f = 1) dissolved in it enters the first reactor and
leaves the second one at rate Qi. Reactant 2 (f = 2) is absorbed in reactor 1 and
reacts there with reactant 1 yielding product 3 (f = 3). The configuration in Fig.5-1
is reduced to that in Fig.5.2-l(l) by choosing reactors j and ^ designated as j = 1
and ^ = 2, respectively. Considering Eqs.(5-18a) to (5-18c) for a'k = 0, yields
that ai = Qi/Qi = P12 = qi2/Ql = 1 while taking Qi as reference flow.
From Eq.(5-la) the state space reads:
SS = [Cf,in,l, Cfi, Cf2] f = 1,..., 3
From Eq.(5-3a) the state vector reads:
S(n) = [Cf,inj(n), Cfi(n), Cf2(n)]
(5.2-2(2b))
(5.2-2(2c))
The probability matrix given by Eq.( 5-27) is reduced, for f = 1,..., 3, to:
p =
Cf,in,l
Cfl
Cf2
Cf,in,l
1
0
0
Cfl
Pin.l
Pll
0
Cf2
0
P12
P22 (5.2-2(2d))
From Eqs.(5-28a), for j = 1, and (5-30a) for ^ = 2, noting that P12 = 1 and
C'f2 = Cf2 , it follows that:
Pin.l=l^l^t pii = l - m A t pl2 = 2At p22=l-^2At (5.2-2(2e))
Considering Eq.(5.2-2(2a)), yields the following relationship:
r(l) = r. = Fj- f j - f j - Tj- kjjCjCj - kjCjCj (5.2-2(2f))
548
where k n indicates the rate constant for reaction i = 1 in reactor 1. However, since
the chemical reaction takes place only in reactor 1, k] should replace k n . For
reactor 1, it follows from Eq.(5-28a) that:
Cfi(n+1) = Cf,in,iPin,l + Cfi(n)pii + Lfi(n) f = 1,..., 3 (5.2-2(2g))
where
Lii(n) = rii(n)At = ri(n)At = - kiCii(n)C2i(n)At
L2i(n) = [^21,1^^21.1^^) + ^*2i(^)]^^ = [1 21,1 21 •" C2i(n)) + r2(n)]At
= [^21,1(^21 - C2i(n)) - kiCjj(n)C2i(n)]At
L3i(n) = r3^(n)At = r3(n)At = kiCii(n)C2i(n)At
For reactor 2:
Cf2(n+1) = Cfi(n)pi2 + Cf2(n)p22 f = 1,..., 3 (5.2-2(2h))
In the numerical solution it was assumed that the reactors are of an identical
volume, thus, |LII = QiA^i = |i2 = QlA^2 = 1 - The transient response of Cn , C21
and C31, i.e. the concentrations in reactor 1 of species 1, 2 and 3 in Fig.5.2-2(1),
is depicted in Fig.5.2-2(2). The effect of i is demonstrated in cases a, b as well as
in cases d, e, and f; the effect of the mass transfer coefficient for absorption, i.e.
^21 1 (for which p = 1 in Eq.5-28a), is depicted in cases b, c. The initial state
vector C(0) = [Cf,in,i(0), Cfi(O), Cf2(0)] was [0, 0, 0] for f = 2, 3 in all cases and
[310-5, 0, 0] as well as [0, 310-5, 0] for f = 1 when exploring the effect of i in
cases a, b and d, e, f. Other conmion parameters were: ki = 10 » C21 = 3-10-5 and
At =10-4.
549
3 10"
2.5 10"
^1.5 10-
1 10" \L
5 10'
. ^ = 10, |i = 100,C = 3 10• C (0) = 0 [i= 100, i = 100, C =3 10' C (0) = 0 5 '^ '^Zl.l l,in.l 1 2 U UrU T ^
LL
\ ^ ]/ y r-<.
"1
/ .1
^ ^
\
y
L
X-
1 2
f = l
1
-
1
-"
(a)
-
~-
"TT^ (b)
0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 0.2 0.25 0.3 t t , |X = 100, ji = 1000, C = 3 10"\ C (0) = 0
3 lQ-5 21,1 l.inj r
2.5 10' L"
^ 2 10" L ^L5 10-
1 10"'
5 10 r6
1 1 -r [/ 1/ 1 1
' 2 ' f= l
3
1 1
' (c) A
-
-
—
1
3.5 10"
2.5 10
0 0.05 0.1 0.15 0.2 0.25 0.3
^1=100, |i =100, C =0,C(0) = 3 10" ^ =10,^1 = 100,C = 0, C (0) = 3 10* r5 ^ ^21.1 l.in.l 1 *_ ^21,1 l.in.l 1
1
\ l
Li- V ^
1/ 1
1 1 1 1 (ej 2 J
—j
_ 3
0 0.01 0.02 0.03 0.04 0.05 0 0.05 0.1 0.15 0.2 0.25 0.3
. 1 = 0, i = 100, C =0,C(0) = 3 10' 3.5 10' ^u ijrui I
3 10"
2.5 10"
^ 2 10"
^ ^ 5 10-
1 10' 1
5 10'
0 0.05 0.1 0.15 0.2 t
Fig.5.2-2(2). Cii, Ci2 and C13 versus t the effect of \i and 1121 1
550
5.2-2(3^ The flow system is shown in Fig.5.2-l(l). It comprises of two reactors and
the chemical reaction given below takes place only in the first reactor. The
following reactions were considered in case 3.13-1 for a closed system:
i = 1: Ai -^ A2 i = 2: A2 -^As (5.2-2(3a))
The carrying fluid, entering reactor 1 at rate Qi, may or may not contain the
reactants or the products. However, in reactor 1 absorption of reactant 1 (f = 1),
i.e. Ai, takes place with the formation of products 2 and 3 (f = 2 and 3). The
configuration in Fig.5-1 is reduced to that in Fig.5.2-l(l) by choosing reactors j
and ^ designated as j = 1 and ^ = 2, respectively. Considering Eq.(5-18a) for a'k
= 0, yields that a i = Qi/Qi = P12 = qi2/Ql = 1 while taking Qi as reference flow.
Thus, Eqs.(5.2-2(2b)) to (5.2-2(2e)) of the previous case are also applicable here
and with respect to Eq.(5.2-2(3a)) the following are the rate equations [55]:
ri = - kiCiC2 r2 = kiCiC2 - k2C2 rs = k2C2 (5.2-2(3b))
For reactor 1, it follows from Eq.(5-28a) that:
Cfi(n+1) = Cf,injPin,l + Cfi(n)pii + Lfi(n) f = 1,..., 3 (5.2-2(3c))
where
Lii(n) = [^lll jACji^Cn) + rii(n)]At = [^nj^C^ - Ci Cn)) + r^WlAt
= [|iiii,i(C*i - Cii(n)) - kiCii(n)C2i(n)]At
L^iCn) = r2i(n)At = r^WAt = kjC^ j(n)C2|(n)At - k2C2i(n)
LjjCn) = r3j(n)At = r3(n)At = k2C2|(n)At
For reactor 2:
Cf2(n+1) = Cfi(n)pi2 + Cf2(n)p22 f = 1,..., 3 (5.2-2(3d))
551
where pin,i, Pii and pi2 are given in Eq.(5.2-2(2e)); p22 = 1- |X2 At. In the
numerical solution it was assumed that the reactors are of an identical volume, thus,
|Xi = Qi/Vi = 1X2 = Q1/V2 = |i. The transient response of C n and C21, i.e. the
concentrations of species 1 and 2 in reactor 1 in Fig.Fig.5.2-l(l), is depicted in
Fig.5.2-2(3) where the effect of |LI (cases a, b and c) and k2 (cases d, e and f) is
demonstrated. In case g the operating conditions are identical to case e where C21
is plotted versus C n . The initial state vector C(0) = [Cf,in,i(0), Cfi(O), Cf2(0)]
reads: [0,0, 0] for f = 1, 3 and [0, 310-10,0] for f = 2. Other conmion parameters
were: ki = 10^, At = 510-4, c^j = 1 and ii^^ | = 10" . As seen, the oscillatory
behavior of Ci 1 depends on |LI and k2.
552
0.0002
0.00015L-
| i = 1, k = 50
U 0.0001
5 10
0.0005,
0.0004
j).0003
^0.0002
0.0001
0|
0
)0001
8 10"
6 10"
4 10"
2 10"
- /
^ = \^^^
1
10, k = 50 2
• n v 1
1 1
1
l _
2
1
(b)
-j
-
-
0 0.2 0.4 0.6 0.8 t
1 10''r
8 10"*
6 10-*
"4 lo--
2 10-*
0
('"'
2
i = 100, k = 50 2
1 1 1 (c)
J
^
-
f 1 1 1 0 0.2 0.4 0.6 0.
-
k
l = 1, k = i ^ 2
1 1
2 . - ' -"*"
f = i
'" m
-
r - - 1 1
0.00035
0.0003
0.00025
0.0002
0.00015
0.0001
5 10'
0
V /
A • /
/ / r
l = l,
1
A i / \ ^ y
k = 150 2
1
.
A/^ /
(e)j
-
/^^--^
0.5 1.5 0 0.5 1.5
0.0007 0.0006 0.0005
J3.0004
u"o.0003
0.0002
0.0001
0 0
p
H ..-'
i = 1, k = 400
1 1 . . . - • • .
1 t=JLH 1
d
L_
^1= 1, k = 150 '^ 2
0.5 1.5 0.0001 0.0002 c 00003
Fig.5.2-2(3). Cii and C21 versus t for the effect of [i and k
553
5>2-2(4) The flow system shown in Fig.5.2-l(l) comprises of two reactors. The
following reactions known as the Brusselator model [60], taking place in the first
reactor, were also considered in case 3.13-4 for a closed system. In reactor 1
absorption of reactant 1 (f = 1), i.e. Ai, takes place with formation of products 2,
3 and 4 (f= 2, 3 and 4).
kj 2 k k
i = 1: Al -> A3 i = 2: 2A3 + A2 ^ 3A3 i = 3: A3 4 A2 i = 4: A3 - t A4 (5.2-2(4a))
The following are the rate equations:
ri = - kjCi r2 = - k2C2C3 + k3C3
r3 = kjCi - 2k2C2C? + 3k2C2C^ - (k3 + k4)C3 r^ = )^^C^ (5.2-2(4b))
For reactor 1, it follows from Eq.(5-28a) that:
Cfi(n+1) = Cf,in,iPin,l + Cfi(n)pii + Lfi(n) f = 1,..., 3 (5.2-2(4c))
where
L,i(n) = [|AiJ jACji i(n) + r,j(n)]At = [|X,, ,(C*, - Cj,(n)) + ri(n)]At
= [^iii,i(Cii - Cii(n)) - k,Cij(n)]At
L2i(n) = r2j(n)At = r2(n)At = [ - k2C2,(n)Cf,(n) + k3C3,(n)]At
L3i(n) = r3i(n)At = r3(n)At = [kjCjiCn) + k2C2i(n)Cf,(n) - (kj + k4)C3j(n)]At
L4i(n) = r4i(n)At = r4(n)At = k4C3i(n)At
For reactor 2:
Cf2(n+1) = Cfi(n)pi2 + Cf2(n)p22 f = 1, •.., 3 (5.2-2(4d))
where pin.i, Pii and pi2 are given in Eq.(5.2-2(2e)); p22 = 1- ^2At. In the
nvimerical solution it was assumed that the reactors are of an identical volume, thus,
Hi = QiA' i = |X2 = QiA^2 = V" The transient response of Ci i, C21 and C31, i.e.
554
the concentrations of species 1, 2 and 3 in reactor 1 in Fig.5.2-l(l), is depicted in
Figs.5.2-2(4a), 5.2-2(4b), 5.2-2(4c). In addition the variation of C31 versus C21
is depicted. The data applied were: the initial state vector C(0) = [Cf,in,i(0),
Cfi(O), Cf2(0)] = [0, 0, 0] for f = 1, ..., 4, ^ij j ^ = 30, |X = 0.06, ki = 1, k2 = 1,
k3 = 2, k4 = 1, Cj 1 = 1 and At = 0.05.
The effect of |l = 0, 0.06, 0.1 and 0.2 is demonstrated in Fig.5.2-2(4a). The
effect of |Lijj J = 30, 5 and 0.5 is demonstrated in Fig.5.2-2(4a), cases c and d for
which IUji 1 = 30, as well as in Fig.5.2-2(4b), cases a to d. The effect of Cti=
1.5, 1 and 0.15 is demonstrated in Fig.5.2-2(4a), cases c and d for which Cj j= 1,
as well as in Fig.5.2-2(4b), cases e to f. Finally, the effect of the reaction rate
constant k2 = 10, 1, 0.7, 0.5 and 0.3 is demonstrated in Fig.5.2-2(4a), cases c and
d for which k2 = 1, as well as in Fig.5.2-2(4c). As observed, the oscillatory
behavior of the concentrations depends on the above operating parameters.
555
t = o t = o
3
2.5
c 2
1.5
1
0.5
0
1
- / - /
1 ':' / - - ' , '
1 = 1
1
= 0.1 1
1
' (e)
—
-
-
1
1 = 0.1
0 10 20 30 t
40 50
U
2,
1.5
1
0.5
01
1
1
^ 1
1
= 0.2 1 1
. f= l
3
1 1
' (g)
^-""""^
1 0 0.5 1 1.5 2 2.5 3 0 0.5 1
t Fig.5.2-2(4a). The effect of [i
556
2.5
U
C , = 1 . 5 1
/
1:/ 1
1 1 2
- H ^
- '3
1 1
' (e)|
J
1
2
1.5
U 1
0.5
0
!
-
c 1
1
11=1-5 1 1
y
1 1
(f)
H
" •
4 6 t
10 0 0.5 1 c 1-5 21
C\=0.15
2.5
20 40 60 80 100 0 t
Fig.5.2-2(4b). The effect of ^ (a to d) and C^^ (e to h)
0.1 ^ 0.2 S i
0.3
557
u
1.6
1.4
1.2|
0.8 0.61 0.4 0.2
k = 2
1
MAL-i r ^^^--<^^'
\r.-''' 1
10
* - -.
1
f = i
' ^
, 2
1
(a)
—'
" ^
-
u
3 | - ;
2
1
0
1 2 t
k = 0.7 2 1 1
^; V, \: '' '•• •-2"
^ f = l
K 3 f 1 1
(c)
-
50 100
50 100
150
10
8
6
^^4
2
0
k = 0.3 2
1
4
/•• .' 2
-.' ,'
T f=i '•
x^ . /
(g)
H
-
t 1 1
u
150
1
0.8
0.61
0.4
0.2
-
-^^^
k = 10 2
1 1 1
1 1 1
(b)
..^^^"^^ -
Fig.5.2-2(4c). The effect of ki
558
5>2-2(5) IMPOSSIBLE REACTIONS CREATING AESTHETIC PATTERNS
The flow system is shown in Fig.5.2-l(l) and comprises of two reactors.
The following "reactions", only partially possible, are based on the Lorenz
equations [85, p.697]. The reactions in case 3.14-3 occur in the present example
only in the first reactor. In this reactor, j = 1, absorption of reactant 2 (f = 2), i.e.
A2, takes place with the formation of "products" 1 and 3 (f = 1 and 3) according to
the following Lorenz "reactions": 10
ri = IOC2 - lOCi for which Ai A2 maybe written
10
r2 = 28Ci - C2 - C1C3 for which no reaction may be realized 8/3
T3 = C1C2 - (8/3)C3 for which A3 Ai + A2 may be written
1
For reactor 1, it follows from Eq.(5-28a) that:
Cfi(n+1) = Cf,in,iPin,l + Cfi(n)pii -f Lfi(n) f = 1, ..., 3 (5.2.2(5a))
where
Lii(n) = rii(n)At =ri(n)At = [10C2i(n) - lOC^^WlAt
L2i(n) = [ 121,1 21 i(n) + r2i(n)]At = [ 21,1( 21 - C2i(n)) + r2(n)]At
= [ 21,1( 21 - C2i(n)) + 28Cii(n) - C2i(n) - Cii(n)C3i(n)]At
L3i(n) = r3i(n)At = r3(n)At = [Cii(n)C2i(n) - (8/3)C3i(n)]At
For reactor 2:
Cf2(n+1) = Cfi(n)pi2 + Cf2(n)p22 f = 1,..., 3 (5.2-2(5b))
where pin,i, p u , P12 and P22 are given in Eq.(5.2-2(2e)). In the numerical
solution it was assumed that the reactors are of an identical volume, thus, |LII =
Ql/Vi = |X2 = Q1/V2 = |X. The transient response of Cn, C21 and C31, i.e. the
559
"concentrations" of species 1, 2 and 3 in reactor 1 in Fig.5.2-l(l), is depicted in
Figs.5.2-2(5a), 5.2-2(5b), 5.2-2(5c).
In addition the variation of C31 versus C21 and C31 versus Cn are depicted,
which reveal very nice patterns from the artistic point of view. This is, by the way,
a mean to generate art from scientific models, which has been demonstrated also in
case 5.2-1(6) above.
The data applied in the computations were: the initial state vector C(0) =
[Cf,in,l(0), Cfi(O), Cf2(0)] = [0, 0.01, 0] for f = 1, 2, 3, ^21,1 = 0' C*i = 0 and At
= 0.01.
The effect of |X = 0, 1,2, 2.07, 3 and 8 is demonstrated in the following in
Figs.5.2-2(5a), 5.2-2(5b), 5.2-2(5c). As seen, for |i = 0 the "concentrations",
which are positive and negative, reveal chaotic behavior as expected from the
Lorenz equations. By increasing \i, i.e. reducing the mean residence time of the
fluid in the reactor, the "concentrations" become positive all the time, and non-
chaotic for |i > 2.
560
n = o
'21 'II
Fig.5.2-2(5a). The effect of |i = 0 and 1
561
H = 2
u
40
30
20
10
0
-II
i = 2.07 1 1
1 1
1 1
J=3
.2
1
1
10 t
15 20
Fig.5.2-2(5b). C n , C21 and C31 versus t, C31 versus C21 and C31 versus C n demonstrating the effect of |i = 2 and 2.07
562
H = 3
Fig.5.2-2(5c). Cii, C21 and C31 versus t, C31 versus C21 and C31 versus Cn demonstrating the effect of |x = 3 and 8
563
5.2-3 COMBINED PROCESSES In the following, combined processes are presented which incorporate several
chemical engineering operations acting simultaneously.
5>2-3(l) - Chemical reaction and heat transfer The flow system shown below comprises two reactors. In the first one the
chemical reaction A—> A2 takes place and simultaneously the heat of reaction is
removed by a cooler. The carrying fluid in which the reacting species are
dissolved, enters the first reactor and leaves the second one at rate Qi.
Fig.5.2-3(1). The flow system
The configuration in Fig.5-1 is reduced to that in Fig.5.2-3(1) by choosing
reactors j and ^, designated as j = 1 and ^ = 2, respectively. From Eq.(5-18a) for
a'k = 0, it follows that a i = Qi/Qi = P12 = qi2/Ql = 1 while taking Qi as
reference flow.
From Eq.(5-la) the state space reads:
SS = [Cf,in,l, Cfi, Cf2, hin,i, hi, h2] f = 1, 2 (5.2-3(la))
and from Eq.(5-3a) the state vector reads:
S(n) = [Cf,in,l, Cfi(n), Cf2(n), hin,i, hi(n), h2(n)] f = 1, 2 (5.2-3(lb))
where S(n+1) is given by Eq.(5-4). The probability matrix given by Eq.(5-27) is
reduced to:
564
P =
Cf,in,l
Cfl
Co
hin.l
hi
h2
Cf,in,l
1
0
0
0
0
0
Cfl
Pin.l
Pll
0
0
0
0
Cf2
0
P12
P22
0
0
0
hin.l
0
0
0
1
0
0
hi
0
0
0
Pin,l,h
Pll,h
0
h2
0 1 0
0
0
Pl2,h
P22,h (5.2-3(lc))
Noting that the process in reactor 1 incorporates the chemical reaction A,—> Aj for which ri = - kiCi as well as a heat transfer process, it follows from
Eqs.(5-28a) and (5-30a) for j = 1 and = 2 that:
Pin,l = m^t pii = l-HiAt pi2 = ^2At P22 = l (5.2-3(ld))
The last probability indicates that the second reactor behaves as a "total collector" of
the reactants and products. If C'f2 = Cf2, i.e. the reactants and products are not
accumulated in the reactor, Eqs.(5-30a) gives P22 = 1 - M 2At.
From Eq.(5-28b) it follows for j = 1 that:
Pin,l.h= Pin.lPT »il ^t Pii,h = Pll
From Eq.(5-30b) it follows for j = 1 and = 2 that:
Pl2,h = PlP2V2At P22,h = 1 - P1P2VzAt
From Eqs.(5-28a) and (5-28b) it follows that:
Lj.j(n) = rfj(n)At
Lf, b(n) = {^hiCp iATj (n)}At - p7i{rf,(n)AH^j}At
(5.2-3(le))
(5.2-3(lf))
(5.2-3(lg))
Thus, from Eqs.(5-28a) and (5-30a) it follows for j = 1, ^ = 2 and f = 1,2 that for
the 1st reactor:
Cfi(n+1) = Cf,in,lPin,l + Cfi(n)pii + Lfi(n)
565
hi(n+l) = hin.ipm,14i + hl(n)pil4i + Lf 14,(11) (5.2-3(lh))
For the 2nd reactor:
Cf2(n+1) = Cn(n)pi2 + Cf2(n)
h2(n+l) = hi(n)pi2j, + h2(n)p22ji (5.2-3(li))
In the case under consideration the following expressions are applicable for f = 1,
2: Lii(n) - rij(n)At - ri(n)At- - k^CjiAt L2i(n) = - Lii(n)
L i i > ) - KiCp.1'^'^0 - Ti (n)]}At- p-H- kiC„(n)AHrt}At
L2i.h(n) = Lii.h(n) (5.2-3(lj)
where AH j is negative for an exothermic reaction. It is also assumed that
Pin.i - Pi - P2 = P as well as Cp 1 - Cp 2 = Cp
yielding from Eqs.(5.2-3(le)) and (5.2-3(10) that
Pm.lJi = f*!' * Pi 1 Ji = P2241 = 1 - »*l t Pl2Ji = t*2At
If all reactors are of the same volume m = M'2 = M- Also for Ah = CpAT it follows
that
hi(n+l) = hin.i + Cp|Ti(n) - Tin,i] (5.2-3(lk)
Substitution of the above expression into Eq.(5.2-3(lh)), considering Eq.(5.2-
3(lj) and the above probabilities, yields:
Ti(n+1) = Ti(n)[l - (m + Hhi)At] + [\i{Tia,i + ^hlTo]At
+ (pCp)-l{kiC„(n)AH,i}At (5.2-3(11)
where AH j is negative for an exothermic reaction.
Similarly, Eq.(5.2-3(li)) yields:
566
T2(n+1) = Ti(n)ji2At + T2(n)(l - ^2At) (5.2-3(lm)
Eqs.(5.2-3(lh)), (5.2-3(li)) for Cfi(n+1) and Ci2(n+1) and (5.2-3(11), (5.2-
3(lm) make it possible to calculate the concentration and temperature distributions
versus time in reactors 1 and 2 for species f = 1,2. In the numerical solution it was
assumed that the reactors are of an identical volume, thus, |Xi = Qi/Vi = |i2 =
QlA^2 = l - The initial state vector S(0) = [Cf,in,i(0), Cfi(O), Cf2(0), hin,i, hi(0),
h2(0)], in terms of temperatures, reads:
S(0) = [Cf,in,l(0), Cfi(O), Cf2(0), Tin,i, Ti(0), T2(0)] for f = 1, 2
In the computations ki(l/min) = exp(17.2-11600/[1.987T(K)]), AHH =
-18000 cal/gr-mole A, p = Igr/cc and Cp = Ical/gr ^C. Other conmion parameters
were: Tin,i = Ti(0) = T2(0) = 25^C and At = 0.0005. The transient response of
Ti(^C) and T2(^C), i.e. the temperatures in reactors 1 and 2, as well as the
concentrations Cn, C21, C12 and C22 versus t, i.e. the concentrations of Ai and
A2 in reactors 1 and 2 are demonstrated in Figs.5.2-3(la) and 5.2-3(lb).
The effect of |Li(l/min) = 0, 10 for a heat transfer coefficient |Lih(l/min) = 100
is depicted in cases a to d; other common parameters are given in the figure. The
effect |ih = 10, 100 for |i = 10 is given in cases c to f. The effect of Cii(0) = 0.2,
0.5, i.e the initial concentration of Ai in reactor 1, is shown in cases c, d, e, h.
The effect of Ci in (0) = 0, 0.2, i.e. the inlet concentration of Ai into reactor 1, is
shown in cases g to j .
567
n = 0, ^ = 100, c (0) = 0.2, c = 0 n 11 l.in
0.2
u
^ \ \
0
i = 0, \i = 100, C 1 1 1
1 1 1
(0) = 0.2,C =0 1 l.m
' ' (b) fl = ll
-
21
f2 = 12, 22 1 1
0 0.1 0.2 0.3 0.4 0.5 t
1= 10, i = 100, C (0) = 0.2,C =0 ^ ^ h 11 l.in
\i = 10, |i = 100, C (0) = 0.2, C = 0 ^ ^h ir ^ l.in
0.2|
U
^"o.i
1 _ f 1 =
/
/
1
11 1 1
/
21, 22 1 1
-
' ' (d)
• l2=12 1
1 1 0 0.1 0.2 0.3 0.4 0.:
t
45
40
35
30
25
t = 1
- /
r\ '•
10,
\ i
^ h=
= 1
1
10,
1
\ 2
1
C (0) = 11
1
r
= 0.2, c 1 1
rr
= 0 .in
(e)
A
-
10, ^^= 10, C (0) = 0.2, C =0
0 0.2 0.4 0.6 0.8 t
0.2 0.4 ^ 0.6 0.8
Fig.5.2.3(la). Ti(oC) and TiC^C), Cn, C21, C12 and C22 versus t, demonstrating the effect of |ii and |ih
568
36 n = 10, Li = 100, C (0) = 0.5, C = 0 ^ '^h 11 l.in
^ ^ ^ = 10, [i,= 100, C, (0) = 0.5, C,. = 0
'—^ ^ ^ 1 ^—7^ fl = l l f 2 = 1 2 " 1
i = 10, i = 100, C (0) = 0.5, C. = 0.2
1 r (i)'
J L 0 0.2 0.4 0.6 0.8 1
t
2.5
2
l = 10, i = 100, C (0) = 0.5, C = 0.2 n 11 l.in
1.5L
u 0.5
01
1
— _
L_
L / -r i
1 1 1 1 (j)j X 1
/ -H
/ f2 = 12/ - J
/ 1 / / -J
/ /
/ J < -.- fl = l l .21,22
/ — t 1 V 1
0 0.2 0.4 0.6 0.8 1 t
Fig.5.2.3(lb). Ti(oC) and T2(«C), Cn , C21, C12 and C22 versus t, demonstrating the effect of Ci(0) and Ci,in
5>2-3(2) - Cooling heat transfer The flow system shown below comprises of three reactors. Cooling or
heating of the entering fluid takes place in reactors 1 and 2.
\ i \ i '^0,2 ^ 0,2
Fig.5.2-3(2). The flow system
569
The configuration in Fig.5-1 is reduced to that in Fig.5.2-3(2) by choosing
reactors j , a and ^, designated as j = 1, a = 2 and ^ = 3, respectively. From
Eqs.(5-18a), (5-18b) for a'k = 0, it follows that p23 = 1. oci2 = 1 + OC21; a i =
Ql/Ql taking Qi as reference flow.
From Eq.(5-la) the state space reads:
55 = [hin,i, hi, h2, hs]
and from Eq.(5-3a) the state vector reads:
S(n) = [hin,i, hi(n), h2(n), hsCn)]
(5.2.3(2a))
(5.2.3(2b))
where S(n+1) is given by Eq.(5-4). The probability matrix given by Eq.(5-27) is
reduced to:
hin.l
hi
h2
h3
hin.l
1
0
0
0
hi
Pin,l.h
Pll.h
P21,h
0
h2
0
Pl2.h
P22,h
0
h3
0 1 0
P23,h
P33,h 1
p =
(5.2-3(2c))
It follows from Eqs.(5-28b), (5-29b) and (5-30b) for j = 1, i = a = 2 and ^ =
3, assuming that the density of the fluid flowing in the system remains constant,
that: Pin,i,h = m At pii,h = 1 - ai2M.iAt pi2,h = ai2^2At
P21,h = a2imAt p22,h = 1 - (1 + a21)R2At P23,h = 3At
P33h=l-^3At (5.2-3(2d))
Lfi.h(n) = lhlCp,lATl (n)At = ^hiCp.ilTj.o " T,(n)]At
Lf2.h(n) = lh2Cp,2AT 2(n)At = ^h2Cp.2[T2.o " T2(n)]At (5.2-3(2e))
In addition:
570
hi(n+l) = hin,ipin,l,h + hi(n)pii,h + h2(n)p2i,h + Lfi,h(n)
h2(n+l) = hi(n)pi2,h + h2(n)p22,h + Lf2,h(n)
h3(n+l) = h2(n)p23,h + h3(n)p33,h (5.2-3(2f))
The following substitution is made for replacing enthalpies by temperatures,
i.e. Ah = CpA; thus:
hi(n+l) = hin,i + Cp{Ti(n) - Tin,i} i = 1, 2, 3
Substitution of the above expression into Eq.(5.2-3(2f)), assuming that C ^ = C 2 = C and considering the above probabilities, yields:
Ti(n+1) = Ti(n)[l - (ai2Hi + |Lihi)At] + T2(n)a2i|iiAt + [^iTin,i + ^hiTo,i]At
T2(n+1) = Ti(n)ai2R2At + T2(n)[l - { h2 + (1 + a2i)^2}At] + |ih2To,2At
T3(n+1) = T2(n)ji3At + T3(n)[l - 113M] (5.2-3(2g)
where ai2 = 1 + 0L2I' Eqs.(5.2-3(2i)) make it possible to calculate the temperature
distributions versus time in reactors 1, 2 and 3. In the numerical solution it was
assumed that m = |i2 = H3 = M. The initial state vector S(0) = [hin,i, hi(0), h2(0),
h3(0)], in terms of temperatures, reads S(0) = [Tin,i, Ti(0), T2(0), T3(0)] =
[50OC, 250c, 250c, 250C]. Other conmion parameters were TI Q = T2,o = lO^C
and At = 0.005. The transient response of TiC^C), T2(oC) and T^i^C), i.e. the
temperatures in reactors 1, 2 and 3, is depicted in Fig.5.2-3(2a). The effect |Ll = 0,
1,10,100 is demonstrated in cases a to d; the effect of ai2 = 0,25 in cases b and e
and of fihi = Mh2 = 0» 10 in cases e and h.
571
i = 0, i = i = 10, a = 0 h r h2 12 hi h2 12
1 1
4 " 3 \ ^ A " •V 1
2
1 1
1 1 (b)
^ -^
1 1 1
0 1 2 3 4 t
32
30
28
p 2 6
22
20
18 (
30
25
^ 2 0 U h^lS
10
5
^ = 10, i =\i =10 , hi h2
1 1 1 1 = 1
" \ 2
—. ^^.3 "^ - - ->-
1 1 1
a = 0 12
1
1
3 0.1 0.2 0.3 0.4 t
1 1 1 1
2
1 i 1 1
0 1 2 3 t
I
1 4
(c)
-
0
(e)
50 [
45
40
35
30
25
20 5
50
45
40
35
30
25
1 20,
\ 0
D
, = 100,n^=H^=10.a^=0
1 1 1 "1 (d)
,' / ] •' ^ 1
/ J 1 1 1 1
0.02 0.04 0.06 0.08 0.1 t
/ / / /
/ /
1 1 1 1 2 4 6 8 1
t 0
Fig.5.2-3(2a). Ti(oC), TiCC) and T3(oC) versus t, demonstrating the effect of )x, ^hi = |Xh2 and ai2
572
5.2-3(3) - Heat transfer in impinging streams Impinging streams were thoroughly treated in chapter 4.5 when studying the
RTD of such systems. The flow system below comprises of four reactors three of
which are equipped with heat exchangers.
in,l
M4
obi-^Mobi " ^ rdo j = l ^ = 2 b = 3
\ l ' 0,1 ^0,2 T Q 2
5=4
T T ^03 ^0.3
4n,3
34
Fig.5.2-3(3). The impinging-stream flow system
The configuration in Fig.5-1 is reduced to that in Fig.5.2-3(3) by choosing
reactors j , a, b and ^, designated asj = l , a = 2, b = 3 and ^ = 4, respectively.
From Eqs.(5-18a) to (5-18c) for a'k = 0, it follows: 1 + a3 = P14 + P34; 1 + a2i =
Pl4 + OC12; a i 2 + OL32 = OC21 + a23; 0C3 + a23 = ^32 + P34. From symmetry
considerations ai2 = CX21 = a23 = CX32 = oc ; thus, P14 = P34 = a3 = a i = 1 while
taking Qi as reference flow.
From Eq.(5-la) the state space reads:
SS = [hin,i, hin,3, hi, h2, h3, h4]
and from Eq.(5-3a) the state vector reads:
S(n) = [hin,i, hin,3, hi(n), h2(n), h3(n), h4(n)]
(5.2.3(3a))
(5.2-3(3b))
where S(n+1) is given by Eq.(5-4). The probability matrix given by Eq.(5-27) is
reduced to:
573
hin,!
hin3
hi
h2
h3
h4
hin.l
1
0
0
0
0
0
hin3 0
1
0
0
0
0
hi
Pin,14i
0
P l U
P21J1
0
0
h2
0
0
Pl2Ji
P22Jh
P3241
0
h3
0
Pin3Jh
0
P23J1
P33J1
0
h4
0
0
Pl44i
0
P344i
P44ai
p =
(5.2-3(3c))
From Eqs.(5-28b), (5-29b) and (5-30b) for j = 1, i = a = 2, i = b = 3 and | =
4, assuming that the density of the fluid flowing in the system remains constant,
follows that:
Pin.lJi = ^ At p i u = 1 - (1 + ai2)mAt p2i4i = a2i|AiAt
Pl2Ji = a l2^2At P224i = 1 - (a2i + a23)^2At P32Ji = a32M-2At
Pin34i = «3^ At P334, = 1 - 034 + a32)^3At P234i = a23^3At
Pl4Ji=Pl4M4At p444,= l-M4At P34Ji = P34M4At (5.2-3(3d))
LfiMn) = l hi P.i Tj (n)At = li .Cp.; Ui.o - Ti(n)]At i = 1,2,3 {5.2-3(3e))
In addition:
hi(n+l) = l^,ipin,i ji + hi(n)pii J, + h2(n)p2i4i + Lfi j,(n)
h2(n+l) = hi(n)pi2ji + h2(n)p224i + h3(n)p32ji + Lf2ji(n)
h3(n+l) = lUn3Pin3Ji + h2(n)p234i + h3(n)p334i + LojiCn)
h4(n+l) = hi(n)pi4ji + h3(n)p344, + h4(n)p444i (5.2-3(30)
The following substitution is made for replacing enthalpes by temperatures,
.e. Ah = CpA; thus,
hi(n+l) = hin,i + Cp{Ti(n) - Ti„.i} i = 1,..., 4 (5.2-3(3g))
574
Substitution of the above expression into Eq.(5.2-3(2f)), assuming that Cp,i = Cp,
hin,l = hin,3 and applying the probabilities in Eq.(5.2-3(3d)) as well as that ai2 =
0C21 = a23 = 0 32 = cx, yields:
Ti(n+1) = Ti(n)[l - { ihi + (1 +a)^i}At] + T2(n)a^iAt + [|LiiTin,i + JXhiTo,i]At
T2(n+1) = Ti(n)a^2At + T2(n)[l - { h2 + 2a|Li2}At] + T3(n)aji2At
+ ^h2To,2At
T3(n+1) = T2(n)a|Li3At + T3(n)[l - {|Xh3 + (1 +a)|i3}At] + [^3Tin,i + ^h3To,3]At
T4(n+1) = Ti(n)^4At + T3(n)H4At + T4(n)(l - |ii4)At (5.2-3(3h))
To obtain the last expression, one should take hin,i = CpTin,i stemming from
Eq.(5.2-3(3g)). Eqs.(5.2-3(3h)) make it possible to calculate the temperature
distributions versus time in reactors 1, 2, 3 and 4. In the numerical solution it was
assumed that |ii = |i2 = |13 = ^4 = M'. The initial state vector S(0) = [hinj, hi(0),
h2(0), h3(0), h4(0)], in terms of temperatures, reads S(0) = [Tinj, Ti(0), T2(0),
T3(0), T4(0)] = [SO^C, 250C, 25oC, 25oC, 25^Cl Other common parameters
were: Ti,o = T2,o = T3,o = lO^C and At = 0.0005. The transient response of
Ti(oC), T2(^C), T3(oC) and T4(oC), i.e. the temperatures in reactors 1, 2, 3 and 4,
is depicted in Fig.5.2-3(3a). The effect of |i = 0, 10, 100 is demonstrated in cases
b, c and d; the effect of a = 0, 10 in cases a and b and of |ihi = V^hi = ^h3 = 0» 10»
100 in cases b, e and f.
575
60
50
G40
^ 3 0
20
10
-
-
a = 0, i = 10, \i^= 10
t i l l 4 . - • • •
i = l , 3
" " - • - . 2
1 1 1 1
(a^
H
-\
-
~
—. '
\~
r
a = 1
1
10, i = 10, \3i^ 10
1 1 1
4. . . - - - - • -
1,3
2
1 1 1
(bj
1
J
A H
0 0.1 0.2 0.3 0.4 0. t
5 0 0.1 0.2 0.3 0.4 0.5 t
30
25
^ 2 0
u o
H"l5
10
a = 1
1
10, i = 0, i =
1 1
i = 4
- . J^ ,3
1 1
10
1
L
(c)
1
-
r\£\ UU
80
60
40
on
-
- '
a = 10, |i = 100, j = 10
1 1 1 1
1 . 2 , 3
1 1 1 1
(ci)|
-
-
0.1 0.2 0.3 0.4 0.5 0 0.02 0.04 0.06 0.08 0. t t
100
^ 8 0
u o ^•"60
40
on
-
a = 10, ^ = 10, ji = 0
1 1 1 1 i - 4
1 , 2 , 3
1 1 1 1
(e2|
H
-
0 0.2 0.4 0.6 0.8 t
30
25
20
15
10 1 0
a = 10, ^ = 10, u = 100 h
r' Lv
1 1
4
1. 3
(f)
-1
1 1 2 , 0.1 0.2 0.3
Fig.5.2.3(3a). Ti(oC), T2(«C),T3(^C) and T4(«C) versus t, demonstrating the effect of |i, a and |ihi = |Lih2 = Hh3
576
5,2-3(4) - Concentration of solutions The concentrator is a 4-stage system where evaporation of the solution takes
place in reactors 1, 2 and 3. The inlet concentration of the solution is Cijn,!-
Fig.5.2-3(4). The concentrator
The configuration in Fig.5-1 is reduced to that in Fig.5.2-3(4) by choosing
reactors j , a, b and ^, designated a s j = l , a = 2, b = 3 and ^ = 4, respectively.
From Eqs.(5-18a) to (5-18c), for a'k > 0, it follows:
1 = p34 + cx'i + oc'2 + a'3 1 = ai2 + oc'i ai2 = a23 + a 2 a23 = P34 + oc'3
where a'l = Q'i/Qi and taking Qi as reference flow. It assumed that a'i = a' (i =
1, 2, 3), thus:
a i2 = 1 - a' a23 = 1 - 2a' P34 = 1 - 3a'
From Eq.(5-la) the state space reads:
SS = [Ci,in,h Cii , C12,, Ci3, C14]
From Eq.(5-3a) the state vector reads:
S(n) = [Ci,in,l, Cii(n), Ci2(n), CnCn), Ci4(n)]
(5.2-3(4a))
(5.2.3(4b))
(5.2-3(4c))
where S(n+1) is given by Eq.(5-4). The probability matrix given by Eq.(5-27) is
reduced to:
577
l.in.l
Cu
Cl2
Cl3
Ci4
Cl,in,l
1
0
0
0
0
Cii
Pin.l
PU
0
0
0
Cl2
0
P12
P22
0
0
Cl3
0
0
P23
P33
0
Ci4
0
0
0
P34
P44
p =
(5.2-3(4d))
From Eqs.(5-28a) to (5-30a), assuming C'14 = C14, i.e. no accumulation in reactor
4, noting that Lfi(n) = 0, i = 1, 2, 3, it follows:
Cii(n+1) = Ci,injPinj + Cii(n)pii Ci2(n+1) = Cii(n)pi2 + Ci2(n)p22
Ci3(n+1) = Ci2(n)p23 + Ci3(n)p33 Ci4(n+1) = Ci3(n)p34 + Ci4(n)p44
(5.2-3(4e))
Pin,i = m At pii = 1 - ai2 [i\M P12 = ai2|i2At
P22 = 1 - OC23 ^2At P23 = a23l 3At
P33 = 1 - P34 3At P34 = p34NAt P44 = 1 " P34 4At (5.2-3(4f))
Eqs.(5.2-3(4e)) and (5.2-3(4f)) make it possible to calculate the concentration
distributions of the salt versus time in reactors 1 to 4. In the numerical solution it
was assumed that 1x1 = 112 = ^3 = ^4 = M- The initial state vector S(0) = [Cijn,!,
Cii(O), Ci2(0), Ci3(0), Ci4(0)] = [1, 0, 0, 0, 0] in cases a to c in Fig.5.2-3(4a)
and [0, 1, 0, 0, 0] in case d; At = 0.005. The transient response of Cu (i = 1, 2, 3,
4), i.e. the salt concentration in reactors 1 to 4, is depicted in Fig.5.2-3(4a). The
effect |Li = 1, 10 is demonstrated in cases a and b; the effect of a = 0.1,0 in cases b
and c. The effect of the initial concentration in reactor 1 is demonstrated in case d.
578
1= 1, a = 0.1, C =1 l.in.l
U
1.5r
1
0.5
0
C.(0) = 0(1 = 1,2,3,4) ii
1 1 1 1
.-'^ - \ 1 = 1 . - . - - ^ ' ' - " /
_ / , ' 2 / 3 .'4
/ / / / / / , ' L i i j i \ L
A
(a)
4 6 t
8 10
i = 10, a = 0, C =1 l.in.l
C (0) = 0 (i = 1,2,3,4)
U
1.5
1
0.5
0
li
1 1 1
i = l
1 /l/l.\
/ / • ' / • •
L - : 1 L
(c)
—
|x = 10, a = 0.1, C =1 l.in,l
C_.(0) =
1
= 0( i =
JL '
= 1,2,3,4)
J .
2 /• 1
h / ^ • / / ' / 3 / 4
n^// -' / / ' / . ' ' \:J- 1 1
1
1
-j (b)
A
0.5 1.5
i = 10, a = 0.1, C =0 l.in.l
C(0) = 1,C.(0) = 0(1 = 2,3,4) 11 ii
0 0.5 1 1.5 0 0.5 1 1.5 t t
Fig.5.2-3(4a). Cn (i = 1, 2, 3, 4) versus t, demonstrating the effect of |LI and a
5>2-3(5) - Electrolysis of solutions-model A [86] The following scheme was suggested as a possible network model to
describe a real electrolytic processes. Reactors 1 and 2 are continuous-flow stirred-tank electrolytic reactors (CSTER), reactor 3 is a reactor for the recycling electrolyte and reactor 4 is collector in which no electrolytic process takes place.
579
Fig.5.2-3(5). The electrolyser flow system
The configuration in Fig.5-1 is reduced to that in Fig.5.2-3(5) by choosing
reactors j , a, b and ^, designated asj = l , a = 2, b = 3 and ^ = 4, respectively.
From Eqs.(5-18a)-(5-18c), for a'k = 0 and taking Qi as reference flow, it follows:
P34 = 1, a23 = a3i = a as well as that ai2 = 1 + OC23 (5.2-3(5a))
where a = Q23/Q1 is the recycle. From Eq.(5-la) the state space reads:
SS = [Ci4n,b Cii, C12,, Ci3, C14] (5.2-3(5b))
where Ci,in,l is the concentration at the inlet to reactor 1 of the species designated
by 1 undergoing electrolysis in reactors 1 and 2. From Eq.(5-3a) the state vector
reads:
S(n) = [Ci,in,l, Cii(n), Ci2(n), Ci3(n), Ci4(n)] (5.2-3(5c))
where S(n+1) is given by Eq.(5-4). The probability matrix given by Eq.(5-27) is
reduced to:
P =
l.in,l
Ci i
C12
Cl3
Ci4
Cl,in,l
1
0
0
0
0
Cii
Pin.l
Pll
0
P31
0
C12
0
P12
P22
0
0
Cl3
0
0
P23
P33
0
Ci4
0
0
P24
0
P44 (5.2-3(5d))
580
From Eqs.(5-28a)-(5-30a), assuming C'i4= C14, i.e. no accumulation in reactor 4 of species 1, it follows:
Cii(n+1) = Ci j[n,l(n)pin,l + Cii(n)pii + Ci3(n)p3i + Lii(n)
Ci2(n+1) = Cii(n)pi2 + Ci2(n)p22 + Li2(n)
Ci3(n+1) = Ci2(n)p23 + Ci3(n)p33 Ci4(n+1) = Ci2(n)p24 + Ci4(n)p44
where (5.2-3(5e))
Lii(n) - ^iijACu i(n)At« finjtC*! - Cii(n)]At- ji^ [0 - Cii(n)]At
Li2(n) - ^i2.iACi2,i(n)At- |ii2.i[Ci2 " C^2^^)]M^ Hj .i O - Cj2(n)]At
mi. i = kMAiA i (112,1 = kMA2A 2 (5.2-3(50)
Wp.i (1/sec) are mass transfer coefficients for the transfer of solute f in process p (=
1 to designate an electrolytic process) from the bulk of the solution to the electrode
in reactor i. f = 1, p = 1 and i = 1 means the mass transfer coefficient for the
transfer of solute 1 in process 1 in reactor 1; f = 1, p = 1 and j = 2 stands for the
mass transfer coefficient for the transfer of solute 1 in process 1 in reactor 2. ku is
the mass transfer coefficient in m/sec, Ai and Vi are, respectively, the electrode area
and the effective electrolyser volume. It is assumed [86] that the electrolytic
process takes place under limiting current conditions, i.e. the solute concentrations
on the surface of the electrode, C*ii = C*i2 = 0. The probabilities are as follows:
Pin,l = JA At Pll = 1 - ai2 mAt pi2 = ai2M'2At
P22 = 1 - ai2fA2At P23 = a23^3At p24 = P24M4At = M4At
R33 = 1 - 031 wAt P31 = a3iM4At P44 = 1 - M4At (5.2-3(5g))
where 031 = 023 = Q23/Q1 = a is the recycle. Eqs.(5.2-3(5e)) to (5.2-3(5g)) make
it possible to calculate the concentration distributions of Cu versus time in reactors i
= 1, 2, 3 and 4. In the numerical solution, fully described by a, [ii, M-n.b ^11.2
and dt, it was assumed that m = ^2 = M3 = M4 = M** 1^^ ^ ^^ ^^ calculations
were based on ref.[86]. The initial state vector S(0) = [Ci^n,!, Cii(O), Ci2(0),
581
Ci3(0), Ci4(0)] = [0, 0.04, 0, 0, 0] in cases a to e in Fig.5.2-3(5a), [0.04, 0, 0, 0,
0] in cases f and g and [0.04, 0.04, 0, 0, 0] in case h; At = 0.1. The transient
response of Cn (i = 1, 2, 3,4) in reactors 1 to 4, is depicted in Fig.5.2-3(5a). The
effect of the recycle a = 0.5, 5 is demonstrated in cases a and b ; the effect of |i =
0.002, 0.02, 0.2 in cases a, c and d; the effect of the mass transfer coefficient
Hi 1,1 = Hi 1,2 = 0.0094, 0.2 in cases b and e as well as in cases f and g in which
^11,1 = ^11,2 = 0.0094, 0, respectively. Note that cases g and h demonstrate
absence of an electrolytic process. The effect of the location of the introduction of
species Ai into the system is shown in cases g and f. In cases a to e, Ai was
introduced initially into reactor 1; in cases f and g it was introduced into the inlet of
reactor 1 where the initial concentration in this reactor as well as in the others was
zero. Only in case h, Ai was introduced both in reactor 1 and continuously at the
inlet to it.
u
o.osp
0.021
0.01
0|
0
p
)
k
Y
a = 0.5, i = 0.02
C =0,C (0) = 1.in.1 i r '
\ l i = ll
1 2 . \
14. - \ > S l iiL
., u = 0.0094 11.1
0.04,C .(0) = 0 (i = 2,3,4)
^^:-^_--:~.. ^ .
1
(aj
-
,—
(b^ a = 5, ^ = 0.02, i = 0.0094 42J 11.1 '
C =0,C (0) = 0.04,C (0) = 0 i.in,i i r ' i r '
0 = 2.3,4)
50 100 150 0
0.04
0.03
0.02
0.01
0
- \
-
-/*
a = 5,
C = 1.in.1
\ l i =
12
13_
^ =
= 0,C
11
_ —
0.002,
,(0) =
i = 0.0094 ^ J 11.1
0.04,C .(0) = 0
(1 = 2,3.4) 1
H
14 A
1 - 50 100 150
a = 5, ^ = 0.2, i = 0.0094 ^^1
(i=2,3,4)
582
u
0.04
0.03
0.02
0.01
0
a = 5, n = 0.02, i = 0.2 ^q
\ C =0,C (0) = 0.04,C (0) = 0 \ i.in.i ' i r ' ' ir ' J \ (i = 2,3,4) 1
- \ i l = ll J
12 \v...,^^^ 14
'— . — . - - • 1 - |- • • • - _ 10
0.025,
0.02
0.015
0.01
0.005
^ 0
15 0
(f)
a = 5, ^ = 0.02, i = 0.0094 J y • 11.1
I; C =0.04, C (0) = 0 (i = 1^) I pi' 1,in,l ' r ' ^ ' - I
_J \ \ I 100 200 300 400 500 t
Op-
0.035 0.03
0.025
0.02
0.015
0.01 0.005
0 0 100 200 300 400 500 600 -100 0 100 200 300 400 500 600
t t
Fig.5.2-3(5a). Cii (i = 1, 2, 3, 4) versus t, demonstrating the effect of a, |LI, |LIII,I and the introduction location of species Ai
C . =0.04,C.(0) = 0 (i = 1-4) 1,in,1 1i
_L I \ \ I
(h)-^
a = 5, ^ = 0.02, Li = 0 n 11,1 '
C =C (0) = 0.04,C (0) = 0| i.in.i i r ' i r ' - j
(i = 2.3,4) I I I 1 r ^
5>2-3(6) - Electrolysis of solutions-model B [86] In this configuration, a CSTER is imbedded between two perfectly-mixed
reactors 1 and 3 in the forward loop. As in model A above, electrolyte recycling is represented by a perfectly-mixed reactor 4 in the feedback loop shown in Fig.5.2-3(6). Electrolysis takes place in reactor 2 and the collector is reactor 5.
Fig.5.2-3(6). The electrolyser flow system
583
From Eqs.(5-18a) to (5-18c), for a'k = 0 and taking Qi as reference flow, it
follows:
P35 = 1, OC12 = a23, CX34 = (X41 = a and ai2 = 1 + QLU (5.2-3(6a))
where a = Q34/Q1 is the recycle. From Eq.(5-la) the state space read:
SS = [Ci,in,b Cii, C12,, Ci3, Ci4, C15] (5.2-3(6b))
where Ci^n,! is the concentration at the inlet to reactor 1 of the species designated
by 1 undergoing electrolysis in reactor 2. From Eq.(5-3a) the state vector reads:
S(n) = [Ci,in,l, Cii(n), Ci2(n), Ci3(n), CuCn), Ci5(n)] (5.2-3(6c))
where S(n+1) is given by Eq.(5-4). The probability matrix given by Eq.(5-27) is
reduced to:
Cl,in,l Cii C12 Ci3 Ci4 Ci5
Cl,in,l
Cll
C12
Cl3
Ci4
Cl5
1
0
0
0
0
0
Pin.l
Pll
0
0
P41
0
0
P12
P22
0
0
0
0
0
P23
P33
0
0
0
0
0
P34
P44
0
0
0
0
P35
0
P55
p =
(5.2-3(6d))
From Eqs.(5-28a)-(5-30a), assuming C'14 = C14, i.e. no accumulation in reactor 5
of species 1, it follows:
Cii(n+1) = Ci,i„,i(n)pi„,i + Cii(n)pii + Ci4(n)p4i
Ci2(n+1) = Cii(n)pi2 + Ci2(n)p22 + Li2(n)
Ci3(n+1) = Ci2(n)p23 + Ci3(n)p33 Ci4(n+1) = Ci3(n)p34 + Ci4(n)p44
Ci5(n+1) = Ci3(n)p35 + CisWpss (5.2-3(6e))
where
Li2(n) = jli2^iACj2 j(n)At = ^,2,i[Ci2 - Ci2(n)]At = fl,2_i[0 - Ci2(n)]At
584
12,1 = kMA2A 2 (5.2-3(6f))
|Xii,2 (1/sec) is the mass transfer coefficient for the transfer of solute 1 in process 1
(electrolysis) in reactor 2. kM is the mass transfer coefficient in m/sec, Ai and Vi
are, respectively, the electrode area and the effective electrolyser volume. It is
assumed [86] that the electrolytic process takes place under limiting current
conditions, i.e. the solute concentrations on the surface of the electrode, C*i2 = 0.
The probabilities are as follows:
Pin,i = m At pii = 1 - ai2mAt P12 = ai2|ii2At
P22 = 1 - CXl2 2At P23 = OC23M'3At
P33 = 1 - 0Ci2H3At P34 = a34^4At P35 = p35^4At = ^sAt
P44 = 1 - cx4m4At P41 = a4im At P55 = 1 - ^sAt (5.2-3(6g))
noting that a i2 = a23 = 1 + OC34 and a4i = a34 = Q34/Q1 = a is the recycle.
Eqs.(5.2-3(6e)) to (5.2-3(6g)) make it possible to calculate the concentration
distributions of Cn versus time in reactors i = 1,..., 5. In the numerical solution,
fully described by a, m, 11,2 and At, it was assumed that m = |i. The initial state
vector S(0) = [Ci,in,b Cii(O), ..., Ci5(0)] = [0, 0.04, 0, 0, 0, 0] in cases a to d in
Fig.5.2-3(6a) and [0.04, 0, 0, 0, 0, 0] in case e and f; At =0.1, 0.5 in cases a to d
and e, f, respectively. The transient response of CH (i = 1, 2, 3) in reactors 1, 2,
3, is depicted in Fig.5.2-3(6a). The effect of the recycle a = 0.05, 5 is
demonstrated in cases a and b; the effect \i = 0.02,0.2 in cases b and c; the effect
of the mass transfer coefficient ii 1,2 = 0.0094 and 0.2 in cases c and d as well as e
and f. The effect of the introduction location of species Ai , i.e. the inlet
concentration to reactor 1, Cijn,! = 0.04, while the initial concentration in all
reactors is zero, is demonstrated in cases e and f for |i 11 2 = 0.0094, 0 which
demonstrate again the effect of |Lii 1,2. Note that in case f no electrolysis takes place
in reactor 2 indicated by |Lii 1,2 = 0.
585
u
(a) a - 0.05, ^ - 0.02, ji - 0.0094
11,2
C = 0, C (0) = 0.04. C = 0 l.in.l 11 li
(i = 2-5)
h-
\ l
\ns^
h ^
» « (bj a - 5, ji - 0.02, \i - 0.0094 |
11,2
C = 0, C (0) = 0.04, C = 0 J i.iii.i i r ' li n
(i = 2-5)
1 1 H 150 0 50 100 150
0.04
0.03
^0.02
0.01
' ' (c)|
a - 5, fi - 0.2, \i - 0.0094 11,2
C = 0, C (0) = 0.04, C = 0 J l.in,l 11 li ^
(i = 2-5)
' ' (d)|
a - 5, ji - 0.02, n - 0.2 11,2
C = 0, C (0) = 0.04, C = 0 J i.in.i i r ' li '
(i = 2-5)
15 0
U
0.04
0.03 -0.02
0.01
-
"" F"
/ ^ jr a -
C
l,in.
1
5
i""
•• 1 1
^^ . ^
H m 0.02, n -11.2
0.04. C = 0 (i =
li
1 i
' (e)|
i = l j 2, 3"1
- j 0.0094
1-5) H
1
i ' 1 « ( f ^
^fn'"^ 1, 2, 3
ly
/ a - 5, ji - 0.02, M. - 0 11,2
C =0.04. C = 0 (i = l-5) 1 l.in.l li
i i i , i , . 0 200 400 J 600 800 1000 0 200 400 600 800 1000
Fig.5.2-3(6a). C n (i = 1, 2, 3) versus t, demonstrating the effect of
a, |i9 M*! 1,2 And the introduction location of species Ai
5.2-3(7) - Simultaneous dissolution, absorption and chemical reaction
In the following, simulation is carried out of a combined process
incorporating dissolution, absorption of species f = 1 takes place into the solution
as well as reacting with species 2 arriving from reactor 2 at flow rate Q12 according
to
586
Ai + A2 ^ A3 for which - n = -12 = 13 = k2CiC2 (5.2.3(7a))
The feed to reactors 1 and 2 may contain species 1 and 2 at concentrations Cf,in,i
and Cf4n,2 where f = 1,2.
Qi
Q2
ri r\ 1
j = l absorption + reaction
i
1 ki db
a: dissol
= 2 ution
^13 do ^ - 3
Fig.5.2-3(7). Flow system for the simultaneous dissolution,
absorption and chemical reaction
It is assumed that the quantities dissolved and absorbed do not change the
flow rate Qi + Q2. From Eqs.(5-18a) to (5-18c), for a'k = 0 and taking Qi as
reference flow, it follows:
P i3= l and a 2 i = a 2 = Q2/Qi.
where a = Q2/Q1. From Eq.(5-la) the state space reads:
SS = [Cf,in,l, Cf,in,2, Cfi, Cf2, Cfs] f = 1, 2
From Eq.(5-3a) the state vector reads:
S(n) = [Cf,in,l, Cf,in,2, Cfi(n), Cf2(n), Cf3(n)] f = 1, 2
S(n) = [Cf,in,l, Cf,in,2, Cfi(n), Cf2(n), Cf3(n)] f = 1, 2
(5.2-3(7b))
(5.2-3(7c))
(5.2.3(7c))
where S(n+1) is given by Eq.(5-4). The probabiUty matrix for f = 1, 2, given by
Eq.(5-27), is reduced to:
587
Cf,in,l Cf,in,2 Cfi Cf2 Co
Cf,in.l
Cf,in,2
P = Cfi
Cf2
Cl3
1
0
0
0
0
0
1
0
0
0
Pin,l
0
Pll
P21
0
0
Pin,2
0
P22
0
0
0
P13
0
P33 (5.2-3(7d))
From Eqs.(5-28a) to (5-30a), assuming C'o = Cf3, i.e. no accumulation of the
species in reactor 3, it follows for f = 1,2 that:
Cfi(n+1) = Cf,in,l(n)Pin,l + Cfi(n)pii + Cf2(n)p2i + Lfi(n)
Cf2(n+1) = Cf,in,2(n)pin,2 + Cf2(n)p22 + Lf2(n)
Cf3(n+1) = Cfi(n)pi3 + Cf3(n)p33 (5.2-3(7e))
where
Lfi(n) = [Hfj jACfi j(n) + rfj(n)]At
Lf2(n) = if2.2ACf2,2(n)At (5.2-3(7f))
In |Xfp,i the following designations are applicable: f = 1, 2 indicate species 1 and 2,
respectively; p = 1, 2 stand for processes of absorption and dissolution,
respectively; i = 1, 2 indicate reactors 1 and 2, respectively. Thus,
L,,(n) = [^ij jACji i(n) + ri,(n)]At = [^l,,,(C*, - CjjCn)) - kjCijCnKji (n)]At
LjiCn) = [r2i(n)]At = - kjC, j(n)C2i (n)]At
Li2(n) = 0
L22(n) = \l22^2^C22^2^TlW = 22,2 22 " C22(n)]At (5.2-3(7g))
The probabilities are as follows:
Pin,l=|ilAt Pin,2 = a2mAt pii = l-M,iAt p2i = a2imAt
P22 = 1 - a2m2At P13 = |X3At P33 = 1" m^i (5.2-3(7h))
588
noting that a2i = a2. Eqs.(5.2-3(7e)) to (5.2-3(7h)) make it possible to calculate
the concentration distributions of Cfi versus time in reactors i. In the numerical
solution, fully described by a2, m, M-ii,!, 22,2* 2, C*ii, C*22 and At, it was
assumed that |Xi = p,, iin j = 1x22,2 = 1. In addition, C*ii = C*22 = 1 in cases a to
e and C*ii = C*22 = 0 in case f. The initial state vector S(0) = [Cfjn,b Cf,in,2»
Cfi(O), Cf2(0), Cf3(0)] = [0, 0, 0, 0, 0] for f = 1, 2 in cases a to e in Fig.5.2-3(7a)
and [0, 0, 1, 0, 0] for f = 1 and [0, 0, 0, 1, 0] for f = 2 in case f; At = 0.002. The
effect of |i = 0, 1, 10 is demonstrated in cases a, b and c; the effect of k2 = 1, 10 in
cases b and d; the effect of a2 = 1, 10 in cases b and e. In case f, no dissolution or
absorption take place; however chemical reaction occurs since the initial
concentrations of A1 in rector 1 and of A2 in reactor 2 were unity. The latter
species was transferred by the flow to reactor 1 undergoing there a chemical
reaction.
589
C , =C , = 1, C (0) = 0 (i = 1-3, f= 1, 2) 11 22 f 1
U
1
0.8
0.6 a
0.4
0.2
0
Li = 0, a = 2
1 - -r
f//" /
r 1 1
= l , k = 1 2
1
' (^
-
—
0
u
1 4
1.2
1
0.8
0.6
0.4
0.2
0
-
1
/
1
/
---
1= 1, a = 10,k =1 ^ ' 2 2 1 i 1
2 V - ^ /
fi = l l
_12
1 1 1
_ ^\
-J
-
- • -=
0
C =C =1,C (0) = 0 0=1-3, f= 1,2) 11 22 fi ^ '
H = 0, a = 1, k =1 2 2 1 1
22
11 ^ —
" / / ^ ' ' 7 / / ' " J/ / r 1 1
' (b)
—
-
1
= C =1,C (0) = 0 ( i= l -3 , f= l ,2 )
^ = l .
1
2 2 _ _
" /
a = 2
1
I L
1, k =10 2
1
" ^
1
(d)
—
0 1 2 3 4 t
C* =C* =0,C (0) = C (0)=1,C(0) = 0 11 22 ir 22 fi
Fig.5.2-3(7a). Cn, C21 and C22 versus t, demonstrating the effect of |Li, k2, and a2
590
NOMENCLATURE
a, b , ..., Z designation of peripheral reactors in Figs.4-1 and 4 - l a .
aj * stoichiometric coefficients of species j in the /th reaction,
aj stoichiometric coefficients of species j .
hi* ^hj heat transfer area to reactors i and j , m^.
api, apj mass transfer area for process p corresponding to conditions
prevailing in reactors i or j , m^.
A, B ZxZ square matrices given by Eq.(2-34).
A^ = fk designates a system with respect to its chemical formula f and
location k in the flow system. Aj = i designates the state of the system (a molecule), i.e. a specific
chemical formula (Chapter 3).
Ai(t) = Ai concentration of a chemical species i at time t (Chapter 2).
Ai(0) initial concentration of species i.
ajk» bjk elements of the matrices A, B.
CA(t), CAO concentration of A at time t and t = 0, respectively.
Cj Chapter 4: concentration of the stimulating input in reactor j in
moles/(m^ reactor); concentration of species j , moles of j/m^» also
designating the state of the system.
Cj(n), Cj(n+1) concentration of species j in the mixture at time interval t and t + At
or step n and n+1, respectively.
C(n+1) concentration vector at step n+1.
Cj concentration of species j in reaction i.
C*2 equilibrium concentration of species 1 absorbed on the surface of the
liquid in reactor 2 corresponding to its partial pressure in the gas
phase above the Uquid.
C* J as above, but for species 1 in reactor 1.
Cfi, Cfj concentration of species f in reactors i or j , kg or kg-mol/m^.
C'fi. C'fj concentration of species f leaving reactors i or j , kg or kg-mol/m^.
These concentrations are not, in general, equal to the concentrations
Cfi, Cfj in the reactor.
591
Cfi(n)
C'fi(n)
C'f^(n)
C'4
Cf,in,j
Cf,in,i
CSTER
Di(%)
I-\ne
Dmax
E
eq.
exact
f
fi, fj
concentration of species f in reactor i at time t or at step n, kg or
kg-mol/m^.
concentration of species f leaving reactor i at time t or at step n, kg or
kg-mol/m^.
concentration of species f in reactor ^, kg or kg-mol/m^.
concentration of species f leaving reactor ^, kg or kg-mol/m^.
This concentration is not, in general, equal to the concentrations
concentration of species f leaving reactor ^ at time t or at step n, kg
or kg-mol/m^.
concentration of the tracer in reactor ^, kg or kg-mol/m^.
concentration of the tracer leaving reactor ^, kg or kg-mol/m^.
This concentration is not, in general, equal to the concentrations
concentration of species f at the inlet to reactor j , kg or kg-mol/m^.
concentration of species f at the inlet to reactor i, kg or kg-mol/m^.
specific heat of the fluid mixture in reactor j , kcal/(kg K); similarly
for reactor i.
continuous-flow stirred-tank electrolytic reactor
deviation in the concentration for the /th pair of data between the
exact solution and Markov chain solution , i.e.,
Di(%) = 100ICexact,i " CMarkov,il/Cexact,i-
mean deviation defined by
'mean )ZPi ^) = (1/H) / D (%) where H is the number of pairs.
i=l considered in the comparison.
maximum value of Di(%).
age distribution defined by Eq.(4-20).
at equilibrium.
exact solution
running index, 1, 2,..., F, for designating the different species.
subscript; designating species f in reactors i or j .
592
fjj probability that, starting from state Sj, the system will ever pass
through Sj. Defined in Eqs.(2-95).
fjk probability that, starting from Sj, the system will ever pass through
Sk. Defined in Eqs.(2-98).
probability that state Sj is avoided at steps (times) 1, 2, . . . , n - 1
where re-occupied at step n. Defined in Eqs.(2-94).
fjk(n) indicates that Sk is avoided at steps (times) 1,..., n-1 and
occupied exactly at step n, given that state Sj is occupied initially.
Defined in Eqs.(2-97).
total number of species or reactants.
specific enthalpy of the fluid in the feed vessels to reactor i, kcal/kg.
specific enthalpies of the fluid in reactors i, j and in collector ^,
kcal/kg or kg-mole.
specific enthalpies of the fluid in reactors i and in collector ^ at time t
or step n, kcal/kg or kg-mole.
specific enthalpies of the fluid leaving reactors i, j and ^, kcal/kg or
kg-mole.
H the rate of supply of Ai in moles/sec from the vapor phase (Ai)gas
into the condensed phase,
j , k integers designating states j and k, respectively.
J, K total number of cities in the extemal circle and in the intemal city,
respectively,
kj reaction rate constant with respect to the conversion of species j in
the /th reaction (in consistent units),
k, kj reaction rate constant (Chapter 3).
kij reaction rate constant for the conversion from state i (species A[) to
state j (species Aj).
kfp,i, kfp j mass transfer coefficients for process p with respect to species f
corresponding to conditions in reactors i and j , m/s.
khi, khj heat transfer coefficient corresponding to the conditions in reactors i
andj, kcal/(sm2K).
kM mass transfer coefficient, m/s.
LHS left hand side.
fjj(")
fjk(n)
F
hin,i hi, hj, h^
hi(n), h^(n)
h'i, h'j, h'
593
L(n) a mathematical expression corresponding to time t or step n.
n designates step n in discrete Markov chains.
NA(t), NAO number of moles of A at time t and t = 0, respectively.
No total number of inhabitants in the state.
Nj(t) number of inhabitants occupying state j (an external city j) at time t.
Nk(t) number of inhabitants occupying state k (an internal city k) at time t.
N® number of reacting species in reaction i.
p(y, T, X, t) probability density function, i.e. probability per unit length.
P(y, T, X, t) transition probability function defined by Eq.(2-185).
p, q constant one-step transition probabilities.
p subscript designating some transfer mechanism such as absorption,
dissolution, etc., or simultaneously several processes. It is assigned
arbitrarily numerical values such as absorption - 1 , dissolution - 2.
P total number of transfer mechanisms.
Pin4> Pin j single step transition probabilities from the state of the feed reactors
(to reactors i and j) to the state of reactors i and j .
Pini h single step transition probability with respect to enthalpy (or
temperature) from the state of the inlet reactor (to reactor i) to the
state of reactor i; similarly for reactor j , i.e., p • j .
p-j ij single step transition probability to remain in the state of reactor j
with respect to enthalpy (or temperature). Py jj single step transition probability with respect to enthalpy (or
temperature) from the state of reactor k to the state of reactor j .
Pj i jj single step transition probability with respect to enthalpy (or
temperature) from the state of reactor k to the state of reactor i. Pj • h single step transition probability with respect to enthalpy (or
temperature) from state of reactor j to state of reactor i.
Pjj Yi single step transition probaHlity to remain in the state of reactor i
with respect to enthalpy (or temperature),
pk, qk one-step transition probabilities which depend on the state k.
Pjk> Pkj one-step probability or the transition probability from state j to state k
(or the opposite) in one step (one time interval) for each j and k;
594
probability of occupying state k after one step given that the system occupied state j before. Defined in Eqs.(2-13,13a) and (4-4,4-4a).
PJE' Pka transition probabilities from reactor j to , from k to , respectively.
Pjk the probability of the transition Aj -> Aj for the ith reaction.
Pjj, ptt probability of remaining in step j during one step; probability of
remaining in reactor .
Pjj(n) probability of occupying Sj after n steps (or at time n) while initially
occupying also this state.
pjk(n) n-step transition probability function designating the conditional
probability of occupying Sk at the nth step given that the system
initially occupied Sj.
Pjk(n,r) probability of occupying state k at step r given that state j was
occupied at step n.
Pjk('Cit) transition probability of a system to occupy state k at time t subjected
to the fact that the system occupied state j at time T.
Pi(t) probability that the number N(t) of events occurred (customers
arrived) is equal to i, given that the service time is t; defined in
Eq.(2-89). Probability of the system to occupy Sj at time t; defined
inEq.(2-112).
Px(t) defined in Eq.(2-119).
Po(t) probability that the system remains at x = 0, i.e. state SQ, until time t.
P one-step transition probability matrix defined by Eq.(2-16).
P(n) a probability matrix containing the elements pjk(n). Defined in
Eq.(2-31).
prob{ Sj} probability of observing event Sj. Probability of occupying state Sj.
prob{ Sk I Sj} conditional probability. Probability of observing an event Sk under
the condition that event Sj has already been observed. Probability
of occupying state Sk under the condition that state Sj has already
been occupied,
prob {SkSj} probability for the intersection of events Sk and Sj or probability
of observing, not simultaneously, both Sk and Sj.
qj(t) rate or intensity function indicating the rate at which inhabitants
leave state Sj (extemal city j), 1/s.
595
Qjk(0 transition probability of the inhabitants from Sj to occupy Sk at
time t.
Qi» Qj volumetric flow rate of the fluid entering reactors i and j , Figs.4-1
and 5-1, m^/s.
Qi» Qj volumetric flow rate of the fluid leaving reactors i and j , Fig.5-1,
m % . This flows do not contain any dissolved material.
Qji and Qy interacting flows between reactors (states) j and i or i and j ,
respectively, Fig.4-1, m^/s. Similarly for Qki and Qik.
Qp external flow into reactor P , m^/s.
Or external flows into the reactors, r = j , a, b , . . . , Z, m^/s.
qi^ volumetric flow rate from reactor (state) i to ^ , Fig.4-1, m^/s.
Similarly, qj^, qk^ and qp^.
Tj = dCj/dt rate of change by reaction of the concentration of species j per unit
volume of fluid in reactor.
rj ^ rate of change by reaction at time t of the concentration of species j in
the ith reaction per unit volume of fluid in reactor,
rj (n) rate of change by reaction at step n of the concentration of species j
in the /th reaction per unit volume of fluid in reactor,
r . reaction rate of species f by reaction m per unit volume of reactor (or
fluid in reactor) j corresponding to the conditions in this reactor;
similarly for reactor i. (xri)
r . (n) as above, at time t or step n; similarly for reactor i.
rfj(n) rate of change at step n by all reaction of the concentration of species
f in reactor j which equals Z^^'f (^) where m = 1, ..., R.
R total number of reactions.
RHS right hand side
RTD residence time distribution.
Ri recycle ratio defined in Eq.(4-22).
Sj, Sk designate events or states j and k, respectively. S stands for state
and the subscript] designates the number of the state.
SS state space. Set of all states a system can occupy.
596
Si(n) occupation probability of state i at time n by the system. Defined in
Eq.(2-21a).
Si(0) initial occupation probability of state i by the system.
S(n) state vector of the system at time n (step n). Defined in Eq.(2-22a).
S(0) initial state vector. Defined in Eq.(2-22).
t, T designates generally time where in a discrete process t designates the
number of steps from time zero, s.
tin mean residence time of the fluid in the reactor, s. Defined in Eq.(4-
26).
mean residence time of the fluid in reactor j , s.
residence time of the fluid in a plug-flow reactor, s.
residence time of the fluid in a plug-flow reactor j , s.
source temperature from which (to which) heat is transferred into
(from) reactor j , K. Similarly for TQJ
temperatures in reactors i and j , K.
temperature at inlet to reactor j , K.
volume of reactor, m .
volume of reactor (or fluid in reactor) j or i, m .
X a prescribed value in Eq.(2-119) indicating the number of events
occurring during the time interval (0, t) = t; x indicates also a
numerical value corresponding to the state of a system, i.e. x = 0, 1,
2, ...
xo initial magnitude of the state.
X(t) a random variable describing the states of the system with respect to
time and referring to Eq.(2-8). It also designates the fact that the
system has occupied some state at time or step t. Referring to
Eq.(2-119) and the following ones, X(t) which is a random
variable, designates the number of events occurring during the time
interval (0, t).
X(n) the position at time or step n of a moving particle (n = 0,1,2, . . . ) .
Xn number of customers in the queue inmiediately after the nth customer
has completed his service.
tmj
tp
tpj
To.i
Ti,Tj
Tinj V
\ ^ i
597
Yn the number of customers arriving during the service time of the nth
customer.
Z total number of events, states, chemical species or reactors that a
system can occupy.
Z(n) size of the jump of the particle at the nth step.
Greek letters
ttji ratio between the flow from reactor j to reactor i and the total flow.
rate Qj. Similarly, ay, aki, aik defined in Eqs.(4-2), (4-6) as well
as Oji, ttkj, ocjki similarly defined.
tti, ak ratio between the flow entering reactor i or k to the total flow rate Qj.
Defined in Eq.(4-12d),
a'i, a'j defined in Eqs.(5-18), (5-18d).
P the mean number of customers being serviced per unit time,
pj^ defined in Eqs.(4-2), (4-6). Similarly, pr^ r = i, j and k. ACf I driving force for the transfer process p with respect to species f at
conditions prevailing in reactor j ; similarly for reactor i replacing
subscript]; kg or kg-mol/m^.
AC^ j(n) as above at time t or step n, kg or kg-mol/m^.
ATi = To,i - T- on reactor i; similarly for reactor j , K. ATj(n) = To,i - T.(n) at time t or step n, K.
AH^^^ heat of reaction m at conditions in reactor j , kcal/kg or kg-mol.
AHrj heat of reaction at conditions in reactor j , kcal/kg or kg-mol.
X rate factor; rate of arrival of customers or the rates events/time or
births/time.
A.i birth rate which is a function of the state Si.
X^ mean occurrence rate of the events which is a function of the actual
state x; mean birth rate.
At time interval, s.
|i , |X volumetric heat transfer coefficients defined in Eq.(5-22) for
reactors i and j , 1/s.
|lj mean recurrence time defined in Eqs.(2-96). |ij (1/sec) is a measure
of the transition rate of the system defined in Eqs.(4-3).
598
|Li defined in Eq.(4-11).
|ix death rate in Eq.(2-158). Mean occurrence rate of the events which
is a function of the actual state x. U. o tt. . volumetric mass transfer coefficient for species f, for mass transfer ^fp,i ^fpo
process of type p (for example: absorption, p = 1; desorption, p = 2; dissolution, p =3; etc.) corresponding to reactors i or j ; defined in
Eq.(5-14), 1/s.
V period.
^ subscript designating the collection reactor for the tracer. The final
reactor in the flow system symbolized as a "dead" or an "absorbing
state" for the tracer for which p^^ = 1.
limiting probabilities. Defined in Eqs.(2-105).
stationary distribution of the limiting state vector. Defined in Eqs.(2-
105a).
density of the content of reactors i, j or ^, kg/m^.
density of the streams leaving reactors i, j or ^, kg/m^.
density of the stream entering reactors i or j , kg/m^.
TCk
7C
Pi.
P'i
Pj'P^
. P'j' P'
Pin,i» Pinj
599
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