Thesis Sgen

8/12/2019 Thesis Sgen

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AGH - UNIVERSITY OF SCIENCE AND TECHNOLOGYFACULTY OF NON-FERROUS METALS

Department of Theoretical Metallurgy and Metallurgical Engineering

Ph.D. THESIS

Title: MINIMIZATION OF ENTROPY GENERATION IN STEADYSTATE PROCESSES OF DIFFUSIONAL HEAT AND MASSTRANSFER

Author: Mgr Jacek Mikoaj atkowski

Supervisor: Prof. zw. dr hab. in. Zygmunt Kolenda

KRAKW, NOVEMBER 2006


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Pozwalam sobie wyraziserdeczne podzikowanie

Profesorowi Zygmuntowi Kolendzie za cenne rady

i za kierownictwo naukowe mojej pracy doktorskiej

I would like to express my sincere gratitude to

Professor Zygmunt Kolenda, for his guidance

and support during the completion of the thesis.


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Table of Content

1. Introduction 7

2. Description of the thermodynamics of the non-equilibrium

processes of mass and energy transfer literature overview 8

2.1 Theoretical background 8

2.2 Calculation of entropy production 12

2.3 Examples of phenomenological description 17

2.3.1 Heat conduction and electric current flow 17

2.3.2 Thermal diffusion 19

2.4 Irreversible, stationary process of heat conduction 20

2.5 The principle of minimum entropy production instationary states 23

2.6 Principle of entropy production compensation 26

3. Aims of the thesis 29

4. Minimization of entropy production and principle ofentropy production compensation in steady state linear

processes of diffusional mass and energy transfer 33

4.1 Heat conduction in a plane-wall 33

4.2 Diffusional mass transfer 39

5. Entropy production minimization in stationary processes

of heat and electric current flows 42

5.1 Non-linear boundary problem without theinternal heat source 45

5.2 Boundary problem with boundary conditions

of the third kind (Sturm-Liouville boundary

conditions) without internal heat source 45

5.3 Boundary problem with simultaneous heatand electric current flows 49

5.4 Methods of calculation of additional internal

heat source and entropy generation rate 51


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6. Methods of solving the boundary problems with

Euler-Lagrange equations 61

6.1 General method of determining the internal heat

source and entropy generation rate 61

7. Solutions for the specific boundary problems 68

7.1 Simultaneous heat and electric current flows 68

7.2 Boundary conditions of the second and thirdkinds - heat conduction coefficient

independent of temperature 87

7.3 Boundary conditions of the third kind

(Sturm-Liouville boundary conditions) - heat

conduction coefficient dependent on temperature 92

8. Applications 96

9. Summary 98

Appendix 1 100

References 102


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Symbols

A - chemical affinity

A, B - integration constantsCi - concentration of i-th componentci -integration constants

deS - entropy exchange with surroundings

diS - entropy production due to irreversibility

DT - thermal diffusion coefficient

Dd - Dufour coefficient

F - Helmholtz free energy

F - thermodynamic force

F -functionG - Gibbs free energy

H - enthalpy

h - heat transfer coefficient

I,i - density of electric current

J - thermodynamic flux

Jk - mass flux of k-th component

Jq - heat flux

Js - entropy flux

Ju - internal energy flux

k - thermal conductivity coefficientkf - thermal conductivity coefficient of fluids

Lik - phenomenological coefficients

n - normal to the surface

ni - number of moles of i-th component

p - pressure

q& - heat flux

vq& - internal heat source

Q& - heat flow

Q -heat

sT - Soret coefficient

S - entropy of the system

s - specific entropy


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T - absolute temperature or dimensionless temperature

Tf -fluid temperature (absolute or dimensionless)

u - specific internal energyU - internal energyv - specific volume

V - volume

W - work

X - thermodynamic force

x, y ,z - cartesian coordinates

Greek symbols

- change

i - chemical potential of i-th component

i - stoichiometric coefficient

ik - stoichiometric coefficient of the k-th component for the i-th

chemical reaction

- extent of reaction - density

- specific electric resistivity

- entropy generation rate (entropy production)

- time

- boundary surface

- function

- electric potential


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1. Introduction

Progress of the human civilization is intrinsically connected with the use

of energy sources. Methods of harnessing the natural energy resources can be

better or worse, more or less effective, cost-conscious or unlimited. It is ratherobvious that we are trying for those methods to be better, more effective and

cost-cutting. These aims are primarily reached through the development of

scientific research and in the topics discussed in this thesis, through the

applications of methods of non-equilibrium thermodynamics. The mainproblem, widely discussed during the last decade, is the issue of diminishing

natural resources, specially the energy resources.

The problem of diminishing energy resources was first discussed in a well

known report Limits of progress (Meadows and others, 1972). The authors of

the report, using various mathematical models, formulated the hypothesis that inthe middle of the 21stcentury there will be a collapse in the development of rich

countries as the result of the overexploitation of the natural resources.

We know well today that those catastrophic predictions have not beenconfirmed. In addition to economic problems, we face today the dangers of

excessive pollution and irreversible destruction of the natural environment as the

result of increasing demand for nonrenewable energy resources. Negativeconsequences of global warming as an effect of human activity are the prime

example.

Thus, we can ask the question are the dangers of our civilizations

progress real as the effect of diminishing natural resources, especially fuel

resources. This question is based on the premise that non-renewable natural

resources are gone. However, the optimistic is the fact that natural energy

resources can be replaced by other renewable sources of primary energy or

nuclear energy, which at the current progress of technology, could become the

sources of unlimited energy.

Even, considered negative, population growth does not seem to be a threatsince each new human being is born with the ability to be creative. It seems as

well that the accumulated use of energy in the industrial countries allows for the

fast progress in technology which in turn results in a decrease of irrational use of

resources.

The following thesis is about the solutions which lead to the decrease in

the exploitation of natural resources and to limiting the dangers to the natural

environment.


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2. Description of the thermodynamics of the non-equilibrium processes ofmass and energy transfer literature overview

In this chapter the fundamental terminology of the non-equilibrium

thermodynamic processes has been presented. The expression describing theentropy production (entropy generation rate) has been derived as the measure

of intensity of the irreversible process. The principle of entropy production

compensation has been discussed as the basis for minimizing entropyproduction. The main non-equilibrium processes and their links have also been

discussed.

2.1 Theoretical background

The fundamental relationship resulting from the classical

thermodynamics, known as the Clausius inequality, describes the change of

entropy dSof the thermodynamic system in the form:

QdS

T

(2.1)

where Q is the elementary heat exchanged between the system and theenvironment and Tis the local temperature (relation (2.1) has a local character).

Relation (2.1) can also be written in the form:

0T dS Q (2.2)

Inequality sign in (2.1) and (2.2) describes the irreversible processes while the

equality is required for the idealized equilibrium or reversible processes.

Deriving , as done by T. de Donder [2], the term for non-compensated heat

Q T dS Q= (2.3)

the Second Law of Thermodynamics can be written as:

0Q


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As the result of de Donders analysis, non-compensated heat is the

measure of spontaneity of the process which consequently leads to general

definition of chemical affinityAfor the spontaneous chemical reaction:

QAd

=

where d= dni/i is called the extent of a reaction, dniis the elementary change

in number of moles of the particular component in the reaction while i is the

stoichiometric coefficient.

In case when the system is doing only the mechanical work, the formulation of

the First Law of Thermodynamics gives:

Q dU pdV = + (2.4)

where dUis the elementary change of internal energy, p is the pressure and dV

is the elementary change of volume of the system.

Combining equations (2.3) and (2.4) results in:

dU TdS pdV Q=

Introducing, from the definitions the expressions for:

enthalpy H: H = U + pV

Helmholtz free energy: F = U TS

Gibbs free energy: G = H TS,

we receive the following set of equations:

dU TdS TdV Ad

dH TdS Vdp Ad

dF SdT pdV Ad

dG SdT VdP Ad

=

= +

=

= +

(2.5)


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Expression (2.3) can be further changed into the expression:

Q QdS

T T= +

(2.6)

From the definitions

e

Qd S

T=

and (2.7)

i

Q

d S T=

(2.8)

equation (2.6) becomes:

e idS d S d S = + (2.9)

Since

QdS

T

then

edS d S (2.10)

which means that always:diS 0

(2.11)

where: diS > 0 for irreversible processes

diS = 0 for reversible processes


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From equation (2.6) we can see that eQ

d ST

= describes the exchange of

entropy between the system and the surroundings and iQ

d ST

= is the entropy

production due to irreversible processes inside the system. In the analysis ofexpression for diS, significant role is played by the local formulation of the

Second Law of Thermodynamics. It states that in every microscopic part of the

system in which irreversible processes take place, the entropy production

(entropy generation rate) diS/d, where is the time, must be positive or:

0id S

d>

(2.12)

It means as well that in a given volume, there can simultaneously take place

processes with positive and negative entropy production but in such a way thatthe positive entropy production is greater from the negative production (within

the absolute value), which insures the validity of inequality (2.12).

Differentiating equation (2.9) gives:

e id S d S dS

d d d = +

(2.13)

where dS/d is the entropy rate of change of the system as the result of the

entropy rate of exchange with the surroundings dSe/d and the entropy

production (entropy generation rate) is

id S

d

=

(2.14)

The general differential equation of the local entropy for any irreversible closed

system is:

s

dsdivJ

d

= +

(2.15)

and


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( )sds

div J svd

= + +

(2.16)

for the open system, wheresrepresents the entropy value per unit volume ,

the density of the substance, Js entropy flow (entropy flux) exchanged withthe surroundings and vis the flow velocity.

The most important element, in the analysis of irreversible processes, is thecalculation of entropy production (rate of entropy production). This value

directly describes the level of irreversibility of the given process. It is important

to mention as well that the entropy production is path dependant and should notbe confused with entropy change.

The condition:

0id S

d

=

(2.17)

directly indicates that, for example, the process along two different paths A and

B such thatA B >

(2.18)

means that path A is more irreversible than path B.

2.2 Calculation of entropy production

The method of calculating the amount of entropy production related todifferent irreversible processes, is based on the entropy balance equations. For

example, equation (2.15) yields:

s

dsJ

d

= +

r

(2.19)


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Neglecting the kinetic energy dissipation and with no external fields, Gibbs

relation:

k k

k

T ds du dn= (2.20)

where kis the specific thermodynamic potential for the k-thcomponent of the

system while dnk represents the elementary change of the amount of the k-th

component, leads to:

1 k k

k

ns u

T T

=

(2.21)

Introducing the mole-number balance equation for the amount of the k-th

component,

kk jk j

j

nJ

= +

r

(2.22)

where ik is the stoichiometric coefficient of k-th component for the j-th

chemical reaction and jis the rate of thej-threaction in volume V,

1 jj

d

V d

=

and the equation of the First Law of Thermodynamics:

0u

uJ

+ =

r

(2.23)

whereJuis the internal energy flux, equation (2.21) takes the following form:

,

1 k ku k jk j

k k j

sJ J

T T T

= +

r r

(2.24)

Introducing from definition,


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j k jk

k

A =

equation (2.24) becomes,

1 j ju k k k u k

k k j

AJ JsJ J

T T T T T

+ = +

r rr r

(2.25)

where the following property was used for the scalar functiong and vectorJ,

( ) ( )g J J g g J = + r r r

Comparing equation (2.25) with the entropy balance equation (2.19) leads to

the expression:

u k ks

k

J JJ

T T

=

r rr

(2.26)

and the expression for entropy production:

10

j jku k

k j

AJ J

T T T

= +

r r

(2.27)

In case of the external electric field with the electric field strength E = ( electric potential) resulting in the flow of electrical current with the current

density I, the following equation is derived

( )1 j jku k

k j

IJ J

T T T T

= + +

rr r

(2.28)


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The derived equation (2.28) is the general expression for the entropy production

in which four irreversible processes were taken into account: heat conduction,

mass diffusion, electric current flow and chemical reactions.

Mathematical structure of equations (2.27) and (2.28) indicates that they be bothwritten in the general form:

k k

k

F J = (2.29)

where Fk is called the thermodynamic force and Jk the thermodynamic flow,

coupled with forces. For the processes discussed above, the forces and flows are

listed in Table 1.

Table 2.1 Forces and flows in entropy generation equations

Irreversible Process Force Fk Flow JkHeat Conduction 1

T

Heat flow Ju

Diffusionk

T

Diffusion current Jk

Electric current flow E

T T

=

r

Current density I

Chemical reactionjA

T Reaction rate

1 jj

d

V d

=

Equations (2.27) and (2.28) play a fundamental role in determining the entropy

production for the processes with known thermodynamic forces and flows. In

general, the amount of entropy production is calculated from the mathematicalmodels for the processes being considered.

In case of cross effects, for example simultaneous heat conduction and electriccurrent flow, the entropy production is described, with a good accuracy, by the

linear relationship between flows and thermodynamic forces, in the form:

k kj j

j

J L F= (2.30)


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where Lkj is called phenomenological coefficients, whose values can be found

experimentally.

Substituting equation (2.30) into equation (2.29), the entropy production takes

the quadratic form:

,

0jk j kj k

L F F= > (2.31)

Because of positive sign of entropy production, a matrixLjkis said to be positive

definite. This means that the two-dimensional matrixLjk,

11 12

21 22

L L

L L

is positive definite if the following conditions are satisfied:

L11> 0, L22 > 0 and (L12+L21)2< 4L11L22

In general, for the matrix

11 12 1

21 22 2

1 2

.......................

:

...........

n

n

n n nn

L LL L

L

L L

=

there are following conditions:

Lkk> 0 for k < 1, n >

and all the main minors are positive definite.

In 1931, Lars Onsager introduced the principle of symmetry of matrix L. Thismeans that for the linearly independent forces and flows, the following relations

hold true:

Ljk = Lkj

(2.32)

These relations are sometimes known as the Fourth Law of Thermodynamics.


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2.3 Examples of phenomenological description

In the following sections, the phenomenological description of several

cross effects such as heat conduction and electric current flow as well as thermal

diffusion will be presented in detail.

2.3.1 Heat conduction and electric current flow

This phenomena is known as thermoelectric effect and its general description isshown in Figure 1.

I

Al Al

Cu Cu JqT T+T T T

Seebeck Effect Peltier Effect

- /T =Jq/I

Fig. 2.1 Thermoelectric effect

Phenomenological equations are as follows:

1

1

q qq qe

e ee eq

EJ L L

T T

EI L L

T T

= +

= +

rr

rr

(2.33)

where index q and erelate to heat flow and current density.

According to Onsagers principle:

Lqq> 0 , Leq> 0 and Lee=Lqe

After rearrangement, equations (2.33) become


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2

2

1

1

q qq qe

e ee eq

T EJ L L

T x T

E T

I L LT T x

= +

=

rr

rr

(2.34)

Comparing expression

2

1qq

TL

T x

with the Fouriers law leads to the relation:

2

qqLk

T=

(2.35)

which for the known coefficient of heat conduction k, allowsfor calculation ofLqq. In a similar way, another relation can be derived:

ee

TL =

(2.36)

where is the specific electric resistance and

0

eq ee

T

L L TT

=

=

(2.37)

where /T is the Seebecks coefficient determined experimentally

similarly to

qe eeL L= (2.38)

where0

q

e T

J

I =

=

is the Peltiers coefficient.


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2.3.2 Thermal Diffusion

The interaction between heat and mass flows produces two effects known

as the Soret effect and the Dufour effect. The equation for entropy productionhas the form:

( ),1 k T p

q k

k

J JT T

=

r

(2.39)

For the two component system, the following relations are obtained:

1 1 11 12

2 2 1

1 1 1 11 11 12

2 2 1

11

11

qq

q q

q

L nJ T L n

T T n n

L nJ T L n

T T n n

= +

= +

(2.40)

where nand mean accordingly the number of moles in the substance and the

partial molar volume. This allows to determine the diffusion and heat

conduction coefficients:

1 1 11 11

2 2 1

2

11

qq

nD L

T n n

Lk

T

= +

=

(2.41)

From the Onsagers principle,

Lq1 = L1q

Introducing the general coefficient of thermal diffusion


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1

2

1

1qT

L TD

T n T

=

the Soret coefficient becomes:

1

2

1 1 1

qTT

LDs

D D T n= =

(2.42)

and the Dufour coefficient is:

1 1 1

11 2 2 1

1

1d qn

D L n T n n

= +

(2.43)

The values of coefficientsT and Dd are determined experimentally what allows

for the calculation of phenomenological coefficientLq1.

Many examples of the applications of linear thermodynamics for irreversible

processes can be found in literature [2].

2.4 Irreversible, stationary process of heat conduction

An important group of irreversible processes is represented by stationarystates. Stationary state is reached by the system when the imposed

thermodynamic forces are constant (independent of time) and this state is

characterized by the constant entropy production. As an example, stationarystate for heat conduction has been discussed. The schematic diagram (infinite

plane-wall) and the geometry of the process are shown in Figure 2. The heat

flows spontaneously from the source with the higher temperature THto a source

with a lower temperature TL. The coefficient of thermal conduction of the plane-

wall is constant.


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T

Hh

- Lh

k

TH

qJr

fluid flow

fluid flow kk with temp. TL

with temp. TL

TH

x

0 L

Fig. 2.2 Boundary value problem of the first kind

Since the heat conduction is the only irreversible process, therefore the entropy

production is1

qJT

= r

(2.44)

If the temperature gradient is only in x direction, the entropy production per unit

length is given by

21 1 ( )( )( )

q qT xx J J

x T x T x = =

(2.45)

The total entropy production is therefore

0 0

1( )

L L

it q

d Sx dx J dx

d x T

= = =

(2.46)

q

r

k = constant


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In case of the stationary state, 0T

t

=

and qJ

r

is constant. The stationary state

also implies that the total entropy of the system is constant:

0e id S d S dSd d d = + =

which means that:

e id S d S

d d =

Integrating equation (2.46) gives

0 0

10

( )

LLq q qi

q

C H

J J Jd SJ dx

d x T x T T T

= = = >

so

, ,i

s out s in

d SJ J

d=

(2.47)

whereJsis the entropy flow.

In general:

is

d Sdiv J

d=

(2.48)

Identical equations describe the mass diffusion while the temperature T is

replaced by the concentration of the k-thcomponent of the substance.


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2.5 Principle of minimum entropy production for stationary states

The expression

k k

k

F J =

which describes the entropy production for irreversible processes, shows how

different thermodynamic flows Jk are coupled with the corresponding

thermodynamic forcesFk.

Let some forcesFk, (k=1,2,..,m)to be at a fixed nonzero value, while leaving

the remaining forces Fk, (k=m+1, ..n), free. In stationary state the

thermodynamic flows corresponding to the constrained forces reaching anonzero constant (k = 1,2,..m), Jk=constant, whereas the other flows are equal

to zero,Jk=0, (k=m+1,..n).

Prigogine [2] has revealed the truthfulness of the following principle:

In a linear regime, the total entropy production in a system subject to flow of

energy and matter, /id S d dV = reaches a minimum value at the non-equilibrium stationary state.

Mathematically,

minimumitV

d SdV

d

= =

(2.49)

Such a general criterion was first formulated by Lord Rayleigh as the principle

of least dissipation of energy.

Consider, for example, a system with two forces and flows that are coupled. The

total entropy production is:

( )1 1 2 2i

t

V

d SJ F J dV

d

= = +


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Let the force F1 remain at a fixed nonzero value. In addition, in a stationary

state, J1=constant 0 and J2 = 0 .The linear phenomenological relations of

flows and forces give:

1 11 1 12 2

2 21 1 22 2 0J L F L FJ L F L F

= += + =

Using Onsagers reciprocal relations L21=L12, the total entropy production

becomes:

( )2 211 1 12 1 2 22 22tV

L F L F F L F dV = + +

In follows from the Prigogines principle that reaches minimum when:

( )22 2 12 12

2 0V

L F L F dVF

= + =

(2.50)

whereF2is the free force.

The entropy production is minimized when the integrand expression in (2.50)is equal to zero, that is

2 2 1 1 2 2 2 0J L F L F= + = .

This result can easily be generalized to an arbitrary number of forces and flows.

Consider the boundary problem of the first kind shown in Figure 2.2. For theone dimensional heat flow, entropy production is given by equation (2.46):

0 0

1( )

L L

it q

d Sx dx J dx

d x T

= = =

Introducing expression:

1q qqJ L x T

=


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equation (2.46) becomes:2

0

1L

qqL dxx T

=

(2.51)where T=T(x).

In can be demonstrated that according to calculus of variation, integral (2.51)

reaches minimum when the Euler-Lagrange equation is satisfied:

0x

d

T d x T

=

(2.52)where

x

TT

x

=

Assuming that [2]

2 2

qq avg L kT kT=

(2.53)

where Tavg is the average temperature of the plane-wall, the Euler-Lagrange

equation gives

10qq

d dL

dx dx T

=

Therefore

1qq q

dL J constdx T

= =

Finally

Tk c o n s t

x

=

which leads to the Laplace differential equation:


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2

2

( )0

d T x

dx=

In case of 3D the following equation is obtained:

2 2 22 ( , , ) 0

T T TT x y z

x y z

= + + =

(2.54)

This classical approach is only valid when the assumption of the average

temperature (equation 2.53) can be accepted. When such an assumption is not

justified, the resulting heat conduction equation takes a different form. This is

discussed in the Chapter 3 - Aims of the Thesis.

2.6 Principle of entropy production compensation

The previous discussions concerning the local formulation of the Second Law of

Thermodynamics can be generalized. Let the thermodynamic system consist of

rsubsystems. The additive property of entropy as the function of state results in

1 2 3 ....... 0ri i i i id S d S d S d S d S = + + + +

wherediSk is the entropy production in the k-thsubsystem.

Local formulation of the Second Law of Thermodynamics requires that thefollowing inequality must be satisfied:

0kid S for every k.

The above inequalities are valid for all systems, not only isolated systems,

regardless of the boundary conditions.

Therefore, the local entropy production can be defined as

( , ) 0id S

x

d

=

(2.55)


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and

( , )i

V

d Sx dV

d

=

where the local entropy production depends on the Cartesian coordinates x.

It has also been shown that the positive sign of the local entropy production does

not preclude at same place the processes with both positive and negative entropy

production as long as the condition (2.55) is satisfied.

Assuming that the subsystem A is not isolated but it can interact with another

subsystem A , it is obvious that for the isolated system consisting ofA andA,

its entropy is

S*= SA + SA

therefore*

' 0A AS S S = +

It can be therefore concluded from the above that if SAis decreasing than SA

must be increasing in such a way that the following inequality is satisfied:

'A AS S > (2.56)

Because of the local formulation of the Second Law of Thermodynamics,

inequality (2.55) applies to each smallest part of the system.

Based of the above considerations, one can formulate the principle of entropy

production compensation:

Entropy production for the thermodynamic system can be decreased in only

such a way if the system undergoes coupled processes.

This principle has important practical applications. It also allows to introduce

optimization and direct comparison methods for various solutions to the process.

For example, the process of synthesis, important in the creation of

macromolecules in living organism, can be considered here [6].

glucose + fructose sucrose + H2O(2.57)


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The changes of entropy in this reaction can be replaced by the analysis of Gibbs

free energy changes G. This change G = + 0.24 eV, which means that,

spontaneous reaction (2.57) is taking place in the wrong direction, from right to

left, which means it leads to decomposition of sucrose.

In order of achieving the preferable direction of the process, the entropy

production compensation principle says that there must be another reaction

taking place in the system, with a negative entropy production (G) so

G + G 0

In living organisms, such a reaction does take place and involves the ADP(adenosine diphosphate) and ATP (adenosine triphosphate) molecules.

ATP + H2O ADP + phosphate

The ATP and ADP molecules play an important role in the process of

metabolism (see [6]).

The change of Gibbs free energy in the above reaction, G = 0.30 eV

Therefore

G + G = - 0.06 eV

The full mechanism of the chemical reactions consists of two component

reactions:

ATP + glucose ADP + glucose 1-phosphate

(glucose 1-phosphate) + fructose sucrose + phosphate

The summary reaction looks as follows:

ATP+ glucose + fructose sucrose + ADP + phosphate

The principle described above also applies to the processes of protein synthesis

from amino-acids and DNA, carrying the genetic information.


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3. Aims of the thesis

The irreversibility of the physical and chemical thermodynamic processes,

taking place in the surrounding Nature, are the reasons for degradation

(dissipation) of all forms of energy, and the measure of this dissipation,according to the Second Law of Thermodynamics, is the increase of entropy or

entropy production. Qualitatively, this is described by the Gouy-Stodola

theorem [1], which states that the lost available work (destroyed exergy) W is

directly proportional to the entropy increase S for all elements taking part inthe process. Mathematically,

0W T S = or

0W T =& (3.1)

where T0 is the temperature of the environment and is called the entropy

production or rate of entropy generation.

Note: the exergy is the maximum available work to be gained from the system

when the conditions of the surroundings are the frame of reference. More on

exergy, see [7].

The above equations directly indicate that the amount of entropy

production is the measure of degree of irreversibility in the system. The energy

lost, described by the Gouy-Stodola theorem, are not possible to be recovered,

therefore the above law is often called the law of destroyed exergy. The typical

irreversible processes are processes with finite gradient values characterizing the

field potential diffusional flow of energy and matter, processes of friction and

turbulent flow, irreversible chemical reactions.

The Gouy-Stodola theorem concludes that minimization of non-linear losses of

exergy is only possible through the minimization of entropy production.As already shown in Chapter 2, entropy production is defined by

id S

d

=

where diSis the entropy change of the system resulting from the irreversibility

of the process, while is time and the Second Law of Thermodynamics shows

that always and in each place of the thermodynamic system:


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30

0

which means that entropy production is path dependant (entropy production is

path dependant and should not be confused with entropy change).

The core of entropy production minimization centers on searching for

methods of conducting the process, so with the known constraints and boundary

conditions, is minimum. The entropy production minimization principle is

fully appreciated in conjunction with the entropy compensation principle, which

appears to be a new, expanded version of Prigogines principle.

It is important to emphasize though the difference between Prigogines

principle and entropy production minimization as two principles are often

understood as equivalent.Prigogines principle allows to determine the parameters of stationary

process as in Chapter 2.5 where the constant thermodynamic force F1gives as

the condition for stationary process the zero value of flowJ2.

Since, as shown in Chapter 2.1, entropy production is path dependant,minimum can be reached in various ways which will be explained below.

Entropy production in the one-dimensional process of heat conduction

(k=constant) , when the assumption that T=Tavg can not be accepted, is

2

2( )( )

( )k dT xx

T x dx =

Calculating the classical minimum of this function

2

2

( )0

( )

d d k dT x

dx dx T x dx

= =

the Euler-Lagrange equation is obtained (Chapter 4.1)

22

2

10

d T dT

dx T d x

=

The solution T=T(x)of the above equation determines the minimum since


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2

20

d

dx

>

The above short discussion illustrates the new and additionalinterpretation of Prigogines principle which should not be consider

equivalent to entropy production minimization principle. It is interesting to see

that Bejan, without explaining, as above, the difference between the two

principles, does not link Prigogines principle with the entropy production

minimization principle [1].

The above principles have been used in this thesis in order to minimize

the entropy production in diffusional flow of matter and heat.

New differential equations were presented, describing the processes ofheat conduction in solids and the equation of diffusional mass transfer has been

obtained. This equation was derived with the use of calculus of variations.

The Euler-Lagrange differential equations were introduced. The in-depth

physical analysis of the additional internal heat source was introduced, as the

element insuring the entropy production minimization. The agreement of the

analysis with the First and Second Laws of Thermodynamics has been shown.

It is very important to emphasize that in this thesis, the classical Prigogines

assumption that

T=Tavg

is not accepted and the entropy production is considered in the most generalform

21

2

0

( )

( )

k T dT dx

T x dx

=

instead of

21

2

0

1( )

avg

dTk T dx

T dx

=

The major difference resulting from deriving the equation for heat conduction

with the use of Prigogines principle has been demonstrated.

The detailed analysis has been devoted to boundary conditions, including the


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32

conditions of Sturm-Liouville type. Equations describe the simple cases when

the physical properties of the system such as the coefficient of heat conduction

or the coefficient of mass diffusion are constant as well as when they are

functions of temperature and substance concentration. The obtained equations

describing the energy and mass transfer are mathematically non-linear due tothe non-linear nature of internal sources of heat and mass. In addition, this non-

linearity is intensified by the nature of physical constants and difussional field

potentials, mentioned above. Finally, the practical applications of the obtained

results have also been discussed.


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4. Entropy production minimization and entropy productioncompensation principle in stationary, linear processes of diffusionaltransfer of mass and energy

In this chapter, the main applications of the entropy minimizationprinciple and compensation of entropy production have been presented for some

specific processes of mass and energy transfer. The discussion centers on thestationary processes. Since the mathematical description is similar, the specific

discussion was presented for the processes of heat conduction in solids with the

boundary conditions of the first kind (first boundary problem).

4.1 Heat conduction through a plane-wall

As clear from the previous discussion, entropy production in the processof heat conduction is given by

term u uX = r r

o

(4.1)

where:

- the thermodynamic force( )1

u

i

X gradT x

=

r

- thermodynamic vector flow ( ) ( )u ik T grad T x= r

For the one-dimensional steady state heat flow (Figure 4.1), we obtain

2

1 ( )

u

dT xX

dxT=

r

( )( )u

dT xJ k T

dx=

r

therefore2

2

( )term

k T dT

T dx

=

(4.2)and


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21

2

0

( )t

k T dT dx

T dx

=

Fig. 4.1 Heat Conduction through a plane-wall

The aim is to look for a such temperature function T=T(x) which ,while

satisfying the boundary conditions, minimizes the entropy production.

The application of calculus of variations (see Appendix 1) leads to Euler-

Lagrange differential equation.

0x

d d

T dx T

=

where

x

dTT

dx=

With the use of (4.2) and assuming that k = const, the above equation becomes

( ) dTdx

q k T= &

( )T T x=

T1

T2

0 1

Boundary Conditions:

x= 0 T(0)=T1

x= 1 T(1)=T2

A=1m2

x < 0 , 1 >

T < T1, T2>


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22

2

10

d T dT

dx T dx

=

(4.3)

whose solution, with the boundary conditions from Figure 4.1., has the form

21

1

( )

x

TT x T

T

=

(4.4)

which describes the temperature distribution along the thickness of the plane-

wall. The entropy production is

21

2

0

( )k T dT dx

T dx

=

which in combination with the above solution for T(x)gives

2

2

1

lnt TkT

=

(4.5)

versus the classical solution:

1 2

2 1

1 1( )ct k T T

T T

=

where it can be easily shown thatc

t t >

Analysis of the equation (4.3) demonstrates that the expression:


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2

( )v

k dTq x

T dx

=

&

(4.6)

is the additional internal heat source , which must be continuously

removed from the system.

Applying the solution (4.4), the following expression is obtained:

2

2 21

1 1

( ) ln 0

x

v

T Tq x kT

T T

=


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The quantities qin and qout can be derived from the relations:

11

0 2

12

1 2

ln

ln

in

x

outx

dT Tq k kT

dx TdT T

q k kT dx T

=

=

= =

= =

&

&

(4.9)

which shows that

2

11

out

in

q T

q T= & &

Relations (4.5) and (4.7), on the other hand, show that

0 0div q and div s< =& &

which is a contradiction to both laws of thermodynamics. Explanation comes by

considering equation (4.3) and its second term as the additional internal heatsource qvresulting in the entropy production which does not exist in the classicalapproach. Substituting (4.8), (4.9) into balance equation (4.11) satisfies the First

Law of Thermodynamics. Integration

1

0

( )

( )

vq x d xT x

&

gives the entropy production2

2

1

ln 0tT

kT

= >

which satisfies the Second Law of Thermodynamics.

One of many applications of the above discussion, has been presented by Bejan

[1] for the process of minimization of liquid helium boil-off in the cooling

system of the superconducting magnet (Fig. 4.3).

The balance equation for heat flow results in

out in i

i

Q Q Q= & & &

Minimization of outQ& leads to the minimization of the helium boil-off rate. The

minimization problem can be formulated as follows:

To determine the optimal distribution of cooling stations (flows iQ& ) which leads

to the minimization of /out fg m Q h= &

& .


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The detailed solution is presented by Bejan [1].

/out fg m Q h= &&

Fig. 4.3 Mechanical support for helium boil-off cooling

4.2 Diffusional transfer of matter

Lets consider the two component perfect solution in which the

concentration of substance dissolved C1is much smaller than the concentrationof the solvent C2, hence C1


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1 1

111 1

1X

T

LJ

T

=

=

Using the relation

1 1,0 1lnT C= +

and assuming one dimensional flow

1

1

1RT C

C

=

and hence

1 1

1

RC

C=

and

1 11 1

1

1J L R C

C

=

from the Ficks Law

1 1J D C=

which means that

11

1

L RD

C

=

finally, the entropy production for the stationary flow is

2

11 1

1

( )dCRD

x X JC dx

= =

and the entropy production for D = constant


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41

21

1

10

1t

dCRD dx

C dx

=

where C1=C1(x).

The minimum of this integral

21 1

1

10 0

1( ) minimum

dCx dx dx

C dx

=

is reached for the function C1(x) satisfying the Euler-Lagrange equation

1

0x

d

C dx C

=

where Cx= dC/dx .

After some rearrangement, the following differential equation is obtained

221 1

2

1

10

2

d C dC

dx C dx

=

which has a general solution

2

1( ) ( )C x Ax B= +

Assuming the boundary conditions of the first kind: for x = 0, C1 = C 1, 0andfor x = 1, C2 = C 2, 0 , the solution looks as follows:

21/2 1/2 1/2

1 1,0 1,0 2,0( ) ( )C x C C C x =

The solutions of several mass transfer processes when the diffusion coefficient

Dis the function of concentration, can obtained in the identical way as for the

processes of heat conduction.


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43

Entropy production minimization

21

2

0

( )t

k T dT dx

T dx

=

leads to the Euler-Lagrange equation

22

2

1 ( )( ) 0

2

d T dk k T dT k T

dx dT T dx

+ =

(5.1)

Comparing with the classical approach

22

2( ) 0

d T dk dT k T

dx dT dx

+ =

equation (5.1) can also be written in the form

2 22

2

1 ( )( ) 0

2

d T dk dT dk k T dT k T

dx dT dx dT T dx

+ + =

(5.2)

The physical interpretation of (5.2) leads to the conclusion that the additional

internal heat source is

21 ( )

( )2

v

dk k T dT q x

dT T dx

= +

&

which satisfies the First and Second Laws of Thermodynamics.

As shown by the experimental results, the coefficient of heat conduction

k = k(T)dependence on temperature also depends on many factors. Hence, the

most appropriate method is the approximation in the given temperature interval.


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44

In general, in practical cases, the temperature interval is not large and permits

for the approximation [1]

1

1

( )( )

n

T xk T k

T

=

(k1 the value of heat conduction coefficient at temperature T1), which is

usually sufficient. An important characteristic is the fact that equation (5.2)

allows to find effective analytical solutions as shown in this thesis.

Finally, the Euler-Lagrange equation (5.1), after dividing by k(T),leads to the

general expression

22

2

1 10

2 ( )

d T dk dT

dx k T dT T dx + =

which can be written as

22

2( ) 0

d T dT F T

dx dx

+ =

(5.3)where

1 1( )

2 ( )

dkF T

k T dT T =

In addition to equation (5.3), the boundary value problem must also contain the

boundary conditions. In this case they are:

1

2

( 0)( 1) .

T x TT x T

= =

= =

where 1T and 2T are known.


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5.2 Boundary value problem with boundary conditions of the third kind(Sturm-Liouville boundary conditions) without internal heat source

The uniqueness of a solution to each boundary value problem is achieved

by setting the boundary conditions. From the physical point of view, differentialequation describes the process inside the thermodynamic system while the

boundary conditions set the interaction mechanism of the system with its

surroundings. In case of the one-dimensional systems, the boundary conditions

applied most often can be classified into three groups.

- boundary conditions of the first kind, or Dirichlet conditions, describe the

functional distribution of the thermal field potential (temperature at theboundary surface) what can be generalized as

( )ix

T x T =

where denotes the boundary surface.

- boundary conditions of the second kind, or Neumann conditions, describe the

case of a given derivative of the field potential in the direction normal to the

boundary surface, what means the following relation is know:

( )T

f xn

=

The solution is then known with the accuracy of the constant.

- boundary conditions of the third kind, or Sturm-Liouville conditions, are given

in the form

( )fT

k h T T n

=

where k heat conduction coefficient of the solid, n normal to the boundary

surface, h coefficient of heat transfer at the boundary surface, Tf fluid

temperature, T- the boundary surface temperature .

It is easy to notice that through the appropriate choice of constants in the thirdkind boundary condition, the first and second kinds can be obtained. The


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47

the term2

( )vk dT

q xT dx

=

&

represents the additional internal heat source.

- the Sturm-Liouville boundary conditions

x = 0

Heat transferred to the boundary surface x = 0.

fluid 1

h1

c o n v e c t i o nq& 0xq =&

T0

T1

x = 0

0convection xq q

==& &

which leads to the following condition:

( )1 0 10x

dTh T T k dx =

= (5.4)

where h1is the coefficient of heat transfer from fluid 1 to the wall atx = 0.


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49

-for the near the wall surface at x=1

2

2 2

2

2

2

0

d

f f

f

k d Td x

d xT

where1

T and2

T describe the temperature distribution, correspondingly in

the near the wall regions of fluid 1 (x=0) and fluid 2 (x= 1) with the thickness of

d1and d2and heat conduction coefficients 1k and 2k .

Since the thickness of the near the wall regions, in turbulent flows, is usually in

the neighborhood of few millimeters, the influence of these additional entropy

production can be ignored.

The numerical analysis focuses therefore on the effect of the heat transfer

coefficients h1and h2on the amount of entropy produced in the plate.

The above discussion goes beyond the established aims of the thesis. However,

the closer analysis demonstrates that including the convection (near the wall)

terms of the entropy production would not be difficult from the methodological

point of view and could be solved with the numerical methods. The simplified

case of the problem discussed above has been presented by Rafois and Ortin [8].

5.3 Boundary value problem with the simultaneous heat and electriccurrent flow

The boundary value problem, generally non-linear, of the simultaneous

heat and electric current flows with the boundary conditions of the first kind is

being discussed here. The Joules heat is dissipated inside within the system. It

is assumed that the heat conduction coefficient and the electric resistivity arefunctions of temperature, k=k(T)and =(T).

The geometry of the system for the adiabatically insulated rod is shown in

Figure 5.3.


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adiabatically insulated rod surfaceT(x=0) = T1 T(x=1) = T2

inq& outq&

x = 0 x = 1

Fig.5.3 Geometry of the adiabatically insulated rod surface

Entropy generation rate is

2 2

2

( ) ( )( )

( ) ( )

k T dT i T x

T x dx T x

= +

where i is the electric current density

Minimization

21 1 2

2

0 0

( ) ( )( ) minimum

( ) ( )

k T dT i T x dx dx

T x dx T x

= +

leads to Euler-Lagrange equation

22 2

2

1 ( ) 1 ( )( ) 0

2 ( ) 2 ( )

d T dk T dT i d T T T

dx k T dT T dx k T dT

+ + =

which can be written in the general form

22

1 22( ) ( ) 0

d T d T F T F T

d x d x

+ + =

additional internal heat source

k=k(T) =(T)


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51

where

1

1 ( ) 1( )

2 ( )

dk TF T

k T dT T =

2

2

( )( ) ( )

2 ( )

i d TF T T T

k T dT

=

Boundary value problem is completed with the boundary conditions

1

2

( 0)

( 1) .

T x T

T x T

= =

= =

Several special cases have been analyzed by the use of various analytical and

numerical methods. The work has been divided into four following situations:

k

Case #1

k - constant - constant

Case #2

1

1

n

Tk k

T

=

- constant

Case #3

k-constant1

1

n

T

T

=

Case#4

1

1

n

Tk k

T

=

oL T

k=

5.4 Methods of calculation of additional internal heat sources andentropy generation rate

The obtained solutions allow to determine the additional (resulting fromthe entropy production minimization) internal hear sources and the entropy

generation rate. Particularly, the derivation of the Bernoulli equation from the

Euler-Lagrange equation allows to derive these values. The difficulties thatmight be encountered during the calculations result from the non-explicit form

of the solution.


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1 - k=constant, boundary conditions of the first kind without the internal heat

sources

Boundary value problem is described by:

- Euler-Lagrange equation

22

2

10

d T dT

dx T dx

=

- boundary conditions

1

2

( 0)

( 1) .

T x T

T x T= == =

where2

( )( )

v

k dTq x

T x dx

=

&

and the total heat source

21

,

0

1( )

( )v t

dTq x k dx

T x dx

=

&

Since

21

1

( )

x

TT x TT

=

and

2 2 2

1

1 1 1

ln ( ) ln

x

T T TdTT T x

dx T T T

= =

therefore


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54

and2

1 ( )( )

2v

dk k T dT q x

dT T dx

= +

&

and the total amount of the heat source

21

,

0

1 ( )

2v t

dk k T dT q dx

dT T dx

= +

&

and entropy generation rate connected with heat source becomes

21 1 ( )

( )( ) 2

dk k T dT x

T x dT T dx

= +

and

21

0

1 1 ( )

( ) 2t

dk k T dT dx

T x dT T dx

= +

The functional relations for T(x) and dT/dx are determined by solutions to

equations (5.6) and (5.7) after prior calculation of the integration constants,while F2(T) = 0.

The total entropy production is

( )

21

2

0

( )( )

t k T dT dxdxT x

=

or

( )

2

1

2

( )

( )

T

t

T

k T dT dT

dxT x =

Solution to Eq.(5.6) can be easily obtained by the substitution


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dTu

dx=

which gives2

2

d T duu

dx dT =

and equation (5.6) takes the form

( ) ( ) 0du

k T F T udT

+ =

(5.8)

where

1 ( )( )

2

dk k T F T

dT T=

The solution of (5.8) is

( )

( )

1

F TdT

k TdTu c edx

= =

which after simplification gives

1ln ( ) ln

21

k T TdTc e

dx

+

=

hence

1 ( )d T

c T k T d x

=

and

1 1 2( )( )

dTc dx c x c

Tk T= = +

Finally


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56

[ ]2

1

32( )

T

t

T

k TdT

T =

3 - k=constant, boundary conditions of the third kind (Sturm-Liouville

boundary conditions) without the internal heat sources.



22

21 0d T dT

dx T dx = (5.9)


1 0 00

2 0,011

0 ( )

1 ( )

xx

xx

dTfor x k h T T

dx

dTfor x k h T Tdx

==

==

= =

= =

(5.10)

The general solution has the form

1

3( )c xT x c e=

hence(5.11)

1

3 1

c xdT c c edx

=

Substitution of expressions (5.11) into the boundary conditions (5.10) gives


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1 1

1 3 1 3 1

1 3 2 3 2

c c

kc c h c

kc c e h c e

+ =

=

(5.12)

where

1 1 0 2 2 0,0andh T h T = =

The values of c1and c3are numerically determined because of the non-explicit

character of equations (5.12).

The additional internal heat source is

2

( )( )

v

k dTq x

T x dx

=

&

and its total value is

21

,

0

1

( ) ( )v t

dT

q x k dxT x dx

= &

Using equations (5.10) leads to

12

3 1( )c x

vq x kc c e= &

and

1

, 1 3( ) (1 )c

v tq x kc c e= &

Entropy generation rate is

2

1

( )( ) 0

( )vq xx kc

T x = = >

&

and is equal to tas divs& = 0.


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59

and after the second integration

2( )

1

F T dT

dT

x cc e

= + (5.16)

Equation (5.16) is given in the non-explicit form.

The additional internal heat source is given by:

21 ( )

( )2

v

dk k T dT q x

dT T dx

= +

&

and

21

,

0

1 ( )( )

2v t

dk k T dT q x dx

dT T dx

= +

&

while T(x)and dT/dx are described by equations (5.13) and (5.14). Examples of

the numerical analysis are provided in Chapter 7.

Entropy production at the additional heat source takes the form

2( ) 1 1 ( )

( ) ( ) ( ) 2

vq x dk k T dT

x T x T x dT T dx

= = +

&

and

21

0

1 1 ( )

( ) 2t

dk k T dT dx

T x dT T dx

= +

Total value of the entropy production comes from the definition


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60

2

1

21

2 2

0

( ) ( )T

t

T

k T dT k T dT dx dT

T dx T dx

= =

and is obtained in the same way as in 2.

5 - Simultaneous heat and electric current flows, k = k(T) , = (T)



22

1 22( ) ( ) 0d T d T F T F T

d x d x + + =

(5.17)

where

1

1 ( ) 1( )

2 ( )

dk TF T

k T dT T

=

2

2

( )( ) ( )

2 ( )

i d TF T T T

k T dT

=


1

2

( 0 )

( 1)

T x T

T x T

= =

= =

(5.18)

It can be easily shown that equation (5.17) can be transformed into the Bernoulli

equation in the form

1

1 2( ) ( )du

F T u F T u

dT

+ =

(5.19)


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61

where

d Tu

d x=

The additional internal heat source is

2 21 ( )

2 2v

dk k T dT i d q T

dT T dx dT

= + +

&

and1

,0

( )v t vq q x dx= & &

In general, because of the non-explicit form of the solution, the numerical

calculations must follow.

Energy generation rate at the additional heat source is

2 2( ) 1 1 ( ) ( )( ) ( )( ) ( ) 2 2

v

q x dk T k T dT i d x T TT x T x dT T dx dT

= = + + +

&

and

21 2

0

1 1 ( ) ( )( )

( ) 2 2t

dk T k T dT i d T T dx

T x dT T dx dT

= + + +

Total value of the entropy production comes from definition

2 2

1 1

12

2

( )T T

t

T T

k T dT i dT dT dT

T dx T dx

= +

where T(x)and dT/dxare obtained from the solution of Bernoulii equation

(5.19) (see next Chapter 6).


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62

6. Methods of solving the boundary problems with Euler-Lagrangeequations

The analysis of the non-linear boundary value problems with internal heat

sources always leads to the non-linear Euler-Lagrange differential equation ofthe second order in the form

22

1 22( ) ( ) 0

d T d T F T F T

d x d x

+ + =

(6.1)

In particular, the functions 1( )F T and 2 ( )F T have the form:

- heat conduction coefficient k does not depend on temperature,

without the internal heat sources

1 2

1( ) , ( ) 0F T F T

T= =

- heat conduction coefficient k = k(T), without the internal heat

sources

1 2

1 ( ) 1( ) , ( ) 0

2 ( )

dk TF T F T

k T dT T = =

- heat conduction coefficient k = k(T), internal heat source in the

form of Joules heat

1

1 ( ) 1( )

2 ( )

dk TF T

k T dT T =

2

2

( )( ) ( )

2 ( )

i d TF T T T

k T dT

=


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63

The specific form (6.1) of the Euler-Lagrange equations, when F1(T)andF2(T)

are both not equal to zero, allows to obtain an analytical solution, usually though

in the non-explicit form.

By substitution dT udx

=

and differentiating2

2

d T du du dT duu

dx dx dT dx dT = = =

after substituting into (5.19) gives

2

1 2( ) ( ) 0du

u F T u F T dT

+ + =

Finally equation (5.19) becomes

1

1 2( ) ( )du

F T u F T udT

+ = (6.2)

Equation (6.2) represents the Bernoulli differential equation which has the

general solution in the non-explicit form:

1 1

22 ( ) 2 ( )

2 12 ( )F T dT F T dTdT

e F T e cdx

= +

(6.3)

Hence

1 1

1

22 ( ) 2 ( )

2 12 ( )F T dT F T dTdT

F T e c edx

= +

(6.4)


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64

and finally in the non-explicit form

1 1

122 ( ) 2 ( )

2 1 22 ( )F T dT F T dT

F T e c e dT x c

+ = +

(6.5)

The integration constants c1and c2must be determined from the boundary

conditions, usually numerically.

The derivations presented above as well as the solutions obtained are new in

the literature of entropy production minimization problems.

The above solution has been used in the analysis of the boundary problems

discussed in chapter 5.

6.1. General method of determining the internal heat source and entropygeneration rate

As seen in the relations derived in chapter 5.4, the integration over the x

coordinate, expression for ,v tq& and t, is very difficult because of the presenceof T(x)and dT/dx, which must be substituted as the functions of x.

The analysis of the equations indicates as well that it is possible to simplifythe calculations by using simple substitutions and the solutions to Bernoullis

equation for /u dT dx= . In the following discussion, as an example, the

calculation method for case 2 will be presented.

- k = k(T) , boundary conditions of the first kind without the internal heat

sources

Expression

21

,

0

1 ( )

2v tdk k T dT

q dxdT T dx

= + &


65/102

65

can be rewritten in the form

1

,

0

1 ( )

2v tdk k T dT dT

q dxdT T dx dx

= + & hence

2

1

,

1 ( )

2

T

v t

T

dk k T dT q dT

dT T dx

= +

&

(6.6)By substitution

( )dT

u Tdx

= =

(6.7)

the integration of (6.6) becomes much easier, as dT/dxis the indirect function of

x (as T=T(x)).Substituting (6.7) into (6.6) yields

2

1

,

1 ( )( )

2

T

v t

T

dk k T q T dT

dT T

= +

&

In the identical way, the calculation for the entropy production resulting from

the additional heat source can be simplified by transforming expression

21 1

0 0

1 1 ( )

( ) ( ) 2v

t

q dk k T dT dx dx

T x T x dT T dx = = +

&

into

2

1

1 1 ( )

( ) 2

T

t

T

dk k T dT dT

T x dT T dx

= +


66/102

66

using, as before,

( )dT

u Tdx

= =

The calculations can be done in the same way as for other boundary problemswithout the internal heat sources.

Total value of the entropy production comes from definition

2

1

2

( )T

t

T

k T d T d T

T d x =

(6.8)

and the solution is identical as in 2.

For the problem of simultaneous heat and electric current flows, the above

analysis requires modification.

Entropy production describing additional heat source becomes

21 2

0

1 1 1( )

( ) 2 ( ) 2 ( )t

dk dT i d T T dx

T x k T dT T dx k T dT

= + + +

which can be written in the form of two integrals

2

1

11

2

0

1 1 1

( ) 2 ( )

1( )

2

T

t

T

dk dT

dTT x k T dT T dx

d dTi T T dT

dT dx

= +

+ +

Transformation above allows, as before, for solving the Bernoulli equation for


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67

( )dT

u Tdx

= =

which after substituting into expression for tleads to the direct expression

2 2

1 1

12

2

( )T T

t

T T

k T dT i dT dT dT

T dx T dx

= +

The method described above is the new concept in literature dealing with

entropy production minimization, calculation of the internal heat sources and

entropy production t. The most important feature of this method is that it canbe applied to any mathematical relationship for k=k(T) and =(T) as the

solution of the governing differential equation is given in the general form.


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68

7. Solutions for the specific boundary value problems

7.1 Simultaneous heat and electric current flows

All the calculations below originate from the general solution (6.4) which

can be easily rearranged into

1

1

2 ( )2

2

2 ( )

2 ( ( )F T dT

F T dT

F T e dTdT

dx e

=

(7.1)

Case#1

The values for k, are both constant

General equation

( ) 02

'1

2

1''

22 =

+

+

dT

dT

k

iT

TdT

dk

kT

is simplified to the form

( )21

'' ' 0T TT

+ =

(7.2)

where2

2''

d TT

dx=

'dT

Tdx

=

2

2

i

k

=


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69

which according to (7.1) has the solution for F1(T)= - 1/T and F2(T)= in the

form:

( )1

222

2

112

2

12 ( )( )22

1

dTT

dTT

dTe dTdT T T c Tdx

e T

= = =

(7.3)

Hence

2

1

2dT

dx

T c T

=

(7.4)

which after the substitution

1

1

1

2w c T

c=

gives

21

2

1

12

1

2

dwdx

c

wc

=

after integration

1

1 2

1

1sin (1 2 ) 2 ( )c T x cc

= +

an explicit solution becomes

1 2

1

1 sin 2 ( )( )

2

c x cT x

c

+ + =

(7.5)


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70

Applying boundary conditions, two algebraic equations are obtained which

allow to find the integration constants c1and c2

1 1 1 21 2 sin 2 ( )c T c c =

and

1 2 1 21 2 sin 2 (1 )c T c c = +

The graphs of T (x) vs. x are shown in Fig. 7.1.

Fig. 7.1 The graph of T(x) vs. x for different values of boundary

temperatures when k=constant and =constant

Local entropy production (x) can also be calculated according to

2 2

2( )

( ) ( )

k dT ix

T x dx T x

= +


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71

it is assumed that2

= 12

iK

k

=

Therefore

[ ]

2

1

2

4 2( )

( )( )

k T c T kx

T xT x

= +

and

1

4( ) 2

( )

kx kc

T x

=

The graph of local entropy production (x) vs. x isshown in Fig. 7.2.

Fig. 7.2 The graph of local entropy production (x) vs. x for different

values of boundary temperatures when k=constant and =constant


72/102

72

The total entropy production tcan also be calculated according to:

1 1

10 0

2

( ) 2 ( )t x dx k cT x

= =

Assuming k = 20 W/mK, the following numerical values for total entropy

production are obtained and shown in Table 7.1.

Table 7.1

Boundary

Temperatures

T1=0.4

T2=0.1

T1=0.8

T2=0.5

T1=1.0

T2=0.6

T1=1.0

T2=0.7

TotalEntropy

Production

t

(W/K)

226.4 63.6 54.8 48.4

The internal heat source

2 21( )

2 2v

dk k dT i d q x T

dT T dx dT

= + +

&

can also be calculated and when k=constantand =constant:

2 2

( )2

v

k dT iq x

T dx

=

&

substituting for (dT/dx)2from Eq. 7.3gives

22

1 1( ) 2 ( 3 22

v

k iq x T cT k kcT

T

= = + &

and


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73

1 2( ) 3 1 sin 2 ( )vq x k k c x c = + + +

&

Finally

{ }1 1

, 1 2

0 0

( ) 2 sin 2 ( )v t vq q x dx k c x c dx = = + + & &

Assuming again k = 20 W/mK, the following numerical values for the total heat

source are calculated and presented in Table 7.2.

Table 7.2

BoundaryTemperatures

T1=0.4T2=0.1

T1=0.8T2=0.5

T1=1.0T2=0.6

T1=1.0T2=0.7

Total Heat

Source

,v tq& (W)

-31 -24.6 -25.4 -24.2

Case #2

Conditions:n

T

Tkk

=

1

1 and = constant

General equation

( ) 02

'1

2

1''

22

=

+

+

dT

dT

k

iT

TdT

dk

kT

becomes


74/102

74

22 1'' ( ') 02 n

nT T

T T

+ + =

(7.6)

where2

1

12

ni T

k

=

which according to equation (7.1) has the solution for

1

2 1

( ) 2

n

F T T

=

and

2 ( ) nF T T

=

in the form:

2 12

2 22

1

2 12 12

2

2 22 (1 )

ndT

T nn n

n nndT

T

e dT T dTdT c TT Tdx T T

e

= = =

(7.7)

Hence1

2

1

21

n

T dTdx

c T

=

(7.8)


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75

Table 7.3 below is the summary of the integrals obtained and the solutions for

different values of n.

Table 7.3n integral solution

-22

1

2( )

dTdx

T T c T =

2

1

2

22 ( )

T c Tc

T

= +

-1

1

21dT dx

T c T=

1

2

1

1 1ln 2 ( )

1 1

c Tx c

c T = +

+

0

2

1

2dT

dxT cT

=

1 21

1 sin 2 ( )( )

2

c x cT x

c

+ + =

Same as in case#1, k,constant

1

1

21

dT dxc T

= 1

2

1

2 1 2 ( )c T cc

= +

k-linear, - constant

2 1 / 2

1

21

T dTdx

c T=

1

1 1

23/2

1 1

sin ( ) 12 ( )

cT T cT x c

c c

= +

3

1

21T dT dx

c T= ( )1 1 22

1

2 21 2 ( )

3

cTcT x c

c

+ = +


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76

The total entropy production can be determined with the described earlier

2 2

1 1

21 1 2

2

0 0

2

2

( ) ( )

( ) ( ) 1

t

T T

T T

k T dT i T dx dx

T dx T

k T dT i T dT dT

dTT dx T

dx

= + =

= +

(7.9)

Applying this method in Case#2yields

2 2

1 1

2 2

1 1

1 1 21

1

2 12

1

2

33 2 2

1 21

1 1

2 (11

2 (1 )

2

1 2 1

n

T T

t n

T T

n

nT Tn

n

T T

Tk c TT i

dTT c T

T T

T

k i T

T c T dT dT T c T

= + =

= +

which provides a general expression which can be calculated numerically for the

specific boundary conditions.


77/102

77

In the section below, a specific solution for n=1 (k-linear function of

temperature, constant,same as Case#4) is examined. Solution

1

2

1

2 1

2 ( )

c T

x cc

= +

results in

2 2

1 2

1

11 ( )

2( )

c x c

T xc

+=

(7.10)or

2 2

1 1 2 1 2

1

1 1 1( )

2 2T x c x c c x c c

c = +

which produces the following graphs for the specific boundary temperatures,

represented in Fig.7.1.

Fig. 7.3 Graph of T(x) versus x for n=1 and for different values of

boundary temperatures when k=linear and =constant


78/102

78

The local entropy production can be calculated according to:

2 2

2

( )( ) ( ) ( )

k T dT ix T x dx T x

= +

which in this case becomes

( )

1

2

1 11

1 121 1

2 1 2 1

( ) 2 1

kT

i k kT

x c T cT T T T T T

= + = +

as2

21

1

12

ni TK

k

= =

and

T1=1(dimensionless temperature)

the following expression is obtained:

11

1

2 2( )

( )

kx c

T T x

=

which is represented graphically in Fig 7.2.


79/102

79

Fig.7.4 Graph of local entropy production (x) versus x for n=1 and for

different values of boundary temperatures when k=linear and

=constant

The total entropy production tcan also be calculated according to:

1 1

11

10 0

2 2( )

( )t

kx dx c dx

T T x

= =

Assuming k1= 20 W/mK, the following numerical values for total entropyproduction are obtained and shown in Table 7.4.

Table 7.4

Boundary

Temperatures

T1=1.0

T2=0.1

T1=1.0

T2=0.3

Total

Entropy

Production

t

(W/K)

139.2 82


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80

The internal heat source

2 21 ( )( )

2 2v

dk k T dT i d q x T

dT T dx dT

= + +

&

can also be calculated, giving

[ ]2

1 11

1 1

1( ) 2 (1

2 2v

k k iq x cT

T T

= +

&

or

[ ]1 111 1

3( ) 1v

k kq x cT

T T

= &

and finally

2 211 2

1

3( ) 1 ( ) 0

2v

kq x c x c

T

= + +


81/102


82/102


83/102

83

Table 7.6 is the summary of the integrals obtained and the solutions for different

values of n.

Table 7.6

n integral solution

-2

3

1

21 3

T dTdx

c T=

1 3 / 2

1

2

1

2sin ( 3 )2 ( )

3 3

c Tc

c

= +

-1

2

1

21 2

dTdx

cT=

( )

1

1 2

1

1sin 2 2 ( )

2cT x c

c = +

0

2

1

2dT

dxT cT

=

1 2

1

1 sin 2 ( )( )

2

c x cT x

c

+ + =

same as in Case#1, k,constant

2

1

2dT

dxT T c

=+

1

2

1ln 2 ( )

1

T cx c

T c

+ = +

+ +

3

2

1

22

dTdx

T T c=

+

2

1 1

2

1

2 21ln 2 ( )

2

c T cx c

Tc

+ + = +


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84

The total entropy can be determined with the use of the method described in

Case#2(equation 7.7)

2 2

1 1

2 2

1 1

2

11

1 1 1

2 1

1

1 2 21 1

11

1 1

2 ( 1) 1

( 1)

( 1)2

2 ( 1)

n

T Tn

tn

T T

T Tn n

n nT T

Ti

kT T n c T dT dTT T T T n c

T n c i Tk dT dT

T T T n c

+ = ++

+ = +

+

Case #4

Conditions:n

T

Tkk

=

1

1 ,k

TL0= (Wiedemann-Franz Law)

General equation

( ) 02

'1

2

1''

22 =

+

+

dT

dT

k

iT

TdT

dk

kT

becomes

22 1'' ( ') ( ) 02

nT T g T

T

+ + =

(7.13)

where

1

2 21 2 0 1

2( )

2

nn i L Tg T n T and

k = =

which according to (7.1) has the solution for

1

2 1( )

2

nF T

T

=

and


85/102

85

1 2

2 ( )nF T n T =

in the form:

( 2) 12

1 2 1 2 ( 2) ln2 2

( 2) 1 ( 2) ln2

2

1 2 2 1 1

2 2 2

1 1

2 2 2

2 ( 2 (

22 ( 2 (

2 2 1

ndT

n n n T T

n n TdT

T

n

n n n

n n n

n n

n n

n T e dT n T e dT dT

dx ee

Tn c

n T T dT n T dT n

T T TT nc ncT

T T

= = =

+ = = = =

= =

Hence:

1

1

2

1

n

n

T dTdx

nc T

=

(7.14)

Note: In case of n=0 (k-constant,linear), the above integral solution can not

be used as F2 = 0 and equation

21'' ( ') ( ) 0T T g T T

+ =

becomes21'' ( ') 0T T

T =

the substitution u=dT/dxleads to the known solution

21

1

x

TT T

T

=


86/102

86

Table 7.7 is the summary of the integrals obtained and the solutions for different

values ofn.

Table 7.7

n integral solution1

1

21

dTdx

cT=

1

2

1

2 12 ( )

c Tx c

c

= +

same as in case #2 (for n=1) k-linear, -constant

2

2

1

21 2

TdTdx

c T=

2

2

11

1 12 ( )

22T x c

cc = +

32

3

1

21 3

T dTdx

c T=

3

1 2

1

21 3 2 ( )

9c T x c

c = +

The total entropy production can be determined with the use of the method

described in Case#2

2 2

1 1

2 2

1 1

1 1 21 0

2 1

1

1 1

1

21 0 11

1 1 1

2 (1 )1

2 (1 )

12

2 1

n

n

T T

t nn nT T

n

T Tn n

n nT T

Tk nc T

T i L TdT dT

T T ncTTT k

T T

ncT i L Tk dTdT

T T k T nc T

= + =

= +


87/102

87

7.2 Boundary conditions of the second and third kinds - heat conductioncoefficient independent of temperature

Case#1 Boundary conditions of the second kind (gradient at x=0)

22

2

10

d T dT

dx T dx

=

2

0 constant

1 (1)

dTx

dx

x T T

= = =

= =

the solution is

1

3( )c x

T x c e=

The graph of T(x) vs. x for dT/dx = -1and different values of T2is shown in Fig.

7.5.

Fig.7.5Graph of T(x) versus x for dT/dx = -1 and for different values of T2.


88/102

88

Local entropy production

( )11

22

2

3 1 1 t22

3

( ) constant =

( )

c x

c x

k dT k x c c e kc

T x dx c e

= = = =

Assuming, as before k = 20 W/mK, the following numerical values for total

entropy production are obtained and shown in Table 7.8.

Table 7.8Boundary

value at x=1T2=0.1 T2=0.2 T2=0.5 T2=0.8 T2=1.0

Total Entropy

Production

t(W/K)

60.9 35.2 14.5 8.5 6.4

Case#2 - Boundary conditions of the second kind (gradient at x = 1)

22

2

10

d T dT

T dxdx

=

1

1

0 ( 0)

1

x T x T

dTx

dx

= = =

= =

the solution is

1

3( )c x

T x c e=

The graph of T(x) vs. xfor different values of1and for T1=1.0 is shown in Fig.

7.6.

In addition, local entropy production

( )11

22

2

3 1 1 t22

3

( ) constant =( )

c x

c x

k dT k x c c e kc

T x dx c e

= = = =


89/102

89

Fig.7.6Graph of T(x) versus x for different values of dT/dx and for T1=1.0

Case#3 - Boundary conditions of the third kind (Sturm-Liouville boundary

conditions).

22

2

10

d T dT

dx T dx

=

0 1 0 0

1 2 1 0,0

0 ( )

1 ( )

x x

x x

dTx k h T T

dx

dTx k h T Tdk

= =

= =

= =

= =

or

0 1 0 1 0 1

1 2 1 2 0,0 2

x x

x x

dTk h T hT

dx

dTk h T h T

dk

= =

= =

+ = =

= =


90/102

90

which in combination with the solution

xcecxT 13)( =

results in the following equation for constant c1:

1

1 2

1 1 1 2( )ch kc e kc h

=

+

where

13

1 1

c

h k c

=

Taking

T0=1.3 , T0,0=0.85 (dimensionless temperatures)h1=h2=20, 200, 2000 W/m

2K

k = 20 W/mK

1= h1T0 = 26, 260, 2600 W/m2K

2= -h2T0,0= -17, -170, -1700 W/m2K

results in the following graph of T(x) vs. xas shown in Fig.7.7.

Fig.7.7 Graph of T(x) vs. x for different values of heat transfer coefficient

h and for T0,0=0.85


91/102

91

Similarly, taking

T0=1.3 , T0,0=0.05h1=h2=20, 200, 2000 W/m

2K

k = 20 W/mK

1= h1T0 = 26, 260, 2600 W/m

2

K2= -h2T0,0= -1, -10, -100 W/m2K

the graphs presented in Fig. 7.8 are produced.

Fig.7.8 Graph of T(x) vs. x for different values of heat transfer coefficient

h and for T0,0=0.05

Local entropy production, as in the previous cases is given by

( )11

22

2

3 1 1 t22

3

( ) constant =( )

c x

c x

k dT k x c c e kc

T x dx c e

= = = =


92/102

92

7.3 Boundary conditions of the third kind (Sturm-Liouville boundaryconditions) - heat conduction coefficient dependent on temperature

( )

21 1

'' ' 02

dk

T Tk dT T

+ =

withn

T

Tkk

=

1

1

becomes

22 1'' ( ') 0n

T T

n T

+ =

(7.15)

Sturm-Liouville boundary conditions are described by the system of two

equations:

1 0 1 0 0

2 1 2 0,01

0 ( )

1 ( )

x x

x x

dTx k h T T

dx

dT

x k h T Tdk

= =

= =

= =

= =

Thus

1 0 1 0 1 0 1

2 1 2 1 2 0,0 2

x x

x x

dTk hT hT

dx

dTk h T h T

dk

= =

= =

+ = =

= =

(7.16)

It can be easily shown that the solution of equation (7.13) is

2

( )nT Ax B= + (7.17)

which when substituted into Sturm-Liouville boundary conditions (7.14) gives:


93/102

93

( ) ( )

2 21

1 1 1

2 21

2 2 2

2

2

n n

n n

Ak B h B

n

A

k A B h A Bn

+ =

+ + =

taking

T0=1.3 , T0,0=0.05

h1=h2 =20 W/m2K

k = 20 W/mK

1= h1T0 = 26 W/m2K

2= -h2T0,0 = -1 W/m2K

and using

[ ] [ ]

2

2 /

1 1

2 / 1

1

2 / 1 2 /12 22

( ) ,

,2

2,

( )

n

n

n

n n

T Ax B

n h BA

k B

AkA B h A B

nB

+

= +

=

+ + =

(7.18)

produces the following graph of T(x) vs. xfor different values of n.

Fig.7.9 Graph of T(x) vs. x for different values of n


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8. Applications

As stated by Bejan [1]:Entropy Production Minimization is the method of

thermodynamic optimization of real systems that owe their thermodynamic

imperfections to heat transfer, fluid flow, and mass transfer irreversibilities.The method combines at the most fundamental level the basic principles of

thermodynamics, heat and mass transfer and fluid mechanics and because of its

interdisciplinary character, EGM is distinct from each of these classicaldisciplines.

Thesis Sgen

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