PCT 301T Modeling(1)
-
Upload
portia-shilenge -
Category
Documents
-
view
218 -
download
1
Transcript of PCT 301T Modeling(1)
-
8/22/2019 PCT 301T Modeling(1)
1/21
Page 1 of21
PCT 301T
MODELING
Modelling is used to study the dynamic behaviour, process design,
model-based control, optimization and predictions of the processes.
Modelling principles
Dynamics models of chemical processes consist of ODE (ordinary
differential equations) and / partial differential equations (PDE), plus
related algebraic equations.
For process control problems, dynamic models are derived using
unsteady state conservation laws.
-
8/22/2019 PCT 301T Modeling(1)
2/21
Page 2 of21
A systematic approach for developing dynamic models
1. State the modelling objectives and the end use of the model.2. Draw the schematic diagram of the process and label all
process variables.
3. List all the assumptions involved in developing the model.
4. Write appropriate conservation equation (mass, component,
energy, and so forth).5. Introduce equilibrium relation and other algebraic equation
(from thermodynamics, transport phenomena, chemical kinetics
etc).
6. Perform degree of freedom.
7. Simplify the model. (Grouping of like terms).8. Classify the inputs as the disturbance or manipulated
variables.
-
8/22/2019 PCT 301T Modeling(1)
3/21
Page 3 of21
Conservation laws
Theoretical models of chemical processes are based onconservation laws such as the conservation ofmass and energy .
Conservation of Mass
Rate of mass accumulation = rate of mass in
rate of mass out
Conservation of Component i
Rate of = rate of rate of + rate of
mass mass in mass out mass
accumulation component i produced
-
8/22/2019 PCT 301T Modeling(1)
4/21
Page 4 of21
Conservation of Energy
Rate of = rate of rate of + net rate of
energy energy in by energy out by heat addition
accumulation convection convection to the system
+ net rate of work performed on the system
by the surrounding
-
8/22/2019 PCT 301T Modeling(1)
5/21
Page 5 of21
Uses of Mathematical Modeling
To improve understanding of the process
To optimize process design/operating conditions
To design a control strategy for the process
To train operating personnel
-
8/22/2019 PCT 301T Modeling(1)
6/21
Page 6 of21
Developm ent of Dynam ic Models
Il lustrat ive Example: A Blending Process
FIRST ORDER SYSTEM RESPONSE FOR UNSTEADY STATE
PROCESS
Example 1
Liquid - level system (Mass storage)
Object ive is to measure the height level (h) of the system and
determine the transfer function.
-
8/22/2019 PCT 301T Modeling(1)
7/21
Page 7 of21
Assumpt ion made: out flow (Q0) is linearly related to the hydrostatic
head of the liquid level through the resistant (R)
Conservat ion equat ion :Mass
Accumulation = Input - output
Degree of freedom analys is: 4 -3 = 1
2 input and 2 output
Q0
h(t)
Rh
Qi
x(t)
-
8/22/2019 PCT 301T Modeling(1)
8/21
Page 8 of21
Accumulation = Input output
outin mmdt
dM
...Eq(1)
0
)( QQ
dt
Ahdi ..Eg(2)
R
hQ
dt
Adhi
. ...Eq(3)
RQhdtdhRA i
..Eq(4)
-
8/22/2019 PCT 301T Modeling(1)
9/21
Page 9 of21
Apply Laplace transformation in Eq (4)
And find the transfer function.
11)(
)(
s
K
RAs
R
sX
sH p
...Eq(5)
Apply the inverse to determine the process response
DO EXAMPLE (B) FROM YOUR NOTES
Process Gain (Kp)
Process time
(tau)
-
8/22/2019 PCT 301T Modeling(1)
10/21
Page 10 of21
Example 2
Consider a perfectly mixed stirred-tank heater; with a single feed
streams and a single product stream, as shown below. Assuming that
the flow rate and temperature of the inlet streams can vary, that the
tank is perfectly insulated, and the rate of heat, Q added per unit timecan vary, heat losses are negligible, density and heat capacity of the
liquid are assumed to be constant. Develop a mathematical model to
find the tank liquid temperature as a function of time and its time
response when the inputs are subjected to the step change.
-
8/22/2019 PCT 301T Modeling(1)
11/21
Page 11 of21
Fig: 1 Stirred tank heating process with constant volume.
Object ive is to measure the temperature (T) of the system anddetermine the transfer function.
-
8/22/2019 PCT 301T Modeling(1)
12/21
Page 12 of21
Assumpt ion made: the outlet and the inlet flow are equal, volume,
density and the heat capacity of the liquid are constant and heat losses
are negligible.Conservat ion equat ion :Energy
Rate of = rate of rate of + net rate of
energy energy in by energy out by heat addition
accumulation convection convection to the system
+ net rate of work performed on the system
by the surrounding
-
8/22/2019 PCT 301T Modeling(1)
13/21
Page 13 of21
WQTfCTfCdt
dHpip .Eq(1)
QTTfCdt
dTCV ipp ...Eq(2)
QTfCfCpTdt
dTCV
ipp
.Eq(3)
QfC
TTdt
dT
fC
CV
p
i
p
p 1
...Eq(4)
-
8/22/2019 PCT 301T Modeling(1)
14/21
Page 14 of21
Q
fC
TT
dt
dT
fC
CV
p
i
p
p
11
Apply the Laplace transformation and find the transfer function
So,Q
sf
V
fCsT
sf
VsT
p
i
1
1
)(
1
1)(
...Eq(5)
Process time
(tau)
Process Gain (Kp2)
Process Gain (Kp1)
-
8/22/2019 PCT 301T Modeling(1)
15/21
Page 15 of21
)(1
)(1
)(21
sQs
KsT
s
KsT
p
i
p
...Eq(6)
Two transfer functions
T.F 1 1)()( 1
sK
sTsT p
i
T.F 2 1)(
)( 2
s
K
sQ
sT p
-
8/22/2019 PCT 301T Modeling(1)
16/21
Page 16 of21
An unsteady-state mass balance for the blending system:
rate of accumulation rate of rate of(2-1)
of mass in the tank mass in mass out
1 2
(2-2)
d Vw w w
dt
-
8/22/2019 PCT 301T Modeling(1)
17/21
Page 17 of21
The corresponding steady-state model
The Blending Process Revisited
For constant density:
1 2
1 1 2 2
0 (2-4)
0 (2-5)
w w w
w x w x wx
-
8/22/2019 PCT 301T Modeling(1)
18/21
Page 18 of21
Equation 2-7 can be simplified by expanding the accumulation term using the
chain rule for differentiation of a product:
Substitution of (2-7) into (2-8) gives:
Substitution of the mass balance in (2-6) for in (2-9) gives:
-
8/22/2019 PCT 301T Modeling(1)
19/21
Page 19 of21
After canceling common terms and rearranging (2-6) and (2-10), a more
convenient model form is obtained:
-
8/22/2019 PCT 301T Modeling(1)
20/21
Page 20 of21
Typical question
A stirred-tank heating process described by)(
1
1)('
1
/1)(' ' sT
s
w
msQ
s
w
m
wCsT i
, is used to
preheat a reactant containing a suspended solid catalyst at a constant flow rate of 1000
kg/h. the volume in the tank is 2 m3, and the density and specific heat of the suspended
mixture are, respectively, 900 kg/m3
and 1 cal/g0C. The process initially is operating
with inlet and outlet temperature of 100 and 1300C. The following questions concerning
process operations are posed:
-
8/22/2019 PCT 301T Modeling(1)
21/21
Page 21 of21
(a) What is the heater input at the initial steady state and the value of K and
TAU?
(b) If the heater input is increased by 30%, how long will it take for the tank
temperature to achieve 99% of the final temperature change?
Assume the tank is at its initial steady state. If the inlet temperature in increased
suddenly from 100 to 1200C, how long will it take before the outlet temperature
changes from 130 to 1350C.