24 CHAPTER 2 MODELING OF CSTR PROCESS 2.1 INTRODUCTION The CSTR process model is needed to understand the dynamic characteristics and for design a proper control scheme. In CSTR, irreversible exothermic reactor concentration and temperature is to be maintained at desired range through inlet coolant flow rate manipulation. The CSTR has either constant or variable mixing condition inside the reactor. In order to identify the model of CSTR, the mass, energy and component balance equation must be properly explained. The derivation of these balance equations needs a background study about the process parameters and their chemical reactions during the mixing. The following section explains the CSTR linear and nonlinear modeling techniques. 2.2 PROCESS DESCRIPTION (LINEAR CONDITION) The schematic diagram of the CSTR is shown in Figure 2.1. The reactant ‘A’ is fed to the reactor with volumetric flow rate q f , molar concentration (or composition) C f and temperature T f . The components inside the reactor are well mixed with a motorized stirrer. Both the reactant A and product B are withdrawn continuously from the reactor with a flow rate, concentration C and a temperature T. To remove the exothermic heat that is
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CHAPTER 2
MODELING OF CSTR PROCESS
2.1 INTRODUCTION
The CSTR process model is needed to understand the dynamic
characteristics and for design a proper control scheme. In CSTR, irreversible
exothermic reactor concentration and temperature is to be maintained at
desired range through inlet coolant flow rate manipulation. The CSTR has
either constant or variable mixing condition inside the reactor. In order to
identify the model of CSTR, the mass, energy and component balance
equation must be properly explained. The derivation of these balance
equations needs a background study about the process parameters and their
chemical reactions during the mixing.
The following section explains the CSTR linear and nonlinear
modeling techniques.
2.2 PROCESS DESCRIPTION (LINEAR CONDITION)
The schematic diagram of the CSTR is shown in Figure 2.1. The
reactant ‘A’ is fed to the reactor with volumetric flow rate qf , molar
concentration (or composition) Cf and temperature Tf. The components inside
the reactor are well mixed with a motorized stirrer. Both the reactant A and
product B are withdrawn continuously from the reactor with a flow rate,
concentration C and a temperature T. To remove the exothermic heat that is
25
generated due to the chemical reaction, coolant is circulated at outer side of
the reactor. A inlet coolant stream with a volumetric flow rate qc , and an inlet
temperature Tcf continuously take out the heat to maintain the desired reaction
temperature.
Figure 2.1 CSTR Process setup
The objective of the controller design is to keep the concentration
(C) and temperature (T) of the product into desired range by adjusting the
inlet coolant flow rate qc(t). The nominal initial parameter settings of the
process considered in this study are given in Table 2.1.
Table 2.1 CSTR Parameters
Process parameter Initial operating condition Inlet feed flow rate (qf) 100 l/min Inlet feed temperature (Tf) 350 K Inlet coolant temperature (Tcf) 350 K Inlet concentration (Cf) 1 mol/l Volume of the tank (V) 100 l Activation energy (E/R) 1104 K Reaction rate constant (Ko) 7.21010 min-1 Heat reaction -2105 cal/mol Liquid density (ρ) 1103 g/l
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2.3 MATHEMATICAL MODELING (LINEAR MODEL)
The following assumptions are made to the linear CSTR process.
1. Exothermic reaction
2. Constant mixing inside the reactor
3. Constant volume and constant parameters
The mass balance equation of the CSTR is expressed in Equation (2.1)
Vdt
dFF inin (2.1)
In flow mass – Out flow mass = Rate of change of mass
Where, Fin - Inlet flow rate,
F - Outlet flow rate,
ρ - Density of the reactor,
ρin - Density of inlet stream and
V - Volume of the tank.
The change of individual components inside the reactor with
respect to time during reaction is identified to find CSTR model. The
component balance equation of the ith component is expressed as in Equation
(2.2)
AAAAinin Cdt
dVKVCFCCF (2.2)
ith component in flow – ith component outflow + ith component value = Rate
of change of ith component
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Negative sign indicates that the CA decreases during reaction.
Assuming that the reaction is BA , i.e., component ‘A’ reacts irreversibly
to form component ‘B’ and the heat generated during reaction is removed
through the coolant flow (qc). The energy balance equation of the reactor is
expressed in Equation (2.3)
( )P in A c P
dTC F T T kVC H q C V
dt (2.3)
where k = /
0
E RTk e
k0 is a pre-exponential factor, E/R is the activation energy, T is the reaction
temperature and R is the gas law constant.
From the mass, energy and component balance equations, the
model describing the rate of change of concentration and temperature in the
system is then given by Equation (2.4) and Equation (2.5)
))((
)(exp1)(
))(
exp()())((
3
2
1
tTTtq
KtqK
tRT
EtCKtTT
V
q
dt
dT
cf
c
c
f
f
(2.4)
)(exp)())(( 0
tRT
EtCKtCC
V
q
dt
dCf
f (2.5)
The CSTR process model derived from Equation (2.4) and
Equation (2.5) shows that, it has exponential terms and product terms. The
derived equations are implemented in MATLAB Simulink to perform open
loop and closed loop analysis. The simulink model of the CSTR process is
shown in Figure 2.2.
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Figure 2.2 Simulink model of CSTR process
The open-loop response of the temperature and concentration when
the inlet coolant flow rate (qc(t)) varies from 85 l/min to 110 l/min is obtained
as given in Figure 2.3 and Figure 2.4 respectively. From the responses, it is
observed that the parameters vary from over-damped to underdamped, which
clearly shows the nonlinear dynamic behavior of the CSTR process.
0 10 20 30 40 50 60 70 800.09
0.1
0.11
0.12
0.13
0.14
0.15
Time in Seconds
Con
cent
rati
on i
n m
ol/l
Concentration in CSTR
Figure 2.3 CSTR Concentration – open loop response
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0 10 20 30 40 50 60 70 80431
432
433
434
435
436
437
438
439
Time in seconds
Tem
pera
ture
in K
Reactor Temperature
Figure 2.4 CSTR Temperature – open loop response
2.3.1 Linearization
The objective of the linearization is to get the model of the process
with the form expressed in Equation (2.6)
DuCxY
BuAxX
(2.6)
The input, output and state of the system are expressed as deviation variable
form given in Equation (2.7) and Equation (2.8)
s
AsA
TT
CC
x
xx
2
1 (2.7)
sTTy
s
AfsAf
fsf
jsj
FF
CC
TT
TT
u
u
u
u
u
4
3
2
1
(2.8)
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Where Tj is the jacket temperature and Tf , CAf , F are the inputs. The jacobian
matrix parameters A, B, C and D are derived as in Equation (2.9) – (2.11)
1
1
2
2
1
2
2
1
1
1
2221
1211
)(sAs
PP
s
P
sAss
KCC
H
CV
UA
V
FK
C
H
KCKV
F
x
f
x
f
x
f
x
f
aa
aaA
(2.9)
Where
s
s
s
s
os
T
KK
RT
EKK
1
exp
PCV
UA
u
f
u
f
b
bB
0
1
2
1
1
12
11 (2.10)
The output matrices are:
C = [0 1]
D = 0 (2.11)
From the initial parameters and state space model, five linear
operating regions are identified around the steady state and the Eigen values
of the each regions are derived to find the stability condition and is shown in
Table 2.2.
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Table 2.2 CSTR Stable operating regions
Operating region Eigen values Stability
CA = 0.0795, T=443.4566, qc= 97 λ1 = -1.0 ; λ2 = 1.5803 Saddle point