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 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product B are withdrawn continuously from the reactor with a flow rate,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

26

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

27

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

29

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)

30

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

CA = 0.0885, T=441.1475, qc= 100 λ1 = -2.3899 ; λ2 = -1.0 Stable

CA = 0.0989, T=438.7763, qc= 103 λ1 = -7.7837 ; λ2 = -1.0 Stable

CA = 0.1110, T=436.3091, qc= 106 λ1 = -24.9584 ; λ2 = -1.0 Stable

CA = 0.1254, T=433.6921, qc= 109 λ1 = -59.8325 ; λ2 = -1.0 Stable

Where CA, T, qc are the linearization points of the CSTR. The

transfer function of the linear regions and its open loop poles are derived and

listed in Table 2.3.

Table 2.3 Open loop poles _ Stability Analysis

Operating region Transfer function Location of

Poles Stability

1 42.15188.5

04351.0882.82

66.0

SS

e s

-2.59±j3.31 Stable

2 22.13931.3

04237.02 SS

-1.97±j3.96 Stable

3 15.11779.2

04118.0665.22

15.0

SS

e s

-1.39±j3.12 Stable

4 205.9728.1

03988.022.22

16.0

SS

e s

-0.864±j3.12 Stable

5 352.77632.0

03844.0221.12

15.0

SS

e s

-0.382±j3.03 Stable

32

For qc=113.25, CA=0.1425 and T=460.125, the transfer function

and poles are obtained as G(s) =69.22948.5

1915.02 SS

and 2.97±j3.79

respectively and system attains unstable state. The result clearly shows that

the system attains unstable state when the coolant flow and concentration

raises above 110 l/min and 0.13 mol/l respectively. From the step input

responses and stability analysis, it is observed that the stable operating region

of the CSTR falls in C(t) - (0,0.13566 ) mol/l & qc (t ) - (0,110.8) l/min.

In CSTR process linear modeling, the operating regions for stability

are selected through the local model networks i.e., work region of the CSTR

divided into N small regions based on state space model, each of which

linearly approximates the local property of the assigned region.

2.4 PROCESS DESCRIPTION (NONLINEAR CONDITION)

The CSTR presented in literatures for control purposes have been

modelled as linear. But in real time most of the reactors are nonlinear in

nature which has high dynamic characteristics. For nonlinear CSTR, mixing

of reactants inside the reactor is non-uniform, resulting that the concentration

at different points are not uniform. The inlet coolant flow passes into the

reactor quickly as a result the concentration changes rapidly than the linear

condition. The temperature inside the reactor varies continuously and

provides high dynamic output characteristics. Hence, the entire CSTR plant

model should be identified for proper control.

2.4.1 Takagi-Sugeno Fuzzy Model

Takagi & Sugeno (1985) introduced a modeling technique to

represent the nonlinear system using local linear models. The linear model of

CSTR is used in each fuzzy rule to describe the nonlinear behavior. The

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overall multi model network is formulated (Tan W et al 2004) by blending the

local linear models through the gating system. Moreover, global performance

of the Local Model Network (LMN) highly depends on the performance of

the local controllers. The gating system, acts as a weighting function and

smoothes the transient response when the set-point changes. The rule

associated with particular local model of the system can be defined as given

in Equation (2.12),

)()(

)()()(

)(.....)(: 1111

txCty

tuBtxAtxTHEN

isMtandZisMtIFZiRule

i

ii

ii

(2.12)

i = 1, 2, ..., r

where {z1(t), z2(t),…,zp(t)} are premise variables, nRtx )( is the state vector, mRtu )( is the input vector, nn

i RA , mn

i RB , nq

i RC are the system

matrices for rule ‘i’, y(t) is the output and ‘r’ is the rule number. The center of

gravity defuzzification method is used to derive the output of the fuzzy model

which can be expressed as in Equation (2.13),

r

i

i

r

i

ii

txCzhty

tuBtxAzhx

1

1

1

1

)()()(

)]()()[(

(2.13)

Where,

i

j

jij

r

i

ZMzw

zw

zwzh

1

1

1

1

11

)()(

)(

)()(

34

Mij(Zj) is the fuzzy membership grade of Zj in Mj1. It is assumed

that Wi(z(t))≥0, i=1,2,…,r and for

r

i

i tzw1

0))(( all ‘t’. Therefore, grade of

membership function is expressed as

N

i

i

i

zh

andzh

1

1)(

];1,0[)(

for all ‘t’

The output of T-S model provides the information about the

nonlinear process from the linear models.

2.4.2 Neural Network Model Identification

Soft computing techniques such as fuzzy based and Neural

Network based nonlinear model identification studied by Anna Jadlovska et al

(2008) and Rankovic et al (2012) are widely preferred to identify nonlinear

process dynamic behavior. These techniques identify the input-output data of

the process where the state equations of the process are indefinite, and the

states are unattainable. The performance of Nonlinear Autoregressive models

with eXogenous input (NARX) model is experimented by Eugen Diaconescu

(2008) and from the results it is concluded that NARX model is able to

capture the nonlinear dynamics of any process. The single input - single

output nonlinear process is described by NN model expressed as in Necla

Togun et al (2012) and is shown in Equation (2.14).

)()](),....,2(),1(),(),....,2(),1([)( kemkykykynkukukufky (2.14)

Where ‘n’ and ‘m’ are the maximum input and output with n≥0, m≥1 and f is

a nonlinear function. The implementation of NN model identification is

explained in Figure 2.5.

35

[

Figure 2.5 Neural Network based nonlinear model identification

The unknown function ‘f’ is approximated by the regression model

shown in Equation (2.15)

m

i

m

ij

n

i

m

j

n

i

m

j

n

i

n

j

kejkyikujicjkyikyjib

jkuikujiajkyjbikuiaky

1 0 1

1 1 0 1

)()()(),()))(),(

)()(),()()()()()( (2.15)

Where, a(i) and a(i,j) are coefficients of exogenous terms, b(i) and b(i,j) are

autoregressive terms, c(i,j) is the nonlinear cross terms. Equation (2.15) can

be written in the following matrix form as shown in Equation (2.16).

TTTTT XCYBUAbyua

mky

ky

ky

][][][.

)(

...

)1(

)(

(2.16)

The output can be expressed as in Equation (2.17)

CUY (2.17)

The flow of NARMA model identification is explained in Figure 2.6

y(k-n)

y(k-1)

y(k-1)

u(k-n)

u(k-2)

u(k-1) y(k) u(k)

y’(k)

Nonlinear plant

Z-1

Z-1

Z-n

NN MODEL

Z-1

Z-1

Z-n

36

Figure 2.6 Nonlinear model identification

The neural network models derived from the input and output data

of the process has poor generalization, over-fitting issues and requires more

training data.

2.4.3 LSSVM Based Model Identification

The LSSVM based on SVM is presented by Suykens (1999) to

eliminate the complex quadratic programming solving procedure. The

LSSVM model uses the following function given in Equation (2.18) to

approximate the unknown function

bxwxy T )()( (2.18)

Where ɸ(x) is a nonlinear mapping from the input space x to a higher

dimensional feature space, w is the weight vector and b is the bias. In LSSVM

the optimization function can be obtained by using the squared loss function

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and equality constraints, which gives the following optimization function as

in Equation (2.19)

2

1

1 1min ( , )

2 2

NT

k

k

J w e w w C e

(2.19)

subject to the equality constraints as given in Equation (2.20)

( ( ) ) 1 0, 1,...,T

k k ky w x b e k N (2.20)

Where Ke is the training error and C is the regularization parameter. The

Lagrangian function given in Equation (2.21) is constructed to solve the

optimization problem

1

( , , ; ) ( , ) { ( ( ) ) 1 }N

T

k k k k

k

L w b e J w e y w x b e

(2.21)

Where αkand b are the Lagrange multipliers to be estimated based on the N

training data set. Optimal condition for the Equation (2.21) can be obtained

through the solution of partial derivatives of L(w,b,e;α) with respect to

w,b,e;α as in the form Equation (2.22)

N

K

N

k

kkkK xCexww

L

1 1

)()(0 (2.22)

Substituting ek and w with αk and b, we can get the following Equation (2.23)

N

k

Kkk bxxKy1

),( (2.23)

Where K and b are the solutions to the linear system and K is the kernel

function satisfying the mercer condition, i.e., represents the mapping of input

38

space x with high dimensional feature space. The solution of the above linear

equation is given in Equation (2.24) and Equation (2.25)

1

0

0

0

0

00

000

00

e

b

w

IYP

IC

Y

PI

I

T

T

(2.24)

Where P=[ φ(x1)Tyl…….φ(xN)TyN ], and Y=[ y1…yN]

1

001

b

ICPPY

YT

T

(2.25)

The classification or regression can be done through LSSVM by

solving the linear equation instead of complex quadratic programming

method. LSSVM uses less training samples, so that it can solve over-fitting

issues and according to finite samples and provides better model performance

with better generalization. In this work, LSSVM based nonlinear model

identification with MLP and linear kernel function is proposed for CSTR

process.

2.5 SUMMARY

In this section, the local linear model of an exothermic CSTR with

a first order reaction is considered for simulation studies. The linear model of

the CSTR is formulated by assuming a constant reactor volume and perfect

mixing in the reactor. The mathematical model of the CSTR under nonlinear

condition is obtained by considering non-uniform concentration and mixing

inside the reactor. The nonlinear model of the CSTR is derived by two

methods viz., T-S multi model from local linear models and second approach

using neural networks and LSSVM based identification. The second one

eliminates the issues involved in local linear models.

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In the following chapter, the adaptive N-PID control design for a

CSTR using GA, SA, PSO, ABC and their hybrid approaches are discussed

briefly.