Tj Project Report-final 1 23

8/3/2019 Tj Project Report-final 1 23

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CHAPTER 1

INTRODUCTION

1.1 INTRODUCTION

In order to develop a model that is mathematical for a non-linear system is a

vital theme in a lot of disciplines of engineering. The models can be used for

a variety of things such as simulations, for the analysis of the behaviour of

the system, for a better understanding of the mechanisms in the system and to

create new processes and new controllers. Two broad methods can be

followed in obtaining the mathematical model. The first one being the

formulation of the model from the first principles by using the laws

governing the system. 'This method is usually referred to as mathematical

modelling. The second method requires the experimental data which is

obtained by exciting the plant and measuring its response. This method is

known as system identificationand is preferred in the case in which the plant

or process involves really complex physical phenomena or in which the plant

shows strong nonlinearities.'[1].

Getting a model which is mathematical and for a system which is complex

might be time consuming as it needs assumptions such as the defining of an

operating point,ignoring some system parameters doing linearization about

that point.etc[6]

Many process models are non-linear. When a model has only one different

equation the solution of that model might or might not be very complicated.

Whereas, when more than one differential equations are present it tends to get

difficult to understand the dynamics of the processes. Thus we tend to

linearise transfer process models. Linearization is beneficial in the following

ways:


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1) Systems that are linear can be analyzed easily using mathematical tools.[3]

2) The behaviour of the system can be studied and analyzed because the

relationship between the process inputs and the outputs are expressed in terms

of process variables and operating conditions.

3) The stability of a linear system is generally easy to investigate.

If one needs to write the model equations in transfer function format it is

necessary to use Laplace transformation as well. Hence to apply Laplace

transform to non linear equations, we need to linearise the equations first.

1.2 AIMS AND OBJECTIVES

1) To study the dynamics of a CSTR process taken for the case study.[5]2) Develop a non linear model by relating a subsystem in the MATLAB

software and to simulate the response of the non linear model of a CSTR

process.

3) To develop linear model of a CSTR process.4) Design of suitable control strategies for the process mentioned above.5) Testing the performance of the controller through time domain

specifications.

1.3 ORGANISATION OF THE PROJECT

The work presented in this report starts with a description of what a

continuously stirred jacketed tank reactor is. This is followed by the

description of the problem that has been taken up. Using this description and

linearization, the material balance equations of the CSTR have been arrived at.

The Modeling has further been completed by the two point method. This is

followed by the control of the process which is done using Cohen-coon and

IMC based control. The time domain characteristics of the closed loop

responses have been tabulated and compared.


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CHAPTER 2

CONTINUOUSLY STIRRED TANK REACTOR

2.1 JACKETED CSTRThe jacketed continuously stirred reactor is a self adaptable process. This

means that the measured process variable (PV) logically tries to reach a steady

operating level if the controller output (CO) and the major disturbances (D) are

held constant for a particular amount of time. As can be seen from Fig 2.1, in a

CSTR a reactant feed flow enters through the top of the vessel. After entering

a chemical reaction tends to convert most of the feed into the desired product

as the material moves gradually through the stirred tank. The stream that exits

from the bottom of the vessel includes the newly created product as well as the

portion of the feed that did not get converted while in the vessel. It is necessary

to keep in mind certain assumptions such as that the residence time which is

(the probability distribution function that describes the amount of time a fluid

element can spend inside the reactor.) or overall flow rate of reactant feed as

well as the product through the vessel are kept constant. The chemical

reaction that occurs in the reactor is exothermic that means that the heat energy

is released as feed gets converted to product.[7]


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Fig 2.1-- Jacketed stirred reactor

2.2 DESCRIPTION OF THE COOLING JACKET

The chemical reaction that takes place releases heat, this energy in turn leads

to the rise in temperature of the material in the vessel. As the temperature

increases the conversion of feed to product happens faster and thus leads to the

release of even more heat. In order to prevent the hotter temperatures from

further increasing the rate of reaction that will eventually produce even more

heat, the vessel is enclosed within an outer shell (jacket). A cooling liquid

moves through the jacket that collects heat energy from the outer surface of the

reactor vessel and carries the heat away along with itself as it exits from the

jacket.[7]

As the flow of the cooling liquid through the jacket increases even more heat is

removed. As this happens the reactor temperature is lowered and thus it slows

the rate of the reaction and decreases the amount of feed that is converted to

product. But when the flow of cooling liquid through the jacket decreases

some of the energy that is given off from the heat-producing reaction

accumulates in the vessel and drives the reactor temperature higher rather than

being carried away with the cooling liquid. This results in an increased

conversion of reactant feed to product.


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As it is known that the reactor has a constant residence time the amount of heat

energy which is given off inside the vessel is directly related to the percent of

feed converted to product. Thus by controlling the temperature in the reactor

the percent conversion to the desired value can be maintained. Because the

vessel is well mixed the temperature inside the reactor is almost the same as

the temperature flowing out the exit stream and therefore our measured

process variable (PV) becomes our reactor exit temperature.[7]

The major disturbance in this jacketed stirred reactor is the cooling liquids

temperature that enters the jacket and that changes over time. Because the

temperature of the cooling liquid entering the jacket changes so does its ability

to remove heat energy from the reactor.

PV = reactor exit stream temperature (measured process variable, oC)

CO = signal to valve that adjusts cooling jacket liquid flow rate (controller

output,%)

D = temperature of cooling liquid entering the jacket (major disturbance,

o

C)

SP = desired reactor exit stream temperature (set point,oC)


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CHAPTER 3

PROBLEM FORMULATION

3.1 PROCESS DESCRIPTION

Fig 3.1 Schematic diagram of a CSTR

The figure above 3.1 shows us a schematic diagram of a CSTR process and

the variables associated with it. In the CSTR a first order irreversible

exothermic reaction A B takes place. "The heat that is produced due to the

reaction is removed by the coolant that flows through the jacket around the

reactor. The transfer characteristics of the above mentioned process are shown

below."[3]


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Fig 3.2 Transfer characteristics of a CSTR

Curve A in the above fig 3.2 describes the amount of heat released by the

exothermic reaction and is shown by a sigmoid function of the temperature (T)

in the reactor. Whereas the heat that is removed from the reaction by the

coolant is shown by a linear function that of the temperature (T) and is

represented by the curve B.[4]

Accordingly we know that when the CSTR is at

a steady state the heat produced by the reaction should be equal to the heat

removed by the coolant. This need gives us the steady states shown in the fig

no and is represented by P1, P2, P3 at the junction of curves A and B. The

steady states P1 and P3 are known as stable while P2 is unstable.


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3.2 PROBLEM DESCRIPTION

The process under our study is a first order exothermic irreversible CSTR

process. A feed of a pure substance say "A" is mixed with a perfect recycle

stream with the recycle flow rate (1-)F.[5] The primary interest is in the

conversion that takes place inside the reactor which is basically the outlet

concentration of component A .Certain assumptions are made so as not to

complicate our model. The assumptions are as follows :

1) The reactor is mixed ideally.2) The heat losses to the environment are ignored.3) Physical properties are assumed constant.

It is also assumed that the level is constant. This is because if the level is

variable the system becomes convoluted as then there is a need to deal with a

total mass balance equation, an energy balance equations and a component

balance equation. By assuming the level to be constant the total mass balance

is eliminated as our inlet flow will be equal to the outlet flow.[4]

Our model equations are as follows:

The component balance for the reactor is given by[2]:

dCA

V = FCAin + F(1-) CAFCAVkeE/RT

CA (3-1)dt

Putting =1 which means that it is zero cycle :

dCA

V = F (CAin -CA)VkeE/RT CA (3-2)

dt


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Similarly the energy balance equation is given by[2]

:

dT

VCp = FCp(Tin-T) + Vke E/RTCAH + Q (3-3)dt

Putting =1 which means that it is zero cycle we get:

dT

VCp = FCp(Tin-T) + VkeE/RT

CAH + Q (3-4)dt

The linearised model which relates changes in the reactor flow to the changes

in the concentration of the component A, is given by[2]

dCA 457.6s + 1

= 6.69*104 (3-5)dF 2.55*105 s2 + 1254.4 s +1

Nomenclature:

The variables which have been used in the model along with their steady state

values:[1]

1) V Reactor Volume = 5m32) CAOutlet Concentration of Component A = 200.13 kg/m33) CAin Inlet Concentration Of Component A = 800 kg/m34) F Total Volumetric Flow Rate = 0.005 m3/s5) k Pre Exponential Constant = 18.75 s-16) E Activation Energy For The Reaction = 30 kj/mol


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7) T Reactor Temperature = 413K8) Tin Temperature of inlet flow = 353K9) Density = 800 kg/m310) Cp Specific Heat = 1.0 kj/kg.K11) H Heat Of Reaction (exothermic) = 5.3 kj/kg12) Q Heat Supplied To The Reactor = 224.1 kj/sec13) R Gas Constant =0.0083kj/mol.K

It can be see from equation (3-2) and (3-4) that both the equations are non

linear. The problem we face with these equations is that we cannot estimate the

behaviour of the outlet concentration CA when the reactor itself is changing

throughout. In order to rectify this we need to linearise the model which in turn

helps us to get an idea of the process behaviour [7]

3.3 SIMULATION OF A CSTR PROCESS

Process simulation in MATLAB simulink is explained to develop a linear and

a non linear model for a CSTR process and to compare them

First a separate exponent block is created this is because the exponent of the

temperature is calculate twice.[6]

This is then created as a subsystem and can be

used further in the material balance and energy balance equations.


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Fig 3.3 Calculation of the exponent

Fig 3.4 the Exponent Part Shown As A Subsystem

Now the mass balance equation has been used to create another subsystem

known as the reactor component balance with inputs as F and T and the output

being CA.


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Fig 3.5 Reactor Component Balance

The structure above closely resembles the equation 1-1 which has been

adjusted in such a way that the derivative dCA/dt becomes the left hand side of

the equation

Fig 3.6 Reactor Component Balance Represented As A Subsystem


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Similarly the energy balance equation 1-3 has been represented as a subsystem

below

Fig 3.7 Reactor Energy Balance

Fig 3.8 Reactor Energy Balance Represented As A Subsystem


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The linear and the non linear reactor model of a CSTR has been shown below Fig 3.9

Fig 3.9 Non linear and linear reactor model of a CSTR process

The Final open loop response is shown below

Fig 3.10 Response of reactor concentration using original and linearised model


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3.4 TWO POINT METHOD:

The Two-Point Method helps to determine the parameters of the PID

controller. With the method one can find the times, t1 and t2 which are the time

when the process amplitude reaches 63.2 % and 28.3 % of the final

steady value respectively.[1]

Fig.3.11 response of the linearised model

From the Fig.3.11 above it is noticed that

t1 = which is the time taken to reach 63.2% of the steady-state value = 2500 sec

t2 = which is the time taken to reach 28.3 % of the steady-state value = 900 sec

= Process time constant = 1.5(t1-t2) =2400 sec

td = Dead-time = t1- = 100 sec

K = Static Gain = = 163793.1


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CHAPTER 4

CONTROL OF THE CSTR PROCESS

4.1 COHEN-COON METHOD OF CONTROL

Using the values above the PID values of the controller are calculated below.

These formulas are based on the cohen-coon method of tuning. Cohen-Coon

were two physicist who designed formulas to tune the parameters for a PID

controller.[2]

These formulas were derived so as to get minimum integral

square error.

Now using the values of , td as obtained above and substituting in the aboveequations we get the following values of KC, I, D :

KC = Proportional Gain = 15

I = Integral time = 0.050

D = Derivative time = 0.2

Putting the values of P, I D obtained above in the process below:

)2

11

4(

)8

13

632

(

)43

4(

p

d

p

pi

pc

T

T

T

TT

TK

4-1

4-2

4-3


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Fig .4.1 Block diagram for the closed loop response of a CSTR process under

PID controller(Cohen-Coon tuning method)

Fig .4.2 Closed loop response of a CSTR process under PID controller(Cohen-

Coon tuning method)


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4.2 INTERNAL MODEL CONTROL (IMC)

Internal Model Control (IMC), was developed by Morari and coworkers.It is

based on an understood process model and thus leads to methodical

expressions for the controller settings. The viewpoint behind IMC is that

control will be reached only if the system encapsulates either completely or

clearly some representation of the process to be controlled.This strategy has

the "POTENTIAL" to achieve the perfect control.The block diagram is shown

below:[6]

Fig.4.3 Block diagram for internal model control

From fig no 4.3 it can be seen that a process model and the controlleroutput

P are used to calculate the model response. This response of the model is

subtracted from the actual response of the system Y. The difference -Y isthen fed into the input of the IMC controller.[5]

Using the block diagram reduction technique the following relation is derived

4-4


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Incase of a special scenario when =G (perfect model) the above equation ca

be reduced to :

Now using the values of , td and K that have been obtained using the two-

point method we calculate the value of the PID controller by using the internal

model control (IMC method).

IMC approximation for a first order +time delay process:

1)()(

sTp

sKpesG

(4-6)

Where

Kp=163793.1, p= 2400 & =100

Now using a first order Pade approximation for dead time:

15.0

15.0

s

s

se

(4-7)

Using equations (4-6) and (4-7)

)15.0)(1)((

15.0)(

ssTp

sKpsG

(4-8)

Putting the values

)1100*5.0)(1)(2400(

1)100(5.01.163793)(

ss

ssG (4-9)

Now factoring out the non-invertible elements

4-5


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)15.0)(1)(()(

ssTp

KpsG

(4-10)

Now the idealized controller is shown by

(4-11)

1.163973

)1100*5.0)(12400(

ss(4-12)

Now we add the filter

= G-1

(s).f(s) (4-13)

Substituting (4-12) into (4-13)

1s

1

1.163973

)1100*5.0)(12400(

ss(4-14)

The corresponding PID is

(4-15)

Using the values obtained in the equations above the formulas mentioned below are

arrived at

Where, = filter factor

p

p

p

p

T2

T

0.5T

)0.5(

0.5T

d

i

pc

T

T

KK4-16

4-17

4-18


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From the two point method we calculated our G(s) as

(4-19)

G(s) = 163793.1e-100

(4-20)2400(s) +1

Now using the values of Kp , p and from equation (4-10) we get:

Kc = 32.24

Ti = 0.1333

Td = 10 ; = 0.2

Fig.4.4 Closed loop response of a CSTR process under PID controller(IMC-

tuning method)


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CHAPTER 5

PROCESS SIMULATION AND RESULTS

5.1 ADDITION OF DISTURBANCES TO THE PROCESS

It is necessary to consider the effect of disturbances on a process. One cannot

simply ignore its effect as disturbances can cause the output of the process to

deviate from the set point by a substantial amount.

Fig.5.1 Process with disturbance

The block labelled Step-1 acts as the disturbance to the system. The simulation

for the fig 5.1 has been shown below:


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Fig.5.2 Closed loop response with disturbance applied at 50 secs for a set point change

(200-400)

From Fig 5.2 it can be seen that the disturbance has been given at a time of 50

seconds and the range of the value of the disturbance is 30 units.

Thefig below (fig no 5.3) shows the regulatory response for the process when

the disturbance is applied at a time of 50 seconds and the range of the value of

disturbance is 30 units. However the difference being that the step input in this

process has been varied from (200-300) .

Fig.5.3 Closed loop response with disturbance applied at 50 secs for a set point change (200-300)


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5.2 TIME-DOMAIN SPECIFICATIONS

Peak time, settling time and percentage overshoot are some of the

characteristics that have been looked at:

1) Settling time is the time that the response takes to reach and remain inside a

particular tolerance level. This level may vary but it is usually taken to be 2%-

5% of the eventual value.[3]

2) Peak time is the time the response takes to reach its highest point at the very

first instant.

3) Overshoot is defined as the output that exceeds the eventual steady-state

output.

4) Percentage overshoot is shown by the subsequent formula :

(4.1)

A = Amplitude at peak time

B = final value of the response

5) Integral square error (ISE) and Integral absolute error (IAE) help to measure

a systems performance. They have been calculated for the process and the

output is shown using MATLAB.


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Fig.5.4 Calculation of ISE and IAE (step input 200-400) using Cohen-coon

From the Fig no 5.4 above it is seen that the ISE and IAE values have been

calculated. This has been done for a process with a step input change of 200-

400 units. A disturbance has been added to the system and the process is being

controlled by a PID controller which has been modelled by the Cohen-Coon

method.

In the figure above the block showing 'Display' gives the ISE whereas the

block which reads 'Display1' gives the IAE.

Below are all the MATLAB simulations which show the closed loop response

with the applied disturbances for various set point changes and various tuning

methods. These responses have been used to calculate the time domain

specifications.


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Fig.5.5 Closed loop response for time domain characteristics (step input

200-400) using cohen-coon

Fig.5.6Closed loop response for time domain characteristics (step input 200-300)

using cohen-coon


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using IMC


using IMC

From the responses obtained above and from the block diagrams it is seen that


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Table.5.1 Time domain specifications, ISE & IAE results using Cohen-coon based control and IMC

based control


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CHAPTER 6

SUMMARY AND CONCLUSION

6.1 WORK DONE

The work done includes:

Introduction: It deals with understanding the dynamics of the process.It gives us knowledge as to why one needs to develop a mathematical

model and linearise certain processes.

CSTR: This describes what a continuously stirred jacketed tank reactoris and tells us how it functions.

Problem Formulation: Used the mathematical model, the materialbalance and energy balance equations to get a linear and a non linear

model.

Modeling: The two point method has been used to get the values ofcertain parameters that are required.

Control: IMC based control and Cohen-Coon based control has beenapplied to get the closed loop responses of the process.

Time Domain Specifications: Various time domain characteristics werecalculated and they help to show which tuning method is more

effective.

The checking of the performance of the controller for various set pointchanges and analysing which control strategy is better suited for the

purpose of control.

6.2 CONCLUSIONFrom the analysis and discussions it is seen that the IMC tuning

method is a better method for controlling the response for the CSTR

process as compared to the Cohen-Coon method. This is because with

the IMC the settling time, ISA & ISE values are smaller as compared

to those of the Cohen-Coon method.


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It is noticed that the with the IMC controller

1) Process variable generally does not overshoot its set point aftera disturbance or set point change.

2) Less sensitive to error made in measuring the dead time.3) Absorbs the disturbance better and passes less of it to the

process.

4) The closed loop time constant is user defined; hence it has one parameter which can help in speeding up or slowing down the

process.

Whereas with the Cohen-Coon controller

1) It can give really bad results if the dead time is measuredincorrectly.

2) Aims for quarter amplitude damping rather than a first orderProcess with dead time.

3) Leads to low damping and high sensitivity in the system.

6.3 SCOPE FOR FUTURE WORKIn the present work an attempt has been made for modeling , and the

control of a CSTR process through simulation using MATLAB

software. The present work can be extended by doing real time

experimental work for a CSTR process and to implement the above

said controllers. .


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REFERENCES

1. Patterson D.W, "Artificial Neural Networks-Theory and Applications",Patience Hill, 2008

2. Bequette, B.W. "Process Control: Modelling, Design and Simulation",Patience Hill, Upper Saddle River

3. N.Kanagaraj, P.Sivashanmugam and S.Paramasivam, "Fuzzy CoordinatedPI Controller: Application to the Real-Time Pressure Control Process",

Hindaai Publishing Corporation,Volume 2008.

4.

Stephanopoulos, George. Chemical Process Control: An Introduction toTheory and Practice. New Delhi: Prentice-Hall of India Private Limited,

2003.

5. Doug Cooper, Robert Rice, Jeff Arbogast, "Cascade vs. Feed Forward forImproved Disturbance Rejection" Presented at the ISA 2004, 5-7 October

2004, Reliant Center Houston, Texas.

6. Takagi S, Sugeno M. "Fuzzy identification of fuzzy systems and itsapplication to modelling and control", IEEE Trans. Systems Man Cybern.,

Vol 15, 116-132

7. Brian Rofell, Ben Betlem. " Linearization of model equations" in ProcessDynamics and Control, 1

stedition, John Wiley and Sons limited, 2006, pp

97-127.

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Tj Project Report-final 1 23