Concordia, Brainard B. (Termpaper)

Republic of the Philippines
Cagayan State University
College of Engineering
Carig Campus, Carig Sur, Tuguegarao City




MODELING AND CONTROL DESIGN OF AN ISOTHERMAL CONTINUOUS STIRRED TANK REACTOR






Submitted by:

BRAINARD B. CONCORDIA
10-7119
Chemical Engineering Student





ChE 71 (PROCESS DYNAMICS AND CONTROL)




Submitted to:

ENGR. CAESAR P. LLAPITAN
Chemical Engineering Instructor






November 10, 2015
Date Submitted
NOMENCLATURE
k1 Rate constant for A B (mol/l-min-1)
k2 Rate constant for B C mol/l-min-1)
k3 Rate constant for 2A D (mol/l-min-1)
rA Molar rate formation of A
rB Molar rate formation of B
rC Molar rate formation of C
rD Molar rate formation of D
CA Concentration of A
CB Concentration of B
CAs Steady State Concentration of A
CBs Steady State Concentration of B






ABSTRACT
This paper presents a comparative analysis performance of conventional controller with three configurations for concentration control of isothermal continuous stirred tank reactor, which is used to carry out chemical reactions in an industry. The specific reaction that the reactor carried out in this study is the Van de Vusse Reaction which is a series-parallel reaction. Isothermal continuous stirred tank reactor is the type of reactor which operates at a constant temperature and this is the type of reactor being used in this study. Mathematical models were developed for the isothermal CSTR. Control strategy was made and PID controller were implemented in controlling the product concentration of the reactor. Time domain analysis of the controller is performed to study the performance of the different controller configuration by the addition of delay and disturbance, it is observed that PID without any delay or disturbances performs best to control the product concentration of isothermal CSTR.









Chapter 1
INTRODUCTION
1.1 Isothermal Continuous Stirred Tank Reactor
Continuous Stirred Tank Reactor (CSTR) is a typical chemical reactor system with complex nonlinear dynamic characteristics. There has been considerable interest in its real time control based on the mathematical modeling. However, the lack of understanding of the dynamics of the process, the highly sensitive and nonlinear behavior of the reactor, has made difficult to develop the precise mathematical modeling of the system. An efficient control of the product concentration in CSTR can be achieved only through accurate model. Developing mathematical models of nonlinear systems is a central topic in many disciplines of engineering. Models can be used for simulations, analysis of the system's behavior, better understanding of the underlying mechanisms in the system, design of new processes and design of controllers. In a control system the plant displaying nonlinearities has to be described accurately in order to design an effective controller. In obtaining the mathematical model, the designer follows two methods. The first one is to formulate the model from first principles using the laws governing the system. This is generally referred to as mathematical modeling. The second approach requires the experimental data obtained by exciting the plant and measuring its response. This is called system identification and is preferred in the cases where the plant or process involves extremely complex physical phenomena or exhibits strong nonlinearities. (M. Shyamalagowri and R. Rajeswari, 2013)
Process control is vital to the automated operation and monitoring complex technical process. These processes represents the unit operation of material transformation. Raw materials are transformed into products, often using other materials and energy. With the aid of process control, processes are monitored and influenced as they happen. This is enabled by the measurement and control of variables such as flow rate, pressure, temperature and concentration.
Controlling CSTR is an attractive issue for control engineers for its dynamics (non-linear). A control mechanism is needed to cancel the negative impact due to nonlinearities which may cause undesired effect in chemical plants. Errors and random disturbances are modelled by using the controller. In search of getting the most linear response PID controllers are widely used. PI and PID controllers are ideal for controlling non-linear process. PID controllers can be used for performance increase with respect to system variations. 2DOF PID controller is can also be used because of its better control on non-linear process (B. Dey et. al., 2014).

1.2 Objectives:
The specific objectives of this term paper are:
To develop transfer function and simulation model for isothermal CSTR.
To create a control strategy for the product concentration control of the isothermal CSTR.
To implement PID tuning to the proposed feedback controller for concentration control and compare when delay and disturbances is added.





1.3 Process Variables
CACBCACB
CA
CB
CA
CB
Figure 1. Schematic Diagram of the CSTR
This term paper will determine the product concentration CB as the result of varying the feed flow, thus the controlled variable is the product concentration, the manipulated variable is the feed flow rate, and the disturbance in consideration is the varying feed flow.
k2k2k1k1The Continuous Stirred Tank Reactor is taken for nonlinearity identification process. It consists of a CSTR carrying out the Vande Vusse reaction scheme described by the following reactions:
k2
k2
k1
k1
A B C (1)k3k3
k3
k3
2A D (2)
Here B is the desired product, C and D are the undesired byproducts, k1, k2 and k3 are the reaction rate constants.


1.4 Mathematical Model of the System
Developing mathematical models of nonlinear systems is a central topic in many disciplines of engineering. Models can be used for simulations, analysis of the system's behavior, better understanding of the underlying mechanisms in the system, design of new processes and design of controllers. In a control system the plant displaying nonlinearities has to be described accurately in order to design an effective controller. In obtaining the mathematical model, the designer follows two methods. The first one is to formulate the model from first principles using the laws governing the system. This is generally referred to as mathematical modeling. The second approach requires the experimental data obtained by exciting the plant and measuring its response. This is called system identification and is preferred in the cases where the plant or process involves extremely complex physical phenomena or exhibits strong nonlinearities.
This reaction describes the chemical conversion, under ideal conditions, of an inflow of substance A to a product B. To simplify the problem the following assumptions are taken:
The liquid in the reactor is ideally mixed.
The density and the physical properties are constant.
The liquid level h in the tank is constant, implying that the input and output flows is equal: Q1 = Q2.
The continuous stirred tank reactor is isothermal, so we do not need an energy balance and can assume that the reaction rate parameters are constant.

Chapter II
REVIEW OF RELATED LITERATURES
2.1 Transfer Function and its Derivation
The overall material balance of the system is given as,
d(Vρ)dt= Fiρ-Fρ F= Fi
And the component material balances are given as,
Component A (Equation 1):
dCAdt= FV CAf- CA- k1CA- k3CA2
Component B (Equation 2):
dCBdt= -FVCB - k1CA- k2CB
Component C (Equation 3):
dCCdt= -FVCC+ k2CB
Component D (Equation 4):
dCDdt= -FVCD+ 12k3CA2
2.1.1 Steady-State Behaviors
Since we are only concerned about CA and CB, we only need Equation 1 and 2.
dCAdt= FV CAf- CA- k1CA- k3CA2
dCBdt= -FVCB - k1CA- k2CB
Assume that F/V (space velocity) is the input variable of interest (V/F is known as the "residence time" or "space time").
Solving for the steady-state for CA
-k3CAs2+ -k1- qrsVCAs+ qrsVCAfs=0
CAs= - k1+ qrsV + k1+ qrsV2+ 4k3 qrsVCAfs2k3 (using the positive root)
Solving for the steady-state for CB
CBs= k1CAsFsV+ k2
Parameter values: (k1 = 5/6, k2 = 5/3, k3 = 1/6, CAfs = 10)
The material balance on A and B of Equations 1 and 2 do not depend on the concentration of C and D components. So it is enough to solve the Equations 1 and 2 to compute the maximum concentration of B. The steady state concentration of A and B with respect to the manipulated variable (Fs/V) is shown in Figures 1 and 2. The component A has positive gain variation in the range of 0.9 to 7.5 (mpl). The component B has both positive and negative variation in the range of 0.4 to 1.3 (mpl).

2.1.2 State-Space Model
The linear state space model is represented as,
x=Ax+Bu
y=Cx+Du

The state variable is represented as,
x=CA-CAxCB-CBx
The output variable is represented as,
y=CA-CAxCB-CBx
The input variable is represented as,
u=FV-FsV

The two dynamic functional equation is represented as,
dCAdt= f1CA,CB,FV= FVCAf- CA- k1CA- k3CA2
dCBdt= f2CA,CB,FV= -FVCB+ k1CA- k2CB

The elements of state space A matrix is found by
Aij= fi xixs,us
The elements of state space B matrix is found by
Aij= fi ujxs,us
The state space model is represented as
A=-FsV-k1-2k3CAs0k1FsV-k2
B=CAfs-CAsFsV-CBs0
C= 01
D= 00
Based on steady state operating point CAs = 3gmol/l, CBs = 1.117gmol/l, Fs/V = 0.5714/min
A= -2.400.83-2.23
B= 70.57-1.1170
C= 01
D= 00
Converting the state space model to transfer function,
G(s) = C (sI – A)-1B
Process transfer function:
gps= -1.117s+3.1472s2+4.6429s+5.3821
Process transfer function with delay:
gps= -1.117s+3.1472e-0.5ss2+4.6429s+5.3821
Process transfer function with disturbance:
gds= 0.4762s2+4.6429s+5.3821

2.2. Control Strategy
2.2.1 Control Design of the Reactor
Since the primary control objective is to control the product concentration of isothermal CSTR. The schematic diagram of the feedback control loop of isothermal CSTR is shown in the figure below.

Figure 4. Schematic Diagram of Feedback Control Loop

Here the CM represents the measurement of concentration and CC represents the concentration controller. The next figure shows the block diagram approach of feedback control scheme.
2.2.2 Feedback Control System
Input FlowInput FlowSensorSensorProcessProcessValveValveActuatorActuatorControllerControllerSet PointSet PointInput flow disturbanceInput flow disturbance
Input Flow
Input Flow
Sensor
Sensor
Process
Process
Valve
Valve
Actuator
Actuator
Controller
Controller
Set Point
Set Point
Input flow disturbance
Input flow disturbance







Figure 5: Block diagram based feedback control approach for concentration of isothermal CSTR
2.3 PID Tuning
The PID controller is designed based on internal model control and stability analysis principles. The proposed controllers are applied to stable transfer function models of isothermal CSTR carrying out Van de Vusse reaction. Simulation results on non-linear model equations of isothermal CSTR carrying out Van de Vusse reaction is given to show the effectiveness of the proposed PID controllers. The performance under model uncertainty is also studied considering perturbation in one parameter at a time. The performance of proposed controllers is compared with the direct synthesis method (Chien et al., 2003).
PID-1.117s+3.1472s2+4.6429s+5.38210.4762s2+4.6429s+5.3821PID-1.117s+3.1472s2+4.6429s+5.38210.4762s2+4.6429s+5.3821
PID
-1.117s+3.1472s2+4.6429s+5.3821
0.4762s2+4.6429s+5.3821
PID
-1.117s+3.1472s2+4.6429s+5.3821
0.4762s2+4.6429s+5.3821







Figure 6: Transfer function based feedback control approach for concentration control of isothermal CSTR
Figure 6 shows the transfer function model of the feedback control scheme for concentration control of isothermal CSTR. This control approach is based on the study of Vishal Vishnoi, Subhransu Padhee and Gagandeep Kaur. The system is tuned using the Ziegner-Nichols criteria of tuning

Chapter III
SIMULATION AND DISCUSSION OF RESULTS
We have applied three configuration of the controllers, PID controller for concentration control, PID controller with disturbances, and PID controller with delay and disturbances.
The values of proportional gain, integral gain and derivative gain of PID controller were based on the study Controller Performance Evaluation for Concentration Control of Isothermal Continuous Stirred Tank Reactor. The PID controller is tuned using the Ziegner-Nichols criteria of tuning and the unit step response of feedback control is shown in Figure 8b.
3.1 Configuration I (PID without delay and disturbance)

Figure 8a. Simulink for feedback control for concentration control
TIMETIME
TIME
TIME
Figure 8b. Simulink result for feedback control for concentration control

Figure 8c. PID Tuner (without delay and disturbance)
Figure 8a and 8b shows PID controller for concentration control and Figure 8c shows the PID tuner. Based on the result for this simulation, the graph shows that the peak overshoot for this simulation is 1.4%, a rise time of 0.468 seconds, a settling time of 2.79 seconds and a peak time of 1.01 seconds. These results will be compared for the result of PID controller with disturbances and PID controller with both delay and disturbance to.
3.2 Configuration 2 (PID + disturbance)

Figure 9a. Simulink for feedback control with disturbance
TIMETIME
TIME
TIME
Figure 9b. Simulink result for feedback control with disturbance

Figure 9c. PID Tuner (with disturbance)
Figure 9a and 9b shows unit step response of feedback control with a disturbance. Due to this disturbance, the peak overshoot increases to 4.06%, a 2.66% difference from the feedback control for concentration control. The graph also shows the rise time to be 0.925 seconds, the settling time to be 2.69 seconds and took 1.04 seconds to reach the peak point. It is also evident in the graph of PID Tuner that the tuned response is faster to reach its stable condition in around 7 seconds compared to the block response that took much longer.
3.3 Configuration 3 (PID + disturbance + delay)

Figure 10a. Simulink for feedback control with disturbance and delay
TIMETIME
TIME
TIME
Figure 10a. Simulink for feedback control with disturbance and delay

Figure 10c. PID Tuner (without delay and disturbance)
Figure 10a and 10b shows unit step response of feedback control with a disturbance and delay while Figure 10c shows the PID tuner. Based on the simulation result, the addition of delay results the peak overshoot of control with disturbance to become lower resulting to 2.59%, a 1.47 reduction. The rise time shows a 0.978 seconds, a settling time of 2.61 seconds and a peak time of 1.03.
The next table will show the simulation of the three conventional controllers together.

Figure 11. MATLAB Simulink Set-up for three different controllers
TIMETIMELegend:Red: PID + disturbanceBlue: PIDPink: PID + disturbance + delayLegend:Red: PID + disturbanceBlue: PIDPink: PID + disturbance + delay
TIME
TIME
Legend:
Red: PID + disturbance
Blue: PID
Pink: PID + disturbance + delay
Legend:
Red: PID + disturbance
Blue: PID
Pink: PID + disturbance + delay
Figure 12. Comparative response of different controller
Figure 12 shows a comparative response of the three different controllers. As shown in the figure, the PID with disturbance get the higher peak overshoot due to the added disturbance in the system. To summarize for easier comparison, a table is shown below.
Parameters/type
Peak overshoot (%)
Rise Time
(sec)
Sett Time
(sec)
Peak Time
(sec)
PID
1.4
0.468
2.79
1.01
PID with disturbances
4.06
0.925
2.69
1.04
PID with disturbances and delay
2.59
0.978
2.61
1.03
Table 1. Transient Response
Table 1 shows the comparative transient response of conventional PID controller. From transient response analysis in table 1, it is evident that PID without any disturbances and delays performs the most satisfactory among the three configurations of controller in concentration control.

















CHAPTER IV
Conclusion
The product concentration control of isothermal continuous stirred tank reactor is presented in this study. Chemical reaction systems and modeling was introduced and the development of simulation for simulation model had been made. The PID controller is tuned using the Ziegler-Nichols criteria of tuning.
From figure 8, the overshoot, rise time, settling time and peak have the values of 1.4%, 0.468 seconds, 2.79 seconds and 1.01 seconds respectively. This values compared to the PID + disturbance or the PID + disturbance + delay is the best in result because it got the least overshoot, the shortest time to reach its peak and the fastest to rise. With this results, disturbances or delays should be avoided as much as possible in designing the control of the system.
With the completion of these study, as a Chemical Engineering Student, we were able to understand and learned the principles of controls and its importance for our profession. We also able to learn to develop mathematical models that can be used for simulations, analyze the system behavior for a certain process and develop a control configuration for it, and better understanding of underlying mechanisms in the system, design of new processes and controller design. Doing this paper also helps to master the use of MATLAB software in understanding the behavior of different control systems and controllers and how to apply them to their respective chemical processes.


APPENDIX
1. Block Diagrams for Simulation for Concentration Control (without delay and disturbance)

Figure 13a. Unit Step Source Block Parameter

Figure 13b. PID Controller Block Parameter

Figure 13c. Process Transfer Function Block Parameter


Figure 13d. PID Tuner Controller Block Parameter



2. Block Diagrams for Simulation for Concentration Control (with disturbance)

Figure 14a. Unit Step Source Block Parameter

Figure 14b. PID Controller Block Parameter

Figure 14c. Process Transfer Function Block Parameter


Figure 14d. Disturbance Transfer Function Block Parameter


Figure 14e. PID Tuner Block Parameter

3. Block Diagrams for Simulation for Concentration Control (with disturbance and delay)

Figure 15a. Unit Step Source Block Parameter

Figure 15b. PID Controller Block Parameter


Figure 15c. Process Transfer Function Block Parameter

Figure 15d. Disturbance Transfer Function Block Parameter


Figure 15e. Delay Transfer Function Block Parameter

Figure 15f. PID Tuner Block Parameter


k1
5/6
k2
5/3
k3
1/6
CAfs
10
CAs
3 gmol/l
CBs
1.117 gmol/l
Fs/V
0.5714/min
Table 2. Parameters Used in the Study

Proportional Gain
0.2
Integral Gain
0.95
Derivative Gain
0.023
Table 3. Values of Gain for PID Controller


REFERENCES
D. Krishna. Tuning of PID controllers for Isothermal Continuous Stirred Tank Reactor. Elixir International Journal, 2012.
M. Shyamalagowri and R. RajeswariModeling. Simulation of Non Linear Process Control Reactor – Continuous Stirred Tank Reactor. International Journal of Advances in Engineering & Technology, September 2013.
Vishal Vishnoi, Subhransu Padhee, and Gagandeep Kaur. Controller Performance Evaluation for Concentration Control of Isothermal Continuous Stirred Tank Reactor. International Journal of Scientific and Research Publications, Volume 2, Issue 6, June 2012.