Journal of Applied and Industrial Sciences, 213, 1 (4): 16-23, ISSN: 2328-4595 (PRINT), ISSN: 2328-469 (ONLINE) Research Article Cascade Control of a Continuous Stirred Tank Reactor (CSTR) 16 A. O. Ahmed 1, G.A. Gasmelseed 2, A. B. Karama 3 and A.E Musa 4 (1) Faculty of graduate studies and research, University of Karary, Khartoum Sudan E-mail: areejomer11@hotmail.com (2) Department of Chemical Engineering, Faculty of Engineering, University of Science and Technology Email: gurashigar@hotmail.com Telephone: +249919634134 (3) Faculty of graduate studies, University of Karary, Khartoum Sudan (4) Department of Leather Technology, College of Applied and Industrial Sciences, University of Bahri, Khartoum Sudan, P.O.Box 166 Corresponding author E mail: ali26w@hotmail.com Telephone: +2499194456 (Received August 1, 213; Accepted October 16, 213) Abstract Cascade control is commonly used in the control of chemical processes to reject disturbances that have a rapid effect on a secondary measured variable, before the primary controlled variable is affected. In this paper, cascade control strategy is used to control the temperature inside a jacketed exothermic continuous stirred tank reactor. The transfer functions of the cascade control loops were identified from the literature [1]. The ultimate gains for the secondary and primary loops were determined using direct substitution, Routh-Hurwitz, Root- locus and Bode methods. To tune the controllers, the secondary controller was first tuned, and then the primary controller was tuned using Ziegler-Nichols technique. The system stability was determined, analyzed and investigated using Routh and Argand diagram. Index Terms : Cascade control, Primary, Secondary loops, Tuning. T I. INTRODUCTION he control of chemical reactors is one of the most challenging problems in control processes. Considering that a CSTR is the heart of many processes, its stable and efficient operation is of paramount importance to the success of an entire process. Many reactors are inherently unstable. The instability appears when irreversible exothermic reactions are carried out in a CSTR (Fig. 1). These reactions tend to produce a large increment in temperature, forcing the rupture of safety and reducing the lifetime of the reactor. The solution to this problem is a temperature control system capable of detecting the rising of the reactor temperature and quickly removing heat from the reactor [2]. As the processes requirements tighten, or in processes with slow dynamics, or in processes with too many or frequently occurring upsets, the control performance provided by feedback control may become unacceptable. Cascade control is a strategy that in some applications significantly improves the performance provided by feedback control [3]. Overview: 1. Cascade control: Cascade control consists of two sensors, two transmitters, two controllers and one final control element. The controlled variable is the reactor temperature. This strategy works as follows: the primary controller looks at the reactor temperature ( ) and decides how to manipulate the jacket temperature ( ) to satisfy its set point. This decision is transmitted to the secondary controller in the form of a set point. The secondary controller in turn manipulates the signal to the valve to maintain the jacket temperature at the set point provided by the primary controller [3]. 2. Mathematical model: Mathematical model is used to investigate how the behavior of a chemical process changes with time under the influence of changes in the external disturbances and manipulated variables. Two approaches are in use: i. Experimental approach: In this case the physical equipment of the chemical process is available to the designer. Consequently, the values of various inputs (disturbance, manipulated variables) are changed deliberately and through appropriate measuring devices it can be observed how the outputs (temperature, pressure, flow rates, concentration) of the chemical process are changed with time. Such procedure is time and effort consuming and it is usually quite costly because a large number of such experiments must be performed [4].
ii. Theoretical approach: It is quite often the design the control system for a chemical process is carried out before the process has been constructed. In such a case it is not possible to relay on the experimental procedure and a different representation of the chemical process is needed in order to study its dynamic behavior. The representation is usually given in terms of a set of mathematical equations whose solution yields the dynamic or static behavior of the chemical process [4]. 3. Ziegler-Nichols (ultimate-cycle method) Many tuning methods have been proposed for PID controllers, such as Ziegler-Nichols method, Cohen-Coon method, Minimum error integral criteria, the systematic Trial method and quarter decay ratio method. In Ziegler-Nichols method the integral time is set to its maximum value and the derivative time to its minimum. The proportional gain is slowly increased until the system begins to exhibit sustained oscillations with a given small step in set point or load change. The proportional gain and period of oscillation at this point are the ultimate gain,, and ultimate period, (Table 1) [6]. Figure 1: Cascade control of a CSTR 4. System stability: Table 1 Z-N PID controller settings, [2] Type of controller P.5 PI.45 PID.6.5.125 Stability is the most important system specification. An unstable system cannot be designed for a specific transient response or steady state error requirements. In control loop the controller must be tuned to obtain satisfactory dynamic behavior of the controlled variable. Dynamically, a system is stable of it is response is bounded for all bounded inputs [5]. Different techniques are available to analyze stability of a closed-loop system: Routh-Hurwitz: This technique allows us to compute the number of roots of the characteristic equation in the right half-plane without actually computing the values of the roots [4]. Root Locus Analysis: Root locus is a graphical representation of the roots of the closed-loop characteristic polynomial as a chosen parameter (proportional gain) is varied from to infinity [4]. Direct Substitution Analysis S is substituted by jω in the closed-loop characteristic equation to find the ultimate gain and the crossover or ultimate frequency [4]. Frequency Response Analysis: i. Bode plots: The magnitude (modulus) and phase angle (argument) are plotted against frequency [6].
Journal of Applied and Industrial Sciences, 213, 1 (4): 16-23, ISSN: 2328-4595 (PRINT), ISSN: 2328-469 (ONLINE) ii. Nyquist plot: The real and imaginary parts of G(jω) can be plotted on the s-plane with ω as the parameter [6]. Location of roots Here, system stability is governed by the location of the roots of the characteristic equation. The system is stable if all the roots lie on the left hand plane of the s-plane. The system is unstable if any of the roots lie on the right hand portion of the s-plane. The system is critically stable if the roots lie along the imaginary axis of the s-plane [5]. II. MATERIALS AND METHODS A jacketed CSTR with exothermic reaction was chosen to study cascade control. Transfer functions of cascade control loop elements were derived using the mathematical model (theoretical approach). Then the block diagram loops were constructed with time constants and gains obtained from literature. Ultimate gains for secondary and primary loops were calculated using different stability determination methods, these are: direct substitution method, Root-locus and bode methods. Z-N tuning method was used to determine the optimum settings of ultimate gains and to calculate the controller adjustable parameters (. Stability investigation was made using Routh and Roots location method. Finally, the offset was determined for a step change of magnitude 5% in the set point. III. RESULTS AND DISCUSSION The Mathematical Modeling of a CSTR: A mathematical description of a CSTR is based on balance equations expressing the general laws of conservation: Energy balance around the jacket: +. (1) Energy balance around the reactor: +.(2) The P-Controller transfer function: =. (3) Transmitter: The simplest form of the T.F. of order lag: a transmitter is a first =. (4) From literature [3] the following transfer functions for the cascade control loop are obtained (Fig. 2 and Fig. 3): =.. (5) =. (6) =.. (7) =. (8) =. (9) =. (1) =.2. (11) =.5. (12 Figure 2: Block diagram of cascade loop
Journal of Applied and Industrial Sciences, 213, 1 (4): 16-23, ISSN: 2328-4595 (PRINT), ISSN: 2328-469 (ONLINE) Figure 3: Block diagram of the reduced cascade loop The open-loop transfer function of the secondary loop: OLTF =... (13) The open-loop transfers function of the primary control loop: OLTF =. (14) The characteristic equation of the primary control loop:.5 + 12.65 + 37.6 S +25 + 6 =..(15) Ultimate gains results: Two cases appear when determining the ultimate gain for the secondary loop (. If the loop is unstable, the value of can be found directly, tuned with Z-N method and used to calculate. If there is no crossover frequency for the secondary control loop (system is unconditionally stable), therefore large values for the gain can be used, which produces a very fast closed-loop response, to compensate for any change in the disturbance arising within the secondary process. Once the value of has been selected, the crossover frequency for the overall openloop transfer function can be calculated. Then the value of the primary controller gain ( can be selected using Z- N. Here the secondary loop was unconditionally stable, hence is taken to calculate (Table 2). Table 2 Ultimate gain results Cascade loop Secondary loop Overall loop Direct sub. method Stable system No cross-over 157 27.4 Root Locus method Stable system No cross-over 155 27.1 Bode method Stable system No cross-over 167.4 28.2 average ---- ---- 159.8 27.56
Phase (deg) Magnitude (db) Imaginary Axis Journal of Applied and Industrial Sciences, 213, 1 (4): 16-23, ISSN: 2328-4595 (PRINT), ISSN: 2328-469 (ONLINE) Using MATLAB, Root locus and Bode plots (Figs. 4, 5, 6) are deduced for the secondary and primary loops: 1.5 Root Locus 1.5 -.5-1 -1.5-4 -35-3 -25-2 -15-1 -5 5 Real Axis Figure 4: Root-locus plot for the secondary loop -1 Bode Diagram -2-3 -4-5 -6-45 -9 1-1 1 1 1 1 2 1 3 Frequency (rad/sec) Figure 5: Bode plot for the secondary loop
Imaginary Axis Journal of Applied and Industrial Sciences, 213, 1 (4): 16-23, ISSN: 2328-4595 (PRINT), ISSN: 2328-469 (ONLINE) Root Locus 4 3 2 1 System: sys Gain: 155 Pole:.116 + 27.1i Damping: -.428 Overshoot (%): 11 Frequency (rad/sec): 27.1-1 -2-3 -4-5 -4-3 -2-1 1 2 3 4 Real Axis Figure 6: Root-locus plot for the reduced primary loop
Phase (deg) Magnitude (db) Journal of Applied and Industrial Sciences, 213, 1 (4): 16-23, ISSN: 2328-4595 (PRINT), ISSN: 2328-469 (ONLINE) 5 Bode Diagram -5-1 -15-2 -9-18 System: sys Frequency (rad/sec): 28.2-27 Phase (deg): -18 1-2 1-1 1 1 1 1 2 1 3 1 4 Frequency (rad/sec) Figure 7: Bode plot for the reduced primary loop
Journal of Applied and Industrial Sciences, 213, 1 (4): 16-23, ISSN: 2328-4595 (PRINT), ISSN: 2328-469 (ONLINE) Tuning using Z-N method: = 159.8, =.2278 fastest response ( is large without affecting the stability). Nomenclature: P 79.9 - - PI 71.9 - PID 95.88 Stability investigation: The Characteristic equation of the reduced primary loop:.5 + 12.65 + 37.6 S +25 + 6* =.5 + 12.65 + 37.6 S +9613 = Using roots location method, the roots are: >> roots ([.5 12.65 37.6 9613]) The values of the roots are: 1.e+2 * -2.5153 -.73 +.1956i -.73 -.1956i All of the roots lie on the LHP (left hand plane), therefore the system is stable. Offset due to a step change in the set point: = = -- =.5 Offset = =.49 IV. CONCLUSIONS The first step in the cascade control of a CSTR should be the use of an appropriate mathematical model of the reactor. From these models, a set of open-loop transfer functions are obtained. from the calculations(direct substitution method) and the root-locus, Bode plots are deduced, It has been shown that the secondary loop is unconditionally stable for all values of the controller gain, while the primary loop is conditionally stable, i.e. there is a range of the controller gain which gives a close-loop stable reactor ( <159.8). After tuning with Z-N method and investigation of the stability using the optimum controller gain the system was found to be stable. The performance of the proposed system is satisfactory as the analysis of control system gives stable system with CSTR: Continuous Stirred Tank reactor : Reactor temp. : Initial reactor temp. : Jacket temp. : Initial jacket temp. : Time constant ( ) : Constant : Controller gain : Primary controller gain : Secondary controller gain : Ultimate gain of the primary loop : Ultimate gain : Ultimate gain of the secondary loop : Ultimate period : Cross-over frequency : Integral time. : Derivative time. Z-N: Ziegler-Nichols method Acknowledgment The authors are indebted to the administration of Karary University, faculty of graduate studies and research for their help and support. REFERENCES [1]. Stephanopoulos, G., (25). Chemical process control, an introduction to theory and practice, Prentice Hall of India Private limited, New Delhi. pp, 41-42. [2]. Thoma M.,(27). Selected topics in Dynamics and Control of Chemical and Biological Process, Springer Science media, Germany. pp, 5-6. [3].Carlos A. (26). Principles and practice of Automatic Process Control, John Wiely and sons Inc, Asia. Pp, 31-315. [4]. Gasmelseed, G.A. (212). Advanced control for graduate students, G-town, Khartoum. pp, 142,174,183. [5]. Abu-Gouk, M. E, (23). Controlling Techniques and System Stability, University of Khartoum Press, Khartoum. pp, 165,166,17. [6]. Chau, C. (21). Chemical Process Control, Reed educational and professional publishing Ltd., San Diego. pp, 81,82.