Ind. Eng. Chem. Res. 1989,28, 1481-1489 Ulrich, G. D.; Milnes, B. A.; Subramanian, N. S. Particle Growth in Flames: 11. Experimental Resulta for Silica Particles. Combust. Sci. Technol. 1976, 14, 243. Wang, C. S.; Friedlander, S. K. The Self-preserving Particle Size Distribution for Coagulation by Brownian Motion. J. Colloid Interface Sci. 1967, 24, 170. Wu, J. J.; Flagan, R. C. Onset of Runaway Nucleation in Aerosol
1481
Reactors. J. Appl. Phys. 1987, 61, 1365. Wu, J. J.; Flagan, R. C. A Discrete-SectionalSolution to the Aerosol Dynamic Equation. J. Colloid Interface Sci. 1988, 123, 339. Received for review October 14, 1988 Revised manuscript received May 8, 1989 Accepted June 15, 1989
PROCESS ENGINEERING AND DESIGN Control of a Multivariable Open-Loop Unstable Process Apostolos Georgiou,? Christos Georgakis, and William L. Luyben* Chemical Process Modeling a n d Control Research Center, Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015
A systematic approach is presented for the design of a control system for a multivariable open-loop unstable process. The closed-loop stability, dynamic performance, and robustness of the multiloop single-input-single-output controllers are established by an effective damping coefficient and its corresponding effective closed-loop time constant. Controller tuning is reduced to a four-step optimization problem. The procedure is illustrated on a complex system of two reactors in series, with a separator and recycle, where an exothermic second-order reaction occurs. The system is highly nonlinear and has three open-loop poles in the right half of the s plane. The effects of variation of reactor size and kinetic parameters are used to explore the robustness of the control system. Performance is evaluated by simulation of the nonlinear mathematical model.
A number of processes in the chemical industry are multivariable and open-loop unstable (e.g., exothermic reactors). There have been many theoretical studies of multivariable controller design techniques for these processes: (1)pole placement methods (see Wonham (1967), Power (1975), Edgar (1976),Friedland (1986)); (2) internal model control (IMC) (see Zafiriou and Morari (1987));and (3) model predictive control for unstable systems (Cheng and Brosilow (1987)). Concerning (1)above, it has been pointed by many authors (see above cited references) that while these methods have found application in other disciplines, for example in the aerospace industry, they have not found wide application in the chemical/petroleum industry. Some of the reasons are (a) complexity, excessive engineering manpower requirements, operator nonacceptance; (b) requirement of measurements (or estimates) of the states (concentration, etc.) that are not available because of physical or economical constraints; ( c ) difficulty in handling integral action which is very important in chemical processes; (d) no guidance to the engineer for tuning these methods on-line (chemical processes are highly nonlinear and usually require on-line tuning); and (e) lack of robustness (sensitivity to model/plant mismatch). Concerning (2) and (3), the recent IMC work (Zafiriou and Morari (1987)) and model predictive control work (Cheng and Brosilow (1987)) are still in the initial stages of design.
* Author to whom correspondence should be addressed. Present address: Chemical Engineering Department, Princeton University, Princeton, N J 08544. 0888-5885/89/2628-1481$01.50/0
Therefore, most multivariable industrial processes are still controlled satisfactorily by simple three-mode (proportional-integral-derivative (PID)) single-input-singleoutput (SISO) controllers. Pure optimization techniques based on minimizing a performance index such as the integral square error (ISE) or integral absolute error (IAE) of the outputs or inputs have been proposed, but these methods have several problems: (a) they do not provide any guidance for how to weight the outputs or the inputs in the performance index; (b) they do not provide any guidance for incorporating important specifications of a controller design (overshoot, settling time, robustness, etc.) in the performance index; (c) many times, these pure optimization procedures find a local o p t i m u m and not a global optimum. We used the IAE optimization procedure (Georgiou et al. (1986)) to tune the SISO controllers, but the performance was unsatisfactory compared to that obtained by the procedure proposed in this paper. We are not aware of any practical systematic procedure for designing multiloop SISO conventionalPID controllers for multivariable open-loop unstable processes. This paper proposes such a procedure. It should be emphasized that multivariable controllers are not considered in this paper. The objective of this paper is to illustrate a controller design for a multivariable open-loop unstable process. The modeling of the multivariable open-loop unstable system is discussed first. This system consists of two jacketed CSTR’s with recycle and separator where an exothermic second-order reaction occurs. The distinctive features of the dynamics of the system are the strong nonlinearity of temperature involved in the kinetics and 0 1989 American Chemical Society
1482 Ind. Eng. Chem. Res., Vol. 28, No. 10, 1989
I'F
1
I
I
F
dj
Figure 1. Schematic representation of the network of two CSTR's with recycle and separator.
the three open-loop poles in the right half of the s plane. Then, employing a multivariable root locus procedure, the extension of the damping coefficient, widely used in the SISO case, to the multivariable systems is shown. A relationship of the closed-loop performance of a process with the effective damping coefficient and effective time constant is demonstrated. Based on these concepts, a four-step optimization procedure to tune the three modes of proportional-integral-derivative (PID) controllers is proposed. The issue of robustness of a control system, namely that stability and acceptable performance be maintained in the face of plant-model mismatch, is very crucial in chemical process control where accurate dynamic models of the process are rarely available. In the final section of this paper, an indication of the robustness of a controller for this type of systems is presented. The effect of variations of the kinetic parameters is used to examine the robustness of the control system. Rigorous nonlinear simulation experiments confirmed that the proposed controller design technique produces a stable, workable, and robust closed-loop system.
Description of the Systems Studied In this study, we consider two different designs of a network of two jacketed CSTR's in series with separator and recycle. This system preserves much of the complexity of industrial reactors. An exothermic, second-order, autocatalytic, irreversible reaction takes place in the reactors, A+B-2B
The schematic representation of the system is shown in Figure 1. The CSTR's have dimensions of a typical industrial reactor. The whole system is open-loop unstable because of the limited heat removal due to the small ratio of area to volume. The metal walls of the reactors and jackets are neglected. The first design (hereafter called dsgnl) consists of two 500-gal reactors, while the second design (dsgn2) consists of two 5000-gal reactors. I t is assumed that mixing is perfect in the reactors and in the cooling jackets. Specific heat, exothermic heat of reaction (-AH), heat-transfer coefficients ( U ) ,and heat-transfer areas are assumed constant. Feed and product flow rates and reactor and jacket volumes are assumed constant. The design parameters are given in Table I. For simplicity, the separator is assumed to be at steady state. For the separation, it is assumed that the ratio of the mole fraction of substance A in the recycle stream to the mole fraction of A in the product stream is equal to 10.
Table I. Design Specifications of the system parameters concn of A in the feed, C A ~ ,kmol/m3 concn of B in the feed, cBo,kmol/m3 feed flow, F , m3/min feed temp, To,K activation energy, E, kJ/kmol weexponential term, kn. l/(min kmol) ieaction heat, -AH, kJjkmol recycle, r ( F 4 / F ) reactor vol, Vi, m3 reactor diameter, D, m reactor height, H, m heat-transfer area, m2 overall heat-transfer coeff, Vi, kJ/(m2 K min) overall heat-transfer coeff, Btu/(h ft2
Systems dsgnl 10 0 0.021 294 5303.4 4.25 X 10l2 6 X lo4 2 1.892 1.06 2.13
dsgn2 10 0 0.21 294 5303.4 4.25 X 10l2 6 X 10' 2
8.0 68.0
18.925 2.292 4.572 37.16 68.0
192.22
192.22
5608.6 4916.6 320.38 325.54 2.673 1.129
55 980 49 246 320.33 325.58 2.678 1.128
325.54 1.613 325.54 0.161 0.3 0.0375 0.083
325.58 1.611 325.58 0.161 5.0 0.135 3.21
0.052
0.9743
294 310.1
294 298
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306
90 3.14 4.18 900 1000
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output heat-transfer rate, Q1, kJ/min output heat-transfer rate, Q2,kJ/min temp of the first reactor, TI, K temp of the second reactor, T2,K concn of the first reactor, CAI, kmol/m3 concn of the second reactor, cAZ, kmol/m3 temp of recycle, Tp, K concn of recycle, cA4, kmol/m3 temp of product stream, T3,K concn of product stream, cA3, kmol/m3 jacket vol, VJi, m3 jacket width, m coolant flow of the first reactor, Fcl, m3/min coolant flow of the second reactor, Fc2, m3/min coolant feed temp, T,,, K coolant exit temp of the first reactor, TCl, K coolant exit temp of the second reactor, TCZ, K molecular weight of reactants, MW specific heat of reactants, cp, kJ/(kg K) specific heat of coolant, cp,, kJ/(kg K) density of reactants, p , kg/m3 density of coolant, pc, kg/m3
The dynamic model consists of three nonlinear differential equations for each reactor (material balance and energy balance for the reaction mass and energy balance for the jacket) and two algebraic equations for the separator. The system dynamic model is given in the Appendix section. The controlled variables are the temperatures, TIand T2 The manipulated variables are the cooling water flows, Fcl and FC2, in the first and second reactors, respectively. Reactor temperature transmitter spans of 10 K are used. A second-order time lag ( T = 0.3 min) is assumed for each transmitter. Jacket water control valves are sized to give three times the normal flow rate when wide open. If either control loop is on manual, the system is always unstable. This makes the design and the tuning of the conventional controller structures a challenging problem.
Controller Design The easiest and most common approach to the design of control schemes for nonlinear multivariable systems is to linearize the modeling equations and to apply standard linear design procedures. The system is described by six nonlinear differential equations. By linearization of these equations around the steady state, a sixth-order linear model may be obtained in the form x = Ax + B u (1) y =cx
Ind. Eng. Chem. Res., Vol. 28, No. 10,1989 1483 Table 11. Eigenvalues of the Open-Loop Systemsa dsgnl dsgn2 real part imaginary part real part imaginary part 0.1841 X lo-' 0.1013 X lo-' 0.587 X -0.1008 X lo-' 0.1841 X lo4 -0.1013 X lo-' 0.587 X 0.1008 X lo-' 0.000 0.4472 X 10-1 O.OO0 0.4134 X lo-' -0.5658 X lo-' O.OO0 -0.5533 x 10-1 0.000 0.000 0.0000 -0.3307 -0.9674 -0.1112 x 10' 0.000 -0.7698 0.000 a The
eigenvalues corresponding to the transmitters are not giv-
en.
Table 111. min 7,, versus R I for Dsgnl Rr min 711 specified ( 0.5 52.85 0.7 1.0 14.35 0.7 1.5 12.10 0.7 2.5 17.35 0.7 3.5 21.10 0.7 4.5 23.35 0.7 Table IV.
Tni
versus R n for Dsan2
RD
7 ~ 1
1.0 2.5 4.5 6.5
3.1 2.7 2.0 1.5
m= f 0.56 0.66 0.70 0.67
where the state vector x, control vector u, and output vector y are
The system has three open-loop poles in the right half of the s plane as shown in Table 11. The standard test of controllability (Ray, (1981))is satisfied. The conventional control structure consists of two PID temperature controllers. Therefore, the feedback controller matrix is diagonal. The transfer function of the PID controller used is
(3) where a = l l l 0 , Kd is the proportional gain, TE is the reset time, and T D is ~ the derivative time (i = 1, 2). If we take into account the transmitters and the controllers, the final closed-loop system becomes 14th order.
Effective Damping Coefficient: Closed-Loop Dynamic Performance The dynamic performance of any system can be deduced by observing the location of the roots of the characteristic equation of the system. There is a quantitative relationship between the location of roots in the s plane and the damping coefficient. Since these concepts can be extended to multivariable sysbms, it is useful to briefly review them. (1)The roots that lie close to the imaginary axis (dominant roots) will dominate the dynamic response since the ones farther out will die out quickly. The larger the radial distance of the dominant roots from the origin, the faster will be the dynamics of the system (smaller time constants). (2) The farther any complex conjugate roots are from the real axis, the more underdamped the system will be. For single-input-ingle-output systems, the design of the controllers based on the location of the closed-loop poles
can be done by the root locus diagram or by the basic pole-placement Bass-Gura formula (Friedland (1986)). Recently, Hwang and Chang (1987) related the leading pole (including damping factor) to the stability robustness of pseudo-second- and third-order systems via the three leading poles of an arbitrary order closed-loop system. In multivariable systems, if all of the state variables are measurable or can be estimated and the system is controllable, the proportional feedback gain matrix can be selected to give any set of closed-loop eigenvalues desired (Wonham (1967)). If not all of the states are available, whith is the case in most practical industrial chemical engineering systems, all the closed-loop poles cannot be arbitrarily selected (Power (1975)). Also, in pole placement techniques, the addition of the integral mode yields an augmented system which is complex for the designer to handle (Edgar (1976)). It is not possible to move all the poles of a complex multivariable system to any desired location with conventional PID controllers. However, the construction of a multivariable root locus provides valuable insights into the effects of the controller modes on the complex dynamics of the closed-loop system. The construction of the root locus for multivariable systems is not directly analogous to SISO systems. In the following paragraph, we present some of the rules of the classical SISO root locus case that do not apply in the multivariable case (for more details see Yagle and Levy (1982) and Tzafestas (1984)). In the SISO case, the rule for the location of root loci on the real axis is very simple: there is a single root locus branch at a points on the real axis if and only if there is an odd number of real poles and zeros located to the right of s. The simplicity of this rule is due to the fact that only one branch of the root locus can lie on the real axis at any given point. However, in the multivariable case, several branches can lie on the real axis at a given point. Thus, the problem in the multivariable case is not only one of determining whether a branch is present, but also one of determining how many branches are present. Since multivariable root loci are branches of an algebraic function (Postlethwaite and MacFarlane (1979)), their behavior is much more unusual than that of SISO root loci. Therefore, unlike the SISO case, knowledge of the pole and zero locations alone is not sufficient for determining the number of loci on the real axis. A branch lying on the real axis can turn around at a branch point and double back on itself. It is generally very difficult to plot the various loci branches for multivariable processes analytically. In this study, the construction of the root locus was done numerically. Another way to represent a multivariable root locus is root contours (Kuo (1985)),but this method is more tedious. Let us initially assume proportional-only controllers with a ratio of the two gains equal to 0.6 (Kca/Kcl= 0.6). By varying the controller gain in the first reactor, we construct a multivariable root locus, which is shown in Figure 2. To make Figure 2 more clear, parts A and B of Figure 2 are redrawn in Figures 3 and 4. Figure 3 represents part A of the multivariable root locus, which is more useful for controller design. The complex root that lies on a radical line drawn from the origin which is closest to the imaginary axis can be considered as giving the effectiue damping coefficient of the closed-loop system. The reciprocal of the radial distance from the origin out to the root is the effectiue time constant. The closed-loop dominant roots for three different choices (sets) of proportional gains are shown in Figure 5. The first choice gives an effective damping coefficient of 0.3 and a real pole very close to the origin. The second and third choices give an effective
1484 Ind. Eng. Chem. Res., Vol. 28, No. 10, 1989 1-t-
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Figure 4. Part B of the multivariable root locus diagram (part B defined in Figure 2). K,, = 0.084.099.
1
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Figure 5. Closed-loop dominant roots of the system (dsgnl) for different combinations of the proportional gains We, proportional , gains; 5; effective damping coefficient; d , radial distance; *, +, . location of the poles for different choices of the gains). 6
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Figure 3. Part A of the multivariable root locus diagram (part A defined in Figure 2).
Figure 6. Comparison of the servo responses of the temperature (Tz) of the second reactor for different choices of the gains (cases A-C defined in Figure 5).
damping coefficient of 0.7. Notice that the second combination has a smaller time constant than the third one. Figure 6 shows the responses of the temperature in the for a 0.5% (1.63K) step change in T, second reactor (T2) set point for the three different sets of controller gains.
The changes in T2disturb the first reactor because of the recycle. Figure 7 gives the responses of the temperature (TI)and the coolant flow of the first reactor. Notice that the control system that has a damping coefficient 0.3 shows more oscillations than the ones that have a damping
,
Ind. Eng. Chem. Res., Vol. 28, No. 10, 1989 1485 320.56
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Figure 7. Comparison of the responses of temperature and coolant flow of the fmt reactor for different combinationsof the proportional gains. (a) Temperature, (b) coolant flow (cases A-C defined in Figure 5).
coefficient of 0.7. Also, the control system with { = 0.7 and smaller radial distance (higher effective time constant) gives a very sluggish response. Therefore, the closed-loop dynamic performance of a multivariable control system can be quantitatively described by the effective damping coefficient and its effective time constant. The concepts of the effective damping coefficient can be extented also to PI or PID controllers. For example, Figure 8 gives the response of PI controllers when they are tuned to give different effective damping coefficients. Also notice that the control system that has the larger damping coefficient requires less control action.
Tuning Procedure A systematic procedure for the tuning of multiloop SISO PID controllers for open-loop unstable multivariable processes is presented in this section. The first step is to calculate the optimum gains. (A) Proportional Mode. The procedure for finding the gains of the controllers, Kci, is as follows: (1)Define RK as the ratio of the tuning gains, RK = Kc2IKcl. (2) With RK fixed, vary Kcl and note the largest value of the damping coefficient. (3) Repeat the above step for different values of RK and select the RK value that gives the largest effective damping coefficient. (4) Choose a Kcl that gives the desired effective damping coefficient, 5: In a more general form, the mathematical formulation is given below. find RK where "K, MRK, Kcl) VRK E R+ (positive real numbers)
VKcl E S+ (stability region)
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9 Figure 8. Comparison of the responses of the first reactor for proportional-integral controllers with different damping coefficients for set point change in temperature Tz.(a) Temperature, (b) coolant flow. (- --) { = 0.4,(-) { = 0.7. 0.9
;
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GAIN OF THE 1ST REACTOR
Figure 9. Effective damping coefficient versus the gain of the fiist controller for different ratios of the gains (dsgnl).
The best effective damping coefficient depends on the system and the expected model uncertainties, as will be shown in the next section. The engineer has to make a conscious decision on the trade-offs between performance and robustness. Figures 9 and 10 illustrate the proposed procedure for dsgnl and dsgn2, respectively. These figures give the effective damping coefficient versus gain of the first con-
1486 Ind. Eng. Chem. Res., Vol. 28, No. 10, 1989 0.6
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5
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4.0
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G A I N OF THE 1ST REACTOR
Figure 10. Effective damping coefficient versus the gain of the first controller for different ratios of the gains (dsgn2). Table V. Final Controller controller settings gains, m3/(min K) dimensionless gains reset times, min derivative times, min
Settinge dsanl 0.0271/0.0271 4.07/3.89 12.1/18.2 0.48/2.16
dsan2 6.318/0.316 14.76/2.43 8.4/37.8 2.0419.18
troller for different ratios of the gains. The optimum ratios are 0.6 for dsgnl and 0.05 for dsgn2. The corresponding gains (Kcl) are 0.0452 (m3/min K) for dsgnl if a damping coefficient f = 0.7 is chosen and 6.316 for dsgn2 if the maximum damping coefficient (f = 0.6) is desired. The dimensionless gains of both designs are given in Table V. The ratio of the gains for dsgn2 is smaller than the ratio found for dsgnl. The difference of the two ratios can be interpreted as a result of the effect of the coolant flow rates on the stability of the systems. In the configuration of the CSTRs considered here, the first reactor is more important from a stability point of view since it has a higher concentration of reactant. In dsgn2, the entrance-exit temperature difference of the coolant in the first reactor is only 4 K; in dsgnl, the difference is 16 K. This implies that the first reactor of dsgn2 requires a much higher coolant flow rate and is much more difficult to control compared to the first reactor of dsgnl. This gives a higher controller gain for the first reactor of the dsgn2 compared to the controller gain of the first reactor for dsgnl. Also, notice from Figures 9 and 10 that, for a given damping coefficient, the gain of the first controller increases as the ratio of the gains decreases. In other words, when the first gain increases, the second gain has to decrease for the system to be stable. Therefore, dsgn2 requires a much smaller ratio of the two gains. Devia and Luyben (1978) have quantitatively shown that the controllability of an exothermic chemical reactor decreases when the reactor size increases. This is also predicted by the effective damping coefficient. Notice (Figure 10) that the maximum value of {that can be achieved in dsgn2 is only 0.6, in contrast to dsgnl which gives { = 0.95 (Figure 9). This shows that dsgn2 is more difficult to control. (B) Integral Mode. For fixed RK and Kcl,reset times, qi, can be calculated by employing a similar procedure: (1) Define the ratio RI = q2/q1. (2) With RI fixed, vary TI^ and note the minimum value of q1that gives a specified damping coefficient. We want to keep q small so that errors are driven to zero quickly.
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However, closed-loop stability decreases as TI'S are decreased, so a minimum limit will be encountered. (3) Repeat the above step for different values of RI and select the pair (RI, min rI1)that has the smallest rI1. The mathematical formulation of the above is similar to eq 4. Table I11 gives results for the calculation of RI and min T~~ for dsgnl. For a specified damping coefficient, f = 0.7, the optimum values are RI = 1.5 and q1= 12.1 min. (C) Derivative Mode. Cheung and Luyben (1979) have shown that derivative action improves the response and stability of a reactor. The proposed procedure for tuning the derivative action is as follows: (1) Set R K , R I , Kcl, and 711 at their optimum values determined above. (2) Define the ratio R D = TD2/TD1. (3) With R D fixed, vary TD1 and note the largest damping coefficient. (4) Repeat the above step for different values of R D and select the pair (RD, TD1) that gives the largest damping coefficient. Table IV shows the effect of TD1 on { for different RD's for dsgn2. The values that give the maximum effective damping coefficient are RD = 4.5 and TD = 2.04 min. Benefits of derivative action can be seen clearly in the larger reactor, e.g., 5000 gal (dsgn2). Figure 11 compares the responses of the temperature (Tl) and coolant flow of the first reactor for PI and PID controllers. Clearly, the PID gives better performance and requires less control action. Table V summarizes the tuning parameters for both designs. (D) Recalculate Proportional and Derivative Modes. Somewhat improved performance can be obtained if new updated values of Kcl and TD1 are calculated simultaneously while holding R K , RII, RD, and rI1constant. (1) With TD1 fixed, vary Kcl and note that the largest value of the effective damping coefficient.
Ind. Eng. Chem. Res., Vol. 28, No. 10, 1989 1487 Table VI.
Tni
versus
T D ~
Kc,
0.5 1.0
7.1 9.1 8.7
1.5
K., for Dsgn2
max l 0.56 0.75 0.77
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(2) Repeat the above step for different values of 7D1 and select that pair (rD1, Kcl) that gives the largest maximum damping coefficient. Table VI gives the updated values of Kcl and 7D1 for dsgn2. The new optimum values are KC1= 8.7 m3/(min K) and 7D1 = 1.5 min, giving a higher damping coefficient ({ = 0.8). The previous values were Kcl = 6.3 m3/(min K), 7D1 = 2.04 min, and { = 0.7. Therefore, a higher damping coefficient can be achieved by an additional iteration to recalculate the gains and derivatives. A summary of the proposed control design synthesis technique is given in Figure 12. As shown in this figure, the engineer has to provide a desired effective damping coefficient for the calculation of the gains. The integral modes usually reduce the damping coefficient. In this case, the engineer has to input another smaller value of the damping coefficient in order to calculate the integral modes. In the next section, we give some engineering guidelines for choosing the desired effective damping coefficient required for the proposed controller design procedure.
Effective Damping Coefficient: Robustness of a Control System The effect of the physical reactor size and the effect of the kinetic parameters on the stability of open-loop unstable reactors have been studied by Devia and Luyben (1978). The scope of this section is to show that the ef-
Figure 13. Degradation of the performance of a control system (two PI controllers, l = 0.4) due to the uncertainty in the heat of reaction (increase 4%) for dsgn2. 0.5% step change in T2wt.(a) Temperature, (b) coolant flow. (-) perfect model, (- - -) model-plant mismatch.
fective damping coefficient can be used as an indication of the robustness (insensitivity to model-plant mismatch) of a control system. In order to test the robustness, the effect of changing kinetic parameters on the performance of different control systems is considered. Let us design two control systems with different damping coefficients. The first system consists of two PI controllers and has a damping coefficient of 0.4. The second system consists of two PID controllers and has a damping coefficient of 0.7. Let us also consider uncertainties in the heat of reaction, increasing it by 4% for dsgn2, and let the manipulated variables go to the new steady state in order to keep the temperatures at the desired values. Figure 13 shows the the performance degradation of the first control system (two PIS, same settings) due to uncertainty introduced in the heat of reaction for a 0.5 step change in T2set point. Figure 14 compares the performance of the two control systems designed for different damping coefficients (0.7 and 0.4) under both cases of perfect model and model-plant mismatch in the heat of reaction. Figure 14a shows the performance of the controller designed for damping coefficient 0.4, while Figure 14b shows the performance of the controller designed for damping coefficient 0.7. Clearly, the controller performance with an effective damping coefficient equal to 0.7 is affected less than that with a damping coefficient of 0.4. The above results give an indication (but not a proof) that the robustness of a multivariable control system is related to the effective damping coefficient in a way that
1488 Ind. Eng. Chem. Res., Vol. 28, No. 10, 1989 a20.60,
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t
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proposed method is quite nominal. The calculation of the eigenvalues and the optimization steps can be done by standard IMSL subroutines.
320.40
Acknowledgment
320.20
The authors gratefully acknowledge the financial support of the member companies of the Chemical Process Modeling and Control Research Center at Lehigh University. We also acknowledge the many useful discussions of open-loop unstable reactors with D. Bartusiak and the valuable comments about the multivariable root locus diagrams with Professor S. Johnson, Lehigh University.
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Nomenclature
TIME (MINI
Acronyms IAE = integral absolute error IMC = internal model control ISE = integral square error
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Figure 14. Effect of the damping coefficient on the robustness of a controller system (dsgn2). 0.5% step change in TZwt, 4% increase in AH. (a) Two PI, .t = 0.4, (b) two PID, { = 0.7. (-) perfect model, (- - -) model-plant mismatch.
is analogous to SISO systems.
Conclusions A fairly simple controller design procedure is presented for controlling a 2 x 2 multivariable open-loop unstable process. This approach is based on the fundamental concepts that the closed-loop performance of any control system can be deduced by observing the location of the poles. The effective damping coefficient and its corresponding effective closed-loop time constant provide valuable insights into the dynamic behavior of the complex closed-loop system. Taking advantage of these concepts and employing a multivariable root locus, the tuning of the controllers was reduced to a simple four-step optimization procedure. The damping coefficient can be used as an indication of the robustness of the multivariable system. The best value of { depends on the system and the expected model uncertainties. Therefore, the proposed controller design technique requires continuous interaction with the design engineer. At each point, the engineer has to make a conscious decision on the trade-offs between performance and robustness. In our studies, we found that control systems with damping coefficients between 0.5 and 0.7 gave good performance and robustness. The proposed technique can be extended to 3 X 3 or 4 X 4 systems by defining additional ratios of the gains, reset, and derivative times and increasing the iterative steps of the optimization procedure. The proposed procedure requires minimum analytical development before employing computer-aided design. The computation horsepower required to implement the
English Letters A = heat-transfer area, m2 A = state space matrix B = state space input matrix B, = transfer function of the ith controller cAI = concentration of A in the ith stream, kmol/m3 cp = specific heat of reactants, kJ/(kg K) cp, = specific heat of coolant, kJ/(kg K) d = radial distance of the dominant complex root from the origin E = activation energy, kJ/kmol F, = flow rate of the ith stream F = flow rate of the feed, m3/min K,, = proportional gain of ith controller ko = preexponential term MW = molecular weight of reactants V I = reactor volume, m3 V,, = jacket volume, m3 U, = overall heat-transfer coefficient, kJ/(m2K min) u = input vector R = gas constant R+ = feasible region of ratios RD = ratio of the controller derivative times RK = ratio of the controller gains RI = ratio of the controller reset times R, = reaction rate for the ith reactor, kmol/(min m3) r = recycle ratio S+ = region of stability of the proportional gains T I = temperature of the ith stream, K T,, = temperature of the ith coolant stream, K x = state space vector XA, = mole fraction of substance A in the ith stream y = output vector Greek Letters LY = derivative lead/lag constant AH = reaction heat, kJ/kmol f = effective damping coefficient p = density of reactants, kg/m3 p, = density of coolant, kg/m3 7Di = derivative time of the ith controller, min qi = reset time of the ith controller, min
Appendix: Dynamic Model for the System Studied The schematic representation of the process is given in Figure 1. First Reactor: material balance for substance A
I n d . E n g . Chem. Res. 1989,28, 1489-1493
energy balance for the reactor Vlpcp dTl/dt = FpcpTo+ rFpcpT4+ Vl(-IIH)Rl (1 + r)FpcpTl - UIAl(Tl - Tcl) energy balance for the jacket vJ1pccp, d T c l / d t
=
+ UlAl(T1 - Tcl) Second Reactor: material balance for substance A v 2 dC~n/dt= (1 + r)FCA1 - (1 + ~)FCA~ - R2V2 energy balance for the reactor V2pcP dT,/dt = (1 + r)FpcpT1 + V2(-AH)R, (1 + r)FpcPT2- U Z A ~ T-ZTC2) energy balance for the jacket FclPcCpcTeo - Fc1PCpcTc1
vJ2pccp,
dTc2/dt
= Fc2PcCp,TCo
- Fc2PCpcTc2 + U2A2(T2 - Tc2)
reaction rate
Separator: Steady State F2= F3+ F4 F4= rF3 F3= F
Literature Cited Cheng, C.; Brosilow, C. Model Predictive Control of Unstable Systems. AIChE Annual Meeting, New York, Nov 1987.
1489
Cheung, T. F.; Luyben, W. L. PD Control Improves Reactor Stability. Hydrocarbon Process. 1979 (Sept), 215. Devia, N.; Luyben, W. Reactors: Size Versus Stability. Hydrocarbon Process. 1978 (June), 119. Edgar, T. F. Status of Design Methods For Multivariable Control. AIChE Symp. Ser. 1976, 72(159), 100. Friedland, B. Control System Design. A n Introduction To StateSpace Methods; McGraw-Hill: New York, 1986; p 230. Georgiou, A.; Georgakis, C.; Luyben, W. L. Design of Practical Multivariable Process Controllers. Tech. Research Progress Report No. 3; Chemical Process Modeling ahd Control Center, Lehigh University: Bethlehem, Sept 1986. Hwang, S. H.; Chang, H. C. A Theoretical Examination of Closedloop Properties and Tuning Methods of Single-Loop P I Controllers. Chem. Eng. Sci. 1987, 42(10), 2395-2415. Kuo, B. C. Automatic Control Systems, 5th ed.; Prentice-Hak New York, 1985. Postlethwaite, I.; MacFarlane, A. G. J. Lecture Notes in Control and Information Sciences; Springer: New York, 1979; p 12. Power, H. M. A New Result on Eigenvalue Assignment by Means of Dyadic Output Feedback. Ind. J. Control 1975,21, 149. Ray, H. W. Advanced Process Control; McGraw-Hill: New York, 1981. Tzafestas, S. G. Multivariable Control. New Concepts and Tools; D. Reidel: Dordrecht, The Netherlands, 1984. Wonham, W. M. On Pole Assignment in Multi-Input Controllable Linear Systems. IEEE Trans. Autom. Control 1967, AC-12,660. Yagle, A.; Levy, B. C. Multivariable Root Loci on the Real Axis. Int. J. Control 1982, 35(3), 491-507. Zafiriou, E.; Morari, M. Robust H,-Type IMC Controller Design Via the Structured Singular Value. ZFAC 1987.
Received for review August 16, 1988 Revised manuscript received April 20, 1989 Accepted June 13, 1989
Scaling Up of a Monolithic Catalyst Reactor with Two-Phase Flow Said Irandoust and Bengt Andersson* Department of Chemical Reaction Engineering, Chalmers University of .Technology, S-412 96 Gothenburg, Sweden
Erik Bengtsson and Mikael Siverstrom E K A Nobel AB, S-445 01 Surte, Sweden
Scaling up of the monolithic three-phase reactor has been studied in the hydrogenation step of anthraquinones for large-scale production of hydrogen peroxide. The influence of flow pattern (bubble flow and slug flow), flow distribution, pressure drop, and mass transfer has been investigated. It has been illustrated that the segmented -gas-liquid flow (slug flow) gives the highest production rate with very small scale-up effects. 1. Introduction Multiphase reactors involving gas, liquid, and solid catalysts have important applications in industrial threephase processes. Due to the presence of three phases, the interfacial transport is of major importance for the reactor design. A novel multiphase reactor, the monolithic catalyst reactor, satisfies certain criteria of a desirable catalytic reactor, such as low pressure drop, high catalyst efficiency, uniform flow distribution, low axial dispersion, and simple reactor design. In this type of reactor, the catalyst consists of a large number of parallel straight channels in which the gas and liquid reactants flow cocurrently. A schematic design of the monolithic catalyst support is shown in Figure 1. Various aspects of this reactor have recently been reviewed by Irandoust and Andersson (1988a). Among several possible flow patterns in the monolith channels are slug flow and bubble flow (Figure 2). In both of these flows, the dispersed gas disturbs the laminar flow
in the liquid phase and forces the liquid to recirculate within the liquid plugs. This recirculation increases the radial mass transfer and decreases the axial dispersion. Hence, slug flow or bubble flow is desirable in monolith channels. The flow properties of Taylor flow in the monolithic catalyst support have been studied by Irandoust and Andersson (1989). The present study is concerned with the industrial application of monolithic catalyst reactors in hydrogen peroxide production. The formation of certain flow patterns, the pressure drop over the reactor, and the reactor efficiency in both bubble and slug flow regimes have been studied. Furthermore, scaling up of the monolithic reactor with slug flow has been investigated with a pilot plant reactor and an industrial reactor. 2. Experimental Section The monolithic support used in this investigation con-
0888-588518912628-1489$01.50/0 0 1989 American Chemical Society
