Ind. Eng. Chem. Res. 1991,30,1541-1547
10.16e* G(n)= (25.33s+ 1)[(4.26)’s2 + 2(4.26)(0.619)s + 11 (118) These results are incorrect because the system is really a sampled-data one, and the ultimate gains and frequencies depend on sampling period. B. Models Obtained by Using Correct SampledData Relationships. Using the equations derived in the last section for sampled-data autotuning, we obtained the following discrete models for the experimental heat exchanger. (1) Model derived from first and second ATV tests: Parameter values found: bl = 1.2319,b2 = 0.3002,b3 = -0.01420,b4 = -0.8504 and p1 = 0.8974 r1 = 27.72 K = 11.84 72 = 2.739 p2 = 0.3345 ~-~(-0.04199z + 0.8504) (119) HG(z)= (z - 0.8974)(2 - 0.3345) 11.84e* (120) ()’ = (27.72s+ 1)(2.739s + 1) (2)Model derived from first and third ATV tests: ~-~(-0.1052 + 0.8505) (121) HG(z) (z - 0.8536)(z - 0.4023) 8.52e*
+
+
(122)
Gb) = (18.95s 1)(3.29s 1) The discrete ATV models can be compared to another discrete model obtained by Ekkinat (1990)using PRBS test with auxiliary information. The transfer function obtained from PRBS was
~-~(-0.0126z + 0.8423) (123) HG(z)= (z - 0.9027)(z - 0.3390) Nyquist plots comparing these three models are given in Figure 5. The two models obtained from the autotune testing match well with the model obtained from PRBS in the area near the crossover point. There are some differences between the models in the lower frequency region. This is not surprising because the information from the three autotune tests is concentrated in the high-frequency region.
1541
Conclusions A new method has been developed to determine transfer function models for open-loop stable processes using information obtained from two autotune tests: ultimate gains at two different ultimate frequencies. Then a least-squares method can be used to calculate the parameters of several simple transfer functions. The best model can be selected by examining parameter values and residuals. The method worked very well on several known simulated systems and was tested successfully on a heat-exchanger experiment. The transfer function derived from the proposed method matched well with that obtained from PRBS tests with auxiliary information. The method is suitable for use with plant data since it provides estimates of the steady-state gain in addition to the time constants. The method has the potential to be very useful for practical identification problems. Acknowledgment Some of the preliminary work on the inaccuracy of the autotune method for getting ultimate gains and frequencies was done by Jose Marti while he was a visiting engineer at the Chemical Process Modeling and Control Center at Lehigh University in 1988. Literature Cited Astrom, K. J.; Hagglund, T. Automatic Tuning of Simple Regulators with Specifications on Phase and Amplitude Margins. Automatics 1984,20, 645-651. Eskinat, E. Use of Auxiliary Information and Nonlinear Methods in Process Identification. Ph.D. Thesis, Lehigh University, Bethlehem, PA, 1990. Li, W. An Extension of the ATV Identification Technique. M.S. Thesis, Lehigh University, Bethlehem, PA, 1990. Luyben, W. L. Derivation of Transfer Functions for Highly Nonlinear Distillation Columns. Ind. Eng. Chem. Res. 1987a, 26, 24S2495. Luyben, W. L. Sensitivity of Distillation Relative Gain Arrays to Steady-State Gains. Ind. Eng. Chem. Res. 1987b,26,2076-2078. Luyben, W. L. Process Modeling, Simulation and Control for Chemical Engineers, 2nd ed.; McGraw-Hill: New York, 1990,pp 662-668. Muhrer, C. A. Private communication, 1989.
Received for review September 4,1990 Revised manuscript received December 17, 1990 Accepted February 7, 1991
Multivariable Global Input/Output Linearized Internal Model Control of a Semibatch Reactor Jaydeva Bhat,
K.P.Madhavan,* and M. Chidambaram
Department of Chemical Engineering, Indian Institute of Technology, Bombay, Powai, Bombay 400 076, India
The control of a semibatch reactor is treated as a multivariable setpoint trajectory tracking problem. Global input/output linearization is used to obtain a decoupled linear system that is placed under internal model control. A coordination strategy is proposed for the operation of the two loops. Adaptive features are incorporated in the strategy to account for uncertain kinetics. The control system has a number of tuning parameters that can be used to obtain performance tradeoff. Introduction A semibatch mode of operation is often used for strongly
exothermic reactions to balance the heat generated due to reaction with the available cooling capacity of the re-
actor. There are three phases in the temperature control problem in a semibatch reactor-startup phase, maintenance phase, and termination phase. In the startup phase, the reagent addition and flow of heating/cooling agent are
0888-5885/91/2630-1541$02.50/00 1991 American Chemical Society
1542 Ind. Eng. Chem. Res., Vol. 30, No. 7, 1991
manipulated so as to obtain a rapid approach to the set point with no overshoot. In the maintenance phase the reagent addition and cooling are maintained at appropriate rates so as to keep the temperature at the set point. The termination phase involves the temperature control of the reactor using coolant flow rate. The reactor control problem usually has specifications such as maximum allowable rate of temperature rise or overshoot. Equally important is the need to complete the reaction in a minimum time without violation of the specified temperature limits. Thus the control of a semibatch reactor poses challenging problems. A rigorous approach to the control problem is to use optimal control theory, which will provide optimal policies in the reagent addition and extent of coolant rates (Jackson et al., 1972; Tsoukas and Tirrell, 1982). Bang-bang and singular control policies may result. One encounters difficulties in implementing optimal control policies due to the sensitivity of control laws in modeling uncertainties and the open-loop form of the control law. A less than optimal approach to the startup problem in a batch reactor is to use the dual-mode control (Shinskey and Weinstein, 1965) where discontinuous control policies of maximum heating and cooling are followed by a switch to conventional PID as the set point is approached. The dual-mode control strategy proposed has not considered explicitly the problem of reagent addition. Lewin and Lavie (1990) have studied the problem of specifying startup temperature trajectories for a batch reactor. The guidelines presented by them do not take into account flexibility in reagent additions. A simplified approach (Liptak, 1986) consists of maintaining as high a rate of addition of reagent as possible such that cooling duty is not stressed beyond a permissible limit close to the maximum cooling rate. This approach is suited for control at a constant temperature and may need tuning during the startup phase. For highly sensitive reactors that are prone to potential runaway conditions, Liptak (1986) has advocated the use of controls that limit the rate of rise of temperature during heatup. Thus to simplify the implementation of control, the startup phase of a semibatch reactor can be viewed as a temperture set-point rate-tracking problem in which a desired set-point trajectory is specified a priori. The reagent addition and cooling rate have to be so regulated that the rate of rise of reactor temperature is close to the desired value. This is a two input/single output problem. Jutan and Rodriguez (1987) suggested a parametric method for a two input/single output control problem arising from the use of dual input for heating/cooling in a batch reactor. It is not easy to translate this approach to the semibatch control problem involving reagent addition and cooling. A detailed study of temperature control of a semibatch polymerization reador was made by Longwell (1986). The strategy of control during the startup phase, maintenance phase, and termination phase was considered. For the temperature control problem, the manipulated variable was liquid reagent addition rate (controlling heat generation rate) or flow of heating/cooling agent (controlling heat removal/addition rate). Longwell (1986) had suggested the use of pressure as an auxiliary controlled variable in addition to temperature. The problems encountered by him in the control of semibatch reactor were as follows: (i) inverse response of the reactor temperature to flow of liquid reagent especially at lower temperatures; (ii) complex problem of coordination of variation of the two manipulated variables, Le., using a pressure control loop to
regulate reaction rate and a temperature control loop to regulate heating/cooling duty; (iii) interaction between the temperature and pressure control loops. The algorithm developed by Longwell was model based but had difficulties in locating the "kickoff" point for initiation of the dual-control policy. In this paper, a strategy for control of a semibatch reactor has been developed, using as a test problem a semibatch reactor that represents in an adequate manner some of the major control problems caused by exothermicity and undesirable inverse response behavior due to lack of coordination in reagent addition and heat removal. In the reactor studied, the heating option for facilitating the startup of the reactor is not considered since with only a cooling option is the possibility of incidence of inverse response accentuated. In the reactor chosen, during the early phase of the reaction, the reactor temperature is relatively insensitive to reagent addition. The subsequent rate of rise of temperature as the reaction gets under way is to a large extent influenced by the concentration of the added reagent in the reactor. Thus with the choice of the concentration of the added reagent in the reactor as the second controlled variable in addition to temperature, it should be possible to have a direct control of the rate of rise of reactor temperature by using reagent addition as the manipulated variable. In accordance with this approach, the original two input/single output (temperature) control problem has been transformed into a two input/ two output problem. The temperature control problem has been treated as a set-point tracking problem for the temperature control loop and set-point rate-tracking problem for the concentration control loop. Important issues are (i) control structure for the two loops, (ii) coordination of operation of the two loops, (iii) choice of the switching point for implementation of rate of temperature control policy, and (iv) effect of set-point trajectory on the temperature response and reaction time. In the control strategy development global input/output linearization (Kravaris and Chung, 1987),which has been used with success for batch and continuous reactors, has been considered. Alvarez et al. (1988) have applied a global linearization technique to a two input/two output continuous free-radical polymerization reactor. For the semibatch reaction considered in the present study, global input/output linearization has been done to get a decoupled system. Internal model control (Garcia and Morari, 1982) has been applied to the decoupled systems. A coordinated control strategy for the two decoupled systems is proposed to obtain tradeoff between the reaction time and the set-point tracking error. Application to a Semibatch Reactor Reactor Control Problem. The semibatch reactor considered involves the synthesis of hexyl monoester of maleic acid according the following reaction: maleic anhydride + hexanolA B hexyl monester of maleic acid + water Because of large heat of reaction (AH= -33.5 MJ/kmol). Hugo and Konczalla (1980) suggested that the reaction may be carried out in a semibatch mode. In the semibatch operation reagent A is melted at first, and then reagent B is added at a regulated rate so that heat generated is matched with cooling capacity. The model equations are the same proposed by Westerterp et al. (1984): dCA/dt = - c ~ u i / V- ~ C A C B (1) dC,/dt = (CBL - CB)Ui/VR - kCACB (2)
Ind. Eng. Chem. Res., Vol. 30, No. 7,1991 1543 dT/dt = (-Afz/pC,)kc~CB
- (T - 3 2 7 ) u l / v ~- U2(T - 327) (3)
dVR/dt =
(4)
The reaction data are given in Table I. The control objective is to complete the reaction as fast as possible without exceeding the maximum allowable temperature of 373 K. Westerterp et al. (1984) indicated that 99% reaction can be completed in 8OOO s by adopting a constant rate of reagent addition and full cooling. In our strategy reagent addition rate and coolant flow rate are chosen as manipulated variables. Reactor temperature and concentration of added reagent in the reactor are chosen as the controlled variables. In our approach global input/output linearization with decoupling is proposed. Each system is of relative order 1 and since
is bounded away from singularity except at the initial conditions, the 2 X 2 system could be transformed by global linearization to a decoupled linear system (Sastry and Isidori, 1989). From an examination of eqs 2 and 3, the input/output variable pairs Ul - CB and U2- T form automatic choices as manipulated controlled-variablepairs. The application of global input/output linearization technique is shown as follows: From eq 2
This equation can be solved for U1,given an input VI:
Similarly from eq 3
(
~ ) ~ C A -C( TB- 327)-Ul - U2(T - 327) = VR V2 - T
B2
(7)
U2 can be calculated as u, =
given an input V2. Equations 6 and 8 are the linearization transformations that result in a decoupled linear system. B1 and B2 can be viewed as tuning constants available to the designer. The internal model controller will be diagonal with elements Bls + l and Bg + l. For incorporating robustness, the controller is premultiplied by a diagonal filter with elements l/(Bl’s + 1) and 1/(Bis + 1). The resulting control structure is shown in Figure 1. For most of the simulation cases considered in the present study, the choices B{ = B1 and B i = B2 are used, leading to a diagonal controller with unity gain. The rationale for this
Table I. Model Parameters and Initial Conditione k = 1.37 X lo1*exp(-12628/7? ms/kg mol 8 -AJY/pCp = 16.92 m3 K/kg mol ,C = 10.1 kg mol/mg, C, = 9.7 kg mol/m3 ,C = 0.0, To= 328 K, Vo = 2.2 ms U,5 O.OOO253 At = 10 s
will be discussed in the next section. The temperature control is considered as a set-point trajectory rate-tracking problem. A temperature set-point trajectory that is similar to the response of a first-order process represents in a qualitative manner the form of allowable rate of temperature rise suggested by Liptak (1986). The time constant of the first-order response can be a tuning parameter. For estimation of the time constant of the temperature set-point trajectory, a useful guideline could be an estimate of the minimum rise time. The calculation of the minimum rise time and safe startup time as proposed by Lewin and Lavie (1990) requires simulation of the semibatch reactor for a specified reagent addition policy. Since such a reagent addition policy was not known a priori, an estimate of the minimum rise time was made using an average value of dT/dt based on average value of C A = 5 kg mol/m3, CB = 1kg mol/m3, and T = 360 K. The estimated rise time was 838 s. A time constant of lo00 s was used in the nominal set point trajectory, TD= 375 - 45 exp(-t/1000). This approach appears to be satisfactory in view of the fact that only a small amount of reaction takes place during the startup phase, and as Lewin and Lavie (1990) have pointed out, there is a very little incentive in adhering too closely to a specified trajectory. Simulation Results. A. Study of Response of Decoupled System to Prescribed Temperature a n d Composition Set-Point Trajectories with All State Variables Measured A set-point profile for composition CB was specified by using the simulation results of Westerterp et al. (1984). Since a constant feed rate of reagent was used by them, the composition profile showed a gradual initial rise to a maximum followed by a slow decrease in CB to a low level. The C B profile showed an upward trend toward the terminal phase of reagent addition. For B2 the value chosen was close to the rise time of the Westerterp temperature response, and the B1 value chosen was much smaller since no constraint was placed on the maximum value of flow rate of reagent. The temperature and composition profiles, reagent addition rate, and coolant rate history are shown in Figure 2. As indicated by Longwell (1986) and Shinskey (19791, the sensitivity of the temperature to reagent addition in the initial phase of the reaction is low because of lower temperature and very low levels of B. A similar situation is also seen toward the end of the reagent addition (in the interval of 4000-5000 s). In view of the low gain of the temperature response to reagent addition, there is likely to be a setpoint error during the startup phase. In the present simulation the set-point trajectory for composition CB was assumed independent of the status of temperature control. It is necessary to coordinate the reagent addition rate with the coolant rate such that the temperature reaches the set point with no overshoot and the reaction completion time is reduced. This aspect of coordination is considered in the following section. B. Coordinated Semibatch Reaction Control with Incomplete Measurement and Heat of Reaction Estimation: For coordination of the two control loops shown in Figure 1,the set point of the concentration control loop is varied, after the reaction has set in, to ensure that dT/dt follows dT,,/dt. The set point of the concentration control loop is varied as per the logic given below.
1544 Ind. Eng. Chem. Res., Vol. 30, No. 7, 1991
I
5P
ELECT
ONTROLLER __c
OUTPUT MAP
PROCESS Tset
,T
__c
"2
ONTROLLER
*CB
TR AN SFOR MAT10 N
Figure 1. Multiinput/multioutput global input/output linearized internal model coordinated control structure.
01
where B = (T - 327)(V,+ Vl/ V d . With such a set-point policy it should be possible to limit the rate of temperature rise within the permissible value with proper tuning of the control loops. The transition (switching point) from constant set-point control to a variable set-point control occu18 automatically. The modified control scheme with coordination is shown in Figure 1. One of the drawbacks of globally linearized control structure is that it requires all the state variables. Further kinetic and heat of reaction parameters may not be accurately known. To overcome these difficulties, the heat evolution rate is estimated from the energy balance equation as advocated by Juba and Hamer (1986). In the Juba-Hamer approach, the heat of reaction is estimated from the unsteady-state heat balance equation by using the current values of flow rates, temperature, and its rate of change. This results in modified control laws:
-
007 -
I
5,006004 -
002
-----
I
370-
m -2 0
z -1 5:
g I: tJl
-1
0
-05
330 I
loo0
2000
3000 TIME
LOOO
,
I
5000
I
I
E
.
6000 6500
SECS
Figure 2. Time response of temperature, manipulated variables and composition for aribitrary profiles in temperature and concentration. B1 = 100,B2 = 3000,B1' = B,, B; = B2: (---) desired profile; (-) actual response.
During the initial phase of the reaction, the temperature response is insensitive to reagent addition and as long as dT/dtlT is less than dTMt/dtlT,the policy is to maintain the set point of Cg, CB+, at a fixed value of C With the reagent addition varied to maintainFB at , C the reactor temperature T begins to rise. As dT/dt(, tends to exceed dT,,/dtlT, the,C point is adjusted such that dT/dtIT matches dT,,/dtlp An important feature of this comparison is that it is done on the basis of values coresponding to a given temperature and not according to a given time. The set point variation is calculated by the expression
[
U, = ( p + B ) - (2'- 327)-
Vu1 R
"L "1
- - (T 1 3 2 7 )
(11) where T = dT/dt. This enables one to adapt the control policy to prevailing reaction rates in the reactor. The simulation of the reactor was done with such a control law using values of B1and Bz lower than those used in section A in an attempt to speed up the response. While the temperature reaches the set point without overshoot, a temperature droop is seen toward the latter phase of the reaction. This is due to the saturation effect in the coolant flow. To overcome this mismatch due to saturation of coolant flow in the temperature loop, input to the internal model (V,) is recalculated from eq 8 corresponding to the saturated value of U,,and this value is used as input to the internal model in the temperature control loop as suggested by Calvet and Arkun (1988). The closed-loop simulation results with these modifications are shown in Figure 3. Good temperature control was observed. One of the main problems in such a model-bawd control is the sensitivity of the control law for uncertainty in the kinetics of reaction. The proposed control strategy is evaluated for a disturbance of f20% in reaction rate. The closed-loop simulation results are shown in Figure 4. For
Ind. Eng. Chem. Res., Vol. 30, No. 7, 1991 1545 -1
1
TIME
SECS.
Figure 5. Temperature time response when noise is added to reactor temperature. B1 = 30, B, = 2100, E{ = B1, = Bz: (---) desired trajectory; (-) actual response.
chosen to match prevailing kinetic parameters. The effect of a noisy temperature measurement on the performance of the proposed control law was studied by a representing the measured temperature T,: Tr = T + ak - O . 8 6 6 ~ ~ - ~ where { a h ) is a zero mean, 0.1 variance random noise. The noisy temperature was filtered to give TF: TF(k + 1) = OAT*&) + 0.2Tr(k) TIME
The value of dT/dt was estimated by using filtered value or TF as
SECS
Figure 3. Temperature, composition, and manipulated variables response for coordinated control. B1= 30, B2= 2100, B1'= B1,BP) = Bz: (- - -) desired response; (-1 actual response. """I
%Ow 360
330
'
3201
0
"
500 lo00
I
2000
I
I
3000 TIME
I
'
4000 5000 SECS
I
I
6000
'
7000 7500
Figure 4. Temperature time response when *20% in reaction rate, B1 = 30,Bt = 2100, B{ = B1, B i = B1: (- -) desired trajectory; (solid line 1) actual respome for +20% in reaction rab, (solid line 2) actual
-
response for -20% in reaction rate. Table 11. Reactor Rsrponre Summary for Mismatch in Model Parameters ( B , = 30, Bt = 2100) model variation in switching 99% reaction points, s completion time, s param param, % nominal 1870 6570 kinetic term -20 2240 7290 +20 1560 5320 heat-transfer -20 1810 6950 coeff
a -20% change in heat-transfer coefficient, the temperature response is not significantly affected. It is found that control laws are robust to variation in kinetics and heattransfer coefficient. A summary of the performance of the semibatch reactor for mismatches in reaction rate and heat-transfer coefficients is shown in Table 11. It can be seen that the proposed strategy has a good adaptive feature with the switching point automatically
D = -d T - T F ( ~-)T F (-~ 1)
dt At The presence of noise causes dT/dt to assume negative values, which a t times can drive the denominator of eq 9 to low values. This can lead to high values for,C resulting in excessive reagent addition. As a robustness measure, the value of dT/dt used in eq 9 is constrained to be above a small positive value (D2 0.0001). Though the temperature response in the presence of noise appears to be satisfactory as shown in Figure 5, the proposed robustness measure retards the reagent addition causing the 99% reaction time to be extended to 7990 s. However, as explained in the next section, the value of B1 can be decreased to hasten the response of the composition control loop such that the reaction completion time can be reduced to lower values. C. Simulation for the Study of Effects of Tuning Parameters: The control strategy described in the previous section has a number of tuning parameters. A simulation study has been undertaken to evaluate the effect of the following parameters on the semibatch reactor response: (i) internal model time constant B1and B2 of the two loops; (ii) use of lead-lag term (B# + l ) / ( B i s + 1)for the internal model controller in the temperature control loop; (iii) time constant of the first-order model for temperature set-point trajectory. The ranges for B1 and B2 were chosen by using the following considerations. The lower limit for Bz was chosen to be equal to the time constant of the nominal temperature set point trajectory, namely, lo00 s. A 10-fold increase in B2was envisaged. Since no limit was specified for the maximum reagent addition rate, the concentration loop was tuned to have a faster response. The upper limit of B1was chosen as 100 s (one-tenth of the minimum limit of B2).The lower limit was chosen as 10 s, allowing for a 10-fold variation of B1. The responses of the reactor for various values of B1 are shown in Figure 6. For wide variation of B1from 100 to 20 s, the temperature response is only marginally affected. The switching point and the 99% reaction time do not
1546 Ind. Eng. Chem. Res., Vol. 30, No. 7, 1991
i:::vv,; Loor--
' I
m
3901
i
3701
X
,
I
I
,
340
330
320
0
I
m'
m
I
,
I
,
,
,/
'
1000
2000
3000
4000
TIME
SECS
5000
6000
7000
Figure 6. Temperature response for various values of tuning parameters B,, B2= 2100, Bl' = B1,E,' = B2: (1)B1= 10; (1') Bl = 10, with limit on D;(2) B, = 20; (3) B, = 100.
Table 111. 'Summary of Reactor Response for Variation in B , and B z switching 99% reaction point, s completion time, s B1 B2 1860 7010 1P 2100 1820 6560 lob 2100 1870 6090 20 2100 1860 6370 100 2100 1590 7800 50 lo00 1750 6330 50 1500 2010 5970 50 loo00 a
With limit on D. With noise and limit on D.
---u2 0002
-
0001
-
0015 -
01
-
UI
~
005 0-
380-
I 1
0
1000
2000
3000 TIME
4000
5000
6000
7000
SECS
Figure 7. Temperature response for various values of tuning parameter B?: (1) BP= 1OOO;B1= 50, B< = B1,E,' = E,; (2) B2 = 1500; (3) B2 = 1oOOo.
show significant variation. However when B1is reduced to 10 s, there is excessive reagent addition leading to inverse response behavior followed by a sudden rise in temperature. Though 99% reaction time is much faster, there is excessive overshoot of temperature above the set point. The use of limiting criterion on D as explained in the previous section is seen to contain the reagent addition and ensure a satisfactory temperature response in the presence of noise. The 99% completion time with the limiting criterion is 7010 s, which is faster than the 99% reaction time for B1= 30 s obtained in the presence of noise with limit on D. The responses of the reactor for variation of B2 are shown in Figure 7. The responses show that it is not desirable to tune the temperature loop for a faster response by using a lower value of B2.For a value of B2near the lower end of the range, namely, B2 = 1000 s, though the temperature response is closer to the set-point trajectory, the reaction is not complete even after 8000 s. This is indicative of the improper tuning of the two loops, resulting in inappropriate utilization of available cooling capacity to accommodate higher levels of reagent addition rates. There appears to be a good incentive for slowing down the response of the temperature loop by increasing the value of B2.Though the temperature response is slowed down slightly, the reaction is driven to completion much faster as can be seen from the summary of the results presented in Table 111. There appears to be a tradeoff between the set-point tracking error and 99% reaction completion time. Though the IMC controller for the temperature control loop has the general structure (Bg + l)/(Bis + 11, the simulation results presented so far used the condition B, = B i . The response of the reactor for the case B i < B2 is shown in Figure 8. For a given value of B2,the use of a lead-lag form of the internal model controller is shown to move the temperature response closer to the specified set-point trajectory. The temperature follows the set point
390
I
TIME
SECS
Figure 9. Temperature responses for faster and slower set point trajectories B1 = 30, Bz = 2100,B< = B1,B,' = BP:(dashed line 1) specified trajectory = 373 - 45 exp(-t/500); (solid line 1) actual temperature response; (dashed line 2) specified trajectory = 373 45 exp(-t/5000); (solid line 2) actual temperature response; (solid line 3) with limit on D.
very closely during the maintenance phase. However, the reaction does not go to completion because of the premature termination of cooling by the temperature control loop preventing further reagent addition. The effect of the time constant of the set-point response on the semibatch reactor response was studied by considering two cases-a slower set-point response than the nominal and a faster set-point response than the nominal (see Figure 9). For the slower set-point response, the set-point tracking error is smaller at the beginning of the startup phase. Because of the variable set-point policy, the reagent addition is reduced drastically as dT/dtlT becomes higher than dT,,/dtl, At a later stage, when rate of temperature response becomes slower than the rate of set-point response, the reagent addition is resumed. Because of low level of C, and lower value of T,the resumption of reagent
Ind. Eng. Chem. Res., Vol. 30,No. 7, 1991 1547 addition leads to an inverse response effect. The inverse response triggers an excessively large amount of reagent addition leading to excessive overshoot of temperature. The use of a limit on dT/dt in eq 9,as proposed earlier, prventa the Occurrence of inverse response and causes the temperature to follow the set-point trajectory. The reaction completion time will be delayed beyond 8OOO s because of slower set-point trajectory chosen. On the contrary, speeding up of the set-point response does not produce such an adverse effect. The temperature response shows a slight overshoot as set-point trajectory is speeded up. The 99% reaction completion time occurs at 5980 s. The simulation results show that the tuning parameters available in the proposed control strategy provide the user with the necessary flexibility in exercising control of the semibatch reactor.
T = first derivative of temperature dT/dtlT = value of dT/dt corresponding to a given T dT,,,/dtlT = value of dT,,/dt corresponding to a given T t = time, s At = sampling time, s U1= reagent addition rate, m3/s U,= coolant flow rate parameter, s-l V1 = input to globally linearizing transformation of concentration loop V , = input to globally linearizing transformation of temperature loop VR = reactor volume, m3 (-AH) = heat of reaction, kJ/kg C, = heat capacity of the reaction mixture Greek Symbols p = density of reaction mixture, kg/m3
Conclusion This paper has addressed the problem of temperature control of a semibatch reactor using a control strategy that tackles the coordination problem of reagent addition and coolant flow. The adaptive feature included in the control strategy has shown the ability to adapt the switch point as per the prevailing reactor conditions. The dynamics of the internal model of the temperature control loop and the set-point response dynamics can be chosen as tuning parameters for obtaining the desired tradeoff between set-point tracking error and reaction completion time. The proposed strategy of reagent addition has certain inherent robustness features. During the initial phase when the temperature response is relatively insensitive to reagent addition, the concentration CB is maintained at a safe value. When the reaction gets under way, the reagent addition rate is dictated by the values of dT,,/dt and dT/dt corresponding to the prevailing temperature. Since dT,, f dt falls off as the set-point is approached, the reagent demand will also get progressively reduced. The robustness is further enhanced by using a limit on D (or estimated dT/dt) in the set-point calculation for CB.The multivariable control strategy presented here can be applied to semibatch reactors of relative order 1 for each output variable with exothermic and inverse response features.
Literature Cited
Nomenclat u r e ar = random noise signal B1,B2= tuning parameters CA,CB = concentrationof A and B in the reactor, kg mol/m3 CBL = concentration of B in the feed stream, kg mol/m3 ,C = constant set point for B Cg, = variable set point for B D = estimated value of dT/dt using TF k = specific reaction rate, m3/kg mol s T = reactor temperature, K TF(k)= filtered value of T at kth sampling instant T,(k) = noisy reactor temperature at the kth sampling instant To = initial reactor temperature, K T,, = set point for the reactor temperature, K
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Receiued for review May 31, 1990 Revised manuscript receiued January 30, 1991 Accepted February 12, 1991
