I n d . Eng. Chem. Res. 1989, 28, 1177-1184
1177
Phillibert, N. G. An Investigation of Copper Mordenite Catalyst for the Reduction of Nitric Oxide with Ammonia. M.S. Thesis, The Univeristy of Massachusetts at Amherst, 1985. Pruce, L. M. Reducing NO, Emmissions at the Burner, in the Furnaces, and after Combustion. Power 1981, 125(1), 33-40. Yamaguchi, M.; Matsushita, K,; Takami, K. R~~~~~ NO, from HNOBTail Gas. Hydrocarbon Process. 1976,55(8), 101-106.
siderable degree of leeway regarding conditions of fluctuating temperature and flow rate. In addition, the dualcatalyst concept would be applicable to other reaction systems in which competing reactions, e.g., partial and total oxidation, result in the conversion versus temperature curve going through a maximum. Registry No. NO, 10102-43-9; NH,, 7664-41-7. Literature Cited
Received for review May 31, 1988 Revised manuscript received March 29, 1989 Accepted May 2, 1989
Nam, I. S. Experimental Studies and Theoretical Modeling of Catalyst Deactivation. Ph.D. Dissertation, The University of Massachusetts at Amherst, 1983.
PROCESS ENGINEERING AND DESIGN Temperature Control of Exothermic Batch Reactors Using Generic Model Control B a r r y J. C o t t and S a n d r o Macchietto* Department of Chemical Engineering and Chemical Technology, Imperial College of Science, Technology and Medicine, South Kensington, London SW7 2AY, England
A new model-based controller for the initial heat-up and subsequent temperature maintenance of exothermic batch reactors is presented. The controller was developed by using the Generic Model Control framework of Lee and Sullivan, which provides a rigorous and effective way of incorporating a nonlinear energy balance model of the reactor and the heat-exchange apparatus into the controller. It also allows the use of the same control algorithm for both heat-up and temperature maintenance, thereby eliminating the need to switch between two separate control algorithms as is the case with today’s more commonly used strategies. A deterministic on-line estimator is used t o determine the amount and rate of heat released by the reaction. This information is, in turn, utilized t o determine the change in jacket temperature setpoint in order to keep the reaction temperature on its desired trajectory. The performance of the new GMC-based controller is compared to that of the commonly used dual-mode controller. Simulation studies show the new controller to be as good as the dud-mode controller for a nominal case for which both controllers are well tuned. However, the new controller is shown to be much more robust with respect t o changes in process parameters and to model mismatch. 1. Introduction The initial heat-up from ambient temperature and the subsequent temperature control of exothermic batch reactors have always proved to be a difficult control problem (Shinskey, 1979). Because the amount of heat released as the reaction mixture is heated up can become very large very quickly, the reaction may become unstable and cause the temperature to run away if the heat generated exceeds the cooling capacity of the reactor. This runaway can obviously cause great risk to plant personnel and equipment and can, even in the best case, result in a loss of the batch. Therefore, careful control of the rate of change of the reactor temperature and minimization of the temperature overshoot is required. On the other hand, from a production point of view, the heat-up should be done as quickly as possible in order to reduce the overall cycle time of the reaction process. Therefore, any control strategy for heat-up must balance the needs of production with those of safety and quality. Traditionally, the problem has been approached through the use of open-loop control theory to establish, a priori, 0888-588518912628-1177$01.50/0
the optimal temperature profiles and of standard feedback control algorithms to achieve these profiles. The control actions needed to bring the reactor contents to the desired setpoint were obtained by solving an optimal control problem with the objective of minimizing the time to reach the setpoint (Shinskey, 1979). The resulting strategies are of the on-off or “bang-bang” type and consist in applying maximum heating until the reactor temperature is within a specified number of degrees of the setpoint and then switching to maximum cooling to bring the rate of temperature change to zero when the temperature has reached its final desired setpoint. At this point, standard feedback controllers can be switched on and used to maintain the temperature. The most commonly used strategy of this type in industry is the dual-mode controller of Shinskey and Weinstein (1965), which uses a standard PID controller for maintaining temperature. The main problem with approaches of this type is that the optimal switching criterion from heating to cooling, usually based on the reactor temperature, is determined a priori and is therefore only valid for a specific range of 0 1989 American Chemical Society
1178 Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989
operating conditions. Because heat-up proceeds in an open-loop manner and no feedback from the reactor is used, there is no allowance for modeling errors or for changes in process parameters. The net result is that these strategies lack robustness, and any deviation in the operating conditions from those used to tune the controller may result in significantly poorer control performance. The use of adaptive control algorithms would appear to offer promising solutions to this problem, and there have been several attempts in this direction. A recent paper by Cluett et al. (1985) is typical of these attempts. They used a single adaptive control algorithm for both heat-up and temperature maintenance but found that the algorithm did not handle the sharp change from the heat-up mode to the temperature maintenance mode very well. They state that adaptation during the heat-up mode “misleads the operation of the adaptive system” and find that, in practice, fully adaptive strategies give poor performance. In the end, they effectively revert back to a dual-mode approach, where the PID controller is simply replaced by an adaptive controller just for the temperature maintenance part of the profile. Therefore, the robustness concerns of Shinskey’s dual-mode controllers also apply to this controller. A more encouraging strategy was proposed by Jutan and Uppal (1984), who used a model-based approach to estimate the current amount of heat being released in the reactor at any given moment in time. This information was used in a feedforward control structure designed to counterbalance the effect of the heat released. In order to compensate for modeling errors and for the lack of a precise estimate of the heat released, they combined the feedforward controller with a feedback controller. Although this approach overcomes many of the problems of the open-loop strategies, the control performance reported by the authors could be improved further. The reactor is not smoothly delivered to the desired temperature, and there is the presence of significant overshoot in the reactor temperature. These effects may be attributed to the linearization necessary to implement the feedback control action and to the manner in which the feedforward and feedback effects are added. This paper presents a new model-based controller design for the heat-up and temperature maintenance of exothermic batch reactors, which is derived from the Generic Model Control (GMC) algorithm of Lee and Sullivan (1988) and which uses the on-line heat-released estimation concept of Jutan and Uppal (1984). GMC has several advantages that make it a good framework for developing reactor controllers: 1. The process model appears directly in the control algorithm. 2 . The process model does not need to be linearized before use, allowing for the inherent nonlinearity of exothermic batch reactor operation to be taken into account. 3. By design, GMC provides feedback control of the rate of change of the controlled variable. This suggests that the rate of temperature change, which as mentioned above is very important in heat-up operations, can be used directly as a control variable. 4. The relationship between feedforward and feedback control is explicitly stated in the GMC algorithm. 5 . Finally and importantly, the GMC framework permits us to develop a control algorithm that can be used for both heat-up and temperature maintenance and therefore eliminates the need for a switching criterion between different algorithms; this should result in a much more robust control strategy.
The paper will begin by outlining the details of the GMC controller design and the on-line method used for estimating the current heat released. The designs of the controller and the heat-released estimator are general in nature and applicable in principle to the temperature control of any exothermic and even endothermic batch reaction systems. A specific reaction/reactor example is presented to demonstrate the tuning and nominal performance of the GMC controller. In order to provide a comparison for the GMC controller, the design of a dual-mode controller is then presented and implemented on the same reactor system. Finally, the performance of the two strategies is compared with respect to changes in process conditions and modeling errors, and the robustness of both controllers is evaluated. 2. Generic Model Controller Design 2.1. Control Algorithm Formulation. The formulation of a Generic Model Controller for temperature control of exothermic batch reactors is quite straightforward. GMC requires a dynamic model of the process written in standard state variable form. The controller is formulated by solving the dynamic process model for the derivative of the controlled variable, x , and letting it equal what is, in effect, a proportional integral term operating on the difference between the current value of x and its desired value, xsP. Hence, the GMC control algorithm can be written as dx/dt = K,(x’”-x)
+ K2 s,”( x ’ P -
X )
dt
(1)
where K , and K z are tuning constants. For temperature control of a batch reactor, a process model relating the reactor temperature, TI,to the manipulated variable, the jacket temperature, Tj, is required. Assuming that the amount of heat retained in the walls of the reactor is small in comparison with the heat transferred in the rest of the system, an energy balance around the reactor contents gives the required model:
Q + uA(Tj dTI _WCP
dt
7’1)
(2)
where W is the weight of the reactor contents, C, is the mass heat capacity of the reactor contents, U is the heat-transfer coefficient, A is the heat-transfer area, and Q is the heat released by the reaction. W and Cp are assumed to be constant at this point. Replacing TI for x and T:P for xsP in eq 1,combining eq 1 and 2, and finally solving for the manipulated variable, Tj, we obtain the GMC controller:
WCP Tj = T , + -(K1(T:P UA
-
TI) +
Tj gives the jacket temperature trajectory required so that the reactor temperature, I”,, follows the desired trajectory defined by the values of the GMC constants, K1 and K z . As written, eq 3 gives the continuous form of the GMC algorithm. In order to use GMC in a discrete system, the integral must be evaluated numerically using the approximation
where A t is the sampling frequency of the controller.
Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989 1179 Therefore, the discrete time version of eq 3 is
Ti@) = TI(k)+
Equation 5 gives not the jacket temperature setpoint, T;p@),but the actual jacket temperature, Tj(k), needed at the next time interval to move the reactor temperature toward its setpoint, T:P. If Tj(k) were used directly as the setpoint, then, because the dynamics of the jacket are not accounted for in eq 5 , the resulting control would be sluggish. Therefore, some form of dynamic compensation of T j ( kmust ) be used. If the dynamics of the jacket are assumed to be first order (a reasonable assumption given the findings of Liptak (1986)), then a difference equation can be used At(TjBP(k) - T.(k-l)) Tj(k)= T,'k-l' + 1 (6) 1 Tj
where T~ is the estimated time constant of the jacket. The jacket temperature setpoint, TrP, can be obtained by simply rearranging eq 6. Therefore, the following dynamic compensator is obtained: Tj(Tj(k) - T.ck-1)) TjBp(k) = T.(k-U + 1 (7) 1 At The solution of eq 5 and 7 gives the actual setpoint value for the jacket temperature controller to be used for the next control interval. 2.2. On-Line Estimation of the Heat Released for the GMC Controller. The success of the GMC temperature controller is largely dependent on our ability to measure, estimate, or predict the heat released, Q, at any given period in time. There are three main techniques of estimating Q on-line as discussed by Juba and Hamer (1986): 1. direct use of detailed kinetic models; 2. deterministic on-line energy balances; and 3. empirical heat-released estimators. For most reaction systems of industrial interest, the first approach often proves not to be feasible because of the lack of good kinetic models. In a rapidly changing business such as fine chemicals, there often is not enough time or financial benefit in carrying our detailed kinetic studies of the reactions. Deterministic on-line energy balances can also have drawbacks. The largest problem is often the assumption that the heat held in the reactor walls is small. If the heat capacity of the reactor walls is not small, then a deterministic energy balance requires the solution of a system of coupled differential equations with several unobservable states such as the wall temperatures (Juba and Hamer, 1986). Furthermore, the number of process parameters increases, and there may be difficulty in obtaining good estimates for all of them. In their paper, Juba and Hamer use an empirically developed discrete-time transfer-function model of the reactor. The model was determined experimentally by simulating heat generation by the injection of steam into a reactor full of water. They then use time series analysis to develop a transfer function relating the reactor temperature to the jacket inlet temperature and the heat generation. The model is then inverted to obtain an estimate of the heat released. This method has the advantage of accounting for all the dynamics of the reactor, but it has the disadvantage that the resulting model is specific to the given reaction/reactor system and must be redet-
ermined for each new system. Juba and Hamer used this approach with good success on their pilot plant reactors. In addition, they point out that the heat-released estimator could easily be formulated as a Kalman filter in order to improve the estimates by making use of the structure of the noise model. Indeed, a recent paper by de Valligre and Bonvin (1989) demonstrate the effectiveness of using nonlinear Kalman filters in the estimation of the heat released. In this work, the second approach, the deterministic on-line energy balance, was used because it is the most general of the three approaches and therefore is most appropriate for the generality of the controller formulation. We minimize the problems of unknown process parameters by choosing to estimate Q / U A rather than Q itself. By solving for Q/UA, the number of parameters needed to be determined is minimized to the single group, WC,/ UA. In addition, WC,/UA is the only parameter left in the GMC control algorithm of eq 5 , so effectively one parameter characterizes both the estimator and the controller. To develop the estimator, eq 2 is solved for Q/UA to give
Although the reactor temperature, T,, and the jacket temperature, Tj,are available through direct measurement, the derivative of the reactor temperature must be estimated on-line from the direct measurements of T,. This can often be difficult because numerical differentiation is very sensitive to measurement errors. The performance of the estimator can be dramatically improved by using a high-order difference equation for calculating the derivative and by using low-pass filters on the measurements used in the estimator to remove the high-frequency noise. In this work, we use a three-term difference equation (Jennings, 1964) and exponential filters with time constants of 1 min on both the temperature measurements and the estimate of Q / UA. These filters were only used for estimation; the measured signals of T, and Tjwere still used directly in the GMC control equation. The full description of the estimator then becomes (9)
WC, dT1Jk) ( Q / U A ) ( k=) --+ TrJk)- TjJk) (12) U A dt
where the superscript f indicates the filtered value of T,, Tj, or QIUA. The estimator described by eq 9-13 can be applied to any reaction/reactor system by changing the parameters W, C,, U , and A to reflect the system. Further simplifications of the estimator are possible. For example, it is often possible, given the reactor dimensions and the density of the reaction mixture, to develop a relationship between Wand A. So, if the jacket only surrounds the side
1180 Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989
The heat- and mass-transfer rates in the reactor are assumed to be high enough so that the system is essentially reaction rate limited. Therefore, the rate of production of C and D is only dependent on the reactant concentrations: Hot
Cold
TC - Temperature Connol
W - Weight
Figure 1. Batch reactor schematic diagram.
of a cylindrically shaped reactor, as shown in Figure 1,the relationship of W / A is given as
w
-=--
A
pirr2h 2irrh
_ -p r 2
(14)
where p is the density of the reaction mixture, r is the radius of the reactor, and h is the height of the reactor mixture. Therefore, the expression for We,/ UQ can be replaced, for this special case, by,
-we, - - -CpPr UA
2U
(15)
The determination of the value of W e , / UA to be used in the estimator and controller depends on whether any of these parameters change significantly over the course of the reaction. If they change very little, W e , / UA may often be determined simply by performing an open-loop step test for the jacket temperature setpoint with a cold charge. At these low temperatures, the reaction rate is practically zero, and eq 2 shows that, when there is no heat released, WC,/UA is merely the time constant of the system. If any of the values change significantly over the course of the reaction, further tests may have to be performed to characterize these changes in We,/ UA. It should be noted that the structure of the estimator obtained in eq 9-13 is very similar to the structure of the empirically derived estimator of Juba and Hamer. They are both low-order difference equations and are based on similar plant measurements. If, for some reason, the approach of Juba and Hamer was preferred for a given application, the empirical estimator would simply replace eq 9-13. 3. Reactor Simulation The reactor simulation used in this work is largely based on a dynamic model developed for the Warren Springs Laboratory (Pulley, 1986). A well-mixed, liquid-phase reaction system is considered, in which two reactions are modeled: reaction 1
A+B-C reaction 2 A+C-D Component C is the desired product while D is an unwanted byproduct, and the general operating objective is to achieve a good conversion of C while minimizing the production of D. Extensive optimization of the reactor conditions was presented in the original reference.
where R1 and R2 are the rates of production of C and D, respectively, and MA, MB,and Mc are the number of moles of components A, B, and C present in the reactor at any given time. The rate constants, kl and k2,are dependent on the reaction temperature through the Arrhenius relation. Both reactions have a large heat of reaction (AHl = -41 840 kJ/kmol, AH2 = -25 105 kJ/kmol), which makes the overall reaction system strongly exothermic. Heating and cooling of the reactor contents is performed through the use of a single-pass jacket system. The values of the physical parameters of the reactor such as volume, heat-transfer coefficients, and area were based on the dimensions of the batch reactor presented by Luyben (1973). Control of the jacket temperature is provided by a temperature controller on the jacket inlet stream. The heat exchangers needed to control this temperature are not modeled but are accounted for by basing the time constant of the jacket temperature response on typical figures given by Liptak (1986). Figure 1 presents a diagram of the reactor system. Simulation work by Pulley (1986) indicates that an initial charge that is equimolar in A and B produces the greatest yield of C. Therefore, assuming the density of the reaction mixture is that of water and given the dimensions of the Luyben reactor, the nominal charge to the reactor was assumed to be 360 kg of A and 1200 kg of B. Furthermore, given some cost function, Pulley determined that the optimal isothermal reaction temperature typically fell in the range 90.0-100.0 "C, so the final reaction temperature was set to 95.0 "C. Finally, the jacket temperature was assumed to be limited to the range 20.0-120.0 "C due to the heat-exchanger capacities, and the reaction mixture was assumed to be at 20.0 "C at time 0. Because measurement errors are always present when working with real equipment, these were included in the simulation by adding noise to all temperature measurements. In order to use an appropriate noise model, time series analysis was used to determine the noise models for several temperature indicators on the pilot plants at Imperial College. A first-order moving average noise model was found to fit the majority of these temperature indicators and was therefore used in this work. A full description of the reactor system and the values of the parameters used is given in the Appendix. 4. Comparison of GMC w i t h Traditional Control
Strategies 4.1. Dual-Mode Control. To provide a standard with which to compare the performance of the GMC controller, the commonly used dual-mode controller (DM controller) was implemented. As mentioned in section 1,dual-mode control is an example of an open-loop heat-up controller followed by closed-loop feedback controller to maintain temperature. It was originally developed by Shinskey and Weinstein (1965) and further discussed by Liptak (1986). The DM controller consists of a sequence of control actions, each one carried out after the reactor has reached a certain condition. The sequence of actions is as follows: 1. Full heating is applied (jacket temperature setpoint, TjsP,set to its maximum value) until the reactor temper-
Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989 1181
A
, ,
Min TjJ
T i Jacket Temperature Setpoint
.-----*.
Reactor Temperature
Figure 2. Relationship of dual-mode controller constants. Table I. Constants Used E , = 4.0 "C TD-1 = 2.5 min PL = 50.0 "C TD-2 = 2.0 min
~
"1' /I
2 8 1 ' l ! / 0 10 28
I
I
38
48
I
,
I
59 b8 18 TIME (min)
88
I
I
I
,
98
180
118
128
Figure 3. Dual-mode controller response for nominal operation.
in Dual-Made Controller K , = 26.25 "C jacket/"C reactor rI = 2.75 min rD = 0.406 min At = 0.2 min
ature, T,, is within E,,, degrees of the desired reactor temperature, T:P. 2. Full cooling (jacket temperature setpoint set to its minimum value) is then applied for TD-1 minutes. 3. The setpoint of the jacket temperature controller is then set to PL, the preload temperature for TD-2 minutes. 4. A reactor temperature controller, typically a PID type, is cascaded to the jacket temperature controller and its setpoint set to T:p. When properly tuned, the dual-mode controller is optimal (i.e., it brings the reactor contents to setpoint in minimum time given the constraints of the heat-transfer system) as maximum heating is applied for as long as possible and then full cooling is applied to bring the reactor temperature to its new setpoint with no overshoot and a rate of change of zero. Figure 2 presents the relationship between the seven dual-mode control parameters. Assuming that the jacket temperature controller is considered separately, there are a total of seven tuning constants to be determined for the DM controller (Em, PL, TD-1, TD-2, and the PID constants of the reaction temperature controller, K,, 71, and 7 ~ ) . The DM controller was tuned in the following manner. First, the PID controller was tuned by performing an open-loop step response test on the jacket temperature controller. Its setpoint was changed from 20 to 30 "C when f i e d with a normal charge, and a first-order-with-deadtime model was fitted to the response. The Cohen and Coon tuning rules were then applied to yield the values of K,, T ~ and , TD. Second, the remaining four constants were determined by running a series of simulations. After each simulation run, the performance of the DM controller was analyzed, and the values of the parameters were changed using the rules outlined by Liptak (1986) in an attempt to improve the controller's performance. After five tuning runs, the response in Figure 3 was obtained, while Table I gives the final values of the tuning constants used. It can be seen that the DM controller performs very well in this nominal case. The reactor temperature is delivered to the desired setpoint with no over- or undershoot and the transition between the open-loop heat-up mode and the PID control mode is achieved without any disruption. The vigorous changes in the jacket temperature setpoint are caused by the high gains used in the PID controllers and the noisy reactor temperature measurements. This could be reduced by filtering the reactor temperature before using it in the PID
-1E
I
I
I
l
1
1
1
l
1
1
I
l
In tuning the GMC controller, because overshoot was undesirable, 5 was set to 10.0. The value of T was obtained by examining the tuning charts given by Lee and Sullivan and recognizing that, with f = 10.0, the controlled variable should cross the setpoint at approximately 0.257. Therefore, to achieve a performance similar to the dual-mode controller, T was set to 80.0 min. Figure 5 presents the control performance of the GMC controller for the nominal case using these tuning constants
1182 Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989
lZ81--1
Setpo i n t 48
Dual Mode
30 2 8 1I 0
18
28
38
48
58
b8
78
88
98
188
118
128
TIME (minl
Figure 5. Generic Model Controllers response for nominal operation. Table 11. Constants Used in GMC Controller = 10.0 r = 0.5 m 7 = 80.0 min p = 1000 kg/m3 C, = 1.8828 kJ/(kg "C) T~ = 1.0 min U = 0.6807 kW/(m2 "C) At = 0.2 min
and the others listed in Table 11. For this nominal case, it was assumed that the value of WC,/UA was known precisely. This assumption will be relaxed in the next section. I t can be seen that GMC provides performance similar to that of the DM controller but with less drastic changes in the jacket temperature setpoint, especially as the reactor temperature approaches the setpoint value. Furthermore, the GMC controller provides all the control actions from heat-up to temperature maintenance without having to change control algorithms. The only drawback of the GMC appears to be the existence of a small amount of offset. Theoretical studies of the GMC controller by Lee and Sullivan show that this offset will eventually be eliminated and the desired temperature will eventually be reached. Again, the relatively vigorous movements of the jacket temperature setpoint after the setpoint has been reached is caused by the use of noisy temperature measurements directly in the GMC controller and could be reduced by the introduction of low-pass filters. 5. Robustness Evaluation 5.1. Robustness Tests. The previous section shows that both the DM controller and GMC controller effectively control the reactor temperature for the nominal operation for which they were tuned. However, it is important to examine the robustness aspects of both controllers with respect to changes in operating and process parameters and with respect to model mismatch. This becomes especially important for exothermic batch reactor as the reactor must always be operated safely in spite of these changes. A full robustness analysis of the Generic Model Control formulation is beyond the scope of this paper. Lee and Sullivan discuss the effects of simple linear model mismatch in their original paper, but the extension of these results to the nonlinear batch reactor system is not simple. Therefore, we decided to investigate the robustness properties of the two controllers through simulation studies. Five tests were made in which the two controllers, tuned for the nominal operation, were used to control an operation where some of the conditions have changed from their nominal value. The first simply involved changing the overall amount of the charge from the nominal 1560
0
18
I
I
,
I
28
38
40
50
b0
,
I
70
88
98
,
I
100
118
, 120
TIME (minl
Figure 6. Responses of controllers for weight change.
kg to 1300 kg. It represents a change in operating conditions that could be caused by a deliberate change in product demand or an accidental failure of the charging system. The second test involves the reduction of the heat-transfer coefficient from its nominal value to one 25% less. This test simulates a change in heat transfer that could be expected due to fouling of the heat-transfer surfaces. The third tests the robustness of the controllers in the face of change in the reaction chemistry. As stated by Juba and Hamer (1986), the sensitivity of a given control strategy to variations in reaction chemistry is of great importance. In this case, the reaction rate of the first reaction was increased to about 1.5 times the original rate. This is also equivalent to the presence of unmodeled reactions. The fourth case combined the last two perturbations in the operation, the decrease in the heat-transfer coefficient and the increase in reaction rate. In each of these four cases, the changes in operating or process parameters all push the reaction system closer to instability, especially the fourth case, and therefore provide good tests of controller robustness. The fifth and final case involved using the same controllers to control an endothermic rather than exothermic reactor. This case represents an extreme case of model mismatch where the sign of the heat released has actually been reversed. 5.2. Weight Change. Figure 6 shows the responses for both the DM controller and the GMC controller. It can be seen that the performance of the DM controller is degraded while that of the GMC controller has remained essentially the same as that for the nominal case. The reason for the DM controller degradation is that the value of E,,, is specified for a full reactor. On a partly filled reactor as is the case here, cooling does not have to be applied as early since there is less thermal inertia. Therefore, cooling is applied as if the reactor were full and hence the undershoot of the reactor temperature. GMC, on the other hand, can account for changes in W directly in the model and therefore does not have to be retuned for each set of conditions. This is a great benefit whenever batch sizes change frequently as a result of changing product demands. 5.3. Heat-Transfer Coefficient Change. Figure 7 gives the responses of both controllers in response to a changed heat-transfer coefficient, U. This change tests the performance of the controllers in light of a change in unmeasured parameters. Therefore, in this test, the GMC controller is used with its original estimate of U. Although both controllers show a change in performance, the performance of the DM controller has degraded much further than that of the GMC controller. In this test, the value of E, for the DM controller is too small and full cooling
Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989 1183 ise,
1
I----
GHC
bel 4e
I
/
I
I
I
,
,
38
48
58
be
78
2 0 ( ' i E
10
20
30
40
5e ba 70 TIME ( m i n )
80
90
108
iie
e
120
10
ze
I
BB
9e
,
I
,
l e e i i ~ 128
TIME h i n l
Figure 7. Responses of controllers for heat-transfer coefficient change.
Figure 9. Responses of controllers for combined changes. 110
im]
............................
.*----"*.
SETPOINT Setpo int
.
.
.
0
10
20
. 30
40
.
50 bB 70 TIME ( m i n )
I
[ ---- GMC
Dual Mode
.
.
.
.
.
80
90
100
110
120
Figure 8. Responses of controllers for reaction rate change.
is begun too late to prevent the reactor temperature from overshooting, because heat can no longer be transferred at as high a rate as in the nominal case. Furthermore, after the maximum reactor temperature has been reached (98.61 "C for DM control, 96.66 "C for GMC), the GMC controller returns the reactor back to setpoint in a much smoother and quicker manner than the DM controller. This situation represents a much more dangerous operation than the previous one, because an overshoot in temperature brings the system much closer to instability. 5.4. Reaction Rate Change. The results of the third test are given in Figure 8. Once again, it can be seen that the DM controller's performance has again deteriorated by changing the reaction rate. The maximum reactor temperature has risen to 101.20 "C. On the other hand, the GMC controller's performance has changed very little when compared with the nominal response. The improvement in performance is provided by the on-line heat-released estimator as it can predict the speed at which heat is being released in the reactor. 5.5. Heat-Transfer Coefficient and Reaction Rate Change. Figure 9 shows the performance of the two controllers for a case when the reaction rate increases as well as the heat-transfer coefficient decreases. This is the most strenuous of the four tests,as both changes force the reactor system toward instability. From Figure 9, it can be seen that the DM controller has not prevented a temperature runaway in this case, whereas the performance of the GMC controller is approximately the same as in Figure 7, where only the heat-transfer coefficient had changed. Therefore, this case confirms the result that the GMC controller is much more robust than the DM controller and, therefore, will provide not only better control
0
10
20
30
48
50 b0 70 TIME ( m i n )
88
90
100
110
120
Figure 10. Responses of controllers for endothermic reaction.
performance but also increase the safety of operation. 5.6. Application to an Endothermic Reaction. As a final demonstration of the robustness of the new controller, the reactor simulation was modified so that the reaction carried out was endothermic rather than exothermic, while still using the nominal controllers. The new heats of reaction used were +20 920 kJ/kmol for the first reaction and +16736 kJ/kmol for the second. Although, as expected, the dual-mode controller's performance suffers greatly, as seen in Figure 10, the GMC controller's performance has remained consistent. The overall response of the GMC controller is slightly slower when compared to the nominal case, but this is largely due to the fact that the jacket temperature setpoint is constrained a t 120 "C, and therefore the amount of heat transfer is limited. The ability of the GMC controller to handle such extreme model mismatch is due to the generality in its formulation. 6. Conclusions A model-based control strategy using the Generic Model Control algorithm was developed and applied to the heat-up and subsequent temperature control in an exothermic batch reactor. GMC provides a method in which nonlinear feedforward and feedback effects can be combined properly. In the nominal case, the resulting controller has been shown to provide similar performance to a well-tuned dual-mode controller. However, the new controller is much more robust with respect to changes in measurable and unmeasured process parameters. Furthermore, because the GMC controller works directly on the rate of the change of the jacket temperature, the additional protective rate of change constraint control strategies such as those described by Liptak (1986) are unnecessary.
1184
Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989
Acknowledgment
k , = exp(kll - kI2/(T,'+ 273.15))
B.J.C. thanks the Association of Commonwealth Universities for financial support in the form of a Commonwealth Scholarship.
k 2 = exp(k21- kZ2/(Tr' + 273.15))
Nomenclature C p = mass heat capacity of reactor contents, kJ/(kg "C) C,,$ = molar heat capacity of component i, kJ/(kmol "C) AH, = heat of reaction for reaction i, kJ/kmol At = sampling frequency of GMC controller, s E , = approach temperature difference for dual-mode controller, "C h = height of reactor, m Kl = GMC controller constant 1 K 2 = GMC controller constant 2 K , = dual-mode controller PID gain, "C jacket/"C reactor k , = rate constant for reaction i, kmol-' s-l k,' = rate constant 1 for reaction i ki2 = rate constant 2 for reaction i Mi = number of moles of component i, kmol MW, = molecular weight of component i, kg/kmol PL = preload temperature of dual-mode controller, "C Q = heat released in reactor, kW p = density of reactor contents, kg/m3 r = radius of reactor, m R, = reaction rate of reaction i, kmol/s T = temperature, "C T = first-order time constant (s) or GMC tuning constant t = time, s TD = dual-mode controller PID derivative time, s TD-1 = length of time full cooling is applied in dual-mode controller, s TD-2 = length of time preload is applied in dual-mode controller, s r1 = dual-mode controller PID integral time, s U = heat-transfer coefficient of reactor, kW/ (m2 "C) V = volume, m3 W = reactor weight, kg x = controlled variable = GMC tuning constant Subscripts 1 = reaction 1 (A B 2 = reaction 2 (A + C
+
A = component A B = component B C = component C D = component D f = filter j =jacket r = reactor
--
W = MWAMA + MWBMB + MWcMc + MWflD Mr =MA + MB + Mc + MD cpr
=
+
( c p ~ ~ A cpBMB+ Cp$C
+ Cp&D)/Mr
v = w/p A = 2V/r Qj = UA(Tj' - Tr' )
Q, = -AHlR1- AH2R2 - - --Qr + Qj dT,' dt
MrCpI
dTJ' - - - FjpjCpJCTj"P- Tj') - Qj dt
T, = T,'
VjpjCpJ
+
- 0.866~(~-"
where dk) is normally distributed with oa = 0.1 "C T.J = T.' + a(k)- 0.866a(k-') J where dk) is normally distributed with ua = 0.1 "C. Physical Properties and Process Data. MWA = 30 kg/kmol, MWB = 100 kg/kmol, MWc = 130 kg/kmol, MWD = 160 kg/kmol, C,, = 75.31 kJ/(kmol "C), Cp, = 167.36 kJ/(kmol "C), C = 217.57 kJ/(kmol "C), C, = 334.73 kJ/(kmol "C), k F = 20.9057, k12 = 10000, kzf = 38.9057, k22 = 17000, AHl = -41840 kJ/kmol, AH2 = -25 105 kJ/kmol, p = 1000 kg/m3, r = 0.5 m, U = 0.6807 kW/(m2 "C), pj = 1000 kg/m3, CpJ= 1.8828 kJ/(kg "C), Fj = 0.0058 kg/s, and Vj = 0.6912 m3. Initial Conditions at t = 0. MAo= 12 kmol, MBo = 1 2 kmol, Mco = 0 kmol, MDo = 0 kmol, T,O = 20 "C, and TjO = 20 "C. Literature Cited
C) D)
Superscripts ' = actual value before addition of measurement noise ( k ) = at the kth time interval sp = setpoint
Appendix: Batch Reactor Model Equations: dn/i,/dt = -R1- R2
Cluett, W. R.; Shah, S. L.; Fisher, D. G. Adaptive Control of a Batch Reactor. Chem. Eng. Commun. 1985, 38, 67-78. de ValliBre, P.; Bonvin, D. Application of Estimation Techniques to Batch Reactors. Part 11. Experimental Studies in State and Parameter Estimation. Comput. Chem. Eng. 1989, 13, 11-22. Jennings, W. First Course in Numerical Methods; The Macmillian Company: New York, 1964. Juba, M. R.; Hamer, J. W. Progress and Challenges in Batch Process Control. In Proceedings of the Third International Conference on Chemical Process Control; Morari, M., McAvoy, T., Eds.; Elsevier Science Publishers: New York, 1986. Jutan, A,; Uppal, A. Combined Feedforward-Feedback Servo Control Scheme for an Exothermic Batch Reactor. Znd. Eng. Chem. Process Des. Dev. 1984,23, 597-602. Lee, P. L.; Sullivan, G. R. Generic Model Control. Comput. Chem. Eng. 1988, 12, 573-598. Liptak, B. G . Controlling and Optimizing Chemical Reactors. Chem. Eng. 1986, May 26, 69-81. Luyben, W. L. Process Modeling, Simulation and Control for Chemical Engineers; McGraw-Hill Book Company: New York, 1973. Pulley, R. A. Batch Process Modelling Club Report CR 2828(CON), 1986; Warren Spring Laboratory, Stevenage, Herb, UK. Shinskey, F. G. Process-Control Systems; McGraw-Hill Book Company: New York, 1979. Shinskey, F. G.; Weinstein, J. L. A Dual-Mode Control System for a Batch Exothermic Reactor. Twentieth Annual ISA Conference, Los Angeles, CA, Oct 4-7, 1965.
Received f o r review July 12, 1988 Accepted January 3, 1989
