Fed-Batch Reactor Temperature Control Using Lag ... - ACS Publications

Jun 22, 2004 - A lag-compensated PI controller is shown to provide effective temperature control. In addition, because the process gain between the ...
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Ind. Eng. Chem. Res. 2004, 43, 4243-4252

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Fed-Batch Reactor Temperature Control Using Lag Compensation and Gain Scheduling William L. Luyben† Process Modeling and Control Center, Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015

This paper studies the control of a fed-batch reactor in which the optimum operation is to maximize the feed flow rate, as limited by jacket heat transfer. The volume in the reactor increases with time, as does the heat-transfer area. Therefore, the feed rate also increases with time, which acts as a load disturbance to the reactor temperature controller. A conventional proportional-integral (PI) controller exhibits an offset from the desired setpoint (“droop”) because of the constantly changing load. A lag-compensated PI controller is shown to provide effective temperature control. In addition, because the process gain between the temperature and feed flow rate decreases during the batch as a result of the increase in volume, increasing the controller gain directly with volume (gain scheduling) improves the performance. Significant decreases in batch times are achieved, resulting in higher productivity. In addition, when there are competing undesirable reactions, higher yields are achieved because the reactor temperature is held closer to the optimum. 1. Introduction The interest in the control of batch reactors has increased in recent years because of the expansion of small-volume specialty chemicals. In the biotechnology area, batch reactors are used on both small- and largescale fermenters because of the inherent superiority of batch fermentation over continuous fermentation in most systems. Many of these batch reactors are “semibatch” or “fedbatch” reactors in which an initial amount of material is placed in the reactor, the liquid is heated to the desired temperature, and then additional feed of fresh reactant is gradually added to the vessel. The result is a time-varying process with variable volume. If heating and/or cooling is achieved by heat transfer from the vessel liquid into a heating/cooling medium in a surrounding jacket, the time-varying volume means that the heat-transfer area is also changing with time. The optimum operation of many fed-batch reactors is an operating strategy that minimized the batch time. This corresponds to feeding the fresh feed into the reactor as quickly as possible. The feed rate is often limited by heat transfer. If the reaction is exothermic, heat must be removed. The rate of heat transfer depends on three factors: 1. The temperature difference between the reaction liquid and the jacket coolant. The latter depends on the coolant flow rate, the inlet coolant temperature, and the heat-transfer rate. 2. The overall heat-transfer coefficient U, which depends on agitator mixing in the vessel and the flow rate of coolant in the jacket. 3. The heat-transfer area. If jacket cooling is used, the effective heat-transfer area in a fed-batch reactor varies during the course of the batch directly with the volume of liquid in the vessel. Of course, it also depends on the diameter of the vessel and the aspect ratio (length/diameter). It should be noted that if the fed†

Tel.: (610) 758-4256. E-mail: [email protected].

batch reactor uses an external heat exchanger with process liquid pumped and recycled back into the reactor, the heat-transfer area will not depend on the volume of liquid in the reactor. This is one of the advantages of an external pumparound system. The literature contains a number of papers that discuss the control of batch reactors, with most in the biotechnology area related to batch fermentation control. Some early papers1,2 studied various types of controllers, optimum batch temperature trajectories, and several heat-removal schemes. The state-of-the-art schemes up to 1986 were reviewed by Juba and Hamer.3 Many papers4-10 dealing with various aspects of fedbatch reactors have appeared in the intervening years, with many of them studying complex adaptive, nonlinear, or model-predictive control. Some recent papers11,12 have discussed the dynamic simulation and control of fed-batch systems. In particular, Wassick et al.11 recently presented a study of an industrial fed-batch polymerization reactor in which a complex nonlinear model predictive controller (MPC) is used to adjust cooling water and feed flows. The objective is to minimize the batch time. Experimental plant data are presented. No comparisons between MPC and a simple proportional-integral (PI) temperature controller are given. The most practical discussion of fed-batch reactor control is given by Shinskey.13 Comparisons are given of the performance of various control modes for ramp setpoint startups. The use of valve position control for synchronizing the manipulation of feed and coolant flows is discussed. The purpose of this paper is to illustrate that effective control of fed-batch reactors is achievable with the use of a simple PI controller with lag compensation to reduce offset in the face of feed flow rate changes during the batch cycle. In addition, gain scheduling is used to account for the change in the process gain during the batch. The effect of the feed flow rate on temperature (the process steady-state gain) varies inversely with the

10.1021/ie0308792 CCC: $27.50 © 2004 American Chemical Society Published on Web 06/22/2004

4244 Ind. Eng. Chem. Res., Vol. 43, No. 15, 2004 Table 1. Parameter and Design Values for the Base Case activation energy (Btu/lb‚mol) ) 30 000 heat of reaction (Btu/lb‚mol of A reacted) ) -20 000 density of the process liquid ) 50 lb/ft3 molecular weight of A and B ) 50 lb/lb‚mol heat capacity of the process liquid ) 0.75 Btu/lb‚°F density of the coolant liquid ) 62.3 lb/ft3 heat capacity of the coolant liquid ) 1 Btu/lb‚°F overall heat-transfer coefficient ) 100 Btu/h‚ft2‚°F jacket thickness ) 4 in.

Figure 1. Fed-batch reactor; circulating jacket cooling water.

volume of liquid in the vessel. The performance of the control structure to parametric variability is explored by changing a number of operating parameters over wide ranges (heat-transfer coefficient, specific reaction rate, heat of reaction, and reactor size). Significant improvements in the yield can also be achieved by the use of lag compensation and gain scheduling in systems with competing reactions, where maintaining the optimum temperature profile suppresses the undesirable reaction. 2. Process Studied The primary process considered features the irreversible, exothermic liquid-phase reaction A f B occurring in a fed-batch reactor. A later section of this paper considers a more complex chemical system with competing reactions A f B and A f C, in which the desired product is B. An aspect ratio (L/D) of 2 is used for the reactor vessel. Heat transfer occurs through the circumferential wall area to a jacket surrounding the vertical walls of the vessel. A circulating jacket water system is assumed, so cooling water in the jacket is perfectly mixed at temperature TJ. 2.1. Batch Operation. An initial “heel” that contains no reactant (z ) 0 at t ) 0, where z is the mole fraction of the reactant in the reactor) is placed in the reactor. The temperature of the reaction liquid is initially 90 °F. It is assumed that the vessel is 10% full at the beginning of the batch cycle. As illustrated in Figure 1, two sources of water are used during the batch. One is hot water at 140 °F, which is used to initially heat up the heel to temperatures where the reaction begins when reactant is fed. The other is cold water at 90 °F, which is used to cool the reactor to remove the exothermic heat of reaction once the reaction lights off. The desired operating temperature of the reactor TR is 140 °F. Split-ranged valves on the hot and cold water are used to set their flows FCW and FHW (ft3/h). The maximum flow rate of each is specified to give a residence time in the jacket of 5 min. The volume of the jacket is the circumferential area of the vessel times the width of the jacket, which is assumed to be 4 in. As the reactor temperature approaches the desired value, feed is introduced into the vessel. The feed is pure reactant A (zo ) 1) with a temperature of 90 °F and a flow rate of F(t) (lb‚mol/h). As the concentration of the reactant in the liquid builds up, the reaction begins to produce heat. In the base case, a heat of reaction of -20 000 Btu/lb‚mol is assumed. Table 1 gives details

of the kinetic and physical parameters used. The specific reaction rate is 5 h-1 at 140 °F. Feed is fed to the system until the volume reaches 95% of the total vessel volume. Then the feed is stopped, and the reactor sits for an additional period of 1 h to consume any unreacted A. The feed valve is sized so that the reactor could be filled in 5 h if the feed flow rate is at its maximum. The objective is to minimize the batch time so as to maximize the production rate. This is achieved by running with maximum cold water flow and feeding in fresh feed as fast as possible while holding the reactor temperature at its desired value. Thus, the feed flow rate is limited by heat transfer. Because the volume is increasing with time, the heattransfer area is also increasing with time. Thus, the feed flow rate to the reactor is initially small but becomes larger and larger during the batch. Figure 2 gives typical time trajectories of the variables during a batch cycle. Notice that the feed flow rate is almost a ramp function, which results in the volume increasing more and more rapidly with time. After about 14 h, the reactor is 95% full, and the feed is cut off. In the base case, a 5-ft-diameter reactor is assumed, giving a total volume of 1470 gal. Other reactor sizes are explored later in this paper. 2.2. Control Structure. As shown in Figure 1, two temperature controllers are used. The first has a setpoint of 120 °F and drives the two split-ranged water valves. When the reactor temperature is low, hot water is added to the circulating jacket water loop. As the reactor temperature approaches the setpoint, the hot water valve closes and the cold water valve opens. A proportional temperature control with a gain KC1 ) 2.5 is used in this controller with a temperature transmitter span of 50 °F. The jacket temperature controller is only used to initially add hot water to heat the reactor and to switch to maximum cold water flow during the rest of the batch. The second temperature controller is a PI controller that manipulates the feed flow rate. The setpoint of this controller is initially set at 90 °F and is ramped up to 140 °F over a period of time. This ramp time is one of the control parameters that can be adjusted. Results for various ramp times and controller tuning constants are presented in a later section of this paper. The valve on the feed has a maximum capacity such that the vessel could be filled in 2 h if the maximum flow were introduced. Different reactor sizes are explored, so the feed valve size is adjusted for each case. A 50 °F temperature transmitter span is used in the temperature loop. It should be noted that manipulating the feed to control the reactor temperature has the potential to introduce some interesting dynamics. If the feed temperature is lower than the reactor temperature, the initial effect of increasing the feed flow rate can be a decrease in the reactor temperature because of sensible heat effects. However, as the reactant concentration

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Figure 2. Conventional PI; ZN tuning.

builds up in the reactor, the reaction rate increases, and eventually the reactor temperature increases. Thus, there can be an “inverse response” in this feed-totemperature loop. 2.3. Controller Types. The performances of several types of controllers are compared. Simple linear PI and proportional-integral-derivative (PID) controllers are studied first. Then “lag-compensated” linear controllers are explored. Finally the use of nonlinear gain-scheduled controllers is examined. As we will demonstrate, the performance of a PID controller suffers from “droop” problems; i.e., the reactor temperature is not driven to the setpoint value. This “steady-state” offset occurs because the system is seeing a persistent load change, which is the increasing feed flow rate. The problem is analogous to the classic servomechanism case where a ramp change in the setpoint produces a constant offset between the setpoint and the controlled variable for a lag process. The problem can be described mathematically by showing that a ramp input (1/s) produces a constant error if a PI controller is used (which has only one s in the denominator). The magnitude of the error varies inversely with the controller gain, which is limited by closed-loop stability considerations. Similar problems occur in plantwide processes when a unit sees a disturbance entering over a long period of time. This case was studied by Belanger and Luyben,14 who recommended the use of a lag compensator. The basic idea is to have a high gain at low frequencies (which reduces the offset for a ramp input) while reducing the gain at frequencies near the ultimate frequency (or near the resonant frequency) so that closed-loop stability is maintained. These ideas are developed in a later section of this paper. Because this batch process is nonlinear, the use of gain scheduling is an obvious aspect to evaluate. Gain scheduling is a practical nonlinear control method that

has been successfully applied to many real industrial problems.13 3. Mathematical Model of a Fed-Batch Reactor The nonlinear model of a fed-batch reactor involves a dynamic total mass balance, an energy balance, and a component balance on the liquid in the reactor, plus an energy balance on the jacket water. Reactor Total Balance:

dVR/dt ) F

(1)

where VR has units of ft3 and F of ft3/h. A volumetric balance can be used because the density is assumed constant. Reactor Component Balance:

(F/M)

d(zVR) ) zoFF/M - kz(F/M)VR dt

(2)

where F is the mass density (50 lb/ft3), M is the molecular weight (50 lb/lb‚mol), and k is the specific reaction rate (h-1)

k ) Re-E/RT

(3)

where E is the activation energy (30 000 Btu/lb‚mol). The value of the preexponential factor R is calculated to give a specific reaction rate ko at 140 °F of 5 h-1 at the base-case conditions. Reactor Energy Balance:

Fcp

d(VRTR) ) ToFFcp - λkz(F/M)VR - UA(TR - TJ) dt (4)

where cp is the heat capacity of the process liquid (0.75 Btu/lb‚°F), λ is the heat of reaction (-20 000 Btu/lb‚mol),

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Figure 3. Determining Ku and Pu.

Figure 4. Setpoint ramps.

A is the jacket area (ft2), and U is the overall heattransfer coefficient (100 Btu/h‚°F‚ft2). Jacket Energy Balance:

never gets up to the desired setpoint. As discussed earlier, this is due to the load change in the feed flow rate. Hot water is used for only a short period of time. Then cold water is brought into the jacket and climbs to near its maximum value. Notice that the jacket temperature climbs to about 135 °F, then drops to about 90 °F, and gradually increases over the course of the batch to 100 °F, despite the fact that the heat-transfer rate Q is increasing. This occurs because the heat-transfer area is increasing. Notice that the concentration of the reactant in the reactor z is quite small (5 mol %) for the base-case specific reaction rate. In later cases with small specific reaction rates, higher concentrations are required. The total batch time with this conventional linear PI controller is 14.61 h for the base-case conditions. The lag-compensated PI controller discussed in the next section reduces the batch time to 13.40 h, which is a 8.3% reduction in time (or a 8.3% increase in capacity). As we show later, the additional use of gain scheduling reduces the batch time to 12.96 h. If temperature signals are not too noisy, the use of derivative action can improve control. Figure 5 gives results for a linear PID controller using Ziegler-Nichols settings. The higher gain gives less offset, and the temperature approached the setpoint more quickly. The result is a smaller batch time (13.78 h) for the PID controller, compared to the PI controller (14.61 h). 4.2. Lag-Compensated PI Controller. The offset can be significantly reduced by using lag compensation. The transfer function of a PI controller is

FJcJVJ

dTJ ) TCWFCWFJcJ + THWFHWFJcJ dt (FCW + FHW)FJcJTJ + UA(TR - TJ) (5)

where cJ and FJ are the heat capacity and density of the water (1 Btu/lb‚°F and 62.3 lb/ft3, respectively). Volumes and Areas:

total volume ) π(DR)2LR/4 total heat - transfer area ) πDRLR A(t) ) VR(t) (total area)/(total volume)

(6)

In the base case, the reactor diameter is 5 ft, the length is 10 ft, and the jacket heat-transfer area is 157 ft2. 4. Results 4.1. Conventional PI and PID Controllers. There are several issues in setting up the temperature controller that manipulates the feed. The first is the tuning question. The second is the ramp rate for the setpoint. Two 1-min first-order lags are used in the temperature controller loop so that realistic tuning is obtained. Figure 3 shows the approach taken. First, a ramp rate is specified (0.5 h to ramp from 90 to 140 °F), using proportional temperature controllers with various gains (KC2). A gain of about 0.6 produces very oscillatory behavior. We assume that this is the ultimate gain and the period of the oscillations is the ultimate period (Ku ) 0.6 and Pu ) 1.2 h). The resulting Ziegler-Nichols settings for a PI controller are KC ) 0.273 and τI ) 1 h. For a PID controller, the settings are KC ) 0.353, τI ) 0.6 h, and τD ) 0.15 h. Then different ramp rates are tested. Figure 4 shows that a slow ramp increases the time it takes to get close to the setpoint, while a fast ramp rate produces more overshoot and more oscillatory response. The 0.5 h ramp is selected. In these tests the controller is proportionalonly with a gain KC2 ) 0.2. Figure 2 gives results for a PI controller. ZieglerNichols settings and a 0.5 h ramp are used. The most interesting feature is the offset in temperature, which

GPI(s) )

τIs + 1 OP(s) ) KC τIs E(s)

(7)

where OP is the controller output, E is the error (SP PV), SP is the setpoint, PV is the process variable, KC is the controller gain, and τI is the controller integral time (hours). For a ramp load disturbance, this controller produces an offset. Lag compensation introduces a lag/lead in the controller

GLCPI(s) )

(

τIs + 1 τLCs + 1 OP(s) ) KC τIs RLCτLCs + 1 E(s)

)

(8)

where τLC is the lag time constant and RLC equals 10.

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Figure 5. Conventional PID; ZN tuning.

Figure 6. Open-loop Bode plots of controllers: PI and LCPI.

This is the reverse of a lead/lag element, used for derivative action, where RLC equals 0.1. Derivative action is used to provide phase angle advance so that the dynamic performance at high frequencies (near the resonant frequency) is improved. Lag compensation is used to provide higher gains at low frequencies where it helps to reduce the offset in the face of a ramp disturbance. Figure 6 gives Bode plots for the two controller transfer functions: eqs 7 and 8. The low-frequency magnitude of the lag-compensated controller is clearly shown to be much higher than that of the PI controller.

However, at higher frequencies near the resonant frequency (about 7 rad/h), the magnitudes and phase angles are almost the same. The best value of the lag time constant τLC should be used. Belanger and Luyben14 recommend the following:

τLC ) 3.64Pu

(9)

Using Pu ) 1.2 h, the suggested value of τLC is 4.4 h. Figure 7 shows the results for several values of τLC. A value of 4 h gives results that are not too sluggish and

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times are reduced. Table 2 summarizes the results for different controller types. Lag compensation helps the PI controller more than the PID controller. 5. Gain Scheduling

Figure 7. Effect of τLC on lag-compensated PI. Table 2. Batch Times for Lag-Compensated PI and PID

conventional lag compensated gain scheduled combined lag compensation and gain scheduling

PI batch time (h)

PID batch time (h)

14.61 13.40 13.52 12.96

13.78 13.07 13.31 12.88

produce little overshoot. The controller used in these runs is PI with Ziegler-Nichols settings. Figure 8 gives the time trajectories of the batch reactor when a lag-compensated PI controller is used. The offset is significantly reduced, leading to a shorter batch time. Figure 9 shows a direct comparison of conventional versus lag-compensated controllers. Results for both PI and PID controllers are shown. In both cases, batch

Figure 8. Lag-compensated PI; τLC ) 4.

The results presented above demonstrate a significant improvement in performance when lag compensation is used. However, a close look at the results for any of the controllers (see Figure 9) reveals a slight increase in the offset during the later stages of the batch cycle when using any of the controllers. This is due to the nonlinear character of the batch reactor. The effect of the feed flow rate on the temperature (the process gain) is larger during the early period of the batch when the volume is small. As the volume increases, the effect of the feed flow rate on the temperature becomes smaller. The reaction of 1 mol of A produces a fixed amount of heat, which heats the liquid in the reactor. The larger the volume, the less the temperature will change for a given quantity of reactant fed. This means that the process gain (TR/F) decreases with volume. This suggests that a “gain-scheduling” nonlinear controller should be used to improve the performance of the system. A simple way to change the controller gain is to make it a linear function of the reactor volume.

KC2 )

Kbase C2

+

VR - Vmin R Vmax - Vmin R R

base (Kmax C2 - KC2 )

(10)

is the Ziegler-Nichols value, the maxiwhere Kbase C2 mum reactor volume is 95% of the total, and the minimum is 10% of the total. For both the PI and PID controllers, the maximum gain is set equal to 5 times the Ziegler-Nichols values. This means the controller gain increases by a factor of 5 over the course of the batch.

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Figure 9. Conventional and lag compensation.

Figure 10. Conventional and lag compensation with gain scheduling.

Figure 10 demonstrates the effectiveness of gain scheduling for several cases. The top graph gives the results for a simple PI controller and a lag-compensated PI controller, each with gain scheduling. The bottom graph gives the corresponding results for PID controllers. Offsets are reduced, and batch times are shorter. The PI batch time for the combined control is 12.96 h, compared to a batch time for a simple PI controller of 14.61 h. The PID batch time for the combined control is 12.88 h, compared to a batch time for a PID controller of 13.78 h. It is worth noting that that there is very little improvement in using PID control over PI control when

both lag compensation and gain scheduling are used. This means that there will be little sacrifice in the performance even though temperature signal noise prevents the use of derivative action. 6. Parametric Variations The results presented above demonstrate a significant improvement in the performance when lag compensation and gain scheduling are used. To see if these controllers can handle changes in process parameters, a number of other cases are explored. These results are for a lag-compensated PI controller, using ZieglerNichols settings with gain scheduling.

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Figure 11. Lag-compensated PI with gain scheduling; ko ) 4/5/10.

Figure 12. Lag-compensated PI with gain scheduling; λ ) -15 000/-30 000.

Figure 13. Lag-compensated PI with gain scheduling; U ) 80/ 120.

6.1. Specific Reaction Rate. The base-case specific reaction rate is 5 h-1 at 140 °F. Values of 4 and 6 h-1 were studied. The preexponential factor was changed to give the desired value of k at 140 °F. Figure 11 compares the three cases. As the specifc reaction rate is reduced, the concentration of the reactant in the vessel must increase to provide the required reaction rate. The lower graph in Figure 11 illustrates this point. The ko ) 10 case has a low concentration of A (2-3 mol %), while the ko ) 4 case has a higher concentration of A (5-9 mol %). The tuning of the temperature controller is not changed from those used in the base case. The temperature controller becomes more oscillatory as the specific reaction rate decreases. This is due to the higher reactant concentration, which increases the process gain. Reducing the specific reaction rate has a strong effect, and the temperature controller would require tuning to achieve lower closed-loop damping coefficients.

6.2. Heat of Reaction. The base-case heat of reaction is -20 000 Btu/lb‚mol. Larger and smaller values (-15 000 and -30 000 Btu/lb‚mol) were studied. Figure 12 gives the results without any retuning of the temperature controller. As expected, the higher the heat of reaction, the longer the batch time because the feed must be fed at a slower rate. Stable temperature control is obtained, but the higher heat of reaction case is quite oscillatory. Controller retuning should be done for this very large change (50% increase) in the heat of reaction. As the heat of reaction gets smaller, the effect of the sensible heat of the cold feed becomes more significant. The batch time is much shorter. 6.3. Heat-Transfer Coefficient. The base-case overall heat-transfer coefficient is 100 Btu/h‚°F‚ft2. Values of 80 and 120 Btu/h‚°F‚ft2 were studied, and the results are shown in Figure 13. As expected, the larger the value of U, the shorter the batch time.

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control resulted in shorter batch times. Improved control can have a much more pronounced effect when the product yield is impacted by holding the reactor temperature closer to the optimum. To illustrate this point, a second process is considered in which there are two reactions:

AfB AfC Product B is the desired product. Component C is an undesirable side product. The specific reaction rate of the second reaction is

k2 ) R2e-E2/RT

Figure 14. Lag-compensated PI with gain scheduling; DR ) 4/6.

The tuning of the temperature controller is not changed from what was used in the base case. Temperature control is still stable, but the smaller heat-transfer coefficient produces more overshoot and more oscillation. These 20% changes in U should be accompanied by some controller retuning. 6.4. Reactor Size. The base-case reactor diameter is 5 ft, giving a vessel with a total volume of 1470 gal. Vessels with diameters of 4 ft (752 gal) and 6 ft (2538 gal) were also studied, which halve and double the reactor holdup. These new dimensions do not, however, halve and double the heat-transfer area. Therefore, the batch times change with the reactor size. The 4 ft reactor batch time is 10.86 h. The 6 ft reactor batch time is 15.12 h. Figure 14 gives the results. The tuning of the temperature controller is not changed from that used in the base case, despite the very large change in volume. Temperature control is still stable, but the larger reactor is more oscillatory. This is because of the smaller areato-volume ratio in the larger reactor. 7. Process with Competing Reactions In the process considered up to this point, there has been a single reaction A f B, and the improvement in

Figure 15. Competing reactions.

(11)

where the activation energy (E2 ) 10 000 Btu/lb‚mol) is lower than that of the desired reaction, so the undesirable reaction is less sensitive to temperature. Thus, temperature control can affect the amount of C produced, i.e., can affect the yield of B. If the temperature is too low (below the setpoint), both reaction rates are small. Raising the temperature increases the specific reaction rate of the desirable more than the undesirable reaction because of the large activation energy of the first reaction. The equations describing the system must be modified to handle the ternary system. There are now two component balances. Reactor Component Balances.

d(zAVR)/dt ) zoF - (k1 + k2)zAVR

(12)

d(zCVR)/dt ) +k2zAVR

(13)

where zA is the mole fraction of A and where zC is the mole fraction of C in the liquid in the reactor. The feed is pure A. The value of the preexponential factor R2 is calculated to give a specific reaction rate k2o at 140 °F of 0.5 h-1, which is 1/10 of the specific reaction rate k of the desired reaction. Reactor Energy Balance. The reactor energy must be modified to account for the two reactions, which we assume to have the same heat of reaction λ.

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Fcp

d(VRTR) ) ToFFcp - λ(k1 + k2)zA(F/M)VR dt UA(TR - TJ) (14)

This process was simulated using a simple PI controller and using a PI controller with lag compensation and gain scheduling. Results are given in Figure 15. The concentration of the undesired component C builds up to about 10% when PI control is used. This represents a yield of B of 88.8%. When a lag-compensated PI controller with gain scheduling is used, the concentration of the undesired component C builds up to only 8.5%, which represents a yield of B of 90.5%. This difference in yield can impact the profitability of the fed-batch process. In addition, the batch time with the PI controller is 14.67 h, while with the lag-compensated PI controller with gain scheduling, the batch time is only 13.0 h. This is an 11% reduction. 8. Conclusion The use of lag-compensated and gain-scheduled PI controllers for temperature control of fed-batch reactors has been investigated. Results demonstrate significant improvements in the performance. Batch times are shorter because of the reduction in the temperature offset from the desired setpoint, and yield improvements are achieved in some chemical systems. Robustness studies illustrate that the controller can tolerate fairly large changes in process parameters. Nomenclature A ) heat-transfer area of the jacket (ft2) cJ ) heat capacity of the coolant (Btu/lb‚°F) cp ) heat capacity of the process (Btu/lb‚°F) D ) process deadtime (h) DR ) reactor vessel diameter (ft) E ) activation energy (Btu/lb‚mol) F ) fresh feed flow rate (lb‚mol/h) FCW ) cold water flow rate to the jacket (ft3/h) FHW ) hot water flow rate to the jacket (ft3/h) GM ) process open-loop transfer function GPI ) PI controller transfer function GLCPI ) lag-compensated PI controller transfer function k ) specific reaction rate (h-1) KC ) controller gain ko ) specific reaction rate at 140 °F (h-1) KP ) process open-loop steady-state gain Ku ) ultimate controller gain LC ) closed-loop log modulus (dB) LO ) open-loop log modulus (dB) LR ) length of the reactor (ft) LCPI ) lag-compensated PI controller M ) molecular weight (lb/lb‚mol) PI ) proportional-integral controller Pu ) ultimate period (h) Q ) heat-transfer rate to the jacket (Btu/h) s ) Laplace transform variable TCW ) cold water temperature (°F) THW ) hot water temperature (°F) TJ ) jacket temperature (°F)

To ) temperature of the feed (°F) TR ) reactor temperature (°F) Vtotal ) total volume of the reactor vessel (ft3) U ) overall heat-transfer coefficient (Btu/h‚ft2‚°F) VJ ) volume of the jacket (ft3) VR ) volumetric holdup of the reaction liquid in the reactor (ft3) z ) reactant concentration in the reactor (mole fraction A) zo ) reactant concentration in the fresh feed (mole fraction A) R ) kinetic preexponential factor (h-1) RLC ) parameter in the lag compensator λ ) heat of reaction (Btu/lb‚mol) F ) density of the process liquid (lb/ft3) FJ ) density of the coolant (lb/ft3) τLC ) time constant of the lag compensator (h) τM ) measurement lag time (h) τI ) controller integral time (h) ωR ) resonant frequency (rad/h) ωu ) ultimate frequency (rad/h)

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Received for review December 30, 2003 Revised manuscript received May 3, 2004 Accepted May 14, 2004 IE0308792