Quantitative Comparison of Temperature Control of Reactors with

Ind. Eng. Chem. Res. 2004, 43, 2691-2703

2691

Quantitative Comparison of Temperature Control of Reactors with Jacket Cooling or Internal Cooling Coils William L. Luyben† Process Modeling and Control Center, Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015

This paper presents a quantitative comparison of the effectiveness of two alternative heat-removal schemes in the control of continuous stirred tank reactors (CSTRs). The chemistry features an exothermic irreversible reaction A f B, and different design levels of conversion are explored. Two mechanisms for removing the heat of reaction are studied: (1) a cooling jacket surrounding the vessel or (2) an internal cooling coil. Controllability is shown to become more difficult as the design conversion decreases. Results demonstrate that internal coil cooling provides better dynamic controllability than jacket cooling in terms of being able to handle larger disturbances. A generic “reactor controllability index” (RCI) is proposed that can be used to assess the dynamic robustness of any CSTR. The RCI is the dimensionless ratio of the temperature difference between the reactor and jacket (or the log-mean temperature difference for coil cooling) and the maximum available temperature difference (reactor temperature minus the temperature of the available coolant). The smaller the RCI, the more dynamically robust the reactor. 1. Introduction The control of chemical reactors is an extremely important issue in many chemical plants. If the reactions are endothermic, the control problem is usually not severe because the reaction will simple slow down if insufficient heat is added. No reaction runaway can occur. Likewise, if the reactions are exothermic and reversible, the control is usually fairly easy because the equilibrium constant decreases as the temperature increases. This built-in self-regulation often prevents reaction runaway. However, if the reactions are exothermic and irreversible, reaction runaways can occur. As we demonstrate in this paper, this tendency increases as the conversion level in the reactor decreases because there is more “fuel” available. There is a rich literature in the area of continuous stirred tank reactor (CSTR) control that goes back over half a century. Pioneering work by Aris and Amundson,1 Harriott,2 and Foss3 are some examples. More recent studies include the work of Bequette,4 Shinnar et al.,5 and Luyben.6-8 Most of these studies assume fairly simplistic heatremoval schemes in which either the heat-transfer rate Q is manipulated directly or the coolant temperature TJ in the jacket is manipulated. Of course, in reality, the manipulated variable is the coolant flow rate. Many papers explore systems with jacket cooling. For example, Luyben6 presents examples of tradeoffs between design and control in single and multiple CSTRs using jacket cooling. Another paper7 studies the tuning of temperature controllers in jacket-cooled CSTRs. There are also a few papers that have explored reactors that are evaporatively cooled (autorefrigeration). The most recent8 demonstrates that the heattransfer area of the condenser must be significantly increased above normal steady-state requirements in order to handle dynamic disturbances. †

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

Figure 1. CSTR with cooling jacket or internal coil.

There is one type of heat-removal system that appears to have been neglected in the process control literature: internal cooling coils. Although coils are commonly used in industry, no papers have been found that explore the dynamics of this type of system. The purpose of this paper is to provide a quantitative comparison of the dynamic effectiveness of jacket and coil cooling. 2. Process Studied The basic process considered features the irreversible, exothermic liquid-phase reaction A f B occurring in a CSTR. The vertical cylindrical vessel is assumed to have an aspect ratio (L/D) of 2. In one design, heat transfer occurs through the circumferential wall area to a cooling jacket surrounding the vertical walls of the vessel. A circulating jacket water system is assumed, so the cooling water in the jacket is perfectly mixed at temperature TJ. The cooling water supply temperature is TCin. In the second design, heat transfer occurs through the walls of a cooling coil that loops around inside the reactor. See Figure 1. Plug flow of the coolant is assumed in the coil, so a log-mean temperature difference is used to determine the heat-transfer rate. The reactor temperature is TR, and the temperature difference at the coil inlet is TR - TCin. The temperature difference at the coil outlet is TR - TCout. The coolant flow rate to the jacket is FJ, and that to the coil is Fcoil.

10.1021/ie030721h CCC: $27.50 © 2004 American Chemical Society Published on Web 04/22/2004

2692 Ind. Eng. Chem. Res., Vol. 43, No. 11, 2004 Table 1. Parameter and Design Values activation energy (Btu/lb‚mol) ) 30 000 heat of reaction (Btu/lb‚mol of A reacted) ) -30 000 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 ) 150 Btu/h‚ft2‚°F jacket thickness ) 4 in. coil pipe diameter ) 3 in. coil loop diameter ) 0.8 vessel diameter coil loop spacing ) coil pipe diameter

Several levels of conversion are studied from 98% to 70%, and the impact of this important design parameter on dynamic controllability is illustrated. Both linear and nonlinear models of the jacket-cooled system are used. The linear model is used to calculate open-loop eigenvalues and to find temperature controller tuning constants. The rigorous nonlinear model is used to test the robustness of the process in the face of disturbances in throughput, temperature controller setpoint, and heat-transfer coefficient. A design procedure for sizing the cooling coil is developed. Because the coil occupies the internal volume, the reactor size must be increased somewhat for a specified conversion. The physical layout of the coil must also be determined: diameter of the loops, spacing between loops, and coil pipe diameter. A nonlinear model of the cooling coil system is developed using a 30-lump model for the cooling coil and is used for robustness tests. Table 1 summarizes the physical properties and design parameters. First-order kinetics are assumed. The specific reaction rate k is 0.5 h-1 at 140 °F. The feed is 100 lb‚mol/h of pure component A at 70 °F. An overall heat-transfer coefficient U of 150 Btu/h‚ft2‚°F is assumed. The activation energy E is 30 000 Btu/lb‚mol, and the heat of reaction λ is -30 000 Btu/lb‚mol of A reacted. The flow rate of the cooling water is given in units of ft3/h. The size of the reactor vessel depends on the design value for conversion. The smaller the conversion, the smaller the vessel. Also, this means there is less heattransfer area available. The reactor operates at 140 °F, and the cooling water supply temperature is 70 °F. Thus, the maximum temperature differential for heat transfer is ∆Tmax ) 140-70 ) 70 °F. Even with an infinite flow rate of cooling water, ∆T can never exceed this limit. If the steady-state design uses a large fraction of this ∆Tmax, the dynamic robustness of the system will be limited and dynamic controllability will be poor, as the results given in this paper illustrate. 3. Jacket-Cooled CSTR Linear and nonlinear models have been taken from the literature6 and are summarized in the appendix. Reactors for different conversion levels are explored. After the steady-state designs are performed, their linear dynamics are studied and nonlinear simulations are used to assess their performance. 3.1. Steady-State Design. The steady-state design procedure for a specified conversion χ, with feed flow rate F, feed temperature T0, and feed composition z0 known, involves the following sequential calculations:

1. Composition of A in the product stream (and in the reactor): z ) z0(1 - χ). 2. Molar holdup in the reactor: Vreaction ) Fχ/k(1 - χ). 3. Volume of the reactor: VR ) Vreaction(M /F). 4. Diameter of the vessel: D ) (2VR/π)1/3. 5. Jacket heat-transfer area: AJ ) (πD)(2D) ) 2πD2. 6. Heat-transfer rate: QJ ) F(z0 - z)(-λ) - FcpM(TR - T0). 7. Jacket temperature: TJ ) TR - QJ/UAJ. 8. Cooling water flow rate: FJ ) QJ/cCFC(TJ - TCin). Table 2 gives results for conversions ranging from 98% to 70%. As conversion decreases, the reactor size and jacket area decrease, the jacket temperature decreases, and the cooling water flow rate increases. The last row of Table 2 gives values of the parameter that we define as the “reactor controllability index” (RCI).

RCI )

TR - TJ TR - TCin

As the dynamic results demonstrate later in the paper, RCI is proposed as a dimensionless parameter that indicates the inherent dynamic controllability of a CSTR. It is a ratio of the temperature driving force used in the design of the reactor to the maximum available temperature driving force. A small ratio means that there is plenty of additional driving force available to handle disturbances and transients. This index reflects changes in the heat of reaction, conversion, heat-transfer area, and coolant supply temperature. It should be particularly useful in addressing controllability issues in the scale-up of reactors. The RCIs given in Table 2 increase as conversion decreases, indicating that control of the low-conversion reactors is worse than that of the high-conversion reactors. 3.2. Dynamics. A. Open-Loop Dynamics. For each of the designs developed above, the open- and closedloop dynamic behavior is explored. The linear dynamic model yields an open-loop transfer function relating the reactor temperature to the coolant flow rate. The units of this transfer function are °F/(ft3/h).

G(s) )

TR(s) FJ(s)

)

a23b3(s - a11) 3

(s + b2s2 + b1 + b0)(τMs + 1)2

The a and b coefficients are defined in the appendix, and τM is a 1-min temperature measurement lag (two lags in series). Figure 2 gives Nyquist plots of this transfer function for cases with different conversions. Because the steadystate gain of this transfer function in negative, a negative controller gain will be required. To make the Nyquist plots look like normal plots for systems with positive steady-state gains, we plot the negative of G(s). Note that the 98 and 96% conversion cases start on the positive real axis and move clockwise, ending with phase angles of -270°. These two cases are open-loop stable, as the open-loop eigenvalues in Table 2 show. For lower conversions, the curves move counterclockwise. These systems are open-loop unstable. Their openloop eigenvalues have negative real parts, as shown in Table 2, which get larger as conversion decreases. This is a quantitative indication that closed-loop control will be more difficult as conversion decreases.

Ind. Eng. Chem. Res., Vol. 43, No. 11, 2004 2693 Table 2. Effect of Conversion on the Reactor with Jacket Cooling conversion (%) 98 VR (ft3) D (ft) z (mole fraction of A) QJ (106 Btu/h) AJ (ft2) TJ (°F) FJ (ft3/h) open-loop eigenvalues K Ca τI (h) (dB) Lmax C RCIb

96

94

92

90

80

70

9800 18.4 0.02

4800 14.5 0.04

3133 12.6 0.06

2300 11.4 0.08

1800 10.4 0.10

800 7.98 0.20

467 6.67 0.30

2.68 2128 131.6 698 -0.115 (0.162i, -8.95 24.7 0.888 4.64 0.120

2.62 1322 126.8 740 -0.0606 (0.304i, -9.75 13.8 0.875 4.85 0.189

2.56 995 122.9 777 +0.0345 (0.369i, -10.5 10.4 0.848 5.30 0.244

2.50 810 119.4 811 +0.131 (0.397i, -11.1 8.59 0.835 5.61 0.294

2.44 688 116.4 844 +0.228 (0.394i, -11.8 7.51 0.823 5.93 0.337

2.14 400 104.4 997 +0.228 +1.23, -15.5 5.73 0.762 7.55 0.509

1.84 280 96.2 1126 +0.059 +2.41, -20.1 5.68 0.705 9.01 0.626

a Dimensionless; temperature transmitter span ) 45 °F; maximum coolant flow rate ) twice the steady-state flow. b RCI ) ∆T design/ ∆Tmax ) (TR - TJ)/(TR - TCin).

Figure 2. Jacket cooling; conversion ) 98//96/94/92/90%.

Figure 3. Jacket cooling; conversion ) 98/90/70%.

B. Closed-Loop Dynamics. The ultimate gain and ultimate frequency are calculated for each case, and the Tyreus-Luyben temperature controller settings are used. Previous work7 has shown that this tuning method works well for these open-loop unstable reactors. Tuning constants are given in Table 2. Using these settings, the closed-loop servo transfer function log modulus curves are shown in Figure 3 for three cases. The maximum in the curve Lmax is an indication of the closed-loop C damping coefficient (damping decreases as the peak

Figure 4. Linear simulation; jacket cooling; SF ) 141; conversion ) 98/90/70%.

increases). These curves and the results given in Table 2 show that the closed-loop damping coefficient decreases as conversion decreases. Because dynamic robustness is inversely related to the damping coefficient, the low-conversion reactors are less robust. Figure 4 gives the response of the linear model for a step change in the setpoint of the temperature controller for the 70, 90, and 98% conversion cases. The reactors are all closed-loop stable, but the systems become more underdamped as conversion decreases. The simulation of the rigorous nonlinear model gives similar results. Figure 5 gives the transient responses of the reactor temperature, jacket temperature, coolant flow rate, and reactor composition for three conversion cases (98, 90, and 70%). The disturbance is a step change of 1 °F in the setpoint of the temperature controller, which occurs at a time equal to 0.2 h. Results are similar to the linear responses, but valve saturation begins to occur. Figure 6 shows how the reactors respond to a 25% increase in feed flow. All of the designs can handle this disturbance, but temperature deviations increase as conversion decreases. If a 50% increase in feed flow is made, shown in Figure 7, a reactor runaway occurs in the 70% conversion reactor. These results verify the predictions that the lower the conversion, the poorer the robustness. Notice that the coolant valve goes wide open at about 2.7 h in the 70% conversion case, and thereafter

2694 Ind. Eng. Chem. Res., Vol. 43, No. 11, 2004

Figure 5. Nonlinear simulation; jacket cooling; SF ) 141; conversion ) 98/90/70%.

Figure 6. Jacket cooling; +25% feed; conversion ) 98/90/70%.

the system is open-loop. Because it is open-loop unstable, the reactor runs away to a very high temperature (the simulation is stopped if the temperature exceeds 160 °F). In Figures 8-10, we look at only the 70% conversion reactor. In all of the runs so far, the maximum coolant flow rate was set at twice the design value. In Figure

8, the coolant valve is sized to provide 4 times the steady-state design flow. A reactor runaway still occurs for the 50% increase in feed flow, but it takes longer to occur. A 60% increase runs away very quickly. The previous responses are for load disturbances. The responses to setpoint changes of different magnitudes are shown in Figures 9 and 10. The 70% conversion

Ind. Eng. Chem. Res., Vol. 43, No. 11, 2004 2695

Figure 7. Jacket cooling; +50% feed; conversion ) 98/90/70%.

Figure 8. Jacket cooling; 70% conversion; coolant valve range doubled; feed changes.

reactor with a coolant valve capacity of twice the design can handle a change in setpoint of up to 143 °F but runs

away for any larger change. If a bigger coolant valve is used (with the controller gain correspondingly reduced),


Figure 9. Jacket cooling; 70% conversion; SP ) 141/142/143/144.

Figure 10. Jacket cooling; 70% conversion; coolant valve range doubled; SP changes.

the reactor can survive a setpoint change of up to 147 °F, as shown in Figure 10. Notice that the valve saturates wide open at about 0.7 h and reactor temperatures exceed 157 °F, a 100% overshoot. The dynamic requirement for coolant flow is 4 times the normal steady state.

In this section, we have looked at CSTRs with jacket cooling. In the next section, instead of using jacket cooling, an internal coil is installed inside the reactor vessel. The dynamic performance of this type of cooling system is quantitatively compared with the dynamic


performance of the jacket-cooled system considered in this section. 4. Internal Coil Cooling There are several issues to be resolved in the design of a cooling coil regarding the physical layout of the coil in the vessel. The diameter of the pipe used to make the coil must be specified. In addition, the diameter of the spiral loop and the spacing between the loops must be specified. Should one coil be used or should several coils be installed in parallel? 4.1. Coil Layout. The coil takes up volume in the vessel, so the size of the reactor will have to be larger to provide the same holdup for the reaction. If the coil volume is too high a fraction of the total volume, the mixing in the reactor may be adversely affected. If a small-diameter pipe is used, the heat-transfer area per unit of coil volume will be large. However, the pressure drop through the coil will be large unless multiple parallel passes are used. Parallel passes can present flow distribution problems. Small-diameter tubes also present mechanical-strength problems. Obviously, the diameter of the loop must be smaller than the diameter of the vessel, and if the diameter of the loop is too small, it will interfere with the agitator. The spacing between the flights of the loop cannot be too small or mixing and heat transfer will be degraded. After several informative discussions with Bert Diemer,7 the following coil specifications are used in this study: 1. The pipe diameter is 3 in. 2. The loop diameter is 80% of the diameter of the vessel. 3. The spacing between the loop flights is the diameter of the pipe (3 in.). 4. Only one coil is used. With these specifications, the reactor/coil system can be designed for a given conversion. 4.2. Steady-State Design. Because the 70% conversion case has the worst performance in a jacket-cooled system, we explore the design of a coil-cooled reactor for this case. A. Calculation of Vessel and Coil Sizes. The first part of the design procedure involves an iterative calculation of the size of the vessel because it contains both reaction mass and coil. The pipe diameter Dcoil is 0.25 ft. 1. Calculate the composition of A in the product stream (and in the reactor): z ) z0(1 - χ). 2. Calculate the required molar holdup in the reactor: Vreaction ) Fχ/k(1 - χ). 3. Calculate the required volume of the reaction mass in the reactor: VR ) Vreaction(M /F). 4. Make an initial guess of the diameter of the vessel: Dguess ) (2VR/π)1/3. 5. Calculate the length of the vessel: Lvessel ) 2Dguess. 6. Calculate the number of coil loops that will fit into a vessel with this length: Nloops ) Lvessel/2Dcoil. 7. Calculate the length of the coil: Lcoil ) 0.8DguessπNloops. 8. Calculate the volume of the coil: Vcoil ) (πDcoil2/4)Lcoil. 9. Calculate the total volume of the vessel, which is the required volume of the reaction mass plus the volume of the coil: Vtotal ) Vcoil + VR. 10. Calculate the diameter of a vessel with this volume: Dcalc ) (2Vtotal/π)1/3.

11. If Dcalc is sufficiently close to Dguess, continue. Otherwise, go back to step 4. Direct substitution works quite effectively in the iteration. 12. Calculate the heat-transfer area of the coil: Acoil ) πDcoilLcoil. 13. Calculate the heat-transfer rate: Qcoil ) F(z0 - z)(-λ) - FcpM(TR - T0). As an example of these calculations, consider the 70% conversion design. Without a coil, the vessel diameter is 6.67 ft and the jacket heat-transfer area is 288.6 ft2. The results of the iterative procedure described above yield a vessel with a diameter of 6.78 ft. The volume of the coil is 22.65 ft3, its length is 461 ft, the number of loops is 27, and the coil heat-transfer area is 362 ft2. This is 25% more heat-transfer area than that of the jacketed reactor, so we expect the dynamic performance to be superior. B. General Relationship between the Jacket and Coil Areas. The ratio of the coil heat-transfer area to the jacket heat-transfer area can be expressed in a very general form if the volume of the coil can be neglected so that the sizes of the vessels are the same in the two alternative reactors. If this assumption is not true, the ratio will be slightly higher because the coil-cooled vessel will be slightly bigger. In the 70% conversion case, the volume of the jacket-cooled vessel is 467 ft3 while the volume of the coil is only 22.6 ft3. We define the ratio of the coil loop diameter to the diameter of the vessel as

β ) Dloop/D If the aspect ratio of the vessel is L/D ) 2 and the spacing between the loops is equal to the diameter of the coil pipe, the number of loops is

Nloops ) 2D/(2Dcoil) Then the length of the coil is

Lcoil ) πDloopNloops ) π(βD)(D/Dcoil) ) πβD2/Dcoil So, the heat-transfer area of the coil is

Acoil ) πDcoilLcoil ) πDcoil(πβD2/Dcoil) ) π2βD2 This result is somewhat remarkable. It shows that the coil area is independent of the diameter of the coil pipe! This is counterintuitive. However, it should be remembered that we are assuming a single coil in the vessel. If we used a small-diameter pipe, we could fit more coils in the reactor than if we used a large-diameter pipe. So small-diameter pipes can produce more heat-transfer area. The heat-transfer area in a jacket-cooled reactor is

AJ ) πDL ) πD(2D) ) 2πD2 So, the ratio of the coil area to the jacket area is simply

Acoil π2βD2 πβ ) ) AJ 2 2πD2 This is another counterintuitive result. It shows that the ratio of the coil area to the jacket area does not depend on the diameter of the vessel!


It is interesting to note that the selection of the diameter of the loops (the β factor) has a direct effect on the area ratio. If the loop diameter is set equal to 2/π, the ratio becomes unity. C. Calculation of the Coolant Flow Rate and Exit Temperature. Once the physical size of the vessel and coil have been determined, a second iterative calculation is required to determine how much coolant is required to remove the known amount of heat. The temperature driving force for heat transfer between the reaction liquid at temperature TR and the coil is the log-mean temperature difference.

∆TLM )

(TR - TCin) - (TR - TCout) TR - TCin ln TR - TCout

(

)

The heat-transfer rate is given by

Qcoil ) UAcoil∆TLM We know Qcoil, U, Acoil, TR, and TCin. Combining the two equations above gives one equation in one unknown, the temperature of the coolant leaving the coil TCout. However, the log term precludes an analytical solution, so an iterative “interval halving” solution method is used. 1. A value of TCout is guessed. It must be less than TR. 2. The value of ∆TLM is calculated. 3. Then a value of the heat-transfer rate is calculated: Qcalc ) UAcoil∆TLM. 4. If this calculated Qcalc is sufficiently close to the known value of Qcoil, the iteration has converged. If not, continue. 5. If Qcalc is greater than Qcoil, increase the guessed value of TCout. 6. If Qcalc is less than Qcoil, decrease the guessed value of TCout. Note that the temperature approach on the “cold end” is known (TR - TCin ) 140 - 70 ) 70 °F). This calculation for the 70% conversion case yields a coolant exit temperature of 127.05 °F and a log-mean temperature difference of 33.81 K. Now the coolant flow rate can be calculated.

Fcoil ) Qcoil/[cCFC(TCout - TCin)] The coolant flow rate for the 70% conversion case is 517 ft3/h. This should be compared with the coolant flow rate required in the jacket-cooled configuration of 1126 ft3/ h, which is twice as much. The RCI for the coil-cooled reactor would be the ratio of the log-mean temperature difference to the maximum temperature difference.

RCI )

∆TLM TR - TCin

For the 70% conversion reactor with coil cooling, the RCI is 33.81/70 ) 0.483. For the jacket-cooled reactor, the RCI is 0.626. Thus, the RCI predicts that the dynamic robustness of the coil-cooled reactor should be better than that of the jacket-cooled reactor. The results given below confirm this prediction. 4.3. Dynamic Simulation. A rigorous nonlinear dynamic simulation of the 70% conversion case with coil cooling is used to assess controllability.

Table 3. Effect of Number of Coil Lumps number of lumps

coil outlet temp (°F)

coolant flow rate (ft3/h)

10 20 30 ∞

124.2 125.6 126.1 127.1

555 531 526 517

A. Dynamic Modeling of the Coil. The reaction liquid in the vessel is perfectly mixed, but there is plug flow of cooling water in the coil. In the steady-state design, a log-mean temperature difference is used. In the dynamic model, the coil is lumped into a number of perfectly mixed sections. The number of these lumps was varied so that the dynamic model at steady-state conditions gives values of the required coolant flow rate and the coil outlet temperature that match reasonably well with the rigorous steady-state values. Table 3 shows that, as the number of lumps is increased, the coolant flow rate decreases and the coil outlet temperature increases. The 30-lump model gives a coil outlet temperature within 1 °F of the rigorous value and a coolant flow rate within 2% of the rigorous value. B. Controller Tuning. Because a linear model of the coil system would be quite complex and high order with a 30-lump coil, the ultimate gain and ultimate frequency are obtain using the relay-feedback test. The dimensionless ultimate gain, using a 45 °F temperature transmitter span and a coolant valve than can provide twice the design flow rate, is 16.8, and the ultimate period is 0.183 h. The Tyreus-Luyben tuning constants for the temperature controller are KC ) 5.26 and τI ) 0.40 h. These settings worked well for the reactor. C. Closed-Loop Results. Disturbances in the feed flow rate and temperature controller setpoint are used to assess the dynamic robustness of the reactor. Figure 11 gives results for large increases in the feed flow rate. The system can handle an 80% increase in feed. However, if the feed flow rate is doubled, the coolant valve saturates wide open at about 2.3 h, and thereafter the reactor runs away. Note that the jacket-cooled reactor could not handle a 50% increase. This clearly demonstrates the superior dynamic robustness. Figure 12 shows that setpoint changes of up to 146 °F can be handled. The jacket-cooled reactor could only handle setpoint changes of up to 143 °F. This is only half the setpoint change that the coil-cooled reactor can handle. For the final test of the robustness of the coil-cooled reactor, we explore the effect of decreasing the overall heat-transfer coefficient U. Figure 13 shows what happens when the feed flow rate is increased 20% at time equal to 0.2 h and there is a simultaneous change in U. A change in U from 150 to 130 Btu/h‚ft2‚°F is handled well. A change to 110 Btu/h‚ft2‚°F can barely be handled. The response is oscillatory, and the coolant valve appears to be approaching saturation. A change in U to 100 Btu/h‚ft2‚°F results in a rapid reactor runaway. 5. Comparison of Steady-State Limitations The dynamic results reported above show that once the coolant valve goes wide open, a runaway occurs. Some important insights about this limitation can be gained by simply calculating the required steady-state cooling flow rate for different disturbances, with the


Figure 11. Coil cooling; 70% conversion; feed ) +40/60/80/100.

Figure 12. Coil cooling; 70% conversion; SP ) 142/144/146/147.

physical size of the vessel fixed. This “rating” type of calculation can define feasible steady-state operating regions. Of course, there is no guarantee that the reactor can be dynamically operated inside these regions.

The 70% conversion case is considered for both the jacket-cooled and the coil-cooled reactors. The reactors in both cases are designed for base-case conditions (F ) 100 lb‚mol/h and U ) 150 Btu/h‚ft2‚°F). This fixes the


Figure 13. Coil cooling; 70% conversion; U ) 150/130/110/100.

Figure 14. Effect of feed for a fixed vessel; jacket or coil cooling; 70% conversion.


Figure 15. Effect of U for a fixed vessel; jacket or coil cooling; 70% conversion.

vessel size and the heat-transfer area. Either F or U is changed, and the required heat-transfer rate is calculated. For the jacket-cooled reactor, the new jacket temperature as well as the new coolant flow rate can be directly calculated. For the coil-cooled reactor, an iterative calculation is required to find the coil outlet temperature that gives the new heat-transfer rate. Then the coolant flow rate can be determined. Figure 14 gives the results of these calculations for both heat-removal systems when the feed flow rate is increased. The top graph shows that the heat-transfer rate increases linearly with the feed flow rate, as expected. The middle graph shows that the jacket temperature in the jacket-cooled reactor and the coil outlet temperature in the coil-cooled reactor must decrease. As these temperatures get closer and closer to the temperature of the inlet coolant (70 °F), the required coolant flow rate increases exponentially, as is shown in the bottom graph. For a given maximum coolant flow rate, there is a maximum feed flow rate that can be handled. Figure 14 clearly shows that the feasible operating condition for the jacket-cooled reactor is much smaller than that of the coil-cooled reactor. Figure 15 gives steady-state feasibility results for changes in the overall heat-transfer coefficient. As U decreases, the jacket and coil outlet temperatures decrease to provide the required increase in the differential temperature driving force so that the same Q can be transferred. This means that the coolant flow rate must increase. Figure 15 shows that the jacketcooled reactor requires larger increases in the coolant

flow rate than does the coil-cooled reactor, indicating that its feasible region is smaller. As an example of the difference between steady-state requirements and dynamic requirements, it is interesting to note that Figure 15 predicts that the coil-cooled reactor should be able to operate with heat-transfer coefficients down to about 100 Btu/h‚ft2‚°F if the coolant valve can increase the coolant flow rate by 100%. However, the dynamic results given in Figure 13 show that the reactor can barely handle a U of 110 Btu/h‚ ft2‚°F. This is a good demonstration that steady-state feasibility is only a “necessary but not sufficient condition.” 6. Conclusion The steady-state design and the dynamic controllability of CSTRs with two alternative types of heat-removal systems have been quantitatively compared. The important effect of the design level of conversion has been demonstrated. The dynamic robustness of the coil-cooled reactor is demonstrated to be superior to that of the jacket-cooled reactor. A generic RCI is proposed that can be used to assess the dynamic robustness of any CSTR. The smaller the RCI, the more dynamically robust the reactor. Dynamic simulation results confirm the effectiveness of this index. Of course, it should be kept in mind that the jacketcooled system is more simple and mechanically easier to design than an internal coil. In addition, we have studied a jacket-cooled reactor using a circulating jacket


water system (perfectly mixed jacket). Some of the benefits of the coil-cooled system could be realized by designing a jacket-cooled system with once-through, plug-flow hydraulics. However, the heat-transfer area would be smaller than that available with an internal coil, so the dynamic performance would not be as good.

G(s) )

TR(s) FJ(s)

(

) ( ) (

)

dz F F z0 z - zk0e-E/R(TR+460) ) dt Vreaction Vreaction

(

)

dTR F F T T ) dt Vreaction 0 Vreaction R UAJ λz k0e-E/R(TR+460) (T - TJ) Mcp cpMVR R

()

()

UAJ FJ FJ dTJ TCin TJ + (T - TJ) ) dt VJ VJ cCFCVJ R Linearizing gives three linear ordinary differential equations

dz/dt ) a11z + a12TR + ... dTR/dt ) a21z + a22TR + a23TJ + ... dTJ/dt ) a31z + a32TR + a33TJ + b3FJ + ... where the constant coefficients are

a11 ) a12 ) -

F h -k h Vreaction

zjEk h R(TR + 460)2

a21 ) a22 ) -

λk h Mcp

UAJ zjEk hλ F h 2 Vreaction VreactionMcp McpR(TR + 460) a23 )

UAJ VreactionMcp

a31 ) 0 a32 ) b3 )

UAJ VJFCcC

TCin - TJ VJ

The overscored variables are the steady-state values around which the equations are linearized. Rearranging to find the open-loop transfer function between the reactor temperature and coolant flow rate and including two first-order temperature measurement lags give

a23b3(s - a11) (s3 + b2s2 + b1 + b0)(τMs + 1)2

where

b2 ) -a11 - a22 - a33

Appendix: Nonlinear Model The equations describing a jacketed, constant holdup, nonisothermal CSTR with first-order kinetics are

)

b1 ) a11a22 + a11a33 + a22a33 - a12a21 - a23a32 b0 ) a12a21a33 - a11a22a33 + a11a23a32 Nomenclature aij ) constant coefficient in the linear model Acoil ) heat-transfer area of the coil (ft2) AJ ) heat-transfer area of the jacket (ft2) bi ) constant coefficient in the linear model cC ) heat capacity of the coolant (Btu/lb‚°F) cp ) heat capacity of the process (Btu/lb‚°F) D ) vessel diameter (ft) Dcoil ) diameter of the coil pipe (ft) Dloop ) diameter of loops of the coil (ft) E ) activation energy (Btu/lb‚mol) F ) fresh feed flow rate (lb‚mol/h) Fcoil ) coolant flow rate through the coil (ft3/h) FJ ) coolant flow rate to the jacket (ft3/h) G ) process open-loop transfer function k ) specific reaction rate (h-1) L ) length of the reactor (ft) Lmax ) maximum closed-loop log modulus (dB) C Lcoil ) length of the coil (ft) M ) molecular weight (lb/lb‚mol) Nloops ) number of coil loops in the reactor Qcoil ) heat-transfer rate to the coil (Btu/h) QJ ) heat-transfer rate to the jacket (Btu/h) RCI ) reactor controllability index s ) Laplace transform variable TCin ) coolant inlet temperature (°F) TCout ) coolant exit temperature from the coil (°F) TJ ) jacket temperature (°F) TR ) reactor temperature (°F) Vtotal ) total volume of the reactor vessel (ft3) U ) overall heat-transfer coefficient (Btu/h‚ft2‚°F) Vcoil ) volume of the coil (ft3) VJ ) volume of the jacket (ft3) VR ) volumetric holdup of the reaction mass in the reactor (ft3) Vreaction ) molar holdup of the reaction mass in the reactor (lb‚mol) z ) reactant concentration in the reactor (mole fraction of A) z0 ) reactant concentration in the fresh feed (mole fraction of A) R ) kinetic preexponential factor (h-1) β ) ratio of the coil loop diameter to vessel diameter λ ) heat of reaction (Btu/lb‚mol) ∆TLM ) log-mean temperature differential driving force (°F) ∆Tmax ) maximum available temperature differential driving force (°F) F ) density of the process liquid (lb/ft3) FC ) density of the coolant (lb/ft3) τM ) measurement lag time (h) τI ) controller integral time (h) χ ) fractional conversion of reactant A

Literature Cited (1) Aris, R.; Amundson, N. R. An analysis of chemical reactor stability and control. Chem. Eng. Sci. 1958, 7, 121.

Ind. Eng. Chem. Res., Vol. 43, No. 11, 2004 2703 (2) Harriott, P. Designing temperature stable reactors. AIChE J. 1962, 8, 562. (3) Foss, A. S. Chemical reaction system dynamics. Chem. Eng. Prog. 1959, 54 (9), 39. (4) Bequette, B. W. Effect of process design on the openloop behavior of a jacketed exothermic reactor. Integr. Des. Control, Proc. IFAC Workshop 1994, 123. (5) Shinnar, R.; Doyle, F. J.; Budman, H. M.; Morari, M. Design considerations for tubular reactors with highly exothermic reactions. AIChE J. 1992, 38, 1729. (6) Luyben, W. L. Tradeoffs between design and control in chemical reactor systems. J. Process Control 1993, 3, 17-41.

(7) Luyben, W. L. Tuning temperature controllers on openloop unstable reactors. Ind. Eng. Chem. Res. 1998, 37, 4322-4331. (8) Luyben, W. L. Temperature control of autorefrigerated reactors. J. Process Control 1999, 9, 301-312. (9) Diemer, R. B. (DuPont Engineering Department). Private communication, Sept 3, 2003.

Received for review September 22, 2003 Accepted March 10, 2004 IE030721H

Quantitative Comparison of Temperature Control of Reactors with

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