Temperature Control of Continuous Stirred-Tank Reactors by

the often encountered conflict between steady-state economics and dynamic controllability. The ... Since its steady-state economic effect is minor, th...
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Ind. Eng. Chem. Res. 2004, 43, 6430-6440

Temperature Control of Continuous Stirred-Tank Reactors by Manipulation of Fresh Feed William L. Luyben Process Modeling and Control Center, Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015

When reactor capacity is limited by heat removal, an often-recommended control structure is to run with maximum coolant flow and manipulate feed flow rate to control reactor temperature (TR r F control). This control scheme has the potential to achieve the highest possible production rate; however, if the feed temperature is lower than the reactor temperature, the transfer function between temperature and feed flow rate contains a positive zero, which can degrade dynamic performance. This paper presents a quantitative study of this process, which is an example of the often encountered conflict between steady-state economics and dynamic controllability. The reactor is a jacket-cooled continuous stirred-tank reactor with an irreversible, exothermic, liquidphase reaction A f B. The effects of feed temperature, design conversion, heat of reaction, heattransfer coefficient, and design production rate are explored. Results indicate that the TR r F control structure can only be used in reactors designed for fairly high conversions and for reactions with moderate heats of reaction. Control may be improved by operating with a feed temperature that is higher than the reactor temperature, despite the possible undesirable steadystate economic disadvantages of such a design. 1. Introduction Reactor control is probably second only to distillation control as a subject of interest in the process control literature. Scores of papers appear every year discussing the many theoretical and practical aspects of reactor control. Most studies consider the reactor in isolation, not as part of an integrated plant. The usual types of reactors considered are continuous stirred-tank reactors (CSTRs), tubular reactors, and batch reactors. All of these various types of reactors are designed to operate either adiabatically or with heat transfer to some cooling or heating medium. There is a rich literature on reactor control dating back many decades. The pioneering studies of workers such as van Heerden,1 Amundson,2 Foss,3 and Perlmutter4 provided a solid foundation for the hundreds of researchers who followed. A small sampling of current activity in reactor control is provided in several recent papers.5-7 The key controlled variable in chemical reactor is temperature. As discussed by Shinnar and co-workers,8 temperature is a “dominant variable” for maintaining stable and effective operation. In most chemical systems, temperature has an exponential effect on reaction rates, so even modest changes in temperature can produce very significant effects. Many studies have illustrated the importance of providing adequate heat-transfer area in reactors so that tight temperature control is achievable. Most of the CSTR control studies have considered systems in which heat is transferred to or from the reacting mass (usually liquid) in the reactor vessel to some cooling or heating medium. Jacket cooling, internal coil cooling, external pumparound cooling, and evaporative cooling are the common methods for transferring heat. The flow rate of the cooling/heating medium is usually the manipulated variable that is changed by a reactor * To whom correspondence should be addressed. Tel.: 610758-4256. E-mail: [email protected].

temperature controller, either directly or through a reactor-to-jacket temperature cascade structure. Disturbances typically include changes in feed flow rate, feed composition, temperature of the cooling/heating supply and catalyst activity. A less commonly encountered control structure is to use feed flow rate to control reactor temperature. The concept is mentioned in several books,9,10 but only qualitative discussions are presented. The reason for using this kind of control structure is to maximize production when reactor capacity is limited by heat transfer. If the flow rate of the cooling/heating medium is set at its maximum value, this maximizes heat transfer. Then the TR r F control structure will feed in as much fresh feed as the system can handle and still maintain the desired reactor temperature. Shinskey9 presents a practical discussion of some of the advantages and problems of this control structure. The use of a “valve-position control” (VPC) structure is recommended. A reactor temperature controller sets the coolant valve position. Then a VPC looks at the position of the coolant valve and adjusts the flow rate of the reactor feed to keep the coolant valve near its wide open position. The temperature of the feed to a CSTR has a fairly minor impact on the steady-state design. Unlike a tubular reactor, in which feed temperature is a vital design parameter (lower inlet temperature requires a larger reactor), the temperature of the feed to a CSTR only affects the sensible heat component of the total heat removal/addition rate. The heat of reaction is typically considerably larger than this sensible heat. Since its steady-state economic effect is minor, the temperature of the feed to a CSTR is usually set by an upstream supply temperature. If the feed is at ambient temperature, it is fed to the reactor at this temperature, despite the fact that the reactor may be operating at a significantly higher temperature for kinetic reasons. If the reaction is exothermic and heat must be removed, the

10.1021/ie0400604 CCC: $27.50 © 2004 American Chemical Society Published on Web 09/01/2004

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z ) zo(1 - χ)

(1)

The reaction rate (lb-mol/h) is first-order in the concentration of reactant A.

R ) k(TR)zVm

(2)

where Vm is the molar holdup (lb-moles) of the reactor, and k is the specific reaction rate (h-1) at temperature TR. Therefore, the molar holdup can be calculated from the molar feed flow rate Fm (lb-mol/h) and the specific reaction rate k.

Vm ) Figure 1. Open loop response for +10% step change in fresh feed flow rate.

colder the feed, the less heat must be transferred. This reduces the amount of coolant required because of two effects. First, the total heat-transfer rate is lower. Second, the exit temperature of the coolant is higher, since a smaller differential temperature driving force is required. However, from a dynamic point of view, the temperature of the feed can have a significant impact on controllability. This is particularly true when the feed is colder than the reactor, because the immediate effect of increasing the feed flow rate can be a temporary decrease in reactor temperature. This is illustrated in Figure 1. The sensible heat of the cold feed initially reduces the reactor temperature. Eventually the increase in fresh feed raises the concentration of the reactant in the reactor, the reaction rate increases, and the reactor temperature begins to increase. This “inverse” or “wrongway” response is represented by a positive zero in the process open loop transfer function. This “nonminimum phase” feature degrades feedback control. Thus, there may be an inherent conflict between steady-state economics and dynamic controllability. The purpose of this paper is to explore this “cold feed” situation. 2. Process Studied The process considered features the irreversible, exothermic liquid-phase reaction A f B occurring in a jacket-cooled CSTR. Reactor temperature is TR. The fresh feed is introduced at a rate F and with a temperature To. If To is smaller than TR, inverse reverse response will occur. A circulating cooling water system is assumed, so the jacket is perfectly mixed to a temperature TJ. The cooling water is supplied at temperature TJO, and its flow rate is FJ. In the usual control structure, the coolant flow rate is set by the output signal from the temperature controller. In the control structure studied in this paper, the coolant flow rate is fixed, and the output signal from the temperature controller adjusts the flow rate of the fresh feed. The steady-state design of the CSTR consists of the following steps: First, the desired conversion χ, the feed flow rate F, the feed composition zo (mole fraction reactant A), reactor temperature TR, and the kinetic and heat-transfer parameters are specified. The concentration of reactant in the vessel is calculated (z mole fraction A) from the desired conversion χ and the fresh feed composition.

Fmχ Fmzoχ Fmzoχ ) ) k(TR)z k(T )zo(1 - χ) k(T )(1 - χ) R R

(3)

The specific reaction rate is

k ) Re-E/R(TR+460)

(4)

where E is the activation energy (30 000 Btu/lb-mole). The value of the preexponential factor R is calculated to give a specific reaction rate k(140) at 140 °F of 5 h-1 at the base-case conditions. The volumetric holdup VR (ft3) is

VR ) VmM/F

(5)

where M is the molecular weight (lb/lb-mole) and F is the density (lb/ft3). Now the diameter and length of the reactor vessel are calculated, assuming an aspect ratio (L/D) of 2 for the reactor vessel.

VR )

πDR2LR πDR2(2DR) πDR3 ) ) 4 4 2 2VR 1/3 DR ) π

( )

(6)

The volumetric feed flow rate F (ft3/h) is

F ) FmM/F

(7)

Heat transfer occurs through the circumferential wall area to a jacket surrounding the vertical walls of the vessel. The jacket heat-transfer area is AJ.

AJ ) πDRLR ) πDR(2DR) ) 2πDR2

(8)

The heat-transfer rate, Q, is calculated from the heat of reaction and the sensible heat,

Q ) (zo - z)Fm(-λ) - cpMFm(TR - To)

(9)

where λ is the heat of reaction (Btu/lb-mole) and cp is the heat capacity of the reaction liquid (Btu/lb-°F). The required jacket temperature is

TJ ) TR -

Q UAJ

(10)

where U is the overall heat-transfer coefficient (Btu/ h-°F-ft2). Finally, the flow rate of cooling water FJ (ft3/ h) is

FJ )

Q cJFJ(TJ - TJO)

(11)

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Table 1. Parameter and Design Values for Base Case Parameters Constant for All Cases reactor temp (°F) ) 140 fresh feed composition (mole fraction A) ) 1 activation energy (Btu/lb-mol) ) 30 000 density of process liquid ) 50 lb/ft3 molecular weight of A and B ) 50 lb/lb-mol heat capacity of process liquid ) 0.75 Btu/lb-°F density of coolant liquid ) 62.3 lb/ft3 heat capacity of coolant liquid ) 1 Btu/lb-°F jacket thickness ) 4 in. temperature measurement lags (two first-order) ) 1 min

dz ) a11z + a12TR + a13TJ + b11F + b12FJ dt dTR ) a21z + a22TR + a23TJ + b21F + b22FJ (15) dt dTJ ) a31z + a32TR + a33TJ + b31F + b32FJ dt The constant aij and bij coefficients are given in eqs 16 and 17.

Parameters Varied (Base-Case Values) χ) conversion (95%) F ) feed flow rate (100 lb-mol/h) To ) feed temp (70 °F) U ) overall heat-transfer coefficient (150 Btu/h-ft2-°F) λ ) heat of reaction (-10 000 Btu/lb-mol of A reacted) Base-Case Steady-State Conditions and Equipment Size VR reactor volume ) 380 ft3 DR reactor diameter ) 6.23 ft AJ jacket heat-transfer area ) 244 ft2 Q heat-transfer rate ) 0.688 × 106 Btu/h TJ jacket temperature ) 121 °F FJ cooling water flow rate ) 216 ft3/h

a11 ) -

F h -k h VR

a12 ) -

zjEk h R(T h R + 460)2

a13 ) a31 ) 0

where cJ is the heat capacity of the cooling water (Btu/ lb-°F) and FJ is the density of the cooling water (lb/ft3). This completes the steady-state design of the reactor for specified values of feed flow rate, feed temperature, conversion, heat of reaction, and heat-transfer coefficient. The effects of these design specifications on both the steady-state design and the dynamics of the system are studied in the following sections.

a21 ) -

λk h McP

a22 ) -

UAJ zjEk hλ F h 2 V V McPR(T h R + 460) R RFcP

a23 )

UAJ VRFcP

a32 )

UAJ VJFJcJ

a33 ) -

2. Mathematical Model of CSTR The nonlinear model of the CSTR involves dynamic component and energy balances for the reaction liquid and an energy balance for the water in the jacket. Constant holdup in the reactor and jacket and constant physical properties are assumed. Table 1 give values of design parameters used as well as steady-state values of variables under base-case conditions. Reactor component balance:

dz Fzo Fz ) - kz dt VR VR

UAJ dTR FTo FTR λkz ) (T - TJ) (13) dt VR VR Mcp FcpVR R

()

(14)

In these nonlinear ordinary differential equations, the nonlinearity comes from the product of variables (for example, FTo) and the exponentially temperature dependent k. The state variables are z, TR, and TJ. The manipulated variables are, in general, F and FJ, but we will use F as the manipulated variable in this study. So the open loop transfer function required to design the temperature controller is TR(s)/F(s). Linearization gives three linear ordinary differential equations,

zo - zj VR

b21 )

To - T hR VR

b32 )

TJ0 - T hJ VJ

(17)

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

G(s) )

Jacket energy balance:

F hJ UAJ VJ VJFJcJ

b12 ) b22 ) b31 ) 0

(12)

Reactor energy balance:

UAJ(TR - TJ) FJ dTJ (T - TJ) + ) dt VJ JO FJcJVJ

b11 )

(16)

TR(s) F(s)

)

c2s2 + c1s + c0 (s3 + b2s2 + b1s + b0)(τMs + 1)2

(18)

where

b2 ) - a11 - a22 - a33 b1 ) a11a22 + a11a33 + a22a33 - a12a21 - a23a32 b0 ) a12a21a33 - a11a22a33 + a11a23a32 c2 ) b21 c1 ) a21b11 - a11b21 - a33b21 c0 ) - a33a21b11 - a11a33b21

(19)

Ind. Eng. Chem. Res., Vol. 43, No. 20, 2004 6433 Table 2. Design Cases

To (°F) conversion (%) -λ (103 Btu/lb-mole) U (Btu/h-°F-ft2) F (ft3/h) VR (ft3) D (ft) AJ (ft2) TJ (°F) Q (106 Btu/h) FJ (ft3/h) polesa

zeros Kub Pu (h)

base case

warm feed

hot feed

lower conversion

70 95 10 150 100 380 6.23 244 121 0.688 216 -11.2 -2.30

120 95 10 150 100 380 6.23 244 116 0.875 305 -12.3 -2.00 ( 0.844 i

170 95 10 150 100 380 6.23 244 111 1.06 416 -13.6 -2.04 ( 1.24 i

70 90 10 150 100 180 4.86 148 111 0.638 248 -13.5 -1.28 ( 2.27 i

+58.4 -11.3 3.6 0.66

-48.2 -11.6 14 0.30

+12.7 -13.4 0.54 1.2

-1.64 +13.4 -10.5 1.9 1.0

higher conversion 70 98 10 150 100 980 8.54 459 130 0.718 193 -9.82 -4.26 -0.372 +13.8 -8.78 6.2 0.92

hot feed and lower conversion 170 90 10 150 100 180 4.86 148 94.4 1.01 665 -21.6 -1.44 ( 3.05 i -48.3 -18.0 6.1 0.32

a Two additional poles at s ) -60 from lags. b Dimensionless: temperature transmitter span ) 45 °F; valve maximum flow 4 times steady-state feed flow; with two 1-min lags.

Table 3. Design Cases

To (°F) conversion (%) -λ (103 Btu/lb-mole) U (Btu/h-°F-ft2) F (ft3/h) VR (ft3) D (ft) AJ (ft2) TJ (°F) Q (106 Btu/h) FJ (ft3/h) polesa

zeros Kub Pu (h)

base case

higher heat of reaction

higher heat of reaction with higher conversion

70 95 10 150 100 380 6.23 244 121 0.688 216 -11.2 -2.30

70 95 20 150 100 380 6.23 244 95.2 1.64 1042 -20.8 -0.889 ( 2.73 i

70 98 20 150 100 980 8.53 459 115 1.70 601 -12.1 -2.56

-1.64 +13.4 -10.5 1.9 1.0

+32.2 -21.3 0.54 1.0

-1.32 +32.7 -11.6 3.4 0.78

higher feed flow

lower U

70 95 10 150 150 570 7.13 320 118 1.03 341 -11.6 -1.91 ( 0.439 i

70 95 10 100 100 980 8.54 459 130 0.718 193 -9.82 -4.26

+13.5 -11.1 1.8 1.0

-0.372 +13.8 -8.78 6.2 0.92

a Two additional poles at s ) -60 from lags. b Dimensionless: temperature transmitter span ) 45 °F; valve maximum flow 4 times steady-state feed flow; with two 1-min lags.

Notice that the b21 coefficient is negative when the feed temperature is less than the reactor temperature. This produces a positive root of the numerator polynomial given in eq 18; i.e., the open-loop transfer function has a positive zero. 3. Results for Various Design Cases Number of case studies were explored. The first column in Table 2 gives parameter values and steadystate conditions for the base case: To ) 70, F ) 100, χ ) 0.95, and U ) 150. The concentration of reactant A in the reactor is 0.05 mole fraction. The fresh feed is pure A (zo ) 1). The base-case reactor has a diameter of 6.23 ft, and the heat transfer rate is 0.688 × 106 Btu/h through a jacket area of 244 ft2. The jacket temperature is 121 °F, and the cooling water flow rate is 216 ft3/h. Tables 2 and 3 give equipment sizes and operating conditions for several design cases in which various parameters are changed from the base case. 3.1. Open Loop Response. Figure 1 shows how feed temperature To affects both the steady-state design and the open loop dynamic response. The disturbance is a

10% step increase in feed flow rate at time equals 0.05 h. Three values of To are used in the figure. It is clear that the lower the temperature, the more the “inverse response”. There is an initial decrease in reactor temperature, but the concentration of A immediately begins to increase. Eventually the resulting increase in the rate of reaction generates enough heat to produce an increase in reactor temperature. It takes about 0.15 h for the reactor temperature to come back up to its original value when the feed temperature is 70 °F. The right two graphs show the effect of feed temperature on the required jacket temperature TJ and the heat-transfer rate Q. The colder the feed, the less heat has to be removed. This means the temperature difference is smaller between the reactor at 140 °F and the jacket, so TJ is higher for colder feeds. 3.2. Dynamic Effect of Feed Temperature. The open loop process transfer function given in eq 18 has two zeros and five poles. Under base-case conditions with a feed temperature of 70 °F, the two zeros are s ) +13.4 and s ) -10.5 (see Table 2). Thus, the process has inverse response. Two of the poles are s ) -60 from

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Figure 3. Effect of feed temperature (no lags).

Figure 2. (A) Root locus plot for 70 °F feed temperature. (B) Root locus plots for 70 and 120 °F feed temperature. (C) Root locus plot for 170 °F feed temperature. (D) Root locus plot for 170 °F feed temperature with lags

the two lags. The other three poles lie in the left half of the s plane (s ) -11.2, s ) -2.30, and s ) -1.64), so the system is open-loop stable under the base-case conditions with real distinct poles lying on the negative real axis. Figure 2A gives a portion of the root locus plot for the system without the temperature lags when feed temperature is 70 °F. The important part of the plot is the two loci near the imaginary axis. The loci start at the poles of the open loop system transfer function (only p1, p2, and p3 are shown). The loci move toward the zeros of the open loop transfer function. Since there is zero at s ) +13.4, the loci move into the right half of the s plane as controller gain increases. Thus, there is an ultimate gain, Ku, that is the limit of closed loop stability. Table 2 gives values of the ultimate gain and the ultimate period of the system (with the temperature lags included). The values of gain are made dimensionless by using a temperature transmitter span of 45 °F and a maximum feed valve flow rate of four times the steady-state value. Figure 2B illustrates the impact of feed temperature on the root locus plot. Loci for two values of To are shown (70 and 120 °F). The higher feed temperature shifts the positive zero from +13.4 to +58.4 (see the second column in Table 2). This much larger positive zero shifts the loci away from the real axis, which increases the ultimate gain (Ku changes from 1.9 to 3.6) and decreases the ultimate period (Pu changes from 1.0 to 0.66 h). This indicates an improvement in dynamic controllability. Notice that two of the poles of the open loop transfer function are now complex conjugates, indicating that the open loop system will be underdamped. Figure 2C gives the root locus plot (without the lags) when the feed temperature is higher than the reactor temperature. As given in the third column in Table 2, the zeros are both negative, so there is no inverse response. Without including the lags, there is no ultimate gain, since the loci lie entirely in the left half of the s plane. However, including the lags (as shown in Figure 2D) yields an ultimate gain of 14, which is much higher than those found for cold feeds.

Figure 4. Effect of feed temperature on ultimate gain and zeros with lags.

Figure 3 shows how several parameters vary as feed temperature is varied. The lags are neglected in these results. The ultimate gain increases as feed temperature increases. The pole nearest the imaginary axis is real for feed temperatures at or below 70 °F. At higher feed temperatures, the pole becomes complex. As expected, higher feed temperatures require lower jacket temperatures and higher heat-transfer rates and cooling water flow rates. A wider range of feed temperatures in considered in Figure 4, with the temperature measurement lags included. The lower graph shows what happens to the zeros of the open loop transfer function as feed temperature changes. As To approaches the reactor temperature (TR ) 140 °F), one of the zeros goes to infinity. Solving for the zeros in eq 18 gives

-c1 ( xc12 - 4c2c0 s) 2c2

(20)

But c2 ) b21 ) (To - TR)/VR, so there is a discontinuity at To ) TR. When b21 is 0, c1 ) a21b11, and c0 ) -a33a21b11. Looking at the numerator in eq 18 and letting c2 ) b21 ) 0 give a single zero at

s)-

UAJ c0 -a33a21b11 F h ) ) -a33 ) c1 a21b11 VJ FJcJVJ

(21)

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Figure 7. Effect of conversion.

Figure 5. (A) Closed loop responses; feed temperatures 70, 90, and 120 °F. (B) Closed loop responses; feed temperatures 150, 160 and 170 °F.

Figure 6. Root locus plot for 90 and 95% conversions.

The upper graph in Figure 4 shows how feed temperature affects the ultimate gain for three values of lag time constant. With no lags, the ultimate gain goes to infinity as the feed temperature approaches the reactor temperature. Increasing the lag time constant reduces the ultimate gain. Figure 5A gives closed loop dynamic responses for three values of feed temperature. The disturbance is a +20% increase in the cooling water flow rate from the steady-state value at time equals 0.2 h. The TyreusLuyben (TL) tuning method is used to calculate the controller gain and integral time from the ultimate gain and ultimate period. As feed temperature is increased, higher controller gains and smaller integral times can be used, so control performance is improved (smaller peak temperature deviations and smaller closed loop time constants).

Figure 5B gives results for a higher range of feed temperatures. The higher the feed temperature, the better the dynamic performance. One unexpected finding in these simulations was that the TL tuning constants gave responses that were too oscillatory. The results shown in Figure 5B have the controller gain reduced from the TL gain be a factor of 3. The reason for this can be seen by looking at the root locus plot for the 170 °F feed temperature case in Figure 2D. Notice that the loci are very close to the imaginary axis, and controller gains anywhere near the ultimate gain will give a closed loop root with a large imaginary part (small damping coefficient). The TL settings are derived for an integrator/deadtime process, which has quite different dynamic characteristics. The locations of the closed loop roots with controller gains equal to the TL gain and one-third of the TL gain are shown in Figure 2D. A frequencydomain maximum closed loop log modulus criterion of +2dB yields a controller gain of one-third the TL gain. 3.3. Dynamic Effect of Design Conversion. The specified conversion has a very significant effect on both the steady-state design and the dynamics of the system. As the desired conversion is increased, the required reactor volume increases. There is a slight increase in the heat-transfer rate, but there is a significant increase in the available heat-transfer area. The net result is a decrease in the required differential temperature driving force between the reactor and the jacket, which gives higher jacket temperatures. As given in Table 2, increasing conversion from the base case of 95 to 98% results in a reactor whose volume increases from 380 to 980 ft3 and whose heat-transfer area increases from 244 to 459 ft2. The heat-transfer rate increases slightly from 0.688 to 0.718 (106 Btu/h). The result is an increase in jacket temperature from 121 to 130 °F. The total potential ∆T for heat transfer is the reactor temperature minus the temperature of the supply cooling water (70 °F). The higher the design jacket temperature, the less of this potential ∆T is being used, so more “muscle” is available to handle disturbances. This means better dynamic control. The root locus plots given in Figure 6 illustrate this point. Loci for two values of design conversion are shown. As conversion increases, the loci move away from the imaginary axis, and the ultimate gain increases. The first, fourth, and fifth columns in Table 2 give results for different conversions. Ultimate gains increase as

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Figure 9. Effect of heat of reaction.

Figure 8. (A) Closed loop responses; To ) 70 °F; conversions 90, 95, and 98%. (B) Closed loop responses; To ) 170 °F; conversions 85, 90, and 95%

conversion increases: 90% f Ku ) 0.54, 95% f Ku ) 1.9, and 98% f Ku ) 6.2. Openloop poles move further from the imaginary axis and become distinct real poles instead of complex conjugates. The positive open loop zero moves further to the right along the positive real axis, which makes the loci circle (moving from the poles to the zero) bigger in diameter. These effects indicate improved control as conversion increases. Figure 7 shows the impact of conversion on several variables for two different values of feed temperature. As conversion increases, ultimate gain, reactor volume, jacket temperatue, and heat-transfer area increase. The open loop pole is complex at low conversions and is quite close to the imaginary axis. As conversion drops below about 80%, the real part of the pole becomes positive, so the system is open-loop-unstable. The improvement in control produced by operating with a higher feed temperature is also illustrated in Figure 7. Ultimate gains are higher, and poles are further to the left in the s plane. Figure 8 gives closed loop dynamic responses for several levels of conversion. The disturbance is a 20% increase in the flow rate of the cooling water at time equals 0.2 h. In Figure 8A, the feed temperature is 70 °F, whereas in Figure 8B, the feed temperature is 170 °F. Dynamic performance improves as conversion increases. For the hot feed case, the TL controller settings had to be detuned by dividing the controller gain by 3 and multiplying the integral time by 2. For cold feed temperatures, conversions below 90% give very small controller gains and large peak devia-

tions in temperature (Figure 8A). For hot feed temperatures, conversions as low as 85% give reasonable controller gains and smaller peak deviations in temperature (Figure 8B). These results illustrate the dynamic advantage of using hot feed. 3.4. Dynamic Effect of Heat of Reaction. The heat of reaction λ has a significant impact on both the steadystate design and the dynamics. The higher the heat of reaction, the more heat Q must be transferred from the reaction liquid into the jacket cooling water. For a given conversion and feed flow rate, the size of the reactor is fixed, so the heat-transfer area is fixed. The increase in Q resulting from an increase in λ requires, for the same area, an increase in the differential temperature driving force ∆T ) TR - TJ. If reactor temperature is constant, jacket temperature must decrease, which increases the required flow rate of cooling water. These effects are illustrated in Figure 9 and Table 3. As the reaction gets more exothermic (λ becomes more negative), the ultimate gain decreases and the dominant pole becomes complex with a real part that gets closer to the imaginary axis. Figure 9 gives results for a range of λ’s with two values of conversion. Designs for lower conversions with high heats of reaction are expected to have poor control performance because of the very small values of controller gain. For example, as λ changes from the base-case value of -10 000 to -20 000 Btu/lb-mole, the ultimate gains drops for 1.9 to 0.54 for the 95% conversion case. The degradation of dynamic performance is demonstrated in Figure 10. The disturbance is a 20% increase in the flow rate of the cooling water at time equals 0.2 h. Closed loop results for three values of λ are shown. Controller tuning used the TL settings. As the heat of reaction increases, the peak deviation in the temperature increases. It is also interesting to note that there is less change in the feed flow rate for the same percentage change in the cooling water flow rate at the higher values of λ. This occurs because the cooling water flow rates under design conditions are higher and the jacket temperatures are lower for the higher values of λ. Therefore, the ∆T driving force is larger and is a bigger fraction of the total available ∆Tmax ) TR - TJO. Changing cooling water flow rate does not change the ∆T much when jacket temperature is already low. Therefore, a +20% change in cooling water flow rate with λ ) -10 000 Btu/

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Figure 10. Closed loop responses with different heats of reaction.

Figure 12. (A) Different design feed flow rates; disturbance in FJ. (B) Different design feed flow rates; disturbance in TJO.

Figure 11. Effect of design feed flow rate.

lb-mole results in a feed rate increase of ∼14%. The same percent change in cooling water flow rate with λ ) -20 000 Btu/lb-mole results in a feed rate increase of only ∼5%. 3.5. Dynamic Effect of Design Production Rate. The design value for the reactor feed flow rate sets the design production rate and establishes the size of the vessel for a given conversion. Reactor volume is directly related to production rate, as is the required heattransfer rate; however, the jacket-area-to-volume ratio becomes smaller as the reactor becomes bigger. Therefore, larger differential temperature driving forces are needed, which means jacket temperature gets lower as the reactor gets larger. Since a larger fraction of the total potential ∆T (TR - TJO) is being used, we would expect dynamic controllability to become worse as the reactor becomes larger. Figure 11 shows the effects of design feed flow rate on several parameters. Results for three levels of design conversion are given. As expected, heat-transfer rate and cooling water flow rate increase while jacket temperature decreases as feed flow rate increases. The ultimate gain and the location of the poles of the transfer function depend much more strongly on conversion than on feed flow rate. Low conversions give low ultimate gains and complex conjugate poles lying closer to the imaginary axis. Comparing the first and fourth columns in Table 3 shows the differences between the base-case design at F ) 100 lb-mol/h and a design at F ) 150 lb-mol/h.

Figure 12 shows the responses of three systems with different design feed flow rates for 95% conversion. The disturbance in Figure 12A is a 20% increase in the flow rate of the cooling water at time equals 0.2 h. It is surprising that there is little difference among these cases with vastly different feed flow rates (and, of course, vessel sizes). The disturbance is a change in cooling water flow rate. A number of other disturbances were tested, for example a change in the cooling water supply temperature from 70 to 90 °F, shown in Figure 12B. Again, the system with this feed flow rate manipulation (F r TR control structure) is unexpectedly insensitive to disturbances. It appears that the ability to adjust the load on the system by making appropriate changes in the feed flow rate provides the system with a significant degree of self-regulation. 3.6. Dynamic Effect of Heat-Transfer Coefficient. The overall heat-transfer coefficient U has a very significant impact on both the steady-state design and the dynamics of the system. Smaller Us require larger ∆Ts that give lower jacket temperatures, as shown in Figure 13. Smaller Us also reduce the ultimate gain and change the poles of the open loop transfer function from negative distinct values to complex conjugates that move toward the imaginary axis. Table 3 gives parameter values for the base-case U ) 150 Btu/h-°F-ft2 and for the case with U ) 100 Btu/h-°F-ft2. Figure 14 gives responses to a cooling water flow rate disturbance over a wide range of U’s with 95% conversion. Peak deviations in temperature are much larger for small values of U.

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Figure 15. Root locus plot when manipulating FJ. Figure 13. Effect of heat-transfer coefficient. Table 4. Controller Tuning Parameters conversion (%) 95

90

85

Figure 14. Closed loop responses for different U’s.

4. Comparison of Alternative Control Structures Up to this point, we have studied the F r TR control structure (manipulation of feed flow rate to control reactor temperature). It is interesting to compare the performance of this structure to the more conventional control scheme of manipulating cooling water flow rate to control reactor temperature (FJ r TR control structure). For the design of the temperature controller with this structure, we need the open loop transfer function between TR and FJ. 4.1. Open Loop Transfer Function. Laplace transforming and rearranging the linear ordinary differential equations given in eq 15 give the required transfer function,

GFJ(s) )

TR(s) FJ(s)

)

c1s + c0 3

2

(s + b2s + b1s + b0)(τMs + 1)2

(22)

where

c1 ) a23b32 c0 ) -a11a23b32

(23)

This transfer function has only one zero, which is negative (s ) -5.25 for base-case conditions). Figure 15 gives the significant portion of the root locus plot for the 95% conversion case. The two poles at s )

Ku Pu (h) Kc τI (h) Ku Pu (h) Kc τI (h) Ku Pu (h) Kc τI (h)

manipulate F structure

manipulate FJ structure

1.9 1.0 0.59 2.2 0.54 1.2 0.17 2.6 0.14 1.5 0.044 3.3

24 0.38 7.6/3 0.83 × 2 15 0.38 4.6/3 0.83 × 2 11 0.40 3.3/3 0.88 × 2

-60 from the lags make the system net fourth-order, so there is an ultimate gain (Ku ) 24) and ultimate period (Pu ) 0.38 h). Controller gain is made dimensionless by using a 45 °F temperature transmitter span and a cooling water valve that can pass four times the steady-state design flow rate. Table 4 gives TL controller tuning constants for several values of conversion for the two alternative control structures. In the simulation discussed below, the TL settings worked well for the F r TR control structure; however, significant detuning was required for the FJ r TR control structure. The TL gains were divided by 3, and integral time was multiplied by 2. 4.2. Comparison for Feed Temperature Disturbance. The standard disturbance used up to this point was a change in cooling water flow rate. Since this variable is used as the manipulated variable in FJ r TR control structure, a different disturbance must be selected. There are several possible choices: feed temperature, feed composition, heat of reaction, catalyst activity, heat-transfer coefficient, or cooling water inlet temperature. The first of these was chosen since it is a common occurrence in many processes and serves the purpose of disturbing the system. The feed temperature To is changed from 70 to 120 °F at time equals 0.2 h. Figure 16A gives results for the F r TR control structure with three conversion cases. As conversion decreases, peak temperature deviations increase because of the much smaller controller gains (see Table 4). The 85% conversion case has a peak temperature of ∼163 °F. The cooling water flow rate is constant at its steady-state design value for each conversion case. Figure 16B gives results for the FJ r TR control structure with three conversion cases. The feed flow rate

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Figure 16. (A) Manipulate F with feed temperature disturbance. (B) Manipulate FJ with feed temperature disturbance.

is constant at 100 lb-mol/h for all cases. The same trends occur: as conversion decreases, peak temperature deviations increase; however, the temperature peaks are smaller than those of the F r TR control structure. The 85% conversion case has a peak temperature of ∼151 °F instead of 163 °F. However, as shown in Figure 16B, significant changes in cooling water flow rate are required to handle the disturbance. For the 85% conversion case, the cooling water flow rate more than doubles. So we are comparing a control structure with a fixed cooling water flow rate with one in which the cooling water flow rate can change. A more fair comparison is to compare the two control structures in the situation in which we want to maximize feed flow rate, but the cooling water flow rate is constrained. Figure 17 gives such a comparison. The reactor is designed for 95% conversion and a feed flow rate of 100 lb-mol/h, which fixes the size of the vessel. The design value of the cooling water flow rate is 216 ft3/h. The results shown in Figure 16B reveal that the cooling water flow rate has to increase to ∼300 ft3/h to handle the feed temperature disturbance in the FJ r TR control structure, so we will assume that the maximum cooling water flow rate is 310 ft3/h. Of course, when we use the F r TR control structure and want to maximize feed flow rate, we would set the cooling water flow rate near this maximum. We assume that the cooling water flow rate is 300 ft3/h for this structure. The resulting feed flow rate is 128 lb-mol/h, with a jacket temperature of 116 °F and reactor composition of 0.0633 mole fraction A. Note that the conver-

Figure 17. (A) Comparison of control structures; one To disturbance. (B) Comparison of control structures; two To disturbances. (C) Comparison of control structures; TJO disturbance.

sion is less than 95% under these conditions, so reactor temperature may have to be increased if a fixed conversion is desired. The FJ r TR control structure would keep the feed at 100 lb-mol/h, so for the first 2 h shown in Figure 17A, we are maximizing the production rate when using the F r TR control structure but are not getting maximum production when using the FJ r TR control structure because we need room to be able to manipulate cooling water flow in the face of disturbances. The feed temperature disturbance occurs at time equals 2 h. The F r TR control structure responds by cutting back on the fresh feed flow rate, eventually lining out at ∼100 lb-mol/h. The peak temperature is ∼146 °F. Cooling water flow rate is constant at 300 ft3/ h. Jacket temperature drops to ∼117 °F. The FJ r TR control structure responds by increasing the cooling water flow rate. The limit of 310 ft3/h is

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quickly reached. Eventually, the cooling water lines out at ∼305 ft3/h. The fresh feed flow rate stays constant at 100 lb-mol/h. The peak temperature is only ∼142 °F, which is lower than for the other control structure. Jacket temperature drops to ∼116 °F. Although the temperature control is better when the FJ r TR control structure is used, it does not maximize production rate because control depends on being able to change the cooling water flow rate, both up and down. So for the first 2 h shown in Figure 17A, the FJ r TR control structure is only feeding 100 lb-mol/h, while the F r TR control structure is feeding 128 lb-mol/h (a 28% increase in production rate). Figure 17B shows the contrast between the two control structures even more strikingly. Now there is an additional change in the feed temperature to 150 °F at time equals 6 h. The F r TR control structure simply cuts back on the feed flow rate and gives stable operation with this much hotter feed. However, the FJ r TR control structure can only drive the cooling water to its limit. Thereafter, reactor temperature is uncontrolled and rises to 147 °F. Figure 17C gives the situation in which the inlet temperature of the cooling water increases at time equals 2 h from 70 to 90 °F. The F r TR control structure reduces the fresh feed flow rate as reactor temperature rises, finally ending up at a new steadystate production rate of ∼93 lb-mol/h. The FJ r TR control structure cannot handle this disturbance. Cooling water flow rate, FJ, saturates wide open, but reactor temperature rises to and stays at about 144 °F. 5. Conclusion The use of fresh feed manipulation to control temperature in a CSTR has been explored. Although this control structure does not give as tight control as the conventional manipulation of cooling water flow, it does achieve the objective of maximizing production rate. It also avoids constraints on heat removal when there are changes in heat-transfer rates or coefficients, feed temperatures, or heats of reaction. The transfer functions for the two types of temperature control are derived and used to explore the effects of several design and kinetic parameters on the open loop and closed loop dynamics. Rigorous nonlinear simulations are used to evaluate control performance over a broad range of parameter values. Manipulating feed flow rate to control reactor temperature appears to be effective for reactors that are designed for high levels of conversion and for reactions in which the heat of reaction is not large. Dynamic problems may be experienced when the temperature of the feed is significantly lower than the operating temperature of the reactor. The economics of this type of system involve primarily the value of maximizing feed flow rates. There is the other aspect of deliberately heating the feed to reduce the dynamic problems. The former would dominate in a capacity-limited situation. The latter would involve issues of energy saving (energy added to heat the feed and more heat removed in the reactor). The economics would be highly dependent on the actual situation in the plant. Nomenclature Nomenclature A ) reactant component aij ) constant coefficient in linear ODE

AJ ) heat-transfer area of jacket (ft2) B ) product component bij ) constant coefficient in linear ODE CSTR ) continuous stirred tank reactor cJ ) heat capacity of coolant (Btu/lb-°F) cp ) heat capacity of process (Btu/lb-°F) DR ) reactor vessel diameter (ft) E ) activation energy (Btu/lb-mole) F ) volumetric fresh feed flow rate (ft3/h) Fm ) molar fresh feed flow rate (lb-mol/h) FJ ) cooling water flow rate to jacket (ft3/h) GM ) process open loop transfer function k ) specific reaction rate (h-1) KC ) controller gain (dimensionless) ko ) specific reaction rate at 140 °F (h-1) Ku ) ultimate controller gain (dimensionless) LR ) length of reactor (ft) M ) molecular weight (lb/lb-mole) Pu ) ultimate period (hours) Q ) heat-transfer rate to jacket (Btu/h) R ) chemical reaction rate (lb-mol/h) s ) Laplace transform variable TJ ) jacket temperature (°F) TJO ) cooling water supply temperature (°F) To ) temperature of feed (°F) TR ) reactor temperature (°F) U ) overall heat-transfer coefficient (Btu/h-ft2-°F) VJ ) volume of jacket (ft3) VR ) volume of reactor (ft3) z ) reactant concentration in reactor (mole fraction A) zo ) reactant concentration in fresh feed (mole fraction A) R ) kinetic preexponential factor (h-1) λ ) heat of reaction (Btu/lb-mol) F ) density of process liquid (lb/ft3) FJ ) density of coolant (lb/ft3) τM ) measurement lag time (hour) τI ) controller integral time (hour)

Literature Cited (1) Van Heerden, C. The characteristics of the stationary state of exothermic processes. Chem. Eng. Science 1958, 8, 133. (2) Aris, R.; Amundson, N. R. An analysis of chemical reactor stability and control. Chem. Eng. Sci. 1958, 7, 121. (3) Foss, A. S. Chemical reaction system dynamics. Chem. Eng. Prog. 1959, 54, 9, 39. (4) Perlmutter, D. D. Stability of Chemical Reactors; Prentice Hall: New York, 1972. (5) Lin, S. W.; Yu, C. C. Design and control for recycle plants with heat-integrated separators. Chem. Eng. Sci. 2004, 59, 53. (6) Shinnar, R. A Systematic methodology for the design development and scale-up of complex chemical processes, the role of control and concurrent design. Ind. Eng. Chem. Res. 2004, 43, 246. (7) Jaisathaporn, P.; Luyben, W. L. Dynamic comparison of alternative tubular reactor systems. Ind. Eng. Chem. Res. 2004, 43, 1003. (8) Arbel, A.; Rinard, I. H.; Shinnar, R. Dynamics and control of fluidized catalytic crackers: 4. the impact of design on partial control. Ind. Eng. Chem. Res. 1997, 36, 747. (9) Shinskey, F. G. Process Control Systems, 3rd ed.; McGrawHill: New York, 1988; p 390. (10) Luyben, W. L.; Tyreus, B. D.; Luyben, M. L. Plantwide Process Control; McGraw-Hill: New York, 1998; p 112.

Received for review February 18, 2004 Revised manuscript received June 25, 2004 Accepted July 16, 2004 IE0400604