Tuning Temperature Controllers on Openloop Unstable Reactors

The feed is pure reactant A (z0 = 1) at a temperature of 70 °F. The mole ... the system is openloop stable for conversions greater than about 95%, bu...
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Ind. Eng. Chem. Res. 1998, 37, 4322-4331

Tuning Temperature Controllers on Openloop Unstable Reactors William L. Luyben* Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015

There have been several methods proposed in the literature for tuning feedback controllers in openloop unstable processes. All of these methods assume a transfer function model is available that consists of a steady-state gain, deadtime, and positive pole. In some methods an additional negative pole is considered. This paper points out that this type of transfer function is often not appropriate for tuning temperature controllers in openloop unstable chemical reactors. There are two reasons for this. First, many chemical reactors used in industry have more than one positive pole. For example, the commonly used jacket-cooled continuous stirred-tank reactor has two positive poles even for the simple reaction A f B. Second, in practical applications we may not have a model, just an operating reactor. Even if a model is available, it is usually much more complex than the simple transfer function used in the literature. Openloop identification is not possible because the process is openloop unstable. Usually the only information available is the ultimate gain and the ultimate frequency, which have been obtained from a closedloop relay-feedback test. In this paper the effect of conversion on openloop instability is explored for jacket-cooled continuous stirred-tank reactor systems. Both linear and nonlinear models are used to study controller tuning. The Tyreus-Luyben (TL) tuning method is demonstrated to give more robust control for this type of system than the ZieglerNichols method. A procedure for obtaining an approximate gain/deadtime/positive-pole model is presented so that the tuning methods proposed in the literature can be used. However, these tuning methods provide performance that is no better than the TL method. Introduction

by a deadtime/positive-pole transfer function.

The occurrence of openloop instability in chemical engineering processes is fairly common. Many chemical reactors with back mixing exhibit openloop instability when exothermic irreversible reactions are taking place. This can occur when heats of reaction are high or when heat transfer areas or heat transfer coefficients are low. Conversion must also be moderate so that there is sufficient reactant present in the reactor to fuel the potential runaway. For a variety of reasons, not all reactors operate at high conversions. The initial reactor in a series of CSTRs almost always has moderate conversions. Many polymerization reactors must limit conversion to avoid viscosity problems and/or to achieve desired polymer properties. Luyben (1993) discussed some of the tradeoffs between steady-state economic design and dynamic controllability for jacket-cooled CSTR systems. Direct conflicts and competing goals often occur. He reported that tuning of the temperature controllers is not a trivial job. Use of the Ziegler-Nichols settings often leads to closedloop systems that are too oscillatory. Frequently, empirical tuning has to be employed to achieve reasonable closedloop damping coefficients. This paper addresses this problem of temperature controller tuning. Since the literature contains a number of papers dealing with the tuning of openloop unstable processes, we might expect that the problem has already been solved. Unfortunately, this is not the case. All of the literature papers assume that the dynamics of the openloop unstable process can be approximated * Author to whom correspondence should be addressed at the following: [email protected].

Kpe-Ds GM(s) ) τos - 1

(1)

In some papers additional lags (negative poles) are also considered.

GM(s) )

Kpe-Ds (τos - 1)(τ1s + 1)

(2)

Recent papers that explore this type of system include Poulin and Pomerleau (1996), Venkatashankar and Chidambaram (1994), Shafiei and Shenton (1994), Rotstein and Lewin (1991), and DePaor and O’Malley (1989). These papers deal only with this deadtime/ positive-pole model of an openloop unstable system. We demonstrate that this type of model is not appropriate for continuous stirred-tank reactors (CSTR) with exothermic irreversible chemical reactions. The most simple of all reactors and reactions is be studied. Gain/Deadtime/Positive-Pole Process Figure 1A gives Nyquist plots for a process with the transfer function given in eq 1. The specific numerical value used is D/τo ) 0.1 When a proportional controller is used, the curve starts at ω ) 0 on the negative real axis and goes to the origin as ω f ∞ with phase angles that initially increase above -180° but eventually decrease rapidly. The frequency where the curve crosses the negative real axis is the ultimate frequency ωu. The

10.1021/ie980078c CCC: $15.00 © 1998 American Chemical Society Published on Web 10/03/1998

Ind. Eng. Chem. Res., Vol. 37, No. 11, 1998 4323

Figure 1. Nyquist plots for deadtime/positive-pole transfer function. A. P and PI control. B. Effect of deadtime.

ultimate gain Ku (or Kmax) is the reciprocal of the magnitude of the openloop transfer function GM(iω) at this frequency. A controller gain of half of the ultimate is used, so the curve crosses the negative real axis at (-0.5, 0). There is also a minimum controller gain. The openloop transfer function has one pole in the right half of the s-plane. Therefore, the Nyquist plot must make one counterclockwise encirclement of the (-1, 0) point for the closedloop system to be stable. There is a controller gain Kmin below which an encirclement does not occur. The minimum gain is the reciprocal of the process gain: Kmin ) 1/Kp. Figure 1A shows that when PI control is used the curves starts with a phase angle of -270° because of the integrator in the controller. The curve gets closer to the (-1, 0) point as the integral time constant τI is decreased. The two curves shown use the ZieglerNichols (ZN settings and the Tyreus-Luyben (TL) settings (Tyreus and Luyben, 1992). The TL settings are farther from the (-1, 0) point so the closedloop system is less oscillatory. Figure 1B shows the effect of changing deadtime on the Nyquist plot. As D increases, the distance on the negative real axis in which the (-1, 0) point must lie gets smaller. This indicates that control becomes more difficult. For deadtimes greater than about 0.6, closedloop stability cannot be achieved using a proportional controller. In the next section, the Nyquist plots of a CSTR system are compared with those shown in Figure 1 and look significantly different. Process Studied A simple first-order exothermic chemical reaction A f B is assumed to occur in a CSTR. Different levels of conversion are considered, which lead to different reactor sizes. The reactor operates at 140 °F and the feed flow-rate is 100 lb mol/h. The feed is pure reactant A (z0 ) 1) at a temperature of 70 °F. The mole fraction of reactant A in the reactor (and in the product stream) is z. The molar holdup in the reactor is VR. The reaction rate is assumed to be first order.

R ) VRkz

(3)

The specific reaction rate k has an Arrhenius temperature dependence with an activation energy of 30 000 Btu/lb mol. At the operating temperature of 140 °F, the value of k is 0.5 h-1. Reactor temperature is T, and jacket temperature is TJ. Cooling water supply temperature TJ0 is 70 °F and overall heat transfer coefficient U is 150 Btu/h ft2 °F. The exothermic heat of reaction λ is -30 000 Btu/lb mol. The density of the reaction mass F is 50 lb/ft3, and the molecular weight M is 50 lb/lb mol. The heat capacity cp is 0.75 Btu/lb °F. The heat capacity and density of the cooling water are cJ ) 1 Btu/lb °F and FJ ) 62.3 lb/ft3. A 4-in. jacket is assumed, so the jacket volume VJ ) 0.333AH, where AH is the circumferential heat transfer area between the jacket and the reactor: AH ) πDL. As conversion decreases, the size of the reactor decreases, as does the area for heat transfer between the reaction mass and the cooling water in the jacket. This requires a larger temperature difference between the reaction mass and the coolant in the jacket. The steady-state design procedure consists of the following steps: 1. The conversion χ and the fresh feed composition z0 are specified, and the composition of reactant A in the reactor z is calculated.

z ) z0(1 - χ)

(4)

2. The molar holdup in the reactor is calculated.

VR )

Fχ k(1 - χ)

(5)

3. Assuming an aspect ratio (L/D) of two, the diameter D and the heat-transfer area AH are calculated.

D ) (2VR/π)1/3

(6)

AH ) 2πD2

(7)

4. The heat-removal rate Q, the jacket temperature

4324 Ind. Eng. Chem. Res., Vol. 37, No. 11, 1998 Table 1. Design Parameters for Different Conversions convn (%)

reactor holdup (lb mol)

openloop eigenvalues

z (m.f. A)

heat-transfer area (ft2)

jacket temp. (°F)

98 96 94 92 90 85 80 75 70

9800 4800 3133 2300 1800 1133 800 600 467

-7.89, -0.175 ( i 0.1626 -8.70, -0.0672 ( i 0.321 -9.41, +0.0392 ( i 0.387 -10.2, +0.145 ( i 0.414 -10.8, +0.252 ( i 0.405 -12.5, +0.518 ( i 0.136 -14.4, +1.35, +0.224 -16.5, +1.97, +0.132 -18.8, +2.57, +0.0589

0.02 0.04 0.06 0.08 0.1 0.15 0.20 0.25 0.30

2128 1322 995 810 688 505 401 331 280

131.6 126.8 122.9 119.4 116.4 109.8 104.4 99.9 96.2

Figure 2. Effect of conversion. A. Openloop eigenvalues. B. Jacket temperature and coolant flow-rate.

TJ, and the cooling water flow-rate FJ are calculated.

Q ) (z0 - z)F(-λ) - cpMF(T - T0) TJ ) T FJ )

Q UAH

Q cJFJ(TJ - TJ0)

(8) (9) (10)

Table 1 gives design parameters with conversions ranging from 98% to 70% for the specific numerical example considered. As we show in the next section, the system is openloop stable for conversions greater than about 95%, but the reactor is openloop unstable when the conversion is lower. Openloop Stability The reactor is described by three nonlinear ordinary differential equations: a component balance on the reaction mass in the reactor and energy balances on the reactor and on the jacket.

( ) ( ) ( ) ( )

F F dz ) z z - kz dt VR 0 VR

(11)

UAH λkz F F dT T0 T(T - TJ) (12) ) dt VR VR Mcp cpMVR

Figure 3. Nyquist plots for various conversions.

()

UAH dTJ FJ (TJ0 - TJ) + (T - TJ) ) dt VJ cJFJVJ

(13)

These equations can be linearized, Laplace transformed and rearranged to give the openloop transfer function between the controlled variable, reactor temperature T, and the manipulated variable, cooling water flow-rate F J.

Ind. Eng. Chem. Res., Vol. 37, No. 11, 1998 4325

Figure 4. Effect of conversion. A. Openloop eigenvalues. B. Jacket temperature and coolant flow-rate. C. Maximum and minimum gains.

GM(s) )

a23b3(s - a11) 3

2

s + B2s + B1s + B0

(14)

where a23 ) UAH/VRMcp, b3 ) (TJ0 - T h J)/VJ, and a11 ) h . The variables with overscores are steady-(F h /VR) - k state values.

B2 ) -a11 - a22 - a33 B1 ) a11a22 + a11a33 + a22a33 - a12a21 - a23a32 B0 ) a12a21a33 - a11a21a33 + a11a23a32

(15)

where a22 ) (-λk h Ezj/McpRT h 2) - (F h /VR) - (UAH/VRMcp), h Ezj/RT h 2), a21 ) (-λk h /Mcp), a33 ) (UAH/VRMcp), a12 ) (k a23 ) (UAH/VRMcp), and a32 ) (UAH/VJFJcJ). Notice that the transfer function given in eq 14 is net second order with a third-order denominator and a firstorder numerator. Thus there are three openloop poles and one openloop zero. Since the model is net second order, there is no ultimate gain. However, in any real process there are always additional lags in the system. To model this, we assume that the temperature measurement dynamics are modeled by two first-order lags

in series with time constants τm. A value of 1 min is used for most cases. The openloop transfer function between the lagged temperature measurement Tlag and the cooling water flow-rate becomes

GM(s) )

(

a23b3(s - a11)

)(

)

1 s + B2s2 + B1s + B0 (τms + 1)2 3

(16)

As we show below, the openloop poles lie in the LHP for reactors that are designed for high conversions. This occurs because these reactors are large and provide large heat-transfer areas. This means that the temperature difference between the reactor and the jacket is small. Therefore, the jacket temperature can be significantly decreased by increasing the cooling water flow-rate, and this permits large changes in the heattransfer rate, resulting in good temperature control. When the reactor is designed for low conversions, two of the openloop poles lie in the RHP. Low conversions require smaller vessels with smaller heat-transfer areas. This requires larger temperature differences between the reactor and the jacket. Since it is now more difficult to decrease the jacket temperature, temperature control is degraded.

4326 Ind. Eng. Chem. Res., Vol. 37, No. 11, 1998

A

Figure 5. Nyquist plots for various conversions.

Figure 2A shows how the openloop poles (openloop eigenvalues) change as the design conversion changes from 98% to 90%. For conversions greater than about 95%, all three of the openloop eigenvalues have negative real parts. However, for lower conversions, two of the eigenvalues are complex conjugates with positive real parts. Figure 2B shows how the jacket temperature and cooling water flow-rates change as the design conversion changes. Figure 3 gives Nyquist plots for reactors with conversions varying from 98% to 90% conversions. Note that when the reactors are openloop stable, the plots start on the positive real axis and move clockwise. When the reactors are openloop unstable, the plots move counterclockwise. These plots should be compared with those given in Figure 1 where the transfer function is a deadtime and one positive pole. It is clear that the Nyquist plots are quite different. The high-frequency portion of the two plots look somewhat similar. We discuss the implications of this later in this paper. Controller Tuning For the reactors that are openloop unstable, there are two poles in the RHP. The Nyquist stability critierion tells us that there must be two counterclockwise encirclements of the (-1, 0) point for the system to be closedloop stable. This can be achieved by designing the controller such that the (-1, 0) point lies between the two points where the Nyquist curves cross the negative real axis. The first intersection occurs at some frequency ωmin, and the minimum controller gain Kmin is the reciprocal of the magnitude of GM(iω) at this frequency. The second intersection occurs at the ultimate frequency ωu and the maximum controller gain Kmax (Ku) is the reciprocal of the magnitude of GM(iω) at this frequency. As we discuss later, finding the second intersection is usually easily done by using the closedloop relay-feedback test. However, finding the first intersection is not straightforward. The process is openloop unstable, so conventional openloop identification methods cannot be employed.

B

Figure 6. Nyquist plots for 70% converstion case with P and PI control. A. Complete. B. Enlarged.

Figure 4 gives results over a wider range of conversions: 90% to 70%. All of the reactors are openloop unstable. For conversions lower than 85%, the two positive poles becomes real (not complex conjugates). Jacket temperatures become lower and cooling water flow-rates become larger as conversion is decreased. Figure 4C gives plots of the ultimate frequency and the minimum and maximum gains. As conversion decreases, the difference between these gains shrinks, indicating increasing difficulty in control. Figure 5 gives Nyquist plots for these reactors. The distance on the negative real axis between the two points of intersection decreases as conversion decreases. This implies that control will be more difficult. Figure 6 gives the Nyquist plots for the 70% conversion case for P and PI controllers. Figure 6A shows that the curve for P control starts at a phase angle of zero, while for PI control the curves start a -90° due to the integrator. Figure 6B gives an enlargement of the important region near the (-1, 0) point. The controller gain is set at half

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case, ZN and TL controller settings were calculated. Various disturbances were then imposed on the process.

Figure 7. Closedloop log modulus plots with PI control.

of the ultimate gain for the P controller. The two curves for the PI controllers use the ZN and TL settings. The ZN tuning curve is closer to the (-1, 0) point, so it is expected to have a lower closedloop damping coefficient. This is also indicated in Figure 7 by the closedloop log modulus plots for two designs (70% and 90% conversions) using PI controllers with ZN and TL settings. The peaks for ZN tuning are quite high (+23 dB for 90% conversion and +34 dB for 70%). For TL tuning, the corresponding peaks are +10 dB and +11 dB, indicating more robust controller tuning. Dynamic Simulation Results To confirm these results, dynamic simulations of the nonlinear reactor were made for reactors designed at the 70% and at the 90% conversion levels. For each

Figure 8. Increase in feed flow-rate.

Figures 8-11 show the results. A 25% step increase in feed flow-rate is made at time equal zero in Figure 8. Responses of the 90% conversion design and the 70% conversion design are shown. The ZN tuning gives tighter temperature control, but at the expense of more oscillatory responses. The maximum deviation in temperature is larger for the 70% conversion case. Note the initial decrease in reactor temperature. This is due to the sensible heat of the cold feed. In Figure 9 the feed flow-rate is decreased by 25%. Now the ZN tuning becomes more oscillatory and produces closedloop instability for the 70% conversion case. This shows the superior robustness of the TL settings. Figure 10 gives results for the 70% conversion case when changes in the temperature controller setpoint are made: increased from 140 to 145 °F and decreased from 140 to 135 °F. Both ZN and TL tunings give stable control, but the ZN response is more oscillatory. Figure 11 shows are series of runs for the 70% conversion design in which the disturbance is a change in the overall heat transfer coefficient. The design value is 150 Btu/h ft2 °F. As this parameter is decreased, lower jacket temperatures and higher cooling water flow-rates are required. When U drops to 120 Btu/h ft2 °F, the process becomes closedloop unstable for both settings, but the TL settings produce a drastic runaway to very high temperatures even though the cooling water valve saturates wide open. This does not happen for the ZN settings because the tighter control prevents the

4328 Ind. Eng. Chem. Res., Vol. 37, No. 11, 1998

Figure 9. Decrease in feed flow-rate.

Figure 10. Changes in temperature setpoint.

reactor temperature from “going over the cliff” into the closedloop unstable region. Use of Literature Tuning Methods To use the tuning procedures proposed in the literature by Poulin and Pomerleau (1996), Venkatashankar and Chidambaram (1994), Shafiei and Shenton (1994), Rotstein and Lewin (1991), and DePaor and O’Malley (1989), we must have a transfer function model consisting of a deadtime, gain, and positive pole. The question is how do we obtain these parameters from either a real operating reactor or a nonlinear model? When a Model Is Available. If we have a model of the reactor, as we have in the example considered in this paper, we could linearize it and generate the open-

loop Nyquist plots. In comparing Figure 1 for the deadtime/positive-pole model with Figure 5B for the CSTR reactor model, we see that the high-frequency portions of the curves are similar. We could determine the frequencies (ωmin and ωu) and magnitudes (|GM(iωmin)| and |GM(iωu)|) where the model curve makes the two intersections with the negative real axis. We could then set the process gain in the approximate model equal to

Kp ) |GM(iωmin)|

(17)

The values of the two other parameters in the approximate model (D and τo) then could be calculated so that the values for the ultimate gain and ultimate frequency are achieved.

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Figure 11. Changes in heat-transfer coefficient. A. U ) 135. B. U ) 130. C. U ) 125. D. U ) 120.

As an example, let us consider the 70% conversion case. We want to find an approximate model, having the form

GM(s) )

Kpe-Ds τos - 1

(18)

that fits the two intersections of the negative real axis. Figure 6B gives the Nyquist plot from the linear reactor model when a P controller is used with a gain equal to half the ultimate gain. The ultimate gain is Ku ) 4.53 and the ultimate frequency is ωu ) 18.7 radians/h. The curve first crosses the negative real axis at about -5 when Kc ) 2.26. So the gain of the approximate transfer function is calculated from eq 17 to be Kp ) 5/2.26 ) 2.21.

Next, we calculate the values of D and τo from the known values of Ku and ωu. At the ultimate frequency, the phase angle is equal to -180° and the magnitude is equal to the reciprocal of the ultimate gain. Since the phase angle of a positive pole is -π + arctan(ωτo), the phase angle of the transfer function given in eq 18 at the ultimate frequency is

arg G(iωu) ) -ωuD - π + arctan(ωuτo) ) -π

(19)

The magnitude of the transfer function given in eq 18 is

|GM(iωu)| )

Kp

x1 + (ωuτo)2

)

1 Ku

(20)

4330 Ind. Eng. Chem. Res., Vol. 37, No. 11, 1998

Figure 12. Comparison of several tuning rules.

Since the values of Kp, Ku, and ωu are known, Equations 19 and 20 can be solved for the two unknowns D and τo.

D)

arctanxKuKP - 1 ωu

τo )

xKuKP - 1 ωu

(21)

(22)

For the numerical example, the calculated values are D ) 0.0668 and τo ) 0.1606. We now have an approximate model, so the methods proposed in the literature can be tested. Two methods are evaluated below.

Figure 13. Effect of measurement lag.

(1) Poulin and Pomerleau (PP): Poulin and Pomerleau (1996) propose a method that requires the selection of a maximum closedloop log modulus peak (Mr in their nomenclature). We found that the value of Mr had to be greater than 10 to yield positive values of reset time. A value of +15 dB was chosen, giving the controller tuning constants KPP ) 1.76 and τPP ) 0.975. (2) DePaor and O’Malley (DO): DePaor and O’Malley (1989) present a figure (their Figure 7) that gives the controller gain and reset time for a given ratio of dead time to time constant. For our numerical example, this ratio is D/τo ) 0.416, giving suggested tuning constant KDO ) 0.905 and τDO ) 0.535. Figure 12 compares four tuning methods (ZN, TL, PP, and DO) for the 70% conversion case with a 20% decrease in feed flow-rate. As we saw earlier, the ZN tuning gives unstable response. The TL and PP tuning results are almost identical. The DO tuning gives a large maximum error in temperature. When A Model Is Not Available. If we do not have a linear model, we must rely on experimental data. The ultimate frequency and ultimate gain can be obtained from a closedloop relay-feedback test. However, we also need to find the point where the Nyquist plot first crosses the negative real axis at the lower frequencies so that the approximate value of Kp can be calculated. It is important to remember that the process is openloop unstable; therefore, openloop testing methods cannot be used. One approach to this problem is outlined below. (1) After conducting the relay-feedback test, set the controller gain at half of the ultimate. (2) Select a frequency that is about half of the ultimate frequency and use direct sine-wave testing with the setpoint of the temperature controller as the sinusoidal input. The results are a magnitude and phase angle of the closedloop system (Lc ) closedloop log modulus and θc ) closedloop phase angle) for the transfer function between the controlled variable and the setpoint (the closedloop servo transfer function).

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Y(s) Y(s)setGM(s)GC(s) 1 + GM(s)GC(s)

(23)

(3) Back calculate the phase angle and the magnitude of the openloop transfer function GM(s) (Lo ) openloop log modulus and θo ) openloop phase angle). This is easily done graphically by using a Nichols chart, or it can be accomplished numerically by using an iterative procedure. (4) If the openloop phase angle is -180°, use the openloop magnitude to calculate Kp. If the openloop phase angle is greater than -180°, decrease the frequency and repeat the sine-wave testing. If the phase angle is less than -180°, increase the frequency. This iterative procedure is time-consuming and subject to the usual errors of experimental testing. It should be remembered that the TL method does not require that an approximate transfer function model be generated. Therefore the use of TL tuning is recommended. Effect of Measurement Lag The important parameter that strongly affects the ultimate gain is the measurement time lag. To demonstrate this, the 70% conversion case was explored with several values of the measurement lag τm. Figure 13 shows how the Nyquist curves and the closedloop log modulus plots change as τm changes from 1 to 3 min. The case considered is the 70% conversion design, and TL settings are used in a PI controller for the Lc plots. It becomes more difficult to achieve closedloop stability as τm increases. Conclusion The differences between the model used in the literature tuning methods and the model of a jacket-cooled CSTR have been demonstrated. All of the papers dealing with openloop unstable systems assume the openloop process can be modeled by a transfer function consisting of a positive pole and dead time. However, chemical reactors typically have transfer functions that are not that simple. This makes the application of the literature-recommended tuning methods difficult because an approximate model of the prescribed form must be developed from either fundamental models, which have been linearized, or from experimental data, which can only be obtained from closedloop tests because of the openloop instability of the system. The important impact of conversion on openloop stability has been discussed. For a given temperature, low conversions yield small reactors with small heattransfer areas, resulting in more difficult temperature control. Reactors operating with low conversions also feature significant concentrations of reactants, which can provide fuel for reactor runaways. The TL tuning method has been shown to give more robust control of jacket-cooled CSTRs than the ZN method. The only information necessary can be obtained from a relayfeedback test. Nomenclature aij ) constant coefficient in linear model A ) reactant component AH ) heat-transfer area B ) product component Bj ) constant coefficient in linear model

b3 ) constant coefficient in linear model cp ) heat capacity of reaction mass cJ ) heat capacity of cooling water D ) deadtime or reactor diameter E ) activation energy F ) feed flow-rate FJ ) cooling jacket flow-rate GC ) feedback controller transfer function GM ) process transfer function k ) specific reaction rate Kc ) controller gain Kmax, Kmin ) maximum and minimum controller gains Kp ) process gain Ku ) ultimate controller gain L ) reactor length Lc, Lo ) closedloop and openloop log moduli M ) molecular weight of reactant and product Q ) heat transfer rate R ) overall reaction rate s ) Laplace transform variable t ) time T ) reactor temperature Tset ) temperature setpoint TJ ) jacket temperature TJ0 ) inlet cooling water temperature T0 ) feed temperature U ) overall heat-transfer coefficient VJ ) jacket volume VR ) reactor holdup Y ) process output Yset ) setpoint z ) mole fraction component A in reactor z0 ) mole fraction component A in feed λ ) heat of reaction F ) density of reaction mass FJ ) density of cooling water θo, θc ) openloop and closedloop phase angle τI ) reset or integral time constant τm ) measurement lag time constant τo ) process openloop time constant χ ) conversion ω ) frequency (radians/h)

Literature Cited DePaor, A. M.; O’Malley, M. Controllers of Ziegler-Nichols type for unstable process with time delay. Int. J. Control 1989, 49 (4), 1273-1284. Luyben, W. L. Trade-offs between design and control in chemical reactor systems. J. Proc. Control 1993, 3 (1), 17-41. Poulin, E.; Pomerleau, A. PID tuning for integrating and unstable processes. IEE Proc. Control Theory Appl. 1996, 143 (5), 429435. Rotstein, G. E.; Lewin, D. R. Simple PI and PID tuning for openloop unstable systems. Ind. Eng. Chem. Res. 1991, 30, 18641869. Shafiei, Z.; Shenton, A. T. Tuning of PID-type controllers for stable and unstable systems with time delay. Automatica 1994, 30 (10), 1609-1615. Tyreus, B. D.; Luyben, W. L. Tuning of PI controllers for integrator/dead time processes. Ind. Eng. Chem. Res. 1992, 31, 26252628. Venkatashankar, V.; Chidambaram, M. Design of P and PI controllers for unstable first-order plus time delay systems. Int. J. Control 1994, 60 (1), 137-144.

Received for review February 4, 1998 Revised manuscript received July 3, 1998 Accepted July 5, 1998 IE980078C