Ind. Eng. Chem. Res. 1999, 38, 4337-4344
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Adaptive Temperature Control of Multiproduct Jacketed Reactors Dwayne Tyner and Masoud Soroush* Department of Chemical Engineering, Drexel University, Philadelphia, Pennsylvania 19104
Michael C. Grady DuPont Marshall Laboratory, 3401 Grays Ferry Avenue, Philadelphia, Pennsylvania 19146
This paper presents a new adaptive cascade temperature control system for jacketed stirred tank reactors in which multiple products are produced. The adaptive control system consists of two cascade input-output linearizing controllers and an inversion-based parameter estimator that estimates the rate of heat production by reactions and the overall jacket-reactor heattransfer coefficient. Without a need for retuning, it can provide tight temperature control in highly exothermic, multiproduct, jacketed, stirred-tank reactors in which the reaction rate and the overall heat-transfer coefficient are unknown and vary significantly. The application and performance of the adaptive cascade control system are shown and compared with those of a proportional-integral (PI) cascade control system by an exothermic chemical reactor with varying overall heat-transfer coefficient. Compared to a PI cascade control system, the adaptive cascade control system exhibits better performance and is much easier to tune. 1. Introduction In the chemical/petrochemical industry, a single reactor is sometimes used for the production of many different types of products. For example, a batch polymerization reactor is often used for the production of different grades of a polymer or different polymers.1 The type and composition of the reactants and products in the multiproduct reactors usually influence strongly the dynamics of the reactors, making their effective control difficult. Model-based control is capable of providing an effective control, but it requires a sufficiently accurate process model, which is usually not available. A solution to this model problem is to use a generic first-principles model with unknown parameters to be estimated online, leading to the synthesis of an adaptive model-based controller. Such a controller can provide the tight control in multiproduct reactors without a need for retuning. Research in adaptive control started in the early 1950s when there was a need for better control of highperformance aircraft.2 Adaptive control was proposed as a way of automatically adjusting the controller parameters in the face of changing aircraft dynamics. While the progress in this area was initially very slow, during the past decade, adaptive control theory has made significant advances and many applications have been reported in chemical engineering, mechanical engineering, robotics, bioengineering, and so on.3-12 Control of such processes has also been studied in the context of robust control of processes with time-varying uncertain variables.13,14 For tight temperature control and reliable monitoring of extremely exothermic chemical reactors, such as bulk polymerization reactors, accurate information on the reaction-heat-production rate and the overall heattransfer coefficient of the reactor/cooling equipment is essential.11,15-18 There are no sensors that can measure these important parameters directly. The rate of heat of reaction strongly depends on the type and purity of * To whom all correspondence should be addressed.
reactants, and the overall heat transfer coefficient falls with fouling and an increase in the viscosity of the reacting mixture. Features such as these have motivated a great number of studies on on-line estimation of these parameters.8,10-12,19-25 Methods used to estimate the parameters on-line include the following: (i) Parameter estimation via state estimation. This method requires a dynamic model for each of the unknown parameters to be estimated. Simple parameter models such as “random walk” and “random ramp” are usually used. Once appropriate parameter models are chosen, a state estimator, such as an extended Kalman filter20,26,27 or a reduced-order observer,25 is used to estimate the process parameters, which appear as a subset of the state variables of the combined process and parameter models. (ii) Parameter estimation via dynamic model inversion.24 This parameter estimator includes a left inverse of the process model and at each time instant calculates least-squared error estimates of parameters by using readily available on-line measurements. Adaptive temperature control of stirred tank chemical reactors has been studied extensively.7,8,10-12,18,21 The control methods proposed and/or used all calculate the manipulated input on the basis of estimates of rate of heat production by reactions and/or the overall heattransfer coefficient. To obtain the estimates, Regnier et al.,10 Ni,18 Wright and Kravaris,12 and Defaye et al.29 used very simple models such random walk, ClarkePringle and MacGregor11 used simple models, and Soroush and Kravaris28 used very detailed models of the parameters. In this paper, we present an adaptive cascade temperature control system for jacketed stirred tank reactors. The control system consists of two cascade inputoutput linearizing controllers and an inversion-based parameter estimator.24 The application and performance of the adaptive cascade control system are shown and compared with those of a PI cascade control system by an exothermic chemical reactor with varying overall heat-transfer coefficient.
10.1021/ie990160n CCC: $18.00 © 1999 American Chemical Society Published on Web 09/30/1999
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solved analytically, which leads to an algebraic equation describing the dependence of T ˜ j on Tj. (v) The heat capacities of the reacting mixture and the jacket fluid do not change significantly with temperature. (vi) The mass of the reacting mixture inside the reactor is constant. (vii) The mass of the jacket fluid in the jacket circulation pipe is constant. For such a reactor system, energy balances for the jacket circulation pipe and the reactor, and a shell energy balance for the reactor jacket yield a mathematical model of the form
T˙ ) p1 + R1(Tj - T)[1 - exp(-R3p2)] + R2 T˙ j ) R4(T - Tj)[1 - exp(-R3p2)] + R5Q T ˜ j ) T + (Tj - T) exp(-R3p2)
}
(1)
where
R1 )
Figure 1. Schematic of the reactor system.
The temperature control problem is described first. The adaptive cascade control system is then synthesized. Finally, the application and performance of the proposed adaptive cascade control system are shown and compared with those of a PI cascade control system. 2. Cascade Temperature Control Problem We consider jacketed stirred tank reactors with the schematic shown in Figure 1. The jacket fluid enters the reactor jacket at a temperature of Tj and leaves the reactor jacket at a temperature of T ˜ j. The temperatures of the reactor inlet and outlet streams are denoted by Ti and T, respectively. The jacket and reactor temperatures are measured at very high sampling frequencies and with almost no time delays. The control problem is to maintain the reactor temperature at a desired value by adjusting the net rate of sensible energy added to the jacket by the cold and hot inlet jacket fluid streams ˜ j) + wc(Tc - T ˜ j)]). Once (i.e. by adjusting Q ) c0[wh(Th - T the net rate of the sensible energy needed is calculated by a controller, one can use coordination rules to calculate the required flow rates of the hot and cold inlet jacket fluid streams (wc and wh). For the purpose of developing a reasonably accurate and simple mathematical model of the processes, we make the following standard assumptions: (i) The reacting mixture inside the reactor is wellmixed. (ii) The jacket fluid inside the reactor jacket is not well-mixed. (iii) There is no heat transfer from the jacket fluid to the surrounding. (iv) The velocity of the heating/cooling fluid inside the jacket is very high, so that a quasi-steady-state assumption holds. This assumption allows one to simplify the partial differential equation that governs the temperature of the jacket fluid inside the jacket to an ordinary differential equation with position as the independent variable. The ordinary differential equation can be
R4 )
w0c0 wrcr(Ti - T) m0 , R2 ) , R3 ) mrcr + msrcs mrcr + msrcs w0
w0c0 1 , R5 ) , m0c0 + msjcs m0c0 + msjcs Q ) c0[wh(Th - T ˜ j) + wc(Tc - T ˜ j)] p1 )
RH V r US , p2 ) mrcr + msrcs m0c0
Approximate values of the parameters R1, ..., R5 can usually be obtained easily. On the other hand, the values of the parameters p1 and p2 vary (often significantly) with time and cannot be measured. The upper and lower bounds on the net rate of sensible energy given to the jacket fluid by the inlet jacket fluid streams (Q) will be denoted by Qh and Ql, respectively. The upper and lower bounds on the inlet jacket temperature (Tj) will be represented by Tjh andTjl, respectively. As we will see in the Example section, Ql and Qh can be expressed in terms of whh and wcl, the upper limits on the flow rates of the inlet jacket fluid streams. 3. Adaptive Cascade Control System The adaptive control system consists of two cascade input-output linearizing controllers and an inversionbased parameter estimator that estimates the rate of heat production by the reactions and the overall jacketreactor heat-transfer coefficient. In what follows, we first synthesize the two input-output linearizing controllers by assuming that the values of the model parameters p1 and p2 are known, and we then design an inversion-based parameter estimator to estimate these two parameters that are indeed unknown. Simultaneous use of the cascade control system and the parameter estimator leads to an adaptive cascade control system. 3.1. Cascade Control System. The cascade control system consists of two input-output linearizing controllers. One (master controller) induces a desired response to the reactor temperature by manipulating the set point of the inlet jacket temperature Tj, and the other (slave controller) induces a desired response to the inlet jacket temperature by manipulating the net rate of sensible energy to the jacket fluid Q.
Ind. Eng. Chem. Res., Vol. 38, No. 11, 1999 4339
3.1.1. Slave Controller Synthesis. According to the process model of eq 1, the relative order (degree) of the inlet jacket temperature Tj with respect to the net rate of sensible energy to the jacket fluid Q is 1 (rTj-Q ) 1): that is, the first time-derivative of Tj depends explicitly on Q. We request the relation between Tj and its set point Tjsp in closed-loop to be governed by the following linear differential equation of order rTj-Q ) 1:
γ2T˙ j + Tj ) Tjsp
(2)
We add integral action to the preceding state feedback and obtain the following master controller:30 T - η1 + p1 + R1(Tjsp - T)[1 - exp(-R3p2)] +R2 γ1 η1 + Tsp - 2T - γ1[p1 - R1T[1 - exp(-R3p2)] + R2] Tjsp ) satTj γ1R1[1 - exp(-R3p2)]
η˘ 1 )
{
}
(8) where
{
where γ2 > 0 is a tunable parameter. Substituting for T˙ j from the model of eq 1 and solving for Q leads to the state feedback
Q)
Tjsp - Tj - γ2R4(T - Tj)[1 - exp(-R3p2)]
(3)
γ2R5
We then add integral action to the preceding state feedback and obtain the following slave controller30 Tj - η2 + R4(T - Tj)[1 - exp(-R3p2)] +R5Q γ2 η2 + Tjsp - 2Tj - γ2R4(T - Tj)[1 - exp(-R3p2)] Q ) satQ γ2R5
η˘ 2 )
{
}
}
(4) where
η˘ 1 )
T - η1 + p1 + R1(Tjsp - T)[1 - exp(-R3p2)] + R2 γ1
η˘ 2 )
Tj - η2 + R4(T - Tj)[1 - exp(-R3p2)] + R5Q γ2
{
}
η1 + Tsp - 2T - γ1[p1 - R1T[1 - exp(-R3p2)] + R2] γ1R1[1 - exp(-R3p2)]
{
{
γ1T˙ + T ) Tsp
}
η2 + Tjsp - 2Tj - γ2R4(T - Tj)[1 - exp(-R3p2)] γ2R5
}
(9)
(5)
where γ1 > 0 is a tunable parameter. Substituting for T˙ from the model of eq 1 and solving for the inlet jacket temperature Tj leads to the state feedback
Tsp - T - γ1[p1 - R1T[1 - exp(-R3p2)] + R2] γ1R1[1 - exp(-R3p2)] (6)
If the slave controller tunable parameter γ2 is chosen to be sufficiently small, then under the slave controller of (4) the inlet jacket temperature Tj will be approximately equal to its set point Tjsp. Thus, under this condition one can take the jacket temperature calculated by (6) as the jacket temperature set point, that is,
Tjsp )
The preceding controller is also a nonlinear model-based PI controller that inherently includes an optimal windup compensator. 3.1.3. Cascade Control System. The slave and master controllers of eqs 5 and 8 together form the input-output linearizing cascade control system:
Q ) satQ
The preceding controller is a nonlinear model-based PI controller that inherently includes an optimal windup compensator. 3.1.2. Master Controller Synthesis. According to the process model of eq 1, the relative order (degree) of the reactor temperature T with respect to the inlet jacket temperature Tj is 1 (rT-Tj ) 1); that is, the first time-derivative of T depends explicitly on Tj. We request the relation between T and its set point Tsp in closedloop to be governed by the following linear differential equation of order rT-Tj ) 1:
Tj )
Tjl, w < Tjl satTj[w] } w, Tjl e w e Tjh, Tjh, w > Tjh
Tjsp ) satTj
Ql, w < Ql satQ[w] } w, Ql e w e Qh Qh, w > Qh
}
Tsp - T - γ1[p1 - R1T[1 - exp(-R3p2)] + R2] γ1R1[1 - exp(-R3p2)] (7)
3.2. Parameter Estimator. To implement the preceding input-output linearizing cascade control system, we need to estimate the values of the two unknown parameters p1 and p2. The parameter p2 can easily be estimated from the temperature measurements by using the algebraic equation in eq 1; that is,
pˆ 2 )
(
)
Tj - T 1 ln R3 T ˜j - T
To estimate the parameter p1, we use the inversionbased parameter estimator presented in ref 24, leading to the following parameter estimator:
T-T ˆ T ˆ˙ ) β1
(
)
pˆ 2 )
Tj - T 1 ln R3 T ˜j - T
pˆ 1 )
T-T ˆ - β1{R1(Tj - T ˆ )[1 - exp(-R3p2)] + Rˆ 2} β1
}
(10) where β1 > 0 is the estimator tunable parameter and
Rˆ 2 )
wrcr (T - T ˆ) mrcr + msrcs i
Note that T ˆ is the output of a first-order low-pass filter whose input is T. Theoretically the smaller the value of β1, the more accurate the estimate of p1. However, in
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The tunable parameter γ1 should be set in the order of the average time constant of the first differential equation of the reactor model of (1). The tunable parameter γ2 is proposed to be set about 10 times smaller than γ2. Theoretically, the smaller the value of β1, the more accurate the estimate of p1. However, in the cases in which the standard deviation of the noise component of the temperature measurement T is high, a higher value of β1 should be used. 4. Application to a Chemical Reactor Example
Figure 2. Parametrization of the adaptive control system.
the cases in which the standard deviation of the noise component of the temperature measurement T is high, a higher value of β1 is desirable to filter the noise component more effectively. 3.3. Adaptive Cascade Control System. The inputoutput linearizing cascade control system of eq 9 together with the parameter estimator of eq 10 forms the following adaptive cascade control system: T-T ˆ T ˆ˙ ) β1 η˘ 1 )
T - η1 + pˆ 1 + R1(Tjsp - T)[1 - exp(-R3pˆ 2)] + R2 γ1
η˘ 2 )
Tj - η2 + R4(T - Tj)[1 - exp(-R3pˆ 2)] + R5Q γ2
Tjsp ) satTj
{
{
Q ) satQ
}
η1 + Tsp - 2T - γ1[pˆ 1 - R1T[1 - exp(-R3pˆ 2)] + R2] γ1R1[1 - exp(-R3pˆ 2)]
}
η2 + Tjsp - 2Tj - γ2R4(T - Tj)[1 - exp(-R3pˆ 2)] γ2R5
( )
pˆ 2 )
Tj - T 1 ln R3 T ˜j - T
pˆ 1 )
T-T ˆ - β1{R1(Tj - T ˆ )[1 - exp(-R3pˆ 2)] + Rˆ 2} β1
} }
(11)
To show the application and performance of the derived adaptive control system, as an example, we consider a continuous stirred tank reactor of the form depicted in Figure 1, in which the following irreversible parallel reactions take place: k1
A 98 U1 k2
A 98 U2 kP
A 98 P where P is the desired product and U1 and U2 are the undesirable side products. The reactor model has the form
C˙ A ) -k1CA3 - k2CA0.5 - kPCA + C˙ P ) kPCA -
[
T(k) η1(k + 1) ) λ1η1(k) +[1 - λ1] + pˆ 1(k) + R1[Tjsp(k) - T(k)]θ(k) +R2(k) γ1 η2(k + 1) ) λ2η2(k) +[1 - λ2]
{
Tjsp(k) ) satTj
{
Q(k) ) satQ θ(k) )
[
Tj(k) γ2
+ R4[T(k) - Tj(k)]θ(k) +R5Q(k)
γ1R1θ(k)
}
η2(k) + Tjsp(k) - 2Tj(k) - γ2R4[T(k) - Tj(k)]θ(k) γ2R5
˜ j(k) Tj(k) - T Tj(k) - T(k)
pˆ 1(k) )
]
}
η1(k) + Tsp(k) - 2T(k) - γ1[pˆ 1(k) - R1T(k) θ(k) +R2(k) ]
ˆ (k)]θ(k) + Rˆ 2(k)} T(k) - T ˆ (k) - β1{R1[Tj(k) - T β1
]
(12) with T ˆ (0) ) T(0), η1(0) ) T(0), and η2(0) ) Tj(0), where λ0 ) exp(-∆t/β1), λ1 ) exp(-∆t/γ1), and λ2 ) exp(-∆t/ γ2).
τ
CP τ
(H1k1CA3 + H2k2CA0.5+ HPkPCA)Vr + T˙ ) mrcr + msrcs w0c0 wrcr(Ti - T) + (T - T) mrcr + msrcs mrcr + msrcs j
A parametrization of the adaptive control system of (11) is depicted in Figure 2. The preceding dynamic system is linear in its states, and thus it can be time-discretized exactly. The corresponding discrete-time adaptive cascade control system has the following form: T ˆ (k +1) ) λ0T ˆ (k) +[1 - λ0]T(k)
CAi - CA
[
(
US w0c0
1 - exp -
T˙ j )
[
)]
(
w0c0 US (T - Tj) 1 - exp m0c0 + msjcs w0c0
)]
(13)
+
c0[wh(Th - T ˜ j) + wc(Tc - T ˜ j)] m0c0 + msjcs
(
T ˜ j ) T + (Tj - T) exp -
)
US w0c0
with the reaction rate constants k1 ) z1 exp(-E1/(RT)), k2 ) z2 exp(-E2/(RT)), and kp ) zp exp(-Ep/(RT)). In this reactor, the reactor-jacket overall heat-transfer coefficient is assumed to decrease linearly with the product concentration according to
U ) U0(1 - 0.15CP) where U0 is the value of the reactor-jacket overall heattransfer coefficient when the concentration of the product in the reactor is zero. This decrease in the overall
}
Ind. Eng. Chem. Res., Vol. 38, No. 11, 1999 4341 Table 1. Parameter Values and Operating Conditions of the Chemical Reactor Example c0 ) 1.75 kJ‚kg-1‚K-1 cr )2.00 kJ‚kg-1‚K-1 cs )0.50 kJ‚kg-1‚K-1 m0 ) 5 kg mr )24 kg msr ) 650 kg msj ) 50 kg w0 ) 0.5 kg‚s-1 wch ) 0.5 kg‚s-1 whh ) 0.5 kg‚s-1 wr ) 0.08 kg‚s-1 τ ) 300 s Ti ) 295.2 K CAi ) 10.0 kmol‚m-3 Vr ) 0.0378 m3 U0 ) 0.5 KJ.m-2‚K-1‚s-1 S ) 2.0 m2 Tc ) 283.2 K Th ) 473.2 K T(0) ) 295.2 K Tj(0) ) 295.2 K CA(0) ) 0.1 kmol‚m-3 CP(0) ) 0.0 kmol‚m-3
heat-transfer coefficient is quite common in many reactors wherein generation of viscous products results in a decrease in the overall heat-transfer coefficient.16,25 The values of the rest of the reactor parameters except for those of the reaction frequency factors and activation energies are given in Table 1. The values of the reaction frequency factors and activation energies are given in ref 28. Steady-state analysis of the reactor shows that, at a steady-state temperature of 400 K, the steady-state concentration of the desirable product is maximum.28 We use the adaptive cascade control system to maintain the reactor temperature at 400 K by manipulating the net rate of sensible energy to the jacket fluid (Q). The dynamic system of eq 13 is used to represent the actual process. We assume that measurements of the temperatures are available every 5 s. An integration step size of 5 s is used in the numerical integration of the differential equations of the model. 4.1. Coordination Rules. Once the net rate of sensible energy Q is calculated by a controller, one can calculate the corresponding flow rates of the inlet jacket fluid streams (wc and wh) according to the following coordination rules:
{ [
Q < Ql Q , Ql e Q e 0 wc ) c0(Tc - T ˜ j) 0, 0
