Ind. Eng. Chem. Fundam. Kirschbaum, F.. "Distillation and Rectification", Chemical Publishing Co., Inc., New York, 1948. Koch Engineering Co., "Koch Sulzer Rectification Columns", Bulletin KS-1, Wichita, KanL Kraybill, R. R., Ph.D. Dissertation, University of Michigan, 1953. Lamourelle, A. P., Sandall, 0. C., Chem. f n g . Sci., 27, 1035 (1972). M e r , W. D., Stoecker, W. D., Weinstein, B., Chem. Eng. Prcg.,71 (Nov 1977). Miller, E. G.,S.B. Thesis in Chemical Engineering, University of Delaware, 1948. Nusselt. W.. 2. Ver. Deut. Ina.. 60. 541 (1916). Portalski, S:, Chem. Eng. Sci: IS,787 (1$63).' Schoenborn, E. M.. Dougherty, W. J., Trans. AIChE, 51, 40 (1944)
389
1980, 19, 389-396
Sherwood, T. K., Holloway, F. A. L., Trans. AIChE, 36, 39 (1940). Shulman, H. L., Ullrich, C. F., Wells, N., Proulx, A. Z.,AIChE J . , 1, 259 (1955). Silvey, F. C., Keller, G. J., Proc. Int. Symp. Dist. (Brighton),Ind. Chem. Eng. Symp. Ser., London, No. 32, 4, 18 (1969). Yanagi, T., paper presented at 1975 International Chemical Plant Engineering Congress, Tokyo, 1975. Yoshida, T., Tanaka, T.. Ind. Eng. Chem., 43, 1467 (1951).
Received f o r review March 3, 1980 Accepted August 4, 1980
Control of Temperature Peaks in Adiabatic Fixed-Bed Tubular Reactors Gerhard K. Giger,' Rajakkannu Mutharasan, and Donald R. Coughanowr Department of Chemical Engineering, Drexei University, Philadelphia, Pennsylvania 19 104
Catalytic fixed-bed tubular reactors can undergo severe temperature excursions, particularly when there is a drop in feed temperature. Several control strategies have been studied which attenuate temperature peaks that would occur in an uncontrolled reactor. A practical strategy is recommended which requires feeding a fraction of the reactants at an intermediate point along the reactor bed. A cantrol algorithm which enables the calculation of the secondary flow is proposed and its performance characteristics are evaluated by dynamic simulations. The proposed method is very effective in attenuating the temperature peaks, thus preventing catalyst deactivation and structural damage to the reactor.
Introduction Temperature control of fixed-bed tubular reactors, which has been the subject of several investigators, presents several complex problems involving temperature and composition wave phenomena. Reactant gases have low thermal capacity compared with the catalyst bed. When the feed temperature decreases, the catalyst a t the entrance which is a t a temperature higher than the feed is cooled by the incoming gas, thus causing less reaction to take place in the entrance section of the reactor. As the reactants move further down the reactor, they gain heat as they come in contact with the hot catalyst bed. Since the gas stream has a higher concentration of reactants (because in the entrance section of the bed less reaction took place as compared to the initial steady state) more reaction takes place than occurred during the initial steady state. Thus, for an exothermic reaction, heat is released in an amount greater than that for the initial steady state and causes the occurrence of a temperature peak. For adiabatic reactors, the peak of the temperature wave increases as the wave moves down the reactor. When external cooling is used for the entire section of the reactor, the peak of the temperature wave can be attenuated to some extent; however, total elimination of the wave is difficult. The magnitude of the temperature peak is a function of heat of reaction, thermal capacities of the bed and the reactant gases, and the interphase heat transfer coefficient between the reactant gas apd the catalyst bed. In several industrial reactors, such temperature peaks are high enough to deactivate the catalyst or cause structural damage to the reactor. In industrial situations, the problem of high temperature peaks is handled by stoppage of reactant flow to the reactor. Current industrial practice Ciba-Geigy, Basel, Switzerland. 0196-4313/80/1019-0389$01.OO/O
is overly cautious and in this paper we present several alternative control strategies which do not require reactor shut-down. The main objective of this paper is to present a practical solution to the important problem of control of temperature peaks, with special emphasis on hydrodealkylation reactors. Several papers have appeared in the past on the temperature control of packed bed reactors. Most of these publications treat the design of control algorithms to mqintain exit temperature and/or concentration a t some desired level. It should be pointed out that controlling the exit temperature need not imply that the temperature within the reactor is maintained within safe limits. For example, in the paper by Strangeland and Foss (1970), their algorithm controls the outlet temperature very closely, but large temperature peaks occur within the reactor. In this paper, the primary goal is to solve the problem of controlling the temperature peaks which occur within the reactor. Previous Literature Dynamics of packed bed heat regenerators were first analyzed by Anzelius (1926) and were later extended to packed bed reactors by Brinkley (1947). Amundson (1956) also studied the dynamics of a packed adiabatic reactor in which radial gradients were included. The temperature and concentration interactions were first included in a model for systematic study of the dynamics of such a reactor by Bilous and Amundson (1956). Strangeland and Foss (1970) used the heterogeneous plug-flow reactor model to simulate the control of an adiabatic reactor. They linearized the equations describing the dynamics about the desired steady state to obtain tbe transfer functions relating the deviations in composition and temperature to the inlet condition of the reactor. They also investigated the feasibility of controlling the outlet 0 1980 American Chemical Society
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Ind. Eng. Chem. Fundam., Vol. 19, No. 4, 1980
state of the reactor through temperature manipulation of a secondary feed stream. Michelson et al. (1973) employed the orthogonal collocation method to derive the state space formulation from the dynamic equations to which Vakil et al. (1973) applied the conventional procedures of Kalman filtering and optimal control to study the feedforward scheme suggested earlier by Strangeland and Foss (1970). Kardos and Stevens (1971) presented a method for controlling packed bed reactors through sectional external cooling. Georgakis et al. (1977a,b,c) studied the control of a homogeneous jacketed tubular reactor with axial dispersion. They successfully designed a Luenberger observer when concentration measurements are not possible. The main emphasis in their work was the stabilization of the unstable state by modal control. Sorenson (1977) applied Kalman filtering and optimal control to an adiabatic packed bed reactor described by a system of linear differential equations which were developed by the application of orthogonal collocation to the original partial differential equations and subsequently linearizing them around the steady state. The method was successfully applied to a pilot plant reactor. Sorenson's method, which involved several measurements of several temperatures, one concentration, and one flowrate, required the manipulation of reactor inlet temperature, reactor inlet concentration, and total flowrate. In a practical situation, one hardly ever has the flexibility of Sorenson's procedure. Pavlica (1970) examined the control of temperature peaks within the reactor through the manipulation of inlet temperature and/or inlet concentration. Although he found that the temperature excursions can be reduced, he was unable to eliminate the peaks unless the flow of reactants in the feed gas was stopped. Control of temperature within the reactor by manipulating inlet concentration may not be applicable in an industrial situation. Since the packed bed reactor is a distributed parameter system, any disturbances in the composition and temperature of the feed propagate through the bed as waves. It has been observed by Boreskov and Slinko (1965), Hoiberg (1969), and Crider and Foss (1966,1968) that the temperature wave and the reactant wave travel at different velocities because of the finite rate of heat transfer between the packing and the reactant fluid. Strangeland and Foss (1970) exploited the interaction between the traveling waves of temperature and concentration to control the outlet temperature and concentration by injecting a secondary reactant stream at an intermediate point along the reactor. In their work, both the concentration and temperature of the secondary stream were used as manipulated variables. However, as they point out, the flowrate of the secondary stream is a more practical manipulated variable. Furthermore, control of temperature of the outlet stream may not mean the control of temperature within the reactor. Since severe temperature excursions within the reactor may cause problems which have been described earlier, the primary emphasis in this paper is to control the temperature within the reactor. The main difference between the approach taken in this paper and those of the previous authors is that the temperature within the reactor is the controlled variable as opposed to the reactor exit temperature. Furthermore, in this paper the nonlinearity of the model is retained as opposed to linearizing, which was resorted to by most of the previous authors.
Reactor Model Based on the assumptions of (a) negligible axial dispersion, (b) negligible intraphase temperature gradients, (c) negligible intraphase and interphase concentration gradients, (d) constant specific heats and heat transfer
coefficient, (e) ideal gas behavior, and (f) no reaction in the vapor phase, one can obtain the following equations to describe the dynamics of an adiabatic packed bed tubular reactor
where the superscript s refers to the solid phase and superscript G refers to the gas phase. The associated boundary conditions are given by at z = 0, t L 0: P = Toc at z = 0, t I0: Ci = Cio (i = 1, 2 ,
..., n,)
a t z L 0, t = 0: T" = Ts,"(z) Application of the method of characteristics to the above equations and the use of the ideal gas law give
(5)
where T and 7 are characteristic time and dimensionless distance. The parameters Us and U , are given by (7)
The parameter U, is the ratio of heat transfer characteristics of the catalyst to its thermal capacity, while U, is the ratio of heat transfer characteristics of the catalyst bed to the heat carried away by the flowing gas. The ratio of Usto U, is proportional to the velocity of the thermal wave during the transient state of the reactor. Dynamic Simulation Procedure Pavlica (1970) made a systematic study of the numerical solution of the model equations presented in the previous section. It was found that no consistent integration procedure was able to give numerical accuracy with an acceptable spatial grid size due to the extreme stiffness of the differential equations. Pavlica (1970) reported a semi-analytical predictor-corrector method which is based on integration of the first two differential equations and evaluation of the resulting integrals by collocation polynomials; such a procedure using an axial grid size of 5 / U, gives excellent accuracy. Consider eq 5, which may be written as follows over the interval 7' I7 I?' + A7 P(7' A?, 7' + AT) = p ( v ' , T' AT)e-"gAv
+
+
+
Ind. Eng. Chem. Fundam., Vol. 19, No. 4, 1980 391 Table I.
Reactor Parameters u, = 0 . 5 8 5 m/s
f = 0.4
$‘ep’=
l o 6 W/(K m 3 )
U0$=1.005 X
l o 4 J/(kmol K )
CpG = 6.237 x
ToG= 8 6 7 K -
reaction 1 Ei
1- A Hri) ‘i
Ci,
~
-
_
reaction 2
3.629 X l o z 2 3.604 X 10’ 9.202 X lo’ k I exp(-E,/RTS)C, 9.337 x 10-3
hi
___-
__ C = 0.794 k m o l / m 3 CH 0.564 k m o l / m 3 4.020 X l o 6 J / ( K in3)
L = 7.01 m
1.112 x 1 0 1 3 2.523 X l o 8 1.394 X l o 8 k exp(- E , / R T’)CH O.’C 4.004 X ~
_
_____II-_.___
units km01)03 J/kmol J /kino1 kmol/(s m 3 ) kmol/m3 m13/(s
cc m
L3
2
3
2
S TO BE CALCULATEC
,
0
.
7-A7 L-
I
KNOWN VALUES
7
7
~(DISTANCE) 7
~ L
~
I
7
DIMEhS ONLFSS L ETol!CE
(G2lC NUMBEQ)
Figure 2. Temperature response of the hydrodealkylation reactor for step decrease in feed gas temperature of magnitude 27.8 K.
Figure 1. Grid structure used in the numerical procedure.
If the function T” (v) is fitted with a second-order polynomial over the domain (7’ - Av,q’ + Av), the above equation can be integrated analytically. A similar procedure may be applied to eq 4 to obtain T”(q’ + Aq). The concentration equations (eq 6) are not stiff and may be integrated using any integration scheme. In this investigation the second-order Adam’s method was used. The grid structure used in the calculations is presented in Figure 1. The characteristic time was taken to be equal to the real time since the slope of the characteristic is very small (Liu and Amundson, 1962). The numerical procedure may be summarized as follows. Procedure. (1)Calculate a first estimate of P(v’+ AT, T’ + AT) using eq 4 from P(v’ + Aq, T’) by the Euler method. (2) Calculate ‘IG(q’ + Aq, T’ + AT)using eq 9. (3) Calculate Ci(q’ + AT, 7’ + A7) using a second-order Adam’s method. (4)The value of P(v’ + Aq, 7’ + A T )is updated using a formulation similar to that of eq 9. (5) Comparison of the earlier and the updated values of T“($ + Aq, T’ A T ) is made against a specified convergence criterion. Calculation is repeated from step 2 if convergence is not reached. In all the calculations a space interval of 0.01 dimensionless unit and a time interval of 20 s were used: Typically, a full transient run required 15 s of CPU on an IBM 370/168 system.
+
Dynamics of the Reactor to Feed Temperature Changes In this study the hydrodealkylation reactor will be considered. The parameters for such a reactor, which are taken from Pavlica (1970), are summarized in Table I. The first reaction is a typical hydrocracking reaction while the second reaction is for the hydrodealkylation of toluene. In a hydrodealkylation reactor, several reactions may take place, but for the purpose of simulation of temperature excursions, only the above two reactions are considered (Pavlica, 1970).
D’MENS ONLESS
STAhCF
I
7
Figure 3. Temperature response of the hydrodealkylation reactor to a ramp decrease in feed gas temperature at the rate of 1.67 K/min. Overall change in feed gas temperature is 27.8 K.
The dynamic behavior of the reactor to a step decrease of 27.8 K in feed temperature is shown in Figure 2. The thermal wave takes about 45 min to traverse the reactor. The highest excursion in temperature, which occurs at the exit of the reactor, is about twice as large as the temperature depression at the inlet. The magnitude of the temperature excursion is mainly a function of the exothermicity of the reaction and the parameter U,. It should be pointed out that the temperature a t a particular location in the reactor initially increases and then decreases and settles to the final steady state. In other words, the initial motion of the process is opposite to the ultimate direction of the process. Such processes have been previously described as inverse response systems (Luyben, 1973). Linear controllers perform poorly for processes exhibiting inverse response. The response of the reactor to a step increase in feed gas temperature is not ELS severe and the magnitude of the temperature undershoot is very small. In Figure 3 the temperature response of the reactor to a ramp change in inlet temperature is shown. The temperature peak is much smaller. In fact, one can show that
392
Ind. Eng. Chem. Fundam., Vol. 19, No. 4, 1980 VIX Y G - 1
REACTANT
d?
REACTOR
+ q T p - -
J?-
L
CZ!NT;13LLEQ
6
Figure 4. Control strategies for temperature control of adiabatic packed bed tubular reactors.
Figure 5. Structure of algorithm A.
the overshoot is a strong function of the rate of change of the feed gas temperature, other things remaining constant. The important fact that must be considered in the design of a control algorithm for such a reactor is that sudden changes in manipulated variable, which may cause a rapid decrease in temperature within the reactor, should be avoided. Control Strategies A simple way to avoid high temperature peaks is to control the feed gas temperature by the use of a heat exchanger, whereby the temperature disturbance of the reactant gas can be kept quite small, thus minimizing excess temperature excursions within the reactor. For a large throughput system such as a hydrodealkylation reactor, a large heat exchanger would be needed, which may be impractical for an existing reactor system. In Figure 4, different control strategies which were considered to prevent excess temperature excursions are illustrated. The first control scheme shown in Figure 4 (top) is to recycle the effluent and mix with the feed gas to control the inlet gas temperature to the reactor. For highly exothermic reactions, the effluent gas is a t a sufficiently higher temperature so that the desired inlet gas temperature is obtained with reasonably small recycle. However, it should be pointed out that the cost of the compressor needed to implement such a control scheme would be prohibitive and hence impractical. The second control strategy which was considered is shown in Figure 4 (middle), where a fraction of the condensed product is injected into the reactor at an intermediate point based upon the measured temperature just prior to the injection location. The quantity of liquid required to be injected for cooling hot gases within the reactor would be small since heat required to vaporize the liquid is large. Furthermore, the operating state of the reactor will not be changed very much from that of the uncontrolled reactor and the size of the pump would be small. However, one would need to modify the mathematical model developed earlier to include the two-phase system dynamics over a short length of the reactor. The model should include dynamics of vaporization and changes of the catalyst activity caused by a liquid film covering the surface of the catalyst. We can simplify the model by considering the reactor to consist of two sections separated by a vaporization section. An estimate of the vaporization time can be made assuming constant diameter droplets and constant heat transfer coefficient. Such a
model should approximate the actual dynamics. Due to uncertainties in the mathematical model for the second scheme, a third strategy was considered as shown in Figure 4 (bottom), where a part of the reactant stream is fed forward into the reactor as a function of inlet gas temperature and an intermediate bed temperature. The main advantage of the third strategy compared to the others considered in this paper is that it could be implemented in an existing reactor system without any large capital investment. In this paper, we consider the secondary stream to be of variable flowrate a t the temperature and composition of the feed gas, while in the paper by Strangeland and Foss (1970), a constant flow of secondary stream of variable temperature and composition was used. The primary objective of the control system is to prevent excessive temperature excursions within the reactor while Strangeland and Foss (1970) considered control of outlet temperature.
Control Algorithms Based on the third strategy in the previous section, three control algorithms, called algorithms, A, B, and C, were developed which are effective in controlling temperature peaks within the reactor. The basic structure of these three algorithms enables the detection of inverse temperature response behavior and the subsequent calculation of secondary flowrate based upon measurement of an intermediate bed temperature and the feed reactant temperature. The basic structure of algorithm A is given in Figure 5. The two measured temperatures are the inlet gas temperature and the bed temperature at a distance 7, from the entrance of the reactor. The bed temperature, T(q,,t) is denoted by T,(t) and is delayed by 7 D time units where is the time required for the temperature wave to travel from the entrance and pass through the temperature sensor location. The difference in temperature, 0, which is given by T,(t) - T,(t - 7,,), gives an indication of the state of the reactor a t the distance 7, from the reactor entrance. If 0 is positive, then the peak of the wave is passing through the location 7, and the desired set point is the measured temperature which existed 7 D time units ago. On the other hand, if 0 is negative, the temperature wave has passed through the location qm and the setpoint is selected to be the current measured temperature of the bed, T,(t). This logic enables us to suppress control action when the temperature is decreasing at the location 7,; the fraction a , which denotes the fraction of the feed that is
Ind. Eng. Chern. Fundarn., Vol. 19, No. 4, 1980 393
0 920r 880
00
3
02
C3
34
05
36
07
08
09
I0
DIMENSONLESS DISTANCE,
F i g u r e 6. Temperature response of the reactor using algorithm A for a step decrease in feed temperature of 27.8 K. Secondary feed = 0.5. inlet at 7, = 0.5; ,yc
6 T I M E t Clv' XJ-ES:
Figure 7. Secondary flow for the three control algorithms used in Figures 6, 9, and 11.
injected at the secondary inlet at v,, is calculated from a heat balance, which may be written
Figure 8. Structure of algorithm B. ! Y
g
,601
From this equation
For practical reasons, the variable CY is restricted to the range of zero to cymax. The response of the control system is shown in Figure 6 and the corresponding secondary flow given by cy in Figure 7. Comparison of Figures 2 and 6 shows clearly that the maximum peak temperature is reduced with this control algorithm. An important feature of the algorithm is that the secondary flow is finite over a time period of only about 7 min and the maximum secondary flow is about 30% of the total flow. If the feed gas temperature undergoes a positive step change, the response of the uncontrolled reactor shows almost no downward temperature excursions. However, the time of temperature transients is still determined from the wave velocity, Us/U,. Algorithm A will not respond to positive temperature changes because CY will become negative and the constraint on cy built into the algorithm, 0 < c y
