Control of the Fluidized Catalytic Cracking Process Using a Simplified

Oct 8, 1996 - In this paper, a linear programming formulation using a simplified model predictive control algorithm is presented. The performance of t...
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Ind. Eng. Chem. Res. 1996, 35, 3581-3589

3581

Control of the Fluidized Catalytic Cracking Process Using a Simplified Model Predictive Controller Rola A. Abou-Jeyab and Yash P. Gupta* Department of Chemical Engineering, Technical University of Nova Scotia, Halifax, Nova Scotia, Canada B3J 2X4

Fluidized catalytic cracking (FCC) is an important process in petroleum refining. This process needs to be maintained close to optimum operating conditions because of fiscal incentives. Methods such as dynamic matrix control have been used for control of this process. However, the constrained optimization problem involved in the control has generally been solved in a piecemeal fashion. For improved control, the optimization problem needs to be solved as a whole, that is, without decomposition. In this paper, a linear programming formulation using a simplified model predictive control algorithm is presented. The performance of the formulation is tested on a mathematical model of the FCC process for set-point changes, disturbance rejection, station failure, and model mismatch. The results show that an improved control performance can be obtained by including the constraints within the optimization problem. 1. Introduction Fluidized catalytic cracking (FCC) is the heart of the modern petroleum refinery. Even a small increase in efficiency pays important dividends because of the large amounts of oil handled. The operation of the FCC process close to the optimal operating point is therefore an important objective. Based on the economics, the real-time optimizer updates the optimal values of the controlled and manipulated variables periodically. At these updates, the objective of the control system is to move the process to the new optimal operating point. At other times, the objective of the control system is to cancel the effect of disturbances on the controlled variables by making minimal changes in the manipulated variables from their optimal values. In addition, the constraints on the manipulated and other process variables need to be satisfied. The number of manipulated variables available is more than the number of controlled variables. However, the control system must be able to deal with a changing number of manipulated variables, as some of these become unavailable due to station failures or become only partially available by reaching their limits. Thus, it is important for the control system to be able to deal with the constraints and varying degrees of freedom on a continuous basis. Because of the suitability of the model predictive control (MPC) approach for process control, methods such as dynamic matrix control (DMC) have been used for control of the FCC process. However, the constrained optimization problem involved in the control of the FCC process has generally been solved in a piecemeal fashion. The problem is initially considered to be unconstrained, and then the constraints are accounted for in ad hoc ways (Prett and Gillette, 1980). This is apparently due to the Shell Oil Company’s patent on QDMC (Garcia et al., 1986) which is a quadratic programming formulation of the constrained optimization problem. The piecemeal solution will, in general, result in a suboptimal solution of the constrained optimization problem. Since efficient algorithms are available for the linear programming (LP) problem, Chang and Seborg (1983) have presented an LP formulation for the solution of the constrained optimization problem. However, when an LP is used * Author to whom correspondence should be addressed. E-mail: [email protected].

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with methods such as the DMC, the size of the constrained optimization problem formed for the FCC process becomes too large to be solved on-line at every sampling instant. The objectives of this paper are: 1. To propose an LP formulation of the constrained optimization problem whose solution is facilitated by using a simplified MPC (SMPC) algorithm. 2. To report its performance on a mathematical model of the FCC process that was developed for testing different control algorithms. The advantages of including the constraints within the optimization problem are shown by comparing the results with two other ways of handling constraints. 2. Catalytic Cracker Model Numerous papers relating to the catalytic cracking process can be found in published literature. They represent various aspects of mathematical modeling and simulation, stability, optimization, and optimal control (McGreavy and Isles-Smith, 1986; Zaho and Lu, 1988; MacFarlane et al., 1993; Arbel et al., 1995). A mathematical model of a modern zeolite catalyst cracking plant was proposed by Lee and Groves (1985) for testing different control algorithms. This model adequately describes the general dynamic behavior of the crackerregenerator system shown in Figure 1 and is much simpler than some of the recent models. Therefore, as a first attempt, this model is used in this study. The model treats the riser cracking reactor as an adiabatic plug flow reactor and is described by the following material and energy balances. dyf dz dyg dz

) -R10tc[COR]Φ0yf2 exp

[

[

-Ef

]

RT0(1 + θ)

]

exp(-Rtcz) (1)

-Ef exp(-Rtcz) RT0(1 + θ) -Eg R20tc[COR]Φ0yg exp exp(-Rtcz) (2) RT0(1 + θ)

) R11tc[COR]Φ0yf2 exp

[

]

λ∆HfFo dyf dθ ) dz T0(FsCp + λFoCp + (1 - λ)FDCp ) dz s 0 D © 1996 American Chemical Society

(3)

3582 Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996 Table 1. Size of Constrained Optimization Problem no. of variables no. of constraints

SMPC

DMC

12 10

3216 1616

3. Proposed LP Formulation As mentioned above, the control objective is to maintain the controlled variables close to their set points by making minimal changes in the manipulated variables from their optimal values. Therefore, the objective function was formulated as follows:

Minimize

Figure 1. Fluid catalytic cracking system.

The model treats the regenerator as a perfectly mixedtank and is described by the following material and energy balances.

d (WCRC) ) Fs(CSC - CRC) - ky∞CRCW dt

(4)

1 + 1.5σ d ky C W (5) (Way∞) ) RA(yi - y∞) dt Mc(1 + σ) ∞ RC d [T (WCps)] ) TriFsCps + TaFaCpa - Trg(FsCps + dt rg ky∞CRCW σ ∆HCO2 FaCpa) - ∆HCO + (6) 1+σ MC

(

)

where

CSC ) Ccat + CRC Ccat ) kc

x

tc

CRCn

( )

exp

-Ecf RT1

(7) (8)

The model equations are included to show the relationship between controlled, manipulated, and load variables. The model involves a large number of other variables and parameters that are not directly relevant to the subject matter of the paper. Since these are described by Lee and Groves (1985), their description and values are omitted to conserve space. The solution of the model involves dealing with the stiffness of the model and the interaction between the reactor and the regenerator. Stiffness arises from the fact that the oxygen content in the model has a small time constant compared to the other variables. To avoid this problem, the oxygen content was assumed to reach steady state instantaneously. Incidentally, this is the only assumption we made in simulating the response of the model using the fourth-order Runge-Kutta method. The dynamic relationships between the controlled and manipulated variables were essentially the same as those obtained by Lee and Groves, who used the LSODE package to handle stiffness. The other assumptions are the same as those made by Lee and Groves. The interaction between the reactor and the regenerator is handled by using the alternating integration technique. The riser reactor is assumed to react to changes so quickly that it takes no time in reaching a new steady state. The reactor equations are integrated first along the entire length of the reactor. The properties of the outlet stream are then fed into the regenerator model that is integrated over a sampling interval of 50 s.

l

J)

Np

m

Nm

w1i∑|ei(k+j)| + ∑w2i∑|ui(k+j-1)| ∑ i)1 j)1 i)1 j)1

(9)

where ei(k+j) ) predicted error in the ith controlled variable at j steps ahead, ui(k) ) deviation in the ith manipulated variable from its optimal value, and other variables are defined in the Nomenclature section. Note that the number of e’s and u’s equals lNp and mNm, respectively. Since in a LP formulation the variables need to be nonnegative, each of these e’s and u’s is replaced by two nonnegative variables, e.g., e ≡ e+ - e-. Therefore, the number of variables (V) in the LP formulation is given by

V ) 2lNp + 2mNm

(10)

There is one equation for each of the e’s that relates it to the u’s, and there is one upper and one lower bound on each of the u’s. Therefore, the minimum number of constraint equations (C) in the LP formulation is given by

C ) lNp + 2mNm

(11)

It can be seen that the size of the constrained optimization problem to be solved on-line at every sampling instant is significantly affected by the values of Np and Nm. In the simplified MPC algorithm (Gupta, 1993), one control move into the future is calculated and the error is minimized at one point P steps ahead, where P is a tuning parameter. The minimization at additional points is considered only if needed. Therefore, Np ) Nm ) 1 in this algorithm, and the size of the constrained optimization problem formed for l ) 2 and m ) 4 is shown in Table 1. For a sampling interval of 50 s, the open-loop response of the FCC settles in about 800 sampling intervals. Therefore, in the DMC algorithm, Np equals 800. Then, even with two control moves into the future (Nm ) 2), the size of the constrained problem formed with DMC is very large, as shown in Table 1. The SMPC algorithm was used in this study because it involves a small-size constrained optimization problem and because its control performance had been reported to be similar to that of the DMC algorithm on other processes. Venugopal et al. (1991) compared the performance of the two algorithms on a pilot-scale circulating fluidized-bed combustor where quadratic programming was considered for the solution of the optimization problem. Gupta (1993) compared the robust stability of the two algorithms in unconstrained situations. By using the SMPC algorithm, eq 1 simplifies to:

Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996 3583

Minimize l

J)

m

w1i|ei(k+Pi)| + ∑w2i|ui(k)| ∑ i)1 i)1

(12)

For an LP formulation, each of the e’s and u’s in the constraint equations is replaced by a difference of two nonnegative variables. In the objective function, a sum of these nonnegative variables is minimized. Thus the LP problem was formulated as follows:

Minimize l

m

J ) ∑w1i[ei (k+Pi) + ei (k+Pi)] + ∑w2i[ui+(k) + +

-

i)1

i)1

ui-(k)] (13) subject to the following modeling equations and constraints

e+(k+P) - e-(k+P) - ∆AP[u+(k) - u-(k)] ) y(k) ySP(k+P) - ∆APu(k-1) + ∆y(k+P) (14) u*(k) e u(k) e u*(k)

(15)

where

∆y(k+P) ) ∆AP+1∆u(k-1) + ∆AP+2∆u(k-2) + ... + ∆AP+N-1∆u(k-N+1) (16) ∆AP+i ≡

[

a1,1,P1+i - a1,1,i a1,2,P1+i - a1,2,i ... a1,4,P1+i - a1,4,i a2,1,P2+i - a2,1,i a2,2,P2+i - a2,2,i ... a2,4,P2+i - a2,4,i

]

(17) Both magnitude and velocity constraints were considered in this study. To keep the number of constraints to a minimum, the velocity constraints on the manipulated variables were satisfied using the same equations that represent magnitude constraints. This was done by resetting the limits at every sampling instant based on the constraint that allowed the smallest move. 4. Control Performance The cracking plant model used in this study assumes that the reactor and regenerator pressures are held constant by appropriate regulators. A regenerator control system is assumed to maintain the catalyst holdup in the regenerator constant. This leaves four manipulated variables available: crude oil feed flow rate Fo; air flow rate Fa; catalyst circulation rate Fs; and feed oil temperature Toil. The principal disturbance that affects the process is the carbon formation tendency of the feed governed by the rate constant kc. The main dependent variables to be controlled are regenerator temperature Treg and regenerator flue gas oxygen content Y∞. The above set of manipulated, disturbance, and controlled variables were selected based on the information given by Lee and Groves(1985). The controller model was developed from the nonlinear model by making step changes in the manipulated variables. One manipulated variable at a time was changed by 10%, and the corresponding responses of the controlled variables were recorded. The nonlinearity in the model was handled through feedback, as is done in the DMC algorithm. N was selected so that the open-

loop response curves had essentially settled at their new steady states. The weighting factors were initially selected to account for the difference in the magnitude of the variables. Then the weights on the controlled variables were increased relative to the weights on the manipulated variables because the deviations of controlled variables from their optimal values are less acceptable than deviations of manipulated variables from their optimal values. The performance of the proposed control algorithm is presented for set-point changes, disturbance rejection, station failure, and model mismatch. The responses obtained are labeled as SMPC in the figures. In all of the figures, the changes considered occur at time equal to zero. The negative time region shows the initial steady-state values of the controlled and manipulated variables. For regulation problems, the results are reported in percentage deviation from the initial “optimal” values. For servo problems, the results are reported in percentage deviation from the new optimal values. In an actual application, the optimal values for the controlled and manipulated variables would be determined by the real-time optimizer using steady-state models. The proposed algorithm would then minimize deviations from these optimal values. In this paper, we selected the set points for the outputs and calculated the corresponding set point “optimal values” for u using the steady-state portion of Lee and Groves’ model. Although commercial control algorithms have certain strategies to improve the piecemeal solution, these algorithms are not available in the open literature. Therefore, the advantages of including the constraints within the optimization problem are shown by comparing the results with the following two approaches from the open literature. In these approaches for handling constraints, the regular unconstrained DMC method was used in this study. The tuning parameters were selected such that both SMPC and DMC algorithms gave essentially the same response for a set-point change of 1% in Treg and 12.5% in Y∞. In other words, one algorithm was not tuned tighter than the other. The move suppression factor, weights, and other parameters used are given in the appendix. Ad Hoc Approach. An ad hoc approach for handling constraints was presented by Prett and Gillette (1980) and was used here. The control moves at every sampling instant were first calculated using all four of the manipulated variables. If one or two manipulated variables reached or violated their upper or lower magnitude constraints, these manipulated variables were considered to be unavailable for manipulation. The unavailable manipulated variables were kept at their limits, and the unconstrained optimization problem was solved again using the remaining manipulated variables. If the calculated moves still violated constraints, these were set at their allowable limits. The responses obtained using this approach are labeled as MPC1 in the figures. Direct Limiting Approach. The control moves at every sampling instant were first calculated using all four of the manipulated variables. Whenever the calculated moves violated magnitude or velocity constraints, these were set at their allowable limits. The responses obtained using this approach are labeled as MPC2 in the figures. Test 1. Set-Point Change. Since the real-time optimizer changes the desired values of the controlled and manipulated variables periodically, the control

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Figure 2. Control performance for step changes in set points.

Figure 3. Control performance for step changes in set points.

performance is presented for step changes in set points. Positive and negative changes in the set points are considered. Figure 2 shows the control performance for a case where the set points of Treg and Y∞ are increased by 1% and 12.5%, respectively. Although the direct limiting approach moves the controlled variables to their new set points, three of the manipulated variables deviate by a large amount from their desired values and eventually saturate. The proposed control algorithm moves the controlled and manipulated variables to the new set points smoothly. In the ad hoc approach the controlled variables take a longer time to settle. The manipulated variables deviate from their new optimal values since there was no provision in this approach for keeping them close to these values.

Figure 3 shows the control performance for a case where Treg and Y∞ are decreased by 1% and 19%, respectively. In this case the direct limiting approach does not work at all as the controlled variables deviate by a large amount from the desired values. This happens because the unconstrained control algorithm tries to change the manipulated variables in directions in which they cannot change. In this case the ad hoc approach does not work well either, as there are large offsets in both of the controlled variables. This happens because the unconstrained algorithm does not know the available directions in which the remaining manipulated variables can change. The proposed control algorithm moves the controlled and manipulated variables to the new set points smoothly.

Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996 3585

Figure 4. Control performance for a load change of +5%.

Figure 5. Control performance for a load change of -5%.

Test 2. Disturbance Rejection. In this test, a step change in carbon formation rate constant kc is considered. The disturbance was treated as an unmeasured one; that is, the effect of the disturbance was not fed forward to the control algorithm. Figure 4 shows the control performance for a step increase of 5% in kc. Although the direct limiting approach keeps the controlled variables close to their desired values, three of the manipulated variables deviate by a large amount from their desired values and eventually saturate. The ad hoc approach regulates Treg but leaves an offset of 4% in Y∞. The proposed control algorithm brings the controlled variables back to their set points by making relatively small changes in the manipulated variables.

Figure 5 shows the control performance for a step decrease of 5% in kc. In the direct limiting approach, the controlled variables again deviate by a large amount from the desired values. This is due to the reasons mentioned in test 1. In the ad hoc approach, the response of Y∞ is good, but there is some offset in the response of Treg. The proposed control algorithm again brings the controlled variables back to their set points by making relatively small changes in the manipulated variables. Test 3. Station Failure. In this test, a station failure, that is, a mechanical breakdown where a manipulated variable is unable to change, is considered. The failed manipulated variable was assumed to be Fa. In the proposed algorithm, the station failure was

3586 Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996

Figure 6. Control performance for a load change of +5% when station Fa has failed.

Figure 7. Control performance for a load change of -5% when station Fa has failed.

handled simply by setting the velocity constraint of the failed variable equal to zero. In the ad hoc approach, Fa was removed from the list of the available manipulated variables. The control moves were calculated using the remaining three manipulated variables, that is, by using a 3 × 2 system. The direct limiting approach did not allow the use of this failed information. The results obtained for step disturbances of 5% and -5% in kc are shown in Figures 6 and 7, respectively. For a step disturbance of 5% in kc the controlled variables again go far off in the case of the direct limiting approach. In the case of the ad hoc approach, although Treg is held close to the set point, Y∞ takes a long time to return to the set point. The proposed

control algorithm brings the controlled variables back to their set points by making relatively small changes in the manipulated variables. For a step disturbance of -5% in kc, the performances of the ad hoc and direct limiting approaches are same. There is a very large (325%) steady-state offset in Y∞. The proposed control algorithm keeps the controlled variables close to their set points by making a relatively small change in Fo. Test 4. Model Mismatch. In this test a mismatch between the process and controller models is considered. To do so, the entire open-loop step response between each manipulated and controlled variable was either increased or decreased by 20%. For the 4 × 2 system considered, there are eight step-response curves. Four

Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996 3587

Figure 8. Control performance for step changes in set points using an imperfect controller model.

Figure 9. Control performance for a load change of +2.5% using an imperfect controller model.

of these eight curves were randomly picked for the increase, and the remaining four, for the decrease. In other words, in the four cases all of the step-response coefficients from 1 to N were multiplied by 1.2, and in the remaining four cases these were multiplied by 0.8. The control actions were calculated using the altered coefficients, and these were applied to the unaltered process model. This was done to simulate a situation where an imperfect model is used to control the process. Figure 8 shows the control performance for a set-point change of 1% in Treg and 12.5% in Y∞. Figure 9 shows the control performance for a step disturbance of 2.5% in kc. For this mismatch, the controlled variables in the case of the direct limiting approach again deviate by a large amount from the desired values. The ad hoc

approach performs quite well except for the steady-state offset in Y∞. The proposed control algorithm moves the controlled variables to their set points by making relatively small changes in the manipulated variables. 5. Other Comments (a) In this study, only the direct constraints on the manipulated variables were considered. If there are constraints on other process variables, these can be translated to constraints on u’s using linearized relationships between these variables. These constraints can then be added to the constraint set. (b) If the constraints are not included in the optimization problem, tasks are assigned to manipulated vari-

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ables that they may not be able to do. For example, the algorithm may calculate a move that the manipulated variable is unable to take because of a constraint. As a result, the control performance suffers. If a manipulated variable that is sitting at a constraint is excluded from the optimization problem, control capacity is still lost by not utilizing the direction in which the manipulated variable could have moved. This is because the optimal solution may switch the direction of the move of this manipulated variable for the changed values of the other manipulated variables. (c) The proposed LP formulation was also tested by increasing the number of points on the output trajectory at which the error is minimized. Two extra points on either side of the original point were added. In one trial these were placed 5 steps from the original point, and in another trial these were placed 10 steps from the original point. However, these increases in the value of Np did not result in a better control performance. (d) The improvement in the control performance is because of the inclusion of constraints within the optimization problem. The SMPC by itself did not provide the improved performance. However, the SMPC allowed the formation of a small-size constrained optimization problem that could be solved easily at every sampling instant.

u*(k) ) lower constraints on the manipulated variables in deviation form u*(k) ) upper constraints on the manipulated variables in deviation form u+, u- ) nonnegative variables in the LP formulation, u ≡ u+ - uV ) number of variables in the LP problem w1i ) weight on the ith controlled variable w2i ) weight on the ith manipulated variable Y∞ ) mole fraction of oxygen in flue gas y(k) ) measured value of the controlled variables at current sampling instant ySP(k+P) ) set points of the controlled variables, P steps ahead ∆y(k+P) ) change in the controlled variables at P steps ahead due to past inputs boldface ) vector or matrix

6. Conclusions

Initial Steady State

The simplified MPC algorithm facilitates the solution of the constrained optimization problem involved in the control of the FCC process by forming a small-size LP problem. The simulation results show that an improved control performance can be obtained by solving the constrained optimization problem as a whole, that is, by not dividing it into two parts. Acknowledgment The financial support provided by the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged. Nomenclature ap ) change in the output, P steps ahead, due to a unit step change in the input C ) number of constraint equations in the LP problem e(k + P) ) predicted error in the controlled variable at P steps ahead e+, e- ) nonnegative variables in the LP formulation, e ≡ e+ - ef ) move suppression factor Fa ) air flow rate Fs ) catalyst circulation rate Fo ) gas oil feed rate k ) current sampling instant kc ) carbon formation rate constant “disturbance variable” l ) number of outputs (controlled variables) m ) number of inputs (manipulated variables) N ) number of sampling intervals in which the open-loop response settles Np ) number of points on each output trajectory at which the error is minimized Nm ) number of control moves into the future calculated for every manipulated variable P ) distance in terms of sampling intervals at which the error is minimized T ) sampling interval (s) Toil ) feed oil temperature Treg ) regenerator temperature u(k) ) manipulated variable in deviation form

Appendix

Parameters m ) 4 T ) 50 s General: N ) 800 l ) 2 Nm ) 1 P1Treg ) 80 P2Y∞ ) 120 SMPC: Np ) 1 Np ) 800 Nm ) 2 f ) 1.007 DMC:

Treg ) 684.2 °C Fa ) 7.4 kg/s Fo ) 12.3 kg/s

Y∞ ) 1.58874 × 10-3 Fs ) 84 kg/s Toil ) 146.9 °C

New Steady State for Set-Point Changes Figures 2 and 8: Treg ) 691.7 °C Fa ) 7.4 kg/s Fo ) 12.3 kg/s

Y∞ ) 1.8367 × 10-3 Fs ) 79.5 kg/s Toil ) 146.9 °C

Figure 3: Treg ) 677.7 °C Fa ) 7.0 kg/s Fo ) 11.8 kg/s

Y∞ ) 1.28 × 10-3 Fs ) 84 kg/s Toil ) 148.9 °C

Magnitude Constraints Velocity Constraints 6.6 e Fa e 7.5 75 e Fs e 84 9 e Fo e 13.6 135 e Toil e 154.4

|∆Fa| e 0.5 kg/s |∆Fs| e 2.3 kg/s |∆Fo| e 0.9 kg/s |∆Toil| e 5.6 °C

Weighing Factors: SMPC: w11 ) 7.914 × 10-3 w12 ) 6294.3 w21 ) 0.061 15 w22 ) 5.414 × 10-3 w23 ) 0.036 89 w24 ) 3.374 × 10-3 DMC: w11 ) 7.914 × 10-3 w12 ) 6294.3 w21 ) 0.061 15 w22 ) 5.414 × 10-3 w23 ) 0.036 89 w24 ) 3.374 × 10-3

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Literature Cited Arbel, A.; Huang, Z.; Rinard, I. H.; Shinnar, R.; Sapre, A. V. Dynamics and Control of Fluidized Catalytic Crackers. 1. Modeling of the Current Generation of FCC’s. Ind. Eng. Chem. Res. 1995, 34, 1228-1243. Chang, T. S.; Seborg, D. E. A Linear Programming Approach for Multivariable Feedback Control with Inequality Constraints. Int. J. Control. 1983, 37, 583-597. Garcia, C. E.; Prett, D. M.; Morshedi, A. M. Quadratic Programming Solution of Dynamic Matrix Control (QDMC). Chem. Eng. Commun. 1986, 46, 73-87. Gupta, Y. P. Characteristic Equations and Robust Stability of a Simplified Predictive Control Algorithm. Can. J. Chem. Eng. 1993, 71, 617-624. Lee, E.; Groves, F. R., Jr. Mathematical Model of the Fluidized Bed Catalytic Cracking Plant. Trans. Soc. Comput. Simul. 1985, 2, 219-236. McFarlane, R. C.; Reineman, R. C.; Bartee, J. F.; Georgakis, C. Dynamic Simulator for a Model IV Fluid Catalytic Cracking Unit. Comput. Chem. Eng. 1993, 17, 275-300.

McGreavy, C.; Isles-Smith, P. C. Modeling of a Fluid Catalytic Cracker. Trans. Inst. Meas. Control 1986, 8, 130-136. Prett, D. M.; Gillette, R. D. Optimization and Constrained Multivariable Control of a Catalytic Cracking Unit. Proceedings of the Joint Automatic Control Conference 1980; Paper WP5-C. Venugopal, S.; Gupta, Y. P.; Basu, P. Predictive Control of a Circulating Fluidized Bed Combustor. Can. J. Chem. Eng. 1991, 69, 130-135. Zaho, X.; Lu, Y. Nonlinear Dynamic Model and Parameter Estimation of Fluidized Catalytic Cracking Unit. IFAC Proc., Ser. 2 1988, 1079-1083.

Received for review August 10, 1995 Revised manuscript received April 22, 1996 Accepted April 23, 1996X IE950499J X Abstract published in Advance ACS Abstracts, July 15, 1996.