A Control Scheme for Packed Bed Reactors Having a Changing Catalyst Activity Profile. on-line Optimizing Control

A control scheme for packed a changing catalyst activity optimizing control

Larry C. Windes and W. Harmon Ray+

bed reactors having profile. II: On-line

Department of Chemical Engineering, University of Wisconsin, Madison, Wisconsin 53706, USA (Received 17 June 1991; revised 27 February 1992)

The on-line optimization of packed bed reactor performance is achieved via a comprehensive control system utilizing a two-dimensional partial differential equation reactor model, non-linear distributed parameter state estimation, sequential parameter estimation, control of the maximum estimated temperature and sequential optimization of economic criteria. Excellent performance has been demonstrated in both closed- loop simulation and real-time experiments with an exothermic wall-cooled reactor for the partial oxidation of methanol to formaldehyde. This scheme has shown robustness to both uncertain model parameters and catalyst decay. Rapid optimization was accomplished for reactor start-up, process disturbances and changes in the economic objective. Single-variable optimization of the maximum reactor temperature (as controlled by the wall temperature) as well as two-variable optimization manipulating both hotspot temperature and feed flow rate are demonstrated.

(Keywords: packed bed reactors; catalyst activity; on-line optimization)

In Part 11 a control scheme for packed bed reactors having changing catalyst activity profiles was developed and tested experimentally. In the present paper, an optimizing strategy for the controller setpoints is presented and demonstrated on the experimental reactor. The optimizing strategy is applied using a rigorous partial differential equation model to efficiently control and optimize an exothermic wall-cooled packed bed reactor. An important part of the control strategy is the non-linear distributed parameter state estimation algorithm which generates rather complete information concerning con- centration and temperature profiles in the reactor based on the measurements and the model (cf Windes et a/.*). The model adapts itself to changing conditions in order to maintain its validity in assigning the appropriate control action. Specifically, on-line sequential parameter estimation compensates for catalyst deactivation during reactor operation. The problem of choosing a setpoint for the controller is addressed through optimizing control. An attempt has been made to develop a gener- ally applicable strategy which is robust to modelling errors and catalyst deactivation, able to control the reactor over a wide range of operating conditions and maxi- mize an objective based on the reactor products.

*Current address: Eastman Chemical Company Research Laborato- ries, Kingsport, TN 37662, USA

Author to whom correspondence should be addressed

0959-1524/92/010043-11 0 1992 Butterworth-Heinemann Ltd

Steady-state optimization of fixed-bed chemical reactors has received extensive treatment in the literature (e.g., see the review by Feyo de Azevedo and Rodri- gues3). Reactor optimization in the presence of slowly decaying catalyst has also been examined (e.g., Ray and Szekely4, Ray5, Kovarik and Butt6 and Buzzi-Ferraris et ~1.). The work with decaying catalyst has considered the reactor to be at steady state with respect to the instantaneous catalyst activity profile. Lee and Lees have applied on-line optimizing control to a fixed-bed reactor system. They used a disciete-time model for the reactor rather than differential equations, controlled the reactor temperature at one fixed position and reduced the reactor system to a 2 x 2 MIMO lumped system. Hamer and Richenberg9 improved upon the previous algorithms and applied on-line optimizing control to a packed-bed immobilized-cell reactor. Eaton and RawlingsO used suc- cessive quadratic programming to optimize a chemical process over a particular time horizon with the process model collocation equations as constraints. Least squares parameter estimation and model identification have been combined with discrete step-wise optimization over a selected time horizon, and this method was applied to a two-CSTR problem. Chen and JosephI applied these results in a simulation study to the on-line optimization of the steady-state model equations for a packed-bed ethylene oxide reactor.

In this paper we develop and demonstrate an optimi-

J. Proc. Cont. 1992, Vol2, No 1 43

Control scheme for packed bed reactors. II: L. C. Windes and W. H. Ray

temperature

controller

-_-.,., reactor or

reactor model T T ~r.r.t*~

meas

T.-m.. - I max Interpolation

I- I max setpoint T(r.z,t) State estimation Optimization

Quadratic :

optimum

Objective Compute

evaluation i exit Jl,JZ,J3 ; at Tl ,T2,T3

I I

T(r,z,tl

Figure 1 Schematic for reactor optimization. Adjustment of reactor setpoint, state estimation and optimal parameter estimation

zing controller for a wall-cooled packed reactor for the partial oxidation of methanol to formaldehyde. The emphasis will be on the maximization of an economic objective such as reactor yield. In some optimization problems there is a trade-off between long term catalyst lifetime and instantaneous reactor yields. However, for the present reactor both catalyst lifetime and reactor yield are favoured by moderate reactor temperatures and no significant conflict of objectives exists. Hence we will optimize an instantaneous objective here.

The main features of the comprehensive optimizing controller are:

A two-dimensional heterogeneous packed bed dynamic reactor model (cf Windes et al.j, Schwe- dock et aLi4). A state estimator which provides good composition and temperature profiles with limited measurements and ensures that the parameter estimation, maximum temperature controller and optimization algorithms are using reasonable reactor compositions and temperatures (cf Windes et al.2). Parameter estimation which corrects the catalyst activity profile based on matching the estimated and measured compositions where the estimated compositions are based on the estimated temperature profile (cf Windes and Ray). Feedback control of the maximum estimated reactor temperature using the jacket wall temperature where the setpoint is supplied externally by the optimization algorithm (cf Windes and Ray). Optimization of an objective based on the values of the components exiting the reactor through manipulation of the maximum temperature set-

point in comparison with the current estimated maximum temperature.

This comprehensive set of control system algorithms are shown in the block diagram of Figure I. Note that only one or two temperatures and a single delayed composition measurement are available to the control system, so that the state and parameter estimation algorithms are essential to allow control and optimization of the reactor. The overall control scheme is designed to operate the reactor efficiently and overcome erroneous heat transfer parameters, erroneous kinetic parameters and time-varying catalyst activity. The details of the process model and experimental reactor (cf Windes et al.13, Schwedock, et al.14), the state estimation algorithm (cf Windes et af.z), the parameter estimator and feedback controller (cf Windes and Ray) are given in earlier papers. Thus only the optimizing controller and its performance are described here.

The optimizing controller algorithm

An important aspect of this work is its experimental application to a system of industrial importance. The example system under study is a tubular packed bed reactor for the partial oxidation of methanol to formaldehyde. An undesirable secondary reaction also occurs: the subsequent oxidation of formaldehyde to carbon monoxide. A mathematical model has been developed which does a good job of representing the packed bed reactor built in our laboratory at University of Wiscon- sin14J5. The two-dimensional heterogeneous model is summarized in the Appendix to Part I, with more details given in Ref. 13.

44 J. Proc. Cont. 1992, Vol2, No 1


The objective function for the packed bed reactor performance was chosen to be a weighted sum of each species entering and exiting the reactor, and is calculated from the exit compositions (Y,, Y,), reactor flow rate (fl and feed mole fraction methanol bi).

+ (Pa + 2Vs - V,)Yc) - F(1 - vi)V(j (1)

where the Vi are the values of the components: V,, formaldehyde out; Vz, methanol in; V,, methanol out; V,, carbon monoxide out; Vs, water out; V,, air in.

The exit compositions are computed by solving two sets of quasi-steady non-linear algebraic material balance equations with a given temperature profile, operating conditions, and set of fixed and estimated parameters. These equations are equivalent to the pseudo- steady-state equations solved at each time step in the dynamic solution of the reactor model. Thus, the non- linear objective (1) may be rewritten as:

J = f( Ym, Yc, F, Yi, V = AT, 0, V> (2)

where the vector of temperatures, T, are at the axial collocation points and 0 is a set of catalyst activity parameters. For the ith objective evaluation:

J, = f(T,, 8, V) (3)

For each J, and T, there is a corresponding maximum reactor temperature l-p

The optimization is based on finding the maximum of a quadratic function approximating the local relation- ship between the value of the objective and the reactor temperature, or Tmax. At regular time intervals (e.g., 1 min), the optimization routine is run and a new maximum temperature setpoint is selected. At each optimization time step, three objective function evaluations are made at reactor temperatures determined from an inter- nally computed temperature optimization step size (6T). The steps in the optimization algorithm are:

1. Compute exit compositions with the current estimated reactor temperature profile, and evaluate the objective function.

2. Perturb the temperature profile by an amount proportional to ST, solve the composition equations at the new temperature profile, and evaluate the objective (2).

3. Calculate the optimal direction and magnitude of the next temperature perturbation, perturb the temperature profile, and evaluate the third objective point.

4. Fit the three points with a quadratic, and compute the new maximum temperature setpoint.

5. Filter the setpoint changes to remove noise when 6 T is small.

6. Calculate the value of 6T to use in the next optimization step.

For the first objective function evaluation, the current estimated reactor temperatures are used with the maximum reactor temperature (assumed to be at the reactor centreline):

T, = T; r, = T,,,,, = max (T(z)]~=,,) (4)

For the second evaluation the temperatures are changed according to ST:

Tz = T, + (T, - T,)hT; I-Z = I-, + (r, - T,)6T (5)

where T, is a vector of wall temperatures with identical elements, T,. The reactor temperatures are perturbed by an amount proportional to the temperature rise at their location in the reactor. Note that 6T is not an explicit function of radial and axial position in the reactor; however, the optimization algorithm could be easily modified to incorporate 6 T as an appropriate function of position.

For the third evaluation, the temperatures are changed by 6T in the direction of increasing objective function:

T3 = TIJmax + sign(L, - Lmin)(TI~max - T,)sT (6)

After three values of the objective have been computed at three different Ti, the new maximum reactor temperature setpoint, rset, is calculated. If the three points can be fitted with a concave quadratic function, then the quadratic is used to find the point rset where J(T) has an optimal value. If the three points are convex, then it is necessary to alter the best of these values of r by 6T according to:

Let = Lmax + sign (Lax - L,)(T,,, - T,W (7)

The setpoint, TSl, is constrained by a fixed maximum allowable reactor temperature.

Now that a value for rset has been calculated, 6T must be adjusted for the step size of the next optimization step. The change in 6T is based upon Twt, r,, Tz and r3 with emphasis on the distance between the current value of r, = T,,,,, and TS,,. Also, an attempt is made to eliminate small superficial changes in the optimum setpoint which introduce noise into the system. Finally, the setpoint is conditioned using a variable time constant first-order filter where the time constant increases as the step size 6T decreases. The new value of the maximum temperature setpoint sent to the controller is:

r set(k) = 6&?tCkm I) + (1 - 4rset

where

w = exp(- c,(lSV + G)) c, = 10, c2 = 0.01 (8)

This algorithm has been tested extensively in simulation and has been used during on-line experiments. The results are given in the next section of this paper. In summary, the on-line optimization performs an effective one-dimensional temperature search. If the reactor is far



Table 1 Optimization experiments

Experiment Time (min) Y, CH,OH Flow (g ss) Sample location (cm) Comments

01 02A 02B 02c 02D 03 04

05

0.040.0 0.0-12.5

12.5-37.5 37.5-62.5 62.5-80.0 0.0-80.0 o.(r1oo.o

80.0 0.040.0

20.0

0.05 0.05 0.05 0.05 0.05 0.05 0.05

0.05

0.7 0.5 0.7 I.0 0.7 0.7

OPT 0.82 0.7

65 65 65 65 65

35,65 65

65

Optimize from T, = 240C Begin at T, = 250C; low flow Flow change 0.5 to 0.7 Flow change 0.7 to 1 .O Flow change 1.0 to 0.7 Poor kinetic and heat transfer par. 2-D optimization from T, = 240, F = 0.5 Flow disturbance 0.66 to 0.82 Optimize with low CO penalty Objective change: high CO penalty

away from the optimum, the algorithm quickly adapts in order to move the temperature rapidly. As the optimum is approached, the search is made on both sides of the optimum order to ensure accuracy. The optimum is determined by evaluation of a quadratic to locate the maximum of the objective. As the optimization settles at steady state, the temperature step size is reduced to the lowest level for which accurate values for the change in the objective function can still be obtained. Small insigni- ficant changes in the optimization variable are sup- pressed, but the step size and setpoint quickly change if the optimum or operating conditions shift. This algorithm is also used for the two-dimensional search when the flow rate is also optimized (cf WindeP).

Simulation and experimental results

Two modes of program operation were used in testing the control and optimization algorithm:

(4

@I

reactor measurements obtained at each time step from a dedicated simulator running in conjunction with the main computer control package; and reactor measurements communicated in real time during experiments.

Option (a) is important because it allows simulation of the closed-loop performance of the control and optimization experiments. A separate but structurally identical mathematical model (referred to hereafter as the simulator) is solved as a subroutine to the main program (the estimation/control package). It serves as the reactor for closed-loop simulation. However, the parameters and solution methods are specified independently, which allows the two models to differ exactly as specified by the user.

All of the experiments shown here involve relatively short-term operation (< 5 h) in which the optimum is achieved and then maintained for only a few hours. During these short time intervals, the catalyst activity is almost constant, and the main emphasis for the parameter estimation is quickly converging to appropriate values. During long-term operation (days, weeks, months) the parameter estimation needs to track the declining activity of the catalyst, and the optimization routine makes compensating adjustments in the manipu-

-1

-Steady state

300 320 340 360 240 250 260 270 2

MaxImum temperature (Cl Wall temperature (Cl

1 .oo

IC Methanol /

0.85 .; -Steady state

t; Time trace I A 1 min interval

1.06

3.05

3.04 -Steady state

1.03 A 1 min interv:

300 320 340 ?

Maximum temperature (Cl Maximum temperature (C)

Figure 2 Optimization simulation 01: time trace of optimization compared with off-line steady state optimum: a, b, objective as a function of temperature; c, d, compositions as a function of temperature

lated variables to change the optimum operating conditions.

A summary of five optimization experiments is given in Table 1. Experiment 01 is a straightforward test of increasing the wall temperature of the reactor to achieve optimum operation as indicated by the estimated exit compositions. The experiment begins at steady state operation at a low wall temperature considerably below the optimum. The best available initial conditions were provided to the model.

Simulation of experiment 01 showed an excellent response and the desired maximum temperature is achieved within approximately the same length of time as required for reactor start-up to a constant setpoint maximum temperature. The final steady state, optimum is:

T max = 324C Tw,,, = 257C



- - - Estimate ...*.._, ... S&point

10 20 30

Time (min)

270 b

265 -

255 -

250 -

245 -

-T wall . . . . . . . . Setpoint

240 I I 0 10 20

Time (min)

Figure 3 Optimization experiment 01: response of maximum temperature control loop

2 a

6 ._

2 3

F I 3 i 2 - Current 0 . . . . . . . . New

Time (min) Time (min)

1.0 Methanol

.tl C 2

F - Estimates LL

0.7 I I 0 10 20

Time (mini Time (min)

0.5

0.4

0.3

0.2

0.1

0

-0.1

I - Estimates 0.02N

Figure 4 Optimization experiment 01: performance of optimization

Exit conversion methanol = 0.946 Exit mole fraction carbon monoxide = 0.046

In order to verify that the true optimum is being located by the on-line optimization algorithm, Figure 2 demonstrates that the optimization algorithm moves the reactor to the optimum found by steady state modelling. Figures 24 show the object function verms T,,,,, and wall temperature at steady state. As time increases, the time trace of the optimization in the space of the objective and

1.5 c

7 1.0

VI

lJl

b 2 L 0.5

0

02

I

I

I I I 20 40 60 80

Time (min)

Figure 5 Input for optimization experiment 02. Mole fraction methanol feed is 0.05

manipulated variable exactly coincides with the steady state optimum. The time trace of the optimization also shows the efficiency of the optimization in moving quickly and directly to the optimum. Each data point represents one minute, so the optimum is reached after 6 min. Figures 2c,d show the exit compositions as a function of the maximum temperature at steady state and during the time of optimization.

Figure 3 shows the excellent temperature control response for experiment 01. The recirculating oil loop heats at the maximum rate for more than 25C and then the wall temperature levels off at the final steady value. The final T,,, is attained within 12 min, with almost no overshoot. The T,,,,, setpoint increased rapidly during the first few optimization steps, but then decreased to near the final value by the time the actual T,,,,, reached the setpoint. The data and estimates for T,,,,, were virtually identical. The performance for the optimization is shown in Figure 4. The objective function increases rapidly and maintains a steady value with no overshoot or oscilla- tions. The optimization step size increases rapidly, which is desirable when large changes are being made. The step



-0 20 40 60 80

Time (min)

300 I I I I 240 I I L 0 20 40 60 80 0 20 40 60 80


Figure 6 Optimization simulation 02: a, b, performance of optimization; c, d, response of maximum temperature control loop

excellent agreement between the experimental data and the estimates.

Experiment 02 optimizes the reactor at different flow rates (cf Figure 5) and consists of four parts (A-D). This experiment uses parameter estimation with a good initial guess for the parameters based on the parameter estimation in previous experiments and off-line data analysis. Part A is at low flow rate, so the optimum wall temperature is low in order to minimize carbon monoxide production and to prevent a severe hotspot close to the

-0.151 ' I 0 20 40 60 80 reactor entrance. Because high conversion occurs at

Time (mini much lower temperatures at low flow rates, the optimum wall temperature increases as the flow increases (02B,

280%

d 02C). Part 02D returns to the flow rate of 0.7 g s-i used

275- in 02B to demonstrate performance when the flow rate

270- decreases, and to observe any differences in the optimum because of updated parameter estimates.

265- Simulation of experiment 02 indicated that the opti- 260- mum reactor temperature is significantly higher as the

255- total flow rate is increased. Figure 6a,b shows that the

250- -T objective function declines by a large amount after

wall changes in the flow rate, but increases rapidly as the _. _ -. .

size then decreases rapidly, switches two to three times between positive and negative steps to more accurately determine the optimum, and then maintains the minimum step size. This appears to be good performance on the part of the automatic step size adjustment. In order for the true optimum to be achieved, the estimated methanol conversion and CO production must be accurate when compared to experimental data. Figure 4c,d show

2F--- III New Current ,--. r-r .: t3

+

Time (min)

80

temperature is readjusted by the optimization program. Note that the optimization step size increases rapidly after the flow change because the reactor is no longer close to the optimum. The step size decreased to the minimum value as the optimum was reached. A flow rate of 0.7 g s-l has the largest value of the objective, and therefore is close to the optimum flow rate. Figure 6c,d illustrates the good maximum temperature control during the optimization. The optimization algorithm moves the exit compositions to within a narrow range of values determined by the objective function for the process and the current selectivity of the reactor.

The results from experiment 02 were similar to the simulation, except that the maximum temperature con- straint was reached at the highest flow rate. The compo-

0.2

0.1

0

-0.1

-0.2

-0.3

J t 2

I I y

20 40 60

Time (min)

Figure 7 Optimization experiment 02: performance of optimization. 1, flow 0.5 to 0.7; 2, flow 0.7 to 1.0; 3, flow 1.0 to 0.7

4% J. Proc. Cont. 1992, Vol2, No 1


a ;- ;----

n ,e--

y

i

,J I

r-7 i

: I

i/f __;-__gf$ :/

0 20 40 60

330

320 t; 0

E 310

2 I $ 300

E

f 290

28a

Time (min)

- max

----Estimate ._. ___.._ Setpoint

w 60 Time (min)

-0.2 r

-0.3- I I I 0 20 40 60 80

Time (min)

260 [d

255

250

245

240

235

------Setpoint

230 0 20 40 60

Time (min)

Figure 8 Optimization simulation 03: a, b, performance of the optimization; c, d, response of maximum temperature control loop

sition data and estimates were in agreement, indicating that the computed optimum was correct. Figure 7 shows the experimental results in the objective function. Because of the definition of the objective function and its weighting factors, the reactor optimization tends to try to achieve similar levels of conversion and carbon monoxide production regardless of flow rate. It is for this reason that the optimum temperatures change dramati- cally for different flow rates.

Experiment 03 demonstrates reactor optimization when there are significant errors in the heat transfer parameters and also the kinetic parameters which are estimated on-line. The experiment begins with steady state reactor operation at moderate conditions and a good initial guess for the reactor temperatures. The fluid radial heat transfer is specified 25% too low, and the reaction kinetics are too fast. The feed mole fraction and total flow rate stay constant throughout the experiment, and the wall temperature is adjusted to maintain the optimum based on updated values of the estimated parameters. As the parameter estimates become more realis- tic, the estimated compositions improve, and the optimizer moves the maximum temperature setpoint toward its true optimum. The heat transfer parameters are not estimated on-line, and therefore this experiment tests the optimization performance in the presence of persistent model mismatch.

Simulation of experiment 03, which begins with erroneous model parameters, demonstrated the state estimation, parameter estimation, control and optimization

Al at exit

Al at Z=O

!;~~I _ _____________________----.---- . .

0 10 20 30 40 50 60 70 80 Time (min)

-Actual ___ Estimates

2400u 35 70 2400wo 2400-o

Axial distance (cm)

Figure 9 Optimization simulation 03: erroneous initial parameter estimates: estimates of catalyst activity (0) compared to actual values (---); comparison between estimated and actual temperature profiles

algorithms working together. The optimum is reached after about 80 min when the kinetic parameters have been updated to reasonably accurate values. Figure 8 shows the objective function, optimization step size and temperature response for the simulation of experiment 03. The temperature increases as the estimated preexpo- nential factors decrease, as shown in Figure 9. The temperature profile is correctly estimated with only mod- erately accurate kinetic information, and the temperature profiles are accurately estimated during most of the experiment (Figure 9) even though the heat transfer parameters in the estimator are incorrect. Figure 10 shows the convergence of the composition profiles. Although the initial guess is poor, they converge rather quickly to accurate profiles.

If correct values of the radial heat transfer are used, the results are similar to the previous case, and indicate that errors in determination of the reactor heat transfer will not adversely affect the optimization if the state estimator is used. Accurate values of the exit composition estimates (and hence, good estimates of the true value of the objective), are obtained after 60 min and the optimization reached the optimum operating point for the input conditions of 03. The final steady operation at T max = 324C Tw = 257C agrees with the optimum

J. Proc. Cont. 1992, Vol2, No I 49


- Actual --- Estimates

Figure 10 Optimization simulation 03: comparison between estimated and actual temperature and composition profiles

obtained in experiment 01 with exact parameters used in the model.

Figures I1 and 12 show the experimental results of experiment 03 and are similar to the simulation results shown earlier. The objective function changes due both to changes in actual reactor operation and changes in the estimated kinetic parameters which strongly affect the estimated performance. The optimization step size indi- cates noticeable adjustments in the optimum during the periods following occurrences of the largest corrections in the parameters. By the end of 80 min, the parameters have attained fairly stable values close to those found in other experiments, and the temperatures are close to those previously identified for most efficient reactor operation. Because of the close convergence of the composition profiles to the data, the optimum at which the reactor finally operates is known to be accurate.

Experiment 04 demonstrates two-dimensional optimization of the reactor. Both r,,,,, (as controlled by T,,,, through the cascade controller) and the total flow rate are manipulated to locate the optimum. In this algorithm, the hot spot temperature (T,,,,,) attempts to follow an optimum ridge based on the current flow rate. The experiment begins with the wall temperature and flow rate at low values (TW = r = 240C F = 0.5 g s-l). A flow rate optimization step is made every 5 min, and the


2, I a

O- , 6 .- Lo

.-

2 a a, 3 -2- Zl

2 .-

E ._ E ._ L

-Current 0

-8. I I I 0 20 40 60 80 -.3k-%-%-- GO


:I; 2651 b Ay:

Figure 11 Optimization experiment 03: a, b, performance of the optimization; c, d, response of maximum temperature control loop

T,,, responds accordingly. After about 80 min, the air flow rate was increased, and the optimization algorithms capability to bring the flow rate back to the optimum was tested.

Two-dimensional optimization of the reactor (experiment 04) was accomplished using both T,,, and total flow rate as the manipulated optimization variables (see Figures 13 and 14). The flow rate setpoint was adjusted every 5 min, and the T,,,,, setpoint was recalculated each minute. The optimization follows a ridge; therefore, the sawtooth appearance of the objective function is a result of perturbations in the flow rate to an immediately non- optimal value, but then achieving better performance after a new optimal temperature is attained. The steady state optimum was:

F = 0.66 g s-, T,,, = 330C T, = 264C

At 80 min the flow rate was intentionally increased to a value considerably greater than the optimum. This disturbance caused a large and rapid decrease in the objective. The optimization algorithm then reduced the flow rate, and there was no appreciable decrease in the steady state value of the objective compared with the earlier steady state. The two-dimensional optimization space near the maximum is flat along the ridge of [T,,,,,,Fj pairs where one of the variables is optimized with the other fixed. Figure 14 compares the estimated exit compositions upon which the optimizations are based with the


Time = 29 and Time = 4 and 16 min 41 and 54 min Time = 66 and 79 min

1 .o

0.8

0.6

0.4

0.2

1 :- I:- -i

q I ,P /I C- -_ 0.10

6 0.08 7-_=;

0.06 t

Axial distance (cm)

Figure 12 Optimization experiment 03: comparison between estimated composition profiles and data points. Times are when composition samples were taken: 1, 4 min; 2, 16 mie; 3, 29 min; 4,41 min, 5, 54 min; 6, 66 min; 7, 79 min

1.2

.g 1.0

z

5 0.8 L

.f 0.6

t;

2 0.4 0

0.2

I I I I

0 20 40 60 80

Time (min)

340

la

s 330 0

f 325

3

p $

320

E c 315

310

3051; ' ' ' ' 0 20 40 60 80 1

Time (min)

_ 0.80

; vI 0.75

; 0.70

2 L 0.65

$ 0.60

?j 0.55

+ 0.50

0.45 1

275

lb

260

255

250

T

245

I!!!-

- wall

------Setpoint

2400 20 40 60 80 1

Time (min)

0.85

20 40 60 80 1

Time [min)

00

00

Figure 13 Optimization experiment 04: a, b, performance of two- variable optimization; c, d, response of maximum temperature control loop

data taken at 12.5 min intervals. The carbon monoxide production shows excellent agreement, and the methanol

Methanol

.; '.OF

f 100

0.851 1 1 1. 1 0 20 40 60 80

Time (min)

0.05 0

0.04 IY'Y!I 0.03 I 0 20 40 60 80

Carbon monoxide

o.06a_

Time (min)

0

Figure 14 Optimization experiment 04: comparison between composition data and estimates at reactor exit

conversion estimates are slightly biased too high. The optimum is 9495% conversion and 4.8-5.0% carbon monoxide production. The initial conditions shown here are at low flow rate and low temperature operation. The optimization achieved similar results for cases with other initial reactor conditions.

The fifth optimization experiment (05) demonstrates the response to a change in the calculation of the objective function. The objective function can possibly include economic change, changes in upstream or downstream operation or external changes such as an air-pollution alert. When the objective function changes, the optimum operating point is adjusted. In this experiment, the penalty on by-product production (carbon monoxide) is

J. Proc. Cont. 1992, Vol2, No I 51


360 a

-T 350-c

max - - - Estimate ---.___

z

ii Setpoint ii

e 340 ii

L? : $

0 10 20 30

Time (min)

Methanol

5 moor------

- Estimates

-..---- Setpoint

2450-

Time (min)

Carbon monoxide

0.06d

- Estimates

o.030-


275

Figure 15 Optimization experiment 05: a, b, response of maximum temperature control loop; c, d, compositions at reactor exit

changed. Experiment 01 is repeated in experiment 05 with a lesser penalty on carbon monoxide production for the first 20 min, and a greater penalty on CO production during the second 20 min period.

Experiment 05 demonstrating performance with changing objective function parameters is shown in Figure 15. At the time of an increase in the penalty on by- product production, the optimization step size increases, reflecting the fact that the reactor is no longer at the optimum. The T,,,,,, T, and mole fraction CO all are decreased. The increased weighting on carbon monoxide required that some conversion be sacrificed in order to cut back the carbon monoxide production. Excellent control and optimization performance were achieved both during the initial period and after the change. It is important to note that the optimization algorithm quickly decreases the T,,,,, setpoint even though the optimization was at steady state at the time the objective function was changed. The optimization can automati- cally and efficiently compensate for changes in any external criteria concerning how the reactor is to be operated. There was excellent agreement between simulation and experiment for this case.

It is interesting to note the data spike at t = 33 min in Figure 1.5. This was caused by a temporary temperature measurement malfunction. The optimizer/controller quickly corrected after this disturbance.

Summary and conclusions

The optimization procedure uses state and parameter estimation and control of the maximum reactor temperature in conjunction with a two-dimensional, non-linear, heterogeneous reactor model. This scheme has shown robustness to uncertain parameters, catalyst decay and measurement time delays. Successful experiments have demonstrated rapid and reliable on-line optimization of the reactor when only one or two temperature points and one composition point are available to the control system.

During on-line optimization, the response of the maximum reactor temperature after start-up or low-temperature conditions was as rapid and well-behaved as in the case of control alone. The optimization effectively han- dled changing flow rates, changing objective functions and a poor initial guess in the estimated parameters. The optimal conditions chosen on-line agreed with those identified afterward through steady state modelling. A two-variable optimization was also accomplished in which both the flow rate and the wall/feed temperature were optimized on-line. The optimal operation is achieved within 20 min for the one-dimensional optimization, within 1 h for the two-variable case, and within l-2 h when there are severe errors in the kinetic parameters to be estimated on line.

The advantage of the optimizing controller presented in this work is that it takes into account all available data on temperature and composition, and updates the model and optimization accordingly. This cannot be done with off-line optimization. In addition, complete flexibility is maintained with respect to allowable inputs, model parameters and objective function parameters. These can be changed at any time, with no need to reoptimize the reactor off-line. In t,his procedure, the manipulated variable setpoints are promptly updated to new optimal values based on current knowledge about the states and parameters.

Acknowledgments

The authors are indebted to the National Science Foun- dation, the Department of Energy, the DuPont Com- pany and the Dow Chemical Company for support of this research.

References

Windes, L. C. and Ray, W. H. J. Proc. Cont. 1992,2, 23 Windes, L. C., Cinar, A. and Ray, W. H. C/rem. Eng. Sci. 1989,44, 2087 Feyo de Azevedo, S. and Rodrigues, G. Chem. Eng. J. 1988,38,9 Ray, W. H. and Szekely, J. Process Optimization, Wiley, New York (1973) Ray, W. H. Advanced Process Control, McGraw-Hill, New York (1981) Kovarik, F. S. and Butt, J. B. Cafal. Rev. -Sci. Eng. 1982,24,441 Buzzi-Ferraris, G., Morbidelh, M., Forzatti, P. and Carra, S. Int. Chem. Eng. 1984,24,441 Lee, K. S. and Lee, W. K. AIChE J. 1985,31,661 Hamer, J. W. and Richenherg, C. B. AlChE J 1988,34,626

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10 Eaton, J. W. and Rawlings, J. B. Comp. Chem. Eng. (1990), 14,469 11 Jang, S. S., Joseph, B. and Mukai, H. AIChEJ. 1987,33,26 12 Chen, C. Y. and Joseph, B. Ind. Eng. Chem. Res. 1987,26, 1924 13 Windes, L. C., Schwedock, M. J. and Ray, W. H. Chem. Eng.

Commun. 1989,78, 1

T, wall temperature (equivalent to coolant temperature)

Y! mole fraction methanol in the feed Y dimensionless mole fraction based on feed mole fraction Z axial direction

14 Schwedock, M. J., Windes, L. C. and Ray, W. H. Chem. Eng. Commun. 1989,78,45

15 Schwedock, M. J. Modelling and Identification of a Catalytic Packed Bed Reactor PhD Thesis University of Wisconsin- Madison, 1983

16 Windes, L. C. Modelling and Control of a Packed Bed Reactor PhD Thesis, University of Wisconsin-Madison, 1986

Greek letters l- maximum temperature at which objective is evaluated 6T temperature profile perturbation e all model parameters rD time delay for composition measurement

Nomenclature Subscripts co carbon monoxide

F total flow rate through the reactor (g s-1) J objective r radial direction

M methanol set setpoint W wall

t time T temperature T, reactor inlet temperature Superscripts T mas maximum temperature in the reactor estimated quantity


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