Optimal operation of a semi-batch reactor by self-adaptive models for temperature and feed-rate profiles

Chemical EngineeringScience, Vol. 47, No. 9-l Printed in Great Britain.

1. pp. 2445-2450.

1992.

rmQ9-2509/92 $5.al+o.o0 Pergamon

OPTIMAL OPERATION OF A SEMI-BATCH REACTOR BY SELF-ADAPTIVE FOR TEMPERATURE AND FEED-RATE PROFILES

Press

Lad

MODELS

Sylvie MARCHAL-BRASSELY (*). Jacques VILLERMAUX (**), Jean-Leon HOUZELOT (**). Jean-Louis BARNAY (***) (*)

Rhone-Poulenc Industrialisation, 24 Avenue Jean Jaur&s, B.P. 166, 69151 Dtcines Charpieu Cedex (France)

(**)

Laboratoire des Sciences du Genie Chimique CNRS-Ecole Nationale Supdrieure des Industries Chimiques-INPL- 1 Rue Grandville-BP 45 l-54001 Nancy Cedex (France)

(***)

RhBne-Poulenc Inc., CN 5266 Princeton, New Jersey 08543-5266

1.

Introduction

(U.S.A.)

Specialty chemical processes frequently involve batch reactor operations for which operating parameters have to be tuned in order to optimize yield, selectivity and operating costs. The main control parameters are the batch duration, the reactant feed rates, the temperature profiles and the initial amount of reactants. When stoichiometry and reaction kinetics are known, methods are available to determine optimal parameters for reactor control (Rippin, 1983). The aim of the present paper is to illustrate a method to develop a fast and very flexible strategy for self-adaptive control of semibatch reactors when such informations are not available. The data and constraints of the problem are as follows : - The initial composition of the batch is known

- It is possible to determine the amount of the main constituents at the end of the batch - Temperature and feed-or-removal rate profiles are available - The total duration of the batch is known

r

I

AUTOMATIC

OFTENDENCY

EXPERIMENT

l l

1 Optimal Contol Design

Initialand final compositions

h4initnization of a technical economical Ctitetion

. Feed or removal flow-rate l

Temperatureprofile

l

Batch duration

MODELS

Stoichiometry Kinetics

I

I l

CONS-lRIJClTON

h I

I

Fig. 1 - General optimization strategy From these informations, it is proposed to construct a self-adaptive tendency model describing the reaction stoichiometry and kinetics. This model is only intended to design a control policy applicable to the reactor in a sequence of operations in order to optimize as fast as possible a selected technical economical criterion. Fig. 1 summarizes the general strategy. The 2445

2446

SYLVIE MARCHAL-BRASSELY

Dl

etal.

(Filippi et al., to implement designed. The et al., 1988, constructing

principle of the algorithm

for model construction has been described elsewhere 1986, 1987, 1989, Marchal-Brassely et al., 1988, 1990). A simple, fast and easy method for optimal determination of temperature and. reactant feed-rates has been principle of this iterative method can be found in references (Marchal-Brassely 1990). The present paper is focused on an application of the strategy for temperature and feed-rate profiles to a real-life industrial example.

2.

Principle models

of the algorithm

for construction

of stoichiometric

and

kinetic

The constituents Aj (j=l, 2, 3.. .L) taking part in the process have first to be defined. One tries t9 represent at best the system behavior by means of a set of R independent reactions (i=l, 2, 3...R) such that R Yj = Yjo + C Vij Xi i=l

(1)

where yj, yjo are the respective final and initial values of the dimensionless mole numbers of Aj, Vij is the stoichiometric coefficient of Aj in reaction i and Xi the extent of reaction i (Villermaux, 1985). R, the Vij and the Xi are unknown but the resul? of N ecperimental runs are available (k=l, 2, 3.. .N) in which initial and final compositions yjo and yj were determined for each constituent. The stoichiometric model is constructed in a progressive way. The reactions are identified in sequence up to the required accuracy and up to the maximum number permitted by number of available runs. Reaction extents and stoichiometric coefficients are simultaneously optimized. The latter are automatically rounded off. After obtaining the stoichiometric model, the reactions are linearly combined in order to come up with a realistic system made of reactions involving the smallest number of constituents as possible. At each step, a simple kinetic model is selected with rate laws involving only the reactants and orders 1 or 2. Pre-exponential factors ko and activations energies Ei are identified in order to represent at best the whole set of experimental runs available at this point. The model obtained in this way makes it possible to predict the optimal conditions for the next run. The data resulting from this new run are then incorporated into the preceding data base and a new stoichiometric and kinetic model is determined in order to optimize the criterion etc. This procedure justifies the term of “self-adaptive” strategy, which is described in more details by Filippi et al. (1989). It must be clearly stated that the main objective is to obtain optimum operation performance and incidentally to improve the model, which is only a means to reach the optimum. 3.

Principle

of the iterative

method

for construction

An algorithm was developed in order to optimize according -

the the the the the

of optimal

profiles

to the situation

batch duration feed duration for reactants amount of initial reactants temperature profile feed-rate profile

The principle of the method is as follows i) One first tries to optimize these quantities by giving them a constant value ii) Then, taking as initial values the preceding results, one searches for the batch duration, the corresponding initial and final temperatures, the feed duration, the corresponding initial and final feed rates. In between, the new temperature and feed-rate profiles are assigned to be straight lines connecting the optimised initial and final values. This means that one single interval is involved during the reaction and the feed periods. Of course, the other parameters are simultaneously optimised.

Optimal operation of a semi-batch reactor

Dl

2447

iii) One then proceeds further in the deformation of the profiles by cutting out reaction and feed periods into two equal intervals. One then optimizes, besides the simple parameters, three temperatures corresponding to the beginning, the middle and the end of the feed period. The initial values are determined from the preceding profiles. The new profiles thus consists of two ramps joining optimised points two by two. This process is continued for a larger number of cutting intervals of equal duration. The larger the interval number, the larger the number of ramps and the closer the profile to the optimal one. However, with many intervals, the optimization time gets longer. An advantage of the progressivity of the approach is that the user may quantify the improvement in the criterion at each stage and he may decide to stop the optimisation process if the gain becomes too small or if the obtained profile is obviously difficult to implement in practice. Very frequently, it has been found that a limited number of intervals (say 3 or 4) was sufficient to cause a significant decrease of the technical economical criterion close to the optimum value. As far as observed in practical situations, this optimum is unique. This algorithm, which produces easy to implement profiles was shown to be much more performing than the standard method based on Pontryagin’s maximum principle (Pontryagin, 1964) in the case of systems where sophisticated profiles were required. The solution of this kind of problems involving the optimization of a fairly high number of parameters requires robust and adapted algorithms. Two performing methods are available. The method of Box complex (Box, 1965) and a sophisticated version of Fletcher’s algorithm with automatic evaluation of derivatives (Comet (Staha, 1973)). Both methods were found suitable to the problem, the second one being nevertheless more robust. An example of profile is shown in Fig. 2. in the case of a reaction progressive deformation of the temperature system especially sensitive to temperature variations. The steep profile finally obtained in step (11) is due to the peculiarity of the chemical system considered here. The profile is progressively approached by an increasing number of segments of equal time duration. The corresponding improvement of the optimization index is given in Fig. 3 for each of these profiles. The slight increase observed locally is due to the fact that the optimization involves different points at every iteration but the procedure finally always converge to a minimum.

1000

0

2con

3um

(4) TC 100 so

7 interv&

b0 M 6Oi-0-d SOL 0

Us) 1000

2wa

60&.-e-d 3J

3000

0

(5)

Ud 1Ooa

2000

,

60SOI

3000

.

0

1000

(‘3

t(3)

,

f

E

2ooa3aooo

lGco2ooo3000

(8)

(7)

ji_Ty, , i;[~~l,j, ~LT,), 0

IWO (91

2000

moo

0

mw (10)

Moo

3oao

0

1000

2000

3000

(11)

Fig. 2 - Progressive construction of a temperature profile in the case of a reaction system sensitive to temperature variations (Marchal-Brassely.

1990)

2448

SYLVIE MARCHAL-BRASSELYet al.

Dl

Fig. 3 - Improvement of the criterion by deformation of the temperature profile shown in Fig. 2 01234567890 Number of intervals

4.

Example

of application

The methodology outlined above was applied to a reaction system of industrial interest whose general features are as follows : A liquid A simultaneously reacts with a liquid B and a gas G to give products R and S and by-products denoted L. The reaction is carried out semibatchwise, B and G being added simultaneously or not to A, the aim is to maximize the amount of R within a minimum amount of time, by possibly manipulating the following parameters, which arc specified at the beginning of each run. - Initial amount of A and G (the amount of B is fixed) - Temperature profile during the run - Feed duration and feed-rate profile of G - Feed duration of B (constant rate) - Total batch duration tr The residual amounts of A, B, G and the formed amounts of R, S are determined at the end of the run. L is deduced from a mass balance. The selected technical-economical criterion is

J=S

(2)

nm is the final number of moles of R at the end of the run, ngg the initial number of moles of B and p is a weighting coefficient chosen once for all at the beginning of the optimisation campaign. The aim of the optimization strategy is to minimize 3. For proprietary reasons, the names of the species cannot be disclosed. For the sake of simplicity, the parameters were optimised in a progressive way. At the beginning of the campaign, only the temperature profile and both batch and G-feed durations were considered optimizable. The values of the parameters for successive runs can be found in Table 1. The corresponding stoichiometric models are collected in Table 2. Figures 4 and 5 respectively show the evolution of criterion J and yield of R during the campaign. The yield and the batch duration (both included into the criterion) clearly exhibit a drastic improvement which is quickly obtained by the 8th run. It is interesting to notice the progressive deformation of temperature and flow-rate profiles. The rising temperature profile obtained in run 3 is due to the limited solubility of G which has been accounted for in the model. 5.

Conclusions

A heuristic method was developed, which is both fast and efficient to identify stoichiometric and kinetic tendency models for reactions taking place in batch or semi-batch reactors. A general control strategy for simultaneous optimization of temperature and feed rate profiles together with usual control parameters was also developed. The method is fairly robust with respect to analytical inacurracies as only convergence to optimal conditions is looked at. The method relying on both algorithms was found very performing by several validations issued from real life industrial problems. On these occasions the fast determination of the optimal operating conditions was especially appreciated_

Dl

Optimal

operation

of a semi-batch

reactor

2449

Table 1 - Optimization sequence (Experimental data, arbitraryunits). Runs 1 and 2 are initiahation experiments. From 3 to 6, the amounts of A, B, G and G-feed rate are fixed_ These constraintsare relaxed for 7 and 8 Amount of rcact2Lnts .N’

1 2

Fe&

I duration

A

G

B

G

a

9

1

1.5

I

Batch duratio

Temperature profile

I

Feed-rate pmfile

T

2.5

LL

t

ILL_

t

T a

9

1

1

3.5 T

3

1

9

a

1,02

0963

IL!_

4

a

9

1

5

a

9

1

123 L

1.28

6

a

9

1

1,1

1.32

7

a

1,339

1

1.05

11 L-

8

1.25 a

1,539

-1

1.04

1.04

Table 2 - Stoichiometric

Tt

105 L

0,8

t

rtl

-

models obtained by the algorithm, after linear combination STOICHIOMEIKIC

RUNS

one

1. 2

two

1. 2 3

MODEL

OBTAINED

model

reaction

A+6B+8G->

1.5R+6S+05L

reactions model

A+6B+8G->2R+6S R->L

1. 2 3. 4 1. 2. 3. 4. 5

two reactions model A+60+00->2R+6S R+G->L

1.2.3.4.5.6

_

andmore

three reactions model but bad result -------> two reactions model A+6B+EG->2R+6S R+G-L

2450

SYLVIE

MARCHAL-BRASSELY

Dl

er al.

Oi

N"ofruns

N” of runs

Fig. 4 - Optimization of the criterion as a function of Fig. 5 - Improvementof run number. The first two runs are initialization optimization process experiments. Run 4 yielded a very high value of J (and a poor value of yield) not represented here but the algorithm made use of the information thus gained and was able to retrieve the optimization process which

the yield of R by the

wenton fromrun5 to 8

In addition the optimization criterion was significantly improved in each case. Another advantage of the strategy is the great flexibility of the procedure with respect to the choice of optimizable parameters, which can be modified during the process, and whose range of variation can be restrictedor broadened according to the observations. Limitations due to the range of operability of industrial equipment can easily be taken into account. In its essence, the method yields shapes of temperature and feed profiles which are well adapted to the reaction/reactor system and easy to implement. Further work is in progress to adapt the strategy to specific problems posed by multiphase and catalytic reactions. REFERENCES Box, M.J., 1965, A new method of constrained optimization and a comparison with other methods. Compt J. 8, 42-52. Filippi; C., Greffe, J.L., Bordet, J., Villermaux, J., Barnay, J.L., Bonte, P., Georgakis. C., 1986, Tendency modelling of semi-batch reactors for optimization and control. Chem. Eng. Sci. 41, 4, 913-920. Filippi, C., Bordet, J., Villermaux, J., Marchal-Brassely, S., Georgakis, C., 1989, Batch reactor optimization by use of tendency models. Comp. and Chem. Eng. 13, l/2, 35-47. Filippi, C., 1987, Commande autoadaptative de reacteurs discontinus. Th&se Institut National Polytechnique de Lorraine Nancy. Kuester, J.L., Mize, J.N., 1973, Optimization techniques with Fortran. MC Graw-Hill Book Company, New York, p 399-411. Marchal-Brassely, S., Villermaux, J., Houzelot, J.L., 1988, Utilisation d’un algorithme autoadaptatifpour la conduite optimale des dacteurs discontinus. SIMO 88, R&ents progr&sen Genie des pro&d& 2, 6,93-98. Marchal-Brassely, S., Villermaux, J., Houzelot, J.L., Georgakis, C., Bamay, J.L., 1989, Une m&hode itkrative efficace d’optimisation des profils de tempdrature et de ddbit d’alimentation pour la conduite optimale des reacteurs discontinus. Toulouse 89, Rkents progr&s en G6nie des pro&d&s 3,9,441-446. Marchal-Brassely, S., 1990, Conduite optimale de reacteurs discontinus. ThBse Institut National Polytechnique de Lorraine Nancy. Pontrvarrin. L-S.. Boltvanskii. V.G.. Gamkrelidze. R.V.. Mishenhenko, E.P., 1964. The mathem&&l theory of opiimal p&es&s. Pergamon gess, New York. Rippin, D.W.T., 1983, Simulation of single and multiproduct batch chemical plants for optimal design and operation. Comp and Chem. Eng. 7,3, 137-156. Staha, R.L., 1973, Constrained optimization via moving exterior truncations. Ph. D. Dissertation, University of Texas, Austin. Villermaux, J., 1985, Genie de la reaction chimique, Conception et fonctionnement des r&cteurs. Technique et Documentation, Lavoisier.