A

Optimal Temperature Control

of Semibatch Polymerization Reactors

H Hinsberger S Miesbach and H J Pesch

1

Research Scientist Clausthal University of Technology Institute of Mathematics

Erzstr D ClausthalZellerfeld Germany

Email hinsb ergmathtuclausthalde

2

Research Scientist Siemens AG Corp orate Research and Development ZFE T SN

OttoHahnRing D Munich Germany

Email stefanmiesbachzfesiemensde

3

Professor of Mathematics Clausthal University of Technology Institute of Mathem

atics Erzstr D ClausthalZellerfeld Germany Email p eschmathtuclausthalde

B

Abstract Recently in a pap er of Chylla and Haase a mo del of a multi

pro duct semibatch p olymerization reactor has b een developed which is repres

entative of those found in the sp eciality chemical pro cessing industry One of

the aims in these pro cesses is to keep a certain reaction temp erature setp oint

in order to t the quality requirements for the p olymer

In the present pap er the optimal solutions of the underlying optimal con

trol problems of the ChyllaHaase reactor which have b een computed by a

new direct multiple sho oting metho d are discussed It can b e shown that

the rst of the two pro ducts for which physical data are given in can b e

controlled along its required constant reaction temp erature setp oint while for

the second pro duct this cannot b e achieved b ecause of certain mathematical and technical reasons

Introduction

Theory and numerical metho ds for the solution of optimal control problems have

reached a high standard There is a wide range of applications the most challenging

of which are from the eld of aerospace engineering and rob otics see for example

the survey pap er

However in chemical engineering only a few problems of this type can b e found to

b e fully describ ed in the literature although it is wellknown that many problems in

chemical engineering can b e formulated as problems of optimal control The reason

for that is that mo deling of chemical pro cesses and providing of appropriate data is

surely more dicult than in many other areas On the other hand new applications

may often provide new challenges even for tried and wellestablished mathematical

metho ds

One of the rare pap ers containing a fully describ ed optimal control problem for

chemical pro cess control was recently published by Chylla and Haase Their

intention was to provide the control community with a complete mo del to test the

pro ducts of their research

Well we have picked up their challenge and present optimal solutions for their

control problems In the present pap er a direct multiple sho oting metho d has b een

used to compute these optimal solutions The solutions obtained help to explain

why for one of two pro ducts sp ecied in there arise diculties in controlling

the pro cess at the prescrib ed constant reaction temp erature which is required to

pro duce an acceptable pro duct with resp ect to its quality

Mathematical Mo del

Sp eciality emulsion p olymers have usually to b e made in a stirred tank reactor in

semibatch pro cesses The reaction temp erature determines the chemical comp osi

tion and particle size distribution of the emulsion p olymer Therefore the reaction

temp erature must b e hold at a constant level to guarantee an acceptable pro duct

It is controlled by the temp erature of the water recirculating through the jacket of

the reactor

For the b enet of the reader the control mo del of the ChyllaHaase reactor

which is based on an extremely oversimplied kinetic mo del describing fairly the

conversion vs time b ehavior in the reactor is summarized as far as it will b e used

in the present pap er

The conversion of the monomer into the p olymer and the energy balance around

the reactor can b e describ ed by the following system of ordinary dierential equa

tions for the mass m kg of monomer and the mass m kg of p olymer in the

M P

reactor and for the reactor temp erature T K

dm

M

in

m R
MW

P M

M

dt

dm

P

R
MW

P M

dt

dT N

dt D

with the abbreviations

in

N m C T T UA T T V T T R H

p M amb j amb P P

M

D m C m C m C

M p M P p P W p W

Here the following partly empirical relations are used

in

m m

M

M

F R i k n n

M M M

MW MW

M M

E

k

fc aT c g

2

2 3

k c exp c f
k k exp

RT

a m

P

aT f

T m m m

M P W

m P m m

M P W

A
B U

B

M P W

h h

f

fc aT c g

2 3

wall

h d exp d c exp c f

wall wall

T T T

j wall

in

kg s for the prescrib ed piecewise constant mass owrate of The notations are m

M

monomer F kmol s for the molar owrate of monomer R kmol s for the rate

M P

of p olymerization n kmol for the number of moles of monomer MW kg kmol

M M

for the molecular weight of the monomer mix i for an impurity factor k s

for the rstorder kinetic constant kg m s for the batch viscosity m kg

W

for the mass of water in the reactor aT and f for auxiliary variables

C kJ kg K for the sp ecic heats at constant pressure H kJ kmol

p MPW P

for the reaction enthalpy at T T H is the heat of p olymerization T K

set P set

for the reaction temp erature setp oint T K for the ambient air temp erature

amb

T K for the average jacket temp erature which will b ecome the control variable

j

U kW m K for the overall heat transfer co ecient A m for the jacket heat

transfer area V kW K for the heat loss to the environment B m for the re

actor b ottoms area P m and B m for the jacket p erimeter and the jacket b ot

toms area resp ectively kg m for the densities of the monomer the p oly

MPW

mer and water h kW m K for the lm heat transfer co ecient and nally

h m K kW for a fouling factor dep ending on the batch number

f

All other quantities are constants dep ending either on the reactor or on two

pro ducts named A and B in Their values rounded from those due to the

English units used in and metric units are given in Tables and

Table Data of the reactor

Symbol Unit Values

T K Winter

amb

Summer

T K T

amb

V kW K

B m

P m

B m

Contrary to the mo del in we drop here the equations for the heating and

co oling system via the recirculation lo op There are several reasons for it Firstly it

is sucient for a rst investigation to choose the average jacket temp erature T as the

j

single control variable This yields the b est p ossible result anyway since the jacket

temp erature can b e controlled by the recirculation lo op only indirectly Secondly

the equations for the recirculationlo op mo del given in the pap er of Chylla and

Haase without any derivation are doubtful since they do not represent an energy

balance b etween reactor and jacket Finally their dierential equations are of delay

and neutral type for which optimal control metho ds are still under development

The optimal control mo del is now completed by giving the ob jective function

and initial andor b oundary conditions In order to guarantee the aforementioned

pro duct quality the deviation of the reactor temp erature from the reaction temp er

ature setp oint must b e minimized over a xed time interval t t Hence we can

f

require

t

f

Z

T t T dt min

set

t

0

The initial conditions result from the recip es for the two dierent pro ducts in

For pro duct A the recip e is as follows i Make an initial charge of monomer

p olymer and water to the reactor at ambient temp erature and at time t ii heat

up the reactor until the reaction temp erature setp oint is reached which determines

a time interval t iii add monomer under a constant feedrate to the reactor

and hold the temp erature setp oint over a given time interval t t iv hold the

temp erature setp oint over another given time interval t t The same recip e is

prescrib ed for pro duct B except that the steps iii and iv have to b e rep eated

once

The data sets for the initial conditions and the time instancies of the recip es are

also given in Tables and

Table Data of the pro ducts A and B

Symbols Unit Values for A Values for B

m kg

M

m kg

P

m kg

W

C kJ kg K

p M

C kJ kg K

p P

in

m kg s

M

MW kg kmol

M

i i 2 random constant in batch

k s

k m s kg

k

E kJ kmol

R kJ kmol K natural constant

c kg m s

c

c

c

a K

H kJ kmol

P

kg m

M

kg m

P

kg m

W

d kW m K

d m s kg

h m K kW

f for batchnumbers

Table Data of the pro ducts A and B Cont

Symbols Unit Values for A Values for B

T K

set

s t t t t

s t t t t t t

f

t s t

s t t t

f

Numerical Results

For the numerical solution of optimal control problems there are basically two well

established approaches the indirect approach e g via the solution of multipoint

b oundaryvalue problems based on the necessary conditions of optimal control the

ory and the direct approach via the solution of constrained nonlinear programming

problems based on discretizations of the control andor the state variables The

application of an indirect metho d is not advisable if the equations are to o complic

ated or a mo derate accuracy of the numerical solution is commensurate with the

mo del accuracy Therefore the easiertohandle direct approach has b een chosen

here Direct collo cation metho ds see e g Stryk as well as direct multiple

sho oting metho ds see e g Bo ck and Plitt b elong to this approach In view of

forthcoming large scale problems we will fo cus here on the direct multiple sho oting

metho d since only the control variables have to b e discretized for this metho d This

leads to lower dimensional nonlinear programming problems

Based on a multiple sho oting metho d for parameter identication in differen

tialalgebraic equations due to Heim a new implementation of a direct multiple

sho oting metho d for optimal control problems has b een developed which enables

the solution of problems that can b e separated into dierent phases In each of these

phases which might b e of unknown length the control b ehavior due to inequality

constraints the dierential equations even the dimensions of the state andor the

control space can dier For the optimal control problems under investigation the

dierent phases are concerned with the dierent steps of the recip es

All the arising nonlinear programming problems for the computations of this

pap er have b een solved by the metho d NPSOL of Gill et al

Since a more detailed description of the multiple sho oting metho d would b e

b eyond the scop e of this pap er we continue with the numerical results The optimal

solutions of the control problems are given for the masses of monomer and p olymer

and for the reactor temp erature of pro duct A in Figs and of pro duct B in

Figs The controls the average jacket temp eratures are compared with

each other in Figs and Note that the initial heating phases are not shown

in the gures

For b oth pro ducts and recip es resp ectively the mass of p olymer increases nearly

linearly during the feed phases of monomer see Figs and and and

Here the reactor must b e co oled down since the reaction releases energy see Figs

and On the other side the mass of p olymer remains nearly constant when no

monomer is fed into the reactor Here the reaction comes to a standstill The loss

of energy to the environment must b e replaced through a heating phase to keep

the reactor temp erature on the prescrib ed reaction temp erature setp oints Figs

and show that the setp oint can b e p erfectly kept for pro duct A while for

pro duct B the reactor temp erature shows considerable deviations from the pre

scrib ed setp oint During the second monomer feed the pro cess b ecomes uncontrol

lable for pro duct B b ecause of the exp onentially increasing viscosity of the reactor

contents and the resulting overexponentially increasing lm heat transfer co ecient

compare the denition of h after Eq This results in a sharp decay of the over

all heat transfer co ecient U towards zero Hence the equation T cannot b e

solved for the control variable T see Eq In technical terms a highly viscous

j

lm forms at the reactor wall and prevents so the heat transfer b etween jacket and

reactor contents

m kg

M

t sec

Fig Mass of monomer vs time for pro duct A

m kg

P

t sec

Fig Mass of p olymer vs time for pro duct A

Conclusion and Outlo ok

Mathematical theory and stateoftheart numerical metho ds p ossess a great ability

in computing optimal solutions for pro cess control in chemical engineering which

T K

t sec

Fig Reactor temp erature vs time for pro duct A

T K

j

t sec

Fig Average jacket temp erature vs time for pro duct A

is until to day not exhausted compared to other elds The investigation of the

optimal temp erature control of a semibatch p olymerization reactor b eing still a

comparatively simple problem might show some of the reasons why mathematics

is not used as it should b e There is a lack of reliable mo dels for chemical pro cess

control which include realistic data Mo deling is surely more dicult as in some

other applications also the mo del of needs some revision and contains several

misprints On the other hand any eort should b e made to develop more reliable

and realistic mo dels to the b enet of b oth chemical engineering and mathematics

It should b e noted that optimal control theory together with numerical metho ds

can provide solutions to problems with more sophisticated ob jective functions e g

minimum time maximum mass maximum pro duct quality etc to those where

inequality constraints for the state as well as the control variables are imp osed to

those which are of considerably larger scale and partly also to those where more

general dynamical equations in particular dierentialalgebraic equations are in

volved Esp ecially large scale problems and dynamical equations of nonordinary

type will b e a challenge for the development of new metho ds in the future Finally

questions related to optimal realtime control metho ds are far from b eing answered

satisfactorily compare The uncertainties known to app ear in chemical pro cesses

are the great challenge here even for nonoptimal standard controllers

Acknowledgement This work was nancially supp orted by the German

Ministry of Research and Technology within the pro ject Nonlinear Dynamics in

Chemical Technology ModelBased Process Control under co de numbers D A

and D A Moreover the authors would like to thank Prof DrIng W Mar

m kg

M

t sec

Fig Mass of monomer vs time for pro duct B

m kg

P

t sec

Fig Mass of p olymer vs time for pro duct B

quardt and DiplIng A Helbig from Aachen University of Technology for fruitful

discussions on the mo deling of semibatch pro cesses

References

Bo ck H G Plitt K J A Multiple Shooting Algorithm for Direct

Solution of Optimal Control Problems Pro c of the th IFAC Worldcongress

Budap est Hungary Vol IX Collo quia

Chylla R W Haase R Temperature Control of Semibatch Polymer

ization Reactors Chem Engng and

Gill P E Murray W Saunders M A Wright M H Users Guide

for NPSOL Version Department of Op erations Research Stanford Uni

T K

t sec

Fig Reactor temp erature vs time for pro duct B

T K

j

t sec

Fig Average jacket temp erature vs time for pro duct B

versity Stanford California Rep ort SOL

Heim A Parameteridentizierung in dierentialalgebraischen Glei

chungen Department of Mathematics Munich University of Technology Mu

nich Germany Diploma Thesis

Pesch H J Oine and Online Computation of Optimal Trajectories in

the Aerospace Field Applied Mathematics in Aerospace Science and Engineer

ing Edited by A Miele and A Salvetti Plenum Publishing Corp oration New

York New York

Stryk O von Numerische Losung optimaler Steuerungsprobleme Dis

kretisierung Parameteroptimierung und Berechnung der adjungierten Vari

ablen FortschrittBerichte VDI Series No VDIVerlag D usseldorf Germany