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