ChemicalEngineering Science, 1973. Vol. 28, pp. 993-1003. Pergamon Pnss. Printed in Great Britain zyxwvutsrqponmlkjihgfedcbaZYXWVUTSR Practical control studies of batch reactors using realistic mathematical models G. MARROQUlNt and W. L. LUYBEN Department of Chemical Engineering, Lehigh University, Bethlehem, Pa, U.S.A. (Received 4 May 1972; accepted 6 August 1972) Abstract - zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA M ost of the optimal control studies of batch chemical reactors have assumed simple, ideal systems where heat removal can be changed instantaneously between zero and some maximum rate. These systems are of low order, usually second, so elegant variational mathematics can be employed to compute optimal time-temperature trajectories. This paper reports results of studies of more realistic systems in which the dynamics of the coolii and heating systems are not neglected, giving systems of much higher order (7th- 10th). Reversible and consecutive exothermic reactions are considered. A variety of practical heat removal systems are studied. Reactor nerformance is found to be reasonablv insensitive to some of the practical departures tram ideality, but trite sensitive to others. INTRODUCTION BATCH chemical reactors, although they occur less frequently than continuous reactors, are still extensively used in the polymer, chemical and pharmaceutical industries because of their intrinsic kinetic advantages in some systems. Their dynamic nature and the wide variations in conditions during each cycle can make their control difficult. Some reactions, such as reversible, consecu- tive and parallel exothermic reactions, are particularly interesting from a control point of view because the time-optimal temperature profile is one that varies continuously during the batch cycle. Denbigh[3] showed how the maximum rate path (MRP) for a reversible reaction A+B could be calculated by differentiating the reac- tion rate expression with respect to temperature and equating to zero. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA T Es-h - = R In (Z&2Ce/ZIEICA) (1) where TMRp = temperature on the maximum path El and Et = activation energies of forward reverse rates Z1 and Z2 = frequency factors of forward reverse rates CA= reactant concentration along MRP CB = product concentration along MRP. rate and and the the The maximum rate path is easily generated by picking a value of CB, calculating CA from the stoichiometric relationship c*= CAV - cc, - Cm) and calculating T- from Eq. (1). The variables CA,, and CBO are initial concentrations. The optimum temperature decreases with time when El is less than Es. Figure 1 shows the maximum rate path for reversible reactions in a phase plane plot. The parameters used are given in Table 1. Figure 1 also shows the equilibrium line, which is ob- tpresent address: Department of Chemical Engineering, National Polykchnic Institute, Mexico. 993 CES Vd. 28, No. 4-A