(693f) Time Optimal Control of Molecular Weight Distribution Using Orthogonal Collocation on Finite Elements and Fixed Pivot Methods

In the present work, a theoretical and experimental investigation on the dynamic optimization of the molecular weight distribution (MWD) for linear free-radical polymerization systems is presented. The MWD calculations were carried out by two different computational methods, namely, the orthogonal collocation of finite elements (OCFE) method and the fixed pivot (FP) technique. The OCFE formulation treats the discrete polymer chain length domain as a continuous one. The chain length domain is divided into a number of finite elements and within each element a number of collocation points are specified from the roots of appropriate by selected orthogonal polynomials. The concentrations of the ?live? and ?dead? polymer chains are approximated by low-order Lagrange interpolation polynomials. The residual equations arising from the population balances for the ?live' and ?dead' polymer chains are assumed to be satisfied exactly at the collocation points. The closure of the overall material balance is ensured regardless of the selected kinetic mechanism. The fixed pivot technique guarantees the correct calculation of any two moments of the MWD. The ?live? and ?dead? polymer chains are assumed to be concentrated at selected representative chain lengths (i.e., discretization points). Specific reaction steps leading to the formation of ?live' and ?dead' polymers, having chain lengths different than the ones corresponding to the selected representative chain lengths, are assigned to the nearest neighboring discretization points in such a way so that any two moments of the chain length distribution (CLD) are exactly preserved. It is shown that for linear polymers, both numerical methods (i.e., OCFE and FP) yield identical results for both narrow and bimodal MWDs. Two time optimal temperature policies that ensure the satisfaction of desired polymer quality specifications (e.g., MWD) are derived through the solution of a dynamic optimization problem, using a sequential optimization approach. The minimization of the objective function (i.e., the normalized square deviation of the MWD from a desired one) is carried out, using the OCFE method for the MWD calculations, for two selected cases; a narrow MWD and a bimodal one. The calculated time optimal temperature trajectories were then applied to a free-radical MMA batch polymerization pilot plant unit for the experimental validation of the time optimal policies. The time optimal temperature profiles were easily maintained using a cascade control system by manipulating the flow rates of two water streams (i.e., a hot and a cold). It is shown that experimental results on MWD, monomer conversion, reactor and jacket temperatures are in excellent agreement with theoretical model predictions.