Minimizing the process time for ultrafiltration/diafiltration under gel polarization conditions

This study examines a generalized ultrafiltration/diafiltration process that is designed to reduce the initial volume of a given process liqueur and to eliminate impurities from the product solution. This theoretical investigation focuses on applications where the permeate flux is given by the gel polarization model. The goal of this paper is to use optimal control theory to determine optimal time-varying diluant addition that minimizes treatment time. We propose a diafiltration model in a dimensionless form with normalized model equations in order to determine general features of optimal diluant utilization strategy. Based on the model, we formulate the optimal control problem and apply the theory of optimal control exploiting the Pontryagin’s minimum principle. We confirm the analytical results by numerical computations using numerical methods of dynamic optimization. We prove that optimal control strategy is to perform a constant-volume diafiltration step at optimal macro-solute concentration that guarantees maximal removal of micro-solute at any time instant. This constant-volume diafiltration step is preceded and followed by optional ultrafiltration or pure dilution steps that force the concentrations at first to arrive to the optimal macro-solute concentration and at last to arrive to the desired final concentrations. Finally, we provide practical optimization diagrams that allow decision makers to determine the optimal diluant control of a given separation task.