Optimal balancing of temporal and buffer costs for ultrafiltration/diafiltration processes under limiting flux conditions

This paper studies the problem of economically oriented optimal operation of an ultrafiltration/diafiltration process that is designed to reduce the initial volume of a given process liquor and to eliminate impurities from the product solution in a batch setup. This theoretical investigation focuses on applications where the permeate flux is given by the well-known limiting flux model and the rejections of micro-solute and macro-solute are assumed to be zero and one, respectively. Unlike previous approaches to the problem, we consider a complex economical objective that accounts for the total operational costs involving both the cost of consumed diluant and processing time-related costs. The optimization problem is formulated as a multi-objective optimal control problem and it is solved using the analytical approach that exploits Pontryaginʹs minimum principle. We prove that economically optimal control strategy is to perform a constant-volume diafiltration step at a given, optimal macro-solute concentration. This constant-volume diafiltration step is preceded and followed by 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. By taking into account the unit prices of both processing time and utilized diluant, we provide a practical algebraic formula that allows decision makers to evaluate the optimal starting point of the constant-volume diafiltration step and to adapt it when considered prices change. Finally, we demonstrate the applicability and achievable benefit of the here presented approach on an industrial-scale case study using literature data.