Optimal Permeability Character for Quasi-Static Countercurrent Dialysis

An important parameter in the design of countercurrent dialyzers is the permeability of the membrane which separates the blood and dialyzate. In this paper, methods of optimization theory are applied to specify membrane permeability along the length of a countercurrent dialyzer operating under quasi-static conditions. Two optimization problems are considered. In the first the performance functional is a quadratic measure of membrane complexity; in the second it includes also a quadratic measure of the deviation of the blood output concentration from a desired (low) level. For both situations it is shown that the optimal membrane is one of constant permeability along its length, a result which is intuitively appealing and has an obvious economic consequence. Formulas for computing the optimal permeabilities are given, and some problems for future research in this area are outlined. A generalization of the principal result is also included, and some deficiencies in the present analysis are discussed.