When should batch ultrafiltration be stopped to maximise profit in the recovery of a non-rejected solute?

• Use of logarithmic Integral to devise a simple expression for process profit in product recovery with batch ultrafiltration.
• Derivation of simple expression for the stopping concentration to maximise profit.
• Proof of no optimum during constant flux operation.

The optimisation of batch ultrafiltration for product recovery is analysed. Assuming limiting flux operation and rejection coefficients of one and zero for the macrosolute and microsolute respectively, an expression is derived for the total profit of the process where that profit is defined as the value of the recovered microsolute minus the operating cost. It is found that in order to maximise the profit, the ultrafiltration should be stopped when the macrosolute has reached a concentration clim/eα where clim is the so-called limiting concentration of the macrosolute and α is a dimensionless number that represents the ratio of operating costs to product value. No optimum exists when the system is operated at constant flux.