Graphical Solution for a Multistage Continuous Feed-and-Bleed Ultrafiltration System
Consider a continuous feed-and-bleed ultrafiltration multistage unit, using the gel polarization model for the limiting flux.[more]
A protein solution is fed into an -stage continuous feed-and-bleed ultrafiltration system at a rate of 1 L/min. The feed enters at 10 g/L and the gel concentration can be taken to be 300 g/L. The mass transfer coefficient is in each stage and the corresponding area equals 0.9 .
You can use a graphical approach developed by Foley  to obtain a value of the retentate concentration that exits the last membrane separation module. The blue curve is the flux curve, which can be thought of as an equilibrium line. The green lines trace the progress of the operation.
This Demonstration displays this approach and gives the value of the concentration factor (i.e. the ratio of the exit concentration to the inlet concentration). It is clear that beyond four stages, improvement of the concentration factor is negligible.[less]
 G. Foley, "Solution of Algebraic Equations in the Analysis, Design and Optimization of Continuous Ultrafiltration," Chemical Engineering Education, 45(1), 2011 pp. 59–64.