A model for transient cross flow filtration in a narrow rectangular domain

• Analytical solutions are developed for transient solution with cake bed deposition and depth plugging.
• Governing equations include a generalized Darcy’s law and distribution function controlling ratio between fouling phenomena.
• Design parameters are used to derive parameter operating limits.
• Comparisons are made between clean filter (no-fouling case) and transient solution.

Analytical solutions for flow rate, pressure, and permeate flux, which vary with time and distance, are developed for a cross flow filter in a narrow rectangular domain with either one or two permeable walls. The governing equations consist of a generalized Darcy’s law combined with flow and continuity equations. The contribution to fouling as either a static surface cake or depth plugging in the media is governed by a distribution function representing the ratio of the two phenomena. The mathematical model and solution for the clean filter case (i.e. no fouling) are presented first, along with design parameters (characteristic length, flow rate, and pressure) that arise from the solutions. The analytical solutions for the no-fouling case yield limiting values of filter length, inlet flow rate, and inlet pressure drop. These bounds also provide reasonable limits for operating parameters in the transient model with fouling. Parameter studies that compare transient effects with fouling to the clean filter case are presented to illustrate the application of the model.