Phenomenological theory of depth membrane filtration

A phenomenological theory of depth membrane filtration (DMF), in which outside-in hollow fiber membranes are used as collectors of colloidal particles, is developed to study the performance of a hollow fiber membrane filter with two product streams, permeate (clarified liquid that passed through semipermeable membrane) and filtrate (clarified liquid due to the collection of suspended particles on the external surface of hollow fibers). The theory is based on the general phenomenological expression for the rate of particle deposition on the membrane surface, in which the deposition rate is proportional to the product of the particle deposition coefficient and the concentration of suspended particles, with the deposition coefficient being an arbitrary function of the mass of deposited particles and permeate velocity. The system of governing equations is solved by the generalized Crank–Nicholson finite-difference method and the approximate method using the averaging of the permeate velocity in the mass conservation equation. The effect of membrane permeability on the filter productivity is studied. It is shown that the approximate solution can be used as a relatively simple and accurate tool to study and design hollow fiber filters for DMF.