A simplified mathematical model for deposition in a filter

We develop a mathematical model to analyze filter fouling behaviors within a simplified two-dimensional model of a porous medium and to predict filter efficiency based on these analyses. The model distinguishes itself through its inclusion of all three fouling mechanisms -- adsorption, blocking, and cake -- and by simulating a moving boundary in the cake. To account for changeable conditions, we vary coefficients that quantify both the tendency of particles to foul the filter and the resulting changes in porosity. We find that increases in cake formation and cake fouling prolong the life of a filter and lead to more purified filtrates, since the cake allows particles to accumulate within it before those particles reach the membrane. These benefits are augmented by a smaller energy cost, as the large particles that comprise the cake hold much higher porosity than does the membrane, thereby necessitating less added pressure to maintain the prescribed constant inlet flux of particles.