Granular filtration of hydrosols and aerosols

We will briefly describe the filtration behavior of hydrosols in a granular filter bed, but first, it is useful to conceptualize the structure of the granular medium. The bed is assumed to consist of a large number of unit bed elements (UBEs) in series (Tien, 1989). Each UBE has a certain type of flow channel, and the granular medium surface acts as a particle collector, collecting particles from the fluid flowing in the channel. The porous medium in each UBE may be represented in a number of ways. 

Very detailed models of granular filtration have been developed using such idealizations of the porous medium (Tien, 1989). A simple aspect of this filtration is that, as the liquid flows down the porous medium, particles are transported perpendicular to the flow by electrostatic force. Brownian difiusion, various surface interaction forces and interception. Gravitational force is also present its 

Pp(4/3)jt/ dtp to arrive at equation (6.2.55), the De term will become De)p (4/3)nr dr. From the description of the De-containing term in (6.2.50h), we conclude that (De)pp(4/3)nr drp will represent the mass rate of disappearance of particles of size tp per unit filter volume we will represent it as (De). Further, for incompressible flow, if we assume that all particles are of the same size Tp, then we can rewrite equation (6.2.55) after a control volume analysis in the manner of equation (6.2.50i) as follows (with Dp replaced by D reflecting dispersion 

If we ignore any dispersion effects and employ only the interstitial velocity in the z-direction, (Vp Vg not being relevant), we get the time-dependent z-directional equation for particle mass concentration, Bp,rp the liquid  

We identify as the volume of particles deposited per unit filter volume, a specific volume based deposit. Then 

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