## Filtration

A mathematical description of liquid drainage from a packed bed by centrifugal forces is essentially the same as that used to describe more conventional gravity or differential-pressure filtration, the primary differences being that the centrifugal force or the pressure generated by the centrifugal force is substituted for the gravitational or differential-pressure terms. As filtration is an extensively characterized field of study, a description of which is beyond the scope of this article, it is recommended that the reader refer to the literature for an in-depth mathematical discussion of both conventional and centrifugal filtration. However, a brief summary of some of the more important parameters that govern flow velocity and pressure drop during centrifugal filtration follow. A simple basket centrifuge is shown schematically in Figure 2. Assuming a constant height of liquid within the basket, the velocity of the filtrate, u, through a given cake thickness, dl, is given by the relation:

where H is the basket height or length (2nr'H is the cross-sectional area of the filter); r' is the distance from the axis of rotation to the inner cake surface; dV/dt is the volume of filtrate passing in time dt; a is the specific resistance of the cake; ¿u is the viscosity of the filtrate; dP' is the pressure drop across a given thickness of filter cake; and dl is a given cake thickness.

The velocity of the filtrate through the cake and underlying filter is thus proportional to the volume of filtrate flow or the pressure differential across the filter cake, and inversely related to the surface area of the filter, filtrate viscosity and cake resistance. Eqn [24] may be rearranged and integrated to determine the total pressure drop across the cake at time t :

where r is the distance from the axis of rotation to the outer cake surface.

If the resistance of the filter is negligible, AP' is equivalent to the centrifugal pressure. A parameter that is widely used to characterize the performance of filtration equipment is the drainage number: