Given the filtration conditions and the specifications of a filter, the common problems to be solved are (1) the amount of the filtrate obtainable in a given time, (2) the amount of the solvent that can be passed through the cake in a given time, and (3) concentration of the recovered material in the wash solvent.

Filtration can be carried out in two ways: (1) constant-pressure filtration, i.e., the filtration rate varies, and (2) constant-rate filtration in which the ? P varies. Two methods are available to treat filtration problems. They are (1) Ruth's equation ^[1] and (2) the Kozeny-Carman ^[2] relation.

Ruth's Equation

To establish Ruth's equation, the following variables are first defined:

• V = filtrate, ft ³

• ? = filtrate density, lb/ft ³

• W = filterable solids, lb

• 

• x = mass fraction of the solids in a slurry

Using these definitions, the following relations are developed:

from which

The Hagen-Poiseuille equation for the laminar flow of fluids in tubes of circular cross section is given by

Assuming Eq. (11-2) applies to the filtrate flow, it can be shown that the rate of filtration is given by

where A	= total filtering area, ft 2
?	= time of filtration, h
?	= viscosity of filtrate, lb/ft h
L	= thickness of cake, ft
? P C	= pressure drop across cake, lbf/ft 2
R C	= resistance of cake = 32 L/ g cD 2 e, h 2 lbf/ft