# Linear Filtration Formulation

The radial filter cake equations derived above can be readily converted to linear filter cake equations by means of the transformation given by:

Thus, application of Eq. 12-102 to Eqs. 12-87 respectively, the following thickness-averaged mass balance equations for the linear cake formation (Civan, 1998b):

Similarly, Eqs. 12-97 through 100, respectively, become (Civan, 1999b):

Eqs. 12-107 through 110 can be solved numerically, subject to the initial conditions given by:

The volume fractions of the filter cake solids and pore fluid can be expressed in terms of the cake porosity, respectively, as

where ɸ is the average cake porosity (cm3/cm3). The following expressions for the small particle volume flux and mass per carrier fluid volume
can be written according to Eqs. 12-94 through 96, respectively, as:

Note that Eqs. 18, 28, and 24 of Corapcioglu and Abboud (1990) correspond to the present Eqs. 12-103, 12-109, and 12-106, respectively, with some differences. Equation 12-103 simplifies to their Eq. 18, assuming pp is constant and substituting Eq. 12-112. The present Eqs. 12-106 and 12-109 simplify to their Eqs. 24 and 28, substituting Eq. 12-113 for Ԑp2l«Ԑl. Also, Corapcioglu and Abboud (1990) did not distinguish between the rates of deposition of small and all (large plus small) particles over the progressing cake surface (i.e., R°p2s and R°ps) and used R°p2s = R°pS in their Eq. 24.

This assumption is not valid because most small particles migrate into the cake and only a little fraction of the small particles can deposit by the jamming process over the slurry side of the filter cake, such as described by Civan (1996). The filter cake is essentially formed by the deposition of the large particles and the deposition of the small particles over the progressing cake surface is negligible. The deposition of the small particles more dominantly occurs within the cake matrix as the suspension of small particles flows through the cake.

Therefore, there is a large order of magnitude difference between the rates of the small and large particles deposition over the filter cake. However, it is more accurate to use Eq. 12-91.

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