# Pressure-Flow Relationships

The slurry carrier fluid flow rate ut(cm3/cm 2-s) can be expressed using the effluent fluid pressure Pe(atm) at the outlet side of the filter, the pressure Pc(atm) at the slurry side cake surface, and the harmonic average permeability of the cake and filter system. Forchheimer's (1901) law of flow through porous media for the linear case is given by

The pressure differences over the filter cake and the porous media can be expressed by integrating Eq. 12-116, respectively, as (Civan, 1999b):

The instantaneous volumetric fluxes and densities of the suspensions of fine particles flowing through the cake matrix and porous formation are assumed the same. Then, adding Eqs. 12-117 and 118, and rearranging and solving, yields (Civan, 1999b) for Darcy flow (βf =βc =0j:

Although the preceding approach yields a reasonably good accuracy, a more rigorous treatment should facilitate

Forchheimer's law (1901) for the radial flow case is given by

where h is the formation thickness. Thus, invoking Eq. 12-125 into Eq. 12-124 yields

The pressure differences over the filter cake and porous media can be expressed by integrating Eq. 12-126, respectively, as (Civan, 1999b):

The densities and instantaneous flow rates of the suspensions of fine particles flowing through the cake matrix and porous formation are assumed the same. Then, adding Eqs. 12-127 and 12-128, and rearranging and solving, yields (Civan, 1999b) for Darcy flow (βf =βc =0):

double Although this approach leads to a reasonably good estimate, a more rigorous approach should employ

The inertial flow coefficient is estimated by the Liu et al. (1995) correlation given by Eq. 12-16.

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