Particle Deposition Rates
The rate of deposition of the particles of the slurry over the slurry side cake surface is assumed proportional to the particle mass flux approaching the filter cake. The rate of erosion of the particles from the slurry side cake surface is assumed proportional to the tangential, excess shear stress above the critical stress necessary for particle mobilization. Therefore, for dynamic filtration involving cross flow, the net mass rate of all particles (large plus small) deposition per unit area of the slurry side cake surface is given by the difference between the deposition and erosion rates as (Civan, 1996, 1998b):
where ut is the carrier fluid filtration flux normal to the cake surface (cm3/ cm 2 • s) and Cpl is the slurry particle concentration expressed as the particle mass per unit volume of the carrier fluid in the slurry. For small particles retention over the slurry side of the cake, an expression similar to Eq. 12-134 can be written as:
in which (Cp2i)slur denotes the mass of small particles per volume of the carrier fluid in the slurry. The net mass rate of deposition of small particles within the filter cake is given by:
which is similar to Eq. 33 of Corapcioglu and Abboud (1990), but the deposition and the mobilization terms are more consistently expressed. The rate expressions given by Eqs. 12-134 through 136 for the deposition of the total (fine plus large) and fine particles of the slurry over the progressing cake surface and the retention of the fine particles of the flowing suspension within the cake matrix can be expressed in terms of the volumetric rates, respectively, as (Civan, 1999b):
In Eqs. 12-10 through 12, the slurry shear-stress, ts, acting over the progressing cake surface is estimated using the Rabinowitsch-Mooney equation (Metzner and Reed, 1955). This equation can be expressed for linear and radial flow cases, respectively, as follows:
where ḱr and n' are the consistency (dyne/cm2/sn') and flow (dimensionless) indices, which are equal to the fluid viscosity, jo, and unit for Newtonian fluids, respectively, and v is the tangential velocity of the slurry over the filter cake surface. For static filtration, v = 0 and therefore T = 0, and the second term on the right side of Eq. 12-134 drops out, leading to an expression similar to Corapcioglu and Abboud (1990) and Tien et al. (1997).
tcr is the minimum slurry shear-stress necessary for detachment of particles from the progressing cake surface. Following Ravi et al. (1992), the critical shear stress necessary for detachment of the deposited particles from the progressing cake surface can be estimated according to Potanin and Uriev (1991) by Eq. 12-6. However, the actual critical stress can be substantially different than predicted by Eq. 12-6, because the ideal theory neglects the effects of the other factors, including aging (Ravi et al., 1992), surface roughness, and particle stickiness (Civan, 1996) on the particle detachment.
Therefore, Ravi et al. (1992) recommend that the critical shear stress be determined experimentally. U(ts-tcr) is the Heaviside unit step function. It is equal to zero when ts < tcr and one for Ts≥tcr. k°d and k°e are the rate coefficients for the total (fine plus large) particles deposition and detachment at the progressing cake surface. k°2 and k°2 are the rate coefficients for the fine_ particles deposition and detachment at the rogressing cake surface. ҟd and ҟe are the cakethickness-average rate coefficients for the deposition and mobilization of the fine particles within the filter cake matrix.
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