Figure 2. Important parameters in cake formation.
The drag that is imposed on each particle is transmitted to adjacent particles. Therefore, the net solid compressive pressure increases as the filter plate is approached, resulting in a decrease in porosity. Referring to Figure 3A, it may be assumed that particles are in contact at one point only on their surface, and that liquid completely surrounds each particle. Hence, the liquid pressure acts uniformly in a direction along a plane perpendicular to the direction of flow. As the liquid flows past each particle, the integral of the normal component of force leads to form drag, and the integration of the tangential components results in frictional drag. If the particles are non-spherical, we may still assume single-point contacts between adjacent particles as shown in Figure 3B.
Consider flow through a cake (Figure 3C) with the membrane located at a distance x from the filter plate. Neglecting all forces in the cake other than those created by drag and hydraulic pressure, a force balance from x to L gives:
The applied pressure p is a function of time but not of distance x. Fs is the cumulative drag on the particles, increasing in the direction from x = L to x = 0. Since single point contact is assumed, the hydraulic pressure pLis effectively over the entire cross section (A) of the cake; for example, against the fictitious membrane shown in Figure 3B. Dividing Equation 1 by A and denoting the compressive drag pressure by ps= F/A, we obtain:
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