Sludges may be reduced in volume by using the unit operation of dewatering to remove excess water. In dewatering using a vacuum (such as using a vacuum filter) or the plate-and-frame press, a cake is formed on the surface of the filter cloth. Thus, these processes of dewatering may be called cake filtration. In cake filtration, the flow of the filtrate in its most elementary form, may be considered as through a tightly packed bank of small crooked tubes across the cake. The rightmost drawing of Figure 7.11 is a schematic section of a filter cake and filter cloth. The general equation of momentum across a filter, Equation (7.15), may be applied across the cake; for convenience, the equation is reproduced below.
Because of pressure, the solids are tightly packed. The first term on the right-hand side of the above equation is the inertial force. Because the cake is tightly packed, this initial force must disappear upon entrance of flow into the cake and the flow is not accelerated; the first term is therefore equal to zero. Thus, the only valid term of the right-hand side of Equation (7.15) when applied to cake filtration is the term 36Ks(1 - n)/(dVsp/p.) and the equation reduces to
SoKi-rVsp dr 2
In the derivation of Equation (7.15), the equation was derived by applying the equation of momentum in the direction of fluid flow. In the present case, however, the equation is applied in the direction opposite to the fluid flow. Hence, the -Ap becomes Ap. Also, in terms of differentials, Ap/l = dp/dl; therefore, dp = (l-rVp dl dr3
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