Although the filtration equipment takes several forms, its operation is basically the same. The solid-liquid feed stream passes through a porous screen that retains the solids but passes the liquid. As the flow continues, a cake builds up on the screen. The cake has a complex pore structure determined by the nature of the solid particles. The structure removes additional particles by a simple straining mechanism. A French geologist named d'Arcy formulated the equation that describes the flow of Newtonian fluids through a porous bed in 1856. Neglecting the gravitational effects on the fluid, a form of d'Arcy equation adopted for filtration can be written as:
Q = K A ?P / ÂµL ---------------------- (1)
Where, Q is the liquid flow rate, K is the permeability of the filter cake, ?P is pressure drop across the cake, A is the filtration area, Âµ is the liquid viscosity and L is filter cake thickness. The equation (1) is valid for constant filter cake thickness. During filtration, however, the cake thickness increases as more and more liquid is filtered. To account for the changing thickness, the d'Arcy equation can be modified to give the differential equation:
dV / dT = KA ?P / ÂµL --------------------- (2)
The value of L changes with the volume of liquid filtered as follows:
L = C V / D A ------------------------------- (3)
Where C is the solid concentration, ie mass of solid per unit volume of liquid, D is the bulk density of the solid deposited; V is the volume of the liquid filtered. Substituting the expression for L into the modified d'Arcy equation gives the following differential equation describing cake filtration:
V dV = (K A2 ?P D / ÂµC) / dT --------------------(4)
This equation can be integrated for various flow conditions encountered by the process engineers. The equation, however, is applicable to only dilute slurries. When the solids volume is more than 5% of the total volume of liquids, filtered, other flow equations must be developed. However, the great majority of filter cake filtration in the process industries do treat slurries having solid content within 5%.
Considering operating parameters, two special cases are industrially important. One is constant rate filtration wherein the rate of feed delivery to the filter stays the same during the filtration cycle. The other is constant pressure filtration, where the feed is delivered under a pressure that does not vary over the cycle.