The dynamics of variable-rate and -pressure filiations can be illustrated by pressure profiles that exist across the filter medium. Figure 7 shows the graphical representation of those profiles. According to this plot, the compressed force in the cake section is:

where p, = pressure exerted on the sludge over the entire cake thickness ps = static pressure over the same section of cake p corresponds to the local specific cake resistance (rw)x. At the sludge-cake interface ps( = pj and p = 0; and for the interface between the cake and filter plate pst = pst and p = p, -p's(. p'sl corresponds to the resistance of filter plate pf, and is expressed where W = rate of filtration (m3/m2-sec).

Figure 7. Distribution of static pressure pst in liquid and p along the cake thickness and filter plate: I, II -boundaries between the cake and sludge at x" and t'; III, IV-boundaries between cake layers or cake and filter plate at x" and t' ; V- boundary line between the cake and filter plate or free surface of filter plate; 1,3-curves p^f^) and p=f(hoc) at t'; 2, 4 -curves ps[=f(hoc) and

Figure 7. Distribution of static pressure pst in liquid and p along the cake thickness and filter plate: I, II -boundaries between the cake and sludge at x" and t'; III, IV-boundaries between cake layers or cake and filter plate at x" and t' ; V- boundary line between the cake and filter plate or free surface of filter plate; 1,3-curves p^f^) and p=f(hoc) at t'; 2, 4 -curves ps[=f(hoc) and

Note that Apr is constant during the operation. Pressure p is also the driving force of the process. Therefore, starting from the governing filtration equations, the general expression for an infinitesimal increment of solid particle weight in a cake of unit of area is xwdq (q = filtrate volume obtained from 1 m2 filtering area, m3/m2). The responding increment dp may be expressed as:

xw is not sensitive to changes in p. In practice, an average value for xw can be assumed. Note that W is constant for any cross section of the cake. Hence, Equation 33 may he integrated over the cake thickness between the limits of p = 0 and p=p, -p'sl, from q = 0 to q = q:

l*xwW

Parameters q and W are variables when filtration conditions change. Coefficient (rw)x is a function of pressure:

The exact relationship can be derived from experiments in a device called a compression-permeability cell. Once this relationship is defined, the integral of the right side of Equation 34 may be evaluated analytically (or if the relationship is in the form of a curve, the evaluation may be made graphically). The interrelation between W and Pi is established by the pump characteristics, which define q = f(W) in Equation 34. Filtration time may then be determined from the following definition:

dq dx

Hence,

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