## Ith

■ Direction of flow of slurry

FIGURE 7.11 (a) Büchner funnel filtration assembly; (b) leaf filter assembly; and (c) mechanics of cake filtration.

If pp is the density of solids, the mass dm in the differential thickness of cake dl is dm = Sodl(1 - n)pp. Solving this for dl and substituting, Equation (7.56) becomes dp = (±zJM dn

In this equation, ¡1 and Vs influence directly the flow characteristics of the filtering water. Since So determines the value of Vs, it also influences directly the flow characteristics of the fluid. All the other factors in the equation are inherent characteristics of the cake. All these cake characteristics may be lumped up into a single term. Call this term as the specific cake resistance a. Hence, in terms of the specific cake resistance, dp is dp = aAAV-dm (7.58)

Examining the various parameters constituting a, as far as p and m as the variables of integration are concerned, they are all constant. Hence, a is constant over these variables of integration. In addition, Vs is also a constant over these variables of integration. Integrating from P2 to Pb

mc is the total mass of solid collected on the filter cloth.

Note that the resistance of the filter cloth has heretofore been neglected. Calling this resistance Rm and including it, the total pressure drop may be written as

7.6.1 Determination of a

Because V is the volume of the filtrate collected at any time t, Vs is (dV/dt)/So; also, expressing mc as cV (where c is the mass of cake collected per unit volume of filtrate), substituting in Equation (7.60), and integrating under the assumption of constant pressure differential operation yields

Vacuum filtration is normally conducted under constant pressure differential; hence, -AP has been made constant. In addition, a may vary over time; hence, its average value a has been used.

Equation (7.61) is an equation of a straight line between t/V and V, whose slope m is given by ¡ic a/[2(-AP)S2o]. By determining this slope from experimental data, the value of a can be determined. The slope may be determined by

¡c n1 and n2 and the number of elements in the respective group. Remember that to fit a straight line into experimental data, the data must be divided into two groups: n1 is the number of elements in the first group and n2 and is the number of elements in the second group.

The laboratory experiment involves using a Buchner funnel by adopting the setup shown in Figure 7.11. Operating the funnel at a constant pressure difference -AP, the amount of filtrate collected is recorded with time. The data collected gives the relationship between t/V and V as called for by Equation (7.61). The cake collected is also weighed to determine c; ¡ is determined from the temperature of the filtrate.

Example 7.7 A Buchner funnel experiment to determine the specific cake resistance of a certain sludge is performed. The results are as shown in the following table. -AP = 51 cm Hg, filter area = 550 cm2, ¡i = 15(10-4) kg/m • s, and c = 0.25 g/cm3. Determine a.