## Info

7.6.2 Design Cake Filtration Equation

In an actual filtration installation, may it be a vacuum filter or a plate-and-frame press, the resistance of the filter medium is practically negligible; it may, therefore, be neglected. Considering this fact, Equation (7.61) may be written as

Since it is important to be able to calculate the amount of dewatered sludge that is finally to be disposed of, for convenience Equation (7.64) may be expressed in a form that will give this amount directly without considering the filtrate. This is done by utilizing the specific loading rate, also called filter yield Lf defined as

From this definition, Lf is the amount of cake formed per unit area of filter cloth per unit of time. In vacuum filtration, the only time that the cake is formed is when the drum is submerged in the tank. In pressure filtration, the only time that the cake is also formed is when the sludge is pumped into the plates. Thus, t in the previous equation is called the form time f. Also, the filters operate on a cycle. Calling the cycle time as tc, tf may be expressed as a fraction f of tc. Thus, t = f = ftc may be substituted in Equation (7.65) and, if this equation is substituted in Equation (7.64) and the result simplified and rearranged, we have

For incompressible cakes, Equation (7.66) is the design cake filtration equation. In vacuum filtration, f is equal to the fraction of submergence of the drum. Also, for the pressure filter, f is the fraction of the total cycle time that the sludge is pumped into the unit.

For compressible cakes such those of sewage sludges, a is not constant. Equation (7.66) must therefore be modified for the expression of the specific cake resistance. The usual form used is a _ ao(-AP)s (7.67)

where s is a measure of cake compressibility. If s is zero, the cake is incompressible and a equals ao, the constant of proportionality. Substituting in Equation

ICCoftc

### 7.6.3 Determination of Cake Filtration Parameters

In design, the parameters Lf, -AP, f, and tc may be specified. | is specified from the temperature of filtration. The value of c may be determined by performing an experiment of the particular type of sludge. To determine ao, take the logarithms of both sides of Equation (7.67), lna _ lnao + sln(-AP) (7.69)

In this equation, s is the slope of the straight-line equation between In a as the dependent variable and ln(-AP) as the independent variable. By analogy with Equation Equation (7.62),

And, i n1 i n1 1X, ln a), - s1X, J[ ln (-AP)], nj 1 ' nj 1 '

The experimental procedure to gather data used to obtain 5 and ao may be performed using the leaf filter assembly of Figure 7.11, although the Buchner experiment may also be used. The leaf filter is immersed in the sludge and a vacuum is applied to suck the filtrate during the duration of the form time tf. For a given -AP, the volume of filtrate over the time tf is collected. This will give one value of a. To satisfy the requirement of Equation (7.69), at least two runs are made at two different pressure drops. From these pairs, the parameters ao and 5 may be calculated.

Example 7.8 A leaf-filter experiment is run to determined ao and 5 for a CaCO3 slurry in water producing the results below. ¡1 = 8.9(10-4) kg/m. s. The filter area is 440 cm2, the mass of solid per unit volume of filtrate is 23.5 g/L, and the temperature is 25°C. Calculate ao and 5.

 -AP (kN/m2) V (L) 46.18 111.67 Time (sec)

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