When sludge is fed to a filter by a positive-displacement pump, the rate of filtration is nearly constant (i.e., dV/dx = constant). During constant-rate filtration, the pressure increases with an increase in cake thickness. Therefore, the principal variables are pressure and filtrate volume, or pressure and filtration time. Equation 9 is the principal design relation, which may be integrated for a constant-rate process. The derivative, dV/dt, may be replaced simply by V/x:
The ratios in parentheses express the constant volume rate per unit filter area. Hence, Equation 24 is the relationship between time t and pressure drop Ap. For incompressible cakes, r0 is constant and independent of pressure. For compressible cakes, the relationship between time and pressure at constant-rate filtration is:
Filtration experiments are typically conducted in pilot scale equipment and generally tests are conducted either at constant pressure or constant rate to determine ax,,, as well as s and Rf, for a given sludge and filter medium. Such tests provide empirical information that will enable the time required tor the pressure drop to reach the desired level for a specified set of operating conditions to be determined. In the initial stages of filtration, the filter medium has no cake. Furthermore, Ap is not zero, but has a value that is a function of the resistance of the medium for a given flowrate. This initial condition can be stated as:
For an incompressible cake (where s = 0), Equation 25 takes the form:
As noted earlier, for thick cakes, the resistance of the filter medium may be neglected. Hence, for Rf= 0, Equation 25 simplifies to:
An increase in pressure influences not only coefficient r0, but the cake's porosity as well. Since the cake on the filter plate is compressed, residual liquid is squeezed out. Thus, for constant feed, the flowrate through the medium will not be stable, but will fluctuate with time.
The weight of dry solids in a cake is:
where x„ = weight of solids in the cake per unit filtrate volume.
The concentration of solids in the feed sludge is expressed by weight fraction c. It is also possible to evaluate experimentally the weight ratio of wet cake to its dry content m. Hence, a unit weight of sludge contains mc of wet cake. We denote y as the specific weight of feed sludge. This quantity contains c amount of solids; hence, the ratio of the mass of solids in the cake to the filtrate volume is:
Thus, from the sludge concentration c and the weight of wet cake per kg of dry cake solids m, Xq can be computed. If the suspension is dilute, then c is small; hence, product mc is small. This means that Xq will be approximately equal to c. According to Equations 29 and 30, the weight ratio of wet to dry cake will vary. Equation 30 shows also that because x„ depends on the product mc, at relatively moderate suspension concentrations this effect will not be great and can, therefore, be neglected. However, when filtering concentrated sludges the above will play some role; that is, at constant feed, the filtrate changes with time.
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