From Equations 76 and 77 we have:

where

The numerator of Equation 79 characterizes the cake resistance. The denominator contains information on the driving force of the operation. Constant K'" (sec/m2) characterizes tile intensity at which the filtration rate decreases as a function of increasing filtrate volume.

Substituting 1/R for W in Equation 78 and taking the derivative with respect to q, we obtain:

The expression states that the intensity of increase in total resistance for cake filtration is constant with increasing filtrate volume. Replacing W by dq/dx in Equation 78 and integrating over the limits of 0 to q between 0 and x we obtain:

Note that this expression reduces to Equation 74 on substituting expressions for Win (Equation 77) and K'" (Equation 79).

Examination of Equations 46, 61, 68 and 80 reveals that the intensity of increase in total resistance with increasing filtrate volume decreases as the filtration process proceeds from total to gradual pore blocking, to intermediate type filtration and finally to cake filtration. Total resistance consists of a portion contributed by the filter medium plus any additional resistance. The source of the additional resistance is established by the type of filtration. For total pore blockage filtration, it is established by solids plugging the pores; during gradual pore blockage filtration, by solid particles retained in pores; and during cake filtration, by particles retained on the surface of the filter medium.

The governing equations (Equations 42, 67, 74 and 81) describing the filtration mechanisms are expressed as linear relationships with parameters conveniently grouped into constants that are functions of the specific operating conditions. The exact form of the linear functional relationships depends on the filtration mechanism. Table 1 lists the coordinate systems that will provide linear plots of filtration data depending on the controlling mechanism.

In evaluating the process mechanism (assuming that one dominates) filtration data may be massaged graphically to ascertain the most appropriate linear fit and, hence, the type of filtration mechanism controlling the process, according to Table 1. If, for example, a linear regression of the filtration data shows that q = f(x/q) is the best linear correlation, then cake filtration is the controlling mechanism. The four basic equations are by no means the only relationships that describe the filtration mechanisms.

Type of Filtration |
Equation |
Coordinates |

With Total Pore Blocking |

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