## Filtermedium Filtration Formulas

In the case of single-particle blockage, we first consider aim surface of filter medium containing Np number of pores. The average pore radius and length are rp and { , respectively. For laminar flow, the Hagen-Poiseuille equation may be applied to calculate the volume of filtrate V' passing through a pore in a unit of time:

Consequently, the initial filtration rate per unit area of filtration is:

Consider 1 m of suspension containing n number of suspended particles. If the suspension concentration is low, we may assume the volume of suspension and filtrate to be the same. Hence, after recovering a volume q of filtrate, the number of blocked pores will be nq, and the number unblocked will be (Np - nq). Then the rate of filtration is:

k' is a constant having units of sec"1. It characterizes the decrease in intensity of the filtration rate as a function of the filtrate volume. For constant V', this decrease depends only on the particle number n per unit volume of suspension. The total resistance R may be characterized by the reciprocal of the filtration rate. Thus, W in Equation 42 may be replaced by 1/R (sec/m). Taking the derivative of the modified version of Equation 42 with respect to q, we obtain:

Comparison with Equation 42 reveals:

Equation 46 states that when complete pore blockage occurs, the intensity of the increase in the total resistance with increasing filtrate volume is proportional to the square of the flow resistance.

In the case of multiparticle blockage, as the suspension flows through the medium, the capillary walls of the pores are gradually covered by a uniform layer of particles. This particle layer continues to build up due to mechanical impaction, particle interception and physical adsorption of particles. As the process continues, the available flow area of the pores decreases. Denoting x0 as the ratio of accumulated cake on the inside pore walls to the volume of filtrate recovered, and applying the Hagen-Poiseuille equation, the rate of filtration (per unit area of filter medium) at the start of the process is:

When the average pore radius decreases to r, the rate of filtration becomes

where

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