# Factors Affecting the Rate of Filtration

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## Chapter: Pharmaceutical Engineering: Filtration

Equation (1) may be used as a basis for the discussion of the factors that determine the rate of filtration.

Factors Affecting the Rate of Filtration

Equation (1) may be used as a basis for the discussion of the factors that determine the rate of filtration.

The concept of a channel with a hydraulic diameter equivalent to the complex interstitial network that exists in a powder bed leads to the equation

= KAΔP/ηL                (11:1)

where Q is the volumetric flow rate, A the area of the bed, L the thickness of the bed, ΔP the pressure difference across the bed, and η the viscosity of the fluid. The permeability coefficient, K, is given by ε3/5(1 - ε)2S02 where ε is the porosity of the bed and S0 is its specific surface area (m2/m3).

### Pressure

The rate of filtration at any instant of time is directly proportional to the pressure difference across the bed.

In cake filtration, deposition of solids over a finite period increases the bed depth. If, therefore, the pressure remains constant, the rate of filtration will fall. Alternatively, the pressure can be progressively increased to maintain the fil-tration rate.

Conditions in which the pressure is substantially constant are found in vacuum filtration. In pressure filtration, it is usual to employ a low constant pressure in the early stages of filtration for reasons given below. The pressure is then stepped up as the operation proceeds.

This analysis neglects the additional resistance derived from the sup-porting septum and the thin layer of particles associated with it. At the begin-ning of the operation, some particles penetrate the septum and are retained in the capillaries in the manner of depth filtration, while other particles bridge the pores at the surface to begin the formation of the cake. The effect of penetration, which is analogous to the blinding of a sieve, is to confer a resistance on the cake-septum junction, which is much higher than the resistance of the clean septum with a small associated layer of cake. This layer may contribute heavily to the total resistance. Since penetration is not reversible, the initial period of cake filtration is highly critical and is usually carried out at a low pressure. The  amount of penetration depends on the structure of the septum, the size and shape of the solid particles, their concentration, and the filtration rate.

When clarifying at constant pressure, a slow decrease in filtration rate occurs because material is deposited within the bed.

### Viscosity

The inverse relation between flow rate and viscosity indicates that, as expected, higher pressures are required to maintain a given flow rate for thick liquids than that necessary for filtering thin liquids. The decrease in viscosity with increase in temperature may suggest the use of hot filtration. Some plants, for example, the filter press, can be equipped so that the temperature of hot slurries can be maintained.

### Filter Area

In cake filtration, a suitable filter area must be employed for a particular slurry. If this area is too small, the excessively thick cakes produced necessitate high pressure differentials to maintain a reasonable flow rate. This is of great importance in the filtration of slurries giving compressible cakes. When clar-ifying, the relation is simpler. The filtration rate can be doubled by simply doubling the area of the filter.

### Permeability Coefficient

The permeability coefficient may be examined in terms of its two variables, porosity and surface area.

Evaluation of the term ε3/(1 - ε)2 shows that the permeability coefficient is a sensitive function of porosity. When filtering a slurry, the porosity of the cake depends on the way in which particles are deposited and packed. A porosity or void fraction ranging from 0.27 to 0.47 is possible in the regular arrangements of spheres of equal size. Intermediate values will normally be obtained in the random deposition of deflocculated particles of fairly regular shape. A fast rate of deposition, given by concentrated slurries or high flow rates, may give a higher porosity because of the greater possibility of bridging and arching in the cake. Although theoretically the particle size has no effect on porosity (assuming that the bed is large compared with the particles), a broad particle size distri-bution may lead to a reduction of porosity if small particles pack in the inter-stices created by larger particles.

Surface area, unlike porosity, is markedly affected by particle size and is inversely proportional to the particle diameter. Hence, as commonly observed in the laboratory, a coarse precipitate is easier to filter than a fine precipitate even though both may pack with the same porosity. Where possible, a previous operation may be modified to facilitate filtration. For example, a suitable particle size may be obtained in a crystallization process by control of nucleation or the proportion of fines in milling may be reduced by carefully controlling residence times. In the majority of cases, however, control of this type is not possible, and with materials that filter only with difficulty, much may be gained by con-ditioning the slurry, an operation that modifies both the porosity and the spe-cific surface of the depositing cake.

In clarification, high permeability and filtration rate oppose good particle retention. In the formation of clarifying media from sintered or loose particles, accurate control of particle size, specific surface, and porosity is possible so that a medium that offers the best compromise between permeability and particle retention can be designed. The analysis of permeability given above can be accurately applied to these systems. Because of the extremes of shape, this is not so for the fibrous media used for clarification. Here it is possible to develop a material of high permeability and high retentive capacity. However, such a material is intrinsically weak and must be adequately supported.

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