The Theories of Filtration

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Chapter: Pharmaceutical Engineering: Filtration

The Theories of Filtration: Flow of Fluids Through Porous Media, Factors Affecting the Rate of Filtration, The Retention of Particles in a Depth Filter, The Conditioning of Slurries, The Compressibility of Cakes, Cake Washing and Dewatering


Two aspects of filtration theory must be considered. The first describes the flow of fluids through porous media. It is applicable to both clarification and cake filtration. The second, which is of primary importance only in clarification, examines the retention of particles in a depth filter.

Flow of Fluids Through Porous Media

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

Q = 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).

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 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.


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.

The Retention of Particles in a Depth Filter

Theoretical studies of particle retention have been restricted to granular media of a type used in the purification of municipal water. The aim is to predict the variation of filtrate quality with influent quality or time and then estimate the effect of removed solids on the permeability of the bed. Such studies have some bearing on the use of granular, sintered, or fibrous beds used for clarifying pharmaceuticals.

The path followed by the liquid through a bed is extremely tortuous. Violent changes of direction and velocity will occur as the system of pores and waists is traversed. Deflection of particles by gravity or, in the case of very fine particles, by Brownian movement will bring particles within range of the attractive forces between particles and the medium and cause arrest. Inertial effects, that is, the movement of a particle across streamlines by virtue of its momentum, are considered to be of importance only in the removal of particles from gases. In liquid-solid systems, density differences are much smaller.

Opportunity for contact and arrest depends on the surface area of the bed, the tortuosity of the void space, and the interstitial speed of the liquid. Since the inertial mechanism is ineffective, increase in interstitial velocity decreases the opportunity for contact and retention of particles by the medium. Therefore, the efficiency of a filter decreases as the flow rate increases. However, efficiency increases as the density or size of the influent particles increases and decreases as the particle size in the bed decreases. Each layer of clean filter is considered to remove the same proportion of the particles in the influent. Mathematically expressed,

where C is the concentration of the particles that enter an element of depth dx. The value of K, which is a clarifying coefficient expressing the fraction of par-ticles that deposit in unit depth of the bed, changes with time. Initially, the rate of removal increases and the efficiency of filtration improves. It has been sug-gested that this is because the deposition of particles in the bed is at first localized and the surface area and tortuosity increase. Later, the efficiency of removal decreases because deposition narrows the pores, reduces convolutions and surface area, and increases the interstitial liquid velocity. The failure of the medium to adequately retain particles or the decrease in permeability and fil-tration rate eventually limits the life of the filter. If deposition is reversible, the permeability and retentive capacity can be restored by vigorous backwashing. Alternatively, the medium should be cheap and expendable.

A mathematical account of the theories of clarification with depth filters is found in the work of Ives (Ives, 1963; Ives, 1962) and Maroudas and Eisenklam (Maroudas and Eisenklam, 1965).

The Conditioning of Slurries

The permeability of an ideal filter bed, such as that formed by a filter aid, is about 7 x 10-13 m2. This is more than 10,000 times the permeability of a pre-cipitate of aluminum hydroxide. Therefore, the modification of the physical properties of the slurry can be a powerful tool in the hands of the filtration engineer. This is called slurry conditioning. Two methods, flocculation and the addition of filter aids, will be discussed here.

Flocculation of slurries is a common procedure in which the addition of flocculating agents is permissible. The aggregates or flocs, which are charac-terized by high sedimentation rate and sedimentation volume, form cakes with a porosity as high as 0.9. Since this is also associated with a decrease in specific surface, flocculation gives a marked increase in permeability. How-ever, such coagulates are highly compressible and are, therefore, filtered at low pressures.

Filter aids are materials that are added in concentrations of up to 5% to slurries that filter only with difficulty. The filter aid forms a rigid cake of high porosity and permeability due to favorable shape characteristics, a low surface area, and a narrow particle size distribution, properties that can be varied for different operations. This structure mechanically supports the fine particles originally present in the slurry. Diatomite, in the form of a purified, fractionated powder, is most commonly used. Other filter aids include a volcanic glass, called “perlite,” and some cellulose derivatives.

Filter aids cannot easily be used when the solids are wanted. Their excellent characteristics, however, lead to their use as a “precoat” mounted on a suitable support so that the filter aid itself forms the effective filtering medium. This prevents blinding of the septum. Precoat methods take several forms and are discussed in the section devoted to filters. A practical account of the prop-erties and uses of filter aids has been given by Wheeler (Wheeler, 1964).

The Compressibility of Cakes

In the theory of cake filtration described above, the permeability coefficient was considered constant. The observation that a cake may be hard and firm at the cake-septum junction and sloppy at its outer face suggests that the porosity may be varying throughout the depth of the cake. This could be due to a decrease in hydrostatic pressure from a maximum at the cake face to zero at the back of the supporting septum. The hydrostatic pressure must be balanced by a thrust, originating in the viscous drag of the fluid as it passes through the cake, transmitted through the cake skeleton, and varying from zero at the cake face to a maximum at the back of the septum equal to the pressure difference. The relation between this compressive stress and the pressure applied across the cake is represented in Figure 11.1.

We have so far considered that no deformation occurs under this stress, that is, the cake is perfectly rigid. No cake, in fact, behaves in this way. However, some, such as those composed of filter aids or, of coarse, isodiametric particles, approximate closely to a perfectly rigid cake. Others, such as cakes deposited from slurries of heavily hydrated colloidal particles, are easily deformed, so the permeability coefficient, until now assumed constant, is itself a function of pressure, hence equation (1) no longer applies.

FIGURE 11.1 Stress distribution in a filter cake.

 This effect can be so marked that an increase in pressure actually decreases the rate of filtration. Most slurries behave in a manner intermediate between these two extremes.

Cake Washing and Dewatering

Cake washing is of great importance in many filtration operations because the filtrate retained in the cake can be displaced by pure liquids. Filtration equip-ment varies in its washing efficiency, and this may influence the choice of plant. If the wash liquids follow the same course as the filtrate, the wash rate will be the same as the final rate of filtration, assuming that the viscosity of the two liquids is the same and that the cake structure is not altered by, for example, peptization following the removal of flocculating electrolytes. Washing takes place in two stages. The first involves the removal of most of the filtrate retained in the cake by simple displacement. In the second, longer stage, removal of filtrate from the less accessible pores occurs by a diffusive mechanism. These stages are shown in Figure 11.2.

Efficient washing requires a fairly cohesive cake, which opposes the for-mation of cracks and channels, which offer a preferential course to the wash liquid. For this reason, cakes should have even thickness and permeability.

Subsequent operations, such as drying and handling, are facilitated by removing the liquid retained in the cake after washing, which occupies from 40% to 80% of the total cake volume. This is achieved by blowing or drawing air through the washed cake, leaving liquid retained only as a film around the particles and as annuli at the points of contact. Since both surface area and the number of point contacts per unit volume increase as the particle size decreases, the effectiveness of this operation, like washing, decreases with cakes composed of fine particles.

FIGURE 11.2 Displacement of filtrate by displacement washing.

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