Filtration is the separation of suspended impurities from liquid or gas by passing the fluid through a porous membrane that retains the particles on its surface or in its pores.

Since separation of liquid-solid suspensions is often different from separation of dust-laden gases, including aerosols, we shall discuss them separately.

In liquid filtration, the suspension passes through a filtration membrane; the suspended particles remain on its surface and in its pores, while the clarified liquid, called *filtrate*, is collected behind the filtration membrane. The separated solid phase, the sediment, forms a continuously growing layer on the filtration membrane surface. It commonly consists of randomly lying particles of different shapes.

Fresh portions of liquid are passed through this layer and then through the filtration membrane. The liquid flow velocities, as a rule, are low, and the flow is laminar and described by the equation

where V is the filtrate volume, m^{3}; F, the filtration surface, m^{2}; t, the time, s; Δp, the pressure difference. Pa; η, the filtrate viscosity, N s/m^{2}. The terms in parentheses represent the *specific pressure drop*—i.e., the pressure drop for a superficial velocity of 1 m/s—for the sediment and filtration membrane, respectively.

The total mass of dry sediment is determined as

where m is the mass of dry sediment per unit filtrate volume, kg/m^{3}; ρ, the filtrate density, kg/m^{3}; C, the mass concentration of particles in suspension, kg/kg; ω is the ratio of wet to dry sediment mass, α is the specific pressure drop of sediment per unit of dry mass; r, the specific pressure drop per unit surface of the filtration membrane.

Integration of Eq. (1) at Δp = const yields

or for specific conditions at a constant rate of filtration

where K_{2} and C' are the constants that are characteristic for these specific conditions.

The above equations are not valid for separation of finely dispersed suspensions of low concentration by loose fibrous filtration partitions when the particles penetrate deep into the filter bed and the sediment is not formed on the filtration membrane surface.

The average specific pressure drop α of the sediment depends on pressure drop:

where α' is a constant value depending on particle shape and size and s is a coefficient of sediment compresibility which varies from 0 to 1. If the sediment consists of solid particles, increasing pressure does not change their size and packing density, i.e., s = 0, and the pressure drop of filtration membrane is low relative to that of the sediment layer, then Eq. (1) takes the form

i.e., at a constant pressure difference Δp the instantaneous rate of filtrate flow is inversely proportional to the quantity of sediment. Where the sediment consists of easily deformable particles, s approaches 1 and

i.e., the instantaneous rate of filtrate flow is independent of pressure. These are extreme cases. Actually, s varies from 0.1 to 0.8. Hence, in filtration of a suspension with solid, granulated or crystalline particles, increasing pressure difference brings about a nearly proportional growth in the flow rate of the suspension being filtered. If flocculent or clay sediments are predominant in the suspension, then increasing pressure difference has almost no effect on the flow rate. Some sediments are compressed, and the rate of filtrate flow is reduced, if a certain critical pressure drop is exceeded.

A special filtering substance is sometimes added to slimy suspensions to improve filtration, and clarification is carried out at a constant rate of filtrate flow (not at Δp = const) because otherwise a large amount of filtrate-contaminating impurity slips through at the beginning and a fast growth of the layer and its pressure drop loss is observed afterwards. It has been shown in practice that in general it is more advantageous to start filtration at a low pressure difference, gradually increasing it as the sediment layer grows. Centrifugal pumps with a steep characteristic operate in much the same fashion.

Equation (1) implies that the rate of filtrate flow is inversely proportional to filtrate viscosity η. Therefore, in order to increase filter capacity it is advisable either to heat the suspension or to add substances to reduce η.

Particle size plays an important role in filtration. Filtration of fine suspensions causes an increase in α' and a fall in the filtration rate. Therefore, it is a good practice to coarsen fine particles, e.g., by coagulation by either heating the suspension or adding coagulants.

In order to achieve the maximum capacity of *batch-type filters* one must know to what thickness the sediment layer should be brought and at what moment regeneration should be started. Such optimization of filler operating regime has been carried out for (1) Δp = const, (2) v = const, and (3) variable v and Δp. Here it was assumed that the complete cycle of filter operation that is a vessel with a filtration membrane and filtrate drain consists of six operations: preparation of the filter, feeding of suspension, filtration, sediment washing, air blowing, and discharging. Filtration, washing, and blowing are basic operations, the rest are auxiliary ones. Thus, the operation period consists of two components t = t_{b} + t_{aux}. The optimization has shown that for Δp = const and a substantial pressure drop of filtration membrane the filter capacity reaches the maximum when the relation between t_{b} and t_{aux}

is valid and for v = const the relation

is valid, where η is the viscosity of suspension liquid phase, N s/m^{2}, and r_{0} are the permeabilities of the filter and the sediment, respectively, m^{−2}; x_{0}, the ratio of the sediment volume to the filter volume; Dp, the pressure difference, N/m^{2}.

Comparison of optimum values of (t_{b} – t_{aux}) for Δp = const and v = const yields √2, i.e., 40% difference.

Optimization of the v = const regime changing over to Δp = const was much more complicated, and the results, as was expected, appeared to depend on the duration of operation under each regime, but on the whole were within the 40% indicated above.

The main distinction between filtration of polluted gases and that of liquid suspensions follows from the large (of the order of 10^{3}) difference in densities of the carried medium (gas) and dust particles. A dust-gas system can be stable only in the case of fine particles of 10 μm in size and smaller. This is known as an aerosol. Coarser particles, up to 100 μm in size and larger, can be present in real conditions, for instance, in industrial gas exhausts, but more rough and primitive filtration devices such as cyclones and settlers are used more frequently than filters for their separation.

Due to the great difference of gas and solid particle densities the latter, even for a high mass concentration, are at a sufficiently large distance from each other that they can be assumed noninteracting between themselves.

In what follows we shall first discuss filtration of aerosols containing solid particles (filtration of mists will be considered later).

*Aerosol filtration* involves two stages, a stationary one and a nonstationary one. The first stage includes particle sedimentation on or in a clean or cleaned-by-regeneration filtration membrane, where the role of the dust layer formed is not significant. Trapping efficiency and pressure drop do not change in time and are determined only by the properties of filtration membrane and settling particles and by the gas flow parameters. The duration of this stage depends on the specific rate of filtrate flow, the dust content of the flow (concentration of solid particles in it), and the extent of dust trapping.

The second, nonstationary, stage begins at the moment when the sediments of the trapped dust clog up pores producing on the surface of filtration membrane a layer of such thickness that it has an appreciable effect on the pressure drop Δp and the efficiency of particle trapping h. The former parameter is growing progressively and the latter, as a rule, first rises and then may drop. A rise in η is attributed to the appearance of an additional filter bed, while a reduction, to the local "pressing through" of the dust as a result of increase in Δp. Variation of Δp and η versus time is depicted in Figure 1.

**Figure 1. Variation of particle trapping efficiency (η) and pressure drop (Δp) with amount of material trapped on a filter.**

Theoretically, filtration of aerosols can be calculated only at the first stage, and even then only by modeling filtration membrane as single-row fibers separated by a space large enough in relation to a particle size. Sedimentation of particles on the fiber (a single cylinder) is schematically shown in Figure 2. In an aerosol flow, the particles follow the streamlines. However, if the latter bend in the filtration zone, the particle path deflects from the streamlines and under inertia forces (coarse particles), tangency (finer particles), or electrical attraction (charged particles) the particles settle out on the fiber surface. These effects also include the Brownian motion of highly dispersed particles and sedimentation of coarse particles by gravity.

Considering that the effects work independently, Devies derived the overall efficiency of sedimentation of particles of a definite size on the cylinder (so-called fractional efficiency) under inertia, diffusion, and tangency

where St and Pe are the Stokes and Peclet numbers, R = r/a, r and a are the radii of a particle and a cylinder (a fiber), respectively. Other formulas are also available for calculation of fractional of efficiency of particle trapping from aerosol flow by fibers. These include the Friedlander, Torgeson and Kirsch-Stechkina-Fuchs formulas. In order to calculate the efficiecy of polydispersed particle trapping one must know their size spectrum and do an appropriate summation. However, we should keep in mind a selective breakthrough that shows up as an anomalous increase in the breakthrough of 0.2-0.3 μm particles. With increasing fiber thickness and flow velocity the anomaly shifts toward coarser particles.

In contrast to suspension and *aerosol filtration*, mist filtration does not give rise to a filter bed produced from the trapped substance, but the liquid clogged in the filtration membrane produces a so-called capillary effect. It involves spreading of clogged droplets with subsequent merging into coarser droplets or production of liquid films on fibers, accumulation of liquid at the sites of fiber interlacing, capillary condensation and adhesion of neighboring fibers due to capillary forces. All this affects the structure of filter beds, hindering gas flow and raising the pressure drop and, thereby, droplet breakthrough as a result of liquid forcing through the filtration membrane.

A particular emphasis should be laid on filtration in air oil filters. They use as the filtration membrane thick (25-100 mm) layers of coarse fibers (from 30 to 250 μm) and wire meshes, punched sheets, paper, and corrugated board impregnated with oil. The efficiency of trapping of the dust particles out of the air flow by the oil film, is higher than that without oil.

The mechanism of particle trapping by aerosol flow filtration through a layer of granular material is similar in outline to that through fibrous material; inertia, diffusion, tangency, and gravity effects work simultaneously much in the same way. The breakthrough K = 1 – η is described by the formula of the type

where δ is the thickness of granular layer, Pe the Peclet number, d_{g} the grain diameter, and k and k_{1} are empirical constants with k_{1} varying within the range –1/3 to –2/3.

Investigations have established that at low flow velocities (up to 0.05 m/s), particles smaller than 0.5 μm in diameter settle out mainly under the action of diffusion, while the coarser particles settle by gravity. Selective breakthrough is inherent in granular as well as fibrous filtration membranes, but the greatest breakthrough is due to the coarse particles, from 0.4 to 0.9 μm in size. Increasing velocity results in increasing breakthrough and with the released particles becoming smaller.

Gas flow in a granular layer pores involves multiple breakup of the main flow and variation of velocity in value and direction. Some ("flow-through") channels have velocities which are substantally higher than the average one and inertia effects are predominant; also, there are stagnation zones with practically zero velocity in which particle sedimentation is due to diffusion and gravity.

At low flow velocities the stream is laminar and the relation between the average velocity and the pressure difference is linear and described by Darcy's equation. As the velocity increases, the laminar flow is disturbed, the importance of inertia effects grows as a result of multiple jet convergence and divergence, and the proportionality factor between velocity and Δp reduces.

The pressure difference for a layer of grains of arbitrary shape may be calculated for higher gas velocities by the formula

where λ and n are respectively the pressure drop coefficient of the layer and an exponent both of which are the functions of the Reynolds number, δ the layer thickness, U the average flow velocity related to the entire filtration surface, ρ_{g} gas density, d_{e} the equivalent grain diameter, ε the porosity, and Ф the coefficient of the grain shape: Ф_{s} = 1 for spheres, Ф_{s} = 0.8 for shingle and sand, and Ф_{s} = 0.5 for coal and coke. For laminar flow n = 1 and formula (12) transforms into the *Kozeny-Karman equation*

where η_{g} is the gas viscosity, N s/m^{2}.

For calculating Δp of granular layers for the laminar and turbulent gas flow use can be made of the Ergun equation.

that does not involve a shape coefficient.

Specific empirical formulas are available in the literature for many specific grain types.

Formulas (11)-(14) pertain to regenerated filtration membranes; as they are clogged by trapped particles, Δp increases.

#### REFERENCES

Perry, R. H. and Chilton, C. H. Eds., (1973) *Chemical Engineer's Handbook*, 5th ed., McGraw-Hill, New York.