Constant incompressible cakes

The constant given the value 5 in equation 1 depends on particle size, shape, and porosity it can be assumed to be 5 for low porosities. Although equation 1 has been found to work reasonably well for incompressible cakes over narrow porosity ranges, its importance is limited in cake filtration because it cannot be used for most practical, compressible cakes. It can, however, be used to demonstrate the high sensitivity of the pressure drop to the cake porosity and to the specific surface of the soHds. 

Constant-Rate Filtration For substantially incompressible cakes, Eq. (18-51) may be integrated for a constant rate of slurry feed to the filter to give the following equations, in which filter-medium resistance is treated as the equivalent constant-pressure component to be deducted from the rising total pressure drop to... 

For incompressible cakes, eoefficient r is constant and independent of pressure. For compressible cakes (s 0) r may be estimated from the expression r = aAP. Substituting for r into the above relation, we obtain ... 

Filter cakes may be divided into two classes—incompressible cakes and compressible cakes. In the case of an incompressible cake, the resistance to flow of a given volume of cake is not appreciably affected either by the pressure difference across the cake or by the rate of deposition of material. On the other hand, with a compressible cake, increase of the pressure difference or of the rate of flow causes the formation of a denser cake with a higher resistance. For incompressible cakes e in equation 7.1 may be taken as constant and the quantity e3/[5(l — e)2S2] is then a property of the particles forming the cake and should be constant for a given material. 

Equation 7.2 is the basic filtration equation and r is termed the specific resistance which is seen to depend on e and S. For incompressible cakes, r is taken as constant, although it depends on rate of deposition, the nature of the particles, and on the forces between the particles, r has the dimensions of L-2 and the units m-2 in the SI system. 

Constant-pressure cake filtration with non-Newtonian suspending fiuids has received considerable attention [Kozicld, 1990]. The average ecific resistance has been found to vary considerably as a fimction of the flow behaviour index N even during the filtration of apparently incompressible materials [Shirato et al, 1977], Later papers extended the analysis to compressible filter cakes and constant rate and variable pressure and rate filtrations [Shirato et al, 1980a, b]. 

The specific resistance value (a) in the case of incompressible cakes assumes a constant value. For compressible cakes, the resistance varies with the pressure according to equation 2.5 ... 

For constant pressure, constant a, and incompressible cake, V and f are the only variables in Eq. (14.2-13). Integrating to obtain the time of filtration in / s. 

Constant-Rate Filtration of Incompressible Cake. The filtration equation for filtration at a constant pressure of 38.7 psia (266.8 kPa) is... 

For incompressible cakes Equation 10.94 can be used directly at different pressures. However, for compressible cakes, the relationship between a and -AP needs to be determined experimentally by performing filtration runs at different constant pressures. Empirical equations may be fitted to... 

The specific cake resistance a should be constant for incompressible cakes but it may change with time as a result of possible flow consolidation of the cake and also, in the case of variable rate filtration, because of variable approach velocity. 

Figure 9.5 Plot of t V = f V) for constant pressure filtration and incompressible cakes...

Figure 9.7 Plot of Ap = f(f) for constant rate filtration, incompressible cake...

Note that the above equation is only valid for incompressible cakes, negligible medium resistance and constant submergence, /. 

As an example of the application of the above analysis for flow through packed beds of particles, we will briefly consider cake filtration. Cake filtration is widely used in industry to separate solid particles from suspension in liquid. It involves the build up of a bed or cake of particles on a porous surface known as the filter medium, which commonly takes the form of a woven fabric. In cake filtration the pore size of the medium is less than the size of the particles to be filtered. It will be appreciated that this filtration process can be analysed in terms of the flow of fluid through a packed bed of particles, the depth of which is increasing with time. In practice the voidage of the cake may also change with time. However, we will first consider the case where the cake voidage is constant, i.e. an incompressible cake. 

During constant pressure drop filtration of an incompressible cake, how does filtrate flow rate vary with time ... 

A leaf filter has an area of 2 m and operates at a constant pressure drop of 250 kPa. The following results were obtained during a test with an incompressible cake ... 

Equation (1), known as the two-resistance filtration model, is a simple expression for describing the filtration of incompressible cakes, the specific cake resistance, a, can be regarded as a constant. In the case of compressible cakes, the effect of variation in cake porosity on spedfic cake resistance must be considered (Tiller and Shirato, 1964). Filtration tests should be performed under diflerent pressure drops to establish an empirical relation between the specific cake resistance, a, and the pressure drop across the filter cake, APe (McCabe et al., 1993) ... 

The same moisture content of the produced cake can be obtained in shorter dewatering times if higher pressures are used. If a path of constant dewatering time is taken, moisture content is reduced at higher pressures with a parallel increase in cake production capacity. This is an advantage of pressure filtration of reasonably incompressible soHds like coal and other minerals. 

Equation 13 is the relationship between filtration time and filtrate volume. The expression is applicable to either incompressible or compressible cakes, since at constant Ap, r and Xq are constant. If we assume a definite filtering apparatus and set up a constant temperature and filtration pressure, then the values of Rf, rQ, n and Ap will be constant. 

Also from (7.3.1.1), rm is the filter media resistance and a is the average specific cake resistance. If the filter cake is incompressible, a is constant for compressible cake a is defined as ... 

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