Incompressible Cake

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. 

Experimental exponents for cake thickness vary from 0.5 to as much as 3.0. The theoretical value of //2 may be approached only by incompressible cakes of a narrow range of sizes. The proper and characteristic value for the mean particle size, d, is difficult to ascertain. In practice, the most finely divided particles, eg, 10—15 wt % of soHds, almost whoUy determine the Hquid content of a cake, regardless of the rest of the size distribution. It seems reasonable to use a d closely related to Hquid content, eg, the 10% point on a cumulative weight-distribution curve. 

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 cake, the pressure distribution and the rate depend on the resistance of the filter medium and the permeability of the cake. Figure L8-150 shows several possible pressure profiles in the cake with increasing filtration rates through the cake. It is assumed that r /i i = 0.8 and /p//i = 0.6. The pressure at / = ri, corresponds to pressure drop across the filter medium Ap, with the ambient pressure taken to be zero. The filtration rate as well as the pressure distribution depend on the medium resistance and that of the cake. High medium resistance or blinding of the medium results in greater penalty on filtration rate. 

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 ... 

For an incompressible cake (where s = 0), Equation 25 takes the form ... 

In case 4, the increasing pressure compresses the cake to such as extent that it actually squeezes off the flow so that as the pressure increases the flow rate decreases. This situation can be compensated for by adding a filter aid to the slurry. This is a rigid dispersed solid that forms an incompressible cake (diatomaceous earth, sand, etc.). This provides rigidity to the cake and enhances its permeability, thus increasing the filter capacity (it may seem like a paradox that adding more solids to the slurry feed actually increases the filter performance, but it works ). 

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. 

Comparing equations 7.8 and 7.24 shows that for an incompressible cake ... 

Before this equation can be integrated it is necessary to establish the relation between r and V. If v is the bulk volume of incompressible cake deposited by the passage of unit volume of filtrate, then ... 

For incompressible cakes, Equation (7.66) is the design cake filtration equation. In vacuum filtration,/is equal to the fraction of submergence of the drum. Also, for the pressure filter,/is the fraction of the total cycle time that the sludge is pumped into the unit. 

Dip coating is analogous to a slip casting process for making ceramic parts. The membrane deposition behavior by slip casting can be described by a theory of colloidal filtration for incompressible cakes [Aksay and Schilling, 1984] and compressible cakes [Tiller and Tsai, 1986). The theory predicts that the thickness of the consolidated layer, L, is given by... 

Incompressible cakes have flow rates that are dependent upon the pressure or driving force on the cake. In comparison, compressible cakes, i.e., where s approaches 1.0, exhibit filtration rates that are independent of pressure as shown below. 

Compressible cakes are composed of amorphous particles that are easily deformed with poor filtration characteristics. There are no defined channels to facilitate liquid flow as in incompressible cakes. 

For incompressible cakes, the filtration rate is directly proportional to the specific cake resistance and the pressure/vacuum, and inversely proportional to the viscosity and cake thickness. The filtration rate is inversely proportional to the ratio of solids to filtrate, while the rate of cake formation is directly related to this ratio. For compressible cakes, the filtration rate is relatively independent of pressure. The more flocculated the solids, the more compressible will be the filter cake. 

For incompressible cakes a is independent of the pressure drop and of position in the cake. 

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