Cake specific resistance

The scale-up of conventional cake filtration uses the basic filtration equation (eq. 4). Solutions of this equation exist for any kind of operation, eg, constant pressure, constant rate, variable pressure—variable rate operations (2). The problems encountered with scale-up in cake filtration are in estabHshing the effective values of the medium resistance and the specific cake resistance. 

The specific cake resistance is the most troublesome parameter ideally constant, its value is needed to calculate the resistance to flow when the amount of cake deposited on the filter is known. In practice, it depends on the approach velocity of the suspension, the degree of flow consoHdation that the cake undergoes with time, the feed soHds concentration, and, most importantly, the appHed pressure drop Ap. This changes due to the compressibiHty of most cakes in practice. often decreases with the velocity and the feed concentration. It may sometimes go through a maximum when it is plotted against soHds concentration. The strongest effect on is due to pressure, conventionally expressed as ... 

An example of the concentration effect on the specific cake resistance is available (12) that reports results of some experiments with a laboratory horizontal vacuum belt filter. In spite of operational difficulties in keeping conditions constant, the effect of feed concentration on specific cake resistance is so strong that it swamps all other effects. 

The benefits of prethickening can be summarized as an increase in dry cake production, reduction in specific cake resistance, clearer filtrate, and less cloth blinding. 

If ah of the nonfiltration operations are grouped together into a downtime, assumed to be fixed and known, an optimum filtration time in relation to p can be derived by optimizing the average dry cake production obtained from the cycle. Eor constant pressure filtration and where the medium resistance R and the specific cake resistance are constant, the fohowing equation appHes ... 

When the medium resistance R is smah compared with the specific cake resistance (, the second term in the above equation becomes negligible and the optimum filtration time becomes equal to downtime p. For any other case, p is always greater than p. It fohows, therefore, that the filtration time... 

It is both convenient and reasonable in continuous filtration, except for precoat filters, to assume that the resistance of the filter cloth plus filtrate drainage is neghgible compared to the resistance of the filter cake and to assume that both pressure drop and specific cake resistance remain constant throughout the filter cycle. Equation (18-51), integrated under these conditions, may then be manipulated to give the following relationships ... 

The symbol Ot represents the average specific cake resistance, which is a constant for the particular cake in its immediate condition. In the usual range of operating conditions it is related to the pressure by the expression... 

Specific cake resistance (porosity) Viscosity Homogeneity of cake as deposited on the filter medium... 

At n = 1 N-s/m, hj, = 1 m and u = 1 m/s, the value r = AP. Thus, the specific cake resistance equals the pressure difference required by the liquid phase (with a viscosity of 1 N-s/m ) to be filtered at a rate u = 1 m/s for a cake 1 m thick. This hypothetical pressure difference is, however, beyond a practical range. For highly compressible cakes, the value ro reaches 10 m or more. Assuming V = 0 (at the start of filtration) where there is no cake over the filter plate, the equation becomes ... 

This expression can be represented graphically in dimensionless form to simplify its use. It is generally expressed as the so-called filtration number , defined as follows E, = /iR, / 2APT3 jr x . The filtration number, E, is dimensionless and varies from zero at Rf = 0 to a large value when there is an increase in the viscosity of the sludge and Rf or a decrease in pressure drop, auxiliary time, specific cake resistance and the ratio of cake volume to filtrate volume. It may be assumed in practice that F(, = 0 to 10. If washing and drying times are constant and independent of filtration time, they may be added directly to the auxiliary time. In... 

To apply these equations, let s consider the following example. Determine a constant rate of filtration and the time of operation corresponding to the maximum capacity of a batch filter having the following conditions maximum permissible pressure difference AP = 9x10 N/m sludge viscosity /r = 10 N-s/m filter plate resistance Rf = 56x 10 ° m specific cake resistance r = 3 X 10 m ° x = 0.333 auxiliary time = 600 s maximum permissible cake thickness h = 0.025 m. The solution is as follows ... 

A further recommendation, depending on the application, is not to increase the pressure difference for the purpose of increasing the filtration rate. The cake may, for example, be highly compressible thus, increased pressure would result in significant increases in the specific cake resistance. We may generalize the selection process to the extent of applying three rules to all filtration problems ... 

Comparative calculations of specific capacities of different filters or their specific filter areas should be made as part of the evaluation. Such calculations may be performed on the basis of experimental data obtained without using basic filtration equations. In designing a new filtration unit after equipment selection, calculations should be made to determine the specific capacity or specific filtration area. Basic filtration equations may be used for this purpose, with preliminary experimental constants evaluated. These constants contain information on the specific cake resistance and the resistance of the filter medium. 

The specific cake resistance r(nr ) depends on particulate bed characteristics e and According to the Carman-Kozeny equation for packed beds (Chapter 2)... 

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 ... 

Both permeability, K, and specific cake resistance, a, depend on the particle shape and size distribution. However, this relationship has not been recognized conclusively. Therefore, information on the crystal shape and size distribution, which can be obtained from experiments on crystallization, cannot be easily translated into the language of filtration characteristics. 

The values of the specific cake resistance, a, and the resistance of the filter medium, Rfm, can be determined from the slope and the intercept of the plot t/V versus V for a given AP (see Fig. 

Specific cake resistance a ranges from 10 m/kg for easily filtered large particles to lo m/kg for gelatinous cakes. The membrane resistance Rp typically equals 10 to lO" m. Both a and Rp depend on the filtering pressure. 

Measurements of filtration rates should be repeated at different pressures or different vacuum levels. This gives information on the influence of pressure on the specific cake resistance. The specific resistance of cakes that are difficult to filter is often pressure-dependent. Thus, use of excessive pressure can result in blocking of the cake, causing filtration to stop. In the case of compressible cakes, information is needed over the whole range of pressures being considered for industrial filters since extrapolation of compressibility beyond the experimentally covered region is always risky. The larger the scale of an experimental filter, the less risky predictions based on the experimental data. 

In conclusion, the following experiments on filtration-washing-deliquoring should be performed to produce data (viscosity of liquids, effective solid concentration, specific cake resistance, cake compressibility, etc.) that are necessary to evaluate times of individual steps of filtration at an industrial scale, i.e. to obtain the proper basis for scale-up of filtration processes measure the filtrate volume versus time make marks on your vacuum flask and take down the time when the filtrate level reaches the mark => no more experiments are needed for preliminary evaluations of filtration properties of slurries initially fines pass the filter medium => recirculate them to the slurry,... 

The overriding factor will be the filtration characteristics of the slurry whether it is fast filtering (low specific cake resistance) or slow filtering (high specific cake resistance). The filtration characteristics can be determined by laboratory or pilot plant tests. A guide to filter selection by the slurry characteristics is given in Table 10.3 which is based on a similar selection chart given by Porter et al. (1971). 

Figure 14.4 Specific cake resistance of several microorganisms measured by dead-end filtration.

Calculate the volume of filtrate versus time relationship, when the specific cake resistance of baker s yeast a and the resistance of the filtering medium are 7 X lO m kg and 3.5 X 10 ° m", respectively. How long does it take... 

---