Variable Pressure and Rate Filtration

Ps = static pressure over the same section of cake 

Parameters q and W are variables when filtration conditions change. Coefficient (ry,)x is a function of pressure  

The exact relationship can be derived from experiments in a device called a compression-permeability cell. Once this relationship is defined, the integral of the right side of Equation 34 may be evaluated analytically (or if the relationship is in the form of a curve, the evaluation may be made graphically). The interrelation between W and Pj is established by the pump characteristics, which define q = f(W) in Equation 34. Filtration time may then be determined from the following definition  

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The dynamics of variable-rate and -pressure filtrations can be illustrated by pressure profiles that exist across the filter medium. Figure 7 shows the graphical representation of those profiles. According to this plot, the compressed force in the cake section is ... 

Variable Rate and Pressure Filtration for Compressible Cakes... 

The punq) characteristic is again required to evaluate variable rate and pressure filtration. The result normally required is a calculation of the time taken to filter a knorvn volume or mass of slurry. The cake depth is also inportant as most filters are of the batch type and this d th must be less than the clearance within the filter. 

Equation (2.19) contains three variables and four constants time, filtrate volume and pressure and filtration area, viscosity, concentration and specific resistance. The last two are constant only if the filter cake is incompressible. The equation can be solved analytically only if one of the three variables is held constant. This reflects the physical mode of operation of industrial filters vacuum filtration tends to be at constant pressure and pressure filtration is often under constant rate, at least until some predetermined pressure has been achieved. Thus the following mathematical models are very relevant to these filtrations. 

Table 2.2 Illustrative table headings for time required to effect a variable rate and variable pressure filtration...

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. 

Variable-Pressure, Variable-Rate Filtration The pattern of this categoiy comphcates the use of the basic rate equation. The method of Tiller and Crump (loc. cit.) can be used to integrate the equation when the characteristic curve of the feed pump is available. 

Because pore sizes in the cake and filter plate are small and the liquid velocity through the pores is low, the flow of filtrate may be considered laminar and Poiseuille s law is applicable. Filtration rate is directly proportional to the difference in pressure and inversely proportional to the fluid viscosity, as well as to the hydraulic resistance of the cake and filter plate. Because the pressure and hydraulic resistances of the cake and filter plate change with time, the variable rate of filtration may be expressed as ... 

The resistance of the cake of accumulated particulates, Rc, is more complicated it is a variable which increases as filtration proceeds resulting in a progressively lower filtration rate at constant pressure. This is due to the continually increasing thickness of the cake and its compaction under the pressurized conditions of filtration. If Rm is defined as above, Rc must also include the effect of pore plugging within the membrane. 

The effect of pressure shown earlier is modified in most industrial flltrations in which cake compressibility usually lies between 0.1 and 0.8. Furthermore, the resistance of the filter reduces the effects of the respective variables. It has been found, however, that an increase in pressure causes a nearly proportionate increase in the flow rate in the filtration of granular or crystalline solids. Flocculent or slimy precipitates, on the other hand, have their filtration rates increased only slightly by an increase in pressure. Some materials have a critical pressure above which a further increase results in an actual decrease in flow rate. 

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 typical sequence of operations in horizontal diaphragm and filter presses is shown schematically in Figure 6.3 (see also Sections 1.4.2.5 and 1.4.2.3). Suspension is fed into the chambers of the press, with either a positive displacement or centrifugal pump to initiate respectively constant rate/variable pressure or variable rate/variable pressure filtration. Cakes are usually formed simultaneously on the two opposing sides of each chamber. In plate... 

In industrial filtration operation carried out under variable pressure and variable rate conditions, the method of Tiller must be employed to integrate Eq. (1) for the general case (Tiller, 1958). 

By operating cycle. Filtration may be intermittent (batch) or continuous. Batch filters may be operated with constant-pressure driving force, at constant rate, or in cycles that are variable with respect to both pressure and rate. Batch cycle can vary greatly, depending on filter area and sohds loading. 

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