Forms of Cake Filtration Equation

Substituting Equation (2.16) into Equation (2.12) gives d/ icVa 

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. 

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In cake filtration, the filter medium acts as a strainer and collects the solid particles on top of the initial layer. A filter cake is formed and the flow obeys the Carman-Kozeny equation for packed beds. 

Resistivity of filter cakes depends on the conditions of formation of which the pressure is the major one that has been investigated at length. The background of this topic is discussed in Section 11.3, but here the pressure dependence will be incorporated in the filtration equations. Either of two forms of pressure usually is taken,... 

The rate of filtrate delivery is inversely proportional to the combined resistance of the cake and filtering medium, inversely proportional to the viscosity of the filtrate, and directly proportional to the available filtering area and the pressure-difference driving force. This statement can be expressed in equation form as... 

Erom this definition, Lf is the amonnt of cake formed per nnit area of filter cloth per unit of time. In vacnnm filtration, the only time that the cake is formed is when the drum is submerged in the tank. In pressure filtration, the only time that the cake is also formed is when the slndge is pumped into the plates. Thns, t in the previons equation is called the form time tf. Also, the filters operate on a cycle. Calling the cycle time as t tf may be expressed as a fraction / of C- Thus, t = tf = fL may be... 

The functional form of the static filtration curve (Figure 33) may be readily derived if it is assumed that the flow of the filtrate through the filter cake is governed by Darcy s law (126). The flow rate of the filtrate dVfe/df is given by Darcy s equation... 

In cake filtration, once a layer of solid particles has formed on the filter medium, its surface becomes the actual filter medium. As the solids get deposited they add to the thickness of the cake whereas the clear liquid passes through. The cake is therefore composed of a bulky mass of particles, among which irregular small channels run. The flow of liquid through the channels is always laminar and can be described by the following Ruth s filtration equation ... 

The filter cake concentration profile relative to actual height from the medium can be obtained from the above if the superficial filtrate flow rate through the medium q has been measured via a modified form of Darcy s law Equation (2.47) ... 

Constant-pressure compressible cake filtration, solving the basic flow and filtration equations on an incremental time basis. Cloth resistance is included, hence the pressure drop forming the filter cake changes according to the total jplied pressure minus that due to the flow rate of filtrate through the filter cloth. This is solved 1 means of a fixed number of iterations. 

From equation (6.24), which accounts for the presence of cake formed during the primary filtration stage, the total cycle time is... 

On continuous filters cake formation is described by the general filtration equation (i.e. the integrated form of equation (6.3) with Ap = constant) to an extent dictated by the time of filtration such that... 

Equation 10.5 can be used to predict the washing curve but the washing rate has to be determined first, from the cake and medium resistance, using the general filtration equation 9.5 for Ap = 5 bar. As the mother liquor and the washing liquid have the same properties, the washing rate is equal to the final rate of filtration when the cake has been formed (i.e. after 0.028 m of the feed suspension has been filtered). The sequence of the calculations is as follows ... 

In our discussion of the filtration process, we had taken the simple-minded view that static filtration will continue until the point at which mudcake thickness reaches the equilibrium thickness derived earlier. At that time, cake growth terminates, but front motions continue as determined by Equation 17-72. This view is approximate and was adopted for discussion only. In reality, the shear stress in the borehole continuously acts on the mudcake as it is formed, so that the interactions between mudcake growth, reservoir Darcy flow, and borehole annular flow can be complicated. We do not pretend to solve this more realistic problem, but we do believe that the principal elements of both static and dynamic filtration processes have been satisfactorily identified. 

In the filtration of small amounts of fine particles from liquid by means of bulky filter media (such as absorbent cotton or felt) it has been found that the preceding equations based upon the resistance of a cake of solids do not hold, since no cake is formed. For these cases, in which filtration takes place on the surface or within the interstices of a medium, analogous equations have been developed [Hermans and Bredee, J. Soc. Chem. Ind., 55T, 1 (1936)]. These are usefully summarized, for both constant-pressure and constant-rate conditions, by Grace [Am. In.st. Chem. Eng. J., 2, 323 (1956)]. These equations often apply to the clarification of such materials as sugar solutions, viscose and other spinning solutions, and film-casting dopes. 

The basic hydrodynamic equations are the Navier-Stokes equations [51]. These equations are listed in their general form in Appendix C. The combination of these equations, for example, with Darcy s law, the fluid flow in crossflow filtration in tubular or capillary membranes can be described [52]. In most cases of enzyme or microbial membrane reactors where enzymes are immobilized within the membrane matrix or in a thin layer at the matrix/shell interface or the live cells are inoculated into the shell, a cake layer is not formed on the membrane surface. The concentration-polarization layer can exist but this layer does not alter the value of the convective velocity. Several studies have modeled the convective-flow profiles in a hollow-fiber and/or flat-sheet membranes [11, 35, 44, 53-56]. Bruining [44] gives a general description of flows and pressures for enzyme membrane reactor. Three main modes... 

Equations (30.32) and (30.33) apply both to continuous vacuum filters and to continuous pressure filters. When R is negligible, Eq. (30.33) predicts that the filtrate flow rate varies inversely with the square root of the viscosity and of the cycle time. This has been observed experimentally with thick cakes and long cycle times with short cycle times, however, this is not true and the more complicated relationship shown in Eq. (30.32) must be used. In general, the filtration rate increases as the drum speed increases and the cycle time diminishes, because the cake formed on the drum face is thinner than at low drum speeds. At speeds above a certain critical value, however, the filtration rate no longer increases with speed but remains constant, and the cake tends to become wet and difficult to discharge. 

The imbalance in the chemical potential of the water in the shale and drilling fluid results in a tendency for water to enter the shale. Equation 110 is applicable to both water- and oil-based drilling fluids. When fij = fjL sh equation 110 gives the well-known expression for the swelling pressure (Psh — P) between the shale and the drilling fluid. The permeability of shales is very low and the rate of filtration into the shale will be below the critical filtration rate (140) and no filter cake will form on the surface of the shale. 

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