Figure 3. Frictional drag on particles in compressible cakes.
The terra ps is a fictitious pressure, because the cross-sectional area A is not equal to either the surface area of the particles nor the actual contact areas In actual cakes, there is a small area of contact Ac whereby the pressure exerted on the solids may be defined as Fs/Ac.
Taking differentials with respect to x, in the interior of the cake, we obtain:
This expression implies that drag pressure increases and hydraulic pressure decreases as fluid moves from the cake's outer surface toward the filter plate. From Darcy's law, the hydraulic pressure gradient is linear through the cake if the porosity (e) and specific resistance (a) are constant. The cake may then be considered incompressible. This is illustrated by the straight line obtained from a plot of flowrate per unit filter area versus pressure drop shown in Figure 4. The variations in porosity and specific resistance are accompanied by varying degrees of compressibility, also shown in Figure 4.
The rate of the filtration process is directly proportional to the driving force and inversely proportional to the resistance.
Because pore sizes in the cake and filter medium are small, and the liquid velocity through the pores is low, the filtrate flow may be considered laminar: hence, Poiseuille's law is applicable. Filtration rate is directly proportional to the difference in pressure and inversely proportional to the fluid viscosity and to the hydraulic resistance of the cake and filter medium. Because the pressure and hydraulic resistances of the cake and filter medium change with time, the variable rate of filtration may be expressed as:
where V = volume of filtrate (m3) A = filtration area (m2) t = time of filtration (sec)
Assuming laminar flow through the filter channels, the basic equation of filtration as obtained from a force balance is:
where Ap =
pressure difference (N/m2) viscosity of filtrate (N-sec/m2) filter cake resistance (m"1) initial filter resistance (resistance of filter plate and filter channels) (m"1) u = filtration rate (m/sec), i.e., filtrate flow through cake and filter plate dV/dx = filtration rate (m3/sec), i.e., filtrate flow rate
Filter cake resistance (Rc) is the resistance to filtrate flow per unit area of filtration. Rc increases with increasing cake thickness during filtration. At any instant, Rc depends on the mass of solids deposited on the filter plate as a result of the passage of V (m3) filtrate. Rf may be assumed a constant. To determine the relationship between volume and residence time t, Equation 5 must be integrated, which means that Rc must be expressed in terms of V.
We denote the ratio of cake volume to filtrate volume as x^. Hence, the cake volume is x^V. An alternative expression for the cake volume is hcA; where hc is die cake height in meters. Consequently:
Hence, the thickness of the cake, uniformly distributed over the filter plate, is:
The filter cake resistance may be expressed as:
where r0= specific volumetric cake resistance (m"2).
As follows from Equation 8, r0 characterizes the resistance to liquid flow by a cake having a thickness of 1 m.
Substituting for Rc from Equation 8 into Equation 5, we obtain:
Filtrate volume, Xq can be expressed in terms of the ratio of the mass of solid particles settled on the filter plate to the filtrate volume (xw) and instead of r0, a specific mass cake resistance rw is used. That is, r w represents the resistance to flow created by a uniformly distributed cake, in the amount of 1 kg/m2. Replacing units of volume by mass, the term r0 Xq in Equation 9 changes to rwxw. Neglecting filter plate resistance (Rf = 0), and taking into account Equation 7, we obtain from Equation 3 the following expression:
At fn= 1 N-sec/m2, hc = 1 m and u = 1 m/sec, r 0 = Ap. Thus, the specific cake resistance equals the pressure difference required by the liquid phase (with a viscosity of 1 N-sec/m2) to be filtered at a linear velocity of 1 m/sec through a cake 1 m thick. This hypothetical pressure difference, however, is beyond a practical range. For highly compressible cakes, r0 can exceed 1012m2. Assuming V = 0 (at the start of filtration) where there is no cake over the filter plate. Equation 9 becomes:
At ^ = 1 N-sec/m2 and u = 1 m/sec, Rf = Ap. This means that the filter plate resistance is equal to the pressure difference necessary for the liquid phase (with viscosity of 1 N-sec/m2) to pass through the filter plate at a rate of 1 m/sec. For many filter plates Rf is typically 1010 m"1 .
For a constant pressure drop and temperature filtration process all the parameters in Equation 9, except V and t, are constant. Integrating Equation 9 over the limits of 0 to V, from 0 to t, we obtain:
Dividing both sides by itr0x0/2A gives:
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, r0 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, r0, fi and Ap will be constant.
The terms in parentheses in Equation 13 are known as the "filtration constants", and are often lumped together as parameters K and C; where:
Hence, a simplified expression may be written to describe the filtration process as follows:
Filtration constants K and C can be experimentally determined, from which the volume of filtrate obtained over a specified time interval (for a certain filter, at the same pressure and temperature) can be computed. If process parameters are changed, new constants K and C can be estimated from Equations 14 and 15. Equation 16 may be further simplified by denoting x0 as a constant that depends on K and C:
Substituting Tq into Equation 16, the equation of filtration under constant pressure conditions is:
Equation 18 defines a parabolic relationship between filtrate volume and time. The expression is valid for any type of cake (i.e., compressible and incompressible). From a plot of V + C versus (t+t0), the filtration process may be represented by a parabola with its apex at the origin as illustrated in Figure 5. Moving the axes to distances C and t0 provides the characteristic filtration curve for the system in terms of volume versus time. Because the parabola's apex is not located at the origin of this new system, it is clear why the filtration rate at the beginning of the process will have a finite value, which corresponds to actual practice.
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