Radial Filtration Formulation
Consider that a slurry is applied to the inner surface of a drum filter and the filtrate leaves from its outer surface. The model developed here is also equally applicable for the reverse operation. The filter cake is located between the filter inner surface radius rw(cm) over which the cake is formed, and the slurry side cake surface radius rw(cm) and its thickness is denoted by h = rw-rc. The external surface radius of the filter from which the filtrate leaves is re(cm) and the filter width is indicated by w(cm), such that the area of the inner filter surface over which the cake is formed is 2nrwԜ. The slurry flows over the cake surface at a tangential or cross-flow velocity of vf(cm/s) and the filtrate flows into the filter at a filtration velocity of uf(cm³/cm³.s) normal to the filter face due to the overbalance of the pressure between the slurry and the effluent sides of the filter. The flowing suspension of particles and the filter cake (solid) are denoted by the subscripts / and s, respectively.
The carrier phase (liquid) and the particles are denoted, respectively, by / and p. Following Tien et al. (1997), the slurry is considered to contain particles larger than the filter medium pore size that form the filter cake and the particles smaller than the pore sizes of the filter cake and the filter medium, which can migrate into the cake and the filter to deposit there. All particles (small plus large) are denoted by p, and the large and small particles are designated by p1 and p2, respectively. Civan (1998b, 1999b) developed the filtration models by considering the cake-thickness averaged volumetric balance equations for
1. The total (fine plus large) particles of the filter cake;
2. The fine particles of the filter cake;
3. The carrier fluid of the suspension of fine particles flowing through the filter cake; and
4. The fine particles carried by the suspension of fine particles flowing through the filter cake.
The radial mass balances of all particles forming the cake, the small particles retained within the cake, the carrier fluid, and the small particles suspended in the carrier fluid are given, respectively, by (Civan, 1998b):
Ρp and Ρ, are the densities of the particles and the carrier fluid (g/cm3). us and ut are the volumetric fluxes of the compressing filter cake and the carrier fluid flowing through the cake (cm3/ cm 2 • s). cp2s and cp2l denote the small particle masses contained per unit volume of the cakeforming particles and the carrier fluid flowing through the cake (g/cm3). t and r denote the time and radial distance (cm), respectively. Rps is the mass rate of particle deposition from the slurry over to the moving cake surface (g/s/cm3) given by:
where R°ps and R°p2s denote, respectively, the mass rates of large and small particles deposition from the slurry over the cake surface (g/s/cm3). R°p2s is usually negligible unless the small particles are retained by a process of jamming of small particles across the pores of the large particles, such as described by Civan (1994, 1996) and Liu and Civan (1996).
The variation of the filter cake thickness (cm) h = rw - rc can be calculated using the variable radius, rc = r c ( t ) , of the slurry side filter cake surface. For many practical applications, it is reasonable to assume that the particles and the carrier fluid are incompressible. The volumetric retention rates of the large and small particles are given, respectively, by:
The volumetric concentration (or fraction) of species / in phase 7, the volume fraction of species i of phase j in the bulk of the cake system, and the superficial velocity of species / of phase j are given, respectively, by:
t denotes the time; ɸ, Ԑp2j, and Ԑp2l are the cake-thickness-average porosity, the fine particle volume fractions of the cake matrix and the suspension of fine particles flowing through the cake matrix, respectively; (Ԑpt)slurry is the volume fraction of the total (fine plus large) particles in the slurry; and (ut)slurry and (ut)flltrate denote the volume fluxes of the carrier fluid entering and leaving the filter cake, respectively. Substituting Eqs. 12-92 to 12-96 into Eqs. 12-87 to 12-90 leads to the following volumetric balance equations, respectively (Civan, 1999b):
Eqs. 12-97 through 100 can be solved numerically subject to the initial conditions given by:
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