Filtration theories

In deriving this equation it is assumed that the cake is uniform and that the voidage is constant throughout. In the deposition of a filter cake this is unlikely to be the case and the voidage, e will depend on the nature of the support, including its geometry and 

Filter cakes may be divided into two classes—incompressible cakes and compressible cakes. In the case of an incompressible cake, the resistance to flow of a given volume of cake is not appreciably affected either by the pressure difference across the cake or by the rate of deposition of material. On the other hand, with a compressible cake, increase of the pressure difference or of the rate of flow causes the formation of a denser cake with a higher resistance. For incompressible cakes e in equation 7.1 may be taken as constant and the quantity e3/[5(l — e)2S2] is then a property of the particles forming the cake and should be constant for a given material. 

It may be noted that, when there is a hydrostatic pressure component such as with a horizontal filter surface, this should be included in the calculation of — AP. 

Equation 7.2 is the basic filtration equation and r is termed the specific resistance which is seen to depend on e and S. For incompressible cakes, r is taken as constant, although it depends on rate of deposition, the nature of the particles, and on the forces between the particles, r has the dimensions of L-2 and the units m-2 in the SI system. 

Relation between thickness of cake and volume of filtrate 

The pressure drop across the filter cake may be related to the filtrate flow by the expression (McCabe et al., 2005)  

physically represents the pressure drop necessary to give unit superficial velocity of filtrate of unit viscosity through a cake containing unit mass of solid per unit filter area. It is related to the properties of the cake by 

If a cake is composed of rigid nondeformable solid particles a is independent of -AP and does not vary throughout the depth of the cake, and is known as incompressible cake. However, if the cake contains nonrigid, deformable solid particles or agglomerates of particles the resistance to flow will depend on the pressure drop and will vary throughout the depth of the cake. In this case the cake is called compressible and an average value of the specific resistance for the entire cake must be used in Equation 10.89. This average specific resistance must be measured experimentally for any particular slurry. 

By analogy with Equation 10.89 the filter medium resistance may be defined by the following relation  

It is reasonable to assume that is constant during any filtration cycle and that it includes the resistance to filtrate flow offered by the filter channels. This being the case. Equations 10.88, 10.89, and 10.91 can be combined to give 

---

There is an additional pressure drop across the cake, developed by electroosmosis, which leads to increased flow rates through the cake and further dewatering at the end of the filtration cycle. The filtration theory proposed for electrofiltration assumes the simple superposition of electroosmotic pressure on the hydraulic pressure drop. 

It does not matter, from the fundamental point of view, how the pressure drop is generated in the filter. In the case of the centrifugal filters there is an additional phenomenon of the mass forces acting on the Hquid within the cake. The conventional filtration theory must be amended to include this effect (2). 

Modem filtration theory tends to prefer the Ruth form of Darcy s law, ie,... 

Conventional filtration theory has been challenged a two-phase theory has been appHed to filtration and used to explain the deviations from paraboHc behavior in the initial stages of the filtration process (10). This new theory incorporates the medium as an integral part of the process and shows that the interaction of the cake particles with the medium controls filterabiHty. It defines a cake-septum permeabiHty which then appears in the slope of the conventional plots instead of the cake resistance. This theory, which merely represents a new way of interpreting test data rather than a new method of siting or scaling filters, is not yet accepted by the engineering community. 

While research has developed a significant and detailed filtration theory, it is still so difficult to define a given liquid-solid system that it is both faster and more accurate to determine filter requirements by performing small-scale tests. Filtration theoiy does, however, show how the test data can best be correlated, and extrapolated when necessary, for use in scale-up calculations. 

Wash Time Cake-washing time is the most difficult of the filtration variables to correlate. It is obviously desirable to use one which provides a single cni ve for all of the data. Filtration theory suggests three possible correlations [Eqs. (18-59) to (18-61)]. These are listed below, beginning with the easiest to use ... 

To put the various contributions to the modeling of codeposition of particles into perspective, not only must the relevant electrochemical literature be reviewed, but also that in the field of filtration. Theories on the mechanisms of capture and encapsulation of particles during metal deposition have only begun to appear in the last two decades. The scope of this section will be to summarize the models published since 1970, and give the basis for their explanation. (A review of the older work can be found in the Ph.D. thesis of C. Buelens [58].) Let us see review the models in chronological order. 

Brown Coal Liquefaction process, 6 849 Brown cyclization product, 21 147 Brownian diffusion, 13 151, 152 Brownian diffusion, in depth filtration theory, 11 339... 

Cake filtration theory, 11 330-337 advanced modeling for, 11 336—337 compactable cake modeling in,... 

Direct particle interception, in depth filtration theory, 11 339 Direct potentiometry, 9 582—585 Direct printing, 9 218 Direct process, silicone synthesis via,... 

Electrostatic fluidized-bed coating, 7 55-56 Electrostatic forces, 9 569, 570 11 800 and adsorbent selectivity, 1 584 in adsorption, 1 583 in solvent-solute interactions, 23 91-92 Electrostatic particle forces, in depth filtration theory, 11 339 Electrostatic precipitators (ESP), 11 714 13 180 23 552 26 699-706 advantages of, 26 700 applications of, 26 701-703, 705t design considerations related to,... 

Research Centers (IUCRC), 24 395 Inelastic mean free path (IMFP), 24 87 Inert fluids, 11 877 properties of, 11 879 Inert gas dilution, 11 456 Inert gases, 13 456 17 376-377. See also Helium- group elements Noble gases narcotic potency and solubility of, 17 377 Inert gas generators, 17 280 Inertial confinement fusion targets, microcapsules as, 16 460 Inertial impaction, in depth filtration theory, 11 339... 

Most penetrating particle size, in depth filtration theory, 11 340-341 Motens, molecular formula and structure, 5 127t... 

Particle boundary location, 18 148 Particle capture mechanisms, in depth filtration theory, 11 339-340 Particle changes, in solid-fluid reactions, 21 344... 

Sedimentation classifiers, 76 619-620 in depth filtration theory, 11 339 in particle size measurement, 78 142-144 in solid-liquid separation, 76 656-657 Sedimentation rate... 

Stokes number (Stk), 22 57, 23 184, 190 in depth filtration theory, 77 340 Stokes Raman scattering, 27 322 Stokes scatter, 76 485-486 Stokes shifts, 20 512 Stomach poison insecticides, 74 339... 

Ruth, B. F. Ind. Eng. Chem. 38 (1946) 564. Correlating filtration theory with practice. 

Ideally, cross-flow microfiltration would be the pressure-driven removal of the process liquid through a porous medium without the deposition of particulate material. The flux decrease occurring during cross-flow microfiltration shows that this is not the case. If the decrease is due to particle deposition resulting from incomplete removal by the cross-flow liquid, then a description analogous to that of generalised cake filtration theory, discussed in Chapter 7, should apply. Equation 8.2 may then be written as ... 

Gel Filtration-Theory and Practice. (1984) Pharmacia Pine Chemicals, Rahms i Lnnd, Uppsala, Sweden. 

The transport behavior of colloids commonly is modeled by colloid filtration theory (CFT) (Yao et al. 1971), which is based on extension of the common advection-dispersion equation. The one-dimensional advection-dispersion-filtra-tion equation is written... 

This filtration theory and a local re-computation of the evolving unit-cell geometry due to deposition of particles (Fig. 13) was employed and a transient filtration model has been derived and tested with very good success against experimental data with ceramic, metallic and fibrous filters (Bissett and Shadman, 1985 Zarvalis et al., 2003). In addition, the same unit-cell-based... 

Filterability of slurries depends so markedly on small and unidentified differences in conditions of formation and aging that no correlations of this behavior have been made. In feet, the situation is so discouraging that some practitioners have dismissed existing filtration theory as virtually worthless for representing filtration behavior. Qualitatively, however, simple filtration theory is directionally valid for modest scale-up and it may provide a structure on which more complete theory and data can be assembled in the future. 

Because the structure of a cake is highly dependent on operating conditions and its history, the Kozeny equation is only of qualitative value to filtration theory by giving directional effects. 

Some data fitted to these equations by Tiller et al. (1979) are in Table 11.8 here the constant k is the same for both a and e, although this is not necessarily generally the case. Unfortunately, these data show that the parameters are not independent of the pressure range. Apparently the correlation problem has not been solved. Perhaps it can be concluded that insofar as the existing filtration theory is applicable to real filtering behavior, the approximation of Almy and Lewis may be adequate over the moderate ranges or pressures that are used commonly, somewhere between 0.5 and 5 atm. 

---