Simple Filter Theory

The flow through a filter is shown schematically in Fig. 12.8. A slurry (a fluid containing suspended solids) flows through a filter medium (most often a cloth, but sometimes paper, porous metal or a bed of sand). The solid particles in the slurry deposit on the face of the filter medium, forrning the filter eake. The liquid, free from solids, flows through both cake and filter medium. Applying 

Bernoulli s equation from point 1 to point 3, we see elevation. There is a slight change in velocity (due 

Here there are two resistances in series with the same flow rate through them. Letting the subscript FM indicate filter medium, we write Eq. 12.25 twice and equate the identical flow rates (see Fig. 12.8)  

This equation describes the instantaneous flow rate through a filter it is analogous to Ohm s law for two resistors in series, so the fj, Ax/k terms are called the cake resistance and the cloth resistance. 

The resistance of the filter medium is normally assumed to be a constant independent of time, so Axlk) is replaced with a, constant a. If the filter cake is uniform, then its instantaneous resistance is proportional to its instantaneous thickness. However, this thickness is related to the volume of filtrate which has passed through the cake by the material balance  

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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. 

Strain, nominal The strain at a point calculated in the net eross section by simple elastic theory without taking into aceount the effeet on strain produced by geometric discontinuities sueh as holes, grooves, filters, etc. 

Observations which could not be accommodated within a simple, limited capacity filter model led to the development of late-selection theories of atten-... 

Eulerian equations for the dispersed phase may be derived by several means. A popular and simple way consists in volume filtering of the separate, local, instantaneous phase equations accounting for the inter-facial jump conditions [274]. Such an averaging approach may be restrictive, because particle sizes and particle distances have to be smaller than the smallest length scale of the turbulence. Besides, it does not account for the Random Uncorrelated Motion (RUM), which measures the deviation of particle velocities compared to the local mean velocity of the dispersed phase [280] (see section 10.1). In the present study, a statistical approach analogous to kinetic theory [265] is used to construct a probability density function (pdf) fp cp,Cp, which gives the local instantaneous probable num-... 

We describe a model that combines rough surface contact mechanics with elastic pad mechanics. The theory is capable of describing planarization in both the classical contact and roughness-dominated regimes (see the solid curves in Fig. 6.7). We then introduce a simple approximation for the asperity filtering regime. 

In the asperity filtering regime, the Greenwood and Williamson theory no longer properly models the local contact pressure since the model contains no notion of asperity width. We describe a simple statistical method for incorporating width effects in two-dimensional polishing. A more sophisticated but more complex approach based on elasticity theory that takes into account asperity shape and the interaction of the asperity with trench structures of similar size can be found in Reference 27. The statistical approach assumes that... 

Filtration, in the most general sense, may be defined as the removal of particles from the aerosol. This occurs either by their attachment to nonaerosol media (walls, vegetation, "fabric filters", etc.) or to larger particles which are subsequently removed. Since particle transport in the gas is intimately involved, a characterization of the gas flow field and the detailed mechanisms of particle kinetic theory near a surface must be invoked. Classically, filtration was treated as the simple adhesion of a single particle to a surface. However, it is now known that after the first particles adhere, subsequent ones tend to be captured by the initial ones to form chains. Impaction of a large particle upon such a chain or other break-off processes can cause resuspension. Thus, filtration is dependent upon properties of the aerosol and gas as a whole [1.9,10]. 

Precipitation is, by far, the most common method for dealing with metal-containing waste. The theory is simple. Some metal salts are very insoluble, precipitation generates these insoluble salts in the waste stream by the addition of an appropriate counter anion the precipitate is then filtered off (scheme 14.2). 

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