Filtration equations for

Traditionally, the average specific cake and medium resistances have been deterrnined from constant pressure experiments and the solution of the basic filtration equation for constant pressure which relates filtrate volume to time. This relationship is, in theory, paraboHc but deviations occur in practice. 

We now turn attention towards the ease of eonstant-rate filtration. When sludge is fed to a filter by means of a positive displaeement pump, the rate of filtration is nearly constant, i.e., dV/dx = constant. During constant-rate filtration, pressure increases with cake thickness. As sueh, the principal filtration variables are pressure ind filtrate volume, or pressure and filtration time. Integrating the filtration equation for a constant-rate process, we find that the derivative dV/dx ean simply be replaeed by V/x, and we obtain ... 

This can be substituted into Equations (2.19) or (2.22) to give the general filtration equation for conopressible cakes. Considering only the pressure drop due to the filter cake the general equation is ... 

Constant-Rate Filtration of Incompressible Cake. The filtration equation for filtration at a constant pressure of 38.7 psia (266.8 kPa) is... 

More recently, Machac and Crha reviewed the progress to date and showed the importance of the correct values of the constants of the power law model, n and K, in the evaluation of filtration experiments with power law fluids. This is because these constants appear in the governing filtration equations. For example, for constant pressure filtration and when the medium resistance can be neglected, the governing equation for a power law fluid characterized by constants n and K (i.e. when shear stress r and rate of shear 7 are related as t = is... 

The scale-up of conventional cake filtration uses the basic filtration equation (eq. 4). Solutions of this equation exist for any kind of operation, eg, constant pressure, constant rate, variable pressure—variable rate operations (2). The problems encountered with scale-up in cake filtration are in estabHshing the effective values of the medium resistance and the specific cake resistance. 

This can be substituted for in the basic filtration equation 4, which can then be solved for the filtration operation in question. 

It is both convenient and reasonable in continuous filtration, except for precoat filters, to assume that the resistance of the filter cloth plus filtrate drainage is neghgible compared to the resistance of the filter cake and to assume that both pressure drop and specific cake resistance remain constant throughout the filter cycle. Equation (18-51), integrated under these conditions, may then be manipulated to give the following relationships ... 

The correlations used are based partly on theoretical consideration and partly on empirical observations. The basic filtration data are correlated by application of the classic cake-filtration equation, aided by various simplifying assumptions which are sufficiently valid for many (but not all) situations. Washing and drying correlations are of a more empirical nature but with strong experimental justification. If steam or thermal diying is being examined, additional correlations are required beyond those summarized below for such applications, it is advisable to consult an eqmpment manufacturer or refer to pubhshed technical papers for guidance. 

Comparative calculations of specific capacities of different filters or their specific filter areas should be made as part of the evaluation. Such calculations may be performed on the basis of experimental data obtained without using basic filtration equations. In designing a new filtration unit after equipment selection, calculations should be made to determine the specific capacity or specific filtration area. Basic filtration equations may be used for this purpose, with preliminary experimental constants evaluated. These constants contain information on the specific cake resistance and the resistance of the filter medium. 

Equation 18 defmes 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+Tq), 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 Tq 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. 

Replacing V by q, and denoting the actual filtration rate (dq/dx) as W, the governing filtration equation may be rewritten for a unit area of filtration as follows ... 

Filtration equation. The basis for the analysis of filters is normally chosen as the unit volume of clear filtrate collected . 

It was shown in Chapter 2 that the simplest models of solid-liquid separation are those based of the Carman-Kozeny equation for filtration in whieh the bed permeability (filtrability), F, may be expressed by... 

Filtration of 300 ml of fermentation broth was carried out in a laboratory-sized filter with a pressure drop of lOpsi. The filtration took 20 min. Based on previous studies, the filter cake obtained from Penicillium chrysogenum was compressible with the exponent 5 in the equation for calculation of filter area equal to 0.5. 

The filtration equation (5.3-21) must be adopted if a centrifuge is used for separation. During centrifugation, the surface of the filter cake accessible for flow decreases since the cake builds up toward the axis of the centrifuge. Therefore, must be substituted by the product AcAim where is the average flow area and Aim is the logarithmic mean area through the cake. This yields ... 

The assumption that the filtration can be treated by calculating the separation around one single collector body and integrating the result for the total bed underlies the scheme represented by Eqs. (3.2.5) and (3.2.6). For many decades research was dedicated to determining p(x) [1], An equation for calculating r (x)... 

The transport of both solute and solvent can be described by an alternative approach that is based on the laws of irreversible thermodynamics. The fundamental concepts and equations for biological systems were described by Kedem and Katchalsky [6] and those for artificial membranes by Ginsburg and Katchal-sky [7], In this approach the transport process is defined in terms of three phenomenological coefficients, namely, the filtration coefficient LP, the reflection coefficient o, and the solute permeability coefficient to. 

Results from constant differential pressure filtration tests have been analyzed according to traditional filtration science techniques with some modifications to account for the cross-flow filter arrangement.11 Resistivity of the filter medium may vary over time due to the infiltration of the ultrafine catalyst particles within the media matrix. Flow resistance through the filter cake can be measured and correlated to changes in the activation procedure and to the chemical and physical properties of the catalyst particles. The clean medium permeability must be determined before the slurries are filtered. The general filtration equation or the Darcy equation for the clean medium is defined as... 

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. 

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. 

Substituting for V from equation 7.43 into equation 7.41, the filtration time for maximum throughput is ... 

Tufenkji N, Elimelech M (2004) Correlation equation for predicting single-collector efficiency in physicochemical filtration in saturated porous media. Environ Sci Technol 38 529-536 Turner BL, Kay MA, Westermann DT (2004) Colloid phosphorus in surface runoff and water extracts from semiarid soils of the western United States. J Environ Qual 33 1464-1472 van Genuchten MT (1980) A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci Soc Am J 44 892-898... 

An efficient and simple kinetic resolution of the racemic Betti base 387 was achieved via its reaction with acetone in the presence of L-(- -)-tartaric acid. When a suspension of racemic 387 in acetone was treated with L-(- -)-tartaric acid, the (A)-enantiomer formed a crystalline L-tartrate salt 389 this was filtered off, and the (iJ)-enantiomer could be isolated as a naphth[l,2-< ]oxazine derivative 388 from the filtrate (Equation 41). Both enantiomers were obtained in excellent yields and ee s. The enantioseparation is presumed to take place via a kinetically controlled N,0-deketalization of the (3)-naphth[l,2-< ]oxazine derivative <2005JOC8617>. An improved method for the enantioseparation of 387 was developed by the reaction of the ring-chain tautomeric l,3-diphenyl-3,4-dihydro-2//-naphth[2,l-< ][l,3]oxazine (41 X, Y = H) and L-(-f)-tartaric acid, yielding the crystalline 389 in 85% yield <2007SL488>. 

While the calibration curves presented in Figure 1 provide an adequate treatment of gel permeation calibration data, they are nonlinear in the extremes of the useful calibration regions. Two methods of data treatment may be used to provide linear representations of gel filtration results. These methods substantially increase the usefulness of the technique. The equations for these methods, developed by Porath (8) and Ackers (9), are as follows ... 

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... 

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