Filtered fluid equation

Equation (6.3.3b) results from assuming a Poiseille flow in a filter s pore of typical radius r, i is the dynamic viscosity of the fluid. Equation (6.3.3c) is a common expression for the electro-osmotic coefficient [13], with d and , respectively, the dielectric constant of the fluid and the -potential of the pore wall. For the time being, we shall assume il> constant (independent of C(x,t)). 

BecaiKe depectinized and filtered apple and grape Juices are Newtonian fluids, equations were derived by Bayindirli (1992,1993) that can be used to estimate their viscosities as a function of concentration and temperature. The equation for viscosity of both apple and grape Juices is of the form ... 

For filter design and performance prediction it is necessary to predict the rate of filtration (velocity or volumetric flowrate) as a function of pressure drop, and the properties of the fluid and particulate bed. This can be achieved using the modified Darcy equation developed in Chapter 3. 

Andrews, A. T. IV., and Sundaresan, S. Closures for filtered two-fluid model equations of gas-particle flows, Manuscript in preparation (2006). 

Ruth, B. F. Ind. Eng. Chem. 27 (1935) 708 and 806. Studies in filtration. III. Derivation of general filtration equations. IV. Nature of fluid flow through filter septa and its importance in the filtration equation. 

The governing equation is therefore identical with that for the irrotational flow of an ideal fluid through a circular aperture in a plane wall. The stream lines and equipotential surfaces in this rotationally symmetric flow turn out to be given by oblate spheroidal coordinates. Since, from Eq. (157), the rate of deposition of filter cake depends upon the pressure gradient at the surface, the governing equation and boundary conditions are of precisely the same form as in the quasi-steady-state approximation... 

The modelling example of the previous section shows that to simplify the general mathematical model of the studied process, the real flow in the filter unit has been considered in terms of its own simplified model. Indeed, it is difficult to understand why we have used a flow model, when in fact, for the flow characterization, we already have the Navier-Stokes equations and their expression for the computational fluid dynamics. To answer this question some precisions about the general aspects of the computational fluids dynamics have to be given. 

Now, let us determine the expression for q. This expression can be readily derived from a material balance using the Reynolds transport theorem. This theorem is derived in any good book on fluid mechanics and will not be derived here. The derivation is, however, discussed in the chapter titled Background Chemistry and Fluid Mechanics. It is important that the reader acquire a good grasp of this theorem as it is very fundamental in understanding the differential form of the material balance equation. This theorem states that the total derivative of a dependent variable is equal to the partial derivative of the variable plus its convective derivative. In terms of the deposition of the material q onto the filter bed, the total derivative is... 

In this equation, p and Vs influence directly the flow characteristics of the filtering water. Since S determines the value of Vs, it also influences directly the flow characteristics of the fluid. All the other factors in the equation are inherent characteristics of the cake. AU these cake characteristics may be lumped up into a single term. Call this term as the specific cake resistance a. Hence, in terms of the specific cake resistance, dp is... 

Microporous membranes will fill their pores with wetting fluids by imbibing that fluid in accordance with the laws of capillary rise. The retained fluid can be forced from the filter pores by air pressure applied from the upstream side. The pressure is increased gradually in increments. At a certain pressure level, liquid will be forced first from the set of largest pores, in keeping with the inverse relationship of the applied air pressure P and the diameter of the pore, d, described in the bubble point equation ... 

When dealing with turbulent flows all the relevant dimensionless numbers are evaluated with the available quantities. For example, in DNS, the fluid and particle instantaneous velocities will be employed, whereas in large-eddy simulation (LES) or in Reynolds-average Navier-Stokes-equations (RANS) simulations the filtered or Reynolds-average values will be used. 

The imbalance in the chemical potential of the water in the shale and drilling fluid results in a tendency for water to enter the shale. Equation 110 is applicable to both water- and oil-based drilling fluids. When fij = fjL sh equation 110 gives the well-known expression for the swelling pressure (Psh — P) between the shale and the drilling fluid. The permeability of shales is very low and the rate of filtration into the shale will be below the critical filtration rate (140) and no filter cake will form on the surface of the shale. 

Now suppose the entire vessel in Fig. 12.1 is filled with some porous solid, such as sand (this vessel now resembles the sand filters frequently used to clarify muddy water). Equation 12.1 still describes the situation exactly as it did before, because it is based on the steady-flow energy balance for a constant-density fluid and it has no built-in assumption that the flow is occurring in an open vessel rather than a porous medium. The significant differences that we will see, if we compare the two situations, are as follows ... 

This filter model considers that particles are transported from the flowing fluid to the filter media by Brownian diflFusion, fluid flow (interception), and settling. The eflFects of each of these mechanisms are assumed to be additive. Happel s (26) equations for flow in a packed bed as used by PfeflFer (27) are assumed in calculating the diflFusion and interception transport mechanisms. 

Because lymph is formed from fluid filtered from the blood, an understanding of transcapillary exchange must be gained first. Usually pressure parameters favor filtration of fluid across the capillary wall to the interstitium (/c) according to the Starling-Landis equation ... 

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