Porous media is typically characterized as an ensemble of channels of various cross sections of the same length. The Navier-Stokes equations for all channels passing a cross section normal to the flow can be solved to give:
Where parameter c is known as the Kozeny constant, which is interpreted as a shape factor that is assigned different values depending on the configuration of the capillary (as a point of reference, c = 0.5 for a circular capillary). S is the specific surface area of the channels. For other than circular capillaries, a shape factor is included:
The specific surface for cylindrical pores is:
Replacing 2/8'/' with shape parameter c and SA with a specific surface, the Kozeny equation is obtained.
Tortuosity t is basically a correction factor applied to the Kozeny equation to account for the fact that in a real medium the pores are not straight (i.e., the length of the most probable flow path is longer than the overall length of the porous medium):
To determine the average porosity of a homogeneous but nonuniform medium, the correct mean of the distribution of porosity must be evaluated. The porosities of natural and artificial media usually are normally distributed. The average porosity of a heterogeneous nonuniform medium is the volume-weighted average of the number average:
The average nonuniform permeability is spatially dependent. For a homogeneous but nonuniform medium, the average permeability is the correct mean (first moment) of the permeability distribution function. Permeability for a nonuniform medium is usually skewed. Most data for nonuniform permeability show permeability to be distributed log-normally. The correct average for a homogeneous, nonuniform permeability, assuming it is distributed log-normally, is the geometric mean, defined as:
For flow in heterogeneous media, the average permeability depends on the arrangement and geometry of the nonuniform elements, each of which has a different average permeability. To explain this, consider the flow into the face of a rectangular element with overall dimensions of height H, width W and length L. Within that rectangular system, consider a series of smaller, parallel rectangular conduits, such that the cross-sectional area of each flow element is A,, A2, A3, etc.
Since flow is through parallel elements of different constant area, Darcy's law for each element, assuming the overall length of each element is equal, is:
The flowrate through the entire system of elements is Q=Qi+Q2+---Combining these expressions we obtain:
This means that the average permeability for this heterogeneous medium is the area-weighted average of the average permeability of each of the elements. If the permeability of each element is log-normally distributed, these are the geometric means.
Reservoirs and soils are usually composed of heterogeneities that are nonuniform layers, so that only the thickness of the layers varies. This means that ((kp)) simplifies to:
If all the layers have the same thickness, then h
where n is the number of layers.
From an industrial viewpoint, the objective of the unit operation of filtration is the separation of suspended solid particles from a process fluid stream which is accomplished by passing the suspension through a porous medium that is referred to as a filter medium. In forcing the fluid through the voids of the filter medium, fluid alone flows, but the solid particles are retained on the surface and in the medium's pores. The fluid discharging from the medium is called the filtrate. The operation may be performed with either incompressible fluids (liquids) or slightly to highly compressible fluids (gases). The physical mechanisms controlling filtration, although similar, vary with the degree of fluid compressibility. Although there are marked similarities in the particle capture mechanisms between the two fluid types, design methodologies for filtration equipment vary markedly. This reference volume concentrates only on process liquid handling (i.e., incompressible fluid flow and processing).
WHAT IS THE DIFFERENCE BETWEEN INCOMPRESSIBLE AND COMPRESSIBLE FLUIDS?
The dependency of liquid volume on pressure may be expressed in terms of the coefficient of compressibility. The coefficient is constant over a wide range of pressures for a particular material, but is different for each substance and for the solid and liquid states of the same material. For liquids, volume decreases linearly with pressure. For gases volume is observed to be inversely proportional to pressure/. If water in its liquid state is subjected to a pressure change from 1 to 2 atm, then less than a 10'3 % reduction in volume occurs (the compressibility coefficient is very small). However, when the same pressure differential is applied to water vapor, a volume reduction in excess of 2 occurs.
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