There are essentially four important physical parameters that characterize a filter media and are used as a basis for relating the characteristics of the material to the system flow dynamics. These are porosity, permeability, tortuosity and connectivity.
We may begin by describing any porous medium as a solid matter containing many holes or pores, which collectively constitute an array of tortuous passages. Refer to Figure 1 for an example. The number of holes or pores is sufficiently great that a volume average is needed to estimate pertinent properties. Pores that occupy a definite fraction of the bulk volume
constitute a complex network of voids. The manner in which holes or pores are embedded, the extent of their interconnection, and their location, size and shape characterize the porous medium.
The term porosity refers to the fraction of the medium that contains the voids. When a fluid is passed over the medium, the fraction of the medium (i.e., the pores) that contributes to the flow is referred to as the egective porosity of the media. In a general sense, porous media are classified as either unconsolidated and consolidated and/or as ordered and random. Examples of un- consolidated media are sand, glass beads, catalyst pellets, column packing materials, soil, gravel and packing such as charcoal.
Examples of consolidated media are most of the naturally occurring
Modeling the pore size in terms of a probability distribution function
enables a mathematical description of the pore characteristics. The narrower the pore size distribution, the more likely the absoluteness of retention. The particle-size distribution represented by
the rectangular block is the more securely retained, by sieve capture, the narrower the pore-size distribution.
rocks, such as sandstones and limestones.
64 WATER AND WASTEWATER TREATMENT TECHNOLOGIES
Materials such as concrete, cement, bricks, paper and cloth are manmade consolidated media.
Ordered media are regular packings of various types of materials, such as spheres, column packings and wood. Random media have no particular correlating factor.
Porous media can be further categorized in terms of geometrical or structural properties as they relate to the matrix that affects flow and in terms of the flow properties that describe the matrix from the standpoint of the contained fluid.
Geometrical or structural properties are best represented by average properties, from which these average structural properties are related to flow properties.
Figure 1. SEM of pores on surface of activated carbon particle.
WHAT FILTRATION IS ALL ABOUT 65
PORE STRUCTURE
A microscopic description characterizes the structure of the pores. The objective of a pore-structure analysis is to provide a description that relates to the macroscopic or bulk flow properties. The major bulk properties that need to be correlated with pore description or characterization are the four basic parameters:
porosity, permeability, tortuosity and connectivity. In studying different samples of the same medium, it becomes apparent that the number of pore sizes, shapes, orientations and interconnections are enormous. Due to this complexity, pore-structure description is most often a statistical distribution of apparent pore sizes. This distribution is apparent because to convert measurements to pore sizes one must resort to models that provide average or model pore sizes. A common approach to defining a characteristic pore size distribution is to model the porous medium as a bundle of straight cylindrical or rectangular capillaries (refer to Figure 2). The diameters of the model capillaries are defined on the basis of a convenient distribution function.
Pore structure for unconsolidated media is inferred from a particle size distribution, the geometry of the particles and the packing arrangement of particles. The theory of packing is reasonably well established for symmetrical geometries such as spheres and cylinders. Information on particle size, geometry and packing theory allows us to develop relationships between pore size distributions and particle size distributions. A macroscopic description is based on average or bulk properties at sizes much larger than a single pore. In characterizing a porous medium macroscopically, one must deal with the scale of description. The scale used depends on the manner and size in which we wish to model the porous medium. A simplified approach is to assume the medium to be ideal; meaning homogeneous, uniform and isotropic.
4- f H
Figure 2. Simplified capillary view of flow through pores.
66 WATER AND WASTEWATER W A T M E N T TECHNOLOGIES
PERMEABILITY AND DARCY’S LAW
The term reservoir description is applied to characterizing a homogeneous system as opposed to a heterogeneous one. A reservoir description defines the reservoir at a level where a property changes sufficiently so that more than a single average must be used to model the flow. In this sense, a reservoir composed of a section of coarse gravel and a section of fine sand, where these two materials are separated and have significantly different permeabilities, is heterogeneous in nature.
Defining dimensions, locating areas and establishing average properties of the gravel and sand constitutes a reservoir description, and is a satisfactory approach for reservoir-level type problems.
The governing flow equation describing flow through as porous medium is known as Darcy’s law, which is a relationship between the volumetric flow rate of a fluid flowing linearly through a porous medium and the energy loss of the fluid in motion.
Darcy’s law is expressed as:
where
A h = Az + - AP + constant P
The parameter, K, is a proportionality constant that is known as the hydraulic conductivity.
Darcy’s law is considered valid for creeping flow where the Reynolds number is less than one. The Reynolds number in open conduit flow is the ratio of inertial to viscous forces and is defined in terms of a characteristic length perpendicular to flow for the system. Using four times the hydraulic radius to replace the length perpendicular to flow and correcting the velocity with porosity yields a Reynolds number in the form:
The hydraulic conductivity K depends on the properties of the fluid and on the pore
WHAT FILTRATION IS ALL ABOUT 67
structure of the medium. It is temperature-dependent, since the properties of the fluid (density and viscosity) are temperature-dependent . Hydraulic conductivity can be written more specifically in terms of the intrinsic permeability and the properties of the fluid.
where k is the intrinsic permeability of the porous medium and is a function only of the pore structure. The intrinsic permeability is not temperature-dependent.
In its differential form, Darcy's equation is:
e = q = - _ _ k dP
A P dx (5)
The minus sign results from the definition of Ap, which is equal to p2 - p,, a negative quantity. The term q is known as the seepage velocity and is equivalent to the velocity of approach v,, which is also used in the definition of the Reynolds number.
Permeability is normally determined using linear flow in the incompressible or compressible form, depending on whether a liquid or gas is used as the flowing fluid. The volumetric flowrate Q (or Q,,,) is determined at several pressure drops.
Q (or Q,) is plotted versus the average pressure pm. The slope of this line will yield the fluid conductivity K or, if the fluid density and viscosity are known, it provides the intrinsic permeability k. For gases, the fluid conductivity depends on pressure, so that
K = K( I + $ )
where b depends on the fluid and the porous medium. Under such circumstances a straight line results (as with a liquid), but it does not pass through the origin;
instead it has a slope of bK and intercept K. The explanation for this phenomenon is that gases do not always stick to the walls of the porous medium. This slippage shows up as an apparent dependence of the permeability on pressure.
68 WATER AND WASTEWATER TFNJATMENT TECHNOLOGIES
EFFECTS OF HETEROGENEITY, NONUNIFORMlTY AND ANISOTROPY ON PERMEABILITY
Heterogeneity, nonuniformity and anisotropy are terms which are defined in the volume-average sense. They may be defined at the level of Darcy's law in terms of permeability. Permeability, however, is more sensitive to conductance, mixing and capillary pressure than to porosity.
Heterogeneity, nonuniformity and anisotropy are defined as follows. On a macroscopic basis, they imply averaging over elemental volumes of radius E about a point in the media, where E is sufficiently large that Darcy's law can be applied for appropriate Reynolds numbers. In other words, volumes are large relative to that of a single pore. Further, E is the minimum radius that satisfies such a condition. If E is too large, certain nonidealities may be obscured by burying their effects far within the elemental volume.
Heterogeneity, nonuniformity and anisotropy are based on the probability density distribution of permeability of random macroscopic elemental volumes selected from the medium, where the permeability is expressed by the one-dimensional form of Darcy's law.
Permeability is the conductance of the medium and has direct relevance to Darcy's law. Permeability is related to the pore size distribution, since the distribution of the sizes of entrances, exits and lengths of the pore walls constitutes the primary resistance to flow. This parameter reflects the conductance of a given pore structure.
Permeability and porosity are related to each other; if the porosity is zero the permeability is zero. Although a correlation between these two parameters may exist, permeability cannot be predicted from porosity alone, since additional parameters that contain more information about the pore structure are needed.
These additional parameters are tortuosity and connectivity.
Permeability is a volume-averaged property for a finite but small volume of a medium. Anisotropy in natural or manmade packed media may result from particle (or grain) orientation, bedding of different sizes of particles or layering of media of different permeability. A dilemma arises when considering whether to treat a directional effect as anisotropy or as an oriented heterogeneity.
In an oriented porous medium, the resistance to flow differs depending on the direction. Thus, if there is a pressure gradient between two points and a particular fluid particle is followed, unless the pressure gradient is parallel to oriented flow paths, the fluid particle will not travel from the original point to the point which one would expect. Instead, the particle will drift.
TORTUOSITY
Tortuosity is defined as the relative average length of a flow path (i.e., the average length of the flow paths to the length of the medium). It is a macroscopic measure of both the sinuosity of the flow path and the variation in pore size along the flow
WHAT FILTRATION IS ALL ABOUT 69
path. Both porosity and tortuosity correlate with permeability, but neither can be used alone to predict permeability. Tortuosity and connectivity are difficult to relate to the nonuniformity and anisotropy of a medium. Attempts to predict permeability from a pore structure model require information on tortuosity and connectivity.
CONNECTIVITY
Connectivity is a term that describes the arrangement and number of pore connections. For monosize pores, connectivity is the average number of pores per junction. The term represents a macroscopic measure of the number of pores at a junction. Connectivity correlates with permeability, but cannot be used alone to
predict permeability except in certain limiting cases. Difficulties in conceptual simplifications result from replacing the real porous medium with macroscopic parameters that are averages and that relate to some idealized model of the medium.
Tortuosity and connectivity are different features of the pore structure and are useful to interpret macroscopic flow properties, such as permeability, capillary pressure and dnpersion.
THE KOZENY EQUATION
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:
70 WATER AND WASTEWATER T R E A l " TECHNOLOGIES
and
2 26,
s, = -
S1I2k
Replacing 2/8" with shape parameter c and SA with a specific surface, the Kozeny equation is obtained.
s = @ S A
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):
2 - c6,3
s - -
t k
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:
m
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:
WHAT FILTRATION IS ALL ABOUT 71
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,, A,, 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=Q1 +a+. . . Combining these expressions we obtain:
A((k)), = A,(k,) +A&) + . . . (164
or
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 thlckness of the layers varies. This means that
(( 4)) simplifies to:
If all the layers have the same thickness, then
12 WATER AND WASTEWATER TREATMENT TECHNOLOGIES
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
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 IOJ %
reduction in volume occurs (the
compressibility coefficient i s very small).
However, when the same pressure differential is applied to water vapor, a volume reduction in excess of 2 occurs.