Porous media number

Figure 9. a) A P-T diagram showing the boiling curves of adsorbed water in porous media (numbers refer to the radius of the pore). The dotted curve is the liquid spinodal curve Sp(L). b) The peperitic (fyke in the lava flow of Pardines (the boundaries are outlined by the thick curve). The horizontal dashed lines indicate the po,sitions of the successive growth pulses of the dyke. 

In the case of a porous medium the characteristic difficulties of the intermediate case can be circumvented by a device originally introduced by Maxwell [16] and later exploited in some detail by workers from Oak. Ridge National Laboratory and the University of Maryland [I ]—[21]. The idea was stated very succinctly by Maxwell as follows. "We may suppose the action of the porous material to be similar to that of a number of particles, fixed in space and obstructing the motion of the particles of the moving systems". ... 

The differential material balances contain a large number of physical parameters describing the structure of the porous medium, the physical properties of the gaseous mixture diffusing through it, the kinetics of the chemical reaction and the composition and pressure of the reactant mixture outside the pellet. In such circumstances it Is always valuable to assemble the physical parameters into a smaller number of Independent dimensionless groups, and this Is best done by writing the balance equations themselves in dimensionless form. The relevant equations are (11.20), (11.21), (11.22), (11.23), (11.16) and the expression (11.27) for the effectiveness factor. 

Porous Media Packed beds of granular solids are one type of the general class referred to as porous media, which include geological formations such as petroleum reservoirs and aquifers, manufactured materials such as sintered metals and porous catalysts, burning coal or char particles, and textile fabrics, to name a few. Pressure drop for incompressible flow across a porous medium has the same quahtative behavior as that given by Leva s correlation in the preceding. At low Reynolds numbers, viscous forces dominate and pressure drop is proportional to fluid viscosity and superficial velocity, and at high Reynolds numbers, pressure drop is proportional to fluid density and to the square of superficial velocity. 

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 maimer in which holes or pores are embedded, the extent of their interconnection, and their location, size and shape characterize the porous medium. 

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. 

Coimectivity is a term that describes the arrangement and number of pore coimections. For monosize pores, coimectivity 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 caimot 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 dispersion. 

Compared with the use of arbitrary grid interfaces in combination with reduced-order flow models, the porous medium approach allows one to deal with an even larger multitude of micro channels. Furthermore, for comparatively simple geometries with only a limited number of channels, it represents a simple way to provide qualitative estimates of the flow distribution. However, as a coarse-grained description it does not reach the level of accuracy as reduced-order models. Compared with the macromodel approach as propagated by Commenge et al, the porous medium approach has a broader scope of applicability and can also be applied when recirculation zones appear in the flow distribution chamber. However, the macromodel approach is computationally less expensive and can ideally be used for optimization studies. 

Here again, the usual porous medium Reynolds number is defined by Eq. (13-12) without the numerical factor (2/3) ... 

At high Reynolds numbers (high turbulence levels), the flow is dominated by inertial forces and wall roughness, as in pipe flow. The porous medium can be considered an extremely rough conduit, with s/d 1. Thus, the flow at a sufficiently high Reynolds number should be fully turbulent and the friction factor should be constant. This has been confirmed by observations, with the value of the constant equal to approximately 1.75 ... 

An expression that adequately represents the porous medium friction factor over all values of Reynolds number is... 

Resin flow models are capable of determining the flow of resin through a porous medium (prepreg and bleeder), accounting for both vertical and horizontal flow. Flow models treat a number of variables, including fiber compaction, resin viscosity, resin pressure, number and orientation of plies, ply drop-off effects, and part size and shape. An important flow model output is the resin hydrostatic pressure, which is critical for determining void formation and growth. 

Fig. 37. Experimental course of the resistance coefficient f as a function of the Re number for narrowly ffistnbnted polystyrene (Mw = 23.6 106 g/mol, c = loo ppm) in toluene and the mean molecular weight after passing through the porous medium...

Figure 6. An idealized scheme for a sedimentary porous medium with pore walls covered by a biofilm. High sulfate reduction rates are maintained even in depths to which sulfate cannot diffuse because of recycling of sulfate within the biofilm. Numbered points (in black circles) denote the following processes I, Respiration consumes oxygen. 2, Microbial reduction of reactive metal Oxides. Reduction of reactive ferric oxides is in equilibrium with reoxidation of ferrous iron by Os. Thus, no net loss of reactive iron takes place in these layers. 3, Microbial reduction of ferric oxides. 4, Sulfate reduction rate (denoted as SRR). 5, Sulfide oxidation, either microbiologically or chemically. 6, Sulfide builds up within the hiofilm, sulfate consumption increases, reactive iron pool decreases. 7, Formation of iron sulfides.

Washing. As ice moves upward through the column, it behaves as an unconsolidated porous medium and carries brine with it, held by viscous and capillary forces. The flow is entirely laminar, because the Reynolds number based on particle diameter is always less than 0.05. The brine is carried thus at the surface and in the fillets between the particles. The downward flowing wash water moves between the particles and mixes with the brine mainly by molecular diffusion. Salt will diffuse from the brine to the wash water. 

Soft biological structures exhibit finite strains and nonlinear anisotropic material response. The hydrated tissue can be viewed as a fluid-saturated porous medium or a continuum mixture of incompressible solid (s), mobile incompressible fluid (f), and three (or an arbitrary number) mobile charged species a, (3 = p,m, b). A mixed Electro-Mechano-Chemical-Porous-Media-Transport or EMCPMT theory (previously denoted as the LMPHETS theory) is presented with (a) primary fields (continuous at material interfaces) displacements, Ui and generalized potentials, ifi ( , r/ = /, e, to, b) and (b) secondary fields (discontinuous) pore fluid pressure, pf electrical potential, /7e and species concentration (molarity), ca = dna/dVf or apparent concentration, ca = nca and c = Jnca = dna/dVo. The porosity, n = 1 — J-1(l — no) and no = no(Xi) = dVj/dVo for a fluid-saturated solid. Fixed charge density (FCD) in the solid is defined as cF = dnF/dV , cF = ncF, and cF = cF (Xf = JncF = dnF/d o. 

The flow through porous media of emulsions, foams, and suspensions can be important in a number of applications ranging from fixed-bed catalytic reactors in the chemical process industries, to flows through soil environments, to flow in underground reservoirs. To understand the flow of dispersions in porous media one needs a knowledge not only of the properties of the dispersion, but also of the porous medium. Pore characterization itself has been reviewed elsewhere [30,416]. 

Natural convective flows in porous media occur in a number of important practical situations, e.g., in air-saturated fibrous insulation material surrounding a heated body and about pipes buried in water-saturated soils. To illustrate how such flows can be analyzed, e.g., see [20] to [22], attention will be given in this section to flow over the outer surface of a body in a porous medium, the flow being caused purely by the buoyancy forces resulting from the temperature differences in the flow. The simplest such situation is two-dimensional flow over an isothermal vertical flat surface imbedded in a porous medium, this situation being shown schematically in Fig. 10.25. 

Derive the value for the Nusselt number for fully developed flow through a porous medium-filled pipe with a uniform heat flux at the wall. 

Knudsen coefficient — The term relates to a particular type of mass transfer of gases through the pores of a specific porous medium. The gas transport characteristics depends on the ratio of the mean free path for the gas molecule, A, to the pore diameter, dpore, which is called the Knudsen number, Kn (Kn = -7 —). 

Although the random field Y(r) is correlated, it takes its values in the space of real numbers R, while the porous medium has to be represented by a discretevalued field fg(r). In order to create such a field from Y(r), one applies a nonlinear filter fg(r) = G(Y(r)), i.e., the random variable fg(r) is the deterministic... 

Let us consider the spatially periodic porous medium consisting of an infinite number of identical unit cells. The spatially periodic medium is subjected to a macroscopic deformation described by the tensor of deformation A, and the local displacement d — A x + d can be decomposed into a macroscopic deformation A x and a microscopic spatially periodic displacement d cf. Poulet et al. (1996). This decomposition introduced into elastostatic Equations (29) and (30) yields... 

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