Governing Equations for Porous Media
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Conservation of Mass
The fluid-saturated porous media can be considered as a two-phase system that contains fluid (f) and solid matrix (sm). According to the continuity equation in the multi-fluid model:
the continuity equation for the fluid phase (f) in the porous media is
where is the intrinsic phase-averaged velocity. The right-hand side is zero, because there is no phase change between the porous matrix and the fluid. If the fluid phase in the porous media can be assumed as incompressible, the volume-averaged density is equivalent to the density of a single phase, and eq. (1) can be simplified as
where is the extrinsic phase-averaged velocity.
For an incompressible flow in a porous media of uniform porosity, the continuity equation is
Conservation of Momentum
Since a macroscopic approach is the most feasible way to model transport, there needs to be a way to model the bulk resistance to the flow caused by the porous zone. In 1856, Henry Darcy experimentally measured the resistance to a steady, one-dimensional, gravitationally-driven flow through an unconsolidated, uniform, rigid, and isotropic solid matrix. He came up with a relationship for the pressure gradients/resistance as a function of the dynamic viscosity, μ_{f}, the extrinsic phase-averaged velocity, and the permeability, K, now known as Darcy’s law.
where the permeability, K, is the resistance to the flow, which is analogous to the thermal conductivity in the Fourier’s law of heat conduction. It has units of meters squared.
Darcy’s law can be expanded into vectorial form for an anisotropic solid matrix.
where the permeability, K, becomes a second-order symmetric tensor. When the porous medium is isotropic (having equal resistance in all directions), the permeability reduces to a scalar (single value), K. Darcy’s law is valid for Re<1, or the creeping flow regime, where the viscous forces dominate. The Reynolds number of the porous media is defined by the mean volumetric velocity, and the average characteristic length scale of the voids, .
In this flow regime, the wall effects are confined to within one or two particle diameters of the wall. Also, in the region where the flow enters the porous zone, the entrance region is confined to approximately three particle diameters. Therefore flow becomes very closely uniform in one main direction, and the walls bounding the porous zone have minimal effect.
As the Reynolds number increases from unity to about 10, the drag smoothly transitions from linear to nonlinear. This linear to nonlinear transition of drag in a porous zone represents the flow moving from the creeping flow to the laminar flow regime. To account for this nonlinearity, when Re>10, Ward (1964) came up with a quadratic drag term dependent of ρ / K^{1 / 2}.
where C_{f} is a dimensionless drag constant, initially thought by Ward (1964) to be a universal constant, 0.55. Since the drag smoothly transitions from linear to nonlinear when 0<Re<10, the significance of the quadratic drag term can be formed as a linear function of Re.
It was later found to vary based on the nature of the porous medium. From the differential momentum equation,
the advective term is . From a scaling analysis, the velocity of the fluid is on the order of the volume weighted average velocity over the porosity, for a single phase, . Therefore, the advective term is on the order of . A scaling analysis between the advective term and the quadratic drag term gives a ratio proportional to , where L is the macroscopic characteristic length scale. The ratio of these two terms is usually very small, therefore the advective term often can be assumed to have a negligible effect on the momentum equations.
An extension of the Stokes drag force on a sphere in an infinite domain is to include the effects of the neighboring spheres, as with the case of packed beds made of spheres. The Stokes drag force is the drag exerted by a single sphere in an infinite domain on a flow with a (negligible inertia). This was accomplished by superimposing the Stokes flow on the Darcy flow, i.e.,
where the effective viscosity denoted by is a function of the dynamic viscosity, the porosity, and the tortuosity of the porous media. The tortuosity is a measure of the connectivity of void space in a porous zone. Equation (13) is called the Brinkman equation, and it is usually noted that it is valid for high porosity, . When the porosity is low, the stresses felt on a fluid in one pore are communicated to the fluid in another pore mainly by pressure because the solid matrix prevents direct viscous interaction of the fluid in separate pores. Even though this equation is only valid for higher porosities, the Laplacian operator of is needed when the boundaries of the porous media affect the flow field.
An empirical momentum equation can be heuristically obtained if Darcy’s law, eq. (4), is combined with the inertial drag component, eq. (7), and Brinkman’s equation, eq. (9).
However, for a more complete understanding, this equation is compared to a volume-averaged momentum equation. If the fluid phase is incompressible, i.e., , the volume-averaged momentum equation for the fluid phase,
from multi-fluid model, becomes
where in the volume-averaged momentum equation from multi-fluid model is replaced by
and X_{f} is replaced by g, i.e., gravity is the only body force. Since the flow in a porous media is usually laminar because of the characteristic diameter, the product of the velocity deviations, , is also small and therefore neglected henceforth. The third term on the right hand side of eq. (11) represents interaction between the solid matrix and the fluid phase.
For an incompressible fluid, the volume average of the stress tensor is
Since the local static pressure is needed in the equation of state for an ideal gas, or for the temperature at a liquid vapor interface, the local pressure is more accurately described as the intrinsic phase average pressure, not the extrinsic phase averaged pressure. If the deviation of pressure within a fluid volume element is small, , then the volume-averaged pressure is
Substituting eq. (12) into eq. (11) and considering eq. (13), the momentum equation becomes
Using the continuity equation, eq. (2), and assuming constant porosity, the momentum equation becomes
The empirical momentum relationships are heuristically related to the volume-averaged momentum equation through reasonable observations. In the second term on the right hand side of eq. (15), the value of represents μ'_{f} in eq. (10). The fourth term on the right-hand side of eq. (14) is replaced by the Darcian relationship and the quadratic drag term. The Darcian drag term comes from the deformation stress caused by the solid/fluid interaction, and the quadratic drag term comes from the pressure coefficient associated with the fluid flowing around the solid matrix.
This equation reduces to eq. (10) if the inertia terms are negligible. In eq. (16), the viscous interaction between the fluid and the solid matrix is replaced by Darcy loss and inertial loss terms. The shear losses between the fluid and the solid matrix are accounted for without having any information on the characteristics of the flow in each pore.
The permeability, K, can be obtained by empirical relations, which are well tabulated for various types of porous media. For packed beds of spherical particles of diameter d, the permeability is
Further details on permeability and porosity of various porous materials, including porous wicks in heat pipes, can be found in Kaviany (1995) and Faghri (1995). For convenience, the porous zone properties for a variety of applications are presented in the following table, including K, , ΔA_{sm} / ΔV and r_{eff}.
Solid Matrix | Porosity | Permeability K (m^{2}) | Effective Pore Radius r_{eff}(m) | |
Black Slate Powder | 0.57 – 0.66 | 4.9×10^{-14} – 1.2×10^{-13} | 7×10^{5} - 8.9×10^{5} | |
Brick | 0.12 – 0.34 | 4.8×10^{-15} – 2.2×10^{-13} | ||
Cigarette | 1.1×10^{-9} | |||
Cigarette filters | 0.17 – 0.49 | |||
Coal | 0.02 – 0.12 | |||
Concrete (ordinary mixes) | 0.02 – 0.07 | |||
Copper powder (hot-compacted) | 0.09 – 0.34 | 3.3×10^{-10} – 1.5×10^{-9} | ||
Cork board | 2.4×10^{-11} – 5.1×10^{-11} | |||
Fiberglass | 0.88 – 0.93 | (5.6 – 7.7)×10^{4} | ||
Hair (on mammals) | 0.95 – 0.99 | |||
Hair felt | 8.3×10^{-10} – 1.2×10^{-9} | |||
Leather | 0.56 – 0.59 | 9.5×10^{-14} – 1.2×10^{-13} | (1.2 – 1.6)×10^{6} | |
Limestone (dolomite) | 0.04 – 0.10 | 2×10^{-15} – | ||
4.5×10^{-14} | ||||
Sand | 0.37 – 0.50 | 2.0×10^{-11} – 1.8×10^{-10} | (1.5– 2.2)×10^{4} | |
Screen, SST, 200 mesh | 0.733 | 5.2×10^{-11} | 58×10^{-6} | |
Screen, Nickel, 50 mesh, sintered | 0.625 | 6.63×10^{-10} | 305×10^{-6} | |
Felt, Sintered, SST | 0.916 | 5.46×10^{-10} | 94×10^{-6} | |
Felt, Nickel, A30 | 0.815 | 3.06×10^{-11} | 120×10^{-6} | |
Powder, Sintered, Nickel | 0.597 | 58×10^{-6} | ||
Beads, Monel, 70-80 Mesh | 0.4 | 7.75×10^{-11} | 96.9×10^{-6} | |
Sandstone (“oil sand”) | 0.08 – 0.38 | 5×10^{-16} – 3×10^{-12} | ||
Silica grains | 0.65 | |||
Silica powder | 0.37 – 0.49 | 1.3×10^{-14} – 5.1×10^{-14} | (6.8 – 8.9) ×10^{5} | |
Soil | 0.43 – 0.54 | 2.9×10^{-13} – 1.4×10^{-11} | ||
Wire crimps | 0.68 – 0.76 | 3.8×10^{-9}– 1.0×10^{-8} | (2.9– 4.0)×10^{3} |
Energy Equation
The derivation of the energy equation in a porous medium is similar to that of continuity and momentum equations. The volume-averaged energy equation
where in the volume-averaged energy equation is replaced by in eq. (18), i.e., the product of the deviations were neglected. is heat transfer from solid matrix to the fluid phase.
By applying the continuity equation (2), eq. (18) becomes
The volume-averaged energy equation in the solid matrix is:
where the convection term is dropped since the solid matrix is in the solid state and has no velocity. The thermal boundary conditions at the fluid-solid interface, ΔA_{sm}, are:
These boundary conditions between the fluid and the solid matrix are what make up the heat transfer term between the fluid and solid, . Equations (21) and (22) are very general, and no assumptions about the thermal equilibrium between the fluid and the solid matrix have been made. If the fluid and the solid matrix are considered to be in local thermal equilibrium, , the energy equations for the solid and the fluid can be added to find the total porous media energy equation:
The energy equation can be further simplified by assuming that the enthalpy is only a function of temperature and constant specific heat:
The effective heat capacity, thermal conductivity and heat generation rates per unit volume, denoted by the subscript “eff,” are defined by
where k_{eff,f} and k_{eff,sm} are, respectively, the effective thermal conductivity of the fluid and solid-matrix in the porous media, and both of them depend on porosity and pore structure in the porous media. When the thermal conductivities of the fluid and the solid matrix are close to each other, k_{f}˜k_{sm}, one can assume that and , and use eq. (26) to evaluate the effective thermal conductivity. In a case where thermal conductivities of the fluid and the solid matrix differ significantly, the following equation should be used (Hadley, 1986):
where
The above energy equation, eq. (24), is only valid when the fluid and the solid matrix are assumed to be in local thermal equilibrium. While local thermal equilibrium is an often used hypothesis for heat transfer in porous media, it is well recognized that local equilibrium between different phases in a system of solid and fluid cannot be achieved when thermal properties for different phases differ widely, or during rapid heating or cooling. The fluid and solid-matrix energy equations need to be kept separate, and this situation is referred to as local thermal non-equilibrium. In this situation, heat transfer from the solid matrix to the fluid phase, , in eqs. (19) and (20) can be modeled as a convective boundary condition, with a heat transfer coefficient of h_{c}. When Fourier’s law of heat conduction is applied, the energy equations for the fluid and solid matrix are
Equations (31) and (32) are more accurate representations of the thermal field in a porous zone. However, they require an empirical relationship for the heat transfer coefficient between the solid matrix and the fluid phase.
Species
The volume-averaged species equation can be readily applied to the fluid phase in the porous media:
Applying the continuity equation (2) and the definition of the mass fraction, the averaged-species equation becomes
References
Alazmi, B. and Vafai, K., 2000, “Analysis of Variants within the Porous Media Transport Models,” ASME Journal of Heat Transfer, Vol. 122, pp. 303-326.
Alazmi, B. and Vafai, K., 2001, “Analysis of Fluid Flow and Heat Transfer Interfacial Conditions between a Porous Medium and a Fluid Layer,” International Journal of Heat and Mass Transfer, Vol. 44, pp. 1735-1749.
Faghri, A., 1995, Heat Pipe Science and Technology, Taylor & Francis, Bristol, PA.
Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA
Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.
Hadley, G.R., 1986, “Thermal Conductivity of Packed Metal Powders,” International Journal of Heat and Mass Transfer, Vol. 29, pp. 909-202.
Kaviany, M., 1995, Principles of Heat Transfer in Porous Media, 2^{nd} ed., Springer Verlag, New York.
Nield, D.A., and Bejan, A., 1999, Convection in Porous Media, 2^{nd} ed., Springer-Verlag, New York.
Ward, J. C., 1964, “Turbulent Flow in Porous Media,” ASCE J. Hyd. Div., Vol. 90, HY5, pp. 1-12.