General Approach

In addition, if the Conduction option is switched on, the heat transfer between the porous solid matrix and the fluid flowing through it is also considered. Therefore, the porous matrix act on the fluid flowing through it via the Si, Siui, and (if heat conduction in solids is considered) QH terms in equations Equation 2 and Equation 3 (see “The Navier-Stokes Equations for Laminar and Turbulent Fluid Flows”), whose components related to porosity are defined as:

where k is the resistance vector of the porous medium (see below), γ is the user-defined volumetric porous matrix/fluid heat transfer coefficient which can dependent on the flow velocity, Tp is the temperature of the porous matrix, T is temperature of the fluid flowing through the matrix, and the other designations are given in “The Navier-Stokes Equations for Laminar and Turbulent Fluid Flows”. In addition, the fluid density in equations Equation 1 – Equation 3 is multiplied by the porosity n of the porous medium, which is the volume fraction of the interconnected pores with respect to the total medium volume.

In the employed porous medium model turbulence disappears within a porous medium and the flow becomes laminar.

If the heat conduction in porous matrix is considered, then, in addition to solving equations Equation 1 – Equation 3 describing fluid flow in porous medium, the equation Equation 1 (see “Conjugate Heat Transfer”) describing the heat conduction in solids is also considered within the porous medium. In this equation the source QH due to heat transfer between the porous matrix and the fluid is defined in the same manner as in the equation Equation 1 but with the opposite sign. The values of γ and C for the porous matrix may differ from those of the corresponding bulk solid material and hence must be specified independently. Density of the solid material is multiplied by the solid volume fraction in the porous matrix, i.e. by (1 – n).

Thermal conductivity of the porous matrix can be specified as anisotropic in the same manner as for the solid material.

The conjugate heat transfer problem in a porous medium is solved under the following restrictions:

• heat conduction in a porous medium not filled with a fluid is not considered,

• porous media are considered transparent for radiation heat transfer,

• heat sources in the porous matrix can be specified in the forms of heat generation rate or volumetric heat generation rate only; heat sources in a form of constant or time-dependent temperature can not be specified.

To perform a calculation in Flow Simulation, you have to specify the following porous medium properties: the effective porosity of the porous medium, defined as the volume fraction of the interconnected pores with respect to the total medium volume. Later on, the Permeability type of the porous medium must be chosen among the following:

• isotropic (i.e., the medium permeability is independent of direction),

• unidirectional (i.e., the medium is permeable in one direction only),

• axisymmetrical (i.e., the medium permeability is fully governed by its axial and transversal components with respect to a specified direction),

• orthotropic (i.e., the general case, when the medium permeability varies with direction and is fully governed by its three components determined along three principal directions).

Then you have to specify some constants needed to determine the porous medium resistance to fluid flow, i.e., vector k defined as , where P, ρ, and V are fluid pressure, density, and velocity, respectively. It is calculated according to one of the following formulas:

• , where ΔP is the pressure difference between the opposite sides of a sample parallelepiped porous body, is the mass flow rate through the body, S and L are the body cross-sectional area and length in the selected direction, respectively. You can specify ΔP as a function of , whereas S and L are constants. Instead of mass flow rate you can specify volume flow rate, v. In this case Flow Simulation calculates . All these values do not specify the porous body for the calculation, but its resistance k only.

• , where ΔP is the pressure difference between the opposite sides of a sample parallelepiped porous body, V and ρ are the fluid velocity and density, respectively, L is the body length in the selected direction. You can specify ΔP as a function of V, whereas L is constant, since V and ρ are calculated.

• , where V is the fluid velocity, A and B are constants, ρ is the fluid density. Here, only A and B are specified, since V and ρ are calculated.

• , where μ and ρ are the fluid dynamic viscosity and density, D is the reference pore size determined experimentally, ε is the porous medium's porosity. Here, only D and ε are specified, since μ and ρ are calculated.

• , differing from the previous formula by the f(Re) factor, yielding a more general formula. Here, in addition to D and ε, f(Re) as a formula dependency is specified: , where φ(Re) is the flow resistance coefficient of a narrow channel.

The specified reference pore size is also used to calculate the turbulence dissipation after the porous medium. In case of the two first formula, the specified pore size is used to calculate the turbulence dissipation after the porous medium only. By default, it is set to 0.00001 m.

To define a certain porous body, specify both the body position in the model and, if the porous medium has a unidirectional or axisymmetrical permeability, the reference directions in the porous body.