## 7.19.2 Momentum Equations for Porous Media

Porous media are modeled by the addition of a momentum source term to the standard fluid flow equations. The source term is composed of two parts: a viscous loss term (Darcy, the first term on the right-hand side of Equation  7.19-1 ) , and an inertial loss term (the second term on the right-hand side of Equation  7.19-1)

 (7.19-1)

where is the source term for the th ( , , or ) momentum equation, is the magnitude of the velocity and and are prescribed matrices. This momentum sink contributes to the pressure gradient in the porous cell, creating a pressure drop that is proportional to the fluid velocity (or velocity squared) in the cell.

To recover the case of simple homogeneous porous media

 (7.19-2)

where is the permeability and is the inertial resistance factor, simply specify and as diagonal matrices with and , respectively, on the diagonals (and zero for the other elements).

FLUENT also allows the source term to be modeled as a power law of the velocity magnitude:

 (7.19-3)

where and are user-defined empirical coefficients.

 In the power-law model, the pressure drop is isotropic and the units for are SI.

Darcy's Law in Porous Media

In laminar flows through porous media, the pressure drop is typically proportional to velocity and the constant can be considered to be zero. Ignoring convective acceleration and diffusion, the porous media model then reduces to Darcy's Law:

 (7.19-4)

The pressure drop that FLUENT computes in each of the three ( , , ) coordinate directions within the porous region is then

 (7.19-5)

where are the entries in the matrix in Equation  7.19-1, are the velocity components in the , , and directions, and , , and are the thicknesses of the medium in the , , and directions.

Here, the thickness of the medium ( , , or ) is the actual thickness of the porous region in your model. Thus if the thicknesses used in your model differ from the actual thicknesses, you must make the adjustments in your inputs for .

Inertial Losses in Porous Media

At high flow velocities, the constant in Equation  7.19-1 provides a correction for inertial losses in the porous medium. This constant can be viewed as a loss coefficient per unit length along the flow direction, thereby allowing the pressure drop to be specified as a function of dynamic head.

If you are modeling a perforated plate or tube bank, you can sometimes eliminate the permeability term and use the inertial loss term alone, yielding the following simplified form of the porous media equation:

 (7.19-6)

or when written in terms of the pressure drop in the , , directions:

 (7.19-7)

Again, the thickness of the medium ( , , or ) is the thickness you have defined in your model.

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