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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)


 S_i = - \left(\sum_{j=1}^3 D_{ij} \mu v_j + \sum_{j=1}^3 C_{ij} \frac{1}{2} \rho \vert v\vert v_j \right) (7.19-1)

where $S_i$ is the source term for the $i$th ( $x$, $y$, or $z$) momentum equation, $\vert v\vert$ is the magnitude of the velocity and $D$ and $C$ 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


 S_i = - \left( \frac{\mu}{\alpha} v_i + C_2 \frac{1}{2} \rho \vert v\vert v_i \right) (7.19-2)

where $\alpha$ is the permeability and $C_2$ is the inertial resistance factor, simply specify $D$ and $C$ as diagonal matrices with $1/\alpha$ and $C_2$, 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:


 S_i = -C_0 \vert v\vert^{C_1} = -C_0 \vert v\vert^{(C_1-1)}v_i (7.19-3)

where $C_0$ and $C_1$ are user-defined empirical coefficients.

figure   

In the power-law model, the pressure drop is isotropic and the units for $C_0$ 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 $C_2$ can be considered to be zero. Ignoring convective acceleration and diffusion, the porous media model then reduces to Darcy's Law:


 {\bf\nabla} p = - \frac{\mu}{\alpha} {\vec v} (7.19-4)

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


\Delta p_x = \sum_{j=1}^3 \frac{\mu}{\alpha_{xj}} v_j \Delta n_x


 \Delta p_y = \sum_{j=1}^3 \frac{\mu}{\alpha_{yj}} v_j \Delta n_y (7.19-5)


\Delta p_z = \sum_{j=1}^3 \frac{\mu}{\alpha_{zj}} v_j \Delta n_z

where $1/{\alpha_{ij}}$ are the entries in the matrix $D$ in Equation  7.19-1, $v_j$ are the velocity components in the $x$, $y$, and $z$ directions, and $\Delta n_x$, $\Delta n_y$, and $\Delta n_z$ are the thicknesses of the medium in the $x$, $y$, and $z$ directions.

Here, the thickness of the medium ( $\Delta n_x$, $\Delta n_y$, or $\Delta n_z$) 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 $1/{\alpha_{ij}}$.



Inertial Losses in Porous Media


At high flow velocities, the constant $C_2$ 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:


 {\bf\nabla} p = - \sum_{j=1}^3 C_{2_{ij}} \left ( \frac{1}{2} \rho v_j\vert v\vert \right ) (7.19-6)

or when written in terms of the pressure drop in the $x$, $y$, $z$ directions:


$\displaystyle \Delta p_x$ $\textstyle \approx$ $\displaystyle \sum_{j=1}^3 C_{2_{xj}} \Delta n_x \frac{1}{2} \rho v_{j} \vert v\vert$  
$\displaystyle \Delta p_y$ $\textstyle \approx$ $\displaystyle \sum_{j=1}^3 C_{2_{yj}} \Delta n_y \frac{1}{2} \rho v_{j} \vert v\vert$  
$\displaystyle \Delta p_z$ $\textstyle \approx$ $\displaystyle \sum_{j=1}^3 C_{2_{zj}} \Delta n_z \frac{1}{2} \rho v_{j} \vert v\vert$ (7.19-7)

Again, the thickness of the medium ( $\Delta n_x$, $\Delta n_y$, or $\Delta n_z$) is the thickness you have defined in your model.


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