Superficial Velocity Formulation

Simcenter STAR-CCM+ models the pressure losses in the porous region by introducing momentum sinks in the momentum equations. Simcenter STAR-CCM+ includes the user-specified volume porosity in the unsteady terms and in the calculation of effective thermal conductivity.

Continuity Equation: The continuity equation that governs the mass balance in a porous medium then reads:
Figure 1. EQUATION_DISPLAY

$\frac{\partial}{\partial t} (\int_{V}^{} ρ χ) d V + \oint_{A} ρ v_{s} \cdot d a = \int_{V}^{} S_{u} d V$

(1840)

where $v_{s}$ is the superficial velocity, $ρ$ is the fluid density, $a$ is the surface area vector, and $S_{u}$ is a user-defined mass source or sink.

Momentum Equation: The momentum equation contains an additional source term that accounts for the resistance to the flow imparted by the porous medium. The porous medium resistance force is defined in terms of the superficial velocity $v_{s}$ and the porous resistance tensor $P$ :

Figure 2. EQUATION_DISPLAY

f_{p} = - P \cdot v_{s}

(1841)

The porous resistance tensor consists of two components:

Figure 3. EQUATION_DISPLAY

P = P_{v} + P_{i} | v_{s} |

(1842)

where $P_{v}$ is the viscous (linear) and $P_{i}$ is the inertial (quadratic) resistance tensor. This source term represents a momentum sink that creates a pressure drop.

The momentum balance for a porous medium is then given by:

Figure 4. EQUATION_DISPLAY

\begin{array}{l} \frac{\partial}{\partial t} (\int_{V}^{} ρ v_{s}) d V + \oint_{A}^{} ρ v_{s} \otimes v_{s} \cdot d a = \\ - \oint_{A}^{} p I \cdot d a + \oint_{A}^{} T \cdot d a + \int_{V}^{} f_{b} d V + \int_{V}^{} f_{p} d V + \int_{V}^{} S_{u} d V \end{array}

(1843)

where $f_{b}$ is the body force vector comprising all other volumetric body forces, $I$ is the identity matrix, and $f_{p}$ is the porous resistance force given by Eqn. (1841).

Energy Equation: The energy equation for a porous medium is given by:
Figure 5. EQUATION_DISPLAY

$\frac{\partial}{\partial t} \int_{V} (χ {(ρ E)}_{fluid} + (1 - χ) {(ρ C_{p})}_{solid} T) d V + \oint_{A}^{} ρ H v_{s} \cdot d a = - \oint_{A}^{} \dot{q} ″ \cdot d a + \oint_{A}^{} T \cdot v_{s} d a + \int_{V}^{} f_{b} \cdot v_{s} d V + \int_{V}^{} S_{u} d V$

(1844)

where $H$ is total enthalpy and the heat flux $\dot{q} ″$ is defined as:
Figure 6. EQUATION_DISPLAY

$\dot{q} ″ = {- k}_{eff} \nabla T$

(1845)

and
Figure 7. EQUATION_DISPLAY

$k_{eff} = χ k_{fluid} + (1 - χ) k_{solid}$

(1846)

where $k_{e f f}$ is the effective thermal conductivity of the porous medium, $k_{fluid}$ is the thermal conductivity of the fluid, and $k_{solid}$ is the thermal conductivity of the solid.

Influence of Porous Resistance on Temperature for Incompressible Flow

When you simulate incompressible flow through a porous medium and you solve for energy, a change in temperature proportional to the user-specified porous resistance takes place.

Consider the following example: a one-dimensional steady incompressible flow of a fluid with constant $C_{v}$ (specific heat at constant volume) through a porous medium with a constant cross-section.

Mass conservation requires that the velocity $v$ is constant. Momentum conservation reduces to a balance between the pressure gradient $d p / d x$ and the porous resistance force $f_{p}$ :

Figure 8. EQUATION_DISPLAY

f_{p} = \frac{d p}{d x}

(1847)

The energy equation reduces to:

Figure 9. EQUATION_DISPLAY

\frac{d H}{d x} = 0

(1848)

and since total enthalpy is $H = C_{v} T + \frac{p}{ρ} + \frac{v^{2}}{2}$ , the equation becomes:

Figure 10. EQUATION_DISPLAY

\frac{d T}{d x} = \frac{- 1}{ρ c_{v}} \frac{d p}{d x} = \frac{- f_{p}}{ρ c_{v}}

(1849)

Hence, it is normal to expect a temperature increase or decrease for an incompressible flow through a porous medium.