Porous Medium¶

Introduction¶

Porous media are used to simulate fluid flow through packed beds. This feature implicitly models a packed structure and predicts the relationship between flow rate and applied pressure drop as a function of the packing structure. This implicit representation avoids the need to resolve a fine solid structure using high resolution simulations. Typical applications include filtration processes, packed bed flow, and separation processes.

The porous medium feature can either be selected for the entire geometry, which is the default, or have a child geometry where it will be active. Users have three options for adding geometry to the porous media parent: (1) via the Add Geometry form, which opens automatically when a new static body parent is created; (2) by importing geometry through the Assembly Explorer; or (3) by importing geometry from a spreadsheet.

For an already-existing parent component, you can add children via the Add Geometry command found on the Context Specific Toolbar. Children geometry can also be extracted from an assembly file.

Flow through the packed bed is modeled using the generalized non-darcy transport model (He, 2019):

\[\frac{\partial \mathbf{u}}{\partial t}+(\mathbf{u} \cdot \boldsymbol{\nabla})\left(\frac{\mathbf{u}}{\phi}\right)=-\frac{1}{\rho} \boldsymbol{\nabla}(\phi p)+v_{\mathrm{e}} \nabla^2 \mathbf{u}-\frac{\phi v}{K} \mathbf{u}-\frac{\phi F_\phi}{\sqrt{K}}|\mathbf{u}| \mathbf{u}+\phi \mathbf{G}\]

Within this model, \(v_e\) is the effective kinematic viscosity, \(\phi\) is the porosity, and \(G\) is the body force induced by external force fields. The third and fourth terms on the right-hand side represent (i) the linear (Darcy) drag force and (ii) the non-linear (Forchheimer) drag force.

The Darcy drag force is a function on the fluid permeability, \(K\), and the molecular viscosity. The Forchheimer drag force also includes a dependence on the structure function, \(F_\phi\). These three properties characterize the porous media packing structure.

For a given viscosity, density, velocity, and fluid permeability, the pressure gradient during steady state flow can be calculated from:

\[\nabla P = -\frac{\nu \rho v}{K} - \frac{F_\phi \rho v^2}{\sqrt{ K }}\]

Where

\begin{align} F_\phi & = \frac{1.75}{\sqrt{150 \phi^3}} \\ & \\ K & = \frac{\phi^3 d^2}{150 (1 - \phi)^2} \end{align}

Property Grid¶

`General`¶

Option

Three frameworks are available for defining the permeability and structure function of the packed bed.

Packed Spheres Ergun

This is the default option. It will use the equation above as the basis for the pressure drop calculation across the bed using local velocities.

Porosity: [-] | Fraction of volume occupied by fluid.
Average Particle Diameter: m | Average particle diameter in the fluidized bed (typically Sauter mean diameter - \(d_{32}\)).

Custom Constant

This option allows the user to adjust the parameters as described in the Porous Medium Review.

Porosity: Value between 0 and 1.
Permeability: m ² | Term that relates flow rate of a fluid through a porous medium to the applied pressure gradient.
Structure Function: Also called the Interia function, this function is related to the porosity.

Custom UDF

This allows the most flexibility in defining the parameters for the Ergun model. The equations below use the experimental work of Ergun to fit \(F_\phi\) and \(K\).

Material Properties UDF

m ² , m ² /s, and dimensionless | This UDF defines a set of custom (and potentially spatially varying) values for the porosity, permittivity, and structure function within a porous media. Four outputs must be defined within the UDF: floating point variables phi, K, f, and nu_e. These output values specify the local porosity, permeability, structure function, and effective viscosity within the porous media. The porosity and structure function are both dimensionless. The specified porosity should be between [0,1]. The permeability has units of m ² and the effective viscosity has units of m ²/s. This is a local UDF, calculated on a voxel-by-voxel basis using the local fluid properties.

Download Sample File: Material Properties

Custom Pressure Drop UDF

User-defined function for pressure drop.

Pressure Drop UDF

Pa/m | This UDF defines a custom pressure gradient across a packed bed. One output must be defined within the UDF: a floating-point variable named dP_L. This output value characterizes pressure drop along the packed bed. This is a local UDF, calculated on a voxel-by-voxel basis using the local fluid properties.

Download Sample File: Pressure Drop

If a Custom Constant or Custom UDF is selected, the following section will launch:

`Advanced`¶

Enable Effective Viscosity

This option allows users to define a local viscosity based on the configuration of the porous media model formulation.

Off: Default.
On: Allows entry of a user-defined effective viscosity.

Effective Viscosity

m ² /s | User-defined viscosity.

Porous Medium Toolbar¶

Context-Specific Toolbar Forms	Description
Add Geometry	The Add Geometry form adds child geometry by importing from external CAD files, extracting from external CAD assemblies, or defining internally using built-in parametric geometry.
Move	The Move form enables three-dimensional rigid body transform of object through free drag or point-to-point snapping.
Rotate	The Rotate form enables three-dimensional rotation of geometry.
Scale	The Scale form enables volumetric scaling of a geometry about a set anchor point.
Mate	The Mate form allows surface-to-surface mating and alignment.
Help	The Help command launches the M-Star reference documentation in your web browser.