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7.19 Porous Media Conditions

The porous media model can be used for a wide variety of problems,

including flows through packed beds, filter papers, perforated plates, flow

distributors, and tube banks. When you use this model, you define a cell

zone in which the porous media model is applied and the pressure loss in the

flow is determined via your inputs as described in Section

7.19.2

. Heat

transfer through the medium can also be represented, subject to the

assumption of thermal equilibrium between the medium and the fluid flow,

as described in Section

7.19.3

.

A 1D simplification of the porous media model, termed the

can be used to model a thin membrane with known velocity/pressure-drop

characteristics. The porous jump model is applied to a face zone, not to a

cell zone, and should be used (instead of the full porous media model)

whenever possible because it is more robust and yields better convergence.

See Section

7.22

for details.

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7.19.1 Limitations and Assumptions of the Porous Media Model

7.19.2 Momentum Equations for Porous Media

7.19.3 Treatment of the Energy Equation in Porous Media

7.19.4 Treatment of Turbulence in Porous Media

7.19.5 Effect of Porosity on Transient Scalar Equations

7.19.6 User Inputs for Porous Media

7.19.7 Modeling Porous Media Based on Physical Velocity

7.19.8 Solution Strategies for Porous Media

7.19.9 Postprocessing for Porous Media

7.19.1 Limitations and Assumptions of the Porous

Media Model

The porous media model incorporates an empirically determined flow

resistance in a region of your model defined as

porous media model is nothing more than an added momentum sink in the

governing momentum equations. As such, the following modeling

assumptions and limitations should be readily recognized:

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Since the volume blockage that is physically present is not

represented in the model, by default

FLUENT

uses and reports a

superficial velocity inside the porous medium, based on the

volumetric flow rate, to ensure continuity of the velocity vectors

across the porous medium interface. As a more accurate alternative,

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you can instruct

FLUENT

to use the true (physical) velocity inside

the porous medium. See Section

7.19.7

for details.

The effect of the porous medium on the turbulence field is only

approximated. See Section

7.19.4

for details.

When applying the porous media model in a moving reference

frame,

FLUENT

will either apply the relative reference frame or the

absolute reference frame when you enable the

Relative Velocity

Resistance Formulation

. This allows for the correct prediction of the

source terms. For more information about porous media, see

Sections

7.19.6

and

7.19.6

.

When specifying the specific heat capacity,

C

P

, for the selected

material in the porous zone,

C

P

must be entered as a constant value.

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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

)

(7.19-1)

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where

is the source term for the

th (

,

, or

) momentum

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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

, respectively, on

specify

and

as diagonal matrices with

and

the diagonals (and zero for the other elements).

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FLUENT

also allows the source term to be modeled as a power law

of the velocity magnitude:

(7.19-3)

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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

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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)

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The pressure drop that

FLUENT

computes in each of the three

(

,

,

) coordinate directions within the porous region is then

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(7.19-5)

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where

are the entries in the matrix

in

Equation

7.19-1

,

are the velocity components in the

,

,

,

, and

are the thicknesses of

and

directions, and

the medium in the

,

, and

directions.

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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

.

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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)

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or when written in terms of the pressure drop in

the

,

,

directions:

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(7.19-7)

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Again, the thickness of the medium (

,

thickness you have defined in your model.

, or

) is the

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7.19.3 Treatment of the Energy Equation in

Porous Media

FLUENT

solves the standard energy transport equation

(Equation

13.2-1

) in porous media regions with modifications to the

conduction flux and the transient terms only. In the porous medium,

the conduction flux uses an effective conductivity and the transient

term includes the thermal inertia of the solid region on the medium:

(7.19-

8)

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where

= total fluid energy

= total solid medium energy

= porosity of the medium

= effective thermal conductivity of the medium

= fluid enthalpy source term

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Effective Conductivity in the Porous Medium

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The effective thermal conductivity in the porous medium,

computed by

FLUENT

as the volume average of the fluid

conductivity and the solid conductivity:

, is

(7.19-9)

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where

= porosity of the medium

= fluid phase thermal conductivity (including the turbulent

contribution,

)

= solid medium thermal conductivity

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The fluid thermal conductivity

and the solid thermal

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conductivity

can be computed via user-defined functions.

The anisotropic effective thermal conductivity can also be specified

via user-defined functions. In this case, the isotropic contributions

from the fluid,

, are added to the diagonal elements of the solid

anisotropic thermal conductivity matrix.

7.19.4 Treatment of Turbulence in Porous Media

FLUENT

will, by default, solve the standard conservation equations for

turbulence quantities in the porous medium. In this default approach,

turbulence in the medium is treated as though the solid medium has no

effect on the turbulence generation or dissipation rates. This assumption

may be reasonable if the medium's permeability is quite large and the

geometric scale of the medium does not interact with the scale of the

turbulent eddies. In other instances, however, you may want to suppress the

effect of turbulence in the medium.

If you are using one of the turbulence models (with the exception of the

Large Eddy Simulation (LES) model), you can suppress the effect of

turbulence in a porous region by setting the turbulent contribution to

viscosity,

, equal to zero. When you choose this option,

FLUENT

will

transport the inlet turbulence quantities through the medium, but their effect

on the fluid mixing and momentum will be ignored. In addition, the

generation of turbulence will be set to zero in the medium. This modeling

strategy is enabled by turning on the

Laminar Zone

option in

the

Fluid

panel

. Enabling this option implies that

is zero and that

generation of turbulence will be zero in this porous zone. Disabling the

option (the default) implies that turbulence will be computed in the porous

region just as in the bulk fluid flow. Refer to Section

7.17.1

for details

about using the

Laminar Zone

option.

7.19.5 Effect of Porosity on Transient Scalar

Equations

For transient porous media calculations, the effect of porosity on the

time-derivative terms is accounted for in all scalar transport equations and

the continuity equation. When the effect of porosity is taken into account,

the time-derivative term becomes

(

,

, etc.) and

is the porosity.

The effect of porosity is enabled automatically for transient calculations, and

the porosity is set to 1 by default.

, where

is the scalar quantity

7.19.6 User Inputs for Porous Media

When you are modeling a porous region, the only additional inputs for the

problem setup are as follows. Optional inputs are indicated as such.

1.

Define the porous zone.

2.

Define the porous velocity formulation. (optional)

3.

Identify the fluid material flowing through the porous medium.

4.

Enable reactions for the porous zone, if appropriate, and select the

reaction mechanism.

5.

Enable the

Relative Velocity Resistance Formulation

. By default, this

option is already enabled and takes the moving porous media into

consideration (as described in Section

7.19.6

).

6.

Set the viscous resistance coefficients (

or

in Equation

7.19-1

,

in

in Equation

7.19-2

) and the inertial resistance coefficients (

Equation

7.19-1

, or

in Equation

7.19-2

), and define the direction

vectors for which they apply. Alternatively, specify the coefficients for the

power-law model.

7.

Specify the porosity of the porous medium.

8.

Select the material contained in the porous medium (required only for

models that include heat transfer). Note that the specific heat capacity,

,

for the selected material in the porous zone can only be entered as a constant

value.

9.

Set the volumetric heat generation rate in the solid portion of the porous

medium (or any other sources, such as mass or momentum). (optional)

10.

Set any fixed values for solution variables in the fluid region

(optional).

11.

Suppress the turbulent viscosity in the porous region, if appropriate.

12.

Specify the rotation axis and/or zone motion, if relevant.

Methods for determining the resistance coefficients and/or permeability are

presented below. If you choose to use the power-law approximation of the

porous-media momentum source term, you will enter the

coefficients

and

in Equation

7.19-3

instead of the resistance

coefficients and flow direction.

You will set all parameters for the porous medium in

the

Fluid

panel

(Figure

7.19.1

), which is opened from the

Boundary

Conditions

panel

(as described in Section

7.1.4

).

Figure 7.19.1:

The

Fluid

Panel for a Porous Zone

Defining the Porous Zone

As mentioned in Section

7.1

, a porous zone is modeled as a special type of

fluid zone. To indicate that the fluid zone is a porous region, enable

the

Porous Zone

option in the

Fluid

panel. The panel will expand to show

the porous media inputs (as shown in Figure

7.19.1

).

Defining the Porous Velocity Formulation

The

Solver

panel contains a

Porous Formulation

region where you can

instruct

FLUENT

to use either a superficial or physical velocity in the

porous medium simulation. By default, the velocity is set to

Superficial

Velocity

. For details about using the

Physical Velocity

formulation, see

Section

7.19.7

.

Defining the Fluid Passing Through the Porous Medium

To define the fluid that passes through the porous medium, select the

appropriate fluid in the

Material Name

drop-down list in the

Fluid

panel

. If

you want to check or modify the properties of the selected material, you can

click

Edit...

to open the

Material

panel; this panel contains just the

properties of the selected material, not the full contents of the

standard

Materials

panel.

If you are modeling species transport or multiphase flow,

the

Material Name

list will not appear in the

Fluid

panel. For

species calculations, the mixture material for all fluid/porous

zones will be the material you specified in the

Species

Model

panel

.

For

multiphase

flows,

the

materials

are

specified

when

you define the phases, as described in Section

23.10.3

.

Enabling Reactions in a Porous Zone

If you are modeling species transport with reactions, you can enable

reactions in a porous zone by turning on the

Reaction

option in

the

Fluid

panel and selecting a mechanism in the

Reaction

Mechanism

drop-down list.

If your mechanism contains wall surface reactions, you will also need to

specify a value for the

Surface-to-Volume Ratio

. This value is the surface

area of the pore walls per unit volume (

), and can be thought of as a

measure of catalyst loading. With this value,

FLUENT

can calculate the

total surface area on which the reaction takes place in each cell by

multiplying

by the volume of the cell. See Section

14.1.4

for details

about defining reaction mechanisms. See Section

14.2

for details about wall

surface reactions.

Including the Relative Velocity Resistance Formulation

Prior to

FLUENT

6.3, cases with moving reference frames used the

absolute velocities in the source calculations for inertial and viscous

resistance. This approach has been enhanced so that relative velocities are

used for the porous source calculations (Section

7.19.2

). Using the

Relative

Velocity Resistance Formulation

option (turned on by default) allows you

to better predict the source terms for cases involving moving meshes or

moving reference frames (MRF). This option works well in cases with

non- moving and moving porous media. Note that

FLUENT

will use the

appropriate velocities (relative or absolute), depending on your case setup.

Defining the Viscous and Inertial Resistance Coefficients

The viscous and inertial resistance coefficients are both defined in the same

manner. The basic approach for defining the coefficients using a Cartesian

coordinate system is to define one direction vector in 2D or two direction

vectors in 3D, and then specify the viscous and/or inertial resistance

coefficients in each direction. In 2D, the second direction, which is not

explicitly defined, is normal to the plane defined by the specified direction

vector and the

direction vector. In 3D, the third direction is normal to the

plane defined by the two specified direction vectors. For a 3D problem, the

second direction must be normal to the first. If you fail to specify two

normal directions, the solver will ensure that they are normal by ignoring

any component of the second direction that is in the first direction. You

should therefore be certain that the first direction is correctly specified.

You can also define the viscous and/or inertial resistance coefficients in

each direction using a user-defined function (UDF). The user-defined

options become available in the corresponding drop-down list when the

UDF has been created and loaded into

FLUENT

. Note that the coefficients

defined in the UDF must utilize the

DEFINE_PROFILE

macro. For more

information on creating and using user-defined function, see the separate

UDF Manual.

If you are modeling axisymmetric swirling flows, you can specify an

additional direction component for the viscous and/or inertial resistance

coefficients. This direction component is always tangential to the other two

specified directions. This option is available for both density-based and

pressure-based solvers.

In 3D, it is also possible to define the coefficients using a conical (or

cylindrical) coordinate system, as described below.

Note that the viscous and inertial resistance coefficients are

generally based on the superficial velocity of the fluid in the

porous media.

The procedure for defining resistance coefficients is as follows:

1.

Define the direction vectors.

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To use a Cartesian coordinate system, simply specify the

Direction-1

Vector

and, for 3D, the

Direction-2 Vector

. The unspecified

direction will be determined as described above. These direction

vectors correspond to the principle axes of the porous media.

For some problems in which the principal axes of the porous medium

are not aligned with the coordinate axes of the domain, you may not

know a priori the direction vectors of the porous medium. In such

cases, the plane tool in 3D (or the line tool in 2D) can help you to

determine these direction vectors.

(a)

porous region. (Follow the instructions in

Section

27.6.1

or

27.5.1

for initializing the tool to a position on an

existing surface.)

(b)

Rotate the axes of the tool appropriately until they are aligned

with the porous medium.

(c)

Once the axes are aligned, click on the

Update From Plane

Tool

or

Update From Line Tool

button in

the

Fluid

panel.

FLUENT

will automatically set the

Direction-1

Vector

to the direction of the red arrow of the tool, and (in 3D)

the

Direction-2 Vector

to the direction of the green arrow.

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To use a conical coordinate system (e.g., for an annular, conical filter

element), follow the steps below. This option is available only in 3D

cases.

(a)

Turn on the

Conical

option.

(b)

Specify the

Cone Axis Vector

and

Point on Cone Axis

. The

cone axis is specified as being in the direction of the

Cone Axis

Vector

(unit vector), and passing through the

Point on Cone Axis

.

The cone axis may or may not pass through the origin of the

coordinate system.

(c)

Set the

Cone Half Angle

(the angle between the cone's axis and

its surface, shown in Figure

7.19.2

). To use a cylindrical coordinate

system, set the

Cone Half Angle

to 0.

Figure 7.19.2:

Cone Half Angle

For some problems in which the axis of the conical filter element is

not aligned with the coordinate axes of the domain, you may not

know a priori the direction vector of the cone axis and coordinates of

a point on the cone axis. In such cases, the plane tool can help you to

determine the cone axis vector and point coordinates. One method is

as follows:

(a)

Select a boundary zone of the conical filter element that is

normal to the cone axis vector in the drop-down list next to the

Snap

to Zone

button.

(b)

Click on the

Snap to Zone

button.

FLUENT

will automatically

Cone Axis

Vector

and the

Point on Cone Axis

. (Note that you will still have to

set the

Cone Half Angle

yourself.)

An alternate method is as follows:

(a)

(Follow the instructions in Section

27.6.1

for initializing the tool to a

position on an existing surface.)

(b)

Rotate and translate the axes of the tool appropriately until the

red arrow of the tool is pointing in the direction of the cone axis

vector and the origin of the tool is on the cone axis.

(c)

Once the axes and origin of the tool are aligned, click on

the

Update From Plane Tool

button in

the

Fluid

panel.

FLUENT

will automatically set the

Cone Axis

Vector

and the

Point on Cone Axis

. (Note that you will still have to

set the

Cone Half Angle

yourself.)

2.

Under

Viscous Resistance

, specify the viscous resistance

coefficient

in each direction.

Under

Inertial Resistance

, specify the inertial resistance coefficient

in

each direction. (You will need to scroll down with the scroll bar to view

these inputs.)

For porous media cases containing highly anisotropic inertial resistances,

enable

Alternative Formulation

under

Inertial Resistance

.

The

Alternative Formulation

option provides better stability to the

calculation when your porous medium is anisotropic. The pressure loss

through the medium depends on the magnitude of the velocity vector of

the

i

th component in the medium. Using the formulation of

Equation

7.19-6

yields the expression below:

(7.19-10)

Whether or not you use the

Alternative Formulation

option depends on

how well you can fit your experimentally determined pressure drop data to

the

FLUENT

model. For example, if the flow through the medium is

aligned with the grid in your

FLUENT

model, then it will not make a

difference whether or not you use the formulation.

For more infomation about simulations involving highly anisotropic porous

media, see Section

7.19.8

.