### 1.12.1. Full Porous Model

The full porous model is at once both a generalization of the Navier-Stokes equations and of Darcy's law commonly used for flows in porous regions. It can be used to model flows where the geometry is too complex to resolve with a grid. The model retains both advection and diffusion terms and can therefore be used for flows in rod or tube bundles where such effects are important.

In deriving the continuum equations, it is assumed that ‘infinitesimal’ control volumes and surfaces are large relative to the interstitial spacing of the porous medium, but small relative to the scales that you want to resolve. Thus, given control cells and control surfaces are assumed to contain both solid and fluid regions.

The volume porosity at a point is the ratio of the volume available to flow in an infinitesimal control cell surrounding the point, and the physical volume of the cell. Hence:

| (1–277)

It is assumed that the vector area available to flow, , through an infinitesimal planar control surface of vector area is given by:

| (1–278)

where: is a symmetric second rank tensor, called the area porosity tensor. Recall that the dot product of a symmetric rank two tensor with a vector is the vector . ANSYS CFX presently allows only to be isotropic; that is, .

The general scalar advection-diffusion equation in a porous medium becomes:

| (1–279)

In addition to the usual production and dissipation terms, the source term will contain transfer terms from the fluid to the solid parts of the porous medium.

In particular, the equations for conservation of mass and momentum are:

| (1–280)

and:

| (1–281)

where is the true velocity, is the effective viscosity - either the laminar viscosity or a turbulent quantity, and is a momentum source which, in addition to other terms such as buoyancy, includes a contribution (where and represents a resistance to flow in the porous medium). This is in general a symmetric positive definite second rank tensor, in order to account for possible anisotropies in the resistance.

In the limit of large resistance, a large adverse pressure gradient must be set up to balance the resistance. In that limit, the two terms on the right-hand side of Equation 1–281 are both large and of opposite sign, and the convective and diffusive terms on the left-hand side are negligible. Hence, Equation 1–281 reduces to:

| (1–282)

Hence, in the limit of large resistance, you obtain an anisotropic version of Darcy's law, with permeability proportional to the inverse of the resistance tensor. However, unlike Darcy's law, you are working with the actual fluid velocity components , which are discontinuous at discontinuity in porosity, rather than the continuous averaged superficial velocity,

#### 1.12.1.1. Heat Transfer Through the Fluid Only

Heat transfer can be modeled with an equation of similar form:

| (1–283)

where is an effective thermal conductivity and is a heat source to the porous medium.

#### 1.12.1.2. Additional Variable Transfer Through the Fluid Only

For the fluid phases:

| (1–284)

where is a volumetric source term to the porous medium, with units of conserved quantity per unit volume per unit time.

#### 1.12.1.3. Heat Transfer Through the Fluid and Solid

The current porous solid heat transfer formulation allows a finite temperature difference between the fluid phases and the solid phase. It is a non-thermal equilibrium model, therefore there are separate energy equations for each phase within the domain (N fluid phases plus one solid phase). In addition, it makes no assumption on the solid material properties.

For the fluid phases:

| (1–285)

For the solid phase:

| (1–286)

where the solid fraction , and the interfacial heat transfer to the fluid from the solid, , is determined using an overall heat transfer coefficient model using:

| (1–287)

is the overall heat transfer coefficient between the fluid and the solid.

is the interfacial area density between the fluids and the solid. For multiphase flows, this concept can be split into a fluid-independent interfacial area density and a contact area fraction between the fluid and the solid:

| (1–288)

where represents the contact area fraction of fluid with the solid. For single-phase flows, .

#### 1.12.1.4. Additional Variable Transfer Through the Fluid and Solid

For the fluid phases:

| (1–289)

For the solid phase:

| (1–290)

where , and the interfacial transfer between the fluid and the solid, , is determined using an overall transfer coefficient model using:

| (1–291)

is the overall additional variable transfer coefficient between the fluid and the solid.

is described in Heat Transfer Through the Fluid and Solid.

#### 1.12.1.5. Time-varying Porosity

For calculations without any heat or Additional Variable transfer through the solid, it is possible to use a porosity that varies with time. For calculations that do include heat or Additional Variable transfer through the solid, this is inadvisable because the term proportional to the rate of change of porosity is omitted from the equations within the solid.

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