### 5.2.1. The Inhomogeneous Model

#### 5.2.1.1. Interfacial Area Density

Interfacial transfer of momentum, heat and mass is directly dependent on the contact surface area between the two phases. This is characterized by the interfacial area per unit volume between phase and phase , known as the interfacial area density, . Note that it has dimensions of inverse length.

Interfacial transfer can be modeled using either the particle or mixture models. These essentially provide different algebraic prescriptions for the interfacial area density.

##### 5.2.1.1.1. The Particle Model

The Particle model for interfacial transfer between two phases assumes that one of the phases is continuous (phase ) and the other is dispersed (phase ). The surface area per unit volume is then calculated by assuming that phase is present as spherical particles of Mean Diameter . Using this model, the interphase contact area is:

| (5–5)

This simple model is modified for robustness purposes in two ways:

- is clipped to a minimum volume fraction to ensure the area density does not go exactly to zero.

- For large (that is, when the assumption of being dispersed is invalid), the area density is decreased to reflect the fact that it should lead to zero as tends to 1.

With these modifications, the area density for the particle model is implemented as

| (5–6)

where

| (5–7)

By default, and take values of 0.8 and 10-7, respectively. In some cases, it may be appropriate to use a different value for ; for example, increasing it to 10-3 provides a very crude nucleation model for boiling a subcooled liquid. is controlled by the parameter `Minimum Volume Fraction for Area Density`.

For non-drag forces, the solver uses a slightly different formulation of area density called the Unclipped Interfacial Area Density. In this formulation, the area density is permitted to go to zero, that is, in Equation 5–7. In addition, the area density is reduced more aggressively as the dispersed phase volume fraction becomes large:

| (5–8)

where:

The value of the exponent can be controlled with CCL. In the singleton particle model, you can also specify a value for the Area Density Reduction Exponent. Setting the exponent to a negative value causes the interfacial area density to be computed using the expressions for the clipped area density described in Equation 5–6 and Equation 5–7, rather than the expression for the unclipped area density.

Non-dimensional interphase transfer coefficients may be correlated in terms of the particle Reynolds number and the fluid Prandtl number. These are defined using the particle mean diameter, and the continuous phase properties, as follows:

| (5–9)

| (5–10)

where , and are the viscosity, specific heat capacity and thermal conductivity of the continuous phase .

##### 5.2.1.1.2. The Mixture Model

This is a very simple model that treats both phases , symmetrically. The surface area per unit volume is calculated from

| (5–11)

where is an interfacial length scale, which you must specify.

By way of example, suppose you have oil-water flow in which you may have either water droplets in continuous oil, or oil droplets in continuous water, in the limits , respectively. Then, a simple model for interfacial area density that has the correct behavior in these two limits is given by:

| (5–12)

and are clipped to be no smaller than permitted by the parameter `Minimum Volume Fraction for Area Density`, which takes a default value of .

Non-dimensional interphase transfer coefficients may be correlated in terms of the mixture Reynolds number and Prandtl number defined as follows:

| (5–13)

| (5–14)

where , , and are the density, viscosity, specific heat capacity and thermal conductivity of the mixture respectively, defined by:

| (5–15)

##### 5.2.1.1.3. The Free Surface Model

The free surface model attempts to resolve the interface between the fluids. If there are just two phases in the simulation, the following equation is used for interfacial area density:

| (5–16)

This area density is clipped to be no smaller than , where is controlled by the parameter `Maximum Length Scale for Area Density`, which takes a default value of 1 m.

When more than two phases are present, this is generalized as follows:

| (5–17)

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