### 5.7.2. Inhomogeneous Interphase Heat Transfer Models

Interphase heat transfer occurs due to thermal non-equilibrium across phase interfaces. The total heat per unit volume transferred to phase due to interaction with other phases is denoted , and is given by:

| (5–122)

where:

| (5–123)

#### 5.7.2.1. Overall Heat Transfer Coefficients

Heat transfer across a phase boundary is usually described in terms of an overall heat transfer coefficient , which is the amount of heat energy crossing a unit area per unit time per unit temperature difference between the phases. Thus, the rate of heat transfer, , per unit time across a phase boundary of interfacial area per unit volume , from phase to phase , is:

| (5–124)

This may be written in a form analogous to momentum transfer:

| (5–125)

where the volumetric heat transfer coefficient, , is modeled using the correlations described below.

#### 5.7.2.2. Particle Model Correlations

For particle model, the volumetric heat transfer coefficient is modeled as:

| (5–126)

Hence, the interfacial area per unit volume and the heat transfer coefficient are required.

More information on interfacial area density calculation is available.

It is often convenient to express the heat transfer coefficient in terms of a dimensionless Nusselt number:

| (5–127)

In the particle model, the thermal conductivity scale is taken to be the thermal conductivity of the continuous phase, and the length scale is taken to be the mean diameter of the dispersed phase:

| (5–128)

For laminar forced convection around a spherical particle, theoretical analysis shows that . For a particle in a moving incompressible Newtonian fluid, the Nusselt number is a function of the particle Reynolds number and the surrounding fluid Prandtl number .

Additional information on models in ANSYS CFX is available in Particle Model Correlations for Overall Heat Transfer Coefficient in the CFX-Solver Modeling Guide. Some additional details for the Interface Flux model are provided below.

- Interface Flux

  The heat flux coefficients for both fluids and the interfacial heat flux value, F 12, from Fluid 1 to Fluid 2 (Fluid 1 is the fluid to appear on the left of the Fluid Pairs list) are specified. F 12 is the rate of heat transfer per unit time per unit interfacial area from phase 1 to phase 2. Hence, the heat transferred to fluid 2 from fluid 1 per unit volume is given by:

  | (5–129)

  may be given as a constant or an expression.

  Typically, will be a function of the fluid 1 and fluid 2 temperature fields, and possibly other variables. In this case, you may accelerate convergence of the coupled solver by also specifying optional fluid 1 and fluid 2 heat flux coefficients.

  | (5–130)

  The solver takes the absolute value of these flux coefficients to ensure that they are positive. This is required for numerical stability. The partial derivatives need not be computed exactly; it is sufficient for the specified coefficients to simply approximate the partial derivatives. Specification of heat flux coefficients only affects the convergence rate to the solution of the coupled heat transfer equations, it does not affect the accuracy of the converged solution.

  For example, the simple model using a heat transfer coefficient multiplied by a bulk temperature difference my be recovered using:

#### 5.7.2.3. Mixture Model Correlations

When using the mixture model, the Nusselt number is defined in terms of a mixture conductivity scale and the mixture length scale:

| (5–131)

For details, see Mixture Model Correlations for Overall Heat Transfer Coefficient in the CFX-Solver Modeling Guide.

#### 5.7.2.4. The Two Resistance Model

There are special situations where the use of an overall heat transfer coefficient is not sufficient to model the interphase heat transfer process. A more general class of models considers separate heat transfer processes either side of the phase interface. This is achieved by using two heat transfer coefficients defined on each side of the phase interface.

Defining the sensible heat flux to phase from the interface as:

| (5–132)

and the sensible heat flux to phase from the interface as:

| (5–133)

where and are the phase and phase heat transfer coefficients respectively. is interfacial temperature, and it is assumed to be the same for both phases.

The fluid-specific Nusselt number is defined as:

| (5–134)

where is the thermal conductivity of fluid , and is the interfacial length scale (the mean particle diameter for the Particle Model, and the mixture length scale for the Mixture Model).

In the absence of interphase mass transfer, you must have overall heat balance . This condition determines the interfacial temperature:

| (5–135)

It also determines the interphase heat fluxes in terns of an overall heat transfer coefficient:

| (5–136)

Hence, in the absence of interphase mass transfer, the two resistance model is somewhat superfluous, as it may be implemented using a user-specified overall heat transfer coefficient.

It is possible to specify a zero resistance condition on one side of the phase interface. This is equivalent to an infinite fluid specific heat transfer coefficient . Its effect is to force the interfacial temperature to be the same as the phase temperature, .

Modeling advice is available in Two Resistance Model for Fluid Specific Heat Transfer Coefficients in the CFX-Solver Modeling Guide.

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