16.5.10 Description of Heat Transfer

The internal energy balance for phase q is written in terms of the phase enthalpy, Equation  16.5-11, defined by

where $c_{p,q}$ is the specific heat at constant pressure of phase $q$. The thermal boundary conditions used with multiphase flows are the same as those for a single-phase flow. See this chapter in the separate User's Guide for details.

The Heat Exchange Coefficient

The rate of energy transfer between phases is assumed to be a function of the temperature difference

where $h_{pq} (= h_{qp})$ is the heat transfer coefficient between the $p^{\rm th}$ phase and the $q^{\rm th}$ phase. The heat transfer coefficient is related to the $p^{\rm th}$ phase Nusselt number, ${\rm Nu}_p$, by

Here $\kappa_q$ is the thermal conductivity of the $q^{\rm th}$ phase. The Nusselt number is typically determined from one of the many correlations reported in the literature. In the case of fluid-fluid multiphase, ANSYS FLUENT uses the correlation of Ranz and Marshall [ 284, 285]:

where ${\rm Re}_p$ is the relative Reynolds number based on the diameter of the $p^{\rm th}$ phase and the relative velocity $\vert\vec{u_p} - \vec{u_q}\vert$, and Pr is the Prandtl number of the $q^{\rm th}$ phase:

In the case of granular flows (where $p = s$), ANSYS FLUENT uses a Nusselt number correlation by Gunn [ 117], applicable to a porosity range of 0.35-1.0 and a Reynolds number of up to $10^5$:

The Prandtl number is defined as above with $q = f$. For all these situations, $h_{pq}$ should tend to zero whenever one of the phases is not present within the domain. To enforce this, $h_{pq}$ is always multiplied by the volume fraction of the primary phase $q$, as reflected in Equation  16.5-103.