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23.5.9 Description of Heat Transfer

The internal energy balance for phase q is written in terms of the phase enthalpy, Equation  23.5-11, defined by

 H_q = \int c_{p,q} d T_q (23.5-82)

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 Chapter  7 for details.

The Heat Exchange Coefficient

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

 Q_{pq} = h_{pq} (T_p - T_q) (23.5-83)

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

 h_{pq} = \frac{6 \kappa_q \alpha_p \alpha_q {{\rm Nu}}_p}{{d_p}^2} (23.5-84)

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, FLUENT uses the correlation of Ranz and Marshall [ 296, 297]:

 {\rm Nu}_p = 2.0 + 0.6 {\rm Re}_p^{1/2} {\rm Pr}^{1/3} (23.5-85)

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:

 {\rm Pr} = \frac{{c_p}_q \mu_q}{\kappa_q} (23.5-86)

In the case of granular flows (where $p = s$), FLUENT uses a Nusselt number correlation by Gunn [ 130], applicable to a porosity range of 0.35-1.0 and a Reynolds number of up to $10^5$:

 {\rm Nu}_s = (7 - 10 \alpha_f + 5 \alpha_f^2)(1 + 0.7 {\rm R... ... 2.4 \alpha_f + 1.2 \alpha_f^2){\rm Re}_s^{0.7} {\rm Pr}^{1/3} (23.5-87)

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  23.5-84.

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