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7.13.4 Heat Transfer Calculations at Wall Boundaries



Temperature Boundary Conditions


When a fixed temperature condition is applied at the wall, the heat flux to the wall from a fluid cell is computed as


 q = h_f (T_w - T_f) + q_{\rm rad} (7.13-8)

where


$h_f$ = fluid-side local heat transfer coefficient
$T_w$ = wall surface temperature
$T_f$ = local fluid temperature
$q_{\rm rad}$ = radiative heat flux

Note that the fluid-side heat transfer coefficient is computed based on the local flow-field conditions (e.g., turbulence level, temperature, and velocity profiles), as described by Equations  7.13-15 and 12.10-5.

Heat transfer to the wall boundary from a solid cell is computed as


 q = \frac{k_{s}}{\Delta n} (T_w - T_{s}) + q_{\rm rad} (7.13-9)

where


$k_{s}$ = thermal conductivity of the solid
$T_{s}$ = local solid temperature
$\Delta n$ = distance between wall surface and the solid cell center



Heat Flux Boundary Conditions


When you define a heat flux boundary condition at a wall, you specify the heat flux at the wall surface. FLUENT uses Equation  7.13-8 and your input of heat flux to determine the wall surface temperature adjacent to a fluid cell as


 T_w = \frac{ q - q_{\rm rad}}{h_f} + T_f (7.13-10)

where, as noted above, the fluid-side heat transfer coefficient is computed based on the local flow-field conditions. When the wall borders a solid region, the wall surface temperature is computed as


 T_w = \frac{(q - q_{\rm rad}) \Delta n}{k_{s}} + T_{s} (7.13-11)



Convective Heat Transfer Boundary Conditions


When you specify a convective heat transfer coefficient boundary condition at a wall, FLUENT uses your inputs of the external heat transfer coefficient and external heat sink temperature to compute the heat flux to the wall as


$\displaystyle q$ $\textstyle =$ $\displaystyle h_f (T_w - T_f) + q_{\rm rad}$  
  $\textstyle =$ $\displaystyle h_{\rm ext} (T_{\rm ext} - T_w)$ (7.13-12)

where


$h_{\rm ext}$ = external heat transfer coefficient defined by you
$T_{\rm ext}$ = external heat-sink temperature defined by you
$q_{\rm rad}$ = radiative heat flux

Equation  7.13-12 assumes a wall of zero thickness.



External Radiation Boundary Conditions


When the external radiation boundary condition is used in FLUENT, the heat flux to the wall is computed as


$\displaystyle q$ $\textstyle =$ $\displaystyle h_f (T_w - T_f) + q_{\rm rad}$  
  $\textstyle =$ $\displaystyle \epsilon_{\rm ext} \sigma (T_{\infty}^4 - T_w^4)$ (7.13-13)

where


$\epsilon_{\rm ext}$ = emissivity of the external wall surface defined by you
$\sigma$ = Stefan-Boltzmann constant
$T_w$ = surface temperature of the wall
$T_{\infty}$ = temperature of the radiation source or sink on the exterior
    of the domain, defined by you
$q_{\rm rad}$ = radiative heat flux to the wall from within the domain

Equation  7.13-13 assumes a wall of zero thickness.



Combined External Convection and Radiation Boundary Conditions


When you choose the combined external heat transfer condition , the heat flux to the wall is computed as


$\displaystyle q$ $\textstyle =$ $\displaystyle h_f (T_w - T_f) + q_{\rm rad}$  
  $\textstyle =$ $\displaystyle h_{\rm ext} (T_{\rm ext} - T_w) + \epsilon_{\rm ext} \sigma (T_{\infty}^4 - T_w^4)$ (7.13-14)

where the variables are as defined above. Equation  7.13-14 assumes a wall of zero thickness.



Calculation of the Fluid-Side Heat Transfer Coefficient


In laminar flows, the fluid side heat transfer at walls is computed using Fourier's law applied at the walls. FLUENT uses its discrete form:


 q = k_f \left ( \frac{\partial T}{\partial n} \right )_{\rm wall} (7.13-15)

where $n$ is the local coordinate normal to the wall.

For turbulent flows, FLUENT uses the law-of-the-wall for temperature derived using the analogy between heat and momentum transfer [ 197]. See Section  12.10.2 for details.


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