Thermal condition of a wall

What happens to heat at an outer wall when heat transfer is on: nothing passes (adiabatic), the gas-side wall face is held at a temperature (fixed temperature), or heat flows through the shell and any insulation to the surroundings by convection and radiation (heat loss).

Last updated 2026-11-03

Theory

With heat transfer the compressible solver the compressible solver solves the energy equation; the wall's temperature condition in 0/T is one of three (OpenFOAM user guide, 2026):

mode	`0/T` condition	inputs
Adiabatic	`zeroGradient` (no heat flux)	none
Fixed temperature	`fixedValue`, $T_w$	Wall temperature
Heat loss	`externalWallHeatFluxTemperature`, mode coefficient	shell, insulation, environment, emissivity, ambient

Heat loss models the wall as a stack of planar layers (shell, then optional insulation) between the gas-side wall face and the environment; no solid region is meshed. The solver is given the layer thicknesses and conductivities, the ambient temperature $T_a$ , the outer convective coefficient $h$ and the emissivity; the boundary condition itself adds the radiation to $T_a$ . The gas-side heat transfer comes from the turbulence wall functions (compressible::alphatJayatillekeWallFunction, turbulent Prandtl number 0.85). The heat flux leaving the gas is equation (1).

The outer $h$ depends on the outer surface temperature, which is not known before the solve:

1. The program estimates a first surface temperature from resistances in series (an assumed gas-side film of 20 W/(m² K), the layers, the outer film), equation (2), and evaluates $h$ there. 2. After the start-up (upwind) phase it re-evaluates $h$ once from the solved wall heat flux: the surface temperature that carries that flux to the surroundings, equation (3), and $h$ at that temperature, written into the restart.

Both values are recorded with the run and shown in the report.

Internal walls (baffles, tube walls inside the muffler) are always adiabatic; only outer walls offer this setting. Quality gate 2 checks the energy balance: the enthalpy the gas loses must equal the heat through the walls within 1 %; with heat transfer the automatic stop also waits for this balance.

Equations

q = \frac{T_w - T_a}{\dfrac{1}{h} + \sum_i \dfrac{t_i}{k_i}} \quad (+\ \text{radiation})

(1)

T_{s,0} = T_a + (T_{gas} - T_a)\,\frac{R_{out}}{1/h_g + \sum_i t_i/k_i + R_{out}}, \qquad R_{out} = \frac{1}{h(T_{s,0}) + h_r(T_{s,0})}, \qquad h_g = 20\ \text{W/(m}^2\text{K)}

(2)

q = \left(h(T_s) + h_r(T_s)\right)(T_s - T_a), \qquad h_r = \varepsilon\sigma\,(T_s^2 + T_a^2)(T_s + T_a)

(3)

where	meaning	unit
$q$	Heat flux leaving the gas, per unit wall area	W/m²
$T_w$	Gas-side wall-face temperature (solved)	K
$T_a$	Ambient temperature (Ambient temperature)	K
$h$	Outer convective heat-transfer coefficient	W/(m² K)
$t_i, k_i$	Thickness and conductivity of layer $i$ (shell, insulation)	m, W/(m K)
$T_s$	Outer surface temperature	K
$T_{gas}$	Inlet gas temperature	K
$h_r$	Linearised radiation coefficient	W/(m² K)
$\varepsilon$	Emissivity of the outer surface	-
$\sigma$	Stefan-Boltzmann constant	W/(m² K⁴)

Default and rule

Default: AdiabaticMeasured
Rule: Adiabatic.
Source: Validation: straight pipe D 0.3 m, L 3 m, 0.157 kg/s of air at 700 K, 1.5 mm stainless shell with a 10 mm wrap (k 0.06), still air, emissivity 0.8. The computed temperature drop is 20.205 K against 19.660 K from an analytic pipe calculation with cylindrical layer resistances (+2.8 %) and 19.053 K with the planar layers the solver uses (+6.0 %); the energy balance closes to 0.017 %. Evaluating $h$ at the gas-side face temperature instead of the outer surface temperature (574 K against 386 K in this case) overestimates the convection by 20 %; equations (2) and (3) therefore use the outer surface temperature. Correlations: Outer environment.

Valid range Adiabatic | Fixed temperature | Heat loss; outer walls, heat transfer on

Planar layers

Curvature of the shell is not resolved: the error grows for a thick insulation on a small pipe (3.1 % between planar and cylindrical resistances in the validation case).

One h update

The outer $h$ is updated once, after the start-up phase; it then stays fixed while the solution settles.

Internal walls

Always adiabatic: heat exchange between two gas streams through a tube wall is not modelled.

References 1

OpenCFD Ltd. (2026). OpenFOAM user guide. openfoam.com. Web page

Theory

Equations

Default and rule

Valid range Adiabatic | Fixed temperature | Heat loss; outer walls, heat transfer on

Related 9

References 1