NAFEMS T2: steady-state conduction with nonlinear Stefan–Boltzmann radiation

Nonlinear heat transferPASS · <0.01%

One-dimensional steady heat conduction along a bar whose hot end is held at 1000 K (726.85 °C) and whose cold end radiates to a 300 K (26.85 °C) environment with emissivity ε = 0.98. Solved via full Newton–Raphson with the exact 4εσT³ radiation Jacobian and co… show full text

Tolerance:: 5%

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Problem statement

Geometry: Prismatic bar L = 100 mm, cross-section 10 × 10 mm; tetrahedral mesh, element size 5 mm.
Material: Conductivity k = 55.6 W/(m·K), Stefan–Boltzmann constant σ = 5.670374419 × 10⁻⁸ W/(m²·K⁴).
Boundary conditions: Surface number 1 (x = 0) fixed at T_A = 726.85 °C (1000 K); surface number 2 (x = 100 mm) radiating with ε = 0.98 to T_∞ = 26.85 °C (300 K); lateral faces insulated.
Measured quantity: Cold-end temperature T_B at x = 100 mm (NAFEMS T2 reference: 927 K = 653.85 °C; exact 1D root: 653.854 °C) and radiated power Q_rad = A·εσ(T_B⁴ − T_∞⁴).

WebCAE Terminal

Paste these commands into the Terminal panel inside WebCAE. They use the same text DSL as the application; run one command per line:

units mm
material E=2e11 nu=0.3 kappa=55.6
primitive box dx=100 dy=10 dz=10
mesh tet size=5
thermal surface number 1 temperature=726.85
thermal surface number 2 radiation=0.98 ambient=26.85
solve thermal
probe temperature x=100 id="TB"

Results comparison

Reference value	T_B = 653.85 °C (927 K)Q_rad = 4.0586 W
Our result	T_B = 653.854 °C (927.004 K)Q_rad = 4.0586 W
Deviation	<0.01%
Tolerance	5%
Status	PASS · <0.01%
Mesh details	tet4, size 5 mm (366 nodes); 3 Newton iterations, residual 2.39e+0 → 4.25e-11 W

Updated: 2026-10-27 · version not recorded