Transient heat conduction: integral energy conservation under time-varying loads

Transient heat conductionPASS · 1.13%

A steel bar subjected simultaneously to time-modulated Dirichlet wall temperature T_D(t), convective ambient temperature T_∞(t), and volumetric heat generation Q(t). Verifies exact discrete thermal energy conservation ΔU = E_gen + E_flux + E_dir − E_conv and t… show full text

Tolerance:: 5%

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Problem statement

Geometry: Prismatic bar 100 × 10 × 10 mm; quadratic tetrahedra (tet10), element size 4 mm.
Material: Steel preset (k = 50 W/(m·K), ρ = 7850 kg/m³, c_p = 460 J/(kg·K)).
Boundary conditions: Initial T₀ = 30 °C; surface number 1 temperature 150 °C with ramp table 0:0.2,20:1; surface number 2 convection h = 80 W/(m²·K), ambient 25 °C with table 0:1,40:3; volumetric heat generation Q = 2 MW/m³ with table 0:1,15:0.25. Δt = 1 s, t_end = 40 s, θ = 0.5.
Measured quantity: Relative energy balance residual |ΔU − (E_gen + E_dir − E_conv)| / E_scale (target 0) and agreement between nodal probes and cross-section averages at t = 20 s.

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 steel
primitive box dx=100 dy=10 dz=10
mesh tet size=4 order=2
thermal surface number 1 temperature=150
thermal surface number 2 convection=80 ambient=25
load heat_generation=2e6
solve thermal transient dt=1 tend=40 t0=30 theta=0.5 amp_temperature="0:0.2,20:1" amp_ambient="0:1,40:3" amp_heat="0:1,15:0.25" probes="10,5,5;90,5,5" frames=5
probe temperature probe=1 time=20 id="p1"
probe temperature probe=2 time=20 id="p2"

Results comparison

Reference value	balance residual = 0 (≤ 1e-9)T(x=10 mm, t=20 с) = 90.902 °C
Our result	residual 2.34e-14T_probe(x=10 mm, t=20 с) = 89.876 °C
Deviation	1.13%
Tolerance	5%
Status	PASS · 1.13%
Mesh details	tet10, size 4 mm (2514 nodes); 40 steps of Δt = 1 с, θ = 0.5, 5 frames

Updated: 2026-10-27 · version not recorded