How to use this calculator
Choose the shape, enter the volumetric heat generation rate, the half-thickness or radius and the conductivity, and either the cooling fluid (h and T∞) or the surface temperature.
Formulas
Plane wallTmax = Ts + q̇L²/(2k), Ts = T∞ + q̇L/h
CylinderTmax = Ts + q̇r₀²/(4k), Ts = T∞ + q̇r₀/(2h)
SphereTmax = Ts + q̇r₀²/(6k), Ts = T∞ + q̇r₀/(3h)
Worked example
Wall 50 mm thick (L = 25 mm), q̇ = 1 MW/m³, k = 15 W/m·K, cooled on both sides by 30 °C fluid with h = 500 W/m²·K (the default)
- q″ = q̇L = 25 000 W/m², so Ts = 30 + 25 000/500 = 80 °C.
- Tmax − Ts = 10⁶ × 0.025²/(2 × 15) = 20.8 °C.
- Tmax = 100.8 °C at the midplane.
Frequently asked questions
Why does the temperature profile curve?
Every layer adds its own heat, so the heat flux grows linearly toward the surface and the temperature profile is a parabola.
Wall insulated on one side?
Use the full thickness as L: the insulated face acts like the midplane of a wall twice as thick.