Conjugate Radiative Heat Transfer Calculator — Conduction + Radiation Wall Solver
Solve 1D transient conduction through a layered wall with a radiative surface balance — ambient sink or gray-enclosure radiosity matrix — plus convection. Stefan-Boltzmann boundary fluxes.
A clear path from inputs to results.
Model how heat moves through a layered wall when both convection and thermal radiation act on it — the “conjugate” problem of a solid exchanging heat with a hot gas and a radiating environment at the same time. It solves 1D transient conduction through your material stack while applying a radiative surface balance, either against a single sink temperature or across a gray-enclosure radiosity matrix.
How to use
- Pick a scenario preset — vacuum radiator panel, enclosure furnace wall, hypersonic thermal shield, or electronics bay — to load representative gas, environment, and sink temperatures plus a convection coefficient.
- Choose the radiation mode: ambient sink (surface radiates to one effective environment temperature) or enclosure matrix (surface-to-surface gray radiosity among several surfaces).
- Select a layer preset (TPS, alloy, or insulation) that sets each layer’s thickness, conductivity k, density, and specific heat for the conduction mesh.
- Set the surface emissivity, convection coefficient h, gas temperature, environment/sink temperatures, and — in enclosure mode — the enclosure coupling factor, surface count, and temperature gradient.
- Set the initial temperature, total simulation time, and time step for the implicit solver.
- Read the steady wall and cold-side temperatures, the convective and radiative surface fluxes, the radiosity/irradiation of each enclosure surface, and the through-thickness temperature profile.
The formula
Reference: Stefan-Boltzmann radiative exchange and the gray-enclosure radiosity method (Incropera & DeWitt, Fundamentals of Heat and Mass Transfer; Modest, Radiative Heat Transfer). 1D transient conduction discretized in conservative finite-volume form with implicit Euler; σ = 5.670374419×10⁻⁸ W·m⁻²·K⁻⁴ (CODATA/NIST).
Worked example
Surface flux balance on a hypersonic thermal shield
- Hypersonic thermal-shield scenario (ambient radiation mode): convection coefficient h = 120 W/m²/K, gas temperature T_g = 1900 K, radiative sink T_env = 320 K, emissivity ε = 0.9, σ = 5.670374419×10⁻⁸. Evaluated at a representative hot-wall temperature T_s = 1200 K.
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The convective flux into the wall is q″_conv = h·(T_g − T_s) = 120 × (1900 − 1200) = 84,000 W/m² = 84.00 kW/m². The radiative surface term is q″_rad = εσ(T_env⁴ − T_s⁴) = 0.9 × 5.670374419×10⁻⁸ × (320⁴ − 1200⁴). With 320⁴ = 1.048576×10¹⁰ and 1200⁴ = 2.0736×10¹², the difference is −2.063114×10¹², so q″_rad = 0.9 × 5.670374419×10⁻⁸ × (−2.063114×10¹²) = −105,287 W/m², i.e. the surface loses 105.29 kW/m² by reradiation. The net surface flux is 84.00 − 105.29 = −21.29 kW/m², meaning at 1200 K the wall is radiating away more than convection delivers and will cool toward its radiative equilibrium.
At T_s = 1200 K: convective input 84.00 kW/m², reradiative loss 105.29 kW/m², net −21.29 kW/m². These are exactly the surface terms the implicit conduction solver applies at the hot boundary each time step.
Interpreting the result
The surface sits where incoming convection balances reradiation and conduction inward. Because radiation scales as T⁴, it dominates at high surface temperature: in the worked example the hot wall already reradiates more than convection supplies, so its equilibrium temperature lies below 1200 K. A higher emissivity sheds more heat and lowers the wall temperature; a higher convection coefficient or gas temperature pushes it up. In enclosure mode the single sink is replaced by a radiosity matrix, so surrounding surfaces that are themselves hot reflect and re-emit energy back, raising the wall temperature relative to the open-sink case.
Assumptions & limitations
- Heat conduction is 1D through the layer stack; lateral spreading and multi-dimensional effects are ignored.
- Surfaces are gray and diffuse with constant emissivity; spectral and specular effects are not modeled.
- In ambient mode the surface radiates to a single effective environment/sink temperature; in enclosure mode view factors come from a simplified symmetric coupling matrix set by the enclosure factor, not a detailed geometry.
- Material properties (k, ρ, cₚ) are constant per layer and independent of temperature.
- The radiative boundary is linearized about the current surface temperature and solved implicitly, so the scheme is stable but accuracy depends on the chosen time step; the radiosity system is solved by Gauss-Seidel iteration to a fixed tolerance.
Frequently asked questions
What does “conjugate” heat transfer mean here?
It means the solid conduction problem and the fluid/radiative boundary conditions are solved together rather than in isolation. The wall’s surface temperature both depends on and feeds back into the convective and radiative fluxes, so the engine iterates the surface balance against the implicit conduction solution each time step.
What is the difference between ambient and enclosure radiation modes?
Ambient mode radiates the surface to a single effective sink temperature — appropriate for a part facing open sky or deep space. Enclosure mode builds a gray-body radiosity matrix J_i = ε_iσT_i⁴ + (1−ε_i)ΣF_ij J_j among several surfaces, so energy bounces between walls before escaping. Enclosures generally run hotter than an open sink at the same emissivity.
Which scenario preset should I start from?
Use the vacuum radiator panel for space-facing panels with weak convection, the furnace wall for strong surface-to-surface enclosure radiation, the hypersonic thermal shield for aggressive combined convective-radiative heating, and the electronics bay for low-temperature conjugate heating in a gray enclosure. Each preset loads representative gas, environment, sink, convection, and emissivity values you can then adjust.
Why does the surface lose heat at 1200 K in the example?
Radiation scales as T⁴, so at 1200 K the reradiated flux εσ(T_s⁴ − T_env⁴) is 105.29 kW/m² while convection only delivers h(T_g − T_s) = 84.00 kW/m². The net −21.29 kW/m² means 1200 K is above the surface’s radiative-convective equilibrium, so it would cool until the two fluxes balance.
How does emissivity change the result?
Emissivity multiplies the radiative flux. Raising it lets the surface shed more heat by radiation (and absorb more incoming radiation), which for a hot, convectively heated wall lowers the equilibrium temperature. Low-emissivity surfaces reradiate poorly and run hotter, which is why high-temperature coatings favour high emissivity.
Is the time step important?
The conduction scheme is implicit, so it stays stable at large steps, but the radiative boundary is linearized about the current surface temperature. Very large steps can under-resolve fast surface transients even though the run will not blow up. Shorten the step when the surface temperature changes quickly to keep the flux balance accurate.
Methodology last reviewed 2026-10-06. Results are informational and not financial or engineering advice — see our disclaimer.