Stefan-Boltzmann Law — Fundamentals of Radiative Heat Transfer and CAE Implementation

Good question. In short, integrating Planck's radiation law over all wavelengths naturally yields T⁴.

Planck's spectral radiance is

Integrating this over all wavelengths $\lambda = 0 \sim \infty$ gives the total radiant energy $E_b$. Using the variable transformation $x = hc/(\lambda k_B T)$, the integral becomes

This definite integral has a fixed value: $\Gamma(4)\,\zeta(4) = 6 \times \pi^4/90 = \pi^4/15$. Hence $E_b \propto T^4$ emerges. The fourth power comes from "the photon density of states being proportional to $\nu^2$ in three-dimensional space" and "the energy average of the Bose-Einstein distribution"; it's a consequence of spatial dimensions and quantum statistics.

Yes. $\sigma$ can be expressed solely in terms of fundamental physical constants:

Since the 2019 SI redefinition, $k_B$, $h$, and $c$ are all defined values, so $\sigma$ also has an exact, fixed value. In CAE textbooks, it's often rounded to $5.67 \times 10^{-8}$, but knowing this is useful for accuracy discussions.

The net radiative heat exchange between two surfaces is determined by the temperature difference of both surfaces:

This is the basic formula for gray bodies. The key point here is that it's the difference of $T^4$, not $(T_1 - T_2)^4$. This nonlinearity is what makes CAE tricky.

For example, let's calculate for a slab ($T_1 = 1200°C = 1473\,\text{K}$) and its surroundings ($T_2 = 30°C = 303\,\text{K}$).

On the other hand, natural convection under the same conditions, with $h \approx 10$ W/(m²K), gives $q_{\text{conv}} = 10 \times 1170 \approx 12$ kW/m². That means radiation is about 18 times greater than convection. In the world of 1200°C, radiation is overwhelmingly dominant.

Exactly. In continuous casting, radiative cooling occurs from the slab surface immediately after it leaves the mold, and the solidified shell grows from the outside. If $\sigma\varepsilon T^4$ is not accurately modeled, the shell thickness prediction will be off, leading to the risk of breakout (molten steel leakage). This is one of the most feared accidents in steel mills.

A practical guideline is surface temperature of 600°C (approx. 900K). Beyond this point, the contribution of radiation increases rapidly. The power of T⁴ is formidable; when the temperature doubles, the radiant energy becomes 16 times greater.

However, in a vacuum, convection is zero, so in spacecraft thermal design, radiation is the only means of heat rejection even at room temperature. The ISS (International Space Station) radiators operate around 280K, determining the heat rejection rate solely by $\sigma T^4$. Because the temperature is low, a large area is required.

There are two main approaches. The most commonly used is conversion to a radiative heat transfer coefficient.

Here, the radiative heat transfer coefficient $h_{\text{rad}}$ is

$h_{\text{rad}}$ is calculated using temperatures from the previous iteration and updated via the Newton-Raphson method. Since it can be added to the convective $h_{\text{conv}}$, it fits easily into the existing heat transfer coefficient framework.

The other is directly incorporating the derivative of T⁴ into the Jacobian.

This is placed into the tangent stiffness matrix of the NR method. Abaqus's *RADIATION and Ansys's SRDOPT internally use this technique. Convergence is fast, but poor initial temperature estimates can cause divergence.

There are three main strategies.

1. Weak Coupling (Sequential): Calculate convection coefficients with CFD → solve heat conduction + radiation with FEM → feed temperature back to CFD. Easy to implement but slow convergence.
2. Strong Coupling (Monolithic): Combine fluid, solid, and radiation into a single equation system and solve simultaneously. Ansys Fluent's S2S (Surface-to-Surface) model uses this approach. Convergence is fast but memory consumption is high.
3. DO/MC Models: When participating media (e.g., combustion gases) are present, solve the Radiative Transfer Equation (RTE) using the Discrete Ordinates method or Monte Carlo method. The S-B law is used as a wall boundary condition.

In practice, "first converge the flow field without radiation, then activate the radiation model" is stable. Turning everything on at once risks divergence.

The basic flow is as follows:

1. Identify Radiation Surfaces: List high-temperature surfaces and the surfaces they "see." Also check for obstructions.
2. Set Emissivity: Check for temperature dependence. Don't blindly trust literature values as they change significantly with oxidation state.
3. Calculate View Factors: Use the solver's automatic calculation (e.g., Hemicube method) or, if available, use analytical solutions for verification.
4. Mesh Consideration: Radiation surfaces have T⁴ nonlinearity, so refine areas with steep temperature gradients (edges, corners).
5. Set Initial Temperature: Use initial conditions close to actual operating temperatures. Starting at 300K will cause the iteration count to explode to reach 1500K.
6. Stepwise Solution: Activate in order: convection → convection + radiation. Don't turn everything on at once.

In Continuous Casting, the slab pulled from the mold passes through the secondary cooling zone. The cooling mode is divided into three regions:

1. Spray Cooling Zone: Forced convection by water spray ($h = 500\sim2000$ W/(m²K)) is dominant. Here, S-B radiation accounts for only about 10-20%.
2. Radiative Cooling Zone: After the spray ends, the slab temperature is still 900〜1100°C. In this section, $q_{\text{rad}} = \varepsilon\sigma(T_s^4 - T_\infty^4)$ accounts for over 80% of cooling.
3. Roll Contact Zone: Conductive contact with guide rolls. Contact area is small but locally important.

In CAE models, Lagrangian tracking along the casting direction (slice method) is common. A 2D cross-section model of the slab is extracted and passed through each cooling zone according to the withdrawal speed. Latent heat of solidification ($L \approx 270$ kJ/kg) is handled by the enthalpy method.

Sharp observation. The emissivity changes over time due to the growth of scale (oxide layer) on the slab surface. Just below the mold, $\varepsilon \approx 0.4$ (metallic luster present), but at the exit of the secondary cooling zone, oxidation progresses and $\varepsilon \approx 0.8$. Inputting this variation via a temperature-dependent table is the best practice in real work. Using a constant $\varepsilon = 0.8$ throughout will overestimate the upstream cooling rate and skew the shell thickness prediction.

Industrial furnaces are a treasure trove of radiative heat transfer. Inside a furnace above 1000°C, 70-90% of the heat input to the workpiece (heating target) is via radiation. Multi-surface radiative exchange between furnace walls, heaters, and workpieces is modeled using the Zone Method (Hottel's zone model) or the Radiosity Method.

Especially in gas-fired furnaces, combustion gases (CO₂, H₂O) themselves participate in radiation, so the S-B law alone is insufficient. The gas emissivity must be calculated using the Weighted-Sum-of-Gray-Gases (WSGG) model, and the Radiative Transfer Equation needs to be solved. Ansys Fluent's DO model and STAR-CCM+'s DO/S2S model support this.