Engineering task and calculation objective
This module calculates thermal radiation of technical surfaces according to Section K1 of the VDI Heat Atlas (VDI-Wärmeatlas, 12th edition 2019), the standard German reference work for heat transfer. Every surface emits radiation according to its temperature and its emissivity, following the Stefan-Boltzmann law; between surfaces at different temperatures a net radiative exchange establishes itself, which depends on the emissivities, the temperatures, and the geometric arrangement of the surfaces — described by view factors.
In practice, the radiation calculation is needed to determine heat losses of hot equipment, piping, and furnace walls, to assess surface temperatures for touch protection, or to correctly capture the radiative share of the total heat transfer alongside convection. The module covers the technically important configurations: parallel plates and gaps, concentric cylinders and annular gaps, enclosed bodies, tube rows in front of walls, as well as special cases such as bores and rectangular geometries.
Anyone who wants to calculate radiative heat exchange needs above all reliable emissivities — depending on material, oxidation, and surface condition they scatter between below 0.1 (bright metals) and above 0.9 (paints, oxidized or soiled surfaces).
Standard and calculation basis: VDI-Wärmeatlas, 12. Auflage 2019
Calculation workflow
- Choose the geometric arrangement: First, the radiation configuration is defined: parallel surfaces or a parallel gap, concentric cylinders or an annular gap, a fully enclosed body, tube rows in front of a wall, or another of the configurations tabulated in the VDI Heat Atlas.
- Define the emissivities: For each participating surface, the emissivity is determined according to material and surface condition (bright, oxidized, painted, soiled) — for technical surfaces the most uncertain input quantity of the entire calculation.
- Determine the view factors: The view factors describe what fraction of the radiation leaving one surface strikes another; closed-form relations are available for the standard arrangements, and summation and reciprocity relations help with composite geometries.
- Calculate the radiative exchange: The net heat flow between the surfaces is calculated from the fourth powers of the absolute temperatures, the Stefan-Boltzmann constant, the emissivities, and the view factors; multiple reflections between gray surfaces are captured via the exchange relations of the respective arrangement.
- Superimpose with convection: For loss and surface temperature calculations, the radiative share — often as an equivalent radiative heat transfer coefficient — is superimposed on the convective heat transfer; at surface temperatures from about 50 to 100 °C upward, both contributions are of comparable magnitude.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| A1 | A1 ε1 T1 | m² |
| A2 | A2 ε2 T2 | m² |
| 1 | A1 ε1 T1 | - |
| 2 | A2 ε2 T2 | - |
| T1 | A1 ε1 T1 | K |
| T2 | A2 ε2 T2 | K |
| Phi_12 | φ12 | - |
| Phi_21 | φ21 φ21 = A1 / A2 ∙ φ12 | - |
| E_1 | E1 E2 | W/m² |
| E_2 | E1 E2 | W/m² |
| Q_12 | Q_12 | W |
| Q_21 | Q_21 | W |
| Length Radius | L R | m |
| Length Radius | L R | m |
| Area Emissivity | F' ε | - |
| Temperature | T | K |
| Emissivity calculated from table | εeff | - |
| Heat flux | Q | W |
| Temperature Angle | T α | RAD |
| Length Height | L h | m |
| Height Length | h l | m |
| Length Length | a b | m |
| Length Length | a b | m |
| Area Emissivity | F' ε | m² |
Worked example
An uninsulated vessel wall with a painted surface (emissivity ε = 0.90) has a surface temperature of 80 °C and stands in a large hall with a wall and ambient temperature of 20 °C. Find the net radiative heat flux and the equivalent radiative heat transfer coefficient — a worked example of a radiation heat transfer calculation.
Given values
| Emissivity ε | 0.90 |
| Surface temperature ϑW | 80 °C (TW = 353.15 K) |
| Ambient temperature ϑU | 20 °C (TU = 293.15 K) |
| Stefan-Boltzmann constant σ | 5.67 · 10⁻⁸ W/(m²·K⁴) |
| Arrangement | Small body in large surroundings (view factor φ = 1) |
Solution
Net radiative heat flux
For a body enclosed by large surroundings:
q̇ = ε · σ · (TW⁴ − TU⁴)
TW⁴ = 353.15⁴ = 1.555 · 10¹⁰ K⁴
TU⁴ = 293.15⁴ = 7.385 · 10⁹ K⁴
Difference: 8.169 · 10⁹ K⁴
q̇ = 0.90 · 5.67 · 10⁻⁸ · 8.169 · 10⁹ ≈ 417 W/m²
Equivalent radiative heat transfer coefficient
Referred to the temperature difference of 60 K:
αS = q̇ / (ϑW − ϑU) = 417 / 60 ≈ 6.95 W/(m²·K)
The radiative share is thus of the same order of magnitude as free convection on a vertical wall — in loss calculations for warm equipment, both contributions must be added.
Result
| Radiative heat flux q̇ | ≈ 417 W/m² |
| Radiative heat transfer coefficient αS | ≈ 6.95 W/(m²·K) |
All values are illustrative. The applicable standard and project-specific boundary conditions remain authoritative.
Frequently asked questions
Why must I never calculate radiation with Celsius temperatures?
The Stefan-Boltzmann law contains the fourth power of the absolute temperature. The calculation must be done in kelvin, and the difference of the fourth powers (T₁⁴ − T₂⁴) must not be replaced by the fourth power of the temperature difference — one of the most common mistakes. For small temperature differences, the exchange can be linearized and expressed as a radiative heat transfer coefficient αS.
How accurately must the emissivity be known?
The emissivity enters the heat flow linearly and scatters enormously for metals: mill-bright aluminum lies at about 0.05, rusted or oxidized steel surfaces at 0.7 to 0.9, paints — almost regardless of color — at about 0.9 to 0.95. To estimate losses conservatively, calculate with the upper value; with bright sheet cladding (aluminum jacketing of insulation), the low emissivity reduces the radiative losses considerably.
What does the view factor state and when is it 1?
The view factor φ₁₂ specifies what fraction of the energy radiated diffusely from surface 1 strikes surface 2 directly; it is a purely geometric quantity. It becomes 1 when surface 1 is completely enclosed by surface 2 — for example a vessel in a large hall. For this important special case, the exchange simplifies to q̇ = ε₁·σ·(T₁⁴ − T₂⁴), provided the surroundings are large compared with the body.
From what point is the radiative share relevant compared with convection?
With free convection on outer surfaces, the convective heat transfer coefficient typically lies at 3 to 10 W/(m²·K); a radiative share of 5 to 7 W/(m²·K) — as results at ε ≈ 0.9 and moderate temperatures — is then of equal rank and must not be neglected. At high temperatures (furnaces, uninsulated hot parts), radiation clearly dominates because of the T⁴ dependence.