Reference ID: MET-D62B | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
Radiation heat exchange between surfaces is the dominant mode of heat loss for equipment and products exposed to large, cold surroundings—most commonly the sky. In food, pharmaceutical, and cryogenic process engineering, the calculation predicts overnight cooling or warming of stored material, sizing of insulation, and risk of condensation or frost. Typical applications include:
Estimating night-time temperature drop of fruit in orchards to decide on frost-protection measures.
Rating radiant coolers and cryogenic shields where convection is suppressed.
Checking surface temperatures of insulated pipe racks to stay above the dew-point.
The method treats each surface as a diffuse-gray emitter/absorber and uses the net-radiation concept, valid when participating media (gases, moisture) are transparent to infrared.
Methodology & Formulas
Convert practical inputs to absolute units
\[
D = \frac{D_{\text{cm}}}{100},\quad A_{1} = \pi D^{2},\quad T_{1} = T_{1,^{\circ}\text{C}}+273.15,\quad T_{2} = T_{2,^{\circ}\text{C}}+273.15
\]
Net radiative heat flow
For a small convex object (surface 1) completely enclosed by a large enclosure (surface 2), the view factor \(F_{12}=1\) and the exchange reduces to:
\[
q = \varepsilon_{1}\,A_{1}\,\sigma\,\bigl(T_{1}^{4}-T_{2}^{4}\bigr)
\]
where
\(\sigma = 5.670 \times 10^{-8}\ \text{W m}^{-2}\text{ K}^{-4}\) (Stefan–Boltzmann constant)
\(\varepsilon_{1}\) = hemispherical emissivity of the object (dimensionless).
Outside these limits, use a radiation–convection coupled model or spectral surface properties.
Evaluate the temperature difference: radiation becomes significant when surface temperatures exceed ~400 °C or when the absolute temperature difference is large.
Check the emissivity of the materials; polished metals (ε ≈ 0.05) contribute little, whereas oxidized steel or refractory (ε ≈ 0.8) dominate.
Compare the calculated radiation heat-flux (σ ε F (T₁⁴–T₂⁴)) with the convection/conduction flux; include radiation whenever it is >20% of the total.
For furnaces, kilns, high-temperature reactors, or any vacuum environment, always include radiation.
For two large, gray, diffuse, parallel plates of equal area separated by a gap that is small compared with their width, the shape factor F₁₂ = 1. If the plates are finite or other surfaces participate, obtain F₁₂ from standard charts or numerical integration; always enforce the summation rule ΣFᵢⱼ = 1 for an enclosure.
Metals: emissivity rises with temperature and oxide layer thickness; freshly cleaned stainless steel ε ≈ 0.15, heavily oxidized ε ≈ 0.8.
Refractories: ε is relatively constant (0.7–0.9) up to 1500 °C; use manufacturer data.
In design, perform a sensitivity study: vary ε ±0.1 and check the impact on heat duty; use the worst-case value for sizing equipment.
Yes, for first-pass hand calculations, use an effective coefficient hrad = 4 σ ε Tₘ³ where Tₘ is the mean absolute temperature; add this to the convection coefficient hconv to give htotal = hconv + hrad. Be aware that this linearization is only valid for small temperature differences; for large ΔT or enclosure problems, solve the radiation balance separately.
Using ambient temperature instead of the temperature of surrounding surfaces for the radiation sink.
Neglecting the fourth-power temperature dependence—linearizing without checking the temperature range.
Assuming gray surfaces when strong wavelength dependence exists (e.g., CO₂/H₂O gas radiation); use band or weighted-sum-of-gray-gases models.
Forgetting to update emissivity when surface conditions change (oxidation, coating, dust).
Worked Example: Radiative Heat Loss from an Apple on a Cold Conveyor Belt
A food-processing plant cools 8 cm apples after a hot-water rinse. Each apple is treated as a small gray sphere (ε = 0.95) that radiates to an enclosing refrigerated tunnel wall at −20 °C. Estimate the net radiation heat the apple gives up while it is at 10 °C.
Knowns
Diameter of apple, D = 8 cm = 0.08 m
Apple surface temperature, T1 = 10 °C = 283.15 K
Surrounding tunnel temperature, T2 = −20 °C = 253.15 K
Apple surface emissivity, ε = 0.95
View factor from apple to tunnel, F12 = 1.0 (apple completely enclosed)
Stefan–Boltzmann constant, σ = 5.67 × 10−8 W m−2 K−4
Step-by-Step Calculation
Compute the apple surface area:
\[ A_1 = \pi D^2 = \pi (0.08)^2 = 0.020 \; \text{m}^2 \]
Convert temperatures to kelvin (already shown in Knowns).
Use the net radiation equation for a small gray body in large surroundings:
\[ q = \varepsilon\, \sigma\, A_1\, F_{12}\, (T_1^4 - T_2^4) \]
