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Heat transfer and thermal systems
Reverse view factor from reciprocity
View factor from surface 2 to surface 1, from the areas A1 and A2 and the known view factor F12 between diffuse surfaces.
- Ref
- HTS-34
- Rev
- v1
- Field
- Heat transfer and thermal systems
- Result
- F21
The reciprocity relation for view factors between diffuse surfaces is A₁F₁₂ = A₂F₂₁, so F₂₁ = (A₁/A₂)F₁₂. It lets you find one view factor from the other and the two areas. Both factors must lie between 0 and 1; a result above one signals inconsistent areas or an inconsistent view factor.
View factor from surface 2 to surface 1, from the areas A1 and A2 and the known view factor F12 between diffuse surfaces. A result above one flags inconsistent geometry or input.
Calculator
- Exchange area, A₁F₁₂ = A₂F₂₁0.15 m²
- Margin below one, 1 − F₂₁0.7
- Area ratio, A₁/A₂0.4
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Of the radiation leaving surface 2, 30 % reaches surface 1, compared with 75 % in the other direction.
- The exchange area A₁F₁₂ = 0.15 m² is the same from either side, which is why the smaller surface sees a larger fraction of the larger one.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| A1 Area of surface 1 | 0.01 to 10 m² | 0.015 | 15 | 14.98 | +10% | +10 % |
| A2 Area of surface 2 | 0.01 to 10 m² | 0.015 | 15 | 14.98 | +10% | −9.09 % |
| F12 View factor from 1 to 2 | 0 to 1 | 0 | 0.4 | 0.4 | +10% | +10 % |
Sweep of View factor from 1 to 2
| View factor from 1 to 2 | View factor from 2 to 1 |
|---|---|
| 0 | 0 |
| 0.125 | 0.05 |
| 0.25 | 0.1 |
| 0.375 | 0.15 |
| 0.5 | 0.2 |
| 0.625 | 0.25 |
| 0.75 | 0.3 |
| 0.875 | 0.35 |
| 1 | 0.4 |
Sweep one variable
Variables and how each one contributes
View factor from 2 to 1
What the equation gives you. Everything else feeds it.
Area of surface 1
The area of surface 1, in m². Raises F₂₁ in direct proportion. Raising it by 10 % (to 0.22) moves the result by +10 %, from 0.3 to 0.33.
Area of surface 2
The area of surface 2, in m². Lowers F₂₁ in inverse proportion. Raising it by 10 % (to 0.55) moves the result by −9.09 %, from 0.3 to 0.2727.
View factor from 1 to 2
The view factor from surface 1 to surface 2. Raises F₂₁ in direct proportion. Raising it by 10 % (to 0.825) moves the result by +10 %, from 0.3 to 0.33.
Worked example with the values above
- Area ratio: A₁/A₂ = 0.2 / 0.5 = 0.4.
- Multiply by F₁₂ = 0.75.
- F₂₁ = 0.3.
Assumptions and limits
- Opaque, diffuse and grey surfaces, isothermal with uniform properties.
- A transparent, non-participating gap, and absolute temperatures in kelvin.
Common mistakes
- Using °C in the fourth-power terms.
- Using the plate formula when the surfaces are not large compared with the gap.
- Forgetting that an emissivity of 1 means a perfect black surface and 0 a perfect reflector.
Statistics and uncertainty
Emissivity values are uncertain by 10 % to 30 % for real surfaces, and temperature errors are multiplied by four in the fourth-power law, so a 2 % error in absolute temperature gives roughly 8 % in flux.
Where it is used
- Furnaces, cryostats, ovens and spacecraft thermal design.
- Choosing shield layers and estimating radiant exposure.
Maths you need
- Stefan–Boltzmann law: E = σT⁴, with σ = 5.670×10⁻⁸ W/(m² K⁴)
- Emissivity: ε, the ratio of emission to that of a blackbody
- View factor: the fraction of radiation leaving one surface that reaches another