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Heat transfer and thermal systems
Exchange between large diffuse grey plates
Net radiant flux between two large parallel opaque diffuse grey plates with opposing faces at T1 and T2 and a transparent gap.
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- HTS-31
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- v1
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- Heat transfer and thermal systems
- Result
- q″ [W/m²]
Between two large parallel opaque diffuse grey plates, the net radiant flux is q″ = σ(T₁⁴ − T₂⁴)/(1/ε₁ + 1/ε₂ − 1). The denominator is the sum of the surface resistances. Black plates (ε = 1) give the largest exchange, σ(T₁⁴ − T₂⁴). Positive flux goes from plate 1 to plate 2, and equal temperatures give none.
Net radiant flux between two large parallel opaque diffuse grey plates with opposing faces at T1 and T2 and a transparent gap. Positive flux goes from plate 1 to plate 2.
Calculator
- Flux between black plates, σ(T₁⁴ − T₂⁴)992.3 W/m²
- Ratio to the black-plate exchange, 1/(1/ε₁ + 1/ε₂ − 1)0.6667
- Equivalent radiation coefficient, σ(T₁ + T₂)(T₁² + T₂²)/(1/ε₁ + 1/ε₂ − 1)6.615 W/(m² K)
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.
- Between plates at 400 K and 300 K the net exchange is 661.5 W/m², 66.7 % of the black-plate value.
- The surface resistances add: for the emissivities used, they cut radiative exchange to 0.667 of the black-body case.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (W/m²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| T1 Temperature of plate 1 | 100 to 2,000 K | −302.4 | 604,500 | 604,800 | +10% | +67.9 % |
| T2 Temperature of plate 2 | 100 to 2,000 K | −603,900 | 964 | 604,800 | +10% | −21.5 % |
| ε1 Emissivity of plate 1 | 0.05 to 1 | 49 | 793.9 | 744.8 | +10% | +8.2 % |
| ε2 Emissivity of plate 2 | 0.05 to 1 | 49 | 793.9 | 744.8 | +10% | +8.2 % |
Sweep of Emissivity of plate 1
| Emissivity of plate 1 | Net radiant flux (W/m²) |
|---|---|
| 0.05 | 49 |
| 0.1688 | 160.7 |
| 0.2875 | 266.2 |
| 0.4062 | 366 |
| 0.525 | 460.5 |
| 0.6438 | 550.2 |
| 0.7625 | 635.5 |
| 0.8812 | 716.6 |
| 1 | 793.9 |
Sweep one variable
Variables and how each one contributes
Net radiant flux
What the equation gives you. Everything else feeds it.
Temperature of plate 1
The temperature of plate 1, in K. Raises the net flux as the fourth power. Raising it by 10 % (to 440) moves the result by +67.9 %, from 661.5 to 1,111 W/m².
Temperature of plate 2
The temperature of plate 2, in K. Raises the opposing term and lowers the net flux. Raising it by 10 % (to 330) moves the result by −21.5 %, from 661.5 to 519.4 W/m².
Emissivity of plate 1
The emissivity of plate 1. A higher emissivity raises the flux. Raising it by 10 % (to 0.88) moves the result by +8.2 %, from 661.5 to 715.8 W/m².
Emissivity of plate 2
The emissivity of plate 2. A higher emissivity raises the flux. Raising it by 10 % (to 0.88) moves the result by +8.2 %, from 661.5 to 715.8 W/m².
Worked example with the values above
- Fourth-power difference: 400⁴ − 300⁴ = 1.75×1010 K⁴; × σ = 992.32 W/m².
- Resistance term: 1/ε₁ + 1/ε₂ − 1 = 1.5.
- q″ = 992.32 / 1.5 = 661.54 W/m².
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