Library / Heat transfer and thermal systems / HTS-40
Heat transfer and thermal systems
View factor between two parallel coaxial disks
Fraction of the radiation leaving the disk of radius r1 that reaches the coaxial parallel disk of radius r2 at separation H, with S = 1 + (H² + r2²)/r1².
- Ref
- HTS-40
- Rev
- v1
- Field
- Heat transfer and thermal systems
- Result
- F12
The view factor between two parallel coaxial disks of radii r1 (emitting) and r2 (receiving), a distance H apart, is F₁₂ = 2u²/[S + √(S² − 4u²)], with u = r₂/r₁ and S = 1 + (H² + r₂²)/r₁². The calculator forms S from the three lengths. Equal disks one radius apart (u = 1, S = 3) give 0.382. It assumes nothing blocks the view between the disks.
Fraction of the radiation leaving the disk of radius r1 that reaches the coaxial parallel disk of radius r2 at separation H, with S = 1 + (H² + r2²)/r1². Facing disks with coincident axes and nothing in between.
Calculator
- Radius ratio, u = r₂/r₁1
- Geometry parameter, S = 1 + (H² + r₂²)/r₁²3
- View factor from 2 to 1, F₁₂(r₁/r₂)²0.382
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.
- 38.2 % of the radiation leaving the first disk lands directly on the second.
- By reciprocity the second disk sends 38.2 % of its radiation to the first; the two agree when the disks are the same size.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| r2 Radius of the receiving disk | 0.01 to 10 m | 5e-5 | 0.99 | 0.99 | +10% | +14.2 % |
| H Axial separation | 0.01 to 10 m | 0.009805 | 0.99 | 0.9802 | +10% | −8.47 % |
| r1 Radius of the emitting disk | 0.01 to 10 m | 0.0099 | 0.5 | 0.4901 | +10% | −5.61 % |
Sweep of Axial separation
| Axial separation (m) | View factor from 1 to 2 |
|---|---|
| 0.01 | 0.99 |
| 0.02371 | 0.9766 |
| 0.05623 | 0.9453 |
| 0.1334 | 0.8752 |
| 0.3162 | 0.7298 |
| 0.7499 | 0.4803 |
| 1.778 | 0.2016 |
| 4.217 | 0.05068 |
| 10 | 0.009805 |
Sweep one variable
Variables and how each one contributes
View factor from 1 to 2
What the equation gives you. Everything else feeds it.
Radius of the emitting disk
The radius of the emitting disk, in m. A larger emitting disk sees the target at a smaller relative size, so the view factor falls for fixed r₂ and H. Raising it by 10 % (to 1.1) moves the result by −5.61 %, from 0.382 to 0.3605.
Radius of the receiving disk
The radius of the receiving disk, in m. A larger receiving disk fills more of the view, so the view factor rises. Raising it by 10 % (to 1.1) moves the result by +14.2 %, from 0.382 to 0.4362.
Axial separation
The axial separation, in m. A larger separation lowers the view factor. Raising it by 10 % (to 1.1) moves the result by −8.47 %, from 0.382 to 0.3496.
Worked example with the values above
- Radius ratio: u = 1/1 = 1; S = 1 + (1² + 1²)/1² = 3.
- Root: √(S² − 4u²) = √(9 − 4) = 2.2361.
- F₁₂ = 2 × 1 / (3 + 2.2361) = 0.38197.
Assumptions and limits
- Blackbody or diffuse grey behaviour as stated, in a vacuum or non-participating medium.
- Absolute temperatures and the exact values of σ, h, c and kB given on the page.
Common mistakes
- Using °C in Planck’s law or Wien’s law.
- Mixing per-metre and per-micrometre spectral units.
- Treating a grey surface as if it followed the blackbody spectrum without considering emissivity.
Statistics and uncertainty
The radiation laws for blackbodies are exact, but real-surface emissivity and its variation with wavelength add uncertainty of 10 % or more, and measured view factors depend on exact geometry.
Where it is used
- Infrared sensing, lamps and radiant heaters.
- Setting up radiosity networks and linearised radiation coefficients.
Maths you need
- Planck’s law: spectral emissive power of a blackbody
- Wien’s law: the peak wavelength is inversely proportional to T
- Radiosity: the total radiation leaving a surface per unit area