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Heat transfer and thermal systems
Mean natural convection on a vertical isothermal plate
Churchill–Chu mean Nusselt number for natural convection on a vertical isothermal plate, with Ra based on the height and the wall-to-ambient temperature difference.
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- HTS-28
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- v1
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- Heat transfer and thermal systems
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- NuL
The Churchill–Chu correlation gives the mean Nusselt number for natural convection on a vertical isothermal plate over the whole laminar and turbulent range: Nu = [0.825 + 0.387 Ra^⅙/(1 + (0.492/Pr)^(9/16))^(8/27)]². Ra is based on the plate height and the wall-to-ambient temperature difference, with properties at the film temperature.
Churchill–Chu mean Nusselt number for natural convection on a vertical isothermal plate, with Ra based on the height and the wall-to-ambient temperature difference. Fluid properties at the film temperature; 0.1 ≤ Ra ≤ 10¹².
Calculator
- Coefficient for air with k = 0.026 W/(m K) on a 0.5 m plate3.169 W/(m² K)
- Nusselt number if Ra rises tenfold122.6
- Ra^⅙ group21.54
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.
- At Ra = 1×108 the mean Nusselt number is 60.95; for air on a 0.5 m plate, that is about 3.169 W/(m² K).
- A tenfold rise in Ra only raises Nu to 122.6, a factor of 2.01, which shows how weakly buoyant convection responds to temperature difference.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| RaL Rayleigh number based on height | 0.1 to 1e12 | 1.094 | 1,104 | 1,103 | +10% | +2.88 % |
| Pr Prandtl number | 0.01 to 1,000 | 25.46 | 83.34 | 57.87 | +10% | +1.27 % |
Sweep of Rayleigh number based on height
| Rayleigh number based on height | Mean Nusselt number |
|---|---|
| 0.1 | 1.094 |
| 4.217 | 1.53 |
| 177.8 | 2.539 |
| 7,499 | 5.102 |
| 316,200 | 12.25 |
| 1.334e7 | 33.82 |
| 5.623e8 | 102.7 |
| 2.371e10 | 331.1 |
| 1e12 | 1,104 |
Sweep one variable
Variables and how each one contributes
Mean Nusselt number
What the equation gives you. Everything else feeds it.
Rayleigh number based on height
The Rayleigh number based on the plate height. Raises the Nusselt number, roughly as the fourth root in the laminar range. Raising it by 10 % (to 1.1×108) moves the result by +2.88 %, from 60.95 to 62.71.
Prandtl number
The Prandtl number. Raises the Nusselt number slightly, with a limited effect at high Pr. Raising it by 10 % (to 0.77) moves the result by +1.27 %, from 60.95 to 61.72.
Worked example with the values above
- Prandtl factor: [1 + (0.492/0.7)^(9/16)]^(8/27) = 1.1942.
- Bracket: 0.825 + 0.387 × 21.544 / 1.1942 = 7.807.
- Nu = 7.807² = 60.949.
Assumptions and limits
- Fluid properties are evaluated at the film or bulk temperature as stated, and are otherwise constant.
- The geometry and flow regime match the correlation range, such as smooth tubes, single cylinders or isothermal plates.
Common mistakes
- Using a correlation outside its Reynolds, Prandtl or Rayleigh range.
- Using the wrong characteristic length, for example plate length instead of tube diameter.
- Taking the properties at the wall or free-stream temperature when the film temperature is intended.
Statistics and uncertainty
Published correlations typically scatter by 10 % to 25 % against experiment, and the property evaluation temperature adds to the error. Use the result as an estimate to be confirmed by test for critical designs.
Where it is used
- Estimating coefficients for heat sinks, ducts, tubes and walls.
- Screening calculations before a detailed model or test.
Maths you need
- Reynolds number: Re = ρUL/μ
- Prandtl number: Pr = μcp/k
- Nusselt number: Nu = hL/k