Library / Heat transfer and thermal systems / HTS-27
Heat transfer and thermal systems
Cylinder crossflow mean Nusselt number
Churchill–Bernstein correlation for the average Nusselt number of a single circular cylinder in crossflow, with Re based on diameter and the upstream velocity and properties at the film temperature.
- Ref
- HTS-27
- Rev
- v1
- Field
- Heat transfer and thermal systems
- Result
- NuD
The Churchill–Bernstein correlation gives the average Nusselt number of a single circular cylinder in crossflow over a very wide Reynolds range: Nu = 0.3 + [0.62 Re^½ Pr^⅓/(1 + (0.4/Pr)^⅔)^¼][1 + (Re/282000)^⅝]^⅘. Re is based on diameter and the upstream velocity; properties are at the film temperature. Use it for Re·Pr > 0.2.
Churchill–Bernstein correlation for the average Nusselt number of a single circular cylinder in crossflow, with Re based on diameter and the upstream velocity and properties at the film temperature. Use for Re·Pr > 0.2.
Calculator
- Re·Pr product (should exceed 0.2)7,000
- Re^½ Pr^⅓ group88.79
- Coefficient for air with k = 0.026 W/(m K) and D = 20 mm69.33 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.
- At Re = 10,000 and Pr = 0.7 the mean Nusselt number is 53.33, which for a 20 mm cylinder in air gives about 69.33 W/(m² K).
- The 0.3 term gives the limit of conduction at very low Re, and the last factor accounts for the change in the boundary-layer behaviour at high Reynolds number.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| ReD Reynolds number based on diameter | 1 to 1e7 | 0.7831 | 9,868 | 9,867 | +10% | +5.42 % |
| Pr Prandtl number | 0.1 to 100 | 23.37 | 314.3 | 291 | +10% | +3.86 % |
Sweep of Reynolds number based on diameter
| Reynolds number based on diameter | Mean Nusselt number |
|---|---|
| 1 | 0.7831 |
| 7.499 | 1.624 |
| 56.23 | 3.935 |
| 421.7 | 10.35 |
| 3,162 | 28.76 |
| 23,710 | 87.08 |
| 177,800 | 318.9 |
| 1,334,000 | 1,568 |
| 1e7 | 9,868 |
Sweep one variable
Variables and how each one contributes
Mean Nusselt number
What the equation gives you. Everything else feeds it.
Reynolds number based on diameter
The Reynolds number based on the diameter. Raises the Nusselt number, roughly as the square root at moderate Re. Raising it by 10 % (to 11,000) moves the result by +5.42 %, from 53.33 to 56.22.
Prandtl number
The Prandtl number. Raises the Nusselt number, roughly as the cube root. Raising it by 10 % (to 0.77) moves the result by +3.86 %, from 53.33 to 55.39.
Worked example with the values above
- Main group: 0.62 √10,000 0.7^⅓ = 55.05; denominator [1 + (0.4/Pr)^⅔]^¼ = 1.1399.
- High-Re factor: [1 + (Re/282000)^⅝]^⅘ = 1.0981.
- Nu = 0.3 + 48.292 × 1.0981 = 53.328.
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