Convection coefficient
Three geometries cover most of what an engineer needs: forced flow over a flat plate, forced flow inside a pipe, and free convection from a vertical surface. In each case the Nusselt number is a correlation in Reynolds or Rayleigh and Prandtl, and the coefficient is Nu·k/L — so the characteristic length is as important as the correlation itself.
| Quantity | Value | Note |
|---|---|---|
| Geometry | Flat plate | characteristic length L |
| Characteristic length | 1.000 m | length or height |
| Velocity | 10.00 m/s | free stream |
| Reynolds | 6.293e+5 | V L / ν |
| Grashof | — | g β ΔT L³ / ν² |
| Rayleigh | — | Gr × Pr |
| Prandtl | 0.707 | fluid property |
| Conductivity | 0.0263 W/m·K | fluid property |
| Nusselt | 659.9 | from the correlation |
| Coefficient h | 17.4 W/m²·K | Nu k / L |
| Heat flux | 347 W/m² | h ΔT at 20 K |
Reynolds is 6.29e+5, in the mixed regime. The coefficient scales with velocity^0.8, so doubling the flow does not double the heat transfer.
Nu = 0.664 Re^0.5 Pr^1/3 · Re below 5 × 10⁵.
Nu = 0.037 Re^0.8 Pr^1/3 · Above 5 × 10⁵.
Nu = 0.023 Re^0.8 Pr^n · n = 0.4 heating, 0.3 cooling.
Nu = 0.59 Ra^0.25 (laminar) · 0.1 Ra^1/3 when turbulent.
A clear path from inputs to results.
Three geometries cover most of what an engineer needs: forced flow over a flat plate, forced flow inside a pipe, and free convection from a vertical surface. In each case the Nusselt number is a correlation, and the coefficient is Nu·k/L.
How to use
- Choose the geometry and the fluid.
- Enter the characteristic length and the velocity.
- Set the temperature difference and the heat-flow direction.
- Read the Nusselt number and the coefficient.
The formula
Reference: Blasius and Colburn for the plate, Dittus-Boelter for the pipe, Churchill-Chu for the vertical plate.
Worked example
Air at 10 m/s over a 1 m plate
- 1 m plate, 10 m/s air at 300 K.
- ΔT 20 K, heating.
- 1
The Reynolds number is 10 × 1 / 1.589 × 10⁻⁵ = 6.29 × 10⁵.
- 2
That is in the mixed regime, so Nu = (0.037 Re^0.8 − 871) Pr^1/3 = 658.
- 3
The coefficient is 658 × 0.0263 / 1 = 17.3 W/m²·K.
- 4
At ΔT = 20 K the heat flux is 346 W/m².
- 5
Doubling the velocity to 20 m/s raises h to about 29 W/m²·K — an increase of 68%, not 100%.
Nu 658 · 17.3 W/m²·K · 346 W/m².
Interpreting the result
The exponent is the practical point: laminar flow scales as V^0.5 and turbulent as V^0.8, so doubling the flow never doubles the heat transfer. That is why a bigger fan has diminishing returns and why the transition to turbulence matters more than the fan setting.
Assumptions & limitations
- Fluid properties at 300 K, which is a film-temperature approximation.
- Smooth surfaces and an unheated starting length of zero.
- Fully developed flow in the pipe.
- No boiling, condensation or large property variation.
Frequently asked questions
Why is the coefficient lower on a longer plate?
Because the boundary layer thickens along the plate, so the average coefficient falls with length. That is why a small heat sink outperforms a large flat one per unit area.
When does natural convection matter?
When there is no forced flow, or when it is very slow. At 20 K of ΔT on a 1 m vertical surface in air the coefficient is about 3 W/m²·K — an order of magnitude below forced convection.
What about a fin?
A fin uses the same coefficient but with an efficiency factor that accounts for the temperature gradient along it. The coefficient is the input, not the answer.
What is the characteristic length?
For a plate, the length along the flow; for a pipe, the bore; for a vertical surface, the height. Getting it wrong is the most common error.
Does the pipe correlation work for gases?
Yes, Dittus-Boelter is valid for 0.7 < Pr < 160 and Re > 10,000, which covers both air and water. Below Re = 2,300 the flow is laminar and Nu is 3.66.
Methodology last reviewed 2026-10-22. Results are informational and not financial or engineering advice — see our disclaimer.