Convective heat transfer is everywhere — cooling a CPU, heating water in a boiler, chilling air in an HVAC coil — and it all hinges on one coefficient: h, the convective heat transfer coefficient. But h is notoriously hard to measure directly. So engineers find it indirectly through the Nusselt number (Nu), a dimensionless group linked to the flow by well-established empirical correlations like Dittus-Boelter. This free Nusselt Number & Heat Transfer Coefficient Calculator computes Nu from Reynolds and Prandtl numbers using the right correlation for your flow, then converts it straight to h — bridging fluid dynamics and heat transfer in one step.
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The Nusselt Number & h Calculator
Enter the Reynolds and Prandtl numbers (or compute Re from the flow), pick a correlation matching your flow regime and geometry, and get the Nusselt number plus the convective heat transfer coefficient h. Consistent SI units throughout.
Nusselt Number & Heat Transfer Coefficient Calculator
What Is the Nusselt Number?
The Nusselt number (Nu) is the dimensionless ratio of convective to conductive heat transfer across a fluid boundary layer:
- Nu = 1 → heat moves by conduction only (no benefit from fluid motion).
- Nu > 1 → convection enhances heat transfer; turbulent flows often reach Nu > 100.
It's one of the most important dimensionless numbers in engineering — the bridge between the flow field and the thermal result.
From Nu to h
The whole reason we compute Nu is to get the convective heat transfer coefficient h, which we can't easily measure but which governs every convection calculation:
where k is the fluid's thermal conductivity and L is the characteristic length (pipe diameter or plate length). Once you have h, the heat transfer rate follows from Newton's law of cooling, q = h·A·ΔT — and h feeds directly into the overall coefficient U used to size exchangers in our Heat Exchanger LMTD calculator.
The Key Correlations
Nu is found from empirical correlations chosen to match your flow regime and geometry:
| Correlation | Formula | Use case |
|---|---|---|
| Dittus-Boelter | 0.023·Re0.8·Prn | Turbulent pipe, Re>104 |
| Sieder-Tate | 0.027·Re0.8·Pr1/3·(μ/μw)0.14 | Turbulent, viscous fluids |
| Gnielinski | (f/8)(Re−1000)Pr / [1+12.7√(f/8)(Pr2/3−1)] | Transition & turbulent, accurate |
| Laminar pipe | 3.66 or 4.36 | Re<2300 |
| Flat plate (lam.) | 0.664·Re0.5·Pr1/3 | External flow, Re<5×105 |
All of them take the Reynolds number and Prandtl number as inputs — which is why Re is the starting point of every convection calculation.
Dittus-Boelter in Detail
The Dittus-Boelter equation is the workhorse of turbulent pipe-flow heat transfer:
- n = 0.4 when the fluid is being heated (wall hotter than fluid)
- n = 0.3 when the fluid is being cooled (wall cooler than fluid)
Laminar Flow: Constant Nu
Here's an elegant result: for fully developed laminar pipe flow, the Nusselt number is a constant — it doesn't depend on Re or Pr at all:
These are exact analytical values from solving the energy equation, not empirical fits. Which one you use depends on your wall boundary condition — a distinction that also matters when setting up a CFD case. Whether your flow is laminar or turbulent is decided by the Reynolds number.
Choosing the Characteristic Length
| Geometry | Characteristic length L |
|---|---|
| Internal pipe flow | Pipe diameter D |
| Non-circular duct | Hydraulic diameter Dh |
| External flat plate | Plate length in flow direction |
Worked Example
Water at 1.8 m/s through a 0.038 m pipe, wall hotter than fluid (heating). Properties at bulk mean temperature: ρ = 985, μ = 4.89×10−4, cp = 4183, k = 0.651.
- Re = ρVD/μ = 985 × 1.8 × 0.038 / 4.89e-4 ≈ 137,800 (turbulent)
- Pr = μcp/k = 4.89e-4 × 4183 / 0.651 ≈ 3.14
- Nu = 0.023 × 137,8000.8 × 3.140.4 ≈ 470
- h = Nu·k/D = 470 × 0.651 / 0.038 ≈ 8,050 W/m²K
Common Mistakes
- Wrong characteristic length. Pipe diameter for internal, plate length for external — never mix them.
- Using a correlation out of range. Dittus-Boelter needs Re>104; below that, use laminar or Gnielinski.
- Wrong n in Dittus-Boelter. n = 0.4 for heating, 0.3 for cooling — not interchangeable.
- Evaluating properties at the wrong temperature. Use the bulk mean (film) temperature for fluid properties.
- Treating Nu as universal. Each correlation has a valid Re, Pr and geometry window.
- Forgetting the ~10–25% scatter. These are empirical fits; don't over-trust the last digit.
- Confusing laminar boundary conditions. Nu = 3.66 (const T) vs 4.36 (const q) are different cases.
Frequently Asked Questions
What is the Nusselt number?
The dimensionless ratio of convective to conductive heat transfer across a boundary layer. Nu = 1 means pure conduction; turbulent flows reach Nu > 100. It links the flow to the convection coefficient h.
How do you calculate the convective heat transfer coefficient?
h = Nu·k/L: find Nu from a correlation, multiply by thermal conductivity k, divide by characteristic length L (pipe diameter or plate length).
What is the Dittus-Boelter equation?
Nu = 0.023·Re0.8·Prn, n = 0.4 heating / 0.3 cooling. Valid for turbulent flow in smooth tubes, Re > 104, 0.6 < Pr < 160.
What is the Nusselt number for laminar pipe flow?
A constant: 3.66 for constant wall temperature, 4.36 for constant wall heat flux — exact analytical values independent of Re and Pr.
Which characteristic length should I use?
Pipe/hydraulic diameter for internal flow; plate length for external flow. The wrong length invalidates the correlation.
Why do different correlations give different Nusselt numbers?
They're curve-fits to different datasets with ~10–25% scatter and specific valid ranges. Gnielinski is more accurate than Dittus-Boelter but more complex. Match the correlation to your flow.
Conclusion
The Nusselt number is the elegant bridge between fluid flow and heat transfer: find it from the Reynolds and Prandtl numbers with the right correlation, then convert straight to the convective coefficient via h = Nu·k/L. Choose the correlation that fits your regime and geometry, use the correct characteristic length, evaluate properties at the film temperature, and respect each correlation's valid range — and you'll get an h you can trust for sizing exchangers, cooling electronics, or validating a CFD model.
Use the calculator above to move from flow conditions to a convection coefficient in seconds, with the correct correlation applied automatically.
For more CFD, heat transfer and fluid mechanics tutorials plus free engineering calculators, explore Free CFD Tutorial. If this tool helped you, please share it with your fellow engineers and students.