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Heat transfer · dimensionless numbersThe Nusselt number (Nu) is the dimensionless ratio of convective to conductive heat transfer. Compute it from its definition Nu = h·L/k, from the Dittus–Boelter turbulent-pipe correlation, or from the laminar flat-plate correlation — and convert Nu back into a heat transfer coefficient h.
Solve in any of the four directions. h is the convective coefficient, L the characteristic length, k the fluid thermal conductivity.
Valid for ReD ≥ 10,000, 0.6 ≤ Pr ≤ 160, L/D ≥ 10, fully developed turbulent internal flow. The tool warns outside these ranges.
Valid for laminar boundary layers: Rex < 5×105, Pr ≥ 0.6.
Approximate values near 20 °C (consistent with Incropera, Fundamentals of Heat and Mass Transfer, App. A). Pr of liquids drops sharply with temperature — treat oil values as order-of-magnitude.
| Fluid | Pr (approx) | Notes |
|---|---|---|
| Mercury | 0.025 | Liquid metal — conduction dominates |
| Air | 0.71 | Gases are ≈ 0.7–1.0 |
| Water | 7.0 | At 20 °C |
| Ethanol | 16 | Light organic liquid |
| Ethylene glycol | 151 | Antifreeze; upper end of Dittus–Boelter range |
| Engine oil | ~10,400 | At 20 °C; range 100–40,000 — outside Dittus–Boelter validity |
| Glycerol | ~11,600 | Very viscous — momentum diffusivity dominates |
The Nusselt number (Nu) is the dimensionless ratio of convective to conductive heat transfer across a fluid layer: Nu = h·L/k, where h is the convective heat transfer coefficient, L a characteristic length and k the fluid thermal conductivity. Nu = 1 means pure conduction; larger values mean convection dominates. Example: h = 500 W/m²K, L = 0.02 m, k = 0.6 W/m·K gives Nu = 500×0.02/0.6 = 16.67. Mode 1 of this calculator solves the definition in all four directions (Nu, h, L or k).
The heat transfer coefficient h (W/m²K) is a dimensional quantity that depends on the fluid, the flow and the geometry. The Nusselt number is its dimensionless form, Nu = h·L/k, which lets engineers compare convection across different fluids and sizes and build general correlations. Boundary with the sibling tool: the Heat Transfer Coefficient Calculator computes h itself (including U-values and unit conversion); this calculator works with the dimensionless Nu and the correlations that predict it, and can convert Nu back to h via h = Nu·k/L.
The Dittus–Boelter correlation predicts the Nusselt number for fully developed turbulent flow inside a pipe: Nu = 0.023·Re^0.8·Pr^n, with n = 0.4 when the wall heats the fluid and n = 0.3 when it cools it. Worked example (Mode 2): Re = 20,000, Pr = 7, heating gives Nu = 0.023 × 20000^0.8 × 7^0.4 = 138.23. With k = 0.6 W/m·K and a 0.025 m pipe diameter, that is h = Nu·k/L = 138.2264×0.6/0.025 = 3317.43 W/m²K.
Use n = 0.4 when the wall is hotter than the fluid (heating: heat flows wall → fluid) and n = 0.3 when the wall is cooler (cooling: fluid → wall). The exponent corrects for how the temperature profile distorts viscosity near the wall. Same flow, different direction: Re = 20,000, Pr = 7 gives Nu = 138.23 for heating but only 113.78 for cooling — about 18% lower, so picking the wrong exponent is a real error, not a rounding one.
Three conditions: Reynolds number Re ≥ 10,000 (fully turbulent), Prandtl number 0.6 ≤ Pr ≤ 160, and pipe length L/D ≥ 10 so the flow is developed. This calculator shows a warning whenever Re < 10,000 or Pr leaves 0.6–160, and labels the regime (laminar below 2,300, transitional 2,300–10,000, turbulent above 10,000). Gases (Pr ≈ 0.7) and water (Pr ≈ 7) fit comfortably; engine oil (Pr in the thousands) does not — the correlation was never validated there.
For laminar flow over an isothermal flat plate the local Nusselt number is Nu_x = 0.664·Re_x^(1/2)·Pr^(1/3), valid for Re_x < 5×10^5 and Pr ≥ 0.6. Example (Mode 3): air-like flow at Re_x = 100,000 with Pr = 0.707 gives Nu_x = 0.664 × 100000^0.5 × 0.707^(1/3) = 187.06. Above Re_x ≈ 5×10^5 the boundary layer transitions and this formula underpredicts — the tool warns you.
The local Nu_x describes convection at one position x along the plate, while the average Nu over 0→L describes the whole plate (what you need for total heat rate). For a fully laminar plate they are linked exactly: the average equals twice the local value evaluated at Re_L, Nu_bar = 2·Nu_x(Re_L). Example: Re_L = 400,000, Pr = 7 gives an average Nu of 1606.67. Mode 3 reports both numbers so you can use whichever your hand calculation needs.
Pr depends only on the fluid and its temperature — the reference table on this page covers the common cases: air ≈ 0.71, water ≈ 7.0 (20 °C), ethanol ≈ 16, ethylene glycol ≈ 151, engine oil ≈ 10,400 and glycerol ≈ 11,600. Two cautions: Pr of liquids falls sharply as temperature rises (oils especially — always use the value at your operating temperature), and Dittus–Boelter is only validated up to Pr = 160, so the oil and glycerol presets will trigger the range warning by design.
None — Nu is dimensionless, so it never needs unit conversion. Only the inputs carry units: h in W/m²K (or BTU/(hr·ft²·°F)), L in m/cm/mm/inch/ft, k in W/m·K (or BTU/(hr·ft·°F)). The calculator converts everything to SI internally; for reference, 100 BTU/(hr·ft²·°F) = 567.83 W/m²K. To get a dimensional coefficient back out, use h = Nu·k/L — Mode 2 does this automatically.
No. Dittus–Boelter is strictly a turbulent correlation; below Re ≈ 10,000 (and certainly in laminar flow, Re < 2,300) it extrapolates badly. For fully developed laminar flow in a circular tube the Nusselt number approaches a constant instead: Nu = 3.66 for constant wall temperature, Nu = 4.36 for constant wall heat flux. For the laminar entrance region use a Graetz-type correlation such as Seider–Tate. Check the regime badge in Mode 2 before trusting any number it produces.
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