Nusselt Number Calculator
The Nusselt number (Nu) is the ratio of convective to conductive heat transfer at a surface. Enter the convective coefficient, characteristic length, and fluid thermal conductivity to compute Nu directly, or pick an empirical correlation for pipe flow, flat plates, or natural convection. Every output updates instantly and the step-by-step panel shows the substituted math.
Formula
Worked example
Water (Pr = 7.0, k = 0.607 W/m·K) flows at Re = 50,000 through a pipe (D = 0.02 m). Dittus-Boelter (heating): Nu = 0.023 × 50000^0.8 × 7.0^0.4 = 0.023 × 2,626 × 2.180 ≈ 131.6. Then h = Nu × k / D = 131.6 × 0.607 / 0.02 ≈ 3,993 W/(m²·K).
What is the Nusselt number?
The Nusselt number (Nu) is a dimensionless ratio that compares the convective heat transfer across a fluid boundary layer to the heat that would be transferred by conduction alone across the same layer. It is defined as Nu = h L / k, where h is the convective heat transfer coefficient in W/(m²·K), L is the characteristic length in metres (inner diameter for pipes, plate length for flat surfaces), and k is the thermal conductivity of the fluid in W/(m·K). A Nusselt number of 1 means convection contributes nothing beyond pure conduction; values in the tens or hundreds signal vigorous convection. Engineers use Nu to compute h once they know the flow conditions and fluid properties, which then lets them size heat exchangers, cooling fins, pipes, and electronic cooling systems.
How to choose the right correlation
The correct correlation depends on the geometry and flow regime. For turbulent flow inside smooth round pipes, Gnielinski (1975) is the most accurate choice: it works from Re = 3,000 (covering the transition zone) up to 5,000,000 and Pr from 0.5 to 2,000. The simpler Dittus-Boelter formula from the 1930s is still widely used in textbooks but is less accurate and only valid for Re above 10,000. For fully developed laminar pipe flow (Re below about 2,300), no Re or Pr dependence exists: Nu is exactly 3.66 for a constant-temperature wall or 4.36 for a constant-heat-flux wall. Flow over a flat plate splits at a critical Reynolds number of about 5 × 10^5: laminar boundary layers use Nu = 0.664 Re^0.5 Pr^(1/3) and turbulent ones use Nu = 0.037 Re^0.8 Pr^(1/3). When there is no pump or fan and the flow is driven entirely by buoyancy (natural convection), the Churchill-Chu correlation covers vertical plates across the full laminar-to-turbulent range using the Rayleigh number Ra.
Reynolds and Prandtl numbers: the key inputs
The Reynolds number Re = rho u D / mu (or equivalently Re = u D / nu) measures the ratio of inertial to viscous forces. Low Re means laminar, orderly flow; high Re means turbulent, chaotic flow with much higher convective coefficients. The Prandtl number Pr = mu cp / k (or Pr = nu / alpha) compares the momentum diffusivity to the thermal diffusivity of the fluid. Fluids with Pr much greater than 1, like oils (Pr 50 to 10,000), have thick thermal boundary layers relative to the velocity boundary layer, while liquid metals (Pr near 0.01) spread heat much faster than momentum. Common room-temperature values: air Pr 0.71, water Pr 6.9, engine oil Pr 200. For natural convection, the Rayleigh number Ra = Gr × Pr combines the Grashof number (buoyancy vs. viscous forces) with Pr and is the single parameter controlling the flow regime.
Deriving the heat transfer coefficient from Nu
Once you have Nu, recover the convective heat transfer coefficient with h = Nu × k / L. For example, air (k = 0.026 W/m·K) flowing in turbulent forced convection with Nu = 200 over a plate of L = 0.3 m gives h = 200 × 0.026 / 0.3 = 17.3 W/(m²·K). That h then feeds directly into Newtons law of cooling: q = h A (T_surface - T_fluid), where q is the heat flow in watts and A is the surface area. This chain from flow conditions to Nu to h to heat flux is the backbone of almost all convective heat transfer design calculations in chemical engineering, HVAC, electronics cooling, and automotive thermal management.
Common Nusselt number correlations
| Correlation | Geometry / regime | Valid range | Formula |
|---|---|---|---|
| Definition | Any | All conditions | Nu = hL/k |
| Dittus-Boelter | Turbulent pipe | Re >= 10,000; 0.6 <= Pr <= 160 | Nu = 0.023 Re^0.8 Pr^n (n=0.4 heating, 0.3 cooling) |
| Gnielinski | Turbulent pipe | 3,000 <= Re <= 5,000,000; 0.5 <= Pr <= 2,000 | Nu = (f/8)(Re-1000)Pr / [1+12.7(f/8)^0.5(Pr^2/3-1)] |
| Laminar pipe (isothermal) | Laminar pipe, constant T wall | Re < 2,300, fully developed | Nu = 3.66 |
| Laminar pipe (heat flux) | Laminar pipe, constant q wall | Re < 2,300, fully developed | Nu = 4.36 |
| Flat plate laminar | External flow over plate | Re < 5×10^5; Pr > 0.6 | Nu = 0.664 Re^0.5 Pr^(1/3) |
| Flat plate turbulent | External flow over plate | Re > 5×10^5; Pr > 0.6 | Nu = 0.037 Re^0.8 Pr^(1/3) |
| Churchill-Chu | Vertical plate, natural convection | Ra < 10^12 | Nu = [0.825 + 0.387 Ra^(1/6)/psi]^2 |
Summary of the correlations available in this calculator. All assume smooth surfaces and developed flow unless noted.
Frequently asked questions
What does a Nusselt number of 1 mean?
Nu = 1 means convection is doing nothing more than conduction would do across the same fluid layer. This happens in quiescent (completely still) fluids or in the theoretical limit of no flow. In practice, any fluid motion raises Nu above 1, sometimes dramatically in turbulent flow.
Which Nusselt correlation should I use for pipe flow?
For turbulent flow (Re above 10,000), Gnielinski is the most accurate choice and is valid down to Re = 3,000 for the transition zone. Dittus-Boelter is simpler and common in textbooks but less accurate, especially near the lower Re limit. For laminar, fully developed pipe flow (Re below 2,300) the Nusselt number is a constant: 3.66 for an isothermal wall or 4.36 for a constant heat flux wall.
What is the characteristic length for different geometries?
For internal flow in a round pipe, use the inner diameter. For a flat plate, use the length in the flow direction. For external flow over a cylinder, use the outer diameter. For natural convection over a vertical plate, use the plate height. For horizontal plates, a common characteristic length is the plate area divided by the perimeter.
How does the Prandtl number affect the Nusselt number?
In the Dittus-Boelter form Nu is proportional to Pr^n (n = 0.3 or 0.4), so doubling Pr from 3.5 to 7 raises Nu by about 23% (for n = 0.4). Fluids with very high Pr, like oils, develop thick velocity boundary layers but thin thermal ones, leading to lower local heat transfer per unit of driving temperature difference. Liquid metals (very low Pr) conduct heat so well that the thermal boundary layer spreads far beyond the velocity layer, and special correlations apply.
What is the difference between local and average Nusselt number?
The local Nusselt number Nu_x describes heat transfer at a single point along a surface, and varies along the flow direction as the boundary layer grows. The average Nusselt number Nu_L (used in most design calculations) integrates the local value over the entire surface. For a laminar flat plate the average is exactly twice the local value at the trailing edge: Nu_L = 0.664 Re_L^0.5 Pr^(1/3), while the local form is Nu_x = 0.332 Re_x^0.5 Pr^(1/3). This calculator gives the average, which is what you need for total heat transfer from a surface.
Sources
- Incropera, F.P. et al. - Fundamentals of Heat and Mass Transfer, 7th ed. (Wiley, 2011)
- Gnielinski, V. - New equations for heat and mass transfer in turbulent pipe and channel flow. Int. Chem. Eng. 16, 359-368 (1976)
- Churchill, S.W. and Chu, H.H.S. - Correlating equations for laminar and turbulent free convection from a vertical plate. Int. J. Heat Mass Transfer 18, 1323-1329 (1975)