Nusselt number calculator
Estimate the Nusselt number and heat transfer coefficient for plates, cylinders, tube banks and pipes.
1Vertical plate
Dh = 4A/P: four times the flow area divided by the wetted perimeter.
Fewer than 10 rows applies the correction C2.
0° is vertical. Valid up to about 60°.
Air at 350 K: 2.08 × 10−5 Pa·s.
Enter C1 and m yourself
Leave both empty to use the Grimison table. Fill in both to override it.
Average Nusselt number Nu
2Fluid properties
Natural convection only. Ideal gas: β = 1/Tf.
Air at 20 °C and 1 atm. For better accuracy, use the properties at the film temperature shown in the result.
Prandtl number Pr
- Kinematic viscosity ν = μ/ρ
- –
- Thermal diffusivity α
- –
- Film temperature Tf
- –
Evaluate the properties at Tf = (Ts + T∞)/2.
3Use it in Simcenter STAR-CCM+
- As a boundary condition. For a surface you do not mesh, such as the outside of a casing, set the wall to a convection condition with this coefficient and the fluid temperature.
- As a check. For a surface you do resolve, compare the wall heat transfer coefficient from your simulation with this estimate. A large gap points to the mesh, the wall treatment or the reference temperature.
- As a first estimate. Correlations give area averages for simple shapes. Real geometry, conjugate heat transfer and mixed convection need the full simulation.
Simcenter STAR‑CCM+ wall boundary
- Thermal Specification
- Convection
- Heat Transfer Coefficient
- –
- Ambient Temperature
- –
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How to use the Nusselt number calculator
1. Pick the geometry and the flow. Nine cases cover natural convection on plates and cylinders, forced convection over plates, cylinders and tube banks, and flow inside tubes and ducts. The sketch shows which length goes into Nu and Re.
2. Choose the options and enter the conditions. Temperatures are absolute, in kelvin. For natural convection only the difference between the surface and the fluid enters the correlation; in every case the temperatures also give the heat flux.
3. Check the fluid properties. The defaults are air at 20 °C. Pick water or enter your own values, ideally at the film or mean temperature shown in step 2.
4. Read the Nusselt number and the heat transfer coefficient. A yellow note appears when your inputs are outside the range the correlation was fitted to. Copy the values, or copy the settings for a convection wall in Simcenter STAR-CCM+.
What the Nusselt number tells you
The Nusselt number compares convective heat transfer at a surface with pure conduction across the same fluid layer: Nu = hL/k. Nu = 1 means the fluid only conducts; larger values mean the flow carries heat away more effectively. Once a correlation gives Nu, the heat transfer coefficient follows as h = Nu k/L, with the same characteristic length L the correlation uses. Typical coefficients (Incropera et al., Table 1.1):
| Process | Typical h, W/(m²·K) |
|---|---|
| Free convection, gases | 2–25 |
| Free convection, liquids | 50–1000 |
| Forced convection, gases | 25–250 |
| Forced convection, liquids | 100–20 000 |
| Boiling and condensation | 2500–100 000 |
Forced convection correlations take the form Nu = C Rem Prn and natural convection correlations Nu = C Ran. The Reynolds number Re compares inertia with viscosity, the Rayleigh number Ra buoyancy with diffusion, and the Prandtl number Pr momentum diffusion with heat diffusion. All three are on the dimensionless numbers page.
Which correlation to use
For each case the calculator uses the correlation recommended by Incropera et al. Nu is an average over the surface and Nux a local value. The derivations, sketches and the full Grimison and duct tables are on the heat transfer coefficient correlations page.
Characteristic lengths. Plates use the length in the flow direction, or the height for a vertical plate. For a horizontal plate, L = As/P, the surface area divided by the perimeter. Cylinders, tube banks and circular tubes use the diameter D; non-circular ducts use the hydraulic diameter Dh = 4A/P.
| Case | Correlation | Valid for |
|---|---|---|
| Vertical plate | Nu = {0.825 + 0.387 Ra1/6 / [1 + (0.492/Pr)9/16]8/27}2 (Churchill and Chu) | All Ra |
| Vertical plate, laminar | Nu = 0.68 + 0.670 Ra1/4 / [1 + (0.492/Pr)9/16]4/9 | Ra ≤ 109 |
| Inclined plate | As for the vertical plate, with g cos θ in Ra | 0 ≤ θ ≤ 60° from vertical; lower side of a heated plate or upper side of a cooled plate |
| Horizontal plate, hot side up | Nu = 0.54 Ra1/4, or 0.15 Ra1/3 | 104–107, or 107–1011 (also a cold plate facing down) |
| Horizontal plate, hot side down | Nu = 0.27 Ra1/4 | 105–1010 (also a cold plate facing up) |
| Horizontal cylinder | Nu = {0.60 + 0.387 Ra1/6 / [1 + (0.559/Pr)9/16]8/27}2 | Ra ≤ 1012 |
| Flat plate, laminar | Nu = 0.664 Re1/2 Pr1/3 (uniform Ts) or 0.680 Re1/2 Pr1/3 (uniform q″) | Re ≤ 5 × 105, Pr ≥ 0.6 |
| Flat plate, laminar then turbulent | Nu = (0.037 Re4/5 − 871) Pr1/3 | 5 × 105 ≤ Re ≤ 108, 0.6 ≤ Pr ≤ 60 |
| Flat plate, turbulent | Nu = 0.037 Re4/5 Pr1/3; for uniform q″ the local Nux = 0.0308 Rex4/5 Pr1/3 | Turbulent from the leading edge, 0.6 ≤ Pr ≤ 60 |
| Cylinder in cross flow | Churchill and Bernstein, or Hilpert: Nu = C Rem Pr1/3 | Re Pr ≥ 0.2; Hilpert 0.4–4 × 105, Pr ≥ 0.7 |
| Bank of tubes | Grimison: Nu = 1.13 C1 (ReD,max)m Pr1/3, × C2 for fewer than 10 rows | 2000 ≤ ReD,max ≤ 40 000, Pr ≥ 0.7 |
| Circular tube, laminar | Nu = 3.66 (uniform Ts) or 4.36 (uniform q″) when fully developed; Sieder–Tate or Hausen with entry length | Re ≤ 2300 |
| Tube or duct, turbulent | Dittus–Boelter: Nu = 0.023 Re4/5 Prn, n = 0.4 heating, 0.3 cooling; Sieder–Tate: 0.027 Re4/5 Pr1/3 (μ/μs)0.14 | Re ≥ 10 000, L/D ≥ 10; Pr 0.6–160 (Dittus–Boelter), 0.7–16 700 (Sieder–Tate) |
| Tube or duct, turbulent (wider range) | Gnielinski: Nu = (f/8)(Re − 1000) Pr / [1 + 12.7 (f/8)1/2(Pr2/3 − 1)], with f = (0.790 ln Re − 1.64)−2 | 3000 ≤ Re ≤ 5 × 106, 0.5 ≤ Pr ≤ 2000 |
| Non-circular duct, laminar | Fully developed Nu for the cross-section, from Incropera’s Table 8.1 | ReDh ≤ 2300 |
Hilpert’s constants, Grimison’s C1, m and C2, and the laminar duct values are looked up automatically. For tube pitches between the tabulated ones, the calculator takes the nearest tabulated point and says so; you can enter your own C1 and m instead.
Fluid properties and the film temperature
Fluid properties change with temperature, and the boundary layer spans the range between the wall and the free stream. For external flow, evaluate μ, ρ, cp and k at the film temperature Tf = (Ts + T∞)/2. For flow in tubes and ducts, use the mean (bulk) temperature Tm; the Sieder–Tate correlations add the viscosity μs at the wall temperature to account for the property change across the boundary layer.
The presets are air at 20 °C and 1 atm (μ = 1.82 × 10−5 Pa·s, ρ = 1.204 kg/m³, cp = 1004.5 J/(kg·K), k = 0.026 W/(m·K)) and water at 20 °C (μ = 1.002 × 10−3 Pa·s, ρ = 998.2 kg/m³, cp = 4182 J/(kg·K), k = 0.598 W/(m·K)). When the film temperature is far from 20 °C, enter properties from a table at that temperature.
Natural convection also needs the thermal expansion coefficient β. For an ideal gas β = 1/T, with T in kelvin at the film temperature; the button next to the field sets it for you. The air preset, 3.43 × 10−3 1/K, is close to 1/(293 K). For liquids, take β from a property table: water at 20 °C has about 2.07 × 10−4 1/K.
Worked example: a heated vertical plate
The calculator’s default case: a 0.5 m tall vertical plate at Ts = 330 K in still air at T∞ = 290 K, with the air preset at 20 °C. The film temperature is 310 K; with the properties at 310 K and β = 1/Tf, h comes out about 3 % lower (3.94 W/(m²·K)).
Pr = cpμ/k = 1004.5 × 1.82 × 10−5/0.026 = 0.703 (1)
Gr = gβ(Ts − T∞)L3/ν2 = 9.81 × 3.43 × 10−3 × 40 × 0.53/(1.51 × 10−5)2 = 7.36 × 108 (2)
With ν = μ/ρ = 1.51 × 10−5 m²/s.
Ra = Gr Pr = 5.18 × 108 (3)
Below 109, so the boundary layer is laminar.
Nu = 0.68 + 0.670 Ra1/4 / [1 + (0.492/Pr)9/16]4/9 = 78.2 (4)
The laminar form, which Incropera recommends for Ra ≤ 109. The all-Ra form gives 100.
h = Nu k/L = 78.2 × 0.026/0.5 = 4.06 W/(m²·K) (5)
q″ = h(Ts − T∞) = 4.06 × 40 = 163 W/m² (6)
Which form? Here the all-Ra form gives 28 % more than the laminar form (100 against 78.2). For Ra ≤ 109 Incropera recommends the laminar form for better accuracy, so the calculator’s “Recommended” setting uses it and shows the all-Ra value for comparison; above 109 it switches to the all-Ra form. Near Ra = 109 the two forms differ by about a third, so the recommended value steps there; for a sweep across 109, choose “All Ra”. The previous version of this page gave 15.2 for the laminar form, because it used Ra1/6 instead of Ra1/4.
Accuracy and limits
Convection correlations are fits to experiments on clean, simple geometries. Expect differences of 10–25 % against measurements: Incropera notes errors as large as 25 % for the Dittus–Boelter equation and recommends the Gnielinski correlation, which the calculator also offers, when you need better accuracy in tubes.
The correlations assume idealised conditions: a constant surface temperature or heat flux, a uniform approach flow and properties evaluated at one temperature. They do not capture surface roughness, free-stream turbulence, entry effects beyond the cases listed, mixed convection (Richardson numbers between about 0.1 and 10) or interaction with nearby surfaces.
When the geometry or the flow departs from these cases, a conjugate heat transfer simulation gives local coefficients and temperatures instead of one average. The correlations remain useful as a first estimate and as a check on the simulation.
Using the result in Simcenter STAR-CCM+
As a boundary condition. On a wall that stands for a surface outside your fluid domain, set the thermal specification to convection and enter the heat transfer coefficient and the ambient temperature from step 3. Simcenter STAR-CCM+ then computes the heat flux from the wall temperature it solves for.
As a check on a resolved simulation. When you mesh the fluid, compare the area-averaged wall heat transfer coefficient with the correlation, using the same reference temperature (T∞ or Tm) in both. Resolve the thermal boundary layer with a first cell at y+ ≈ 1 rather than relying on thermal wall functions; the y+ calculator sizes that cell. For liquids with Pr well above 1, such as water and oils, the conductive sublayer is thinner than the viscous one, roughly by Pr−1/3, so check that the first cells resolve it too.
References and changelog
References
F. P. Incropera, D. P. DeWitt, T. L. Bergman and A. S. Lavine, Fundamentals of Heat and Mass Transfer, 6th ed., Wiley, 2007: chapters 7 (external flow), 8 (internal flow) and 9 (free convection), including the Hilpert, Grimison and duct tables.
S. W. Churchill and H. H. S. Chu, “Correlating equations for laminar and turbulent free convection from a vertical plate”, Int. J. Heat Mass Transfer 18, 1323–1329, 1975, and “Correlating equations for laminar and turbulent free convection from a horizontal cylinder”, Int. J. Heat Mass Transfer 18, 1049–1053, 1975.
S. W. Churchill and M. Bernstein, “A correlating equation for forced convection from gases and liquids to a circular cylinder in crossflow”, J. Heat Transfer 99, 300–306, 1977.
R. Hilpert, “Wärmeabgabe von geheizten Drähten und Rohren im Luftstrom”, Forschung auf dem Gebiete des Ingenieurwesens 4, 215–224, 1933.
E. D. Grimison, “Correlation and utilization of new data on flow resistance and heat transfer for cross flow of gases over tube banks”, Trans. ASME 59, 583–594, 1937.
F. W. Dittus and L. M. K. Boelter, University of California Publications in Engineering 2, 443, 1930.
E. N. Sieder and G. E. Tate, “Heat transfer and pressure drop of liquids in tubes”, Ind. Eng. Chem. 28, 1429–1435, 1936.
H. Hausen, “Darstellung des Wärmeüberganges in Rohren durch verallgemeinerte Potenzbeziehungen”, Z. VDI Beiheft Verfahrenstechnik 4, 91–98, 1943.
V. Gnielinski, “New equations for heat and mass transfer in turbulent pipe and channel flow”, Int. Chem. Eng. 16, 359–368, 1976.
Review
Reviewed by Christoffer Johansson, M.Sc., Senior Application Specialist at Volupe. Version 2.0.
Changelog
October 2026: rebuilt as a live calculator. Automated checks confirm the same results as the previous version, except for these fixes:
– The laminar Churchill and Chu form for vertical and inclined plates now uses Ra1/4; the old Ra1/6 gave values three to five times too low.
– The hydraulic diameter of a non-circular duct can be entered; the old field was locked, so the turbulent duct result was always 0.
– For uniform heat flux with a turbulent boundary layer, the result is the turbulent local value; the old page showed the laminar average.
– Grimison uses the Reynolds number at the maximum velocity between the tubes, and Hilpert picks its constants from the Reynolds number.
New: heat transfer coefficient and heat flux, validity notes, the Gnielinski correlation, a mixed boundary layer, laminar duct values, a water preset and Simcenter STAR-CCM+ settings.
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Reference
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Questions about the Nusselt number
Calculator reviewed by
Christoffer Johansson
Senior Application Specialist, M.Sc.
Questions about heat transfer? We reply within one working day.
Ask a technical question →
What is a typical Nusselt number?
It depends on the flow. Pure conduction gives Nu = 1. Natural convection on a 0.5 m plate in air gives about 80–100, air at 5 m/s along a 1 m plate about 340, and turbulent water flow in a pipe several hundred. Nu grows roughly with Re1/2 to Re4/5 in forced convection and with Ra1/4 to Ra1/3 in natural convection.
How do I get the heat transfer coefficient from the Nusselt number?
Use h = Nu k/L, with the thermal conductivity k of the fluid and the same characteristic length L the correlation uses: the plate length or height, the diameter, or the hydraulic diameter. The calculator shows h next to Nu and lets you copy it.
At which temperature should I evaluate the fluid properties?
For external flow, at the film temperature (Ts + T∞)/2. For flow in tubes and ducts, at the mean fluid temperature Tm, with the wall viscosity μs at Ts where Sieder–Tate asks for it. Step 2 of the calculator shows the temperature to use.
Is my flow laminar or turbulent?
On a flat plate, transition starts near Rex ≈ 5 × 105. In a pipe, flow is laminar below Re ≈ 2300 and usually turbulent above about 4000; Dittus–Boelter and Sieder–Tate need Re ≥ 10 000, while Gnielinski covers 3000 to 5 × 106. On a vertical plate in natural convection, the boundary layer turns turbulent near Ra ≈ 109. The calculator warns you when the regime you picked does not match these values.
How accurate are Nusselt number correlations?
Expect 10–25 % against measurements on the idealised geometry each correlation was fitted to; Incropera notes errors as large as 25 % for Dittus–Boelter. Real parts add roughness, entry effects, free-stream turbulence and neighbouring surfaces, so treat the result as an estimate.
Can I use the result in a CFD simulation?
Yes, in two ways: as a convection boundary condition on walls you do not mesh, with the heat transfer coefficient and the ambient temperature, and as a sanity check on a simulation that resolves the boundary layer. For the check, compare area averages and use the same reference temperature.
Can Volupe help me set up heat transfer in Simcenter STAR-CCM+?
Yes. Our application engineers help with conjugate heat transfer, wall treatment for thermal boundary layers and validation against correlations or measurements. Ask a technical question and we reply within one working day.
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