Heat transfer coefficient correlations
Nusselt correlations for natural and forced convection on plates, cylinders, tube banks, pipes and ducts, with validity ranges.
The Nusselt number
The Nusselt number, Nu = hL/k, tells you how much convection enhances heat transfer compared with conduction alone. Here h is the convective heat transfer coefficient, L a characteristic length and k the thermal conductivity of the fluid. CFD engineers use it to estimate h, either as a boundary condition for a heated or cooled flow or to check simulation results. This page collects widely used empirical correlations for Nu, with their validity ranges.
Calculate it: the Nusselt number calculator implements the correlations on this page.
In equation form:
Nu = hL / k (1)
Most correlations give the average Nusselt number Nu, defined in the same way with the average coefficient h; some also give the local value at a position on the geometry.
The other inputs are usually the Reynolds number Re, the Prandtl number Pr, the Rayleigh number Ra and the Grashof number Gr. The dimensionless numbers page explains them.
Natural convection
These correlations apply to purely buoyancy-driven (free) convection, where the Nusselt number is a function of the Rayleigh and Prandtl numbers:
Nu = f(Ra, Pr) (2)
The Rayleigh number is the product of the Grashof and Prandtl numbers, Ra = Gr Pr. The Grashof number relates buoyant to viscous forces:
Grx = gβ(Ts − T∞)x3 / ν2 (3)
where g is the gravitational acceleration, β the thermal expansion coefficient, Ts the surface temperature, T∞ the bulk temperature, x the characteristic length and ν the kinematic viscosity. The Prandtl number relates momentum diffusivity to thermal diffusivity:
Pr = ν / α = cpμ / k (4)
with the thermal diffusivity α = k/(ρcp) and the dynamic viscosity μ. The Rayleigh number indicates whether natural convection occurs: below a critical value there is no flow and heat moves by conduction alone; above it, convection dominates. In most engineering problems the critical value lies between 106 and 108.
Vertical surface
A hot vertical surface (Ts > T∞) develops a free-convection boundary layer, as the heated, lighter fluid near the wall rises against gravity (Figure 1).
Churchill and Chu proposed this correlation for the entire range of Ra:
NuL = {0.825 + 0.387 RaL1/6 / [1 + (0.492/Pr)9/16]8/27}2 (5)
For purely laminar flow, typically RaL ≲ 109 on vertical walls, this form can be slightly more accurate:
NuL = 0.68 + 0.67 RaL1/4 / [1 + (0.492/Pr)9/16]4/9 (6)
RaL ≲ 109
Equations 5 and 6 were developed for isothermal surfaces but also work well for a uniform heat flux, where Ts varies along the wall. In that case base NuL and RaL on the surface temperature at mid-height, Ts(L/2).
Inclined surface
On an inclined surface, gravity is no longer parallel to the wall, so buoyancy acts both along and normal to it. The boundary layer differs between the two sides and depends on whether the plate is hot or cold relative to the surroundings (Figure 2: a) hot plate, b) cold plate).
The boundary layer is sensitive to the angle, but for inclinations θ ≤ 60° from the vertical, Equations 5 and 6 can be used with g replaced by g cos θ.
Horizontal surface
On a horizontal surface, buoyancy acts only normal to the wall. As for the inclined plate, the boundary layer differs between the two sides and depends on whether the plate is hot or cold (Figure 3: a) hot plates, b) cold plates).
Several correlations exist; these are the ones suggested by McAdams.
Upper surface of a hot plate, or lower surface of a cold plate:
NuL = 0.54 RaL1/4 (7)
104 ≲ RaL ≲ 107
NuL = 0.15 RaL1/3 (8)
107 ≲ RaL ≲ 1011
Lower surface of a hot plate, or upper surface of a cold plate:
NuL = 0.27 RaL1/4 (9)
105 ≲ RaL ≲ 1010
The best accuracy comes from defining the characteristic length as
L = As / P (10)
where As is the surface area and P its perimeter.
Long horizontal cylinder
Around a heated horizontal cylinder the boundary layer grows from the bottom (θ = 0) to the top (θ = 180°), where it forms a plume (Figure 4).
Of the many correlations, one of the most widely used is by Churchill and Chu, covering a wide range of Rayleigh numbers:
NuD = {0.60 + 0.387 RaD1/6 / [1 + (0.559/Pr)9/16]8/27}2 (11)
RaD ≲ 1012
Forced convection
These correlations apply where forced convection dominates the heat transfer and the boundary layer. The Nusselt number is then a function of the Reynolds and Prandtl numbers:
Nu = f(Re, Pr) (12)
Flat plate in parallel flow
Figure 5 shows a boundary layer growing from the leading edge of a flat plate in parallel flow.
Isothermal plate. For steady, incompressible, laminar flow with constant properties, no viscous dissipation and dp/dx = 0, the local Nusselt number at a point x ≤ xc is
Nux = 0.332 Rex1/2 Pr1/3 (13)
Pr ≳ 0.6
Under the same conditions, the average Nusselt number up to x ≤ xc is twice the local value:
Nux = 2Nux = 0.664 Rex1/2 Pr1/3 (14)
Pr ≳ 0.6
Once the flow has turned turbulent, the local Nusselt number is
Nux = 0.0296 Rex4/5 Pr1/3 (15)
0.6 < Pr < 60
For mixed laminar and turbulent boundary layer conditions, the average Nusselt number can be approximated by
Nux = [0.664 Rex,c1/2 + 0.037(Rex4/5 − Rex,c4/5)] Pr1/3 (16)
where Rex,c is the critical Reynolds number at transition. When L ≫ xc, the laminar part is negligible and Equation 16 simplifies to
Nux = 0.037 Rex4/5 Pr1/3 (17)
Equation 17 is also a good approximation when the boundary layer is turbulent over the whole plate.
Plate with uniform heat flux. For a laminar flow, the local Nusselt number is approximately
Nux = 0.453 Rex1/2 Pr1/3 (18)
Pr ≳ 0.6
For a turbulent flow it is approximately
Nux = 0.0308 Rex4/5 Pr1/3 (19)
0.6 ≲ Pr ≲ 60
The average Nusselt number over a plate with laminar flow and uniform heat flux is
Nux = 0.68 Rex1/2 Pr1/3 (20)
Cylinder in cross flow
Another common case is a single long cylinder in cross flow (Figure 6).
Churchill and Bernstein proposed a correlation for the average Nusselt number that covers wide ranges of ReD and Pr. It is valid for ReD Pr > 0.2:
NuD = 0.3 + 0.62 ReD1/2 Pr1/3 / [1 + (0.4/Pr)2/3]1/4 × [1 + (ReD/282 000)5/8]4/5 (21)
ReD Pr > 0.2
Other correlations use different expressions for different Reynolds numbers to account for the moving separation point. A widely used one is Hilpert’s:
NuD = C ReDm Pr1/3 (22)
Table 1 gives C and m. Evaluate all properties in Equation 22 at the film temperature Tf = (Ts + T∞)/2.
Table 1: constants for Equation 22 (Hilpert).
| ReD | C | m |
|---|---|---|
| 0.4–4 | 0.989 | 0.330 |
| 4–40 | 0.911 | 0.385 |
| 40–4000 | 0.683 | 0.466 |
| 4000–40 000 | 0.193 | 0.618 |
| 40 000–400 000 | 0.027 | 0.805 |
Bank of tubes in cross flow
Many industrial heat exchangers, such as tube coolers and boiler heating coils, have bundles of tubes in cross flow, usually a) aligned or b) staggered (Figure 7).
The heat transfer coefficient on the first row is similar to that of a single cylinder. It increases deeper into the bundle and usually stabilises after four or five rows. For bundles of ten or more rows (NL ≥ 10), Grimison proposed a correlation for air, usually extended with a Prandtl number factor for other fluids:
NuD = 1.13 C1 ReD,maxm Pr1/3 (23)
NL ≥ 10, 2000 < ReD,max < 40 000, Pr ≥ 0.7
Evaluate the fluid properties at the film temperature Tf = (Ts + T∞)/2.
ReD,max is based on the maximum velocity inside the bundle. For aligned tubes it occurs at A1 and, for incompressible flow, is
Umax = ST / (ST − D) · U∞ (24)
For staggered tubes it occurs at A1 or A2. If the rows are spaced so that 2(SD − D) < (ST − D), it occurs at A2 and is
Umax = ST / [2(SD − D)] · U∞ (25)
Otherwise it occurs at A1 and Equation 24 applies. Table 2 gives C1 and m for a range of tube arrangements.
Table 2: constants C1 and m for Equation 23 (Grimison).
| SL/D | ST/D = 1.25 | ST/D = 1.50 | ST/D = 2.00 | ST/D = 3.00 | ||||
|---|---|---|---|---|---|---|---|---|
| C1 | m | C1 | m | C1 | m | C1 | m | |
| Aligned | ||||||||
| 1.25 | 0.348 | 0.592 | 0.275 | 0.608 | 0.100 | 0.704 | 0.063 | 0.752 |
| 1.50 | 0.367 | 0.586 | 0.250 | 0.620 | 0.101 | 0.702 | 0.068 | 0.744 |
| 2.00 | 0.418 | 0.570 | 0.299 | 0.602 | 0.229 | 0.632 | 0.198 | 0.648 |
| 3.00 | 0.290 | 0.601 | 0.357 | 0.584 | 0.374 | 0.581 | 0.286 | 0.608 |
| Staggered | ||||||||
| 0.600 | – | – | – | – | – | – | 0.213 | 0.636 |
| 0.900 | – | – | – | – | 0.446 | 0.571 | 0.401 | 0.581 |
| 1.000 | – | – | 0.497 | 0.558 | – | – | – | – |
| 1.125 | – | – | – | – | 0.478 | 0.565 | 0.518 | 0.560 |
| 1.250 | 0.518 | 0.556 | 0.505 | 0.554 | 0.519 | 0.556 | 0.522 | 0.562 |
| 1.500 | 0.451 | 0.568 | 0.460 | 0.562 | 0.452 | 0.568 | 0.488 | 0.568 |
| 2.000 | 0.404 | 0.572 | 0.416 | 0.568 | 0.482 | 0.556 | 0.449 | 0.570 |
| 3.000 | 0.310 | 0.592 | 0.356 | 0.580 | 0.440 | 0.562 | 0.428 | 0.574 |
For fewer than ten rows, correct the average Nusselt number from Equation 23 with a factor C2:
NuD|NL < 10 = C2 NuD|NL ≥ 10 (26)
NuD|NL < 10 = 1.13 C1C2 ReD,maxm Pr1/3 (27)
Table 3 gives C2 for different numbers of rows.
Table 3: correction factor C2 for Equation 27, fewer than ten rows.
| Rows, NL | Aligned | Staggered |
|---|---|---|
| NL = 1 | 0.64 | 0.68 |
| NL = 2 | 0.80 | 0.75 |
| NL = 3 | 0.87 | 0.83 |
| NL = 4 | 0.90 | 0.89 |
| NL = 5 | 0.92 | 0.92 |
| NL = 6 | 0.94 | 0.95 |
| NL = 7 | 0.96 | 0.97 |
| NL = 8 | 0.98 | 0.98 |
| NL = 9 | 0.99 | 0.99 |
When the fluid temperature changes a lot through the bundle, |Tin − Tout| large, the Grimison correlation may not suffice. Then use the correlation by Zhukauskas, with all properties at the mean of the inlet and outlet temperatures; see the literature for details.
Flow inside a circular tube
Inside a tube, the Nusselt number depends on the flow regime and on the wall boundary condition. Figure 8 shows how the thermal boundary layer develops in a heated pipe with a) uniform surface temperature and b) uniform surface heat flux.
Laminar flow. In fully developed laminar flow the Nusselt number is constant, independent of ReD, Pr and position, but its value depends on the wall condition. The analytical local values are:
NuD = 3.66 (28)
Uniform surface temperature, Ts = constant
NuD = 4.36 (29)
Uniform surface heat flux, q″s = constant
In Equations 28 and 29, evaluate the thermal conductivity k at the local mean temperature
Tm = 2 / (umrmax2) ∫0rmax uTr dr (30)
where rmax is the inner tube radius and um the mean velocity, which for an incompressible fluid is
um = 2 / rmax2 ∫0rmax u(r, x) r dr (31)
For a uniform surface temperature with the velocity and thermal boundary layers developing together (a combined entry length), Sieder and Tate proposed for the average Nusselt number
NuD = 1.86 [ReD Pr / (L/D)]1/3 (μ/μs)0.14 (32)
Ts = constant, 0.48 < Pr < 16 700, 0.0044 < μ/μs < 9.75
This correlation is meant for tubes where the entry region matters. Whitaker defined that limit as
[ReD Pr / (L/D)]1/3 (μ/μs)0.14 ≳ 2
Below this limit the flow can be treated as fully developed and Equation 28 is a good approximation for the whole tube. For a tube with an unheated starting length, where the velocity profile develops before the thermal boundary layer (a thermal entry length), the average Nusselt number can be estimated with Hausen’s correlation:
NuD = 3.66 + 0.0668(D/L)ReDPr / {1 + 0.04[(D/L)ReDPr]2/3} (33)
Evaluate all properties in Equations 32 and 33, except μs, at the average mean fluid temperature Tm = (Tm,in + Tm,out)/2.
Turbulent flow. For small to moderate temperature differences Ts − Tm, the Dittus–Boelter equation is a common approximation:
NuD = 0.023 ReD4/5 Prn (34)
n = 0.4 for heating (Ts > Tm), 0.3 for cooling; 0.7 ≤ Pr ≤ 160, ReD ≳ 10 000, L/D ≳ 10
For large temperature variations, Sieder and Tate recommend
NuD = 0.027 ReD4/5 Pr1/3 (μ/μs)0.14 (35)
0.7 ≤ Pr ≤ 16 700, ReD ≳ 10 000, L/D ≳ 10
In Equations 34 and 35, evaluate all properties except μs at Tm. Both work for uniform surface temperature and for uniform heat flux. More accurate, but more complex, correlations exist, such as those of Petukhov and Gnielinski; see the literature.
Non-circular duct
For turbulent flow, the circular-tube correlations also apply to non-circular ducts if the tube diameter is replaced by the hydraulic diameter
Dh = 4Ac / P (36)
where Ac is the flow cross-section and P the wetted perimeter. The heat transfer coefficient varies around the perimeter and approaches zero in sharp corners, so Equations 34 and 35 give the average around the periphery.
For fully developed laminar flow the Nusselt number can be found analytically, as for the circular tube, but it depends on the shape. Table 4 gives values for common cross-sections.
Table 4: Nusselt numbers for fully developed laminar flow in ducts, NuD = hDh/k.
| Cross section | b/a | Uniform q″s | Uniform Ts |
|---|---|---|---|
| Circle | – | 4.36 | 3.66 |
| Square | 1.00 | 3.61 | 2.98 |
| Rectangle | 1.43 | 3.73 | 3.08 |
| Rectangle | 2.00 | 4.12 | 3.39 |
| Rectangle | 3.00 | 4.79 | 3.96 |
| Rectangle | 4.00 | 5.33 | 4.44 |
| Rectangle | 8.00 | 6.49 | 5.60 |
| Parallel plates | ∞ | 8.23 | 7.54 |
| Parallel plates, one side insulated | ∞ | 5.39 | 4.86 |
| Equilateral triangle | – | 3.11 | 2.47 |
References and changelog
The correlations above are a selection of those in the literature. For details or alternatives, see the references.
References
[1] F. P. Incropera and D. P. DeWitt, Fundamentals of Heat and Mass Transfer, 5th ed., Wiley, 2002.
[2] J. H. Lienhard IV and J. H. Lienhard V, A Heat Transfer Textbook, 5th ed., 2020.
[3] Wikipedia, Nusselt number, accessed February 2023.
Author and review
Written by Johan Bernander, M.Sc., Application Specialist at Volupe. Reviewed by Christoffer Johansson, M.Sc., Senior Application Specialist.
Changelog
October 2026: new layout. The equations are now text instead of rendered images, and the constant tables (Hilpert, Grimison, row corrections, non-circular ducts) are real tables. The correlations are unchanged.
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What is the Nusselt number?
A dimensionless number, Nu = hL/k: the heat carried by convection divided by what conduction alone would carry across a fluid layer of thickness L. Nu = 1 therefore means conduction only. Values range from about 4 in laminar pipe flow to several thousand in strongly turbulent flow. Correlations give Nu from the Reynolds and Prandtl numbers (forced convection) or the Rayleigh number (natural convection), and from Nu you get h = Nu k / L.
What is a heat transfer coefficient correlation?
An empirical relation, fitted to experiments, that gives the Nusselt number for one geometry and flow regime. Forced convection correlations usually take the form Nu = C Rem Prn, natural convection ones Nu = C Ran. The heat transfer coefficient then follows from h = Nu k / L.
At which temperature should I evaluate the fluid properties?
For external flows, use the film temperature Tf = (Ts + T∞)/2 unless the correlation says otherwise. For flow in pipes and ducts, use the mean bulk temperature of the fluid. Each section above states the temperature its correlation was fitted with.
Is my case natural, forced or mixed convection?
Compare buoyancy with inertia through Gr/Re2. Well below 1, forced convection dominates; well above 1, natural convection dominates. Around 1 both matter, and neither type of correlation alone is reliable.
How accurate are these correlations?
Expect errors of up to about 20–25 % inside the stated ranges, and more outside them. Use the correlations for quick estimates, boundary conditions and sanity checks, not as a replacement for a validated simulation.
Can I use a correlation to check a CFD result?
Yes. Compute the average Nusselt number from the matching correlation and compare it with the simulation for the same geometry and conditions, using the same reference temperature and length. A large difference usually points to the near-wall mesh, the wall treatment or the turbulence model.
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