Convection and Heat Transfer Coefficients

Newton’s law of cooling, Reynolds, Prandtl and Nusselt numbers, forced flow in pipes and over surfaces, and natural convection.

On this page

  1. Newton’s law of cooling
  2. Reynolds, Prandtl and Nusselt numbers
  3. Forced convection in a pipe
  4. Forced convection over surfaces
  5. Natural convection
  6. Typical values and boiling and condensation
  7. FAQ

Newton’s law of cooling

When a fluid moves past a surface at a different temperature, heat flows between them at a rate set by the convection coefficient h. It lumps together all the complicated physics of the thin boundary layer where the fluid’s speed and temperature change.

Q̇ = h A (Ts − T∞)

h is not a material property. It depends on the fluid, its speed, the shape and size of the surface and whether the flow is laminar or turbulent. For a real problem you calculate it from a correlation for the Nusselt number, or take it from a table of typical values.

Reynolds, Prandtl and Nusselt numbers

  • Reynolds number Re = ρ v L ÷ μ: the ratio of inertia to viscosity. It decides whether the flow is laminar or turbulent (the same Re as in Continuity, Bernoulli and the Energy Equation).
  • Prandtl number Pr = μ cp ÷ k: how quickly momentum spreads compared with heat, a property of the fluid.
  • Nusselt number Nu = h L ÷ k: the dimensionless coefficient. It says how much better convection is than pure conduction through the same fluid layer.
FluidPrandtl number
Air (20 °C)0.71
Water (20 °C)7.0
Water (80 °C)2.2
Engine oil (40 °C)about 1 000
Liquid metals0.01 to 0.03

Correlations have the form Nu = C Rem Prn: find Re and Pr, calculate Nu, and then h = Nu k ÷ L. The fluid properties are taken at the film temperature (the average of surface and fluid) or the bulk temperature, as the correlation states.

Forced convection in a pipe

For fully developed turbulent flow in a smooth pipe (Re above 10 000, 0.6 < Pr < 160, length over 10 diameters) the Dittus–Boelter equation is the standard:

Nu = 0.023 Re0.8 Prn

Worked example: water in the DN80 line

  • The 10 L/s in the 77.9 mm pipe (from Pipe Friction) has Re = 162 784; water at 20 °C has Pr = 7.01 and k = 0.597 W/m·K.
  • Heating the water: Nu = 0.023 × 162 7840.8 × 7.010.4 = 740, so h = Nu k ÷ D = 5 673 W/m²·K.
  • With the pipe wall 20 K hotter than the water, each meter transfers h π D ΔT = 27.77 kW. At half the flow, h falls with v0.8 to about 3 258 W/m²·K.

Worked example: air in a duct

  • 5 m/s of air at 20 °C in a 200 mm duct: Re = 66 412, Pr = 0.71, k = 0.0258 W/m·K.
  • Nu = 0.023 × Re0.8 × Pr0.4 = 144, so h = 18.6 W/m²·K: 304 times lower than water in a pipe of similar size, which is why air-cooled heat exchangers are so much bigger than water-cooled ones.

Forced convection over surfaces

  • Flat plate, laminar (ReL below 5 × 10⁵): NuL = 0.664 ReL½ Pr⅓ (average over the plate length L). Turbulent: NuL = 0.037 ReL0.8 Pr⅓.
  • Cylinder in crossflow (Churchill–Bernstein): NuD = 0.3 + 0.62 Re½ Pr⅓ ÷ [1 + (0.4 ÷ Pr)⅔]¼ × [1 + (Re ÷ 282 000)⅝]⅘, valid for Re Pr above 0.2.
  • Sphere, tube banks, impinging jets have their own correlations in the heat transfer texts; use the one for your geometry.

Worked example: a hot plate and a hot rod in a fan’s air stream

  • Plate: air at 3 m/s over a 0.5 m × 0.5 m plate (flow along 0.5 m) at 60 °C in 20 °C air. ReL = 99 619 (laminar), Nu = 0.664 × 316 × 0.89 = 187, h = 9.6 W/m²·K. One side loses h A ΔT = 96 W.
  • Cylinder: a 25 mm rod in air at 5 m/s: Re = 8 302, Nu = 48.3, h = 49.9 W/m²·K.
  • In both cases h is about 15 to 30 W/m²·K: typical for forced air at a few m/s. To raise it, increase the speed (h grows roughly as v0.5 to v0.8) or use a better fluid.

Natural convection

With no fan, warm fluid next to a hot surface is lighter and rises, drawing cooler fluid in: natural (free) convection. The driving number is the Rayleigh number, Ra = g β ΔT L³ ÷ (ν α), with β = 1 ÷ T for a gas. For a vertical plate over the whole range the Churchill–Chu equation works:

NuL = { 0.825 + 0.387 Ra1/6 ÷ [1 + (0.492 ÷ Pr)9/16]8/27 }²

Worked example: a vertical panel in still air

  • A 0.3 m high panel at 50 °C in 20 °C air (film temperature 35 °C; ν = 16.45 mm²/s, α = 23.29 mm²/s, Pr = 0.71).
  • Ra = g β ΔT L³ ÷ (ν α) = 9.807 × (1 ÷ 308.15) × 30 × 0.3³ ÷ (0.00001645 × 0.00002329) = 67 297 071.
  • Nu = 54.2 and h = Nu k ÷ L = 4.9 W/m²·K: the usual 3 to 8 W/m²·K for air at modest temperature differences. Radiation adds about the same again (see Thermal Radiation), so total h is about 10 to 12.

Typical values and boiling and condensation

Typical convection coefficients h (W/m² K)1101001 00010 000100 000Natural convection, gases2 to 25Forced convection, gases25 to 250Natural convection, water50 to 1 000Forced convection, water500 to 10 000Forced convection, oils50 to 2 000Boiling water2 500 to 100 000Condensing steam5 000 to 100 000Typical ranges for design. The coefficient depends on the fluid, the speed, the geometry and the temperaturedifference: calculate it from a correlation when it matters.
Typical heat transfer coefficients. Boiling and condensing fluids transfer heat extremely well because of latent heat.

Boiling and condensation involve a change of phase, which moves a great deal of energy (the latent heat) at nearly constant temperature, so h reaches 5 000 to 100 000 W/m²·K. Nucleate boiling on a heated surface follows a curve with a peak (the critical heat flux, about 1.1 MW/m² for water at atmospheric pressure) beyond which a vapor film forms and the surface temperature jumps: the “burnout” that destroys boilers and heaters. Condensing steam on a clean tube is dominated by the film of condensate.

Related

FAQ

Why is h for air so much lower than for water?

Air has a low conductivity (0.026 against 0.6 W/m·K), a low density and a low specific heat, so it carries less heat away for the same speed. Water in a pipe typically reaches 1 000 to 10 000 W/m²·K, air 10 to 100.

Does a faster flow always help?

Up to a point: h rises roughly as v⁰·⁸ in turbulent pipe flow but the pressure drop and pumping power rise as v² to v³. For electronics cooling, the fan noise limits the speed.

Where do I take the fluid properties?

At the film temperature for external flow and natural convection, and at the bulk (mean) temperature for flow in a pipe, as the correlation states. For most air and water work the variation is small.

How accurate are these correlations?

Typically ±20 percent, sometimes ±30. That is enough for sizing, but add margin, and test or measure where a temperature limit is critical.

References

  • Incropera, F. P., et al., Fundamentals of Heat and Mass Transfer, Wiley (Dittus–Boelter, flat plate, Churchill–Bernstein, Churchill–Chu).
  • 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, 1975; Churchill, S. W. and Bernstein, M., J. Heat Transfer, 1977.
  • Dittus, F. W. and Boelter, L. M. K., Univ. of California Publications in Engineering, 1930.
  • Çengel, Y. A. and Ghajar, A. J., Heat and Mass Transfer, McGraw-Hill; VDI Heat Atlas.

Last reviewed: October 2026. Correlations are accurate to about 20 percent.