Convection is heat carried between a surface and a moving fluid. Newton’s law of cooling, q = h·A·(T_s − T∞), lumps everything about the flow into one number, the heat transfer coefficient h. It is not a property of the fluid: it depends on the fluid, its speed, the geometry and whether the flow is laminar or turbulent, and it is found from experiments or from Nusselt-number correlations.
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Newton’s law of cooling
Right at a wall the fluid is still, so heat crosses the first thin layer by conduction. The moving fluid beyond it sweeps that heat away, and the faster and more turbulent the flow, the thinner the layer and the larger h. Newton’s law of cooling is the definition of h that wraps this up: q = h·A·(T_s − T∞).
The range is wide. Free convection in still air gives h of a few W/(m²·K) up to about 25; forced air, roughly 10 to a few hundred; forced water, hundreds to tens of thousands; boiling and condensation, higher still. As a thermal resistance, R = 1/(h·A), it adds in series with conduction through a plane wall.
Finding h from correlations
Correlations give h through the Nusselt number, Nu = h·L/k, a dimensionless ratio of convection to conduction in the fluid. For forced convection, Nu depends on the Reynolds number, Re = V·L/ν, and the Prandtl number Pr, a fluid property. For free convection, the Rayleigh number replaces Re.
Each correlation states the geometry, flow regime and range it was fitted for. Laminar flow along a flat plate uses Nu = 0.664·Re^½·Pr^⅓ for Re below about 5 × 10⁵. Fully developed turbulent flow in a tube uses Dittus–Boelter, Nu = 0.023·Re^0.8·Pr^n. Fluid properties are evaluated at the film temperature (T_s + T∞)/2 for external flow, or at the mean fluid temperature for flow in a tube. Expect correlations to be accurate to roughly ±10–25%.
Key formulas
Formulas used on this page, with the textbook each is cited from
Formula
Expression
Where
Cited from
Newton’s law of cooling
q = h·A·(T_s − T∞)
h = heat transfer coefficient (W/(m²·K)), A = surface area (m²), T_s = surface, T∞ = fluid temperature
Incropera et al., Fundamentals of Heat and Mass Transfer
Nusselt and Reynolds numbers
Nu = h·L/k, Re = V·L/ν
L = characteristic length (m), k = fluid conductivity (W/(m·K)), ν = kinematic viscosity (m²/s)
Incropera et al., Fundamentals of Heat and Mass Transfer
Laminar flat plate, average
Nu_L = 0.664·Re_L^½·Pr^⅓
Re_L < 5 × 10⁵, Pr ≥ 0.6, properties at film temperature
Incropera et al., Fundamentals of Heat and Mass Transfer
Dittus–Boelter (turbulent tube flow)
Nu_D = 0.023·Re_D^0.8·Pr^n
n = 0.4 heating the fluid, 0.3 cooling; Re_D > 10 000, 0.6 ≤ Pr ≤ 160
Çengel & Ghajar, Heat and Mass Transfer
Worked examples
Example: Measuring h on a heated test plate
Given
A fan blows room air at 22 °C across a 0.2 m × 0.3 m aluminium plate with an electric heater bonded to its back.
The back and edges are insulated. At steady state the heater draws 150 W and the exposed face reads 78 °C. Radiation from the polished face is neglected.
Find
The average convection heat transfer coefficient.
Solution
At steady state, all heater power leaves by convection from the one exposed face.q = 150 W, A = 0.2 × 0.3 = 0.06 m²
Temperature difference between surface and air.T_s − T∞ = 78 − 22 = 56 K
Rearrange Newton’s law of cooling.h = q / (A·ΔT) = 150 / (0.06 × 56) = 44.6 W/(m²·K)
Answer: h ≈ 45 W/(m²·K), in the expected range for fan-driven air. This is how h is measured in practice; the uncertainty is mostly in the heat lost through the "insulated" back and by radiation.
Example: Forced air over a warm flat plate
Given
Air at 25 °C flows at 4 m/s along a 0.5 m long, 0.3 m wide flat plate held at 75 °C.
At the 50 °C film temperature, take ν = 18.2 × 10⁻⁶ m²/s, k = 0.0280 W/(m·K) and Pr = 0.704, interpolated from standard property tables for air at 1 atm.
Find
Whether the boundary layer stays laminar.
The average heat transfer coefficient.
The heat lost from one face.
Solution
Reynolds number at the trailing edge. It is below 5 × 10⁵, so the flow is laminar over the whole plate.Re_L = 4 × 0.5 / 18.2 × 10⁻⁶ = 110 000
Average Nusselt number from the laminar flat-plate correlation.Nu_L = 0.664 × 110 000^½ × 0.704^⅓ = 196
Heat rate from one face.q = 11.0 × (0.5 × 0.3) × (75 − 25) = 82 W
Answer: The flow stays laminar (Re ≈ 1.1 × 10⁵), h ≈ 11 W/(m²·K), and one face loses about 82 W. That is a quarter of the value measured in the first example, because the fan there makes the flow much more disturbed than a smooth 4 m/s stream.
Common mistakes
Treating h as a fluid property. h depends on geometry, speed and flow regime as well as the fluid. Find it for the situation, not from a table of fluids.
Using a correlation outside its range. Check Re, Pr and the geometry against the correlation’s stated limits before using it.
Using the solid’s k in the Nusselt number. Nu = h·L/k uses the conductivity of the fluid, not the wall. The Biot number is the one that uses the solid.
Evaluating properties at the wrong temperature. Use the film temperature for external flow and the bulk mean temperature for internal flow.
Ignoring radiation from hot surfaces. Near room temperature in still air, radiation can be as large as free convection. Add it unless the surface has a low emissivity.
In industry
Frequently asked questions
What is a typical heat transfer coefficient for air?
Roughly 2–25 W/(m²·K) for free convection and 25–250 W/(m²·K) for forced convection. Correlations give the value for a specific case.
What is the difference between free and forced convection?
In forced convection a fan, pump or wind drives the flow. In free (natural) convection, buoyancy drives it: fluid near a hot surface becomes lighter and rises.
What is the Nusselt number?
The ratio of convective to conductive heat transfer across the fluid layer at a surface, Nu = h·L/k. Nu = 1 would mean the fluid only conducts.
How is the heat transfer coefficient different from the overall coefficient U?
h covers one fluid-to-surface film. U combines every resistance between two fluids, including both films and the wall.