Newton's Law of Cooling

A compact, empirical model for convection at a surface–fluid interface.

foundationsheat transferconvection

⏱️ About 18 min

Why can the same hot surface cool slowly in still air but extremely fast in flowing water? Newton’s law of cooling captures that difference in one coefficient: h.

💡

The big idea: Convection heat transfer is often modeled with Newton’s law of cooling, where the heat flux is proportional to a temperature difference and an empirical coefficient h that depends on flow and geometry.

🎯 By the end, you'll be able to

Write Newton’s law of cooling for heat flux and heat rate
Explain why h is not a material property
Estimate order-of-magnitude h values for common situations
Compute a convection heat rate from h, area, and temperatures

📎 Helpful to know first

Transient Conduction & Lumped-Capacitance Analysis

What convection does (and why we use a coefficient)

Convection is heat transfer between a surface and a moving (or buoyancy-driven) fluid. Unlike pure conduction inside a solid, convection depends strongly on the flow near the surface (mixing, turbulence, boundary layers) and on geometry (plates, pipes, fins, cylinders).

Because the detailed fluid mechanics can be complex, we often represent convection with an empirical coefficient h, measured or estimated from correlations and experiments.

\[ q'' = h\,(T_s - T_\infty) \]

Newton’s law of cooling (heat flux form).

\[ \dot{Q} = h\,A\,(T_s - T_\infty) \]

Newton’s law of cooling (heat rate form).

🔑 What is h?

The convection coefficient h (W/(m²·K)) is an effective measure of how easily heat crosses the thin near-wall region of fluid. It is not a material property like k (thermal conductivity). It changes with velocity, turbulence level, surface roughness, orientation (natural convection), and characteristic length.

Typical h ranges (order of magnitude)

These values are rough, but useful for quick estimates:

Air: free convection ≈ 5–25 W/(m²·K); forced convection ≈ 10–200 W/(m²·K).

Water: forced convection ≈ 500–10,000 W/(m²·K).

The big jump from air to water largely reflects water’s higher thermal conductivity and the fact that liquid flows can sustain very thin thermal boundary layers at modest velocities.

✨ Sign convention and interpretation

If the surface is hotter than the fluid (T_s > T_∞), then q'' is positive in the direction from the surface to the fluid: heat leaves the surface. If the surface is colder, the sign flips and the fluid warms the surface.

From heat flux to heat rate

Heat flux q'' has units of W/m². Multiply by area A to obtain heat rate Q̇ (W). For nonuniform surfaces, h and temperature can vary with position, but this lesson uses the common uniform/average approximation.

📝 Worked example: A flat electronic enclosure wall (area A = 0.30 m²) is at T_s = 60°C in air at T_∞ = 25°C. A fan produces an average convection coefficient h = 35 W/(m²·K). Estimate the convective heat loss rate.

Use Newton’s law of cooling: Q̇ = h A (T_s − T_∞).
Compute temperature difference: ΔT = 60 − 25 = 35 K.
Compute: Q̇ = (35 W/(m²·K)) (0.30 m²) (35 K) = 367.5 W.

✓ 368 W

✏️ Practice: A warm plate of area 0.20 m² is at 50°C in a room at 20°C. If h = 12 W/(m²·K), what is the convective heat transfer rate from the plate?

Solution

Newton’s law: Q̇ = h A (T_s − T_∞).
ΔT = 50 − 20 = 30 K.
Q̇ = 12 × 0.20 × 30 = 72 W.

Check your understanding

1. In Newton’s law of cooling, which statement about h is most accurate?

h is an empirical coefficient capturing near-wall transport; it varies with flow regime, velocity, geometry, and fluid properties.

2. If T_s < T_∞, then Newton’s law q'' = h(T_s − T_∞) gives:

A negative (T_s − T_∞) indicates heat transfer from the hotter fluid to the cooler surface.

3. A typical range for forced convection of water is closest to:

Forced convection in water commonly yields h in the hundreds to thousands W/(m²·K).

✅ Key takeaways

Newton’s law of cooling models convection as q'' = h(T_s − T_∞).
The coefficient h is empirical and depends on flow, geometry, and fluid properties.
Typical h values are much larger for forced convection and for liquids like water.
Heat rate follows Q̇ = hAΔT for a uniform average approximation.

➡️ To understand what sets h, we next look at the near-wall flow structure that controls transport: the boundary layer.

Want to test yourself on this? Try the Chemical Aptitude test →