Conduction, convection, radiation — Thermodynamics & Fluids

What you'll learn

Compute conduction through a wall
Calculate convective heat transfer with Newton's law of cooling
Calculate net radiation from a surface using kelvin temperatures
Decide which heat transfer mode matters in a situation

Compute heat flow by conduction through a wall, convection from a surface and radiation to the surroundings.

Why it matters

Heat moves in three ways, and nearly every engineering heat problem mixes them: a furnace wall conducts heat, hot air convects it away, and a glowing element radiates it. Engineers size insulation, cooling fins, electronics heat sinks, kilns and building envelopes by putting numbers on these three modes. This lesson gives you one equation for each and shows when each one matters.

Conduction: heat through a solid

Heat diffuses through a material from hot to cold. For steady, one-dimensional conduction through a flat wall, Fourier's law gives:

q = k · A · (T_hot − T_cold) / L

q is the heat transfer rate in watts (W).
k is the thermal conductivity in W/(m·K): copper ≈ 400, steel ≈ 45–50, concrete ≈ 1.0–1.8, brick ≈ 0.7, glass ≈ 0.8–1.0, wood ≈ 0.12–0.16, mineral wool ≈ 0.035–0.04, still air ≈ 0.026.
A is the area (m²) and L the thickness (m).

A temperature difference in kelvin equals the same difference in °C, so you can use either for ΔT.

Worked example 1. A brick wall (k = 0.72 W/(m·K)) is 0.2 m thick and 10 m² in area. The inside face is at 20 °C and the outside face at 0 °C.

q = 0.72 × 10 × (20 − 0) / 0.2 = 720 W

Convection: heat carried by a moving fluid

When a fluid flows past a surface, it carries heat away (or brings it in). Newton's law of cooling wraps all the messy fluid behaviour into one coefficient h:

q = h · A · (T_surface − T_fluid)

h is the convection coefficient in W/(m²·K). Rough ranges: still air (free convection) 2–25, forced air 25–250, forced water 100–20,000, boiling or condensing water 2,500–100,000.

h is not a material property: it depends on the fluid, its speed, the geometry and whether the flow is laminar or turbulent. Engineers get it from correlations based on dimensionless numbers (Nusselt, Reynolds, Prandtl) that you meet in a full heat transfer course.

Worked example 2. A 2 m² hot plate at 80 °C sits in a fan-driven airflow at 20 °C with h = 25 W/(m²·K).

q = 25 × 2 × (80 − 20) = 3,000 W

Radiation: heat as electromagnetic waves

Every surface above absolute zero emits thermal radiation; no medium is needed (that is how the Sun heats the Earth). A surface exchanging radiation with large surroundings loses net heat:

q = ε · σ · A · (T_s⁴ − T_surr⁴)

σ = 5.67 × 10⁻⁸ W/(m²·K⁴), the Stefan–Boltzmann constant.
ε is the emissivity, between 0 and 1: polished aluminium ≈ 0.05, oxidised steel ≈ 0.8, paint, brick and skin ≈ 0.9.
Temperatures must be absolute (kelvin), because of the fourth power.

Worked example 3. A 1 m² painted surface (ε = 0.9) at 350 K faces surroundings at 300 K.

q = 0.9 × 5.67e-8 × 1 × (350⁴ − 300⁴) ≈ 352 W

Radiation grows very fast with temperature: at 1,000 K the same surface would lose over 50 kW per square metre. At room temperature it is often comparable to free convection, so do not ignore it.

Which mode matters?

Inside a solid: conduction only.
Surface to a moving fluid: convection, plus radiation if the surface is hot or the convection is weak.
Across a vacuum (spacecraft, vacuum flasks): radiation only.
Most real problems are in series: inside air → wall → outside air. The next lesson handles that with thermal resistances.

Common mistakes

Using °C in the radiation equation. Use kelvin: 77 °C is 350.15 K.
Mixing up k and h. k is in W/(m·K) and belongs to a material; h is in W/(m²·K) and belongs to a flow situation.
Using mm for thickness with k in W/(m·K). Convert 200 mm to 0.2 m.
Forgetting that rates are in watts. Energy over time is rate × time: 720 W for 24 h is 17.3 kWh.

Recap

Conduction: q = k A ΔT / L (Fourier).
Convection: q = h A ΔT (Newton).
Radiation: q = ε σ A (T_s⁴ − T_surr⁴), in kelvin (Stefan–Boltzmann).
Use real property data from handbooks or manufacturers for real work; these values are typical, not design values.

Try it

On the Do tab, write Fourier's law in Python and compare conduction losses through different wall materials.

Key terms (6)

Fourier's law: For steady one-dimensional conduction through a flat wall, q = k·A·(T_hot − T_cold)/L.
thermal conductivity: k, a material property in W/(m·K) that measures how readily heat conducts through it.
Newton's law of cooling: q = h·A·(T_surface − T_fluid), the convective heat transfer rate between a surface and a fluid.
convection coefficient: h, in W/(m²·K); it depends on the fluid, its speed, geometry and flow regime, not just on a material.
emissivity: ε, between 0 and 1, the fraction of ideal blackbody radiation a surface emits at the same temperature.
Stefan–Boltzmann constant: σ = 5.67 × 10⁻⁸ W/(m²·K⁴), the constant in the radiation equation.