Heat Transfer: Conduction, Convection & Radiation

Touch a metal spoon left in hot soup and burn your finger. Stand near a fire and feel warmth across open air. See the shimmer above a hot road on a summer day. Three different physical mechanisms, three different names — conduction, radiation, convection — but one underlying principle: energy moving from where it's concentrated to where it isn't, because that's the direction entropy increases.

Each mechanism has its own equation and its own conditions. Conduction dominates in solids; convection in fluids; radiation is the only one that works through a vacuum (which is how the Sun's energy reaches Earth). In most real engineering problems — insulating a building, designing a heat sink, calculating heat loss — you need to know which mechanism dominates and which you can safely ignore.

Conduction: Fourier's Law

In conduction, faster-moving (hotter) molecules collide with slower-moving (cooler) neighbours, transferring kinetic energy without bulk material flow. The rate of heat flow by conduction through a material is:

where Q/t is the rate of heat transfer (W), k is the thermal conductivity (W/m·K), A is the cross-sectional area (m²), ΔT is the temperature difference (K or °C), and d is the thickness (m). This is Fourier's Law of heat conduction.

Materials with high k are good conductors (metals); materials with low k are good insulators:

Still air is one of the best thermal insulators (k = 0.025 W/m·K) — the basis of double glazing (trapped air layer), wool clothing (traps air between fibres), and cavity wall insulation. Metals conduct heat so well because free electrons carry energy as well as charge — the same electrons responsible for electrical conductivity explain why good electrical conductors are also good thermal conductors (Wiedemann-Franz Law).

Worked example: Heat loss through a window

A single-pane glass window (k = 1.0 W/m·K, thickness 4 mm = 0.004 m, area 1.5 m²) has indoor temperature 20°C and outdoor −5°C (ΔT = 25 K).

Nearly 10 kW through one window — which is why double glazing (with an air gap) reduces this dramatically. With a 16 mm air gap: Q/t = 0.025 × 1.5 × 25 / 0.016 = 58 W — 160× less heat loss.

Convection: Bulk Fluid Movement

Convection transfers heat by the mass movement of a fluid carrying thermal energy. Two types:

Natural (free) convection: driven by buoyancy. Hot fluid expands, becomes less dense, rises. Cool fluid sinks to replace it, creating convection currents. Examples: hot air rising above a radiator, ocean thermohaline circulation, atmospheric weather patterns, Earth's mantle convection driving tectonic plates.

Forced convection: a pump, fan, or external pressure drives fluid movement, enhancing heat transfer far beyond natural convection rates. Examples: car cooling systems (water pump circulating coolant), fan-assisted ovens (even heat distribution), heat exchangers in power stations.

Newton's Law of Cooling approximates convective heat loss from an object at temperature T to surroundings at T_∞:

where h is the convective heat transfer coefficient (W/m²·K) — which depends strongly on fluid velocity, fluid properties, and surface geometry. Typical values: natural convection in air h ≈ 5–25 W/m²·K; forced convection in water h ≈ 500–10,000 W/m²·K. This is why blowing on hot food cools it fast — forced convection dramatically increases h.

Radiation: Stefan-Boltzmann Law

All objects at temperatures above absolute zero emit electromagnetic radiation — primarily infrared for everyday temperatures. This radiation requires no medium and travels at the speed of light. The total power emitted by a surface:

where ε is the emissivity (0 ≤ ε ≤ 1, dimensionless), σ = 5.67 × 10⁻⁸ W/m²·K⁴ is the Stefan-Boltzmann constant, A is the surface area (m²), and T is the absolute temperature (K). The T⁴ dependence means radiation increases steeply with temperature — doubling T increases power by 16×.

Emissivity measures how efficiently a surface radiates relative to a perfect blackbody (ε = 1). A polished metal mirror has ε ≈ 0.02 (poor radiator); matte black paint has ε ≈ 0.97 (near-perfect radiator). Good absorbers are good emitters — and poor absorbers (reflective surfaces) are poor emitters. This is why space suits are silver (low ε, minimises radiative heat loss in space) and solar thermal collectors are matte black (high ε, maximises absorption).

The net radiation heat transfer between an object (temperature T) and its surroundings (temperature T_s):

Worked example: Human body radiation

A person (ε = 0.97, skin area A = 1.8 m², T = 310 K) in a room at T_s = 293 K:

A resting person radiates ~100 W of net heat — which is why a room full of people warms up quickly, and why thermal radiation is important in building energy calculations.

Wien's Displacement Law: Peak Wavelength

The peak wavelength of thermal radiation shifts with temperature:

At room temperature (293 K): λ_peak = 9.9 μm — mid-infrared, invisible. At the Sun's surface (5,778 K): λ_peak = 502 nm — green-yellow visible light. This is why incandescent light bulbs glow orange-white (filament at ~2,700 K peaks in near-infrared, with some visible); why stars' colours reveal their surface temperatures; and why thermal cameras detect people at night (body heat peaks in the infrared).

The Three Mechanisms

Heat transfers from hot to cold by three mechanisms: conduction (through direct contact, molecule to molecule), convection (via bulk fluid movement), and radiation (electromagnetic waves requiring no medium). In practice, all three often occur simultaneously.

Conduction: Q = kAΔT/d

Fourier's Law of conduction: the rate of heat flow through a material is:

where k = thermal conductivity (W/m·K), A = cross-sectional area (m²), ΔT = temperature difference (K), d = thickness (m). Thermal conductivity values: copper 401 W/m·K, aluminium 237, glass 1.0, wood 0.1–0.2, air 0.026, aerogel 0.015. Poor conductors (low k) are good thermal insulators. U-values for building materials (W/m²·K) are derived from this formula: U = k/d.

Worked Example: Heat Loss Through a Window

Single-glazed window: A = 2 m², d = 4 mm = 0.004 m, k_glass = 1.0 W/m·K, ΔT = 15°C (inside 20°C, outside 5°C).

Double glazing with a 10 mm air gap: k_air = 0.026 W/m·K → P = 0.026 × 2 × 15/0.010 = 78 W. Nearly 100× less heat loss — explaining the dramatic energy saving from double glazing.

Convection

Convection transfers heat through bulk fluid movement. Natural convection: warm fluid rises (lower density), cool fluid sinks, creating circulation cells. Forced convection uses fans or pumps to move fluid past a surface. The rate depends on fluid velocity, viscosity, and thermal properties — quantified by the Nusselt number and heat transfer coefficient h: P = hAΔT. Central heating radiators use both: radiation (lower contribution than the name implies) and convection as warm air rises from the surface.

Radiation: Stefan-Boltzmann Law

All objects above 0 K emit thermal radiation. The power radiated by a perfect black body:

where σ = 5.67 × 10⁻⁸ W·m⁻²·K⁻⁴ (Stefan-Boltzmann constant), A = surface area, T = absolute temperature. The T⁴ dependence means small temperature increases give large power increases: doubling T increases radiated power 16-fold. Net radiation between an object at T and surroundings at T_0:

where ε is emissivity (1 for a perfect blackbody, <1 for real surfaces). Human skin ε ≈ 0.98 (near-perfect radiator). The Sun (T ≈ 5,778 K) radiates most intensely in visible light; Earth (T ≈ 288 K) radiates in infrared — the basis of the greenhouse effect.

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