Heat Transfer

Three modes of heat transfer

Mode	Mechanism	Where it occurs	Rate equation
Conduction	Molecular collisions, lattice vibrations	Solids, stationary fluids	Q/t = -k·A·dT/dx (Fourier's law)
Convection	Bulk fluid motion	Liquids and gases	Q/t = h·A·ΔT (Newton's cooling)
Radiation	Electromagnetic waves	Anywhere (including vacuum)	Q/t = ε·σ·A·T⁴ (Stefan-Boltzmann)

Conduction

Heat flows from high to low temperature regions in a solid (or stationary fluid):

Q/t = k · A · (T_hot − T_cold) / L

For a wall of thickness L, area A, with temperatures T_hot and T_cold on each side. k is thermal conductivity (W/(m·K)).

Material	k (W/(m·K))	Notes
Diamond	1000-2000	Best natural conductor
Silver	406	Best metal conductor
Copper	385	Common conductor
Aluminum	205	Used in heat sinks
Iron	80	—
Glass	0.8	Poor conductor
Brick	0.7	—
Wood	0.15	Reasonable insulator
Fiberglass insulation	0.043	—
Air (still)	0.024	Excellent (if convection prevented)
Aerogel	0.012	Among lowest known

Convection

Bulk motion of fluid carries heat. Newton's law of cooling:

Q/t = h · A · (T_surface − T_fluid)

where h is the convective heat transfer coefficient (W/(m²·K)). h depends on flow conditions, fluid type, and surface geometry.

Setting	h (W/(m²·K))
Free convection in air	5-25
Forced convection in air	25-250
Free convection in water	50-1,000
Forced convection in water	50-20,000
Boiling water (vigorous)	2,500-100,000
Condensation of steam	5,000-100,000

Radiation

Stefan-Boltzmann law:

Q/t = ε · σ · A · T⁴

where:

• ε — emissivity (0 to 1; 1 for blackbody).
• σ = 5.67 × 10⁻⁸ W/(m²·K⁴) — Stefan-Boltzmann constant.
• A — surface area.
• T — absolute temperature in Kelvin.

For a body in surroundings at T_env:

Q_net/t = ε · σ · A · (T⁴ − T_env⁴)

Note T⁴ scaling — small changes in T produce large changes in radiation. Doubling T → 16× radiation power.

Real-world heat transfer

System	Dominant mode(s)
Coffee cup cooling	Convection (steam) + conduction (cup) + slight radiation
Solar heating Earth	Radiation (only — vacuum of space)
Laptop CPU cooling	Conduction (heat sink) + forced convection (fan)
Iron heating water on stove	Conduction (pot) + convection (water) + radiation (small)
Skin losing heat in cold	Radiation + convection + evaporation (sweat)
Thermos bottle	Suppresses all three modes
Spacecraft thermal control	Radiation only
Volcano cooling lava flow	Radiation (surface, very hot) + convection (atmosphere)

JavaScript — heat transfer calculations

// Conduction through a wall
function conductionRate(k, area, T_hot, T_cold, thickness) {
  return k * area * (T_hot - T_cold) / thickness;
}

// 10 m² window, 4mm thick, glass k = 0.8, indoor 22°C, outdoor 0°C
console.log(`Single-pane heat loss: ${conductionRate(0.8, 10, 22, 0, 0.004).toFixed(0)} W`);
// Wow, 44,000 W — but real windows have much smaller temperature gradient through air films
// Realistic U-value ~6 → 10 m² × 22°C × 6 = 1320 W

// Convection cooling
function convectionRate(h, area, T_surf, T_fluid) {
  return h * area * (T_surf - T_fluid);
}

// 1 m² wall at 30°C in 20°C room with natural convection h ≈ 10
console.log(`Wall convection: ${convectionRate(10, 1, 30, 20)} W`); // 100 W

// Radiation
const SIGMA = 5.67e-8;  // W/(m²·K⁴)

function radiationRate(emissivity, area, T) {
  return emissivity * SIGMA * area * Math.pow(T, 4);
}

// Person: ε ≈ 0.95, area ≈ 1.8 m², T ≈ 33°C surface (skin) = 306 K
console.log(`Person radiating at 33°C: ${radiationRate(0.95, 1.8, 306).toFixed(0)} W`);
// ~850 W (huge in absolute terms!)

// Net radiation in environment
function netRadiation(emissivity, area, T_object, T_env) {
  return emissivity * SIGMA * area * (Math.pow(T_object, 4) - Math.pow(T_env, 4));
}

// Person at 306 K in 295 K (22°C) room
console.log(`Net radiation loss: ${netRadiation(0.95, 1.8, 306, 295).toFixed(0)} W`);
// ~120 W net (comfortable; close to metabolic rate ~100 W)

// Equilibrium temperature for spacecraft
function spacecraftTemp(absorptivity, emissivity, solarFlux = 1361, area_facing = 1, area_total = 6) {
  // Q_in = α · S · A_facing
  // Q_out = ε · σ · T⁴ · A_total
  const Q_in = absorptivity * solarFlux * area_facing;
  const T = Math.pow(Q_in / (emissivity * SIGMA * area_total), 0.25);
  return T;
}

// White paint α = 0.2, ε = 0.9
console.log(`White spacecraft: ${spacecraftTemp(0.2, 0.9, 1361, 1, 6).toFixed(1)} K`);
// ~285 K (~12°C)

// Black absorber α = 0.95, ε = 0.95
console.log(`Black spacecraft: ${spacecraftTemp(0.95, 0.95, 1361, 1, 6).toFixed(1)} K`);
// ~330 K (~57°C — too hot)

// Series resistance for layered walls
function totalU(layers) {
  // U = 1/total resistance; layer i has R_i = thickness/k
  const R_total = layers.reduce((sum, [thickness, k]) => sum + thickness/k, 0);
  return 1 / R_total;
}

// Wall: 1cm gypsum (k=0.17), 10cm fiberglass (k=0.04), 1cm wood (k=0.15)
console.log(`Wall U: ${totalU([[0.01, 0.17], [0.10, 0.04], [0.01, 0.15]]).toFixed(2)} W/(m²K)`);
// ~0.4 W/(m²·K) — reasonable insulated wall

Where heat transfer matters

• Building design. Insulation, windows, HVAC, R-values, U-values for walls and ceilings.
• Electronics cooling. Heat sinks (conduction + convection), liquid cooling, thermal interface materials.
• Power generation. Steam turbines, nuclear reactors, fuel cells — all involve massive heat transfer.
• Industrial. Heat exchangers, distillation columns, ovens, furnaces, refining.
• Climate science. Earth's energy balance — solar input vs IR re-emission. Greenhouse effect (extra IR absorption).
• Aerospace. Spacecraft thermal control, reentry heating, jet engine combustion chamber design.
• Cooking. Pan choice (heat conductivity), oven design, microwave (radiative + dielectric).

Common mistakes

• Confusing thermal conductivity with heat capacity. Conductivity (k) — how fast heat travels through. Capacity (c) — how much heat to raise T. Different properties; both matter.
• Forgetting all three modes coexist. Real situations involve multiple modes; one might dominate but others contribute. For accuracy, account for all.
• Using temperature in Celsius for radiation. Stefan-Boltzmann uses absolute T (Kelvin). Celsius gives wildly wrong answers.
• Treating "still air" as a good conductor. Air's low k (0.024) makes it a great insulator IF convection is prevented. Otherwise, air movement transfers heat fast.
• Ignoring T⁴ in radiation. Doubling T gives 16× radiation power. Hot objects radiate far more than warm ones.
• Confusing emissivity and absorptivity. Kirchhoff's law: at thermal equilibrium, ε = α (good emitter = good absorber at same wavelength). White surfaces: low both visible AND IR. Need wavelength-specific values for accurate analysis.