Heat Transfer:
Conduction, Convection & Radiation

1. Three Modes Overview

Every heat-transfer problem involves one or more of these three mechanisms. They differ fundamentally in the medium required and the governing equation:

2. Conduction & Fourier's Law

Jean-Baptiste Joseph Fourier showed in 1822 that the heat flux through a material is proportional to the temperature gradient and the material's thermal conductivity:

The total heat flow rate through an area A of thickness L is:

Thermal resistance behaves exactly like electrical resistance — series layers (wall + insulation) simply add: R_total = Σ(Lᵢ/kᵢAᵢ).

3. Convection & Newton's Law of Cooling

When a fluid moves past a solid surface, it carries heat away from (or toward) that surface. Newton's law of cooling parameterises this with a convective heat transfer coefficient h:

The coefficient h is not a material constant — it depends on fluid velocity, geometry and flow regime (laminar vs. turbulent). It is determined via the dimensionless Nusselt number Nu = hL/k_fluid, which correlates with the Reynolds (Re) and Prandtl (Pr) numbers.

Natural vs. Forced Convection

In natural (free) convection the fluid motion is driven by buoyancy — warm fluid rises, cool fluid sinks. Typical h values range from 5–25 W/(m²·K) in air. Forced convection (a fan blowing over a heat sink) raises h to 25–250 W/(m²·K). Liquid cooling (water, oil) reaches 100–20 000 W/(m²·K) because liquids have much greater thermal capacity per unit volume.

4. Radiation & Stefan-Boltzmann Law

Every body above absolute zero emits electromagnetic radiation. Josef Stefan (1879) and Ludwig Boltzmann (1884) derived the total emitted power for a perfect black body:

Real surfaces are not perfect black bodies. Their emissivity ε (0 ≤ ε ≤ 1) scales the emission: Q̇ = εσAT⁴. Net heat exchange between two parallel surfaces is:

Wien's Displacement Law

The peak wavelength of emission shifts with temperature: λ_peak · T = 2.898 × 10⁻³ m·K. At 5778 K (Sun) λ_peak ≈ 500 nm (green light). At 300 K (room temperature) λ_peak ≈ 9700 nm (mid-infrared) — invisible to the human eye but detected by thermal cameras.

5. Heat Diffusion PDE

Combining Fourier's law with the conservation of energy in a continuum gives the heat equation — one of the most important PDEs in physics:

In 1D, the discretised finite-difference update is:

6. JavaScript Implementation

The following solves the 1D heat equation on a rod using an explicit finite-difference scheme, plus a lumped-body cooling model with all three modes combined.

// 1D heat diffusion — explicit finite differences
class HeatRod {
  /**
   * @param {number} n      - number of spatial nodes
   * @param {number} alpha  - thermal diffusivity [m²/s]
   * @param {number} dx     - node spacing [m]
   */
  constructor(n, alpha, dx) {
    this.T   = new Float64Array(n).fill(300); // K
    this.T2  = new Float64Array(n);
    this.n   = n;
    this.alpha = alpha;
    this.dx  = dx;
    // max stable dt: alpha*dt/dx² ≤ 0.5
    this.dt  = 0.4 * dx*dx / alpha;
  }

  step() {
    const { T, T2, n, alpha, dx, dt } = this;
    const r = alpha * dt / (dx * dx);
    // interior nodes
    for (let i = 1; i < n - 1; i++) {
      T2[i] = T[i] + r * (T[i+1] - 2*T[i] + T[i-1]);
    }
    // Dirichlet boundary: fix left end hot, right end cold
    T2[0] = T[0];
    T2[n-1] = T[n-1];
    T2.copyWithin(0);
    T.set(T2);
  }

  // Neumann BC: zero gradient (insulated end) at index i
  insulate(i) {
    const mirror = i === 0 ? 1 : this.n - 2;
    this.T[i] = this.T[mirror];
  }
}

// Lumped body: conduction + convection + radiation cooling
function lumpedCool(T0, Tamb, params, totalTime, dt) {
  const { m, Cp, A, h, epsilon, k, L } = params;
  const sigma = 5.67e-8;
  let T = T0, t = 0;
  const history = [{ t, T }];
  while (t < totalTime) {
    const qConv = h * A * (T - Tamb);          // W
    const qRad  = epsilon * sigma * A * (T**4 - Tamb**4); // W
    const qCond = k * A / L * (T - Tamb);     // contact to substrate, W
    const dT = -(qConv + qRad + qCond) / (m * Cp) * dt;
    T += dT;
    t += dt;
    history.push({ t, T });
  }
  return history;
}

// Example: a steel sphere (r=0.05m) cooling from 600K in 300K air
const sphere = {
  m: 4.1,          // kg = ρ·V = 7900·(4/3π·0.05³)
  Cp: 490,         // J/(kg·K)
  A: 0.0314,       // m² = 4π·0.05²
  h: 15,           // W/(m²·K) natural convection in air
  epsilon: 0.8,    // oxidised steel
  k: 0,            // no conduction path (suspended)
  L: 1             // ignored when k=0
};
const curve = lumpedCool(600, 300, sphere, 3600, 1); // 1h @ 1s steps
console.log(`After 1 hour: ${curve.at(-1).T.toFixed(1)} K`);

7. Combined Cooling Model

Real objects lose heat via all three modes simultaneously. A hot metal plate suspended in air loses heat by:

• Natural convection — warm air rises from both surfaces (dominant below ~500 K)
• Thermal radiation — significant above ~400 K, dominant above ~800 K
• Conduction through any mounting brackets or thermal pads

The relative importance changes dramatically with temperature. At 300 K a blackened surface in still air loses roughly 60% by convection, 40% by radiation. At 1000 K radiation accounts for ~85% of total heat loss.

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