Fourier's Law & the Heat Equation

The intuition: heat slides downhill

Touch a metal spoon left in a pot and your fingers find the truth instantly: heat moves from hot to cold, and it moves faster the steeper the temperature difference. That single sentence is Fourier's law. There is no "cold flowing in" — cold is just the absence of heat, and energy always drains from the warm end toward the cool end, like water running downhill.

The key word is gradient. Heat doesn't care about the temperature itself; it cares about how sharply temperature changes across space. A wall with 20 °C inside and 0 °C outside leaks heat. Make the wall thinner and the same 20-degree drop happens over a shorter distance — the gradient is steeper, and heat pours out faster. Make the wall thicker and the gradient flattens, and the leak slows. The temperature difference alone doesn't set the flow; the difference per unit distance does.

Now run the clock forward. Heat leaving the hot regions and arriving at the cold ones smooths the temperature profile over time. Sharp spikes flatten, dents fill in, and everything drifts toward a single uniform temperature. That smoothing process — written as a partial differential equation — is the heat equation, and it is the same mathematics that governs how ink disperses in water, how a stock-price model spreads, and how a CPU sheds the heat of a billion transistors.

Fourier's law — the flux equation

Joseph Fourier wrote it down in 1822. In its full vector form:

q = −k ∇T

where:

• q is the heat flux — energy crossing a unit area per second, in watts per square metre (W/m²). It's a vector: it has a direction.
• k is the thermal conductivity of the material, in W/(m·K). Copper has k ≈ 400; air ≈ 0.026; aerogel ≈ 0.013.
• ∇T is the temperature gradient — the spatial rate of change of temperature, in K/m.
• The minus sign encodes the direction: heat flows opposite to the gradient, i.e. from hot toward cold.

In one dimension (heat through a wall or along a rod) it collapses to the form most people meet first:

q = −k (dT/dx)        // flux per unit area
Q = k·A·(ΔT) / L      // total power through a slab

That second form is the practical engineering version: total heat-flow power Q (in watts) through a slab of area A and thickness L with a temperature difference ΔT across it. It's the equation behind every insulation R-value, every heat sink, and every thermos.

From flux to the heat equation

Fourier's law tells you the instantaneous flow given a temperature profile. To predict how that profile evolves, add one more ingredient: conservation of energy. For a tiny volume, the rate that its stored thermal energy changes equals the net heat flowing in, which is minus the divergence of the flux:

ρ·c · ∂T/∂t = −∇·q          // energy balance for a volume element
            = ∇·(k ∇T)       // substitute Fourier's law

If the material is uniform (constant k), pull it out and divide through by ρc:

∂T/∂t = α ∇²T               // the heat equation

α = k / (ρ·c)               // thermal diffusivity, m²/s

Here ρ is density (kg/m³), c is specific heat capacity (J/(kg·K)), and ∇²T is the Laplacian — the divergence of the gradient. The Laplacian measures curvature: where the temperature profile is a peak (curving down, ∇²T < 0), the point cools; where it's a valley (curving up, ∇²T > 0), it warms. Heat literally chases curvature out of the profile until it's flat. That's why the heat equation is the archetypal diffusion and smoothing equation.

Thermal diffusivity — the speed of cooling

The single number that controls how fast a body responds is the thermal diffusivity α, not the conductivity k. Conductivity tells you how much heat flows at steady state; diffusivity tells you how fast a change propagates. A material can conduct well yet respond slowly if it has a lot of thermal "mass" (high ρc) to charge up.

Material	k (W/m·K)	α (m²/s)	Time to diffuse 1 cm
Copper	~400	1.1×10⁻⁴	~0.9 s
Aluminium	~237	9.7×10⁻⁵	~1.0 s
Steel (mild)	~50	1.3×10⁻⁵	~8 s
Glass	~1.0	5×10⁻⁷	~3.3 min
Water	~0.6	1.4×10⁻⁷	~12 min
Wood (oak)	~0.17	1.3×10⁻⁷	~13 min
Air (still)	~0.026	2.0×10⁻⁵	~5 s

The "time to diffuse 1 cm" column uses t ≈ L²/α. Notice air conducts almost nothing (tiny k) yet diffuses temperature quickly (large α) — because it has almost no thermal mass. That's the secret of insulation: still air is a great insulator not because temperature changes spread slowly through it, but because so little energy flows. Wood and water store heat and conduct it poorly, so their interiors lag badly.

Why thickness costs you time²

Because the heat equation has one time derivative and two space derivatives, dimensional analysis forces a single scaling law: the distance heat spreads in time t is the diffusion length

L ≈ √(α·t)     ⇔     t ≈ L² / α

Time grows as the square of distance. This one fact explains a huge amount of everyday thermal behaviour:

• Cooking. A 2 cm-thick steak takes about 4× longer to cook through than a 1 cm one — not 2×. Doubling thickness quadruples the time for the centre to come up to temperature.
• The ground. The daily temperature swing penetrates only ~10 cm into soil; the yearly swing reaches a few metres, which is why cellars stay cool in summer and frost lines sit below buried pipes.
• Quenching steel. Thin parts harden through; thick parts stay soft in the core because heat can't escape the centre fast enough.

A worked example: cooling a copper sphere

Take a 5 cm-diameter copper ball heated to 200 °C and dropped into a 20 °C room with a surface heat-transfer coefficient h ≈ 15 W/(m²·K) (natural air convection). First check whether the inside stays uniform — the Biot number:

L_c = V/A = r/3 = 0.025/3 = 0.0083 m   // characteristic length of a sphere
Bi  = h·L_c / k = 15 × 0.0083 / 400 ≈ 3.1×10⁻⁴

Bi ≪ 0.1, so the copper's interior is essentially isothermal — we can use the lumped Newton-cooling shortcut instead of the full PDE. The time constant is:

τ = ρ·c·V / (h·A) = ρ·c·L_c / h
  = 8960 × 385 × 0.0083 / 15 ≈ 1.9×10³ s ≈ 32 min

T(t) = T_env + (T₀ − T_env)·e^(−t/τ)

So the ball cools to 1/e of its excess (the 180 °C above-room excess drops to ~66 °C, so the ball sits at ~86 °C actual) in about 32 minutes. Swap the copper for glass (k ≈ 1) and the Biot number jumps to ~0.12 — now the surface cools faster than the core can keep up, the inside lags, and you must solve the full heat equation to get the temperature at the centre right.

Newton's law of cooling vs the full equation

Aspect	Newton's law of cooling	Full heat equation
Equation	dT/dt = −h(T − T_env)	∂T/∂t = α∇²T (+ boundary conditions)
Tracks	One bulk temperature	Full temperature field T(x, y, z, t)
Valid when	Biot number Bi = hL/k < 0.1	Always (it's the underlying physics)
Internal gradient	Assumed negligible (isothermal body)	Resolved explicitly
Best for	Thin / high-conductivity objects (a coin, a wire)	Thick / low-conductivity bodies (a roast, soil, a wall)
Solution	Single exponential decay, e^(−t/τ)	Sum of decaying Fourier modes, each ∝ e^(−α k² t)
Cost	One ODE — pencil and paper	PDE — separation of variables or numerics

The deepest connection: solve the heat equation by separation of variables and the temperature profile decomposes into Fourier sine/cosine modes (this is literally why Fourier invented Fourier series). Each spatial mode of wavenumber k decays as e^(−αk²t). Sharp features (large k) die out fastest, smooth features (small k) linger — which is exactly why any initial temperature pattern relaxes into a smooth, slowly-fading shape.

Where Fourier's law shows up

• Buildings. Insulation R-values, double glazing, and thermal-bridge calculations are all Q = kAΔT/L bookkeeping. A typical home loses several kilowatts through walls and windows in winter.
• Electronics. A modern CPU dumps 100–250 W through a die a few square centimetres across; heat sinks, heat pipes, and thermal-interface paste exist to keep ∇T (and therefore the junction temperature) under control.
• Geophysics. Earth's geothermal gradient (~25–30 °C/km near the surface) plus rock conductivity gives the planet's heat flow (~47 TW globally). The same equation dates the cooling of the crust and of cooling magma bodies.
• Cooking & food. Sous-vide times, roast resting, and freezing/thawing all hinge on the t ≈ L²/α scaling.
• Manufacturing. Welding heat-affected zones, quenching and annealing, plastic injection-mould cooling cycles.
• Beyond heat. The identical mathematics is Fick's law for chemical diffusion, the Black–Scholes equation in finance (which becomes the heat equation after a change of variables), and image-blurring (Gaussian smoothing) in computer vision.

Common misconceptions & edge cases

• "Conductivity tells you how fast it cools." No — diffusivity α = k/ρc does. A thick aluminium block conducts beautifully yet has so much thermal mass it takes a while to change temperature.
• "Heat is a fluid that flows." Fourier's caloric-era language survives, but heat is energy transfer, not a substance. The "flux" is real; the "caloric fluid" is not.
• "Cold flows into the hot object." Only heat flows, always from hot to cold. "Cold" is just less heat.
• "The minus sign is optional." It's essential — it's what makes the equation obey the second law of thermodynamics (heat never spontaneously flows up a gradient).
• "The heat equation handles radiation and convection too." Pure Fourier conduction is one of three transfer modes. Radiation scales as T⁴ (Stefan–Boltzmann) and convection moves the fluid itself; both usually enter only as boundary conditions on the conduction problem.
• "Information travels instantly." The parabolic heat equation does predict infinitesimally fast far-field response — an unphysical artefact. Real heat propagates at finite speed; the hyperbolic Cattaneo–Vernotte correction fixes it, but the error is negligible for ordinary engineering.
• "Run the equation backward to recover the past." You can't, stably. Diffusion destroys information; the backward heat equation is famously ill-posed, which is why you can't un-blur a photo perfectly.