Temperature Distribution: How Heat Spreads Through a Solid

The equation that governs how heat spreads

Heat moves through a solid by conduction: energy hops from hotter, more energetically vibrating atoms to cooler neighbouring ones without any bulk motion of material. The governing law, Fourier heat-conduction law (1822), says the local heat flux is proportional to the local temperature gradient — heat flows from hot to cold, and it flows faster where the temperature changes more steeply:

q = -k · ∇T          (Fourier's law: heat flux vs. temperature gradient)

∂T/∂t = α · ∇²T      (the heat equation, combining Fourier's law with energy conservation)

k = thermal conductivity (W/m·K)
α = k / (ρc_p) = thermal diffusivity (m²/s)

Combining Fourier's law with conservation of energy gives the heat equation, a partial differential equation that describes how a temperature field T(x,t) evolves over both space and time. It is mathematically identical in form to the diffusion equation that governs how a dye spreads through still water or how a gas diffuses through a room — heat, in this sense, literally diffuses.

Steady state versus transient: two very different problems

When a rod's ends are held at fixed temperatures long enough, the temperature field stops changing with time — the ∂T/∂t term vanishes — leaving the much simpler Laplace equation, ∇²T = 0. For a simple 1D rod with no internal heat sources, this forces the steady-state temperature to vary linearly between the two end temperatures, regardless of the material's conductivity — conductivity affects how much heat flows and how long equilibrium takes to reach, but not the final steady-state shape. The transient problem — how the temperature field evolves on the way to that equilibrium, starting from some initial distribution — is the harder, time-dependent version, and it is where the full heat equation with its ∂T/∂t term actually matters.

Analytical solutions and their limits

For simple geometries — an infinite rod, a slab with fixed boundary temperatures, a sphere cooling in a uniform environment — the heat equation has exact closed-form solutions, often built from Fourier series (an ironic full-circle, since Joseph Fourier developed his eponymous series specifically to solve this equation) or from the error function for semi-infinite domains. These solutions are invaluable for validation but only work for idealised shapes with simple, unchanging boundary conditions. Real components — a fin with variable cross-section, a rod that is also convecting heat to surrounding air, an object with a spatially varying material — need numerics.

The finite difference method: turning calculus into arithmetic

The finite difference method (FDM) replaces the continuous rod with a discrete chain of points spaced Δx apart, and replaces the second-derivative term in the heat equation with a simple three-point average:

∂²T/∂x²  ≈  ( T[i+1] - 2·T[i] + T[i-1] ) / Δx²

explicit update:
T_new[i] = T[i] + α · Δt/Δx² · ( T[i+1] - 2·T[i] + T[i-1] )

stability requires:  α · Δt / Δx²  ≤  0.5   (1D explicit FDM)

That stability bound is not a suggestion — the explicit scheme above genuinely blows up into oscillating nonsense if the time step is too large relative to the spatial spacing, a form of the same CFL condition that limits time steps in fluid and wave simulations. Halving the spatial spacing to double the accuracy quarters the maximum stable time step, since Δx appears squared in the bound, which is why fine-resolution transient heat simulations can get expensive quickly. Implicit schemes trade a more expensive linear solve at each step for unconditional stability, letting engineers take much larger time steps when only the final steady behaviour matters and the fine time-resolution of the transient is not the point.

Fins: why heat sinks have that shape

A cooling fin loses heat both by conducting it along its own length and by convecting it away from its surface into the surrounding air the whole way along, which makes the governing equation for a fin an extra term richer than the plain rod case — a balance between conduction along the fin and convective loss from its sides. Solving it shows temperature decays roughly exponentially away from the fin's base, which is exactly why heat sink fins are made thin and numerous rather than few and thick: more surface area per unit volume moves more heat into the air, even though a thinner fin conducts less heat along its own length.

Frequently asked questions

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