Heat Conduction

What you're looking at

A metal rod has its two ends clamped to heat reservoirs at fixed temperatures — say a hot left end and a cold right end. Heat flows through the rod from hot to cold by conduction: fast-jiggling atoms at the hot end jostle their neighbours, passing energy down the line without the material itself moving. The colour shows the temperature at each point, and the graph below plots the temperature profile T(x).

Fourier's law

Heat flux — the rate energy flows through a cross-section — is proportional to the temperature gradient: steeper temperature differences drive faster heat flow.

Good conductors like copper have a large k and move heat readily; insulators like wood and glass have a small k and barely conduct. Switch materials and watch how quickly (or slowly) the rod responds.

The heat equation

Combining Fourier's law with energy conservation gives the equation the simulation actually solves, the heat (diffusion) equation:

It says each point's temperature drifts toward the average of its neighbours — so bumps smooth out and gradients relax. We integrate it with a simple finite-difference scheme across the rod.

Reaching steady state

With the ends held fixed, the rod eventually stops changing: heat flows in the hot end and out the cold end at the same steady rate. In that steady state the temperature profile is a perfectly straight line from one end temperature to the other (the dashed reference line). The flux is then constant everywhere, q = k·(T_hot − T_cold)/L.

Things to try

Start from a Hot spike in the middle of a cool rod and watch the bump spread out and fade — pure diffusion. Compare copper and wood reaching steady state. Change an end temperature mid-run and watch the whole profile re-adjust toward the new straight line.