Heat Flow Through Solids
When one end of a metal bar is heated, the temperature change doesn't appear instantly at the other end. Heat conducts through the bar atom by atom — energetic atoms vibrate and transfer kinetic energy to their neighbors. In metals, free electrons carry most of the thermal energy, which is why good electrical conductors are also good thermal conductors. This simulation visualizes the 1D heat equation as temperature evolves in real time.
Fourier's Law
Joseph Fourier established in 1822 that heat flux q (energy per unit area per unit time) is proportional to the temperature gradient: q = -k(dT/dx). The proportionality constant k is thermal conductivity — a material property ranging from ~0.02 W/(m·K) for aerogel to ~400 W/(m·K) for copper. The negative sign means heat flows from hot to cold, satisfying the second law of thermodynamics.
Thermal Diffusivity
While conductivity k tells you how much heat flows, thermal diffusivity α = k/(ρcₚ) tells you how fast temperature changes propagate. A material can have high conductivity but also high heat capacity, making it slow to change temperature. Diffusivity captures this balance. High-α materials like copper reach equilibrium quickly; low-α materials like concrete or brick store heat for hours — which is why masonry buildings stay cool in summer.
Engineering Applications
Thermal conduction analysis is essential in electronics cooling (preventing chip overheating), building insulation design, metallurgical heat treatment (quenching rate determines microstructure), and spacecraft thermal management. The heat equation, solved numerically with finite differences or finite elements, is one of the most computed PDEs in engineering practice.