Transient Conduction

Two clocks, one slab

When a body cools, two processes race. Heat has to conduct from the core out to the surface (resistance ≈ L/k), and then convect from the surface into the air (resistance ≈ 1/h). The Biot number is just the ratio of those two: Bi = hL/k. It tells you which clock is slow — and the slow one controls the show.

When the inside keeps up: lumped capacitance

If Bi < 0.1, conduction inside is so fast compared to convection outside that the whole body stays essentially uniform — no internal gradients worth mentioning. Then you can throw away the PDE and use a single exponential: T(t) = T∞ + (Tᵢ − T∞)·e^(−t/τ) with τ = ρcV/hA. That's the dashed blue curve. Pick copper with gentle airflow and watch it sit right on top of the full solution.

When it lies

Crank up h, fatten the slab, or switch to glass or brick, and Bi climbs past 1. Now the surface dumps heat faster than the core can resupply it, so a real gradient opens up — the surface is cold while the center is still glowing. The lumped model, which assumes one uniform temperature, can be off by hundreds of degrees. The lumped-error readout shows the gap in real time. This is why you quench-harden steel: the surface cools fast, the core stays hot, and the resulting stresses are the whole point.

How it's solved

Explicit finite differences in time on a grid from the insulated centerline to the convective surface. The surface node carries a convection energy balance; the timestep is held just under the Fo(1+Bi_cell) ≤ ½ stability limit so the march never blows up. Dimensionless time is the Fourier number Fo = αt/L² — when Fo ≈ 1, the thermal wave has crossed the slab.