Biot Number

Named for Jean-Baptiste Biot, whose 1804 experiments on heated bars preceded Fourier's theory by two decades, this is the number that decides whether you may treat a cooling object as a single lump. It compares the resistance to getting heat out of the body $(L_c/k)$ with the resistance to getting it off the surface (1/h). Below Bi = 0.1 the interior is within a few percent of uniform and the lumped-capacitance method is legitimate; above it, the centre lags the skin badly and you need a chart, a series solution or a finite-element model.

The characteristic length is volume divided by surface area, not the diameter: for a sphere that is r/3 and for a long cylinder r/2, and forgetting the factor is the classic error that pushes a valid problem out of the lumped regime on paper. Worked example: a 20 mm steel billet, $L_c$ = 0.01 m, k = 45 W/(m·K), quenched in air at h = 100 gives Bi = 0.022, comfortably lumped. Quench the same billet in agitated water at h = 5000 and Bi = 1.1, so the surface transforms while the core is still glowing, which is precisely the metallurgy that hardening exploits, and precisely why quench cracks happen.