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Biot Number

Also known as Bi

Bi=hLck\mathrm{Bi} = \frac{h L_c}{k}

Worked example: Steel billet, h 100, Lc 10 mm, k 45 → Bi 0.0222 — press Try an example to run it live, then adjust anything.

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Biot Number explained

kLchBi

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 (Lc/k)(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, LcL_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.

Biot Number formula

Bi=hLck\mathrm{Bi} = \frac{h L_c}{k}
Where
  • Bi\mathrm{Bi}= Biot number
  • hh= Film coefficient (W/(m²·K))
  • LcL_c= Characteristic length (m)
  • kk= Solid thermal conductivity (W/(m·K))

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