How this instrument works
The Biot number compares two thermal resistances: how hard it is for heat to leave a body's surface by convection, against how hard it is for heat to move through the body itself by conduction. Bi = h·Lc ⁄ k puts the convective coefficient h and characteristic length Lc on top, the solid's own conductivity k on the bottom. A steel forging cooling in oil and a foam cup cooling in still air can sit in the same fluid conditions and land on wildly different Biot numbers, because the number is really a statement about the material, not the fluid.
The characteristic length Lc is usually the ratio of the body's volume to the surface area it is losing heat through, V ⁄ As, not a linear dimension pulled off a drawing. For a sphere of radius r that reduces to r ⁄ 3; for a long cylinder cooling on its curved surface it is r ⁄ 2; for a thin slab cooling from both faces it is half the thickness. Getting Lc from the actual heat-loss geometry, rather than guessing at 'the size', is where most hand calculations of Bi go wrong before they even reach the division.
Below Bi ≈ 0.1, the internal conduction resistance is small enough that the whole body can be treated as one uniform temperature at every instant — the lumped-capacitance assumption that turns a partial differential equation in space and time into a single exponential decay in time alone. Push Bi well past 1 and the surface is losing heat far faster than the interior can resupply it; a real temperature gradient develops inside the part, and no single number can describe its state — that regime needs a spatial conduction solution, not this instrument.
- Enter the convective heat transfer coefficient, W ⁄ (m²·K) — how fast heat leaves the surface into the surrounding fluid; a quenching oil sits far higher than still air.
- Enter the characteristic length in mm, cm, or m — volume divided by surface area for the actual shape, not a linear dimension read off a drawing.
- Enter the thermal conductivity, W ⁄ (m·K) of the solid itself; this is a property of the part, not of the fluid around it.
- Read the Biot number. Below about 0.1 the lumped-capacitance model applies; above roughly 1, expect a real temperature gradient inside the part.
Worked example — quenching a 1 cm steel pin
A metallurgist quenches a small steel pin in an oil bath that gives a convective coefficient of h = 50 W/(m²·K) at the surface. The pin's characteristic length works out to Lc = 0.01 m, and the steel has a thermal conductivity of k = 15 W/(m·K). The Biot number is Bi = h·Lc ⁄ k = (50 × 0.01) ⁄ 15 = 0.5 ⁄ 15 = 0.0333333333333, exactly one-thirtieth.
Because 0.0333 sits comfortably under the 0.1 threshold, the metallurgist can model the pin's cooling with a single exponential decay curve instead of solving the heat equation across its cross-section — a full spatial simulation would return essentially the same core and surface temperature at every timestep. Had the quench instead used a fiercer coefficient of h = 500 W/(m²·K), the same pin would return Bi = 0.333, ten times higher and past the lumped-capacitance limit, meaning the surface would cool measurably faster than the core.
Questions
What counts as a low Biot number?
Below about 0.1. At that point the resistance to conduction inside the body is small enough compared with the resistance to convection at its surface that temperature differences inside the part become negligible, and the lumped-capacitance model — one uniform temperature decaying exponentially with time — stays accurate to within a few percent.
How do I choose the characteristic length Lc?
Use volume divided by the surface area actually exposed to the fluid, Lc = V ⁄ As. For a sphere that reduces to r ⁄ 3, for a long cylinder cooling on its curved surface it is r ⁄ 2, and for a slab cooling from both faces it is half the thickness. Picking a linear dimension instead of V ⁄ As is the most common source of a wrong Biot number.
What happens when the Biot number is high?
A high Biot number, roughly above 1, means the surface loses heat faster than conduction inside the solid can resupply it, so a real temperature gradient builds up between the surface and the core. The single-temperature lumped model breaks down, and predicting the part's response needs a spatial solution — a finite-difference or finite-element conduction model — rather than one exponential curve.
Is the Biot number the same thing as the Nusselt number?
No, and confusing them is a common mistake. Both compare a convective effect to a conductive one, but the Nusselt number divides by the fluid's thermal conductivity to describe convection at the surface, while the Biot number divides by the solid's own conductivity to describe conduction inside the body — two different materials, on opposite sides of the same surface.
Does the Biot number stay constant during a real quench?
Only if h and k stay constant, and in a real quench they often do not. The convective coefficient h can swing sharply as boiling regimes change around a hot part entering oil or water, and k shifts somewhat with temperature too, so engineers often recompute Bi at a few points through the process rather than trusting one value for the whole cooldown.
Can the Biot number exceed 1?
Yes, with no fixed upper limit. A copper block, with high conductivity, stays well under 0.1 in most cooling, but a poorly conducting material such as glass or a thick polymer part cooled by a fast-moving fluid can reach Biot numbers of 10 or more, meaning conduction inside the part is the dominant bottleneck and its surface tracks the fluid temperature almost immediately.