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Biot Number Calculator

Enter the convection coefficient, length and conductivity.

How it is calculated

The Biot number compares heat carried away at the surface with heat conducted inside the body: Bi = h × Lc / k. Here h is the convective heat transfer coefficient (W/m²·K), Lc is the characteristic length in metres (usually volume divided by surface area) and k is the thermal conductivity of the solid (W/m·K). Bi has no unit and is shown with four decimals. The tool then tests the lumped-capacitance rule of thumb: below 0.1 it answers "Yes (Bi < 0.1)", at 0.1 or above "No (Bi ≥ 0.1)". Conductivity must be greater than zero, and all three fields must be filled, otherwise no result appears. Nothing is converted, so Lc must already be in metres.

Worked example

A stainless-steel cube with 6 cm sides cools in moving liquid. Its characteristic length is volume / surface area = 0.06³ / (6 × 0.06²) = 0.01 m. Take h = 500 W/m²·K and k = 15 W/m·K. Then Bi = 500 × 0.01 / 15 = 0.3333. That is above 0.1, so the tool answers "No (Bi ≥ 0.1)": temperature differences inside the cube cannot be ignored and the simple lumped model is not reliable here.

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Common mistakes

  • Entering Lc in millimetres or centimetres. A 6 cm cube needs 0.01 m; typing 10 or 1 makes Bi a thousand or a hundred times too large.
  • Using the wrong conductivity. k belongs to the solid, not to the surrounding fluid; the fluid's value gives a meaningless Bi.
  • Taking the full thickness or the radius as Lc. The usual characteristic length is volume / surface area (a third of the radius for a sphere), which is smaller and can flip the verdict.
  • Treating Bi < 0.1 as a guarantee. It is a rule of thumb for the lumped model, and h can differ enormously between still air and a quench bath.
  • Entering k = 0 for an "insulator". Conductivity must be above zero, so the tool shows no result; use a small positive value instead.

About this calculator

The Biot number (Bi) is a dimensionless ratio used in transient heat transfer: Bi = h·Lc/k, where h is the convective heat transfer coefficient at the surface, Lc is a characteristic length (typically volume/surface area), and k is the thermal conductivity of the solid. It compares how quickly heat moves through the material's interior to how quickly it's carried away at the surface.

When Bi < 0.1, temperature within the object stays nearly uniform as it heats or cools, which justifies the much simpler “lumped capacitance” model that ignores internal temperature gradients entirely. This calculator checks that condition for you, which is why it's a standard first step for mechanical and chemical engineers analyzing quenching, heat treatment, and cooling problems before deciding whether a full conduction analysis is even necessary.

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