Biot Number Calculator

The Biot number (Bi) is a dimensionless quantity used in heat transfer analysis. It represents the ratio of the thermal resistance inside a solid body to the thermal resistance at its surface (convective resistance). A small Biot number means the interior of the body heats or cools at nearly the same rate as the surface, while a large Biot number indicates a significant temperature gradient inside the material.

The Biot number is calculated as Bi = h · Lc / k, where h is the convective heat transfer coefficient (W/m²·K), Lc is the characteristic length of the solid (m), and k is the thermal conductivity of the material (W/m·K). The characteristic length is typically defined as the ratio of the body's volume to its surface area (V/A).

Lumped system analysis (also called the lumped capacitance method) is considered valid when the Biot number is less than or equal to 0.1 (Bi ≤ 0.1). Under this condition, the temperature variation within the solid is small enough to be neglected, and the entire body can be treated as having a single uniform temperature at any given time.

A Biot number greater than 1 indicates that the resistance to heat conduction inside the solid is greater than the convective resistance at the surface. This means there will be a significant temperature gradient within the material, and the lumped system approximation is not valid. More detailed analysis (such as using the Heisler charts or finite element methods) is required.

The characteristic length Lc is defined as the volume-to-surface-area ratio (V/As). For a sphere of radius r, Lc = r/3. For a long cylinder of radius r, Lc = r/2. For a plane wall of half-thickness L (heated from both sides), Lc = L. For a cube with side length a, Lc = a/6.

Although both are dimensionless and share a similar-looking formula, the Biot number and Nusselt number are distinct. The Biot number uses the thermal conductivity of the solid and describes the ratio of internal conduction resistance to surface convection resistance. The Nusselt number uses the thermal conductivity of the fluid and describes the ratio of convective to conductive heat transfer in the fluid.

Jean Baptiste Biot (1774–1862) was a French physicist, astronomer, and mathematician. He made significant contributions to science including work on the polarization of light and the Biot-Savart law in electromagnetism. The Biot number in heat transfer is named in his honor.

Yes, the Biot number is especially important in transient (time-dependent) heat conduction. It determines whether a simplified approach — the lumped system analysis — can be used to model how a solid heats or cools over time. When Bi > 0.1, spatial temperature variation must be accounted for using methods such as the one-term approximation or numerical solutions.