Biot Number Calculator
The Biot number calculator determines whether lumped capacitance analysis can be used for a heat transfer problem. Biot number (Bi) is the ratio of convective heat transfer at the surface of a solid to the conductive heat transfer within the solid. When Bi is at or below 0.1, the solid can be treated as thermally lumped (uniform temperature), greatly simplifying the transient heat transfer analysis to a single exponential equation. When Bi exceeds 0.1, the internal temperature gradients are significant and a more detailed conduction analysis is required. This tool is essential for engineers analyzing quenching operations, heat treatment of metals, cooling of electronic components, and temperature response of sensors.
Biot number formula
Bi = h * Lc / k
Lc = Volume / Surface Area
Sphere: Lc = r / 3
Cylinder (long): Lc = r / 2
Flat plate (both sides): Lc = L / 2
Biot number interpretation
- Bi less than or equal to 0.1: lumped capacitance valid, uniform temperature in solid.
- Bi = 0.1 to 1: moderate internal resistance, approximate methods may apply.
- Bi greater than 1: significant temperature gradients inside solid; use conduction solution.
- Bi greater than 100: surface temperature equals fluid temperature; surface resistance negligible.
Biot number: frequently asked questions
What is the Biot number?
The Biot number (Bi) is a dimensionless number in heat transfer representing the ratio of thermal resistance inside a solid to the resistance at its surface. Bi = h * Lc / k, where h is the convective heat transfer coefficient, Lc is the characteristic length, and k is the thermal conductivity of the solid.
When is lumped capacitance analysis valid?
Lumped capacitance analysis assumes the solid has uniform temperature throughout (no internal temperature gradient). This is valid when Bi is less than or equal to 0.1. At Bi below 0.1, the internal thermal resistance is much smaller than surface resistance, so temperature gradients inside the solid are negligible.
What is the characteristic length for Biot number?
The characteristic length Lc = Volume / Surface Area for Biot number calculations. For a sphere: Lc = r/3. For a long cylinder: Lc = r/2. For a flat plate (one side): Lc = L (thickness). For a cube: Lc = a/6. Using the correct characteristic length is critical for accurate Biot number results.
What does a high Biot number mean?
A Biot number much greater than 1 means the internal thermal resistance dominates. Temperature gradients inside the solid are significant and cannot be ignored. Full transient heat conduction analysis is required using the one-term series approximation or numerical methods. This is common for thick solids or materials with low thermal conductivity.
What is the time constant for lumped capacitance?
When Bi is less than 0.1, the temperature decays exponentially: (T - T_inf) / (Ti - T_inf) = exp(-t / tau), where tau = rho * V * Cp / (h * A) is the time constant. A larger mass or lower h gives a longer time constant (slower cooling).
Official sources
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology.