Biot Number

What is Biot Number?

The Biot number (Bi) is a dimensionless quantity used in heat transfer analysis. It compares the internal thermal resistance within a solid body to the external convective thermal resistance at its surface. Named after the French physicist Jean-Baptiste Biot, this number helps engineers determine whether a simpler lumped capacitance analysis is sufficient or if a more detailed spatial conduction analysis is needed.

The Biot number is defined as the ratio of internal conduction resistance to external convection resistance. When Bi is small (less than 0.1), the temperature gradient inside the body is negligible, and the lumped capacitance method provides accurate results. When Bi is large (greater than 1), significant temperature gradients exist within the body, requiring a full conduction analysis.

Biot Number Formula

The Biot number is calculated using the following formula:

Bi = h × L / k_s

Where:

• Bi — Biot number (dimensionless)
• h — Heat transfer coefficient (W/m²·K)
• L — Characteristic length, typically volume divided by surface area (m)
• k_s — Thermal conductivity of the solid (W/m·K)

How to Use the Biot Number Calculator

Select the variable you want to solve for from the dropdown menu. Enter the known values in the appropriate fields and the calculator will compute the result instantly. You can solve for:

• Biot Number (Bi) — When you know the heat transfer coefficient, characteristic length, and thermal conductivity
• Heat Transfer Coefficient (h) — When you know Bi, characteristic length, and thermal conductivity
• Characteristic Length (L) — When you know Bi, heat transfer coefficient, and thermal conductivity
• Thermal Conductivity (k) — When you know Bi, heat transfer coefficient, and characteristic length

The characteristic length is typically the body's volume divided by its surface area (V/A). For a sphere this equals r/3, for a long cylinder r/2, and for a flat plate half the thickness.

Interpreting the Biot Number

The value of the Biot number tells you which heat transfer analysis method is appropriate:

• Bi < 0.1 — Lumped capacitance method is valid. The internal conduction resistance is much smaller than external convection resistance. Temperature gradients inside the body are negligible (less than 5% error).
• 0.1 ≤ Bi < 1 — Moderate internal resistance. Lumped capacitance may have limited accuracy. Consider using a full spatial analysis for better precision.
• Bi ≥ 1 — Significant internal temperature gradients exist. A full spatial conduction analysis is required. The lumped capacitance method is not valid.

Example Calculation

Example: A steel billet (k = 45 W/m·K) with characteristic length L = 0.03 m is quenched in oil with h = 300 W/m²·K. What is the Biot number?

Bi = h × L / k = 300 × 0.03 / 45 = 9 / 45 = 0.2

Since Bi = 0.2 > 0.1, the lumped capacitance method is not strictly valid. A spatial conduction model is needed for accurate temperature predictions.

Applications of Biot Number

• Metal Quenching — Determining whether a heated part cools uniformly or develops internal thermal stresses
• Food Processing — Validating pasteurization models by checking if food items heat uniformly
• Electronics Cooling — Verifying that small components reach steady state quickly enough for lumped thermal analysis
• Casting and Forging — Deciding whether to use simplified or detailed thermal models for solidification simulations

Common Mistakes

• Wrong characteristic length — For a sphere it is radius/3 (V/A), not the diameter or radius. Using the wrong dimension changes Bi by a factor of 2 to 6.
• Applying lumped capacitance when Bi exceeds 0.1 — Above this threshold, internal temperature gradients become significant.
• Confusing thermal conductivities — Bi uses the solid's conductivity in the denominator, not the surrounding fluid's conductivity.