Biot Number Calculator
This Biot number calculator works out the dimensionless Biot number for a solid body exchanging heat with a surrounding fluid, and tells you whether the simple lumped capacitance model applies. Enter the convective heat transfer coefficient at the surface, the characteristic length of the body, and the thermal conductivity of the material, and the calculator returns the Biot number along with a plain verdict on the modelling assumption. The Biot number, written Bi = h Lc / k, is the ratio of the resistance to heat conduction inside the object to the resistance to heat convection at its surface. When it is small, heat moves through the interior far more easily than it leaves the surface, so the whole body stays at nearly one temperature as it heats or cools, and you can model the transient with a single exponential curve rather than solving the full heat conduction equation. When it is large, the inside cannot keep up with the surface and steep internal temperature gradients form. The usual rule of thumb is that below 0.1 the lumped capacitance model is accurate enough for engineering work. The characteristic length is normally the volume of the body divided by its surface area. The defaults describe a small aluminium part in moderate airflow. Change the inputs to match your own problem, and see the worked example below for the full arithmetic.
With a Biot number of 0.0125, below 0.1, the lumped capacitance model is valid: internal temperature gradients are small and the object can be treated as a single uniform temperature.
The characteristic length is usually the body volume divided by its surface area. Keep all inputs in SI units so the Biot number stays dimensionless. Estimate only.
How it works
The Biot number is the surface convection resistance compared with the internal conduction resistance, written Bi = h Lc / k. The numerator h times Lc captures how strongly the surface exchanges heat over the size of the body, and the denominator k captures how quickly heat spreads through the material. Divide the two and the units cancel, leaving a pure number. A low Biot number means conduction wins and the interior stays uniform, which is the condition for the lumped capacitance model. A high Biot number means convection at the surface outpaces conduction inside, so a temperature gradient builds through the body and a full transient conduction solution is needed.
Worked example
Take a heat transfer coefficient of 50 W/m²K, a characteristic length of 0.05 m, and a thermal conductivity of 200 W/mK, roughly an aluminium part in moderate airflow. The numerator is 50 times 0.05, which is 2.5. Dividing by the conductivity of 200 gives a Biot number of 0.0125. Because that is well below 0.1, the lumped capacitance model is valid and the part can be treated as a single temperature as it cools. If the material were instead a poor conductor with k of 1.5, the Biot number would rise to about 1.67, far above 0.1, and internal gradients could no longer be ignored.
Related calculators
- Heat Transfer Calculator: conduction, convection and radiation heat flow.
- Heat Conduction Calculator: conductive heat flow through a wall.
- Reynolds Number Calculator: laminar or turbulent flow.
- Specific Heat Calculator: heat energy from mass and temperature change.