Free Biot Number Calculator

What the Biot Number Reveals

The Biot number calculator is a free online tool designed for heat transfer analysis. It instantly computes the Biot number, a dimensionless quantity that compares the thermal resistance to heat flow within a body to the resistance at its surface. In practical terms, the Biot number answers a key question: how uniformly will the interior temperature change when the surface is heated or cooled?

When the Biot number is much smaller than 1, heat is conducted into the interior much faster than it can be transferred across the surface, so the body’s temperature remains nearly uniform throughout. When the Biot number is much greater than 1, the surface heat transfer is far more efficient than internal conduction, leading to a steep temperature gradient inside the material.

Heat Transfer Mechanisms

To understand the Biot number, one must consider two simultaneous processes. The first is the heating (or cooling) of the surface itself, which depends on the heat transfer coefficient $h$ (in W/(m²·K)). The second is the conduction of heat from the surface into the rest of the material, governed by the thermal conductivity $k$ (in W/(m·K)). The Biot number directly compares these two processes:

• If heat transfer at the surface is slower than internal conduction (Bi ≪ 1), the body heats uniformly.
• If surface heat transfer is faster than internal conduction (Bi ≫ 1), a large temperature difference develops across the body.

This tool functions essentially as a heat transfer calculator and a thermal conductivity calculator combined, using the Biot number formula to give insight into the uniformity of heating.

Biot Number Formula

The mathematical expression for the Biot number is:

$$\mathrm{Bi} = \frac{h \, L_c}{k}$$

where:

• $h$ – heat transfer coefficient at the surface [W/(m²·K)],
• $L_c$ – characteristic length of the body [m],
• $k$ – thermal conductivity of the material [W/(m·K)].

The characteristic length is defined as the ratio of the volume $V$ to the surface area $A$ through which heat is transferred:

$$L_c = \frac{V}{A}$$

Because it is a ratio of quantities with consistent units, the Biot number itself is dimensionless; its magnitude alone reveals the thermal behavior of the system.

Example Calculation

A practical example helps illustrate the concept. Consider a copper pot filled with water. The water has a thermal conductivity of approximately $k = 0.7$ W/(m·K). The heat transfer coefficient between the copper and the water is about $h = 13.1$ W/(m²·K). If the water depth—taken as the characteristic length—is $L_c = 15$ cm (0.15 m), the Biot number becomes:

$$\mathrm{Bi} = \frac{13.1 \times 0.15}{0.7} \approx 2.807$$

With a Biot number greater than 1, a noticeable temperature difference exists between the bottom and the top of the water: the warmer bottom and cooler top are not at the same temperature. If we reduce the water depth (say, to 5 cm), the characteristic length decreases, and the Biot number drops below 1, indicating a much more uniform temperature distribution.

Using the Calculator

The online Biot number calculator streamlines these calculations. By entering the heat transfer coefficient, the thermal conductivity, and the characteristic length (or volume and area), the tool instantly returns the Biot number. It serves as a practical heat transfer analysis tool for students, engineers, and anyone studying thermal behavior.