Definition
The Biot number (symbol Bi) is a dimensionless quantity used in heat transfer to compare the internal thermal resistance of a solid body to the external thermal resistance due to convection at its surface. It is defined as
$$ \mathrm{Bi} = \frac{h,L_c}{k} $$
where h is the convective heat‑transfer coefficient (W m⁻² K⁻¹), L₍c₎ is a characteristic length of the body (typically volume divided by surface area, m), and k is the thermal conductivity of the material (W m⁻¹ K⁻¹).
Overview
The Biot number provides a criterion for assessing whether temperature gradients inside a solid can be neglected during transient heat‑transfer analyses. When Bi ≪ 1 (commonly Bi < 0.1), the resistance to heat conduction within the object is much smaller than the resistance to heat convection at its surface; the body behaves as a “lumped system,” and its temperature can be approximated as spatially uniform. Larger Bi values indicate that internal conduction resistance is comparable to or exceeds the external convection resistance, requiring detailed spatial temperature analyses (e.g., using the heat‑diffusion equation).
In engineering practice, the Biot number is employed in:
- Lumped capacitance method for transient heating or cooling.
- Design of thermal insulation and heat‑exchanger components.
- Evaluation of the validity of simplified thermal models.
Etymology/Origin
The number is named after the French physicist and mathematician Jean‑Baptiste Biot (1774 – 1862), who made significant contributions to optics, electromagnetism (e.g., the Biot–Savart law), and heat‑transfer theory. The use of his name for this dimensionless parameter dates to early 20th‑century heat‑transfer literature.
Characteristics
| Property | Description |
|---|---|
| Dimensionless | No units; ratio of two thermal resistances. |
| Typical range | 0 ≤ Bi ≤ ∞; values < 0.1 justify the lumped‑capacitance assumption, 0.1 ≤ Bi ≤ 10 indicate moderate internal resistance, and Bi ≫ 1 requires detailed conduction analysis. |
| Dependence on geometry | The characteristic length L₍c₎ varies with shape: for a slab, L₍c₎ = thickness/2; for a sphere, L₍c₎ = radius/3; for a cylinder, L₍c₎ = radius/2. |
| Relation to other dimensionless numbers | Often considered alongside the Fourier number (Fo = αt/L₍c₎²) when analyzing transient conduction, and the Nusselt number (Nu = hL/k) when evaluating convective heat transfer. |
| Physical interpretation | Bi = (convection resistance)⁻¹ / (conduction resistance)⁻¹; a high Bi implies the surface exchanges heat with the environment much faster than heat can be conducted through the material. |
Related Topics
- Fourier number (Fo) – dimensionless time parameter for transient conduction.
- Nusselt number (Nu) – dimensionless convective heat‑transfer coefficient.
- Stefan number (Ste) – ratio of sensible to latent heat in phase‑change problems.
- Lumped system analysis – simplified thermal model applicable when Bi < 0.1.
- Thermal conductivity (k) – material property governing internal heat conduction.
- Convective heat‑transfer coefficient (h) – parameter describing heat exchange between a surface and a fluid.
- Heat diffusion equation – governing differential equation for temperature distribution in solids.
The Biot number remains a fundamental tool in thermal‑engineering analysis, guiding the selection of appropriate modeling approaches for transient heat‑transfer problems.