Biot Number Calculator

The Biot number Bi is a dimensionless ratio comparing the conductive resistance inside a solid body to the convective resistance at its surface. It is defined as Bi = h ⋅ L_c ÷ k, where h is the convective heat transfer coefficient W/(m²⋅K), L_c is the characteristic length m, and k is the solid's thermal conductivity W/(m⋅K). When Bi < 0.1, the internal temperature gradient is negligible and the lumped capacitance method applies. This simplifies transient conduction analysis from a partial differential equation to an ordinary one. Getting this threshold wrong leads to errors exceeding 5% in predicted cooling or heating times, which matters in quench-hardening steel, food safety pasteurization, and electronics thermal management.

This calculator accepts direct inputs or computes L_c = V ÷ A from standard geometries (plane wall, long cylinder, sphere). The tool approximates steady boundary conditions and uniform h. For geometries with non-uniform convection or internal heat generation, a full finite-element approach is required. Pro tip: in forced convection scenarios, always verify h against published correlations for your flow regime before trusting a lumped analysis.

The Biot number is defined as the ratio of internal conductive resistance to external convective resistance at the surface of a body:

Where h = convective heat transfer coefficient W/(m²⋅K), L_c = characteristic length m, k = thermal conductivity of the solid W/(m⋅K).

The characteristic length L_c is the ratio of the body's volume to its surface area:

For standard geometries this simplifies to:

Lumped capacitance criterion: if Bi < 0.1, the temperature within the body can be assumed spatially uniform during transient heat transfer. This reduces the governing PDE to an exponential decay ODE with time constant τ = ρ c_p Vh A_s.