What is the Biot number and how does it determine the analysis approach for transient conduction?

The Biot number (Bi = hL_c/k) compares external convective resistance to internal conductive resistance, where L_c is characteristic length (volume/surface area). When Bi < 0.1, internal resistance is negligible, and the lumped capacitance method applies (uniform temperature throughout the body). When Bi > 0.1, temperature gradients within the body are significant, requiring spatial analysis using Heisler charts or numerical methods. For a sphere being quenched (h=1000 W/m²-K, k=200 W/m-K, D=50mm): Bi = 1000×0.0083/200 = 0.04, so lumped analysis is valid. This criterion is crucial for hardening, annealing, and thermal processing design.