Dimensionless groups in transient conduction The ratio of surface convection resistance to internal conduction resistance is termed as:

Introduction / Context:
Dimensionless numbers compare competing physical mechanisms. In transient conduction and lumped-capacitance analysis, the Biot number assesses the relative importance of internal conduction versus external convection.

Given Data / Assumptions:

• Internal characteristic length L_c (typically volume/surface area for bodies).
• Convection coefficient h at the surface and thermal conductivity k within the solid.
• Thermal resistances: R_conv = 1/(hA) and R_cond ≈ L_c/(kA).

Concept / Approach:
Biot number Bi is defined as h L_c / k. It can be interpreted as the ratio of internal conduction resistance to surface convection resistance (R_cond / R_conv). Small Bi (≪ 1) implies negligible internal temperature gradients (justifying lumped analysis); large Bi indicates significant internal gradients.

Step-by-Step Solution:
R_cond / R_conv = (L_c/(kA)) / (1/(hA)) = h L_c / k = Bi.Hence Bi compares internal conduction to surface convection effects.Although some texts phrase the ratio oppositely (R_conv/R_cond), the conventional named number associated with these resistances is the Biot number.

Verification / Alternative check:
Lumped capacitance criterion uses Bi < 0.1 to justify spatially uniform temperature within the solid.

Why Other Options Are Wrong:
Grashof relates buoyancy to viscous forces in free convection; Stanton concerns convective heat transfer in boundary layers; Prandtl links momentum and thermal diffusivities; Nusselt is a dimensionless temperature gradient or hL/k, not a resistance ratio as posed.

Common Pitfalls:
Mixing up Biot and Nusselt; remember Bi uses a solid's L_c while Nu uses a fluid's flow length scale.

Final Answer:
Biot number