Lumped System Analysis

Assess when the spatial variation of temperature is negligible, and temperature varies nearly uniformly with time, making the simplified lumped system analysis applicable.

Solve the transient conduction problem in large mediums using the similarity variable, and predict the variation of temperature with time and distance from the exposed surface.
Lumped system analysis provides great simplification in certain classes of heat transfer problems without much sacrifice from accuracy.
When a solid body is being heated by the hotter fluid surrounding it (such as a potato being baked in a oven), heat is first convected to the body and subsequently conducted within the body. The Biot number is the ratio of the internal resistance of a body to heat conduction to its external resistance to heat convection. Therefore, a small Biot number represents small resistance to heat conduction, and thus small temperature gradients within the body.
Lumped system analysis assumes a uniform temperature distribution throughout the body, which will be the case only when the thermal resistance of the body to heat conduction (the conduction resistance) is zero. Thus, lumped system analysis is exact when Bi=0 and approximate when Bi >0. Of course, the smaller the Biot number, the more accurate the lumped system analysis.
Biot Number, \[Bi = \dfrac{h L_c}{k} = \dfrac{\dfrac{L_c}{K}}{\dfrac{1}{h}} = \dfrac{\mathrm{Conductive\ resistance\ within\ the\ body}}{\mathrm{Convection\ resistance\ at\ the\ surface\ of\ the\ body}}\]
It is generally accepted that lumped system analysis is applicable if Bi \(<\) 0.1