Approximation for small Biot numbers

A simple calculation for the heating or cooling of a body of any shape is possible for the limiting case of small Biot numbers (Bi — 0). This condition is satisfied when the resistance to heat conduction in the body is much smaller then the heat transfer resistance at its surface, cf. section 2.1.5. At a fixed time, only small temperature differences appear inside the thermally conductive body, whilst 

We simply assume that the temperature of the body is only dependant on time and not on the spatial coordinates. This assumption corresponds to Bi = 0 because A — oo, whilst the heat transfer coefficient a A 0. We apply the first law of thermodynamics to the body being considered in order to determine the variation of temperature with time. The change in its internal energy is equal to the heat flow across its surface 

If a body of volume V has constant material properties g and c, then it holds that 

For the example of the plate we show that (2.199) corresponds to the exact solution for Bi = 0 with a / 0. We investigate further, for which Biot numbers not equal to zero this simplified calculation of the temperature change according to (2.199) can be applied to, whilst still obtaining sufficiently high accuracy. 

This agrees with (2.199), because for the plate of thickness 2R, V/A = R. The same method can be used to show that for other shapes of solids (2.199) agrees with the exact solution for Bi = 0, corresponding to A — oo, with q 0, If however, with a finite value for A the Biot number would be zero when a = 0, it follows from (2.171) that = 1. The insulated plate keeps its initial temperature o, or in other words no equalization of the temperatures o and s takes place. 

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