A solution for small times

For the heating or cooling of a plate dealt with in section 2.3.4.3 and written in dimensionless variables according to (2.158) and (2.159), we obtain 

This solution allows for a clear interpretation. The terms in the curly brackets correspond to the temperature pattern in a semi-infinite solid. Its surface lies at r+ = — 1 for the first term, and it stretches itself out in the positive redirection. The second term belongs to a semi-infinite body which stretches out from its surface at r+ = +1 in the negative redirection. The subtraction of these two temperature profiles from the initial temperature d+ = 1 yields the symmetrical temperature profile for the plate shown in Fig. 2.32, which satisfies the condition di)+/dr+ = 0 at r+ = 0. 

The solution (2.189) is only valid for times, for which the contribution of the first curly bracket to the surface temperature at r+ = +1 is negligibly small. Therefore 

This means that the solution given in (2.189) is generally applicable to situations where t+ 0.2. 

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