The temperature profile at which the entropy production is minimum has been computed here for the case of a metal bar heated at both ends by two difference heat sources. It is not the linear profile corresponding to the steady state but it is a profile exponentially increasing from the cold to the warm source.
The entropy production for any temperature profile T(x) is:
where K is the heat diffusion coefficient and is considered constant.
Varational calculus shows that the entropy production is an extremum for T(x) satisfying:
the solution of which is simply
Such profile fits the profile found for the family of broken linear profiles (see
Entropy production in a metal bar for steady and non-steady states). Fig. 1 shows the present profile (dashed blue), the profile found for the family of broken linear profiles found previously (dotted red), and the linear steady-state profile (dashed black):
| Figure 1 |
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