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Pedagogy

The Thermal Fin "Tfin" Problem

Forward Problem

Authors: G Rozza & AT Patera

in collaboration with DBP Huynh, NC Nguyen and, previously, S Sen & S Deparis

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**Q1** As the Bi number increases, how does the temperature
distribution in the fin change — in particular as regards
variations in *x*_{1} and *x*_{2}? When would you expect the classical
1D fin approximation to yield reasonable predictions for the temperature field in the fin?

**Q2** Assume that the spreader plate is isothermal: what does
the classical 1D fin approximation then predict for *T*_{av}(*μ*)? How accurate is this approximation in the limit that
(*i*) *κ * is large (and hence there is little
temperature drop in the base/spreader), and (*ii*) Bi is small (hence the temperature distribution in the fin is largely
1D)

**Q3** Can you construct a simple lumped/analytical model that
will be reasonably valid even for *κ * not too large? (*Hint*:
consider a "constriction" resistance for the base/spreader and a
classical 1D fin resistance placed in series.) How accurate is
your approximation over the full range of parameters?

**Q4** Are the results for large Bi of interest, and why or
why not?