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Pedagogy

A Transient Heat Treatment Problem

Forward Problem

Authors: G Rozza & AT Patera

in collaboration with DBP Huynh & NC Nguyen

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**Q1** As the Bi number increases, how does the temperature
distribution in the railroad rail section change — in particular
as regards variations in *x*_{1}; and *x*_{2}? When would you expect a
constant distribution for the temperature field in the section?

**Q2**
Analyze how the thickness of the web connecting the top
and the bottom of the section is influencing the temperature
distribution and the role played by the base of the railroad
section.

**Q3**
Can you construct a simple lumped/analytical model that
will be reasonably valid even for Bi not too large so that you can
build an approximate solution of type
*u*(**μ**) = (1 – exp(*K*(**μ**) *t* ))? What (geometrical and physical) quantities

are in $K$? How do these quantities
influence the time scale of the thermal treatment? How accurate is
this approximation? For thinner webs connecting the top and bottom
of the section can you see a separate time scaling?

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