Rapid Reliable Solution
of the Parametrized Partial Differential Equations
of Continuum Mechanics and Transport

Pedagogy

A Transient Heat Treatment Problem
Forward Problem

Authors: G Rozza & AT Patera
in collaboration with DBP Huynh & NC Nguyen


 

Q1 As the Bi number increases, how does the temperature distribution in the railroad rail section change — in particular as regards variations in x1; and x2? 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?

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