Heat Transfer
posted by Farant .
Derive the general heat conduction equation in cylindrical coordinates by applying the first law to the volume element.

For transient situations with no heat generation and azimuthal symmetry, see the derivation of Eq. 228 at
http://wwwme.nchu.edu.tw/Enter/html/lab/lab516/Heat%20Transfer/chapter_2.pdf
In this case, T depends upon time (t) and r only.
There is a more complicated equation where there are variations with length z and aximutual angle phi.