Find a (steady state) temperature distribution in a partially submerged pipe sho

Question

Find a (steady state) temperature distribution in a partially submerged pipe shown. Temperature of air = -20^0, fluid (inside) = +10 degree. Temperature in soil: Changes from +2^0 at the bottom of the pipe to the air temperature at the surface, as a linear function of the radial angle. Pocket of water, at 0 degree as shown When using Fourier series in angular coordinate theta, retain the number of terms that provides reasonable accuracy. Comment on: how this number changes with the radial coordinate. Plot lines of equal temperature. Sketch zones where temperature gradients are highest (in these zones, mechanical stresses, caused by temperature gradients, will be the highest)

Explanation / Answer

solution:

1)here radial temperature distrubution is given by

dTm/dm=2m

on soving we get that

Tm=m^2

equation heat conduction becomes

d/dr(r*dT/dr)=-2

so on solving we get that

T=-2r+c1+c2

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