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    Heat Equation with Circular Symmetry : Total Heat Energy, Flow of Heat Energy and Equilibrium Temperature Distribution

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    8. Heat Equation with Circular Symmetry. Assume that the temperature is circularly symmetric:

    u u(r,t), where r^2 x^2 | y^2. Consider any circular annulus a ≤ r ≤ b.

    a) Show that the total heat energy is r π f^b_a cpurdr.
    b) Show that the flow of heat energy per unit time out of the annulus at r b is: (see attachment for equation).
    A similar results holds at r = a.
    c) Assuming the thermal properties are spatially homogenous, use parts (a) and (b) to derive the circularly symmetric heat equation without sources: (see attachment for equation)
    d) Find the equilibrium temperature distribution inside the circular annulus a ≤ r < b if the outer radius is insulated and the inner radius is at temperature T.

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    Solution Preview


    Right now I do not have access to a scanner to work this problem through but essentially what you are doing in the proof is looking at Gausses divergence theorem and its equivalent form to a surface in the first part of the proof. I found some lecture notes that go through this step by step plus a derivation of the heat equation for polar coordinates. I will have access to a scanner tomorrow and can work the problem by ...

    Solution Summary

    Heat Equation with Circular Symmetry, Total Heat Energy, Flow of Heat Energy and Equilibrium Temperature Distribution are investigated.