# Two-dimensional, steady-state conduction occurs in a hollow

Two-dimensional, steady-state conduction occurs in a hollow cylindrical solid of thermal conductivity k = 16 W/m ∙ K, outer radius ro = 1 m, and overall length 2zo = 5 m, where the origin of the coordinate system is located at the midpoint of the centerline. The inner surface of the cylinder is insulated, and the temperature distribution within the cylinder has the form T(r, z) = a + br2 + c In r + dz2, where a = 20°C, b = 150°C/m2, c = - 12°C, d = - 300°C/m2 and rand z are in meters.

(a) Determine the inner radius rj of the cylinder.

(b) Obtain an expression for the volumetric rate of heat generation, q(W/m3).

(c) Determine the axial distribution of the heat flux at the outer surface, qr (ro' z). What is the heat rate at the outer surface? Is it into or out of the cylinder?

(d) Determine the radial distribution of the heat flux at the end faces of the cylinder, qz(r, + zo) and qz(r, - zo). What are the corresponding heat rates? Are they into or out of the cylinder?

(e) Verify that your results are consistent with an overall energy balance on the cylinder.

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