A spine attached to a wall maintained at a uniform temperature Tb has the shape of a circular cone with base diameter b, height L and a concave parabolic profile, and it is exposed to a fluid at a uniform temperature T?. (a) Assuming that the thermal conductivity k of the material of the spine and the heat transfer coefficient h are constants, and that the variation of the temperature in the r direction is negligible, develop an expression for the steady-state temperature distribution T(x) in the spine. (b) Obtain an expression for the rate of heat loss to the surrounding fluid.

In addition to the questions asked, compare the fin efficiency for the cone shaped fin to that of a solid circular rod fin that has the same volume and same length. If the purpose of the extended surface is to increase heat flow from a system, do you recommend using the cone shaped fin if manufacturing processes and costs are the same?

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You need to start by setting up a differential equation for the heat flow - have you done that?