How Do You Derive the Temperature Distribution Equation for a Conical Fin?

[Mir17]

Homework Statement

Derive a differential equation (do not solve) for the temperature distribution in a straight conical fin. Assume one dimensional heat flow. This equation is assumed to be 1-D steady state conduction.

Homework Equations

For this problem, we can use the generalized fin equation. Please see the attached image of the equation because I do not know how to use the equation editor on here.

The Attempt at a Solution

For the conical fin problem I understand that the cross sectional area is non-uniform and it changes with position. I need to simplify the generalized fin equations using the assumptions of a conical shape fin and then acquire an ordinary differential equation for temperature distribution. I am unsure of any other assumptions I can use for conical fins to further simplify the generalized equation into an ordinary differential equation.

 

[Chestermiller]

If R₀ is the radius at the base of the cone, and L is the height of the cone, what is the local radius at x as a function of x, R₀, and L? What is the local cross sectional area? What is the local surface area between x and x+dx?