Drive an expression for heat transferred by a finite extended rod of uniform cross-section area with perfect insulated free end .

Assumptions:

Explanation:

Use the appropriated heat transfer equation (cylindrical coordinates),  

After separating variables and integrating we obtained the following expression,

After repeating the procedure state above we obtained the following expression,

According to the image attached, the corresponding boundary values for the system are

Then, the temperature distribution in the rod along r axis corresponds to,

After deriving the function, we obtained the expression that follows,

Evaluating the previous expression at the surface of the rod,

The heat transferred by the finite extended rod corresponds to,

Replacing the derivate term, we obtained the expression that follows,

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