A Circular Fin Protruding from a
Hot Wall

Point of Interest: r/r_o = 0.25
and z/L = 0.5

It is given that the two-dimensional steady-state temperature distribution of a circular fin is:

For r/r_o = 0.25, L/r_o = 5, and z/L = 0.5, the above equation reduces to

     
     

where the first four zeros of J_o(α_n) are given as:
α₁ = 2.405, α₂ = 5.520, α₃ = 8.654, and α₄ = 11.792

The next step is to determine the Bessel functions using five terms of the power series expansion:

For α₁ = 2.405, α₂ = 5.520, α₃ = 8.654, and α ₄ = 11.792, the Bessel functions and hyperbolic sine functions are summarized in the following table:

The dimensionless temperature at r/r_o = 0.25, L/r_o = 5, and z/L = 0.5 becomes

Is the approximation for the Bessel functions good enough by only including five terms of the power series expansion? Use the simulation in this section to check the accuracy. Suppose that the first four zeros of J_o(α) are not given in the assumption, at least how many terms in the power series expansion do you need in order to obtain a reasonable solution?