A Circular Fin Protruding from a
Hot Wall

Fins are often used as a medium to increase heat transfer. Joseph, an engineer, is interested in using a circular fin with radius r_o and length L to cool a hot wall. The wall temperature at the base of protruding fin is T_o. The fin is cooled by a fluid stream at . Based on constant thermal conductivity and an infinite heat transfer coefficient assumption, the two-dimensional steady-state temperature distribution of the fin is,

  

where , and α_n's are the positive roots of J_o(α_n) = 0 for n = 1, 2, 3, ... Joseph would like to find the temperature value at a particular point on the fin.

Determine the value of θ/θ_o for r/r_o = 0.25, L/r_o = 5, and z/L = 0.5.

• J_o(α_n), the Bessel function, is zero when
  α₁ = 2.405, α₂ = 5.520, α₃ = 8.654, and
  α₄ = 11.792
• Determine the values of the Bessel functions
  J_o(α_nr/r_o) and J₁(α_n) based on the power series expansion with at least five terms (Note: This is the approach only if tables and graphs of Bessel funtion values are not available).