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MATHEMATICS - CASE STUDY

    Introduction


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 ro and length L to cool a hot wall. The wall temperature at the base of protruding fin is To. 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 αn's are the positive roots of Jo(αn) = 0 for n = 1, 2, 3, ... Joseph would like to find the temperature value at a particular point on the fin.

     
    Questions

   

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

     
    Approach

   
  • Jo(αn), 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
    Jo(αnr/ro) and J1(αn) 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).