MATHEMATICAL ANALYSIS OF TRIANGULAR FIN:

Triangular fin.

The generalized differential equation for temperature distribution is
…………………………………………….(1)

Considering the straight fin of triangular profile, since x =0 is chosen at the fin tip and the maximum thickness at the base x =w is 2d1 then yx = d1(x/w) and the general differential equation (1) reduces to

This is a distinguish form of Bessel’s differential equation and on Comparing it with the generalized Bessel’s equation and solution. We determine its general solution as,

Where Jo and Yo are zero order Bessel’s functions of the first and second order respectively.
The integration constants C1 and C2 are to be determined such that T satisfies the required boundary conditions,
T=TO at x= w, dT/dx=0 at x=0
It is seen that J1(0)and Y1(0)=-8,
So applying the secondary conditions as,

We find that C2=0 and the general solution must then be represented by

The evaluation of C1 by application of the first boundary Condition leads to the particular solution for T which reads,

The rate of heat dissipation from the surfaces of the straight triangular fin is,

For x=(0 ==>4), T0=100 0C,N=.754

We calculated the value of temperature gradient from the above expression.