MATHEMATICAL ANALYSIS OF RECTANGULAR FIN
Considering the one dimensional fin exposed to a surrounding fluid at a temperature T . The temperature of the base of the fin is T .

One dimensional conduction and convection through a rectangular fin.
Be making an energy balance on an element of the fin of thickness dx
Energy in left face = Energy at right face +Energy lost by convection
Energy in left face,
Energy out at right face, = -kA |x+dx

Energy lost by convection = hPdx
Here,
A = cross sectional area of fin
P= perimeter of fin
The energy balance yields, (
Let

Then the equation becomes,

One boundary condition is
At x=0
The other boundary condition depends on physical situation. Several cases may be considered.
Case 1: The fin is very long and the temperature at the end of the fin is essentially that of the surrounding fluid.
Case 2: The fin is of finite length and loses heat by convection from its end.
Case 3: The end of fin is insulated so that dT/dx == 0 at x =L
CONSIDERING CASE 2:
Temperature distribution along the fin,

Where,


At base, X=0

Now, heat transfer rate at X = 0,