Lastly we will calculate the fin of a triangular profile, or that of a wedge. It is possible to reduce this to that of a wedge, but the equations are beyond the scope of this project. Further information on wedge shaped fins can be found in a paper published by D.R Harper and W.B.Brown in the 1922 National Advisory Committee on Aeronautical Technology Report #158, on page 677.
Start with Newton's equation of cooling as always.
Here we substitute our equation of a line. It is important to note here that the diagram above is NOT backwards. We just made a small convention to put the heat source at x=L and the tip of the fin at x=0. Thus this will simplify our profile equation to the one shown below.
The solution involves a complicated Bessel function of the first kind.
We have the initial condition knowing the temperature at x=L.
Now we calculate the total heat across the bar.
The equation for profile area.
Here we have total heat / initial
temperature excess. This gives a simple formula, that is fairly easy to
Note, we have replaced the complicated expressions inside the Bessel functions with a u.
However, we need to take a derivative with respect to y0, so we will need to substitute back in to get a y0.
Here we take the derivative to find the maximum value for total heat / initial temperature excess.
Use a numerical solver because Maple cannot come up with a closed form answer due to the Bessel functions.
Now that we have our most favorable u, we will use that to generate the characteristic length, height, and profile area.
Here we see that although a lot of heat is lost, the temperature of the fin as a whole is still quite high at the end.
Heat Conduction in the Steady State