Optimizing Precharged Cylinders

[Matt H.]

Originally Posted by Ether

I wonder if that is a valid assumption. The kick happens so fast I would think adiabatic would be a better approximation than isothermal. The explosive expansion cools the air and it takes time to absorb heat from the cylinder walls.
~

This is probably a fairly valid assumption so I'll give the new calculation a try!

For this we have to introduce the new constant y the adiabatic index (note it's really supposed to be a gamma, but until CD offers support for Greek letters I'll take what I can get.)

In air y=7/5
For an ideal adiabatic process
P*V^y=constant
which implies
P=P_int*d^(7/5)/(l^(7/5)) As above A cancels
so
P=P_int*(d/l)^(7/5)

Now as before
F=P_int*A*(d/l)^(7/5)
W=the integral over l of P_int*A*(d/l)^(7/5) from d to L
or in a simplified notation
W=P_int*A*d^(7/5)*the integral over l of l^(-7/5) from d to L

which gives
W=(5/2)*P_int*A*(d^(7/5))*(d^(-2/5)-l^(-2/5))

Because it is late the graphing calculator method shows that the energy output is maximized when d=.431*L.


The true answer probably lies somewhere between the two answers.

A note:
The assumption that no additional air flows into the cylinder during the process probably isn't justified. This will lower the optimal precharged length because it slightly increases the pressure as the cylinder expands.
A solution to the problem which takes this into account become exceedingly complicated very quickly (i.e. I don't think I can do it). If anyone would like to try it using Bernoulli's equation and a .32 cv I would love to see that worked out.