Natural Science Forum / Physics / Research / April 2008

I'm trying to convince a friend of mine that his "revolutionary energy
saving" scheme is physically ridiculous.  He doesn't take at face
value my claim that the 2nd Law of Thermodynamics prevents it from
working as well as he wants.  My explanation falters because I haven't
done the maths to show the ridiculousness of his plan. I'll be seeing
him again in a week, and I hope to have a written-out explanation of
where the energy goes in his device so I can walk him through the
maths without being all hand-wavy.

In doing the maths, I've come to realize that there is a part of
thermodynamics I don't understand well enough to model.  I've checked
my old college physics texts, I've used Google to the best of my
ability, and before I came here I peeked at sci.physics (and decided I
probably wasn't going to get usable help there).

Here's what I'm trying to model:  At one point in his device he has a
cylinder filled with nitrogen gas (well, air, which is close enough to
N2 for the level of precision I'm computing at).  He compresses it
with a piston, and at peak compression injects a small quantity of
liquid nitrogen (at 77K) into the cylinder, using the high temperature
of the compressed air to vaporize the liquid nitrogen, thereby raising
the pressure in the cylinder, developing power as the piston is pushed
down.  His claim is that the large expansion ratio of LN2 will allow a
large amount of power to be developed.  My claim is that the extreme
cold of the LN2 will chill the gas in the cylinder such that there
isn't that large of a pressure increase -- and I wouldn't be surprised
if there was a net energy loss in the overall cycle.

My method of analysis falters at the injection of the LN2.

Here's what I've got:

Using the ideal gas law PV=nRT I can solve for n=PV/RT to get the
number of moles of N2 in the cylinder at the beginning of the cycle.
I can also, using the relationship PV=(\gamma-1)U, solve for U=PV/
(\gamma-1) to get the internal energy of the gas.  For the compression
stage, I can use the relationship PV^\gamma=P'V'^\gamma to compute the
new pressure at the smaller volume, then either use PV/T = P'V'/T' or
P'V'=nRT' again to get the temperature.  From there, I can compute the
new U' = P'V'/(\gamma-1), and \delta U = U'-U to get the energy needed
to compress the gas.

The next step in the cycle would be to inject a small amount of LN2.
My approach to this is to assume that \delta n moles of LN2 has an
internal energy of u, so after injection the cylinder would contain n'
= n+ \delta n moles of N2, have an internal energy of U'' = U'+u, a
volume of V', a pressure of P''=U''/((V'(\gamma-1)), and a temperature
T''=n'R/P''V'.

The only unknown is the internal energy u of \delta n moles of LN2.
Using U=nRT/(1-\gamma), and \gamma=1.4, I get the internal energy of
gaseous N2 at 77K to be 1.6kJ/mole. Wikipedia says the heat of
vaporization is 5.6kJ/mol, which would mean the internal energy of LN2
at 77K is -4.0kJ/mole, which is unrealistic.  Obviously I'm doing
something wrong.  The obvious answer is that \gamma is temperature
dependent and the internal energy of gaseous N2 at 77K is not as low
as 1.6kJ/mole.  But I haven't been able to find a figure for it
anywhere on line.

So am I on the right track?  If so, how do I compute the internal
energy from the LN2 injected into the cylinder?
Any help would be appreciated.