With the time it would take to reach any kind of appreciable pressure increase, I think just pumping a few extra times would be more energy efficient.
Tried it. Long time ago. I dont think it worked. I'm going to try it again, but what worries me is that to get enough heat to make a noiceable distance improvement, it might start to melt the tank.
Oh, and is "B" "b"?
P[tank] * V[tank] / (Ff / A[barrel]) = V[barrel]
V[barrel] = pi * r^2 * h[barrel length]
P[tank] * V[tank] / (Ff / A[barrel]) = pi * r^2 * h[barrel length]
P[tank] * V[tank] / (Ff / A[barrel]) / (pi * r^2) = h[barrel length]
That's about as far as I got. You can just measure the friction force of the dart in a particular barrel by starting it at the top of your barrel (pointed downwards), throwing a really light string (massless? ) through it, and attaching that string to a weight of known mass. Something like 200g is nice. Just start the dart at the top, release it, start a timer, and stop the timer just as it exits the barrel. Really crappy kinematic technique, sure, but I was tired when I thought of this, and still am. Either way, you can calculate the acceleration of the dart by:
2 * d / t^2 = a
Where d is the length of the barrel sample, t is your time, and 2 is in fact the numeral "two." So:
Force of gravity - Force of Friction = mass * acceleration
You have the acceleration, you can measure the full mass of the moving system (dart, wieght, and all), and you know that the force of gravity is just that mass times 9.8m/s/s, so there's your friction force.
What worries me about that first equation? Well, for some odd reason I thought that you could compare the force exerted by the pressure with the friction force, despite the fact that the non-static friction force is constant, versus the pressure force, which decreases on a rather sharp curve.
Perhaps we need to work with energies. And something cooler than an ideal gas law. Regardless, it's all moot until someone proves it.