No, the last graph is the Fanno Flow equation, but with the Mach number input to the Fanno equation limited by the output of the Siegel equation.... The blue line on the upper graph is the Siegel equation by itself.... I have not graphed the Fanno Flow data by itself, as I don't see how Mach 100 makes sense, you said yourself they are unrealistic.... However, your results from your M100 column could certainly be graphed.... Here is a chart I have prepared showing the three current ideas which vary with pressure....I didn't use the trendline equations, so as not to introduce any small errors, I went back to the original data that Scott sent me for Gamma, and the data I got from the Peacesoft and Wolfram Calculators for the Speed of Sound and Air Density, for every 500 psi up to 6000.... The Fanno Flow is using Scott's equation, using Mach 100 (to max. the velocity), and the Fanno / Siegel data is using the output of the Siegel equation as the limiting Mach, which is then used in the Fanno Flow equation.... The Pressure / Mass curve is plotted by using the equation developed by Steve_in_NC for maximum velocity based on pressure and density, but is corrected to use the actual Density values at each Pressure (because air is not an Ideal Gas over 2000 psi), and using 1/3 of the mass to incorporate the KE of flow in a tube, which has that factor in it, as explained in my post # 517....I have no idea if any of these theories are valid, but presenting them all in one chart may be useful.... All these curves are plotted using zero pellet mass.... The Fanno and Siegel equations are based on Gamma while the other is based on Density.... both of which vary with Pressure....Bob
You need to work with the speed of sound in the propellant gas at the beginning of the cycle, which for 5000 psi air is 1615 fps.... Five times that is 8075 fps, but that isn't the output of the Siegel equation, because k = 1.736 at that pressure, so Siegel's equation yields 1615 x 2 / (1.736-1) = 4389 fps.... Siegel's equation does NOT give the Mach number relative to the expanding and cooling gas behind it, but to the starting point.... Most light gas guns (which were the reason Siegel developed his equation) do NOT start with air at high pressure, they start, in fact with Helium at 1 atmosphere (although it may be heated).... If you start with air at 1 atmosphere, k = 1.4, and the maximum Mach relative to that starting air is Mach 5.... With Helium, where k = 1.67, it is Mach 3, but of course the SoS in Helium is higher to start with....If you note, I am using the SoS for the starting gas, just as Siegel's equation is intended to be used.... I have never seen any experimental results quoted that exceeded Siegel's equation, so I think it is a valid starting point for the maximum Mach number used for Fanno Flow.... Neither of us can prove it, it's just another possibility.... I still don't see how you can develop Mach 100, as you suggest....Bob
CO222 on the Yellow is going to (easily) join the 2000+ club, IMO.... providing he gets the 9mm projectile weight down to the 8 gr. proposed for the test....http://www.network54.com/Forum/79537/message/1461804297/Maximum+airgun+velocity+-+building+the+new+hypergunBased on Lloyd's previous shot data, my prediction is 2300+ at the Chorny, assuming it is about the same distance away as Lloyd's.... Two versions of Lloyd's spreadsheet give 2312 and 2308 based on Lloyd's 1745 fps .22 cal shot, and 2362 and 2354 based on his 2162 fps .278 cal shot.... Steve_in_NC is predicting (revised to) 2206 at the muzzle and 2163 at the Chrony, based on resetting one of his formulas for his 2259 fps maximum possible recently proposed.... It will be extremely interesting if the shot breaks that new 2259 maximum.... which I think entirely likely, at least at the muzzle.... Incidently, using my version of Steve's equation, adjusted for 1/3 of the air mass, with an 8 gr. bullet @ 4350 psi, 0.354" barrel that is 59" long (Va = 5.81 in^3) yields a maximum velocity of 2389 fps.... The air density I used is the average from Peacesoft & Wolfram @ 4350 psi = 323 kg/m^2 = 81.7 gn/in^3....v = sqrt ( 2 x P x Va / 12 / [W + D x Va / 3 ] / 225218 ) = sqrt ( 2 x 225218 / 12 ) x sqrt ( P x Va / [ W + D x Va / 3] ) = 193.7 sqrt ( 4350 x 5.81 / [ 8 + 81.7 x 5.81 / 3 ] ) = 193.7 sqrt ( 25274 / 166.2 ) = 2389 fpsThe actual velocity should be less because the reservoir is only about 1.8X the barrel volume, which will cause a pressure loss during the shot.... which is allowed for in Lloyd's spreadsheet calculations....Bob
Lloyd, I missed your prior posts about tonight's test.... Congratulations on completely putting to rest all the nonsense about heating, and pulling off yet another shot at well over 2000 fps....Bob