In reading the post by Ben Jordan regarding the SCTA's rhetoric about a gas
bottle being an rocket, I decided to investigate further. My goal was to
find out how much thrust could be produced by a standard "K" bottle which
can store 240 cubic feet of any gas at approximately 2200 psig.
Bottom line: "K" bottle of with a broken off valve: Hydrogen = 175.03 pounds
of thrust
Nitrogen = 174.12 " " "
Oxygen = 173.8 " " "
The reason is that the lighter gas, hydrogen has a much higher sonic
velocity than the other 2 gasses. The interesting part is that the K bottle
at approximately 100 pounds in weight would accelerate at approximately 1.75
g's and reach a speed of 56 ft per second at the end of the first second.
The kinetic energy at the end of the first second would be approximately
4800 ft-lbs.
The conclusion I reach from this is that both the SCTA and Ben Jordan are
wrong. The force is much greater than Mr. Jordan's claimed value of around
30 pounds or so and far less that the 2000 pound rocket indicated by the
SCTA. What it seems to point out to me is the need for urban legend type
data to be removed from the rule book or supplemental rule book text and
real numbers used. I would suggest that the SCTA give thought to having one
of the Southern California Universities thermodynamics classes use this
problem to generate real data for the rule book, rather than use anything I
develop. After reading the supplemental rules for hydrogen use, I can find
no real fault with them (as if anyone asked me to look anyway!!). While I am
only a technically interested party (curious at any rate) I do not see why
hydrogen fueled vehicles could not compete in any of the car classes without
undue safety concerns. In fact, those who run a N2 pressurized nitrous
system pose exactly the same safety concerns as those running H2.
Of course, I could be wrong.
end of my thoughts on the subject.
mayf.
no need to read further...details of the analysis follow below.
At this point it does not matter whether or not the gas is Hydrogen (H2),
Nitrogen (N2), Oxygen (O2) or any of the other gasses. What matters is the
molecular weight of the molecule, the internal pressure and the temperature
of the stored gas. Also important is the ambient pressure, although that is
only important if the pressure inside the tank is low enough to prevent
sonic flow at the broken off valve. I looked up the molecular weight of each
of the gasses listed above: H2 =2.016; N2 =28.016; O2 =32, to slide rule
accuracies. The universal gas constant is 1544 in the foot-pound-second
system. The gravitational constant is 32.174 lbm-ft/lb-sec^2. The ratio of
specific heats for the gasses are: H2 = 1.41; N2 = 1.40; O2 = 1.39. The
stored gas is assumed to be the same temperature as the ambient air
temperature and for worst case ambient pressure I assume a value of 12.6
psig, similar to the atmospheric pressure at Bonneville. I also looked up
the densities ot the gasses used above they are: H2 = 0.0888974 grams per
liter; N2 = 1.25068 g/l; O2 = 1.4277 g/l.
A standard "K" bottle, at 240 cubic feet capacity (under pressure) holds
6796 liters. Therefore the amount of gas of each type in the "K" bottle is
H2 = 609.877 g; N2 = 8499 g; O2 = 9703 g. At a conversion of 454 grams per
pound the mass of gas of each type in a "K" bottle is; H2 = 1.34 pounds; N2
= 18.7 pounds; O2 = 21.372 pounds. As can be seen, the difference is
significant between H2 and the other gasses.
The critical pressure ratio of 12.6/2200 indicates that the flow rate is
sonic, Mach 1, at the exit of the broken valve. In order to determine the
maximum flow rate and hence maximum thrust produced the temperature of the
flowing gas at the "nozzle" needs to be calculated. As before, these
equations can be found in most fluid mechanics text books. The critical
pressure ratio is: H2 = 0.5266; N2 = 0.528; O2 = 0.5299.
The notation I use is 1 = inside the bottle, 2 is the "nozzle" exit, 3 is
ambient conditions at Bonneville in August (P = 12.6 psig, Temp = 95 Deg F).
The bottle has been sitting at the B'ville conditions so the internal gas is
also initially at 95 deg F. T1 = T3 = 460 + 95 = 555 deg R. The calculated
gas temperature at the "nozzle" exit is: for H2, T2 = 460.6 deg R; for N2,
T2 = 462.5 deg R; for O2 T2 = 464.4 deg R.
The velocity of the gas at the the exit of the "nozzle" is equal to the
sonic velocity of the gas at the temperature of the gas at the exit. The
equation for the sonic velocity can be found in any Thermodynamics text.
When I plugged in all the numbers for each of the gasses at the calculated
exit temperature, I get the following sonic velocities: H2 =4000 ft/sec; N2
= 1071.5 ft/sec; O2 = 1001.0 ft/sec.
The density of the gas in the "nozzle" throat can be like wise found. These
are using standard equations from thermodynamics and gas dynamics texts. The
air pressure aat the exit is 12.. psig and the t3emperature is that
calculated above for the nozzle. For hte gasses in question the density at
the nozzle is: H2 = 0.00015986 slugs; N2 = 0.00221245 slugs; O2 = 0.00251566
slugs.
Now, another assumption: The size of the nozzle. I looked at my welding
bottles and given the size of the valve, I suspect that the throat diameter
may be around 5/8 of an inch or so. Ao that cross sectional area, in square
feet is 0.00213 square feet.
The mass flow rate for each of the gasses can be determined now. This is
area times density times velocity: H2 = 0.00136 slugs/sec; N2 = 0.00505072
slugs/sec; O2 = 0.005365 slugs/sec.
Now, the bottom line. Thrust can now be calculated for the first second,
which is what I was interested in to begin with. Thrust is mass times
velocity. And the uiversal gravity constant has a part in this also. But, I
digress. H2 = 175.03 lbf; N2 = 174.12 lbf; O2 = 173.8 lbf. All for the first
second. Of course, the mass in the tank is less for the next second and
thish entire analysis has to be redone for each succeding timne increment.
But the bottom line is that the force decays off fairly rapidly.
So for a bottle of gas that weighs in at round a 100 pounds then the
acceleration would be about 1.75 g's. In the first second of bottle flight
it would go about 56 ft and would be traveling at about 56 ft/second. The
kinetic energy in the first second is about 4900 ft-lbs.
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