The Projectile Velocity of an Air Cannon

Transcription

1 WJP, PHY381 (211) Wabash Journal of Physics v4.3, p.1 The Projectile Velocity of an Air Cannon Z. J. Rohrbach, T. R. Buresh, and M. J. Madsen Departent of Physics, Wabash College, Crawfordsville, IN (Dated: May 6, 211) Pressurized air cannons are inexpensive, safe, and ipressive, therefore aking the excellent for studying projectile otion in an undergraduate laboratory. However, in order to study projectile otion, the exit velocity of the projectile fro the air cannon ust be known, and this exit velocity is a function of the internal ballistics of the cannon. Several theoretical odels for the internal ballistics of an air cannon have been proposed, but experiental data on exit velocity as a function of initial conditions have been sparse. We have constructed an air cannon, and we provide experiental data in stark disagreeent with the existing odels along with a new odel to explain the.

2 WJP, PHY381 (211) Wabash Journal of Physics v4.3, p.2 Cannons have been around for over a illenniu and any attepts have been ade to correctly describe the trajectory of their projectiles. Cannons are powered by gas expansion in the barrel which causes the projectile to be thrown forward by the force of the expanding gas and play a signicant role in illustrating various physics properties such as recoil [1], conservation of oentu [1], the work-kinetic energy theore [2], and air drag [3], aong others. These effects can be illustrated with a relatively siple air cannon. The use of an air cannon as opposed to a rifle or other firear has several advantages. First, it creates a safer lab environent. But also, the internal dynaics of an air cannon are uch sipler and easier to understand because they do not rely on gas expansion driven by an unpredictable explosive echanis. The dynaics of the projectile s otion after it leaves the barrel of the air cannon are well understood; however, these calculations rely on an accurate knowledge of the initial conditions of the syste. This experient seeks to understand the conditions that give the exit velocity. a.) x A b.) L V A c.) V valve L A d L FIG. 1. Three odels of the expansion of gas in an air cannon. (a) An explosive gas expands adiabatically [4]. This odel is described by Eq. (3). (b) Gas fro a pressurized reservoir expands isotherally [5]. See Eq. (5). (c) The expansion of the gas is liited by a valve with a finite flow factor. This is the odel advocated by this paper. The basic concept behind an air cannon is that a reservoir of volue V is pressurized at P. This reservoir is connected to a long barrel of cross-sectional area A loaded with a projectile of ass. As gas in the reservoir expands, the pressure accelerates the projectile

3 WJP, PHY381 (211) Wabash Journal of Physics v4.3, p.3 along the length L of the barrel at which point the projectile exits the cannon at a speed v. We would like to find the exit velocity v as a function of these initial conditions. To do so, two odels have been previously proposed. In the first, illustrated in Fig. 1(a), the air fro the reservoir (of volue V = Ax ) expands quasistatically and adiabatically [4]. Assuing no air leakage, there are three forces on the slug (illustrated in Fig. 1). There is a force AP (x) due to the pressure fro the gas, an opposing force AP at fro the atosphere, and a linear friction force f. By Newton s Second Law, these add up to give F = dv dt = v dv dx = AP (x) AP at f, (1) where P at is atospheric pressure and f is a ter of constant friction. Since we are assuing adiabatic expansion, we know that P (x)(v + Ax) γ = P V γ, where γ = 7/5 for diatoic gasses such as air. Thus, in the adiabatic case, we have ( ) v dv dx = A P V 7/5 (V + Ax) P 7/5 at f, (2) which yields an exit velocity at x = L of ( v = (P 5 ( ) ) ) 2/5 V V 1 2ALP at 2Lf. (3) AL + V The second odel, illustrated in Fig. 1(b) takes the expansion of gas to be quasistatic and isotheral. Following the derivation given in ref. [5], we know that P (x)(v + Ax) = P V. Plugging into Eq. (1) gives or v = v dv ( ) dx = A P V V + Ax P at f. (4) 2 ( ( P V ln 1 + AL ) ) ALP at Lf. (5) V Note that this equation differs slightly fro the result offered by ref. [5]. This is because we are not aking the siplifying assuption that P P at. We find that in both of these equations, the frictional ter f serves as siply a horizontal offset for the odel. Because we find that the application of a very sall P P at ejects the slug and because we see fro our data in Fig. 3 that our data are not horizontally offset, we take f.

4 WJP, PHY381 (211) Wabash Journal of Physics v4.3, p.4 A real pneuatic air cannon ust have a valve between the reservoir and the barrel in order to allow pressurization of the reservoir before the firing of the projectile. While one could in theory iagine a perfect valve that does not have any appreciable effect on the air that flows past it, this is in practice hard to realize. A reasonable valve type is a solenoidactuated diaphrag valve, which has an associated flow coefficient C v. Such valves have been used on pneuatic cannons on Mythbusters and the Punkin Chunkin contest [6]. The flow rate Q through a valve with coefficient C v is a function of the pressure drop across the valve [7]. Therefore, it is unreasonable to ignore the effect of the valve, since the pressures on either side of the valve are not the sae. We propose a new odel, shown in Fig. 1(c), that takes into account the flow rate through the valve. The flow rate Q (a quantity given in units of olecules/s differentiated fro Q which is in L/in at STP) is a function of the pressure in the reservoir P (t) and the pressure in the barrel P b (t). It is given to be [7] (.471)BC v P (t) 1/G g T, P (t) 2P b (t), Q(t) = ( ( )) ( ) BC v P (t) P b(t) 1 P b(t) /G 3 P (t) P (t) g T, P (t) 2P b (t), where G g = 1 is the specific gravity of air, T is the teperature in the reservoir, and B is a proportionality constant to convert between Q and Q. We are assuing that teperature is constant, which is an approxiately valid assuption given our experiental data (we only easure a 1 K teperature drop in firing the cannon). Using the Ideal Gas Law, the gas inside the tank and outside the tank is characterized by the equations (6) P (t)v = N(t)k B T, (7) P b (t)a (d + x(t)) = N b (t)k B T. (8) The nuber of olecules in the tank and barrel are governed by the flow of olecules between the through the valve, dn/dt = Q and dn b /dt = Q, and, by Newton s second law, the position function, as above, is governed by d 2 x/dt 2 = A(P b (t) P at )/. These differential equations, when cobined with the initial conditions, can be nuerically solved to give v = dx/dt as a function of initial pressure P. We use an ASME-Code Horizontal Pressure Tank with a volue of V = 4.196±.1 L (all easureents given to a 95% confidence interval). We attach a pressure sensor (Oegadyne

5 WJP, PHY381 (211) Wabash Journal of Physics v4.3, p.5 Solenoid-Actuated Diaphrag Valve To Air Copressor Manual Valve Therocouple V Pressure Sensor h Slug Barrel A = π D 2 / 4 l d x L Photogates A P b (t) f Slug A P at FIG. 2. A scheatic of the air cannon. The tank is a reservoir with a volue V = ±.1 L, initially charged to pressure P. The tank is discharged using a diaphrag valve with C v = 2.8. The pressurized air then propels the slug, with a height of h = ±.58 c, a distance of L = ±.8 c out of the barrel of cross sectional area of A = ±.26 c 2. Model PX39-1GV), therocouple (Oega Model TC-K-NPT-E-72), solenoid-actuated diaphrag valve (Granzow Model 21HN5KY16-14W) with C v = 2.8, and air intake hand valve to the tank as shown in Fig. 2. The diaphrag valve opens when a current of 44 A activates a solenoid in the valve. This air rushes into the attached barrel a 1.95-c diaeter, c long 34/34L sealess stainless steel threaded pipe and forces the slug out. The exit velocity is easured by two photogates positioned l = 24.6 ±.9 c apart. We collect data siultaneously fro the therocouple, pressure sensor, and photogates at a sapling rate of 1 khz. The photogate trigger has an internal clock of 1 khz. Data acquisition is triggered when an aeter, connected to the solenoid actuator circuit, reads an increasing current across 5 A. We loaded the cannon with an acetal copolyer cylindrical plastic slug of ass = 9.6 ±.48 g, height h = ±.58 c, and diaeter D = ±.58 c. The diaeter of the slug was such that it just fit into the barrel of the cannon. We tested whether air could escape fro around the edges of the slug by closing the diaphrag valve and attepting to load the cannon. The slug was sufficiently air tight that it built back pressure when we tried to insert it.

6 WJP, PHY381 (211) Wabash Journal of Physics v4.3, p.6 Our loading procedure is to open both the diaphrag valve and the hand valve to ake sure that there is not a pressure built up between the slug and the tank. Then, we place the slug in the barrel and use a steel rod to slowly push the slug back a predeterined length L h = ±.8 c. We then close the solenoid-actuated diaphrag valve before reoving the rod. We use an air copressor to pressurize the tank to the desired initial pressure P, close the hand valve, and wait as long as two and a half inutes until the reading on the digital pressure gauge stabilizes. After this, we fire by opening the diaphrag valve. Using this loading procedure, we collected data for the exit velocity v of the slug as a function of initial reservoir pressure P. These data are shown in Fig. 3(c). We also did the sae thing for a slug of approxiately twice the length and ass (specifically, = 19.4 ±.48 g and h = ±.58) as shown in Fig. 3(a) and an aluinu slug ( = 19.9±.48 g and h = ±.58 c) as shown in Fig. 3(b). It is clear that our data are in gross disagreeent with both the adiabatic and isotheral odels. However, we are in uch better agreeent with a nuerical calculation of the valve flow odel presented above. However, we would expect our odel to be an upper bound, given that we are not taking into account the higher-order dissipative effects of turbulent flow and air drag in the cannon. One such way to incorporate drag ay have been preposed by ref. [8], in which the authors consider the fact that the colun of air outside the slug needs to be accelerated in addition to the slug itself. The first-order linear friction ter has also not been included in our odel. However, as discussed above, we only expect this to introduce a horizontal offset that our data do not show. We also took data for different initial positions L of the projectile in the barrel. These data are shown in Fig. 4. Our odel is given credence by the fact that it approxiately qualitatively atches our data. We believe that the discrepancy coes fro the fact that we are not taking into account the effects of turbulent flow or pressure wave propagation tie. In conclusion, we have shown that both the adiabatic and isotheral expansion odels are not consistent with the aterial liits of a real air cannon fired by eans of opening a valve. This is because the assuption undergirding both odels that the air pressure in the reservoir is the sae as the pressure in the barrel is difficult to eet given the necessity of a valve with associated flow coefficient C v. We put forward an alternative odel that still serves as an upper bound for exit velocity of the slug that is uch closer to the experiental

7 WJP, PHY381 (211) Wabash Journal of Physics v4.3, p.7 15 a.) v s 1 5 b.) P kpa 15 v s 1 5 c.) 2 4 P kpa v s P kpa FIG. 3. (color online.) Exit velocity as a function of initial reservoir pressure of (a) a plastic slug with = 9.6 ±.48 g and h = ±.58, (b) a plastic slug with = 19.4 ±.48 g and h = ±.58, and (c) an aluinu slug with = 19.9±.48 g and h = ±.58 c. The pink curve is the isotheral odel and the blue curve is the adiabatic odel. The two odels are close to each other because the teperature drop associated with the adiabatic expansion is so sall. Our data disagree with both. The orange curve is a nuerical plot of our new odel, which provides a uch ore reasonable upper bound to our data.

8 WJP, PHY381 (211) Wabash Journal of Physics v4.3, p v s L c FIG. 4. Exit velocity as a function of initial position L of a slug in the barrel. This slug has = 9.6 ±.48 g and h = ±.58. The pink curve is the isotheral odel and the blue curve is the adiabatic odel. The two odels are close to each other because the teperature drop associated with the adiabatic expansion is so sall. Our data disagree with both. The orange curve, which is in approxiate qualitative agreeent with our data. data. In the future we hope to use this odel to accurately predict the final position of a projectile fired fro an air cannon. [1] B. Taylor. Recoil experients using a copressed air cannon. The Phys. Teacher (26). [2] K. Tsukaoto and M. Uchino. The Blowgun deonstration experient. The Phys. Teacher (28). [3] H. R. Kep. Trajectories of projectiles in air for sall ties of flight. A. Association Phys. Teachers. 55 (12), (1987). [4] C. E. Mungan. Internal Ballistics of a pneuatic potato cannon. Eur. J. Phys (29). [5] M. Denny. Internal Ballistics of an Air Gun, The Phys. Teacher (211). [6] Solenoid-actuated diaphrag valves have been used in air cannons by, for exaple, Discovery Channel s Mythbusters in Episode 61 (suarized on the unaffiliated web page http : //kwc.org/ythbusters/26/9/episode 61 deadly straw priar.htl).

9 WJP, PHY381 (211) Wabash Journal of Physics v4.3, p.9 Also, the ore efficient butterfly valves have been used in construction of air cannons in the Punkin Chunkin World Chapionship (see http : // 1/air cannon sends pupkins 37 feet). [7] Swagelok Copany. Valve Sizing Technical Bulletin. MS-6-84-E. downloads/webcatalogs/en/s 6 84.pdf (27). [8] E. Ayars and L. Buchholtz. Analysis of the vacuu cannon. A. J. Phys. 72 (7), (24).