Thermodynamics Air Cannon

FIn summary, the problem is to find appropriate values for the initial and final pressure and volume to fire a spherical pellet with a radius of 8.75 mm at a maximum velocity of 100 m/s, using an adiabatic expansion process. The equations used are the adiabatic expansion equation and the work equation. After an iterative process, the values of initial pressure, volume, and final volume are found to be 420465 Pa, 10^-5 m^3, and 2.79x10^-5 m^3 respectively. The barrel length is calculated to be 0.0744 m. The minimum final pressure is 1 atm and a length of 0.75 m is chosen
  • #1
RazerM
6
0

Homework Statement


1.75 J is required to fire a spherical pellet of radius 8.75 mm at a maximum velocity of 100 m/s

Find appropriate values for the inital pressure and volume and final pressure and volume


Homework Equations


Adiabatic Expansion
[tex]P_{1}V_{1}^{\gamma}=P_{2}V_{2}^{\gamma}[/tex]

[tex]\gamma=1.40\qquad P_{2}=10^5\text{ Pa}[/tex]

[tex]W_{\text{total}}=\frac{P_{1}V_{1}-P_{2}V_{2}}{\gamma-1}[/tex]

[tex]
\begin{align*}
W_{\text{total}} &= W_{\text{atm}}-W_{\text{useful}} \\ \frac{P_{1}V_{1}-P_{2}V_{2}}{\gamma-1}&=10^{5}(V_{2}-V_{1})-1.75
\end{align*}
[/tex]

The Attempt at a Solution


[tex]
\begin{align*}
V_{2}&=V_{1}\left( \frac{P_{1}}{P_{2}} \right)^{\frac{1}{\gamma}}\\
V_{2}&=V_{1}\left( \frac{P_{1}}{10^{5}} \right)^{\frac{1}{1.4}}\\
\end{align*}
[/tex]
It is then possible to rearrange work equation into
[tex]V_{1}\left( [P_{1}+4\cdot 10^{4}]-\left[ 1.4\cdot 10^{5}\left( \frac{P_{1}}{10^{5}} \right)^{\frac{1}{1.4}} \right] \right)=0.7[/tex]

[tex]\text{Then using an arbitary guess for }V_{1}\text{ and an iterative process to find }P_{1}\text{ I found }[/tex]
[tex]
V_{1} = 10^{-5}\text{ m}^{3}[/tex]
[tex]
P_{1} = 420465\text{ Pa}[/tex]
[tex]
V_{2} = 2.79\cdot 10^{-5}\text{ m}^{3}
[/tex]
Giving a ridiculous barrel length of
[tex]
\begin{align*}
L &= \frac{V_{2}-V{1}}{\pi r^{2}} \\
L &= 0.0744\text{ m}
\end{align*}
[/tex]


I'm not asking for a full solution; just a hint as to how to find out values that satify the criteria of work and have a reasonably small V_1 so that there can be many shots for say a 0.001m^3 tank
 
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  • #2
The pellet travels down a cylindrical barrel of length L whose cross-sectional area is [itex]A = \pi r^2[/itex] where r = 8.75 mm. So the final volume is LA. The minimum final pressure is 1 atm. Choose a reasonable length for the barrel, say .75 m. Find an initial volume and pressure such that the work done in adiabatically expanding to volume LA is equal to 1.75 J.

AM
 
Last edited:

What is a Thermodynamics Air Cannon?

A Thermodynamics Air Cannon is a device that uses the principles of thermodynamics to launch objects, typically using compressed air as the propelling force.

How does a Thermodynamics Air Cannon work?

A Thermodynamics Air Cannon works by compressing air in a chamber and then releasing it quickly, creating a burst of high pressure that propels an object out of the cannon. This is achieved through the conversion of potential energy (stored in the compressed air) into kinetic energy (movement of the object).

What is the purpose of a Thermodynamics Air Cannon?

The purpose of a Thermodynamics Air Cannon is to launch objects at high speeds and distances, typically for recreational or experimental purposes. It can also be used for practical applications such as cleaning or removing debris.

What factors affect the performance of a Thermodynamics Air Cannon?

The performance of a Thermodynamics Air Cannon is affected by factors such as the amount of compressed air used, the size and shape of the cannon, the weight and shape of the projectile, and the angle at which it is launched.

Are Thermodynamics Air Cannons safe to use?

When used properly and with caution, Thermodynamics Air Cannons are generally considered safe. However, it is important to follow safety guidelines and wear protective gear when operating the cannon. It is also important to use appropriate projectiles and not aim the cannon at people or animals.

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