The Hyperloop is a hypothetical new fast transport system between cities, which works by launching pods that carry people through a very low air pressure tunnel. The tunnel and Hyperloop pod are both cylindrically symmetric. If the radius of the pod is too close to the radius of the tunnel, then the air flowing through the cracks will become supersonic, generating shockwaves and disrupting the smooth operation of the Hyperloop. The pod helps prevent this by sucking up and compressing 0.49 kg/s of air through the front of the pod as it travels, but this still means some air must flow around the pod.
If the radius of the tunnel is 1.1 m, the speed of the pod is 300 m/s and the air in the tunnel is at a pressure of 99 Pa and temperature of 293 K, then what is the maximum radius of the pod in m that will keep the air flow relative to the pod below the speed of sound?
- Treat air as incompressible. This isn't true at high mach number, but it will make the solution easier.
- The molar mass of air is \(\mu= 29\) g/mol.
- Neglect the effects of gravity and viscosity.
- Assume the air flow is perfectly laminar.