Some ideas from Elon Musk’s original white paper on the Hyperloop:
The vehicle is streamlined to reduce drag and features a compressor at the leading face to ingest oncoming air for levitation and to a lesser extent propulsion. Aerodynamic simulations have demonstrated the validity of this ‘compressor within a tube’ concept. (Page 14)
4.1.3. Compressor
One important feature of the capsule is the onboard compressor, which serves two purposes. This system allows the capsule to traverse the relatively narrow tube without choking flow that travels between the capsule and the tube walls (resulting in a build-up of air mass in front of the capsule and increasing the drag) by compressing air that is bypassed through the capsule. (See P. 17 for details of the mechanics)
4.2. Tube
The main Hyperloop route consists of a partially evacuated cylindrical tube that connects the Los Angeles and San Francisco stations in a closed loop system (Figure 2). The tube is specifically sized for optimal air flow around the capsule improving performance and energy consumption at the expected travel speed.
The expected pressure inside the tube will be maintained around 0.015 psi (100 Pa, 0.75 torr), which is about 1/6 the pressure on Mars or 1/1000 the pressure on Earth. This low pressure minimizes the drag force on the capsule while maintaining the relative ease of pumping out the air from the tube. The efficiency of industrial vacuum pumps decreases exponentially as the pressure is reduced (Figure 13), so further benefits from reducing tube pressure would be offset by increased pumping complexity.
(P. 24)
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Overcoming the Kantrowitz Limit
Whenever you have a capsule or pod (I am using the words interchangeably) moving at high speed through a tube containing air, there is a minimum tube to pod area ratio below which you will choke the flow. What this means is that if the walls of the tube and the capsule are too close together, the capsule will behave like a syringe and eventually be forced to push the entire column of air in the system. Not good.
Nature’s top speed law for a given tube to pod area ratio is known as the Kantrowitz limit. This is highly problematic, as it forces you to either go slowly or have a super huge diameter tube. Interestingly, there are usually two solutions to the Kantrowitz limit – one where you go slowly and one where you go really, really fast.
The latter solution sounds mighty appealing at first, until you realize that going several thousand miles per hour means that you can’t tolerate even wide turns without painful g loads. For a journey from San Francisco to LA, you will also experience a rather intense speed up and slow down. And, when you get right down to it, going through transonic buffet in a tube is just fundamentally a dodgy prospect.
Both for trip comfort and safety, it would be best to travel at high subsonic speeds for a 350 mile journey. For much longer journeys, such as LA to NY, it would be worth exploring super high speeds and this is probably technically feasible, but, as mentioned above, I believe the economics would probably favor a supersonic plane.
The approach that I believe would overcome the Kantrowitz limit is to mount an electric compressor fan on the nose of the pod that actively transfers high pressure air from the front to the rear of the vessel. This is like having a pump in the head of the syringe actively relieving pressure.
It would also simultaneously solve another problem, which is how to create a low friction suspension system when traveling at over 700 mph. Wheels don’t work very well at that sort of speed, but a cushion of air does. Air bearings, which use the same basic principle as an air hockey table, have been demonstrated to work at speeds of Mach 1.1 with very low friction. In this case, however, it is the pod that is producing the air cushion, rather than the tube, as it is important to make the tube as low cost and simple as possible.
That then begs the next question of whether a battery can store enough energy to power a fan for the length of the journey with room to spare. Based on our calculations, this is no problem, so long as the energy used to accelerate the pod is not drawn from the battery pack.
This is where the external linear electric motor comes in, which is simply a round induction motor (like the one in the Tesla Model S) rolled flat. This would accelerate the pod to high subsonic velocity and provide a periodic reboost roughly every 70 miles. The linear electric motor is needed for as little as ~1% of the tube length, so is not particularly costly.
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Can it Really be Self-Powering?
For the full explanation, please see the technical section, but the short answer is that by placing solar panels on top of the tube, the Hyperloop can generate far in excess of the energy needed to operate. This takes into account storing enough energy in battery packs to operate at night and for periods of extended cloudy weather. The energy could also be stored in the form of compressed air that then runs an electric fan in reverse to generate energy, as demonstrated by LightSail.
http://www.spacex.com/sites/spacex/files/hyperloop_alpha-20130812.pdf