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I would like to know how I could estimate the friction caused by air flowing at a speed of 1000 km/h in a tube. The tube could be made out of any material, which makes it possible to calculate, preferably plastics, steel or Teflon.

Edit: The tube has a form of the Large Hadron Collider in Cern just with a circumference of 10000 km instead of 27 km.

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How serious are you about that particular velocity? For ordinary air this is near the speed of sound, and the usual, simple formalism needs to be augmented pretty heavily. – dmckee Nov 27 '11 at 23:32
@dmckee Thank you very much for your question. I know that this is a high speed, but it is the speed between 1000 and up to 1500 km/h that I am interested in. – I_like_traffic_lights Nov 27 '11 at 23:44
That crosses mach 1.0. Tough regime that I don't even know how to advice you on. For lower speeds, you compute the Reynolds number to determine the laminar/structured/turbulent flow situation and look up the answer in some Handbook or another (your local University library has it). Maybe you can do the same around mach one, but I don't know what books to look in. – dmckee Nov 27 '11 at 23:47
@dmckee Thanks again, for simplicity let's say that that we stay below Mach 1 (1224 km/h). I have also updated the question to add a detail. – I_like_traffic_lights Nov 28 '11 at 0:00
@I_like_traffic_lights: you sound like a mad bomber. Why would you be interested in this problem? – Ron Maimon Nov 28 '11 at 0:42
up vote 2 down vote accepted

As @dmckee and others have pointed out, this is not a straightforward problem to solve, helped of course by your choice one of the most difficult flow regimes (transonic) ;-)

(For what it's worth this is more of an engineering question than a physics question....)

Relevant parameters you will need to define are:

  • the diameter of the pipe
  • the state of the air at the beginning of its flow (any two of pressure, temperature and density)
  • whether the pipe is insulated or not (the question here is what happens to the heat produced by the drag, is it lost to the surroundings or is it kept in the airflow)

Some interesting background reading are (this is where the pipe loses heat to the surroundings and the temperature of the air is constant over the length of the pipe) and also (the pipe is insulated, so the temperature of the air rises along the length of the pipe).

Understanding Rayleigh and Fanno flow will give you lots of interesting insightsand will tell you about the qualitative nature of the effects (e.g. in Fanno flow, the Mach number of the air will be 1.0 along the entire length of the pipe, the temperature will rise, the velocity will rise, and the pressure and density will fall). But if you want to know specifically what the friction is, that will depend on additional items (surface roughness, etc) and it might be hard to find engineering data that is applicable to your regime of interest.

I will confess some curiosity to the application or idea behind the question. Your ring is a quarter the diameter of the earth and your flow speed is (roughly) the rotational speed at the equator.

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In normal engineering situations, you can ballpark the pressure drop pretty well by equating one velocity head to 40 pipe diameters. We ordinarily wouldn't apply this when you're so close to the speed of sound, but with your velocity of 280 m/s you're looking at a velocity head of around 50 kPa. If you're tube is 2.5 meters in diameter, you therefore lose 50 kPa for every hundred meters of pipe, for a total pressure drop of 500 MPa. Calculating your power losses by (flow)x(pressure) gives you a power of 750 GW, which is approximately the total electric generating capacity of the United States.

For the sake of comparison, the flux capacitor used to power the Delorean in Back to the Future requirede only 1.21 Gigawatts, and even that for only a very short period of time.

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SORRY. I do not think so. If you're tube is 2.5 meters in diameter, you therefore lose 50 kPa for every hundred meters of pipe, for a total pressure drop of 500 MPa. – user10417 Jul 9 '12 at 14:34

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