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
http://en.wikipedia.org/wiki/Rayleigh_flow (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 http://en.wikipedia.org/wiki/Fanno_flow (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.