The advantage of using a ring main for air distribution is not that you can take advantage of the air flowing through both sides of the ring (which could potentially make the pipe size smaller). The advantage of a ring main is that a portion of the ring can be shut off for maintenance and still keep most of the distribution system live. In fact this makes the pipe size potentially larger because you need to allow for the situation where the air has to flow the "long way around" when the shorter route is shut off.
I would not recommend using Hardy-Cross or any other network based process. You should use the same process as you would for your Xmas tree design, taking into account the actual flow length when various sections are shut off.
The most difficult part is trying to work out what the design flow rates should be. If you assume that all take-offs are being used at full capacity all the time it could make the pipe very large. But you usually cannot ignore the possibility that several take-offs will be used at the same time. This same design choice will have been made when specifying the air compressor capacity, so you may be able to get some help from that process.