1- Air jet should flow tangent to the ring against the rotation velocity vector in order to obtain the highest possible relative velocity. A deflector should increase the contact area between jet and ring outer surface if not the cooling surface will be rather small since at such jet velocities the Coanda effect will be reduced.

2- Pressure ratio indicates an over critical air jet speed for which you should determine the profile of the boundary layer and allows an estimation of the convection coefficient.

3- Convection area depends on the angle over which the deflector leads the jet, flow depends on the temperature you will allow to the air at end of contact.

I you want only to compute the heat loss then use the conduction equation for heat applied to the ring. Due to the fact that it is a high conductive material you could consider temperature field uniform and a steady state condition.

If you want to compute the air mass flow use the equations for an adiabatic flow through a nozzle.

Since you did not react as other do I presume that you did not understand what I wrote.

To help you here the result: with k=109 W/(m*K) for brass

the heat flow is

Q= 50.7 W/mm length of your cylinder.

I hope now you have all what you needed.

Thats aa great answer and very clear @ Nick Hamburger

One more question Mr. Nick

you said that the cooling effect will be more if the nozel is Tangent to the moving object in an opposite direction can you explain that a bit