I wonder whether there is a good equation to estimate gas (say, air) flow rate through a needle valve?

Right now I know the pipe diameter (1/4") and the outflow pressure (20 psi).

I'd talk to the vendor and see if they have a Cv value for the needle valve because you need that.  Once you have that, you can use a variety of methods:

- VOGT has one for their manual valves
- Fisher and other control valve manufacturers would have flow equations for their valves (For Fisher, you'd have to assume a C1 factor to convert Cv to Cg but Cg is 'likely' about 30*Cv).  The flow rates won't be exactly the same but they'll be in the same ball park.

Your needle valve vendor might even have an equation for you.  Now, if you want the flow with the valve only partially open, your vendor is going to have to give you some indication of the needle valve's characteristic curve (linear, quick opening, equal percentage, etc).

If the 1/4" piping connected to the valve has a significant pressure drop, you'll need to include its effect as well to estimate the flow rate.

you needle valve catalog as a flow equation and a list of valve coefficient data that will solve your problem.

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