You're going to need to get your hands on thermodynamic charts (aka. psychometric chart, thermodynamic diagram, phase diagram, mollier diagram) for nitrogen, and maybe air, if you don't have them already. They come in several forms:
- p-T Pressure-Temperature
- p-v Pressure-Volume
- p-h Pressure-Enthalpy
- T-S Temperature-Entropy
- h-S Enthalpy-Entropy
All have their uses and some will be more useful in specific applications. For gas compression and liquifaction p-v and T-S(or p-h) are the most useful respectively, although as a disclaimer I'll note that it's bee a long while since I've used either. All of these charts are also pretty daunting the first time you see them and it might take a bit of time before you know your way around them. You're going to need to know how to plot thermodynamic cycles on them (which is essentially what you use them for) as a means of determining the conditions required/produced, and you're also going to need to know what to do when your cycle enters the 2-phase envelope.
Thanks to the wonders of the internets you can look at
other people's mathmatical solutions for similar cases. This also has the only chart for nitrogen I've seen online so far (fig.2 is a p-h diagram).
In this instance, LN2 is produced at atmospheric pressure by pressurising air from 1 x atmospheric (~0.1MPa) to 150 x atmospheric (~15MPa) in a 5 compressor cascade, and then cooling it to 160 Kelvin (-113 C) before passing it through the throttle. This converts 7% by mass of the nitrogen passing through the throttle to liquid.
The same result could have been achieved at a lower pressure, but it would have required a lower temperature too - this is something you will have to take into account, as you're going to be limited by what temperatures and pressures you can achieve.