Ok, here's my design/analysis of the cycle:
The effect you are trying to harness is the Joule-Thomson effect, via a throttling process, which seems at first glance like it makes sense but is actually a pretty strange bird.
https://en.wikipedia.org/wiki/Joule–Thomson_effect
One might think from harnessing our intuition that PV=nRT (the ideal gas equation) implies that a reduction in pressure will have a corresponding reduction in temperature, but it actually isn't that simple because volume is also a variable in this equation. What you actually get in an ideal gas reducing pressure reduces volume by the same proportion and no change in temperature results.
The J-T effect is thus a
real gas effect, based on the non-ideal behavior of gases. This turns out to be important for our example because of the resulting strangeness:
sometimes when you expand a gas the temperature drops, but sometimes it rises. As it turns out, for air (well, nitrogen at least), we run into this region and that prevents a single-stage
freezer from working using air. So rather than complicate this by trying to design a two- stage system, I'm going to punt and just design a
refrigerator, not a
freezer. Turns out, the wiki article does most of the work for us in this sample problem:
https://en.wikipedia.org/wiki/Joule-Thomson_effect#Throttling_in_the_T-s_diagram
The sample shows that if you start with 300K air at 200 bar(!) and expand it to 1 bar, you get 270K air. If we assume the refrigerator needs the air to be supplied 5K below the resulting fridge temperature, that's perfect. If we assume a refrigerator uses half the energy of a freezer and provides half the cooling (because its warmer), that means
@Asymptotic's example would be a 287.5 watt power input for 300 W (1022 BTU) of cooling. Airflow required based on the 5K delta-T and the specific heat of air is 178 standard cubic meters per hour (105 CFM). Note: I believe
@Asymptotic's vortex cooler capacity is rated versus ambient temperature, not versus freezer/fridge temperature, which is why I our results are out of line (why think the 35 CFM quoted is way low).
Now, 178 standard cubic meters per hour compressed to 200 bar is a monster compressor. Here's the thermodynamics:
https://en.wikipedia.org/wiki/Compressor#Effect_of_Cooling_During_the_Compression_Process
Let's assume roughly isothermal compression 300k (in reality it will start at the fridge temp, but I don't think that's a significant difference). The input power required for compressing 178 m3/hr from 1 to 200bar is 29 kW.