Your statement leaves me with a lot of questions. I have used cryo fluid to cool a compressor with inter cooler issues, its not ideal.
First does your company have standards for cryogenic piping design? If not, CGA or EIGA are the best source for cryo work. Maybe ask your liquid Nitrogen Supplier for some data. There is a whole can of worms here....
If you are using US units, you can find the proprieties relative to water for flow characteristics, but that's assuming your not having mix phase flow by the time the fluid gets to the nozzle. This will cause vapor lock issues depending on system design, which will cause erratic cooling. To make this work best you will need sub cooled Nitrogen, well insulated pipe (maybe VJ pipe), appropriately designed pipe work and, the flow is steady. If you have this, the calculation is text book.
There are charts for flash of cyro fluids through nozzles depending on the delta P, this will give you an approximation of liquid carry over. From there you will have an idea of the mixing of your hot nitrogen with the remaining liquid to complete the vaporization.
Just to recap, the problem is pretty text book, however field application can be a challenge with mix phase flow. I don't know your application/budget but I ended up using a Valtek mark-one instead of a nozzle, and had two thermo couples which controlled the valve off a target temp.
