Q:

Liquid Nitrogen Expansion to Desired Temperature

I want to take liquid nitrogen in a pressurized container and spray it through a hose onto objects in a room.  
The container has a pressure regulator so I can adjust pressure as desired.  The requirement is that a particular temperature is reached on exit from the hose.  
I understand that the final temperature will of course be room temperature but that is after reaching equilibrium.  I desire a temperature above liquid but still very cold no where near room temp upon exiting the hose (T_desired or T_3).  
There would be a phase change from liquid to gas, and then a gaseous expansion to the desired temperature that can be achieved by regulating the pressure valve.
What are the equations for this process, particularly if I want to know the volume of nitrogen gas emitted from the hose at the desired temperature?
Note: This is for a practical application so the answer can contain simplifying assumptions, but please list them if used.  Ex. Constant volume as liquid.
Edit: If temp cannot be regulated by the pressure valve alone, then I would like to know what temp it is at, as well as the volume.
This container is a good example: princetoncryo.com/cylinder/cryo-cyl/cryo-cyl-35l-lp.html

A:

This problem has certainly been solved in practice in a number of applications - cryosurgery, cryogenic treatment of metals, and cryogenic cooling of x-ray crystallography samples, but the approach will depend on the application.
The flow of liquid nitrogen is always always unstable and fluctuating, because it is usually impossible to prevent small amounts of boiling from generating unpredictable amounts of gas.  If this is unacceptable, you will need to boil the liquid first in a vessel or phase separator, and then pass it through the nozzle.
If you intend for the liquid to be boiled in the hose by external heating or contact with ambient air, you may be disappointed.  Droplets of liquid will remain in the stream because of the Leidenfrost effect - poor thermal contact between the droplets and the wall of the hose.  The result will be a spray of droplets mixed with gas.
You might therefore need a proper heat exchanger - using a long thin pipe or perhaps a metal mesh, to ensure the liquid is fully boiled.  Ambient heat can be quite adequate, and is routinely used in boil-off heat exchangers, provided you don't mind large amounts of ice and water condensing on the outside.  I wouldn't worry about condensation of oxygen - as long as this occurs in an open environment the oxygen will not build up in dangerous amounts.  Oxygen buildup can certainly be dangerous if it occurs inside an enclosed container.
If you need a controlled temperature, then I agree that a heater and temperature sensor are required.  A pipe exposed to the environment will gradually cool down as it becomes insulated by ice.
The dewar will have two outlets - one for liquid (unpredictable as I explained above) and one for gas.  The gas outlet is very useful, but is not fully insulated, so there will be a minimum temperature below which it will not go.  You say that the container has a pressure regulator - but you need to clarify this.  Does it have a built-in boiloff device to maintain the pressure?  If not, you will find that extracting liquid or gas depressurises the dewar.
Before you do any of that, however, you need to say how big a room you are going to be doing this in, what size of dewar you are using, and what rate of flow you are hoping to use.  I would like to carry on discussing with a StackExchanger who is still breathing air with at least a bit of oxygen in it.
Sorry if this answer is disappointingly unmathematical - you need to define the physical regime first, before equations can be usefully applied.

