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.
From the first expression we see that we can vary $V$ and $P$ such that $T$ remains constant. But from $TV^{γ-1}=\text{const.} \implies T=\frac{\text{const.}}{V^{γ-1}}$, since $V$ has varied, the temperature should not remain constant.
The above statement is wrong.
The ideal gas equation is as follows:
$$PV = nRT$$
If a process is isothermal, then $PV$ must remain constant.
If a process is adiabatic, it must satisfy $PV^{\gamma} = \text{const.}$
The above two conditions cannot be satisfied together.
Therfore, an isothermal process cannot be adiabatic process and vice versa.
Best Answer
Refrigerant is NOT an ideal gas, so the ideal gas law is inappropriate in this application. The temperature of the refrigerant is related to the pressure that the refrigerant is experiencing through the Antoine equation. See https://en.wikipedia.org/wiki/Antoine_equation. As the refrigerant flows through the expansion valve, it experiences a significant drop in pressure. Due to this, the refrigerant is superheated on the low pressure side of the expansion valve, and it boils as a result. The heat required for boiling comes from the refrigerant itself, so the temperature of the refrigerant rapidly drops as boiling occurs, and this temperature drop does NOT require any heat transfer to or from the environment. Thus, on the low pressure side of the expansion valve, there is a mixture of low pressure vapor and low pressure liquid that goes to the evaporator.