13. Can a nozzle completely condense a gas?

            Consider a converging nozzle discharging gas from a very large tank. Let  be the outside pressure and  be the tank pressure with  > . If  is continually decreased the flow rate will continually increase until  reaches a “critical” value . Decreasing  below will not increase the flow rate. For an ideal gas and the critical pressure  is

The maximum flow velocity is , which is the local speed of sound. . Since  we have

Now the maximum flow velocity  in terms of the inlet conditions is

           In a straight nozzle the flow velocity will increase if the nozzle area increases beyond the throat, will remain constant if the area remains constant (neglecting friction losses), and will remain constant if the area decreases. In this last case the flow rate will decrease. The maximum density is 0.640 .

            If the nozzle longitudinal axis curves after reaching sonic velocity then it may be possible to increase the density beyond 0.640 , and increase the flow velocity. Consider a small increment of path along a curved nozzle. Assume the curved nozzle produces the thermal separation and the gas at the outer radius has a velocity increase and the gas at the inner radius has a velocity decrease then the gas readjusts to the equilibrium Maxwell-Boltzmann distribution of speeds. This results in a thermal velocity which varies linearly with the radius. Now using this linear variation of thermal velocity with radius, the mass continuity, linear momentum conservation (in two directions since the flow is two dimensional), angular momentum conservation in one direction, and energy conservation there is a specific value of nozzle area at the end of this increment which is determined subject to these equations. This area is smaller than the entrance area to this section of nozzle. Further, the exit velocity may be greater than the entrance velocity. If this process is continued the flow velocity may approach the mean speed of the background gas and the density may increase by an order of magnitude! Possibly a complete condensation will occur.