Converging-Diverging Nozzle

[Jonny6001]

Hello,

I am interested in the boundary condition effects on a converging-diverging nozzle.

If you have a certain inlet pressure and temperature to a supersonic converging-diverging nozzle which exits into a the outlet pressure, is there a limit to the velocity that you can generate at the nozzle exit?

I mean as the gas velocity speeds up in the diverging section the temperature and pressure reduce. If the local nozzle static pressure gets too low relative to the exit pressure then you over-expand the gas and shocks form inside the nozzle.
Does this mean that the inlet and exit pressures limit the maximum exit gas speed that can be reached? And short of increasing the inlet pressure or decreasing the exit pressure no further velocity increase can be had? Would increasing the inlet temperature with the same inlet pressure increase the available expansion?

Thank you.

 

[boneh3ad]

Assuming you are using a thermally and calorically perfect gas, then the maximum velocity that can be achieved from a given reservoir condition depends entirely on the total temperature in that reservoir. You can start with the energy equation and the definition of enthalpy and derive the following relation:
$$u_{\text{max}}^2 = \dfrac{2a_0^2}{\gamma -1}$$
where $a_0 = \sqrt{\gamma R T_0}$ is the speed of sound based on the total temperature in the reservoir $T_0$ and $\gamma$ is the ratio of specific heats and $R$ is the specific gas constant.

Of course, that is the maximum theoretical velocity regardless of nozzle geometry. If you have a more specific nozzle geometry in mind you can get more information. For example, for a given nozzle, the maximum mass flow rate that you can get through the nozzle is based on the point where the nozzle becomes choked and can be shown based on conservation of mass to be
$$\dot{m} = \dfrac{p_{01}A^*}{\sqrt{T_{01}}} \sqrt{\dfrac{\gamma}{R}}\left( \dfrac{2}{\gamma+1} \right)^{\frac{\gamma +1}{2(\gamma-1)}}$$
where $p_{01}$ and $T_{01}$ are the total pressure and temperature respectively in the reservoir and $A^*$ is the throat area.

Other flow variables like the velocity for a given nozzle geometry (most importantly $A_e/A^*$) are relatively easily calculable, though it is slightly more involved than above.