# Converging-Diverging Nozzle

by Jonny6001
Tags: convergingdiverging, nozzle
 PF Gold P: 1,507 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.