## Isentropic Flow Through Nozzles

In a converging-diverging nozzle a large fraction of the thermal energy of the gases in the chamber is converted into kinetic energy. As will be explained, the gas pressure and temperature drop dramatically and the gas velocity can reach values in excess of two miles per second. This is a reversible, essentially isen-tropic flow process and its analysis is described here. If a nozzle inner wall has a flow obstruction or a wall protrusion (a piece of weld splatter or slag), then the kinetic gas enery is locally converted back into thermal energy essentially equal to the stagnation temperature and stagnation pressure in the chamber. Since this would lead quickly to a local overheating and failure of the wall, nozzle inner walls have to be smooth without any protrusion. Stagnation conditions can also occur at the leading edge of a jet vane (described in Chapter 16) or at the tip of a gas sampling tube inserted into the flow.

Velocity

From Eq. 3-2 the nozzle exit velocity v2 can be found:

This equation applies to ideal and non-ideal rockets. For constant k this expression can be rewritten with the aid of Eqs. 3-6 and 3-7. The subscripts 1 and 2 apply to the nozzle inlet and exit conditions respectively:

This equation also holds for any two points within the nozzle. When the chamber section is large compared to the nozzle throat section, the chamber velocity or nozzle approach velocity is comparatively small and the term v\ can be neglected. The chamber temperature T\ is at the nozzle inlet and, under isentropic conditions, differs little from the stagnation temperature or (for a chemical rocket) from the combustion temperature. This leads to an important simplified expression of the exhaust velocity v2, which is often used in the analysis.

It can be seen that the exhaust velocity of a nozzle is a function of the pressure ratio P\/P2, the ratio of specific heats k, and the absolute temperature at the nozzle inlet as well as the gas constant R. Because the gas constant for any particular gas is inversely proportional to the molecular mass 93i, the exhaust velocity or the specific impulse are a function of the ratio of the absolute nozzle entrance temperature divided by the molecular mass, as is shown in Fig. 3-2. This ratio plays an important role in optimizing the mixture ratio in chemical rockets.

Equations 2-14 and 2-15 give the relations between the velocity v2, the thrust F, and the specific impulse Is; it is plotted in Fig. 3-2 for two pressure ratios and three values of k. Equation 3-16 indicates that any increase in the gas temperature (usually caused by an increase in energy release) or any decrease of the molecular mass of the propellant (usually achieved by using light molecular mass gases rich in hydrogen content) will improve the perfor-manace of the rocket; that is, they will increase the specific impulse Is or the exhaust velocity v2 or c and, thus, the performance of the vehicle. The influences of the pressure ratio across the nozzle pl/p2 and of the specific heat ratio k are less pronounced. As can be seen from Fig. 3-2, performance increases