300 200 100 0 Pressure, psia

300 200 100 0 Pressure, psia

Exit

Nozzle inlet

Exit

— Throat Nozzle inlet

FIGURE 3-3. Typical variation of cross-sectional area, temperature, specific volume, and velocity with pressure in a rocket nozzle.

area is inversely propportional to the ratio v/V. This quantity has also been plotted in Fig. 3-3. There is a maximum in the curve of v/V because at first the velocity increases at a greater rate than the specific volume; however, in the divergent section, the specific volume increases at a greater rate.

The minimum nozzle area is called the throat area. The ratio of the nozzle exit area A2 to the throat area A, is called the nozzle area expansion ratio and is designated by the Greek letter e. It is an important nozzle design parameter.

The maximum gas flow per unit area occurs at the throat where there is a unique gas pressure ratio which is only a function of the ratio of specific heats k. This pressure ratio is found by setting M = 1 in Eq. 3-13.

The throat pressure pt for which the isentropic mass flow rate is a maximum is called the critical pressure. Typical values of this critical pressure ratio range between 0.53 and 0.57. The flow through a specified rocket nozzle with a given inlet condition is less than the maximum if the pressure ratio is larger than that given by Eq. 3-20. However, note that this ratio is not that across the entire nozzle and that the maximum flow or choking condition (explained below) is always established internally at the throat and not at the exit plane. The nozzle inlet pressure is very close to the chamber stagnation pressure, except in narrow combustion chambers where there is an appreciable drop in pressure from the injector region to the nozzle entrance region. This is discussed in Section 3.5. At the point of critical pressure, namely the throat, the Mach number is one and the values of the specific volume and temperature can be obtained from Eqs. 3-7 and 3-12.

In Eq. 3-22 the nozzle inlet temperature Tx is very close to the combustion temperature and hence close to the nozzle flow stagnation temperature T0. At the critical point there is only a mild change of these properties. Take for example a gas with k = 1.2; the critical pressure ratio is about 0.56 (which means that pt equals almost half of the chamber pressure px)\ the temperature drops only slightly (Tt — 0.917^), and the specific volume expands by over 60% (V, = 1.61 K,). From Eqs. 3-15, 3-20, and 3-22, the critical or throat velocity vt is obtained:

The first version of this equation permits the throat velocity to be calculated directly from the nozzle inlet conditions without any of the throat conditions being known. At the nozzle throat the critical velocity is clearly also the sonic velocity. The divergent portion of the nozzle permits further decreases in pres sure and increases in velocity under supersonic conditions. If the nozzle is cut off at the throat section, the exit gas velocity is sonic and the flow rate remains a maximum. The sonic and supersonic flow condition can be attained only if the critical pressure prevails at the throat, that is, if Pi/Px is equal to or less than the quantity defined by Eq. 3-20. There are, therefore, three different types of nozzles: subsonic, sonic, and supersonic, and these are described in Table 3-1.

The supersonic nozzle is the one used for rockets. It achieves a high degree of conversion of enthalpy to kinetic energy. The ratio between the inlet and exit pressures in all rockets is sufficiently large to induce supersonic flow. Only if the absolute chamber pressure drops below approximately 1.78 atm will there be subsonic flow in the divergent portion of the nozzle during sea-level operation. This condition occurs for a very short time during the start and stop transients.

The velocity of sound is equal to the propagation speed of an elastic pressure wave within the medium, sound being an infinitesimal pressure wave. If, therefore, sonic velocity is reached at any point within a steady flow system, it is impossible for a pressure disturbance to travel past the location of sonic or supersonic flow. Thus, any partial obstruction or disturbance of the flow downstream of the nozzle throat with sonic flow has no influence on the throat or upstream of it, provided that the disturbance does not raise the downstream pressure above its critical value. It is not possible to increase the throat velocity or the flow rate in the nozzle by further lowering the exit pressure or even evacuating the exhaust section. This important condition is often described as choking the flow. It is always established at the throat and not the nozzle exit plane. Choked flow through the critical section of a supersonic nozzle may be derived from Eqs. 3-3, 3-21, and 3-23. It is equal to the mass flow at any section within the nozzle.

TABLE 3-1. Nozzle Types

Subsonic

Sonic

Supersonic

Throat velocity Exit velocity Mach number

Pressure ratio

Shape

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