**Unformatted text preview: **48 Rocket Propulsion Elements At optimum expansion the ideal exhaust velocity 02 is equal to the effective
exhaust velocity 6 and, from Equation 3—15, it is calculated to be 6000 ft/sec.
Therefore, the thrust F and speciﬁc impulse can be determined from Equations 242 and 2—2.
F = v2w/g = 6000 x 2.2/32.2 = 410 lb
1, = c g = 6000/32.2 = 186 see A number of interesting deductions can be made from this example.
Very high gas velocities (over a mile per second) can be obtained in rocket
nozzles. The temperature drop of the combustion gases through a rocket nozzle is appreciable. In the example given the temperature changed
through 201017 in a relatively short distance. This is not surprising, for the increase in the kinetic energy of the gases is derived from a decrease
of the enthalpy, which in turn is roughly proportional to the decrease in
temperature. Because the exhaust gases are still very hot (1990°R) when leaving the nozzle, they contain considerable energy not available for
conversion into kinetic energy of the jet. The required nozzle area decreases to a minimum (at 164-lb per in.2
pressure) and then increases again. Nozzles of this type (often called De
Laval nozzles after their inventor) consist of a convergent and a divergent
section. From the continuity equation the area is inversely proportional
to the ratio 0/ V. This quantity is also plotted in Figure 3~4. There is a
maximum in the curve 0/ 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 letter 5. e = Ag/A, (3—18) For any isentropic steady ﬂow process, such as occurs in rocket nozzles, the weight ﬂow can be computed from the continuity equation, the isentropic relations, and the nozzle gas velocity (Equations 3—2, 3—6, and
3—15) between any section at and the ﬁrst nozzle inlet section. w = Ampl /§[ iﬁ [(ﬁf/k — (Lm)(k+1)/k]}‘/é
R \ 1711 P1 P1 The maximum gas ﬂow per unit area occurs at the throat, and a unique
gas pressure corresponding to this maximum ﬂow will exist. This throat
pressure p, for a maximum ﬂow in an isentropic expansion nozzle can be
found by differentiating Equation 3—19 and setting the derivative equal to zero.
P; 2 ‘k/tk~1)
P1 _ (k + 1) <3_19) (3~20) T Nozzle Theory and Thermodynamic Relations 49 The throat pressure for which the isentropic weight ﬂow is a maximum
is called the critical pressure. The ﬂow through a given rocket nozzle with
given inlet condition is less than the maximum possible, if the pressure
ratio is larger than that deﬁned by Equation 3~20. Rocket thrust chambers
usually have suﬂicient chamber pressure to attain the critical pressure
at the throat. Various values of the critical pressure ratio for different
values ofk are shown in Table 3—1. At the point of critical pressure, the values of speciﬁc volume and the
temperature can be obtained from the isentropic relations and Equation 3—20:
)1/(k—D ) (3—22) V: k (3—21) Tl ll , +1
/1( 2 ' 2
T.
‘ik —+ 1 From Equations 3—15, 3‘20, and 3—22. the critical velocity or throat
velocity v, is obtained. (3—23) 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 temperature is T, and Equations 3—9 and 3~23
are identical. Therefore, the critical throat velocity u, is always equal to
the local acoustic velocity a for ideal nozzles in which critical conditions
prevail; the Mach number at the throat of an ideal rocket is unity. The
divergent portion ofa nozzle permits a further decrease in pressure and an
increase in velocity above the velocity of sound. If the nozzle is cut oﬁ"
at the throat section, the exit gas velocity will be sonic. The sonicand
supersonic ﬂow condition can be attained only if the critical pressure
prevails at the throat, that is, if (oz/p1 is equal or less than the quantity
deﬁned by Equation 3—20. There are, therefore, essentially three types
of nozzles: subsonic, sonic, and supersonic. They are described in Table
3—3. The supersonic nozzle is the one which is of interest to the rocket
engineer. The ratio between the inlet and exit pressures in all rockets is
sufﬁciently large to induce supersonic ﬂow. Only if the chamber pressure
drops below approximately 32 lb per square inch absolute is there any ...

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