SAMPLE - 48 Rocket Propulsion Elements At optimum expansion...

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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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