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An approximate value of this velocity can also be obtained from the throat velocity and Fig. 3-4. The ideal propellant consumption for optimum expansion conditions is m = F/v2 = 5000/2605 = 1.919 kg/sec

The specific volume at the entrance to the nozzle equals

Vl = RTi/Pi = 355.4 x 2800/(2.068 x 106) = 0.481 m3/kg

At the throat and exit sections the specific volumes are obtained from Eqs. 3-21 and

The areas at the throat and exit sections and the nozzle area ratio A2/A, are

An approximate value of this area ratio can also be obtained directly from Fig. 3-5 for k = 1.30 and p\/p2 = 811.2. The exit temperature is given by

Thrust and Thrust Coefficient

The efflux of the propellant gases or the momentum flux-out causes the thrust or reaction force on the rocket structure. Because the flow is supersonic, the pressure at the exit plane of the nozzle may be different from the ambient pressure and the pressure thrust component adds to the momentum thrust as given by Eq. 2-14:

The maximum thrust for any given nozzle operation is found in a vacuum where /?3 = 0. Between sea level and the vacuum of space, Eq. 2-14 gives the variation of thrust with altitude, using the properties of the atmosphere such as those listed in Appendix 2. Figure 2-2 shows a typical variation of thrust with altitude. To modify values calculated for optimum operating conditions (p2 = Pi) for given values of px,k, and A2I A„ the following expressions may be used. For the thrust,

At the throat and exit sections the specific volumes are obtained from Eqs. 3-21 and

The areas at the throat and exit sections and the nozzle area ratio A2/A, are

A, =mV,/v, = 1.919 x 0.766/1060 = 13.87 cm2 A2=mV2/v2 = 1.919 x 83.15/2605 = 612.5 cm2 6 = a2/A, = 612.5/13.87 = 44.16

T2 = T1(p2/p1f-m = 2800(0.002549/2.068)0'2307 = 597 K

For the specific impulse, using Eqs. 2-5, 2-18, and 2-14,

If, for example, the specific impulse for a new exit pressure p2 corresponding to a new area ratio A2/A, is to be calculated, the above relations may be used.

Equation 2-14 can be expanded by modifying it and substituting v2, v, and V, from Eqs. 3-16, 3-21, and 3-23.

Atvtv2

AtPl

The first version of this equation is general and applies to all rockets, the second form applies to an ideal rocket with k being constant throughout the expansion process. This equation shows that the thrust is proportional to the throat area At and the chamber pressure (or the nozzle inlet pressure) p\ and is a function of the pressure ratio across the nozzle p\/p2, the specific heat ratio k, and of the pressure thrust. It is called the ideal thrust equation. The thrust coefficient CF is defined as the thrust divided by the chamber pressure pl and the throat area A,. Equations 2-14, 3-21, and 3-16 then give c = ÂA2 Pi /M2

The thrust coefficient CF is a function of gas property k, the nozzle area ratio e, and the pressure ratio across the nozzle p\/p2, but independent of chamber temperature. For any fixed pressure ratio p\/p$, the thrust coefficient CF and the thrust F have a peak when p2= p3. This peak value is known as the optimum thrust coefficient and is an important criterion in nozzle design considerations. The use of the thrust coefficient permits a simplification to Eq. 3-29:

Equation 3-31 can be solved for CF and provides the relation for determining the thrust coefficient experimentally from measured values of chamber pressure, throat diameter, and thrust. Even though the thrust coefficient is a function of chamber pressure, it is not simply proportional to p\, as can be seen from Eq. 3-30. However, it is directly proportional to throat area. The thrust coefficient can be thought of as representing the amplification of thrust due to the gas expanding in the supersonic nozzle as compared to the thrust that would be exerted if the chamber pressure acted over the throat area only.

The thrust coefficient has values ranging from about 0.8 to 1.9. It is a convenient parameter for seeing the effects of chamber pressure or altitude variations in a given nozzle configuration, or to correct sea-level results for flight altitude conditions.

Figure 3-6 shows the variation of the optimum expansion (p2 = Pi) thrust coefficient for different pressure ratios p\/p2, values of k, and area ratio e. The complete thrust coefficient is plotted in Figs 3-7 and 3-8 as a function of pressure ratio p\/p^ and area ratio for k = 1.20 and 1.30. These two sets of curves are useful in solving various nozzle problems for they permit the evaluation of under- and over-expanded nozzle operation, as explained below. The values given in these figures are ideal and do not consider such losses as divergence, friction or internal expansion waves.

When pi/pi becomes very large (e.g., expansion into near-vacuum), then the thrust coefficient approaches an asymptotic maximum as shown in Figs. 3-7 and 3-8. These figures also give values of CF for any mismatched nozzle (p2 / Pi), provided the nozzle is flowing full at all times, that is, the working fluid does not separate or break away from the walls. Flow separation is discussed later in this section.

Characteristic Velocity and Specific Impulse

The characteristic velocity c* was defined by Eq. 2-18. From Eqs. 3-24 and 3-31 it can be shown that c* = IhA _ /sfo = _£_ __<JkRT\ (3-32)

It is basically a function of the propellant characteristics and combustion chamber design; it is independent of nozzle characteristics. Thus, it can be used as a figure of merit in comparing propellant combinations and combustion chamber designs. The first version of this equation is general and allows the determination of c* from experimental data of m, pi, and At. The last version gives the maximum value of c* as a function of gas properties, namely k, the chamber temperature, and the molecular mass 9Ji, as determined from the theory in Chapter 5. Some values of c* are shown in Tables and 5-5. The term c*-efficiency is sometimes used to express the degree of completion of the energy release and the creation of high temperature, high pressure gas in the chamber. It is the ratio of the actual value of c*, as determined from measurements, and the theoretical value (last part of Eq. 3-32), and typically has a value between 92 and 99.5 percent.

Using Eqs. 3-31 and 3-32, the thrust itself may now be expressed as the mass flow rate times a function of the combustion chamber (c*) times a function of the nozzle expansion CF),

FIGURE 3-6. Thrust coefficient CF as a function of pressure ratio, nozzle area ratio, and specific heat ratio for optimum expansion conditions (p2 = pi)-

p tp

FIGURE 3-6. Thrust coefficient CF as a function of pressure ratio, nozzle area ratio, and specific heat ratio for optimum expansion conditions (p2 = pi)-

FIGURE 3-7. Thrust coefficient CF versus nozzle area ratio for k = 1.20.

-j FIGURE 3-8. Thrust coefficient CF versus nozzle area ratio for k = 1.30.

Some authors use a term called the discharge coefficient CD which is merely the reciprocal of c*. Both CD and the characteristic exhaust velocity c* are used primarily with chemical rocket propulsion systems.

The influence of variations in the specific heat ratio k on various parameters (such as c, c*A2/At, v2/v„ or Is) is not as large as the changes in chamber temperature, pressure ratio, or molecular mass. Nevertheless, it is a noticeable factor, as can be seen by examining Figs. 3-2 and 3^1 to 3-8. The value of k is 1.67 for monatomic gases such as helium and argon, 1.4 for cold diatomic gases such as hydrogen, oxygen, and nitrogen, and for triatomic and beyond it varies between 1.1 and 1.3 (methane is 1.11 and ammonia and carbon dioxide 1.33). In general, the more complex the molecule the lower the value of k\ this is also true for molecules at high temperatures when their vibrational modes have been activated. The average values of k and 9JJ for typical rocket exhaust gases with several constituents depend strongly on the composition of the products of combustion (chemical constituents and concentrations), as explained in Chapter 5. Values of k and 9JJ are given in Tables 5^1, 5-5, and 5-6.

Example 3-4. What is percentage variation in thrust between sea level and 25 km for a rocket having a chamber pressure of 20 atm and an expansion area ratio of 6? (Use k= 1.30.)

SOLUTION. At sea level: Pi/Pi = 20/1.0 = 20; at 25 km: = 20/0.0251 = 754 (see Appendix 2).

Use Eq. 3-30 or Fig. 3-8 to determine the thrust coefficient (hint: use a vertical line on Fig. 3-8 corresponding to A2/A, = 6.0). At sea level: CF = 1.33. At 25 km: CF = 1.64.

Under- and Over-Expanded Nozzles

An under-expanded nozzle discharges the fluid at an exit pressure greater than the external pressure because the exit area is too small for an optimum area ratio. The expansion of the fluid is therefore incomplete within the nozzle, and must take place outside. The nozzle exit pressure is higher than the local atmospheric pressure.

In an over-expanded nozzle the fluid attains a lower exit pressure than the atmosphere as it has an exit area too large for optimum. The phenomenon of over-expansion for a supersonic nozzle is shown in Fig. 3-9, with typical pressure measurements of superheated steam along the nozzle axis and different back pressures or pressure ratios. Curve AB shows the variation of pressure with the optimum back pressure corresponding to the area ratio. Curves AC and AD show the variation of pressure along the axis for increasingly higher external pressures. The expansion within the nozzle proceeds normally for the

Divergence

Divergence

FIGURE 3-9. Distribution of pressures in a converging-diverging nozzle for different flow conditions. Inlet pressure is the same, but exit pressure changes. Based on experimental data from A. Stodala.

initial portion of the nozzle. At point / on curve AD, for example, the pressure is lower than the exit pressure and a sudden rise in pressure takes place which is accompanied by the separation of the flow from the walls (separation is described later).

The non-ideal behavior of nozzles is strongly influenced by the presence of compression waves or shock waves inside the diverging nozzle section, which are strong compression discontinuities and exist only in supersonic flow. The sudden pressure rise in the curve ID is such a compression wave. Expansion waves, also strictly supersonic phenomena, match the flow from a nozzle exit to lower ambient pressures. Compression and expansion waves are described in Chapter 18.

The different possible flow conditions in a supersonic nozzle are as follows:

1. When the external pressure p3 is below the nozzle exit pressure p2, the nozzle will flow full but will have external expansion waves at its exit (i.e., under-expansion). The expansion of the gas inside the nozzle is incomplete and the value of CF and Is, will be less than at optimum expansion.

2. For external pressures p3 slightly higher than the nozzle exit pressure p2, the nozzle will continue to flow full. This occurs until p2 reaches a value between about 25 and 40% of /?3. The expansion is somewhat inefficient and CF and Is will have lower values than an optimum nozzle would have. Shock waves will exist outside the nozzle exit section.

3. For higher external pressures, separation of the flow will take place inside the divergent portion of the nozzle. The diameter of the supersonic jet will be smaller than the nozzle exit diameter. With steady flow, separation is typically axially symmetric. Figs. 3-10 and 3-11 show diagrams of separated flows. The axial location of the separation plane depends on the local pressure and the wall contour. The point of separation travels downstream with decreasing external pressure. At the nozzle exit the flow in the center portion remains supersonic, but is surrounded by an annular shaped section of subsonic flow. There is a discontinuity at the separation location and the thrust is reduced, compared to a nozzle that would have been cut off at the separation plane. Shock waves exist outside the nozzle in the external plume.

4. For nozzles in which the exit pressure is just below the value of the inlet pressure, the pressure ratio is below the critical pressure ratio (as defined by Eq. 3-20) and subsonic flow prevails throughout the entire nozzle. This condition occurs normally in rocket nozzles for a short time during the start and stop transients.

The method for estimating pressure at the location of the separation plane inside the diverging section of a supersonic nozzle has usually been empirical. Reference 3-4 shows separation regions based on collected data for several dozen actual conical and bell-shaped nozzles during separation. Reference 3-5 describes a variety of nozzles, their behavior, and methods used to estimate the location and the pressure at separation. Actual values of pressure for the over-expanded and under-expanded regimes described above are functions of the specific heat ratio and the area ratio (see Ref. 3-1).

The axial thrust direction is not usually altered by separation, because a steady flow usually separates uniformly over a cross-section in a divergent nozzle cone of conventional rocket design. During transients, such as start and stop, the separation may not be axially symmetric and may cause momentary but large side forces on the nozzle. During a normal sea-level transient of a large rocket nozzle (before the chamber pressure reaches its full value) some momentary flow oscillations and non-symmetric separation of the jet can occur during over-expanded flow operation. Reference 3-4 shows that the magnitude and direction of transient side forces can change rapidly and erratically. The resulting side forces can be large and have caused failures of nozzle exit cone structures and thrust vector control gimbal actuators. References 3-5 and 3-6 discuss techniques for estimating these side forces.

When the flow separates, as it does in a highly over-expanded nozzle, the thrust coefficient CF can be estimated if the point of separation in the nozzle is known. Thus, CF can be determined for an equivalent smaller nozzle with an exit area equal to that at the point of separation. The effect of separation is to increase the thrust and the thrust coefficient over the value that they would have if separation had not occurred. Thus, with separated gas flow, a nozzle designed for high altitude (large value of e) would have a larger thrust at sea level than expected, but not as good as an optimum nozzle; in this case separation may actually be desirable. With separated flow a large and usually heavy portion of the nozzle is not utilized and the nozzle is bulkier and longer than necessary. The added engine weight and size decrease flight performance. Designers therefore select an area ratio that will not cause separation.

Because of uneven flow separation and potentially destructive side loads, sea-level static tests of an upper stage or a space propulsion system with a high area ratio over-expanded nozzle are usually avoided; instead, a sea-level test nozzle with a much smaller area ratio is substituted. However, actual and simulated altitude testing (in an altitude test facility similar to the one described in Chapter 20) would be done with a nozzle having the correct large area ratio. The ideal solution that avoids separation at low altitudes and has high values of CF at high altitudes is a nozzle that changes area ratio in flight. This is discussed at the end of this section.

For most applications, the rocket system has to operate over a range of altitudes; for a fixed chamber pressure this implies a range of nozzle pressure ratios. The condition of optimum expansion (p2 = Pi) occurs only at one altitude, and a nozzle with a fixed area ratio is therefore operating much of the time at either over-expanded or under-expanded conditions. The best nozzle for such an application is not necessarily one that gives optimum nozzle gas expansion, but one that gives the largest vehicle flight performance (say, total impulse, or specific impulse, or range, or payload); it can often be related to a time average over the powered flight trajectory.

Example 3-5. Use the data from Example 3-4 (px = 20 atm, e = 6.0, k = 1.30) but instead use an area ratio of 15. Compare the altitude performance of the two nozzles with different e by plotting their CF against altitude. Assume no shocks inside the nozzle.

SOLUTION. For the e = 15 case, the optimum pressure ratio pi/p¡ = P\¡Pi, and from Fig. 3-6 or 3-8 this value is about 180;/>3 = 20/180 = 0.111 atm, which occurs at about 1400 m altitude. Below this altitude the nozzle is over-expanded. At sea level,Pi/Pi = 20 and pi = 1 atm. As shown in Fig. 3-10, separation would occur. From other similar nozzles it is estimated that separation will occur approximately at a cross-section where the total pressure is about 40% of p3, or 0.4 atm. The nozzle would not flow full below an area ratio of about 6 or 7 and the gas jet would only be in the center of the exit area. Weak shock waves and jet contraction would then raise the exhaust jet's pressure to match the one atmosphere external pressure. If the jet had not separated, it would have reached an exit pressure of 0.11 atm, but this is an unstable condition that could not be maintained at sea level. As the vehicle gains altitude, the separation plane would

Altitude, m

FIGURE 3-10. Thrust coefficient CF for two nozzles with different area ratios. One has jet separation below about 7000 m altitude. The fully expanded exhaust plume is not shown in the sketch.

Altitude, m

FIGURE 3-10. Thrust coefficient CF for two nozzles with different area ratios. One has jet separation below about 7000 m altitude. The fully expanded exhaust plume is not shown in the sketch.

gradually move downstream until, at an altitude of about 7000 m, the exhaust gases would occupy the full nozzle area.

The values of CF can be obtained by following a vertical line for e = 15 and e = 6 in Fig. 3-8 for different pressure ratios, which correspond to different altitudes. Alternatively, Eq. 3-30 can be used for better accuracy. Results are similar to those plotted in Fig. 3-10. The lower area ratio of 6 gives a higher CF at low altitudes, but is inferior at the higher altitudes. The larger nozzle gives a higher CF at higher altitudes.

Figure 3-11 shows a comparison of altitude and sea-level behavior of three nozzles and their plumes at different area ratios for a typical three-stage satellite launch vehicle. When fired at sea-level conditions, the nozzle of the third stage with the highest area ratio will experience flow separation and suffer a major performance loss; the second stage will flow full but the external plume will contrast; since p2 < p^ there is a loss in Is and F. There is no effect on the first stage nozzle.

Example 3-6. A rocket engine test gives the following data: thrust F = 53,000 lbf, propellant flow m = 208 lbm/sec, nozzle exit area ratio A2/A, = 10.0, atmospheric

During flight During seaievel static tests

During flight During seaievel static tests

FIGURE 3-11. Simplified sketches of exhaust gas behavior of three typical rocket nozzles for a three-stage launch vehicle. The first vehicle stage has the biggest chamber and the highest thrust but the lowest nozzle area ratio, and the top or third stage usually has the lower thrust but the highest nozzle area ratio.

FIGURE 3-11. Simplified sketches of exhaust gas behavior of three typical rocket nozzles for a three-stage launch vehicle. The first vehicle stage has the biggest chamber and the highest thrust but the lowest nozzle area ratio, and the top or third stage usually has the lower thrust but the highest nozzle area ratio.

pressure at test station (the nozzle flows full) /?3 = 13.8 psia, and chamber pressure Pi = 620 psia. The test engineer also knows that the theoretical specific impulse is 289 sec at the standard reference conditions of Pi = 1000 psia and Pi = 14.7 psia, and that k = 1.20. Correct the value of the thrust to sea-level expansion and the specific impulse corresponding. Assume the combustion temperature and k do not vary significantly with chamber pressure; this is realistic for certain propellants.

SOLUTION. The actual pressure ratio was Pl/Pi = 620/13.8 = 44.9; the ideal pressure ratio at standard conditions would have been equal to 1000/14.7 = 68.0 and the actual pressure ratio for expansion to sea level would have been 620/14.7 = 42.1. The thrust coefficient for the test conditions is obtained from Fig. 3-7 or from Eq. 3-30 as Cf = 1.52 (for P]/Pi = 44.9, e = 10 and k = 1.20). The thrust coefficient for the corrected sea-level conditions is similarly found to be 1.60. The thrust at sea level would have been F = 53,000 (1.60/1.52) = 55,790 lbf. The specific impulse would have been

The specific impulse can be corrected in proportion to the thrust coefficient because k, T, and therefore c* do not vary with px \ Is is proportional to c if m remains constant. The theoretical specific impulse is given for optimum expansion, i.e., for a nozzle area ratio other than 10.0. From Fig. 3-6 or 3-7 and for pi/p2 = 68.0 the thrust coefficient is 1.60 and its optimum area ratio approximately 9.0. The corrected specific impulse is accordingly 255 (1.60/1.51) = 270 sec. In comparison with the theoretical specific impulse of 289 sec, this rocket has achieved 270/289 or 93.5% of its maximum performance.

Figs. 3-10 and 3-11 suggest that an ideal design for an ascending (e.g., launch) rocket vehicle would have a "rubber-like" diverging section that could be lengthened so that the nozzle exit area could be made larger as the ambient pressure is reduced. The design would then allow the rocket vehicle to attain its maximum performance at all altitudes as it ascends. As yet we have not achieved a simple mechanical hardware design with this full altitude compensation similar to "stretching rubber." However, there are a number of practical nozzle configurations that can be used to alter the flow shape with altitude and obtain maximum performance. They are discussed in the next section.

Influence of Chamber Geometry

When the chamber has a cross section that is larger than about four times the throat area (Ax/At > 4), the chamber velocity v\, can be neglected, as was mentioned in explaining Eqs. 3-15 and 3-16. However, vehicle space or weight constraints often require smaller thrust chamber areas for liquid propellant engines and grain design considerations lead to small void volumes or small perforations or port areas for solid propellant motors. Then v\ can no longer be neglected as a contribution to the performance. The gases in the chamber expand as heat is being added. The energy necessary to accelerate these expanding gases within the chamber will also cause a pressure drop and an additional energy loss. This acceleration process in the chamber is adiabatic (no heat transfer) but not isentropic. This loss is a maximum when the chamber diameter is equal to the nozzle diameter, which means that there is no converging nozzle section. This has been called a throatless rocket motor and has been used in a few tactical missile booster applications, where there was a premium on minimum inert mass and length. The flight performance improvement due to inert mass savings supposedly outweighs the nozzle performance loss of a throatless motor. Table 3-2 lists some of the performance penalties for three chamber area ratios.

Because of this pressure drop within narrow chambers, the chamber pressure is lower at the nozzle entrance than it would be if AxjAt had been larger. This causes a small loss in thrust and specific impulse. The theory of this loss is given in Ref. 3-7.

TABLE 3-2. Estimated Losses for Small-Diameter Chambers

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