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to the first term (the momentum thrust) in Eqs. 2-14, 3-29, and 3-30 and not to the second term (pressure thrust).

A small nozzle divergence angle causes most of the momentum to be axial and thus gives a high specific impulse, but the long nozzle has a penalty in rocket propulsion system mass, vehicle mass, and also design complexity. A large divergence angle gives short, lightweight designs, but the performance is low. There is an optimum conical nozzle shape and length (typically between 12 and 18 degrees half angle) and it is usually a compromise which depends on the specific application and flight path.

The bell-shaped or contour nozzle (see Figs. 3-12 and 3-13) is probably the most common nozzle shape today. It has a high angle expansion section (20 to 50°) right behind the nozzle throat; this is followed by a gradual reversal of nozzle contour slope so that at the nozzle exit the divergence angle is small, usually less than a 10° half angle. It is possible to go to large divergence angles immediately behind the throat (20 to 50°) because the high relative pressure, the large pressure gradient, and the rapid expansion of the working fluid do not allow separation in this region unless there are discontinuities in the nozzle contour. The expansion in the supersonic bell nozzle is more efficient than in a simple straight cone of similar area ratio and length, because the wall contour is designed to minimize losses, as explained later in this section. For the past several decades most of the nozzles have been bell shaped.

A change of flow direction of a supersonic gas in an expanding wall geometry can only be achieved through expansion waves. An expansion wave occurs at a thin surface, where the flow velocity increases and changes its flow direction slightly, and where the pressure and temperature drop. These wave surfaces are at an oblique angle to the flow. As the gas passes through the throat, it undergoes a series of these expansion waves with essentially no loss of energy. In the bell-shaped nozzle shown in Fig. 3-14 these expansions occur internally in the flow between the throat and the inflection location /; the area is steadily increasing like a flare on a trumpet. The contour angle 6¿ is a maximum at the inflection location. Between the inflection point I and the nozzle exit E the flow area is still increasing, but at a diminishing rate, allowing further gas expansion and additional expansion waves. However, the contour of the nozzle wall is different and the change in cross-sectional area per unit length is decreasing. The purpose of this last segment of the contoured nozzle is to have a low divergence loss as the gas leaves the nozzle exit plane. The angle at the exit 0e is small, usually less than 10°. The difference between 0¿ and 6e is called the turn-back angle. When the gas flow is turned in the opposite direction (between points I and E) oblique compression waves will occur. These compression waves are thin surfaces where the flow undergoes a mild shock, the flow is turned, and the velocity is actually reduced slightly. Each of these multiple compression waves causes a small energy loss. By carefully determining the wall contour (by an analysis that uses a mathematical tool called the method of characteristics), it is possible to balance the oblique expansion waves with the oblique compression waves and minimize the energy loss. The analysis leading to the nozzle contour is presented in Chapter 20.33 of Ref. 3-3 and also in Refs. 3-8 to 3-11; it is based on supersonic aerodynamic flow, the method of characteristics (Ref. 3-1), and the properties of the expanding gas. Most of the rocket organizations have computer codes for this analysis. The radius of curvature or the contour shape at the throat region have an influence on the contour of the diverging bell-shaped nozzle section.

The length of a bell nozzle is usually given a fraction of the length of a reference conical nozzle with a 15° half angle. An 80% bell nozzle has a length (distance between throat plane and exit plane) that is 20% shorter than a comparable 15° cone of the same area ratio. Ref. 3-9 shows the original presentation by Rao of the method of characteristics applied to shorter bell nozzles. He also determined that a parabola was a good approximation for the bell-shaped contour curve (Ref. 3-3, Section 20.33), and parabolas have actually been used in some nozzle designs. The top part of Fig. 3-14 shows that the parabola is tangent (0,) at point I and has an exit angle (0e) at point E and a length L that has to be corrected for the curve 77. These conditions allow the parabola to be determined by simple geometric analysis or geometric drawing. A throat approach radius of 1.5 r, and a throat expansion radius of 0.4 r, were used. If somewhat different radii had been used, the results would have been only slightly different. The middle set of curves gives the relation between length, area ratio, and the two angles of the bell contour. The bottom set of curves gives the correction factors, equivalent to the X factor for conical nozzles, which are to be applied to the thrust coefficient or the exhaust velocity, provided the nozzles are at optimum expansions, that is, p2 = Pi-

60% length 70% length 80% length 90% length 100% length

10 20 30 40 Expansion area ratio e

60% length 70% length 80% length 90% length 100% length

60% length 70% length 80% length 90% length 100% length

10 20 30 40 Expansion area ratio e

Percent of length of a 15 deg half-angle conical nozzle with same area as bell shape

TABLE 3-4. Data on Several Bell-Shaped Nozzles

Area Ratio 10 25 50

Correction factor A 0.9829 0.9829 0.9829 80% Bell Contour

Length" 6.45 11.94 18.12

Correction factor A 0.985 0.987 0.988

Approximate half angle at inflection point and exit 25/10 30/8 32/7.5 (degrees) 60% Bell Contour

Length" 4.84 9.96 13.59

Correction factor A 0.961 0.968 0.974

Approximate half angle at inflection point and exit 32.5/17 36/14 39/18 (degrees)

"The length is given in dimensionless form as a multiple of the throat radius, which is one.

Table 3^1 shows data for parabolas developed from this figure, which allow the reader to apply this method and check the results. The table shows two shortened bell nozzles and a conical nozzle, each for three area ratios. It can be seen that as the length has been decreased, the losses are higher for the shorter length and slightly higher for small nozzle area ratios. A 1 % improvement in the correction factor gives about 1 % more specific impulse (or thrust) and this difference can be significant in many applications. The reduced length is an important benefit, and it is usually reflected in an improvement of the vehicle mass ratio. The table and Fig. 3-14 show that bell nozzles (75 to 85% length) are just as efficient as or slightly more efficient than a longer 15° conical nozzle (100% length) at the same area ratio. For shorter nozzles (below 70% equivalent length) the energy losses due to internal oblique shock waves become substantial and such short nozzles are not commonly used today.

For solid propellant rocket motor exhausts with small solid particles in the gas (usually aluminum oxide), and for exhausts of certain gelled liquid propel-lants, there is an impingement of these solid particles against the nozzle wall in

FIGURE 3-14. Top sketch shows comparison sketches of nozzle inner wall surfaces for a 15° conical nozzle, an 80% length bell nozzle, a 60% length bell nozzle, all at an area ratio of 25. The lengths are expressed in multiples of the throat radius r„ which is one here. The middle set of curves shows the initial angle 0,- and the exit angle 6e as functions of the nozzle area ratio and percent length. The bottom curves show the nozzle losses in terms of a correction factor. Adapted and copied with permission of AIAA from Ref. 6-1.

the reversing curvature section between / and E in Fig. 3-14. While the gas can be turned by oblique waves to have less divergence, the particles (particularly the larger particles) have a tendency to move in straight lines and hit the walls at high velocity. The resulting abrasion and erosion of the nozzle wall can be severe, especially with the ablative and graphite materials that are commonly used. This abrasion by hot particles increases with turn-back angle. If the turnback angle and thus also the inflection angle 6>, are reduced, the erosion can become acceptable. Typical solid rocket motors flying today have values of inflection angles between 20 and 26° and turn-back angles of 10 to 15°. In comparison, current liquid rocket engines without entrained particles have inflection angles between 27 and 50° and turn-back angles of between 15 and 30°. Therefore the performance enhancement caused by using a bell-shaped nozzle (high value of correction factor) is somewhat lower in solid rocket motors with solid particles in the exhaust.

The ideal bell-shaped nozzle (minimum loss) is long, equivalent to a conical nozzle of perhaps 10 to 12°, as seen in Fig. 3-12. It has about the same length as a full-length aerospike nozzle. This is usually too long for reasonable vehicle mass ratios.

Two-Step Nozzles. Several modifications of a bell-shaped nozzle have evolved that allow full or almost complete altitude compensation; that is, they achieve maximum performance at more than a single altitude. Figure 3-15 shows three concepts for a two-step nozzle, one that has an initial low area ratio A2/At for operation at or near the earth's surface and a larger second area ratio that improves performance at high altitudes. See Ref. 3-5.

The extendible nozzle requires actuators, a power supply, mechanisms for moving the extension into position during flight, fastening and sealing devices. It has successfully flown in several solid rocket motor nozzles and in a few liquid engine applications, where it was deployed prior to ignition. Although only two steps are shown, there have been versions with three steps; one is shown in Fig. 11-3. As yet it has not made the change in area ratio during rocket firing. The principal concerns are a reliable rugged mechanism to move the extension into position, the hot gas seal between the nozzle sections, and the extra weight involved.

The droppable insert concept avoids the moving mechanism and gas seal but has a potential stagnation temperature problem at the joint. It requires a reliable release mechanism, and the ejected insert creates flying debris. To date it has little actual test experience. See Ref. 3-12.

The dual bell nozzle concept uses two shortened bell nozzles combined into one with a bump or inflection point between them, as shown in Fig. 3-15. During ascent it functions first at the lower area ratio, with separation occurring at the inflection point. As altitude increases and the gas expands further, the flow attaches itself downstream of this point, with the flow filling the full nozzle exit section and operating with the higher area ratio at higher performance. There is a small performance penalty for a compromised bell nozzle

Second nozzle exit segment ' in stored position

Extendible nozzle with two segments

Second nozzle exit segment ' in stored position

Center line nozzle exit segment (fixed to chamber)

nozzle exit segment in deployed position after moving aft

Center line nozzle exit segment (fixed to chamber)

nozzle exit segment in deployed position after moving aft

Droppable insert (mechanisms for holding moving, or releasing the inserts are not shown)

Droppable insert (mechanisms for holding moving, or releasing the inserts are not shown)

Chamber

Chamber

Protrusion or hump in contour

Dual bell nozzle

Protrusion or hump in contour

FIGURE 3-15. Simplified diagrams of three altitude-compensating two-step nozzle concepts.

FIGURE 3-15. Simplified diagrams of three altitude-compensating two-step nozzle concepts.

contour with a circular bump. To date there has been little experience with this concept.

Nozzles with Aerodynamic Boundaries

The group of two-step nozzle concepts described above corresponds to the performance represented by upper portions of the two fixed area ratio nozzle curves shown in Fig. 3-10; the performance of a continuously varying nozzle with full altitude compensation is shown by the dashed curve. When integrated over the flight time, the extra performance is important for high velocity missions such as the single stage to orbit application. The three nozzles shown on the right side of Fig. 3-12 offer full altitude compensation and are discussed next. Refs. 3-5 and 3-8 give more information.

The plug nozzle or aerospike nozzle has an annular doughnut-shaped chamber with an annular nozzle slot. An alternate version has a number of individual small chambers (each with low area ratio short nozzles, a round throat, and a rectangular exit) arranged in a circle around a common plug or spike. The outside aerodynamic boundary of the gas flow in the divergent section of the nozzle is the interface between the hot gas and the ambient air; there is no outer wall as in a conical or bell-shaped nozzle. As the external or ambient pressure is reduced during the ascending flight, this gas boundary expands outward, causes a change in pressure distribution on the central spike, and allows an automatic and continuous altitude compensation. The aerospike contour with the minimum flow losses turns out to be very long, similar in length to an optimum bell nozzle as shown in Figs. 3-12 and 3-13. The mass flow per unit exit area is relatively uniform over the cross section and the divergence losses are minimal.

If the central plug is cut off or truncated and the wall contour is slightly altered, then the nozzle will be very short, as shown in Fig. 3-13; it will have some internal supersonic waves and will show a small but real loss in thrust compared to a nozzle with a full central spike. The pressure distribution and the heat transfer intensity vary on the inner contoured spike wall surface. Figure 8-14 shows a typical pressure distribution over the contoured spike surface at high and low altitudes.

The pressure in the recirculating trapped gas of the subsonic region below the bottom plate also exerts a thrust force. The losses caused by the cut-off spike can be largely offset by injecting a small amount of the gas flow (about 1% of total flow) through this base plate into the recirculating region, thus enhancing the back pressure on the base plate. The advantages of the truncated aerospike are short length (which helps to reduce the length and mass of the flight vehicle), full altitude compensation, no flow separation from the wall at lower altitudes, and ease of vehicle/engine integration for certain vehicle configurations.

The linear aerospike nozzle is a variation of the round axisymmetric aerospike nozzle. Basically, it is an unrolled version of the circular configuration. It is explained further in Chapter 8.2.

In the expansion deflection nozzle (Fig. 3-12) the flow from the chamber is directed radially outward away from the nozzle axis. The flow is turned on a curved contour outer diverging nozzle wall. The nozzle has been shortened and has some internal oblique shock wave losses. The hot gas flow leaving the chamber expands around a central plug. The aerodynamic interface between the ambient air and gas flow forms an inner boundary of the gas flow in the diverging nozzle section. As the ambient pressure is reduced, the hot gas flow fills more and more of the nozzle diverging section. Altitude compensation is achieved by this change in flow boundary and by changes in the pressure distribution on the outer walls.

Multiple Nozzles. If a single large nozzle is replaced by a cluster of smaller nozzles on a solid motor (all at the same cumulative thrust), then it is possible to reduce the nozzle length. Similarly, if a single large thrust chamber of a liquid engine is replaced by several smaller thrust chambers, the nozzle length will be shorter, reducing the vehicle length and thus the vehicle structure and inert mass. Russia has pioneered a set of four thrust chambers, each with 25%

of the total thrust, assembled next to each other and fed from the same liquid propellant feed system. This quadruple thrust chamber arrangement has been used effectively on many large Russian space launch vehicles and missiles. As seen in Fig. 3-13, this cluster is about 30% shorter than a single large thrust chamber. The vehicle diameter at the cluster nozzle exit is somewhat larger, the vehicle drag is somewhat higher, and there is additional engine complexity and engine mass.

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