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Axial distance

Thrust chamber contour

FIGURE 8-8. Typical axial heat transfer rate distribution for liquid propellant thrust chambers and solid propellant rocket motors. The peak is always at the nozzle throat and the lowest value is usually near the nozzle exit.

Thrust chamber contour

FIGURE 8-8. Typical axial heat transfer rate distribution for liquid propellant thrust chambers and solid propellant rocket motors. The peak is always at the nozzle throat and the lowest value is usually near the nozzle exit.

high values are for the nozzle throat region of large bipropellant thrust chambers and high-pressure solid rocket motors. The lower values are for gas generators, nozzle exit sections, or small thrust chambers at low chamber pressures.

Cooling of Thrust Chambers

The primary objective of cooling is to prevent the chamber and nozzle walls from becoming too hot, so they will no longer be able to withstand the imposed loads or stresses, thus causing the chamber or nozzle to fail. Most wall materials lose strength and become weaker as temperature is increased. These loads and stresses are discussed in the next section. With further heating, the walls would ultimately fail or even melt. Cooling thus reduces the wall temperatures to an acceptable value.

Basically, there are two cooling methods in common use today. The first is the steady state method. The heat transfer rate and the temperatures of the chambers reach thermal equilibrium. This includes regenerative cooling and radiation cooling. The duration is limited only by the available supply of pro-pellant.

Regenerative cooling is done by building a cooling jacket around the thrust chamber and circulating one of the liquid propellants (usually the fuel) through it before it is fed to the injector. This cooling technique is used primarily with bipropellant chambers of medium to large thrust. It has been effective in applications with high chamber pressure and high heat transfer rates. Also, most injectors use regenerative cooling.

In radiation cooling the chamber and/or nozzle have only a single wall made of high temperature material. When it reaches thermal equilibrium, this wall usually glows red or white hot and radiates heat away to the surroundings or to empty space. Radiation cooling is used with monopropellant thrust chambers, bipropellant and monopropellant gas generators, and for diverging nozzle exhaust sections beyond an area ratio of about 6 to 10 (see Fig. 8-2). A few small bipropellant thrusters are also radiation cooled. This cooling scheme has worked well with lower chamber pressures (less than 250 psi) and moderate heat transfer rates.

The second cooling method relies on transient heat transfer or unsteady heat transfer. It is also called heat sink cooling. The thrust chamber does not reach a thermal equilibrium, and temperatures continue to increase with operating duration. The heat absorbing capacity of the hardware determines its maximum duration. The rocket combustion operation has to be stopped just before any of the exposed walls reaches a critical temperature at which it could fail. This method has mostly been used with low chamber pressures and low heat transfer rates. Heat sink cooling of thrust chambers can be done by absorbing heat in an inner liner made of an ablative material, such as fiber-reinforced plastics. Ablative materials are used extensively in solid propellant rocket motors and will be discussed further in Chapters 11 and 14. The analysis of both of these cooling methods is given in the next section of this chapter.

Film cooling and special insulation are supplementary techniques that are used occasionally with both methods to locally augment their cooling capability. All these cooling methods will be described further in this chapter.

Cooling also helps to reduce the oxidation of the wall material and the rate at which walls would be eaten away. The rates of chemical oxidizing reactions between the hot gas and the wall material can increase dramatically with wall temperature. This oxidation problem can be minimized not only by limiting the wall temperature, but also by burning the liquid propellants at a mixture ratio where the percentage of aggressive gases in the hot gas (such as oxygen or hydroxyl) is very small, and by coating certain wall materials with an oxidation-resistant coating; for example iridium has been coated on the inside of rhenium walls.

Cooling with Steady-State Heat Transfer. Cooled thrust chambers have provisions for cooling some or all metal parts coming into contact with hot gases, such as chamber walls, nozzle walls, and injector faces. Internal cooling passages, cooling jackets, or cooling coils permit the circulation of a coolant. Jackets can consist of separate inner and outer walls or of an assembly of contoured, adjacent tubes (see Figs. 8-1 and 8-9). The inner wall confines the gases, and the spaces between the walls serves as the coolant passage. The nozzle throat region is usually the location that has the highest heat-transfer intensity and is therefore the most difficult to cool. For this reason the cooling jacket is often designed so that the coolant velocity is highest at the critical regions by restricting the coolant passage cross section,

Injector i

Injector i

Top view without manifold

FIGURE 8-9. Diagram of a tubular cooling jacket. The tubes are bent to the chamber and nozzle contours; they are formed hydraulically to give a variable cross section to permit the same number of tubes at the throat and exit diameters. Coolant enters through the inlet manifold into every other tube and proceeds axially to the nozzle exit manifold, where it then enters the alternate tubes and returns axially to go directly to the injector.

Exit (Section C)

Top view without manifold

FIGURE 8-9. Diagram of a tubular cooling jacket. The tubes are bent to the chamber and nozzle contours; they are formed hydraulically to give a variable cross section to permit the same number of tubes at the throat and exit diameters. Coolant enters through the inlet manifold into every other tube and proceeds axially to the nozzle exit manifold, where it then enters the alternate tubes and returns axially to go directly to the injector.

and so that the fresh cold coolant enters the jacket at or near the nozzle. While the selection of the coolant velocity and its variation along the wall for any given thrust chamber design depends on heat-transfer considerations, the selection of the coolant passage geometry often depends on pressure loss, stresses, and manufacturing considerations. An axial flow cooling jacket, or a tubular wall, has a low hydraulic friction loss but is practical only for large coolant flows (above approximately 9 kg/sec). For small coolant flows and small thrust units, the design tolerances of the cooling jacket width between the inner and outer walls or the diameters of the tubes, become too small, or the tolerances become prohibitive. Therefore, most small thrust chambers use radiation cooling or ablative materials.

In regenerative cooling the heat absorbed by the coolant is not wasted; it augments the initial energy content of the propellant prior to injection, increasing the exhaust velocity slightly (0.1 to 1.5%). This method is called regenerative cooling because of the similarity to steam regenerators. The design of the tubular chamber and nozzle combines the advantages of a thin wall (good for reducing thermal stresses and high wall temperatures) and a cool, lightweight structure. Tubes are formed to special shapes and contours (see Figs. 8-1 and 8-9), usually by hydraulic means, and then brazed, welded, or soldered together (see Ref. 8-3). In order to take the gas pressure loads in hoop tension, they are reinforced on the outside by high-strength bands or wires. While Fig. 8-9 shows alternate tubes for up and down flow, there are chambers where the fuel inlet manifold is downstream of the nozzle throat area and where the coolant flow is up and down in the nozzle exit region, but unidirectionally up in the throat and chamber regions.

Radiation cooling is another steady-state method of cooling. It is simple and is used extensively in the low heat transfer applications listed previously. Further discussion of radiation cooling is given in the Materials and Fabrication subsection. In order for heat to be radiated into space, it is usually necessary for the bare nozzle and chamber to stick out of the vehicle. Figure 8-18 shows a radiation-cooled thrust chamber. Since the white hot glowing radiation-cooled chambers and/or nozzles are potent radiators, they may cause undesirable heating of adjacent vehicle or engine components. Therefore, many have insulation (see Fig. 8-15) or simple external radiation shields to minimize these thermal effects; however, in these cases the actual chamber or nozzle wall temperatures are higher than they would be without the insulation or shielding.

Cooling with Transient Heat Transfer. Thrust chambers with unsteady heat transfer are basically of two types. One is a simple metal chamber (steel, copper, stainless steel, etc.) made with walls sufficiently thick to absorb plenty of heat energy. For short-duration testing of injectors, testing of new propellants, rating combustion stability, and very-short-duration rocket-propelled missiles, such as an antitank weapon, a heavy-walled simple, short-duration steel chamber is often used. The common method of ablative cooling or heat sink cooling uses a combination of endothermic reactions

(breakdown or distillation of matrix material into smaller compounds and gases), pyrolysis of organic materials, counter-current heat flow and coolant gas mass flow, charring and localized melting. An ablative material usually consists of a series of strong, oriented fibers (such as glass, Kevlar, or carbon fibers) engulfed by a matrix of an organic material (such as plastics, epoxy resins or phenolic resins). As shown in Fig. 14-11, the gases seep out of the matrix and form a protective film cooling layer on the inner wall surfaces. The fibers and the residues of the matrix form a hard char or porous coke-like material that preserves the wall contour shapes.

The orientation, number and type of fiber determine the ability of the composite ablative material to withstand significant stresses in preferred directions. For example, internal pressure produces longitudinal as well as hoop stresses in the thrust chamber walls and thermal stresses produce compression on the inside of the walls and tensile stresses on the outside. We have learned how to place the fibers in two or three directions, which makes them anisotropic. We then speak of 2-D and 3-D fiber orientation.

A set of strong carbon fibers in a matrix of amorphous carbon is a special, but favorite type of material. It is often abbreviated as C-C or carbon-carbon. The carbon materials lose their ability to carry loads at about 3700 K or 6200 F. Because carbon oxidizes readily to form CO or C02, its best applications are with fuel-rich propellant mixtures that have little or no free oxygen or hydroxyl in their exhaust. It is used for nozzle throat inserts. Properties for one type of C-C are given in Table 14-5.

Ablative cooling was first used and is still used extensively with solid propellant rocket motors. It has since been successfully applied to liquid propellant thrust chambers, particularly at low chamber pressure, short duration (including several short-duration firings over a long total time period) and also in nozzle extensions for both large and small thrust chambers, where the static gas temperatures are relatively low. It is not usually, effective for cooling if the chamber pressures are high, the exhaust gases contain oxidative species, or the firing durations are long.

Repeatedly starting and stopping (also known as pulsing) presents a more severe thermal condition for ablative materials than operating for the same cumulative firing time but without interruption. Figure 8-10 shows that for small pulsing rockets, which fire only 4 to 15% of the time, the consumption or pyrolysis of the ablative liner is a maximum. This curve varies and depends on the specific duty cycle of firings, the design, the materials, and the pauses between the firings. The duty cycle for a pulsing thruster was defined in Chapter 6 as the average percent of burning or operating time divided by the total elapsed time. Between pulsed firings there is a heat soak back from the hot parts of the thruster to the cooler outer parts (causing them to become softer) and also a heat loss by radiation to the surroundings. At a duty cycle below 3%, there is sufficient time between firings for cooling by radiation. At long burning times (above 50%) the ablative material's hot layers act as insulators and prevent the stress-bearing portions from becoming too hot.

Ablative Pyrolysis

Burning time/total time

FIGURE 8-10. Relative depth of pyrolysis of ablative material with different duty cycles using many short-duration thrust pulses for a small liquid propellant reaction control thrust chamber of 20 lbf thrust.

Burning time/total time

FIGURE 8-10. Relative depth of pyrolysis of ablative material with different duty cycles using many short-duration thrust pulses for a small liquid propellant reaction control thrust chamber of 20 lbf thrust.

Depending on the design, the thrusters with duty cycles between 4 and 25% have the most severe thermal loading.

It is often advantageous to use a different cooling method for the downstream part of the diverging nozzle section, because its heat transfer rate per unit area is usually much lower than in the chamber or the converging nozzle section, particularly with nozzles of large area ratio. There is usually a small saving in inert engine mass, a small increase in performance, and a cost saving, if the chamber and the converging nozzle section and the throat region (up to an area ratio of perhaps 5 to 10) use regenerative cooling and the remainder of the nozzle exit section is radiation cooled (or sometimes ablative cooled). See Fig. 8-2 and Ref. 8^.

Film Cooling

This is an auxiliary method applied to chambers and/or nozzles, augmenting either a marginal steady-state or a transient cooling method. It can be applied to a complete thrust chamber or just to the nozzle, where heat transfer is the highest. Film cooling is a method of cooling whereby a relatively cool thin fluid film covers and protects exposed wall surfaces from excessive heat transfer. Fig. 8-11 shows film-cooled chambers. The film is introduced by injecting small quantities of fuel or an inet fluid at very low velocity through a large number of orifices along the exposed surfaces in such a manner that a protective relatively cool gas film is formed. A coolant with a high heat of vaporization and a high boiling point is particularly desirable. In liquid propellant rocket engines extra fuel can also be admitted through extra injection holes at the outer layers of the injector; thus a propellant mixture is achieved (at the periphery of the chamber), which has a much lower combustion temperature. This differs from film cooling or transpiration cooling because there does not have to be a chamber

Layer of relatively cool gas

Graphite insert

Layer of relatively cool gas-

FIGURE 8-11. Simplified diagrams of three different methods of forming a cool boundary layer.

Layer of relatively cool gas

Cool burning solid propella or insulator

Graphite insert

Layer of relatively cool gas-

Liquid propellar injection

Cool burning solid propella or insulator

Annular zone of extra fuel

Annular zone of extra fuel

Liquid propellar injection

FIGURE 8-11. Simplified diagrams of three different methods of forming a cool boundary layer.

cooling jacket or film-cooling manifolds. In solid propellant rocket engines this can be accomplished by inserting a ring of cool-burning propellant upstream of the nozzle, as shown in Fig. 8-11 or by wall insulation materials, whose ablation and charring will release relatively cool gases into the boundary layer.

Turbine discharge gas (700 to 1100°C) has also been used as a film coolant for uncooled nozzle exit sections of large liquid propellant rocket engines. Of course, the ejection of an annular gas layer at the periphery of the nozzle exit, at a temperature lower than the maximum possible value, causes a decrease in a specific impulse. Therefore, it is desirable to reduce both the thickness of this cooler layer and the mass flow of cooler gas, relative to the total flow, to a practical minimum value.

A special type of film cooling, sweat cooling or transpiration cooling, uses a porous wall material which admits a coolant through pores uniformly over the surface. This technique has been successfully used to cool injector faces in the upper stage engine (J-2) of the moon launch vehicle and the Space Shuttle main engine (SSME) with hydrogen fuel.

Thermal Insulation

Theoretically, a good thermal insulation layer on the gas side of the chamber wall can be very effective in reducing chamber wall heat transfer and wall temperatures. However, efforts with good insulation materials such as refractory oxides or ceramic carbides have not been successful. They will not with stand differential thermal expansion without cracking or spalling. A sharp edge on the surface (crack or flaked-off piece of insulator) will cause a sudden rise in the stagnation temperature and most likely lead to a local failure. Asbestos is a good insulator and was used several decades ago; because it is a cancer causing agent, it is no longer used. Coating development efforts with rhenium and other materials are continuing.

Insulation or heat shields have been successfully applied on the exterior of radiation-cooled thrust chambers to reduce the heat transfer to adjacent sensitive equipment or structures. With hydrocarbon fuels it is possible to form small carbon particles or soot in the hot gas and that can lead to a carbon deposit on the gas side of the chamber or nozzle walls. If it is a thin, mildly adhesive soot, it can be an insulator, but it is difficult to reproduce such a coating. More likely it forms hard, caked deposits, which can be spalled off in localized flakes and form sharp edges, and then it is undesirable. Most designers have preferred instead to use film cooling or extra high coolant velocities in the cooling jacket with injectors that do not create adhesive soot.

Hydraulic Losses in the Cooling Passage

The cooling coil or cooling jacket should be designed so that the fluid adsorbs all the heat transferred across the inner motor wall, and so that the coolant pressure drop will be small.

A higher pressure drop allows a higher coolant velocity in the cooling jacket, will do a better job of cooling, but requires a heavier feed system, which increases the engine mass slightly and thus also the total inert vehicle mass. For many liquid propellant rockets the coolant velocity in the chamber is approximately 3 to 10 m/sec or 10 to 33 ft/sec and, at the nozzle throat, 6 to 24 m/sec or 20 to 80 ft/sec.

A cooling passage can be considered to be a hydraulic pipe, and the friction loss can be calculated accordingly. For a straight pipe,

where A/7 is the friction pressure loss, p the coolant mass density, L the length of coolant passage, D the equivalent diameter, v the average velocity in the cooling passage, and / the friction loss coefficient. In English engineeering units the right side of the equation has to be divided by g0, the sea-level acceleration of gravity (32.2 ft/sec2). The friction loss coefficient is a function of Reynolds number and has values betwen 0.02 and 0.05. A typical pressure loss of a cooling jacket is between 5 and 25% of the chamber pressure.

A large portion of the pressure drop in a cooling jacket usually occurs in those locations where the flow direction or the flow-passage cross section is changed. Here the sudden expansion or contraction causes a loss, sometimes larger than the velocity head v2/2. This hydraulic situation exists in inlet and outlet chamber manifolds, injector passages, valves, and expansion joints.

The pressure loss in the cooling passages of a thrust chamber can be calculated, but more often it is measured. This pressure loss is usually determined in cold flow tests (with an inert fluid instead of the propellant and without combustion), and then the measured value is corrected for the actual propellant (different physical properties) and the hot firing conditions; a higher temperature will change propellant densities or viscosities and in some designs it changes the cross section of cooling flow passages.

Chamber Wall Loads and Stresses

The analysis of loads and stresses is performed on all propulsion components during their engineering design. Its purpose is to assure the propulsion designer and the flight vehicle user that (1) the components are strong enough to carry all the loads, so that they can fulfill their intended function; (2) potential failures have been identified, together with the possible remedies or redesigns; and (3) their masses have been reduced to a practical minimum. In this section we concentrate on the loads and stresses in the walls of thrust chambers, where high heat fluxes and large thermal stresses complicate the stress analysis. Some of the information on safety factors and stress analysis apply also to all propulsion systems, including solid propellant motors and electric propulsion.

The safety factors (really margins for ignorance) are very small in rocket propulsion when compared to commercial machinery, where these factors can be 2 to 6 times larger. Several load conditions are considered for each rocket component; they are:

1. Maximum expected working load is the largest likely operating load under all likely operating conditions or transients. Examples are operating at a slightly higher chamber pressure than nominal as set by tolerances in design or fabrication (an example is the tolerance in setting the tank pressure regulator) or the likely transient overpressure from ignition shock.

2. The design limit load is typically 1.20 times the maximum expected working load, to provide a margin. If the variation in material composition, material properties, the uncertainties in the method of stress analysis, or predicted loads are significant, a larger factor may be selected.

3. The damaging load can be based on the yield load or the ultimate load or the endurance limit load, whichever gives the lowest value. The yield load causes a permanent set or deformation, and it is typically set as 1.10 times the design limit load. The endurance limit may be set by fatigue or creep considerations (such as in pulsing). The ultimate load induces a stress equal to the ultimate strength of the material, where significant elongation and area reduction can lead to failure. Typically it is set as 1.50 times the design limit load.

4. The proof test load is applied to engines or their components during development and manufacturing inspection. It is often equal to the design limit load, provided this load condition can be simulated in a laboratory. Thrust chambers and other components, whose high thermal stresses are difficult to simulate, use actual hot firing tests to obtain this proof, often with loads that approach the design limit load (for example, higher than nominal chamber pressure or a mixture ratio that results in hotter gas).

The walls of all thrust chambers are subjected to radial and axial loads from the chamber pressure, flight accelerations (axial and transverse), vibration, and thermal stresses. They also have to withstand a momentary ignition pressure surge or shock, often due to excessive propellant accumulation in the chamber. This surge can exceed the nominal chamber pressure. In addition, the chamber walls have to transmit thrust loads as well as forces and in some applications also moments, imposed by thrust vector control devices described in Chapter 16. Walls also have to survive a "thermal shock", namely, the initial thermal stresses at rapid starting. When walls are cold or at ambient temperature, they experience higher gas heating rates than after the walls have been heated. These loads are different for almost every design, and each unit has to be considered individually in determining the wall strengths.

A heat transfer analysis is usually done only for the most critical wall regions, such as at and near the nozzle throat, at a critical location in the chamber, and sometimes at the nozzle exit. The thermal stresses induced by the temperature difference across the wall are often the most severe stresses and a change in heat transfer or wall temperature distribution will affect the stresses in the wall. Specific failure criteria (wall temperature limit, reaching yield stress, or maximum coolant temperature, etc.) have to be established for these analyses.

The temperature differential introduces a compressive stress on the inside and a tensile stress on the outside of the inner wall; the stress 5 can be calculated for simple cylindrical chamber walls that are thin in relation to their radius as s = 2XE A 77(1 -v) (8-12)

where X is the coefficient of thermal expansion of the wall material, E the modulus of elasticity of the wall material, AT the temperature drop across the wall, and v the Poisson ratio of the wall material. Temperature stresses frequently exceed the yield point. The materials experience a change in the yield strength and the modulus of elasticity with temperature. The preceding equation is applicable only to elastic deformations. This yielding of rocket thrust chamber wall materials can be observed by the small and gradual contraction of the throat diameter after each operation (perhaps 0.05% reduction in throat diameter after each firing) and the formation of progressive cracks of the inside wall surface of the chamber and throat after successive runs. These phenomena limit the useful life and number of starts or temperature cycles of a thrust chamber (see Refs. 8-5 and 8-6).

In selecting a working stress for the wall material of a thrust chamber, the variation of strength with temperature and the temperature stresses over the wall thickness have to be considered. The temperature drop across the inner wall is typically between 50 and 550 K, and an average temperature is sometimes used for estimating the material properties. The most severe thermal stresses can occur during the start, when the hot gases cause thermal shock to the hardware. These transient thermal gradients cause severe thermal strain and local yielding.

A picture of a typical stress distribution caused by pressure loads and thermal gradients is shown in Fig. 8-12. Here the inner wall is subjected to a compressive pressure differential caused by a high liquid pressure in the cooling jacket and a relatively large temperature gradient. In a large rocket chamber,

Liquid coolant

Liquid film

Distance from thrust chamber axis

FIGURE 8-12. Typical stresses in a thrust chamber inner wall.

Liquid coolant

Liquid film

Typical temperature distribution

Neutral axis (zero stress)

1. Stress due to thermal expansion gradient across wall only

2. Stress due to pressure differential across wall only

3. Resultant stress (sum of curves 1 and 2) with no yielding and constant modulus of elasticity

4. Actual stress in wall with yielding at hot gas side and changing modulus of elasticity

5. Yield stress distribution across wall (varies with temperature)

Distance from thrust chamber axis

FIGURE 8-12. Typical stresses in a thrust chamber inner wall.

such as is used in the Redstone missile, the wall thickness of the nozzle steel way may be 7 mm and the temperature differential across it may readily exceed several hundred degrees. This temperature gradient causes the hot inner wall surface to expand more than the wall surface on the coolant side and imposes a high compressive thermal stress on the inside surface and a high tensile thermal stress on the coolant side. In these thick walls the stress induced by the pressure load is usually small compared to the thermal stress. The resultant stress distribution in thick inner walls (shown shaded in the sample stress diagram of Fig. 8-12) indicates that the stress in the third of the wall adjacent to the hot gases has exceeded the yield point. Because the modulus of elasticity and the yield point diminish with temperature, the stress distribution is not linear over the yielded portion of the wall. In effect, this inner portion acts as a heat shield for the outer portion which carries the load.

Because of the differential expansion between the hot inner shell and the relatively cold outer shell, it is necessary to provide for axial expansion joints to prevent severe temperature stresses. This is particularly critical in larger double-walled thrust chambers. The German V-2 thrust chamber expanded over 5 mm in an axial and 4 mm in a radial direction.

Tubes for tubular wall thrust chambers are subjected to several different stress conditions. Only that portion of an individual cooling tube exposed to hot chamber gases experiences high thermal stresses and deformation as shown in Fig. 8-17. The tubes have to hold the internal coolant pressure, absorb the thermal stresses, and contain the gas pressure in the chamber. The hottest temperature occurs in the center of the outer surface of that portion of each tube exposed to hot gas. The thermal stresses are relatively low, since the temperature gradient is small; copper has a high conductivity and the walls are relatively thin (0.5 to 2 mm). The coolant pressure-induced load on the tubes is relatively high, particularly if the thrust chamber operates at high chamber pressure. The internal coolant pressure tends to separate the tubes. The gas pressure loads in the chamber are usually taken by reinforcing bands which are put over the outside of the tubular jacket assembly (see Fig. 8-1 and 8-9). The joints between the tubes have to be gas tight and this can be accomplished by soldering, welding, or brazing.

When a high-area-ratio nozzle is operated at high ambient pressure, the lower part of the nozzle structure experiences a compression because the pressure in the nozzle near the exit is actually below atmospheric value. Therefore, high-area-ratio nozzles usually have stiffening rings on the outside of the nozzle near the exit to maintain a circular shape and thus prevent buckling, flutter, or thrust misalignment.

Aerospike Thrust Chamber

A separate category comprises thrust chambers using a center body, such as a plug nozzle or aerospike nozzle. They have more surface to cool than ordinary thrust chambers. A circular aerospike thruster is described in Chapter 3 and shown schematically in Fig. 3-12. Here the diameter of the exhaust flow plume increases with altitude. A linear version of a truncated (shortened) aerospike thrust chamber is currently being developed with liquid oxygen and liquid hydrogen as the propellants; see Refs. 8-7 and 8-8. An experimental engine; assembly (XRS-2200) with 20 cells and two hydrogen-cooled, two-dimensional, curved ramps is shown in Figs 8-13 and 8-14. Each individual small (regen-eratively cooled) thrust chamber or cell has its own cylindrical combustion chamber, a circular small nozzle throat, but a rectangular nozzle exit of low area ratio. The gas from these 20 rectangular nozzle exits is further expanded (and thus reaches a higher exhaust velocity) along the contour of the spike or ramp. The two fuel-cooled side panels are not shown in these figures. The flat surface at the bottom or base is porous or full of small holes and a low-pressure gas flows through these openings. This causes a back pressure on the flat base surface. This flow can be the exhaust gas from the turbopumps and is typically 1 or 2% of the total flow. The gas region below this base is enclosed by the two gas flows from the ramps and the two side plates and is essentially independent of ambient pressure or altitude. Two of the XRS-2200 engine drive the X-33 wing shaped vehicle aimed at investigating a single stage to orbit concept.

The thrust F of this aerospike thrust chamber consists of (1) the axial component thrusts of each of the little chamber modules, (2) the integral of the pressures acting on the ramps over the axially projected area Aa normal to the axis of the ramps, and (3) the pressure acting over the base area ^base-

Here 9 is the angle of the module nozzle axis to the centerline of the spike, m is the total propellant flow, v2 is the exhaust velocity of the module, A2 is the total exit area of all the modules, p2 is the exhaust pressure at the exit of the module, and p} is the ambient pressure at the nozzle exit level. These expressions are a simplified version of the thrust. Not included, for example, is the negative effect of the slipstream of air around the engine (which causes a low-pressure region) and the friction on the side plates; both actually decrease the thrust slightly. For each application there is an optimum angle 9, an optimum ramp length, an optimum ratio of the projected ramp area to the base area, and an optimum base pressure, which is a function of the base flow.

The local gas pressures on the ramps are influenced by shock wave phenomena and change with altitude. Figure 8-14 shows a typical pressure distribution on a typical ramp surface and the flow patterns for low and high altitude. The hot gas flows coming out of the cell nozzles are turned into a nearly axial direction by multiple expansion waves (shown as dashed lines), which cause a reduction in pressure. At lower altitudes the turning causes compression shock waves (represented as solid lines), which causes local pressures to rise. The compression waves are reflected off the boundary between the hot gas jet

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