## Sample Thrust Chamber Design Analysis

This example shows how a thrust chamber is strongly influenced by the overall vehicle system requirements or the mission parameters and the vehicle design. As outlined in the Design Section of Chapter 10 and in the discussion of the selection of propulsion systems in Chapter 17, each engine goes through a series of rationalizations and requirements that define its key parameters and its design. In this example we describe how the thrust chamber parameters are derived from the vehicle and engine requirements. The overall system requirements relate to the mission, its purpose, environment, trajectories, reusability, reliability, and to restraints such as allowable engine mass, or maximum dimensional envelope. We are listing some, but not all of the requirements. It shows how theory is blended with experience to arrive at the initial choices of the design parameters.

Here we define the application as a new upper stage of an existing multistage space launch vehicle, that will propel a payload into deep space. This means continuous firing (no restart or reuse), operating in the vacuum of space (high nozzle area ratio), modest acceleration (not to exceed 5 g0), low cost, moderately high performance (specific impulse), and a thrust whose magnitude depends on the payloads, the flight path and acceleration limits. The desired mission velocity increase of the stage is 3400 m/sec. The engine is attached to its own stage, which is subsequently disconnected and dropped from the payload stage. The payload stage (3500 kg) consists of a payload of 1500 kg (for scientific instruments, power supply, or communications and flight control equipment) and its own propulsion systems (including propellant) of 2000 kg

(for trajectory changes, station keeping, attitude control, or emergency maneuvers). There are two geometric restraints: the vehicle has an outside diameter of 2.0 m, but when the structure, conduits, certain equipment, thermal insulation, fittings, and assembly are considered, it really is only about 1.90 m. The restraint on the stage length of 4.50 m maximum will affect the length of the thrust chamber. We can summarize the key requirements:

Desired velocity increase Aw Maximum stage diameter Maximum stage length Maximum acceleration

Uppermost stage to an existing multistage launch vehicle 3500 kg

3400 m/sec in gravity free vacuum 1.90 m 4.50 m 5 2 n

Decisions on Basic Parameters. The following engine design decisions or parameter selection should be made early in the design process:

Propellant combination Chamber pressure Nozzle area ratio

Feed system, using pumps or pressurized tanks Thrust level

From a performance point of view, the best propellant combination would be liquid oxygen with liquid hydrogen. However, this bipropellant would have a low average specific gravity (0.36), a very large liquid hydrogen tank, and would cause an increase in vehicle drag during ascent. It would have some potential problems with exceeding the allocated stage volume, hydrogen mass losses, and the vehicle structure. The lower stages of the existing launch vehicle use liquid oxygen with RP-1 fuel with an average specific gravity of about 1.014, and the launch pad is already equipped for supplying these. The new stage is limited in volume and cross section. Because of these factors the propellant combination of liquid oxygen and RP-1 (a type of kerosene) is selected. From Fig. 5-1 we see that the theoretical specific impulse is between 280 and 300 sec, depending on the mixture ratio and whether we use frozen or shifting chemical equilibrium in the nozzle flow expansion. This figure also shows that the maximum value of the characteristic velocity c* is reached at a mixture ratio of about 2.30, which is a fuel-rich mixture. We select this mixture ratio. Its combustion temperature is lower than the mixture ratios with higher values, and this should make the cooling of the thrust chamber easier. We will see later that cooling may present some problems. Based on universal experience, we select a value of Is part way (about 40%) between the values for frozen and shifting equilibrium, namely 292 sec at the standard chamber pressure of 1000 psi or 6.895 MPa, and a nozzle big enough for expansion to sea level. From Fig. 5-1 and Table 5-5 we find the molecular mass to be 23 kg/kg-mol and the specific heat ratio k to be about 1.24. Later we will correct this value of Is from this standard reference condition to the actual vacuum specific impulse of the thrust chamber.

Next we will select a chamber pressure, a nozzle area ratio and a feed system concept. Historically there has been favorable experience with this propellant combination at chamber pressures between 400 and 3400 psia with nozzle area ratios up to about 40 with both gas generator cycles and staged combustion cycles, giving proof that this is feasible. The following considerations enter into this selection:

1. Higher chamber pressures allow a smaller thrust chamber and (for the same nozzle exit pressure) a shorter nozzle cone with a smaller nozzle exit diameter. The thrust chamber is small enough for a toroidal tank to be built around it, and this conserves stage length. This not only saves vehicle space, but usually also some inert mass in the vehicle and the engine. Figure 8-23 shows the relative sizes of thrust chambers for three chamber pressures and two nozzle area ratios (e of 100 and 300). The nozzle length and exit diameter cannot exceed the values given in the requirements, which, as can be seen, rules out low chamber pressure or high area ratio. The dimensions shown are calculated later in this analysis.

D100 = 0.60 m D300 = 1.05 m p, = 4.826 MPa (700 psia)

^100

3.32 m

FIGURE 8-23. Comparison of thrust chamber sizes for three chamber pressures and two nozzle area ratios (100 and 300).

2. The heat transfer rate is almost proportional to the gas density, which is proportional to the chamber pressure, as shown by Eq. 8-21 and 8-23. On some prior thrust chambers there have been problems with the formation of solid carbon layer or deposits either inside the cooling jacket (increasing wall temperatures) or on the inner walls of the combustion chamber (the solid can flake off and cause burnout). This favors a lower chamber pressure.

3. Concern over leak-free seals for both static and dynamic seal increases with chamber pressure, which in turn causes all feed presures also to increase.

4. A feed system using pressurized gas is feasible, but its inert masses of tanks and engine are favorable only, if the chamber pressure is very low, perhaps around 100 psia or less. The tanks for propellants and pressurizing gas become very heavy and the thrust chamber will be very large and exceed the dimensional restraints mentioned above. We therefore cannot use this feed system or very low chamber pressures.

5. If we use a pump feed system, the power needed to drive the pumps increases directly with chamber pressure px. In a gas generator engine cycle this means a slightly reduced performance as the value of px goes up. For a staged combustion cycle it means high pressures, particularly high pressure hot gas flexible piping, and a more complex, heavier, and expensive engine. We therefore select a gas generator cycle (see Fig. 1-4) at a low enough chamber pressure, so that the thrust chamber (and the other inert hardware) will just fit the geometrical constraints, and the engine inert mass and the heat transfer will be reasonable.

For these reasons we pick a chamber pressure of 700 psia or 4.825 MPa and an area ratio of 100. With further analysis we could have picked pi more precisely; it could be somewhat lower or higher. Next we correct the specific impulse to the operating conditions using a ratio of thrust coefficients. We can use Eq. 3-30 or interpolate between Figs. 3-7 and 3-8 for a value of k = 1.24. The reference or standard condition (see Fig. 3-6) is for a pressure ratio p\/p3 of 1000/14.7 = 68, which corresponds to an area ratio of about 8. Then (Cf)standard = 1.58. For the actual high-altitude operation the pressure ratio is close to infinity and the nozzle has an area ratio of 100; we can determine the thrust coefficient by interpolating k = 1.24. The result is (Cy)vacuum = 1.90. The new ideal specific impulse value for a chamber threshold of 700 psia and a nozzle area ratio of 100 is therefore 292 x (1.90/1.58) = 351.1 sec. In order to correct for losses (divergence, boundary layer, incomplete combustion, some film cooling, etc.) we use a correction factor of 0.96 giving a thrust chamber specific impulse of 337.1 sec. The engine uses a gas generator and this will reduce the engine specific impulse further by a factor of 0.98 or (/j)engine = 330.3 sec or an effective exhaust velocity of 3237 m/sec.

Stage Masses and Thrust Level. An estimate of the stage masses will next be made. We assume that the inert hardware (tanks, gas, generator, turbopumps, etc.) is about 7% of the propellant mass, which is conservative when compared to existing engines. In a full-fledged engine design this number would be verified or corrected once an estimated mass budget becomes available. From Eq. 4—7

Solve for mp = 7639 kg. The final and initial masses of the stage are then 4023 kg and 11,002 kg respectively.

The maximum thrust is limited by the maximum allowed acceleration of 5g0. It is Fmax — m0 a= 11,002 x 5 x 9.8 = 539,100 N. This would become a relatively large and heavy thrust chamber. Considerable saving in inert mass can be obtained if a smaller thrust size (but longer firing duration) is chosen. Since this same thrust chamber is going to be used for another mission where an acceleration of somewhat less than 1.0 g0 is wanted, a thrust level of 50,000 N or 11,240 lbf is chosen. The maximum acceleration of the stage occurs just before cutoff; it is a = F/mf = 50,000/4023 = 12.4 m/sec2 or about 1.26 times the acceleration of gravity. This fits the thrust requirements. The following have now been determined:

Propellant Liquid oxygen and liquid kerosene

Thrust 50,000 N or 11,240 lbf

Chamber pressure 700 psia or 4.826 MPa

Nozzle area ratio 100

Specific impulse (engine) 330.3 sec

Specific impulse (thrust chamber) 337.1 sec Engine cycle Gas generator

Usable propellant mass 7478 kg

Propellant Flows and Dimensions of Thrust Chamber. From Eq. 2 6 we obtain the propellant mass flow m = F/c = 50,000/3200 = 15.625 kg/sec

When this total flow and the overall mixture ratio are known, then the fuel flow riif and oxidizer flow m0 for the engine, its gas generator, and its thrust chamber can be determined from Eqs. 6-3 and 6-A as shown below.

tkf = m/(r + 1) = 15.446/(2.3 + 1) = 4.680 kg/sec m0 = mr/(r + 1) = (15.446 x 2.30)/3.30 = 10.765 kg/sec

The gas generator flow mgg consumes about 2.0% of the total flow and operates at a fuel-rich mixture ratio of 0.055; this results in a gas temperature of about 890 K.

The flows through the thrust chamber are equal to the total flow diminished by the gas generator flow, which is roughly 98.0% of the total flow or 15.137 kg/sec.

The duration is the total effective propellant mass divided by the mass flow rate lb = mplmp = 7478/15.446 = 484.1 sec or a little longer than 8 minutes The nozzle throat area is determined from Eq. 3-31.

A, = F/ipiCp) = 50,000/(4.826 x 106 x 1.90) = 0.005453 m2 or 54.53 cm2

The nozzle throat diameter is Dt — 8.326 cm. The internal diameter of the nozzle at exit A2 is determined from the area ratio of 100 to be D2 = VT00 x D, or 83.26 cm. A shortened or truncated bell nozzle (as discussed in Section 3.4) will be used with 80% of the length of a 15° conical nozzle, but with the same performance as a 15° cone. The nozzle length (from the throat to the exit) can be determined by an accurate layout or by L = (D2 - £>,)/(2tan 15) as 139.8 cm. For an 80% shortened bell nozzle this length would be about 111.8 cm. The contour or shape of a shortened bell nozzle can be approximated by a parabola (parabola equation is y2 = 2px). Using an analysis (similar to the analysis that resulted in Fig. 3-14) the maximum angle of the diverging section at the inflection point would be about 6,• = 34° and the nozzle exit angle 6e = l°. The approximate contour consists of a short segment of radius 0.4r, of a 34° included angle (between points T and I in Fig. 3-14) and a parabola with two known points at I and E. Knowing the tangent angles (34 and 7°) and the y coordinates [ye = r2 andj,- = r, + 0.382 r, (1 - cos 0,-)] allows the determination of the parabola by geometric analysis. Before detail design is undertaken, a more accurate contour, using the method of characteristics, is suggested.

The chamber diameter should be about twice the nozzle throat diameter to avoid pressure losses in the combustion chamber (Dc — 16.64 cm). Using the approximate length of prior successful smaller chambers and a characteristic length L* of about 1.1m, the chamber length (together with the converging nozzle section) is about 11.8 inch or 29.9 cm. The overall length of the thrust chamber (169 cm) is the sum of the nozzle length (111.8 cm), chamber (29.9 cm), injector thickness (estimated at 8 cm), mounted valves (estimated at 10 cm), a support structure, and possibly also a gimbal joint. The middle sketch of the three thrust chambers in Fig. 8-23 corresponds roughly to these numbers.

We have now the stage masses, propellant flows, nozzle and chamber configuration. Since this example is aimed at a thrust chamber, data on other engine components or parameters are given only if they relate directly to the thrust chamber or its parameters.

Next we check if there is enough available vehicle volume (1.90 m diameter and 4.50 m long) to allow making a larger nozzle area ratio and thus gain a little more performance. First we determine how much of this volume is occupied by propellant tanks and how much might be left over or be available for the thrust chamber. This analysis would normally be done by tank design specialists. The average density of the propellant mixture can be determined from Eq. 7-1 to be 1014 kg/m3 and the total usable propellant of 7478 kg. Using densities from Table 7-1 the fuel volume and the oxidizer volume can be calculated to be 2.797 and 4.571 m3 respectively. For a diameter of 1.90 m, a nearly spherical fuel tank, a separate oxidizer cylindrical tank with elliptical ends, 6% ullage, and 2% residual propellant, a layout would show an overall tank length of about 3.6 m in a space that is limited to 4.50 m. This would leave only 0.9 m for the length of the thrust chamber, and this is not long enough. Therefore we would need to resort to a more compact tank arrangement, such as using a common bulkhead between the two tanks or building a toroidal tank around the engine. It is not the aim to design the tanks in this example, but the conclusion affects the thrust chamber. Since the available volume of the vehicle is limited, it is not a good idea to try to make the thrust chamber bigger.

This diversion into the tank design shows how a vehicle parameter affects the thrust chamber design. For example, if the tank design would turn out to be difficult or the tanks would become too heavy, then one of these thrust chamber options can be considered: (1) go to a higher chamber pressure (makes the thrust chamber and nozzle smaller, but heavier), (2) go to a lower thrust engine (will be smaller and lighter), (3) store the nozzle of the upper stage thrust chamber in two pieces and assemble them once the lower stages have been used and discarded (see Fig. 8-19; it is more complex and somewhat heavier), or (4) use more than one thrust chamber in the engine (will be heavier, but shorter). We will not pursue these or other options here.

Heat Transfer. The particular computer program for estimating heat transfer and cooling parameters of thrust chambers will depend on the background and experience of specific engineers and rocket organizations. Typical computer programs divide the internal wall surface of the chamber and nozzle into axial incremental axial steps. Usually in a preliminary analysis the heat transfer is estimated only for critical locations such as for the throat and perhaps the chamber.

From Fig. 5-1 and Eq. 3-12 or 3-22 we determine the following gas temperatures for the chamber, nozzle throat region, and a location in the diverging exit section. They are: Tx = 3600 K, T, = 3243 K, and Te = 1730 K at an area ratio of 6.0 in the diverging nozzle section. The chamber and nozzle down to an exit area ratio of 6 will have to be cooled by fuel. For this propellant combination and for the elevated wall temperatures a stainless steel has been successfully used for the inner wall material.

Notice that beyond this area ratio of about 6, the nozzle free stream gas temperatures are relatively low. Uncooled high temperature metals can be used here in this outer nozzle region. Radiation cooling, using a material such as niobium (coated to prevent excessive oxidation) or carbon fibers in a nonpor-ous carbon matrix, is suitable between an area ratio of 6 and about 25. For the final large nozzle exit section, where the temperatures are even lower, a lower cost material such as stainless steel or titanium is suggested. Ablative materials have been ruled out, because of the long duration and the aggressive ingredients in the exhaust gas. The gas compositions of Figs. 5-2 and 5-3 indicate that some free oxygen and hydroxyl is present.

We now have identified the likely materials for key chamber components. The best way to cool the radiation cooled exit segment of the nozzle (beyond area ratio of 6) is to let it stick out of the vehicle structure; the heat can then be freely radiated to space. One way to accomplish this, is to discard the vehicle structure around the nozzle end.

As in Fig. 8-8, the maximum heat transfer rate will be at the nozzle throat region. A variety of heat transfer analysis programs are available for estimating this heat transfer. If a suitable computer program is not available, then an approximate steady-state heat transfer analysis can be made using Eqs. 8-15 to 8-19 and the physical properties (specific heat, thermal conductivity, and density) of RP-1 at elevated temperatures. The film coefficients of Eqs. 8-23 and 8-25 are also needed. This is not done in this example, in part because data tables for the physical properties would take up a lot of space and results are not always reliable. Data from prior thrust chambers with the same propellants indicate a heat transfer rate at the nozzle throat region exceeding 10 Btu/in.2-sec or 1.63 x 107 W/m2.

The RP-1 fuel is an unusual coolant, since it does not have a distinct boiling point. Its composition is not consistent and depends on the oil stock from which it was refined and the refining process. It is distilled or evaporated gradually over a range of temperatures. The very hot wall can cause the RP-1 to locally break down into carbon-rich material and to partially evaporate or gasify. As long as the small vapor bubbles are recondensed when they are mixed with the cooler portions of the coolant flow, a steady heat transfer process will occur. If the heat transfer is high enough, then these bubbles will not be condensed, may contain noncondensable gases, and the flow will contain substantial gas bubbles and become unsteady, causing local overheating. The recondensing is aided by high cooling passage velocities (more than lOm/sec at the throat region) and by turbulence in these passages. A coolant flow velocity of 15 m/sec is selected for the throat and 7 m/sec in the chamber and nozzle exit segment.

The material for the cooling jacket will be stainless steel to resist the oxidation and erosion of the fast moving, aggressive hot gas, which contains free oxygen and hydroxyl species. The cooling by fuel will assure that the temperatures of this stainless steel are well below its softening temperature of about 1050 K.

The construction of the cooling jacket can be tubular, as shown in Figs. 8-1 and 8-9, or it can consist of milled channels as shown in Figs. 8-2 and 8-17. The cross section of each tube or cooling channel will be a minimum at the throat region, gradually become larger, and be about two or more times as large at the chamber and diverging nozzle regions. The wall thickness (on the hot gas side) should be as small as possible to reduce the temperature drop across the wall (which reduces the thermal stresses and allows a lower wall temperature) and to minimize the yielding of the material that occurs due to thermal deformation and pressure loads. Figure 8-12 shows this behavior, but for a thick wall. Practical considerations such as manufacturability, the number of test firings before flight, the deformation under pressure loads, the temperature gradient and dimensional tolerances also enter into the selection of the wall thickness. A thickness of 0.5 mm and a cooling velocity of 15 m/sec have been selected for the throat region of the cooling jacket and cooling velocities of 7 m/sec in the chamber and the cooled nozzle segment. Milled slots (rather than tubes) have been selected for this thrust chamber.

The selection of the number of milled slots, their cross sections, and the wall thickness is a function of the coolant mass flow, its pressure, wall stresses, wall material, and the shape of the channel. Figure 8-24 and Table 8-6 describe the channel width and height for different numbers of channels and different locations. The fuel coolant flow is diminished by the gas generator fuel flow (0.293 kg/sec) and is about 4.387 kg/sec. For this flow and a cooling velocity of 15 m/sec in the throat region the cumulative cross-sectional area of all the channels is only about 3.62 cm2. The cooling velocity is lower in the chamber and nozzle regions and the cumulative channel flow area will be larger there. The variables are the number of channels, the thickness of the hot wall, the rib thickness between channels, the cooling velocity, the gas temperature, and the

Hot gas side Wall of inner wall thickness

FIGURE 8-24. Segment of cooling jacket with milled channels and an electroformed outer wall.

Hot gas side Wall of inner wall thickness

FIGURE 8-24. Segment of cooling jacket with milled channels and an electroformed outer wall.

 Throat Section