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NASA TECHNICAL NOTE NASA TN D-8078
s*a
A RAPID METHOD FOR OPTIMIZATION
OF THE ROCKET PROPULSION SYSTEM
FOR SINGLE-STAGE-TO-ORBIT VEHICLES
Charles H. Eldred and Sabina V. Gordon
Langley Research Center
Hampton, Va. 23665
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION • WASHINGTON, D. C. • JULY 1976
https://ntrs.nasa.gov/search.jsp?R=19760019167 2020-06-21T00:29:56+00:00Z
1 Report No 2 Government Accession No
NASA TN D-80784 Title and Subtitle
A RAPID METHOD FOR OPTIMIZATION OF THE ROCKETPROPULSION SYSTEM FOR SINGLE -STAGE -TO -ORBITVEHICLES
7 Author(s)
Charles H. Eldred and Sabina V. Gordon
9 Performing Organization Name and Address
NASA Langley Research Center
Hampton, Va. 23665
12 Sponsoring Agency Name and Address
National Aeronautics and Space AdministrationWashington, D.C. 20546
3 Recipient's Catalog No
5 Report DateJuly 1976
6 Performing Organization Code
8 Performing Organization Report No
L-1022610 Work Unit No
790-40-07-06
11 Contract or Grant No
13 Type of Report and Period Covered
Technical Note
14 Sponsoring Agency Code
15 Supplementary Notes
16 Abstract
A rapid analytical method for the optimization of rocket propulsion systems is presentedfor a vertical take-off, horizontal landing, single-stage-to-orbit launch vehicle. This methodutilizes trade-offs between propulsion characteristics affecting flight performance and enginesystem mass. The performance results from a point-mass trajectory optimization programare combined with a linearized sizing program to establish vehicle sizing trends caused bypropulsion system variations. The linearized sizing technique was developed for the class ofvehicle systems studied herein. The specific examples treated in this paper are the optimiza-tion of nozzle expansion ratio and lift-off thrust-to-weight ratio to achieve either minimumgross mass or minimum dry mass. Assumed propulsion system characteristics are highchamber pressure, liquid oxygen and liquid hydrogen propellants, conventional bell nozzles,and the same fixed nozzle expansion ratio for all engines on a vehicle.
17 Key Words (Suggested by Author(s))
Reusable launch vehicleRocket propulsionSingle -stage -to-orbit vehicleLinearized sizing of rocket vehiclesRocket launch trajectories
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Unclassified
18 Distribution Statement
Unclassified - Unlimited
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20 Security Classif (of this page) 21 No of Pages 22 Price'
Unclassified 31 $3.75
For sale by the National Technical I nformation Service, Springfield, Virginia 22161
A RAPID METHOD FOR OPTIMIZATION
OF THE ROCKET PROPULSION SYSTEM FOR
SINGLE-STAGE-TO-ORBIT VEHICLES
Charles H. Eldred and Sabma V. GordonLangley Research Center
SUMMARY
A rapid analytical method for optimization of rocket propulsion systems is presentedfor a vertical take-off, horizontal landing, single-stage-to-orbit launch vehicle. Thismethod utilizes trade-offs between system characteristics affecting flight performanceand engine system mass. The performance results obtained from a point-mass trajec-tory optimization program are combined with a linearized sizing program to establishvehicle sizing trends caused by propulsion system variations. Although this method isapplicable and adaptable to a wide range of flight vehicle systems, the example presentedis restricted to a single class of vehicle and propulsion systems. The linearized sizingtechnique was developed for this class of systems. The resulting sizing program issmall enough to be readily adaptable to a desk top programmable calculator and, becauseof its explicit solution, is fast and precise. The specific example treated in this paper isthe optimization of nozzle expansion ratio and lif t-off thrust-to-weight ratio to achieveeither minimum gross mass or minimum dry mass. Assumed propulsion system char-acteristics are high chamber pressure, liquid oxygen and liquid hydrogen propellants,conventional bell nozzles, and the same fixed nozzle expansion ratio for all engines on avehicle. Optimization results show a strong trend toward vehicles having low enginemass rather than high propulsive preformance, with optimum values of expansion ratio inthe range of 40 to 45 and optimum lift-off thrust-to-weight ratios in the range of 1.20 to1.35.
INTRODUCTION
Single-stage-to-orbit (SSTO) systems are being studied as potential launch vehicleconcepts for the time period following space shuttle. On the basis of projected advance-ments in technologies and possible changes in mission requirements, such systems areenvisioned either as shuttle replacements or in complementary mission roles. If the shut-tle system is assumed to have a 15-year operational life, a new replacement system will
be required around 1995. With 8 to 9 years allowed for design and development lead time,the period available for technology preparation is only slightly more than 10 years.
Recent studies at the Langley Research Center have been directed toward definingthe necessary and high-yield areas of technology preparation. An integral part of thisprocess is the definition and preliminary design of attractive system concepts, which arethen used to measure the effects of technology variations. One of the most crucial tech-nologies for such systems is that of propulsion, which is the principal subject of thispaper.
Some method of total system synthesis is required to compare the various propul-sion system concepts and to identify many of the trade-offs within a given propulsion con-cept. The extent of synthesis detail utilized in such studies varies over a wide range, themost detailed methods requiring large computer resources. The method described in thispaper emphasizes a simplified synthesis which can reduce the resource requirements forsuch propulsion system trade studies.
The key to this simplified synthesis is a linearized sizing analysis. Martin, in ref-erence 1, uses a linearized sizing analysis to optimize the phasing points of multiphaseSSTO propulsion systems and to compare various types of propulsion concepts on thebasis of ideal velocity. The implicit assumption is that the effect of propulsion systemvariations on the ascent trajectory is small in comparison with the variations due to otherfactors. In the present study, a similar type of sizing analysis is used in conjunction withactual trajectory performance for analyzing trade-offs between engine systems mass andflight performance for a selected engine concept. The sizing analysis is configured for avertical take-off, horizontal landing (VTOHL) SSTO with all rocket propulsion. Eithergross mass or dry mass may be used as a figure of merit. The fact that the interfacebetween the sizing program and the trajectory program is simple allows a different tra-jectory program to be substituted if desired.
The example presented herein is the optimization of engine nozzle expansion ratioand lift-off thrust-to-weight ratio to achieve either minimum gross mass or minimum drymass. The propulsion system uses rocket engines operating at high chamber pressureswith liquid hydrogen and liquid oxygen propellants and conventional bell nozzles, all noz-zles on a given vehicle having the same fixed expansion ratio. There are other potentiallyattractive rocket propulsion schemes for VTOHL SSTO application, such as the use of amixture of fixed expansion ratios and dual-position nozzles; however, these are morecomplex than the example scheme, both from considerations of vehicle integration andoperation and from the standpoint of propulsion system optimization.
SYMBOLS
Cgcj residual propellant mass coefficient, ratio of residual propellant mass tomp,sd
ascent propellent mass.mP,a
Csv reserve propellant mass coefficient, ratio of reserve propellant mass tomp,sv
ascent propellant mass, — — —mp,a
E mam engine system mass coefficient, ratio of installed engine system weight
to thrust,
G glide -reuse systems mass coefficient, ratio of glide -reuse systems mass tonip
entry mass, — — , see equation (10)me
g standard free-fall acceleration, 9.80665 m/s
rp
Is engine specific impulse, -=—, snig
K tankage systems mass coefficient, ratio of tankage systems mass to totalmKmam propellant mass, - — -mp,a + mp,sd + mp,sv
m engine propellant mass flow rate, kg/s
m^ vehicle dry mass, mj + mj^ + mg + rriQ, kg
mg engine systems mass, kg, see equation (9)
me vehicle entry mass, kg, see equation (13)
mj fixed mass, kg, see equation (7)
mo glide -reuse systems mass, kg, see equation (10)
mo- vehicle gross mass, kg, see equation (11)o
niT£ tankage systems mass, kg, see equation (8)
mpay payload mass, kg
nip a mass of ascent propellants. including in-flight losses, kg
nip 0 mass of orbital maneuvering propellants, kg
nip r mass of reaction control system (RCS) propellants for entry, kg
mp sd mass of main propellant residuals, kg
mp sv mass of mam propellant performance reserves, kg
N vehicle system thrust-to-weight ratio, ™mgg
q free-stream dynamic pressure, Pa
T rocket engine thrust, N
a angle of attack, deg
ra ascent propellant mass ratio, ratio of gross mass to mass at ascent trajectorynig.
main engine cut-off, m g - m p ) a
Fo orbital maneuvering propellant mass ratio, ratio of initial orbital mass tome + mp o
entry mass, —me
Fr entry RCS propellant mass ratio, ratio of entry mass to landing mass,me
me - mp,r
AV ideal velocity change, m/s
engine nozzle expansion ratio
Subscripts:
main ascent propellant
orbit maneuvering system (OMS) propellant
entry reaction control system (RCS) propellant
si sea level or lift-off conditions
vac vacuum
Abbreviations:
LH2 liquid hydrogen
LINSIZ linear sizing computer program
LOX liquid oxygen
ODIN optimal design integration, computerized design integration system
OMS orbital maneuvering system
POST program to optimize simulated trajectories, trajectory computer program
RCS reaction control system
SSME space shuttle main engine
SSTO single-stage-to-orbit
TPS thermal protection system
VTOHL vertical take-off, horizontal landing
METHOD OF ANALYSIS
Overview
Propulsion system optimization for a vehicle flight system should include such con-siderations as (1) engine performance and mass characteristics; (2) propellant and tankagerequirements; (3) flight trajectory and performance; (4) integration of the propulsion sys-tem with the vehicle (installation, aerodynamic effects, center of gravity); and (5) overallsystem synthesis or sizing. For cases in which the range of propulsion system variablesunder consideration is restricted in some way, a simplified treatment of one or more ofthese factors may be appropriate. The method of analysis presented herein applies sim-plifications to (4) and (5) above. Propulsion system variations are assumed not to impactconfiguration features such as vehicle shape, aerodynamic coefficients, vehicle balance(center of gravity), etc. Also, system sizing changes are assumed to be adequately rep-resented by a combination of linear relationships. However, for both assumptions, thisdoes not preclude the need for a reference vehicle design which adequately defines andaccommodates all the above considerations.
A principal feature of this method of analysis is the use of a simplified sizing pro-gram which incorporates considerations (1), (2), and (5); has the form to permit an explicitsolution; and is small enough to be run on a desk top programmable calculator. Calibra-tion of this sizing program against the characteristics of a more detailed design assuresproper sizing behavior.
Ascent trajectory performance, which is assumed to be a driving sizing require-ment, is computed with a generalized point-mass trajectory optimization program. Tra-jectory performance is determined for various combinations of the propulsion systemcharacteristics. Because the effects of integrating the propulsion system with the vehicleare assumed not to vary the overall aerodynamic characteristics, it is possible to com-pute all trajectories by use of the characteristics of a single vehicle (excluding the pro-pulsion system variations of interest). These trajectory results are then used as inputsfor the sizing program. In this way, the combined trajectory flight performance andvehicle sizing effects of propulsion system variations are evaluated.
Sizing Analysis
A linearized sizing program was used to determine vehicle sizing for each combina-tion of propulsion system characteristics evaluated for trajectory performance. Thisprogram, designated LINSIZ, was implemented on a desk top programmable calculator inabout 275 program steps. This linearized method is similar to the approach used in ref-erence 1; however, more details in the propulsion area are provided in the present study.This sizing approach simplifies the overall vehicle system to 11 mass categories:
1 - Mam engine ascent propellant
2 - In-flight propellant losses
3 - Ascent propellant reserves
4 - Ascent propellant residuals
5 - QMS (and RCS) orbital propellants
6 - RCS entry propellant
7 - Payload
8 - Fixed items
9 - Tankage systems
10 - Engine systems
11 - Glide-reuse structures and systems (wings, TPS, landing gear, etc.)
Mass groupings that are used in this study are defined as follows:
Dry mass Categories 8, 9, 10, and 11
Entry mass Dry mass plus categories 4, 6, and 7
QMS initial mass Entry mass plus category 5
Burnout mass Entry mass plus categories 3 and 5
Gross mass Burnout mass plus categories 1 and 2 (all categories)
Categories 1 to 7 are self-evident; however, categories 8 to 11 require some expla-nation. Vehicle fixed mass includes such items as crew, crew provisions (cabin andenviromental control), payload bay structure, payload provisions, and avionics, which areindependent of propulsion system variations and which are basic to the orbital mission ofthe vehicle. Tankage systems include basic tanks and associated structure and plumbing,which are sized by the total ascent propellant requirements. Engine systems includeengines, plumbing, pumps, gimbals, thrust structure, etc., which are sized by the thrustlevel and propellant flow rate requirements. The glide-reuse category includes wings,tail, TPS, landing gear, etc., which are sized by the entry mass.
For each mass category, a linear mass relationship appropriate to a VTOHL SSTOvehicle is used:
Ascent propellant, including m-fhght losses,
ra - 1= — - m
Reserves and residuals
mp,sv + mp,sd = (csv + csd)mp,a
OMS propellant
mp,o = (ro - !)me
where
(4,
RCS entry propellant
mp,r = rr'
me
where
Fr * e i l o a > i (6)
is a useful approximation.
Fixed mass
mf = Fj + F2mpay (7)
where Fj is a fixed mass constant (in kilograms), F2 is a fixed mass coefficient(fraction of payload mass), and F2mpav reflects the mass of payload accommodation'sto allow the scaling of fixed mass with payload variations. (For the type of enclosed baywith full-length doors assumed in this study, F2 appears to have a value of about 0.6.)
8
However, since payload is not varied in this study, fixed mass is simply mf, and equa-tion (7) is not used.
Tankage systems
mK = K(mp>a + mp)Sd + mpjSV) (8)
Engine systems
mE = EslNslmg (fl)
Glide-reuse systems
mG - Gme (10)
The definition of gross mass gives the following expression:
mg = mp,a + mp,sv + mp,sd + mp,o + mp,r + mpay + mf + mK + mE + mG (n>
Substitution of equations (2) and (3) gives
nig = nip a(1 + Cgv) + mp a^sd + (^o ~ l)me + mp r + mpay + mf + mK + mE + mG (1^)
From the definition of entry mass
me = mp,acsd + mp,r + mpay + mf + mK + mE + mG ^13^
equation (12) can be simplified to
mg = mp>a(l + Csv) + r0me (14)
Substitution of equations (5) and (10) into equation (13) gives
me = 1 (mp,acsd + mpay + mf + mK + m
1 -LL^-L.Grr
Then substituting equation (15) into equation (14) gives
mg. =O
csv) + p—: (mp,acsd + mpay + mf + mK1 - r_ - - G
(16)
Also, substitution of equations (8) and (9) now gives
mn- = mp,a i i c i r°°sd ,F 1
1 r " G
F IT0^
rr - irr
- G
\-> OTT 4- (_/ c
—^- (mpay + mf + EslNslmg)l . _l^-i - G
(17)
And finally, substituting equation (1) and solving for the gross-mass gives
(mlm]
rr G
pay ' mfj r ,1 r " Gr1 r
_ ^o^sd -^0^(1 + Csv + Cscj)l+ i i •ov r 1 r i1 1 r - 1 c i i r - 1
GT"1 "P1 r xr
(18)
Equation (18) is the key equation to the explicit sizing solution. Once the grossmass is determined, all the other mass characteristics can be established by using thefollowing sequence:
Ascent propellant, including in-flight losses, by using equation (1)
Reserves and residuals by using equation (2)
Entry mass by rearranging equation (14) to give
m - csv)(19)
10
OMS propellant by using equation (3)
RCS propellant by using equation (5)
Tankage systems by using equation (8)
Engine systems by using equation (9)
Glide-reuse systems by using equation (10)
Potentially, any of the inputs to equation (18) could be used as design variables fora sizing study; however, in applying this sizing analysis to the problem of propulsion sys-tem optimization, the inputs fall into three general categories:
(1) Design assumptions and requirements
Payload mpay
OMS requirements To or AVO and Is o
RCS requirements Tr or AVr and Is r
Propellant coefficients Csv. Cscj !
(2) Common mass factorsFixed mass ~ ~ / nif
Glide-reuse coefficient G
Tankage coefficient K
(3) Design variablesEngine factors e, ES±, and Is a
Thrust-to-weight ratio NSJ
Ascent propellant mass ratio Ta
After the trajectory performance is determined for an assumed combination of e
and Ns}, the resulting ascent mass ratio Fa, along with the appropriate engine mass fac-tor Esi, is evaluated by the sizing program. The resulting gross mass and dry massare utilized as the overall indicators of propulsion system optimality.
Within the limitations of the linearized sizing, validity of the results rests primarilywith the proper selection of values for the three dry mass coefficients E, G, and Kand the fixed mass mj. In order to establish these coefficient values, a point-designvehicle system is needed for which there are detailed mass information and sizing trend
data. It is desirable to duplicate both the absolute sizing level and the sizing trends suchas those associated with variations in mass ratio.
11
Trajectory Analysis
The purpose of the trajectory analysis is to identify the variations in ascent perfor-mance, as measured by the ascent mass ratio Fa, which result from variations in char-acteristics of the rocket propulsion system. To do this, optimized ascent trajectoriesare computed for each propulsion system configuration by use of a generalized programcalled POST (program to optimize simulated trajectories) described in reference 2.
POST is a discrete parameter targeting and optimization program having the capa-bility to target and optimize point-mass trajectories while satisfying equality and inequal-ity constraints. Because of the generality of the simulation and optimization routines,POST can be used to solve a wide variety of problems for a powered or unpowered vehicleoperating near a rotating oblate planet. These problems can include ascent, reentry, andorbital transfer trajectories. In this study, POST is utilized for powered ascent trajec-tories which satisfy various in-flight constraints, meet orbital insertion requirements,and optimize the vehicle flight controls in order to minimize the ascent propellant require-ment. Typical in-flight constraints include limits on dynamic pressure q, product ofdynamic pressure and angle of attack qa, and vehicle acceleration. Insertion conditionsmay include altitude, flight-path angle, and velocity.
Interface requirements between the sizing program and the trajectory analysis aresuch that another trajectory program could be substituted for POST if desired. Vehiclesimulation inputs such as aerodynamics, planform loading, in-flight constraints, and con-trol modes must be representative of the vehicle being synthesized in the sizing analysis.These inputs are not varied within the trajectory analysis. The inputs that are variedbetween the different trajectory cases are the propulsion system characteristics. Thesizing-trajectory interface requires that the propulsion system weights used in the sizingbe consistent with the flight performance characteristics. The principal output of the tra-jectory analysis is the ascent propellant mass ratio Fa, which is used as a sizing input.
Propulsion system characteristics are input as total vacuum thrust, total mass flowrate, and total nozzle exit area. Effective thrust due to back pressure losses is calculatedin the program. In addition to main propellant flows, in-flight propellant losses (due tosuch things as boiloff and auxiliary power generation) can be simulated by the addition ofa zero-thrust, zero-exit-area "engine." The in-flight losses, which are then included inthe ascent propellant requirement, have been taken into account in this manner.
EXAMPLE
The example presented in this study is the optimization of the rocket engine nozzleexpansion ratio e and lift-off thrust-to-weight ratio Ngl for a VTOHL SSTO vehicleusing identical multiple engines with conventional bell nozzles. All nozzles on a given
12
vehicle have the same fixed expansion ratio. Optimization is on the basis of either mini-mum gross mass or minimum dry mass. A Langley Research Center (LaRC) in-housepreliminary design provided a point-design vehicle for calibration of the linearized sizingmodel and for the trajectory analyses. This point-design vehicle was based on missionrequirements similar to the space shuttle and on assumed 15-year advancements beyondthe space shuttle level in two basic technology areas, structures and propulsion. Trajec-tory and sizing analyses were performed for a matrix of e and Nsj combinations, withe ranging from 30 to 80 and NSJ ranging from 1.15 to 1.50.
Design Requirements
Basic space shuttle design requirements were utilized extensively for this study(ref. 3). Although it is unlikely that a post-shuttle vehicle would be designed for the samemission needs as the shuttle, the shuttle offers the best defined set of requirements avail-able at this time. The sizing mission used was the 29 500-kg payload launched from theKennedy Space Center (KSC) into a 28.5° inclined orbit. A 4.6-m-diameter by 18.3-m-long payload bay is provided. The orbital maneuvering system provided AV = 229 m/s,which was sufficient for near-Earth maneuvers and entry deorbit. The OMS provides forall maneuvers beyond the 93-km by 185-km insertion orbit provided by the main propul-sion system. The entry cross range requirement is established at the shuttle level of2037 km. A reaction control system is provided for attitude control during the initialphases of reentry.
Technology Assumptions
The assumed development schedule for the SSTO has the advantage of a 15-yearadvancement in technology beyond the level for the shuttle. Significant gams should resultfrom existing programs in the areas of structures and propulsion.
Structures. - The information presented in reference 4 indicates that a 25-percentstructural mass reduction from current technology is possible for a vehicle developed inthe period from 1985 to 1990. Consequently, a 25-percent structural mass reduction froma space shuttle reference has been assumed in the areas of the body, wing, tail, tanks, andlanding gear. No mass reductions are assumed for the propulsion systems or for thesupporting subsystems.
Propulsion.- The engine characteristics used in this study are based on space shut-tle main engine technology, which incorporates a high chamber pressure (20 MPa) andLOX-LH2 propellants. A LOX-LH2 mixture ratio of 7:1, as proposed in reference 5, isassumed instead of the SSME value of 6:1. This variation appears to offer advantagesover the SSME for SSTO application because the vehicle sizing advantages due to theincreased propellant bulk density (about a 9-percent increase) more than offset the effects
13
of the somewhat lower specific impluse (about a 1.5-percent decrease). However, theoverall differences between these two mixture ratios are not very large.
Engine characteristics used in this study are shown in figures 1 and 2. The varia-tion of specific impulse with expansion ratio e is shown in figure 1. This figure isbased on the reference point for LOX-LH2 presented in reference 5 (a mixture ratio of7:1 and e = 200) and on characteristics of minimum-surface-area nozzles for variationsin e presented in reference 6. The curve for a mixture ratio of 6:1 with the SSMElocated at e = 77.5 is shown for comparison. Figure 2 shows the variation of the enginesystem mass coefficient E with the expansion ratio for thrust levels at both vacuum andsea level. The sea level mass coefficient Esj is the one used in the sizing program.These mass characteristics were derived from weight reports and trade studies for thespace shuttle and reflect the SSME operating at maximum power.
Point-Design Vehicle
A point-design vehicle derived from LaRC in-house studies of an SSTO system wasused for the trajectory evaluation and also to establish coefficients for the linearized siz-ing program. This design was generated in Langley Research Center studies which uti-lized the ODIN system (ref. 7). Vehicle mass properties and other characteristics aresummarized in table 1 and the physical layout is shown in figure 3. For this vehicle, ewas 80 and Nsi was 1.347. The ascent mass ratio Fa determined from trajectoryanalysis was 7.3342. The gross mass was 2.15 Gg, and the reference theoretical wingarea was 1087 m^. Design requirements and assumptions for the point-design vehiclewere consistent with those previously discussed, with two minor exceptions. First, theOMS mass ratio Fo was insufficient in the point-design vehicle to give the desired AV> iof 229 m/s, and second, the value of Is assumed for the main engines was slightly higherthan that assumed for this study, as shown in figure 1. This resulted in a somewhat lowerascent mass ratio Fa than would be required for the engines with the lower value of Is.Corrections for both these factors are incorporated in the sizing analysis.
Sizing Analysis
Tables 2 and 3 illustrate the use of the point-design vehicle to calibrate the linearsizing program. The mass categories of the point-design vehicle were redistributedaccording to the linear sizing categories. The fluid categories were all well defined, how-ever, the four categories included in dry mass required some judgment. Assignment ofthe major vehicle components were straightforward, with two possible exceptions: Basicstructure and Growth. Seventy-five percent of the basic structure (item 3.1) was assignedto fixed mass to account for payload accommodations, which include the payload bay,structural supports, and doors. Growth (item 18.0) was proportioned among the four dry-
14
mass categories based on the mass level without growth. For many of the smaller sub-systems whose categories were not obvious, category assignments were based on corre-lations of sizing trends with available trend data. In this current example (and probablytypical in general), the best sizing trend correlation between LINSIZ and ODIN wasachieved by placement of these smaller subsystems in the fixed mass category. The sum-mation of masses under each category as shown in table 2 provides the information neededto establish constants and coefficients for the sizing program. On the basis of this massdistribution, the following input values for the calibration vehicle were established:
N s l = 1.347
Ta = 7.3342 (e = 80, Is,a,vac = 453 s)
T0= 1.03667
Fr = 1.0168
Csv = 0.002
C s d= 0.007
mpay = 29 500 kg
m{ = 32 600 kg
K = 0.0240
G = 0.296
The resulting calibration vehicle from the LINSIZ program is compared with thepoint -design vehicle in table 3. All categories agree to within 1 percent. Figure 4 com-pares the effect on gross mass of varying the ascent mass ratio as predicted by LINSIZand ODIN. The agreement is within 1 percent in the gross mass range of interest for thisstudy. However, significant errors are generated farther from the calibration point. The
15
LINSIZ gross mass is higher than the ODIN gross mass at mass ratios above the calibra-tion point and is lower below the calibration point. As discussed in the preceding para-graph, manipulation of the fixed mass category allows a degree of control on these sizingtrends, with increases in fixed mass tending to decrease the sensitivity of gross mass tochanges in mass ratio.
The baseline vehicle for this sizing study is based on the point-design and calibra-tion vehicles but incorporates the corrections for the OMS velocity change AV and spe-cific impulse Is of the mam engines, as noted earlier. The following changes weremade to the LINSIZ inputs:
Nsl = 1.350
Ta = 7.45365 (e = 80, Is,vac = 449.35)
ro = 1.05207 (AV0 = 229 m/s, ISjO = 460 s)
The resulting vehicle is about 11 percent heavier than the calibration vehicle and isshown in the last column of table 3. The only sizing inputs varied from the baseline vehi-cle for the analysis of e and Nsi effects are Nsi, Fa, and Es\.
Trajectory Analysis
A typical ascent trajectory is shown in figure 5. The trajectory simulation was setup to minimize the ascent propellant requirements while satisfying in-flight constraintsand orbital insertion requirements. Controls used are pitch angle for the first 30 s andthe angle of attack a thereafter. These are implemented as linear functions of time ineight different segments over the ascent trajectory. The in-flight constraints are maxi-mum ascent acceleration, maximum dynamic pressure q, and maximum qa. Ascentacceleration is limited to 3g by continuous engine throttling after the 3g level is reached.Dynamic pressure is limited to 48 kPa as an inequality constraint. The no-wind valueof qa is limited to 120 kPa-deg by controlling the angle of attack in the region of highdynamic pressure. (This qa value was found to be optimal in ref. 8.) Aerodynamiccoefficients were obtained from a representative Phase B space shuttle orbiter. Orbitalinsertion requirements, which are based on the perigee of a 93- by 185-km orbit, areimplemented as equality constraints for altitude and flight-path angle. Velocity is the tra-jectory termination parameter.
In addition to main propellant flows, in-flight propellant losses (due to such thingsas boiloff and auxiliary power generation) were simulated. This amounted to a total loss
16
of 0.17 percent of the gross mass expended at a constant rate over the time of the ascentburn. Ascent propellant requirements, which are described by the main propellant massratio Fa, include these in-flight losses.
POST iterates the trajectory until convergence criteria which indicate that a satis-factory solution has been found are met. Convergence criteria include both constrainttolerances and measures of optimahty. Typical tolerances on converged solutions are2 m on insertion altitude and 0.004° on insertion flight-path angle.
RESULTS AND DISCUSSION
The method outlined in this paper for the optimization of SSTO rocket propulsionsystems is easy and rapid to use once the basic trajectory simulation and sizing programconstants are set up. The only input changes required between cases for the trajectoryprogram are derived from the selection of expansion ratio e and lift-off thrust-to-weightratio Ng}. These changes are input as total vacuum thrust, total propellant flow rate,and total nozzle exit area. The input changes required for the sizing program are simply(1) lift-off thrust-to-weight ratio NSJ, as used in the matching trajectory case; (2) enginemass coefficient Esi, corresponding to the selected expansion ratio; and (3) ascent massratio Fa, as determined by the trajectory.
Results of the ascent trajectory analysis for the example are shown in figure 6. Amatrix of cases was evaluated with the expansion ratio ranging from 30 to 80 and the lift-off thrust-to-weight ratio varying from 1.15 to 1.50. (The expansion ratio of 80 repre-sents an approximate limit for stable nozzle flow at sea level conditions.) The perform-ance map of figure 6 utilizes sliding scales for both e and Ns^ in order to have scalesfor both variables. Significant reductions in propellant requirements are possible at thelarger expansion ratios and at the higher thrust-to-weight ratios. However, increases inboth expansion ratio and thrust-to-weight ratio are at the expense of increasing enginemass.
Results of the sizing analysis are shown in figures 7 and 8. Figure 7 is a map ofthe dry-mass change relative to the baseline vehicle (e = 80, Nsi = 1.35). Minimum drymass is achieved at e = 40 and Nsj = 1.20, a 15.9-percent reduction from the baselinevehicle. Details of this vehicle are presented in tables 4 and 5. A comparison of fig-ures 6 and 7 shows nearly a complete inversion of results between trajectory performanceand dry mass. The highest trajectory performance (lowest ascent propellant mass ratio)results in the heaviest dry-mass vehicle, whereas the minimum dry mass occurs in theregion of lowest trajectory performance. It is apparent that the engine mass penaltyrequired to achieve the low ascent propellant mass ratios more than offsets the benefitsof the low mass ratios.
17
Figure 8 is a map of the gross-mass change relative to the baseline vehicle. In thiscase, minimum gross mass is achieved at e = 45 and Nsi = 1.35, a 9.6-percent reduc-tion from the baseline vehicle. Details of this vehicle are presented in tables 4 and 5.This vehicle provides somewhat higher trajectory performance than the minimum-dry -mass vehicle; however, as before, the highest trajectory performance results in the heavi-est vehicle. The optimum occurs at a fairly low expansion ratio and a moderate thrust-to-weight ratio; therefore, a strong trade-off between trajectory performance and enginemass is indicated.
Comparison of total propulsion system mass in tables 4 and 5 (that is, the sum oftankage systems and engine systems)-indicates that the propulsion system mass is on theorder of 50 percent of the dry mass and that it therefore has both strong direct and resiz-ing effects. The propulsion system mass is reduced from 51.7 percent of the dry massfor vehicle 1 to 48.4 percent for vehicle 2 and to 49.1 percent for vehicle 3. As a resultof resizing, the propulsion system mass of vehicle 1 is reduced 21 percent for vehicle 2and 17.5 percent for vehicle 3. Another important measure of vehicle optimality is pay-load fraction, shown in table 5, based on either dry mass or gross mass. The payloadfraction based on vehicle dry mass increased from 11.7 percent for vehicle 1 to 14.0 per-cent for vehicle 2 and 13.5 percent for vehicle 3.
Results of this example indicate the strong influence of trade-offs between enginesystem mass and flight trajectory performance for this class of vehicles. These resultsapply only to the VTOHL SSTO vehicle utilizing the assumed propulsion scheme (allengines on a vehicle have the same fixed expansion ratio). The optimum vehicles tend toaccept significant flight performance penalties due to low expansion ratios and low lift-offthrust-to-weight ratios in order to achieve the benefits of low engine mass coefficients.This illustrates the importance of engine mass coefficient E as a technology area forthis class of rocket propulsion systems.
These results also indicate the suitability of this method — linearized sizing com-bined with optimized trajectory performance — for the optimization of propulsion sys-tems when the range of variables is appropriately limited. As illustrated in the example,rocket propulsion systems are characterized by conflicting flight performance and vehi-cle sizing considerations and therefore require this type of combined performance-sizinganalysis for optimization. The linearized sizing analysis provides adequate sizing char-acterization while also offering the potential for considerable savings in analysis timeand/or cost over other sizing analyses. However, because of the linearizing assumptions,care must be taken to provide calibration with a suitable vehicle system and to limit theextent of sizing perturbations.
There are a number of other propulsion schemes and operational modes of potentialinterest for SSTO application. Other rocket propulsion schemes include mixed expansion
18
ratios, dual-position nozzles, and dual fuels. The basic assumptions behind this methodof analysis are quite general; therefore, all these propulsion-operational schemes areadaptable to this type of analysis - some require only slight modification of the inputswhile others may require modifications to the sizing equations. The computation of sen-sitivity information is another feature of this method associated with the explicit sizingsolution. Small perturbations to the sizing imputs can be used to derive sizing sensitivi-ties in order to show the impact of technology changes.
CONCLUDING REMARKS
A method has been developed for the analytical optimization of the rocket propulsionsystem for a vertical take-off, horizontal landing, single-stage-to-orbit launch system.This method combines the performance results obtained from a point-mass trajectoryoptimization program with a linearized sizing program to establish vehicle sizing trends.This linearized sizing program, which simplifies the overall system to 11 mass categories(4 dry mass categories), has an explicit solution form and is small enough to offer sub-stantial computational savings over many other sizing programs.
Results are presented for a vehicle system utilizing rocket engines with high cham-ber pressure, liquid oxygen and liquid hydrogen propellants, conventional bell nozzles, andthe same fixed expansion ratio for all engines on a vehicle. Both expansion ratio and lift-off thrust-to-weight ratio were optimized to achieve vehicles with either minimum grossmass or minimum dry mass. Results show that propulsion system mass is a dominantvehicle sizing factor. The trades between engine system mass and trajectory perform-ance show a strong trend toward low engine mass. Optimum values of expansion ratiofell in the range of 40 to 45; optimum values of lift-off thrust-to-weight ratio fell in therange of 1.20 to 1.35.
With proper care in its application, both in establishing the required sizing inputsand in limiting the range of perturbations, this method should give quite satisfactoryresults. Also, the method is adaptable to many other flight vehicle/propulsion systemconcepts.
Langley Research CenterNational Aeronautics and Space AdministrationHampton, Va. 23665May 19, 1976
19
REFERENCES
1. Martin, James A.: A Method for Determining Optimum Phasing of a Multiphase Pro-pulsion System for a Single-Stage Vehicle With Linearized Inert Weight. NASATN D-7792, 1974.
2. Brauer, G. L.: Cormck, D. E.; Habeger, A. R.; Peter sen, F. M.; and Stevenson, R.:Program To Optimize Simulated Trajectories (POST). Volume I - FormulationManual. NASA CR-132689, 1975.
3. Malkin, M. S.: Space Shuttle/The New Baseline. Astronaut. & Aeronaut.,, vol. 12, ,no. 1, Jan. 1974, pp. 62-78.
4. Brooks, George W.: Developing Structures Technology for the Day After Tomorrow.Astronaut. & Aeronaut., vol. 11, no. 7, July 1973, pp. 56-66.
5. Beichei, Rudi: Propulsion Systems for Single-Stage Shuttles. Astronaut. & Aeronaut.,vol. 12, no. 11, Nov. 1974, pp. 32-39.
6. Ahlberg, J. H.; Hamilton, S.; Migdal, D.; and Nilson, E. N.: Truncated Perfect Nozzlesin Optimum Nozzle Design. ARS J., vol. 31, no. 5, May 1961, pp. 614-620.
7. Rau, Timothy R.; and Decker, John P.: ODIN: Optimal Design Integration System forSynthesis of Aerospace Vehicles. AIAA Paper No. 74-72, Jan.-Feb. 1974.
8. Martin, James A.: Optimal Payload Ascent Trajectories of Winged Vehicles. J.Spacecraft & Rockets, vol. 11, no. 12, Dec. 1974, pp. 860-862.
20
TABLE 1.- POINT-DESIGN VEHICLE - MASS PROPERTIES
VTOHL SSTO with LOX-LH2 propellant; reference wing area,1087 m2; Nsl = 1.347; e = 80; and Is?vac = 453 s
Item
1.02.03.0.1
.2
.3
.4
4.0
5.0
6.0
8.0
9.0
10.011.012.013.014.015.018.0
20.021.022.023.024.0
25.026.027.0
.1
.2
Component category
Wing groupTail groupBody group (total mass, 62 515 kg)
Basic structureThrust structureLOX tanksFuel tanks
Induced environmental protectionLanding, docking, recoveryPropulsion - ascentPropulsion - auxiliaryPrime powerElectrical, conversion and distributionHydraulic, conversion and distributionSurface controlsAvionicsEnvironmental controlPersonnel provisionsGrowth
Dry mass
PersonnelCargoRCS reservesRCS propellantResiduals
Entry mass
OMS propellantReserve fluidsMam propellant (total mass, 1 855 998)
In-flight lossesAscent
Gross mass
Mass, kg
23 7173 454
19 5755 538
12 29625 10635 882
8 73561 291
3 8251 7753 4033 1713 3052 0211 857
790
15 475231 216
705
29 48368
4 53613 064
279 072
10 224
3 725
3 7251 852 2732 149 019
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29
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31
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