Preliminary Sizing Methodology for Hypersonic Vehicles

Armand J. Chaput Genera I Dynam ics, Fort Worth, Tx

AIAAIAHSIASEE AIRCRAFT DESIGN, SYSTEMS AND OPERATIONS MEETING

September 14-1 6, 1987lSt. Louis, Mo

Copyright O 1988 by General Dynamics Corporation All rights reserved

Published by the American Institute o f ~eronautics and Astronautics with permission

PRELIMINARY SIZING METHODOLOGY FOR HYPERSONIC VEHICLES

Armand J. Chaput* General Dynamics Fort Worth, Texas

Abstract

- Cycle specific impulse (Sec) Simplified design and analysis relationships are developed which are suitable for conceptual-level sizing and synthesis oi hypersonic vehicles. The relationships are developed from point design studies and normalized for generalized vehicle design applications. First and second-order polynominal curve-fit expressions describe the normalized, aerodynamic and propulsion performance parameters. Geometry is represented by simple shapes and areas. Mass properties are approximated by unit weights and factors. A simplified atmosphere and equations of motion complete an equation set which can be iterated and solved using elementary programming

- Vehicle specific impulse (Sec)

- Incremental weight ratio (Dimens- ionless)

- Horizontal tail area ratio (Dimen- sionless)

- Landing gear gross weight fraction

- Other fluids fraction (Dimensionless) techniques.

- Propellant packing factor (Dimen- sionless)

Nomenclature

- Coefficient defined by Equation 37 (Dimensionless) - Fuel reserve fraction (Dimensionless)

- En ine projected inlet frontal area ( F t 5

- Ratio of actual to approximated fuselage wetted areas

- Wing aspect ratio (Dimensionless)

- Wing span (Ft)

- Drag due to lift (Dimensionless)

.- Minimum drag coefficient (Dimensionless)

- Lift coefficient (Dimensionless)

- Wing theoretical root chord (Ft)

- Wing theoretical tip chord (Ft)

- Nominal fuselage diameter (Ft)

- Vehicle drag (Lb)

- Fuselage "photographic" scale factor (Dimensionless)

- Vertical tail area ratio (Dimen- sionless)

- Fuselage length (Ft)

- Vehicle lift (Lb)

- Lift-to-weight ratio (Dimensionless)

- Mach number (Dimensionless)

- Number of integration intervals (Dimensionless) - Total energy parameter ( ~ t 2 / ~ e c 2 )

- Engine scale factor (Dimensionless) - Free-stream dynamic pressure ( ~ b l ~ t 2 )

- Stoichiometric fuel-to-air ratio (Dimensionless) - Takeoff dynamic pressure ( ~ b / F t z )

- Propellant fraction (Dimensionless)

- Net thrust (Lb)

- Fuselage radius at station x, (Ft)

- Wing exposed pianform area (Ft2)

- Horizontal tail planform area (Ft2) - Net specific thrust (Sec)

- Vertical tail planfdrm area (Ftz) - Standard acceleration due to gravity (Ft/~ecZ)

- Theoretical wing reference area (Ft2) - Altitude (Ft)

- Total vehicle wetted area (Ft2) - Initial incremental altitude (Ft)

- Fuselage wetted area (Ft2)

- Horizontal tail wetted area (Ft2)

- Final incremental altitude (Ft)

* Deputy Program Mana er National Aero-Space bane (NASP) Chairman, AIAA Aircraft Design TC, Associate Fellow - Vertical tail wetted area (Ftz)

- Wing wetted area (Ft2)

- High-speed propulsion system thrust- to-weight ratio (Dimensionless)

- Low-speed propulsion system thrust- to-weight ratio (Dimensionless)

- Vehicle takeoff thrust-to-weight ratio (Dimensionless)

- Fusela e structure unit weight S (Lb/Ft

- High-speed propulsion system unit weight (Lb)

- Horizontal tail structure unit weight ( ~ b / ~ t 2 )

- Vertical tail structure unit weight ( ~ b / ~ t 2 )

- Wing structure unit weight ( ~ b / ~ t 2 )

- Free-stream velocity (Ft/Sec)

- Initial incremental velocity (Ft/Sec)

- Final incremental velocity (Ft/Sec)

- Initial ascent velocity (Ft/Sec)

- Final ascent velocity (FtISec)

- Fuel tank volume ( ~ t 3 )

- Engine airflow (LbISec)

- Propellant flow ra te '(Lb/Sec)

- Vehicle weight (Lb)

- Empty Weight (Lb)

- Fuel weight (Lb)

- Finai mission weight (Lb)

- Fuselage structure weight (Lb)

- High-speed propulsion system weight (Lb)

- Horizontal tail structure weight (Lb)

- Initial mission weight (Lb)

- Low-speed system propulsion weight (Lb)

- Payload weight (Lb)

- Propellant weight (Lb)

- Reserve propellent weight (Lb)

- Vertical tail structure weight (Lb)

- Wing structure weight (Lb)

- Gross takeoff weight (Lb)

- Design takeoff wing loading ( ~ b l F t 2 )

- Initial incremental weight (Lb)

- Final incremental weight (Lb)

- Fuselage station (Ft)

- Free-stream pressure relative to sealevel (Dimensionless)

- Coolant contribution t o specific thrust (Sec)

- Takeoff and initial acceleration fuel fraction required

- Coolant flow fraction of equivalence ratio (Dimensionless)

- Incremental velocity change (Ft/Sec)

- Free-stream absolute temperature relative to sealevel (Dimensionless)

- Wing taper ratio (Dimensionless)

- Average propellant density (LbIFt3)

- Fuel equivalence ratio (Dimensionless)

Backmound

After a 20-year hiatus during which only a few stalwarts kept capabi l i t i es alive, t h e country is once again embarked on the development of airbreathing hypersonic vehicles. For t h o s e involved in t h e design and development of these vehicles, coming back up to speed in hypersonics has been a relatively rapid process. This is due in large part t o the impetus provided by major new hypersonic programs such as the National Aero-Space Plane (NASP). However, because of national security considerations and resource limitations, only a few designers will be directly involved in these efforts. How, therefore, do those who a re not participants in such programs develop t h e insights and skills required t o execute a design much less even know where to s ta r t?

Fortunately, there a r e a number of excellent t e x t s available in specific hypersonic disciplines. Of these the most useful f o r design purposes a r e in the field of aerodynamics. Excellent propulsion and structure-related texts a r e also available, but they are usually oriented toward functional specialists and are sometimes difficult t o reduce t o vehicle design practice. Almost non- ex i s ten t a r e t e x t s on hypersonic veh ic le design. However, there a re many excellent design studies from the 60's and early 70's which provide data on specific vehicle concepts. Although dated, these studies provide excellent baselines from which contemporary applications can be developed. In addition, the design data presented can be normalized and used t o synthesize a wide range of state-of-the-art derivatives.

Analvtical Approach

Specific baseline vehicle design d a t a is reduced t o normalized form using parametrics which model primary design drivers (thrust, drag, and weight) as functions of size, shape and trajectory. Other drivers such as volume, cooling requirements, and maneuver loads can also be included a s required. The combined e f f e c t s of the individual p a r a m e t r i c s a r e l inked by s i m p l i f i e d a tmospher ic and vehicle per formance models, and evaluated numerically using elementary programming

techniques (an iterative, multi-variable version of the sizinglsynthesis procedure described in this paper executes on a programmable 16K pocket computer).

Application of the parametrics to vehicles which may vary significantly from the baseline are accommodated by changes in geometric inputs or by adjusting design parameters (i.e. modifying structural unit weights to capture the impact of a structural design change). Other changes, such as technology advancements can be incorporated as multipliers (i.e. a percentage reduction in structural unit weight to represent advanced material substitution) and appropriate sizing and performance impacts determined. More complicated changes such as a fuel or engine cycle change require development of new parametrics which can be added to an appropriately modularized computational procedure.

Baseline Approach

Minimum requirements for a baseline vehicle are (1) it should be representative of the configuration type under consideration, and (2) it should be described by a relatively complete technical data package. A conventional wing-body-tail design (see Figure 1) was selected as the example for this paper because of the interest in this configuration type and the completeness of its technical documentation. It was, however, only one of a number of hypersonic transport (HST) confieprations evaluated in Reference 1 which could have been selected.

The baseline HST is powered by 4 hydrogen-fueled Pratt & Whitney turbo-ramjet study engines. They are mounted

external to the basic vehicle structure and require some wing-forebody compression a t hypersonic speeds to achieve design objectives (Figure 2) . The basic characteristics of the vehicle are as follows:

LH2 Fuel Weight ( W F J = 196.646Lb

Payload 1 Wp.4 y) - 200passenqer-s. baggage and cargo

Engine Thrust - 86.800 Lb (Unins ta l l~d . (each) Sealeuel Static)

A geometric description of the baseline is contained in Table 1. Weights are contained in Table 2. Figure 3 shows the design mission profile and the constraints which determined the limits of the operational envelope. Because of the prevailing interest in accelerator type vehicles, however, the synthesislsizing procedure was coded and applied only up to the end of the ascent portion of the design mission profile (M=6.0. h=103.000 Ft) . Cruise, descent, loiter and landing requirements were characterized by a single, post-ascent fuel reserve.

Vehicle Geometry Relationships

Simple geometric shapes are used to represent major components of the baseline vehicle. In order to maintain geometric similarity with the baseline, fuselage geometry changes are limited to "photographic" scaling. The winglempennage and fuselage, however, are scaled independently in order to decouple planform and volume effects. Simple wetted and/or planform area ratios are

/

AIRFRAME I TANK ;

SEC A A 75' -.

6061 t I

Figure 1 Baseline Hypersonic Transport (From Reference 1)

--/B!!l - I 1- i

BOUNDARY LAYER PLOW BOUNDARY LAYER OUCT

/ FWD MOUNT TURNBUCKLES

SECONDARY ACTU

BOUNOARY CAYER BLEED HOLES / ~RIMARY ACTUATOR / \ EXPANSION JOINT

RAMP NO. 4 VEHICLE THERMAL TITANIUM 16 AL. 4VI PRESSURE SHELL RAMP NO. 3 PROTECTION SYSTEM

Figure 2 HST Propulsion System

Table 1 BASELINE HSl' GEOMETRY

Wine Reference Area (SREF)

Span ( 6 )

Root Chord (C,)

Tip Chord ( C t )

Aspect Ratio (AR)

Exposed Area tS ~ . y p WING)

Taper Ratio ( A )

Horizontal Tail I I Area (S H T )

Vertical Tail

used to correct baseline parameters for scale effects. Major deviations from the baseline configuration require careful assessment of input parameters and may result in rederivation in some areas.

1207 Ft2

Area ( S V T )

Fuselage

Length ( 0 Nominal Diameter (dl

Total Volume (VTOT;IL)

Tank Volume ( V T A I V K )

Wetted Area (SWET)

Provulsion

Engine Frontal Area (AF)

Fuselage Definition - For preliminary sizing purposes, reasonable accuracy can be achieved by analytically approximating the vehicle fuselage as a series of cones and cylinders as shown in Figure 4. The fuselage is assumed to contain three fuel compartments which are described by forward and aft truncated cone sections and one connecting cylindrical center section. Cone shapes form the forward and aft fuselage closures. Non-fuel fuselage volume allocations can be modeled directly or approximated by appropriately reduced fuel packing factors. In addition, limits can be placed on any dimensions (i.e. fixing the size of the forward cone to prevent the crew compartment from scaling below reasonable limits.

1079 Ftz

314.5 Ft

20.9 Ft 83418 Ft3

46416 Ft3

19863 Ft2

113 Ft2

Handbook procedures provide the following generalized equations for the approximated wetted areas and volumes:

Table 2 BASELINE HST WEIGHTS (Lb)

Structure

Wing

Horizontal Tail

Vertical Tail

Fuselage (inc. tanks)

Pro~uls ion

Engine & Acc

Air Induction

Nacelles & Structure

Landing Gear

Subsvstems

Fuel

Pressurization & Lubrication

Secondary Power & Elect Distribution

Controls

Subtotal (Type 1)

Instr. 8 Avionics

Environmental Control

Passenger & Cargo Provisions

Crew Provisions

Subtotal (Type 2)

Emptv Weight

Pavload & Crew

Fluids Fuel

Residuals & Other Fluids

Losses

GROSS TAKEOFF WEIGHT

Pack ing f a c t o r s provide c o r r e c t i o n s f o r t a n k s h a p e variations, voids, and non-optimums. Similarly, wet ted a r e a multipliers a r e applied t o co r r ec t fo r non-circular shapes and proturberances (i. e. passenger compar tmen t s and engine cowls).

Baseline vehicle dimensions were sca led f rom Figure 1 a s follows:

Sta t ion .m a

The following values were ca lcula ted using Equation 1:

T h e r a t i o s of t h e s e v a l u e s t o t h o s e c o n t a i n e d in Re fe rence 1 values were used t o ca l cu l a t e t h e wet ted a r e a (kswEr) and packing f a c t o r ( k P F ) corrections:

DYNAMIC PRESSURE

SONIC BOOM

5067-5 MACH NO.

Figure 3 HST Tra j ec to ry Design Constraints

FUSELAGE

I I

AFT TANK FUSELAGE

X. x, x, X, X, X,

h-.ROOT CHORD ~,--d EMPENNAGE

u TIP CHORD

Figure 4 Vehicle Geometry Approximations

Note that these factors not only provide corrections for simplified geometric approximations, but can be used as design parameters to evaluate such effects as vehicle density and configuration-dependent wetted area-to- volume variations. Other volume requirements and sensitivities such as those associated with specific propulsion or subsystem concepts can a lso be incorporated as sizing parameters.

Wina Definition - A simple, trapezoidal shape as shown in Figure 4 is used to approximate wing planform geometry in terms of wing span (b), root chord (C,), and taper ratio (l):

Exposed wing area is approximated by:

Wing wetted area is approximated at twice the exposed planform area.

I f desired, wing volume available for fuel can be included for a nominal airfoil shape and thickness ratio. However, for hydrogen powered vehicles such as the baseline, thin wings do not provide adequate volume to justify their use for fuel containment.

Equations 2 and 3 yield the approximations for the baseline:

SREF = 6600 Ft2 SEXP WING = 4035 Ft2

The latter value compares to the Reference 1 value of 3,880 ~ t 2 . The differences are due to the approximate nature of the equations used. The effect, however, can be nearly mitigated by using the calculated value to derive wetted-area dependent wing parameters (i-e. exposed wing unit weights).

Emoennaae Definition - Horizontal and ver t ica l empennage planform areas are expressed as percentages of the wing reference area. Wetted areas a re approximated by doubling the planform areas. The generalized equations are as follows:

S H T = k HT S REF

S v T = h S VT REF

where, for the baseline case:

Geometry Scaling/Resizing - Wing-empennage size required is defined by wing loading at a maximum weight condition as follows:

where the prime (I) notations reflect the area and takeoff weight of the resized vehicle and

W L = Design Wing Loading

Changes in wing area influence sructural weight, the effects of which can be iterated using a relatively straightforward takeoff weight driven "inner-loop" procedure. Wing area, however, also affects minimum drag and drag due to lift which have influence over the entire mission and require iterations on fuel volume to achieve convergence. Multiple "outer-loop" iterations over the complete mission, therefore, will be required. The convergence criteria is established in the form of percentage fuel remaining (positive or negative) a t mission completion.

To maintain consistency with baseline characteristics, fuselage resizing is limited to "photographic" scaling; that is, all fuselage dimensions are scaled equally. The scaling parameter can be approximated by the cube root of the ratio of required-to-available fuel volume such that:

and

where

Again, the prime ( ' ) notations represent rescaled dimensions to be used for a subsequent iteration.

Weight Relationshim

Vehicle mass properties are broken down into categories consisting of structural, propulsion, landing gear, subsystems, payload (inclutling crew) and fuel weights.

Structural Weights - Fuselage, wing and empennage weights are assumed to be primarily wetted area dependent. Unit Weight Factors ( U W F ) are used to estimate these area-dependent weights as follows:

U W F , were calculated for the baseline from Tables 1 and 2 as follows:

Note that U W F w r N ~ is derived using the calculated value from Equation 3, and that wing and empennage area is approximated as twice that of the appropriate planform.

Propulsion Weights - Two propulsion system types a re normally associated with hypersonic vehicles; low-speed (for takeoff and acceleration t o ramjet speeds), and high- speed (for acceleration from supersonic to hypersonic speeds and to sustain cruise). Due t o the different requirements on the two systems, it is useful t o size and weigh each separately. The following sizing equations a re used:

where

ToiWo = Vehicle Takeoff Thrust-to-Weight Ratio

(T!W)L~S = Low-Speed System Thrust-to-Weight Ratio

ESFHss = Engine Scale Factor

UWFHss = High-speed Engine Weight for ESFHss = 1.0

The following values were derived or given for the baseline vehicle:

In developing the above values, i t was necessary t o allocate weights between the low-speed or high-speed systems. For simplicity, a'nominal 75% / 25% ratio was used based on the primary function of the component. For example, 75% of the variable-geometry/air-induction system which is required for hypersonic flight (but also functions as a low-speed inlet) is allocated t o the high- speed system weight. The turbojet which is used only a t lower speeds (but provides a flow path for the ramjet) had only 25% of its weight charged to the high-speed system. Weights which could not be uniquely associated with either the low or highspeed system were allocated on a 50/50 basis. The allocations resulted in the following weight breakdown:

Subsvstem Weights - Simple ratios a re used t o define subsystem weights. Some systems, however, a re assumed t o be driven primarily by empty weight, while others will be primarily gross weight driven. The control system is an example of a subsystem which is assumed to be sized primarily by gross weight, while the environmental control system is assumed to be driven mostly by empty weight. The subsystem factors arbitrarily defined as Type 1 and Type 2, respectively, for the baseline vehicle are:

Table 2 shows subtotals which dep ic t t h e assumed breakdown by type. Landing gear factors a r e defined by the sum of the nose and main gear which were calculated as follows for the baseline:

Fluid Weivhts - Fluids include primary propellants, reserves, residuals, and o ther fluids. Propellants, reserves and residuals a r e defined in t e rms of tank volume available, fuel density and packing factor. Other fluids a re defined as fixed percentages. The following equations define the various fluid types:

The following numerical values were derived or given for the baseline:

The large reserve propellant fraction (kRES) results from simulating only the takeoff and ascent segments of flight as described previously. Note that the propellant packing factor (kPF) has already been calculated (see Fuselage definition). Note also th2t if multiple propellants a re used (i.e. oxidizers), that P p ~ o p will become an iteration variable due t o density differences which drive volume requirements.

Fixed Weights - This category includes crew, payload and provisions which do not scale with vehicle size. The following Reference 1 values were given:

Crew & Provisions - 1,275 Lb Cargo & Passengers - 48,000 Lb

For simplicity, the values a re summed t o yield a single "payload" weight:

W p A = 49,275 Lb

Aerodvnamic Relationships

It is beyond the scope of this paper t o cover t h e derivation and calculation of generalized hypersonic vehicle aerodynamic relationships. Lnstead, simplified relationships based only on weight (W), wing reference a r e a (SREF), to ta l wet ted a rea ( S W E ~ ) , and dynamic pressure (q) are used.

Lift - The forces perpendicular t o the flight path vector - generally considered significant a re wing/body lift and the normal components of thrust, weight and centrifugal acceleration. Accurate calculation of these fo rces requires implementation of a guidance algorithm which is also beyond the scope of this paper. However, for relatively small angles-of-attack, reasonable accuracy can be achieved by calculating a nominal correction factor in the form of a velocity-dependent multiplier derived from analysis of a baseline vehicle.

The multiplier is derived by performing a second-order fit of lift-to-weight ratio (LIW) to velocity (V) as shown in Figure 5. For the baseline vehicle, the resulting equation 1s:

LIW = 0.967 - 5.833 x lo-%' - 2 . 2 6 2 ~ lo-' v2 (12)

This equation predicts a zero-lift required condition which differs somewhat from more exact formulations; but nonetheless, it yields relatively good accuracy over much of the high-speed envelope. Significant errors, however, will occur a t low speeds where angle-of-attack effects cannot be ignored. Errors will also be incurred on

very high-acceleration trajectories. It is important, therefore, t o check the validity of the equation when ascent accelerations vary significantly from the baseline.

- Vehicle drag is assumed t o consist of two .04

components expressed in coefficient form as follows:

1.00 .03

- 0 REFERENCE 1 DATA 3 - I CURVE FIT APPROX. o o

0.98 - .02 -

0.96 - 1 - -

W 0.94 - .a0

0 1 2 3 4 5 - M A C H NUMBER ( M I 5067-8

0.92 - Figure 6 Baseline Minimum Drag Approximation 0

0.90 0 1000 2000 3000 4000 5000 6000

5067.7 V (Ft I Sec)

Figure 5 Lift-To-Weight Approximation

Cow,, is modeled a s a non-linear function of Mach number, while CD, is approximated as a relatively well- behaved function of .M and CL. Other drag components, including trim, can be modeled as scaler multipliers (i-e. CD,,, = 5%).

Engine-related drag is accommodated by the propulsion related equations.

The aerodynamic coefficients for the baseline vehicle a re shown in Figures 6 and 7. C D ~ ~ ~ ~ is approximated by first and second-order functions over discrete Mach number segments in order to simplify the non-linear equations. Simple numerical curve-fitting techniques resulted in the following expressions:

Linear and second-order functions of Mach Number (M) and lift coefficient (CL), respectively, yield a simplified relationship for drag due to lift. It was derived using a two-variable, least-squares f i t t o the da ta in Figure 7.

For scaling purposes, CD is assumed t o vary linearly with total wetted area as%'ilows:

SYMBOLS REF. 1. DATA - CURVE FIT

5067-9 CD, Figure 7 Baseline Drag-Due-To-Lift Approximation

The reference a rea rat ios a r e included to maintain geometric consistency for substitution into Equation 12. Once again, the ( ' ) notation identifies a resized but "photographically" similar configuration (the exception being the 'Idecoupled" wing area variations previously described). CDL is assumed t o remain constant, since basic wing parameters (i.e. aspect ratio and sweep) a re unchanged.

Different propulsion models a re required for low-speed (takeoff and acceleration t o supersonic speeds) and high- speed (ramjet and/or scramjet) systems. The former typically a re adaptations of existing turbomachines and their performance can be approximated accordingly. Ramjets and scramjets , however, a r e fundamentally different cycles and other methodology must be applied. Fortunately, this methodology is relatively s t raight- forward and is computationally less complicated than for turbomachines (at conceptual levels). This is because ramjets and scramjets depend on forward vehicle motion t o compress incoming air prior t o combustion rather than on turbomachinery. Consequently, their performance levels tend t o be primarily airflow-dependent. Airflow, however, is a function of many design and performance variables; the most dominant of which a r e inlet area, f ree

stream dynamic pressure and vehicle attitude (primarily angle-of-attack). This introduces additional computa- tional complexity.

Turbomachinery performance is also a strong function of airflow, but it is more dependent on throttle setting and is less sensitive to vehicle attitudes.

Airflow Characteristics - First-order approximations of both low and high-speed airf low a r e developed by correlating airflow (W,) per unit engine frontal area (AF) times q as shown in Figure 8. Note that these data from the baseline HST cover a wide range of dynamic pressures (from a few hundred psf to over 2000), but still normalize in a relatively well-behaved fashion over the Mach range. This is particularly t rue f o r ascent where maximum power th ro t t l e set t ings a r e maintained and vehicle at t i tude changes a re small. Note also that the engine size parameters AF and ESFHss for t h e baseline a r e related by the equation:

- - CURVE FIT

\'FULL POWER

/ \ 1 H l G H u DESCENT (Idle) = o\ -

Q -/ -

PART POWER LOW CY ASCENT (Full Power) DESCENT

0J 1 I I I 1

0 1 2 3 4 5 5067.10 MACH NUMBER (MI

Figure 8 Baseline Vehicle Airflow Parametr ic

During descent, considerable throttling occurs for t h e baseline mission and substantial angle-of-at tack (a) excursions a re experienced. And, although the baseline da ta in Reference 1 is not sufficiently comprehensive t o rigorously verify t h e relationship, it does show t h e influence of (a) on W, independent of throttle setting. The d a t a also demonstrates the relative insensitivity of turbomachinery performance t o (a), and that ascent- descent variations a r e primarily due to differences in throt t le setting. Note also that the airflow parameter (Wa;qAF) clearly shows the effect of ramjet transition both on ascent and descent.

Since this paper addresses only the ascent portion of flight where angle-of-attack variations a r e small and power s e t t i n g s a r e c o n s t a n t , ( a ) and t h r o t t l i n g relationships are not presented. They were derived, however, and found t o be computationally manageable. Nonetheless, for the ascent phase, reasonable accuracy is achieved by ignoring these e f f e c t s and cor re la t ing (Wa/qAF) with .M only, a s shown in Figure 8. For s implici ty , t h e cor re la t ion is def ined by s e p a r a t e polynomial approximations over discrete Mach ranges as follows:

Fuel Flow Characteristics - Fuel flow (iVF) and Airflow (W,) a re related by the following relationship:

where

pa = Stoichiornetric Fuel-to-Air Ratio ( = .0293 for H2)

Q = Equivalence Rat10

Optimum cycle efficiency generally occurs a t or near Q = I (i.e. stoichiometric combustion). Throttling and/or cooling requirements, however, can result in lean ( @ < I ) or rich (@>I) mixture operations. The baseline HST speed regime in Reference 1 is relatively low and no excess cooling flows a r e required, excep t f o r c ru i se (i.e. equivalence ratio remains constant a t Q = I for the entire ascent phase). For vehicles which fly a t higher speeds (above approximately iM=lO), a Q schedule as a function of M and q may be required t o accurately model ascent performance.

T h r u s t C h a r a c t e r i s t i c s - T h r u s t a v a i l a b l e is approximated by a s p e c i f i c t h r u s t (FSP) and (Q) parametric where FSp by definition is given as:

where

Fy = Net propulsive thrust (i.e. gross thrust less ram drag less installation losses)

The p a r a m e t r i c h a s been found t o be r e l a t i v e l y insensitive t o q variations and t o vary primarily with M. Furthermore, the effects a re approximately linear over the ranges Q <I and Q > I .

The characteristic Mach Number variation is shown by t h e ascent d a t a in Figure 9. The small differences between ascent and descent show the validity of the linear parametr ic relationship for @ < I . For @ > I , however, Fsp varies according t o a different linear relationship.

Equivalence ratios a re typically greater than unity in speed regimes where more cooling flow is required than can be accommodated by s toichiometr ic combustion. Because of airflow limits, t h e excess fuel does not combust , but i t does have a t h r u s t con t r ibu t ion attributable to the momentum of the excess fuel. This thrust contribution primarily is a function of injection temperature, which for the case of hydrogen, can be assumed t o be near engine material temperature limits (about 2000"R). Under these temperature conditions, each additional equivalence ratio of coolant contributes about 13 seconds of specific thrust, or:

techniques for the other regimes will be similar; the only unique modeling requirement being for throttled turbojet

where

For purposes of comparison, this relationship translates into a specific impulse for the coolant of about 445 seconds.

Note that in Figure 9, the performance of the turbojet low-speed system can also be represented by a F S p function. Its throttled performance, however, is more complex than the ramjet, and does not reduce to a simple @ relationship. If a rigorous performance simulation is required for throttled turbomachinery, more complex modeling techniques will be needed.

TURBOJET , , R A M J E T I

- 0 ,'-\&...-- A I FULL POWER

w7RT PO WE"^ 0 ASCENT ( 6 1 1 - DESCENT ($<<I1 -

- -CURVE F I T APPROX.

" 0 1 2 3 4 5 6

~067.11 M A C H NUMBER (MI Figure 9 Baseline Vehicle Specific Thrust (Fd

Characteristics

Pro~ulsion Scaling - The parametric nature of the propulsion relationships presented allow thrust t o be calculated directly from LVa and AF. Engine frontal area required for the resized vehicle can be determined a number of ways. One is t o maintain a fixed ratio of vehicle cross section-to-engine capture area. Another way is t o maintain a prescribed thrust-to-drag ratio a t a given design point.

Engine weights (and volumes if available) can also be assumed t o vary linearly with engine size as shown in Figure 10. Note, however, that the baseline scaling relationship is highly dependent on engine design and mater ials and does not apply universally. I t does, however, provide a benchmark value against which technology trades can be performed.

Performance Modeling

Performance is modeled separately fo r each of the following baseline HST f l ight regimes described in Reference 1:

1. Takeoff and initial climb 2. Ascent 3. Cruise 4. Descent 5. Loiter 6. Landing

performance.

3oJ I I 1 I 100 120 140 160 180 2

5067.12 ENGINE FRONTAL AREA (FT2) Figure 10 Engine Weight Scaling

Takeoff and Initial Climb - This mission segment typically sizes wing area and the low-speed system. A number of parametrics a re available which correlate performance in this regime. Figure 11 is an example which is based on handbook da ta from over 50 military and civil aircraft. It enables a first-order approximation of wing loading ( W L = q T ~ O C L ~ , ~ ) and ToiWo required as a function of take-off ground roll.

lU000 1 M I N I M U M / /

DEMONSTRATED

H S T (REF 11

0 100 200 300 400 500 600

Figure 11 Takeoff Performance Parametric

Fuel consumption f o r t akeof f and ini t ia l c l imb is estimated a s fixed percentage of vehicle gross weight. Although a simplified assumption, three fundamentally different size and weight configurations from Reference 1 were eva lua ted and al l required fue l quant i t i es equivalent to almost exactly 1.7% of gross weight to reach M=0.8 (a nominal ascent trajectory starting point).

The takeoff sensitive sizing parameters for the baseline HST were calculated or given as follows:

Simplified relationships can be developed to model all six regimes but, as described previously, only takeoff and ascent a r e quantified in this paper. The modeling

TOIWO = 0.65

AW7.io = 0 0 1 7 WTl0

Ascent - Ascent performance can be modeled with good accuracy using a simplified form of the basic accelerator equation

where over the range from State 1 t o State 2

A V l -2 = Change in velocity (FtlSeci

FV = Average net propulsive force (Lb i

- D = Average Vehicle Drag ( L b )

- WF = Average Fuel Flow (LblSec)

W I = Initial Weight ( L b )

The approximate equality symbol is used in Equation 28 for simplicity, a s it uses t h e ini t ia l value of t h e subscripted terms for the summation in lieu of the more exac t average value over t h e interval . Also, t h e simplified equation excludes a l t i tude (i.e. po ten t ia l energy) effects. This omission can be the source of some error and a correction factor is presented in the section t o follow.

Regardless of t h e approximate nature of the equation, its fo rm (and subsequent c o m p u t a t i o n ) is s impl i f i ed considerably by defining summation intervals in terms of a fixed gross weight variation (k) where:

w2 = Final Weight ( L b ) I t can also be shown that the number of summation intervals is related t o vehicle fuel fraction by t h e

Equation 22 can be expressed in a more familiar form by following relationship: applying the notation:

and Altitude Correction - The simplified a c c e l e r a t o r equat ion neg lec t s t h e change in p o t e n t i a l energy associated with altitude variations. When the corrections a r e included, the equation for a single sum mat ion interval

(24) can be approximated by the following specific energy formulation:

This yields expressions which a r e standard forms for rocket applications: F.v 2 (32)

v : + Z g h 2 = / g ( + j L n i k ) + v,I + 2 g h ,

X less familiar form which is consis tent with t h e propulsion parametrics presented herein results from combining Equations 19 and 20 which yields:

FsP , a, D, and W , are trajectory variables, and a series of incrementa l c a l c u l a t i o n s c a n a p p r o x i m a t e t h e complete trajectory and yield the following form of Equation 27:

where

Note that when the bracketed term is large compared t o 2gAh, the error of omission is negligible. Early in the flight where climb rates a re high, the error introduced can be substantial. The problem with this e f f e c t , however, is not the form of the equation, but rather that it requires definition of an altitude-velocity profile and iteration t o solve. Also, a velocity-altitude profile is cumbersome for design definition purposes. A more meaningful definition is one based on parameters such as q and M, which have physical significance for sizing vehicle structure and systems. This can be done using al ternate formulations which correlate q and .W with velocity and altitude. One formulation derives from an ideal (gas) atmosphere which gives by definition:

where

S = PIP 0

0 = tlt 0

and Po and to are sealevel standard atmospheric pressure and temperature, respectively.

Simplified Atmosphere - q and M, by definition, a re given as follows:

The equations can also be expressed in the forms:

Typically, a s tandard atmosphere is d e s c r i b e d by discontinuous S and 0 functions expressed algebraically for discrete altitude ranges. It is possible, however, t o derive a single approximate relationship which covers the range from sea level t o 200K feet. The expression can be derived using curve f i t t ing techniques, and is given without proof as follows for an approximate form of a 1962 Standard Atmosphere:

where

- REFERENCE 1 DATA - - - LINEAR APPROXIMATION

> (2.112 + 0.05183 a - .004043 a-) (36) 5067-15

MACH NUMBER (M) h = a - 8

Figure 13 Baseline Vehicle Trajectory Approximation

a = 9.4113 - 21.521Ln (610) (37)

Figure 12 shows the correlation between the approximate and exact atmospheres.

200

180 - 0 PREDICTED h

LY Y

g 1"- 5

1 140 - - LINE OF PERFECT n CORRELATION

,& 120 - - \ /// hpred = a - e2.112 + . O W 3 a

WHERE 6' a = 9.4113 - 21.824

5067-14 ALTITUDE (hl - 1000 FT

Figure 12 Standard Atmosphere Approximation vs Exact Pormulation

Traiectorv Modeling - The previously derived thrust, drag, and atmosphere approximations a re used to predict velocity and altitude changes over constant incremental rat ios of weight change. An approximate q=f ( M ) schedule for the baseline vehicle trajectory is shown in Figure 13.

The simulation p ta r t s a t initial point 1 where all vehicle (i.e. 2'. D, LIW. W,, 4, etc.) and atmospheric variables are known. The energy s t a t e associated with the next velocity and altitude condition (left side of Equation 32 is calculated as follows:

Next, an iterative procedure is used t o calculate the altitude, velocity condition which meets t h e input trajectory definition ( q = f ( M ) ) . The iterative procedure required for solution is executed as follows:

Estimate new h, (any value)

Calculate corresponding V2 (Equation 38)

From h2, determine 8, (From Standard Atmospheric Tables e.g.Reference 2)

From V2 and 82, calculate M2 (Equation 34)

From M,, determine q2 (From input trajectory schedule)

From qz and V2, calculate d / € J 2 (Equation 35)

From Ci2/02 calculate new hz* (Equation 36)

Compare h2 and h2*

1. If difference meets convergence criteria, exit

2. If not, se t h2 = h2* and return to Step B.

Typically, convergence occurs in 3 t o 5 cycles. Upon convergence, a new al t i tude and velocity is defined which, in turn, enables specific thrust, airflow, drag and equivalence ratio t o be recomputed from the previously developed parametrics. A third energy s t a t e can then be predicted and another altitude and velocity iterated. This process continues until a prescribed final velocity is achieved. At this point, the takeoff and ascent segments have been completely modeled. Figure 14 shows the results obtained for t h e baseline compared t o those presented in Reference 1. The differences, although acceptable for preliminary sizing purposes, a r e due primarily t o the simplified Figure 13 approximation of the relatively complex q=f ( M ) schedule defined by Figure 3. Had the baseline flown a less complicated t ra jec tory , a much b e t t e r cor re la t ion would have resulted. This has been verified by comparisons with exact three-degree-of-freedom solutions fo r vehicles flying constant q trajectories.

Vehicle Convergence and Resizing - Upon completion of the ascent phase (or mission completion in the case of a full simulation), the volume of fuel expended plus all required reserves is compared to t h e volume of fuel available. A new vehicle scale f a c t o r required is

calculated using Equation 8. If the volume change required is within predetermined accuracy limits, the simulation is considered converged. If not, the vehicle is "photographically" resized per Equations 6 and 7. m

4501 0 SYNTHESIS PREDlCTlON 1 = 425 2 0 1 2 3 4 . 5 6

5067.16 M A C H NUMBER (MI Figure 14 Comparison of Trajectory Prediction Results

High-speed engine rescaling can be performed by using a number of criteria to include a specified design thrust-to- drag ratio, a constant vehicle frontal-to-engine capture area ratio, or others. The low-speed system is scaled separately based on design takeoff or landing thrust-to- weight ratio, acceleration rate or other criteria. In either event, an i terat ive sizing/weight procedure is required, and the synthesis loop is repeated until all parameters a re within prescribed convergence criteria.

Applications

The simplified synthesis procedures described are suitable for performing preliminary sizing studies of vehicles which differ from t h e initial baseline. For example, technology effects can be evaluated by changing input variables such as wing unit weight, engine thrust-to- weight ratio, volume-to-wetted area ratio, and others.

Figure 15 shows an example of a sizing study performed t o quantify the effects of engine size (reflected in terms of design thrust-to-drag ratio) on a variable-sweep wing version of the baseline (all vehicles were sized for equal percent ascent fuel comsumptions). The variable sweep wing benefits were approximated by increasing takeoff wing loading t o a value which can be achieved a t reduced wing sweep angles (125 ~ b / ~ t 2 ) . The penalties were modeled by increasing wing unit weight from 4.68 to 10.6 ~ b / ~ t : ! (values obtained from Reference 1 design study). The predicted gross weight impact is substantially less than that of the variable sweep-wing version of the baseline described in Reference 1. This version, however, had a number of other design differences which were responsible for the additional weight growth.

1 SYMBOLS - SYNTHESIS PREOlCTiON 1 "."

14.0 5001!25 1 . 115 1.k I.& 1.r5

5067-17 DESIGN THRUST-TO.ORAG RATIO (FN , D) @ M = 1.2

Figure 15 Variable Sweep Variant Engine Sizing 5 tuay

Conclusions

Simple linear and second-order parametrics can be used t o model the important thrust, drag, weight and fuel flow characteristics of a typical hypersonic vehicle. These equations a r e sufficiently tractable that they can be coded, including iteration and resizing loops t o provide the individual engineer with a pocket version o i a preliminary synthesis code. Furthermore, by appropriate modifications to the synthesis inputs and/or algorithms, a range of configuration and technology alternatives can be evaluated.

References

1 Jar let t , F.A., "Performance Potential of Hydrogen Fueled, Airbreathing Cruise Aircraft," Volume 2 - Phase I Studies, General Dynamics Convair Division, GD/C-DCB66-004/2, May 1966.

2 Prat t & Whitney,Aircraft, " Aeronautical Vest- Pocket Handbook," Sixteenth Edition, June 1977