for an aerospace plane, part 2: data, … an aerospace plane, part 2: data, tables, and graphs 1,2...

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AERO-AS_RONAUTICS REPORT NO. 248

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73/8

OPTIMAL TRAJECTORIES

FOR AN AEROSPACE PLANE, PART 2:

DATA, TABLES, AND GRAPHS

by

m

A. MIELE, W. Y. LEE, AND G. D. WU

=

z

(NASA-CR-IBT84B) OPTIMALAN AERUSPACE PLANE. PART

AND GRAPHS (Rice Univ.)

TRAJECInRIES FUR

2: DATA, TA_LES,

94 p CSCL OIC

G3/05

N91-16')II

unclas0329316

1

RICE UNi_RSITY

1990

https://ntrs.nasa.gov/search.jsp?R=19910006698 2018-06-17T10:12:24+00:00Z

AERO-ASTRONAUTICS REPORT NO. 248

OPTIMAL TRAJECTORIES

FOR AN AEROSPACE PLANE, PART 2:

DATA, TABLES, AND GRAPHS

by

A. MIELE, W. Y. LEE, AND G. D. WU

RICE UNIVERSITY

1990

r

i

Optimal Trajectories

for an Aerospace Plane, Part 2:

Data, Tables, and Graphs 1,2

by

A. Miele 3, W. Y. Lee 4, and G. D. Wu 5

AAR-248

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1Portions of this material were presented by the senior author

at the 1990 American Control Conference, San Diego, California,

May 23-25, 1990.

2This research was supported by NASA Langley Research Center

Grant No. NAG-I-1029 and by Texas Advanced Technology Program

Grant No. TATP-003604020.

3Foyt Family Professor of Aerospace Sciences and Mathematical

Sciences, Aero-Astronautics Group, Rice University, Houston, Texas.

4post-Doctoral Fellow, Aero-Astronautics Group, Rice University,

Houston, Texas.

5Graduate Student, Aero-Astronautics Group, Rice University,

Houston, Texas.

ii AAR-248

Abstract. This report is a follow-up to Ref. 1 and presents

data, tables, and graphs relative to the optimal trajectories for

an aerospace plane. A single-stage-to-orbit (SSTO) configuration

is considered, and the transition from low supersonic speeds to

orbital speeds is studied for a single aerodynamic model (GHAME)

and three engine models.

Four optimization problems are solved using the sequential

gradient-restoration algorithm for optimal control problems: (PI)

minimization of the weight of fuel consumed; (P2) minimization of

the peak dynamic pressure; (P3) minimization of the peak heating

rate; and (P4) minimization of the peak tangential acceleration.

The above optimization studies are carried out for different

combinations of constraints, specifically: initial path

inclination either free or given; dynamic pressure either free or

bounded; tangential acceleration either free or bounded.

Key Words. Flight mechanics, hypervelocity flight,

atmospheric flight, optimal trajectories, aerospace plane,

sequential gradient-restoration algorithm.

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i. Introduction

This report is a follow-up to Ref. 1 and presents data,

tables, and graphs relative to the optimal trajectories for an

aerospace plane. A single-stage-to-orbit (SSTO) configuration is

considered, and the transition from low supersonic speeds to

orbital speeds is studied.

The aerodynamic configuration is the general hypersonic

aerodynamics model example (GHAME). For the engine model, three

options are considered: (EMI) this is a ramjet/scramjet

combination in which the scramjet specific impulse tends to a

nearly-constant value at large Mach numbers; (EM2) this is a

ramjet/scramjet combination in which the scramjet specific

impulse decreases monotonically at large Mach numbers; (EM3) this

is a ramjet/scramjet/rocket combination in which, owing to

stagnation temperature limitations, the scramjet operates only at

M < M,; at higher Mach numbers, the scramjet is shut off and the

aerospace plane is driven only by the rocket engines. Here,

M, = 15 is a threshold Mach number.

With the above understanding, we study four basic

optimization problems: (PI) minimization of the weight of fuel

consumed; (P2) minimization of the peak dynamic pressure; (P3)

minimization of the peak heating rate; and (P4) minimization of

the peak tangential acceleration. These optimization problems are

solved for different combinations of constraints imposed on the

initial path inclination 70, the dynamic pressure q, and the

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tangential acceleration a T . Specifically, Y0 can either be free

or given (Y0 = 0); q can either be free or bounded (q ! 1500

ibf/ft2);and a T can either be free or bounded (a T ! 3ge). A bound

on the heating rate Q (for example, Q ! 150 BTU/ft2sec) is not

imposed because it can be satisfied or nearly satisfied

indirectly if the dynamic pressure bound is satisfied.

The notations used in this report are identical to those

of Ref. i. Therefore, a list of symbols is omitted.

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2. Data

The following data are used in the numerical experiments on

optimal trajectories.

2.1. Spaceplane. For the aerospace plane, the initial weight

(weight at the end of the turbojet phase) is W 0 = 290000 ibf; the

reference surface area (wing area) is S = 6000 ft2; the lower

bound on the angle of attack is _£ = -2.0 deg; the upper bound on

the angle of attack is _u = 12.0 deg.

The aerodynamic configuration is the general hypersonic

aerodynamics model example (GHAME). For this configuration, Fig. 1

shows the drag coefficient CD, the lift coefficient C L, and the

lift-to-drag ratio E = CL/C D versus the Mach number M and the

angle of attack _.

2.2. Engines. For all engine models, the inclination of the

thrust with respect to the aircraft reference line is

= 0.0 deg; the lower bound for the power setting is B£ = 0; the

upper bound for the power setting is Bu= i.

Engine model EMI is a ramjet/scramjet combination with

combustor cross-sectional area S e = 400 ft 2. For B = 1 and for

engine model EMI, Fig. 2 (ramjet) and Fig. 3 (scramjet) show the

thrust T, the specific impulse Isp, and the fuel rate T/Isp

(weight of fuel consumed per unit time) versus the Mach number M

and the altitude h.

Engine model EM2 is a ramjet/scramjet combination with

combustor cross-sectional area S e = 400 ft 2. For 8 = 1 and for

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4 AAR-248

engine model EM2, Fig. 2 (ramjet) and Fig. 4 (scramjet) show the

thrust T, the specific impulse Isp, and the fuel rate T/Isp

versus the Mach number M and the altitude h.

Engine model EM3 is a ramjet/scramjet/rocket combination

with ramjet/scramjet combustor cross-sectional area S e = 400 ft 2

and maximum rocket thrust T,= 189200 ibf. For B = 1 and for

engine model EM3, Fig. 2 (ramjet), Fig. 5 (scramjet), and Fig. 6

(rocket) show the thrust T, the specific impulse Isp, and the

fuel rate T/Isp versus the Mach number M and the altitude h.

Note that engine model EM2 differs from engine model EMI as

follows: in EMI, the specific impulse tends to a nearly constant

value at large Mach numbers; in EM2, the specific impulse

decreases monotonically at large Mach numbers.

Also note that engine model EM3 differs from engine model

EM2 as follows: in EM2, the scramjet operates up to orbital

speeds; in EM3, the scramjet operates only at M _ 15; at higher

Mach numbers, the scramjet is shut off and the aerospace plane is

driven only by the rocket engines.

2.3. Physical Constants. The radius of the Earth is assumed

to be r e = 0.2093E+08 ft = 6378 km. The Earth's gravitational

constant is _ = 0.1409E+17 ft3/sec 2. The sea-level acceleration

of gravity is ge = 32.20 ft/sec 2.

2.4. Atmospheric Model. The atmospheric model used is the

US Standard Atmosphere, 1976 (Ref. 2). In this model, the values

of the density are tabulated at discrete altitudes. For

-- 5 AAR-248

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intermediate altitudes, the density is computed by assuming an

exponential fit for the function p(h). This is equivalent to

assuming that the atmosphere behaves isothermally between any two

contiguous altitudes tabulated in Ref. 2.

2.5. Initial Conditions. The initial time is the end of the

turbojet phase and the beginning of the ramjet phase. At t = t O ,

we assume that

x 0 = 0 ft,

h 0 = 42004 ft = 12.8 km,

V 0 = 1936 ft/sec,

y0 = free or Y0 = 0.0 deg,

W 0 = 290000 ibf.

2.6. Final Conditions.

(la)

(Ib)

(ic)

(id)

(le)

For engine models EMI and EM2, the

final time is the end of the scramjet phase; for engine model EM3,

the final time is the end of the rocket phase. At t = tf, we

assume that

xf = free, (2a)

hf = 262467 ft = 80.0 km,

Vf = 25792 ft/sec,

(2b)

(2c)

6 AAR-248

yf = 0.0 deg,

Wf = free.

These conditions correspond to orbital speed.

(2d)

(2e)

1 :

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7 AAR-248

3. Problems

We study four basic optimization problems: (PI) minimum fuel

weight; (P2) minimum peak dynamic pressure; (P3) minimum peak

heating rate; (P4) minimum peak tangential acceleration.

In addition to the initial conditions (i) and the final

conditions (2), we'consider the following supplementary constraints:

(A) T0 = free, q = free, a T = free; (3a)

(B) T O = free, q ! 1500 lbf/ft 2, a T _ 3.0 ge; (3b)

(C) T O = 0.0 deg, q = free, a T = free; (3c)

(D) T O = 0.0 deg,q < 1500 ibf/ft 2, a T ! 3.0 ge" (3d)

Concerning the heating rate Q, we do not consider a bound

of the form Q < 150BTU/ft2sec because it can be satisfied or nearly

satisfied indirectly if the dynamic pressure bound is satisfied.

The following terminology is self-explanatory: Problem (PIA)

is Problem (PI) subject to conditions (A); Problem (PIC) is Problem

(PI) subject to conditions (C) ; Problem (P3A) is Problem (P3)

subject to conditions (A); and so on.

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• Results

Numerical solutions for the optimization problems of Section 3

were obtained by means of the sequential gradient-restoration

algorithm (SGRA, Refs. 3-4) employed in primal form. We note

that there are four basic performance indexes [(Pl),

four combinations of supplementary constraints [(A),

and three engine models [EMI, EM2, EM3]. This leads to a total

of 48 optimization problems to be solved. A cross section of the

solutions obtained is shown in Tables 1-12 and Figs. 7-9 of this

report•

Tables 1-3 present summary results for groups of problems

and list the following quantities: the weight of fuel consumed;

the peak dynamic pressure; the peak heating rate; the peak

tangential acceleration; the initial path inclination; the time

duration of each segment of the trajectory; and the final time.

Tables 4-12 present detailed results for particular problems

and list the following quantities: the weight of fuel consumed

during the ramjet phase, the scramjet phase, and the rocket phase

as well as the total weight of fuel consumed; the flight time of

the ramjet phase, the scramjet phase, and the rocket phase as

well as the total flight time; the peak heating rate and the peak

dynamic pressure; the peak tangential acceleration, the peak

normal acceleration, and the peak total acceleration; the values

of the state variables at the beginning and at the end of each

segment of the trajectory•

(P2), (P3) , (P4)],

(B) , (C) , (D)] ,

-- 9 AAR-248

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Figures 7-9 present detailed results for particular problems

and show the following quantities as functions of the time: the

altitude, the velocity, the path inclination, and the weight; the

angle of attack and the power setting; the heating rate, the

dynamic pressure, the tangential acceleration, and the total

aerodynamic load; the Mach number, the specific impulse, and the

thrust.

Regardless of how a particular optimal solution is generated,

the jud_oment of its engineering usefulness depends on whether

the following conditions are satisfied or nearly satisfied:

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(i) Y0 relatively small; (4a)

(ii) q < 1500 Ibf/ft2; (4b)

(iii) Q < 150 BTU/ft2sec; (4c)

(iv) a T <_ 3g e. (4d)

To arrive at this judgment, it is appropriate to subdivide the

solutions of the problems studied into three groups.

Group G1 includes the solutions of Problems (PIA), (P2A),

(P3A), (P4A). These are solutions of the four basic optimization

problems obtained for engine model EMI and for constraints of

Type (A). Hence,y 0 is free, the dynamic pressure q is unconstrained,

and the tangential acceleration a T is unconstrained; the heating

rate Q is also unconstrained. Inspection of Table 1 and Tables 4-7

shows that none of the solutions of Group G1 satisfies the

requirements (4).

i0 AAR-248

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Group G2 includes the solutions of Problems (PIA), (PIB),

(PIC), (PID). These are solutions of the minimum fuel weight

problem obtained for engine model EMI and for constraints of

Type (A), (B), (C), (D), respectively. Inspection of Table 2,

Table 4, and Tables 8-10 shows that only one of the solutions of

Group G2 nearly satisfies the requirements (4). This is the

solution (PID).

Group G3 includes the solutions of Problem (PID) for engine

models EMI, EM2, EM3, respectively. Inspection of Table 3, Tables

10-12, and Figs. 7-9 shows that all of the solutions of Group G3

satisfy or nearly satisfy the requirements (4). In percentage of

the initial weight W 0 (weight at the end of the turbojet phase),

the minimum fuel weight is 34.3% for engine model EMI, 44.3% for

engine model EM2, and 60.7% for engine model EM3.

Assume now that the weight of fuel consumed during the

turbojet phase is 5% of the take-off weight. One concludes that,

in percentage of the take-off weight WT0, the minimum fuel weight

is 37.6% for engine model EMI, 47.1% for engine model EM2, and

62.7% for engine model EM3.

To sum up, if engine model EM2 is the one closer to reality,

the SSTO mission appears to be feasible; obviously, its ability to

deliver payloads into orbit can be improved via progress in the

areas of aerodynamic properties and specific impulse properties.

On the other hand, if engine model EM3 is the one closer to reality,

the SSTO mission appears to be marginal, unless substantial

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progress is achieved in the areas of aerodynamic properties and

specific impulse properties. Under the latter scenario, alternative

consideration should be given to studying the feasibility of

a two-stage-to-orbit (TSTO) mission.

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References

I. MIELE, A., LEE, W. Y., and WU, G. D., Optimal Trajectories

for an Aerospace Plane, Part i: Formulationf Results r and

Analysis, Rice University, Aero-Astronautics Report No. 247,

1990.

2. NOAA, NASA, and USAF, US Standard Atmosphere_ 1976, US

Government Printing Office, Washington, DC, 1976.

3. MIELE, A., and WANG, T., Primal-Dual Properties of

Sequential Gradient-Restoration Algorithms for Optimal

Control Problems, Part i r Basic Problem, Integral Methods in

Science and Engineering, Edited by F. R. Payne et al,

Hemisphere Publishing Corporation, Washington, DC, pp.

577-607, 1986.

4. MIELE, A., and WANG, T., Primal-Dual Properties of

Sequential Gradient-Restoration Algorithms for Optimal

Control Problems r Part 2_ General Problem, Journal of

Mathematical Analysis and Applications, Vol. 119, Nos. 1-2,

pp. 21-54, 1986.

-- 13 AAR-248

Table I. Unconstrained solutions, engine model EMI,

various performance indexes, constraints of Type (A).

Quantity Problem Units

(PIA) (P2A) (P3A) (P4A)

(W0-W f)/W 0 0. 337 0. 347 0. 357 0. 550

max (q) 1540 999 1157 3751

max (Q) 165 161 98 495

max (a T )/ge 9.1 5.2 4.0 i. 1

n

ibf/ft 2

BTU/ft2sec

u

Y0 42.0 50.0 40.4 38.3 deg

T 1 34 54 48 144 sec

_2 409 475 731 704 sec

8f 443 529 779 848 sec

Wf = W 2 and ef = e2 for engine model EMI.

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Table 2. Constrained solutions, engine model EMI,

minimum fuel weight, various constraint combinations.

Quantity Problem Units

(PIA) (PIB) (PIC) (PID)

(W0-Wf)/W 0 0.337 0.340 0.339 0.343

max(q) 1540 1112 1765 1500

max(Q) 165 148 200 153

max(aT)/g e 9.1 3.0 13.7 3.0

m

ibf/ft 2

BTU/ft2sec

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_0 42.0 39.4 0.0 0.0 deg

T 1 34 55 34 55 sec

T 2 409 498 335 487 sec

ef 443 553 369 542 sec

Wf = W 2 and 8f = e2 for engine model EM1.

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15 AAR-248

Table 3. Effect of the engine model, Problem (PID),

minimum fuel weight, constraints of Type (D).

Quantity Engine model

EMI EM2 EM3

Units

(W0-W f)/W 0 0. 343

max (q) 1500

max (Q) 153

max (a T )/ge 3.0

0.443 0.607

1425 1500

157 110

3.0 3.0

ibf/ft 2

BTU/ft2sec

70 0.0

T 1 55

T 2 487

T 3

0f 542

0.0 0.0 deg

44 57 sec

472 97_ sec

- 277 sec

517 431 sec

Wf = W 2 and 8

Wf = W 3 and @

f = 82 for engine models EMI, EM2.

f = 03 for engine model EM3.

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Table 4A. Results for Problem (P1A), engine model EM1,

minimum weight of fuel consumed,

7o=free, q=free, aT=free.

Quantity Dimensionless

value

Reference

value

Dimensional Units

value

Ramjet fuel consumption

Scramjet fuel consumption

Total fuel consumption

0.3687E-01

0.2999E+00

0.3368E + 00

0.2900E + 06

0.2900E + 06

0.2900E + 06

0.1069E+05 lbf

0.8697E+05 lbf

0.9767E+05 lbf

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Ramjet flight time

Scramjet flight time

Total flight time

0.6573E-02

0.7920E-01

0.8577E-01

0.5161E+04

0.5161E+04

0.5161E+04

0.3393E÷02 sec

0.4088E +03 sec

0.4427E÷03 sec

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Peak heating rate

(8 = 125.9 sec)

Peak dynamic pressure(8 = 23.4 sec)

0.1099E+01

0.1027E+01

0.1500E+03

0.1500E+04

0.1648E+03 Btu ft "2 sec" 1

0.1540E +04 lbfft 2

E :

=

m

Peak tangential acceleration(8 = 20.7 sec)

Peak normal acceleration

(8 = 62.5 sec)

Peak total acceleration

(8 = 20.7 sec)

0.9053E+01

0.9298E+00

0.9384E+ 01

0.3220E+02

0.3220E+02

0.3220E+ 02

-20.2915E+03 ftsec

-20.2994E+02 ft sec

-20.3022E+03 ft sec

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Table 4B. Results for Problem (P1A), engine model EM1,


7o = free, q = free, a T = free.


value

Reference

value

Dimensional Unitsvalue

Ramjet initial altitude

Ramjet initial velocity

Ramjet initial path inclination

Ramjet initial weight

0.1600E + 00

0.7506E--01

0.7328E+00

0.1000E+01

0.2625E + 06

0.2579E+05

0.5730E+02

0.2900E ÷ 06

0.4200E+05 ft

-10.1936E+04 ftsec

0.4199E+02 deg

0.2900E+ 06 lbf

Scramjet initial altitude

Scramjet initial velocity

Scramjet initial path inclination

Scramjet initial weight

0.3629E+00

0.2625E+00

0.2157E+00

0.9631E + 00

0.2625E+06

0.2579E+05

0.5730E+02

0.2900E + 06

0.9525E+05 ft

-10.6771E+04 ftsec

0.1236E+02 deg

0.2793E+06 lbf

Scramjet final altitude

Scramjet final velocity

Scramjet final path inclination

Scramjet final weight

0.1000E+01

0.1000E+01

0.0000E + 00

0.6632E + 00

0.2625E+06

0.2579E+05

0.5730E+02

0.2900E+06

0.2625E+06 ft

-10.2579E+05 ft sec

0.0000E÷ 00 deg

0.1923E÷06 lbf

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18 AAR-248


minimum peak dynamic pressure,


Quantity Dimensionless Reference Dimensional Unitsvalue value value




0.3726E--01

0.3095E+00

0.3467E÷00

0.2900E+06

0.2900E + 06

0.2900E ÷ 06

0.1080E+05 lbf

0.8975E+05 lbf

0.1006E+ 06 lbf

Ramjet flight time


Total flight time

0.1047E--01

0.9204E--01

0.1025E + 00

0.5161E÷04

0.5161E+04

0.5161E+04

0.5404E+02 sec

0.4751E+ 03 sec

0.5291E+03 sec

Peak heating rate(8 = 196.6 sec)

Peak dynamic pressure(6 = 0.0sec)

0.1073E+01

0.6661E+00

0.1500E + 03

0.1500E + 04

0.1609E+03 Btu ft "2 see 1

0.9991E+03 lbfft "2

= =



(8 = 0.0 sec)


(/9 = 0.0sec)

0.5227E+01

0.5379E+01

0.5462E+01

0.3220E+02

0.3220E+02

0.3220E+02

-20.1683E+03 ftsec

-20.1732E+03 ftsec

-20.1759E+03 ftsec

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___ 19 AAR-248


minimum peak dynamic pressure,



value

Reference

value

Dimensional Units

value





0.1600E + 00

0.7506E-01

0.8735E+00

0.1000E+01

0.2625E+06

0.2579E+05

0.5730E+02

0.2900E + 06

0.4200E+ 05 ft

-10.1936E+ 04 ft sec

0.5005E+02 deg

0.2900E+ 06 lbf





0.3726E+00

0.2581E+00

0.1603E + 00

0.9627E + 00

0.2625E + 06

0.2579E+05

0.5730E + 02

0.2900E + 06

0.9780E+05 ft

-i0.6657E+04 ft sec

0.9184E+01 deg

0.2792E+ 06 lbf

--7





0.1000E+01

0.1000E+01

0.0000E ÷ 00

0.6533E + 00

0.2625E+06

0.2579E+05

0.5730E+02

0.2900E + 06

0.2625E+06 ft

-I0.2579E+05 ft sec

0.0000E+ 00 deg

0. 1894E+06 lbf

20 AAR-248


minimum peak heating rate,


Quantity Dimensionless Reference Dimensional Unitsvalue value value




0.3556E--01

0.3215E + 00

0.3570E+00

0.2900E + 06

0.2900E+06

0.2900E + 06

0.1031E+05 lbf

0.9323E+05 lbf

0.1035E+06 lbf

Ramjet flight time


Total flight time

0.9369E--02

0.1416E+00

0.1509E + 00

0.5161E+04

0.5161E+04

0.5161E+04

0.4836E+ 02 sec

0.7307E+03 sec

0.7791E+03 sec

r


Peak dynamic pressure(8 = 12.1sec)

0.6549E+00

0.7711E+00

0.1500E+03

0.1500E+04

0.9823E+02 Btu ft "2 sec "1

0.1157E+04 lbfft "2

=

Peak tangential acceleration(e = 33.9 sec)


(8 = 136.0 sec)


(8 = 9.2 sec)

0.3968E+01

0.6961E + 00

0.3994E+01

0.3220E+02

0.3220E+02

0.3220E+02

-20.1278E+03 ftsec

-20.2242E+02 ftsec

-20.1286E + 03 ft sec

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Table 6B. Results for Problem (P3A), engine model EM1,minimum peak heating rate,

7o=free, q=free, aT-free.

Quantity Dimensionless Referencevalue value

Dimensional Units

value





0.1600E + 00

0.7506E-01

0.7054E + 00

0.1000E÷01

0.2625E + 06

0.2579E+05

0.5730E+02

0.2900E + 06

0.4200E+ 05 ft

-10.1936E+04 ft sec

0.4042E÷02 deg

0.2900E+06 lbf





0.3658E+00

0.2519E+00

0.1694E + 00

0.9644E+00

0.2625E ÷ 06

0.2579E+05

0.5730E+02

0.2900E+06

0.9600E+05 ft

0.6498E+04 ft sec 1

0.9704E+01 deg

0.2797E+06 lbf





0.1000E+01

0.1000E + 01

0.0000E + 00

0.6430E+00

0.2625E+06

0.2579E+05

0.5730E+02

0.2900E+06

0.2625E+06 ft

-10.2579E+05 ft sec

0.0000E+ 00 deg

0.1865E+06 lbf

22 AAR-248


minimum peak tangential acceleration,


Quantity Dimensionless Reference

value value

Dimensional Units

value

w




0.4512E-01

0.5051E+00

0.5502E+00

0.2900E + 06

0.2900E + 06

0.2900E + 06

0.1309E+05 lbf

0.1465E+06 lbf

0.1596E+06 Ibf

Ramjet flight time


Total flight time

0.2788E--01

0.1363E + 00

0.1642E + 00

0.5161E+04

0.5161E+04

0.5161E+04

0.1439E+03 sec

0.7036E+03 sec

0.8475E+03 sec

: :

w_J



0.3302E+01

0.2501E+01

0.1500E + 03

0.1500E + 04

0.4953E+03 Btu ft "2 sec "1

0.3751E+04 lbfft 2



(8 = 126.6 sec)


(8 = 126.6 sec)

0.1060E+01

0.2109E+01

0.2361E+01

0.3220E+02

0.3220E+02

0.3220E + 02

-20.3413E+02 ftsec

-20.6792E+02 ft sec

-20.7601E+02 ftsec

m

w

m

23 AAR-248


minimum peak tangential acceleration,



value value

Dimensional Units

value





0.1600E + 00

0.7506E--01

0.6690E + 00

0.1000E+01

0.2625E+06

0.2579E + 05

0.5730E+02

0.2900E + 06

0.4200E+ 05 ft

-10.1936E+04 ft sec

0.3833E+02 deg

0.2900E+ 06 lbf





0.3678E+00

0.2578E+00

0.1701E+00

0.9549E+00

0.2625E+06

0.2579E + 05

0.5730E+02

0.2900E + 06

0.9655E+05 ft

-10.6651E+04 ft sec

0.9745E+01 deg

0.2769E+06 lbf

==





0.1000E+01

0.1000E+01

0.0000E + 00

0.4498E+00

0.2625E+06

0.2579E+05

0.5730E+02

0.2900E + 06

0.2625E+ 06 ft

-10.2579E+05 ft sec

0.0000E +00 deg

0.1304E+06 lbf

... 24 AAR-248

Table 8A. Results for Problem (P1B), engine model EM1,


7o=free, q_ 1500 psf, aT<3.0 ge"


value value

Dimensional Units

value




0.3944E-01

0.3005E + 00

0.3399E+00

0.2900E + 06

0.2900E + 06

0.2900E + 06

0.1144E+05 lbf

0.8713E+05 lbf

0.9857E+ 05 lbf

Ramjet flight time


Total flight time

0.1076E-01

0.9640E-01

0.1072E+00

0.5161E+04

0.5161E+04

0.5161E+04

0.5554E+02 sec

0.4976E+03 sec

0.5531E+03 sec


Peak dynamic pressure(8 = ll.7sec)

0.9879E+00

0.7410E+00

0.1500E+03

0.1500E + O4

0.1482E+03 Btu ft 2 sec "I

0.1112E+04 lbfft "2



(8 = 48.3 sec)


(0 = 15.0 sec)

0.3000E+01

0.1642E+01

0.3724E+01

0.3220E÷02

0.3220E+02

0.3220E+ 02

-20.9660E+02 ft sec

-20.5287E+02 ft sec

-20.1199E+03 ftsec

,.. 25 AAR-248

Table 8B. Results for Problem (P1B), engine model EM1,


To=free, q< 1500 psf, aw-_ 3.0 ge"


Dimensional Units

value





0.1600E + 00

0.7506E--01

0.6886E + 00

0.1000E+01

0.2625E + 06

0.2579E+05

0.5730E÷02

0.2900E ÷ 06

0.4200E+05 ft

-10.1936E+04 ft sec

0.3945E+02 deg

0.2900E+ 06 lbf





0.3775E÷00

0.2621E + 00

0.1665E + 00

0.9606E + 00

0.2625E÷06

0.2579E+05

0.5730E+02

0.2900E + 06

0.9908E+ 05 ft

-10.6759E+04 ft sec

0.9538E+01 deg

0.2786E+ 06 lbf





0.1000E÷01

0.1000E+01

0.0000E ÷ 00

0.6601E+00

0.2625E+06

0.2579E+05

0.5730E+02

0.2900E+06

0.2625E+06 ft

-10.2579E+05 ft sec

0.0000E+ 00 deg

0.1914E+06 lbf

w

w

--- 26 AAR-248

Table 9A. Results for Problem (P1C), engine model EM1,


9,0=0.0 deg, q=free, aT=free.


Dimensional Units

value

w




0.3859E--01

0.3005E+00

0.3391E+00

0.2900E+06

0.2900E+06

0.2900E + 06

0.1119E+05 lbf

0.8716E+05 lbf

0.9835E+ 05 lbf

Ramjet flight time


Total flight time

0.6615E-02

0.6490E-01

0.7152E-01

0.5161E+04

0.5161E+04

0.5161E+04

0.3414E+02 sec

0.3350E+03 sec

0.3691E+03 sec



0.1333E+01

0.1177E+01

0.1500E+03

0.1500E+04

0.1999E+03 Btu ft "2 sec "1

0.1765E+04 lbfft 2

L

w

Peak tangential acceleration

(0 = 60.9 sec)


(0 = 2.0sec)


(0 = 60.9 sec)

0.1373E+02

0.5376E+01

0.1373E+02

0.3220E÷02

0.3220E÷02

0.3220E÷ 02

-20.4420E+03 ftsec

-20.1731E+03 ftsec

-20.4422E+03 ft sec

w

27 AAR-248

Table 9B. Results for Problem (P1C), engine model EM1,


70=0.0 deg, q=free, aT=free.


value value

Dimensional Units

value





0.1600E + 00

0.7506E-01

0.0000E + 00

0.1000E + 01

0.2625E+06

0.2579E+05

0.5730E+02

0.2900E + 06

0.4200E+ 05 ft

-10.1936E÷04 ft sec

0.0000E÷00 deg

0.2900E÷06 lbf





0.3466E + 00

0.2626E ÷ 00

0.2367E+00

0.9614E+00

0.2625E+06

0.2579E+05

0.5730E+02

0.2900E + 06

0.9097E+05 ft

-10.6773E+04 ft sec

0.1356E+02 deg

0.2788E+06 lbf





0.1000E + 01

0.1000E + 01

0.0000E + 00

0.6609E+00

0.2625E+06

0.2579E+05

0.5730E+02

0.2900E+06

0.2625E+06 ft

-10.2579E+05 ft sec

0.0000E÷00 deg

0.1917E+06 Ibf

w

w

-- 28 AAR-248

Table IOA. Results for Problem (P1D), engine model EM1,minimum weight of fuel consumed,

7o=0.0 deg, q_ 1500 psf, awS3.0 ge"


value value

Dimensional Unitsvalue

m




0.4079E-01

0.3022E+00

0.3430E+00

0.2900E + 06

0.2900E + 06

0.2900E + 06

0.1183E+05 lbf

0.8764E+05 lbf

0.9947E+05 lbf

Ramjet flight time


Total flight time

0.1057E-01

0.9437E-01

0.1049E+00

0.5161E+04

0.5161E+04

0.5161E+04

0.5457E+02 sec

0.4871E+03 sec

0.5417E+03 sec



0.1018E+01

0.1000E+01

0.1500E÷03

0.1500E+04

0.1528E+03 Btu ft "2 sec -1

0.1500E+04 lbfft 2



(8 = 1.1 sec)


(8 = 2.7 sec)

0.3000E+01

0.4994E+ 01

0.5532E+01

0.3220E+02

0.3220E÷ 02

0.3220E+02

-20.9660E+02 ft sec

-20.1608E+03 ftsec

-20.1781E+03 ftsec

m

29 AAR-248

Table 10B. Results for Problem (P1D), engine model EM1,


70=0.0 deg, q_ 1500 psf, aT<3.0 ge"


Dimensional Units

value





0.1600E+00

0.7506E-01

0.0000E + 00

0.1000E + 01

0.2625E+06

0.2579E+05

0.5730E+02

0.2900E + 06

0.4200E+ 05 ft

-10.1936E+04 ft sec

0.0000E+00 deg

0.2900E+06 lbf





0.3729E+ 00

0.2618E+00

0.1717E+00

0.9592E + 00

0.2625E+06

0.2579E+05

0.5730E+02

0.2900E + 06

0.9786E+05 ft

-10.6752E+04 ft sec

0.9837E+01 deg

0.2782E+06 Ibf





0.1000E + 01

0.1000E+01

0.0000E + 00

0.6570E+00

0.2625E+06

0.2579E+05

0.5730E+02

0.2900E + 06

0.2625E+06 ft

-I0.2579E+05 ftsec

0.0000E+00 deg

0.1905E+06 lbf

_ 30 AAR-248

Table 11A. Results for Problem (P1D), engine model EM2,


70=0.0 deg, q< 1500 psf, aT<3.0 ge o


value value

Dimensional Units

value




0.3624E--01

0.4073E + 00

0.4435E+00

0.2900E + 06

0.2900E+06

0.2900E + 06

0.1051E+05 lbf

0.1181E+06 lbf

0.1286E+06 lbf

Ramjet flight time


Total flight time

0.8617E-02

0.9150E--01

0.1001E+00

0.5161E+04

0.5161E+04

0.5161E+04

0.4447E+02 sec

0.4723E+03 sec

0.5167E+03 sec

!



0.1050E+01

0.9500E÷00

0.1500E ÷ 03

0.1500E+04

0.1575E+03 Btu ft 2 sec "1

0.1425E+04 lbfft 2



(8 = 2.7 sec)


(8 = 2.7 sec)

0.3000E+01

0.5449E+01

0.5909E+01

0.3220E+02

0.3220E+02

0.3220E+ 02

-20.9660E+02 ft sec

-20.1755E+03 ftsee

-20.1903E+03 ftsec

-- 31 AAR-248

Table 1lB. Results for Problem (P1D), engine model EM2,


7o=0.0 deg, q< 1500 psf, aT<3.0 ge'


value value

Dimensional Units

value

w





0.1600E + 00

0.7506E-01

0.0000E + 00

0.1000E+01

0.2625E + 06

0.2579E+05

0.5730E+02

0.2900E + 06

0.4200E+ 05 ft

.i0.1936E+04 ft sec

0.0000E+00 deg

0.2900E+06 lbf





0.3415E+00

0.2371E+00

0.9738E-01

0.9638E+00

0.2625E+06

0.2579E+05

0.5730E+02

0.2900E + 06

0.8964E+ 05 ft

-10.6115E+04 ft sec

0.5580E+01 deg

0.2795E+06 lbf





0.1000E + 01

0.1000E+01

0.0000E + 00

0.5565E+00

0.2625E + 06

0.2579E+05

0.5730E+02

0.2900E+06

0.2625E+06 ft

-I0.2579E+05 ft sec

0.0000E+00 deg

0.1614E+06 lbf

i

w

-- 32 AAR-248

Table 12A. Results for Problem (P1D), engine model EM3,


70=0.0 deg, qs 1500 psf, aTS 3.0 ge •


value value

Dimensional Units

value



Rocket fuel consumption


0.4407E-01

0.1562E+00

0.4070E + 00

0.6073E + 00

0.2900E+06

0.2900E + 06

0.2900E + 06

0.2900E + 06

0.1278E+05 lbf

0.4531E+05 lbf

0.1180E+06 lbf

0.1761E+06 lbf

Ramjet flight time


Rocket flight time

Total flight time

0.1098E--01

0.1886E-01

0.5367E-01

0.8351E-01

0.5161E+04

0.5161E+04

0.5161E+04

0.5161E+04

0.5667E+02 sec

0.9735E+02 sec

0.2770E+03 sec

0.4310E+03 sec

w


Peak dynamic pressure

(8 = 43.1 sec)

0.7333E+ 00

0.9998E+00

0.1500E+03

0.1500E+ 04

0.1100E+03 Btu ft "2 sec "1

0.1500E+04 lbfft "2



(8 = 2.3 sec)


(8 = 2.3sec)

0.3000E+01

0.5171E+01

0.5643E+01

0.3220E+02

0.3220E+02

0.3220E+02

-20.9660E+02 ftsec

-20.1665E+03 ftsec

-20.1817E+03 ftsec

w

!

-- 33 AAR-248

Table 12B. Results for Problem (P1D), engine model EM3,


7o=0.0 deg, q< 1500 psf, aw_3.0 ge"


value value

Dimensional Units

value





0.1600E + 00

0.7506E--01

0.0000E+00

0.1000E+01

0.2625E+06

0.2579E+05

0.5730E+02

0.2900E ÷ 06

0.4200E+ 05 ft

-10.1936E+04 ftsec

0.0000E÷ 00 deg

0.2900E+ 06 lbf





0.3507E÷00

0.2656E+00

0.1480E + 00

0.9559E+00

0.2625E+06

0.2579E+05

0.5730E+02

0.2900E+06

0.9205E+05 ft

-10.6849E+04 ft sec

0.8479E+ 01 deg

0.2772E+06 lbf

Rocket initial altitude

Rocket initial velocity

Rocket initial path inclination

Rocket initial weight

0.6295E+00

0.6289E+00

0.7083E--01

0.7997E+00

0.2625E + 06

0.2579E+05

0.5730E+02

0.2900E+06

0.1652E+06 ft

-10.1622E+05 ftsec

0.4058E+01 deg

0.2319E+06 lbf

Rocket final altitude

Rocket final velocity

Rocket final path inclination

Rocket final weight

0.1000E÷01

0.1000E+01

0.0000E ÷ 00

0.3927E+00

0.2625E+06

0.2579E+05

0.5730E+02

0.2900E + 06

0.2625E+ 06 ft

-10.2579E+05 ft sec

0.0000E+ 00 deg

0.1139E+06 lbf

p

-- 34 AAR-248

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