FOR AN AEROSPACE PLANE, PART 2: DATA, … an Aerospace Plane, Part 2: Data, Tables, and Graphs 1,2...
Transcript of 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
=
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(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
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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.
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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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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
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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)
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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)] ,
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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).
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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.
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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
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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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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 :
=
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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,
minimum weight of fuel consumed,
7o = free, q = free, a T = free.
Quantity Dimensionless
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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Table 5A. Results for Problem (P2A), engine model EM1,
minimum peak dynamic pressure,
7o=free, q=free, aT=free.
Quantity Dimensionless Reference Dimensional Unitsvalue value value
Ramjet fuel consumption
Scramjet fuel consumption
Total fuel consumption
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
Scramjet 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
= =
Peak tangential acceleration(8 = 39.5 sec)
Peak normal acceleration
(8 = 0.0 sec)
Peak total acceleration
(/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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Table 5B. Results for Problem (P2A), engine model EM1,
minimum peak dynamic pressure,
7o=free, q=free, aT=free.
Quantity Dimensionless
value
Reference
value
Dimensional Units
value
Ramjet initial altitude
Ramjet initial velocity
Ramjet initial path inclination
Ramjet initial weight
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
Scramjet initial altitude
Scramjet initial velocity
Scramjet initial path inclination
Scramjet initial weight
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
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Scramjet final altitude
Scramjet final velocity
Scramjet final path inclination
Scramjet final weight
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
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Table 6A. Results for Problem (P3A), engine model EM1,
minimum peak heating rate,
7o=free, q=free, aT=free.
Quantity Dimensionless Reference Dimensional Unitsvalue value value
Ramjet fuel consumption
Scramjet fuel consumption
Total fuel consumption
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
Scramjet 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 heating rate(8 = 380.8 sec)
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)
Peak normal acceleration
(8 = 136.0 sec)
Peak total acceleration
(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
Ramjet initial altitude
Ramjet initial velocity
Ramjet initial path inclination
Ramjet initial weight
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
Scramjet initial altitude
Scramjet initial velocity
Scramjet initial path inclination
Scramjet initial weight
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
Scramjet final altitude
Scramjet final velocity
Scramjet final path inclination
Scramjet final weight
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
Table 7A. Results for Problem (P4A), engine model EM1,
minimum peak tangential acceleration,
7o=free, q=free, aT=free.
Quantity Dimensionless Reference
value value
Dimensional Units
value
w
Ramjet fuel consumption
Scramjet fuel consumption
Total fuel consumption
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
Scramjet 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
Peak heating rate(8 = 467.6 sec)
Peak dynamic pressure(8 = 464.1 sec)
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
Peak tangential acceleration(8 = 478.1 sec)
Peak normal acceleration
(8 = 126.6 sec)
Peak total acceleration
(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
Table 7B. Results for Problem (P4A), engine model EM1,
minimum peak tangential acceleration,
7o=free, q=free, aT=free.
Quantity Dimensionless Reference
value value
Dimensional Units
value
Ramjet initial altitude
Ramjet initial velocity
Ramjet initial path inclination
Ramjet initial weight
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
Scramjet initial altitude
Scramjet initial velocity
Scramjet initial path inclination
Scramjet initial weight
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
==
Scramjet final altitude
Scramjet final velocity
Scramjet final path inclination
Scramjet final weight
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,
minimum weight of fuel consumed,
7o=free, q_ 1500 psf, aT<3.0 ge"
Quantity Dimensionless Reference
value value
Dimensional Units
value
Ramjet fuel consumption
Scramjet fuel consumption
Total fuel consumption
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
Scramjet 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 heating rate(8 = 224.7 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
Peak tangential acceleration(8 = 122.7 sec)
Peak normal acceleration
(8 = 48.3 sec)
Peak total acceleration
(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,
minimum weight of fuel consumed,
To=free, q< 1500 psf, aw-_ 3.0 ge"
Quantity Dimensionless Referencevalue value
Dimensional Units
value
Ramjet initial altitude
Ramjet initial velocity
Ramjet initial path inclination
Ramjet initial weight
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
Scramjet initial altitude
Scramjet initial velocity
Scramjet initial path inclination
Scramjet initial weight
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
Scramjet final altitude
Scramjet final velocity
Scramjet final path inclination
Scramjet final weight
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,
minimum weight of fuel consumed,
9,0=0.0 deg, q=free, aT=free.
Quantity Dimensionless Referencevalue value
Dimensional Units
value
w
Ramjet fuel consumption
Scramjet fuel consumption
Total fuel consumption
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
Scramjet 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
Peak heating rate(0 = 84.4 sec)
Peak dynamic pressure(0 = 25.6 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)
Peak normal acceleration
(0 = 2.0sec)
Peak total acceleration
(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,
minimum weight of fuel consumed,
70=0.0 deg, q=free, aT=free.
Quantity Dimensionless Reference
value value
Dimensional Units
value
Ramjet initial altitude
Ramjet initial velocity
Ramjet initial path inclination
Ramjet initial weight
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
Scramjet initial altitude
Scramjet initial velocity
Scramjet initial path inclination
Scramjet initial weight
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
Scramjet final altitude
Scramjet final velocity
Scramjet final path inclination
Scramjet final weight
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"
Quantity Dimensionless Reference
value value
Dimensional Unitsvalue
m
Ramjet fuel consumption
Scramjet fuel consumption
Total fuel consumption
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
Scramjet 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
Peak heating rate(8 = 210.4 sec)
Peak dynamic pressure(8 = 13.1 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
Peak tangential acceleration(8 = 186.1 sec)
Peak normal acceleration
(8 = 1.1 sec)
Peak total acceleration
(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,
minimum weight of fuel consumed,
70=0.0 deg, q_ 1500 psf, aT<3.0 ge"
Quantity Dimensionless Referencevalue value
Dimensional Units
value
Ramjet initial altitude
Ramjet initial velocity
Ramjet initial path inclination
Ramjet initial weight
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
Scramjet initial altitude
Scramjet initial velocity
Scramjet initial path inclination
Scramjet initial weight
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
Scramjet final altitude
Scramjet final velocity
Scramjet final path inclination
Scramjet final weight
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,
minimum weight of fuel consumed,
70=0.0 deg, q< 1500 psf, aT<3.0 ge o
Quantity Dimensionless Reference
value value
Dimensional Units
value
Ramjet fuel consumption
Scramjet fuel consumption
Total fuel consumption
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
Scramjet 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
!
Peak heating rate(8 = 226.3 sec)
Peak dynamic pressure(8 = 10.2 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
Peak tangential acceleration(8 = 205.0 sec)
Peak normal acceleration
(8 = 2.7 sec)
Peak total acceleration
(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,
minimum weight of fuel consumed,
7o=0.0 deg, q< 1500 psf, aT<3.0 ge'
Quantity Dimensionless Reference
value value
Dimensional Units
value
w
Ramjet initial altitude
Ramjet initial velocity
Ramjet initial path inclination
Ramjet initial weight
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
Scramjet initial altitude
Scramjet initial velocity
Scramjet initial path inclination
Scramjet initial weight
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
Scramjet final altitude
Scramjet final velocity
Scramjet final path inclination
Scramjet final weight
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,
minimum weight of fuel consumed,
70=0.0 deg, qs 1500 psf, aTS 3.0 ge •
Quantity Dimensionless Reference
value value
Dimensional Units
value
Ramjet fuel consumption
Scramjet fuel consumption
Rocket fuel consumption
Total 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
Scramjet 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 heating rate(8 = 139.4 sec)
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
Peak tangential acceleration(8 = 148.2 sec)
Peak normal acceleration
(8 = 2.3 sec)
Peak total acceleration
(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,
minimum weight of fuel consumed,
7o=0.0 deg, q< 1500 psf, aw_3.0 ge"
Quantity Dimensionless Reference
value value
Dimensional Units
value
Ramjet initial altitude
Ramjet initial velocity
Ramjet initial path inclination
Ramjet initial weight
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
Scramjet initial altitude
Scramjet initial velocity
Scramjet initial path inclination
Scramjet initial weight
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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