Upload
others
View
2
Download
0
Embed Size (px)
Citation preview
='_ 2.,
I_,_/---x,_.,li5-
Statistical Methods for Rapid Aerothermal Analysisand
Design Technology .........
Douglas DePriest,
Carolyn Morgan
: ;_ Hampton Un_hrers]_
Hampton, VA 23668
January 31, 2002
Abstract
The cost and safety goals for NASA's next generation of reusable launch vehicle (RLV)
will require that rapid high-fidelity aerothermodynamic design tools be used early in the
design cycle. To meet these requirements, it is desirable to establish statistical models
that quantify and improve the accuracy, extend the applicability, and enable combined
analyses using existing prediction tools. The research work was focused on establishing
the suitable mathematical/statistical models for these purposes. It is anticipated that the
resuIfing models can be incorporated into a software tool to provide rapid, variable-
fidelity, aerothermal environments to predict heating along an arbitrary trajectory. This
work will support development of an integrated design tool to perform automated thermal
protection system (TPS) sizing and material selection.
Introduction
Recent design experience with NASA's X-37 has demonstrated the need for considering
higher fidelity aerodynamic heating early in the design cycle. In the case of X-37, the
vehicle shape was optimized for aerodynamic performance and resulted in severe
aerodynamic heating that forced costly redesign of the nose and wing surfaces and
lowered flight margins. The availability of higher fidelity aerothermal analysis earlier in
the design cycle could have prevented these problems.
Development of this technology impacts NASA's goal of reduced cost by enabling faster
and more optimized design cycles. Utilizing higher fidelity analyses earlier in the design
will avoid the delay and expense of late design changes. Optimizing the thermal-
structural and TPS design in the process will minimize vehicle weight leading to lower
launch cost. The technology described is applicable across all next generation
systems/architectures for any vehicle configuration and includes metallic and ceramicTPS and hot structures.
There will be two phases to this effort and they are model development and model
validation. The initial phase will be to explore and/or develop the
statistical/mathematical methods that can be used to transform the point wise aeroheating
predictions of current tools to yield complete aerothermal environments through a
trajectory corridor. The approach is intended to identify statistical/math models that
best characterize and/or model a set of sphere stagnation data. Once several acceptable
models have been identified, they will be tested in the second phase to see if they are able
to predict heating values within the normal trajectory of corridor values.
2
Description of Sphere Stagnation Data Sets
There were two worksheets for analysis; one containing the full trajectory space (1269
measurements) and the second with measurements only within the entry corridor (138
measurements). For these data sets, a sphere shape configuration is used and the
measurements are taken at the stagnation point on the sphere. For simplicity, a sphere of
one foot radius is assumed. Generally, as the sphere radius decreases, the heat ratemeasurements will increase.
For each case, eleven (11) variables are labeled on the worksheets. The first three
variables (altitude, velocity, and wall temperature) are considered the basic independent
variables. The other variables are derived from these basic three variables. For example,
the density, pressure, and temperature are direct functions of altitude. The Mach number,
dynamic pressure, Reynolds number, and energy variables are combinations of the 3
basic variables, e.g. dynamic pressure equals density*velocity*velocity. The responsevariable of interest is heat rate.
The heating rate values in the spreadsheet were generated using MINIVER-tape
calculations. The datasets contain values for velocity and altitude but no angle-of-attack
values because of the sphere assumption.
Analysis Methods
Our goal is to begin with some rudimentary analysis on these type datasets to explore the
behavior and relationships, correlations, summary statistics, graphics including contour
plots and 3-D plots, robust and residual analyses. Several regression models were
investigated including multiple linear regression, polynomial regression, step-wise, best
subset, quadratic and cubic regression models as well as some nonlinear regression
models. Regression analysis allows one to model the relationship between a response
variable and one or more predictor variables. One of the useful features of a regression
model is that it can be used to predict or estimate a future response value based on a
given set of values of the predictor variables.
Regression analysis results usually include the following: regression equation, predictor
table, summary statistics, ANOVA Table, list of unusual observations, contour plots, 3-D
plots and residual plots. To appropriately use the t-test, F-test and associated confidence
intervals, the data are assumed to meet Certain .........conditions. These include (1) the residuals
(error component) are assumed to be normally distributed, (2) variation is constant
(homoscedasity) and (3) independence. The study of unusual patterns in the residuals
through residual analysis may indicate underlying weaknesses in the model.
s
in
=
Z L _
Data Exploration
Table 1 in the Appendix lists the 138 data cases used to develop a predictive
model relating the response variable, heat rate, to the 10 predictor variables shown. Due
to the large quantity of data, the full trajectory data set will note be included in the
appendix. It should be noted that unless otherwise indicated tables and figures will appear
in the Appendix. The summary statistics for the independent and response variables areprovided in Tables 2 and 2a.
Prior to the model building activities, graphical methods were used to help
identify any underlying relationships between the variables being studied. Matrix plots
of cross-graphs of the variables are provided in Tables 3 and 3a. The plots highlight
• Quadratic relationship between heat rate and altitude, temp
• Strong positive linear relationship between mach and velocity, mach and altitude,
velocity and altitude
• Exponential relationship between heat rate and Reynolds number.
On the other hand, Tables 4 and 4a consist of the Correlation Matrix that specifies the
Pearson correlation and corresponding p-values. Given the inherent relationships
between the independent or predictor variables, a principal components analysis was
performed to help define a set of orthogonal variables so that the first principal
component accounts for the largest possible amount of the total dispersion in the data, the
......second principal component accounts for the second largest possible amount of the total
dispersion in the data, etc. This would be beneficial in helping to identify a subset list of
candidate predictor variables for the analysis. Tables 5a and 5b show the results of the
_alysis using the correlation matrix of the predictor variables as input to the principal
components analysis.
Another method that was used to identify a subset list of candidate predictor variables is
best subset selection. Table 6 shows the results of this analysis for the corridor data set.
Best subset selection identified altitude, velocity, mach, dynamic pressure and Reynolds
as the top five prediction variables.
Classification and Regression Tree (CART) based models are exploratory techniquesfor uncovering structure in data that are used for:
• developing prediction rules that can be rapidly evaluated
• screening variables
• assessingthe adequacy of linear modeis .....
• summarizing large data sets for both classification and regression problems.
Tables 7a and 7b show the results of the CART _alysis and Figure 1 shows the resulting
CART tree. CART selected velocity, mach, altitude, energy and dynamic pressure as the
primary prediction variables. The tree indicates that for those data cases with velocity
measurements]ess than 13750, the predicted Values for heat rate_e generally below
thirty. Furthermore, if values ofmach are below 7.7, then the predicted heat rate is
approximately 3.909.
The classical regression methods are often used to obtain models for prediction. The
challenge is the development of the best mathematical expression to describe in somesense the behavior of a random variable of interest as a function of one or more
independent or predictor variables. The classical regression techniques however make
several strong assumptions about the underlying data, and the data can fail to satisfy these
assumptions in several different ways as indicated the Analysis Methods section.
In the case where there are one or more outliers in the data or the data may not be fitted
well by any straight line, robust regression methods come into play. These methods
minimize the effect of the outliers and can be useful in helping to identify the outliers inthe data.
Scatterplot smoothers are useful tools for fitting arbitrary smooth functions to a scatter
plot of data points. The smoother summarizes the trend of the response as function of the
predictor variables. All of the above analysis methods are used to explore the given datasets.
Results
Several approaches were used in the exploration and identification of
statistical/mathematical models for the given data sets including the classical multiple
regression, classification and regression trees (CART), and other advanced analysis
methods. One of the interesting results concerns a comparison of some initial multiple
regression model types using only the independent predictor variables for both full andcorridor data sets. These results are summarized below in Table A that includes Data,
Coefficient of Determination R 2, Adjusted R 2, fit standard error S, F-statistics and the
model type.
Table A: Comparison of Model Types for Full and Corridor Data using Three (3)
Independent Predictor Variables
Data R 2 Ad i R 2 Std Error F-statistic Model Tv_e
Full 48.3% 48.2% 196.5 394.15 Linear
Corridor 85.2% 84.9% 8.72 257.21 Linear
Full 84.4% 84.4% ....... _!08.0 1140.45 Quadratic
Corridor 97.6% 97.5% 3.54 895.68 Quadratic
Full 85.6% 85.5% 103.9 937.21 Cubic, Interact
Corridor 99.7% 99.6% 1.34 4751.89 Cubic, Interact
................................... ..-w ..................... ......-.. ....................................
- :-:- i.:̧
In reviewing Table A, an obvious conclusion is that the regression models appear to be
more appropriate for the corridor data than the full trajectory data for all model types.=
5
For everymodeltypecomparisonin TableA, therearehigherR2andadjustedR2valuesandlowerstandarderrorvaluesfor thecorridordataset. A morein-depthanalysiswasconductedon thecorridordatasetasit morerealisticallysimulatespossibleentrytrajectoryandheatingratesof anexperimentalspacevehicle. The structuredapproachthatwasusedfor model identificationconsistsof thefollowing: (1) analyzethesummarystatisticsresultsfor errorsandconsistency,(2) conductpreliminaryanalysisusingonlythe independentpredictorvariables,(3) usegraphicalmethodsto identify underlyingrelationships,(4) employmethods(BestSubsetSelection,PrincipalComponents,etc..)thataid in identifying mostlikely additionalpredictorvariablesand(5) specifyamodelusing classicalregression,classificationandregressiontrees(CART) andotheradvancedstatisticalmethods.Theresultsaresummarizedin TableB belowthatincludesrank, (R2),adjustedR2,fit standarderror(S),F-statisticsandthemodeltype. Additional analysisandresultsonthenumber1,number2 andnumber6 rankedmodelsareprovidedin theAppendixin Tables8 to 16and Figures2 to 12. Regressionmodelranked6 hasallpredictorvariablesexceptaltitude,velocity andTwall.
Table B: Identificationof ModelTypesfor theCorridorDataSet
Rank R 2 Adi R 2 Std Errgr F-,statistic Model
1 99.8 99.7 1.130 4140.87 Quad&Interacts (5V)
2 99.7 99.6 1.348 4751.89 Cubic, Interact. (3V)
3 97.6% 97.5% 3.54 895.68 Quad& Inter (3V)4 96.77 .... 4.1533 483.15 CART
5 99.4 99.3 1.851 1665.11 Cubic WO Indep (7V)
6 99.4 99.4 1.724 3843.80 Linear (6V)
7 99.1% 99.0% 2.209 2798.68 Linear 1 (5V)
8 85.2% 84.9% 8.72 257.21 Linear (3V)
Conclusions
A number of methods were considered in thi_s analysis including classical multiple
regression, polynomial regression, classification regression trees (CART), principal
components, correlationmatrix and resicl_! analysis. Several graphical methods were
_iSed in model development and assessing model adequacy. In addition, several
• tec_iques were used to screen/identify underlying relationships. For example the matrix
plot in Tables 3 and 3a suggested the inclusion of quadratic and interaction terms in our• modelsl Whereas, principal components and best subset selection were used to screen
and identify the main predictor variables. All of these methods helped to guide us in the
..... selection of the predictor variables ino_ models. Using all of the above methods,
Several promising candidate models have been developed that may be used to predict the
response variable, heat rate for the entry corridor data set. In the next phase of our
research, validation of the adequacy of our models and other advanced methods will be
explored.
6
References
1. Bowles, J.V. and Henline, W.D., "Development of Aerothermodynamic Environments
Database for the Integrated Design of the X-33 Prototype Flight Test Vehicle",AIAA 98-0870.
2. Montgomery, Douglas C.. (2001). Design and Analysis of Experiments, 5 th edition.John Wiley & Sons, New York, NY
. Meyers, R. H. and D. C. Montgomery (1995). Response Surface Methodology."
Process and Product Optimization Using Designed Experiments. John Wiley & Sons,New York, NY.
. Thompson, R.A., "Automated TPS/Structural Sizing Method", (Personal
communication), September 1999.
5. Wurster, K.E.; Riley, C.J.; and Zoby, E.V., "Engineering Aerothermal Analysis forX34 _e_al Protection System", Journal of Spacecraft and Rockets, Vol. 36, 1999,
pp. 216-227.
APPENDIX
O0
@
@
ii
o
o
0r,,,j
W_Wi
w!w!
O0
+i+!+[+_WI_ W
:7 :i
+!+_+!+i+r+T+
w
i --
- z
elCl
O
0mmI..,I-,o
°,,
[..
W W W _ wiw
]
CN
w
J _
L_
oN
rJ_
rj_.m
C,_ l_ • C_
• r_ C_ •0 _ 0 C_
E"I_j GO _ 0 C_
• 00 0 •r_ '_ 0 C_
0 Lf_ • (_J• (_] 0 •
0 _ 0 Lf_
o
®
_ o 0 0 _O
0 0 • ['_• L_ O •
it') 0 0 £_J
_C_ O 0 0 _O
0 O • (_• 0 0 •
0 C_ 0 _
!I°o. Q° o. o_oo _ o° c_
oI--I c_]
_F
H
oj_
c_
S
_ C_ I/3 r'--
d co u') _ _0
if3 co ° 0• I'-- co °
("4
E'I_ I o _D CD _.0
• C) CD °
C_ ,-I Lr) co
(',4 c_
_J CD CD CD ['--CD C) • C")
° C_ 0 °
CD _,4 C_ C_
_I (:D _ u9 C',I
,--I
_0
_J
o
0 (_D ° CO
• C_ 0 •0 C) C_ ["-
Lr_ c_ CD ,--IC,J 03 ,--I
C_ 0 CD ['--
0 C) ° CO• 0 C_ °
(D CO 0 C'_U') C_
o° o° °. o°_ o
®
_6
• _,--I
(/)
!:!!!!_zI "_ _I 0
0 -_0 > 0 0
"(3
.i
I I
0 0Lr_ U'_
o
D
411
m
g
Q
o
g
I I
o 0L_ Lr_r-- cM
i"4'
1
0L_
,D
I
0u_
o,I
,I,
dI,
Q
Q
t
0O
o
l,
1
oJ
o,I
T_ _ _T ¸ :
=
0
@@
Illlll|
!lelllll@
Illlilll
r_
0
....-
m _._
%
w
<
00
00
I I --1
0 0L_ LO
0 0LO
-7--
O
b-
I
I
n
I
m
,C
0 0
cO
2
0
omul
0mul
m
QO
kl Lr) CO
! •
o o
'0
b") _ t.O 0
0 0 0 0I
kl O_ 0 OJ 0 COCXl 0 I._ 0 Of)
I_ O0 0 Lr) 0 _0
I_t 0 0 0 0 0I
o00
0
,m_ o _ F_ _ o _ oo o T-I o o ,_o oE-I _ o c_ o F_ 0 _ 0
C_ 0 0 0 0 0 (_ 0I
0",'-I
I
F_
U
c_
@uleel
0O
Om
14
M"0,-i0
14
c_
u
X
m o.qP o
o o
o O t-I Oo_ O 00 O
o o O O
'_ O u') o a_ O_0 o G0 o r.I O
o _ O _ O!
O O O O O O
_IP O GO O 1-t O i_ O
(N O _t O _ O _ O
oooooooo
Cq O GO O _ O i"_ O _ Oi.-4 O O O O O O O i-I O
O O O O O O O O O OI
4J.H
4.1.HUO
Ii
oo_o_ooo_oooI
O O O O O O 00 O O O O_ O O'_ OO O O O _ O O O ¢N O O"1 O _ OO O O O _ O f_ O ¢_1 O O'1 O _ O
I
'1:1 O O _ O Ch O _ O O O _ O 0"_ O O O_1 O O I-I O _ O _ O O1 O 1-1 O _D O _ O
O O I_ O I_ O O O IZ_ O _D O _ O _P O
_ o _ o o o o o c; o o o c; c; o o oI I I i 1 I
0",'14_
q_ i
4.1
4.1
,,'-4
m
.oo
E0o
u_0
o
oO_E
('_ o o o o o ¢xl _-_
0 O 0 0 0 0 0 O 0
-_ _ 0 _ _ 0 -_ _ 0
> O • > O (D > O"O _ _ 1,4 "El 1,4
W-I D._ k,-i D.4 _ 1_,0 _ 0 _ 0
"O O -_ _ O -_ "Cl O -,_
f,4 ,-I _ kl ,-4 .ID 1,4 ,-tco O z_ co o zl co O zl
O D o D O zl
o_ _ 04
O_
0
@
m
-- eJ_l
I=I
,I, II,
m
L
°.
.,-I
0,1
0
0
U i
i_ o_ o
• •
0 0
I.) i
I,D ._. 0
_D O0
o ol.J
I.}
o oI3 I I
0U
U
_ O'3
• _;0
U 1 I
0
0
_ 0 _ _ _
I t I
od dd_I
0 0 ot
oooooooI
1_- _ _ CI_ 0
_, O_ ,e_ _ 0 ¢_1
ooo oooI I I I
oooooooI I I I I I
oooo ooI I i I
oI"_ ,--t OQ Oh ['_ _ kD _1
,--_ _--I _ _ I"_ u'} k_
OQ ¢Q _ _ I'Q erl _ I_
oooooooi U
0 0 0 0
I I
o
i --=-
Table 6: Best Subset Selection
Response is Heat Rate
Vars
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
R-Sq
65 4
58 8
85 1
84 8
97 5
93 7
98 7
98 6
99 3
99 1
99 4
99 4
99 5
99 5
99 6
99 6
99 7
R-Sq (adj)
65.2
58.5
84.8
84.6
97.4
93 6
98 7
98 6
99 3
99 0
99 4
99.3
99.5
99.5
99.6
99.6
99.7
C-p S
2E+04 13.233
2E+04 14. 446
6624.2 8.7306
6731.8 8.7998
1006.9 3.5947
2710.5 5.6821
455.9 2.5858
487.3 2.6546
192.0 1. 9155
296.7 2.2086
131.5 1.7236
167.0 1.8394
106.5 1.6362
113.7 1.6618
53.2 1.4300
70.5 1.4997
I0.0 1.2341
AV D DR
le e yeE
tlTn nyn
iowsT ne
tcaieM or
uiltmaPlg
dtlypcrdy
ey,,,hes,
X
X
XX
X X
X X X
X X X
XX XX
XX X X
XX X XX
XX XXX
XXX XXX
XX X XXX
XXXX XXX
XX X XXXX
XXXX XXXX
XX XXXXXX
XXXXXXXXX
21
z?
lli,,l
0
@@
N
ot_
oo
o
oo _o _o _ o
o
o
0 01 ,_ 0
I:I _ • a-_ ¢)0 _ cx_ .,-_
•H ,-_ I ,--I
I:1 o (_
•_1 .,-1 r-.- ,.
4J q)
•,-I I 0
oq
E
0
<
@@
mm
-k
oo
• o_1 °Lr') E" _<
k.O
L_
L--
o
L.
[..
Oq_
_D
al
otRt
al
Q_
v..,,q
_Dom
.4
q
m
7
7
P
_m
C_
_D
v
L_
0
• I
II
i
C
In
L_
00
_-_ _ o o co Lt_ 0 0 0
0 0 0 0 0 O_ 0 0 0 tO
o o o o o o o o o o
_-_ co t_ o _ 0_ k0 _ ko _x_ _-_
¢_I t_ _ _-_ o 0o u'1 _ 0I I I I
0 u_ r-q CO u') kD o 0 _ oU _D O_ kD Oa r_ _-_ o 0 co o
o o o o 0 0 o o[z_ c_ o c_ o o o 0 0 o_/] , o • o 0 0 0 c_
o o o 0 0 0 00 i •
o o o o o
o'_ 00 r_ _ u_ o'_ _ o _ 00Lr_ _ O_ _ Oq O_ _ 0 _'_ 0
0 -- o_ _-_ r_ o 0 i-_ o r--- oo_ _ _ o or) o o o _ oco _ _ r-I 0 0 0 O,1 0
0 0 0 0 0 0 0 0i C_ 0 0 C_ C_ C_
o o 0 0 o o 0
I o 0 0 0 o
I I
0
t_C'q
II
A
,r=t
v
I
r_
II
O_I
II
[:j
>
eN
eJml
O0
i
U1,4
0r_
0oo
0
O_oo
L_
O0L_O0
O0
T--I
0 _--I
_ 0"_
0"_
400E_
L_
r_
O
r_
00
m
X X
-r-IK.D C'xlC_I _.--I00xl K.O C.xlK.O Cx.l
_'-4 CXl CXl C'q m O Oq Cxl m mI I I I I I
-IJ
rj
c'q o _ u'3 o_ _o Lr_ O Lr300 1,4
•rl _D Lr3 CXl C"3 C,,I _D ,--_ _D '_ O
f._ O O O O O O O O O O
Lf_ CXl O_1_ KD Lf300 U'_ O_1 _D
O',1 '_ '_"_ I Oq CXl m I
_ O O O CO l'_ O O O O"_ O_
C,'3 ,--_ L_ I_ D'30"_ C,"} U_ 1..... ,--I
_ C'_ Lr') '_ C'} CXl ,--_ _ Cxl ,--_ CA1
"_I "_'I-I-I
'_ o 0 0 L_ 0 0 0 L_ L_ 0
r-_ o'_ o _3 ,_1_ KD Lf3 KD
r_
@@
_0o_@@
0
E
04_a@@
e_
e_
O0m
CO
13IB
O0
n_®CO
(1) n_
-r
E_
v
O0wm
7-
E
E
0
I I0 0
Xo_mnb_J=l '-
I0
0
LO!
"olm
ry
0o
_.-.3
r_
oo
33 °O.
cI..
CO
Z
Im
E0z
i@
\
@@
I I I I
...._ _oS _u_
I
!
°'_
!
L_
e_
I
°=
• • i|."• • 0 °
"° _. .°_ o0 O _ •
gO I •
OQ
00
• I
":'..
• j •
._N" •i° •
O •
_000, • •
I
t r I I J i _ i
....-_ np _i i!i ii ii-2 ! _!i!i! -_ ii_ i_
_ o
_ o
D
_ 0_m
-- o
I
rl T_
_ IL _
i •
__v _0
o
=
d_t
c_
I
0000
0 00 00 0urJ 0
, 11001gA
Lo.
000140
0
0 "(30
0143
00
v
Table 11: Regression Analysis for Heat Rate
(5V Corridor Data)
The regression equation is
Heat Rate, BTU/ft2/s = - 815 - 2.71 Altitude, kft
+ 0.0521 Velocity, ft/sec + 3.59 Temp, deg R
- 1.45 Dyn. Pres. ib/ft2 +0.000648 Reynolds per
0.00774 Alt*2
- 0.00314
-0.000052
+0.000002
ft +
Temp*2 -0.000000 Reyn*2 -0.000101Alt*Vel
Vel*Temp +0.000055 Vel*DyP +0.000283 Temp*DyP
DyP*Reyn
Predictor
Constant
Altitude
Velocity
Temp,
Dyn. Pre
Reynolds
Alt*2
Temp*2
Reyn * 2
Alt*Vel
Vel*Temp
VeI*DyP
Temp*DyP
DyP*Reyn
Coef
-814.9
-2.7076
0.052053
3.5875
-1.4525
0.0006481
0.007743
-0.0031392
-0.00000000
-0.00010130
-0.00005249
0.00005516
0.0002828
0.00000222
SE Coef T P
187.1 -4.36 0.000
0.4786 -5.66 0.000
0.005991 8.69 0.000
0.5892 6.09 0.000
0.3884 -3.74 0.000
0.0001496 4.33 0.000
0.001294 5.98 0.000
0.0005719 -5.49 0.000
0.00000000 -5.65 0.000
0.00001424 -7.12 0.000
0.00000845 -6.21 0.000
0.00000750 7.36 0.000
0.0006029 0.47 0.640
0.00000035 6.42 0.000
S = 1.130 R-Sq = 99.8% R-Sq(adj) = 99.7%
33
_r
t,===
gm_
0r_
r_
<oe
r_
rj)r._
CI
UN
0m
000
I
0
(3O
0
r-I
0• C"3
CO °O0 r-IC,,Iu'3
0
c_ r=.lo
•rl ,-I
Pl .,-I _1_ 4-1
® 0
"E
rj_
III
r_
x x x
• _ _ _ ° °? ? _._
_ o o o o o o _ _
• • • o • J
F_
m
i
=
F_
0
o_
=
omml
U)i
im
n,'_,
E__R
II
-r"
Oo')
Ef I0 0
t0
- (N
B _
-- 0
-'T
I
I
iI
n_
N
L_
e-
e_Z o_
e pnnmq
i
El,,n
0Z
go
\go
e
go •
I I t t I
I n
-- o
!
I
u
I
wu
Nll
I,.,,.
epml Jl
.C: ..-.
_'-r
i
i ,._
=I
e_
a
Iii
ii
t
• l •• • ,6 • •
Oo gO • •
I ° ,.'.. .• • _O •
O I• ee,, •
• k •
elI
A'omIO_
I
:o
• •
• • •
• •
O =O O_ ° j• •
Ii
l i I I I Il I I
_ npE_ _P_PmI_
_0I-,.
o(D
o
8
_oc_
_ o
-o
I
O_
Table 14: Regression Analysis Without Alt., Vel &Twall
_. i _ m
The zeg=eisAon equation _s
Haat D_te,
Mach
ib/ft2
i . i . ._ mm --
BTU/ft2/S _ 241 - 0.0266 Mach*3 + 1.68 Mach*2 - 33.2
-0.000030 DynPress*3 + 0.0133 DynPress*2 - 2.99 Dyn. Pros
-0.000000 Energy*2 +0.000117 Energy, ft3/soc
- 43.5 Pressure, ib/ft2 +26874067 Density, slugs/ft3
- 0.0589 Temp, deg R +0.000453 Reynolds per ft
Prodictor
Constant
Mach*3
Math*2
Math
DynPzess
DynPress
Dyn, Pre
EnergY2
Energy,
Pressure
Densi_,
Te_p,
Reynolds
Coaf SE Coof T P
241.19 61.32 3,93 0.000
-0.026629 0.004115 -6.47 0.000
1.6772 0.2470 6.79 0.000
-33.223 4.995 -6.65 0.000
-0.00002972 0.00000566 -5,25 0.000
0.013323 0.002475 5.38 0.000
-2.9908 0.5574 -5.37 0.000
-0,00000000 0.00000000 -3.2.94 0.000
0.00011696 0,00001058 11,06 0.000
-43.51 23.52 -1.85 0.067
26874067 16016537 1.68 0.096
-0.05891 0.08058 -0,73 0.466
0,0004529 0.0001233 3,67 0.000
S = 1.851 R-Sq = 99.4% R-Sq(adj) = 99.3%
i . _ _ 81 mu
39
;. 4
000
0
0
- <O0
mml
_=
m
mm
%.0
k.O
L._
00 c_l
cx! 00o _ O'h
ookP _0
U1,4
0t/]
0
H
0
0
r
r_
ejil
r_
0
O0
{/I I I I I I I I
4-) 0 0 0 _-I _ 0 0 P-'- U_ 0 0 0 Oh _0
• J -- • • J • • • • e o
4-) Cxl _ _ 0 CXl _ Oq CXl (_1 _--I U'_ 0 0.,1 _--I
(D
ql ,"4
""I 4..I
-_ .,_
4,1 4,_
0 0
L_
N
=lid
I,.
olil
=
"0im
G)
(--"_'T"
0._
E_(-
(/)li
"1-
0
I
i
_JLBnbeJ_-i
I0
n
-'T
I
I
!
_L_I
(-q
i
,IN
Z o '_• f'l ""
EL.,
_De_
r
\k
1 1 t I
%
II
Ii
I
--cO
--0
I
-- £%1I
_o')!
I
_ I.OI
I
i
cO
"oII
n-
=
_J
=ejII
• • I
oOo i o _ • •
• • ° io • ° 0
• • i •
• ', • •
0 IO I • •
• O! •
• _i• • I iI
I
• I I I I I !
P
,,
• • II
• Ie •I
• 0 0 I i • •
OIO Oil• • •
I
° ° i,. S
l I
J_O • •
• Io •
o
I
• _ • • •e • I ee •
e • •
• I _Oglm
i • • Ii
t
I I I I i i
c_ ,- 0 "T c_ coI !
II I
_o
_o
-o