stage turbines are considered v, ith the number el desien variables ...... Grossman, B., Mason, W.H., Watson, L.T., and Haftka,. R.T., " ... http://src.doc.ic.ac,.
____________g-_ AIAA 2000-3242
Preliminary Design Optimization Turbine For Rocket Propulsion
For A Supersonic
Nilay Papila _),. Wei Shvy _).. Lisa Griffin _::_, Frank Huber {_ and Ken Tran '-_)
t, University ,2_NASA :' Riverbend _' Boeing
of Florida, Marshall
Gamesvil/e,
Flight
Design
Center,
Services,
- Rocketdyne
FL Huntsville,
Palm
Division,
Beach Canoga
AL Gardens, Park,
FL
C,,\
36th AIAA/ASME/SAE/ASEE . Joint
Propulsion
Conf_erence
16-19 July 2000 / Huntsville,
and Exhibit Alabama
I
For permission
to copy or republish, contact the American Institute of Aeronautics and Astronautics, 1801 Alexander Bell Drive, Suite 500, Reston, Virginia 20191-4344.
AIAA-2000-3242
PRELIMINARY
DESIGN
OPTIMIZATION
FOR
ROCKET
Nilay
Papila',
" Department
Wei
Shyy',
of Aerospace +NASA
" Boeing
Griffin',
Engineering, Flight
Design
Services,
- Rocketdyne
Frank
Huber*
Mechanics
of Florida,
Marshall
: Riverbend
TURBINE
FOR
PROPULSION
Lisa
University
A SUPERSONIC
Palm
Division.
Tran §
& Engineering
Gainesville, Center,
and Ken
Science
FL
Huntsville,
AL
Beach
Gardens.
Canoga
Park.
FL CA
ABSTRACT
In this
stud,,',
we present
preliminary
design
propulsion
system
stage
turbines
variables
with and
the
in the
,_ptimization
required
tv,'o-level
domain
is to balance
"['he accuracy this
,,,trategy
investigation
is
single
stage
p_+tll-ly under _,i,_,mficant
thc
engine
[_ortlon
of
discharge ratio
_t
working
t]uid.
issues
related
the
technique, +I)OF+I the
of
and
has also
criteria
_mpact
the
_+n
been
construction
the
,\
tRSMI
is
,,r
' Stage
Reaction
T,.,
' Input
Temperature
_'+ p!tcil W
' Pitch Speed ' Weioht
of maximizing
design
base
and
Increment
Efficiency
is much
1. IN'FRODUCTION
[1_
";upersonlc
Io_,t at the stage
and
it
tectmoi_gleS
in
Optimizing
the
rocket
a multistage is
desirable
cchniclucs
CADC#'IIllC#_[S
are
being
propulsion
turbine
to
efficient
and
tbi_, ta:.,k. In ucneral,
are needed,
actively
community.
is a labor-intensive
develop
to t, ndertake
>ptimlzation
task effective
tv,,3 types
uf
namely.
that
significant
cIflect;vcness
turbine
Tuvestigated
hvdrooen
exhibit
Fraction
Our
It is demonstrated
the data
_1
,ho\_n
to
turbine
>,hown
refined
\merican
cases
it is
increased
;:/_) -
cases
on efficiency
and
optimization
c,,trespt>nding
¢_f the , 13)
diameter.
77, W and
both
composite
whereas
x_ith
the
for
(Eqn.ll)
s=O)
turbine.
optimization
the results
improving
= DxRP3I
D is meanline
of
based
it=l.
>,nze
\'I'_,
where
values
parameters,
ct>mpc>ite
composite
optimization, different and d, are considered
The
W-W+, J
efficiency
capacity. based
the
represents
runs
for single-stage
based
When
functicm
payload
increment based
t_ptimum
and
rib is the baseline
weight.
,.llealtlilw
6 shows
and
s=l)
are
the
and the
calculated
only,
table
find
Different
the corresponding
Table
based
this
Excel
maximum
with
¢t=l,
weight
the
[tic
tt>lh+v,l ng manner.
Xpay =el x tOOx< ll-rlhl-!
in
for dpay
Apav
case on
the optimization shov;n
lunctlotl
is
The
corresponding 5%
local
it=l.
(rms-
to
()ptimization
and
error
cases.
optimization
s=l),
based
used
all
Apay
3by
of the objective
is for
based
for the
evaluated
with
determined.
based
It=O,
t_ptimization
I11_
together
77, W and
2hal-order
136 be
or minimum
to avoid
function
represents
d = 9"d_.d.
x_av
r/,
and 3-
28-unknown
square
constraints
tried
of
(Eqn. s=OL
can
mean
obtained
are
values
results
.\nothcr
for
the and
fit
of 77, W and Apay
parameters
(t=]. according desirability
are
values
desirability where
are
are
2-stage the
root
a given
points
optimum
_>ptlmum =
the of
the maximum
point
starting
be defined
as
there constructing
for
quality
with
design
d
obtained
4 and 5 for 1-,2-,
case,
approximations
to find
function
the desirability
surfaces
in Tables
comparing the adjusted error) shown in Table 5.
optimum
which
response
for
78
The
RSM-based _9)
and
the
W, and Apay
handle
a straightforward
desirability'
for 17, i.e., all, can
turbine
can
response
surfaces.
referred
overall
task
a composite
response
surface
maximizing
surface
optimization building
individual
is
minimizing
The
DISCUSSION
if required. The
In
AND
designed
space
r/ &
Apav
with
of the response
design
design
tot and
space
If
spaces
((t=l.
s=O),
based
optimization
spaces,
substantial
and Astronautics
are
ts based
shown
details
in Table
on the optimal (t=O, cases.
s=l)
& (t=l,
With
improvement
9 -
values
these of the
AIAA-2000-3242 response surface fit accuracy is observed also for 3-stage (Tables 13 and 14). Based on the results can be made: (i)
lii)
obtained,
tbr 2-stage
the follo'xing
and
observations
In order to determine how the optimum solution for 3pay changes as the weighting constants, t and s, are changing, three designs of (t=l, s=O), (t=O, s=l) and (t=l, s=l) denoted
To ascertain required predictive capability of the RSM. a two-level domain refinement strategy has been adopted. The accuracy of the predicted optimal design points based on this approach is shown to be satisfactory. For Apay-based
optimization,
the 2-stage
turbine
as
..Qt,E2, and
candidate vertex Plane as follows.
=/3(I_+).i
where
"_ a2_'2
points
D.3,respectively, to define
are selected
a plane referred
as
as of
(15)
2 h- O_3_"_ 3
E c_, = 1and 0 _
as
the
DlIeScnt
rcsp_;n:.-,c
>Lll-lat..'c
For both x_eight onb, _ptlmizatlon _,'=/ ,=t)) and efficiency only vpumization el=t), s=]), as expected, the single-_tage design gives the smallest wmght, l-he efficient\ a[,_o improves a:-, the number of stages mcrease_, tgonse surfaces generated for r/, W and Apay by using 1990-data generated by face centered composite design and 249-data selected by OA method. This table illustrate.,, tilat the fidelity of the response surface generated fl_r design space of 249 data, based on ,+rthogonal arrays, are comparable with that of 1990 data based on the face centered criterion. The response surface models arc al.so ab.',es.,,ed bv u_,mg 78-test data to determine the predicttxe accuracy of these models. Table IS presents that the testing tins-errors of response surfaces generated are 1.65% tbr r7 and 0.96% for W using 249-data, and 1.67c_ for r/and 1.21% for W using 1990-data. The results ot optimization based on Apay and composite desirability tunction of 77 & W with 249-data ,,elected by orthogonal arrays are shown in Table 19 for tim original design space and in Table 20 for the refined design space. When these results are compared with the results of 1990-data presented in Tables 7 and 13, it is observed that the optimum r/, W and Apay are largely consistent. However, it is also observed from Figure 7 which shows the comparison of the design variables for optimization based _n i=lpayl, some of the design variables are different even though optimum r/, W and Apay are consistent. This .,,hows that there are multiple points in the design space which yield comparable performance. Nevertheless. it remains true that the twostage turbtne is most >ratable from a payload point of view.
6 Institute of Aeronautics
and Astronautics
AIAA-2000-3242 7. CONCLUDING
REMARKS
In summary, the result of the RSM computation indicates that indeed the efficiency rises quickly from 1 stage to 2 stages but the increase is much less pronounced with 3 stages. A 1-stage turbine performs poorly under the engine balance boundary condition. A significant portion of kinetic energy is lost at the turbine discharge of the lstage design due to high pressure ratio and high energy content, mostly hydrogen, of the working fluid. Adding a 2 nd stage recovers most of that wasted energy resulting in much better efficiency for a 2-stage turbine. An extra 3 'd stage only improves the efficiency slightly. Understandably, the turbopump weight also increases substantially from 1 to 3 stages even though the 3-stage turbine diameter is smaller. The smaller diameter is the direct outcome of higher RPM that is the result of lower exit annulus area satisfying the constraint defined tot AN'. ]'he exit annulus area is smaller because of the lower pressure ratio per stage. However. the 3-stage turbine is much longer and requires a change in bearing configuration that adds significantly into the overall wei,,ht The optimum 2-sta_e turbine resultine from the RSM optimization is consistent with a design produced by an experienced engineer; namely, that most of the work is done bv the 1" stage at very h_w reaction. By varying from 1- to 2-to 3-stages, _e _bscr'ce lhaI while the size of the training data increase naturally \t.ith the number ot design variables, the actual need is case dependent. Furthermore, it seems that the selection of the data distribution can be more critical than the data size. Present investigation has also demonstrated that the criteria for selecting the data base exhibit significant impact on the cfficienc,v and effectiveness of the constructum of the response surface. S. AC KNOWIA'2DG
I,?,MENT
This present work is supported by NASA Marshall Space Flight Center IGrant #: NAG8-1251). We have also rcceu,,ed _aluable comments fr(',rn t-'r_fessor t),,aphael [taftka o1 the University of Fh>rida. 9. REFERENCES 1Myers, R. H. and Montgomery, D. C., Response Surtace Methodology - Process and Product Optimization Using Designed Experiments, New York: John Wiley & Sons, Inc., 1995. "JMP version 3. Statistics Institute Inc., 1998. _Sloan, J., "Airfoil Low P,evnolds Science Thesis,
And
and Wing
_pmber University
Graphics
Planlorm
Guide.
Optimization
Flieht Vehicles." ot Elorida, 1998.
American
Master
SAS
ti)r _I
4Madsen, J.I., "Design Devices," Philosophy University, 1998. SMicrosoft
Corporation
Optimization of Doctorate
(1985-1996).
of Internal Flow Thesis, Aaiborg
Microsoft
Excel 97.
6Madsen, J.I., Shyy, W. and Haftka, R.T., "Response Surface Techniques for Diffuser Shape Optimization," accepted for publication in AIAA Journal, 1999. VTucker, P.K., Shyy, W. and Sloan, J.G.,
"An Integrated
Design/Optimization Methodology for Rocket Engine Injectors," AIAA/SAE/ASME/ASEE 34th Joint Propulsion Conference, Paper No. 98-3513, July 13-15, 1998. SPapila, N., Shyy, W., Fitz-Coy, N. and Haftka, R.T., "Assessment of Neural Net and Polynomial-Based Techniques for Aerodynamic Applications," AIAA 17th Applied Aerodwmmics Conference, Paper No. 99-3167, 1999. "Shyy, W., "Fucker, P.K. and Vaidyanathan, R., "Response Surface and Neural Network Techniques for Rocket Engine Injector Optimization," A1AA/SAE/ASME/ASEE 35th Joint Propulsion Conference, Paper No. 99-2455. June 20-24, 1999. "_Unal, R., Lepsch, R. A., :md McMillin, M. L., "Response Surface Model Building and Multidisciplinary Optimization using D-Optimal Designs," 7'/' A 1AA/USA F/NA SA/ISSMO Symposium on .Ihdtidisciplinat T Analysis amt Optunizatlon, Paper No. 98-4759. September 2-4. 1998. t:Unal,
R., Braun.
R. D., Moore.
A.A., and Lepsch.
R.A..
'Response Surface Model l?_uilding l rsing Orthogonal Xrravs for Computer Experiments," 19 'h Ammal International Cotg_,'rence _g the lmernattonal Society of Parametric Analysis, New Orleans, Louisiana, May 2230. pp. 13. 1997. t:Unal. R., Braun. R. I3.. Moore. A. A, and Lepsch, R.A., "Design Optimization _n Cost Basis Using Taguchi's Orthogonal Arrays" Proceedings of the Annual American Society for Engineering Management (ASEM) Conference, October, 1996. _3Unal, R., and Dean, E. B., "Design For Cost And Quality: The Robust Design Approach," http ://m ij U11O. larc. nasa. go v/pap/robdes/robdes, htm l, 1995. t_Unal, R., and Dean, Design Optimization Overview," Proceedings
7 Institute of Aeronautics
and Astronautics
E. B., "Taguchi Approach to for Quality and Cost: An Of the International Society of
AIAA-2000-3242 Parametric Analysts 13 th Annual Orleans, Louisiana, May 21-24, 1991.
Cottference,
New
_5Dean, E. B., "Taguchi Methods from the Perspective Competitive Advantage," http://akao.larc.nasa.gov/dfc/ tm.html, 1995.
tSOwen, A., "Orthogonal Arrays for: Computer Experiments, Visualization, and Integration in high dimensions," http://src.doc.ic.ac, uk/public/pub/Mirrors/ lib.stat.cmu.edu/designs/owen.readme, 1994.
of
tVLadson, L.S.,
Z°Venter,
1. Design
Space for Single-Stage
Turbine
A. D., Jain,
A., and Ratner,
G.,
Haftka,
R.T.,
and
Starnes,
J.H.,
(All geometric
design
variables
are normalized
by the baseline
values) Variable Mean Diameter.
Table
Lower Limit 0.50
Upper Limit 1.50
Speed, RPM Blade Annulus Area. A ..... Vane Axial Chord. c, Blade Axial Chord, Cb
0.70 0.70 0.39 0.26
1.30 1.30 1.71 1.14
Stage Reaction,
0.0%
50%
2. Design Space for 2-Stage
D
sr
l'urbine
tAll geometric
design
Variable Mean Diameter, D Speed, RPM Blade Annulus Area. A;,,m 1" Blade Height (';4 of Exit Blade), h_ 1_tVane Axial Chord. c,i 1_tBlade Axial Chord. %1 2 ''a Vane Axial Chord, c,.2 2 "a Blade Axial Chord, %2 1_tStage Reaction, srl 2_ Stage Reaction, sr2 I st Work Fraction, wft
American
Institute
M.,
Jr.
"Construction of Response Surface Approximation for Design Optimization," AIAA Journal, 36(12), pp. 22422249, 1998.
_7Trosset, M. W., and Torczon, V., "Numerical Optimization Using Computer Experiments," NASA CR201724 ICASE Report No.9738, pp. ! 6, August 1997.
Table
Waren,
"Design and Testing of a Generalized Reduced Gradient Code for Non-linear Programming," ACM Transactions on Mathematical Software, Vol. 4, No. 1, pp.51-56, 1978.
_6Balabanov, V.O., Giunta, A.A., Golovidov, O., Grossman, B., Mason, W.H., Watson, L.T., and Haftka, R.T., "Reasonable Design Space Approach to Response Surface Approximation," Journal of Aircraft, 36(1), pp. 308-315, 1999.
8 of Aeronautics
variables
are normalized
Lower Limit 0.50 0.70 0.70 0.90 0.39 0.26 0.21 0.17 0.0% 0.0% 50%
and Astronautics
bv the baseline
Upper Limit 1.50 1.30 1.30 1.50 1.71 1.14 1.41 1.13 50% 50% 85%
valuest
AIAA-2000-3242 Table3.Design Space for3-Stage Turbine (All geometric design variables arenormalized bythebaseline values) Variable Mean Diameter,
LowerLimit
Speed, RPM Blade Annulus Area, A._n i st Blade Height (% of Exit Blade), I st Vane Axial Chord, cvl 1 st Blade Axial Chord, Cbl 2 "d Vane Axial Chord. cv2 2 n'l Blade Axial Chord, c_2 3 rd Vane Axial Chord, c,3 3 r'j Blade Axial Chord. oh3 I st Stage 2 "d Stage 3 rd Stage 1 st Work 2 "d Work
hi
Reaction, sr, Reaction, sr2 Reaction, sr3 Fraction, wf_ Fraction, wf2
Table
1-Stage 2-Stage
0.50 0.70 0.70 0.90 0.39 0.26 0.21 0.17 0.21 0.17 0.0% 0.0% 0.0% 40% 30%
D
4. Response
Surface
No. of Design Parameters 6 11
3-Sta_e
Summary
No. of Coefficients 28 78 136
"Fable 5. The quality of the Second-Order Response Surface obtained IMean values ot tl. Wand Apay are normalized
I -Stage
2-Stage
3-Stage
1.50 1.30 1.30 1.50 1.71 1.14 1.41 1.13 1.41 1.13 50% 50% 50% 80% 10%
tor 1,. _ and 3-Stage
15
UpperLimit
Turbine No. of Desien Points 76 1990 2235
for 17, W, and Apay 1.2 and 3-Stage Turbine by the baseline values_
R: Ra" rms- ,'rrc_r Mean R-" t),a: rms- _rror Mean
r1 0.998 0.997 2.5()f} 0.57 0.995 !).994 1.31 '.;_ 0.78
W 0.999 0.999 0.82 f:;0.60 0.996 0.996 2.56c4 0.86
Apay 0.998 0.997 4.09¢_ -0.43 0.995 0.995 9.58% -0.24
R2 I),a: rms- ,';rot Mean
0.989 0.988 2.05¢ 0.89
0.989 0.988 2.90_ 1.41
0.994 0.994 8.4'.;'_ -0.26
American
Institute
9 ot Aeronautics
and Astronautics
AIAA-2000-3242 Table6.Optimization based onApay (All geometric
design
variables
and composite desirability function of 7"/and W for single-stage and output parameters are normalized by the baseline values)
Wopt
Apayopt
0.766
0.731
-0.214 -0.193
Error % of mean
0.797 2.9
0.733 0.3
4.8
RSM(t= I, s=O) Meanline Error % of mean
0.399 0.383 1.8
0.407
-0.611
0.402 t
-0.623 2.7
RSM(t=O, s= 1) Meanline Error % of mean
0.781 0.797
0.762 0.762 0.03
-0.216
RSM(t= 1, s= 1) Meanline Error % of mean
0.702 0.718 3.1
0.583
-0.261 -0.239
_'loDt
RSM(Apay) Meanline
2.2
-0.199 3.8
0.588 0.8
I
turbine
D
RPM
1.181
0.975
1.166
1.706
0.880
0
0.502
1.284
0.699
0.394
0.264
0.5
1.260
0.915
1.300
1.575
0.880
0
0.895
1.284
0.699
1.706
0.264
0
Aalul
Cv
Cb
sr
5
Table 7. Optimization based on Apm and composite desirability function of r/and W for 2-stage turbine for original design space (All geometric design variables and output parameters are normalized by the baseline values)
[ RSM(Apav) : _
floor
Wopt
Apayopt
1.10 , 1.1_.
1.05 1.05
i
0.11 0.14
]
I
0.00
RSbl(t=l.
[
0.66
-0.34
0.65
0.65
0.35
1.10
0.60
4.00
Meanline Error %of mean[ RSM(t=O.
0.65
s=l)
Meanline
0. ll 113
1.10
0.14
] [
0.41
13.50
IError '_ot"meanI 0.30 I
1.70
Error %of mean[ 3.30 RSM(t=I..s=I)] 0.99 Meanline 0.991
RPM
Aann
hi
Cvl
c,,2
Cbt
%2
srt
srz
Wn
0.99
1.14
1.50
1.57
0.97
0.71
0.68
0
0.50
0.85
1
1.16
10.60
Error ' _of mean I 2.50 s=0;
D
]
0.50
t i
1.27
] 0.70 1
1.24
i
{).92
]
t.27
I ,:0 :k-I "_0 ]
o.84 0.03 0.03 '""'i 0.85
1.50
1.71
t
1.76
11 i
1.30__ .............. l 50
1.44
1.06
t).92
1.13
0.7l
0.79
a 0.85 : I 13 ....12__1_
t 71 _[7(,
Lttl 0.50
0
0
0.50
Ill 0
t10
0.40
Table 8. Optimization based _n Apay and composite desirability function of 71and W for 3-staee turbine for original design space IAII geometric design variables and output parameters are normalized by the baseline values)
"lq,_o t
Wop
I
Apayopt
RSM (Apay) Meanline Error %of mean
1.24 1.21 4.72
1.62 1.57 3.52
0.14 0.10 14.59
RSM (t=l, s=O) Meanline Error ',;_of mean
0.85 I_.83 2.13
1.13 1. t3 0.34
-0.23 -0.26 14.39
RSM (t=O, s=l; .Meanline Error %of mean
1.26 123 4.27
1.74 1.69 3.10
0.12 0.09 13.54
RSM (t=l. r=/; Meanline -. , Error %of mean
IO8 i.10 2.16
1.33 1._4 0.59
0.05 0.04 4.11
:\merlcan
10 Institute _)t Aeronautics
and Astronautics
0.50
0.80
0.50
AIAA-20(X)-3242
RSM(z._pay) RSM (t=1, s=O) RSM (t=O, s=l) RSM (t=l, s=l) Table
D
RPM
Aa_
hi
c,,l
c¢_
e,3
ebl
Cb2
Cb3
1.07 0.50 1.19 0.91
1.07 1.28 0.96 1.29
0.98 0.70 1.20 0.70
1.50 0.90 1.50 1.50
1.59 1.71 1.44 1.71
1.09 1.76 1.06 0.62
0.87 1.41 0.78 0.78
0.56 0.73 0.56 0.17
1.02 1.41 0.99 0.21
0.89 0.17 0.90 1.13
9. Upper and Lower Limits (All geometric
D
RPM
srt 0 0.5 0 0
of the Design Parameters of the refined designed spaces design variables are normalized by the baseline values)
A_,m
ht
cv,
c,,2
Cbl
st2 0.5 0.5 0.5 0
st3 0.5 0.5 0.5 0
Cb2
Srl
sr2
wta
Max
1.50
1.30
1.30
1.50
1.71
1.76
0.92
1.13
0.50
0.50
0.85
Min
0.50
0.70
0.70
0.90
0.39
0.26
0.21
0.17
0.00
0.00
0.50
Refined
Max
1.26
1.05
1.20
1.50
1.71
1.11
0.79
0.79
0.05
0.50
0.85
(Apay) Refined
Min Max
1.06 0.60
0.93 1.30
1.08 0.76
1.44 1.50
1.44 1.71
0.81 1.76
0.64 0.92
0.60 1.13
0.00 0.50
0.45 0.50
0.82 0.54
It=l,
Min
0.50
1.21
0.70
1.44
1.57
1.61
0.85
1.03
0.45
0.45
0.50
Refined
Max
1.34
0.98
1.30
1.50
t.63
1.20
0.76
0.90
0.05
0.05
0.82
¢t=O, s=l)
Min
1.14
0.86
1.24
1.44
1.36
0.91
0.62
0.70
0.00
0.00
0.75
Refined
Max
1.00
1.30
0.76
1.50
1.71
1.76
0.92
1.13
0.05
0.05
0.54
!t=l.
Min
0.80
1.21
0.70
1.44
1.57
1.61
0.80
1.03
0.00
0.00
0.50
s=l)
Table
Original Refined Refined Refined Refined
(Apay) (t=1. s=O) (t=O, s=l) (t=l. s=/)
Refined IApay) Refined (t=l, s=O) Refined (t=O, s=l) Refined It=l. s=l)
0.3 0.3 0.3 0.1
10. Center Coordinates of the refined designed spaces for 2-stage turbine IAII ,.zeometric desi,,n variables are normalized by the baseline valuesl
D
RPM
A:,._
h_
c_
c_2
0 1.16 0.50 1.24 0.91
0 0.99 1.27 0.92 t.27
0 1.14 0.70 1.30 0.70
0 1.50 1.50 1.50 1.50
0 1.57 1.71 1.50 1.7I
0 0.97 1.76 t.06 1.76
Cb_ 0 0.71 0.92 0.69 0.85
Cb2
sr_
st2
we,
0 0.68 1.13 0.80 1.13
0 0 0.5 0.0 0
0 0.5 0.5 0.0 0
0 0.85 0.5 0.78 0.50
"Fable 1 I. 1 rpper and l_ower Limits _)f the l)csign Parameters of the refined designed spaccs tall geometric design x ariables are normalized by the baseline values_
Original
0.6 0.4 0.6 0.6
w f2
for 2-stage turbine
Original
s=O)
wft
for 3-stage turbine
Max Min Max Min
D 1.50 0.50 1.17 0.97
RPM 1.30 0.70 1.13 1.01
A_ 1.30 0.70 1.05 0.92
h_ 1.50 0.90 1.50 1.44
c_t 1.71 0.39 1.71 146
c,,2 1.76 0.26 1.24 0.9-,
c,,_ 1.41 0.21 0.99 0.75
ct, I 0.73 0.17 0.62 0.51
cl__ 1.41 0.21 1.14 0.90
ct,_ 1.13 0.17 0.99 0.79
sr, 0.5 0.0 0.05 0.00
srl 0.5 0.0 0.50 0.45
sr;l 0.5 0.0 0.50 0.45
wf I 0.8 0.4 0.62 0.56
Max Min Max Min Max
0.60 0.50 1.29 1.09 1.00
1.30 1.22 1.02 0.90 1.30
0.76 0.70 1.26 1.14 0.76
0.96 0.90 1.50 1.44 1.50
1.71 1.57 1.60 1.34 1.71
1.76 1.61 1.18 0.88 0.74
1.41 1.29 0.93 0.69 0.88
0.73 0.68 0.60 0.49 0.23
1.41 1.29 1.10 0.86 0.33
0.26 0.17 0.99 0.80 1.13
0.50 0.45 0.05 0.00 0.05
0.50 0.45 0.50 0.45 0.05
0.50 0.45 0.50 0.45 0.05
0.44 0.40 0.62 0.56 0.64
0.3 0. t 0.30 0.26 0.30 0.28 0.30 0.26 0.14
Min
0.80
1.23
0.70
1.44
1.57
0.44
0.64
0.17
0.21
1.03
0.00
0.00
0.00
0.58
0.10
American
Institute
11 of Aeronautics
and Astronautics
AIAA-2000-3242 Table12.Center Coordinates oftherefined designed spaces for3-stage turbine (All geometric design variables arenormalized bythebaseline values) D 0 1.07 0.50 1.19 1.00
Original Refined (Apay) Refined (t=l, s=O) Refined (t=O, s=l) Refined(t=l, s=l)
RPM 0 1.07 1.28 0.96 1.00
Am 0 0.98 0.70 1.20 1.00
h1 0 1.50 0.90 1.50 1.20
Cvl 0 1.57 1.71 1.44 1.05
Cy_
Cv_
Cbl
Cb2
Cb_
0 1.06 1.76 1.06 1.06
0 0.85 1.41 0.78 0.85
0 0.56 0.73 0.56 0.45
0 0.99 1.41 0.99 0.85
0 0.90 0.17 0.90 0.68
sr! 0 0.0 0.5 0.0 0.3
sr2 0 0.5 0.5 0.5 0.3
sr3 0 0.5 0.5 0.5 0.3
wfl 0 0.6 0.4 0.6 0.6
wf2 0 0.3 0.3 0.3 0.2
Table 13. Optimization based on Apay and composite desirability function of 7?and W for 2-stage turbine for refined design space (All geometric design variables and output parameters are normalized by the baseline values) rlo_t
W_
1.13
1.04
0.15
1.13
1.04
0.15
0.03
0.02
0.16
RSM(t= 1, s=O) Meanline Error %of mean
0.65
0.65
-0.35
0.65 0.00
0.65 0.00
-0.35 0.01
RSM(t=O,
s=l)
1.15
1.10
0.15
Error %of mean
1.15 0.02
1.10 0.01
0.15 0.09
1.00 i 1.00
0.85 0.85
0.04 0.04
RSM
(Apay)
IMeanline Error % of mean
Meanline
Apayovt
D
RPM
Aim
hi
1.08
s=l)
1.44
Cv2
Cbl
CB2
Srl
0.79
0.71
0.62
0.1
0.5
0.92
1.13
0.5
0.5
1.12
1.02
0.50
1.27
0.70
1.50
1.71
1.76
1.23
0.93
1.30
1.50
1.31
0.88
0.71
0.73
0.1
1.58
0.92
1.01
0
sr2
Wn
0£
0
05
0.8
r
1
RSM(t=I, Meanline
1.50
Cvl
0.91
1.27
0.701
.50
1
1.71
0
f 1
0.51
[Error '/(of mean Table 14. Optimization based on Apay and composite desirability function of r/and W fl)r 3-stage turbine lor refined design space (All geometric design variables and output parameters are normalized by; the baseline values) 'lqoot
Wopt
Apayopt
RSM (Apay) Meanline Error %of mean
1.20 1.21 0.22
1.54 1.54 0.13
0.11 0.11 0.41
RSM (t=l, s=O) Meanline F_rror C+of mean
0.82 0.82 0.04
1,13 1.13 0.02
-0.27 -0.27 0.31
RSM (t=O, s=l) Meanline Error %of mean
1.24 1.23 1.39
1.75 1.72 1.29
0.09 0.09 1.35
RSM (Apay) RSM(t= I, s=O) RSM(t=O, s= 1)
D 1.03 0.50 1.20
, 1.11 1.27 0.94
Aa 0.92 0.70 1.26
ht 1.50 0.96 1.47
1.46 1.71 1.35
0.94 1.76 0.88
c,.3 0.75 1.41 0.69
Cbl 0.51 0.73 0.54
Cb2 0.90 1.41 0.86
• L
American
Institute
12 of Aeronautics
and Astronautics
Cb3 0.79 0.17 0.80
srl 0.04 0.5 0
sa'2 : _;;Brgl 0.5 0.5 0.62 0.5 0.5 0.4 0.45 0.45 0.6
wt'2 0.28 0.3 0.3
AIAA-2000-3242 Table15.Optimization summary for 1,2and3-stage turbinewithresponse surface inoriginaldesign space (All outputparameters arenormalized bythebaseline values)
Apay (t=l,
s=O)
(t=O, s=l)
(t=l,
Table
s=l)
qo_,t
Woof
1-stage 2-stage 3-stage
0.77 1.10 1.24
0.73 1.05 1.62
-0.21 0.11 0.14
1-stage 2-stage
0.40 0.65
0.41 0.66
-0.61 -0.34
3 -stage l-stage 2-stage 3-stage l-stage 2-stage
0.85 0.78 1.10 1.26 0.70 0.99
i. 13 0.76 1.10 1.74 0.58 0.85
-0.23 -0.22 0.11 0.12 -0.26 0.03
3-sta_e
1.08
1.33
0.05
16. Optimization
summary for 1.2 and 3-stage turbine with response surface in refined design (All output parameters are normalized by the baseline values)
l-stage Apay
(t=l,
2-stage 3-sta_e 1-stage 2-stage 3-staoe l-stage 2-stage 3-sta_e l-stage
s=O)
It=O, s= l )
(t= 1. s= 1 )
Table
2-sta_e
17.The quality
Apayovt
floor
Wovt
Apayov,
0.77 1.13 1.20 0.40 0/05 0.82 0.78 1.15 1.24
0.73 1.04 1.54 0.41 0.65 1.13 0.76 1.10 1.75
-0.21 0.15 0.11 -0.6 l -0.35 -0.27 -0.22 0.15 0.09
0.70
0.58
-0.26
1.00
0.85
0.04
of the Second-Order Response Surface obtained tor r/, W and Apay of 2-Stage 1990-data tFCCD criterion) and 249-data (OA crtteriom (Mean
1990-data
249-data
values of r/, W and Apay are normalized
by the baseline
values)
RRa: rms- error Mean
r1 0.995 0.994 1.31% 0.78
W 0.996 0.996 2.56% 0.86
Apay 0.995 0.995 9.58% -0.24
Rz Ra 2 rms- error Mean
0.995 0.992 2.128% 0.89
0.998 0.997 0.826%) 0.92
0.994 0.992 20.68% -0.11
.\merican
Institute
13 of Aeronautics
and Astronautics
space
Turbine
for
AIAA-2000-3242 Table18.Testing oftheSecond-Order Response Surface obtained forrl andWof2-Stage Turbinefor1990-data (FCCDcriterion) and249-data (OAcriterion)with78-testdata # of design
points
# of test data
rms-error
for rl (%)
rms-error
for W(%)
249
78
1.65
0.96
1990
78
1.67
1.21
Table 19. Optimization based on Apay and composite desirability function of q and W for 2-stage turbine original design space with 249-data (OA criterion). (All geometric design variables and output parameters normalized by the baseline values)
Wopt
for are
I /xpayoot
1.04
[
0.]3
1,02
I
2.76
0.83
I
0.00
Meanline Error %of mean RSM(t=O,
s= 1)
Meanline _ RSM(t=I. Meanline
s=l)_ I 0.96
76
0._21
1.i3
0.5
0.0
0.9
Error '5 of mean_ Table 20. Optimization based on Apay and composite desirability function of q and W t'or 2-stage turbine tot refined design space for 249-data (OA criterion). (All geometric design variables and output parameters are normalized by the baseline values) D rlopt I Wopt RSM I@ay) t 1.13 Meanline ' 1.12 Error % of mean RSM(t= 1, s=O) Meanline
0.63
I I
1.02 1.02 0.01
-0.38
RSM(t=O,
s= 1)
1.10
RSM(t= 1, s= 1) Meanline Error %of mean
0.15 0.37
0.64
t0"00311 0.0002
Error %of mean
j
1.03
0.16
-0.38
mean
Meanline
!
0.64
Error _of
c,2c,,ic,2 I
RPM
[Apayopt 1.10
t.50
1.05
I
1.57
0.53
1.27
0.70
0.90
0.391
0.26
0.931
t.30 i
1.50
1.57
0.70 t
0.0002 [
0.15
1.14
1.10
0.15
0.005 0.97
0.001 0.84
[ 0.029 0.02
0.98 0.08
0.84 0.05
0.02 6.08
I
t 09o t 039
American
institute
14 of Aeronautics
and Astronautics
_t ]
0.85
0.51
o,o,1o,
AIAA-2000-3242 Selecti_h 6_ 1 Desig_,..._J _-_
I Normalization -if required
Function Approxtmatmn
1-',,
Cross-Validation (ModelTesting)
] ]
Optimization
Figure
1. Function
Approximation
and Optimization
A
i
Flow Chart
A
'
_]" :_ .T.o I 1 .,-.}..,"-t'-" :'::
/
o"1
:,C,
Figure
2. Face Centered
!
I
_!_
('t_mpos.te
I
X_
Designs
(FCCD)
for 3 Design
o,
American
15 Institute of Aeronautics
and Astronautics
Variables
AIAA-2000-3242
E]fect ot # of Turbine
Stages
( normalized
1.3
for Optimum E_ficiency
by the baseline
Optimum Diameter
value)
( normalized
1.20
1.2
i
I
I i
,4,--
1.1
by the baseline value)
1.15
1
•s
1.0
I /'
WO.9
S
,m 1.10
I i
'
r •
I
/
0.8
•s
t 05
"3,
0.7 0.6
1,00 0
1
2 # ol Stage
EYfect of # ol Turbine ( normalized
3
Stagesfor
4
1
2 # of Stage
OptimumWeignt
by the baseline
Optimum RPM
value)
1.7
1 20 ,
.4'
15
3
( normalized by the baseline valuel - .......................................
115
/ / / s
13
110
/ / /
E
:s a_ 1.05 rr
/"
s
s
100
09
I i
/
/
0 95
II /
07
0.90
05 0
1
2 # of Stage
E_fect of _ of Turbine Payload Q_
Qnange
3
0
4
1
Optimum AnnuLus Area
Stages for Optimum Incremental
( normalized
by the baseline
2 # of Stage
normalized
vaiuet
by the baseline
value)
1 20
.................................
_15 :5 01
/
110
-_.%,
I
_
/
o,o i
_
0
>.
.......
s
1
t 05
,,
.
_
C
2
3
4
.1,00
i I
r_
T4-O 1 "d
x x
-%
095
/ /I
0 9O
/ //
_- -0 2 :?
:3 85 0.80
-0.3
0
Figure
3. [:ftect
of the number output
(A[I geometric
o[ turbine
parameters: design
1
:
# of Stage
stage
on optimum
t t, tL', :md Apay
variables
and output
design
calculated parameters
parameters;
tk)r Apay-based are normalized
16 American
Institute
o[ Aeronautics
and
Astronautics
2 # o! Stage
D, RPM.
3
and A.,,,,, and optimum
optimization by the baseline
values)
4
AIAA-2000-3242 Bfect of # of Turbine Stages for Optimum Bficiency 0.9
i
i
( normalized by the baseline value)
Optimum Diameter ( normalized by the baseline value)
0.600
0.8 0.7 0.6
m
0.550
I s
J
i i
0.5
I i
J
J
i
i o 0.500
i
i
ff
0.4
0.450
0.3 i
0.2
0.400 0
1
# of 2Stage
1
2
3
# of
IRfect o! # of Turbine Stagesfor Optimum Weight
Optimum ( normalized
( normalized by the baseline value)
by
3
4 ;
3
4
3
4
Stage
RPM the
baseline
value)
1 3O
1.2
#,
11
1 29
i
1.0
i i i I
0.9
1 28
I i I 0,_
i
___ 0.8
i i
;_0.7
127 0.6
1 126
I
0,5 I i I1"
04
I 25
03
o
1
2 # of Stage
3
0
2 = Of
Ootimum
E_fect of n of Turbine Stages lor Oplimum Pay_oacl ( normalized by the baseline vatue) 1
i
2
( normalized 071
3
Stage
Annulus bY
lne
Area baselme
value}
...................
4
-02 070
-_ -03 _ -04
• ........
->,,- .......
-_
/
G..
./ /
-,'_, 5
/
/ I / /
-C.6
/
II /
0 69
-0.7
0 # of Stage
Figure
4. Effect
of the
number
of turbine
stage
output
parameters:
on optimum
geometric
design
variables
design
parameters;
tT. W, and Apay
for Weight-based iAIl
1
2 _ of Stage
i
optimization
and output
parameters
(t=/.
Institute
of Aeronautics
and A ..... and optimum
s=d)
are normalized
17 American
D, RPM.
calculated
and Astronautics
by the baseline
values)
AIAA-2000-3242
E]fect of # of Turbine Stages 1,3
( normalized
Optimum Diameter
for Optimum Efficiency
by the baseline
value) .;7
( normalized
1.30
by the baseline
value)
"1
1.2 1.25
1.1 i /
o
i I t_
1.0
7"
!
I /
0.9
1.20
7" s
0.8
II' 1.15
0.7
1 0
1
# of _tage
3
2 # of Stage
4
3
4
Optirru m RPM ETfect of # of Turbine Stagesfor ( normalized
]
!
Optimum Weight
by the baseline
( normalized
value)
18
by the baseline
value)
# 094
16 / / //
14 /
Z
>
//
I I
/
O..
/
/
n"
/
t.2
0.92
/
/
1.0
f
0.8
m" 0.90
0.6 0
1
_fect
2 # of Stage
of # of Turbine Stages ( normalized
020
3
for Optirrum
by the baseline
o
4
1
2 # of Stage
Optimum Annulus
Payload
value)
( normalized .....................................
135
3
4
3
4
Area
by the baseline
value)
015 010 1.30
/
0.05 #1
-_ 0.00
........
/
1
90.05
.....
2
/I
3
i I
1.25
(2/I
-010
/ I
-0.15
/ /t
-0.20
li t
120 0
-025 # of Stage
1
_
2 # of Stage
J
Figure
5. Effect of the number
(All geometric
design
of turbine stage on optimum design parameters; _+utput parameters; r/, W. and Apay calculated Ibr q-based optimization (t=O, s= I) variables and output parameters are normalized
t+-
American
Institute
18 of Aeronautics
and Astronautics
D, RPM, and A ......and optimum
by the baseline
values)
AIAA-2000-3242
cz-plot forApay
Response Surface tbr 2-Stage (normalized by the baseline value )
Turbine
(t= 1,s=0) (t=l,s=l)
(t=0.s= 1 )
Dpay _
•
0.118547 0.108876 0.100127 0.0825071 0.0646329 0.0441748 0.00650685 -0.0415149
i_-0.0755047 -0.112671 -0.167669
-0279187 -0.224047 -03322
(t= 1,s=O)
Figure 6.6z-Plot tor Apay normalized by the baseline values ) Response Surface for 2-Stage Turbine based on composite desirability function optimization (Effect ot values of t and s on optimum Apay )
Amer|can
Institute
_9 of Aeronautics
and Astronauttcs
AIAA-2000-3242
Optization
Based on Payload Increment for Original Design Space E}----t
1 O 0.8 +
,..- 0.6
[]
,.L.'
[]
._c 0.4 O a
0.2 []
¢ 0 _3 :_ -0.2
2
4
3
rq
5
6
7
9
10
1!1
_-0.4 _ -0.6 0
z
-0.8
O
Design Variables
ia) Ori,_,inal
Optization 1.0
0.6
Space
Based on Payload Increment for Refined Design Space ..........
--'
;
0.8 4-
Desi,,n
(DV)
O
,
[]
O
40
_
"7 O4 _ i, O O
:::
O oOA
o.o 2
> -0.2
........ 4 5
s
6
7
_
Z3
_o
FCCD
O
-0,4 E -0.6 ,
© Z
9
; _
0
-0.8 -1.0 Design Variables
i b)Refined
Figure
7. Comparison data
(DV#1:
of the IDc,_ign Variables
¢FCCD)
D, DV#2:
RPM,
and 249-data DV#3:
IOA)
Amn DV#4:
Design
(DV)
Space
for Optimization I__)rboth
Original
h_, DV#5:
bJ.scd Design
c,.i, DV#6:
sr,_, and DV#11:
on Payload Space
and
c,.2, DV#7:
wn)
q,
20 ,-\mencan
Institute
of Aeronautics
and Astronautics
Increment Refined
Cbl, DV#8:
Design
(Apay_
u,,,ing
1990-
Space
Cb2, DV#9:
sq,
DV#10:
