may 1983 perfect bell nozzle parametric and optimization
TRANSCRIPT
NASAReferencePublication1104
May 1983
Perfect Bell NozzleParametric andOptimization Curves
J. L. Tuttleand D. H. Blount
|¥
{NASA-F2-1 I0_) PFPFFC_ BI_I[ $C77I_EA_AMF_IC AND OPTI_7_TIC_ CEF_. _ (N,_S_)32 p HC AC3/MF AOI CSC[ 21H
H II_,0
25lh _r¥1958-1983
NASAReferencePublication1104
1983
NASANational Aeronauticsand Space Administration
Scientific and TechnicalInformation Branch
Perfect Bell NozzleParametric andOptimization Curves
J. L. Tuttleand D. H Blount
George C. Marshall Space Flight CenterMarshall Space Flight Center, Alabama
TABLE OF CONTENTS
INTRODUCTION .................................................................
NOZZLE DESIGN• ° ° - ............................................................
NOZZLE DATA........... ° .....................................................
NOZZLE OPTIMIZATION .............
CONCLUSIONS ..................................................................
REFERENCES° ..................................................................
Page
1
1
2
2
3
26
|¥
pJ_CEDING PAGE BLANK NOT FILMED
iii
Figure
A.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
LIST OF I LLUSTRATIONS
Title Page
Illustration of optimum nozzles for given constraints ................................ 6
Perfect nozzle shapes ........................................................ 7
L/D t versus A D ( 10 < A D < 180) .............................................. 8
L/D t versus A D (100 <_ A D < 2000) ............................................. 9
L/Dt versus A D (1000 < AD < 6100) .......................................... 10
As/A t versusA D (10<A D<150) ............................................. 11
As/A t versus A D ( 1O0 < A D < 3000) ........................................... 12
As/At v_rsus AD (1000 < AD < 6100) .......................................... 13
THETA versus Ac/A t ........................................................ 14
Ae/A t versus Pe/Pc ( 10 < A D < 80) ............................................. 15
Ae/A t versus Pe/Pc (80 < A D < 6 i 00) ........................................... 16
ETA nversusA D(IO<A D< 180) ............................................. 17
ETAn versus AD (100 < AD < 1480) ............................................ 18
ETA n versus A D ( 1000 < A D < 6100) ........................................... 19
A D versus Ae/A t for maximum ETA n ........................................... 20
L/D t versus Ae/A t for maximum E'FA n .......................................... 21
A D versus Ae/A t for minimum As/A t nozzles ...................................... 22
L/D t versus Ae/A t for minimum As/A t nozzles ..................................... 23
A D versus Ae/A t for minimum L/D t nozzles ....................................... 24
L/D t versus Ae/A t for minimum L/D t nozzles ..................................... 25
|V'
iv
AD
Ae/A t,e
As/A t
Y/R t
L/R t
L/D t
Pe/Pc
ETA n
7
Cfg
Cfn
Cfi
THETA
DEFINITION OF SYMBOLS
Ratio of design exit area to throat area
Ratio of exit area to throat area
Ratio of contour surface area to throat area
Ratio of radial distance from centerline to throat radius
Ratio of axial distance from throat to throat radius
Ratio of axial distance from throat to throat diameter
Ratio of exit pressure to chamber (total) pressure
Nozzle efficiency
Gas specific heat ratio, gamma
Two-dimensional thrust coefficient without friction
Two-dimensional thrust coefficient with friction
One-dimensional isentropic thrust coefficient
Angle between contour and centerline (deg)
|¥
V "!
REFERENCE PUBLICATION
PERFECT BELL NOZZLE PARAMETRIC A, PTIMIZATION CURVES
!¥
INTRODUCTION
Parametric nozzle performance data for bell nozzles are often required in tile analysis and tradestudies of rocket engines. In order to get accurate comparisons, it is very important to have a consistent
set of data which covers the entire range of area ratios and nozzle lengths allowable. Although these
data are available (in fragmented form) over a fairly wide range of area ratios, new high perfornlance/high pressure space engine applications require these data at area ratios well above those considered in
the past. This document presents design data for untruncated nozzle expansion area ratios from 10 to
6100 for a specific heat ratio, (_,) of 1.2 and includes curves which permit the optimization of nozzlesfor maximum thrust coefficient within a given length, surface area or area ratio.
NOZZLE DESIGN
The computer code documented in Reference 1 was used to perform the calculations for this
report. The nozzle contours are constructed using tile "method of characteristics" which is detailed in
many texts such as Reference 2 and are of the wind tunnel or "perfect" nozzle type. These perfectbell nozzles are two-dimensional axisymmetric (i.e., all properties can be specified by two dimensions:
axial distance downstream from the throat and radial distance from the centerline). They have the
imposed constraint that the flow field velocity vectors at the nozzle exit (untruncated design area ratio,
A D) be of uniform magnitude and parallel to the centerline. Three thrust coefficients are calculated by
tile code for vacuum external conditions. They are the gross thrust coefficient, Cfg, which is found by
integrating the momentum and pressure force across the throat and then adding tile thrust due to
pressure on the nozzle wall; the net thrust coefficient, Cfn, which is found by calculating the wall shear
force (drag) caused by friction and subtracting it from the gross thrust; and the ideal thrust coefficient,
Cfi, which is found by evaluating the one-dimensional isentropic flow equations at the given area ratio.
A nozzle efficiency factor ETA n , is defined as Cfn/Cfi and is used to convert ideal thrust coefficients
(and thereby one-dimensional isentropic specific impulse) to the actual two-dimensional axisymmetriccase with friction. Since the wall friction is only applied to find the net thrust coefficient after the
frictionless nozzle contour has been determined, it does not affect the contour or surface area.
A description of the input required by the nozzle design code is shown in Table I. The actual
input values used to generate the data for the curves are shown in Table 2. Slight changes in the numberof points on the centerline and exit radius were necessary to successfully execute the code for different
ranges of the design area ratio, A D. A corner expansion was chosen since it yields shorter nozzles than
ones with a radius downstream of the throat for any given area ratio. Although the friction factor would
depend on wall smoothness and the Reynolds number which is a function of combustion product density,nozzle size, and the viscosity of the combustion products, a value of 0.003 Fanning (0.012 Darcy-
Weisbach) was chosen as representative. Large, high pressure nozzles would have a little lower and small,low pressure nozzles a little higher friction factor.
NOZZLE DATA
The nozzle contours produced by the design procedures above are shown in Figure 1 where the
radial distance from the centerline, Y, and the distance downstream from the throat, L, are normalized
by the throat radius, R t. The untruncated design area ratio, A D, represents the area ratio where uniform,
parallel flow has been achieved for that contour. Since these nozzles are very long and derive very little
thrust from the nearly parallel walls near their exit, the nozzles are nearly always truncated to a smaller
area ratio as discussed later. The contours for AD'S of 600 and larger are of course not shown to
completion in this figure and the somewhat erratic values are the results of convergence tolerance as the
code iterated to obtain the desired input values. It should be noted that only the A D need be specified
to define a nozzle contour and only two of the three variables Y/R t (which is x/7-), L/R t and A D need
to be specified to fully define a truncated nozzle. To enable a comparison of the performance of the
different contours, lines of constant net thrust coefficient, Cfn, (which includes friction) and wall surface
area normalized to throat area, As/A t, are superimposed. Although Figure 1 gives a good visualization
of the nozzle contours and performance comparison it is sometimes difficult to accurately read data fromit.
Figures 2 through 8 give improved resolution and interpreting capability. Figures 9 and 10present the internal wall pressure normalized by chamber pressure (i.e., total pressure since the calcula-
tions are for isentropic flow). This is the pressure that is integrated along the wall to find the wall thrust
contribution. The important nozzle efficiency, Cfn/Cfi, which was discussed above in the nozzle design
section is presented in Figures 11 through 13. Note that the efficiency has a maximum value for each
area ratio. Since the one-dimensional, frictionless, ideal thTust coefficient is constant for a given area
ratio, there is an A D contour which produces a maximum thrust for a given area ratio. In other words,
a set of nozzles exists which produce a maximum thrust for a given area ratio. This is covered furtherin the next section.
|
NOZZLE OPTIMIZATION
An Optimization Process is used to determine where to truncate the full nozzle contour to obtain
maximum thrust (i.e., max Cfn) within a given constraint such as nozzle area ratio (or diameter), surface
area, or length. This optimization is best visualized by considering an enlarged section of Figure I shown
in Figure A. Point "a" on the figure, which is the point where the net thrust coefficient, Cfn, line is
tangent to a line of constant Y/R t (or area ratio, E), is the optimization representing the exit point of a
nozzle contour yielding maximum thrust for a given area ratio or diameter. This is the optimization
mentioned in the preceding section and as discussed there is also the point of maxinmm nozzle efficiency
as seen in Figures I 1 through 13. These points have been replotted on Figures 14 and 15 to enable
nozzles of this set to be easily defined. Similarly. point b, tile point at which the Cfn line is tangent to
the constant surface area line, As/A t, represents the optimization of maximum thrust for a given surface
area. This set is given in Figures 16 and 17. Nozzles of maximum thrust for a given length are repre-
sented by point C and curves for this set are shown in Figures 18 and 19. These minimum length
optimized nozzle will be of a slightly different contour than the Rao minimum length nozzles [31
because the Rao nozzles do not have the added constraint which requires uniform parallel flow at the
2
untruncateddesignarea ratio. The actual difference in performance at a given length will be very small
between the two design methods, and the family of perfect bell nozzles has the advantage of including
data for the other optimized sets of minimum surface area and maximum efficiency for a given area
ratio as well as any other nozzles up to the untruncated wind tunnel nozzle. Point "d" on Figure A
represents the most thrust obtainable from any given nozzle contour, A D. When the nozzle contour extends
beyond this point, wall friction forces become greater than the pressure forces netting a negative thrust
contribt:tion which decreases the total thrust. This set of nozzles is not of practical interest, however,
because the set of nozzles represented by point "a" are both shorter and have smaller diameters for thesame thrust.
CONCLUSIONS
Use of the optimization curves allows selection of nozzle contnurs with certain desired charac-
teristics and comparison of nozzles of one optimum set to others. Minimum length perfect Bell nozzles
are similar in contour and performance to the minimum length nozzles designed by the Rao optimizationmethod.
Variable
FCON
R
XNSL
CYLIlT
XNI
XNE
DLX
ATOL
PITOL2
FM
CF!
DRAG1
TOLl
PA
TABLEl.
ORIGINAL PAGE
OF POOR QUALITY
BELL NOZZLE DESIGN PROGRAM INPUT DESCRIPTION
PERFECT GAS MODEL OPTION
GAMMA = 1.2
Default Value Description
1.0
0.0
0.0
0.01
20.0
60.0
0.1 _
0.0005
0.005
1.005
0.003
0.011
0.00001
0.0
FCON = 1.0 for two-dimensional
axisymmetric flow
FCON = 0.0 for two-dimensional plane flow
Normalized radius of circle ( R/R t) forcircular expansion
R = 0.0 for corner expansion
Number of streamlines to be calculated
Ratio of cylinder radius to throat radius
Number of points on starting line
Number of points on exit Mach line
Maximum distance between adjacent Math
lines along the cylinder normalized by
throat radius (AX/Tt)
Iteration tolerance on design area ratio
Maximum increment of tan 0 for corner or
circular expansion
Starting line Mach number for uniform flow
Incompressible coefficient of skin friction
along nozzle contour
Thrust loss at the throat
Iteration tolerance for interior point
Ambient pressurc ratio (Pa/Pc)
4
=J<>
m.qZ
>.,=_
<Z<Z©
<N
m.©
Z<
<
<m.,
:J
NNOZ
mm
U.,
c_m.,
...Imm<
o
,<
c_ c_f_
r'-- r'--
I d
(_NAL PAQ£ I_POOR QUALrI'_
C-'I_,_"" '_' "" ;_',__'".;_l_. J_ .'.... .'i'
OF PC'_2 .... "_'"
o
0
E
E
0
0
0
[./.,
/
\
6
_-r........7H---t
...... i .....
' I
I t _
-h-_ -k._..
t
t,,ri
L_.k-....
• Ii..... -L-
+...... +.__
i
....1--.-I
r.... t--'--
k-....
-j: ', i
: I .
] ........:................t..............]-TI .......r--/--1-----_-f -t-- --" -t-, _T- --k-.... :_-t- _i !_,#_' _i!_i__
d'_ ....- ................. -- ......... .-..
i-"
Iu-
r _ i i i "
.I8.IU.
+ .....
....... . ,.f-
t__
..__
' !iiji!.... i,.t;1, .... , t/,---
L__]ii__ 7T1771T.......
O.0
tl
t, !
" i
_2;2
i ........ j......... _+......." 7
-TT-', _ .......
i:-_ ':
.....- ti!iIi!li! i!!!•t- -.-
_i]]i ....i........i"
:i "i i
.................¢!iiii/ll)!iii)
0
6_
|
! I
1
I
r
, 4,, t'''
ORIGIN,O,L P_L_; r-;
OF POOR QUALI'i'Y
I " I , • 1 ' "1' =I
, i
.,,_ _, __ ;_ t_ .J,._. _L
o R =oo o I _1
Ii t _ v jl Ill : '
i i rl "=$tT T i i- _ ---
_' i, I i_L._ {: L ' I_4lI1_T'7.' i .. t .tt ti;li _i!71
i .If,- _ l.j, /l;i: ll+jil:l! _ _m!"r_ J l'"i::li/17:/t:_," i ; -- :,.#'. I. ilI
_ ii
I ; i i i il j !JI/i:::!:lII_..... , ,., : : ilf..i..:lli
,,, . I,i i I:1_1/ r17
i [ '
' :,t t i !ti.l:iiii.Jlnrl:l<, _',," i] I 1.:i1 ii I _:_!!l.;.l: 7i
t
!
i
C
<
,-.1
LT.
I¥
IIIII I Ill
OR/GINAL pp_ ,._OF POOR QUALITY
<ra_
r_
|V'
_GI79
I0
|
ORIGir_AL P/_G_ ,rgOF POOR QUALITY.
i I
I
I
I
t
' ! ! ......... | _TI_1----i_--l_l - - 1- .... , --_
:: I .........
-" .... T-,..........r......................-_"E'ff_- I' ....:4-,
........ -l-----t----k-............i.......!.......!-........:................._.................i......._........_................+........._............................. ' ' " i I '' I''' ! ' ' lt_ "_ ........
.... ,..,,1-_._...........I-..............-I _-_.,-!-.................................I _' ' :t........, ._--!, ,:_......-,
! W,,ts_ k l 1_II, : [L, i!!.... t-1 ..............T-..... 1 i :.............i........+,.......:.....1........I
, , r i I i i ' : i .... ;.I
,,,,,,<
.<-<
,,,4
L:_
!'V'
II
lifted
ORIGINAL p_i_. ;_j
OF POOR QUALITY1 .... i , -r--- V r......
• I i i i .....
: -- _ _- -- -+. -0_0 0 0 0: : ! _ _ i i : _ _ _ 0 0 0 O0 _ ........ "-
i : : l:.:..i z:i:i i :. i........ llllllllll.ll;lll_, _
• I ' : " I_ _ ' " -"-_
' ' ' --4I : ,, i _z, , , ,: fill _r--
.... i <r.O i I ....... " .... _......
:I _ _ ,_i, :' i _ ! i: . -r ...
, Ii.....I _ : ! ,...._t_ ..............-_..............: _._ _. . ........,_.... ..... 7 _71
, 17 ,i :i..... ii ' i . : i ' i i -_T,--:-._-_-_[--_7--li-,-, .........i ._ ..,_ ...,_,, .I; /i # 1_ ----+-_
'; !: i .... --,--4-" IIT-T:TT-,T-I: .............._-7
!: :, " i: I. l -'-"'-'1 !! i-L./L_ 1__--'-__._____-._-_I .... :: ! . , _,,,,,'"'I . -# t #1 I _ i ,_... , ', '. _i,. _ F; ' _ - ...... _-----:-----÷-_t .....I.-#-4--_+- .-
. . ,i " !" :' :_...: :....... ,i _.___........I!_.!. •i
L_,_"" ,-'. i! .... ,. i i .. ....i. .....
i
i
" i ' :, ....
:_! !c_ ' _c_-'i_7•__._,o_--,_-....,,__
_'i' l' _'' _ _----._----4-----4
<
w¢.)>
-....,..
<
,,6
-I
l,,i.
tl,/
12
ORIGINAL PAGE J_OF POOR QUALITY
|¥
OF POOR Q'JAL,TY
0 0
: . : : : : :
0
0000
OOO
00
0
0
.<
<:
< >\ <
L.
11f
VJ.3H.L
14
OOgum
llllllll11111111NIIiltlNIIiI!I
....Ill!Itlilllli*I I!IIHI'
P.L:::',ttlt I I t
+,l _t,,I
:ii!
:::" tIH'.
2:[ .L.
X:i
!Tii11ii
+.L,,
,i.....! E
i):
.....22
1* ii _ t i I
1._, .lo _.°
fit:k
ORIGI,'IA_ Fi'-,_ i.;';OF POOR QUALITY
)1'1,Nil.... dHJ
)i_: ,,:j
ii:i !_iiI
!iI: !iiili*i
;;i;
41 ;,;;
211
iiii_ii!̧_'iill-
i_I
xi
]'iI
I "
!:iT !
:i L/.
.t. !illI-1-:I I"I5t--t ! :
:i; :P-':
t: .....
.÷ ..
•..***
4.*
!l!lllliJ IIf lIl Itllt)l!t ! ""!)!II!i_ !11 )illii!!i!i"_i'.it !ii ii! i))_.
2',: i',_ ii_ ;, ! i ;;:ti'. _ ', :','.:',i ::l ;II ; ', _ ', ;;iii_ _ [::_: ,;: t:l_. !,. : : _:: ',:: :P"t _:: l:' : : ' ', ,'',;t; ; ', 'i:;: it; i;i i:I'ti:i:t::::
,i::t.:_ :t; .,::I::;::;;::",l ,,, ;ii , ; ; ; l+ ........
.... '!: ,;., ......
,i,!_N_ I_:::
if! _,
i! :]i*_:i:,ii!;_:!I ' -"i:,;I;i! !!L!l! : l
:i: i!!!l "
I!iiiii;:?iilI?:i; tl, :
t!!!i!!!!!Ii--; : I!!":---L--2--%-
_; :: 2LLIL:.2 ' --S 7 I _ ..... --
iSii::::_!i!ii!I::7-i:,:.ii_:i-'.:::::i! :--!:L! 7TT-_T
!iTT::"":i':! !-_'"")::::::: :--
..........:[:-'::t! !I _i -
..........::::::',::1
..........!i!i If::! ",';:'::H:: 0
IV/oy
UC1.\
CI..>.
<
<
Li.
15
!1t'
Nil/iNn mmmmm mm m nmmmmmm mmmm
OF POOR QUALII'Y
¥
ii!i!!
0
0000
000
00
_V/eV
o
16 .q
• ii
ORK$1NAL PP.C_. _
OF POOR QUALflY
<
P.
>
<k-,
,...;
I..,,,t
L,I.
17
|
I /lI II I I
18
ORIGINAL PA3_ [_l
OF POOR QUALITY
<el)
¢,1
<
Lu
|
it
n m m wm m mm mm mm mmmmmm m m',-" m/ mlmmm
-- .J
_r'_.l I _T
--+ ......_ ........_-........+......... ! i .........I.......+ ........i........i ..............i.......i ............
' ' .-.i I........I I ! I I..... t
•.............. J
! .......I
i _ I, 0_ '--_I _ i i ....'
I.,,_.
_._. i i . i.......!rl
i i---_ ....._.... : : +: . . ......_............ <
i ; i 1 , _ i : '<_,. ,.......7......__.......!.......i.....i..... _ ....I ! i i ! i
I i i i . ! I :
_"---| ....... ..... *,I....... _i..... _I....... _--"-----_i !........_-,.........,-........... ,IT.... -i-i Z+........
,--t: ........"I...............i......".....i.......i-......i---_......_.......... ,-....+-C9I ',I _ i i ! i j ! i _
-I......!......f..........+-......!.......+.............+.......+............'-.......' ' I [ _ I i ! ' i !
I-1 , i ,_ I , :
: i i ! _ j I-.
.....i-........I...................:.............._..................•.....lZ
.....-I-----_---+........+.......!.......................-+........_,.---_i i i : _ ! !
.................-_..........._........._.............._..........._.-'i.•+-
i i ! i !
, I i ! i _ ! i , L<o---_ I---+---i---l---i----!.....+-------+......+.....i--..---,-........+-.......+-........I-......_.........i.........
_-_--I--_--÷ ....I--+--+.....t---+......._......._......._......I.......:........._---_. I. il ! il ! i ! , _ i ! i _ i + i
.... _---4-.---_----l-.....I-......+.....* ......._.......,.. _........._........+........_........i .....I- ....L. ;......;t / / 1 i i i i ! ; i ', ; ! / _ ': !
......l,, l,,, i,, !, ,i, i ,,!,,, i,, !, i _, ;, i ;, !.__....+ .........
__, .............................. LL i. , : _....... _,_[ I t i ! .......i "" I :, i ........_ .....+-:.......i........!.....l_..._, : ......._-_......._,
' ....."..................... -i........f........i.......4-....+ +• _....._.............I -.--ii I i I i. _ i I i I :
<-
{.-
F,
19
!I,/
OriGINAL PAeE iS
OF POOR qUALT'-_
0000
-1" t/ I • t ! .................. I
I
'i
I o°
............. .:!.:: ..... ::'' l
"- - t-'" ............................
"- =====================::,:i::::;
..... " .....t....I,;_........
.......... , :11 ":: • ' :i .. ...... I: : il .... :
_ ............. :_ ;:: .... .. /:i:.,: .'. !!1 i: :
I................. ::t: ,.: .... : I:i'LF ' _'" "' ";I .... 1 ................ ::: : :::::::::::::::::::::::::' '
+ _........,l::::'::::x:::::_........... ...................... ,..: ...." ::_I;:;: ::::::. :: I..l .1:1:,1::::1:::l,. :......... :-...:_ ,.:: ..............
• I". I ;: ;i I ' : ":I':' ....... I : '
_" :;:;'::;:
_4.- • tTi't i'i't...............i r, _ t! t ,t .... tl t , , .
•- (:IV
<
e<
<
uJ
#._3
_o
<-{D
<
<
4
2O
00
_/7
91V'
21
h--d ....
22
IIII
,+:,'I
:X:I
N
i+,t
X:;I
;:I1
pu4
-,4
.... i
!i!
!!!:_._
.... i
;1Lgl
!Ill
r:.,t
t:+'l
++++3',:+'I
+++b+,l
0000
000
'"+"' Ill:IIIiIi i;++::ti+ x+
+I++ _M
:+If
lilt:I: _,_
++++ ++1
+++_ +i+
+::+t++++ili!:i:
++t+
C)¥
00
0r-
<-
E
-o B
\
_d
L--
I
00
.... ;;
!!!! :i:tl
:<i
0
_(]/7
ORICdN._,LPA,,2-_;_QUALITY 0
000
000
0
\
0=
<-
B
s.-
r_m
L_
23
If
ORtGIII_,I.. F_'Z _SoF poor QuALrrY
!
0000
iLii,il
i-i-,,.... 0....
: :,!i
,n ........ n,,,,,./,,I.... o....,.,o..,
:,.; h;
:1: i
2"'!11'
:i.,° ........ ,.,_,. ,.,,,.... i ....
,, tnllp
._-.4
-,-4!
.... i ....
,i ........ 0,.,
.... 0.,,
.... 0....,, ....
.... 0....,0,.,.... a-.
.... 0....
.... , ,
.... o....
T;!'._!I
.... i ....
.... F/
,,llr
i i ilJl
L,I!
i_iii
h_
!i!i
711
i<i
t_Pt
iv
:L
.ii,
,,h
i_i*'F*
i!ii iI ii_!! ,I,t_- _-,, f!t! fi '
"i!
_; ii!', i,
i _ !_ _
._ ,,,
I::,:I',iI'#II"r",.i.-.i ._._
',_
...,_
......::::_:1
........21
: i!i
.... I
<! :!!!
,o,q
......... ,, ,,,,q
' LL
.... i
l:i:
(IV
00
TI_ o.........-l._i!H-_P._I'--
z_:!lili!l,!'.i:',!il;;;; ; ;; : ;;; T
!i! i;,i : _iiiiii i
'''liP''' +*I ) _ I_ i 0
_ill [[[_
__![!!!!_![i::!i[[!
!i_i!!!' i;i!!!!ii!i
................ I0
"=TFF T-'TT_T_:_fi: :_i ; ;.i;_l
i
:_!:l:_i!! _:i!!!i_iI, , :';:':Iill
::ii]! i,'iiiii!iill
_,_ i I I I , t l
_ i n ' I , ' I ' .,11 t I I I t I
i ;'1',:11 ...... _'*'
I'!:11'. ! ! .1'.1!1! 1: _I ,_.
0pm
<C
<C
o
=_
<-
;>
<
kl.
24 "1
00
0
_/7
0000
00o
00
0
\e
<
©
-J
<-
<
..J
m
..a
25
|
o
.
°
REFERENCES
United Technologies, Pratt and Whitney Aircraft Group: Bell Nozzle Design Program. NASA Con-
tract No. NAS9-2487, June 30, 1964.
Shapiro, Ascher H." The Dynamics and Thermodynamics of Compressible Fluid Flow. New York,
The Ronald Press Company, 1953.
Rao, G. V. R." Exhaust Nozzle Contour for Optimum Thrust. Jet Propulsion, Vol. 28, No. 6,
June 1958, pp. 377-382.
| ,V'
26