UNCLASSIFIED AD NUMBER AD907326 NEW LIMITATION CHANGE TO Approved for public release, distribution unlimited FROM Distribution authorized to U.S. Gov't. agencies only; Administrative/Operational Use; NOV 1972. Other requests shall be referred to Advanced Ballistic Missile Defense Agency, Huntsville, AL 35807. AUTHORITY ABMDA/AL ltr, 1 May 1974 THIS PAGE IS UNCLASSIFIED
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UNCLASSIFIED AD NUMBER - apps.dtic.mil · ieI A 'Ale'JIJL SA~Tht''ir-N I Y Te "'Tht 4L ~' ~ ~ j. A ~ ' * ... lbm/in 3 Pt Density of throat -insert, lbm/in3 m Yield strength ofmotor
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UNCLASSIFIED
AD NUMBER
AD907326
NEW LIMITATION CHANGE
TOApproved for public release, distributionunlimited
FROMDistribution authorized to U.S. Gov't.agencies only; Administrative/OperationalUse; NOV 1972. Other requests shall bereferred to Advanced Ballistic MissileDefense Agency, Huntsville, AL 35807.
AUTHORITY
ABMDA/AL ltr, 1 May 1974
THIS PAGE IS UNCLASSIFIED
AD072 -. W~~f~l'9ieI 'Ale'JIJL A SA~Tht''ir
- Y N I
Te "'Tht 4L
~' ~ ~ j. A ~ ' *
SCOMPUTE~RcRg FOR tN
AND PE--RFO ,R M NdE ANAY..~SOID PROPELN RcTMOTT'OR
November 1972-1
FV I
B ROWN ENGIN EERING ,5,-a,
ATELEDYNE
BROWN ENGINEERINGRESEARCH PARK
HUNTSVILLE, ALABAMA 35807
-(205) 536-4455JWX (810) 726-2103
February 7, 1973Ditibution limited to U.S. Gov'IN agenctes onTn-
z A1luation; 1 2 FEB 1973. Other requestsf-Ll1s documont -must be refOrrold to
Director AecAdvanced Ballistic Missile Defense AgencyDepartment of the Army
P. 0. Box 1500Huntsville, Alabama 35807
Attention: RDMH-S/Mr. Carmichael
Subject: Contract DAHC60-69-C-0037
Dear Sir:
In accordance with Sequence No. G003 -of the DD Form 1423 for thesubject -contract, Teledyne Brown Engineering is submitting seven (7)copies of its Technical Report MS-ABM.DA-1683, "Computer Programfor Sizing and Performance Analysis of Solid Propellant Rocket Motors".This report was verbally approved. Distribution is being made per theDD Form 1423.
Distribution: DirectorAdvanced Ballistic Missile Defense AgencyDepartment of the ArmyCommonwealth- Building1320 Wilson BoulevardArlington, Virginia 22209Attention: RDMD-AO (1 copy)
Defense Documentation CenterCameron Station-Alexandria, Virginia 22314Attention: Documentation Center :(2 -copies)
Commanding GeneralU. -S. Army Safeguard System CommandDepartment of the ArmyP. -0- Box 1500Huntsville, Alabama 35807Attention: SSC-CS -(w/o copy)
Defense Contract Administration Services =OfficeFifth Floor, Clinton Building-210.9 West Clinton-Avenue
Huntsville, Alabama 35805Attention: Mr. D. W. VanBrunt (w/o copy)
f1I
f
, - -
TECHNICAL REPORTMS-ABMDA-1683
COMPUTER PROGRAM FOR SIZING-AND
PERFORMANCE ANALYSIS OF SOLIDPROPELLANT ROCKET MOTORS
L_ By
j J. L. Thurman
November 1-972
1Prepared For
-U.S. ARMY ADVANCED BALLISTIC MISSILE DEFENSE AGENCYDEPARTMENT OF THE ARMY
HUNTSVILLE, ALABAMA
FContract No. DAHC60-69-C-0037
fPrepared By
MILITARY SYSTEMSTELEDYNE BROWN ENGINEE RING
HUNTSVILLE, ALA-BAMA
i
ABSTRACT
IA FORTRAN digital computer program was developed which is
generally applicable for preliminary design tradeoff analyses of solid-
propellant rocket-motors for a variety of applications. The pro-
i gram computes rocket motor length, propellant and inert component
weights, mass fraction, burn-time, specific impulse, -ideal stage burn-
jout velocity, and -other performance parameters as a function of -input
motor diameter, chamber pressure, thrust, payload- mass, propellant
i ballistic properties, propellant- web fraction, port-to-throat area ratio,
and -other required-parameters. Either conical or contoured nozzle
I exit geometries -may be analyzed. Options are available for considering
a head-end propellant web and inert residual slivers.
I
SIo Approved: Approved:
G.R. Cuinn, Ph.D. -S. M. Gilbert, Ph.D.Manager ManagerMissile Requirements and System Synthesis DepartmentTechnology Branch
The weight of the forward head insulation is computed-from the-
following empirical relationship (Ref. 9):
Whi =1. 93-(100) Di
(3-29)-
0.7854 + 03925 .n PD-" tb CD'
where
3-13
3.2.3 Ignitor Boss
The weight of the -ignitor boss, W ib on the forward head is
estimated from the empirical expression (Ref. 9):
mWib 1. 13 PD D At-30)
f 3. 2.4 Ignitor
The weight of the ignitor, Wign, is estimated from the follow-
I ing empirical relationship which was derived through correlation ofreported ignitor Weights (Ref. 4)-for previously designed-and fabricated
11 rocket motors:
ign b]
Equation 3-31 was found to produce an excellentfit of reported igix,4-rLweight data for rocket motors with an average burning surface area
ranging between 35 00 and- 44000 in2z.
3.2.5 Cylindrical Case Structure
f The weight-of the cylindrical portion of the motor case strtic-
ture, Wcs-,- is computed from the relationship
Wc _ L D2- D (3-32)iCS 4 m 4M c
3-14
3.2.6 Cylindrical Case Liner
The weight of the liner adjacent to the cylindrical motor c ,
Wcl, -is determined from
W D L TI c P1 (3-33)
For the computer -program described herein, a typical liner density,
P -of 0.06 Ibm/ins is employed
3.2.7 AftHead Membrane
The weight--of the aft head membrane- structure, Wahs' is
estimated from the following empirical relationship (Ref. 9):
Ii
I Wahs 0.25?l -,Wahs -o , D - m + +"
0.3925 ____
X. XDM' [0O.7854 + In-1 2At13.z :(3-34)
} 13.2.8 Aft Head Insulation
The weight of the aft head insulation, Whi is scaled as 1. 54
times that of the previously computed forward head insulation as
+ recommended by Reference 9.
* f 3-15
3. 2. 9 Aft Head Nozzle Attachment
The weight of the aft head nozzle attachment boss, Wab, is
estimated using -the following relationship (Ref. 9):
W 10.z6 P D Z (3-35)wab D M (
3.2. 10 Nozzle
Either conical or contoured nozzle designs may be analyzed
using this -program. Conical nozzles of the type illustrated in Figure
3-2 are designed using -nondocumented procedures developed by
Mr. A. R. Maykut of the U.-S. Army Missile Command (MICOM) at
Redstone Arsenal The validity of these procedures -has been proven at
Redstone Arsenal through numerous design analyses and reproductions
of previous operationalnozzle designs. Contoured nozzle length and
weight, when applicable, are scaled from a corresponding design of
a conical nozzle with equivalent throat area and fixed exit cone half-
angle of 17.5 degrees using procedures described in Reference 10. A
more detailed description of the contoured- nozzle scaling procedure is
[i presented later.
Discussing first the conical -nozzle analysis, the entrance wall
thickness, Te, of the nozzle at the aft head attachment radius is compu-
ted fromI ro PDop D (3-36)
e 0o cos4n-
The parameter r usually is governed by the size of the aft head
] opening which is required for insertion and extraction of the propellant
mandrel. For this computer program, the nozzle entrance cone half-
angle, 4, is fixed at 60 degrees.
3-16
- ,
The thickness computed by Equation 3-36 is compared with a specified
minimum value which is dictated by fabrication constraints, and the
larger of the two values is used in further calculations.
The next step in the nozzle analysis is the determination of the
required insulation thickness at the nozzle entrance. The procedure
which is employed in sizing-the insulation thickness is described in
Reference 5. A thermally thin structural wall is assumed to be pro-
tected from the combustion- 'gases by a refractory insulation. The
temperature of the exposed insulation surface is assumed -to be equivalent
to that of the adjacent combustion gases, T c , and the temperature rise
of the protected structure during motor operation is assumed to be
i much smaller than T . The -predicted temperature rise ofithe struc-
tural wall during time t is computed from the -general expression
c. P. T.
AT T . Tf- (3-37)c P c
where 0 is--the time constant (_Ti 2 / 2 ai). The function f(t/e) is reported
in Reference 5 as
t -t 1 Z (-I)n + l
t + + (exp nz (3-38)r6 7r n=l n e
Equation 3-37 cannot be solved directly for Ti because of the implicit
relationship-of ri with the series function f(t/8). Therefore, an itera-
tive solution is accomplished -in which Equation 3-37 is solved- repeti-
tively to compute values of predicted temperature rise for progressively
improved estimates of 'ri. The least value of Ti for which the predicted
structural temperature rise is maintained below a specified limit of
10°R is selected for design application.
3-17
The first 20 terms are utilized in the series evaluation of f(t/0).
Ablation -of the insulation is not accounted for in the determination of
the required insulation- thickness. Therefore, the selected thickness
should be conservative. An approximate, but nonconservative, means
of accounting for ablation would be to substitute the insulation ablation
temperature for T in Equation 3-37.c
The weight of insulation in the nozzle entrance cone, Wei, is com-
puted from
W ( T o, r-rt T. . -(3 -39)i .COS + r ° lro - rwei sin o Le otr eop e
The weight of the nozzle entrance cone structural material, Wes, is
computed from-
r )-cos + -r p
Yes (Te 2(3-40)We sin op +(ie
The weight of the nozzle throat insulation is computed from the relation
Wti IT-p Te rt + Tie (cos=-+ cos a) + r a t](2ktrt) sintea (3-41)L(N)The thickness of the throat structural shell, Tts, is sized-from the
Sfollowing expression which accounts for the reductioninpressure load-
ing of the structural shell due- to the load carried by the throat insert
material:
I r D- Et ___
T -s r - (3-42)-ts rn sE t
n--s
; ) 3-18
The design pressure PD in Equation 3-42 should rigorously be based
upon a gas pressure whose value lies between the expected values at
the throat and in the combustion chamber, since the inner surface of
the throat insert is exposed to pressures in this range. However, for
design conservatism, the value of PD used in this program-is computed
as- 1.5 times the chamber pressure. The corresponding weight of the
throat structural shell, Wts, is determined from
[ro ~t o)]Wt - Tt r + T. (cosp+ cos-a)+ ra tste ra + (Cos b + -co sc
S2 k rt sin ((-43t t ( - 3
From -geometrical considerations, the weight of the -throat
insert, Wt, is established as
W irp kt--r [ -sin(+at t
-0.667k sin3 ) 1+a - 5sin (:+a) (3-44)
The thickness -of the exit -cone structure at the downstream end of thenozzle throat insert, Txs, is sized from
P rD es~X
T • (3-45)xs oCos a;n
The design pressure PDx is defined-as 1.5 times -the local pressure in
the nozzle exit cone. The structural thickness computed from Equa-
tion 3-45 is compared with a specified =minimum value which is dictated-
by fabrication- constraints,_ and the larger of the-two values is selected-
for design application. The corresponding weight of the exit cone
structure, Wxs, is computed from
: 3-19-
'77
w iT ct oxs_ s 3 n(8 (erat)F rat +(Ti +7
XS S) e XS
4 ~t+r.+TS~os£KIr+ rie co )+ r Ti+Tmn)
-(rat + Ti COS-az (rat +le- cs)-( +.e) -(re I(3
The-weight of the nozzle exit-cone insulation, W.,i is- determined from
W. ITeT ictnra + Te +re) rrt (3-47)
The-weight -of the attachment- -boss, VlI which is an.-integral part of
7'the nozzle structure,- -is estimated -from
W M= 1rr T -p- .(3 -48)Ynb op e-s
kFrom thje previously-computed- nozzle -component weights, the -total
nozzle weight, W -is- computed as the- sumn
W=W .+W- +W + W +W +W + W.+W -(3-49)-n ei es ti ts t xs- xI_ ib-
The-nozzle length, L ,for conical entrance and- exit geometries,
may be computed from
Entrance Throat Exit
L (r -rctn + r k(s in4 + sina +( - r ~Itn a . (5)n opot0 t t \re at)J
Equations 3-36 -through 3.-5O were derived for nozzles with
conical entrance and- exit geometries.
3-20-
3am
However, most modern-day rocket motors which are designed .or
flight operation employ nonconical contours which are optimized for
improved performance and reduced length. Because of the difficulty
associated with detailed design of a contoured nozzle, the simplified
empirical procedure described in Reference 10 for approximating
contoured nozzle weight and length from a corresponding conical
nozzle design was employed in this program,
To calculate contoured nozzle weight using t',e previously
described approximate method, the following assumptions are made:
* The weights of conical and contoured nozzles areidentical from the forward attachment boss to anexpansionr~atio downstream of the throat of 3.-0 for
nozzles of equal throat area. (Nozzle shape withinthis region is relatively independent of downstreamshape.)
0 The nozzle weights downstream of an exit expansionratio of 3A-are identical for contoured and conicalnozzles of equal surface area,
The two types of nozzles are assumed to have identical throat areas,
chamber presures, -burning durations, and propellant flame tempera-
ture. The family of contoured nozzles for which this procedure applies
is described by O. J. Demuth in Reference 11.
To determine the conical nozzle surface area which is equiva-
1 lent to that of a contoured nozzle having the same throat area, anequivalent conical nozzle expansion ratio, Ee , is derived from prei-
ously computed contoured nozzle data. This equivalent expansion
ratio is defined as -that of a conical nozzle with a 17. 5-degree exit
cone half-angle which -has the same surface area as a particular con-
toured nozzle of the same throat area. Contoured nozzle weights are
then computed from-the previously described conical nozzle equations
(Equations 3-36 through 3-50) when the applied value of the inside radius
{ t3-21
..........
of the throat exit plane, re, is based on the equivalent expansion ratio,
ce. and the input nozzle exit cone half-angle is taken as 17. 5 degrees.
The curve of Fe as a function of c (Ref. 10), which is incorporated
into this computer program, is presented in Figure 3-4. Figure 3-4
applies for a Demuth-type contoured nozzle with an initial divergence
half-angle (immediately downstream of the throat) of 32. 5 degrees
and an expansion ratio at Lhe point of parallel flow of 30. A review of
previously designed contoured nozzles (Ref. 4) revealed that this
particular exit shape is generally applicable for relatively large tactic-
cal rocket motors. Of course, flight nozzles are truncated at expan-
sion ratios slightly less than that required for parallel flow with little
performance penalty. Additionally, Figure 3-4 is based on a ratio of
combustion gas specific heats of -1. 2 and a ratio of throat radius of
curvature to throat radius of 1.2.
Direct calculation of the length of a contoured nozzle is again
quite difficult. To simplify the calculations in this program, contoured
nozzle length is related empirically to a 17.- 5-degree conical nozzle of
the same throat area and expansion ratio. In determining contoured
nozzle length, the length-from the -throat -to -the exit plane is first cal-
culated as for a 17. 5-degree conical nozzle. This length is then multi-
plied by an empirically derived factor, Lcont/Lcon, for the appropriate
contoured nozzle expansion ratio. The curve of Lcont/Lcon as a func-
tion of e (Ref. 10), which is employed in the computer program de-
scribed herein, is presented in Figure 3-5. The curve shown in
Figure 3-5 is for the same Demuth-type nozzle shape as discussed
previously for nozzle weight calculations. The corrected length between
the throat and the nozzle exit plane is then added to the entrance length,
between the forward attachment boss and the throat, to determine the
total contoured nozzle length.
3f, 3-22
40
36 _
32
28
24
oN 20
U
4 16C4J
c"
-V ~ ~ ~ ~ L 12 - ______
= 1.2
r /rt 1.24- 0 a = 32--.55*
c = 30 for Parallelflow-
32550 4812 16 20 :24 :28
Contoured-Nozzle Expansion Ratio, 0
FIGURE 3-4. EQUIVALENT CONICAL EXPASION RATIO AS A FUNCTIONj OF CONTOURED -NOZZLE EXPANSION RATIO
3-23
14 ~- ----
'101
0 _
00
.1.2---- C
J. 1Oo -___ 32.550_
c00 o
43.2
The -previously described procedure could be extended for
analyzing other Demuth-type exit shapes, with different values of
a. and 4 for parallel flow, by incorporating additional curves of feI/
and L /L as a function of 4 from Reference 10.cont/ -con
3.2.11 Motor Attachment Skirts
The weight of the forward attachment skirt, Wfs,- is estimated
from the following relationship recommended by Reference 9:g max W 021-(L C D/Pi PL m +1 (3-51)
W [0. Dcfsm- E DEc D_
The maximum longitudinal acceleration--of the stage under considera-
tion, gmax, is -assumed to- Qccur at stage burnout:
g = F/Wbo (3-52)
Now, at this stage of the analysis, W is unknown, since its valuebo -
depends upon-the undetermined weights of the attachment skirts and
interstage structure. However, these weights are usually relatively
small in comparison with the remaining inert component weights.
Therefore, an iterative solution is accomplished in which gm isax.
first estimated using a value of Wbo which includes only the previously
computed weights of the motor case, nozzle, ignitor, liner-, insula-
-tion, and the -tage payload. This value of gmaxi is utilized- in-
the calculation of the attachment skirt and interstage weights. The
value of g -is then recalculated using the estimated skirt and inter-1.
-stage weights included in W bo This iteration- is continued until the
3-25
computed values of Wbo (or gmaxi) on successive -iterations are
equivalent within a specified tolerance. At this point, the correct values
of the attachment skirt and interstage weights are considered to be
established.
The weight of the aft attachment skirt, Was, is estimated from-
/gmaxi 1 W °SWas 0. 0055 Dm nE
-(3 -53)
05 LC ++110)
Dm + "
In the -evaluation of Was, -the value of gmaxi.i1 must -be estimated from-
a separate analysis of stage i-l, and Wois estimated as part of the
previously described iteration on Wbo.
S3. 2. 12 InterstageStructure
The interstage structural weight, Wint, is estimated, when re-
quired:, from-the following relationship which is recommended by
i Referen,,e 9:Dm
Wint =0. 155 Dm + Lni+i + Lint)
maxi WPL _215 (LC + Dm+ 05(354)
Eint-D + 11
IThe evaluation of gmaxi and Wint is included in the previously described
-iteration on Wbo.
-- 3-26
3. 2. 13 Rocket Motor Length
Rocket motor length between the nozzle exit plane and the for-
ward head, Lm, is computed by summing the constituent lengths.
Lm= L h + Ic + Lah + Ln -(3-55)
The length of the forward head, Lfh, is determined from
14-h Dm/2P (3-56)
and the length of the aft head, Lah, is computed- from
Pm (12rop- 2 0.5
Lah ~ - Dm(3-57)i-
3.3 ROCKET MOTOR PERFORMANCECHARACTERIZATION
Sufficient preliminary design information has now-been computed
-from -the previously described analysis to -permit prediction of rocket
motor performance. After setting the sum of the-previously computed
-inert component weights equal to Wjp, the gross stage weight, Wo, is
computed from
Wo WIP + WpL+ p . (3-58
j. 3-27
- - -_ _ _ _ _ - -- ,- --- ' - - - - -. . . . "-9
Stage mass fraction, p., is defined as
Wp_ (3-59)IL Wo - WPL
Rocket motor mass fraction, X, is defined as
Wp. (3-60)Wo - Win t - WpL
The delivered specific impulse, Ispd, is computed from
sPd =Cf (3-61)
The predicted ideal velocity increment produced by the stage, neglect-
ing drag and gravity losses, is computed from
In Ebo (3-62)AV I e-Isp d c nWb-o
Thrust-to-weight ratios at ignition, Aign , and burnout, Abo, are com-
puted from Equations 3-63 and 3-64, respectively:
Aign F/W 0 (3-63)
Abo = F/Wbo . (3-64)
This completes the description of the rocket motor design
J analysis -as it is- programmed for the digital computer. The program
is formulated in such a manner that parametric tradeoff analyses of
{ pertinent design variables may be readily accomplished by stacking
input cases back-to-back. Moreover, an automatic variable adjusting
procedure, which would determine the required value of a specific
3-28
independent variables (e.-g., motor diameter, chamber pressure,
propellant web fraction, etc. ) to produce a desired motor performance
or design characteristic (e.g., mass fraction, motor length, or
velocity increment) may be easily incorporated into the program if
desired.
3U
i '. 3=29-
4. PROGRAM APPLICATION
4.1 METHODOLOGY
The computer program described herein may be utilized to
generate both point and optimized preliminary designs of solid propellant
rocket motors through the proper selection and manipulation of appropri-
ate input variables. Typical design problems and corresponding methods
for using the program to solve these problems are presented in Table 4-1.
The computer program contains no automated variable optimization
scheme. Therefore, the optimization- of pertinent design variables must
be accomplished using "brute force" techniques wherein graphs are
plotted of computed performance and/or weight characteristics which
result from manual perturbation of the variables of interest. Optimum
values of design variables -which yield either maximum performance or
minimum weight characteristics are then selected =from these graphs.
For example, Problem 1 of Table 4-1 Illustrates -the technique which
would- be employed to establish the optimum chamber pressure for a
minimum weight-design, assuming all other input variables are fixed.
j Problem- Za of Table 4-1 illustrates how the designer would
determine the optimum motor diameter and corresponding chamber
pressure which would be required to provide a given motor length.
Similarly, Problem Zb illustrates the technique which would -be used to
determine the optimum motor length and corresponding chamber pressure
which would yield a given motor diameter. The latter two examples are
frequently encountered in -the design of tactical and- strategic -missiles
which must be contained within fixed- envelope dimensions.
Problem 3 illustrates how the program would be employed to
evaluate the variation of stage impulsive velocity with motor diameter,
expansion ratio and payload weight.
44-1i
U_ .0
OCLc4. 4-b 41 4.1 41t2
5 s z:; a,= 4.L
4J- - i 4534 -a-4.a 4C2 ;I0 0.1 .u 0)a.~ .00 ~s.a.V 0454. LL -a 0- ' a S.- E
4- -. - S. a4j-2-
v5', 0 451.4n0. CO u - q on - u -- V1 -A. 0- U
-h j
coo xM
x- I- "
E w
-Z.
4. WCom 0 o_ _J. .'. .l 01 *-.
10 =-I zS'U 2,~a) F ) u,31-1'E 4'u Z-,- 1C- ' O -O ' 4% mL +- >'-I 'd
x,4 x -4 OJp CV - -L '0--Ia. ~ L c.-l- > .Cr d
4104 - .C - -a EW0.W - 2-_ (U- -.-2 >1 .. E >lS *- 0j I-I *.aID' - 4. 0~ W X'. 4seo- mn (V in C 4'S wc 's o ~ S4o L- .- n~n-W'' - 4EC ' u
t.J- -'04. ST '-'li ULA':~ ;2 E -' 0~ 0S. a- -v On 0 L4-A 4'~ 4L 0. S_ Rs- (DC n = ; 3 C_ '0,;--14-2 om 0 (1 --0 41 U~ s_ W40.C. _o.C LA
*.j 0 I-in Lu ~ 2Oi wE~' -0. 4j vAWW E,4-s. 4jich.-WS.-- 'i -W * V a-.-- dLiZ ~ ~ 4 4J5> Xs W6 4LjUC'l C60.5 ..- -u s_ (D - -,-4J-a.- 4UC Ua.-C O D Cn*-U C. e5 *U.. 5 2 -L.-. > = -a in- a) -'d u s. 4j 0o " 4)Cl 1 411us0 V - o in *. Q0 Ms~J' 4 m W -- a.WcouW 'u 0
E r . .- WWW0 C.- 2''u-in. 1, C .:C'. S E *.,u l0in in >L in' 4-. 1. -U. :C UA.CV i 4 ' - '4U 4 >4 3: 4' nn~i LA4UJ1 0Li 'a
uJ W-0 W . 0 5 s.-Cn --U WE a0iW. cN00..Z0 L0U . 10~.- 4-' Mi. -0- 39- O L.- J_. os>' 4-.,F . sri- Ci . - iv L.Oiu tA M_ e 45 C - 4O-civi o L0 ~ ~ ~ ~ I a.1 It>-1 a. a= L..-- >-0 a. 1 - *-,0,i a '0L 6 _ _ _o _ _ _ _ _-r c
4-': : LoM(D4-QC C)>u ( 4. q - 03
2u-' w- -o w 0
mI 417
r 0,0-aoi i 0,.C1
io 2W -WC IC.- oU 4j Wa -> Ia=U ) z .. ( c'4' 0 D. - >1 o -I-a.-q* C '-' rA'4 01 0 -,d-
o: -. C _3 CL 'a 20' (A"0 .'' L: 4-64 2. aI a.O s_~ U,
o 5 44~~. -4 ~ 4) W 4. U 'UI--.o *iU.;- 2 L I.n 245 .on, *0 0 5
cc u. 4O s-S W 0 C. EN -2 4' ' 0~ 4."W
LE .0 E m 4-c IV 5.M ~ W4'24' W +A 0 M- 0 .- - W 0 4
.02~~ t- -v-D o. 444 - 00 W _ I.-0. 4 W 42-.- 0L )~ 20. M .'- "4 4.D'. -4.~ 44 - 0 0 C
04C d)> .'U L' m 347, 4f MA 4 0 4 . IW 1 c L -- iEMcL-XmML ML W CL 4-POL.L WLM a
cu' wL. 0. 0_ C 22 0a).L4xOL440. 32' W 'v- EC 04o 2 4' -- Li E.- 0 - J- U c >,2 4-O C .*4 IO U ~ L-~-- J r4~W 'W 4S 4-' W 3C-. 4 >La IA 'WW.- 5 .UxW
41W Wi 2-> r= 4LA> E5 G~' '-- Li WO.C =U2 .0 '4 1n2.o
C W'U. -3: L UWC 14) .U LOWO C W'. 1.n'uC4V 4CA'u4D oL= 0+.- a, 2.- .W 4 *--)( W.- U 0.' 4.'- Z-C. (; 4JCW
wL d)-i 4) wL 4.n.VL-- .UV W > F UV 5.3 = ~ 20 3O >n 20 UL 0 UW 45
g, . _-1oJ1
Problem 4 illustrates how the sizing program described herein
may be used in conjunction with a stage optimization routine to develop
a near-optimum preliminary design of the propulsion components of a
multistage vehicle. A typical stage optimization routine, which was
computerized by Mr. J-. H. Dobkins of Teledyne Brown Engineering, is
described in Appendix B. Results from this routine, which is entitled
the Missile Optimization Program (MOP), have been demonstrated to
agree closely with those from other more elaborate analyses. Since
MOP requires ideal impulsive velocity as input, the designer must
estimate velocity losses which will result from drag and gravitational
effects. Additionally, input values of mass fraction and specific -impulse
for each stage must be specified. Minimized gross vehicle weight and
* optimized propellant weight distributions for a prescribed number of
stages are then computed using MOP for the ranges of-payload weights
and ideal impulsive velocities of interest. For selected point values of
ideal impulsive velocity and payload weight, the designer computes
corresponding values of required total impulse and thrust (for a pre-
scribed burn time) for each stage. The sizing program may then be
utilized to establish optimum diameters (or lengths), -chamber pressures
and expansion ratios for all of the stages as- discussed previously.
Following the preliminary design of a single or multistage ,ehicle,
the evaluation of its predicted flight performance is usually desireG.
The -trajectory analysis which is 'equired to evaluate flight performance
is beyond the -scope of this investigation. However, vehicle design
characteristics, e.g.-, weights, -mass flow -rates, thrust levels,J [envelope dimensions, which are desired from the sizing program may
be input into a trajectory simulation computer program to predict
flight performance. The trajectory simulation program described in
:Reference 12 is typical of many applicable programs which are available.
4-3
4. z SAMPLE PROBLEM
To demonstrate the validity of the computer program, a sample
problem was processed which consisted of an analytical reproduction of
the Thiokol TX-354-3 (Castor II-A) rocket motor design described in
Reference 13. This motor was chosen for discussion herein because
of the ready availablity of Unclassified documentation describing its
design-and performance characteristics in detail. Other rocket motor
designs described in Reference 4, e.g., Spartan, were also analyzed
successfully using this program, but the classified nature of their design
characteristics precluded their discussion herein.
The cross-sectional arrangement of the propellant grain in the!TX-354-3 rocket-motor is illustrated in Figure 4-1. The grain design
is basically a cylindrical port type which is segmented by two radial
slots, as shown in Figure 4-1. A propellant web fraction, Fw, of 0. 76-was
deduced- from the published geometric characteristics of the propellant
charge. The TX-354-3 nozzle design (Figure 4-2) utilizes a conical
expansion geometry, a graphite -throat insert, and requires no entrance
structure. The nozzle is designed for operation-at near-vaccum condi-
tions. AISI 4130 steel is utilized as the structural material for the
rocket motor chamber and nozzle body.
The required-input parameters corresponding to the TX-354-3
rocket-motor were -translated into a set of motor design- and- performance
characteristics using the computer program described herein. The
computer printout of the input and- output parameters of the sample
problem analysis -is--presented in Figure 4-3. For comparison, published
values (Ref. 13) of pertinent design and performance parameters of
the TX-354-3 motor are included in parentheses adjacent -to the corre-
sponding computer values. Comparison of the published and computed'
[ ,44,
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. . . .40 . . . . N .' ..- * .0
-- a3 , -- a 1
0 w~n nlen I7t. 0u1 4*000 u14.4 W.. 42nf to .3; n
a 4S~l141 407, 0'.44 IV0 r -1.0 0200 *4 .q.001 44 000 -44,~ o r* 0 3 4 9 744 . 4 1 * 4. 4 ' . -
values reveals certain discrepancies which must be discussed. For
example, the computer nozzle weight of 170. 5 Ibm is much less than
the published value of 539 Ibm. This large discrepancy is difficult to
explain without a knowledge of the details of the original structural
analysis from which the nozzle was designed. However, it may be
speculated that bending stresses from predicted lateral flight loads
could have dictated larger required structural thicknesses than those
which resulted from the pure hoop stress equations employed in this
computer program. (The credibility of the nozzle design procedures
employed herein was demonstrated through the almost exact analytical
reproduction of the existing Spartan booster nozzle.) The only other
conceivable explanation for this discrepancy is that relatively large
design safety factors (>> 1.5) could have been employed in the original
-design analysis of the TX-354-3 nozzle.
The only other significant discrepancies between published and
computed motor design parameters are in total motor length, total
inert parts weight, and mass fraction. The discrepancy between the
published motor length of 245 inches and the computed value of 242. 6 inches
results primarily from the neglect in the computer program of the reported
lengths associated with the pyrogen and nozzle attachment bosses (Figure
4-1). The discrepancy between the published total inert parts weight of1523 Ibm and the computed value of 811.4 ibm results primarily from
the nozzle weight deficit (368. 5 lbm), the chamber weight deficit (92 ibm),
and the weight deficit caused by neglect of the additional insulation weight
required by the two radial slots in-the propellant. Additionally, the
discrepancy between the published- mass fraction of 0.-84 and the computed
value of 0. 91 results directly from the previously described discrepancy
in total inert parts weight.
L -4-8
Additional test cases, covering a variety of motor configurations,
should be processed in the future to clearly establish the accuracy and
general applicability of the computer program.
4-9
£
5. PROGRAM OPERATING INSTRUCTIONS
The previously described design equations were coded into a
FORTRAN computer program consisting of approximately 530-cards.
Multiple cases may be submitted simultaneously by stacking the- input
cards back-to-back. Only the -input parameters which change need be
added to complete each new case. The remaining input parameters
always revert to the values used in-the preceding case.
5.1 INPUT DATA
All program input variables are read into the array A(i), i =
1, 27. A card-by-card description of the inputs, including symbol
definition-and format, is presented--below.
S Card 1, Format (I2)
A Code word KTR
A KTR is the number of input variables (one per -card) tobe read following this card. At the end of the -last case,input KTR as 99 to end-the job.
-0- Cards 2 through 28, Format (IZ, El5. 8)
A These 27 cards contain the 27 input variables which are
read into the array A(i). On each of the 27 cards, theproper index, i, is punched into columns 1 and--2, andthe corresponding element A(i) is punched into columns3 through 17, using the format given above.
A The 27 inputs, A(i), i = 1, 27, and the equivalentprogram variables are defined in Table 5-1.
* Card 29, Format (I2)
A Code word KTR
A This is the same code word as described above. -Ifanother case is to-follow, KTR is the number of inputelements A(i) to be -changed from the values used-inthe preceding case. If no more cases follow, KTRshould be punched as 99.
5-L
TABLE 5-1. INPUT VARIABLE DEFINITIONFOR CARDS-2 THROUGH 28
EQUIVALENTPROGRAM
ELEMENT VARIABLE DEFINITION UNITS
A(l) DM Rocket motor outs.ide diameter in.
A(2) PC Chamber pressure psia
A(3) F Required thrust F
A(4) FW Pronellant web fraction --(Ratio of web thickness topropellant outside radius)
A(5) APT Port-to-throat area ratio --
A(6) PAM Ambient pressure psia
A(7) TC Propellant flame temperature OR
A(8) RHOP -Propellant density lbm/in 3
A(9) GAM -Combustion gas ratio of specific ---heats
A(lO) ATB Pressure-coefficient in pro-j pellant-burning rate law
A(I1) AMW Molecul-ar weight of combustion l bm/moleL gas
A(12) AN Pressure -exponent in propel-lant ---burning rate law
Insert -MUT Poi sson-'s ratio -0.12SIGT Allowable tensile 3,355 psi
stress
Insert 4 ( RHOSL Density 0.02 lbm/in 3~Sl ivers
NOTES:Typical values for 4130 steel.
j 2 Typical values for FM 5048 silica/phenolic.
3 Typical values for ATJ graphite.
4 Typical value for phenolic microballoon.
5-8
6. REFERENCES
1. Weisbord, L., "A Generalized Optimization Procedurefor N-Staged Missiles", Jet Propulsion, March 1958
2. Stone, M. W., "Grain Design Handbook", Internal Memo-
randum, Rohm and Haas Company, Redstone Arsenal-,Alabama, October 4, 1956
3. Stone, M.W., "The Slotted-Tube Grain Design", Re-port No. S-27, Rohm and Haas Company, Redstone Arsenal,Alabama, December, 1960
4. "Rocket Motor Manual", CPIA/M1, Chemical PropulsionInformation Agency, Johns Hopkins University, SilverSpring, Maryland, May 1969
5. Barrere, M., et. al., Rocket Propulsion, Elsevier Pub-lishing Company, New York, N. Y.-, 1960
6. Sutton, G. P., Rocket Propulsion Elements, John Wiley andSons, New York, N. Y., Second Edition, June 1958
7. "Program for Preliminary Design of Internal BurningSolid Propellant Rocket Motors", Thiokol Chemical Cor-poration, !Huntsville, Alabama
8. Bruhn, E.:F. , Analysis and Design of Flight VehicleStructures, Tri-State Offset Company, Cincinnati, Ohio,1965
9. Alexander, R. V., "Summary Report-of Program Beta, ADigital Computer Program for High-Speed Analysis ofPerformance, Preliminary Design, and Optimization-ofU Solid Rocket Vehicles", Technical Memorandum No.176-SRP, Aerojet General Corporation, Sacramento,California
10. Threewit, T. R., et. al., "The Integrated Design ComputerProgram and ACP--103 Interior Ballistics ComputerProgram", STM-180, Aerojet General Corporation,Sacramento, California, December 1964
11. Demuth, O.J. , and M.J. Ditore, "Graphical Methods forSelection of Nozzle Contours", presented at the SolidPropellant Rocket Research- Conference, Princeton Univer-sity, Princeton, N.-J. 28-29 January 1960
FII
6-I
12. Boehni, B. W., Rand's Ommibus -Calculator of the Kinematicsof Earth Trajectories, Prentice-Hall, Inc., Englewood Clifts,N. J., :1964
13. "Technical Description of the TX-354-3 Rocket Motor,Report No. 57A-66, Thiokol Chemical Corporation,Huntsville, Alabama, April 7, 1967
-V
11
Ii-
k [A 6-2
APPENDIX A
FORTRAN LISTING OF SIZING PROGRAM
A complete listing of the FORTRAN statements is furnished inthis appendix as a reference for making any desired changes to thesource deck of the solid rocket motor sizing program.
t
11
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A--13
APPENDIX B
DESCRIPTION OF MISSILE OPTIMIZATION PROGRAM (MOP
B. 1 ANALYSIS
I The following analysis is based primarily on the work of
Weisbord (Ref. 1) with modifications to simplify automation and nn.: .. 1
changes to fit current usage.
The ideal impulsive velocity, AV I , achievable for any roclkc
with constant specific impulse for each stage is given in Equation 1.
wol WozAV = C In + C2 In .bo---- (1)
Wbol WboZ
where
Wo, - gross start weight for the ith stage, ibm
Wboi - burnout gross weight for the ith stage, ibm
I C, - gcIsp, ft/sec
I For any given mission, AV I can be estimated from Equation 2.
j tb
AV I = AVact + g sin 0 dt + AVD (2)
I 0
4 The Missile Optimization Program was developed by Mr. J. H.
Dobkins of Teledyne Brown Engineering.
B-1 .
where
AVact - actual impulsive velocity, ft/sec
0 - angle of velocity vector above local horizon, rad
AVD - drag losses, ft/sec
t - time, sec
tb - burn time, sec
For large missiles drag losses usually amount to less than 5 percent
and are not considered important for preliminary analyses. Howc ,c2-
for smaller, high-velocity vehicles this factor becomes more c~r1,
and must be estimated.
The gravity loss -term is zero for horizontal launch and is (gt) io,
a vertical launch. Thus it-is strongly trajectory dependent and son~e
approximation -between the limits must be -made.
Each stage consists of motors, propellant tanks, misceli c n
hardware, guidance, etc. The mass fraction (XMFi) and structu:7
fraction (XMSi ) of stage i are defined in Equations 3 and 4.
XMF i = Wp/(Woi - WPL i ) (3)
XMSi = WS/(Wo i - WPL, ) (4)
where
WPi - weight of propellant, lbm
Wsi - weight of stage - empty, Ibm
WPL i - weight of stage payload (upper stagesplus payload), Ibm.
B-Z
Mass fraction data can be estimated from past experience. Structural
fraction is calculated within the program from the mass fraction and
thus need not be input.
The burnoixt weight of any stage may be written in the form of
Equation 5.
Wbo i =Woi+ + XMS i ( W o i - oi+ 1
= XMSi W0 i + (1 - XMSi) Wo i + l (5)
and Equation-1 -becomes
N Woii AV I = Ci In (6a)i= 1 XMSi W0 i + (1 - XMSi) Woi+l
or
0 ~I~n Fwi0+I AV1 (6b):i= 1XMSi W0 i + (1 - XMSi) W
The problem requires that Wo. be minimized for a given AV,.TI
- Therefore, the Wo, become variables and the maximum or minimum
take-off weight is achieved when
aWo,o 0 i=2, N (7)
-i B-
B-3
Since it is difficult to explicitly write these differentials, note
that Equation 6 is of the form
f (Woi) = o(8)
and the partials can be written
awo- af/aw0 ia Woi af/aWo I
from the chain rule for differentiation. Equation 9 implies that if
t Equation 7 is to be satisfied then
-= 0 (10)-aWoi aw0 iI
must also be satisfied. The process yields for i 2, 3