JPRS ID: 8627 TRANSLATION STATICS OF ROCKET MOTOR SYSTEMS BY VE. B. VOLKOV, T.A. SYRITSYN AND G. YU. MAZIN

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- ~ dF II ' I ! SYSTEMS
6Y YE. 6. VOLKOV. T. A. SYR I TSYN ANO ti. YU. MAZ I N
2i AUGUST i9T9 CFOUO~ i OF 3
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JPRS L/8627 F~OR OF~iC(AL USE UNLY
~ 21 August 1979
Transl.ation
STATICS OF ROCKET MOTOR SYSTEMS
By
Ye. B. Volkov, T. A, Syritsyn and G. Yu. Ma~jn
FBIS FOREIGN BROADCAST INFORMATION SERVICE
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JPRS L/8627
2~. August 1979
STATICS OF ROCKEt MOTOR SYSTEMS
Moscow STATrKA Z D~NAM~KA RAKETNYKH DVZGATEL'NYKH USTANOVOK V
DVUKH KNZGAKH ~"5tatics and Dynamics of Rocket Motor Systems in
Two Volumes") in Russian 1978 signed to press 17 Feb 78 pp 2-221
[Volume One ("5tatics") of book by Ye. 8. Volkov, T. A. Syritsyn
and G. Yu. Mazin, Mash3nestr~yeniye Publ3shers, ].550 copies, 222
pages] ~
CONTENTS PAGE
Pref ace 5
Section I. Static Chara~,:teristics of Liquid Propellant Rocket
Motors
f
Chapter 1. General Description of Motors 7
1.1. Classification and Designs of Liquid-Propellant Rocket Motors 7
1.2. Engi,ne Characteristics 15
1.3. Characteristics of an Engine Without Generator Gas After-
burning 21
1.4. Optimal Combust:ion Chamber Presaure in a Motor With a Gas
Pressurization Supply System 25
1.5. Limiting Combustion Chambe~: Preseures in an Engine WiCh
Generator Gas Afterburning 28
Chapter 2. Methods of Analyais and Calculation of Static Character-
istics 38
2.1. Disturbances rf Operating Cnnditions 38
2.2. Methods of An~.].~?sis and Cale~zlation of Effect of Disturbances
on Engine Characteristics 43
Chapter 3. Static Equations of Motor Components 47
�s- .
3.1. Combustion Chamber and Gas Genarator Equation 47
3.2. Pump Equations 49
- j - ' II - USSR - A FOUO~
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3.3. Turbin~ Equatiions 51
3.4. Supply Line ~:quations S4
'3 ~ 5. Pre~.~urp AcnumulnCnr Equations 57
3.6. Thrust CharAC!:~ristics 58
Chgprer 4. Engine Static Characteristice
4,1. General 5olution , 61
4.2. Model of I'ropulsion Syatem Wieh Gas Presaurization Supp~y '
SysCem ' 62
4.3. Mode1 of Engine Without GeneraCor Gas Afterburning 65
4.4. MoCor Witt~ Generator Ggs Afterburning (G-L) 67
4.5. Motor With Generator Gas AfCerburning (G-G) 71
4.6. Synthesized ~ngine Characteristics 76
4.7. Maximum Propulsion System Running Time 78
ChapCer S. SCatiyCical Analysis of Precision of Motor Operation ~Q
5.1. Laws o� DisCribuCion of Operation Parameters 84
5.2. Statistical Characteristics of Precision g9
5.3. CondiCions of Engine Efficiency 92 ~
5.4. Regression Analysis of Precision 99
ChapCer 6. Tuning and Ad~usting Motors 107
6.1. Tasks and Methods of Tuning and Adjuating 107
6.2. Individual Tuning and Ad~ustment 109
6.3. StaCistical Tuning and Ad~ustment 118
6.4. Comparison of Methods of Tuning and Ad~ustment 119
Section II. SCatic CharacCeristics of Solid-Propellant
Rocket Motors 123
Chapter 7. Operating Charac~eristics of Solid-Propellant Racket
Motors � 123
7.1. Solid Rocket Propellants and Principal Designs of Sol�d-
Propellant Rocket Motors 123
7.2. Empirical Law of Rate of Combustion of Solid Rocket
Propellants Under Static Conditions 128 ~
7.3. Law of Change of Propellant B urni n g Surface on a Time
Axis 131
7.4. DeterminaCion of Operating Parameters and Characteristics
of Solid-Propellant Rocket Motors With Zero-Dimensional
Statement of the Problem 135
7.5. Determination of Operating Parameters of Solid-Propellant
Rocket Motors With One-Dimensional Statement of the Prob-
lem 141
7.6. Determination of Operating Parameters of Solid-Propellant
Rocket Motors With Charges With Step-Wise Change in Flow
Passage Cross-Sectional Area 148
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Chnp~4r g. Uevigtiong of Uperating Paramerere of Solid-Prope~.lanC
Rocknt Motor~ in the Vicinity of Specified Condi~ions 152
8.1. Ite~.atione for Deviations of Operating Parametera of Solid-
PropellanC Rocker Motora ~.n Vicini~y o� Specified Condi-
~ tions for a Zero-D~menaional Variant~ 152
8.2. Itel~tions for Devia~iona in OperaCing Parametera of Solid-
Propellant Rocketi Motora 3n the ~nvirons of Specified Con- ~
diCions in the Case of a One-Dimensional Solution 156
8.3. Selection o� Optimal pk and ak valuea 162
_ Chapter 9. Factors bisturbing Operating CondiCione of So1id-
Propel].ane Rocket Motora 168
9.1. General Survey of Disturbing Factors 168
9.2. 7nfluence of G-Loadings on OperaCing Conditions of So11d-
_ I'rqpellanC Rocket Motors lh9
9.3. Nozzl,e Ero.sion 173
9.4. Disturbances of Operating Conditiona of a Solid-Propellant
Rocket Metor and Its OuCpuC Characteristics Connected With
Removal of Thermal ProCection Materials 176
9.5. Figuring Heat Losaes and Incomplete Fuel Combustion 178
Chapter 10. Influence of Charge Initial Temperature on Character-
istics of a Solid-Propellant Rocket Motor, Motor Tuning
' and Ad~usCment 184
10.3.. Relationship BeCween Characteristics of a solid-Propellant
Rocket Motor and Charge Initial Temperature 184
10.2. Ob~ecttves and Means of Tuning and Ad~ustment of a Solid-
Propellant Rocket Motor 187
10.3. Tu~ing of a Solid-PYopellant Rocket Motor Nozzle Co Con-
stant Pressure 189
10.4. Tuning a Solid-Propellant RockeC Motor Nozzle to Constant
Thrust 192
10.5. Tuning a Solid-Propellant RockeC Motor to Constant Flow
RaCe 194
10.6. Ca~ses of Nonuniformity of Charge Temperature Field and
Its Equalizing Time 196
10.7. Influence of Nonuniformity of Charge Temperature Field on
Solid-Propellant Rocket N;otor Operating Conditions 199
Section III. Static Characteristics of Hybrid Rocket Motors 206
Chapter 11. Designs and Features of Operation of Hybrid Rocket
Motors 206
11.1. Designs of Hybrid Rocket Motors 206
11.2. Propellant Combustion in a Hybrid Rocket Motor 21~+
11..3. Hybrid Rocket Motor Combustion Chamber Fquations 219
11.4. Equadvns of Liquid Component Supply System Equipment 222
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Chapeer 1.2. Statiic Characteriatics o� Hybrid Rocket Motora 226 -
12.~. Influence of ExCernal and Intierngl Factors (Disturbances) 226
~ on Operating Parametera of Hybrid Rocket Motors
12.2. TUning and Ad~ustment of a Hybrid Itocket Motor 236
Bibliography 240
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~ Vr ~ rv.~~W NNI.1 NA1VIr
PUBLICATION DATA
Engliah title ~ STATICS OF ROCKET MOTOR SYSTEMS
Rus$ian title s STATIKA I DTNAMIKA RAKETNYKH
DVIGATEL'NYKH USTANOVOK V DViJKH KNIGAKH _
Author (s) : Ye.B.Volkov, T.A. Syriteyn, G.Yu.Mazin
EdiCOr (s) ;
Publishing House : Mashinostroyeniye
Place of Publication ; Moscow
DaCe of Publication ; 1q78
Signed to preas : 17 Feb 78
. Copies : 1550
_ COPYRIGHT ; Izdatel'stvo "MashinQStroyeniye", 1978
4a
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I
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- PREFACE
One of the most important Casks in rocke~-engine Cheory is ~he calculation
of the parameters of their operating conditiona. These parameters are
determined by the specific features of the engine design and the character-
istics of a11 engine componenCs. DeCermination of parameters, especially
- for enginea of complex design, constitutes a fairly difficult Cask.
A1Chou~h the rocket engines employed at the present time are highly diversified,
Cheir operation is based on a number of identical processes. In connection
with this Chere exists much in common in the characCeristics of engines,
even of different types, and correspondingly in the methods of calculating
Chese characteriseics.
This book is a firsC attempt Co present from a unified viewpoint methods of
aiialysis and calculation of Che parameters and characteristi~cs of steady-;;Cate
operating processes of rocket engines operating on liquid, solid and hybrid
fuels. The identical nature of the goals and methods of such an analysis
for engines of differing types is illustrated by presentation of the material
of its principal sections. At the same time there of course also exist (and
this is 'reflected in the book) features of calculation of parameters which are
specific for each type of engine. This is connected with dif�erences in the
design and operating conditions of different engines, as well as the fact
that methods of calculating parameters were el~borated separately for each
type of engine.
Calculation of the parameCers and characterisCics of a rocket engine is
based on the points of tt~eory of calculation of the components and elements
com~rising an en~ine. It is assumed that the reader is acquainted with
these points, and therefore they are presentFd in brief and only where this
is necessary in order to elucidate the specific features of solving the main
problem determination of the parameters of an entire engine as a whole.
The quantities and their numerical values characterizing engines are in con-
formity with the International system of units (SI). Parameters and engine
design layouts are based on published foreign materials.
Section III was written by Ye. B. Volkov, Section I by T. A. Syritsyn, and
Section II by G. Yu. Mazing.
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The auChor~ would ~.ike to exprese their thanka to Doctor of Technical Sci-
ence~ Professor M. F. Dyunze for his valuable commenta made on edit3ng this
book, and they would like to expresa thanks in advance Co the readers �or
critical commenes on the aubstance and method of~presentation of the
maCerials.
Please send a11 comments and remarks in care of the following addresa:
Moscow, B-78, Pervyy Ilasmannyy pereulok, 3, izdatel'stvo MashinoaCroyeniye.
~
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Section I. STATIC CHARACTERr5TICS OF LIQUID-P1tOPELLANT ,
ROCK~T MOTORS
Chapter 1. GENERAL DESCRIPTION OF MOTORS
1.1. Classitication and Uesigngof Liquid Propellant Rocket Motors
ZhRD, that is, engines burning liquid rocket fuel, are widely util~ized today
in rockeCs and space hardware. `
In the general case a ZhRD consists of the following:
_ a combustion chamber, in which fuel or gas generation producCs are
transformed as a result of chemical reactions into combustion products, which
generate a reaction force when escaping from the noazle;
a supply system, which includes that equipment which feeds the propellant
components from the storage tanks to the combustion chamber;
automatic control equipment those devices which contrnl engine
operation, ad~ustment and servicing operations.
Liquid-propellant rocket moCors or engines are components of th~a propulsion ~
system.
A propulsion system contains one or several ZhRD, propellant tanks, uniCs
for producing Cank pressure, fuel and oxidizer lines from tanks to engines,
and auxiliary devices.
A detailed classification of motors based on various aCtributes can be
found in [1].
Rocket engines are subdivided into two groups by type of propellant feed: ZhRD
with gas pressurization feed, and ZhRD with pumped feed.
In a ZhRD with a gas pressurization system, the fuel components are fed to
the combustion chamber by expulsion from the propellant storage tanks by
gases the pressure of which exceeds pressure in the combustion r_hamber.
Figure 1.1 contains a diagram of a ZhRD with a gas pressurizatic~n supply
system.
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. S~
J ~
Y
: ~
Figure 1.1. Diagram of ZhRD With a Gas PreasurizaCion Supply System
Key:
1. Combustion chamber 4. Pressure reducer
2. ShuCoff valves 5. Compressed-gas tank
3. Tanks conCaining propellant
ccmponents
~
The supply system contains a compressed-gas Cank or pressure accumulator and
automatic conCrol devi~es ensuring.a specified pressure in the tan~;. The
source of gas providing the requisite pressure in the tanks can be compressed-
p,as tanks a gas pressure accumulator (GAD), a liquid-reactant gas generator,
ahAD), or a gas generator with a solid fuel charge a solid-reacCant gas
generator (PAD). Compressed-gas pressure accumulators are extensively
utilized due to their simplicity of design, operating process and high degree
of reliability. Propellant is fed from the tanks to the combustion chamber
by pressure drop
I~o-PK > 0.
The requisite pressure in the tank is determined from the relation
Pa = PK -I- ,F,~ ~pr - nFl p~
Ghere ~pi hydraulic resistance of the lines, valves, cooling jacket and in-
:jectors; nHp hydrostatic pressure.of the column of liquid; n-- calculated
�~xia1 load factor.
An advantage of ~his method of propellant fe~d over a pumped system lies in a
comparative design simplicity. But at the same time employment of a gas
pressurization supply system results in heavier tanks, since they must stand
up to in~ernal pressure exceeding pressure in the combustion chamber
P6~Pk=1.3-i.7.
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A volume of propellanC which is approximaeely equal to the volume of the
Canks is determ~.ned by relaCion
Psp
VT ~ ~vPr' (I~1)
where P-- engine thrust; Tp operaCion time; Iy specific thrugC impulse;
pT propellant denaity.
Consequently, with increase in thrust impulse (PTp) there is an increase in
vol~ime of propellanC, mass of propellanC tanks, and the required quantiCy of
gas to drive the combuatibles from the tanks into Che combustion chamber.
Figure 1.2 shows relation T~T~P~, for which specific engine mass
(mA~ y= I~IAIP, MA mass of primed engine) is identical wiCh a gae
pressurizaCion and pump-fed system. Therefore motors wiCh a gas pressurization
supply system can compete successfully wiCh moCors with a pump supply gystem
only with modest Chrust impulses.
t; c
SO
~k0
A
JO
YO
1D
6
0 10 20 JD 40 SO 60 70 80 P,KN
Figure 1.2. Areas of Application of ZhRD With a Pump and Gas Pressuriza-
Cion Fuel Feed System
Key:
A-- pump system B-- Gas pressurization system
In moCors with a gas pressurization supply system, pressure in the com-
bustion chamber does not exceed 2.0-3.0 MPa. At the same time we know that
for a given fuel the specific thrust impulse can be increased only by in-
creasing the ratio of nozzle expansion. With limited pk, a value greater
than this can be obtained only by reducing pa. In connection with this,
motors with a gas nressurization supply system can be employed for operation
in a vacuum (when pa can be reduced) and in cases where large thrust im- -
pulses are not required.
riotors with a gas pressurization supply system are employed for the follow-
ing: course correction and attitude change, docking and undocking space-
craft, and less frequently as propulsion motors for upper rocket stages.
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In ~hRD wiCh ~ pump ~upply ~y~tern, combugCibl~~ ar~ f~d by purnp~ from th~
Cank~ to th~ combugtion rh~mb~r. T~nk pre~sur~ in theg~ mnCOrg ig in~ig-
nifi.canC, ~n~uring Cgnk ~Cabi~3~y and c~viC~Cionl~~g pump np~r~C3on.
7.hRb wtt~ a pump ~upply gy~Cem gre gubdivid~d intio twe grdup~ on rh~ ba~ig
af uCili~ation of warking subsCanc~: 2hRD wiChouC Afe~rburnin~ of ga~
gen~r~ei~n product~ in th~ combugtion Chamber, ~nd zh~U wieh gft~rburning
nE g~ner~Cor ga~ in Che combu~eion ch~mber.
~igur~ 1.3 cone~ins ~ block diagr~m of ~ motor withouC ~fterburn~ng of
gener~Cor gag.
_ -
=
t
~ t
r
7 '
5
~N S
rigure 1.3. Aiagram of 2hRD Without Afterburning of Generator Gas
Key:
1. Cas generator 4. Oxidizer pump
2, Turbine 5. Combustion chamber
3. I~'uel pump 6. Automatic control devices
7. ExhausC nozzle
in tliese motors generator gas, after passing ehrough the turbine, is
exhMUSted externally. In view of the fact that the turbines are not
cooled, the generator gas Cemperature is low and Che gas which is ejected
inCo the atm~sphere past the turbine possesses unexpended energy. With a
~~ressure increase in the combustion chamber, there is an increase in
��equired turbine output and generator gas consumption, which in turn leade
~u a loss in the specific impulse produced by the combustion chamber and
exhaust nozzles. Therefore motors without afterburning of generator gas
have an optimal size pk, exceeding of which results in a decrease in
specific impulse. Actual values pk for moters without generator gas after-
burning range between 5.0 and 15 MPa.
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xox o~~~ct~. us~ oNr~Y ;
ZhRU with ~feerburning of g~nerator gas make it po~eible more Fu11y Co
, utiliz~ fue~ ~nergy to obCain ~pecifin impule~, ~ihce in ehi~ in~tance
Che entire fue1, with an ~pt~.mal ratio of components~ ie fed into the
combu~Cion chamber.
MoCor~ wieh gfterburning of generator gae ar~ subdivided in~o three Cypest
ZhRD w3~Ch afterburn~.ng of ox3dizing generaCor gas;
ZhitD with ~fterburning of reducing gen~rator gas;
ZhRD with ewo ga~ g~ner~torg.
In rhe fir~t two arrangemenr~ gas generator gae and a liquid component are
�ed into the~combustion chamber, and there�ore euch motor arr~ngemenea are
degignated G-L.
In motor~ w~,Ch ewo gas gener~tore, gas generaCor gasea are fed into Che ~
combustion chamber, and Cherefore Chey are deeignated G-G.
~igurE 1.4 contains a block diagram of a G-L motor wiCh oxidizing gas
~enerator.
_ -
~ r
J
s
~ .s
~
i~'iRure 1.4. D3agram of ZhRD WiCh Afterburning of Oxidizing Generator Gas
Key:
1. Gas generator 4. Oxidizer pump
- 2. Turbine 5. Pressure chamber
3. Fuel pump 6. Automatic control componenta
7. Gas l~ne
In motors with an oxidizing gas generator, the entire oxidizer is fed into
the gas generstor, bypassing the combuetion chamber. The greater part of the
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run urrL~lcw uoL u?~~~,i ,
fuei i~ fed ineo ehe combuetiion chamber, w~.tih a le~e~r amount baing fed
ineo rha g~s generatior. In moeors w~.Ch ~fC~rburning of rgducing ga~~ the
entir~ fu~1 and a porGion o� tha oxidizer is f~d tio the gas generator.
Gagificee~.on of ~uel takee p~gce in Che ga~ genera~or with a�1 or afel. . -
A�C~r th~ ~eneraCor gge ~.g appli~d eo the turbin~, it passeg into Che com-
buation chamber, where there occurs fue~, mixing wiCh the liquid component
and combuseion wirh a~l.
~3gure 1.5 conCaina a block d~.agram o� a motor operatiing on a G-G errange-
menC.
-
_
~ s r
s r
i ~
~
d
9
1.5. Diagram of G-G 2hRD
Key:
1. Gas generator for driving 5. Gas generator for driving
turbine oxidizer pump turbine
2. Fuel pump-driv~ Curbine 6. Oxidizer pump-drive turbine
3. Fuel pump 7. Oxidizer pumn .
4. Reducing generator gas line 8. Oxidizing generator gas line
9. Combustion chamber
T}ie propellant components are fed,to the combustion chamber by two independent
turbopump units, each of which has a~as generator. This arrangement makes
jt possible to select an optimal pump rpm and to reduce the weight of the
supply system.
Motors with afterburning and generator gas, due to the absence of losses in
Che specific impulse with e~ection of generator gas, which possesaea con-
siderable energy, make it possible substantially to increase pk and thua to
increase engine economy.
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With g pre~~ur~ ~.ncrea~e ~.n th~ combua~ion ch~n?ber, how~v~r, tih~ mag~ o�
gupply ~yseem componenti~ incre~~~~ ~ub~tgntia],ly, and diEficulC~.~~ ~ri~e in
provid~ng r~liable cool.~ing and 3n obe~~.ning seal~. -
Zn addition, motorg with a�eerburn~.ng af generaCor ga~ gre moYe complex
than those without a~rerburn~.ng, due Co eub~Cantiial pres~ures in tha equip-
ment and the pra~enc~ of ~ g~~ f~~d lin~ b~tw~~n ~urbin~ and combu~ti,on
chamber, and tihoy have a high epecific ma~~. Therefor~ in selectiing a
generator gas motior arrangement (motor w~.th or without aftierburning), it
is essenC~ai to proceed from the purpoee and efficiency of motor employmene
on th~ vehicle to be propelled.
itocket velocity at tihe end of the powered aegment of flight i~ detarmin~d
with the relat3on
n
UK ~ ~y ~n NKr - Ov~, (1.2)
where �K � M?~~�? Mo - MK ~ MT; Mo rockeC launching masg; Mk
mass of structure; Av~ total velocity losses cauaed by the gravitational
atrraction of the earth, air reaiatance and other factors; ~v~/vk~0.18-0.27;
i-- number of rocket gCagea; MT propellanC mase.
~quation (1.2) in differential form appeara as follows:
d=-~~ ~ dly - dMK lk* (1.3)
nK I
y M'` M
~-n
�
K
The engine mags is a component part of the mass of the structure, and there-
fore ~quality dMk=dMA applies. For a given velocity (distance of flighC)
vk~const, equation (1.3) can be written as follows:
dMA=_ M"" d~yln ( )
Mn MA(NK-1) ly NK� 1.4
This last equation determines equivalent change in engine masa and epecific
thrust impulae.
If the relative change in specific impulae is 0.01, the equivalent change in
engine mass is
dM ~ M~�K
MA = 0~01 Ma(PK- i) 111
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I~'or conCemporery values of uk and Mk/MA an increase ~.n epecific impul~e by
1~~ in influence on vk or range is ~quiv~].enti eo reduci.ng eng3ne ma9~ by _
10-Z5Y.
'1'hu~ ir noC enough to ~sximate eh~ degree of ideality of a motor ~o~ely
on ehe ba~i~ of ~pecific impul~e or eng~.ne mase; one muee exam~ne eh~
~ggre~aGe of the~.r influenc~ ~n e rocketi's cheraceeris~ic~~
in Ch~ g~gumption thar ChrugC-eo-weight coefficient uk ig optimal~ with ~
~quaeion (1.4) one can deCermine Che Qquivalent change in engin~ maes for
~h~ firgC gnd eecond rocket stagea in relat~.on to eCage thruee.
a
Calculationg have ghown thar �or �irae and second sCage motors~ there is an
incre~ee in Che value of the engine maes equivalent with an increase in
~tage thrust.
~or qecond stage motors, however, the engine mass equivalent ie less than
for firsC seage engineg. There�ore any measures for firae-stage enginea
which promote an increase in specific 3mpulae are expedient, even with~an
increase in engine mass. ,
~or rocket upper-stage motors the expediency of auch measures ahould be
evalueCed taking into account change in masa. Proceeding from Che above,
we can state Chat advisable for the first rocket stages, especially if
hig}i Chrusts are required, is employment of engines with afCerburning,
- ~ince such engines make it posaible subatantially to boost Iy, in apite of
a certain increase in mgss in comparison with enginea without afterburning.
For rocker upper-stage motors, with small thrust valuea, specific impulse
los:~es connected with cooling, greater complexity of design and increased
mass in comparison with motors without afCerburning are not always com-
~ pensated by gain in specific impulae. In addition, in rocket upper-stage
motors high degrees of expansion of gsses in the nozzle (high specific
thrust impulses) can be obtained for small pk, which is characteriatic of
motors without afterburning. There~ore in some instances it is advisable
to employ motors without generator gas afterburning for rocket final stages. ~
5election of an engine.arrangement with generator gas afterburning is
deCermin~d by the energy potential of the given layouC, magniCude of thrus~,
and other parameters.
Witt~out going into detail for t3~e present on the energy potential of engine
]ayouts, we shall perform a qualitative comparison of different arrangements.
All other conditions being equal, it is desirable to have a high generator
gas temperature. At the same time tt~e maximum generator gas temperature is
limited by the heat resistance of the material of which the turbine blading
compc~nents are made, and for oxidizing gas compriaes 1,000�K, and for
reducing gas 1,300-1,400�K. When high-boiling components are employed as
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ro~ o~~zcr.n~, us?: oN~Y
fuc~l (NDM(;, ~~1cuh~1, ~ec), le id advl~abla tn employ nxidizing ~fCc~rburnl.ng, ,
'ChtH i~ due to Che fact Chnt generaCor gae Cemperature in thie arran~emanC
ig lnwer and ~he gaeification producee do no~ contain sooe resing, which
can fo~l turbine blading and alter characreristics~
A motnr w~Ch oxidizing gas ~fC~rburn3ng weigha lesa than wiCh reducing g~~
afCerburning due ~o ~he absenne of additional cooling manifolds and the
small siz~ of conCrols mounted in the fuel gas generetior line.
Ce is noe r~dvisgble Co E~mploy motore wieh a reducing gas generator when
1ow-bdiling components (hydrogen, etic) gre employed as fuel, aince in Chis
insCance with equal pk there can be lower preesures beyond the pumps.
Yf engine shutdown Cakes place in two etages, and the aecond atiage operates
on a gag generator arrangement, in ordex to obtain high final gCage apecific
thrust impulse it is advisable to employ a reducing arr~ngement, aince
(RT)g>(RT)ok�
1.2. Engine Charr~cCerisCics
1.2.1. ClassificaCion of CharacterisCica
The relationship beCween thrust and apecific thruat impulse and the principal
factors which change under operating conditiona are called engine character-
isCics.
The most importanC fact~rs which change in the process of operation are ambienC
pressure and consumption of propellant componenta. During a rocket's flight
Chere occurs change in the altitude aC which the engine operaCes. Ambient �
pressure changes in conformity with change in altitude. Relation pHspH(H) is
based on standard atmosphere figures. Calculations indicate that if one
employs InCernational Standard Atmosphere (ISA) tables, one can assume pH~O at
altitudes greater than 30 km. This assumption produces an error of less than
4~.
The dependerice of khrust and specific impulse on ambient pressure, with con-
stant fuel consumption and ratio of propellant components, is called al-
titude characteristic.
During engine operation there occurs change in operating conditions due to
change in consumption of propellant components.
The relationship between thrust and specific impulse on the one hand and
propellant consumption on the other, with a specified ratio of propellant
components, is called throCtle charateristic.
In the general case thrust and specific thrust impulse are depe~ndent not
only on consumption and ambient pressure but also on many other factors
as well. These relationships will be examined in subsequent chapters.
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Un~ clif[~r~nd,~~ between cnmbu~eion ch~mb~r characteri~tic~ and chgracter-
t~Cict~ oE Ch~ engine a~ n whole. If in en engine tha enCir~ propellnnt
Kupply ip proc~~~ed inCo combueCion productis and ~.e e~eneed 3nto thd anr
virnnmene only ~hrough the combueCion chamber (motiore with a ga~ praeauriza-
eion ~upply ~y~tiem and moCorg with generator gaA afterburni~g)., Che al-
Citud~ and thrnttle ch~racCer~.~tic~ of ehe combust~.on chamber will al~o
be engi.ne characteriaCice.
initi~l relaGiong needed for con~tructing characCeriseics include equatione
o� specific thrueC impul~e
!y = ~al y,,, - (l.b)
and Chruat
, P a ~P~~Y~ nm - Fapw? (1.6)
where ~I coefficient of comp].eCeness of specific thruet impuls~; Iy~n
CheoreCical value o� gpecific Chrust impulae in a vacuum;
/y,~~ ~d.~ ~m ; F~ nozzle throat area; wg velocity of outflow of
combuseion products fYOm tihe nozzle.
ror r~~const quantitica Iy and ~I can be asaumed conatant for all conditions,
and error will not exceed 3X.
1.2.2. CombusCion Chamber CharacterisCica
A1Citude characteristic. The term "altitude characteristic" came about dur-
ing Che operation of the firat rockets, which were launched from the ground
and reached a certain altitude. At the presenC time flying vehicles, in-
cluding rockets, are launched not only from the ground but also under
waCer (the Polaris missile), from a specific altitude in the atmosphere, and
Erom other planets. Therefore, while retaining the Craditional term al-
, titude characteristic, we shall examine change in combusCion chamber para-
m�:ters in relation not to altitude of flight but rather ambient preasure.
It follows from equations (1.5) and (1.6) that with supersonic flow of gas
t.~ a rocket n~~:.zle, when exhaust velociCy w8 is indepnndent of external
conditions, ~,pecific impulse and thrust are linearly dependent on ambient
pressure an~' decraase with an increase in pH.
F!.gurc 1.6 shows relation Iy~fl(pH) and P~f2(pH). When presaure at the
~iozzle exiC is less than ambient pressure, a compression wave may form in
the nozzle, and the linear relationship between thrust and speciiic impulse
~n the one hand and pg on the other will be disrupted. When the shock
wave enters the nozzle the thrust taken �from the inner duct of tl.z conr
bustion chamber begins to increase, and intensity of thrust reduction
decresses with an increase in pg. The nature of change in altitude
characteristic during nozzle operation with a shock wave is accompanied by
a patCern of shock wave movement into the nozzle.
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/y~ ~ I
~n ,
~
~n
P
~ , ~~K /N
~'i$ure 1.6. Ralationahip Between Thrust and Speci�ic Thruat impul~e, and
Ambient Preseure
In Figure 1.6 the daehed line ehowa Che chgracCer of altitude characCeristic
change with nozzle exhaust without a shock wave, while the aolid line showa
Che aleiCude characteristic for shock-wave flow with pH~pck� Shock-wave con-
diCions are obaerved in f3rst-stage rockeC engines~ pressure at the nozzle
exit of which is fairly low from the condition of obtaining mean maximum
specific impulae in the powered segment of flight (pg 0.5 MPa).
During operation at low altitudes ambient presaure is aufficienC for the ~
shock wave to enter deep into the nozzle. It follows from Che above that
calculation of specific impulse and thrust in a vacuum, based on the~
resulte of ground tests, with the presence of a shock wave, cannot bQ per-
formed with the normal formulas, aince in this case one would be determining
thruet and specific impulse noC in a vacuum but wiCh specific back preasure.
In order to determine the altitude characteristic, enginea must be tested on
special test beda which make it poasible to establiah the required degree of
vacuum at the nozzle, preventing operation under ahock-wave conditions.
It is convenient to examine the altitude characteriatic in relative quanti-
ties
8P ~ p~ -1' = fap~ ,
P~
b!y ~ ~r~ ~Y. n t r ~ /y~;,~'
i
One readily notes that relative changea in thrust and specific impulse are
identical. We shall transform relations dP and dIy by substituting
m�pkFkp/s, and we obtain
8P a U~r sa !
~tP p~, ~ ~.7~
where �
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~ ~a ~ ~ RTK ~
f~ ~ ~ 9 6 ~
K
b ~ Y?~ ~ :
n-- combu~tion products expanaion polytropic curve indicatior. When exam~n-
ing engine characterietics one ignores the relationehip between specific
pre~sure ~mpulee S and combusCion chamber preeaur'e. Thie assumpeion is '
equivelenC ~o asguming independence of diseociaCion of combustion producta
Erom pressure. This aeaumption is not rough within a narrow range of
change of pk. 'The e~fect of the influence of pk on s also diminiahed due
to Ch~ face that wiCh a change 3n pk temperature Tk and gas constant R of th~
combu~tion products changQ in oppoaiee directiona end approx~mately to an
identical degree, that is, RTk~f~nk~�
~igures 1.7 and 1.8 ahow relations gp ~ 8P (f~~ p,c) for a given
ambient pressure.
It follows from an analysis of the grapha (figurea 1.7 and 1.8) that SP
increases for combuation chambera wiCh a greater degree of nozzle expansion,
while when f~'const at reduced operaCing conditions~ that is, with
reduced pressure values in the combustion chamber.
~K
/j
~ '
t~ ~
~igure 1.7. Relationship Between dP and Degree of Nozzle Expansion
Pk1~Pk2~Pk3 ,
'?1~rottle characteri~tics are plotted in relation to flow rate or pressure
in the combustion chamber.
Since , when Fkpsconst, ih=cpk and the type of
characteristics P(pk) and P(m) is identical,
they can oe differentiated only by acale.
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~'OR O~~ICLAL US~ ONLY ~
L
d'P -
fe~
e~
r~
~
/~r
~'igure ]..8. Relationship Between dP and Presaure in Combustion Chamber ,
f~l'f~2~�~3
Lee us examine the throttle characteristics for ,two conditions of motor
ueilizaCion ppa0 (vacuum) and pH at altitude. Thruat in a vacuum is deter-
mined by the relation
pn ~~o Fopa~ ~ (1.8)
Thrust at altitude
P = P� - F,p�. (1.9)
With a change in pk, pressure at nozzle exit pg also changes proportionally,
and therefore quantity FaJFkp , which determines ratio pg/pkp ~ remains un-
changed. Therefore with an increase in pressure, as was indicated above,
RTk and exhauat velocity w8 do not change. ConsequenCly, following eub-
sCituCion in equation (1.8) of relation fi~cpk, we obCain
pn � ~pPK~ : (1.10)
where
. Cp � ~a ~Q i n ~ p . .
It follows from equation (1.10) that Chrust in a vacuum is linearly
dependent on pk.
Thrust at arbitrary altitude P=cppk-Fapx also changes proportionally to
pressure in the combustion chamber, differing from thrust in a vacuum only
by constant (for conditions of comparison) negative term FgpH.
Specific thrust impulse in a vacuum ia
~r.n~ m . (1.11)
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Aftier sub~Citiuting in equation (1.11) relat~.ong (1.10) and ~cpk, we obCa~.n
f y, n e`~ d const. ~
Conaequently~ specific ChrusG impulse 3n g vacuum ie independent of preg-
~ure in ehe combusti~on chamber (flow rate).
5pecific thrust impulae
1 y m I y,,, - ~pK" ( l.12)
with g pres~ure change in Che combuseion chember varies :Ln a more complex .
manner (hyperbolic relaCion).
The difference between Iy~~ and IY is equal to ~PK" and dependa'on pk.
When pk.,0 this difference also approaches zero, and when pk.?a, the specific
Chrust impulse asymptoCically approaches specific thrust impulse in a
vacuum.
P'
''J
~ ~ Pa �
/ P
l~n
1 ~
y
/
? P~.mtn
-faPiv
Figure 1.9. 1'hrottle Characteristic Curve of a Combustion Chamber
Figure 1.9 shows a throttle characteriatic curve of a combustion chamber.
't,~e tangent of the angle of inclination of straight lines P(pk) and Pn~pk~ �
identical and equal to specific thrust impulse in a vacuum. Point
(pk~0, P=-FgPg) is formal and necessary only in order to plot the character-
istic curve.
The combustion chamber when pk~0 does not generate thrust, let alone negative.
This is due to the fact that relations (1.5) and (1.6) are correct only for
a certain region of change of pk. Beginning with a certain value pk~ min
when pk _ ~ l.
~ ~ \~k~ ~g i btkt tg~/ ~
l 1 I
C ~ ~9a C F~ - F~ x ) ~ .
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qn coeffici~nt ti~king 3.~t~d ~ccoune pre~~ur~ decreage due to a fi~~.re number
of blxdes (qn�]..1-1.3); a coe�ficienti which takes intio account pregeure
lo~~~~ in overcoming hydraulic resisCance ~.n the pump blad~.ng; DZ, D~~ b2,
bL, ~ 2, p 1-- geomatric characteri.eCics of pump blading.
~ollowing differenC3aCion of equaeions (3.8) and (3.9) and gimple transforma-
tions Caking into conaideration inequal3tiea px ~ po, we obtain expressions
for coefficients of influence
B`nM -4- ~-m~
2Apn~ Bnm P ~
~pN~ n _ PH ~ aPN~ m ~ r
APn! Cms ~o , D d
pN
bpN~ p~~_-...~. i, bPp~ Do ~ pH ~ bpp, p�1 i
Px
aN~ n� aPN~ n' aN~ in � 1'+' aPN~
brv, n� bpH~ P"" bN~ Pv �
bN~ D~ bPN~ Di 6N'~ ~I - I.
Coefficients of infLuence in pump equations can be determined from the pump
flow rate characteristic and similarity relations.
From the pump pressure-flow rate characteristic, which can be obtained as a
result of tests on pumps, we have
a ~ tB aH, (3.10)
px. Pe
where ~ H-- angle of inclinaCion of the tangent to the pump characCeristic -
curve aC point pH.
We know from general pump theory a correlation which determines similar con-
d3tions [3]:
Pa=~n)~p l~~/'P~~(n n P 0). (3.11)
Differentiating equation (3.11) taking into account relation (3.10), we have
ap~, n a 2-~ tgtC~;
bPM. P= I- m tga,,:
On
bpN.o,=2-3 ~ tga,,.
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3.3. Turbine EquaCions
Turbinea employed in ZhRD as pump-drive tu~bines are usually impulse turbines
wiCh velocity stages or presaure turbines. Most frequenCly impulse Curbines
are employed in engines without generatior gas afterburning, and pressure
~ turbines in enginea with afterburning of generator gas.
The principal relation deacribing turbine operation is a Curbine power equa-
tion. Power is determined by gas flow rate (preasure), adiabatic work per-
formed by gas expansion in the turbine, and efficiency, that is,
Na � mTLyt~l r (3.12)
Following linearization of relation (3.12) we obtain a turbine equaCion
written in variationa
8NT � 8m~ BL~A -I- 8t~r (3.13)
We shall datermine variables gmT, BLan and bt~T~ ~ which enter equations
(3.13); for this we shall utilize relation (3.10).
The rate of gas flow through a turbine depends on the type type of gas flow
in the blading and is determined by turbine type. For supercritical flow
conditions (impulse turbines) the rate of gas flow through the turbine is
determined by relation
rit 6 ~x~ prrFc
r - R7,T ~
and variation of flow rate SmT = 8pr~ - 0,58RTT SFc� The capability
flf gas to perform work RTT is determined by the ratio of propellant components
:.n the gas generator K and is practically independent of pressure. Relation
F.TT=f(K) is determined as a result of thermodynamic calculation and can be
- d.~scribed by the correlation
BRTT = K aRrT
RTT dK '
Substituting the last correlation in the equation for d mT, we obtain
8"4 = 8prr a~t. x8K aF~~ (3.14) .
where
, K aRTT
a"'r~ K - ~`5 fjTT aK �
For subcritical flow, characteristic of pressure or reaction turbines, gas
flow rate through the turbine is determined by relation
~ x-I-1
F~ x2x i RTT ( p~T 1 x_~ p~T ` x , (3.15)
rr 1 rr 1
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wh~r~ FC nozzle rhro~~ ~g r~~, p~,~ pr~egur~ ati eurbina no~~l~ exiC.
pre~~ur~ ~t Curbin~ in1~C p~~ d~pgnda on th~ d~gr~~ of rurbin~ ragctiion
~
- K-~ tx=T
A~*�"Prr P-(i-p)Cp~, " ~ (3.16)
wh~r~ degre~ of turb3n~ reaction; p2T gae presgure aC tiurbine out-
l~t.
begr~~ o� turbin~ r~action ie d~Cermin~d by a~emiempirical r~lation of
type I5j
~ P~a~b~~~~ i (3.17)
wher~ g, b, c-- experimental consCants; u-- turbin~ biade tip epeed;
It~ ~ ;
cl gae v~locie,y at nozzle exie. Variat~.on of gas flow rate through th~
turbine ie obtained from equarion (3.15)
8ritt ~ 8F~ BP~r a,;,t, K8K -~4~- aT (8pir - aprr)? (3. l8)
where
t
Z(~~*~~' -(x-~,~) rPml .
p~r x
~ rr ~ ~
~a ~ ~ .
, x r ` %
2K ( p 1 -Ip I
\ / \ !
Following linearization of equations (3.16) and (3.17) and their common
, s-,lution, we obtain
bPit ~ a, (8n - 8ci) Q~bPir ~aaPrr+ ~3.19)
x-1 �
x ~
bP I p'-T
k ~(Ptr)
ai~ x+~ C : .
ihere x_~
_ p~ r P:, " ~ ,
~ prr ) p
liq ~ C ; Q~ ~ C ;
' _ t
b~ P f b( l-{- 2C ( 1 1~
l \ 1 \ 1 J
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N~+~
~
~`~'P~-(~-~Cp~) x �
Ad3abaeic work by ggs in Che ~urbina
(3,20)
L,~ ~ RT 1- ~1 " �r `r
k-I Cprr /
~o11ow3ng 13ne~rization of equ~tion (3.20) we obtain
BL,A 28c~ � pc~ K8K - (apn - 8prr~~ ~3,21)
wh~ra
� t
~
(K- I) r.P.~t "
~L~ K � 2a,;~*, K; ~T ~ x_~ � '
2K I - (~-1 x
\ prc !
Turbine ef f iciency is determined by parameter u/cl thaC ie, t~* ~ i(u/cl),
A variation of Curbine efficiency can be presented in Che form
a~t � ~8n - a~~), ~s.~~~
where
~n = -u/----~c
- a'u .
~T a ~ ~ ~
5ubstituting relatione (3.18), (3.19), (3.21) and (3.22) in original
equetion (3.13), we obtain reaction turbine power variation equation
A.Yt ~ bF~ an+T, v~~BPrr aNf,~ o,*aPh aNT. �8n an?T, xaK~ (3.33)
where
.
an?*. vr~ ~ 1 - acr (1- a,~) - ~ a'�--~
~ -1 ~ ;
a,a~ -
QNt' Ptr a ~ ~ I ~ ~4i
\
aN=~ n = ~t -~-1~i ,
aH ~ tr ~ a~ , K(1- ac*a~j.
, *
For an impulse turbine, characteristic o f which are critical or super-
critical flow conditions, a turbine power variation can be obtained from
equation (3.23). In this case condition
aT=~~~t=0+
applies, and the coefficients of influence in equation (3.23) will be deter-
mined by the relations:
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~n+t, p~~ 1; anr,, n ~ ~pn;
aNr~ p~* p; aN~~ x�' a,;,t~ K� ,
3.4. Supply Line CqugG~.ons
ZhRD equ~.pment (pumps, turbines, gas genaratiors, combu~tion chamber~) con-
eain duct~ Chrough which liquid or gae flow~, and eng~.ne componene~ are
linked by hydraul~.c or gas lines. Therefore tihe pr3ncipal etage~ of
~peratin~ proces~es~ which detiermine the characteriet~.c~ of an engine ae ~
whol~, take place both in flow ductg and lines. Let us examine Che ~taeic
charac~eristic~ of hydraulic and gas lines.
3.4.1. Hydraulic Linee
Hydraulic linea link the pumpa with combuetion chamber and gas generator '
(pressure lineg) gnd propellanC Canks wieh pumpg (sucCion linee). nependences
nf pressure differenCial, that is, hydraulic resisCance, on rate of flow and
density of propellant and geometric dimeneions of Che 13ne consritute the
sCatic characteriatics of all lines. In Che general form a staCic
characteristic of a line ia deacribed by funcCional relgtion
Op - ~p (in~ P~ D).
I'ressure losses ~ p in the line are formed of line and local loases.
t,in~ losses are determined by hydraulic losses, caused primarily by friction `
t~y the. conveyed liquid against the wall of the line, and are characterized
by the relation
oaT~ a ~ p~ ~ ~a.~4~
where'~ i-- coefficienC taking into account liquid friction against the line
walls; wi velocity of liquid movement; 1, d~ line length and .
equivalent diameter respectively.
Pressure losses for overcoming local resistances are proportional to dynamic
pressure
arr~
OPM, r � er ~ ~ (3.2b)
where coefficient of local resistances.
?.ocal resistances in hydraulic lines can be constant and variable.
Engine lines consist of a number of sections, which differ from one another
in geometric dimensions, and may contain several different local constant
and variable hydraulic resistances. ,
~ 54
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T~tal 1n~HQ~ Edr ehe c~nCixe l~,ne ~ detiermined by th~ eum of ail componanCs '
n n
~pr ~ t~1 �pf, spM � �aM.1~ (~,~s~
VelociCy of the conveyed 1i.quid in ~ sece~.on of tiha line ie detarmin8d by
~qq~tion ~f di~conC3nuiey
tui s .
SubstiCut~ng the 1~st relation in equation (3.26), we obtain
SPt m R p; OPM p ~ (3,27)
n
where R a~- ~ 7~t ~~d totial coefficient of line losses;
e~ t ,t
n
total coefficient of 1oca1 loeses.
~
If we designate presaure at line inlet p1, and outlet p2, the line equation
will be written in the form
sp a' Pi - P~ �~R E~ P~ (3,28)
Following lfnearization and traneformatione, we obCain a line equa~ion in
the form
r ~p m. 28rit - ap 6er~ ~R bea t8E? (3.29)
where
b~, R-1- ~ t bev. 1- .
In a number of cases, for analysis of characteristics it is necessary to
have variation's not of presaure differential buC preseure at line outlet.
For theae cgses the line equaCion is wriCten in Che form
~s ��o., o ap~ - ao~, ~8m -I- ba, eaP - b aR - b
a, R p� t~~ (3.30)
where ~ . p-
aI~~A~ ~ ~ ~ ~0i. M � Z =p i f1 a ~ ~
py pi P~~ 0 P!
~p
a ~
0~~ R ~ f/p~~ E a
P!
3.4.2. Gas Lines
In propulaion systems with generator gas afterburning, the combustion
chamber ia connected to the turbine by a gas line which has gas resiatances
on the injectora and other elements.
55
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~
FOR ~~FICIAL US~ ONLY
A ga~ lin~ equn~ion~ ~~mi~.ar to a hyd~~ul~.c line ~quaCion, ie wri.tten in
the form '
~r~Pk~~r p ~ (3,31)
where reeiaeanC~ coeffici~nC; p~~ average gas 8~nsiCy in ~he ~,aa.
Average g~s dens,lty can be determined from the relat~.on
(3,32)
. Prr ~ TK ~
wh~re T2~ gas Cemperature beyon~d the turbine.
Ga~ temperature beyond the turbine d~ffere from temperature in ~he gae
ganeraCor due to the work performed by the ga~ in the turbine. In order to ~
d~termin~ ga~ temperature drop in the turbine, we ehall utilize known rela-
t ion
~a ~T rr - T~r~ � ~1t. (3.33)
From equation (3.31) we have
8 ~P~. � 28m - aprr~ ~3.34)
where
~
Opn=Pst-AK.H 8~pr~�~BP:r-e-~aPK~
Accepting asaumption RZT~Rk=R from equation (3.32) we obtain
~rr � a en~~ - ~~~t -h T~ aT't T:t + TK BTK. (3.35)
~r~m equation(3.33), assuming cp=const, we obtain
Trr 87' --T'r_ - BT~T 2&i Br~T. (3.36) .
7'rr - 7'~T r~ 7'rr - T:r
~ollowing common solution of equations (3.34)-(3.36) and taking into account
relations (3.21) and (3.22), we finally obtain gas line equations:
a ep~. _~r ~ am ~ a~P~~. X~BK~ Qppn, K�81r - QpPn~ nali~ (3.3?) -
aso � TK ~ K� arrr ,
n 2~T~r -4~ TK) rrr~ ~rrr Tn, +
ae . K- ~ T rr K' aTK .
pro 2 ~T~r ~ TKI ~rK~ ~rK = TK ~ ?
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aAp ~ n ~ ~
rr'~ ~q~
re ~ ~ ,
K'~ K" r~ei.o nf propellanC component~ in the combustion chamber ~nd g~~
gen~rator respectiv~ly.
3.5. t~re~~ure Accumulator ~qu~tiona
tn propuleion eystem~ with gas preseurization eupply ~ystems~ the propellane
component~ are forced from the Canke by preesure accumulator~. The ~nost
common are gae pr~ssur~ a~cumulatiore and eolid fue~. pres~ure accwnulaCor~
(GAD and PAD), In both eases preseure i~ the tanke, by megn8 of which the
pres~ure force i~ generated, ~.s determ~ned by the characteristics of the ac-
cumulatore.
3.5.1. Gas Preasure Accumu~ator
Cas ia fed from a tank under high preesure into a gaa pr~esure reducer,
in which pressure ie reduced to the required 1eve1, and from which it ia
fed into the tank to force out propellanti. ~'hue pressure abov~e the aurface
of the propellant in the tank will be determined primarily by precieion
of reducer performance and the characteriatica of the gae accumulator. IniCial
gas presaure i.n the tanks ig determined by the conditione of tan~C filling.
~ina1 presaure in the compreased gas tank ehould be greater than eupply
pregsure p~ by the magnitude of minimum preasuxe drop in th~i teducer ~ pp,
requisite to ensure normal reducer operation. Q pp i dete~mined by reducer
deaign and dependa on supply presaure. At high supp~ pressurea
,
Op~ ~ Ipp, where pp reducer outlet preasure~
1~0.25... 0.4.
Taking into account gas orificing and ite expanaion in the tank in the procesa
of pressurization, the gas preasure accumulator opere~t3on equation hae the
form
~P6 "I' ~Op~ V
~u ~ Pr. ~V ra,~t - ~ PsVe, (3.3$)
where V~83 volume of compressed g~s tanke; p~~H initial gas pressure
in compresaed gas Cank; V6 volume of propellant tank;
x-~
r� Kl x . T ~
cl ~ C Pn N/ ' ~f ~ T_~' .
The values of coefficienta cl and c2 are determined by the polytropic curve
index and pressure differential p~.k/p~~p a~d are indicaCed in Table 3.1.
Table 3.1.
R ( 10 7 4 4
or. K
ti 0~55 0~60 0,70 0,82
e~ 0~7b 0~80 0~87 0,90 ~
57
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~qu~tion (3.38) i~ r~~olv~d in relation to tank pr~~~ur~t
wher~ Fe b~r~ a_ b~p~, (3~39) .
~ ~
~ ~ b, ,
b ~ ~ J. ' ~
ci ~'e~ ~
Pr~~~~ra ~e reduc~r outlet is det~rmined by reducer d~si$n and preseure at
reducpr in1eC. ~he value of maximum pr~~sure deviation at reducer ouClet
from nominal (initial value) ~pp~pp-ppo is etatad in the data aupplied by
the reducer manufacturar. Ten presgure variation with a gae pregsure
accumulatcr hae Ch~ form
8Pe boe~ v~, Ndpr, - bve, ApAPP~ (3,40)
wh~re
b ~ b_
r. N, b D~ b!~ ~
pa~ P~. N p6' p ~ .
3.5.2. 5olid Fuel Preaeure Accumulator
With employmenC of a golid fuel or cartridge presaure accwnulator, fuel is
forced from the tanka by aolid fuel combuation producte. Uti.lizing equation
of sCate and balance of arrival of aolid propellant gases into the propellant
tank and rate of flow from the propellant tanks~ one can obtain a tank pres-
sure equaCion (1~:
pa = ~Va+P
~ ,
where F~ propellant charge combustion aurface; u-- rate of cartridge
combustion; p density of solid propellant; fo reduced solid propel-
1znC force; coefficient of solid propellant gas energy losses in the
tanks.
The last equation, written in variations, has the form
8ps = 8Fn 8u aPn 8f
o- 8Va - 8t�. (3.41)
s.6. Thrust Characteriatics
Thrust and specific thrust impulse are propulsion system output character-
istics, which determine its economy and ballistic capabilities.
Propulsion system thrust is determined by the relation P~mIy. '
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~Ott O~FIG~At, U~~ ONLY
In var~ae3on~ eh~ l~~e ~qu~Cion 3~ wri.tt~n in the form
8P 8m -p oly, (3,42)
3pecifia thrugt impulse, ae we know from theory of QngSne~, i~ dee~rm3n~d
by the qual~ty of organi:atiot~ of the oper~tion pro~~~~, ~atio of prdp~ll~n~
component~, eombu~eioe chamb~r p~~~sur~~ and degra~ of expaneion of ga~e~:
~y � (PK~ K~ ~J~
/ wh~re
PK
In variation~ gpecific thruet impulee is written in Che form
8/y ~ a~, x8K ~I- a~, oMBpK a~~ K~ (3.43)
where coeff3ciente of influenc~ ~I~~ are obCained frnm relation
x a~r .
a/~ a ,r +
a
XY-- are determined from the reeult~ of Chermadynamic calculation (1).
Equation (3.43) is valid for an engine with generator gas afterburning.
For an engine without genereror gas afterburning specific thrust impulae
is determined by relation (1.15)
~r ~ f r, ~ ( ~ e~), ~ (3.44)
where .
~~~~~rt _ ~r.o.el
m~ l /r. ~
besignating ~.~/m ~ Q~ ~re obtain engine apecific thruet impulse ,
variation
dly s a~, i 8~r. r'F Q~, i a9 a~, ia ~a~r. o. (3.95)
where
Ql. !r ~ ` (1 - q); .
r
a~~ ~ m~(1r. K~~r. o. e~~
~y.o.c
' ol. lu, e~ Tr Q�
59
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~ A Ghruet equation for ~n eng~.ne wiChouC gen~rator gae afCerburning i~
,
p'Pk~'~o . c �
A vari~Cion of the engine thru~t equation
dP s ap, pKBPK ap? po, oBPo, (3,4a)
wher~ ~
aP~ PK ~K + aP~ Pp. Q~
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Chapter k. ENGINE 3TATIC CHARACT~FtISTICB (DET~ttMINED PROBLEM)
4.1. t3eneral Solution
At the gtage of designing a propuYeion syetem it ie neceseary to evaluaCe
the effect of dieturbances on oper~ting parametiere in order to elaborate
engineering-design meaeures to eneure the dasired precieion and reliability.
Static character3atics make 3t poseible to eetablish Che mutual influence of
componente during the3r concurreue operation with apecific design
characteristics of motora, propellant components, and ambient conditione.
Input data far obtaining staC3c characteristics are the followingt
propuleion syetem des3gn and nominal (by epecifications) engine
operation parameter values;
composition and characteristics of diaturbing factora;
propulsion syatem component equations written in variations.
In the general case the method of calculating static characteriatics con-
sists in the following. For a concrete propulsion system design one for-
mulates pump and turbine outputs, preasures and flow rates balance equations.
After aubstituting component and line equations in theae balance relations,
one obtains a system of equationa which is writCen in standard form:
N JM
~ E a,,ayi = b,~ KaJCK} , (4.i~
where j y~ variations of operating process parametere; a xk variations
of d3sturbing factors; number of equation in the system, il1, L; ~~1, N;
Kel, M.
Syatem of equations (4.1) can be rewritten in matrix form
A~By~~B~BxI, (4.2)
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all a1! al! ~ ~ ~ Q1N
where A~ a!~ ~aa au ~ Qw m~erix of equation co~ffici~nCg on iti~
oL, a~, a~ , . , a~r ~eft ~i.de~
bti btt , , , btM
e~, b~~ b~! biM m~tr3x of ~quation coefficienee on iCe
' ' ' ' ' ' right ~ide;
bi~ bL~ , , , b~
ayl bx,
by~ ax~ ~tricee of variationa of propulsion
by ~ dx ~yeeem characterieeics gnd dieturbance
variations.
. ,
8yN 8xM
Lquetion (4.2) i~ resolved relative to any operating procesg characCeristi.c
~
- ~ ey A-~B ~ dx ta.a~
Syetem of equatione (4.1) can be formulated for propuleton By~Ceme in euch
a manner Chat roatrix A is quadratic [1~NJ. In Chis case aquation (4.3) ie
resolved ~n determinants
~
8b~ ~ ~ ~ x8xx~ ~4.4)
wh~re C~~ k-- coefficient of influen~ce of xk diaturbance on operating
procesg parameter y~, wh~ch ie defined as folloWa:
e~, x ,
~~,K� s ~
G-- determinant of matrix A; k-- additional determinant obtained by ,
substituting in matrix A, in place of the column corresponding to y~
pzrameter, the colwnr~ from matrix B,corresponding to xk disturbance. . -
4.2. Model of Propulaion SysCem With Gas Pressurization Supply Syatem
Figure 1.1 contains a diagram of a propulafon syatem.
~alance equations for this engine layout have the following form:
oxidizer system equation
Pa. w~ a PK DPo~ - nPa~No~~ (4.5)
fuel syetem equation
P6. e=' PK OPr - nPrHr~ (4.6)
62
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wher~ n(?~H statiic pre~eure o� p~opellane componenti column; n-- cgl-
cularod coe~ficient of rocket axial G-loading.
SincQ Hi and n chattge in flight~ gtat3c presaure is also a variable quan~iCy.
When calculari.ng d~viatiione of propulsion system characteriseice, for
def~n~tene~~ one aseumes atatic preseure for the inieial or final moment of
engine oper~?tion.
Ag~ume the propul~ion gyeCem hae a gae preasure accumulator; then following
linearization of equetione (4.5~ 4.6) and aubetitiution of relaeiona (3.40),
(3.29) and (3.2), we obtain a ayetem written in etandard form:
. ,
8pK - ApK~ moK ~oK - QpK~ m~ dm~ - B~Kp;
a~ 8pK 2 oaoK BmoK ~ p6~ oKbaa, oK~ Ar. N BPr, N-
- Ab. oK6D6. oKpp 8Pp -(enaK + enn) ~PoK -
~pOKbApOK~ ~OK ~ON~ ~4~~~
PK ~Pu 'I" ~ ~p~ 8i?tr = Pd~ ~vd. r~ Ar. M aPr. N-
" Ad. r6P6. r~ Pp 8Pp -(sa~ + oP~T> ~r -
~ ~p~bepr' ~ .
MaCrix ~A~ of CoefficienCs With Operation Characteristice
8pK bm~ ain~
OK~ ( pK, n~r
~ ~�Q � I a �
vk ~ ~voK o
~ I 0 I 2Apr
Matrix ~B) of Coefficienta During Disturbances
0~ FKp 8p~ I 8p~ I aP~, I b~ I ~,r
-1 0 I 0 0 0 0
I I I
G OP~ ~1- SP~* ~ I' ac~P6. oK � PP - OP~AeDoK~ ~oKl 0
0 I 0 ~ Opp Op~ I- a6. r606. P~ Pp I ~ -~De'~ep~.~ t~
I
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The mnin gy~tem determinant is
L1 A 4 L1
poN~Pr 2Pe (OPraaK~ moK +~PoKaDk, mr~' (4,81
Ag an example we shall ob~ain additi~.ona1 detierm~.nanCs for dieturbarices
~Pac? ~Pr? S~K~;
'epK, FKp - 4 e`poK sA~~ .
epK, po~ ~ z ep~ e"n~r~, mo~~
~p~~ p~ ~ 2 ~p~ ~p~ap~~
~moK' FKp ~ ~pK ~P~~
~MOK~ poK ~ ~poK l~ ~p~ +pKapK. "'r~' ,
emOK' Pr r~ ~p pKOPK, mr'
Am~~ pKp = 2pK ~poK;
.
~mr, poK _ PK ~poK~pK~ m~~
l1~c~ p~ ~ Ap~ ~2 ApoK'~' pKapK~ ro~~'
In conformity wiCh relation (4.4), propulsion system equations in variations
will be written in the form ~
aPK � Cp~.'~v ~Kp CpK~ PoK ~oK CpK, p~ aP~?
~ s C~at' rr~ ~Kp'~' CmoK~ PoK ~oK "y" CmoK� praP~'
~ (4.9)
� C~irr. fkp ~Kp'f' CM~. PoK ~oK Crii~, p~aPr~
~ arc = a~;~,K - a~~.
Coefficients of influence C~~ i are determined with equation (4.4) and depend
on nominal engine parameters.
For combustion chamber pressure, for example, coefficients of influence are
determined by relations ~
. C~ F = ~ ' +
. ~ . ~ i o,~i~ (~K + ) evoK + IR -I-110Pr )
t
~''o~~ pok -
` 2 (17 U ( 1 + 1 ' (4.10) -
FC PK \ ~PoK 1~ ~Pr )
1
COK� Pr - + 1 �
~ (K ~ ) -4- PK ~ - 1
nP,~ e~. ~
64
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.
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As follows from Che relationa, coefficienCs of ~.n~luence are determined by the
valuQa of nominal pressures in the combustion chamber, presaure differential
and ratio o� propellant coroponents. The character of change in coefficienra
of 3n�luence with change in nominal combuetion chamber presaure (ratio of
propQllant cdmpon~nts and pressure differentials ara aseumed conetanC) ia ih~
dicated in Figure 4.1.
. ,
~r
N~+!)
~ .
1) `.~N,~' ~�"r
/K
~ `'~I F~
Figure 4.1. RelaCionship Between Coefficients of Influence and CombusCion
� Chamber Pres'sure
Thus with an increase in nominal combustion chamber pressure, Che influence of
disturbances and in particular change in nozzle throat area and density of
propellant components diminishes.
As an example we shall specify the numerical values of coefficients of in-
fluence for the propulsion system of the second stage of an (Eybl Stor)
- rocket with a gas pressurization supply system [3].
The propulsion system has the following nominal parameters:
propellant NDMG + HN03
combustion chamber pressure pk, Mn/m2 1.42
. raCio of propellant components K 2.8 -
pressure in fuel tank p~.~r , Mnfm2 2.6
pressure in oxidizer tank, p 6~~, Mn/m2 1.96
The quantities in the determinants have the following values:
~PoK = P6. oK - pK = l,96 - 1,~2 = 0,54 Mx/K~:
~Pr = P6. r-PK = 2,6 - 1,42 = 1,18 Mx/w~.
1~ 2,8 .
' apN ~ +it~ - K ~ = q~g ~ _ ~~72 +
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~ ~ ,
^ . QpN, mr ffi ~
~ ~ ~ ~ ~ ~
A+~ 4~pok ~P~ ~ 2PK \~P~~PK~ mcK ~Pok~p�, in~,~ ~
~ 4~0,64~~1,18 ~ 2~ 1,42 (1,18~0,72 0,64~0,28) ~ b,39; ,
~pK ~ FKp ~ -4 SPok,Pr ~ - 4 ~0~54, ~ I ~ 18 e - 4,6b;
~pK, paN ~ z epoK sp~aPK, ~ s~0~54~ 1,18~0~T2 ~ 0~~~
~pK~ p~ c, 2~po~.eP~opK~M~ = Z�0~64~1~18~0,28 ~ 0,36;
. ~"'oK~ pkp ~ 2 aprPK 2~ 1~18~ 1~~2 ~ 9~94; . .
' � . . Amo~~ PoN ~poN ~2 ~Pr PKop~~ , .
- ~ 0,54 (4�1,18-~-1,~2~0,28) ~ 1,~9; .
~moK, P~ ~P~PKapK, m~ e--1,18~ 1,~2~Oi28 n-- 0,47;
S;n~~ pKp C 2APoKpK ~ 2~O~b4~ 1~42 s. 1~53;
� . . ~
eMP' poK n r~P~KpK�nK, m~ - 0,64 � 1~42 ~ 0~72 0,6b;
~m~, p~ ~ Opr DPoK -I' PKapK, mr~ = 1,18 (2�0,54 -{-1,42�0,28) ~ 1,~6.
Coefficients of influence C~~ i are contained in Table 4.1.
Table 4.1
Coefficient BPKP ( 8p0K ~0~ ,
- ~1~ -0,47 0, I M 0~067
~mpK O,6Z 0,28 -0,08T
, 0,28 -0,09b 0,33
4.3. Model of Engine WiChou~ Generator Gas AfCerburning
Figure 1.3 contains a diagram of an engine wiChout generator gas after-
burning. For an engine of this design, balance equations (of compatibility
of components) written in variatinns, are Che following. The equation of
balance of outputs is
s
NTBNt = ~ NM,~6Nx.i+ ' . (4.11)
where j = ox, r. ' ~
Equations of balance of pressures in oxidizer and fuel lines from pumps to
combustion chamber:
66 ~ ~
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~Ok UF~ LC I AL ll;i); ONl ~Y
~
,
~ ' pr~ oK BpH, oK = Prr aPK ~I- ~PoK 8 ~Pok~ '(4,12)
Px, r BpN~ r= prr 8pK ~I- ~pr 8 ~pr.
- ~quation~ of belance of preeeures in gae generaCor 13nest
~ r� M
PM: oK apN, oK a Prr 8prr -I- DI~oK 8 ~puK~ (4.13)
pN, r a
pN,,r ~ prr B
Prr - I- O Pr 8 ~ P~�
Equations of balance of flow rates:
-r � s~ r . ~ -r: . ~
�moK ~oK ~ I/1oK8moK, moK~oK~ (4~ 14)
r . ~ ~s~
m~ 8m~ ~ llir ~Ilir mr ~r
5ubetituting in equaeions (4.11)-(4.13) equation (4.14) and equationa of
componenCe (3.2), (3.6)~ and (3.29), following eimple Cransformations we
obtain a aystem of five equatione, which in maCrix form appeare as followa: ~
A ~ay~ mB ~dx~~
BPac
bm;~
8m~
I ay dm;K i ( d, ~IN. r ~
where ~ bmc '
bn ' -
bF~
bF,~
The elements of matrix ~A~ are contained in Tab1e 4.2, and the elementa
of matrix ~B~ are in Table 4.3.
(See Table 4.2 and Table 4.3 on the following two pages).
4.4. Motor With Cenerator Gas Afterburning (G-L)
Figure 1.4 contains a diagram of a motor with afterburning of (oxidizing)
generator gas.
Equation of output balance:
s
IVT 8NT =~i v�~ BNp~. l b)
67
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~ ~o~ o~rict~, usr orrLY
Tab1a 4.2.
. ~ 4 ~ k ~ , ~
I a . d ~ ~ ,
r ~ ~
~ I~ ~ ~ ~ ~
I~~ I~ ic~ ~~i id~
I ~ � .
_ _ ~
~d ~
c
I I ~ ~ N
� ~ ~
~~r ~2 4
, q d' ~
~ ~ ~
M ~
~I I~ ~ ~ ~ ~O �
I~M ~ ~ i~
I
~ ~
. ~ ~ t
~ ` ~ ~ ~
~ ~
~ o . a , ~
~ ; ~c ~ d' n'~ ~ ~
~ ~ ~ ~ 4
Q z" IZ~ V " ~
~ tl ~ ~ o ~
IZM ~ ~
. I
I
� y
IL I ~ ~ 'd � 1~ t
~ ~ N ~ ~
i�E' n n" ~ a ' o eo .
I~~ ~ ~I~
. I� ao
M
� C1.
i
I
O � .
~ . . .
1 CL
~t a � o ~ � o ~ ~
a N ~E ~ �E
.o ~o`~ I tl4 ~i o
~c
IZ � ~ q ~ . 4 ~
~ ~ ~ I~~o ~
. ~
IE
68
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i
~OR OFF~CIAt ttS~ ONLY '
e ~ ly I" � �
Tab1e 4.3. ~
_ `
~yh e e o e
_
O O O O 1q4
a
e o e ~$g e
- r r ~
, O O (44 O O
~ '
a
o ~q~ e e e
a
- - -
4`
~ ~a 4~
e ~ e
9
14~ I4~
~ ~
4� 4~ Q~
o �g
9~ ~ �
~ ~
~ ~ ~ o~
, r~e ~ e'~ ~
` . . ~ y
$a e o ~ 4d e i ed
I I
~
$o a 14 c 1~?d o
. I I
M
~ '~R O O O O
~ '
~ IZ ~ O O O
O ~
. ;a ~ (
_ ~ �
a`
y ~ 4
~ o o � d
o~ -0p ~y ~
li Q ~ 4
~o I
I �
x
� ~ qo
v'~ 'aY ' I
o~ a a
,o; � o ~ C
~ ~a" Qd
0
12 o Y
0
ia ~os
~ I
69
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~OFt O~~ICrAL U5~'dNLY
Eqnations of balance o~ gas generator 11ne preesures;
pN~ eK BPN. oK ~ prr 8prr -4- ~Pok 8 ~PoK+ (4,16)
pN, r BPN, r a Prr 8prr -I- Sp~ 8 ~p~.
~quaeions of combustion chamber line presaure balance:
pN. r BpN. r~ PK 8pK Opc 8 Opr~ (4.17)
P~r 8pst ~ PK 8pK -I' ~pre 8 ~Pre~ (4.18)
Equgeion of fuel flow raee balance:
, � - - � . ~4,19)
iri~8iit~ ~ I=?i~8/fir m~8m~�
After substiCuting equation (4.18) in (4.15), uCilizing the relation for
componenta (Ch. 3), system (4.15)-(4.19) reducea to a system of four
equationa, for which the matrix elements have the form
~oK
. b~oK ~Pr
.
B~N. ac
I by I� bm~ ~ ~ bx ~ m : .
8n
BFKD ~
The elements of maCricea ~A~ and ~B~ are contained in tables 4.4 and 4.5.
G~yT, A~
Z1 - aNT~ PrraP~~~ maK + ptr ~PKapK, ~p~
+ ~P~~ ~�Prr' ^~ac + a~pr~. K�J I + aNT ~ K":
� ZZ a Pi~ CPKaQK~ in~ - ~Pr~Apro. K'~~
e~.
Z~ _ oNr' prr�prr, m~ �Nr prr 'pn ~��rr~ - as~n. x�) ~
~Pri
~ ~ aN*~~+ Ptr aNT~ pt ~~Or�~ � `
.
Mr . fi~
9r'.K~-~ 9r,ra �
, m~ . mr
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Tab1e 4.4
.
Iin~ Ont~ 8m~ bn
tlA~ - li1r, ac x~7~'r - Ap. r Z~~r ~~At - Fj TVM pN~ n
~
" � -
pr, o~pr. a~ Mat
-"prr~~~~, - 0 � Frraar~. M~ Pe. aeaP~. ac, n
- ~ ~PoK
~
~ ~ ~ ~ ~ ~'~pTf i ~
- Prr~p~r~ moK pa. r Pe, r' p~. raoM. t~ n
~aPrr
. _
pe, rapM, r~ "'r
'Q ~ ~ p a r
PK p~, m~ K p~ m~ 2 Apc9rr ~ Pw. raox. n
K
- 2 OprQr. K
4.5. Motor With Generator Gas AfCerburning (G-G)
~igure 1.5 contains an engine diagram.
We ahall deaignate the parameters of an oxidiz~.ng gas generator Z~ and
reducing gas generator Zr .
Output balance equaCion
8~ a BNe.OK~
8N* = BNM. r� (4.21j
Gas generator line presaure balance equation
pM. oK aPx. Prr aPrr ~PoK a sd~~; ~
Pn. oK aPx. oK = Prr aprr ~poK a ~~ri (4.23)
pN. ~ BPr. r=~Prr 8pcr Ap~ a~Pri (4.24) -
Pp. r 8px. r~ 8P'~rr 8p~r -F epr 8 Apr. (4.2~
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~OEt O~FICIAL U5E ONLY ~
~ ~ i c~
~ o 0 0 ~
Tgble 4.5. ~ o 0 oJ
~ -
' o o ,o~ c
~O , y
t~4
r.r. e + .
M/
~ 4~ � i,
O ~ O ~.p~
. 1q 1~
G
`~i C O O ~
. G O
L L
~ ~ ~ ~
c~ .
(Z~ 1
Qo q
o
t
S~ ,~O � �
i~ ~ ~
h ` .
$d o lo It! 1~
~ ~ '
~g o ~~d o 0
. ~
w
w ; .
� o 0 0
~ '
O '
O ~
~ ~Z~ O O O
o I
OV 04 I
4
Z. 4 ~ , 4 ~ .
m Q
$ ~ ~ � ~ '4. ~
Iza ~ i
I
-
~ ~
~ no
0
0
_
~s~ aZ ,pY O O
�L O
O .
I~ ~ (
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CombueCion chamber gae ducC balanca equation
~~�r d~�r'* pK 8PM `1' ~P~�r~ d Ap~~i (~.26)
- a~ir " P~ epK -h ~P~~ 6~p~~~ N~~
F1ow ratie balance aquatidna
moK BmoK ~ m01t d~K ~ mON 8mOR1 (~.28)
rtt~ bntr ~ m~ m~ (~.l~
Sys~em (4.20)-(4.29) reducee to a sys~em o� six equatione, tihe matrix
elemen~s of which have the form
~ �o
BatoK ,
~oK
, ~8y~~ 8mr
8rn~ ~
8n~
8K�
1 The elements of matrices ~A~ and ~B~ are contained fn tables 4.6 and 4.7.
Table 4.6.
AmaK BmoK bin~ Ae~~ 8no bnr
Qit Qt~ et~ as~ oi~ ~
au M~ a:a 0 u:~
a~l ou a~ 0 ai~ 0
c~l a~ 0 ou a~~ 0
a~i 0 aN aw 0 a~~
0 aa Qa Q~~ ~ ~~o
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rOlt OF~ICIAL US~ pNLY �
4~
.
'~able 4.7. a o 0 0 0
'a
- -
� O ~ O O C O
' O O O O O
, e~. ~w~'
e 1~y~~ �M
O O o ~a C?
~ ~
.
, o o �~g o 0 0.
_ 'Or
. p~o
- -
~x tl~" tlz o 0 0 0
~ b`IP~ ~ ~I~'~'
~
~t o ~ o 0 0 0
�b
~ ~ o 0 0 o O
40 40
L L
0 0 0 0 0 ~ ~a�
~O y~
14 It~
( (
~
� ~ ~
~ 4~ 4
. d O O ~ ~ O O
~ 1~ Iq
~ ~
'6 I4
i. p4~ i.
,g` o " o o c~a o1�a
.~~tl ~ ~ I
~a
I I
a� ~
v
~ I ~
a o
~ p o4 ,pY 1 ~n; Ia o p -
-0Zm a o ; ~
I i~ ~a
~ ~ 74
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a11 aN*' A~~apPC' moK'}'9o (~aNT~ p~
pK~ M~ -aNd~ oK~ mq~ "t'
~
K� aP~.
j P1r aNT~ par C' ~ aApo~~ N, + N~,R~?
a
a~ qr N.= r o � p-}- epo 1- o � ~
t9 o p~r ~ AK~ mq~ K r~~ ~?x. oK~ "~an
po~o
q~ d a N--T 0 0 . ~
13 N*' p~r pcr' M~ P~r [9~ CPK ~N~ M~ pon~
t
~p"r~ ~ K, ~ asp~~~ R~~ ;
9r
al~ a aNT' Pir \pKavk~ in~ - 0~
aep~ ~ K�
. ~P~.
at6 N,�r~ n� - ~dNa. oK~ n� P~ a~p~~' n�i
' al~ �
a r r qp .
a~t � N "p~,~ (PeapK~ '"oK'~' ~pr~a0pe~~ K) '
\
a � 90
~ a~~ � aN*, P~rapp~. MaK ~ NT~ pZr ~ X
X I pK�pK~ moK 'I' ~P~s ~~-f' aep~.~ Kl 1'~' aN=~ K~i
~ /J
a r r
~ ~ 9~ ~ ~pKapK ~ m~ - Opre, - aN2~ m'
~0 NT ~ p~ ~ i
~
Q'14-a r r a r ~"~'a ~ ~"_X �
Nt~ APf pPC' ^~r Ns� p'ir P~ �
X [PKapK, m~ AP~~ (1 -{-aep", ~11 -uNT~ K~ -a~�x. r~ '^~4~'
~ JJ
Q31 - px. OKapq. OK' T~KQO - P~r~ o o'- 2~QoK~
PPt' "~oK
~32 - pe . oK90apr. oK ~ ^~pc :
~--pfr� o o~
pCI'' mr
~35 - Px. oKap~ n~:
~~1 = Pe. o~0uh. oc� m~:
a~2 = a41 -Prra ~ . ~ -2Apr~~
pfT' moK
~4~ - - PCf a ~ ~ :
pf't' ~'r '
a~b ~ Q3bi Qbl ~ ` A('T ~ ~p~QTI' ~
75
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~ ,
a69 " pr~ ~aPr. r~ m~9~ - osi - Z~�; -
_ r ~
db1 PM, r~pw r~ n~r~0
a6A �Q pN, rapw M~~
Q69�~PpfA01' �r i
. ~ rr~ "',n
- p~ pt, rap~ +~~i
pM pr, r~pN~ mr
r_ p~raprr~ m~ ~Op~~ ,
~eA pN. ~ap~ Kr�
4.6. Synthes3zed Engine Charateriatica
As a reault of solu~ion of matrix equations for a epecific engine, we
obtain equ~Cions o� the type
8y
j = ~Cy~xlBxr,
which link an operating parameter variation with deviations of dieturbing
factors from atandard or nominaX valuea.
Table 4.8 is construcCed for a motor, and depicCa the degree of sensitivity
of operating parameters to varioua disturbancea.
Table 4.8.
bx!
~oK ~r ~RKp e~
a~~ .
aQit . CpK. PoK ~'pK, p~ ' CpK~ PKp . L`pK~ ~
8P Cp~ PpK CP~ ~ CA~ ~Kp . . . , Cp~ ~ ,
b/y C/~ PoK Cl ~ or CI. FKp C/~ ~
. . . . . ~ .
. . . .
8n Cn~ P~ Cru 0~ Cn~ AKp Cm .
For a specific engine design, coefficients of influence Cyi~ Xi are the
essence of the number, since they are determined only by ttle nominal
cperating parameters.
Table 4.8 enables vne to determine for specified values of disturbing
factors ~ xi the values of operating parameters and to obtain input data
for tuning and ad~usting a propulsion system and for designing control
sysCems, as well as elaboration of operating processes, for example, con-
ditions of thermostatic control.
~ 76
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The value o� an operat~ng parame~er for variation d'y~ is determined from
relarion
~,=~,~i +8y,).
In addi,Cion, the c~lculation reenlte contained in Table 4.8 enable one to
eatimate the aeneitiviCy o� Any operaGing parameCeY to a given disturbance
and to elgborate measures for compensation for the strongast dieturbances.
Engine equn~iona written in variaeiona make iC possible to synthesize
dimenaional circuita o� componenta and engine for rhe purpose of optimal '
distribution of tolerancea on geomeeric dimensiona of componenta.
In general manufacturing tolerancea for the fabrica~ion of engine atructurAl
componenta are selected from a number of conflicCing conditions.
Following are the principal demands impoaed on Colerances: design com-
patibility o� components, producibility, coat (the amaller the tolerance,
the greater 'the cosG of manufacture), and prec3sion. In general one can
s~ecify in a motor a small number of geome.*.ric dimensions of componenta
which significantly affect the value of the principal operating parameters
pk, P, Iy, K, which determine economy, precision and operational reliability.
These dimensions include nozzle throat area FkP, pump impeller diameter DH
and turbine rotor wheel diameter DT, diameters of hydraulic resistances
(ad~usting disks, 3ets D~),, plus several others.
For analysis of tolerances as determining parameters, one can select combustion
chamber pressure or specific thrust impulse. Utilizing Table 4.8, one can
write an accuracy equation for any of the specified parameters:
Sy = ~ Cy, x~8x~? (4.30)
where
X! ~ FKp+ ~M+ Dr
In calculating tolerances one can make the following assumptions regarding
the law of sutmnation of particular deviations dxi, which can be selected fmn ex~dn~-
Cion of three cases. The worst case is wher~ change in all components only
increases full deviation of the operating parameter from Che desired value.
A statistical case is iahere changes in ~`xi are viewed as random quantities,
and ~y is a random quantity. This case will be examined in general form in
the following chapter. And finally, one can conaider a composite case.
When calculating the worst case, we rewrite equation (4.30) in the form
n
sy ~ E_~ l I d~, ~4.ai~
�
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wt~erh j-- maximum allowable daviation; d~ maximum deviaCion of dimen~ion.
Thiy meChod is uC~.lized when the number of examined dimenaions dx~ is emall.
One can determine d~ as follows from equation (4.3~). Maximum al~.owable
devi~tion j and sensieivity Ci are known. Let the absolute values of par-
ticular deviaCiona ICi.~d3 be identica~. for all elements, that is, we assume
uniform d3stribution o� particu~.ar deviations.
From Che raCio of max3.mum allowable deviation ~ to the number of elements n,
we deCermine particular deviaCions
n � ( Ct I dt� (4.32)
From Chis we calculate component tolerance
dt ~ n I I C~ I~ (4.33)
Thus if the corresponding sensitivity is sma11, the dimension of a component
'will have a greater tolerance, and vice versa.
4.7. Maximum Propulsion System Running Time
The principal external factor influencing propu3.sion system characteristics
is temperature of propellant components. This is due to the fact that
propellant characterist3cs (density, viscosity, enthalpy) are dependent on
temperature, and a change in these characteristics leads to deviations in
flow rates, thrust, and other engine characteristics.
It is convenient to characterize the influence of temperature of propellant
components on propulsion system parameters by maximum running time, that is,
propulsion system running time to total exhaustion (depletion) of ~ne of
the propellant components.
As a consequence of the fact that the dependence of physicochemical
cl,aracteristics on temperature differs for oxidizer and fuel, maximum
propulsion system running time for oxidizer and fuel differs in magnitude
and is determined by evident relationships:
_ MoK ~ _ M~ ~ (4.34)
oK - mr
� mOK
�~here ~iok, M~. masses of propellant components in a rocket's tanks.
In view of ehe fact that propellant mass and flow rate are dependent on
Cemperature, maximum propulsion system running time, all other conditions
being equal, is also determined by the temperature of the propellant com-
'ponents.
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Zn a linear approximak~.on, we stiall deCermine variation of maximum time
with the equation
8~,a ~ 81N~ _ 8�ioK~
8~'~ ~ 8M~ - 8ritr, (4.35)
where
. 8y y�' ToK, ~'r, ~'1oK~ Njr~ jnoK~ m~.
Variations of flow rates of propellanC components are deCermined in
solving the problem of influence of external and internal factora (Table 4.8):
Bm,K = ~:nioN~ poK $PoK Cm~~ prb~~i
(4.36)
bm~ = Cmr~ PoKaPoK `I- Cm~~ prSp~�
The dependence of density of propellant components on temperaCure is deter-
mined by the relation
PJ � PJ -F- ~T1-' TI)~ (4.37)
where j= ox, r; temperature coefficient; T~ standard (nominal)
temperature of propellant component.
Table 4.9 contains the characteristics of several propellant components.
Table 4.9.
1 � X~MHq~CK~11 p, K~ npR ~~r
H~aeexne Kownoxexra ~opryna � M ~ w�
. 288 K
A30TN8A I(HGIOT2 4 HNO~ ~ 63 1b20 -1,66
4er~pexoxNCb asoTa 5 N 0 92 14~U -1.98
TeTpBHNTPOM0T8H 6 C(I~O~~ 196 1650 -1.7
KNCnopo,q 7 0~ 32 1140 ---4~4
(90 K)
IIepexecb Bo,qopoAa 8 H 0~ 34 1460 -1 ~6
~rop 9 ~i 38 1510 --3~2
Kepocex 10 Cz?~siHl~~so 100 ~ 834~ -1~15
17(Y~�)
Bo,qopoA 1 H= 2 710 -4,3
CnNpr ~87HJI~ 12 CyHbOH 46 789 -1,32
H~IMr 13 ~ ~CHa)sN~H~ s0 785 -1,0
rHj(P03NHCHJ~8T1/{ (NH9)aHzO 50 1030 -l,l
rNpp03HH 1 S N9H~ 32 ~0~~ -]~Z
CnNpr Merenoaa~ 16 32 791 -1,14
79
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K~y to ~rabie 4~9 on preceding paget
L. Componene 9. ~~.uorin~
2. Chemiaal �ormulg 10. Kerosene
3. kp, gr 11. Kydrog~n
m3 12. Alcohol (ethyl)
4. NiCric acid 13. NDMG
5. Nitrogen tetroxide 14. Hydraxine hydrate
6. TeCranitromethane ~.5. Hydrazine
7. Oxygen 16. Methyl alcohol
g. Hydrogen peroxide ~7� Conditional
~or noncryogenic propellanC componenta, when aeparate thermosCatic c~nt:ol
i~ not employed, ehe following condition is met:
T~ ~ T~ .
gnd in thig case '
aP~ = ap~, rBT~ ~ (4.38)
where
T
ap~~ r ~ P/ ~
Equation (4.36), taking into account (4.38), is wriCten in the form
8m�K = B�K8T' (4.39)
8/it~ = B~BT,
whcre
� o~ + ~ ~ ,
BOK =~~mOK' ~ON PON ~m~K~ p~ Pr
(4.40)
Br ~ CC~ -�K -~-C� T.
m~~ poK PoK ~nr, pr P~
We sh~ll evaluate Che signs and relationship between coefficients B~. The ~
following qualitative relations occur for noncryogenic propellant components:
PoK ~ Pr~ p=" = 1,8 = 2,2~ ~ox ~ ~r ~ ~
Pr
~~oK ~~r oK ~ 1_ I,~;
' r
~'mI~ Pr ~ ~'m~~ Gj C
I('mo~c' PoK I~ I C~oK� pr
- I C~"r� po~c ( C I C~"r' P~ I'
, 80 ,
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Cm~ ds 2,2 -t- 2~i
. moK, p~
C.
Mp~ 0~ 2~6+ 18~
C^~~~ ~oK
nn the basis of ehe above relaCiona, it fo~.lows from equaCione (4.40) that
r~nk ~ Q~ 8~ ~ U and ( BOIC ~ ef I~
D
Thug witih an incr~ase in the temperatur~ of propellant components prnpellant
flnw rgCes decreaee, whereby the degree of change in oxidizer flow rate ig
lese than thati of tihe fuel. Variatiion of mass fueling of propellant com-
ponenr~ dep~nds on tl~e meChod of fueling.
a) ~ueling by volume.
Wieh fueling by volume, the mass of the fueled prope].lanC is
,
M ~ Vp, (4.41)
where V--- volume of propellant pumped on board.
Since wiCh fueling by volume the volume of propellant pumped on board
depends on the temperature, taking (4.38) into account, equation (4.41) has
Che form
bM K = c,Kar;
where 8M~ = L~BT, ~ (4.42)
L~ _ i .
It follows from the correlation of quantities and P~ that ~
L~ < Q; ~ LoK I< I Lr
Substituting (4.42) and (4.39) in equation (4.35) we obtain variations of
maximum time for fueling by volume
~o ~ ~LoK - BoK) bT ~ (4.43)
az~ ~ ~L~ - e~~ aT.
Since B~ ~ 0 and L~ t 0, the sign of the coefficient with ~Q is determined
by the correlation between L~ and B~. Results of propellant calculations
' indicate thar for motors there occurs relation
~ L~ ~ i ~ BI ~ and ~ LoK - BoK 0; ( L~ - Br 0.
It follows from this that with fueling by volume, with an increase in tem-
perature of the propellant components, maximum running time for oxidizer
and fuel diminishes.
81
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~
~
. ~
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, b) ~ueling by mase.
WiCh fusling by mase, the mass nf tihe propellant pumped on board ia in-
dependent oE remperatiure~ tiheC is,
sMo = 8MM ~ o. '
In Chis case it followe from equation~ (4.35) and (4.39) that
8~'0 ~ - f~oKBT;
ar~' ~ _ B~ar. (4.44)
S~nce I BoK ~ B~ ~ and BoK < 0; B~ < 0~ with an increase in
the temperature of propellanC componenta, with fueling by masa Che maximum
running rime increasea, whereby �or fuel this 3ncrease is greater than for
oxidizer.
Figure 4.2 shows qualitative relation ~~mf(T) for bo~h typee of �ueling.
J . L
j t' ,
y~ L, M
IIK
/
~IIf
~ > >
Figure 4.2. Dependence of Maximum Running Time on Propellant Temperature
Moet�
- - - - - -
M~
T T
Figure 4.3. Dependence of Fuel Remaind~ on Temperature
It follows from an analysis of Figure 4.2 that when T~ T ma:imum propulsion
system running time is determined by the oxidizer supply, while when T~ T
it is determ~ned by the fuel supply.
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This meane thgC when rhe engine ie ~hut down undet~ condit~.one T�T, un-
congumed components r~main nn boatid rhe rocket~
The mase of the unconsumed propellant depende on remperature and ia ea-
timated by quantity
Mo~* ~ ~ ~'oK - I m~~ (4.45)
where ~~ok when T~! T, ~~r when T~ T.
Relation ~4.45) ia ehown in F~.gure 4.3, where M ~ guaraneeed fuel reserves.
The engine is ~hut down by an automatic preaet range control unit.
GuaranCeed reserves are enaured in order to elimina~e the posaibility of
premature engine ehuCdown due to exhaustion of one of the propellant com-
ponenCs.
Guaranteed fuel reserve refers to the minimal quanCity nf propellant
gbove standard which enaures, with a apecified probability under gii operat-
ing conditions, engine shuCdown by the automatic preset range control unit.
In order to reduce fuel residue MocT~ and conaequently passive rocket mass,
control systems are employed, which under all conditiona ensure simultaneous
tank emptying, thaC is, condiCion
'~oK-'~r = 0 and Mo~T�0.is fulfilled.
r
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Chapter 5. STATZSTICAL ANALY5I5 OF P1tECISION OF
MOTOR OPERATION
5.1. Lawe of bistribut3on of Operation ParameCers '
For analysis and c:alculati3on of precision and reliability of engine operation,
it is essential to know the lawe of diaCribution of operation parametera and
their statietical characCeristics. .
In pracCice one very frequently encountera normal distribution. This is due
to the �act Chat it is Che maximum d3atributiion toward which other distribu-
tions approach. In addition, it follows from Lyapunov's theorem thaC dis-
Cribution of a sum of a sufficiently large number of independent or alightly-
dependenC random quantities with random distribuCions approaches normal dis-
tribution. .
Densiey of normal distribution ia described by relation
~P ~y) _ ~,~Y2n eXp ~ ~
Therefore when processing statistical realizations one first of all tests
the assumption of normal distribution. Testing of distribution is a task of
statistical verification of hypoCheses, that is, tesCing of the adequacy of
empirical laws of distribution of samples obtained durings tests to theoretical
normal disCribution. The procedure of testing the law of distribution is as
follows.
Actual data on the aggregate of operation parameter values are determined as
~ result of tests and engine operation. If we designate the magnitude of
r,easurement of operation parameter y~,then or the aggregate of all measure-
,.:ents we can construct variation s erie s~y~~ , for which sequence
yl~ Y2 ~'yn is valid. For formalization of calculations, we make the
_ series denser by breaking it down into separate intervals, and in each interval
all values are replaced by one, corresponding to the middle of the interval.
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yRfyJ
y
Figure 5.1. Histogram of bistribuCion
~ Ineerval length is determined by relation [18]
e~ oux - min
~ 3~2 oQ N ~
For each interval one calculateR frequency ~
P*~=N ~
where m~ number of ineasuremenCs falling ~vithin interval .j; N-- number of
all measuremenCa.
As a resulC of this processing we determine an empirical distribuCion
function comprisiag accumulated frequencies in intervals
F* (y3 E P*~ ,
~�1
and for each interval
y~ = 2 fy, + y,+~)�
On the distribution function we plot a hi~stogram (Figure 5.1) or empirical
diatribuCion density
= F ~(bi)
b '
Adequacy of empirical distribution to normal is tested on Che basis of
statistical criteria which have been well elaborated in mathematical
sta~tistics. The Pearson criterion is most convenient for calculations [28j.
Criterionx 2 is based an statistical distribution
x~ _~(m
f- NP j)/NP j . (5.1)
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in the gseumpCion Chat p~rt p*~ of ~.ncid~nce df random quanCi~y y~ in cer-
tnin interval jL~_1, L~] ig deCermined direcCly by sample ~y~~ , while
� e~rim~te of probabi~ity of occurrence of evene P*~P[L~_~ ~~ ~p~~�Prr.A-Prr>0;
~Crn ~ T~t.A - T~T>
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wh~re gubecript ,q de~igngCeg gllow~ble parameter values prescribed by
t~chnicnl gp~cificationg.
A11 the enumerated parametera are random quantiCie~ for a epecified point
in time.
In analyzing geaei~Cical characCeriatics, Q y~ and m~,~ are derermined for
them.
Allowabie par~meter value~ are prescribed by technicQl apecificaCione~ and
therefore Chey can be considered d~t~rmin~d, Chae ig, a0. In the
general case, when y~ and yi A are random quanCitieg, and undpr the condi- .
tion that they have normal igws of diatribution, the probability of fulfill-
mene of condition of pfficiency will be determined by the equation
M
p (V ~I > j ~P (V d~p a ~?5 -I- m (z~), (5.13j
0
where ~~~(?j~ disCribution function of ~l' ; t~ (z~ Laplace function;
Q~ (-z) ~ -Q~ (i); z~ = m,~~/v,p~.
Since an efficiency functiott is a auperpoaiCion of two normal functio.~~
~ (Y~ ) and ~ (Y~A) , then
~l � mylA - "~~1?
Y ~ ~ ~s. i4~
~W/ ~ Q~/ + Q~/A - ~~Q~~Q~~A~
where j~ pK, Pcr, K, Tcr+ /
r�
If there is no correlation between y~ and y~A ~f~ ~ then
~
Q~~ ~ VY~ ~ Y~A.
Obvious conaequences follow from relation (5.13):
a) when m~ m ; ~D ~Z~) = 0 and P~~~ > 0~ ~ 0~5~
r~ v~A
b) when m~~ < m~~A; ~(z~~ and P('Y~ ~ ~ ~~5~
c) when "t~~ ) "t~~A' m (tl) < 0 and p ~ ~ 0,5.
5.3.3. Condition of Reliable Combustion Chamber Cooling
As an example we shall examine the condition of combuation chamber wall
temperature efficiency.
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~'Olt U~i~'TC [AL lISN: dNi,Y
'Cn ehc general ce~e Che combuetion chamber ~f a ZhitD con~isCs o� twa con-
eenCrin ahell~. A propelLanC componenr move~ b~tween th~ ouC~r and inner
shell, gbsorbing h~~e f rdm ehe inn~r ehell, which ig heat~d by combugrion
products.
With ung~tiefacCnry ~ooling, Che inner wgLl of the combugCion chamber cgn
burn ;:hrough, wieh gub~equenC failure. ~
A condition of reliable cooling is
Q~ Qn Qor ~ Qa~ ~~.l~i~
wh~re Qg quantiCy of h~aC accumulaeed by the syatem dur.ing operation;
Q~ quantiey of heae conducted to the wall; Qor quatttity of heat
remov~d by liquid coolant from the wall; Q~p allowable (crirical) quanCi-
ty of h~at accumulated by tihe wall maCerial.
puantities Q in the general case are random funcCions of the wall co-
ordinaCes an~ operating time. In addition, Q~ depends on other random
argumenCs.
~or example, Q~ depends on rate of flow, ratio of propellant componentg
and other facCors. QoT depends on rate of flow and thermophysical proper-
ties of the cooling propellant component and iCs flow conditions in the
coolant ~acket; Q~ is determin ed by Che mechanical properties of the
material.
We can demonstrate that condition (5.15) is equivalent to the following
T ~T~ or
+V r= T~ (~r~ x~ J~ t) T(z~ y~ Z) > 0, (5. l6)
where T(�) inner shell heating temperature; TA shell critical
temperature.
The process of heat exchange in the combustion chamber takes place as
frllows. Heat from the combustion products is conveyed to the walls as a
result of convective and radiant heaC exchange, spreads in the walls due
to heat conductivity, and is further conveyed to the liquid coolant.
Gas wall temperature T is the principal characteristic on which one ~udges
~ooling system efficfency.
"otal specific heat flow under steady-sCate c,perating conditiens, transferred
~o the liquid [5]:
T~-T*
9~ = t ~ a� 9s ~x~~
+ + j~
where
7'M = 7'e: ~ 9s ~ ~ = T.x -f - O T = T,~ (z) ,
~M+~x
s
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~ ~Olt nFF'zCIAL U5 ~ nNLY ~
Cemperatur~ of liquid;
a~ m ~C~O,A~()..1,84 d ar ~x~
coefficient of h~at tranefer from gases Co wall;~ ,
a~ a 27C,~ ( 0 1*.~ ~P~~~.e a a~ ~X~ ~
!
coefficient of heat Cransfer from wall to liquid; 1, ~ wall
thickness and coefficient of heat conducCivity of m~terigl; D~ equivalent
diameter;
~ To e~O~I~r ~
T~ calculated eemperaCure, taking into account heat transfer by radia-
- tion; Tp gae st`agnation temperature; x-- coordinate coinciding with com-
bustion chamber getzeratrix, figur.ed from inducCion manifold.
For stead~-gtate operating conditions and specified combustion chamber
geomerry, f0 is independent of x, and wall temperature changes insignificant-
ly lengthwise.
'Cherefore change in convective l~eat flow along combustion chamber length
is determined chiefly by chamber diameter (D'1�82) and reaches maximum
value in the nozzle Chroat area.
Wall temperature depends on thickness, and tt can be determined as follows:
T'r � T a-~T
r. R- T x. a= 7' ~X~ ~5.17)
where 0~ z~ S-- wall thickness coordinate; Tx. n:3 T,~ a* � wall
svrface temperature on coolant side;
r~ n= Tx 9~ Wall surface temperature on gas side.
Allowable wall temperature value T~ is approximately determined ~ _
TA~T~.A-x~Tr.A-T:,~, (5.18)
where T~.p~ and TX~-- allowable wall temperature value on the gas and
coolant side respectively;7G delta function, X=0 when z=O;x =1 when
z= Quantity TX~q is determined by temperature of thermal decomposition
or co~lant liquid film boiling Cemperature. Probability of reliable com-
bustion chamber wall cooling is determined by inequality
P(~>0)~P~T,~(t)-T(x,z)>0~. (5.19)
Determination of P(~ T~ 0) is performed as followa.
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Several points muat be selected on the combustion chsmber surfgce. However~
since a maximum heat flow ~ake~ place in Che nozzle throar area, one csn
limit examinat3on to checking reliable cooling of ~ust this areg. On the
z axis (wa11 thickness) one should examine two points: z=U and zm s. :
In this cgse (5.19) breaks dnwn inro two:
, pl ~ p [TA(0) - T (xKP, 0) > OJ = P ~~r~ > 1 (5,20)
Ps ~ p(TA(8) - T(xKp~ a) > 01 ~ p~V~r~ > 1
'The sought value P(~ i~0) is detiermitted by formula (5.12), which for N=2
assumes the form
P~V ~r > 0) = Pl �PQ (P~, m- p1Ps) n aresln r,pT~,,~T~, (5.21)
r'or normal distribution ,
P~ = P (~pr~ > ~ 0~5 m (x~), (5.2'2)
Where (z~) Laplace function;
TA~ - T~
z~ � (5.23)
r TAI + ~rI
Tx! ~ ~'r. ~ 7'~ = Tz, ~
QTA~ = QTr. A' when z=0; Q~AI ~ Qrx. ~ when z= d` ;
Tr.A~ Tx,A~ Qr~,A' Q%x.~ " are deCermined from ~:eference or experimental
d~ra. T~ and 6 T~ are determined from heat exchar.ge equations with fixed
vatues xi and zi.
qy (8 - s)
T~ = T ~X~ Z) = Tx. n / J . ~ (5.24)
~
s
QT~ � QTx. n + C 8 j~ a4~ + ` ~ R~ + L ~a ~ z~ ~ ~~r
7'~ -1- Tx
9~ � - ~
where a a ~
r ac
QQ _ ~QTr vT~ !Y qi -I- ~ a~ o~~ -I- oa= :
s'- a a a~ A 7~ a~R
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M ~
~ ct~r.~i
a~ ~ ~ .
. 7~~ m~u erCOrf~~ari T* ~ 1~sx ~
a Q~ e~~~T 6~rr r .
r o u~, ~3 +
vT* � or~ a~nr,
The quanti~iea in the above relaCiona conCain mean valuea T~ and aCandArd
deviation a 2T . Some of Chem are primary and are obtained from reference
data, while ott~ers in turn constiCuCe functions of. random argumentis fi, To,
w, etc.
The correlation coefficient between functions ~iTl and ~i T2 is determined
from relation
r �r,Qr~r~T~`f'OT~n�r.~t~T~A~T.~ (5.25)
~6r,, +Cr~ ~ �
Y�~r~+�~T~1~'r~A~
Ta determine quantities r and r we employ expressions for
T and TA . TLT2 T1A ' T2~0?
rr~. r~ ~ Q Q (a~ar, a~e -f- . . .
r~ ri
where
~
al - ( dTe 1 \ ~To / - 1~8 ~a=,~ ;
~s � ~ aa~ ~ ~ a
a~ ~ ~ et~ . .
5.4. Regression Analysis of Precision
The methods of analysis of statistical characteristics presented in the
preceding sections do not always make it possible to establish an
analytical relationship between engine parameters.
For example, it is extremely difficult to establish an analytical relation- -
ship between specific thrust impulse, combustion chamber pressure, and
propellant component ratio. In such cases it is sufficient to link system ~
input (k, pk) with its output Ik, without examining the intermediate
physical processes. For this one employs a method of regression analysis,
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~
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based on descripCion of aurface of aysCem "O~tpuC" reaponse to "InpuC"
in cereain vecror space. LeC ua assume we muat determine the dependence
of index y(operaCion parameCer) on aeveral factora ~x~,~ . The type of
function ;?=y {x~} is not known in advance. The funcCion ie represenCed by
a seri~a
y�' bo F~ bcxr F~ br/xcx~ -I- Fi buxi T� (6.26)
wherp bi, bi~ coefficients of,regression, determining the degree of ~in-
fluence of factor xi and their interaction xix~ on output index y.
Det~rmingtion of regression coefficienCs b~, bi~ is Che principal prob- ~
1em which ie solved in regression analyais. Coefficients of regresaion are
derermined on the basis of experimental reaults. The method of regres$ion
analysis is w idely employed for experimental esCablishment of int~rrelation-
_ ships between characteristics of complex systema. There exist a number of
reserictions on their employment, however, Che principal of which are the
following.
1. Results of ineasurements should follow the normal law of diaCribution.
2. Error sCandard deviaCions should be constant.
For motors operattng under ateady-state condiCions, the ubove-enumerated
condiCions are met.
Most frequenCly multiple linear regression is employed in analyzing moCor
static characteristics.
Let us assume that a priori data enable us Co state that the input
regression equation has Che form
J = bo -f- b,x~ 62xa . . . b~k, (5.2?~
The adequacy of equation (5.27) to actual operation is determined from the
rPSUlts of the experiment shown in Che Table 5.4.
Table 5.4.
y I xl I x~ I x~ I x~ I... I xK I n
ili I xli I z,i ~ x91 ( x~l I... I x,h ni
y~ i xi: I xu I x~ I x~: I x,q I n~
. . . I . . . I . . . ( . . . . . . I . . . I . . . I . . .
yN I xvy x~ x~ I x~N I I x~ nN
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To solve ene adequacy problem we musC determine etia~isCical eaCimatiea o�
coefficienCe b~, according ~o resultis nf obaervaCions (Tab~.e 5.4).
In conformity with Table 5.4, we construce a aystem of equations ~
Goxo~ -1- blxli -I- bax~l . bkxkl yl~ -
box~, blx~9 bsxa~ . . . b~~ � ~~t (5.28)
boxoN b1x,N bax,N . . . 6~k~?? = yn?~ ,
where x~~=1.
~sC{mates of coefficienCs bi are performed utilizing the method of leasC
aquares. In conformity wiCh Chis method, Co obtain an optimal approxitnation
of regression equation (5.27) to experimenCal data (Table 5.4), the follow-
ing condition must be meC: '
N
S = E n (y boxo - b~xl - . . . _ b~,~' ~ min.
To meet this condition one should squate to zero parCial derivaGives by bi,
as a result of which one obtains system of equations
a
o ~ - 2 Fi n [y - ~bo bixi . . . 6~it)l =
� ab = - 2 ,~,r n [y - (bo blxl . . . bx~) x~1= 5.29
~ ~ ~
as _ - 2 ~ a f y - (bo blzl -i- . . . bxk~ z~] = 0. -
~bk -
From system of equations (5.29) it is not difficult Co obtain a aystem of
equaCions with reference Co bi:
bo F.~ n b~ F+ nz~ bz ~i nX~ -I- . . . -I- bk ~
i ~k = ~ ny; -
bo ~ bi ~ ~i bz ~ nx,zz . . . bk ~ nx~xk =
. . . . . . . . . F+ nxly; . . . . (5.30)
bo ~k b~ F.~ nxixk bz ~ nxsxk � � � b~ ~ ~k =
. - Fi nXky
From the first equation of system(5.30) foilows
bo = ~ - bizl - b=x, . . . - b,kx,~, (5.31)
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. ;
where - ~ nb . ~ ~r .
y~ ~n+ xt~ ~n ? '
' Caking (5.31) into accounr, rEgression equaCion (5.27) can be written
� ~J b~ ~X~ - x~) b~ ~X9 - x9) -I- . Gk ~xk - xk). (6.32) ;
If we designate e~ = y-- y~ ~x = x-- x~ equation (5.32) will
~
assume the form ;
ey = bl Axl 69 ~x, . . . + bk ex~. ~~.3a~ ;
;
Taking into account Che relations of (5.31),~~ysCem (5.30) is reduced to '
the form ~
bl ~ n Oxi -F- bs E n ez, eX, -E- ,
bk ~/i ~X10X,y =~j n eXl e y;
f
bi ~ n Ox~ Ox1-f- bn ~ n Axz t'
- -I- bk E n ex, eX~ n ex, ey; (6.34) '
:
~Ji ~ n Ox10xk b, E n eX, eXk
bk Fi n Axk =~j It ~xk ~y. 4'
. i
rt
i
We shall insert into system (5.34) paired correlation coefficients ~
~
a~ n Ax~ Ox~ . ~ a ~x~ ey
rx~' xl NoXioX~ ~~y, xt = NQ~IQy ~
wh~re
~�l,k? %=1,k, ~~1, N=~n,
we obtain
,
- b16x1 62Qx1Qx=/xl~ x~ � . . bkQx~QxkrX1~ rk ;
,
= rz~ fIQx1~Y: s
. ,
bzQz . . . bkQx ~X rx , X ~ c
b1Qx,6x,rX,. x~ , : ~ s R
= rz~~ YQx~QY~ (5.35) i
:
� � � � � � � ~ � � � � � � � ~ � � � � 4 .
6ivXivXkrX1. xk 620x=QxR/x~~ sk . . . fjkQxk - ,
_ ~zk. yQxk~Y� ~
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Coef'ficiert,w bi nf ~ysrem (5.35) are determined by the ratio
' b! � s~ ~ (5.36)
- where
s
Qxl Qxl~x~~xl, x' Ux1UxkJ'Xi~ xA
4
e ~ ~x~ . . . 0'x'~xk/x~~ z~ .
. ~ . . . . . . . . . . . .
Q~
xk
A~ addit3onal determinant.
Coefficienta of regression bi depend on correlation coefficients ry X and
root-mean-s~uare deviations. The adequacy of regression equation (~.2~) or
(5.33) is tested by Fisher's criterion [28]
f' _ ~ ~ ~~.3~~
o~
where
N
Q~ _ ~j ~b/ - ltt~)~
a '
If value F, obtained wiCh formula (5.37), is less than critical value Fkp,
with a specified confidence coefficient, the regr~ssion equation is
adequate to a real process.
Values Fkp are contained in Table 5.5: kl=N-1; k2=k1-1 when~!=0.1.
We shall examine the method of calculating regression coefficients with a
concrete example. Let us assume that as a result of tests we have ob-
tained an experimental field (pk, T) of combustion chamber preasures and
propellant component temperaCure as indicated in Figure 5.4.
Table 5.5. ~
i I s I a ~ 4 I a I ~ io ~ ia
I 161,0 199 216,7 224 230 237 242 246
j 18,b~ 19.0 19~1 19,2 19,3 f9,3 19,4 19,4
3 10~1 9.b 9,3 9,1 9,0 8,9 8,8 8,7
4 7~? 6,9 6,6 6,4 6,2 6.1 b,9 5,8
b 6,6 b,8 5,4 b,2 b,0 4,9 4,3 4,6
7 8,6 4,7 4,3 4, t 3,9 3,8 3~6 3,5
10 4,9 4~1 3,7 3,6 3,3 3,1 3,0 2,8
15 4,5 3,7 3,3 3,0 2,9 2~7 2,? 2,4
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A regression equation is soughC in Che form ~
pK = bo'+ 61r,
To reduce ~he vo~.ume o� calculations it is convenienC to break down the
gxes of pres~ures and temperature into in~ervals p pk and ~ T and to
present the resul.ts in dimension].eas quantitiea p~ and T'.
P~r,~Mnd!
, � ,
a �
16 . ~ � ~ -
f0 ~
~ ~ �
� ~ � `
l9
iSJ t 1J 29J' ~'K
Figure 5.4. Experimental Relation pk@pk (T)
One selects base values pk~o, Tp, for which one adopts nominal quantities or
midpoints of ineasurement inCervals. Let pk~on15.25 MPa, Ta5 K. The value
of interval hpk=0.5 MPa, hT=10 K. Then dimensionless quantities p'k and
T' are expressed with integers 0, 1, 2... m.
Experiment resulCs are contained in Table 5.6.
The number of experimenCal points in the intervals is indicated at the inter-
section of the lines and columns. '
~tean dimensionless quantities
PK _ ~j ~"P" = 38 = 0,053; T~ _ ~ N T~ _ ~ - 0,184.
Mean dimensional pK = pK, o+ pKhpK = 15,25 0,053 � 0,5 ~
= 15,28 MIIa, T= To T'hT = 5-~- 0,184� 10 = 23,4 K.
Root-mean-square deviations of dimensionless variables
Fj nP,c ~p�~~ 48 - 0,053~= 1,12;
_ QpK = N - _ rK ~
~ =Y
104 �
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~T, ~ ~ Z nr ~T ~T,~~ ~ 7 ~ 77
- Y~- 0~ 183' m. l~4;
a~,,K = hpKvpK Q I, I2 ~0,5 = 0,56 MIIa;
~r = hrQr~ = 10� 1,4 ~ 14 K.
Table 5.6.
r~
T9 I-1 I O I 1 ~ R
~
PK ~PK ST npK nPKpK �PK ~pK~~
~~b~ 10-40 40-30
-2 I~ 14-14,b ( 0 I 0 I 0 I 1 I 2 I 3 I-6 12
--1 I 14,5-1b I 0 I 0 I 3 I 3 I 3 I 9 I 9 I 9
0 15-15,b 2 3 1' 5 3 14 0 0~
~ ~ ~ 1
1 I Ib,b-16 I 2( 2 I 2 I l I 0~? I 7 7
2 I 16-16,b ( 3 f 1( 0 I 1 I 0 I b I 10 20
nT 7 6 6 11 8( N=38I ~'=2I ~=48
nTT' -14 -6 0 I 11 !6 ~ 7 I
nT (T')' 28 I 6 I 0 11 I 32 I~ 77 I I
l
_ np~ pKpxT ~ I-16 I-4 I 0 I-2 I=14 I~ 36) I
9
Correlation coefficient
~'j nv,c~TPKT~-NAKT~ -36-38~O~b3�0~184 0~61
,
P ~ T NQp, Qr.
K - 38--1,12�1,4 �
K
From system (5.35) coefficient bl is determined as follows:
6~ � ~vK. r' QT = -0,61 ~-=-46 = - 0,024.
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In r.onformity witih (5.32) we have pk=pk + b~ (T-T) or
pK = 15,28 0,024 (T 23,4) = 16,84 - 0,024T, (5.38)
As was Co be expecCed, Che dependence of combustion chamber preseure on fuel
temperature is slight. , _
We shall verify the adequacy of equation (5.38) Co an actual process ac-
cording to Fisher's criCerion ~ -
s
2j ~PKJ`-p,c)~
Qp~.' b_~ _ ~48~0,212;
1~= ~ - o'~~ =037.
pK '
From Table 5.5, when k1=k2=4 we find Fkpffi6.4. Since F~'Fkp, the regression
equation is adequate to an actual proceas.
r
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roR orrr.czA~ usL ocrLY
- Chapter 6. TUNING AND ADJUSTING MOTORS
6.1. Tasks and Methods of Tuning and Ad3usting
As a result of the operation of internal disturbing factors Chere occurs
deviation of the operation parameters o� a concrete motor from nominal
values, that is, reproducibility of operating conditions is disturbed.
Variants of operation process parameters leads to an increase in the field
of rocket tra~ectory dispersion, decreased accuracy and range. In addition,
an increase in parameters spread also leads to narrowing of the area of
stable operation and complicates rocket conCrol as a consequence of the
occurrence of additional disturbing factors.
To compensaCe for the influence of internal disturbing factors and to in-
crease rocket accuracy and range, engines are Cuned and ad~usted to nominal
rating, that is, nominal values of the principal engine operation parameters
are secured by selecting the requisite characteristics of components (con-
trollers, throttles, reducers, ~ets, etc) contained in Che propellant com-
ponent line.
Tuning and adjustment designates a complex of calculation-experimental
work ensuring securement of specified operation parameters by means of one-
time influence on several engine characteristics. Tuning and adjustment
should optimally ensure meeting tactical-technical requirements placed on an
engine in respect to principal parameters.
Since engine operation is characterized by a substantial number of parameters,
- tuning and adjustment ensure acquisition of specified values only of those
~arameters which affect a motor's economy, efficiency, and controllability.
The concrete tasks of tuning and adjustment are determined by the specific
peculiarities of the design and by the motor's specific purpose and operating
conditions.
For example, if an engine is designed for a booster, its tuning and ad~ustment
do not involve any particular labor, since in this instance tough demands
are not imposed on precision of engine operation.
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FOCt d~~~CIAL ti5~ dNLY
if dn engine i~ de~ign~d ~dr b~1li~eic mi~~il~g~ which ghduLd b~ highly
~~GUrat~ in hitCit~~ eh~ eargee, high d~mandg are impoghd nn Cuning ~nd
r~d~u~CmenC. Tni~ i~ prtmarily for tiwn reagon~.
A I~l~;l~ drbr~~ nP v~rinnc~ in rgrio nf prop~118nC compnn~nCa re~u1CH in
decrenH~d ~Cdnomy (~p~nifiC Chru~C impul~e) and incre~d~d gugr~rir~ad fu~l
regidu~ in the Cnnkg, ~nd ~ongequ~nCly ~n incr~~g~ in mag~ gt launch.
With ~ 1~rge u~yi~nc~ of t~r~l propellanC coneumpCion (thru~C), rockeC c~nrrol
~ is made diffic~~1C ~g ~ cnn~~quenc~ of th~ occurrence of additSonal dig-
turhnncei~ affece~ing Ch~ conCrol gygCem.
'Ch~ ~cdnomy of ~n ~ngine i~ characterized by epeci�ic ehru~t impulge~ which
~
for n given d~gign nnd ep~cified prnpellant is determined primarily by com-
n~~ri~n chbmber pre~~ure and ratio af prop~llanC componene~.
Thux Hecurement of ehe fnllowing condition coneCituCes the firsC end~basic ~
tnak nf euning ~nd ~d~ugtment:
,
p~i ~ PK; K~ ~ k',
whpre j engin~ number.
'Che c~ndieions of efficiency depend on engine design; there is a large num-
her of possible engine layouts. In a motor ~,�ith a pumped aupply syaCem~
turbtne efficiency ig ~ecured with tuning and adjuatment. Thia is due to
the t'aCC that gas with a high temperature and pressure is acting on the
turbine blades.
5ince the turbine blades are not cooled, they can burn out wiCh a gas tem-
perAture arid pressure deviation from the apecified values. The temperature
of the gas passing through the turbine blades is determined by the ratio
of propellant cnmponents in the gas generator.
T~~erefare the second task of tuning and adjugtment is that of fulfilling
condition
K"~ eK"
.
In multiple co~rbust�on chamber propulsion systems there occurs a thrust
~Pread relative to the axis of the rocket (eccentriciCy of thrusC) due to
~ spread of operation parameters. As a result the rocket becomes difficult
o control in Che flight tra~ectory. Therefore the third Cask of tuning
ind adjnstment is securement of the following condition:
P~-P,
where j-- number of combustion chamber.
In addition to the above-enumerated tasks, in certain cases there may be
other parCicular tuning and adjustment tasks, such as ensuring a specified
turbine shaft rpm, specified pressure in the tanks, etc.
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~t~k n~'~~CIAL USF, dNLY
U~p~nding on eh~ f~~eur~~ of prdpul~ion gyee~m d~gign~ condiCinng of Cuning
' gnd ud~uAem~ne e~n bh a~ follewe:
e) e~a~ ~ a??~~ = o, artM o;
d) 8K' ~ 0, bpK ~ 0.
in eh~ firg~ vgrignt combuetion Gh~mb~r pre~~ur~ i~ noe dir~ctly ~d3use~d.
How~ver~ du~ to g~cur~m~nC of nomin~l rge~~ of flow, it ig clu~~ to rae~d
preg~ur~. Th~ m~Chod of gdju~Cm~nt i~ ~el~cted taking inGa ~acounG th~
fegeur~~ of ehp propul~ion ~y~Cem, its epeCific purpoee and ehe gdopC~d
te~ting ~y~tem. IC cgn be individual and ~Catigtic~l~ dep~nding on Che
m~Chod of obCaining inpuC daCa for tuning and ad~u~tmenC.
InpuC daCa for individual tuning and ad~ugtm~nt ar~ re~ulCg of ghop te~Cg on
all pngine componentg and g~sembli~g, a~ well ge propulsion gygtem C~gC~.
~ Input dgtg fnr gtgtigtical ad~ugtm~nt gre the re~ulti~ of tegtg on prac~ding
engin~~~ gv~raged nver the full number of Cestg.
Tuning and ad~ugtmen~t i~ broken down by procees cycle into "KTI+KVI" ad~ugt-
menC and ad~ugtmenC with KVI. `
Th~ firgt type of tuning and adjuatment includea the performanc~ of proc~ae
monitoring tests (KTI) of each motor on tesC beds~ nnd selective moniCoring
rests (KVI) of a motor aelected ae random from a batch~ which ia tested to
determine that principal chgracterietics are in conformity with the require-
metttg of technical epecifications.
"KTI+KVI" tuning and ad~uatment is individual as a rule and conCains the
following stages:
motor tuning and ad3ustment for performing KTI;
process roonitoring Cests;
tuning ad~ ustmenC;
selective monitoring tests.
Tuning and adjustment with KVI can be both individual and statistical.
6.2. Individual Tuning and Adjustment
6.2.1. Input Data for Tuning and Adjustment
Input data for tuning and adjustment include apecified (nominal) operation
parameters and component shop test results. �
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Nnmin~l velues ere ~pecified for eho~e parameeera tio be ad~u~C~d.
`Ther+e param~eer~ includ~ tihe folinaing:
~ombu~tian ~h~mb~r preeaur~ pk;
prnpell~nC componpnt raCes of flow intio combuetion chamber m~;
ratia of propellgnC componenC~ in ga~ ggnerator K".
In gddiCinn to eh~ abov~-~num~rgeed param~eerg, exCernal di~Curbancea are
~igo ~ppcified for ~nsuring identiical cond~.Cions for tuning and ad~u~Cing
all motnrs: pre~~ur~~ at pump inlet po.~~ po.~ and propellant com-
ponene d~n~ieie~ P ok gna �
~~llowing manuf~cture, each compon~ne i~ eub~ected to indEipendene (eh~p)
Ce~Cg, during which individual characteri~tics are determined.
1. Hydr~ulic charact~rigCics. Tegting determine~ preseure logaes:
a) in line~ from pumpa to combuaCion chamber Ap~~ ~pr;
b) in combustion chamber cooling ~acket Ap~;
_ a
c) in lines from pumps to gas generator d~ ox ~ 11 p r f
d) valve hydraulic regigtances Op~,~; DpK,~�
The values of all indicaCed losses are determined froa� the results of
running x liquid through, usually water, and liquid flow rate during this
test is specified from the condiCion of equality o.~ hydraulic resistances
_ with waCer gnd the gctual propellant component, and is determined with
the formulg
. ~
m.~mK ~ ,
where mg, flow rates of water and propellant component respectively;
P B~ pk "�ensity of water and component reapectively.
2. Prgssure-flow rate characteristics of pumps at nominal angular vslocities
'H=pH~m, n).
?s a result of pump testing, one dete~mines coefficients A, B, C in pressure
characteristic equation pH=AnZ+nnm�Cm +p0 or quantity tg aH~ as well as
efficiency 1~ N. ~
3. Turbine characteristics rl and r2 in pawer equation
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~O[t Ok'F~CtAL US~ ONLX
N ~1
r Prrn r~ ~ ,
1
6.2.2. lndividugl muning ~nd Ad~uetment of a Motor WiChout Generator Ca~
AfC~rburni.ng
Ag a~on~~qu~nce of th~ fgce thgC in a motor witihouti generator gas ~fCer-
burning there i~ no direce link b~twe~n combu~eion ehamber and turbine
driving g~ner~tor, gd~u~tmenti of combu~rion Ch~mber and gas generaCor can
b~ performed ~equentially.
In Chig cage ad~usCment prior to KTI ie broken down inCO two sCagee:
ad3usCmenr of engine pgram~ter for fulfilling condition
8m,~ = dmr ~ 0
~ by pl~cing throCCle disks in ehe combugtion chamber linea;
turbine-pump ad~u5tment ( a K"=0) by gd~usting control devices or
insCalling jets in the gas generator lineg.
Combustion Chamber Tuning and Ad~ustment
'To fulfill condieion BmoK - am~ ~ 0 it is essential that
the pressures generated by the pumps heve ehQ following values:
. `
PM. oK PK Gi ~pr oK " po. oK~
� (6.1)
pr. r- PK L'j ~ptr - Po. r?
where ~ epr, Op'J ~pK, Opp, ~ total losses f rom pumps
to combustion chamber.
P~
P,v �
i
PM ~ ~
~N
n
l7J Ill NI
~igure 6.1. Similar Pump Conditions
111
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A~ g consequence of th~ in�luenae of v~rioue internal dieCurbing fgceur~
on rhe pumps, ehe requir~d preesureg ae the pump outlets (6.1) will noti
Coincide with nominal values; otherwise there would be no need for gd~uet- _
ment.
'Phus the r~quir~d pre~sures p~ can be obta3ned for nominal flow raees with
differing rpm.
_ 'Th~ r~quir~d rpmg nok ~nd nY*, can be determined by two meChods.
1. ~quations of similgrity. We know from theory of pumps thaC the L�ollow-
ing relations are valid for similar conditions:
P~ ~ pH ~n~/n)'~ (6.2)
m' ~ m ~n'/n)~ . . (6.3) ,
Line of similar conditions BC ia contained in H'i~.~re 6.1. For tranaition along
curve n*tlconsC to the point of nominal conditions, one can uCilize linear
approximation
PM ~ PN -I- (m - m) tg a.~ (6.4)
where
. tg p~N aPM .
_ a,;~
~
M_,;,
Substituting equation (6.4) in relation (6.2), taking into accou~t (6.3),
wE obtain
.
~ Pu = ~pM -I- C 1- n ) 1g ~r �M~ C n . (6.5)
n
In a linear approximation, in the vicinity of point A one can write
dp�
nPN = e,~,
where
OP~�P�-P~~ An~n-n�;
ap" is determined from equaCion (6.5) for nominal conditions, that is,
when n-n*;
da ~ n (2pM'I"mtgaM~, � thus,
.
2~~~tga~ .
n' n 1- p" - D' (6.6)
~ T
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~o~ o~~zcir~ us~ orrLY
From formule (6.6) one detiermine~ r~qu~.r~d nnk and n~, eneuring p~~ok ~nd
r.
2. PuMp charact~rietia lf the equaCion for pressure at pump our].eC is
gpecified, the required rpm cgn be d~Cermitt~d frnm obvioua ~quation
PN
j ~ pK -I- ~ ~p~, ~ - pol � Alnl ~ -I- B~ni ml - f - c~m~ 'I- Ay. ~6.?) .
Resolving equation (6.7) relative to n~, we obC~in
- E3~et~ t~(B/m/)~ 4~C~mj - pa Ap~~ - 2p~1 A/
ni ~ zA~ , (6.8)
where j ~ o~
As a consequence of the �act that the characteristice of the lines and pumpg
are differen~C, as was already noted, n~~nok. At the same time condiCion
n~k~zn~- ahould be met in the eurbopump unit, where z-- reduction gear ratio
(in a turbopump unit without a reduction figure z~l). The greater of
quantities ~*kand n~ is selected. Assume that nok n~. Then the pre~sure
produced by the oxidizer pump will be equal to Che required p ~ and
pressure at the fuel pump ouClet will be greater ehan required~p~*~ p~~~
(Figure 6.2), and rate of flow will be greater than nominal,
.
d ~'~~n~~ P
p
~ aow P~ rd f
PM
~ ~ .
P~iwr'
n~
. ~
?
r ~iN "~r
Figure 6.2. Graphic Determination of Qp~ _
In order to obtain nominal fuel flow rate it is necessary to throttle down
the fuel pump, that is, place a hydraulic resistance, a throttle disk, in
the line beyond the pump.
The excessive pressure (throttle effect) which is absorbed by the throttle
disk is determined as
OPu~ = PM~~-PM. r~ (6.9)
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wher~ p is ca~.culaeed wirh equation (6.ii), while p~*r equgtion
(6.7) w~ti~i n~nak.
The diameter of Che throttle diek of the throC~led pump is deCermined from
tihe relation
~ 2 Y nk� Y~P oaw' (6.10)
' where k-- coefficienC ~f Chermal expansion; coef�icient of flow
rate.
Coef�icients k and ~It are determined experimentally.
A�ter installing ehe Chrottle diek, tihe combuaCion chamber Cogether wieh ~
lines and pumps are ad~ueted to nominal pk and k'. The motor, however,
has not yet been ad~usted. Required pk and k~ ahould be aecured by
corresponding turbopump and gas generator operating conditiona.
Ad~usCing Turbopump and Gas GeneraCor
Under sCeady-state conditions operaCions for tuning and ad~usting combustion
chamber paramet~rs to nominal valuea should be performed with observance of
a basic requirement balance of turbine and pump outputs, that is,
N* _ ~ JIIh~. (6.11)
- Oxidizer and fuel pump ouCpuC are determined on Che basia of the results of
combustion chamber tuning and ad3ustment and pump Ceats
NN~ a pN~'~`~ . ~6. ~ 2~
p/~N/
- Turbine power is determined by the relation
~ l~"~ (6.13)
1VT = Prrn' l~i - ~
where rl and r2 are determined from turbine tests.
~ubstituting (6.12) and (6.13) in equation (6.11), we obtain required gas
,enerator pressure ensuring nominal rates of flow into Che combustion
chamber
. - . -
px. oK"toK + pr. r~r ~ � .(6.I4)
p~~ PrnM. r n. r_ r,n.
POK~11. OK 1
~ ~ YRTrr .
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Requi~i.re pxeesure dropg in tihe ~g~ ~en~~~tor line~ are det~rmined from
presgure balance equaC~.on
M~ ~ ~
OPok = pN, oK prr~ ~
- ' ~ (6.15)
~Pr ~ pti~ r - F~r~
JeCs (rhroCele resist~nceg) are placed in the gas generator line for ad~ust-
ment purpoaes, wiCh presgure drop on the ~eta detexmined by the relations
M~ A
en~. oK = epn~ - epoK?
~P~c. r = dp~ -
- enrr
where OpoK~ Op~ gas generator line hydraulic resiatancea, deCermined
by tesCs.
Engine tuning and ad~ustment to ~pecified parameter valuea is completied with
placemenC of ~ets in Che gas generator line.
6.2.3. 7lming and Ad~ustment of a Motor with Generator Gas Afterburning
In motors with afterburn3ng of generator gas there is a direcC link between
operating processes in the combustion chamber and in Che gas generator.
Therefore the separaCe method of combustion chamber and gas generaCor ad-
~ustment is not applicable here. A characteristic condition of motor tutting
and adjustmenC for motors wiCh generator gas afterburning ia
~'pk�~, ~K�~.
()ne can influence operation in such motors by means of changing the
hydraulic resistance of three lines: the combustion chamber fuel oxidizer line
(depending on afCerburning arrangement) and two gas generator lines. Since
a control (RKS control) is placed in one of Che gas generator lines, the
control element is ad3usted insCead of placing a throCtle disk (~et).
As a result of solving the problem of influence of external and internal
factors, we have knowledge of correlations which detarmine the link
between ad~usted parameters, disturbances and line hydraulic resistances,
for which we take coefficients of hydraulic resistances of the throttle
disks and controls:
ax = g; ck. x,aX~ + ck. ~Ka~oK. W+ ~W. ~a~W.
+ Ck. bp, a~P. o; ~s: is~
8PR COK. t~SX( CpK~ ~K. W~OK. W'+' NpK' ~tll. P~b.!(. P~
~ CPK. ~P o~Sp. o~
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where ~W, i~-- coeffici.enes of line hydraulic reaistignce.
Since we hgve Cwo ad~ustment condiCions ~bpN ~ 0, 8K = 0) gnd
Chree poseible actione
B~oK, w? b~w, r~ 8~p. o~ one of Che lines is
not throCtled.
As a rule one adoptis as unthroCtled line the line of that component which ie
fed entirely to the ggs generator.
For oxidizing afeerburning, for example, the oxidizer ~.ine ia not throCeled ~
- (g~oK, ~ 0) , and ~or a reducin~ arrangement the fuel line (8~,,,, U)~
Fulfi~.ling in system (6.16) ad~uatment condition (s;~.~0, a Km0) and
assuming, for example, g~o,c. W~ 0~ we deCermine unkiiown d~ Ly~h and
d~ .
p.o
~t bvKCK~ tp. o" bKCPK~ Zp. o,
"bW. r ~ Q r
~e b!'(CPK~ ~W: r_" bPKCK~ tp. o ~6.
~+Sep~ o � Q ~
where
e~ cK, eW, ~pK, ea o- t~. ~K, ti,,. o~
6x cK,X~aXt; bpK ~pK. x~ax~. ~
tw (sl
From coefficientis of hydraulic resistances it is not difficult to
proceed to throttle disk pressure drops
~ bw. r~~w, r~ ~-T" "bw. r~?
.
~Pw. r � ~m, r - �
Pr
h.2.4. Process Monitoring Tests and Tuning ~?d~usCment _
In order to check engine efficiency and to ensure that operation parameters
meet the requirements of technical specifications, each tuned and ad~usted
motor is sub~ected to process monitoring tests.
KTI involves measuring a number of operation parameters, including those
for which tuning and adjustment were performed.
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When the va~.ues of th~ ad~uated parame~ere (pk and K) ere not wieh~.n eh~
l~.m~ts ~pecified by tiechnica~. specificaCions, Che moCor is retuned in order
to ensure nominal values during operatiion or KVI. Tuning ad~ustiment follow-
ing KTI ie also necessary becau~e moCors which have been sub~ected to process
monieoring tieses are Caken dovm; some of tihe componente are replaced with new
ones, which leade ~o change ~.n characteristiico.
Followin~ are input dgta for engine tuning and ad~ustment;
megsured valuea nf ad~ustiment parameters pk, K; ~
measurement of preasure losses in mo~or lines with replacement of
componenta ~
eepoK~ ~on~c
values of required lossea on dieka and ~eta obtained during initial
ad~ ustment dp .
For definieeness we ahall examine reeuning and read~ustment of a moCor with
afterburning of oxidizing generator gas. In this case the following are
retuning input data:
nK, rtN? eePoK; eeP~; OpW, ePo, o�
The following system of equations is written relative to the parameters Co
be ad3 usted
~'~t ` PK. x~ � QAK~ Sp~,~ o~~pP. o. K- ~Pp. o~
~~PK' pW. P~~Pw. r. K- OPw. r~ ~
~ aPK~ OPu. K~~~oK ~ aPK~ Sp~00Pri
~X - KN~ ~ aK~ ApP. o~DpP. o. K- OI~P. o~ ~ ~s. ~ 8~
[1K, Aw. r~~pw, r, x- OPw, r~
+aK. epoK~~PoK+aK, epraop~,
where ~pP,o,,c~ ~Pm.r, K new values of pressure losses in the control
element and on the throttle disk.
System (6.18) is resolved in relation to 4pi~r�.k and ,4 Pp.o.k~
~P~. Ap~ o�Op~. o. K~ aPK. P~n. rePw. r. K� dAK~
aR. so ~n . o. K+ aK, p OP~. K~ dK, (s. i s~
where p ~.r .
aAK - V'~ PK. N) -I- aax. evp, oePP. o~
~�apK~ pc~. r~Pm. r- aPK~ ~poK~~aaK - apK~ op~eea~;
dK Ku~ aK. Ap~,. o~Pp. o-~' aK. ePm. rdpw. aK. eAoK~~poK�
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Valuca dk, dk are determinad by nomina~. paramete~g and r~su~.tia of m~~sure-
mentie du~3ng KmI.
We finally have from syetem (6.19)
dK asdaK ,
ePp~ dpK~ App. o~ r aN~ App~ o~
~6.2~~
_ dpK - a~dK
~pm. r. K� p~ak~ PW. r~ apK~ P~~,
where ~
Q' = aK--=--pu?' r ; = apK ~ AD~' � ~
aPK~ PW~ ~ ~K. ADp~ o ,
Selective monitoring testa (KVI) are performed following motor readjustmen~.
One randomly selecCed motor from a batch is sub~ected to KVI. If tests on
this motor indicate thati operaCion parameCers aYe within Che limiCs specified
by technical specifications, the entire batch is accepted; otherwise it ia
re~ected.
6.3. SCsCigtical Tuning and Ad~ustment _
Analysis of results of Cuning and ad~ustment indicates that with stable
production and a tesCed engine design, pressure differentials and dimensions
of the throttling cross section of throttle disks and jeCs lie within very
narrow limits for a11 motors of a gfven type. Therefore one can tune and
ad~ust moCors in batches with disks of identical dimensions, constituting
a mathematical expectation from all preceding motors. In addition, this
method of tuning and ad~ustment does not require performance of KTI, and
therefore iC is comparatively simple, economical, and is the only possible
method of ad~usting motors which cannot be taken down.
Foe application of the sCatistical method, it is necessary to ad~ust the
first N motors according to the individual tuning and adjustmenC method.
The number of motors N tuned and adjusted individually depends on stability
of production and degree of finished developmenC of design. The moment of
transition from individual to statistical Cuning and ad3ustment is determined
as follows. ~
Ior N tuned and adjusted motors one determines the mathematical expectation
.,f throttle disk area or coefficient of hydraulic resistance
N Jp
Fj F�il ~,j ~m!
M~FW~1.~ 1 N Cr M lbmJ~ a~N (6.21)
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Individu~~. tuning and ad~ustment ie performed for N+1 motiors~ as a result of
which one determines ~~,Tp, and the moCor ie ee~ted (KVY or KTI).
Aa g reault of Che Cestg one detiermines deviati.one of ad~ueted parametere
from those spacified by tiechnical gpecificaCione r'pk,N ,fK~ . Two equationa
cgn be writiten fnr aKN ,~"pk: f
BPK. N�~ CrK, x~8xt -I- Cp~, e,o~~W~ (6.22)
sxN ~ E ~K, x,axl ~x, E,~,a�W. ~s.23~
In like manner one wriCes equationa for ~pk~~ and ~K~, which will be,
with staCiatical tuning and ad~ustmenC,
saK. ~pK, x,aX, ?J CpK, eWM ~B~W~~. ~s.~4~
sx~ ~ s cK, x,aX, ~ cK, t~W,t. ~ ~s.~6~
The first term of equations (6.22)-(6.25) ia an unknown quantity and
determines the diecrepancy between the required and expected (following
statistical tuning and ad~ustment) value of the parameter to be ad~usCed.
If we eliminate Cerms EC~~ X~x, from equationa (6.22), (6.24) and (6.25),
we obtain
aPK. aPx. b+ Z ~vK. eW, IM t~W,] - a~W, Tp); (s.2s>
aK~ = aKr ~ ~j Ck. (M [a~,L~l a~~,~ *~l , (s.2r~
~ If values ~pk~~ attd ~K~ do not go beyond the limiC established by the
technical specifications, all motors following N+1 can be tuned and adjusted
by the statistical method.
In the process of manufacture, accuracy of tuning and ad~ustment is verified
with KVI, and when necessary it is adJusted as indicated above.
6.4. Comparison of Methods of Tuning and Adjustment
'I'wo methods of tuning and ad~ustment are examined: KTI + KVT and KVI.
With the first method all engines are sub~ected to selective monitoring
tests. After these tesCs tuning adjustment fs performed: each motor is
disassembled, components are inspected for defects, discovered defects are
corrected, defective components and single-use parts are replaced, components
are Created, and the moCor is reassembled for performance of KVI and delivery
to the customer.
- Thus with KTI not only precision of tuning and ad~ustment is tested but also
the efficiency of each motor; hidden defects are spotted, which are corrected
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eti the manufac~uring etage, which in ehe final analysis leads to increased
r~ligbiliCy. This is eapec3glly important at early s~ages of development
and m~nufactiure of newly-designec~ motore.
'Che mcrhod oE tuning and ad~usrm~nt with KTT. is highly precise. However~
~longside ndvantiages, this procedure of tuning and ad~ustmenC also po~-
sesseg some ahorrcomings. They include ehe following:
in~CigC~.on of moeor wear even before it goes into service;
considerable economic expendiCures on Cesting, disassembly and
assembly of _zch motor;
impoasibility o� taking down undisassembl.able motors, eCc.
Proceeding �rnm Che above, one can conclude Chat it is advisable to per-
form Cuning and ad~ustment with KTI on newly-deaigned motors at early
stages o~ manuf~cture. This meChod of tiuning and ad~usCment ma,kes it pos-
sible to ~bCain a f airly large volume of sCaCietical material e?~senCial for
evaluating reliability, to spoC latent def ec~s at early stage~ of manufacture
in each motor and to correcC them.
In the process of engine improvement and stabilization of production one
can eliminaCe KTI and perform tuning and ad~ustment with KVT (sCatistical
Cuning and ad3usCment).
LeC us perform an economic comparison of both methoda.of tuning and ad~usC-
ment. ~
KTI + KVI Tuning and Adjustment Program
All motors wiChout exception are subj ected to KTI. From a batch of n* units,
one motor is subjected to KVI with subsequent inspection for defects and
destrucCion or utilization other than designated. The size of batch n* depends
on the stage of production and experience of the manufacturer. At early
st~ges of production the baCch size is smaller than for producCion in full
swing.
Tuning and Ad~ustment Program With KVI
Out of a batch of n motors one is sub~ected to KVI with subsequent in-
spection for defects and destruction. The size of batch n is determined by
~he same factors as for n*. It is also obvious that at early stages of
~,roduction n~ n*. In the process of improving production the size of
batch n increases and approaches n*.
Let us introduce the following designaCions:
N-- volume of moCor series, number of units;
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n* volume of baCch for KVI with Cuning and ad~uatment with KTI;
n-- volume of baCch for KVI;
cT cost oF KTI on one motor;
cg eoaC of KVI for one mo~or;
cn cost of disassemb~.y, inspection for defects and assembly of
motor following KTI;
- c-- cost of engine manufacture.
Cost of moto,r tuning and ad3ustment on KVI + KTI:
~KT~i ~~TN ~n~ N ~~BN -I~ ca) n . � (6.28)
Cost of tuning and ad~ustment on KVI program: ~
~KBH = (~BH -F~ ~AI n . (6.29)
Relative cost:
~ K H- n C~ -I- n" (6.30) .
where
~ ~ ~nt -4-
~su -h '
~
s a
6'
~
~
C4o1~
Z
1
~ 4 8 >2 ~6 n
Figure 6.3. Relationship Between Cost of Tuning and Ad~ustment and Size
of Sample
Let us estimate quantity c. With KVI a motor is tested for service life
and conformi~ty between principal characteristics and the requirements of
Cechnical specifications. Motor operation and accuracy of tuning and ad-
~ustment are checked from KTI results. Consequently the cost of performing
KVI is greater than the cost of KTI due to a difference in operation time
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end pr.op~il~nC Elow rate, c~ y c~ FoY an ~pproxima~e analy~i~ on~ cnn
.~HHUme C~y~ ~C~u � -
'~'li~ CdNC nf mne~r di~~~~~rnbly ig 1~~~ thgn eh~ co~C of mgnufgcCur~. Nnw-
~ver~ with irCr~ag~ in compl~xiCy o� moror de~igm m~d wiCh employment of
aggr~Kgive propcliant componenC~, th~ co~t of disa~sembly epproaches rh~
~o~t nf m~nufaCture.
~ar mn~~rg for which on~ ~gn p~rform KTI wiChouti di~a~~embly~ c~�0.
qu~nCiey ~ c~n vary wiehin limie~ 0.15 ~ c~ 1.
~i~ure 6.3 ~hawg r~l~eion c~f(c, n) when n~20.
tr. fdllnw~ from ~n cutalysie of ~igure 6.3 Chat with gn increase in volume
of ~~mple n(~g th~ p1gnC gain~ ~xperi~nce and know-how), euning and ad-
ju~tm~n~ with KTI i~ ~conomically dieadvanCageous~ eince cKTN Y ~KBN �
The volume df gample n~ wher~by outlayg on boCh programg aYe identicgl, ig
deeermin~d frnm ~quation (6.30), if we aseume condiCion c~l~
n�
n~, rn......~
~ ~ .
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' 3ecrion II. STATLG CHARAC'~~1tI3~ICS OF 30tID-P1tOPEttANT ROCKET
MOTOR3
Chapter 7. OP~~ATING CHARACT~RiSTiC3 OF SOLID-
PROP~tLANT ROCKt:T MOTOtt3
7.1. Solid Rocket Propellants and Principal Designs of Solid-PropellanC
Rocket Mntora
At the present time two principal type~ of golid rockeC propellgnte (TRT)
are employed in rocket hardware: balli~tite~ and compogiCe. '
Bglligtite propellanCs are chemically based on organic compounds conCaining
the oxygen-rich nitro- or nitrate groupa. 1'hus in a ballistite propellant
both the fuel ~C~ and ~K~ and oxidizer ~0~ are contained within the etruc-
Cure of the same molecule. c?ne of th~ main components of ballistite
propellanr, which determine ita mechanical strUCture~ is nitrocelluloee a
product of nitration of cellulose. A second mandatory component is a
solvent (plasticizer). Nitrocellulose fonas, together with the solvent~ a
plastic, from which various-shaped chargea are made by the continuoue mold-
ing method.
Low-volatility solvents are employed in rocket propellants: nitroglycerin~
dinitrate diethylene glycol, and dinitrotoluene [22J. Propellants based on
theae solvents are designaCed ballistite propellants. The above-liated
solvents, just as nitrocellulose, are active propellant components. Among
the solvents, the higheat energy characteriatics are posseased by nitro-
glycerin, which for thia reason is employed conaiderably more frequently than
the others. In addition to theae two principal components, additives are
added to a ballistite propellant~ Which enaure the propellant~s stability
during storage, stability of combustion, which increase or retard rate of com-
bustion, as well as additives to aid in the manufacturing procese.
Composite solid propellants constitute a mechanical mixture (aggregate), con-
sisting chiefly of finely-ground mineral oxidizer and an organic fuel-
b~nder. Most frequently ammonium perchlorate ie used as oxidizer in modern
7'RT [22), while potaseium perchlorate and ammonium nitrate are used less
frequently. Polyurethane, polybutadiene and other polymers are employed as
fuel-binder. Light metals are frequently added to compoeite propellant in
order to improve its energy characteristics magneaium, aluminum [22].
Composite propellant constitutea a viscous mass after mixing the compeaents.
Composite propellant charges are made by free casting or injection molding
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m~thad~~ ~ind~r polym~rixat3on e~k~~ pl~c~ during h~ating, wieh th~ forma-
tion of a~olid prop~ll~nt block.
Modified b~11i~Cit~ 'T~tT wiCh m~ch~nicgl inclu~ion~ of miner~l oxidiz~r, ~x-
pln~iv~~ wieh ~ po~ieiv~ oxygen balance gnd metaliic fuei occupy en inter-
m~di~t~ pd~i~ion betw~~n the two princ3.pa1 types of TItT.
A1i ~alid ra~k~e 1~rdp~llr~nC~ burn in p~rall~l lay~ti~ in ~uch a mannpr Chat
eh~ burning gurfgce (combu~tiion fr~nt) occupie~ aC e~ch gucceeding poinC in
eim~ ~ pd~ition equidigtanC from tihe proc~~ding po~i.eion.
5ince lin~gr burning rgee, thati i~, rat~ of mov~ment of the combaetion frnnt
intd ehe charge, comprig~~ ~~v~ral mm/~ for mod~rn TttT, les~ frequently in
~he nrd~r of 20 tn 30 mrn/~, in order Co ~neure the de~ired gas generation
, nn~ mu~t employ charges wiCh radial burning~ tihe burning surface o� which is
di~eribue~d alon~ the entire length of the motor.
Considernbly l~~g fr~quently one employ~ charges which burn from the end,
e~t~t~d on the lateral gurf ace with g noncnmbu~tible material (~igt~re 7.1).
.
r
- ~ � r s .
-�I
r �
Figure 7.1. Solid-Propellant Rocket Motor With End-Burning Charge
Key:
1. Case 3. Igniter '
2. Charge 4. Coating
5. Nozzle
B B e.
~ f t ~ ~ f
~
_ ~ ,
Figure 7.2. Solid-Propellant RockeC Motor With InserC Charge
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Key eo Figur~ ~.2 on pr~c~ding p~g~t
1. C~~~ 3. Igniter
2. Cherg~ 4. Diaphra$m
5. Nozxle
~ ~ t~ 4 ' ~ f 6-6
~
~ ,
Pigure 7.3~ Solid-Prop~llant Rocket Motor With Case-Bonded Charge
Kpy:
1. Ca~e 4. Adhesion layer
2. Charge 5. Nozzle
3. Igniter 6. Ineulation
There are two ba~ic designa for eolid-propellant rocket motors with radial
burning:
solid-propellant rock~t nators with fr~e filling, that is~ with an
inserted charge (Figure 7.2); _
solid-propellant rocket motora With a case-bonded charge (Figure 7.3).
The first of these is typical for solid-propellant motorg with ba:.ligtite
propellant, and the aecond of compoaite-propellant motora.
In a solid-propellant rocket motor with free filling, the charge consiate of
separate grains plgced in Che combuation chamber with a gap. Following are
disadvantages of this arrangement:
low coefficient of combustion chamber propellant filling;
contact between burning gas and motor case along ite entire inner
surface, which requires either a thick-walled case or thick insulation;
the necessity of employing epecial devices to secure the charge in
" the chamber and to prevent charge components from ejecting during combuetion.
Th~ above-listed drawbacks result in poor mass characteristics of an RUTT
[solid-propellant rocket motorj with an inserted charge, which limits the
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ar~n of ~e~ uei~i~gtion eo in~tiane@~ wh~r~ Ch~ d~ci~iv~ ro~.~ i~ play~d by
~impliCiCy of d~~ign, 1ow ~o~e nf m~nuf~ctUr~ ~nd ~implicitiy of operaeion.
'Chig grr~ng~m~ne i~ ~xr~n~iv~ly ~mpldy~d in unguid~d rockeC pr~~~cCile~
nnd in b~~npC~r rock~eg ~f vnriou~ d~gigna~ion.
Whc~n fu~lin~ ~n ItU'1"C wieh compo~iC~ prop~llane~ ~trong adh~~Lon of cl~argh
eo mo~or ca~~ c~n b~ g~cur~d durin~ ch~rging. In ~ueh a moCor eh~ ch~rg~
burn~ ~1on~ eh~ ine~rt?~1 ~~vitiy gurfa~~, ~nd ~h~ ~~~e m~terial i~ proteetpd
from eh~ ~ombugeion product~ by the entir~ Chickn~~~ of eh~ propellant
eh~rg~. Cone~cti b~tw~~n Che hoC gag~~ an~ ca~~ occur~ AC the very end or
in rh~ fin~1 ~e~g~ of chargp burning~
~ ~ ~ s . _
~ ~ p ~ -�1
~
. .
, x
n
~
~igure 7.4. 5olid-~ropellane itock~t MoCor Wieh Coated InserC Charge
Key:
1. C~se 5. Obturator
2. Charge 6. Insulation
3. Coating 7. Igniter
4. 5upport 8. Nozzle
Advantagps of this design include the following:
high coefficienC of chamber propellant filling;
reliable protectian of the motor case surface by the enCire thick-
ness of the charge, which makes it possible for the case to be thin-walled,
utilizing lighCweight high-strength materials (titanium, plastics,
aluminum alloys);
absence of special devices for securing ehe charge in the motor.
t~tsadvantag~s of this arrangement include relative complexity of loading,
:~g well as complexity of charge inspection (flaw detection) during
.�aintenance. However, in view of the significant advanCages, and partic-
ularly due to good mass characteristics, Chis arrangement is widely em-
ployed :~n rockets and has become the basis for large-diameter rockets with
considerable engine burning time.
An intermediate position is occupied by an arrangement With a free-inserted
coated charge (Figure 7.4). Rubber obturator 5, against which the charge
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r~ge~, pr~v~nte g~e from pae~in~ Chrough the ~nnul~r gap berwe~n charg~
and nngin~ ca~e. A~ a con~equQnc~ of ehi~ a~tiagnan~ zon~ form~ 3,n Ch~
nnnular ~ap, which i~ fiZ~.~d w3th gas~~ thtiough a e1o t on th~ front ~AC@
of th~ charg~ in rhe initi~l period of ~ng3na nperation. Neat exchangp in
tih~ seggnant zone t~ke~ pl~c~ with 1ow inten~ity, a~ a con~equence of
wh~.ch thermal proti~ceion of ehe case alon~ Ghe 1engCh of th~ charge cgn
b~ ~e~ur~d with a compara~ively thin in~u~.at3on coaCing.
itDTT opergtion br~akg down into a principal, operat~.ng period and Cr~naition
period. The traneition per~od~ includg ehe following: burn 3niCiaeion,
pre~eure drop following charge burnup or ~s a regulti of burn inCerruption,
~nd tr~n~3eion from one mode Co gnotiher for a dual-mode engine. Tranaition
peri~d~ comprige a small percentage of Cotal RDTT operat~.ng Cime. During
the principal period RDTT working parametere change comparatively elowly
and amoothly in conformity with change in charge burning gurface and flow
pas~age crose ~ectional area ae a conaequence of propellant burnup. The
proces~e g taking place in an RDTT during thie period can be viewed r~e
qua~i-~Ceady-~Cat~. In this volume we ehall examine only Che quasi-~Cendy-~CaGe
period of R~TT operation.
The principal factore determining the working parametere and thrust
characterigtics of an RbTT during thie period are the following:
propellant composition, which deCermines its energy characteriatics;
_ law governing propellant burning;
shape of charge, area of its burning surface, and the law governing
its change on a time axis;
nozzle throgt area;
features of engine gas-dynamic route, which determine the ~
character of chamber inCernal gas flows.
The methods of calculating operation parameCers and etatic characteristics
of an RDTT also change in conformity with the specific features of an en-
gine's gas-dynamic path.
The first method is utilized for a motor wiCh an end-burning charge, as well
as for motorg with radial burning charges with insignificant pressure
drops along the charge. In these cases it is possibl~ to coneider preasure
and other gas physical parameters consCant for Cotal engine ullage, that is,
Co solve a problem aCated zero-dimensionally.
The second method is used when a substatttial pressure drop is established
along the charge and it is necessary Co take into consideration changes in
thermodynamic parameters lengthwise along the motor, examining one-dimen-
sional gas flow.
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3ome ~CudieA r~commanded more rigorn~~ ~olu~ione Caking into accounC twn-
and thr~~-dtm~n~ionnliry of ga~ flow in RDT'C [23~ 24~, but tihe~g ~olueion~
~re unwi~ldy nnd er.e littl~ ~uiCad for ~naLy~i~ of ~Cntic char~cCari~tic~
ki~~r7~.
Solueion of ehe problem ~tated one-d~.men~ionally breake down in turn inro
two variantg.
'~he �ir~e variant encompa~sa~ radial burn3ng charges with lengthwise con-
~tant �low pas~age cro~~ ~eceional area.
The ~econd varianti covers charge~ with lengChwise vari~ble croge secGional
area. In practice one mo~ti frequently observes etep-wise change ~n flow-
. pgs~gge cros~ gecCional area at Che boundary of two sectione of a charge of
differing geometry, such as in a g1oC charga at tihe boundary of the alot
and cylindrical secCion; also included here ~re multit3er and multisection
char~ea with intermediate spaces.
We should noCe ehae the reaults of one-dimensiong~. and zero-dimensional
solurions converg~ wiCh an increas~ in motor �low paseage croas aecCional
greas as a reault of charge burnup. Therefore the above aolution variants
can stand side by side in performing calculaCions for one and the same en-
~tne.
7.2. ~mpirical Law of Rate of Combustion of So'lid Rocket Propellant~
Under Static Conditions
Linear velociCy of burning of a solid rocket propellant is defined asthe
velocity of diaplacement of the burning aurface inCo the cha~ge. Since
rocket propellants burn in parallel layers, the direction of burning
velocity always coincides with the normal to the burr.ing au~face.
Linear burning velocities of modern rocket propellants under RI)TT condi-
tions range from 0.3-0.5 mm/s to 40-50 mm/s [26]. High burning velocitiea
are desirable for charges in uncontrolled rocket pro~ectiles and fo~r
rockeC boosters, as well as end-burning charges for suatainer engine.
Low burning velocities are necessary for ensuring exCended operatir~g time
for sustainer engines with internally and radially burning charges, ss well
- as �or solid-propellant gas generators (presaure accumulatora) with ex-
' Cended operaCing time.
~~ropellant t~urning rt~te u is determined by its physicochemical character-
;stics, combusCion cl~amber pressure pk, rate of gas flow across the burning
~urface, initial charge temperature TH, as well as G loade acting on Che
charge during burning.
The composition of the propellanC and the process of its manufacture exert
considerable influence on the rate of burning. For i~itroglycerin ballistite
propellants quantity u increases with an increase in nitroglycerin content.
128
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'~h~ CondiCion~ of ~h~ mo].ding procees ~xer~ cer~ain i.nf~uence. ~or com-
po~i.C~ propel~anCs u i~ derarm~.ned by the rype of oxidiz~r and grain ~ize.
Thp r~t~ of burn~.nq
can b~ chan~ed gubgtantially by catalystg added in
gmal~. quant~tiee tio the propellane.
OE rhe b~~e-known nxidizere~ the greaeest burning ra~e produced by
poease~.um perchlor~ee, and ehe slowesC by ammonium n~.eraee [26].
At C11~ presene Cime Cher~ do not exi~C rigorous theoretical meChods for
nulculnCing ehe burning raee of TItT. ~~eabliahmenr o� such meChodg ie
made difficult by the complexity of tihe burning mechanism of solid rocket
propellantg, iCs multiiple-eCgge characCer, and by the parCicipa~i.on of
a large number of physical and chemical factors. Therefnre in calculating
Che operatiion p~rameCers of an ItDTT one utilizes an experimental 1aw of
TRT burning, that is, an experimental relationehip between the linear
rate of burning and principal deCermining parameCera in Che form
u = uioft ~p) fa ~U) fa ~TH)~ ~7.1)
where functions fl, f2 and f3 are usually assumed independent of one another.
We shnll designate u~~uZ~f2(v)fg(TH). _
Let us examine the rel.ationahip between burning raCe and preasure umfL(r)�
For ba.llistite propellante, in Che low preasure range (to 30-40�105 Pa),
the reiationship between burning rate and preasure is expresaed by the
formula
u = ulp~~ . (?.2)
which in interior ballistics is called the exponential law of combustion. -
With an increase in presaure the exponential relaCionship transiCions to a
linear relaCionship
u = u 1 (1 bP). ~ (?.3)
The linear combustion law is valid for pressures from 4 x 106 Co 2 x 10~ Pa.
In the pressure inCerval from 3 x 106 to 15 x 1U6 Pa one can utilize both
the exponential and linear rel~Cions with approximately equal accuracy.
We shall subsequently designate quantity ulp, determined by propellant com-
position, unit burning rate. The law of combustion of composiCe propellants
is usually expressed by formulas of the same type as for ballistite propel-
lants. Taken for separate pressure intervals, they approximate an ex-
perimental curve with acceptable accuracy.
The value of exponent N for modern rocket propellants varies between 0.1 and
0.85. Higher values of y are characteristic of ballistite propellants. For
composite propellants the burning rate is dependent on pressure to a lesser
degree.
129
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5ummerfield, on tihe bgsis of a aimple physical model of combus~ion of com-
pn~i.C~ propell~nCs, propoeed the fo~.~.owing rel~t~.on for deCermining Cheir
hurning rat~:
u p ~ ~ (?.4)
Ln this equaeion a and b are conetants which are independent of preasure,
expresaing the relationahip between burning rate and varioue phyai.cochemical
parnmeters.
Zn practice coefficieneg a and b are deCermined from experiment, and 5ummer-
field'g law is transformed in practiice into an emp3rical relation, which for
pure composite compositinns (without metallic additivea) can be utilized for
calcul~Cing burning rate acrosa a broad range of pressurea from 1 x 105 to
1 x IQ Pa. For calculaC3ng RD?fT operations, it is expedient Co present
Summerfield's relation 3n the form
u = Q-~~ . (7.4a)
We should point to one of the possible deviaCiona from the above-examined
laws of combustion, Che so-called "plaCeau" effect observed during com-
bustion of propellants with the addition of various lead compounds. Within
a certain range of pressures the burning rate for such propellants ia in-
dependenC of pressure (y��~0) .
The burning rate of a propellanC increases during gas flow along the burning
surface of a charge. The increase in propellant burning rate is caused by
increased flow of heaC from gas to propellant with increased turbulence of
gas flow in Che vicinity of the burning surface. In the literature this
phenomeron is called erosion or turbulent burning. A number of inves~igators
point to the existence of a certain threshold rate of flow, beginning
with which Che erosion effect occurs.
C~nsideration of this phenomenon in calculating RDTT operaCion is performed
by means of a correlation funcCion which constitutes the ratio of rates of
propellant burning wiCh a gas flow and in a calm environment:
fs ~U~ � u~~ �
ulo
At Che present time this relation ia usually presented in the form
ft ~v) � 1-~-1to (v - U~p~ (7.5)
~r in a function of dimensionless velocity of flow
fs (U) � 1 /r~ - 7~�p)~ (7.6)
where v~yp, ~~p - so-called threshold velocity of flow; k~, k~ coef-
ficient of erosion; when
130
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~'~~'np t
~,C ~np } l l~~~ C+
The burning ~atie of rh~ ma~oritiy of so11d prop~l.lanCe dependa to a substan-
tial degree on the initial temperatiure of the charge. The fol].owing formulas
are moat frequen~ly utii~.ized ~o expreae thie relation:
uw � B _ (T B T~n,) ? (7.T)
W~ a 8~ ~TN'rxN~~ ~7~8~
where Tx charge temperature for which a burning rate ia determined; THN "
Cemperature adopeed as atandard; u~T and u1N unit burning ratea at
specified and standard temperature reapectively. B and D-- physicochemical
consCantia which are individual characteriatics of ~ given kind of fuel.
Formulas (7.7) and (7.8), in spite of an external di��erence, are maChematical- ~
ly identical expressiona under the condition that one and the same value TN
:t~ used in boCh relaCions. The relationahip between the cons~ants is
eatablished as follows: 1
D=B .
For rockeC propellants known from Che literature, constant D rangea from 0.001
to 0.004 [26]. The upper value applies to postwar balliaCite propellants
with a high temperature relaCion, while the lower value applies to com-
posite propellants.
TRT burning rate is also affected by stressed state of the charge and
G loads to which a rockeC is sub~ected in flight. However, since these
factors are of a random character, we shall examine Cheir influence together
with the influence of other factors diaturbing RDTT operation conditions.
7.3. Law of Change of Propellant Burning Surface on a Time Axis
The burning surface of a'charge is a aecond importanC factor in gas forma-
Cion in an RDTT. Depending on the nature of change in burning surface on
a Cime axis, we distinguish charges of progressive shape (surface increases
during burning), degressive shape and charges with a constant burning sur-
face (neutral charge). The ratio of burning surface S to the initial value
~ of this surface Sp is called the charge progressiveness characteristic
Selection of charge shape should ensure Che character of change in pressure
and consequenCly engine thrust on a time axis in conformity with the required
flight program.
If flight conditions do not require changes in engine thrust across a broad
range, a charge with a constant burning surface is preferable. In practice
such charges are employed considerably more frequently than the others.
131
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Solid r~cketi propellan~s burn ~.n peral~.e~ layers~ which 3n moeC case~
m~k~~ ir pnspiU~e, guided by elamenCary geomeCric coneidnra~ion~, to c~l-
cu~.nta tn gdvanc~ change in burning eurfeae a~ a funct3on of Cha p~r-
ccnt~~~ nf Che burned portion of Che charge
Wc~lu?,
that is, to detiermine r~l~tiior? .
We sha~l note that Chi~ is possible only with a unifnrm field of propellanti -
burning raCes, thati is, when the 1lnear burning rate ~.e iden~3ca1 for all
points in a charge.
CompleCe charge burning Cime ie detiermined by thickness of the burning
web e1, which consCitutes ite smallest linear dimension in the principal
direction of burning.
SomeCimeg in place of relation ~(t~) one uCi~.izes equivalenC relation r(z),
where z=e/e1 relative Chickttess of the burned charge layer correaponding
Co given value ~ .
As is indicated by etudies, dependence ~ on z can be expressed for all
charge shapes by a polynomial of the type
= klx (1 kQZ kazs -I- . . (7.9)
where coefficienCs kl, k2, k3, etc reflect the geometric propertiea of a
given charge.
Obviously with rigorous observance of constancy of burning surface p? ~1,
=z. In actuality, however, some variability of surface during burning
is observed for the majority of chargea which are considered neutral.
Let us examine some general patterns of behavior of charge surface during
burning. �
In the general case the perimeter of the burning aurface ~~.in a cross or
meridional section of a charge consists of areas of smooth curves and points
where they intersecC surface fracCures. Figure 7.5 contains typical
variants of change in surface shape during burning of various charge elements.
If two ad~acent surface sections, intersecting, forra an angle of less than
180�, in the prucess of burning the angle shifts together with the burning
s~irface but remains constant. If that same angle is greaCer than 180�, the
apex of the angle rounds off in the process of burning.
ror example, the conical surface of an end-burning charge inevitably
degenerates during burning into a spherical surface. In order to preserve
Che anQle of conicity it is necessary to place along the axis of the charge
a rod of propellant with a higher burning rate u2 ~ ul, which satisfies
condition
sin a = u
~
_ 132
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. ~
,
a~ b )
' � o~
~
u
a,
� ,c(~ ' dl
,
~igure 7.5. Characteristic Variants of Change in Perimeter of Burning for
Various Charge ElemenCs
Key: ~
a. ExCernal angle burning d. ReCention of cone-ahaped
b. Internal angle burning crater with utilizaCion
c. Development of cone-shaped of a lead rod (u2 y ul);
~ crater in end burning e. Burning of a corner with
charge coated face
(Figure 7.5 d). A similar rounding off of the angle apex will be observed
by the points of intersection of the burning surface with the inhibitory
coating (Figure 7.5 e). .
In order to determine the behavior of a more co~plex charge contour, we
shall examine a certain fracture-free elementary region of a curvilinear
aection (Figure 7.6). We shall assume axis x is the principal direction
of combustion propagation.
133
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.
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! .
. ~ /
eGf E
~c
� dx
~
. x
~igure 7.6. CombusCion of a Curvilinear Charge Section
During Cime d~ Che combusCion �ront w311 move in direction x by amount
dx~u/sin a~dC, where
aaarctgdx, and therefore the axial velocity of su'rface
elemenC displacement ia equal to
ux = dz ~ u V ~ + . ' (7.10~
dt ~r
y/~ x
tf the principal direction of propagatiion of combustion coincides with axis y,
- Chen
dy/dt = uY 1-f- (d~~' I Oa)
it is obvious that with axial burning constancy of rates dx/dt for all
poinrs wi11 constituCe a condition of charge state stability, for otherwise _
the section shape will deform in the process of burning.
Relations (7.10) and (7.10a) assume special imporCance when calculating ,
change in charge surface with a nonuniform field of burning rates. Field
n~nuniformity may be caused by changes in initial temperature or propellant
chemical composition wiChin the charge. The current position of the charge
ssrface is no longer determined thereby by initial geometry of the charge
and its changes according to the law of combustion in parallel layers. In
a nonuniform field of velocities there may occur distortion of rectilinear
prcfiles, sharpening of angles greater than 180� and blunting of angles
less than 180�. These changes can be taken into consideration on the basis
~f differential relaCions (7.10) and (7.10a) with uCilization of local burning
rate u.
In some cases, with deformation of the initial prof~.le in an inhomogeneous
Eield of propellant burning rates, a new stable profile may form. Profile
stability is secured, for example, by the fact that change in burning rate ~
on the y coordinate (with axial burning) is compensated by corresponding
change in angle e~ . This case will be examined in greater detail in
Chapter 10.
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a) ,
, . ,
~
Y
~ 2 f
<
4 ~I
Figure 7.7 Shapes of RDTT Charges
Key:
a. Shapes of insert charge 2. Star
propellant elements (graina) 3. Wagon wheel
b. Shapes of case-bonded ' 4. Slotted charge
charges
_ For the ma3ority of radial burning charge .shapes, the required progression
_ characteristic is ach ieved by selecCion of a ratio of degressively and
progressively burning antagonistic surface elements.
Antagonistic elements can be represented in each charge section, such as
in a star, or placed along Che length of the charge, constituting essential-
- ly separate charges with a differing character of combustion (slotted
charge).
The most typical charge shapes are presented in Figure 7.7.
7.4. Determination of Operating Parameters and Characteristics of Solid-
Propellant Rocket Motors With Zero-Dimensional Statement of the
Problem
With steady-state operation the motor material balance equation is written in ~
the form
135 -
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. mQ m~ -Ir m,~ (7,11)
wh~rr ml pQr-~econd grrivel nf mg~e of ga~e~ due to prop~llant burning~
vr~1~i~111r~Cion of charge coating mat~rial end ca~e~in~ulation co~Cing; m~
~i~r-H~c:nnd flow r~e~ of gaAes th rough the nozzle; m2 me~s consumptioa of
~ ~nd filling rhe ~p~c~ evacuated during charge burnup.
in e~timgting t~rm m w~ ~hall proc~ed from Ch~ g~~umpCion that it~ qu~ntiey i~
enCirely deC~rmin~d ~y g~~ yi~id from burning of rockeC prop~llant . Chgng~ in
~~g yt~ld due Cd th~rmal de~CruaCion of Che charge cogting gnd combu~tion
chamb~r ingeallaCion play~ g role of adju~t~nent to th~ principal qu~ntiCy~
which wp ehg11 examin~ eeparately. With thig 8seumption and with utilization
of th~ ~xpon~nCial lgw of combu~tion
mi ~ Pt"~pKS~ ~7.12)
wh~re p T-- dengity of rockeC propellanC; S-- burning ~urface; pk
gCe~dy-etate pr~s~ure in the chamber.
Per-eecottd ggs flow Chrough the nozzle is expreased by formula
~ ~`AA`FK� (7.13)
m~ ~xRTK .
Ii~rh ~ C-- coefficient of nozzle flow raCe; ~kp nozzle throat area;
t
2 ~
A~ ~k+?~ 'k~T'
RTk tt~ermodynamic value of rocket propellant force; a;, coefficient of _
losse~, taking into account decrease in propellant force due to thermal
loeses and chemical incompletenesa of combustion.
T'he quantity of gasea used to fill area vacated during combustion per unit
~ of time Q Wcek wiCh sCeady-sCate pressure pk~ ie determined Aa
~ pR sm�K,
xRT~
~quation (7.11), with substitution of expresaione for mi~ m2 and m~~ as-
sumes the fo rm
~ 1 ~`~p"F"p (7.14)
PruiPKs 1` PtX
TK / a xRT~ .
We shall estimate the second term in the p~renthesea~ asauming
pK ~ 40 ~ 10' I7a. xRT,~ a 10' ,Q~c~ P, i-1600 kg/m3.
We sh$11 obtain pt- ~ 0.~3? ~ that ie ~ 0.3X With reap~ct to 1. `
136
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Ignnring Chi~ qugnCiey~ ~dlving ~qu~Cinn (7.14) r~].~Cive eo pk, w~
find
i
Pk ~ ~-~S~~r . (7.1 b)
~ KD
Utilizing Che ~bov~ r~lation~~ w~ ~hall pxamine the condition~ of ~tg~~.c
~tabiliCy of lt~~'~ op~raCion. Figure 7.8a ~howe chgnge in quantitiee m~ and -
mL for Chp cas~ < 1; ~igur~ 7.8b ehow~ ch~nge in th~ game ch~r~cteri~Cic~
for eh~ c~g~ v> 1. Th~ poine of inCergecCion of curv~~ mC and m~ cor-
r~~pond~ tn ~t~gdy-etiate pre~~ur~ in the engine. If foti any reaeon engine
pra~~ure ri~~g above steady-eCaCe~ in Che cgae of y~ 1, flow rate will be
great~r then ~as arrival ~nd pr~~~ure will drop until it become~ equal to
Pk.ycT� ~en preesure drope below pk~y~~, gae arrival exceede flow rate
~nd pre~~ur~ wi11 riee until it r~ache~ the level pk~y~~. Thu~ when y~ 1
RD'~T op~rgtion ig ~taCically ~table. When y> 1 any insignific~nt engine
pr~~gur~ devi~eion from pk~y~~ l~adg to further movement away from the
point r~f ~quilibrium proces~~ eieher in the direction of unlimited preasure
incr~~se on in the direceion of a presaure drop ~o zero.
.
~n~
/
I ~ m~ I me
I ~ ' ~ ~ ~
I j ( ( ~ ~ ~
I I i ~
~ ~ ~ I I I I I I
( 1 I I 1~ I I ~
I? I I I 1
~ic~t~ P /ns ycn~ / /R~t~r /
s) b) c)
~igure 1.8. Conditions of Static preasure Stability in a Solid-Propellant
Rocket Motor .
Key�
.a. ~o r the exponentinl comUustion c. ~or binon~ial linear combustion
law when y~ 1 lgw
b. For the exponential combustion
la~+ when y ~ 1
The magnitude of steady-state pressur~e wiCh the linear binomial combustion
laW is determined by relation
1
P~ = ~qFKp ~ .
-b
Prsut~ VXR7'K
In this case the condition of atatic pressure etabiliCy (see Figure 7.8c) is
expressed by inequality
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P~Su~o~ < ~0-~
x k'
~ox 5ummerfield'e ].aw pk.ycT ie determined by the formula
S a 1a~
pK � C Kp - 'S/ ~
If w~ proce~d from the general relation for a burning propellent aurface
5�5p pk should change on a time axis in conformity with change in
ch~racterietic ~(1~). Utilizing the 1r~CCer~ one can obtain for the
principal period of motor operati~n relaCion pk In order to Cranei-'
tion to relaCion pk (t) one muet utfl9.ze a gas arrival equation. For the
exponenCial combusCion 1aw
rso� ~ ) u y . (7.16)
Fro~n thie we obtain e working formula for convereion
et = p,~~~pK �i~' c~.~7~
where pk~ j~ maan presaure and regular burning rate characteriatic
valuea in interval d~ ; Q t-- correaponding Qt~+time interval.
Knowing the magniCude o� engine presaure in ateady-sCate operation~ one can
calculate auch engine output parameters as rate of gas flow through the
nozzle, thrueC and specific thruet.
b
RaCe of gas flow is calculated with formula (7.13).
To determine engine thrust one can utilize relation
p meva PaFn - PrFa�
We shall convert it inCo formulas which we ahall utilize in subsequent deri-
v ations. for this we sha11 replace the first two terms on the right eide
with a monomial, which includes one of the known gas-dynamic functiona.
Utilizing function f(7?), the equation can be rewritten in the form
P = Q~PKF~ - P.E~: ~7.18)
1
f � (1 C 1- -k~ 1~',~ ,
where Q-~ a coefficient characterizing loeses of total preasure in d~e
RDTT nozzle. Utilizing function 2(7~), we obtain
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p � ~ xcmc~v2 PrFe~ (7,19)
wher~ pKo ~'~'~TK critic~l veloci~y; x ~ a coefficien~
characterizing nozzle loeses.
Z (~l) ~ �
Subetitiuting in formula (7.19) expanded expreesions for m~ and akp, we
obtein
t ~ ~ ~
p � 2 ~ ~P~PKFMoZ ~~.I - P.F.. (7,19a)
The principal virtue of relations (7.18) and (7.19)~ alongeide their
simplicity, ig the fact that with their utilization calculaCions are eim-
plif ied by employing tabular gas-dynamic functione f(~ g) and 2(~ g). It
is not difficult to find argument from gae-dynamic table8 according to
ratio Fkp/F~*q (n g), where q(1?g~ is also a tabular gas-dynamic func-
tion.
We ehall utilize relation (7.19) to obtain a formula for epecific thruat im-
pulse. Divi:ding both parte of the equalitiy by and eubstiruring an ex-
panded expresaion for d~~ in ehe laet term~ We obrain
t
/r u' Y ~ j/'~K 2XcZ ~~ol C ~ ~ ~ ) ~F~~ P~'- . (7.20~
In view of the triviality of the aecond term in the bracketa~in calculating
RDTT spec~fic impulae nozzle loes coefficient ~ is ugually applied to
the entire brackets.
Nozzle loss coe�ficient x ~ characterizes the diatinction of the actual
process taking place in the nozzle from the ideal process on which deriva-
tion of the above obtained relationa is based and Wt?ich preaupposes one-
dimensional isentropic flow of an ideal gas in the nozzle. Usually this
coefficient ia presented in the form of a product:
x~ II x~~. (?.21)
where x ~i factors resulting from individual types of loeses.
' The principal types of nozzle losses include the folloWing:
1. Diaperaion losses, connected with nonuniformity of nozzle flow,
that is, with the presence of a radial velocity component at the nozzle exit;
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2. two-phaee los~ee witih condeneed particlas in the combusCion
producte;
3. friction lossee; ~
4. thermal loaeee connected with heati diseipetion intio ehe nozzle
walle.
Diepereion loesea are eaeily determined by ceiculation.
~or conical nozzlea coefficienti of loases of Chis Cype ie aqual Co
xc~ ~ COS~ ~ ~
wh~re d 1-- half-angle of nozxl~ op~ning.
For shaped nozzles
xc~ C 1-~- COS ~ a, ~
where r( p,~l nozzle opening half-anglea at the beginning of the bell
mouth and in the exit aection respecCively.
Calculation determinaCion of the remaining types of losaea involves a nua~-
ber of difficulties. Thie applies in particular to two-phaee loeses, which
for RDTT with metallized propellante are the principal type of specific inr
pulse loeses. This is due to inadequate atudy of the mechaniem of inter-
action beCween condensed particlea and gas, the nature of the proce~ees of
particle coagulation and fractionation and, chiefly, to a lack of reliable
data on dimensions of condensed particlea and the spectrum of their distribu-
tion. Two-phase loeses increase in direct proportion to Che mass per-
centage of condensate in combuation products and decrease with an increase
in nozzle aize in proportion to the aquare of the diameter of Che parCicles.
The magnitude of theae loesea for a nozzle of inedium size, with average
particle aize of 2-4 microns in the combustion chamber~rangea from 2 to 4X.
Thermal losses for large engines with an inaulated nozzle aurface do not ex-
ceed fractions of a percent. For small motora with uninsulated nozzlea the
magnitude of thermal lossea increasea to several percent.
Fr.i:tion losses depend substantially on the state of the nozzle surface.
T~~e development of surface roughness during nozzle heat eroaion is accom-
- panied by an increase in these loeses, which may comprise 1-2X.
Study [19] lists standard values of nozzle apecific impulae losses for an
RDTT burning aluminized propellant, obtained on the basis of experimenta
with a motor with a thrust nf 230 kg:
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Teble 7.
1 K~n~opa~ notspr Z CplAM~IIN~/!A~ NN~
nOTepN N3�3a 38fl8~AW88NNA KONJ~lN~Hp09eHHOA
~PENOA36NH04NONHM@ ROi'!PN 4 ~~0
CIOTEpII HA 'tQ@NNE 5 1,0 .
rIOTlpN N8 KONA~HCetlH10 6 0~3
nOtlpN tlA CN841tN yfUIOTNlNqA ~ ~~Z
RoaNwo narepN 8 I 6~0
Key:
1. Category of lossee 4. RecombinaCion loss~s
2. Avergge magnitude of loesea 5. Friction loeses
3. Loee~ae due to condeneed 6. CondeneaCion lossea
phase delay 7. Shock wave losaes
8. To~al loaeas
When utilizing a flush nozzle additional loases occur, the magnitude of which
is determined primarily by the degree of nozzle peaetration into the com-
bustion chamber and content of inetal additive in the propellan~. Accord~ng
to ~8], with a degree of penetration from 20 to 60X theae lossea comprise
approximately 0.4X for a composite propellant with SX aluminum; they in-
crease to 1-1.2X for propellant with 21X aluminum.
7.5 Determination of Operating Parameters of Solid-Propellant Rocket Motora
With One-Dimensional Statement of the Problem (Chargea With Lengthwise-
Conatant Flow Passage Croes-Sectional Area)
An increase in Che coefficient of combustion chamber propellant filling for
the purpose of improving a motor's masa characteristic ~ leads to a situation
where at the initial stage of charge combuation the rate of gae flow along
its surface increasea sharply. Thia ia accompanied by erosion effect, that
is~ an increase in rate of propellant burning, reaulting in a preasure in-
crease in the motor. In addition, there occurs a substantial preseure drop
along the charge. In other worde~ with the adopted working presgure in the
motor (pressure at nozzle inlet) there is an increase in preasure at the
forward face, from which in this instance one must proceed in performing motor
case strength calculations.
Uttder these conditions, the previously-examined zero-dimensional
solution becomes unsound. The following problem solution in a one-dimensional
statement is suitable to an equal degree for charges of various shapes: a
charge of cylindrical single-cavity grains, including a single-grain variant,
for a star charge and its modificattons, charges of teleacopic and crosa
shape, a"wagon wheel" charge, etc, under the condition that the chamber flow
passage cross sectional area remains constant lengthwise on the charge. With
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a ,
a
x dx
L
Figure 7.9. Calculation Diagram of Charge W3th Lengthw3se-ConsCant Flow
Passage Cross SecCional Area
ehig condition Che divereiCy of charge shapes can be reduced to the diagram
contained in Figure 7.9. For eimplicity of calcutaeions, we shall firet
examine a charge wit4r a blind forward end. Subsequently it will be shown in
~ (7.6) how end burning can be conaidered with thie arrangemenL.
We ehgll examine change in thermodynemic paYameters along the meCor's
gas path, proceeding from the forward end to the nozzle.
1. Region "d-k," bounded by the charge enda
We ahall assume that gas temperature rem~ins constant within the engine
cAVity, equal to the temperature of propellant combusCion. Change in
parameters of gas flow in aection "d-k" will be determined by two equations:
quantity of movemenC and continuity.
The quantiCy cf movemenC equation for a conatant section cavity assumes
the form
ritv-}- pF= const ~ 1~,. ~'P~�const~Ip .
Ir. the initial channel aection when va0 Ip~p~ Fk.
Fc.r any section
~ + pF - const ~ ~7.22) ~
wherE
k-I ~
t ~ ~ r~ + ; ~ (7.23)
.
gas-dynamic function;
= dimensionlesa velocity.
~ ~x~
, `
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Equation (7.22) doe~ noe coneider the ~orcae of gae friction on the charg~
surface. EeC ~aaCion of thie facCor in ehe region of Reynolds numbere '
characteri~tic for flow along a charge (Re w?lOS-106), indicates thaC Chese
lo~~~e comprise ae the 3nitial etage of combuetion'noe nwre th~n 2-5X of ~
pres~ure drop along the charge caused by gae fiow. It follows f rom the `
equation of quentity of movementi thae static preseure along the cavity
changes ae -
P � P~' (7.24)
To derive a continuity equation we aha11 specify an elementar~? burning
region of extent dx (eee Figure 7.9). Change in flow rate through the
cavlty croas secCion in region dx is equ~]. to gas arrival in this eection:
_ ~
~ ~ Pr n~u~of~ ~P) ft dx. ~7.26) .
We ehall present a general expression for gae flow raee in tihe form
m ~ PrUF ~ ~ Fv.
Taking inCo consideration relation (7.24), we obtain
. _ . . _
m ~ xR Fa~r (l~). (7.26)
Differentiating (7.26) and subatituting the result in (7.25), we obtain
. xR~ Fa,~v ~ (~)1 } d~ ~ pt II~iofz ~P) f~ ~u) dx. (7~27)
We shall designate the braces on the left side of the equality by
Subatituting on Che right aide of the equality expreseions for tl (p) in
conformity with (7.2) and f2 (p) in conformity with (7.6), dividing ttie
variablea and integrating, we obtain:
for the erosion burning spction
~ .
~(1~) d11 _ prtTru o RTK ~X - X 7 28
, J (~~~y ~ ~ k~? - l1~p)) FKQKpp~ ~ � ^p~~ ( ~ ~
. - ~v
where x*p coordinate of the section in which dimenaionless velocity
~ reeches a value ~ ~,p;
� for a nonerosion combustion section
!?1 dA _ IIru e RTK X._ "(?.29)
Ir (~)1" FK~~~ y .
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Combining equatione (7.28) and (7.29) and eubstituting an expreseion for
akp on Che right-hand side~ we obCain
where ~ � pKpr`~ Y xRTK ~ (7.30)
n
Q'1(~) ~r - ~ (1l) d~ ~ p d~, .
J ~~I~~ ~ ~ k7? ~np~I ~ ~
~np
- S~ ~~r x-- charge lateral burn3ng eurfaca.
Ftgure 7.10 contains a graph of function for p~0.15; kT~1.0-5.0 ~
r,rid Y ~0.3-0.7.
a~(~,) ~ . .
aJJ r ~ ~ ti ~
-~-}K,~-~5
- _ ~ ~ZO ` ,
/ ~ } ~t
~ ~ / ~ ~ ~ - /~,~'w"~'S ,
i~ I ~'~/r,~,~~0
~ Ci
q30 I~ , v ,
~
~ ~
, ~
gzs a~s
o, ~
Q~p ' g45
.
o q ~ a2 ,t ~
R1s
o, ~ qt aa qy as gs ~
Figure 7.10. Grapha of Function ~(7~ ) With p~0.15
Key:
a. v=0.3 c. v=0.7
b. v~0.5
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~
. ; .
We sha11 noCe Chat the complex 3~?fFk conaCitu~es U. A. Pobedonoetsev~s
cri~erion for a finitie charge eecrion.
The obrained xelations make it poaeible tio deCermine all parameCere of
interesC to ue in section ",q-K" (see F3gure 7.9).
The presaure drop along tihe charga is deCermined as
Op~.K ~ pA - pK m pA [1 - r (~N~1; _ ~7.31) ~
~
We aha11 note thaC when ~~0.5, a aimplified expreseion can be uaed to ~
compute gas-dynamic functicn r
r = 1 - . (?.32)
With valuea 7~ =S we obtain
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i"
�
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l?ma ~ PrutosdPK (Ku (7,38)
Ga~ flow rhrough Che nozz].e can be expreased ~ust a~
~ Aa0 (7,39)
1n~ xRTK .
Equat~.ng Che righe-hand parta of equatione (7.38) and (~.39), we obtain
i .
PK = [ rSd (Ku -I~ ~1 uto YxRTK n (~K11~ ~ (7.40)
ontpeFKp J
The obtained relation essenCially compriaes a modification of formula
(7.15) for p~essure in the cg8e of a zero-dimenaional variant, but
taking inCo accoune in integral form change in rate of burning and pres-
gure through the engine.
An expression for pk can glso be obtained taking into account (7.24) from
relaeion (7.30):
~
r.}- I-v
PK � l Fk (SKl k xRTK, ~(~K). (?.41)
One can obtain fmm ec}~alities (7.40) and (7.41) a relation linking dimension-
less velociCy in the exit aecCion of the charge cavity with the correla-
Cion between cavity and nozzle throa~ areas.
FK ~ ac~o (7.42)
~NP 9 ~~x) C~ K~ ) ,
In Che process of engine operation ~ea Fk increases as a consequence of
propellant burnup, and the percentage share of erosion coflbustion in total
gas generaCion contiuuously decreases, becoming zero when ~k ~ p.
At the same time there occurs equalization of pressure along t~e length of
the combustion chamber. Relation (7.15) becomea applicable thereby. Thia
relation can be utilized to determine mean indicated preasure pk in Che
engine, aubatituting in it time-averaged values S~p and ul Then one
can determine from formulas (7.15) and (7.30) the maximum am~unt average
engine presaure p ~~/pk on a time axis is exceeded, where p~~ ~gX
maximum pressure '~n the forward part of the motor after it entere quasi-
steady-state mode. Charge geometric parameters at the moment of attainment
of pA ~gX can be assumed equal to their initial value. If we assume
" utev I~~o~
we obtain from formulas (7.15), (7.30) and (7.42)
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" ;
.
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.
t
t r
k-I z ~ .
P.~ ~ ~_~~~_~r
k~ , ~7,43)
~ 1 ~ o~('~
In d~Cormining engine thrust correepond3ng to Che momenC of attainment of
P mgx~ one can proceed from relation (7.19a)~ bearing in mind tihat one
m~st eub~tiCuCe in this relat~on fu11 preaeure at nozzle inlet.
t
p~ 2 2 ~'~cQ~~Ka Z(~w) - pxFa~
~k'1'~~ n(~K)
7.6. DeCermination of OperaCing Parameeera of Solid-Propellant RockeC
MoCnrg With Charges WiCh SCep-Wise Change in ~'low Passage Croas-
SecCional Area
Charges with flow passage croas aectional area changing step-wiae along _
the length of the engine are in widespread use. They can be aubdivided
to two basic groups:
monoblock charges consisting of two aectiona with differing
geometry;
secCional charges, consisting of spveral sections (tiera) divided
by intermediate spaces. -
~ ,R
K~ d
f ft
"igure 7.11. Diagram of Charge With Step-Wise Change in Flow Passage
Cross-Sectional Area
A slotted charge consisting of a cylindrical ansi slotted section can serve
as an example of charges of the first group. An example of a aectional
charge is the propellant charge powering the Titan-3C booster, consiating
af five short elements with a cylindrical cavity, burning on the surface
~~f the ends and cavity.
We shall first examine an RDTT with a charge consisting of sectians with
differing geometry. Figure 7.11 contains a diagram of such a charge.
We shall examine here a charge consisting of two sections, although the
relations obtained below can also apply to more complex variants. The
charge cavity cross sectional area changes abruptly at the boundary of the
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~ections, buC it remaina constant in each. We shall deeignate in the ~
vicinity of Che boundary two test sections 1 and 2, in the upper and lower
secrions. From the cont3nuity equations writiten for these aecrians we
have
9 ~~a) � 9 ~ -F ' , (7.44)
Relative losses o� total presaure during'sudden expans3on (compresaion)
are determined aa
ePo ~ ~ k + 1 ~,n .
' Pos
where coeff icient of hydraulic lossea equa]. Co [ 29 ]~c,~ = 2(1 - F' )
~
during comprgsaion;
~P,~,~ = F~ - 1 ' during expansion.
ConsequenCly, ,
9 ~ 4 F ~ 1 - ~ k + 1 ~~n (?.45) .
~ and dimensionless velociCy at the beginning of the second section ~ 2 is a
function of ~ 1 and the ratio of areas F1/F2.
An equation of quantity of gas movement for section "1-k" can be wriCten in
the form
~KvK -E- PHF~) _ ~m~vi -I- P~F~) -I- P~ ~F~ F~~
where m1 and mass gas flow rates in sections 1 and k.
Expressing complete impulse flow through function r(~), we obtain
PKF~ _ P~F~. Pi i ~ -
~r~K, - r~~3~ F ~~1 1~.
\
For a random section in area "1-k" one can write
p Fl ~ F= -1 ~ , (?.96)
r(~) - pl F~ l r(~i) Fi
We shall designate
b = P~ . ~ -F F' --1, � (?.47)
F~ L r (~i) Fi
Then static pressure in any section af anea "2-k" will be determined as
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~
FOR 01~FICIAL V8R nNi.Y
p ~ ~r (7,4b)
{telntt~n (7.48) i~ anelogoue eo C7.24)~ but in pigce df p~ it coneeina ~
~ combin~d parameCer b .
~ In orde~ tid obtain ~ r~l~tion for gag grrivai in ~eceion "2-k," w~ ~hall
, utiltze ~qugtion (7.27), ~ub~rituting in it etatic pre~sure dee~rmin~d by
fnrmul~ (7.48).
pividin~ the variableg and ineegrgting the lefC-hand ~id~ frdm 712 te p~k~
nnd the,righe-hand eid~ from x~ to xk~ we obtain
m(~K) - m(~) a F~B�S~ RTK, (7,49)
tt~lneion (7.49) differe from Che previously derived (7.30) in thaC Che dif-
f~rence of valueg of funcCiong ~j (~1) enti~rs th~ Left-h~nd side~ and
parameCer g in place of p~ ittCo the right-hand side.
Ggg flow rgte in gection k is dete:wnined �rom relaCion (7.33). Kydraulic -
losses in Che prenozzle apace are calculaCed as in 7.5. It ie more con-
venient to perform calculation of engine parameCers in Che following
~equenc~.
1. Specifying preasure at forward end p~ , one determi~s and pl
on t}~e basis of relations (7.3A) and (7.24) .
2. With formula (7.47) we calculate parameter S, and then find ~ k ~
nnd pk with (7.45) and (7.48)~ (7.49).
3. We determine flow ratea cbk and mT, and Chen ~6~ as well.
With ~ k we calculate the coefficient of losses of complete
~ p:essure in the prenozzle space.
5. On Che basis o: total flow and presaure pk, we obCain the re-
quired nozzle throaC area.
Nomograms can be constructed on the basis of such calculations for propel-
lanr with apecified characteristics, enabling one to solve a direct prob-
lem to determine engine pressure on the basis of apecified engine and
charge geomeCry.
'Che specif ic features of calculating the operating parametera of a section
charge reduce to figuring hydraulic lossea in the spaces between sections
and figuring initial velocity at section inlet when c~lculating gas genera-
tion in the section cavity.
With a large intermediafce space and narrow cavity in the above-lying
section (tier), calculation of hydraulic losses is performed on the basis
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of r~iari~n (7.36). Yf tihe ~~ceiun~ Are ~eparAt~d by n~rtiow eran~vers~ ~].oCa,
eh~ following re].~rinn i~ r~commend~d in [29] for dee~rmining loeee~ of com~
plere prp~~ur~ in Che ~loet
aa~a~ ~ - ~ ~ e~,
wh~r~ p 7~ difference of dimeneionlese velocieies at elot inlet and out-
1et. -
if flow rate at gecCion cavity inleti ~Bx is known~ initig]. velo~ity ~ex is
obCeined frdm the condieion
x ~ 1~?TK
b t~xl ! ~c.z K ' (7.50) .
wherp ~ gx coefficient of stream narrowing at cavity inlee.
With this initial condition, change in etaCic pYeeAUre, according to
equation (7.22), followe the relation
P�,~ "Ir(~)� (7.b1)
Consequently the relation (7.49) in thie case aseumes the form
, ~ ~~1- ~ (~exl m ~t=~ I ~ XRT~ r ~7.6Z~ .
where ' p,,~ ~
p a .
Obviously relations (7.50)~ (7.51) and (7.52) muat be followed in solvi~ng
the problem examined in 7.5 with a burning forward end. If T gX Che
value of function ~(f~gX) can be calculaCed with the relation
3 3-v~n 1/?v-~~.
y 2v~ 1/ v-~
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l~ux ui~rllilew uoG u?vw
Chapter 8. D~NIATION3 OF OPERATING PARAMETER3 Ot~
SOLID-PROPLLLANT ROCItET MOTOlt3 IN THL VICINITY
OF 3PECIFILD CONDITIONS '
8.1. Relation~ for Ueviations of Operating Parameters of Solid-Propeliant
ltocket Motore in Vicinity of Specified Conditicme for a~ero-
Dimensional Variant
We shall utilize formula (7.15) to derive relaCioes for daviation of RDTT
operating parametere. Taking its logarithm~ differenCiating it~ end sub-
sequently replacing differentials with finite parameter increments~ we ob-
tain . . -
p ~ ! ~ L ~ P~r
i e (Rr~1 _ e~ _ "F~ J' .
'F ~ ~o
For simplifying calculations and subsequent concluaions it is expedient to
proceed to relative deviationa of parameCers e~xi~ Q xi/xi�
The relation for relative engine preaeure deviation aseumea the form
~PK @ [a~,+ ~s+a~* -~-Bx-I- ~ 6 (RTK) a~~ - aF,~] .
. (8.1)
Examining variations of quantiCies ul and RTk we ahall distinguiah com-
ponents, one of which resuits from variants in the characteristica of the
propellant praper, while the other is determined by ahange in initial
charge temperature.We at~a11 examine small deviations of initial charge
cemperature from its expected value, whereby one determines nominal en-
gine characteristics. Change in ItDTT thrust parameters within a broad
range of temperatures, as well as methods of engine tuning and adjueCment
making it poasible to reduce this change to a minimum. will be examined
in Chapter I0. The charge temperature deviations examined in thia deriva-
tion play the role of errora in determining expected charge temperature
both during engine tuning and adjusCment and for a motor without tuning and
adjusCment. It ie asaumed t~ereby that temperature is conetant throughout
the entire volume of the charge. Consequently, ,
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a~,, ~ 8u, ~ u~ BTN~ ~8,z~
BR.TK ~ 8 (RrK) a--~ ~ BTN, (8,3)
H~r~ eh~ ~ymbal N cla~re~ variat~on Components regulting Erom vgrience of
Ghnrnceeriy~ic~ of the prdpellant prdper~ ~ con~pquenc~ of random ch~ng~~
in rhemic~l edmpo~itinn, d~vi~tiong in Ch~ manuf~cruring pracr~~~ chang~~ in
prnpcrCieg during gCnreg~, etc. Sub~eiCuting (~.2) gnd (8.3) in (8.1) gnd
utillzing r~l~tinns (10.1) ~nd (10,2) (~e~ Chapt~r 10) for d~termining
deriv~eiv~g, w~ obt~in
8pK ~ ~ ~8u~ 8S -I- 8Pr 8x
8~~ b~~a ~ 8 (Rrk1-~- ~nl D) T�BTp, . (8.4)
Prnc~~ding in like mann~r wiCh equ~tion (7.~~), we obCain
~ 8PK -E- BF~p 8~pe ~ 8X - d ~Rr~) - mT�BT,,,
5~bsttCuCing here the value of d'pk from the preceding formula, we obtain
~ am~ bu t-}- 8S 8pt z ax - ~aF~,
~a~~ + 2 8 C~TK) -I- (vm D) TMbTM ~ . ~a~~
To determine thrust variation we shall utilize equation (7.19a), rewriting
it in the form
~
P-f- FaP. ~ 2~ k~. X~ro~PKF~cZ (~a~~
where x a coefficient characterizing losses in the supezeonic portion
of the nozzle.
Sequentially taking the logariti~m and differentiating this equation, we ob-
tain
(F~p1
AP pp AFa _ aXe y~~ ~p,c + AFKp ~ BZ Fa
~i~u Xe ~Pe PK FKp v/ F'Kp ~ Z~
~ Fa
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RaCio ~kp/~a ig equ~1 to th~ value of ga~-dynamic funceinn q(~ ge a
con~equ~nea of which eh~ la~ti e~rm c~n b~ r~wr~.ee~n g~
a_a~~~~e1 ~e
~,1_
4~( _1_ z t~o) ' ,
In order Co obtgin th~ ~impl~et analytical ~nLueion,~we ~hgll ~mploy an
apprc~ximating ~~1aCion which linke Ch~ num~ricgl valu~s of funcCione q(7~)
~nd ~(~1) in th~ range of velociti~~ ~ g of ineer~~t tio ue. It i~ evident
from the graphe z~f (q), plorted for the super~onic flow region with variou~ .
valu~e of k ehat within a�airly broad range of change in ~ g one can
~ regort to a linear approximaeion
Z � a � b9 (8.6)
For exampl~~ when k=1.25 in the range ~~1.7-2.4, if we aeaume gr1.425 and ~
b~0.50, th~ error of approximation doea not exceed 1.6x.
UCilizing relation (8.6) and proc~eding eo r~lative quantiCi~e, we obtain
8P = C 1-f- p�)~ 8~c 8xc 8P~ -I- BFKo -
. - - a
d - 8 I4 (~)1 - ~a BFa. (8.7)
9~(
?a1 - b .
We ahall introduce deaignation Yv~
SubstiCudng in (8.7) an expresaion for arpk, we obtain
bp au,+aPT+~+ ~ 8x+
~ 8 (RTK) - vBT~ -I- (U -4- m) TMBTu - v8~~v -
b(1- y1 8(9 ~~o~l - YdgFo -I- ~1-I- Yv~ aX~. (8.8)
_ a _b
q (~?e)
In order to obtain an expression for dIy it is necessary to subtract (8.5)
t rom (8.8) : a1r s 8P ~ i~C a,~, ~Vul ~ w~~T~
V/I~ ~
1 ~
~ y~ (aX -1- 8 ~~~)1- y'?~o,, (8~~ -I- BFKv) -
, ~1-~' Yv~ a b~ 8 I9 ~~a~l - YvBFo
9 (~?o)
(M -4~ Yv -4- M( v)
+ ~ - ~ T.BTM ( i -I- Yv) bx~. ~s.s~
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dX~ in Curn can b~ repreeenCeA in Ch~ form of a~um og var3at3nn~ of in-
dividual Cype~ ~f lo~a~g in Che euper~onic portiion of the nozzl~t
n
a~ r a~,~ ~e. ~o~
Of thee~, only diepersion 1o~~ee can be expressed by eimple anglyCical
relaeiona:
bX~, ~ tg ~aboc
in Ch~ ca~~ nf ~ conic~l nozzle;
8~~ tg a0. 4�t (a?ba~ -I- ao6ao)
in the case of a shaped nozxle.
Calculated determingtion of variatiinng of other Cypes of logees in Che
guperaonic portion of ehe noxzle encounter~ subst~nCial difficulties.
IC is atated in [19~ thaC different kinde of losees of specific impulse,
as loeses due to two-phase flow, due to recombination procesee~, heat
dissipation into the nozzle wa11g gnd losees in the boundary layer can be
connected witih engine pregsure and with nozzle dimen~ions as followe:
I
~
~ ~~---~;3' ~
_ where Rkp radius of critical secCion.
Congequently, the relative variance of these losaes with apecifiQd con- �
ditione of flow in the nozzle can in a first approximation be conaidered
as
8~~ - o, t ~aFKP + 28aK).
Coefficienta of system of equations (8.1)-(8.10) are figured for a epecific
motor on the basis of its known characteristics. For specified external
and internal disCurbances orxi, one can determine from a solution Co the
system of equaCions deviaCiona of engine operating parameters fy~ from
their nominal values, and one can obtain coefficients of influence
a~
~zi as deviations of individual engine operating parametera by a unit
of deviation of each of the disturbing factors xi. Maximum deviations of
principal operating parameCers are determined with the following formulas:
i
aPK. nP = L.~ 8X~ ;
~ ~ .
- am~. np � ~ (da aX~ ;
. ~
6P~p= zP 8x!)~~
r
bly. np = , ~ \ d ~t,~ .
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8u1, 8S, 8p*, ~8 (R7'�), 8X, 8cp~, 8F�p. conetiieuee disturbancee e~x~, ene~ring ~
into the formu'1a~ for ~pk. Qp and ~~hc. n p, tlietiutibanc~e 8Fn? 8xc~ b~9 ~~ln)1
above gnd beyond ehe onea enumeratied above~ also entier into Che formula
Eor ~ p~,p end ~iy~ np. Detiivarive~ ,
8 8rn~ aP a~y
. ~xt ~ ~xi ~ ~
con~tieute coe�ficienCe of influence determined fYOm the eolutiion of eyetem
(8.1)-(8.10) r8laCive eo the specified dieCurbing factore.
8.2. Relations �or Deviation~ in Operad.ng Parametera of So1id-Propellanti
Itocketi Motor~ in the Environe of Specified Conditions in the Case of
a One-Dimenaional Solution
Anglytical determination of deviations of RDTT operating pgrametera in the
environs of specified (nominal) condit3ons in the given caea are based on
soluCion of a system of linearized equations obtained as a resulC of trans-
formaCion o� relaCions (see 7.5).
Ag in Che preceding paragraph, we sha11 presenC deviaCiona of operating
parameterg and charging parameters of RDTT in relaCive~quanCitiea.
Dimensionless coefficients with relative deviations (varlations) are given
in the designationa previously adopted 3n Chapter 1.
LogarithmicaIly varying equation (7.30), which linka presaure at the forwerd
end of the engine with charging parametera and combined gas-dynamic function
~(f1) and proceeding Co relaCive deviationa, we obtain
- aP~ a' ~ CBPr -F' 8u~o B.Sd - 8FK bx -1.
-f- ~ 8 (RTK) - m~j ~"i ~m ~1~~~ 8~~~ '
We shall presenC this relation in the following form:
aa�"aP~ ~Bp* ba8u~o -1- b~a.Se
where ~BFK 6~6X 63 rb ~RT K) -I- a38~ � p, (8.11)
a3"=-1~ 6~a~6a~b,si = ~ ~ .
~a~- � b~aab3ra l 1 ~ .
a ~_y' 2 l-v ,
~ ~d ~
0.1 = ~~K~ d~ ~ ~ ~ ~ y �
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A r~~ativQ change in pre~~ur~ drop ~long the charg~ can be pr~senr~d in
th~ form
onA opK
aepA,~m~_ o~,
Ur~lizing reletions (7.24) and (7.31), we obtia~.n
e epA, K~'i =
r(~ BpA. K ~~"'~~j 6P~
or
a~!'K8 Sp~,, K-~- aAAKBPA a�~" KaPK ~ ~8.12)
wh~re ~ -
- P 1
.
a~,;'K ~ -1; a~','K ~ ~
v~ r(11K)
a~~ K � ~ .
In like manner we obtain from equation (7.31)
BApA, K = 8p~ - i ( K) ~ (~K)18~K
or
aK�KB~pA, K-h aKASpA a~8~ p, (8.13)
where Q~ m ~ ~ Qn~ ,a 1 ~ vK ~ - I - r ( ~ c~' � ,
Logarithmicly varying the equation of gas flow through the nozzle and
proceeding to relaCive deviations, we obtain
a,n~ = 8~~ + + aP~ + aFKp - ax - 2 8 (RTK) -
. ` ~ ~n~~`� n~ a~`K
or
a~ ~a~?k -I- bK. o~~ ~BQ~ �K. ~BPK h~. eBFac
bK. ~~X b~ ~8 (RTK) ~ 0, (8.14)
where
aK ~a-1; bK.c� ~.e� aK.c~`bK.o�
bK.C ~ uK.C � ~ ~ i
aK. t = - ~ ~n~~~~ n � .
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~
Differentia~ing an expreseion for ~ after traneformations we obCain
2 -1- ~ (n (~)1 n-~j 8~.
BarQ m- k ~ t (8.1 b)
LoggriChmicall.y varying Che gas �low equation for the char$e caviCy exit .
section and proceeding Co relaCive deviatione, we obtain
8mN ~ 8FN -I- 8PK 2 8x - s (RTK) -I- y-=
~j ~ I?J ~~~18~
or
~
aK KBmK bKBFK a~BPK -f - 6K8x 6K rB~,~RT K1-f - bKB~, ~ 0, (8.16)
where
' m
aN 6 ao 1 bx 6Rr t. .
1 K a K� 1i K�L K �3~ r~ f
bK � y ~ ~ ~ [y
In like manner, Cransforming the equaCion of gas generation from the
charge end surface, we obtain
amT = aPr + asT -I- 8u,o v6pK
or '
aT b~k -f- ~,~PT -E- bT*8S: 6t8uio a=BpK ~ p~ (8.17)
where ar"'=---1~ ~m~~bs~l; ~tm+y~
We can transform as follows the last equation sums of gas generated
from the end surface and cavity surface:
.
8m~ = 8mT SmK .
� m~
Since
mK ~ PTu~oSsPKKu~
~'r a P'~toS:PK~
nk = p~1oSsPK ~ Ku ~ ~ ~
we obtain �
trtK Kv . mr I
=S
?~e K~ -F' S ~ ~c Kw ~ r
where S ~ ST/Ss,
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In t~.nal form we obCain . , . ; . , '
A~~ ~BI/iQ uQ *Blrit aQ KSmK ~ 0; (8,18).
where
.
(jmC ~ 1' QmT ~y ' nik ct .
� ~ � Ku -F,~ ' a� Ku ~
We have obtained a cloaed sys~em of eight equatiione. Fo~lowing are un-
knowns in Chis syaCem:
8~, 8pk, 8~p~, K, 8ar~, 8mt, BinK, 8rit~,
External and internal disturbances are manifeated in Che form
8u,o~ 8A~? BFK~ BFKP~ 8x, 8(RTK)~ &Ss? bsT~ 8~~�
The obtained system of equaCions makea it poes3ble for the epecified dis-
Curbances to find deviations of the principal parameCers of the in-chamber
process and gas flow Chrough the nozzle.
In order to obCain Chrust variations we must utilize equaCion (7.43),
transforming it as was done in 8.1.
We sha11 obCain the following:
ap = ( i + ~P) {a~~ + 8~~ + saK -I- BFKn - ~ ~a~~~ aa~K} -
- yPBFa.
Value Spk is placed here from the soluCion of system of equations (8.11)-
(8.18).
Variations of specific thrust impulse are obtained as
a~Y = ap - a,n~.
For brevity of presentation in deriving a system of equations, we have not
examined the,individual variation components u10 and fiTk, which can be in-
troduced into the equations as was shown on 8.1.
We sha11 note that the relationship beCween variations ~pk and the dis-
turbances producing them can be established by another method, logarithmicat-
ly varying equation (7.40).
Table 8.1 contains coefficients of influence calculated for RDTT withti~e
following characteristics;
v= 0,3; ~K = 0,3; kx = 2,0~ ~l~p = 0,15~ k= 1,2; S= 0~05�
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'Cnbl~ 8.1.
OTNJIOMlNt1A flInAMlTpO~ M~ ! MNN4y OTIta011lNNII
On~Myu~eHUe ~ 2 roaMyu~~aw~ro ~~Krop~
EAA DpK 81~t Am~
8u~o 1,429 1,449 0 1~~29
8pr 1,429 1,429 0 1,449 '
a(xRrN) 0,714 0,714 0 0~414
8~K -0,410 -2,383 -1~08b -2~370
BFKp -1,019 -0,934 -1~a66 -1,9~1
6S6 l,384 -1,460 1~047 1~470
8~c -1,020 -0,93b -1~068 -1,9~1
8Sr 0,04b 0,041 0~047 0~041
Key. .
~1. Disturbance 2. Deviations of parameCers per uniC
of deviation of the diaturbing
factor
The following conclusions can be drawn from the calculaCion resulta:
1. Disturbances ~u10~ ~p T and d`(x RTk), do not influence quanCity
~ k, produce in the case under examination the same relaCive deviaCions d'p
and d~m~ as in Che case of a zero-dimenaional aolution. Coefficients of in-
fluence for preasure prove to be equal Co 1/1- Y for t~P T and dulp, and for
~f(x RTk) comprise
_ 1 ~ for pressure and 2( t, for rate
2 I-v
of flow.
2. A different character of influence is observed for disturbances
connected with change in ~ k. Influence of Fk, eT FkP and o`Sd is manifeated
most strongly.
An additional relationship between coefficients of influence is seen from
the numerical results nf all calculations, including those contained in
Table 8.2:
a~ _ i a~
- aFKP - i - v - aFK .
This relation can be obtained analytically if one logarithmically varies
formula (7.40).
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'C~bl~ 8.2.
1 OTNAQNlHY! OpA H~ lJ~HNNyy OTKAOMINNR ~0lMYtl(~IOU~~P01blIlTOp~ 'Z
DOlNyUIlMM~
~.o,oa s..o,t ~..o,a
b~K 1,202 1,183 -0~877
~FKp -f1~427 -0~44g --0~652
8 * n~~ O,OtB 0,104
Key:
1. Disturbgnce 2. Deviation ~p per unit of die-
eurbing facCor deviation
3. The in�luence of di~turbances e~ Fk~ ~S Fkp and d Sd increasea
with a decrease in characeeristic S~ ehar ie, with a decreaee in the role
o� end burn~ng in thg proce~g o� gas formation. This ia evident from
Table 8.2, which cone~ine the reeults of c~lculatione �or variante di�-
�ering by qugntity S, but with common value~ of the remaining characCeri~tics;
y~ ~~~3~ ~'K ~ k~? ~ 2?4~ 1~np = 0~ 15~ k~ 1~2.
At Che same Cime the i.nfluence o� deviaCion arST increases.
4. The i.nfluence of deviationa dFk and orS~ on quanCity dpa in-
creases substantially with an increaee in ~ k, which can be aeen from the
figures in tables 8.1 and 8.2, obtained for ~ km0.3 and T k~0.5. The in-
fluence of deviations d'Fkp and e~`ST decreases with an inc~eaye in ~ k.
5. '.Che value of erosion coefficient k~ influences in the same
direcCion as ~ k. Table 8.3 contains results of calculations for variants
differing in their values with common characteristics
v- ~~3~ 7~K = 0,3i 71,~p � U~ 15~ k= 1,2; S= 0.05.
Table 8.3.
~ 1 OTIIJIOMfNN! 8pA N~ lJ~NNM4y OTKAONlNNN lO~Yytt(~IOtt(!t0 ~~KtOp~ ~
BO~ItrtalMM!
R~.. i,a I R~ ~~,o k~.. ~,a
8FK 0,360 ~~4I0 0.454
aFKp 1~069 1~019 0~97b
~6 1~381 1,384 1~386
0~04T 0,045 0~042
Key:
1. DisCurbance 2. Deviation orp~q per unit of disturbing
factor deviation
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run urrt~trw uor~ u?vLi
8.3. Select~.on of Opt~mal pk and ~k Valuee
SelecCion of opeimal RDTT desi,gn parameterg and operating cond~.tiong in
~ngine design and ~ngineering hag tihe ul.timaCe ob~ect~.ve oE nbtaining
ehh leape launch m~~e value m~ for a veh~cle with a epeci�ied paylogd
i m~~~~ nnd pc~rfhrmance dnta (range~ gpeed and alriCude).
'~he initial masa o~ a rockeC with RDTT is c~etermined in general korm as
mn, k ~ I
m0~ I-�KU~l-a)-f~ ~ 8,19
where N k= a/m~ relative fuel reserve; q-- coefficient of engine mass
perfection; k-- a coefficienC taking into accuunC the mass of connecting
asaemblies, lines, aerodynamic elemenes and auxilixry devices.
Consequent~.y, deCermination of ehe condition of minimum value of m~ boils
dnwn to examination fox the extremum of product r k(1+ d), which is the
function of a large number of ballistic, gas-dynamic and structural
p~rameCers. Performance of this task developa into a multilevel in-
vestigaCion which goes far beyond the limiCa of tfie Copic under in-
vestigation. WiChin Che framework of our presentation we sha11 limit our-
selves Co a brief analysis of the influence of two parameters on the mass
chnracteristics of the vehicle engine operating pressure pk and
quantity ~ k, which constituCes one of Che mosC imporeant characCeristics
of loading conditiuns. With this approach the condition of the extremum
is written in the form
- aa K
apK ~ (1-}- a) = Ot
aa (8.20)
a~K ~ 0.
We shall examine derivative
a~ �
a�K _ a,K. a~,,
~PK a~,, aPK ~
where Che first factor of the right side characterizes change in required
fuel reserve with change in specific impulse Iy when vk,=consC. Taking in-
to account gravitational losses, which for vertically-launched rockets
comprise the bulk of velocity losses in the powered segment of flighC,
a�x 1 ln (1 - �K) - a~6�K $.21) ~
, ~
d1~nK.econ~t) ~Y ~ - b
1- �K +1
where 1~ initial thrust-weight ratio
~ ps .
, .
8~?~0
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a and b-- coeff~.cients of approximaCion, dependenti on ~he control
program, rhaC iA, on the type o� function A~9 (~1). ~or examp~.e, for
Eunction A(~) in [27], a~0.10, b~0.689.
tE in the first approx~.mation we proceed from Taiolkovskiy's formula,
thc~n
~d~"_~__ ~ -N" lt1(I _ �K)~
~ ~ ,
which also follows from relation (8.21), when n=~.
DerivaCiv~' a Iy/ a pk expreases the relationship bet~veen specific impulae
and combuation chamber pressure, which was examined in general form in
Chapter 3. accord~.ng to the approximation dependence of RDTT specific
impulse on pressure obtained in [4] for standard composite propellants
when pk~(30-70)�105 Pa,
~ = 0,76 - O,OOGpK.
Let us examine the relationship to pressure of engine mass perfection co-
efficient c~ =mk~~ /w. Without going into a.~detailed analysis of the
components of engine structure mass mk,~ we shall represent it as the
sum of two terms:
A� mN. 9-~- mrt. 9~
where mH~~ mass of the structure's load-bearing elements, for which
the forces af engine internal pressure are the principal type of load;
m.R,,~ mass of structure elements indifferent to internal pressure
(heat shielding components, nozzle inserts, diaphragms, etc). Quantity
mn.~ is determined chiefly by the mass of the cylindrical engine case
and end plates. Mass of the cylindrical case is
mr. K = nDHPKktL~K?
where DH--- engine bore; P k-- density of material; L-- length of
charge; kL coefficient taking into account difference in charge and
thrust chamber lengths; a~~ relative wall thickness, equal to
P~
- e -
K - 2Q �
a
Here p'm calculated pressure; Q B-- ultimate strength of case
material.
The mass of two end plates of elliptical shape without insulation and
nozzle cutouts with a semiaxis ratio of 1:2 is determined with the follow- _
ing f~rmula [4]:
mR = 8 7ZDNpKORi
where a~-- relaCive thickness of end plate wall.
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_ _
Hcnc~nfdrth we ~hgll aegume ap ~ 4 k . Consequently mgep mH~~ can ba
predenC~d i.n ~ fir~e ~pproximation a~
~ ~kL~~ 6l
rrtN, ~ ~ kmnn~MPK K ~ ~
- 3 /
wlirre I, chnrg~ 1~ngth in g~ug~s; kmH coefficlent Caking into gc-
cnunr mn~y of gddieional ~lements also contained in mH~~ (faseeners,
thresd~d cnnnection~, eCc).
periveCive ap wi~h specified charge mass and length ig deCermined ag
da I dmN, ~ nDN 6 I
a~ = w T~ kniN w PK ~k~L kp. (8,22)
wliere kp~p~m/pk coef�icient of calculaeed preseure exceeding engine
working pressure.
Let us exnmine the sCruceure of coefficienC kp:
kp = ~ 1-~- 6P,~ p
~ ~ ~ (8.23)
N,1cCor pk~/PkN ~haracCerizes change in pk with change in initial charge
tem~~c~ruture in the specified temperaCure range of engine utilization:
'('~~N i'fHT. Calculation of ratio pkT/PkN is examined in Chapter 10.
ft;~tto (p~ max~pk~ is figured with formula (7.43).
~nctor (1+ ~p,p~ ) takes into ar.count maximum variance of quantity p in
- the environs of its nominal value. Maximum variance dpq p wit~ the
assumption thaC the principal disturbances engendering it are of a random
nature and are governed by Che law of normal distribution:
, BpA. nD ~ Y~ aui Su,o)~ -f- 6PT~ -I- ~ a~-BFR~~
+ ~ ~x 8X ~ ~ a cRrK~ a ~RTK~~~'~ ~ a~p"B--bSs~
'
-t' ~ asT -F~ ~ a~~ a~~~ . (8.24)
ap,~ ~pA
Here t~u10, ~N T, etc values of disturbances; a~te ' dp~ , ete co-
efficients of influence of disturbances on quantity p~ , determined from
solution of (8.11)-(6.18) as numbers indicating deviation ~p~, caused by
a single disturba?ice of a given type. p is a reserve strength factor
adc;pted on the basis of experience in eng3ne development or determined for
specified reliability taking into account normal distribution of wall
thickness, material strength characteristic, and quantity p~ .
~
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in ~el~ceing ~n enginp wnrking pregsure �rnm exCr~mum condieione hk (1+ d) ~
it is also necessary ~o consider possibiliCies of abnorm~l burning and
engine operation in~tabiliey when pk drope below a certain limit deCer-
mined exper3mentally for each propellant Pkmin� This condition can bQ
presenred in the �orm
PK aP~, nv~ ~ PK m~n~ (8~2~)
where tT pk~ ~ p-- maximum spread of pk around its nominal value with Che
lowest utilization Cemperature TH min�
Quantity pk~ ~ p~ according to 8.1, is calculated with the formula
BpK~ np � 1~ y X'
x ~(8u~)' -I- -I- (8Pt)' -h 4 (~X)' (8 (~K)1' -4~ f8~~)� (6FKp)' .
(8.26)
In case the value of pk obeained from the condition of the optimum
fails to satisfy condition (8.25), in spiCe of Che condiCion of optimiza-
tion it is necessary to proceed from a higher value of pk~ which should
ensure reliability of engine operation at the lower pressure limit. ~
Probability of normal engine operaCion aC the lower working pressure limit
ig deterniined as
Bep (PK pK n110 ~ � ~ ~Z~i
_ where C~ (Z) Gaus.q probability integral;
Za PK-'Pu m~n .
QO min '
Qp min = YQpK~
en�
Here v min root-mean-square deviation of pk, calculated on the basis
of solu~ion (8.1); Q~p.~ k~r root-mean-square error with which allow-
able value pk min is determined experimentally.
It follows from the above-examined relations that static characteristics
play an important role in selection of optimal engine pressure.
The second equation of system (8.20) establishes in general form a link
between coefficient a and gas-dynamic parameCer ~ k. Velocity coef-
ficient ~ k influence~quantity a in two directions:
accord~ng to relation (7.43), coefficient ~ k, with specified
value pk, determines ratio p~/pk, and consequently calculated value
p'm as well, that is, an incr~ ase in ~ k should lead to a heavier
strucCure and increase in a;
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according to relation (7.30), quantiry ~ k, with epecified value
p~a , dQterminee charg~ flow peg~ege cro~~ sectional area ~k~ gnd con-
- H~quenCly the coefficiene of engina propellant filling.
S~lection o� e higher value ~ k leade to a decrease in ~k~ and consequently
to a d~creaeo in a. ,
~ollowing thea~ conclu~ione~ derivati,ve a
a can b~ wriCten in the �orm
d~K
a� a aa aa ~ ~
aa .~.,1 ~ 8,27
� a,ca""'_1 ~ a~ ~ .
Here ~=ST~~kaM caefficient of propellant filling of thrust chamber cross
aectional area FkBn �
Tt~i~ coefficient can b~ preaented in the form _
I - FN
Hence E ~ ~w ~
a~ ~ i
a~' - F~.N e~~''
Derivative dTK f aF~ is determined from soluCion (8.2).
Derivative ~a/8e is determined on the basis of ana].ysis of masses and
dimensions of RDTT structural elements.__r
eti �
i
I
I
~~l /R/II ~A
Figure 8.1. Depel~dence of Coeffieient of RDTT Mass Perfection on ~ k
Solution of equation (8.27) enables one to establish an opCimal value
x k.o T� which with the adopted level pk ensures minimum a(Figure 8.1).
In the region lying to the left of d k~~,~.T, the influenceaE low denaity
of engine propellant filling predominates with low values of ~ k and ~
correspondingly high Fk. In Che region to the right of T k.o7! T the
factor of more heavily weighCing Che structure due to an increase in the
ratio p~ m/Pk With an increase in ~ k predominates. At the point of the
optimum, the influence of these oppositely-acting factora balances, secur-
ing a~in�
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Wc! Nl~nll nnCc thnt in plaee ~f argumc~nC ~ k one aan utilix~ robedono~e~ev~~
c:rit~rlnn x~ s~~fk, Alnce according to (7.3~)~ there exi~t~ an unambiguoug
cnrr~~pond~nce between x and ~ k. This wi11 lead to change in the form
~ of notaCion nf Che ~bove-examined relationa, without altering thair Bub-
gtgnce.
Link~d to opCimal values pk and 7~,k is auch an imporeant itDTT
characteri~tic a~ ultimae~ operaCing time ~ , which is defined ae the
fuel burnup Cime by the thickne~g of the bur~~ng web. Nominal value p
i~ determined ag
'~na'�` T ~
~ u~ (z) ~P (Z)I" dZ
where z~e/~1 relative thickness of burned propellanC lay~r.
Here functions p(Z) and ul (7.) take ineo coneideration changea in pr~s-
sure and uniC burning rate, corresponding to currenC value Z. ~rosion
effecC at Che initiaCion of burning, as we11 as change in ul in web C~ick-
n~ss, due to nonuniformity of temperarure field, ia taken into account
by funcCion ul (Z).
Variation of ultimate engine operaCing time is determined as
8~~p m bel - 6u1 - vBpK,
where a el relative deviation of thicknesa of burning web, caused by
manufacturing errora; ~ ul variation of unit burning raCe; ~ pk
variation of inean indicated engine pressure.
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; . .
Chapter 9. FACTORS DISTURBING OPERATING CONDITIONS
OF SOLID-PROPELLANT ROCKET MOTORS
9.1. General Survey of Disturbing FacCors
FacGors cauaing disturbance of RUT'r operating conditiona and deviation
of operating parametera from epecified figures are extremely divereified
in their rature and man3�eatation. ~~me are connected with variance in
loading parameters and random proceasea during engine operation. Others
constitute factors not taken into account in deriving principal relations
for calculating RDTT operating characteriatica. The influence of these
factora in comparison with the main factors enCering into calculated
relations is ama11, and from methodological considerations should be
figured in the form of corrections, that is, diaturbances of calculated ~
operaCing conditions.
Disturbing factors are divided into external and internal, based on
mode of manifestation.
External disturbing factors include the following:
deviations of charge initial temperature, cau3ed by changes in
ambient temperature and accompanied by changes in rate of fuel burning,
its energy characteristic RTk and denaity p T:
external load facCors which alter the conditions of charge burn-
ing and which diarupt ita conCinuity.
Following are internal disturbing factors:
deviation of charge geometry within the limits of process allow-
ances in its manufacture;
variance in propellant burning rates and energy characteristics,
caused by deviations in its composition and manufacturing process from
standards;
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d~vigCion~ of nozzle ehrogC ~rea gnd dimen~ion~ o� axh~u~C bell .
mouth fr~m nomin~~ v~lue wiChin t;he limitie of nozzle and nozzle insert
mnnufacturing Colerances;
- def~ctg in nharge mechan~.cgl sCructure (cracka~ pores)~ occurring
during charge manu�acture or during ~tiorage and transport~
The above-enumerated internal disturbing faceors are connected with errora
of manufecture and mgnufacturing process deviat~ons. In addition inCernal -
disturbing factora are in operation which are caused by phenomeng attend-
ing Che RDTT op~raCing proceas. These facCore include Che following:
change in nozzle rhroat area as a consequence of heat eroeion or, on
the contrary, as a conaequence of elagging (obliteration) by condensed
propellant combustion products;
variance in h~at losaes due to variabiliCy of procesaes of heat
eranafer from combustion producta to the engine's inCerior surface;
variance in completeness of propellant combustion as a conaequence
of instabiliCy of combustion conditiona, as we11 as mechanical e~ection
of incompletely-burned charge particlea;
change in flow rate of propellant combustion products and their
Chermodynamic characteristics d~se to masa removal of insulation and non-
propellant motor components.
The above is a liat of the principal factors causing deviaCion in RDTT
- operaCing parameters f rom their calculated (standard) values.
One should bear in mind that the influence of one and the same disturbing
factor can be manifested in several directions, causing various dis-
turbances which in turn deCermine variance of RDTT operating character-
istics. For example, change in initial charge temperature gives rise to
disturbances
o~lt1 ~~s1'~ d~~X~1 ~P t . etc.
The enumerated factors are far from equivalent in degree of effect on
RDTT operating conditions. We should emphasize first of all the tempera-
ture factor, which significantly exceeds all other factors in its con-
sequences, a fact which makes it necessary to devote a separaCe chapter
to examination of this factor. Sections in this chapter deal with the
most significant of the other factors. I
9.2. Influence of G-loadings on Operating Conditions of Solid-Propellant
Rocket Motors
G-loadings of various direction, to which a rocket is sub~ected in flight,
can influence RDTT operating conditions through disturbances of various
kinds, the principal disturbances among which are the following:
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cheng~ in rgtie o� fuel burning;
change in burning eur�ace ae a consequence of formae~.on of cracks
gnd d3erupe3on of charge-case adhea3on; .
chnnge in thruse chamber �low paesage croea seceional areae es a
congeque~tce of charge deformatiion;
change in nozxle flow raCe coefficient. -
9.2.1. Change in Propel~ant Burning Rate Under G-loadings
As experiment indicaCes, the burning rate of eolid propellant increeses
in a stresaed atate. ~
It. Ye. Sorkin [24] linka this with atress tensor influence on the rate
of condenaed phase breakdown, equating this phenomenon with dispersion
af propellant microparCiclea from the aurface into the gaseous phase. IC
is possible that an incxease in the propellant burning rate in a
- etressed state ie cauaed by Che appearance o� a neCwork of microcracka,
rhat is, increase in the effective propellanC burning surface.
It is proposed that the influence of charge deformation on burning rate
be taken into account by means of introducing into Che law governing TRT
combustion a correction factor [24]:
s ~ I fi ~
where ~ deformation of elongation; 7~ experimental coefficient;
n exponent.
In calculating disturbances of RDTT operating conditiona one should
assume Che follow3ng:
C~u~ L:'~ ~
Change in burning rate from centrifugal G-loadinga was discovered in
developing rockets and missiles with RDTT which turn in flight on their
longitudinal axis. Studies of TRT combuation in motors mounted on a
centrifuge or on a rotating test baich indicated thaC all types of rocket
propellants are sub~ected to the influence of centrifugal G-loadings on
rate of burning, but this influence is stronger on propellanta wiCh _
metal additives and begins to be manifested at a low level of G-loading
L121� ~
Change in the burning rate during rotation is caused by the thermal ef-
fect of condensate particles on the charge burning surface, Craters are
formed on this surface, at the peaks of which globules of condensate are
located. Formation of craters leads to increased eff ective charge
surface and consequently to increase in the mass burning rate. At the
same time there is an increase in the linear rate of propellant burning.
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'~hig tgkes place a cone~quence of Che fgcC thgC combusCion of inetal
pxrticleg take~ plac~ ne~r ehe surface~ which leada to an increase
in he~C flow toward Ch~ surfac~. Zntieng~.fication of heae transfer from
gg~~~ eo prop~llgnti i~ ~l~d ~chi~ved due to ehe heat conductivity of ~he
conCacCing condensaC~ parricleg. Tn the final ~nalysi~ ehe effeceive
burning rate (thaG ie, in relaeion eo flat burning eurface) increases.
A similar mechaniam of interaction between condensaCe and combuseion
surface ie manifested not only in meCal].ized compneite propellants but
also in propellant~ of any composiGion, inc~.uding ball~.etite, since the
cnmbusCion producta of any TRT conCein a certain quantity of solid
particles.
The authors o� (23] propose thaC the coefficient of increase of
propellant burning rate wiCh centrifugal G-loadings k~ be determined
from anelyCically obtained equation
kp (kp 1) ~ j/'kn(ka 1) ~ pzr
_ where D2= ~Ik/~ r-- raCio of coefficienCs of heat conductiviCy of
propellant combustion particles and gaseous producte; D1 dimensionless
parameter consCituting a complex aggregate of physicochemical characCer-
istica of propellant and combuation producta, which is determined
experimenCally; N G-loading facCor.
Change in burning rate under G-loadings is determined by the magniCude
and orientation of the acceleration vector relaCive to the burning sur-
face; maximum change occurs when the centrifugal force vector is directed
perpendicular to rhe burning surface and into the charge. With an increase
in G-loading the burning rate increases, asympCotically approaching its
upper limit: .
kpnp �~y-~?~� ka = Z (1 -I-1~4Ds ly. .
When calculating RDTT operating condition disturbances one ehould assume
6u~c=k~-1.
9.2.2. Change in Charge Burning Surface Under G-loadings As a Consequence
of FormaCion of Cracks
In Che process of operation a rocket propellant chazge is affected by
loads of various kind, which begins during manufacture and is contin:ed
at subsequent stages of utilization during tralnsport, storage under
variable temperature conditions, loading, etc. Accumulation of the
influence of applied loads can lead to the formation of cracks. Sig-
nificant defenCs in charge structure occurring in the process of manu-
facturing and storage can be discovered by means of crack detection in-
spections and can serve as a reason for removing a motor from use.
However, detection of small defects is difficult.
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Rgpid prea~ure-loading of a charge during engine ignition and impoai-
Cion on the charge of various-directed G-loadinga during fl.ighe can lead
to Che growth of prior-formed small cracke and the appearance of new
cracks at point~ where the mechanical structure of the charge weakena.
Crnck formation in bonded charges of compoeitn propellant ia connecred
wiCt~ tihe existence of tensile atresaea. For inaerted charges the oc-
currence of cracks 3s possible when the charge strikes the diaphragm with
a sharp increase in axial 1oad3ngs.
IC would be erroneous Co assume thet the aurface of any crack conaCituCes
a direct addition Co Che toCal charge burning aurface. The condiCiona of
flame propagation in narrow cracks and channels were the sub~ect of
special experimental atudies [11]. It was established Chat there exists
a certain threshold crack width below which flame propagation deep into
Che crack becomes impossible, and consequently inclusion of the crack
- surface in the overall burning surface as we11. Following are the
principal �actors determining Chreshold crack width: composition of Che
propellant, its burning rate, presaure, initial charge temperature.
Threshold crack width decreases wiCh an increase in pressure and
propellant burning rate as well as with an increase in oxidizer contznti.
- It increases with an increase in oxidizer average particle diameter.
If flame penetrates into a crack, Che crack may continue to develop as
a consequence of the development of overpressure within the crack, as
well as from stresses applied to the propellant caused by G-loadings.
Tensile stresses acCing in a direction croaswise to the crack are the
most dangerous for crack growth.
In determining deviations of engine operating conditions caused by the
occurrence of cracks in a~tT charge, one must take into consideration
crack existence time, figured from the moment the crack is opened by the
combustion front to complete burnup of the propellant layer in which the
crack is located. If during this time engine pressure reaches a new
level corresponding to the increased burning surface, consideration of
the influence of a crack, manifested in the form of disturbance ~S, is
effected on the basis of the relations in Chapter 8. If crack exiatence
time is significantly less than Cransient process time, investigation of
such disturbances passes into the area of RDTT dynamic characteristics
examined in the second part of this study.
9.2.3. Change in Thrust Chamber Flow Passage Cross Sectional Areas With
Axial G-loading
A composite propellant charge bonded to the case seeks to displace toward
the nozzle under the effect of pressure drop forces along its length
~p k and axial G-loadings. This is hindered by forces of adhesion between
propellant and case, as well as inCernal propellant cohesion forces. With
considerable charge length and G-loadings there occurs an appreciable
axial creep by the propellant mass, which leads to change in engine flow
passage cross sectional areas. Change in cross sectional area varies.
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along the length of the charge. The conaequences of thie change also
differ. An increase in flow passage crosa seceional areas in Che for-
ward parC of rhe charge, where gas velocitiies are eignif3.cantly below~
the erosion combustion threshold, does not produce aubseantial changes
in conditiona of propellant burning. On the conCrary, a decrease in
flow paesage croae aectional are~s in that portion o~ the charge ad-
~acent to ~he nozzle, where eroaion combuation occura, leads to in-
tensification of the eroaion effect and can have a aubatanC3a1 influence
on engine operat3ng characteristics.
In other words, when calculating diaturbances of RDTT operating conditions,
in Chis ir~sCance one can proceed from change in cross sectional area of
gas-dynamic passage f~ Fk, estimated from charge deformation in the lower
portion, facing Che nozzle.
Change in nozzle flow rate coefficient with centrifugal G-loadinga is
due to nonunifarmity of distribut3on of pressure of the slowed flow in
the nozzl~ inlet section. This effect is obaerved at high rpm, and its
consideraCion becomes substantial only for spin-stabilized pro~ectiles.
9.3. Nozzle erosion
In an RDT~ Che nozzle is the most heat-stressed strucCural assembly. The
critical section region the nozzle throat is particuXarly heavy and
is the mo~t sub3ected Co erosion.
Heat erosion of the nozzle throat area resulCs in decreased engine pres-
sure as we11 as Chrust and specific thrust 3mpulse. A decrease in P and
. Iy takes place both as a consequence of a drop in pressure and a decrease
in the da/dkP ratio, since heat eros3on of the nozz].e exit section is
usually insignificant in comparison with Fkp erosion.
Following are the principal factors determining the magnitude of nazzle
erosion:
1) gas flow parameters (gas Cemperature and density, composition
of combustion products);
2) thermal sCate of nozzle surface;
3) nozzl~ material;
4) duration of engine operation.
In estimating expected nozzle throaC area erosiony one must differentiate
among the following principal events.
1. Nozzles on motors with brief charge burning time, operating under
Cransient heating conditions, fabricated of inetals with a melting point
_ below the propellant burning temperature (T~,~ Tk);
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2. Nozzle inserts of heat-reaietiane maeer~.als (var3ous forms of
graphite, CungsCen, molybdenum) wiCh a melCing po~.nt (sublimaC~on)
above the propellant burning temperatiure (T~~ y Tk).
We sha~l initia~~.y examine Che f~.ra~ case.
As is indicgted by an analyais of experimental data on eroaion o�
ttozzles fabricated of inetals with a relatively low melting poinC (atain-
less and chrome ateel, heat-resisCanC ateel, eCc), significanC nozzle
' eroaion 3s obaerved in those cases when temperature on Che inCerior
surface of Che nozzle ehroat reaches a certain critical 1eve1 T~~g, close
to the metal's melting poinC.
DuraCion of nozzle operation without aignificanC eroaion of the Chroat
area will be deCermined by the time required to reach temperature T~~g.
The meChod of estimating quantity.~.~ is preaented in [22]. Here we
sha11 limit ourselves to an overall~a~proximate relation for p, ob-
tained on Che basis of the specified method:
Tnp ~ I,42 (In 9c. s)~~a a9 , (9.1)
Here
. TK-Tc~s ,
ec.s� ~
TK-TH
Tg initial temperature of nozzle material;f1 , c, p-- coefficients of
heae conductivity, heat capacity and densiCy of nozzle material respective-
ly; heat transfer coefficienC.
The formula is valid for the region of Biot criterion values Bi=0.4-4.0.
It follows from Che formula that nozzle erosionless operaCing time
decreases sharply with an increase in propellant burning temperature Tk
and heat transfer coefficient a, which in turn is deCermined by the level
of working pressure in the engine. Time ~ ap increases with an increase
in set ~ c' thermal acCivity of the maCer3a1. As a consequence of
this, low-carbon steel is a preferable material for nozzles of this type,
with a coefficient of thermal conductivity several times greater than
for he~t-resistant steels.
Let us proceed to the second case. When utilizing nozzle inserts of heat-
resistant materials with a melting (sublimation) point higher than the tem-
perature of the propellant combustian products, chemical erosion is the
main cause of nozzle throat erosion, Solid propellant combustion
products contain a number of compounds and elements which at high tem-
peratures can enter into chemical reaction with the nozzle ir~sert material,
forming oxides of carbon or metals. As is indicated by studies, under
condiCions of an RDTT, water vapor and carbon dioxide possess a high
degree of chemical activity [2]. For exampZe, the following reactions
with graphite are possiblP:
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- C-~- C0, 2C0; C-}- H s0 CO H s,
'Che 1.inear rate of chemical erosion i~ deCarmined by rhe fo~lowing
ralation (22]:
~ ~C~
Mit
u, c~~ ~j ~ttK ~ (9.?)
where ~,k coefficient of convecti.ve heaC emiseion; cp mean thermal
c~pacity of combuetiion producte in ~the r~gion of the boundary layer;
Cei concenCration of i ox3diz~ng componenC of combuaCion producCm at
rhe core of the f1ow; MB~, Mgi molecular magsea of the oxidized and
nxidizing components (elements); K-- atoichiometric coefficienC in the
eyuation of oxidizer reaction with an oxidizing component; p 8-- densiCy
of insert maeerial; coefficienC taking into account the etaCe of
the insert surfac~ and the structure of its material.
In view of the fact Chat at the pregent time coefficient ~ is deCermined
only experimenCally, this relation does noti enable one to obtain reliable
calculate~ figures. EsCablishing a direct relaCionship beCween raCe of
eroeion and coefficient of convecCive heat emission, however, it opens up
Che way to modeling o~ ~rosion proceases. Since according to the Bartz
relatior ~ k ~,p0.8dkp� , the rate of nozzle eroaion on a full-scale
specimen can be deCermined from the test figures on a model version as
~
u: u~NOj~ ( ~ )O'B ( dN(~ MO )O,~ ~
PK. MOA Kp
where pK.MOA, dKp,,,oA pressure and t:hroat area diameter for the
model.
During nozzle erosion considerable surface roughness may occur as a can-
sequence of nonuniform removal of material. This leads to additional
losses of specific thrust impulse due to friction. According to ex-
perimenCal data [33], these losses may comprise 0.5-1.0% of nominal
value.
ToCal drop in specific impulse due to nozzle erosion as a consequence of
drop in pressure, decrease in expansion raCio and development of roughness
can constitute a considerable quanCity from its initial value, correspond-
ing to initiation of engine operation.
In some cases in the process of engine operaCion there occurs a decrease
in nozzle throat area (nozzle obliteration) as a consequence of deposition
of condensed phase on the nozzle surface. This process can be cyclic.
Periods of deposition of condensate can alternate with periods of con-
densate removal, when the force of gas-dynamic resistance exceeds the
force of Yts cohesion with Che nozzle wall. ~
Methods of calculating condensate deposition on nozzle walls have been
little elaborated up to the present time.
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9.4. DisCurbances of OparaCing Cond3t~.one of a Soli.d-Propellant Rocket
MoCor ~nd Its Output CharacCerieCice ConnecCed WiCh ltemoval of
Thermal Prntection MaCerials
Acrive thermal protection of the motor structure is exteneively utilized
in i2UTT with extended operaCing time, proCection based on absorption of
a sub~tiantigl percentage o� the heat ~ransferred to the aurface with
Ureakdown gnd removal of heat shield~.ng material. Polymer-based heat
shielding materials (TZM), mosC frequently reinforced plastics, are
employed For Chis purpose. Under the effect of heat there takes place
breakdown of the organic bond (rubber, epoxy or phenolic resins, eCc),
wieh the formation o� ga~es and coke reaidue. The latter forms, to-
gether with a filler, a porous carbonized layer, which subsequently,
inCeracting with the stream of propellant combustion products, is carried
away from the surface of the protective coaCing. Thus as a result of
the Chermal action of the main flow of gases on Che protecCive coating,
tl~ere occurs compleCe removal of a certain layer of the coating and
formation of a carbonized layer, which retaine solid pyrolysis products
and filler. The relationship between thickness of the removed and
carbonized layers is deCermined by the characteristics of Che material
and the concrete condiCions of proCective coating operation.
F'ollowing are th e principal consequences of breakdown and removal of Che
protective coating:
1) change in the composition and temperature o� propellant com- -
bustion producCS at nozzle throat section inlet; -
2) change in flow rate of g&ses emerging from the thrust chamber
and at nozzle exit;
3) change in passive weight of ~he sCrucCure.
In chemical composition protective coating componenCs sub~ected to thermal
destrucCion are close to the fuel-binder of composite fuels. One can as- ~
sume that the decomposition products of these coating components will
affect the equilibrium composition of the gaseous phase of TRT combustion '
products in a first approximation just as change in the content of fuel-
binder in the composition of propellant, which can be expressed by Che
following relation:
8 (RTK)r = d (RTK) R~,K 8in~~ (9.3)
dm~
where m~. relative content of fuel-binder in the propellant; d'm~ "
change in this percentage due to TZM [heat protection coating].
Since all rocket propellants known from the liCerature are compositions
with a negative oxygen balance, quantity ~(RTk) is negative.
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The degree of influence of the examined factor dependa on the degree to
which the composition of a given propellant differa from atoichiometric,
since quantity
d~RTK~ increasea in a direction away from
dm~ stoichiometry toward excesaive binder.
This also determines the contradictory nature of data in the literature on
the influence of removal of protective coating on quantity RTk and specific
Chrust impulse.
The influence of the filler, which proceeds together w~th products of de-
composiCion from Che removed coaCing layer, is different. Alongside a given
thermal effect of reactions taking place in the flow core, as a rule its
entry will be accompanied by an increase in condensed phase content:
J~r~ \ f N
a(RTK~N dm ~"I Kl R,,K U/nN~ ~9.'~!
K
where mH relative percenCage share of condensed phase; dmg in-
crease in this share due to inflow of condensed coating removal products.
Relative change 8(RTK)~ = 8(RTK)r S(RT�), will be manifested
in change in pressure in the engine, thrust and specific thrust impulse.
The simplest 'thing is to take into consideration the influence of addirional
mass TZP removal products in prior-derived relations by means of vari~tion
d'p T, which, applied to this problem, we can present in the form
SPr = BPT. -f- Bm~PK~ . (9.5)
where ~P T~ variation of density of the principal propellant; dm~m R/mT
coating mass protection in relation to consumption of principal
propellant at operating pressure pk.
Let us examine how RDTT principal characteristics for a zero-dimensional
solution change due to removal of in-chamber heat protection coating.
According to relations (8.4), (8.5), and (8.9), we obtain
tbPK~Y. n= 1~ y[bm~PK + 2 b~RTK~n, :
(bnk)y. n = ~ ~ y [ SmnPK 2 b ~RTK)n, + (9.G1
1 v ~
~sjYlY. n= ~ YP v alil~Pe 12 ~ 1YP v~ s~RTK
1~~
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A number oF inv~gtiigatinrs hgve endeavnred ro inCroduce gr? independent
~pccl�ic impul~~ of TzP ~nd burned nonpropellanC maCerigl~, determining iC
from LngKeg of. ~DTT ~pecific impul~e. Nowever, the values of Chis -
c:t~~rucCeri~tic obtained by th~m range wiChin broad limiCa from 50 Co 200 e,
vnrying in r~1~Cion Co the propellant oxygen balance [19, 35J. For
exnmpLe~ for a gpececraft RDTT burning g propellanC wiCh IY~290, and w~.Ch
'C7.1~ removal compriging 0.5-1.59~ of propellant mase, independent apecific
co~ring impulse comprised ~50 s[35J. ~Consequpnrly, such a TZP characCer-
~ igric ig noC universal and ie of limited, purely cognitive interese.
9.5. ~iguring Heat Loeses attd Incomplete Fuel CombusCion
Lo~ses from heat Cransfer into the moCor case and from incomplete com-
bu~tion ~re figured, in determining RFTT operating characCerisCics, with
correction facror X to fuel force RT~, determined experimentally by
c~lorimetric method or on th~ basis nf thermodynamic calculaLione.
A gener.al exp~psrsion for determining coefficienC of heat losses in an RDTT
}ias the form
Se
~ a (TK r~. es
x~i- ,
rnrHo
w}~ere ~h~ q~anCiCy of fuel burning per unit of time; H~ total en-
thalpy of a unit of mass of combustion producCs; T~~g temperature of.
_ tliru~t chamber interior surface contiguous to gases; a-- coefficient
oE heat trasnfer; 5gp thruKC chamber interior surface area.
- Parameters a and T~~g change both along the chamber surface contiguous
to the gases and in time, causing char.ge in coefficient X on a time axis.
T;.e magnitude of the coefficient of thermal losses and the relaCions
d~..ermining it are quite different for a motor without insulation with an
- insert charge and for a motor with interior-insulaCed case. Therefore
tt~c~se two instances are examined separately.
9~5.1. Determination of Heat Losses for RDTT Without Heat InsulaCian
_ 'Ct~ is instance is characterized by tfie trarsi~it nature of heat losses through-
o~t ttze extent of the entire operating period, as well as the significant
mlgnitude of these losses, caused by the large surface in contact with the
~z~es. The bulk of heat losses (f rom 70 to 9U~) usually occurs on the
r_ylindrfcal surface of the thrust chamber. Proceeding from the determining
role of convective heat exchange in an RDTT, the heat transfer coefficient '
can be represcnted as ~0,8
a = Ka ,z ? -
r
1~8
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wh~re m'm/~k mags fLow of ggs per unir o� nharg~ flnw p~~~~g~ cro~~
secCiongl gre~; d r eff~ctive rhermal eecCion diameCer; lta g
coefficient taking ineo accounC ehe Chermophy~icsl properti~s of ehe gas.
Chgnge in charge flow paeeage crosa ~ectional area in tiim~ can be expr~ssed
wirh tihe relaCion
FK = FNO ~ ~ -I- bt~),
where ~kp iniCial flow pgesage croeg s~cCional area; ~i relatiive
percenCage ahare of burned fuel;
e , S,~
~ �
m ' ` e ~ a FKOY
We shall assume change in masa flow of gas along Che length of the charge
Co be linear
� ~ x
mx = m~ ~ .
We shall also agsume dr FK'~,
An expanded expression for determining thermal losaes in the cylindrical
secCion of the chamber assumes the form
t �
~ (TK - Te. r ~x~ ~l~ .tA,e~
KTnDK ~ (9.7)
x= 1- F~o
mc~~L ' Ho I ~-4~ ~h (~~1~~9
where quanCity T~~g (x, t) is determined f or each chamber zone with co-
ordinate x by solving the problem of wall transient heak conductivity wiCh
a variable value a. In view of the unwieldiness of solution, the practical
value of equation (9.7) for obtaining quantitative resulCs is very limited.
However, iC enables one analyCically to establish the character of change
in time of thermal lo~ses in the engine.
If we assume T~~B (x, t) for the present point in time to be the average
value T~~g along the length of the chamber, and if we apply a power of 1
with the binomial in the denominaCor, we obCain the relation
x = 1 A T ~ . (9.8) .
Ya. M. Shapiro (26], performing calorimetric measurements of heat losses
on a model motor with interruption of combustion, obtained the experimental
relation
x_1_ a
~ + bl,~ . (9.9)
He obCained the following for the conditions of the performed er.periments: -
B=0.30; bl=5. Consequently the experimer.r confirms analytical relation
(9.8). In formula (9.9) change Tk-T~~B in time is taken into considera-
tion indirectly by selection of coefficienC bl which is not equal to
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.
Che valu~ b~~/(1-s) in formula (9.8)~ but is somewhat higher~ as could
b~ expact~d. mhus according to (9.9) when 8~0.30; b~5~ Che value of X
in eh~ procesa o~ engine operation varies from 0.7 to 0.95~ and correepond-
ingly he~r losses, initially comprieing 30r, decllne to 59' by tihe end of
EtD7"r operetion.
Since iniCiation o� steady-stia~e condit~.ona for such moeore mkes place
whett'~P~ 0.1, the mggnitude of Chermal losees and their variance in the ini-
tial period of nperaeion may exert a eubatanti~l influence on the aCability
of working characterisCics and parCicularly on quantiCy pk max�
~ollowing are the principal causee of variance of thermal losses in the
initial period:
random changea in the character of presaure increase in the chamber
when entering conditions leading to a aharp dispersion_of the time-average
coefficient of thermal Cransfer by change in quantity m;
change in the s',:aCe of Che chamber surface due Co deposition of con-
densate (sooC, metal o~:ides), gummy depoaie residues, etc.
9.5.2. Determination of Thermal Losses for an RDTT With Removable Heat
' Protection CoaCing
Heat losses for a motor with a bonded charge burning on the cavity surface
and on the end, with a heat protection coating on the nozzle-adjacent por-
tion of Che case and on Che upper end plate, decrease sharply in comparison
wirh uninsulated RDTT and comprise fractions of a percent in steady-state
mode. We should also note that removal of coating material is accompanied
by regeneration of the heat transferred into the co ating, minus thaC per- �
:entage absorbed during endothermic pyrolysis reactions. These phenomena
are taken into account in determining disturbances connected with removal
of TZP in conformity with the method described in 9.4.
Proc~sses taking place in the coating at the initiation of engine operation
reyuire special examinatiqn. During heating in a reinforced plastic,
a c~early-marked pyrolysis f ront is formed, which leav~ behind a carbonized
layer. Initially there occurs displacement of the pyrolysis front deep into
the material. Subsequen*_ly, when carbonized material comes into contact
with high-temperaCure gases on the surface, material begins Co be carried
aF�ay from the surface. The rates of displacement of the pyrolysis front
u~ and the surface of coating removal u~~g equalize with time. After this
t'e thickness of the carbonized layer contained between Chese sur.faces
remains constant in time. Consequently the quantity of heat accumulated
in the coating also remains consCant. However, as long as us remains
greater than u~.g, the thickness of the carbonized layer increases.
The quantity of heat accumulated in a carbonized layer per unit of area wi~.l
total
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' ~06 ' .
Q.K ~ f Pos~oa ~T - T N) dx?
d
where p a .znd co~ denaiCy and heat capacity of the carbonized materigl;
T-- loca~ material temperature; Lo6 Chickneae o� carbonized ],ayer.
Irrevereib],e thermal lossea connec~ed wiCh accumulaCion of heat in the
carbonized layer in the initial period, per un~.t of time per unit of coat~.ng
arp~, will be as follows:
dQ,K 1 dL
~ Po6~o6 ~T c. e T ~
where T~.g temperature of removal eurface; TS temperaCure on pyrolyais
f rox?C .
With the paesage of time dLo d/dC approaches zero, causing quantity
dQek/dt also Co approach zero.
One should bear in mind that with eome charge ahapes, such as a slotted
charge, new secCions of coating involved in heat and maes exchange with
the hot flow of gases are continuously being exposed during burning in
the area o� the slots, as a consequence of which thermal losses connected
with accumulation of heat 3.n the carbonized coating layer accompa~y the
entire period of charge combustion. ~
In view of the triviality of heat losses in RDTT with removed hpat proCec-
tion coating, it is expedienC to effecC consideraCion of thia factor in
the form of ~(RTk).
9.5.3. ConsideraCion of Incompleteness of Solid Propellant CombusCion
Following are the most typical causes of incomplete propellant combustion
in an RDTT:
incompleteness of the ~hemical reacCions taking place in the gaseous
phase;
incomplete combustion of inetay. particlea in the Chrust chamber;
e~ection of charge particles through the nozzles. ~
Incomplete combustion in the gaseous phase usually occurs with low engine
pressures. The following flame zone reactions, characteristic of combustion
of ballistite propellants, are highly sensitive to decreased pressure in the
engine:
2C0 2N0 = 2C0, N,;
2Ha -I- 2N0 - 2H=0 N,.
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The raCe of theee r~aceions ia governed by a dependence whicli i~ common to
bimoLecular reacti~na:
Tc
~"t " K*ClC"
where CL and C2 concentrations of reacting aubsCances; C3 con-
cenCrarion of substance forming during reaction.
Proceeding to relative concentraCiona C~C/P , we obtain
dc, ~ - K~C,C,, (9.10)
dt
Sinc~ these reactions produce a gubaeantial quantity of the heat r~leased
during combustion (up Co 50ti in the case of balliatite propellanta of
average caloricity), their incompleCeneas can have a subatantial effect on
quantity RTk. �
_ Cor reactions which take place in the gaseous phase and poasess a differing
order of magnitude, on the whole completeness of combustion is deCermined
by exponential relation
Xc~ = 1 - e-ke~ (9.11)
where t Cime. '
When utilizing relaCion (9.11), one must substitute in it the total time in
Che thrusC chamber of er~ch separate forming gas porCion
L
dz -
tnD - ~ v '
~l -
where xi distance from the area of specific gas generation to the foxward
end; L-- total chamber length (from end plate to nozzle throat area); v--
local axial velocity of gas flow in Che specified section.
For the motor as a whole we obtain
� L
dz
xer ~ 1- t~ r g! eX p - X J ~ r
. z~
ahere fr relative change in quantity RTk due to flame a~ne reaction; gi
ass attraction of i for~eation in total combustion products discharge;
K-- coefficient of chemical reaction, determined by pressure and tem- ~
peraCure.
The same apprnach can be utilized in determining incompleteness of com-
bustion of inetal particles. In this case the chemical constant ahould
take into account influence of particle diameter, concentration of
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oxidizing componentis in ehe gaseous medium~ and the metial's abi~.ity tio be
chemically acCivaCed. In conformiCy wiCh the figures on cnmbu~tion raCe of
aluminum contained in [38]~ one can repreeent in an approximate Faehion com-
ponen~ XM~ resulting from incompleee aluminum combustion:
' L
F dx
XM�tlM~gtKJ u '
c~ 9 ~
where Ka3~103 Cok relarive concentration of oxidizing reagents 3n
d~ '
propellant combustion products; dT average diameter of aluminum particles,
in microns; fM relative change in quantity RTk due to combustion of
aluminum.
E~ection of particles of unburned propellant ia usually connected wiCh
breakdown of charge elements in the final stage of combuation. It is
manifested mdat sCrongly during u~ilization of thin-webbed charges.
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Chapter 10. INFLUENCE OF CHARGE INITIAL TEt~'ERATURE ON~CHARACTER-
IS~ICS OF A SOLID-PROPELLANT ROCKET MOTUR, MOTOR
TUNING AND ADJUSTMENT
Among the factors disrupting RDTT operating conditiona, change in initial
charge temperature in Che process of rocket operation occupies a special
place. Greater attention to the temperature factor is dictated by the fact
that in practice it produces the most profound changes in engine operating
conditions, which go far beyond Che bounds of individual random deviations -
from the specified conditions. At the same time in most cases it is pos-
sible to take into account in advance Che consequences of this facCor and to
limit its influence by means of special measures: by ad~ustment or thermo-
sCatic control of the motor prior to launch.
In examining the quesCion of the effect of charge initial temperature on
engine operating characCeristics, one ahould emphasize two instances:
1) initial charge temperature is practically constant throughout the
entire thickness of the burning web; . ~
2) initial charge temperature varies within significant limits through
th~ thickness of Che web.
Each of these instances is examined in this chapter. `
Problems connected with RDTT tuning and adJustment are examined applicable -
to the first case. However, the obtained relations can also be applied
to the case of temperature varying through the thickness of the web witr.
u+:ilization of the concept of effective charge temperaCure, which is separate-
ly selected for each temperature field realization.
1J.1. Relationship Between Characteristics of a Solid-Propellant Rocket Motor
And Charge Initial Temperature
ChapCer 7 contained various expressions of the empirical relationship between
TRT burning rate and charge temperature. In order to simplify the mathematical
calculations connected with determining RDTT operating parameters at different
~ charge temperatf~res, it is more convenient to utilize exponential relation
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u1TH � u1TN8~ ~T-rn?),~ (10,1)
Hence�orCh we qhall omit eubscripr "H" in this chapter witih the curren~
charge ~empergtua�e value, Eor the purpose of symbol simplification.
Selection of temperature TN~ adopted ae standard, ia determined by the
~pecific features of the motor being deaigned. For a motor deaigned for
use acrosa a~road range of remperatures, tiemperatiure TK min can be
adopted as TN, corresponding to Che lower limit of motor utilization under
the moet adverse condiC3ons of propellanC combustion, th~t is, a case
designated as standard for ensuring atability cf charge burning.
In this chapter, for simplification of mathematical calculat~ons, we have
also assumed TNmTH min�
The difference in selecCion of standard temperature has no effect on the
magnitude of power D, but leads tn a disparity of values u1N. If we assume
that standard temperature TIN corresponds to u~N, then, adopting T~I as
standard temperature, one musti adopt in formula (10.1) the following as ,
~oefficient of burning rate:
~N""N1
u1N = u1Ne r r .
Initial char~e temperature also affecCs propellanC burning temperature. For
ballistiCe propellants of average caloric3ty, the heat capacity of which is
approximately equal to that of the combustion products, change in com-
bustion temperaCure is numerical].y equal to change in initial charge tem-
pe,taCure [ 26 .
T
K~r~ = TK (N) `F' Ta - TN.
' WiCh a change of TN from -50 to +50�C, product RTk will change by 3.5-4.5%.
For composite propellants with a higher combustion temperature, relative
change RTk with temperature will be smaller. According to experimental
data obtained for a spacecraft RDTT [35], this change will comprise only 2%.
Consequently the dependence of propellant force RTk on initial charge
temperature is relatively slight. On analo~y with relation (10.1) we can
write
RTK ~T~ = RTK ~N~ �ez"` (T-Ttv), (10.2) .
The ~alue of coefficient m in the exponent varies in relation to the
propellant's.energy characteristics, comprising approximately 0.0003 for
ballistite propellants with low caloricity, and approaching 0.0002 for
composite propellants [35, 26].
We shall initially determine how RDTT operating characteristics vary with
charge initia:. temperature in rhe absence of adjustment and cantrol.
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UCilirlnq fnrmul~~ (7.15) and (7~40) and ~ub~tituting in them the depend~nces
of. r.nrobuHtion rate (10.1) and propellanC force (~.0.2) on initial ct~grge Cem-
~~4rf~ture, we obtain for engine working preaeure aC temperaCure 7'
~ t
m*D
PT � C~~~NPtSKu 1~xRTK cN~ 1~"� e.~~r" rn?) . (10.3)
~cAFKp' ~
'The ~Ir:~r factor expresses the magnitude of combuation chamber pressure pN at
standard (nominal) CemperaCure TN.
Consequently, relaCive presaure change in a noncontroll&b1e motor with
change in charge temperature will compriae
m-I.D ~T -T
N~ ~ ~ ~.38~
- =e ~
PN
Lt follows from formula (1~.3) that the relative pressure change with an
increase in charge temperature is independent of charge ioading parameters
buC is determined by Cemperature differenCial and conatants D, m, v. We
sha11 note that wiCh a specified value of constants D and m, chamber pres-
sure at small values of v is less dependent on charge temperature and, on the
contrary, high values of v increase the temperature dependence. The resulCs
of calculations of pT/pN for two propellants are contained in Table 10.1.
Table 10.1.
?(apaxTepHCTeKa tonneee
1 pTIPN pTlpN 1-F~ X tT/iN
� i v
Heperynapyen+a~ pATT 2
0,0038 I 0,69 I 3.19 I 3~31 I 1 I 0,318
0,0014 0,4 1,27 1,28 1 0,800
PATT~ pEfYJINPYOMHA N8 t10CT0AHCTBO A8H11CHHA 3
0,0038 I 0,69 I 1 I 1,395 I 1,432 I 0,710
0,0014 0,4 1 1,135 1~155 0.883
P11TT~ p2fYAHpyeM61~ N2 ilOCTOAHCTBO TAM! 4�
0,0014 I 0,49 I 0,7b6 I 1 ( 1,363 ( 0 981
Key:
1. Propellant characteristic 3. RDTT controlled for constant pressure
2. Nonemtrollable RDTT 4. RDTT controlled for constant thrust -
Note: TN=-40�C; T=+50�C.
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Relative change in ~hruat due ro temperature will.compr~.se
pr Pr acf (1~a1- P~~Inr , (10.4)
pn? a~~ c d) - au~nrv
C onsidering Che amallnesa of the second Cerm of tihe difference, one can in
' a first approximat3on assume
pr pr ~ e m ~ (T-TN)
PN PN ~
The linear rate of fuel combustion at temperaCure T is determined as
~ b-}~mv
uT ~ uiTPT = u1NeT
~ ~T-rN).
Charge burning Cime at temperaCure T: �C T~el/uT.
Relative change in burning t:Lme will be .
D-~-mv
,~N _ ur = e 1-v ~''-''N) , ( )
10.5
Let us determine haw specif~,c thrust impulse changes wiCh initial char.ge
temperature. Substituting in formula (7.20) dependences on temperature
for propellant force (10.2) and for working pressure (10.3), we obtain
~T~ - ~ k 2k, 7L ~~KTK)N e"` ~T-TN) 2xcZ(~o) -
t
_ / k ~ \ F - m-I-o
~ 2 a Px 1-v (T -T N) (1 O.6)
~ ~cFKP P~w e ~
where pkN and (RTk)N pressure and propellant force at standard temperature
TN.
10.2. Ob3ectives and Means of Tuning and Ad3ustment of a Solid-Propellant
Rocket Motor
Prelaunch adj~stneri~of an RDTT is at the present time the principal procedure
employed in regulG,ting the thr.ust parameters of this motor. Tuning and ad-
~ustment eliminates Co a significant degree the influence of the main causes
of instability of an RDTT the depenlence of TRT combustion rate on charge
' temperature and difference in burning rates of charges produced from dif-
ferenC batch~s of propellants.
Depending on the ;r.ared ob~ective, one distinguishes motor ad~ustment for
constant pressure, constant thrust, azd constant flow rate. Ad~ustment for
constant flow rate is characteristic fnr RDTT utilized for auxiliary pur-
poses as a gas generator. When utilizing an RDTT as a main propulsion unit,
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ad~ustment for conatant pre~sure and tihruat is moeti typica~.. ItDTT ad~ust-
menC for consCAnt presaure throughouC the enCire speci�ied range of engine
util~zaCion teroperatures makes it posaible substantially to reduce maximum
pressure (pm) max~ Which is an input quantity ~.n making angine serength ca1-
~~,inrione. This reduces engine masa characteriatic a and achievea a'd~:crease
In rnckeC grose lxunching mase with specified range and payload.
` ~ ~
Figure 10.1. Di.agram of Device for SmooCh Change in Nozzle Throat Area
Ad~ustment of' an RDTT for constant thrust makes iC possible at a11 charge
temperatures for ballistic-type rockets to ensure constant parameters of
the powered segment of flight and thus to facilitate securement of a high
degree of accuracy of target impact. For rockets with a f~ight mode close
to cruise, thrust adjustmenC makes it possible to maintain the required
flighC condit3ons and Co avoid speed going below or above the values
specified by the flight program.
In the case of small motors, adjustment can be performed in conformity
~ with ambient temperature. For large motors, however, due to the thermal
iner~ia of the charge, its temperaCure cati differ substantially from am-
bient temperature, and motor tuning and ad~ustment should be performed in
conformity with the readings of the teTM:perature sensors contained in the
moCor.
The simplest means of RDTT ad~ustment is change in nozzle throat area in
conformity with initial charge temperature and propellant charge performance
spAcifications unit burning rate specified for the given propellant
batch. In ~ome instances release valves can be employed.
' The simplest device for changing nozzle throati area is a set of replaceable
nozzles. Replaceable nozzle inserts can be used in plac~ of replaceable .
nozzles in order to shorten the time required to ready a motor for launch.
Sometimes multiple-nozzle units are employed, with alternate nozzles
c~vered by diaphragms. When a certain pressure is exceeded the diaphragms
are cut, total throat area increases, and further engine pressure increase
~ ~s prevented.
Smooth adjustment of nozzle throat area in conformity with charge temperature
is achieved with utilization of a throttle (Figure 10.1) which can displace
along the axis of the nozzle. A throttle can be moved into place manually -
or with the aid of a mechanical drive. There are a number of self-adjusting
nozzle designs, where the throttle is moved into position automatically,
without inCervention by servicing personnel. Some of them are described in [26]. ,
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'1'he devicea mentiioned above make it poesible to Cune an eng~ne not only
on iniria~. charge eemperature but also on oeher parametera as well, devia-
tions of which from nomina~. value are known pr~.or to launch.
10.3. Tuning of a Solid-Propel].ant Rocket Motor Nozzle Co Con~aCanC Preasure
- R~Caining in an RDTT constiant preasure or presaure varying within ama11
limits throughout the entiire ~emperature range of engine utilizaCion makes iC
po~sible to reduce standard preasure on the basis of which engine wa11 thick-
nese is deterininad, and thus to improve its weight characteriatica. The gain
in structural weight achieved Chereby will depend on the one hand on Che
characeeriatics of the propellant uCilized, thaC is, on the posgible pressure
drop in the absence of control, and on the other hand on the weight o� addi-
tional components and devices by means of which nozzle throat area is varied.
The throat area of one or aeveral nozzles, in a multiple-nozzle variant,
aZtered during tuning and ad~ustmenC, can be repreaented as the sum of a
certain constant component Fkp N, equal to throat area at s~andard charge
temperature Z~N, and variable component ~kp. Deaignating X=Fkp/Fkp N, we
obCain
FKn r� FKp N( l-~- X). (10,7)
We shall first examine ad,justment of nozzle throat area solely on initial
propellant charge temperature.
Substituting (10.7) in formula (10.3), we obtain
t
C~D'~m~ ~T -T NI ~-v
pr = Pnr [ ~ X J ~ (10.8)
In order to ensure constant pressure with varying propellant charge tem-
perature, it is necessary to satisfy condition
I X = C~~+m~ ~T-TN). ~ 10.9~
With equatioi? (10.9) one can find the nozzle throat area required with the
given temperature
(D+"') (T-TN~ (IO.IO)
FKp T = FKP Ne
For a propellant with a low temperature dependence, expanding the exponential
factor into a series and discarding terms of a second order of smallness,
we obtain:
1-}- X= 1-i- (D m) (T - TN) or X�~-~p m) ~7~ - 7'rv).
If the nozzTe throat area is simultaneously ad~usted according to actually
measured combustion rate deviation ~ul from the standard value for a given
batch of propellant charges, one should substitute in formula (10.3) in
place of u1N
U1N \ I ~ N/- u1N ~ 1"~ SUI~'
, lg9
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Required nozzZe ehroat area wi~.~ be de~ermined as
~'Hp r ~ FKn N (1-~- SUl) @~~~m) (T -TN~. (10,11)
For ama11 bu valuea for propellants with a low Cemperatiure dependence:
X - 8u -4- + m) 8T.
Tuning and ad~usCment Co any parameter with deviation known prior to launch
is performed in like manner.
Relative change in operating tiime of a motor ad~usted for constant pressure
is equal to
Tr = e � (r-rN~. (10.12)
tN up
It follows from a comparison of formulas (10.5), and (10.12) that in a
rnotor with constant pressure maximum variance of time with change in
propellant charge temperature 3.s less than in an uncontrolled motor.
Let us see how engine thrust varies in relation to temperature when ma3n-
taining conatant pressure if the discharge area remains constant.
Utilizing relation (7.18), we obtain
PT _ Qcrf (~ar) - Px!~ (10.13)
- PN acNf ~~aN) - Pe/PK ~
In order to utilize this relation, one must establish a link between
quanCity X and change in gas-dynamic function f("~ a).
Since
9 ~~'aN~ � -FKp N .
FQ '
FKc r .
9 ~~'or) = Fa ~ ~
9 ~~ar) = 9 (~arv) ~1-I- X)� (10.14)
Since quantity q(~ aN) is assumed specified, relation (10.14) makes it
possible to determine value q('~ aT) from quantity X, after which one can
determine f C~ aT) �
Table 10.1 contains calculation figures for two propellants, which indicate
t}at maintaining constant engine presst~re by ad~usting FkP involves sig-
niFicant thrust variation. Prelaunch tt:Prmostatic engine control is the
- only possible method of motor ad~ustment whereby conditions of co~stant
pressure and thrust coincide.
In some cases, for rockets of the simplest design, in place of ad~ustmer_t
for constant pressure it is sufficient to restrict pressure variation with-
in certain limits, so that on the one hand it never drops below the level
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pmin' gu~ranteeing stable propellanC combustion, and on the other hand
does noC rise above pmax, ~olerated by ~Cructuxal atrengCh. This can be
achieved by ~raduated ad~uaCment of Fkp with a aeC o~ interchan;eable
nozzlea, riozxle inserts or nozzle plugs (wi~h a multiple-nozzle veraion).
~ F
� ~�'pmi~
fM/ n~4sx I
. p.pm aP,~~ ~ -
~ ~ ~f I , I I I
~ ~ npmt~ ~ 11l
i i p ~ ~ i ~
I~~h
~-t-o~~d'~
t'-r--~"'~ pT ~ I con~o
j I ~co~~o I
Ic~~.,~ , co~AO ~ I I I ~
co~nc
T~ l'j rffi r T,y Tj rr T~ 1
d~
Figure 10.2. Diagram of Selection of Throat Areas of Interchangeable Nozzles
Key:
a.--Whef? ad,justing RDTT to con- b. When ad~usting RDTT to constant
stant pressure (pk=const) thrust (P=const)
1. Nozzle
A diagram of a selection of throat areas of interchangeable nozzle devices
and establishment of an operating range of temperatures for each of them is
contained in Figure 10.2. We shall adopt as a figuring unit a nozzle throat
area providi~g maximum pressure p~in at temperature TN. In order to ensure
that this pressure remains constant with temperature, with continuous ad-
3uatment, the nozzle throat area should follow relation (10.9). The
curve of relative increase in throat area ~kp max was plotted in conformity
with this relation. Now we shall plot a curve of variation in F~kp min in
order to ensure constant pressure pmax in a motor with continuous corrective
ad~ustment. The curve is constructed on the same relation (10.9), but the
starting-poi~t va_lue (F'~kpN~min at temperature TN wi11 be that which is deter-
mined by ratio
Pmin \1~-v
~FKp N~min - ( nmsx / ~
We shall run from a point on the upper curve corresponding to p=p~in with
T=TN, horizontal segment $kP=const=l to the point of intersection with the
lower curve. The point of intersection will determiae the upper temperature
boundary of employment of the first interchangeable device (nozzle) TI. We
begin the horizontal segment for the second nozzle with a curve corresponding
to pmin, but not from a point with coordinate TI but rather a point lying
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~ bF !01 ' I ! ;
BY YE. 6. VOLKOV, T. A. SYR I TSYN AND Ci. YU. MAZ I N

2i AUGUST i979 C FOUO ~ 3 OF . 3
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5-10�C to thc lefe of ie, in drder to en~urd overlep of CempergCure rgngeg .
oC ~mpldym~ne of int~rch~ngpable noxx~.eg.
N'rom Che ~~i~ri~~ v~lu~s'~kp det~rmined from Che curve, the nbgoluCe ehro~t
urea v~1u~~ ~re nbtained by mulCiplying ~l;p by (~kp N), cglculge~d fdr
pregel.eceed ~ngin~ chgrg~C~rigCi~~ ~t temp~rgCure TN gnd with Ch~ adopted
vglue pmin�
10.G. Tuning ~ 5olid-I'ropellgne !tockee Moeor Nnzzle en Congtant ThrusC
Wc~:~h~ll fir~t ~x~mine Che case wh~re tuning ~nd ad~u~emenC i~ performed by
chdnging the nnzzle ChrnaC area with a con~Cant nozzlp exit ~rea.
Utilizing rel~tion (7.16), Che condition of congCgnC CArugC at vgrioug
propellant chnrge temperaeures can be wriCt~n in the form
Q~rf (~ar)Pr ~ QcNf (~~n?)pN~
whence
pr _ Q~Nf (11av1 (10.15)
PN QcTf (~aT) ~ '
Substit~ting p~/pN from (10.8), we obCain
e ~�y (r-TN) a ~ 1 ~ X~ ~ ~ ~ owI (tiaN~ (10.16)
ocT~ ~~oT) ~
7'}ie ot~tained equation can be so~ved by two methods.
- 't't~e first method consists in utilizing the relaCion for T obtained by means
~f loKariChmic operaCion (10.15):
T~ T ~ ln t X lnf �`"i (~�"i1 1
N~- p,~ m p.{. m L Qe7'/ (f~aT) J~
5p~cifying valueq X for a propellant with known characteristics D, m and y,
one can construcC relation T=f (X).
A second method of soluCioii is based on the fact that there exiats a rela-
tionship between Che values of gas-dynamic functions f(~) and q(il) in
ti~c region 7l ~1.8-2.5, expressed by an approximating relation of the type
f ~ k~ (9 (~)1".
With proper selection of kg and n, approximation error does not exceed 1%. f
tn the range yL a�1.8-2.5 when k=1.15-1.25, exponent n is close to 0.9.
192
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~'rnm Ch~ approxim~Cing reletiion we obtain
l 1~a ) ~ ~ 1~a ) ~ (10.17)
( r) 9 ( r ( t )n .
Sub~tituting the obtained result in (10.16), w~ obCa~n
to+m?~r-r ~ ?_v
1-~- X~ e
-nT~ ( Q~r 1-~n'tT~ (10. l8)
~QcN/ ~
If in Che firgC gggumption we aeaume ~~r , then
(D-Hm~
1-~- X ~ e -n c . (10.18a)
It~lative engine pregeure change when tuning a nozzle Co conatanC thrugt ie
_ deCermined by the relaCion obCgined from formulga (10.16), (10.17) and
(10.18a):
~ ~~~m~ ~r`rN~
~ e ----~-~~yr _
pN . (10.19)
Listed below are Che resulCs of calculations f~;.~ a motor tuned for consCanC
thrust in the range T=-40... +60�C. The following were assumed in the cal-
culation: y- 0,4; D~ 0,0014; m= 0,0002; TN = -
_--4U� C? F~IFKv = 6,26; v~r = v~N,
'Cable 10.2.
T. �C -40 -12,4 -~f2,9 I -F33,3 .}.b6
X 0 0~1 0,2 0,3 0~4
PT/PN ~ 0~92 0,85 0,79 0,74 `
It is evident from tables 10.1 and 10.2 that working preasure in a motor
tuned for constant thrust increases with a decrease in initial propellant
charge temperature in the degree to which this is necessary in order to
cor,~pensate for a decrease in rate of propellanC combustion with temperature.
It follows from a comparison of lines 1 and 3 in Table 10.1 that relative
pressure change in a motor with constant thrusC, when Y=0.4 is approx-
imately the same as in an uncontrolled motor, but the maximum pressure
values in Chese motors are located at the opposite ends of the operating
temperature range.
For propellants with a high temperature relation (high valuea of D), when
Cuning a nozzle for constant thrust the throat area muat be varied within
broad limits. With an increase in D, there is also an increase in preasure
differential pp/pT in the preselected temperature range. A high value of
193
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expnn~ne y in Che giv~n case i~ a favorable �actor which ensureg the re-
qui~lre effecti of gd~usCment with gmall changea in nozzle flow passage croe~
~ectional area and working pr~s~ure in th~ tnueor figures in Table 10.1
for 'y n0,4 and 0.69).
Wh~n ineerchgngp~blp nozrle insertg or nozzles are emplnyed in plnce of ~
coneinuoug vari~eion of nnzzle Chroat erea for the purpns~ of tuning, the
mpehod of selecting ingerts and the range of their ueilizaCion proveg
_ anxlogous eo that which was examined in chapter secCion 10.3. -
In the cage of employment of interchnngeable inserts, specifying allowable
thruge v~riatiion limies Pmax"Pmin~ one plots for these 1imiCs curves of
coneittuous chang~ in nozzle throat area within the preselecCed temperature
rgnge ~ccording to relaCion (10.18), and then on~ egtablishea, in confr?rmiCy
with the diagram in 10.2), the temperature interval of utilization of the
individual inserts.
Wieh utilizaeion of interchangeable nozzles, if Che raCio ~q /~kp remains
constant for Chem, f(~ A)=const and equationa (10.18) and (10.19) assume
the Eorm
(~+m) (r-rN
. ~+Xae v ~
Pr ~ e D~ ~T_TN~
pN
Relative change in propellant charge burning time '
TT uN ~ ~-m (T-Tly~ ' -
zN - uf '
10.5. Tuning a Solid-Propellant Rocket Motor to Constant Flow Rate
At the present time an RDTT in rocket equipment is extensively employed as
a generaCor for gas utilized in the most diversified systems, such as
auxiliary propulsion units, servodrives, hot-gas gyrosystems, as well as
supercharging, additional gas feed, stage separation, and buoyant rescue
eq�ipment inflation systems. 5olid-propellant gas generators are frequently
employed to drive turbines powering on-board electric generators (9). Many
cc these gas-using devices impose rigid requirements on constancy of flow
r1te. We shall examine possible ways of solving thia problem.
Substituting in the flow rate formula (7.13) dependences of pressure in an
uncontrolled motor and propellant force on propellant charge temperature,
we obtain a formula which indicates how propellant consumption variea with
initial charge Cemperature in the absence of tuning:
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"ir ~ e �_~m ( r-'rn+)
mN ;
If a conatancy of gae flow to Che gae-driven dev3ce is achieved by dumping
gas from a receiver in the line in which pressure is maintiained constane,
relaeiv~ gas loeaes in relation to initial temperaCure are as follows:
D~-vm
Km, ~ m--~N ~ e^~ ~ r-rN) _ 1.
mN
Consequently, with this method of regulating gas f1ow, the sCore af fuel
in Ch~ generaCor musC be increased by Kml timea over the minimum requ:taite
q~antity calculaCed for temperature TN.
Uumping of gas direcCly from Che gas generator through a constant-pressure
valve ensures propellant charge combuation at Che same presaure at all
tiemperatures. With an unknown Chroat area of the nozzle Chrough which gas
enters the line to the driven device, gas flow fluctuates within Che limits
nf variation with Cemperature of quanCiCy that is,
'"r ~ e (r-rnr)
' ` mN / paa ,
In view of the amall value of constant m for rocket propellants, gas flow -
can be considered practically constant. However~ actual pro~ellant con-
sumption, taking inCo account dumping inCo the aCmosph~re, will vary as
Km~ ~ rT _ eD (T-TN) ~
mN
Relative dumped gas flow will be
~ mcf _ e~ (T -T N) _ 1 ~
mN _
increasing with an increase in initial propellant charge temperature. Burn-
ing time will decrease simultaneously with an increase in propellant charge
temperature.
' ConsequenCly selection of thickneas of burning web with this mode of tuning
should be performed for the preselected gas generator operation time for
the highest propell~nt charge temperature, while the required burning
surface should be selected according to the preselected consumpCion for the
lowest charge temperature.
Thus the charge burns incompletely at minimum temperature during operation
of the gas-driven device; at maximum temperature the charge burns completely,
_ but a substantial portion of the generator gases will be discharged into
the atmosphere. The actual propellant aupply should be specified at ltm2 times
the quantity required by the gas-driven device.
195
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linpr.oductive pr.opellant coneumptiion can be avoided when tuni.ng the ga~
~eneratnr nozzle Co constanti seC p~~,,.
1~RT,c Per�orming Che substitutions
which we have employed in the preceding chapter subdivietnns,we shall nb-
Cain ~ relaCinn of the requisite change with eemperatiure of the gas
generaeor relative throat area
D Vnv (T,_rN).
~+}(`~8
Cas flow rate and charge combustion time remain atricCly conaCant with any
iniCial charge temperature.
10.6. Causes of Nonuniformity of Charge Temperature Field and Ita
Equalizing Time
During rocket operation ie is possible that initial charge temperature TH
will differ significanCly from ambienC Cemperature TA. Such a difference
in temperaCures may occur, for example, as a result of airlifting rockets
large distances from one climate zone Co Pnother. With a large-size propellant
charge and initial Cemperature drop of aeveral dozen degreea, from several
hours to several days are required to reach equal charge and air temperatures.
Therefore in practice Chere is the possibility *_hat an RDTT will be fired
prior to establishment of a temperature equilibrium between propellant
charge and environment, with a substantial nonuniformity of propellant
charge temperature field.
When a rockeC is continuously sited at a launch position or on a launcher
en route, not in a container, under atmospheric condiCions, nonuniformity _
of propellant charge temperature can be caused by the daily fluctuations in
ambient temperature. The magniCude of these fluctuations depends on the
time of year and climatic conditions. For large propellant charges thermal
relaxation time may prove to be greater Chan the period of air temperature
flucCuaCions, which lead to the occurrence of a temperature gradient through
th~ thickness of the charge.
No:~uniformity of propellant charge Cemperature field, a direct oonsequence of
which is change in uniC burning rate through the Chickness of the charge,
leads to def~rmation of the burning surface in the process of motor operation. _
In other words the posi~ion ~nd shape of the combuation surface as well as
its toCal area with t~ nonuniform charge temperaCure field, considered for
any given point in timAr may di.ffer substantially from that which is deter-
mined by geometric calculatio~s performed on *he basis of the hypothesis
propellant burni~g in parallel la~yers. First of all this affects both
tne magnitude of inean values of RDTT thrust parameters and their extremal
values, which determine max3mum rocket G-loading and motor case strength.
It is difficult to foresee in advance what temperature conditions are the
most difficult.
196
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With a uniform remperature field of a propellanC charge in an unconCrolled
~DTT, maximum presaure is achiev~d aC maximum charge temperaCure. WiCh a
nonun3form temperature field an additional factor appeare increase in
burning gurface ae a consequence of ite deformaeion during combuetion; a
faceor deCermined by tempergture d3.fferential Chrough the thickness of the
propellant charge but not by rhe mean temperature value. It ie not known
in advance which of these factora change in combustion rate averaged on -
the propellant charge volume or char.ge in burning aurface area exerta
~ gtronger influence and whether pressure determined by maximum surface is
greater Chan pressure determined by maximum temperature.
Possible change in RDTT ba113stic and thrusC pa~ameters wiCh temperature
differentials in the propellant charge should also be taken ineo conaidera-
tion duYing RDTT tuning (prelaunch ad~ustmenC) [26].
In view of this f~ct iC seemg advisable firsC of all to examine eatimate
relations which determine propellant charge temperature fie].d equalization
time and which limit the region where the nonuniformity of this field must _
- be taken inCo account.
Then the most typical instances of deformation of burning surface and the
influence of Chis factor on RDTT sCatic characCeriatics will be examined.
As studies indicate, duration of change in rocket propellant charge tem-
perature field with constant ambient Cemperature Tp and heat tranafer co-
efficient a is determined primarily by the regular mode stage. At this
sCage change in Cemperature for all points in a propellant charge on a Cime
axis follows a simple expnnential relation, that is, temperaCure eimplex
natural logariChm
T-T~
e ~ TM - T~
for any point in a propellant charge with Cemperature T will change according
to the linear law. Quantity ~
ms = dt (tn A),
called cooling (heating) rat~, will be the same for all points in a charge,
as well as for its mean mass temperature fi.
Then the time to reach charge average Cemperature Tp will be specified as
Tp = m~ In TA_To . (10.20)
Relation (10.20) enables one to cietermine Che time required to reach average
charge tEmperature Tp, clnse to TA and assumed equilibrium. It also enables
one to solve the inversE problem: from the specified charge time under given
temperature conditions, to deCermine the average charge temperatur~ reached
by this time, end from its comparison with TA, to estimate the degree of non-
uniformit,y of the charge's temperature field.
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HeaCing (cooling) rate m,~ ia determined with the formula
mt ~ Cpr ~ (10.21)
where d-- coefficienti of heat tiransfer �rom ehe environmene eo the motor -
surface; c-- propellane apecific hegt; rA_ Tn
. ,rA~
T criterion of
nonuniformity of propellant charge temperatur~ field; F-- exterior lateral
motor surface; W-- volume of propellanC charge.
For charges of tubular shape, which includes sloCted charges,
m (t - M~) RN (10.22)
F �
where Rg outer charge radius; Rgg cavity radius;
M~RgHfRH. _
Criterion '~J is determined by the Biot criterion Bi~ K RH/~ T and Che
geometric shape of the propellant charge, but is not dependent on its abaolute
dimensions, which makes it possible to obtain from an experiment on a model
or from Cemperature field calculations value ~i for all geometrically similar
propellanC charges.
On the basis of Cemperature field calculations for charges of tubular ahape,
we determined the values of criterion '~i for variants with different co-
efficients M (Figure 10.3).
w
~o '
o.e ~
0,6
4S~ ~~~Z .
M-Q 4
M~q6
O, t M ~
, 0 J 4 S 6 ~ B 9!0 t0 JO 40 SO 60 70 s0 90 100eL
Ftgure 10.3. De;~endence of Criterion of Nonuniformity of Temperature Field
~ f~~r a Propellant Charge With a Cylindrical Cavity on Biot's
Crit~�rion
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Cu1cul~Cing Biot's criterion with prespecified conditiiong o� heaC exchange
between motor t~nd environment, we can obtain from Che graph correeponding
vnLue ~ and then from formula (10.21) obtain, taking inCo account rela-
tion (10.22), m,~, requisite �or eaCimating temperaturQ equalization tiime.
A ma~or advantage of the above-examined relations is Che fact Chat ~hey
enable one tn obta3n an estimatie of Cemperature field nonuniformiey, avoid-
ing excessively unwieldy calculations cotinected with determining the tem-
perature field proper in the dynamics of iCs change.
10.7. Influence of Nonuniformity of Charg'e Temperature Field on Solid-
Propellant Rocket Motor Operating Conditiona
5ince with a nonuniform tempera~.ure field the burning rate varies from one
section of the propellant charge surface to anoCher, in order to calculate
preseure in formula (7.15), in place of product u1S, one must substitute
Che following inregral gas formation characteristic:
r = E esul ~r, x,
where ul (r, x, a relation characterizing change in unit combustion
rate by cylindrical coordint~tes r, x, In those cases where temperature
drop on the propellant charge radius is determining, iC would seem possible
to limit oneself to taking i.nto account change in velocity in this direction.
Then, taking into account relation (10.1), one can write
x
r ~ Fj ~Stl~NCD AT (r) ~
where 4 T(r) a function expressing change in temperature difference T-TN
on the charge radius.
For a relative pressure change in the process of propellant charge combustion
we obtain
~
~ C~ ~Sev e 7' (?)1 t=v ~
nk, (rr l~=y 1 r
Pko To / esen nr ~
0
where r ~ and ri values of gas formation characteristic for the initial
and i poinCs in time.
Thus calculution of RDTT characteristics with a nonuniform propellant charge
temperature field is complicated by the fact that for each point in time the
combustion surface must be broken down into sections, within which the com-
bustion rate can be assumed constant, corresponding, according to (10.1), to
the average propellant temperature in that section. Displacement of the
combustion front in each of these sections in the r_ourse of a shart time
interval is assumed to occur at a constant pressure. Distortion of the com-
bustion front is taken into account according to relation (7.10). A new en-
gine pressure value is determined for the end of the time interval, in con- _
formity with the reached magnitude of combustion surface and with new
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diHtribuCion of burning ratea on the surface. Time interval (charge Chick-
nn~s ~lnterval) is salected so Chal� change in propellant charge temperature
wi~hin Che int~rval of combustion surface displacement does not exceed 3-5�C.
Examining Che influence of propellant charge temperature field nonuniformity
on RDTT operatiing conditions, one can specify three typical cases:
1) end-burning charges, with axial direcCion of burning normal to
the plane of the maximum temperature gradient;
2) radial burning charges on wh3ch the combusCion front aC any point
in time coincides with the isotherm (telescopic charge, cylindrical single-
cavity grain);
3) radial-burning charges with a combustion front which does not coin-
cide with Che isoCherms in the charge (sCar, s1oCCed charge).
10.7.1. End-Buxning Charge
We shall first examine a charge of the simplest shape a cylinder coated on
Che laCeral surface and burning on the end facing ehe nozzle. We shall
assume Chat Che temperaCure field is symmetrical and thaC Cemperature
changes only on the charge radius. To avoid ambiguity, we shall consider an
exCerior-cooled charge. For simplicity of calculations we shall assume
that temperature TN has been reached on the exterior surface of the charge,
and temperature T1 is maintained on the charge axis. Analysis of temperature
fields on the basis of j26] indicates that in the majority of cases, cor-
responding Co maximum Cemperature differenCial on the propellant charge
radius, the Cemperature profile is close to parabolic, that is, one can
assume: ~
AT(r)=T-TN=(Tl-TN)(1 -RH,' (10.23)
~
where r-- radius of a random point; T-- temperature at this point (Figure
10.4),
Distribution of rates of propellant combustion on Che radius will be expressed
by the relation
u = u~eM c1=''~pv~
where -
119 - D(Tl - TN); r= r/R,,.
higher burning rate on the axis of the charge will in time lead to deforma-
tion of the combustion front, which is initially flat, into a crater.
If a crater is present the local axial burning rate at that point in the
profile at a distance r from the axis of the charge will be expressed,taking
into account relation (7.10), as ~
- ~x ~ u1NeM ~1-~~~Pv Y 1 + \ az a~/ax (10.24)
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rok o~rzctnL us~ o~LY
X 0, S
T
04 M~D?S
R2
q>5
o,~ q ~
0,Y ~s
0,1
7~0 q8 q6 q~ qZ 0 -qY -0,4 -g6 -qd -1,0~
Figure 10.4. Stable Crater Profiles in an End-Burning Charge wiCh Nonuniform
TemperaCure Field
Gas generation from the burning surface per uniC of time will be expressed
as
� F
mr = ~ Ptux dF~ (10.25)
6
where dF=2 ~jr dr projection of burning surface element onto a plane
perpendicular to the axis of the charge.
Substituting (10.24), (10.25) and (7.15) into material balance equation
(7.11), we obtain an expression determining the current engine pressure value:
t
1 ' � 1=y
PK = 2nRHPrY~XRTK te~N r EM (~-~')f ~ 1 \ ~ ~ .
~~AFKp tj
, (10.26)
With a charge of considerable length, there occurs stabilization of the
crater profile in the process of combustion. The condition of stability of
- crater profile is expressed by constancy of the axial component of burning
raCe on the charge radius uX (r)=const. -
e-Mr~ = d'~~ (10.27)
Y ~ + ~
where x=x/RH.
~1 201
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Solv�tng equation (10.27), we obtain a relation which determinea combustion _
profile
r 10.28
Z m_ r~~4Mi~ - l~~~j l~l. ~ ~ ,
1~
Since for a atabilized combustion prof3le uXsconst �or the entire surface,
we obtain ,
mr = prur,ST~
where uTl burning rate on the axis of the propellant charge. ,
10.7.2. Radial-Burn3ng Charge With Isothermal Ori~ntation of Burning
Surf ace
As an illustraCion we shall examine a propellant charge conaisting of one
end-coated cylindrical grain with initial cavity radiue rp and initial
exterior radius RH. We shall assume temperature distribution on the radius
corresponding Co relation (10.23). Then ~he gas formation function for any
point in time will assume Che form
I' ~ ~LIIIN I/CM ~1=.s) + ReM
~ where r-- current cavity radius value; L-- grain length; R-- current
value of exCerior grain radius;
i ~ r/R; R = R/R�. ~
Relative pressure change in the process of charge combustion will be
t
- M ~~-~l~ M ~~-R~~ ~
. PKt ~r~ -I-Rr~ (10.29)
P~co t ro~M ~1-ru) ~
where i~ = rr._1-~- ~r; Ri =~r-i - OR�
rn view of a difference in rates of displacement of the outer and inner
surfaces, intervals Ar and AR are not equal to one another. Change in in�~~
terior radius ~r for a certain time interval Ot will be
e~ = erR� = u,NeM ~~`~~~p� 0~.
During this same time the exterior radius will decrease by amount
eR = eR � RH = u~NeM ~I-R~~Py Ot, .
c~nsequently, A't _ eM (10.30)
ARr
Specifying a given displacement interval for one of the surfaces, such as ~r,
the displacement interval of the second surface should Ue determined from
_ formula (10.30).
202
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~ ' I
The Cime interve~ corre~ponding to an independenC surface diaplacement in-
terv~l will be
~t = Dr~RN ,
.
, . ~1N�M ~1 r~)pKl
A charge of this shape~ representing with a homogeneous temperature field a
classic example of a neuCral combustion charge, will diaplay a degressive
character of combustion if temperature increases toward the outer surface,
and progressive if it decreases in that direction.
10.7.3. Radial-Bur~ning Charges T,Jith a Combustion Front Interaecting the
Isothern~
The simplest vartant for Chis case is a slotted charge in which in the ~
area of the slot the combustion front is initially orthogonal to the
isoth::rms, while in the cylindrical secticn it always has an isothermal ~
orientation. DistribuCion of CemperaCures on the radius in both secCions
can be assumed identic:al, since the slots, in view of their radial orienta-
tion and narrow width, not cause a subatantial temperature field dis-
tortion.
The gas formation function for a given point in time will consist of four
components:
r= ru + r,~ + rT + r~.
characterizing gas formation; ~
rp, on the surface of the cylindrical cavity; f'yr on the surface
of the slots; r T-- on the surface of the burning end (or ends); /'~c
on the surface of the slot web.
~
2 Naamt,oMQ "
0
_1J
%
-~e�
-~s� .>o
_~?a
_90
f ,
, T~M/l0~ld o
+7 ~yp�~Q1 C
~ 3
f10�
+f1,
f1J�
3 rt,�nrpamypd 14, 7�C
Figure 10.5. Temperatu�re Field of Charge With Star-Shaped Cavity With
Maximum Temperature Differential Through the Thickr.ess of the Web
203
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Key to I~'igure 10.5 on preceding~page:
1. Decrease in percentiage of un- 2. IsoCherm
burnec~ propellanC particlea 3. TemperaCure
with a negaCive temperature
~ ump
i B~ f
i ~
~ A
I
/
i
Figure 10.6. Change in Combustion Front With Nonuniform Temperature F3eld
of Charge With Star-Shaped Cavity
If we include the slot web area in the end area and figure the web perimeter
with correction factor ~tY 0.6, with the above temperature diatribution on the
radius of the charge, the function of gas formation in a first approximation
will be expressed as
r
. u~ = 2n (ro e) eM [L - L~ 7~e 2n (~o e~ ( ~ _
R ~
2nL,~ j eM ~~=r'~ cos (d! �r) dr 2n f re~ f~=�') dr,
r
where L-- total length of charge; L~-- lengCh of slot section;
r~rp+~;
e-- thickness of burned layer of propellant detercnined for the cylindrical
section; dl�r local angl.e between combustion front and direction of
radius run through slot sector angle point.
Calculatior, for a charge with star-shaped cavity and its modifications
proves to be more complex. Figure 10.5 gives an idea of the character of
the temperature field of such a charge. In this case one must consider
~ 204
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temp~rature change both on radius r and by coordinaCe~angle er . Non-
untformiCy of Che temperaeure f3eld of a charge with a aCar-shaped
cavity leads to change in its progressiveneag characCeristics and per-
cenCage of degresaive reaidue, as ia shown by the d~.agram in Figure 10.6~
- The dashed-line cur~e corresponds to the poeition of Ctie combuetiion
front at a temperature which is constant through the entire charge sec-
- tion.
With a temperaCure diminishing from the periphery to the center of the ~
charbe, as a consequence of lag in burning rate on the axis of a sCar
point, the combusCion fronC will assume position A on reaching po3nC 1.
The burning s~ur~ace in this instance will be greaCer than with uniform
temperature distribution, and the percentage of degressive residue will
increase.
On the other hand, at a temperature which increasea toward Che center of
the charge, burning on the axis of a star point will run ahead of burning
- in the middle of a pro~ection, as a consequence of which when the com-
bustion front passes through point 1 it wi11 occupy position B, which cor-
responds to a smaller surface quanCity. We include below ca~culated -
figures for a star-cavity charge wiCh a diameter of 152 mm [25] with maximum
possible temperature drop.
Chan e in ercenta e of de ressive char e residue % T`-~~ TA -+2~� C;
8 P g $ 8 ~ Tp =-I-21� C TN =-29� C
Change in burning time, % �
-I-7,96 ~,36
-}-q1,5 -39,2
~
205
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Section IZr. 3TA'rZC Ct~ARACTERISTICS 0~ NYB1tIU ItOCKLT MO~ORS
Chgpter 11. D~SIGNS AND ~E/.7'Utt~S OF OPERATION 0~ HYBRID ROCit~T MOTORS
' 11.1. Deaigne of Hybrid ttocket MoCore
A hybrid rockee moeor burning a two-component propellant contains the follow-
ing:
combugtion chamber containing solid component cherge;
Cank with liquid componenti;
equipment of system for feeding liquid propellant componenC into the
combu~tion chamber;
avComatic cnntrol elements, with the aid of which motor operaCion is
controlled (launch, shutdown~ CransiCi.on from one mode to another, conCrol,
etc).
- Classification of GRll [hybrid rocket nwtors], ~uet as of other eypeg of
rocket motors, can be performed according to various a~tributea they can
be differentiated, for example, by function, magnitude of thrust, potential.
number of firings and ehutdowna, etc. These classification atCributes are
ineignificant, however, for analysis of atatic characteristics. More im-
portant in this regard are differences among GRD in featurea of the devicea
nf the system for feeding liquid component to the combuation chamber.
Practically all types of corresponding ayaCems of liquid-propellant~rocket
motors can be used to feed liquid propellant componQnt inta the combustion
chamber of a GRD. In confon~ity with this, one can designate the following
GRD by type of liquid component supply syatem:
with a gas pressurization supply syatem;
wiCh a pump aupply system.
GRD wiCh a gas pressurization liquid component eupply system in turn can be
classified by type of device boosting pressure in the tank during motor
206
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oppr~Cion (pr~~~ur~ ~ccumu~.geor). ('ht ~n~logy wi.eh 2hRD, ~n~ c~n cnngider -
po~aibi~ u~il.izgtion of Ch~ f~llowing in hybrid motor~t
gir (gg~) pre~~ur~ ~ccumul,aeor (VA~);
cgr~ridge pre~~ure accumulator (~AD);
hybrid pr~~~ur~ ~ecum~lgti~r (GAD).
gncri~d ~~mpr~~~~d gag i~ uegd tio force prop~Ylant from the tank wiCh ~n
~ir pr~~~ur~ accumulator. If a motior employ~ a eartiridg~ or hybrid pr~~-
gurp accumulatnr, fu~l combu~tion producti~ ar~ fed ineo th~ tank (~olid
propellant in th~ fireC inRCance and liquid-eolid in tih~ ~~cond).
~i~urc~ 11.1 containe a diagr~m of a GRD wirh a gas pr~s~urix~tion eystem
nf f~eding 3.iquid componenC and ~n air pres~ure accumulator.
W1i~n Che moCor i~ ignired, velve 2 open~, and compr~~ged g~g ig f~d Chrough
a pregsur~ reducer to diaphrggm 4. After p~npCrgting the diaphragm, ge~
fi11~ th~ fr~~ ~pac~ in the ~ank. Diaphragm 6 bureC~ und~r Che increas~ed
preg~ure of rhe liquid compnnent; th~ componenC pgge~~ through open valve 7
to Che combuetion chamber and them inCo the charge caviCy. The compnnenCg
ignite (eith~r gelf-igniCion or, if the pair of componenCg do not gpontanenu~-
ly ~ombu~t they are ignited by $n ouCside source), and the motor enter~
th~ r~quired operating mode. The motor ig ehut down by closing valve 7~ am
a reault of which liqnid component is no longer fed to the co~bustion
chamb~r.
f
t
J
~
S
6
~
d
Figure 11.1. biagram of GRD With Gas Pressurization Liquid Propellant Com-
ponenr Supply System
Key: .
1. Compressed gas tank 5. Tank with liquid propellant com-
_ 2, 7. Valve ponent
3. Pressure reducer 8. Combustion chamber
4, 6. Diaphragms
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r
r
a
~
a -
~
~
~
d
9
f!
. _ _
~igur~ i1.2. biagr~m of Hybrid Pr~~~ur~ Accumul.ator
Key:
1. ~oCtled gag 4, 6, 9. Diaphragn?~
2. Valve 5. Pr~seure accumulator tank
3. Prea~ure reduc~r 7. Valve
8. Pressure accumulator chamber
10. Liquid propellane tank
A GitD cartridge preegure accumulator may not differ in design and operating
featurea from gnalogoug devices used by liquid-propellant rocket motors and
therefore will not be discuesed.
~'igure 11.2 containg a diagram of a typical hybrid preggure accumulator.
As i~ evident from a comparieon of figures 11.1 and 11.2, in this in~tance
the pressure accumulator essentially comprises a amall hybrid motor from
the combustion chamber of which combustion products enter the main engine'$
li;uid-propellant component tank. This Cype of pressure accumulator is more
complex in arrangement than a VAD~ buC ite utilization makeg it possible in
pr.:nciple to improve engine maga characteristics by reducing Che mase of
the compr~sgpd-ga~ tank. .
A Cttb with a pumped liquid component supply syatem can incorporate the fol-
lowing arrengemeets:
cloaed;
open.
In the first inatance gae, after passing through the Lurbine, is fed into
the combustion chamber and is subaequently ejected from the main exhaust
nozzle together ~+ith the combustion produc~. In the second arrangement
gas, after pasaing through the turbine, is ejected into the atmoaphere
through special exhaust nozzles.
208
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A CRD, ~u~e liquid-prop~].l~ne moror~~ egn ~mploy varioug ~yp~~ of eurbin~
ga~ g~n~rge~r~. dn th~ ba~i~ of thi~ ntirribuC~, GRU with ~ pump~d li.quid
prapell~ne compon~ne ~upply ~y~C~m ~r~ ~ubdivid~d into mneor~t
with a~ingle-componenC ga~ g~n~araeor (hydrog~n p~roxid~, hydraz~n~,
etc);
with ga~ tapped fror.? th~ main combu~rion chemb~r;
wiCh eurbine-driving g~~ form~d in ~~peeial chamb~r in which ~ ewo-
compon~t~t propellant burn~ (for ex~mple~ th~ ~~~n~ compon~ne~ in th~ m~in -
~ombu~tion ch~mb~r).
On~ ~hould exp~ce Gitn eurbopump uniCg tn be ~impler than tiho~e of a zhRD,
~inc~ th~y mu~t ~upply only one liquid component.
Ju~t in liquid-prop~llant rock~C motore, a~om~whaC el~vatpd tank pr~~~ur~
mu~e b~ provid~d for GEtU with a pump~d eupply ~yet~m, in ord~r Co ~n~ur~
gt:bility gnd to mgintgin c~vitation-fr~e pump operation. PracCically a11
Chose eyp~~ of d~vi~s~ which are employ~d for the gam~ purpose in 1lquid-
prop~ll~ne rock~t reoCnrg can b~ ug~d to boo~t tank pregeur~.
f
!
_ J
4
s
.
p 6 `
e ~
jr
Pigure 11.3. Closed CFtD Arrangement
Key:
l. Tank 6. Combustion chamber reactor
2. Pump 7. Combuation chamber
_ 3. Gas generator reactor A. Peroxide feed to chamber �
4. Turbine 6. Peroxide feed to gas generator
5. Gas generator pump B. Steam-bas feed to tank
Steam-gas feed to chamber
209
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~
.~...r- !
i
s ~
y
s
A
A
-
~igure 11.4. Open GRD ArrangemenC
Key:
1. Tank 5. T~nk pr~~~ur3~~Ciofl gas
2. Pump generaCor
3. Turbine A. Movement of liquid component
4. Turbine gas generator 6. Gas movem~aC
Figureg 11.3 and 11.4 preeent two layoute as e~camplee of poesible GRD
arr~ngementg with a pumped liquid component supply sy~t~am~ differing in
th~ principle of utilization of epent turbine gas and type of ga~
~ aenerator employed. Figure 11.3 contains a diagram of a GRD aith a closed �
liquid component feed system and a gas generatoY eaiploying a eingle sub-
~tance, ~rhich is also a propellant component. Thie arrangement can be
utilized~ for example, When hydrog~n peroxide is the liquid component.
'fhe diagram does not indicate the automatic control devices which control
motor op~ration, and in particular motor igniCion, !a the procees of vhich
~ additioaal measures a~ust be taken to accelerate turbine buildup to che
required operating conditions, to provide preliminary tank presaurizatioe,
etc.
~=gure 11.4 contains an open GRD arrangement vith a gae generator employing
hyorid (so~id-liquid) propellant conaisting of the same components as the
motor propellant. As in the precedina instance, the diagre~a does not
shoW automatic control devices.
Specific Chrust impulse of a GRD With a pump liquid component supply syatem,
juat as for Zh1tD of similar arrangements, dependa on the type of supply
syetem. In an open-arra~gement motor thruaC P~Pk+Po.c~
210
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wh~r~ P~ ehru~C ~@n~~~~~d by ehr~ eotnbu~~ion ~h~tnb~t~; Pa~c ehruge
g~n~r~t~d by eh~ eurbinp ~xh~u~e t~o~zi~.
if w~ d~~ign~e~ finw ehr~ugh ~h~ enmb~~eion ehamb~r noa~1~ ~nd turbin~
~xh~u~e no~~l~~ wi~h ~fi~ and r~~pect3v81y, ~pecific ~ngine impui~~
T' P~~e ~ NMR ~Y~ e. cMd~ Q~
~ ~ ~s
H~r~ I~,~~ ~nd apeeiffe ehruet impul~@g of combus~iofl chamb~r and
eurbin~ naz~1~~; ~i~~~fik'~o.c�
t~~~~gn~ting eo~ffici~n~ af ~p~eific thra~e impul~~ 1o~~~g in ~~upply ~yg-
t~m aith
~ .e.e
~I ~ ~11t~ \1' ,y ~
a~ c~n nbtgin Ir = ~l - g~).
Ag a~~ ~h~wn ~arlier, qu~nty ~ i i~ conn~ce~d With th~ magnitud~ of combu~eion
chamb~r pr~~~~r~ p~; eh~ higher thig pr~g~ur~, the greater i~ ehat is~
th~ ~r~~t~r Ch~ 1o~g to the turbin~ nozzl~g. Taking into coneid~ration the
f~et that ~p~cific chemb~r impul~~ increa~e~ with an incr~a~~ in pre~gur~
pk, dne cen ~~t~bif~h that function ~y(pk) hag for ~hRU a maxim~m in the
r~gion of preg~ur~~ pk ~~00�105 Pa. Por GRD incr~age in mas~ with an in-
cr~ag~ in ch~mber pregsure ~hould b~ considQrably more appr@ciabl~ than
for liquid-prop~llan~t motora, 9ince the chamber 3$ ~ub~tantially larger (it
containg ~~o1id prop~llgnt component charge). Therefor~ a mov~ to closed
degigns, that is, into th~ area of high chamber preeeureg~ for GRD should
produce 1~~~ gain in total characteristice (energy, maes)~ in connection
ai~h ahich employment of cloged de,~igng i~ l~g~ ~fficient h~re.
p Mo~ ~P,q .
l
!
=I
~igure 11.5. Regions of Lfficient Eroployment of V:srious Types of Supply
Sy~temg
Key:
I. Region of gas pressurization II. Region of pump systems
gystems
One can evaluate in approximately the same manner the correlation betWeen
regions of efficient employment of gas preesurization and pump liquid
211
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prnpcll~ne cnmpnn~ne ~upply py~e~m~. ~or liquid-prop~L~gne rockee moeor~
tt ~H rxpc~dt~nt re ~mploy gar~ pr~~gurizgCinn ~y~Cemg wiC:h low Chruge im-
p~ilr~c valutw t~I~~ (~C mo~or burning Cime) ~~inc~ wieh an in~regse in
impul~~ i tl~~r~ i~~n~ncr~a~~ ~,n eank volume and Che influQnc~ of eheir
n~a~~ on ov~rall m~gg chgr~cC~ri~ticg. Thi~ alsa app11~~ Co GltU, buC ~ince
in ~ CRD eh~ e~nk cone~ing oniy a portion of ehe p~opell~ne, Che per~enCg~~
~hnr~ af the gupply ~ygr~m m~s~ in CoCgl ~ngine ma~g wi11 be 1~~~ eh~n
far ~ zh~tD, ~nd Ch~ r~;;;idn of ~Pfici~nC utili.z~tion of gag pr~g~urization -
~ygt~m~ ~hould b~ brd~d~r for a GRI) than �or liquid-prope].lenC moCorg
(~igur~ 11.5).
- Certnin diff~r~n~~~ in GRD l~yout~ may also be cau~ed by the gpecific
d~gi~n f~atur~~ ~f th~ir combu~tion chambere. The chamber~ in Che above
GRU df~gramg ar~ of the ~impl~~t type uncooled, and with liquid propellant
cdmpan~ne f~d only through Ch~ h~ad of th~ chamber. Obvio~gly Che facC of
a liquid prapell~n~ compon~nr makpg it po~~ible to cool the chamber in the
sgme m~nn~r g~ i~ p~rfdrm~d in ZhitU chamb~r~. Since thaC portinn of the
ch~mber in whieh the solid propellane component charg~ ie plgced can be
ghi~ld~d frorn heating by the propellant layer and eh~ dim~nsions of
a Gttll chamber are considergbly greater than Choee of a ZhRD chamber (wiCh
~qua1 thrugt), one c~n a~~ume thaC cooling only the unprotected portion uf
~he chamber and ttozxle i~ reagon~ble or possible (~igure 11.6).
~
~
Figure 11.6. GitD Chamber Cooling Diagram
. ~ "~a~re
1�
.
f,~
rigure 11.7. Diagram of GRD Chamber With Afterburning
'Che liquid propellant component can be fed inCo the chamber not only through
the head of the chamber but also into the region forward of the nozzle,
that is, beyond the charge (Figure 11.7). In this case propellant com-
bustion takes place in two atages. The first stage takes place in the
charge cavity, followed by "afterburning'in the nozzle-adjacent cavity of
212
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Che ~ombu~tion product~ form~d ~.n th~ ch~rge cavitiy. In conformiCy witih th~.~,
Ch~ ehgmber d~gign di~grg?m~eci in ~'igur~ 11.7 ie someeimes called a deaign
wiCh ~fCerburning.
Henceforth we ghall employ the following deeignation for euch a chamber:
m~ per-aecond coneumpt3on of solid propellant component;
m* per-gecond conaumpCion of 1~.quid component;
~~~r rhat part of the per-second liquid component flow enCer-
ing the head o� the chamber;
~*~p tihat portion ~f Che per-second liquid component flow en-
tering Che afterburner.
, . ~L �
m~ � m~. r T m~. A�
Toral propellant consumption m~~ m~ + m~.
Propellant conaumption relatiion: K m m�~/m*.
Just as for ZhRD, specific Chrust impulae ie determined by the ratio of
components and pressure differential between chamber and nozzle exit:
Iy~I~1(' P
Various propellanC pairs can be employed in GRb.* Many possible composi-
tions have been proposed. Practically all corresponding components of
propellants used in modern ZhRD can be employed as liquid oxidizers or
fuels. Nitrs~es or perchloraCes of some elements (sodium, potassium,
lithium, etc) or groups (ammonium, hydraz~.ne, nitronium, eCc), for example,
can be employed as solid oxidizers. A wide range of substances can be
employed as~solid fuels polymeric compounds, rubber, hydrides of inetals,
(aluminum, lithium, beryllium), etc [7J. GRD based on a liquid oxidizer
and solid fuel are considered preferable in a number of indices. Such
GRD ar~ sometimes called straight-design motors.
* Employment of two components in a GRD is the moat typical. Foreign
sources, however, have contained reports of GRD designs employing a
three-componenC propellant; two liquid components (oxidizer and liquid
hydrogen, for example) wauld be fed into Che combustion :hamber in
order Co improve a motor's energy characteristics.
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Figure 11.8. Poasible Shapea of G1tD Propellant Charge Croas Sections
In gddition to other facCors, the characCeristics o� a G1tD are der,ermined
by thc shape and dimensions o� the solid componenC charge. PublicaCions "
have discussed employmenti of char~es of various shape. The simplest ia
the cylindrical charge wieh a single cavity (Figure ii.8). A charge of this
shape, however, possesses a relaeively ama11 combustion surface and, in
addition, does not ensure (and thia may prove to be easential) consCancy
of surface during motor ope~ration. Figure ii.8 also shows other possible
charge cavity cross secCional ahapes.
The principal characteristics of GItD, ~ust as other types of rocket motors,
are Chrust, specific Chrust impulse, and duraCion of operation. With a
given propel.lanC and engine design, theae characteristics are deCermined
by per-~second cnmpon~nt flow and consumption rates, which in turn are
dete~:mined by the de~ree to which the values of the moCo~s numerous design ~
parameters and operating conditions correspond Co Che design or rated
values.
The most important factors in this respect, which influence GRD character-
istics, are the following:
deviations from raCed values of design parameters o� the liquid com-
ponent supply system;
deviations of combustion chamber dimensions, and particularly the
nozzle throat area;
deviations of dimensions (length, cavity cross sections) of the
solid component charge;
deviations of solid component burning rate;
deviations of propellant component temperature.
~hese factors should be primarily considered in analyzing the static
~haracteristics o� a GRD. -
11.2. Propellant Combustion in a Hybrid Rocket Motor
The processes of combustion of a solid-liquid propellant in the conditions
of a GRD chamber are extremely complex and unique. IC is the features
precisely of these processes which distinguish hybrid rocket motors to the
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greaeegt degree frnm zh1tD and 1tDTT. In order to evaluate and ca~.culHCe rhe
characCeriatics o� a GRD, ~us~ as the characCerie~ics of motor~ operating
on a so~.id propellant, one muaC know the relations datermining the linear
raee of combustion of rhe solid propellanC component, ~hat is, Che rate of
displacemen~ of ~he combustion front into the charge in a direction per-
pendicular td its sur�ace. These relations ara diaCinguighed from Choae
which are atilized to deCermine ehe combu3tion rate o� 1tDTT propellants,
which is connected with the featurea of the procesa of combuation of GRD
propellants and Che compositiou of Chese propellants. As was noted above,
ItDTT propellanta contain both fuel and oxidizing elements in a ratio en-
sur3ng aC all t3mes 3ndependent (w3thout the parCicipation of additional
components) combusCion of Chese propellants. In connection with Chis, com-
busxion terminates entirely in the layer directly ad~acent to the surface
of the charge. Therefore in the ma~oriey of cases the atate and parameters
of the envirbnment ad~acent to tihe burning surfACe does not exert sig- .
nificant influence on the raCe of burning of RDTT propellant. Processes "
tgke place dif.ferenCly under conditions of combusCion of a GRD hybrid
propellane. The solid component contains a large surplus of fuel or
oxidizing elements and frequently is incapable of independent combustion.
Requisite f.or combuation, Chat is, reactions of oxidation of fuel elements
with release of heat, is contact between the subsCance of which the so~.id com-
ponent consisCs and the substiance of the liquid component. Usually this con-
_ tact and the following oxidation reactions take place in a zone above the
surface of the solid componenC. This zone is the combustion zone proper.
But in order for sCeady-staCe combusCion to be maintained, it is essential
that new doses of components continuously enter this zone.
1 2
----~-s J
~
. ~
- - s
s _
, , y.
� Figure 11.9. Diagram of Combustion of Hybrid Propellant
Key:
l. Direction of movement of liquid 4. Zone of chemical reactions
component 5. Mixing zone
2. Heat flow 6. Surface of gasification
3. Zone of termination of re~c- 7. Zone of solid component
tions and equalization of com- heating
, position .
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l~igure 11.9 containa a diagram of ste~dy-etate conditiona of hybrid
propellnnt. Hegt flowe from the combuetion aone proper to Che surface
nf thh ~olid component, as a reault of which Chie componene becomes
i'lCt1C4d. When the surface o� the componenC r~aches a certain temperai:ure,
g~eificneion begins. The procesa of gasif.ication can tiake place in a
vnrying manner, depending on the compoeition of the solid componenC. zt -
can involve melC3ng with subsequenC vaporization of a liquid film, sublima-
rion (transition of a solid substance to a gaseous substance without
intermediat~ transformation into a liquid) o~ pyrolysis (chemical decomposi-
Cion with formatiion of a gaseous substance). Gasification products proceed
from the surface into Che combuation zone.
Upon entering Che combustion chamber, the liquid component is broken up into
fine droplets by in~ectors. The droplets are heated and vaporized by the
h~~t released in the combustion zone. Thus fuel and oxidizer enter
oxidation reacCions in gaseous form. These reacCiona begin at certain
local componenC ratio values which differ from the average. Then, in the
process of movement by combustion products through Che charge cavity toward
the nozzle-ad~acenC space, these reactions Cerminate and there occurs
equalization of Che composition of the gas mixture to the computed value,
corresponding to Che selected propellant raCio.
As �ollows from what we have stated about the character of combustion wf a
hybrid propellant, this combusCion constitutes an aggregate of complex
physicochemical processes. It is therefore more correct to apply to Che
solid component not the terms "combustion" and "rate of combustion" but
rather the Cerms "gasification" and "rate of gasification." In addition
one can conclude that the determining factors for rate of gasification are
primarily those facCors which influence intensity of delivery of heat to
the surface of the solid component. The rate of gasification should also
be determined by the thermophysical properties of the component proper (for
example, melting and evaporation heat and temperature, heat conductivity,
etc). Heat is transferred to the surface of the solid component from the
c~mbustion zone by means of radiation and convection. The process of heaC
transfer is highly complex, buC the factors which primarily determine its -
intensity are obvious. They include the properties (pressure and density)
of the gas flowing over the surface of the solid component, the rate of
gas movement along the surface, as well as temperature in the combustion
zone, which in turn is determined by the composition of the propellant
pair employed.
There have been numerous studies of the processes of cambustion of hybrid
oropellant, aimed at establishing relationships for determining the rate
of gasification of the solid component. Most frequently these relations
are in the following form:
u = utPK ~Pv)~A�C+ (11.1)
where ul coefficient taking into account the properties of the propel-
lant component, including their temperature: for a given pair of components
u1=u1 (T); pk, p, v-- pressure, density and velocity of gases over the
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qurface of tihe solid components; A-- coeff~.c~.ent taking ~nCo ~ccounC
oCher propertiea of ehe combuot3on prnducea; v, a- coeffic~.enre.
Desi.gnating u1 (T) Aa=u~ (T), we obCain
u~ u~ cT~ pK cpv~~~ c~ ~.2~
Usually for pracCica~. needa re1aC~.on (11.2) is eatabliahed experimentally.
Coefficienea ~V and ~i and function uZ (T) are determined thereby from
experience. We sha11 note that i� we disregard the ~.nfluence of velocity
of gas movement on the rate of gasification, thaC ie, if we assume ~~0,
_ Chen relation (11.2) assumes the form
u a ut (T) PN+
which corresponds to one of the most common Cypes of solid propel~.ant com-
bustion rate formula. Taking i,nto account that according to Che gas flow
conrinuity equation pv=m/F, where c~ per-second gas flow rate through the
section; I~' croes sectional area, we obtain from formula (11.2)
u � u~PK ~ F ~ ~ . (11.3)
From this follows a number of conclusions which are important for analysis
of the operating process in a GRD combustion chamber. Per-second rate of
gas flow thfough a cross section of the caviCy increases with progression
along the charge as a consequence of the generation of additional mass
during gasification of the solid component. Consequently, with uCiliza-
tion of a charge in3tially containing a caviCy of constant (lengChwise)
section, the rate of gasification also increases in the direction of gas
movement, which should lead to nonuniform gasif3caCion along the charge
and to change from the iniCial shape of the cavity.
Calculation of GRD static characteristics taking into account change in rate
of gasification along the length of the charge is difficult. At the same
time sufficient accuracy of solution for a number of problems can be ob-
Cained by utilizing average rate of gasification, which is defined as the
_ rate of gasificaCion which produces the same total (along the entire
cavity surface) gas formation as actually occurs, Chat is, taking into
account change in rate.
Let us determine the average rate for the case of employment of a
cylindrical cavity.
If cavity diameter is D, charge length L, density of the solid component
pt, then gas formation of the entire charge wi11 be _
L
mT = nDpr ~ u dl, rAe u= u(~. ,
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By deCermination of average ratie, gae formatiion can also ba determin~d
in Che fortn ~ nDLpru~p,
Cnnsequentl.y, ~
u~p~~~ udl. (11.4)
The rgte of gagification in the initial croas section o� Che charge cavi~y,
where rate of gas flow is equal tio liquid componenC flow rate fh* according
' to relation (~.1.3), 3s determined ae
,
uo�u~PK~~~p�
We shall designaee ~
Y1�~ m~� ,
mf ~'=71t
UCilizing Chese relationa from (11.4), we can obtain [7] the following
expression for average rate:
ucp � uo ~ (1-' - ~ ~ -a ' (11.5)
' ~~-V') -1
Or, ~esignating
K (V~~ a +h ( ~ - ) (11.6)
~ +~~-a_'~ ?
we f ini thaC `
u~p = uoK c~a, a~. c~,.~~
Following are typical for GRD: 0~3 0,6; 0~2 0,3.
K(~Q, = 1,1 1~2 and, consequently, the average rate of gasifica-
tion differs from the least (on entry into the charge cavity) by (10... 20)~.
~ We must note that in the process of operation af a GRll the propellant charge
in which initially had a constant cavity cross section, there occurs
. "equalizaCion" of rates. The further the section from the beginning nf
Che cavity, Che higher the rate in that section and the greater Che degree
ro which its area grows, which in conformity with (11.3) should cause
a decrease in rate with an increase in time. Therefore the above-established
:orrelation between average and minimum rates of gasification is maximum and
uecreases with cont:lnued motor operation.*
* We should draw attention to Che fact that cavity shape also changes at the
same time from cylindricalit becomes conical with a complex prof ile. -
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'1'h~ basic pn;~sibiliey of nne of Che merhods of G1tD contirol algo fnl~.nwa frnm
formula (11.3). If a moeor employs a combustion chamber with 1.iquid cnm-
ponent fed both tihrough Che chambex head into the charge cavi.ty and into the
nozzle-ad~acenC space beyond the cavity (see ~igure 11.7), obvious~y c~oC Che
enCire flow rar~ of Lhe ~.iquid component is deCermining for rgre of
~gstf.icaCion, buC only Chat parr of the flnw which goes Chrough Che chamber
ghead. Therefore, A1Cering Che distribution o� liquid component �low ra~e
between the Cwo above-stated componenGs, one can af�ecC charge gas forma-
tion to a certa3n degree and thus influence such princ~.pal motor
characeeriseics as propellanC consump~ion and compoaition (raCio of com-
ponenCs), which in turn determine motor ~hrust and specific thrust impulse.
The gbove-noted feaCure of a GRD wieh a chamber in which liquid propellant
component is fed not nnly into ehe charge cavity but also bypassing it ineo �
rhe nozzle-ad~acent space, should be considered when determining Che sbatic
- characCeristiics of a GRD of Chis design. ~
11.3. Hybrid Rocket MoCor Combustion Chamber Equations
The system of equaCions describing the relationship betwaen the chart?cCer-
istics of a hybrid rocket motor under steady-state operating condiCi~nq
contains, in the general form, equations of:
combustion chamber; -
liquid component supply system equipment;
motor thrust characCeristics.
Systems of linearized moCor equipment equations, which make it possible to
determine deviations of parameters from their nominal values, can be
utilized to solve the ma~ority of problems pertaining to examination and
determination of the static aharacteristics of GRD, just as in the case of
solving anaingous problems pertaining to ZhRD.
GRD CombusCion Cha;.~ber Equations
CRD comb~lstion chamber equations should inCerrelate deviations in chamber
pressure and fuel component flow and consumption rates with deviations of
the principal factors influencing these parameters, that is, with deviations
of charge dimensions, density of solid component, temperature of the com-
ponents, etc. '
Let us examine a combustion chamber of the most general design, with com-
bustion product afterburning (see Figure 11.7). `
We shall designate in addition: T~ and TT temperatures of the liquid
and solid components; K~T stoichiometric ratio of rates of component
consumption; d-- excess liquid component ratio; ~_~M.~/m* ratio
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af liquid ~nmpnnenti r~e~ of flow Chrough Ch~ h~gd of eh~ cnmbugtion Ch~mb~r
ed irH roegl t'1ow rae~; R~nd Tk g~~ consCane end C~mper~Cur~ oE com-
buHtidn produce~ on enery into Ch~ nozxl~; n polyCropic ~xp~ngion ~x-
� p~nr.nr; ~f C-- ad~ffici~nC of lo~~~~ during gag movemant Chrough eh~e nozzle.
np~rnrion itt th~ combugCion chamber und~r motor ~t~ady-~tati~ op~rating con-
ditinng is degcribed by Che fnllowing gy~Cem nf r~1~Cion~:
~~ng~rv~tidn of m~Ctier ~quation:
.
m~ ~ m* m~. ~-I- m,~, x~ (11,8)
equation of ggg flow raee through the nozzle: ~
nt~ c, ~Pc 6 rn) AcFKa ~ ( I I.9)
equation~ of flow rate-con~umpCion raCios:
K~aK~*~ ~ ; ~ (11.10)
. (il.ll)
equation of charge gae formation for a aingle-cavity grain:
~ L m~4 b
mT ~ f nDptur d~ j nDPtuiPK dl~ (11.12)
w}ierc mi ga~ flow raCe in ~ given secCion;
~ ~ mt ~ (m~. ~ ~r)�
~rom (].1.12) one can obtain
. ~ ~
~ ~ (4on~ -61 ~l - uip:LO~-~ppK rit*~,P01J~=p - mM. r? (11.13)
equation of influence of temperature of propellant components and
the ratio of their flow rates-consumption on parameters of the gas mix-
ture:
RTK ~ R7'K (a. TM, Tt)~ ~ (11.14)
equation of the influence of Cemperature of the solid component on
the raCe of its gasification: ulsul (TT)�
ConcurrenCly solving equations (11.9), (11.10) and (11.14), linearized
in the environs of nominal conditions, we obtain the first of GRD chamber
equations: '
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~K'dp~ ~ "'~Nt* ~ dp ?~Il1t 6~""�BFMp
bK~B~Pa ~N"`D7'x ~KT87'r ~ o, ( l 1~ l6)
wh~r~ dim~n~ionl~~g ~d~ffici~neas*
aK~~�bk"p~bKn~~,t+ aK~~ ~'.~.~~~ar '
c~,~ ~ 1 T~y B RT 1, aN ~
RfiK 1-~- k
a(~ r* ~ I .,I.~.. d (RT__~~I~. ,
R7 K da ' ~ R~`K ~`t
From equations (11.12) and relation ul~ul (TT) we ohte3n Che second CRD
chamb~r equaCion (ga~ formation equation):
a~ r8r?~r Qr M, ram~, a~"aPK b~~BUt
br�,t8pr b~ dL bDBD ~~*8Tr m 0, (11.16)
wher~ co~fficiQnt~
a~'* ~ 1 ~ a~ ~ G, 1 - f (~P? ~ f (~P+
b~ ~ b~* ~ b~' ~ f(~? K~ ~ b� ~ i_~ j(4?~
* a ~i. _aa..Tu.1.. '
~ ~ f ~~Pr
In expreasions for co~fficients function
f(~P~ - ~ ~~K~'-a~. (~t.17)
Prom GKU chamber equaCions (11.15) end (11.16), as ~ particular case cor-
responding to K=0, rt=0 and ~3 =0, one easily obtains solid propellanC
thrust chamber equationa:
.
* Designations a have been adopted for coefficiente with deviations in
operating parameters and proceae characteristics, b-- for coefficienta
with deviaCions in sCrucCural dimensions and characteriatics, c-- for
coefficients with deviations in environmental factors affecting the process.
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; 8pM - dmr BFNv 6tpc - a R~~ 8Tt ~ 0~
(11,18) -
T ~1u
v8p~ - 8ri~1 8u~ 8pf 8D 8L ~ aT 8T* ~ 0.
11.4. I.qur~tten~ a[ Liquld Componene 3upply 5ydC~nu Cquipm~ne
'Th~g~ includ~ Ch~ followin~ ~quaei.on~= C~rbin~; pump~; Curbin~ ~as generator;
p~~~~ur~ ~ccum~ilgtor; p~dp~l~~ne ~nd g~?s lin~~; ~ec.
dn~ c~n ~xp~cC ~h~e a11 ~he~~ compon~nt~ ~nd ~l~m~nt~ of a GRD liquid
proppllane compon~nt ~upply ~y~C~m wi11 b~ ~imilar eo corr~~ponding
- ~quipm~n~ of liq~id-prop~llant rocketi motor ~upp],y ~y~tpm~, which ~hould
gl~n be r~fl~nt~d in gimil~rity of th~ corregponding~eq~ation~. Trangf~rm-
ing previous ~qu~Cion~, tgking inCo accounr Chi~ ~quation of ZhttD equip-
m~nt, to a form which i~ more conven~ene and mor~ frequently uCilized fot�
hybrid moeor equipm~ne, w~ obtain the followieg equation~.
Turbin~ ~qu~eion~
An already noted~ the mogt probabl~ in ~ Gltn ig employment of liquid com-
ponent gupply ~ygtem open layoaCe, eh~ turbineg of which can operate with
lgrge pre~gure differentialg and in connectiion with thie can be deeigned
ag congtant-pr~s~ure turbines.
tn this case the ba~ic eurbine equation~which relateg d~viationg in turbin~
output with deviationg in working medium flow rate and oCher parametere
inEluencing output has the form
ar"8NT aT rr8mrr a"~8n aR~' 8(Rr~~ -4-
bTKp''BFKp. t-~. bt,. ~gF.. b~`~1~ ~ 0, (11.19)
where
Rr A
aTrc=,~aN~b'T'~~ 1; Qf~A*; aT ''a 1--,~;
6T Kv. r~- bi.. ~,a 2(!�~.^.~1_ ~
u
2rct -1
�t ~T B ~ (4't ~ ~ - Pr cos act -
r ~
Af ~ 1- ~r Pr cos a~ - �'rx -h �t'Vr cos ~BT
~ B*~Y~T-2x~QTcosal-}-x'. .
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Pump ~qu~Cion~
Pump ~qu~eion~ d~e~rmin~ eh~ r~l~eian~hip b~~w~~n d~vi,~eion~ in pump out-
pu~ ~nd hr~~~ur~ ~nd variou~ f~cror~ ~ff~cCfng th~~~ prin~ip~l pu~np
p~r~m~e~rgs
dMp"d ~pM dN "dlitN Apdn bM
�8ffo cp"'d~* ~ 0~ (11,40)
aN"AtJ~ aN"8i~, oNBn bN'6Ho GN~M ~~BP~ .
~11,~1)
wh~r~ co~ffici~ne~
pM~~ bM ~ es 1 i Qr M~~lt.t1 ~ pN ~y 2.~ �6 MII ~
tg ap t~ a�'
CMM +a~s ~ - ~ ~ QN~ ~ bN bH~ a' ~ ~
QNN ~ t ~.CLL ~
~h ' �
~g aN ~g ~ ~
, Qn ~ 2 t ~n t a ~ A a, ~
~ _ ~.n''
N ~g a^ ~g Qp , CA~` ~g ~ t~ aM ~
d!~ pump manufacCuring error ae regard~ preesure (determined during
pump teet~ under nominal opergting conditione).
'Che quantiti~g controlled in the pump equaCions and the coefficiente of
theg~ equations are linked with experimental pump pre8a~re characterietica
(Figure 11.10) plotted in coordinateg
. ~!i. ~ ~
where QE{ volumetric per-second liquid flow rate and Hn pre~~ur~
gen~rgted by the pump .
~QM ~ ~~P~~ 1`! = ~P~P,~~�
Points A(on,the preeaure characteristic curve) and B(an the efficiency
line) correapond to nominal condicions.
~ -
M
~ A
~ ~
d
$
a.A
~
. Q
~ n
Figure 11.10. Pump Pressure Characteristic Curve
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Lin~ ~quaeion~
Th~ hydr~uli~ 1in~ ~qu~Cion 1ink~ d~vi~tion~ in fiow r~ee ~nd 1in~ pr~~~ur~
drop with d~viation~ in d~a~iCy of th~ 13quid ~nd dim~n~ion~ (gnd ~hgp~)
af eh~ hydr~uiic 1in~t
,
QMM~/1lM l1M
Mg dPM bM ~FM ~A~~Pat ~ 0, (11.~2)
whpre
QMM~Z; QAjM~-b~~C~~-~~
in ~h~ 1in@ ~qu~eien w~ d~~ignate with ~M a~ynthe~ized ~in~ r~gi~~ence co-
~Efici~nt which take~ inCo ~ccount all re~i~tanc~g throughout the iine.
G~D ~e~ 1ine~ ghould be Eigured wieh epecial ~q~~t3ons only in those ca8e~
wh~re ga~ flow ret~~ ~nd pre~~ur~ 10~~~~ in these lin~~ ar~ ~ufficiently
i~r~~, which eccur~, for ~xampi~, in cioaad-1~youe n~otor~. Sinc~ ~uch
d~~ign~ ~r~ 1ieC1~ prob~bl~ in aRD applic~eiong, h~nceforth geg line equa-
tion~ wi11 not b~ geparat~ly con~ider~d.
Prp~gur~ Accumulator EquaCiong
pr~g~ure accumulaCor ~qu~tiong ehould roake it possible to determine tank
preggure changp.
A gag pressure accumulator equation is written on analogy aith the equetion
Eor a 2hRD ~a~ pressure accumulator~ in the form
aoeBPa 60~. ~8p~, M`~' ba��dPc~ m 0, (11,23)
where p~ preggure in the propellant component tank; p ~~p initial
pre~sure in th~ accumulator tank; pp pressure at reducer outlet.
Tha valuea of coefficients in equation (11.23) can be determined ~ust as
for equation (3.40), which degcribes the operation of a ZhRD gag presgure
~crumulator (see 3.5).
The eype of carCridge preesure accumulator equation depend~ on the operating
mod~ aE th~ PAD [cartridge presgure accumulator~. If the accumulator ia
degigned for gupercritical gag floa from the chamber~ PAD equations are
w;itten in the form of an RDTT equation, that ig, in the form of equationa
(t1.18). Prom theae equationa one obtaine gas flow rate deviation, which
~an be utilized for determining deviation in presaure p 6 With the tank gas
mgss gtate equation.
If preggure in the PAD can be assumed equal to pressure in the tank. PAD
equationa are s~rritten in the forcn of (3.41).
And finally, if a hybrid presaure accumulator is employed, ita operation is
described by its rnm syatem of equetiona, ahich mekea it poaeible to determine
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eh~ Eiou ree~ ~nd p~r~m~g~r~ af ga~ ~n~~ring Eh~ mo~or e~nk. Thi~ ~y~e@m in-
~iud~~ ~h~mb~r ~qu~eion~ in eh~ form (1i.15) ~nnd (],1.iG). pr~~~ur~
~h~ng+~ in eh~ t~nk t~ ~ofln~e~~d wieh ch~ng~ in g~~ floa r~e~ through ~n
~qu~eion of ~e~r~.
Turbin~ G~~ G~n~raeor Eq~ation~
Wh~n ~xgminifl~ tnogor ch~r~~~@rigCic~, ch~ng~~ in rgt~ of gag fiow ineo th~
eurbin~ ~nd moenr ~ffiei~ncy ~houid b~ obta3ned from ga~ gee~rator ~qua-
eiong.
~ar ~ g~n~r~tor op~rating on ~~oiid-eomponent propell~nt, th~ ~qu~tiong
~r~ ifl eh~ form of PAll ~qugtiong wiCh gupercriticai gas floW, that i~, th~
Eorm (11.1~).
Ev~rything ~e~~~d abov~ for gn analogou~ preaeur~ accumulator appiiee to ~
~eneraCor operating on a hybrid propellant.
If ~ GttD degi~n employg ~]~qud n~tvpmp~tl~ne g~n~r~r~r, hydrogan p~roxid~~ for
~x~mpi~, chang~ in 8a~ flow rate to th~ tiurbin~ ie equgl to chenge in
p m p~11~n~ (peroxide) flow rate,
bm~r ~ BntN~o~ ~ .
Equation~ of peroxide feed i~to the reactor ahould be ~ritten to deteraiine
dmH2O2�
Chang~ in gag capability to perform Work ie cgueed by changea in peroxid~
concentration and temperature, and can be obtaieed With the relation
$ (R77rr ~ f IdKH~o~ ~ BTH.o~1~
Equations of Thrugt Characterigtics
Deviations i,n motor thrugt and specific thrust im~ulse are obtained from
equations of thrust characteriatics. 'These equations can be borroWed from
2hRb theory in the form of equations (3.45) and (3.46).
Conaidering thaC for GRD, Which a8 a rule have only one propellant pump
and relatively loW combugtion chember presgure, required gas flow rate to
the turbine Will b~ subatantially less than for a ZhRD of equal thrugt,
_ in theee equations one can digregard terms Which take into account devia-
tiong in thrust and thru~t impul~e of the turbine exhauat nozzles.* In this
ingtance deviation in thrust is equal to deviation of combustion chambet -
pressure.
r
* Chapter subgection 1.2 stated that for a ZhRD aithout afterburning relative
gas f1oW rate through the turbine exhaust nozzles comprises (2-8)X of total
f1oW rate. Taking this into account, one can expect that in hybrid motorg
this conaumption rate aill be in the order of 1X~ Which results in extremely
little influence of change in the thruat characteristics of the turbine ex-
haust nozzles on change in thruat characteristics of the motor as a whole.
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Chaprer l2. grATiC CHARACT~RiSTiCS OF EtYYBRiD ROCKET MOTORS
12.1. Infiuence of Bxternal and Internal Factore (Di~turbancas) on Operating
Parameters of Hybr~d Rocketi Motore
Ju~~ a~ for liquid or eolid propellant rocket motore, (iAD operating con-
dition8 dep~nd on quiee A number of factore. DeCermination of devigtiona of
op~ration paramet8re with chang~ in the~e faceore (ar, as they eay, with
the occurrence of dieturbancee) constitutee ona of Che principal taeka of
inve8tigat3on of motor etatic characteristica.
In Che majority of practic~l cagea thig tesk can be accomplished with utiliza-
tion of a eystem of ataeic equations of GRD equipment in the amall devia~:ions '
specified above.
We gh~ll examine ~WO examplea of eatimate of the influence of varioue dis-
turbances on the operating conditione (parametera) of a hybrid rocket
motor.* '
1. Influence of Diaturbances on OperaCin$ Conditions of a GRD With a Gas
Pressurization Liquid Prapellant Component Supply System
Figure 12.1 containg a diagram of a GRD. This motor employs a combu~tion
ct.amber of the most genernl type, With liquid component being fed both into
the charge cavity and into the nozzle-ad~acent epace (afterburner). Divi-
aion of the floa of liquid component (floa rate 16~) into two parts
(~x,~. and d~x~A ) takea place at point N. Ignoring pressure loases in the
charg~ cavity, that is, aeauming that presaure in the afterburner ie equal
to preasure at the head of the combustion chamber, and designating this
pressure by pk, We determine that preasure dropa in the line branchea from
~J ro the afterburner and head of the combustion chamber are squal, thae is,
* The material for these examples Was borrowed from a book by Ye. B. Volkov,
G. Yuo Mazin, and Yu. A. Shishkin, "Raketnyye dvigateli na kombinirovannom
toplive" [Combined Propellant Rocket Motors], Mashinostrnyeniye, 1973.
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(~PN"R~m~~ r~~~pN-~~m~, A~ epN'~M~ ~ ~ 2,
N ~n~
~
,~e
Figure 12.1. D3agram of GRD With Gas Presaurization Liquid Fuel Component
, Supply SygCem
The pr~saure acamulatnr, which forces propellant component from the Cank,
is designated in Figure 12.1 in general form. In order noC to complicate
analysis of GRD characteristics by including in them accumulator characteristcs,
we shall place tank preasure devistion dp6 among rhe disturbing factors
epecified for calculating deviations in motor parameters. In this case the
system o� equations for the GRD in the diagram will be as follows:
1. cKKBpK aK *bnt,~ a**dri~r 'E' 6K~PaFK~ bK~BtPc "
2. a T T~ Q1 r8~~. �t"8PK -f - br a~ '4' br ~~t -F
-1' ~8pr b~80 0; .
3. ~t~~~8rtt~ om,~. ramx. cmM. ~x~ A a 0~
1~ fR M
4. Q~Bmx a~
6-Nb s pa-N + bM ~N~- N~ ~M"`~P* ~ -
b. a~~. r~M. ~M N~Rb SPN-k b~'K. r~N-K. r"~'
+~�~`~a,~ ~ o~
12.2
8. o~ ~rtl~c. R-}- cM N_Ra APIy-R 'F' bMN',c. ~~N-K. A~, ~
+ = o~ ~
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7. G~p~~ L1 pd-N AaN NBpN ba6 NBpd e -
8. ati
N~; +~8 ~pn?-k Q~ ABPN -I' anl~ wBpK b'
9, aK bK a H*bmf -i~ uK ~8m,~ ~ ~12. 2)
l0. alydly aj 8K ~ 0;
11. a,p, 8P a,P,"8pK ~ 0, ~ �
Zn syeCem of equationa (12.2):
equaCions (1) and (2) are equat3ona of GRD chamber with component fed .
tio the head of ehe chamber and af~erburner, written in the form (~.1.15) and
(11.16);
equation (3) was abt~ined from relation mM ~ m,~,~ m~c,A and
ie a liquid component flow rate balance equaCion. In this equation
aM* am*. r~~~ aM* A~, . I
where ip ~ m*,rlm~.
equations (4)-~(6) are propellant line equations; the f irst of Chem '
appliea to line segment "tank-point N," Che second to line segmenC
"poinC N-chamber head," and Che third Co aegmenC "point N-afterburner";
equations (7) and (8) are equations of presaure balance on theae same
line segments taking into accounC (12.1); in Cheae equations pN presaure ,
at point N; Che co~fficienCs comprise:
aaPNN =aNNkk=-1~ 6~N = ~ i
�pa-N
Q6NN PN ~ aN k ~ PN ~
~ Op~-nr avN-~r
PK At
a~y_k = - e_ ;
pn?-K
equation (9) was obtained from ratio K~x /ml;
QK=QKT~-QK*~'~ .
equations (10) and (11) are equations of thrust characteristics; in
these equaCions:
R a~ 1k
a~~al=-aT"~ 1: a/=-~~/ ~y ~ ' -
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Ag i~ ~videnC from sysC~m of equaC~.ons (12.2), operaCion n� a GEtll~ even of
Chc exCramely simple arrangnment ~akan for analysis~ and with a numbc:r nf
Himplifying nq~umpCions (presyure accumulatior equations are noti congider~d;
iti is as~umed ehaC Chere are no preasure changes along tihe snlid propellanC
ch~r~e caviey, tihere ia no conCrol or ad~ueCmenC, eec)~ ia deacribed by a
gy~ttm of equarions which ie eign~ficanCly more complex Chan eo].id-propellant
moCor equaCions and which approximately corresponda in complexity to a sys-
Cem of equations for a ZhRD with a aimilar propellant supply ~ygtem.*
Sysrem of equations (12.2) is a closed syetem~ wt?ich contains ii unknown
deviations of GRD operating conditions parameters from eheir nominal
.
va ues. . . .
~ 8p,~, 8m,~, 8mr, 8m*, 8m*, A,
BpN~ 8 ~pa-N, 8 ~pN_k, 8K~ 8~y, 8P,
The following deviaCions are examined in this inatance as diaturbances
which cause change in mutor operating conditions:
deviation in dimensions of solid componenti charge ~L, p1D, and
coefficienC in the law of its rate oF gasification ~u1;
deviation in size of nozzle Chroat area and coefficient of loyses
f ~'kP. O~j~;
deviation in propellant component densiCies dP T, o~P~";
_ deviaCion in tank pressure oTp6 , due Co error in aacumulator
operation or change in G-loais acting on the liquid component during rockeC
flight;
deviation in coefficients of hydraulic losses in the liquid com-
ponent supply lines
a~,6-N~ B~,N-Kr~ a~,N-KA�
The values of the coefficients in the equations of system (12.2) are deter-
mined primarily by the values of the following dimensionless coefficients,
indices and ratios specified for rated (nominal) motor operaCing conditions:
v, P6 ~ PK .
OP6-N ~PN_k
* The system of equations for a ZhRD wi~th a gas pressurization propellanC _
supply sysCem would be mcre complex than system (12.2) by inclusion of
additional equations which describe operation of the feed line for the
second liquid component, but on the other hand it would be distinguished by
greater simplicity of combustion chamber equations.
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tn nddi~ion~ ChQ valuQe of some coeEf.icientq are detiermined by Che vr~lua
of derivative funcCions RTk (T~, T~) and I~ (K) at poi.ntie corresponding
Co nominal GRD aperating conditions. '
Solueion of system (12.2) wiCh preselected valuae of pargmeCere of nominal _
operating condit3one and adopted diaxurbance valueg makes ie poss~.bl~ to
deCerm3ne a~.l deviations of motor characteristiics. As,an i1lusCratinn, Table
12.1 conCaina the results of calculaCian of coefficients of influence of
various d~.sturbances on principal GRD parameters.*
r
~ Table 12.1.
1 ' 2 KO~Q~N411lHTd liIMIINNN J~JIA I1~p~MQTpO~
noii~o , HoawyWeHUa
p OPK Oin* I einr 8 K b1Y
1 dFKp -0,410 -{-0,630 -~-0~394 -}~0,236 -0,047 '
2 8~~ -0,410 -~-0,630 -}-0~394 -~-0,236 -0~047
3 6ui -I-0,073 -0,113 -~-1 ~000 -l ~ I 13 -0,222 :
4 8pT -}-0,073 -0,113 -}-1 ~Q00 -1 ~ 1 ! 3 -0~22Z
b dL -}-0,073 -0,113 -}-l~000 -1,113 --0,222 ;
6 8D -0,22 -{-0,034 -0~300 -{-0,334 =0,06? ~
7 8pM -~-0,192 -~-0,205 -F0~128 -~-0,07? -0~016
8 8 -}~0,782 -~-0,835 -~-O~b22 -~-0,313 -0,062 .
9 8~-N -0,044 -0,048 -0~030 -0,018 -0,004 ,
10 S~y_K. A -0,025 -0,038 -I-0,039 -0,077 -0,015
11 d~v_x. r -0,122 -0, l 19 -0,13? -~-0,018 -0,004
Key: 2~ Coefficients of influence for
1. DisCurbances
parameters
_ ,
~ * Coefficient of influence is defined as a number which indicates the
magnitude of motor parameter deviation caused by a single disturbance, ~that
. is quantity bpK~BFKQ~ am*/bFKp , etc.
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For calculaeione we have aseumed
pK ~ 40~ io~ n~; R~~; o,s~; v~ o; y~ m o,e; :
~pd_N " 3� 10~ CIa ~ ~PN-A ~ 10~ 10~ Ra;
R a~y ~ o,~, a f RTK) a(RT,e ~ a~tl~~ ~ o.
]Y ~ ~l~ T ~c * ~
In calculating deviationa in specific thrust impulae ~Iy it was assumad that
a deviation in the coeffic3ent of ratio of components K in either direction
from nominal value resulte in an impulse decrease; ~his is equivalent to
Che assumption that nominal conditione correapond to maximum value Ty.
The calculation results conCained in Table 12.1 enable one to draw certain
conclusions on the charACteriatics of a GRD laid out as specified. .
An increase in nozzle throat area causea, ~uat se in an RDTT, a decreaee in
chamber pressu~e, but to a lesser degree, which is due to an increase in
liquid component flow raCe with a decrease pk~ that is~ the presence of ~n
additional fa`ctor opposing change in chamber preasure. �
_ In the cited example gas formaCion of the charge also increases toge~her ~
wiCh an increase in the raCe of consumption of Che liquid propellant com-
ponent. This occurs because we have adopted a law of gasification whereby
the rate of gas formation ia independent of presaure (Y �0) and at the
same time depends significantly on liquid component flow rate (R m0.65).
An increase in charge length, denaity of the solid component, or its rate ~
of gasification (coefficient ul) leads to one and the same conaequences:
the rara of consumption of the solid component increases, which leade ii
~ turn ti~ increased combustion chamber presaure and, as a conaequence, to a
certai:~ decrease in liquid component flow into the chamber. Differing-
.sign changes in conaumpCion of propellant components produce aignificant
change in the coefficient of conaumption ratio and corresponding change in
specific thrust impulse.
Deviation in charge cavity diameter dif�erently affects motor parameters.
An inerease in diameter causes a reduced rate of gasification which ia more
appreciable than the increase in gas fo~rmation surface, and therefore solid
componenC consumption decreases. This results in a combuation chamber f
pressure drop, but to a~}.esser degree than decrease in charge gas formation.
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An increase in tank pregaure quite appreciably affeces motior parametara.
I~ ngu~ee gn increase in liquid component flow rate, which leade eo in-
craa~nd chaYge gas formation and cort~bustion chamber pressure.
Changes in rhe coefficients of ~.ine hydraulic resietances produce in mosC
cases opposite-aign changes in all operation parameters. An excepCionis
~ the case of change in hydraulic resiatance in the line through which
Liquid component is fed ~o the aftierburner. With an increase in Chis
reaiatgnce thera can be an increase (that is, a like-aign change) in charge
gas formation. Thia is due ~o tihe fact that in thie case, in apite of a
decrease in overall liquid componenti flow rate (~~n~ ~0), its flow into the
charge caviCy may increase, which ].euds to an increase in the solid com~
ponent gasification rate. Thia feaCure of a GRD of the deaign under dis-
cuas~on can be utilized to perform motor tun3ng and ad~uarment. ,
2. Maximum Burning Time !
It follows from Table 12.1 that conaumption of GRD propellant components
is deCermined by Che denaity of Che liquid component. A denaity increase
_ by 1% in our i1lusCrat3on produces an increase in Che masa consumption of
both components: liquid by approximately~ 0.2%, and aolid by 0.13x. The densiCy
of Che liquid component is deCermined by its temperature, and therefore
propellant component consumpCiona,are dependent on temperature. This
_ produces change in other motor parameCers, including maximum burning time.
We sha11 designate w3th MT and M~ masses of solid and liquid propsllant
componenCs in a motor prior to ignition (mass of the solid com~onent
charge and mass of liquid propellant on board). Component burnup time with '
consumption rates in~ and m.r wi11 be
MT and ~y
TT~ m T~�
r m*
and their ~E~viations with consumpCion deviations
. ~
dT* = 8MT - BmT; 8i* = 8MM - 8m,~.
232
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An analysis oE Che influence o� varioua factore on engine burning tiime can
be performed on the basis of th~ae relations. ,
In Che case we are exam3ning, that is, when only the temperature of ~he ~
liquid compo~nen~ can change (3n the process of motor atorage),
8Ms = ~dM* ~ 0
and 8~T BmT~ 8t,~ m- 8tit,~. ~ ;
With an increase in the temperature of the 13quid component, ita density ,
decreases, as do mas5 flow and consumption ratefs, in conformiCy with Table
12.1. Consequently, a temperaCure increase increr~aes component conaump-
tion time, and the 3ncrease is more rap~d for the liquid component (0.2X) ~
than for the solid (O.J.3~). Change in maximum motor burning time for our ~
example is i1lusCrated in Figure 12.2~
~ r 1
~raa~a~ xawnoyaHm
= 2
Te6~701I~. NOM/10NtKII!
~ T
Figure 12.2. RelaCionship Between GRD Propellant Gomponent Consumption
Time and Temperature of the Liquid Component
Key: ~
1. Liquid component 2. Solid component
If the motor is designed r..o operate at a temperature T='T~it is natural to ~
assume that for this temperature: Tr = T,~.
When ' T> 7' ~t,~ > TT, and when T< T'tM
Under these conditions liquid component consumption time will be the time
limiting normal (~~ith utilization of both components) motor operation at
temperature$ below the rated figures, while at temperatures above the
rated figure it will be charge consumption time.
3. Influence of Disturbances on Operating Conditions of a GRD With a
Pumped Liquid Component Supply System ~
As an example we have specified deviations in the operating parameters of
an open-arrangement GRD with a hXbrid propellant turbine gas generator ~
. i
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(~igur~ 12.3). Motor con~rol cgp~biliCy i~ noC figur~d in. Liquid com-
pen~nt flow frnm eh~ e~nk through th~ pump (~hp) ae po~,nt 2 i~ di~tribut~d _
intc~ th~ f~llo~ing eompnn~nC~: flow tio ehe g~~ g~ner~eor ~nd flow ed
Ch~ cdmbu~tion ch~mb~r ~hx . Ae po3nt 1 flow in Curn divide~ into two
flowg to th~ Chamb~r head ~nd to th~ ~fe~rburn~r fi~~p . A~ in d~~
prec~ding ~x~mp].~~ i.t i~ ~~gum~d that pre~gure r~maing unchgnged glong Ch~
1Qngth of rhe ~hgrge c~viCy, Ch&C i~~
~ ~ ep1.K.A - sp~-~~
~p ~�K, r
~ ~
. ~N
~i~ ~~a
t
f
~ ~
~igure 12.3. Diagram of GRD With a Pumped Liquid Propel.lsnt Component
Supply System
~ Key:
1. Point of division of flow of 2. Takeoff point where liquid ~
liquid component to com- component is fed to gag
bustion chamber generator
Table 12.2 contains the results of calculation of coefficients of in-
fluence of various disturbances on the principal parametera of the
motor under discussion.
hor calculations we have asaumed the following:
" pK = 1Q0 � 10' Iia; Pr = 70 � 10� tIa? Pa � 5,5 � lOs CIa;
~p~N = 0~5 ~ 106 I1a; ~pN_, ~ 20 � 10' Iia; ~p~_K =10 � 10~ ila;
IC = b; lCrr ~ 40; ac,~ = 0,97; v~ 0; ~p = 0,8; 0,65; At ~ 0~75;
aN 0,25~ g~~ R aK � 0,2.
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'C~b1~ 1~.2.
1 Z KOf~NRMlHTY ~ANMHNN
~ ~OiMy1qlNNM
no nop. ep~ ei;~* emr eK I e~y
~ aFkp -~,ooo --o,oia o .-o,oia ..-0,004
-1,000 -0,018 0 -0,018 -0,004
3 O,178 0,003 -}-1,066 -1,083 -0,2t2
4 8u1 0,198 0,003 -}�I~OfiB -1,083 �-0,212
6 8Pf ~O,178 0~003 -1-~~0~ -1~063 -0,212
8 8D -O,Ob3 -0~001 -0,3Z1 ~{-0,320 -0,084
7. BFKp�r O,100 0,118 -h0,069 0,041 -0,008
8 BP,n -I-0,30T 0~32b -~-0,20b 0,120 -0,024
9 rr ~}�0,100 O,IlB -I-0,08~ 0,041 --0,008
~p d ~ 0,016 0~017 O,OII 0,006 --0,001
11 8u~~ ~0,016 ~0,01y ~0,011 -}~0,006 --4,OOl
12 r -I-0,016 0,017 0,01 I-f-0,006 -4,001
-O~OOb -0~005 -0~063 -0~002 0
14 8Ho --0~341 -0,363 --0,227 -0,136 -0,027
IB 8~;, -{-1~023 -1-1,084 -}-O,GBb -f-0,409 -0~082
ig a~r 1,023 -~-1 ~084 -}~0,68b -I-0,409 -0,082
I7 8 -~-0,030 -I-0~032 -}~0,020 -f-0,012 -0,002
-0~003 --0,003 -0,002 --0,001 0
-0~012 -0,012 -0,008 -0,004 ~--O,OOI
Zp 8 .
~ 0 -}-0,002 -E0,001 -f-0~001 0
21 8 l.k. r -0,010 -}-O,OOI -0,062 -I-0,062 -0,012
-~-0,002 0 . -~-0,012 --0,012 -0,002
~ 8 t-tr -O~t4b -O~Ibb -0,097 -O,ObB -0,012
~4 bp,~ -}-0,974 -}-1~041 -I-0,65 -}-0,390 -0,078
Key:
1. DiaC~trbances 2. Coefficienta of influence
The reaction of a GRD with liquid component pumped aupply to the effect of
various diaturbances differs from that which occurred in a motor with a
gas pressurization sysCem. For example, change in nozzle Chroat area in a
GRD with a pump system producea practically the same combuation chamber
pressure change, while in the preceding example the corresponding inN
fluence factor was only 0.41. In a motor with a pump supply syeCem com-
ponent consumption raCes change little, while they increased quite ap-
preciably for a motor with a gas pressurization syatem. These differencea
are connected with the fact that combustion chamber preasure changes in
the motor arrangements we have examined differently influence the liquid
component consumption rate in the first of the arrangements being com-
pared this influence was significantly greater. For this same reason one
also observes a difference in the reaction of motors to identical changes
in a number of other factors, such as the dimensions of the solid propellant
charge in the thrust chamber and coefficient ul for this charge.
For GRD with a pumped supply system the influence of tank pressure change
on mQtor parameters is much less than fpr a motor with a gas pressuriza-
tion sysCem, since in this case this pressure is of itaelf small and plays
a minor role in producing the overall pressure which determines the flow
rate of tY~e liquid component.
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Chang~g in den~iCies of th~ propell~nC compon~ntg ~xerti qualitatively
Id~nCicgl infli~enc~g on the opergCing pgrameCer~ of motord nf boCh tiype~~
~lel~nugh qu~nCiCativ~ly ehe co~ffieienes o~ in�luence differ.
~nfluenc~~ which ~re ch~race~rigCic nnly of GItD wieh g l~quid component
pumped gupply ~y~Cem include deviaCione in charactprigeics gnd dimendiong
of rurbin~, pump, ggg generator and hydraulic line which feedg liquid com-
pon~nt Co Che gen~rator (numberg 7-16~ 19 in Table 12.2). MoCor paz~gmetere
~re mosr strongly influenced by deviaeions in pump and Curbin~ e�ficiency.
Changes in ~Eficiency produce changes ~.n ouCput expended on p~mping liqutd
component, which produces a change in raCe of f1ow, and as a consequence
algo in all oeher motor operation parameeera.
One'~ ~Ceention i~ drgwn by Ch~ relaeively little influence on motor
parameters by chgngea in the dim~nsions of the gas generator goLia com-
ponent charge. This feature of the motor under digcussion ie cottnected
with the fact thaC consumption of solid componenC in the gag generator com-
prises a very amall pare of total gas conaumed on the turbine
Krr a .'~.~'1~0'
+~trr
Among deviaCions in coefficienes of hydraulic resistances, the greatest
influence on motor operation parameters is exerted by deviation in Che
coefficiene of resiatance of the line through which liquid compon~nt
is fed to the gas generator. An increase in this registance leada to a
decrease in flow of gas into the turbine, a decrease in Curbine output and~
as a consequence, decreased propellant componenC flow and consump*.ion rate
as well as combustion chamber pressure.
Deviations in flow and consumption rate~ of liquid and solid components
depend on the density of the liquid component unequally. Therefore in the
given GRD arrangement change in temperature of the liquid component may
lcad to burnup of the propellant components at different times.
NonsimulCaneous burnup of the propellanC componenta can also occur in the
remaining cases, where under the influence of any disturbancea there occur
appreciable deviations in the coefficient of ratio of consumption rates
with constant charge masa. As is evident from Table 12.2, such values of
~SK occur under the effecC of deviations dul, dpT, d~H, dnT, etc.
The existence of substantial deviations in the coefficient of consumption
~aCe rat~ios is also undesirable becauae it causes a reduction in specific
thrust impulse.
12.2. Tuning and Adjustment of a Hybrid Rocket Motor
General statement of the problem of tuning and adjusting a GRD does not
differ from its statement as applied to liquid or solid-propellanC rocket
236
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m~tnrg. Conerol d~vic~~ Cgn b~ utiliz~d eo correet ~om~ d~viaeion~ ~f
mnCdr opr.r~eing condieiong fr~m Ch~ r~C~d figuree. Bue gv~n for control-
lnblc~ mht~r~ tt i~ oxtremely de~irable to reduce the vartanc~ in pgrameter~
wl~tr.h wnuld hr.cur wiehout control. Thie ig connected with the fact th~t
intrdductinn df g coneroi ~y~Cem compl3cgt~~ ~ moeor, waii th~ f~ct
eh~e control canno~ ~iimin~te a11 harmful d~vi~eion~ of moeor ch~r~ct~ti-
i~~i~g ~nd i~ ~agier to ~ff~cC wiCh 1~~~ v~rianc~ in ch~ract~ri~eic~. In
nunnecCion wiCh ehis it i~ alw~ys d~~ir~bl~ C~ r~duc~ eo ~ minimum v~ri~n~~
in princip~l moeor p~r~m~C~rg, wh3ch ig ~1~o the principal ob~~cCive of
moeor tuning ~nd ad~ugtment. nepending on eh~ function, de~ign feaCur~~
~nd eondition~ of ~mploym~nC of a GRD~ Cuning and gd~uetm~nt can have Che
purpoge of minimi~ing the epregd of variou~ parameeer~. They includ~
firgt ~nd foremost th~ co~fficiene of ratio o� propellant componenC con-
~umpeion raCeg. n~vi~Cidn~ in thi~ param~t~r noC nnly r~ducp specific
ehrust impulse buC glso 1~ad, gs wa~ indicaC~d ~bov~~ to nnn~imulCan~ou~
exhaustinn nf ehe prop~llanC componente~ that ie~ ~~e~ntially lead Co g
decr~gqe in the quantiCy of propellant which c~n be prnducCively utilized.
Cnmbustion chamber pressure is ~ aecond parem~ter a decre~~ed gpr~ad of
which can be achieved by tuning and adjusCmenC. If tuning eleo minimizeg
variancee in the coefficient of propellanC con~umption rate ratio and cam-
bustion chgmber preagure, this is also achieved by decreasing varience in
thrust, which is simultaneously dependent on combustion chamber pre~eure
(propellant consumption) and gpecific impulse (ratio of propellanC com-
ponents).
~or a GRb with a pumped liquid component gupply syatem and a two-compon~nt
gas generator, on analogy with a ZhRb, it is highly desirable to reduce the
variance in the coefficienC of propellant component consumption rate ratio
in the gas generator. Deviations in this coefficient from the raCed value
not only change turbine output but can also have an adverse effect on
operating conditions of the gas generator proper, and particularly the
turbine blad~s.
~p6
/
.
~'~M�i
~igure 12.4. Relations for GRD Tuning and Adjustment
beviations in characteristics of various motor components can be selected '
as influences with the aid of which tuning and adjustment is performed,
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d~p~nding on tl~~ motior d~~ign. Ie i~ imporCant thaC eh~e~ d~viation~ can
h~ nd~uut~d fnirly ~a~ily and pr~ci~~ly during tuni~g and ehaC ~he ceaf-
f Iriri~tr+ ~~I' thelr in~luc!ncu en ehe tdrgat motor psrameter.~ po~~a~N rhr~
r~yutHltc mugni~ud~.
A~y~t~m of GRD ~t~Ci~ ~quationg of ~h~ typ~ pr~eented abov~ 3g ~dopted
rh~ bagi~ of tuning calcul~eion~. The ~y~t~m is ~olv~d r~lativ~ to
qu~ntiti~~ ~dop~~d ~g ~h~ influ~nce~ ~d~ugted in ehe tuning proc~~~. Thpn
on~ introduc~g ineo thi~ ~olution the vaiu~e of tho~~ di~turbanceg which
~hould b~ compen~ated for during tuning, ~nd then ot~~ fi~ureg the requisite
valu~~ of th~ controlled influenceg.
L~e ug con~ider ig ~n ~xampl~ tuning and adguetmen~ of a GRD w~th a~a~
pr~~~urix~eion liquid compon~nC ~upply uy~tem~ a diagram of which i~ con-
egin~d in ~igur~ 12.1. W~ ~h~11 ~dapt g~ tuning and ad~u~tm~nt tagkg
~limingtion of deviatinn~ of combuetion chacaber p~eesur~ end co~fficient
of prnpellant aompon~nt r~tio, that is, we sha11 etate that ae g zegulC
of tuning end adjugtment we ~hould achievc~ the following: aPk� aK~O.
5ince motor tuning and ad~uatm~nt ghould be perfarmed an two parameteYS,
it i~ ~lso necegeary Co have Cwo influencing characterietics. Let us
gdopt a~ such ci~argcteristica deviation in tank preesure and deviationin
hydraulic registance on the line through which the liquid component is
fed to the combustion chamber head~ that is, deviationa dp~ and rf'N-~.�
As ig evident from Table 12.2, preciaely these influences have relatively
larg~ coefficientg of in�luenc~ on principal motor parametera, and at the
same time they can be easily achieved from an engineering atandpoint.
Assuming in system (12.1) dpk=0; dKsO and considering dp~ and d~~~.
unknown, we obtain a system of 19 equations with an equal number of un-
knowns. In order to solve the gystem, the valuee of those deviations of
cxternal and internal factors which must be compensatpd for as a result
of tuning and adjusCmenC ghould be prespecified. Just as when applied to
: GhRD, the problem can be solved either with utilization of statiatical
dar~ e~r With employmenC of the results of teat~ ott individual motor com-
ponents. For GRD which are designed to be employed under different tem-
perature conditiona~ tuning and adjusement can be performed in two stages.
Deviations in design parametera and characteristics of aolid propellant
charge '~BFKp~ gul~ a~N_,~ , etc) are established at the plant where
tt~e motor is manufactured. These quantities are put into the syatem of
~quations and, solving this system, one obtains
apd ~ fod taP~) = fo6 cT ~ i 2.3~
a e~N-~ ~ fe ~ f~ (T,~).
238
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Thaa~ ral~eion~ d~t~rmin~ eh~ value~ of eh~ peremet~r~ nf euning and ad-
ju~tm~ne in eh~ funcCion of liqui,d compon~ne t~mp~raCure w3th eho~e
devi~eion~ of oth~r di~turbances wh3ch occur i.n the given motor (~igure
12.4). tf eh'~ compon~nC Cemp~raCur~ prior to tnotor igni~ion ig known~
thhn det~rmintng from (12.3) the requieite valuee for tank pressure Cr~ducer
~d~ur~rment) ~nd hydraulic loe~es (ad~u~tmene of cont~o11ab1~ hydraulic
rt~t~+eunCc), one can Ai~o reduce to ~ minimum d~viation~ in th~ coef-
fiei~nt of camponent con~umpeion r8te ratio and comb~~eion chamber preseure
snd, together with th~~e, ae indicaCed above, motor thru~t a~ well. -
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,
BIBLIOCiRAPHY
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240 T
FOR OFFICIl~L USE ONLY
~
g
i~
. . ~ . . . . . . . . . . . . _.a . . x a_ G.
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COPYRIGHT: Izdatel'stvo "Mashinostroyeniye", 1978
3f?24
CSO: 8144/1234 - END -
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