zowJd

IN:

Yang,

J.C.,et

al.,Editors,

SolidPropellant

Gas

Generators:

Proceedingsof

the1995

Workshop,

NISTIR

5766,June

28-29,1995,

32-44Pp,

1995

A.

L

32

I 1 .

ACKNOWLEDGEMENTS

l I would like to express my deep appreciation for various solid propellant researchprojects supported by many U.S. Government organizations (such as ARO, OhJR,AFOS~ NWC, BRL, AFPL) as well as many other research projects supported byvarious industrial companies (such as TC, Aerojet, Olin, RRC, Wyle Lab, ARC).Through these projects and many related works, I have gained useful knowledge and

mu experience in solid-propellant combustion and ignition processes.l I would and like to thank Professor Martin Summetileld, series editor-in-chief, for the

opportunity for me to serve as co-editor of Fundamentals of Solid-PropellantCombustion, printed as Vol. 90 of AI&% Progress Series, November 1984.

l I would also like to thank many of my co-workers and graduate students at ThePennsylvania State University. Through their collaboration and dedication manyinteresting and important research results were obtained.

l Thanks to Dr. John C. Yang and Dr. William L. Grosshandler of NEST for theirinvitation for my participation in this workshop.

Pennsylvania State University

COMMENTS ON WORKSHOP TOPIC

L

1

e It is very exciting to see that solid propellants are bring considered for gas generatorapplication in fire extinguishment.

e Great Engineering Challenge! !

w-h

l-ennsylvania State University

1

GENERAL BACKGROUND OF SOLID PROPELLANTS

(1) SOLID STATE SUBSTANCES WHICH CONTAIN BOTH OXIDIZERS AND FUEL

INGREDIENTs

(2) ABLE TO BURN IN ABSENCE OF AMBIENT AIR OR OXIDIZERS

(3) NORMALLY USED TO GENERATE HIGH-TEMPERATURE COMBUSTION PRODUCTS FOR

PROPULSION PURPOSES

(4) CLASSIFIED INTO TWO DIFFERENT TYPES (HOMOGENEOUS AND HETEROGENEOUS)

BASED ON DIFFERENCES IN THEIR PHYSICAL STRUCTURE

35

CLASSES OF PROPELLANTS

* Homogeneous

Uniformphysical structure.Fuel and oxidizer are chemically bondedtogether.Major constituents are nitrocellulose (NC)and nkroglycerine (NG).Also referred to as double-base propellants.

o Heterogeneous

Non-untiormphysical structure.Polymeric fuel binder and crystallineoxidizers.Also referred to as composite propellants,

36

.FUNDAMENTALSOFSOLID-PROPELLANTCOMBUSTION

Edited byKenneth K. KUOThe Pennsylvania State UniversityUniversity Park. Pennsylvania

Martin SummerfieldPrinceton Combustion Research Laboratories. inc.Monmouth Junction, New Jersey

Volume 90Progress inAstronautics and Aeronautics

Martin Sumrnerfield, Series Editor-in-ChiefPrinceton Combustion Research Laboratories. inc-Monmouih Junction, New Jersey

Publkhed by the American Institute of Aeronautics and AswonauIti. Ix.1633 Broadway, New York. N.Y. 10019

37

i-

TaWe of Contents

Quakers.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .~j

Prehee...

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xtts

.Ackrswlcdgmettts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix

Chapter 1. Survey of Rocket Prouel!antsand Their Combust . .ion C~ . . . . . . . . . . . . . . . . . . . . . . . . . . 1

bv IV.Kubo(aPerformance of a Solid-Propellant Rocket Motor. . . . . . . . . . . . . . . . . . . . . .4Stable Combustion ofa Rocket Motor . . . . . . . . . . . . . . . . . . . . . . . . . . ...12Temperature Sensitivity of Burning Rate. . . . . . . . . . . . . . . . . . . . . . . . . ...13Thermochemical Properties of Propellant ingredients . . . . . . . . . . . . . . ...16Thermochernieal Properties of Propellant CombustionRoducts . . . . . . ...20Combustion Processes of Various Types of Sotid Propdkmts.. . . . . . . . ...26

Double-Base Propellants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Ammonium Perchlorate Composite Pro@laMs . . . . . . . . . . . . .. . ...28Composite Modified Double-Base Propehms.... . . . . . . . . . . . . . ...29Nitrarnine Composite Propellants.. . . . . . . . . . . . . . . . . . . - . . . . ...-32

Burning Rate Controlling Factors of Solid Prqxibt s.. . . . . . . . . . . . . ...33Combustion Characteristics Required for Sotkl propellants and -

eral Descri@ion of the CornbuQon wave. . . . . . . . . . . . . . . . . . . ..33Heat-Transfer M.t%hartisn?in Combustion Waves . . . . . . . . . . . . . . ...36Heat Transfer in Scdid Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...38Heat Transfer in Gas Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..~Reaction Ratein Gas Phase . . . . . . . . .. - . . . . . . . . . . -- . . . . . . ...--43Burning Rate of Solid Propellant Calculated by a Simplified

Gas-Pitase Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..-....47

38

Chapter 2. Chemistm of ~ tion and CombwstA d Asm=ium-Perchlorate-Based PropellanU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

bv K.Kishore ond V.Goyathri-.

Chcmistwoflgnition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...55Sequenceofignition . . . . . . . . . . ...--....-............. . ..55TheoriesofIgnition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...57RoleofOxidizerand Bier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . .58EffectofPressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..@EffectofOxidifingAt-~e. . . . . . . . . . . . . . . . . . . . . . . . . . . . ..61lgnitionofCompositePr~ll~tFueis byHC104 Vapor . . . . . . . . ...63Preignition Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... ...~EffectofCatalystson lgnilion.. . . . . . . . . . . . . ..1 . . . . . . . . . . . . ...66

ChcmistryofCombustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...68lntrodttctiontoCotnbustionMechanism . . . . . . . . . . . . . . . . . . . . . ...68SurfaceReactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Subsurface R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...75&s+haseReaetions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...94EffectofCatalysts on Propellant Combustion . . . . . . . . . . . . . . . . . ...98

FutureResearch Dir@!ions . . . . . . . . . . . . . . . - . . . . . . . . . . . . . . . . . . ...106

Chapter3. TheThermalBehav iorofCyclotrimethylenetnnitmmine(RDX)and CyclotetmmethYlenete!mrntramine (HEX) . . . . . . . . . ..121

by T. L. BoggsCrystallography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...123SublimationofHMX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126DecompositionoftheSolidRDXand HEX . . . . . . . . . . . . . . . . . . . . . . . . 127Liquefaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...142DceompositionofLiquid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...144Pyrolysis of HMX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150ShoekTubeStudies . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . ..157IgnitionofHMX . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . - . . . . . .. . . 160Self-Deflagrationof HMXandRDX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

Self-Deftagration RateasaFunetion ofPressure . . . . . . . . . . . . . . ...161SeM-Def?agratiost RateasaFunction ofPressureand initial

Sam#eTemperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164SurfaceStructurcofSe!f-Deflagrating HEX....................165

Chapter4. ChemistryefN~efiterand Nitramine PropeWnts.. ..l77bvR.A.Fifer

DecompositionofNfiroceiiti&.. . . . .1 . . . . . . . . . . . . . . . . . . . . . . . ..-181Kinetics ofNitrocellui~ bm~sition. . . . . . . . . . . . . . . . . . . . ...182Productsand hfccitanistn ofNkrocellulose Decomposition. . . . . . ...184Plasticizers andStabilizcrsin Nitroccllulose Decomposition . . . . . ...189

CatalysisofNitratcEsterPropcUants . . . . . . . . . . . . . . . . . . . . . . . . . . ...191DeeompositionofNitramines... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...202

KinctiesofHMXand RDXMomPosition . . . . . . . . . . . . . . . . . . ...203ProduetsandMechanism ofHMXandRDX Pyrolysis . . . . . . . . . ...207CatalysisofNitraminePropellants . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

FlameZoneChemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..- . . . ..219.

39

..

Chapter 5. Solid-Propellant Ignition Theories and Experiments . . ...239by C. E. Hermuncc

lntrodttetion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...241Radiam Energy Ignition %mrces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246Shock Tubeaad O&erlgnitionE..periments . . . . . . . . . . . . . . . . . ...259

Theoretical Modeis of Sdid-Propelkmt Ignition and Related Theory . . ...266Comm-entsoutiudonM etImds. . . . . . . . . . . . . . . . . . . . . . . . . . ...275Sdi&Phase Rrnion Mcchanisrn Theory . . . . . . . . . . . . . . . . . . ..w. .278Theory of Midort by Heterogeneous Reaction with External Oxidizer 281Gas--Phase Thcpry of Solid-Propellant Ignition . . . . . . . . . . . . . . . ...285

Gas-Phase Theory-Shock Tube Cas~. . . . . . . . . . . . . . . . . . ...286Gas-Phase Theory-Radiant Heat Input. . . . . . . . . . . . . . . . . ...289

Chapter 6.. Flame Spreading ttnd OveraIl Ignition Transient . . . . . . . .305bvM. K umur ond K. K. Kuo

Ignition and Thrust Tra&ients in Solid-Propellant Rocket Motors . . . . ...309Ignition Devices or Inters . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .39Physical Processes &minglgnitionTrartsient . . . . . . . . . . . . . . . . . ...312I@ition Tra*t M~hand Experiments . . . . . . . . . . . . . . . . . . . ..3l5

Flame SpreadhtgwcrWProPellantsa ndFuels . .. . . . . . . . . . . . . . ...327Flame-Spread& Mechanisms . . . . . . . . . . . ...-................327FIatne&.wca&gtika ndExperiments . . . . . . . . . . . . . . . . . ...330

Flame Sgweadkg imto S&d-Propellant Cracks and Flaws . . . . . . . . . . . . . . 339Physical Proecs=s Atrimg Flame Spreading and Combustion in I%-

pelkttlt Defects . . . . . . . . . . . . . . . ..-. .....-.....-.-.......s~Theories and Ex@mmt son Flame Spreading in Propellant Cracks..341

40

.Chapter 7. Steady-$by G. Lengcllk, ~. Bizot, J. Dutcrque, andJ. F. Trubert--

General Behavior and Flame Structure of Homogeneous Propellants . . ...367 ?reSent5 use+~(

371 in fbrwi&t/On on eBurning Rate Laws: lnftucnccof Additives. . . . . . . . . . . . . . . . . . . . . . . . . .Detailed Study of the Combustion Mechanisms in the Con&nscd Phase . . . 375

# obe s?hcture andDegradation of the Propellant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..W . ,Condensed-Phase Degradation Gases . . . . . . . . . . . . . . . . . . . . . . . . . 384 Cgmbu.stfofi Medtqtii;+Condensed-Phase Behavior of the Additive . . . . . . . . . . . . . . . . . . . . .387

Detailed Study of the Combustion Mechanisms in the Gas phase . . . . . . . . 387 G? ~0~ ogeeeousPrimary and Secondary Ftamc Rcgimcs. . . . . . . . . . . . . . . . . . . . . ...389 p ropei/4R fs, ,+1e AV@.lPrimary Flame Structure with/without Additives . . . . . . . . . . . . . ...392 4

Mechanisms of Super-Rate and Plateau or MesaEffects . . . . . . . . . . . . . . . 398 0 t ~u~ PP ~4~?s aJW/

Chapter 8. Steady-State Burnin@JComposite PropellarMs /%& w MPSQ 6)@-------- -.. .under Zero Cross-Flow SMsatlon . . . . . . . . . . . . . . . . . . . . . . . . . . . .-409

l

by K. N. R. RamohalliVarious CombustionModels of Composite Solid Propellants . . . . . . . ...415

The Guirao-Williams Model for AP Combustion . . . . . . . . . . . . . . ...420jOeS ~eta;led

The Granular Diffusion Flame Model. . . . . . . . . . . . . . . . . . . . . . ...427 treat~ent # w?.; uThe Hcrmancc Model for Propellant Combustion . . . . . . . . . . . . . . . . 434 ~ode/~ de ve(vp~dThe Bcckstcad-Dcrr-Price Model or the Modci of Multipie Flames . ..443

-& hebv9enQ0*2Petite Ensemble Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...451 .The Statistical Flame Description . . . . . . . . . . . . . . . . . . . . . . ...458Summary of the PEM Equation (Applied to Each

pm~e IIahts

PseudopropeIlant) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465Future Dcvelopmcnts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471

Chapter 9. Combustion of Metalizcd Pr~p&t: . . . . . . . . . . . . . . . . 4nby E.W.Price

Metals as Fuel Ingredientsin Propellants . . . . . . . . . . . . . . . . . . . . . . . . . . 479 * Metal and Oxide Properties

n. ex~eRS:Ve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 ~eSG1-lptJUhS ad

Results of Controlled Eiperimcnts with Aluminum . . . . . . . . . . . . . . . . . . . 484Propellant Combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49(5 int-pddiowEffect of Aluminum on Propc!lant Burning Rate . . . . . . . . . . . . . . . . . . . . SOI OF tie combusti~Combustion Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503Future Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 10.

505 p~enomena 0fErosive BumiRg of Sd

hetd~edpropd~%id Propellants . . . . . . . . . . . . . . . . 515

byM. K. R&an and K. K. KuoThcorctical Approaches to Erosive Burning . . . . . . . . . . . . . . . . . . . . . . ..S21

Cbssifscationof Erosive=lngThcorim . . . . . . . . . . . . . . . . . . ...521 Cotiers lJartouSModels Based on Phaomcnologicsd Heat-Transfer Theories . ...522 %eoret;e~[ ~ud~ /!sModels Based on Integral Boundary-Layer Analysis. . . . . . . . ...523Models Based on a Modification of the Propellant

Combustion Mechanism . . . .,26 ;;ok ;;;::~~

. . . . . . . . . . . . . . . . . . . . . . . . . . .

Models Based on a Chemically Reacting Boundary-Lay=Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...527 Cate qorie S based

Other Modc}s . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...-........528Recent Thcorctical App$oachcs . . . . . . . . . . . . . . . . . . . . . . . . . . . ...529 upok tie hetioiSummary of Theoretical Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541 @ a~prt7tkcJ .

Expcrimcntai Workon Erosive Burning . . . . . . . . . . . . . . . . . . . . . . . . ...541Preseh erus:wcExpcrimcntal Methods . . . . . . . . ..- . .--.......-..--......-.--541 .

Rcccnz Expcrimcntal work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...559 $Urtttn ~tia,rl~= !Summary of Expcrimcntal Work..... . . . . . . . ...-.............563 9

Important Results of Erosive Burning Studies. . . . . . . . . . . . . . . . . . . . . ..575 dtenOkefia, G~C/Generally Observed Effects of Various Parameters . . . . . . . . . . . . ...575 Physical Mechanism of Erosive Burning Phenomena. . -...........588 ~a$; C .9ec4an;Sqj

Rccommcndation for Future work.... . . . . . . . . . . . . ...-.-.........590.

41.

:.,-

Chapter 11. Transient Burning of Solid Propellants . . . . . . . . . . . ...599by K. K. Kuo, J. P. Gore, and M. Sumrnerfield

Mechanism of Transient Burning... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602Mathematical Description of the Transient Burning Phmomma. . . . . . ...606Description of the Existing Modcis . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....614

dp/dt Modcis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...622Flame Description Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...624Zeldovich Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .631UnsteadyStateGas-PhaseModels . . . . . . . . . . . . . . . . . . . . . . . . . . ..635

ExperimentalWork inTransient Combustion ofSolidPro@lants . . . . ...635Experimcnta} ApparatusUsedby Various Researchers . . . . . . . . . ...636ResultsofParamctricStudics... . . . . . . . . . . . . . . . . . . . . . . . . . . . ..648

Future Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...651

chap$m12. Extinetio=l%eories andExp&me rots . . . . . . . . . . . . . . . 641byL.DeLuco

Technical f3ackgrotmd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...666LiteratureSurveyon Dynamic Extinction . . . . . . . . . . . . . . . . . . . . . . . . . . 669

Theoretical Results on Dyrtamic Extinction by Fas!Depressurization .670Experimental Results on Dynamic Extinction by Fast

Dcpressurization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...--.674Dynamic Extinction by Fast Deradiation . . . . . . . . . . ..- . . . . . . . . .. WCI.

Extinction by Other Qucn.ching Tmhniques. . . . . . . . . . . . . . . . . . . . . . ...685Injection ofl%mel nhibicors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .68sHeat Sink

. . . . . . ..- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6s7MiscclIaneoos Qumching Techniques . . . . . . . . . . . . . . . . . . . . . . ...688

hlathematical Formuia&mof lheProblem. . . . . . . . . . . . . . . . . . . . . . . ..6s9Nonlinear Burning .%bility of Solid Propellants. . . . . . . . . . . . . . . . . . . . . 702

Nonlincw Sta&BurningS tabili[y. . . . . . . . . . . . . . . . . . . . . . . . . . ..7o7Nonlimea-z Dynamic Burning Stability.. . . . . . . . . . . . . . . . . . . . . . ..7Is

Numerical anci Experimmal Validation . . . . . . . . . . . . . . . . . . . . . . . . . ..7I9Coochrsionsm dFutureWork... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...725.

42

,.

(-Chapter 13. Expcrimcntal Observations of Combustion instability. ..733

b-vE. JV.Price ~i5-SSt?s expert-General Featur= of Instability

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Gains andlosscs . . . . . . . . . . . . ~j~ ie~~all o~~fz)-v~c(:. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Pr~ Contribttting to Stability . . . . . . . . . . . . . . . . . . . . ....... J741 Comb& S 1*VR iltI~dil{?~Combustion Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742Measuring Com&t&Rmponse . . . . . . . . . . . . . . . . . . . . . . . . . . . . 749 1 n nxket ~o&s.Particulate Damping

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Other Gains and Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ;? COUt?f% tec~n;~~esRocket Motor Instabiliijes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...759

Bulk Mode Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...759 for h Mf?qSt4Wmefi~Transverse Modclnstability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765 of comb Qdie)bAxiai Mode Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...767Fluid DynamicaUy Excited Oscillations . . . . . . .

. . . . . . . . . . . . . . ...774 ~e$~onse $fTxfIbasEffect of Propellant Characteristics on Combustion instabilky. . . . . . . . . . 775 ~df;OQS ~~~s ofSummary and R~~ti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...778

M stabi I;tfw, 8#4~tChapter 14. Thcorctimi AdyQs efco91ms$kmhs2aM@y....... . 791 of p~*$el!Qnt

by J. S. ?ien Chwa cte~;stics onLinear Analysis of Wave Motion... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .79S

Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795 tit& stm ;J&J;if~Acoustic Energy ina Sound Field . . . . . . . . . . . . . . . . . . . . . . . . . . ...7%Acoustic Admittance Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798Linear Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...800

Acoustic Amplification: Propellant Response Function . . . . . . . . . . . . ...805Estimate of Time Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...805 dds N;*Qua.sisteady Gas-Phase and Unsteady SoIid-Phase Models . . . . . . . . . 808 &O@:Caj QndslsNonsteady Gas-Phase Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816 7Velocity Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819 @ combustfo~Others . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . s20 ~fi~ab)l~t

Acoustic Damping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...821Nozzte Response 821 Jnc/kd; Y. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Particle Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...823 71

i%eqrothers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824 -+;* ~~