cqy9273 - UNT Digital Library/67531/metadc...techlibrarykafb,nm b nacarme57c11...
42
265 57+2 ‘% c Opy RM E57C11 ------ - .=--- ._. _ “-”’-c~_’@iis ---” “-=..., +d=m+=, +., .. -.. —-~%. -J ~— ~-.. -— — RESEARCHMEMORANDUM “-”:= THERMAL RELATIONS FOR TWO -PHASE EXPANSION WITH PHASE EQUILIBRIUM AND EXAMPLE FOR COMBUSTION PRODUCTS OF BORON -CONTAINING FUEL By LeonardK.Tower LewisFlightPropulsion Laboratory Cleveland. Ohio/~,./b.2 “} ;>,.a,.~.: ..... ...”.””=”-’ /.. ~ >. j ~~&~:+”:7 .”..”.”’’..” ,..L- ,. “f ,00 ...., .● *.,*.!..-*”* ..................’”’”’”’ ...-. ---- ,., ....- : :: “.., NATtONALADVISORY COMMITTEE FOR AERONAUTICS WASHINGTON May 20,1957 . , . ... ,“ r HADCADJ’573848 .s -—. .
Transcript of cqy9273 - UNT Digital Library/67531/metadc...techlibrarykafb,nm b nacarme57c11...
26557+2 ‘% cOpy
RM E57C11
------- .=--- ._. _“-”’-c~_’@iis---”“-=..., +d=m+=,+., ..
-..—-~%. - J~— ~-.. -—
—
RESEARCHMEMORANDUM“-”:=THERMAL RELATIONS FOR TWO -PHASE EXPANSION WITH PHASE
EQUILIBRIUM AND EXAMPLE FOR COMBUSTION PRODUCTS
OF BORON -CONTAINING FUEL
By LeonardK. Tower
LewisFlightPropulsionLaboratoryCleveland.Ohio/~,./b.2 “}
;>,.a,.~.:........”.””=”-’/.. ~
>. j~~&~:+”:7.”..”.”’’..”,..L-,. “f,00....,.●*.,*.!..-*”*..................’”’”’”’...-. ----
,., ....- ::: “..,
NATtONALADVISORY COMMITTEEFOR AERONAUTICS
WASHINGTONMay 20,1957
.
,
. ...,“
r
HADCADJ’573848.s
-—. .
TECHLIBRARYKAFB,NM
b
NACARM E57C11
NATIONALADVISORYCOMMITI’EEFOR
RESEARCHMMORANDUM
IlllllllullllllllllllllllcI143733
AERONAUTICS
THERMALRELATIONSFORTWO-PHASEEXPANSIONWITHPHASEEQUIL18R3TJMAND
EXAMPLEFORCOMBUSTIONPROIXJCTS@’BORON-CONTAIMNGFUEL
By LeonardK. Tower
SUMMARY
Jetenginefuelscontainingboronformboricoxide(B203]whenburned.Thissubstancecanexistbothas a liquidanda vaporathightemperatures.Ifa portionof theB20
tcondensesduringexpansion,the
performanceoftheenginecanbe affecedappreciably.Thephysicallypossibleeffectof condensationonperformanceisoftenbracketedbyassumingthatexpansionoccurseitherwithno changeincompositionorwithequilibriumcondensation.
Onemethodof calculatingtheisentropicexpansionprocessisbymeansof equationsrelatingpressure,temperature,anddensity.Commonlyusedequationsfromengineeringthermodynamicsrelatingthesevariablesarenotsuitedto theanalysisofan isentropicprocessinvolvingequi-libriumcondensation.Thisreportpresentsequationsrelatingtempera-ture,pressure,and.densityforan isentropicprocesswithequilibriumcondensation.
Theimportanceofusingequationswhichproperlydescribethisproc-essisemphasizedby a calculationdiscussedhereinoftheisentropicvariationofpressurewithtemperatureinvolvingcombustionproductsofboron.At thechosencondition,thecoefficientfortheisentropicvari- ‘ationofpressurewithtemperaturewas5.8withnophasechangeassumedand12.9withcondensationassumedtooccur.
A procedureisgivenforintegratingtheequationsinvolvingpres-sure,temperature,anddensityinorderto analyzean expansionprocess.An exampleintheformof a problemconcerningtheuseofethyldecaboraneina ramJetengineservesto illustratethisprocedure.
INTRODUCTION
Thelargeheatingvaluesofboron-containingfuelsofferthepossi-bilityof improvementintheflightrangeofair-breathingjetengines.Thedegreeof improvementoverhydrocarbonfuelsdependsinpartbothonthethermalproperties_o~.thecomb~stion~oductsandonthenatureof
- ~dl.
2 NACARM E57C11
theexpansionprocess.Theboron-containingfuelsdifferfromconven-tionalfueisbecausetheyproduceboricoxide(B@3),whichcanexistsimultaneouslyas a vaporanda liquid.If condensationoftheoxideoccursduringtheexpansionprocess,an appreciableeffectuponthrustmaybe encounteredundercertainconditions.
Thedegreetowhichcondensationoccursinactualexpansionprocessesdependsupontherateof condensationrelativetothetimeofexpansion.Intheabsenceofinformationoncmdensationratesitiscustomaryto —
brackettheregionofprobableperformance-withcalculationsthatassume geitherinfinitelyfastor infinitelyslowcondensationrates.Theseare @designatedrespectivelyas expansionwithequilibriumcondensationandexpansionwithno condensation.
Expansionswithno condensationarereadilyhandledandunderstoodby meansofelementarythermodynamicconcepts.Expansionswithequilib-riumcondensationaremoredifficulttohandle.Itis importantthatperformancecalculationsinvolvingtheassumptionofphaseequilibriumbe baseduponcorrectequationsandprocedures.
Thepurposeofthisreportisto definecertainpropertiesforanexpansionprocessinvolvingcombustionproductswithequilibriumconden-sation.Thesepropertiesare
[11 thevariationofpressurewithdensity
at constantentropy@P/bp)sj2 thevsriationofpressurewithtempera-tureat constantentropy(W/M)sj (3)thevsriationofstreamvelocityat constantentropy(dtJ)s.A numericalintegrationof thesetermspro-videsthepressure,temperature,density}velocity}andlocalMachnumberat anypointduringtheexpanstin.A proce,@xreforintegratingthesetermsisgiven,togetherwithanexampleoftheanalysisofa specificexpansioncondition.
w
4
Theequationsandprocedurehereinomittheeffectofdissociationof gases.Dissociativereactionsgenerallyassumeimportanceonlyabove40000R atpressuresencounteredinhigh-speedflight.Forpracticallyallconditionswhereboron-carbon-hydrogenfuelsmightbeburnedinaLr-breathingjetengines,saturationofthegasesintheexhaustnozzlewillnotoccuruntila temperaturelessthan4000°R occurs.
Previousreportspresentothermethodswherebytheexpansionprocesswithphaseequilibriumcanbe analyzed.Reference1 containsa generalprocedureby whichthenozzleoutletvelocitycanbe obtainedforspeci-fiednozzleinletconditionsandexpansionratio.Isentropicexpansioncanbe consideredforsystemswithcompositionseitherfrozenor inphaseandchemicalequilibrium.Inaddition,reference1 presentsa procedure ,fordeterminin~dissociation.
combustiontemperatureswhichi.nclu&stheeffe& ofK —
●
NACARM E57C11 3
.
Becauseoftheeffortrequiredinapplyingtheproceduresofrefer-ence1 to specificperformancecalculations,theauthorsofreference2developeda simplifiedmethodforfuelscontainingboron,csrbon,andhydrogenburnedinair. Thethermalpropertiesforcombustionproductsof certainstoichiometricallyburnedfuelsareaddedto obtainpropertiesforfuelscontainingboron,carbon,andhydrogeninanyproportionandatanyequivalenceratio.Thepropertiesareintabularformandalsointheformofpressure-enthalpycharts.
Thechartsofreference2 includetheeffectofbothvaporizationandphasechange.Inregionswheretwophasesarepresent,thechartscanproperlybe usedforthespecificfuelsandequivalenceratiosforwhichtheywerecalculated.Thetablesofreference2 neglectdissocia-tionofgases,buttheycanbe usedto includetheeffectofphasechange.It iscorrectto addpropertiestabulatedforcombustionproductsofspecificfuelsto obtainthepropertiesforanyboron-carbon-hydrogenfuelwhenequilibriumphasechangeoccurs.However,thedeterminationof conditionsatmanypointsintheexpansionprocessinvolvesnumeroustrialanderrorcalculations.Furthermore,the~ch numbercannotbedeterminedina mannerconsistentwiththeassumedprocessof equilibriumcondensationfromdatacontainedinthetables.
Theequationsandprocedureinthepresentreportprovideametkdalternativetothoseofreferences1 and2 forcalculatingtheexpansionprocesswithequilibriumcondensationofB203. Inanalyseswkre manypointsduringtheexpansionaredestied$t~s methodmaYPZ’OVemorecon-venientthantheothers.Moreover,theequationspresentedhereingreatlyaidinvisualizingthe
Sincethepurposeusefulinanalyzingan
effectofcondensationontheexpansionprocess.
DISCUSSION
ofthisreportistoisentropicexpansion
equilibrium,thesepropertiestillf-tistbe-.ationofpressurewithdensityisdefinedas
where
definethermalpropertiesprocesswithtwophasesinlhted. Theisentropicvari-
{1]
(2)
4
and
NACARM E57C11
()=+V‘fr’ ‘Vfr
SymbolsaredefinedinappendixA. Thedefinitionsof ~ andthelastexpressionaredescribedmoreexplicitIy inappendixB. Theisentropicvariationofpressurewithtemperatureis PN
where
Thedifferentialexpansioncanbe
-+
)~ %2 -‘fr+ 1—.- Xv RT
6 =xv A&J
1+~— -~RT
formofthegeneralenergyequationforisentropicexpressedas
(UdU)s= - ‘f dT = - &
Thespeedof soundis
C’mandMachnumberismerely
M= u/c
(4)
(5)
u
d(lnP) (6} -
(7)
(8)
Derivationsofequations(2),(5),and(6}arecollectedinappendixC.
Equations(2)and(5)for y and s,respectively,arerigorouswherecondensationistheonlychangeincomposition.H thecompositionisassumedtobe frozenduringexpansionso_thatnophasechangeoccurs,thosetermsinequations(2)and(5)involvingtheheatofvaporization~ arezero. Equations(2)and(5)become’thefamiliarexpressions
.-
r ‘ Y-fr= (=ph-+fr (9) ‘
.
NAOARME57Cll
& ‘%= (Z#-Jfrfromwhich
_ rf~efr rfr- 1
5
(lo)
(11)
isderived.
Whenvaluesof T and e computedforequilibriumphasechangearecomparedto Tfi and efr forfrozencomposition,themarkedeffectofcondensationontheflowprocessisemphasized.As an extremeexample,considerthecombustionproductsofbaon atan equivalenceratioof 0.9,a pressureof 1 atmosphere,anda temperatureof 3960°R (nottheflametemperature).Assumethatthecombustionproductsarenotdissociatedatthistemperaturebutthatphaseequilibriumexists.Theratioofgaseousto liquidB203isthen2.102at thiscondition.Ill?e~ansionisassumedto occurwithno condensation(frozencomposition),I_frand e=fromequations(9)and(10)sre1.2083and5.802,respectively.If equi-S libriumcondensationisassumedduringexpansion,T and e fromequa-tions(2)and(5)are1.1234and12.939,respectively.
.Valuesof y- and & calculatedfromequations(2)and(5)include
theeffectof condensation,whilethepresenceof dissociatedgasesisignored.By meansofthemethodpresentedinreference1,bothphasechangeandrecombinationof dissociatedgasescanbe evaluated,althoughgreatereffortis involvedthanintheuseofequations(2)and(5). Fortheexampleunderconsideration,T is1.1214and e is.13.091ifbothphasechangeandrecombinationof dissociatedproductsareincluded.Thesevsluesagreereasonablywellwiththevaluesneglectingdissociationcalculatedlyuseofequations(2)and(5). Forthecombustionproductsofmostboron-containingfuels,saturatedmixturesofB#3 vaportowhichequations(2)and(5)areapplicablecanexistonlyunderconditionswheredissociationissmall.
By meansofequations(2),(5),and(6),expansionprocessesinvolv-ingequilibriumphasechangecannowbe analyzedwiththeassumptionthatgaseousproductsarenotdissociated.Forsmallintervalsof theexpan-sionprocess,pressureandtemperaturedonotchangegreatly.Both T
I and e canthenbe treatedas constantsintheinterval.Equations(1],(4),and(6)canthenbe integratedforthissmallintervalto givetheseequations:
Pz/Py= (PJPYV- (12]
NACARM E57C11
J
pz/py= (@j)e (13)s
U;- U;=~(~-Tz) (14) —
Thesecomrmtationsarerepeatedforthenum%erof intervalsintowhichtheexpa&ionisdivided,-with
An exsnmleis includedto
yandc ‘recalculated
EXAMPU3
illustratethemannerin
foreachintervai”;a
whichtheequa-tionsdevelopedhereinareusedtoanalyzea one-dimensionalexpansionwithphaseequilibrium.Theexampleshows”theeffectwhichtheassump-tionofequilibriumphasechangehasontheexpansionprocess.
Considerthefollowingproblem:a raqjetengineburningethyldeca-boranefueloperatesat 60,000feet,a flightMachnumberof4.0,a~ anequivalenceratioof 0.6. Inletanddiffuserpressurerecoveryis0.443,andcombustionefficiencyisl.CO. CombustorinletMachnumberis0.175.FlameholderWag isneglected.Theexhaust-nozzleoutlettemperatureandvelocityaredesiredforisentropicexpansionto ambientpressurewith .equilibriumcondensation.Alsodesiredarenozzlecontoursrequiredfor(1)constantchangeofvelocitywithdi.stancealongthenozzleand(2)constantchangeofMachnumiberwithdistancealongthenozzle.
.“
ChartsandprocedurediscussedinappendixD wereusedto determineconditionsatthecombustoroutlet.Thecombustiontemperatureandpres-surewerefoundtobe 4400°R and4.524atmospheres,respectively(1atm= 2116lb/sqft). Dissociationof gaseswasneglectedinthiscalculation.Sinceno condensedB203waspresentat thecombustoroutlet,theconditioninthenozzleatwhichcondensationbecamepossiblewasfoundbya procedureinappendixD. Forconvenienceindiscussion,thisconditionwherecondensationcantheoreticallybeginisreferredto asnozzlestation1. Subsequentstationsforthestepwisecalculationinthecondensedregionarereferredto as 2,3,4, andsoforth.Eightstepswereusedto calculatethatportionof thee~ansionprocesstiterthetheoreticalsaturationpoint.ThenozzleinletwasdesignatedasstationC, thenozzleoutletas stationD. Nozzlestation9 thuscorre-spondsto thenozzleoutlet,stationIl. —
Valuesfor P, T,U, C)M) T) ~d e forstation1 on-d ‘e pre-sentedintableI. ThedetailedcalculationsshownintableII *e dis-
c
cussedmorefullyinappendixD. By followingtheheadingsoftable11thereadercanmakesimil=calculationsforotherconditionsandother -fuelscontainingboron,cabon,and@drogen.
NACARME57Cll
ObserveintableI thatthesameconditionof static
7
twostations,1 andl(a),sreshownto havetemperatureandpressure.Thetemperature
at station1 wasassumedtobe i~initesimal~abovethetemper~turewherecondensationstartsinthisparticulsrcase. Thatat stationl(a)waaassunedtobe infinitesimallybelowthecondensationcondition.Sharpdiscontinuitiesin y and & resultbecauseofthesuddenappearanceoftermscontaining+ inequations(2)and(5)withtheonsetof conden-sation.ThestreampropertiesP,T, and U arethesameat stations1andl(a). However,thesonicvelocityandMachnumberjumpbecauseofthediscontinuityin T withtheonsetof condensation.
Betweenstations1 andl(a)thereisa 1~-percentincreasein cforequilibriumcondensationoverthevalueforno condensationat thesametemperatures.Thereisan 8-percentdecreasein T forthesamecircumstances.
FromthedataoftableI werecomputedthenozzlecontoursrequired
[1for 1 constantchangeofvelocitywithdistancealongthenozzleaxisand 2 constantchangeofMachnuniberwithdistance.Thenozzlewasassumedtobe of circularcrosssectionand2 feetlong. Thenozzleradiusrequiredtopass1 poundofmixturepersecondis shownplottedagainstaxialdistanceforbothcasesinfigure1. Alsoshowninfigure1 isthenozzlecontourforconstantincreaseofvelocitywithdistanceifno condensationwereto occur.
Forthecaseof constantMachnumberincreasea procedureinappendixD avoidsa discontinuityinthenozzlecontourbetweenstations1 andl(a).Sucha discontinuitywouldoth-ise resultfromthesuddenincreaseinT causedbytheonsetof condensation.Betweenthenozzleinlet(stationC) andstation1 thevalueof cfr usedincomputingstreamvelocityfromequation(14)isassumedtobe 5.15,while T& isassumedtobe1.24inequation(7)forthespeedof sound.TheflowareasandMachnumbersbetweenstationC andthesaturationconditioncalculatedusingthesevaluesof yfr and &fr assumeno dissociationofgases.Incor-poratingtheeffectofdissociationbetweenthesestationschangestheflowsreasandMachnumbersveryslightlyinthisparticularexample.
LewisFlight~opulsionLaboratoryNationalAdvisoryCommitteeforAeronautics
Cleveland,Ohio,hkrch15,1957
8 NACARM E57C11
APPENDIXA
SYM80LS
specificnozzlearea,sqft/(lb/see)A
a
c;
constantinexpressionformolarentropy
constant-pressurespecificheatof substance,Btu/(lbmole)(%)
localspeedof sound,ft/secc
con&ant-pressurespecificheatof combustionproductsat constantcomposition,Btu/(lb)(OR)
Cp
constant-volumespecificheatof combustionproductsat constantcomposition,Btu/{lb}(OR)
gravitationalconstant,32.17ft/sec2
“
totalenthalpyof combustion-productconstituent(sumof chemicalenergyandsensibleenthalpy),Btu/lbmole
heatofvaporization,Btu/lbmole.
totalenthalpy(sumof chemicalenergyandsensibleenthalpy),Btu/lb
—
heatof combustionoffuel,Btu/llof%
J
2
M
conversionfactor,778.16ft-lb/Btu
distancealongnozzleaxis,ft
Machnumber,ratioof streamvelocityto localspeedof sound
molecularweight,lb/lbmolem
%
n
nf
meanmolecularweightofcombustionproducts,lb/lbmole
numberofmoles
numberofmolesof combustion-productconstituentresultingfromburningfuelstoichiometricdlyin1 poundof air
t
staticpressureof combustionproducts,atm(1atm= 2116lb/sqft) .P
NACARM E57C11
c
P
R
s
s
T
u
v
w
x
r
&
partialpressureof combustion-productconstituent,atm
gasconstant,1.98718
entropyof substance,
entropyofmixtureof
Btu/(lbmole)(OR)
Btu/(lbmole)(%)
materials,Btu/%
statictemperature,%
axialstresmvelocity,ft/sec
volumeofmixtureof combustionproducts,cuft
weight,lb
numberofmolesof constituentdividedby totalmolesof gasandvapor
coefficientforvaxiation@ pressurewithentropy
coefficientforvariationofpressurewithentropy
arithmeticaverageof c overtemperature
density,lb/cuft
densityatconstant
temperatureat constaht
interval
equivalenceratio,actualfuel-airratiodividedby stoichiometricfuel-airratio
Subscripts:
a air
B combustorinlet
c combustoroutletornozzleinlet
c condensedphase
D nozzleoutleta
f fuel
fr no changeincomposition(frozencomposition)
10 NACARM E57C11
J
v=1,2,3
noncondensinggasesonly,vaporof condensingmaterialexcluded
allgases,includingvaporofcondensingmaterial-
allmaterials,includingconde~edpk=
constantpressure
constantentropy
stoichiometric
constanttemperature
vaporphaseof condensingmaterial
adjacentstationsinflow
atomsofboron,hydrogen,andcarbon,respectively,infuelformulaBaHPC~
stationsinnozzleaftercondensationbegins
—
alllflm
NACARM E57Cll
APPENDIXB
XL
.
4
RELATIONSFORONE-DIMENSIONALISENTROPICEXPANSIONllIZCH
NO CONDENSATIONANDNO DISSOCIATION
Thisappendixdiscussesthedefinitionsof T and e fortheisen-tropic~ansion oftwo-phasesystemswithno condensation.Similarmaterialhasbeenpresentedelsewhere(ref.3),butthetopicisdis-cussedherebecausesomeofthetermsderivedarerequiredintheanaly-sisof equilibriumcondensationinappendixC.
Considera mixtureof idealgasescontainingfinelydividedcondensedmaterialtoolargeforBrownianmotion.TheidealgasIawappliedtothegasesinthemixtureis
Pv = RJTZni (Bl)
Thedensityofthemixtureincludingthecondensedphaseti
p =mlp@7 (B2] ‘
providedthevolumeofthecondensedphaseisnegligiblecomparedto thatofthegas. Substitutingequation(B2)inequation(Bl)gives
(B3)
Tk term ~nJ#@nf canbe regardedas ~, themeanmolecularweightofthemixtureof gasesandcondensedmaterial,as contrastedwiththemeanmolecularweightofthegasesalone,whichis definedas 2nim@nf.
Thederivative(aP/aT)sisfoundbysionforreversibleentropychange
Tds=dh-;
E compositionisfixed(nocondensation)betweenparticlesandgas,
consideringthegeneralexpres-
a’ (J34)
andthereisthermalequilibrium
Set ds equalto zero. Thencombineequationsobtain
(B@
(B3),(B4),and(B5)to
12 NACARM E57C11
●
(B6) “
Equation(B6)canbe expressedsomewhatdifferently.Letthemolefractionforallconstituentsincludingthecondensed@ase be definedasXk= nk/&i. Thenumberofmolesofcondensedmaterialisnotincludedinthesummationhi. Forthisreason,~Xk= 1 + Xc. Thisconvention,usedthroughouttheremainderofthisreyort,gives
(B7)
Thederivative@P/~p)s canbe foundby differentiatingtheloga-rithmofequation(B3)withrespectto densityat constantentropywith2nkmk constantfora givenweightofmixture:
$3S’:+$R9($9S+
Forfrozencomposition(B8)gives
(~Zni/ap)siszero.
az~()2: ~s (B8)
Combiningequations(B6)and
Thus,forfrozencomposition
..—
(B9) -
(B1O)
Thenumeratorof e~ression(B1O)is CP andthedenominatoris CV fora unitweightofthemixtureofgasesandcoiidensedmaterial.Thedefini-tionof
Yforthecaseoffrozencompositionisthusextendedby equa-
tion(B1O to includethesituationof smallparticlesinvelocityandtemperatureequilibriumwiththegas.
Equation(B1O)expressedintermsofmolefractionis
(Bll)
.
b
.
NAC!ARM E57Cll
4Equations(B7)
.and(Bll)canbe combinedto give
Yf~Efr ‘~‘fr
Onlyby specifyingfrozencompositionistheobtainedas givenby equation-(B12).
1?)
(B12]
relationbetween& and T
14
APPENDIXC
NACARM E57cll
.
RELATIONSFORONE-DIMENSIONALIXZNTROPICEXPANSIONWITH
PHASECHANGE~ ONECONSTITUENTANDNO DIS-&M31ATION
Inthisappendix,equations(2),(5],and(6),for T, e,and dU,respectively,axederived.Thesee~ressionspertainto two-phaseisen-tropicexpansionwithphaseequilibrium. +
— Nd
Derivationof e
It isconvenienttotofind y. Fora fixed
wherethesummationover
derivee first.Theresultcanthenbe usedmassofmaterialwithgasesbehavingideally,
s = Znk~ (cl)
k termsincludesthecondensedphaseanditsvaporaswellasthenoncondmsinggases.
Forthenoncondensinggasesandthevaporphaseofthecondensingmaterial,collectivelyrepresentedby thesubscripti,
.
(q i ~TJsi=T -Rlnpi+ai (C2) “—
Forthecondensedphasetheentropycanbe considerednearlyindependentofpressure.Hence(seeref.4),
(C3)
Inan isentropicprocess
ds=o =h#i~ +~~dnk (C4}
—
Representingboth Sk and nk asfunctionsofthetemperatureandpressureofthesystemyields
—
(C5) -
NACARM E57C11
and
amlllsJ‘ik=(a,m+(a)pd’
Combiningequations(C4),(C5),and (C6)gives
If equation(C7)libriumwiththe
15
%s(2),+%(2),“s=“- (C7]
istobe valid,theparticlesmustbe inthermalequi-gasand
Thederivativesareequations(C2)and(C3)
vapor.
nowevaluated.Firstfind ~nk(ask/~p)T.From
(9(-~($),and(&S@P)T = O. Thepartialpressureofof idealgasesis
Pi = Pnl/2ni
where
Xni=~+Zng
(C8)
a constituentina mixture
(C9)
(Clo}
Assumethatneitherng nor Z% changesforthenoncondensinggases.Differentiatethelogsrithmof equation(~) forthevaporphaseandcombinetheresultwithequation(C1O)to obtain
(CQ
However,thepartialpressureofa saturatedvaporbehavingas an idealgasisa functionof temperatureonly. TherelationisknownastheClausius-Clape~onequation(seeref.4). Inreference4 itisshowntkt (apv/aqT+O ifthemolarvolumeof theliquidbecomesnegligiblecomparedto thatof itsvapor.Equation[CU.)thengives
16 NACARM E57C11
●
()
a% %3r T=- ~
(C12).
where & = nv~~i. FcIrthenoncondensinggases,differentiationofequation-(C9)&d substitutionof
(~)&~p 1=-+pgp Tp
Finally,since(~v~P)T is
(~)
a‘nk p ~
equation(C12)yield
‘~’ = & (C13)
~“%zero, .—
‘-:*
Substitutingfor Zng fromequation(C1O)resultsin
(C14)
me term ~sk(hk/@T ofeq-tion(C7)i6co~iderednext. ~molenumbersof onlythecondensedphaseanditsvaporarqassumedtochange.Therelationbetweenmolesof liquidandmolesofvaporIsd~=- *.. Then
(C15)
But
~- Sc=AH+/T (C16)
Equation(C16),togetherwithequation(C12),whensubstitutedinequa-tion(C17jgitis
whichcanbe written
.
.
F@ -XJ
—
●
(C17).
NACARM E57C11 17
Theterm Z~(&@T)P of equation(C7}isnowconsidered.Differenti-atethelogarithmof equation(C9)forthevaporphasewithrespecttotemperature:
Thisyieldsfinally
.
Fora liquidsurfaceequationis
inequilibriumwithits
% %-%—=—dT RT2
vapor,theClausius-Clapeyron
(cm)
ifthemolarvolumeoftheliquidisnegligiblecomparedtothatof itsvapor(seeref.4). Thisexpressionisapplicabletovaporsbehavingasidealgases.3Ttheradiioftheliquidsurfacessre.nottoosmall,thevalueof AHv canbe takenasthatof a flatliquidsurface.Forin-stance,thesurfaceenergyof a dropas smallas 2.2x10-6centimeterisnegligible(ref.5). Theratioofvaporpressuresofthesmalldropandtheflatliquidisa functionof surfaceenergyofthesm.11drop(ref.4). Thereforeparticlesas smallas 2.2X10-6centimeterhavenearlythesamevaporpressureastheflatsurface.Substitutingequation(C20]inequation(C19)resultsin
()?lnv nv{AHv/RT}~p=T(l-~)
Becauseonly ~ and ~ change,
(C21)
(C22)
Substitutionof equations(C16)and(C21)inequation(C22)gives
me term ~tIk(a~aT)p Of e~Ui3tiOn(C7) C$3nequations(C2)and(C3),
-
(C23)
nowbe found.From
18 NACARM E57C11
and
foranyconstituent.From
-&)@PgTp
()asc?iT p = ~c9c/T 1equation(C9)
= -&&#-’)p= -#%)p
.
.
(C24}
fornoncondensinggases.!l?hise~ression,togetherwithequation(C18)andequations(C24)}uponsummationyields. —— .-.
()
asZnkmp=
Substitutingequations(C14),(C17),yields
~n@~)k/T (C25)
(C23),and(C25)Inequation(C7).
Let Xk = q/Zni foranyconstituentincludingthecondensedparticles;then
In appendixBandcondensed
it isshownthatifthecompositionofa mixtureor gasesparticlesisfrozen(nochangeincomposition)
w-
amm
●
✎
NACAM E57C11
Rewriteequation(C26)as
.
.
xv
()ap T — (~/RT)2‘fr+l. XvE‘%~~= 1+
+ (QRT)1-(5)
Observethat *r inequation(5)mustbe evaluatedby insertionofthosevaluesof the Xkts requiredforphaseequilibriumat thetemperatureandpressuxeinquestion.
Equation(5)wasderivedontheassumptionthatthemixturewassaturatedsothatB#3 condensed.H themixtureis superheatedor ifexpansionisfrozen,noB203condenses,andtheheatofvaporizationisnotrecovered.Equation(10)intheD193USSIONresults.
Derivationof y
Considerequation(lM),thedifferentialformof theideal.gaslaw:
($)S’:+$%)S+H%)S
Fromequation(C1O)writethelastderivativeinequation(B4)as
Since ~ = ~(T,P),
(*):(a4a@)s
(B8)
(C27)
(C28)
NOW,(aP/aT~shas been evaluated in equations (C26) and (5). It c~nbe
expressed as (aP/aT)s= &p/T andtheotherderivativesofequation(C27)aregivenbyequations(C12)and(C21).Withthesesubstitutionsequa-tion(C27)becomes
()
&niTS=
iv(AI@T} ]()w-xJ-m+&$ (C29)s
20
Also>
NACARM E57Cll
.
.
—
MBHmb
6$s = ($33($)s
T(Y)‘z ps
(C30]
Substitutingequations(C29)and(C30)intoequation(B8)andsolving +for(bP/bp)sgives — _kFm
()aP P.TS= xv
{
(C31)
p T-
Eliminatinge by means
()$s=
Finally
or
ofequation(5)givqs
=]
++%%
Thespeedof soundina regionofphasechangeisthen
(C32)
.
.
(C33)
(C33a)
(C34)
.
.
NACARM E57C11 21
coi%d+
b
Equation(C33)wasderivedby assumingthattheB203presentcon-denses.. Ifthemixtureissuperheatedor ifexpansionisfrozen,theterms ~/(1 - Xv)and ~/RT(l - ~) inequation(C31)bothvanish,sincetheyresultonlyfromconsideringcondensation.Equation(9)intheDL%USSIONthenreplacesequation(C33).
DifferentialFormofGeneralEnergyEquation
Foranadiabaticprocessthegeneralener~ equationis
(U2/%J)+ (h/Znk~)= Constant (C35)
Thiscanbe differentiated
Foran isentropicprocess,
to give
IHU+w=o@
equations(Bl)and
()
RT2ni!??$s=7
(C36)
(B4)give
(C37)
Combiningequations(C36)and(C37)gives.
Butif ~ isdefinedas Z~m@ni, equation(C38)becomes
~ d(lnP)(UdU)s = - ~
Anotherformof thisexpressionresultsfromwithequation(4):
(UdU)s = - ~ dT
(C38)
(C39)
combiningequation(C39)
(~~o)
.
22 NACARME57C11
APPENDIXD
CALCULATIONOFTHE
Thecalculationoftheexampleconsistsofthreeparts.First,thecombustoroutletconditionsaredetermined.Second,theconditionsinthenozzleatthetheoreticalsaturationpointarefeud. Third,theconditionsthroughtheremainderofthenozzlewithequilibriumcondensa-tionaredeterminedby a seriesof stepwisecalculationsbasedupontheequationsdevelopedherein.
.
—
.
CombustorOutletConditions
Thecompositionoftheburnedmixture,neglectingdissociation,isfoundasfollow. Thechemicalformulaofa fuelormfiure offuelscontainingboron,carbon,andhydrogencan%erepresentedasBa~CX,where a, j3,and X areintegersor decimals.Theatomfractionsarethen: forboron,a/(a+ ~ -t-k)jforhydrogen,~/(u-I-~ I-h}jforcarbon,A/(a-l-p+x). Theseatomicratiosandthetriangularchsrtoffigure2areusedto findthemoles~@3 ofB203,n~~ ofH202andn&2 ofCOZ
formedbyburningthefuelstoichiometricallyincompositionforallequivalenceratiosles8than
@~205 moles B203/lbair
1 poundofair. The .stokhiometricisthen. .—
@~20 molesH20/lbair
~n~o molesC02/l?Jair2
0.00726(1-@)moles02/lbair
0.02740molesN2/lbati
Thestoichiometricfuel-airratiousedincomputing4 canbefoundbymeansoftheatomfractionandfigure3.
Forethyldecaborane(B10C2H18),thefuelconsideredintheexample,theatomratiosare0.333forboron,0.6~ forhydrogen,and0.067forcarbon.Fromfigure2 thestoichiometricmolenumbersarefoundtobe:U$203,O*oo~9jn&20,0*O~66jand% 0.00103.Fromfigure3 the2’ :stoichiometricfuel-airratioisfoundtobe 0.0778.
.-.
.
NACARM E57CIll 23
Ifthecombustiontemperatureislessthanthesaturationtempera-tureforB203vaPor,bothcondensedandgaseousB#3 willbe Present=Thecriterionforthepresenceofthecondensedphaseis .s8follows:
where
Thevaporpressureis
+ %2Cl+ %02) + 0-00726(~
plottedagainsttemperature
- @) + 0.02740
infigure4. If
(Dl)
(D2)
in-equality(Dl)indicatesno condensedmaterial,theamunt ofB@3 vaporis merely
Iftheinequalityindicatescondensedmaterial,theamwnt ofB#3 vaporis
. where
5i2knv=p-pv
andtheamountof condensed
Theamount of noncondensing
(DEJ
B203
‘c“=
ismerely
gasesandB203vaporis
Thetotalenthalpyoftheburnedmixturefora fuelcontainingboron,carbon,andhydrogenisthen
()
Wfl+~hc’ ‘c@)B203,c+ %@%)B203,v‘@n~20@?)H20‘@%02@)C02 +
0.00726(1-@)(H~)02+ 0.02740(F$)N2Btu/lbair (D8).
NACARM E57C1124
Totalelude
enthalpiesof thechemicalenergy.
fortheothermaterials
.constituentsareplottedinfigure5. Fey in-DataforB203are~romreference6,whiledata
.
arefromreference1. --
Conservationofenergyacrossthecombtistorisexpressedas .—
(II9)b
Enthalpyof airis giveninfigure6. Etstfmlpiesof somefuelsof inter- .8esttakenfromreference2 areasfollows:““
BoronDiboraneEthyldecaboraneHydrocarbonfuel
Pentaborane
Formula
BB2E6B~oc2H18CxHa
B5H9
Phase: Tempera-Iture,
OR
Crystal--------Gas - 536.7Liquid 536.7Liquid
Liquid.1 536.7
AssignedI
:
enthalpy,%>Btu/lb
28,84336,57536,26320,000
.33,828\
Enthalpiesofotherfuelscontainingboron,cerbon,andhydrogeninknown .proportionsandhavingexperimentallydetewinedor empiricallycalculatedheatsof combustioncanbe foundfromtheequation .
~=Ahc+ ;n[ 1(n’~)B203+(n’~)H20+(nt~)C02- 0.~0726(~)02
~ St
Forheatsof combustion determined at 298.16°K (536.7°R),(1$)02is24.642Btu/lbmole,(@)coz iS 4.026Btu/lbmole,(~)B203(c~stal)is91.605Btu/lbmole,and(@H2@apor) is24.642Btu/lbmole. --
CombustionpressurePc andvelocityUc requiredtoevaluateex-pressions(Dl),(D3),and(D9]arefound%y”-rneansofthemomentumequation
(D1O)
NACARM E57C11 25
andthecontinuityequation
i![+W4$B“da. (Dll)
wherethestatic-pressurelossdueto flameholderandcombuetorwallfrictionisrepresentedby theratio PyPB. ‘I!hemeanmolecularweight~ ofthecombustionproductsat stationC canbe expressedas
%, c = 69.64(XC+ ~) + 28.016XH20+ 44.01~02 + 32%2 + 28.016XN2
(D12)
Also,(R/ma)Bis0.06888forthetircompositionassumedherein.
Thedeterminationofthecombustiontemperaturerequiresthatequa-tions(IN),(D1O),and(Dll)he satisfied.Furthermore,if inequality(Dl)showsthepresenceof condensedB203,theproperamountsof ~and ~ determinedfromequations@4], (D5),and(D6)mustbe usedinequation(D9).Thesolutionfora singlecombustiontemperaturemayrequireseveraliterations.
FortheexamplebeingconsideredthecombustorinlettemperatureTB wasfoundtobe 1590°R, whiletheinletpressurewas4.747atmos-pheres.By iterationofequations(D9),(D1O),and(Dll)theflametem-peratureTc wasfoundtobe 4400°R neglectingdissociation.VelocityUc was980feetpersecond,andpressurePC was4.524atmospheres.Sinceinequality(Dl)revealedthatno condensedB203waspresent,thesaturationconditionforB.#3maybe reachedintheexpansionprocess.
ConditionsinNozzleat Saturation
Thepressureandtemperatureatthesaturationpointcanbe foundfromconditionsatthecombustoroutlet.ForthecombustiontemperatureTc andtheparameter4~203 pfI/2nkjtheratio P1/Pc isred frOUIfigure7. T& ratio T1/Tc canbe calculatedfromequation(13)using
c of 5.15. Thesubscript1 refersto theconditionwheresatura-~on isreached.Fortheexampleherein@~203 Pc/Znkis0.199.Thisvalue,togetherwiththecombustiontemperatureof44COoR, determinesaP1/Pc of0.637anda T1/Tc of 0.916.ThUS PI and T1 ~e 2.8=
atmospheresand4032°R, respectively.
~ theinequality(Dl)showsthepresenceof condensedB@3 atthecombustoroutletconditions,figure7 cannotbe used. Thederivationoftheequationfromwhichfigure7 isconstructedappearsat theendofthisaypendti.
26 NACARM E57C11
ExpansionwithPhaseEquilibrium.
TheequationsdevelopedinappendixesB andC ai@usedto calculate_______conditionsthroughtheregionofphasecha~e inthenozzle.Thedetailedcalculationprocedureis illustratedintableII. Thereader,by follow-ingtheheadingsoftable11,canmakesimilarcalculationsforisentropicexpansionwithphasechange.AlternateproceduresareprovidedintableIIforthosecaseswherethemixtureissuperheatedandthosecaseswherethemixtureissaturated.A testisprovidedfordeterminingwhichcase *“ispresentatanycondition. -~.— .—. m
At theprecise-pointwherethecondens.&tioncancommence,thecom-putationcanbe madeeitherway. Thetempe~atureand-pressureinrows1andl(a)oftable11for”theexamplearesuchthatthgtestforcondensedB203showsthistobe theconditionwheresaturationisreached.Thisis
—
a consequenceofhavingusedfigure7 to determinethesaturationpointfortheexample.Row1 wasworkedassuming~thatthetestforcondensedmaterialshowedthemixturetobe veryslightlysuperheated.Rowl(a),forthesametemperatureandpressureasrow1,wasW5rkedwiththeassumptionthat~hemixture&s verysl@htlybelowthepointtion. Column15oftable11 canbe expressedas
of satura-.—
+- --
(D13) -
The variation of the constant-pressure specific heats_.C:tureforthemroductsisgiveninfimre 8. Inspectionof
withtempera-column15 in
tableII show;thatsumma~ion(D13)=ouldhaveb~enapproximatedby a—
valueof about5 overtheenti~erangeof conditions~~ thetable.-VaJ-ue6of ~/RT forB203,column16,areshowninfiwe 9 as a functionof
—
temperature.
CalculationofNozzleI!ontours
Considerthecaseof constantchangeofvelocitywithMachnumber.Thedistancefromthenozzleinletto anystationrIfora nozzle2 feetinlengthis i —.
(Uv- L@h=& .
Forthecaseof constantchangeofMachnumberwithlengththerea discontinuityinMachnumberat thepointof condensation.Thisiscircumventedby calculatingthedistance% asfollows
.
is .—
NACARM E57C11 27.
DerivationofEquationUsedto ConstructFigure7
If inequality(Dl)showsno condensedB203atthecombustoroutlet,thepartialpressureofB203inthenozzlecanbe representedat anypointdowntothesaturationconditionby
At thesaturationpoint(station1)thepartialpressureisequaltothepartialpressureofthevaporinequilibriumwiththecondensedphase.Thelattercanbe representedapproximatelyoverquitea rangeof temper-
s ature by89 14.87- 68,300/Tl~ e
~ Equatingthesee~ressionsforthesaturation.
pointgives
‘%203pl 14.87- 68,300/Tl.
% ‘e(D14)
AssumethatcompositiondoesnotchangebetweenstationC andstation1.It isthenpossibleto definea mean Ffr forthisintervalsuchthat
Tc/Tl= (p@l) l/FroEquation(D14)canthenbe rearrangedas
*%203PC PC 14.87-
%.~e
A constantvaluefor ~fr of5.15waschosenin constructingfigure7.
.
28 NACARM E57C11
1.
2.
3.
4.
5.
6.
.
REFERENCES
Huff,VearlN.,Gordon,Sanford,andMmrell,VirginiaE.: General.
MethodandThermodynamicTablesforComputation~fEquilibriumCom-—
positionandTemperatureofChemicalRSactions.:-NACARep.1037, .1951. (SupersedesNACATNts2113and””2161.)
Hall,EldonW.,andWeber,RichardJ.: TablesandChartsforThermo-dynamicCalculationsInvolvingAiran@FuelsContainingEoron,Carbon,Hydrogen,and@gen. NACARM E56B27j1956. ~
a!Maxwell,W.R.,Dickinson,W.,andCaldin,E.F.: Adiabatic~Fansionof a GasStreamContainingSolidPsrt:cles.AircraftEng.,vol.XVIII,no.212,Oct.1946,pp.350-=1.,
Glasstone,Samuel:ThermodynamicsforChemists.D.VanNostrandCo.,Inc.,1947.
Setze,PaulC.: A StudyofLiquidBoroK’OxideparticleGro~h Ratesina GasStreamfora SimulatedJetEngineCombu%tor.NACARME55120a,1957.
Eetze,PaulC.: A ReviewofthePhysicalandThermiqmamicofBoricOxide.NACARM E57B14,1957;””
l?roperties .
.
NAC!ARM E57C11.
.
28
.
w
TABLEI. - PROPERTIESOFEXPANDINGCOMBUSTIONPROIKJCTSAT POINTS
WITHINREGIONCll?BORICOXIDE(B@3)CONDENSATION
Tozzle3ta-tion,
l-l
1l(a)234
5k78L9
staticstaticpres- temper-sure,ature,P, T,ah ‘?R
2.885 40322.885 40321.817 38511.144 3670.720 3486
.454 3293
.286 3084
.180 2858
.113 2622
.07122396
Veloc-Speedity, ofu, Soundj
f’t/secc,ft/sec
2705 29102705 27873634 27094325 26364880 2568
5341 25075738 2#56090 23786409 22876689 .2191
Machnumber,
M
0.9296.9706
1.3411.6411.900
2.1302.3472.5652.8023.053
Coefficientofpressure-densityvariation,
T
1.2531.1491.1511.1561.166
1.1831.2071.2301.2441.250
Coefficientofpressure-te.mperaturevariation,
e
4.94710.0489.5848.9788.090
7.0456.0715.4085.1154.993
%ozzleoutlet,stationD.
-.
Ttie 1t2-ip p>
atm
2 3
T, ~
[11
mki(l)n
~)n-1~
%
+
1 e.affil(a) ;.8&23 1:1444 .m
}.127 —- ----—---- --—-- -——--- [email protected] 0.04ss 0.127 2.754 0.cm155 0 o;03%?23%?7--.--— —— --.m9 1.759
.OC.9 0,Ooul .KW4 ------.—— ---
.0245.C61fl .012.s
1.1195 .CC074 .Cca61 .W ----.— ——--- .0215 .0235.crm25 .no75 .mo44 .Wul .0341.6 ---_— —----- .o129 .0326
.00289 .Milll .Co3z2 .Om.s5 .m394 ----— ------- .0385
.WJ7m.0392
.22SM .CcoM3.CDW115
.cm147 .03s50 -----— -—---.179aa .ccao22
.CQ25 .oa5.001$5 .03374 -.--—- —-----
.LwX1124.0L%M52 .04Fz
.U299 .omcXm7 .oo1540 .CE53T.?4—-—- ------- JW&o .0454.CnXWOl .onlsB .woom40 .W15425 .m~ -.--— -—---- .0460
16 17 I 18 19 19 20 I 21 I w 22Ir(s):(4) 1P (3)<(4) If (3]z (4) Ir (S)<(4)
l-~> (9)(16)2 1- (9)T (W+iim) (:)!ig.9) 1 + (la) #qt4T&r A
4032 0.1Q74032 .127s831 .Ow96m .054342.9 .0s16
3293 .0199SO04 .Ola2858 .w-mmz? .03S32398 .2931
5 .4546 .226
.124: .11s9 .c1712
T=xle 11ta-Ip X%o,
~
, I
1
I0.0704
l(a) .07942 .Om?, .Osll
;:: ,::,~
.0627
.06i?8: .m2E9 .o&?$
.0176O.fmz?
.o176 .06%2,0177 ,0234,Olm .0243,0121 .Oe.50
jll “:%
.OIES .0s61Olm .0661.01= .0e61
,7788 4.947,7768 4.947‘7657 5.Ow79sl 8.077,Eo21 5.121I & ,,,
.-” ‘S.13Tmos 5.MSelm 5.101,W.2s 5.054,ma 4.09a
c7-.7.6W.s.9.4%
!0.%!2.60km?7.lo!9.95H--——----0.9S1 0.7668.mm .Slm.9766 .407’s.9en .esm.!,,,
.s255 ~.lml
.2975 .0666
.994S6 .0160
.9990!3.00298
.9s992 .LW350
—----10.040U.5849,97’98.090
4.947 -—-1.W591.03301.0220l.olso
r.ti1.0J251.QM6l.ml1.0
-- —--0.17s6.1645.Mm.ls54
t
., ..:“ .Imt.1740.1872.1261.Km3
---— 1.2331,1421.1511.1561.16&. ..!.;.lmg
1:2s01.2441.280
---—--—---—
------—-----—-“—
I.wi8.OnB.4025.1154.99s
132
g
0.926.9.97
1.341l.eu1.KQ
24 -
&26
m,Q)n.1-(%
0.02936.meo2.Olwo.OMS.oom84
----— ---—-9.Q609.2618.54
---ml181124
-J2.1s2.3412.5S52.-S.om
.msalB
.msw?
.m2275
.Oolsm
.Oolm’a
7.527s0.sssB.739S5.261SS.uu
193
E%
U?8
qw”,,,
, , 4 .4248 , ,
.20
G
jA
~ .18 r
~ IConstantchange of Mach number
$ ,16//
with distant e; equilibrium
3
/ /condeneat ion
r\ / /
A
I \
; .1, -
8 /
d / /Conetant change of velocity /
~ ,12 with diztance; equili.b- , /rim condensation
$ / ‘
$.10
\
/
gf
‘Com3tant change of/ / velocity vith dietmc e;
g / no CQndeU8ati.CiU
j .08 / ‘/
k
: / ‘ / < ~
s/- / -
z+ -. 7 d — -
.060 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0
Distance along nozzle, ft.
Figure L - Nozzle contour for conztant chmge of velocity with distance anfi for conetant changeof Mach number with distance. Remjet engine at 60,000 feet and flight kch number of 4. O;ethyldecaborene fuel; equivalent e rat lo, 0.6.
..
1,
.50
1.
Atm fmctluaof bmcm in fuel
Fly’eey*ti -c-, ~dMde,d~f camEUbYst@b@ri Ctlmnlugofftwl.
wN
8W?.7 ‘ ‘
. I
.s0
4248 , ,
1
~ fraction of bca.m in fiml “--... ..- ---
Figure 3. - Etoichimetric flla.1-a~ratio far fuel CIJ~tii* brRm, Cabq @ ~og,~.
-air ratio,Chio!mtric
,..
* ,,,11
m4. - Vm$iatlm of bcmlo dam (Fl#JJ vapor prom- Wltb tow+’ratam . (Data - l’ef. 6.)
I I
8 . 87ZV ‘), I 1, ,,
,
NACARME57C11
.
.
8
.
.
35
Temperature, %
(a) Boric oxide.
Figure5. - Vsriatkm of total enthalpy per mole with temperature for COMbUBtlO?kpKdUOtccmatituents.
u-m
w0)
‘ii’
im 1400 18C0 ZMXl Moo SW m m 42~ 4EO0 5cm 54(XI 5ElcaTeqmratmre, %
(b) !&ter, cerbon dioxide, oxygen, ad nitrogen.
F@m 5. - concluded.Vhiationof total entklpy F rnle tith tmatnrera co&mstio&pmWct constltuemk.
&,8?W ‘ ‘
2
4 , , . 4248 ,
Temperature, %7
me 6. - Variation of total enthalpy of ati with temperature.
38 NACARM E57C11
.
.
.’60
.031----3400 36CCI 3800 4QO0 4200 .... 4403 4600 4600
TemperatureJ ‘R=W .-
.—
Figure7. - Chertfor determining cotiditioneat saturation temperaturefrom known combuetor outlet condition.
.
NACARM EqCll
.
3’-’J
,
w
.
.
39
(a) Boric oxide.
Fi~e 8. - Variation of molar con6tant-praasure epeolfia heat with temperature for combu8tlon-produetcon.9tituents.
-..Tempsmtme, %
(b)Cnrbm tiide, water,WW=?J,d nitIw@h
Fig.me8. - CowJu4e3. VariatiOn~ * ~ nstant-KcaB6ma SWCiflc h-t with t~atura for c.ahution-~
Constitualte .
.“, ,,1
a%’ % ‘
Temperature, %
Flgum 9. - %riatim of AE@ with tapemb.zm f= boric oxide,
* . . ● a
