the theory of flammability limits - conductive convective wall losses
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8/9/2019 The Theory of Flammability Limits - Conductive Convective Wall Losses
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469
yLimits
allLosses
MENTOFTHEINTERIOR
y
gDirector
Publ
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atalogedasfollows:
ylimits.Conductive-convective
enching.
BureauofMines; 8469)
8469-
.Title.II.Series:United States.Bu-
vestigations;8469.
628.9'22]79-607931
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ATURE
aofa tube.
ofa particle.
reffectiveheatcapacitypergram.
mof solidparticle.
onofdustina dust-airmixture.
ofacoaldustatwhichits maximumSoccurs.
.
diameterorquenchingdistance.
caldustparticle.
htedaverageparticlediameter.
ration.
aparticle.
freepathfor molecularcollisions.
berforwall-lossquenching.
yexpandingflame.
alparticle.
aminarburning velocity.
hevelocityofaflame frontrelativeto
urningvelocityforquenchingbynatural
urning velocityforquenchingbyflamestretch
oyancy(processa).
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gnationlimitburningvelocity.
ocityforconductive-convectivewall-loss
ocityforconductive-convective,
cingbyinertpowders.
rningvelocityforacoaldust-air flamewhichoccurs
edgases.
urnedgases.
egas withrespecttotheparticle.
ualparticle.
n.
ss.
effectivediffusivity.
whichequalsthe numberofflame-zone
mefrontfromwhichconductivelosses
tlyinfluencethepropagationprocess
nessforradialheat conductionlossesto
reffectivethermalconductivity.
ofthegas.
reffectivedensityofreacting,combusti-
omprisingthedustparticle.
ngtimeforaparticlein theflamefront.
ngtimeforthegas intheflamefront.
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rei
ewall-losslimit:quenchingdiameters3
efornearlimitflamepropagation6
sdimensionsandboundaryconstraintsonthe
ability8
rtpowders14
ts;particlelagintheflamefront;the
ensitiesfor thewall-lossquenchingofa
e3
bustionandbuoyancyforcefields7
zonestructureforhorizontalpropagationina tube8
tsinmethane-airmixturesforthethree
ationintubesofvaryingdiameter,
velocitiesatthoselimit concentrations.9
gvelocitiesfor(a)buoyant-convective
ssquenching,asafunctionoftube
ctionsofflamepropagation12
oflimit velocities,(S)+(S ),with
ningvelocity11
ernalwallquenching"ofmethane-air
16
rnalwallquenching"ofcoaldust-air
ading17
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BILITYLIMITS
allLossesandThermalQuenching
gvelocities,formulatedinanearlier Bureau
appliedtotheproblemofflame propagation
The limitburningvelocityforconductive-
ching(processb)is (Su)b=~-,wheretheproper
nstant,Pe,isdeterminedmainlybytheratioof
ato flame-zonecrosssectionalarea.Thisratio
narearelatestotheshapeof theflamefrontand
dbyboundaryconditions.Thecomparisonof(Su)b
mitvelocitiesforsystemsmixedbynaturalconvec-
e influenceoftubedimensionsandboundary
rearthlylimits forthethreedirectionsof
sedbyinertpowdersisshownto besimilarly
mitburningvelocity,(Su)b.Thetube'ssurface-
simplyreplacedbythepowder'ssurfaceareaper
blemixture.Thermallossquenchingbythese
criticalPecletconstantsomewhathigherthan
dtheproblemis complicatedbythefiniteheat
andparticlelageffectsinthe flamefront.
econceptoflimitburningvelocitieswas
tativetheoryofflammabilitylimits.It wasshown
ocessescandissipatepowerfromacombustionwave
ationatsome characteristicallylowlimitvelocity.
sandonecomplicationwereinvolved:(a)free,
mist,PittsburghResearchCenter,BureauofMines,
enthesesrepresentitemsinthelist ofreferences
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onductive-convectivewalllosses;(c)radiation;
emixing(thecomplication);and(e)flowgradient
wasshownthatforpremixedgases,the "normal,"
agation,asconventionallymeasured,involves
teractionofprocessesaande.
tion,extinctioniscausedbytheascendanceof
tingthroughthemechanismofflamestretch.An
e,emanatingfromaconvectivelyrisingflamekernel,
aswhosemotionis alsoinfluencedbybuoyancy.
kernel,thecoldsurroundingsmustmoveoutward,
athofburned-gaskernel.In theequatorialregions,
ngsis downward,asrequiredbythenetbuoyancy
pagationthusoccursintoavelocitygradientinthe
but finite,propagationvelocitytheflameis
ncy-inducedflows.Forhorizontalpropagation,
simply derivableasthebalancebetweenthe
ndbuoyancyforces,andis(Su )a=[2agpb/pu]1'3.
twasshownthattheblowofflimit velocitywas
formulaewereshowntoproperlypredictboth the
ceofthelean limitforavarietyof fuelsandthe
elimitwithincreasinggravitationalacceleration.
veryexistenceof"normal"limitsofflammability
siscausedbythe competitionbetweencombustion
thatthiscompetitionresultsinthe presenceofa
gvelocityatthelimit. Flamepropagationcan
dealburningvelocityexceedsthelimitvelocity.
burningvelocitiesarelessthanthelimit velocity
yandthushavezerovaluesfor theirrealburning
ntinuity,orminimum(limit)valueforthe
wninasubsequentpublicationtohavea profound
ureofdiffusionflames(11). Theconceptledto
mestructurewhichis atvariancewiththetradi-
chmoreconsistentwithrecentdata(15).
onsiderthesecondcompetitiveprocess:
walllosses.Anequationisderivedfor thelimit
ctive-convectivewalllossquenching,(Su)b.
ttothe conceptofacriticalPecletnumberfor
htubes,whichconcepthasalreadybeenshowntobe
valuesforquenchingdiameters.Themagnitudeof
wsone toassessrealisticallytheinfluenceof
true"limitmeasurements.Withfurtherdevelopment,
abletothe problemofthethermalquenchingof
particles.Suchparticlescanbeconsideredas
withintheflammablemixture.
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ECTTVEWALL-LOSSLIMIT:QUENCHINGDIAMETERS
ghtubesoffinitedimensionswilllosecombus-
dingsby heatconductionthroughthetubewalls
colderthan theburnedgas).Thisloss process
turegradientsinthegasnearthe wallandaquenched
edinfigure1.For large-diametertubes,the
isfarremovedfromthecentralregionsand doesnot
actualburningvelocityinthecentralregions.
ersdiminish,theradialtemperaturegradientsinthe
convergeinwardandsoonbegintoinfluencethebulk
allypropagationisquenchedatsomefinitetube
enchingdiameter.Thesenonadiabaticlossproc-
vectorsperpendiculartothepropagationdirection.
lossesnotonlyarein the"rim"regionswhere
ewallbutalso comefromtheburnedgasregions
iedbyvon ElbeandLewis(21),Spalding(20),
erlad(17),andGersteinand Stine(6),andthe
studieshavebeensummarizedinasurveyby
equationmaybeobtainedbythe followingelemen-
l,flat,laminar,flame-frontthatis propagating
ustionpowerdensityinthepropagationdirectionis
nsitiesfor thewall-lossquenchingofaflame
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ombustionpowerdensity3is,ineffect,the
mtheburnedtothe unburnedgasthatmaintainsthe
rateinacoordinateframemovingattheburning
powerdensityisapproximatelyequaltotherate
enthalpy(bythecombustionreactions)perunit
wnin figure1,thispowerdensityis axialand
opagationalongthetubeaxis.
s,processb competeswiththepropagation
onductive-convectiveheatflowdensitytothe
(Tb-Tu)= (Tb-Tu)Nu.Thisheatflowis the
halpyfromtheburnedgasper unitareaoftube
agationofa planarflamefrontina circular
gas thatisactivatedbythecombustionpower
sectionalarea,irr0. Forthecompetingradial
theareainvolvedisthe flame-zonecontact
sanadditionalperimeterareain theburnedgases
Thetotal areainvolvedfortheheatlossflux
x,whereg isadimensionless,geometricwallloss
sourcepoweristherefore
heatlosspoweris
ssbis obtainedwhentheidealcombustion
asheatloss power.Equatingsourcepowerand
= a/(Su).dealandot= ,gives
gNu)1/2a/r0.(1)
nchingdiameter,dq,atthelimit offlamepropa-
creasingdiameterissimply2r0 ,andthelimit
Su)b, onehas:
gnizableasacriticalPecletnumber,Pe,and
ionlessquantitythatcorrelatesthe measured
nceptsandadefinitionof(Su)idealarepresentedin
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withthecombustionandtransportpropertiesofthe
owever,itshouldberecognizedthatone isnothere
berinthe conventionalsense.Theconventional
f convectivetoconductiveheattransfertoa
adofasinglesurface,thereare different
ximatelyorthogonallyorientedrelativetoone
flow"reflectedinSu [or(Su)ideal]relatesto
mburnedtounburnedgasthatis necessarytomain-
agationrateatSu.It istheflow"eigenvalue"
pagation.Theconductivewalllossreflectedin
irectionofflamepropagation.Asindicatedin
of equation2,thePecletconstantistheratio
heflame-zonesource(requiredtomaintainpropa-
erateofenergytransferin theorthogonaldirec-
boundarylayeratthe wall.Themechanismofenergy
nanddiffusionforboth directions.AvelocitySu
allynoconvectiveheatbeingtransportedtoany
hatflow.
ofequation2, theaxialcombustionsource
ticflamepropagationwascomparedwiththeradial
l.Inreality,asoneapproachesthelimit, flame
gationbecomesnonadiabatic,andaxialheatlosses
ttothe coolerunburnedgases.Thusinasystem
quenching,theorthogonalitybetweenpropagation
odissappear.Someof thesegeometriccomplexi-
ortly.Nevertheless,theireffectisquantifiable
3-factor.
mind,equation2indicatesthat steady-state
yifthe Pecletnumberexceedssomecriticalvalue;
stionsourcepowerexceedstheradialpowerloss.
thecriticalPecletnumberis 60forcircu-
withtheearliercalculationsofvonElbeand
easuredPevaluesof14 to18fordownwardpropaga-
tes,theratio oflossareato flameareais
ncethecriticalPecletconstantshouldbe0.71
man'splatevalueswouldcorrespondtotubular Pe
dberecognizedthatmeasuredvaluesare gener-
shortchannelsoffinitel ength,whereasthe
calculationsthatapplytotubesofinfinitelength.
seffectivequenchersthantubesof infinite
tofindthemeasuredquenchingdiameterssys-
hetheoreticalones.Otherfactorsinvolvingthe
flowconvergenceatthetubeinletshould also
alueofthecriticalPecletnumberforwall-quenching.
entialvalidityofequation2by varyingSuanda
mescontainingvariousinertgases.Potterand
widerrangeoffuel-oxidizercombinationsand
uredependenceofdwas accuratelypredictedby
oftheratioa/Su,as equation2requires.
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tion,theabsolutevalueofthecritical Peclet
ratioofthecontactperimeterloss areatothe
larea.Fortheassumedplanarflamefrontin a
on,thatratiois2gAx/rQ= 23a/r0Su.Notethat
urceareavariesinverselywith Su.Anexact
afactor,g, isdifficulttomake.Its accurate
etailedsolutiontotheflamepropagationequation
ncertainfactoris thuslumpedintothePeclet
ydeterminedempirically.
ver,thattheaboveratiois necessarilyinflu-
ezoneshape,and thattheoverallbalancemaybe
genceandconvectioneffects.Flameshapeiscon-
,radialboundaryconstraints(tubeshape),axial
sedversusopen-endedignition),andselectivedif-
cellularflamestructures).Theseprocessesmust
ecritical Pecletnumberatthewallquenching
cetheprecisedetails oftheflowstructure
frontandnearthewalls,and theymayaddaflame-
m.
HAPEFORNEARLIMITFLAMEPROPAGATION
gationbehaviorofnear-limitmixturesclearly
cesinboththeflameand theflowstructurefor
agation.Levy'sstudies(13),for example,showed
tureissphericalduringupwardpropagationin tubes
dsto bemaintainedevenduringflamequenching;but
ationintubes,thecurvaturediminishes,andatextinc-
yflat.TherecentobservationsofSapko,Furno,
uctureofnearlimitflamesin averylargescale
atthesamebuoyancy-induceddistortionsoccurin
aryconstraints.Inthecompositionregionbetween
dlimits,the"normal"sphericalflameisreplaced by
lshape.
sphericalflamekernelandassignstoit a
accelerationvector(8-ll_),theobserveddistortion
dexpect.Thatis, thebuoyantaccelerationvector
ntatthetop ofthesphere,andhencethe entire
atestheupwardpropagationvelocity(neglectingthe
emoment).Atotherpointsin theupperhemisphere,
sdiminishedbythe cosineoftheanglebetweenthe
ward-directed,buoyantaccelerationvector.Asimple
esultantflame-frontdisplacementshowsthatthe
maintainsphericalcurvatureintheupperhemi-
mekernelexpandsand rises.
n,ontheotherhand,thebuoyantacceleration
to theoutwardnormalpropagationvectoratthe
sdeceleratespropagationinthelowerhemisphere
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verybottomofthesphere,theentirevector
decelerationreducesthedownwardpropagationveloc-
owerhemispherethedecelerationisdiminished
eneteffectfordownwardpropagationisto reduce
aflatteningflame-zone,forasthe flamekernel
ttendsto becanceledbyitsbuoyantdeceleration.
tgeneratesapproximatelythecorrectoverallflame
agation,eventotheextentofreversingthecurva-
frontto dimpleinwardatthebottomofthe sphere
ybeginstoexceedthegravity-freeflamespeed(18).
uchchangesinflameshapemustnecessarily
etconstant.Considerfirstpropagationfromthe
anarignition.Neglectbuoyancyforthemoment
agateat Suintotheinitiallystatic,cold gas.
low structuregeneratedbythecombustionforce
itinteractswiththewallboundaryconstraints.
bedbyJost (12),whoshowedthatpropagationof
anarwavesoonbecomesimpossible.Thegas
omtheunburnedside mustadjusttoaPouiseille
edside,andi tmustalsouniformlyaccelerateto
thedirectionperpendiculartothe flamefront.
ossibletosatisfybothconstraintsacrossaflat
inglytheflamefronttakesonaparaboloidshapein
,andvorticesappearinthecoldgaszonejust ahead
onofflamecurvatureis independentofbuoyancy.
mplexitiesthatareobservedinthe presenceof
figures2and 3,whichdepictspropagationina
esincombustionandbuoyancy
o
e.
y,
tion
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ndflame-zonestructurefor
tube.
flu-
nd
.
rdflowofcoldgas(9-11).Thisconvectivecell motion
velopingparaboloidflameshape,asdepictedin fig-
extendsto accumulatehot,burnedgasesnearthe
nburnedgasesnearthebottom.Thebuoyancyvortex
ces)propagationnearthetopand diminishestherate
ttom.
eshape andwidthoftheflamefront are
ehorizontallimitisapproached,a conditionis
onswherethe componentofthebuoyantretardation
heflamefrontjust balancesthetrueflamespeed.
einthe lowerregion,andtheflameissaid to
ube.Atthehorizontallimit, flamepropagation
arthetop ofthetubeis finallyquenchedby
alllossesandbyflame-stretchflowgradienteffects.
e-zonethicknessthatmightbe expectednear
depictedinfigure3.Theflame-zonethickness
enedbyconvectiveanddiffusiveflowsnearthe
eisaflame zonewideningnearthetopofthe tube
uenchedboundarylayeratthewall.Themaximum
eshouldbejustbelowthe topofthetube,where
as narrowest.Thereisnecessarilyaflowgradient
vectivevortexinallregionsof theflamezone.
dby balancingthebuoyancyforcecouplethat
vortexagainstthecombustionforce(8).
RATUSDIMENSIONSANDBOUNDARYCONSTRAINTS
SOFFLAMMABILITY
asurementsformethane-airmixturesintubesof
ysummarizedinfigure4. Alsoshownaretheburn-
tsofAndrewsandBradley(1),GunterandJanisch (7),
,andEgertonandThabet(4_) .Theleancomposition
ionareshownasupward-directedarrows.Thearrow
thediametersof thetubeswithinwhichthelimits
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mits
asurementsinmethane-air
ctionsof flame
yingdiameter,
velocitiesat
.
ws,
l
s
e
ws
m-
al
e
u.
m
its
ed
e
-
end.
b-
me
d
0percent(^2).This ispresumablythelean
on,intheabsenceof wallcomplications,usinga
oncriterion.Thesoftcriterionisthe detection
flamepropagationthatextendsfar beyondthe
estimatethelimit burningvelocityforwall
iamtube,usingaconsistentvalue(8) foraof
e= 25(itsmeasuredvalue)gives(Su)b= 6cm/sec.
o(Su)a,thehorizontallimit velocityforquench-
Onewouldthus concludethatwalllossesina
mpariblesignificancetobuoyancyindetermining
emaygo abitfurther.ConsiderthemeasuredSu
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(theuppercurveinfigure 4)andthemeasured
ntmethanein the2.5-cm-diamtube(h2.5in
mitcompositionupwarduntil itintersectsthe
ngvelocityof12cm/sec.Is itacoincidence
ltothe sumof(Su)aand(Su)b?Note alsothat
ufor theuandh valuesintubesoffixed diameter
f3cm/secbetween(Su)a>e.fand(Su)a .Oneis
asimpleadditiveeffect:in thepresenceofboth
thehorizontallimitswouldcorrespondto the
)
ouldreplace (Su)aby(Su)a,e^.
on3 agreeswiththedatais indicatedin
nlimitsin tubesofvaryingdiametersaretaken
.(Su)bis thencalculatedforthevaryingtube
5, asabove.Themeasuredlimitcompositiongives
chistakenfromtheburning velocitydataof
EgertonandThabet(4).The(Su)a valuefor
akenas6cm/sec,while the(Su)ae/|.valuefor
enas3cm/sec.Comparisonofthe lasttwocolumns
ation3isapproximatelycorrect.However,onecan-
he precisevaluesforeachcasec onsideredinview
ntaluncertainties.Forexample,themeasuredlimit
geofwatervaporcontents,varyingfrommethane-
o mixturesthatwerefullysaturatedatambient
itcompositionswereforclosed-endignition,
n.ThemeasuredSuvaluesare takentobeequal
goodassumptionformixturesabove7 percent
velocitiesareexceedinglydifficultto measurein
methane,preciselybecauseofthese competing
rthatinorder toobtainaccurateleanlimit
dependentofwalllosseffects,onemustsatisfythe
(Su)a>e^.If,for example,onesets(Su)b= 0.1
equirement,thenthe flammabilitytubeforhori-
sshouldhaveatubediameterin excessof20cm.
orintroducedbyusinga 10-cmtubeisatmost 0.1
elativeerror of1to2 partsin50.Clearly,
ametertubesare "truer"earthlylimitsthanthose
tertubes.Ultimately,thesizerequiredtoobtainthe
ythe accuracydesiredbytheinvestigator.
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hesumof limitvelocities,(Su)a+(Su)b,
burningvelocity
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nt.Thiscouldalterthe proportionalityconstant
antly.Nevertheless,forsmallerdimension,near
bsenceofupper boundaryconstraints,equation4
ardstagnationlimit.Asbefore,thelimitis given
=(Su )a^+(Su )b.Now,however,thelimitveloci-
dprocessbcontainanr0-dependence.(Su)bvaries
(Su)adependsonr0 .Thesum(Su)a^+ (Su)bnow
pendence.Thisisshowninfigure 5,wherethesum
wardpropagationiscontrastedwiththesums for
pagation.Inthelattertwocases,as tubediameters
a"true"limitcondition,andthelimitvelocity
ependentoftubedimensionsanddependsonly on
n,thesituationisclearlymorecomplicated.
ueforlarge r0,thereisonlya broadminimumin
.This transitionregionisin facttheregion
ementsofdownwardlimitsaremade.Forthe curves
-
ting
y
an
y
p-
ue
t
n
s
nis
y
Su
n
mitburningvelocitiesfor
enching,and(b)wall-
onoftubediameterfor
epropagation.
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bouttheleanestvalueobservedfor thedownward
re5predictsthat downwardlimitsshouldshift
onsincreaseford0 >20cm.Thisis inmarkedcon-
aviorforthe upwardandhorizontalcurves.
edatatoindicatethat theformulaforthebuoyant
ngflamekernel,whichwasusedto deriveequa-
earevirtuallynodataon thedownwardflame
ofdiameterssignificantlylargerthan20 cm.
ardcurveoffigure5 predictthatdownwardlean
rfortheselargerdimensions.Forexample,the
50-cm-diamtubeis13.5cm/sec.Fromfigure4,
wardlimitof6.2 percentmethaneforthe50-cm-diam
ube,onepredictsadownwardlimitof 6.5pctmeth-
er,thedownwardlimitshouldreachthe stoi-
tina tube1,000cmindiameter.
oclaimthat suchprojectionsareunrealistic.
sofbuoyancy.Theyresultfromthe operational
mepropagation.Thetubeisnecessarilyconstrained
oryreferenceframeasthecenterof flameprop-
tvelocity.Yet,toan observeratalargedis-
nition,the effectisrealinthat theflamefront
e combustiblemediumisofinfinitevolumeandif
opentothesurroundings.
atsuggeststhattheeffectis indeedarti-
ontainerisclosed,thenregardlessof the
ntrise velocitywillnecessarilyapproachzero
gwaveapproachestheclosedtopofthe container.
mncannolongerrise bybuoyancy,downwardpropaga-
dtheflamefrontcanthen reachanobserversituated
Suchan effectisindeedobservedforpropaga-
e.
hapeofa fireballinanN2-dilutedmethane-air
dburningvelocityofabout9cm/seccorrespondstothat
xture.Its motionandshapearedepictedbySapko
1+Themixturedoesnotpropagateat allinthe
thefireballdiameterexceeds10to20cm.This
odistort markedlyandtodimpleinwardatits
softhe distortedfireballcontinuestoriseupward
it approachestheclosedtopofthespherical
tactsthetop ofthesphere,itsbuoyantvelocity
opagationinthedownwarddirectionbecomespossi-
servedtoestablishaflatflamefront that
ternaryonecontaining6.9percentCH^j, 65.8percent
hichis equivalenttoabinarymixtureof5.8
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wnwardandcompletelyconsumestheremainderofthe
the12-ft-diamsphere.
wardpropagation,(Su)a^,+(Su)b^.,infigure5,
ptionthattheflammabilitytubeis opensothat
e coldgasflowsrequiredtocompensateforthe
eburnedgascolumn.Asthe burnedgascolumn
gasexitsfromthetop andentersfromthebottom.
datthetop,equation4 soonbecomesinapplica-
tcanno longerbeastagnationlimit.If thetube
egasis ignitedatthatclosedend,then thecold
fthewaveis drivenbytheburnedgasexpansion
nditionatthetopofthe tube,andtherequired
cityprofileinthe coldgasventingfromthe
ownwardlimitwouldpresumablybegovernedby
amestretch.Thesum(Su)a ^+(Su)bwouldthen
mptoticlimitinthe6-cm/secrange.
cyeffectfordownwardflamepropagationappears
perimentalmethod:themeasureddownwardlimit
dentonwhetherthesystemisopenorclosedwith
s.Butitseems hardlycorrecttoconsiderthe
rtifact,whenitis preciselytheprocessresponsi-
fthe limitsofflammabilityinthefirst place.
NERTPOWDERS
problemofflamepropagationinnarrowtubes,
tion1,whichdefinesthe limitburningvelocity
esto thetubewalls.Itis givenby:
etnumberfor wallquenching.Nowforacircular
sr, theinternaltubesurfaceareatowhichcom-
rr.Thesourceof combustionenergyisthegas
sevolumeisirr02.The surfacetovolumeratiois
isthisratiothat determinesthecriticaldimen-
micsourcestrengthisovercomebythe thermallosses
f thissurfacetovolumeratio,thelimit burning
boundaryis notanexternalonebutan internal
s pulverizedintoapowderinsucha waythatthe
ratioispreservedinthesurfaceareaofpowderper
gas.Inthis sense,thequenchingeffectcausedby
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wdertoaflammablemixturemaybeviewedasthe
mbustionwaveby"internalwalls."
autiousinapplyingthisconcept,forthereis
tweentheinternalsurfaceareaofthepowderand
ctwall.The wallboundarycanconductheattoa
niteheatcapacity;howeverthe"internalwall"
tcapacity.Wewillevaluatethefiniteheatcapac-
tionsinthe followingsection.
d,it isassumedthattheseinertpowderwalls
onductive-convectivewalllossesin thesamewaythat
eraninertdustof soliddensityp, consistingof
eterdp,dispersedin thegasmixtureata mass
.Thenumberdensityofparticlesisn =6Cm/irdp3pp.
r unitvolumeofgasis mrdp2,andhencethesur-
tvolumeof flammablegasis
(7)
oequation6and settinga=0.55cm /secgives
owthe limitburningvelocityforquenching
nertparticlesintheflammablemixture.
nmind(to bediscussedinthefollowingsec-
asthelimitburningvelocityfor inertparticle
causeoftheadditivityprinciple(Eq.3),(Su)'b
reductionintheidealburningvelocitycausedby
owderin theflammablemixture.Asystemiscom-
ertpowderwhenthatreductionin burningvelocity,
bybuoyantquenching,isequalto theidealvalue;
)a =(Su)ideal.However,iftheflameis not
fthermallossestotheinert additiveisstillfelt
anditsreal burningvelocityisreducedby(Su) 'b.
withrespecttoCmgives
u)'b foragivenCmwiththe reductionin
yS,andsolvingfor Pegives
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vestudiedthe effectofaddedrockdust(CaC03)
dersontheburningvelocityofmethaneair mixtures.
e2,andtheresultantvaluesfor Pecalculated
on10.The additionofthepowderedinertdusts
lquenchingbyheatconductiontotheadditive,but
seinemissivityofthe burnedgascausedbythe
ncreaseresultsinradiativelosses,whichare
diativeloss limitvelocity(Su)c.Themagnitude
8cm/secdependingonthedustconcentrationand
bothlosseffectsarepresentsimultaneously,it
ssmallvalue of(Su)cfromthemeasuredASU
heeffectthatisattributableto (Su)balone.
ersforinertquenchingaveragetoaboutPe =100.
somewhathigherthanthe theoreticalwallloss
one cannotbetooimpressedwiththeresult that
wasnoless effectivethanthe9umparticle.
sfor"internalwallquenching"
yinertpowders
10.
fectofinternalquenchingbyinertparticles
SmootandHorton (19)dataontheburning velocity
eirdatashowthat burningvelocitiespeakat
hatdependmarkedlyonparticlesize.Forfine
ngvelocity,(Su)(max),occursatrelativelylow
reasforcoarserduststhemaximumappearsatamuch
.(Su)(max)issignificantlysmallerforthe
rehigherdustloadings.
orrelatesto thefactthattheoverallrate
esemaximainS ,islimitedby therateofpyrol-
ecoalparticles.All peakvaluescorrespond
a stoichiometricconcentrationofcombustible
ecoarserthedust,themoredifficultit isfor
atilizeduringits passagethroughtheflame
cle,thesmalleris thefractionofitscom-
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strequiredtogenerateastoichiometricconcentration
orthecoarsecoals,onlytheshallowshellnear
pestcornerscandevolatilizeintimetocontribute
hefinercoals,mostof thecoalparticledevolati-
.
massof coal(ortheremainingcharresidue)
powdertowhichcombustionenergyislost.The
adingis seeninthedata asareductionof
thePecletconstants,calculatedfromequation10,
tioninSu(max)by theexcessinertdust.Forthe
for"internalwallquenching"of
xcesscoalloading
10.
hequenchingofexcesscoaldustin the
whichisquiteclosetothe valuescalculated
ngofmethane-airmixturesbyrockdustandalumina
FECTS;PARTICLELAGINTHEFLAMEFRONT;
LAYER
thederivationof equation8,thatrelatedto
naddedpowder.Equation6, fortube-wall
ndertheassumptionthatheatwascontinuouslybeing
undarylayerandthroughtheinteriorsurfaceto a
pacity.Ineffect,itwasassumedthat thecold
edcoldat Tuduringandafterthe passageof
sassumptionwasincludedinthesubsequentderiva-
ever,thewallboundaryhasafiniteheatcapacity
volumeit contains),thenthetemperaturegradient
edheatlossto thewallwouldnotbe maintained.
ternalwalls"ofinertparticles,wheretheir
ausetheirwalltemperaturetorise significantly
eof theflamefront.Iftheparticlemass isso
risesas fastasthegastemperaturerises within
mallossesare notmaintained,andthegradient
stanceitis notthesurface-to-volume
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portant,butonlyits heatcapacityperunitvol-
tothemassconcentrationCm,andindependentof
a"smallparticle"is thecaseofaninert gas
etoconsiderthisextremecasein ordertobe
equation8. TheN2particlesaddedtoa flammable
s rapidlyasthecombustionproducts,andsincethey
sing equation8tocalculateinertingrequirements
esult.Thiscanbe demonstratedasfollows:
etheinert gasismixedonthe molecularscale,
ldbe themoleculardiameterofthenitrogenmol-
olid) Ndensityforpand typicalvaluesfor
dto thepredictionthataPstoichiometricmethane-
dat aCmvalueofonly 1mg/1.Thepredictionis
eN2moleculesarenotcoldwallsthat remaincool
e,butrathertheir heatcapacityissosmallthat
creasinggastemperature.Datashowthatthemix-
aCmof500mg/1,whichcorrespondstosome
heabsurdresult of1mg/1wasobtainedbecause
ndits rangeofvalidity.The1mg/1valuecouldbe
culehadaninfiniteheatcapacityandcouldthus
t wasimbeddedinreactinggaswhosetemperature
s physicallyimpossible,sinceeachmoleculehas
.
volumeof1 mg/1ofN2istrivial compared
unitvolumeof thereactingmixturewhosetotal
00mg/1.Thus,theadded1mg/1of N2hasatrivial
etemperatureortherate offlamepropagation.
500mg/1ofaddedN2to reducetheflametempera-
eof2,200 Ktoits limitvaluenear1,500K (8).
nalinsightintothefallacyjust consideredby
ofCm- 500mg/1andsubstitutingthetypicalvalues
the"effectiveparticlediameter"atwhichquench-
ogenwere tobeconsideredasaninert powder.
value isclearlymuchlargerthanthe molecular
nificantlylargerthanthemeanfreepathfor
1ym.Thislatter conditionisnecessaryforthe
fdensity,heatcapacity,andthermalconductivity
ngsin thefluidcontinuum.Clearlycontinuumequa-
quations1 and8,andhencetheirvalidityis
L.Thisconditionisviolatedbychoosing
theN2moleculeasthe particlediameter.
erthis particleheatcapacityandheating
To whatextentcaninertparticlesaddedtoa
wtheincreasinggastemperaturesintheflamefront,
agthegastemperatures?
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particleoffinitemasssubjectedto alami-
emperatureriseisgiven by:
ume,citsheat capacitypergram.andAis its
minarheatflux totheparticleis equatedtothe
rdensity,Scp (T-T ).Thisgives:
-T).
mof particleandgasareapproximatelyequal
echaracteristic timeconstantforparticle
ontgives:
aracteristicheatingtime isT= a/Su2.
heatingtimetoflamegasheatingtime is
clearlytheparticletemperatureshould
perature.AT-ratioof lessthanunityis aphysi-
wouldviolateour continuumconstraintthat
wouldmeanthattheparticletemperatureshould
ggastemperatureofthe combustingsystem.In
oor lessthanunity),theparticlec annotbe
all,andonlyitsmassor totalheatcapacityis
effect.If, ontheotherhand,theT-ratio is
nequation8 isapplicable,andtheparticleshould
ngflamefrontpassage.Settinga=0.55 cm2/sec,
articleandgasdensities,and usingthelimit
cm/sec,oneobtainsr- 0.5ymfora T-ratioof
smaller)particleshould closelyfollowthe
rewithintheflamefront,andequation8 wouldnot
hand, foralOym-diamparticletheT-ratiois 10
peratureriseoftheparticlewill beanorderof
atofthegas. ThusalOym-diam(orlarger)particle
allduringflamefrontpassage,andequation8should
sconsideredintables 2and3werei ntheappli-
sthekinematicsofinert particlemotionin
citly assumedinderivingequations8and11
he flamefrontwouldinstantaneouslyfollowthe
attheir velocitieswerealwaysequaltothegas
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rthevalidityofthatassumption.Thequestionis
First,thereis thequestionoftheheattransfer
theparticle.Theconductive-convectiveheat
oasolidsurface(or thereverse)isproportional
whichis proportionaltotheratioof particle
yerthickness.Forthedimensionsandvelocitiesof
tothe squarerootoftheReynoldsnumber,/Re.
relativevelocities,Reis smallandNu->1. This
sumedinthederivationofequation11,and only
boundarylayerwhosedimensionswerecomparableto
consideredtobeimportantinthe heattransfer
articleshouldlagthegas flow,Nucanexceed
rheattransfertotheparticle.If thereis
ofgaswithrespecttothe particle,theboundary
smallerthantheparticledimensions.Actually,for
tiesofinteresthere,Nuis neversignificantly
firstproblemisofminor importance.
er,isthesecondreasonforconsideringparti-
elags thegasflowsothat itstimeoftravel
gnificantlylongerthan thatofthegas,then its
heflame frontwillbehigherthan itsinitial
staticmixture.Ifparticleslagin theaccelerating
eflamefrontandtheir effectismagnified.
icsofparticlemotioninorderto estimatethis
ceonasphere ofradiusr, whichisimbeddedin
ngwithauniformrelativevelocityuwithrespect
Stokes'law:
ectionoftherelativevelocityu.A moreaccu-
or(1 +3/16Re),whichwouldincreasethedrag
hemostextremecase.Wehereignorethesecond
rmula.Nowin acoordinatesystemmovingwiththe
elocity,gasand particlesareinitiallyatrest
andbothenter theflamefrontatthe velocityS.
aversesthroughthe flamefrontandreactstogive
ityisSb=Supu/pb.The detailsoftheacceleration
ombustionforcedependonthe detailsoftheflame
andprobablyreasonablyaccurate,toassumea
svelocityas ittraversesacrosstheflamefront.
x= 0),(vg)0=Suwhile att= Tgas=a/Su2
= Supu/pb.Thisgives
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ocityuponentering theflamefrontisalso
u)0 =(vg)0-(vp)0=0.At anytimet,while
amefront(thatis,t
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kedly.Itsexitvelocityfromtheflamefrontwould
gasvelocity.Thus,althoughit andthegasenter
sec,thegasexitsatS -180cm/secwhereasthe
60 cm/sec.Theparticlevelocityhasthusonly
cityhasincreasedbya factorofsix.Thissame
wouldbepresentevenat thelimitvelocityfora
ameter.Theeffectofthisparticlelag isasig-
ectiveconcentrationofthedustinthe flame
ethatboththefiniteheatcapacityeffect
particledrageffectconsideredabove,tendto
e-sizedependencepredictedbyequation8.Thefinite
thequenchingeffectivenessofverysmallparti-
ticalsize inthe1-10ymrange,they cannolonger
cledrag effectcausesthecoarserparticlesto
flowof theflamefront,andasthey "pileup"
ionincreases.Thisincreasedeffectiveconcentra-
orthecoarserparticles'lowersurfacearea.Both
particle-sizedependenceofthequenchingbehavior
sizedthattheseconsiderationsapplyonlyto
-> 1.Forsuchinertparticlestheboundarylayer
od,and heattransportisbypure conduction.If,
tinertand cangeneratevolatilesduringitspass-
t,thenits effectivenessmaybemagnifiedintwo
ationis endothermic,thenclearlyitseffec-
gnified,andit wouldtendtoremaincolderfor
ation11 wouldthenunderestimateitst-ratio,
inasacoldwalldownto smallerparticlesizes.
neratesvolatileduringitspassagethroughthe
ferbetweenthegasand theparticleisnolonger
theboundarylayer.Instead,thedevolatilizing
rroundingflamegasesonadimensionalscalethat
tudelargerthantheboundarylayerthicknessfor
e.Assumingthatthegasesare inertorinhibit-
convectivecooling(orchemicalinhibition)that
pureconductionprocessusedto deriveequations2
ogytothis argumentfortheadditionofgases:
moleculeswithmanydegreesofmolecularmotion
ertingagentsthansimpleratomicordiatomic
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gvelocities,whichwaspreviouslyappliedto
chingbybuoyancy(processa),appearsto beapplica-
hermalquenchingofflamesbyheatlosses toexter-
nallyadded inertpowders.Forwalllosses
tyis(Su)b =aPe/2r0,andinthe presenceof
sses,asimpleadditivityprincipleappearsto apply.
th lossprocesses,thelimitofflamepropagation
f limitvelocities,(Su)a+(Su)b,is justequalto
ofthemixture,(Su)ideal(whichis measuredinthe
).ThevalueofthecriticalPecletconstant,Pe,
pearstobe sensitivetoflameshapeandboundary
he rangePe-25 to60appearstoapply tomost
roblemof downwardflamepropagationinopen
dependenceontubediameter.Thisunusualdepend-
odefinea downwardflamepropagationlimitthat
toftubesize. Downwardflamepropagationlimits
sualdependenceonboundaryconstraints,namely,
openorclosed.Thesespecialcomplicationsare
rhorizontalflamepropagation.Forthosetwo
ation,"true"limitsareobservablesolongas the
largeenoughthat(S) (S) .
owderaddition,alimitvelocityisobtainable
83PeCm/ppdp.MeasuredcriticalPecletconstants
addedinert powdersareintherange Pe-75-175.
acityeffects,theequationfor(Su)'bis applicable
10ymin diameter.Ananalysisofthekinematics
eleratingflamefrontwasalso presented.It
articlesabove50umin diameter,asignificant
hisparticlelageffectincreasestheparticle's
heflame front.Boththefiniteheatcapacity
effectlimitthe applicabilityoftheabove
dtoreduceormoderatethe predicted
nce.
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radley.TheBurningVelocityofMethane-Air
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