The Theory of Flammability Limits - Conductive Convective Wall Losses

8/9/2019 The Theory of Flammability Limits - Conductive Convective Wall Losses

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469

yLimits

allLosses

MENTOFTHEINTERIOR

y

gDirector

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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


28/31

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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The Theory of Flammability Limits - Conductive Convective Wall Losses

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