New Determination of the laminar burning velocity and the Markstein … · 2002. 11. 6. · Powder Technology 122 2002 222–238Ž. Determination of the laminar burning velocity and

Ž .Powder Technology 122 2002 222–238www.elsevier.comrlocaterpowtec

Determination of the laminar burning velocity and the Markstein lengthof powder–air flames

A.E. Dahoea,b,), K. Hanjalicc, B. Scarlettaa Department of Chemical Engineering, Delft UniÕersity of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands

b Department of Engineering, UniÕersity of Cambridge, Cambridge, United Kingdomc Department of Applied Physics, Delft UniÕersity of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands

Received 11 August 2000; accepted 2 October 2000

Abstract

This work deals with the determination of the laminar burning velocity and introduces the Markstein length of powder–air mixtures. Apowder burner was used to stabilize laminar cornstarch–air dust flames and the laminar burning velocity was determined by means oflaser Doppler anemometry. The dust concentration was varied from 0.26 to 0.38 kg my3. The measured laminar burning velocities werefound to be sensitive to the shape of the flame. With the same dust concentration, parabolic flames were found to have a laminar burning

Ž y1 y1 .velocity, which was almost twice that of a planar flame ca. 30 cm s for the latter as compared with ca. 54 cm s for the former .From this discrepancy and the flame curvature, the Markstein length could be determined. It was found to have a value of 11.0 mm. ThisMarkstein length was subsequently used to correct the measured laminar burning velocities at various dust concentrations in order toobtain the unstretched laminar burning velocity. The unstretched laminar burning velocity lies between 15 and 30 cm sy1 and is thoughtto be a property of the dust and of the concentration.q2002 Elsevier Science B.V. All rights reserved.

Keywords: Dust explosion; Burning velocity; Markstein length; Flame propagation; Laser Doppler anemometry

1. Introduction

The severity of a dust explosion is a function of a widevariety of parameters. Some of these parameters representproperties that are specific to the chemical properties ofthe exploding dust–air mixture, while others reflect thesensitivity of the explosion to the flow properties of thedust cloud. Due to the absence of a comprehensive descrip-tion of the transient combustion behavior of any particular

Ždust–air mixture under arbitrary conditions pressure, tem-.perature, flow properties , it is common practice to mea-

Ž 3sure the explosion severity in laboratory test vessels 1-m.vessel, 20-l sphere and to predict what would happen if

the same mixture exploded in an industrial plant unit. Thisis currently done by means of the well-known cube-root-

w xlaw 1–3 . The maximum rate of pressure rise measured inthe test vessel is multiplied by the cube root of the testvolume to yield a K value, which is assumed to be aSt

volume invariant dust explosion severity index. The dustexplosion severity of the same mixture in a plant unit is

) Corresponding author. Tel.:q44-1223-332603; fax:q44-1223-332662.

Ž .E-mail address: [email protected] A.E. Dahoe .

predicted by dividing thisK value by the cube root ofSt

the volume of the plant unit. The resulting dust explosionŽseverity dictates the design basis for safety protection e.g.

.explosion relief venting, explosion suppression . In thisapproach, known as the VDI methodology, it is assumedthat laboratory test data can be considered to be applicableto accidental explosions in plant units and to represent aconservative case even when the actual industrial circum-stances are not reproduced in the laboratory experiments.

Already from the beginning, the VDI methodology wasw xquestioned by a number of researchers. Eckhoff 4 pointed

out that the cube-root-law was no more than an approxima-tion of a single realization of the explosion pressure curve.

w xBased on earlier work by Nagy et al. 5 on the pressuredevelopment during the course of explosions in sphericalvessels, he discussed the conditions under which the cube-root-law is a valid approximation of the dust explosionseverity, and the conditions under which the cube-root-lawmay be used to scale dust explosion severities, measured inlaboratory test vessels, into dust explosion severities thatone might expect in larger industrial vessels.

ŽFirst, the mass burning rate i.e. the product of theburning velocity, the flame area, and the density of the

.unburnt mixture which is to be consumed by the flame

0032-5910r02r$ - see front matterq2002 Elsevier Science B.V. All rights reserved.Ž .PII: S0032-5910 01 00419-3

( )A.E. Dahoe et al.rPowder Technology 122 2002 222–238 223

has to be the same in both the test vessel and the industrialvessel at the moment when the rate of pressure rise reachesits maximum value. This condition is only fulfilled whenboth vessels are spherical, ignition occurs at the center ofboth vessels, changes in the pressure and temperature ofthe unburnt mixture ahead of the flame have the sameeffect on the burning velocity, and the turbulent flowfields, as well as changes in the flow fields during thecourse of an explosion, are identical in both vessels. Inpractice none of these requirements are fulfilled becausethe changes in the pressure and temperature of the unburntmixture are not the same in differently sized vesselscontaining identical mixtures, and dust explosion severitytesting is done in laboratory vessels under conditions ofsignificant but unknown turbulence. Moreover, test resultsfrom laboratory vessels are applied to industrial situationswith unknown turbulence. Based upon the fact that dustexplosion severity increases with increasing turbulenceintensity, it is presently assumed that if the turbulenceintensity in the laboratory test vessels is made high enough,laboratory test results yield conservative estimates of whatmay happen during an explosion in a plant unit. This,however, may lead to unacceptable over-estimations insituations where the turbulence levels in industrial practiceare much lower than those created in laboratory test ves-sels, but also to under estimations of the explosion severityunder circumstances where additional turbulence is gener-

w xated by the explosion itself. Tamanini et al. 6 demon-strated that worst case predictions by means of the VDImethodology may underestimate the dust explosion sever-ity when the turbulence varies at the time of the explosion.

Secondly, the thickness of the flame must be negligiblewith respect to the radius of the vessel. An inherent flaw ofthe cube-root-law is that it does not take the phenomenonof flame thickness into account. As the flame thicknessbecomes appreciable with respect to the apparatus radiusŽ .i.e. )1% , the cube-root-law becomes increasingly inac-

w xcurate in predicting the maximum rate of pressure rise 7 .If for example chemically identical mixtures, under physi-cally identical conditions, are ignited to deflagration in twodifferently sized vessels, application of the cube-root-lawto the maximum rate of pressure rise, measured in the twovessels yields differentK values. Since many powdersSt

have a flame thickness that is appreciable with respect tothe radius of the 20-l sphere, and even with respect to theradius of the 1-m3 vessel, the cube-root-law may not beconsidered as generally valid for the prediction of dustexplosion severity.

In order to overcome the limitations associated with thecube-root-law, several models have been proposed by otherresearchers. Unlike the cube-root-law, which takes thesingle instant of the rate of pressure rise measured in a testvessel to predict a single instant of the rate of pressure riseduring an industrial explosion, these so-called integralbalance models are capable of predicting the entire pres-sure evolution during an explosion. More importantly,

since their derivation is based on fundamental relationshipsbetween the pressure development and the mass burningrate at any instant, the effect of mixture composition,pressure, temperature, and turbulence on the transient com-bustion process can be taken into account explicitly. Exist-ing models of this kind are those of Bradley and Mitche-

w x w x w xson 8 , Nagy and Verakis 9 , Perlee et al. 10 , andw x w x w xChirila et al. 11 , Bradley et al. 12 , Tamanini 13 andw xDahoe et al. 7 . Ideally, these models enable engineers to

predict the behavior of explosions, and hence the explosionseverity, under industrial circumstances when the behaviorof the mass burning rate is known from laboratory experi-

Ž .ments and the varying turbulent aerodynamic parametersare known for the industrial circumstances. In the case ofpremixed gases, the burning velocity and the flame thick-ness are recognized as fundamental measures of the driv-ing force behind the combustion process and these quanti-ties have been used with success to model the massburning rate.

When burning velocities and flame thickness are usedas key parameters in integral balance models, a distinctionis made between a laminar burning velocity and a laminarflame thickness, on the one hand, and a turbulent burningvelocity and a turbulent flame thickness on the other. Thereason for making this distinction is that, when a flame isstabilized in a laminar flow of combustibles, it establishesitself at a fixed position in the flow field and its surfaceremains smooth. In other words, it inherits the laminarbehavior of the flow field. Moreover, the velocity at whichthe cold reactants enter the flame zone in the normaldirection, the laminar burning velocity, appears to be amixture specific property. It reflects the sensitivity of thecombustion process to changes in the chemical composi-tion, fuel concentration, oxygen content, particle size, pres-sure and temperature of the approaching flow of reactants.By contrast, when a flame is trapped within a turbulentflow of combustibles, it inherits the turbulent nature of theflow field: the turbulence of the approaching flow continu-ously distorts the flame and ceaselessly shifts its positionin space between certain geometrical boundaries. As aresult, the surface area of the instantaneous laminar flamechanges in a chaotic manner, which is determined by theturbulence of the flow field. Owing to the fact that therelevant time scales of the fluid structures that composethe turbulent flow field are much larger than the chemicaltime scale of the instantaneous combustion zone, the geo-metrical boundaries between which the instantaneous flamefront shifts its position are identified as a turbulent flamethickness. Due to the enhancement of heat and masstransfer by turbulence, the turbulent flame zone propagateswith a turbulent burning velocity which is greater than thelaminar burning velocity. The local consumption of reac-tants at any particular portion of the flame surface, how-ever, occurs within a zone whose width is equal to thelocal laminar flame thickness and at a rate which isdetermined by the local laminar burning velocity. It is for

( )A.E. Dahoe et al.rPowder Technology 122 2002 222–238224

this reason that the turbulent burning velocity and theturbulent flame thickness are always expressed in terms ofa combination of the laminar flame propagation parameters

Žand the turbulence features of the flow field i.e. root-.mean-square velocity and length scale .

While the laminar burning velocity and the laminarflame thickness of a premixed gas are considered asfundamental mixture properties, their application to dustclouds appears to be controversial. This controversy stemsfrom the fact that, while on a macroscopic scale, a dustcloud may seem like a homogeneous mixture of fuel andoxidizer, on a microscopic scale it consists of a number ofdiscrete particles immersed in a continuum of the oxidizer.With solid fuel particles of a density of 1000 kg my3, anda particle size of 15mm, one finds an interparticle spacingof about 150mm when the solid fuel concentration isequal to 500 g my3. With the interparticle spacing being

Ž5–10 times the particle size dust explosions typicallyy3.involve dust concentrations between 125 and 1000 g m ,

a combustible dust cloud consists of a number of discreteparticles that are separated by a distance, which are ordersof magnitude larger than the molecular mean-free path. Asa result, physical and chemical properties that determinethe magnitude of the laminar burning velocity may not beconsidered as volume average mixture properties.

w xBradley and Lee 14 pointed out that dust clouds dopossess a laminar burning velocity and a laminar flamethickness similar to the ones used in the modeling of gasflames if the fuel particles emit appreciable amounts ofvolatile components. As the particle temperature increasesin the preheat zone of the flame, volatile components beginto be emitted and mix with the surrounding oxidizer. Thisleads to the formation of a primary reaction zone that islargely sustained by gaseous combustion. There are theheat release in the primary reaction zone and the subse-quent conduction of heat into the unburnt mixture thatdetermine the magnitude of the laminar flame propagationparameters. In comparison with premixed gases, the lami-nar burning velocity and the laminar flame thickness of adust cloud additionally depend on the rate of evolution ofvolatile components, the mixing of these volatile compo-nents with the oxidizer that surrounds the particles, thecoupling of the particle and gas phase oxidation and theradiative energy exchange between the flame and theunburnt mixture.

The emission of volatile components and the mixingprocess continues into the reaction zone and is largelycompleted before the end of the reaction zone. Withlignites, bituminous coals, vegetable grains and powderedfoodstuffs, the remaining char burns in the hot oxygenbeyond the preheat zone. Particles of less than 1mm canbe oxidized entirely in the primary reaction zone and theeffect of the chemical heat release of the oxidation of theseparticles on the laminar burning velocity is additive to thatof the volatile species. For particle sizes greater than 10mm, the oxidation of char in the primary reaction zone is

slow in comparison with the gaseous combustion process.In this case, the oxidation of char within the primaryreaction zone hardly affects the laminar burning velocity.However, when the oxygen concentration beyond the pri-mary reaction zone is high enough, the subsequent oxida-tion of the char particles may result in the formation of asecondary reaction zone. The presence of a secondaryreaction zone where the combustion of char occurs mayincrease the laminar burning velocity because of the addi-tional heat release. When gas phase reactions have such apredominant influence on the laminar flame propagation ofdust clouds, then it is furthermore anticipated that theenhancement of this burning velocity due to the effects ofturbulence is similar to what occurs with entirely gaseous

w xpremixed flames 15 . In spite of the similarity with thecombustion of premixed gases and the importance oflaminar flame propagation parameters in the modeling ofdust explosions, our knowledge of these flame character-istics lags behind in what is already known for gas flames.

It is the purpose of this paper to demonstrate that thesame approach which has been successful in the study ofgas flames can also be applied to gas–solid mixtures.Hence, the present study deals with the determination ofthe laminar burning velocity of dust–air mixtures. Corn-starch was chosen as the model material because of itsconsistent composition and particle size, its high volatile

w xcontent, and its use by previous researchers 15,16 . In thisexperimental work, laminar cornstarch flames were stabi-lized by means of a powder burner and the laminar burn-ing velocity at the center of the flame surfaces was deter-mined by performing velocity measurements in the flowfield along the symmetry axis. The velocity measurementswere achieved by means of laser Doppler anemometry.The laminar burning velocity of cornstarch–air mixtureswas determined with dust concentrations from 0.26 to 0.38kg my3. The determination of the unstretched laminarburning velocities from the measured laminar burningvelocities requires the use of an additional parameter,namely, the Markstein length. The role and significance ofthis quantity will be clarified in the next section.

The laminar burning velocity of cornstarch–air mixtureswas measured by previous researchers by means of othermethods. The results obtained in this work are comparedwith the results obtained by two previous investigations

w xnamely, those of Proust and Veyssiere 16 and Bradley etw xal. 15 .

2. The laminar burning velocity and the Marksteinlength

In order to clarify the aim and structure of the presentstudy of cornstarch–air mixtures, it is helpful to considerthe determination of the laminar burning velocity of pre-mixed gases.

A common way to measure the laminar burning veloc-Žity of premixed gases is by means of a Bunsen burner see

( )A.E. Dahoe et al.rPowder Technology 122 2002 222–238 225

.Fig. 1 . The device essentially consists of a tube thatserves as a mixing chamber for the fuel and the oxidizer.When the combustible mixture is ignited, a stationaryflame surface establishes itself at a small distance abovethe tube. The detachment of the flame is caused by the factthat the heat losses to the rim sustain a narrow regionwhere the temperature is so low that combustion cannotoccur. The flame anchors itself in that position by adaptingits shape to changes in the velocity of the oncomingmixture. Depending on the magnitude and the spatialdistribution of the exit velocity of the unburnt mixture, the

Ž .flame may assume a variety of shapes see Fig. 1 . If theexit velocity, z, is less than twice the laminar burningvelocity, the flame will have a parabolic shape. At greaterexit velocities, the flame assumes a conical shape and at

Žstill higher velocities e.g. five times the laminar burning.velocity the shape becomes hyperbolical and ‘tip-

blowthrough’ may occur.Burning velocity measurements with the Bunsen burner

are often conducted with conical flames. The method relieson a relationship between the laminar burning velocity of aconical flame and the flow speed at the burner exit. Strictlyspeaking, each point of the flame surface has a radius ofcurvature in the direction normal to the unburnt mixture.However, if the conical flame is sufficiently large, asurface element of the cone mantle may be regardedlocally as an oblique planar combustion wave as depictedby Fig. 2. The dashed line represents a surface element ofthe boundary where the temperature rises just above theinitial temperature of the unburnt mixture,T , due to heatu

conduction from the combustion wave. The associateddensity change beyond the dashed line is indicated by thedeflected streamlines. In the case of a planar flame thestreamlines intersect the flame perpendicularly and theseparation between any two adjacent streamlines remainsconstant along the flame surface provided that the incre-ment of the corresponding stream function is the same. Inother words, the tangential velocity to the surface element,

Ž . Ž .Fig. 1. Bunsen burner with different flame shapes: a conical flame, bŽ . Ž .parabolic flame, c low velocity button shaped flame, d hyperbolic

flame with ‘tip-blowtrough’.

Fig. 2. Decomposition of the velocity of the unburnt mixture.

z , is zero and does not change along the flame surface,t,f

and the unburnt mixture enters the flame perpendicularlywith a uniform velocityz . The laminar burning velocityf,n

of the surface element may therefore be related to thevelocity at the tube exit by measuring the cone anglea

and by decomposing the exit velocityz along the dashedŽ .line into a normal and a tangential componentz andz .n t

Then,

r S 'r z sr zsinasr z 1Ž .u uL u n u u ,f f

and hence

S szsina . 2Ž .uL

However, when this methodology is applied to the centerof the flame tip, and there is nothing wrong in doing so, itis found thatS sz becauseas90 8. Thus, the laminaruL

burning velocity at the center of the flame tip is found tobe a factor 1rsina greater than that of the cone mantleand it is evident that different laminar burning velocitiesexist within the same combustible mixture. This impliesthat the laminar burning velocity may not be regarded as afundamental mixture property unless it is ‘normalized withrespect to flame shape’ in some way. This issue has been

w xaddressed by a number of researchers 17–31 .The increased laminar burning velocity at the center of

the flame tip is associated with two phenomena namely,flame stretch and flame curvature. The flame tip is anarrow region where the cone mantle develops curvature.As the oncoming unburnt mixture reaches the flame tip, itis heated by the lateral parts of the flame, and the uniformvelocity profile is therefore distorted. The unburnt mixtureno longer enters the flame perpendicularly and has atangential velocity component. The tangential velocity also

( )A.E. Dahoe et al.rPowder Technology 122 2002 222–238226

has a gradient along the flame surface. A laminar flamewithin a nonuniform flow field with a sufficiently strongvelocity gradient along its surface is subjected to tangentialstrain and therefore develops curvature. If a curved flame

Žis concave with respect to the unburnt mixture upper part.of Fig. 3 , adjacent points traveling along the flame surface

Ž .move closer together negative flame stretch in the tan-gential direction. Conversely, if the flame is convex to-

Ž .wards the unburnt mixture lower part of Fig. 3 , adjacentŽpoints on the flame surface move further apart flame

.stretch in the tangential direction. The convergence andthe divergence of the streamlines along the flame surfacein Fig. 3 illustrate how the local flame structure is beingmodified by the strain. It is evident that the mass flow perunit area of a concave flame exceeds that of a planar flameand for a convex flame it becomes less.

Although flame curvature is invariably coupled tononuniformities of the velocity field, it must be realizedthat even in the absence of strain, it may significantly alterthe local laminar burning velocity. A flame surface servesas a local sink for reactants and a local source for heat.Increasing the diffusion rate of the reactants to the flameincreases the rate of heat release and hence, tends toincrease the flame temperature. An increase of the conduc-tion heat flux from the flame sheet into the oncomingunburnt mixture tends to reduce the flame temperature.However, changes in the conductive heat flux are coupledto changes in the diffusive flux of the reactants towards theflame surface. When a planar flame develops curvature,the heat conducted from one location is convected toanother location at the flame surface. A convex bulgeconducts heat into the oncoming unburnt mixture, as shownin the lower part of Fig. 4, and this heat is convectedtowards parts of the bulge away from the center. This leadsto an increased laminar burning velocity at the lateral partsof the flame and to a lower laminar burning velocity at thecenter. In the case of concavity, shown in the upper part ofFig. 4, the heat conducted by the lateral parts of the bulgeis convected towards the center and the effect on thelaminar burning velocity is reversed.

In the case of weakly strained flames, the influence offlame stretch on the laminar burning velocity of a curved

Ž w x.Fig. 3. Flame stretch in a nonuniform flow field after Ref. 21 .

Fig. 4. Heat conduction, reactant diffusion and convective heat transfer inŽ w x .the preheat zone of a curved flame after Ref. 32 , p. 358 .

flame is taken into account by expressing the latter asw xfollows 17,p,22,32,p,357 ,

LLoS s 1q S . 3Ž .uL uL

RR

Here So denotes the laminar burning velocity of anuL

unstretched flame and the quantityLL is known as theMarkstein length.RR is the radius of curvature of the flamesheet, defined as the reciprocal of the mean curvaturew x33,p,136 . It is taken to be positive for convexity towardsthe burnt mixture and negative for convexity towards theunburnt mixture.

w xThe Markstein length,LL , introduced by Clavin 21 forgaseous fuels, is a mixture specific constant with a magni-tude of the order of the flame thickness and serves as ameasure of the sensitivity of the laminar burning velocityto the influence of flame shape modifications. Neither itstheoretical nor its experimental evaluation is easy, andmuch remains to be learned about its precise functionaldependence on the chemical and transport properties of aspecific mixture. It is for this reason that the Williams–Clavin formula is currently accepted as an adequate rela-

Ž .tionship to be used in conjunction with Eq. 3 in order todescribe the response of the laminar burning velocity tochanges of the flame shape. The Williams–Clavin formulaw x20,23,34 is given by,

LL 1 1 Ze Ley1 1ygŽ . Ž .Mk' s ln qo ž /d g 1yg 2gL

=ln 1qxŽ .Ž .gr 1yg

d x . 4Ž .Hx0

This formula was derived for a two reactant mixture with asingle-step overall reaction rate, a large activation energy,a constant thermal conductivity, a constant kinematic vis-cosity, and a constant specific heat. In this equationMk'

o Ž 2.ŽLLrd denotes the Markstein number,Ze' E sRT TL a f f.yT the Zeldovich number,Le'lrC ID the Lewisu P

( )A.E. Dahoe et al.rPowder Technology 122 2002 222–238 227

Ž .number andgs T yT rT . With g typically lying be-f u f

tween 0.8 and 0.9, this formula predicts thatLL is 2.0–2.6times the thickness of an unstretched laminar flame,d o, ifL

Les1. More sophisticated expressions have also beenw xdeveloped. Clavin and Garcia 28,35 for example, ex-

Ž .tended Eq. 4 to include the temperature dependence ofw xthe thermal and molecular diffusivity. Rogg and Peters 31

Ž .derived an analytical expression similar to Eq. 4 byperforming a theoretical analysis on a weakly strainedstoichiometric methane–air flame using a reduced three-step mechanism with six reactants. They rejected the sim-plification of a global single step reaction involving onlytwo reactants, but retained the assumption that the diffusiv-ities are independent of the temperature.

When laminar flames are subjected to severe stretch,Ž .Eq. 3 is no longer valid and one must resort to an

equation that invokes two Markstein lengths: one for strain,LL and one for curvature effects,LL . The stretch rate,s,s c

of a surface element,A, in a strained fluid is defined as,

1 dAss . 5Ž .˙

A dt

It consists of two additive contributions,s and s . The˙ ˙s c

first includes the influence of strain and the second in-volves the effect of curvature. At any point of a flamesurface, these contributions are related to the velocity ofthe flow field, z, and the normal unit vector of the surfacein the direction of the unburnt mixture by the universalexpression,

ss ynn:=zq=Pz q So=Pn . 6Ž .˙ uL^ ` _ ^`_

ss sc

Here So is the magnitude of the unstretched laminaruLw xburning velocity vector. Clavin 21 has shown that when a

planar laminar flame with a thicknessd o is distorted intoL

a bulge of a sizeL4d o, then the local laminar burningL

ŽFig. 5. Experimental Markstein numbers measured by Quinard dataw x.taken from Ref. 21 .

Ž w x.Fig. 6. Wrinkled flame burner after Ref. 28 .

velocity at each point can be related to the local stretchrate as,

So yS LL 1 dAuL uL 2s qOO e . 7Ž . Ž .o o ž /S S A dtuL uL

Here ´sd orL is a small number. This equation can beL

rewritten into,

1 dAo 2S yS sLL qOO e 8Ž . Ž .uL uL ž /A dt

So yS sLL sqOO e 2 . 9Ž . Ž .˙uL uL

It is then obvious that the Markstein length is intended toserve as a proportionality ratio between the change of thelaminar burning velocity and the stretch rate. In order toreconcile this result with the general expression for the

Ž .stretch rate Eq. 6 , researchers have decided to expressLL s as a linear combination of quantities that account for˙

Fig. 7. Markstein lengths of methane–air mixtures from numerical experi-Ž w x.ments from Bradley et al. 24 .

( )A.E. Dahoe et al.rPowder Technology 122 2002 222–238228

Fig. 8. Markstein numbers of methane–air mixtures from numericalŽ w x.experiments from Bradley et al. 24 .

the separate effect of variables such as strain rate, flamecurvature, and pressure, each having its own Marksteinlength. Thus,LL s was replaced byLL s qLL s q . . . and˙ ˙ ˙s s c c

hence,

So yS s LL s qLL s q . . . qOO e 2 . 10Ž . Ž .˙ ˙Ž .uL uL s s c c

Ž . Ž . Ž .Combination of Eqs. 6 , 8 and 10 leads to the equationwhich is to be used in case of severe flame stretching,

w x w x oS sLL nn:=zy=Pz q 1yLL =Pn S 11Ž .uL s c uL

ow xmS sLL nn:=zy=Pz q 1q k qk LL SŽ .uL s 1 2 c uL

12Ž .LLc ow xmS sLL nn:=zy=Pz q 1q S . 13Ž .uL s uLRR

Ž . Ž .It is seen that Eq. 13 may be simplified to Eq. 3 whenthe influence of strain is negligible. In the step from Eq.

Ž . Ž . Ž12 to Eq. 13 , use has been made of the relationship seeŽ . Ž . w x.Equations 9.38.7 and 9.41.8 of Ref. 36 ,

=Pnsy k qk . 14Ž . Ž .1 2

Here k and k denote the minimum and maximum1 2

curvature of an arbitrary surface. The mean curvature isdefined as the sum of these quantities and its reciprocalvalue is considered to be the radius of curvature,RR, of aflame surface.

The Markstein length of several gaseous fuels wasinvestigated by a number of researchers and the mostillustrative of these investigations will be mentioned here.

w xQuinard 21,28 used a wrinkled flame burner to experi-mentally determine the Markstein length. The study in-volved hydrogen–, methane–, ethylene–, and propane–airmixtures as seen in Fig. 5. The measurement techniquewill be described here to illustrate how the Marksteinlength was determined. The wrinkled flame burner, shownin Fig. 6, consists of three sections: a settling chamber, aconvergent section and a rectangular burner head. Thecombustible mixture is made laminar in the settling cham-ber, and an initial uniform velocity profile at the entranceof the convergent section is ensured by grids. At the exit ofthe convergent section there is an array of water cooledtubes. These tubes cause laminar perturbations in the ve-locity profile and as a result, the flame takes the form of atwo-dimensional sinusoidal sheet. At the exit of the burner,a water-cooled grid decouples the hot combustion productsfrom the cold ambient air. The curvature of the flame frontwas obtained by means of a photo-diode array and byfitting a portion of the digitized picture to a sinusoid of anappropriate wavelength. The local burning velocity wasmeasured by means of laser doppler anemometry. Thecurvature and velocity measurements were performed at

Fig. 9. The powder burner set-up used to measure the laminar burning velocity of cornstarch–air mixtures by means of laser Doppler anemometry.

( )A.E. Dahoe et al.rPowder Technology 122 2002 222–238 229

Fig. 10. Positioning of the measurement location.

the minima and maxima of the flame surface where thestrain and curvature systematically have opposite signs andequal magnitudes. Here, also the tangential component ofthe flow velocity is zero and practically no horizontalvelocity gradient exists.

w xBradley et al. 24 solved the conservation equations formass, momentum, species and energy for inwardly ori-ented spherical laminar methane–air flames, initiated by aninstantaneous ignition around an outer spherical boundary.The reduced kinetic mechanism adopted was that of Mauss

Ž .and PetersC with 18 species and 40 reactions, in which1

OP , HOP , HO P , H O , PCH, PCH , PCHO, PCH O,2 2 2 2 2

and PCH are steady-state species. As these ‘numerical3

implosion experiments’ resolved the details of the flowfield and the curvature at any particular location of theflame front, the separate Markstein lengths,LL and LL ,s c

could be determined over a wide range of equivalenceratios. These quantities and the associated Markstein num-bers are shown in Figs. 7 and 8.

3. Experimental set-up

The experimental set-up is shown in Fig. 9 and consistsof two main parts: a powder burner to create stabilizeddust flames and a laser Doppler anemometer for measuringthe flow velocity at various locations inside the flamezone.

The powder burner consists of a glass tube in which thecombustible particles are fluidized together with a certainamount of glass beads. Before each experiment the tubewas filled with a mixture of approximately 100 g of 44–88mm glass beads and about 300 g of cornstarch. The tubehas a length of 51.5 cm, an internal diameter of 4.8 cm anda burner head, with a 28.5-mm internal diameter burnerrim, is mounted onto the tube exit. The distributor is aporous plate and the flow rate is regulated by means of aflow meter. Since the pressure drop across the fluidizedbed is impossible to predict exactly, and since the flowmeter is calibrated at atmospheric pressure, it cannot beused to control the flow rate. Hence, the exact flow ratehad to be calculated for each flame from the velocities atthe tube exit as measured with the laser Doppler anemome-

Ž .ter see Fig. 10 .Figs. 11 and 12 show the radial profile of the measured

vertical velocity component at two different heights abovethe burner head, namely, at a distance of 2.5 and 6.3 mm.Unlike the parabolic profiles commonly observed at theexit of the Bunsen burner, the velocity profiles at thepowder burner exit were observed to be flat. The reasonfor this is that the dust cloud enters the burner head as a

Žlaminar, plug flow due to the uniform velocity distribution.created by the fluid bed and the length to diameter ratio of

Žthe burner head is too small diameter 28.5 mm, length 55.mm for a parabolic profile to be formed. In the case of the

Bunsen burner, the length to diameter ratio is sufficientlylarge to allow the development of Poiseuille flow.

At greater heights, there is a zone where the initiallyuniform velocity distribution shows a change that makes it

Fig. 11. Radial profile of the vertical velocity component at 2.5 mm above the burner head.

( )A.E. Dahoe et al.rPowder Technology 122 2002 222–238230

Fig. 12. Radial profile of the vertical velocity component at 6.3 mm above the burner head.

necessary to consider two possible causes. The first possi-ble cause is the behavior of a pressure gradient. Inside theburner head, the pressure gradient is oriented along thesymmetry axis. As soon as the fluid exits the burner head,it expands and the pressure gradient points away fromcenter line. As a result, the flow changes its direction inthe outer parts of the jet, and the vertical velocity compo-nent becomes smaller.

The second possible cause may be sought in the mo-mentum diffusion, away from the laminar jet and into theambient air due to the shear forces. The distance acrosswhich this alters the velocity profile can be estimated bymeans of,

'ds p Õt , 15Ž .

where d denotes the penetration depth,Õ the kinematicviscosity and t the time during which the momentumtransfer occurs. At an exit velocity of 30 cm sy1, a

stationary fluid element needs about 0.02 s to cross adistance of 6.3 mm. With these values and a kinematicviscosity of 10y5 m2 sy1, one finds a penetration depth of0.8 mm, which does not account for the observed distor-tion of the velocity profile at a height of 6.3 mm. At largerheights, however, this effect would become an importantfactor and, together with the effect of the pressure gradi-ent, the laminar jet may entrain ambient air into the flame.

The laser Doppler anemometer used in this work mea-sures only one velocity component and consists of thefollowing components: a laser, a beam splitter, a photo-multiplier tube, a high voltage power supply and a correla-tor. The laser, the beam splitter and the photomultipliertube are mounted on a type RF 340 optical bench, suppliedby Malvern Intruments. The laser is a Melles Griot type05-LHP-927 He–Ne laser with an output power of 50 mW.It emits a light beam with a wavelength of 632.8 nm andobtains its energy from a Melles Griot 05-LPL-944-080power supply.

Fig. 13. The autocorrelation function.

( )A.E. Dahoe et al.rPowder Technology 122 2002 222–238 231

Table 1Ž .Optimal values of the parameters of Eq. 16

Parameter Value"S.E.y1Ž .a 11.5"2.0 =101y3Ž .Õ 970.1"3.8=10

y3Ž .s 20.6"7.5 =10Õ

y2Ž .m 23.8"2.5 =10y6Ž .r 92.9"9.8 =10

y6 Ž .s 7.919=10 fixedy1Ž .b y1.8"2.0=101

For a 95% confidence interval, multiply S.E. by 1.9793.

The primary laser beam is divided into two equalintensity beams by means of the beam splitter, which canbe rotated around an axis that coincides with the centerline of the primary beam. The separation between the twonew beams can be varied by adjusting the position and theorientation of the optical crystals inside the beam splitter.In this work, the separation between the two beams was setto 20 mm and a converging lens with a focal length of 250mm was used to cross the beams in order to form theprobe volume. Hence, the fringe spacing was equal to7.919=10y6 m. When a particle crosses the probe vol-ume, it scatters the light of both incident beams, and thescattered light is collected in the backscatter mode by anEMI 9863 KBr100 photomultiplier. Before reaching thephotomultiplier, the scattered light passes through a zoom-lens, a 200-mm pinhole and an optical filter that admitslight of narrow range around 632.8 nm. The high voltageŽoptimal signals were obtained in the voltage range be-

.tween 1.2 and 2.5 kV was supplied by an EMI PM 28Bpower supply. The photomultiplier tube converts the opti-cal signal into a time series of TTL signals with a repeti-tive pattern of which the cycle time corresponds to themodulation frequency of the optical signal. The electricalsignals given off by the photomultiplier are subsequentlyprocessed by a 64-channel Malvern K7025 correlator,which computes the autocorrelation function of the re-ceived electrical signals.

w xIt can be derived 37 that the autocorrelation function,Ž .G t , has the following form,

2 2 2Õ t m

G t sa exp y 1qŽ . 1 2ž / 2r

=

22 ps t 2p ÕtŽ .Õ

exp y cos qb . 16Ž .12 ž /ž / ss

In this equation:a denotes a scaling factor;Õ denotes the1

velocity component of the scattering particle, perpendicu-lar to the bisection of the angle formed by the crossinglaser beams;s denotes the standard deviation ofÕ; t

Õ

denotes the separation time;m denotes the fringe visibility;r denotes the effective radius of the measuring volume;s

denotes the fringe spacing; andb denotes the contribution1

of the background signal.The correlator was continuously operated in a specific

program,ldamenu. This program collects the autocorrela-tion functions and performs a rudimentary form of datavalidation before storage in a digital computer.

v The program normalizes the autocorrelation functionon the basis of the highest recorded value and checkswhether the highest normalized value occurs in the firstfew channels of the correlator.

v The program verifies that the normalized autocorrela-tion function has a decreasing trend.

At each measuring location in the flow field, at least sixvalid autocorrelation functions were gathered, and the

Ž .velocity was determined by fitting Eq. 16 to these datasets. Fig. 13 shows an example of a fitted autocorrelationfunction, and the optimal values of the parameters of Eq.Ž .16 are shown in Table 1.

4. Determination of the laminar burning velocity andthe Markstein length

w xDeshaies and Cambray 38 performed an experimentalstudy on the laminar burning velocity of propane–oxygen–nitrogen mixtures with a nonuniform flow field. The exper-imental set-up used by these authors is shown in Fig. 14.The reactive mixtures flowed from a cylindrical tube,where different kinds of honeycomb structures and damp-ing grids are positioned and impinges on a flat stagnationsurface. After ignition, axi-symmetric flames with a posi-tive, negative and zero curvature were seen to stabilizebelow the stagnation plate. Laser Doppler velocimetry wasused to measure the local velocity of the fluid and lasertomography to visualize the shape of a meredian section ofthe flame front. The visualization of the flame front was

Fig. 14. Schematic diagram of the experimental set-up used by Deshaiesw xand Cambray 38 .

( )A.E. Dahoe et al.rPowder Technology 122 2002 222–238232

w xFig. 15. Structure of the flow field measured by Deshaies and Cambray 38 .

accomplished by seeding the reactive mixture with smalloil droplets that evaporate upon entering the hot zones ofthe flame. Prior to entering the flame zone, these dropletsserve as scatterers of the laser sheet light, and after evapo-ration there are no scatterers present. The meridian sectionof the flame, which appears as a transition region betweenthe bright unburnt mixture and the dark flame zone, wasrecorded by means of a video camera and the pictureswere digitized and stored as grayscale images.

Deshaies and Cambray distinguished four zones in theŽ .flow field of the flames Fig. 15 : the flow field of the

unburnt mixture which is not influenced by the heat of theflame, the preheat zone of the flame, the reaction zone, anda final zone with the combustion products. In the firstzone, the vertical velocity component was found to de-crease owing to the presence of the stagnation plate at adownstream position in the flow field. The decrease of thevertical velocity component continues until the heat of the

Fig. 16. Mass throughput of the fluidized bed as a function of the velocity at the center at the lowest measuring location above the burner head.

( )A.E. Dahoe et al.rPowder Technology 122 2002 222–238 233

Ž .Fig. 17. The height of the flame tip in case of planar flames, the center of the flame surface relative to the burner head for the flames investigated in thiswork.

flame accelerates the fluid. This minimum velocity marksthe upstream boundary of the preheat zone. Deshaies andCambray fitted a straight line to the velocity profile in thefirst zone and extrapolated this information to what thevelocity would be, in the absence of heat effects, at theposition where the flame temperature attains its highestvalue. This velocity was taken as the laminar burningvelocity by these authors.

In the present work, eight cornstarch dust flames wereŽ .investigated and these are specified here as flames A–H .

Ž .At low flow rates flames A–C , the dust flames wereobserved to stabilize as a flat flame, very close to the

Ž .burner rim. At higher flow rates flames E–H the flamewas found to stabilize further away from the burner rimand its shape changed into a parabola. Between theseextremes there seems to be more than one stable situationŽ .flame D : the flame showed a tendency to change itsshape continually between the parabolic shape and the

Žplanar shape, with a preference for the former moderately.curved . The associated mass throughput and particle ve-

locities measured with the laser Doppler anemometer areshown in Fig. 16, and it is seen that this transitional

behavior of the flame occurs when the exit velocity at theburner head is equal to approximately 0.5–0.55 m sy1.Fig. 17 shows the height of the flame tip above the burnerhead of the flames investigated.

Figs. 18 and 19 show the profiles of the vertical veloc-Ž . Ž .ity component measured in flames C and D . With the

dust flames it is also possible to discern a flow fieldstructure as depicted by Fig. 17, and it is interesting to seethat the flow velocity increases further in the after-burningzone. This indicates that the laminar combustion process ofcornstarch–air mixtures may involve multiple flame zones.Only the first combustion zone, the actual flame, is thetruly premixed flame. The after-burning zone is consideredto be a diffusion flame where the burning of char takesplace, as mentioned in the introduction of this paper and

w xpointed out by Bradley and Lee 14 .The laminar burning velocity of the dust flames was

taken to be equal to the minimum velocity that marks theupstream boundary of the preheat zone for two reasons.The first reason is that an extrapolation like the oneapplied by Deshaies and Cambray would lead to unaccept-ably low and even negative values. The second reason is

Fig. 18. Vertical velocity component at the center line of flame C.

( )A.E. Dahoe et al.rPowder Technology 122 2002 222–238234

Fig. 19. Vertical velocity component at the center line of flame D.

that it is desirable to relate the propagation velocity of theflame to known properties of the combustible mixtureinvolved. Since the mixture properties undergo changes inthe preheat zone that cannot be described by means of asimple extrapolation from the mixture properties prior tothe preheat zone, such an extrapolation leads to an unde-fined, and perhaps nonexistent, frame of reference. Inorder to find the minimum value of the vertical velocitycomponent and to overcome the errors introduced by thescatter in the measured data, a third order polynomial was

w xfitted to the initial part of the velocity profiles 39 and theresulting laminar burning velocities are shown in Fig. 20.

The laminar burning velocities shown in Fig. 20 appearto be clustered according to the shape of the flame front.The more the shape deviates from a planar flame towards aparabolic flame, the higher the experimental burning veloc-ity becomes. At a dust concentration of 0.33 kg my3 thelaminar burning velocity is observed to increase by afactor of about two when the flame shape changes from

planar to parabolic. The reasons for this behavior are theflame curvature and strain due to nonuniformities in the

Ž .velocity field which were discussed in Section 2. Eq. 13was used to find the value of the unstretched laminarburning velocity.

Since the pressure gradient at the burner exit is smalland there is no stagnation point in the flow field, thecontribution of the first term on the right hand side of theabove equation is considered to make a negligible contri-bution to the experimental value of the laminar burning

Ž .velocity. Hence, the equation may be simplified to Eq. 3 ,taking only the effect of the flame curvature into account.

Ž .The first step in using Eq. 17 to find the unstretchedlaminar burning velocity is to find the Markstein length,LL , from the experimental data at a dust concentration ofc

0.33 kg my3 in Fig. 20. The idea behind this correction isŽ .that, since the dust concentration is the same in flames B ,

Ž . Ž . Ž .C and G , there is a Markstein length such that Eq. 17must give the same unstretched laminar burning velocity

Fig. 20. The laminar burning velocity,S , of flames A–H.uL

( )A.E. Dahoe et al.rPowder Technology 122 2002 222–238 235

Table 2The laminar burning velocityS , the radius of curvatureRR and theuL

unstretched laminar burning velocitySo of flames A–HuL

oŽ .Flame Dust conc. S RR m SuL uLy3 y1 y1Ž . Ž . Ž .kg m cm s cm s

A 0.26 22.8 0.3125 22.0B 0.33 29.4 0.5 28.8C 0.33 30.0 2.5 29.0

y2D 0.29 38.0 3.11=10 28.1y2E 0.21 33.1 1.12=10 16.7

y3F 0.27 41.4 8.7=10 18.3y2G 0.33 54.5 1.35=10 30.0

y3H 0.38 48.8 7.3=10 19.5

Ž .for all three flames. The radii of curvature see Table 2Ž . w Ž . Ž .xwere determined by fitting a parabola,a s s x s , y s ,

to a few points of the flame profile which was obtained byphotographing the flame and by employing the parametricform of the parabola as1

1 yY sŽ .s . 17Ž .3r2X2RR 1qy sŽ .

Ž .Application of this procedure and Eq. 17 to obtain thesame unstretched laminar burning velocity,So , for flamesuLŽ . Ž .B and G resulted in a Markstein length,LL , of 11.0c

mm. Although the Markstein length is known to depend onthe composition of the combustible mixture, the lattervalue was subsequently used to calculate the unstretchedlaminar burning velocity of all flames, and the results areplotted in Fig. 21. The unstretched laminar burning veloci-ties are between 15 and 30 cm sy1.

Fig. 22 shows a comparison between the unstretchedlaminar burning velocities obtained by means of the pow-

w xder burner and the results of Proust and Veyssiere 16 .The data band reported by these authors was obtained byobserving the motion of upwardly propagating laminarcornstarch–air flames in a tube. The parabolic dust flameswere allowed to propagate from the open end of the tubeŽ . Ž .at the bottom up to the closed end at the top inquiescent cornstarch–air mixtures of various dust concen-trations. The laminar burning velocity was taken as thedifference between the velocity of the flame contour andthe velocity of the unburnt mixture ahead of it, obtainedfrom photographic records of the moving flame. None of

1 2 Ž . w Ž . Ž .xIf a is a regular curve inR with a s s x s , y s , then theŽ . w xcurvature,k s , of a is given by 40 , p. 11

xX s yY s y xY s yX sŽ . Ž . Ž . Ž .k s s .Ž . 3r2X2 X2x s q y sŽ . Ž .In the present work the flame profile was parametrized as

x s s sŽ .a s .Ž . 2½ y s sb qb sŽ . 0 2

the resulting laminar burning velocities was corrected forflame stretch and flame curvature.

Although our results and those of Proust and Veyssiereappear to be of the same magnitude, an interpretation ofFig. 22 in terms of flame stretch and flame curvature putsthings in a different perspective. It was mentioned inSection 2 that when a flame front bulges convexly withrespect to the unburnt mixture, it propagates with a lowerburning velocity than that of a planar flame. The movingflames studied by Proust and Veyssiere were parabolic and

Žconvex with respect to the unburnt mixture see the pho-w x.tographs presented by these authors 16 . A correction of

the laminar burning velocity of a parabolic flame by meansof the radius of curvature and the Markstein length accord-

Ž .ing to Eq. 3 changes the measured laminar burningŽ Ž .velocity by a factor of 1.5 to 2 see flames D–H in Table

.2 . In their case the associated unstretched laminar burningvelocities would be between 40 and 50 cm sy1. This isabout twice the unstretched laminar burning velocitiesobtained in our experiments using the powder burner. Thereason for this discrepancy must be sought in the fact thatbuoyancy may have a different effect on moving flames ina tube than in the case of a stationary flame. When a flamepropagates in the upward direction from the open end of atube, the flame enters the unburnt mixture at the down-stream side with a velocity that is assisted by the buoyancyforce. As a result, the velocity difference between themoving flame and the moving reactants ahead of it, whichis the measured burning velocity, may be larger than in thecase of the powder burner, where the unburnt mixtureenters a stationary flame from the downstream side.

w xBradley et al. 15 determined the laminar burningvelocity of a 0.260 kg my3 cornstarch–air mixture byigniting turbulent dust clouds to deflagration in a fan-stirredbomb and found a laminar burning velocity of 12 cm sy1.The turbulent burning velocity was determined at various,known, turbulence levels. Next, the ratio between theturbulent burning velocity and the laminar burning veloc-ity, S rS , was plotted against the ratio between theuT uL

root-mean-square value of the turbulent velocity fluctua-tions and the laminar burning velocity,ÕX rS . From thisrms uL

plot, the laminar burning velocity could be determined byŽmeans of extrapolation to a zero turbulence level i.e.

X .Õ ™0 . This extrapolation is based on the assumptionrms

that the generalized correlation for the turbulent burningvelocity of gaseous fuels,

nXS ÕuT rmss1qC , 18Ž .ž /S SuL uL

is also applicable to the turbulent burning velocity of dustclouds. In this equation,C is a parameter that depends onthe length scale of the turbulence andn is known as thebending exponent. For stoichiometric propane–air mix-

( )A.E. Dahoe et al.rPowder Technology 122 2002 222–238236

Fig. 21. The unstretched laminar burning velocity,So , of flames A to H.uL

w xtures for example, Trautwein et al. 41 observed thatCs3.31 andns0.48.

At a dust concentration of 0.260 kg my3, Bradley et al.Ž y1.found a laminar burning velocity 12 cm s which is

about half the unstretched laminar burning velocity ob-Ž y1tained by means of the powder burner 22 cm s ; see

.Table 2 . This difference can be attributed to the fact thatthe results obtained by means of the fan-stirred bomb werenot corrected for the effect of flame stretch and flamecurvature. The turbulent burning velocities used in theextrapolation were obtained at a flame radii of 20, 25, 30

and 35 mm. As the flames grow in the outward directionafter point ignition, they are convex with respect to theunburnt mixture and, as discussed in Section 2, the laminarburning velocity is less than that of a planar flame. Whenthese flame radii are compared with the radii of curvaturein Table 2, it is seen that a correction by means of theMarkstein length leads to an unstretched laminar burningvelocities of about 1.5 times the measured laminar burningvelocity. With this correction, one obtains an unstretchedlaminar burning velocity of 18 cm sy1 which is closer tothe value of 22 cm sy1 found with the powder burner.

Fig. 22. A comparison between the unstretched laminar burning velocity,So , of flames A–H and laminar burning velocities measured by Proust anduLw xVeyssiere 16 .

( )A.E. Dahoe et al.rPowder Technology 122 2002 222–238 237

5. Conclusions

v Laminar burning velocities of cornstarch–air mixturesŽ y3.of various dust concentrations 0.26–0.38 kg m have

been measured by stabilizing laminar dust flames on apowder burner. The measured laminar burning velocitywas found to be sensitive to the shape of the flame. Whenthe dust concentration was kept the same, parabolic flameswere found to have a laminar burning velocity which wasalmost twice the laminar burning velocity of a planar

Ž y1flame ca. 30 cm s for the latter as compared with ca. 54y1 .cm s for the former; see Fig. 20 and Table 2 .

v From the discrepancy mentioned under the previousitem the flame curvature Mark–Stein length,LL , of lami-c

nar cornstarch–air flames could be determined, and it wasfound to have a value of 11.0 mm. With this Marksteinlength, the measured laminar burning velocities at variousdust concentrations could be corrected in order to obtainthe unstretched laminar burning velocity of cornstarch–air

Žmixtures as a function of the dust concentration see Fig..21 . The unstretched laminar burning velocities are be-

tween 15 and 30 cm sy1.v The development of damaging pressures due to explo-

sions are predicted by means of integral balance models.With gas explosions in particular, the turbulent burning

Ž .rate is commonly incorporated by means of Eq. 18 or asimilar expression and it is evident that inaccuracies in thelaminar burning velocity invariably result in large errors.Since the Markstein length of cornstarch–air mixtures ismuch larger than the Markstein length of methane–air

Žmixtures 11.0 mm for the former as compared with 0.1,.0.2 mm for the latter; see Fig. 7 errors due to the use of

laminar burning velocities which are not corrected bymeans of the Markstein length are more severe for dust–airmixtures than for purely gaseous premixtures. Hence,knowledge of the Markstein length is of crucial importancein the modeling of dust explosions.

List of symbolsA Ž 2.Flame area mCP Ž y1 y1.Constant pressure specific heat J mol KEa Ž y1.Activation energy J molKSt Volume normalized maximum rate of pressure

Ž y1.rise of a dust explosion Pa m sn Bending exponent in generalized turbulent burn-

Ž .ing velocity correlation –R Ž y1 y1.Universal gas constant J mol Ks Ž y1.Stretch rate sss Ž y1.Stretch rate due to strain ssc Ž y1.Stretch rate due to curvature sSuL Ž y1.Laminar burning velocity m sSuT Ž y1.Turbulent burning velocity m sT Ž .Absolute temperature KTf Ž .Flame temperature KTu Ž .Temperature of the unburnt mixture Kz Ž y1.Velocity vector m s

ÕXrms Root-mean-square value of the velocity fluctua-

Ž y1.tions m s

Greek lettersg Ž .Heat capacity ratio –d Ž .Penetration depth mdL Ž .Laminar flame thickness mÕ Ž 2 y1.Kinematic viscosity m sl Ž y1 y1.Thermal conductivity W m Kr Ž y3.Density kg m

Other symbolsID Ž 2 y1.Diffusion coefficient m sLL Ž .Markstein length mLLs Ž .Strain Markstein length mLLc Ž .Curvature Markstein length mRR Ž .Radius of flame curvature m

Dimensionless groupsLe Ž .Lewis number –Mk Ž .Markstein number –Ze Ž .Zeldovich number –

References

w x Ž .1 National Fire Protection Association NFPA , Venting of deflagra-tions, 1988, NFPA 68, 1988 edition.

w x Ž .2 Verein Deutcher Ingenieure VDI , Pressure venting of dust explo-sions, 1995, VDI 3673.

w x3 W. Bartknecht, Dust Explosions: Course, Prevention, Protection,Springer, 1989, Translation of Staubexplosionen by R.E. Bruderer,G.N. Kirby and R. Siwek.

w x4 R.K. Eckhoff, Dust Explosions in the Process Industries, 2nd edn.,Butterworth and Heinemann, 1996.

w x5 J. Nagy, E.C. Conn, H.C. Verakis, Explosion development in aspherical vessel, Report of investigations 7279, United States De-partment of the Interior, Bureau of Mines, Washington, 1969.

w x6 F. Tamanini, Turbulence effects on dust explosion venting, PlantrŽ . Ž .Operations Progress 9 1 1990 52–61.

w x7 A.E. Dahoe, J.F. Zevenbergen, S.M. Lemkowitz, B. Scarlett, Dustexplosions in spherical vessels: the role of flame thickness in thevalidity of the ‘cube-root-law’, Journal of Loss Prevention in the

Ž . Ž .Process Industries 9 1 1996 33–44, March.w x8 D. Bradley, A. Mitcheson, Mathematical solutions for explosions in

Ž .spherical vessels, Combustion and Flame 26 1976 201–217.w x9 J. Nagy, H.C. Verakis, Development and Control of Dust Explo-

sions, Marcel Dekker, New York, 1983.w x10 H.E. Perlee, F.N. Fuller, C.H. Saul, Constant-volume flame propaga-

tion, Report of investigations 7839, United States Department of theInterior, Bureau of Mines, Washington, 1974.

w x11 F. Chirila, D. Oancea, D. Razus, N.I. Ionescu, Pressure and tempera-ture dependence of normal burning velocity for propylene–air mix-tures from pressure–time curves in a spherical vessel, Revue

Ž . Ž .Roumaine de Chimie 40 2 1995 .

( )A.E. Dahoe et al.rPowder Technology 122 2002 222–238238

w x12 D. Bradley, M. Lawes, M.J. Scott, E.M.J. Mushi, Afterburning inspherical premixed turbulent explosions, Combustion and Flame 99Ž .1994 581–590.

w x13 F. Tamanini, Modeling of turbulent unvented gasrair explosions,Ž .Progress in Aeronautics and Astronautics 154 1993 3–30.

w x14 D. Bradley, J.H.S. Lee, Proceedings of the First International Collo-quium on the Explosibility of Industrial Dusts, vol. 2, 1984, pp.220–223.

w x15 D. Bradley, Z. Chen, J.R. Swithenbank, Burning rates in turbulentŽfine dust–air explosions, Proceedings of the 22nd Symposium Inter-

.national on Combustion, The Combustion Institute, 1988, pp.1767–1775.

w x16 C. Proust, B. Veyssiere, Fundamental properties of flames propagat-ing in starch dust–air mixtures, Combustion Science and Technol-

Ž .ogy 62 1988 149–172.w x17 G.H. Markstein, Nonsteady Flame Propagation, Pergamon, 1964.w x18 S.H. Chung, C.K. Law, An invariant derivation of flame stretch,

Ž .Combustion and Flame 55 1984 123–125.w x19 P. Clavin, F.A. Williams, Theory of premixed-flame propagation in

Ž .large-scale turbulence, Journal of Fluid Mechanics 90 1979 589–604.

w x20 P. Clavin, F.A. Williams, Effects of molecular diffusion and ofthermal expansion on the structure and dynamics of premixed flamesin turbulent flows of large scale and low turbulence, Journal of Fluid

Ž .Mechanics 116 1982 251–282.w x21 P. Clavin, Dynamic behavior of premixed flame fronts in laminar

and turbulent flows, Progress in Energy and Combustion Science 11Ž .1985 1–59.

w x22 M. Matalon, On flame stretch, Combustion Science and TechnologyŽ .31 1983 169–181.

w x23 D. Bradley, A.K.C. Lau, M. Lawes, Flame stretch as a determinantof turbulent burning velocity, Proceedings of the Royal Society of

Ž .London, Series A: Mathematical and Physical Sciences 338 1992359–387.

w x24 D. Bradley, P.H. Gaskell, X.J. Gu, Burning velocities, Marksteinlengths, and flame quenching for spherical methane–air flames: a

Ž .computational study, Combustion and Flame 104 1996 176–198.w x25 S.M. Candel, T.J. Poinsot, Flame stretch and the balance equation

Ž .for the flame area, Combustion Science and Technology 70 19901–15.

w x26 C.K. Law, Dynamics of stretched flames, Proceedings of the 22ndŽ .Symposium International on Combustion, The Combustion Insti-

tute, 1988, pp. 1381–1402.w x27 G.I. Sivashinsky, Instabilities, pattern formation, and turbulence in

Ž .flames, Annual Reviews on Fluid Mechanics 15 1983 179–199.w x28 G. Searby, J. Quinard, Direct and indirect measurements of Mark-

Ž .stein numbers of premixed flames. Combustion and Flame 82 1990298–311.

w x29 V.P. Karpov, A.N. Lipatnikov, V.L. Zimont, Flame curvature as adeterminant of preferential diffusion effects in premixed turbulent

Ž .combustion, Progress in Astronautics and Aeronautics 173 1997235–250, Chap. 14.

w x30 V.P. Karpov, A.N. Lipatnikov, P. Wolanski, Finding the Marksteinnumber using the measurements expanding spherical laminar flames,

Ž .Combustion and Flame 109 1997 436–448.w x31 B. Rogg, N. Peters, The asymptotic structure of weakly strained

Ž .stoichiometric methane–air flames, Combustion and Flame 79 1990402–420.

w x32 F.A. Williams, Combustion Theory, 2nd edn., Addison-Wesley Pub-lishing, 1985.

w x33 D.C. Kay, Theory and Problems of Tensor Calculus, Schaum’sOutline Series, McGraw-Hill, 1988.

w x34 K.N.C. Bray, N. Peters, Laminar flamelets in turbulent flames, in:Ž .P.A. Libby, F.A. Williams Eds. , Turbulent Reacting Flows, Aca-

demic Press, 1994, pp. 63–113, Chap. 2.w x35 P. Clavin, P. Garcia, The influence of the temperature dependence of

diffusivities on the dynamics of flame fronts. J. de Mec. Theoretique´ ´Ž .et Appliquee 2 1983 245–263.´

w x36 R. Aris, Vectors, Tensors, and the Basic Equations of Fluid Mechan-ics, Dover Publications, New York, 1989.

w x37 C.A.P. Zevenhoven, Particle Charging and Granular Bed Filtrationfor High Temperature Application, PhD thesis, Delft University ofTechnology, Delft, The Netherlands, December 1992, Delft Univ.Press.

w x38 B. Deshaies, P. Cambray, The velocity of a premixed flame as afunction of the flame stretch: an experimental study, Combustion

Ž .and Flame 82 1990 361–373.w x39 A. Vleeshouwers, The propagation of gas and dust flames, Part A:

The experimental determination of the laminar burning velocity ofcornstarch by means of laser doppler anemometry. Part B: Evalua-tion of density profiles in propagating methane–air flames in aconstant volume vessel by means of interferometry. Master’s thesis,Delft University of Technology, Division of Particle Technology,February 1997.

w x40 A. Gray, Modern Differential Geometry of Curves and Surfaces,Studies in Advanced Mathematics, CRC Press, 1993.

w x41 S.E. Trautwein, A. Grudno, G. Adomeit, The influence of turbulenceintensity and laminar flame speed on turbulent flame propagationunder engine like conditions, Proceedings of the 23rd SymposiumŽ .International on Combustion, The Combustion Institute, 1990, pp.723–728.