Available online at www.sciencedirect.comProceedings

Proceedings of the Combustion Institute 33 (2011) 1219–1226

www.elsevier.com/locate/proci

of the

CombustionInstitute

On the critical flame radius and minimumignition energy for spherical flame initiation

Zheng Chen a,*, Michael P. Burke b, Yiguang Ju b

a State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Aerospace Engineering,

College of Engineering, Peking University, Beijing 100871, Chinab Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA

Available online 7 August 2010

Abstract

Spherical flame initiation from an ignition kernel is studied theoretically and numerically using differentfuel/oxygen/helium/argon mixtures (fuel: hydrogen, methane, and propane). The emphasis is placed oninvestigating the critical flame radius controlling spherical flame initiation and its correlation with the min-imum ignition energy. It is found that the critical flame radius is different from the flame thickness and theflame ball radius and that their relationship depends strongly on the Lewis number. Three different flameregimes in terms of the Lewis number are observed and a new criterion for the critical flame radius is intro-duced. For mixtures with Lewis number larger than a critical Lewis number above unity, the critical flameradius is smaller than the flame ball radius but larger than the flame thickness. As a result, the minimum igni-tion energy can be substantially over-predicted (under-predicted) based on the flame ball radius (the flamethickness). The results also show that the minimum ignition energy for successful spherical flame initiationis proportional to the cube of the critical flame radius. Furthermore, preferential diffusion of heat and mass(i.e. the Lewis number effect) is found to play an important role in both spherical flame initiation and flamekernel evolution after ignition. It is shown that the critical flame radius and the minimum ignition energyincrease significantly with the Lewis number. Therefore, for transportation fuels with large Lewis numbers,blending of small molecule fuels or thermal and catalytic cracking will significantly reduce the minimum igni-tion energy.� 2010 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

Keywords: Spherical flame initiation; Critical flame radius; Minimum ignition energy; Lewis number

1. Introduction

Flame initiation is one of the most importantproblems in combustion research and it plays animportant role in the performance of combustionengines. Understanding flame initiation is thereforenot only important for fundamental combustion

1540-7489/$ - see front matter � 2010 The Combustion Institdoi:10.1016/j.proci.2010.05.005

* Corresponding author. Fax: +86 10 6275 7532.E-mail address: cz@pku.edu.cn (Z. Chen).

research but also essential for better control of fuelefficiency, exhaust emissions, and idle stability inengine operation. It is well known that successfulignition depends on the amount of energy in theform of heat and/or radicals deposited into a com-bustible mixture. If the energy is smaller than theso-called minimum ignition energy (MIE), theresulting flame kernel decays rapidly because heat/radicals conducts/diffuse away from the kernel andthe dissociated species recombine faster than theyare generated by chemical reactions within the

ute. Published by Elsevier Inc. All rights reserved.

1220 Z. Chen et al. / Proceedings of the Combustion Institute 33 (2011) 1219–1226

ignition kernel [1–4]. Furthermore, experiments[5–7] and simulations [23,24,29,30] (see also refer-ences in Ref. [1]) demonstrated that there is a criticalflame radius for spherical flame initiation such thatflame kernels that can attain this critical radius resultin successful ignition. Despite extensive researcheffort over many decades on flame initiation, itremains unclear what length scale (i.e. the criticalflame radius) controls flame initiation and how it isrelated to the MIE.

In the classical thermal-diffusion theory, theminimal radius of the developing flame kernelfor successful flame initiation was related to thequenching distance or the flame thickness [2–4].To explain their measurements of the MIE for dif-ferent mixtures, Lewis and von Elbe [2] proposedthat the spark heated the surrounding mixture andensured continued flame propagation. The mini-mum size of this heated “spark kernel” wasthought to be related to the quenching distanceor the flame thickness, such that the MIE was pos-tulated to be the spark energy necessary to heatthe kernel to the adiabatic flame temperature [2].Similarly, based on the thermal-diffusion theory,Zeldovich [4] proposed that the critical lengthcontrolling spherical flame initiation was the flamethickness such that the MIE was proportional tothe cube of the flame thickness.

Unfortunately, the above models could onlyphenomenologically describe the spark ignitionsince fuel consumption and thus mass diffusionwere not considered. A more accurate descriptionof flame ignition that included the effect of prefer-ential diffusion of heat and mass (i.e. the Lewisnumber effect) was proposed later by Zeldovichbased on studies of adiabatic flame balls [4]. A dif-fusion-controlled stationary flame ball with acharacteristic equilibrium radius – the flame ballradius – was found to exist via asymptotic analysis[4]. However, stability analysis [8] showed thatadiabatic flame balls were inherently unstable: asmall perturbation would cause the flame eitherto propagate inwardly and eventually extinguish,or to propagate outwardly and evolve into a pla-nar flame. The unstable equilibrium flame ballradius (instead of the flame thickness) was there-fore considered to be the critical length controllingspherical flame initiation such that the MIE wasproposed to be proportional to the cube of theflame ball radius [4–6].

Recently, He [9] studied mixtures with largerLewis numbers and found that a quasi-steady prop-agating spherical flame with radius less than theflame ball radius can exist when the Lewis numberis sufficiently large. It was concluded that flame ini-tiation for mixtures with large Lewis numbers wascontrolled not by the radius of stationary flame ballbut instead by a minimum flame radius for the exis-tence of self-sustained propagating sphericalflames. However, the relation between the mini-mum ignition energy and the critical ignition length

scale was not examined, and the radiative loss effectwas not considered in Ref. [9]. When radiation wasincluded [10], similar results were also found evenfor mixtures with Lewis number near unity, andthe minimum ignition energy was shown to bestrongly affected by Lewis number for both adia-batic and non-adiabatic cases.

The discussion above suggests that it is neces-sary to understand the controlling length scale(the critical flame radius) for spherical flame initi-ation in order to rigorously determine the mini-mum ignition energy. Furthermore, since MIEwas found to be strongly affected by the Lewisnumber [11], the effect of preferential diffusion ofheat and mass transfer must also be examined.Thus, the objective of the current study is toaddress the following questions: (1) is the criticalflame radius that controls spherical flame initia-tion the same as the flame thickness or the flameball radius, (2) how is the critical flame radiusrelated to the flame thickness and flame ballradius if they are not the same, (3) how is the crit-ical flame radius related to the MIE, and (4) howdoes preferential diffusion of heat and mass affectthe critical flame radius as well as the MIE?Below, we first provide a summary of theoreticalresults so as to provide a unified interpretationof the role of critical flame radius and the effectof preferential diffusion on spherical flame initia-tion. This is followed by detailed numerical simu-lations of transient spherical flame initiations ofdifferent fuel/oxygen/helium/argon mixtures (fuel:hydrogen, methane, and propane) to demonstratethe validity of the theoretical results.

2. Theoretical analysis

In our previous study [10], spherical flame ker-nel evolution (with and without an external ignitionsource at the center) was investigated analyticallybased on the quasi-steady assumption of flamepropagation in the flame-front attached coordi-nate. The theory developed in Ref. [10] will beutilized here with the main results briefly summa-rized below.

For adiabatic propagating spherical flames, thefollowing algebraic system of equations for flamepropagating speed U (normalized by adiabaticplanar flame speed), flame radius R (normalizedby adiabatic planar flame thickness), and flametemperature Tf (normalized by adiabatic flametemperature increase) is obtained [10]:

T f R�2e�URR1R s�2e�Utds

� Q � R�2e�UR ¼ 1

LeR�2e�ULeRR1

R s�2e�ULesds

¼ expZ2

T f � 1

rþ ð1� rÞT f

� �; ð1Þ

where Le, Z and r are, respectively, the Lewisnumber, Zeldovich number, and the ratio of

Z. Chen et al. / Proceedings of the Combustion Institute 33 (2011) 1219–1226 1221

burned to unburned gas densities [10]. The igni-tion power is provided as a heat flux at the center[10]. (Steady-state energy deposition is employedin order to achieve an analytical solution. Other-wise, an analytical solution cannot be obtained[10]. However, as will be demonstrated by thenumerical results, this simplification is effectiveto gain qualitative understanding of the physicsin a broad parameter range.) By solving Eq. (1)numerically, the relations for the flame propagat-ing speed, flame radius, and flame temperature, aswell as the existence of different flame regimes fordifferent Lewis numbers and/or ignition powerscan be obtained [10]. It should be noted that theeffects of radiative loss [10,12,13] are not consid-ered in this study but are part of our intended fu-ture work. In the following, the critical flameradius, the minimum ignition power, and the ef-fects of preferential diffusion will be investigatedbased on Eq. (1).

As suggested in Refs. [5–9], the critical condi-tions for a spherical flame to propagate in aself-sustained manner control the spherical flameinitiation. Therefore, the conditions for the exis-tence of a propagating spherical flame are firstinvestigated for cases without ignition powerdeposition at the center (Q = 0). Figure 1 showsthe flame propagating speed as a function of flameradius for mixtures with different Lewis numbersand Q = 0. The Zeldovich number, Z = 10, andthe thermal expansion ratio, r = 0.15, are fixedfor all the theoretical results except those inFig. 5. It is seen that for each mixture at a givenLewis number, there is a critical flame radius,RC, above which the flame can successfully prop-agate outward and eventually become a planarflame. On the other hand, there exists no quasi-steady solution below the critical flame radius.Therefore, to successfully initiate a spherical

Normalized flame radius, R

Nor

mal

ized

flam

ep

ropa

gatin

gsp

eed,

U

10-1 100 101 1020

0.4

0.8

1.2

1.6

Le=0.5

0.7

1.01.5

22.5

O

O

O O

O

O

Fig. 1. Flame speed as a function of flame radius formixtures with different Lewis numbers and Q = 0 (thecritical flame radius for each case is denoted by a circleat the corresponding minimum flame radius).

flame, the ignition source must be large enoughto sustain a flame to a radius beyond RC. The crit-ical radius is shown to increase significantly withthe Lewis number. This is due to the fact thatthe positive stretch rate (proportional to theinverse of flame radius) of the propagating spher-ical flame makes the flame weaker at higher Lewisnumbers [14,15].

The flame ball radius (RZ, corresponding toU = 0 in Fig. 1) was commonly considered to bethe minimum radius below which a spherical flamecannot propagate outwards in a self-sustained man-ner [5,6,8]. However, Fig. 1 shows that RC is equal toRZ only when Le < 1.36, and that RC < RZ forLe > 1.36. Therefore, for mixtures with Lewis num-bers larger than a critical Lewis number Le* = 1.36,the stationary flame ball radius, RZ, is not the mini-mum radius for the existence of propagating spher-ical flames, and the critical flame radius controllingspherical flame initiation is neither the stationaryflame ball radius nor the flame thickness.

We now consider cases in which an externalenergy flux is deposited in the center of a quiescentmixture and examine how the ignition power cor-relates with the critical flame radius. Figure 2shows the flame propagating speed as a functionof flame radius at different ignition powers formixtures with Le = 2. For Q = 0, only a C shapedflame branch for U–R exists (which is also shownin Fig. 1 for Le > Le* = 1.36), and there is a crit-ical flame radius, RC, at the turning point, and aflame ball radius, RZ, at U = 0. At a low ignitionpower, Q = 0.5, there is a new flame branch (leftbranch) of U–R solution curve at small flame radiiwith the flame propagating speed decreasing shar-ply to zero (flame ball solution). On the leftbranch, the maximum possible flame radius isdefined as the lower critical flame radius, R�C ,and the flame ball solution is defined as the lowerflame ball radius, R�Z . It is seen that R�C ¼ R�Z forQ = 0.5. The C shaped flame branch (rightbranch) is shown to be slightly shifted to the leftside due to the ignition power deposition. Onthe right branch, the corresponding upper criticalflame radius, RþC , and the upper flame ball radius,RþZ are defined in the opposite way. It is seen thatthe left and right branches move towards eachother when the ignition power increases. Whenthe ignition power is larger than a critical value(equal to 0.968 for Le = 2), the two branchesmerge with each other, resulting in new upperand lower branches. A spherical flame can there-after propagate outward along the upper branchU–R correlation and a successful spherical flameinitiation can be achieved. Therefore, this criticalignition power is defined as the minimum ignitionpower, Qmin. It should be noted that Fig. 2 showsthat there might be two flame propagating speedsolutions for a given flame radius and ignitionpower. However, the fast flame speed solutionis stable while the slow one is unstable, which

Normalized flame radius, R

Normalizedflamepropagatingspeed,U

0 10 20 30

10-3

10-2

10-1

100

Q=00.5

Q=0.5

0.96

0.960.97

0.97

1

Le = 2 .0

2

1

: RZ

: RZ

: RC

: RC+

-

-

+

Fig. 2. Normalized flame propagating speed as a func-tion of flame radius at different ignition powers.

1222 Z. Chen et al. / Proceedings of the Combustion Institute 33 (2011) 1219–1226

corresponds to an unrealistic solution from theo-retical analysis and thus cannot be realized bynumerical simulation [10].

The changes of the upper and lower critical flameradii and flame ball radii with the ignition power areshown in Fig. 3 for Le = 2.5 as well as Le = 2.0. It isobserved that the upper and lower flame ball radii,RþZ and R�Z , are both strongly affected by the ignitionpower. However, for the critical flame radii, R�Cmonotonically increases with Q, while RþC remainsalmost constant. The lower critical flame radiusand the lower flame ball radius are shown to bealmost the same (R�C � R�Z ). According to the defi-nition of Qmin given above, the minimum ignitionpower for successful flame initiation is reachedwhen RþC ¼ R�C (denoted by the dashed lines inFig. 3). In Refs. [5,6,8], the minimum ignitionpower is defined as the power at which RþZ ¼ R�Z .Figure 3 shows that the minimum ignition powerdefined according to RþZ ¼ R�Z (Q0min ¼ 1:048 forLe = 2 and Q0min ¼ 2:53 for Le = 2.5) is higher

Ignition power, Q

FlameballradiusRZ,criticalflameradiusRC

0 1 2 3100

101

102

Qmin = 2.05

Qmin = 0.968

Le = 2.5

Le = 2.0 RC

RZ

RZ

RZ

RC

RC

+

-

+

+

+

- RC-

RC-RC

+

RZ-

orRZ-RZ

+ or

Fig. 3. Change of upper and lower critical flame radiiand flame ball radii with the ignition power.

than the Qmin defined based on the critical flameradius (Qmin = 0.968 for Le = 2 and Qmin = 2.05for Le = 2.5). Therefore, for mixtures with Lewisnumbers larger than the critical Lewis number,Le* = 1.36, the minimum ignition power isover-predicted based on the flame ball radius.Only for mixtures with Lewis number less thanLe* = 1.36 is the minimum ignition power basedon the critical flame radius the same as that basedon the flame ball radius.

It is noted that the critical flame radius, RþC ¼ R�C ,at the minimum ignition power, is nearly the same asthe critical flame radius, RC, at zero ignition powerdeposition since RþC remains almost constant for dif-ferent Q (Fig. 3). Comparing the critical flame radiusand the flame ball radius at Q = 0 reveals that theminimum ignition power is over-predicted usingthe flame ball radius instead of the critical flameradius. Figure 4 shows the variation of the criticalflame radius and the flame ball radius (both normal-ized by the flame thickness) with Lewis number.They are both shown to depend strongly on theLewis number. Three different regimes in terms ofthe Lewis number are observed. In regimes I andII, with Lewis number less than Le* = 1.36, the crit-ical flame radius is the same as the flame ball radiusand it is smaller/larger than the flame thicknesswhen the Lewis number is below unity (regime I)/above unity (regime II). In regime III, withLe > Le*, the critical flame radius is shown to besmaller than the flame ball radius but larger thanthe flame thickness. The difference between the crit-ical flame radius and the flame ball radius/flamethickness is of order-one magnitude when Le >2.0. As a result, the minimum ignition power willbe substantially over-predicted/under-predictedbased on the flame ball radius/flame thickness formixtures with large Lewis numbers. Therefore, theflame ball radius can be considered as the flame ini-tiation controlling length only for mixtures withLe < Le* while the flame thickness only forLe = 1.0.

Le

RCorRZ

0.5 1 1.5 2 2.510-1

100

101

102

RC

RZ

RC=RZ=1

I II III

Le=Le*Le=1

Fig. 4. Change of the critical flame radius and the flameball radius with the Lewis number.

Z. Chen et al. / Proceedings of the Combustion Institute 33 (2011) 1219–1226 1223

To reveal the correlation between Qmin and RC,the minimum ignition power, Qmin, and the cubeof the critical flame radius, R3

C, for mixtures withdifferent Lewis numbers (Le = 1.4–2.5) and Zel-dovich numbers (Z = 10, 13) are plotted inFig. 5. Both Qmin and R3

C are shown to dependstrongly on the Lewis number. Moreover, it isobserved that the minimum ignition power variesalmost linearly with the cube of the critical flameradius, i.e. Qmin � R3

C . Therefore, the minimumenergy deposition for successful spherical flameinitiation is proportional to the cube of the criticalflame radius. If the minimum ignition power isplotted against the cube of flame ball radius orflame thickness, the correlation would not be lin-ear since the flame ball radius/flame thickness ismuch larger/smaller than the critical flame radiuswhen Le > Le* = 1.36.

The above results show that the relationshipamong the critical flame radius, the flame ballradius, and the flame thickness strongly dependson the Lewis number and that the minimum igni-tion power varies almost linearly with the cube ofthe critical flame radius. These results wereobtained from theoretical analysis. One limitationof this analysis is that the ignition energy deposi-tion is modeled as a boundary condition in thecenter [10]; under most realistic conditions, it isdeposited as a function of time and space. More-over, the theoretical analysis is constrained byquasi-steady, constant-density, and infinite activa-tion energy assumptions. In the next section, wepresent results from direct numerical simulations(with detailed chemistry) that include all of theabove-mentioned effects neglected in the theoreti-cal analysis in order to test the applicability of thetheoretical results under more realistic conditions.As shown below, the simulations qualitativelyconfirm the results obtained from the theoreticalanalysis.

Z = 10

Cube of the critical flame radius, RC3

Minimunignitionpower,Q

min

0 500 1000 1500 2000 25000

0.5

1

1.5

2

Z = 13

2.0

1.4

1.6

1.7

1.8 = Le

1.9

1.51.4

2.5

2.0

1.9

1.8

1.71.6

1.5

Le = 2.1

2.3

2.4

2.2

Fig. 5. Minimum ignition power and cube of the criticalflame radius for mixtures with different Lewis numbersand Zeldovich numbers.

3. Numerical simulation

A time-accurate and space-adaptive numericalsolver for Adaptive Simulation of UnsteadyReacting Flow, A-SURF, has been developed[19] and used to study spherical flame initiationand propagation. The conservation equations ofone-dimensional, compressible, multi-compo-nent-diffusive, reactive flow in a spherical coordi-nate are solved using the finite volume method[16,19]. The convective flux, diffusion flux, andstiff chemistry are calculated by MUSCL-Han-cock scheme, central difference scheme, andVODE solver, respectively [16,19]. The thermody-namic and transport properties as well as thechemical reaction rates are evaluated by CHEM-KIN packages [17] incorporated into A-SURF.A-SURF has been successfully used in our previ-ous studies on propagating spherical flames [18–22]. The details on the governing equations,numerical schemes, and code validation can befound in Refs. [16,19].

In all the simulations, the computationaldomain is 0 6 r 6 100 cm and a multi-level,dynamically adaptive mesh [16,19] with a mini-mum mesh size of 8 lm is used. Zero-gradientconditions are enforced at both inner (r = 0) andouter (r = 100 cm) boundaries. At the initial state,the homogeneous mixture is quiescent at 298 Kand atmospheric pressure. Flame initiation isachieved by spatial dependent energy depositionfor a given ignition time [23]

_qignit ¼E

4pr3igsig=3

exp � p4

rrig

� �6� �

if t < sig

0 if t P sig

8<:

ð2Þwhere E is the total ignition energy, sig, the dura-tion of the energy source, and rig, the ignition ker-nel radius. It is noted that the duration of thesource energy and the ignition kernel size both af-fect the MIE [1,2,24]. In this study, since theemphasis is on the correlation between the MIEand the critical flame radius and on the Lewisnumber effect on spherical flame initiation, boththe ignition kernel size and time are kept constantwith sig = 200 ls and rig = 200 lm, respectively.The values of sig and rig are chosen according tothe simulation results reported in Ref. [23] whichpresented the dependence of the MIE on the igni-tion kernel size and time.

Simulations utilizing detailed chemical mecha-nisms for fuel/O2/He/Ar mixtures (fuel: H2, CH4

and C3H8) have been conducted. For H2/O2/He/Ar mixtures, the recent mechanism developed byLi et al. [25] is employed. For CH4/O2/He/Arand C3H8/O2/He/Ar mixtures, GRI-MECH 3.0[26] and San-Diego Mechanism 20051201 [27]are used, respectively. The mixture compositionis specified in the form of

L b(mm)

0

0.5

1

1.5

270% He50% He25% He0% He

(a)

(b)

1224 Z. Chen et al. / Proceedings of the Combustion Institute 33 (2011) 1219–1226

0:3u

uþ rsFuelþ rs

uþ rsO2

� �þaHeþð0:7�aÞAr;

ð3Þwhere u is the equivalence ratio and rs is the stoi-chiometric oxygen-to-fuel molar ratio (rs = 0.5 forH2, 2.0 for CH4, and 5.0 for C3H8). The volumet-ric fraction of fuel and oxygen is fixed to be 30%,while that of inert diluents, helium and argon, isfixed to be 70%. With the increase/decrease ofthe helium/argon fraction, the thermal diffusivityof the mixture increases, resulting in a higher Le-wis number [18] while the global activation energyas well as the adiabatic flame temperature remainsnearly unchanged, resulting in the same Zeldovichnumber. Therefore, different amounts of helium(a = 0%, 25%, 50% and 70%) can be used hereto investigate the Lewis number effect on sphericalflame initiation.

Figure 6 shows the flame radius evolution fordifferent ignition energies for H2/O2/He/Ar atu = 2.0 and a = 0%. The MIE for this mixture,Emin = 0.165 mJ, and for all other mixtures wascalculated by the method of trial-and-error withrelative error below 2%. It is observed that aself-sustained propagating flame can be success-fully initiated only when the ignition energy isabove the MIE. By plotting the flame propagatingspeed, Sb = dRf/dt, as a function of flame radius,results (not shown here due to space limitation)similar to the theoretical predictions shown inFig. 2 (except the right and lower branches whichcannot be calculated from transient numericalsimulation) are obtained. In the numerical simula-tion, the critical flame radius cannot be defined inthe same way as that in the quasi-steady theoreti-cal analysis. However, Fig. 2 shows that the criti-cal flame radius is almost the same as the flameradius at which the minimum propagating speedoccurs with the minimum ignition power deposi-tion (Q = 0.97). Therefore, the critical flame

Time, t (s)

Flameradius,R

f(cm)

0 0.001 0.002 0.0030

0.1

0.2

0.3

0.4

0.165

E=0.2 mJ

0.17

0.18

0.160.15

H2 : O2 : Ar =24% : 6% : 70% vol.

Critical radius: RC = 0.11 cm

Fig. 6. Spherical flame initiation for H2/O2/Ar mixturesat different ignition energies.

radius in the transient numerical simulations isdefined as the radius corresponding to the mini-mum propagating speed for MIE deposition. Asimilar definition was also used in experimentalmeasurements [7]. In Fig. 6, the critical flameradius, RC = 0.11 cm, occurs at the inflectionpoint for E = Emin = 0.165 mJ.

To investigate the preferential diffusion effecton spherical flame initiation, numerical simula-tions on the initiation of H2/O2/He/Ar flames atdifferent equivalence ratios and different amountsof helium dilutions were conducted. The burnedMarkstein length (Lb), the critical flame radius(RC), and the MIE (Emin) of different H2/O2/He/Ar mixtures are shown in Fig. 7. The burnedMarkstein length, Lb, is obtained from linearregression of the flame propagating speed,Sb = dRf/dt, and the flame stretch rate, K = (2/Rf)(dRf/dt) [15,21,22]. Figure 7a shows that theburned Markstein length increases with thehelium fraction at each equivalence ratio. It is wellknown that the burned Markstein length increaseswith the Lewis number [14,15]. Therefore, as men-tioned before, the Lewis number also increaseswith helium fraction. Figure 7a also shows thatthe burned Markstein length increases with theequivalence ratio for a fixed helium fraction.Therefore, the Lewis number of the H2/O2/He/Ar mixture increases with both the helium frac-tion and the equivalence ratio. Consistent with

RC(mm)

0

0.5

1

1.5

2

E min(mJ)

1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

ϕ

(c)

Fig. 7. Burned Markstein length (a), critical flameradius (b), and minimum ignition energy (c) of H2/O2/He/Ar mixtures.

Cube of the critical flame radius, RC3 (mm3)

Minimum

ignitionenergy,E

min(mJ)

0 1 2 3 4 50

0.5

1

1.5

H2

C3H8

CH4

He % vol. = 0, 25, 50,70 %

Stoichiometric Fuel+O2 = 30 % vol.Inert gas: He + Ar = 70 % vol.

Fig. 9. Minimum ignition energy and cube of the criticalflame radius of different fuel/O2/He/Ar mixtures.

Z. Chen et al. / Proceedings of the Combustion Institute 33 (2011) 1219–1226 1225

the theoretical analysis on the Lewis number effect(Fig. 5), Fig. 7b and c show that the critical flameradius and the MIE of the H2/O2/He/Ar mixturesalso increase with the helium fraction and theequivalence ratio. These results are also consistentwith previous studies on the Lewis number effecton the critical flame radius [7] and the MIE [11].

To reveal how the critical flame radius is cor-related with the MIE, Fig. 8 shows the MIE as afunction of the cube of the critical flame radiusfor H2/O2/He/Ar mixtures with different equiva-lence ratios and different helium fractions. Simi-lar to the theoretical results (Fig. 5), thetransient numerical simulation also shows thatthe MIE changes almost linearly with the cubeof critical flame radius instead of the flame thick-ness or the flame ball radius, demonstrating a lin-ear correlation: Emin � R3

C . Therefore, the lineardependence of the MIE on the cube of the criticalflame radius reveals that the critical flame radiusdefined in this study is the controlling lengthscale for spherical flame initiation and the MIE.Figure 8 also shows that the slope of Emin–R3

Cdecreases with the increase of the equivalenceratio. This is because the activation energyincreases while the adiabatic flame temperaturedecreases with the equivalence ratio for richH2/O2/He/Ar flames, which results in a largerZeldovich number at higher equivalence ratio.Therefore the numerical simulation demonstratesthat the slope of Emin–R3

C decreases with the Zel-dovich number, which is the same as the conclu-sion drawn from the theoretical analysis (seeFig. 5). As discussed above, the ignition energyis specified differently in theory and simulation– it is given as a boundary condition at the centerin the theoretical analysis and as a volumetric,time-dependent heat source in the simulation.Therefore, the comparison between the theoreti-cal and numerical results yields qualitativeinstead of quantitative agreement.

Cube of the critical flame radius, RC3 (mm3)

Minimum

ignitionenergy,E

min(mJ)

0 2 4 6 8 10 12 140

0.2

0.4

0.6

0.8

1= 1.5= 1.8= 2.0

ϕ

ϕϕ

12

3

4

1

2

3

4

1

2

3

4

Case vol.% He1 0%2 25%3 50%4 70%

Fig. 8. Minimum ignition energy and cube of the criticalflame radius of different H2/O2/He/Ar mixtures.

To further demonstrate the validity of the the-oretical results, simulations for other fuels (CH4

and C3H8) have also been conducted. Similarresults to those of H2/O2/He/Ar are obtained.Figure 9 shows that the linear correlation,

Emin � R3C, also works for CH4/O2/He/Ar and

C3H8/O2/He/Ar mixtures. Therefore, the numeri-cal simulation demonstrates the validity of thetheoretical results in that the MIE is proportionalto the cube of the critical flame radius instead ofthe flame thickness or the flame ball radius.

4. Concluding remarks

Spherical flame initiation is studied usingasymptotic analysis and detailed numerical simu-lations. The critical flame radius that controlsspherical flame initiation is found to be differentfrom the flame thickness and the flame ball radius.Depending on the Lewis number, there are threedifferent flame initiation regimes: in regime I withLe < 1.0, the critical flame radius and the flameball radius are the same and are smaller than theflame thickness; in regime II with 1.0 < Le < Le*,the critical flame radius and the flame ball radiusare also the same but are larger than the flamethickness; while in regime III with Le > Le*, thecritical flame radius is smaller than the flame ballradius but larger than the flame thickness. There-fore, the minimum ignition energy can be substan-tially over-predicted (under-predicted) based onthe flame ball radius (the flame thickness) for mix-tures with large Lewis numbers. A linear relation-ship between the minimum ignition energy/powerand the cube of the critical flame radius is demon-strated by both theory and simulation. Moreover,the Lewis number effect is shown to play a veryimportant role in spherical flame initiation. Thecritical flame radius and the minimum ignitionenergy increase significantly with the Lewis

1226 Z. Chen et al. / Proceedings of the Combustion Institute 33 (2011) 1219–1226

number. Therefore, for fuels with much higherthermal diffusivity than fuel mass diffusivity, lar-ger ignition energy is needed to initiate a self-sus-tained propagating premixed flame.

In previous experimental studies on the MIE inthe literature (see Ref. [1] and references therein),not only the Lewis number but also the globalactivation energy and the flame temperature var-ied with mixture composition. Therefore, thosedata cannot be used to demonstrate directly theeffects of Lewis number on the ignition and thelinear correlation between MIE and the cube ofthe critical flame radius (Figs. 5, 8 and 9 show thatthe linear correlation depends on the activationenergy). Nevertheless, the experimental data [28]showed that for lean mixtures, the MIE increasesdramatically from methane to ethane, propaneand butane, which is consistent to the presentresults of the effect of the Lewis number. In orderto further validate the theoretical results in thisstudy by experiments, measurements of the MIEand critical flame radius of different fuel/O2/He/Ar mixtures should be conducted, for which theLewis number can be modified by varying thehelium/argon fraction such that the global activa-tion energy as well as the adiabatic flame temper-ature remain nearly unchanged.

Acknowledgements

The work at Peking University was supportedby National Natural Science Foundation of China(Grant NO. 50976003) and State Key Laboratoryof Engines at Tianjin University (Grant NO.K2010-02). The work at Princeton Universitywas supported by the Air Force Office of ScientificResearch (AFOSR) Plasma Assisted CombustionMURI research program under the guidance ofDr. Julian Tishkoff and the US Department ofEnergy, Office of Basic Energy Sciences as partof an Energy Frontier Research Center on Com-bustion (Grant NO. DE-SC0001198).

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