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Study of Methane Propagating Flame Characteristics Using

PDF-Monte Carlo Model and Reduced Chemical Kinetic Scheme

Ridha Ennetta,* Marzouk Ljili, and Rachid Said

Ionized and ReactiVe Media Studies (EMIR), Preparatory Institute of Engineers Studies of Monastir(IPEIM), AVenue of Ibn El Jazzar, 5019 Monastir, Tunisia

ReceiVed January 17, 2009. ReVised Manuscript ReceiVed April 25, 2009

The main purpose of this work is to simulate turbulent premixed and expanding flame in an adiabaticcombustion chamber without taking into account flame-walls interactions. The turbulence is supposed to beisotropic and homogeneous with no decay or decreasing, but spatially correlated. The chemistry is representedby a four-step scheme of methane-air combustion. A one-dimensional simulation is considered because ofthe spherically symmetric of the problem. Ignition occurs in the middle of the domain thanks to an energysource such a spark. The simulation, based on Monte Carlo scalar probability density functions (PDF) transportmethod, is used under different equivalence ratios (ER) and different turbulence intensities ( u). We haveplaced our emphasis on some flame characteristics such as the flame mean radius, the turbulent flame radius,the flame propagation velocity, and the flame brush thickness. The results of our simulations, carried out insimilar conditions to some available experiments, are in good agreements. Indeed, we notice that the flame

radius is enhanced by ER at constant u, and by u at constant ER. The flame brush thickness shows a quickgrowth at the first stage and a moderate growth in the intermediate regime, but a bending effect is observedin the final stage of the so-called fully developed flame.

1. Introduction

Due to their fundamental importance for premixed combus-tion theory, turbulent flame characteristics were a subject of alarge number of numerical investigations for many decades.1

Reliable and flexible computational modeling is the key inachieving the objectives of such investigations. Direct numericalsimulation (DNS), Large eddy simulation (LES), and Reynolds-

averaged Navier-Stokes (RANS) are widely considered as threeprincipal approaches in computational turbulent combustion.2

DNS provides a very accurate, model-free representation of theunsteady evolution of turbulent flows. However, applicationsare limited by the computational power.3-5 LES has been thesubject of much modern research and is increasingly becomingmore popular.6 RANS is the most popular approach forengineering applications.7

Various models have been proposed to describe turbulentpremixed flames.8 Probabilistic approaches have proven to beeffective in this regard.3,6 This approach follows from the

definition of the fine-grained density function.9 The ensembleaverage of the fine-grained density function is considered andis termed the probability density function (PDF).10

The primary advantages of probability methods are: (i) theyprovide closed-form representation of chemical source terms,and (ii) they are applicable to both premixed and non-premixedflames.

The probability density functions (PDF) methods have beenpopular since 1970s.11,12 Development of the Lagrangian MonteCarlo particle methods13 has enabled PDF calculations to beconducted of a variety of flame configurations.14

The main purpose of this work is to simulate a premixedturbulent flame in a constant volume vessel. The adoptedconditions are similar to those realized in many experiments.The Monte Carlo scalar PDF transport method that is the basisof our numerical simulation has been used for few decades bymany authors to simulate ignition time and the flame growth inturbulent combustion case.15-18 Nowadays, this method con-tinues to be efficient for calculating flame characteristics taking

* To whom correspondence should be addressed. Phone: (+216)98487430;

fax: (0216)73500277; E-mail: ridha_ennetta@yahoo.fr, Ridha.Ennetta@isetjb.rnu.tn.

(1) Lipatnikov, N.; Chomiak, J. Progr. Energ. Combust. Sci. 2002, 28,174.

(2) Givi, P. Spectral and Random Vortex Methods in Turbulent ReactingFlows. In Turbulent Reacting Flows; Libby, P. A., Williams, F. A. Eds.;Academic Press: London, UK, 1994; Ch 8, 475-572.

(3) Pope, S. B. Proc. Combust. Inst. 1990, 23, 591612.(4) Poinsot, T.; et al. Prog. Energy Combust. Sci. 1996, 21, 531576.(5) Vervisch, L.; Poinsot, T. Annu. ReV. Fluid Mech. 1998, 30, 655

691.(6) Givi, P. AIAA J. 2006, 44 (1), 1623.(7) Pope, S. B. Advances in PDF Methods for Turbulent Reactive Flows.

In AdVances in Turbulence X; Andersson, H. I., Krogstad, P. A. Eds.;CIMNE: 2004; pp 529-536.

(8) Veynante, D.; Vervisch, L. Prog. Energy Combust. Sci. 2002, 28(3), 193266.

(9) Lundgren, T. S. Phys. Fluids 1967, 10 (5), 969975.

(10) OBrien, E. E. The Probability Density Function (PDF) Approachto Reacting Turbulent Flows. In Turbulent Reacting Flows, Topics in AppliedPhysics; Libby, P. A., Williams, F. A., Eds.; Springer-Verlag: Heidelberg,1980; Ch 5, 44, 185-218.

(11) Dopazo, C.; OBrien, E. E. Combust. Sci. Technol. 1976, 13, 99112.

(12) Pope, S. B. Combust. Flame 1976, 27, 299312.(13) Pope, S. B. Annu. ReV. Fluid Mech. 1994, 26, 2363.(14) Bilger, R. W.; et al. Proc. Combust. Inst. 2005, 30, 2142.(15) Haworth, D. C.; Pope, S. B. J. Comput. Phys. 1987, 72, 311346.(16) Galzin, F. Contribution la modlisation de la combustion dans

les moteurs allumage command. Ph.D. Thesis, University of Rouen:France, 1996.

(17) Fruchard, N. Lallumage dans les moteurs essence: une modli-sation et des applications. Ph.D. Thesis, University of Rouen: France, 1995.

(18) Borghi, R.; Champion, M. Modelisation et theorie des flammes;TECHNIP: Paris, France, 2000.

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into account turbulence and chemistry interaction using reducedchemical kinetic mechanisms.19,20 For our case, turbulence issupposed to be homogeneous and isotropic, and its character-istics17 are those of the frozen k- model. Moreover, theturbulence does not present any decay or dissipation ratedecrease because we suppose that the calculation domain is anopen space without flame-walls interactions, nut turbulentvelocities are spatially correlated, and this fact makes our studydifferent from classical PDF methods.

In the present study, chemistry is described by the four-stepchemical kinetic mechanism of Jones and Lindstedt.21 Themixture is composed of methane as fuel and air as oxidizer.

The results of our simulations are in good agreements withsome available experimental data obtained in fan-stirred spheri-cal bombs (experimental apparatus and measurement methodsare well described in literature30-32,43,44). We have notice anenhancement, by the equivalence ration (ER) and the turbulentintensity (u), of the flame radius and the turbulent velocity.The flame brush thickness shows a quick growth at the firststage, when the turbulent velocity grows too, and a moderategrowth in the intermediate regime after the turbulent speedreaches its asymptote, but a bending effect is observed in thefinal stage of the so-called fully developed flame.

2. Theoretical Basis

2.1. Formulation. The equations that determine our problemare, respectively:

the mass conservation equation written in spherical coor-dinates, whose solution gives us the expansion velocities dueto temperature gradient between hot products and fresh gas:

the Lagrangian joint PDF transport equation:

where

and

the perfect gas equation:

where F is the density, ur is the Favre averaged radial velocitycorresponding to the expansion velocity due to the temperature

gradient between hot products and fresh gas. In eq 2, and uare, respectively, a scalar vector and a velocity vector in thephysical space and to which correspond respectively the vectorsof random values, and Vin the conditional space. Fu, is thePDF of velocities and scalars. The change terms that character-ized the stochastic process Ai and R present the following terms:ij that is the strain tensor, Fi is the stirred force per volumeunity, P is the pressure, Jk

R represents the diffusive fluxes, R isthe reaction rate, and, finally, SR is the source term. In eq 3, T

is Favre averaged temperature, and M is the molar mass of themixture.

2.2. PDF-Monte Carlo Method. The use of PDF constitutesa potential solution to describe the evolution of turbulent reactiveflows in which fluctuation terms need statistical treatment. Thetype of PDF that we use in this work is the evolution PDF(transported PDF) called Popes method.15 This method uses aMonte Carlo particle solver, and the form of the PDF may freelyevolve.

It can be highlighted that the high dimensionality of underly-ing PDF scalar transport equation requires Monte Carlostochastic solution methods. Monte Carlo method evokes therepresentation of the PDF with a whole of elements distributed

throughout the flow field, and from which the moments ofinterest may be calculated. In the Lagrangian case, the elements(particles) are free to roam the physical domain as dictated bythe hydrodynamic field, and the composition of the elementschanges only due to mixing and reaction.22

The calculation domain is divided into a given number ofcells Nc. Initially, each one contains Ni particles. These particlesmove in the domain thanks to the following velocities:

gas expansion velocity due to temperature gradient betweenburned gases (hot products) and fresh gases.

turbulent diffusion where a correlation velocity deducedfrom turbulence spectrum is respected.

2.3. Simulations Parameters. The physical environment ofthe current computation is a one-dimensional (1D) spherically

symmetric domain with changes only in the radial directionbeing considered. The premixed methane-air mixture is initiallyat rest, and flame initiation is triggered by a localized depositof thermal energy generated by a hot source (spark, laserignition, pyrotechnic devices, etc.), making the temperature growhighly in this region,17 as is the case in internal combustionengines or other such devices. The computational environmentin the radial direction is composed of Nc ) 500 cells each witha specified size r. Initially, each cell contains a certain numberof particles (Ni ) 1000 for example). Each particle has a velocityand some themochemical variables, such as concentration andtemperature. The pressure is held constant throughout thecalculation at 1 atm.

The numerical procedure tracks each fluid particle during eachtime step to account for convective transport by the mean flowand by turbulence in space and in time. Chemical and thermalcomposition of particles changes due to chemical reaction andmolecular mixing.

To detect more precisely the flame front evolution, we havechosen a cell size (r ) 0.25 mm) and a calculation time step(t) 0.1 ms) which remain constant along the time and overallthe domain. These lasts parameters are chosen from literature.Indeed, in a recent work, Raman et al.23 have performed somenumerical tests to determine the optimal global time step, theoptimal space step, and to estimate the increase in computation

(19) Ren, Z.; Pope, S. B. Combust. Flame 2004, 136, 208216.

(20) Garrick, S. C.; and Interante, V. 9th Int. Symp. Flow Visualisat.,Heriot-Watt University: Edinburgh, 2000; 1-10.(21) Jones, W. P.; Lindstedt, R. P Combust. Flame 1988, 73, 233249.

(22) Lajili, M.; et al. Prog. Comput. Fluid Dynamics 2008, 8 (6), 331341.(23) Raman, V.; et al. Combust. Flame 2004, 136, 327350.

F

t+

1

r2

(r2Fur)

r) 0 (1)

t(F(_ )fu_,_ (V_,_ )) +

i)1

3

xi[F(_ )Vifu_,_ (v_,_ )] )

-i)1

3

Vi[F(_ )Ai|V_,_ fu_,_ (V_,_ )]

-R)1

N

R

[F(_ )R|V_,_ fu_,_ (V_,_ )]

(2)

Ai )1

F(ij

xj-

P

xj+ FFi); i,j ) 1, 3

R )1

F(Jk

R

xk+ FR + SR); R ) 1, N

Pj ) FRT

M(3)

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load with the number of particles per cell and consequently thenumber of particles in the whole domain. It can be concludedthat the tracking time scales linearly with particles number, sizegrid, and global time step. Moreover, it is found for themethane-air studied that a time step of 1.0 10-4 s yieldsconsistent results for both reacting and nonreacting cases.23

Turbulent length scale and turbulent time scale values are equalto those used in experimental cases and given by the followingformulas:

The following input values are required for the simulation: timestep t, cell size r, turbulent kinetic energy k, dissipation ofturbulent kinetic energy , normalized mean fuel concentration,Cfini, ignition energy x, pressure, ignition time tignit, ignition gaprignit, initial number of particles in each cell Ni, adiabatic flametemperature Tadiab, and initial temperature Tini.

2.4. Chemical Kinetics. Detailed chemical kinetic descrip-

tions of hydrocarbon combustion may require the tracking ofhundreds of chemical species and thousands of reaction steps.For the foreseeable future, CPU time and computer memorylimitations will prohibit implementation of fully detaileddescriptions of combustion chemistry into CFD simulations ofcombustion hardware.24-26 As a result, it is important tominimize this number while retaining essential features of thedetailed chemistry.

Reduced chemical kinetic mechanisms, which can representimportant aspects of the behavior of these detailed mechanismsusing few enough scalars that they can be implemented intoCFD simulations, offer large potential improvement in themodeling of practical combustion devices.27 The feasibility ofthese mechanisms in the simulation of internal combustionengines (ICE) was demonstrated in a previous study.28

In the present study, the four-step reaction mechanism ofJones and Lindstedt21 was chosen thanks to its good results in1D propagating flame21,29 and in ICE simulations:28

3. Results and Discussion

In this section we try to study more precisely the turbulentflame front characteristics such as the flame mean radius, the

flame turbulent velocity, and the flame-brush thickness, whichrepresents a great challenge for the turbulent combustioncommunity. The predicted values are compared to the experi-mental results realized by many authors. The geometry studiedwas a parallelepiped vessel with a constant volume. Ignitionoccurs in the middle of the domain then a flame kernel growsspherically.

3.1. Flame Mean Radius. The flame mean radius (Rf) isdefined by Lecordier,30 in 2D configuration, as the radius of a

circle that contains the same surface of burned gas ( Sb).

However, in our case (1D configuration), the flame mean radiusis defined as the distance between the ignition center and theflame front position. The latter is determined when finding apitchfork of temperature in the vicinity of 600 K, whichcorresponds to silicon oil vaporisation temperature in theexperimental case.

Figures 1 and 2 show the flame mean radius evolutionfunction of time for different turbulence intensities and differentequivalence ratios. We can see that the predicted values foundby our simulations are in good agreements with the experimentalones.

In order to study the equivalence ratio (ER) effect on flame

mean radius, one maintains the turbulence intensity u constantand varies the ER. The effect of this last parameter on flame

(24) Montgomery, C. J.; et al. 46th AIAA Aerospace Sciences Meetingand Exhibit; AIAA: Reno, Nevada, 2008; p 1014.

(25) Glassman, I.; Yetter, R. A. Combustion, 4th ed.; Elsevier: 2007.(26) Skevis, G.; Goussis, D. A.; Mastorakos, E. Int. J. Altern. Propul.

2007, 1(2/3), 216-227.(27) Montgomery, C. J.; et al. J. Propul. Power2002, 18 (1), 192198.(28) Ennetta, R.; Hamdi, M.; Said, R. Therm. Sci. 2008, 12 (1), 4351.(29) Truffin, K. Modelling of Methane/Air Flames Using Detailed and

Reduced Kinetic Schemes; CERFACS, INP Toulouse: France, 2001.(30) Lecordier, B. Etude de linteraction dune flamme prmlange avec

le champ arodynamique, par association de la tomographie Laser et de laP.I.V. Ph.D. Thesis, University of Rouen, France, 1997.

(31) Bradley, D.; et al. Combust. Flame 1994, 99, 562572.(32) Hainsworth, D. Study of Free Turbulent Premixed Flames. M.Sc.Thesis, Massachusetts Institute of Technology, USA, 1985.

lt ) 0.164k

3/2

(4)

t ) 0.3k

(5)

CH4 +1/2O2 f CO + 2H2

CH4 + H2O f CO + 3H2

H2 +1/2O2 T H2O

CO + H2O T CO2 + H2

Rf ) 1Sb (6)

Figure 1. Flame mean radius evolution vs time.

Figure 2. Flame mean radius evolution vs time.

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mean radius is shown in Figures 1 and 2. It is worth notingthat for lean flames the flame mean radius is quite parabolic.However, as we increase ER to stoichiometry; the flame meanradius becomes strongly parabolic. Also, Figure 1 shows thatthe effect of turbulence intensity is more significant than theER one, on the flame mean radius.

These results are in perfect agreement with those announcedby Bradley31 and Hainsworth.32

3.2. Flame Propagation Velocity. The flame propagation

velocity is defined as the differential of the flame mean radiusversus time. Figures 3-5 compare the flame propagationvelocities found by our simulations and experimental resultspresented by Abdel-Gayed et al.,33 Kobayashi et al.,34 andAldredge et al.35 There is a very good agreement betweenexperimental and simulation results.

Our comparisons were made with different experiments. Eachone describes a certain stage of flame propagation. Descriptionof the different stages of the turbulent flame will be discussedin details in Section 3.3.

It is obviously clear that under Kobayashi experimental34

conditions, the flame propagation is in a transient regime

characterized by a constant propagation velocity (but at the same

time a growing flame width). This phenomenon seems to be in

contradiction with Abdel-Gayed et al. experiments33 in the

spherical bomb, which only describe the initial stage of theturbulent flame characterized by a growth in both the flame

speed and width. However, in our case, weve prolonged

simulations to enough time until obtaining a constant turbulent

flame speed. So, St values are taken when the propagation

velocity reaches its asymptote. Thus, our comparison with the

experimental results of Kobayashi et al.34 is legitimate.

In this following section, we place our emphasis on equiva-

lence ratio and turbulence intensity effects on flame propagation

velocity.

Concerning equivalence ratio effect on flame propagation

velocity, it has been known for many decades, in laminar

combustion regimes, that the flame burning velocity is maximal

when the equivalence ratio is around stoichiometry value.36,37In recent work, Bradley38 affirmed that for turbulent case the

flame propagation velocity, which strongly depends on burning

velocity, increases with ER. These declarations are justified in

Figure 4. Also, we can remark that in the case of lean flames

(ER ) 0.9), the flame propagation velocity increases slowly.

To discover the effect of the turbulence on turbulent premixed

flame characteristics, we have represented the flame velocity

ratio St/SL versus u/SL (see Figures 3 and 5). It is obviouslyclear that the flame propagation velocity increases with turbu-

lence intensity. These results are confirmed by theory. In fact,

a high level of turbulence intensity makes microscale mixing

more efficient and turbulence diffusivity stronger, so the flame

will be much faster (see Figures 3 and 5.)It is worth noting that this big scatter of experimental data

(see Figure 3) is a widely recognized fact.1 It is attributed to

both the experimental error and inadequate or incomplete

theoretical assumptions.32,39

We can affirm that, at weak and moderate turbulence, the

flame-burning velocity ratio varies function of the dimensionless

turbulence intensity, responding to Damkohlers model in the

case of n ) 1. Thus, the expression takes the following form:

(33) Abdel-Gayed, R. G.; Bradley, D.; Lawes, M. Proc. R. Soc. Lond.,Ser. A 1987, 414, 389413.

(34) Kobayashi, H.; et al. 26th Symp. (Int.) Combust.; The CombustionInstitute: Pittsburgh; PA, 1996; pp 389-396.(35) Aldredge, R. C.; et al. Combust. Flame 1998, 115, 395405.

(36) Metgalchi, M.; Keck, J. C. Combust. Flame 1980, 38, 111129.(37) Konnov, A. A.; Dyakov, I. V. Combust. Flame 2004, 136, 371

376.

(38) Bradley, D.; et al. Combust. Flame 2003, 133, 415430.(39) Lapatnikov, A. N.; Chomiak, J. Prog. Energy Combust. Sci. 2005,31, 173.

Figure 3. Normalized turbulent velocity vs normalized rms turbulentvelocity in spherical flame.

Figure 4. Dependences of turbulent flame speed on rms turbulentvelocity.

Figure 5. Normalized turbulent velocity vs normalized rms turbulentvelocity.

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This formula is similar to that given by Gulder40 for weakReynolds number.

3.3. Flame Brush Thickness. Figure 6 shows the flame-brush thickness t (called also flame width) evolution vs time.This flame width is defined by:

where c ) (T - Tu)/(Tb - Tu) is the progress variable definedas a normalized temperature (u for unburned and b for burnedgases).

The agreement between simulation and experimental results

is satisfactory. The asymptotic tendency of t, affirmed byGalzin16 and Liptanikov and Chomiak,1 and observed byMoreau,41 is predicted by our simulation. Although, Karlovitz42

and Scurlock43 have pointed out that the increase oft is mainlycontrolled by the turbulent diffusion law, whereas flamepropagation reduces t and it can reach approximately constantvalues after the development phase. However, in some experi-ments, such as engines, for example, this constancy oft canbe caused by other effects (e.g., turbulence decay or wallinfluence) rather than by reaching the regime of turbulent flamepropagation characterized by a fully developed t.

Indeed, Zimont, in his last book,46 affirms that the first initialstage of flame kernel formation is characterized by the non-

equilibrium of both small-scale and large-scale wrinkles, inaccordance with the Taylor theory.47 Thus, we notice anincreasing of the brush width t like utand relatively fast growthof St.

Concerning the intermediate stage, it is marked by thequasiequilibrium of the small-scale wrinkles while the largescales still practically nonequilibrium. In fact, we have in thiscase an intermediate asymptotic state. So, these transient flamesare characterized by nearly constant flame speed and at the same

time by increasing brush width, and that is why they are calledintermediate steady propagation (ISP) flames.

As for the final steady stage, both large- and small-scalewrinkles are in equilibrium; it is the final asymptotic state. It issteady state (SS) flames analyzed in many publications (usuallyin the form of the classical 1D stationary turbulent flame).Indeed, this stage cannot be reached by experiments because itneeds a long period time of flame study, a thing that is notpossible experimentally. However, by simulation and especiallyby Monte Carlo simulations, this study is possible with currentCPU capacities.

4. Conclusion

Turbulent premixed combustion was numerically simulatedin an adiabatic constant volume vessel. Simulation was basedon the PDF-Monte Carlo method. Turbulence was supposed tobe homogeneous and isotropic. The chemistry was describedby a four-step reaction mechanism of methane combustion.Confrontation with many experimental results gave satisfactoryagreements, especially on mean flame radii, turbulent propaga-tion velocity, and flame-brush thickness.

The current model correctly predicts trends such as the effectof increasing turbulence intensity and equivalence ratio on thepremixed methane-air flame propagation characteristics, suchas the mean radius and the turbulent speed. Indeed, we have

notice an increase of the mean flame radius and the turbulentspeed when ER or u increases, but the effect of turbulenceintensity was more significant.

Also, simulations show that turbulence intensity is thedominant factor in the determination of the turbulent flamespeed.

After ignition and during the first stage of the flamepropagation, both the turbulent velocity and the brush widthare growing. After that, a bending effect was observed in theturbulent flame speed evolution (characterizing the second stageof the flame propagation called also the ISP regime). Then, PDF-Monte Carlo simulations demonstrate that the flame brushthickness has a tendency to be constant at increasing time (this

stage is called the SS regime). Experiments were unable to reachthis last regime. Thus, the PDF-Monte Carlo method could bea very useful tool to explain theoretical prediction of this regime.

Finally, it will be very interesting, in a next work, to studythe propagation flame characteristics by more consideringturbulence-chemistry interactions. Also, it would be useful totake into account more detailed chemistry allowing us toevaluate pollutant emissions.

EF9000456

(40) Gulder, O. L. Combust. Flame 2000, 120, 407416.(41) Moreau, P. Aerospace Science Meeting, AIAA Paper 1977, 77, 49.(42) Karlovitz, B.; et al. J. Chem. Phys. 1951, 19, 541547.(43) Scurlock, A. L.; Grover, J. H. Proc. Combust. Inst. 1953, 4, 645

658.(44) Renou, B.; et al. 18th Int. Colloq. on the Dynamic of Explosions

and ReactiVe Systems: Seattle, Washington, USA, 2001.(45) Atashkari, K. Towards a general correlation of turbulent premixed

flame wrinkling. In Engineering Turbulence Modelling and Measurements;Rodi, W., Laurence, D. Eds.; Elsevier: Amsterdam, 1999; Vol. 4, pp 805 -814.

(46) Zimont, V. Kolmogorovs legacy and turbulent premixed combus-tion modelling. In New DeVelopments in Combustion Research; WilliamJ. Carey Inc, Ed.; Nova Science Publishers: New York, 2006; Ch 1, pp1-93. (47) Taylor, G. I. Proc. R. Soc. Lond., Ser. A 1935, 151, 465478.

St/SL ) 1 + C1u/SL (7)

t )1

max |c|(8)

Figure 6. Dimensionless mean flame brush thickness (t/Lt) vsdimensionless time (t/t).

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