CH4 mixtures and comparison with experimental results

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Assessment of the method for calculating the Lewis number of H2/CO/CH4 mixtures and comparison with experimental results Denis Lapalme*, Romain Lemaire, Patrice Seers
Department of Mechanical Engineering, Ecole de technologie sup
erieure, 1100 Notre-Dame Ouest, Montr
eal, QC, H3C 1K3, Canada

article info

abstract

Article history:

This paper investigates the various techniques used in the literature to calculate the

Received 2 November 2016

effective Lewis number of two-component (H2/CO and H2/CH4) and three-component fuels

Received in revised form

(H2/CO/CH4 and H2/CO/CO2) over a wide range of equivalence ratios (0.6
4
1.4) under

5 January 2017

laminar flame conditions. The most appropriate effective Lewis number formulation is

Accepted 17 January 2017

identified through comparison with experimentally extracted Lewis numbers (Le). The

Available online xxx

paper first identifies the proper methodology to extract the experimental Le from the burned Markstein length of an outwardly propagating flame. Second, the different meth-

Keywords:

odologies for the calculation of the effective Le are presented and compared to experi-

Effective Lewis number

mental results for H2/CH4 and H2/CO mixtures. Based on the experimental results, it is

Syngas combustion

shown that the calculation of the effective Le of mixtures can be divided into a three-step

Outwardly propagating flame

procedure depending on the equivalence ratio: (1) calculation of the Le for each fuel and the

Markstein length

oxidizer; (2) use of the Le mixing rule; and (3) assessment of the necessity or not of combining the fuel's and oxidizer's Lewis numbers. The paper shows that, in rich mixtures, the oxidizer Le needs to be taken into account. Lastly, the methodology is validated for H2/ CO/CH4 and H2/CO/CO2 fuels. © 2017 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.

Introduction Combustion is a process that relies profoundly on transport phenomena. Through heat transfer, the heat released by the flame is used to preheat the reactants to the activation temperature. Meanwhile, mass transfer ensures that the flame is continuously supplied with reactants. The imbalance of thermal diffusivity to mass diffusivity is known as nonequidiffusion [1]. It is represented by the Lewis number (Le), which is defined as the ratio of the thermal diffusivity to the

mass diffusivity of the deficient reactant. Nonequidiffusion impacts the behavior of the stretched flames in many ways, such as on the minimum ignition energy and flame kernel [2e4] or on the determination of flame speed [5,6]. In particular, flame's stability characteristics depend on nonequidiffusion through a mechanism referred to as diffusionalethermal instability. While the flame is unconditionally unstable when Le is below the critical value Le* (slightly smaller than 1 and thus often approximated as Le* ¼ 1) [7], diffusion helps to stabilize the flame when Le > Le*. This instability mechanism is the cause of the apparition of

* Corresponding author. E-mail address: [email protected] (D. Lapalme). http://dx.doi.org/10.1016/j.ijhydene.2017.01.099 0360-3199/© 2017 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved. Please cite this article in press as: Lapalme D, et al., Assessment of the method for calculating the Lewis number of H2/CO/CH4 mixtures and comparison with experimental results, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/ j.ijhydene.2017.01.099

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Nomenclature A Flame area, m2 A1 Factor of mixture strength, e Specific heat, J kg1 K1 Cp Dij Binary mass diffusivity, m2 s1 Di,mix Mixture-averaged mass diffusivity, m2 s1 (dT/dx)max Maximum temperature gradient, K m1 Ea Activation energy, J mol1 k Thermal conductivity, W m1 K1 Lb Burned Markstein length, m Unburned Markstein length, m Lu Flame thickness calculated based on the l fG temperature gradient, m Flame thickness calculated based on the kinetic l fK definition, m Le Lewis number, e Effective Lewis number, e Leeff Combination of Le associated to the fuel and Lef =O2 oxidizer, e Fuel Lewis number, e Lefuel Individual Lewis number of each component i Lei (fuel i or O2 as the limiting reactant), e Lei with diffusion towards the whole air/fuel Lei,a mixture, e Lei with diffusion towards air, using the quantity Lei,4 of air required to maintain 4 of the overall mixture, e Lei with diffusion towards air, using all the air Lei,4i available in the overall mixture, e Diffusion-based mixing formulation of Lei, e LeD Heat-based mixing formulation of Lei, e LeH Volume-based mixing formulation of Lei, e LeV Oxidizer Lewis number, e LeO2 cells on lean hydrogen and rich propane flames [7e10]. These cells wrinkle the propagating flame front, which creates a selfaccelerating flame [7], induces turbulence in the unburned mixtures, and leads to engine knock [11]. The correct interpretation of many combustion phenomena therefore relies on the precise calculation of the Le which is a key parameter to discriminate between stable and unstable flame fronts. While the definition of Le in a single-fuel mixture is quite straightforward, defining Le in a multifuel mixture is not trivial since the diffusivity characteristics of each fuel have to be considered. Because there is no consensus on the matter, three formulations of an “effective” Lewis number (Leeff) have been reported in the literature. Bouvet et al. [12] performed an experimental investigation examining the validity of the three Leeff formulations in lean mixtures (equivalence ratios of 0.6 and 0.8) of H2/CO, H2/CH4, H2/C3H8, and H2/C8H18. Using the theoretical link proposed by Chen and Ju [13] between the burned Markstein length (Lb) and Le, Bouvet et al. [12] compared the experimentally determined Markstein lengths to the Markstein lengths calculated using each formulation of Leeff. They identified a qualitatively accurate formulation for the prediction of Le in H2/alkane mixtures, while no Leeff formulation has been found to adequately fit the results obtained with H2/CO mixtures.

LeBM LeChen LeLaw qi Qi R R0 Sb S0b Su S0u t Tad Tu Yi Yi,mix Ze

Experimental Le extracted based on the work from Bechtold and Matalon [32], e Experimental Le extracted based on the work from Chen et al. [5,44], e Experimental Le extracted based on the work from Law and Sung [51], e Nondimensional heat release, e Heat of reaction, J kg1 Flame radius, m Universal gas constant, J mol1 K1 Laminar flame speed of the burned gas, m s1 Unstretched laminar flame speed of the burned gas, m s1 Laminar flame speed of the unburned gas, m s1 Unstretched laminar flame speed of the unburned gas, m s1 Time, s Adiabatic temperature, K Unburned mixture temperature, K Mass fraction of species i in the fuel blend, e Mass fraction of species i in the air/fuel mixture, e Zel'dovich number, e

Greek symbols a Thermal diffusivity, m2 s1 a Factor accounting for the expansion ratio, e k Stretch rate, s1 rb Density of the burned mixture, kg m3 ru Density of the unburned mixture, kg m3 s Flame expansion ratio, e 4 Equivalence ratio, e F Ratio of excess-to-deficient reactants, e Mole fraction of species i in the fuel blend, e ci

Over the last few years, there has been a marked interest for hydrogen-based fuels, since hydrogen can reduce the emission of pollutants from combustion by being a noncarbon-based source of energy and by extending the lean operating limit [14e16]. Many studies have been devoted to the evaluation of the combustion characteristics and performance of such fuels in gas turbines [17,18] and spark ignition engines [19,20]. A precise characterization of combustion properties of hydrogen-based fuels, including Le, is of importance in theoretical and numerical analyses [21e24]. The goal of this paper is thus to focus on investigating syngas Le while broadening the investigation of Bouvet et al. [12] by (1) assessing the validity of the three Leeff formulations using two additional theoretical equations linking Le to the Markstein length; (2) expanding the investigation from lean to stoichiometric and slightly rich conditions; (3) evaluating the impact of CH4 and CO2 addition to H2/CO. This broad investigation is performed at equivalence ratios (4) ranging from 0.6 to 1.4, with respect to the rich flammability limit of CH4. The volumetric content of H2, CO and CH4 was varied from 0 to 100%,1 while the CO2 content was 1

2% of H2 (5% at 4 ¼ 0.6) was required to ensure the combustion of pure CO.

Please cite this article in press as: Lapalme D, et al., Assessment of the method for calculating the Lewis number of H2/CO/CH4 mixtures and comparison with experimental results, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/ j.ijhydene.2017.01.099

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limited to 20% to prevent the apparition of cells on the flame front. The first part of the paper details the formulations used for the calculation of Leeff. The second part presents the experimental setup and details the extraction of the experimental Le. The third part compares Leeff to experimental results and discusses more than 10 methods used to calculate Leeff. Finally, the main conclusions are presented.

Lewis number Herein, Le is defined based on the components of the mixture, while the experimental Lewis number is extracted from outwardly propagating flames, as presented in Section Experimental Lewis number. The various definitions of Le are also presented.

By definition, Le (see Eq. (1)) is the ratio of the thermal diffusivity to the mass diffusivity of the deficient reactant, which is defined as the fuel or oxidizer species for lean and rich mixtures, respectively. In Eq. (1), a is the thermal diffusivity of the fueleair mixture and Dij is the mass diffusivity of the deficient reactant i towards j at the temperature of the freestream reactant. a Dij

(1)

In most analyses of fueleair mixtures, the reactants are assumed to be scarce compared to nitrogen [25]. Therefore, j represents N2 and Dij is taken as the binary diffusion of the reactant towards N2 because of the abundance of the latter. This probably holds true for alkanes due to their high molar air-to-fuel ratios, but this is not the case for H2 due to its low molar air-to-fuel ratio. Therefore, the mixture-averaged coefficient of mass diffusion (Di,mix) was also calculated using Eq. (2) where Y is the mass fraction and c the molar fraction of each specie, s, in the mixture [26,27]: 11

0 Di;mix

O2 faces multiple fuels, LeO2 is typically calculated for the mixture as a whole. It is known, however, that the Le of pure rich CH4 and pure rich H2 are dissimilar [28,29], meaning that LeO2 is different as a function of the fuel. In this paper, LeO2 is calculated according to the three methods presented in Section Calculation of Lei and it is proposed to calculate LeO2 considering O2 diffusing towards each fuel i. In our study, once each fuel and oxidizer Lei were known, they were coupled (separately) according to the different methods reported in the literature and summarized in Section Formulations of Le for multifuel mixtures to determine the fuel and oxidizer Lewis numbers. Finally, in some instance, we proposed correcting the mixture Le to obtain a Leeff taking into account the influence of the excessive reactant (fuel or O2, depending on the equivalence ratio). This method is presented in Section Combination of the fuel with the oxidizer.

Calculation of Lei

Lewis number

Lei ¼

3

C B N
B X cs C C ¼ 1  Yi;mix B C B @ s¼1 Dis A

(2)

ssi

A detailed comparison of Le calculated using Dij and Di,mix (not presented herein for brevity) revealed that the latter yields the best agreement with the experimental results for CH4/air, H2/air, and especially H2/CO mixtures. Therefore, Di,mix is recommended and used throughout this paper for the calculation of Le. When two or more fuels are blended together, such as in syngas, and when combustion takes place under lean conditions, calculating Le is more complex because of multiple deficient reactants. Such mixtures require calculating the Le of each single fuel i, hereafter Lei. These Lei can be calculated according to three methods presented in Section Calculation of Lei. On the other hand, there is only a single deficient reactantdthe oxidizer O2din rich mixtures. Therefore, even if

In most studies [8,12,30e32], Lei is calculated by dividing a of the whole mixture (i.e. all fuel species and air) by the Di,mix of fuel i with respect to the rest of the mixture. Herein, this approach is identified as Lei,a. The second approachdproposed by Tang et al. [33]dconsists in calculating ai and Di,mix for each fuel i as if the fueleair mixture only consisted of fuel i and air (and any diluents). Moreover, ai and Di,mix are calculated at 4 representative of the proportion of air available to consume each fuel i, as expressed by Eq. (3): ðc =c Þ 4i ¼  i. air actual cf cair

(3)

stoich

where cf and cair are the mole fractions of all fuels and air, respectively. This methodology, which will be referred to as Lei,4i, is used in the literature [34e36], but always in relation to the heat-release parameter qi of the heat-based formulation presented in Section Formulations of Le for multifuel mixtures. This paper extends the use of this methodology to volume-based and diffusion-based formulations, also presented in Section Formulations of Le for multifuel mixtures. Furthermore, this paper also proposes using this methodology for the oxidizer (demonstrated with two-fuel mixtures) by calculating the Le of the oxidizer for each fuel before combination with a mixing rule from Section Formulations of Le for multifuel mixtures. Lastly, the third approach to estimate Lei, named Lei,4, considers the mixture of each fuel i and air, but at the overall mixture's equivalence ratio. The quantity of diluents associated with each fuel i is proportional to its concentration in the fuel mixture.

Formulations of Le for multifuel mixtures An important objective of this paper is to identify the best way to calculate Le in a multifuel mixture, which depends on how each Lei are weighted in order to provide the mixture's Leeff. The three formulations proposed in the literature are described below and their respective equations are summarized in Table 1.

Please cite this article in press as: Lapalme D, et al., Assessment of the method for calculating the Lewis number of H2/CO/CH4 mixtures and comparison with experimental results, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/ j.ijhydene.2017.01.099

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Overall Le calculation procedure

Table 1 e Effective Lewis-number formulations. Name

Formula

Volume-based (V)

LeV ¼

Diffusion-based (D) Heat release-based (H)

Xf

c Lei i¼1 i

LeD ¼ Pf Pf LeH ¼ 1 þ

a

cD i¼1 i ij

q ðLei  i¼1 i Pf q i¼1 i

1Þ

ð5Þ ð6Þ ð7Þ

The first method consists of a volume-based formulation (LeV) (see Eq. (5)) proposed by Muppala et al. [31]. It defines LeV with a linear average formulation based on the volume fraction (ci) of the fuels in the fuel mixture. It is worth noting that Muppala et al. [31] concluded that this method is too simplified for their turbulent flame speed modeling. A second formulation, referred to as the diffusion-based formulation (LeD) (see Eq. (6)), is issued from the work of Dinkelacker et al. [32], who postulated that a local enrichment of fuel can be expected from laminar flame curvature. The resulting overall reaction-rate enhancement due to curvature was modeled with an effective mass diffusivity, which is then used with the mixture's a to obtain LeD. The third formulation was rigorously derived from the asymptotic theory in Law et al. [8]. It is a weighted average of the Lei based on the nondimensional heat release (qi) (see Eq. (7)) associated with the consumption of each fuel in the mixture, where qi is defined as: qi ¼

Qi Yi Cp Tu

(4)

with Qi and Yi being the heat of reaction and the mass fraction of fuel i, respectively, and Cp the specific heat of the upstream fluid at the unburned temperature Tu. This heat-based (LeH) formulation has been used extensively in two- and three-fuel blends [12,32e35,37e41].

Fig. 1 illustrates the possible interactions of the various definitions for determining the Leeff of a given mixture with the above methodology. First, a Lei method is chosen and applied to each fuel (Fig. 1 upper-left) and to the oxidizer (Fig. 1 upperright). Then the Lei are combined by using one of Eqs. (5e7) to get a Lefuel (Fig. 1 middle-left) and a LeO2 (Fig. 1 middle-right). The Leeff can be directly equal to the Lefuel (or LeO2 ) or can require using Lef =O2 (Fig. 1 bottom) as this study will show.

Properties calculated by numerical simulation The different fuel, oxidizer, and mixture properties required to calculate the equations in Section Experimental procedures were obtained with the CHEMKIN-PRO package [44]. The unburned and burned density (ru and rb) as well as the adiabatic flame temperature (Tad) were calculated with the EQUIL code [45], while the flame speed (S0u ) was computed with the PREMIX code [46]. For the latter, the multicomponent and thermal-diffusion (Soret) effects were included in the calculations. Continuations were used to gradually expand the computational domain by decreasing the GRAD and CURV options, ultimately set at 0.015. This is tighter than what can be observed elsewhere [47,48] and provided solutions with at least 700 grid points. These properties were computed with two versions of the GRI-Mech 3.0 kinetic mechanism (53 species, 325 reactions). A modified version of GRI-Mech 3.0, designed to yield better flame speed predictions for syngas containing small concentrations of CH4 [11,49], was specifically used when the fuel contained between 1% and 40% of CH4. The original GRI-Mech 3.0 [50] was used for all the other mixtures. Using numerical values, the Zel'dovich number was calculated as per Eq. (9): 
 Ze ¼ Ea ðTad  Tu Þ R0 T2ad

(9)

0

in which R is the universal gas constant and Ea is the activation energy expressed according to Eq. (10):

Combination of the fuel with the oxidizer The analysis conducted by Addabbo et al. [25] showed that Le of the fueleair mixture is the combination of the Le of the fuel and oxidizer (referred to as Lef =O2 ), where the fuel's Le is calculated with one of Eqs. (5e7) and the oxidizer's Le is LeO2 , as defined in Section Calculation of Lei. While Bouvet et al. [12] observed that using Lef =O2 worsened the predicted results, their investigation was limited to lean mixtures. Since the mixtures tested herein cover lean, stoichiometric, and rich mixtures, the Lef =O2 proposed in [25,42] and used in [38,41,43], was reassessed in this study and is given by Eq. (8): Lef =O2


 ðLeexc  1Þ þ Ledef  1 A1 ¼1þ 1 þ A1

(8)

where Leexc and Ledef are the Lewis numbers of the excessive and deficient reactant, respectively. The parameter A1 represents the measure of the mixture strength and is given by A1 ¼ 1 þ Ze(F e 1). Here, F is defined as the ratio of the mass of excess-to-deficient reactants in the fresh mixture relative to their stoichiometric ratio (F ¼ 1/4 for 4
1 and F ¼ 4 for 4 > 1) [34,35]. Finally, Ze is the Zel'dovich number defined in the next section.


 Ea ¼ 2R0 vln ru S0u vð1=Tad Þ

(10)

Ea represents the sensitivity of the laminar burning rate to variations in the flame temperature. In this paper, this sensitivity was evidenced by slightly changing the concentration of N2 in the mixture while keeping the blend and the equivalence ratio unchanged as in Sun et al. [51].

Experimental procedures Presentation and validation of the experimental setup The experiments were conducted in a cylindrical constantvolume chamber with an internal diameter of 300 mm and a length of 330 mm. An optical access to the chamber was provided by two opposite and transparent sapphire windows of 40 mm in radius located on the flat sides of the vessel. A vacuum is created in the chamber, which is then filled with the different gases (purity levels: H2 e 99.999%, CO e 99.5%, CO2 e 99.9%, CH4 e 99.9%, air e hydrocarbon free) until the

Please cite this article in press as: Lapalme D, et al., Assessment of the method for calculating the Lewis number of H2/CO/CH4 mixtures and comparison with experimental results, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/ j.ijhydene.2017.01.099

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Fig. 1 e Overview of the effective Lewis number calculation procedure.

volume of each gas required to create the desired premix mixture was reached. The volumetric flow rates of the gases were measured with separate mass-flow controllers (OMEGA FMA 5423 for air; FMA 5414 otherwise; accuracy of ±1.5% full scale, repeatability of ±0.5% full scale), and then thoroughly mixed in the gas line before being sent to the chamber. The flow rates were adjusted to ensure that the required volume of each gas was reached in the same time lap. All experiments were performed at an initial temperature of 298 K and at 1 atm. Each mixture was tested thrice and the average is reported in this paper. The mixture was spark-ignited at the center of the vessel by two tungsten electrodes, creating an outwardly propagating flame (OPF) that quenches when touching the walls of the chamber. Flame propagation was imaged with a Z-type Schlieren photography setup comprising a pair of 108 mm diameter mirrors and recorded by a high-speed camera (Phantom V9.1) operating at 9800 images per second with a resolution of 352  352 pixels2. Flame development was relatively spherical, although bulges were visible where the flame was in contact with the electrodes. Hence, once the bulges are withdrawn, the flame front can be represented by the equation of a circle with a radius R. We adopted the flame-edgedetection methodology applied in [52,53]. It consists of (i) a background subtraction to reduce the noise; (ii) similar to [2], a withdrawal of a p/7 rad triangular area of the flame on each side of the two electrodes to eliminate the bulges; (iii) edge detection of the flame disk localized based on the maximum intensity of the post-subtraction image; and (iv) identification of the smallest circle bounding the flame disk, yielding R.

These steps were repeated for all the acquired images, giving the evolution of the flame radius as a function of time t, and thus reflecting the flame speed of the burned boundary Sb as per Eq. (11): Sb ¼

dR dt

(11)

Because of such flame motion and of its intrinsic curvature, the outwardly propagating flame is subjected to a temporal change of the flame surface element, or stretch, at a rate given by Eq. (12): k¼

1 dA 1 8pRdR 2 dR 2 ¼ ¼ ¼ Sb A dt 4pR2 dt R dt R

(12)

The unstretched flame speed S0b is then obtained by extrapolating the stretched flame speed Sb to a zero stretch rate k ¼ 0. The sensitivity of Sb to the stretch rate variation is expressed by a parameter, unique for each mixture, called the Markstein length (Lb) [51]. The study of Chen [54] demonstrated that extrapolation with nonlinear model 1 (NM I, see Eq. (13)) is to be preferred for mixtures with positive Lb, while extrapolation with nonlinear model 2 (NM II, see Eq. (14)) is for mixtures with negative Lb. These criteria for the extraction of S0b and Lb have been adopted in this paper. Sb ¼ S0b 

2S0b Lb R


 2S0 Lb lnðSb Þ ¼ ln S0b  b RSb

(13)

(14)

Please cite this article in press as: Lapalme D, et al., Assessment of the method for calculating the Lewis number of H2/CO/CH4 mixtures and comparison with experimental results, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/ j.ijhydene.2017.01.099

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Note that the calculations were limited to flames with radii between 6 and 30 mm to limit the disturbances caused by ignition [55] and to avoid wall interference [56]. The flameradius range was further reduced if cellular instabilities occurred in order to keep only the stable part of the flame propagation. The unstretched unburned laminar flame speed ðS0u Þ was retrieved from S0b with Eq. (15): S0u ¼

S0b s

(15)

where s ¼ ru =rb is the thermal expansion ratio calculated with Chemkin.

Fig. 2 shows the good agreement between the experimental burning velocities and the predictions with both versions of the GRI-Mech 3.0 as stated above (the mean deviation between experimental and modeled values being of only 8.9% on average). The burning velocity in mixtures with high H2 content (cH2 0.6) at 4 ¼ 0.6 is, however, slightly underestimated. Fig. 3 presents the Lb of selected mixtures along with their standard deviations represented as error bars which illustrate the high repeatability of the measurements performed in this study. For CH4 flames, the scatter of Lb in rich mixtures is noticeably larger than in lean mixtures [52,57e59]. Except at 4 ¼ 0.6, Lb presents good agreement with the lower range of the measurements reported in the literature. Reported

Fig. 2 e Comparison of experimental (symbols) and numerical (lines) laminar burning velocities. A: H2/CH4 flames; B: syngas flames (the number beside SH representing the H2/CO ratio).

Fig. 3 e Burned Markstein length of selected mixtures. Please cite this article in press as: Lapalme D, et al., Assessment of the method for calculating the Lewis number of H2/CO/CH4 mixtures and comparison with experimental results, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/ j.ijhydene.2017.01.099

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measurements for pure H2 show a large scatter [51,57,58,60]. Results from this study reveal good agreement with the midhigh range of the scatter. Results from H60M40 (60% H2 and 40% CH4) [57,58] and lean syngas [12] show that the present measurements also present an overall good agreement for 2fuel mixtures.

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propagating flame [5,54] (thus named LeChen herein) and was employed by Bouvet et al. [12].

1 

Lb Ze Ze  1 2 2 slf

 LeChen ¼

(20)

Flame thickness Experimental Lewis number Le is a parameter representing the transport of the freestream reactants. Upstream from the reaction zone, however, the reactants experience preheating in the transport zone and produce many intermediate species through intermediate reactions. These species have different diffusivities which, according to Law and Sung [28], could manifest their influence through some nonequidiffusive phenomena. As a result, three methods were identified to experimentally determine Le from a flame. The first method, expressed by Eq. (16), was presented by Law and Sung [28], who performed an integral analysis of the stretch flame and obtained a Le (named LeLaw herein). They used this method to extract the Le of H2-air and C3H8-air mixtures over a wide range of equivalence ratios. This method is a function of the Markstein length of the unburned mixture (Lu), s, Ze, the flame thickness (lf), and a, which is defined by Eq. (17). Note also that Lu in Eq. (16) comes from Lb with Eq. (18). LeLaw



  2ð1  1=sÞ Lu 1a ¼1þ þ ð1  1=sÞ  Zeð1=sÞð1  aÞ lf 1  1=s

a ¼ 1 þ ln

Lu ¼



 1 1 þ 1  expð1Þ s s

 Lb 1 þ a  lf s s

(16)

(17)

(18)

The second method was presented by Bechtold and Matalon [42,61] and followed the asymptotic theory expanded to exhibit an explicit dependence on 4. It is therefore valid for near-stoichiometric conditions as well and was used in Jomaas et al. [61] to extract the Lewis number of H2-air and C2H2-air mixtures over a wide range of equivalence ratios. This Le, subscripted as BM, is expressed by Eq. (19): LeBM ¼ 1 þ

  1   Lb 2 2Ze pffiffiffi 1
pffiffiffi s  1  ln s þ1  pffiffiffi 2 lf sþ1 s1 (19)

Yet this expression gives the limiting values, as the entire flame is view as an interface [62]. Recently, Giannakopoulos et al. [62] provided a systematically-derived solution uniformly valid across the entire flame zone, in which the flame properties must be calculated on an isotherm sufficiently close to the burned side of the flame. The Le calculated using this methodology were observed to be very similar to the values given by Eq. (19). Therefore, the more straightforward method of Bechtold and Matalon [42,61] is used in this study. The third expression presented by Eq. (20) derives from an integral analysis of the fundamental flame parameters originally performed for a counterflow flame [1]. It was recently used by Chen in theoretical analysis of the outwardly

The formulations above for the extraction of the experimental Le depend on the laminar-flame thickness (lf), for which two definitions are commonly used. The first definition (Eq. (21)), which comes from the kinetic analysis (hence identified as lfK ) [1,63], is widely used [3,8,11,13,64] and was chosen by Bouvet et al. [12] in their assessment of Leeff. In that definition and based on Law et al. [8], the ratio of thermal conductivity to heat capacity (k/Cp) was assessed for the unburned gas mixture at the average temperature between the inlet and the adiabatic-flame temperature. lfK ¼

k ru Cp S0u

(21)

That notwithstanding, some authors [28,51,65] consider that lf resulting from this definition is uncertain because of the arbitrariness of the temperature at which k/Cp is assessed and because it depends on S0u . Hence, they prefer to use a lf based on the gradient method (lfG ) presented by Eq. (22). lfG ¼

Tad  Tu ðdT=dxÞmax

(22)

In Eqs. (21) and (22), S0u , Tad, and the maximum gradient of temperature (dT/dx)max obtained from the temperature profile solution were extracted from the PREMIX calculations. Both methods are compared and assessed during the extraction of the experimental Le in Section Experimental Le determination.

Experimental Le determination Due to the dependence of the three methodologies used for the extraction of the experimental Le on the definitions of lf, a total of six expressions have been tested, as a first step, in H2/ air, CH4/air, and CO/air mixtures. For each methodology, Lb has been determined experimentally while s, Ze and lf were derived from Chemkin solutions.

H2/air mixtures The results for H2 are presented in Fig. 4-A, where both definitions of lf are used in combination with Eqs. (16e20), while Fig. 4-B only presents results close to the theoretical limits. These limits are depicted by the short horizontal lines located on the left and right sides of the figures and represent the plausible limit values of Le for extremely lean and extremely rich mixtures assessed with the mixtureaveraged diffusion of H2 and O2 in the mixture. The lean limit Le calculated herein is in concordance with Hawkes and Chen [66], while the lean and rich limits are similar to values reported by Law and Sung [28]. To exemplify the impact of the experimental determination of Lb, Fig. 4-B also

Please cite this article in press as: Lapalme D, et al., Assessment of the method for calculating the Lewis number of H2/CO/CH4 mixtures and comparison with experimental results, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/ j.ijhydene.2017.01.099

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Fig. 4 e Extracted Lewis number for H2/air mixtures with lean and rich limits. A: results using the kinetic and gradient definitions of the flame thickness; B: results close to the limits.

shows the extracted Le values based on Lb of [57,60] representing the range of values found in the literature. As the experimental results herein fall within this range for a given method of extraction (LeBM), it also shows the consistency of the experiment results reported in this paper. Fig. 4-A shows the clear impact of the choice of the definition of lf on Le. When lfK (Eq. (21)) is used with LeLaw and LeBM, it yields results far above the limit under rich conditions and negative Le at 4 ¼ 0.6. Even if Le obtained with LeChen is only slightly below the limit at 4 ¼ 1.4, the exponential increase of LeChen throughout the range of 4 analyzed here foretells higher-than-limit Le for richer mixtures. For these reasons, lfG is preferred and is used in Fig. 4-B, allowing comparison of the predictions of Eqs. (16e20). Clearly, LeChen is in close agreement with the two other expressions for lean mixtures, but peaks at 4 ¼ 1.0 before decreasing with increasing 4, while it is expected that Le would peak for rich mixtures. Therefore, this formulation is rejected for the calculation of Le of H2 mixtures. At 4 ¼ 1.4, LeLaw and LeBM seemed to have plateaued under the rich limit, while others [28,51] approached the rich limit when 4  3.0. Since LeBM in combination with lfG is closer to the rich limit, such a formulation therefore appears specifically recommended for H2/air mixtures and was adopted herein for the following.

Fig. 5 e Extracted Lewis numbers of CH4/air mixtures. LeBM and LeLaw used with lfK yielded Le much higher than the rich limits and are therefore not shown.

literature thus strengthening the consistency of the experimental data reported in this work.

CH4/air mixtures

CO/air mixtures

Fig. 5 presents the experimental Le for CH4/air mixtures, along with their lean and rich limits. The lean limit of CH4 is similar to that reported by Hawkes and Chen [66], but slightly higher than the 0.955 reported by Dinkelacker et al. [32]. The lean and rich limits of CH4/air mixtures show that CH4 is slightly sensitive to 4, and that the Le remains close to 1. It is clear from Fig. 5 that LeLaw and LeBM yielded Le much higher than the rich limit, especially for rich mixtures. On the other hand, LeChen extracted using lfG (solid diamond) is below the lean limit, in addition to being nearly insensitive to 4. Overall, LeChen in combination to lfK (hollow diamond) offers the best agreement with respect to the limits, both in trend and in absolute value. Finally, as was done earlier, the Lb of [52,57] were used to illustrate again that the results obtained herein fall in the range of Le extracted from the

The limits of CO/air mixtures presented in Fig. 6 show that the Le of CO is close to unity and insensitive to 4. This result was expected, since CO and O2 have relatively similar molar weights and comparable LennardeJones parameters for length and energy, which result in quite close mass diffusivities. Acetylene, whose molar weight is also similar to that of O2, was also shown to have Le close to unity and insensitive to 4 [61]. This insensitivity of the Le of CO towards 4 is globally well determined by the three methods of extraction as long as lfG is used. The lean limit of the figure is slightly lower than the value of 1.09 of Hawkes and Chen [66], while no comparatives were found for the rich limit. The small rise of experimental Le at 4 ¼ 1.4 above the limit could be associated with the presence of H2 in the flame (2%), since this deviation is in line with the sensitivity of H2 towards 4. All the CO mixtures presented

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Fig. 6 e Extracted Lewis number for CO/air mixtures. LeBM and LeLaw used with lfK yielded Le much higher than the rich limits and are therefore not shown.

in Fig. 6 required a small amount of H2 (2% at all 4 except for the leanest mixture, for which 5% was needed) to ensure ignition and combustion, which is taken into account during the Le extraction. Most extraction methods yielded Le lower than the limits. LeBM presents the best agreement with the limits, both in trend and in absolute value, and is therefore preferred for CO/air mixtures in the following. Overall, the above results show that the difference in the fuel's diffusion behavior implies that distinct methods must be used for the experimental determination of Le from the flame response. These methods are summarized in Table 2.

Results and discussion for Leeff The previous section validated the extraction methodology for determining the experimental Le. In this section, two- and three-component fuels are used to compare the different methodologies listed in Section Lewis number in calculating mixture Leeff. Comparing the obtained results to experimental Le makes it possible to identify the proper method to be used for calculating Leeff.

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experimental Lb are necessary to calculate the experimental Le, they are also used to identify this transition point. Fig. 7 presents Lb for different 4 and for different proportions of H2 in CH4. For a given 4, Lb decreased rather linearly from pure CH4 to up to 80% H2 (60% H2 at 4 ¼ 0.6), followed by an uptick towards pure H2. Many authors [12,58,67] reported this “restabilization” in which the Lb of pure H2 was higher than in a mixture with a low concentration of CH4. According to Okafor et al. [67], re-stabilization starts when H2 reaches 70% of the H2/CH4 blend. It is also at this concentration that the flame speed truly starts to respond to the addition of H2 [68]. Based on the re-stabilization and the methodologies of extraction identified in Table 2, LeChen was used with lfK for mixtures containing 0
H2
70%, while LeBM in combination to lfG was preferred for mixtures with 70 < H2
100%. Based on the above results, the experimental Le of H2/CH4 mixtures were compared to the various ways of obtaining Leeff introduced in Section Lewis number (see Fig. 8, in which 4 increases from top to bottom). The left column assesses the influence of using different definitions of Lei (Section Calculation of Lei) in combination to different mixing rules (Eqs. (5e7)), while always using Lef =O2 (Eq. (8)), in which the LeO2 was computed as Lei,a presented in Section Calculation of Lei. The results show that, for H2/CH4 mixtures, the choice of the fuel's Lei does not have a meaningful impact on a given mixing-rule methodology, but the mixing rule methodology used to calculate Leeff is the determining factor. The volumebased formulation LeV (Eq. (5)) best brings out the general trend of the experimental Le, as was observed by Bouvet et al. [12]. This formulation provides a good qualitative match to the uptick towards high concentrations of H2 noted at 4 of 0.8 and 1.0, as well as the preceding linear decrease at 4 ¼ 0.8 and stagnation at 4 ¼ 1.0. The heat-based formulation (Eq. (7)) caused an increase in Leeff when H2 was added to CH4, contrary to the experimental trend. Leeff from the diffusion-based formulation (Eq. (6)) first decreased before increasing to the 60% mark, but with a behavior closer to a parabola than an uptick. The impact of considering or not LeO2 in the calculation of Leeff is presented in the right column of Fig. 8. Since the

Methane/hydrogen flames Table 2 indicates that the experimental Le of CH4 and H2 should be extracted with different methods, therefore suggesting that, when both fuels are mixed, there must be a point of transition for the methodology to be used. Because the

Table 2 e Methodology for the extraction of the Le based on the diffusion behavior of investigated mixtures. Fuels

Extraction methodology

H2; CO

LeBM used with lfG

CH4

LeChen used with lfK

Fig. 7 e Markstein lengths of the H2/CH4 mixtures.

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Fig. 8 e Lewis number of H2/CH4 mixtures. 4 increases from top to bottom. Mixing rules are compared in the left column while the need for the f/O2 combination is analyzed in the left column. (The reader is referred to the web version of this paper for the interpretation of the color-coded lines). best results were achieved with LeV, this formulation was used in combination with Lei,4i (Eq. (3)) for each fuel. For brevity and clarity in the figures, results from the definition Lei,4 are not shown as it did not improve the results. The

LeO2 shown in Fig. 8 (right column) was calculated first by considering that O2 diffuses toward the whole mixture (black curves) and is thus a Lei,a. The results show two different behaviors:

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Fig. 9 e Lewis number of H2/CO mixtures. 4 increases from top to bottom. Mixing rules are compared in the left column while the need for the f/O2 combination is analyzed in the left column. (The reader is referred to the web version of this paper for the interpretation of the color-coded lines). Please cite this article in press as: Lapalme D, et al., Assessment of the method for calculating the Lewis number of H2/CO/CH4 mixtures and comparison with experimental results, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/ j.ijhydene.2017.01.099

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1) For 4 ¼ 0.6 and 1.2, a better agreement was achieved when Lef =O2 was ignored (solid curves). At 4 ¼ 0.6, the rather linear decrease to low experimental Le was best reproduced with the uncombined formulation, which also accurately matches the small exponential increase observed at 4 ¼ 1.2. As Lef =O2 was designed to mix the abundant and deficient reactant, it is therefore not surprising that Lef =O2 worked best in mixtures near stoichiometry, as will be shown next, and thus must be avoided far from stoichiometry, which is in concordance with Bouvet et al. [12]. 2) At 4 ¼ 0.8 and 1.0, a better prediction of the experimental Le was obtained when using the Lef =O2 (Eq. (8) e dashed curves). At 4 ¼ 0.8, this fueleoxidizer combination revealed the uptick with increasing H2 concentration, which is otherwise missed. Meanwhile, at 4 ¼ 1, the deficient reactant was undetermined and only Lef =O2 was used. When Lef =O2 was used, the Lei of the fuel and oxidizer were calculated independently of the deficient reactant before being combined, yielding a Leeff able to vary with the fraction of H2 in the mixture. This behavior reproduces the trend observed experimentally. The results herein suggest that the Lef =O2 should be used in the range 0.8
4
1 and that 4 ¼ 0.8 was close to the tipping point from far to near stoichiometry on the lean side. So far, only the LeO2 definition in which O2 diffuses simultaneously towards all the others species of the mixture (LeO2 ;a ) was analyzed. Because Lei,4i proved to be the best for the fuel Le, it was also used for the oxidizer and yielded one LeO2 ;4i for each fuel i. Since LeV was identified as performing best for mixing the fuel's Lei, it was also used to combine the individual LeO2 ;4i . Fig. 8 (right column) shows that, for 4 ¼ 0.8 and 1.0, using Lef =O2 with either of the LeO2 definitions is key to qualitatively reproducing the experimental trends. It is thus concluded that LeO2 is required in the near stoichiometric range, where it is needed in the Lef =O2 definition for 4 ¼ 0.8 and 1.0, and in slightly rich mixtures (4 of 1.2), where Leeff only depends on the diffusion of O2 and thus of LeO2 (red (in the web version) curves on Fig. 8 e bottom of right column). For the leanest mixture tested herein (4 of 0.6), Leeff depended solely on fuel diffusion. Fig. 8 (right column) shows that, for all cases, LeO2 ;4i yielded results more similar for Leeff than LeO2 ;a and a decision was made to withhold making a conclusion about which methodology should be used until the H2/CO mixtures were analyzed.

Hydrogen/carbon monoxide flames Fig. 9 presents the Leeff of the H2/CO/air mixtures, along with the experimental Le. Since the experimental Le for both H2 and CO were extracted with LeBM using lfG , only this methodology was used. The left column of Fig. 9 examines the influence of the methodology of the different fuel Lei in combination with Lef =O2 in which the LeO2 was computed as Lei,a, presented in Section Calculation of Lei. Fig. 9 (left column) with 4 ¼ 0.6 and 0.8 (top two figures) shows that the experimental Le decreased rather linearly with the addition of H2. This behavior was not reproduced by LeD, regardless of the definition of Lei. For the same 4, LeH and LeV were influenced by the method used to calculate the Lei of

each fuel. The experimental behavior is well reflected by either LeH or LeV when combined with Lei,4i or Lei,4, while Lei,a did not reproduce the experimental trend. For 4 of 0.6 and 0.8, Lei,4i was in closer agreement with the experimental results and, for the sake of unity with H2/CH4 mixtures and for its ease of use, LeV is recommended. Fig. 9 (right column) shows how calculating LeO2 influences the results. As presented in Section Methane/hydrogen flames, the use of Lef =O2 (dashed curves) must be pursued only for near-stoichiometric mixtures (0.8
4
1.0). Lef =O2 was less accurate in predicting the strong linear decrease of Le at 4 ¼ 0.6. In rich mixtures (last two figures of the bottom of Fig. 9  right column), the experimental Le first evidenced a linear increase when H2 was added to CO, followed by a large uptick as the mixture approached pure H2. Lef =O2 accurately predicted the experimental Le up to the uptick, but discrepancies appeared afterwards. Section Methane/hydrogen flames (H2/CH4 mixtures) provided no conclusive definition of LeO2 (LeO2 ;a or LeO2 ;4i mixed through LeV) for use, while the bottom graphs of Fig. 9 (right column) show that, in rich mixtures, LeO2 ;a did not provide the uptick towards pure H2. The use of LeO2 ;4i , however, presented an exponential increase predicting rather qualitatively the experimental Le. Still, discrepancies can be noted for “pure” CO at 4 ¼ 1.4 (2% H2 required for the combustion, as discussed in Section CH4/air mixtures) and for 80% H2 at 4 ¼ 1.2. At 4 ¼ 1.0, the nonlinear

Table 3 e Procedure to calculate the Lewis number. 4 < 0.8

0.8
4
1.0

4  1.2

Calculation of Lei

Lei,4i

Lei,4i and LeO2 ;4i

LeO2 ;4i

Calculation of Leeff

LeV

LeV fuel combined with LeV O2 through Lef =O2

LeV

Table 4 e Composition of the three-component blends tested. H2/CO

H2

CO

CO2

CH4

Equivalence ratio

2/3 2/3 2/3 1.5 4 1.5 4 1.5 4 4 1/4 1/4 2/3 1/4 1/4 1.5 1.5 2/3 4 4 2/3 1/4

32 24 8 36 48 48 16 12 32 64 8 16 12 16 18 54 48 36 64 72 32 16

48 36 12 24 12 32 4 8 8 16 32 24 48 64 72 36 32 54 16 18 48 64

e e e e e e e e e e e e e e 10 10 20 10 20 10 20 20

20 40 80 40 40 20 80 80 60 20 60 60 40 20 e e e e e e e e

0.8 0.6 1 1.2 1 1 0.6 0.8 0.8 1.2 1.0 1.2 0.8 0.6 0.6 0.8 0.8 1 1 1.2 1.2 1.4

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Fig. 10 e Comparison of the Leeff calculated according to the methodology summarized in Table 3 with the experimental Le for A: all the blends including those of Table 4; B: H2/CO blends; C: H2/CH4 blends.

behavior and uptick towards “pure” CO was also best revealed by LeO2 ;4i , which is therefore the recommended definition for LeO2 . Overall, the results show that the experimental Le of H2/ CO/air mixtures were best predicted using the LeV based on Lei,4i formulation. In the near stoichiometric range, the Lef =O2 is required to mix the deficient and abundant reactants' Le, which are calculated with LeV using Lei,4i for both the fuel and O2. In leaner and richer mixtures, there was a clearly deficient reactant and the Lef =O2 combination is not required. Under lean mixtures (4 < 0.8), only the combination of LeV with Lei,4i (fuel species as deficient reactant) is needed. In rich mixtures, Leeff is obtained by calculating Lei,4i for the oxidizer mixed by LeV. Table 3 summarizes the procedure for all 4 values.

predicted Leeff did not match the experimental Le, suggesting that further investigation is needed for a mixture of these two components.

Conclusion Methodologies for calculating the Lewis number in laminar flames of multifuel mixtures were investigated for lean and rich mixtures. As the validation required the experimental Le, the investigation also covered the extraction methodology from outwardly propagating flames using the Markstein length, as proposed in the literature [5,28,42]. As for the experimental determination of Le, the following conclusions have been reached:

H2/CO/CH4 and H2/CO/CO2 Up to this point, the determination of a methodology has been approached with two-component mixtures (as summarized in Table 3), in order to yield a Leeff over a wide range of equivalence ratios. In view of fully investigating the proposed methodology, further tests were performed with threecomponent blends (see Table 4). To confirm that the proposed methodology provided the best agreement with the experimental values and not only respected the physical trend, an analysis of the RMS of the studentized residuals of all the results was performed and confirmed the choices presented in Table 3. Fig. 10 compares the experimental Le to the Leeff calculated for all the mixtures tested herein and following the recommendation summarized in Table 3. Based on the results presented before, the experimental Le was extracted with LeBM using lfG , except when cCH4 >30%, in which case, LeChen with lfK was used. Overall, Leeff agreed quite well qualitatively with most of the experimental results, with the strongest disagreement observed for rich mixtures of H2/CO/CH4 blends. Finally, H2/CO2 mixtures (results not presented) were also tested according to the different methodologies above, but the

 The extraction method with LeBM yielded the best results in flames of H2, CO, and their blends, as well as in CO2-diluted flames.  Using LeChen yielded the best results in flames of CH4 and in H2/CH4 blends where H2 < 70%. For higher concentrations of H2, the LeBM method is recommended. Three steps are needed to determine the Leeff of a mixture, depending on the equivalence ratio: (1) choose the Lei associated with each fuel and the oxidizer; (2) choose a mixing rule, and (3) use or not a fueleoxidizer combination. The most qualitatively accurate methodology was identified based on H2/CH4 and H2/CO mixtures and were then validated using three-component mixtures representative of syngas. The main conclusions drawn in that respect are:  The volume-based mixing rule (LeV) was found to be the most accurate for all considered blends, including syngas.  Lei has to be calculated as if the mixture consisted solely of the volumes of air and fuel i, and thus as Lei,4i.  Even if O2 is a single deficient reactant in rich mixture, the authors propose calculating the oxidizer's Lei considering

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that O2 diffuses into each fuel i and then combining each LeO2 ;4i according to the LeV rule to obtain the Leeff.  The combination of the fuel and oxidizer Le (Lef =O2 ) is required solely in near-stoichiometric mixtures (0.8
4
1.0). The former alone is required in leaner mixtures; only the latter is required when 4  1.2. Finally, one should note that the above mixing rules for the calculation of the effective Lewis numbers of mixtures are to be applied to laminar flames while other mixing rules for turbulent flames may be more relevant (e.g. [32,69]).

Acknowledgement This research was supported by the Fonds de recherche Na
bec. ture et technologies of the government of Que

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Please cite this article in press as: Lapalme D, et al., Assessment of the method for calculating the Lewis number of H2/CO/CH4 mixtures and comparison with experimental results, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/ j.ijhydene.2017.01.099

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 7 ) 1 e1 5

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Please cite this article in press as: Lapalme D, et al., Assessment of the method for calculating the Lewis number of H2/CO/CH4 mixtures and comparison with experimental results, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/ j.ijhydene.2017.01.099