Study and characterization of the instabilities generated in expanding spherical flames of hydrogen/methane/air mixtures

Study and characterization of the instabilities generated in expanding spherical flames of hydrogen/methane/air mixtures M. Reyes a,* , R. Sastre a , F.V. Tinaut b , J. Rodrı ´guez-Fern andez c a Department of Energy and Fluid Mechanics Engineering, University of Valladolid, Paseo del Cauce s/n, 47011, Valladolid, Spain b CMT-Motores T ermicos, Universitat Polit ecnica de Val encia, 46022, Valencia, Spain c Universidad de Castilla la Mancha. ETS Ingenierı´a Industrial (Edificio Polit ecnico), Avda. Camilo Jos e Cela,3, 10371, Ciudad Real, Spain highlights Origin and nature of instabilities developed in spherical hydrogen/methane flames. Differentiate instabilities of hydrodynamic and thermal-diffusive origin. Instability peninsulas calculated to establish stability limits in each mixture. Lewis effective number determines the shape and location of instability peninsulas. article info Article history: Received 11 March 2022 Received in revised form 3 May 2022 Accepted 8 May 2022 Available online xxx Keywords: Combustion instabilities Thermal-diffusive and hydrodynamic effect Peninsula of instability Cellularity Growth rate of instabilities Schlieren photography abstract In the present work an analysis of the origin and nature of intrinsic instabilities in combustion processes with different hydrogen/methane mixtures is developed. These expanding spherical flame front experiments have been developed in a cylindrical constant volume combustion bomb, which allows recording the process through Schlieren photography method. The stability study in combustion processes has a great importance to assure their security and control, since the understanding of flame instabilities is necessary for improving the internal combustion engines performance. To carry out this mentioned study, a review of the concepts and parameters used in spherical flame front instabilities research is first proposed, as well as a physical explanation of each concept and the relations among them. Additionally, a methodology that aims to determine the influence of the fuel mixtures in the origin and development of the flame front instabilities is suggested. Moreover, the intrinsic effects of the combustion process, such as the thermal-diffusive and the hydrodynamic effect, are separately studied, including their individual contributions to the growth rate of instabilities which allows to determine combustion nature and to obtain the instability peninsula of each fuel mixture. *Corresponding author. E-mail address: [email protected] (M. Reyes). Available online at www.sciencedirect.com ScienceDirect journal homepage: www.elsevier.com/locate/he international journal of hydrogen energy xxx (xxxx) xxx https://doi.org/10.1016/j.ijhydene.2022.05.063 0360-3199/©2022 The Author(s). Published by Elsevier Ltd on behalf of Hydrogen Energy Publications LLC. This is an open access article under the CC BY-NCND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article as: Reyes M et al., Study and characterization of the instabilities generated in expanding spherical flames of hydrogen/methane/air mixtures, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2022.05.063 Finally, this methodology includes a qualitative study of the cellularity phenomenon (when the instabilities develop all over the flame front), considering the parameters which influence on this phenomenon. ©2022 The Author(s). Published by Elsevier Ltd on behalf of Hydrogen Energy Publications LLC. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/). Introduction Nowadays, there is an energy problem worldwide based on the one hand, on a practically exclusive dependence on fossil fuels derived from petroleum, and, on the other hand, on the effect that their use has on the environment due to the emissions generated in their combustion process. It is estimated that the transport sector is responsible for almost 30% of total CO 2 emissions in the European Union (EU), with this gas being the main cause of the greenhouse effect. A possible solution is the replacement of these CO 2 -producing fuels with others with low or no carbon content such as hydrogen (H 2 )or ammonia (NH 3 ), being more sustainable from the environmental point of view. Prior to the introduction of any fuel mixture in a conventional engine, its behavior must be studied by isolating it from other external factors that may affect the performance of combustion, such as the turbulence that occurs in an engine due to the movement of the pistons. Constant volume combustion bombs (CVCB) allow the analysis of mixtures in laminar regime, thus facilitating the study of their nature. The above approach is at the base of this research work, in which laminar combustion of spherical flames of mixtures of H 2 / CH 4 /air with different percentages of hydrogen (H 2 ) are characterized in a CVCB with cylindrical geometry. A fundamental aspect in the characterization of fuel combustion is the study of its stability, which consists of the analysis of the development of the instabilities inherent to the combustion process: origin, nature (thermo-diffusive or hydrodynamic, mainly), growth, etc. These can alter the morphology of the flame and affect the burning velocity. There is a growing interest in understanding and controlling the unstable behavior of cellular flames [1] and the transition to cellularity in expanding spherical flames [2e4]. There are several ways to characterize a combustion process depending on the method and resources used, but all of them aim to determine the most significant parameters that define how a combustion is developing: thermodynamic conditions, burning velocity, instabilities, and flame morphology among others. The Schlieren technique is used in this work to record the flame front development. This technique is based on the light deflection caused by density gradients [5]. It was first employed by Toepler in 1864 [6]. The Schlieren method is widely used for the study of flames in a combustion process since it allows their recording with a great clarity [7]. Thanks to the use of this technique, it is possible to study the morphology [8,9], instabilities and cellularity of the flame front, as well as its burning velocity [4,5,8,10] and, therefore, characterize the fuel mixture considered [11,12]. Spontaneous perturbations appear in flames due to various phenomena such as acoustic vibrations or pressure waves [4]. Intrinsic combustion effects cause that the amplitude of the waves on the flame front decreases or increases depending on the dominant action of those effects. The result of each effect in the flame front is the instability, independently of their stabilizing or destabilizing character [13,14]. The phenomena that can produce instabilities in the flame front (in a laminar regime) of a premixed combustion are three: volume forces, hydrodynamic effects and thermo-diffusive effects [15]. Volume forces are relevant when a less dense fluid moves towards a denser one in the opposite direction to that of a volume force (for example, gravity). Then, there appears an instability due to volume forces (also known as RayleighTaylor [16]). In a flame, this occurs when the combustion products (with lower density) advance towards the fresh mixture (higher density) upwards (contrary to gravity); the flame front is then considered as a discontinuity of densities. This instability type, compared with the other two, has the lower influence on the development of the flame. The second instability considered is the hydrodynamic one (Darrieus-Landau, DL), which is associated with the thermal expansion or density difference between the unburnt and burnt gases in a combustion process, delimited by the flame front [13,14]. The hydrodynamic effect generates destabilizing instabilities since they tend to intensify all of the flame front perturbations significantly larger than the flame front thickness (d l )[15,17]. Hydrodynamic instability can also be stimulated by reduced flame thickness and high pressures [18]. In addition, turbulent propagation speed can be enhanced by the onset of the hydrodynamic instability [19,20]. The thermal-diffusive effect is the cause of the third instability type that affects the combustion process in a significant way. This instability does not have a unique stable or unstable character since it involves thermal and molecular diffusivities, that have opposite results [15,21e24]. The molecular diffusivity affects the perturbations in such a way that it increases them (destabilizing contribution); on the contrary, the thermal diffusivity tends to attenuate the flame front (stabilizing contribution). This effect can be represented quantitatively by the dimensionless Lewis number (Le), which is the ratio between thermal diffusivity (a) and molecular diffusivity (D). Destabilizing thermal-diffusive instability occurs when the Le is smaller than a critical value (when the molecular diffusivity dominates the process), and cracks and cells develop international journal of hydrogen energy xxx (xxxx) xxx2 Please cite this article as: Reyes M et al., Study and characterization of the instabilities generated in expanding spherical flames of hydrogen/methane/air mixtures, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2022.05.063 from the beginning on the flame front surface; otherwise, the flame should be unwrinkled at first, although after a certain time, needed for the hydrodynamic instability to appear, wrinkles can be visible [25]. In addition to the latter, the influence of the flame stretch must be considered in spherical flames. This parameter includes strain and curvature effects due to the flame geometry and always tends to attenuate the instabilities in the combustion beginning (intrinsically stable character). It should be studied together with the nature of the mixture and, specifically, the thermo-diffusive effect [24,26] represented by the Lewis number (Le) or better, the effective Lewis number (Le eff ), see Refs. [21,24,27,28], which is a weighted average of the reactants Lewis numbers. The critical radius and the cellular radius are studied in literature to characterize the onset of instability on flames and the cellular phenomenon, respectively [24,29e33]. The dimensionless flame radius is the Peclet number (Pe), which is the ratio between the instantaneous radius and the flame thickness. Many definitions of critical radius can be obtained in literature, Bradley et al. [34] defined the critical radius at the instant of the appearance of the first cracks in the flame front. Bechtold and Matalon [2] identified the critical radius, for flames with effective Lewis number bigger than a critical value, as the radius beyond which the flame is unstable due to the dominance of the hydrodynamic effect when instabilities are no longer dampened, although visually it is not appreciable. This is the definition of critical radius adopted in this work, associated with the critical Peclet number (Pe cr ). Another definition refers to the flame radius when the flame is visibly unstable due to the development of cellularity. This radius can be called critical cellular radius, also utilized by Bradley et al. [34]. From this radius, the cellularity increases the flame front surface, which produces an augmentation in the apparent combustion burning velocity (or better, combustion rate). In the works of Jiang et al. [35] and Tinaut et al. [5] this definition for the critical radius is applied. The understanding of hydrogen/methane combustion and flame instabilities is necessary for improving internal combustion engines performance. Previous works have been focused on the burning velocity study of hydrogen/methane mixtures [36e42]. Other studies have focused on the engine combustion [43e45]. Various researchers have investigated the cell formation in hydrogen/air flames and in mixtures of hydrogen with other hydrocarbons flames [46e49], indicating that hydrogen addition increased hydrodynamic and thermaldiffusive instabilities. Additionally, an understanding of the formation and origin of cellular instabilities in hydrogen/ methane-air flames should be considered further. Jiang et al. [50] studied the cellular structure of methane/hydrogen/air flames in a spherical combustor, for various equivalence ratios and hydrogen fractions, and studied a correlation between the cellular structure and pressure. Kim et al. [51] studied cellular instabilities on expanding spherical propagation hydrogen/aire, methane/air and propane/air flames. They characterized the size of the cell, observing bigger cells due to hydrodynamic instability than the generated by the thermal-diffusive. The critical Peclet number is also affected by the initial pressure, decreasing with the increment of pressure. Zhang et al. [52] made a flame dynamics analysis of natural gas enriched with hydrogen in premixed turbulent combustion, obtaining an increment of the wrinkled structure with the turbulence intensity and with the hydrogen ratio. Cellular instabilities enhance the global propagation rate of laminar flames and have also been studied by many researchers [30,47,53e55]. Kwon et al. [56] studied hydrogen flames at elevated pressures to identify the effects of thermal expansion ratio, flame thickness, Lewis number, and stretch rate on the generation of hydrodynamic and diffusionalthermal instabilities in the flame front. Jiang et al. [50] investigated cell formation in hydrogen/methane premixed flames and obtained a correlation between cellular structure of the flame front and pressure. Sun et al. [30] studied the cellular instabilities of hydrogen-air premixed flames at different initial conditions, showing that, for lean hydrogen/air flames, the cellular instabilities are dominated by the thermaldiffusive instability, and for stoichiometric and rich hydrogen/air flames, the cellular structure is influenced by the hydrodynamic instability. Okafor et al. [57] have developed an experimental and theoretical investigation of cellular instabilities in lean hydrogen/methane mixtures in a constant volume combustion chamber, obtaining an increment of the cellular instability and self-acceleration with the increment of the hydrogen content and mixture pressure. Hu et al. [47] observed that the increment of hydrogen in methane/ hydrogen/air mixtures promotes cellularity in lean flames. Law et al. [58] studied the outward propagation and development of surface cellular instability of spark-ignited spherical premixed flames of mixtures of hydrogen, hydrocarbon, and air, showing that propane substitution moderates cell formation due to both diffusionalethermal as well hydrodynamic instabilities. Smallbone et al. [59] carried out studies of flame instability, laminar and turbulent velocity of hydrogen/ air mixtures in a fan stirred combustion bomb. Wang et al. [60,61] investigated the flame front structure of syngas premixed flames. Vu et al. [1] investigate the cell formation in hydrogen/methane/monoxide eair premixed flames. Jiang et al. [50] studied the influence of pressure in the development of a cellular structure. In present work an analysis of the origin and nature of intrinsic instabilities in spherical expanding flames of hydrogen/methane blends is developed, because there is a lack of data in literature. These spherical flame front experiments have been performed in a cylindrical constant volume combustion bomb (CylCVCB), which allows image recording through Schlieren photography method and recording the instantaneous pressure. With the methodology presented in this research, it is possible to obtain the stability maps of different combustion mixtures, which corresponds with the images directly obtained. To carry out this research, a review of the concepts and parameters used in the study of instabilities in spherical flames is first proposed, as well as a physical explanation of each concept and the relations among them. After that, the experimental installation and methodology are explained, and finally experimental results are presented for mixtures of methane/hydrogen with different percentage of hydrogen (from 0 to 100%), for two different fuel/air equivalence ratios, and initial pressure and temperature constant in all the experiments. international journal of hydrogen energy xxx (xxxx) xxx 3 Please cite this article as: Reyes M et al., Study and characterization of the instabilities generated in expanding spherical flames of hydrogen/methane/air mixtures, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2022.05.063 Stability analysis in spherical flames To characterize an unstable spherical flame, it is important to differentiate between the instant at which instabilities are originated (at the first stages of the combustion or in the subsequent instants of the development of the combustion process) and whether or not the evolution of the instability leads to the development to a cellular flame front. The growth rate of spherical flame instabilities is defined as the increase in time of the relative amplitude of the perturbation (ratio between the amplitude and the instantaneous radius of the sphere) formed on the surface of the flame [14] as the joint effect of the hydrodynamic and thermaldiffusive instabilities. When the flame radius has reached a certain value r 0 (significantly greater than the thickness of the flame, dl) the amplitude Aof the perturbation can be non-dimensionalised with rto give a. Once perturbed variables are expanded in a spherical harmonic series, atakes the following form (adapted from Ref. [4] for coherence with the subsequent expressions for the terms appearing in it): a¼a0RuþdlU r,ln R(1) where a0is the initial dimensionless amplitude when r¼r 0 ; R¼r/r 0 is the dimensionless flame radius; dlis the flame front thickness (dl¼a=Sl, where ais the thermal diffusivity of the mixture and Slthe laminar velocity [10]). uis associated with hydrodynamic instability [4,21,29,32]; it is a dimensionless parameter (usually denoted as uDL in its equivalent in flat flames), that only depends on the thermal expansion sand the dimensionless wavenumber n.Uis associated with thermaldiffusive instability [29]; it is also dimensionless and depends on the thermal expansion, Lewis number and Zeldovich number (b) (see below). The dimensionless wavenumber n quantifies the number of waves, with a lwavelength, that fit in a circumference of radius rn¼2pr l, as illustrated in Fig. 1. The growth rate of an instability is 1 =aðda =dtÞ[4,21], hereinafter identified by the letter S, will adopt different forms depending on the case. The general function of SðtÞis taken from the research of Addabbo, Bechtold and Matalon [21,32], collected in the study of Lapalme et al. [24] and represented in equation (2). SðtÞ¼1 a da dt ¼1 r dr dt udl rU¼1 r dr dt uU Pe(2) In equation (2) it is possible to see the contributions to the instability grown rate due to the hydrodynamic (u) and thermo-diffusive ( U Pe) effects [29]. Hydrodynamic effect is always positive (contributing to instability), but the thermodiffusive effect can increase or reduce the first, depending on the value of U: U¼Q1þbLeeff 1 s1Q2þPrQ3(3) Note that Uconsists of three contributions, each accompanied by a Qicoefficient. The physical meaning of each of them is as follows: the term Q1brings the influence of the thermal conductivity (stable character); the second term bLeeff 1 s1Q2represents the effect of the molecular diffusion (stable or unstable) and the term PrQ 3 (Pr is the Prandtl number) brings the effect of the viscosity (stable) [62]. The coefficients Q 1 ,Q 2 and Q 3 are given in Eqs. (4)e(6). Q1¼g1 sDn4ðsþ1Þþsn3ð2uþ5Þþn2us 2s2þs1 þnsðs73usuÞ2sð1þuÞ(4) Q2¼g2ðs1Þ 2D2n4þn3ð2us þ2uþ10s3Þþn22su2þð5s1Þu þ3s2s22þnsu2ð14sÞ14s2þ1uþ39s8s2 2su2þ4uþ3 (5) Q3¼2nðn21Þðs1Þ sD½ðnþ2Þð ~ kðxÞg3Þ3ð ~ lðxÞ1Þ (6) where D¼2auþb2a. The definitions of coefficients g1;g2 and g3appear in the second column of Table 1. It can be seen that the three gidepend on the expansion ratio sand a function ~ kðxÞwhich represents the thermal conductivity expressed in terms of the dimensionless temperature T=Tu (i.e., the flame temperature Tover the unburned gas temperature Tu) and scaled by its value in the unburned gas. In order to obtain explicit expressions of githat still retain the main dependence on s, different possibilities of the variation of ~ kðxÞ can be considered: constant (order of 1), variable as ffiffiffi T p, and variable as T. The expressions of gifor the three possible dependences of ~ kðxÞappear in the additional columns of Table 1. In the first case ( ~ k(x)¼1), with the hypothesis of constant transport properties, thermal conductivity is a constant and the influence of viscosity on the stability of the flame is minimal [21]. It is worth mentioning the formulation made by Bradley and Harper in 1994 [4] with the assumption of constant properties. Coefficients Q i are also detailed in Bradley [3] which describes four stages in the propagation of a laminar spherical flame, these are: (1) stable laminar propagation, (2) cracking of the flame front and cell formation, (3) propagation of the cellular flame, (4) turbulent propagation. Fig. 1 eFlame radius, wave amplitude and wavelength. international journal of hydrogen energy xxx (xxxx) xxx4 Please cite this article as: Reyes M et al., Study and characterization of the instabilities generated in expanding spherical flames of hydrogen/methane/air mixtures, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2022.05.063 The effect of the stretch cannot be studied separately because the nature of the mixture must be considered and, specifically, the thermo-diffusive effect reflected in the ratio between the effective Lewis number Le eff in a mixture, and the critical Lewis number of the combustion (Le eff* ). Equation (7) shows this latter parameter, where bis the Zeldovich number; as for Q 1 and Q 3 , they must be evaluated for the critical radius or Peclet number according to equations (4) and (6). Leeff*¼1s1 bQ2ðQ1þQ3PrÞ(7) As it was explained before, if in a fuel mixture Le eff <Le eff* ,it means that the thermo-diffusive effect is destabilizing, and instabilities grow from the beginning. On the contrary, in combustions with Le eff >Le eff* the thermo-diffusive effect is stabilizing and, therefore, the only effect capable of increasing instabilities is the hydrodynamic one. It is possible to obtain the curves for the growth rate of the instability Swhen the values of Eq. (2) are represented versus the wavenumber nor alternatively the wavelength of the perturbations lfor a particular flame front radius or time [2,4,64], as can be seen in Fig. 2, where it is possible to see two regions: stable and unstable. Both zones are separated by the value of S¼0, the stability limit for a spherical flame. The dotted line corresponds with the critical radius r cr from which there are certain wavelengths whose growth rate is positive (unstable flame). Experimental apparatus and procedure Experimental setup In this research work, experiments were conducted in a cylindrical constant volume combustion vessel (CylCVCB) [5], with the following dimensions: diameter 114 mm and height 135 mm, to investigate the combustion development and in particular, instabilities and the onset of cells in the flame surface, see Fig. 3. The combustion chamber was designed with two optical windows made of fused silica on the sidewalls of the cylinder. The installation is instrumented to measure instantaneous pressure during the combustion (piezoelectric pressure transducer Kistler type 7063) and to register the flame development with a high-speed Schlieren photography system: flame images were recorded using a high-speed camera Phantom V210 at 7000 frames per second (resolution 832 800 and exposure time of 10 ms). Hydrogen, methane and air are individually introduced from pressure tanks using a partial pressure method, and an ignition system starts the combustion at the center of the combustion chamber (for more information about the experimental installation see Ref. [5]). An algorithm is used to process the images obtained with the Schlieren technique (high-speed camera), to obtain the time evolution of the flame radius and to study the cellular structure which appears as a wrinkling of the flame surface. Radius evolution is obtained through frame by frame processing, after background removal, binarization and thinning transformation to highlight the borders between cells, as it is explained in detail in Ref. [5], obtaining the instantaneous position of the flame front (with a random sample consensus algorithm programmed in Matlab) and identifying the cells that appear in the flame during de combustion development. Instantaneous pressure is registered inside the combustion chamber, and it is analyzed by means of a two-zone diagnosis model, with the aim of obtaining burning velocity, Lewis number and effective Lewis number (Le and Le eff ), thermal expansion s, instability growth rate S, and the rest of the parameters necessary to study the instability grown rate and their coefficients, explained in detail in previous sections. Table 1 eAnalytical expressions that define the coefficients g i and practical as a function of the thermal conductivity [21,63]. ~ kðxÞ¼1 ~ kðxÞ¼ ffiffiffiffi T p~ kðxÞ¼T g1s s1Z s 1 ~ kðxÞ xdx s s1ln s2s ffiffiffi s pþ1 s g21 s1Z s 1 ~ ~ kðxÞ xlns1 x1dx 1 s1Z0 ∞ ln½1þðs1Þezdz 4 s1fffiffiffi s p1ln½0:5ðffiffiffi s pþ1Þg 1 g31 s1Z s 1 ~ kðxÞdx 12ðs3=21Þ 3ðs1Þ sþ1 2 Fig. 2 eInstability growth rate curves for a given flame radius ras a function of the perturbation wavelength. international journal of hydrogen energy xxx (xxxx) xxx 5 Please cite this article as: Reyes M et al., Study and characterization of the instabilities generated in expanding spherical flames of hydrogen/methane/air mixtures, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2022.05.063 Methodology In this research work, mixtures of H 2 /CH 4 with different percentages of hydrogen (varying from 0, 20, 50, 80 and 100%) are analyzed to determine the influence of hydrogen addition on the stability of a flame, specifically on the origin and nature of the instabilities intrinsic to a combustion process. To isolate the effect of hydrogen from other possible influencing variables, two groups of experiments with stoichiometric and lean fuel/air ratio (F¼1 and F¼0.7) have been selected; with an initial pressure of 0.1 MPa, and an initial temperature of 323 K. The first group (F¼1) has been studied to identify the influence of the hydrogen in the flame stability analysis (stability limits, combustion nature, instabilities, and cellularity). The aim of the second group is to study the impact of lean equivalence ratio on the stability analysis, as well as the effect of the hydrogen content, by comparing the results with those of stoichiometric equivalence ratio. The stability study in flames of H 2 /CH 4 eair mixtures is made by examining four fundamental characteristics of a spherical flame: stability limits related to perturbations wavelength l, flame nature, instabilities growth rate Sand the cellularity phenomenon. In this section, it is briefly explained the meaning of each one of them and their main goals. These four characteristics are developed and applied to the selected experiments in the section corresponding to results. Stability limits related to perturbations wavelength (l) To study the stability limits related to the wavelength of the perturbations (l) it is useful the graphical representation of the stability limits of a flame, which is called stability curve or instability peninsula [13]. Fig. 4i shows an example of this chart where the wavenumbers for which instabilities growth rate is zero (S¼0) are plotted versus the Peclet number. Similarly, instead of the Peclet number, the flame radius could also be utilized (this is employed in the results section of this paper). These values conform a region that contains the wavenumbers of all the perturbations that experience an increment of their amplitude at a certain flame Peclet number (or radius) since their growth rate is positive (S>0); therefore, it is called the unstable region (the blue one). The other area is the stable region, in which the growth rate of the perturbations is negative (S<0) and consequently the waves are attenuated. It should be noted that the nose of the peninsula corresponds to the critical Peclet number (or critical flame radius) from which the flame enters in the unstable region. For a Peclet number larger than the critical value, there is a series of growing perturbations defined by their wavenumbers (wavelengths) between n max (or l min ) and n min (or l max ), while all those which are outside this range are decreasing. Fig. 4ii shows a growth rate curve for a given Peclet number in which these limits are marked. These maximum and minimum values fulfill the following condition for each Peclet number (and associated flame radius): S¼0¼1 a da dt ¼1 r dr dt udl rU0uU Pe ¼00Pe ¼U u(8) Fig. 4iii contains a joint representation of the information of the instability growth rate curve for a given Peclet number and the instability peninsula. Fig. 3 eSchematics of the experimental setup. international journal of hydrogen energy xxx (xxxx) xxx6 Please cite this article as: Reyes M et al., Study and characterization of the instabilities generated in expanding spherical flames of hydrogen/methane/air mixtures, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2022.05.063 Flame stability nature. A combustion is considered intrinsically unstable when the flame front instabilities appear and grow already from the beginning of the process, i.e., if the instability growth rate is positive (S>0) at the first states of the combustion. On the contrary, if the instabilities begin to grow only after the flame radius has reached a certain value (even if it is small), a combustion is considered stable. Therefore, the combustion stability character is defined by the instabilities growth rate in the first stages of the process. As mentioned before, the hydrodynamic effect has always a destabilizing character. However, the thermal-diffusive effect could be stabilizing (counteracting the hydrodynamic effect together with the flame stretch) or destabilizing (increasing the hydrodynamic effect), meaning that the combustion stability nature is strongly dependent on this second effect. This section aims to characterize the combustion nature and the morphology of a stability curve derived from an unstable combustion. With the objective of identifying the stability nature of a flame, the value of its effective Lewis number must be compared with the value of its critical effective Lewis number. If the effective Lewis number value is greater than the critical one (Le eff >Le eff* ) the combustion is stable at the beginning, its instability peninsula is located to the right of the vertical axis, the instabilities have a hydrodynamic source and start to grow only once the Peclet number has a value higher than 0. Otherwise, when Le eff ;<Le eff* , the combustion has an unstable nature, the instabilities arise from the thermal-diffusive effect already from the beginning (in addition to the hydrodynamic contribution. Following the previous approach, the limit between the stable and unstable combustion is well marked by the effective Lewis number whose critical value is, in general, close to 1. In Fig. 5 two instability peninsulas for different Lewis effective numbers are presented. When the Lewis number is equal to its critical value (Le eff ¼Le eff* ), the peninsula nose touches the vertical axis, i.e., the critical Peclet number of the combustion is equal to zero (broken line in Fig. 5). When the Lewis number is lower than its critical value (Le eff >Le eff* ), the peninsula is displaced to the right in the figure. Instabilities growth rate and relative contributions of hydrodynamic and thermal-diffusive effects. As previously mentioned, the influences of hydrodynamic and thermal-diffusive contributions are reflected on the instabilities growth rate (see Eq. (2) and Eq. (3)). The thermaldiffusive effect has more influence at the primary stages of the combustion process, since it is inversely proportional to the Peclet number ( U Pe). When the flame radius increases, this effect loses importance, and the growth rate is dominated by the hydrodynamic effect (u). Consequently, the more stabilizing power the thermal-diffusive contribution has, the later the instabilities due to the hydrodynamic contribution appear. According to the above, regardless of the combustion Fig. 4 eSchemes of the stability curve. Fig. 5 eInstability peninsulas for two mixtures with different values of Le eff . international journal of hydrogen energy xxx (xxxx) xxx 7 Please cite this article as: Reyes M et al., Study and characterization of the instabilities generated in expanding spherical flames of hydrogen/methane/air mixtures, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2022.05.063 nature, all the flames would visibly develop instabilities in a combustion chamber with an infinite size. Cellularity development. The cellularity phenomenon is the visible consequence of the full development of the instabilities in a flame front. At a certain radius, the waves amplitudes are so big that they instantly break down the flame front structure in such a way that cells are spread all over the flame surface in a perceptible way, which can be recorded by the photographic schlieren technique. A qualitative analysis of the flames can be carried out through the study of those images in order to establish the moreand less-determinant parameters on the apparition and development of the cellularity phenomenon. Results of the stability of hydrogen/methane flames In this section results of the stability of hydrogen-methane flames are presented for different percentages of hydrogen in the flame mixture and for two different fuel/air equivalence ratios: stoichiometric and lean mixture (F¼0.7). For all cases, the values of the expansion ratio sare calculated for each time. The values of each mixture effective Fig. 6 eInstability peninsulas. international journal of hydrogen energy xxx (xxxx) xxx8 Please cite this article as: Reyes M et al., Study and characterization of the instabilities generated in expanding spherical flames of hydrogen/methane/air mixtures, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2022.05.063 Lewis number are the corresponding to the critical radius for the stable ones and to the initial radius for those unstable, always in the unburned side. The critical Lewis number is calculated according to Eq. (7). Stability limits In Fig. 6 the stability maps versus the Peclet number (left) and the flame radius (right) are presented for all the experiments developed. Comparing these two representations, it can be removed the flame thickness influence. In Fig. 6i and 6ii stability curves are presented for the experiments developed at stoichiometric fuel/air equivalence ratio for different hydrogen content (0%, 20%, 50%, 80% and 100%). It can be seen that both peninsulas of the mixtures with 50% H 2 for F¼1are at the left of the chart with the lowest value of the critical Peclet number or flame radius (corresponding to their noses). According to the above mentioned, this means that the instabilities grow almost from the combustion beginning but this does not correspond necessarily with an earlier apparition of a cellular flame, as explained later. The peninsulas obtained for the rest of fuel compositions are different depending on if they are plotted versus the Peclet number or flame radius. It can be seen in Fig. 6i and 6ii that the percentage of hydrogen affects significantly to the stability limits but not in a proportional way. Comparing the maps plotted versus the Peclet number and versus the flame radius (Fig. 6i and 6ii) it is possible to see the influence of the flame front thickness, especially for the combustions with 80% and 100% of hydrogen, since they have bigger Peclet numbers for the same radii (thinner flame front thickness). In Fig. 6ii it can be observed that the development of instabilities occurs at smaller radii for the medium and high hydrogen content mixtures (50, 80 and 100% of H 2 ) than for those with high methane content (20 and 0% of H 2 ). Contrasting Fig. 6i and 6ii shows that the flame thickness decreases with increasing hydrogen content: for the same Peclet number a smaller radius is obtained. Instability peninsulas versus Peclet number for combustions under lean-fuel conditions are presented in Fig. 6iii, 6v. First thing to notice is that the experiments with low hydrogen content are slightly displaced to the left compared to those with stoichiometric-fuel conditions (i.e., slightly smaller values of the critic Peclet number). As for the medium and higher hydrogen content mixtures, their peninsulas are completely different since now they are unstable combustions from their origin. The morphology of this curves is just a line, meaning that the growth rate curves (S¼fðnÞ) only have one root, i.e., only one value of the wavenumber (n) fulfils S¼0 (instead of the two that the stable combustions have with a peninsula shape); the unstable region corresponds, consequently, with the upper part of the graph shown in Fig. 6v and the stable region is the inferior part. When the instability peninsulas are represented versus the flame radius (Fig. 6iv and 6vi) the peninsulas for the lower hydrogen content mixtures are displaced to the right, i.e., their critical radius are bigger compared to those of the stoichiometric case. This suggest that mixtures with 0 and 20% of H 2 (i.e., methane is dominant) are more stable at lean conditions than at stoichiometric conditions. For high and medium hydrogen contents (dominant hydrogen) the tendency is the same, independently of if the instability curves are represented versus the Peclet number or flame radius (see Fig. 6v and 6vi). Flame stability nature The comparison of the effective Lewis number, Le eff , (calculated with mixture properties) with the critical value Le eff* (calculated by means of Eq. (7)) provides the key to know if a mixture is intrinsically stable or not. For this reason, the values of these two parameters, in each combustion process, are plotted together in Fig. 7. All the stoichiometric mixtures studied in this research work are stable in their origin, since all the Lewis numbers represented in Fig. 7 (i) are bigger than its critical value (Le eff >Le eff* ). Comparing this Figure with the instability peninsulas (Fig. 6) it can be said that the closer the two Lewis number are, the more to the left the peninsula is and the less stabilizing capability the thermal-diffusive effect has. The evolution of the Lewis number and its critical value based on the hydrogen content can be related to the dominant fuel in each case. Traditionally, for mixtures of methane/ hydrogen/air, has been accepted that hydrogen is dominant when its content is 50% and higher, while in case of a lower hydrogen content methane is dominant. This can be seen in the research of burning velocities of this types of mixtures carried out by Reyes et al. [36]. These results are compatible with the experimental behavior observed for stoichiometric mixtures in this study. It can be seen in the stability maps (Fig. 6), especially in those represented versus the flame radius (Fig. 6ii and 6iv) or in the Lewis numbers (Fig. 7i), that the shape of Fig. 7 eLe eff and Le eff* versus the percentage of hydrogen in the fuel mixture. international journal of hydrogen energy xxx (xxxx) xxx 9 Please cite this article as: Reyes M et al., Study and characterization of the instabilities generated in expanding spherical flames of hydrogen/methane/air mixtures, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2022.05.063 giCoefficients of Eqs. 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Int J Hydrogen Energy 2021;46(17):10494e505. international journal of hydrogen energy xxx (xxxx) xxx 17 Please cite this article as: Reyes M et al., Study and characterization of the instabilities generated in expanding spherical flames of hydrogen/methane/air mixtures, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2022.05.063