Burning Behaviour of High-Pressure CH₄-H₂-Air Mixtures

Abstract

Experimental characterization of the burning behavior of gaseous mixtures has been carried out, analyzing spherical expanding flames. Tests were performed in the Device for Hydrogen-Air Reaction Mode Analysis (DHARMA) laboratory of Istituto Motori—CNR. Based on a high-pressure, constant-volume bomb, the activity is aimed at populating a systematic database on the burning properties of CH₄, H₂ and other species of interest, in conditions typical of internal combustion (i.c.) engines and gas turbines. High-speed shadowgraph is used to record the flame growth, allowing to infer the laminar burning parameters and the flame stability properties. Mixtures of CH₄, H₂ and air have been analyzed at initial temperature 293÷305 K, initial pressure 3÷18 bar and equivalence ratio ϕ = 1.0. The amount of H₂ in the mixture was 0%, 20% and 30% (vol.). The effect of the initial pressure and of the Hydrogen content on the laminar burning velocity and the Markstein length has been evaluated: the relative weight and mutual interaction has been assessed of the two controlling parameters. Analysis has been carried out of the flame instability, expressed in terms of the critical radius for the onset of cellularity, as a function of the operating conditions.

1. Introduction

No matter the claim and the quest for new answers to the needs of an energy-thirstier society, thermal conversion systems will play a key role in the energy supply chain. This awareness, along with the urgent need of CO₂ reduction (currently the main driver of technology advance), has at least two direct consequences: the introduction in the energy cycle of CO₂-neutral fuels (biomass-derived) and the design/development of more efficient conversion systems. The latter point can be anticipated to have a larger practical impact than the former, given the majority of energy sources will be represented by fossil fuels for many years to come.

Sometimes placed in the category of “alternative” fuels, methane (CH₄) is indeed a hydrocarbon, sporting the peculiarity of the lowest C/H ratio, if compared to other fuels: by trivial reasoning, the less the carbon, the less the CO₂. This feature has boosted a renewed attention to CH₄, which represents the main constituent of natural gas. CH₄ has been used with heat engines for a long time, even if with the status of a “niche” solution: as a fuel for internal combustion (i.c.) engines, it offers some advantages over gasoline, having wider flammable limits and better anti-knock characteristics, at the cost of a lower flame speed. A feasible opportunity to overcome the limits of CH₄ is offered by H₂: adding H₂ to CH₄ (or natural gas) can improve the flame speed/stability and lower the lean operating limit. The overall effect is to extend the stable operation map to extreme conditions (e.g., high EGR).

Crucial for the development and design of i.c. engines and gas turbine combustors is the knowledge of laminar combustion properties: they offer the basis for modelling and simulation of flame-turbulence interaction. Data on the combustion properties of pure gaseous fuels are widely available in the literature [1,2,3,4,5,6,7,8,9,10,11,12], but hardly in a systematic form; moreover, data for multi-component fuels at high pressure are even scarcer: filling this gap is the scope of the Device for Hydrogen-Air Reaction Mode Analysis (DHARMA) project, aiming at generating a comprehensive and coherent grid of data on the combustion properties of CH₄ and H₂, obtained in conditions as close as possible to those of actual engines.

The project relies on an optically accessible constant-volume bomb (static P ≤ 20 MPa), where test are carried out on spherical expanding flames; the operating conditions (P₀, T₀), the equivalence ratio, the relative composition of the mixture, the ignition energy can be varied in a meaningful range. A shadowgraph setup, based on a c.w. laser source and a CMOS camera, is used to follow the flame growth with high time and space resolution.

The present work was carried out in the above-described framework and deals specifically with the effect of the initial pressure and of the H₂ amount on the combustion properties of CH₄-H₂-air mixtures.

The initial pressure varied between 3 and 18 bar; the H2 percentage varied from 0% to 30% (vol.): the equivalence ratio $\varphi$ was 1.0. These conditions were selected to match the operating range of automotive internal combustion engines.

The experimental tests yielded the (unstretched) laminar burning velocity and the Markstein length in each condition. These flame characteristics constitute a necessary input for the development and validation of theoretical models of laminar flame propagation. In addition, the interaction between laminar burning velocity and stretch (expressed by the Markstein length) allows to highlight the relation between flame and turbulence within the wrinkled thin laminar flamelet regime of premixed turbulent flames. These theoretical models, in turn, represent the basis for the design and testing of i.c. engines and/or combustors.

Other than the laminar burning characteristics, the phenomenon of flame instability has been explicitly analyzed, offering further details on flame properties. Under certain conditions, instability may wrinkle and break the flame front into small combustion cells: the effective flame front area increases, and the combustion process accelerates. Flame cellularity can be considered as an intermediate regime between laminar and turbulent combustion. Since its relevance in real combustion systems cannot be overlooked, the dependence of cellularity on both pressure and H₂ amount was suitably investigated.

2. Experimental Setup and Procedures

The general arrangement of the experimental layout is shown in Figure 1: a detailed description is given in [13]. The heart of the DHARMA laboratory is a constant-volume test reactor, made of stainless steel (AISI 316): the cylindrical chamber (i.d. = 70 mm, h = 90 mm, aspect ratio = 1.29) is rated for a maximum pressure of ≤20 MPa (static).

A total of 6 optical accesses are available: the larger viewports (d = 65 mm) are located normal to the chamber axis, providing nearly full access to chamber bore; smaller diameter ports are positioned on the chamber side, along two orthogonal axes. Hi-grade quartz windows (85 mm diameter, 30 mm thick) are installed in the main ports, the smaller side ports can be fitted either with quartz windows (49 mm diameter, 20 mm thick) or with a variety of stainless steel adapters (i.e., transducers, electrodes, sampling ports, etc.).

Four additional service ports are available, e.g., for the intake of the combusting mixture and the vent of the exhaust gases.

The mixture is ignited with an automotive inductive ignition system (energy ≤ 60 mJ), which was characterized recording the time evolution of the voltage drop and the current flow between the electrodes: this allowed to evaluate the power and energy release. Being known the electrical behaviour of the coil, the energy of discharge can be set in the range 0÷60 mJ, adjusting the time of charge (dwell time). The spark discharge takes place in the center of the chamber between two pointed-tip tungsten electrodes (dia. = 1 mm, gap = 1 mm).

Figure 1. Layout of the experimental apparatus.

Figure 1. Layout of the experimental apparatus.

A high-frequency dynamic pressure transducer detects pressure history during the combustion events. A metal-shielded, type K thermocouple is used to monitor the temperature of the gases, save for the combustion phase.

The gas handling system was designed to prepare combustible mixtures of variable composition with high accuracy, spanning a range of initial pressures, which included values of relevance in spark-ignition engine operation.

High purity gases (CH₄: 99.9995%, H₂: 99.999%, dry air: 99.999%, N₂: 99.9995%) are used to prepare the mixtures, relying on the partial pressures method [2]: the amount of each gas is metered by a solenoid valve, controlled by a high-resolution (100 MHz) counter/timer board installed in the main computer. The pressure is monitored by a high-accuracy pressure transmitter (0–30 bar, accuracy ±0.08% FS). The gas supply system allows one to prepare combustible mixtures up to 30 bar. After each test, the system is vented, purged with N₂ and pumped down to 10⁻² mbar. All the devices operate with a high degree of automation, to maximize safety and repeatability of the tests.

The entire lab conforms to current safety standards on the use of combustible gases, and is provided with interconnected gas leak sensors, cylinder cut-off devices and forced venting systems.

A parallel-beam direct shadowgraph layout [14] was set up for the analysis of spherical expanding flames. A Diode-Pumped Solid-State c.w. laser (2W @532nm) is used as the light source. High-resolution, time-resolved image acquisition is accomplished by means of a CMOS camera (Photron SA-5, 1024 × 1024 pixel, 1,000,000 fps, shutter time ≥368 ns), interfaced to an independent workstation.

3. Evaluation of Laminar Parameters

3.1. Theory

Time-resolved, shadowgraph images of the spherical expanding flame allow to evaluate the laminar burning parameters, according to a well-known approach [2,5,7,8,9,10,12,15,16]. The time evolution of r_u (the flame radius on the unburned gas side) is obtained through frame-by-frame processing, assuming the luminous front in the shadowgraph corresponds to the radius on the unburned gas side [16].

The stretched flame speed V_s can then be evaluated as:

$$V_s=\frac{d{r}_u}{dt}$$

(1)

The obtained speed includes the stretch effects associated to the propagation of a flame surface, which experiences curvature and flow dynamic strain [17,18,19,20,21,22]. The flame stretch α is defined as the relative rate of change of the flame area: for a spherically expanding laminar flame it can be expressed as:

$$\alpha =\frac{1}{A}\frac{dA}{dt}=\frac{2}{r_u}\frac{d{r}_u}{dt}=2\frac{V_s}{r_u}$$

(2)

The relationship between flame speed and stretch has been thoroughly investigated. It depends on the mathematical model and the associated hypotheses. A detailed analysis leads to the nonlinear formulation [23,24]:

$${\left(\frac{V_s}{V_{s0}}\right)}^2\ln {\left(\frac{V_s}{V_{s0}}\right)}^2=-\frac{2L_b\alpha }{V_{s0}}$$

(3)

where V_s0 is the unstretched flame speed and L_b is the burned gas Markstein length.

For weakly stretched flames, Equation (3) reduces to the linear formulation of [17,21]:

$$V_s=V_{s0}-L_b\cdot \alpha$$

(4)

By plotting V_s against α, it is possible to estimate (i) the unstretched flame speed V_s0 as the value assumed by V_s at α = 0, and (ii) the Markstein length L_b as the gradient of the best straight line fit. The Markstein length indicates how and to what extent the flame is influenced by the stretch. Positive L_b are associated to flames with speed decreasing with stretch (which are stable), while in the case of negative L_b the flame speed tends to increase with stretch, becoming unstable; moreover, the magnitude of L_b indicates to what extent the flame propagation is influenced by the stretch.

The unstretched flame speed V_s0 refers to the limiting case of a plane flame front, with infinite radius and negligible curvature: in such a case, V_s0 can be related to the unstretched laminar burning velocity u_l0 through the following relation:

$$u_{l0}=V_{s0}\frac{{\rho }_b}{{\rho }_u}$$

(5)

where ρ_b is the density of burned gases and ρ_u the density of unburned gases. Equation (5) holds true in the hypotheses of perfect gases, isobaric, adiabatic and equilibrium conditions.

The laminar flame thickness δ_l is a characteristic length, rather than an actual physical dimension: it’s meant to describe the extension of a reaction zone and different definitions have been proposed, based on the balance of mass, energy or momentum. In the current work, the hydrodynamic thickness has been used:

$${\delta }_l=\frac{{\nu }_u}{u_{l0}}=\frac{{\mu }_u}{{\rho }_u\cdot u_{l0}}$$

(6)

where ν_u and μ_u are respectively the kinematic and dynamic viscosity of the unburned mixture [6].

3.2. Data Analysis and Processing

An image processing routine has been implemented to infer the flame radius r_u from the shadowgraph data: for each frame, the flame contour is traced and the area of the projected flame ball is evaluated; the radius is estimated as that of a circle of equal area to the flame. The time evolution of the flame radius r_u offers the basis for the evaluation of the laminar parameters (u_l0 and L_b).

In the present work, a linear dependence was assumed between the flame speed and the stretch, Equation (4): the validity of this postulation was corroborated “a posteriori” by the results.

The usual approach for the linear analysis consists of a series of steps, which makes use of Equations (1),(2),(4) and (5): a polynomial fit of r_u data is first performed; differentiation of the fit allows to obtain the stretched flame speed V_s defined in Equation (1); being known V_s and r_u, the stretch rate α can be evaluated after Equation (2); according to Equation (4), linear-fit extrapolation of the flame speed to α = 0 gives the unstretched flame speed V_s0, while the slope of the fit allows to estimate the burned gas Markstein length L_b. The critical step in this approach resides in the polynomial differentiation, which is highly sensitive to the choice of the original data interval: this may introduce noise and lead to distorted profiles of V_s, affecting both V_s0 and L_b.

At least two different approaches have been proposed in the literature [25,26] for an analytical solution, allowing to obtain both the unstretched flame speed and the Markstein length, directly from the time evolution of the flame radius. Following the method of Tahtouh et al. [26], Equations (1) and (4) can be rearranged as:

$$\frac{dr}{dt}=V_{s0}-2L_b\frac{dr}{rdt}$$

(7)

The solution to Equation (7) is:

$$r\left(t\right)=2L_{b}W\left(Z\right)$$

(8)

where W is the Lambert function, and Z is given by:

$$Z=\frac{e^{\frac{V_{s0}\cdot t+C_1}{2L_b}}}{2L_b}$$

(9)

C₁ is a constant to be determined.

The values of V_s0 and L_b (along with C₁) are obtained minimizing the following equation:

$${{\displaystyle \sum _1^N\left(r_u\left(t\right)-r\left(t\right)\right)}}^2={\displaystyle \sum _1^N{\left(r_u\left(t\right)-2L_{b}W\left(Z\right)\right)}^2}$$

(10)

where r_u is the experimentally measured radius. The Lambert function W is a multi-valued function [27]: if Z is real, for −1/e ≤ Z < 0 W(Z) can assume two possible real values. The branch satisfying the condition −1 ≤ W(Z) is termed the principal branch and denoted W₀(Z). The branch satisfying the condition W(Z) ≤ −1 is called the alternate branch and denoted W₋₁(Z). For L_b > 0 (that is Z > 0), the principal branch of the Lambert function (W₀) must be used, while for L_b < 0, W₋₁(Z) should be used. This method delivers directly V_s0 and L_b for each test case. The stretched flame speed and the stretch rate can be obtained differentiating Equation (8) with respect to time.

4. Laminar Burning Properties

The following results refer to the combustion behaviour of CH₄ and of CH₄-H₂ mixtures, with a H₂ fraction of 20% and 30% in volume: these values optimize the trade-off between additional technical complexity and performance benefits, allowing for a direct utilization of CH₄-H₂ mixtures in current natural gas engines [28,29]. The initial temperature T₀ varied in the range 293÷305 K, while the initial pressure P₀ was set at 3, 6, 12 and 18 bar (abs.). The study was limited to stoichiometric air-fuel mixtures (equivalence ratio $\varphi$ = 1.0), since it was targeted to the development of automotive internal combustion engines, operating with closed-loop three-way catalyst.

Time-resolved shadowgraph images of spherical expanding flames were used to infer the laminar flame parameters. As stated earlier, laminar analysis can be meaningfully carried out in the constant-pressure phase only, before the chamber pressure shows a sensible increment. Moreover, as stated by Bradley et al. [6], the early stages of the flame kernel growth are affected by the spark energy release, and cannot be taken into account in the evaluation of laminar flame properties.

The resulting measurement time-window starts after the ignition disturbances are over, and ends when the pressure shows an appreciable rise. In the present tests, spark energy was set at 20 mJ (electrical). Since the ignition system is inductive, only a small fraction of this energy is released in the breakdown phase (<10 ns), which (due to the high power peaks) drives the early kernel growth and induces likely perturbations in the flame [30,31]. Given the small spark energy, the extent of the ignition disturbances can be expected to be accordingly limited.

Analysis of the flame radii in the early stages revealed no discernible evidence of ignition-induced effects: nonetheless, pending further investigation, a conservative criterion was applied, based on the morphology of the flame: flames with an aspect ratio (major axis/minor axis of the fitting ellipse) lower than 0.98 were discarded. Typical value for the minimum radius was about 4.5 mm.

As stated above, the maximum radius was obtained applying the constraint of constant pressure: typical value was ~9.5 mm, corresponding to ~27% of the chamber radius. This value meets the (r_u < 0.3r_chamber) criterion suggested by Burke et al. [32] to avoid the effects of confinement in a cylindrical chamber.

Figure 2 shows three sets of frames, each set coming from a single combustion event: they illustrate the flame development in the constant-pressure phase, at P₀ = 3 bar. The combustion behaviour is compared for CH₄ and H₂-CH₄ mixtures with a H₂ content of 20% and 30% (vol.). Recording speed was 7000 fps (143 µs between consecutive frames) with a shutter speed of 1 µs; optical magnification ratio was 3.27:1, resulting in a spatial resolution of 15.3 pixel/mm. Flames appear smooth and virtually free of wrinkles; the crack shown for the 30% mixture is the self-similar development of initial ignition disturbances. The effect of H₂ addition can be appreciated in the increased size of the flame ball at a given time.

Figure 2. Flame propagation in the isobaric phase for stoichiometric CH₄ and H₂-enriched mixtures (20% and 30% vol.) at 3 bar and room temperature. Window dia. 65 mm.

Figure 2. Flame propagation in the isobaric phase for stoichiometric CH₄ and H₂-enriched mixtures (20% and 30% vol.) at 3 bar and room temperature. Window dia. 65 mm.

The output of shadowgraph image processing is shown in Figure 3: the time-evolution of the flame radius is compared for CH4 and CH4-H2 mixtures, at different starting pressure: each set of data corresponds to a single, time-resolved combustion event, which has been selected as representative of the test point (P₀, T₀, % H₂, $\varphi$). The useful range for data analysis is highlighted in colour for each case: grayed-out data were discarded in the extraction of laminar parameters.

As mentioned earlier, the analytical solution delivers V_s0 and L_b directly. The time evolution of V_s can anyway be evaluated differentiating Equation (8). This allows one to trace the behavior of the stretched flame speed against α, as shown in Figure 4 for the same test cases and conditions of Figure 3. Figure 4 clearly shows the effect of the starting pressure and of the Hydrogen amount on the flame behaviour: increasing the pressure has a detrimental effect on the flame speed; on the contrary, the addition of H₂, even in limited amounts, has always a positive effect on the flame speed.

Figure 3. Time evolution of flame radius. Stoichiometric CH₄ and H₂-enriched mixtures (20% and 30% vol.) at room temperature. Starting pressure as indicated.

Figure 3. Time evolution of flame radius. Stoichiometric CH₄ and H₂-enriched mixtures (20% and 30% vol.) at room temperature. Starting pressure as indicated.

The unstretched laminar burning velocity u_l0 was calculated with Equation (5). In a recent paper [33], Varea et al. proposed a new approach for the measurement of u_l0, which bypass the evaluation of the expansion factor. The authors highlighted the drawbacks of the approach generally used in the literature, which requires the evaluation of the ratio ρ_u/ρ_b: this can introduce errors if any of the hypotheses behind Equation (5) are not satisfied. In the case of CH₄, they concluded that the “old” method delivers reliable results. In the present work, the expansion factor was obtained from the properties of the reactant species and of equilibrated adiabatic products, evaluated with the GASEQ code [34].

Figure 4. Evolution of the flame speed versus flame stretch. Same cases as Figure 3.

Figure 4. Evolution of the flame speed versus flame stretch. Same cases as Figure 3.

The u_l0 values for CH₄ are summarized in Figure 5 as a function of initial pressure P₀. As the starting pressure is increased from 3 to 18 bars, a decrease of the laminar burning velocity can be observed; u_l0 at 18 bar is about 40% of 3 bar. The comparison is proposed with the results from other research groups [3,4,7,8,9,10,11]. Data at high pressures are less readily available than atmospheric pressure: the present results for u_l0 show a fairly good agreement with most of published data, save for the values of Halter, which appear slightly lower.

The variation of the burning velocity with initial pressure and temperature can be expressed by the simple empirical correlation [8,10]:

$$u_{l0}=u_{l0,r}{\left(\frac{P_0}{P_{0,r}}\right)}^{\beta }{\left(\frac{T_0}{T_{0,r}}\right)}^{\gamma }$$

(11)

where u_l0,r is the laminar burning velocity at the reference temperature T_0,r and pressure P_0,r. In the present study, only the pressure dependence was evidenced.

The parameter β was optimized in the range 3÷18 bar, using 3 bar as the reference pressure. As shown in Figure 6, the results are well described by Equation (11): the coefficient of determination R² is >0.99. In the case of CH₄ the value of β is −0.45, which is bracketed between the findings of Gu et al. [8] and Halter et al. [10]. CH₄-H₂ mixtures show a minor reduction in the magnitude of β, which is proportional to the amount of H₂.

Figure 5. Unstretched laminar burning velocity as a function of the starting pressure, for stoichiometric CH₄ at room temperature.

Figure 5. Unstretched laminar burning velocity as a function of the starting pressure, for stoichiometric CH₄ at room temperature.

Figure 6. Effect of initial pressure on the unstretched laminar burning velocity. Solid lines represent fits obtained by Equation (11): values of β as indicated.

Figure 6. Effect of initial pressure on the unstretched laminar burning velocity. Solid lines represent fits obtained by Equation (11): values of β as indicated.

The effect of the addition of H₂ to CH₄ is twofold: the laminar burning velocity not only increases but becomes also less sensitive to the initial pressure.

The Markstein length is shown in Figure 7 as a function of the starting pressure. In the case of CH₄ at lower pressure, L_b is positive, corresponding to a flame speed which increases as α decreases. As the pressure is increased, the Markstein length becomes smaller, and eventually negative: this result is in agreement with the findings of Gu et al. [8], who reported on CH₄ flames up to 10 bar. The effect is probably related to the decrease of flame thickness associated to the pressure increment: this reduces the effect of curvature in counteracting the hydrodynamic instabilities.

Figure 7. Burned-gas Markstein length as a function of the starting pressure, for stoichiometric CH₄ and H₂-enriched mixtures at room temperature.

Figure 7. Burned-gas Markstein length as a function of the starting pressure, for stoichiometric CH₄ and H₂-enriched mixtures at room temperature.

The addition of H₂ to CH₄ has the anticipated effect [6] of lowering the Markstein length, since the mixture thermal equidiffusivity is altered.

Due to the combined effects of initial pressure and Hydrogen content, high-pressure CH₄-H₂ flames are characterized by the lowest values of L_b.

The obtained results confirm the selected procedure for data analysis as appropriate. Since −0.2 ≤ L_b ≤ 0.2 mm, and the minimum flame radius used in the image processing is ~4.5 mm, the ratio |L_b/r_{u min}| results lower than ~0.045. According to Halter et al. [24], this legitimate the use of the linear approach, as the relative error respect to the nonlinear approach is negligible.

5. Flame Stability and Cellular Structures

The burned-gas Markstein length L_b expresses how and how much the stretch rate affects the flame propagation. Aerodynamic stretch, induced by flame curvature, interacts with the propensity of the flame to develop instability (hydrodynamic and/or thermo diffusive). In this sense, L_b can be considered an indicator of flame instability. Interestingly enough, the Markstein length is evaluated from the expanding flame in the constant-pressure regime, when the flame is smooth and there is little or no distortion (see e.g., Figure 2). In other words, L_b has a predictive nature.

Evidence of flame instability, if any, appears afterwards, depending on the experimental setup. The flame travelling the chamber volume may go through various morphological alterations, characterized by the appearance and branching of large cracks and, eventually, fine cellular structures [35,36,37,38,39,40,41]. Figure 8 shows three sets of frames at selected times after the spark: they were obtained in the case of pure and H₂-enriched CH₄ mixtures, ignited at P₀ = 6 bar. Recording conditions were as in Figure 2.

Figure 8. Evolution of flame instability for stoichiometric CH₄ and H₂-enriched mixtures (20% and 30% vol.). P₀ = 6 bar, T₀ = 298 K. Window dia. 65 mm.

Figure 8. Evolution of flame instability for stoichiometric CH₄ and H₂-enriched mixtures (20% and 30% vol.). P₀ = 6 bar, T₀ = 298 K. Window dia. 65 mm.

It is worth stressing that these images refer to late stages of flame propagation, when the combustion process is no more isobaric.

At moderate values of the initial pressure (P₀ = 6 bar), the CH₄ flame keeps its laminar shape until it approaches the chamber walls. The addition of H₂ to the mixture, even in limited amount (20%), corresponds to the appearance of large cracks on the flame surface. This effect gets more striking as long as the flame travels the chamber, with the cracks being suddenly integrated by cellular structures [36,40].

As shown in Figure 7, when the initial pressure is increased, the Markstein length decreases, suggesting that flame instability should increase. This is confirmed by Figure 9, which reports three sets of frames obtained for mixtures ignited at P₀ = 12 bar: no matter the mixture composition, in all the cases the flame exhibits deviations from a smooth, laminar morphology. These phenomena take place earlier in the process (that is at smaller radii), if compared to P₀ = 6 bar. The typical sequence of large cracks followed by smaller, uniformly distributed “cells” is always present; if the amount of H₂ is increased, cellular structures show up at earlier times.

Figure 9. Evolution of flame instability for stoichiometric CH₄ and H₂-enriched mixtures (20% and 30% vol.). P₀ = 12 bar, T₀ = 298 K. Window dia. 65 mm.

Figure 9. Evolution of flame instability for stoichiometric CH₄ and H₂-enriched mixtures (20% and 30% vol.). P₀ = 12 bar, T₀ = 298 K. Window dia. 65 mm.

As suggested by Law et al. [39], two characteristic instants can be identified: the first is related to the branching of large cracks across the flame surface, the second to the sudden appearance of cells, almost uniformly over the same surface, with a characteristic size much smaller than the large cracks. Following the convention of Law et al. [39], we chose to adopt the second instant as representative of the flame loosing stability: this allows to define the critical radius r_cr for the onset of instability.

Figure 10 reports the values of r_cr as a function of the starting pressure, for different mixture compositions. All the flames at P₀ = 3 bar show a laminar behaviour, hence the lack of data points for this case.

At P₀ = 6 bar, CH₄ flames show the appearance of cellular structures when the flame approaches the chamber wall: these points are greyed-out, since they are likely influenced by additional phenomena, like flame-wall interaction.

These results show that increasing the pressure, the critical radius gets smaller: the resulting picture is of a flame that departs earlier from the laminar regime, and mostly develops under a wrinkled regime. Adding H₂ to CH₄ further reduces r_cr (i.e., increases instability), even if the effect gets less marked as the pressure is increased: adding 30% of H₂ to CH₄ reduces r_cr by 24% at 12 bar and by 21.5% at 18 bar.

The morphological analysis appears to confirm the trends from the flame stretch analysis. As shown in Figure 11, the critical radius r_cr exhibits a reasonable correlation with the Markstein length L_b.

As far as the observed instabilities are concerned, it is well established that flame cellularity can be hydrodynamic and thermo-diffusional in nature (if body-force effects are absent).

Figure 10. Critical radius for the onset of flame instability as a function of the starting pressure, for stoichiometric CH₄ and H₂-enriched mixtures at room temperature.

Figure 10. Critical radius for the onset of flame instability as a function of the starting pressure, for stoichiometric CH₄ and H₂-enriched mixtures at room temperature.

Figure 11. Critical radius for the onset of flame instability versus Markstein length.

Figure 11. Critical radius for the onset of flame instability versus Markstein length.

Hydrodynamic instability is caused by the density difference across the flame. It is expressed by the thermal expansion ratio of the burned to the unburned mixtures, and is always present for all flames [38].

Thermo-diffusional instability originates from the diffusive disparity of heat conduction from the flame and reactant diffusion towards the flame; it may not be present, if the flame is equidiffusive [21]. The mixture parameter representing the effect of non-equidiffusion is the global Lewis number, Le, defined as the ratio of the heat diffusivity of the mixture to the mass diffusivity of the limiting reactant to the abundant inert [20,21,22]. Values of Le > 1, =1 and <1 correspond to flames which are diffusionally stable, neutral of unstable, respectively.

In the case of a spherical expanding flame, cellular instability is initially suppressed by the strong curvature-induced stretch, associated with the corresponding small flame radii. As the flame expands, the stretch decreases, until a condition is reached in which cell development can no longer be suppressed and cells suddenly appear over the flame surface. This state depends on the combined effects of the hydrodynamic and diffusional–thermal instabilities.

The above-described transition can be expressed in terms of the critical Peclet number, obtained scaling the critical radius r_cr by the laminar flame thickness δ_l. An expression has been proposed of Pe_cr as the sum of two terms, corresponding to hydrodynamic and diffusional–thermal instabilities [40].

Being known r_cr and L_b, the comparison can be drawn between the Markstein number (defined as L_b/δ_l) and Pe_cr: the results are shown in Figure 12.

Actually, in the present conditions, expressing the comparison in terms of non-dimensional parameters does not add to a better understanding of the phenomena.

Figure 12. Critical Peclet number versus Markstein number.

Figure 12. Critical Peclet number versus Markstein number.

As shown in Figure 12, the almost constant value of Pe_cr suggests the flames are equidiffusive in nature [40], yet negative values of Ma stand for Le < 1: such values correspond to flames where the deficient reactant is the more diffusive one, so that thermo-diffusive instability (of cellular nature) is promoted. Then one should expect Pe_cr to increase with Le, but data show a reversed trend.

The described behaviour prevents a straightforward identification of the role of the hydrodynamic and thermo-diffusive terms in the global Pe_cr. Pending a deeper theoretical analysis, a possible explanation of this apparent incongruity could reside in the different conditions in which L_b and r_cr have been evaluated: while the former is a parameter obtained in the isobaric phase of flame development, the latter was measured later, when the global pressure in the chamber was increasing. The unpredictability of the results could be due to the scaling by δ_l, since this parameter is meaningfully defined in the isobaric phase.

6. Conclusions

The effect of the initial pressure and of the H₂ content on the combustion of CH₄-H₂-air mixtures was investigated studying spherical expanding flames. Laminar burning parameters (unstretched burning velocity, burned-gas Markstein length) and flame instability characteristics (critical radius for the onset of cellularity) were evaluated for mixtures with different H2 percentage (0%, 20% and 30% vol.), equivalence ratio $\varphi$ = 1.0, initial temperature T₀ = 293÷305 K and initial pressure P₀ = 3÷18 bar.

The obtained results allowed to assess quantitatively the effect of pressure and of H₂ amount: the laminar burning velocity was found to decrease with the former and increase with the latter. The flame stability decreased when any of the two parameters was increased.

The laminar burning velocity and the Markstein length were found to be in good agreement with published data, wherever available. As a matter of fact, the present results extend the data published so far in literature, as far as high-pressure, stoichiometric H₂-enriched mixtures are concerned.

In agreement with the observed behaviour of the Markstein length, the results on flame cellularity highlight the increasing relevance of the phenomenon at high pressures and/or for significant concentrations of hydrogen.

The main findings can be summarized as follows:

• If the initial pressure is increased, the laminar burning velocity decreases and the Markstein length becomes negative. In the case of CH₄, when P₀ is raised from 3 to 18 bar, u_l0 decreases by ~40%. In the same conditions, L_b shifts from about 0.20 to −0.07.
• If H₂ is added to CH₄, the burning velocity increases. At P₀ = 3 bar, u_l0 increases by ~25% if the amount of H₂ in the mixture is 30%. The Markstein length of H₂-enhanced mixtures is shifted to smaller values, if compared to pure CH₄, and becomes negative at P₀ > 6 bar.
• The behaviour of u_l0 vs. P₀ follows a power law, with exponent β = −0.45 for CH₄. The magnitude of β decreases with the amount of H₂.
• The critical radius for the onset of cellularity was found to decrease when either the pressure or the H₂ content in the mixture were increased. At P₀ = 3 bar, regardless of the amount of H₂, no sign of instability was observed; at P₀ > 6 bar, the flame get increasingly unstable with the amount of H₂. The critical radius shows a good correlation with the Markstein length.

The present findings offer a quantitative frame of reference for the burning properties of CH₄-H₂, highlighting the specific features of the combustion process in condition typical of i.c. engines.