The effect of dilution on the diffusive-thermal instability of the rich premixed hydrogen deflagration
Introduction
The emergence of pulsating diffusive-thermal instabilities in rich premixed laminar hydrogen-air flames can occur for freely propagating combustion waves [1], [2], [3], [4], [5], [6], [7], [8], spherically expanding flames [9], [10], burner stabilized flames [11], [12] as well as in micro-flow reactors with controlled temperature of the wall [13].
The case of freely propagating deflagration is the most well studied by means of numerical analysis using detailed reaction models (see e.g. Ref. [8]). For normal ambient conditions the predicted concentration limit for the onset of diffusive-thermal oscillations is in the range of 7.16–8.47 in terms of equivalence ratio, ϕ, and 75–79% in terms of H2 content in a fresh mixture e.g., see Ref. [14] for more detailed overview. Uncertainty in the estimation of critical parameters is caused by the sensitivity of the results to the specific reaction mechanism used in numerical calculations, as demonstrated in Ref. [8]. The predicted characteristic frequency of oscillations ranges from tens to hundreds of Hz. The onset of diffusive-thermal oscillations is also possible for leaner mixtures [8]. However this significantly changes the characteristic critical parameters (pressure and frequency). For example, for mixtures with ϕ∼4 the critical pressure increases above 10 atm, while the frequency of pulsations shifts to the kHz spectrum.
In spherically expanding rich hydrogen flames [9], [10] the diffusive-thermal instabilities manifest themselves in bulk oscillations or the formation of dissipative structures i.e. spirals and target waves. A similar behavior can be found for butane [10] and n-butanol [15] mixtures with air/oxygen/helium. In both cases the nature of this behavior is the high Lewis number leading to pulsating diffusive-thermal instabilities. For hydrogen flames, the Lewis number greater than one is achieved for rich mixtures, while for heavy hydrocarbons - for lean mixtures. The neutral boundary for the onset of pulsating instabilities in the plane of parameters, pressure vs. equivalence ratio, is located close to the critical parameters for the emergence of diffusive-thermal instabilities of deflagration waves [9]. The loss of stability occurs at elevated pressures and characteristic frequencies of oscillations are of the order of kHz for ϕ∼4 and P∼20 atm.
It is demonstrated in Refs. [1], [9] that in the case of freely propagating flames (both planar and spherical) the onset of the flame pulsations can be qualitatively described by the Sivashinsky criterion [16] relating the Lewis number of the deficient component of the mixture (oxygen) and the overall Zel'dovich number as (Le−1)Ze = κc, where κc is estimated in Ref. [16] by using the activation energy asymptotics as . In order for instability to occur, the Lewis number should be greater than one and the Zel'dovich number should be sufficiently large. Heat exchange with the burner can significantly promote the onset of instabilities in the case of burner stabilized combustion fronts [12]. It is possible to encounter flame oscillations for mixtures with a Lewis number less than one, and, moreover, a stationary planar flame can become completely unstable for all values of the Lewis number [17]. This fact may simplify the possible direct experimental observation of the diffusive-thermal oscillations for burner stabilized flames. However, pulsations of the combustion front may also occur [11], [12] once the flame approaches the surface of a burner close enough. In this case the onset of instabilities occurs for a wider range of equivalence relations and pressures with frequencies of oscillations of the order of several kHz and higher. This may complicate the experimental investigation of the critical parameters of the diffusive-thermal instability.
In the configuration with the micro-flow reactor with controlled temperature of the wall it is also possible to observe the diffusive-thermal pulsations. In particular in Ref. [13] it is demonstrated that the normal flame solution branch becomes unstable as the flow rate is decreased due to the diffusive-thermal (DT) oscillations. This causes the emergence of periodic flame oscillations, chaotic regime of pulsations, mixed mode oscillations etc. Periodic and mixed mode regimes are observed experimentally for methane [18] and ethylene [19]. As for the hydrogen air mixtures it is shown in Ref. [13] that instabilities occur if the equivalence ratio and pressure are sufficiently increased.
It is well known that besides the diffusive-thermal instabilities other types of instabilities may occur in combustion such as hydrodynamic instability, which occur due to the jump in density across the reaction front. In the presence of mass forces, the buoyancy driven instabilities may also emerge. Interaction of instabilities of different types often occurs in combustion systems. This leads to a complex behavior of the combustion wave, which is an interesting phenomenon and it has to be a subject of separate studies.
In all the configurations discussed above the characteristics of the diffusive-thermal instabilities are closely related to the corresponding behavior in freely propagating flames. As a result, these configurations share the same difficulties associated with direct experimental investigation of the diffusive-thermal instabilities of rich hydrogen flames. Furthermore, the emissivity of hydrogen flame front weakens with increasing the equivalence ratio. Since the OH radical content decreases, widely employed techniques for flame diagnostics, chemiluminescence and laser induced fluorescence, become less effective. The flame sensitivity to the perturbations imposed by experimental setup grows, which may lead to preliminary quenching. Therefore, in order to study this phenomena experimentally it is desirable to find appropriate conditions and reduce the equivalence ratio of the mixture when the onset of the oscillatory regime of the flame propagation occurs. A straightforward decrease of the equivalence ratio leads, however, to the increase of the critical pressure and of the frequency of pulsations and thus complicates the required experimental setup. The conflicting requirements for the parameters of the experiment in terms of equivalence ratio on one hand and pressure/time resolution of imaging equipment on the other hand motivates the search for appropriate conditions, which would allow to relax these contradictions.
In this work, we numerically investigate the effect of the mixture dilution with different inert species on the characteristics of diffusive-thermal instabilities of freely propagating deflagration waves in a model with a detailed reaction mechanism and mixture averaged molecular diffusion. Although many efforts have been made to investigate the influence of various factors such as equivalence ratio, pressure and the rate of dilution on velocity, Markstein length, emergence of the cellular instabilities of the hydrogen flames [20], [21], [22], [23], [24], [25], [26], as far as we know, there is no systematic study of the effect of dilution of the hydrogen-oxygen mixture on the characteristics of pulsating diffusive-thermal instabilities. This is subject of the current work.
Section snippets
Mathematical model and numerical integration
The mathematical model and numerical scheme for integration of the detailed system of the governing equations used in this work has been reported in e.g. Refs. [27], [28]. It is based on the numerical simulation of the one-dimensional system of conservation equations using the detailed chemistry mechanisms and detailed transport models.
Previously, a number of detailed mechanisms were considered to study the influence of chemical kinetics on the onset of instabilities see e.g., Refs. [8], [12].
Methodology
In this work we choose additives which do not influence the chemistry directly via additional reaction pathways, but which only influence the kinetics indirectly via dilution effects and changes of the flame dynamics properties such that they rather shift the stability limits towards the desired values of pressure and equivalence ratios.
It is expected that the diluent can modify the density, specific heat and transport properties (temperature and species diffusion) of the fresh mixture, as well
Results and discussion
In this section, we undertake an analysis of the effect of the dilution of a fresh mixture with inert species on the properties of combustion waves, such as burning temperature, flame speed and stability.
In what follows, the initial temperature of the mixture is assumed to be T0 = 298K, and the pressure is either normal or elevated. The equivalence ratio for most calculations is fixed at ϕ = 4, unless otherwise specified. These moderately rich conditions are normally achievable in experiments
Conclusions
In this manuscript the onset of diffusive-thermal instabilities of freely propagating combustion waves in rich hydrogen-oxygen mixtures diluted with inert gases was investigated. An emphasis was put on the effect of the degree of dilution on the characteristics of pulsating instabilities, such as critical pressure and the period of oscillations.
It is found that the dilution reduces the critical pressure for the onset of flame oscillations, while the period of oscillations can be increased by an
Acknowledgments
The authors acknowledge the support from DFG-RFBR grant number 17-53-12018 (RFBR)/ BY 94/2-1 (DFG). VVG acknowledges the support of TSNIIMASH through the ‘s-Flame’ project, RFBR grant numbers 17-01-00070.
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