Flame Balls Research Articles

Nonlinear time-fractional differential equations in combustion science

Abstract The application of Fractional Calculus in combustion science to model the evolution in time of the radius of an isolated premixed flame ball is highlighted. Literature equations for premixed flame ball radius are re-derived by a new method that strongly simplifies previous ones. These equations are nonlinear time-fractional differential equations of order 1/2 with a Gaussian underlying diffusion process. Extending the analysis to self-similar anomalous diffusion processes with similarity parameter ν/2 > 0, the evolution equations emerge to be nonlinear time-fractional differential equations of order 1−ν/2 with a non-Gaussian underlying diffusion process.

E215 微小重力場におけるふく射の影響を受ける極低伸長率対向流予混合火炎及びFlame ballとの相関(OS-9:燃焼の最近の進展(5))

E215 微小重力場におけるふく射の影響を受ける極低伸長率対向流予混合火炎及びFlame ballとの相関(OS-9:燃焼の最近の進展(5))

Existence of radial stationary solutions for a system in combustion theory

In this paper, we construct radially symmetric solutions of a nonlinear non-cooperative elliptic system derived from a model for flame balls with radiation losses. This model is based on a one step kinetic reaction and our system is obtained by approximating the standard Arrehnius law by an ignition nonlinearity, and by simplifying the term that models radiation. We prove the existence of 2 solutions using degree theory.

The structure of lean hydrogen-air flame balls

An analysis of the structure of flame balls encountered under microgravity conditions, which are stable due to radiant energy losses from H2O, is carried out for fuel-lean hydrogen-air mixtures. It is seen that, because of radiation losses, in stable flame balls the maximum flame temperature remains close to the crossover temperature, at which the rate of the branching step H+O2→OH+O equals that of the recombination step H+O2+M→HO2+M. Under those conditions, all chemical intermediates have very small concentrations and follow the steady-state approximation, while the main species react according to the overall step 2H2+O2→2H2O; so that a one-step chemical-kinetic description, recently derived by asymptotic analysis for near-limit fuel-lean deflagrations, can be used with excellent accuracy to describe the whole branch of stable flame balls. Besides molecular diffusion in a binary-diffusion approximation, Soret diffusion is included, since this exerts a nonnegligible effect to extend the flammability range. When the large value of the activation energy of the overall reaction is taken into account, the leading-order analysis in the reaction-sheet approximation is seen to determine the flame ball radius as that required for radiant heat losses to remove enough of the heat released by chemical reaction at the flame to keep the flame temperature at a value close to crossover. The results are relevant to burning velocities at lean equivalent ratios and may influence fire-safety issues associated with hydrogen utilization.

Experimental and computational study of the transition to the flame ball regime at normal gravity

Ultra lean flames propagating in a tube of 12.2mm internal diameter in mixtures of (40mol% H2+60mol% CH4) fuel gas with air have been studied experimentally, using OH Planar Lased Induced Fluorescence (PLIF), and by numerical simulations, for equivalence ratios between 0.30 and 0.50. Hypothetical OH LIF signals were reconstructed from the calculated data and compared with measurements. The simulated OH LIF intensity distributions are found to be similar to the measured ones in the experimental flames for equivalence ratios 0.31⩽φ⩽0.44. Within this range, when the equivalence ratio was decreased down to the lean limit value, the shapes of the simulated and experimental flame fronts changed from relatively large egg-shaped to small, about 3mm diameter, nearly spherical flame balls. Simulations showed the presence of a recirculating flow in near-limit flames. The front of the lean limit flame balls resides entirely within the volume of this recirculating flow. The flame front was nearly uniform in lean limit flame balls and became increasingly non-uniform with the departure from the limit, with an eventually vanishing bottom front. The experimental and simulated flammability limits lie far below the theoretical limit for a planar flame, and simulated flame temperatures exceed adiabatic temperatures, indicating strong preferential diffusion (Lewis number) effects in these flames. These effects are qualitatively analyzed based on the behavior of measured and simulated flame parameters. It is suggested that preferential diffusion in non-uniform flames, well-removed from the lean flammability limit, is primarily controlled by flame stretch, while in near-uniform limit flame balls it is dominated by a curvature effect similar to that in microgravity flame balls.

On the critical flame radius and minimum ignition energy for spherical flame initiation

Spherical flame initiation from an ignition kernel is studied theoretically and numerically using different fuel/oxygen/helium/argon mixtures (fuel: hydrogen, methane, and propane). The emphasis is placed on investigating the critical flame radius controlling spherical flame initiation and its correlation with the minimum ignition energy. It is found that the critical flame radius is different from the flame thickness and the flame ball radius and that their relationship depends strongly on the Lewis number. Three different flame regimes in terms of the Lewis number are observed and a new criterion for the critical flame radius is introduced. For mixtures with Lewis number larger than a critical Lewis number above unity, the critical flame radius is smaller than the flame ball radius but larger than the flame thickness. As a result, the minimum ignition energy can be substantially over-predicted (under-predicted) based on the flame ball radius (the flame thickness). The results also show that the minimum ignition energy for successful spherical flame initiation is proportional to the cube of the critical flame radius. Furthermore, preferential diffusion of heat and mass (i.e. the Lewis number effect) is found to play an important role in both spherical flame initiation and flame kernel evolution after ignition. It is shown that the critical flame radius and the minimum ignition energy increase significantly with the Lewis number. Therefore, for transportation fuels with large Lewis numbers, blending of small molecule fuels or thermal and catalytic cracking will significantly reduce the minimum ignition energy.

Experimental study of lean flammability limits of methane/hydrogen/air mixtures in tubes of different diameters

Lean limit flames in methane/hydrogen/air mixtures propagating in tubes of internal diameters (ID) of 6.0, 8.9, 12.3, 18.4, 25.2, 35.0, and 50.2 mm have been experimentally studied. The flames propagated upward from the open bottom end of the tube to the closed upper end. The content of hydrogen in the fuel gas has been varied in the range 0–40 mol%. Lean flammability limits have been determined; flame shapes recorded and the visible speed of flame propagation measured. Most of the observed limit flames in tubes with diameters in the range of 8.9–18.4 mm had enclosed shape, and could be characterized as distorted or spherical flame balls. The tendency was observed for mixtures with higher hydrogen content to form smaller size, more uniform flame balls in a wider range of tube diameters. At hydrogen content of 20% or more in the fuel gas, limit flames in largest diameters (35.0 mm and 50.2 mm ID) tubes had small, compared to the tube diameter, size and were “lens”-shaped. “Regular” open-front lean limit flames were observed only for the smallest diameters (6.0 mm and 8.9 mm) and largest diameters (35.0 and 50.2 mm ID), and only for methane/air and (90% CH 4 + 10% H 2)/air mixtures, except for 6 mm ID tube in which all limit flames had open front. In all experiments, except for the lean limit flames in methane/air and (90% CH 4 + 10% H 2)/air mixtures in the 8.9 mm ID tube, and all limit flames in 6.0 mm ID tube, visible flame speeds very weakly depended on the hydrogen content in the fuel gas and were close to- or below the theoretical estimate of the speed of a rising hot bubble. This observation suggests that the buoyancy is the major factor which determines the visible flame speed for studied limit flames, except that last mentioned. A decrease of the lean flammability limit value with decreasing the tube diameter was observed for methane/air and (90% CH 4 + 10% H 2)/air mixtures for tubes having internal diameters in the range of 18.4–50.2 mm. This effect has been attributed to the stronger combined effect of the preferential diffusion and flame stretch in narrower tubes for flames which resemble rising bubble.

Generalized flame balls

We consider generalized flame balls which correspond to stationary spherical flames with a flow of hot inert gas, either a source or a sink, at the origin. Depending on the flow, these flames can have positive, zero, or negative burning speeds, with zero speeds characterizing the Zeldovich flame balls. A full analytical description of these structures and their stability to radial perturbations is provided, using a large activation energy asymptotic approach and a thermo-diffusive approximation. The results are also complemented by a numerical study. The number and stability of the generalized flame balls are identified in various regions of the l-M-h 0 space, where l is the (reduced) Lewis number, and M and h 0 the flow rate and its enthalpy at the origin, respectively. It is typically found that, when the flow is a source, there is a maximum value of the flow rate M ax depending on l and h 0, above which no stationary solutions exist, and below which there are two solutions characterizing a small stable flame ball and a large unstable flame ball; the implications of these results to the problem of ignition by a hot inert gas stream are discussed. When the flow is a sink, however, there is typically a single unstable solution, except for sufficiently large values of the Lewis number and large negative values of M, where three flame balls exist, the medium one being stable. Finally, the relation between the flame speed, positive or negative, and the flame curvature, small or large, is discussed.

A new type of steady and stable, laminar, premixed flame in ultra-lean, hydrogen–air combustion

Ultra-lean, hydrogen–air mixtures are found to support another kind of laminar flame that is steady and stable beside flat flames and flame balls. Direct numerical simulations are performed of flames that develop into steadily and stably propagating cells. These cells were the original meaning of the word “flamelet” when they were observed in lean flammability studies conducted early in the development of combustion science. Several aspects of these two-dimensional flame cells are identified and are contrasted with the properties of one-dimensional flame balls and flat flames. Although lean hydrogen–air flames are subject to thermo-diffusive effects, in this case the result is to stabilize the flame rather than to render it unstable. The flame cells may be useful as basic components of engineering models for premixed combustion when the other types of idealized flames are inapplicable.

A hypothetical burning-velocity formula for very lean hydrogen–air mixtures

Very lean hydrogen–air mixtures experience strong diffusive-thermal types of cellular instabilities that tend to increase the laminar burning velocity above the value that applies to steady, planar laminar flames that are homogeneous in transverse directions. Flame balls constitute an extreme limit of evolution of cellular flames. To account qualitatively for the ultimate effect of diffusive-thermal instability, a model is proposed in which the flame is a steadily propagating, planar, hexagonal, close-packed array of flame balls, each burning as if it were an isolated, stationary, ideal flame ball in an infinite, quiescent atmosphere. An expression for the laminar burning velocity is derived from this model, which theoretically may provide an upper limit for the experimental burning velocity.

Interactions of flame balls

Flame ball interactions are numerically investigated in a reaction–diffusion system characterized by single-step Arrhenius kinetics and radiative heat losses. It is found that the interactions of two neighbouring flame balls are characterized by two distinct regimes – a repulsion regime and an attraction regime, depending upon the separation distance. The two regimes join at a critical separation distance, which corresponds to an unstable equilibrium state. For supercritical separation distances, the two flame balls repel and drift apart from each other; whereas for sub-critical separation distances, they move towards each other and eventually merge into a single stationary flame ball. In this connection, flame ball interactions are found to exhibit a qualitatively reverse character in comparison with the well-known van der Waals curve which characterizes intermolecular forces.

Theoretical analysis of the evolution from ignition kernel to flame ball and planar flame

Dynamics of flame kernel evolution with and without external energy addition has been investigated analytically and numerically. Emphasis is placed on the effects of radiation heat loss, ignition power and Lewis number on the correlation and transition between the initial flame kernel, the self-extinguishing flame, the flame ball, the outwardly propagating spherical flame and the propagating planar flame. The present study extends previous results by bridging the theories of the non-adiabatic stationary flame balls and travelling flames and allowing rigorous consideration of radiation heat losses. The results show that the effects of radiation heat loss play an important role in flame regimes and flame transition and result in a new isolated self-extinguishing flame. Furthermore, it is found that radiation heat losses significantly increase the critical ignition radius and result in three different dependences of the minimum ignition power on the Lewis number. Comparisons between the results from the transient numerical simulation and those from the quasi-steady state analysis show a good agreement. The results suggest that prediction of flame initiation without appropriate consideration of radiation is not acceptable.

On a Model of Flame Ball with Radiative Transfer

In this paper, we derive an equation for the growth of a flame ball for a free boundary combustion model with radiative transfer. The equation for the radiative field is given by the linearized Eddington equation. We then study the mathematical properties of this equation of growth and carry out numerical computations in order to discuss the stability or instability of steady flame balls.

Instability in a flame ball problem

In this paper we give a rigorous mathematical proof of the instability of stationary radial flame ball solutions of a three dimensional free boundary model which models the combustion of a gaseous mixture with dust in a microgravity environment.

Audio–video fire-detection of open fires

In the scope of the development of a video-based fire detection system, investigations on the dynamic behaviour of open flames have been made [Schultze. Untersuchungen zur Audio-Video-Detektion von offenen Bränden, Diploma thesis, Duisburg, 2003]. Standardised fires were recorded with different audio and video systems and their flow and flickering analysed. The analyses show that the flickering of flames is not only viewable in the video data, but also detectable in the acoustic recordings. Using a block-based motion estimator it is possible to trace the movements of flames. Due to the high velocity of flame movements this motion estimation is however only successful in high speed recordings. Therefore a simplified flow analysis has been developed in order to trace the characteristic pumping movement of flame balls in turbulent open flames. The simplified flow analysis is applicable on usual video recordings and should be considered for implementation in an automatic flame detection system.

Flame Balls for a Free Boundary Combustion Model with Radiative Transfer

We study radial flame ball solutions of a three‐dimensional free boundary problem (FBP), which models combustion of a gaseous mixture with dust in a microgravity environment. The model combines diffusion of mass and temperature with reaction at the flame front, the reaction rate being temperature dependent. The radiative flux due to the presence of dust enters the equation for the temperature in the form of a divergence term. This flux is modeled by Eddington’s radiative transfer equation. The main parameters are the dimensionless opacity and the ratio of radiative and thermal fluxes. We prove existence of spherical flame ball solutions for the FBP. Bifurcation diagrams are obtained, exhibiting the multiplicity of solutions. Singular limit cases of the parameter values are also discussed.

An adaptive multiresolution method for combustion problems: application to flame ball–vortex interaction

In this article, we present numerical simulations of flame ball–vortex interactions using adaptive multiresolution methods. The numerical scheme for the convection–diffusion–reaction equations modelling this problem is based on a finite volume discretization coupled with a discrete multiresolution analysis. The grid in physical space is adapted dynamically to track the evolution of the solution in scale and space. Time integration is done by an explicit Runge–Kutta scheme. In space we use second-order centered schemes. The implementation is based on a graded tree data structure, which improves both CPU and memory performances, as no fine gridding is required in regions where the solution is smooth. To illustrate the features and the efficiency of the method, we compute several flame ball–vortex interactions and study the role played by the fluid flow on the evolution of the flame ball. We observe the roll-up of the flame ball around the vortex into a snail-like structure. We also put into evidence the flammability limit of the flame ball in function of both vortex and radiation intensities.

Investigation of the early stages in laser-induced ignition by Schlieren photography and laser-induced fluorescence spectroscopy

Laser ignition has been discussed widely as a potentially superior ignition source for technical appliances such as internal combustion engines. Ignition strongly affects overall combustion, and its early stages in particular have strong implications on subsequent pollutant formation, flame quenching, and extinction. Our research here is devoted to the experimental investigation of the early stages of laser-induced ignition of CH4/air mixtures up to high pressures. Tests were performed in a 0.9-l combustion cell with initial pressures of up to 25 bar with stoichiometric to fuel-lean mixtures using a 5-ns 50-mJ 1064-nm Nd:YAG laser. Laserinduced fluorescence (LIF) was used to obtain two dimensionally resolved images of the OH radical distribution after the ignition event. These images were used to produce an animation of laser ignition and early flame kernel development. Schlieren photography was used to investigate the laserinduced shock wave, hot core gas, and developing flame ball. We extend existing knowledge to high-pressure regimes relevant for internal combustion engines.

Lifetimes of flame balls dragged by model turbulent flows: Role of velocity gradient fluctuations.

An isolated combustion spot-known as a flame ball (FB)-is considered while it is advected by a turbulent flow of a lean premixture of such a light fuel as hydrogen. A Batchelor approximation for the surrounding Lagrangian flow is made. This in principle gives one an access to the FB lifetime t(life) and to its response to the ambiant Lagrangian rate-of-strain tensor g(t), by means of a nonlinear and forced integro-differential equation for the current FB radius. For a diagonal g(t) deduced from random Markov processes of the Ornstein-Uhlenbeck type, or linearly filtered versions thereof, extensive numerical simulations and approximate theoretical analyses agree that (i) flame balls can definitely live for much longer than their time of spontaneous expansion/collapse; (ii) large enough values of t(life) are compatible with Poisson statistics; (iii) the variations of <t(life)> with the characteristics of g(t) mirror the latter's statistics, more precisely that of trace(g(2)). Open problems, dealing with a nondiagonal g(t), ignition-related transients and/or collective effects, finally are evoked.

On self-drifting flame balls

A three-dimensional (3D) reaction-diffusion model for premixed flames with radiative heat losses is studied numerically. In accordance with previous analytical predictions and two-dimensional (2D) numerical simulations, it is shown that cellular flames occurring in low Lewis number premixtures can propagate at heat-loss rates greater than the maximum that extinguishes the planar flame. At sufficiently high heat losses, the flame interface breaks up into separate self-drifting flame balls while a significant portion of the fuel remains unconsumed.