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Radiation Heat Transfer in Particle-Laden Gaseous Flame: Flame Acceleration and Triggering Detonation

In this study we examine influence of the radiation heat transfer on the combustion regimes in the mixture, formed by suspension of fine inert particles in hydrogen gas. The gaseous phase is assumed to be transparent for the thermal radiation, while the radiant heat absorbed by the particles is then lost by conduction to the surrounding gas. The particles and gas ahead of the flame is assumed to be heated by radiation from the original flame. It is shown that the maximum temperature increase due to the radiation preheating becomes larger for a flame with lower velocity. For a flame with small enough velocity temperature of the radiation preheating may exceed the crossover temperature, so that the radiation heat transfer may become a dominant mechanism of the flame propagation. In the case of non-uniform distribution of particles, the temperature gradient formed due to the radiation preheating can initiate either deflagration or detonation ahead of the original flame via the Zel'dovich's gradient mechanism. The initiated combustion regime ignited in the preheat zone ahead of the flame depends on the radiation absorption length and on the steepness of the formed temperature gradient. Scenario of the detonation triggering via the temperature gradient mechanism formed due to the radiation preheating is plausible explanation of the transition to detonation in Supernovae Type Ia explosion.


Introduction
Notoriously, while studying combustion in gaseous mixture, the radiation of hot combustion products is usually not important, as the radiation absorption length in a gaseous mixture is very large, so that the gaseous mixture is almost fully transparent for the radiation and therefore the radiation heat transfer does not influence the flame dynamics. For example, the photon mean free path in the atmosphere at pressure P 1atm  is about tens meters because of very small The situation changes drastically if the gaseous mixture is seeded with fine inert particles, which absorbed and heated by thermal radiation and then transfer the heat by conduction to the surrounding gas. In this case the gas temperature ahead of the flame lags that of the particles and the radiation preheating causes either acceleration of the flame or non-uniform temperature distribution with a proper temperature gradient, formed ahead of the flame, trigger either new deflagration or detonation via the Zel'dovich gradient mechanism. In the present paper we investigate the influence of the radiation preheating for the particle-laden hydrogen-oxygen and hydrogen/air flames. Scenario of the radiation preheating resulting in the triggering detonation can be plausible mechanism explaining deflagration-to-detonation transition in the thermonuclear Type Ia supernovae (SN Ia), which still remains the least understood aspect of the SN Ia explosion phenomenon. In the case of non-uniform distribution of particles, which is typical e.g. for dust deposits layers ("methane-air detonation" in coal dust), the time of the radiative heating is longer. If time of the flame arrival to the boundary of denser particles layer, where the radiation is noticeably absorbed, is long enough, the maximum temperature within the temperature gradient established due to the radiative preheating may exceed the crossover value. In this case either new deflagration or detonation can be ignited via the Zel'dovich's gradient mechanism [1,2]. What kind of combustion regime is ignited in the "distant" particle seeded cloud depends on the radiation absorption length and on the steepness of the formed temperature gradient.
It is known, that uncontrolled development of detonation poses significant threats to chemical storage and processing facilities, mining operations, etc. [3][4][5], while controlled detonation initiation can be a potential application for propulsion and power devices [6,7].
Detonations may or may not develop depending on the ability of a particular mixture composition to sustain detonations, and on the ability of flames to accelerate and produce shocks that are strong enough to ignite detonation. Study of the premixed flames and detonations arising and propagating in the particle-laden gaseous combustible mixture is important for the understanding of unconfined vapor cloud explosions and accidental explosions in many industrial processes associated with the risk of dust explosions, and for better performance of rocket engines using fluid or solid fuels, see e.g. [8,9]..
Majority of the previous studies used a one-step chemical reaction model and were mainly focused on the deflagration-to-detonation transition (DDT) in a gaseous combustible mixtures in attempt to understand nature of the detonation formation. Although significant progress has been made in the understanding of the flame dynamics, the nature of the transition to detonation still remained highly uncertain because a one-step chemical model allows ignition at any temperatures, so that the results of such studies were often remained questionable. In the 1980s-1990s, several groups used a one-step Arrhenius chemical model and asymptotic methods for high activation energy to examine effects of the radiation on the flames propagating in a gas mixture seeded by solid particles [10,11]. The flame propagating in the presence of the uniformly dispersed inert solid particles has been considered with and without account of radiative heat transfer [12][13][14][15][16][17][18]. Coal combustion research has been focused mainly on two aspects of practical interest: the production of volatiles due to thermal decomposition of coal dust and char combustion [19][20][21]. The combustible volatiles can react and release energy, which in turn may contribute to the heat-up of the particles, enhance the combustion energy release due to energy feedback mechanism resulting in an explosion. For the coal-dust suspension air filling the coal-fired burners and for rocket engines using the solid or fluid fuels as well as for coal-fire mining safety both the ignition and combustion evolution are of paramount importance. Effect of radiation transfer on a spray combustion can be of interest for practical cases such as diesel engines, gas turbine combustors etc.
A combustible mixture can be ignited by electrical sparks, or by thermal heating. The ignition capability of an electrical spark varies with fuel concentration, humidity, oxygen content of the atmosphere, temperature, and turbulence, requiring about 0.01-0.03mJ depending on the mixture reactivity. In contrast, radiation-induced ignition typically requires much larger amounts of energy to be released in the mixture. Direct thermal ignition of gaseous combustible mixture by absorption of radiation causing a rapid increase in temperature at least up to 1000K is possible by focusing a high power laser radiation and has been demonstrated both theoretically and experimentally [22,23]. However, ignition at low power levels is unlikely because of a very large length of absorption of the combustible gases at normal conditions.
In the present study effects of thermal radiative preheating is considered for the flames propagating in a two phases composite comprising of gaseous combustible mixture and inert particles. Recent experiments have shown that the dust cloud flame propagation is strongly influenced by the thermal radiation [24][25][26][27]. The effects of the radiation preheating is investigated for the hydrogen-oxygen and hydrogen-air flame. The gaseous phase is assumed to be transparent for the radiation, while solid particles absorb and reemit the radiation.
Different scenarios are considered depending on spatial distribution of the suspended particles and the laminar flame velocity in a pure gaseous mixture. It the case of uniform spatial distribution of the particles the thermal radiation emitted from the hot combustion products is absorbed by the particles ahead of the flame resulting in the radiation preheating, which in turn causes the increase of the flame velocity. It is shown also that the radiative heat flux from the primary particle-laden flame may generate secondary explosion ahead of the flame in the distant particle cloud. This phenomenon is demonstrated for the non-uniform spatial distribution of particles, when the radiation absorbed far ahead of the flame creates a nonuniform temperature distribution in the unburned mixture. If maximum temperature ahead of the flame rises up to the crossover value before the flame arrival to this location, then either new deflagration or detonation can be ignited via the Zeldovich gradient mechanism.
The paper is organized as follows. In Section 2 we present the mathematical model used to study the problem in question. A simple model explaining the principal features of the radiation preheating, and numerical study of the influence of the radiative preheating on combustion wave velocity for a uniform spatial distribution of the particles is presented in Section 3. Triggering of deflagration or detonation ahead of the flame depending on the steepness of temperature gradient created by the radiative preheating in the case of nonuniform particle distribution is considered in Section 4. We conclude in Sec. 5. Appendix A presents validation and thorough convergence and resolution tests of the numerical scheme used in the present studies.

Formulation of the problem; Governing equations
We The standard notations are use: P ,  , u , are pressure, mass density, and flow velocity of the The changes in concentrations of the mixture components due in the chemical reactions are defined by the solution of system of chemical kinetics The right hand parts of Eq. (8) contain the rates of chemical reactions for the reactive species.
We use the standard reduced chemical kinetic scheme for hydrogen/oxygen combustion with the elementary reactions of the Arrhenius type and with pre-exponential constants and activation energies presented in [28]. The equations of state of the fresh mixture and combustion products are taken with the temperature dependence of the specific heats, heat capacities and enthalpies of each species borrowed from the JANAF tables and interpolated by the fifth-order polynomials [29]. The transport coefficients were calculated using the gas kinetics theory [30]. The gaseous mixture viscosity coefficients are where i i n n   is the molar fraction, is the viscosity coefficient of ispecies, (2,2)   -is the collision integral which is calculated using the Lennard-Jones potential 30, î m is the molecule mass of the i-th species of the mixture, i  is the effective molecule size. The thermal conductivity coefficient of the gas mixture is The binary coefficients of diffusion are A detailed description of the transport coefficients used for the gaseous phase and calculation of diffusion coefficients for intermediates has been published previously, see e.g. [31][32][33].
The dynamics of solid particles is considered in continuous hydrodynamic approximation.
The interaction between particles is assumed negligibly small for a small volumetric concentration of particles, so that only the Stokes force between the particle and gaseous phase is taken into account. Then, the equations for the phase of suspended particles are: Where p T -temperature of the particles, -is the thermal radiation heat flux absorbed and reemitted by the particles. The heat transferred from the particle surface to surrounding gaseous mixture is p pg where 2 pg p P,p p0 2r c / 3 Nu     is the characteristic times of the energy transfer from the particle surface to surrounding gaseous mixture, P,p c and p0  are specific heat and the mass density of the particle material, Nu is the Nusselt number (see e.g. [34]).
For a one-dimensional planar problem the equation for the thermal radiation heat transfer in the diffusion approximation is [35,36]: where the radiation absorption coefficient is is the radiation absorption length. The particle-laden mixture is considered optically-thick so that the radiation energy flux emitted from the flame, which is mainly determined by the particle concentration is assumed to be equal to the blackbody radiative heat source, implemented by Belotserkovsky and Davydov [38]. It was further modified [39] so that a high numerical stability of the method is achieved if the hydrodynamic variables are transferred across the grid boundary with the velocity, which is an average value of the velocities in neighboring grids. The improvement of overall modified solver implemented in [39] provides the second order in space that differs from the original first-order method [38]. The modified CPM solver was thoroughly tested and successfully used for modeling knock appearance in SI-engines [39,40] and to study the flame acceleration in tubes with non-slip walls, the transition form slow combustion to detonation [31][32][33] and other problems of transient combustion, e.g. ignition of different combustion modes [41][42][43]. The system of chemical kinetics equations represents a stiff system of differential equations, and it was solved using standard Gear's method [44]. The developed algorithm was implemented using the FORTRAN-90. Convergence and resolution tests presented in Appendixes A were carried out to verify that the observed phenomena are correctly caught remaining unchanged with increasing resolution.

Radiation heat transfer and flame propagation for uniformly dispersed particles
We consider the hydrogen-oxygen flame propagating in the mixture with uniformly dispersed identical solid spherical particles of radius p r 0.75 m   , density of the particle material, The gaseous phase temperature ahead of the flame increases due to the heat transferred from the particles to the surrounding unburned mixture Taken into account that p T T  , and 2 p p L 1/ r N   , we obtain from (18) and (19)   ,p The characteristic time of the radiation preheating of the Lagrangian particle is approximately the time of its arrival to the flame front, f t L/ U  . Taking this into account, the maximum temperature increase of the unburned mixture close ahead of the flame can be estimated as: It can be readily observed that the maximum temperature increase due to the radiation preheating does not depend on the radiation absorption length (see Fig.1). This is due to the fact that although the local radiant heat flux absorbed by the particle is larger for smaller absorption length, but for a larger absorption length particles absorb the radiant heat flux over a longer time until their arrival to the flame front. This difference in time for smaller and larger absorption lengths compensates the lesser local heating for a larger absorption lengths.
Taking into account that the adiabatic flame temperature is less than it is in a pure mixture due to dilution by the inert solid particles, and that the photons emitted from the flame front are produced within the radiative layer near the flame front of the finite thickness, the effective temperature of the radiation emitted from the flame surface can be estimated as b,eff T 2700K  . Then, the maximum temperature increase caused by the radiation preheating can be estimated as T 160K   , which is in a good agreement with the numerical simulation shown in Fig.1 and Fig.2. Fig. 1 shows temporal evolution of the maximum gaseous The velocity and structure of a laminar flame in the classical theory of combustion by Zel'dovich and Frank-Kamenetskii [45] is defined in the approximation of the linear thermal conduction equation. For the one-dimensional problem the temperature distribution (for constant thermal diffusivity) is where g g g g , p / c const      is the coefficient of gaseous thermal diffusivity. It follows from Eq. (23) that heat propagates at the distance g x 4 t   . Taking this into account, it is straightforward to obtain an order-of-magnitude estimates for the speed and width of the laminar flame, which propagates as a result of thermal conduction heat transfer: This estimate is based on the consideration that the characteristic heat diffusion time scale is much longer than the characteristic time of the heat release in the reaction R  , otherwise any small disturbances of the combustion wave will diffuse away resulting in the flame extinguishing.
Another well known conclusion of the classical combustion theory is that in the approximation of linear thermal conduction and if the coefficient of thermal diffusivity is equal to the coefficient of diffusion of reagents, the total enthalpy is constant inside of the flame, which implies similarity of the temperature and the density (concentration of reagent) distributions in the flame front structure. Contrary to the classical theory if the radiative heat transfer becomes dominating process, the heat propagation is defined by the nonlinear heat conduction equation. The total enthalpy of the unburned mixture and combustion products remains equal but the total enthalpy is not constant inside of the flame and there is no similarity of the temperature and the density distributions. In the case of dominating radiation heat transfer the temperature distribution is considerably different compared to that given by Eq. (23). The temperature distribution in the planar thermal wave for the nonlinear radiation thermal conductivity (22) is In this case the reaction can be ignited within the preheat zone, which size by the order of

The radiative preheating of non-uniformly dispersed particles: Ignition of deflagration and detonation ahead of H 2 /O 2 flame
Consider a non-uniform distribution of the particles shown schematically in Fig. 6.
Concentration of the particles immediately ahead of the flame front (gap 2 in Fig.6) is relatively low so that the radiation absorption length here is much larger then the gap width.
Below we assume that the "gap" between the flame and the left boundary of the particles cloud is transparent for the thermal radiation so that the radiant heat flux is absorbed only in the layer 3. If time of the flame arrival to the boundary of the layer 3 is long enough, so that temperature of the particles and surrounding mixture can rise up to the value suitable for ignition before the flame arrival, which is about 1ms for H 2 /O 2 flame (corresponding width of the gap is about 1cm), then the maximum temperature (T g in Fig.6) within the temperature gradient established due to the radiative preheating exceeds the crossover value. What kind of combustion regime is ignited via the Zel'dovich's gradient mechanism in the denser dust cloud depends on the radiation absorption length and, correspondingly, the steepness of the formed temperature gradient. The ignition starts when the temperature exceeds the crossover value, which is for hydrogen/oxygen at 1atm is 1050±50K. Figure 6. Scheme of the radiation preheating of the gaseous mixture inside the gas-particles cloud ahead the flame front. 1-high temperature combustion products; 2 -"gap" with lower concentration of particles; 3 -cloud of particles; T g -temperature of the radiative preheated gas.
The temperature gradient established due to the radiative preheating of the mixture in the gas-particle cloud depends mainly on the radiation absorption length and it is also influenced by the gas expansion during the heating. Since the characteristic acoustic time in the preheat zone is much smaller then time of the radiative heating up to crossover temperature, the pressure is equalized within the region heated by the radiation, and the temperature gradient is formed at the constant pressure 0 P P 1atm   . Classification of the combustion regimes in hydrogen/oxygen and hydrogen/air mixtures initiated by initial temperature gradient via the Zel'dovich's gradient mechanism has been studied in [41,42] using a detailed chemical kinetic models.  According to the classification of combustion regimes initiated by the initial temperature gradient in hydrogen oxygen at 1 atm [41,42], such temperature gradient can ignite a deflagration. The time evolution of the gaseous temperature profiles presented in the middle frame of Fig. 8 shows development of the spontaneous reaction wave on the formed temperature gradient, which then transits into a deflagration wave. The dashed line in the upper frame of Fig. 8 shows the initial number density profile of the particles and the solid line shows the particles number density profile formed at the instant 0 t 900 s   prior to the ignition, when the temperature gradient with maximum temperature T* 1050  К is formed (the first temperature profile in the middle frame). The calculated evolution of the pressure profiles in the bottom frame of Fig. 8 indicates a small variation of pressure during the formation of deflagration. A more gentle temperature gradient can be formed either in the cloud with smaller concentration of the particles, or for the cloud with a properly diffuse interface instead of the stepwise particles number density distribution. In the latter case the radiation absorption length varies along the diffusive cloud interface resulting in the formation of a smooth temperature profile with a shallow temperature gradient capable to initiate either fast deflagration or detonation. Examples of the particle clouds with diffuse interface and the calculated temperature profiles caused by the radiative preheating are shown in Figs.9 and 10. The upper frame in Fig. 9 shows the initial (dashed line) number density of the particles, which drops linearly on the scale 1cm from its maximum value 7 3 p N 2.5 10 cm    . The diffuse boundary of the particles cloud is smeared during the radiation preheating due to the expansion of the gas and at the instant 0 t 1600 s   prior to the ignition, when the temperature gradient with maximum temperature T* 1050  К is formed it is shown by solid line. The middle frame in Fig. 9 shows the calculated temporal evolution of the gaseous temperature profiles. It depicts the development of spontaneous reaction wave on the formed temperature gradient of the scale 0 (T * T ) / dT / dx 8cm     , which then transits into the fast deflagration behind the outrunning shock in the vicinity of the gas-particles layer boundary.
The weak shock waves outrunning the deflagration are seen in the bottom frame of Fig. 9. The initiation of a detonation by the initial temperature gradient requires more shallow gradient. According to [41,42] the minimum scale of the initial linear temperature gradient in hydrogen/oxygen mixture at normal conditions ( 0 P 1atm  ) and T* 1050  К at the top of the gradient needed for the detonation initiation is 0 (T * T ) / T 20cm    . Figure 10 presents results of the simulation for the layer with initial diffuse boundary with the particles number density, which drops linearly on the scale of 10cm from its maximum value 7 3 p N 2.5 10 cm    . The upper frame in Fig. 10 shows the initial particles number density (dashed line) and the formed diffuse boundary of the particles cloud smeared due to gas expansion during the thermal radiation at the instant 0 t 4980 s   prior to the ignition (solid line). Temporal evolution of the gaseous temperature during the detonation formation in the vicinity of the diffusive layer boundary is depicted in the middle frame in Fig.10. The temporal evolution of the temperature profile corresponds to the development of spontaneous reaction wave, its coupling with the shock wave and the formation of the detonation wave.
Temporal evolution of the pressure profiles, corresponding to the formation of shock wave, its coupling with the spontaneous reaction wave and formation of strong shock corresponding to a detonation wave is seen in the bottom frame of Fig. 10.

Discussion and conclusions
The present study demonstrates that the radiative heat transfer in a particle-laden gaseous for cosmology. Observations using Type Ia supernovae as standard candles have revealed that the expansion rate of the universe is accelerating and have led to the discovery of dark energy [47][48][49]. Because of their extreme and predictable luminosity, SN Ia are extensively used as standard candles to measure distances and estimate cosmological parameters critical for our understanding of the global evolution of the Universe. To improve these measurements, we need comprehensive theoretical and numerical models of SN Ia that describe details of the explosion and connect them to observed characteristics of SN Ia, such as spectra and light curves. But the way in which Type Ia supernovae explode is not completely understood. The current leading paradigms for the explosion is scenario of the deflagration-to-detonation transition. There is increasing evidence that a detonation is needed to explain majority features of the SN Ia explosion. Nowadays the best modeling describing majority of the observed SN Ia events is provided by the so-called delayed detonation model [50][51][52], which imply a phase of subsonic thermonuclear burning (deflagration) during which the star expands and a phase of a detonation, which burns remaining fuel on timescales much shorter than the timescale of the explosion. The paradigm of the delayed detonation models is consistent with the theoretical models [53,54] and with theoretical conclusion [55] that a detonation in a strongly Fermi-degenerated matter is unstable against 1D pulsations at densities higher than 7 3 2 10 g / cm  and it becomes stable at lower densities near the star surface. There has been numerous attempts both analytical and numerical [50][51][52][53][54][55][56][57][58][59][60] to explain the detonation formation in SN Ia explosion. However, after many years of studies a fundamental question what is the mechanism of deflagration-to-detonation transition (DDT) in the supernova Type Ia events still remains the least understood aspect of the SN Ia explosion phenomenon (see e.g. [59]).
Type Ia supernovae begins with a white dwarf (WD) near the Chandrasekhar mass that ignites a degenerate thermonuclear runaway close to its center and explodes having initial During the late deflagration phase the radiant flux produced by the radioactive decays of 56 56 56 Ni Co Fe   increases considerably in course of the star incineration by the expanding deflagration wave. Absorption of the radiant energy flux in the outer layers of the star may produce a preconditioned region with the shallow temperature gradient such that detonation can be ignited via the Zel'dovich gradient mechanism.

Appendix A: Code validation, resolution and convergence tests
The thorough convergence and resolution tests were carried out to verify that the observed phenomena of the radiation preheating and impact of the radiation heat transfer in particleladen flame were correctly caught remaining unchanged with increasing resolution.   The problem considered in Section 4 demands more robust parameters of the computational setup. The most demanding is the case where the detonation arises as a result of auto-ignition inside the hot-spot. The consequence of the processes taking place in such a case should be appropriately resolved. As it was shown in [41,42] the detonation caused by the Zel'dovich gradient mechanism arose via the following scenario: 1) the spontaneous combustion wave were formed, 2) as the spontaneous wave decelerated the pressure wave formed behind its front overran it forming the shock wave, 3) detonation established after the transient process involving flame acceleration in the flow behind the outgoing shock wave. Figure A3. Comparison of calculated data (solid line) with analytical solution (dashed line) for discontinuity decay in air for different driver pressures (p 1 ) at normal ambient conditions (p 0 =1atm, T 0 =300K).
Due to this sequence of the events one should appropriately resolve the combustion waves propagating through reacting medium at initially elevated temperature (~1000K, see figures in Section 4) and on the background of elevated temperature and pressure behind the shock front. Besides, the coupling of the reaction wave and shock should be resolved taking into account that the flame thickness is much larger than the width of the shock front. According to this we performed a number of resolution tests for the flames propagating at elevated temperatures and pressures and interacting with the compression and shock waves that can arise in the compressible flow. Therefore to obtain the converged solution for the detonation initiation problem one should use a finer resolution from the very beginning. The hydrogenoxygen flame thickness at normal conditions is about 0.24mm and one should resolve it with not less than 20 cells to avoid unphysical couplings with the shocks that are usually are smoothed over 5-6 cells even by the high-order schemes with limiters. This means that one should resolve flame structure at the higher temperatures and use finer meshes, as it is seen in Fig. A2, which shows resolution tests for different ambient temperatures with grey signs for acceptable range of convergence. We also additionally performed the tests of how shocks are reproduced with chosen numerical method. For example, Fig. A3 shows quantitative reliability of the computational method, demonstrating the comparison of calculated data (solid line) with analytical solution (dashed line) for discontinuity decay in air for different driver pressures (p 1 ) at normal ambient conditions (p 0 =1atm, T 0 =300K). Figure A4. Stoichiometric hydrogen-oxygen burning velocities versus the ambient temperatures. Solid line shows the extrapolation of the experimental data obtained in [57] and signed with squares. Diamonds represent the calculated data with error bars determined by the temperature rise behind the compression wave emerged from the ignition kernel. Figure A4 shows the burning velocity-temperature dependence. One can observe that agreement between calculated data and the extrapolation of the experimental data using the global reaction order 2 74 n .  is satisfactory good. The error bars for calculations are due to the lack of information about the ignition conditions. The ignition conditions in the calculations were such that the compression wave arose and outran from the ignition zone preheating the mixture ahead the forming flame front that could cause the discrepancy with the experimental data [61]. For the model considered in the present paper the resolution was taken with at least 48 computational cells per flame front at normal conditions. It agrees well with the results obtained previously in [41,42]. The performed resolution and convergence tests have shown that the processes in the considered problem setup are well resolved with the chosen method and chosen grids. There is almost no influence of scheme diffusion and all the scales and values are reproduced with high enough accuracy.