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

Transcription

1 Radiation Heat Transfer in Particle-Laden Gaseous Flame: Flame Acceleration and Triggering Detonation M. A. Liberman 1 *) M. F Ivanov 2, A. D. Kiverin, 2 1 Nordita, KTH Royal Institute of Technology and Stockholm University Stockholm, Sweden 2 Joint Institute for High Temeratures of RAS, Moscow, Russia 27 March 2015 Abstract In this study we examine influence of the radiation heat transfer on the combustion regimes in the mixture, formed by susension of fine inert articles in hydrogen gas. The gaseous hase is assumed to be transarent for the thermal radiation, while the radiant heat absorbed by the articles is then lost by conduction to the surrounding gas. The articles and gas ahead of the flame is assumed to be heated by radiation from the original flame. It is shown that the maximum temerature increase due to the radiation reheating becomes larger for a flame with lower velocity. For a flame with small enough velocity temerature of the radiation reheating may exceed the crossover temerature, so that the radiation heat transfer may become a dominant mechanism of the flame roagation. In the case of non-uniform distribution of articles, the temerature gradient formed due to the radiation reheating 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 reheat zone ahead of the flame deends on the radiation absortion length and on the steeness of the formed temerature gradient. Scenario of the detonation triggering via the temerature gradient mechanism formed due to the radiation reheating is lausible exlanation of the transition to detonation in Suernovae Tye Ia exlosion. Keywords: Flame; Radiation; Detonation; Particles; Ignition; Gradient * ) Corresonding author at Nordic Institute for Theoretical Physics (NORDITA): Tel: address: mliber@nordita.org (M. A. Liberman) 1

2 1. Introduction Notoriously, while studying combustion in gaseous mixture, the radiation of hot combustion roducts is usually not imortant, as the radiation absortion length in a gaseous mixture is very large, so that the gaseous mixture is almost fully transarent for the radiation and therefore the radiation heat transfer does not influence the flame dynamics. For examle, the hoton mean free ath in the atmoshere at ressure P 1atm is about tens meters because of very small (10 10 )cm values of the Thomson scattering and "bremsstrahlung" cross section rocesses. Therefore the contribution of radiation to the heat transfer is negligibly small. If the flame roagates in a tube out from the closed to the oen end, the radiation heat losses of the hot combustion roducts cause a relatively modest cooling of the burned roducts resulting in a modest decrease of ressure behind the flame front, which is negligible comared to the thermal conduction heat losses to the tube walls. In the traditional theoretical combustion the heat is transferred by the molecular gaseous thermal conduction and/or convection while the radiation heat transfer is negligible because the energy transferred by the radiant heat flux contributes far too small to the mechanism of combustion wave roagation and does not influence the flame velocity. The situation changes drastically if the gaseous mixture is seeded with fine inert articles, which absorbed and heated by thermal radiation and then transfer the heat by conduction to the surrounding gas. In this case the gas temerature ahead of the flame lags that of the articles and the radiation reheating causes either acceleration of the flame or non-uniform temerature distribution with a roer temerature gradient, formed ahead of the flame, trigger either new deflagration or detonation via the Zel'dovich gradient mechanism. In the resent aer we investigate the influence of the radiation reheating for the article-laden hydrogen-oxygen and hydrogen/air flames. Scenario of the radiation reheating resulting in the triggering detonation can be lausible mechanism exlaining deflagration-to-detonation 2

3 transition in the thermonuclear Tye Ia suernovae (SN Ia), which still remains the least understood asect of the SN Ia exlosion henomenon. In the case of non-uniform distribution of articles, which is tyical e.g. for dust deosits 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 articles layer, where the radiation is noticeably absorbed, is long enough, the maximum temerature within the temerature gradient established due to the radiative reheating 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" article seeded cloud deends on the radiation absortion length and on the steeness of the formed temerature gradient. It is known, that uncontrolled develoment of detonation oses significant threats to chemical storage and rocessing facilities, mining oerations, etc. [3-5], while controlled detonation initiation can be a otential alication for roulsion and ower devices [6, 7]. Detonations may or may not develo deending on the ability of a articular mixture comosition to sustain detonations, and on the ability of flames to accelerate and roduce shocks that are strong enough to ignite detonation. Study of the remixed flames and detonations arising and roagating in the article-laden gaseous combustible mixture is imortant for the understanding of unconfined vaor cloud exlosions and accidental exlosions in many industrial rocesses associated with the risk of dust exlosions, and for better erformance of rocket engines using fluid or solid fuels, see e.g. [8, 9].. Majority of the revious studies used a one-ste chemical reaction model and were mainly focused on the deflagration-to-detonation transition (DDT) in a gaseous combustible mixtures in attemt to understand nature of the detonation formation. Although significant rogress has been made in the understanding of the flame dynamics, the nature of the transition to detonation still remained highly uncertain because a one-ste chemical model allows ignition 3

4 at any temeratures, so that the results of such studies were often remained questionable. In the 1980s-1990s, several grous used a one-ste Arrhenius chemical model and asymtotic methods for high activation energy to examine effects of the radiation on the flames roagating in a gas mixture seeded by solid articles [10, 11]. The flame roagating in the resence of the uniformly disersed inert solid articles has been considered with and without account of radiative heat transfer [12-18]. Coal combustion research has been focused mainly on two asects of ractical interest: the roduction of volatiles due to thermal decomosition of coal dust and char combustion [19-21]. The combustible volatiles can react and release energy, which in turn may contribute to the heat-u of the articles, enhance the combustion energy release due to energy feedback mechanism resulting in an exlosion. For the coal-dust susension 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 aramount imortance. Effect of radiation transfer on a sray combustion can be of interest for ractical cases such as diesel engines, gas turbine combustors etc. A combustible mixture can be ignited by electrical sarks, or by thermal heating. The ignition caability of an electrical sark varies with fuel concentration, humidity, oxygen content of the atmoshere, temerature, and turbulence, requiring about mJ deending on the mixture reactivity. In contrast, radiation-induced ignition tyically requires much larger amounts of energy to be released in the mixture. Direct thermal ignition of gaseous combustible mixture by absortion of radiation causing a raid increase in temerature at least u to 1000K is ossible by focusing a high ower laser radiation and has been demonstrated both theoretically and exerimentally [22, 23]. However, ignition at low ower levels is unlikely because of a very large length of absortion of the combustible gases at normal conditions. 4

5 In the resent study effects of thermal radiative reheating is considered for the flames roagating in a two hases comosite comrising of gaseous combustible mixture and inert articles. Recent exeriments have shown that the dust cloud flame roagation is strongly influenced by the thermal radiation [24-27]. The effects of the radiation reheating is investigated for the hydrogen-oxygen and hydrogen-air flame. The gaseous hase is assumed to be transarent for the radiation, while solid articles absorb and reemit the radiation. Different scenarios are considered deending on satial distribution of the susended articles and the laminar flame velocity in a ure gaseous mixture. It the case of uniform satial distribution of the articles the thermal radiation emitted from the hot combustion roducts is absorbed by the articles ahead of the flame resulting in the radiation reheating, which in turn causes the increase of the flame velocity. It is shown also that the radiative heat flux from the rimary article-laden flame may generate secondary exlosion ahead of the flame in the distant article cloud. This henomenon is demonstrated for the non-uniform satial distribution of articles, when the radiation absorbed far ahead of the flame creates a nonuniform temerature distribution in the unburned mixture. If maximum temerature ahead of the flame rises u 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 aer is organized as follows. In Section 2 we resent the mathematical model used to study the roblem in question. A simle model exlaining the rincial features of the radiation reheating, and numerical study of the influence of the radiative reheating on combustion wave velocity for a uniform satial distribution of the articles is resented in Section 3. Triggering of deflagration or detonation ahead of the flame deending on the steeness of temerature gradient created by the radiative reheating in the case of nonuniform article distribution is considered in Section 4. We conclude in Sec. 5. Aendix A 5

6 resents validation and thorough convergence and resolution tests of the numerical scheme used in the resent studies. 2. Formulation of the roblem; Governing equations We consider a lanar flame roagating from the closed to the oen end of a duct. The governing equations for the gaseous hase are the one-dimensional, time-deendent, multisecies reactive Navier-Stokes equations including the effects of comressibility, molecular diffusion, thermal conduction, viscosity and detailed chemical kinetics for the reactive secies H 2, O 2, H, O, OH, H 2 O, H 2 O 2, and HO 2 with subsequent chain branching, roduction of radicals, energy release and heat transfer between the articles and the gas hase. The system of equations for the gaseous hase is u t x 0, (1) Yi Yi 1 Yi Yi u D i t x x x t xx u St ch, (2) u u P (u u ), (3) t x x x E u E (Pu) xxu T T t x x x x x Y Y (u u ) k x x k t ch St k k hk DkT hk u cp,q, (4) PR Tn R Y TT R Y, (5) B B i i i i mi i h ct ct hy, (6) k k v v k k k k 4 u 3 x, (7) xx 6

7 The standard notations are use: P,, u, are ressure, mass density, and flow velocity of the gaseous mixture, Y i i / - the mass fractions of the secies, 2 E u /2 - the total energy density, - the internal energy density, R B - is the universal gas constant, m i - the molar mass of i-secies, Ri R B /mi, n - the gaseous molar density, ij - the viscous stress tensor, cv cviyi - is the constant volume secific heat, c vi - the constant volume secific i heat of i-secies, h i - the enthaly of formation of i-secies, (T) and (T) are the coefficients of thermal conductivity and viscosity, D i(t) - is the diffusion coefficients of i- secies, Y/ t - is the variation of i-secies concentration (mass fraction) in chemical i ch reactions, mn is mass density of the susended articles, N the articles number density, u, r, - the m velocity, radius and mass of the sherical article, St m /6 r Stokes time, Q interhase thermal exchange source, c P, the constant ressure secific heat of the article material. The changes in concentrations of the mixture comonents due in the chemical reactions are defined by the solution of system of chemical kinetics dyi F i (Y 1,Y 2,...Y N,T), i 1,2,...N dt. (8) The right hand arts of Eq. (8) contain the rates of chemical reactions for the reactive secies. We use the standard reduced chemical kinetic scheme for hydrogen/oxygen combustion with the elementary reactions of the Arrhenius tye and with re-exonential constants and activation energies resented in [28]. This reaction scheme for a stoichiometric H 2 /O 2 mixture has been tested in many alications and to a large extent adequate to comlete chemical kinetics well describing the main features of the H 2 /O 2 combustion. The comuted thermodynamic, chemical, and material arameters using this chemical scheme are in a good 7

8 agreement with the flame and detonation arameters measured exerimentally. For examle, for P0 1. 0bar we obtain for the laminar flame velocity, the flame thickness and adiabatic flame temerature U f 12m/s, Lf 0.24mm, Tb 3012K, corresondingly, for the exansion ratio (the ratio of the density of unburned gas and the combustion roducts) u / b 8.36, and for temerature and velocity of CJ-detonation TCJ UCJ 2815m / s. 3590K, The equations of state of the fresh mixture and combustion roducts are taken with the temerature deendence of the secific heats, heat caacities and enthalies of each secies borrowed from the JANAF tables and interolated by the fifth-order olynomials [29]. The transort coefficients were calculated using the gas kinetics theory [30]. The gaseous mixture viscosity coefficients are 1 i i i 2 i i i 1, (9) where n n i i is the molar fraction, 5 ˆmkT is the viscosity coefficient of i - i i 2 (2,2) 16 i i secies, (2,2) - is the collision integral which is calculated using the Lennard-Jones otential 30, ˆm i is the molecule mass of the i-th secies of the mixture, i is the effective molecule size. The thermal conductivity coefficient of the gas mixture is 1 i i i 2 i i i 1. (10) Coefficient of the heat conduction of i-th secies i i c i /Pr can be exressed via the kinematic viscosity i and the Prandtl number, which is taken Pr The binary coefficients of diffusion are 8

9 D 2kTmm ˆ ˆ mˆ mˆ 3 i j i j 1 ij 2 (1,1) * 8 ij (T ij ), (11) where ij 0,5 i j, T * * ij kt/ ij, * * * ij i j ; * are the constants in the exression of the Lennard-Jones otential, and (1,1) ij is the collision integral similar to (2,2) 30. The diffusion coefficient of i-th secies is D (1C )/ D. (12) i i i ij i j A detailed descrition of the transort coefficients used for the gaseous hase and calculation of diffusion coefficients for intermediates has been ublished reviously, see e.g. [31-33]. The dynamics of solid articles is considered in continuous hydrodynamic aroximation. The interaction between articles is assumed negligibly small for a small volumetric concentration of articles, so that only the Stokes force between the article and gaseous hase is taken into account. Then, the equations for the hase of susended articles are: Where N Nu t x u u u u u t x St T T 2r N t x c 0 (13) 2 4 u Q 4T qrad P, 0, (14) T temerature of the articles, 2 r 2 4 N 4 T qrad. (15) - is the thermal radiation heat flux absorbed and reemitted by the articles. The heat transferred from the article surface to surrounding gaseous mixture is Q (T T)/, (16) g 9

10 where 2r c / 3 Nu is the characteristic times of the energy transfer from the 2 g P, 0 article surface to surrounding gaseous mixture, c P, and 0 are secific heat and the mass density of the article material, Nu is the Nusselt number (see e.g. [34]). For a one-dimensional lanar roblem the equation for the thermal radiation heat transfer in the diffusion aroximation is [35, 36]: dqrad 4 d 1 dx dx 3 4 T q rad, (17) where the radiation absortion coefficient is 1/L r N, and 2 L1/ r N is the 2 radiation absortion length. The article-laden mixture is considered otically-thick so that the radiation energy flux emitted from the flame, which is mainly determined by the article concentration is assumed to be equal to the blackbody radiative heat source, q (x X ) T, where 4 rad f b W / m K is the Stefan-Boltzmann constant, and T b is temerature of the combustion roducts. The calculations were carried out for stoichiometric hydrogen-oxygen and hydrogen-air mixtures at initial ressure P0 1atm with a small solid inert sherical articles susended in the gaseous mixture. For simlicity the articles are assumed to be identical with the mass density of the articles, mn much smaller than the gas density, mn / 1, so that there is only one way of a momentum couling of the articles and the gaseous hase. The numerical method used in the resent studies is based on slitting of the Eulerian and Lagrangian stages, known as Coarse Particle Method (CPM). For the first time it has been develoed by Gentry, Martin and Daly [37] and afterwards was modified and widely imlemented 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 10

11 neighboring grids. The imrovement of overall modified solver imlemented in [39] rovides the second order in sace that differs from the original first-order method [38]. The modified CPM solver was thoroughly tested and successfully used for modeling knock aearance in SI-engines [39, 40] and to study the flame acceleration in tubes with non-sli walls, the transition form slow combustion to detonation [31-33] and other roblems of transient combustion, e.g. ignition of different combustion modes [41-43]. The system of chemical kinetics equations reresents a stiff system of differential equations, and it was solved using standard Gear s method [44]. The develoed algorithm was imlemented using the FORTRAN-90. Convergence and resolution tests resented in Aendixes A were carried out to verify that the observed henomena are correctly caught remaining unchanged with increasing resolution. 3. Radiation heat transfer and flame roagation for uniformly disersed articles We consider the hydrogen-oxygen flame roagating in the mixture with uniformly disersed identical solid sherical articles of radius r 0.75 m, density of the article material, 3,0 1g / cm, and concentrations N ; ; cm -3, that corresonds to the radiation absortion lengths: L 1, 2, and 4cm. As it was mentioned above, the articles ahead of the flame absorb the thermal radiation, their temerature increases and they transfer the heat to the surrounding gas mixture. The characteristic time scales of the roblem for the chosen arameters are 2r / 9 3 s, g 1s ; the 2 St 0 g g characteristic gas-dynamic time scales are L f /U f 20 s and L/Uf 1ms. Since the characteristic time of energy transfer between the articles and the gaseous hase g is much smaller than the characteristic gas-dynamic time scales, the temeratures of the articles and the gaseous hase are aroximately equal, T T. The mass loading arameter values are 11

12 / 0.2; 0.1 and 0.05 for L 1, 2 and 4cm, corresondingly. Because g / 1, the momentum couling of the articles and the gaseous hase is small, and g also is small the influence of the articles on the flame dynamics. Thus, only the heating of articles by the absorbed radiation and corresonding heating of the gas mixture ahead of the flame will influence the flame dynamics. The resence of neutral solid articles is similar to the dilution of combustible mixture with inert gas decreasing the adiabatic temerature behind the flame. This effect is also small for (c / c ) 1. This choice of the arameters allows, V,g us to distinguish the effect only of the radiation reheating on the flame dynamics. Taken into account that the stationary flow is established during the time of the order L / U f, it is straightforward to obtain an estimate for the maximum temerature increase ahead of the flame due to the radiation reheating in the laden-article mixture. In the coordinate system co-moving with the flame front the unburned mixture with susended articles flows toward the flame with the normal laminar flame velocity U f. The thermal radiation is areciably absorbed by the articles, which are located at x flame front, and their temerature can be estimated from the energy balance L ahead of the dt x U t c 4 f 2 b dt L g, c T ex r N (T T),. (18) The gaseous hase temerature ahead of the flame increases due to the heat transferred from the articles to the surrounding unburned mixture dt dt T T g (19) Taken into account that T T, and L1/ r N, we obtain from (18) and (19) 2 dt 4 1 x Uf t c 1 T, b ex dt L L, (20) 12

13 where g / g c V,g / cp,. The characteristic time of the radiation reheating of the Lagrangian article is aroximately the time of its arrival to the flame front, t L/U. Taking this into account, the maximum f temerature increase of the unburned mixture close ahead of the flame can be estimated as: (1 e ) Tb b Uf c (1 ) ( c c V,g)Uf T T (21) It can be readily observed that the maximum temerature increase due to the radiation reheating does not deend on the radiation absortion length (see Fig.1). This is due to the fact that although the local radiant heat flux absorbed by the article is larger for smaller absortion length, but for a larger absortion length articles absorb the radiant heat flux over a longer time until their arrival to the flame front. This difference in time for smaller and larger absortion lengths comensates the lesser local heating for a larger absortion lengths. Taking into account that the adiabatic flame temerature is less than it is in a ure mixture due to dilution by the inert solid articles, and that the hotons emitted from the flame front are roduced within the radiative layer near the flame front of the finite thickness, the effective temerature of the radiation emitted from the flame surface can be estimated as Tb,eff 2700K. Then, the maximum temerature increase caused by the radiation reheating 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 temoral evolution of the maximum gaseous temerature increase at the distance 2mm ahead of the flame front calculated for the radiation absortion lengths L 1, 2 and 4cm. One can see that the stationary values of temerature in the gaseous mixture ahead of the flame are established during t L/Uf0, where f0 U is the normal laminar flame velocity in ure gas mixture. Fig. 2 shows the rofiles of the gaseous hase temerature, which are established after the stationary flow was settled, calculated for the radiation absortion lengths, L=1, 2, and 4cm. 13

14 Figure 1. Time evolution of the gaseous temerature during radiative reheating at the distance 2mm ahead the flame front for different radiation absortion lengths, L=1, 2, and 4cm. On the x-axis time is in units L/U. f T, K L=1cm L=2cm L=4cm (x-x f ), cm Figure 2. Temerature distribution ahead of the flame front calculated for uniformly distributed susended articles and for different radiation absortion lengths, L=1, 2, and 4 cm. Thin line corresonds to the temerature rofile for the laminar flame in a ure stoichiometric hydrogen/oxygen mixture. Absortion of the thermal radiation emitted from the hot combustion roducts by the articles ahead of the flame results in the radiation reheating, which in turn results in the increase of the flame velocity. Fig. 3 shows the increase of the flame velocity relative to the unburned mixture due to the radiation reheating of the gaseous mixture ahead of the flame calculated for the radiation absortion lengths L=1, 2 and 4cm. Recall, that when flame roagates from the closed end of a duct, the unburned gas ahead of the flame moves to the oen end with the velocity u ( 1)Uf, where u / b is the density ratio of the unburned u and burned 14

15 b fuel, resectively [45, 46]. The velocity of the flame with resect to the tube walls is U fl Uf and with resect to the unburned gas the flame velocity it is f U. Due to the radiative reheating of the reacting gaseous mixture ahead of the flame, the flame velocity increases. Asymtotically, the flame velocity goes to the normal laminar flame velocity for L, which corresonds to the ure gaseous mixture. The alicability of the model limits values of L from below, L form L r ( 0 / g). 0.5cm, which follows from inequality 1 written in the Figure 3. Temoral evolution of the flame velocity for the flame roagating through the gasarticles cloud with uniformly susended micro-articles for different thermal radiation absortion lengths L. The flame velocity is normalized on the normal laminar flame velocity U f0 in ure gas mixture. Time is in units L/U f0. The radiation reheating time and the temerature increase ahead of the flame attained due to the radiation reheating is larger for lower flame velocities and/or for a lower initial ressures. For a small enough velocity of the flame (as well for smaller gaseous hase density) the maximum temerature due to the radiation reheating may exceed the crossover temerature, when the endothermic reaction stage asses to the fast exothermic stage. In this case the radiative heat transfer may dominate the gaseous thermal conduction mechanism of the flame roagation. In the aroximation of radiation thermal conductivity, the coefficient of radiation thermal conductivity is defined as 15

16 3 Rad 16 T L / 3. (22) 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 aroximation of the linear thermal conduction equation. For the one-dimensional roblem the temerature distribution (for constant thermal diffusivity) is 1 2 T(x, t) ex( x / 4gt), (23) 2 t g where g g / gcg, const is the coefficient of gaseous thermal diffusivity. It follows from Eq. (23) that heat roagates at the distance x 4 gt. Taking this into account, it is straightforward to obtain an order-of-magnitude estimates for the seed and width of the laminar flame, which roagates as a result of thermal conduction heat transfer: Lf g R, U f g / R. (24) 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 aroximation of linear thermal conduction and if the coefficient of thermal diffusivity is equal to the coefficient of diffusion of reagents, the total enthaly is constant inside of the flame, which imlies similarity of the temerature and the density (concentration of reagent) distributions in the flame front structure. Contrary to the classical theory if the radiative heat transfer becomes dominating rocess, the heat roagation is defined by the nonlinear heat conduction equation. The total enthaly of the unburned mixture and combustion roducts remains equal but the total enthaly is not constant inside of the flame and there is no 16

17 similarity of the temerature and the density distributions. In the case of dominating radiation heat transfer the temerature distribution is considerably different comared to that given by Eq. (23). The temerature distribution in the lanar thermal wave for the nonlinear radiation thermal conductivity (22) is 2 2 1/3 T(x) T 1 x / x. (25) 0 0 In this case the reaction can be ignited within the reheat zone, which size by the order of magnitude is the radiation absortion length L. Corresondingly, the flame thickness is by the order-of-magnitude equal to the radiation absortion length, i.e. L fr L, and the flame velocity can be estimated as U f,rad (L/ R) (L/L f0)uf0 Uf0, where L f0 and U f0 are the width and velocity of the laminar flame in ure gaseous mixture, resectively. It is clear that the flame velocity for the dominating radiative heat transfer significantly exceeds the laminar flame velocity in a ure gas mixture. As soon as combustion has been initiated by the rimary article-laden flame, the combustion generates secondary exlosions ahead of the flame in the article-laden mixture, and such combustion will look like a sequence of thermal exlosions, and can be accomanied by a strong increase in ressure. Presumably such scenario is likely to occur in coal dust exlosion where combustible volatiles can react and release energy, which in turn may contribute to the heat-u of the article and combustible volatiles can react and release energy, which enhance the energy release. As an examle of the influence of the radiation reheating for the flame with lower laminar velocity in a ure gas mixture, we consider the hydrogen/air flame for the same arameters of the uniformly disersed articles as in Fig. 1. In this case the radiation reheating time is longer, because the flame velocity is aroximately 6 times smaller than for the hydrogen oxygen flame. Because of the lower adiabatic flame temerature ( T b(h2air) 2100K ) and larger density of the gas mixture the radiative reheating is only 17

18 aroximately 1.3 times greater than for the hydrogen-oxygen, however the flame velocity increased in this case considerably times. Figure 4. Temoral evolution of the gaseous temerature during radiative reheating at the distance 2mm ahead the flame front for H 2 /O 2 (solid line) and H 2 /air (dashed line) flames for radiation absortion lengths, L=1cm. On the x-axis time is in units L/U. f U f0 t/l Figure 5. Temoral evolution of the H 2 /O 2 (dashed line) and H 2 /air (solid line) flame velocity for the conditions in Fig.4. The flame velocities are normalized on the corresonding normal laminar flame velocity U f0 in ure gas mixture. Time is in units L/U f0. Fig. 4 shows the time evolution of the gaseous temerature ahead the flame front for H 2 -O 2 and H 2 -air flames for the radiation absortion length, L=1cm. The corresonding increase of the flame velocity relative to the unburned mixture due to the radiation reheating shown in Fig.4 for H 2 -air flame and for H 2 -O 2 flame is resented in Fig.5. 18

19 For the thermal radiation heat transfer to become a dominating rocess the necessary condition is that the characteristic time of ignition ahead of the flame to be small comared with the time the original flame travels through the radiation reheat zone, ind(t cr ) L / Uf. It is unlikely that this condition can held for fast H 2 /O 2 and H 2 /air flames, but it is likely ossible for a very slow flame, such as e.g. methane/air flame. All the same, in the case of non-uniform distribution of articles (shown schematically in Fig. 6), the time of the radiative heating can be long enough to rise temerature in the mixture ahead of the flame above the crossover value. The result of the radiation reheating is an inhomogeneous in sace temerature distribution formed in the unburned mixture ahead of the flame with the steeness of the temerature gradient determined by the thermal radiation absortion length. If the maximum temerature within the temerature gradient established due to the radiative reheating of the gas-article mixture exceeded the crossover value, then deending on the steeness of the temerature gradient either deflagration or detonation can be ignited via the Zeldovich's gradient mechanism [1, 2]. 4. The radiative reheating of non-uniformly disersed articles: Ignition of deflagration and detonation ahead of H 2 /O 2 flame Consider a non-uniform distribution of the articles shown schematically in Fig. 6. Concentration of the articles immediately ahead of the flame front (ga 2 in Fig.6) is relatively low so that the radiation absortion length here is much larger then the ga width. Below we assume that the "ga" between the flame and the left boundary of the articles cloud is transarent 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 temerature of the articles and surrounding mixture can rise u to the value suitable for ignition before the flame arrival, which is about 1ms for H 2 /O 2 flame (corresonding width of the ga is about 1cm), then the maximum temerature (T g in Fig.6) within the temerature 19

20 gradient established due to the radiative reheating exceeds the crossover value. What kind of combustion regime is ignited via the Zel'dovich's gradient mechanism in the denser dust cloud deends on the radiation absortion length and, corresondingly, the steeness of the formed temerature gradient. The ignition starts when the temerature exceeds the crossover value, which is for hydrogen/oxygen at 1atm is 1050±50K. Figure 6. Scheme of the radiation reheating of the gaseous mixture inside the gas-articles cloud ahead the flame front. 1- high temerature combustion roducts; 2 - "ga" with lower concentration of articles; 3 cloud of articles; T g -temerature of the radiative reheated gas. The temerature gradient established due to the radiative reheating of the mixture in the gas-article cloud deends mainly on the radiation absortion length and it is also influenced by the gas exansion during the heating. Since the characteristic acoustic time in the reheat zone is much smaller then time of the radiative heating u to crossover temerature, the ressure is equalized within the region heated by the radiation, and the temerature gradient is formed at the constant ressure PP0 1atm. Classification of the combustion regimes in hydrogen/oxygen and hydrogen/air mixtures initiated by initial temerature gradient via the Zel'dovich's gradient mechanism has been studied in [41, 42] using a detailed chemical kinetic models. Figure 7 shows the calculated temoral evolution of the gaseous temerature and the article mass density rofiles during the reheating and establishing of the final temerature 20

21 gradients at t0 900 s, when the maximum temerature raised u to the crossover value. The calculations were erformed for the initial stewise density of articles, for articles of radius r 1 m, with the maximum concentration of articles N cm within the gasarticles layer. The scale (T* T 0) / dt / dx 1cm of the temerature gradients in Fig. 7, which are formed near the left boundary of the article-gas cloud is close to the value of the radiation absortion length L1/ r N 1.2cm. 2 Figure 7. Time evolution of the gaseous temerature (a) and the mass density of the susended solid articles (b) rofiles during radiative reheating inside the gas-articles cloud ahead of the roagating flame. Profiles are shown with the time intervals of 50s. For initial 7 3 stewise articles density rofile, N cm, r 1 m. According to the classification of combustion regimes initiated by the initial temerature gradient in hydrogen oxygen at 1 atm [41, 42], such temerature gradient can ignite a deflagration. The time evolution of the gaseous temerature rofiles resented in the middle frame of Fig. 8 shows develoment of the sontaneous reaction wave on the formed temerature gradient, which then transits into a deflagration wave. The dashed line in the uer frame of Fig. 8 shows the initial number density rofile of the articles and the solid 21

22 line shows the articles number density rofile formed at the instant t s rior to the ignition, when the temerature gradient with maximum temerature T* 1050 К is formed (the first temerature rofile in the middle frame). The calculated evolution of the ressure rofiles in the bottom frame of Fig. 8 indicates a small variation of ressure during the formation of deflagration. Figure 8. Temoral evolution of the gaseous temerature rofiles (middle frame) and ressure rofiles (bottom frame) during the slow combustion wave formation in the vicinity of the margin of the gas-articles cloud far ahead the roagating flame front, t s, 50 s. The uer frame shows the distribution of articles mass density: the initial stewise density rofile (dashed line) and density rofile at time instant t 0 rior to the ignition (solid line). A more gentle temerature gradient can be formed either in the cloud with smaller concentration of the articles, or for the cloud with a roerly diffuse interface instead of the stewise articles number density distribution. In the latter case the radiation absortion length varies along the diffusive cloud interface resulting in the formation of a smooth 22

23 temerature rofile with a shallow temerature gradient caable to initiate either fast deflagration or detonation. Examles of the article clouds with diffuse interface and the calculated temerature rofiles caused by the radiative reheating are shown in Figs.9 and 10. Figure 9. Temoral evolution of the gaseous temerature (middle frame), articles mass density (uer frame) and ressure (bottom frame) rofiles during the fast combustion wave formation behind the outrunning shock in the vicinity of the margin of the gas-articles cloud ahead of the original roagating flame: t s, 50 s. The initial linear density rofile of width 1.0cm (dashed line), density rofile at t 0 rior to the ignition (solid line). The uer frame in Fig. 9 shows the initial (dashed line) number density of the articles, which dros linearly on the scale 1cm from its maximum value N cm. The diffuse boundary of the articles cloud is smeared during the radiation reheating due to the exansion of the gas and at the instant t s rior to the ignition, when the temerature gradient with maximum temerature T* 1050 К is formed it is shown by solid line. The middle frame in Fig. 9 shows the calculated temoral evolution of the gaseous temerature 23

24 rofiles. It deicts the develoment of sontaneous reaction wave on the formed temerature gradient of the scale (T* T 0) / dt / dx 8cm, which then transits into the fast deflagration behind the outrunning shock in the vicinity of the gas-articles layer boundary. The weak shock waves outrunning the deflagration are seen in the bottom frame of Fig. 9. Figure 10. Temoral evolution of the gaseous temerature rofiles (middle frame) and ressure rofiles (bottom frame) during the detonation formation in the vicinity of the gasarticles cloud boundary ahead the roagating flame: t s, 4 s. The uer frame shows the distribution of articles mass density: the initial linear density rofile of width 10.0cm (dashed line) and density rofile at time instant t s rior to the ignition (solid line). The initiation of a detonation by the initial temerature gradient requires more shallow gradient. According to [41, 42] the minimum scale of the initial linear temerature gradient in hydrogen/oxygen mixture at normal conditions ( P 0 1atm) and T* 1050 К at the to of the gradient needed for the detonation initiation is (T* T 0) / T 20cm. Figure 10 resents 24

25 results of the simulation for the layer with initial diffuse boundary with the articles number density, which dros linearly on the scale of 10cm from its maximum value N cm. The uer frame in Fig. 10 shows the initial articles number density (dashed line) and the formed diffuse boundary of the articles cloud smeared due to gas exansion during the thermal radiation at the instant t s rior to the ignition (solid line). Temoral evolution of the gaseous temerature during the detonation formation in the vicinity of the diffusive layer boundary is deicted in the middle frame in Fig.10. The temoral evolution of the temerature rofile corresonds to the develoment of sontaneous reaction wave, its couling with the shock wave and the formation of the detonation wave. Temoral evolution of the ressure rofiles, corresonding to the formation of shock wave, its couling with the sontaneous reaction wave and formation of strong shock corresonding to a detonation wave is seen in the bottom frame of Fig Discussion and conclusions The resent study demonstrates that the radiative heat transfer in a article-laden gaseous flame can considerably influence the overall icture of the flame roagation. It is shown that deending on the satial distribution of the susended articles, the radiative reheating can considerably intensify the gaseous burning, leading to an increased flame velocity and can romote formation of the temerature gradients, which can trigger off new combustion regimes ahead of the rimary flame via the Zel'dovich gradient mechanism with ossible triggering of a detonation. The erformed numerical simulations demonstrate the lausibility of radiation reheating as the rincial effect of the combustion intensification and in some cases initiation of detonation in the gaseous fuel, where relatively low concentration of susended solid articles or any other substance can absorb the radiative heat flux and rise temerature of the fuel 25

26 ahead of the flame. The resented results show that the thermal radiative reheating lay a significant role in determining the regimes of combustion in two-hase reacting flows. It should be emhasized that this study is a necessary rerequisite aiming to show rincile hysics and role of the radiative reheating, which can be imortant for understanding different combustion henomena. The radiative reheating and the radiative heat transfer can be imortant for understanding different combustion henomena at terrestrial conditions and in astrohysics. The conditions under which a reactive two-hase mixture can ignite and roduce a heat release are imortant in different areas of fire safety. In articular, the reheating of the combustible mixture ahead of the flame, due to the absortion of thermal radiation emitted from the flame by the susended articles, which results in flame acceleration and triggering of detonation, is a lausible rout in order to identify the nature of dust exlosion. The danger of dust exlosion exists in rocesses that are accomanied by the formation of clouds of fine dust articles. These events are common risks in the coal, metallurgy, chemicals, wood, hydrogen and hydrogen-based technologies and other industries. Recent exeriments have shown that the dust cloud flame roagation is strongly influenced by the thermal radiation [24-27]. Triggering a detonation by the temerature gradient formed ahead of the flame due to the radiation reheating, considered in the resent study, can be lausible scenario of the transition to detonation in the thermonuclear Tye Ia suernovae (SN Ia) exlosion. Tye Ia suernovae have received increased interest because of their imortance as standard candles for cosmology. Observations using Tye Ia suernovae as standard candles have revealed that the exansion rate of the universe is accelerating and have led to the discovery of dark energy [47-49]. Because of their extreme and redictable luminosity, SN Ia are extensively used as standard candles to measure distances and estimate cosmological arameters critical for our understanding of the global evolution of the Universe. To imrove these measurements, we 26

27 need comrehensive theoretical and numerical models of SN Ia that describe details of the exlosion and connect them to observed characteristics of SN Ia, such as sectra and light curves. But the way in which Tye Ia suernovae exlode is not comletely understood. The current leading aradigms for the exlosion is scenario of the deflagration-to-detonation transition. There is increasing evidence that a detonation is needed to exlain majority features of the SN Ia exlosion. Nowadays the best modeling describing majority of the observed SN Ia events is rovided by the so-called delayed detonation model [50-52], which imly a hase of subsonic thermonuclear burning (deflagration) during which the star exands and a hase of a detonation, which burns remaining fuel on timescales much shorter than the timescale of the exlosion. The aradigm 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 ulsations at densities higher than g/cm and it becomes stable at lower densities near the star surface. There has been numerous attemts both analytical and numerical [50-60] to exlain the detonation formation in SN Ia exlosion. However, after many years of studies a fundamental question what is the mechanism of deflagration-to-detonation transition (DDT) in the suernova Tye Ia events still remains the least understood asect of the SN Ia exlosion henomenon (see e.g. [59]). Tye Ia suernovae begins with a white dwarf (WD) near the Chandrasekhar mass that ignites a degenerate thermonuclear runaway close to its center and exlodes having initial radius RWD 8 10 cm. Such a wide range of the length scales necessitates to use models of the infinitely thin flame (see [59] for a review of exlosion scenarios) that limits sufficiently the understanding of the transient henomena. In the context of thermonuclear burning of SN Ia, combustion initially roceeds in the deflagration mode from the center of SN Ia with the velocity and width of the laminar flame Uf 7 10 cm/s and Lf 2 10 cm. Possible new mechanism of DDT in SN Ia exlosion may be associated with detonation triggering due to 27

28 radiation reheating similar to the scenario of the detonation ignition considered in Sec.4. During the late deflagration hase the radiant flux roduced by the radioactive decays of Ni Co Fe increases considerably in course of the star incineration by the exanding deflagration wave. Absortion of the radiant energy flux in the outer layers of the star may roduce a reconditioned region with the shallow temerature gradient such that detonation can be ignited via the Zel dovich gradient mechanism. Acknowledgements This work was suorted in art by Ben-Gurion University Fellowshi for senior visiting scientists (ML), by the Research Council of Norway under the FRINATEK, Grant (ML), and by the Program of Russian Academy of Sciences "Basic roblems of mathematical modeling" (MI, AK). We acknowledge the allocation of comuting resources rovided by the Swedish National Allocations Committee at the Center for Parallel Comuters at the Royal Institute of Technology in Stockholm, the National Suercomuter Centers in Linkoing and the Nordic Suercomuter Center in Reykjavik. One of the authors (M.L.) is grateful to Axel Brandenburg and Nils E. Haugen for useful discussions during writing this aer. 28

29 Aendix A: Code validation, resolution and convergence tests The thorough convergence and resolution tests were carried out to verify that the observed henomena of the radiation reheating and imact of the radiation heat transfer in articleladen flame were correctly caught remaining unchanged with increasing resolution. Figures A1 and A2 reresent the results of the convergence test for a flame roagating through the ure gaseous stoichiometric hydrogen-oxygen mixture at normal conditions ( T 0 300K, P0 1atm) and for the mixture with susended micro articles (for the uniform articles distribution, corresonding to Figs.1-3 ( L 2cm) in Section 3. In the latter case the combustion velocity and the flame structure is fully defined by the state of the gaseous mixture just ahead of the flame front. As the temerature in this region does not exceed K and the ressure remains constant the convergence is for almost the same fine resolution as in the case for a ure gaseous mixture (see Fig. A1). As it was mentioned in Sec.3 in case of considerably smaller flame velocities the reheating is more efficient close to the flame front. In such conditions one should resolve flame structure at the higher temeratures and use finer meshes (see Fig. A2). The meshes were taken to resolve the structure of the flame front with 6, 8, 12, 24, and 48 comutational cells, corresonding to the comutational cell sizes: = 0.1, 0.05, 0.025, 0.01 and 0.005mm, resectively. The accetable quantitative convergence was found for resolution of 24 comutational cells er flame front (see Figs. A1, A2). Therefore the resolution 24 and 48 comutational cells was tyically used for solving the roblem in Section 3. As it is seen from Fig. A2, which shows the convergence tests for three different initial temeratures ( T 0 293, 600, and 1000K, corresonding to curves 1, 2, 3 in Fig.3) the flame dynamics at the elevated temeratures can be resolved only with a finer resolution than at the lower temeratures. Therefore to obtain the converged solution for the detonation initiation roblem one should use a finer resolution from the very beginning. 29

30 Figure A1. Resolution test for normal velocity of stoichiometric hydrogen-oxygen flame at normal ambient conditions ( T 0 300K, 0 1atm ). U f flame velocity reroduced with the comutational cell of size ; T b adiabatic temerature of the combustion roducts; u / b exansion ratio, index c corresonds to the converged values; Burning velocity is resented for two cases: without articles (emty signs) and for gaseous mixture with susended articles (filled signs). Figure A2. Resolution tests for different ambient temeratures (1-293K; 2-600K; K). Filled signs show accetable range of convergence. U f is a normal flame velocity reroduced with cell size. The roblem considered in Section 4 demands more robust arameters of the comutational setu. The most demanding is the case where the detonation arises as a result of auto-ignition inside the hot-sot. The consequence of the rocesses taking lace in such a case should be aroriately resolved. As it was shown in [41, 42] the detonation caused by 30