Effects of thermal radiation heat transfer on flame acceleration and transition to detonation in particle-cloud flames

Transcription

1 Effects of thermal radiation heat transfer on flame acceleration and transition to detonation in article-cloud flames M.A. Liberman 1 M.F Ivanov, A. D. Kiverin, 2 1 NORDITA, AlbaNova University Center, Roslagstullsbacken 23, SE Stockholm, Sweden 2 Joint Institute for High Temeratures, Russia Abstract The current work examines regimes of the hydrogen-oxygen flame roagation and ignition of mixtures heated by radiation emitted from the flame. The gaseous hase is assumed to be transarent for the radiation, while the susended articles of the dust cloud ahead of the flame absorb and reemit the radiation. The radiant heat absorbed by the articles is then lost by conduction to the surrounding unreacted gaseous hase so that the gas hase temerature lags that of the articles. The direct numerical simulations solve the full system of two hase gas dynamic time-deendent equations with a detailed chemical kinetics for a lane flames roagating through a dust cloud. It is shown that deending on the satial distribution of the disersed articles and on the value of radiation absortion length the consequence of the radiative reheating of the mixture ahead of the flame can be either the increase of the flame velocity for uniformly disersed articles or ignition either new deflagration or detonation ahead of the original flame via the Zel'dovich gradient mechanism in the case of a layered article-gas cloud deosits. In the latter case the ignited combustion regime deends on the radiation absortion length and corresondingly on the steeness of the formed temerature gradient in the reignition zone that can be treated indeendently of the rimary flame. The imact of radiation heat transfer in a article-laden flames is of aramount imortance for better risk assessment and reresents a route for understanding of dust exlosion origin. Keywords: Thermal radiation, Flame acceleration, Detonation, Dust exlosion, Hydrogen safety * ) Corresonding author: mliber@nordita.org 1

2 1. Introduction Hydrogen has emerged as an imortant fuel in a number of diverse industries as a means of achieving energy indeendence and to reduce emissions. Nowadays, when hydrogen technologies and fuel cells are enetrating the market in a number of alications, extensive research is still needed for effectively addressing the high-risk technological barriers in a recometitive environment. Wide sread deloyment and use of hydrogen and hydrogen-based technologies can occur only if hydrogen safety issues have been addressed in order to ensure that hydrogen fuel resents at least the same level of hazards and associated risk as conventional fuel technologies. Hazard identification is the necessary ste to ensure the full and safe utilization of hydrogen in either hydrogen safety engineering or risk assessment. The urose of the hazard identification is to identify all events that can affect facility oeration leading to a hazard to individuals or roerty. Since hydrogen is an extremely flammable and easily detonable gas when mixed with air over a wide range of comosition, exlosion hazards associated with the roduction, transortation and storage of hydrogen must be resolved to a sufficient confidence level and the key challenges facing the future widesread use of hydrogen are safety-related issues. The hazardous otential of hydrogen-air and hydrogen-oxygen mixtures has been extensively studied and a huge number of exerimental, theoretical and numerical studies insired by their imortance for industrial safety had been taken in attemt to understand nature of the exlosion [1-9]. Although most accidental exlosions are deflagrations, in the worst-case scenario, the flame acceleration can lead to deflagration-to-detonation transition (DDT). Deending on the mixture characteristics, such as concentrations, temerature, ressure and flow geometry, combustion rocess can undergo strong flame acceleration and deflagration-to-detonation transition (DDT). These regimes are characterized by high burning rates and consequently by high ressure loads. The resulting detonation is extremely 2

3 destructive, can induce ressures u to or above 10 MPa, and therefore can have esecially catastrohic consequences in a variety of industrial and energy roducing settings related to hydrogen. Since the discovery of detonation more than 150 years ago, a huge number of exerimental, theoretical and numerical studies had been taken in attemt to understand nature of the transition from deflagration to detonation (see e.g. [11, 12] and references within). These studies are insired by their imortance for industrial and nuclear ower lants safety as well their otential alication for micro-scale roulsion and ower devices [13, 14]. Desite many years of substantial achievements in the area of the flame acceleration and DDT, still many secific asects of the roblem remain unclear. Deendence of the otential danger of the combustion rocess aeared to be very sensitive to the geometrical conditions of the rocesses, mostly to the confinement and to the congestion of the volume. Currently a unified hysical model and corresonding numerical instrument which can be used over the entire range of henomena is not available. Numerous combustion models are usually addressing only secific regime or henomenon and are alicable only in their domain of validity. The hazardous otential of hydrogen-oxygen and hydrogen-air mixtures has been extensively studied assuming a erfect mixture of hydrogen fuel and oxidant. Since the ioneering studies by Shchelkin, Zeldovich, Oenheim and their co-authors [15-18] there has been a continuous efforts aiming to elucidate a reliable hysical mechanism exlaining DDT. From the beginning there was widely sread oinion that turbulence lay a key role in the flame acceleration and DDT. A common belief was that a fast flame acceleration and the transition to detonation can occur only for strongly turbulent flames. Since the very first DDT studies it was known that the resence of obstacles increases the flame acceleration and shortens considerably the run-u distance. The exeriments demonstrated that a flame accelerates more raidly toward the oen end of a duct if it asses through an array of 3

4 turbulence-generating baffles. This resumably was the reason why the first attemts to exlain DDT were associated with turbulent flames and were based on the assumtion that DDT might occur only in the case of turbulent flames. Channels with rough walls or obstacles are often used to study DDT since it is believed that in this case the run-u distance is more or less fixed and controlled by turbulence [19, 20]. However, recent large scale exeriments [21] with different kind of mixtures (C 2 H 4 -air, CH 4 -air, C 3 H 8 -air, H 2 -air) suggest that the selfacceleration mechanism of the flame may be much better reresented by flame instabilities than by turbulence build-u. All the same DDT is easily observed in channels with smooth walls [22-24] and in thin caillary tubes [25]. The first exlanation of the flame acceleration in tubes with no-sli walls before the DDT occurred was given by Zel'dovich [26]. In his detailed analysis of the Shchelkin s exerimental results Zel'dovich has ointed that turbulence is not a rimary factor resonsible for flame acceleration in a smooth-walled channel and sequential detonation formation. Exlaining the nature of the flame acceleration in the DDT events Zel'dovich emhasized that the flame acceleration in a tube with no-sli walls is due to stretching of the flame front caused by the flame interaction with a nonuniform velocity field of the ustream flow, while turbulence lays a sulementary role if any deending on the exerimental conditions. Although the qualitative icture of the DDT is more or less clear, however a quantitative theory and the hysical mechanism of DDT are still oorly understood and requires better theoretical and hysical interretation. With the advance in scientific comuting, research has been shifted towards the use of comutational aroaches. Nowadays, numerical simulations can rovide a qualitative icture of the basic rocesses from ignition and flame acceleration u to the transition to detonation. The reviews [11, 12] summarize the numerical efforts undertaken in the ast decades to understand the deflagration-to-detonation mechanism in a highly reactive gaseous mixtures (e.g. hydrogen/air, acetylene/air) using a one-ste Arrhenius chemical model. The conclusion 4

5 drawn from these studies was that the mechanism of DDT is the Zeldovich gradient mechanism [27] involving gradient of reactivity. However, as an unsteady rocess, DDT involves multile rocesses of vastly different scales. Among them, comlex chemical reactions lay a first-order controlling role for gaining scientific insight into the mechanism of DDT [28]. The numerical study of the Zeldovich gradient mechanism using a detailed chemical reaction models for hydrogen/oxygen and hydrogen/air [29, 30] has shown that the minimum scale of the temerature gradient (the length-scale of the temerature inhomogeneity) caable to initiate detonation exceeds size of the hot sots formed in the unreacted material ahead of accelerating flame by orders of magnitude. Recent 2D and 3D numerical simulations of the flame acceleration and DDT in hydrogen-oxygen mixtures that have taken into account a detailed chemical kinetics have revealed an adequate mechanism of DDT [24, 31-34]. An imortant roblem of hydrogen safety is connected with leakage of the hydrogen and its further exlosion. Hydrogen release might occur on storage, transort and handling. Once a flammable mixture forms, it can be ignited by a variety of uncontrolled means. An ignition is likely for all forms of releases due to the wide range of flammability and low ignition energy. The destructive ower of the resulting exlosion deends on the volume of the reactive mixture, its comosition, and the geometries of confinement. In most ractical cases ignition arises from a small area of combustible mixture, as an accidental ignition e.g. from an electric sark, or any local heating and starts as a laminar flame. The flame evolution can result in substantial flame acceleration, deending on the geometries of confinement, e.g. friction of the wall results in the flame front stretching, obstacles may result in flame turbulization, etc. Mechanisms of ignition by transient energy deosition and different scenarios of a detonation initiation were investigated in [35]. 5

6 Most of studies on dust exlosions have been erformed to examine the characteristics or the indices of dust exlosions in a closed vessel [9]. The fundamental mechanisms of flame roagation in dust susension, however, have not been sufficiently studied and the limited exerimental results known in the literature are often contradict each other. In articular, the influence of radiative heat transfer on the rate of flame roagation and duct cloud exlosions is not yet fully understood. Combustion of the hydrogen-oxygen and hydrogen-air mixtures was studied largely at standard environmental conditions, however new technologies often dictate substantial variation of the conditions which have to be taken into account for the safety analysis. Among others, the fundamental roerties of hydrogen mixtures with dusts of fine articles have to be considered. Non-uniformities of the gas distribution and small, micron-size solid articles susended in the gas mixture can considerably affect regime of combustion in some cases leading to the strong flame acceleration and DDT. However, only limited amount of the exerimental data are available on the behavior of the hydrogen in the resence of susended articles related to the roblem of dust exlosion. 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 mixtures in attemt to understand mechanism of the detonation formation [36]. More than 50 years ago Essenhigh and Csaba [37] have indicated that radiation heat transfer lays a very significant role in the behavior of a dust cloud s lanar flame. In earlier studies the flame dynamics affected by the radiative reheating has been investigated using asymtotic methods and a one-ste Arrhenius chemical model with high activation energy [38-40]. Majority of research has been focused coal combustion with two asects of ractical interest: the roduction of volatiles due to thermal decomosition of coal dust and the ignition conditions [41-45]. 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 6

7 in an exlosion [45, 46]. 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. Although there is an obvious lack of radiation measurements in large scale remixed systems related to exlosions and the actual level of thermal radiation emitted from the flame remains conjectural, recent exeriments have shown that the dust cloud flame roagation is strongly influenced by the thermal radiation [47, 48]. The data resented ermits an assessment of the lausibility of combustion initiation due to forward thermal radiation. Although some models were develoed recently, they were only for secific dusts (mainly for a coal dust flames) and might not be alicable generally [43, 48-50]. Identities of the imortant flame rocess articularly the controlling regimes of the flame roagation rocess, the knowledge of the role of the radiation heat transfer is highly demanded due to the raised likelihood to meet such conditions in the accidental conditions. In the resent aer we consider the influence of thermal radiation emanating from the hot combustion roducts on the rate of flame roagating in hydrogen gaseous mixture susended with fine articles. Numerical and exerimental studies show that hydrogen accumulation leads either to a stratified distribution of concentration or to the formation of a homogeneous layer if the convective flows at the to of the enclosure are high enough. Thus, we consider two ossible scenarios: homogenous and a stratified (layers) nonuniform distribution of the susended articles. In the case of non uniform disersion of the articles, which is tyical for e.g. dust layers [7, 9, 43-45], the time of the radiative heating is longer and the radiative reheating can be sufficient to ignite the surrounding combustible mixture ahead of the flame via the Zel'dovich gradient mechanism. In the latter case the ignited regime of combustion can be either deflagration or detonation deending on the radiation absortion length, which in turn determines steeness of the temerature gradient in the non uniform temerature 7

8 distribution formed within the reheat zone. The effects of thermal radiation heat transfer on flame acceleration and transition to detonation in dusty-laden hydrogen mixtures analyzed theoretically and using high resolution numerical simulations for combustible materials whose chemistry is governed by a detailed chemical reaction model for chain-branching reactions. 2. Problem setu. Physical and numerical model We consider a lane flame roagating from the closed to the oen end of the duct in a article-cloud hydrogen-oxygen mixture. For a ure gaseous flames the radiation emanated by high temerature combustion roducts is not imortant because the radiation absortion length in a gaseous mixture at normal conditions is very large, so that the gas is usually considered as fully transarent for the radiation and the radiation heat transfer does not influence the flame dynamics. For examle, at normal ressure P=1atm the cross sections of the Thomson scattering and "bremsstrahlung" rocesses in the air are very small, (10 10 )cm, so that the mean free ass of a hoton is more than tens meters. The radiation heat losses of the high temerature combustion roducts are small comared with the gaseous thermal conduction heat transfer to the tube walls and can be neglected. In the traditional theoretical studies of combustion the heat is transferred by the conduction and/or convection while the radiation heat transfer is negligible because the energy transferred through radiation is generally too small to affect the velocity of combustion wave. The situation changes drastically if the gaseous mixture is seeded by the articles. Particles are absorbed the radiant heat flux emanated from the flame, their temerature increased and then they lost heat by conduction to the surrounding unreacted gaseous hase, so that the gas hase temerature lags that of the articles. The radiation reheating of the gas ahead of the flame affects the flame dynamics resulting in the increase of the flame velocity or in the ignition of either deflagration or detonation in the "distant" article seeded layer ahead of the flame in the case of a nonuniformly disersed article-cloud. 8

9 2.1. Governing equations 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, gas-articles momentum and energy exchange and detailed chemical kinetics for eight 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 governing equations for the gaseous hase are 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 where we use standard notations: P,, u reresent ressure, mass density and flow velocity of the gas hase, Y i i / is the mass fractions of the secies, 2 E u /2 is the total energy density, is the internal energy density, R B is the universal gas constant, m i is the 9

10 molar mass of i-secies, Ri R B /mi, n is the gas molar density, ij is the viscous stress tensor, cv cviyi is the constant volume secific heat, c vi is the constant volume secific i heat of i-secies, h i is 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 is the articles number density, u, r, m are velocity, radius and mass of the sherical article, St m /6 r is the Stokes time, Q is thermal energy exchange between the gas hase and the articles, the constant ressure secific heat of the article material. c P, is The changes in concentrations of the mixture comonents due in the chemical reactions are defined by the solution of equations 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. The reaction mechanism is the standard reduced chemical kinetic scheme for hydrogen/oxygen consisting of 19 elementary reactions of the Arrhenius tye with reexonential constants and activation energies resented in [51]. 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 agreement with the flame and detonation arameters measured exerimentally. For P bar we obtained for the laminar flame velocity, the flame thickness, adiabatic flame temerature and the exansion coefficient (the ratio of densities of the unburned gas, u, and the combustion roducts, b ): Uf 12m/s, Lf 0.24mm, 10

11 Tb 3012K, u / b 8.36, corresondingly, and for temerature and velocity of CJdetonation TCJ 3590K, UCJ 2815m / s. The transort coefficients were calculated using the gas kinetics theory [52]. 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 52, 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 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) 52. The diffusion coefficient of i-th secies is D (1Y)/ D. (12) i i i ij i j 11

12 A detailed descrition of the transort coefficients used for the gaseous hase and calculation of diffusion coefficients for intermediates is described in detail in [24, 31-33]. 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 [53]. The solid articles are modeled using continuous hydrodynamic aroximation. The article-article interactions is negligible for a small volumetric concentration of the articles and only the Stokes force between the article and the gaseous hase must be taken into account. Thus, the equations for the 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 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 density of the article material, Nu is the Nusselt number [54]. For a one-dimensional lane roblem the equation for the thermal radiation heat transfer in the diffusion aroximation is [55, 56]: 12

13 dqrad 4 d 1 dx dx 3 4 T q rad, (17) where the radiation absortion coefficient is radiation absortion length. 1/L r N, and 2 L1/ r N is the 2 It should be noticed that although the emissivity of the gaseous combustion roducts is relatively low, however even small concentration of articles increases the emissivity considerably. From theoretical oint of view the article-laden mixture is otically-thick if the radiation absortion length is small comared with the domain size. Radiation emanates from both gaseous combustion roducts and susended articles, which are at high temeratures and hence high radiation levels are exected. The thermal radiation flux emanating from both high temerature combustion roducts with seeded articles deends on the emissivity of the burned volume, which relates to the concentration of articles in combustion roducts and was assessed to be close to theoretical values of blackbody radiation for a burned gas temerature [47]. Thus, we assume the blackbody radiative flux emitted from the flame front, q (x X ) T, where 4 rad f b W / m K is the Stefan-Boltzmann constant, and Tb 3000K is temerature of the hydrogen-oxygen combustion roducts. The calculations were carried out for stoichiometric hydrogen-oxygen at initial ressure P0 1atm with an identical micron-scale inert sherical articles susended in the gaseous mixture. The mass loading arameter was taken small, / 1, so that there is only one way of a momentum couling of the articles and the gaseous hase. Therefore, only heating of the articles by the radiation and the heat transfer from the articles to the gas hase ahead of the flame will influence the flame dynamics. This choice of the arameters allows us to distinguish the effect of the radiation reheating on the flame dynamics. 13

14 2.3. Numerical algorithms High resolution numerical simulations were erformed to study the effect of thermal radiation heat transfer and the interaction of a stoichiometric hydrogen-oxygen and hydrogen -air flames with a dust cloud comosed of either uniformly disersed or stratified (layered) articles. An exhaustive descrition of the convergence and resolution tests of the numerical method have been erformed in [24, 31-33]. The governing equations are solved using the method develoed originally by Gentry, Martin and Daly [57] and afterwards modified and imlemented for various hydrodynamic roblems by Belotserkovsky and Davydov [58]. The method is based on slitting of the Eulerian and Lagrangian stages and known as Coarse Particle Method (CPM). The algorithm was further modified by Liberman et al. [59] 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 second order in sace solver was thoroughly tested and successfully used for modeling knock aearance in SI-engines [59, 60], to study the flame acceleration in tubes with non-sli walls and DDT [31-33] and various roblems of transient combustion, e.g. ignition of different combustion modes [29, 30, 35]. The system of chemical kinetics equations reresents a stiff system of differential equations and was solved using standard Gear s method [61]. The develoed algorithm was imlemented using the FORTRAN-90. An additional 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 article-laden flame were correctly caught remaining unchanged for increased resolution. The main secific feature of the reacting flows is the resence of the reaction fronts that must be resolved fine not only to diminish the scheme viscosity influence but to resolve their structures and interactions with the comression and shock waves that can arise in the comressible flow. The fine grids according to the hysical limitations were used. To resolve 14

15 structure of the hydrogen-oxygen flame front which thickness is about 0.24mm at normal conditions the grid cells less than 0.01mm were used to avoid unhysical coulings with the shocks that are usually smoothed over 5-6 cells even by the high-order schemes with limiters. The convergence and resolution tests were carried out to verify that finer grid resolutions, u to 0.005mm rovides quantitative reliability of the method for calculating the flames roagating at elevated temeratures and ressures and interacting with the comression and shock waves that can arise in the comressible flow. In articular it was demonstrated that the comutational method reroduces with high accuracy analytical solution for discontinuity decay in air for different driver ressures and a so-called Blast wave test by Woodward and Colella [62]. 3. Effect of radiation reheating for uniformly disersed articles 3.1. The flame acceleration due to radiation reheating We consider the hydrogen-oxygen flame roagating in the mixture with uniformly disersed identical solid sherical articles of radius r 0.75 m, and the article mass density 3,0 1g / cm. The thermal radiation emitted from the flame is absorbed and reemitted by the articles ahead of the flame and the intensity of the radiant flux decreases exonentially on the scale of the order of the radiation absortion length L1/ r N. The 2 time evolution of the temerature at the distance just 2mm ahead the flame front shown in Fig.1 was comuted for different radiation absortion lengths: L=1, 2, and 4cm corresonding to the concentration of articles: N ; ; cm -3. The maximum temerature of the articles ahead of the flame and the maximum increase of the flame velocity achieved during the characteristic hydrodynamic time of the order t L/U f0, where U f0 is the normal laminar flame velocity in ure gas mixture. It seen from Fig.1 that the maximum temerature of the radiation reheating deends weakly on the radiation absortion 15

16 length. Although the local radiant heat flux absorbed by the article is larger for smaller absortion length, but in the case of a larger absortion length articles absorb the radiant heat flux over a longer time until their arrival to the flame front. This difference for smaller and larger absortion lengths comensates the lesser local heating for a larger absortion lengths. 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. Fig. 2 shows the rofiles of the gaseous hase temerature which are established after the stationary flow was settled for the radiation absortion lengths, L=1, 2, and 4cm f0 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. The flame velocity increase with resect to the unburned gas due to the radiation reheating (Fig.2) is shown in Fig. 3. Notice that for a flame roagating from the closed end of a duct, 16

17 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 b gas [63, 64]. The velocity of the flame with resect to the tube walls is UfL Uf and with resect to the unburned gas the flame velocity is U f. Figure 3. Comuted time evolution of the flame velocity for the 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 radiant heat absorbed by the articles is then lost by conduction to the surrounding unreacted gaseous hase and the gas hase temerature lags that of the articles. The characteristic time scales of the roblem. For the arameters used in simulations the characteristic time scales are: 2r / 9 10 s, 2 St 0 g g 2r c / 3Nu 0.3 s, 2 g P, 0 L /U f 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 gas-dynamic time scales L/U f, the temeratures of the articles and the gaseous hase are aroximately equal, T T. The value of mass loading arameters are: / 0.2; 0.1 and 0.05 for L 1, 2 and 4cm, corresondingly. Because / 1, the momentum couling of the articles and the gaseous hase is small. Thus, only the radiation heating of articles and corresonding 17

18 reheating of the gas mixture ahead of the flame influence the flame dynamics. The resence of inert articles is similar to dilution of combustible mixture with inert gas, decreasing the adiabatic temerature behind the flame. This effect is small for (c / c ) 1. This choice, V,g of the arameters allows us to distinguish the effect 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 of the gas ahead of the flame. In the coordinate system co-moving with the flame front the unburned mixture with susended articles flows toward the flame with the laminar flame velocity at x U f. The thermal radiation is areciably absorbed by the articles, which are located L ahead of the flame front, and their temerature is defined by the energy equation 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 gas dt dt T T g (19) Taken into account that T T for g L/Uf we obtain from (18) and (19) dt 4 1 x Uf t c 11/ T, b ex dt L L, (20) where g / g c V,g / cp,. The characteristic time of the radiation heating is aroximately time of the Lagrangian article arrival to the flame front, t L/U. Then, the maximum temerature increase f achieved in the mixture close to the flame front can be estimated as: 18

19 (1 e ) Tb b f V,g f V,g T T (21) U ( c c ) U ( c c ) It can be readily observed from (21) that the maximum temerature increase due to the radiation reheating does not deend on the radiation absortion length (see also Fig.1). The maximum temerature increase given by (21) is in a good agreement with comuted values, taking into account that the adiabatic flame temerature is smaller due to gas 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 Structure of the flame; radiation dominated regime of the flame roagation It follows from Eq. (21) that the effect of the radiation reheating is enhanced for a flame with lower normal laminar flame velocity or for a lower initial gas density. The comuted increase of the gas temerature and the corresonding increase of the flame velocity of the H 2 -O 2 flames roagating at the initial ressures 1.0atm, 0.3atm and 0.2.atm are shown in Fig.4(a, b). Figure 4 (a, b). Radiation reheating at the distance 2mm ahead the flame front (a) and the flame velocity increase (b) for different initial ressures of the for H 2 /O 2 mixture: solid line - P0 1atm, dashed line - P0 0.3atm, dashed-doted line - P0 0.2atm. The radiation absortion length is L=1cm. On the x-axis time is in units L/U f0. 19

20 Fig. 5(a, b) shows the time evolution of the maximum temerature and the corresonding increase of the hydrogen-air flame velocity in comarison with these values for hydrogenoxygen flame for initial ressures 1atm and radiation absortion length L 1cm. It is seen that for slower H 2 -air flame the reheating is more effective and the flame velocity increased much more than for the H 2 -O 2 flame. 2.5 U f, m/s U f0 t/l Figure 5. (a): Time 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, P0 1atm. On the x-axis time is in units L/U f0. 5(b): The flame velocity increase for the conditions of Fig.5(a) for H 2 /O 2 (solid line) and H 2 /air (dashed line) flames for radiation absortion lengths, L=1cm, P0 1atm. 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. It is interesting to note that in the framework of a one-ste chemical model the width of a flame decreases with the increase of the flame seed. Indeed, the velocity and structure of a laminar flame in the classical combustion theory [63, 64] is defined in the aroximation of the linear equation of thermal conduction, which solution in a one-dimensional case is 1 2 T(x, t) ex( x / 4gt), (22) 2 t g 20

21 where g g / gcg, const is the thermal diffusivity coefficient. It follows from Eq. (22) that during time t the 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 due to thermal conduction: Lf g R, U f g / R. (23) These exressions are based on the consideration that the characteristic heat diffusion time scale is much longer than the characteristic time R of the heat release in the reaction, otherwise any small disturbances of the combustion wave will diffuse away resulting in the flame extinguishing. Thus, we arrive to the conclusion that LU f f g. On the contrary, the increase of the flame velocity caused by the radiation reheating is accomanied by an increase of the flame front width. Fig.6 shows comuted width of the flame deending on the radiation absortion length. Large values of L corresond to ure gas mixture with the flame front width Lf 0.24mm for H 2 -O 2 at P 1atm. 0.3 L f, mm L, cm Figure 6. Width of the hydrogen-oxygen flame versus the radiation absortion length, P0 1atm. According to (21) for the flame with small enough velocity the maximum temerature of the unreacted gas ahead of the flame can exceed the ignition threshold. For a hydrogenoxygen this is the crossover temerature, when the endothermic reaction stage asses to the fast exothermic stage. In this case radiation may dominate the conduction heat transfer and 21

22 thermal radiation makes a decisive contribution to the overall energy transort. As an examle Fig. 7(a, b) shows the comuted time evolution of the maximum temerature increase ahead of the flame in methane-air with susended articles and the corresonding flame velocity increase due to the radiation reheating for the radiation absortion length L 1cm. Because of much smaller velocity of the flame in methane-air, the radiation reheating time is much longer than for H 2 -O 2 or H 2 -air flames. In the calculations the reduced chemical reaction mechanism for methane-air combustion develoed in [65] was used T, K U f /U f U f0 t/l Figure 7(a, b). Radiation reheating at the distance 2mm ahead the methane-air flame (a) and the flame velocity increase (b) at initial ressure P0 1atm (solid line) and P0 0.5atm (dashed line ). The radiation absortion length is L=1cm. On the x-axis time is in units L/U f0. The flame structure changes considerably if radiation dominates heat transfer rocess. In the framework of a one-ste chemical model, the total enthaly is constant inside of the flame which imlies similarity of temerature and concentration of reagent distributions in the flame structure. This conclusion in the classical combustion theory obtained in the aroximation of linear thermal conduction and when the coefficient of thermal diffusivity equals to the 22

23 diffusion coefficient of reagents. On the contrary, if the radiation becomes dominating rocess, the heat roagation is defined by the nonlinear equation of heat conduction. In the diffusion aroximation the coefficient of radiation thermal conductivity is 3 Rad 16 T L / 3. (24) In this case 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 similarity of the temerature and the density distributions. Solution of the thermal conduction equation for the nonlinear thermal conductivity (24) is much more gentle than that given by (22) 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 f)uf Uf, where L f and U f are the width and velocity of the laminar flame given by Eq. (23). If radiation dominates the flame velocity exceeds significantly 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 in the mixture ahead of the flame. Such combustion wave will look like a sequence of thermal exlosions and can be accomanied by a strong ressure increase. The necessary condition for the radiation to become dominating mechanism of a flame roagation is that the characteristic time of ignition in the mixture ahead of the flame to be small comare with the time the original flame assing through the radiation reheat zone, ign (T cr ) L / Uf. It is unlikely that this condition is fulfilled for fast H 2 -O 2 or H 2 -air flames, but it is ossible for a very slow flame, such as e.g. methane-air flame, where the rocess can be additionally enhanced due to volatile combustion of coal dust is involved. In order to have 23

24 a notable effect ignition have to be formed well ahead of the advancing flame, thus relatively long time scale of the radiation reheating and length scales are essential. This is unlikely for a fast hydrogen-oxygen flame with uniformly disersed articles, however it is ossible for non-uniform distribution of susended articles. 4. Ignition of deflagration and detonation by the radiation reheating for non-uniform concentration of articles Non-uniform distribution of articles in a dust-air cloud may be formed when fine articles are raised either in the rocess of hydrogen leakage or by an exanding gas cloud or as result of a weak local exlosion. In coal mines it is a well established henomenon that the ressure wave of a weak methane exlosion can diserse dust deosits leading to the formation of an non-uniformly susended articles forming a layered dust-air cloud. We consider a non-uniform distribution of the articles concentration shown schematically in Fig.8. Figure 8. 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. Concentration of the articles immediately ahead of the flame (ga 2 in Fig. 8) 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 mixture is transarent for the radiation so that the radiant heat flux is fully absorbed in the layer 3. If 24

25 time of the flame arrival to the boundary of the layer 3 is long enough, so that the temerature of the articles and the surrounding mixture can rise u to the ignition value before the flame arrival, which is about 1ms for H 2 /O 2 flame, then the maximum temerature (T g in Fig. 8) within the temerature gradient established due to the radiative reheating exceeds the crossover value which is for hydrogen/oxygen at 1atm is 1050±50K. The corresonding width of the ga is in the range of a few centimeters. What kind of combustion regime is ignited via the Zel'dovich's gradient mechanism in the denser layer deends on the radiation absortion length and, corresondingly on the steeness of the formed temerature gradient (a) T g, K 0.1 (b) ρ, kg/m x, cm Figure 9. Temoral 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 stewise articles density rofile, 7 3 N cm, r 1 m. The temerature gradient, which is 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 modified during the exansion of heated u mixture. Since the characteristic acoustic time in the reheat zone is much smaller then the time of the radiative heating u to crossover 25

26 temerature, the ressure is equalized within the region heated by the radiation, and the temerature gradient is formed at the almost constant ressure P P0 1atm. Classification of the combustion regimes in hydrogen-oxygen and hydrogen-air mixtures initiated by the initial temerature gradient via the Zel'dovich's gradient mechanism [27] has been investigated by Liberman et al. [29, 30] using a detailed chemical kinetic models. ρ, kg/m (a) (b) 1.3 T, K , atm x, cm 1 Figure 10. 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 0 900s, 50s. 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). Figure 9(a, b) shows the comuted time evolution of the gaseous temerature (9a) and the articles density rofile (9b) during the reheating when the maximum temerature raised u to the crossover value and the final temerature gradient is formed at t 0 calculations were erformed for the initial stewise density of articles of radius r 900 s. The 1 m with the maximum concentration of the articles within the gas-articles layer N cm. The scale of the temerature gradient in Fig. 9, which is formed near the left boundary of the article-gas cloud is (T* T 0) / dt / dx 1cm, which is close to the 26

27 value of the radiation absortion length L1/ r N 1.2cm. According to the classification 2 of combustion regimes [29, 30] initiated by the initial temerature gradient in hydrogen oxygen at 1 atm, this temerature gradient can ignite a deflagration. The time evolution of the gaseous temerature rofiles resented in the middle frame of Fig. 10 shows the develoment of the sontaneous reaction wave for the same conditions as in Fig. 9 and the formation of a deflagration wave. The dashed line in the uer frame of Fig. 10 shows the initial rofile of the article concentration while the solid line shows the articles number density rofile formed at the instant t s just rior to the ignition, when the temerature gradient with the maximum temerature T* 1050 К is formed (the first temerature rofile in the middle frame). The rofiles in the bottom frame of Fig. 10 indicate a small variation of ressure during the formation of deflagration wave. ρ, kg/m (a) (b) 1.8 T, K , atm x, cm 1 Figure 11. 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). For a article-gas cloud with a diffuse instead of the stewise left interface a smoother temerature gradient is formed during the radiation reheating. In the latter case the radiation absortion length varies along the diffusive article-cloud interface resulting in the formation 27

28 of the temerature rofile with a smoother temerature gradient caable to initiate either fast deflagration or detonation. Examles of the article-gas cloud with diffuse interface and the comuted temerature rofiles caused by the radiative reheating are shown in Fig. 11 and Fig. 12. The uer frame in Fig. 11 shows the initial (dashed line) number density distribution of the articles, which dros linearly on the scale 1cm from its maximum value N cm. The diffuse interface of the article-cloud is smeared during the radiation reheating due to the exansion of the article-gas cloud. The temerature gradient with maximum temerature T* 1050 К is formed at the instant t s shown by solid line in Fig. 11. The middle frame in Fig. 11 shows the comuted time evolution of the gaseous temerature rofiles after t s. 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 article-cloud boundary. The weak shock waves outrunning the deflagration are seen in the bottom frame of Fig. 11. According to [29, 30] 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 which can initiate a detonation is (T* T 0) / T 20cm. This is ossible for the article-cloud shown in Fig. 12 where on the left diffuse interface of the articles number density dros linearly on the scale of 10cm from its maximum value N cm. The uer frame in Fig. 12 shows the initial articles number density rofile (dashed line) and the formed articles number density due to the exansion during the radiation heating at the instant t s rior to the ignition time (solid line). Time sequences of the gaseous temerature rofiles during the detonation formation in the vicinity of the diffusive interface is deicted in the middle frame in Fig

29 ρ, kg/m 3 T, K 0.1 (a) (b) 20 10, atm x, cm Figure 12. Temoral evolution of the gaseous temerature rofiles (middle frame) and ressure rofiles (bottom frame) during the detonation formation in the vicinity of the gas-articles 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). One can observe the develoment of sontaneous reaction wave, which then is couling with the shock wave and finally due to the ositive feedback between the reaction and ressure ulses the detonation wave is formed. Time evolution of the ressure rofiles, corresonding to the formation of the shock wave and its couling with the sontaneous reaction wave, the enhancement and formation of a strong shock corresonding to the transition to detonation is shown in the bottom frame of Fig Conclusions The resent study has been erformed in order to examine the role of the thermal radiation emitted from the flame roagating in a article-gas cloud mixture and to examine the ignition of different combustion modes due to the thermal radiation reheating of the unburned art of the cloud ahead of the original flame. The gaseous hase is assumed to be transarent for the thermal radiation, while the susended inert articles ahead of the flame 29