Modeling flame propagation of micron-sized iron dust particles in media with spatially discrete sources

Modeling flame propagation of micron-sized iron dust particles in media with spatially discrete sources

Fire Safety Journal 69 (2014) 111–116 Contents lists available at ScienceDirect Fire Safety Journal journal homepage: www.elsevier.com/locate/firesa...

1MB Sizes 0 Downloads 34 Views

Fire Safety Journal 69 (2014) 111–116

Contents lists available at ScienceDirect

Fire Safety Journal journal homepage: www.elsevier.com/locate/firesaf

Modeling flame propagation of micron-sized iron dust particles in media with spatially discrete sources Mehdi Bidabadi a, Mohammadhadi Hajilou a, Alireza Khoeini Poorfar a,n, Sina Hassanpour Yosefi a, Saeedreza Zadsirjan b a School of Mechanical Engineering, Department of Energy Conversion, Combustion Research Laboratory, Iran University of Science and Technology (IUST), Tehran, Iran b School of Aerospace Engineering, Amirkabir University of Technology, Tehran, Iran

art ic l e i nf o

a b s t r a c t

Article history: Received 30 June 2013 Received in revised form 22 July 2014 Accepted 16 August 2014

In this study, an attempt has been made to investigate the flame propagation of micron-sized iron dust particles numerically in media with spatially discrete sources. A thermal model based on a diffusioncontrolled regime of iron dust particles is generated to estimate the aspects of flame propagation in heterogeneous media. Flame propagation speed in various dust concentrations has been studied. Also in this case, obtained results for different particle diameters have been compared. Furthermore, flame speed as a function of particle size is investigated and a comparison between rich and stoichiometric dust concentrations has been made. Minimum ignition energy as a function of dust concentration for different particle sizes has been studied. The obtained results show a reasonable compatibility with the existing experimental data. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Flame propagation Iron dust cloud Heterogeneous combustion Thermal model Minimum ignition energy Discrete heat sources

1. Introduction Dust explosion has been a recognized threat to humans and industries for the last 150 years [1–6]. The occurrence of three fatal combustible dust explosions within one year in 2003 prompted the U.S. Chemical Safety and Hazard Investigation Board (CSB) to commence a broader study on the extent, nature and prevention of combustible dust fire and explosion hazards. The board presented its preliminary findings after reviewing the combustible dust incidents of the past 25 years [7]. Thus, a precise knowledge of dust explosions' hazards are essential to estimate the consequences of a dust explosion. Eckhoff [3] has introduced two types of dust particle flames. The first, the Nusselt type, is controlled by diffusion of oxygen to the surface of individual, solid particles, where the heterogeneous chemical reaction takes place. In the second type or in case of volatile flames, the rates of gasification, pyrolysis and devolatilization are the controlling processes and the chemical reaction takes place mainly in the homogeneous gas phase. Combustion of metallic materials is a challenging scientific subject that also has significant practical applications. Since this phenomenon is a highly exothermic chemical reaction,their use in n

Corresponding author. Tel.: þ 98 21 77 240 197; fax: þ 98 21 77 240 488. E-mail address: [email protected] (A.K. Poorfar).

http://dx.doi.org/10.1016/j.firesaf.2014.08.002 0379-7112/& 2014 Elsevier Ltd. All rights reserved.

many industrial operations is significant and efficient. Combustion of iron dust cloud is experimentally investigated in some works. Iron is regarded as a non-volatile metallic fuel and the oxidation process takes place as a heterogeneous surface reaction. The major characteristic feature of iron combustion is that iron burns heterogeneously in air. It means that the combustion reaction occurs at the surface of iron particles and no flame is observed in the gaseous oxidizer phase [8]. Mafi [9] presented a noble and comprehensive model for the combustion of iron particles. In 2012, Bidabadi and Mafi [10] proposed an analytical model of combustion of a single iron particle burning in the gaseous oxidizing medium. In this work, it has been demonstrated that once the particle surface is ignited, the combustion process is started and particle begins to burn. In this stage, the particle temperature rises from the ignition temperature to the final maximum temperature at the burnout time. Moreover, Bidabadi and Mafi [8] investigated time variation of combustion temperature and burning time of a single iron particle. Recently, the fundamental aspects of premixed flame propagation in micro-iron dust particles were studied by Broumand and Bidabadi [11]. Their analysis allows for the investigation of the effects of particle size and dust concentration on the combustion characteristics of iron dust clouds. The aforementioned study explained that for micron-sized and larger iron particles, the combustion time is proportional to d2.

112

M. Bidabadi et al. / Fire Safety Journal 69 (2014) 111–116

Furthermore, Bidabadi and Mafi [12] presented a mathematical model for temperature profile and flame speed across the combustion zones propagating through iron dust clouds. They demonstrated the obtained gas temperature distribution in different flame zones in the channel and also flame speed changes in terms of particles' radius, equivalence ratio, and channel width in both lean and rich mixtures. Tang et al. [13] experimentally performed the so-called argon/ helium test to determine the modes of particle combustion in iron dust flames. Also, Tang et al. [14] experimentally studied the quenching and laminar flames propagating in a fuel-rich suspension of iron dust in air in a reduced-gravity environment provided by a parabolic flight aircraft. In addition, Sun et al. [15] experimentally examined the structure of flames propagating through metal particle clouds and the behavior of particles near flames. Propagating diffusion fronts in reactive, heterogeneous media consisting of two spatially separated phases are common in many fields such as chemical kinetics, combustion,and biology, [16]. Goroshin et al. [17] studied the effects of the discrete nature of heat sources on flame propagation in particulate suspensions. They have illustrated this phenomenon by comparing flame speeds calculated both from continuous and discrete models in lean aluminum and zirconium particle–gas suspensions. They have reported in their work that lower flame speeds and a weaker dependence of the speed on oxygen concentration are predicted by the discrete flame model. Tang et al. [18] investigated the effect of discreteness on heterogeneous flames and propagation limits in regular and random particles. Also Mendez et al. [19] studied the speed of reaction-diffusion fronts in spatially heterogeneous media. Nomura and Tanaka [20] introduced a simple dust cloud model for the theoretical predictions of dust explosions. Their model could envisage the combustion of dust particles dispersed uniformly in air, where it has been presumed that flame propagation begins with the ignition of the central particle and develops radially with spherical symmetry. However, in their research, they considered flame propagation in one-dimensional direction such as in a tube. In the present model, flame propagation resulting from the combustion of solid fuel—iron— is obtained in three dimensions based on the superposition's rule in the Cartesian coordinates compared to the polar coordinate used in their paper. Most importantly, their simulation was not for a specific solid fuel and was presented in a generic form. In the present study, the effects of particle size and dust concentration on the flame propagation of micron-sized iron dust particles in media with spatially discrete sources are studied numerically. A 3-D thermal model in Cartesian coordinates based on the viewpoint of discrete heat sources has been utilized. As mentioned by Bidabadi and Mafi [8–10], in the combustion of a micron-sized iron dust cloud, the diffusion-controlled regime is taken into consideration. Flame propagation speed as a function of dust concentration has been investigated. In addition, variations of flame speed with particle diameter for a fuel-rich iron–air suspension have been shown. Also the minimum ignition energy as a function of dust concentration in different particle diameters has been studied. It can be seen that the results of the presented model are in good agreement with the experimental data.

2. Thermal model The mechanism of the combustion of dust clouds is a very complex process. The difficulty in their study is due to various processes such as: heating, evaporation, mixing with oxidizer, ignition, burning and quenching of particles in the dust cloud. In the study of flame propagation in dust clouds, particle size and dust concentration play very important roles. Also, the interaction

between the particles in the mixture always makes the dust combustion an unstable process. Heat transfer is the dominant phenomenon in the process of flame propagation in dust clouds. A thermal model based on heterogeneous combustion in threedimensional space which relies on the following assumptions has been generated: 1. The particle is spherical and the flame diameter remains constant and is equal to the particle diameter. 2. The thermal properties of the medium and particles are independent of temperature. 3. There is an equal and constant space between the particles distributed in the dust cloud. 4. A constant rate of energy release is considered during the combustion of a single particle. 5. The radiation heat transfer in the iron dust cloud is neglected and only conductive heat transfer has been considered.

A relation for burning time of a single micron-sized iron particle is presented by Glassman and Yetter [21]. As mentioned earlier, the mode of the combustion of micron-sized iron particles is diffusion-controlled. Also, iron undergoes a heterogeneous combustion in oxygen. The burning time of a single iron particle in a diffusion-controlled regime,t bdif f , can be obtained from the following relation t bdif f ¼

ρp d2p;0

8ρ1 DO2 ;1 lnð1 þ νY O2 ;1 Þ

ð1Þ

where ρp is the iron particle density, dp;0 is the initial diameter of the iron particle, ρ1 is the ambient gas density, DO2 ;1 is the mass diffusivity of oxygen into the air, ν is the mass stoichiometric index of combustion of iron, and Y O2 ;1 is the mass fraction of oxygen in the ambient gas far from the particle surface. When the ignition system provides the minimum amount of energy to the dust cloud, the temperature of some particles is increased to the ignition temperature. As these particles start to burn, they act as a heat source in the dust cloud system and cause the temperature of the surrounding region to rise. The temperature rise in the other particles is calculated as the sum of thermal effects from the burned and burning particles. In case a highenough temperature is provided, the combustion process will proceed to the other particles as shown in Fig. 1. The temperature increase of particles in the preheated zone as a result of only the conduction heat transfer mechanism is expressed based on the superposition principle. To model the single-particle combustion and the time–place temperature distribution of its domain, the energy equation in spherical coordinates is used   1 ∂ 2 ∂T a ðr; tÞ 1 ∂T a ðr; tÞ r ¼ : ∂r α ∂t r 2 ∂r

ð2Þ

Here T a ðr; tÞis Tðr; tÞ T 1 , and T 1 is the ambient temperature. The boundary and initial conditions of the above equation are also shown below 9 8 k A ∂ T ðr; tÞ ¼ q_  Heavisideðτ  tÞ; @r ¼ r p > > > > = < p ∂r a : T a ð1; tÞ ¼ 0 > > > > ; : T a ðr; 0Þ ¼ 0 The heaviside function is defined as ( HðnÞ ¼

0;

@n o 0

1;

@n Z 0

)

M. Bidabadi et al. / Fire Safety Journal 69 (2014) 111–116

113

Table 1 Thermophysical properties of iron particle and air used in calculations. Property

Value

Unit

Ref.

ρp Cp Kp T f lame T ignition T1 ρ1 DO2 ;1 ν Y O2 ;1

7860 447 80.2 1990 850 300 1.1614 22.5  10  6 0.2846 0.233

kg=m3 J=kg K W=m K K K K kg=m3 m2 =s – –

[25] [25] [25] [14] [8,14] Assumed [25] [25] [10] [10]

3. Results and discussion Fig. 1. The spatial distribution of particles in dust cloud: layer n  1 (burned particles), layer n (burning particles), and layer nþ 1 (preheating particles) by Bidabadi et al. [23].

Here q_ is the rate of heat release of a single particle from its surface during the burning time which is defined as below [22] q_ ¼ Akp ðT f T 1 Þr p 1 :

ð3Þ

The space–time temperature distribution of particles has been obtained through the whole domain by Bidabadi et al. [23] as stated below: 2 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi13 rp ðr  r p Þ2 A ðr  r p Þ2 A5  Heavisideðt  τÞerf [email protected] T a ðr; tÞ ¼ ðT f  T 1 Þ 4erf [email protected] r 4αt 4αðt  τÞ

ð4Þ T s ¼ ∑ ∑ ∑ T a ði; j; kÞðr i;j;k ; t ig;i Þ: i

j

ð5Þ

k

Ta is the space–time distribution of temperature around a single burning particle and beyond, and Ts is the total effect of burning and burned particles which is indicative of the temperature of medium fluid around a particle in the preheated zone. T1 ¼300 K and Tf ¼ 1990 K [14] are the considered values. Table 1 shows other properties of iron particles and properties of air. The space between the target particle and each particle placed at i, j, k is presented by r i;j;k ¼ L

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 i2 þ j2 þ k

ð6Þ

where L is the space between two adjacent layers which is defined by the following equation by Bidabadi et al. [23]: L ¼ ðπ dp ρp =6C d Þ1=3 3

ð7Þ

where ρp is the particle density, Cd is the dust cloud concentration and dp is the particle diameter. The flame propagation speed is defined as the ratio of the space between two adjacent layers to the difference of their ignition times [24]. As mentioned earlier, since micron-sized particles are being dealt with, the formulation presented by Glassman and Yetter [21] is utilized here. Fig. 1 shows the spatial distribution of particles in the dust cloud. The ignition time of a single particle in a layer can be the representative of the ignition time of the mentioned layer. The particle is assumed to be positioned at the origin of the local coordinate system [23].

A computer code has been generated to calculate the burning speed of each layer after the release of the energy from the ignition system. As soon as the minimum amount of energy to the dust cloud is provided, the temperature of the particles is increased to the ignition temperature and combustion takes place. The logic and algorithm of the program are shown in Fig. 2. With regard to the criterion used for igniting the next layer of particles, it should be said that following the release of energy, at first, the temperature of the first layer caused by the ignition system at the considered location is calculated. When the temperature of the first layer of particles reaches the ignition temperature, it is recorded as the ignition time of the first layer;and the calculations continue to find the ignition times of the other layers. Beyond the first layer (n 4 1), the preheating of layers is influenced by the burning of the preceding layers in addition to the ignition system. Thus, when a layer's temperature reaches the ignition temperature, the relevant time is recorded as the ignition time of that layer. Flame propagation speed is determined by dividing the distance between two adjacent layers L by the difference between ignition times of these two layers. If we assume that conduction is the only mechanism of flame propagation, the rise of a particle's temperature will be a function of thermal diffusivity α. The higher this value, the sooner the adjacent layers reach the ignition temperature and the flame propagation speed increases. The results have been compared with the experimental works of Sun et al. [15] and Tang et al. [14]. In the experiments of Sun et al. [15], the diameters of most iron particles prior to the experiment were distributed from 1 to 5 μm based on the data supplied by a manufacturing company, and also measured using a scanning electron microscope. In the experiments performed by Tang et al. [14], a fuel-rich suspension of iron dust in air in a reduced-gravity environment provided by a parabolic flight aircraft was investigated. The combustion process is represented by a one-step irreversible reaction of the form vF ½F þ vO2 ½O2  -vproduct ½P, where F, O2 and P denote fuel, oxygen and product, respectively, and the quantities vF , vO2 , and vproduct denote their respective stoichiometric coefficients. Fig. 3 illustrates the flame propagation speed with the dust concentration for an iron particle diameter of 5 μm. The flame velocity predicted by the model is the laminar burning velocity of a planar front because of the fact that the laminar burning velocity is the speed at which a plane flame front will propagate into a stationary, quiescent flammable mixture of infinite extent [26]. According to Fig. 3, with a rise in the value of dust concentration from nearly 0.4 kg/m3 to 1 kg/m3, values of the flame propagation speed increase more rapidly. The obtained results form the presented model show reasonable agreement with the experimental data provided by Sun et al. [15].

114

M. Bidabadi et al. / Fire Safety Journal 69 (2014) 111–116

Fig. 2. Flowchart for estimation of the ignition time of iron particles.

Flame propagation speed as a function of dust concentration for different particle diameters has been shown in Fig. 4. As it is apparent in this figure, for smaller particle sizes, with an increase in dust concentration, flame speed tends to increase more rapidly and it has larger values in comparison with larger particle diameters. Also, with a rise in dust concentration and for a certain value of that, the difference between values of flame speed in different particle sizes tends to grow. Furthermore, one can conclude that as the particle diameter increases, the rate of changes in the values of flame propagation speed decelerates and it can be seen that for

larger particle sizes, the difference between their values of flame speed becomes smaller. The propagation speed in terms of particle diameter is studied in Fig. 5. It shows that the results of the presented model and the experimental data provided by Tang et al. [14] are in good agreement. It should be noted that the experimental data is provided for a fuel-rich iron–air suspension with equivalence ratioϕ ¼ 1:43  1:90. In the presented model in Fig. 5, the equivalence ratio is considered to be 1.63. Also, the stoichiometric dust concentration is taken to be 0.725 kg/m3 which is presented by Goroshin et al. [27]. Fig. 5 shows that for iron particle diameters smaller than 10 μm, the flame speed rises dramatically when the particle diameter decreases. But after a certain value, the rate of changes begins to slow down and as is clear, for diameters beyond 25 μm, the value of flame propagation speed tends to have very small changes. Fig. 6 illustrates the comparison of flame propagation speed as a function of particle size between rich and stoichiometric dust concentrations. For a considered particle size, the flame front speed at stoichiometric dust concentrations has lower value compared to the rich dust concentration. For smaller particle diameters, the difference between the values of flame speed of the two aforementioned cases is fairly considerable, but for larger diameters, there seems to be a convergence between these two cases. According to Eqs. (1), (6) and (7), it is noteworthy to point out that when dealing with small particles in a certain layer and in the case of rich dust concentration, there are more particles available and the distances between the particles inside the layer are less than in the case of stoichiometric dust concentration; therefore, the heat release from one layer to another is more, resulting in higher values of flame propagation speed. However, in case of larger particle sizes, the effect of mass concentration becomes less important, since in a certain layer and in the two aforesaid cases, it takes more time to ignite a layer with larger particle sizes and more time for the energy wave to be transferred from one layer to another, consequently lower values for burning velocity can be observed. The minimum ignition energy (MIE) for a layer in the presumed model as a function of dust concentration for different particle sizes is demonstrated in Fig. 7. This figure represents an important feature in fire safety science. As can be observed, with a rise in dust concentration, minimum ignition heat decreases. The amount of energy after a certain value of dust concentration reaches a point where no significant changes in the minimum ignition heat can be seen. This is due to the saturation condition that would be generated in the medium. Therefore, when the dust concentration of a given particle size reaches a specific value, the MIE maintains the same level and no considerable changes will occur afterwards. Moreover, one can conclude that for rich dust concentrations, the minimum amount of heat needed to ignite a certain layer in the presented model is lower than that in the case of stoichiometric and lean dust concentrations. It means that at higher dust concentrations, since there is more fuel available, less energy is required to ignite a layer. Also, it can be observed that as the particle size decreases, the value of minimum ignition energy reduces, i.e. the larger the particle, the more is the energy needed for ignition.

4. Conclusions In this paper, a thermal model has been generated so as to simulate the flame propagation of micron-sized iron dust particles in media with spatially discrete sources. A computer code has been made in order to study the effects of dust concentration and

M. Bidabadi et al. / Fire Safety Journal 69 (2014) 111–116

Fig. 3. Comparison of experimental measurements of flame propagation speed with the obtained results from the presented model.

Fig. 4. Flame propagation speed as a function of dust concentration for different particle diameters.

Fig. 5. Flame speed as a function of particle size for a rich mixture, with φ ¼ 1:64.

Fig. 6. Flame speed in terms of particle size for stoichiometric and rich dust concentrations ðφ ¼ 1:64Þ.

115

116

M. Bidabadi et al. / Fire Safety Journal 69 (2014) 111–116

Fig. 7. Minimum ignition energy as a function of dust concentration in different particle diameters.

particle size on flame propagation speed. This study is aimed for better prediction of different aspects of dust explosions and increase in safety systems. Flame propagation speed as a function of dust concentration is investigated. It is observed that with the increase of dust concentration, flame speed tends to rise and its rate of changes for smaller particles is higher than that for larger sizes. Furthermore, flame propagation speed in terms of particle size for fuel-rich iron–air suspension is studied. It is shown that as the particle size increases, the value of flame speed decreases and after a certain diameter, the rate of flame speeds' changes tends to decelerate. Also, it is demonstrated that in case of stoichiometric dust concentration, i.e. ϕ ¼1, flame speed has lower values with the increase of particle size in comparison with the rich dust concentration. Minimum ignition energy as a function of dust concentration for different particle diameters is illustrated. It is observed that minimum ignition energy tends to have lower values as the particle size decreases. Moreover, it can be concluded that for rich dust concentrations, the minimum amount of energy needed to ignite a certain layer in the presented model is lower than that in case of stoichiometric and lean dust concentrations. Results from the presented model are in good agreement with the experimental data. References [1] G.H. Pedersen, R.K. Eckhoff, Initiation of grain dust explosions by heat generated during single impact between solid bodies, Fire Saf. J. 12 (1987) 153–164. [2] B.S. Varshney, S. Kumar, T.P. Sharma, Studies on the burning behaviour of metal powder fires and their extinguishment: part 1—Mg, Al, Al–Mg alloy powder fires on sand bed, Fire Saf. J. 16 (1990) 93–117. [3] R.K. Eckhoff, Dust Explosion in the Process Industries, second ed., Butterworth Heinemann, Oxford, 1997. [4] A.E. Dahoe, K. Hanjalic, B. Scarlett, Determination of the laminar burning velocity and the Markstein length of powder–air flames, Powder Technol. 122 (2002) 222–238. [5] G. Joseph, C.S.B. Hazard investigation team, combustible dusts: a serious industrial hazard, J. Hazard. Mater. 142 (2007) 589–591. [6] W. Gao, R. Dobashi, T. Mogi, J. Sun, X. Shen, Effects of particle characteristics on flame propagation behavior during organic dust explosions in a half-closed chamber, J. Loss Prev. Process Ind. 25 (2012) 993–999. [7] G. Joseph, A. Blair, J. Barab, M. Kaszniak, C. MacKenzie, Combustible dusts: a serious industrial hazard, J. Hazard. Mater. 142 (2007) 589–591.

[8] M. Bidabadi, M. Mafi, Time variation of combustion temperature and burning time of a single iron particle, Int. J. Therm. Sci. 65 (2013) 136–147. [9] M. Mafi, Modeling of Combustion of Iron Particles (Ph.D. thesis), Iran University of Science and Technology, 2012. [10] M. Bidabadi, M. Mafi, Analytical modeling of combustion of a single iron particle burning in the gaseous oxidizing medium, Proc. Inst. Mech. Eng. Part C 227(5) (2012) 1006–1021. [11] M. Broumand, M. Bidabadi, Modeling combustion of micron-sized iron dust particles during flame propagation in a vertical duct, Fire Saf. J. 59 (2013) 88–93. [12] M. Bidabadi, M. Mafi, Analytical investigation of temperature distribution and flame speed across the combustion zones propagating through an iron dust cloud utilizing a three-dimensional mathematical modeling, Korean J. Chem. Eng. 29 (8) (2012) 1025-1037. [13] F.D. Tang, S. Goroshin, A.J. Higgins, Modes of particle combustion in iron dust flames, Proc. Combust. Inst. 33 (2011) 1975–1982. [14] F.D. Tang, S. Goroshin, A.J. Higgins, J. Lee, Flame propagation and quenching in iron dust clouds, Proc. Combust. Inst. 32 (2009) 1905–1912. [15] J.H. Sun, R. Dobashi, T. Hirano, Structure of flames propagating through particle clouds and behavior of particles, in: Proceedings of the 27th Symposium (International) on Combustion, The Combustion Institute, USA, 1992, pp. 2405–2411. [16] J. Xin, Front propagation in heterogeneous media, SIAM Rev. 42 (2000) 161–230. [17] S. Goroshin, J.H.S. Lee, Y. Shoshin, Effect of the discrete nature of heat sources on flame propagation in particulate suspensions, in: Proceedings of the 27th Symposium (International) on Combustion, The Combustion Institute, USA, 1992, pp. 743–749. [18] F.D. Tang, A.J. Higgins, S. Goroshin, Effect of discreteness on heterogeneous flames: propagation limits in regular and random particle arrays, Combust. Theory Model. 13 (2009) 319–341. [19] V. Mendez, J. Fort, H.G. Rotstein, S. Fedotov, Speed of reaction-diffusion fronts in spatially heterogeneous media, Phys. Rev. E 68 (2003) 041105. [20] S. Nomura, T. Tanaka, Theoretical analysis of dust explosions, Powder Technol. 71 (1992) 189–196. [21] I. Glassman, R.A. Yetter, Combustion, 4th ed., Elsevier, Burlington, 2008. [22] H. Hanai, H. Kobayashi, T. Niioka, A numerical study of pulsating flame propagation in mixtures of gas and particles, Proc. Combust. Inst. 28 (2000) 815–822. [23] M. Bidabadi, S. Zadsirjan, S.A. Mostafavi, Propagation and extinction of dust flames in narrow channels, J. Loss Prev. Process Ind. 26 (2013) 172–176. [24] S. Goroshin, J.H.S. Lee, Y. Shoshin, Effect of the discrete nature of heat sources on flame propagation in particulate suspensions, Proc. Combust. Inst. 27 (1998) 743–749. [25] F.P. Incropera, D.P. DeWitt, Fundamentals of Heat and Mass Transfer, John Wiley & Sons, New York, 1996. [26] D. Drysdale, An Introduction to Fire Dynamics, second ed., Wiley, Chichester, England, 1998. [27] S. Goroshin, J. Mamen, L. Lee, K. Sacksteder, Ground-based and microgravity study of flame quenching distance in metal dust suspensions, in: Proceedings of ICDERS (2005).