Journal of Loss Prevention in the Process Industries 43 (2016) 138e146 Contents lists available at ScienceDirect Journal of Loss Prevention in the Process Industries journal homepage: www.elsevier.com/locate/jlp Modeling quenching distance and ﬂame propagation speed through an iron dust cloud with spatially random distribution of particles Mehdi Bidabadi a, Mohammad Mohebbi a, Alireza Khoeini Poorfar a, *, Simone Hochgreb b, Cheng-Xian Lin c, Saeed Amrollahy Biouki a, Mohammadhadi Hajilou a a b c Combustion Research Laboratory, School of Mechanical Engineering, Iran University of Science and Technology (IUST), Narmak, 16846-13114, Tehran, Iran Department of Engineering, University of Cambridge, Trumpington Street, Cambridge, CB2 1PZ, United Kingdom Department of Mechanical and Materials Engineering, 10555 W. Flagler Street, EC 3445, Florida International University, Miami, FL 33174, USA a r t i c l e i n f o a b s t r a c t Article history: Received 19 October 2015 Received in revised form 4 March 2016 Accepted 18 May 2016 Available online 20 May 2016 In this research combustion of iron dust particles in a medium with spatially discrete sources distributed in a random way has been studied using a numerical approach. A new thermal model is generated to estimate ﬂame propagation speed and quenching distance in a quiescent reaction medium. The ﬂame propagation speed is studied as a function of iron dust concentration and particle diameter. The predicted propagation speeds as a function of these parameters are shown to agree well with experimental measurements. In addition, the minimum ignition energy has also been investigated as a function of equivalence ratio and particle diameter. The quenching distance has been studied as a function of particle diameter and validated by the experiment. Considering random distribution of particles, the obtained results provide more realistic and reasonable predictions of the combustion physics compared to the results of the uniform distribution of particles. © 2016 Elsevier Ltd. All rights reserved. Keywords: Flame propagating speed Quenching distance Iron Random particle distribution Heterogeneous combustion 1. Introduction Dust explosions are a recognized threat to humans and property for the last 150 years (Eckhoff, 2003). Many combustible dusts allow a ﬂame to propagate through the fuel particles if dispersed as a cloud in air and ignited, in a manner similar to (though not identical to) the propagation of ﬂames in premixed fuel oxidant gases (Proust, 2006). Such dusts include common foodstuffs like sugar ﬂour, cocoa, synthetic materials such as plastics, chemicals and pharmaceuticals, metals such as aluminum and magnesium, and traditional fuels such as coal and wood (Abbasi and Abbasi, 2007). One of the earliest recorded and the most serious of the accidents triggered by dust explosion occurred at Leiden, the Netherlands, on 12 January 1807 (Abbasi and Abbasi, 2007). Similar disasters induced by metal dust occurred in 2011 in which three iron dust ﬂash ﬁres occurred over a period of ﬁve months and killed ﬁve workers at the Hoeganaes Corp. facility in Gallatin, Tennessee. These three events are examples of hazard identiﬁcation and * Corresponding author. E-mail address: [email protected] (A.K. Poorfar). http://dx.doi.org/10.1016/j.jlp.2016.05.018 0950-4230/© 2016 Elsevier Ltd. All rights reserved. general drift in the management of safety barriers. Most of these accidents occurred during maintenance activities. They could have been prevented if the risk status of the system had been known. A comprehensive review of dust explosions cases and causes was presented by Abbasi and Abbasi (2007). Accidental dust explosions are highly undesirable in any plant, yet an explosion hazard always exists wherever dusts are produced, stored or processed, whenever a threshold quantity of powdered ﬂammable material is present in the air. With the advancement of powder technologies for materials processing, and the increase of powder handling processes, hazard assessment and the establishment of preventive methods for dust explosions have become more important from the view point of industrial loss prevention. Therefore, a correct understanding of the combustion mechanism is necessary in order to minimize the probability of occurrence of such events in the future. The combustion of metallic particles used in a variety of industrial applications is a highly exothermic event. Basic mechanisms of combustion of such two-phase mixtures are not well understood because of two major problems facing the research in combustion of dust particles (Hanai et al., 2000). The ﬁrst is the complex nature of processes involving the physical and chemical properties of the fuel. The second relates to the size, shape and spatial distribution of the particles. Iron is regarded as a non- M. Bidabadi et al. / Journal of Loss Prevention in the Process Industries 43 (2016) 138e146 volatile metallic fuel, and the oxidation process takes place as a heterogeneous surface reaction. The major characteristic feature of iron combustion is that it burns heterogeneously in air: the oxidation reaction occurs at the surface of iron particle, and no ﬂame is observed in the gaseous oxidizer phase. Iron particles do not evaporate during the combustion process, and the combustion product, iron oxide, remains in the condensed phase. Sun et al. (1990), Sun et al. (1998) experimentally examined the combustion zone propagating through an iron particle cloud and the process of iron particle combustion. They have demonstrated that the burning time of an iron particle is proportional to the nondimensional diameter when its diameter is small; as the iron particle diameter becomes larger, the burning time increases with a power of the non-dimensional diameter. Sun et al. (2006) experimentally studied the concentration and velocity proﬁles of iron particles across up- and downward ﬂame propagation in the vicinity of the combustion zone. Bidabadi et al. (2010) proposed a mathematical model based on utilizing the Lagrangian equation of motion and the effective thermophoretic, gravitational and buoyancy forces acting on the particles in order to represent the velocity proﬁle of the micro-iron dust particles. Bidabadi and Maﬁ (2012, 2013), theoretically investigated the evolution in combustion temperature and burning time of a single iron particle in air, and proposed an analytical model that agreed with the experimental ﬁndings. Beach et al. (2007) investigated the combustion of iron nanoparticles as a potential alternative fuel, in which the burning time of iron particles was calculated for both spherical and disc shaped iron particles using the heat balance and chemical kinetic theories. There are two general approaches to model dust combustion: the continuous or macroscopic approach and the discrete or microscopic approach. From the microscopic viewpoint, the propagation of the ﬂame front is inherently unsteady, as it migrates from particle to particle (Mukasyan et al., 1996; Rogachev et al., 1994). The spatial distribution of particles strongly inﬂuences the ﬂame propagation (Tang et al., 2009a,b). In contrast, in the traditional continuous or macroscopic approach to model particles in a gaseous suspension, the discrete nature of the heat sources is averaged to yield a mean propagation speed. Continuous models of dust combustion cannot usually capture the lean ﬂammability limit concentration or the threshold particle diameter, dust concentration and heat release. Indeed, it can be demonstrated that the ﬂammability limit depends on spatial distribution of particles (Rashkovskiy et al., 2010). A new thermal model has been generated to estimate the ﬂame propagation speed for micron-sized iron particles under various dust concentrations and sizes in air. This discrete heat source method provides a dust combustion model, from ignition process to the ﬁnal state, including steady ﬂame propagation, ﬂame quenching and explosion. In the current research paper, the combustion of iron particles distributed uniformly in space is studied. As a further improvement to the model, a random distribution of particles is used in the governing equations to predict the ﬂame features, such as; ﬂame propagation speed, minimum ignition energy, and quenching distance. The model starts by considering single particle combustion to obtain a space-time temperature distribution. The ensemble reacting front in the suspended dust combustion is then considered using the superposition principle to include the effects of surrounding particles. All the burned and burning particles are considered as heat sources, and the channel walls are assumed to behave as heat sinks. Finally, the ﬂame propagation speed and quenching distance in the narrow channel are determined. 139 2. Discrete thermal model The combustion of dust clouds is a complex process, involving particle heating, evaporation, intermixing with oxidizer, ignition, burning and quenching of particles. Particle size and dust concentration and distribution of the particles clearly play very important roles. Reaction-diffusion phenomena have been modeled extensively in homogeneous media where the reactants are distributed continuously in space. In a homogeneous system, the heat source term does not depend on spatial coordinates and a solution can be obtained by solving a set of scalar, ordinary differential equations. However, in heterogeneous media, the reactants form a separate phase within a diffusive medium causing the reaction to occur locally around the boundaries or inside the sources. Unlike homogeneous media, the reaction is localized at the position of the heterogeneities and the heat source term depends explicitly on the coordinates of the reacting sources in the domain (Tang et al., 2009a,b). 2.1. Uniform distribution In the uniform distribution approach, dust particles are assumed to be uniformly dispersed in air as shown schematically in Fig. 1. The ignition system provides the minimum necessary energy to the dust cloud, raising the temperature of some particles to the ignition temperature. As these particles start to burn, they act as a heat source and cause the temperature of the surrounding region to rise. The temperature rise in the other particles is calculated as the linear superposition of thermal effects from the burned and burning particles. Particle ignition is assumed to take place once the particle reaches a minimum temperature Tig ¼ 850 K (Tang et al., 2009a,b), and the combustion process propagates to other particles. The mixture is assumed to be stagnant, so that the temperature increase of particles in the preheated zone is assumed to be an exclusive result of conduction heat transfer through the gaseous medium (Bidabadi et al., 2013). The thermal model generated in this study for the uniform particle distribution model is based on heterogeneous combustion in three-dimensions. The model relies on the following assumptions: 1. Each particle is spherical in shape, and the associated ﬂame diameter remains constant and equal to the particle diameter (Sun et al., 1990). 2. No oxide layer surrounds the iron particles. 3. The thermal properties of the medium and particles are independent of temperature. Fig. 1. Spatial distribution of particle in a uniform dust cloud: layer n 1 represents (burned particles), layer n (burning particles), and layer n þ 1 (preheating particles). 140 M. Bidabadi et al. / Journal of Loss Prevention in the Process Industries 43 (2016) 138e146 4. The Biot number for the burning iron particles is negligible, so particles have a single temperature (Bidabadi and Maﬁ, 2012). 5. There is an equal and constant spacing between the particles in the uniform model. 6. The reaction of a particle proceeds at a ﬁxed rate, releasing a proportional rate of energy. 7. The micron sized iron particle is assumed to burn in a diffusioncontrolled regime (Bidabadi and Maﬁ, 2012, 2013), corresponding to inﬁnitely fast reaction rate relatively to diffusion. The combustion process is represented by a one-step irreversible reaction form as: nF F þ nO2 O2 /nP P (1) where F, O2 and P denote fuel, oxygen and product, respectively, and the quantities nF, nO2 , and nP denote the respective stoichiometric coefﬁcients. The stoichiometric chemical reaction equation between Iron and Oxygen is deﬁned as follows: 3Fe þ 2O2 /Fe3 O4 (2) The burning time of a single iron particle in a diffusioncontrolled regime (t), is explained by Glassman and Yetter (2008) and can be evaluated from the following expression: t¼ rp dp;0 2 8r∞ Do2 ;∞ ln 1 þ nYo2 ;∞ (3) where all variables are deﬁned in the nomenclature section. The heat transfer in single-particle combustion, and the temperature distribution throughout the domain is modeled as a purely conductive process in spherical coordinates, for the temperature rise Ta ¼ T T∞ relative to the ambient temperature T∞: 1 v 2 vTa ðr; tÞ 1 vTa ðr; tÞ r ¼ vr a vt r 2 vr (4) where r is the radial distance from the origin of the coordinate system, t is time, and a is the gas phase diffusivity. The boundary and initial conditions for the equation above are: kp A v Ta ðr; tÞ ¼ q_ H ðt tÞ; vr at r ¼ rp ; (6) The solution to equation (4) for temperature rise was obtained through the whole domain by Bidabadi et al. (2013) as: Ta ðr; tÞ ¼ Tf T∞ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 r rp 4erfc r 4at r 2 p sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2ﬃ3 r rp 5 Hðt tÞerfc 4aðt tÞ (8) Ta is the space-time distribution of temperature around a single burning particle and beyond, 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 and r is the radial distance, rp is particle radius, t is time, t is the burning time and Erfc is the complementary error function. The solution for any point in the domain is assumed to be given by the linear superposition of the solutions for all burning and burned particles which is presented in equation (8). The spacing between the target particle and each particle placed at i, j, k is presented by: ri;j;k ¼ L qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ i2 þ j2 þ k2 (9) where L is the spacing between two adjacent layers and determined from the mass dust concentration cd: L¼ pd3p rp !1=3 (10) 6Cd The resulting ﬂame propagation speed is deﬁned as the ratio of the space between two adjacent layers to the difference of their ignition times in the discrete domain (Goroshin et al., 1998). S ¼ L tig;nþ1 tig;n (11) The ignition time of a single particle in a layer is assumed to be representative of its layer ignition time. Every experimental apparatus in dust combustion uses an igniter. To study the effect of this component, the energy release is assumed to be sudden and it is presented by Dirac delta function in the present paper. The igniter’s distance to the nearest layer is assumed to be L and the energy release occurs in a plane parallel to layers. The 1D energy equation for the igniter presented in Cartesian coordinates is also given by the heat equation: 1 vTa;ig ðx; tÞ ¼ a vt vx2 Ta;ig ¼ T T∞ where q_ is the rate of heat release of a single particle during the burning time, and H is the Heaviside function. The rate of heat release is assumed to behave as (Hanai et al., 2000): Ta ri;j;k ; tig;i i¼1 j¼1 k¼1 (5) q_ ¼ AkP Tf T∞ rp1 n X n X n X v2 Ta;ig ðx; tÞ Ta ð∞; tÞ ¼ 0; Ta ðr; 0Þ ¼ 0; Ts ¼ The boundary conditions of this equation are: Ta;ig ðx; 0Þ ¼ 0 k v 1 T ðx; tÞ ¼ Q dðtÞ vr a;ig 2 at x ¼ 0 (13) where x is the distance of a particle in layer of n from the igniter’s origin, equal to x ¼ nL and Q is the total energy released from igniter and considered to be the minimum energy that needed to ignite ﬁrst layer and d(t) is the Dirac delta function. The solution to Eq. (12) yields: Ta;ig ðx; tÞ ¼ (7) (12) 2 Qa 1 x pﬃﬃﬃﬃﬃﬃﬃﬃ exp k 4at pat (14) Equation (14) is assumed as an initial condition for the calculation of ﬂame features, such as; ﬂame propagation speed and quenching distance. M. Bidabadi et al. / Journal of Loss Prevention in the Process Industries 43 (2016) 138e146 preheating of the layers is inﬂuenced by burning of the preceding layers, in addition to the ignition system. 2.2. Random distribution The model discussed in the previous section is limited to a system with particles distributed uniformly and statically in a reacting medium. In reality, dust particles may not be distributed regularly in space, so that a random placement with a given mean distance can be more realistic. Therefore, in order to create a reaction media with randomly distributed particles, a number of nodes Ne uniformly distributed within the medium is considered. Each empty node shows the location where a particle can be placed. The particles in the model are randomly distributed, generating a dust cloud with N particles. The domain size is constant and equal to the one in the uniform distribution model. The variable b deﬁnes the fraction of ﬁlled locations: b¼ N Ne (15) For uniform distribution of particles, b ¼ 1, and for random distribution, b < 1. In order to maintain the same overall mass concentration as in the uniform distribution model, a modiﬁed distance between two adjacent layers, Lr, is deﬁned as: Lr ¼ bpd3p rp !1=3 6Cd (16) so that the total volume is (NeL3r ) and the total particle mass is (NrpVp), and therefore the total mass concentration per unit volume is maintained. The random model employs the same governing equations used in the case of uniformly distributed particles. An algorithm illustrating how the ignition times are obtained is shown in Fig. 2. Following the energy release by the ignition system, the temperature of the ﬁrst layer at the considered location is calculated. When the temperature of the particles in the ﬁrst layer reaches the ignition temperature, it is recorded as the ignition time of the ﬁrst layer, and the calculations are continued to ﬁnd the ignition times of the other layers. Beyond the ﬁrst layer (n > 1), Fig. 2. Flowchart for calculating the ignition time of iron particles. 141 2.3. Quenching distance The analysis of ﬂame propagation or extinction in narrow channels can take into consideration the heat loss to the walls. At ﬁrst, combustion of single particle is studied to obtain a space-time temperature distribution. The generated heat diffuses to the preheat zone through conduction. Based on the superposition principle, the space-time temperature distribution of particles, as heat sources, and the heat loss to the walls, as heat sinks, are the two parameters that affect the temperature of the preheat zone. The particles ignite when their temperature reach the ignition point. The ﬂame propagates in a narrow channel with inﬁnite length and constant width. If the ﬂame quenches in the channel, the channel width is deﬁned as equal to/or less than the quenching distance. The burned and burning particles are heat sources. In this model, the particles are structurally placed within the spacing shown in Fig. 1. The spacing between two adjacent layers is calculated by Eq. (10). Because of symmetry, a single particle of a layer can be representative of the whole layer to determine the ignition time of all the particles of that layer. This particle is assumed to be positioned at the origin of the local coordinate system. The ﬂame propagates through the layers and ignites them one by one, and their temperature is represented as Tsource; p ¼ n X n X n X T ri;j;k ; tig;i (17) i¼1 j¼1 k¼1 where the ignition time of layer i is indicated as tig,i, and the local temperature deﬁnes whether the particle has ignited at the given temperature. 2.3.1. Heat sink The present model estimates the quenching distance of a speciﬁc mixture by means of two parameters: dust concentration and channel width. In this model, the walls play the role of heat sinks, as described in Fig. 3. The heat transfer to walls by the thermal conductivity of the mixture is considered to be one-dimensional and perpendicular to wall surface. A layer in the preheat zone is considered as a lumped capacitance mass with uniform temperature. The heat loss causes the reduction in the temperature of the particles, and all the particles in one layer are ignited at the same time. Walls are at temperature Tw, the lumped capacitance average temperature of layer is Tlayer and the channel width is D. The rate of heat loss to the walls of the channel is obtained from the work done by Bidabadi (1996). Fig. 3. Schematic of lumped capacitance assumption for layers and one-dimensional heat transfer to walls. The quenching distance is obtained for a parallel ﬂat plates conﬁguration. 142 q_ loss ¼ M. Bidabadi et al. / Journal of Loss Prevention in the Process Industries 43 (2016) 138e146 2kA Tlayer Tw Table 1 Thermo-physical properties of iron particles and air used in calculations. (18) D=2 The total heat loss of one layer in the preheat zone, from time zero to time tr is: Ztr Qloss ¼ q_ loss dt (19) 0 Using the lumped capacitance assumption, the temperature reduction of that layer is expressed as Tsink ¼ rQCloss . pV The temperature reduction over a time tr becomes: Tsink ¼ 4k rCp D2 Ztr Tlayer Tw dt (20) 0 2.3.2. Sink and sources superposition Because the heat equation is linear in temperature, it is again possible to calculate the total effect in the preheat zone as the algebraic sum of the temperatures of all the heat sinks and sources. Tlayer ¼ Tsource Tsink Tlayer ¼ n X n X n X T ri;j;k ; tig;i i¼1 j¼1 k¼1 (21) 4k rCp D2 Ztr Tlayer Tw dt 0 (22) The present study uses an iterative method to obtain and calculate the integral part in Eq. (22). When the temperature of the target particle of layer n þ 1 in the preheat zone reaches the ignition temperature, this layer starts to ignite and burn. The calculations assume the values of atmospheric pressure of P ¼ 101 kpa, initial ambient temperature of T∞ ¼ 300 K and the adiabatic ﬂame temperature of Tf ¼ 1990 K (Tang et al., 2009a,b). Iron dust ﬂames were studied experimentally in the context of laminar ﬂames propagating in suspensions of iron particulates by Mamen (2006) and Tang et al. (2009a,b). Their experiments were performed in a reduced-gravity environment onboard a parabolic ﬂight aircraft to minimize particle settling and buoyancy-induced convective ﬂows that cause ﬂame disruptions. The experiment consisted of producing a suspension of iron particulates inside a glass tube at the initial atmospheric pressure and ambient temperature and initiating a propagating ﬂame at the open-end of the tube. Quenching plate assemblies forming rectangular channels with variable widths were installed inside the tube as depicted in ﬁgure. Pass and quench events across the channel were used to ﬁnd the quenching distance. Flame propagation was recorded by a high-speed digital camera. The iron dusts used by (Tang et al., 2009a,b) are (Alpha Aesar, Atlantic Equipment Engineers, Inc.), were characterized using the Mie scattering technique with a Malvern Mastersizer 2000E system; the arithmetic mean particle size of the powders used, are given in Table 1. 3. Results and discussion Fig. 4 shows the variation of the mean ﬂame propagation speed as a function of dust concentration for iron particles of 5 mm diameter, for cases where iron particles are uniformly or randomly distributed in reaction media. Each realization was assigned a Value Particle properties Cp 447 kp 80.2 rp 7860 1990 Tf Tig 850 n 0.2846 Air properties Do2 ;∞ 22.5 106 0.233 Yo2 ;∞ r∞ 1.1614 T∞ 300 Unit Ref. J/kg K W/m K kg/m3 K K e Incropera and DeWitt (2002) Incropera and DeWitt (2002) Incropera and DeWitt (2002) Incropera and DeWitt (2002) Tang et al. (2009a,b) Bidabadi and Maﬁ (2013) m2/s e kg/m3 K Incropera and DeWitt (2002) Bidabadi and Maﬁ (2013) Incropera and DeWitt (2002) Assumed different random distribution of particles, containing the same overall concentration and the total number of particles. According to Fig. 4, under uniform distribution, increasing the dust concentration results in an approximately linear increase in ﬂame propagation speed. The ﬂame speed under random distribution using a speciﬁc dust concentration depends on the particular distribution of particles. The ﬂame speed increases with dust concentration. The variation in the ﬂuctuation in ﬂame speed values is seen to increase with the value of b, which characterizes the level of nonuniformity. The results are compared with the experimental work of Sun et al. (1998). The model results for both uniform and random distribution show good agreement with the published experimental data, regardless of the value of b. The simulations slightly overestimate the ﬂame propagation speed in general, but deviate from the experimental results at low concentrations. The role of particle diameter on the ﬂame propagation speed is shown in Fig. 5, for three different diameters, as a function of dust concentration. The increase in particle diameter leads to a decrease in ﬂame velocity and a decrease in the relative ﬂuctuation in the simulations. This is due to the fact that larger particles require much more energy to be ignited as compared to smaller ones, and that the inter-particle distance of larger particles is lower than that of smaller particles. Fig. 6 illustrates the behavior of ﬂame speed as function of particle diameter for a particular overall equivalence ratio. The results for uniform and random distributions in the current research paper are compared with the experimental data reported by Tang et al. (2009a,b) for a fuel-rich iron/air suspension with equivalence ratio f ¼ 1.43e1.90 with fuel particles having speciﬁc surface area, ranging from 0.013 to 0.175 (m2/g) in a reduced-gravity environment. The results for the uniform and individual realizations of the random distribution model are shown as a function particle diameter. Clearly the results for each realization depends on the spatial distribution of particles. Similarly to Figs. 5 and 6 shows that an increase in particle diameter leads to a lower ﬂame speed. This is a result of the fact that larger particles take longer to reach the speciﬁed ignition temperature. The obtained results from the present model are in very good agreement with the experimental ﬁndings (Tang et al., 2009a,b). Fig. 7 shows how the minimum ignition energy increases nearly linearly as a function as a function of particle diameter for a mixture with a stoichiometric dust concentration. As the particle size increases, the assumed distance between neighboring particles (L) will rise according to Eq. (15) and the minimum ignition energy required to ignite the mixture should increase nearly linearly. The variation of the MIE for a layer in a uniformly distributed medium is illustrated in Fig. 8 as a function of the equivalence ratio for a particle diameter of 10 mm. The minimum ignition energy M. Bidabadi et al. / Journal of Loss Prevention in the Process Industries 43 (2016) 138e146 143 Fig. 4. Flame propagation speed as a function of iron dust concentration for the uniform and random distribution of particles for dp ¼ 5 mm, T∞ ¼ 300 K, P ¼ 101 kpa and different values of b: a) b ¼ 0.9, b) b ¼ 0.64, c) b ¼ 0.32. decreases strongly for lower concentrations, f < 1. Beyond an equivalence ratio of unity, the rate of reduction in MIE decreases for larger concentrations. The reason is that with the rise of dust concentration the assumed distance between of neighboring particles reduces and leading to a lower energy for ignition of dust cloud. Fig. 9 shows the variation of quenching distance in terms of various iron particle diameters. By increasing the value of particle diameter, the amount of energy release per unit volume is decreased during the combustion process therefore the preheating time is increased for the layers, and the ﬂame propagation speed is decreased. The reduction in preheating time means the reduction of heat loss. Therefore, a bigger particle diameter leads to a higher heat loss and quenching distance, as shown in Fig. 9. This prediction 144 M. Bidabadi et al. / Journal of Loss Prevention in the Process Industries 43 (2016) 138e146 Fig. 5. Flame propagation speed as a function of dust concentration for different particle diameters for the uniform and random distribution of particles for T∞ ¼ 300 K, P ¼ 101 kpa. Fig. 6. Flame propagation speed as a function of particles diameter for a rich mixture, with 4 ¼ 1.63, T∞ ¼ 300 K, P ¼ 101 kpa. Fig. 7. Minimum ignition energy as a function of particles diameter for 4 ¼ 1, T∞ ¼ 300 K, P ¼ 101 kpa and uniform particle distribution. was validated by the experimental data for two different iron particle diameters reported by Mamen (2006). Fig. 10 demonstrates how an increase in dust concentration results into smaller quenching distance. The amount of energy release per unit volume increases with dust concentration during the combustion process. Therefore, the ratio of heat release to heat losses to the wall decreases, so that the ﬂame propagation speed is increased and the quenching distance becomes narrower. 4. Conclusion In the present study, ﬂame propagation of iron dust particles in a medium with spatially discrete sources is numerically investigated, both for a uniform as well as for a random distribution of particles for a given concentration and diameter. In addition, the variation of quenching distance is also considered as a function of the same variables. M. Bidabadi et al. / Journal of Loss Prevention in the Process Industries 43 (2016) 138e146 145 Fig. 8. Minimum ignition energy as a function of equivalence ratio for dp ¼ 10 mm, T∞ ¼ 300 K, P ¼ 101 kpa and uniform distribution. Fig. 9. Experimental validation of the quenching distance of various iron particle diameter for 4 ¼ 1, T∞ ¼ 300 K, P ¼ 101 kpa. Fig. 10. Quenching distance of 9 mm iron particles as a function of dust concentration in T∞ ¼ 300 K, P ¼ 101 kpa. The simulation results show that the ﬂame propagation speed for of both uniform and random particle distributions are the same on the mean. Both sets of simulations are able to reproduce the behavior of experimental data with both diameter and concentrations rather well. The ﬂame propagation speed in iron-air suspensions decreases as the particle size increases signiﬁcantly. More energy is required for the ignition of larger particles relative to the smaller ones as compared to the smaller ones. The calculated minimum ignition energy is shown to increases with an increase in particle size and decrease with bulk equivalence ratio for selected cases using uniform distribution. Finally, the calculated quenching distances appear to capture reasonably well the threshold for quenching distances, providing an effective model both for ﬂame propagation as well as quenching of dust clouds. As in combustion physics of dust cloud, particles are distributed randomly in the reaction 146 M. Bidabadi et al. / Journal of Loss Prevention in the Process Industries 43 (2016) 138e146 medium, therefore taking into account this phenomenon in the governing equations, the predicted results are more realistic. The presented thermal model can be developed to consider particle size distribution to enhance the obtained results in comparison with the experimental data. Nomenclature A Cd d D DO2 H i,j,k kp L r t tr T P N Ne Q Qloss q_ loss Lr Area (m2) Dust concentration (kg/m3) Diameter (m) The channel width (m) Oxygen diffusivity coefﬁcient (m2/s), Heaviside function Components of Cartesian coordinate Conduction coefﬁcient (W/m K) Distance of two adjacent particles or layers (m) Radial distance (m) Time (s) The relative time between the ignition of two neighbor layers Temperature (K) Pressure (Pa) Total number of particles Total number of empty location before distributing particles Energy released from igniter (j) The total heat loss of one layer (j) The rate of heat loss to the walls (j) The modiﬁed distance between two adjacent layer (m) Greek symbols a Thermal diffusivity (m2/s) r Density (kg/m3) t Particle burning time (s) 4 Equivalence ratio Subscripts a Burning zone f Flame g Ambient gas ig Ignition p Particle w Wall ∞ Ambient property References Abbasi, T., Abbasi, S., 2007. Dust explosionsecases, causes, consequences, and control. J. 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