## Servicios Personalizados

## Articulo

## Indicadores

- Citado por SciELO

## Links relacionados

- Similares en SciELO

## Compartir

## Latin American applied research

##
*versión impresa* ISSN 0327-0793

### Lat. Am. appl. res. vol.40 no.1 Bahía Blanca ene. 2010

**ARTICLES**

**Quenching distance and laminar flame speed in a binary suspension of solid fuel particles**

**M. Jadidi ^{1}, M. Bidabadi^{2} and A.Sh. Shahrbabaki^{1}**

^{1} *Department of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran 16844, Iran.*

^{2}

*Department of Mechanical Engineering, Iran University of Science and Technology, Combustion Research Laboratory,*

Narmak, Tehran 16844, Iran.

Narmak, Tehran 16844, Iran.

*e-mail: jadidi_m@yahoo.com, bidabadi@iust.ac.ir, Ab_shabani@iust.ac.ir*

** Abstract -** A mathematical combustion model of non-adiabatic dust flame is developed for a suspension of two monosize metal powders. Depending on the combustion properties of particles at different concentrations, the flame structure may display either an overlapping or a separated configuration. In the present study the heat loss term, which is assumed to be linearly proportional to temperature difference, is added to the energy conservation equation. The laminar flame speed and quenching distance are obtained by solving the energy equation in each zone and matching the temperature and heat flux at the interfacial boundaries. Calculated values of flame speed adequately agree with experimental data.

* Keywords* - Dust Combustion; Binary Suspension; Non-Volatile Solid; Laminar Flame Speed; Quenching Distance.

*Nomenclature*

a | thermal conductivity of gas, m^{2}s^{-1} |

B | dust concentration |

c_{p} | specific heat of gas |

c_{s} | specific heat of solid |

d | diameter of the quenching channel |

m | mass of particle |

p | combustion time ratio of the second dust to the first dust |

Q | heat of reaction per unit mass of fuel |

Nu | Nusselt number, |

r | particle radius |

T | temperature |

v | flame speed |

W_{F} | heat source term |

x' | spatial coordinate |

x | independent variable that is related to the spatial coordinate x'as x= |

y | nondimensional coordinate |

Z | distance between two combustion zones |

*Greek Symbols*

a | heat transfer coefficient between gas and channel walls |

F | equivalence ratio |

? | heat loss parameter |

nondimensional flame speed | |

? | heat transfer coefficient, |

µ | nondimensional dust concentration |

? | nondimensional temperature, T/T_{u} |

? | density |

t_{c} | combustion time of individual separated particle at initial oxygen concentration |

? | ratio of the characteristic particle heat exchange time and combustion time of the particle |

*Subscripts*

g | gas |

i | conditions at ignition points |

st | stoichiometric |

s | solid fuel particles |

u | characteristics of unburned mixture at x= - 8 |

**I. INTRODUCTION**

Because of their high combustion enthalpy, metal powders are often used as additives in propellants, explosives, or pyrotechnics. The combustion stability in rocket engines has also been improved by the addition of metal powders to propellants (Markstain, 1963; Gordon *et al.*, 1968). Natural and industrial combustible dusts mostly have a wide domain of particle size distribution. Given that the ignition temperature and combustion rate of an individual dust particle are functions of particle size, real dust suspensions have a complex, multistage flame structure. Research on multistage dust combustion was initiated by Goroshin *et al*. (2000) who theoretically and experimentally investigated the combustion of a suspension of two monosize powder. They developed a simple analytical model for the adiabatic flame in a fuel-lean binary suspension, which permits the analysis of flame speed and structure as a function of the dust composition and combustion properties of individual particles. In the experimental section of the study, the flame speed in a binary suspension of aluminum and manganese powders was investigated by observing the laminar stage of flame propagation in a semi-open vertical tube. Their proposed model correctly predicted dependence of the flame speed on mixture composition (mass ratio of manganese and aluminum dusts in suspension) and the mixture composition at the limit of flame propagation (Goroshin *et al.*, 2000). This model has been either expanded or tested experimentally in a number of studies. Similar to and building on the experiment of Goroshin *et al.* (2000), Risha *et al.* (2005) examined the flame characteristics of bimodal nano and micro-sized aluminum particle/air laden flows using a Bunsen-burner type apparatus. Boichuk *et al.* (2002) examined combustion of dual-fuel particle-laden flows for aluminum-boron dust mixtures. In an analytic study, Huang *et al.* (2007) investigated extensively the effects of various parameters such as the particle composition, equivalence ratio and particle size on the burning behavior of bimodal nano-micro sized aluminum particle/air mixtures. In this study, the flames were assumed to consist of several different regimes, including preheat, flame, and post flame zones for fuel-lean mixtures. The results showed that at low percentages of nano particles in the fuel formation, the flame exhibits a separated spatial structure with a wider flame regime but at a higher loading of nano particles, an overlapping flame configuration is observed. Findings show that whereas a small particle at low pressure may burn under kinetically controlled conditions, a large diameter particle at high pressure may possibly burn under diffusion controlled conditions (Huang *et al.,* 2005). Recently, Huang *et al.* (2009) theoretically studied the combustion of aluminum particle dust in a laminar air flow under fuel-lean conditions. A wide range of particle sizes at nano and micron scales were explored. The flame speed and temperature distribution was obtained by numerically solving the energy equation in the flame zone, and with the particle burning rate modeled as a function of particle diameter and ambient temperature. With the decrease of particle diameter from the micron to the nano range, the flame speed increased and the combustion transited from a diffusion-controlled to a kinetically-controlled mode. They concluded that no universal law of flame speed had existed for any range of particle sizes.

Nano particle combustion has attracted researchers' interest in recent years, and a number of studies specifically on the combustion of aluminum nano particles have been conducted (e.g. Rai *et al.,* 2006; Yetter *et al.,* 2009; Levitas*,* 2009). However, since the model proposed in this article is in the diffusion-controlled mode, it is more adequate to be used for and limited to the study of micro-sized particles.

Alongside aluminum, other combustible dusts particles including Iron (Tang *et al.,* 2009), Boron (Dreizin *et al.,* 1999), Titanium (Molodetsky *et al.,* 1998), Zirconium, Manganese, Magnesium, Coal, Wood, Lycopodium and Maize starch (Eckhoff, 2003) can be mentioned. The model presented in this paper so far has been used for aluminum, manganese, and iron particles (Goroshin *et al.* 1996; Goroshin *et al.,* 2000; Huang *et al.,* 2005; Huang *et al., *2007; Tang *et al.,* 2009), and since other particles have different flame structures and combustion mechanisms, this model is not suitable for them. For example, Seshadri *et al. *(1992) proposed a model for the combustion of Lycopodium and organic particles, which included vaporization term, and Yeh and Kuo (1996) showed that boron particles have a two-stage combustion and their own specific model.

Nonetheless, no model has thus far been proposed to predict the quenching distance at binary suspension. In 1996, Goroshin and his associates (Goroshin *et al.*, 1996) proposed a model to predict the quenching distance in the monosize aluminum combustion, which showed a good agreement between the theoretical quenching distance and flame speed with the experimental results. In the present study, in order to propose a new model, the equations proposed in two studies of fuel-lean mixtures of non-volatile solids conducted by Goroshin and his associates (Goroshin *et al.*, 2000; Goroshin *et al.*, 1996) are combined, that is, the heat loss term to the wall, which has been neglected in the previous binary suspension studies, is added to the energy equations. Finally, the flame speed is compared with the experimental results of Goroshim* et al*. (2000) study; and the quenching distance for binary suspension including two different types of metal powder with the same average diameters is introduced. Since the flame model in a binary suspension which is developed in this paper is based on simple descriptions of monosize dust flame (Goroshin *et al.*, 1996) and flame in binary suspension (Goroshin *et al.*, 2000), these two models are explained briefly here.

**A. Flame Speed in Monosize Suspension (Goroshin et al., 1996)**

In a mono-dispersed aluminum particle-laden flow, the flame usually consists of three distinct zones (Fig. 1): preheat, flame, and post flame zone (Goroshin *et al.*, 1996; Huang *et al.,* 2005). In the preheat zone, the reaction rate is negligibly small. Particles are heated by the surrounding gas until their temperature reaches the ignition point. In the flame zone, particles are ignited and totally consumed. The third zone is the post flame zone where the temperature of the gas and particles decreases asymptotically to the ambient temperature at infinity. The flame speed, quenching distance, flammability limits and temperature distribution are derived by solving the energy equation in each zone and matching the temperature and heat flux at the interfacial boundaries.

**Figure 1:** Structure of the dust flame in monosize suspension

The major approximations and assumptions are: (1) the dust cloud consists of uniformly distributed alumi num particles with air; (2) the gravitational effects andheat transfer by radiation are neglected (Goroshin *et al.*, 1996). The quenching diameter of a channel is much less than the free path length of radiation in an unburned dust suspension. Therefore, the radiation emitted by the flame front and post flame zone passes the preheat flame zone almost without extinction and is absorbed almost entirely by the channel walls. In addition, the total emissivity of the aluminum flame is apparently low, due to the very small radiation absorption coefficient of the submicron aluminum oxide particles (Plass, 1965; Goroshin *et al.,* 1996). The negligible role of radiation heat transfer in a small scale Bunsen-type aluminum dust flame was proven by Goroshin *et al.* (1996); (3) the particle velocity is approximately equal to the gas velocity; (4) collisions and interactions between burning particles are neglected; (5) the Biot number is very small, suggesting a uniform temperature distribution within each particle (Huang *et al.,* 2005; Huang *et al.*, 2007); (6) the thermal conductivity of the gas is a linear function of temperature: ?=?* _{u}* (

*T*/

*T*) (here the index "

_{u}*u*" is denoting unburned mixture) (Seshadri

*et al.,*1992; Goroshin

*et al.*, 1996; Goroshin

*et al.,*2000); (7) The reaction rate during diffusive combustion is considered to be constant and equal to an average value (

*m*

_{s}/*t*), (

_{c}*m*is the mass of particle and

_{s}*t*is the burning time), (Goroshin

_{c}*et al.*, 1996; Goroshin

*et al.,*2000; Huang

*et al.*2005; Huang

*et al.,*2007); (8) Ignition point occurred when the particle temperature is close to the auto ignition temperature of a single particle,

*T*(the ignition temperature during a slow, quasistationary heating) (Goroshin

_{s}^{i}*et al.*, 1996; Huang

*et al.,*2007); (9) Due to the low fuel concentration, we can neglect the total heat that is consumed by particles in the preheat zone and by the condensed reaction products in the post flame zone; (10) Because of the excess of the oxygen in a lean mixture and weak dependence of oxygen diffusivity on ambient gas temperature, the total particle burning time in a flame front is close to the burning time of a single particle

*t*, also the reacting dust do not compete in oxygen consumption and, therefore, heat conduction is the only channel of their interaction. Goroshin

_{c}*et al.*(1996) showed that the governing heat diffusivity equation in this problem can be transformed into a linear form by introducing an independent variable

*x*that is related to the spatial coordinate

*x'*as

*x*=

_{}. In order to model the heat loss, it has been supposed that this term is linearly proportional to the temperature difference of gas and the wall. This represents loss through heat conduction from the dust-air mixture to the wall, which is assumed to be maintained at a constant downstream temperature (Goroshin

*et al.*, 1996).

With these assumptions, the gas phase governing equations for mass and energy conservation for the problem illustrated in Fig. 1 can be written as follows (Goroshin *et al.,* 1996):

(1) |

(2) |

and gas is supposed to be ideal and isobaric

(3) |

heat transfer coefficient between gas and flat channel is computed in the following way (Goroshin *et al.,* 1996):

The particle temperature can be found from the equation describing inert particle heating in the preheat zone:

(4) |

and in a dimensionless form, we have (Goroshin *et al.*, 1996):

(5) |

The dimensionless energy conservation equation for the particle phase is:

(6) |

Here a nondimensional gas temperature *?* is defined as *?=T/T _{u}* and nondimensional solid temperature

*?*is defined as

_{s}*?*=

_{s}*T*/

_{s}*T*(Subscripts "

_{st}*s*", "

*g*" and "

*st*" are denoting solid, gas and stoichiometric, respectively),

*y*is a nondimensional coordinate defined as

*y=x/v*

_{u}*t*(where

_{c}*v*is the laminar flame speed). Parameter is a nondimensional flame speed, =

_{u}*v*

^{2}_{u}*t*/a

_{c}*(where a*

_{u}*=?/c*

_{u}*?*

_{p}*). ?*

_{u}*, c*

_{u}*,*

_{p}*T*and

*T*are density, specific heat, temperature of the gas, and temperature of solid, respectively. Parameter

_{s}*µ*is a nondimensional dust concentration defined as:

*µ*=

*BQ*/c

*?*

_{p}*(*

_{u}*T*-

^{i}_{s}*T*),

_{u}*B*is dust concentration,

*Q*is the heat of reaction,

*T*is the ignition temperature during a slow, quasistationary heating and

^{i}_{s}*?*is nondimensional ignition temperature of solid particles. The heat loss parameter is the ratio of particle combustion time to heat transfer time in the channel (),

^{i}_{s}*d*is the width of the channel,

*Nu*is Nusselt number and

*b'*is equal to 8 (Goroshin

*et al.*, 1996). Where

*r*is the radius of fuel-particle and is a ratio of the characteristic particle heat exchange time and combustion time of the particle.

The boundary conditions for the above differential equations are as follows:

(7) |

By solving these energy equations in each flame zone and by matching the heat fluxes and resultant temperatures obtained from these solutions on the boundary of each zone, the algebraic equation for the nondimensional flame speed () is found to be:

(8) |

where

(9) |

**B. Flame Speed and Structure in the Binary Suspension (Goroshin et al., 2000)**

In order to describe the combustion of a suspension of two monosize powders, an analytical model was constructed (Goroshin *et al.*, 2000). The schematic representation of all possible configurations of the stationary flame front in binary suspension is presented in Fig. 2 (the lower step corresponds to the number "1" dust). The position of the boxes corresponds to the position of combustion fronts of corresponding dusts. In spite of having a higher ignition temperature, the particles of the second dust can heat up faster and may ignite first if their characteristic heat exchange time is smaller than that of the first dust.

**Figure 2. **Possible flame configurations in binary dust suspension

In a binary suspension, parameters *µ _{i}* (i=1, 2) are the dimensionless mass dust concentrations of corresponding dusts, andis the dimensionless flame speed. In addition, parameteris the combustion time ratio of the second dust to the first dust. For the sake of convenience, the number "1" is assigned to the dust with a lower particle ignition temperature (Goroshin

*et al.,*2000; Huang

*et al.,*2007).

The heat transfer equations for all flame configurations shown in Fig. 2 can be written in the same form as Eqs. (5), (6) for a monosize suspension in the adiabatic condition:

(10) |

The inert heating of particles before their ignition is governed by

(11) |

(12) |

The boundary conditions are identical with that of Eq. (7) but expanded for Z and Z+p boundaries.

By solving the energy equation for each zone and matching the resultant temperature and heat flux at the interfacial boundaries, the algebraic equations for the flame speed and the distance Z between the ignition points of the first and second dust can be obtained (Goroshin *et al.*, 2000). For the flame with overlapping combustion zones (Z < 1, configuration 2, Fig. 2) and for flame fronts with separated combustion zones (Z > 1, configuration 1, Fig. 2), the algebraic equations defining flame speed and parameter Z are presented in the Ref. (Goroshin *et al.*, 2000). Also, algebraic equations can be used to define the flame speed in the case when the second dust ignites first (configuration 3 and 4, Fig. 2). To do this, the dusts simply have to be renumbered (12) (Goroshin *et al.*, 2000).

**II. THEORETICAL ANALYSIS**

**A. Formulation of the Problem**

An analytical model is presented to include heat transfer to wall in a suspension of two monosize metal powders and obtain the quenching distance and flame speed in non-adiabatic condition. The heat loss term, which is similar to the term in section (I.A), is added to the binary suspension (section (I.B)). Flame structure is assumed as represented in Fig. 2. Heat transfer equations of gas phase for all the configurations of Fig. 2 are obtained as follows:

(13) |

Solid phase equations are the same as Eqs. (11) and (12), and will remain unchanged.

The heat loss parameter is obtained by relation of (). The algebraic equations for the flame speed and the distance Z between the first and second dust ignition points can be obtained by solving the energy equation for each zone and matching the resulting temperature and heat flux at the interfacial boundaries. For flame fronts with separated (Z > 1, configuration 1, Fig. 2) and overlapping (0 < Z < 1, configuration 2, Fig. 2) combustion zones the equation defining flame speed is:

(14) |

where

(15) |

In order to obtain the unknown parameters, four equations are required. In each of the separate and overlap configurations, two equations are needed to determine the flame speed and the distance between two ignition points. Since the above Eq. (14) is valid for both overlap and separate configurations, two more equations are required as follows:

(16) |

Note that the explicit forms of *f*_{i} are different from the overlapping and separated flame configurations. Eqs. (14), (16) must be solved simultaneously for flame speed and the parameter Z, which falls in the range of Z > 1 for a separated flame and 0 < Z < 1 for an overlapping flame (Huang *et al.*, 2007).

In addition, the above Eqs. (14), (16) can be used to determine flame speed and parameter Z, when the second particle cloud ignites first (configurations 3 and 4, Fig. 2). To do this, the dusts simply have to be renumbered (12) (Goroshin *et al.*, 2000). Equations' inputs, such as particle diameter, ignition temperatures, and burning times, are the same as that of Goroshin's model (Goroshin *et al.*, 2000).

**B. Results and Discussions**

The nondimensional flame speed () and a corresponding flame front structure obtained from Eqs. (14-16) for heat loss parameter ? equal to 0.01 are mapped in Fig. 3 in relation to mixture composition µ_{1}/µ_{2} and initial mass concentration of the first dust in the suspension (different values of the parameter µ_{1}). The results show that when the amount of heat loss is non zero, generally two values of flame speed, the lower one of which is unstable (Goroshin *et al.*, 1996; Zeldovich *et al.*, 1985), are possible. The second dust does not ignite, unless the final flame temperature of the mixture is higher than the ignition temperature of the second dust (dotlines in Fig. 3). Since the specific heat of the solid phase in lean dust mixtures is neglected, flame speed in this region is not influenced by the presence of the second dust (Goroshin *et al.*, 2000). The speed remains constant until the flame temperature reaches the ignition temperature of second dust. It starts to increase by the ignition of the second dust. The distance between two combustion zones (Z) decreases by the increase of mass concentration of the second dust. For some values of µ_{1}and µ_{2}, the separated configuration of the flame changes to the overlapping configuration; and hence, Z > 1 reduces to Z < 1. At high initial dust concentrations (e.g. µ_{1} > 2), the first dust alone can provide a flame temperature higher than the ignition temperature of the second dust. In this case, the combustion fronts may overlap even at the lowest concentrations of the second dust.

**Figure 3.** Dependence of the nondimensional flame speed in the binary suspension on the ratio of dust concentrations

In Figs. 4 and 5, the loss effects can be seen obviously. Fig. 4 refers to µ_{1}and shows the flame speed profile for different values of heat loss parameter (?). This parameter increases while the flame speed decreases (Goroshin *et al.*, 1996) and the distance between two combustion zones (Z) increases. The theoretical model shows that a separated flame is much thicker than an overlapping flame.

**Figure 4**. Dependence of nondimensional flame speed in the binary suspension on the ratio of dust concentrations under different values of the heat loss parameter ?

**Figure 5.** Dependence of nondimensional flame speed in the binary suspension on the ratio of dust concentrations under different values of the heat loss parameter ?

Figure 5 refers to µ_{1} = 1.67 and shows that until flame temperature reaches the ignition point of second dust, the flame speed remains constant. By the increase of heat loss parameter, the flame speed decreases and the distance between two combustion zones increases, in accordance with Fig. 4.

The theoretical values of the quenching distance are obtained from the dimensionless flame speed curves (Fig. 3). The quenching condition occurs at the critical point of the dimensionless flame speed curve (bifurcation point) (Goroshin *et al.*, 1996). The quenching distance as a function of µ_{2}/µ_{1}in binary suspension is shown in Fig. 6.

**Figure 6.** Dependence of the quenching distance in the binary suspension on the ratio of dust concentrations

From the theory it can also be seen that for larger particles, the quenching distance is larger than that of smaller particles (Goroshin *et al.*, 1996). As shown in Fig. 3, in regions where flame speed is constant and not influenced by second dust, until the flame temperature reaches the ignition point of second dust and this ignition leads to the increase of flame speed and hence the decrease of quenching distance, the quenching distance, as represented by dotlines in Fig. 6, remains constant (since the concentration of the first dust is constant and heat loss to particles is neglected) as well and is equal to the quenching distance related to the first dust. Since the increase of flame speed leads to the increase of generation rate and the quenching distance is obtained from the equality of generation and heat loss rates, the quenching distance should decrease so as to increase the heat loss rate. Increasing the first dust concentrations, leads to the increase of speed and hence the decrease of the quenching distance.

In Figs. 7 and 8, the relative flame speed and the distance of two combustion zones, obtained from this theory, are compared with the experimental data of the combustion of monosize aluminum-manganese particles (Goroshin *et al.*, 2000). As could be seen, the change trend of the analytical results is similar to the experimental data and these two correspond well. As it was mentioned before, the increase of heat losses leads to the decrease of flame speed and the increase of parameter Z.

**Figure 7. **Comparison of the theoretically predicted (lines) under different values of the heat loss parameter *h* and experimentally observed (crosses) relative flame speeds in binary suspensions of manganese and aluminum powders

**Figure 8.** Theoretically predicted flame structure under different values of the heat loss parameter ?.

**III. CONCLUSIONS**

In this paper, a simple analytical model of binary suspension of two monosized particles with heat losses was studied. Since there exists heat loss to walls in the reality (non-adiabatic), ignoring this term does not represent a real case. For a binary particles dust cloud, the flame structure may display either an overlapping or a separated configuration, depending on the combustion properties of particles at different concentrations and the amount of heat losses to wall. In order to model the heat loss, it has been supposed that this term is linearly proportional to the temperature difference of gas and the wall (Goroshin *et al.*, 1996). The quenching distance was obtained by including the heat loss term to heat transfer equations of gas phase. An acceptable agreement in terms of flame speed was found between the theoretical results and the experimental data. Flame speeds with heat losses reported in this study were slightly lower than that of Goroshin *et al. *(2000) and the flame speed decreased reasonably when heat loss parameter increased. Also, when the amount of heat loss increases, the distance between two combustion zones Z is increased.

**REFERENCES**

1. Boichuk, L.V., V.G. Shevchuk and A.I. Shvets, "Flame propagation in two-component aluminum-boron gas suspensions", *Combust. Explos. Shock Waves*, **38**, 651-654 (2002). [ Links ]

2. Dreizin, E.L., D.G. Keil, W. Felder and E.P. Vicenzi, "Phase Changes in Boron Ignition and Combustion", *Combust. Flame*, **119**, 272-290 (1999). [ Links ]

3. Eckhoff, R.K. *Dust explosions in the process industries*, extended 3^{rd} ed., Gulf Professional Publishing/Elsevier, Boston (2003). [ Links ]

4. Gordon, A.S., C.M. Drew, J.L. Prentice and R.H. Knipe, "Techniques for the study of the combustion of metals", *AIAA J.*, **6**, 577-583 (1968). [ Links ]

5. Goroshin, S., M. Bidabadi and J.H.S. Lee, "Quenching distance of laminar flame in aluminum dust clouds", *Combust. Flame*, **105**, 147-160 (1996). [ Links ]

6. Goroshin, S., M. Kolbe and J.H.S. Lee, "Flame Speed in a Binary Suspension of Solid Fuel Particles", *Proc. Combust. Inst.*, **28**, 2811-2817 (2000). [ Links ]

7. Huang, Y., G.A. Risha, V. Yang and R.A. Yetter, "Analysis of Nano-Aluminum Particle Dust Cloud Combustion in Different Oxidizer Environments", *43 ^{rd} Aerospace Sciences Meeting and Exhibit*,

*AIAA*, Reno, Nevada, 2005-0738 (2005). [ Links ]

8. Huang, Y., G.A. Risha, V. Yang and R.A. Yetter, "Combustion of bimodal nano/micron-sized aluminum particle dust in air", *Proc. Combust. Inst.*, **31**, 2001-2009 (2007). [ Links ]

9. Huang, Y., G.A. Risha, V. Yang and R.A. Yetter, "Effect of particle size on combustion of aluminum particle dust in air", *Combust. Flame*, **156**(1), 5-13 (2009). [ Links ]

10. Levitas, V.I., "Burn time of aluminum nanoparticles: Strong effect of the heating rate and melt-dispersion mechanism", *Combust. Flame*, **156**, 543-546 (2009). [ Links ]

11. Markstain, G.H., "Combustion of metals", *AIAA J.*, **1**, 550-562 (1963). [ Links ]

12. Molodetsky, I.E., E.P. Vicenzi, E.L. Dreizin and C.K. Law, "Phases of Titanium Combustion in Air", *Combust. Flame*, **112**, 522-532 (1998). [ Links ]

13. Plass, G.N., "Temperature Dependence of the Mie Scattering and Absorption Cross Sections for Aluminum Oxide", *Appl. Opt.*, **4**(12), 1616-1619 (1965). [ Links ]

14. Rai, A., K. Park, L. Zhou and M.R. Zachariah, "Understanding the mechanism of aluminum nanoparticle oxidation", *Combust. Theory Modell.*, **10**, 843-859 (2006). [ Links ]

15. Risha, G.A., Y. Huang, R.A. Yetter and V. Yang, "Experimental investigation of aluminum particle dust cloud combustion", *43 ^{rd} Aerospace Sciences Meeting and Exhibit*,

*AIAA*, Reno, Nevada, 2005-0739 (2005). [ Links ]

16. Seshadri, K., A.I. Berlad and V. Tangirala, "The structure of premixed particle-cloud flames", *Combust. Flame*, **89**, 333-342 (1992). [ Links ]

17. Tang, F., S. Goroshin, A. Higgins and J. Lee, "Flame propagation and quenching in iron dust clouds", *Proc. Combust. Inst.*,**32**, 1905-1912 (2009). [ Links ]

18. Yeh, C.L. and K.K. Kuo, "Ignition and combustion of boron particles", *Prog. Energy Combust. Sci.*, **22**, 511-541 (1996). [ Links ]

19. Yetter, R.A., G.A. Risha and S.F. Son, "Metal particle combustion and nanotechnology", *Proc. Combust. Inst.*, **32**, 1819-1838 (2009). [ Links ]

20. Zeldovich, Ya.B., G.I. Barenblatt, V.B. Librovich and G.M. Makhviladze, *The Mathematical Theory of Combustion and Explosions*, Consultants Bureau, New York, (1985). [ Links ]

**Received: November 1, 2008. Accepted: February 17, 2009. Recommended by Subject Editor Eduardo Dvorkin.**