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Flame speed in a binary suspension of solid fuel particles
Proceedings of the Combustion Institute, Volume 28, 2000/pp. 2811–2817
FLAME SPEED IN A BINARY SUSPENSION OF SOLID FUEL PARTICLES SAMUEL GOROSHIN, MASSIMILIANO KOLBE and JOHN H. S. LEE Department of Mechanical Engineering McGill University 817 Sherbrooke St. West Montreal, Quebec, Canada H3A 2K6
Most natural and industrial combustible dusts have a wide distribution of particle sizes. Yet, the majority of experimental data on flame propagation in dust clouds are given in relation to some average particle size, and all known theoretical models of dust combustion consider only monosize suspensions. Since the ignition temperature and combustion rate of an individual dust particle are functions of particle size, the flame in real dust suspensions has a complex, multistage structure. As a first step toward understanding multisize dust combustion, the combustion of a suspension of two monosize powders (that in general can be also of different chemical nature) is investigated in the present work theoretically and experimentally. A simple analytical model developed for the flame in a fuel-lean binary suspension permits flame speed and structure to be analyzed as a function of the dust composition and combustion properties of individual particles. The flame speeds predicted by the binary model were compared with flame speeds calculated from a model of monosize dust flame using various average particle size representations. It is shown that averaging of the particle size in general fails to correctly predict the flame speed over the wide range of the binary dust compositions. The flame propagation speed in a binary suspension of aluminum and manganese powders was investigated experimentally by observing the laminar stage of flame propagation in a semi-open vertical tube. The model correctly predicts 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.
Introduction With the exception of a few special dusts (e.g., lycopodium), most dusts used in experiments have a continuous spectrum of particle sizes over a specific size range. Yet, the experimental data on the combustion parameters of dust suspensions (flame speeds, flammability limits, etc.) are commonly given only in relation to some average particle size. This makes it difficult to qualitatively compare experimental results from different sources as every particular dust has a unique particle size distribution. The polydisperse nature of real dusts also hampers the comparison of the results of theoretical (which, in most cases, consider monosize suspensions) and experimental investigations. While combustion of volatile, polydisperse sprays has been numerically investigated in several works (for example, Refs. [1,2]), flame propagation in a non-volatile multisize dust suspension has yet to be addressed. The combustion and ignition characteristics of individual fuel particles are functions of particle size. Therefore, the flame in a suspension of real polydisperse dust has a complex multistage spatial structure analogous to the flame structure in homogeneous systems with parallel reactions [3,4]. As a first step
toward the understanding of phenomena of multistage dust combustion, the present work analyzes combustion of a simple binary suspension formed by the dispersion of a mixture of two monosize powders. In the general case considered below, these powders cannot only be of different particle sizes but, moreover, can be of a different chemical nature. Thus, we will also be able to address a practically important question on how the addition of a reactive (low combustion time) component can influence the flame speed in an otherwise low-burning-rate suspension. The theoretical predictions made by this model are validated experimentally in the present work by observing flame speeds in aluminum/manganese suspensions. In order to obtain an analytical solution of the problem, we have restricted our consideration to fuel-lean mixtures of non-volatile solids. We assume that the predominant mechanism of flame propagation in such suspensions is molecular heat conductivity [5,6]. We also assume that the combustion products of both dusts are solid and do not interact chemically with each other. Due to the excess of oxidizer, the reacting dusts do not compete in oxygen consumption and, therefore, heat conduction is the only channel of their interaction.
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冦
0, ⳮ⬁ ⬍ y ⬍ 0 d2h dh ⳱ W ⳮlj(hsi ⳮ 1), 0 ⱕ y ⱕ 1 2ⳮj dy dy 0, y ⬎ 1
(1)
y → ⳮ⬁, h ⳱ 1; y ⳱ 1, h ⳱ h⬘; dh/dy|0Ⳮ ⳱ dh/dy|0ⳮ
Fig. 1. Structure of the dust flame in monosize suspension.
Theoretical Analysis Flame Speed in the Monosize Suspension The model of a flame in a binary suspension developed in this paper is based on a simple description of a monosize dust flame that was previously used by the authors to study dust flame quenching [5]. Our conjecture of the flame structure in monosize suspensions is illustrated in Fig. 1. In the preheat zone (ⳮ⬁ ⬍ x ⬍ 0), the reaction rate is negligibly small and the particles are heated by the gas. The difference in temperature between the dust particles and the gas progressively increases due to inertia in heat exchange. At some instant, the amount of heat generated by the reaction on a particle surface becomes larger than the heat flux from the hot gas, and the particle ignites. The transient period from ignition to stable particle combustion (controlled by oxygen diffusion) is short in comparison to the total particle combustion time sc. We also suppose that at the ignition point, the particle temperature is close to the autoignition temperature of a single particle Tis (the ignition temperature during a slow, quasi-stationary heating [5]). The reaction rate during diffusive combustion is considered to be constant and equal to an average value, ms/sc (where ms is the mass of the particle). As was shown in Ref. [7], the equation governing heat diffusivity 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 x⳱
x⬘
冮
0
(q/qu)dx
and by assuming that the thermal conductivity of the gas is a linear function of temperature: k ⳱ ku(T/ Tu) (here the index u is denoting unburned mixture). With these assumptions, the governing heat diffusivity equation and the boundary conditions for the problem illustrated in Fig. 1 can be written in a dimensionless form:
Here, nondimensional gas temperature h is defined as h ⳱ Tg/Tgu; nondimensional coordinate y, as y ⳱ x/tusc (where tu is the flame speed). Parameter j and parameter l is a nondimensional flame speed and a nondimensional dust concentration, respectively, defined as j ⳱ t2usc/au, l ⳱ BQ/cpqgu(T1s ⳮ Tu) where au ⳱ k/cpqgu; B is dust concentration; Q is the heat of reaction; and subscripts s and g, denote solid particles and gas, correspondingly. The inert heating of the particle in the preheat zone (x ⬍ 0, Fig. 1) is described by the equation dhs (h ⳮ hs) ⳱ dy n
(2)
y → ⳮ⬁, h ⳱ hs ⳱ 1; y ⳱ 0, hs ⳱ hs1 Here n ⳱ [(r2csqs/3aucpqg)]/sc is a ratio of the characteristic particle heat exchange time and combustion time of the particle. Parameter n is close to unity for nonvolatile fuel and Lewis number (Le) ⳱ 1. By solving heat transfer equation 1 in each two flame zones and by matching the heat fluxes obtained from this solution on the boundary of the preheat and combustion zone (x ⳱ 0), the algebraic equation for the nondimensional burning velocity can be found.
冢hh
j⳱l
i s i
ⳮ 1 [1 ⳮ exp(ⳮj)] ⳮ 1
冣
(3)
The ratio of the gas and flame temperatures at the ignition point that is present in equation 3 can be found from the solution of linear equation 2: (h is ⳮ 1)/(hi ⳮ 1) ⳱ (1 Ⳮ jn)
(4)
And the final equation that completely defines the flame speed in a monosize suspension is j ⳱ l(1 Ⳮ jn) [1 ⳮ exp(ⳮj)]
(5)
The same approach that was used above to obtain flame speed in monosize suspension will be used below to analyze a stationary flame in a binary dust suspension. Flame Speed and Structure in the Binary Suspension In comparison to monosize suspension, the number of parameters that characterize a binary dust suspension doubles. As in a monosize suspension, parameters li (i ⳱ 1,2) are the dimensionless mass
FLAME SPEED IN A BINARY SUSPENSION
Fig. 2. Possible flame configurations in binary dust suspension. Lower step marks the combustion front of the first dust (lower particle autoignition temperature); higher step marks the combustion front of the second dust (higher particle autoignition temperature). Step width indicates combustion zone of corresponding dust.
dust concentrations of corresponding dusts, and j ⳱ t2usc1/au is the dimensionless flame speed. We also introduce a new parameter, p ⳱ sc2/sc1, that is the ratio of combustion times of the first and the second dusts. For convenience, we will also assign the number 1 to the dust with a lower particle ignition temperature. All possible configurations of the stationary flame front in binary suspension are schematically shown 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. Despite having a higher ignition temperature, the particles of the second dust can heat up faster and may ignite first if their characteristic time of heat exchange is smaller than that of the first dust (i.e., when n1 ⬎ n2p, configurations 3 and 4 in Fig. 2). The heat transfer equation for all flame configurations shown in Fig. 2 can be written in the same form as equation 1 for a monosize suspension: d2h dh ⳮ j ⳱ dy2 dy
(6)
冦
0, combustion is absent ⳮjl1 (his1 ⳮ 1), only the first dust burns W ⳮjl2 (his2 ⳮ 1)/p, only the second dust burns ⳮjl1 (his1 ⳮ 1) ⳮ jl2 (h1s2 ⳮ 1)/p, both dusts burn
The inert heating of particles before their ignition is governed by dhs1 (h ⳮ hs1) ⳱ dy n1 dhs2 (h ⳮ hs2) ⳱ dy n2 p
(7)
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Equations 6 and 7 form a complete system sufficient not only to obtain flame speed j, but also to find the distance Z between the ignition points of the first and second dusts (see Fig. 2) that defines the flame structure. The algorithm of finding the solution is the same as in the problem of a monosize suspension. The flame is divided into several zones where the source term in equation 6 is constant. The obtained solutions are matched on the borders between zones to determine the unknown constants in the general solution of differential equation 6. The algebraic equations defining flame speed and parameter Z are then found by matching heat fluxes on the same borders. For the flame with overlapping combustion zones (Z ⬍ 0, configuration 2, Fig. 2) the obtained solution can be written as (hig2 ⳮ 1) j ju Ⳮuⳮ ⳮ ln u ⳱ 1 (hig1 ⳮ 1) l1 l1 u⳱
1 ⳮ exp(ⳮjp) j/l1ⳮ 1 Ⳮ 2(ⳮj) l2 (hig1 ⳮ 1) (his2 ⳮ 1) ⳮ i pl1 (hg2 ⳮ 1) (hs1 ⳮ 1)
Z ⳱ (ln u)/j
冧
Z⬍0
(8)
For flame fronts with separated combustion zones (Z ⬎ 0, configuration 1, Fig. 2) the equation defining flame speed is
冢l1
(1 ⳮ exp(ⳮjp) ⳱ jp
1
Z ⳱ (ln u)/j
ⳮ
hig1 ⳮ 1 l1 hig2 ⳮ 1 l2 Z ⬎ 1
冣
冧
(9)
Equations 8 and 9 can also be used to define the flame speed in the case when the second dust ignites first (configurations 3 and 4, Fig. 2). For this, we simply have to renumber the dusts (1 ↔ 2). Equations 8 and 9 have to be supplemented by expressions connecting the unknown gas temperature at the point of particle ignition with the known autoignition temperature of a single dust particle. These expressions can be easily obtained by solving linear differential equations 7 (they are too cumbersome to show here in their open form): hig1 ⳱ f1(his1, j, l1, l2, p, n1)
(10)
hig2 ⳱ f2(his2, his1, j, l1, l2, p, n1, n2, Z) For slow flames, when the characteristic heating time of the particle is much less than the particle residence time in the preheat zone, that is, when for the first dust jn1 K 1 and for the second dust j(n2p ⳮ Z) K 1, the temperatures of the particle and the gas at the ignition points coincide: hig1 艑 his1, hig2 艑 his2.
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Fig. 3. Dependence of the nondimensional flame speed j in the binary suspension on the ratio of dust concentrations. Ignition temperatures of the corresponding dusts are his2 ⳱ 7, his1 ⳱ 4, and parameter p ⳱ 0.3.
It is interesting to note that the flame speed, defined by equation 9 (separated combustion zones), is actually the speed of the flame propagating through a suspension of the second dust preheated by combustion of the first dust. In other words, combustion of the second dust controls the speed of the flame propagation. The distance between two combustion zones is automatically adjusted in such a way that the heat input provided by the second dust will push the flame through the first dust until it reaches the flame speed equal to the flame speed through the second dust. By analogy with regimes observed in homogeneous mixtures with multistep reactions [3,4], we can name such a mode of flame propagation as a control regime. The opposite of a control regime is a merging regime, in which not only combustion zones but also preheat zones of both combustion fronts overlap: Z ⬍ 1 and jZ K 1. Unlike homogeneous mixtures, where the transition between the control and merging regimes is rapid [4], in dust suspensions this transition occurs gradually through an intermittent transient regime when combustion zones simply overlap (Z ⬍ 0). The flame speed in the merging and transient regimes is defined by equation 8. The nondimensional flame speed and a corresponding flame front structure obtained from equations 8–10 are mapped in Fig. 3 in relation to mixture composition (l2/l1 ratio) and initial mass concentration of the first dust in the suspension (different values of the parameter l1). If the final flame temperature of the mixture is lower than the ignition temperature of the second i dust (l2 Ⳮ l1 (his1 ⳮ 1)/(hs2 ⳮ 1) ⬍ 1), the second dust does not ignite (shaded area in Fig. 3). As we
neglect the specific heat of the solid phase in lean dust mixtures, the flame speed in this region is not influenced by the presence of the second dust. The speed remains constant until the flame temperature reaches the ignition temperature of the second dust (the right border of the shaded area). It starts to increase when the second dust ignites. At the beginning, when the second dust concentration is low, the distance between combustion fronts is large and the influence of the second dust on the flame speed is minimal. However, as the concentration of the second dust increases, the distance between the two combustion fronts decreases, and at some dust concentrations (depending on concentration of the first dust), their combustion fronts start to overlap. At large initial dust concentrations (e.g., l1 ⱖ 2), the first dust alone can provide a flame temperature that is greater than the ignition temperature of the second dust. The combustion fronts in this case may overlap even at the smallest concentrations of the second dust. We will now apply this model to the problem of flame propagation in suspension of two dusts of different particle sizes but of the same chemical nature. For the sake of simplicity, we will assume that the ignition temperatures for both dusts are equal. (This is true, for example, for aluminum dust where the particle ignition temperature is a weak function of particle size as it is basically defined by the melting of the aluminum oxide protecting film [8].) The behavior of the flame speed and flame front structure is analyzed as a function of the composition of the suspension (mass percent ratio of each dust), while the total mass concentration of the suspension (B) is kept constant. The result of the analysis is shown in Fig. 4 for dusts with a particle size ratio equal to 4 that gives, in accordance to equation 5, the same ratio of flame speeds (V ⬃ 1/R) in the corresponding monosize suspensions. In the same figure, we have also plotted flame speeds calculated from equation 5 (equation for the flame speed in a monosize suspension) using different representations of the average particle size in binary suspension. One representation uses an arithmetic mean particle radius r10 ⳱
兺 rini冫兺 ni
the other uses the mass mean particle size [15] r43 ⳱
兺 r4i ni冫兺 ri3ni
For binary suspension, mass mean particle radius can be calculated using the expression r43 ⳱ r1 A Ⳮ r2 (1 ⳮ A) where A is the mass percent of one of the dusts. The arithmetic mean particle radius is
FLAME SPEED IN A BINARY SUSPENSION
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residence heating time of the larger dust. However, higher first dust concentrations also provide higher heating temperatures that will shorten time needed for the large dust to reach the ignition temperature. Experimental Dusts and Apparatus Due to the effects of gravity (particle sedimentation, natural convection), the experiments on flame propagation in dust suspensions are restricted in the normal laboratory environment to small particle sizes, low dust concentrations, and relatively fastburning fuels [6]. It is difficult to obtain sufficient quantities of dusts that have narrow and non-overlapping particle size distributions in the small particle size range (r ⬍ 10 lm) required for experiments affected by gravity. Thus, in the present experiments on flame propagation in binary dust suspensions, we used fine industrial dusts of different chemical natures, namely aluminum and manganese, but with a close particle-size distribution Fig. 4. Dependence of the normalized flame speed (A) and flame structure (B) on composition of the binary suspension of dusts with different particle sizes. Total mass concentration of the suspension remains constant, dusts are chemically equivalent, and the particle size ratio is 4:1.
r01 ⳱ (r1n1 Ⳮ r2n2)/(n1 Ⳮ n2) where n1 ⬃ A/r31 and n2 ⬃ (1 ⳮ A)/r32. As can be seen from the flame speed plots in Fig. 4A, for the problem of flame propagation the arithmetic mean radius gives a completely erroneous representation of binary suspension. The mass mean particle size gives a good speed prediction only when the mass concentration of the smaller dust is less than 50%. In general, we failed to find an average size that can match the flame speed predicted by the theory over the whole range of mixture compositions. The corresponding structure of the flame (parameter Z) is shown in Fig. 4B as a function of the suspension composition. Due to a larger inertia in heat transfer, dust that is comprised of larger particles reaches the ignition temperature when the combustion of smaller dust is already completed (Z ⬎ 1). It is interesting to note that the distance between ignition points of the first and the second dusts (Z) reaches maximum at about 50%–50% composition. The non-monotone behavior of Z is the result of competition between the two factors that influence the second dust heating. On one hand, the increase in concentration of small particles (first dust) leads to a higher flame speed and therefore reduces the
r32 艑 3–5 lm The choice of aluminum was obvious as its ignition and combustion characteristics are relatively well known, and the flame propagation in aluminum dust clouds has been studied extensively by the authors and in literature [9,10]. In order to verify the theoretical model, the other dust should have sufficiently different (lower) flame propagation speed and also lower ignition temperature. After the testing several dust suspensions (Fe, Cr, Mn), manganese was chosen due to the higher flame temperature and better fluidizing characteristics of the available powders. The stoichiometric concentration of the aluminum dust suspension in air is about Bst ⳱ 310 g/m3, and the adiabatic flame temperature Tad ⳱ 3540 K. The stoichiometric concentration of the manganese/air suspension is about 830 g/m3 (MnO is a main final product) and Tad ⳱ 2766 K. Since the theory is limited to fuel lean mixtures, the total dust concentration in our experiments with binary mixtures did not exceed 250 g/m3. Stabilized Bunsen-type dust flames have been used in our recent work to measure burning velocities in aluminum-rich dust clouds [10]. For fuellean dust suspensions, this relatively accurate method is difficult to apply, as the lean dust flame is very unstable. Thus, for the purposes of the present work, we observed laminar dust flames propagating in tubes. In the present experiments on the measurement of flame propagation speeds, we employed the same apparatus that was used previously in our quenching distance experiments [5]. The details of the dust dispersion and optical dust monitoring system are also
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Fig. 5. X ⳮ t diagrams of the experimentally observed flame propagation modes through dust clouds in a semiopen tube.
TABLE 1 Flame speeds measured in Mn/Al suspensions Flame Speed at B ⬇ 250 g/m3
Mixture Composition Al 33% Mn/67% Al 50% Mn/50% Al 67% Mn/33% Al Mn
52 Ⳳ 5 cm/s 43 Ⳳ 5 cm/s 23 Ⳳ 3 cm/s No flame propagation below B ⳱ 350 g/m3 Flame propagates only in pure oxygen at B ⬎ 400 g/m3
described elsewhere [5,10]. The optical dust monitoring system was calibrated (see also Ref. [5] for the calibration procedure) with the same binary dust mixtures that were used in flame experiments. The dust was ignited at the upper open end of the Pyrex tube (length ⳱ 120 cm, i.d. ⳱ 50 mm) by an electrically heated tungsten wire. Immediately after ignition, the gas flow was abruptly cut off by a solenoid valve, causing the flame to propagate downward into the tube of the now-quiescent dust cloud. The optical signal that indicates the dust concentration was recorded by the computer data acquisition system, and propagating flames were filmed by a video camera through a set of dense neutral filters. Experimental Observations Flame propagation through dust suspensions in tubes is a complex, non-steady phenomenon similar to non-steady gas flame propagation in tubes observed by Gue´noche [11]. As for gas flames, acoustic instabilities play a major role in flame turbulization in dust suspensions. The peculiarities of the flame
acoustic coupling in dust suspensions was experimentally studied by one of the authors [12,13]. In general, three major scenarios can be observed for dust flame propagating from the upper open end to the closed end of the tube, as shown in x ⳮ t diagram of flame propagation in Fig. 5. Right after ignition, a laminar flame with constant speed and a parabolic shape is observed in cases A and B (Fig. 5). In case A, acoustic oscillations begin at about 20–30 cm from the open end of the tube. After propagating in an oscillating mode for about 5–7 cm, the flame quenches. In case B, propagation of the flame is identical to case A before the onset of acoustic oscillations. In this case, however, after slowing down during the oscillating stage, the flame then rapidly accelerates, reaching speeds of about 1–2 m/s. It slows down again in the middle of the tube, producing several large-amplitude, violent pulsations, and accelerates again rapidly, reaching speeds in excess of 5 m/s. In some cases (scenario C), the flame starts to accelerate right after ignition and reaches a speed of about 1–2 m/s after the first 10–20 cm of propagation. Only the flame speed during the laminar stage of flame propagation was recorded for the purposes of the present work. We have also assumed that the observed laminar flame speed is proportional to the burning velocity. (This at least was verified for aluminum dust suspensions in which burning velocities that were measured with Bunsen burner at different oxygen concentrations [10] were proportional to laminar flame speeds in tubes of similar mixtures [5].) Three different binary suspensions with compositions 33% Mn/67% Al (mass percent), 50% Mn/ 50% Al, and 67% Mn/33% Al and pure Al and Mn suspensions have been tested. The observed average flame propagation speeds in different suspensions at a concentration of about 250 g/m3 are presented in Table 1. As shown in Table 1, the flame did not propagate in 67% Mn/33% Al suspension at dust concentrations lower than 350 g/m3. At higher dust concentrations, the observed flame was not laminar. In pure manganese/air suspensions we did not observe flame propagation even at very high dust concentrations, and only the manganese/oxygen suspension was able to support a flame. The normalized (by flame speed in pure aluminum suspension) experimental flame speed data are compared in Fig. 6 with similarly normalized theoretical predictions. In these calculations, we suppose that the ignition temperature of aluminum particles is about 2300 K [8] and that of manganese particles is about 1200 K [14]. The ratio of the combustion times of equal size manganese and aluminum particles is estimated to be about p ⳱ 3.5 in accordance to the law of particle combustion in the diffusion mode and the ratio of their specific densities and stoichiometric coefficients.
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flames in the binary dust suspension can be extended into a description of the multisize dust combustion. The future microgravity experiments on dust combustion planned by the authors and other investigators will permit verification of the developed theories in a wide range of particle sizes and for different fuels. Acknowledgments This work is supported by the Canadian Space Agency and NASA under contract no. 9F007-9-6042/001/SR. The authors are also grateful to Dr. Andrew Higgins for the useful discussion and comments on theoretical and experimental results presented in this paper.
REFERENCES
Fig. 6. (A) Comparison of the theoretically predicted (lines) and experimentally observed (crosses) relative flame speeds in binary suspensions of manganese and aluminum powders. (B) Theoretically predicted flame structure.
As can be seen in Fig. 6, the model correctly predicts the behavior of the flame speed as a function of dust composition. It also predicts that the binary flame (when both dusts burn) exists only for mixtures with an aluminum concentration higher than 30% (flame temperature higher than 2300 K). When the aluminum concentration in the mixture approaches this point, the manganese and aluminum combustion fronts began to separate (Fig. 6B). At lower than 30% aluminum concentration, aluminum cannot ignite and plays the role of an inert additive. Conclusion In conclusion, we would like to emphasize the results of the present work which indicate that representing real polydisperse dusts by some average particle size is not adequate for flame propagation models. Instead, a flame model that takes into account the complex spatial structure of the polydisperse combustion has to be developed. The simple theoretical approach used in the present work (neglecting oxygen consumption and specific heat of the solid phase in fuel-lean dust suspensions) to describe
1. Tambour, Y., Combust. Flame 44:15–28 (1985). 2. Greenberg, J. B., and Shpilberg, I., Atom. Sprays 5:507–523 (1995). 3. Zeldovich, Ya. B., Barenblatt, G. I., Librovich, V. B., and Makhviladze, G. M., The Mathematical Theory of Combustion and Explosions, Consultants Bureau, New York, 1985, p. 411. 4. Volpert, V. A., Khaikin, B. I., and Khudyaev, S. I., The Problems of Technological Combustion, Vol. 1, Chernogolovka, USSR, 1981, p. 110 (in Russian). 5. Goroshin, S., Bidabadi, M., and Lee, J. H. S., Combust. Flame 105:147–160 (1995). 6. Goroshin, S., and Lee, J. H. S., in Fifth International Microgravity Combustion Workshop, NASA Publications CP-1999-208917, Cleveland, OH, 1999, pp. 123– 126. 7. Seshadry, K., Berlad, A. L., and Tangirala, V., Combust. Flame 89:333–342 (1992). 8. Friedman, R., and Macek, A., Combust. Flame 6:9–19 (1962). 9. Ballal, D. R., Proc. R. Soc. London A 385:21–51 (1983). 10. Goroshin, S., Fomenko, I., and Lee, J. H. S., Proc. Combust. Inst. 26:1961–1967 (1996). 11. Gue´noche, H., Non-Steady Flame Propagation (G. H. Markstein ed.), Macmillan, New York, 1964, p. 107. 12. Goroshin, S., Shevchuk, V. G., and Ageev, N., Combust. Expl. Shock Waves 17:595–600 (1981). 13. Aslanov, S. K., Shevchuk, V. G., Kostyshin, Yu. N., Boichuk, L. V., and Goroshin, S., Combust. Expl. Shock Waves 29:163–169 (1993). 14. Breiter, A. L., Mal’tsev, V. M., and Popov, E. I., Combust. Expl. Shock Waves 14:558–570 (1976). 15. Reist, P., Aerosol Science and Technology, McGrawHill, New York, 1993, p. 26.
FLAME SPEED IN A BINARY SUSPENSION OF SOLID FUEL PARTICLES SAMUEL GOROSHIN, MASSIMILIANO KOLBE and JOHN H. S. LEE Department of Mechanical Engineering McGill University 817 Sherbrooke St. West Montreal, Quebec, Canada H3A 2K6
Most natural and industrial combustible dusts have a wide distribution of particle sizes. Yet, the majority of experimental data on flame propagation in dust clouds are given in relation to some average particle size, and all known theoretical models of dust combustion consider only monosize suspensions. Since the ignition temperature and combustion rate of an individual dust particle are functions of particle size, the flame in real dust suspensions has a complex, multistage structure. As a first step toward understanding multisize dust combustion, the combustion of a suspension of two monosize powders (that in general can be also of different chemical nature) is investigated in the present work theoretically and experimentally. A simple analytical model developed for the flame in a fuel-lean binary suspension permits flame speed and structure to be analyzed as a function of the dust composition and combustion properties of individual particles. The flame speeds predicted by the binary model were compared with flame speeds calculated from a model of monosize dust flame using various average particle size representations. It is shown that averaging of the particle size in general fails to correctly predict the flame speed over the wide range of the binary dust compositions. The flame propagation speed in a binary suspension of aluminum and manganese powders was investigated experimentally by observing the laminar stage of flame propagation in a semi-open vertical tube. The model correctly predicts 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.
Introduction With the exception of a few special dusts (e.g., lycopodium), most dusts used in experiments have a continuous spectrum of particle sizes over a specific size range. Yet, the experimental data on the combustion parameters of dust suspensions (flame speeds, flammability limits, etc.) are commonly given only in relation to some average particle size. This makes it difficult to qualitatively compare experimental results from different sources as every particular dust has a unique particle size distribution. The polydisperse nature of real dusts also hampers the comparison of the results of theoretical (which, in most cases, consider monosize suspensions) and experimental investigations. While combustion of volatile, polydisperse sprays has been numerically investigated in several works (for example, Refs. [1,2]), flame propagation in a non-volatile multisize dust suspension has yet to be addressed. The combustion and ignition characteristics of individual fuel particles are functions of particle size. Therefore, the flame in a suspension of real polydisperse dust has a complex multistage spatial structure analogous to the flame structure in homogeneous systems with parallel reactions [3,4]. As a first step
toward the understanding of phenomena of multistage dust combustion, the present work analyzes combustion of a simple binary suspension formed by the dispersion of a mixture of two monosize powders. In the general case considered below, these powders cannot only be of different particle sizes but, moreover, can be of a different chemical nature. Thus, we will also be able to address a practically important question on how the addition of a reactive (low combustion time) component can influence the flame speed in an otherwise low-burning-rate suspension. The theoretical predictions made by this model are validated experimentally in the present work by observing flame speeds in aluminum/manganese suspensions. In order to obtain an analytical solution of the problem, we have restricted our consideration to fuel-lean mixtures of non-volatile solids. We assume that the predominant mechanism of flame propagation in such suspensions is molecular heat conductivity [5,6]. We also assume that the combustion products of both dusts are solid and do not interact chemically with each other. Due to the excess of oxidizer, the reacting dusts do not compete in oxygen consumption and, therefore, heat conduction is the only channel of their interaction.
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冦
0, ⳮ⬁ ⬍ y ⬍ 0 d2h dh ⳱ W ⳮlj(hsi ⳮ 1), 0 ⱕ y ⱕ 1 2ⳮj dy dy 0, y ⬎ 1
(1)
y → ⳮ⬁, h ⳱ 1; y ⳱ 1, h ⳱ h⬘; dh/dy|0Ⳮ ⳱ dh/dy|0ⳮ
Fig. 1. Structure of the dust flame in monosize suspension.
Theoretical Analysis Flame Speed in the Monosize Suspension The model of a flame in a binary suspension developed in this paper is based on a simple description of a monosize dust flame that was previously used by the authors to study dust flame quenching [5]. Our conjecture of the flame structure in monosize suspensions is illustrated in Fig. 1. In the preheat zone (ⳮ⬁ ⬍ x ⬍ 0), the reaction rate is negligibly small and the particles are heated by the gas. The difference in temperature between the dust particles and the gas progressively increases due to inertia in heat exchange. At some instant, the amount of heat generated by the reaction on a particle surface becomes larger than the heat flux from the hot gas, and the particle ignites. The transient period from ignition to stable particle combustion (controlled by oxygen diffusion) is short in comparison to the total particle combustion time sc. We also suppose that at the ignition point, the particle temperature is close to the autoignition temperature of a single particle Tis (the ignition temperature during a slow, quasi-stationary heating [5]). The reaction rate during diffusive combustion is considered to be constant and equal to an average value, ms/sc (where ms is the mass of the particle). As was shown in Ref. [7], the equation governing heat diffusivity 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 x⳱
x⬘
冮
0
(q/qu)dx
and by assuming that the thermal conductivity of the gas is a linear function of temperature: k ⳱ ku(T/ Tu) (here the index u is denoting unburned mixture). With these assumptions, the governing heat diffusivity equation and the boundary conditions for the problem illustrated in Fig. 1 can be written in a dimensionless form:
Here, nondimensional gas temperature h is defined as h ⳱ Tg/Tgu; nondimensional coordinate y, as y ⳱ x/tusc (where tu is the flame speed). Parameter j and parameter l is a nondimensional flame speed and a nondimensional dust concentration, respectively, defined as j ⳱ t2usc/au, l ⳱ BQ/cpqgu(T1s ⳮ Tu) where au ⳱ k/cpqgu; B is dust concentration; Q is the heat of reaction; and subscripts s and g, denote solid particles and gas, correspondingly. The inert heating of the particle in the preheat zone (x ⬍ 0, Fig. 1) is described by the equation dhs (h ⳮ hs) ⳱ dy n
(2)
y → ⳮ⬁, h ⳱ hs ⳱ 1; y ⳱ 0, hs ⳱ hs1 Here n ⳱ [(r2csqs/3aucpqg)]/sc is a ratio of the characteristic particle heat exchange time and combustion time of the particle. Parameter n is close to unity for nonvolatile fuel and Lewis number (Le) ⳱ 1. By solving heat transfer equation 1 in each two flame zones and by matching the heat fluxes obtained from this solution on the boundary of the preheat and combustion zone (x ⳱ 0), the algebraic equation for the nondimensional burning velocity can be found.
冢hh
j⳱l
i s i
ⳮ 1 [1 ⳮ exp(ⳮj)] ⳮ 1
冣
(3)
The ratio of the gas and flame temperatures at the ignition point that is present in equation 3 can be found from the solution of linear equation 2: (h is ⳮ 1)/(hi ⳮ 1) ⳱ (1 Ⳮ jn)
(4)
And the final equation that completely defines the flame speed in a monosize suspension is j ⳱ l(1 Ⳮ jn) [1 ⳮ exp(ⳮj)]
(5)
The same approach that was used above to obtain flame speed in monosize suspension will be used below to analyze a stationary flame in a binary dust suspension. Flame Speed and Structure in the Binary Suspension In comparison to monosize suspension, the number of parameters that characterize a binary dust suspension doubles. As in a monosize suspension, parameters li (i ⳱ 1,2) are the dimensionless mass
FLAME SPEED IN A BINARY SUSPENSION
Fig. 2. Possible flame configurations in binary dust suspension. Lower step marks the combustion front of the first dust (lower particle autoignition temperature); higher step marks the combustion front of the second dust (higher particle autoignition temperature). Step width indicates combustion zone of corresponding dust.
dust concentrations of corresponding dusts, and j ⳱ t2usc1/au is the dimensionless flame speed. We also introduce a new parameter, p ⳱ sc2/sc1, that is the ratio of combustion times of the first and the second dusts. For convenience, we will also assign the number 1 to the dust with a lower particle ignition temperature. All possible configurations of the stationary flame front in binary suspension are schematically shown 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. Despite having a higher ignition temperature, the particles of the second dust can heat up faster and may ignite first if their characteristic time of heat exchange is smaller than that of the first dust (i.e., when n1 ⬎ n2p, configurations 3 and 4 in Fig. 2). The heat transfer equation for all flame configurations shown in Fig. 2 can be written in the same form as equation 1 for a monosize suspension: d2h dh ⳮ j ⳱ dy2 dy
(6)
冦
0, combustion is absent ⳮjl1 (his1 ⳮ 1), only the first dust burns W ⳮjl2 (his2 ⳮ 1)/p, only the second dust burns ⳮjl1 (his1 ⳮ 1) ⳮ jl2 (h1s2 ⳮ 1)/p, both dusts burn
The inert heating of particles before their ignition is governed by dhs1 (h ⳮ hs1) ⳱ dy n1 dhs2 (h ⳮ hs2) ⳱ dy n2 p
(7)
2813
Equations 6 and 7 form a complete system sufficient not only to obtain flame speed j, but also to find the distance Z between the ignition points of the first and second dusts (see Fig. 2) that defines the flame structure. The algorithm of finding the solution is the same as in the problem of a monosize suspension. The flame is divided into several zones where the source term in equation 6 is constant. The obtained solutions are matched on the borders between zones to determine the unknown constants in the general solution of differential equation 6. The algebraic equations defining flame speed and parameter Z are then found by matching heat fluxes on the same borders. For the flame with overlapping combustion zones (Z ⬍ 0, configuration 2, Fig. 2) the obtained solution can be written as (hig2 ⳮ 1) j ju Ⳮuⳮ ⳮ ln u ⳱ 1 (hig1 ⳮ 1) l1 l1 u⳱
1 ⳮ exp(ⳮjp) j/l1ⳮ 1 Ⳮ 2(ⳮj) l2 (hig1 ⳮ 1) (his2 ⳮ 1) ⳮ i pl1 (hg2 ⳮ 1) (hs1 ⳮ 1)
Z ⳱ (ln u)/j
冧
Z⬍0
(8)
For flame fronts with separated combustion zones (Z ⬎ 0, configuration 1, Fig. 2) the equation defining flame speed is
冢l1
(1 ⳮ exp(ⳮjp) ⳱ jp
1
Z ⳱ (ln u)/j
ⳮ
hig1 ⳮ 1 l1 hig2 ⳮ 1 l2 Z ⬎ 1
冣
冧
(9)
Equations 8 and 9 can also be used to define the flame speed in the case when the second dust ignites first (configurations 3 and 4, Fig. 2). For this, we simply have to renumber the dusts (1 ↔ 2). Equations 8 and 9 have to be supplemented by expressions connecting the unknown gas temperature at the point of particle ignition with the known autoignition temperature of a single dust particle. These expressions can be easily obtained by solving linear differential equations 7 (they are too cumbersome to show here in their open form): hig1 ⳱ f1(his1, j, l1, l2, p, n1)
(10)
hig2 ⳱ f2(his2, his1, j, l1, l2, p, n1, n2, Z) For slow flames, when the characteristic heating time of the particle is much less than the particle residence time in the preheat zone, that is, when for the first dust jn1 K 1 and for the second dust j(n2p ⳮ Z) K 1, the temperatures of the particle and the gas at the ignition points coincide: hig1 艑 his1, hig2 艑 his2.
2814
FIRE—Ignition and Flame Spread
Fig. 3. Dependence of the nondimensional flame speed j in the binary suspension on the ratio of dust concentrations. Ignition temperatures of the corresponding dusts are his2 ⳱ 7, his1 ⳱ 4, and parameter p ⳱ 0.3.
It is interesting to note that the flame speed, defined by equation 9 (separated combustion zones), is actually the speed of the flame propagating through a suspension of the second dust preheated by combustion of the first dust. In other words, combustion of the second dust controls the speed of the flame propagation. The distance between two combustion zones is automatically adjusted in such a way that the heat input provided by the second dust will push the flame through the first dust until it reaches the flame speed equal to the flame speed through the second dust. By analogy with regimes observed in homogeneous mixtures with multistep reactions [3,4], we can name such a mode of flame propagation as a control regime. The opposite of a control regime is a merging regime, in which not only combustion zones but also preheat zones of both combustion fronts overlap: Z ⬍ 1 and jZ K 1. Unlike homogeneous mixtures, where the transition between the control and merging regimes is rapid [4], in dust suspensions this transition occurs gradually through an intermittent transient regime when combustion zones simply overlap (Z ⬍ 0). The flame speed in the merging and transient regimes is defined by equation 8. The nondimensional flame speed and a corresponding flame front structure obtained from equations 8–10 are mapped in Fig. 3 in relation to mixture composition (l2/l1 ratio) and initial mass concentration of the first dust in the suspension (different values of the parameter l1). If the final flame temperature of the mixture is lower than the ignition temperature of the second i dust (l2 Ⳮ l1 (his1 ⳮ 1)/(hs2 ⳮ 1) ⬍ 1), the second dust does not ignite (shaded area in Fig. 3). As we
neglect the specific heat of the solid phase in lean dust mixtures, the flame speed in this region is not influenced by the presence of the second dust. The speed remains constant until the flame temperature reaches the ignition temperature of the second dust (the right border of the shaded area). It starts to increase when the second dust ignites. At the beginning, when the second dust concentration is low, the distance between combustion fronts is large and the influence of the second dust on the flame speed is minimal. However, as the concentration of the second dust increases, the distance between the two combustion fronts decreases, and at some dust concentrations (depending on concentration of the first dust), their combustion fronts start to overlap. At large initial dust concentrations (e.g., l1 ⱖ 2), the first dust alone can provide a flame temperature that is greater than the ignition temperature of the second dust. The combustion fronts in this case may overlap even at the smallest concentrations of the second dust. We will now apply this model to the problem of flame propagation in suspension of two dusts of different particle sizes but of the same chemical nature. For the sake of simplicity, we will assume that the ignition temperatures for both dusts are equal. (This is true, for example, for aluminum dust where the particle ignition temperature is a weak function of particle size as it is basically defined by the melting of the aluminum oxide protecting film [8].) The behavior of the flame speed and flame front structure is analyzed as a function of the composition of the suspension (mass percent ratio of each dust), while the total mass concentration of the suspension (B) is kept constant. The result of the analysis is shown in Fig. 4 for dusts with a particle size ratio equal to 4 that gives, in accordance to equation 5, the same ratio of flame speeds (V ⬃ 1/R) in the corresponding monosize suspensions. In the same figure, we have also plotted flame speeds calculated from equation 5 (equation for the flame speed in a monosize suspension) using different representations of the average particle size in binary suspension. One representation uses an arithmetic mean particle radius r10 ⳱
兺 rini冫兺 ni
the other uses the mass mean particle size [15] r43 ⳱
兺 r4i ni冫兺 ri3ni
For binary suspension, mass mean particle radius can be calculated using the expression r43 ⳱ r1 A Ⳮ r2 (1 ⳮ A) where A is the mass percent of one of the dusts. The arithmetic mean particle radius is
FLAME SPEED IN A BINARY SUSPENSION
2815
residence heating time of the larger dust. However, higher first dust concentrations also provide higher heating temperatures that will shorten time needed for the large dust to reach the ignition temperature. Experimental Dusts and Apparatus Due to the effects of gravity (particle sedimentation, natural convection), the experiments on flame propagation in dust suspensions are restricted in the normal laboratory environment to small particle sizes, low dust concentrations, and relatively fastburning fuels [6]. It is difficult to obtain sufficient quantities of dusts that have narrow and non-overlapping particle size distributions in the small particle size range (r ⬍ 10 lm) required for experiments affected by gravity. Thus, in the present experiments on flame propagation in binary dust suspensions, we used fine industrial dusts of different chemical natures, namely aluminum and manganese, but with a close particle-size distribution Fig. 4. Dependence of the normalized flame speed (A) and flame structure (B) on composition of the binary suspension of dusts with different particle sizes. Total mass concentration of the suspension remains constant, dusts are chemically equivalent, and the particle size ratio is 4:1.
r01 ⳱ (r1n1 Ⳮ r2n2)/(n1 Ⳮ n2) where n1 ⬃ A/r31 and n2 ⬃ (1 ⳮ A)/r32. As can be seen from the flame speed plots in Fig. 4A, for the problem of flame propagation the arithmetic mean radius gives a completely erroneous representation of binary suspension. The mass mean particle size gives a good speed prediction only when the mass concentration of the smaller dust is less than 50%. In general, we failed to find an average size that can match the flame speed predicted by the theory over the whole range of mixture compositions. The corresponding structure of the flame (parameter Z) is shown in Fig. 4B as a function of the suspension composition. Due to a larger inertia in heat transfer, dust that is comprised of larger particles reaches the ignition temperature when the combustion of smaller dust is already completed (Z ⬎ 1). It is interesting to note that the distance between ignition points of the first and the second dusts (Z) reaches maximum at about 50%–50% composition. The non-monotone behavior of Z is the result of competition between the two factors that influence the second dust heating. On one hand, the increase in concentration of small particles (first dust) leads to a higher flame speed and therefore reduces the
r32 艑 3–5 lm The choice of aluminum was obvious as its ignition and combustion characteristics are relatively well known, and the flame propagation in aluminum dust clouds has been studied extensively by the authors and in literature [9,10]. In order to verify the theoretical model, the other dust should have sufficiently different (lower) flame propagation speed and also lower ignition temperature. After the testing several dust suspensions (Fe, Cr, Mn), manganese was chosen due to the higher flame temperature and better fluidizing characteristics of the available powders. The stoichiometric concentration of the aluminum dust suspension in air is about Bst ⳱ 310 g/m3, and the adiabatic flame temperature Tad ⳱ 3540 K. The stoichiometric concentration of the manganese/air suspension is about 830 g/m3 (MnO is a main final product) and Tad ⳱ 2766 K. Since the theory is limited to fuel lean mixtures, the total dust concentration in our experiments with binary mixtures did not exceed 250 g/m3. Stabilized Bunsen-type dust flames have been used in our recent work to measure burning velocities in aluminum-rich dust clouds [10]. For fuellean dust suspensions, this relatively accurate method is difficult to apply, as the lean dust flame is very unstable. Thus, for the purposes of the present work, we observed laminar dust flames propagating in tubes. In the present experiments on the measurement of flame propagation speeds, we employed the same apparatus that was used previously in our quenching distance experiments [5]. The details of the dust dispersion and optical dust monitoring system are also
2816
FIRE—Ignition and Flame Spread
Fig. 5. X ⳮ t diagrams of the experimentally observed flame propagation modes through dust clouds in a semiopen tube.
TABLE 1 Flame speeds measured in Mn/Al suspensions Flame Speed at B ⬇ 250 g/m3
Mixture Composition Al 33% Mn/67% Al 50% Mn/50% Al 67% Mn/33% Al Mn
52 Ⳳ 5 cm/s 43 Ⳳ 5 cm/s 23 Ⳳ 3 cm/s No flame propagation below B ⳱ 350 g/m3 Flame propagates only in pure oxygen at B ⬎ 400 g/m3
described elsewhere [5,10]. The optical dust monitoring system was calibrated (see also Ref. [5] for the calibration procedure) with the same binary dust mixtures that were used in flame experiments. The dust was ignited at the upper open end of the Pyrex tube (length ⳱ 120 cm, i.d. ⳱ 50 mm) by an electrically heated tungsten wire. Immediately after ignition, the gas flow was abruptly cut off by a solenoid valve, causing the flame to propagate downward into the tube of the now-quiescent dust cloud. The optical signal that indicates the dust concentration was recorded by the computer data acquisition system, and propagating flames were filmed by a video camera through a set of dense neutral filters. Experimental Observations Flame propagation through dust suspensions in tubes is a complex, non-steady phenomenon similar to non-steady gas flame propagation in tubes observed by Gue´noche [11]. As for gas flames, acoustic instabilities play a major role in flame turbulization in dust suspensions. The peculiarities of the flame
acoustic coupling in dust suspensions was experimentally studied by one of the authors [12,13]. In general, three major scenarios can be observed for dust flame propagating from the upper open end to the closed end of the tube, as shown in x ⳮ t diagram of flame propagation in Fig. 5. Right after ignition, a laminar flame with constant speed and a parabolic shape is observed in cases A and B (Fig. 5). In case A, acoustic oscillations begin at about 20–30 cm from the open end of the tube. After propagating in an oscillating mode for about 5–7 cm, the flame quenches. In case B, propagation of the flame is identical to case A before the onset of acoustic oscillations. In this case, however, after slowing down during the oscillating stage, the flame then rapidly accelerates, reaching speeds of about 1–2 m/s. It slows down again in the middle of the tube, producing several large-amplitude, violent pulsations, and accelerates again rapidly, reaching speeds in excess of 5 m/s. In some cases (scenario C), the flame starts to accelerate right after ignition and reaches a speed of about 1–2 m/s after the first 10–20 cm of propagation. Only the flame speed during the laminar stage of flame propagation was recorded for the purposes of the present work. We have also assumed that the observed laminar flame speed is proportional to the burning velocity. (This at least was verified for aluminum dust suspensions in which burning velocities that were measured with Bunsen burner at different oxygen concentrations [10] were proportional to laminar flame speeds in tubes of similar mixtures [5].) Three different binary suspensions with compositions 33% Mn/67% Al (mass percent), 50% Mn/ 50% Al, and 67% Mn/33% Al and pure Al and Mn suspensions have been tested. The observed average flame propagation speeds in different suspensions at a concentration of about 250 g/m3 are presented in Table 1. As shown in Table 1, the flame did not propagate in 67% Mn/33% Al suspension at dust concentrations lower than 350 g/m3. At higher dust concentrations, the observed flame was not laminar. In pure manganese/air suspensions we did not observe flame propagation even at very high dust concentrations, and only the manganese/oxygen suspension was able to support a flame. The normalized (by flame speed in pure aluminum suspension) experimental flame speed data are compared in Fig. 6 with similarly normalized theoretical predictions. In these calculations, we suppose that the ignition temperature of aluminum particles is about 2300 K [8] and that of manganese particles is about 1200 K [14]. The ratio of the combustion times of equal size manganese and aluminum particles is estimated to be about p ⳱ 3.5 in accordance to the law of particle combustion in the diffusion mode and the ratio of their specific densities and stoichiometric coefficients.
FLAME SPEED IN A BINARY SUSPENSION
2817
flames in the binary dust suspension can be extended into a description of the multisize dust combustion. The future microgravity experiments on dust combustion planned by the authors and other investigators will permit verification of the developed theories in a wide range of particle sizes and for different fuels. Acknowledgments This work is supported by the Canadian Space Agency and NASA under contract no. 9F007-9-6042/001/SR. The authors are also grateful to Dr. Andrew Higgins for the useful discussion and comments on theoretical and experimental results presented in this paper.
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Fig. 6. (A) Comparison of the theoretically predicted (lines) and experimentally observed (crosses) relative flame speeds in binary suspensions of manganese and aluminum powders. (B) Theoretically predicted flame structure.
As can be seen in Fig. 6, the model correctly predicts the behavior of the flame speed as a function of dust composition. It also predicts that the binary flame (when both dusts burn) exists only for mixtures with an aluminum concentration higher than 30% (flame temperature higher than 2300 K). When the aluminum concentration in the mixture approaches this point, the manganese and aluminum combustion fronts began to separate (Fig. 6B). At lower than 30% aluminum concentration, aluminum cannot ignite and plays the role of an inert additive. Conclusion In conclusion, we would like to emphasize the results of the present work which indicate that representing real polydisperse dusts by some average particle size is not adequate for flame propagation models. Instead, a flame model that takes into account the complex spatial structure of the polydisperse combustion has to be developed. The simple theoretical approach used in the present work (neglecting oxygen consumption and specific heat of the solid phase in fuel-lean dust suspensions) to describe
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