Reaction of a Particle Suspension in a Rapidly-Heated Oxidizing Gas

DOI: 10.1002/prep.201400269

Reaction of a Particle Suspension in a Rapidly-HeatedOxidizing GasMichael Soo,*[a] Samuel Goroshin,[a] Jeffrey M. Bergthorson,[a] and David L. Frost[a]

1 Introduction

A suspension of reactive solid fuel particles dispersedwithin hot oxidizing gases may be formed, for example, fol-lowing the detonation or deflagration of a metallized ener-getic material or during an accidental explosion in a coalmine or the process industries [1]. In the first example, thereaction time for the energetic material is typically muchshorter than the characteristic time for heat exchange be-tween the particles and the combustion products; hencethe particle suspension may be treated as if it were sud-denly exposed to a hot oxidizing atmosphere. Measures ofthe performance of the energetic material, such as thework done by the expanding products and blast wave foran explosive, or the specific impulse of a propellant, willdepend critically on the subsequent reaction delay andcombustion time of the solid-fuel suspension. The overallsystem performance may be tailored by choosing appropri-ate particle sizes and other system parameters as guidedby the available experimental data and predictions fromtheoretical models.

Historically, most of the information on solid-fuel reactivi-ty has been obtained from laboratory experiments usingrelatively large single particles, with sizes ranging from tensto hundreds of micrometers. In the experiments, the parti-cles are typically either injected into an oxidizing flow pro-duced by a hydrocarbon flame [2], or heated by the gasbehind a reflected shock within a shock tube [3]. The ex-

perimental results are then used to construct models thatare extended to predict the combustion behavior of muchsmaller particles and of dense particulate suspensions thatare common in practical particle-fuel systems. The underly-ing assumption made when extrapolating to practical ener-getic materials is that the combustion physics on which themodels are built do not change with particle size or con-centration. However, the validity of these assumptions istheoretically questionable and difficult to justify, especiallyin light of recent experiments using nano-sized particles[4, 5] and with large particle concentrations [6–8].

Experiments in which a small amount of aluminumpowder was injected into a shock tube and heated by a re-flected shock have indicated that the aluminum particlecombustion regime shifts from being predominantly diffu-sion controlled to a kinetically-controlled reaction regimewhen the particle size is reduced to a value below aboutten micrometers [9]. In contrast with isolated single parti-cles, the reaction of a particle suspension modifies the am-

[a] M. Soo, S. Goroshin, J. M. Bergthorson, D. L. FrostDepartment of Mechanical EngineeringMcGill University817 Sherbrooke St. WMontreal, Quebec, Canada*e-mail : [email protected]

Abstract : The reaction of a suspension of solid particles ina rapidly-heated oxidizing gas is relevant to metalized ex-plosives and propellants, as well as to combustion of solidfuel-particle suspensions in premixed-gaseous-fuel cloudsencountered in accidents within the mining and process in-dustries. A simplified model is considered, using a con-stant-volume approximation, which assumes that non-vola-tile particles react heterogeneously via a one-step surfacereaction. The resulting unified particle reaction rate in-cludes both kinetic and diffusive reaction resistances. It isshown that the onset of the chemical reaction in a rapidlyheated particulate suspension may occur by two differentphysical mechanisms. The first mechanism, realized ina dilute suspension of particles, is defined by the ignition

of a single particle, i.e. , by the critical phenomenon associ-ated with the rapid transition from a kinetically- to diffu-sively-limited reaction regime. The second mechanismdominates the reaction onset in a dense particulate sus-pension and occurs in a similar manner to the reactiononset in a rapidly-heated homogeneous gas mixture, wherethe highly-activated reaction occurs in an explosion-likemanner after some time delay and preheating. Unlike thesingle particle ignition phenomenon, the second mecha-nism lacks criticality and is not limited to particles abovea certain size. The interplay between these two reaction-onset mechanisms leads to a nontrivial dependence of thetotal reaction time on the particle size and solid-fuel con-centration within the suspension.

Keywords: Particle combustion · Ignition · Burning time · Metalized explosive

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bient gas environment by reducing the concentration ofoxidizer and increasing the temperature, which in turn in-fluences the particle combustion rate [10]. The combustionregime for a particle suspension with a given initial particlesize may change as the particles burn out and may alsodiffer from the combustion regime of an isolated particlewith the same size.

Recent experimental observations of flames in hybridcombustible gas-solid fuel mixtures have shown that thecombustion physics of the solid suspension depends criti-cally on the initial mass concentration of the solid fuel [7] .For the case of aluminum particles, the particle combustionregime within a methane-air-aluminum flame changes rap-idly when the particle concentration reaches a criticalthreshold value of about 150 g m�3 [7] , shifting from a rela-tively slow oxidation regime, characteristic of an isolatedsingle particle within the same gaseous environment, toa relatively fast reaction regime associated with a flamefront. An analogous transition is also observed in suspen-sions of iron powder in methane-air mixtures indicatingthat the phenomenon is of a general nature and is not fuelspecific [8] . These results also suggest that the commonpractice of introducing externally-defined parameters, suchas ignition temperature, ignition delay, and burn time, ob-tained from single-particle experiments into reactionmodels for metallized energetic compositions may be inad-equate for cases in which a dense particle suspension isformed.

The goal of this paper is to use a simple, transparentmodel to describe the characteristics of the combustion offuel particle suspensions and to illustrate how distinct com-bustion phenomena may arise that do not occur duringthe combustion of a single particle. The model does notrepresent a comprehensive description of the combustionof a specific fuel, but rather demonstrates, in general terms,how the mass concentration of a solid fuel can influencethe reaction behavior and combustion regime of the parti-cle suspension with different particle sizes. To accomplishthis, it is assumed that the particles undergo a purely heter-ogeneous reaction with a gaseous oxidizer through one-step surface Arrhenius kinetics, which avoids the complica-tions introduced by phase transitions, multistage kinetics,and the complex thermodynamics typical of “real” systems[11]. Through this analysis, it is theoretically shown that thegeneral terms ignition temperature, combustion time, and ig-nition delay, adapted from combustion models for largesingle particles, may be inadequate to describe the com-plex phenomena associated with the combustion of parti-cle suspensions and for small particle sizes.

The presented work illustrates the diverse combustioncharacteristics associated with adiabatic, constant-volumecombustion of a stationary particle suspension for a particu-lar choice of values for the reaction rate and other physicalparameters associated with the solid and gaseous compo-nents. The effect of applying a rate of heat loss is also ex-amined to illustrate how the combustion of a suspension

might behave in a real system subject to, for example, ex-pansion cooling. However, systematic parametric analysis ofpossible combustion phenomena that may arise in differentexperimental situations following detonation and deflagra-tion events is beyond the scope of the presented work.

2 Model Formulation and Results

2.1 Kinetic-Diffusion Reaction Rate

The general treatment of heterogeneous reactions gov-erned by the competition between transport and kineticrates, and the associated thermal regimes and criticalnature of regime change, is thoroughly investigated in theclassical work of Frank-Kamenetskii [12] and further ana-lyzed in Vulis [13]. A similar approach is applied here ina simplified form to analyze and interpret different reactionregimes for the case of a single particle and a particulatesuspension.

As stated above, it is assumed that oxidizer consumptionrate by chemical reaction on the particle surface is de-scribed by a single-step first-order Arrhenius reaction, i.e. ,

dmdt¼ Cs A k ð1Þ

k ¼ k0 exp

��Ea

RT s

�ð2Þ

Here, Cs is the concentration of the oxidizer at the parti-cle surface, A is the particle surface area, and k is the kineticreaction rate, which depends on the activation energy Ea

and the particle temperature Ts. Following the assumptionmade by Frank-Kamenetskii of quasi-static equilibrium [12] ,the concentration of the oxidizer at the particle surface Cs

is found from the balance between the rate of consump-tion of the oxidizer by the chemical reaction in Equation (1)and the rate of transport of the oxidizer to the particle sur-face by molecular diffusion and convection, i.e. ,

where b is the mass-transfer coefficient between a particleand the gas, per surface area, and C0 is the volumetric con-centration of the oxidizer in the flow. By introducing the ex-pression for Cs from Equation (3) into Equation (1), the uni-fied reaction rate, taking into account both the kinetic reac-tion and diffusion “resistances”, may be written as

dmdt¼ A C0 b Da* ð4Þ

where Da* = k/(k +b). The normalized Damkçhler parame-ter Da* is related to the traditional Damkçhler number,Da = k/b, by the expression Da* = Da/(1 + Da). The value of

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Da* is close to unity when the reaction rate is controlled bythe diffusion of the oxidizer (i.e. , the diffusion regime,where k @ b) and approaches zero when the rate of particleoxidation is determined by Arrhenius surface reaction rate(i.e. , the kinetic regime where b @ k).

For the sake of simplicity, the contribution of the Stefanconvective flow [12] to the heat and mass transfer betweenthe particle and the gas is neglected. The Stefan flow isnegligible when the concentration of the oxidizer in theflow is low and/or the volume of reacted gas oxidizing onparticle surface is closely matched by the volume of thegaseous reaction products formed. In the absence of theStefan flow, the mass and heat exchange coefficients be-tween a particle and the gas take the following simpleform:

b ¼ Sh D2r

ð5Þ

h ¼ Nu l

2rð6Þ

where the Nusselt (Nu) and Sherwood (Sh) numbers are cal-culated assuming spherical particles, which are stationaryrelative to the gas (Nu = Sh = 2). At Lewis numbers close tounity, and under the particle lumped-capacitance assump-tion, the profiles of the temperature field and oxidizer con-centration around a particle are similar in all oxidation re-gimes, as shown in Figure 1.

In the diffusion-limited stationary reaction regime, theconcentration of the oxidizer at the particle surface is closeto zero, and the particle temperature is close to the maxi-mum adiabatic flame temperature of the stoichiometricmixture [13]. In the kinetically-limited regime, where diffu-sion is much faster than the rate of chemical reaction, theoxidizer concentration at the particle surface is close to theconcentration within the bulk flow and the temperaturedifference between the particle and gas does not exceedone characteristic temperature interval, DT�RTg0

2/Ea

[13, 14].

2.2 Ignition of a Single Particle

The transition from the kinetically-limited to the diffusion-limited reaction regime occurs as a discontinuous jump ata critical particle temperature and is the classic definition ofthe ignition of a particle. The corresponding critical gastemperature is denoted the ignition temperature. The re-verse discontinuous transition from diffusion-limited to ki-netically-limited reaction is called extinction. The Semenov

Figure 1. Temperature and oxidizer concentration profiles adjacentto the particle surface for different combustion regimes.

Figure 2. (left) Semenov diagram of the single particle ignition and extinction. (right) dependence of the particle temperature (Ts) on gastemperature (Tg) illustrating disappearance of the critical ignition and extinction points below some critical particle size (unstable regimesare marked with dashed lines). The temperatures are scaled by the activation temperature Ta = Ea/R.

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Particle Suspension in a Rapidly-Heated Oxidizing Gas


thermal ignition diagram, shown in Figure 2 (left) adaptedfor the kinetic/diffusive particle reaction rate, qualitativelyillustrates both the critical ignition and extinction points fora single particle and the subsequent transition into stableregimes. It is important to note that once ignition occurs(i.e. , the jump to the diffusion-limited burning regime), theparticle will also undergo extinction and return to the ki-netically-limited burning regime when it reaches a criticalsize due to particle burnout. This is not reflected in theclassical particle ignition theory, which is only formulatedto determine ignition temperatures and stationary statesand ignores the changing particle mass with time [15].With the inclusion of time-dependent particle burnout, forlarge initial particle sizes, the mass of the extinguished par-ticle is negligibly small in comparison to the initial mass.However, for decreasing initial particle sizes, extinctionoccurs more rapidly after ignition, leading to a non-negligi-ble mass of the extinguished particle which burns in the ki-netically-limited regime.

There is a critical particle size, for which the ignition andextinction temperatures coincide, illustrated in Figure 2(right), leading to the existence of a limit where the particleis unable to ignite at any gas temperature. The breakdownof the ignition criticality is derived from the particle ignitionconditions and reflects the dependence of the particle tem-perature on the gas temperature for the classical stationarymodel for particle ignition. Below this critical size, there isno ignition temperature as the particle temperature is a mo-notonically increasing function of the gas-phase tempera-ture and is not significantly different from the gas tempera-ture which is expected in a purely kinetically-limited com-bustion regime.

The particular set of numerical values for the thermody-namic and transport constants of the gas and solid phaseused to calculate the curves in Figure 2 (right) are shown inTable 1 and are also used in the subsequent examples inthis paper. The values of the thermodynamic parametersare typical for metal fuels, and with the particular values ofthe reaction constants chosen, both the diffusion and kinet-ic combustion regimes occur with micrometer-sized parti-cles, where the molecular mean free path is still muchsmaller than the particle radius and, therefore, Knudsen dif-fusion effects can be neglected.

The ignition temperature calculated from the balance be-tween the rates of heat production and heat loss and their

derivatives is plotted in Figure 3 as a function of particlesize. It is evident that the particle ignition temperature in-creases with decreasing particle size, reflecting the fact thatthe heat loss per unit particle surface area increases propor-tionally to 1/r. For the set of numerical parameters chosen,the heat loss rate exceeds the heat production rate for anyparticle temperature when the particle size is below a criti-cal size of about 1.9 mm, and ignition, or a transition to dif-fusion-limited combustion, becomes impossible.

If a particle below this limiting size is injected into a hotoxidizing gas it will heat slightly above the gas temperatureand will undergo oxidation within the kinetic reactionregime without igniting. The time history of the tempera-ture and radius of a single particle is plotted in Figure 4 forthree different particle sizes corresponding to differentcombustion regimes. The smallest particle undergoesa stable kinetically-limited oxidation process (Figure 4A).The largest particle ignites leading to a stable diffusivecombustion regime with Da* near to unity, which thenquenches when the particle size is less than 10 % of the ini-tial value (Figure 4C). For an intermediate particle diameter,an intermediate transient regime occurs in which the parti-cle ignites, but extinguishes prior to reaching a fully-devel-oped diffusive-combustion regime (Figure 4B).

Table 1. Numerical values of the gas and solid fuel parameters.

Property Value Property Value

l 0.0238 W m�1 K�1 T0 300 Kcg 1010.08 J kg�1 K�1 cs 978.56 J kg�1 K�1

D 2.01 � 10�5 m2 s�1 k0 70 m s�1

1g 1.172 kg m�3 1s 2700 kg m�3

Tg0 2100 K g 1.12C0 0.27 kg m�3 q 3.10 � 107 J kg�1

a 2.01 � 10�5 m2 s�1 Ea/R 15000 K

Figure 3. Dependence of the ignition temperature on particle size.Note that below a critical value of the particle radius of 1.9 mm ig-nition is impossible.





The length of the kinetically-limited reaction “tail” de-pends exponentially on the ambient gas temperature and,thus, is usually very long for single particles or for a dilutesuspension in which the self-heating of the mixture is negli-gible. In the transient combustion regime observed in Fig-ure 4B, the time for complete oxidation of the particle inthe kinetic regime is usually much longer than for largerparticles.

It should be noted that the particle size defining theborder between the diffusion- and kinetically-limited com-bustion regimes is not only a function of the reaction pa-rameters but also depends on the heat exchange coeffi-cient between the particles and gas. As an example, thedetonation of metalized explosives typically results in muchlarger gas-particle slip velocities, in comparison with metal-ized propellants that undergo deflagration, resulting ina shift of the border between the two regimes towardslarger particle sizes [16] .

2.3 Reaction of a Dense Particulate Suspension

The dynamics of the solid-fuel reaction change completelyif the initial concentration of the solid fuel is sufficient tosignificantly influence the gas temperature through theparticle heating and subsequent heat release by the parti-cle reaction. The effective reaction rate derived in the previ-ous section is given in Equation (7). The algebraically re-duced conservation of energy equations describing thechange of the particle temperature Ts by heat from the re-action and convective heat loss to the gas is given in Equa-

tion (8) and its changing radius r due to mass consumptionis given in Equation (9). These must be supplemented withequations describing the time dependent temperature ofthe gas Tg heated convectively by the particles shown inEquation (10) and the depletion of the oxidizer mass con-centration C due to the reaction shown in Equation (11).

The governing equations are written with a constantvolume approximation and the assumption that the pres-

Figure 4. Time histories of the temperature (scaled to the activa-tion temperature Ta), radius and normalized Damkçhler parameterfor a single particle with different sizes injected into a hot oxidizinggas flow illustrating the different possible reaction regimes: (a)a small particle (1 mm radius) that oxidizes within the kineticregime, (b) an intermediate sized (7 mm) particle that ignites andquenches, and (c) a large particle (20 mm) that ignites and burnswithin the diffusive regime.





sure increase does not change the reaction kinetics or theheat and mass transfer constants. The exclusion of the gas-dynamic effects that are specific to a particular experimen-tal situation allows us to focus primarily on the effect ofparticle mass concentration B in the combustion dynamicsof the suspension.

The time histories of the particle temperature, radius,concentration and the normalized Damkçhler parameterare determined by numerically integrating Equations (7)–(11) with the assumption that the initially cold particle sus-pension is suddenly exposed to a hot gas. The results areshown in Figure 5 for the same three particle sizes consid-ered for the reaction of a single particle in Figure 4. For thepurposes of this illustration, we assume that the fuel sus-pension is lean with a mass concentration of 200 g m�3,corresponding to about two-thirds of the stoichiometricvalue and with an initial gas temperature of 2100 K, whichis characteristic of hydrocarbon flames.

As seen from Figure 5A for a small particle size, in con-trast with a single particle, the dense suspension of kineti-cally-reacting particles undergoes a rapid reaction onsetand a corresponding sharp temperature increase. Althoughthis behavior has the appearance of an ignition event, therapid rise of the reaction rate of the small particles withinthe suspension lacks criticality as the value of the Damkçh-ler parameter does not exceed 0.2 and, thus, the combus-tion process remains close to the kinetically-limited reactionregime. The nature of the sharp reaction onset in this caseis analogous to that of a reactive gas mixture heated underadiabatic conditions. After a certain delay time duringwhich the mixture slowly oxidizes, the reaction proceedsrapidly in an explosion-like manner reflecting the self-accel-erating nature of Arrhenius kinetics at high activation ener-gies and large heats of reaction. The adiabatic reactiondelay time in this case can be estimated using the samewell-known formula for gaseous mixtures if the reactiondelay time is significantly longer than the particle heatingtime [14].

For large particles, which have the ability to ignite andburn within the diffusion-limited regime, there is a complexinterplay between the onset of the reaction due to the par-ticle cloud effect and the individual particle ignition (Fig-ure 5C). The calculations indicate that, for any given particleconcentration, the cloud effect leading to a reduction inthe reaction delay diminishes with increasing initial particlesize, where eventually the suspension combustion willbehave in a similar fashion to the single particle. For sus-pensions of intermediate-sized particles, the diffusive-com-bustion regime is followed by extinction and kinetically-controlled combustion of the remaining fuel similar to thecase of a single particle (Figure 5B). However, unlikea single particle, the transition to the kinetically burningbranch of the suspension does not result in a long kinetic-reaction particle-burnout “tail” but proceeds rapidly due tomuch higher gas temperatures.

Figure 5. Reaction of an instantaneously heated suspension ofsolid fuel particles with a solid fuel concentration of 200 g m�3 withthree different particles sizes, resulting in combustion regimes thatare (a) kinetically-limited, (b) intermediate, and (c) diffusively-limit-ed.





An important parameter for practical particle combustionapplications is the sum of the delay and reaction timeswhich indicates how rapidly the solid fuel energy is re-leased for a particular energetic system. This is denoted forsake of brevity as the “combustion time”. The combustiontime calculation is limited to the point when 95 % of theparticle radius is consumed in order to exclude the dispro-portional contribution of the long, kinetic-reaction “tail”during which a relatively small amount of fuel is consumedafter particle extinction. The dependence of the combus-tion time on particle size is plotted in Figure 6 for differentsolid fuel concentrations.

From Figure 6, the overall combustion time of a singleparticle initially rises with particle radius, but then sharplydrops after the initial particle size crosses a critical thresh-old at which point the particle burns primarily in a diffu-sion-limited regime. After this drop, the combustion timeslowly increases for increasing particle sizes although this isnot readily apparent on the semi-log vertical scale onFigure 6. The fraction of the particle mass that burns in thefast diffusive regime increases with particle size within thissize range, compensating for both the increased heatingand reaction times. With an increase in fuel concentrationwithin the suspension (and therefore gas temperature), theadvantage of the diffusion-combustion regime diminishesand, for high particle concentrations, the overall combus-tion time increases monotonically with particle size.

The same interplay between kinetically- and diffusively-limited combustion regimes at different fuel concentrationsexplains why the combustion time dependence with re-spect to concentration intersects for two different particle

sizes, as shown in Figure 7. At low fuel concentrations and,thus, lower gas temperatures, the large particles have anadvantage. They can ignite and burn faster in the diffusionregime than kinetically-reacting particles at lower tempera-tures. At high fuel concentrations and, thus, high gas tem-peratures, the advantage of the diffusive combustion disap-pears and small particles react more rapidly due to theirhigher specific surface area.

Figure 7 also exhibits an initial sharp drop in the overallcombustion time with an increase in the fuel concentrationfor both particle sizes. The combustion time for a suspen-sion of 4 mm particles reacting primarily in the kineticmode decreases by about an order of magnitude in con-trast with a single particle. The exponential dependence ofthe reaction delay time on the gas temperature, which isproportional to the fuel concentration, is a clear demon-stration of the “cloud” or “collective” effect on the combus-tion process. The drop in combustion time will be evenstronger if the increase of the heat and mass transfer coeffi-cients with temperature and pressure are taken into ac-count. Accounting for the “cloud effect” may explain theextremely low combustion times observed experimentallyfor aluminum particles burning within the products of met-alized explosives that are difficult to reconcile with themuch longer times derived from experiments with singleparticles [17]. The recently proposed hypothesis [18, 19]that the short reaction times are due to the cracking of theprotective alumina shell by a shock wave resulting in fasteroxidation kinetics cannot explain why combustion times ofthe same order are observed during the detonation of alu-minum suspensions in air where the shock pressure is

Figure 6. Dependence of combustion time on particle size at dif-ferent solid fuel concentrations.

Figure 7. Dependence of combustion time on concentration ofthe solid fuel for particle sizes of 4 and 10 mm.





lower by almost three orders of magnitude [20]. In addi-tion, it is well known that protective oxide films can swiftlyre-establish on the metal surface if the conditions are notoptimum for the particle ignition [21].

2.4 Reaction of a Suspension in a Cooling Gas

In a real experimental situation, the system is non-adiabaticand the temperature of the gas decreases with time due toheat loss or may fall sharply due to the expansion of thegas products as in the case of a detonation event. Model-ing the complex flow dynamics of explosive formulations isoutside the scope of this paper but, in the context of thepresent model, the general considerations of what mayoccur during the combustion of a suspension in sucha system can be examined through the addition ofa simple convective cooling rate of the gas phase as shownin Equation (12) which is a modified form of Equation (10),i.e. ,

Here, the hw is an arbitrary heat transfer coefficient, andT0 is the ambient temperature. The addition of the coolingrate in this simple form is not connected to any particularexperimental situation, but serves to qualitatively demon-strate the type of analysis that can be performed to findoptimum particle sizes for a given particle loading. The in-troduction of the cooling rate affects the ability of the par-ticles to react to completion before the gas temperaturedrops below the point at which the reaction rate is negligi-ble. The cooling of the gas may also lead to a longer reac-tion onset time, in which case the particles will never reactwithin the given time frame. Among several parameters, in-cluding the initial gas temperature and reaction constants,the major factors affecting the reaction completenesswithin a given time are the particle size and mass concen-tration of the fuel within the suspension. Although the par-ticle size has long been recognized as a critical parameter,the fuel concentration within the suspension is typically ne-glected. To analyze this situation, a criterion is set such thatif burnout of 95 % of the particle radius does not occurwithin the assigned time frame, then it is considered to beunburned. The resulting combinations of particle size andconcentration for which particle burnout occurs (shaded re-gions) are plotted in coordinates of particle size vs. massconcentration in Figure 8 for three different cooling rates.Particle sizes below 1 mm were not considered in the calcu-lation.

This is by no means a comprehensive analysis of themodel as the set reaction time window and cooling ratesare arbitrary, but it does show that the outcomes are notnecessarily intuitive. The resulting particle size/concentra-tion combinations for both the lowest and intermediatecooling rates indicate that at lower concentrations, it is ac-tually more beneficial to use larger particle sizes since the

smaller, more kinetically-limited particles do not burn tocompletion. In the intermediate case, at low concentra-tions, there is an isolated “island” of large particle sizes thatburnout. This island continually shrinks with increasingcooling rate until, for a certain range of concentrations,there will only be a very small range of optimum particlesizes. In the case of the highest cooling rate, it is seen thatonly relatively small particle sizes have the ability to reactto completion.

3 Conclusions

The primary conclusion of the presented work is that theonset of the chemical reaction in a rapidly heated particu-late suspension may occur by two different physical mecha-nisms. The first mechanism, realized in a dilute suspensionof particles, is defined by the ignition of a single particle,i.e. , by the critical phenomenon associated with the rapidtransition from a kinetically- to diffusively-limited reactionregime. The second mechanism dominates the reactiononset in a dense particulate suspension and occurs in a sim-ilar manner to the reaction onset in a rapidly-heated homo-geneous gas mixture, where the highly-activated reaction

Figure 8. The transformation of the range of particle sizes andconcentrations (shaded area) where 90 % of the solid fuel mass isburned within 500 ms with a 20 % and 40 % increase of the coolingrate.





occurs in an explosion-like manner after some delay andself-heating. Unlike the ignition phenomenon, the secondmechanism lacks criticality and is not limited to particlesabove a certain size. The interplay between these two reac-tion onset mechanisms leads to a nontrivial dependence ofthe total reaction time on the particle size and fuel concen-tration within the suspension.

Symbols and Abbreviations

a – Thermal diffusivity [m2 s�1]b – Mass transfer coefficient [m s�1]g – Stoichiometric indexl – Thermal conductivity [W m�1 K�1]1g – Gas density [kg m�3]1s – Solid density [kg m�3]A – Particle surface area [m2]B – Particle concentration [kg m�3]cg – Gas specific heat capacity [J kg�1 K�1]cs – Solid specific heat capacity [J kg�1 K�1]Cs – Oxidizer concentration at surface [kg m�3]C – Oxidizer concentration [kg m�3]C0 – Initial oxidizer concentration [kg m�3]D – Mass diffusivity [m2 s�1]Da – Damkçhler numberh – Heat transfer coefficient [W m�2 K�1]hw – Cooling coefficient [s�1]k – Arrhenius rate [m s�1]k0 – Pre-exponential factor [m s�1]m – Oxidizer mass [kg]Nu – Nusselt numberq – Heat of reaction [J kg�1]r – Particle radius [m]r0 – Initial particle radius [m]Sh – Sherwood numberTa – Activation temperature [K]Tg – Gas temperature [K]Ts – Particle (solid) temperature [K]T0 – Ambient temperature [K]

Acknowledgments

Special thanks to Keishi Kumashiro for his help in the formulationof the governing equations. Support for this work was provided bythe Defense Threat Reduction Agency under contract HDTRA1-11-1-0014 (program manager Suhithi Peiris), the NSERC CREATE onClean Combustion Engines, and a McGill Engineering DoctoralAward.

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Received: November 9, 2014Revised: March 30, 2015

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M. Soo,* S. Goroshin, J. M. Bergthorson,D. L. Frost

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Reaction of a Particle Suspension ina Rapidly-Heated Oxidizing Gas





Reaction of a Particle Suspension in a Rapidly-Heated Oxidizing Gas

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