a correlation for burn time of aluminum particles in the transition regime

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Proceedings of the Combustion Institute 32 (2009) 1887–1893

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of the

CombustionInstitute

A correlation for burn time of aluminum particlesin the transition regime

Patrick Lynch, Herman Krier, Nick Glumac *

University of Illinois at Urbana-Champaign, 3001 Mechanical Engineering Laboratory MC-244,

1206 West Green Street, Urbana, IL 61801, USA

Abstract

A study of the combustion times for aluminum particles in the size range of 3–11 lm with oxygen, car-bon dioxide, and water vapor oxidizers at high temperatures (>2400 K), high pressures (4–25 atm), andoxidizer composition (15–70% by volume in inert diluent) in a heterogeneous shock tube has generateda correlation valid in the transition regime. The deviation from diffusion limited behavior and burn timesthat could otherwise be accurately predicted by the widely accepted Beckstead correlation is seen, forexample, in particles below 20 lm, and is evidenced by the lowering of the diameter dependence on theburn time, a dependence on pressure, and a reversal of the relative oxidizer strengths of carbon dioxideand water vapor. The strong dependence on temperature of burn time that is seen in nano-Al is notobserved in these micron-sized particles. The burning rates of aluminum in these oxidizers can be addedto predict an overall mixture burnout time adequately. This correlation should extend the ability of mod-elers to predict combustion rates of particles in solid rocket motor environments down to particle diametersof a few microns.� 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

Keywords: Aluminum combustion; Transition regime; Burn time

1. Introduction

Aluminum is a very energetic material thatburns with a variety of oxidizers, notably in solidrocket motors (SRMs). There has been a recenttrend towards smaller particle sizes to improveenergy release rate and reduce the required in-rocket residence time in SRMs.

Particles above 30 lm have been well studiedand have been most often observed to burn in adiffusion limited regime. In this regime, aluminumvaporizes and then burns in the gas phase, gener-

1540-7489/$ - see front matter � 2009 The Combustion Institdoi:10.1016/j.proci.2008.06.205

* Corresponding author. Fax: +1 217 244 6534.E-mail address: [email protected] (N. Glumac).

ating the product aluminum oxide (Al2O3). Diffu-sion limited droplet theory [1] predicts the time ittakes for a particle to complete combustion isdependent on the initial diameter squared, hencethe d2 law. Additionally, the combustion time isindependent of ambient pressure (for fixed ambi-ent oxidizer mole fraction) and relatively indepen-dent of ambient temperature as well.

Several deviations from the strict diffusion lim-ited structure have been observed, even for largeparticles. Early experimental results by Friedmanand Macek [2] showed that the diameter depen-dence had an exponent of 1.2–1.5, significantlysmaller than 2. For particles below 20 lm, Parret al. [3] observed an exponent much smaller than1. Over the years, various reasons have been given

ute. Published by Elsevier Inc. All rights reserved.

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1888 P. Lynch et al. / Proceedings of the Combustion Institute 32 (2009) 1887–1893

to raise or lower this exponent, including convec-tive effects, the formation and accumulation of thecondensed phase product aluminum oxide, thetransition towards a kinetically limited structureat small diameters, and the agglomeration andfragmentation of particles. Beckstead [4] presentsa widely accepted correlation for aluminum invarious oxidizers, temperatures, pressures, andsizes, which works very well for particles largerthan 20 lm. This correlation predicts a weak tem-perature and pressure dependence, uses n = 1.8for the diameter dependence and predicts theburning rate of Al in H2O being about half thatof Al in O2 and that of Al in CO2 being aboutone fifth of the rate in O2.

The combustion of very small metal particleshas also been studied [5], especially in recent years,and these are expected not to follow the classicaldiffusion limit. Both quasi-steady as well as trans-port limited assumptions may not be valid. In thequasi-steady kinetic limit, the limiting factorbecomes the kinetics of the reaction rather thanthe transport, and the burn time is proportionalto d (not d2) and to 1/P [6]. Additionally in thiskinetic regime, the ambient temperature affectsnot only the ignition but the burning rate of alu-minum to a far greater extent than is seen in largerparticles [5]. It is commonly assumed that at smallparticle sizes and low pressures (enhanced diffu-sive transport), the diffusion limited approxima-tion will fail. In general, transition towards thekinetic limit will be experimentally observed as aweakening of the n exponent in the dn law, anenhanced pressure dependence on the burningrate, and possibly an enhanced temperaturedependence as well. For small Al particles, how-ever, the quasi-steady approximation will likelyfail before the kinetic limit is reached. Thoughmany studies have addressed transient particlecombustion, there is no clear consensus on howburn time is expected to differ when the quasi-steady assumption for Al combustion is no longervalid.

This paper presents a systematic study of Alparticles burning in the transition regime, wherethe Beckstead correlation begins to fail. For parti-cles in this regime (we cover the range of 3–11 lmdiameter and 4–25 atm), we present data fromwhich an alternative correlation is derived, allow-ing for the more accurate prediction of combus-tion times of Al in H2O, O2, and CO2 in thisintermediate size range.

2. Experimental approach

The heterogeneous shock tube facility at theUniversity of Illinois generates a high tempera-ture, high pressure environment ideal for ignitingsmall clouds of particles. The driven section (thetest section) is 8 m long and 8.9 cm internal diam-

eter. Other relevant dimensions and descriptionsof this facility can be found in previous publica-tions [7–10]. Through the pressure ratio of the dri-ver and driven sections, a strength selectableshock can produce a combustion environmentfor approximately 2 ms in this shock tube. Vari-ous compositions of test gases can be used, nota-bly oxidizers of aluminum: O2, CO2, and H2O inmixtures with Ar or N2.

For each test, the tube is evacuated, flushed,and filled with the desired test gas, typicallybetween 20% and 70% by volume oxidizer withthe balance argon. The driven section of the shocktube is heated to approximately 335 K in order tokeep the small partial pressures of water in thevapor phase and prevent it from condensing intodroplets. Temperatures exceeding 3000 K andpressures up to 30 atm are achievable.

Figure 1 shows a schematic of the operation ofthe shock tube. Approximately 2 mg of aluminumparticles are injected into the tube from the side ata port 1.44 m upstream from the end wall of thetube. Through the rupture of a double diaphragmsection, the shock is quickly fired while the cloudof particles is suspended in the test gas. The cloudof particles drifts towards the end wall as the gasis accelerated behind the incident shock. When theshock reflects against the end wall, it meets andstops the particles in the test section where theyreact in the high temperature and high pressurezone. Optical access to the burning particles isacquired through fiber optics placed along the testsection in the places predicted by a particle trajec-tory model. This model has been used by us previ-ously [8] to predict the position of assumedlyspherical particles. The fibers are filtered for486 nm light and passed to a photodiode. The alu-minum monoxide (AlO) Dm = 0 B–X transition iscentered at 486 nm, and the presence of this alu-minum combustion intermediate is used to char-acterize a burn time. There are no interferencesin this range that would systematically affect theburn time estimation [9].

In order to characterize the diameter depen-dence of particles, two different size distributionsof aluminum were studied. The first was size dis-tribution ‘‘H-2” purchased from Valimet Inc.;these particles are nearly spherical with a distribu-tion centered nominally at 2 lm. The second sizewas a sample sieved from a �325 mesh 99.999%Al powder purchased from Alfa Aesar. The sam-ple was sieved through a 10 lm sieve, but did notpass through a 5 lm sieve. Scanning ElectronMicrographs of these particles showed they wereless spherical than the other sample and with amass averaged diameter of approximately 11 lm.The 11 lm particle micrographs are included inFig. 2, and the size distributions are shown inFigs.3 and 4. The H-2 sample median is approximately0.9 lm and the number average is approximately1.2 lm; however, as emitted light intensity is

Fig. 1. Schematic of operation of heterogeneous shock tube.

Fig. 2. Scanning Electron Micrograph of aluminumsieved through 10 lm sieve.

Fig. 3. Size distribution of Valimet ‘‘H-2” aluminumpowder.

Fig. 4. Size distribution of ‘‘5–10 lm” aluminumpowder.

P. Lynch et al. / Proceedings of the Combustion Institute 32 (2009) 1887–1893 1889

dependent upon the mass of the particles, themass average (3.1 lm) is the appropriate averag-ing of the particles. Because of this cubic averag-ing, even though a very small number ofparticles occupy the tail of the distribution, thetail accounts for most of the mass, as well as theenergy release, and it dominates the signalsobtained by the photodiodes.

Burn times for particles were based on the 10–90% area method of the AlO emission, as used inprevious studies. The burn times have a small cor-

rection (less than 10%) to account for the motionof particles across the field of view of the fiberoptics while they are being stopped by thereflected shock [8]. In order to isolate the kineticsof each individual oxidizer, the contribution to theburning rate of the oxygen dissociated from theprimary oxidizers is also subtracted from theburning rates for carbon dioxide and water vapor.The dissociation and burning rate from these sec-ondary oxidizers can be upwards of a 15% correc-tion, which is dependent upon the compositionand pressure of the primary oxidizer, with higheroxygen concentrations for higher water vaporand carbon dioxide concentrations and for lowerpressures causing higher corrections. At these ele-vated temperatures, the burning rates for molecu-lar oxygen, atomic oxygen, and the hydroxideradical (OH) were taken as the same, and otherspecies (including carbon monoxide) were ignoredbecause of their low concentrations.

3. Results

Figures 5 and 6 show the variation of burntime with composition of oxidizer for the threeprimary oxidizers at 8.5 atm and 2650 K for the3 lm and 11 lm sample, respectively. As has beenseen previously in this size range, for low concen-

Fig. 5. Burn time vs. composition for 3 lm aluminum at2650 K and 8.5 atm.

Fig. 6. Burn time vs. composition for 11 lm aluminumat 2650 K and 8.5 atm.

Fig. 7. Burn time vs. pressure for 3 lm aluminum at2650 K. The O2 and CO2 experiments are at 40%oxidizer in Ar, while the H2O is at 50% in Ar.


trations, below 45% oxidizer, aluminum burningin the oxygen burns the quickest, followed bythe carbon dioxide, and finally the water vapor.The reversal at the lower concentrations comesas kinetic effects become important, while therapid diffusion of water vapor compared to CO2

for example, becomes less critical. The data how-ever appear to show an intersect in the 45–65%range where the reaction rate may revert to theO2, H2O, CO2 order predicted by previous corre-lations. This dependence contrasts the Becksteadcorrelation which does not predict as strong adependence on concentration, nor does it predictthe oxidizer strengths changing with respect tothe other oxidizers.

The relatively large error bars represent statis-tical variation in the burn time from several shotsand multiple measurements in each shot, not mea-surement error. Typically, the most repeatableburn times (from shot-to-shot as well as at differ-ent cloud locations during each shot) are obtainedfrom the region just behind the brightest part ofthe burning cloud for several cm until the intensityhas vanished. This choice of region is critical inthe data gathering process. The initial injectedcloud contains both isolated particles and agglom-

erates, and some powder hits the chamber walls.After the passage of the shocks the agglomeratedparticles decelerate less rapidly than the individualparticles, and these end up at the front (down-stream end) of the burning cloud. Emission fromthese particles must be rejected. Likewise, parti-cles from the walls are entrained into the flowafter the passage of the shock, and these have adifferent thermal history than the particles thatare always outside the boundary layer. Emissionfrom these particles, which will end up at theupstream end of the burning cloud, must also berejected. By appropriately selecting the region ofthe burning cloud that consists of single particlesaccelerated in the free-stream, the spurious effectsof agglomeration and wall impacts are avoided.

While water vapor may have the largest diffu-sion coefficient, it is not the first oxidizer to exhibita transition away from diffusion limited behavioras can be seen by analyzing the pressure depen-dencies on the burn time as shown in Figs. 7and 8 for the 3 and 11 lm samples, respectively.Oxygen, for example, has a pressure dependencethat approaches sb � P�1 for both the larger andsmaller particles. On the other hand, the pressuredependence of carbon dioxide is rather indetermi-nate. Previous work for 10 lm particles suggestssb � P0.3 [8], but the uncertainty in the exponentdoes not preclude a 0 or even slightly negativeexponent. With water vapor as an oxidizer, theburn time increases with pressure.

The increase of burn time with pressure that isseen with carbon dioxide and water vapor oxidiz-ers likely happens for two reasons. The first is thatradical recombination, which increases with pres-sure may deplete the radical pool and slow thereaction. This effect is more pronounced in thetransitional regime than in the diffusion limitedregime because the kinetics are becoming impor-tant. The other reason discussed previously [11]and proposed by other investigators [12] is the sig-nificant increase of vaporization temperature ofaluminum with pressure. With an increased

Fig. 8. Burn time vs. pressure for 11 lm aluminum at2650 K. The O2 and CO2 experiments are at 40%oxidizer in Ar, while the H2O is at 50% in Ar.


vaporization temperature, there is less excessenthalpy available to react in the gas phase. Ourprevious work [9,13] suggests that the peak AlO(i.e. gas phase) temperature for small aluminumparticles during combustion is about 3200 K (eventhough the adiabatic flame temperature is signifi-cantly higher). This temperature is fairly indepen-dent of composition, pressure, and temperature.In the vicinity of 10–15 atm for example in watervapor, the vaporization temperature of the parti-cle surpasses 3200 K, and thus the mode of com-bustion may undergo a significant change wherethe surface of the particle is no longer boiling.This effect, if true, would undoubtedly decreasethe combustion rate at higher pressures.

The remaining indication of the transition isthe diameter dependence on the burn time. Inorder to characterize the diameter dependence,the size distribution was accounted for usingBazyn’s methods [14]. A diameter dependent inlength and intensity profile is created for each par-ticle size in the two distributions, and are summedweighted by their frequencies. The burn timesfrom the resultant profiles are then fit to theexperimental data and a diameter dependence nis determined for each set of conditions. The

Fig. 9. Diameter exponent n vs. composition of primaryoxidizer at 8.5 atm and 2650 K.

resulting diameter dependencies are plotted vs.mole fraction of primary oxidizer in Fig. 9 andpressure in Fig. 10.

One curve nicely fits the diameter dependencefor all three oxidizers across the mole fractionrange from 0 to 1. This exponential starts nearthe value of 2 when the mole fraction of oxidizeris small and the burn times are long. Rather thandecaying towards 1, the dependence decaystowards zero. In fact most of the values are belown = 1, which is the theoretical kinetic limit. Suchlow values of n have been seen previously in thislab [8] and by other researchers [3] and shouldnot be a surprise given that the experimental valueof n, even in the diffusion limit, is below 2. Quasi-steady theory predicts that the transition from dif-fusion to kinetic limits should depend very weaklyon oxidizer mole fraction [15]. Therefore, theobserved strong dependence suggests that thequasi-steady approximation may fail when burntimes are fast for small particles used here.

Additionally, there is a weak but noticeablepressure dependence seen in the diameter expo-nent in the three oxidizers as can be seen inFig. 10, it ranges from n � P�0.2 to n � P�0.4

and thus the value of �0.3 was chosen for all threeoxidizers. This dependence is counterintuitive aspressure would serve to reduce the diffusion andshould inhibit the transport and raise rather thanlower the diameter exponent. However, in thetransition region, several competing phenomenaappear to be active, and thus it is unclear how use-ful intuition based solely on idealized limitingbehaviors is.

Based on these results we propose the correla-tion in Eqs. (1)–(3) along with the coefficientsfound in Table 1, where sb is the burn time inmicroseconds, P is the pressure P0 is 8.5 atm,XOX is the mole fraction of primary oxidizer,and d is the diameter in microns. The startingpoint was the relation for the diameter exponentn, which contained the previously discussed pres-sure and composition dependence. From there,

Fig. 10. Diameter exponent vs. pressure at 2650 K. TheO2 and CO2 experiments are at 40% oxidizer in Ar, whilethe H2O is at 50% in Ar.

Table 1Coefficients to be used with Eq. (1)

Oxidizer a0 (ls) a1 a2

O2 200 0.5 �0.5CO2 500 0.6 0.3H2O 86 �1.7 0.75

Table 2Results of the test of mixtures of oxidizers

Mixture sb (ls)

Pure compositions50% CO2 58050% O2 250

Components of the mixture28% CO2 98021% O2 640

Overall mixtureUsing Eq. (3) 390Experimental 384.4


the overall pressure dependence, oxidizer compo-sition dependence and leading coefficient wereobtained by least squares fit of all the data usingan assumed power law. It is important to notethat pressure contributes to the correlation bothin the pressure term, as well as the pressure depen-dence of n; thus while the pressure coefficient a2

for H2O shows an increase sb � P0.75, the overallvalue is closer to sb � P0.5 due to the pressuredependence on the diameter exponent. The corre-lation agrees well with the experimental data ascan be seen in Fig. 11. For mixtures, we recom-mend a first-order approximate approach of addi-tion of the burning rates (inverse of burn time) asshown in Eq. (3), where the weighting of the oxi-dizer composition is handled explicitly within theindividual burn time Eqs. (1) and (2).

sb ¼ a0X a1OX

PP 0

� �a2

dn ð1Þ

n ¼ 2 expð�4:3X OXÞPP 0

� ��0:3

ð2Þ

sb;mix ¼Xn

i¼1

1

sb;i

!�1

: ð3Þ

Equation (3) was applied to a composition of20% O2, 30% CO2, and 50% Ar by volume andwas experimentally tested for several shots at2650 K and 8.5 atm, using the 11 lm aluminum.The results are shown in Table 2. A compositionof 50% CO2 with the remainder Ar at these condi-tions has a burn time around 580 ls, while a 50%

Fig. 11. Burn time predicted by the correlation vs.experimental burn time of all data points in this study.

O2 composition at the same condition has a burntime less than half that at 250 ls. As burning ratesadd, these two values should be the upper andlower bounds of this mixture of CO2 and O2 total-ing 50% oxidizer. When shocked, the dissociationat the high temperature causes the composition toshift to about 21% O2, 28% CO2, 49% Ar, andabout 2% relatively slow reacting CO which isignored. Based on the burn time (and burningrate) of these oxidizers, even though the composi-tion contains more CO2, one would expect theoverall rate to be closer to that of oxygen. FromEq. (3), the predicted burn time of the mixturewas around 390 ls, and the experimental burntime was around 384 ls. This is perhaps fortuitousagreement, but the trends agree with intuition andlend validity to the concept of simply adding theburning rates in this first-order scheme. Agree-ment within 20% was reached with the 3 lm pow-der at this condition as well.

There are at least three effects that areneglected in our analysis. The first effect is thetemperature dependence of the burning rate.Although the increase in temperature has beenshown to increase the burning rate for nano-Al,which burns below the ignition temperature ofmicro-Al [5], this effect is not yet seen in micro-Al. With aluminum particles on the 5 lm scalein water vapor for instance, the temperaturedependence was nearly non-existent as long asthe sample was fully ignited, which matches nicelywith the Beckstead correlation, even though thepressure dependence has been shown to be quitedifferent [9].

A second neglected effect would be the diame-ter’s effect on the diameter exponent. There is cer-tainly dependence here, but it is impossible totrack over our narrow size range, especially con-sidering the size distributions of particles used inour samples. For now, we must assume that theexponent is constant for all particles in the sam-ple. If there were a way to ensure an even moretightly controlled, size-selectable distribution, forexample all particles within one micron of a centerpeak, then a diameter dependence on the diameterexponent could be tracked, but it is a dauntingchallenge.


The final neglected effect is that of convectionon the burning rate. In the shock tube, the parti-cles are not stationary during combustion, andtypically decelerate during combustion. Relativevelocities depend on the conditions, but trajectorymodeling suggests that at ignition (T � 2300 K),the particles have Reynolds numbers in the rangeof 3–40, with velocities relative to the gas in therange of 20–300 m/s. During combustion, parti-cles continue to decelerate, and by the end of com-bustion, velocities in the range of cm/s or lowerare expected. The effect of convection on the burn-ing rate and ignition delay is unknown, butremains of significant concern. By using condi-tions behind the reflected shock that are close tothe ignition temperature, the convective effectcan be minimized since ignition is delayed, givingthe particles a chance to slow down. Under theseconditions, (e.g. Tambient = 2400 K), similar burntimes are observed, suggesting that the convectiveeffects may be minimal, but no detailed study hasyet been undertaken.

4. Conclusions

Based on a large dataset of heterogeneousshock tube tests, a correlation for aluminum par-ticles in the 3–11 lm size range for oxidizers H2O,CO2, and O2 for pressures in the 4–25 atm rangeat ambient temperatures of 2400–3000 K has beengenerated. Significant differences in the sensitivityof the burn rate with respect to particle diameter,pressure, and oxidizer content are observed, ascompared to the Beckstead correlation whichholds well for larger particles. This correlationshould extend the ability of modelers to predictcombustion rates of particles in SRM environ-ments down to particle diameters of a fewmicrons.

Acknowledgments

This research was sponsored by the DefenseThreat Reduction Agency under contractsHDTRA1-07-1-0011 and DAAE30-1-9-0080(Program managers William Wilson and SuhithiPeiris) and the Office of Naval Research under

contract N00014-08-1-0072 (Program managersJudah Goldwasser and Daniel T. Tam). Thanksalso go to Maurine Vogelsang, Priya Ghandi, JeffMason, Armando Bernal, and David Joyce fortechnical assistance. The Scanning ElectronMicrographs were taken in the Frederick SeitzMaterials Research Laboratory Central Facilities,University of Illinois, which are partially sup-ported by the U.S. Department of Energy underGrant DEFG02-91-ER45439.

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a correlation for burn time of aluminum particles in the transition regime

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