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Numerical Simulation of Quasi-Steady, Single Aluminum Particle Combustion in Air

Yongjun Liang and Merrill W. BecksteadDepartment of Chemical Engineering

Brigham Young UniversityProvo, Utah 84602

Abstract

A two-dimensional numerical model of aluminumparticle combustion in air is presented. This modelincludes nine reactions: two surface, three gaseous, onedissociation and three condensation. The initial combustionof an aluminum particle, free falling in air, shows thereaction is similar to the diffusion controlled combustionprocess due to the fast reaction between aluminum vaporand oxygen. The flame zone thickness is about three timesthat of the particle radius and at the flame zone thetemperature flattens due to the dissociation of aluminumoxide. The position and temperature of the flame arestrongly influenced by the aluminum oxide condensationprocess.

Introduction

Adding aluminum particles to solid propellantcan increase the motor's specific impulse and oftendamp acoustic combustion instability. Compared tohydrocarbon droplet combustion, aluminumcombustion is a complicated process which is noteasily modeled. First, in aluminum combustion, thegas phase combustion products condense to liquidaluminum oxide. This condensation dominates thecombustion process and contributes considerably tothe amount of heat released during combustion.Second, condensed aluminum oxide can deposit onthe particle surface to form an oxide cap whichdistorts the distribution of gasification velocity,temperature and other quantities around the particle.Third, the dissociation of the condensed productmaintains the flame temperature fairly constant atthe gasification temperature of the aluminum oxide.

In a very early study, Burtlett et al (1963)studied the combustion of aluminum particles in amethane-oxygen flame and postulated that thecombustion of aluminum particles was notnecessarily controlled by a gas diffusion mechanism,but, instead, it was controlled by the diffusion ofaluminum vapor through the oxide shell.Fragmentation and other violent events wereattributed to the rupturing of the shell. Drew et al(1967) investigated the ignition and combustion of

aluminum particles in H2-O2, CO-O2 andcyanogen- O2 flames. Jetting, spinning, andfragmentation were observed and were attributed tothe asymmetric burning conditions imposed by theformation of an oxide cap. Prentice(1964) haspublished results for flash and laser ignitedaluminum particles burning in N2-O2, Ai- O2 andCO2 - O2 (dry and wet) and found thatenvironments containing large amounts of N2,H2O and CO2 favored the accumulation of oxideproduct on the surface of the burning particles.

Aluminum combustion models have beendeveloped since the 1950's and 60's. The simplestaluminum particle combustion model is the d-squared law which states that the burn time isproportional to the diameter of the particle squared.Gremyachkin et al (1975) proposed a model wherethe reaction was assumed to occur at the surface ofthe particle. This model was constructed in an effortto explain fragmentation and hollow oxide spheres.Brzustowski and Glassman (1964) were among thefirst to suggest that aluminum burns in the vaporphase. They stated that a metal will burn in thevapor phase if the boiling point temperature of ametal is lower than that of its oxide. Brzustowskiand Classman's model included many of the sameassumptions as hydrocarbon droplet combustionmodels. Following their work, other vapor phasecombustion models of aluminum particles weredeveloped (e.g. Law, 1973; Turns et al, 1987; Brooksand Beckstead, 1995). All of these models havefocused on calculating the burning time and flametemperature, but could not predict the distributionsof physical quantities nor processes such ascondensation and deposition. The postulatedcombustion mechanisms were much simplified,using global kinetics.

In aluminum combustion, an important processis the condensation where considerable heat isreleased. Hermsen and Dunlap (1969) used classicalhomogeneous nucleation theory to describe thiscondensation process, but the condensation in

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Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.

aluminum combustion is different from the classicalcase. In aluminum combustion the condensationoccurs with a chemical reaction. In addition, sinceno gaseous aluminum oxide has been observed inexperiments, the quantity of supersaturation inhomogeneous nucleation theory is difficult todetermine. Drew, et al(1964) found that thedeposition of the aluminum oxide on the particleduring combustion can contribute to the formation ofa hollow oxide sphere. Often the resulting oxidesphere has a diameter of similar size to the originalparticle. Turns et al (1987) considered oxidedeposition in their theoretical model. King (1978)studied aluminum particle combustion in CO2 - N2 ,in which the particle was stationary. His onedimensional simulation did not consider finitechemical reaction and deposition. In his studycondensation was assumed to reach completion onan infinitely thin surface. Bucher, et al (1996a) havemeasured the temperature and species distributionaround an aluminum particle free falling in air. Inthis experiment the measured particle surfacetemperature was between 2300 and 2400 K.However in Dreizin's (1996) experiment, the particlewas superheated initially, apparently causing thesurface temperature to exceed the aluminum boilingtemperature. Obviously the particle surfacetemperature strongly depends on the initial conditionof the aluminum particle and the ignition laserpower used to ignite the particle. Also, Bucher et al(1996a) gave a list of possible reactions of thecombustion of aluminum in air. Due to thecomplexity of combustion process, the chemicalkinetic data for most reactions are not available. Thenumber of reactions, or which reactions, are neededto describe the combustion process is still an openquestion.

In this work, the two-dimensional combustionof a single aluminum particle free falling in air isstudied numerically, including the physical processesof aluminum evaporation, finite gaseous reactions,surface reactions, aluminum oxide condensation,dissociation of liquid aluminum oxide and theformation of an aluminum oxide cap.

Development of Physical Models:

In this study, the following has been assumed:1) The particle is spherical2) The change in particle diameter is small during

combustion (due to oxide deposition)3) Flow around the particle is laminar

4) The flame has spherical symmetry5) The local homogenous flow(LHF) model is

applicable to the liquid aluminum oxide.

Combustion mechanism:

According to the analysis of the combustionprocess and the chemical kinetic data available atthis point, a nine-step mechanism was chosen asbelow.

Surface reactions:Al,,* ———^ Al,(*)

>A120(S)

Gas phase reactions:Al(g) + O A/0 + O

AlO + O2 > AIO2 + OWhere

*3 = 9.76 x 1013 exp(-8%) cm3mol'ls

k. = 4.63 x 1014 exp(-10°0%,) cet al, 1996a)

' l ' 1

-O2+MDissociation reaction:

'2AJ0 + -0,

->A/203(;)

>A/203(;)

(Rl)

(R2)

(R3)

(R4)

"' (Bucher

(R5)

(R6)

(R7)

(R8)

<R9)

Condensation:

2A/0 + -0.

A/,0+0,

AIO2 + AIO2

Here we assume the surface reaction (R2) is diffusioncontrolled and k7 - ks = k9 = d)cond .The implication of the assumptions for thecondensation reactions will be discussed next.

Condensation model:

The condensation of oxiae during aluminumcombustion is accompanied by a chemical reaction.A two-step process is assumed as shown below.

»AJ203(/) (RIO)The first step is a chemical reaction process and canbe described by an Arrhenius expression. Thesecond step is a pure condensation process and can

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be described by classical homogeneous nucleationtheory. With this two-step process the chemicalreaction is dependent upon the physical condensationto a liquid oxide because the existence of gaseousoxide has been questioned. For the first reaction stepthe rate expression is

(1)», = krCamCb

nFor the second condensation step

*>2 = Cjcon (2)

Where rcm is the nucleation rate. Fromhomogeneous nucleation theory (Zettlemoyer, 1969)We can write

con

And

n{ - n, exp

(3)

(4)

Where m is the mass of a molecule, p is the liquiddensity, a is the surface tension of a flat liquidsurface, at is the condensation coefficient, ni isthe number of critical size clusters per unit volume,v is the volume per molecule in the liquid state and Sis the supersaturation. The total rate for the two-stepcondensation process is

1 Clrr f^f* r C°f**- *-'/"'VJ'rm^n^n ' *VW^I*\'H ,_.

rm<a2

Next it is assumed that the denominator inEquation (5) does not change significantly duringthe condensation. So the equation reduces to

®cond = KrconCamCb

n (6)Where K becomes an empirical constant. InEquation (4) the supersaturation S has a large effecton the condensation process. For a typicalcondensation process

S = —— (7)Poo

Where p is the partial pressure of the vapor insystem and px is the vapor pressure of thecondensed phase. In the aluminum combustionprocess, p is zero so Equation (7) can not be used inEquation (6). Instead Equation (8) has been used todetermine S.

5 = 1PAI

(8)

Where /?, is the partial pressure of species i, and i=A10, AIO2, A12O.

Deposition model:

Many experiments show that because of theformation of aluminum oxide cap on a combustingaluminum particle surface, the final particlediameter may change only little. For this reason,changing particle diameter has not been consideredat present. We assume that the oxide depositsuniformly on the particle surface and migrates to thedownstream side to coalesce into an oxide cap.

(9)

Figure 1. Deposition on a sphere

The deposition height h^ can be described by theequation

—— = 0n

Where V is the volume of the cap calculated fromthe A/203(;) distribution.

Mathematical models

Governing Equations:

The general form of the governing equation can bewritten as

aIn spherical coordinates,V- (pVf) = ̂ l-(pr\^

r orsn

(10)

Oft 0 !)

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For continuity equation0>=l,r«=0 and 5^=0 (13)

For r-direction momentum equationO=w r , r,j,=u.and

dp pul 1 dl 2 ffu\ 1 8 \ . ad(u,\\S. = -—+*—*-+—,—— r u—H +————— u s m f f — — -dr r r dr\ dr J sine del f f r ^ r J ]

For 9-direction momentum equation<3?=ue , F^nand

i dp pu u. i g ( . „ ci/.^ i5.,,= — — + ̂ r " +— ———— //sm0— - +—* rSe r Ssmedey del r2si

_r 2 <» r [ r

For species conservation equations

0= Yit T^pDt «*A120W)) ,TAk0w)= —° 'A

and S* = «, (16)Where ^ is the mass fraction of species i, <7 A is theSchmidt number (with <J A =0.5)and

i= Al, AlO, A12O, AIO2 ,02,O, A/203{/) ,

To insure a balance of the mass, the calculated

diffusion velocities, Vt = — —- VYt , are corrected

by a uniform velocity vector to keep

For the energy equation(17)

<D=T,pm

V-(kVT) and

D, (19)

Equation (10) is solved in non-dimensional form.

Boundary conditions:

(1) Inlet condition:ue = sin<9, M* = -cos<9, 702 = 0.233,

7^ =0.767, Yt=0.(i*02,N2), r=l,«.=«,(0 (20)Where, the inlet velocity ux(t) (or particle fallingvelocity when the coordinate system is not movingwith the particle) is governed by

(21)

Where the last term on the right hand side isintegration on the particle surface to consider thecontribution of the evaporation to the particlemovement.(2) Outlet condition:

= 0dr

(3) Symmetrical condition:

—— = 0, ue = Q,(0 = Q,K)

(4) Particle surface interface condition:energy balance:

8T.dr ~ Q2™2AIO Qdep =

mass balance:

mvap AlO

M ALO———M AlO

species balance:YAIO = 0

_ PAI MAI(18) M(S)

Where ft is the body force of unit mass of species i -. ^_Introducing non-dimensional quantities

* ur . ue . tum , r * p fffMoo ^oo AD AD Pa, ' fff

p M

M +fh i M*°*l * MAIO

SS

(22)

(23)

(24)

(25)

(26)

(27)

(28)

(29)

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And

In above equations £?2is the reaction heat ofreaction (R2), Qdep is the deposition heat release,mAlo is the mass consumption rate of A1O inreaction (R2), A/zv is the latent heat, mmp is the

evaporation rate of Al, ur is the radial velocity of

gas on the particle surface, and M, is the molecularweight of species i. The transport andthermodynamic properties are calculated using theCHEMKEN transport and thermo-dynamicpackage(Kee et al, 1992 ).

Numerical method:

In this study a staggered grid system is usedwhere the velocities are defined at the controlvolume surface, and scalar quantities are defined atthe center of the control volume. Non-uniform gridsare used in order to improve the accuracy. The fullyimplicit SIMPLER algorithm (Patankar,1980) isused to solve the partial differential equations inwhich the QUICK scheme is used.

The role in constructing a non-uniform gridQUICK scheme is to always use two upwind nodesand one downwind node. For example, consider thecontrol volume i belowWhen ue > 0,

1-1 i e j+i

And

0

Figure 2. Construction of QUICK scheme when Ue>Q

(31)• 4*,

When ue < 0,

-f—0

ei

i

J+l— 1 ———

*1i + 2

——— ^X2

2*2-3*,

(32)In this study, when the calculated temperature

exceeds the Al2O3(l) dissociation temperature(4000K) the dissociation reaction (R6) occurs whichkeeps the flame temperature, Tflamet at the

A12O3<1) dissociation temperature. Here, the rate

constant, k6, for reaction (R6) is not used. Instead,the dissociation rate of A1O (the generation rate ofA1O from reaction (R6)), (D^io > *s use^ ^ a

variable, so, when the calculatedtemperature Ty exceeds 7^,^, Equation (33 ) issolved to determine d>Mlo to keep Ty = T^^.

(33)

Results and discussion:

Dreizin(1996) found that the combustion ofaluminum particles free falling in air has threestages: stable, strong oscillation and weakcombustion stage. The current study investigates thefirst stage of combustion. In this study, an initialparticle diameter of 210 urn has been assumed. Thesurrounding air temperature is 300 K, and theparticle falling velocity is O.OOlm/s. An unsteadysimulation will be presented in the future.

For this study, the grid is uniform in the 6direction with 71 nodes. In the r direction the grid isnot uniform and has 80 nodes extending to 60 timesthe particle radius. The smallest distance betweengrids near the particle surface is about 0.0001 r0.The half of the gas phase grid used in this study isshown in Figure 4.

Figure 3. Construction of QUICK scheme when Ug < 0

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Figure 4. Half of the computation domain

Since the particle velocity is quite small, thecalculated results are almost symmetric,consequently distributions will not be compared fordifferent 6 positions.

In order to check the accuracy of the numericalmethod used in this study, the flow around a non-reacting sphere was studied. Figure 5 shows thecalculated wake length using two different finitedifferencing schemes. It can be seen that theQUICK scheme is more accurate than the hybridscheme. In the region of Re<80, the wake lengthcalculated using the QUICK scheme agrees well withthe experimental data (Kalra and Uhlherr, 1971).The Reynolds number for a combusting aluminumparticle, free falling in air, is less than 1. For thisreason, the QUICK scheme is adequate for thisstudy.

0.0 100.0 200.0Re

300.0

4500

20000 5 10 15 20 25 30 35 40 45 50 55 60

r/r0

Figure 6. Temperature vs. radial non-dimensional distance

In Figure 6, the calculated combustiontemperature reaches the dissociation temperature atabout r/, =5 and extends to about r/ _=8. The

thickness of the dissociation zone is about threetimes that of the particle radius. In this dissociationzone the temperature is kept to the dissociationtemperature of liquid aluminum oxide ofapproximately 4000 K. Figure 7 shows thisdissociation process.

f 8-

Ir-oS. 6-S>£ 5-

S 4 -

J;:

5 1 "n -

Figure 5. Non-dimensional wake length vs. Reynolds number_QUICK, —Hybrid, .Experiment

5 10 15 20 25 30 35 40 45 50 55 60

r/r0

Figure 7. Dissociation rate of aluminum oxide vs. radialnon-dimensional distance

The condensation rate of liquid aluminumoxide is shown in Figure 8.

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0 5 10 15 20 25 30 35 40 45 50 55 60

r/r0

Figure 8. Condensation rate vs. radial non-dimensional distance

We can see the condensation determines thelocation of the flame and the temperaturedistribution and occurs very rapidly in a narrowregion. Figure 9 and Figure 10 show the distributionof the calculated species. Figure 9 shows an overallview while Figure 10 is an enlarged view of theflame zone.

coexistence zone of Al and O2 is considerablynarrow and located in the outer edge of the flamezone which shows that the diffusion controlledcombustion mechanism in theoretical models(Brzustowski and Glassman, 1964; Law, 1973;Turns et al, 1978; Brooks and Beckstead, 1995) arereasonable.

Figure 10. Mass fraction of species vs. radial non-dimensionaldistance

Figure 9. Mass fraction of species vs. radial non-dimensionaldistance

Figure 11 shows a plot of the reaction rates ofsome species. The reaction of aluminum with oxygenis observed to occur only in the narrow flame zone.From Figure 9 we can see that nitrogen diffuses tothe particle surface; therefore, reactions of nitrogenwith aluminum or aluminum oxides might bepossible which may promote observed disruptivecombustion (Bucher et al 1996b). Between theparticle surface and the flame zone the mostimportant species is A1O. A1O is produced in theflame zone and diffuses back to the particle surfacereacting with the liquid aluminum to form A12O.In the flame zone there are two reactions of speciesA1O. One is to form AIO2 and another is thecondensation reaction. Figure 10 shows that theformation of AIO2 is the dominant process.Because of these two reactions, the concentration ofA1O at the outer edge of flame goes to zero. The

The distribution of Al and O2 are like that ofdiffusion flame combustion. They both areconsumed completely in the flame zone. The

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r/r0

Figure 11. Reaction rate vs. radial non-dimensional distance

condensation reaction (Bucher et al, 1996a)AIO2+AIO——»A/203(/) can't occur at that

location which will let a lot of AIO2 exist in thefinal combustion product. Therefore , we assumeAIO2 condenses through reaction (R9) and AIO2

will be finally condensed. Species A/203(/) ismainly produced in the flame zone and diffuses tothe particle surface and deposits on the particlesurface. From the results we can see that next to thecondensation rate the most important quantity toinfluence the condensation is the O2 concentrationwhich is different from the classical condensationprocess.

Conclusions

The two-dimensional numerical simulation ofaluminum particle combustion in air shows that 2nine reactions combustion mechanism, thealuminum oxide condensation and deposition modeland the aluminum oxide dissociation modelproposed in this study are reasonable. The results ofthe simulation indicate that the condensation ofaluminum oxide dominates the combustion processand contributes considerably to the amount of heatreleased during combustion. Due to the dissociationprocess of liquid aluminum oxide there is atemperature plateau region. Also, the results indicatethat aluminum particle combustion is limited by thechemistry kinetics, but approaches a diffusioncontrolled process.

References

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Brooks, K. P. and Beckstead, M. W., "Dynamics of aluminum combustion", J. PropulsionPower 11 (4), 1995,pp.769-780.

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Prentice, J. L., "Aluminum droplet combustion:rates and mechanisms in wet and dry oxidizers",NWC TP 5569, April 1974.

Turns, S. R., Wong, S. C. and Ryba, E.,"Combustion of aluminum-based slurryagglomerates", Combust. Sci. Tech. 54, 1987,pp.229-318.

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