COMBUSTION OF ZIRCONIUM IN OXYGEN AT HIGH … Bound...bustion rates dependent on the burning...

Philips J. Res. 35, 390-440, 1980 R1029

COMBUSTION OF ZIRCONIUM IN OXYGENAT HIGH PRESSURES

by E. FISCHER

AbstractThe combustion of zirconium droplets in oxygen at pressures from 0.3 barto 9 bar is investigated. Experimental results obtained from spectroscopieand high-speed photographic measurements are compared with model cal-culations based on a rate equilibrium model for the processes at the dropletsurface. The various phenomena observed in the experiments ~ dropletgrowth in the ignition phase, corona formation, expansion of the coronaand droplet inflation at the end of the steady-state combustion ~ arediscussed and explained within the framework of the theoretical model. Anat least semiquantitative understanding is achieved for all the phenomenawhich could not be explained in the equilibrium models used by now. In thelast chapter the modifications are discussed, which are necessary if themodel is applied to the combustion in flash bulbs.

1. Introduetion

The combustion of metal particles in oxygen has long been used for gen-erating short intense flashes of light. Metals suitable for such an explosivecombustion àre in particular iron, aluminium, magnesium, most of the rare-earth metals and last but not least zirconium and hafnium. The latter twoform a special group within the metals capable of a fast combustion. Most ofthe metals, when heated above a critical ignition temperature, evaporate andform a reactive gas mixture in which the oxidation takes place. The reactionenergy is transported to the metal surface by conduction or radiation tosustain the evaporation. Zirconium and hafnium show a quite different modeof combustion. Because of the very low vapour pressure of these metals thereaction is shifted to the solid (resp. liquid) metal surface or even to theinterior of the particles which may be covered by an oxide layer: While for agas-phase reaction the evaporation of the metal is the rate limiting process,there are a number of processes, which may control the reaction rate atdifferent stages of combustion, when the reaction is shifted from the gas to theliquid phase.

As in the combustion of zirconium and hafnium the reaction energy isdirectly transfered to the liquid particle, the surface temperature is extremelyhigh. Depending on the oxygen pressure, temperatures up to 5000 K have been

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Combustion of zirconium in oxygen at high pressures

reached. Due to this property the combustion of zirconium and hafnium is theoutstanding mechanism for generation of intense visible radiation by chemicalreactions. As the physics and chemistry is similar for zirconium and hafnium,we have confined our investigations to the combustion of zirconium, becausethis metal is of greater interest from the economic point of view. It is used inall nonelectrical photographic flash bulbs and for many pyrotechnicalapplications.It has been mentioned above that various mechanisms may limit the com-

bustion rates dependent on the burning conditions and the state of the com-bustion. There are some phenomena associated with such changes of thelimiting rates, that can scarcely be understood at first sight. Only a detailedinvestigation and numerical comparison of all the transport processes andreactions which might play a role can lead to an at least qualitative under-standing of the observed phenomena.

Before we start with the description of the physical model of the com-bustion, a short outline of the experimental observations shall be given, whichhave been gathered by several authors during the last decade.

2. Experimental observations

To start a spontaneous combustion of zirconium in oxygen, the materialmust be heated to IOOOK-I400K, dependent on the size and surface proper-ties of the specimen and on the oxygen pressure. In most experiments smallwires, shreds or foils are heated electrically 1,7) or by laser pulses 2-6). Oncethe ignition temperature has been reached the reaction rate becomes so fastthat it exceeds the heat losses. The temperature rapidly rises to the meltingpoint at which one or more droplets are formed, which then run up to astationary burning temperature within part of a millisecond. For pressures upto 1 bar this ignition phase has been investigated by Kettel ') with electricallyheated wires of 0.1 mm diameter, which at the melting point disintegrate intodroplets of 0.2 to 0.4 mm diameter. At higher pressures there is no markeddifference in this heating process. The temperature increase is steeper,however, so that at 8 bar the rise time from the melting point to the final tem-perature is about 0.1 ms (for d = 0.2 mm).

Immediately before the droplets reach their final temperature the radius ofthe luminous surface is increased by a factor of 1.2 to 1.7. The interpretationof this increase is completely antithetic in the work of Kettel1), who attributesit to the formation of gas bubbles inside the droplet, and the papers byMarshall and Pellett 6) and Maloney 8), who assume a luminous flame zonearound the original droplet. We will come back to this point later in thispaper.

E. Fiseher

The next step of combustion is the formation of a corona of condensing'zirconium oxide around the burning droplet. This corona first appears as aluminous fog layer of increasing radius, but within a millisecond or part of it(depending on the oxygen pressure) the distance between the droplet and thecorona tends to a stationary value. In the boundary of the corona the oxideparticles begin to form larger droplets (fig. la). The distance betweenthe corona and the surface of the original droplet is independent of the dropletsize and it decreases with increasing pressure. At low pressures the luminosityof the corona is low compared with that of the zirconium droplet. Most of theradiation emitted from the corona is scattered light from the parent droplet.At higher pressures (> I bar), when the reaction rates increase and the coronacomes closer to the droplet surface, the emission of the corona itself becomesmore and more important (fig. lb), so that at pressures above 5 bar the coronacan scarcely be distinguished from the original droplet (fig. le).The shape of the corona is exactly that of the parent droplet, which means

that if the spherical form of a droplet is disturbed by a collision with anotherparticle, the form of the corona changes, too. Even if the droplet is in rapidmotion, the corona around a spherical droplet remains spherical. If, however,larger oxide particles are formed in the corona they are transported to thewake of the flow. This effect has been studied by Nelson et al. 2) for freefalling droplets. In this case the oxide particles coalesce in the wake and forma follower droplet, which may become so large, that it touches the parentdroplet. The two droplets then mix up, the corona vanishes for a short periodof time and then it is newly formed. This cycle may be repeated several timesuntil the droplet has sucked in so much oxide that the combusion extinguishes.

At higher velocities the oxide particles no longer form a follower droplet,but leave the corona in the wake of the flow. The droplet leaves behind a"vapour trail" which rapidly cools off so that it can be seen only as anabsorbing streak (fig. 2).

If the burning droplet is at rest, its diameter decreases almost linearly withtime, while the oxide particles in the corona grow larger. The distancè betweenthe droplet surface and the corona remains constant, however. This behaviouris typical for combustion at pressures up to about 3 bar. For higher pressuresanother phenomenon is observed in some, but not in all cases: After a fewmilliseconds of burning time the diameter of the corona, which is nearly asluminous as the droplet itself at these pressures, begins to grow. At the sametime the corol}a cools down, so that it becomes invisible within a few milli-seconds (fig. 3). This "blowing-off" does not appear for all. droplets. Smalldroplets are more likely to show this phenomenon than larger ones and thetime from ignition to "blowing-off" is shorter for small droplets.

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(a)

(b)

Fig. I. (a) Zirconium droplets burning in oxygen at 0.3 bar, (b) 3 bar and (c) 9 bar (magnification:40: I).

(c)

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Fig. 2. Shadow of rapidly moving droplets with vapour trails of Zr02• Time interval betweenphotos is 0.5 ms (magnification: 30: I).

Another effect is the blow-up of the whole zirconium droplet, which mayoccur at a rather late stage of combustion (fig. 4). This effect, too, is notobserved for all droplets, but depends on the "burning history". It is favouredby the presence of gaseous impurities in the oxidizing atmosphere or in thedroplet material. The influence of nitrogen on the blow-up process has beenstudied in detail by Meyer et al. 3,5). They found that, depending on thenitrogen content of the gas mixture, the results were solid oxide spheres, gas


Fig. 3. "Blow-off" effect: successive states of a droplet burning at 9 bar (magnification: 40: 1).Time interval between successive images is 0.4 ms.

filled bubbles or small dust particles resulting from exploded bubbles. But alsothe combustion in pure oxygen yields all these combustion products, even ifthe metal is carefully degassed. The various products which are found in thedebris of the combustion in a combustion flash bulb are shown in fig. 5. Thedebris was extracted from the combustion chamber shortly before the end ofthe combustion. There are some unburned metal particles left, which can be

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Fig. 4. Inflation of a falling zirconium droplet burning in oxygen with small admixture ofnitrogen. Photograph from ref. 3 (pressure = 0.8 bar, magnification: 6: I). Time intervalbetween successive images is lA ms.

identified by the strong reflection in the micrograph of the ground particlesthat were embedded into resin. Most of the burned material is found in theform of solid oxide spheres with pores and sometimes with large cavities in thecenter. Besides there are a lot of "egg shells", which are fragments of bubblesthat have been crushed during the extraction or embedding procedure. (Thedark spheres seen in the picture are holes in the resin mass resulting from par-ticles which have fallen out of the embedding material during the grindingprocess.) The oxide particles of irregular shape are fragments of some largerparticles destroyed in the extraction process.

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Combustion of zirconium in oxygen at highpressures

Fig. 5. Micrograph of debris extracted from the combustion chamber shortly before the end ofcombustion. White color is unburned zirconium, light gray is zirconia, dark gray is embeddingresin and black is pores and holes. Large black circles are places where zirconia spheres havefallen out of the resin (magnification: 80: I).

E. Fischer

A theoretical model, that claims to be realistic must account for all thevarious phenomena described above: the expansion of the droplets, the for-mation and stability of the corona, the "blowing-off" occurring at higherpressures and the formation of bubbles at the late stages of combustion. Itmust yield the correct surface temperatures and combustion rates dependenton the oxygen pressure and the droplet radius.

In our attempt to set up such a theoretical model we were successful intoobtaining an accurate quantitative description of some of the phenomena (for.instance the determination of the stationary combustion temperature), othereffects are only explained qualitatively, because for the material properties ofzirconium and its oxides at temperatures between 3500 K and 5000 K only

_ . rough estimates can be obtained, especially for the accommodation coef-ficients on the droplet surface and the diffusion data of oxygen and zirconiummetal versus liquid zirconia. Other phenomena, as for instance the expansionof the luminous surface, are scarcely accessible to detailed experiments, whichcould definitively rule out the flame theory of Marshall and Pellett 6) andMaloney 8), so that the explanation given in this paper remains somewhatspeculative. The various phenomena mentioned above are discussed more indetail in the next sections.

3. Stationary combustion model

The experiments of all the authors mentioned above have shown that theignition phase, characterized by a rapid rise of temperature, an enlargementof the luminous surface and by the formation of the corona, is followed by acomparatively long period characterized by a constant temperature and anapproximately linear decrease of the droplet radius. It is obvious that thisperiod should be most easily accessible to a quantitative theoretical model.Measurements of the stationary temperature have been carried out by twomethods: Kettel ') has determined the temperature from the density of high-speed photographic films, using incandescent tungsten wires for calibration.His results are in agreement with other measurements, if the unknown totalemissivity is set to about 0.7 to 0.8 instead of the value 0.55 which is claimedby Kettel. Measurements by quotient pyrometry have been carried out byDelhaas 9) using the wavelengths Àl = 433 nm and À2 = 671 nm. The problemwith such measurements is that the intensity of the emitted light of an isolateddroplet is too low to allow space-resolved measurements. The integral radia-tion, however, may contain light emitted from the corona or ZrO band radia-tion. Thus an error of about 200 K must be admitted in the experimentalresults. Within this error margin our results obtained from intensity measure-ments at Àl = 396 nm and À2 = 700 nm agree with those of Delhaas. For pres-

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4400

T 5000(K)

t4800

4600

4200

4000

0.4 0,6 0.8 1 2 4 6 8__ p(bar)

0,1 0.2

Fig. 6. Steady-state combustion temperature in dependence on oxygen pressure. Measured points(by a part with errors bars): I-O-l Kettel (ref. 1), I )( IDelhaas (ref. 9), 0 Nelsonand Levine, ref. 2, I • I author. The two lines represent the theoretical results for a =100 urn (solid) and a = 50 urn (dashed).

sures above 5 bar our measurements were corrected for band radiation at396 nm, which is negligible at temperatures below 4500K. The experimentaldata are put together in fig. 6, where in addition a theoretical curve is given,resulting from the model described below.

Another quantity which is accessible to measurements is the droplet radiusand its change with time. One question must, however, be answered first: isthe luminous surface which we see on high-speed photographic films identicalwith the droplet surface, or is there a flame zone around the droplet? Marshalland Pellett 6) who plead for the flame hypothesis argue that the boundary ofthe droplet becomes more and more fuzzy with time. They take the differencein density. between two photos - one taken at be beginning of the com-bustion, the other one taken at a later point of time - as the density caused byradiation from the flame zone. The same effect has been observed in ourexperiments (fig. 7), but it could unobjectionably be attributed to increasingfuzziness of the photo, caused by the motion of the droplet away from thefocal plane. There are at least three arguments for this interpretation.(1) The fuzziness increases steadily. By comparison of the slope of the

density profile with that obtained from an incandescent wire inclined withrespect to the focal plane the velocity of the droplets perpendicular to this

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In (lila)

t 3

2

50 100 150 200 250--- r Iurn)

Fig. 7. Photographic density profiles of successive images of a droplet burning in oxygen at 1 barpressure.

plane could be estimated. The velocity distribution is exactly the same asthe one observed within that plane.

(2) The increase in fuzziness is not the same for all particles. There are a fewparticles which show no increase at all.

(3) Some particles are found which appear to be fuzzy at first and becomesharp at later points of time.

Marshall and Pellett's 6) observation of a fuzziness increasing with pressurecan be easily explained by the fact that the influence of convection, and con-sequently the velocity of the droplets perpendicular to the focal plane,increases with pressure.It should therefore be assumed, that the luminous surface is the surface

of the liquid droplet. Of course this demands another explanation for theincrease of the droplet radius immediately before the stage of stationary com-bustion is reached. We williater come back to this explanation. Assuming thatthe internal growth of the droplet has come to an end when the phase of lineardecrease of the radius begins, the change of the radius can be taken as ameasure of the evaporation rate. If there exists thermodynamic equilibrium atthe droplet surface, the reaction rate depends on the composition of the sur-face layer. The observation, that the temperature remains constant over a longperiod of time therefore implies that the surface composition does not change,

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so that the evaporation rate is equal to the reaction rate. Though this assump-tion, on which most theoretical models published by now 1,4,6,7) are based,leads to sufficient accuracy in the determination of the combustion tempera-ture, these models cannot tell the whole truth. If the rate of reaction equals theevaporation rate, there is no reason to be found, why a droplet whould stop itsstationary combustion and cool down to end in a solid and sometimes porouszirconia sphere (fig. 8a) or blow up to form a zirconia bubble with a shellthickness of only a few microns (fig. 8b).Thus we must abandon the assumption of equilibrium between reaction and

evaporation. The decrease of the droplet radius can no longer be regarded as ameasure of the reaction rate but only as a measure of the evaporation rate,and even that only with the restrietion that the internal state of the dropletremains unchanged. At low pressures Kettel ') reports a unique relationbetween the change of radius and the oxygen pressure.

In our experiments at higher pressures (1 to 9 bar) such a relation could notbe found. For the highest pressures (p >7 bar) even the droplets formed bythe material of one wire in one experiment showed different rates of decrease.The measurements at 1 bar, 3 bar and 5 bar showed a reasonable repro-ducibility, but the evaporation rate depends on the actual size of the droplet.This effect increases with pressure. At 1 bar it is significant only for smalldroplets (radius < 70 urn), so that it can be understood that the effect was notfound by Kettel, who has investigated droplets of 100 urn to 200 urn radius.The accuracy of the experimental results decreases with increasing pressure, asthe droplet is obscured more and more by radiation from the corona. In addi-tion, due to the enhanced convection velocities and to the "blow-off" effectmentioned above, only few data points could be obtained for p > 3 bar" Thusfrom the experimental results a quantitative agreement with some theoreticalmodel should not be expected, but the data should at least allow a qualitativecontrol.

3.1. Basic equationsAs has been mentioned in the last section, a really stationary state of com-

bustion does not exist, because the composition and the radius of the dropletschange with time. There is, however, a rather long period of time in which atleast the rate-controlling processes remain unchanged. The time constants forchanges of droplet radius, surface temperature and droplet composition canbe regarded as very large compared with those of the reactions or the evapora-tion, so that an equilibrium is established at the liquid surface of the dropletand its corona. The model presented here is based on a chemical rateequilibrium at the droplet surface and the assumption of stationary diffusion

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(a)

(b)Fig. 8. Products of zirconium combustion. (a) Compact zirconia sphere with small pores(magnification: 250: I). (b) "Egg shell", solidified skin of a zirconia bubble (magnification:250: I).

(1)


processes in the gas between the droplet and its corona. The diffusion proces-ses within the burning droplet are described by a step model.

Let us first consider the processes at the surface of the liquid droplets,which are formed within a small fraction of a millisecond as soon as themelting point of the wire material is reached. At this time the droplet can beregarded as a homogeneous sphere of zirconium with small admixtures ofZ:r02 resulting from the oxide layer which was formed in the heating processof the wire. For the experiments, in which the wire is heated to the meltingpoint by a current pulse, the heating time is short, so that the initial state ofthe droplets can be regarded as pure zirconium. There are two possibleprocesses that limit the reaction rate at this stage of combustion. One limit isthe gas kinetic rate of oxygen molecules impinging upon the droplet surface:

(P02 is the oxygen pressure, m02 is the mass of the oxygen molecules, Tthe gastemperature and k the Boltzmann constant). To estimate the maximum pos-sible rate this value must be multiplied by some accommodation coefficient a02

to account for the fact that not all oxygen molecules remain long enough atthe surface to allow subsequent chemical reaction or diffusion into the metal.

The other limit to the reaction is the rate at which the oxygen molecules canreact with the surface atoms of the droplets. According to Kofstad 10) thisreaction process is proportional to the square root of the oxygen pressuremultiplied by an exponential term containing the activation energy of theprocess:

RR = C exp ( - Ei kT) VP02 • (2)

The numerical values of C and Ei can be derived from measurements of thereaction rate carried out by Kettel '), The results confirm the pressure depen-dence of eq. (2). The numerical form is

(3)

where P02 is measured in bar. The measurements of Kettel cover the tem-perature range from 1600K to the melting point. Certainly it is not quite cor-rect to extrapolate the formula above the melting point or even to' tempera-tures in the range of 4500K, but as no other data are available the best way isto use these data and to look if reasonable results can be obtained with them.In the calculation of the stationary combustion the magnitude of the surfacereaction rate is not critical anyway, because at these high temperatures thekinetic rate is lower all over the pressure range considered. The only point

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where RR comes in is to determine the rise time of temperature in thebeginning of the combustion.

Of course formula (3) holds only for a clean metallic surface. During theheating process, however, more and more of the surface gets covered by oxidemolecules, so that the rate of surface reactions will be reduced. Instead,however, oxygen molecules or rather atoms may diffuse through this oxidelayer and react within the liquid droplet. If the surface layer reaches a criticalthickness, the diffusion through this layer may become the rate-determiningfactor. The diffusion coefficient of oxygen in Zr02 at high temperatures is notknown. An estimate of the critical surface layer thickness can be obtainedfrom the experiments, however. Once the critical thickness is reached thereaction rate is no longer sufficient to maintain a constant burning tempera-ture. The decreasing temperature leads to a reduction of the evaporation ofZr02 from the surface and thus to an accelerated growth of the oxide layer, sothat the temperature falls very quickly at this point. Thus we can estimate thediffusion coefficient from the time which is necessary to reach the point wherethe rapid cooling begins. In the numerical calculations we have used the con-dition that the maximum rate is

R = RR for d < do

~ ~do

R = RR - for d > dod

where d is the thickness of the oxide surface layer and do is a constant whichmust be adjusted to the experiments.

Whether the thickness of the oxide layer grows or shrinks during the com-bustion process, depends on the balance of the reaction rate, the evaporationrate and the rate at which oxide molecules are dissolved in the metal core. Thechange of d is described by

dd mzro2- = --[R(I +x) - (l-x)Rn-Rv].dt {!Zr02

(5)

In this equation x is the molar fraction of oxide molecules dissolved in themetal core, (I - x) Rn is the diffusion rate between the oxide shell and themetal core, Rv is the rate of evaporation. The unknown coefficient Rn can beestimated from the blow-up effect mentioned above. Details on this phenom-enon will be given later. The change of the molar fraction x is

dx 3mzr- = - [Rn(l - xY +Rx(1 + x)].dt a{!Zr

(6)

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This formula is correct only if the thickness of the oxide layer is small com-pared with the droplet radius and the oxide content of the core is small. Thecorrections that have to be made, if these conditions are not fulfilled are easyto calculate, but are omitted here to keep the expression clear.

The change of the droplet radius with time follows from the evaporationrate:

da mZr02-=--Rv.dt {lZr02

To determine the net evaporation rate Rv the concentrations of Zr02' ZrO, Zrand O2 in the gas phase near the droplet surface must be known. As long asthe total collision rate at the surface is large compared with the net evapora-tion rate, the partial pressures should obey the thermodynamic equilibriumrelations. For use in the numerical calculations the equilibrium constantsknown from literature (JANAF 11» have been approximated by the analyticalexpression (all pressures given in bar):

(7)

PZr VP02/PzrO = exp(24.684 - 143075/T)

PZrO VPo2/Pzro2 = exp(24.956 - 121007/T).

(8)

(9)

While the assumption of thermodynamic equilibrium appears to be satisfied atthe surface for those species which contain zirconium, the dissociation ofoxygen will be well below its thermodynamic equilibrium level, because a largefraction of the oxygen molecules impinging upon the surface is consumed intothe droplet. Therefore it appears to be more reasonable to leave atomicoxygen completely out of consideration in the thermodynamic calculations.

Of course eqs (8) and (9) are not sufficient to determine the four unknownpartial pressures. One additional condition we find in the pressure balance.Estimating the magnitude of the. oxygen flux velocity from the experimentaldata, we find that this velocity does not exceed the range of a few meters persecond, Le. well below the velocity of sound, so that the total pressure isapproximately constant:

P = P02 + PZr +PZrO +PZr02 = constant. (10)

The fourth condition, which determines the composition of the gas phase,results from the diffusion equilibrium between the droplet and the corona. Wewill come back to this condition below. Once the composition of the gas phaseat the surface is known, the net evaporation rate can be calculated. The dy-namic equilibrium at the surface is determined by the evaporation rate of zir-conium and zirconium oxide from the liquid and condensation of Zr or Zr02vapour onto it. If the surface consists of pure Zr02' the evaporation rate is

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Rvo = PVz.o,l V2n mzro2 kT. (11)

where PVz.o, is the saturation vapour pressure of Zr02' If only the surfacefraction (1 - Xd) is covered by Zr02 the rate is reduced by this molar fraction.Instead the corresponding molar fraction must be added for evaporation ofZr. The condensation rates of Zr and Zr02 result from equations analogous to(11) with only the saturation pressures replaced by the actual pressures. Thusthe net evaporation rate is

Rv = RZr02 + RZr

with (12)(1 - Xd)

RZr02 = V2 kT (Pvz.o, - PZr02)'nmzro2

XdRZr = V2n mZ

rkT (Pvzr - PZr).

For the saturation vapour pressures (in bar) of Zr and Zr02 the analyticexpressions

PVZr = exp (14.817 - 70784/T) (13)

andPVZ.o, = exp(17.168 -77612/T) (14)

have been used (approximation to tables given in ref. 11). In the quasi-steady-state combustion the composition of the gas phase changes only slowly so thatall the material that evaporates from the surface must leave the surface regionby diffusion against the incoming stream of oxygen. On its way the zirconiummetal vapour and the ZrO will be transformed to Zr02 as soon as it reacheszones of lower temperature. Condensation of Zr02 will not occur, however,until the molecules have undergone a certain number of collisions with oneanother to form small clusters first and then larger condensate droplets.A detailed theoretical investigation of the condensation process shall not begiven here, because a gas kinetic approach to the determination of the clustergrowth and diffusion would introduce so many new unknown parameters intothe calculations that the results would be of no practical use. Even very simpli-fied models as proposed e.g. by Turkdogan 12)all contain adjustable factorswhich are applicable only in a limited range of pressures and droplet sizes.From the experiments it can be deduced that the distance between thedroplet and the corona is independent of the droplet size and is proportionalto the inverse square root of the total pressure. The distance is approximatelygiven by

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c - a = o.ol/Vi (cm). (15)

The partial pressures of the individual species in the gas between the dropletsurface (r = a) and the corona (r = c) can only be determined if the reactionvelocities in the gas phase are known. As data on these reactions are missing,we must content ourselves with a rather rough approximation for the deter-mination of the particle fluxes in this zone. The following simplifying assump-tions are made.(1) Reactions take place only at the liquid surface of the droplet and of the

condensed corona particles.(2) The diffusion coefficients of Zr, ZrO and Zr02 against oxygen are equal.(3) The partial pressure of Zr02 is equal to the saturation vapour pressure

at r = c.Denoting the sum of the particle flux densities of all the zirconium-con-

taining species as jl, the flux of oxygen molecules by jo and the correspondingpartial pressures by PI and Po, these flux densities must obey a diffusion lawof the form

dpI 1 . .dr = nD (PI Jo - Po}1), (16)

where n is the total particle density and D the diffusion coefficient. Thequantity nD does not depend on the pressure. The numerical value used in thecalculations was fitted to the experimental results. If no chemical reactionstake place in the gas phase, the divergence ofthe fluxes j, and j, is zero, so thatfor spherical symmetry the flux densities can be expressed by

A Bi, = "2' jo = - "2 .

r r(17)

Introducing the total pressurep = Po + PI> eq. (16) can be written in the form

dp1 1dr = - nDr2 [(A - B)PI - pA]. (18)

With the boundary conditions at the droplet surface and at the corona eq. (18)can be solved. It yields not only the radial pressure profiles but in addition arelation between the constants A and B, the missing condition to determinethe equilibrium at the droplet surface.

The diffusion flux densities are connected to the reaction and evaporationrates by the continuity conditions at the surface.(1) The total amount of zirconium evaporating from the surface must be

carried away by diffusion:A"2 =Rv.a

(19)

B P +.lp--R R _ Zr 2ZrOR2 - R + Zr V·a P1

So far we have only considered the rates of the processes that govern the par-ticle fluxes and productions inside and outside the zirconium droplets. Thetemperatures of the surface, of the surrounding gas and of the corona havebeen regarded as parameters of the solution. To treat the combustion problemas a whole the flux and production of energy must of course be determinedsimultaneously with the particle flux balance.

In the energy balance at the surface of the droplet the most important termsare heat production by chemical reactions and heat consumption by evapora-tion. In addition radiation, conductive heat losses to the surrounding gas andconduction into the droplet must be considered. The latter process can betreated in a very simple way, because the thermal conductivity within thedroplet is so high, that even in the rapid heating process at the beginning ofcombustion the droplet can be regarded as isothermal. (An estimate for adroplet of 100 urn diameter has shown that at no point of time the tem-perature in the center differs by more than 50K from that of the surface.)

The energy produced by chemical reactions consists of two fractions. Onone hand there are the reactions in which liquid Zr is transformed into liquidZr02 at the surface of the droplet or inside it. On the other hand gas-phasereactions may release energy to the droplet if its surface acts as a partner in atriple collision. The following reactions have to be considered in the energybalance.

(20)

E. Fischer

(2) The oxygen flux supplied by diffusion is consumed into the droplet bypart only. The rest is used to establish the chemical equilibrium in the gasphase near the droplet surface. Thus we get for the diffusion flux of O2:

(1)(2)(3)(4)(5)

Zr(l) + O2 -> Zr02(l) + 1.72x 10-18 WsZr(l) -> Zr(g) - 0.98 X 10-18 WsZr02(l) -> Zr02(g) - 1.04X 10-18 WsZr(g) + i02 -> ZrO(g) + 1.05X 10-18 WsZrO(g) + i02 -> Zr02(g) + 0.61 X 10-18 Ws.

(21)

Denoting the five reaction enthalpies by flH1 to DJ/5 the energy balance canbe written in the form

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The last term contains the radiation loss (s is the relative effective emissivity,a the Stefan-Boltzmann constant) and the conductive loss to the gas.The temperature gradient at the surface must be determined from the

energy balance in the surrounding gas, which again is coupled to the balancein the corona. Assuming that no reactions take place in the gas phase, theprocesses supplying energy to the corona are condensation of Zr02 andoxidation of ZrO and Zr. Smaller fractions are also contributed by absorptionof radiation from the droplet and heat conduction. This energy is carried offby conduction and by radiation from the corona itself. The heat capacity ofthe corona is small compared with that of the droplet, so that we may neglectit in the balance:

(23)

(The flux densities j denote the values at the droplet surface.) Radiativeheating of the corona has not been included in this equation, as this effect issmall compared to the transport of reaction energy, and a meaningful estima-tion of the absorptivity of the corona is scarcely possible. The same holds forthe emissivity es. This term has been included only to check the influence ofradiation on the results in the numerical calculations, when e: is variedbetween its theoretical limits (0 < ec < 1). In equation (23) the symbolr = c ± 0 denotes the values at the inner and outer boundary of the corona.As we have regarded the thickness of the corona as zero, the temperaturegradient must be taken as an unsteady function at r = c.

The temperature profile in the gas is determined only by heat conduction.Introducing the heat flux potential

T

S = f ÀdT (24)Too

the solution of the heat flux equation is

s, K2S = - for r >c and S = - + Ks for r<c. (25)r r

The constants Ks to Ka follow from the continuity of S at r = c:

s; K2 Tc- = - + Ka = f À dT,c c Too

from the boundary value at r = a:

(26)

E. Fischer

Ta

J ÀdTa Toe

(27)

and from the condition (23) in which the temperature gradients can be ex-pressed by

and À(()T) =" - -c2 •ur r=c+O(28)

Equation (23) contains the corona temperature explicitely as well, so that thesystem of eqs (23) to (28) can only be solved by iteration. Also the solution ofthe diffusion equation (17) depends on Tc via the boundary condition at r = c,so that all the eqs (1) to (28) must be solved simultaneously in an iterativeprocedure to find the time dependence of temperature, radius and composi-tion of the zirconium droplet for a given set of initial conditions. To obtainrealistic results the main problem is to find reasonable data on the variousmaterial properties that enter the set of equations given above. A critical viewof these data is given in the next section.

3.2. Material properties

As has been shown, the combustion of zirconium droplets is controlled bytransport and reaction processes within the liquid droplet, at its surface and inthe surrounding gas. Thus material and transport properties of gas and liquidenter the solution as well as the rate coefficients at the surface. Unfortunatelyonly part of the data are known from literature or can be estimated with highaccuracy. The other data must be fitted to the experiments, but due to thelarge number of fitting parameters there is some uncertaintly in finding thecorrect set of parameters. The processes which control the steady-state tem-perature of the surface, are comparatively well-known, and on the other handthis temperature can be measured with reasonable accuracy. The error limitsof these measurements have been discussed above.

Though a large number of processes influence the energy balance at the sur-face, the stabilization of the temperature primarily results from the balance oftwo processes, the increase of energy losses from the surface by evaporationof zirconium oxide and the decrease of the kinetic collision rate of oxygenmolecules with the surface when the partial pressure of oxygen diminishes.The chemical equilibrium data (eqs (8) to (14» can be regarded as ratheraccurate, so that the only uncertainty results from the unknown accommoda-tion coefficient of oxygen at the surface. Putting a02 to unity leads to tempera-

410 PhlllpsJournal of Research Vol.35 No.6 1980

PhillpsJournal of Research Vol.3S No.6 1980 411


tures that are up to 500K above the measured ones. The best fit to theexperiments is found with a02 of the order of 0.1 with decreasing tendency tohigher temperatures. The numerical calculations have been carried out witha02 = 0.1. The accommodation coefficients for Zr and Zr02 molecules havebeen set equal to one. The influence of these coefficients on the balance issmall, however, as they only come in, if there are deviations from thermo-dynamic equilibrium at the surface. Heat conduction to the gas and evenradiation from the surface are less important in the stationary energy balance,so that the equilibrium temperature is scarcely affected by the trapping ofradiation caused by an optically thick corona layer or by additional irradia-tion from other burning droplets.

At the equilibrium point the reaction rate approximately equals the evapo-ration rate, so that a reasonably good estimation of the surface temperaturecan also be found from an equilibrium model in which the two rates areassumed to balance exactly. The small deviations from this equilibrium,however, which are influenced by all the processes mentioned above, deter-mine the changes of composition of the droplet and of its surface layer and arethus responsible for the final fate of a droplet; they must all be considered tofind out why one droplet becomes an oxide bubble while another will end as asolid sphere.

During the quasi-stationary combustion stage the kinetic collision ratelimits the reaction at the surface. The temperature rise at the beginning of thecombustion is controlled by other processes. Due to its high activation energythe reaction of oxygen molecules at the surface is kinetically restrained at lowtemperatures, so that the reaction rate given in eq. (3) controls the heatingprocess. The accuracy of this formula is high below the melting point, butapplying it to liquid droplets may induce some errors. On the side of the lossprocesses evaporation is not important at low temperatures. Instead nowradiation and thermal conduction losses become dominant. In the determina-tion of the radiative losses the unknown surface emissivity limits the accuracyof the calculations. A relative emissivity of 0.7 to 0.8 appears reasonable for ahot oxide surface. This as in agreement with the value 0.7, which must be usedto fit Kettel's temperature 1) measurements to the theoretical and to otherexperimental data. In the numerical calculations the value IJ = 0.7 has beenused.

The conductive losses are subject to the largest error margin. Theoreticaland experimental data on the thermal conductivity of oxygen are availableonly up to 2000 K (ref. 13). Between room temperature and 2000K the con-ductivity increases approximately linearly with temperature. At higher tem-peratures a steeper slope should be expected due to transport of electronic

E. Fischer

excitation and dissociation energy. This fraction of the conductivity has beenapproximated by an exponential term taking the dissociation energy of oxygenfor the activation energy. Of course, this approximation holds only for lowdegrees of dissociation, because at·still higher temperatures, when the gas iscompletely dissociated, the conductivity decreases. This behaviour is well-known for nitrogen, where experimental data from are measurements areavailable 14). The exponential approximation was first tested with the nitrogendata, after which the data were reduced to oxygen, changing the activationenergy and the mean energy transported in the excited states by the ratio of thedissociation energies. This leads to the analytic approximation

À = T[1.3Sxl0-4 + 60exp(-60000/n1 (Wm-1K-1). (29)

Using this approximation reasonable results are obtained for pressures above1 bar, but at low pressures the heat losses would exceed the energy gain byreactions for small droplets before they reach the evaporation equilibrium. Assuch an effect is not observed in the experiments, the transport of intrinsicenergy in excited levels must have been overestimated in eq. (29). Electronicexcitation must be below its equilibrium level. An estimation of the dis-sociation rate given by Kettel ') shows that at pressures below 1 bar thevelocity of the oxygen flow towards the droplet is too high to allow the equi-librium to by established. The exponential term of eq. (29)must therefore bereduced by a pressure-dependent factor. In the calculations a reduction factor1/(1 + O.S/p) (p in bar) has been used.According to the very crude approximation used for the thermal con-

ductivity, not more than an order of magnitude estimate can be obtained fromthe theory for the rise time of temperature, but it is sufficient to establishtrends and possible limits for reaching steady-state combustion.

The initial temperature rise and the stationary temperature at which adroplet burns can be calculated with sufficient accuracy from the data givenabove. To answer the question of how long a particle can remain at this tem-perature, the changes in the composition of the surface and of the droplet'sinterior must be known. As we have mentioned above, the evaporation rate ofoxide from the surface is not exactly equal to the reaction rate during thestationary combustion. Thus the thickness of the oxide layer may change. Onthe one hand it may grow larger or smaller dependent on the sign of thequantity (A - B) in eq. (18); on the other hand the thickness will be steadilyreduced by diffusion of oxide molecules into the metal core of the droplet. Thediffusion rate Ro (eq. (5» can be estimated from the composition of dropletsthat are sucked out of the combustion chamber a few milliseconds afterignition. But as most droplets are split into many smaller ones, when extracted

412 Phlllps Journal of Research Vol.35 No.6 1980

PhilipsJournal of Research Vol. 35 No.6 J980 413


(a)

(b)Fig. 9. Particles extracted from the oxygen chamber during combustion. (a) Metal sphere withlow content of Zr02 (magnification: 800: 1). (b) Fragment of a large particle with metal core(white) and oxide surface layer (gray) (magnification: 800 : 1).

E. Fischer

from the burning chamber in a liquid state, and since the incorporation ofoxygen goes on during the cooling process, this method gives only a crudeestimate. Most metal particles that have been extracted from the burningchamber show a dendritic structure (fig. 9a) with an oxide content of only a.few percent. Other particles, which have solidified as a whole, clearly show upthe oxide surface layer (fig. 9b). From these facts we can deduce that thediffusion rate can neither be so high that most of the oxide is dissolved into themetal core, nor so low that diffusion is negligible. The rate coefficient shouldtherefore be of the order of 1020 cm" S-I. This value is confirmed by theobservations of the "blow-up" effect, to which we will return later on.As has been mentioned above, the second quantity that determines the

growth of the oxide surface layer is the difference between the reaction rateand the rate of evaporation. This quantity depends on the rate equilibrium atthe surface and on the diffusion rates between the surface and the corona.Thus in addition to the equilibrium relations at the droplet surface we needdetailed information on the processes in the surrounding gas to obtain quan-titative statements about the growth of the oxide layer. The diffusion coef-ficient of zirconium oxide against oxygen at high temperatures is not known.An estimation from hard-core molecular data for the product of particledensity and diffusion coefficient can be obtained from the formula (see ref. IS):

3 V 2nkT 1nD = 16 --;;:; nUli' (30)

where f.l12 is the reduced mass and nUli is the collision cross-section. In-serting the effective molecular radii of oxygen and Zr02 we find a valuenD = 6 x 1017 liT (cm-I çl). Besides the errors inherent in the hard-coreapproximation a further uncertainty results from the clustering of the oxidemolecules on their way to the corona. Thus the numerical value given aboveshould only be regarded as a crude estimate. An experimental check would bepossible by measuring the time at which the oxide layer reaches a critical thick-ness, so that diffusion of oxygen through this layer becomes the rate deter-mining process. At this point of time a rapid decrease of the surface tempera-ture is observed. Unfortunately the diffusion coefficient of oxygen throughzirconium oxide at temperatures of about 4000 K is also unknown, so that thecritical thickness of the oxide layer remains as another adjustable factor in theinterpretation of the experimental results. In the numerical calculations wehave therefore treated the value given above for nD as fixed and have adjustedthe diffusion rate in the liquid surface layer to the experiments. The maximumpossible reaction rate has been assumed to be given by eq. (3) and above acritical layer thickness do to be reduced by the diffusion process by a factor

414 PhilipsJournal of Research Vol.35 No.6 1980


d/d« according to eq. (4). To get the cooling point into the right order ofmagnitude, the value of do had to be chosen in the order of 0.1 urn.

With the set of equations and the material data given above, the com-bustion of isolated zirconium droplets can be described" not very accurately,of course, but in such a way that all the various phenomena occurring duringthe combustion process can be understood at least qualitatively. Using thenumerical results obtained from the model calculations these phenomena arediscussed more in detail in the following sections.

4. Stages of the combustion process

4.1. IgnitionThe ignition of zirconium wires or shreds can be initiated by Ohmic heating,

by a laser pulse or some other intense irradiation or by contact with someother preheated piece of matter. Dependent on the mechanism, on the shapeof the zirconium specimen and on its surrounding conditions the temperaturemay rise more or less quickly. As soon as the melting point is reached,however, there are only a few parameters left that determine the furtherevaluation. The size and initial composition of the spherical droplets, formedat the melting point are responsible for all effects, which are observed at laterstages of combustion. Within a general picture of zirconium combustion theformation of liquid droplets should be the starting point. The ignition phasethen covers the time from droplet formation to the point at which the maxi-mum surface temperature is reached. During this period of time, besides therise time itself, two effects are of interest: the growing of the droplets and theformation of the corona.

4.1.1. Temperature rise time

The rise time has been calculated from the basic equations given in the lastsection. In these equations we have assumed that the characteristic time ofchanges in temperature is slow compared to the diffusion time and to the timeconstant of heat conduction within the droplet.The time constant for diffusion is given by

[2

'D=-·D(31)

Using the distance between the stationary corona and the droplet (eq. (15» forthe characteristic length l and the diffusion coefficient according to eq. (30),we obtain a value 'D = 4 IlS at 4000K and 10 IlS at the melting point. The risetimes which we have calculated for droplet radii from 30 urn to 100 urn and

PhilipsJouroal of Research Vol.35 No,6 1980 415

E. Fischer

pressures from 0.3 bar to 8 bar are all in the range from 30 us to 300 us, sothat the quasi-stationary approximation appears justified with respect to thediffusion.

In the energy balance of the droplet surface (eq. 22) the interior of thedroplet has been treated as isothermal, so that the heat consumed by thedroplet is tna3 gm Cp CöT/"Dt)a. For an exact solution the energy balanceinside the droplet must be solved with the right-hand side of eq. (22) as aboundary condition for the heat flux into the droplet. The energy balance inspherical coordinates has the form

(32)

(Àm is the thermal conductivity of the liquid metal). The characteristic time forheat conduction into the droplet can be estimted from this equation

(33)

For a droplet of a = 50 urn the time T,1. is of the order of 0.1 ms, which iscomparable to the total temperature rise time. An exact solution of eq. (32)has therefore been carried out with the simplified boundary condition:

(34)

which of course is valid only up to temperatures a few hundred degrees belowthat of stationary combustion. Results have shown that even at high pressures(8 bar) the temperature differences inside the droplet are small (max. 200K fora droplet of a = 50 urn) during the entire heating time. Thus the approxima-tion in the left-hand side of eq. (22) is justified even during the ignition phaseof the droplet. Calculated temperature rise times - here defined as the periodof time from droplet formation to the point when 990/0of the maximum tem-perature is reached - are given in fig. 10, dependent on droplet radius and gaspressure. For comparison some experimental points are given which have beenestimated from high-speed photography. The error limit of these points isabout 0.2 ms, the interval between two subsequent frames.

4.1.2. Droplet .gr owi ng

As has been mentioned before, approximately at the time when the finaltemperature is reached the zirconium droplets begin to grow. The radius ofthe luminous surface increases by a factor of between 1.2 and 1.7. In section 2we have discussed why this growth should be regarded as an increase of the

416 PhillpsJouroal of Research Vol.35 No.6 1980


-r(ms)

.p=lbaro p =5 baro p= 8 bar

0.4

0.3 bar

0.6

0.2

0~-+----~---r---.----~---r---440 50 60 70 80 90 100

Fig. 10. Temperature rise time from melting point to maximum temperature plotted versusoxygen pressure and particle size. Measured points are given with error bars.

liquid surface and not as some kind of optically thick flame. There must besome process by which the droplets or parts of them get a foamy structurewhen the temperature exceeds a certain limit. The internal pressure in theliquid must exceed the outside pressure. To form a bubble inside a liquid thevapour pressure must counteract the outside pressure and, in addition, thesurface tension. Bubbles can only be formed if there are nuclei of some criticalsize, so that the difference between the saturation vapour pressure and theoutside pressure exceeds the surface tension. The critical size decreases as thepressure difference increases. Once the bubbles are formed they will growfurther with a growing rate limited only by the rate at which the energy ofvaporization can be supplied 16) or, in a mixture of fluids, by the total amountof the vaporizable component.

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E. Fischer

The surface temperature of a burning zirconium droplet is determined bythe rate equilibrium described in sec. 3. If the surface of the droplet is coveredby an oxide layer, the oxide vapour pressure is close to its equilibrium value.Due to the high oxygen pressure, however, the metal vapour pressure will befar below its equilibrium value. If a bubble is formed inside the droplet, whereoxygen can only exist to an amount described by the equilibrium relations (8)and (9), the pressures of Zr02 and Zr will both be at their equilibrium values,reduced of course by the molar fractions at which they are mixed in the fluid.Thus the total internal pressure can be expressed as

Pi = xPVzr02 + (1 - x)Pvzr +PVzrO + P02 (35)

with PZrO and P02 resulting from eqs (8) and (9), Pi reaches its maximum nearx = 0.5, the outside pressure corresponds approximately to x = 1. But as PVZris smaller than PVZr02 there exists a second value Xcrit at which Pi equals theoutside pressure. This value is about 1.5070 at P = 1 bar and 7% at 8 bar,always at the maximum combustion temperature. The ratio of the maximuminternal pressure and the outside pressure at Pmax is shown in fig. 11. Two

Pmax-P-

t 3......-....../ "I "I ~

/I

2 II

O+---~-.---.--~--r--.---r~o 2 468_ p(bar)

Fig. 11. Ratio of maximum internal pressure to outside pressure at steady-state burning condition(----a = 100 urn, ----a = 50 urn) .

.418 Phlllps Journalof Research Vol.35 No.6 1980


processes can be found by which zones of material can be formed in which themolar fraction Xcrit is considerably exceeded. On the one hand such zonesresult from the formation of the droplets: before the zirconium specimenreaches its melting point it will be covered by an oxide layer. As the meltingpoint of the oxide is higher than that of the metal, the melting process willstart inside the specimen, the solid oxide layer will be crushed and small oxideparticles will be distributed in the forming droplet. Due to the slow diffusionof Zr02 in Zr (see sec. 3.2) these particles will survive as oxide-enriched zonesfor a long time in the combustion process. The second source of zones inwhich the oxide molar fraction is high is obviously the combustion itself. Oncethe droplet has formed and the convection inside the droplet has faded out, anew oxide surface layer is formed. Of course, somewhere in the boundarybetween the metal core and the oxide layer the molar fraction x must assumeany value between zero and one, so that the conditions for bubble formationare always fulfilled if only the temperature is high enough. In principle thebubbles in this boundary layer could grow to an arbitrary size as sufficientmaterial can be supplied, but the size will soon be limited as oxygen mustdiffuse through this bubble layer to sustain the combustion. It is evident thatan exact mathematical formulation of the growth process is of no practicaluse, as too many factors in the heating process can influence the bubble for-mation. There is the oxygen content, which depends on the heating processand eventually on the method of fabricating the zirconium, there may beimpurities in the metal, which serve as nuclei for bubble formation; and theremay be some gas content in the metal, which makes the bubbles grow larger.Therefore we should confine ourselves to a qualitative understanding of thegrowth effect. The experiments show that the growth process comes to an endwithin about 1 ms. This time scale shows no significant pressure dependence.For pressures greater than 3 bar, however, it cannot exactly be measured asthe droplet is obscured by the radiation of the corona, so that its size can bemeasured only before the formation and after the "blowing-off" of thecorona. A typical example of the time dependence of the droplet size is shownin fig. 12 for combustion at 1 bar. Figure 13 shows the diameter of theluminous image at 9 bar, where droplet and corona can be measured inde-pendently only at the later stages of combustion.

4.1.3. Corona formation

Simultaneously with the growth of the droplet, evaporation of oxide fromthe surface begins to stabilize the surface temperature. The evaporating oxidediffuses against the incoming stream of oxygen into regions of lower tempera-tures, where it condenses first into larger clusters and finally into small

Philips Journol of Research Vol.35 No.6 1980 419

E. Fischer

a ••(IJm) • •

t • •100 • •I>

•80 • • • • •

•60 •

•••

40

>

0 2 4 6 8 10 12 14 16_t(ms)

Fig. 12. Change of droplet size with time. (Before first point and after last point blackening of thephotographic film is too weak to allow measurement.) Oxygen pressure is 1 bar.

4 6_t(ms)

Fig. 13. Change of droplet radius (a) and corona radius (c) for two droplets at 9 bar. Dropletradius could only be measured when the corona was blown off.

c Iurn)cturn)

t 140

120

100

80

60

40

0 2

a

420 Phlllps Journal of Research VoJ.35 No.6 1980


droplets. We will not go into the details of the theoretical description in thispaper, but only briefly outline the essential aspects of a model, which explainsthe formation and stability of the corona. Assuming that no non-gaseousimpurities exist near the burning droplet, the condensate particles must beformed by homogeneous nucleation 17) in the gas. The formation of clusterscan then be described by simple gas kinetic formulas, which together with thediffusion equations yield the size distribution of clusters dependent on thedistance from the droplet. The results show that the concentration of isolatedmolecules falls off very rapidly at a certain distance from the surface, which isidentified with the corona distance. Once larger droplets have been formedtheir position is stabilized. As long as the flux of oxygen is approximatelyequal to the flux of Zr02 vapour, the Stokes force of the oxygen flow whichwould tend to move the oxide particles back to the surface is balanced by themomentum transfer from molecules condensing onto the particles in thecorona. During the ignition phase the oxygen flux exceeds the vapour fluxfrom the surface so that the condensing oxide particles cannot find a stableposition and will first show up as a structureless fog layer around the droplet,which is then transformed into the stationary corona within less than 1 ms.

4.2. Steady-state combustionsThe ignition phase, which covers a time of at most one or two milliseconds

in the parameter range considered here, is followed by a comparatively longperiod of time in which no dramatic effects can be observed from outside. Thetemperature remains constant within the measuring accuracy and the dropletradius shows an approximately linear decrease. To calculate the surface tem-perature in this period the simplest way would be to solve the equations givenin the last section with the condition (i)T/bt)r=a = O. There is, however,another quantity that may change drastically during this stage of combustion:the thickness of the oxide layer which covers the droplet. The growth velocityof this layer may vary and even change sign when slight changes of the tern:'perature occur. To get the whole story of combustion of a particular dropletwe must determine the time-dependent solution of the basic set of equations.There are three quantities that are continuously changing: the droplet radius,the oxide layer thickness and the composition of the droplet core. Thesequantities have been calculated together with the temperature for variousoxygen pressures and droplet sizes (0.1 bar <p < 9 bar, 30 llII1<a < 200 j.LIIl).To account for the ZrO-bubble formation inside the droplets the mass densityhas been assumed to be reduced to 25070 of the density of liquid zirconium.Figure 14 shows the calculated temperature dependence on droplet size andoxygen pressure. As can be seen from the graph, the temperature is nearly

PhJlipsJournal of Research Vol.35 No.6 1980 421

E. Fischer

Bbar

T 5bar(KJ 3bar

2bar

1bar

0.5bar

0.25bar

3500

3000

0.1bar

40 60 BO 100 120 140 160 180-a(lJmJ

Fig. 14. Steady-state combustion temperature in dependence on oxygen pressure and dropletradius (theoretical model).

independent from the particle size at high pressures, whereas at low pressures(p < 1 bar) a noticeable increase is found with increasing droplet radius. Thatthis effect could not be found in Kettel's experiments 1) appears obviousregarding the measuring accuracy of ± 200K.

A comparison with experimental data is given in fig. 6, where the pressuredependence of the burning temperature for a = 100 J.UTI is shown togetherwith temperature measurements reported by several authors. In the pressurerange between 0.3 bar and 3 bar the measurements agree very well with thetheoretical model. There is a systematic deviation at very low pressures whichmay be caused by the fact that the accommodation coefficient of oxygen at thesurface has been assumed to be constant at all temperatures. At temperaturesabove 3 bar the experimental data are very inaccurate, because molecularbands from the gas phase and continuous radiation from the corona mix upwith the radiation of the droplet surface. In the theoretical curve an additional

422 Phlllps Juurnal of Research Vol.35 No.6 1980


uncertainty comes in as well. The problem is that at high temperatures theequilibrium model described in the last chapter leads to a multivalued solu-tion. The solution of the equilibrium relations for a given surface temperatureleads to a third-order algebraic equation with one or three real roots. Thecoefficients of the third-order equation may change by orders of magnitudewithin a very small interval of temperatures and thus change the solution fromunique (T< Ti) to three-valued (Ti < T< T2) and back to unique (T> 7;.).Solving the energy balance and plotting dT/dt at the droplet surface against T,we find S-shaped curves (fig. IS). For T> 7;. there is a unique solution with

t -- (î)-----.__CD ............-_ <,......._ ......_ <,<,

...................... <,

dTdi

Fig. IS. Schematic representation of the function dT/dt. The function is three-valued between 11and 72.Curve 1: dT/ dt = 0 at 10 < 1;. Stable equilibrium at 10.Curve 2: dT/dt = 0 at 10 between 11 and 72. dT/dt < 0 at 7;. Equilibrium at 10 may be changed

by external perturbations.Curve 3: dT/dt = 0 at 10 between 11 and 72. dT/dt > 0 at T2• No stable equilibrium possible

(reactive boiling).

dT/dt < O. For low temperatures dT/dt is always positive. If it changes signbelow Ti (curve 1), there is no problem in the determination of the stationarytemperature, because starting at low temperatures a stable equilibrium will bereached as soon as dT/dt becomes zero. Even if dT/dt = 0 is reached in theinterval between Ti and ~ (curve 2 in fig. 15) the equilibrium will be reached.The only difference is that if the equilibrium is disturbed by some influencefrom outside - this may be an impact with another burning droplet or oxideparticle or the bursting of a major gas bubble inside the droplet - the burning

Philips Journalof Research Vol.35 No.6 1980 423

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E. Fiseher

conditions may change from one branch of the curve to another, where dT/dtis negative. Thus, if the equilibrium point is between 11 and 12, suddenchanges of the burning conditions must be expected. According to our set ofdata the condition T> 11 is reached between 1 bar and 2 bar, dependent onthe particle size.If the pressure is increased above about 8 bar the situation is changed again.

At this pressure the zero position approaches 12. For higher pressures dT/ dt ispositive at T; (curve 3), so that when this point is reached the state must jumpto another branch where dT/dt is negative. The temperature should go downto 11, where the situation is reversed. It cannot be expected, of course, that thestate of the surface will exactly follow the path described above. It is morecorrect to say that the rate equilibrium, which has been assumed in the model,must break down at these pressures. Different points of the surface may befound at different states around the temperature To (fig. 15): The surface canbe regarded as being in a state of "reactive boiling" where bubbles of Zr orZrO vapour are formed at different points of the surface which prevent theoxygen from reaching the liquid droplet and thus reduce the total rate ofsurface reactions. Whether this picture ofthe burning mechanism is true couldnot be proved in the experiments. Such an investigation would necessitate anaccurate space-resolved measurement of the spectral emission to determinethe distribution of Zr, ZrO and Zr02 in the vapour phase.

An additional indication of a changing burning mechanism in the pressure. range of 8 bar can be found from the decreasing rate of the droplet radius.As has been mentioned in sec. 3, the experimental results are no longerreproducible at these high pressures. Unfortunately a direct determination ofda/dt is not possible, because the droplet is totally obscured by the corona. Aswe know, however, from the experiments at lower pressures that the distancebetween the corona and the droplet remains constant in the stationary com-bustion stage, the relation da/dt = de/dt should hold. The droplet radius canbe estimated from the corona radius according to eq. (15). The dependence ofda/dt on a as calculated from the theoretical model is shown in fig. 16together with experimental points measured at 1 bar, 3 bar and 5 bar. Thepoints determined from the measurements at 8 or 9 bar are not shown. Theydo not fit into the picture at all. They scatter all over the range between thetheoretical curve (extrapolated from lower pressures) and da/dt = O. In somecases the rate of decrease is abruptly changed (see fig. 13). Thus it must beconcluded that the quantity measured indirectly by the method describedabove is not identical with da/dt. In sec. 4.1.3 we have mentioned that the rateof evaporation must approximately be equal to the reaction rate if the coronais to find a stable position. If the reaction is shifted from the liquid surface


do-dt

(ern 5-1)

t

2

03bor

20 40 60 80 100--ollJm)

Fig. 16. Function da(a)/dt a steady-state combustion temperature. Measured points: 0 1 bar,• 3 bar, + 5 bar.

to the gas phase, less momentum is transferred to the corona by the con-densing particles, so that the corona will grow slightly.This effect should just be expected if bubbles of unburned Zr and ZrO are

formed at the surface. The decrease of the size of the corona"will always beless than expected from the equilibrium model. A quantitative theoreticalprediction is not possible of course, as the boiling process depends on toomany parameters: on the distribution of bubbles in the liquid droplet, on thedistribution of condensed particles in the corona and on instabilities in theoxygen flow.

Though quantitative results for pressures above 8 bar cannot be obtainedfrom the model calculations, at least this limit and the reason for the poorreproducibility can be understood from the model. For pressures up to 8 barthe numerical results are in good agreement with the experiments. In par-ticular the strong increase of the evaporation rate at small droplet radii isfound correctly from the calculations. This effect was not to be expected from

Phllips Journal of Research Vol.35 No.6 1980 425

E. Fischer

an extrapolation of the low pressure measurements done by Kettel '), whofound the evaporation rate to be independent of the droplet size. As can beseen from fig. 16, this approximation is valid only in the very limited regionwhich was covered by Kettel's experiments (p < 1 bar, a > 100 urn).

Besides the droplet temperature and droplet radius, there are two otherquantities that depend on time, but which are not accessible to a direct experi-mental investigation: the thickness of the oxide layer that covers the dropletsurface and the composition of the droplet core. Though these quantities donot influence the temperature and the evaporation rate of steady-state com-bustion, they are essential for the final evaluation of the droplet, which isdiscussed in the next section.

4.3. Extinction

If metal combustion is used for light generation, the objectives are not onlyto obtain a high combustion temperature and a rapid combustion but also tohold a burning droplet in the active combustion state as described in the lastchapter until all material has been reacted into liquid or solid oxide particles.As in most diffusion-dominated reactions, however, after a certain period oftime the concentration of gaseous, liquid or solid reaction products willrestrain the oxygen flux so much that the active combustion changes to aslower oxidation at lower temperature. In the combustion of zirconium thiseffect occurs if the oxide layer on the droplet surface exceeds a critical thick-ness. As this point the diffusion of oxygen through the oxide layer becomesthe rate-limiting process. If this critical thickness is already reached beforeevaporation of the oxide becomes the dominant cooling mechanism, thedroplet will not reach the active combustion state at all, but will run through atemperature maximum and then cool off slowly, while a continuous oxidationtakes place inside the droplet. In our experiments such "duds" were onlyobserved with large droplets at low pressures or if larger amounts of impuri-ties were contained in the metal or in the oxygen filling. Normally, this meansthat if the droplet surface consists of pure metal at the moment of formationand if no non-reactive gas components hinder the chemical reaction, theheating proces is so rapid that thermal and radiative losses are negligible. Notuntil evaporation comes up, the energy production will be balanced. At thistime the surface will be covered by an oxide layer. Depending on the burningconditions this initial layer will grow further or it will shrink. As has beenmentioned before, the difference between the reaction rate and the evapora-tion rate is small compared with the rates themselves. Thus it may last somemilliseconds - at pressures below 1 bar even some ten milliseconds - untilthe oxide layer reaches a critical size or until the initial layer is consumed if

426 Phlllps Journni of Research Vol.35 No.6 1980


dd/dt is negative. As during such a long period of time the droplet size and thecomposition of the core may markedly be altered, dd/dt may even changesign. The numerical results show that, dependent on the initial droplet sizeand oxygen pressure, all three types of d(t) curves can be found, those withdd/ dt positive all the time as well as those with dd/ dt changing from positiveto negative approximately when the equilibrium temperature is reached andalso those in which dd/ dt changes sign twice.

a=l,()

a=50llm

2 6-t(ms)

Fig. 17. Thickness of the oxide surface layer versus time for droplets of various radii burning inoxygen at 2 bar pressure.

Figure 17 shows the time dependence of the layer thickness for droplets ofvarious radii burning in oxygen at 2 bar pressure. All three types of curves arefound in the plot. The curves for the 60 urn and 70 urn particles would inter-sect with the zero line if not the reaction equilibrium was changed when theoxide layer falls below a certain thickness. This minimum value has beendenoted do in the description of the theoretical model. The 60 urn and 70 urncurves have been plotted only up to the point where do is reached. For thefurther path of the curve the theory gives an approximately constant value justbelow do (0.7 do< d < 0.9 do). The temperature should show a further increasein this combustion mode.

According to the theory, a droplet which has once reached this mode ofcombustion should remain in this state until the entire droplet has been con-sumed. Droplets burning this way - similar to the combustion of aluminium,but on a much higher temperature level - have not been observed in the

Phllips Journalof Research Vol.35 No.6 1980 427

E. Fischer

experiments, however. Thus there must be a mechanism that prevents thedroplet from reaching this state of combustion.

An interpretation can easily be found, if we remember the explanation thathas been given for the volume growth in the ignition phase. Besides the ZrObubbles in the droplet core, which are mainly responsible for the volumegrowth, an additional bubble layer should exist at the boundary between theoxide layer and the metal core. Due to the surface tension the pressure insidethe bubbles exceeds the outside pressure. If now the oxide layer is reducedmore and more, the bubble layer becomes unstable and expands into theincoming gas. A wave of ZrO leaves the droplet surface, blowing the coronaaway from the droplet. The oxygen flux to the droplet is abruptly stopped andwith it the energy production at the surface. The droplet cools down and theflux of ZrO is automatically reduced. In principle it should be possible thatafter blowing off the corona in a wave of ZrO the droplet resumes its steady-state combustion. But as the atmosphere around the droplet contains mainlyZrO and not O2, the ZrO will first react with the cooled surface and form anew oxide layer. When finally enough oxygen reaches the surface to continuethe combustion, the thickness of the oxide layer already exceeds the criticalpoint where diffusion through this layer determines the reaction rate. Thedroplet cools down - in the same way as the smaller or larger droplets do -until it finds a new equilibrium state, where the rate of energy productioninside the droplet balances the losses by radiation and conduction.

The curves in fig. 17 should of course not be taken as an accurate quan-titative statement about the limiting radii for the "blow-off" effect. There aretoo many parameters in the model, for which only rough estimates have beenused in the calculations. Besides, the initial conditions are somewhatidealized. It is assumed that the droplets immediately take up their sphericalshape as soon as the melting point has been reached and that internal con-vection is completely absent.· Such convective effects would give rise to anenhanced dissolution of the oxide into the droplet core. Another reason for areduced thickness of the oxide layer may be interaction with other droplets. Ina collision of two droplets some fraction of the corona will usually fall backinto the droplet. This again leads to an internal motion of the liquid and thusto a reduction of the oxide layer. Due to these effects larger or smaller dropletsmay reach the "blow-off" state too.

The same situation we find if we look at the pressure range in which theoxide layer falls below a critical thickness (fig. 18). Again we see that thereis only a small range of pressures (from 1 bar to 2 bar for a droplet witha = 70 JlIIl) limited in both directions, in which the critical thickness isreached. Of course the situation here is the same as mentioned above for the

428 Philip, Journal of Research Vol.35 No.6 1980


d(IJm)

t/

II

/11

6

4

bar

O+---.---.---.---~--.-~o 2 4

__ t(ms)

Fig. 18. Thickness of the oxide surface layer versus time for droplets of 70 urn initial radiusburning at various oxygen pressures.

droplet size: by some external effects the layer thickness may be reduced sothat it falls below the critical value. Thus "blow-off" mayalso occur outsidethe range calculated from the model if there exists any set of conditions forwhich dd/ dt is negative. At pressures smaller than about 1 bar dd/ dt ispositive for droplets of any size, so that instabilities of the corona are not tobe expected. At pressures above the upper limit, however, "blow-off" maywell occur, especially for those droplets which have been formed by coagula-tion of two or more smaller ones in the ignition stage of combustion.The predictions of the model calculations have been confirmed qualitatively

by the experiments with the restrietion however, that the critical pressurerange appears to be shifted to somewhat higher values. At 1 bar "blow-off"was not found in the experiments. At 3 bar it occurred for most droplets andin the measurements at 8 and 9 bar a few droplets with expanding corona wereobserved. The numerical and theoretical results can be summarized asfollows.(1) "Blow-off" is not found in the pressure range up to 1 bar.(2) Between 1 bar and 5 bar it is observed for most droplets.(3) Above 5 bar only part of the droplets show the expansion of the corona.

Most of these particles have undergone a collision or fusion with otherdroplets at the beginning of their life. ,

(4) "Blow-off" is most likely to occur in the case of droplets between 50 urn

PhllIps Journalof Research Vol.35 No.6 1980 429

E. Ftscher

and 80 J.IlIl radius. Mostly it begins in the interval between 30 ms and 6 msafter ignition. The time is shorter for small droplets.

The final developments of the droplet can be calculated from the theoreticalmodel only if no "blow-off" occurs. The expansion of the ZrO vapour emittedfrom the surface layer is certainly not an equilibrium process, so that theinitial conditions for a further calculation of the droplet temperature, radiusand composition cannot be derived from the state before the "blow-off". Theexperiments show, however, that when an equilibrium state is re-established,the oxide layer covering the surface must be so thick that active combustioncannot be restored. The droplet cools off as if the "blow-off" had not hap-pened. During the steady-state combustion stage the droplet temperature isnot exactly constant, but because of the changing droplet and corona radii itshows a weak decrease which is well below 100K per ms. As soon as diffusionthrough the oxide layer gains controlover the reaction, the cooling ratechanges to values of 1000-2000 K/ms within some tenths of a millisecond.

A quantitative measurement of the decay time was not possible in ourexperiments, because by heating of shreds or wires always several droplets areformed which vary considerably in size and thus cool down at different times.Measuring the decrease of the colour temperature of the complete set ofdroplets will therefore always exhibit some mean value of the temperaturesand thus show a much slower decrease than a single droplet. An estimate ofthe cooling rate can be obtained from the high-speed photographic films.Measuring the transmission of the droplet images on subsequent frames wefind that the blackening is constant during the steady-state combustion stageand then falls from near saturation to zero within about one millisecond for adroplet with a radius of 66 J.IlIl (and p = 1bar) at the end of steady-state com-bustion. The temperature limit at which black body radiation would cause anobservable blackening of the film was proved to be (2800 ± 200) K. This leadsto a cooling rate of (1.7 ± 0.5) KI J.Ls,which is in good agreement with thetheoretical model calculations. Tests at other droplet radii and oxygenpressures confirmed the theoretical predictions.

With the beginning of the cooling process the droplet radius, which showeda linear rate of decrease during steady-state combustion, decreases veryrapidly. For the example given above the droplet radius dependent on time isplotted in fig. 12. From the figure it can be seen that within the last milli- .second in which the droplet is observable; the radius decreases by a factor of1.5. This is approximately the same factor as we find for the droplet growth atthe beginning of the combustion. This is exactly what is to be expected, as wehave attributed the volume growth to the formation of ZrO bubbles inside thedroplet. As soon as the temperature falls below the value at which the internal

430 Phillps Jour na1of Research Vol.35 No.6 1980

PhilipsJournal of Research Vol. 35 NO.6 1980 431


pressure equals the outside pressure the bubbles must vanish if they consistonly of ZrO. If there are impurities in the metal or in the oxygen, which formgaseous compounds at the temperatures prevailing inside the droplet, thesesubstances will remain as bubbles inside the cooling droplet, leading to poresand cavities in the solidified zirconia sphere (see figs 5 and 8a). The same mayhappen, in the case of large droplets, if the temperature difference between thedroplet core, in which the reactions take place, and the surface becomes solarge that the surface layer already solidifies before the ZrO-bubbles in thecenter vanish. In this case the result is a hollow sphere of zirconia (fig. 19).

Fig. 19. Micrograph of à hollow zirconia sphere (magnification: 250: 1).

The very thin "egg shells" which are found in the combustion products(fig. 8b) cannot be ascribed to this cooling process, because their formationrequires an additional growth of the droplet volume by a factor of the order often before the solidification of the surface. A theoretical explanation for thiseffect is given in the next section.

4.4. Bubble formationAccording to the model considered in the foregoing the diffusion of Zr02

molecules into the metal core of the droplet is so slow that the average con-

E. Fischer

centration of Zr02 in the core is still very low at the time when the dropletcools down. Only in the regions where oxide particles have been includedduring formation of the droplet, the concentration of Zr02 is high enough toform small ZrO bubbles that lead to the droplet growth mentioned in sec. 4.1.If, however, in the history of a droplet some event has happened by which theoxide concentration in the core has been increased - this may be a collision orfusion with another droplet or the capture of a larger oxide particle from thecorona - the critical oxygen content which is necessary for bubble formationmay be exceeded in a larger region of the core. Thus if the small bubblesformed at ignition have sufficiently coagulated, so that the surface tension isnot too high to prevent further vaporization, an explosive growth of thebubbles occurs, which is controlled only by the rate at which the energy forvaporization can be supplied. In this phase an enhanced energy support ismade possible by the bubble formation itself which increases the reactive sur-face and reduces the effective oxide layer thickness. Thus, in spite of the largeamount of energy which is necessary for the formation of ZrO, the tempera-ture decreases only slightly. If enough unburned material is present the processgoes on until the bubble bursts into small droplets, which immediately solidifyand can be found as small porous spheres in the combustion debris (fig. 20).

Fig. 20. Micrograph of small porous zirconia spheres resulting from explosion of a bubble (mag-nification: 250: 1).

432 Philips Journalof Research Vol. 35 NO.6 1980


If the skin of the bubble is strong enough to hold the internal pressure untilall the metal is transformed to ZrO or Zr02' the blow-up process will stop, butat the same time the temperature will fall very abruptly dropping below themelting point of Zr02 in such a short time that the ZrO in the bubble cannotreact with the incoming oxygen before the skin of the bubble completely solid-ifies and forms and "egg shell" as shown in fig. 8b. During the cooling processfurther oxygen will of course diffuse into the shell and replace the ZrO.

Finding a quantitative theoretical answer to the question of whether adroplet will blow up under certain initial conditions appears hopeless withinthe framework of the model applied here. There are too many unknownparameters and material constants to allow a quantitative determination ofthe oxide content of the core and the time dependence of the internal bubblesize distribution. The numerical calculation can only give an additional con-firmation of whether the right order of magnitude of the material constantshas been chosen. Figure 21 shows the result of such a test calculation. Onecurve exhibits the minimum oxide concentration needed for ZrO bubble ex-pansion in the limit of vanishing surface tension, the other shows the averageoxide content of the core, calculated with the step model described in sec. 3,both for a droplet with a = 70 urn at p = 0.3 bar. The pattern of the curves

2 ·4 6___ t(ms)

Fig. 21.0 Time dependence of the core combustion. Solid line: molar fraction of Zr02 in the core(a = 70 JJ.l11. P = 0.5 bar). Dashed line: minimum Zr02 content for inflation.

I I8 I I

xzro2I II I

(%) I I

tI II I

6 I II fI II fI J

4 t Xm1n -----/<, --------

PhllIps Journalof Research Vol.35 No.6 1980 433

E. Fischer

is typical also for higher pressures and radii. The minimum content drops to avalue of 3 to 80/0 in the ignition phase. Then it slightly rises according .to theslightly decreasing droplet temperature. When the temperature drops at theend of the steady-state combustion, the critical value shows a rapid increaseuntil bubble formation becomes completely impossible when the pressure Pmax

(see sec. 4.1) drops below the outside pressure. The actual oxide content showsan approximately linear slope. Just at the end of steady-state combustion itcomes close to the critical value. Other parameters may even lead to slightlyintersecting curves, which would denote blow-up if the initial small bubbleshave coagulated sufficiently at that time. Though one cannot trust the state-ment that blow-up will really occur at these parameter values, the resultsclearly indicate that the material constants are at least of the right order ofmagnitude to allow bubble formation. Furthermore the curves show thatsmall oxide particles of only a few percents of the droplet volume are alreadycapable of bringing the two curves to interseet if such a particle falls into theburning droplet. Thus it can be understood that bubbles and compact oxidespheres can be formed simultaneously in one experiment from droplets ofequal initial size.

The formation of bubbles can be influenced by the presence of gaseousimpurities inside the droplets or in the oxygen. Meyer et al. 3,6) have studiedthis effect in detail for oxygen atmospheres containing up to five percents ofnitrogen. The highly reproducible experimental conditions - isolated free fal-ling droplets of predefined size - allowed them to define criticallimits for theoccurrence of stable or exploding bubbles. The formation of stable bubbleswas found only at N2/02 ratios between 0.01 and 0.04 for droplets with aninitial diameter of 525 J.U11 at a pressure of 625 torr. Above this limit bubblesalways exploded.Doping the zirconium with various gases leads to an enhanced formation

of bubbles, but in this case a more steady increase of the droplet radiusis observed, which begins already when the droplet has just reached itsmaximum temperature. Figure 22 shows the combustion of droplets con-taining bromine at an oxygen pressure of 1 bar. Similar effects were observedfor chlorine, fluorine and less pronounced for N2 and H20..Within the framework of our model the observed facts can easily be ex-

plained. If the zirconium contains impurities that do not form liquid or solidcompounds either with the metal or with oxygen at the temperature of steady-state combustion, these admixtures will diffuse into the small ZrO bubblesformed in the ignition phase and thus .enhance the internal pressure in thesebubbles. The bubbles also grow further after the point is reached at which allZr02' initially incorporated in the droplet, is transformed into ZrO vapour.

434 PhilipsJournal of Research Vol.3S No.6 1980


Fig. 22. Slow inflation of droplets induced by doping the zirconium wire with bromine. Timeinterval between photos I ms (p = I bar, magnification: 35 : 1).

When the bubbles have reached a size at which the diffusion of oxygenthrough these bubbles hinders the chemical reaction, the droplet will begin tocool down, while the bubble growth goes on, so that the final product issimilar to that obtained by the more explosive bubble formation which isobserved with pure metal and pure oxygen or 02/N2 mixtures. In principle theeffect of nitrogen in this case is the same as described above, but at thebeginning of the combustion the concentration at the droplet surface is so lowthat it cannot be expected to have much influence. The gas flux to the surfaceleads to a steady increase of the nitrogen concentration at the surface, as the

Philips Journalof Research Vol. 35 NO.6 1980 435

E. Fischer

oxygen is consumed by reactions while the nitrogen does not participate in anychemical process. It can only be removed by diffusion into the droplet. Withsome time delay, therefore, in spite of the low concentration a remarkableflux of nitrogen will penetrate the droplet and increase the pressure in theinternal bubbles and thus lead to an additional growth. Thus when the con-centration of Zr02 in the core reaches the value necessary for blow-up, bubblenuclei of sufficient size are more likely to be present than without nitrogen.

Once the expansion of the droplet has started, the nitrogen barrier at thesurface can easily diffuse through the bubble skin into the interior and thusholds the internal pressure above the outside pressure down to temperatures atwhich the partial presure of ZrO can no longer balance the outside pressure.By means of suitable admixture of nitrogen the pressure balance can besteered in such a way that the bubble is stable down to the melting point. Thusthe formation of "egg shells", which depends on many accidental parametersin the combustion in pure oxygen, can be controlled by small admixtures ofother gases.The combustion model presented here certainly needs some further

improvements to achieve an accurate description of the complete combustionprocess. Better data on reaction rates and diffusion coefficients need to bederived for the processes at the droplet surface as well as inside the dropletitself and in the surrounding gas. The model for the transport processes insidethe droplet must be refined, as the step model used here can only give verycrude estimates of the transport rates. The model does, however, have oneessential advantage over the equilibrium models published hitherto: it gives anat least qualitatively consistent picture of all the various events that mayhappen at some point in the burning history of a zirconium droplet, and forthe steady-state combustion agreement is obtained with the experiments for allparameters that are accessible to a quantitative determination.

5. Application to flash bulbs

The investigations described in the foregoing are concerned only with thecombustion of isolated droplets in an oxygen atmosphere at constant pres-sure. The only point where the influence of the surroundings comes in is thepossible change of initial conditions by collisions with other droplets or oxideparticles. In technical applications, however, these idealized conditions do notprevail. In photographic flash bulbs - the main application for combustionat high pressures - the zirconium is existent as a dense package of zirconiumwool, the single fibers being small shreds with diameters of the order of15-25 J.Lm.In contrast to the experiments where wires are heated uniformly tothe melting point, in a flash bulb the shreds are heated locally by the impact of

436 Phillps Journalof Research Vol.35 No.6 1980


incandescent particles from an initial fuse explosion. Thus the shreds melt atsome point and liquid droplets run along the shreds until they come to the endor eventually to a crossing point of two shreds. To form a droplet of 200 urndiameter a shred of about 1 cm length must be consumed. Though the meltingvelocity of the shreds is of the order of 200 to 250 cm/s 18,19) the formationtime of a 200 urn droplet should be 4 to 5 ms. During this time interval thedroplet is connected to the cold shred, so that owing to the additional coolingit will not reach the final temperature until the shred is completely consumed.Most of the oxide formed in this initial combustion phase is transported intothe droplet by convection, so that the steady-state combustion will start withan oxide content of the droplet which is higher than that of an electricallyheated droplet. The oxide is distributed more uniformly, however, so that theexpansion of the droplets should be lower. This is in agreement with theresults of Kettel19) who finds a ratio of 1.5 between the volume of the dropletand the volume of the consumed shred. Due to the rapid motion along theshred the droplets will have a high velocity when they have reached steady-state combustion. Thus much of the oxide willieave the corona. It first formsvapour trails behind the droplets and then smears out into a dense fog all overthe bulb volume (fig. 23).

The formation of a thick oxide layer on the droplet surface which leads tocooling down at a rather early stage of combustion is less probable in flashbulbs than for isolated droplets, since on the one hand the steady-state com-bustion starts with a comparatively thin oxide layer and on the other hand theoxide cover may be destroyed by collisions or by fusion of droplets. Thus thedroplets remain in the state of active combustion for a longer period of time.Due to the higher oxygen content of the core it is to be expected that nearly alldroplets reach the critical core composition for bubble formation. Indeedblow-up is frequently observed (fig. 24) at the later stages of combustion, butnot nearly as frequently as one would expect from the droplet composition.This may be due to the fact that because of the more uniform oxide distributionthe initial foamy structure is less pronounced and consequently large cavitieswhich can serve as starting points for bubble formation occur les frequently.

From the photos of the "blow-up". effect (figs 4 and 24) it can be seen, that,although the period of time from blow-up to cooling-down or explosion lastsonly about 1 ms, the surface of the droplet is increased so much that theintegral radiation output is higher or at least equal to that of a droplet whichreaches the end of steady-state combustion without inflation. Thus at leastone of the goals of flash bulb research - the reduction of the total com-bustion time - could be reached if it were possible to steer the bubble for-mation so that all droplets are inflated in a rather early stage of combustion.

Philips JournnI of Research Vol. 35 No.6 1980 437

E. Fischer

Fig. 23. Combustion of zirconium shreds in a flash bulb (PHILlPS PFC4). Photos were taken at2 ms, 6 ms, 10 ms and 14 ms after ignition (magnification: 4: I).

a

c

438

b

d

PhilipsJournalof Research Vol.35 No.6 1980


a

c

e

b

d

f

Fig. 24. "Blow-up" effect in a flash bulb. The droplets in the center inflate within about I ms andthen vanish. (Photo a is an enlargement of the lower part of fig. 23c. Time interval between suc-cessive photos is 2/3 ms.)

PhilipsJournal of Research Vol. 35 NO.6 1980 439

E. Fischer

Acknowledgements

The author wishes to thank Mr H. G. Ganser for performing the experi-ments and Dr R. Schäfer for many helpful discussions.

Philips GmbH Forschungslaboratorium Aachen, September 1980

REFERENCES1) F. Kettel, The oxidation of zirconium at high temperatures, Philips Res. Repts 28, 219,

1973.') L. S. Nelson and H. S. Levine, The combustion ofzirconium droplets in oxygen / rare gas

mixtures, High Temp. Sci. 1, 163, 1969.S) R. T. Meyer and L. S. Nelson, The role of nitrogen in the formation of microbubbles

during the explosive combustion of zirconium droplets in N./O. mixtures, High Temp. Sci.2,35, 1970.

4) L. S. Nelson, H. S. Levine, D. E. Rosner and S. C. Kurzius, Combustion of zirconiumdroplets in oxygen / rare gas mixtures - Kinetics and Mechanism, High Temp. sci. 2, 343,1970.

5) R. T. Meyer and W. G. Br eil an d, Nitrogen thermochemistry during the combustion ofzirconium droplets in N2/Ó2omixtures, High Temp. Sci. 4, 255, 1972.

6) R. L. Marshall and G. L. Pellett, Vapor-phase combustion phenomena when Zr dropletsburn in oxygen, Proc. Spring Meeting Western States Section, The Combustion Institute,China Lake, Calif., USA, 1969.

7) K. M. Maloney, Kinetics of the zirconium-oxygen active combustion system, High Temp.Sci. 3, 445, 1971.

8) K. M. Maloney and T. C. M. Pillay, The active combustion mechanism of single AI and Zrstrands in oxygen as determined by high speed photography, Combustion and Flame 18, 337,1972.

9) D. H. W. Delhaas, private communication.10) P. Kofstadt, High temperature oxidation of metals, Wiley, New York, 1966, p. 244.") JANAF-Thermochemical Tables, Dow Chemical Company, Midland, USA, 1971.I') E. T. Turkdogan, The theory of enhancement of diffusion-limited vaporization rates by a

convection-condensation process, Trans. Metal!. Soc. AIME (USA) 230, 740, 1964.IS) H. J. M. Hanley and J. F. Ely, The viscosity and thermal conductivity coefficients of dilute

nitrogen and oxygen, J. Phys. Chem. Ref. Data 2, 735, 1973.14) J. Uhlenbusch, Zur Bestimmung der Transport Koeffizienten eines Plasmas aus

gemessenen Bogendaten, Habilitationsschrift, TH Aachen, 1966.15) J. O. Hirschfelder, C. F. Curtiss and R. B. Bird, Molecular theory of gases and liquids,

Wiley, New York, 1964, p. 527.16) T. G. Theofanous and P. D. Pat el, Universal relations for bubble growth, Int. J. Heat

Mass Transfer 19, 425, 1976.17) F. F. Abraham, Homogeneous nucleation theory, Academic Press, New York, 1974.18) D. H. W. Delhaas and J. G. M. A. Ponsioen, private communication.19) F. Kettel, private communication.

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