R831 Philips Res. Repts 28, 219-244, 1973

THE OXIDATION OF ZIRCONIUM AT HIGHTEMPERATURES

by F. KETTEL

AbstractThe oxidation of zirconium is investigated for two regions: the initialstep of oxidation and stationary combustion. The reaction of zirconiumwith oxygen has been studied using thin zirconium wires. Given aninitial reaction temperature between 1000 and 1400 K and an oxygenpressure of 20 to 700 Torr the reaction leads to a practically adiabaticrise in temperature of the zirconium wire. During this initial range ofoxidation the reaction rate can be determined from this temperaturerise and is described by the equilibrium of oxygen adsorption at thesurface. Above the melting point of the zirconium the wire disintegratesinto droplets, whose temperature rises to a point at which it becomesstationary. In this second stage of combustion a corona around thedroplets is formed by combustion products. The extent of this conden-sation zone, the temperature and the loss in weight of the droplet havebeen determined as a function of the oxygen pressure. The results of theexperiments can be described by a reaction model based on the assump-tion that zirconium vapour already reacts at the surface of the dropletto form zirconia (Zr02)' At low oxygen pressure (p < 10 Torr) thefinal temperature is determined by the thermal equilibrium with thereaction of oxygen dissolution in the zirconium droplet, while at highpressure (p > 10 Torr) the temperature is limited by the saturationvapour pressure of zirconium.

1. Introduetion

The kinetics of the oxidation of zirconium have been the subject of numerousinvestigations 1) in the temperature range from room temperature up to1300 oe. These investigations deal with the oxidation behaviour after the for-mation of a compact oxide film. In that case the rate of oxidation will be limitedby a solid-state diffusion (Wagner theory). The initial oxide formation has notbeen examined.More recently the combustion of zirconium in oxygen was also investigated

with respect to processes taking place in photographic flashbulbs. However,these investigations were limited to the spectrum of the emitted light 2) and toa general consideration of the thermodynamics of the reaction 3). For a studyof the oxidation kinetics, the processes in flashbulbs are too complex.Nelson et al.4-8), Marshall and Pellet 9) and Maloney and Pillay 10.11) have

described in a number of papers the combustion of individual zirconium drop-lets in the range of stationary combustion and at the final stage of the reaction.The present work covers both the initial stage of oxidation up to the melting

point, and the subsequent formation and combustion of droplets. These resultswill be presented in secs 3 and 4 respectively.

220 F.KETIEL

2. Experimental method

A zirconium wire of 0·1 mm diameter is clamped over a length of 10 mm in a300-cm3 chamber filled with oxygen. A current pulse lasting 0·1 ms heats thewire up to a temperature of about 1000 to 1400K at which oxidation proceedsso strongly that the energy of the reaction exceeds the heat losses.The temperature of the wire is recorded photoelectrically (fig. I). The experi-

mental set-up is calibrated by comparison with a tungsten radiator. The signalof the receiver is converted to the temperature T, of an equivalent black-bodyradiator. To obtain the true temperature the emissivity of the radiator shouldbe known. After linearization of the signal the set-up is able to measure thetemperature T, and the change in temperature with time dTs/dt.In order to measure dTs/dt, the upper limiting frequency of the set-up had

to be limited to 1 kHz, which is still sufficient to cover regions of continuousrise of temperature.When the melting point of the zirconium is exceeded the wire disintegrates

into individual droplets. Their development and temperature is recorded by aHycam high-speed camera and measured by densitometric determination of theblackening (fig.2), which is calibrated by simultaneous recording of four stand-ard tungsten radiators at different known black-body temperatures Ts• Thuseach photograph, as represented in figs 10 and 11, shows four dots which serveas references for temperature calibration. The radiation of the droplets isreduced by neutral density filters at a value at which their blackening is of thesame order as the calibration points (0·4-1·5).The droplet diameter and its changes were determined by measurements of

the films under a microscope. These measurements were in some case supple-mented by measurements of the blackening across the droplets with a micro-densitometer having a round aperture of 30 {Lmin diameter.

5 Il Combustion0-._ .........----1 I--_C:=h chamber

200pF Receiver

Flt"p-flop

Fig. 1. Experimental arrangement for measuring the rate of reaction of zirconium wires inoxygen.

OXIDATION OF ZIRCONIUM AT HIGH TEMPERATURES 221

Tungsten [ampsrI1

1*11I/,\~====~====~~

High-speedcamera

Process

LogEs

,,--"'--'\,,--\r.--,--/--;I'--rl'-//.

s

Fig. 2. Experimental arrangement for the observation of the combustion of zirconium drop-lets and for the determination of their temperature.

3. The oxidation of zirconium wires up to the melting point

3.1. The energy balance of the wire

The energy balance of the wire per unit length can be described by

dTtn: a

2CZr eZr - + Wrad + Wcond + R = 0,

dt(1)

in which a is the radius of the wire, CZr eZr is the specific heat per cm", andT is the average temperature over the cross-section of the wire.The first term, n a

2 eZr eZr dTjdt, is the gain in energy of the wire per second;Wrad and Wcond are the losses by radiation and conductance of heat respectively.If the reaction is started at sufficiently high temperature, which is the case forour experiments, the radiation and conductance losses can be neglected in com-parison with the reaction energy. This reaction energy R = L1H dNjdt is thetotal amount of energy generated or consumed by reactions and phase trans-formations, L1H the enthalpy of the reaction or transformation per mol (nega-tive in the case of exothermic reactions), and dNjdt the number ofmoles react-ing or transformed per second and per cm length.The oxygen entering the wire will in part diffuse and be incorporated at inter-

stitial sites in the metal, and in part it will, on exceeding the solubility limit of

222 F. KETIEL

oxygen in ee-Zrat the surface (29 at %), lead to the formation of a mixed phasect-Zr+ Zr02' So long as at the surface of the wire the phase boundarya-Zr + Zr02-Zr02 is not reached, the reaction between Zr and O2 is a sur-face reaction. The reaction energy in the energy balance (1) is then given by

where Joz is the reaction rate, determined by the rate of oxygen flowing in thesurface. At 1600 K the reaction enthalpy iJHR = -258 kcal mol-i, both forthe incorporation at interstitial sites (ce-Zr phase) and for the formation ofzirconia (c-Zr + Zr02 phase) 12.13).The only enthalpy change which is notentirely negligible compared with this high value is the heat of fusion, whichamounts to 4·89 kcal mol-1 14).This means that, neglecting a temperature dependence of iJHR, the reaction

rate is directly proportional to dT/dt:

a CZr eZr dTJ, ----°z - 2iJHR dt

(2)

Therefore measuring dT/dt yields the reaction rate Joz' with CZr = 8·75 . 10-2

cal g-1 deg-1 and eZr = 6·5 g cm-3 respectively 14). In the case of a con-stant emissivity Joz is also proportional to dTs/dt.The reaction rate generally depends on the oxygen pressure, the temperature

and the time. Due to the great volume of the reaction chamber compared withthe quantity of zirconium, the O2 pressure remains constant during the reac-tion. For surface reactions the reaction rate does not depend explicitlyon thetime. For the present experimental method the reaction rate in this case onlyis coupled with the time via the temperature.

3.2. Experimental resultsFigure 3 shows, as function of the time, the temperature Ts and the rise in

temperature dTs/dt. The curves were recorded at an oxygen pressure of 100 Torrin the chamber and with different initial temperatures.

Figure 4 shows the curves of temperature and time at different filling pres-sures of oxygen and the same initial temperature. .In fig. 5 the determination of T, and of dTs/dt was extended up to a tem-

perature of T, R:i 2500 K. Up to about this temperature the wire is still extanteven though the melting point of zirconium has already been exceeded.The melting point of zirconium is shown by the plateau in the tempera-

tureftime curves at T,= 1970 K. From this black-body temperature and amelting temperature of 2125 K one can then calculate a total emission coef-ficient of e = 0·7. The endothermic process of fusion manifests itself also inthe deep indentations of the dTs/dt curves. Figure 5 shows indentations in the

OXIDATION OF ZIRCONIUM AT HIGH TEMPERATURES 223

20G0

100-Torr02o 1S(start) = 1400KX Ts(start) = 1260K+ Ts(start)= 1190K

Ts(K)

1'750

4.105

dTSëif

. (Kis)

2

10G0~--~--~--~--~--~--~~~0o 234567-t(ms)

Fig. 3. Rise in temperature of zirconium wires in consequence of their reaction with oxygenat different initial temperatures.

2000l's 1S(startJ=1300K 200Torr

(K) 1900

t 1800

1700

1600

2 3 4-t(ms)

5

Fig. 4. Rise in temperature of zirconium wires on account of their reaction with oxygen atdifferent oxygen pressures.

224 F.KETTEL

20-TorrO2Ts(start) =7370K

2500r-------------------~--~3.lOs

dT,;dt

(Kis)

Ts(K)

r 2000

10000'---'-----'-----'---'---...L_---'---___..--'---_.__-_j 0

2 I; 6 8 10_t(ms)

Fig. 5. Rise in temperature of a zirconium wire because of reaction with oxygen.

dTs/dt curves even above the melting point, and these must also be interpretedas endothermic reactions at those temperatures.

The curves of fig. 3 show different regions. Due to the limitation in frequencyresponse, the results for t < 0·6 ms should not be taken into account. Theadjacent region, which extends up to a temperature T; = 1700 K, shows dif-ferent time dilatations of the three curves. Here first of all a change is likelyin the total emission coefficient during the formation of a tarnish film.

Figure 6 shows the behaviour ofthe temperature oftwo wires heated in argon

1785K ....

1730 K ~

10mV/cm

i----I~.0·5ms/cm

Fig. 6. The black-body temperature of a clean and of a superficially oxidized zirconium wireat the same true temperature,

- - ..._-~-~~-----;-------------------------------

OXIDATION OF ZIRCONIUM AT HIGH TEMPERATURES 225

with equal, very high power. The température/time curve ofthe first wire showsno change of the emissivity. The second wire, the surface of which had pre-viously been oxidized, initially shows a temperature 55 K higher which fallsas the tarnish film dissolves in the zirconium (this recording was made with anupper limiting frequency of 16 kHz).The previous history of the zirconium wires also has a marked effect on the

process of oxidation in this region of up to 1700K. The first maximum of thedTs/dt curves in fig. 3 at 1'0, 1·3 and 1·5ms attains considerably higher valuesin the case of wires previously tempered for extended periods in a high vacuum.If, however, the wires are enriched with oxygen (20-25 at %), the first maximumdisappears almost entirely.

Only at temperatures above 1700 K do the relationships become clearer. Thethree curves of fig. 3 can here be made to coincide by displacements along thetime axis. This means that the rate of oxidation does not depend on the timebut only on the temperature and the pressure of oxygen. The rate of reactioncan here be determined by means of the energy balance (eq, (2».In fig. 7 the rate ofreaction, J02, is plotted on a logarithmic scale against the

reciprocal of the temperature at different oxygen pressures. The relationship islinear from 1700K up to the melting point. From the slope of the straight linesplotted on these scales the energy of activation of the reaction is found to be33 kcal mol-1 at pressures above 100 Torr. At Iow pressures the straight linesindicate a higher energy of activation, e.g. 38 kcal mol-1 at 20 Torr.

1.Io-2'r------------,

t

2G-Torr----7------J - p'-Vbrmk7'

200100

60

20

1.10.40~.5~---0~.5~5~--~0~.6~~

__ ~~~O(K-?

Fig. 7. The rate of reaction of zirconium wires as a function of lIT at different oxygen pres-sures.

226 F. KEITEL

x

Ts=1850K

x

2

00 10 20 30

Fig. 8. The rate of reaction of zirconium wires as a function of the square root of the oxygenpressure.

Figure 8 shows the rate of reaction as a function of the square root of theoxygen pressure for two different temperatures. Both curves exhibit good pro-portionality to P02 1/2.

3.3. Discussion of the experimental results

Since the rate of reaction in the range of 1700 K to the melting point is notan explicit function of the time, a process of diffusion in an oxide film mustbe excluded as the rate-limiting step of the reaction. Furthermore, the P021/2

dependence ofthe reaction rate, as shown in fig. 8, suggests that a dissociative-oxygen-adsorption equilibrium is established in this temperature region.Ehrlich 15) described this process by physical adsorption of O2 molecules atthe surface, dissociation of these molecules at chemisorption sites, and irrevers-ible incorporation (or a subsqeuent reaction) of chemisorbed oxygen succes-sively.For the values of pressure and temperature considered here, the following

conditions are fulfilled:(1) The lifetime ofthe molecules in the absorbed state is very short (at 1000 K:

7: F::j 10-9 s). Therefore the thermodynamic equilibrium of the physicaland chemical adsorption is reached in a very short time. The change intemperature in this time is negligibly small.

(2) The reaction rate J02 of the incorporation of oxygen is small compared tothe kinetic rate of the gas flux given by

OXIDATION OF ZIRCONIUM AT HIGH TEMPERATURES 227

(3)

where P02 is the oxygen pressure, m02 the mass of the molecule and k isBoltzmann's constant. This is shown in fig. 7.

(3) The ratio of occupied to vacant sites of chemisorption is small comparedwith unity for the temperatures considered here.

For these conditions the reaction rate according to Kofstad 16) is

J02 = const. Po/12 exp (- Elkt). (4)

The energy of activation E is the summation of the energies of activation Elof the various reaction steps.This relation corresponds, in respect of dependence on pressure and tem-

perature, to the experimental findings in the temperature range between 1700 Kand the melting point, as represented in figs 7 and 8.Towards lower pressures of oxygen or higher temperatures, departures ofthe

rate of oxidation from the behaviour expressed by (4) is to be expected withrespect to both the pressure and the temperature dependence 17). Finally at asufficiently high temperature or sufficiently low pressure the rate of oxidationwill be equal to the kinetic rate of the gas flux given by relation (3):

(5)

At an oxygen pressure of 10-3 Torr and temperatures between 1580and 1830 KGebhardt et a1.18) found from gravimetrie results that the oxidation was inaccordance with eq. (5).The fact that the oxidation behaviour in the range from 1700K up to the

melting point can be represented quite well by eq. (4) implies that here thetotal emission coefficient hardly changes and that the energies of activation Elalso remain constant. The formation of a tarnish film would lead to a changein these two magnitudes.A possible change in the total emission coefficient has already been referred

to. The change of the temperature as a function of time will not then be pro-portional to the rate of reaction.The energies of activation, El> depend on the state of the surface, the orien-

tation of the crystals, and the phase present at the surface. The differences inoxidation behaviour below 1700K between the differently pretreated wires willhave to be attributed to such influences.The phase present at the surface can, at the temperatures considered and

with continuous uptake of oxygen, pass through the regions

f3-Zr --+ (a + (3)-Zr --+ ce-Zr --+ ce-Zr+ Zr02 --+ Zr02•

The O2 concentration at the surface of the wire and hence the phase present

228 F.KETTEL

at the surface is calculated by solution of the diffusion problem

bc = ~~( r D bC),bt r br ör

where c is the oxygen concentration. With the boundary conditions

D (bC) = r-;br r=a

and the starting condition Ct= 0 = 0, and with a diffusion coefficient D depend-ing on the temperature, and hence on the time, this problem leads with thetime transformation

• = J Ddto

to the solution applicable to short times (D t/a2 ~ 1) *):

1 fJ.(.') lItC - = -·-f~ dr: + -fJ. dtr-Q n1!2 D(-c') (-c- -c')1/2 2a 02

o 0

(6)

in respect to the concentration of oxygen at the surface.Figure 9 represents the evaluation of this integral, with a diffusion coef-

ficient 19)D = 5·4exp (-50·8. 103/RT) cm- S-1

and the experimental result from fig. 3 with the initial temperature of 1400K.

Cr=a

(mol02 )cm3

3.10-2r------------------,-21 Zr02

4·7.10 1------

t 2

(Ix+p)-Zr

------------------~~2 3 t.

--t(ms)Fig. 9. Variation, in time, of the oxygen concentration in the surface of a zirconium wireduring reaction with oxygen.

*) The solution of e(r,.) is found by Laplace transformation of the differential equation,by expanding the modified Bessel functions in the transformed solution for large valuesof the argument, and retransformation.

(X -Zr

OXIDATION OF ZIRCONIUM AT HIGH TEMPERATURES 229

Although this curve is of only qualitative significance, as in a quantitativedescription each phase region would have a different diffusion coefficientassigned to it, the trend of the curve is nevertheless quite characteristic. Initiallythe entire region from f3-Zr to ee-Zr+ Zr02 is traversed very rapidly. Changesin the total emission coefficient and the potentials El affect the reaction.In the region from 2·4 to 3·6 ms, corresponding in fig. 3 to the range from

1700K to the melting point, there is hardly any change in the surface concen-tration of oxygen. Here the reaction can therefore be described by (4) withconstant energies of activation and a constant emission coefficient (s = 0·7).Since the rate of oxidation in this region, according to fig. 8, is proportionalto P02 1/2, a mixed phase of «-Zr + Zr02 will be present at the surface if thepressure of oxygen is high, and an ee-Zrphase if the oxygen pressure is low.Possibly this accounts for the difference in the energy of activation of the rateof reaction between high and low oxygen pressures in fig.7.A zirconia phase at the surface, which would lead to diffusion-controlled

oxidation, is not obtained.During the process of fusion at practically constant temperature,

(è:Jc/'ör)r=a = Jo2/D remains nearly constant. The surface concentration ofoxygen increases again.The identations in the curve of dT./dt in fig. 5, above the melting point,

must be attributed to the endothermic reaction 20)

t Zr + t Zr02 = ZrO (g).

At a temperature of 2700 K (T. = 2500 K) the thermodynamic data yieldfor this reaction a vapour pressure of ZrO of a few Torr. A cine record of theprocess with the help of a schlieren arrangement showed at this instant a striationaway from the wire, which could have been caused only by a foreign gas becausethe arrangement was not sensitive enough to record changes in density of theoxygen.

3.4. Conclusions

In the temperature region from 1700 K to the melting point, the observedP02 dependence of the oxidation rate on the oxygen pressure and a temperaturedependence which obeys an Arrhenius equation strongly suggests that dis-sociative-oxygen-adsorption equilibria are involved. In this region a mixedphase of C(-Zr+ Zr02 will be present at the surface if the oxygen pressureis high, and an C(-Zrphase for low oxygen pressure.

Below 1700K the pretreatment of the wires is observed to play an importantrole. The interpretation of these results is therefore assumed to be hindered bythe fact that the entire region from f3-Zr to C(-Zr+ Zr02 is traversed rapidly.Also the emission coefficient is probably not constant.

230 F. KETTEL

The observed endothermic reactions above the melting point are attributedto the reaction Zr + Zr02 -+ 2 ZrO (g).

4. Droplet formation and combustion at the stationary temperature

4.1. Experimental observations

At T,= 2500-2700 K the wire disintegrates into small droplets with diametersof 200 [Lm to 400 [Lm (fig. 10). These droplets move at different velocities indifferent directions, their temperature rises to a stationary, final value. The risein temperature of these droplets to the final temperature, and hence the rate ofreaction, could not be followed precisely on account of too small a temporalresolving power (4 frames/ms) and too inaccurate a detennination of the tem-perature. From the overall rise in temperature which, at an oxygen pressure,for example, of 200 Torr, takes place in 1 ms, it is possible to estimate a meanrate of reaction of 1 . 10-2 mol cm-2 S-l.

At the final temperature the distribution ofthe oxygen in the droplet ofliquidzirconium will be largely homogeneous on account of the mixing of the liquidand the high coefficient of diffusion. The need of energy, producing adiabaticheating from the initial temperature to the final temperature, including thelatent heat of fusion, leads to a calculated concentration of oxygen of1 . 10 - 2 mol cm- 3 in the droplet, corresponding to a composition of aboutZrOO·3·

Fig. 10. Droplet formation of a zirconium wire of 0·1 mm diameter.

OXIDATION OF ZIRCONIUM AT HIGH TEMPERATURES 231

As the final temperature is being reached, a corona around the droplet(fig. 11) appears within the time of a few hundred [Ls. Initially the coronaappears as a luminous fog layer without any structure. In the further courseof the reaction even bigger radiant particles develop in the corona. Obviouslycondensation of the product of combustion, Zr02, takes place in the corona.

Shortly before the final temperature is reached, the diameter of the dropletincreases rapidly to a maximum value, which then decreases slowly as thereaction proceeds.

Fig. 11. Zirconium droplet with corona.

Figure 12 shows an example of the variation of the temperature and diameterof a droplet and of the diameter of the corona.

4.l.1. Droplet temperature

During the time of observation (up to 60 ms) the final temperature of thedroplet under constant Oj-pressure conditions remained constant, at leastwithin the measuring accuracy. In the range of droplet diameters here observed(200-400 [Lm) it was independent of the diameter of the droplet.

The relation between the oxygen filling pressure and the final temperatureof the droplet is shown in fig. 13. The temperature plotted is T" as found fromthe measurements of blackening. In this temperature region a total emissioncoefficient of O·5 to 0·6 has been assumed. The course of the true temperature,calculated with e = 0,55, has been plotted also. This curve practically coincideswith the curve of the vapour pressure of zirconium.

Fig. 12. Variation, in time, of the diameter of the corona, of the diameter of the droplet,and of the temperature of the droplet on reaching the final temperature and during stationarycombustion.

100010----------------.,

p(TOff)

t 100

10

3500. 4000 4500-T(K)

Fig. 13. The stationary final temperature of zirconium droplets on combustion in oxygen, asa function of the oxygen pressure. Ts is the measured black-body temperature, T is the truetemperature, evaluated with 8 = 0·55.

OXIDATION OF ZIRCONIUM AT HIGH TEMPERATURES 233

Figs 14 and 15. Burning droplets of zirconium, viewed in transmitted light.

4.1.2. The coronaThe temperature of the particles in the corona could not be determined by

measuring the blackening, since in the available time of observation, theirdiameter generally reached values of only 10 to 30 [Lm. Figures 14 and 15,however, which show the droplet and the corona against the transmitted lightof a foreign source used to illuminate the combustion chamber, clearly revealthat the temperature within the corona is considerably lower than the tem-perature of the droplet.

234 F. KE'ITEL

In a number of cases we were also able to observe a coalescing of the con-densate within the corona into a satellite droplet and its penetration into thecentral droplet. For the observation of several repetitions of this process, de-scribed by Nelson et aI.4-8), the time availablewas obviously not long enough.The penetration of the condensate in the corona into the droplet had also beenfollowed in the above-mentioned schlieren arrangement: after a short periodof violent vibrations of the droplet there was suddenly a strong erruption of agas, which could again be attributed to the reaction t Zr + t Zr02 -+ ZrO(gas).

Measurement of the diameter of the corona showed that at constant fillingpressure the distance between the corona and droplets of various thicknessesremained constant. With decreasing filling pressure of oxygen the distancebetween droplet and corona increased. Figure 16 shows the reciprocal distanceof the corona as a function of the filling pressure.

1 100r----------------------d-a(cm' 80I 60

~2a

20

°0~-,:':-00c:--2-:-'0'-::O--::3:':-00c:--4.,..0-:-0-50J...0-6"..0-O-:-l700_p(Torr)

Fig. 16. The reciprocal of the distance to the corona as a function of the oxygen pressure.

4.1.3. Loss of mass of the droplet

In the time elapsing between the formation of the droplet at about 2700 K,and just before the temperature reaches its stationary value, the droplet diam-eter is observed to remain at a constant value do. Subsequently the diameterofthe droplet began to increase to a maximum value d"" far above the measuringerror (about 5%) of the diameter of the droplet. This increase was up to 80%of the diameter do.The slow decrease in the diameter of the droplet during the reaction at the

stationary final temperature was determined fairly accurately by measurementsof 100 to 150 pictures. During the time of observation the droplet diameterdecreased linearly with time.Table I shows the initial diameter do, the maximum diameter dn" and the

decrease, as a function of time, of the radius of the droplet, dajdt, for differentdroplets at the same filling pressure of 200 Torr.

OXIDATION OF ZIRCONIUM AT HIGH TEMPERATURES 235

TABLE I

do (urn) dm (urn) -dajdt (cm S-l)

220 370 2.20.10-1300 410 1·27 . 10-1370 530 1·40. 10-1220 350 1·85. 10-1

The values of dajdt exhibit differences of up to 70%. No correlation wasfound between dajdt and the initial diameter or the maximum diameter.

Marshall and Pellett 9) ascribed the increase in the diameter of the dropletas observed by them to the appearance of a luminous layer (ZrO) around thedroplet. The appearance of this layer as a zone of reaction was considered tobe coupled with the appearance of the corona and was thought to manifestitself as a flattening of the steepness of the edges of the blackening at themargin of the droplet.In the measurements of the present work, the increase in the droplet occurred

preponderantly before the appearance ofthe corona. Measurement ofthe profileof blackening of the droplets at do and at the value of d shortly before theappearance of the corona showed the same steepness of the edges.We attribute the increase in the droplet diameter to swelling brought about

by the formation of small bubbles of zirconium vapour within the droplet. Atthe final temperature the vapour pressure of zirconium, according to fig. 13, isabout equal to the external pressure. This representation was confirmed byliquid droplets formed in argon by tungsten wire. The vapour pressure of thetungsten at the melting point is about 10-2 Torr. If the test was made in aprevacuum (10-2 Torr), marked swelling of the droplet and, on cooling, cor-responding shrinkage could be observed. At an argon pressure of 100 Torr nochange in diameter could observed.In accordance with this representation we attribute to the droplet a reduced

density

(dO)3

ered = eZr dm '

with

The loss of mass of the droplet then becomes

d (4n ) daered- -a3 = ered4na2-.

dt 3 dt(7)

236 F.KETIEL

Irrespective of how mass is lost, whether as Zr, ZrO, or Zr02 vapour, the massflux of the constituent Zr at the surface of the droplet, r = a, is *)

da(jzr)a = -(!red dt (g cm-2 S-1) (8)

and its molar flux is(!red da

(Jzr)a = - _- - (mol cm? S-1).Mzr dt

If the molar flux (9) is calculated from the values of table I with the reduceddensity, a value is obtained which shows differences of only 7% for the dropletsof different thickness.Thus the flow rate (JZr)a was determined at different filling pressures by the

three measurements do, dm and dafdt. The filling-pressure relation is shown infig. 17.

(9)

6.1O-3·r--------------,

(Jzr)a 5

(mol)crnè s 4

t 32

O~~-~-~-~-~~-~o 100 200 300 400 500 600 700-p(Torr)

Fig. 17. Molar rate of loss of zirconium droplets on combustion in oxygen, as a functionof the oxygen pressure.

4.2. Discussion of the experimental resultsThe experimental results of Nelson 6) concerning the conservation of the

mass of zirconium in droplet + corona after different periods of combustionjustify the suggestion that all the zirconium leaving the droplet occurs as Zr02in the corona.

*) jk and Jk indicate respectively the overall mass flux and molar flux of the constituent k,and correspondingly hand J, indicate respectively the overall mass flux and molar fluxof the component i. The overall flux can be divided into a component displaced with themass average velocity v = :E et vde and a diffusion component hD or JtD:

i, = et Vt = et v + jtD

J, = Ct Vt = Ct v + JtD C

The sum of the mass-diffusion currents must vanish:

:EhD =:E Mt JtD = o.

_._--._-~----------------- ......---- .................."""'!'!'---OXIDATION OF ZIRCONIUM AT HIGH TEMPERATURES 237

In the region beyond the corona, r > d, there is an inflow of only oxygen inan amount at least stoichiometrie to (JZr)d' If we do assume the stoichiometrieamount, then at the point of r = d, the corona, 'we have

a2

(Joz)d = ct Jo + JOz)d = - - (JZr)a,d2

(10)

or, converting to mass flux,

é(J'oz) = - - M (Jzr).

d d2 Oz a

Because of the absence of divergence of the mass flux of the constituent O2 inthe region of r > d we find

a2jOZ = - 2:Moz (JZr)a = (jo +joz) = (e V)r>d'

r(11)

Writing c = p/RT for the concentration, and M = xoMo + xozMoz (x =mole fraction) for the mean molecular weight we find

M02 Pe =-(1 +xo)-2 z RT

and

2 RTa2-v = - (JZr)a.

1+ xOz P r2

With the values, applicable at an oxygen filling pressure of 700 Torr, of(Jzr)a = 5. 10-3 mol cm-2 s-1, a = 2. 10-2 cm and d = 3 . 10-2 cm, forexample at a temperature of 3000 K, at r = d (xo2 Rj 1), this leads to

(12)

v = -550 cm S-l,

With respect to the dissociation of oxygenk1

O2 + 02+t20 + O2,

k-lin which the rate of reaction is well known 21):

k1 Rj 1013 exp(-Q/RT) cm" mol-1 S-l

(Q = 118 kcal mol "! = 02-dissociation energy), a flow rate of this magni-tude represents such a great reduction of the dissociation in terms of thethermodynamic-equilibrium value that even at considerably higher tempera-ture a dissociation can still be neglected.In the region between droplet and corona, a < r < d, a vapour ____;Zr, ZrO,

Zr02 - flows towards the corona. Reactions between the vapour of Zr or

238 F. KETTEL

ZrO and molecular oxygen imply a dissociation of oxygen. These reactionswill also require an energy of activation of the order of 118 kcal mol-1• Thecollision yield, exp(- QfRT), at 4000 K will then be about 10- 6. The numberof collisions per second in the small volume between r = a and r = d andthe number of collisions per second against the surface of the droplet arenearly of the same order. But since in the case of wall collisions we can assumea collision efficiency of about unity, the reactions will preferentially take placevia heterogeneous reactions.Furthermore the observation that the temperature of the corona is con-

siderably lower than that of the droplet excludes a major production of energywithin the space a < r < d.The energy balance for the droplet is

(13)

where A is the heat conductivity.The total energy by reactions is given by - ~ iJHI (JI)a, where (JI)a is the

flux at r = a of the gaseous component i produced by the reaction with theenthalpy zlHI' The various reactions are:

(1) (JZr)a Zr(l) --*Zr (g): iJHzr = 136 kcal mol-i,(2) (JZrO)a: Zr(1)+t02--*ZrO(g): iJHzro RiOkcalmol-i,(3) (Jzr02)a: Zr (1)+ O2 --*Zr02 (g): iJHzr02 = -98 kcal mol-I,

The energy losses 8 a Ta4 = a Ts

4 are calculated from the measured black-body temperature. The measured flux (JZr)a of the constituent Zr is

(JZr)a = (JZr)a + (Jzro)a + (Jzr02)a'

Then the energy balance is satisfied only for the case (Jzr)a I':::;j (Jzr02)a' Thethird reaction predominates. The first and second reactions require a largepositive value of À (dTfdr)a. This leads to either an unrealistic high heat con-ductivity or an extremely high temperature in the zone between droplet andcorona.The conclusion is that the reaction of a vapour of zirconium or zirconium

monoxide to produce gaseous zirconia already takes place at the surface ofthe droplet. Any condensation of Zr02 vapour at the droplet surface must beexcluded on account of the high ZrO vapour pressure by the reactionZr (1) + Zr02 (1) --*2 ZrO (g). This reaction has been observed after pene-tration of a satellite droplet (Zr02) into the parent droplet (Zr).If it is assumed that J Zr02= JZr and that the oxygen flux is stoichiometrie,

i.e. J02 = _JZr, then with div JI = 0 in the region a < r < d,

(14)

OXIDATION OF ZIRCONIUM AT HIGH TEMPERATURES 239

Neglecting external forces, pressure gradient and diffusion thermoeffects theenergy balance in the space a < r < d is given by

1 d ( dT) é dT-- r2 A.- = _(Jzr)aLlCp-;r2 dr dr r2 dr

(15)

LlCp = Cp Zr02 - C; 02 is the difference of specific heat per mol.With the boundary condition at r = a,

-LlHzr02 (pr)a = e a Ta4 - A. (dTjdr)a,

we obtain the following expression for the temperature field between dropletand corona:

LlHZr02 (Jzr)a + e a Ta4 [ ( LlCp r - a) ]T= Ta + exp (pr)a--- - 1 . (16)

(Jzr)a LIc, A. ria

Taking for A. an average value of 5.10-4 cal cm-1 S-1 deg-\ ê = 0'55,andthe other values from experimental results, the values for the temperature ofthe corona Td shown in table II are obtained.

TABLE II

p (Torr) Ta (K)

700 4700400 4470200 4260100 407040 3880

28502780275027502750

For all filling pressures we therefore find a temperature of the corona lyingclose to the melting point of zirconia.The transport of O2 and Zr02 in the zone of diffusion a < r < d is

governed by the diffusion equation

a2- (JZr)a = c D grad x02'r2

where c is the concentration in mol cm- 3 and x02 is the mol fraction of 0.2(X02 + XZr02 = 1).Since the diffusion coefficient is proportional to T3/2 and inversely propor-

tional to the pressure, c D becomes independent of the pressure and propor-tional to Tl/2. The integration is performed with a constant mean value of c D.With the boundary condition in the corona

(X02)d R::l 1

240 F.KETIEL

one obtains the mol fractions of oxygen and zirconia in the diffusion zone:

1 (1 1)Xo = 1- - (Jzr) a2 - - -z cD a r d'(17)

Xzroz = _1_ (Jzr)a a2 (~_ ~).cD r d

With respect to the oxygen mol fraction at r = a the condition

(xoz)a ~ 0

must be imposed. According to the measured values of (pr)a and (d - a) atall filling pressures a value of the diffusion coefficient, at N.T.P. (1 atm, roomtemperature), of Do Ri 0·23 cm" S-1 is required for (xo2)a = O.If an exothermic reaction is to occur at the surface of the droplet, the con-

ditionpzrOz

or (xzro2)a = -- ~ 1p

must be satisfied. However, this also fixes an upper limit for the temperature,since if the pressure of the constituent Zr at the surface exceeds the filling pres-sure by only a small amount, then the resulting pressure gradient would preventany flow of oxygen to the surface. In its place a flow outward from the dropletsurface, which could only be Zr vapour, would occur. Thus the upper limitfor the temperature can be described by the pressure relation

pz/ ~ 1,P

where p is the filling pressure and PZr S the saturation vapour pressure of Zr.The temperature measurements of fig. 13 (8 = 0'55) show that at filling

pressures above 20 Torr this upper limit of the temperature is reached:

P Ri pz/· (18)

In this model describing the combustion at the stationary final temperaturethe dissolution of oxygen into the droplet is neglected. This reaction can beneglected when the oxygen partial pressure at the droplet surface falls to theequilibrium value of the liquid droplet.We have estimated that the oxygen content of the droplet will, at the final

temperature, be about 1 . 10-2 mol/cm", corresponding to a composition ofabout ZrOO'3' At equilibrium, the pressure of oxygen above this liquid willbe very small, while the zirconium-vapour pressure will roughly correspondto that of pure zirconium.For high filling pressures, when the upper limit of temperature is reached,

OXIDATION OF ZIRCONIUM AT HIGH TEMPERATURES 241

the oxygen partial pressure at r = a is very small. Itmay be justified to neglecta gas-uptake reaction.At low filling pressures (p < 10 Torr), however, the upper limit of the tem-

perature according to (18) will not be reached.If at the surface of the droplet the reaction

Zr (1) + O2 -+ Zr02 (g)

is to take place, with fJHRa = -98 kcal mol-I, then we put for the mol frac-tion (xzr02)a:

( )_ (Pzr02)a "" PZr S

XZr02 a - ,...., -- •

P P(19)

If the saturation vapour pressure of the zirconium is less than the fillingpressure, the droplet will also take up oxygen in accordance with a flux enteringthe droplet:

(I. I) _ (P02)a _ (X02)a02 a - (2 'TC m02 k Ta)1/2 - P (2 'TC m02 k Ta)1/2 . (20)

The reaction enthalpy for this flux is fJHRI = -258 kcal mol-i. Now theenergy balance for r = a becomes

The flux follows from div J1 = 0:

a2JZr02 = czr02 vzr02 = """2 (Jzr02)a,

r

The solution of the diffusion equation (c D = const.) with the boundary con-dition (X02)d = 1, gives the :flux (Jzr02)a:

(Jzr02)a = (J021)a (xzr02)a X

X exp ( __ 1 (Jo/)a d - a) [1- exp ( __ 1 (Jo/)a d - a)J-i.cD dia cD dia

(22)

If (J021)a from (20) and (xzr02)a from (19) are substituted (x02 + xzro2 = 1),this equation leads to the :flux(J zro2)a as a function of the temperature and thefilling pressure. In conjunction with the energy balance it is then possible togive the final temperature as a function of the filling pressure.

242 F. KETIEL

With d -+ 00,

(dT) _Ta_To

À - =-Àdr a a

À = 5.10-4 cal cm=' S-1 deg-\ Do = 0·23 cm" S-1.

Figure 18 shows the relation between filling pressure and final temperature inaccordance with these equations: at very low filling pressures the final tem-perature is determined by the equilibrium with the gas-uptake reactionLlHR

I o-;». and at a high filling pressure by the vapour-pressure curve.In a cine recording at a filling pressure of 2 Torr it was possible to follow

a droplet during combustion for 40 ms. During this time there was not acorona round the droplet nor did a measurable change in droplet diameteroccur. The temperature of the droplet was below 3000 K.

15:,---------------,-----,

5

p(Torr)

1 10

2500 3000 3500-T(K)

4000

Fig. 18. The final temperature ofzirconium droplets on combustion in oxygen at lowpressure.

4.3. ConclusionsThe initial increase of droplet diameter, just before the final, stationary tem-

perature is reached, is attributed to the formation of small Zr-vapour bubbleswithin the droplet, which results in a reduced effectivedensity. Taking this intoaccount leads to the experimental determination of the molar flux of Zr which.leaves the droplet, and which is observed to be independent of the dropletdiameter.The reaction Zr (1) + O2 -+ Zr02 will predominantly take place at the

droplet surface. The dissociation of O2 on its way to the surface can be ne-glected. The upper limit of the temperature, which depends on the oxygen pres-sure, is determined from the saturation pressure of the component with thehighest vapour pressure. If P02 > 10 Torr, the temperature limit is determined

OXIDATION OF ZIRCONIUM AT HiGH TEMPERATURES 243

by the vapour pressure of Zr. A calculation of the temperature distributionbetween the droplet and the corona suggests that, contrary to the temperatureof the droplet, the corona temperature is practically pressure-independent.For low oxygen pressure: P02 < 10 Torr, no corona is formed. The droplet

will take up oxygen, and this reaction enthalpy (LJHR = -258 kcal mol-I)together with the rate of impingement of O2 molecules determines the tem-perature dependence on P02 in this region.

4.4. Concluding remarks

The mechanism of combustion at the stationary final temperature is de-scribed in terms of mass and energy balance with the help of a simple model.The momentum balance is replaced by the assumption of P = const., and theequations of the reactions are reduced to the statement P PI:! Pz:, (Po2)a PI:! O.This model can represent only a rough approximation: if the partial pressure

of oxygen at the surface is very low, then with a molar flux of the order of themeasured (JZr)a the velocity of flow becomes so high that the assumption ofP = const. is no longer justified. For P = const. the velocity of flow must bekept below the velocity of sound and the partial pressure of oxygen must beof the order of several Torr. In that case an oxygen-uptake reaction possiblyresults.

For more-detailed statements about the reaction we lack, above all, data onthe equilibrium vapour pressure of liquid zirconium containing various amountsof oxygen.Regarding the processes taking place in the zone of condensation, the model

restricts itself to the assumption of (X02)d Rj 1. The formation of criticalnuclei in the zone of transport, the fixing of the corona at a fixed distancefrom the central droplet, and the radiation of energy by condensed particlesof increasing size and decreasing number have been left out of account.

Acknowledgement

The author is indebted to Mr H. Körver for his assistance with the experi-ments and their evaluation, and to Dr H. Hörster for valuable discussions.

Philips Forschungslaboratorium Aachen, April1973

244 F. KEITEL

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bustion Institute, China Lake, California, 1969.10) K. M. Maloney, to be published in High Temp. Science.11) K. M. Maloney and T. C. M. Pillay, to be published.12) Gmelins Handbuch der anorg. Chemie, Zirconium, 8th edition, 1958,p. 209.13) G. Boureau and P. Gerdanian, High Temperatures, High Pressures 2, 681, 1970.14) All thermodynamical data weretaken from Janaf, Thermochemical Data, Dow Chemical

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