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Nucleation and growth of oxide particles in metal vapour flames
NUCLEATION
AND GROWTH OF OXIDE PARTICLES METAL VAPOUR FLAMES
IN
ROBERT W. HERMSEN AND ROGER DUNLAP United Technology Center, A Division of United Aircraft Corporation, Sunnyvale, California A set of equations describing the condensation of aluminium oxide droplets from an atmosphere containing metal vapour and oxygen has been formulated and programmed for solution on a high-speed digital computer. The initial formation of the particles is described by classical nucleation theory, particle growth takes place by diffusion of metal and oxygen to the particle surface followed by reaction on the surface to form condensed oxide. Starting with a given mass fraction of metal vapour, condensation takes place both by nucleation of new particles and growth of all the particles in the system until the partial pressure of metal vapour decreases to the equilibrium value. The time required tor condensation and the resulting particle size distribution have been calculated tor dillerent inmai metal vapour concentrations and temperatures.
dominant unless extreme care is taken to remove all foreign particles which might otherwise be present in the system. In a metal vapour flame, however, homogeneous nucleation rates are very rapid and there may be few foreign particles present that are suitable as nucleating centres for the metal oxide. Thus the homogeneous mechanism is of greater importance and may dominate the nucleation process. Following nucleation, growth takes place by diffusion of the condensing species to the droplet surface, condensation on the surface and transfer of the latent heat away from the droplet. This is not an ordinary condensation process in the sense that it represents only a phase change, but it also may involve heterogeneous chemical reactions at the phase boundary 4. Reactions are required for the condensation of alumina since no vapour molecule having the same composition exists. Heterogeneous reaction On oxide particle surfaces also appears to be important in the condensation of beryllia and magnesia. This paper reports the results of an analytical investigation of the homogeneous nucleation and growth of alumina droplets that was conducted in order to gain some ir~sight into the resulting particle size distributions and the relative importance of the nucleation and growth processes. Two cases were analysed, nucleation
Introduction
IN MANY respects the combustion of particles of the light metals aluminium, beryllium and magnesium in a hot oxidizing atmosphere is similar to the classical vapour diffusion mechanism of hydrocarbon fuel droplets. One distinguishing feature of the combustion of these metals, however, is the formation of a condensed oxide product in the form of finely dispersed particles, or smoke. Furthermore, because the bond strengths of the gaseous oxide species of the metals are relatively weak, a major portion of the heat of combustion results from the condensation process. Although some condensation occurs on the relatively cool surface of the burning metal particle, experimental evidence indicates that a substantial amount of oxide is formed by nucleation and subsequent growth of liquid droplets in or near the gasphase flame zone1- 3. Nucleation of the metal oxide droplets may take place either by the heterogeneous mechanism, in which foreign particles already present in the system act as nucleating centres, or by the homogeneous mechanism, in which droplets larger than the critical nucleus size are formed by statistical fluctuations in the gas phase. In commonly observed condensation processes occurring at ambient temperature and below, the heterogeneous mechanism is ordinarily 253
ROBERT W. HE
254
and growth in an atmosphere of constant temperature and composition, and the transient problem in which the temperature and composition of the atmosphere in a closed system change during condensation. Pdthough neither of these idealized cases is truly representative of a metal vapour flame, the results indicate some important features of the condensation process and should aid in understanding the overall mechanism of metal combustion. Particle Nucleation The rate of homogeneous nucleation of alumina droplets was calculated from the equation 16~a3V 2
I = (2rurtm) ~ v S ( p ° / k r ) 2 exp
]
- 3(~-T-~-St j
where tr is the surface tension and r is the volume per molecule of liquid alumina, m is the mass of an aluminium atom, pO is the equilibrium partial pressure of aluminium in alumina vaporization products, and S is an effective supersaturation. Equation I is a modified form of the classical nucleation equation given by Frenkel s, and expresses the number of embryonic droplets per unit time per unit volume that pass through the nucleation free energy barrier and grow to larger sizes. The radius ofdroplets at this critical point is given by r* = 2 a r / k T InS
VoL 13
SEN AND ROGER DUNLAP
[2]
For a simple condensation process involving no chemical reactions the supersaturation, S, is simply the ratio of the partial pressure of the condensing species to the equilibrium vapour pressure. In the condensation of alumina, however, several vapour species and chemical reactions may be involved and S will depend on the unknown reaction path and the reacting species concentrations. Since our present understanding provides little or no basis for choosing the correct reaction path it was assumed that S = p/pO where p is the partial pressure of aluminium in the gas and pO is the equilibrium partial pressure defined above. This assumption could potentially tead to large errors in the nucleation rate, and its possible effect on the results will be discussed in a later section.
Both equations I and 2 neglect the effect of surface curvature on the surface tension and the effect of the translational and rotational entropy of the nuclei. In obtaining equation 1 it was further assumed that only collisions of metal atoms with critical embryos contributed to the nucleation rate and that the accommodation coefficient for these collisions was unity.
Particle Growth An equation for the growth rate of a droplet in a supersaturated atmosphere has been obtained by Gyarmathy 6 who considered diffusion of the condensing species to the droplet surface, and transfer of the latent heat of condensation back to the gas. Neglecting non-continuum effects, Gyarmathy's solution applied to the growth rate of an alumina droplet is (1 P'
r*/r) In S +
p
where r is the instantaneous particle radius, p, M and i,. are respectively the density, molecular weight and heat of vaporization of alumina, n is the number of moles of vapour resulting from the vaporization of one mole of alumina, × is the thermal conductivity of the gas phase, P is the total pressure, p is the sum of ,,he partial pressures of the condensing species and D is an appropriate mean diffusion coefficient. Equation 3 has the form of a driving force divided by the sum of two resistances, a heat transfer resistance proportional to 12/u, and a diffusional resistance proportional to l/pD. Over the range of conditions investigated in this work the diffusional resistance was always less than 0.01 times the heat transfer resistance and was neglected in the calculations.
Particle Distribution as a Function of Time Considering Nucleation and Growth
Consider a system containing aluminium vapour and oxygen plus inerts, occupying a volume V and initially containing no particles. Particles of radius r o are formed by nucleation at a rate I V where l(S, T), the nucleation rate
June 1969
255
N U C L E A T I O N A N D G R O W T H OF OXIDE PARTICLES IN METAL V A P O U R FLAMES
given by equation 1, is a function of the temperature and supersaturation. Once formed the particles grow at a rate #(S, T, r) which is given by equation 3 and is a function of the instantaneous particle radius as well as temperature and supersaturation. Let f(r, t) be the normalized particle frequency distribution such that f(r, t) dr is the number fraction of particles with radii between r and r + dr at time t. Normalization requires that
S t e a d y system In a system with constant composition and temperature the supersaturation and volume will also be constant: therefore I V = constant
and f is a function of r only, having the torm
~fff ( r , t ) d r = 1
By equating the accumulation of particles in the class r - , r + dr to the difference between particles growing into and out .of this class it is easily shown that fmust satisfy the differential equation ?(ND ?(Nf~) I- 0 ?t ~'r
change continuously during condensation according to the conservation equations. Each of these cases is analysed below.
[4]
with initial condition
i(r) = (A/r)(1 - r*/r) where A
0 rAttl 0
fir, 0 + ) = 6(r - r o)
for
r < ro
I .2
~(I m
where
for r < ro forro~< r < r , , for r>~ rm
-r*,r)
--
r~)
+ r*lr.,
+ r .2 in :(r,. Nit) = .fro IV dt
nqT.RT 2 In S/pMI~
[7]
where r,,. the radius of the largest particles in the distribution, is a function of time given implicitly by the integrated form of equation 6
I V = i'(S, 7", r0t fifo, t) Nit)
fir, t ) = 0
=
Under these constraints, the following solution for fir, t) satisfying equation 4 and the initial and boundary conditions may be obtained flr, t ) =
and boundary conditions
[6]
[5]
is the total number of particles in the system. and 6 is the Dirac 6 function. In general, S and T will vary with time as metal vapour and oxygen are removed and latent heat is added to the gas by the condensation of alumina. The system may also be free to exchange matter and heat with its surroundings by diffusion and convection. However, two limiting cases may be defined that do not require a complete description of these external transport processes. First is the steady system in which exchange with the surroundings exactly balances the consumption of oxygen and metal vapour and the heat generation within the system so that composition and temperature remain constant. Second is the closed system in which no external exchange takes place so that composition and temperature
-
_
r 0
)
r*),(r o -r*):
=
At
[8]
In order toavoid the singularity in equation 8 occurring at ro = r* the initial radlus must be chosen slightly larger than r*, that is roir*-
1 =~:>
0
The singularity arises because according to the classical theory of nucleation droplets of radius r* are in equilibrium with the supersaturated vapour and therefore have zero growth rate. Equation 1 gives the rate at which droplets with radii slightly larger than r* are formed. The characterization of these embryonic droplets by an arbitrary c as is done here is justified irom a practical standpoint since the time required for the droplets to grow from ro to a radius r >> r* is short relative to the total growth time. The calculations were performed taking t; = 10-3: however, the actual choice oft: (from 10-2 to 10 -'~) did not influence the results.
Vot 13
ROBERT W. HEKM'TEN AND ROGER DUI~AP
256
shape with maxima occurring at rm, the largest particle radius. Distributions after 0-3 and 1-0 msec for T = 3000CK and p = 0-1 atm are shown in Figure 2. Although the distribution becomes broader, corresponding to the increase in rm, and the maximum decreases, the triangular shape does not change with time.
1-5 0~ E
0
/.,. ¢J
F1. 2
Closed system
Assuming constant heat capacities the temperature increase resulting from condensation may be obtained from the energy equation and is given by
0
~0-9
E
r ' = (1 + C,%)/(1 + C2%)
~x 0-6
[9]
where T' is the temperature divided by its initial value, % is the particle mass fraction and C , = I,,/CgTo
03
C2 = C d C g -
I
0
i
0'2
;
i
,
'
i
0-~. 0"6 Ti rne~ msec
i
0"8
1
i
'1-0
FIGURE !. Maximum particle radius versus time in a steady system. A - - T = 3000 K. p = 0-1 atm: B - - T = 3500 K, p=0.1atm:C--T=3000 K , p = 0 - 0 5 a t m . P = 1.0atm
Curves of rm versus t calculated from equation 8 are shown in Figure 1 for three different sets of conditions. Values of l., ng and pO were obtained as functions of T from equilibrium vaporization calculations for alumina based on the JANAF thermochemical data 7, and u was estimated from the calculations of Brokaw 8. The density and surface tension of alumina were taken as 3-5 g/cm 3 and 700 erg/cm 2 respectively 9. Since over the range of interest r m >> r o, the growth curves are nearly parabolic. The particle growth rate increases slowly with increasing metal vapour concentration owing to tl~e logarithmic dependence of A on supersaturation. Increasing temperature causes a decrease in growth rate but the effect is small since the increase in equilibrium partial pressure, and resulting decrease in S, is almost cancelled by the increase in T 2. Normalized particle frequency distributions calculated from equation 7 are triangular in
2.4 I ¸
t =0-3reset i
i/i ¢o
i
U
f
E
r
C
t-lmse
1"2 r-
0-6
0
0-4
0"8
1.2
1-6
Particte radius,microns FIGURE 2. Normalized number frequency distribution after 0-3 and 1.0 msec. Steady system at T = 3000"K,p = 0.1 atm. P = 1.0 atm
June 1969
The supersaturation decreases as condensation takes place owing to the depletion of condensing species and to the increase in equilibrium vapour pressure associated with the temperature rise. The increase in vapour pressure may be described by the integrated Clasius-Clapeyron equation p°(T) = p°(To)exp [C3(1 - I/T')]
i. 10]
was projected over a short time interval. Repetition of this procedure over sufficiently short time intervals on a high-speed digital computer was used to calculate the particle size distribution at various times during the condensation process. The number of particles added to the system over a short time interval, At, is given by t+At
AN-
where
S IVdt
[13]
t
C3 = I,,M/nRTo
Equation l0 may be combined with a material balance to give S "-- S O where
257
NUCLEATION AND GROWTH OF OXIDE PARTICLES IN METAL VAPOUR FLAMES
-
C5(xp
3
,-
So is the initial supersaturation and C4 = 2(Mo/M) (P/Po) C5 = n(Mo/M)
Constant total pressure will be assumed so that the volume of the system is given by v
=
Vo(l
-
c5%)r'
[12]
where Vo is the initial volume. In equations 11 and 12 it has been assumed that the gas phase obeys the perfect gas relations and has constant molecular weight. Equations 1 4 and 9-12 together with the initial and boundary conditions represent a complete mathematical description of the problem. Sin,:e ? is an implicit function of time, through its dependence on T and S, an analytical solution of equation 4 could not be obtained. Instead, a numerical method was employed to calculate fir, t) and the time required for complete condensation, that is, the time for S to approach unity. The method involved adding particles to the system, in accordance with the instantaneous nucleation rate, and following the growth of each particle in time. At any instant during the process, summation over all particles determined the total condensed mass from which the instantaneous temperature and supersaturation was calculated: The temperature and supersaturation determine the instantaneous nucleation rate and particle growth rates from which the system growth, in particle number and size,
where I and V are given by equations l and 12 respectively.-In reality these particles are distributed continuously over the time interval" however, in the present scheme this continuous nucleation process was approximated by a series of small-step increases in the number of particles. Thus, at time t a number of nuclei equal to (IV~ At was added to the system and this class of particles grew during the interval At together with the classes formed previously. No particles were added during the time interval. The length of the time interval was determined by requiring that the nucleation rates at thebeginning of two successive intervals differ only by a small prescribed amount, generally between 5 and 15 per cent. The growth ofeach particle class in the system during the time interval At was computed from the integrated form of equation 6 -
+ It*), [I,-,t,+
- I,.,),]
+ (r~, 2 In [{(r,),+ n, - (r~,}/{(r,), - (r*),}] = AAt
[14]
This equation is strictly valid if T 2 InS is constant during the time interval At. This was found to be a very good approximation for time intervals chosen on the basis of the change in I Vexcept in a few cases for which the time interval was shortened so that T 2 l nS did not vary by more than five per cent. Since equation 14 is implicit in particle radius it was solved by trial using the interval halving method. The radius at t + At was determined to an accuracy of better than one part in ten thousand to prevent error accumulation. In order to avoid the singularity in equation (14) for (ri}, = r,* t'i~z nucleus size was taken to be 1-01 r* for purposes
ROBERT W . HERMSEN A N D ROGER D U N L A P
258
of ca|cuLation. Several calculations using a starting radius of 1.001 r * showed no effect of decreasing the starting radius on the final size distribution or on the time required for condensation. Initial calculations showed that after condensation was 20-30 per cent complete on a mass basis the nucleation rate had decreased from its initial value by at least five orders of magnitude and continued to decrease rapidly. Thus, the particle size classes that needed to be accounted for could be limited to a reasonable number by cutting off nucleation after some point in the calculation. In practice this required accounting for between 50 and 100 classes, which was also sufficient to give smooth particle size distributions. After nucleation was stopped the growth of each particle class was calculated until S approached unity and no further condensation took place. During this phase of the calculation the time increment was chosen so that T 2 InS changed by less than five per cent between successive time intervals. The frequency distribution, fir.t) was calculated at any desired time from the number of particles in each class and their size according to the relation f{(r, + r,+ ,)/2} ---
ANd(r,- r,+
1-15]
70
"T
o
50 t -106 p sec
c
30 20
t:11.2/
10 •
0
9'0 t
11"18
8"0 1"15 7.0 k.
60
1.12 n
E
E O
- 50 109
~-o U3
30
C
106 ~ cl
E
2"0 1.0 1,00 10 20 30 40 50 60 70 80 90 100 110 Time, mlcrosec FIGURE 4. S u p e r s a t u r a t i o n and t e m p e r a t u r e versus time in a closed system at P = !.0 arm and initially at T = 3400 K, p = 0.1 arm
6O c
of alumina particles from an atmosphere at 1 atm total pressure with initial conditions of T = 3400°K and p = 0.1 atm. The two distributions shown are for times of 11-2 and 106 lasec corresponding respectively to about eight per cent and nearly 100 per cent completion of condensation. Figures 4 and 5 show the variation of supersaturation, temperature and nucleation rate with time during condensation for the same conditions.
1 03
,)N
Normalized frequency distributions are shown in Figure 3 at two times during the condensation
U~
VoL 13
0.1
), 02 0-3 Partlcl.e radius,microns
0-L
FIGURE 3. N o r m a l i z e d n u m b e r frequency distribution after 11,2 and 106 lasec. Closed system at P = !-0 atm and initially at T = 3 40ff K, p = 0- i a t m
A range of initial tempciatures and aluminium partial pressures were studied, and although the final average particle size and the time required for condensation varied over a wide range, the shape of the distribution curves and the variation of S, T and I in time were all similar to Figures 3-5. Final maximum alumina particle radii versus initial temperature are shown in Figure 6 for initial aluminium partial pressures of 0.05, 0-10, and 0-20 atm. Because of the narrow spread in final size distribution the average final radii are only a few per cent
Jtme 1969 12 !
'-'10 I/1
";' 8
E
"-6
B~
I
0
2
Z,
i
L
1
I
6
8
10
12
t
14
16
18
20 22
fime.microsec
FIGURE 5. N u c l e a t i o n rate versus time in a closed system at P = !.0 a t m and initially at T = 3400 K. p = 0.1 arm
I0° 0L u
6
E
13 104 "
6 ,m
~J ~"
259
NUCLEATION AND GROWTH OF OXIDE PARTICLES IN METAL VAPOUR FLAMES
initial temperature. Examination of the numerical calculations revealed that this behavioar was a direct consequence of the highly nonlinear dependence of nucleation rate on initial temperature. For example, at p = 0.1 atm the initial nucleation rate decreased only by a factor of three when the initial temperature was changed from 2 500°K to 3 090°K. However, at initial temperatures of 3250 °, 3400 ° and 3500°K the nucleation rate had dropped by factors of 10 -2, 10 -7 and 10 - 1 6 respectively. This drastic decrease in nucleation rate caused fewer total particles to form per unit condensable mass and resulted in a significant increase in average particle size. Since the particle surface area available for condensation was also reduced, the time required for condensation showed a corresponding increase. In view of the importance of the nucleation rate in determining the final particle size and condensation time it is of interest to review the assumptions implicit in equation 1. The effect of the assumption that only metal atom collisions contribute to the nucleation rate, even though
2
10¸2[
o_ 10-2
....
2300 2500 2700 2900 3100 3300 351)0 3700 Initial t e m p e r a t u r e , °K
I03I
FIGt:RE 6. Final m a x i m u m particle radius for a l u m i n a c o n d e n s a t i o n in closed system at P = !-0 a t m
smaller than those shown in Figure 6. The time required for the supersaturation to fall from its initial value to a value of 1.05, where condensation is essentially complete, is shown in Figure 7 as a function of initial temperature for the same three gas compositions. Discussion Probably the most striking feature of"the results of this study is the dependence of final particle size and condensation time on initial temperature shown in Figures 6 and 7. Below some critical temperature, which depends on initial aluminium partial pressure, the results are insensitive to initial temperature. At higher temperatures, however, both particle size and condensation time increase very rapidly with increasing
E_10s
Po
E
/
i-.--
10-7[ I0-8 2L00
.............. 2800 3000 3200 3z.00 36L)b
2 600
Initial temperature,°K FIGURE 7. T i m e for s u p e r s a t u r a t i o n to tall f r o m initial value to i.05 in closed system at P = l-0 a t m
260
ROBERT W. HERMSEN AND ROGER DUNLAP
collisions of other species must make a contribution, is reduced by the assumption of completely effective collisions. Although little is known about the effect of surface curvature on surface tension, its effect on nucleation rate is probably small. More serious are neglect of the translational and rotational entropy of the nuclei, and the assumption that the supersaturation may be expressed in terms of metal vapour partial pressure alone. Translational and rotational entropy terms have been shown to cause a significant increase in the magnitude of the nucleation rate in some cases although the functional dependence on supersaturation is not significantly changed ~°. The assumption regarding the supersaturation was compared with several alternatives, each based upon a different possible reaction path for alumina condensation. These alternative schemes would lead to final particle sizes and condensation times somewhat lower than those sho,~n in Figures 6 and 7; however, the curves would retain their characteristic shape. The droplets, which are initially very small, begin their growth in the free molecule r6gime. At this time equation 2, which neglects noncontinuum effects, overestimates the growth rate. The particles quickly grow to larger sizes, however, and the time spent in the noncontinuum r6gime is short compared to the subsequent growth time in the continuum r6gime. Thus, little error will result in the final particle sizes and condensation times. Because of errors resulting from the assumptions discussed above the actual values of final particle size and condensation time reported are probably not much more accurate than an order of magnitude. The qualitative dependence of particle size and condensation time on initial temperature and metal partial pressure shown in Figures 6 and 7 is correct, however, and points out the importance of nucleation in controlling the condensation process. More accurate calculations will depend upon a greater understanding of nucleation processes involving chemical reactions and the particular reaction path for alumina condensation. The particle size frequency distributions for both the steady and closed systems were
VoL 13
skewed to favour larger particles. In the closed system the distribution became narrower as time progressed and finally represented a maximum spread of only 10 to 20 per cent in particle size when condensation was complete. On the other hand, size distributions that have been measured for alumina particles formed either in aluminium vapour combustion 1~ or in the combustion of aluminized solid propellants 12 are all reasonably broad and skewed toward smaller particle sizes. The systems studied in this work represent two idealized limits of the actual situation existing in a metal vapour flame. The steady system requires that the transport of heat and mass exactly balance the heat released and mass removed by condensation and does not allow for the removal of particles. The closed system, on the other hand, does not allow any transport of heat or mass across the system boundary and requires that the initial supersaturation was achieved instantaneously. The supersaturation and temperature histories in many practical situations will probably be between the two limits. Additional factors that were beyond the scope of this study will also influence the oxide particle size in the case of a burning metal particle. First, the growth of an oxide nucleus will be determined by the time it spends in the flame zone as well as the distribution of supersaturation and temperature that it experiences during this time. Nuclei formed at the inner boundary of the combustion region have the opportunity to grow for a longer time before being convected away from the flame than nuclei formed at the outer edge of the flame which have little opportunity to grow. Brownian motion will result in a distribution of residence times for nuclei formed at the same point in the flame. In addition, Brownian motion may strongly influence the final distribution of oxide particle sizes by promoting collision and coalescence of the liquid droplets. The present study has shown that an important factor in the complete model of a burning aluminium particle is the alumina nucleation rate, which has a strong influence on final particle size in the range of temperatures expected for aluminium combustion.
Juno 1969
NUCLEATION AND GROWTH OF OXIDE PARTICLES IN METAL VAPOUR FLAMES
The authors are indebted to Mr R. L. Carlson who assisted with the numerical calculations. Financial support was provided by the Propulsion Division, Directorate of Engineering Sciences, Air Force Office of Scientific Research under Contracts AF 49(638t-1202 and AF 49(638)-1489. (Received July 1968; revised October 1968) References t METAL COMBUSTIONSTUDY GROUP, "Aluminum particle combustion progress report', Tech. Progr. Rep. No. 415. NOTS TP 3916, U.S. Naval Ordnance Test Station (April 1966) 2 BRZUSTOWSKI, T. A, and GLASSMAN, I. Hcterogeneotts Combustion (Wolfhard. H. G., Glassman, I., Green, L., Jr, Eds), p !i 7. Academic Press: New York (1964) 3 CASSEL, H. M. and LIERMAN, I. Combustion & Flame. 6. 153 (1962)
261
4 MARKSTEIN, G. H. Ninth Symposium (International) on Combustion, p 137. Academic Press: New York (1963) 5 FRENKEL, J. Kinetic Theory 0/" Liquid3, p 397. Oxford University Press: London (1946) 6 GYARMATHY,G. Z. angew. Math. PILl's. 14, 280 (1963) "7 STULL, D. R. et al. J A N A F Thermochemical Tables. Dow Chemical Company: Midland, Michigan (August 1966) 8 BROKAW, R. S. Alignment charts for transport properties, viscosity, thermal conductivity and diffusion coefficients for non-polar gases and gas mixtures at low density', N A S A TR R-81 (1961) 9 KINGERY, W. D. J. Amcr. ceram. Soc. 42. 6 (1959) ~o LO~HE, J. and POUND, G. M. J. chem. Phrs. 36, 2080 ( ! 962) ~ HE~S~N, R. W. "Vapor phase combustion of beryllium and aluminum', UTC 2183-FR, United Technology Center, Contract AF 49(638)-1489 (March 1968) ~2 CROWL C. T. et al. "Dynamics of two-phase flow in rocket nozzles', UTC 2102-FR. United Technology Center, Contract NOw-64-0506-c (September 1965)
AND GROWTH OF OXIDE PARTICLES METAL VAPOUR FLAMES
IN
ROBERT W. HERMSEN AND ROGER DUNLAP United Technology Center, A Division of United Aircraft Corporation, Sunnyvale, California A set of equations describing the condensation of aluminium oxide droplets from an atmosphere containing metal vapour and oxygen has been formulated and programmed for solution on a high-speed digital computer. The initial formation of the particles is described by classical nucleation theory, particle growth takes place by diffusion of metal and oxygen to the particle surface followed by reaction on the surface to form condensed oxide. Starting with a given mass fraction of metal vapour, condensation takes place both by nucleation of new particles and growth of all the particles in the system until the partial pressure of metal vapour decreases to the equilibrium value. The time required tor condensation and the resulting particle size distribution have been calculated tor dillerent inmai metal vapour concentrations and temperatures.
dominant unless extreme care is taken to remove all foreign particles which might otherwise be present in the system. In a metal vapour flame, however, homogeneous nucleation rates are very rapid and there may be few foreign particles present that are suitable as nucleating centres for the metal oxide. Thus the homogeneous mechanism is of greater importance and may dominate the nucleation process. Following nucleation, growth takes place by diffusion of the condensing species to the droplet surface, condensation on the surface and transfer of the latent heat away from the droplet. This is not an ordinary condensation process in the sense that it represents only a phase change, but it also may involve heterogeneous chemical reactions at the phase boundary 4. Reactions are required for the condensation of alumina since no vapour molecule having the same composition exists. Heterogeneous reaction On oxide particle surfaces also appears to be important in the condensation of beryllia and magnesia. This paper reports the results of an analytical investigation of the homogeneous nucleation and growth of alumina droplets that was conducted in order to gain some ir~sight into the resulting particle size distributions and the relative importance of the nucleation and growth processes. Two cases were analysed, nucleation
Introduction
IN MANY respects the combustion of particles of the light metals aluminium, beryllium and magnesium in a hot oxidizing atmosphere is similar to the classical vapour diffusion mechanism of hydrocarbon fuel droplets. One distinguishing feature of the combustion of these metals, however, is the formation of a condensed oxide product in the form of finely dispersed particles, or smoke. Furthermore, because the bond strengths of the gaseous oxide species of the metals are relatively weak, a major portion of the heat of combustion results from the condensation process. Although some condensation occurs on the relatively cool surface of the burning metal particle, experimental evidence indicates that a substantial amount of oxide is formed by nucleation and subsequent growth of liquid droplets in or near the gasphase flame zone1- 3. Nucleation of the metal oxide droplets may take place either by the heterogeneous mechanism, in which foreign particles already present in the system act as nucleating centres, or by the homogeneous mechanism, in which droplets larger than the critical nucleus size are formed by statistical fluctuations in the gas phase. In commonly observed condensation processes occurring at ambient temperature and below, the heterogeneous mechanism is ordinarily 253
ROBERT W. HE
254
and growth in an atmosphere of constant temperature and composition, and the transient problem in which the temperature and composition of the atmosphere in a closed system change during condensation. Pdthough neither of these idealized cases is truly representative of a metal vapour flame, the results indicate some important features of the condensation process and should aid in understanding the overall mechanism of metal combustion. Particle Nucleation The rate of homogeneous nucleation of alumina droplets was calculated from the equation 16~a3V 2
I = (2rurtm) ~ v S ( p ° / k r ) 2 exp
]
- 3(~-T-~-St j
where tr is the surface tension and r is the volume per molecule of liquid alumina, m is the mass of an aluminium atom, pO is the equilibrium partial pressure of aluminium in alumina vaporization products, and S is an effective supersaturation. Equation I is a modified form of the classical nucleation equation given by Frenkel s, and expresses the number of embryonic droplets per unit time per unit volume that pass through the nucleation free energy barrier and grow to larger sizes. The radius ofdroplets at this critical point is given by r* = 2 a r / k T InS
VoL 13
SEN AND ROGER DUNLAP
[2]
For a simple condensation process involving no chemical reactions the supersaturation, S, is simply the ratio of the partial pressure of the condensing species to the equilibrium vapour pressure. In the condensation of alumina, however, several vapour species and chemical reactions may be involved and S will depend on the unknown reaction path and the reacting species concentrations. Since our present understanding provides little or no basis for choosing the correct reaction path it was assumed that S = p/pO where p is the partial pressure of aluminium in the gas and pO is the equilibrium partial pressure defined above. This assumption could potentially tead to large errors in the nucleation rate, and its possible effect on the results will be discussed in a later section.
Both equations I and 2 neglect the effect of surface curvature on the surface tension and the effect of the translational and rotational entropy of the nuclei. In obtaining equation 1 it was further assumed that only collisions of metal atoms with critical embryos contributed to the nucleation rate and that the accommodation coefficient for these collisions was unity.
Particle Growth An equation for the growth rate of a droplet in a supersaturated atmosphere has been obtained by Gyarmathy 6 who considered diffusion of the condensing species to the droplet surface, and transfer of the latent heat of condensation back to the gas. Neglecting non-continuum effects, Gyarmathy's solution applied to the growth rate of an alumina droplet is (1 P'
r*/r) In S +
p
where r is the instantaneous particle radius, p, M and i,. are respectively the density, molecular weight and heat of vaporization of alumina, n is the number of moles of vapour resulting from the vaporization of one mole of alumina, × is the thermal conductivity of the gas phase, P is the total pressure, p is the sum of ,,he partial pressures of the condensing species and D is an appropriate mean diffusion coefficient. Equation 3 has the form of a driving force divided by the sum of two resistances, a heat transfer resistance proportional to 12/u, and a diffusional resistance proportional to l/pD. Over the range of conditions investigated in this work the diffusional resistance was always less than 0.01 times the heat transfer resistance and was neglected in the calculations.
Particle Distribution as a Function of Time Considering Nucleation and Growth
Consider a system containing aluminium vapour and oxygen plus inerts, occupying a volume V and initially containing no particles. Particles of radius r o are formed by nucleation at a rate I V where l(S, T), the nucleation rate
June 1969
255
N U C L E A T I O N A N D G R O W T H OF OXIDE PARTICLES IN METAL V A P O U R FLAMES
given by equation 1, is a function of the temperature and supersaturation. Once formed the particles grow at a rate #(S, T, r) which is given by equation 3 and is a function of the instantaneous particle radius as well as temperature and supersaturation. Let f(r, t) be the normalized particle frequency distribution such that f(r, t) dr is the number fraction of particles with radii between r and r + dr at time t. Normalization requires that
S t e a d y system In a system with constant composition and temperature the supersaturation and volume will also be constant: therefore I V = constant
and f is a function of r only, having the torm
~fff ( r , t ) d r = 1
By equating the accumulation of particles in the class r - , r + dr to the difference between particles growing into and out .of this class it is easily shown that fmust satisfy the differential equation ?(ND ?(Nf~) I- 0 ?t ~'r
change continuously during condensation according to the conservation equations. Each of these cases is analysed below.
[4]
with initial condition
i(r) = (A/r)(1 - r*/r) where A
0 rAttl 0
fir, 0 + ) = 6(r - r o)
for
r < ro
I .2
~(I m
where
for r < ro forro~< r < r , , for r>~ rm
-r*,r)
--
r~)
+ r*lr.,
+ r .2 in :(r,. Nit) = .fro IV dt
nqT.RT 2 In S/pMI~
[7]
where r,,. the radius of the largest particles in the distribution, is a function of time given implicitly by the integrated form of equation 6
I V = i'(S, 7", r0t fifo, t) Nit)
fir, t ) = 0
=
Under these constraints, the following solution for fir, t) satisfying equation 4 and the initial and boundary conditions may be obtained flr, t ) =
and boundary conditions
[6]
[5]
is the total number of particles in the system. and 6 is the Dirac 6 function. In general, S and T will vary with time as metal vapour and oxygen are removed and latent heat is added to the gas by the condensation of alumina. The system may also be free to exchange matter and heat with its surroundings by diffusion and convection. However, two limiting cases may be defined that do not require a complete description of these external transport processes. First is the steady system in which exchange with the surroundings exactly balances the consumption of oxygen and metal vapour and the heat generation within the system so that composition and temperature remain constant. Second is the closed system in which no external exchange takes place so that composition and temperature
-
_
r 0
)
r*),(r o -r*):
=
At
[8]
In order toavoid the singularity in equation 8 occurring at ro = r* the initial radlus must be chosen slightly larger than r*, that is roir*-
1 =~:>
0
The singularity arises because according to the classical theory of nucleation droplets of radius r* are in equilibrium with the supersaturated vapour and therefore have zero growth rate. Equation 1 gives the rate at which droplets with radii slightly larger than r* are formed. The characterization of these embryonic droplets by an arbitrary c as is done here is justified irom a practical standpoint since the time required for the droplets to grow from ro to a radius r >> r* is short relative to the total growth time. The calculations were performed taking t; = 10-3: however, the actual choice oft: (from 10-2 to 10 -'~) did not influence the results.
Vot 13
ROBERT W. HEKM'TEN AND ROGER DUI~AP
256
shape with maxima occurring at rm, the largest particle radius. Distributions after 0-3 and 1-0 msec for T = 3000CK and p = 0-1 atm are shown in Figure 2. Although the distribution becomes broader, corresponding to the increase in rm, and the maximum decreases, the triangular shape does not change with time.
1-5 0~ E
0
/.,. ¢J
F1. 2
Closed system
Assuming constant heat capacities the temperature increase resulting from condensation may be obtained from the energy equation and is given by
0
~0-9
E
r ' = (1 + C,%)/(1 + C2%)
~x 0-6
[9]
where T' is the temperature divided by its initial value, % is the particle mass fraction and C , = I,,/CgTo
03
C2 = C d C g -
I
0
i
0'2
;
i
,
'
i
0-~. 0"6 Ti rne~ msec
i
0"8
1
i
'1-0
FIGURE !. Maximum particle radius versus time in a steady system. A - - T = 3000 K. p = 0-1 atm: B - - T = 3500 K, p=0.1atm:C--T=3000 K , p = 0 - 0 5 a t m . P = 1.0atm
Curves of rm versus t calculated from equation 8 are shown in Figure 1 for three different sets of conditions. Values of l., ng and pO were obtained as functions of T from equilibrium vaporization calculations for alumina based on the JANAF thermochemical data 7, and u was estimated from the calculations of Brokaw 8. The density and surface tension of alumina were taken as 3-5 g/cm 3 and 700 erg/cm 2 respectively 9. Since over the range of interest r m >> r o, the growth curves are nearly parabolic. The particle growth rate increases slowly with increasing metal vapour concentration owing to tl~e logarithmic dependence of A on supersaturation. Increasing temperature causes a decrease in growth rate but the effect is small since the increase in equilibrium partial pressure, and resulting decrease in S, is almost cancelled by the increase in T 2. Normalized particle frequency distributions calculated from equation 7 are triangular in
2.4 I ¸
t =0-3reset i
i/i ¢o
i
U
f
E
r
C
t-lmse
1"2 r-
0-6
0
0-4
0"8
1.2
1-6
Particte radius,microns FIGURE 2. Normalized number frequency distribution after 0-3 and 1.0 msec. Steady system at T = 3000"K,p = 0.1 atm. P = 1.0 atm
June 1969
The supersaturation decreases as condensation takes place owing to the depletion of condensing species and to the increase in equilibrium vapour pressure associated with the temperature rise. The increase in vapour pressure may be described by the integrated Clasius-Clapeyron equation p°(T) = p°(To)exp [C3(1 - I/T')]
i. 10]
was projected over a short time interval. Repetition of this procedure over sufficiently short time intervals on a high-speed digital computer was used to calculate the particle size distribution at various times during the condensation process. The number of particles added to the system over a short time interval, At, is given by t+At
AN-
where
S IVdt
[13]
t
C3 = I,,M/nRTo
Equation l0 may be combined with a material balance to give S "-- S O where
257
NUCLEATION AND GROWTH OF OXIDE PARTICLES IN METAL VAPOUR FLAMES
-
C5(xp
3
,-
So is the initial supersaturation and C4 = 2(Mo/M) (P/Po) C5 = n(Mo/M)
Constant total pressure will be assumed so that the volume of the system is given by v
=
Vo(l
-
c5%)r'
[12]
where Vo is the initial volume. In equations 11 and 12 it has been assumed that the gas phase obeys the perfect gas relations and has constant molecular weight. Equations 1 4 and 9-12 together with the initial and boundary conditions represent a complete mathematical description of the problem. Sin,:e ? is an implicit function of time, through its dependence on T and S, an analytical solution of equation 4 could not be obtained. Instead, a numerical method was employed to calculate fir, t) and the time required for complete condensation, that is, the time for S to approach unity. The method involved adding particles to the system, in accordance with the instantaneous nucleation rate, and following the growth of each particle in time. At any instant during the process, summation over all particles determined the total condensed mass from which the instantaneous temperature and supersaturation was calculated: The temperature and supersaturation determine the instantaneous nucleation rate and particle growth rates from which the system growth, in particle number and size,
where I and V are given by equations l and 12 respectively.-In reality these particles are distributed continuously over the time interval" however, in the present scheme this continuous nucleation process was approximated by a series of small-step increases in the number of particles. Thus, at time t a number of nuclei equal to (IV~ At was added to the system and this class of particles grew during the interval At together with the classes formed previously. No particles were added during the time interval. The length of the time interval was determined by requiring that the nucleation rates at thebeginning of two successive intervals differ only by a small prescribed amount, generally between 5 and 15 per cent. The growth ofeach particle class in the system during the time interval At was computed from the integrated form of equation 6 -
+ It*), [I,-,t,+
- I,.,),]
+ (r~, 2 In [{(r,),+ n, - (r~,}/{(r,), - (r*),}] = AAt
[14]
This equation is strictly valid if T 2 InS is constant during the time interval At. This was found to be a very good approximation for time intervals chosen on the basis of the change in I Vexcept in a few cases for which the time interval was shortened so that T 2 l nS did not vary by more than five per cent. Since equation 14 is implicit in particle radius it was solved by trial using the interval halving method. The radius at t + At was determined to an accuracy of better than one part in ten thousand to prevent error accumulation. In order to avoid the singularity in equation (14) for (ri}, = r,* t'i~z nucleus size was taken to be 1-01 r* for purposes
ROBERT W . HERMSEN A N D ROGER D U N L A P
258
of ca|cuLation. Several calculations using a starting radius of 1.001 r * showed no effect of decreasing the starting radius on the final size distribution or on the time required for condensation. Initial calculations showed that after condensation was 20-30 per cent complete on a mass basis the nucleation rate had decreased from its initial value by at least five orders of magnitude and continued to decrease rapidly. Thus, the particle size classes that needed to be accounted for could be limited to a reasonable number by cutting off nucleation after some point in the calculation. In practice this required accounting for between 50 and 100 classes, which was also sufficient to give smooth particle size distributions. After nucleation was stopped the growth of each particle class was calculated until S approached unity and no further condensation took place. During this phase of the calculation the time increment was chosen so that T 2 InS changed by less than five per cent between successive time intervals. The frequency distribution, fir.t) was calculated at any desired time from the number of particles in each class and their size according to the relation f{(r, + r,+ ,)/2} ---
ANd(r,- r,+
1-15]
70
"T
o
50 t -106 p sec
c
30 20
t:11.2/
10 •
0
9'0 t
11"18
8"0 1"15 7.0 k.
60
1.12 n
E
E O
- 50 109
~-o U3
30
C
106 ~ cl
E
2"0 1.0 1,00 10 20 30 40 50 60 70 80 90 100 110 Time, mlcrosec FIGURE 4. S u p e r s a t u r a t i o n and t e m p e r a t u r e versus time in a closed system at P = !.0 arm and initially at T = 3400 K, p = 0.1 arm
6O c
of alumina particles from an atmosphere at 1 atm total pressure with initial conditions of T = 3400°K and p = 0.1 atm. The two distributions shown are for times of 11-2 and 106 lasec corresponding respectively to about eight per cent and nearly 100 per cent completion of condensation. Figures 4 and 5 show the variation of supersaturation, temperature and nucleation rate with time during condensation for the same conditions.
1 03
,)N
Normalized frequency distributions are shown in Figure 3 at two times during the condensation
U~
VoL 13
0.1
), 02 0-3 Partlcl.e radius,microns
0-L
FIGURE 3. N o r m a l i z e d n u m b e r frequency distribution after 11,2 and 106 lasec. Closed system at P = !-0 atm and initially at T = 3 40ff K, p = 0- i a t m
A range of initial tempciatures and aluminium partial pressures were studied, and although the final average particle size and the time required for condensation varied over a wide range, the shape of the distribution curves and the variation of S, T and I in time were all similar to Figures 3-5. Final maximum alumina particle radii versus initial temperature are shown in Figure 6 for initial aluminium partial pressures of 0.05, 0-10, and 0-20 atm. Because of the narrow spread in final size distribution the average final radii are only a few per cent
Jtme 1969 12 !
'-'10 I/1
";' 8
E
"-6
B~
I
0
2
Z,
i
L
1
I
6
8
10
12
t
14
16
18
20 22
fime.microsec
FIGURE 5. N u c l e a t i o n rate versus time in a closed system at P = !.0 a t m and initially at T = 3400 K. p = 0.1 arm
I0° 0L u
6
E
13 104 "
6 ,m
~J ~"
259
NUCLEATION AND GROWTH OF OXIDE PARTICLES IN METAL VAPOUR FLAMES
initial temperature. Examination of the numerical calculations revealed that this behavioar was a direct consequence of the highly nonlinear dependence of nucleation rate on initial temperature. For example, at p = 0.1 atm the initial nucleation rate decreased only by a factor of three when the initial temperature was changed from 2 500°K to 3 090°K. However, at initial temperatures of 3250 °, 3400 ° and 3500°K the nucleation rate had dropped by factors of 10 -2, 10 -7 and 10 - 1 6 respectively. This drastic decrease in nucleation rate caused fewer total particles to form per unit condensable mass and resulted in a significant increase in average particle size. Since the particle surface area available for condensation was also reduced, the time required for condensation showed a corresponding increase. In view of the importance of the nucleation rate in determining the final particle size and condensation time it is of interest to review the assumptions implicit in equation 1. The effect of the assumption that only metal atom collisions contribute to the nucleation rate, even though
2
10¸2[
o_ 10-2
....
2300 2500 2700 2900 3100 3300 351)0 3700 Initial t e m p e r a t u r e , °K
I03I
FIGt:RE 6. Final m a x i m u m particle radius for a l u m i n a c o n d e n s a t i o n in closed system at P = !-0 a t m
smaller than those shown in Figure 6. The time required for the supersaturation to fall from its initial value to a value of 1.05, where condensation is essentially complete, is shown in Figure 7 as a function of initial temperature for the same three gas compositions. Discussion Probably the most striking feature of"the results of this study is the dependence of final particle size and condensation time on initial temperature shown in Figures 6 and 7. Below some critical temperature, which depends on initial aluminium partial pressure, the results are insensitive to initial temperature. At higher temperatures, however, both particle size and condensation time increase very rapidly with increasing
E_10s
Po
E
/
i-.--
10-7[ I0-8 2L00
.............. 2800 3000 3200 3z.00 36L)b
2 600
Initial temperature,°K FIGURE 7. T i m e for s u p e r s a t u r a t i o n to tall f r o m initial value to i.05 in closed system at P = l-0 a t m
260
ROBERT W. HERMSEN AND ROGER DUNLAP
collisions of other species must make a contribution, is reduced by the assumption of completely effective collisions. Although little is known about the effect of surface curvature on surface tension, its effect on nucleation rate is probably small. More serious are neglect of the translational and rotational entropy of the nuclei, and the assumption that the supersaturation may be expressed in terms of metal vapour partial pressure alone. Translational and rotational entropy terms have been shown to cause a significant increase in the magnitude of the nucleation rate in some cases although the functional dependence on supersaturation is not significantly changed ~°. The assumption regarding the supersaturation was compared with several alternatives, each based upon a different possible reaction path for alumina condensation. These alternative schemes would lead to final particle sizes and condensation times somewhat lower than those sho,~n in Figures 6 and 7; however, the curves would retain their characteristic shape. The droplets, which are initially very small, begin their growth in the free molecule r6gime. At this time equation 2, which neglects noncontinuum effects, overestimates the growth rate. The particles quickly grow to larger sizes, however, and the time spent in the noncontinuum r6gime is short compared to the subsequent growth time in the continuum r6gime. Thus, little error will result in the final particle sizes and condensation times. Because of errors resulting from the assumptions discussed above the actual values of final particle size and condensation time reported are probably not much more accurate than an order of magnitude. The qualitative dependence of particle size and condensation time on initial temperature and metal partial pressure shown in Figures 6 and 7 is correct, however, and points out the importance of nucleation in controlling the condensation process. More accurate calculations will depend upon a greater understanding of nucleation processes involving chemical reactions and the particular reaction path for alumina condensation. The particle size frequency distributions for both the steady and closed systems were
VoL 13
skewed to favour larger particles. In the closed system the distribution became narrower as time progressed and finally represented a maximum spread of only 10 to 20 per cent in particle size when condensation was complete. On the other hand, size distributions that have been measured for alumina particles formed either in aluminium vapour combustion 1~ or in the combustion of aluminized solid propellants 12 are all reasonably broad and skewed toward smaller particle sizes. The systems studied in this work represent two idealized limits of the actual situation existing in a metal vapour flame. The steady system requires that the transport of heat and mass exactly balance the heat released and mass removed by condensation and does not allow for the removal of particles. The closed system, on the other hand, does not allow any transport of heat or mass across the system boundary and requires that the initial supersaturation was achieved instantaneously. The supersaturation and temperature histories in many practical situations will probably be between the two limits. Additional factors that were beyond the scope of this study will also influence the oxide particle size in the case of a burning metal particle. First, the growth of an oxide nucleus will be determined by the time it spends in the flame zone as well as the distribution of supersaturation and temperature that it experiences during this time. Nuclei formed at the inner boundary of the combustion region have the opportunity to grow for a longer time before being convected away from the flame than nuclei formed at the outer edge of the flame which have little opportunity to grow. Brownian motion will result in a distribution of residence times for nuclei formed at the same point in the flame. In addition, Brownian motion may strongly influence the final distribution of oxide particle sizes by promoting collision and coalescence of the liquid droplets. The present study has shown that an important factor in the complete model of a burning aluminium particle is the alumina nucleation rate, which has a strong influence on final particle size in the range of temperatures expected for aluminium combustion.
Juno 1969
NUCLEATION AND GROWTH OF OXIDE PARTICLES IN METAL VAPOUR FLAMES
The authors are indebted to Mr R. L. Carlson who assisted with the numerical calculations. Financial support was provided by the Propulsion Division, Directorate of Engineering Sciences, Air Force Office of Scientific Research under Contracts AF 49(638t-1202 and AF 49(638)-1489. (Received July 1968; revised October 1968) References t METAL COMBUSTIONSTUDY GROUP, "Aluminum particle combustion progress report', Tech. Progr. Rep. No. 415. NOTS TP 3916, U.S. Naval Ordnance Test Station (April 1966) 2 BRZUSTOWSKI, T. A, and GLASSMAN, I. Hcterogeneotts Combustion (Wolfhard. H. G., Glassman, I., Green, L., Jr, Eds), p !i 7. Academic Press: New York (1964) 3 CASSEL, H. M. and LIERMAN, I. Combustion & Flame. 6. 153 (1962)
261
4 MARKSTEIN, G. H. Ninth Symposium (International) on Combustion, p 137. Academic Press: New York (1963) 5 FRENKEL, J. Kinetic Theory 0/" Liquid3, p 397. Oxford University Press: London (1946) 6 GYARMATHY,G. Z. angew. Math. PILl's. 14, 280 (1963) "7 STULL, D. R. et al. J A N A F Thermochemical Tables. Dow Chemical Company: Midland, Michigan (August 1966) 8 BROKAW, R. S. Alignment charts for transport properties, viscosity, thermal conductivity and diffusion coefficients for non-polar gases and gas mixtures at low density', N A S A TR R-81 (1961) 9 KINGERY, W. D. J. Amcr. ceram. Soc. 42. 6 (1959) ~o LO~HE, J. and POUND, G. M. J. chem. Phrs. 36, 2080 ( ! 962) ~ HE~S~N, R. W. "Vapor phase combustion of beryllium and aluminum', UTC 2183-FR, United Technology Center, Contract AF 49(638)-1489 (March 1968) ~2 CROWL C. T. et al. "Dynamics of two-phase flow in rocket nozzles', UTC 2102-FR. United Technology Center, Contract NOw-64-0506-c (September 1965)