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Computational analysis of coagulation and coalescence in the flame synthesis of titania particles
Powder Technology 118 Ž2001. 242–250 www.elsevier.comrlocaterpowtec
Computational analysis of coagulation and coalescence in the flame synthesis of titania particles Tue Johannessen a,) , Sotiris E. Pratsinis b, Hans Livbjerg a a
Department of Chemical Engineeringr ICAT, Technical UniÕersity of Denmark, DTU, Building 229, DK-2800 Lyngby, Denmark b Institute of Process Engineering, Swiss Federal Institute of Technology, CH-8092 Zurich, Switzerland Received 3 February 2000; received in revised form 19 September 2000
Abstract A method of combining computational fluid dynamics with a mathematical model for the particle dynamics has been applied to simulate experimental data from the synthesis of TiO 2-particles in diffusion flames. Parameters of the coalescence kinetics are estimated by fitting the model predictions to the measured specific surface area of the product particles. The estimated kinetics can be used to predict the surface area and aggregate structure of the particles for a wide range of synthesis conditions. The regular equation for the rate of coagulation is modified to take into account the effect of dilution. The accuracy of the results, especially the degree of aggregation, i.e. the aggregate size, is highly dependent on the inclusion of this effect. When the dilution is accounted for, the predicted aggregate sizes Žnumbers of primary particles per aggregate. compare well with reported data based on small-angle X-ray scattering measurements. q 2001 Elsevier Science B.V. All rights reserved. Keywords: Flame; Titania; Nanoparticle; Coalescence; Simulation; Aggregate
1. Introduction Flame processes are presently used for the manufacture of several nano-structured commodities, e.g. titania catalysts and pigments, carbon black and fumed silica w1x. As our insight into the fundamentals of flame processes improves, the flexibility of these processes and their ability to produce nanoparticles and powders with high specific surface areas are likely to lead to novel, tailor-made materials such as catalysts and membranes. However, to improve the control of particle characteristics, there is a need for accurate modeling and simulation of particle formation. With the improved capacity of computational fluid dynamics ŽCFD. packages, a detailed modeling of the temperature and velocity fields within the flame has come within reach. Integrated with realistic models for the particle dynamics, a complete model for flame-based particle generators can be set up w2x. Recent works on particle modeling in flames have employed a coagulation–coalescence model for particle dynamics combined with either an assumed or measured
)
Corresponding author. Tel.: q45-45-883288; fax: q45-45-932906. E-mail address: [email protected] ŽT. Johannessen..
temperature profile Že.g. Refs. w3,4x.. In the present study, we use the commercial CFD-code FLUENT to simulate the gas temperature and velocity profiles of the burners. Once the flow, temperature and composition fields from the CFD simulations are known, the particle model can be integrated along a number of characteristic trajectories. The model is validated by comparison with experimental results from Pratsinis et al. w5x on the synthesis of TiO 2 in diffusion flames.
2. Experimental The experimental data concern TiO 2 particles synthesized in a laboratory burner by combustion of TiCl 4 in a diffusion flame w5x. The burner is a multiple-concentric-tube burner which is fed by methane, air, and argon gas saturated with TiCl 4 vapor at a controlled temperature. The experimental work consists of an investigation of the specific surface area and morphology of the particles for different flame configurations and flow rates of the feed gases. Four different flame configurations were employed Žcf. Fig. 1. with a constant flow rate of methane, 0.312 lrmin, and a constant flow of argon through the
0032-5910r01r$ - see front matter q 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 2 - 5 9 1 0 Ž 0 0 . 0 0 4 0 1 - 0
T. Johannessen et al.r Powder Technology 118 (2001) 242–250
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3.1. Particle dynamics The modified monodisperse model describes the particle dynamics by the variation of three variables, the number concentration of particles Ž N ., the average surface area Ž a., and solid volume Ž Õ . of the single particlesraggregates. The rate of change of N, a and Õ are affected by coagulation, coalescence and particle dilution caused by dispersion. To include the effect of dilution caused by eddy dispersion in the flame, the original monodisperse model by Kruis et al. w3x was modified by a dilution factor f D calculated from the CFD-data by Johannessen w6x: m PM fD s Ž 2. m PM ,o where m PM is the mass fraction of particles in the gas mixture, and subscript o denotes the conditions at the start of the precursor jet. With the dilution effect included, the rate of change of the particle number concentration and the corresponding growth rate for the volume of one particle, Õ, become w6x: Fig. 1. The four different flame configurations used by Pratsinis et al. w5x.
TiCl 4-bubbler, 0.25 lrmin. Three different air flow rates were applied: 2.5, 3.8 and 5.5 lrmin. For all flames, the TiCl 4 precursor enters the flame through the central nozzle. Flames A and B are double diffusion flames, in which the annular feed port of methane has air on both sides, while C and D are single diffusion flames. Flame A has a comparatively high jet velocity, since the central tube contains both the precursor gas and the combustion air. The effluent particles are recovered on a glass fibre filter and their specific surface area, SA , is measured by multipoint nitrogen adsorption ŽGemini 2360, Micromeritics. at 77 K using the BET equation. Assuming spherical particles, one can calculate a BET equivalent primary particle size, d BET , from: d BET s
6
rp SA
Ž 1.
where rp is the bulk solid density. For further details of the experimental work, the reader is referred to Pratsinis et al. w5x.
3. Theoretical The simulation method comprises two main elements: a computation of the flow and temperature fields in the flame by a commercial CFD software package, and a subsequent integration of a monodisperse, Lagrangian model for the formation and transport of particles within the flow fields. The CFD model of the diffusion flame is presented in detail in Johannessen et al. w2x and Johannessen w6x and shall only be briefly outlined here.
dN
1 N d fD s y rg b N 2 q dt 2 fD dt
Ž 3.
and dÕ
1 s
dt
ž
1
N 2
rg b N 2 Õ s
/
1 2
rg b NÕ
Ž 4.
where N is the number of particles per unit mass of gas, rg is the gas density, b is the collision frequency function and t is the elapsed time along a particle trajectory. The second term on the right-hand side of Eq. Ž3. denotes the rate of concentration change due to the dilution of the central jet by the surrounding particle free gas. The use of a concentration based on mass of gas is expedient because it is invariant to the thermal expansion of the gas. The collision frequency kernel b is calculated by the Fuchs equation w3,7,8x, which covers the whole particle size range from the free molecular to the continuum regime. The particle collision radius, rc , needed for the calculation of b , is equal to the true radius for a solid sphere. The collision radius depends on the fractal dimension when the particles are shaped like dendritic aggregates of coagulated primary particles. The collision radius for an aggregate consisting of n p identical primary particles is calculated by Matsoukas and Friedlander w9x: rc s r p
Õ
ž / Õp
1
1
Df
Df
s rp Ž n p .
Ž 5.
where Õp is the volume of the primary particles, all assumed to be identical. The fractal dimension is assumed to be Df s 1.8, a value often observed for cluster–cluster coagulation w10x. This value was verified also by small-angle X-ray scattering for various flame generated powders
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by Hyeon-Lee et al. w11x including the employed data here by Pratsinis et al. w5x. The net rate of change for the surface area of an aggregate, a, consists of two terms: the rate of area growth due to coagulation and the rate by which the surface area becomes diminished by sintering or coalescence: da
1 s
dt
1
ž
rg b N 2 a y rs Ž a,Õ,T .
/
N 2
Ž 6. dp s
3.2. Coalescence of aggregates The rate of coalescence of a particle, consisting initially of two separate, spherical particles contacting each other, can be approximately described by Koch and Friedlander w12x: rs s
1
Ž a y as .
tf
Ž 7.
where a is the surface area, Õ the volume of the particle and as is the surface area of a solid sphere with the same volume as the sintering particle. The characteristic coalescence time, t f , depends on the sintering mechanism, which may be, e.g. viscous flow, lattice diffusion, surface diffusion or grain boundary diffusion Žcf. e.g. Ref. w13x.. The expressions for t f , however, all yield to the common form:
t f s k 0 d pm
T
exp
T0
EA
1
R
T
ž ž
1 y T0
//
Ž 8.
where EA is an activation energy Že.g. for the diffusion coefficient in the case of diffusive coalescence. and d p is the initial diameter of the two contacting primary particles. The pre-exponential term, k 0 d pm , is the characteristic sintering time at the temperature T0 . The power of d p , m, ranges from 1 to 4 depending on the diffusion mechanism. Several attempts have been made to extend Eq. Ž7. for more complicated shapes w3,14–16x. We use the points-ofcontact ŽPOC. model of Johannessen w6x Žcf. Ref. w2x.. When the particles are relatively compact, this model follows the rate expression of Eq. Ž7.. Aggregates of two or more primary particles are assumed to consist of equally sized, spherical primary particles. Sintering is assumed to be localized to the vicinity of the points of contact between primary particles following Eqs. Ž7. and Ž8.. Therefore, the net rate of surface area reduction for an aggregate is obtained by multiplying the local sintering rate by the number of contact points in the aggregate. The resulting sintering rate expression becomes w2x:
°
Ž ay as .
t Ž r s~ f
d p)
;
.
s
¢Ž
n p y1 .
ž
d p) s
During the coalescence of an aggregate, the primary particle size increases and the number of contact points decreases. The number of primary particles in an aggregate n p , the diameter d p , the surface area a p and the volume Õp of one primary particle are computed from the particle volume Õ and surface area a by straightforward geometrical considerations:
ž
6 Ž Õr2.
p
1r 3
/
for n p F 2
0.41a p Ž 2 a p y a2 P p ,s . s Ž n p y1 . t f Ž dp . t f Ž dp .
/
ž
/
for n p ) 2
Ž 9.
6Õ a
p ,
Õp s
6
d p3 ,
np s
Õ Õp
,
a p s p d p2
Ž 10 .
The number of contact points is assumed equal to n p y 1, which is equivalent to assuming chain-like aggregates. This is a reasonable assumption for aggregates with relatively high fractal dimension Ž Df - 2.. In some cases, the actual number of contact points for n p particles might be higher than n p y 1. A more complex relation between n p and the actual number of contact points might be derived from TEM-images. In the POC model, the coalescence of an aggregate is completely described by the expression for the coalescence of two primary particles and the number of contact points in the aggregate. There are no restrictions on the form of the expression for the characteristic sintering time. The POC model ensures a correct asymptotic behavior for the final stage of coalescence when the whole particle is dense w6x. The transition from AaggregateB to Adense particleB, at n p s 2, occurs without discontinuity in the sintering rate expression Žcf. Eq. Ž9... 3.3. Computational fluid-particle dynamics The flow, temperature and composition fields in the flame are computed by the CFD code, Fluent w17x. A structured grid is made for the computational domain selected to be the space that is confined radially by a coaxial silica glass shield surrounding the flame and extending axially from 1 cm below the burner mouth to 15 cm above. The flame model comprises the time-smoothed equations of change together with the differential Reynolds stress turbulence model with nonequilibrium wall functions. The rate of methane combustion is calculated as the limiting value of a regular Arrhenius expression for the single-step methane combustion w18x and the rate calculated by the Magnussen and Hjertager w19x turbulent combustion model. Radiant energy absorption and irradiation are incorporated in the enthalpy balance. In the primary CFD simulations, the transport of particles within the flame is computed like that of a pseudo gas component, PM, which has been assigned a negligibly small molecular diffusivity. Hence, the calculated variation of the PM concentration within the flame is identical to that of the particles, assuming that the transport of particles occurs solely by convective flow and eddy diffusion. This is reasonable, since the eddy diffusion rate of particles by far exceeds their Brownian diffusion rate. Turbulent eddies
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will transport particles in ensembles. Therefore, the rate of coagulation is still dominated by Brownian motion within the eddy. The influence of turbulence on the coagulation rate is insignificant for sub-micron particles in systems with little or moderate degree of turbulence w20x. The CFD-model of the diffusion burner is similar to that of Johannessen et al. w2x, who provide further details about the model calculations. The simulation results are exported to a separate Fortran code, which performs the following calculations. The flow of particulate matter is divided into N characteristic trajectories following the mean flow of particulate matter as computed for the pseudo gas component PM. In this way, the path of the trajectories is determined by both convective gas flow and lateral eddy diffusion, the influence of which is significant. The values of temperature, gas velocity, dilution ŽEq. Ž2.., and time elapsed for the particles are mapped from the CFD simulation results onto each point of the trajectories. The Lagrangian particle model of Eqs. Ž3., Ž4. and Ž6. is integrated along each trajectory and the primary particle size, the number of primary particles per aggregate and the aggregate collision cross section are calculated as a function of distance above the burner to the outlet of the process w2x. The following assumptions and approximations are made for the simulations: v
The oxidation of TiCl 4 is assumed to occur instantaneously when the gas temperature exceeds a given value TD , converting all precursor molecules to free TiO 2 AmonomerB molecules.
v
v
245
The subsequent formation and growth of particles occur by coagulation caused by Brownian collisions between particles with the monomers as starting point and leading to a steadily growing average particle size. The notion of a collision limited growth, even for the monomers, is based on the assumption that dimeric and larger clusters are stable—a plausible assumption due to the extreme supersaturation of the monomers at the time of their formation.
4. Results and discussion 4.1. Flame structures and temperature contours For the various flame configurations, the difference in flame structure is clearly visible in the computed temperature contours for flames A, B, C and D Žair flow rate s 3.8 lrmin. shown in Fig. 2. For all four flame types, the precursor enters in the central tube. However, flames A and B are double diffusion flames with a reaction front between the methane and air on the innermost side and between the methane and the external, still air. Flames C and D only have a single flame front between the methane and the air in the outermost annulus. Flames C and D have almost identical appearances. The only difference is a slightly higher velocity in the central tube for flame D since the methane and precursor-gas flow in the central tube. The appearance of flame A is evidently quite different from the other flames due to its much higher velocity in
Fig. 2. Temperature contour plots for the flames A, B, C and D Žair flow rate s 3.8 lrmin.. Two particle trajectories, enveloping 10% and 90% of the total particle flow, respectively, are superimposed on each contour plot. The right part of the figure shows simulations of the outlet primary particle size of each trajectory of the flames on the left-hand side.
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the central jet, which contains the mixture of both the air and the precursor gas. The maximum flame temperature is located on the jet axis for flames C and D, but off the axis for A and B. Molecular and especially turbulent diffusion is responsible for the dispersion of particulate matter in the radial direction. Fig. 2 shows, superimposed on the temperature contours, the trajectories enveloping 10% and 90%, respectively, of the total particle flow. Evidently, trajectories close to the axis have different time-temperature histories than the peripheral ones. Fig. 3 shows the temperature as a function of residence time along the centre axis for flames A through D. The high linear gas velocity in the central tube of flame A yields short residence times for the A-trajectory as well as a low T-maximum. Flames B, C and D all have comparable residence times, but only C and D have a high T-maximum at the centre-line. From Fig. 2, we can see that the trajectories with higher temperatures are located away from the centre-line for flames A and B, whereas the opposite is the case for C and D. 4.2. Least-squares fitting of the specific surface area For all experimental conditions, CFD simulations are carried out and read into the Fortran code. For each case, five trajectories are computed and the corresponding values of residence time, temperature and dilution factor are stored for the Lagrangian integrations.
Initial values of the coalescence parameters are assigned in the programme, which subsequently performs a multiparameter fit by minimizing the sum of squared differences between the simulated and measured specific surface area, i.e.: 2
SSQ s Ý Ž ln Ž SA ,meas . y ln Ž SA ,model . . i
Ž 11 .
i
The logarithmic differences are used to ensure that experimental points with highly different values of the specific surface area are equally weighted in the fitting procedure. The initial conditions for the particle volume, surface area and number concentration are calculated from the assumption that the solid oxide initially consists of TiO 2 monomers, the size of which is computed from the molecular volume of titania. The precursor vapours are assumed to form TiO 2 monomers instantly at a given decomposition temperature, TD s 340 K, approximately 20 K above the inlet temperature. The value of TD affects the final particle characteristics insignificantly, but the chosen value minimizes SSQ of Eq. Ž11.. It is plausible that the decomposition of the precursor takes place downstream from the jet opening where water and oxygen are present and where the temperature has increased. By fitting the model to the measured specific surface areas, the following values of the two coalescence parameters k 0 and EA of Eq. Ž8. are estimated: k 0 s Ž 5.5 " 7.5 . = 10 27 my4 s and EA s 148 " 35 kJrmol
Fig. 3. Simulated temperature along the centre symmetry axis of the titania flames A, B, C and D Žair flow rates 3.8 lrmin. shows the influence of flame configuration on the timertemperature history. The double diffusion flames A and B have much lower centre-line temperatures and in addition, flame A has a very short residence time caused by air and carrier-gas being present in the jet-tube. The single diffusion flames, C and D, both show high centerline temperature maximum.
These estimates are based on the following values of the constants in Eq. Ž8.: m s 4 and T0 s 1400 K. The estimated standard errors shown with the parameter estimates are rather large, due to some correlation between the two parameters. However, the estimated standard error in predicting the specific surface areas by the model is 30%, which is quite satisfactory in view of the considerable number of experiments covering a large range of process conditions and leading to a significant variation of the measured specific surface area values as shown in Fig. 4. In general, the experimental trend is well reproduced by the simulations which are also plotted in Fig. 4. The obtained level of activation energy for TiO 2 sintering can be compared to other values reported in the literature, e.g. Kobata et al. w21x and Xiong and Pratsinis w22x. The reported activation energies are 258 w21x, 31.4 w22x and 148 kJrmol Žthis study.. When the resulting characteristic coalescence times are compared over the entire temperature span, there is a considerable difference. A plausible explanation is that the three studies of TiO 2-synthesis use different reactors and the procedure of obtaining t f also varies. Both Kobata et al. w21x and Xiong and Pratsinis w22x
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Fig. 4. The specific surface area for TiO 2 powder produced in different flames ŽPratsinis et al. w5x. for the experimental conditions specified in Section 2 and Fig. 1. The curves show simulations obtained by fitting the pre-factor and activation energy of the coalescence mechanisms to the experimental data.
Fig. 5. The variation of temperature along the trajectories for flame B Žair flow rates 3.8 lrmin.. The simulations are carried out with five trajectories for each flame.
used furnace reactors, while the present study is based on a diffusion flame reactor. In addition, Kobata et al. w21x measured the sintering rate of TiO 2 for temperatures between 1123 K and 1473 K and Xiong and Pratsinis w22x studied furnace temperatures from 1300 K to 1700 K, whereas temperatures in this study are closer to 2000 K, at least for some of the flames. Therefore, the complicated nature of coalescence of nano-particles is certainly a subject for further study. Phase transformations Žamorphous– anatase–rutile. during the process also make it unlikely that the coalescence can be described by an expression based on a single activation energy. Fig. 2 shows the computed effluent primary particle size for different trajectories for the four depicted flames. Flames C and D yield the smallest specific surface areas and, hence, the largest primary particle sizes. For a given flame, the primary particle size only varies moderately between different trajectories, but for flames C and D it increases towards the symmetry axis, whereas it decreases for flames A and B. The primary particle size is not only determined by the rate of coalescence, but also by the rate of coagulation, since primary particle growth by coalescence cannot proceed beyond the formation of a spherical particle. Hence, diminishing both the rates of coagulation and coalescence is instrumental in increasing the specific surface area. This can be achieved by keeping the precursor concentration low and the time during which the particles are exposed to the high temperature short. This explains the high specific surface area of particles from flames A and B, which have a shorter residence time at the high temperature than those of C and D as seen from trajectory T,t-curves in Fig. 3. Furthermore, the dilution of the particles caused by eddy diffusion is more efficient for the double diffusion flames A and B than for C and D. Dilution obviously diminishes the rate of coagulation, which in some Žouter. regions of the flame becomes the limiting factor in primary particle size.
For flame B2, the five temperature trajectories are shown in Fig. 5 with the height above the burner as an independent variable. Since it is a double diffusion flame, the position of the highest temperature is shifted away from the axis Žtrajectory 5.. Fig. 6 shows the variation of the number of primary particles per aggregate along each trajectory. Especially for the centre trajectory ŽNo. 1., there is an enormous increase of n p just above the burner due to the slow coalescence rate at the initially low temperature. In addition, the innermost trajectory is slowly diluted, which results in a high coagulation rate. The physical interpretation of this is that once the monomers are formed from the decomposition of the precursor, large AamorphousB aggregates of partly coalesced monomers are formed. This can be compared with an initial formation of a large snowflake, which subsequently transforms into an ice cube. The rapid decrease in n p occurs when the temperature increases and initiates coalescence. The coalescence causes n p to drop several orders of magnitude after the peak has been reached. This is related to primary particle growth as seen in Fig. 7. However, close to the
4.3. Simulated eÕolution of particle characteristics along the trajectories
Fig. 6. The variation of the number primary particles per aggregate along the trajectories for flame B Žair flow rates 3.8 lrmin. based on the fitted coalescence parameters.
T. Johannessen et al.r Powder Technology 118 (2001) 242–250
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Fig. 7. The variation of the primary particle size along the trajectories for flame B Žair flow rates 3.8 lrmin. based on the fitted coalescence parameters.
particle collection point Ž12 cm above the burner., n p increases again when the temperature has dropped and the coalescence rate is low. Therefore, the increase in n p is caused by coagulation without coalescence close to the outlet. The concentration of particles decreases and therefore n p increases. A comparison of Figs. 5–7 shows the importance of temperature for controlling the rate of coalescence. The rapid decrease in n p appears to occur when the trajectory temperature exceeds approximately 900 K during the heating period. During the cooling period, the growth of the primary particles Žcf. Fig. 7. stops below a temperature in the range of 1000–1300 K depending on the trajectory number. 4.4. Aggregate sizes: number of primary particles per aggregate Hyeon-Lee et al. w11x used small-angle X-ray scattering ŽSAXS. to study the particleraggregate structure of the flame-synthesized TiO 2-particles by Pratsinis et al. w5x. For the diffusion flame reactor in this study, they report the number of primary particles per aggregate for the data analyzed here. Table 1 summarizes the results: flame A has the highest value of n p , consistent with the fact that the smallest primary particles are found in flame A and that aggregates therefore must consist of numerous smaller
primary particles. For flame B, n p is somewhat smaller, whereas flames C and D produce non-agglomerated, large particles. TEM images of particlesraggregates that illustrate the difference between flame-type can be seen in Pratsinis et al. w5x. Table 1 also shows the predicted values of n p for four ŽA, B, C and D. flames based on the simulations of this study. The predicted values show exactly the same trend as the SAXS data. The actual level of agglomeration computed by the model of course depends on the height above the flame. As an example, the particles in flame C will consist of single particles right above the maximum temperature of the flame but, at 5–10 cm higher above the flame, they will have agglomerated slightly to two or more large particles per agglomerate. However, the relative degree of agglomeration can be compared by normalizing each model prediction with the flame with the lowest degree of agglomeration Ži.e. flame C ´ a factor of 2.2.. The comparison of the normalized predicted values Ž64, 21, 1, 1.3. to the experimental Ž67, 29, 1, 1. is excellent. Therefore, we can conclude that the accuracy of the simulated flow-patterns and mixing effects of feed-streams in the flames is satisfactory. The model results are significantly affected by including dilution in the model Žcf. Eq. Ž3... The last column in Table 1 shows the predicted degree of agglomeration in the flame when dilution in not accounted for. The rate of coagulation is highly overestimated and the model predicts the aggregates with values of n p almost an order of magnitude higher than when dilution is incorporated. However, the simulated specific surface areas are almost unaffected by the dilution effects, since the simulations with dilution included are still in the region of coalescence controlled primary particle growth.
5. Concluding remarks and further research The Lagrangian coagulation–coalescence model for the particle dynamics combined with a CFD simulation model of the flame provide useful means for analyzing particle synthesis in diffusion flames. By fitting an effective sintering rate, the model can reproduce the measured specific
Table 1 Number of primary particles per aggregate, n p , as a function of flame configuration Flame configuration
Data from Hyeon-Lee et al. w11x
Model prediction
Present study Normalized Žwith flame C.
Model prediction without accounting for dilution Žcf. Eq. Ž3..
Flame A Flame B Flame C Flame D
67 29 1 1
140 46 2.2 2.8
63.6 Ž140r2.2. 21 Ž46r2.2. 1 Ž2.2r2.2. 1.3 Ž2.8r2.2.
617 907 23 26
The calculations of the present study are based on an air flow-rate of 3.8 lrmin.
T. Johannessen et al.r Powder Technology 118 (2001) 242–250
surface area with a satisfactory accuracy for a wide range of burner conditions with a fourfold variation of the particle surface area. In addition, the model provides a unique insight into the detailed processes of particle generation. This is useful in the optimization of burner construction and combustion conditions. With great accuracy, the degree of agglomeration in the process can be predicted and it has been shown that including the effect of dilution in the coagulation rate is essential for the model. Considerable further work is needed since the model neglects the influence of particle radiation on the gas temperature, the transport of particles by thermophoresis and changes of particle densities during phase transformations. The inclusion of the first two mechanisms in the model is the subject of the on-going work and, once implemented, it might affect the values of the fitted parameters. With respect to coalescence of nanoparticles, there is an obvious need for improved models. This is clear from the large variation of activation energies for coalescence previously reported from different sources. Since it is known that there are phase transformations from amorphous to anatase and rutile, it might be necessary to study coalescence expressions with different levels of activation energies corresponding to the different solid phases.
Notation Particleraggregate surface area Žm2 . a as Surface area of a completely coalesced Žspherical. aggregate Žm2 . D Particle diffusion coefficient Žm2rs. Aggregate fractal dimension Df dp Primary particle diameter Žm. EA Activation energy ŽJrmol. Dilution factor fD Parameter in expression for the characteristic coak0 lescence time Žmy4 rs. Mass fraction of species i mi Mass fraction of the PM pseudo gas component m PM N Particle number concentration per unit mass of gas Žkgy1 . Number of primary particles per aggregate np p Pressure ŽPa. r Radial coordinate Žm. R The gas constant rc Particleraggregate collision radius Žm. rp Primary particle radius Žm. SA Specific surface area Žm2rg. t Time Žs. T Temperature Parameter in expression for the characteristic coaT0 lescence time, 1400 K TD Decomposition temperature for precursor ŽK. Particleraggregate solid volume Žm3 . Õ Õp Primary particle volume Žm3 .
Greeks a b m r rp tf
249
Fraction of total particle mass flow Collision frequency function Žm3rs. The gas viscosity Žkgrmrs. Gas density Žkgrm3 . Particle density Žkgrm3 . Characteristic coalescence time Žs.
Acknowledgements This work was supported by a grant from the Danish Council for Technological Research.
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w18x W. Bartok, A.F. Sarofim, Fossil Fuel Combustion, Wiley, New York, 1991. w19x B.F. Magnussen, B.H. Hjertager, On mathematical model of turbulent combustion with special emphasis on soot formation and combustion,16th Symp. ŽInt’l. in Combustion, Cambridge, MA, Aug. 15–20, 1976.
w20x K. Christensen, The Formation of Submicron Particles from the Combustion of Straw, PhD thesis, Dept. of Chem. Eng., Technical University of Denmark Ž1995.. w21x A. Kobata, K. Kusakabe, S. Morooka, Growth and transformation of TiO 2 crystallites in aerosol reactor, AiChE J. 37 Ž1991. 347–358. w22x Y. Xiong, S.E. Pratsinis, J. Aerosol Sci. 24 Ž1993. 283–300.
Computational analysis of coagulation and coalescence in the flame synthesis of titania particles Tue Johannessen a,) , Sotiris E. Pratsinis b, Hans Livbjerg a a
Department of Chemical Engineeringr ICAT, Technical UniÕersity of Denmark, DTU, Building 229, DK-2800 Lyngby, Denmark b Institute of Process Engineering, Swiss Federal Institute of Technology, CH-8092 Zurich, Switzerland Received 3 February 2000; received in revised form 19 September 2000
Abstract A method of combining computational fluid dynamics with a mathematical model for the particle dynamics has been applied to simulate experimental data from the synthesis of TiO 2-particles in diffusion flames. Parameters of the coalescence kinetics are estimated by fitting the model predictions to the measured specific surface area of the product particles. The estimated kinetics can be used to predict the surface area and aggregate structure of the particles for a wide range of synthesis conditions. The regular equation for the rate of coagulation is modified to take into account the effect of dilution. The accuracy of the results, especially the degree of aggregation, i.e. the aggregate size, is highly dependent on the inclusion of this effect. When the dilution is accounted for, the predicted aggregate sizes Žnumbers of primary particles per aggregate. compare well with reported data based on small-angle X-ray scattering measurements. q 2001 Elsevier Science B.V. All rights reserved. Keywords: Flame; Titania; Nanoparticle; Coalescence; Simulation; Aggregate
1. Introduction Flame processes are presently used for the manufacture of several nano-structured commodities, e.g. titania catalysts and pigments, carbon black and fumed silica w1x. As our insight into the fundamentals of flame processes improves, the flexibility of these processes and their ability to produce nanoparticles and powders with high specific surface areas are likely to lead to novel, tailor-made materials such as catalysts and membranes. However, to improve the control of particle characteristics, there is a need for accurate modeling and simulation of particle formation. With the improved capacity of computational fluid dynamics ŽCFD. packages, a detailed modeling of the temperature and velocity fields within the flame has come within reach. Integrated with realistic models for the particle dynamics, a complete model for flame-based particle generators can be set up w2x. Recent works on particle modeling in flames have employed a coagulation–coalescence model for particle dynamics combined with either an assumed or measured
)
Corresponding author. Tel.: q45-45-883288; fax: q45-45-932906. E-mail address: [email protected] ŽT. Johannessen..
temperature profile Že.g. Refs. w3,4x.. In the present study, we use the commercial CFD-code FLUENT to simulate the gas temperature and velocity profiles of the burners. Once the flow, temperature and composition fields from the CFD simulations are known, the particle model can be integrated along a number of characteristic trajectories. The model is validated by comparison with experimental results from Pratsinis et al. w5x on the synthesis of TiO 2 in diffusion flames.
2. Experimental The experimental data concern TiO 2 particles synthesized in a laboratory burner by combustion of TiCl 4 in a diffusion flame w5x. The burner is a multiple-concentric-tube burner which is fed by methane, air, and argon gas saturated with TiCl 4 vapor at a controlled temperature. The experimental work consists of an investigation of the specific surface area and morphology of the particles for different flame configurations and flow rates of the feed gases. Four different flame configurations were employed Žcf. Fig. 1. with a constant flow rate of methane, 0.312 lrmin, and a constant flow of argon through the
0032-5910r01r$ - see front matter q 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 2 - 5 9 1 0 Ž 0 0 . 0 0 4 0 1 - 0
T. Johannessen et al.r Powder Technology 118 (2001) 242–250
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3.1. Particle dynamics The modified monodisperse model describes the particle dynamics by the variation of three variables, the number concentration of particles Ž N ., the average surface area Ž a., and solid volume Ž Õ . of the single particlesraggregates. The rate of change of N, a and Õ are affected by coagulation, coalescence and particle dilution caused by dispersion. To include the effect of dilution caused by eddy dispersion in the flame, the original monodisperse model by Kruis et al. w3x was modified by a dilution factor f D calculated from the CFD-data by Johannessen w6x: m PM fD s Ž 2. m PM ,o where m PM is the mass fraction of particles in the gas mixture, and subscript o denotes the conditions at the start of the precursor jet. With the dilution effect included, the rate of change of the particle number concentration and the corresponding growth rate for the volume of one particle, Õ, become w6x: Fig. 1. The four different flame configurations used by Pratsinis et al. w5x.
TiCl 4-bubbler, 0.25 lrmin. Three different air flow rates were applied: 2.5, 3.8 and 5.5 lrmin. For all flames, the TiCl 4 precursor enters the flame through the central nozzle. Flames A and B are double diffusion flames, in which the annular feed port of methane has air on both sides, while C and D are single diffusion flames. Flame A has a comparatively high jet velocity, since the central tube contains both the precursor gas and the combustion air. The effluent particles are recovered on a glass fibre filter and their specific surface area, SA , is measured by multipoint nitrogen adsorption ŽGemini 2360, Micromeritics. at 77 K using the BET equation. Assuming spherical particles, one can calculate a BET equivalent primary particle size, d BET , from: d BET s
6
rp SA
Ž 1.
where rp is the bulk solid density. For further details of the experimental work, the reader is referred to Pratsinis et al. w5x.
3. Theoretical The simulation method comprises two main elements: a computation of the flow and temperature fields in the flame by a commercial CFD software package, and a subsequent integration of a monodisperse, Lagrangian model for the formation and transport of particles within the flow fields. The CFD model of the diffusion flame is presented in detail in Johannessen et al. w2x and Johannessen w6x and shall only be briefly outlined here.
dN
1 N d fD s y rg b N 2 q dt 2 fD dt
Ž 3.
and dÕ
1 s
dt
ž
1
N 2
rg b N 2 Õ s
/
1 2
rg b NÕ
Ž 4.
where N is the number of particles per unit mass of gas, rg is the gas density, b is the collision frequency function and t is the elapsed time along a particle trajectory. The second term on the right-hand side of Eq. Ž3. denotes the rate of concentration change due to the dilution of the central jet by the surrounding particle free gas. The use of a concentration based on mass of gas is expedient because it is invariant to the thermal expansion of the gas. The collision frequency kernel b is calculated by the Fuchs equation w3,7,8x, which covers the whole particle size range from the free molecular to the continuum regime. The particle collision radius, rc , needed for the calculation of b , is equal to the true radius for a solid sphere. The collision radius depends on the fractal dimension when the particles are shaped like dendritic aggregates of coagulated primary particles. The collision radius for an aggregate consisting of n p identical primary particles is calculated by Matsoukas and Friedlander w9x: rc s r p
Õ
ž / Õp
1
1
Df
Df
s rp Ž n p .
Ž 5.
where Õp is the volume of the primary particles, all assumed to be identical. The fractal dimension is assumed to be Df s 1.8, a value often observed for cluster–cluster coagulation w10x. This value was verified also by small-angle X-ray scattering for various flame generated powders
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by Hyeon-Lee et al. w11x including the employed data here by Pratsinis et al. w5x. The net rate of change for the surface area of an aggregate, a, consists of two terms: the rate of area growth due to coagulation and the rate by which the surface area becomes diminished by sintering or coalescence: da
1 s
dt
1
ž
rg b N 2 a y rs Ž a,Õ,T .
/
N 2
Ž 6. dp s
3.2. Coalescence of aggregates The rate of coalescence of a particle, consisting initially of two separate, spherical particles contacting each other, can be approximately described by Koch and Friedlander w12x: rs s
1
Ž a y as .
tf
Ž 7.
where a is the surface area, Õ the volume of the particle and as is the surface area of a solid sphere with the same volume as the sintering particle. The characteristic coalescence time, t f , depends on the sintering mechanism, which may be, e.g. viscous flow, lattice diffusion, surface diffusion or grain boundary diffusion Žcf. e.g. Ref. w13x.. The expressions for t f , however, all yield to the common form:
t f s k 0 d pm
T
exp
T0
EA
1
R
T
ž ž
1 y T0
//
Ž 8.
where EA is an activation energy Že.g. for the diffusion coefficient in the case of diffusive coalescence. and d p is the initial diameter of the two contacting primary particles. The pre-exponential term, k 0 d pm , is the characteristic sintering time at the temperature T0 . The power of d p , m, ranges from 1 to 4 depending on the diffusion mechanism. Several attempts have been made to extend Eq. Ž7. for more complicated shapes w3,14–16x. We use the points-ofcontact ŽPOC. model of Johannessen w6x Žcf. Ref. w2x.. When the particles are relatively compact, this model follows the rate expression of Eq. Ž7.. Aggregates of two or more primary particles are assumed to consist of equally sized, spherical primary particles. Sintering is assumed to be localized to the vicinity of the points of contact between primary particles following Eqs. Ž7. and Ž8.. Therefore, the net rate of surface area reduction for an aggregate is obtained by multiplying the local sintering rate by the number of contact points in the aggregate. The resulting sintering rate expression becomes w2x:
°
Ž ay as .
t Ž r s~ f
d p)
;
.
s
¢Ž
n p y1 .
ž
d p) s
During the coalescence of an aggregate, the primary particle size increases and the number of contact points decreases. The number of primary particles in an aggregate n p , the diameter d p , the surface area a p and the volume Õp of one primary particle are computed from the particle volume Õ and surface area a by straightforward geometrical considerations:
ž
6 Ž Õr2.
p
1r 3
/
for n p F 2
0.41a p Ž 2 a p y a2 P p ,s . s Ž n p y1 . t f Ž dp . t f Ž dp .
/
ž
/
for n p ) 2
Ž 9.
6Õ a
p ,
Õp s
6
d p3 ,
np s
Õ Õp
,
a p s p d p2
Ž 10 .
The number of contact points is assumed equal to n p y 1, which is equivalent to assuming chain-like aggregates. This is a reasonable assumption for aggregates with relatively high fractal dimension Ž Df - 2.. In some cases, the actual number of contact points for n p particles might be higher than n p y 1. A more complex relation between n p and the actual number of contact points might be derived from TEM-images. In the POC model, the coalescence of an aggregate is completely described by the expression for the coalescence of two primary particles and the number of contact points in the aggregate. There are no restrictions on the form of the expression for the characteristic sintering time. The POC model ensures a correct asymptotic behavior for the final stage of coalescence when the whole particle is dense w6x. The transition from AaggregateB to Adense particleB, at n p s 2, occurs without discontinuity in the sintering rate expression Žcf. Eq. Ž9... 3.3. Computational fluid-particle dynamics The flow, temperature and composition fields in the flame are computed by the CFD code, Fluent w17x. A structured grid is made for the computational domain selected to be the space that is confined radially by a coaxial silica glass shield surrounding the flame and extending axially from 1 cm below the burner mouth to 15 cm above. The flame model comprises the time-smoothed equations of change together with the differential Reynolds stress turbulence model with nonequilibrium wall functions. The rate of methane combustion is calculated as the limiting value of a regular Arrhenius expression for the single-step methane combustion w18x and the rate calculated by the Magnussen and Hjertager w19x turbulent combustion model. Radiant energy absorption and irradiation are incorporated in the enthalpy balance. In the primary CFD simulations, the transport of particles within the flame is computed like that of a pseudo gas component, PM, which has been assigned a negligibly small molecular diffusivity. Hence, the calculated variation of the PM concentration within the flame is identical to that of the particles, assuming that the transport of particles occurs solely by convective flow and eddy diffusion. This is reasonable, since the eddy diffusion rate of particles by far exceeds their Brownian diffusion rate. Turbulent eddies
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will transport particles in ensembles. Therefore, the rate of coagulation is still dominated by Brownian motion within the eddy. The influence of turbulence on the coagulation rate is insignificant for sub-micron particles in systems with little or moderate degree of turbulence w20x. The CFD-model of the diffusion burner is similar to that of Johannessen et al. w2x, who provide further details about the model calculations. The simulation results are exported to a separate Fortran code, which performs the following calculations. The flow of particulate matter is divided into N characteristic trajectories following the mean flow of particulate matter as computed for the pseudo gas component PM. In this way, the path of the trajectories is determined by both convective gas flow and lateral eddy diffusion, the influence of which is significant. The values of temperature, gas velocity, dilution ŽEq. Ž2.., and time elapsed for the particles are mapped from the CFD simulation results onto each point of the trajectories. The Lagrangian particle model of Eqs. Ž3., Ž4. and Ž6. is integrated along each trajectory and the primary particle size, the number of primary particles per aggregate and the aggregate collision cross section are calculated as a function of distance above the burner to the outlet of the process w2x. The following assumptions and approximations are made for the simulations: v
The oxidation of TiCl 4 is assumed to occur instantaneously when the gas temperature exceeds a given value TD , converting all precursor molecules to free TiO 2 AmonomerB molecules.
v
v
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The subsequent formation and growth of particles occur by coagulation caused by Brownian collisions between particles with the monomers as starting point and leading to a steadily growing average particle size. The notion of a collision limited growth, even for the monomers, is based on the assumption that dimeric and larger clusters are stable—a plausible assumption due to the extreme supersaturation of the monomers at the time of their formation.
4. Results and discussion 4.1. Flame structures and temperature contours For the various flame configurations, the difference in flame structure is clearly visible in the computed temperature contours for flames A, B, C and D Žair flow rate s 3.8 lrmin. shown in Fig. 2. For all four flame types, the precursor enters in the central tube. However, flames A and B are double diffusion flames with a reaction front between the methane and air on the innermost side and between the methane and the external, still air. Flames C and D only have a single flame front between the methane and the air in the outermost annulus. Flames C and D have almost identical appearances. The only difference is a slightly higher velocity in the central tube for flame D since the methane and precursor-gas flow in the central tube. The appearance of flame A is evidently quite different from the other flames due to its much higher velocity in
Fig. 2. Temperature contour plots for the flames A, B, C and D Žair flow rate s 3.8 lrmin.. Two particle trajectories, enveloping 10% and 90% of the total particle flow, respectively, are superimposed on each contour plot. The right part of the figure shows simulations of the outlet primary particle size of each trajectory of the flames on the left-hand side.
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the central jet, which contains the mixture of both the air and the precursor gas. The maximum flame temperature is located on the jet axis for flames C and D, but off the axis for A and B. Molecular and especially turbulent diffusion is responsible for the dispersion of particulate matter in the radial direction. Fig. 2 shows, superimposed on the temperature contours, the trajectories enveloping 10% and 90%, respectively, of the total particle flow. Evidently, trajectories close to the axis have different time-temperature histories than the peripheral ones. Fig. 3 shows the temperature as a function of residence time along the centre axis for flames A through D. The high linear gas velocity in the central tube of flame A yields short residence times for the A-trajectory as well as a low T-maximum. Flames B, C and D all have comparable residence times, but only C and D have a high T-maximum at the centre-line. From Fig. 2, we can see that the trajectories with higher temperatures are located away from the centre-line for flames A and B, whereas the opposite is the case for C and D. 4.2. Least-squares fitting of the specific surface area For all experimental conditions, CFD simulations are carried out and read into the Fortran code. For each case, five trajectories are computed and the corresponding values of residence time, temperature and dilution factor are stored for the Lagrangian integrations.
Initial values of the coalescence parameters are assigned in the programme, which subsequently performs a multiparameter fit by minimizing the sum of squared differences between the simulated and measured specific surface area, i.e.: 2
SSQ s Ý Ž ln Ž SA ,meas . y ln Ž SA ,model . . i
Ž 11 .
i
The logarithmic differences are used to ensure that experimental points with highly different values of the specific surface area are equally weighted in the fitting procedure. The initial conditions for the particle volume, surface area and number concentration are calculated from the assumption that the solid oxide initially consists of TiO 2 monomers, the size of which is computed from the molecular volume of titania. The precursor vapours are assumed to form TiO 2 monomers instantly at a given decomposition temperature, TD s 340 K, approximately 20 K above the inlet temperature. The value of TD affects the final particle characteristics insignificantly, but the chosen value minimizes SSQ of Eq. Ž11.. It is plausible that the decomposition of the precursor takes place downstream from the jet opening where water and oxygen are present and where the temperature has increased. By fitting the model to the measured specific surface areas, the following values of the two coalescence parameters k 0 and EA of Eq. Ž8. are estimated: k 0 s Ž 5.5 " 7.5 . = 10 27 my4 s and EA s 148 " 35 kJrmol
Fig. 3. Simulated temperature along the centre symmetry axis of the titania flames A, B, C and D Žair flow rates 3.8 lrmin. shows the influence of flame configuration on the timertemperature history. The double diffusion flames A and B have much lower centre-line temperatures and in addition, flame A has a very short residence time caused by air and carrier-gas being present in the jet-tube. The single diffusion flames, C and D, both show high centerline temperature maximum.
These estimates are based on the following values of the constants in Eq. Ž8.: m s 4 and T0 s 1400 K. The estimated standard errors shown with the parameter estimates are rather large, due to some correlation between the two parameters. However, the estimated standard error in predicting the specific surface areas by the model is 30%, which is quite satisfactory in view of the considerable number of experiments covering a large range of process conditions and leading to a significant variation of the measured specific surface area values as shown in Fig. 4. In general, the experimental trend is well reproduced by the simulations which are also plotted in Fig. 4. The obtained level of activation energy for TiO 2 sintering can be compared to other values reported in the literature, e.g. Kobata et al. w21x and Xiong and Pratsinis w22x. The reported activation energies are 258 w21x, 31.4 w22x and 148 kJrmol Žthis study.. When the resulting characteristic coalescence times are compared over the entire temperature span, there is a considerable difference. A plausible explanation is that the three studies of TiO 2-synthesis use different reactors and the procedure of obtaining t f also varies. Both Kobata et al. w21x and Xiong and Pratsinis w22x
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Fig. 4. The specific surface area for TiO 2 powder produced in different flames ŽPratsinis et al. w5x. for the experimental conditions specified in Section 2 and Fig. 1. The curves show simulations obtained by fitting the pre-factor and activation energy of the coalescence mechanisms to the experimental data.
Fig. 5. The variation of temperature along the trajectories for flame B Žair flow rates 3.8 lrmin.. The simulations are carried out with five trajectories for each flame.
used furnace reactors, while the present study is based on a diffusion flame reactor. In addition, Kobata et al. w21x measured the sintering rate of TiO 2 for temperatures between 1123 K and 1473 K and Xiong and Pratsinis w22x studied furnace temperatures from 1300 K to 1700 K, whereas temperatures in this study are closer to 2000 K, at least for some of the flames. Therefore, the complicated nature of coalescence of nano-particles is certainly a subject for further study. Phase transformations Žamorphous– anatase–rutile. during the process also make it unlikely that the coalescence can be described by an expression based on a single activation energy. Fig. 2 shows the computed effluent primary particle size for different trajectories for the four depicted flames. Flames C and D yield the smallest specific surface areas and, hence, the largest primary particle sizes. For a given flame, the primary particle size only varies moderately between different trajectories, but for flames C and D it increases towards the symmetry axis, whereas it decreases for flames A and B. The primary particle size is not only determined by the rate of coalescence, but also by the rate of coagulation, since primary particle growth by coalescence cannot proceed beyond the formation of a spherical particle. Hence, diminishing both the rates of coagulation and coalescence is instrumental in increasing the specific surface area. This can be achieved by keeping the precursor concentration low and the time during which the particles are exposed to the high temperature short. This explains the high specific surface area of particles from flames A and B, which have a shorter residence time at the high temperature than those of C and D as seen from trajectory T,t-curves in Fig. 3. Furthermore, the dilution of the particles caused by eddy diffusion is more efficient for the double diffusion flames A and B than for C and D. Dilution obviously diminishes the rate of coagulation, which in some Žouter. regions of the flame becomes the limiting factor in primary particle size.
For flame B2, the five temperature trajectories are shown in Fig. 5 with the height above the burner as an independent variable. Since it is a double diffusion flame, the position of the highest temperature is shifted away from the axis Žtrajectory 5.. Fig. 6 shows the variation of the number of primary particles per aggregate along each trajectory. Especially for the centre trajectory ŽNo. 1., there is an enormous increase of n p just above the burner due to the slow coalescence rate at the initially low temperature. In addition, the innermost trajectory is slowly diluted, which results in a high coagulation rate. The physical interpretation of this is that once the monomers are formed from the decomposition of the precursor, large AamorphousB aggregates of partly coalesced monomers are formed. This can be compared with an initial formation of a large snowflake, which subsequently transforms into an ice cube. The rapid decrease in n p occurs when the temperature increases and initiates coalescence. The coalescence causes n p to drop several orders of magnitude after the peak has been reached. This is related to primary particle growth as seen in Fig. 7. However, close to the
4.3. Simulated eÕolution of particle characteristics along the trajectories
Fig. 6. The variation of the number primary particles per aggregate along the trajectories for flame B Žair flow rates 3.8 lrmin. based on the fitted coalescence parameters.
T. Johannessen et al.r Powder Technology 118 (2001) 242–250
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Fig. 7. The variation of the primary particle size along the trajectories for flame B Žair flow rates 3.8 lrmin. based on the fitted coalescence parameters.
particle collection point Ž12 cm above the burner., n p increases again when the temperature has dropped and the coalescence rate is low. Therefore, the increase in n p is caused by coagulation without coalescence close to the outlet. The concentration of particles decreases and therefore n p increases. A comparison of Figs. 5–7 shows the importance of temperature for controlling the rate of coalescence. The rapid decrease in n p appears to occur when the trajectory temperature exceeds approximately 900 K during the heating period. During the cooling period, the growth of the primary particles Žcf. Fig. 7. stops below a temperature in the range of 1000–1300 K depending on the trajectory number. 4.4. Aggregate sizes: number of primary particles per aggregate Hyeon-Lee et al. w11x used small-angle X-ray scattering ŽSAXS. to study the particleraggregate structure of the flame-synthesized TiO 2-particles by Pratsinis et al. w5x. For the diffusion flame reactor in this study, they report the number of primary particles per aggregate for the data analyzed here. Table 1 summarizes the results: flame A has the highest value of n p , consistent with the fact that the smallest primary particles are found in flame A and that aggregates therefore must consist of numerous smaller
primary particles. For flame B, n p is somewhat smaller, whereas flames C and D produce non-agglomerated, large particles. TEM images of particlesraggregates that illustrate the difference between flame-type can be seen in Pratsinis et al. w5x. Table 1 also shows the predicted values of n p for four ŽA, B, C and D. flames based on the simulations of this study. The predicted values show exactly the same trend as the SAXS data. The actual level of agglomeration computed by the model of course depends on the height above the flame. As an example, the particles in flame C will consist of single particles right above the maximum temperature of the flame but, at 5–10 cm higher above the flame, they will have agglomerated slightly to two or more large particles per agglomerate. However, the relative degree of agglomeration can be compared by normalizing each model prediction with the flame with the lowest degree of agglomeration Ži.e. flame C ´ a factor of 2.2.. The comparison of the normalized predicted values Ž64, 21, 1, 1.3. to the experimental Ž67, 29, 1, 1. is excellent. Therefore, we can conclude that the accuracy of the simulated flow-patterns and mixing effects of feed-streams in the flames is satisfactory. The model results are significantly affected by including dilution in the model Žcf. Eq. Ž3... The last column in Table 1 shows the predicted degree of agglomeration in the flame when dilution in not accounted for. The rate of coagulation is highly overestimated and the model predicts the aggregates with values of n p almost an order of magnitude higher than when dilution is incorporated. However, the simulated specific surface areas are almost unaffected by the dilution effects, since the simulations with dilution included are still in the region of coalescence controlled primary particle growth.
5. Concluding remarks and further research The Lagrangian coagulation–coalescence model for the particle dynamics combined with a CFD simulation model of the flame provide useful means for analyzing particle synthesis in diffusion flames. By fitting an effective sintering rate, the model can reproduce the measured specific
Table 1 Number of primary particles per aggregate, n p , as a function of flame configuration Flame configuration
Data from Hyeon-Lee et al. w11x
Model prediction
Present study Normalized Žwith flame C.
Model prediction without accounting for dilution Žcf. Eq. Ž3..
Flame A Flame B Flame C Flame D
67 29 1 1
140 46 2.2 2.8
63.6 Ž140r2.2. 21 Ž46r2.2. 1 Ž2.2r2.2. 1.3 Ž2.8r2.2.
617 907 23 26
The calculations of the present study are based on an air flow-rate of 3.8 lrmin.
T. Johannessen et al.r Powder Technology 118 (2001) 242–250
surface area with a satisfactory accuracy for a wide range of burner conditions with a fourfold variation of the particle surface area. In addition, the model provides a unique insight into the detailed processes of particle generation. This is useful in the optimization of burner construction and combustion conditions. With great accuracy, the degree of agglomeration in the process can be predicted and it has been shown that including the effect of dilution in the coagulation rate is essential for the model. Considerable further work is needed since the model neglects the influence of particle radiation on the gas temperature, the transport of particles by thermophoresis and changes of particle densities during phase transformations. The inclusion of the first two mechanisms in the model is the subject of the on-going work and, once implemented, it might affect the values of the fitted parameters. With respect to coalescence of nanoparticles, there is an obvious need for improved models. This is clear from the large variation of activation energies for coalescence previously reported from different sources. Since it is known that there are phase transformations from amorphous to anatase and rutile, it might be necessary to study coalescence expressions with different levels of activation energies corresponding to the different solid phases.
Notation Particleraggregate surface area Žm2 . a as Surface area of a completely coalesced Žspherical. aggregate Žm2 . D Particle diffusion coefficient Žm2rs. Aggregate fractal dimension Df dp Primary particle diameter Žm. EA Activation energy ŽJrmol. Dilution factor fD Parameter in expression for the characteristic coak0 lescence time Žmy4 rs. Mass fraction of species i mi Mass fraction of the PM pseudo gas component m PM N Particle number concentration per unit mass of gas Žkgy1 . Number of primary particles per aggregate np p Pressure ŽPa. r Radial coordinate Žm. R The gas constant rc Particleraggregate collision radius Žm. rp Primary particle radius Žm. SA Specific surface area Žm2rg. t Time Žs. T Temperature Parameter in expression for the characteristic coaT0 lescence time, 1400 K TD Decomposition temperature for precursor ŽK. Particleraggregate solid volume Žm3 . Õ Õp Primary particle volume Žm3 .
Greeks a b m r rp tf
249
Fraction of total particle mass flow Collision frequency function Žm3rs. The gas viscosity Žkgrmrs. Gas density Žkgrm3 . Particle density Žkgrm3 . Characteristic coalescence time Žs.
Acknowledgements This work was supported by a grant from the Danish Council for Technological Research.
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