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Droplet and Particle Dynamics during Flame Spray Synthesis of Nanoparticles† Martin C. Heine and Sotiris E. Pratsinis* Particle Technology Laboratory, Institute of Process Engineering, Department of Mechanical and Process Engineering, ETH Zu¨ rich, 8092 Zu¨ rich, Switzerland
Particle formation by flame spray pyrolysis is investigated theoretically and compared with particle size distribution data of ZrO2 production at 100-300 g/h. Spray droplet size distributions as well as droplet and gas velocities are measured by phase Doppler anemometry along the axis and radius of the solid-cone-type spray. Multicomponent droplet combustion is simulated, accounting for droplet polydispersity, transport, and evaporation. The dynamics of the product particle size distributions are investigated, accounting for coagulation and sintering by population balance equations. The effect of the initial droplet size and liquid feed rate on the evolution of the product particle size and polydispersity is analyzed. Introduction Flame spray pyrolysis (FSP) is one of the most promising techniques for synthesis of a broad spectrum of sophisticated inorganic nanoparticles such as complex mixed oxides and spinels,1 lasing materials,1 nanoZrO2,2 nano-CeO2,3,4 ZnO quantum dots,5 Pt/Al2O3 catalysts,6 and even metal alloys.7 FSP overcomes the limitations of gaseous precursors that are required by the industrially established gas-fed flame reactors for the synthesis of nanostructured commodities.8 As a result, FSP is used by a number of industrial and academic laboratories throughout the world. In FSP, the precursor is dissolved in a fuel of solvent (e.g., ethanol and toluene) that is dispersed through a nozzle. The resulting solid-cone spray is ignited and sustained by a pilot flame. Particles are made with the spray flame, where droplet evaporation, combustion, aerosol formation, coagulation, sintering, and even surface growth occur in parallel.9 Though some systematic studies on understanding the role of the FSP process parameters in the characteristics of product particles have been carried out9 and even its modest scale-up to kilograms per hour has been reported,10 this promising technique has been treated rather empirically so far. Essentially little is known on how the multicomponent FSP droplets are converted to particles because in situ measurement of the flame and particle characteristics is rather challenging in these droplet-particleladen flames. Such data are needed to understand some of the intriguing FSP results such as the formation of solid or hollow Al2O3 by switching the dispersionoxidant gas from oxygen to air11 and homogeneous or inhomogeneous nano-Bi2O3 or CeO2 by controlling the composition of the FSP precursor solution.3,12 Before even this, however, a quantitative understanding the role of droplet evaporation on the solid product particle size is needed for FSP process design and operation. Recently, by accounting for coagulation and sintering during FSP and through neglect of the particle poly† In honor of the 60th birthday of Prof. Milorad (Mike) P. Dudukovic. * To whom correspondence should be addressed. Tel.: +41-1-632-3180. Fax: +41-1-632-1595. E-mail: pratsinis@ ptl.mavt.ethz.ch.
Figure 1. Experimental setup for droplet size and velocity measurements by PDA in the FSP process for the synthesis of ZrO2 nanoparticles.
dispersity and droplet evaporation, a good agreement between the predicted and measured average ZrO2 particle diameters was obtained.13 Here a detailed droplet-particle population balance model is developed at nonisothermal conditions. In that way, the role of droplet evaporation in the characteristics of product particles is investigated. Phase Doppler anemometry (PDA) is used to measure gas and droplet velocities and droplet size distributions. Model predictions are evaluated by a detailed comparison to the evolution of the measured particle size distributions. Experimental Section Figure 1 shows the setup of the particle-producing spray flame and the two-dimensional phase Doppler anemometer (TSI Inc., Seattle, WA). The PDA operated with an argon ion laser (Innova 70) is used for measuring droplet size distributions and velocities in the flame spray. The fiber-optic-based system was configured for first-order refracted light with the receiver located 30° off the transmitter axis and operated at approximately 100 mW measured at the probe volume. The droplet
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detection range is between 1 and 100 µm. At each measurement location, about 40 000 droplets are recorded. Small droplets are present in the investigated flames at most locations, making additional tracer particles obsolete. Stokes number analysis shows that droplets with a diameter of 4 µm have a response time of 10 µs,14 which is small enough to follow the gas-phase flow accurately.15 Droplets smaller than 4 µm are filtered from the PDA raw data to obtain the flame gas velocity. The apparatus shown in Figure 1 consists of a stainless steel external-mixing gas-assisted nozzle (970/ 4-S32; Schlick-Du¨sen, Gustav Schlick GmbH + Co., Germany). The liquid precursor flows through the inner capillary tube (i.d. ) 0.5 mm) and is dispersed by 50 L/min of O2 that passes through the annular gap, which is adjusted to a pressure drop of 1 bar across the nozzle tip. The nozzle is surrounded by two stainless steel annuli having inner and outer diameters of 9-9.5 and 10-12.5 mm, respectively. These annuli form a diffusion flame when 2 L/min of CH4 flows through the inner annulus and 4.5 L/min of O2 flows through the outer annulus. Additional sheath O2 (15 L/min) is fed through a sintered-metal plate ring with inner-outer diameters of 28-50 mm surrounding the previous outer annulus. Precursor zirconium n-propoxide [ZP, Zr(C3H7O)4, 70 wt % in n-propanol] is dissolved in ethanol, resulting in precursor solutions of 0.5 M that are fed to the nozzle at 27.1 (flame 1) and 81.1 mL/min (flame 2), resulting in ZrO2 production rates of 100 and 300 g/h. Particles are selected with a commercial jet filter (FRR 4/1.2; Friedli AG, Oberburg, Germany) containing four poly(tetrafluoroethylene) (PTFE, Teflon) coated Nomex baghouse filters, which are cleaned periodically by airpressure shocks.13 Temperature-dependent liquid- and gas-phase properties of ethanol and propanol are used.16 The vapor pressure of ZP is determined with the ClausiusClapeyron equation17 for a constant evaporation enthalpy of 83.7 kg/mol18 and a normal boiling temperature of 208 °C. Because not all properties of ZP are available in the literature, the liquid- and gas-phase heat capacities as well as heat conductivity and viscosity in the gas phase are estimated by using the corresponding properties of dodecane.16 Because of the small molar fraction of ZP, inaccuracies in these four properties hardly affect subsequent calculations. Theory Single-Component Droplet Evaporation. Once the spray is formed, single-component droplet evaporation is calculated following work by Abramson and Sirignano,19 which is accurate even for high evaporation rates because it accounts for the Stefan flow effect on the mass transfer and a nonunity Lewis number. The heat and mass transfer are described by modified Nusselt and Sherwood numbers. Assuming spherical, noninteracting droplets at all times and neglecting the gas solubility in the droplet, gas-phase heat and mass transfer are in a quasi steady state, heat radiation is negligible, and chemical reactions are absent within the droplet boundary layer.19 Assuming a uniform droplet temperature, Td, it changes as
dTd Q˙ d ) dt C h m pF
(1) d
The change in the droplet position is given by
dz/dt ) ud
(2)
while the droplet velocity is
dud 3 CD Fg,∞ ) |u - ud|(ug,∞ - ud) dt 4 dd Fd g,∞
(3)
consistent with work of Faeth.20 Note that eq 3 differs from the original expression of Abramson and Sirignano19 by a 1/4 factor.21 The change of the droplet diameter used by Abramson and Sirignano19 is extended to account for a temperature-dependent liquid density:22
dd ∂Fd dTd ddd 2m ˘ )2 dt 3F πF d d ∂Td dt
(4)
d d
The changes in droplet mass and heat transport are
h gddSh* ln(1 + BM) m ˘ ) πFjgD ˘ Q˙ d ) m
[
C h pF(T∞ - Td) - ∆HV BT
]
(5) (6)
with
yFs - yF∞ Sh0 - 2 ; BM ) FM 1 - yFs
(7)
C h pF(T∞ - Ts) Nu0 - 2 ; BT ) FT ∆HV + Q˙ d/m ˘
(8)
Sh* ) 2 +
Nu* ) 2 +
F(B) ) (1 + B)0.7
pFsMW ln(1 + B) (9) ; yFs ) B piMWi
∑i
Multicomponent Evaporation of a ZP Precursor Solution. The above single-component model is extended to multicomponent droplet evaporation, which is commonly encountered during FSP. Each component vaporizes at a different rate, creating concentration gradients inside the droplet. When the characteristic time for liquid-phase mass diffusion, τM, is much smaller than the droplet lifetime, τd, the most volatile component inside the droplet will evaporate rapidly and the concentrations of the other components will increase. Once the most volatile component evaporated, the next most volatile one evaporates, and so on. For τM . τd, evaporation is so fast that steep concentration gradients will form in a small boundary layer close to the droplet surface while diffusion in the bulk liquid is negligible. As a result, the overall droplet composition will not change during evaporation.21 The liquid-phase diffusion coefficient of species i is23
Di,liq ) kbT/6πµi,liqRgyr
(10)
while the liquid-phase diffusivity of a mixture Di,j can be estimated with the Darken equation17 from the pure solvent diffusivities by weighting them with their mole fractions. For a characteristic diffusion length of half a droplet diameter, the characteristic diffusion time is then given by
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τM ) (d/2)2/Di,j
(11)
For τM . τd, the evaporating mass flux has the same relative droplet surface composition (xi,s) as the bulk liquid (xi,d), which can be expressed as21
xi,s ) xi,dpfuel/ptot
(12)
where pfuel is the sum of all droplet species vapor pressure in the gas phase at the droplet surface. By using an average Spalding mass-transfer number, B h M, and an average thermal conductivity, λhg, the total vaporization rate, m ˘ tot, can be determined21 from eq 5. The change of mass of each droplet component is given by
˘ tot dmd,i/dt ) -xi,sm
(13)
dd )
( ) md,i
∑ π i F
dt
1/3
with
∫0∞q(z) dz ) 1
Ni
dA
where md,0 is the initial droplet mass. For a gas-phase reaction (e.g., combustion) faster than droplet evaporation, the gas-phase concentration of the evaporating species is negligible. New product (e.g., zirconia) molecules are formed from precursor (e.g., ZP) molecules and are treated as the smallest stable particles. By using eq 15, the particle formation rate can be described for the particle number, I˙ N,1, and surface area, I˙ A,1, for a given liquid precursor mass flow rate, m ˘ d, fed into the flame:
dt
(16)
I˙ A,1(z) ) amI˙ N,1(z)
(17)
where am is the surface area of a product molecule and NA is Avogadro’s constant. Similar to eq 16, the overall stoichiometry of the combusting precursor components
ZP: C12H28O4Zr + 18O2 ) ZrO2 + 12CO2 + 14H2O Propanol: C3H8O + 4.5O2 ) 3CO2 + 4H2O Ethanol: C2H6O + 3O2 ) 2CO2 + 3H2O
(18)
is used to calculate the species number flux of O2 and reaction products H2O and CO2 as well as the gas-phase mass and volume fluxes to obtain the gas composition and mass flow rate.
imax
2 ∑ j)1
j-i
βi,jNj - Ni
βi,jNj ∑ j)i
(19)
where Ni is the particle number density (no./m3) and β is the collision frequency function (m3/no./s) for Brownian coagulation given by the Fuchs interpolation function from the free-molecular regime to the continuum regime.28 For coagulation of agglomerates, a fractal dimension of Df ) 1.8 is used.29 The change in the particle (agglomerate) surface area concentration accounting for coagulation and sintering is (see the Appendix):
(15)
dm ˘ d ωZPNA m ˘ dq(z) ωZPNA ) dz MWZP MWZP
1
2j-i+1βi-1,jNj + βi-1,i-1Ni-12 ∑ 2 j)1 i-1
d,i
md(z+∆z) - md(z) ∆zf0 md,0∆z
I˙ N,1(z) ) -
i
i-2
) Ni-1
(14)
Precursor Conversion and Particle Formation. Once the liquid jet is completely atomized, the evaporating precursor mass flux and composition can be calculated as a function of the height above the nozzle, z, by neglecting further droplet breakup or collisions. The density function of the normalized mass evaporation rate of the liquid, q (1/m), is defined as
q(z) ) - lim
|
dN
while the resulting droplet diameter is
6
Particle Dynamics. Ceramic particle dynamics in flames are determined typically by coagulation and sintering.8 The evolution of the size distribution is determined, in principle, by following, at least two particle properties, e.g., volume and area.24 These calculations, however, tend to be quite demanding,25 so simplified solutions are sought. A sectionalization of the size distribution with a geometric volume spacing of 2 gives solutions to the coagulation equations that are good compromises between computational speed and accuracy.26 For example, the error in the geometric standard deviation of the asymptotic self-preserving distribution for Brownian coagulation is only about 3%.27 In that way, the change of the particle number concentration is26
|
i-2
) i
βi-1,jwi,j(AjNi-1 + NjAi-1) + ∑ j)1 i-1
βi-1,i-1Ai-1Ni-1 imax
(1 - wi+1,j)AjNi] - Ai
βi,j[wi+1,jNjAi ∑ j)1 1
βi,jNj ∑ τ j)i
(Ai - NiRi,s) (20)
s,i
where Ai is the total agglomerate surface area in section i and w accounts for the degree of sintering between two colliding agglomerates i and j. Equation 19 is based on the gas volume that is not conserved during the process. Evaporation of the liquid precursor, chemical reaction, air entrainment, and particle condensation changes the gas composition and mass and volume fluxes. Furthermore, the strong temperature gradients change the gas volume, as well. To account for this, the equations are expressed in terms of the number flux by relating Ni to the gas volume flux N˙ i ) NiV˙ gas (no./s) and to the surface area flux A˙ i ) AiV˙ gas (m2/s), while the collision frequency is adapted by β˙ i,j ) βi,j/V˙ gas (1/no.). Particle formation by chemical reaction I˙ N,i and I˙ A,i are determined from the droplet evaporation equations (16) and (17) and are added to the right-hand side of eqs 19 and 20, respectively. In aerosol reactors where the gas velocity is nonuniform, it is more convenient to carry out the calculation in space, not in time. The chain rule can be used: d/dt ) (d/dz) (dz/dt) ) (d/dz) cgas, where cgas is the gas velocity to convert the time, t, to distance in eqs 19 and 20.
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The collision diameter of an agglomerate is given by Kruis et al.:29
dc,i ) dp,i(vi/vp,i)1/Df
dp,i ) 6vi/ai (21)
and
where vp,i (m3) is the volume of a spherical primary particle in section i while vi and ai are the agglomerate volume and surface area, respectively. The numberbased geometric standard deviation of a particle size distribution is defined according to Vemury et al.27 To be consistent with experimental results from thermophoretic sampling and transmission electron microscopy (TEM) image analysis,13 particles smaller than 1.6 nm are neglected. Following Mueller et al.,13 for ZrO2 the characteristic sintering time for grain boundary diffusion is30,31
τs ) 0.001703
RT(dp/2)4 wDbγΩ
(22)
where R is the universal gas constant, w ) 5 × 10-10 m is the grain boundary width,31 γ is the surface tension,32 Ω ) 2.019 98 × 10-5 m3/mol is the molar volume of zirconia, and Db is the grain boundary diffusion coefficient for ZrO2:
Db ) D0 exp(-E/RT)
(23)
where D0 is the grain boundary diffusion preexponential factor and E is the activation energy: D0 ) 9.73 × 10-7 m2/s33 and E ) 2.33 × 105 J/mol. The predictions of the sectional population balance model will be discussed in terms of the Sauter mean particle diameter:14 imax
Nidi3 ∑ i)1
dp )
(24)
imax
Nidi ∑ i)1
2
and the geometric number-based standard deviation defined as34
{[
σgn ) exp
∑ i)1
]} 1/2
imax
Ni(ln di - ln dgn)2 imax
(
Ni) - 1 ∑ i)1
(25)
where dgn is the geometric number mean diameter: imax
ln dgn )
Ni ln di ∑ i)1 imax
(26)
Ni ∑ i)1
Results and Discussion Validation of the Model. The droplet evaporation model was validated by comparing the numerical solution from eqs 1-3 and 13 with Figure 2a-g of Abramson
Figure 2. Gas velocity profiles of spray flame 1 (triangles) and flame 2 (circles) measured along the centerline by PDA and the corresponding lines of regression (flame 1, solid line; flame 2, broken line).
and Sirignano.19 To validate the particle dynamics model, the standard conditions in Table 2 of Kruis et al.29 were used and compared to the results summarized in Figure 3 by Tsantilis et al.35 Excellent agreement was found for both droplet simulations and particle dynamics. Droplet Dynamics. Multicomponent droplet evaporation depends on the characteristic time scales for the liquid-phase diffusion, τM, and droplet lifetime, τd. For flame 1, a 10-µm droplet has a characteristic diffusion time of τM ) 5 ms compared to a droplet lifetime of τd ) 0.1 µs. For droplets in the range of 50 µm, liquid-phase diffusion is still about 1 order of magnitude slower than evaporation. For flame 2 (Table 1), temperatures are higher, leading to faster evaporation so that the ratio τM/τd increases further. These estimates show that liquid-phase diffusion in the interior of the droplet is essentially negligible, consistent with work by Lasheras et al.36 Figure 2 shows the PDA-measured gas velocity profiles along the centerline for flame 1 (triangles) and flame 2 (circles). Regression lines are fitted through the measured axial velocities for flame 1 (solid line) and flame 2 (broken line) and are used for the subsequent simulations. Increasing the liquid feed rate from 27.1 mL/min (flame 1) to 81.1 mL/min (flame 2) reduces the axial velocity at z ) 0.5 cm by 25 m/s because more liquid has to be accelerated by the 50 L/min dispersion gas. Up to z ) 1.5 cm, the gas velocity increases, reaching maxima of 135 and 107 m/s for flames 1 and 2, respectively. This increase is probably caused by the expansion of the gas volume from precursor combustion. Above z ) 1.5 cm, the axial velocity of flame 1 decreases, showing an asymptotic decay of approximately 1/x, as is typically observed for turbulent cold jets.37 At z ) 4.5 cm, in flame 1 a gas velocity of 68 m/s was measured at a data rate of 5 Hz compared to the maximum value of 24 kHz at z ) 0.75 cm, indicating that most droplets have evaporated at this point, consistent with Oh et al.,38 who showed for a 17.8 mL/min kerosene spray atomized with 32 L/min air that after z ) 5 cm about 85% of the droplets were evaporated. At z ) 5-7 cm, only 500 instead of 40 000 droplets were measured at data rates below 1 Hz. At 7 cm, the lowest velocity of 45.7 m/s was measured. From this point on, the regression line was extrapolated using the above asymptotic
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Figure 4. Radial distribution of the Sauter mean droplet diameter and the droplet data rate at z ) 2.5 cm (solid lines) and z ) 5 cm (broken lines) showing the solid-cone spray of flame 2.
Figure 3. (a) Droplet Sauter mean diameter evolution of spray flame 1 (triangles) and flame 2 (circles) measured along the centerline by PDA. The inset shows the corresponding droplet mass distributions for both flames at z ) 1 cm. (b) Evolution of the centerline droplet mass distribution along the axis of flame 2 from z ) 2 to 5 cm. Table 1. PDA and FSP Process Conditions production rate [g/h] precursor flow rate [mL/min] dispersion gas flow rate [L/min] maximum flame velocity [m/s] visible flame height [cm] maximum height for PDA data rate > 1 kHz [cm] maximum height for PDA measurements [cm] maximum temperature13 [K]
flame 1
flame 2
100 27.1 50 135 13.0 2.0
300 81.1 50 107 37.0 7.5
7.0
27.5
2765
2610
assumption. Primary particle growth stops13 at about z ) 10 cm, as is discussed later in Figures 8-11, so inaccuracies in the extrapolated gas velocity hardly affect the particle calculations. For flame 2, the gas velocity decreased more slowly and almost leveled off at about 65 m/s before the typical asymptotic decay can be observed above z ) 10 cm. The gas velocity decreases with increasing liquid flow rate because the same moments of the constant 50 L/min dispersion gas suspend more liquid. Figure 3a compares the measured Sauter mean droplet diameter, dd, evolution along the axis of flame 1 (triangles) and flame 2 (circles). For both sprays, two droplet zones can be distinguished. Up to z ) 1 cm
(breakup zone),39 both dd values decrease and reach minima of 15.4 and 21.5 µm respectively for flames 1 and 2 because liquid ligaments break up to form single droplets. Within this zone, droplets might be nonspherical, especially at z ) 0.5 cm, which is closest to the nozzle, leading to a reduced accuracy of the PDA measurement. In the droplet evaporation zone (z > 1 cm), the spray is fully developed, and an increase of the Sauter mean droplet diameter can be observed. The evolution of the droplet mass distribution in flame 2 is shown in Figure 3b. From z ) 2 cm (solid line) to z ) 5 cm (broken line), evaporation of small droplets shifts the distribution to larger droplet sizes, while the tail of the distribution above 60 µm represents an increasing mass fraction of the remaining droplets.39 Besides evaporation, the mean droplet diameter might also be affected by radial droplet mixing by droplet dispersion, droplet coalescence, or changes in the refractive index of the droplets by evaporation. Increasing the liquid feed rate from 27.1 mL/min (flame 1) to 81.1 mL/min (flame 2) leads to an increase of the droplet diameter by about 6 µm at z ) 1 cm because more liquid has to be atomized with the same amount of dispersion gas. This can be seen by the inset of Figure 3a, which shows the size distributions from both sprays at z ) 1 cm. For flame 1, the biggest droplets measured are about 60 µm, while for flame 2, some droplets even above 70 µm are present. Likewise, the peak of the distribution shifts from 14 to 20 µm for flames 1 and 2, respectively, at z ) 1 cm. Figure 4 shows the radial distribution of the Sauter mean diameter, dd, and droplet data rate at two heights for flame 2. Close to the centerline at z ) 2.5 cm (solid line), dd is almost constant (26-29 µm) and droplets are measured at a high data rate, between 13 and 16 kHz (right ordinate), indicating high droplet concentrations. Between r ) 0.4 and 1 cm, dd decays to 11 µm, where data rates are 2 orders of magnitude smaller compared to the plateau in the spray center. This shows that the centerline conditions represent well the region of major droplet flux within solid-cone sprays.39 At z ) 5 cm (broken lines), the droplet data rate on the centerline dropped to 5% compared to z ) 2.5 cm, mainly because of rapid evaporation of the smallest droplets. Additionally, the spray spreads in the radial direction by the dispersion gas, leading to a smooth radial droplet size profile with dd ) 34 µm on the centerline and a maximum of dd ) 42 µm at r ) 0.6 cm. The data rate is at a maximum at r ) 0.6 cm (1.3 kHz), probably because
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Figure 6. Assuming monodisperse sprays with initial diameters of 10 µm (thick lines) and 40 µm (thin lines) for the conditions of flame 1, the evolutions of the Sauter mean primary (broken line) and mass equivalent agglomerate (solid line) particle diameter are shown. Larger droplets lead to slower precursor conversion and to smaller primary particle sizes.
Figure 5. Evolution of the normalized droplet diameter d/d0 for the temperature and velocity profile for different initial droplet diameters, ranging from (a) 10 to 60 µm (flame 1) and (b) 30 to 110 µm (flame 2). The thick solid lines indicate the measured axial temperature profiles for flame 1 (a, triangles) and flame 2 (b, circles).13
of the radial temperature decay that results in slower droplet evaporation at r ) 0.6 cm compared to the centerline. Effect of the Droplet Size on Particle Dynamics. To simulate droplet evaporation and particle formation, the initial droplet size distributions are specified as the ones in the inset of Figure 3a. Initially, the measured mass mean droplet velocities of 100 and 66 m/s are used for flames 1 and 2, respectively. Parts a and b of Figure 5 show that flame temperatures measured by Fourier transform IR13 increase rapidly along z (flame 1, z ) 0.5 cm, T ) 2610 K; flame 2, z ) 2.5 cm, T ) 2765 K). The liquid precursor exits the nozzle with an initial average axial velocity of 2.3 and 6.9 m/s for flames 1 and 2, respectively, when assuming plug flow at the nozzle tip. Simulations with the proposed droplet evaporation model showed that the residence time within the breakup zone (z < 1 cm) is sufficiently long to heat up the liquid; however, during breakup, the spray is very dense, so that evaporation effects are negligible. As a result, an initial droplet temperature of 410 K is chosen, which is close to the boiling temperature of the mixture at z ) 1 cm. Figure 5a shows the evolution of the normalized droplet diameter d/d0 calculated for single-component ZP droplets with diameters between 10 and 60 µm for flame 1 using the regression line for the temperatures of Mueller et al.13 (Figure 5a, triangles) and the mea-
sured gas velocity profiles (Figure 2). The droplet lifetime and traveling distance increase significantly with increasing droplet size.21 Droplets with an initial diameter of 10 µm vaporize in 0.091 ms within 1 cm, while 30-µm droplets travel 9 cm within their lifetime of 1.05 ms (Figure 5a). The total traveling distance of 40-µm droplets is 15 cm, which exceeds the visible flame height of 13 cm (Table 1), and about 20 wt % of the precursor evaporates in a region with temperatures below 1500 K. Larger droplets can escape through the flame, although reduced in size, e.g., from 60 to 10 µm, which may lead to partially oxidized, inhomogeneous, and even hollow particles.12 Figure 5b shows the same results for the hotter flame 2 and more concentrated ZrO2 aerosol. Increasing the liquid feed rate by a factor of 3 (flame 2, Figure 5b) increases the visible flame height from 13 cm (flame 1) to 37 cm (flame 2). Higher flame temperatures and slower cooling rates result in faster droplet evaporation in flame 2. Consequently, the total traveling distance of 30- and 50-µm droplets from flame 1 to flame 2 reduces from 9 to 7.1 cm and from 23.2 to 17.6 cm, respectively. Droplets up to 70 µm evaporate completely within the visible flame. This shows that increasing the FSP liquid feed rate results in increased droplet/particle residence times at high temperature. This facilitates complete droplet evaporation, resulting in solid particles. This is consistent with Tani et al.,11 who observed the conversion of hollow to solid alumina particles when they replaced air with pure oxygen as the dispersion gas of FSP. Air gives lower maximum temperatures than oxygen because the inert nitrogen takes away heat of combustion. Clearly, looking at the droplet distributions of the inset of Figure 3a and the calculations of Figure 5, one can see that complete droplet evaporation takes place in these flames, which produced solid ZrO2 particles.13 Figure 6 shows the evolution of the primary (broken line) and agglomerate (solid line) particle diameters for monodisperse droplets of initially 10 µm (thick lines) and 40 µm (thin lines) diameters in flame 1. The 10-µm droplets evaporate completely between 1 and 2 cm (Figure 5a), resulting in ZrO2 particle formation by coagulation and sintering. At about z ) 7 cm, agglomerates (Figure 6) start to form, and at z ) 14 cm, the primary particles reach their final size of 16.6 nm.
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Figure 7. Number-based geometric mean standard deviation for monodisperse initial droplet sizes of 10, 20, and 40 µm for flame 1. Only particles larger than 1.6 nm are represented, and the results are plotted until primary particle and agglomerate standard deviations vary by more than 1%. For droplets up to 20 µm, the geometric standard deviation converges toward the selfpreserving value of 1.514 once all of the droplets evaporated.
The 40-µm droplets evaporate more slowly, so complete evaporation is only achieved at z ) 15 cm (Figure 5a). Here droplet evaporation and particle dynamics overlap considerably, so ZrO2 particle formation takes place by coagulation-sintering during continuous release of the precursor, which is a source of new ZrO2 particles. Above z ) 10 cm, where the flame temperature drops below 1500 K (Figure 5a), the average particle diameter decreases, reaching a minimum at about 15 cm because newly formed particles reduce the average diameter while sintering is slow. When new particle formation stops because all droplets have evaporated (z ) 15 cm), then dp increases again. The onset of agglomeration is at about z ) 6 cm, while the final primary particle size of 9 nm is only reached at z ) 17 cm. This shows that bigger droplets lead to the formation of smaller primary particles because the precursor is released later, so that an increasing number of particles miss the hottest area of the flame and nucleate further downstream at lower temperatures, where the characteristic time of sintering is long, resulting in agglomerates that are smaller than those from smaller droplets. Figure 7 shows the evolution of the particle numberbased geometric standard deviation σgn for initial droplet diameters of 10, 20, and 40 µm in flame 1. Results are only shown up to the onset of agglomeration, where the primary particle and agglomerate size deviate by less than 1%. From this point on, only coagulation dominates particle growth, leading to the self-preserving distribution. Droplets of dp ) 10 µm have evaporated at 2.05 cm, so the particle size distribution rapidly approaches the self-preserving limit of spherical particles in the free molecular regime with a standard deviation of 1.514 for the employed section spacing of 2.27 In the case of 20µm droplets, however, the distribution is significantly broadened until after z ) 4.8 cm droplets vaporized completely and the distribution converges to that selfpreserving limit. For an initial droplet diameter of 40 µm, however, the precursor is still released after the onset of agglomeration and the model predicts broader, bimodal particle size distributions and does not reach the self-preserving distribution within this time. This
Figure 8. Density function of the normalized mass evaporation rate, q (eq 15), for flame 1 (solid line) and flame 2 (broken line). For comparison, the corresponding precursor evaporation rates are plotted on the second ordinate. Smaller liquid feed rates (flame 1) lead to faster precursor release and to a steeper gradient of q.
is consistent with Mueller et al.,13 who showed that, for higher liquid feed rates, where droplet sizes are larger, broader primary particle size distributions are measured. Influence of the Droplet Size Distribution on Particle Dynamics. Figure 8 shows the density function of the normalized mass evaporation rate, q, as defined in eq 15 and the total evaporation of the precursor for flame 1 (solid line) and flame 2 (broken line). The initial droplet size distribution at z ) 1 cm (Figure 3a) is represented by sections ranging from 1 to 95 µm with a constant spacing of 0.5 µm. By neglect of droplet-droplet interactions because their concentration is too low for coagulation (about 3 × 104 no./cm3 at z ) 2.5 cm), results have been obtained by superimposing single droplet calculations according to the mass fraction of each section. For flame 1, the evaporation density function q drops rapidly as the flame temperature decreases and the droplet surface area shrinks by evaporation. For flame 2, the precursor release is almost constant up to z ) 3 cm before the evaporation rate decays asymptotically. On the second ordinate, it can be seen that the total precursor evaporation is slower for flame 2 because there is more liquid in larger droplets compared to flame 1. Mueller et al.13 reported that thermophoretic sampling is only possible at locations above z ) 7.5 and 10 cm for flames 1 and 2, respectively; otherwise, the TEM grids are destroyed by the burning droplets. This is consistent with Figure 8, where it is shown that at these points about 95% of the droplet mass evaporated. Figure 9 shows the calculated Sauter mean primary particle size evolution for flame 1 (solid) and flame 2 (broken). These are compared to the data of Mueller et al.13 for flame 1 (triangles) and flame 2 (circles). For flame 2, the combustion reaction is understoichiometric, so air entrainment has to be accounted for. Experiments showed that the spray flame cannot be operated with air instead of O2 as the dispersion gas because the resulting spray flame is unstable. This indicates that air entrainment is just enough to provide the spray flame operated with O2 dispersion gas with sufficient oxygen, while for air, the flame is O2-lean. Because the flame is shielded by oxidant gas from the sheath gas (O2), continuous entrainment of 0.7 mol/m/s O2 is accounted for in flame 2, which is just enough to keep it from being understoichiometric. For flame 1, the total
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Figure 9. Calculated Sauter mean primary particle diameters for ZrO2 made in flame 1 (solid line) and flame 2 (broken line) shown in comparison to experimental data13 (flame 1, triangles; flame 2, circles).
gas flow rate (298 K) after combustion is only 77 L/min compared to 170 L/min for flame 2,13 leading to a decrease in the spray cross-sectional surface area of about 40%, resulting in air entrainment of 0.4 mol/m/s O2. For flame 1, a drop of the average primary particle diameter can be seen at about z ) 12 cm (Figure 9), caused by evaporation from the large tail of the droplet size distribution as shown in Figure 6, where late nucleation of monomers reduced the average primary particle diameter for initial droplet sizes of about 40 µm. Parts a and b of Figure 10 show the calculated particle size distribution at z ) 10 and 15 cm compared to the experimental values for flame 1 obtained by Mueller et al.13 For comparison, the simulation results have been normalized by neglecting particles smaller than 1.6 nm because this was the lower detection limit for the TEM image analysis. It is shown that the shapes of the distributions obtained by the experiments (triangles) and simulations (line) agree well while the model overestimates the average particle size by 0.5-1.5 nm. Simulations predict that continuous nucleation leads to a bimodal distribution of primary particle sizes consistent with that of Tsantilis et al.35 Because the onset of agglomeration occurs further downstream for flame 2, experimental size distributions (circles) at z ) 10, 15, 20, and 40 cm (Figure 11a-d)
can be compared with simulations (line). For all three heights, the calculated and measured particle distributions for primary particles agree well with the experiments, given the complexity of the process. For the small tail of the size distribution, simulations slightly underestimate the number concentrations obtained by the experiments. Simulations show that for increasing z the first mode of the distribution below 1 nm decreases constantly as a result of collisions with primaries from the second mode that are shifting toward larger primary particle sizes. At z ) 40 cm (Figure 11d), the calculated primary particle size distribution is narrowed artificially by the assumption of a monodisperse primary particles size distribution per volume interval.35 In general, the calculated evolution of the primary particle diameter agrees well with the experimental data (Figures 10 and 11). Clearly, accounting for droplet evaporation and particle polydispersity improves the agreement with data over monodisperse models,13 though there is substantial computational effort that gives, however, process insight. Summary and Conclusions The velocity profiles and droplet size distributions of two different flame sprays have been investigated with PDA measurements, giving a better understanding of the FSP process for the synthesis of nanoparticles. With increasing height above the burner nozzle, the average droplet size increases while the droplet concentration decreases until both vanish, indicating rapid droplet evaporation. Higher liquid feed rates at constant dispersion gas flow rate lead to lower gas flame velocities. Along with the increased flame height, this increases the high-temperature residence time of droplets or particles, leading to larger product particles. The coupling between droplet evaporation, combustion, particle formation, and growth has been studied for the synthesis of ZrO2 particles by using a detailed droplet evaporation model based on work by Abramzon and Sirignano19 combined with a sectional population balance model accounting for nucleation, coagulation, and sintering. The spray droplet size influences both the primary particle size and polydispersity. Larger droplets evaporate more slowly than smaller ones, leading to prolonged particle formation and smaller primary but larger agglomerate particle size, thus broadening the product particle size distribution. Once
Figure 10. Simulated primary particle size distributions (lines) compared to the experimental results13 (triangles) for flame 1 at (a) z ) 10 cm and (b) z ) 15 cm.
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Figure 11. Simulated primary particle size distributions (lines) compared to the experimental results13 (circles) for flame 2 at (a) z ) 10 cm, (b) z ) 15 cm, (c) z ) 20 cm, and (d) z ) 40 cm.
all droplets are evaporated, the geometric standard deviation of the particles converges to the self-preserving limit. As long as droplets are present, the particle size distribution is bimodal. When droplets are too large (40 µm for the conditions of flame 1) to evaporate completely in the hot region of the flame, some particles nucleate late and miss the highest temperature region of the flame, reducing the average primary particle diameter of the product. When droplet evaporation and particle polydispersity are accounted for, comparisons between the calculations and experimental data show very good agreement for the average particle size distribution and reasonably good agreement for the particle size distribution. The latter can be further improved with detailed knowledge of the fluid mechanics of the flame spray. Acknowledgment This research was supported, in part, by the Swiss Commission for Technology and Innovation (KTI) and the Swiss National Science Foundation. We gratefully acknowledge discussions with and assistance from R. Jossen during this project. Appendix Derivation of the Agglomerate Surface Balance for a Geometric Spacing of 2. The sectional number and surface balance for aerosol processes accounting for aggregation and sintering based on the geometric spac-
ing of 2 have been used by Tsantilis et al. (Appendix B)35 and Mu¨hlenweg et al. (eq 9).25 The fact that two colliding aggregates can have different degrees of sintering is approached differently. Two partially sintered aggregates of section j < i - 1 with surface aj and volume vj ) 2j-1v1 and i - 1 with surface ai-1 and volume vi-1 ) 2i-2v1 collide to form a new aggregate, which is represented by an imaginary section s of surface area as ) aj + ai-1 and volume vs ) vj + vi-1. Because this section does not exist in the geometric spacing, aggregate s has to be divided into the two neighboring sections i and i - 1 by conserving the number and volume:
vs ) xvi + (1 - x)vi-1; x ) 2j-i+1
(27)
where 0 < x < 1 represents the number fraction of the aggregate that will be represented by section i. The volume fraction 0 < w < 1 represented by section i is then given as
wi,j )
xvi 2j ) j-1 vs 2 + 2i-2
(28)
The aggregate surface area is spited like the aggregate volume because the primary particle size should be identical for both fractions, the ones represented by sections i and i - 1. The gain of the total surface area in section i by the collision described above is
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|
dAi
gain
dt
j,i-1
i-2
)
∑ j)1
βi-1,jNjNi-1wi,jas ) i-2
βi-1,jwi,j(AjNi-1 + NjAi-1) ∑ j)1
(29)
and the loss in section i - 1 due to collision with aggregate j is
|
dAi-1
loss
dt
j,i-1
i-2
βi-1,jNjNi-1ai-1 + ∑ j)1
)-
i-2
βi-1,jNjNi-1(1 - wi,j)as ∑ j)1 i-2
βi-1,j[wi,jNjAi-1 - (1 - wi,j)AjNi-1] ∑ j)1
)-
(30)
with Ai ) Niai. To use eq 30 for the surface balance, the index has to be shifted from i - 1 to i as the loss of surface area in section i is needed:
|
dAi dt
loss
i-1
)
j,i
βi,j[wi+1,jNjAi - (1 - wi+1,j)AjNi] ∑ j)1
(31)
The other two coagulation terms (gain due to collision of two aggregates from section i - 1 and loss in section i due to collision with a larger particle) in the surface balance are not affected by the presence of partially sintered aggregates because splitting of a newly formed aggregate between the neighboring sections does not happen for those cases.26 The total surface balance is then given by
dA dt
|
i-2
) i
βi-1,jwi,j(AjNi-1 + NjAi-1) + ∑ j)1 i-1
βi-1,i-1Ai-1Ni-1 imax
(1 - wi+1,j)AjNi] - Ai
βi,j[wi+1,jNjAi ∑ j)1 1
βi,jNj ∑ τ j)i
(Ai - NiRi,s) (32)
s,i
which is consistent with Appendix B of Tsantilis et al.35 in the absence of sintering. Notation a ) surface area [m2] A ) total agglomerate surface area [m2/m3] A˙ ) total agglomerate surface area flux [m2/s] c ) flame gas velocity [m/s] B ) Spalding transfer number CD ) drag coefficient Cp ) heat capacity [J/kg/K] D ) diffusion coefficient [m2/s] d ) droplet diameter [m] E ) activation energy [J/mol] F ) correction factor ∆Hv ) latent heat of vaporization [J/kg] I˙ A ) nucleation source term for the total particle surface area [m2/m/s] I˙ N ) nucleation source term for the particle number [no./ m/s] kb ) Boltzmann constant [J/K]
m ) mass [kg] m ˘ ) mass flux [kg/s] MW ) molecular weight [kg/mol] N ) agglomerate number density [no./m3] N˙ ) agglomerate number flux [no./s] Nu0 ) Nusselt number Nu* ) modified Nusselt number p ) pressure [Pa] Q˙ ) heat-transfer rate [J/s] q ) density function of the normalized mass evaporation rate [1/m] R ) radius [m] R ) universal gas constant [J/mol/K] Sh0 ) Sherwood number Sh* ) modified Sherwood number t ) time [s] T ) temperature [K] u ) axial velocity [m/s] w ) volume correction factor w ) grain boundary width [m] x ) mole fraction liquid [mol/mol] Y ) mole fraction gas [mol/mol] z ) position [m] Greek Symbols β ) collision frequency [m3/no./s] β˙ ) modified collision frequency [1/no.] γ ) surface tension [N/m] λ ) thermal conductivity [W/m/K] µ ) viscosity [Pa s] F ) density [kg/m3] σgn ) geometric number-based standard deviation τ ) characteristic time [s] ω ) mass fraction [kg/kg] Indices A ) surface area b ) grain boundary c ) collision d ) droplet F ) fuel g ) gas gyr ) gyration h ) heat liq ) liquid m ) mass m ) mass equivalent M ) mass transfer N ) number p ) primary particle proj ) projection s ) condition at the droplet surface T ) heat transfer tot ) total ZP ) zirconium n-propoxide
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Received for review October 8, 2004 Revised manuscript received February 2, 2005 Accepted February 7, 2005 IE0490278
