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FUEL PROCESSING TECHNOLOGY ELSEVIER Fuel Processing Technology 39 (1994) 319-336 Dynamics of pyrogenous fumes Richard C. Flagan Division of Chemistry and Chemical Engineering 210-41, California Institute of Technology, Pasadena, CA 91125, USA (Received 19 April 1993; accepted in revised form 24 February 1994) Abstract Theoretical and experimental investigations of the dynamics of aggregate aerosols have been conducted and led to the development of models to describe the kinetics of agglomeration. The agglomeration rate is determined by the mobilities of the agglomerate particles and by their collision cross sections. Agglomerate particles coagulate more rapidly than do dense particles due to their larger collision cross sections. Experimental studies have been conducted to explore a number of aspects of the evolution of aggregate aerosols. The structure-drag relationship has been explored by image analysis of transmission electron micrographs of mobility classified aggregate particles. Structural rearrangements during sintering of aggregates have been ex- plored by heat treating aggregate particles while they are still entrained in the carrier gas. Particles are observed to retain the appearance of fractal clusters of smaller primary particles throughout coalescence that leads to many primary particles from the original particle being incorporated into a single primary particle in the sintered agglomerate. This observation raises serious questions about the inference of mechanisms of particle growth from structure measure- ments alone. 1. Introduction During pulverized coal combustion, a fume of submicron particles is formed from the mineral matter in the parent coal. Studies of the variation in chemical composition with particle size have revealed that much of the submicron fume is formed from volatilized coal ash I-1-3]. The formation and evolution of the ash fume is governed by homogeneous nucleation, condensation, and coagulation. Vapors of refractory species nucleate relatively early in the combustion process. Coagulation of those fine particles results in a size distribution that is approximately log normal. More volatile species remain in the gas phase until after the nucleation has taken place. Condensation on the surfaces of both the fume and the larger residual ash particles results in the enrichment of the fine particles with volatile, and frequently toxic trace species. The resultant concentration of heavy metals in the size interval between 0.1 and 1 ~m may 0378-3820/94/$07.00 © 1994 Elsevier Science BN. All rights reserved SSDI 0378-3820(94)00036-S

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FUEL PROCESSING TECHNOLOGY

ELSEVIER Fuel Processing Technology 39 (1994) 319-336

Dynamics of pyrogenous fumes

R i cha rd C. F l a g a n

Division of Chemistry and Chemical Engineering 210-41, California Institute of Technology, Pasadena, CA 91125, USA

(Received 19 April 1993; accepted in revised form 24 February 1994)

Abstract

Theoretical and experimental investigations of the dynamics of aggregate aerosols have been conducted and led to the development of models to describe the kinetics of agglomeration. The agglomeration rate is determined by the mobilities of the agglomerate particles and by their collision cross sections. Agglomerate particles coagulate more rapidly than do dense particles due to their larger collision cross sections. Experimental studies have been conducted to explore a number of aspects of the evolution of aggregate aerosols. The structure-drag relationship has been explored by image analysis of transmission electron micrographs of mobility classified aggregate particles. Structural rearrangements during sintering of aggregates have been ex- plored by heat treating aggregate particles while they are still entrained in the carrier gas. Particles are observed to retain the appearance of fractal clusters of smaller primary particles throughout coalescence that leads to many primary particles from the original particle being incorporated into a single primary particle in the sintered agglomerate. This observation raises serious questions about the inference of mechanisms of particle growth from structure measure- ments alone.

1. Introduction

During pulverized coal combustion, a fume of submicron particles is formed from the mineral matter in the parent coal. Studies of the variation in chemical composition with particle size have revealed that much of the submicron fume is formed from volatilized coal ash I-1-3]. The formation and evolution of the ash fume is governed by homogeneous nucleation, condensation, and coagulation. Vapors of refractory species nucleate relatively early in the combustion process. Coagulation of those fine particles results in a size distribution that is approximately log normal. More volatile species remain in the gas phase until after the nucleation has taken place. Condensation on the surfaces of both the fume and the larger residual ash particles results in the enrichment of the fine particles with volatile, and frequently toxic trace species. The resultant concentration of heavy metals in the size interval between 0.1 and 1 ~m may

0378-3820/94/$07.00 © 1994 Elsevier Science BN. All rights reserved SSDI 0 3 7 8 - 3 8 2 0 ( 9 4 ) 0 0 0 3 6 - S

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320 R.C. Flagan / Fuel Processing Technology 39 (1994) 319 336

allow disproportionate amounts of these species to escape collection, even by the best of gas cleaning systems.

Flagan and Friedlander [1] first modeled the evolution of the ash particle size distribution in pulverized coal combustion beginning with the hypothesis that the fine particles resulted from homogeneous nucleation and grew primarily by coagulation. They predicted much more distinct peaks in the submicron size range than had been observed at that time. Improved instrumentation has verified those predictions, and shown that the situation can be even more complex than their simple model indicated. In some cases multiple peaks are seen in the size distribution of the submicron fume particles [4]. This could occur relatively late in the cooling of the combustion products when a second vapor becomes sufficiently supersaturated to undergo homo- geneous nucleation in spite of the large numbers of fume particles produced in the initial nucleation burst. If heavy metals are responsible for this additional nucleation event or if it occurs before the heavy metals condense, further enrichment of the fine particles with heavy metals could result.

A comprehensive theoretical treatment of the aerosol dynamics of pyrogenous fumes requires a number of extensions of the classical descriptions. Rigorous descrip- tions of the coagulation of dense, spherical particles are available [5-7], but fume particles are rarely spherical. The material involved tend to be refractory, so high temperatures are required to achieve complete coalescence. Flame temperatures may not be hot enough to melt some materials, so coalescence is not always achieved. Even with systems that can melt the particles in the primary reaction zone, coagulation during the cooling or quench process can form agglomerates. To predict the dynamics of the fumes produced when coalescence is rate limiting, the structure and dynamics of the resulting aggregates must be understood.

Pyrogenous fumes, including soot, coal ash, and synthetic fumes (TiO2, SiO2, etc.) exhibit a common structure, namely agglomerates of approximately equiaxed par- ticles (spherules) with sizes of a few tens of nanometers. This common structure has been attributed to the physical process of agglomeration [8]. The densities of pyro- genous fume agglomerates decrease with increasing size. Forrest and Witten observed that the aggregate mass varies as a power law of the size of that aggregate, i.e.,

m o o r ° (1)

where D has been called the mass fractal dimension which generally has a value below the Euclidean dimension of 3. Pyrogenous fumes typically have fractal dimensions below 2.

Computer simulations of aggregate formation yield similar structures and provide insights into the formation mechanisms [9]. Relatively dense particles (D ~ 2.4) are produced when individual spheres diffuse to the surface of the growing aggregate (diffusion limited aggregation, DLA). Lower density aggregates (D ~ 1.8) result from coagulation of like-sized aggregates (cluster-cluster aggregation, CCA). Hence, the simulations suggest that pyrogenous fumes grow by the CCA mechanism. These structure calculations make one very important assumption that is not valid for pyrogenous fumes, namely that the structure of the aggregate does change due to fusion spherules under the driving force of surface tension. This may be expected to

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increase the fractal dimension of the aggregates as they age, although direct measure- ments of this effect are still lacking.

Most diffusion limited aggregation and cluster-cluster aggregation models have focused on the structure of the particle, primarily through computer simulations that employed simplistic models of particle aerodynamics and collision cross sections, usually modeling the particle transport with either continuum or free molecular regime descriptions. Pyrogenous fume particles are frequently comparable to the mean-free-path of the gas molecules in size, so neither continuum nor kinetic regime models are strictly valid. Direct measurement of aerodynamic parameters for aggre- gate particles is a prerequisite for the development of a more rigorous understanding of such aerosol systems. To understand the distribution of volatilized trace elements with respect to particle size, it is further necessary to elucidate the mechanisms of mass transport to the surfaces of transition regime aerosol aggregates.

This paper describes theoretical and experimental investigations of the structures and aerodynamics of aggregate pyrogenous fume particles in which the particles are modeled as fractal agglomerates. It reviews previous theoretical and experimental work on the dynamics of the aggregate particles.

2. Dynamics of the particle size distribution

The evolution of an aerosol is described in terms of the particle size distribution function, although chemical composition, phase, and particle morphology are also important properties to be controlled and predicted. One can describe particle size in terms of the numbers of molecules that comprise individual particles. This leads to a discrete representation of the particle size distribution, where the concentration of particles containing g monomers is denoted N 0. While it is convenient to describe the initial formation of particles from the vapor phase, the discrete representation rapidly becomes unwieldy as particle size increases. To describe particles larger than a few nanometers in size, one generally resorts to a continuous representation of the size distribution. If the total number of particles per unit volume of gas is N(t), the number with masses between m and m + d m is

dN = n(m, t) dm (2)

Aerosol models seek to describe the evolution of n(m, t) under the action of a number of mechanisms, i.e.,

nucleation \ dt /'condensation

+ \(dn(m't)~-~ //surface ~- (dn(m,t)]__ (3) reaction \ dt /'coagulation"

The discussion that follows briefly describes some of the key mechanisms and the rate expressions that have been derived to predict the evolution of the aerosol, leading to

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322 R.C. Flagan / Fuel Processing Technology 39 (1994) 319-336

the development of a functional representation of Eq. (3) that is known as the general dynamic equation for aerosols.

2.1. Nucleation

A supersaturated vapor is inherently unstable, and tends to form the condensed phase either by condensation onto existing surfaces or by homogeneous nucleation. The rate of phase transformation is determined by kinetic and transport processes. Here we examine the process of homogeneous nucleation or direct formation of the condensed phase from the vapor phase. The theory of homogeneous nucleation traces its roots back to the work of Gibbs on the thermodynamics of small clusters, and to the early works of Becker and Doering, Volmer, and Zeldovich (see [7]). It still remains a controversial area due to the many assumptions required in its derivation. The key assumptions of the classical theory of homogeneous nucleation are: - Clusters grow by monomer addition. - The rate of monomer arrival at the cluster surface is that determined by kinetic theory, i.e., the monomer flux is

f l _ pl ~/2nml k~ T'

where Pl and ml are the partial pressure and mass of the monomer, respectively, and k~ is the Boltzmann constant.

- A steady-state distribution of cluster sizes is assumed. - The surface free energy a of all sizes of clusters is assumed to be the same as that of the bulk material. This so-called capillary approximation is the most tenuous of the assumptions in the classical nucleation theory.

Clusters smaller than a critical size are more likely to evaporate than grow, while larger clusters will tend to grow. According to the classical theory, the number of molecules in the critical cluster is

32~a3vx z g* - (4)

3(kTln S) 3'

where va is the monomer volume, S = p~/p~ is the saturation ratio, and k is the Boltzmann constant. The net rate of passage of particles through the region near g* is the nucleation rate. The steady-state nucleation rate,

J = flao.nCe(g*)Z (5)

is the product of the rate of monomer arrival to the surfaces of clusters of the critical size, the concentration of clusters at the bottleneck size in a hypothetical constrained equilibrium with the monomer in the supersaturated state,

6(367tv2)l/3g. ~ nee(g* ) = N] exp g* lnS ~ - j , (6)

and a nonequilibrium factor Z that accounts for differences between the steady-state cluster distribution and the constrained equilibrium distribution.

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R.C. Flagan / Fuel Processing Technology 39 (1994) 319-336 323

The classical theory breaks down if the saturation ratio is too large, leading to a small critical cluster size, g*. The nucleation rate depends strongly on the surface free energy, a, of g*-sized clusters, but tr is generally known only for the bulk material. Knowledge of the size dependence of a would be needed to develop a nucleation model that would overcome this problem. At even higher supersaturations, the dimer will be more stable than the monomer. With no barrier to nucleation, all clusters will tend to grow.

When the condensible species are produced by gas phase reactions, e.g., the oxidation of the volatile SiO to the more refractory SiO2, clusters may grow by surface reactions in addition to or rather than condensation. Katz and Donohue [10] have extended the classical theory of homogeneous nucleation to include the contri- butions of surface reactions to the cluster growth process.

2.2. Particle growth

Once particles have been formed by homogeneous nucleation, they grow by the combined effects of vapor deposition and coagulation. The nature of the particles produced depends very strongly on the relative rates of these processes and on the properties of the material at the reaction temperature. The literature on aerosol dynamics is vast, so the reader is referred to a number of texts incorporating the subject for presentation of the fundamentals of the subject [5-7, 11-].

Most derivations of the dynamics of aerosol systems make a critical assumption that is only rarely valid for combustion fumes, namely that the particles are spherical throughout the growth process. This assumption is strictly valid only when the process is operated at temperatures in excess of the melting point of the product material. At lower temperatures, aggregate particles formed by collisions of solid particles coalesce or sinter together at finite rates. Since the particles are nonspherical, their diffusivities and collision cross sections, and the rates of mass transfer to their surfaces can differ dramatically from the values for dense spheres. The structures of aggregate particles are determined by the competition between coagulation and coalescence of the particles. If the time scale for coalescence is short compared to the time between collisions, the particles will evolve as spheres. On the other hand, extremely slow coalescence will lead to the formation of low density aggregates. Coalescence or sintering at intermediate rates will lead to the formation of hard agglomerates.

The dynamics of aerosol populations are described using the so-called general dynamic equation for aerosols, [7]

Dnm(log mp, t) K(mp Dt = - fit, r~)n(mp - fit, t)n(ff~,t) dr~

- ½n(m, t) K(m, r~)n(~, t) d ~ + S(m*, t)6(m - m*) m*

+ ~m(I(m, t)n(m, t)) (7)

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324 R.C. Flagan / Fuel Processing Technology 39 (1994) 319 336

The first term on the right-hand side accounts for the formation of particles of mass m by agglomeration of particles of smaller mass, the second term accounts for the loss of particles of mass m by coagulation with particles of any other size, and m* is a lower bound to the particle size distribution. The continuous representation of the particle size distribution is an acceptable approximation only for particles sufficiently large compared to the molecular mass that the difference between a cluster of size m and size m + ml does not introduce appreciable errors in the continuous distribution. The appearance of particles at that size from below is generally represented using a nuclea- tion model, hence the source term S(m*,t), although hybrid representations that couple a discrete representation of the smallest clusters with a continuous representa- tion of the distribution of larger particles have been developed 1-12]. The fourth term on the right-hand side accounts for particle growth by vapor deposition or shrinkage by evaporation.

2.2.1. Collision frequency function Previous evaluations of the collision frequency function K(ml, mj) have assumed

dense spherical particles. In order to account for agglomerate particles, a modified form for the coagulation frequency expression can be derived in which the fractal cluster dimensions are substituted into the usual derivation for spheres at key points. To put the work on aggregate particles into perspective, we first briefly review of the prior work on spherical particles.

The collision frequency function K(ml, m j) can be derived rigorously for spherical particles that are either very large compared to the mean-free-path of the gas molecules, 2, i.e., continuum regime particles, or particles that are small compared to 2, the so-called Knudsen or free molecular regime particles. The particle size can conveniently be characterized in dimensionless form as the Knudsen number, Kn = 2/a, where a is the particle radius. The continuum regime corresponds to small Knudsen numbers, Kn ,~ 1, while the free molecular regime corresponds to large Knudsen numbers, Kn >> 1. Coagulation in the continuum regime is modeled as diffusion of the two particles until they approach to the point that they contact one another. The collision frequency function is then proportional to the product of the collision radius (which equals the sum of the radii of the two particles) and the pair diffusivity 912 = ~x + ~2, i.e.,

Kc(mi,mj) = 4=(~i + ~j)(ai + a t) (8)

From the perspective of coagulation, it is the Knudsen number based upon the particle mean-flee-path, ld = 2 ~ 2/g~ 2, that determines the onset of deviations of the collision frequency function from the continuum result. Here, c12 = (~2 + ~2)1/2. The particle diffusivity is related to its aerodynamic drag through the Stokes-Einstein relationship,

6~tpa - (9)

kTCc(Kn)

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R.C. Flagan/ Fuel Processing Technology 39 (1994) 319-336 325 where Cc is a slip correction factor that depends on the Knudsen number based on the gas mean-free-path, 2. In the continuum limit

2kT{ 1 1 x 1/3 Kc(mi,mj) = ~ \ m ~ + ~ ) ( m l + m]/3) (lO)

The coagulation frequency function for equal sized particles in this limit is indepen- dent of particle size, i.e., Kc(m, m) = 8kT/31~.

In the free molecular limit, particles move along ballistic trajectories over distances that are larger than the particles in question. The collision frequency then depends on the relative thermal speed of the particles and the projected cross sectional area, i.e.,

Ktm(mi,mj) = re(a1 + a2)2(12

\(3~ )~1/6 (6kT)'/2 ( l p ~ , ' - m~ 1~'/2 m'/3)2 = - - ~ = + - - (m/l/3 + j mj/

(11)

For equal sized particles in the free molecular regime, K(Dp, Dp)= 8(3kT) 1/2 m1/6 = 8(6kT/pp)l/Erp. pp2/3 ""P

Coagulation of spherical particles at sizes intermediate between the free molecular and continuum regimes is well described by a collision frequency function originally derived by Fuchs [5] using the so-called flux matching method wherein the flux of particles calculated using the continuum transport expression is matched to that predicted for free molecular transport. Fig. 1 shows the collision frequency function for particles of density 2300 kg m-3 in atmospheric pressure air at 300 K and 1500 K, expressed in terms of the radii of the coagulating spherical particles. The Brownian coagulation coefficient exhibits its lowest value for particles of equal size. The maximum value of K(rp,rp) occurs at rp ,,~ 20 nm at 300 K and rp ~ 50 nm at 1500 K. The increase in this transitional size results from the increase in the mean-free-path with temperature. The collision frequency is higher for particles of different sizes than it is for similarly sized particles because the large collision area of the large particle is combined with the high diffusivity of the small particle.

The assumption that particles remain spherical as they coagulate is valid for low viscosity liquids, a condition that is frequently invalid for combustion fumes. Rather than coalescing rapidly after they coagulate, such particles frequently retain the structures of the individual particles that have coagulated, forming low density aggregates. The structures of the aggregates influence the rates of particle growth.

The structures of these particles are determined by random sequence of aggregation events, with relatively little rearrangement taking place after the collision except by solid state diffusion or viscous sintering. The primary particle size is determined by the material properties and the conditions of particle formation. Below some critical size, the particles coalesce so rapidly that the particles achieve their theoretical density, but at larger sizes coalescence proceeds more slowly than coagulation, leading to aggre- gate formation. For any particular system, the primary particle size is relatively uniform, even though a broad range of particle sizes may be present.

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326 R.C. Flagan/ Fuel Processing Technology 39 (1994) 319-336

10 .8 , , , . . . . . , . . . . . . . . , . . . . . . . . , . . . . . . .

1 0 .9

10 "1o

10 "11

,..E

......... 300 K 1500 K

\ \rml = 1000 n m

_-~/1°°.~"~" \ I O ' ; ',

r m l = rm2

1 I I I . . . . . ] . . . . . . . . I . . . . . . . . .

1 0 .9 1 0 .8 1 0 .7 1 0 s 1 0 s

rrn 2 (rn)

Fig. 1. Collision frequency function for coagulation of spherical particles of radius r,.2 with smaller particles of radius r,.~. Particle density is 2300 kg /m 3. Calculations are performed for gas temperatures of 300 K (solid curves) and 1500 K (dashed curves).

The coagulation rate for agglomerate particles differs from that for dense spheres for two reasons: (i) the low density aggregates experience larger aerodynamic drag forces than do compact spheres, and hence diffuse more slowly; and (ii) the aggregate particles present larger targets for collision. These two factors have opposite influen- ces on the collision frequency. The collision frequency function calculated using the model of Rogak and Flagan 1-13] is plotted in Fig. 2 for aggregates with fractal dimension 1.8 and primary particle radii ranging from 1 nm to 100 nm. The trends in each data set are similar to those observed for spherical particles, i.e., (i) the collision frequency function for equal sized particles approaches an asymptotic value at small Knudsen number (large radius); (ii) the collision frequency function for equal sized particles peaks near Knudsen number of unity and then decreases with decreasing particle size or increasing Knudsen number; (iii) the collision frequency function for particles of different sizes increases with increasing size difference.

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R.C. Flagan / Fuel Processing Technology 39 (1994) 319-336 327

1 0 "e

1 0 .9

1 0 - lo

. . . . . . . . r 1 = 100 n m

. . . . . . . . r 1 = 10 Rm

- - r 1 = 1 nm

1 0 "11 ' u )

~g

1 0 "12

~E g-

~E

~ 1 0 "13

1 0 "14

1 0 " i s

1 0 "is

1 0 .9 1 0 "e 10 "7 1 0 "s 1 0 "s

rm2 (m)

Fig. 2. Collision frequency function for coagulation of aggregate particles of mass equivalent radius rm2 with smaller particles of mass equivalent radius rmt. The particles are mass fractal with fractal dimension D I = 1.8 for r,, > r~, the primary particle radius. The material density is 2300 kg/m 3. Calculations are performed for a gas temperature of 300 K.

2.3. Vapor deposition The transport of vapor species to the surfaces of aerosol particles depends on the

size of the aerosol particle relative to the mean-free-path of the gas molecules, i.e., it depends on the Knudsen number characteristic of vapor diffusion to the particle,

2AB K h a n --

rp

where 2An is the mean-free-path of vapor species A in background gas B. Obviously, the dependence on the mean-free-path introduces some ambiguity in the description of mass transfer to agglomerate particles since the particle can be characterized with a range of length scales. Restricting our attention to spherical particles for the time being, we may examine the transport processes in three domains of Knudsen numbers. For particles in the free molecular regime, Kn >> l, the flux of vapor to the particle

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328 R.C. Flagan / Fuel Processing Technology 39 (1994) 319 336

surface is the product of the particle surface area and the effusion flux,

Ja.k = 4~r~ [A]6A (12) 4

where [A] is the concentration of vapor A, and C A is the mean molecular speed of A. The total arrival rate is the product of the flux JA and the external surface area of the particle. Trace species that condense on dense, free-molecular-regime particles will be enriched on the smaller particles, with the concentration of the condensate in the fine particles scaling as rp 1

When Kn <~ 1, the particle is said to be in the continuum size regime, and mass transfer to the particle's surface is described by the diffusion equation. In this case the net molar rate of transfer to the surface of a spherical particle becomes,

JA,c = 4 ~ A n r p ( C A - - CA.o~) (13)

The concentration of condensed trace species on continuum-regime particles thus scales as rp 2

At intermediate Knudsen numbers, mass transfer to spherical particles cannot be rigorously derived. Instead, approximate correction factors are used to calculate the vapor flux from the continuum regime flux, i.e.,

JA = Ja, c fl(KnAn) (14)

A commonly used correction factor is that proposed by Fuchs and Sutugin,

1 + KnAB fl(KnAB) = 1 + 1.333KnAB + 1.71KnZaB (15)

For the low density agglomerates produced when a rapid burst of nucleation is followed by coagulation without significant coalescence, the very little of the area of the agglomerate is shadowed by outer portions of the agglomerate and the available surface area is very close to the total surface area of the particle. If the particles are much smaller than the mean-free path and the fractal dimension is below about Df ,~ 2, the vapor deposition rate will be nearly that predicted if one assumes that all of the surface area is equally accessible. Indeed, on the other hand, denser agglom- erates will exhibit a lower available surface area. Meakin et al., and Koch and Friedlander [14, 15] have explored the mass transfer to particles of arbitrary fractal dimension theoretically, estimating effectiveness factors for mass transfer to agglomer- ate particles. Few experimental data are available in which the mass transfer has been measured directly, and those measurements were made on agglomerates with relat- ively low fractal dimensions, on the order of 1.8. Mass transfer and projected area equivalent diameters were found to be very close in that special case [16]. No rigorously derived or fully validated correction for aggregate particles is available.

2.4. Numerical solutions o f the coagulation equation

Solution of the general dynamic equation is performed with varying degrees for rigor using a number of approaches and codes. Discrete representations of the particle

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R.C. Flagan / Fuel Processing Technology 39 (1994) 319-336 329

size distribution, while most precise at the finest scales, are ill-suited to the description of the dynamics of an aerosol whose particles may have masses ranging over twelve or more orders of magnitude. The most rigorous of the codes which numerically integrate over the particle size distribution in continuous form are very computer intensive due to the vast range of sizes and time scales that enter most aerosol dynamics problems, e.g., 1-12]. For these reasons, considerable effort has been invested into the development of approximate models that facilitate detailed calculations while limiting computational demands.

The dynamics of an aerosol undergoing coagulational growth afford important simplifications in many cases. Under certain conditions, the Brownian coagulation equation yields similarity solutions that allow the complete size distribution to desolve from knowledge of the total number and mass concentrations of aerosol particles. The particle masses m are first normalized by the mean particle mass ffz = c~/Noo, where ~b is the total aerosol mass per unit volume of gas, and Noo is the total number of particles per unit volume. If the coagulation kernel for particles of masses nqr/1 and rfir/2 can be written as

K(rfiql,rhq2) = t h ~ K ( q t , q 2 ) , (16)

the kernel is said to be homogeneous of degree y, and the coagulation equation may be solved using similarity techniques [17, 18].

To obtain the similarity solution, it is assumed that the size distribution n(v, t) is proportional to a time-invariant, or self-preserving, distribution ~(n), i.e.,

U~(t)~k(q) n(m, t) - (17)

where r /= m/lh. Substituting these variables into the coagulation equation and inte- grating over all particle sizes, the rate of change of the total number of particles is

dt = N2 (18)

oo oo

= f fK(ql ,r lz)@(ql)O(q2)dr/1 dr/2 (19)

0 0

and ~/(n) is obtained by solving the scaled coagulation equation. Since x and ~/are independent of time, Eq. (18) can be solved analytically in the case of constant aerosol volume ~b. For ~: < 1 one finds,

[ l--Y I'/"-I) No~(t) = N ~ - 1(0) -4- T ~ ) ' K t (20)

For large t, the time variation of the number concentration reduces to

Noo ~ t - I / ( 1 - y ) (21)

where

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330 R.C. Flagan/FuelProcessing Technology 39 (1994) 319-336

Originally derived for spherical aerosol particles, the theory of the self-preserving particle size distribution has recently been extended to describe the evolution of fractal aerosol aggregates. This similarity transformation allows one to transform the general dynamic equation into an ordinary differential equation for the total number of aerosol particles, with the mean size being determined from that number and knowledge of the total mass of aerosol particles.

The next level of sophistication is the transformation of the GDE into a set of differential equations for moments of the particle size distribution [19-21]. These models can faithfully reproduce many of the results of the more precise numerical solutions to the general dynamic equation, although treatment of distributions con- taining more than one peak, or mode requires considerable knowledge about the nature of the particle size distribution.

Attempts to develop models that can recover the full particle size distribution without the computational overhead of integrating over the entire particle size distribution have produced a number of solutions to approximate discrete equations. Generally, the particle size domain is divided into sections. In the earlier models of this genre [22-25-], all of the particles in a given section are represented by a single particle size that is some mean size for the section. Kinetic coefficients (coagulation frequency functions, condensation rates, etc.) representative for that size are then applied to the processes that cause particles to grow from one section to another. The sections representing the smallest particle sizes contain few particle sizes may be described with reasonable accuracy. At larger particle sizes, however, the sections are generally more widely spaced. This leads to poorer approximations, although the approximations inherent in the model are rarely documented. [26]

The sectional approach was put on a firm theoretical basis by Gelbard and Seinfeld [27] who developed a sectional code in which the particle mass distribution is represented as constant within a section. Transport of mass from one section to another by coagulation, condensation, evaporation, or other processes is described by coefficients that are determined by integration over the appropriate bounds in the exact general dynamic equation. The coefficients are evaluated so that aerosol mass is conserved exactly and, in the limit if infinitesimal section widths, the sectional model reduces to the exact general dynamic equation. The use of finite section widths reduces the integro-differential GDE to a manageable set of ordinary differential equations. Computational efficiency is attained by evaluating integrals necessary to determine the kinetic coefficients only once. Thereafter, the coefficients are determined by interpolation on a table of predetermined values. Additional interpolations may be required to take variable gas composition, particle density, or particle structure into account.

A number of important extensions have been made to the original sectional code. One that may be critically important to the modeling of the synthesis of ash particles is the development of the multi-component sectional code, MAEROS [27]. With the assumption that all particles within a section have the same composition, MAEROS makes it possible keep track of variations of chemical composition with particle size. A second important extension is the development of the discrete- sectional representation [28, 29]. These codes make it possible to describe

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R.C. Flagan / Fuel Processing Technology 39 (1994) 319 336 331

the evolution of particles from the molecular scale to the sizes of interest for combus- tion fumes, without overly taxing computational resources.

An important assumption in the algorithms for solving the general dynamic equation is that all particles of a given size have the same composition. While some efforts to a more general description of the composition-size distribution have been reported, a general purpose code that allows description of the full composition-size distribution remains unavailable.

The dynamics of aerosol aggregation can be simulated by any of these methods if the appropriate collision frequency function is incorporated into the general dynamic equation for aerosols. The densities of aggregate particles decrease with decreasing primary particle size. This increases the collision cross section of particles with fine microstructures, thereby increasing the collision frequency function. Both experi- mentally and using the self-preserving particle size distribution model, Matsoukas and Friedlander [30] showed that aerosol aggregates with fine microstructures grow more rapidly than do those with coarser microstructures (larger primary particles), a clear result of this dependence of the collision frequency function on the primary particle size.

A more detailed model of the evolution of the aggregate size distribution has been developed by Rogak and Flagan [13]. Aggregate particles form when the time between coagulation events is smaller than the characteristic time for coalescence. Since the time for coalescence increases rapidly with increasing primary particles size, the sizes of the primary particles produced in a given environment are relatively uniform. Rather than attempting to predict that size, Rogak and Flagan [13] modeled growth from below the critical size for coalescence to aggregation by arbitrarily assuming that all smaller particles were dense spheres, while all larger particles were assumed to be aggregates of the same fractal dimension. Since aggregate particles coagulate more rapidly than dense particles, coagulation accelerates once aggregates begin to form. The discrete-sectional code of Wu and Flagan [31] was adapted to do these calculations. The conditions simulated correspond to the experimental condi- tions used by Okuyama et al. [32] for TiO2 synthesis by titanium tetraisopropoxide pyrolysis. It was assumed that the chemical reactions forming the nucleating phase proceed very fast compared to the time required to form the primary particles, so that initial condition was a high concentration of particles in the smallest size range.

Fig. 3 compares the resulting particle size distributions calculated assuming fractal particles experimental observations. Theoretical results are also shown for dense spheres. Since the experimental measurements were made by differential mobility analysis, the size distributions are presented in terms of the mobility equivalent size. At the earliest times, the particles were smaller than the ultimate primary particle size and, hence were fully dense. The distribution in this case has a single peak, with the typical shape of the self-preserving particle size distribution function that results as the long time limit to aerosol coagulation. At later times, the distribution exhibits two peaks or, at least, a shoulder indicating the transition between coalescent growth and agglomerate formation. The measured particle size distributions have a similarly broad structure, suggesting that the particles may have been near that transition at the point where the size distribution was measured.

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1011

1 0 ~°

E 1 0 9

1 0 s z "ID

1 0 7

106 1 10 100 1000

Dmobility (nrn)

Fig. 3. Dimensionless mobility distribution evolution for agglomerate particles. Data are the mobility distribution measurements of Okuyama et al. (1986) for TiO 2 formation from TTIP vapor in nitrogen at 873 K and 1 atm. pressure.

3. Experimental studies

Experimental investigations of the dynamics of aggregate aerosols have been conducted using model systems in order to generate the data that are needed to validate and improve upon the models presented above. In these studies, agglomerate aerosols were generated in controlled laboratory systems. To understand the relation- ships between particle structure and coagulation kinetics, an understanding of the structure-aerodynamic drag relationship is needed. Hence, the studies began with detailed characterization of the particle structure and measurement of the drag that acts upon the particles that have been so characterized. In addition, experiments have been performed that probe the change in the particle structure due to sintering. Two model aerosols were used in these studies. The first was elemental silicon produced by thermal pyrolysis of silane gas. The second system involved production of TiO2 fumes by pyrolysis of titanium tetraisopropoxide vapor. Both aerosols were generated in laminar flow aerosol reactors. Particle drag and sintering measurements were made while the particles were still entrained in the carrier gas flow.

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The particles for these studies were generated by introducing silane gas into a hot laminar-flow reactor. Rapid pyrolytic decomposition of the silane gas initiated rapid homogeneous nucleation of ultrafine silicone particles. Particle growth by coagula- tion proceeded rapidly. Where larger agglomerates were desired for sintering studies, the particles were allowed to coagulate for 20 and 30 s at ambient temperature, before being introduced into a second furnace while still entrained in the gas flow by another system, agglomerate. TiO2 aerosols were produced by mixing of a hot gas with the reactant vapor to promote rapid heating and reaction.

Particle structure and aerodynamic drag were measured using a combination of techniques [33]. The aerosol was neutralized by exposure to bipolar gas ions pro- duced using a radioactive source in the neutralizer. They were then classified accord- ing to their electrical mobilities in a differential mobility analyzer (DMA). Because most of the charged particles carry only one charge, the mobility classification provided a direct measure of the drag forces acting on the particles. The particle size distribution was measured by scanning the voltage used to separate particles accord- ing to their mobilities, and counting the particles in the aerosol using an on-line condensation nuclei counter (CNC). Particles were collected by thermophoretic deposition directly onto transmission electron microscope grid for subsequent analy- sis using an image processing system. For particles with fractal dimension of about 1.8, the aerodynamics drag on the particle scaled with the projected area of the particle through the transition regime.

3.1. Particle coalescence

Present aerosol dynamic models treat the coalescence of aggregate particles in a very simplistic manner, if at all. The final component in a comprehensive description of the evolution of aggregate oparticles will be a model of the morphological changes that occur as the particles coalesce in the hot combustion environment. This requires an understanding of the rates of coalescence or sintering of two spheres in contact with one another, a problem that has been described in the ceramics literature 1-34]. Coalescence of agglomerating aerosols has identified as a rate controlling process determining the fine structure size in flame synthesized silica 1-35, 36]. More quantita- tive predictions have been made using sintering theory by Helble and Sarofim [-37] in a study of coal ash fumes. By changing the composition of the fume particles to alter the viscosity, the role of particle coalescence was unambigously demonstrated. Koch and Friedlander [38] have used sintering models to describe the evolution of an aerosol by following the change in the surface area of particles as they coalesce. In spite of these advances, no model presented to date fully captures the complexity of the rearrangements of agglomerate particles as they coalesce or sinter together.

Wu 1,31] observed that particles retain their apparent agglomerate structures as they coalesce. Silicon particles were produced by silane pyrolysis at low temperature. The highly agglomerated particles were diluted to suppress further agglomeration and then heat treated while still entrained in the inert carrier gas. The densified particles were collected on filters and examined by electron microscopy and BET gas adsorp- tion methods. Both of these approaches probe the microstructures of the particles.

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Because of the way the particles were collected and processed, no information about the agglomerate structure is available. Comparison of the agglomerates before and after heat treatment reveals particles comprised of roughly spherical primary particles of fairly uniform size, although that size increased dramatically with heat treatment. The fact that the agglomerate structure is preserved while the "primary particle" mass increases by more than an order of magnitude raises serious doubts on the use of micrographs or fractal dimension to infer the growth mechanisms of pyrogeneous fumes. Efforts to reconcile these observations with sintering theory have only recently met with qualitative success, due primarily to the limited data on the solid phase transport properties available for this material. Hence, a comprehensive picture of the structural evolution of a coagulating, sintering aerosol are not possible at this time.

To understand the dynamics of aggregation in aerosol systems, including the growth of particles by cluster deposition at temperatures far below the melting point, a description of agglomerate sintering that rigorously describes the entire process is needed. Not only must the geometrical constraints of the early stage sintering models be relaxed, the combined effects of the various mechanisms must be taken into account. Better experiments are also needed to validate the theory, ideally this would involve the sintering of doublets of equally sized particles. After developing and validating the theory for this idealized model system, extension to more complex aggregate particles can be undertaken with confidence. Although silicon has been an interesting proving ground, materials for which better property data are available need to be investigated. Experiments with different materials are needed to allow the dynamics of aggregate particles that sinter by different mechanisms to be explored. These experiments are continuing in the Caltech program.

4. Summary and conclusions

This paper has explored the dynamics of aggregate aerosols produced when particulate materials are formed from the vapor phase. Theoretical investigations have been conducted to elucidate the factors that determine the dynamics and properties of such aerosols. Aggregate particles are characterized as mass fractal structures. Models have been developed that describe the relationship between par- ticle structure and aerodynamic drag and the collision frequency. Aggregates are seen to coagulate more rapidly than dense particles of the same mass due to the larger collision cross sections of the aggregates. Simulations of aerosol growth using this model of aggregate aerosols yield good qualitative agreement with the trends in the form of the particle size distribution observed in previous experiments.

Experimental studies have been conducted to explore a number of aspects of the evolution of aggregate aerosols. The structure-drag relationship has been explored by image analysis of transmission electron micrographs of mobility classified aggregate particles. For the low fractal dimensions of the aggregates studied (Dr ~ 1.8), the drag on the particle scales well with the projected area. This result holds through most of the transition regime even though it is expected to be strictly valid only for particles much smaller than the mean free path of the gas molecules.

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Structural rearrangements during sintering of aggregates have been explored by heat treating aggregate particles while they are still entrained in the carrier gas. Particles are observed to retain the appearance of fractal clusters of smaller primary particles throughout coalescence that leads to many primary particles from the original particle being incorporated into a single primary particle in the sintered agglomerate. This observation raises serious questions about the inference of mecha- nisms of particle growth from structure measurements alone.

Even though most studies of combustion fume formation have been performed using models based upon spherical particles, those studies have revealed important features of the aerosol. Quantitative predictions of the particle size and composition distribution, of the efficacy of particle removal equipment, and of the influence of combustion modifications on those properties will require models that better repre- sent reality. This paper has outlined some of the developments in that direction, although much remains to be done. Models have been developed, but they have not yet been validated. The fundamental processes that govern combustion fume forma- tion and evolution are amenable to direct experimental study. Data from carefully designed experiments that probe individual mechanisms (coagulation, condensation, coalescence, etc.) will provide a basis for validating and refining existing models or developing new ones as required. More complex experiments will be needed to probe coupled mechanisms. Quantitative predictions of combustion fume characteristics should then become both possible and practical.

Acknowledgement

This work has been supported by the U.S. Department of Energy University Coal Research Program under Gran t Number DE-FG22-90PC90286.

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