Novel Features of Aerosol Coagulation in Nonisothermal Environments

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Novel Features of Aerosol Coagulation in Nonisothermal Environments Daniel E. Rosner*,† and Manuel Arias-Zugasti‡ † ‡

Chemical & Environmental Engineering Department, Yale University, New Haven, Connecticut 06520-8286, United States Departamento de Física Matem
atica y de Fluidos, Universidad Nacional de Educaci
on a Distancia, Apdo: 60141, 28080 Madrid, Spain ABSTRACT: Local supersaturations and the rapid homogeneous nucleation of often subcooled nanometric particles occur when vapors from a high-temperature source diffuse into adjacent regions of much lower temperature. While earlier experimental and theoretical studies were focused on the macroscopic mass transfer rate consequences of such localized condensation (Turkdogan and Mills (1964), Epstein and Rosner (1970)), we identify and describe here, from a broader perspective, several expected novel features of aerosol coagulation dynamics in nonisothermal environments, including the expected particle size “spectra”. In the simplest (limiting) case, differential thermal particle drift down the local temperature gradient dominates the Brownian diffusioncontrolled collision rate, and, for a population of thermally conductive mist droplets coagulating by this mechanism, we have calculated and displayed (e.g., Rosner, D. E.; Arias-Zugasti, M. Thermophoretically-dominated aerosol coagulation. Phys. Rev. Lett. 2011, 106, 015502.) the expected coagulation-aged “self-preserving” droplet size distribution. However, in most applications the more familiar Brownian diffusion contribution to the coagulation rate must also be included, and we summarize here our recent analysis of this combined case, which enables another test of the frequently made assumption of “additive rate constants” and leads to quasi-self-preserving droplet size distributions. Because fully miscible mist droplets may also be distributed with respect to composition (and, hence, thermal conductivity) we take this opportunity to outline how these, and more general phoretic mechanisms (including thermocapillary flow), can be incorporated into the present formulation.

r 2011 American Chemical Society

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vector r r T that counts, irrespective of its direction. Note that the r T can be viewed as the positive “material frame-invariant” r r T)1/2. Hence, we are presenting a more general root: (r rT 3 r nonlinear “constitutive law” for the coagulation rate constant, that satisfies the requirement that all observers would agree upon. It is also noteworthy that this present contribution, again dealing with heat transfer effects on suspended particle motion, actually bears some thematic relation to our contribution to the Churchill 70 issue of I&ECR (Rosner et al.7). There, and in a more detailed paper by Mackowski et al.,8 we showed that in an environment with a spatially uniform volume-averaged temperature, thermal boundary layers adjacent to (radiation-cooled, or laser-heated) individual particles can modify near-continuum aerosol coagulation frequencies and evolving particle size distributions (PSDs). Of course, we should anticipate situations in which both of these distinct thermophoretic mechanisms operate simultaneously. )

1. INTRODUCTION, MOTIVATION The nucleation of condensable vapors often occurs because of supersaturations achieved in the presence of appreciable temperature gradients (see, e.g., references 1
3, and Katz and Ostermier4). Yet, such data have previously been analyzed on the basis of homogeneous nucleation and coagulation rate theories based on the (usually unstated) presumption that “what counts” is the local temperature and not the spatial gradient. The validity of this fundamental assumption has motivated our recent theoretical studies of the coagulation-rate consequences of spatially nonuniform carrier gas temperature. Remarkably, we have recently discovered that temperature gradients in many applications should be able to modify aerosol coagulation rates and resulting particle size “spectra”. The purpose of this paper is to present: (a) our theoretical approach to these anticipated phenomena (section 2), (b) criteria for their likely importance, (c) some chemical/environmental engineering implications of our recent findings, and (d) an outline of future necessary or desirable extensions. Other goals of this presentation are to motivate future experimental verification and extensions of these predictions. It is perhaps of interest to also mention that our claim that the local rate of aerosol coagulation in an isotropic gas mixture can be influenced by the magnitude of the local temperature gradient appears to violate a fundamental principle of the thermodynamics of linear irreversible processes (TIP), viz.: a vectorial process should not be able to “couple” with a scalar rateprocess—a principle usually ascribed to P. Curie (see, e.g., de Groot and Mazur5 and Rosner and Arias-Zugasti6). However, our present treatment falls outside the domain of linear TIP because we find (section 2) that it is the magnitude of the local

2. PARTICLE (MIST) COAGULATION DYNAMICS IN NONISOTHERMAL GASES 2.1. Occurrence of Conductive Mist Nucleation within Thermal Boundary Layers. Preliminary calculations suggested Special Issue: Churchill Issue Received: December 7, 2010 Accepted: May 19, 2011 Revised: May 17, 2011 Published: May 19, 2011 8932

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~ (eq 2) as a function of Knudsen number (gas mfp/particle diameter), for particle/gas Figure 1. Dimensionless particle thermophoretic diffusivity R Fourier thermal conductivity ratio kp/kg = 0.1, 1, 10, 102, 103, 104 and ¥ (after Talbot et al.15 and Rosner and Arias-Zugasti3), the arrow indicates increasing value of the ratio kp/kg in the principal figure.

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that the effects considered here are actually encountered in metallurgical mists/fumes, especially for high temperature operations carried out above atmospheric pressure. But even for micrometer-sized metal aerosols produced at ca. 1 atm pressure we have estimated that the thermophoretic coagulation rate constant (section 2.2.2) can exceed that for Brownian motion by over 1 decade. An early example of this type is provided by the laboratory experiments of Turkdogan and Mills1—who reported condensation-enhanced diffusion-controlled evaporation rates (ca. 3-fold) of levitated, inductively heated Fe(l) þ Ni(l) alloy droplets into 350 K He at p = 1 atm. (For a simple approximate theory to account for these evaporation rate enhancement factors, “bypassing” details of the local nucleated particle size distributions, see ref 9). Droplet diameters were ca. 0.64 cm and Tw values were near 2400 K, implying that nominal temperature gradients were ca. 0.6 K/micrometer (values close to those obtained locally in many combustion operations). In this case the (pressure-insensitive) particle/gas thermal conductivity ratio kp/kg values were also probably of the order of 102. Thus, despite the fact that Kn-values probably exceeded unity (see section 2.2.1), our presently estimated TP-coagulation rate constant for particles of, say, 0.3 and 3 μm was over 20 times that expected for Brownian diffusion acting alone. Moreover, a typical Peclet number of the type Vdriftd/D, (readily shown to be equal to the product of the particle thermal diffusion factor and the relative change in temperature over one particle diameter) was of order 103, thus providing an experimental situation of practical interest where the recently introduced process of thermophoresis induced coagulation3 is relevant. The consequences of both mechanisms (Brownian diffusion and differential thermophoresis) acting together are considered in section 2.2.4, summarizing a much lengthier treatment by Arias-Zugasti and Rosner.10 2.2. Coagulation Dynamics of Mist Populations of Uniform Composition (w/o Capillary Effects). 2.2.1. Criteria for Thermophoretic Effects on Aerosol Coagulation. As will be demonstrated below, thermophoretic effects on aerosols coagulating in the presence of a temperature gradient are likely to rT , become dominant when, in addition to (C1) large local r the following somewhat less intuitive additional conditions are also met, viz.: the suspended particles have (C2) a much greater effective Fourier thermal conductivity than that of the carrier gas;

i.e.: kp/kg > 102, (C3) mean diameters considerably larger than the prevailing gas mean-free-path; i.e. Kn < O (10
2), (C4) an adequate “spread” in diameters. As summarized below (section 2.2.3), these expectations have been verified based on the predicted time-evolution of initially log-normal continuous distributions of spherical particles, having specified a range of initial spread-parameters σg,0 centered about 2 and sufficiently small population-mean Knudsen numbers (based on Sauter mean diameter, d32). 2.2.2. Approximate Rate Constant for Differential Thermal Drift; Size-Dependence Outside of the Free-Molecule Limit. Our present formulation is based on the phenomenon of sizedependent particle thermophoresis, when the underlying mechanism of isolated particle phoresis outside of the free-molecule (Waldmann and Schmitt11) regime is the “thermal creep” boundary condition of ideal gas kinetic theory (see, e.g., Loyalka,12 Rosner and Papadopoulos,13 Davis and Schweiger14). First, we define a dimensionless thermophoretic particle “diffusivity”, ~  RTDP/νg, by the vector expression written here as: R ~ νg 3 ð
r Vp ¼ R rln TÞ

ð1Þ

where νg is the carrier gas momentum (kinematic) diffusivity, and RT (also dimensionless) plays the role of an effective (Soretlike) thermal diffusion factor. For a spherical particle with intrinsic Fourier thermal conductivity kp and diameter dp the ~ has been found to depend on particle size (via decisive factor R the Knudsen number: Kn = mfp/dp) and thermal conductivity ratio kp/kg, in accord with the simple semiempirical relation15 (plotted in Figure 1) ~  R

2Cs 3 ððkg =kp Þ þ 2Ct KnÞ 3 C ð1 þ 6Cm KnÞ 3 ð1 þ 2ðkg =kp Þ þ 4Ct KnÞ

ð2Þ

Inspection of this result (for any particular finite kp/kg-ratio) ~ , the feature central to the reveals that the size-dependence of R coagulation mechanism considered here, peaks at intermediate Kn and vanishes in both the “free-molecule” (fm) and continuum (c) limits. (Ironically, motivated by our interest in a combustiondriven process for the production of optical waveguide fiber preforms, we previously considered the relative importance of this mechanism for the near free-molecule limit case of 8933

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completeness we note here that Brownian diffusion acting alone (e.g., in the absence of a temperature gradient) leads to the familiar rate constant: βB, 12 ¼ 2πðD1 þ D2 Þðd1 þ d2 Þ   2kB T C1 C2 ¼
ðd1 þ d2 Þ 3μ d1 d2

ð6Þ

where D1 and D2 are the corresponding Brownian diffusion coefficients, and C(Kn) is the Cunningham
Millikan
Stokes-drag correction factor (C f 1 in the continuum limit), often wellapproximated by (see, e.g., Friedlander,18 (eq 2.21))   
A3 CðKnÞ ¼ 1 þ 2Kn A1 þ A2 exp ð7Þ Kn ~12 = β12/βref, as a Figure 2. Dimensionless coagulation rate constant, β function of participating particle diameter ratio d2/d1 for several values of their common thermal conductivity ratio: kp/kg. Case shown, Kn1 = 10
2.

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micrometer-sized doped glass microdroplets in combustion products near p = 1 atm.16,17 In that environment Brownian coagulation (eq 6) dominated and thermophoresis played a dominant role “only” in determining the microdroplet deposition rates to an adjacent cool surface. As emphasized in section 2.2.1 thermophoretically modified coagulation rates are more likely to become important for thermally conductive particles in the nearcontinuum (small Kn) regime.) Inspection of the limiting value ~ ¥, reveals that, at most, ~ for kp/kg f ¥, written below as R of R ~ ∼ Kn ∼ dp
1 (locally). Provisional values of all dimensionless R molecule/surface interaction coefficients (Cs, Ct, Cm and the Ai (i = 1, 2, 3) appearing in the Cunningham-Millikan Stokes drag correction, C(Kn) (eq 7), were taken from ref 18. Our estimate of the collision rate “constant” (with units m3/s) associated with the differential thermophoretic drift of particles of diameters d1 and d2 is obtained by adapting Smoluchowski’s estimate for the collision frequency of non-Brownian particles overtaking one another in a dilute suspension laminar shear flow, as summarized by Friedlander.18 This method leads to the simple result: π βTP, 12 ¼ ðd1 þ d2 Þ2 V 1
V 2 ð3Þ 4 ~12  Combining eqs 2 and 3 led us to define a nondimensional β β12/βref where π ~ ¥, 1 νg rln T βref  d12 R ð4Þ 4 Accordingly, we express the dimensionless binary collision rate constant in the following compact form:   ~1
R ~ 2j d2 2 jR ~ ð5Þ β TP, 12 ¼ 1 þ 3 ~ R ¥, 1 d1 which, for kp/kg and Kn in the range of maximum interest (see below) is not especially sensitive to either parameter. Figure 2 ~12 for the intermediate shows the interesting d2/d1 behavior of β (transitional) case: Kn1 = 10
2. In section 2.2.3 we briefly explore the interesting consequences of this coagulation rate constant when introduced (as a “kernel”) into a Smoluchowski-type population balance integro-differential equation. However, for comparison and

2.2.3. Thermophoretically “Dominated” Coagulation; SelfPreserving PSDs. To examine some of the distinguishing features of thermophoretically dominated coagulation we inserted this coagulation rate constant into the Smoluchowski integrodifferential equation for the evolution of a continuous number density distribution function (say, n(v,t), where v is the particle volume: (π/6)d3). In particular, we examined the predicted timeevolution of initially log-normal continuous distributions of spherical particles, having specified the initial spread-parameter σg,0 = 2 and Knudsen number (based on Sauter-mean diameter, d32). We employed efficient numerical methods, combining a quadrature(based) method of moments (QMOM) (McGraw19) with orthogonal collocation (Arias-Zugasti20). As a check, when we inserted the Brownian diffusion rate constant (eq 6 with C = 1) we recovered the now-familiar Brownian self-preserving PSD for compact spheres in the continuum limit (Figure 3a). With our thermophoretically dominated kernel (eqs 1
3) the long-term PSDshape is modified, as also shown in Figure 3a for the particular cases Knref = 10
2, kp/kg = 3  102. In Figure 3b we show the “cumulative” self-preserving PSDs corresponding to pure Brownian coagulation (gray line) and thermophoretically dominated coagulation (solid line). This figure shows that the thermophoretically dominated self-similar PSD is associated with a larger fraction of particles below v, and on the other hand has a much longer tail for large particle volumes, which can also be inferred from the results for skewness (see Figure 4c). These mist population evolution simulations displayed some of the essential “fingerprints” of TP-driven coagulation, which included nonmonotonic approaches to the asymptotic values of spread, skewness, and flatness (Rosner and Arias-Zugasti3). In particular, the longterm geometric mean spread parameter σg,¥ (at 2.74) was noticeably larger than that obtained (2.28) for the continuumlimit isothermal Brownian coagulation case. 2.2.4. Arbitrary Pe12; Non-Additivity of Rate Constants. In the case of different coagulation processes acting together, the (questionable) assumption of rate constant additivity is in widespread use in the literature, although this simplifying assumption is generally not rigorous. In the work summarized here a presumably more accurate approach has been used—viz., the rate constant for combined Brownian (B) þ thermophoretic (TP) coagulation has been obtained by solving the corresponding convection þ diffusion equation in the immediate vicinity of a target particle. The diffusion term is a consequence of the participant particle Brownian motion and the convective term represents the superimposed relative thermophoretic drift between the 8934

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Figure 3. (a) Predicted self-preserving coagulation-aged PSD for the case of thermophoretically dominated binary coagulation with kp/kg > 10 (thin solid lines, after Rosner and Arias-Zugasti3); comparison with asymptotic PSD shape for continuum isothermal Brownian coagulation (thick gray line computed with present numerical scheme, and solid dots after Pratsinis and Pratsinis37—for aggregate fractal dimension =3). (b) Corresponding cumulative self-preserving coagulation-aged PSD.

particles, approximated here by a rectilinear thermophoretic drift (i.e., neglecting creep flow hydrodynamic interactions between the particles). As a consequence of this approximation, the convective-diffusion equation that determines the coagulation rate is the same as the one for the combined Brownian þ differential sedimentation coagulation case (Simons et al.21), exchanging their sedimentation velocities by the corresponding thermophoretic velocities. Thus, the combined coagulation rate constant is given by β12 ¼ ðβB, 12 þ βTP, 12 Þ 3 FðPe12 Þ

ð8Þ

where the correction factor F(Pe) is21 FðPeÞ ¼

Inþ1=2 ðPe=2Þ 4π=Pe ¥ ð
1Þn ð2n þ 1Þ Knþ1=2 ðPe=2Þ 4 þ Pe n ¼ 0

∑

ð9Þ

where I and K are the modified Bessel functions and Pe is the Peclet number, defined as the ratio between the relative thermophoretic and Brownian diffusion velocities )

V1
V2 ðD1 þ D2 Þ=ða1 þ a2 Þ )

Pe12 

ð10Þ

As a consequence of the low convergence rate of the series in eq 9 and the extremely large numbers involved when the Peclet number becomes large, the numerical evaluation of eq 9 for large Pe is a time-consuming mathematical task (Sajo22). For that reason, the convenient approximate least-squares formula recently provided by Sajo23 has been used for our present calculations. 2.2.5. Characteristic Features of Such Coagulation-Aged Populations; Spread, Symmetry, and Peakedness; Insensitivity to kp/kg. A significant number of simulations has been performed for several values of the Peclet number between the pure Brownian and pure thermophoretic coagulation limits, considering both the simple addition of both kernels and the more accurate combined kernel introduced in eq 8. In both cases the transition between the pure Brownian coagulation asymptote (Pe f 0) and the pure thermophoretic coagulation asymptote (Pe f ¥) is nonmonotonic, exhibiting nontrivial behavior for intermediate values of the Peclet number (see Figure 4).

As a typical example, the results shown in Figure 4 correspond to the time evolution of an initially log-normal distribution of particles, with a particle/gas thermal conductivity ratio kp/kg = 103 and initial Knudsen number Kn = 10
2 (based on the Sauter mean diameter d32 of the initial distribution). These results have been computed using QMOM with 6 abscissae (i.e., a total of 12 moments of the distribution has been tracked). It has been observed that as soon as kp/kg > 10, the results obtained were almost insensitive to thermal conductivity. As could be expected, the difference between the results obtained with the simple additive kernel and the results obtained with the combined kernel given by eq 8, peaks at intermediate values of the Peclet number. In this respect, the most remarkable difference found corresponds to the time, tf, needed to reach the quasi-self-preserving (QSP) droplet size distribution function. As can be seen in Figure 4a, where tf is shown in units of the reference time defined by pure Brownian coagulation (tref,B), the time needed to reach the QSP PSD for intermediate Peclet numbers is considerably smaller than would have been expected by the simple addition of both coagulation kernels. Regarding the shape factors of the QSP droplet size distribution function, that is, geometric standard deviation, skewness, and kurtosis (all of them in terms of particle volume to average particle volume ratio: v/v), the differences between both kernels are appreciable for intermediate values of Pe, but remain moderate (see Figure 4b
d, where broken lines correspond to the additive kernel, and dots joined by solid straight lines correspond to the combined kernel given by eq 8). On the other hand, both kernels show a nonmonotonic transition between both asymptotes, with a nontrivial behavior for Pe near 1. In this regard, we believe that the apparent discontinuity observed in Figure 4 for Pe between 0.6 and 1 is an artifact caused by the condition used to stop the calculation. As a consequence, the results corresponding to that range of Pe-values are considered as nonconclusive, and have been shaded in Figure 4. The range of intermediate values of the Peclet number is also where the largest geometric standard deviations σg (Figure 4b), skewness γ1 (Figure 4c), and kurtosis β2 (Figure 4d), are found. On the other hand the range Pe ≈ O (10
1) is where the smallest departures from a log-normal self-preserving size distribution are found, although these discrepancies are quite significant for all values of Pe. This is shown in Figure 4 in terms of the 8935

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Figure 4. Results for combined thermophoretic þ Brownian coagulation as a function of the Peclet number. The dashed lines correspond to results obtained with the simple addition of both Brownian and thermophoretic kernels. The dots joined by straight lines correspond to results obtained with the combined kernel given by eq 8. The gray lines correspond to pure Brownian or pure thermophoretic coagulation. (a) time needed to reach the QSP distribution function; (b) geometric standard deviation in terms of particle volume; (c) skewness and (d) kurtosis both in terms of particle volume. Skewness and kurtosis are also shown as a multiple of the corresponding best-fit value for a log-normal population.

symmetry-related parameter skewness, γ1, (Figure 4c) and the peakedness-related parameter kurtosis, β2, (Figure 4d), which are also shown as a multiple of the corresponding results obtained for a least-squares-fitted log-normal self-preserving distribution function (see subfigures shown in Figure 4c and 4d). We remind the reader that all the QSP droplet size distribution shape factors shown here correspond to particle volume (not to particle diameter).

3. NECESSARY GENERALIZATIONS: CURRENT AND FUTURE STUDIES In view of the simplifying assumptions we have introduced, as well as current interest in more complex environments, it is now possible to foresee necessary extensions, some straightforward, some less so. 3.1. Dynamics of Binary Aerosol Mist Coagulation in a Temperature Gradient. Except in the free-molecule limit,

aerosol mist droplets in a nonisothermal carrier gas will drift at different speeds not only because of size differences but also because of differences in their Fourier thermal conductivities (cf. Figure 1). (For the present we assume that supplementary changes in the gas/liquid molecular accommodation coefficients Cs, Ct, and Cm and the Millikan-Cunningham drag parameters A1, A2, and A3 are subordinate to the effects of changing droplet Fourier conductivity. To explore this previously overlooked coagulation mechanism, we have initiated studies of the interesting model problem of coagulating mists comprising two initially pure but chemically distinct fully miscible components—such as would result from the gradual micromixing of two initially pure

~12 = β12/βref, as a Figure 5. Binary coagulation rate constant “surface”, β function of participating particle diameter ratio d2/d1 and Fourier thermal conductivity ratio k2/k1. (“Trough” shown corresponds to the locus of vanishing relative thermophoretic velocity for the condition: Kn1 = 3  10
2, kp,1/kg = 10).

mists in an environment with a strong local temperature gradient (perhaps 105 K/m). For simplicity, we start with the bivariate (BV) case in which liquid composition and droplet size (e.g., volume) are sufficient to define the “state” of each member of the evolving population of “non-Brownian” internally well-mixed 8936

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Figure 6. Fourier thermal conductivity of homogeneous LiF þ KF solutions as a function of composition for T = 1000, 1100, 1200, 1300, and 1400 K (after Smirnov et al.24).

spherical particles—with a mean diameter large compared to the prevailing carrier-gas mean-free-path. Clearly, the required thermophoretically dominated two-particle collision frequency function: β(v1,v2;ω1,ω2), can be calculated from eqs 1 and 3 provided the droplet thermal conductivities are known as a function of composition. The expected consequence for the collision rate constant is illustrated in Figure 5, for the particular case of kp,1/kg values of order 10 and Kn1,ref = 3  10
2. To fix ideas, the dependence of kp on droplet composition is displayed in Figure 6 for the particular case of LiF(l) þ KF(l), for which the required property measurements are extensive (see, e.g., Smirnov et al.24). In this relatively nonvolatile system volume changes upon mixing are negligible and liquid viscosities are probably low enough to facilitate rapid intradroplet mixing. Because, in this system, there is more than a 2-fold change in thermal conductivity, Figure 5 leads us to expect appreciable effects on the evolution of the joint PDF of volume and composition. By these means we can examine the consequences of thermal conductivity variations for the evolution of the continuous BVdistribution function n(v,ω;t). This function is governed by a nonlinear Smoluchowski-type population-balance equation (integro-PDE). Constant mist mass fraction numerical simulations are being carried out using the above-mentioned Gaussian quadrature-based method of moments (QMOM), as extended into the bivariate domain (Wright et al.25). Now the initial condition is that of two individual log-normal mist populations, each of uniform but different composition (e.g., LiF(l) and KF(l)), sharing the same nonisothermal gaseous environment. Instructive comparisons can then be made with a “univariate” treatment that completely ignores differences in droplet thermal conductivity. Our goal is to understand the operational/environmental conditions under which this differential thermophoretic mechanism participates in, if not dominates, the overall coagulation process. These coagulation calculations may be of Separations interest because coagulation would inevitably act to degrade the performance of any scheme based on exploiting k-variations in a common imposed temperature gradient. 3.2. Role of Surface Tension-Driven (Thermocapillary) Microdroplet Phoresis: “Competition” with Thermophoresis. Our use of “solid body” thermophoresis results for predicting microdroplet transport in a temperature gradient requires further discussion because it is well-known that intradroplet microflows

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can be driven by surface tension gradients. Such flows can retard (or even reverse!) microdroplet motion in nonisothermal environments (see, e.g., Subramanian and Balasubramaniam26). Our present results (section 2 and refs 3 and 10 ) can perhaps be viewed as applicable to engineering environments in which such liquid/vapor interfaces are “contaminated”, retarding any surface tension gradient-driven flows. (Recall that the use of gas/liquid experimental results (e.g., oil droplet drag in low density gases) to describe gas/solid situations is common in aerosol science/ technology, where contaminated liquid surfaces are avoidable only if extraordinary precautions are taken). That said, the consequences of Marangoni-driven flows for coagulating mist populations are certainly of theoretical interest and this topic is suggested for further study. Such extensions are beyond the scope of our present paper, which can therefore simply be regarded as applying to situations for which the viscosity ratio: μl/μg is large (say on the order of 10
1000), but not “too large” (cf. the effect on the coalescence time, section 3.5). 3.3. Extensions to Mists Suspended in Dense Vapors; Sooting High-Pressure Diffusion Flames; Nonisothermal “SASP”. Despite its technological importance, the subject of particle thermophoresis outside of the domain of ideal gas kinetic theory remains an open question (see, e.g., refs 27 and 28). For small departures from ideality, as is relevant to soot evolution in high pressure laminar diffusion flames, the virial EOS approach of Rosner and Arias-Zugasti28 appears to be promising, but remains to be developed. Enskog-like extensions to higher carrier vapor molecular volume fractions (>O (10
1)) will probably be needed to properly deal with supercritical fluid particle processing applications Rosner and Arias-Zugasti6—especially in the deliberate presence of simultaneous temperature gradients. 3.4. Capture Efficiency Considerations for Droplets of Comparable Size. In the language of cloud physics, we have been assuming “unit collision efficiency (fraction)”—as might be appropriate for binary collisions for which the average relative kinetic energy is insignificant compared to typical droplet surface energies (i.e., negligible Weber number). However, especially for the encounter of two spheres of comparable size under nearcontinuum conditions, it should be expected that our present coagulation rate constant estimate (eq 3) requires improvement— with inevitable consequences for the evolving droplet size distributions. Interesting examples of such effects may be found in the extensive emulsion literature (see, e.g., Wang and Davis,29 Ismail and Loewenberg,30 and Rother31). 3.5. Further Generalizations: Immiscible Liquids, Aggregate Formation, Competing Processes: Nucleation, Growth, etc. Of future interest will also be the dynamics of “encapsulated” particles (comprising viscous non-miscible liquids). From the viewpoint of thermophoretic-drift in a nonisothermal ideal gas, such “structured” (shell þ core) microdroplets would have a calculable effective thermal conductivity. Presumably, the outer annulus properties would be heavily weighted and would also determine the above-mentioned gas/liquid surface molecular coefficients Cs, Ct, Cm, A1, A2, A3 in the Knudsen transition regime). Our initial goal will be rational predictions of the coagulation rates of such encapsulated viscous immiscible liquid populations in nonisothermal ideal gases. We also remind the reader that in the treatments above we have presumed that the postcollision rate process of coalescence and spheroidization are rapid compared to the rate of binary encounters that lead to coagulation. However, as emphasized by Friedlander,18 high liquid viscosity and/or lower temperatures 8937

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Industrial & Engineering Chemistry Research often leads to aggregate formation—with dramatic changes in effective collision diameter and thermophoretic behavior. In this connection, our previous studies of the thermophoretic properties of “fractal-like” aggregates (see, e.g., Rosner et al.,32 Filippov et al.,33 Zurita-Gotor and Rosner34,35) suggest that their insensitivity to size and morphology will rarely lead to augmented coagulation in a strong temperature gradient. (Ironically, it is this insensitivity that led to the success of the now widely used experimental technique of “thermophoretic sampling” (Dobbins and Megaridis,36 Rosner et al.32)). Lastly, while we have deliberately focused attention here on instructive limiting cases in which the coagulation of preexisting microdroplets occurs in a strong local temperature gradient, in important engineering applications one must deal with the simultaneous role of additional rate and transport processes, including not only Brownian diffusion (section 2.2.4) but also particle nucleation (“birth”) and growth (from the surrounding supersaturated vapor). The present study and its immediate extensions should facilitate such calculations by now providing rational collision rate expressions which reduce to familiar limiting cases in the absence of these competing phenomena.

4. CONCLUSIONS AND IMPLICATIONS Previous studies of aerosol coagulation dynamics have presumed that the relevant coagulation rate “constant”, β12, is dependent only on the local temperature and not its spatial gradient. Yet, in many “high-intensity” applications, bursts of particle nucleation occur in regions experiencing rather large temperature gradients, sometimes approaching 1 K/micrometer. For this reason we have recently turned our attention to the simultaneous role of particle size-dependent thermophoresis and near-continuum regime Brownian diffusion in shaping “coagulation-aged” (asymptotic) aerosol particle size distributions. This paper has presented a brief summary of this new line of research and identified several extensions of current and future interest. The extreme case of initial thermophoretic “domination” was a natural starting point (Rosner and Arias-Zugasti3), but our present methods/results (section 2.2.4) apply to more general situations often encountered for smaller temperature gradients (