3266
Ind. Eng. Chem. Res. 1995,34, 3265-3277
Translational Brownian Diffusion Coefficient of Large (Multiparticle) Suspended Aggregates Pushkar Tandon and Daniel E. Rosner* High Temperature Chemical Reaction Engineering Laboratory, Department of Chemical Engineering, Yale University, New Haven, Connecticut 06520-8286
Aggregates (composed of large numbers of “primary“ particles) are produced in many engineering environments. One convenient characterization is the fractal dimension Df, the exponent describing how the number of primary particles in each aggregate scales with radial distance from its center of mass. By viewing each ensemble of aggregates of fixed size N as a radially nonuniform but spherically symmetric “porous solid”body, we describe a finite-analytic, pseudocontinuum prediction of the drag, and corresponding translational Brownian diffisivity, for a fractal aggregate containing N ( ~ 1 primary ) particles in the near-continuum (Kn> 1) spherical “primary” particles, our idealized mathematical model is based on the following assumptions: Al. The (ensemble-averaged) aggregate behaves like a quasispherical “porous” granular solid with positiondependent porosity, 6 , associated permeability, x , and nearly uniform sized granules (“primary” particles). A2. Brinkman’s equation defines the flow field inside the porous aggregate while the Stokes equation describes the flow field outside the aggregate. A3. Darcy permeability, x, is nearly isotropic and a function of local porosity only. A4. The local Knudsen number is much smaller than unity, even inside the aggregate. (However, see section 3.6.) A5. The ambient (host) fluid and aggregate may be approximated as isothermal; i.e., each primary particle has nearly the same steady state temperature regardless of its position within the aggregate. Comments on the validity and relaxation of some of these underlying assumptions will be postponed to section 3.2. 2.2. Aggregate Structure for Dr 5 3. For an aggregate whose &rt relation is of the power-law form, R r t B(r/RIP, where Ntrf is the total (expected) number of primary particles contained within a sphere of radius r, and the prefactor p is a dimensionless number discussed below, we have the basic relation:
-
which describes the (expected) number of primary particles (of radius R1) within a shell of thickness dr located a t radius r from the aggregate center of mass. This, in turn, implies that the local solid fraction, @W, is given by
- -
We note that, for Dt 3, @ B so that in this limit the dimensionless factor 5,’ may be identified with the value ofthe solid fraction in the uniform porosity limit; Le., B = @fim = 1 - qim, which, for a random loose packing of uniform size impenetrable spheres, will be about 0.6.3 However, since, in general, p is not independent of Dr, this limiting argument only fixes its order of magnitude, not its exact value.4 Beyond some radius RmaXwe treat
Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995 3267 the domain as primary particle-"free"; i.e., we treat r = as though it were a sharp boundary of the large N aggregate. A reasonable choice for R m , is (Rosner and Tandon, 1994)
R,,
where R, is the well-defined gyration radius of the assembly of "primary" particles about its point center of mass and R+Rm,j = N . The porosity function, e++, for this aggregate model can thus be written E
= elimx 0.4
r
permeability relation based on the Kozeny-Carman relation, but corrected for its inaccurate low 4 behavior. Our drag results using the modified Kozeny-Carman permeability expression results do not depart noticeably from those using Happel law (section 3.2). A single scalar stream function, v, is now introduced in eqs 2.5 and 2.6. In spherical coordinates, the relationship between radial and angular velocities, U r and ue,and the stream function, v , are
-1 9 u, = r2 sin e
(2.8a)
IR,
(2.8b) In terms of this stream function eqs 2.4 and 2.5 transform to
E = 1
r > R,,
(2.4)
2.3. Translational Brownian Diffusion Coefficient of F'ractal Aggregate (Rit*: 1). The ensembleaveraged fractal aggregate is viewed as a rigid permeable sphere; with its porosity having the radial dependence defined by eq 2.4. We consider incompressible Newtonian creeping flow outside the quasispherical aggregate. Stokes equation and the overall mass balance defining such a flow are p div grad u = gradp
r > R,,
(2.5a)
div u = 0
r
> R,,
(2.5b)
where u is the velocity vector, p is the thermodynamic pressure (Rosner, 19861, and p is the Newtonian fluid viscosity. For reasons discussed in section 3.2 the corresponding equations governing such a flow inside the aggregate will be5 taken to be
+ div grad u = grad p
r < R,,
(2.6a)
div u = 0
r
R,,
(2.12a)
e)]sin28
where A, B , C,D,E , F,G, and H are eight constants of integration and 5' r/& is the appropriate dimensionless radius. Using the conditions specified in eq 2.11, we find that C = -1 and D = 0. Knowing the stream function, the local radial and angular component of velocities, as well as the normal and tangential component of stress tensor (n,,and nre),are computed as outside the aggregate:
In the low 4 limit this x relation is seen t o behave like (2Rd2/(l8r$), as required for noninteracting Stokes spheres. However, since corrections to this result may be important in the domain of moderate void fraction (Happel, 1958), we have also examined an alternate
(2.9b)
A u, = u cos e[7 u,=--
u s 2i n e [
+ B - 11
:: 1
---+--2
3268 Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995 Rmni
inside the aggregate, in a region of constant permeability:
u, = -
U sin 0 -E + W + G($sinh T[7
cosht
- -l?-j
coshi
sinhi
sinhE
9
n
Figure 3. Discretization nomenclature for implementation of “finite-analytic” method (FAM)to solve numerically for the viscous flow inside a large, quasispherical multiparticle aggregate of radially nonuniform porosity.
fluid stresses is also satisfied a t the outer surface (see, eg.,Rosner (1976,1986,1990))of the aggregate, i.e., a t r = Rms. Satisfying the above conditions a t each node,
along with the boundary conditions discussed earlier, give us the values of constants of integration in each zone i (A, B, Ei,Fi, Gi,Hi;i = 1, ..., n). The flow field thus generated is used to calculate the drag force on a spherical surface containing essentially all of the aggregate, Le., We solve for the entire flow field inside the fractal aggregate and outside it using a finite-analytic method outlined below (see, e g . , Rosner (1986) and Rosner and Tandon (1994)). The solution to eq 2.9 as reported in eq 2.12 is for a (locally)constant permeability. However, in the aggregate permeability x depends on r through the local porosity, d r j , of the medium (as discussed in section 2.2). Thus, to construct the flow field in our representation of the “fractal” aggregate, we assume that the effective porous medium consists of a central core of constant porosity, E I ~ , , enveloped by successive concentric annular shells. Thus, following the interesting precedent of Ooms et al. (1969), the porosity (and therefore permeability) is assumed to be constant within each of these shells but differs from shell to shell (in accord with xCc$j- evaluated at the arithmetic mean radius of each shell), as given by eq 2.4. The analytic solution as given by eq 2.12b holds within each of these shells for local value of permeability, while eq 2.12a describes the flow field outside the aggregate, i.e., for r > R,, where R,, is calculated using eq 2.3 (Rosner and Tandon, 1994). As illustrated in Figure 3, the aggregate has an inner core and an outer region which is divided into n zones (shells) and n nodes. At each node i ( i = 1,2, ...,n - 11, we impose continuity of both velocity and contact stress in the viscous fluid, i.e.,
FD =
h2zd$
de Rmm2sin 8 (n,, COS 8 -
An alternate way of calculating the drag on the aggregate is to sum the contributions to the body force acting on the fluid, i.e., (2.16) It is straightforward t o show that the drag calculated using eqs 2.15 and 2.16 are equivalent.6 The translational Brownian dfisivity of the aggregate in the continuum (low Knudsen number) regime is then evaluated using the Einstein relation (Bird, Stewart, and Lightfoot, 1960) as (2.17) where k g is the Boltzmann constant, T is the absolute temperature, and the ratio FdU is the so-called friction coefficient, or inverse aggregate mobility.
[ ( ~ r ) r = r ~ l z o n e == i [(ur)r=rLlzone=i+l
3. Results and Discussion [ ( ~ ~ ) r = r , l z o n e == i [(~~)r=r,lzone=I+~
[(~r,9)r=r,1zone=i
= [ ( ~ t , e ) ~ = , , l ~ ~ ~ ~ = i + (2.14) l
Continuity of the normal and tangential velocities and
3.1. Drag on a “Fractal” Sphere and Corresponding Brownian Diffusivity. In Figure 4, we show the calculated drag force as a function of number of primary particles, N , for various values of the fractal dimension, Df.We report our results as dimensionless drag force, 4?+N,LI+ (actual drag normalized with respect t o Stokes drag force 6~@m,U ( d D , r e f ) , on an
Ind. Eng. Chem. Res., Vol. 34,No. 10, 1995 3269 1
0.95
0.8 0.75
‘
10
’
’ ’
’’
,’
100
l O o 0 0 1OOOOO 1E+006
loo0
N Figure 4. Predicted drag force (normalized by the Stokes drag on an impermeable sphere of radius Itm=) on an aggregate of fractal dimension Dfcomprised of N primary particles.
impermeable sphere with identical outer radius Rm, and velocity U in the same host fluid), Le.,
(3.1-1) For Df = 3, corresponding to the case of a quasispherical solid with constant porosity, 6, and permeability, x, we recover the result obtained by Brinkman (1947) and Debye and Bueche (1948), uiz.,
where K RmJ&. Ironically, this result was developed to represent the behavior of a random coil polymer molecule in solution-however, the approximate spatiallyaveraged permeability is somewhat ambiguous in such a case. Yet, by introducing the notion of an effective , show below that (“stretched”)value of K , written K ~ E we this functional form can serve as the basis of a useful correlation formulas to predict fractal aggregate drag and Brownian diffusion coefficient. Our spatially variable porosity results are predicted for particular fractal dimensions that have been observed in aggregated systems of engineering interest. The aggregate-aggregate ballistic case, corresponding to a fractal dimension of 1.8, has been observed in a wide variety of systems, including that of carbonaceous soot in rich hydrocarbon fuellair flames (Megaridis and Dobbins, 1990; Koylu et al., 1995a); however, most individual aggregates are probably ”too stringy“ (azimuthally nonuniform) to be adequately treated using our quasispherical approximation. Of course, these metastable aggregates can go through partial restructuring ( e g . , due to coalescence, surface diffusion, Brownian motion of primary particles, etc. (see, e g . , Tandon and Rosner (1995) and Cohen and Rosner (1995)) and, accordingly, intermediate Df values have been reported, e g . , Df= 2.18 for small silver aggregates studied by Schmidt-Ott (1988) and for colloidal silica aggregates by Aubert and Cannel1 (1986). The fractal dimension of 2.5 (close to that obtained in aggregate-monomer ballistic simulations (Meakin and Witten, 1983) has been reported for metal oxides produced by spray-precipitatiordoxidation
in the seeded premixed laminar flame experiments of Matsoukas and Friedlander (1991). The normalized drag force, atN,D+, on the aggregate (actual drag normalized with respect to the Stokes drag force on a solid impermeable sphere of identical radius Rmm) is seen to decrease as the aggregate becomes more open (i.e., as the fractal dimension decreases). With an increase in size of the aggregated particle (for a given fractal dimension) @+N,Drf increases and approaches unity; Le., the drag approaches the Stokes drag on a solid (impermeable) sphere with an identical outer r a d i ~ s .The ~ hydrodynamic radius calculated using our numerical procedures mentioned in section 2 appears t o be about 20% higher than those reported by Chen et al. (1987)for a simulated cluster of fractal aggregates with Df = 2.1.1° Fbgak and Flagan (1990) have used the method of reflections to estimate the hydrodynamic radius of fractal aggregates, and their reported values were close t o those reported by Chen et al. (19871, who used a similar method t o calculate drag on a set of simulated aggregates. The reason for the discrepancy between our result and theirs may lie in the fact that the morphology and structure of ensemble of aggregates they studied is rather different from the ones we consider. As pointed out by Rogak and Flagan (1990) and Neimark et ul. (19951, there are limitations to characterizing such aggregates by a single fractal dimension alone, even if one is only interested in orientation-averaged diffusivities. We are now extending our effective porous medium methods to incorporate additional morphological information. Paradoxically, Chen et al. (1987)report the ratio of hydrodynamic (mobility) radius to the maximum radius (or ratio of mobility radius to the radius of gyration) t o be independent of aggregate size, a result evidently a t variance with the predictions of our present pseudocontinuum model (Figure 4). Our method can be viewed as an extension of the approach of Debye (1948)l’ and Brinkman (1947) while the point-drag “Stokeslet” method of Rogak and Flagan (1990) is an extension of KirkwoodRiesmann (1948)11theory. It is not clear which method is more accurate (Happel and Brenner, 1986), but Stokeslet methods are not likely to be accurate or efficient for compact aggregates. In this limit, our methods, combined with a relevant empirical permeability law (Cozeny-Carman, section 3.2) are expected to provide better estimates of the drag on these aggregates. In the other limit, when aggregates are open and highly permeable, the Happel permeability law realistically describes the flow through the aggregate, and our methods, combined with this law, are expected t o generate accurate results provided real (ensembles of) aggregates can be treated as quasispherical (see section 3.2 and Neimark et al. (1995)). In any case, the simplicity of the present predictiodcorrelation methods implies that any future systematic corrections found t o be necessary in some domain of N , Df,Kn2R1,... parameter space could be readily applied to these rational “base-case” estimates. Equation 2.16 and the definition of the normalized drag reported in Figure 4 imply that the aggregate Brownian diffusiuity will be smaller than the StokesEinstein Brownian diffusivity of each of its “primary spheres” by the factor
DdD1 = (l/4?+N,Dfj)(N/j3)-1’Df
(3.1-3)
where 9, the abovementioned normalized drag, can be appreciably less than unity. On this basis we have constructed the instructive summary plot, Figure 5,
3270 Ind. Eng. Chem. Res., Vol. 34, No. 10,1995
‘m
0.95
0:
.
t 0.75 0.75
’
a
0.8
0.85
0.9
0.95
1
(P)Md
1
100
10
1000 1
0
0
~ 1E+006 1 ~
N
Figure 5. Predicted dependence of Brownian diffusivity of a quasispherical multiparticle aggregate (normalized by the corresponding Stokes-Einstein Brownian diffisivity of each primary particle) of fractal dimension Dfand aggregate size N (log-log scale) in the continuum (Kn 300, below which their results are systematically smaller than ours. Drag results using our finite-analytic method, but with their permeability law, are found to be within 6% of our results (for N > 40 and Df > 2). Aggregates found in coagulating sytems, in general, are comprised of a large number of so-called “primary” particles of different sizes (Albagli et al., 1994). Moreover, the “primary“ particles themselves are sometimes microporous particle restructured aggregates of the real primary particles, in principle allowing some flow through them. Our methods may need to be generalized to embrace such situations, eg., to predict the sensitivity inside the of our drag results to the known pdf(dl,eff,...) aggregate. Although a large number of aggregates show the characteristic of having their porosity increase with distance from their center of mass, others have giant pores located anywhere inside the aggregate through which significant flow “channeling“takes place. To the extent that an ensemble of large aggregates can be represented realistically by a rigid porous sphere model, our methods/predictions/correlationsare expected to be accurate. However, we take the view that our model and convenient correlation methods (section 3.1) provide useful “base-case” estimates. In the (likely) event that further research indicates the need for systematic corrections in some domain of the Df, N , Knm, ... parameter space, they cadshould be applied to the relations presented in section 3.1. Regarding the realism of this spherically symmetric model, it has also been reported that “small”aggregate-
aggregate encounters lead to larger aggregates for which the gyration radii are not same in all directions. Botet and Julien (1986) and Hentschel (1984) report an asymptotic gyration radius of ca. 2. Moreover, electron micrographs of soots sampled from diverse systems usually reveal a wide variety of nonspherical aggregate morphologies, often rather “stringy“ in character (Ulrich, 1984; Lahaye and Prado, 1981; Neimark et al., 1995). If necessary, our “porous object” model could be generalized to deal with the asymmetric character of large aggregates (Williams and Loyolka, 1991; Payne and Pell, 1960) but the advantages of our finite-analytic numerical method in some of these cases may no longer be available. Due to our interest in ensemble-averaged and orientation-averaged transport properties, these extensions have been postponed and will be taken up, as required, in future studies. Even though the simple scaling law d In &/d In r = constant Df has been found t o apply down to very small aggregate sizes (Schmidt-Ott, 1988,1990) clearly, our pseudocontinuum mathematical model (in which each aggregate is treated as a quasispherical porous object with x1I2 *:2 R d should not be expected to apply to very small aggregates, especially when Df < 2 (say, aggregates comprised of less than about 30 primary particles). The methods discussed above are strictly valid only for very small Knudsen numbers, even inside the aggregate; Le., the gas mean free path, I,, must be much smaller than even the diameter of a primary particle (lg *: %I > 1 (when calculated for the relevant that fractal dimension of the aggregate population). The more general and daunting problem of predicting the evolution of the joint pdf, pdffN,Df,Rlf, for the aggregate population is beyond the scope of this preliminary paper but will inevitably be necessary in developing the emerging field of sol reaction engineering (SRE). 3.6. Internal Rarefaction Effects. Provided the ambient gas mean-free path, I,, remains sufficiently small compared to 2R,, for the aggregate, it should be possible t o estimate the incipient effects of gas rarefaction15 by allowing for rarefaction effects on the permeability x, within each concentric shell comprising the large aggregate. ks a first approximation, this has been done by merely introducing the Stokes-Cunningham drag (slip) correction factor (inferred strictly for an isolated sphere, hence, valid in the limit 4 0); i.e., we assume
-
(3.5-2)
where @+NP+ is the dimensionless function displayed in Figure 4. The collision radius for each aggregate is generally somewhat smaller than our “maximum” radius of the aggregate due to inevitable “interpenetration” when both aggregates are characterized by Df -= 3. Considering the simplest (degenerate) case in which both participating aggregates (A,B) are characterized
x % xc*SCF-tKne&
(3.6-la)
where xc is the previously used (continuum limit) permeability and SCF is the Stokes-Cunningham slip correction factor for drag on an isolated sphere, i.e.,
SCFfKnf = 1
+ 2Kn(1.257 + 0.4 exp(-
E)) (3.6-lb)
Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995 3273 and KneR is calculated using following interpolation formula:
0 C 4 I4rcp (3.6-2) which has the necessary limiting behavior, i.e., the relevant dimension (to which 1, must be compared) changes from 2R1 in the limit 4
0.94
t.'
0.91
/
9-25
0.98
0.96 3
I
0.94
0.92
-
0.9
'2.0.88 0.88
,
I,'
0.84
0.82
where DN is the diffusion coefficient calculated above for a fractal aggregate containing N primary particles, each of mass ml. Now consider a primary particle-laden flow characterized by the Stokes number Stkl. If the primary particles aggregated in the same flow t o form fractal N-mers, then our results show that the governing Stokes number would increase in accord with
where the presence of the factor [B+N,Drfl-' is seen to increase StkN but slightly reduce its sensitivity to N. For example, this equation implies that if one wanted to form a dense granular coating, the formation of weakly bound aggregates could be used to not only cause impaction (which would raise the capture efficiency, especially if Stkl was subcritical; see Konstandopoulos and Rosner (1995))but also cause impaction at incident kinetic energies able to cause aggregate-aggregate breakup (raising both the capture fractions and the solid
I
0,-zr,
io
I
I
100
loo0
I
A
loo00 1OOOOO l E + W
Figure 7. Predicted dependence on the Knudsen number (Knm,) of the normalized drag force on a large aggregate of fractal dimension&comprised of N primary particles. Effect of "internal" rarefaction in reducing the aggregate drag and increasing the corresponding Brownian diffusivity.
fraction 4 in the green granular deposit). A further analysis of this rather interesting situation will be the subject of a follow-up paper (Konstandopoulos and Rosner, 1995). 4. Conclusions and Future Work
We have developed and illustrated a convenient pseudocontinuum "hydrodynamic"method to predict the translational Brownian diffusion coefficient of a quasispherical aggregate comprised of a large number ( N ) of primary particles, with fractal dimension D f . Applied t o large ensembles of fractal aggregates, our methods are expected to be mostly accurate in the regime 2 5 Df 5 3, except for the fact that we impose an isotropic permeability to represent an actual permeability which
3274 Ind. Eng. Chem. Res., Vol. 34,No.10,1995
is inevitably somewhat anisotropic. However, this approximation may prove t o be acceptable in the continuum regime even if it requires improvement in the Knudsen (free-molecule) regime (see, e g . , Rosner and Tandon (1994),Tassopoulos and Rosner (19911,and section 2.3). As an important corollary of this calculation of drag and translational diffision coefficient, we also derive an improved estimate of the important coagulation rate constant for unequal size aggregates of fractal dimension less than 3 in the near-continuum (Kn > 1, Df 2 2) has also been developed to predict the restructuring kinetics of a multiparticle fractal aggregate undergoing either Brownian rearrangement or surface energy driven viscous flow collapse including the associated evolution of its fractal dimension, “primary”particle size, and transport properties (Tandon and Rosner (1995);see, also, Cohen and Rosner (1995)). It should now be possible to adopt a similar approach t o estimate these large aggregate transport properties (Brownian diffisivities, thermophoretic diffisivities) in the importanP free-molecular regime and, ultimately, in the more difficult Knudsen transition regime. When combined with rational methods to predict the restructuring kinetics of multiparticle aggregates, the above results can be applied in controlling the synthesis of valuable powders (Xing et al., 1995,19961,interpreting in situ measurements of aggregate population behavior in suspensions, and predicting deposition rates from flowing aggregate-laden gases (Rosner and Tassopoulos, 1989) in many situations of technological importance (gas cleaning, coating formation, fouling, ..). The simplicity of the present predictiodcorrelation methods (section 3.1) implies that any future systematic corrections found to be necessary in some domain of the N , Df, Knm,,.... parameter space could be readily applied to these rational, “base-case” estimates.
.
Acknowledgment This transport theory paper is dedicated to each of the authors of Transport Phenomena on the 35th anniversary of its publication. This remarkable book has strongly influenced the teaching of undergraduate and graduate Transport Processes at Yale University since one of us (D.E.R.) joined the ChE faculty in 1969. Indeed, much of its philosophy permeates D.E.R.’s Transport Processes in Chemically Reacting Systems, which first appeared in 1986 (second edition in preparation). The research summarized in this paper was principally supported by the US AFOSR (under Grant AFOSR 94-1-0143) as part of our long-range program on multiphase chemically reacting flows, and sol reaction engineering (SRE).We also gratefully acknowledge SRE-related support of the Yale HTCRE Laboratory by
our Industrial Affiliates (ALCOA, Shell, SCM-Chemicals, GE, and DuPont) and helpful discussions with J. L. Castillo, R. D. Cohen, P. Garcia-Ybarra, A. G. Konstandopoulos, U. 0. Koylu, A. V. Neimark, and Y. Xing.
Nomenclature Df = fractal dimension; i.e., the exponent d In nld In r Dtrans,agg = translational Brownian diffusion coefficient Drat,%, = rotational Brownian diffisivition coefficient of an aggregate FD = drag force f = function defined by eq 3.1-2 g = function defined by eq 3.5-3 k g = Boltzmann constant Kn = Knudsen number (eg.,Knm, = ld(2Rd) I, = mean free path of the gas N = total number of primary particles in the aggregate (Figure 2) = cumulative number of “primary“particles up to radius r N = mean number of N in aggregate population p = pressure r = radius (measured from aggregate center of mass); Figure 2 R1 = radius of primary sphere; Figure 2 R,, = maximum (outer) radius; defined by eq 2-4 R, = inner core radius Figure 3 R,A = effective collision radius of aggregate A Re = Reynolds number; U(2Rm,)!v T = temperature u = velocity vector U = host fluid velocity far upstream of aggregate (relative to aggregate center of mass)
iv
Greek Letters /3 = filling factor (eq 2.1) x = Darcy permeability; defined by eq 2.7 p = dynamic viscosity of host fluid = local void fraction within aggregate structure K = R , d h (when x is constant) 4 = local solid fraction (1 - €1 v = momentum diffisivity ple (kinematic viscosity) of Newtonian carrier fluid JC = stress tensor in Newtonian fluid, or 3.14159... t = torque E = normalized distance from the center of the aggregate; defined as rf& (when x is constant) 11 = stream function e = complement of latitude angle 5 = defined by eq 3.5-4 Miscellaneous BSL = Bird, Stewart, and Lightfoot c = pertaining to collision, continuum, or core div = divergence operator eff = effective value of FAM = finite-analyticmethod fm = free-molecular g = pertaining to host gas grad = spatial gradient operator Z = differential operator (eqs 2.9,2.10) lim = limiting value pdf = probability density function rcp = random closed packing rot = rotational SCF = Stokes-Cunningham (slip correction) factor SRE = sol reaction engineering
Ind. Eng. Chem. Res., Vol. 34,No. 10,1995 3276 trans or tr = translational 1 = pertaining to “primary“ particles < > = orientation-average -f j= argument of a function
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Footnotes (1) The word “primary“ here (and throughout this paper) should not be taken to mean “original”, since at any point in a reactor particles that appear to be the primary particles comprising the aggregates are usually themselves the results of growth a n d or coalescence of the real (much smaller) primary particles (see, eg., Cohen and Rosner (1995) and Xing et al. (1995,1996)). (2)-Fractal dimension may thus be defined as d In R/d In r, where N is the total number of primary particles contained in an imaginary sphere of radius r drawn about the aggregate center of mass (see eg., Figure 2 and Meakin and Witten (19831, Mountain and Mulholland (1984), Dobbins and Megaridis (19871,and Schmidt-Ott et al. (1988,1990)). While Df appears to be a “robust” characterization of aggregates in a wide variety of “sooting“systems (Megaridis and Dobbins, 1990; Koylu et al., 1995a), it may not be a s u m i e n t descriptor to define the requisite orientation-dependent transport properties-especially for Df < 2 or aggregate restructuring kinetics (Cohen and Rosner, 1995; Neimark et al., 1995). (3) Indeed, in general, eq 2.1 applies only outside of a uniform porosity “core” of radius R,, where
5 = (EEfL)
1/(3-D~)
R,
3 1 - elim
This equation indicates that the core radius is necessarily not much larger than R1; Le., quite small on the scale of the overall aggregate size. (4) The experimental value of /3 for Df = 1.8 is found to be 1.35 (Koylu et al., 1995). (5) See Castillo and Garcia-Ybarra (1994) and Tam (1969) for a critical discussion of the self-consistency of using Brinkman’s equation when xu2 is not small, cf. Rmm. (6) The surface integral Is(nn)dS can be converted to the volume integral using the Gauss-Ostrogradski theorem (Bird et al., 1960), and on substituting the expression for the stress tensor in terms of pressure and velocity gradient components and using Brinkman’s equation, we obtain the anticipated result. (7) The function, AK),for small values of K behaves like
f ( K 3 = (2/9)K2[1- (4/15)~’+ ...I while, for large values of K ffK3
= 1 - (1/K) f
...)
(8)This strategy was also used to obtain a successful correlation
for the internal effectiveness factor of a large fractal aggregate (Rosner and Tandon, 1994). (9) In the range of N and Df values involved in Figure 4 we find that the exponent a In Bla In N never exceeds about 0.025. (10)Wiltzius (1987)has used dynamic light scattering experiments to infer the hydrodynamic radius of silica aggregates, obtaining values even smaller than those reported by Chen et al. (1987). However, his results probably reflect only some average for a polydispersed aggregate population. (11) These methods were used to estimate the translational friction coefficient on polymer coils in solution. (12) It is interesting to note that this quantity arises naturally in an asymptotic analysis of the problem of porous sphere drag for slowly varying (spatially) Darcy permeability (see Castillo and Garcia-Ybarra (1994)). (13)For a straight chain (Df = 1) we formally define (Eq. 2.3) ( R m a h , = l E (3)’’Rgym
Ind. Eng. Chem. Res., Vol. 34,No. 10,1995 3277 (14) Thus, the Df= 2 free-molecule regime calculations of Mulholland et al. (1988) indicate thatgb(O.5)= 0.86. On this basis alone, in the absence of additional information one might provisionally assurneghfC3 = 1- 0.57(1 on the interval 0.4 5 6 5 1,corresponding to the preliminary estimate that g b GS 0.79 for Df= 1.8 aggregates; i.e., interpenetration is expected to reduce the effective aggregate collision radius (below Rmm) by about 21% in the free-molecular limit. Firmer conclusions about the effective collision radius of fractal aggregates, including the effects of ambient gas mean free path, will evidently have to wait the outcome of size-selective aggregate coagulation experiments (eg., Drayton and Flagan (1994)) and further computer simulations. (15)The complete free-molecule limit would also be tractable (using Monte Carlo simulation methods), but we are, as yet, unaware of such a treatment'results. (16) Especially for soot aggregates in high-temperature, atmospheric (or subatomspheric) pressure flames.
(17) In this 1994 paper, the relationship between aggregate maximum radius and radius of gyration should be baaed on aggregate moment of inertia about its center of mass (point), rather than about one of its axes. This eliminates the factor (3/2)"* and leads to our present relation between R,, values eq 2.3. and R;, Received for review December 19, 1994 Revised manuscript received August 7 , 1995 Accepted August 9, 1995" IE940747W
Abstract published i n Advance ACS Abstracts, September
15, 1995.
