Capture rate consequences of multispherule aggregate formation in gases—combined roles of direct interception and interspherule momentum “shielding”

ABSTRACT

Our recent work on the consequences of multispherule cluster aggregate (CA) formation and deposition-rates on much larger solid targets has emphasized the decisive role of “momentum-shielding” in determining aggregate “mobility” compared to N isolated spherules in the same gaseous environment—an effect analogous to the drag-reduction advantages experienced by birds electing to move “in formation.” The extent of “momentum shielding” is conveniently quantified via a dimensionless function: S_mom(N;Kn₁, aggregate structure), which facilitates predicting the deposition-rate consequences of aggregation in aerosol flow systems when the cluster deposition mechanism is dominated by either: (i) isothermal convective-diffusion (C-D), (ii) thermophoresis (T-P) or: (iii) inertial impaction (I-I). Significantly, isothermal C-D was found to be the only transport-mechanism leading to aggregation-induced reductions in spherule deposition rates on large targets (cf. isolated spherules present at the same mainstream spherule volume fraction). However, we demonstrate here that, for aggregate deposition on sufficiently small solid targets—e.g., fibrous filter elements with diameters of O(10 μm)—even these reductions, which exceed one decade for N = O(10³), can be overcome by the mechanism of “direct-interception” (D-I) associated with nonzero effective aggregate size, without the need to invoke either inertial impaction or thermophoresis. This is especially true for Diffusion-Limited (i.e., “open”) CAs (with D_f = 1.8) at gas pressures such that the constituent spherules are near the continuum (Kn₁ << 1) limit. Our present analysis and numerical illustrations exploit the fact that direct-interception is expected to play a negligible role for the capture of individual (dense) nanospherules (perhaps comparable in size to the prevailing gas molecule mean-free-path) but the underlying theory, exploited, extended, and illustrated here, was developed with the help of initial capture rate experimental data for much larger diameter (but unaggregated) aerosols on single filter fibers in low Re crossflow. With such small diameter targets, we demonstrate that this “interception” augmentation for large CAs can occur even for the limiting case of rcp D_f = 3 aggregates, before the expected onset of CA-inertial effects–i.e., Stk_N << Stk_crit, where, for Re = O(1), Stk_crit is also O(1). A simple method is also presented for predicting interception-modified spherule deposition rates in the presence of log-normal type aggregate size distributions.

Copyright © 2018 American Association for Aerosol Research

EDITOR:

• Yannis Drossinos

1. Introduction/motivation

Using what is now known about the mobility of large multispherule cluster-aggregates (CAs), we recently showed (Rosner and Tandon 2018) that, for uncharged aerosol deposition on much “larger” stationary targets, isothermal convective-diffusion (C-D) is the only transport mechanism (compared to thermophoresis (T-P) or inertial impaction (I-I)) for which aggregation in the mainstream (at constant spherule volume fraction, φ_s,∞ << 1) that will cause a reduction in total spherule deposition rate across the entire Knudsen number range—(including the continuum limit [Kn₁ << 1—which requires elevated pressures for sub-100 nm diameter spherules]). Moreover, this reduced deposition rate (cf. isolated spherules in the same environment) occurs despite the impressive drag-reduction associated with what we have called multisphere “momentum shielding” occurring within such large aggregates. The net result is that the decisive Brownian diffusivity ratio: (D_N/D₁)^2/3 is always less than unity—even in the continuum limit.

However, we demonstrate here (Sections 3 and 4) that, for aerosol deposition on sufficiently “small” targets—i.e., those not much larger than the effective size of large individual cluster aggregates themselves, CAs can deliver their spherules more rapidly than isolated spherules due to the mechanism of “direct interception”—even well before the onset of helpful inertial impaction effects. As a practical example, this situation is expected to occur in what is called “depth-filtration” for aerosol particle removal using “fibrous” filters. As reviewed in Friedlander (2000), such fibers (often glass or polymer) can have diameters in the 1–10 micrometer range, and often form a “matte” with solid fractions less than 0.1 and surface areas/volume in excess of ca. 30,000 m²/m³. By applying and extending what has been learned about such capture by both C-D and direct interception (D-I) using micron-sized dense aerosols (Lee and Liu 1982), we demonstrate below that for large CAs, interception can lead to aggregate:monomer spherule deposition rate ratios in excess of unity without the need for either inertia or (thermo-)phoresis. The physical assumptions underlying our present theoretical approach are first enumerated in Section 2, leading to the rather tractable quantitative interrelations exploited in Section 3. To illustrate our suggested methods, typical predictions (for large (N = O(1000)) fractal-like aggregates comprised of, say, 50 nm radius spherules with D_f = 1.8 [i.e., “DLCAs”] as well as more “compact” aggregates, depositing on, say, 10 micron diameter filaments) of FA:monomer deposition rate ratio (DRR) as a function of Kn₁ are presented/discussed in Section 4. While uncertainties remain about which effective aggregate radius of the (mobility-? gyration-? max-?) actually best describes interception on a particular solid surface, the implications of this work for controlling (maximizing or minimizing) particle deposition on solid targets in aerosol flow systems are summarized in Section 5, which concludes this work.

2. Underlying assumptions

When large aggregate inertial impaction and/or phoresis (e.g., thermophoresis) are negligible transport mechanisms, our analysis of isothermal convective-diffusion (C-D) deposition for large cluster aggregates (or their constituent spherules) on “small” solid targets immersed in a flowing carrier gas is based on the following defensible assumptions:

• A1: While the target dimension (e.g., filter fiber radius, R_t) is now assumed to be not much larger than to the effective collision radius of the CA, it remains very large on the scale of the radius R₁ of the CA's constituent spherules. On this basis, we can invoke the simplification that isolated spherule capture (from a nonaggregated mainstream) will be negligibly influenced by “interception” whereas CA capture rates in the same flow environment will be appreciably augmented by their “finite” (i.e., non-“zero”) size.

• A2: The Pe >> 1 “additive” (C-D)+(D-I) analysis (Lee and Liu 1982) (which adequately described micron-sized unaggregated spherical aerosol particle capture by comparable diameter filaments in Re ≤ O(1) continuum laminar cross-flow) can be applied to our present situation by assuming that large cluster aggregates have an effective interception radius, R_N,eff, comparable to their familiar gyration radii.

• A3: The N >> 1 cluster aggregates and, for comparison, their isolated spherules (N = 1), experience the same internal flow field in a matte of prescribed solid “targets” (here micron-sized cylindrical fibers) under the same gas flow conditions (area-averaged gas velocity, density, and viscosity).

• A4: While Stk_N is sub-critical for “pure inertial impaction”¹ we will also assume here that Stk_N is small enough to neglect what we have called significant “inertial enrichment” (of the local particle concentration in the vicinity of the target forward stagnation region [Fernandez de la Mora and Rosner 1981, 1982]). This effect is a consequence of the fact that the suspended particle “phase” (viewed as an interpenetrating “fluid”) is “compressible” even though the carrier gas velocity is sufficiently subsonic to be considered an “incompressible” fluid. Based on the low-Re/high Pe calculations included in Fernandez de la Mora and Rosner (1982) for both isolated spheres and cylinders, the ability to neglect “inertial enrichment” of the local particle concentration will probably require that Stk_N be less than ca. 0.2.

If we define the relevant dimensionless “interception parameter,” , by the size ratio: R_N,eff/R_t, then our present assumptions imply that while << 1, ≥ O(0.1). Moreover, in terms of the Stokes number governing inertial impaction (Friedlander 2000), our present neglect of aggregate inertial impaction implies Stk_N < Stk_crit, where Stk_crit for Re = O(1) flows in the absence of interception (“point-mass” particles) is itself O(1) (Rosner 2000). It follows from A1 and A4 that in effect we are assuming: Stk₁ << Stk_N << Stk_crit (Rosner and Tandon 2018). These simultaneous conditions will be seen to be satisfied in our illustrative calculations for the “initial” spherule capture rate of, say, N = 1000 DLCAs by a representative 10 micron diameter fiber filter with a solid fraction of 0.05 (Figure 1, Section 4). For cluster aggregates comprised of N-spherules, the corresponding result is simply: Stk_N = S_mom(Kn₁,N; CA-structure)*Stk₁, where S_mom is the dimensionless momentum shielding function introduced in Rosner and Tandon (2017, 2018) and shown plotted (for DLCAs) in Figure 2. Thus, we consider a “clean” (initially unloaded) filament. The time-evolution of η_N will ultimately be strongly influenced by the presence of previously captured CAs—violating our Assumption 3. Dealing with aggregate capture from a log-normal distribution of aggregate sizes via a rational, closed-form approximation will be briefly described in the Appendix.

Figure 1. Predicted spherule deposition rate ratios, DRR, in the presence and absence of Direct Interception (D-I) when the principal deposition mechanism is isothermal Convective-Diffusion (C-D). Cases shown: Capture of DLCAs comprised of N = 1,000 spherules, each of radius R₁ = 50 nm. For the D-I augmentation: (Section 3) “target” diameter d_t = 10 microns and R_N,eff = 1.19 R_N,gyr (unless otherwise specified). Polydispersed (ASD-) results (shown dashed) correspond to Equation (A10[A10] ) using spread values: σ_g(Kn₁) as estimated in Equation (A2a[A2a] ).

3. Analysis

Lee and Liu (1982) presented a Pe >> 1 “additive” diffusion (C-D) + interception (D-I) analysis which quite successfully described large, single-size (unaggregated) spherical aerosol particle capture by only somewhat larger diameter cylindrical filaments in Re = O(1) continuum (Kn << 1) laminar cross-flow. The relevant experiments covered the range of interception parameters: and “solidities” (fiber solid fractions): 0 < φ_t < 0.15, with Kn_t < 10⁻³. Implicit in their treatment was the presumption that Stk values were small enough to be neglected. Based on the subsequent review by Friedlander (2000) of these and earlier data, Stk values were probably below ca. 0.37.

If we apply their recommended expression for the single fiber capture fraction, η_cap, to both multispherule cluster aggregates (size N) and isolated sub-100 nm diameter primary spherules (N = 1) and neglect interception for the isolated spherules on micron-sized filaments (Assumptions A1, A3: <<< 1), then we find that the capture fraction ratio: η_cap,N/η_cap1, can be simplified to the following two-term expression² :[1] where:[2a]

Pe₁ ≡ U d_t/D₁ and K_C is the continuum Kuwabara cell model hydrodynamic factor:[2b]

The first term in Equation (1[1] ), i.e., (D_N/D₁)^2/3, was, in fact, the basis of our recent analysis (Rosner and Tandon 2017, 2018) of the expected aggregate: monomer spherule DRR for “pure convective-diffusion” on large solid targets—leading to the abovementioned conclusion that with C-D alone CAs would always deliver spherules less efficiently than their constituent isolated spherules. For sufficiently large N aggregates (e.g., N >> 300), we exploited effective porous medium methods (Tandon and Rosner 1995; Rosner and Tandon 2017, 2018), using adjusted sphere model (ASM) to embrace the Knudsen transition regime. Efficient computational methods for smaller aggregates, of interest in many applications and to validate ASM, have been recently been developed and illustrated by Corson et al. (2017).

The second term in Equation (1[1] ) now provides a rational estimate of the expected interception effect on DRR when the targets (in this case cylindrical fibers) are no longer much larger than the aerosol particles (here N >> 1 multispherule CAs) being collected. To exploit the simplicity of Equation (1[1] ) for predicting realistic spherule DRRs for multispherule aggregates in the presence of “direct interception,” it remains for us to select a reasonable “effective interception radius,” R_N,eff, for a fractal-like aggregate containing N-spherules. On physical grounds, we expect R_N,eff to be somewhere between the FA mobility radius (Sorensen 2011), R_N,mob (which is also used to determine D_N,eff in the first term) and the “maximum radius,” R_max: R₁*[N/β]^1/D_f, perhaps approaching the latter as D_f approaches 3. Because it is well-defined and simple to evaluate for a wide range of alternative structures, it is tempting to select the intermediate CA “gyration” radius—i.e., :R₁*[N/k₀]^1/D_f for fractal-like aggregates of “size” N. In their Brownian dynamics simulations of aggregate capture by fibers, Balazy and Podgorsky (2007) presumed that R_N,max would be an appropriate effective interception radius. As suggested by Thajudeen et al. (2014), this choice is likely to overestimate R_N,eff for D_f < 2. Indeed, using computationally intensive Monte-Carlo/Brownian-dynamics statistical simulation techniques designed to answer this question, Thajudeen et al. (2014) have shown that, for the widely studied case of “diffusion-limited cluster aggregates” (i.e., D_f = 1.8 with k₀ = 1.3), the choice (1.19)*R_gyr,N best-described their numerical results and the soot filtration experiments of Kim et al. (2009). Accordingly, Figure 1 of Section 4 will be based on this choice—along with the illustrative choices: N = 1000, and R_t = 5 micrometers. However, instructive DRR-values (at Kn₁ = 1) for the abovementioned alternative choices for R_N,eff (as mentioned above) are also included in Figure 1 for completeness (Rational yet eminently tractable DRR-estimates for the more frequently encountered situation of a mainstream ASD which is log-normal here with a median N_g,∞-value of 1000 will be postponed to Appendix 1).

Of course, it is possible that the most relevant CA-interception radius, appearing as the parameter above, has a systematic dependence on surface- and spherule-material properties—i.e., behavior which, to our knowledge, has yet to be systematically studied. In any case, we believe that our present estimates are sufficient to demonstrate the consequences of interception for increasing cluster-aggregate C-D deposition rates, not only to the level expected for isolated spherules in the same environment, but to even greater spherule collection rates—and without the “need” for inertial augmentation or supplementary phoresis.

4. Results and discussion

The illustrative results shown in Figure 1 demonstrate that when the targets (here 10 micron diameter filaments) are not much larger than the mainstream cluster aggregates, the mechanism of “direct interception” is able to increase target capture efficiencies well above those associated with isolated-spherule isothermal convective-diffusion—even without the “need” for particle inertial effects, or supplementary phoresis. (Recall that, for the C-D mechanism, aggregate capture by much larger targets was notably less efficient than in the case of isolated [unaggregated] spherules in the same environment [Rosner and Tandon 2018]—as indicated in Figure 1 by the curve marked “C-D w/o Interception” [calculated via the first term on the RHS of Equation (1[1] )]). The illustrative results shown in Figure 3 indicate that, for our 10 micron diameter fiber targets, interception will also bring DRR above unity not only if D_f = 2.1 (RLCAs) but also for the limiting case of CA “compactness”: rcp D_f = 3. For the cases of RLCAs and rcp D_f = 3 CA, we consider that the effective interception radius, R_N,eff, is given by (1.3)*R_gyr,N and R_max, respectively. The methods of Appendix 1 could be used to further generalize these illustrative results to the case of aggregate size distributions of known spread, as was done for DLCAs in Figure 1.

With Equation (1[1] ) in mind, we should also note that most of the indicated reduction in the DRR with D-I displayed at large Kn₁ is not due to reduced momentum shielding (Figure 2) but rather a reduction in the dimensionless coefficient B associated with its [C_slip(Kn₁)]^−2/3-dependence via D₁. Because for large Kn₁, the Cunningham–Millikan factor approaches 1.612*Kn₁ the D-I contribution to DRR will, therefore, decay like Kn₁^−2/3 (which would have a slope of −2/3 on log-log coordinates [like that chosen for Figure 1] if the cycle size were equal).

Figure 2. Predicted behavior of large aggregate “momentum shielding function”: S_mom(Kn₁;N,D_f) used to estimate (D_N/D₁)^2/3; after Rosner and Tandon (2018); Continuum-limit values (1/η_mom) based on permeable-sphere model; dependence on Knudsen number (via ASM) and total spherule number (N >> 1) for DLCA (D_f = 1.8) fractal aggregates (dashed extensions based on test of ASM for Kn_N << 1).

When the target size is deliberately small enough so that the aggregate interception parameter is large enough to exceed ca. 10⁻¹, we encounter the following inconsistency in extending Lee and Liu (1982) Pe >> 1 (C-D)-theory to apply (via our assumption A2) to FAs (especially with D_f < 2). As mentioned in Rosner and Tandon (2018; Section 6.2), the 2/3 exponent that appears on the C-D term of Equation (1[1] ) follows from the presumed Sc >> 1 disparity between the Brownian “sub”-layer thickness and the momentum (-defect) thickness, δ_mom, associated with the viscous fluid flow past the target. But, if the effective physical size of the FA is no longer small on the scale of δ_mom (which is O(R_t) when Re_t < O(1)), then this exponent should be altered in accordance with . This is but one sign of an inevitable “coupling” between the C-D and D-I mechanisms—associated with the need to impose the Brownian diffusion layer boundary condition at a distance of R_N,eff from the target solid surface (Friedlander 2000). However, because in our present illustration (Figure 1), we find that (DRR)_D-I >> (DRR)_C-D for Kn₁ < 10, this coupling-correction is not expected to alter our principal conclusion about the effect of interception on the aggregate: monomer deposition efficiency “competition.”

We also remark that interception-modified spherule DRR-values formally calculated as outlined in Section 3 are shown “dotted” in Figure 1 when Kn₁ exceeds ca. O(10) for the following reason. The Kuwabara flow field (within the fibrous filter), which underlies the work of Lee and Liu (1982) exploited here, will itself exhibit rarefaction effects when Kn based on R_t exceeds ca. 10⁻². Therefore (even though for N >> 1 the aggregate radius R_N,eff will far exceed R₁) when aggregate interception becomes important and Kn₁ exceeds ca. O(10), nonnegligible “slip-flow” effects within the filter matte should also be expected because Kn_t = (R₁/R_t)*Kn₁. But, for simplicity, we have not corrected the Kuwabara “hydrodynamic factor” (Equation 2b[2b] ) for its expected dependence on Kn_t (Yeh and Liu, 1974). Therefore, only DRR-values shown in Figure 1 for Kn₁ << 10 should be considered self-consistent. Yet, the Kn₁ < 1 portion of Figure 1 serves to make our principal point, viz.: irrespective of “momentum shielding,” the “interception” mechanism for N >> 1 aggregate capture is indeed sufficient to overcome the advantage that “isolated spherules with C-D alone” previously exhibited for capturing/delivering spherules on/to much larger targets from a flowing gas.

While the Lee and Liu (1982) formulation exploited here has been experimentally confirmed up to ca. = 0.2 (for dense spherical particles), formal application of our present results for N >> 1 aggregate capture to situations with _N ≥ O(1) will become problematic even for FAs with Kn₁ ≥ O(1) because of the expected effects each such particle would have on the local flow field as it approaches its solid “target.” With our present choice of R_N,eff ( = 1.19 R_N,gyr for DLCAs) all points plotted in Figure 1 “with interception” correspond to _N ≈ 0.48—which may already be marginal. (In this connection, we note that the nonaggregated numerical calculations reported earlier by Yeh and Liu [1974] for Stk = 0, Pe >> 1 formally displayed the simple analytical ²/(1+ )-dependence up to = 0.4 when Kn_t < 10⁻². Significantly, in that study, no attempt was made to correct for “near-wall” modifications to the effective local Brownian diffusion coefficient, D_eff.) We will have to leave to future study what the limits of validity of the abovementioned ²/(1+)-dependence actually are when applied to the “direct-interception” of such cluster aggregates. For the present, this simple dependence certainly facilitates our extension to predict DRR-values for (C-D)+(D-I) (with results shown dashed in Figure 1) when the mainstream contains a distribution of aggregate sizes (see the Appendix). These “polydispersed” (D-I) augmentations are seen to be especially large in the near “free-molecule” (fm) limit because the spread of such “coagulation-aged” aggregate populations is expected to increase with Kn₁ (Rosner and Tandon 2017). Because present mathematical models make no attempt to alter the arriving particle “drag” law in the immediate vicinity of the solid (fiber) surface, ironically our predicted capture rate results may be more accurate when applied to multispherule cluster aggregates (which are likely to facilitate local fluid (carrier gas) “drainage,” than for impermeable/compact particles, especially when Kn_R1 = (mfp)/R₁ < O(1).

5. Implications/conclusions

Our recent theoretical studies (Rosner and Tandon 2018) showed that, in the absence of either appreciable particle inertia or phoresis (e.g., thermophoresis), large cluster aggregates will normally deliver their constituent spherules to a much larger deposition target less efficiently than if the mainstream spherules remained isolated (i.e., unaggregated)—and this would be true for isothermal convective-diffusion (C-D) at all ratios of gas molecule mean-free-path to spherule radius—including very small (continuum-limit) Kn₁-values. However, as now illustrated in Figures 1 and 3 (for Cluster Aggregates with N = O (1000)), we now find that this disparity can be overcome for characteristic target dimensions (e.g., filter fiber radii near 5 microns) that are small enough to become nearly comparable to the effective radii of the prevailing large N cluster aggregates—i.e., by exploiting the additive effect of “direct interception.”

Figure 3. Comparison of interception-modified (and unmodified) spherule capture rate ratios for three disparate types of N = 1,000 cluster aggregates, from “open” (D_f = 1.8) CAs to “compact” (but gas permeable) rcp D_f = 3 CAs (D_f = 3 limit based on Ergun permeability [Rosner 2000] and using ASM to estimate dependence of S_mom on Kn₁).

Thus, when assessing the suspended spherule deposition-rate consequences of mainstream aggregation for sufficiently small “targets,” the additional mechanism of “interception” must be included. A rational, yet quite tractable, method for accomplishing this has been suggested (Section 3), with the principal remaining uncertainty being the relevant choice of effective Cluster-Aggregate “interception radius” for CA collisions with a comparatively smooth solid surface. For our present purposes, a reasonable first-estimate seems to be a multiple of O(1) applied to the well-defined CA “gyration” radius—e.g., :R_g,N = R₁*;(N/k₀)^1/D_f for FAs (i.e., R₁*[0.864 N^0.556] for DLCAs). While the recent Brownian-dynamics numerical simulations of Thajudeen et al. (2014) point to a correction factor of 1.19 for DLCAs (used in constructing Figure 1)—and ca. 1.3 for RLCAs (D_f = 2.1; considered in Figure 3), further research will probably be needed to improve the generality of such predictions. However, these present estimates appear to be sufficient to demonstrate the consequences of interception for increasing cluster aggregate-delivered (by isothermal C-D) spherule deposition rates, not only to the level expected for isolated spherules in the same environment, but possibly to even greater spherule collection rates—without the “need” for inertial impaction or supplementary phoresis. Moreover, an extension to facilitate such predictions for the more general case of (local) mainstream log-normal type aggregate size distributions is summarized in the Appendix.

Accounting for the simultaneous presence of convective-diffusion, subcritical Stk_N “inertial enrichment” (for, say, 0.2 < Stk_N < Stk_crit(Re_t); cf. our Assumption 4) and “direct interception” is beyond the scope of the work presented here, and tractable methods to embrace such interacting (generally nonadditive) particle transport mechanisms will inevitably be needed. Because these interacting physical phenomena may become important for aerosols comprised of materials of high intrinsic mass density, especially near the condition of maximum aggregate “penetration” (i.e., minimum capture fraction) for a low φ_t fibrous filter, this problem is of more than “academic” interest.

Nomenclature

B	=	multiplier of interception term (Equation (1[1] ))
d	=	diameter
D	=	Brownian diffusivity
Df	=	fractal dimension
Cslip	=	Cunningham-Millikan slip factor (Kneff-dependent) (Rosner and Tandon 2018)
k	=	exponent defined by ∂ ln (Smom)/ ∂ ln N:see Equation (A3[A3] )
k0	=	prefactor (relating N to (Rgyr/R1)Df)
Kn	=	Knudsen number (mfp/R1) (unless otherwise specified)
Knt	=	Knudsen number (mfp/Rt)
KC	=	Kuwabara hydrodynamic factor in the continuum limit (Equation (2b[2b] ))
	=	mass-based aerosol deposition rate
N	=	number of spherules in fractal-like aggregate (FA)
n	=	exponent defined by Equations (A7[A7] ) and (8[A8] )
p	=	Pressure (units:1 bar(105 Pa) unless otherwise specified)
Pe	=	Peclet number: Udt/D = Re * Sc
Re	=	Reynolds number: Udt/ν
	=	interception parameter: RN,eff/Rt
R	=	radius (of spherule, CA or target)
Rgyr	=	aggregate gyration radius, R1.(N/k0)1/Df
RN,mob	=	mobility radius of FA containing N spherules; ≈ N*R1*ηmom,c/Cslip(Kn1*ηmom,c/ηPA)
Rmax	=	maximum radius of FA; R1*[N/β]1/Df
RN,eff	=	effective interception radius of aggregate (Section 3)
Sc	=	Schmidt number for particle diffusion in carrier gas, νg/Dp
Smom	=	“momentum shielding factor” for cluster aggregate (Figure 2); = (1/ηmom,c)*[Cslip(KnN)/Cslip(Kn1)]
Stk	=	Stokes number governing inertial effects on target deposition rates tp/(Rt/U)
tp	=	particle stopping time; (Rt/U)*Stk
U	=	carrier gas velocity

Greek

β	=	prefactor relating N to (Rmax/R1)Df
δ	=	boundary layer thickness
φ	=	local solid fraction
ηcap	=	capture fraction
ηPA	=	normalized, orientation-avgd projected area of CA ≡ <<PA>>/[N.(πR12)]
ηmom	=	normalized continuum-limit drag of FA (cf. N- individual spheres)
μ	=	Newtonian viscosity of carrier gas
ν	=	momentum diffusivity (kinematic viscosity) of carrier gas
ρ	=	mass density
σg	=	geometric mean spread parameter in LN pdf

Subscripts and superscripts

c	=	continuum limit (Kn1 << 1)
crit	=	critical (threshold) value (for hypothetical “point-mass” particles)
eff	=	effective
fm	=	free-molecule
g	=	pertaining to geometric mean (for LN pdf)
N	=	pertaining to a CA with N spherules
p	=	pertaining to aerosol particles (aggregated or otherwise)
mob	=	mobility (drift velocity per unit force)
molec	=	pertaining to the molecular gas
mom	=	pertaining to momentum transfer
mono	=	pertaining to monodispersed (single size)
o	=	evaluated immediately upstream of the filter (or “area averaged” value within the filter)
poly	=	pertaining to polydispersed (LN ASD) case
t	=	pertaining to solid “target”(deposition site)—here cylindrical filaments
1	=	pertaining to N = 1 (single, isolated spherule)
∞	=	in the local mainstream (upstream of target) gas
“	=	reckoned on a “per unit (target) area” basis

Acronyms and abbreviations

ASD	=	aggregate size distribution (assumed log-normal)
ASM	=	adjusted sphere model; KnN,eff = Kn1*(ηmom,c/ηPA)
CA	=	cluster aggregate (comprised of N equal size spherules)
C-D	=	isothermal convective diffusion
D-I	=	direct interception
DLCA	=	diffusion-limited cluster aggregate (Df = 1.8)
DRR	=	deposition rate ratio (aggregate:isolated spherule) on cylindrical target of radius Rt
FA	=	fractal (-like) aggregate
LN	=	log-normal
mfp	=	molecular mean free path in carrier gas
mono	=	pertaining to “mono-dispersed” case (single size)
<<PA>>	=	projected area (orientation-averaged) of CA
pdf	=	probability density function
poly	=	pertaining to “poly-dispersed” case (distribution of sizes)
rcp	=	random close packing of hard spheres (i.e., φ = 0.64)
RLCA	=	reaction-limited cluster aggregate (Df = 2.1)
T-P	=	thermophoretic

Operators

O( )	=	order-of-magnitude
< >	=	volume-averaged
<< >>	=	orientation-averaged
	=	average value of ()

Acknowledgments

It is a pleasure to acknowledge the timely and helpful comments of our Yale/SEAS colleague: J. Fernandez de la Mora.

Additional information

Funding

This work was partially supported by the Industrial Affiliates of the Yale/ChE Sol Reaction Engineering Research Group.

Notes

¹ In our present notation, the Stokes number governing momentum nonequilibrium for suspended isolated spherules flowing past a target, i.e., Stk₁ (≡ t_p,1/(R_t/U), can be conveniently written: where Re_t is the Reynolds number based on U_o and (2R_t) and C_slip is the Kn₁-dependent Cunningham–Millikan mobility correction factor for the effects of gas rarefaction.

² Without comment, Friedlander (2000) reports Lee and Liu's single filament capture fraction for η_cap but divided by the (local “void” fraction) factor: (1− φ_t). However, this particular alteration clearly leaves the ratio (within the same filter matte) η_cap,N/η_cap1, rewritten as in our Equation (1[1] ), unaltered. Only if one considers applications involving nanofiber filters (Ahn et al. 2001) would it be necessary to generalize Equation (1[1] )—depending, of course, on the magnitude of the relevant spherule radius, R₁.

Appendix: Aggregation effects on spherule deposition rates in the presence of both interception and mainstream aggregate polydispersity

Subject to our underlying assumptions regarding interception and isothermal convective-diffusion (Section 2), Equation (1[1] ) of Section 3 adequately describes the aggregation effect on spherule deposition rate when all CAs in the mainstream have the same “size” N. We briefly describe here how this result can be used to quantify the aggregate: monomer spherule deposition rate ratio, DRR, when the mainstream contains a Log-normal ((LN)-) distribution of aggregate sizes—characterized by the median value N_g,∞ (>>1) and the spread parameter σ_g,∞. Thus, if Equation (1[1] ) provides for the “monodispersed” mainstream case, how can we generate the corresponding result: for the “polydispersed case?

As discussed in greater detail in Rosner and Tandon (2018), which included the mechanisms of C-D, TP, or I-I but in the absence of “direct interception” effects, this will require evaluating the quadrature on the RHS of the interrelation:[A1] where:[A2] is the number-mean aggregate “size” in the local mainstream. Suppose that the mainstream D_f = 1.8 coagulation-aged aggregates have a normalized ASD, pdf(N), which is log-normal with a spread parameter adequately described by:[A2a]

From the additive structure of Equation (1[1] ) of Section 3, we now see that the evaluation of Equation (A1[A1] ) can be broken into the sum of two quadrature expressions. But, the first of these (for “pure C-D”) has already been worked out in closed-form (Rosner and Tandon 2018) by exploiting the fact that [D_N/D₁]^2/3 can be adequately represented using a power-law fit to ({S_mom(N;Kn₁,D_f)/N}^2/3, where S_mom(N;Kn₁,D_f) is the useful dimensionless “momentum shielding function” introduced in Rosner and Tandon (2017) and exploited in Rosner and Tandon (2018). Indeed, if we write:[A3] [A4]

To facilitate the numerical evaluation, our S_mom earlier results (Rosner and Tandon 2018) for DLCAs are reproduced in Figure 2, and typical k-values for N = O(1000) can be estimated from the fit:[A5]

If we now provisionally use the estimate R_N,eff = 1.19*R_N,gyr (Section 3) then, for DLCAs, we would expect:[A6] and the second quadrature appearing in Equation (A1[A1] ) above could readily be completed numerically. However, if we represent the simple function: ²/(1+) by a local power-law (with exponent n), then we can write:[A7] where we readily find:[A8]

In this way, we are led to the following potentially useful rational closed-form approximation to the second quadrature—i.e.:[A9] where B(φ_t, Pe₁) is defined by Equation (2a[2a] ). Then:[A10]

Because the fct is always concave upward, this “local power law” approximation is expected to lead to a useful lower bound to —the accuracy of which will improve for ASDs with smaller spread-parameters. We note that when (σ_g → 1 then (A10[A10] ) reduces to Equation (1[1] ), as it must. These methods were used to superimpose such results on Figure 1. Our current Kn₁-dependent estimates for the expected DRRs (including the interception effect for capture by 10 micron diameter fibers) when there is a LN distribution of DLCAs in the local mainstream and N_g,∞ = 1000. Coupled with realistic assumptions for the ASD-spread vs. Kn₁, similar results could also be obtained for the more “compact” aggregate morphologies considered in Figure 3.

We note that, in the presence of “direct interception”, an ASD-spread (σ_g > 1.0) further increases the expected spherule DRR at all Kn₁-values—especially in the presence of significant rarefaction effects because of the increased expected spreads of such populations (Equation (A2a[A2a] )).