**3. Particulates characterization by shape, irregularity and charge in flow conditions**

The Knudsen (Kn) variable is the mean free path length (λ, distance) to particle physical diameter (d) ratio, Kn > 0.01 < 10; and the slip correction for frictional drag velocity is a reciprocal function of the Knudsen variable [53]. The aerodynamic diameter, *D*ae, to Kn flow regime transition relationship applies to non-homogenous air viscosity in flow conditions [16]; and the direct particle density (*ρ*p) to dynamic shape factor (*Χ*) relationship (*ρ*p/*Χ*) is applied to normalize the volume equivalent diameter (*d*ve). Inaddition to particle morphology and density, the other independent variables for elutriator filtering are particle flow and charge, and include the Millikan apparatus [16].

Certain relationships are known: i) the volume equivalent diameter (*d*ve) is the flow voids-adjusted mass equivalent diameter (*d*me δ) to shape factor adjusted for standard density; ii) aerodynamic diameter (*d*ae, *d*a) is the shape increases with increasing particle density and the diameter with electrical charge (*d*m) is larger than the *d*a; iii) particle density (*ρ*p) increases semi-exponentially with the dynamic shape factor, *X*, which is for irregular particles with no flow voids [16]; and iv) the ratio of the *d*<sup>a</sup> to *d*ve (*d*a/*d*ve) increases between continuum and free molecule air flow types (transition regime) for particulates with shape factor-normalized particulate density <sup>&</sup>gt; 1 (*ρ*p/*<sup>X</sup> <sup>ρ</sup>*0; *<sup>ρ</sup>*0, 1.0 g/cm<sup>3</sup> ) ratios, and it decreases in transition flow for particulates with <1 density. The electrical mobility diameter (*d*m) is greater than the volume equivalent diameter (*d*ve) for internal void-containing particles, irregular non-sphere and aggregate particulates with voids in-between.

As per the above discussion, i) for less irregular particulates (*X* < 1) there is a smaller aerodynamic diameter as per a decrease in the *d*a/*d*ve ratio over the flow

regime Knudsen-Weber (Kn) path length (*λ*), and there is an increase in the aerodynamic diameter with more shape irregularity (*X* > 1). In Ref. to the effective particle density (*ρ*eff II) defined as the *measured* particle mass (*m*p) divided by (i.e. normalized to) 0.125 π-adjusted spherical volume (*d*<sup>m</sup> 3 ), the relationship between the internal flow voids-adjusted particle density (*ρ*p) and the dynamic shape factor is *ρ*p/ *X*, in which case an inverse relationship holds between the particle dynamic shape factor and effective particle density [15], where particle density is considered constant (*k*). There is also a relationship between the electrical mobility diameter (*d*m) and the particle shape with the *d*<sup>m</sup> increased with more particulate matter irregularity (*X*, dynamic shape factor > 1–2.5) [15], which is consistent with aggregation or agglomeration via ionic or non-ionic colloidal interactions, and in which case, there is an infra-*log* linear (saturable) increase in particles diameter (*d*p) with increasing particle number (*N*pp) consistent with overlapping of particles in aggregates (in solution) and agglomerates (upon condensation).

As there are lesser than expected boundary effects in the gaseous medium, there is an altered frictional drag (*F*d) relation, the Navier-Stokes function with variables, coefficient of viscosity (η) and velocity of spherical particle (*R*v) is modified [53], where a correction factor is applied for air velocity slip due to a lower *F*<sup>d</sup> in real air flow. For imperfectly spherical nanoparticulates, the determination of particle aerodynamic equivalent diameter (*D*ae) is based on the baseline normalized measure, the particle density to standard density ratio (*P*p/*P*o; g/cm<sup>3</sup> fract) [16]; the effective dynamic shape factor (*Χ*'; *F*p/*F*me) for particle is the ratio of the resistance (drag) force of the actual particle to that of its mass equivalent diameter (*d*me), which is the diameter for a non-spherical irregular particle un-adjusted for internal flow voids as compared to the volume equivalent diameter (*d*VE); and the fraction of internal voids correction factor (*δ*) is for particulates that possess internal voids and/or exterior irregularity with voids. Furthermore, the determined effective agglomerate diameter of irregularly shaped nanoparticles follows a non-linear relationship for its impaction along the respiratory tree during normal inspiratory flow of 30 liters per minute.

Inaddition to the direct relationships between particle size and settling velocity (*V*s) or sedimentation time (*t*s), relative humidity and hydration-based exponential growth, several other aspects have been further characterized by study of virus aerosols and droplets with particle number and volume size concentration and emission small and large particle distribution (B1, B2; Q1, Q2 > 5 μm) comparison modeling of pore size threshold limits of facial PPE (fitted masks) to larger diameter particles [54], which build on earlier works on determining of flow void- adjusted aerodynamic equivalent diameters (*D*ae) from mass equivalent (*D*me) of a 0.125x volume spherical particle as reference.
