On the Effective Density and Fractal–Like Dimension of Diesel Soot Aggregates as a Function of Mobility Diameter

Abstract

A new technique is proposed for the assessment of aggregate morphology based on combined information of aerodynamic and mobility size distributions. Instead of formulating a complex inverse problem having aggregate morphology as unknown, the actual problem is separated to two stages. The aerodynamic distribution is determined directly by the electric low pressure impactor (ELPI) data whereas the morphology is independently assessed by matching the distribution arises from ELPI and from scanning mobility particle sizer (SMPS). The advantage of this approach is that it can estimate different fractal dimension for different aggregate size. The approach is applied to soot aggregates from three different diesel engines. In all cases, a non-monotonic behavior of fractal dimension versus aggregate size is observed. In particular, the fractal dimension initially decreases and then increases (passing through a minimum) as the mobility diameter increases.

Appendix: Detailed calculations of the CDF fitting procedure

1.1 Effective density, ρ_e calculation

The first soot aggregate descriptor, effective density ρ_e, is defined by Maricq and Xu [2] by a relationship that compares the two different measures of equivalent particle size, namely the aerodynamic particle diameter d_a and the mobility particle diameter d_b. Such definition is described by Eq. A1.

$$ \frac{d_a}{d_b}=\sqrt{\frac{SCF\left({d}_b\right)}{SCF\left({d}_a\right)}}\cdot \sqrt{\frac{\rho_e}{\rho_0}} $$

(A1)

where SCF(d_a) and SCF(d_b) are the aerodynamic and mobility Cunningham slip correction factor respectively, while ρ₀ is the unit density (= 1 g/cm³).

Cunningham slip correction factor is defined as:

$$ SCF=1+\frac{2\lambda }{d}\left({A}_1+{A}_2{e}^{\frac{-{A}_3\cdot d}{\lambda }}\right) $$

(A2)

where, λ is the mean free path, A₁ = 1.257, A₂ = 0.4, and A₃ = 0.55.

Mean free path is defined as:

$$ \lambda =\frac{\mu }{\rho}\sqrt{\frac{\pi MW}{2 RT}} $$

(A3)

where, μ is the fluid’s dynamic viscosity, ρ is the fluid’s density, MW is the fluid’s molecular weight, R the ideal gas constant, and T is the fluid’s temperature in SI units.

Knudsen number is defined as:

$$ Kn=2\lambda /d $$

(A4)

As is clear from definitions A2 to A4, SCF is a Knudsen number and mean free path depended factor. Therefore, the ρ_e definition (A1) includes parameters that are all functions of the soot particle diameter:

$$ \mathrm{SCF}\left({d}_{\mathrm{a}}\right)\kern0.5em \mathrm{SCF}\left({d}_{\mathrm{b}}\right) $$

$$ \mathrm{Kn}\left({d}_{\mathrm{a}}\right)\kern0.5em \mathrm{Kn}\left({d}_{\mathrm{b}}\right) $$

$$ {\uplambda}_{\mathrm{a}}\kern0.5em {\Lambda}_{\mathrm{b}} $$

where λ_a is the aerodynamic mean free path for air dynamic viscosity and density at the conditions applied in each ELPI stage and λ_b is the mobility mean free path for air dynamic viscosity and density at atmospheric conditions, while Kn(d_a) and Kn(d_b) are the aerodynamic and mobility Knudsen numbers respectively.

In order to determine the soot aggregate effective density with the use of (A1), we need to know the experimental mobility diameter distribution of the sampled aerosol. CDFs are calculated from SMPS and ELPI measurements and after normalization they give dimensionless, experimental mobility CDF_b and aerodynamic CDF_a respectively. Solution of Eq. (A5) results in a particle diameter distribution of “fitting factor” f (Fig. 2): f_i is calculated for each one of the particle diameters that corresponds to ELPI’s impaction stages. Such calculations result to a triplet of experimental mobility diameter d_b, aerodynamic diameter d_a and fitting factor f based on the measurements of SMPS and ELPI.

$$ CD{F}_a\left({d}_a\right)= CD{F}_b\left(f{d}_a\right) $$

(A5)

Taking into account that experimental mobility particle diameter—according to the above fitting—should be equal to the theoretical mobility particle diameter; we solved the system of Eqs. (A2) to (A4) and calculated the effective density, ρ_e particle distribution for the cases I and II, as described in Section 3.

1.2 Fractal-like dimension, D _f calculation

Naumann [20] introduced the COSIMA analysis in order to calculate the fractal-like dimension, D_f using the effective density, ρ_e. According to this analysis mobility equivalent radius, R_b is a function of the fractal-like dimension D_f (Eq. A6).

$$ {R}_b={h}_{KR}\cdot {R}_{geo}=\left(-0.06483{D_f}^2+0.6353{D}_f-0.4898\right){R}_{geo} $$

(A6)

where, h_KR is the Kirkwood–Riseman ratio accounting for shielding effects and hydrodynamic interactions and R_geo is the geometric radius of fractal particles. The latter is defined as:

$$ {R}_{geo}={R}_0{\left[{f}_v{\left(\frac{R_m}{R_0}\right)}^3\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${D}_f$}\right.} $$

(A7)

where, R₀ is the radius of primary particle of soot aggregate and equals to 16 nm, f_v is a volume filling factor and equals to 1.43 and R_m is the mass equivalent radius as defined at Eq. (A8) for the already calculated effective density and the experimental mobility radius distribution.

$$ {R}_m={\left(\frac{{R_b}^3{\rho}_e}{\rho_0}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.} $$

(A8)

Solving the system of Eqs. (A6) and (A8), mobility particle distribution of fractal-like dimension D_f(d) is calculated for all engines and operating conditions measured (Fig. 5).