We present the combined power law and log-normal distribution
(PL

Particle size distribution is the most important characteristic of nanoparticles, as it controls their deposition to the human respiratory system, their behavior in the atmosphere, and the properties of engineered nanoparticles. The rates of several aerosol processes, such as condensation, coagulation, and deposition, are affected by particle size; thus, the particle size distribution controls also the evolution of the aerosol. While the rates of the aerosol processes depend on the particle size, different particles within a particle size mode have different rates of aerosol processes and, thus, they evolve with different rates. This causes also the shape of the size distribution to evolve. Because particle size distributions usually contain particles with the diameters of several orders of magnitude rather than being monodisperse, i.e., equally sized, an accurate representation of aerosol properties and evolution requires that particle sizes are expressed as distributions. Due to a high count of particles with different sizes, shapes, and compositions within a volume of interest, computational costs to model them separately are extremely too high. Therefore, aerosol models typically model one or more parameter of the size distribution, such as particle number or mass concentration of the total particle size range or of several size ranges separately. Simplifications made for size distributions in aerosol models cause unrealistic shapes for the distributions.

Methods that model a particle size distribution the most realistically are
sectional methods, in which the size distribution is split into separate size
sections. The accuracy of a sectional model can be controlled by the number
of the size sections. Increasing the number of sections increases accuracy,
but the computational cost is also increased. In multidimensional
simulations, such as in computational fluid dynamics (CFD) and in climate
simulations, computational efficiency is a key property of the model.
Simulations involving inverse modeling

Sectional methods vary depending on the conserved property of the aerosol.
Only a single property, e.g., particle number, particle surface area, or
particle mass concentration, can be conserved in the simulation but other
properties will suffer from numerical diffusion, which is seen as the
overestimation of the non-conserved properties

Other approaches to model the particle size distribution are methods based on
the moments of the distribution

The general dynamic equation (GDE) for the number concentration of a size
section

The combined PL and LN distribution model (PL

The formulation of the PL distribution originates from
Eq. (

Three moments required in the modeling of the PL distribution with
parameters

The reconstruction of the distribution parameters from the moments

The LN distribution is expressed by the equation

The combined particle distribution is modeled as the superposition of the PL
and the LN distributions:

Left pane shows examples of power law distributions with different
values of the slope parameter

Intermodal processes between the PL and the LN distributions.
Particles larger than

A schematic presentation of the connections between the distributions is
shown in Fig.

The coagulational transfer remains the only process initiating the formation
of the LN distribution if the condensational transfer is neglected.
Therefore, in the case of low particle number concentration, i.e., low
intramodal coagulation rate, the formation rate of the LN distribution is
slow; thus, the combined distribution would be mainly in a power law form.
However, in realistic particle formation events, log-normal features in the
size distribution are widely observed

The general dynamic equation for a particular moment

Intramodal processes. New particle formation forms particles with
the diameter of

New particle formation is modeled by a term

The sizes of a newly formed particle (

Condensation rate [

If all the parameters in Eq. (

When the mass growth rate is calculated from the vapor concentrations and the
properties of the vapor and the particles using
Eqs. (

Coagulation is modeled as intramodal coagulation within the PL distribution
and within the LN distribution and as intermodal coagulation from the PL
distribution to the LN distribution. The coagulation terms derived from the
equations of

The integrals in Eqs. (

The losses due to coagulation of the particles in the PL

The losses to walls due to diffusion of particles are considered the
depositional losses. They are modeled with the method of

The effect of particle losses on the PL distribution is seen as decreased

The intermodal coagulation is included together with the intramodal
coagulation in the coagulation terms (

Particles with the diameter higher than the cut diameter,

Considering a time step of

The number of particles in the PL distribution to be transferred to the LN
distribution due to the condensational transfer in the time step of

The PL

The capability of the PL

The diameter of a newly formed particle was assumed to be a constant,

Theoretical test cases were used to compare the PL

Input parameters for the test cases. Case names with Atm have the
parameter sets related to atmospheric particle formation and the Exh case
related to particle formation occurring in vehicle exhaust.

The input parameters of the test cases are presented in
Table

The Atm1 case includes simultaneous new particle formation, condensation,
intramodal-, and intermodal coagulation. For the Atm2 case, depositional
losses were also added. The deposition coefficient

The applicability of the PL

Size-dependent condensational growth rate of sulfuric acid-water
particles with the sulfuric acid vapor concentration of

Number (

The Exh case represents simultaneous new particle formation, condensation,
intramodal and intermodal coagulation, coagulational losses, and
depositional losses occurring in diesel vehicle exhaust inside the ageing
chamber of a laboratory sampling system. The values

Because the test cases are purely theoretical, the need for constructing
log-normal features to the distributions through the condensational transfer
artificially is minimal. In the Atm4 case, a time-dependent new particle
formation rate suggests using the condensational transfer, but, according to
the analysis of the shapes and the moments of the distributions, the output
is not very sensitive to the value of

The mobile aerosol chamber is a Teflon bag with the dimensions of

The particle formation event measurement was performed at a street canyon
measurement site of Helsinki Region Environmental Services Authority (HSY)
located in Mäkelänkatu, Helsinki, Finland. The street had dense traffic
during the measurement in 22 April 2015. The chamber was firstly filled with
urban air and, once filled, the air sample was sucked with the measurement
devices located in the mobile laboratory vehicle. The details of processing
the experimental data to obtain the moments (

Obtaining the values for the new particle formation rate,

Time series for the new particle formation rates in the chamber
event that produce the measured concentrations,

Coagulation within the nucleation mode was included in the simulations, but
the coagulational losses to the background mode were neglected because
low number concentration of the background mode would have a minor effect only on the nucleation
mode. The particles formed in this case are possibly multicomponent due to
the origin of the vapors, the new particle formation rate seems to vary
significantly with time, and the measured distributions are wide (GSD up to 2). Therefore, a high value for the condensational transfer factor

Firstly, inverse modeling was performed using the PL

Time series for the condensational growth rates in the chamber event
that produce the measured concentrations,

It can be seen from Fig.

The effect of the choice for the lowest particle diameter,

The effect of the choice for the value of the condensational transfer factor,

Geometric mean diameter (GMD) and geometric standard deviation (GSD) of the nucleation mode in the chamber event, obtained through inverse modeling, using different models.

Figure

To examine the accuracy and computational cost of the PL

The accuracy of the PL

All the other input parameters were the same as those used with the inverse
modeling. The simulations used to examine the model accuracies provide also
the possibility of comparing the computational costs of different models,
because all the simulations were run using the same computer (Intel Core
i5-3470 processor at 3.2

Particle size distributions at the ends of the test case simulations
produced by different models. The input parameter sets are shown in
Table

Figure

The effect of the depositional losses can be seen as a decreased

Relative errors,

Figure

The total computing time of the Atm5 case with the PL

The relative errors of the moments (

Contour plots of particle distributions measured by the Airmodus
Particle Size Magnifier (PSM), TSI Ultrafine Condensation Particle Counter (CPC), and TSI Nano Scanning Mobility Particle Sizer (Nano-SMPS) and
simulated by different models in the chamber event. The value of 0.8 was used
for

Particle size distributions in the chamber event 378 and
978

Number (

Computational costs of different models and relative errors of
number (

Particle size distributions 978

Contour plots of particle distributions simulated by the PL

Particle size distributions obtained from the FS400, the LN, and the PL

Table

The development of

Relative errors (

Figure

The combined power law and log-normal distribution (PL

The PL

Considering the same computing time as the PL

The Hermite–Gauss quadrature

The degree of the quadrature developed here,

The aerosol sample was measured using Airmodus PSM,
TSI Ultrafine Condensation Particle Counter (CPC), TSI Nano-SMPS, TSI Engine Exhaust Particle Sizer (EEPS), and
Dekati Electrical Low-Pressure Impactor (ELPI+). PSM in fixed saturator flow
setting detects particles with the diameters of higher than about
1.6

Initially, the aerosol in the chamber consisted of a background aerosol mode
with CMD of 15

This work was funded by the Maj and Tor Nessling Foundation (project number 2014452), by Tampere University of Technology Graduate School, and by the Finnish Funding Agency for Technology and Innovation (Tekes) as a part of the CLEEN MMEA program. Authors acknowledge the personal of the Air Protection Group at the Helsinki Region Environmental Services Authority (HSY) for enabling the measurement campaign at the HSY's street canyon measurement site in Helsinki.Edited by: V.-M. Kerminen