the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Inversely modeling homogeneous H_{2}SO_{4} − H_{2}O nucleation rate in exhaust-related conditions

### Jenni Alanen

### Marja R. T. Palmroth

### Topi Rönkkö

### Miikka Dal Maso

The homogeneous sulfuric acid–water nucleation rate in conditions related to vehicle exhaust was measured and modeled. The measurements were performed by evaporating sulfuric acid and water liquids and by diluting and cooling the sample vapor with a sampling system mimicking the dilution process occurring in a real-world driving situation. The nucleation rate inside the measurement system was modeled inversely using CFD (computational fluid dynamics) and the aerosol dynamics code, CFD-TUTMAM (Tampere University of Technology Modal Aerosol Model for CFD). The nucleation exponents for the concentrations of sulfuric acid and water and for the saturation vapor pressure of sulfuric acid were found to be 1.9±0.1, 0.50±0.05, and 0.75±0.05, respectively. These exponents can be used to examine the nucleation mechanisms occurring in exhaust from different combustion sources (internal combustion engines, power plant boilers, etc.) or in the atmosphere. Additionally, the nucleation rate can be expressed with the exponents as a function of the concentrations of sulfuric acid and water and of temperature. The obtained function can be used as a starting point for inverse modeling studies of more complex nucleation mechanisms involving extra compounds in addition to sulfuric acid and water. More complex nucleation mechanisms, such as those involving hydrocarbons, are observed with real vehicle exhaust and are also supported by the results obtained in this study. Furthermore, the function can be used to improve air quality models by using it to model the effect of sulfuric acid-emitting traffic and power generation on the particle concentration in urban air.

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_{2}SO

_{4}− H

_{2}O nucleation rate in exhaust-related conditions, Atmos. Chem. Phys., 19, 6367–6388, https://doi.org/10.5194/acp-19-6367-2019, 2019.

Airborne particles are related to adverse health effects (Dockery et al., 1993; Pope et al., 2002; Beelen et al., 2014; Lelieveld et al., 2015) and various effects on climate (Arneth et al., 2009; Boucher et al., 2013). In particular, adverse health effects are caused by the exposure to vehicle emissions which increase ultrafine particle concentration in urban air (Virtanen et al., 2006; Johansson et al., 2007; Pey et al., 2009) in the size range with high probability of lung deposition (Alföldy et al., 2009; Rissler et al., 2012).

Vehicles equipped with internal combustion engines generate nonvolatile particles (Rönkkö et al., 2007, 2014; Sgro et al., 2008; Maricq et al., 2012; Chen et al., 2017); however, volatile particles are also formed after the combustion process during exhaust cooling (Kittelson, 1998; Lähde et al., 2009), i.e., when the exhaust is released from the tailpipe. Thus, volatile particles are formed through the nucleation process; hence, they are called nucleation mode particles here.

An important characteristic of fine particles is the particle size distribution, as it determines the behavior of particles in the atmosphere and particle deposition to the respiratory system. Modeling studies provide information on the formation and evolution of exhaust-originated particles in the atmosphere (Jacobson et al., 2005; Stevens et al., 2012). To model the number concentration and the particle size of the nucleation mode, the governing nucleation rate needs to be known.

The detailed nucleation mechanism controlling particle formation in cooling
and diluting vehicle exhaust is currently unknown (Keskinen and Rönkkö, 2010). The
nucleation mode particles contain at least water, sulfuric acid
(H_{2}SO_{4}), and hydrocarbons
(Kittelson, 1998; Tobias et al., 2001; Sakurai et al., 2003; Schneider et al., 2005). Therefore, it is
likely that these compounds are involved in the nucleation process, but, on
the other hand, some of them can end up in the nucleation mode through the
initial growth of the newly formed clusters. The most promising candidate for
the main nucleating component in the particle formation process occurring in
diesel exhaust is H_{2}SO_{4}, as it has been shown that the
H_{2}SO_{4} vapor concentration in vehicle exhaust
(Rönkkö et al., 2013; Karjalainen et al., 2014), fuel sulfur content
(Maricq et al., 2002; Vogt et al., 2003; Vaaraslahti et al., 2005; Kittelson et al., 2008), lubricating oil
sulfur content (Vaaraslahti et al., 2005; Kittelson et al., 2008), and the exhaust
after-treatment system (Maricq et al., 2002; Vogt et al., 2003) correlate with nucleation mode number concentration, at least in the cases when the test vehicle has
been equipped with an oxidative exhaust after-treatment system. The sulfur
contents of fuel and lubricating oil are connected to the H_{2}SO_{4}
vapor concentration in the exhaust because the combustion of
sulfur-containing compounds produces sulfur dioxide (SO_{2}) that is
further oxidized to sulfur trioxide (SO_{3}) in an oxidative exhaust
after-treatment system (Kittelson et al., 2008), and SO_{3} finally
produces H_{2}SO_{4} when coming into contact with water (H_{2}O) vapor
(Boulaud et al., 1977).

Particle formation due to H_{2}SO_{4} in real vehicle exhaust plumes and
in laboratory sampling systems has been previously simulated by several
authors
(Uhrner et al., 2007; Lemmetty et al., 2008; Albriet et al., 2010; Liu et al., 2011; Arnold et al., 2012; Li and Huang, 2012; Wang and Zhang, 2012; Huang et al., 2014),
but all of them have modeled nucleation as binary homogeneous nucleation
(BHN) of H_{2}SO_{4} and water. Other possible nucleation mechanisms
include activation-type (Kulmala et al., 2006), barrierless kinetic
(McMurry and Friedlander, 1979), hydrocarbon-involving
(Vaaraslahti et al., 2004; Paasonen et al., 2010), ternary
H_{2}SO_{4}–H_{2}O–ammonia (Meyer and Ristovski, 2007), and ion-induced
nucleation (Raes et al., 1986) mechanisms. The choice of binary homogeneous
H_{2}SO_{4}–H_{2}O nucleation in studies involving vehicle exhaust is
mainly made because it has been the only nucleation mechanism for which an
explicitly defined formula for the nucleation rate (*J*) can be presented
(Keskinen and Rönkkö, 2010). An explicit definition is required when the nucleation
rate in cooling exhaust is modeled, as the nucleation rate has a steep
temperature dependency, according to theory (Hale, 2005) and experiments
(Wölk and Strey, 2001). The nucleation rate of BHN is derived from classical
thermodynamics; thus, the theory is called the classical nucleation theory
(CNT). The nucleation rate according to the CNT is explicitly defined as a
function of H_{2}SO_{4} and H_{2}O vapor concentrations
([H_{2}SO_{4}] and [H_{2}O]) and temperature (*T*). The derivation of
the CNT contains, however, a lot of assumptions, and it is thus quite
uncertain (Vehkamäki and Riipinen, 2012). The largest uncertainty rises from the
capillarity approximation; i.e., the physical properties of small newly formed
critical clusters can be expressed as the properties of bulk liquid
(Wyslouzil and Wölk, 2016). Comparing experimental and theoretical nucleation
rates, the CNT underestimates the temperature dependency (Hung et al., 1989) and
overestimates the sensitivity of *J* to [H_{2}SO_{4}]
(Weber et al., 1996; Olin et al., 2014). These discrepancies entail that theoretically
derived nucleation rates need to be corrected with a factor, ranging in
several orders of magnitude, to agree with experimental nucleation rates.

Conversely, the nucleation rates of the other nucleation mechanisms are typically modeled as (Zhang et al., 2012)

where *k* is an experimentally derived coefficient and *n* is the nucleation
exponent presenting the sensitivity of *J* to [H_{2}SO_{4}]. According to
the first nucleation theorem (Kashchiev, 1982), *n* is also connected to
the number of molecules in a critical cluster; however, due to assumptions
included in the theorem, *n* is not exactly the number of molecules in a
critical cluster in realistic conditions (Kupiainen-Määttä et al., 2014). The
value for *k* is typically a constant that includes the effect of *T* and
[H_{2}O], i.e., relative humidity (RH; Sihto et al., 2009; Stevens and Pierce, 2014).
A constant coefficient can be a satisfactory approximation in atmospheric
nucleation experiments, where *T* and RH nearly remain constants. However,
*T* and RH in cooling and diluting exhaust are highly variable; thus, a
constant coefficient cannot be used. The nucleation exponents, *n*, for
H_{2}SO_{4} obtained from the atmospheric nucleation measurements
(Sihto et al., 2006; Riipinen et al., 2007) and from the atmospherically relevant
laboratory experiments (Brus et al., 2011; Riccobono et al., 2014) usually lie between 1
and 2; these are much lower than the theoretical exponents (*n**≳*5,
Vehkamäki et al., 2003).

The first step in examining nucleation mechanisms, other than the CNT, in
vehicle exhaust using experimental data was performed by
Vouitsis et al. (2005). They concluded that nucleation mechanisms with *n*=2,
including the barrierless kinetic nucleation mechanism, can predict nucleation
rates in vehicle exhaust. Later, Olin et al. (2015) and Pirjola et al. (2015)
focused on obtaining nucleation rates inversely; i.e., an initial function for
*J* acts as an input to the model and is altered until the simulated particle
concentration and distribution correspond with the measured ones. These
modeling studies are based on the experiments
(Vouitsis et al., 2005; Arnold et al., 2012; Rönkkö et al., 2013) where the exhaust of a diesel
engine was sampled using a laboratory setup containing an engine dynamometer
and a diluting sampling system (Ntziachristos et al., 2004).

Inverse modeling is a preferable method in obtaining nucleation rates in a
diluting domain over the method based on calculating *J* by dividing the
measured number concentration with an estimated volume of a nucleation
region because the volume of a nucleation region also depends on *n*. In the
case of inverse modeling, there is no need to estimate the nucleation region
because the model simulates *J* at every time step, in a model using temporal
coordinates, or in every computational cell, in a model using spatial
coordinates. Pirjola et al. (2015) modeled the dilution system with an aerosol
dynamics model using temporal coordinates and concluded that hydrocarbons
could be involved in the nucleation mechanism, and *n* lies between 1 and 2.
However, because particle formation in diluting vehicle exhaust involves
strong gradients in temperature and the concentrations of the compounds
involved, information in spatial dimensions is also required to fully
understand the particle formation process. For this reason, Olin et al. (2015)
simulated aerosol dynamics using computational fluid dynamics (CFD) and
concluded that *n* is 0.25 or 1, depending on whether solid particles acting
as an condensation sink for sulfuric acid are emitted or not, respectively.
These values are very low compared to other studies and to the first
nucleation theorem that restricts *n* to at least 1. Values below unity imply
that there can be other compounds involved in the nucleation mechanism in
addition to H_{2}SO_{4}.

Ammonia (NH_{3}) involved in H_{2}SO_{4}–H_{2}O nucleation
(ternary H_{2}SO_{4}–H_{2}O–NH_{3} nucleation) has a notable
effect if the H_{2}SO_{4} concentration is low and the NH_{3}
concentration is high (Lemmetty et al., 2007; Kirkby et al., 2011). The H_{2}SO_{4}
concentration in the atmosphere is low enough for the effect of NH_{3}
to be relevant (Kirkby et al., 2011), but in vehicle exhaust, higher
H_{2}SO_{4} concentrations make the effect of NH_{3} probably
negligible. However, more recent vehicles are equipped with the selective
catalytic reduction (SCR) system which decreases nitrogen oxide emissions
but, on the other hand, increases NH_{3} emissions. Therefore,
NH_{3} can be involved in the nucleation process occurring in vehicle
exhaust of vehicles equipped with the SCR system (Lemmetty et al., 2007). The
SCR system was not included in the experiments of Arnold et al. (2012) and
Rönkkö et al. (2013) mentioned earlier; thus, other compounds involved in the
nucleation process in those experiments are more likely to be hydrocarbons
than NH_{3}.

In this paper, an improved aerosol dynamics model, CFD-TUTMAM (Tampere University of Technology Modal Aerosol Model for CFD), based on our previous model, CFD-TUTEAM (Tampere University of Technology Exhaust Aerosol Model for CFD), which is described in Olin et al. (2015), is presented. The main improvement in the model is its capability to model the initial growth of the newly formed clusters modally using our novel representation of the particle size distribution, the PL+LN (combined power law and log-normal distribution) model described in Olin et al. (2016).

Laboratory experiments designed for nucleation rate modeling purposes are
presented in which the examination of the nucleation rate was aimed towards
pure H_{2}SO_{4}–H_{2}O nucleation instead of nucleation associated
with some unknown compounds existing in real vehicle exhaust. Although the
pure binary nucleation seems not to be the principal nucleation mechanism in
real exhaust (Saito et al., 2002; Vaaraslahti et al., 2004; Meyer and Ristovski, 2007; Pirjola et al., 2015),
neglecting the unknown compounds is reasonable at this stage of nucleation
studies because the knowledge of the nucleation mechanism of the pure binary
nucleation is still at a very low level, and it should be examined more to
better understand the nucleation process in real exhaust. Adding only one
additional compound to nucleation experiments would cause one additional
dimension to the measurement matrix of all changeable parameters considered
and would thus increase the complexity of the experiments. Similarly, adding
the concentration of an additional compound to inverse modeling, the
complexity and the computational cost of the simulations would increase
significantly. Therefore, it is reasonable to begin the inverse modeling
studies using only the pure binary nucleation mechanism. Additionally,
although there are studies suggesting that other compounds are involved in
the nucleation process in real vehicle exhaust, it has not yet been directly
shown that the nucleation rate would be lower or higher with the absence of those
compounds. Comparing the experiments with pure H_{2}SO_{4}–H_{2}O
nucleation to the experiments with real exhaust can provide information on
that.

The pure H_{2}SO_{4}–H_{2}O nucleation was generated by evaporating
H_{2}SO_{4} and H_{2}O liquids and using the dilution system that
mimics a real-world dilution process of a driving vehicle
(Ntziachristos et al., 2004). A similar principle of generating H_{2}SO_{4}
by evaporating it from a saturator has been used in the study of
Neitola et al. (2015), where the concentrations of H_{2}SO_{4} and
H_{2}O and temperatures were kept in an atmospherically relevant range.
In this study, they were kept in a vehicle exhaust-relevant range; thus, the
output is an explicitly defined formula for the H_{2}SO_{4}–H_{2}O
nucleation rate in exhaust-related conditions. The formula is in the form of

which is based on the formula hypothesized by Olin et al. (2015), but with an
additional exponent *m*_{sa} for the saturation vapor pressure of
sulfuric acid (*p*_{sa}^{∘}) to also take temperature into
account. In Eq. (2), *n*_{sa} and
*n*_{w} represent the nucleation exponents for [H_{2}SO_{4}] and
[H_{2}O], respectively. The exponents may also depend on the
concentration levels, but due to the unknown dependency, only constant values
are considered in this study.

The formulation obtained from this study helps in finding the nucleation
mechanisms occurring in real vehicle exhaust or in the atmosphere. Similarly,
it can be used to examine particle formation in coal-fired power plant
exhaust, which is also known to contain H_{2}SO_{4} (Stevens et al., 2012).
For example, the values of the nucleation exponents obtained in this study can
provide information on the nucleation mechanisms because the values differ
with respect to different nucleation mechanisms. Another use of the
formulation is in improving air quality models by using it to model the
effect of sulfuric acid-emitting traffic and power generation on the particle
concentration in urban air.

Laboratory experiments were designed to enable the examination of the effects
of three parameters ([H_{2}SO_{4}], [H_{2}O], and *T*) on the
H_{2}SO_{4}–H_{2}O nucleation rate. The experimental setup is
presented in Fig. 1.

## 2.1 Artificial raw exhaust generation

The artificial raw exhaust sample was generated (the top part of
Fig. 1) by evaporating 98 % H_{2}SO_{4} liquid and
deionized Milli-Q water. H_{2}SO_{4} was held in a PTFE container, and
water was held in a glass bottle. The liquids were heated to temperatures
*T*_{sa} and 43 ^{∘}C, respectively, which determine the
concentrations in the gas phase theoretically through the saturation vapor
pressure. Dry and filtered compressed air was flown through the evaporators
and mixed before heating to 350 ^{∘}C; 2.7 % of carbon dioxide
(CO_{2}) was also mixed with a sample to act as a tracer to determine
the dilution ratio of the diluters. CO_{2} was selected because it has
no effect on the particle formation process and because it exists in real
exhaust as well.

The computational domain in the CFD simulation shown in the bottom part of
Fig. 1 begins before the sample enters the porous tube diluter (PTD); thus, the
concentrations of H_{2}SO_{4} and H_{2}O, temperature, pressure
(*p*), and flow rate need to be known at that point due to the requirement of
the boundary conditions in the CFD simulation. *T* and *p* were measured at
that point, [H_{2}O] was calculated from the measured RH, and the flow
rate was calculated from the dilution ratio of the PTD
with the aid of measured CO_{2} concentrations.

The temperature of the raw sample was 243 ^{∘}C and the mole fraction
of H_{2}O (*x*_{w}) was 0.036, on average. The temperature before
the PTD was lower than the heater temperature, 350 ^{∘}C, because the
sample cooled in the sampling lines, but the temperature of 243 ^{∘}C
corresponds well with the temperature of real exhaust when released from the
tailpipe. In NTP (normal temperature and pressure) conditions,
*x*_{w}=0.036 corresponds with $\left[{\mathrm{H}}_{\mathrm{2}}\mathrm{O}\right]=\mathrm{9.0}\times {\mathrm{10}}^{\mathrm{17}}\phantom{\rule{0.125em}{0ex}}{\mathrm{cm}}^{-\mathrm{3}}$. The mole fractions in real diesel or gasoline
exhaust range between 0.06 and 0.14, but the values higher than 0.036 with
this experimental setup were not used because a more humid sample caused the
water vapor to condense as liquid water in the sampling lines.

The temperature of the H_{2}SO_{4} evaporator, *T*_{sa}, was varied
between 85 and 164.5 ^{∘}C which correspond with the mole fractions
(*x*_{sa}) between $\mathrm{2.2}\times {\mathrm{10}}^{-\mathrm{7}}$ and $\mathrm{1.1}\times {\mathrm{10}}^{-\mathrm{5}}$ in the
raw sample. In NTP conditions, this range corresponds with the [H_{2}SO_{4}]
values between $\mathrm{5.7}\times {\mathrm{10}}^{\mathrm{12}}\phantom{\rule{0.125em}{0ex}}{\mathrm{cm}}^{-\mathrm{3}}$ and $\mathrm{2.8}\times {\mathrm{10}}^{\mathrm{14}}\phantom{\rule{0.125em}{0ex}}{\mathrm{cm}}^{-\mathrm{3}}$. These concentrations are higher than concentrations
in real vehicle exhaust (typically between 10^{8} and
10^{14} cm^{−3}) because particle formation was not observed with
the concentrations below $\mathrm{5.7}\times {\mathrm{10}}^{\mathrm{12}}\phantom{\rule{0.125em}{0ex}}{\mathrm{cm}}^{-\mathrm{3}}$. However, with
real vehicle exhaust, in the same sampling system used here, particle
formation has been observed even with the concentration of $\mathrm{2.5}\times {\mathrm{10}}^{\mathrm{9}}\phantom{\rule{0.125em}{0ex}}{\mathrm{cm}}^{-\mathrm{3}}$ (Arnold et al., 2012), indicating that other compounds
are
involved in the nucleation process.

The determination of [H_{2}SO_{4}] in the raw sample in our experiment was
not straightforward due to the uncertainties involved in the measurement of
[H_{2}SO_{4}]. The detailed information on measuring it, using a nitrate-ion-based (${\mathrm{NO}}_{\mathrm{3}}^{-}$-based) chemical ionization atmospheric pressure interface
time-of-flight mass spectrometer (CI-APi-TOF; Jokinen et al., 2012) and ion
chromatography (IC; Sulonen et al., 2015), is described in the Supplement.
Estimating [H_{2}SO_{4}] theoretically through the saturation vapor
pressure in the temperature of *T*_{sa} provides some information on
the dependency of [H_{2}SO_{4}] on *T*_{sa} in the raw sample.
However, the absolute concentrations cannot be satisfactorily estimated,
firstly because diffusional losses of H_{2}SO_{4} onto the sampling lines
between the H_{2}SO_{4} evaporator and the PTD are high and uncertain and
secondly because measuring H_{2}SO_{4} is generally a challenging task
due to high diffusional losses onto the walls of the sampling lines between
the measurement point and the measurement device. High diffusional losses are
caused by a high diffusion coefficient of H_{2}SO_{4}. Additionally, a low
flow rate from the H_{2}SO_{4} evaporator (0.5 slpm) increases the
diffusional losses before the measurement point. The diffusional losses
before the measurement point, according to the equations reported by
Gormley and Kennedy (1948) and to the humidity-dependent diffusion coefficient of
H_{2}SO_{4} reported by Hanson and Eisele (2000), are 98 % if the walls of
the sampling lines are assumed to be fully condensing. However, some parts in the
sampling lines have high concentrations of H_{2}SO_{4} with high
temperatures, especially with high *T*_{sa} values. Therefore, these
lines are probably partially saturated with H_{2}SO_{4}, which can act to
prevent H_{2}SO_{4} condensation onto the walls. Thus, the actual
diffusional losses are estimated to be between 0 % and 98 %, and they can
also depend on *T*_{sa} and on the saturation status of the sampling
lines during a previous measurement point. In conclusion, the determination
of [H_{2}SO_{4}] in the raw sample was done through inverse modeling using
measured particle diameter information (see Sect. 4.5).
The output of the concentrations from inverse modeling denotes the
diffusional losses of 43 %–95 %
depending on *T*_{sa}.

## 2.2 Raw exhaust sampling system

The sampling system used to dilute and cool the raw exhaust, presented in the bottom part of Fig. 1, was a modified partial flow sampling system (Ntziachristos et al., 2004) mimicking the dilution process occurring in a real-world driving situation. It consists of a PTD, an aging chamber, and an ejector diluter. The PTD dilutes and cools the sample rapidly, which leads to new particle formation. The aging chamber is used to grow the newly formed particles to detectable sizes and to continue the nucleation process. The ejector diluter is used to stop the particle formation and growth processes and to obtain the conditions of the sample required for measurement devices.

Dilution air used with the PTD and the ejector diluter was filtered
compressed air. The ejector diluter used only dry (RH≈3.6 *%*) and unheated (*T*≈20 ^{∘}C) dilution air, but the
dilution air for the PTD was humidified (RH_{PTD}=2 %–100 %) and heated
(*T*_{PTD}=27.5–70 ^{∘}C). Humidifying the dilution air of the PTD was done by directing the
compressed air flow through a container filled with deionized Milli-Q water.
RH_{PTD} and *T*_{PTD} are the variable parameters
used in examining the effect of [H_{2}O] and *T* on *J*, which represent
the conditions of the outdoor air acting in a dilution process in a
real-world driving situation. The range of *T*_{PTD} represents higher
temperatures compared to the temperature of the outdoor air, but lower
temperatures were not used because 27.5 ^{∘}C was the coldest
temperature available with the laboratory setup with no cooling device.

In this experiment, the residence time in the aging chamber was made adjustable by a movable sampling probe inside the aging chamber. The sampling probe was connected to the ejector diluter with a flexible Tygon hose. The residence time from before the PTD to after the ejector diluter was altered within a range of 1.4–2.8 s. Using a movable probe to alter the residence time has only a minor effect on the flow and temperature fields compared to altering the residence time with changing the flow rate in the aging chamber. Maintaining constant flow and temperature fields when studying the effect of the residence time is important because variable fields would alter the turbulence level and temperatures in the aging chamber, both having effects on the measured particle concentration and thus causing difficulties in separating the effect of the residence time from the effect of turbulence or temperature on measured particle concentrations.

The dilution ratio of the PTD was controlled by the excess flow rate after
the aging chamber and calculated by the measured [CO_{2}] before the PTD
and after the aging chamber. The dilution ratio was kept at around 20 in all
measurements. The dilution ratio of the ejector diluter was controlled by the
pressure of the dilution air used with the diluter and calculated also using
CO_{2} measurements. The calculated dilution ratio was around 10.
Because the dilution ratios varied between different measurement points, all
the aerosol results are multiplied by the total dilution ratio, thus making
the results comparable.

## 2.3 Particle measurement

Particle number concentration and size distribution were measured after the
ejector diluter using Airmodus PSM A11 (Airmodus Particle Size Magnifier A10 using Airmodus Condensation Particle Counter A20 as the particle counter), TSI
CPC 3775 (Ultrafine Condensation Particle Counter), and TSI Nano-SMPS (Nano
Scanning Mobility Particle Sizer using TSI CPC 3776 as the particle counter).
The PSM and the CPC 3775 measure the particle number concentration
(*N*_{PSM} and *N*_{CPC}) by counting particles with diameters
larger than ∼1.15 nm (PSM) or ∼2.15 nm (CPC 3775). The
*D*_{50}-cut size (the particle diameter having the detection efficiency of
50 %) of the PSM can be altered by adjusting its saturator flow rate
within the diameter range of 1.3–3.1 nm. Additionally, the CPC 3775 has the
*D*_{50}-cut size of 4.0 nm, and the CPC 3776 has the *D*_{50}-cut size of 3.4 nm. The detection
efficiency curves of the particle counters used are presented in
Fig. 2. The Nano-SMPS measured, with the
settings used in this experiment, the particle size distribution within the
diameter range of 2–65 nm; however, particles with
diameters smaller than ∼6 nm are weakly detectable due to very low
charging efficiency of the radioactive charger, low detection efficiency of
the particle counter, and high diffusional losses inside the device for very
small particles. Nevertheless, using the data from the different saturator
flow rates of the PSM together with the data from the CPC 3775, information
on the particle size distribution around the range of 1.15–6 nm is also
obtained.

Due to particle number concentrations that are too high for the PSM, aerosol measured
with the PSM and the CPC 3775 was diluted with a bridge diluter. It dilutes
the concentration of larger particles (*D*_{p}>10 nm) with the
ratio of 250, but the dilution ratio increases with decreasing particle size
due to diffusional losses to the ratio of 1200
(*D*_{p}=1.15 nm) finally. The dilution ratio was measured with
aerosol samples with the count median diameters (CMDs) of 2–25 nm. The
ratio of the sampling line length and the flow rate of the bridge diluter, a
partially unknown variable, used in the diffusional losses function reported
by Gormley and Kennedy (1948), was fitted to correspond with the dilution ratio
measurement results; the obtained dilution ratios are presented in
Fig. 3.

## 2.4 Measurement sets

By varying [H_{2}SO_{4}] of the artificial raw exhaust sample and
[H_{2}O] and *T* of the dilution air separately and measuring the
aerosol formed in the sampling system, the effects of the parameters on *J*
can be examined. The effects of the parameters are included in
Eq. (2) simply, with the exponents *n*_{sa},
*n*_{w}, and *m*_{sa}. To obtain these three yet unknown
values, at least three parameters were required to be varied in the
experiments. Nevertheless, a fourth parameter, the residence time, was also
varied to provide some validation for the obtained exponents. [H_{2}O]
and *T* of the dilution air were varied simply by humidifying and heating the
dilution air flowing to the PTD and measuring RH and *T* from the dilution
air. Varying [H_{2}SO_{4}] of the raw sample was done by varying
*T*_{sa}, and the values for [H_{2}SO_{4}] in the raw sample were
obtained through inverse modeling.

The varied conditions of the measurements are presented in
Table 1, where all the measurement points are divided
according to the main outputs (*n*_{sa}, *n*_{w},
*m*_{sa}, and $\partial J/\partial t$) that measurement sets were
designed to provide. Examining the effect of temperature (*m*_{sa})
was performed with the measurements of two types: varying *T*_{PTD}
while keeping RH_{PTD} as a constant (set 3a) and varying
*T*_{PTD} while keeping the mole fraction of H_{2}O in the
dilution air of the PTD (*x*_{w,PTD}) as a constant (set 3b). The
time dependence of the nucleation rate ($\partial J/\partial t$) or, in the
other words, the diminishment rate of *J* in a diluting sampling system, is
mainly the product of the exponents *n*_{sa} and *m*_{sa} in
the following way: [H_{2}SO_{4}] decreases steeply due to dilution, losses
to walls, and condensation to particles, resulting in diminishing *J* with the
power of *n*_{sa}; simultaneously, *T* decreases due to dilution and
cooling of the sampling lines, resulting in strengthening *J* with the power
of *m*_{sa}. Examining the diminishment rate provides validation for
the relation of *n*_{sa} and *m*_{sa} obtained from the
simulations. We waited 2–40 min for the particle size distributions to
stabilize after the conditions were changed between the measurement points.
When the particle formation process was satisfactorily stabilized,
measurement data for each measurement point were recorded for 5–40 min,
depending on the stability of the particle generation.

Figure 4 represents examples of particle size
distributions measured with different H_{2}SO_{4} evaporator temperatures,
*T*_{sa}. The PSM+CPC data are calculated using the number
concentrations measured with different saturator flow rates of the PSM and
with the CPC 3775, i.e., with different *D*_{50}-cut sizes. To properly
compare the data measured with different dilution ratios and sampling line
lengths, the comparison requires backwards-corrected data; i.e., all data in
the figure are corrected with the dilution ratio of the bridge diluter and
with the diffusional losses caused by the sampling lines between the ejector
diluter and the measurement devices. However, correcting the distributions
backwards from the measured data to the distributions after the ejector
diluter is not simple because it requires the shapes of the distributions
within the whole diameter range to be known. The data of the PSM and the
CPC 3775 cannot always provide real size distributions because the cumulative
nature of the method using particle counters as the size distribution
measurement can suffer from noise in the measured concentration. For example, the
PSM+CPC data with *T*_{sa}=157.2 ^{∘}C shown in
Fig. 4 imply that the concentration could increase
with decreasing particle size, but the placing of the data points can be
caused by the noise in the measured concentrations. On the other hand, the
data imply that there are no particles smaller than ∼2.5 nm in
diameter, but the data of the smaller particles can be invisible due to the
noise in the measured concentrations (see the Supplement for the detailed
uncertainty estimation of the size distributions). Hence, the unknown
concentration of the particles smaller than ∼2.5 nm in diameter can
have a significant effect on the total number concentration after the ejector
diluter calculated from the measured data because these particles play the
major role in the effect of the diffusional losses in the sampling lines and
in the bridge diluter. Due to these uncertainties, the backwards-corrected
data (denoting the distributions right after the ejector diluter) are not
used when comparing the measured results with the simulated results later in
this paper. Nevertheless, the backwards-corrected data are used when
presenting the distributions from all the aerosol devices together because
the distributions cannot be presented without correcting them backwards due
to different particle losses in the sampling lines of the different devices.

It can be observed that, though the Nano-SMPS data are in a nearly log-normal
form, there are also size distributions in the PSM+CPC diameter range.
Particles generated with lower *T*_{sa} are lower in concentration and
smaller than ones with higher *T*_{sa}; also a higher fraction of
particles are in the PSM+CPC diameter range with lower *T*_{sa}. The smaller diameter edges of
the log-normal size distributions measured by the Nano-SMPS do not connect
with the distributions measured by the PSM and the CPC 3775 due to the weak
detection efficiency of very small particles by the Nano-SMPS. Thus, the
smaller diameter edges of the measured log-normal size distributions are not
accurate. Similar disagreements of the data from these devices have also been
observed elsewhere, both in exhaust-related (Alanen et al., 2015; Rönkkö et al., 2017)
and in atmospherically related studies (Kulmala et al., 2013). By examining the
combination of the size distributions measured by the PSM and the CPC 3775
and the size distributions measured by the Nano-SMPS, the real size
distributions are not in a log-normal form. The detailed uncertainty
estimation of the measured distributions and discussion on this disagreement
can be found in the Supplement.

The particle number concentrations measured with the highest saturator flow
rate of the PSM (*N*_{PSM}), i.e., the particles with diameters larger
than ∼1.3 nm, and the diameters with the average mass
(${D}_{\stackrel{\mathrm{\u203e}}{m}}$) of measurement set 1 are presented in
Fig. 5. ${D}_{\stackrel{\mathrm{\u203e}}{m}}$ values are calculated using
the size distributions measured with the combination of the PSM, the
CPC 3775, and the Nano-SMPS, which are corrected with the diffusional losses
in the sampling lines. Fig. 5 consists of data measured on two different
days. It can be observed that *N*_{PSM} increases steeply with
increasing [H_{2}SO_{4}]_{raw} with lower
[H_{2}SO_{4}]_{raw} values, but the steepness decreases with
an increasing [H_{2}SO_{4}]_{raw} due to an increasing self-coagulation
rate. With lower [H_{2}SO_{4}]_{raw} values, the slope of
*N*_{PSM} versus [H_{2}SO_{4}]_{raw} in a log–log scale,

is approximately 10 but decreases to approximately 0.4 with decreasing
[H_{2}SO_{4}]_{raw}. The slope of *J* versus [H_{2}SO_{4}] is,
by the definition of *J* (Eq. 2),

which is also the nucleation exponent for [H_{2}SO_{4}]. The slope
${n}_{{N}_{\mathrm{PSM}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\mathrm{vs}.\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}[{\mathrm{H}}_{\mathrm{2}}{\mathrm{SO}}_{\mathrm{4}}{]}_{\mathrm{raw}}}$ can
provide a rough estimate of the slope *n*_{sa}, but due to the other
aerosol processes, especially coagulation, having effects on the particle
concentrations, the estimated slope can differ a lot from the real
*n*_{sa} in the nucleation rate function. The slope at higher
[H_{2}SO_{4}]_{raw} values is usually decreased due to
coagulation, and the slope at lower [H_{2}SO_{4}]_{raw} values can
be increased due to decreased particle detection efficiency of smaller
particles. Therefore, *n*_{sa} is expected to be within the range
of 0.4–10. Additionally, the estimated slope can also differ from
*n*_{sa} because ${n}_{{N}_{\mathrm{PSM}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\mathrm{vs}.\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}[{\mathrm{H}}_{\mathrm{2}}{\mathrm{SO}}_{\mathrm{4}}{]}_{\mathrm{raw}}}$ is based on [H_{2}SO_{4}]
in the raw sample rather than the value of [H_{2}SO_{4}] in a specific
location: [H_{2}SO_{4}] decreases from the concentration in the raw sample
by several orders of magnitude during the dilution process.

The effect of humidity on the particle concentration (set 2) is shown in
Fig. 6. The slope of *N*_{PSM} versus
RH_{PTD} in a log–log scale,

is roughly between 0.1 and 0.2. The slope ${n}_{{N}_{\mathrm{PSM}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\mathrm{vs}.\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}{\mathrm{RH}}_{\mathrm{PTD}}}$ nearly equals the slope of
*N*_{PSM} versus [H_{2}O]_{PTD} (${n}_{{N}_{\mathrm{PSM}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\mathrm{vs}.\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}[{\mathrm{H}}_{\mathrm{2}}\mathrm{O}{]}_{\mathrm{PTD}}}$) because *T*_{PTD} is
nearly a constant. The slope ${n}_{{N}_{\mathrm{PSM}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\mathrm{vs}.\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}[{\mathrm{H}}_{\mathrm{2}}\mathrm{O}{]}_{\mathrm{PTD}}}$ corresponds with the slope
*n*_{w}, with the same uncertainties as those involved with the slopes
${n}_{{N}_{\mathrm{PSM}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\mathrm{vs}.\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}[{\mathrm{H}}_{\mathrm{2}}{\mathrm{SO}}_{\mathrm{4}}{]}_{\mathrm{raw}}}$ and
*n*_{sa}. Nevertheless, the effect of decreased particle detection is
not involved because, in this case, particle size has only a weak dependency
on RH_{PTD}. Additional uncertainty in estimating
*n*_{w} arises from the origin of H_{2}O vapor in the system,
which is both the dilution air and the raw sample. Because [H_{2}O] in
the raw sample was kept constant, it has a higher effect on the total
[H_{2}O] with lower values of RH_{PTD}; thus, the
estimated *n*_{w} is lower than the real *n*_{w} in the
nucleation rate function.

The effect of *T*_{PTD} can be observed in
Figs. 6 and 7. Lower
temperatures result in higher concentrations of *N*_{PSM}. However,
the examination is problematic because keeping RH_{PTD} as
a constant while increasing *T*_{PTD} (set 3a) increases
[H_{2}O], which results in lower *N*_{PSM} with lower
temperatures. Therefore, keeping *x*_{w,PTD} as a constant (set 3b) is
better for examining *m*_{sa}. One of the measurements with
*T*_{PTD}=50 ^{∘}C is, however, a significant outlier in set 3b.
Estimating the exponent *m*_{sa} from the slope in
Fig. 7 is not straightforward because temperature
is also included in the concentrations with exponents that are still unknown.

The effect of the residence time on the particle concentrations is presented
in Table 2. With
*T*_{sa}=135.5 ^{∘}C, the ratio of *N* with the residence times
of 1.4 s and with the residence time of 2.8 s is below unity, but it is above
unity with higher temperatures. The ratio below unity signifies that the
nucleation process is still not diminished at the time of 1.4 s; e.g., the
ratio of 0.74 denoting 74 % of particles is formed within the time range
of 0–1.4 s, and the remaining 26 % is formed within the time range of 1.4–2.8 s.
With higher temperatures, the ratio is above unity because self-coagulation
begins to decrease the number concentration, especially at the later times
where the number concentration is the highest. The nucleation process may
continue after 1.4 s, but it cannot be easily seen with higher temperatures.
Because coagulation has no effect on the mass concentrations (*M*), the
ratios of *M* measured with the combination of the PSM, the CPC 3775, and the
Nano-SMPS with the residence time of 1.4 s and with the residence time of
2.8 s are near unity with higher temperatures. The effects of particle growth
and wall losses, however, have effects on the ratios, too. The temperature
with which the coagulation process would eliminate the effect of the
nucleation process, resulting in the number concentration ratio of unity, is
near 142 ^{∘}C.

Every measurement point presented in Table 1 was simulated with the model consisting of four phases: (1) the CFD simulations for solving the flow and the temperature field of the sampling system, (2) the CFD-TUTMAM simulations for solving the aerosol processes in the sampling system, (3) correcting the particle sizes decreasing rapidly in the dry ejector diluter, and (4) calculating the penetration of the particles due to diffusional losses in the sampling lines after the sampling system and the detection efficiencies of the particle counting devices.

## 4.1 CFD model

The CFD simulations to solve the flow and the temperature fields for every simulation case were performed with a commercially available software, ANSYS Fluent 17.2. It is based on a finite volume method in which the computational domain is divided into a finite amount of cells. Governing equations of the flow are solved in every computational cell iteratively until sufficient convergence is reached. In this study, the governing equations in the first phase are continuity, momentum, energy, radiation, and turbulence transport equations.

The computational domain in the CFD simulations is an axial symmetric geometry consisting of the PTD, the aging chamber, and the ejector diluter (Fig. 1). An axial symmetric geometry was selected over a three-dimensional geometry due to high computational demand of the model and a nearly axial symmetric profile of the real measurement setup. The domain was divided into $\sim \phantom{\rule{0.125em}{0ex}}\mathrm{8}\times {\mathrm{10}}^{\mathrm{5}}$ computational cells, of which the major part was located inside the PTD, where the smallest cells are needed due to the highest gradients. The smallest cells were 20 µm in side lengths and were located in the beginning of the porous section, where the hot exhaust and the cold dilution air meet.

In contrast to our previous study (Olin et al., 2015), the ejector diluter was
also included in the computational domain, though it has only a minor effect
on nucleation (Lyyränen et al., 2004; Giechaskiel et al., 2009). Because the ejector
diluter has a high speed nozzle that cools the flow locally to near
−30 ^{∘}C, including it in the domain provides partial validation for
*m*_{sa} in the following way: if too high a value for *m*_{sa}
were used, nucleation would be observed in the ejector diluter, in contrast to the former studies. The internal fluid inside the sampling
lines is modeled as a mixture of air, H_{2}O vapor, and H_{2}SO_{4}
vapor. The sampling lines are modeled as solid zones of steel or Tygon, and
10 cm of the external fluid, modeled as air, is also included in the domain
to simulate natural cooling of the sampling lines.

Flow rate and temperature boundary conditions for the simulated sampling
system were set for the each simulation case to the measured values. Due to
steady-state conditions and high computational demand, all governing
equations were time averaged; thus, the simulations were performed with a
steady-state type. Turbulence was modeled using the SST-*k*-*ω* model,
which is one of the turbulence models used with a steady-state simulation. It
produced the most reliable results of the available steady-state turbulence
models based on the pressure drop in the porous section. Turbulence, however,
can play a significant role in the wall losses of the vapors and the
particles in the regions where the turbulence level is high. In this sampling
system, the turbulence level is high in the upstream part of the aging
chamber where the diameter of the sampling line increases steeply. Validating
the suitability of the turbulence model for this geometry would require a
measurement of, for example, solid seed particle concentrations after and before the
sampling system without any aerosol processes, such as nucleation,
condensation, and coagulation. However, that kind of measurement has not been
performed yet.

## 4.2 CFD-TUTMAM

The main functionality of the CFD-TUTMAM based on the previous aerosol model, CFD-TUTEAM, is described by Olin et al. (2015). However, because the measured distributions are not in a log-normal form, the inclusion of the PL+LN model (Olin et al., 2016) was beneficial. The PL+LN model simulates the initial growth of newly formed very small particles by modeling the particle size distribution with the combination of a power law (PL) and a log-normal (LN) distribution. Newly formed particles are first put to the PL distribution, after which they are transferred to the LN distribution by particle growth.

The CFD-TUTMAM adds three governing equations per distribution (denoted by
*j*) to the CFD model using a modal representation of the particle size
distribution; i.e., the distributions are modeled by three variables: number
(${M}_{j,\mathrm{0}}={N}_{j}$), surface-area-related (${M}_{j,\mathrm{2}/\mathrm{3}}$), and mass (*M*_{j,1})
moment concentrations. *M*_{j,1} values are further divided into different
components in a multi-component system. Due to small particle size and low
particle loading, the aerosol phase has only a minor effect on the gas phase
properties. Therefore, continuity, momentum, energy, radiation, and
turbulence transport equations can be excluded from the computation after the
flow and temperature fields are solved, and only gas species equations and
the aerosol model equations are solved. The governing equation of the aerosol
model for the concentration of a *k*th moment of a distribution *j* is

where ** u**,

*ρ*

_{f}, and ${\stackrel{\mathrm{\u203e}}{D}}_{j,k,\mathrm{eff}}$ are the fluid velocity vector, the fluid density, and the

*k*th-moment-weighted average of the particle effective diffusion coefficient, respectively. The last terms in Eq. (6) represent source terms for nucleation, condensation, coagulation, and intermodal particle transfer. In this study, aerosol is modeled with two distributions: a PL distribution (

*j*=PL) and an LN distribution (

*j*=LN). In this study, two gas species equations, which model the internal fluid mixture as the mass fractions of H

_{2}O and H

_{2}SO

_{4}, are built in the CFD model, but the opposite numbers of the source terms of nucleation and condensation are added to them to maintain the mass closure of the species.

After each iteration step of the CFD-TUTMAM simulation, the parameters of the
distributions are calculated for every computational cell by using three
moment concentrations. The parameters for the PL distribution are the number
concentration (*N*_{PL}), the slope parameter (*α*), and the
largest diameter (*D*_{2}). The smallest diameter (*D*_{1}) has a fixed value of
1.15 nm, which is the smallest detectable particle diameter with the devices
used. The density function for the PL distribution is

where *β*_{0} is a function

The parameters for the LN distribution are the number concentration
(*N*_{LN}), the geometric standard deviation (*σ*), and the
geometric mean diameter (*D*_{g}). An analytical solution exists for
the reconstruction of the parameters from the moment concentrations for the
LN distribution but not for the PL distribution; thus, it is solved
numerically. A numerical solution is obtained by using the
Levenberg–Marquardt iteration algorithm, in contrast to a slower method using
a pre-calculated interpolation table described by Olin et al. (2016).

The nucleation source terms in Eq. (6) for different moments are

where *J* is the nucleation rate as in Eq. (2) and
${m}_{\mathrm{sa}}^{*}$ and ${m}_{\mathrm{w}}^{*}$ are the masses of H_{2}SO_{4} and
H_{2}O in a newly formed particle. The value of *D*_{1}=1.15 nm
was chosen for the diameter of the newly formed particles. A particle of this
diameter is in equilibrium with water uptake in the temperature of 300 K and
in the relative humidity of 22 % if the mass fraction of H_{2}SO_{4} in
the particle is 0.71. This constant value is used with nucleation, though the
mass fraction would vary between 0.5 and 1 if the whole temperature and
humidity range were considered, but the major part of nucleation occurs in
the conditions with the equilibrium mass fraction of near 0.71. This mass
fraction and particle diameter correspond with a cluster containing 5.7
H_{2}SO_{4} molecules and 12.4 H_{2}O molecules.

Diffusion, condensation, and coagulation are modeled as described in Olin et al. (2015), and intermodal particle transfer is modeled as described in Olin et al. (2016). Condensation is modeled with the growth by
H_{2}SO_{4}, which immediately follows the water uptake until the
water equilibrium is achieved. The water equilibrium procedure is also
described in Olin et al. (2015). The coagulation modeling
includes intramodal coagulation within both distributions and intermodal
coagulation between the distributions.

Intermodal particle transfer includes condensational transfer and
coagulational transfer from the PL distribution to the LN distribution. In
contrast to a constant condensational transfer factor *γ* of the PL+LN
model described in Olin et al. (2016), a function of *α*,
*D*_{1}∕*D*_{2}, and *k* is used in the CFD-TUTMAM due to more complex particle
growth modeling. The function used here is

The functional form of *γ* is derived so that the condensational
transfer eliminates the effect of increasing *α* by the condensation
process and also tries to keep *α* positive because a PL distribution
with a negative *α* in combination with an LN distribution represents a
distribution with a nonphysical local minimum between the distributions.
The form of *γ* also restricts *α* from increasing too much, which would
cause numerical difficulties. Particles are not lost or altered during the
intermodal particle transfer; it only controls the ratio of particles
represented in the PL distribution and in the LN distribution. Higher values
of *γ* result in a lower *N*_{PL}∕*N* ratio.

Deposition of particles and condensation of vapors onto the inner walls of
the sampling lines have a direct effect on the aerosol concentrations at the
measurement devices. The particle deposition was modeled by setting the
boundary conditions for the aerosol concentrations at the walls to zero,
which represents deposition driven by diffusion and turbulence. Condensation
of H_{2}O and H_{2}SO_{4} vapors onto the walls was modeled by
setting the boundary conditions for the mass fractions of H_{2}O and
H_{2}SO_{4} at the walls to saturation mass fractions in an aqueous
solution of H_{2}SO_{4}, in contrast to the simpler method in the previous
study (Olin et al., 2015). The simpler method caused H_{2}SO_{4} to be
completely non-condensing onto the walls because the saturation ratio of the
pure vapor never exceeded unity. Instead, the method using the saturation
mass fractions in the solution induces some condensation because the vapor
pressure of a hygroscopic liquid over an aqueous solution is lower than over
a pure liquid. This method also provides smoother behavior of the boundary
conditions on the walls. The method is, however, strongly dependent on the
chosen activity coefficient functions of the vapors, which have large
differences between each other due to their exponential nature. Activity
coefficients used here are based on the values reported by
Zeleznik (1991). However, due to the exponential and non-monotonic nature of
activity coefficients, they cause numerical difficulties in CFD modeling;
thus, a monotonic van Laar-type equation fitted by Taleb et al. (1996) from the
data of Zeleznik (1991) was used.

## 4.3 Dry particle model

The main trend of the RH inside the sampling system is an increasing trend due to
decreasing temperature. This results in an increasing water uptake rate during
the particle growth process, which can be modeled by the condensation rate of
H_{2}O that is simply the condensation rate of H_{2}SO_{4} multiplied
by a suitable factor (the water equilibrium procedure described by
Olin et al., 2015). However, when the sample enters the ejector diluter,
the RH decreases rapidly due to dry dilution air, but the growth process by
the condensation of H_{2}SO_{4} still continues. This results in
an increasing H_{2}SO_{4} amount in the particles but a rapidly decreasing
H_{2}O amount, which cannot be modeled with the water uptake model.
Hence, the particles after the ejector diluter simulated by the CFD-TUTMAM
contain incorrectly too much water.

All the simulated particle size distributions outputted by the CFD-TUTMAM
were corrected to correspond with the water amount that would be in the conditions
after the ejector diluter ($T\approx \mathrm{23}{}^{\circ}\mathrm{C}$ and
RH≈3.6 *%*). These conditions are mainly caused by the
conditions of compressed air directed to the ejector diluter. Additionally,
the particle size measurement device (Nano-SMPS) used room air, having nearly
equal conditions as compressed air, as the sheath flow air. Dry sheath flow
air also dries particles rapidly inside the device. The theory behind the dry
particle model is the same as the theory behind the water uptake model in the
CFD-TUTMAM, but the drying process is significantly faster and in the opposite
direction, in contrast to the water uptake connected to the condensation rate
of H_{2}SO_{4} in the CFD-TUTMAM. Figure 8 represents
examples of particle diameters in different humidities; e.g., a particle with
the diameter of 40 nm in the RH of 60 % shrinks to the diameter of 30 nm
when sampled with the ejector diluter.

## 4.4 Penetration and detection efficiency model

The particle size distributions outputted by the CFD-TUTMAM and corrected with the dry particle model were also corrected according to the penetration and detection efficiency model. Particle penetration in the sampling lines between the ejector diluter and the measurement devices was calculated with the equations of Gormley and Kennedy (1948). All the internal diameters of the used sampling lines were large enough to keep the flows laminar to minimize the diffusional losses. The penetration-corrected size distributions were multiplied by the detection efficiency curves presented in Fig. 2 to simulate the measured number concentrations by the PSM and the CPC 3775 and the measured size distribution by the Nano-SMPS.

## 4.5 Inverse modeling

The simulated number concentrations measurable by the PSM with different
saturator flow rates and by the CPC 3775 and the simulated size distributions
measurable by the Nano-SMPS were compared with the measured ones during
inverse modeling. The exponents *n*_{sa}, *n*_{w}, and
*m*_{sa} were altered until the simulated and the measured variables
corresponded satisfactorily in all simulated cases. The proportionality
coefficient *k* in Eq. (2) is unknown and depends on
the exponents. Because the value of *k* affects the nucleation
rate magnitude directly, it was obtained by fitting until the simulated and the
measured number concentrations corresponded.

Due to the uncertainties involved in the measurement of
[H_{2}SO_{4}]_{raw} (see the Supplement), the boundary conditions
for [H_{2}SO_{4}] in the CFD-TUTMAM simulations could not be set
initially. Hence, [H_{2}SO_{4}]_{raw} was also considered to be a fitting
parameter. It was estimated by comparing the aerosol mass concentrations
because it has a direct effect on the particle sizes but also affects
*J*. Inverse modeling of the vapor concentrations is possible due to the
condensational growth of particles. In conclusion, the inverse modeling
requires fitting all the five parameters (*n*_{sa}, *n*_{w},
*m*_{sa}, *k*, and [H_{2}SO_{4}]_{raw}) to obtain the
function for *J*. The first four parameters were fitted in a way in which they have
the same value for every simulation case, but the last parameter,
[H_{2}SO_{4}]_{raw}, was fitted in every simulation case
separately. In the simulations related to the measurement sets 2–4,
*T*_{sa} was not altered between the measurement points; therefore,
the value of [H_{2}SO_{4}]_{raw} in the simulations was constant.
Because only one parameter was fitted separately, only one of the outputs,
the aerosol number or mass concentration, could correspond with the measured
value exactly. In this study, the number concentration was chosen as the main
output, where the correspondence of the number concentration is preferred over the correspondence of the
mass concentration because the nucleation process is connected more directly to
the number concentration.

The uncertainties involved in modeling turbulence and the condensation of the
vapors onto the walls affect the number and mass concentrations in the
measurement devices. Nevertheless, these uncertainties become partially
insignificant because *k* and [H_{2}SO_{4}]_{raw} are considered
to be fitting parameters, which partially neglect uncertainly modeled losses of
particles and vapors.

In this section, the outputs of the simulations performed using the nucleation rate function with the best correspondence between the measured and the simulated data are described firstly. Finally, the used nucleation rate function is presented.

## 5.1 Sulfuric acid concentrations

Figure 9 represents the comparison of the inversely modeled
[H_{2}SO_{4}]_{raw} with the theoretical concentrations. The
simulated concentrations vary between 0.05 and 0.57 times the theoretical
concentrations, where the lowest values are observed with lower
*T*_{sa} values, probably due to the effect of increasingly saturating
H_{2}SO_{4} liquid onto the sampling lines with higher temperatures that
can decrease the diffusional losses onto the sampling lines. All values lie
between the theoretical level assuming full diffusional losses and the
lossless theoretical level. A weak agreement of the simulated concentrations
with 0.15 times the theoretical curve can be seen, which implies the
diffusional losses of 85 % onto the sampling lines between the
H_{2}SO_{4} evaporator and the PTD. Results and involved challenges of the
additional [H_{2}SO_{4}]_{raw} measurements are presented in the
Supplement.

## 5.2 Particle size distributions

Examples of measured and simulated particle concentrations and size distributions of measurement set 1 are presented in Fig. 10. Figure 10a and c represent the concentrations measured or measurable with the PSM and the CPC 3775. Because the concentrations decrease with an increasing cut diameter in the case with ${T}_{\mathrm{sa}}=\mathrm{102}{}^{\circ}$C (Fig. 10a), particle size distribution exists within this diameter range, which is also seen in the simulated data. However, the concentration measured with the cut diameter of 3.1 nm is twofold compared to the simulated one, implying that the real distribution is not a pure PL+LN distribution or that the shape of the distribution is modeled incorrectly near the diameter of 3.1 nm. Conversely, in the case with ${T}_{\mathrm{sa}}=\mathrm{157.2}{}^{\circ}$C (Fig. 10c), the concentrations are in the same level, which implies no size distribution within that diameter range.

Figure 10b and d represent examples of
measured and simulated Nano-SMPS data. The case with
*T*_{sa}=102 ^{∘}C (Fig. 10b) represents an example of one of the worst
agreements of measured and simulated size distributions. While the simulated
total number concentration agrees with the measured one in that case, the
particle diameter is underestimated with the factor of ∼1.6. The
disagreement is discussed later in this section. Conversely, in the case with
*T*_{sa}=157.2 ^{∘}C (Fig. 10d), the distributions agree well, except
that the model predicts higher particle concentration in the diameter range
of 2.5–7 nm. This disagreement can be due to lower particle detection
efficiency of the Nano-SMPS than that included in the inversion algorithm of
the device (see the Supplement). This is not included in the penetration and
detection efficiency model and is thus not seen in the simulated
distributions. Because the detection efficiency curve of the CPC 3776 is
included in the model, the simulated size distributions measurable with the
Nano-SMPS decrease steeply with a decreasing particle diameter near the
particle diameter of *D*_{50}=3.4 nm. The sharp peak at the diameter
of ∼ 20 nm in the simulated distribution in Fig. 10d is caused by the nature
of the PL+LN model where the PL distribution ends at the diameter of
*D*_{2}≈20 nm. While Fig. 10 represents the
data at the measurement devices, Fig. 11 represents
the example distributions after the ejector diluter. From the latter figure,
the PL distribution is seen as a whole, starting from the diameter of
*D*_{1}=1.15 nm.

The requirement of the PL+LN model can be observed from
Fig. 12, in which the particle number concentrations and
sizes of a single simulation case with different values of
[H_{2}SO_{4}]_{raw} are presented. With low values of
[H_{2}SO_{4}]_{raw}, both *N* and ${D}_{\stackrel{\mathrm{\u203e}}{m}}$ behave
discontinuously if only the LN distribution is simulated: particles are first
small and in a low concentration when [H_{2}SO_{4}]_{raw}
increases and then suddenly rise to higher levels. This is, however, not
seen with the PL+LN model, which has smoother behavior. Therefore, by
simulating with the LN distribution only, it is impossible to produce, for example,
a size distribution with $N={\mathrm{10}}^{\mathrm{4}}\phantom{\rule{0.125em}{0ex}}{\mathrm{cm}}^{-\mathrm{3}}$ or
${D}_{\stackrel{\mathrm{\u203e}}{m}}=\mathrm{3}\phantom{\rule{0.125em}{0ex}}\mathrm{nm}$ with this simulation setup, whereas with the
PL+LN model, it is possible.

## 5.3 Particle concentrations and sizes

Figure 13 represents the comparison of the
simulated and the measured *N*_{PSM} and ${D}_{\stackrel{\mathrm{\u203e}}{m}}$ values
after the ejector diluter. The black dots in Fig. 13a correspond well
with the measured concentrations because they represent the cases for which
*N*_{PSM} was obtained by fitting the value of
[H_{2}SO_{4}]_{raw}. The red dots deviate more from the 1:1 line
because they represent all the other cases, the *N*_{PSM} values of
which originate from the simulations, for example, those simulated with different
RH_{PTD}, *T*_{PTD}, or residence times.
Nevertheless, all the simulated *N*_{PSM} values correspond with the
measured values relatively well. The optimal scenario would be that all the
*N*_{PSM} values would correspond exactly with the measured values,
but that would imply that the exponents *n*_{w} and *m*_{sa} in the
nucleation rate function can be modeled exactly with constant values within
the concentration and temperature ranges of this study. However, it is not
expected that the constant exponents would represent the nucleation
rate function in all concentration and temperature ranges exactly.

The black dots in Fig. 13b
correspond moderately with the measured ${D}_{\stackrel{\mathrm{\u203e}}{m}}$ values. It can be
observed that the points do not lie on a straight 1:1 line perfectly;
instead they form a slightly curved line on which simulated particle sizes
are overestimated near 10 nm but underestimated in small particle sizes.
There are several issues which can cause this discrepancy: (1) the exponent
*n*_{sa} varies with [H_{2}SO_{4}], (2) there is a problem in calculating
${D}_{\stackrel{\mathrm{\u203e}}{m}}$ from the measurement data, (3) there is a problem in estimating a
proper *N*_{PL}∕*N* ratio in the PL+LN model, and (4) there is uncertainty in
simulating the condensation process. The most possible explanation is
the first because according to the CNT, *n*_{sa} decreases with increasing
[H_{2}SO_{4}]. This can be seen as overestimated particle sizes in
mid-ranged particle sizes because smaller particle sizes would require lower
[H_{2}SO_{4}]_{raw}, but that would cause underestimated
*N*_{PSM}. To overcome the underestimated *N*_{PSM} in
mid-ranged [H_{2}SO_{4}] values, *k* should be increased in mid-ranged
[H_{2}SO_{4}] values, which indicates decreasing *n*_{sa} with
increasing [H_{2}SO_{4}]. The second point can explain at least the
discrepancy of the lower values of ${D}_{\stackrel{\mathrm{\u203e}}{m}}$ because calculating
${D}_{\stackrel{\mathrm{\u203e}}{m}}$ from the measured PSM, CPC 3775, and Nano-SMPS data is not
straightforward, especially with the lower values of ${D}_{\stackrel{\mathrm{\u203e}}{m}}$ in
which the distributions measured by the Nano-SMPS are cut from the smaller
diameter edge due to very low detection efficiency. Therefore,
${D}_{\stackrel{\mathrm{\u203e}}{m}}$ calculated from the measurement data may be overestimated
with the lower values of ${D}_{\stackrel{\mathrm{\u203e}}{m}}$. This is also seen as long error
bars towards left, especially for ${D}_{\stackrel{\mathrm{\u203e}}{m}}$ values smaller than
10 nm (see the Supplement for details). However, by comparing the measured
and the simulated size distributions with *T*_{sa}=102 ^{∘}C in
Fig. 11 (measured ${D}_{\stackrel{\mathrm{\u203e}}{m}}=\mathrm{3.6}\phantom{\rule{0.125em}{0ex}}\mathrm{nm}$,
simulated ${D}_{\stackrel{\mathrm{\u203e}}{m}}=\mathrm{2.8}\phantom{\rule{0.125em}{0ex}}\mathrm{nm}$), it can be seen that the larger
diameter edges of the distribution do not correspond satisfactorily either,
which implies that the first point is the most possible explanation. Conversely, the
discrepancy of the higher values of ${D}_{\stackrel{\mathrm{\u203e}}{m}}$ can be partially
explained by the third point because simulating those cases with the LN distribution
only, even higher values of ${D}_{\stackrel{\mathrm{\u203e}}{m}}$ are outputted. That implies that the
PL+LN model underestimates the *N*_{PL}∕*N* ratio. The *N*_{PL}∕*N*
ratio is controlled by the value of *γ*; the proper functional form of
which is still under development in the PL+LN model. The last point can
also explain the discrepancies, but the direction of a discrepancy could be in
one way or another. The red dots follow mainly the same curve as the black
dots, with the exception of four cases in which the values of
${D}_{\stackrel{\mathrm{\u203e}}{m}}$ are clearly overestimated. These cases belong to
measurement set 3 and have high *T*_{PTD}. This discrepancy raises the last
point because there are clearly some uncertainties involved in the
condensation process modeling when *T*_{PTD} is high. It can be
related, for example, to the activity coefficient function of H_{2}SO_{4} because
too low an activity coefficient would cause too low a vapor pressure of
H_{2}SO_{4} at the surface of a particle, which would cause
particles that are too large.

Table 3 represents the ratios of the simulated *N*
and *M* with the residence times of 1.4 and 2.8 s. The simulated ratios
follow the same behavior as the measured ratios: with a low *T*_{sa}
value, the ratios are below unity, and with higher *T*_{sa} values, the
ratio of *N* increases but the ratio of *M* stays near unity. The ratios with
a low *T*_{sa} value correspond with the measured values, but
according to the simulations, the ratio of *N* does not increase with
increasing *T*_{sa} equally with the measured ratios. This implies that the
coagulation rate is underestimated in the model, but the reason for that is
unknown. The temperature with which the coagulation process would eliminate
the effect of the nucleation process, resulting in the number concentration
ratio of unity, is near 148 ^{∘}C (near 142 ^{∘}C according to
the measurements).

## 5.4 Nucleation rate function

The nucleation rate function with the best correspondence between the measured and the simulated data having a type of Eq. (2) used in the simulations has the parameters presented in Table 4 and is thus

where the concentrations are given in the inverse of cubic centimeters, the saturation vapor
pressure in pascals, and the nucleation rate is outputted in the inverse of cubic
centimeters times the inverse of seconds (cm^{−3} s^{−1}).
This function was applied within the environmental parameter ranges presented
in Table 5. The ranges can be considered to be the
ranges within which Eq. (11) is defined. However,
because the major part of the nucleation occurs when [H_{2}SO_{4}] is high
(nearer to the upper boundary than to the lower boundary), a wrong
formulation of *J* in the [H_{2}SO_{4}] values lower than 2×10^{11} cm^{−3} would have only a minor effect on the model outputs.
Therefore, an alternative range with 2×10^{11} cm^{−3} as a
minimum boundary for [H_{2}SO_{4}] is a more credible range within which
the obtained function for *J* produces reliable results.

Because *p*_{sa}^{∘}(*T*) has a nearly equal exponential form with
the saturation vapor pressure of H_{2}O (*p*_{w}^{∘}(*T*)),
*p*_{sa}^{∘}(*T*) can be expressed approximately using
*p*_{w}^{∘}(*T*), with

Hence, the magnitude of *J* remains as in Eq. (11) if
it is expressed with *p*_{w}^{∘}(*T*) using the form

or with both *p*_{sa}^{∘}(*T*) and *p*_{w}^{∘}(*T*)
using, for example, the form

or a different form,

The exponent *n*_{sa}=1.9 is in agreement with the former nucleation
studies related to vehicle exhaust (Vouitsis et al., 2005) or to the atmosphere
(Sihto et al., 2006; Riipinen et al., 2007; Brus et al., 2011; Riccobono et al., 2014), where *n*_{sa}
lies usually between 1 and 2. The exponent *n*_{sa}=1.9 corresponds
best to the kinetic nucleation theory (McMurry and Friedlander, 1979) where
*n*_{sa}=2. Estimating *n*_{sa} from the measured particle
number concentration provided the slope ${n}_{{N}_{\mathrm{PSM}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\mathrm{vs}.\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\left[{\mathrm{H}}_{\mathrm{2}}{\mathrm{SO}}_{\mathrm{4}}\right]}=\mathrm{0.4}$–10. The exponent *n*_{w}
estimated from the measurement data is ${n}_{{N}_{\mathrm{PSM}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\mathrm{vs}.\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}{\mathrm{RH}}_{\mathrm{PTD}}}=\mathrm{0.1}$–0.2, which is remarkably
lower than the inversely modeled exponent *n*_{w}=0.5. The slope of
*N*_{PSM} versus *T*_{PTD} of measurement set 3b in
Fig. 7 is

but the inversely modeled exponent *m*_{sa}=0.75 corresponds with the
slope of −27, which is remarkably more negative than ${n}_{{N}_{\mathrm{PSM}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\mathrm{vs}.\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}{T}_{\mathrm{PTD}}}$ due to the same uncertainties as involved
with the slopes ${n}_{{N}_{\mathrm{PSM}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\mathrm{vs}.\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\left[{\mathrm{H}}_{\mathrm{2}}{\mathrm{SO}}_{\mathrm{4}}\right]}$ and
${n}_{{N}_{\mathrm{PSM}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\mathrm{vs}.\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}{\mathrm{RH}}_{\mathrm{PTD}}}$. In
conclusion, inverse modeling provides, significantly more accurately, the
exponents over the method based on the measurement data only.

The nucleation rate was the highest in the PTD, where the hot sample and the cold
dilution air met. The major part of nucleation occurred in the beginning part
of the aging chamber. No noticeable nucleation occurred in the ejector
diluter, though the temperature reaches −30 ^{∘}C locally, which is in
agreement with the former studies. It provides partial validation for the
obtained *m*_{sa} value.

Homogeneous H_{2}SO_{4}–H_{2}O nucleation rate measurements using the
modified partial flow sampling system mimicking the dilution process
occurring in a real-world driving situation were performed. The aerosol
formed in the diluting and cooling sampling system was measured using the
PSM, the CPC 3775, and the Nano-SMPS. The particle size distribution near the
detection limit of the Nano-SMPS showed clear disagreement with the PSM and
the CPC3775 data, with major underestimation of the smaller particles and
distortion of the size distribution shape due to the limitations involved in
detecting small particles with simultaneous nucleation and particle growth
using the Nano-SMPS. Thus, the data without the PSM and the CPC 3775 would
unrealistically suggest the log-normal shape for the size distributions.

The measurements were simulated with the aerosol dynamics code CFD-TUTMAM
using the nucleation rate, which is explicitly defined as a function of
temperature and the concentrations of H_{2}SO_{4} and H_{2}O.
Equation (2) was used as the functional form of
nucleation rate. The parameters for Eq. (2) which
resulted in the best prediction for particle number concentrations and size
distributions were *n*_{sa}=1.9, *n*_{w}=0.5, and
*m*_{sa}=0.75, thus providing the nucleation rate function
Eq. (11) (or any of
Eqs. 13–15). As discussed in
Sect. 5.3, the obtained exponent
*n*_{sa}=1.9 may be slightly overestimated in high concentrations and
slightly underestimated in low concentrations. Estimating these exponents
using only the measured particle concentrations resulted in markedly higher
uncertainties when compared to modeling them inversely using the CFD-TUTMAM
code.

The raw sample was generated by evaporating H_{2}SO_{4} and H_{2}O
liquids. The concentration of H_{2}SO_{4} was controlled by adjusting the
temperature of the liquid, *T*_{sa}. The boundary condition for
H_{2}SO_{4} concentration, [H_{2}SO_{4}]_{raw}, was handled as
a fitting parameter for correspondence between the simulated size distributions and the
measured ones. Particle sizes were small with low *T*_{sa}, and the
size distributions were not in a log-normal form. Therefore, using the PL+LN
model to represent the size distributions in the CFD-TUTMAM was necessary.

In these measurements, particle formation was not observed with the
H_{2}SO_{4} concentrations below $\mathrm{5.7}\times {\mathrm{10}}^{\mathrm{12}}\phantom{\rule{0.125em}{0ex}}{\mathrm{cm}}^{-\mathrm{3}}$ at
exhaust condition temperatures. However, with real vehicle exhaust, in the
same sampling system used here, particle formation has been observed even
with the concentration of $\mathrm{2.5}\times {\mathrm{10}}^{\mathrm{9}}\phantom{\rule{0.125em}{0ex}}{\mathrm{cm}}^{-\mathrm{3}}$
(Arnold et al., 2012). This indicates that the nucleation rate of the binary
H_{2}SO_{4}–H_{2}O nucleation mechanism is lower than the nucleation
rate in real exhaust. Therefore, the binary H_{2}SO_{4}–H_{2}O
nucleation cannot be fully controlling the particle formation process;
instead, other compounds, such as hydrocarbons, existing in real exhaust are
likely to be involved in the process as well, which is in agreement with the
former exhaust-related nucleation studies
(Saito et al., 2002; Vaaraslahti et al., 2004; Meyer and Ristovski, 2007; Pirjola et al., 2015; Olin et al., 2015).

The obtained exponent *n*_{sa}=1.9 is in agreement with the former
nucleation studies related to the atmosphere or vehicle exhaust
(*n*_{sa}=1–2) and corresponds best with the kinetic nucleation
theory. However, the effects of [H_{2}O] and *T* obtained here may
differ from the former studies because the effects are not extensively
studied in them. The functional form, and especially the values of the
nucleation exponents for the homogeneous H_{2}SO_{4}–H_{2}O
nucleation rate obtained in this study, helps in finding the currently unknown
nucleation mechanism occurring in real vehicle or power plant boiler exhaust
or in the atmosphere. It also provides the starting point for inverse
modeling studies used to examine the hydrocarbon-involved
H_{2}SO_{4}–H_{2}O nucleation mechanism, which is likely occurring in
real vehicle exhaust. It can also be used to improve air quality models by
using it to model the effect of H_{2}SO_{4}-emitting traffic and power
generation on the particle concentration in urban air.

Data are available upon request from the corresponding author (miska.olin@tuni.fi).

The supplement related to this article is available online at: https://doi.org/10.5194/acp-19-6367-2019-supplement.

MO, JA, TR, and MDM designed the experiments, and MO and JA carried them out. MO analyzed the measurement data, developed the model code, and performed the simulations. MRTP designed the IC analysis. MO prepared the paper, with contributions from all co-authors.

The authors declare that they have no conflict of interest.

The authors thank CSC and TCSC for the computational time. We also thank Prof. Mikko Sipilä from the University of Helsinki for lending the chemical ionization inlet for the atmospheric pressure interface time-of-flight mass spectrometer, the tofTools team for providing tools for mass spectrometry analysis, and M. Sc. Kalle Koivuniemi for ion chromatography measurements.

This research has been supported by the graduate school of Tampere University of Technology and the Maj and Tor Nessling Foundation (grant no. 2014452).

This paper was edited by Neil M. Donahue and reviewed by two anonymous referees.

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