Population balance equations, derived from the aerosol general dynamic equation, have been used to estimate particle formation rates (Cai and Jiang, 2017; Kulmala et al., 2012), particle growth rates (Kuang et al., 2012), and the effect of transport on aerosol particle size distribution (Cai et al., 2018). In this study, we use the population balance method to estimate particle number emissions into a column extending from the ground to the top of the atmospheric mixing layer (ML). The time evolution of particle number concentration in size bin i (Ni) in this column can be described as 1ddt(Ni×MLH)=Ei+JGRin,i-JGRout,i-Scoag,i-Sdepos,i. Here Ei (m-2s-1) represents emissions to the size bin i and JGRin,i and JGRout,i describe the growth into and out of the size bin i. Scoag,i and Sdepos,i describe the losses of particles in the size bin i due to coagulation and deposition. The time derivative of the column number concentration can be divided into two terms: the first one is dNidtMLH, which describes the change in the column particle number concentration due to processes directly affecting particle number concentration Ni, and the second term is NidMLHdt, which describes the dilution of the concentration Ni, due to an increase in mixing layer height (MLH) in the morning.

By reorganizing Eq. (1) and writing out all the terms, emission Ei is obtained from 2Ei=dNidtMLH-MLH×GRin,i×NGRin,iΔDp,GRin,i+MLH×GRout,i×NGRout,iΔDp,GRout,i+MLH×CoagSi×Ni+MLH×DRi×Ni+NidMLHdt. Here Ni is the number concentration of particles in the size bin i. GRin,i is the growth rate of particles growing into the size bin i; NGRin,i is the number concentration of particles able to grow into the size bin i in the studied time step (tstep), which is calculated based on GRin,i, and ΔDp,GRin,i is the size range of those particles. Correspondingly, GRout,i is the growth rate of particles growing out of the size bin i, NGRout,i is the concentration of particles growing out of the size bin i in tstep, and ΔDp,GRout,i is their size range. CoagSi is the coagulation sink for particles in size bin i, caused by larger particles, and DRi is the loss rate of particle in the size bin i due to wet and dry deposition.

For the smallest size bin (i=1), the term describing the growth into the size bin is omitted, and thus the emissions calculated for the first size bin also include the flux of growing particles from below the lowest considered size. These particles can originate from primary emissions but also from atmospheric NPF. We omit the first growth term for the smallest size bin for two reasons: (1) to include the effect of atmospheric clustering on particle production and (2) because the measured concentrations of the smallest particles, needed for calculating the flux of particles growing into the size bin, contain large uncertainties. Overall, one should note that applying Eq. (2) to determine particle number emissions includes many assumptions. In the next section, we discuss these assumptions and their validity for our data set from Beijing.

We applied the introduced method to estimate particle number emissions in Beijing, China, using measurements performed at the measurement station of Beijing University of Chemical Technology (BUCT) during January 2018–March 2019. The station is located in the western part of Beijing (39∘56′31′′ N, 116∘17′50′′ E), about 150 m southwest of the closest busy road and 550 m west of the 3rd Ring Road of Beijing. The location of the measurement site is shown in Fig. 1 with respect to urban Beijing and its surroundings. The urban region with high population density (Fig. 1b) and high emissions of PM2.5 and different trace gases (NOx, CO, and SO2) based on emission inventories (Fig. A1) extends ∼20 km west, ∼100–200 km east, and ∼50 km north and south of our site.

For particle size distribution data, we used data measured with a diethylene glycol scanning mobility particle sizer (DEG-SMPS; Cai et al., 2017; Fu et al., 2019; Jiang et al., 2011) and a custom-made particle size distribution (PSD; Liu et al., 2016) system. The DEG-SMPS measures particle sizes between 1 and 6.5 nm (electrical mobility diameter) and the PSD system particle sizes between 3 nm and 10 µm, using a combination of a homemade nano-SMPS (3–55 nm, electrical mobility diameter), a homemade long-SMPS (25–650 nm, electric mobility diameter), and a TSI 3321 aerodynamic particle sizer (0.55–10 µm, aerodynamic diameter). We corrected particle diffusion losses, bipolar charging efficiency, multiple charging, and detection efficiency, when inverting the size distribution data. To obtain the final size distribution for the size ranges where different instruments overlap, we calculated the weighted average of size distributions measured with different instruments. The days when the whole particle size distribution was not measured reliably due to instrument malfunctioning were disregarded. The final corrected data set includes 136 d of particle size distributions between 1 nm and 10 µm, covering the months from October to May. For more details of the particle size distribution measurements performed at the BUCT station, see Zhou et al. (2020).

Based on the particle size distribution data, we classified the days into days with an NPF event and days without an event. A day was classified as an NPF event day if an appearance of a new mode of sub-10 nm particles and the further growth of this mode was observed and it was not clearly linked to particle emissions from traffic.

The MLH was obtained from ceilometer measurements (CL51, Vaisala Inc, Finland) of the optical backscattering by applying a three-step idealized profile (Eresmaa et al., 2012). Because ceilometer data were not available for every day with particle size distribution data, we calculated the average diurnal cycles of the MLH for NPF event days and nonevent days and used them when applying Eq. (2). This is justified as we study the average diurnal cycle of particle number emissions, instead of their day-to-day variation.

For GR we used a constant value of 3 nmh-1 for all the size bins, which corresponds to a typical GR of particles between 3 and 7 nm at the station during the measurement period (Zhou et al., 2020). To describe the losses of particles by coagulation scavenging, we calculated the coagulation sink (CoagS) for each size bin i from the particle size distribution data, based on the coagulation coefficients between particles in size bin i and larger particles (Kulmala et al., 2001).

When applying Eq. (1) to our data set, we calculated particle number emissions to 22 particle size bins with the lower limit Dp and the upper limit Dp×4/3, between 2.0 nm and 1.1 µm. After calculating particle number emissions for each day, we determined the average diurnal cycle of particle number emission size distributions separately on NPF event days and nonevent days.

We compared the emissions determined with our method to those calculated with the GAINS model (Paasonen et al., 2016). The GAINS emissions were retrieved from the model web page (https://www.iiasa.ac.at/web/home/research/researchPrograms/air/PN.html, last access: 14 February 2020, providing calculated emissions for years 2010, 2020, and 2030; IIASA, 2016) for the grid cell of 0.5∘ × 0.5∘ in which the center of Beijing is located. We used emissions calculated for the year 2010 based on the results of Paasonen et al. (2016) that indicate that the emissions for the year 2010 have less uncertainties associated with them than the corresponding values for the year 2020. In addition, to gain insight into the effects of particle transport and the source area of our method, we utilized emissions of PM2.5, NOx, CO, and SO2 obtained from the MIX emission inventory (Li et al., 2017), which is the combined result of the best-available regional-scale emission inventories in Asia. The MIX inventory used here describes emissions for the year 2010 on a 0.25∘ × 0.25∘ grid, and the data are available online (http://www.meicmodel.org/dataset-mix.html, last access: 11 June 2020). In this study, the emissions of different trace gases are used to describe the general activity levels of different kinds of combustion sources, which also emit particles.

During the measurement period, 44 % of the days were classified as NPF event days. Figure 2 presents the average diurnal cycle of the MLH and its time derivative (dMLH/dt) on NPF event days and nonevent days. On both NPF event days and nonevent days, the MLH starts to increase after 06:00 LT in the morning and reaches its maximum around 15:00. However, on NPF event days the MLH reaches clearly higher values (the maximum height ∼2200 m) than on nonevents days (the maximum height ∼820 m), and thus the time derivative of the MLH is larger on NPF event days. Note that the time derivative is shown only for the mornings, when MLH increases, causing dilution of particle concentrations.

The average diurnal variation in particle number size distribution on NPF event days and nonevent days is shown in Fig. 3. On nonevent days particle concentrations between ∼ 6 and 150 nm exhibit clear maxima during morning (06:00–12:00) and evening (17:00–23:00) hours. This is caused by emissions of particles from traffic and possibly other sources and the growth of the emitted particles. On NPF event days, primary particle emissions can also be observed, but the time evolution of the particle size distribution is dominated by the appearance of a high number of sub-5 nm particles between about 08:00 and 17:00 and their growth to larger sizes. One should note, though, that the growth of all sub-5 nm particles, especially those appearing in the afternoon, cannot be observed at the measurement site. This causes difficulties when estimating particle number emissions for NPF event days, as discussed in the next section.

We calculated the average diurnal cycle of particle number emission size distributions separately for NPF event days and nonevent days (Fig. 4). On nonevent days the time evolution of particle number emissions looks reasonable. The emissions at almost all studied sizes are highest during the morning (06:00–12:00) and evening (17:00–22:00), which probably is, at least partly, linked to particle emissions from traffic. The connection to different sources and the differences in particle emissions between different sizes are discussed in more detail in the next sections.

On NPF event days, the time evolution of particle number emission size distributions looks less plausible. A strong production of sub-3 nm particles by atmospheric NPF can be observed during the day, as expected. However, on NPF event days we also see a clearly higher production of particles larger than 3 nm (∼3–5 and ∼7–20 nm) than on nonevent days, simultaneously or immediately after particles are produced to the smallest size bin (∼2–3 nm). This indicates that our calculations are unable to accurately describe particle dynamics in NPF events, and therefore the contribution of NPF can also be observed at sizes larger than 3 nm. There can be several reasons for this. For example, a higher particle formation rate at the higher levels of the boundary layer could lead to an increasing particle concentration with increasing diameter, when more numerous particles from above would be transported to the measurement site and detected after their initial growth during the transportation. In addition, the results can be affected by time- and size-dependent variation in particle GR (see Sect. 3.5). Another possible reason is measurement uncertainties, which can be expected to be highest at the smallest sizes and around the sizes where the particle size distribution instrument changes (see Sect. 2.3). The calculated particle emissions for NPF event days also look unreliable because of the distinct minimum visible between 5.5 and 7.2 nm. The minimum is likely mainly caused by not all sub-6 nm particles growing to larger sizes, as discussed in Sect. 3.1. Therefore, when we subtract the term describing the growth into the bin of 5.5–7.2 nm (see Eq. 2), we end up with emissions that are too small or even negative. In addition, the change in the instrument around that size range may also affect the calculated emissions. Finally, the differences in calculated emissions on NPF event days and nonevent days can also be partly due to differences in the prevailing wind direction on event and nonevent days (see Sect. 3.5). Overall, due to the difficulties in describing particle dynamics on NPF event days, we focus on determining particle number emissions on nonevent days. Determining the exact contributions of primary particle emissions and NPF to particle number concentrations on NPF event days requires further work, and it will be a subject of future study.

To investigate the variation in particle number emissions in more detail, we determined the diurnal cycle of particle number emissions for different size ranges (Fig. 5a on a linear scale and Fig. A2 on a logarithmic scale). We also studied the diurnal cycle of the boundary layer burden of nitrogen oxides (NOx), which is calculated as the product of the NOx concentration and MLH and which roughly represents the diurnal variation in NOx emissions. As shown by Fig. 5b, the estimated NOx emissions have a maximum around 09:00, linked to morning traffic, while they do not have a clear afternoon or evening maximum, likely due to fast photochemical loss of NOx (Lu et al., 2019). Cai et al. (2020) used EMBEV-Link (Link-level Emission factor Model for the BEijing Vehicle fleet; Yang et al., 2019) to estimate the diurnal cycle of PM2.5 emissions at our measurement site. According to the modeling results, PM2.5 emissions originating from gasoline vehicles in urban Beijing start to increase before 06:00 in the morning, reach the first maximum around 07:00–08:00 and the second maximum around 17:00–18:00, after which they decrease to lower nighttime values. However, the modeled PM2.5 emissions from diesel vehicles are highest at night (Cai et al., 2020).

Figure 5a shows that the particle emissions to the smallest studied size bin (∼2–3 nm), which also includes the growth of the particles from smaller sizes, increase in the morning, reach a first maximum just before noon, and show two other peaks around 14:00 and 16:00. The noontime maximum, which is also observed on NPF event days (figure not shown), suggests that formation of sub-3 nm particles by clustering of vapor molecules can take place on nonevent days, but because the growth of particles to larger sizes is not seen, it is not defined as an NPF event. Weak production of sub-3 nm particles can also be observed in the average diurnal cycle of particle number concentrations on non-NPF event days (Fig. 3). In addition to atmospheric clustering, it is possible that some of the sub-3 nm particles originate from traffic (Rönkkö et al., 2017).

The emissions to the size range between 3 and 6 nm are highest between 08:00 and 12:00 and around 14:00 and 17:00 (Fig. 5a). The morning maximum coincides with the morning maximum of estimated NOx emissions (Fig. 5b), suggesting that traffic contributes to particle emissions into this size range. The importance of traffic emissions is also supported by the fact that the diurnal cycle of emissions is roughly similar to the diurnal cycle of modeled PM2.5 emissions from gasoline vehicles in Cai et al. (2020), which have maxima around 07:00–08:00 and 17:00. In addition, clustering of atmospheric vapors and the following growth to 3–6 nm sizes can contribute to the calculated emissions to this size range, as atmospheric clustering seems to also occur on nonevent days. This is further supported by our analysis in Sect. 3.5.

The emissions to the size ranges of 6–30 nm and 30–100 nm have quite similar diurnal cycles with the first maximum between 08:00 and 12:00 and the second, slightly higher maximum after 18:00 (Fig. 5a). The morning maxima indicate particle emissions from traffic to these size ranges too. The fact that the evening maxima are higher than the morning maxima suggests either higher emissions from traffic at these size ranges at this time of the day or possible contributions from other emission sources (see the discussion in the next section).

The emissions to the largest size range (100–1000 nm) are low overall, exhibiting one clear maximum around 10:00 and another, much less pronounced, one around 18:00 (see Fig. A2). Although the morning maximum could be related to emissions from traffic, the fact that it is much more distinct than the evening maximum suggests that it may be partly caused by overestimating the effect of dilution due to an increase in the MLH in the morning. As discussed in Sect. 2.2, it is unlikely that the concentrations of particles larger than 100 nm always decrease with increasing MLH, as assumed in Eq. (2).

To gain more insight into particle emissions at different sizes, we studied the average particle number emission size distributions at different times of the day: early morning (06:00–08:00), late morning (09:00–11:00), evening (18:00–20:00), and after midnight (00:00–02:00; Fig. 6; see also Fig. A3). Clear differences between the size distributions at different hours can be observed, indicating the production of particles from different sources.

Strong production of the smallest (Dp<3 nm) particles is observed at 09:00–11:00 (Fig. 6), which is likely connected to atmospheric cluster formation, as discussed above. The production of particles of this size is moderate also in the early morning and evening and non-negligible even at night. Recently, atmospheric NPF in Beijing was suggested to start with clustering between sulfuric acid and an amine (Deng et al., 2020), and thus this is likely the main mechanism for the observed formation of sub-3 nm particles. This mechanism is stronger during the day, due to photochemical production of sulfuric acid, but it is possible that these clusters also form in the nighttime. On the other hand, traffic emissions may also contribute to the production of sub-3 nm particles, as dilution and cooling of traffic exhaust has been shown to produce a high number of sub-3 nm particles (Rönkkö et al., 2017).

The size distributions of particle number emissions show a maximum around 10 nm at all times (Fig. 6). The diurnal cycle of emissions into this size range (Figs. 4 and 5) indicates that this maximum is likely caused by traffic emissions. This is supported by laboratory measurements showing that traffic exhaust contains nucleation mode particles (Rönkkö et al., 2007; Shi and Harrison, 1999), which in some conditions have a mode diameter of ∼10 nm (Rönkkö et al., 2017). In addition, in roadside measurements of 1–1000 nm particle number concentrations, particle modes of around 1–3 and 10 nm have been observed in urban and semiurban background conditions (Hietikko et al., 2018; Rönkkö et al., 2017).

At sizes between ∼15 and 50 nm, the emissions are clearly highest at 18:00–20:00 (Fig. 6). Although traffic likely contributes to emissions into this size range, high emissions in the evening can indicate the contribution of some other source, such as cooking activities. The contribution of cooking emissions at this time is supported by studies applying PMF analysis to chemical composition and particle size distribution data from Beijing, which have found cooking-related factors peaking around 19:00–20:00 (Cai et al., 2020; Hu et al., 2017; Liu et al., 2017). In a study by Cai et al. (2020), the cooking-related particle number size distribution factor had a GMD of ∼50 nm. In studies focusing on cooking emissions, Chinese cooking has been found to typically produce particles with the mode diameter ranging from 20 to 100 nm (Zhao and Zhao, 2018).

There is a weak maximum visible in the particle size distribution around 100 nm at 09:00–11:00, also seen as a separate shoulder in the logarithmic emission size distribution at 06:00–08:00 (Fig. A3). As discussed above, this maximum may be related to traffic but can also be due to overestimation of the dilution effect for larger particles. Generally, the emissions at sizes larger than 100 nm are low, and particle number emissions around our measurement site seem to be dominated by emissions of smaller particles, especially those in nucleation mode (Dp<30 nm).

We compared our results to annual particle number emissions determined for an approx. 50 km × 50 km grid cell around downtown Beijing with the GAINS model. This was done by calculating the annual sum of the emissions to different size bins, based on particle number emissions determined for non-NPF event days. It should be noted, though, that we used the GAINS emissions calculated for the year 2010 and the number emissions have likely changed since then.

Figure 9 shows that the annual particle number emission size distributions obtained with the two methods are clearly different (see also Fig. A8). In the GAINS model, the particle emissions have a unimodal distribution with a peak at ∼50 nm, while our calculated annual emissions show multiple peaks and clearly higher particle emissions below 60 nm than in GAINS (note that the smallest size bin in GAINS is 3–10 nm). However, at sizes above 60 nm, the two methods agree remarkably well.

The large grid size in GAINS partly explains the lower emissions below 60 nm. Our measurement site is located close to two busy roads, and thus the contribution of traffic emissions to the observed emission size distribution can be expected to be higher than to the more regional-scale emissions obtained from GAINS. Paasonen et al. (2016) also suggested that the emissions of particles with diameters below 30 nm are underestimated in GAINS, because the experimentally determined emission factors for many sources include only particles that are nonvolatile (after heating) and/or particles larger than 10 nm in diameter.

When calculating the total annual particle number emissions to the sizes between 3 and 1000 nm, our method gives clearly higher particle number emissions (1.1×1017 m-2) than GAINS (1.4×1016 m-2). Although the values of particle number emissions determined with our method should not be considered exact, due to the assumptions of the method and contribution of atmospheric cluster formation (see Sects. 2.2 and 3.5), the vast difference between our calculations and the GAINS model highlights the need for increased understanding of anthropogenic particle number emissions, especially for sizes smaller than 60 nm. However, the similarity of the emissions at sizes above 60 nm from GAINS and our method gives confidence in the ability of both methods to yield reasonable estimates for particle number emissions. It also suggests that the emissions obtained with our method originate from an area of approximately the same size as the chosen grid size of GAINS, i.e., 50 km × 50 km, which is consistent with our estimation in Sect. 3.5.1.