The formulation of the PL distribution originates from Eq. (), where the power of Dp is allowed to vary: dNdln⁡DpPL=NPLαD2α-D1αDpα,D1≤Dp≤D2,α≠0NPLln⁡(D2/D1),D1≤Dp≤D2,α=00,otherwise, where NPL is the total particle number concentration, α is the slope parameter, D1 is the smallest diameter, and D2 is the largest diameter of the PL distribution. In this form, the PL distribution has four parameters, which leads to numerical challenges for the reconstruction of the distribution parameters from four moments. Nevertheless, by fixing one parameter, only three moments are required to be modeled and the reconstruction will simplify. Here, the value of D1 is fixed to the diameter of a newly formed particle, which is also physically sensible because that value is not expected to vary significantly; in atmospheric particle formation, the value is about 1.5±0.3nm .

Three moments required in the modeling of the PL distribution with parameters NPL, α, and D2 are, in this article, number, NPL, surface area, SPL, and mass, MPL, concentrations: NPL=NPL,SPL=∫-∞∞spdNdln⁡DpPLdln⁡Dp=s1NPLαα+2D2D1α+2-1D2D1α-1,MPL=∫-∞∞mpdNdln⁡DpPLdln⁡Dp=m1NPLαα+3D2D1α+3-1D2D1α-1, where sp and mp are the surface area and the mass of a particle, respectively, and s1 and m1 are the surface area and the mass of a newly formed particle, respectively. All particles are assumed to be spherical. Equations () and () have singularities at α values of -3, -2, and 0. In those cases, the equations have different formulations, and, from now on, the singularity equations are not shown here due to the fact that α will never equal a singularity value precisely in a simulation. To model the composition of particles can be done by separating the mass concentration to different components using the assumption that the particles are internally mixed; i.e., the composition does not vary with particle diameter. Modeling of the particle composition is, however, outside of the scope of this article.

The reconstruction of the distribution parameters from the moments NPL, SPL, and MPL is performed as follows. The zeroth moment NPL is already one of the distribution parameters, but SPL and MPL are not. The latter are converted to the system of equations of two unknown variables α and d=D2/D1: MPLNPL1m1=αα+3dα+3-1dα-1MPLSPLs1m1=α+2α+3dα+3-1dα+2-1. However, there is no analytical solution for this system of equations, but solving two variables numerically is sufficiently fast for this purpose. A pre-calculated interpolation table is used in the numerical solution, with which a more rapid calculation is obtained. The interpolation table increases the memory cost of the model, but as the table is unique (independent on temporal or spatial coordinate) it needs to be stored in one memory location only.

The LN distribution is expressed by the equation dNdln⁡DpLN=NLN2πln⁡σexp⁡-ln⁡2Dp/Dg2ln⁡2σ, where NLN is the total particle number concentration, σ the geometric standard deviation, and Dg the geometric mean diameter of the LN distribution. The LN distribution is also modeled as three moments, NLN, SLN, and MLN. Following the method of , the reconstruction of the distribution parameters from the moments can be performed using the equations NLN=NLN,Dg=6-2/3π-5/6ρ2/3NLN-5/6SLN3/2MLN-2/3,ln⁡2σ=ln⁡62/3π1/3ρ-2/3NLN1/3SLN-1MLN2/3, where ρ is the particle density.

The combined particle distribution is modeled as the superposition of the PL and the LN distributions: dNdln⁡DpPL+LN=dNdln⁡DpPL+dNdln⁡DpLN. Figure  presents examples of the PL+LN distribution. PL distributions with different values of α are shown in the left pane; N=106cm-3, D1=1.5nm, and D2=5nm are equal in all four distributions. The right pane shows the PL distribution with values N=106cm-3, α=1, D1=1.5nm, and D2=3nm, the LN distribution with values N=5×105cm-3, Dg=4nm, and σ=1.1, and the combination of them.

A schematic presentation of the connections between the distributions is shown in Fig. . Particles in the PL distribution, formed by new particle formation and grown by condensation and coagulation (Fig. ), are transferred to the LN distribution through three intermodal processes: coagulational transfer, intermodal coagulation, and condensational transfer. The coagulational transfer is accounted by intramodal coagulation, i.e., self-coagulation, which is basically an intramodal process, but in this model it is used to initiate the LN distribution by transferring the coalesced resultant particles larger than D2 to the LN distribution. The coagulational transfer is described in more detail in Sect. . After the LN distribution is initiated, particles of the both distributions begin to collide intermodally (the intermodal coagulation). In that case, the resultant particles are always assigned to the LN distribution, which is thought to consist of larger particles than the PL distribution.

The coagulational transfer remains the only process initiating the formation of the LN distribution if the condensational transfer is neglected. Therefore, in the case of low particle number concentration, i.e., low intramodal coagulation rate, the formation rate of the LN distribution is slow; thus, the combined distribution would be mainly in a power law form. However, in realistic particle formation events, log-normal features in the size distribution are widely observed . This is due to the fact that the aerosol processes have normally time- and size-dependent rates and that the particles can be multicomponent and due to the intramodal coagulation that eventually results in self-preserving log-normal distribution . The model described here connects the formation of the LN distribution with the intramodal coagulation only. Therefore, LN features can be generated artificially to the PL+LN distribution by transferring some of the particles from the PL distribution to the LN distribution. This transfer is calculated through condensation (the condensational transfer). Particles that are to be grown beyond the diameter D2 are transferred to the LN distribution by the condensational transfer instead of keeping them in the PL distribution and increasing the value of D2. The condensational transfer is described in more detail in Sect. .

New particle formation is modeled by a term npfX=J(t),X=NPLJ(t)s1,X=SPLJ(t)m1,X=MPL0,X=NLN, SLN, or MLN, where J(t) can be calculated, e.g., through any nucleation theory, in which J(t) depends also on vapor concentrations and temperature, for example. However, finding the correct formulation for J(t) is outside of the scope of this article; thus, we decided to use formation rate as an input parameter that can be either a constant, J, or a time-dependent function, J(t). Additionally, inverse modeling is done to obtain J(t) from measured particle number concentrations.

The sizes of a newly formed particle (D1, s1, and m1) can be obtained from nucleation theories, but they are assumed to be constants here. In the case where J(t) suddenly drops to zero but condensation still continues, concentration of particles with diameters around D1 would subsequently decrease down to zero due to the growth of newly formed particles to larger diameters. Therefore, D1, as a parameter of the PL distribution, should be a variable to model the distribution accurately. However, this would change the distribution back to a more complex four-parameter distribution that is outside of the scope of this article. In that case or with smoothly decreasing J(t), α has a tendency to increase.

Condensation rate [kgm-3s] of vapor v on a particle distribution, PL or LN (denoted with i), can be modeled as condMi=∫-∞∞dmp,vdtdNdln⁡Dpidln⁡Dp, where dmp,vdt is the mass growth rate of a particle [kgs-1] due to vapor v : dmp,vdt= 2π(Dp+Dv)(Dp+Dv)(Cv,∞-Cv,p)⋅Kn+10.377Kn+1+43ϕ(Kn2+Kn), where Dv is the diameter of a vapor molecule. Dp and Dv are the diffusion coefficients of a particle and of a vapor molecule, respectively. Cv,∞ and Cv,p are the mass concentration of the vapor in the far-field and over the particle surface, respectively. Kn and ϕ are the Knudsen number and the mass accommodation coefficient, respectively. The concentration Cv,p is Cv,p=ΓvCv,satexp⁡4SmvkBTρDp, where Γv, Cv,sat, and mv are activity, the saturation concentration, and the molecule mass of the vapor, respectively, S is surface tension, kB is the Boltzmann constant, and T is temperature.

If all the parameters in Eq. () do not depend on the spatial location, as is the case in a one-dimensional simulation, the mass growth rate can be considered a function of time and the particle diameter only. Here, the mass growth rate (single-component case) is expressed using condensational growth rate g(t,Dp): dmpdtt,Dp=dmpdDp⋅dDpdtt,Dp=π2ρDp2g(t,Dp). Hence, the condensation rate for a particle distribution becomes condMi=π2ρ∫-∞∞Dp2g(t,Dp)dNdln⁡Dpidln⁡Dp, which has an analytical solution for the both distributions, PL and LN, when g(t,Dp) can be expressed with a polynomial of Dp. The mass growth rate is proportional to Dp2 when the following conditions are met: (1) the particle size is in free-molecular regime, (2) Dp≫Dv, 3) Cv,∞≫Cv,p. The last one applies when the particle size is large or when the vapor has low saturation vapor pressure. Since particle sizes near the molecular size are modeled in this article, only the first condition applies satisfactorily. Nevertheless, this article concentrates mainly in cases where the mass growth rate is assumed to be proportional to Dp2. Additionally, a single test case, where the mass growth rate is calculated using Eqs. ()–(), is presented. The main point in this article is not to provide the correct formulation for g(t,Dp), but rather to compare different models, and additionally to perform inverse modeling to obtain g(t) from the time evolution of measured aerosol size distributions. Due to the assumption of the proportionality of the mass growth rate, the condensational growth rate becomes size-independent, and finally, the condensation terms used in Eq. () become condX=0,X=Ni2πg(t)∫-∞∞DpdNi,X=Siπ2ρg(t)∫-∞∞Dp2dNi,X=Mi where dNi is an abbreviation of dNdln⁡Dpidln⁡Dp. The analytical solution for Eq. () is condX=Xg(t)⋅0,X=NPL2D1α+2α+1dα+1-1dα+2-1,X=SPL3D1α+3α+2dα+2-1dα+3-1,X=MPL0,X=NLN2Dgexp⁡-32ln⁡2σ,X=SLN3Dgexp⁡-52ln⁡2σ,X=MLN when α is not -3, -2, or -1.

When the mass growth rate is calculated from the vapor concentrations and the properties of the vapor and the particles using Eqs. ()–(), it rarely can be expressed with a polynomial of Dp, unless polynomial fits are done for the function. However, if the vapor concentrations or the other properties are allowed to vary during the simulation, fits for the mass growth rate function may become inconvenient. In that case, the integral in Eq. () cannot be solved analytically. Therefore, numerical integration is required, in which Eq. () is calculated in a form of condMi=π2ρ∑j=1nDj2g(t,Dj)dNdln⁡Dpiln⁡Dj+1Dj, where Dj is the particle diameter of the size section j used in numerical integration when the particle diameter range is split into n sections. Computational cost of numerical integration is, however, higher compared to analytical solution of the integrals. Therefore, Gaussian quadratures are used here to reduce the associated computing time; they provide the optimal particle diameters and their weights for efficient evaluation of the integrals. The details of the Gaussian quadratures are described in Appendix .

Coagulation is modeled as intramodal coagulation within the PL distribution and within the LN distribution and as intermodal coagulation from the PL distribution to the LN distribution. The coagulation terms derived from the equations of are coagNPL=-12∫-∞∞∫-∞∞β(Dp,Dp′)dNPLdNPL′-∫-∞∞∫-∞∞β(Dp,Dp′)dNPLdNLN′coagSPL=-12∫-∞∞∫-∞∞2sp-sp32+sp′3223⋅β(Dp,Dp′)dNPLdNPL′-∫-∞∞∫-∞∞spβ(Dp,Dp′)dNPLdNLN′coagMPL=-∫-∞∞∫-∞∞mpβ(Dp,Dp′)dNPLdNLN′coagNLN=-12∫-∞∞∫-∞∞β(Dp,Dp′)dNLNdNLN′coagSLN=-12∫-∞∞∫-∞∞2sp-sp32+sp′3223⋅β(Dp,Dp′)dNLNdNLN′+∫-∞∞∫-∞∞sp32+sp′3223-sp′⋅β(Dp,Dp′)dNPLdNLN′coagMLN=∫-∞∞∫-∞∞mpβ(Dp,Dp′)dNPLdNLN′, where β(Dp,Dp′) is the coagulation coefficient of particles with the diameters of Dp and Dp′ calculated with the equation β(Dp,Dp′)=2π(Dp+Dp′)(Dp+Dp′)f(Kncoag), where f(Kncoag) is the transition regime function of , f(Kncoag)=1+Kncoag1+2Kncoag+2Kncoag2, and Kncoag is the Knudsen number for coagulation, Kncoag=4(Dp+Dp′)(Dp+Dp′)c¯2+c¯′2. c¯ and c¯′ are the mean thermal velocities of particles with the diameters of Dp and Dp′.

The integrals in Eqs. ()–() cannot be solved analytically in the transition regime because Eq. () cannot be presented in a polynomial form. Therefore, the integrals are calculated numerically or by using quadratures in the same manner as with the condensation term described in Appendix .

The losses due to coagulation of the particles in the PL+LN distribution to the background distribution excluded from the PL+LN distribution are considered the coagulational losses. Particles in the background distribution are assumed to be significantly larger than the particles in the PL+LN distribution. Therefore, the particle diameters of the background distribution can be approximated with a single diameter value, e.g., CMDbg (count median diameter). According to , the coagulation coefficient will then become β(Dp,CMDbg)≈β(D1,CMDbg)DpD1lbg, where lbg is the exponent depending on CMDbg. The value of lbg ranges between -2 and -1 . The coagulational loss term, e.g., for a number concentration is lossNicoag=-Nbg∫-∞∞β(Dp,CMDbg)dNi≈-Nbgβ(D1,CMDbg)D1-lbg∫-∞∞DplbgdNi, in which the last integral can be solved analytically. The analytical solutions for the coagulational loss terms are lossXcoag=-XNbg⋅β(D1,CMDbg)αα+lbgdα+lbg-1dα-1,X=NPLβ(D1,CMDbg)α+2α+2+lbgdα+2+lbg-1dα+2-1,X=SPLβ(D1,CMDbg)α+3α+3+lbgdα+3+lbg-1dα+3-1,X=MPLβ(Dg,CMDbg)exp⁡12lbg2ln⁡2σ,X=NLNβ(Dg,CMDbg)exp⁡12lbg2+2lbgln⁡2σ,X=SLNβ(Dg,CMDbg)exp⁡12lbg2+3lbgln⁡2σ,X=MLN when α is not 0 or -lbg.

The losses to walls due to diffusion of particles are considered the depositional losses. They are modeled with the method of , in which the deposition rate of particles in a test chamber is λ=1V∑wAwu, where V is the volume of the chamber, Aw is the surface area of the wall w, and u is the deposition velocity of particles. A simple approximation for the deposition velocity is used here: u∝Dp-1, which is valid for particles smaller than 100 nm according to . The depositional loss term, e.g, for a number concentration, now becomes lossNidep=-∫-∞∞λdNi=-kdep∫-∞∞Dp-1dNi, where all effects, except the effect of the diameter, are included in the deposition coefficient kdep. The last integral can be solved analytically, from which the depositional loss terms become lossXdep=-Xkdep⋅D1-1αα-1dα-1-1dα-1,X=NPLD1-1α+2α+1dα+1-1dα+2-1,X=SPLD1-1α+3α+2dα+2-1dα+3-1,X=MPLDg-1exp⁡12ln⁡2σ,X=NLNDg-1exp⁡-32ln⁡2σ,X=SLNDg-1exp⁡-52ln⁡2σ,X=MLN when α is not 0 or 1.

The effect of particle losses on the PL distribution is seen as decreased α. In the trivial case, as in Eq. (), α becomes less than zero when kdep>g. This effect is due to increased losses with increasing particle diameters because larger particles have longer residence times from the moment since their formation. However, Eq. () counteracts in this effect by decreasing the deposition velocity with increasing particle size, but with small g, the effect of increased residence time dominates over the effect of decreased deposition velocity. Additionally, α is further decreased due to coagulational losses.

The intermodal coagulation is included together with the intramodal coagulation in the coagulation terms (coagX) seen in Eqs. ()–(). The coagulational (transferXcoag) and condensational (transferXcond) transfer are modeled as follows.

Particles with the diameter higher than the cut diameter, Dcoag=D23-Dp′31/3, form particles with the diameter higher than D2 after coagulating with a particle having a diameter of Dp′, assuming full coalescence (Fig. ). Those resultant particles are transferred from the PL distribution to the LN distribution, because their particle diameters will correspond with the form of a LN distribution rather than a PL distribution, which can be observed using a highly accurate sectional model. The coagulational transfer terms are negative for the PL distribution and positive for the LN distribution to conserve the moments, and they are expressed as transferNPL=-transferNLN=-12∫-∞∞∫ln⁡Dcoag∞β(Dp,Dp′)dNPLdNPL′,transferSPL=-transferSLN=-12∫-∞∞∫ln⁡Dcoag∞sp32+sp′3223β(Dp,Dp′)dNPLdNPL′,transferMPL=-transferMLN=-12∫-∞∞∫ln⁡Dcoag∞mp+mp′β(Dp,Dp′)dNPLdNPL′, which are calculated using the quadrature or numerical integration as in the case of the coagulation terms.

Considering a time step of Δt in a Lagrangian simulation, particles with the diameters larger than Dcond=D2-gΔt will grow due to condensation to have the diameters larger than D2 (Fig. ). Modeling condensation only, the value of D2 at the next time step would increase to D2+gΔt. The condensational transfer is used to transfer the particles in the PL distribution with the diameters of Dcond<Dp<D2 to the LN distribution. However, if the condensational transfer were modeled fully, D2 would never increase and all condensation would affect the LN distribution only. In that case, the distributions would separate from each other. For this reason, the effect of condensational transfer is dampened using a factor γ as a multiplier in the condensational transfer equations. The factor can obtain values between zero and unity, and it describes how the particles will be distributed between the PL and the LN distributions. The value γ=0 produces a distribution that will be mainly in a power law form; the value γ=1 produces a log-normal distribution only. To choose a suitable value for γ for a simulation, the user should consider how well the aerosol formation event follows the approximations of the theory described here. The value 0 is suitable only when the aerosol processes follow the theory exactly. To simulate a realistic particle formation event, the value has to be increased towards unity using the following guidelines. The more the following conditions are met, the higher γ should be used: (1) particle formation or growth are multicomponent processes; (2) the particle formation rate or the condensation growth rate vary significantly with time; (3) the condensational growth rate varies significantly with the particle size; (4) the background aerosol acting as a coagulation sink does not remain in a nearly constant state during the time domain of the simulation; (5) particle sizes in the background aerosol are not significantly higher than in the PL+LN distribution; (6) the depositional losses cannot be approximated with as simple form as described here, e.g., in the case of complex geometry or turbulent flow. In real atmospheric particle formation events, γ should rarely has the value of less than 0.5, which can also be used as an initial guess should figuring the previous guidelines problematic. If the shapes of the distributions to be modeled are initially known, the value of γ can be adjusted to obtain a proper model output, e.g., in the case of inverse modeling. The factor γ can also be considered a time-dependent function or a spatially varying variable, but here we concentrate only to constant values of γ because the theory behind the value of γ is currently unknown.

The number of particles in the PL distribution to be transferred to the LN distribution due to the condensational transfer in the time step of Δt is NPL→LN=γ∫ln⁡Dcondln⁡D2dNdln⁡DpPLdln⁡Dp. Considering infinitesimally small time step (Δt→0), Dcond approaches D2 and NPL→LN approaches γgΔtD2dNdln⁡DpPL, Dp=D2. The transferred amounts for S and M are obtained in the same approach as in Eq. (), and they are negative for the PL distribution and positive for the LN distribution. Hence, the term for the condensational transfer becomes transferXcond=γg(t)D2dNdln⁡DpPL, Dp=D2⋅-1,X=NPL-s2,X=SPL-m2,X=MPL+1,X=NLN+s2,X=SLN+m2,X=MLN, where s2 and m2 are the surface area and the mass of the particle with the diameter of D2. The condensational transfer does not alter the moments of the total distribution because particles are not altered in the transfer, it only transfers the particles between the distributions; therefore, the value of γ has a minor effect only on the moments but a noticeable effect on the shape of the PL+LN distribution.

Obtaining the values for the new particle formation rate, J(t), and for the condensational growth rate, g(t), for the particle formation event, occurred in a mobile aerosol chamber, was performed through inverse modeling. A time domain, starting from 152 s before switching the UV lights on and ending at 1663 s after switching the UV lights on, including 13 Nano-SMPS (Nano Scanning Mobility Particle Sizer) measurement scans, was simulated using different values for J(t) and g(t). Following the approach of , the least squares method was used to minimize the errors of the concentrations N, S, and M at 13 time moments which represent the middles of the Nano-SMPS scans. The values for J(t) and g(t) were assumed constants within a time step of a Nano-SMPS scan, 150 s. The condensational growth rate was assumed also size-independent due to the lack of knowledge of the vapors participating in the condensation process. ODE45 solver was used in the simulations, and it provides the time steps that are sufficiently short to keep the result from altering more than 1 % compared to a previous time step but sufficiently long to keep the total computing time convenient.

Coagulation within the nucleation mode was included in the simulations, but the coagulational losses to the background mode were neglected because low number concentration of the background mode would have a minor effect only on the nucleation mode. The particles formed in this case are possibly multicomponent due to the origin of the vapors, the new particle formation rate seems to vary significantly with time, and the measured distributions are wide (GSD up to 2). Therefore, a high value for the condensational transfer factor γ is expected to produce the best results using the PL+LN model. A constant value γ=0.8 was used in the simulations because it produces the results that are the closest to the results of highly accurate sectional models, in this case. Due to a high value of γ, the coagulational transfer would had a minor effect only, and therefore it was neglected in the simulations. The depositional losses were modeled using the deposition coefficient kdep=3780nmh-1 which is obtained by fitting the simulated number concentrations with the measured ones after particle formation and growth were quenched (t>1500s).

Firstly, inverse modeling was performed using the PL+LN model. The time series of J(t) and g(t), which produced the most corresponding concentrations compared to the measured ones (Fig. ), are presented in Figs.  and . Secondly, inverse modeling was performed using both an FS model having 400 size sections between 1.6 and 100 nm (FS400) and an MC model having 100 size sections between 1.6 and 100 nm (MC100), separately. The computing times of the FS400 and the MC100 models are significantly longer than of the PL+LN model; therefore, the time series of J(t) and g(t), obtained using the PL+LN model, were used as initial guesses when inverse modeling was performed using the FS400 or the MC100 model in order to reduce the computing times. The associated computing time of the automatic inverse modeling procedure using the PL+LN model was approximately 2 orders of magnitude shorter than that of the FS400 or the MC100 model, which implies that a significant improvement in the computing time can be obtained using the PL+LN model in the case of inverse modeling. Inverse modeling was also performed using the computationally more efficient LN model.

It can be seen from Fig.  that there are only minor differences in the values of J(t) between the different models used in inverse modeling. The PL+LN model seems to need higher new particle formation rates compared to the accurate models, the FS400 and the MC100 models. Conversely, the LN model seems to need lower new particle formation rates. These denote that the PL+LN model has a tendency to underestimate the number concentrations and that the LN model has a tendency to overestimate the number concentrations, because the output number concentrations of the models are nearly equal (Fig. ). The values of g(t), seen in Fig. , have also only minor differences between the different models. The values of J(t) and g(t) before the UV lights (t<0s) seem unphysically high. That is caused because the simulation begins with no particles at t=-252s but the measurement data include some nucleation mode particles at that time, even though the background aerosol distribution was subtracted from the measured distribution. All the particles are not subtracted because the subtraction was done with a purely log-normal distribution, which is exactly not the case for measured data. Therefore, the simulation time range -252s<t<-152s was used to produce the measured nucleation mode at the time of the first Nano-SMPS scan, t=-152s.

The effect of the choice for the lowest particle diameter, D1, for the PL+LN model was also examined by performing inverse modeling with D1=1nm and D1=3nm, in addition to D1=1.6nm. Approximately 40 % higher values for J(t) were needed to produce the measured concentrations when D1=1nm was used compared to D1=3nm. This deviation occurs because smaller particles have higher losses, and with higher value for D1 the smallest particles do not exist. The situation is the same for sectional models, because the smallest particle size needs to be chosen for them too. The choice for D1 does not have significant effect on the condensational growth rate, in this case, due to size-independent g(t).

The effect of the choice for the value of the condensational transfer factor, γ, for the PL+LN model was also examined by performing inverse modeling with the values between 0 and 1, in addition to γ=0.8. With the values between 0.4 and 0.9, the times series of J(t) obtained from inverse modeling deviate within 3 % on average. However, with the values outside of that range, the deviation increased up to 36 %. The highest new particle formation rates are needed when the value of γ is low because the PL distribution dominates the number of particles, which leads to more small particles, which have high losses. Only a minor effect (the deviations within 4 %) was seen on the time series of the condensational growth rate when the value of γ was altered between 0 and 1. In conclusion, choosing the value of 0.5 for γ when a better guess of its value is lacking, as mentioned earlier, would produce reasonable results.

Figure  presents the time series of GMD and GSD of the nucleation mode, obtained from the different models used in inverse modeling and from the measured data. It can be observed that the models output these variables relatively well compared to the measured data, although they were not selected as the variables, of which errors are to be minimized, in inverse modeling. However, underestimations of GSD are seen with the LN model. The measured values of GMD and GSD before the UV lights are inaccurate due to the assumption of a log-normal background aerosol distribution mentioned before.

To examine the accuracy and computational cost of the PL+LN model, the simulations using different models were performed using equal time series of J(t) and g(t) and equal time stepping. The time series obtained using the FS400 model are considered the best estimates to produce the measurement results due to the highest number of size sections modeled. Therefore, they were used as the time series for all the models used here, PL+LN, LN, FS400, MC100, FS35, and MC10. FS35 and MC10 denote a fixed-sectional model having 35 size sections and a moving-center fixed-sectional model having 10 size sections between 1.6 and 300 nm, respectively. These section number were used because they provide approximately the same computing times as the PL+LN model. Higher upper diameter limit for the FS35 and the MC10 models were chosen due to higher numerical diffusion associated to these models, which causes higher concentrations in large particle diameters compared to more accurate models. The time series of J(t) and g(t) used here are presented as fits in Figs.  and . The fitted functions were used rather than the time series because the time series with sharp edges would produce distributions with sharp edges as well, which would be unphysical. Here, the time series have sharp edges due to a very limited number of data points (Nano-SMPS scans) within the time domain. It is obvious that because the fits are not very near the time series, the outputs of the models will not be very near the measured data. Nevertheless, the fits were used because the purpose here is to examine the accuracy and the computational cost of the PL+LN model, which is done against the highly accurate models, the FS400 and the MC100 models. This comparison is the most properly done when all the models have the same functions for J(t) and g(t). The time domain to be simulated was split into 7953 time steps for all the models. The time splitting was obtained from the ODE45 solver used with the FS400 model. The time steps had the lengths of between 0 and 0.5 s, the shortest ones being in the beginning of the time domain.

The accuracy of the PL+LN model is examined by comparing the relative errors (δX) of the moments, N, S, and M, and the variables, GMD and GSD. The reference distributions used in the comparison are the distributions produced by the FS400 model and by the MC100 model, which are considered the models that most accurately conserve the number and the mass concentrations, respectively. The distributions from the FS400 model are used as the reference distribution when calculating δX for all the moments and variables, with the exception of the mass concentration, M, for which the MC100 model is used as the reference model. Because the distributions produced by the sectional models are considered here the correct ones rather than the measured distributions, the modeled distributions are used as the reference distributions. In this manner, the differences of the model outputs are caused by the models itself, e.g., due to numerical diffusion or some simplifications used in the model, not by how accurately they correspond with the measured data. The accuracies of the LN, FS35, and MC10 models are also examined.

All the other input parameters were the same as those used with the inverse modeling. The simulations used to examine the model accuracies provide also the possibility of comparing the computational costs of different models, because all the simulations were run using the same computer (Intel Core i5-3470 processor at 3.2 GHz) and had equal time stepping and, therefore, equal number of computations of the general dynamic equation per a moment or a size section. Additionally, the sensitivity of the value of γ was examined using also the values of 0.1, 0.5, and 0.9, in addition to the value of 0.8.