Comparison of discrete, discrete-sectional, modal and moment models for aerosol dynamics simulations

Figures & data

Table 1. A summary of different aerosol dynamic models.

Model name	Overall description	Remarks	References
Weighted residuals model	The PSD is assumed in terms of a complete set of functions	Piecewise polynomial is set as the basis of PSD	(Gelbard and Seinfeld 1978; Sandu 2006)
Sectional model	PSD is divided into different sections	Computationally less expensive than discrete model; Less accurate than discrete model; Numerical diffusion exists	(Gelbard et al. 1980; Geng, Park, and Sajo 2013; Harrington and Kreidenweis 1998; Kommu, Khomami, and Biswas 2004; Pirjola et al. 1999; Seigneur et al. 1986; Talukdar and Swihart 2004; Tsantilis, Kammler, and Pratsinis 2002; Zhang et al. 1999)
Discrete model	PSD is divided into discrete sizes and the increment volume of the two neighboring discrete sizes is the volume of the monomer	Most accurate; Computationally expensive	(Frenklach and Harris 1987; Friedlander 2000; Gelbard and Seinfeld 1979; Landgrebe and Pratsinis 1989; Lehtinen and Kulmala 2003; Smith, Wells, and Kraft 2018)
Moment model	Lognormal method of moments; Quadrature method of moments	Computationally fast; Consider the spread in the distribution	(Frenklach and Harris 1987; Harrington and Kreidenweis 1998; Lin, Sethi, and Biswas 1992; McGraw 1997; Pirjola et al. 1999; Pratsinis 1988; Seigneur et al. 1986; Suh, Zachariah, and Girshick 2001; Talukdar and Swihart 2004; Williams 1986; Yamamoto 2014; Zhang et al. 1999, 2018)
Discrete-sectional model	Combination of discrete model and sectional model; The small particle size region is represented by discrete model, while the rest of the region is represented by sectional model	Accuracy and computational efficiency are better than sectional model or discrete model; Numerical diffusion exists	(Biswas et al. 1997; Chadha et al. 2017; Gao et al. 2017; Kommu et al. 2004; Landgrebe and Pratsinis 1990; Moniruzzaman and Park 2006; Wu and Biswas 1998; Wu and Flagan 1988)
Modal model	Assume monodisperse for each mode; 2n + 1 ordinary differential equations (ODEs) need to be solved	Computationally fast; Total surface area of the particles changing with time could be obtained	(Ackermann et al. 1998; Brown, Kauppinen, Jokiniemi, Rubin, and Biswas 2006; Jeong and Choi 2003; Kommu et al. 2004; Kruis et al. 1993; Schild et al. 1999; Suriyawong, Chen, and Biswas 2010; Xiong, Akhtar, and Pratsinis 1993; Xiong and Pratsinis 1993)
Nodal model	Solve the GDE between n nodes (with geometric scaling)	Does not assume PSD, and gives the spread as well; Assumes fixed nodes	(Mukherjee, Prakash, and Zachariah 2006; Prakash et al. 2003)
Size, and composition resolved models	The general dynamic equation is discretized for both particle size and chemical composition	Particle compositions are represented by the combinations of mass fractions	(Zhu, Sartelet, and Seigneur 2015)

Table 2. Modeling plan.

Modeling plan
I. Model development	Mathematical description of discrete, discrete-sectional, moment and modal models Mathematical expression of aerosol dynamics considered in our simulation Model limitation
II. Software working principle	Flowchart and guide of AAQRL-ADM
III. Model verification	Simulate PSD by considering the effects of Pure coagulation Nucleation and coagulation Condensation Nucleation, condensation, and coagulation Comparison of numerical efficiency of different models
IV. Application	Simulate PSD in a furnace aerosol reactor coupling with flow temperature and velocity

Figure 1. The schematic of the modal model (a) general description of modal model (b) assigning a volume to its adjacent nodes.

Figure 2. Working principle of AAQRL-ADM software.

Figure 3. Graphical User Interface (GUI) of AAQRL-ADM (https://github.com/AAQRL/aerosol_dynamic_models; DOI: 10.5281/zenodo.3571284) software. (a) Main window; (b) Discrete model window.

Figure 4. The particle size distribution (PSD) after the final time for (a) monodisperse case and (b) lognormal case.

Table 3. Comparison of ensemble parameters by different models.

(a) Monodisperse
Parameters	Initial (monodisperse)	Discrete	Discrete-sectional	Modal	Moment
Ntot (#/cm^3)	1 × 10^10	2.04 × 10^9	2.04 × 10^9	2.04 × 10^9	2.04 × 10^9
dpg (nm)	1	1.49	1.49	1.46	1.52
σg	1	1.34	1.34	1.35	1.31
(b) Lognormal
Parameters	Initial (log-normal)	Discrete	Discrete-sectional	Modal	Moment
Ntot (#/cm^3)	1 × 10^10	1.13 × 10^9	1.14 × 10^9	1.13 × 10^9	1.23 × 10^9
dpg (nm)	3	5.50	5.51	5.23	5.62
σg	1.2	1.43	1.43	1.47	1.33

Figure 5. Comparison of discrete, discrete-sectional, moment and modal modes when considering both coagulation and nucleation effects (a) simulated PSD at t = 2 × 10⁻³ s, (b) simulated PSD at t = 0.1 s, (c) total number concentration (N_tot) vs. time, and (d) mean diameter (d_p,mean) vs. time.

Figure 6. PSD vs. particle size for condensation case (a) depleting vapor concentration (b) constant vapor concentration.

Table 4. Comparison of ensemble parameters for particle condensation.

(a) Depleting vapor concentration.
Parameters	Discrete-sectional	Modal	Moment
Ntot (#/cm^3)	1.00 × 10^5	1.00 × 10^5	1.00 × 10^5
dpg (nm)	35.9	34.4	35.0
σg	1.16	1.37	1.16
(b) Constant vapor concentration
Parameters	Discrete-sectional	Modal	Moment
Ntot (#/cm^3)	1.00 × 10^5	1.00 × 10^5	1.00 × 10^5
dpg (nm)	50.9	50.0	49.0
σg	1.14	1.09	1.13

Figure 7. Simulation results of simultaneous nucleation, condensation and coagulation case (a) N_tot vs. t, (b) saturation ratio (S) vs. t, and (c) mean diameter (d_p,mean) vs. t.

Figure 8. Computational time and V_tot vs. discrete number for pure coagulation case with initial PSD of lognormal distribution. (open square: V_tot; closed square: CPU time cost; dash line: fitting curve for CPU time cost).

Table 5. Comparison of computational time cost of different models.

Models	Pure coagulation case with monodisperse initial PSD
	Model parameters	Computational time cost (s)	Computer property
Discrete	Discrete number: 100	5.30 × 10^–1	1 CPU core
Discrete-sectional	Discrete number: 100, Sectional number: 119	1.17 × 10^1	1 CPU core
Modal	Model number: 8	2.60 × 10^1	1 CPU core
Moment	–	5.2 × 10^–2	1 CPU core
Models	Pure coagulation case with lognormal initial PSD
	Model parameters	Computational time cost (s)	Computer property
Discrete	Discrete number: 20,000	1.29 × 10^5	16 CPU cores
Discrete-sectional	Discrete number: 300, Sectional number:119	1.05 × 10^2	1 CPU core
Modal	Model number:8	1.60 × 10^1	1 CPU core
Moment	–	6.6 × 10^–2
Models	Nucleation and coagulation
	Model parameters	Computational time cost (s)	Computer property
Discrete	Discrete number: 8000	1.29 × 10^4	8 CPU cores
Discrete-sectional	Discrete number: 200, Sectional number: 263	1.20 × 10^3	1 CPU core
Modal	Model number: 12	4.65 × 10^2	1 CPU core
Moment	–	5.13 × 10^0	1 CPU core

Figure 9. Schematic of (a) the furnace tube and (b) the computational domain (L = 83.5 cm; R = 0.95 cm).

Figure 10. Gas temperature profile along z-axis in the furnace tube.

Figure 11. Comparison of simulated PSD by discrete-sectional, modal, and moment models at different positions.

Table 6. Comparison of ensemble parameters by discrete-sectional, modal, and moment models for different positions.

(a) z = 20 cm
Parameters	Discrete-sectional	Modal	Moment
Ntot (#/cm^3)	3.34 × 10^8	2.86 × 10^8	5.93 × 10^8
dpg (nm)	44.1	49.6	46.2
σg	1.40	1.55	1.31
(b) z = 40 cm
Parameters	Discrete-sectional	Modal	Moment
Ntot (#/cm^3)	1.64 × 10^8	1.14 × 10^8	2.66 × 10^8
dpg (nm)	54.3	67.2	60.6
σg	1.39	1.56	1.31
(c) z = 60 cm
Parameters	Discrete-sectional	Modal	Moment
Ntot (#/cm^3)	1.12 × 10^8	6.69 × 10^7	1.70 × 10^8
dpg (nm)	61.4	80.2	70.5
σg	1.39	1.56	1.31
(d) z = 83.5 cm
Parameters	Discrete-sectional	Modal	Moment
Ntot (#/cm^3)	9.90 × 10^7	4.42 × 10^7	1.20 × 10^8
dpg (nm)	68.0	92.1	79.2
σg	1.38	1.56	1.30

Table B1. Dimensionless coagulation and condensation coefficients in free molecular and continuum regimes for moment model.

Parameter	Free molecular regime	Continuum regime
Coagulation coefficient for 0th moment	${\xi }_{FM}=r_g^{\prime 1/2}{b}_0\left[\exp \left(\frac{25}{8}{ln}^2{\sigma }_g\right)+2\mathrm{e}\mathrm{xp}\left(\frac{5}{8}{ln}^2{\sigma }_g\right)+\exp \left(\frac{1}{8}{ln}^2{\sigma }_g\right)\right]$ rg’ is dimensionless particle geometric mean radius, rg’ = rg/r1, where r1 is monomer radius; b0 = 0.633 + 0.092σg^2 − 0.022σg^3 (Lee, Chen, and Gieseke 1984).	${\xi }_C=K\left[1+\exp \left({ln}^2{\sigma }_g\right)+1.257\left({Kn}_1/r_g^{\prime }\right)\times \exp \left(\frac{1}{2}{ln}^2{\sigma }_g\right)(1+\exp \left(2{ln}^2{\sigma }_g\right))\right]$ K = 2kBTn1,satτ/3μ, where Kn1 is monomer Knudsen number and Kn1 = λ/r1.
Condensation coefficient for 1st moment	${\eta }_{FM}=v_g^{\prime 2/3}\mathrm{e}\mathrm{xp}\left(2{ln}^2{\sigma }_g\right)$ vg’ is dimensionless particle geometric mean volume, vg’ = vg/v1.	${\eta }_C=\frac{4{Kn}_1}{3}{v}_g^{\prime 1/3}\mathrm{e}\mathrm{xp}\left(\frac{1}{2}{ln}^2{\sigma }_g\right)$
Condensation coefficient for 2nd moment	${ϵ}_{FM}=v_g^{\prime 2/3}\mathrm{e}\mathrm{xp}\left(8{ln}^2{\sigma }_g\right)$	${ϵ}_C=\frac{4{Kn}_1}{3}{v}_g^{\prime 1/3}\mathrm{e}\mathrm{xp}\left(\frac{7}{2}{ln}^2{\sigma }_g\right)$
Coagulation coefficient for 2nd moment	${\xi }_{FM}=r_g^{\prime 1/2}{b}_2\mathrm{e}\mathrm{xp}\left(\frac{3}{2}{ln}^2{\sigma }_g\right)$ $\left[\exp \left(\frac{25}{8}{ln}^2{\sigma }_g\right)+2\mathrm{e}\mathrm{xp}\left(\frac{5}{8}{ln}^2{\sigma }_g\right)+\hspace{0.17em}\text{exp}\hspace{0.17em}\left(\frac{1}{8}{ln}^2{\sigma }_g\right)\right]$ b0 = 0.39 + 0.5σg − 0.214σg^2 + 0.029σg^3 (Lee, Chen, and Gieseke 1984)	${\xi }_C=K\left[1+\exp \left({ln}^2{\sigma }_g\right)+1.257\left({Kn}_1/r_g^{\prime }\right)\times \exp \left(-\frac{1}{2}{ln}^2{\sigma }_g\right)(1+\exp \left(-2{ln}^2{\sigma }_g\right))\right]$

Figure D1. Self-preserving particle size distribution calculated by moment, modal, discrete and discrete-sectional models, compared with analytical solution (Vemury and Pratsinis 1995).

Gelbard, F., and J. H. Seinfeld. 1978. Numerical solution of the dynamic equation for particulate systems. J. Comput. Phys. 28 (3):357–375. doi:10.1016/0021-9991(78)90058-X.

 Web of Science ®Google Scholar

Sandu, A. 2006. Piecewise polynomial solutions of aerosol dynamic equation. Aerosol Sci. Technol. 40 (4):261–273. doi:10.1080/02786820500543274.

 Web of Science ®Google Scholar

Gelbard, F., Y. Tambour, and J. H. Seinfeld. 1980. Sectional representations for simulating aerosol dynamics. J. Colloid Interface Sci. 76 (2):541–556. doi:10.1016/0021-9797(80)90394-X.

 Web of Science ®Google Scholar

Geng, J., H. Park, and E. Sajo. 2013. Simulation of aerosol coagulation and deposition under multiple flow regimes with arbitrary computational precision. Aerosol Sci. Technol. 47 (5):530–42. doi:10.1080/02786826.2013.770126.

 Web of Science ®Google Scholar

Harrington, D. Y., and S. M. Kreidenweis. 1998. Simulations of sulfate aerosol dynamics part ii: Model intercomparison. Atmos. Environ. 32 (10):1701–1709. doi:10.1016/S1352-2310(97)00453-6.

 Web of Science ®Google Scholar

Kommu, S., B. Khomami, and P. Biswas. 2004. Simulation of aerosol dynamics and transport in chemically reacting particulate matter laden flows. Part II: Application to CVD reactors. Chem. Eng. Sci. 59 (2):359–371. doi:10.1016/j.ces.2003.05.010.

 Web of Science ®Google Scholar

Pirjola, L., M. Kulmala, M. Wilck, A. Bischoff, F. Stratmann, and E. Otto. 1999. Formation of sulphuric acid aerosols and cloud condensation nuclei: An expression for significant nucleation and model comparison. J. Aerosol Sci. 30 (8):1079–1094. doi:10.1016/S0021-8502(98)00776-9.

 Web of Science ®Google Scholar

Seigneur, C., A. B. Hudischewskyj, J. H. Seinfeld, K. T. Whitby, E. R. Whitby, J. R. Brock, and H. M. Barnes. 1986. Simulation of aerosol dynamics: A comparative review of mathematical models. Aerosol Sci. Technol. 5 (2):205–222. doi:10.1080/02786828608959088.

 Web of Science ®Google Scholar

Talukdar, S. S., and M. T. Swihart. 2004. Aerosol dynamics modeling of silicon nanoparticle formation during silane pyrolysis: A comparison of three solution methods. J. Aerosol Sci. 35 (7):889–908. doi:10.1016/j.jaerosci.2004.02.004.

 Web of Science ®Google Scholar

Tsantilis, S., H. Kammler, and S. Pratsinis. 2002. Population balance modeling of flame synthesis of titania nanoparticles. Chem. Eng. Sci. 57 (12):2139–2156. doi:10.1016/S0009-2509(02)00107-0.

 Web of Science ®Google Scholar

Zhang, Y., C. Seigneur, J. H. Seinfeld, M. Z. Jacobson, and F. S. Binkowski. 1999. Simulation of aerosol dynamics: A comparative review of algorithms used in air quality models. Aerosol Sci. Technol. 31 (6):487–514. doi:10.1080/027868299304039.

 Web of Science ®Google Scholar

Frenklach, M., and S. J. Harris. 1987. Aerosol dynamics modeling using the method of moments. J. Colloid Interface Sci. 118 (1):252–261. doi:10.1016/0021-9797(87)90454-1.

 Web of Science ®Google Scholar

Friedlander, S. K. 2000. Smoke, dust, and haze: Fundamentals of aerosol dynamics. Topics in chemical engineering. New York: Oxford University Press.

 Google Scholar

Gelbard, F., and J. H. Seinfeld. 1979. The general dynamic equation for aerosols. Theory and application to aerosol formation and growth. J. Colloid Interface Sci. 68 (2):363–382. doi:10.1016/0021-9797(79)90289-3.

 Web of Science ®Google Scholar

Landgrebe, J. D., and S. E. Pratsinis. 1989. Gas-phase manufacture of particulates: Interplay of chemical reaction and aerosol coagulation in the free-molecular regime. Ind. Eng. Chem. Res. 28 (10):1474–1481. doi:10.1021/ie00094a007.

 Web of Science ®Google Scholar

Lehtinen, K., and M. Kulmala. 2003. A model for particle formation and growth in the atmosphere with molecular resolution in size. Atmos. Chem. Phys. 3 (1):251–257. doi:10.5194/acp-3-251-2003.

 Web of Science ®Google Scholar

Smith, A. J., C. G. Wells, and M. Kraft. 2018. A new iterative scheme for solving the discrete Smoluchowski equation. J. Comput. Phys. 352:373–387. doi:10.1016/j.jcp.2017.09.045.

 Web of Science ®Google Scholar

Lin, W. Y., V. Sethi, and P. Biswas. 1992. Multicomponent aerosol dynamics of the Pb-O2 system in a bench scale flame incinerator. Aerosol Sci. Technol. 17 (2):119–133. doi:10.1080/02786829208959565.

 Web of Science ®Google Scholar

McGraw, R. 1997. Description of aerosol dynamics by the quadrature method of moments. Aerosol Sci. Technol. 27 (2):255–265. doi:10.1080/02786829708965471.

 Web of Science ®Google Scholar

Pratsinis, S. E. 1988. Simultaneous nucleation, condensation, and coagulation in aerosol reactors. J. Colloid Interface Sci. 124 (2):416–427. doi:10.1016/0021-9797(88)90180-4.

 Web of Science ®Google Scholar

Suh, S.-M., M. Zachariah, and S. Girshick. 2001. Modeling particle formation during low-pressure silane oxidation: Detailed chemical kinetics and aerosol dynamics. J. Vac. Sci. Technol. A 19 (3):940–951. doi:10.1116/1.1355757.

 Web of Science ®Google Scholar

Williams, M. 1986. Some topics in nuclear aerosol dynamics. Prog. Nucl. Energy 17 (1):1–52. doi:10.1016/0149-1970(86)90041-7.

 Web of Science ®Google Scholar

Yamamoto, M. 2014. A moment method of the log-normal size distribution with the critical size limit in the free-molecular regime. Aerosol Sci. Technol. 48 (7):725–737. doi:10.1080/02786826.2014.922161.

 Web of Science ®Google Scholar

Yu, S., R. Mathur, J. Pleim, D. Wong, R. Gilliam, K. Alapaty, C. Zhao, and X. Liu. 2014. Aerosol indirect effect on the grid-scale clouds in the two-way coupled WRF-CMAQ: Model description, development, evaluation and regional analysis. Atmos. Chem. Phys. 14 (20):11247–11285. doi:10.5194/acpd-13-25649-2013.

 Web of Science ®Google Scholar

Yu, S., P. S. Kasibhatla, D. L. Wright, S. E. Schwartz, R. McGraw, and A. Deng. 2003. Moment‐based simulation of microphysical properties of sulfate aerosols in the eastern United States: Model description, evaluation, and regional analysis. J. Geophys. Res. 108 (D12):4353. doi:10.1029/2002JD002890.

 Web of Science ®Google Scholar

Biswas, P., C. Y. Wu, M. R. Zachariah, and B. McMillin. 1997. Characterization of iron oxide-silica nanocomposites in flames: Part II. Comparison of discrete-sectional model predictions to experimental data. J. Mater. Res. 12 (3):714–723. doi:10.1557/JMR.1997.0106.

 Web of Science ®Google Scholar

Chadha, T. S., M. Yang, K. Haddad, V. B. Shah, S. Li, and P. Biswas. 2017. Model based prediction of nanostructured thin film morphology in an aerosol chemical vapor deposition process. Chem. Eng. J. 310:102–113. doi:10.1016/j.cej.2016.10.105.

 Web of Science ®Google Scholar

Gao, Q., S. Li, M. Yang, P. Biswas, and Q. Yao. 2017. Measurement and numerical simulation of ultrafine particle size distribution in the early stage of high-sodium lignite combustion. Proc. Combust. Inst. 36 (2):2083–2090. doi:10.1016/j.proci.2016.07.085.

 Web of Science ®Google Scholar

Landgrebe, J. D., and S. E. Pratsinis. 1990. A discrete-sectional model for particulate production by gas-phase chemical reaction and aerosol coagulation in the free-molecular regime. J. Colloid Interface Sci. 139 (1):63–86. doi:10.1016/0021-9797(90)90445-T.

 Web of Science ®Google Scholar

Moniruzzaman, C. G., and K. Y. Park. 2006. A discrete-sectional model for particle growth in aerosol reactor: Application to titania particles. Korean J. Chem. Eng. 23 (1):159–166. doi:10.1007/BF02705709.

 Web of Science ®Google Scholar

Wu, C.-Y., and P. Biswas. 1998. Study of numerical diffusion in a discrete-sectional model and its application to aerosol dynamics simulation. Aerosol Sci. Technol. 29 (5):359–378. doi:10.1080/02786829808965576.

 Web of Science ®Google Scholar

Wu, J. J., and R. C. Flagan. 1988. A discrete-sectional solution to the aerosol dynamic equation. J. Colloid Interface Sci. 123:339–352. doi:10.1016/0021-9797(88)90255-X.

 Web of Science ®Google Scholar

Ackermann, I. J., H. Hass, M. Memmesheimer, A. Ebel, F. S. Binkowski, and U. Shankar. 1998. Modal aerosol dynamics model for Europe: Development and first applications. Atmos. Environ. 32 (17):2981–2999. doi:10.1016/S1352-2310(98)00006-5.

 Web of Science ®Google Scholar

Brown, D. P., E. I. Kauppinen, J. K. Jokiniemi, S. G. Rubin, and P. Biswas. 2006. A method of moments based CFD model for polydisperse aerosol flows with strong interphase mass and heat transfer. Comput. Fluids 35 (7):762–780. doi:10.1016/j.compfluid.2006.01.012.

 Web of Science ®Google Scholar

Jeong, J. I., and M. Choi. 2003. A simple bimodal model for the evolution of non-spherical particles undergoing nucleation, coagulation and coalescence. J. Aerosol Sci. 34:965–976. doi:10.1016/S0021-8502(03)00067-3.

 Web of Science ®Google Scholar

Kruis, F. E., K. A. Kusters, S. E. Pratsinis, and B. Scarlett. 1993. A simple model for the evolution of the characteristics of aggregate particles undergoing coagulation and sintering. Aerosol Sci. Technol. 19 (4):514–526. doi:10.1080/02786829308959656.

 Web of Science ®Google Scholar

Schild, A., A. Gutsch, H. Mühlenweg, and S. Pratsinis. 1999. Simulation of nanoparticle production in premixed aerosol flow reactors by interfacing fluid mechanics and particle dynamics. J. Nanopart. Res. 1 (2):305–315.

 Web of Science ®Google Scholar

Suriyawong, A., X. Chen, and P. Biswas. 2010. Nano-structured sorbent injection strategies for heavy metal capture in combustion exhausts. Aerosol Sci. Technol. 44 (8):676–691. doi:10.1080/02786826.2010.485589.

 Web of Science ®Google Scholar

Xiong, Y., M. K. Akhtar, and S. E. Pratsinis. 1993. Formation of agglomerate particles by coagulation and sintering—Part II. The evolution of the morphology of aerosol-made titania, silica and silica-doped titania powders. J. Aerosol Sci. 24 (3):301–313. doi:10.1016/0021-8502(93)90004-S.

 Web of Science ®Google Scholar

Xiong, Y., and S. E. Pratsinis. 1993. Formation of agglomerate particles by coagulation and sintering—Part I. A two-dimensional solution of the population balance equation. J. Aerosol Sci. 24 (3):283–300. doi:10.1016/0021-8502(93)90003-R.

 Web of Science ®Google Scholar

Mukherjee, D., A. Prakash, and M. Zachariah. 2006. Implementation of a discrete nodal model to probe the effect of size-dependent surface tension on nanoparticle formation and growth. J. Aerosol Sci. 37 (10):1388–1399. doi:10.1016/j.jaerosci.2006.01.008.

 Web of Science ®Google Scholar

Prakash, A., A. Bapat, and M. Zachariah. 2003. A simple numerical algorithm and software for solution of nucleation, surface growth, and coagulation problems. Aerosol Sci. Technol. 37 (11):892–898. doi:10.1080/02786820300933.

 Web of Science ®Google Scholar

Zhu, S., K. N. Sartelet, and C. Seigneur. 2015. A size-composition resolved aerosol model for simulating the dynamics of externally mixed particles: SCRAM (v 1.0). Geosci. Model Dev. 8 (6):1595–1612. doi:10.5194/gmd-8-1595-2015.

 Web of Science ®Google Scholar

Lee, K., J. Chen, and J. Gieseke. 1984. Log-normally preserving size distribution for Brownian coagulation in the free-molecule regime. Aerosol Sci. Technol. 3 (1):53–62. doi:10.1080/02786828408958993.

 Web of Science ®Google Scholar

Vemury, S., and S. E. Pratsinis. 1995. Self-preserving size distributions of agglomerates. J. Aerosol Sci. 26 (2):175–85. doi:10.1016/0021-8502(94)00103-6.

 Web of Science ®Google Scholar