By Konstantin Volkov
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53–62, 1984.  S. E. Pratsinis, Simultaneous nucleation, condensation, and coagulation in aerosol reactors, J. , vol. 124, no. 2, pp. 416–427, 1988.  M. M. Williams, On the modified gamma distribution for representing the size spectra of coagulating aerosol particles, J. , vol. 103, no. 2, pp. 516–527, 1985.  R. McGraw, Description of aerosol dynamics by the quadrature method of moments, Aerosol Sci. , vol. 27, no. 2, pp. 255–265, 1997.  D. L. Marchisio and R. O. Fox, Solution of population balance equations using the direct quadrature method of moments, J.
Contributing factors include the increase in nanomaterial generation during product/device manufacturing, handling and use , consumptions of conventional and new tobacco products [2– 4], as well as nanoparticle pulmonary drug development [7, 8]. Of great concern are the toxic health risks and therapeutic benefits induced by the inhalation. This is mainly because the small sizes facilitate ultrafine PM transport into deeper lung airways, translocation to the bloodstream crossing body membrane barriers, thus affecting organs and tissues in both pulmonary routes and systemic regions.
The operator splitting errors for the case of nucleation and condensation show that splitting errors for different moments (except for �0) are comparable with each other. Splitting between nucleation and condensation has marginal impact on accurately predicting the moments. This is in great contrast to the prediction of �0, where splitting between nucleation and coagulation has significant impact. To construct a good operator splitting scheme for the general case of nucleation, coagulation, and condensation the first priority is to consider the splitting error for �0.