A new analytical solution for solving the population balance equation in the continuum-slip regime

Generalized TEMOM Scheme for Solving the Population Balance Equation

“…Figure 1 indicates that as the initial geometric standard deviation increases, the accuracy of all investigated moment models decreases. This further verifies that the initial geometric standard deviation is a critical factor affecting the accuracy of methods of moments (Otto et al 1999;Yu et al 2015b). In addition, for initial geometric standard deviations of 1.200 and 1.350, the Error% for the QMOM involving an integer moment sequence decreased as the number of quadrature points increased, which GENERALIZED TEMOM is consistent with that observed for the generalized TEMOM scheme; however, for initial geometric standard deviations of 1.600 and 1.800, all QMOMs with an integer moment sequence yielded the same results, suggesting that the number of quadrature nodes has no effect on the accuracy for the zeroth moment when the initial geometric standard deviation is relatively high.…”

“…When the initial geometric standard deviation was set to 1.800, the generalized TEMOM with f D 4 exhibited superior accuracy to all the other methods; however, when the geometric standard deviation was increased, the Error% increased for all the methods of moments. This also indicates that the condition of the initial size distribution influences the accuracy of all the methods of moments (Yu et al 2015b). Except for the generalized TEMOM scheme with f D 3, the fractal moments, including m 1/3 and m 2/3 , must be generated by further executing the closure models for moments, as described in Section 4.1.…”

“…However, a recent study (Yu et al 2015b) reported that this method has shortcomings in both the scope of application and the accuracy for the fractional moment at the initial stage. The geometric standard deviations in the free molecular regime and continuum regime were reported to be limited to 1.601 and 1.699, respectively (Yu et al 2015a), which substantially weaken the competitiveness of the TEMOM.…”

Asymptotic Solution of Moment Approximation of the Particle Population Balance Equation for Brownian Agglomeration

“…[18] There are three asymptotic values for dimensionless particle moment, but only one value is effective according to the mathematical and physical constraints, e.g., for aerosol particles, the zeroth moment must decrease while the second moment must increase with the time when its dynamical process is solely dominated by Brownian motion (Xie and He 2013;Xie 2014;Yu et al 2015), then…”

A bimodal moment method model for submicron fractal-like agglomerates undergoing Brownian coagulation

“…Compared to QMOM, TEMOM exhibits a significant advantage in efficiency because it only requires solving three ordinary differential equations (ODEs) for the first three moments, whereas in the QMOM, six ODEs are required to obtain the same moments (Xie, Yu, & Wang, 2012;Yu & Lin, 2009a, 2009bYu et al, 2011). Since the TEMOM has been firstly proposed in 2008 (Yu et al, 2008), it has been further developed for solving the population balance equation in terms of both numerical and analytical solutions (Xie, 2015;Yu & Lin, 2009b;Yu, Zhang, Jin, Lin, & Seipenbusch, 2015;Yu, Lin, Cao, & Seipenbusch, 2015). But the further study for its capacity to resolve the bimodal or multimodal aerosol problems remains an unresolved issue (Liu & Gu, 2013;Yu & Seipenbusch, 2010).…”