(724a) Solving the Population Balance Equation: A Novel Quadrature Method

Moment models are based on specific, discrete or continuous, parametric representations for the probability density function (pdf) and involve/entail the recovery of the underlying pdf when a finite set of its moments is given; this is the essence of the classical (truncated) moment problem. Several moment-based approaches for pdf-reconstructions have been proposed in the mathematical literature [1-4], and in particular various adaptations have been formulated as specialized solution methods for the population balance equations (PBEs). Among the moment closure methods for PBEs the quadrature-based moment methods (QBMM), which are based on the Gauss quadrature assumption, have gained ground [5-6]. In particular, the quadrature method of moments (QMOM), introduced by McGraw (1997) [7] is based on the basic assumption that the underlying pdf is represented by a weighted sum of delta functions, in which the quadrature nodes and weights are determined by inverting a finite set of its moments.

However, numerical instabilities may occur when nodes lie outside the support interval, or their values are close, or when weights have negative values, leading to solutions that are not physically realizable [8-10]. None of the current methods are capable of dealing with these problems. In this work we present a new method based on Gauss quadrature assumption which refines the standard QMOM formulations and guarantees that the nodes are always lie in the support and the reconstruction of the distribution preserves positivity. We discuss cases where the standard QMOM fails and demonstrate the ­stability of the new method.

References

[1] Mead, L. R., & Papanicolaou, N. (1984). Maximum entropy in the problem of moments. Journal of Mathematical Physics, 25(8), 2404-2417.

[2] Athanassoulis, G. A., and P. N. Gavriliadis. "The truncated Hausdorff moment problem solved by using kernel density functions." Probabilistic Engineering Mechanics 17, no. 3 (2002): 273-291.

[3] John, V., I. Angelov, A. A. Öncül, and D. Thévenin. "Techniques for the reconstruction of a distribution from a finite number of its moments." Chemical Engineering Science 62, no. 11 (2007): 2890-2904.

[4] Gavriliadis, P. N., & Athanassoulis, G. A. (2012). The truncated Stieltjes moment problem solved by using kernel density functions. Journal of Computational and Applied Mathematics, 236(17), 4193-4213.

[5] Yuan, C., Laurent, F., & Fox, R. O. (2012). An extended quadrature method of moments for population balance equations. Journal of Aerosol Science, 51, 1-23.

[6] Madadi-Kandjani, E., & Passalacqua, A. (2015). An extended quadrature-based moment method with log-normal kernel density functions. Chemical Engineering Science, 131, 323-339.

[7] McGraw, R. (1997). Description of aerosol dynamics by the quadrature method of moments. Aerosol Science and Technology, 27(2), 255-265.

[8] Li, D., Li, Z., & Gao, Z. (2019). Quadrature-based moment methods for the population balance equation: An algorithm review. Chinese Journal of Chemical Engineering, 27(3), 483-500.

[9] Vikas, V., Hauck, C. D., Wang, Z. J., & Fox, R. O. (2013). Radiation transport modeling using extended quadrature method of moments. Journal of Computational Physics, 246, 221-241.

[10] Su, J., Gu, Z., Li, Y., Feng, S., & Xu, X. Y. (2008). An adaptive direct quadrature method of moment for population balance equations. AIChE journal, 54(11), 2872-2887.