Local fixed pivot quadratue method of moment for bubble population balance equation including coalescence and breakage

Abstract

Population balance equation as an essential tool to describe micro-behaviors and resulting bubble size distribution has received considerable attention in scientific and engineering fields. Numerical solution is the only choice in most cases due to its complexity. However, it is almost impossible for the existing numerical methods to predict both bubble size distribution and its moments exactly. In this work, a new numerical method basing on the idea of short time Fourier transformation, namely local fixed pivot quadrature method of moment, is proposed for bubble coalescence and breakage. A continuous summation of Dirac Delta function as trial functions in the local domain and monomials as the weighted functions to conserve the local moments were adopted. The moments and the bubble size distribution were constructed based on the moments in the local domain. Numerical tests including pure coalescence, pure breakage and coalescence and breakage combined processes showed that both the moments and bubble size distribution were predicted accurately. A special algorithm was used to solve the vandermonde linear system, with which the influence of the ill-conditioned feature of coefficient matrix on the numerical accuracy can be avoided. In theory any number of moments can be tracked with the new method. Moreover, with it one can solely track the bubble size distribution or the moments depending on the concrete application.

Advances in numerical methods for the solution of population balance equations for disperse phase systems

“…The relative errors for the moments are comparable with those with QMOM, and the numerically predicted particle size distribution agrees well with the analytical counterpart [18] . Figure 8 shows the relative errors for the first six moments for combined aggregation and breakage processes with a constant aggregation kernel, linear breakage kernel, uniform daughter distribution and exponential initial particle size distribution using LFPQMOM and volume-based QMOM.…”

“…Therefore, it is difficult to track a large number of moments using SQMOM. To resolve these problems, Su et al [18] proposed a new method, namely local fixed pivots quadrature method of moments, for aggregation and breakage processes. This method has been shown to offer accurate predictions for both the moments and the particle size distribution.…”

“…(34) can be solved using a special algorithm [47] for the Vandermonde linear system to obtain the weights. Su et al [18] adopted Laguerre Gaussian abscissas as pivots for the last sub-domain and Chebyshev Gaussian abscissas as pivots for other sub-domains in order to achieve high numerical accuracy.…”

“…These methods are discussed in detail here, and are compared in terms of numerical accuracy, computational expense, and ease of implementation. Special attention is given to moment methods, such as quadrature method of moments (QMOM) [11] and direct quadrature method of moments (DQMOM) [12] along with five recently developed methods, namely the modified quadrature method of moments (M-QMOM) [13] , adaptive direct quadrature method of moments (ADQMOM) [14] , fixed pivot quadrature method of moments (FPQMOM) [15] , moving particle ensemble method (MPEM) [17] and local fixed pivot quadrature method of moments (LFPQMOM) [18] .…”