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

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.