Abstract: In bubbly flow systems, bubbles may break or coalesce due to bubble?bubble or bubble?fluid interactions in presence of turbulence. In general, the population balance equations (PBEs) need to be solved to model the bubble size distribution (BSD). Two methods for solving the PBEs, i.e., the classes method and the quadrature method of moments (QMOM), were compared in this work. In the classes method, BSD was represented through a finite number of bubble classes, and coalescence rates and breakup rates were transformed into birth and death rates for each class. QMOM solved the equations of lower-order moments of BSD, instead of tracking the representative bubble size. A three-dimensional simulation of two-phase flow (air?water) was performed for a cylindrical gas?liquid bubble column reactor operated at high superficial gas velocities, and the computational fluid dynamics (CFD) were coupled with PBEs. An Euler-Euler two fluid model approach with an RNG k?? model of turbulence model was used. The predictions of the classes method (20 bins), QMOM with four moments and QMOM with six moments were compared with the experimental data in literature. The results showed that both the classes method and QMOM can reasonably predict the time-averaged gas volume fraction, liquid velocity profiles, mean diameter and bubble size distribution. Nevertheless, compared with the classes method, QMOM can save 2~3 times of computational resources. And for the QMOM four moments suffices were used to accurately describe the evolution of the gas phase. In fact, the results found by using the QMOM with four moments and the QMOM with six moments were very similar. In addition, the continuous bubble size distribution was reconstructed by using the low-order moments of QMOM. The predicted BSD was similar to that of the classes method, demonstrating that QMOM was a more efficient method than classes method when the PBM was coupled with CFD simulations for bubbly flow systems.

Key words: computational fluid dynamics, population balance modeling, bubble column, bubble size distribution, quadrature-based moment method

摘要： 采用CFD-PBM耦合方法，对高表观气速下的气?液鼓泡塔进行模拟，得到气含率、轴向液速及气泡尺寸。系统性对比求解群平衡方程(PBE)的不同方法：离散法(20 bins)和QMOM(包括四阶QMOM和六阶QMOM)。模拟结果与文献中的实验数据的对比结果表明，离散法和QMOM均能合理预测气含率、轴向液速、平均气泡大小和气泡尺寸分布。但QMOM比离散法节约2~3倍的计算成本。对于QMOM，使用四个矩能准确描述气相的演化。使用四阶QMOM和六阶QMOM得到的结果非常相似。利用QMOM的低阶矩可以快速有效地重构出单峰气泡尺寸分布。

关键词: 计算流体力学, 群平衡模型, 鼓泡塔, 气泡粒径分布, 矩方法