A quadrature-based moment method for the approximate solution of the generalized population balance equation (GPBE) governing the evolution of the joint size-velocity number density function (NDF) of a particle population is formulated and tested. The proposed method relies on the third-order hyperbolic conditional quadrature method of moments developed for velocity distribution transport. This approach is combined with the conditional quadrature method of moments to incorporate the dependency of the NDF on particle size, leading to an efficient, stable, quadrature method that uses an analytical solution to determine the size-conditioned velocity moments. The incorporation of source terms accounting for aggregation, breakup, and collisions, as well as acceleration terms such as gravity and drag, is performed using a realizable ODE solver. The approach is then demonstrated by considering zero-dimensional cases to verify the correct integration of the source terms. A set of one-dimensional cases involving droplet evaporation and coalescence is used to validate the velocity-dependent source terms. A two-dimensional case of crossing jets of particles with different sizes is used to demonstrate the proposed method in the case of a polydisperse flow with inertial particles. All work has been implemented in the open-source framework OpenQBMM, based on OpenFOAM®.

制定并测试了一种基于正交矩的方法，用于控制粒子群的联合尺寸-速度数密度函数 (NDF) 演化的广义种群平衡方程 (GPBE) 的近似解。所提出的方法依赖于为速度分布开发的矩的三阶双曲条件正交方法运输。这种方法与矩的条件正交方法相结合，以结合 NDF 对粒子尺寸的依赖性，从而形成一种高效、稳定的正交方法，该方法使用解析解来确定尺寸条件速度矩。使用可实现的常微分方程求解器合并考虑聚集、分裂和碰撞的源项，以及重力和阻力等加速度项。然后通过考虑零维情况来验证该方法以验证源项的正确集成。一组涉及液滴蒸发和聚结的一维案例用于验证与速度相关的源项。在具有惯性粒子的多分散流的情况下，使用具有不同尺寸的粒子的交叉射流的二维情况来演示所提出的方法。所有工作都在基于 OpenFOAM® 的开源框架 OpenQBMM 中实现。