In the present paper, we present a method for solving the population balance equation (PBE) with the complete range of kinetic processes included: namely aggregation, fragmentation, nucleation and growth. The method is based on the finite volume scheme and features guaranteed conservation of the first moment by construction, accurate prediction of the size distribution, applicability to an arbitrary non-uniform grid, robustness and computational efficiency which is instrumental for coupling with computational fluid dynamics (CFD). The treatment of aggregation is based on the previous work by Liu and Rigopoulos (2019). An analysis of the aggregation terms in the PBE is made, and the source of conservation error in finite element/volume methods is elucidated. It is subsequently shown how this error is overcome in the present method via a coordinate transformation applied to the aggregation birth double integral resulting from the application of the finite volume method. The contributions to the birth term are delineated and their corresponding death fluxes identified. An aggregation map is then constructed for mapping birth and death fluxes, thus allowing the finite volume method to operate in terms of fluxes and achieve conservation of mass. The method is then extended to fragmentation, for which a map is also constructed to represent the birth and death fluxes. In the implementation, the aggregation and fragmentation maps are pre-tabulated to allow fast computation. It is also shown how the method can be coupled with a total variation diminishing (TVD) scheme for the treatment of growth with minimal numerical diffusion. The method is validated with a number of test cases including analytical solutions and numerical solutions of the discrete PBE for aggregation (theoretical and free molecule/Brownian kernels), fragmentation, aggregation-fragmentation and aggregation-growth. In all cases, the method produces very accurate results, while also being computationally efficient due to the pre-tabulation of the maps and the simplicity of the algorithm carried out per time step.

在本文中，我们提出了一种求解种群平衡方程（PBE）的方法，其中包括完整的动力学过程：聚集、破碎、成核和生长。该方法基于有限体积方案，其特点是通过构造保证一阶矩守恒、尺寸分布的准确预测、适用于任意非均匀网格、鲁棒性和计算效率，这有助于与计算流体动力学耦合。差价合约）。聚合的处理基于 Liu 和 Rigopoulos (2019) 之前的工作。对 PBE 中的聚合项进行了分析，并阐明了有限元/体积法中守恒误差的来源。随后显示了如何在本方法中通过将坐标变换应用于由有限体积法的应用产生的聚合出生双积分来克服该误差。描述了对出生期限的贡献，并确定了它们相应的死亡通量。然后构建一个聚合图来绘制出生和死亡通量，从而允许有限体积方法根据通量进行操作并实现质量守恒。然后将该方法扩展到碎片化，为此还构建了一个地图来表示出生和死亡通量。在实现中，聚合和碎片映射是预先制表的，以允许快速计算。还展示了该方法如何与总变异减少 (TVD) 方案相结合，以处理具有最小数值扩散的生长。该方法通过大量测试用例进行验证，包括离散 PBE 的解析解和数值解，用于聚合（理论和自由分子/布朗核）、碎裂、聚合-碎裂和聚合-生长。在所有情况下，该方法都产生了非常准确的结果，同时由于地图的预先制表和每个时间步执行的算法的简单性，计算效率也很高。聚合-碎片化和聚合-增长。在所有情况下，该方法都产生了非常准确的结果，同时由于地图的预先制表和每个时间步执行的算法的简单性，计算效率也很高。聚合-碎片化和聚合-增长。在所有情况下，该方法都产生了非常准确的结果，同时由于地图的预先制表和每个时间步执行的算法的简单性，计算效率也很高。