The prediction of the particle-size distribution (PSD) of the particulate systems in chemical engineering is very important in a variety of different contexts, such as parameter identification, troubleshooting, process control, design, product quality, production economics etc. The time evolution of the PSD can be evaluated by means of the population balance equation (PBE), which is a complex integro-differential equation, whose solution in practical cases always requires sophisticated numerical methods that may be computationally tedious. In this work, we propose a novel technique that tackles this issue by using an analytical time-stepping procedure (ATS) to resolve the PSD time dependency. The ATS is an explicit time integrator, taking advantage of the linear or almost linear time dependency of the discretized population balance equation. Thus, linear approximation of the source term is a precondition for the ATS simulations. The presented technique is compared with a standard variable step time integrator (MATLAB ODE15s stiff solver), for practical examples e.g. emulsion, ageing cellulose process, cooling crystallization, reactive dissolution, and liquid-liquid extraction. The results show that this advancing in time procedure is accurate for all tested practical examples, allowing reproducing the same results given by standard time integrators in a fraction of the computational time.

在各种不同的环境中，例如参数识别，故障排除，过程控制，设计，产品质量，生产经济性等，化学工程中颗粒系统的粒径分布（PSD）的预测非常重要。可以通过人口平衡方程（PBE）来评估PSD的大小，该方程是一个复杂的积分微分方程，在实际情况下，其求解总是需要复杂的数值方法，这些方法可能在计算上很繁琐。在这项工作中，我们提出了一种新颖的技术，可通过使用分析时间步长过程（ATS）解决PSD时间依赖性来解决此问题。ATS是一个显式的时间积分器，它利用离散人口平衡方程的线性或几乎线性时间依赖性。从而，源项的线性逼近是ATS模拟的前提。将本发明的技术与标准可变步长时间积分器（MATLAB ODE15的刚性求解器）进行比较，以用于实际示例，例如乳液，老化的纤维素工艺，冷却结晶，反应性溶解和液-液萃取。结果表明，该时间提前过程对于所有测试的实际示例都是准确的，从而可以在计算时间的一小部分内重现标准时间积分器给出的相同结果。和液液萃取。结果表明，该时间提前过程对于所有测试的实际示例都是准确的，从而可以在计算时间的一小部分内重现标准时间积分器给出的相同结果。和液液萃取。结果表明，该时间提前过程对于所有测试的实际示例都是准确的，从而可以在计算时间的一小部分内重现标准时间积分器给出的相同结果。