Kostoglou M, Karabelas A J
Division of Chemical Technology, Department of Chemistry, Aristotle University, 54124 Thessaloniki, Greece.
J Colloid Interface Sci. 2006 Nov 15;303(2):419-29. doi: 10.1016/j.jcis.2006.08.005. Epub 2006 Aug 9.
The nonlinear fragmentation population balance formulation has been elevated in recent years from a prototype for studying nonlinear integro-differential equations to a vehicle for analyzing and understanding several physicochemical processes of technological interest. The so-called pure collisional fragmentation, which is the particular mode of nonlinear fragmentation induced by collisions between particles, is studied here. It is shown that the corresponding population balance equation admits large time asymptotic (self-similarity) solutions for homogeneous fragmentation and collision functions (kernels). The self-similar solutions are given in closed form for some simple kernels. Based on the shape of the self-similar solutions the method of moments with Gamma distribution approximation is employed for transient solution (from initial state to establishment of the asymptotic shape) of the collisional fragmentation equation. These solutions are presented for several sets of parameters and their behavior is discussed rather extensively. The present study is similar to the one has already been performed for the case of the much simpler linear fragmentation equation [G. Madras, B.J. McCoy, AIChE J. 44 (1998) 647].
近年来，非线性破碎种群平衡公式已从研究非线性积分 - 微分方程的一个原型提升为一种用于分析和理解若干具有技术意义的物理化学过程的工具。本文研究了所谓的纯碰撞破碎，它是由颗粒间碰撞引起的非线性破碎的一种特殊模式。结果表明，对于均匀破碎和碰撞函数（核），相应的种群平衡方程存在大时间渐近（自相似）解。对于一些简单的核，自相似解以封闭形式给出。基于自相似解的形状，采用具有伽马分布近似的矩量法来求解碰撞破碎方程的瞬态解（从初始状态到渐近形状的建立）。给出了几组参数的这些解，并对它们的行为进行了相当广泛的讨论。本研究类似于已经针对简单得多的线性破碎方程情形所开展的研究[G. 马德拉斯，B.J. 麦科伊，《美国化学工程师学会杂志》44 (1998) 647]。
