Exact solutions of population balance equation

Morphological population balance modelling of the effect of crystallisation environment on the evolution of crystal size and shape of para-aminobenzoic acid

“…Whilst the theoretical solution of the MPB equation can only be obtained for some ideal (simple) cases, numerical solution methods can provide the most convenient and available approach. Lin et al (Lin et al, 2016) developed an invariant method of moments to obtain analytical solution of a PB system, but this method could only be used for solving a one-dimensional homogeneous PB equation with size independent growth rate. Therefore, many other different numerical solution methods have been developed, for example, method of characteristics (Gunawan et al, 2004, Sotowa et al, 2000, moment of classes (David et al, 1995, Puel et al, 2003a, high resolution discretisation schemes (Gunawan et al, 2004, Ma et al, 2002, Wan et al, 2009, method of lines (Gerstlauer et al, 2001), finite-element schemes (Gerstlauer et al, 2006), moving grid techniques (Kumar and Ramkrishna, 1997), hierarchical solution strategies based on multilevel discretisation (Pinto et al, 2007, Sun andImmanuel, 2005), cell-ensemble method (Henson, 2005), Monte Carlo methods (Yu et al, 2015), etc.…”

Symmetries and exact solutions of the population balance equation with growth and breakage processes, using group analysis

“…It is difficult to find the infinitesimal generators of Equation () by directly applying the developed Lie group analysis method 12,13 to the original equation (). However, the admitted scaling group of Equation () is investigated by use of the scaling group analysis 8,12‐14 in this section. Let us consider the following scaling group where a is an arbitrary real group parameter, and equation …”

“…According to the algorithm 8,12‐14 and scaling group () the admitted infinitesimal generator of Equation () can be given by …”

“…On the other hand, the obtained symmetries can also be used to construct the self‐similar solutions. Such as the literatures 14‐16 …”

“…Conversely, one can use the developed Lie group analysis 12,13 to deal with symmetries of integro‐differential equations. In recent years, the developed Lie group analysis 12,13 was applied to solve integro‐differential, stochastic and delay equations, and so on 14‐25 . The main obstacle of this method is in finding the general solutions of the determining equations, approaches of solving determining equations of integro‐differential equations depend on the studied equations, there is no a general approach for solving determining equations of integro‐differential equations 14‐26 …”