Solution of population balance equations in applications with fine particles: Mathematical modeling and numerical schemes

T.T. Nguyen, F. Laurent, R.O. Fox, M. Massot
2016 Journal of Computational Physics  
The accurate description and robust simulation, at relatively low cost, of global quantities (e.g. number density or volume fraction) as well as the size distribution of a population of fine particles in a carrier fluid is still a major challenge for many applications. For this purpose, two types of methods are investigated for solving the population balance equation with aggregation, continuous particle size change (growth and size reduction), and nucleation: the extended quadrature method of
more » ... oments (EQMOM) based on the work of Yuan et al. (J. Aerosol Sci., 51:1-23, 2012) and a hybrid method (TSM) between the sectional and moment methods, considering two moments per section based on the work of Laurent et al. (Commun. Comput. Phys., accepted, 2016). For both methods, the closure employs a continuous reconstruction of the number density function of the particles from its moments, thus allowing evaluation of all the unclosed terms in the moment equations, including the negative flux due to the disappearance of particles. Here, new robust and efficient algorithms are developed for this reconstruction step and two kinds of reconstruction are tested for each method. Moreover, robust and accurate numerical methods are developed, ensuring the realizability of the moments. The robustness is ensured with efficient and tractable algorithms despite the numerous couplings and various algebraic constraints thanks to a tailored overall strategy. EQMOM and TSM are compared to a sectional method for various simple but relevant test cases, showing their ability to describe accurately the fine-particle population with a much lower number of variables. These results demonstrate the efficiency of the modeling and numerical choices, and their potential for the simulation of real-world applications. 2 problem with nucleation and growth. Moreover, to our knowledge, such methods have never been reported for cases where the particle size is decreasing through a continuous process. A different kind of method, the only one that will be called sectional here (even if some of the previous ones are also called sectional in the literature), is based on a closure through a continuous reconstruction of the NDF inside each section. This reconstruction can be constant [25, 26, 27] 50 or affine [28] . When considering sprays, sectional methods are also called "Eulerian multi-fluid methods" [26] or "one-size moment" method (OSM), and they are developed after reduction of the internal variables to only size thanks to velocity moments and a mono-kinetic closure. The corresponding model is a finite-volume method. It was shown to be first-order accurate in the pureevaporation case [29] and exhibited first-order numerical accuracy for the investigated cases, taking 55 into account the collisions (coalescence) [30] . Moreover, in this approach, an affine reconstruction of MUSCL type was tested in the pure-evaporation case. However, if its order of accuracy is higher, the effective accuracy is not much improved compared to the first-order method, except with a large number of sections. Then, as with other discretized methods, sectional methods lead to an accurate prediction of the NDF with a large enough number of sections. However, for many applications, 60
doi:10.1016/j.jcp.2016.08.017 fatcat:x6qetqir4be7lkbvx5rte6ipca