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

Abstract

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 moments (EQMOM) based on the work of Yuan et al.[52]and a hybrid method (TSM) between the sectional and moment methods, considering two moments per section based on the work of Laurent et al.[30]. 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.

Modelling of Bubbly Flow Using CFD-PBM Solver in OpenFOAM: Study of Local Population Balance Models and Extended Quadrature Method of Moments Applications

“…The EQMOM approach approximates the unknown NDF with a weighted sum of smooth, non-negative kernel density functions δ σ (L, L α ) [36,57]:…”

“…The 2N α quantities W αγ , called secondary weights and abscissae, respectively, are computed using the standard Gaussian quadrature formulae for known orthogonal polynomials to the kernel NDF [36,57]. β α 1 γ 1 α 2 γ 2 is the aggregation kernel for the bubbles of size L α 1 γ 1 and L α 2 γ 2 ; a αγ is the breakage kernel for the bubbles size of L αγ ; and b αγ represents the daughter distribution function.…”

New developments of the Extended Quadrature Method of Moments to solve Population Balance Equations

“…More recently, the Extended Quadrature Method of Moments (EQMOM) was proposed 30 as a new approach which is more stable than the previous ones, and yields either continuous 31 or discrete NDFs depending on the moments [1,14,15]. EQMOM has been implemented in 32 OpenFOAM [16] for the purpose of PBE-CFD coupling.…”

“…The previous core procedure [1] will be recalled before describing how it can be shifted 38 toward the new -cheaper-approach. Both implementations will be compared in terms of 39 computational cost (number of required floating-point operations) and run-time.…”

“…The Chebyshev algorithm was found to be the stablest one of the three [1,25] proximates n(ξ) as a weighted sum of Dirac distributions:…”