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Introduction
Published in Wioletta Podgórska, Multiphase Particulate Systems in Turbulent Flows, 2019
and the mean drop size is proportional to the maximum stable drop size. Often such information is insufficient. When particle size distribution is required, or the evolution of particle size distribution in time and space is required, the population balance equation (PBE) is used. The population balance equation allows one to predict other properties of particles, for example drop or bubble composition, or crystal shape. The general information on population balance, including methods of PBE solution, is provided in Chapter 2. The terms of PBE are identified in more detail in Chapters 6 and 7. Examples of multivariate PBE are also presented in these chapters. When information on spatial distribution of the population is of importance, the population balance equation is coupled with CFD models. The main approach for modeling multiphase flows includes Eulerian–Eulerian modeling, discussed in Chapter 5. In this chapter, the Lagrangian–Eulerian model and volume of fluid approach are also briefly discussed.
Nanoparticles in Fluids
Published in Chun Huh, Hugh Daigle, Valentina Prigiobbe, Maša Prodanović, Practical Nanotechnology for Petroleum Engineers, 2019
Chun Huh, Hugh Daigle, Valentina Prigiobbe, Maša Prodanović
with s=2R/false(x+yfalse), where R is the center-to-center distance between the interacting nanoparticles. In Equation (3.14) the factor UT is the total interaction energy, which results from the repulsive and attractive forces described within the framework of the XDLVO theory (Equation 3.1). The population balance equation (Equation 3.12) is a partial differential equation (PDE) that can be solved by various numerical methods, such as the method of moments (Vollmer and Raisch 2006) and of the moving pivot (Kumar and Ramkrishna 1997).
The characteristics of gas-liquid dispersive mixing and microbubble generation in turbulent adjustable jet flow field
Published in Journal of Dispersion Science and Technology, 2023
Xingyu Ai, Xiaolei Cai, Jiaqing Chen, Guodong Ding, Shun Guan, Yipeng Ji
PBM is employed in this work to model the dynamic characteristics of group distribution of bubble including coalescence and breakup. The population balance equation can be expressed as follows: where Dc and Bc describe the death rate and the birth rate of bubbles of volume V due to aggregation, respectively; DB and BB represent the death rate and birth rate of bubbles of volume V due to breakage, respectively. The breakage and aggregation of bubbles cause their birth and death, which are defined as follows: where a is the aggregation frequency; Bc and Dc describe the aggregations under the Kernel model and DB and BB refer to the breakages under the Kernel model.
Modeling of silica synthesis in a laminar flame by coupling an extended population balance model with computational fluid dynamics
Published in Aerosol Science and Technology, 2023
Malamas Tsagkaridis, Stelios Rigopoulos, George Papadakis
A population balance equation (PBE) is employed to describe aerosol dynamics. Particle morphology is an important factor and its prediction is one of the objectives of the present article. A two-dimensional formulation (Koch and Friedlander 1990; Xiong and Pratsinis 1993) introduces an additional dimension such as particle surface area, but its Discretized form is too expensive for coupling with CFD. A two-PBE formulation employs two sets of PBEs, one for the number density of particles and another one for number of primary particles (Rogak 1997) or particle surface area (Tsantilis and Pratsinis 2000); such an approach is computationally feasible and able to incorporate morphology effects, but still requires twice the number of equations as compared with the one-PBE approach. A monodisperse model (Kruis et al. 1993) has also been proposed, although this entails several assumptions. In the present section, we first describe the monodisperse and two-PBE approaches employed in our study. Subsequently, a new formulation is proposed that, while employing one PBE, incorporates elements of the two-PBE approach with respect to the description of morphology.
Computational Fluid Dynamics–Population Balance Modeling of Gas–Liquid Two-Phase Flow in Bubble Column Reactors With an Improved Breakup Kernel Accounting for Bubble Shape Variations
Published in Heat Transfer Engineering, 2020
Weibin Shi, Jie Yang, Guang Li, Yuan Zong, Xiaogang Yang
The bubble size distribution is determined by employing the population balance model with a consideration of bubble coalescence and breakup. Bubbles are divided into several size groups with different diameters specified by the parameter deq,i and an equivalent phase with the Sauter mean diameter to represent the bubble classes. In this study, 16 bubble classes with diameters ranging from 1 mm to 32 mm are applied based on the geometric discretization method where Vi = 2Vi-1. The population balance equation is expressed by Eq. (1), where ni is the number density for ith group, is the mass average velocity vector and Si is the source term.