The Population Balance Equation

Abstract

The chemical engineering community began the first efforts that can be associated with the concepts of the population balance in the early 1960s. The familiar application of population balance principles to the modeling of flow and mixing characteristics in vessels was formally organized by Danckwerts [18]. Certain distribution functions were then defined for the residence times of fluid elements in a process vessel. The residence time distribution function give information about the fraction of the fluid that spends a certain time in a process vessel. Himmelblau and Bishoff [37] describe how the residence time and other age distributions are defined, how they can be measured, and how they can be interpreted. This chapter focuses on the derivation and use of the general population balance equation (PBE) to describe the evolution of the fluid particle distribution due to the advection, growth, coalescence and breakage processes.The historical derivation of the general population balance equation for countable entities is outlined. Two fundamental modeling frameworks emerge formulating the early population balances, in quite the same way as the kinetic theory of dilute gases and the continuum mechanical theory were proposed deriving the governing conservation equations in fluid mechanics. A third less rigorous approach is also used formulating the population balance directly on the macroscopic averaging scales, an analogue to the multiphase mixture models. A fourth less computationally demanding approach is to formulate a moment form of the population balance equation.Considering dispersed two-phase flows, a few research groups adopted a statistical Boltzmann-type equation determining the rate of change of a suitable defined distribution function. Performing the Maxwellian averaging integrating all terms over the whole velocity space to eliminate the velocity dependence, one obtains the generic population balance equation. Rigorous closures are required for the unknown terms resulting from the averaging process. However, by employing the conventional Chapman-Enskog approximate expansion method solving the Boltzmann-type equation this theory provides rational means of understanding for the one way coupling between microscopic particle physics and the average macroscopic continuum properties. Moreover, to represent complex problem physics an optimized choice of H. A. Jakobsen, Chemical Reactor Modeling,

A population balance model for large eddy simulation of polydisperse droplet evolution

“…The basic formulation of PBE's has been extended by Hulburt & Katz (1964), Randolph (1964) and Ramkrishna (1985) to include additional phenomena such as breakup, coalescence, nucleation, condensation, etc. PBE's can be classified into four main categories (Jakobsen 2014): Method of moments, multi-class method, method of weighted residuals and stochastic methods. In this paper we use the multi-class formulation to study the evolution of polydisperse liquid droplets in a surrounding fluid.…”

Coupling Eulerian‐Lagrangian method of air‐particle two‐phase flow with population balance equations to simulate the evolution of vehicle exhaust plume

“…As a result, due to the evolution of two phase dynamic process in space with time, the property of each particle could also change simultaneously . The coupling of dynamic models to describe the trajectory of a particle system in time and space and to use population balance equations (PBEs) to describe the evolution of particle size spectrum could be the key to accurately describe system characteristics …”