Methods for solving the various population balance formulations are presented and explained. The methods are presented progressively based on the kinetic and transport processes involved. In terms of methodology, the solution methods for the kinetic part of the population balance equation (PBE) are classified into several families: analytical/similarity, moment, discretisation and Monte Carlo methods. Methods for solving coupled computational fluid dynamics (CFD) – PBE problems are also presented. For each method, the advantages and disadvantages that determine its suitability for certain classes of problems are discussed.

The derivation and formulation of the population balance equation (PBE) is presented in this chapter. Various formulations such as the discrete, continuous, multidimensional and coupled PBEs are presented under a unifying framework and related to the problems that they can be applied to. The spatially dependent PBE and its coupling with fluid dynamics is also discussed.

Appendices include the basic equations involved in coupling the population balance equation (PBE) with fluid flow, heat and mass transfer (Appendix A), the implementation of the conservative finite volume discretisation method (Appendix B), the derivation of the probability density function (PDF) transport equation (Appendix C) and the derivation of the stochastic field equation (Appendix D).

Appendices include the basic equations involved in coupling the population balance equation (PBE) with fluid flow, heat and mass transfer (Appendix A), the implementation of the conservative finite volume discretisation method (Appendix B), the derivation of the probability density function (PDF) transport equation (Appendix C) and the derivation of the stochastic field equation (Appendix D).

The population balance is introduced as an approach for modelling problems involving a population of particles with a distribution of one or more properties. Numerous applications are identified. The general methodology of applying the population balance in four basic steps is introduced. Basic concepts such as distributions, choice of distributed variables, kinetic and transport processes and the coupling of the population balance with fluid dynamics, are also introduced.

Models for the kinetic and transport processes in the population balance equation (PBE) are discussed. Since the range of population balance problems is very wide, the focus is on the general forms of models and their incorporation into the PBE. More specific examples are drawn from the fields of aerosols and crystallisation that feature in the case studies of Chapter 6. The determination of model parameters from experimental data, or the inverse problem, is also discussed.

Appendices include the basic equations involved in coupling the population balance equation (PBE) with fluid flow, heat and mass transfer (Appendix A), the implementation of the conservative finite volume discretisation method (Appendix B), the derivation of the probability density function (PDF) transport equation (Appendix C) and the derivation of the stochastic field equation (Appendix D).

The application of the theory and methodology presented in the previous chapters for formulating and solving the population balance equation (PBE), as well as its coupling with fluid flow and computational fluid dynamics (CFD), is here demonstrated via three case studies. The first case study is about synthesis of silica nanoparticles in a laminar flame. The second one involves soot formation in laminar and turbulent flames. The third one is about precipitation of barium sulphate crystals in a turbulent T-mixer flow. In each case, the deployment of the population balance methodology is presented in an educational manner, following the four main steps outlined in Chapter 1.

Appendices include the basic equations involved in coupling the population balance equation (PBE) with fluid flow, heat and mass transfer (Appendix A), the implementation of the conservative finite volume discretisation method (Appendix B), the derivation of the probability density function (PDF) transport equation (Appendix C) and the derivation of the stochastic field equation (Appendix D).

The interaction of turbulent flow with the population balance is treated in this chapter. Approaches are described for integrating the population balance equation (PBE) within turbulent flow simulations based on Reynolds-Averaged Navier-Stokes (RANS), large eddy simulation (LES) and direct numerical simulation (DNS). The focus is on kinetic processes in non-inertial particles. The closure problem and the various unclosed terms that appear in the PBE in turbulent flow are discussed. Subsequently, the fundamental concepts and formulations of presumed and transported probability density function (PDF) methods for addressing the turbulence-PBE interaction are presented and explained. Stochastic numerical methods for solution of the PBE-PDF equation are also discussed.