Macroscopic Continuum Mechanical Population Balance

The formulations of the population balance equation based on the continuum mechanical approach can be split into two categories, the macroscopic- and the local instantaneous population balance equation formulations. The macroscopic approach consists in describing the evolution of a number density in time and space of several groups or classes of the dispersed phase properties. 

In this section the macroscopic population balance formulation of Prince and Blanch [102], Luo [79] and Luo and Svendsen [80] is outlined. In the work of Luo [79] no growth terms were considered, the balance equation thus contains a transient term, a convection term and four source terms due to binary bubble coalescence and breakage. 

The number density variable can be related to the fundamental number density distribution function/ by the following definition  

The source terms are assumed to be functions of bubble size / bubble number density m and time t. The birth of bubbles of size di due to coalescence stems from the coalescence between aU bubbles of size smaller than di. Hence, the birth rate for bubbles of size di, Bc,u can be obtained by summing all coalescence events that form a bubble of size di. This gives  

---

To close the population balance problem, models are required for the growth, birth and death kernels. In the kinetic theory context, as distinct from the continuum mechanical approach, the continuum closure may be considered macroscopic in a similar manner as in the granular theory treating macroscopic particle properties. 

In the following sections four alternative approaches for deriving population balance equations are outlined. The four types of PBEs comprise a macroscopic PBE, a local instantaneous PBE, a microscopic PBE, and a PBE on the moment form. Two of these population balance forms are formulated in accordance with the conventional continuum mechanical theory. 

---