Population balance equation, mass transfer

To close the present derivation of the continuum population balance equations, one needs to simplify the last two terms on the left-hand side of Equations (A-23) and (A-24). These terms describe various mechanisms of mass and/or bubble transfer among the regions defined by the characteristic functions (A-2)-(A-4). 

Analysis of Mass Transfer or Reaction in Dispersions with the Population Balance Equation... 

The work discussed in this section clearly delineates the role of droplet size distribution and coalescence and breakage phenomena in mass transfer with reaction. The population balance equations are shown to be applicable to these problems. However, as the models attempt to be more inclusive, meaningful solutions through these formulations become more elusive. For example, no work exists employing the population balance equations which accounts for the simultaneous affects of coalescence and breakage and size distribution on solute depletion in the dispersed phase when mass transfer accompanied by second-order reaction occurs in a continuous-flow vessel. Nevertheless, the population balance equation approach provides a rational framework to permit analysis of the importance of these individual phenomena. 

The deterministic population balance equations governing the description of mass transfer with reaction in liquid-liquid dispersions present a framework for analysis. However, signiflcant difficulties exist in obtaining solutions for realistic problems. No analytical solutions are available for even the simplest cases of interest. Extension of the solution to multiple reactants for uniform drops is possible using a method of moments but the solution is limited to rate equations which are polynomials (E3). Solutions to the population balance equations for spatially nonhomogeneous dispersions were only treated for nonreacting dispersions (P4), and only a simple case was solved for a spray column (B19). Treatment of unmixed feeds presents a problem. 

The remaining chapters in this book are organized as follows. Chapter 2 provides a brief introduction to the mesoscale description of polydisperse systems. There, the mathematical definition of a number-density function (NDF) formulated in terms of different choices for the internal coordinates is described, followed by an introduction to population-balance equations (PBE) in their various forms. Chapter 2 concludes with a short discussion on the differences between the moment-transport equations associated with the PBE and those arising due to ensemble averaging in turbulence theory. This difference is very important, and the reader should keep in mind that at the mesoscale level the microscale turbulence appears in the form of correlations for fluid drag, mass transfer, etc., and thus the mesoscale models can have non-turbulent solutions even when the microscale flow is turbulent (i.e. turbulent wakes behind individual particles). Thus, when dealing with turbulence models for mesoscale flows, a separate ensemble-averaging procedure must be applied to the moment-transport equations of the PBE (or to the PBE itself). In this book, we are primarily... 

By using CFD, the fluid flows can be taken into closer examination. Rigorous submodels can be implemented into commercial CFD codes to calculate local two-phase properties. These models are Population balance equations for bubble/droplet size distribution, mass transfer calculation, chemical kinetics and thermodynamics. Simulation of a two-phase stirred tank reactor proved to be a reasonable task. The results revealed details of the reactor operation that cannot be observed directly. It is clear that this methodology is applicable also for other multiphase process equipment than reactors. 

The source terms for the bubble numbers are due to breakage and coalescence of bubbles, and mass transfer induced size change. Other sources (such as formation of small bubbles through nucleation mechanisms) were neglected in this study. The discretized population balance equation can then be written in the following form... 

In addition to liquid-liquid systems, the population balance equation (PBE) has been applied to crystallization, grinding, interphase heat and mass transfer, multiphase reactions, and floatation. 

The governing equations to model the particle formation dynamics are identical for the gas and Uquid phase, whereby of course, the reaction mechanisms and the kinetic parameters (e.g., for the mass transfer) differ. The population balance equation (PBE) can be considered as the master equation for formation of particles by both top-dovm and bottom-up methods (Eq. 5) ... 

The mathematical modeling of polymerization reactions can be classified into three levels microscale, mesoscale, and macroscale. In microscale modeling, polymerization kinetics and mechanisms are modeled on a molecular scale. The microscale model is represented by component population balances or rate equations and molecular weight moment equations. In mesoscale modeling, interfacial mass and heat transfer... 

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