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 appli-cation of population balance principles to the modeling of flow and mixing charac-teristics 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 statis-tical 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 conven-tional Chapman–Enskog approximate expansion method solving the Boltzmann-type equation this theory provides rational means of understanding for the one way cou-pling 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, DOI: 10.1007/978-3-319-05092-8_9, 937 © Springer International Publishing Switzerland 2014 938 9 The Population Balance Equation distribution function definition considering multiple internal coordinates in an envi-ronment of a particular state may be necessary, and novel closure required, making the approach rather demanding theoretically. The procedure sketched above has much in common with the granular theory of solid particles, outlined in Chap. 4. However, the majority of research groups within the chemical engineering community adopted an alternative approach based on a generalization of the classical continuum mechanical theory developing the population balance equation. On the other hand, the contin-uum theory gives only an average representation of the dispersed phase considering macroscopic scales several orders of magnitude larger than the microscopic particle dimensions and does not provide any information on the unresolved mechanisms that can be utilized formulating the population balance closures. In most cases, the balance principle is applied formulating a transport equation for the distribution func-tion on the integral form. Generalized versions of the Leibnitz-and Gauss theorems are then required transferring the integral balance to the differential form. Thereafter, the governing local instantaneous continuum mechanical equations on the differen-tial form are averaged to obtain a model formulation representing tractable volume and time resolutions. However, the averaging process also give rise to additional closure requirements. In the past, the time after volume averaging procedure was frequently used, being consistent with the time-after volume averaged multi-fluid model formulation discussed in Sect. 3.4.4. The third group of population balances are formulated on integral form directly on the averaging scales and then converted to the differential form employing the extended Leibnitz-and Gauss theorems. In this formulation, the closures are purely empirical parameterizations based on intuitive relationships rather than sound scientific principles. The population balance concept was first presented by Hulburt and Katz [39]. Rather than adopting the standard continuum mechanical framework, the model derivation was based on the alternative Boltzmann-type equation familiar from clas-sical statistical mechanics. The main problems investigated stem from solid particle nucleation, growth, and agglomeration. Randolph [105] and Randolph and Larson [106], on the other hand, formulated a generic population balance model based on the generalized continuum mechanical framework. Their main concern was solid particle crystallization, nucleation, growth, agglomeration/aggregation and breakage. Ramkrishna [103, 104] adopted the generalized continuum mechanical concepts of Randolph [105] and Randolph and Larson [106] to derive the population balance equation. Moreover, Ramkrishna [104] also discussed a more general stochastic the-ory to inquire into the statistical foundation of the population balance equation. In this statistical analysis Ramkrishna [104] used statistical concepts such as probabil-ities and expectancies to derive a set of product density equations. In this framework the PBE is seen as one of the product density equations in the given set. Since this theory is not common background among engineers, this route is not further explored in this book. Ramkrishna [103, 104] did investigate biological populations as well as numerous studies of chemical engineering problems have been reported. Similar PBE modeling approaches have also been used in the theory of aerosols in which the gas is the continuous phase [27, 136], in chemical-, mechanical-and