The population-balance equation PBE

The PBE is a simple continuity statement written in terms of the NDE. It can be derived as a balance for particles in some fixed subregion of phase and physical space (Ramkrishna, 2000). Let us consider a finite control volume in physical space O and in phase space with boundaries defined as dO. and dO., respectively. In the PBE, the advection velocity V is assumed to be known (e.g. equal to the local fluid velocity in the continuous phase or directly derivable from this variable). The particle-number-balance equation can be written as 

If the Reynolds-Gauss theorem is applied to Eq. (2.12) and the integrals on the boundary of the control volume are written in terms of integrals over the control volume itself, it is straightforward to obtain 

Note that, although we treat x and f in the same manner, they are in fact different types of vectors. The vectors x and v are the standard vectors for position and velocity used in continuum mechanics. The internal-coordinate vector on the other hand, is a generalized vector of length N in the sense of linear algebra. 

Equation (2.14) must be coupled with initial conditions given for the starting time and with boundaries conditions in physical space O and in phase space O. Analytical solutions to Eq. (2.14) are available for a few special cases and only under conditions specified by some very simple hypotheses. However, numerical methods can be used to solve this equation and will be presented in Chapters 7 and 8. The numerical solution of Eq. (2.14) provides knowledge of the NDE for each time instant and at every physical point in the computational domain, as well as at every point in phase space. As has already been mentioned, sometimes the population of particles is described by just one internal coordinate, for example particle length (i.e. f = L), and the PBE is said to be univariate. When two internal coordinates are needed, for example particle volume and surface area (i.e. = (v, a)), the PBE is said to be bivariate. More generally, higher-dimensional cases are referred to as multivariate PBEs. Another important case occurs when part of the internal-coordinate vector is equal to the particle-velocity vector (i.e. when the particles are characterized not by a unique velocity field but by their own velocity distribution). In that case, the PBE becomes the GPBE, as described next. 

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If random scission occurs together with end-chain scission (but at a different rate) then defining x by dx/dt = k(i the population balance equations (PBEs) for these combined processes are... 

Considering a single growth direction with one characteristic length L, and a well-mixed crystallizer with growth and nucleation as the only dominating phenomena the population balance equation (PBE) has the form... 

In mathematical terms the population balance equation (PBE) is classified as a non-linear partial integro-differential equation (FIDE). Since analytical solutions of this equation are not available for most cases of practical interest, several numerical solution methods have been proposed during the last two decades as discussed by Williams and Loyalka [209] and Ramkrishna [151]. 

Continuous MSMPR Precipitator. The population balance, which was put forward by Randolph and Larson (1962) and Hulbert and Katz (1964), provides the basis for modeling the crystal size distribution (CSD) in precipitation processes. For a continuous mixed-suspension, mixed-product-removal (CMSMPR) precipitator with no suspended solids in the feed streams, the population balance equation (PBE) can be written as (Randolph and Larson 1988)... 

To predict the evolution of the droplet (floe) size distribution is the central problem in emulsion stability. It is possible, in principle, to predict the time dependence of the distribution of droplets (floes) if information concering the main subprocesses (flocculation, floe fragmentation, coalescence, creaming), constituting the whole phenomenon, is available. This prediction is based on consideration of the population balance equation (PBE). 

This is the population balance equation (PBE). Its principal form was first given by Smoluchowski (1916). In Eq. (4.1), denotes the aggregate number concentration, while Kjj is a kinetic constant related with the collision between clusters of mass i and j, and Etj is the sticking probability of the sub-clusters. Commonly, it is assumed that any successful collision (i.e. surface contact) leads to a stable particle bond and that Ey can be, thus, set to 1. 

The same authors also presented an example of the use of the population balance equation (PBE) (distribution of biomass m) coimected to the multi-zone/CFD model. This example is in several respects relevant for the assessment of the modeling approach. The coupling of the integro-differential equation of the population balance is a numerical challenge, which can nowadays be tackled within the environment of a CFD approach, albeit without consensus on the proper closure assumptions. Still, the computational effort for the numerical solution of the population balance embedded in the multizonal model is extensive, and it is difficult to extend this approach to multiple state variables necessary for dynamic metabolic models. This is an important argument to favor the alternative method of an agent-based Lagrange-Euler approach discussed in Section 3.5. 

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. 

Dividing by (AxAyAz) and allowing the individual dimensions Ax, Ay and Az to shrink to zero in the limit, we get the population balance equation (PBE) in cartesian coordinates ... 

The population balance equation (PBE) was outlined in Chap. 9. A frequently employed form the PBE with diameter as the only inner coordinate (property) can be deduced from (9.62). For clearity, this form of the PBE is listed below ... 

In mathematical terms the population balance equation (PBE) is classified as a non-linear partial integro-differential equation (PIDE). In the PBE (12.308) the size property variable ranges from 0 to oo. In order to apply a numerical scheme for the solution of the equation a first modification is to fix a finite computational domain. The conventional approximation is to truncate the equation by substitution of the infinite integral limits by the finite limit value max- The function/(, r, t) denotes the exact solution of the exact equation. It might be assumed that the solution of the truncated PBE is sufficiently close to the exact equation so that the two solutions are practically equal. Hence, the solutions of both forms of the PBE are denoted by fiC, r, t). 

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) ... 

In the more sophisticated treatments, spatial inhomogeneities are modeled by connected zones or regions of Space that are assumed to be homogeneous. The resulting population-balance equation (PBE) does not explicitly account for local variations in die flow field. 

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... 

To follow the dynamic evolution of PSD in a particulate process, a population balance approach is commonly employed. The distribution of the droplets/particles is considered to be continuous in the volume domain and is usually described by a number density function, (v, t). Thus, n(v, f)dv represents the number of particles per unit volume in the differential volume size range (v, v + dv). For a dynamic particulate system, undergoing simultaneous particle breakage and coalescence, the rate of change of the number density function with respect to time and volume is given by the following non-linear integro-differential population balance equation (PBE) [36] ... 

The most important property for the characterization of particles is particle size. Randolph and Larson (36) pointed out that As no two particles will be exactly the same size, the material must be characterized by the distribution of sizes or particle-size distribution (PSD). If only size is of interest, a single-variable distribution function is sufficient to characterize the particulate system. If additional properties are also important, multivariable distribution functions must be developed. These distribution functions can be predicted through numerical simulations using population balance equations (PBE). 

The one-dimensional aggregation-only population balance equation (PBE) described by Eq. 3.20 with the kernel model given by Eq. 3.24 can be reduced to a set of ordinary differential equations as follows ... 

In order to account for formation of dimers, trimers, and larger aggregates, von Smoluchowski proposed in his pioneering work [73] the following system of population balance equations (PBE) ... 

However, we did not only want to analyze our data by experiments but to use them for a predictive modelling of particle formation. This was realized by population balance equations (PBE) which have been solved by using the commercially available program PARSIVAL by CiT GmbH [81]. For experimental details and specifications of the measurement devices the reader is again referred to the literature [14, 64, 65]. The most important findings will be summarized in the following. 

It is noted that all the terms in the continuum population balance equation have common units, (l/m s[m]). By discretizing the continuous bubble number density in groups or classes, a PBE formulation equivalent to the discrete macroscopic framework is obtained. 

This book provides a consistent treatment of these issues that is based on a general theoretical framework. This, in turn, stems from the generalized population-balance equation (GPBE), which includes as special cases all the other governing equations previously mentioned (e.g. PBE and BE). After discussing how this equation originates, the different computational models for its numerical solution are presented. The book is structured as follows. 

The derivation of regular patterns of cellular behavior that can be assigned to certain defined subpopulations (or single cells) from measured data necessarily involves modeling approaches that allow assignment of mechanistic (e.g., metabohc) models to subpopulations with individual parameters or other variations. Further, such models must allow the tracking of individual cell s dynamic behavior even when they fluctuate between different populations. Two principal approaches can be distinguished systems of partial different equations (population balance systems, PBE) and stochastic cell ensemble models (CEMs)... 

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. 

Since population balance equations are multi-dimensional, their calculations are tedious and hence a lot of research has been focused on the model order reduction methods. One of the most common and efficient reduction methods is the method of moments. There are other methods for solving PBEs such as the discretization methods [101], method of characteristics, successive approximation[102], etc. However, the method of moments is commonly used and is described below. 

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