Population balance models description

A new deterministic population balance model with distributed cell populations has been presented. The model is based on the unstructured approach of Mohler et al. [4]. Concerning outer dynamics the present model is equivalent to the unstructured model which proved to be sufficient to predict virus yields for different initial conditions [4]. The characteristics of the inner dynamics can be simulated except of the decrease of fluorescence intensity at later time points. The biological reasons for this effect are unclear. Presumably there are more states that have to be considered during virus replication like intercellular communication, extent of apoptosis or specific stage in cell cycle. Future computational and experimental research will aim in this directions and concentrate on structured descriptions of the virus replication in mammahan cell culture. 

There is a growing interest in what is, somewhat misleadingly, called multidimensional population balance models. One example of a 2D PB model is the description of a granulation process where not only the particle size distribution with time, but also the fractional binder content is predicted by the model. The binder (liquid) content of the granules governs the agglomeration process. 

Instead of assigning different shear rates, he employed different breakage rate expressions for the two zones. The problem of coupling population balance models with fluid flow models has received some attention recently and coupled PB-CFD models have been developed for a wide variety of processes such as fluidization [70], gas-liquid reactions in bubble columns [71] and nanoparticle synthesis in flame aerosol reactors [72]. Complete description of aggregation in turbulent environments requires simultaneous solution of basic balance equations for mass, momentum, energy and concentration of species present along with population balances for particles/aggregates of different size classes. 

Although evidence exists for both mechanisms of growth rate dispersion, separate mathematical models were developed for incorporating the two mechanisms into descriptions of crystal populations random growth rate fluctuations (36) and growth rate distributions (33,40). Both mechanisms can be included in a population balance to show the relative effects of the two mechanisms on crystal size distributions from batch and continuous crystallizers (41). 

Theoretical representation of the behaviour of a hydrocyclone requires adequate analysis of three distinct physical phenomenon taking place in these devices, viz. the understanding of fluid flow, its interactions with the dispersed solid phase and the quantification of shear induced attrition of crystals. Simplified analytical solutions to conservation of mass and momentum equations derived from the Navier-Stokes equation can be used to quantify fluid flow in the hydrocyclone. For dilute slurries, once bulk flow has been quantified in terms of spatial components of velocity, crystal motion can then be traced by balancing forces on the crystals themselves to map out their trajectories. The trajectories for different sizes can then be used to develop a separation efficiency curve, which quantifies performance of the vessel (Bloor and Ingham, 1987). In principle, population balances can be included for crystal attrition in the above description for developing a thorough mathematical model. 

The development and refinement of population balance techniques for the description of the behavior of laboratory and industrial crystallizers led to the belief that with accurate values for the crystal growth and nucleation kinetics, a simple MSMPR type crystallizer could be accurately modelled in terms of its CSD. Unfortunately, accurate measurement of the CSD with laser light scattering particle size analyzers (especially of the small particles) has revealed that this is not true. In mar cases the CSD data obtained from steady state operation of a MSMPR crystallizer is not a straight line as expected but curves upward (1. 32. 33V This indicates more small particles than predicted... 

The SAXS/TGA approach has been demonstrated to be a useful technique for time-resolution of porosity development in carbons during activation processes. Qualitative interpretation of the data obtained thus far suggests that a population balance approach focusing on the rates of production and consumption of pores as a function of size may be a fruitful approach to the development of quantitative models of activation proces.ses. These then could become useful tools for the optimization of pore size distributions for particular applications by providing descriptions and predictions of how various activating agents and time-temperature histories affect resultant pore size distributions. 

Chen [74] derived with the help of a population balance equation a mathematical mechanistic model, which enables the description and prediction of the droplet size distribution arising from emulsion and suspension processes in the loop reactor. 

The interfacial and turbulence closures suggested in the literature also differ considering the anticipated importance of the bubble size distributions. It thus seemed obvious for many researchers that further progress on the flow pattern description was difficult to obtain without a proper description of the interfacial coupling terms, and especially on the contact area or projected area for the drag forces. The bubble column research thus turned towards the development of a dynamic multi-fluid model that is extended with a population balance module for the bubble size distribution. However, the existing models are still restricted in some way or another due to the large cpu demands required by 3D multi-fluid simulations. 

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... 

Alternatively, the complete population balance can be solved dynamically using efficient ODE solvers [70, 71], The versatile commercial software PREDICI can solve population balances that describe polymerizations with coordination catalysts and many other polymerization mechanisms [72]. In this approach, the complete microstructural distributions are modeled, leading to a detailed description of the polymer microstructure. 

Alopaeus, V., Koskinen, J., and Keskinen, K.I., Simulation of the Population Balances for Liquid-Liquid Systems in a Nonideal Stirred Tank, Part 1 Description and Qualitative Validation of the Model, Chem. Eng. Sci. 54 (1999) pp. 5887 -5899. 

Modeling is probably the tool of excellence for engineers (Chapter 9). It is used to simulate the reaction and the process system in order to shorten the time for development. It is based on models that can be physical or chemical, semi-empirical or empirical, descriptive or more fundamental. To describe the development of the molecular weight distribution upon reaction, moment methods or equations based on population balance are often used. 

The mathematical modeling of these polymers are based on either population balances [24-31] or Monte Carlo methods [32-35]. The output of these models includes a detailed description of the polymer architecture (sol MWD, gel fraction, cross-linking points, long chain branching). The models are rather complex and will not be discussed here. Nevertheless, for cases of practical interest, the average molecular weights can be calculated with a modest computational effort [36]. 

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