Population balance simulator

The population balance simulator has been developed for three-dimensional porous media. It is based on the integrated experimental and theoretical studies of the Shell group (38,39,41,74,75). As described above, experiments have shown that dispersion mobility is dominated by droplet size and that droplet sizes in turn are sensitive to flow through porous media. Hence, the Shell model seeks to incorporate all mechanisms of formation, division, destruction, and transport of lamellae to obtain the steady-state distribution of droplet sizes for the dispersed phase when the various "forward and backward mechanisms become balanced. For incorporation in a reservoir simulator, the resulting equations are coupled to the flow equations found in a conventional simulator by means of the mobility in Darcy s Law. A simplified one-dimensional transient solution to the bubble population balance equations for capillary snap-off was presented and experimentally verified earlier. Patzek s chapter (Chapter 16) generalizes and extends this method to obtain the population balance averaged over the volume of mobile and stationary dispersions. The resulting equations are reduced by a series expansion to a simplified form for direct incorporation into reservoir simulators. 

J. Bronic, P. Frontera, F. Testa, B. Subotiae, R. Aiello, and J.B. Nagy, Study of Zeolite A Crystallization from Clear Solution by Hydrothermal Synthesis and Population Balance Simulation. Stud. Surf. Sci. Catal., 2001, 135 (Zeolites and Mesoporous Materials at the Dawn of the 21st Century), 358-365. 

P-29 - Study of zeolite A crystallization from clear solution by hydrothermal synthesis and population balance simulation... 

The energy laws of Bond, Kick, and Rittinger relate to grinding from some average feed size to some product size but do not take into account the behavior of different sizes of particles in the mill. Computer simulation, based on population-balance models [Bass, Z. Angew. Math. Phys., 5(4), 283 (1954)], traces the breakage of each size of particle as a function of grinding time. Furthermore, the simu-... 

Ramkrishna, D., 1981. Analysis of population balance - IV. The precise connection between Monte Carlo simulation and population balances. Chemical Engineering Science, 36, 1203-1209. 

To our knowledge, this is the first time that an emulsion copolymerization model has been developed based on a population balance approach. The resulting differential equations are more involved and complex than those of the homopolymer case. Lack of experimental literature data for the specific system VCM/VAc made it impossible to directly check the model s predictive powers, however, successful simulation of extreme cases and reasonable trends obtained in the model s predictions are convincing enough about the validity and usefulness of the mathematical model per se. 

Just like in the context of simulating solids suspension, one may wonder whether much may be expected from just sticking to the two-fluid approach combined with population balances. A better way ahead might rather be to combine population balances with LES, while proper relations for the various kernels used for describing coalescence and break-up processes could be determined from DNS of periodic boxes comprising a certain number of bubbles (or drops). The latter simulations would serve to study the detailed response of bubbles or drops to the ambient turbulent flow. 

A prior model was described by Marshall et. al. (36), but this did not Include the hemlhydrate population balance. A version of the present model was given by Steemson et. al. (37), centred around a simulation package developed for alumina plant modelling. This version Included hemlhydrate dissolution but used an assumed correlation for the dissolution rate. 

While mechanistic simulators, based on the population balance and other methods, are being developed, it is appropriate to test the abilities of conventional simulators to match data from laboratory mobility control experiments. The chapter by Claridge, Lescure, and Wang describes mobility control experiments (which use atmospheric pressure emulsions scaled to match miscible-C02 field conditions) and attempts to match the data with a widely used field simulator that does not contain specific mechanisms for surfactant-based mobility control. Chapter 21, by French, also describes experiments on emulsion flow, including experiments at elevated temperatures. 

Coalescence and redispersion models applied to these reaction systems include population balance equations, Monte Carlo simulation techniques, and a combination of macromixing and micromixing concepts with Monte Carlo simulations. Most of the last two types of models were developed to... 

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. 

These models require information about mean velocity and the turbulence field within the stirred vessels. Computational flow models can be developed to provide such fluid dynamic information required by the reactor models. Although in principle, it is possible to solve the population balance model equations within the CFM framework, a simplified compartment-mixing model may be adequate to simulate an industrial reactor. In this approach, a CFD model is developed to establish the relationship between reactor hardware and the resulting fluid dynamics. This information is used by a relatively simple, compartment-mixing model coupled with a population balance model (Vivaldo-Lima et al., 1998). The approach is shown schematically in Fig. 9.2. Detailed polymerization kinetics can be included. Vivaldo-Lima et a/. (1998) have successfully used such an approach to predict particle size distribution (PSD) of the product polymer. Their two-compartment model was able to capture the bi-modal behavior observed in the experimental PSD data. After adequate validation, such a computational model can be used to optimize reactor configuration and operation to enhance reactor performance. 

Instead of arbitrarily considering two bubble classes, it may be useful to incorporate a coalescence break-up model based on the population balance framework in the CFD model (see for example, Carrica et al., 1999). Such a model will simulate the evolution of bubble size distribution within the column and will be a logical extension of previously discussed models to simulate flow in bubble columns with wide bubble size distribution. Incorporation of coalescence break-up models, however, increases computational requirements by an order of magnitude. For example, a two-fluid model with a single bubble size generally requires solution of ten equations (six momentum, pressure, dispersed phase continuity and two turbulence characteristics). A ten-bubble class model requires solution of 46 (33 momentum, pressure. 

Lo [51] simulated two- and three phase isothermal non-reacting stirred tanks with two downward pumping 45o pitched -blade disc turbines and one curved-blade impeller at the bottom. Four or six baffles were placed at equal distance around the vessel wall. An Eulerian multiphase-population balance (MUltiple-SIze-Group, MUSIG) model was used as implemented in CFX. Turbulence of the continuous phase was modeled by the standard k-e turbulence model, and an algebraic relation was used for the particle induced eddy vis-... 

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