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2020 | OriginalPaper | Chapter

15. Compartmental Population Balances by Means of Monte Carlo Methods

Authors : Gregor Kotalczyk, Frank Einar Kruis

Published in: Dynamic Flowsheet Simulation of Solids Processes

Publisher: Springer International Publishing

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Abstract

Stochastic simulation techniques for the solution of a network of population balance equations (PBE) are discussed in this chapter. The application of weighted Monte Carlo (MC) particles for the solution of compartmental PBE systems is summarized and its computational efficacy in form of a parallel GPU implementation is pointed out. Solution strategies for coagulation, nucleation, breakage, growth and evaporation are thereby presented. An application example treats the simultaneous coagulation, nucleation, evaporation and growth encountered during particle production through the aerosol route. Furthermore, the simulation of a compartmental network is discussed and parallel simulation techniques for the transport of weighted MC particles are presented. The proposed methodology is benchmarked by comparison with a pivot method for a variety of test cases with an increasing degree of complexity. Simulation conditions are identified, for which conventional, non-weighted MC simulation techniques are not applicable. It is found, that the specific combination of a screen unit with tear-streams cannot be simulated by conventional methods, termed ‘random removal’, and make thus other techniques—like the here introduced merging techniques necessary.

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previous chapter Numerical Methods for Coupled Population Balance Systems Applied to the Dynamical Simulation of Crystallization Processes

next chapter Modeling, Simulation and Optimization of Process Chains

A constant nucleation rate \(R_{N}\) is assumed, so that newly introduced simulation entries have the weight \(W_{0} = R_{N} \cdot \Delta \tau_{MC}\) and a predefined diameter \(d_{0}\). For the simulation has been set: \(R_{N} = 10^{14} \,\frac{1}{{{\text{m}}^{3}\,{\text{s}}}}\), \(d_{0} = 3\,{\text{nm}}\). A monodisperse population with an initial concentration of \(10^{17} \,\frac{1}{{{\text{m}}^{3} }}\) has been used as start condition, the initial MC particles are equally weighted. The temperature was set to 300 K and the particle density to \(1\,\frac{\text{g}}{{{\text{cm}}^{3} }}\). The simulated time was ca. 25.8 s, which is 500 times the characteristic time needed to reach the self-preserving distribution [28] due to coagulation.

Special modeling is necessary in order to capture the changes of the ‘book kept’ entries due to other non-coagulation processes—this might or might not be possible, depending on the specific process being modelled. Additionally, the tracking of the changes might prove more expensive than the application of the time-driven MC methods without book-keeping.

In order to use the GPU efficiently, larger particle numbers (like e.g. 10000) have to be divided into data blocks consisting of e.g. 512 particle numbers. In the here presented implementation, particle numbers that are multiples of 512 are considered.

The four case studies comprise: (1) a flowsheet without recycle stream and screen units, (2) a flowsheet with one recycle stream and without sieve unit, (3) a flowsheet with one sieve unit and without recycle stream and (4) a flowsheet with a recycle stream and a sieve unit.

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Title: Compartmental Population Balances by Means of Monte Carlo Methods
Authors: Gregor Kotalczyk
Frank Einar Kruis
Publisher: Springer International Publishing
Book: Dynamic Flowsheet Simulation of Solids Processes
Print ISBN: 978-3-030-45167-7

Electronic ISBN: 978-3-030-45168-4

Copyright Year: 2020
DOI: https://doi.org/10.1007/978-3-030-45168-4_15