Chapter 15
Compartmental Population Balances
by Means of Monte Carlo Methods
Gregor Kotalczyk and Frank Einar Kruis
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.
Nomenclature
b
Breakage rate [s
−1 ]
d g
Geometric mean diameter [m]
C dist
Compare distance on GPU memory (integer) [–]
d
∗
Kelvin diameter [m]
d
Diameter of particle [m]
E i, j
Merging error of particles i and j [–]
G. Kotalczyk · F. E. Kruis (B)
Institute of Technology for Nanostructures (NST),
University of Duisburg-Essen, Duisburg, Germany
e-mail: einar.kruis@uni-due.de
G. Kotalczyk
e-mail: gregor.kotalczyk@uni-due.de
© Springer Nature Switzerland AG 2020
S. Heinrich (ed.), Dynamic Flowsheet Simulation of Solids Processes,
https://doi.org/10.1007/978-3-030-45168-4_15
519
Compartmental Population Balances
by Means of Monte Carlo Methods
Gregor Kotalczyk and Frank Einar Kruis
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.
Nomenclature
b
Breakage rate [s
−1 ]
d g
Geometric mean diameter [m]
C dist
Compare distance on GPU memory (integer) [–]
d
∗
Kelvin diameter [m]
d
Diameter of particle [m]
E i, j
Merging error of particles i and j [–]
G. Kotalczyk · F. E. Kruis (B)
Institute of Technology for Nanostructures (NST),
University of Duisburg-Essen, Duisburg, Germany
e-mail: einar.kruis@uni-due.de
G. Kotalczyk
e-mail: gregor.kotalczyk@uni-due.de
© Springer Nature Switzerland AG 2020
S. Heinrich (ed.), Dynamic Flowsheet Simulation of Solids Processes,
https://doi.org/10.1007/978-3-030-45168-4_15
519