15 Compartmental Population Balances by Means of Monte Carlo Methods
533
10
9 simulation particles with a diameter of 0.5 μm, if all particles represent the same
number concentration of real particles.
Traditional solutions of this problem encompass the discretization of the particle
property into bins [39], which would render the combination with the developed
growth/evaporation algorithm impossible, and constant-number approaches [40],
which are known to produce a high level of statistical noise [41].The application
of merging schemes [42] (i.e. the approximation of the properties of several simulation particles by one simulation particle) would pose an attractive alternative.
However, the vast amount of newly resulting particles makes a lot of merging steps
necessary, leading to large computing times. A recent constant-number method has
been presented [41], which renders the breakage event of one particle into many fragments by a single particle. The size of the fragment is selected stochastically, the use
of many simulation particles leads to the correct distribution of fragment sizes. This
scheme is only able to render the parts of the particle size distribution which represent
high number concentrations of the particles. Furthermore, only binary breakage can
be described.
We developed an alternative approach [43] for the derivation of the breakage
scheme by resorting to the argument that the breakage of many MC-particles with
equal properties and weights has to lead to a particle size distribution (PSD) of
fragments, which is described by the corresponding breakage kernel. It allows to
formulate any probability distribution function (PDF) with which the new volume of
the simulation particle is selected by adjusting the statistical weight of the resulting
fragments depending on 1) the selected particle properties, 2) the used PDF and 3)
the given breakage density function. This newly proposed scheme encompasses the
already introduced SWA schemes, especially a number-based (NB, named SWA1
in [41]) and volume-based (VB, named SWA2 in [41]) breakage scheme, and it
makes novel formulations possible: the low volume scheme (LV), which renders
preferably fragment particle sizes at the lower end of the size spectrum, and the
combination of LV with the NB (NB-LV) or VB (VB-LV). Exemplary simulation
results are shown in Fig. 7. It can be seen that the SWA methods (NB and VB) are
only able to render large particle sizes, and that LV, NB-LV and VB-LV are able to
render the whole spectrum of particle sizes. Smaller noise levels are found for VB
and specific VB-LV schemes, making both more suitable for prolonged simulations
than the other presented methods. The LV based simulation method fails to predict
the correct PSDs for longer simulation times. For this reason, the combinations of
LV with VB or NB are needed, in order to ensure the correct shapes of the PSDs for
longer simulation times. The combination ratio R ∈ (0, 1) between the LV and NB
leads to different schemes, while lower ratios R lead to a higher representation of
low-volume MC particles, they also lead to higher noise levels: the setting R = 0.6
leads to more statistical noise than R = 0.9, as it can be seen in Fig. 7. The adaptive
resetting of the factor R in order to avoid the systematical errors, as it is shown in
Fig. 7 for the LV scheme is briefly discussed in [44].
The required simulation times are listed in Table 3, 10
5 simulation particles are
required in order to ascertain a computational accuracy of less than 1%. (I.e. the
533
10
9 simulation particles with a diameter of 0.5 μm, if all particles represent the same
number concentration of real particles.
Traditional solutions of this problem encompass the discretization of the particle
property into bins [39], which would render the combination with the developed
growth/evaporation algorithm impossible, and constant-number approaches [40],
which are known to produce a high level of statistical noise [41].The application
of merging schemes [42] (i.e. the approximation of the properties of several simulation particles by one simulation particle) would pose an attractive alternative.
However, the vast amount of newly resulting particles makes a lot of merging steps
necessary, leading to large computing times. A recent constant-number method has
been presented [41], which renders the breakage event of one particle into many fragments by a single particle. The size of the fragment is selected stochastically, the use
of many simulation particles leads to the correct distribution of fragment sizes. This
scheme is only able to render the parts of the particle size distribution which represent
high number concentrations of the particles. Furthermore, only binary breakage can
be described.
We developed an alternative approach [43] for the derivation of the breakage
scheme by resorting to the argument that the breakage of many MC-particles with
equal properties and weights has to lead to a particle size distribution (PSD) of
fragments, which is described by the corresponding breakage kernel. It allows to
formulate any probability distribution function (PDF) with which the new volume of
the simulation particle is selected by adjusting the statistical weight of the resulting
fragments depending on 1) the selected particle properties, 2) the used PDF and 3)
the given breakage density function. This newly proposed scheme encompasses the
already introduced SWA schemes, especially a number-based (NB, named SWA1
in [41]) and volume-based (VB, named SWA2 in [41]) breakage scheme, and it
makes novel formulations possible: the low volume scheme (LV), which renders
preferably fragment particle sizes at the lower end of the size spectrum, and the
combination of LV with the NB (NB-LV) or VB (VB-LV). Exemplary simulation
results are shown in Fig. 7. It can be seen that the SWA methods (NB and VB) are
only able to render large particle sizes, and that LV, NB-LV and VB-LV are able to
render the whole spectrum of particle sizes. Smaller noise levels are found for VB
and specific VB-LV schemes, making both more suitable for prolonged simulations
than the other presented methods. The LV based simulation method fails to predict
the correct PSDs for longer simulation times. For this reason, the combinations of
LV with VB or NB are needed, in order to ensure the correct shapes of the PSDs for
longer simulation times. The combination ratio R ∈ (0, 1) between the LV and NB
leads to different schemes, while lower ratios R lead to a higher representation of
low-volume MC particles, they also lead to higher noise levels: the setting R = 0.6
leads to more statistical noise than R = 0.9, as it can be seen in Fig. 7. The adaptive
resetting of the factor R in order to avoid the systematical errors, as it is shown in
Fig. 7 for the LV scheme is briefly discussed in [44].
The required simulation times are listed in Table 3, 10
5 simulation particles are
required in order to ascertain a computational accuracy of less than 1%. (I.e. the