15 Compartmental Population Balances by Means of Monte Carlo Methods
525
The second case W max = W min , which is not shown in Fig. 2, is also described
with the resolution s f = W min . Both particle species are described by only one MCparticle, so that only one MC-particle can be found after the coagulation. In order to
apply the constant number scheme, the weight of the simulation entry representing
this particle is divided by two and the particle properties are stored in both positions.
The thus derived coagulation kernel β
(w) is easier to calculate than the originally introduced β
( f p) —a speed up of the simulation up to 10% could be noticed.
Due to its symmetric form, computational advantages for the implementation of the
inverse method can be expected, as only half of the computations of the β
(w) kernel are necessary. The simulation results of particle coagulation for the newly estimated coagulation kernel β
(w) could be found to be as accurate as the β
( f p) -kernel
results (which show excellent agreement with the solution produced by means of the
Discrete-sectional-method in the first place) within the MC-stochastic noise [23].
2.2 Nucleation
Homogeneous nucleation is a mechanism that leads to the formation of new particles,
which have to be included among the simulation entries. Constant number simulationschemes sum up all possible algorithms, which update—somehow—the simulation
properties, but keep the number of the used simulation entries constant. Keeping the
number of simulation entries constant ensures a constant level of stochastic accuracy
and makes a simple prediction of needed computational resources possible. Figure 3
shows possible constant-number nucleation algorithms. They can be used to model
the inclusion of the nucleation particles or particles included by other processes: like
breakage or transport.
The random removal algorithm has been introduced Lin et al. [24] in the framework of the concept of a ‘constant number Monte Carlo simulation’ which is based
on the not-weighted particle scheme. The algorithms applying the merging step are
based on the weighted-particles scheme and the concept of the merging error. They
cannot be used for non-weighted MC simulations. The merge-List is created each
100 merge-steps and contains 100 simulation entries with low-weights.
2.2.1 Merging
The concept of ‘merging’ of simulation entries is proposed in [23]: if two simulation
entries with exactly the same properties are merged, the resulting representation of
the particle size distribution will not change and all the physical processes will be
described in the same way. If the simulation entries differ slightly in their properties,
a small error will be introduced.
525
The second case W max = W min , which is not shown in Fig. 2, is also described
with the resolution s f = W min . Both particle species are described by only one MCparticle, so that only one MC-particle can be found after the coagulation. In order to
apply the constant number scheme, the weight of the simulation entry representing
this particle is divided by two and the particle properties are stored in both positions.
The thus derived coagulation kernel β
(w) is easier to calculate than the originally introduced β
( f p) —a speed up of the simulation up to 10% could be noticed.
Due to its symmetric form, computational advantages for the implementation of the
inverse method can be expected, as only half of the computations of the β
(w) kernel are necessary. The simulation results of particle coagulation for the newly estimated coagulation kernel β
(w) could be found to be as accurate as the β
( f p) -kernel
results (which show excellent agreement with the solution produced by means of the
Discrete-sectional-method in the first place) within the MC-stochastic noise [23].
2.2 Nucleation
Homogeneous nucleation is a mechanism that leads to the formation of new particles,
which have to be included among the simulation entries. Constant number simulationschemes sum up all possible algorithms, which update—somehow—the simulation
properties, but keep the number of the used simulation entries constant. Keeping the
number of simulation entries constant ensures a constant level of stochastic accuracy
and makes a simple prediction of needed computational resources possible. Figure 3
shows possible constant-number nucleation algorithms. They can be used to model
the inclusion of the nucleation particles or particles included by other processes: like
breakage or transport.
The random removal algorithm has been introduced Lin et al. [24] in the framework of the concept of a ‘constant number Monte Carlo simulation’ which is based
on the not-weighted particle scheme. The algorithms applying the merging step are
based on the weighted-particles scheme and the concept of the merging error. They
cannot be used for non-weighted MC simulations. The merge-List is created each
100 merge-steps and contains 100 simulation entries with low-weights.
2.2.1 Merging
The concept of ‘merging’ of simulation entries is proposed in [23]: if two simulation
entries with exactly the same properties are merged, the resulting representation of
the particle size distribution will not change and all the physical processes will be
described in the same way. If the simulation entries differ slightly in their properties,
a small error will be introduced.