15 Compartmental Population Balances by Means of Monte Carlo Methods
535
3.1 Time Step Control for Compartmental PBE Networks
The combination of simultaneous processes rendered by the PBE solver poses a
challenging task, hence the characteristic time-scales for the corresponding processes
may differ in several orders of magnitude and change vastly during the simulation.
Although the developed algorithms for the breakage, coagulation and growth (resp.
evaporation) already adapt to the optimal time-step for each single process (in the
absence of other processes), the simulation of the combined coagulation, nucleation
and growth is driven by the discrete coagulation processes providing an inherent MC
time step τ MC . This is a computationally advantageous setting, if the inherent growth
step τ G is smaller than the coagulation step τ MC . The opposite case, in which τ G is
bigger than τ MC , forces the simulation to use much smaller time steps τ
G and thus to
use much more computationally demanding growth steps. The incorporation of the
breakage as a third process with an inherent time step τ B may force the simulation
to reset this step to a lower value τ
B in a coagulation driven implementation, as well.
This situation becomes even more complicated, if the PBE is solved for different compartments and a particulate flow between the compartments is simulated.
Consider, for example two compartments, in which coagulation takes place, so that
compartment 1 has an intrinsic MC coagulation time step τ
(C1)
MC , while the second
compartment has the inherent coagulation time step τ
(C2)
MC , the additional particulate
flowrate between compartment 1 and compartment 2 might make the update of the
particle populations due to the transport of particles each time interval τ Tr necessary.
It depends on the specific modelled application, whether the minimum of these three
time steps has to be used, or some other minimal time step τ min can be applied. In
all cases, there exists the need, to perform, at least in one of the two compartments,
a smaller time step than the one intrinsically provided (τ
(C1)
MC and/or τ
(C2)
MC ).
The application of time-driven MC methods [48] allows the setting of a variable
time step, but this time step has to be set proportional to the intrinsic step in order
to avoid systematical errors [49], although smaller values are allowed. The main
disadvantage of this approach is, however, its computational costs, hence all possible
coagulation pairs have to be checked for coagulation during the suggested time step—
special book-keeping methods [50] might help to address this problem for the single
simulation of coagulation but their usage is not possible
2 in the context of a PBE
network modelling multiple simultaneous processes.
We have developed the concept of ‘fractional MC time steps’ [51], in order to
address this problem. In the scope of this approach, we modified the fast GPU
acceptance-rejection algorithm [25] in such a way, that an additional stochastic probability is formulated, whether the particles coagulate or not if a smaller time step than
the intrinsic MC AR-time step is needed. We compared our methodology (marked
2 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.
535
3.1 Time Step Control for Compartmental PBE Networks
The combination of simultaneous processes rendered by the PBE solver poses a
challenging task, hence the characteristic time-scales for the corresponding processes
may differ in several orders of magnitude and change vastly during the simulation.
Although the developed algorithms for the breakage, coagulation and growth (resp.
evaporation) already adapt to the optimal time-step for each single process (in the
absence of other processes), the simulation of the combined coagulation, nucleation
and growth is driven by the discrete coagulation processes providing an inherent MC
time step τ MC . This is a computationally advantageous setting, if the inherent growth
step τ G is smaller than the coagulation step τ MC . The opposite case, in which τ G is
bigger than τ MC , forces the simulation to use much smaller time steps τ
G and thus to
use much more computationally demanding growth steps. The incorporation of the
breakage as a third process with an inherent time step τ B may force the simulation
to reset this step to a lower value τ
B in a coagulation driven implementation, as well.
This situation becomes even more complicated, if the PBE is solved for different compartments and a particulate flow between the compartments is simulated.
Consider, for example two compartments, in which coagulation takes place, so that
compartment 1 has an intrinsic MC coagulation time step τ
(C1)
MC , while the second
compartment has the inherent coagulation time step τ
(C2)
MC , the additional particulate
flowrate between compartment 1 and compartment 2 might make the update of the
particle populations due to the transport of particles each time interval τ Tr necessary.
It depends on the specific modelled application, whether the minimum of these three
time steps has to be used, or some other minimal time step τ min can be applied. In
all cases, there exists the need, to perform, at least in one of the two compartments,
a smaller time step than the one intrinsically provided (τ
(C1)
MC and/or τ
(C2)
MC ).
The application of time-driven MC methods [48] allows the setting of a variable
time step, but this time step has to be set proportional to the intrinsic step in order
to avoid systematical errors [49], although smaller values are allowed. The main
disadvantage of this approach is, however, its computational costs, hence all possible
coagulation pairs have to be checked for coagulation during the suggested time step—
special book-keeping methods [50] might help to address this problem for the single
simulation of coagulation but their usage is not possible
2 in the context of a PBE
network modelling multiple simultaneous processes.
We have developed the concept of ‘fractional MC time steps’ [51], in order to
address this problem. In the scope of this approach, we modified the fast GPU
acceptance-rejection algorithm [25] in such a way, that an additional stochastic probability is formulated, whether the particles coagulate or not if a smaller time step than
the intrinsic MC AR-time step is needed. We compared our methodology (marked
2 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.