15 Compartmental Population Balances by Means of Monte Carlo Methods
523
implementation of the coagulation in the framework of a compartmental network,
as well as the transport between single compartments is discussed in the section
afterwards, where the modelling of multiple compartments is applied.
2 Weighted Monte Carlo Particles for the Solution
of the Population Balance Equation
The use of weighted simulation particles (a particle with weight w represents w
real particles within a given reactor volume) has several advantages: it allows to
describe the interaction between simulation particles having different concentrations
coming from different cells or compartments [14, 20, 21]. It can also be used as a
tool to control the number of simulation particles (e.g. to gain numerical accuracy).
In the following, some techniques will be introduced to solve the single mechanisms
presented in Eq. (1) by the application of weighted MC particles.
2.1 Coagulation
The correct description of the coagulation rates for the weighted particles, especially
for a complex coagulation scheme, like the one introduced Zhao et al. [22] shown in
Fig. 2, poses a great difficulty.
The authors [22] presented the ‘ficticious particle theory’ which leads to the
following modified coagulation kernel:
β
( f p)
=
2W j max
W i , W j
W i + W j
β
(3)
The weights of the particles are denoted by W i and W j . The coagulation kernel β
describes the coagulation of the original (non-weighted) system—which might be
the Brownian kernel for the free molecular regime, etc. The resulting coagulation rate
is asymmetric, making a distinction necessary, whether particle i coagulates with j
or vice versa. This definition is 1) difficult to understand conceptually and 2) difficult
to extend on other process—like nucleation or transport of fictitious particles.
We developed in [23] the concept of the stochastic resolution which describes
each coagulation in the frame-work of equally weighted MC-particles, where each
MC-particle describes s f real particles. The value for the parameter s f can be set
arbitrarily. Figure 2 shows that the setting s f = W min = min
W i , W j
leads to the
correct description of the coagulation-scheme. The scaling factor s f depends on the
chosen coagulation pair, so that different coagulation-events are described in different
stochastic resolutions.
523
implementation of the coagulation in the framework of a compartmental network,
as well as the transport between single compartments is discussed in the section
afterwards, where the modelling of multiple compartments is applied.
2 Weighted Monte Carlo Particles for the Solution
of the Population Balance Equation
The use of weighted simulation particles (a particle with weight w represents w
real particles within a given reactor volume) has several advantages: it allows to
describe the interaction between simulation particles having different concentrations
coming from different cells or compartments [14, 20, 21]. It can also be used as a
tool to control the number of simulation particles (e.g. to gain numerical accuracy).
In the following, some techniques will be introduced to solve the single mechanisms
presented in Eq. (1) by the application of weighted MC particles.
2.1 Coagulation
The correct description of the coagulation rates for the weighted particles, especially
for a complex coagulation scheme, like the one introduced Zhao et al. [22] shown in
Fig. 2, poses a great difficulty.
The authors [22] presented the ‘ficticious particle theory’ which leads to the
following modified coagulation kernel:
β
( f p)
=
2W j max
W i , W j
W i + W j
β
(3)
The weights of the particles are denoted by W i and W j . The coagulation kernel β
describes the coagulation of the original (non-weighted) system—which might be
the Brownian kernel for the free molecular regime, etc. The resulting coagulation rate
is asymmetric, making a distinction necessary, whether particle i coagulates with j
or vice versa. This definition is 1) difficult to understand conceptually and 2) difficult
to extend on other process—like nucleation or transport of fictitious particles.
We developed in [23] the concept of the stochastic resolution which describes
each coagulation in the frame-work of equally weighted MC-particles, where each
MC-particle describes s f real particles. The value for the parameter s f can be set
arbitrarily. Figure 2 shows that the setting s f = W min = min
W i , W j
leads to the
correct description of the coagulation-scheme. The scaling factor s f depends on the
chosen coagulation pair, so that different coagulation-events are described in different
stochastic resolutions.