492
R. Ahrens et al.
After having performed this jump, the crystal e j is removed from the ensemble and
the crystal ˜
e i has to be placed in an appropriate way in the ensemble, i.e., one has to
assign an appropriate position to ˜
e i . For designing a stable method, it is proposed in
[52] to choose the new position y of a crystal that emerged from coagulation of the
crystals
m i , x i
and
m j , x j
stochastically, distributed according to the probabilities
P(y = x i ) =
m i
m i + m j
, P(y = x j ) =
m j
m i + m j
,
i.e., to use the center of mass in the probabilities. Similarly as for the growth, the
total rate of agglomeration jumps is the sum of all individual agglomeration jump
rates of pairs of crystals
λ aggl (E) =
1
2N E
N E
i,j=1
κ agg (m i , m j ).
The involvement of two crystals in an agglomeration jump is random with the probabilities
P(e i and e j chosen for agglomeration) =
κ agg (m i , m j )
2N E
.
Ensembles of crystals might be changed also by insertion of crystals in the flow
domain. This process affects usually only a few mesh cells. Crystal insertion is modeled by so-called inception jumps, i.e., each ensemble in mesh cells, where crystals
are injected, is equipped with an additional jump rate λ in (E) and a corresponding
jump, which adds a new crystal to the ensemble.
2.4.3 Coupled Simulation with a Splitting Scheme
Our basic approach for developing a code for solving the PBS (1.1)–(1.4) numerically
consisted in coupling two separate codes: one designed for simulating the Computational Fluid Dynamic (CFD) equations (1.1)–(1.3) with deterministic methods, and
the other one designed for simulating crystal interactions with stochastic methods.
For this purpose, we used the in-house codes ParMooN [24, 68] for the CFD part
and Brush [53] for the SPS part.
The complete simulation procedure is sketched in Fig. 6. In each time instance,
first the CFD equations are solved and then the population balance equation (1.4) with
the SPS method. In order to couple the two codes, an interface was developed and
implemented that is responsible for the data transfer between the codes. Other major
extensions of Brush that were necessary include the simulation of the transport of
the crystals in three dimensions, the implementation of the sedimentation model, the
implementation of crystal-wall interactions, and the implementation of routines for
assigning the crystals to mesh cells. For more details, it is referred to [4].
R. Ahrens et al.
After having performed this jump, the crystal e j is removed from the ensemble and
the crystal ˜
e i has to be placed in an appropriate way in the ensemble, i.e., one has to
assign an appropriate position to ˜
e i . For designing a stable method, it is proposed in
[52] to choose the new position y of a crystal that emerged from coagulation of the
crystals
m i , x i
and
m j , x j
stochastically, distributed according to the probabilities
P(y = x i ) =
m i
m i + m j
, P(y = x j ) =
m j
m i + m j
,
i.e., to use the center of mass in the probabilities. Similarly as for the growth, the
total rate of agglomeration jumps is the sum of all individual agglomeration jump
rates of pairs of crystals
λ aggl (E) =
1
2N E
N E
i,j=1
κ agg (m i , m j ).
The involvement of two crystals in an agglomeration jump is random with the probabilities
P(e i and e j chosen for agglomeration) =
κ agg (m i , m j )
2N E
.
Ensembles of crystals might be changed also by insertion of crystals in the flow
domain. This process affects usually only a few mesh cells. Crystal insertion is modeled by so-called inception jumps, i.e., each ensemble in mesh cells, where crystals
are injected, is equipped with an additional jump rate λ in (E) and a corresponding
jump, which adds a new crystal to the ensemble.
2.4.3 Coupled Simulation with a Splitting Scheme
Our basic approach for developing a code for solving the PBS (1.1)–(1.4) numerically
consisted in coupling two separate codes: one designed for simulating the Computational Fluid Dynamic (CFD) equations (1.1)–(1.3) with deterministic methods, and
the other one designed for simulating crystal interactions with stochastic methods.
For this purpose, we used the in-house codes ParMooN [24, 68] for the CFD part
and Brush [53] for the SPS part.
The complete simulation procedure is sketched in Fig. 6. In each time instance,
first the CFD equations are solved and then the population balance equation (1.4) with
the SPS method. In order to couple the two codes, an interface was developed and
implemented that is responsible for the data transfer between the codes. Other major
extensions of Brush that were necessary include the simulation of the transport of
the crystals in three dimensions, the implementation of the sedimentation model, the
implementation of crystal-wall interactions, and the implementation of routines for
assigning the crystals to mesh cells. For more details, it is referred to [4].