14 Numerical Methods for Coupled Population Balance Systems …
489
2.4 A Stochastic Method for Simulating the Crystal Size
Distribution
This section describes a stochastic particle simulation (SPS) method for computing a
numerical approximation of the crystal size distribution f whose behavior is modeled
by the population balance equation (1.4). This method can be applied successfully
for the simulation of problems given in complex spatial domains.
The basis of the SPS method that is utilized in our simulations is the method
proposed in [52, 53]. This method had to be extended by all features that are caused
from the movement of the crystals in the spatial domain: convective transport in
three dimensions, sedimentation, crystal-wall collisions, and the coupling with the
deterministic methods for solving the other equations (1.1)–(1.3) of the PBS. The
algorithms from [52, 53] include convective transport in one dimension, growth,
and coagulation (collision growth). With respect to the first and third feature, the
method is based on two classical algorithms. The first one is Bird’s direct simulation
Monte-Carlo algorithm for the Boltzmann equation [8] that proposes an approach to
handle the convective transport part with a splitting method. The second algorithm
is the Gillespie algorithm [28, 29] that models the coagulation via stochastic jump
processes. One of the original contributions from [53] is a stochastic algorithm for
simulating crystal growth via a surface reaction model.
Altogether, the splitting scheme applied in the SPS method consists of two parts:
the convective transport of crystals, discussed in Sect. 2.4.1, and Markov jump processes for simulating growth, agglomeration, and insertion of crystals, described in
Sect. 2.4.2. Section 2.4.3 presents the complete algorithm that simulates the PBSs
(1.1)–(1.4).
2.4.1 Convective Transport of Stochastic Computational Crystals
The spatial domain Ω is triangulated by a triangulation consisting of mesh cells
K j , j ∈ {1, . . . , N }. Each mesh cell contains a crystal ensemble E j . In the stochastic
method, computational crystals (particles) are considered that represent an ensemble
of physical crystals (particles). For simplifying the notion, the computational crystals
will be called just ‘crystals’ in the following.
Consider a spatial mesh cell K and a crystal ensemble (K, E), where each crystal
e i in E possesses a spatial and an internal coordinate e i = (x i , m i ), with x i ∈ K and
m i ∈ Ω int . The complete ensemble E with N E crystals is given by E = (e 1 , . . . , e N E ).
Let Δt be a constant splitting time. First of all, the flow field u from the Navier–
Stokes equations (1.1) is responsible for the transport of crystals. Second, crystals are
also moved by sedimentation with the sedimentation velocity u sed . In the convection
step, each crystal e i is transported along the trajectories of u + u sed
x i −→ x i + Δt (u(x i ) + u sed (x i )) .
(2.8)
489
2.4 A Stochastic Method for Simulating the Crystal Size
Distribution
This section describes a stochastic particle simulation (SPS) method for computing a
numerical approximation of the crystal size distribution f whose behavior is modeled
by the population balance equation (1.4). This method can be applied successfully
for the simulation of problems given in complex spatial domains.
The basis of the SPS method that is utilized in our simulations is the method
proposed in [52, 53]. This method had to be extended by all features that are caused
from the movement of the crystals in the spatial domain: convective transport in
three dimensions, sedimentation, crystal-wall collisions, and the coupling with the
deterministic methods for solving the other equations (1.1)–(1.3) of the PBS. The
algorithms from [52, 53] include convective transport in one dimension, growth,
and coagulation (collision growth). With respect to the first and third feature, the
method is based on two classical algorithms. The first one is Bird’s direct simulation
Monte-Carlo algorithm for the Boltzmann equation [8] that proposes an approach to
handle the convective transport part with a splitting method. The second algorithm
is the Gillespie algorithm [28, 29] that models the coagulation via stochastic jump
processes. One of the original contributions from [53] is a stochastic algorithm for
simulating crystal growth via a surface reaction model.
Altogether, the splitting scheme applied in the SPS method consists of two parts:
the convective transport of crystals, discussed in Sect. 2.4.1, and Markov jump processes for simulating growth, agglomeration, and insertion of crystals, described in
Sect. 2.4.2. Section 2.4.3 presents the complete algorithm that simulates the PBSs
(1.1)–(1.4).
2.4.1 Convective Transport of Stochastic Computational Crystals
The spatial domain Ω is triangulated by a triangulation consisting of mesh cells
K j , j ∈ {1, . . . , N }. Each mesh cell contains a crystal ensemble E j . In the stochastic
method, computational crystals (particles) are considered that represent an ensemble
of physical crystals (particles). For simplifying the notion, the computational crystals
will be called just ‘crystals’ in the following.
Consider a spatial mesh cell K and a crystal ensemble (K, E), where each crystal
e i in E possesses a spatial and an internal coordinate e i = (x i , m i ), with x i ∈ K and
m i ∈ Ω int . The complete ensemble E with N E crystals is given by E = (e 1 , . . . , e N E ).
Let Δt be a constant splitting time. First of all, the flow field u from the Navier–
Stokes equations (1.1) is responsible for the transport of crystals. Second, crystals are
also moved by sedimentation with the sedimentation velocity u sed . In the convection
step, each crystal e i is transported along the trajectories of u + u sed
x i −→ x i + Δt (u(x i ) + u sed (x i )) .
(2.8)