14 Numerical Methods for Coupled Population Balance Systems …
491
2.4.2 Modeling of Growth, Coagulation, and Crystal Insertion by
Markov Jump Processes
The crystals are allowed to interact with each other only within their current ensemble.
In particular, crystals do not have to meet in the same point in space in order to
agglomerate, it is enough for them to be contained in a common mesh cell.
Growth, agglomeration, and insertion of crystals are modeled with Markov jump
processes. These processes are described in this section, following [52, 53], in terms
of the so-called ‘stochastic weighted algorithm’. For further technical details, it is
referred to [52, 53].
Starting at some time t ∈ [0, t end ), the system stays in the state E(t) for an exponentially distributed waiting time τ , P(τ ≥ s) = exp(−λ(E)s). Here, λ(E) is the
waiting time parameter that is the sum of the individual rates of all jumps that are
possible in E(t). This parameter is the sum of the growth jump rate λ grow (E) and the
agglomeration jump rate λ aggl (E):
λ(E) = λ grow (E) + λ aggl (E).
First, the simulation of crystal growth will be described. The growth term as it
stands in the population balance equation (1.4) is a transport term along the internal
coordinate. The rationale behind a stochastic simulation of this term by Markov
jump processes is the interpretation of crystal growth as crystal surface growth via
a chemical reaction. One can derive a relation between the growth rate G(c, T ) and
the corresponding reaction rate, e.g., see [5]. A crystal growth jump has an impact
on just one crystal e i . Given a growth height Δm i , the state of e i is changed by
e i = (x i , m i ) −→ (x i , m i + Δm i ) =: ˜
e i .
The crystal e j for which the next growth jump occurs is chosen with the probability
G(c, T , m j )
Δm i
λ grow (E)
−1 .
(2.10)
In our implementation of the SPS method, c and T are assumed to be constant in K
in expression (2.10). The total rate for the growth jumps in E is given by
λ grow (E) =
N E
i=1
G(c, T , m i )
Δm i
.
In agglomeration jumps, two crystals e i and e j , with i < j, are involved. Such a
jump has the form
e i , e j −→ (ξ(x i , x j ), m i + m j ) =: ˜
e i .
491
2.4.2 Modeling of Growth, Coagulation, and Crystal Insertion by
Markov Jump Processes
The crystals are allowed to interact with each other only within their current ensemble.
In particular, crystals do not have to meet in the same point in space in order to
agglomerate, it is enough for them to be contained in a common mesh cell.
Growth, agglomeration, and insertion of crystals are modeled with Markov jump
processes. These processes are described in this section, following [52, 53], in terms
of the so-called ‘stochastic weighted algorithm’. For further technical details, it is
referred to [52, 53].
Starting at some time t ∈ [0, t end ), the system stays in the state E(t) for an exponentially distributed waiting time τ , P(τ ≥ s) = exp(−λ(E)s). Here, λ(E) is the
waiting time parameter that is the sum of the individual rates of all jumps that are
possible in E(t). This parameter is the sum of the growth jump rate λ grow (E) and the
agglomeration jump rate λ aggl (E):
λ(E) = λ grow (E) + λ aggl (E).
First, the simulation of crystal growth will be described. The growth term as it
stands in the population balance equation (1.4) is a transport term along the internal
coordinate. The rationale behind a stochastic simulation of this term by Markov
jump processes is the interpretation of crystal growth as crystal surface growth via
a chemical reaction. One can derive a relation between the growth rate G(c, T ) and
the corresponding reaction rate, e.g., see [5]. A crystal growth jump has an impact
on just one crystal e i . Given a growth height Δm i , the state of e i is changed by
e i = (x i , m i ) −→ (x i , m i + Δm i ) =: ˜
e i .
The crystal e j for which the next growth jump occurs is chosen with the probability
G(c, T , m j )
Δm i
λ grow (E)
−1 .
(2.10)
In our implementation of the SPS method, c and T are assumed to be constant in K
in expression (2.10). The total rate for the growth jumps in E is given by
λ grow (E) =
N E
i=1
G(c, T , m i )
Δm i
.
In agglomeration jumps, two crystals e i and e j , with i < j, are involved. Such a
jump has the form
e i , e j −→ (ξ(x i , x j ), m i + m j ) =: ˜
e i .