486
R. Ahrens et al.
The principal form of the equations to be solved is as follows. Let ˆ
f n = f n−1 , solve
in the first step
∂ t ˆ
f +
u n + u sed,n−1
· ∇ ˆ
f = 0 in (t n−1 , t n ) × Ω
(2.2)
for all y ∈ Ω int . Then, set ˜
f n−1 = ˆ
f n , solve
∂ t ˜
f + ∂ m
G(c n , T n ) ˜
f
= F agg (u n , c n , T n , ˆ
f n ) + F break (u n , c n , T n , ˆ
f n ) in (t n−1 , t n ) × Ω int (2.3)
for all x ∈ Ω, and set f n = ˜
f n . There are several modifications of this basic operatorsplitting scheme for population balance equations, in particular to perform the steps
in a different order, e.g., see [3, 25, 27]. Equation (2.2) is usually a transport equation
with dominating convection, such that one has to utilize a stabilized discretization,
see Sect. 2.3.2. Also (2.3) is a transport equation, but the growth of the crystals
might be sufficiently slow such that one can apply some standard discretization. The
operator splitting introduces an additional splitting error which does not spoil the
optimal order of convergence for low order finite element methods [25].
As already mentioned at the beginning of this section, the definition of the population balance equation (1.4) for the crystal size distribution in a higher-dimensional
domain is a major challenge for the simulation of population balance systems. A
popular way to avoid this issue is the consideration of the first moments of the crystal size distribution, as proposed the first time in [33], where the so-called Method
of Moments (MOM) was derived. The kth moment of the crystal size distribution is
given by
M k =
∞
0
m
k f dm, k = 0, 1, 2, . . . .
(2.4)
It will be assumed, that f is zero for m ≤ m min and m ≥ m max , i.e., that there are
a minimal and a maximal mass for the crystals. Hence, the domain of integration
in (2.4) can be restricted to this interval. On the one hand, the first moments are
often important in practice because they correspond to physical quantities, like the
number of crystals (0th moment) or the mass of crystals (3rd moment). But on the
other hand, the reconstruction of the crystal size distribution from its moments is a
severely ill-posed problem and it is hard to design stable algorithms [34].
Multiplying (1.4) with m
k , integrating with respect to the internal coordinate,
commuting this integration with differentiation in time and with respect to the external
variable yields an equation for the kth moment
∂ t M k + (u + u sed ) · ∇M k =
m max
m min
m
k S dm, k = 0, 1, 2, . . . ,
(2.5)
R. Ahrens et al.
The principal form of the equations to be solved is as follows. Let ˆ
f n = f n−1 , solve
in the first step
∂ t ˆ
f +
u n + u sed,n−1
· ∇ ˆ
f = 0 in (t n−1 , t n ) × Ω
(2.2)
for all y ∈ Ω int . Then, set ˜
f n−1 = ˆ
f n , solve
∂ t ˜
f + ∂ m
G(c n , T n ) ˜
f
= F agg (u n , c n , T n , ˆ
f n ) + F break (u n , c n , T n , ˆ
f n ) in (t n−1 , t n ) × Ω int (2.3)
for all x ∈ Ω, and set f n = ˜
f n . There are several modifications of this basic operatorsplitting scheme for population balance equations, in particular to perform the steps
in a different order, e.g., see [3, 25, 27]. Equation (2.2) is usually a transport equation
with dominating convection, such that one has to utilize a stabilized discretization,
see Sect. 2.3.2. Also (2.3) is a transport equation, but the growth of the crystals
might be sufficiently slow such that one can apply some standard discretization. The
operator splitting introduces an additional splitting error which does not spoil the
optimal order of convergence for low order finite element methods [25].
As already mentioned at the beginning of this section, the definition of the population balance equation (1.4) for the crystal size distribution in a higher-dimensional
domain is a major challenge for the simulation of population balance systems. A
popular way to avoid this issue is the consideration of the first moments of the crystal size distribution, as proposed the first time in [33], where the so-called Method
of Moments (MOM) was derived. The kth moment of the crystal size distribution is
given by
M k =
∞
0
m
k f dm, k = 0, 1, 2, . . . .
(2.4)
It will be assumed, that f is zero for m ≤ m min and m ≥ m max , i.e., that there are
a minimal and a maximal mass for the crystals. Hence, the domain of integration
in (2.4) can be restricted to this interval. On the one hand, the first moments are
often important in practice because they correspond to physical quantities, like the
number of crystals (0th moment) or the mass of crystals (3rd moment). But on the
other hand, the reconstruction of the crystal size distribution from its moments is a
severely ill-posed problem and it is hard to design stable algorithms [34].
Multiplying (1.4) with m
k , integrating with respect to the internal coordinate,
commuting this integration with differentiation in time and with respect to the external
variable yields an equation for the kth moment
∂ t M k + (u + u sed ) · ∇M k =
m max
m min
m
k S dm, k = 0, 1, 2, . . . ,
(2.5)