14 Numerical Methods for Coupled Population Balance Systems …
485
• finite volume methods
– Scharfetter–Gummel method [59]; improved upwind but still quite diffusive,
– FCT [70].
The assessments provided above are based mostly on our experience from [37].
2.3.3 The Population Balance Equation
After having discretized the temporal derivative in (1.4), one obtains an equation for
f n in a four- or even higher-dimensional domain. But this difficulty is not the only
one for solving the population balance equation. There is a transport operator on the
left-hand side of (1.4) whose discretization requires special techniques, and on the
right-hand side there are integral operators whose efficient evaluation is complicated,
in particular for the first term of the agglomeration (1.5).
First of all, there are several principal ways for designing a scheme for computing
a numerical approximation of f n :
• solve an equation in the high-dimensional domain Ω × Ω int , where the left-hand
side is discretized with some appropriate discretization based on FDM, FEM, or
FVM, the so-called direct discretization,
• apply an operator-splitting scheme that deals first with an equation in Ω and after
this with an equation in Ω int ,
• utilize a momentum-based method to transform the population balance equation
to a system of equations in a three-dimensional domain,
• apply a stochastic method for solving (1.4).
The first approaches will be discussed briefly in the following whereas the last
approach is presented in detail in Sect. 2.4.
Utilizing the first approach, the direct discretization, is computationally demanding. One issue is that usual CFD codes do not support four- or higher-dimensional
domains. Using an implicit approach, then the system matrix becomes comparatively dense, compared with 3d, and the question of an appropriate solver for the
linear systems of equations arises. The left-hand side of (1.4) is a transport operator,
which can be considered as a limit case with vanishing diffusion of the convectiondominated operators from the energy and concentration balances. The discretizations
mentioned for the convection-dominated operators in Sect. 2.3.2 can be applied also
for the transport operator of the population balance equation. In addition, one needs
a numerical method for evaluating the integral terms on the right-hand side of (1.4),
see Sect. 3.2 for a discussion of this topic. Direct discretizations of 4d population
balance equations can be found, e.g., in [12, 13, 60], and of a 5d population balance
equation in [40].
Operator-splitting schemes for population balance equations in the form mentioned above were proposed in [25], see also [26]. Motivations for this proposal are
efficiency, the possibility to use software that is designed for domains in usual dimensions, and the possibility to apply different discretizations for the different equations.
485
• finite volume methods
– Scharfetter–Gummel method [59]; improved upwind but still quite diffusive,
– FCT [70].
The assessments provided above are based mostly on our experience from [37].
2.3.3 The Population Balance Equation
After having discretized the temporal derivative in (1.4), one obtains an equation for
f n in a four- or even higher-dimensional domain. But this difficulty is not the only
one for solving the population balance equation. There is a transport operator on the
left-hand side of (1.4) whose discretization requires special techniques, and on the
right-hand side there are integral operators whose efficient evaluation is complicated,
in particular for the first term of the agglomeration (1.5).
First of all, there are several principal ways for designing a scheme for computing
a numerical approximation of f n :
• solve an equation in the high-dimensional domain Ω × Ω int , where the left-hand
side is discretized with some appropriate discretization based on FDM, FEM, or
FVM, the so-called direct discretization,
• apply an operator-splitting scheme that deals first with an equation in Ω and after
this with an equation in Ω int ,
• utilize a momentum-based method to transform the population balance equation
to a system of equations in a three-dimensional domain,
• apply a stochastic method for solving (1.4).
The first approaches will be discussed briefly in the following whereas the last
approach is presented in detail in Sect. 2.4.
Utilizing the first approach, the direct discretization, is computationally demanding. One issue is that usual CFD codes do not support four- or higher-dimensional
domains. Using an implicit approach, then the system matrix becomes comparatively dense, compared with 3d, and the question of an appropriate solver for the
linear systems of equations arises. The left-hand side of (1.4) is a transport operator,
which can be considered as a limit case with vanishing diffusion of the convectiondominated operators from the energy and concentration balances. The discretizations
mentioned for the convection-dominated operators in Sect. 2.3.2 can be applied also
for the transport operator of the population balance equation. In addition, one needs
a numerical method for evaluating the integral terms on the right-hand side of (1.4),
see Sect. 3.2 for a discussion of this topic. Direct discretizations of 4d population
balance equations can be found, e.g., in [12, 13, 60], and of a 5d population balance
equation in [40].
Operator-splitting schemes for population balance equations in the form mentioned above were proposed in [25], see also [26]. Motivations for this proposal are
efficiency, the possibility to use software that is designed for domains in usual dimensions, and the possibility to apply different discretizations for the different equations.