14 Numerical Methods for Coupled Population Balance Systems …
483
cooling crystallization. In batch simulations, the size-dependent residence time leads
to narrow crystal size distributions compared to a uniform crystal residence time.
2.3 Brief Survey on Numerical Methods for Solving a PBS
Let the time interval be decomposed into subintervals [t n−1 , t n ], n = 1, . . . N , with
0 = t 0 < t 1 < · · · < t N = t end and let the (numerical) solution u n−1 , T n−1 , c n−1 , f n−1
at the time instance t n−1 be given. Then, one has to apply some time stepping scheme
to compute the (numerical) solution at t n . This section provides a brief survey on
methods that are proposed in the literature for computing a numerical solution at t n .
Since a monolithic approach for solving the PBS, which computes all unknown
functions together from (1.1) to (1.4), is computationally too demanding, the PBS is
split into several parts and these parts are solved consecutively.
2.3.1 The Navier–Stokes Equations
Because the velocity appears in all equations and there is no back coupling of the
other unknowns to the flow field, it is a straightforward idea to solve first the Navier–
Stokes equations (1.1). These equations can be solved monolithically or decoupled
by a so-called projection scheme. As temporal discretization, often first or second
order time stepping schemes are used, like the Euler schemes, the Crank–Nicolson
scheme, or the backward difference formula of order 2 (BDF2). The nonlinear term
in the momentum balance can be treated implicitly, semi-implicitly, or explicitly.
The semi-implicit approach is called implicit-explicit (IMEX) scheme. Usual spatial
discretizations include finite element methods (FEM), finite volume methods (FVM),
or, for simple domains, finite difference methods (FDM). A detailed description of all
these approaches is far beyond the scope of this paper. Many of them are described,
within the framework of FEMs, in [35, Chap. 7].
The situation becomes more complicated if the flow is turbulent. There is no mathematical definition of turbulence, but a good physical description is that a turbulent
flow contains a wide range of physically important scales. In particular, there are
many small scales that cannot be resolved on affordable grids and, consequently,
that cannot be simulated. Standard discretizations cannot cope with this situation
since they try to simulate all important scales. Simulations with such discretizations
usually blow up in finite time. Since neglecting the small scales leads to physically
incorrect numerical simulations, an approach is needed to model the impact of the
unresolvable scales onto the resolvable scales. This approach is called turbulence
modeling. In the literature, many turbulence models are proposed, e.g., see [54, 58],
and turbulence modeling is still an active field of research. There is no turbulence
model that can be considered to be the best one.
At the end of this step, u n is known and it can be used in the other equations of
the PBS.
483
cooling crystallization. In batch simulations, the size-dependent residence time leads
to narrow crystal size distributions compared to a uniform crystal residence time.
2.3 Brief Survey on Numerical Methods for Solving a PBS
Let the time interval be decomposed into subintervals [t n−1 , t n ], n = 1, . . . N , with
0 = t 0 < t 1 < · · · < t N = t end and let the (numerical) solution u n−1 , T n−1 , c n−1 , f n−1
at the time instance t n−1 be given. Then, one has to apply some time stepping scheme
to compute the (numerical) solution at t n . This section provides a brief survey on
methods that are proposed in the literature for computing a numerical solution at t n .
Since a monolithic approach for solving the PBS, which computes all unknown
functions together from (1.1) to (1.4), is computationally too demanding, the PBS is
split into several parts and these parts are solved consecutively.
2.3.1 The Navier–Stokes Equations
Because the velocity appears in all equations and there is no back coupling of the
other unknowns to the flow field, it is a straightforward idea to solve first the Navier–
Stokes equations (1.1). These equations can be solved monolithically or decoupled
by a so-called projection scheme. As temporal discretization, often first or second
order time stepping schemes are used, like the Euler schemes, the Crank–Nicolson
scheme, or the backward difference formula of order 2 (BDF2). The nonlinear term
in the momentum balance can be treated implicitly, semi-implicitly, or explicitly.
The semi-implicit approach is called implicit-explicit (IMEX) scheme. Usual spatial
discretizations include finite element methods (FEM), finite volume methods (FVM),
or, for simple domains, finite difference methods (FDM). A detailed description of all
these approaches is far beyond the scope of this paper. Many of them are described,
within the framework of FEMs, in [35, Chap. 7].
The situation becomes more complicated if the flow is turbulent. There is no mathematical definition of turbulence, but a good physical description is that a turbulent
flow contains a wide range of physically important scales. In particular, there are
many small scales that cannot be resolved on affordable grids and, consequently,
that cannot be simulated. Standard discretizations cannot cope with this situation
since they try to simulate all important scales. Simulations with such discretizations
usually blow up in finite time. Since neglecting the small scales leads to physically
incorrect numerical simulations, an approach is needed to model the impact of the
unresolvable scales onto the resolvable scales. This approach is called turbulence
modeling. In the literature, many turbulence models are proposed, e.g., see [54, 58],
and turbulence modeling is still an active field of research. There is no turbulence
model that can be considered to be the best one.
At the end of this step, u n is known and it can be used in the other equations of
the PBS.