14 Numerical Methods for Coupled Population Balance Systems …
487
with
S = F agg (u, c, T , f ) + F break (u, c, T , f ) − ∂ m (G(c, T )f ) .
System (2.5) is a closed system for a finite number of moments only in special cases,
e.g., if there are no agglomeration, no breakage, and special growth functions.
For the case that a closure of (2.5) cannot be found, we consider for simplicity
only the growth term on the right-hand side of (2.5). Applying integration by parts
and using that f vanishes at m min and m max , this term can be reformulated as follows
−
m max
m min
m
k
∂ m (G(c, T )f ) dm =
m max
m min
km
k−1 G(c, T )f dm
=
m max
m min
˜
G(c, T )f dm, k ≥ 1,
with the new growth function ˜
G(c, T ) = km
k−1 G(c, T ). Note that this integral still
contains the unknown crystal size distribution f . The principal idea of the Quadrature Method of Moments (QMOM) proposed in [49] consists in approximating this
integral by some quadrature formula
m max
m min
˜
G(c, T )f dm ≈
N
i=1
ω i ˜
G(m i ),
(2.6)
where N is the number of quadrature points, which is prescribed by the user, ω i are
the weights of the quadrature rule and m i are the nodes (quadrature points, abscissas).
Then, at time instance t n , one considers the system of equations for the moments
∂ t M k + (u + u sed ) · ∇M k =
m max
m min
˜
G(c n , T n )f n−1 dm, k = 0, . . . , 2N − 1, (2.7)
where the left-hand side has still to be discretized appropriately and the right-hand
side is approximated with (2.6).
To keep the quadrature error in (2.6) as small as possible, the weights and abscissas
should be chosen such that the optimal order (2N − 1) of the numerical quadrature is
obtained. Several algorithms are available for this purpose. In [41], it is shown that the
long quotient-modified difference algorithm (LQMDA) behaves better than two other
algorithms concerning stability and efficiency. For computing the optimal weights
and abscissas, the knowledge of f is not necessary, but only of the first 2N moments
of f . Thus, for the first time step n = 1, one can use the known initial condition of
f for computing the right-hand side in (2.7) such that the first 2N moments at time
487
with
S = F agg (u, c, T , f ) + F break (u, c, T , f ) − ∂ m (G(c, T )f ) .
System (2.5) is a closed system for a finite number of moments only in special cases,
e.g., if there are no agglomeration, no breakage, and special growth functions.
For the case that a closure of (2.5) cannot be found, we consider for simplicity
only the growth term on the right-hand side of (2.5). Applying integration by parts
and using that f vanishes at m min and m max , this term can be reformulated as follows
−
m max
m min
m
k
∂ m (G(c, T )f ) dm =
m max
m min
km
k−1 G(c, T )f dm
=
m max
m min
˜
G(c, T )f dm, k ≥ 1,
with the new growth function ˜
G(c, T ) = km
k−1 G(c, T ). Note that this integral still
contains the unknown crystal size distribution f . The principal idea of the Quadrature Method of Moments (QMOM) proposed in [49] consists in approximating this
integral by some quadrature formula
m max
m min
˜
G(c, T )f dm ≈
N
i=1
ω i ˜
G(m i ),
(2.6)
where N is the number of quadrature points, which is prescribed by the user, ω i are
the weights of the quadrature rule and m i are the nodes (quadrature points, abscissas).
Then, at time instance t n , one considers the system of equations for the moments
∂ t M k + (u + u sed ) · ∇M k =
m max
m min
˜
G(c n , T n )f n−1 dm, k = 0, . . . , 2N − 1, (2.7)
where the left-hand side has still to be discretized appropriately and the right-hand
side is approximated with (2.6).
To keep the quadrature error in (2.6) as small as possible, the weights and abscissas
should be chosen such that the optimal order (2N − 1) of the numerical quadrature is
obtained. Several algorithms are available for this purpose. In [41], it is shown that the
long quotient-modified difference algorithm (LQMDA) behaves better than two other
algorithms concerning stability and efficiency. For computing the optimal weights
and abscissas, the knowledge of f is not necessary, but only of the first 2N moments
of f . Thus, for the first time step n = 1, one can use the known initial condition of
f for computing the right-hand side in (2.7) such that the first 2N moments at time