14 Numerical Methods for Coupled Population Balance Systems …
505
where F
+
agg (f , m) denotes the source term and F
−
agg (f , m) denotes the sink term
of the agglomeration process. This definition of the source term does not account
for any particles forming with a property larger than the maximum m max . The sink
term, however, allows a particle to disappear, when it agglomerates with another
particle to one with property larger than m max . Hence, technically particles may be
lost over time if m max is too small. The choice of m max should reflect this consideration.
The two key ingredients toward the proposed efficient evaluation of these integrals
are a discretization of the property space Ω int on a uniform grid and a separable
approximation of the agglomeration kernel,
κ agg (m, m
) ≈
M
ν=1
α
ν
(m) · β
ν
(m
)
(3.2)
for a moderate separation-rank M ∈ N of the kernel κ agg . This allows to simplify the
convolution-type integral of F
+
agg (f , m) to a sum of M multi-dimensional convolution
integrals,
F
+
agg (f , m) =
1
2
m 1
0
· · ·
m d int
0
M
ν=1
α
ν
(m − m
)β
ν
(m
)f (m − m
)f (m
) d m
=
1
2
M
ν=1
m 1
0
· · ·
m d int
0
φ
ν
(m − m
)ψ
ν
(m
) d m
(3.3)
with φ
ν
(m) := α
ν
(m)f (m) and ψ
ν
(m) := β
ν
(m)f (m).
Analogously, the sink term results in
F
−
agg (f , m) =
m max
0
· · ·
m max
0
M
ν=1
α
ν
(m)f (m)β
ν
(m
)f (m
) d m
=
M
ν=1
φ
ν
(m) ·
m max
0
· · ·
m max
0
ψ
ν
(m
) d m
.
(3.4)
In particular, m-dependent terms have been factored out of the integral which reduces
the complexity to evaluate F
−
agg (f , m).
3.2.1 Discretization of the Property Space
In order to evaluate the integrals in (3.3) and (3.4) numerically, a suitable discretization of the property space Ω int to discretize f (m) is introduced. One first defines a
uniform tensor grid G by choosing the number of degrees of freedom per property,