8 Flowsheet Simulation of Integrated Precipitation Processes
279
mass balance inside the mixing zones described above can be expressed by:
∂n i,m(x,t)
∂t
−
∂G
x, t, S i , n i,m
∂x
= h i,m
x, t, S i,m , n, X
(19)
h i is the internal particle net formation rate in which all particle formation processes
except of growth are summarized. The model substitutes the PSD by a sum of multidimensional Dirac functions:
f(ξ; x, t) =
N
α=i
ω α (x, t)δ[ξ − −ξ α (x, t)]
(20)
where N is the number of delta functions, ω is the weight of the node α. And each
dirac function can be written as:
δ[ξ − −ξ α (x, t)] =
N S
j=1
δ
ξ j − −ξ j α (x,t)
(21)
and ξ α is the property vector of node α with dimensionality j. Introducing this
approximation into a general PBE with the internal coordinate ξ:
∂f
ξ j
∂t
+
∂
∂x i
u i |ξ j f
ξ j
−
∂
∂x i
D x ∂f
ξ j
∂x i
= S ξ
ξ j
(22)
where u i |ξ is the mean velocity conditioned on the property value ξ and S ξ (ξ)
is the source term. By rearranging, the following moment transport equation for
two-dimensional calculations can be derived:
N
α=1
(1 − k − l)ξ 1
k
α ξ 2
l
α a α + kξ 1
k−1
α
ξ 2
l
α
b 1α + lξ 1
k
α ξ 2
l−1
α b 2α
=
N
α=1
k(k − l)ξ 1
k−2
α ξ 2
l
α C 11α + 2klξ 1
k−1
α ξ 2
l−1
α C 12α
+l(l − 1)ξ 1
k
α ξ 2
l−2
α C 11α
+ S
(N)
kl
(23)
with a and b and C k S
(N)
k abbreviating:
a α =
∂ω α
∂t
+
∂
x i
(u i ω α ) −
∂
∂x i
D x
∂ω α
∂x i
(24)
b 1α =
∂(ξ 1 α ω α )
∂t
+
∂
x i
(u i ξ 1 α ω α ) −
∂
∂x i
D x
∂ξ 1 α ω α
∂x i
(25)