INAUGURAL–DISSERTATION zur Erlangung der Doktorwürde der ...

26 2. Mathematical Modeling
The Eqs. (2.26) – (2.28) are closed by modeling the source terms, i.e., a n , b n and c n ,
using the physical models to account for effects of droplet evaporation, forces on droplet,
coalescence and breakup.
These source terms are calculated through the moment
transformation of phase-space terms, which yields the following linear system
∫ [
P k,l = r k v l − ∂(Rf) − ∂(Ff) ]
∂r ∂v + Γ f + Q f drdv. (2.29)
The exact form of the DQMOM linear system relies on the choice of moments, and it
can be generated from
∫
[ ∂f
r k v l ∂t + ∂(vf)
∂x
]
drdv =
+
+
N∑
n=1
N∑
n=1
N∑
n=1
(1 − k)r k nv l 1
1,n v l 2
2,n v l 3
3,n a n
(k − l 1 − l 2 − l 3 )r k−1
n v l 1
1,n v l 2
2,n v l 3
3,n b n
r k nv l 1
1,n v l 2
2,n v l 3
3,n (l 1 v −1
1,nc 1,n + l 2 v −1
2,nc 2,n + l 3 v −1
3,nc 3,n )
+ δ k0 u l 1
1 u l 3
2 u l 3
3 ψ, (2.30)
where ψ is the evaporative flux, and u 1 , u 2 , and u 3 are three components of the gas
velocity. The complete linear system is formed by combining Eqs. (2.29) and (2.30),
which consists of 5N +1 unknowns, a n , b n , c n and ψ. To obtain a solution for this linear
system, the moments are chosen in a way that the resulting coefficient matrix is nonsingular.
Previous validation studies of DQMOM, and comparison of its performance
with QMOM have demonstrated that by using two-node closure (N = 2) approximation
for f is sufficient to track the lower order moments with small errors [48, 49, 54].
Increasing the number of nodes, N, to three (N = 3) have improved the results, and
in general the evaporation and coalescence terms can be accurately approximated with
N = 2–4 [48, 49, 54, 60]. In the present work, a three-node closure is used, i.e., N is
set to be 3, and the corresponding moment set is chosen as [60, 137] k ∈ {1, ..., 2N};
l ∈ {0, 1}, where l is composed of three components l 1 , l 2 , and l 3 . The chosen set of
k and l values conserves the mass and momentum of droplets, and these values are
found to give non-singular source terms matrix [60]. Along with these moments set,
the calculation of the source terms from the linear system requires the mathematical
formulation for the evaporation, forces on droplet and droplet–droplet interactions,
which enter as sub-models and these sub-models.
discussed and mathematical formulation is given in Subsection 2.4.
The sub-models are individually
The evaporation term in Eq. (2.29) can be simplified by evaluating the integral on
the R.H.S, which is given as
∫
− r k v l ∂(Rf)
∂r
= −(r k v l Rf)| r=∞
r=0 + k
∫ ∞
0
r k−1 v l ∂ (Rf) dr. (2.31)
∂r