INAUGURAL–DISSERTATION zur Erlangung der Doktorwürde der ...

28 2. Mathematical Modeling
the interior of the moment space M = {M(0), M(1), ..M(N)}. This is a standard constrained
optimization problem where the constraints are to satisfy the given moments.
In the ME method, following the moments satisfaction condition, below equation is
the explicit representation of the ME approximation,
f ME
M (x) ≡ exp ( −Σ N j=0 ξ j x j) , (2.36)
where the coefficients ξ 0 , ξ 1 ...ξ N are the Lagrange multipliers, and N is the number of
moments. These coefficients are computed based on the condition of minimizing the
following convex potential:
∆ ≡
∫ rmax
r min
[
exp
(
−Σ
N
j=0 ξ j x j) − 1 ] dx + Σ N j=0 ξ j M(j). (2.37)
The stationary points of Eq. (2.37) are given by ∂∆
∂ξ i
equation
∫ rmax
≡ 0, which yields the following
r min
x i exp ( −Σ N j=0 ξ j x j) dx ≡ M(i). (2.38)
The solution of the above equation gives ξ i , and substitution of these ξ i in Eq. (2.36)
yields the required NDF. The above equation can be solved numerically using a Newton
method, with the initial guess as ξ ≡ (− ln M(0)/(r min − r max )), 0, ...0), and updated
ξ’s are estimated by
Here H is the Hessian matrix defined by H i,j ≡
ξ + ≡ ξ − H −1 (M − 〈X〉 ξ
). (2.39)
∂∆
∂ξ i ∂ξ j
≡ 〈x i+j 〉 for i, j ≡ 0, 1, ...N, and
〈X〉 ξ
≡ (〈x 0 〉 ξ
, ... 〈 x N〉 ) is the vector of approximated moments, which are expressed
ξ
as
〈 〉 ∫ rmax
x
k ≡ x i exp ( −Σ N ξ j=0 ξ j x j) dx. (2.40)
r min
The numerical procedure to implement this approach is same as done by Mead and
Papanicolaou [66] and Massot et al. [98], where a double-precision 24-point Gaussian
quadrature method very efficiently produces the required accuracy for 〈 x k〉 ξ . More
details about the derivation of this method and numerical solution procedure are given
by Mead and Papanicolaou [66].
As the systems of interest in the present study are water spray in quiescent air or
nitrogen as well as PVP/water spray in quiescent air, currently the gas phase is not
fully coupled with DQMOM transport equations but its inlet properties taken from
the experiment are used to compute the droplet motion and evaporation.
2.4 Single Droplet Modeling
This section presents the physical processes due to gas–liquid and droplet–droplet interactions,
namely, droplet heating and evaporation, forces acting on the droplet, droplet