INAUGURAL–DISSERTATION zur Erlangung der Doktorwürde der ...

2.3. Euler – Euler Approach 27
With the assumption that the maximum droplet size is a finite value, the above equation
can be further simplified as
∫
− r k v l ∂(Rf) = δ k0 ψv l + k
∂r
∫ ∞
0
r k−1 v l ∂ (Rf) dr, (2.32)
∂r
where δ k0 is the Kronecker delta, which is defined as δ k0 = 1 if k = 0 and δ k0 = 0 for
any other k value. The quantity ψ = Rf(0) is the evaporative flux, which is a point
wise quantity of the NDF representing the number of droplets having zero size. This
quantity in DQMOM is computed by weight ratio constraints, which are introduced by
Fox et al. [60] where ψ is treated as an additional variable along with a n , b n and c n ’s.
These ratio constraints of weights, radii and velocities [60] are given by
( ) ( )
D wn D rn
= 0;
= 0; (2.33)
Dt w n+1 Dt r n+1
where j is the index for three velocity components.
( )
D vj,n
= 0, (2.34)
Dt v j,n+1
Fox et al. [60] show that the
estimation of evaporative flux via weight ratio constraints is found to give acceptable
results in a stationary one-dimensional configuration. However, Fox et al. [60] suggests
that this calculation procedure is found to pose problems in the case of complicated
distribution functions [64].
In the current study, this is addressed by implementing the maximum entropy (ME)
principle proposed by Mead and Papanicolaou [66] for water and PVP/water spray
flow in air, which estimates the evaporative flux through reconstruction of the droplet
distribution using its moments.
The principle of maximum entropy in the problem of moments is that the distributions
that satisfy the given moment set (also called as constraints), the most likely
or least biased probability density function is the the one whose statistical entropy is
a maximum. This formulation allows the determination of a number density function
from the limited amount of information such as few known moments of a distribution
[66]. The implementation of this method to compute ψ is explained by Massot et
al. [98].
The ME method is first introduced by Mead and Papanicolaou [66] to compute
a distribution for the given moment set based on the maximization of the following
Shannon entropy from the information theory [66],
H[f] ≡ −
∫ rmax
r min
f(x) ln f(x)dx. (2.35)
Mead and Papanicolaou have proven that there exists ME distribution satisfying the
above entropy principle [66] for the case when the vector of moments M belongs to