INAUGURAL–DISSERTATION zur Erlangung der Doktorwürde der ...

3.3. Numerical Performance 53
where β ∈ α, α + 1, ...2N − α − 1. These quantities, σ α,β , are calculated by initializing
and a 0 = M(1)/M(0), b 0 = 0. The recurrence relation is
σ −1,α = 0, (3.26)
σ 0,α = M(α), (3.27)
σ α,β = σ α−1,β+1 − a α−2 σ α−1,β − b β−1 σ α−2,β , (3.28)
where the tridiagonal matrix components are given as
a α = σ α,α+1
− σ α−1,α
, (3.29)
σ α,α σ α−1,α−1
σ α,α
b α = . (3.30)
σ α−1,α−1
Here, the values of a α are the diagonal elements and b α are the upper and lower diagonal
elements of the symmetric tridiagonal matrix. The eigenvalues of this matrix are the
abscissas (droplet radii, velocities) where as the corresponding eigenvectors are the
weights (number densities). More details about derivation of this algorithm is given
by Gautschi [194], and example calculations are given by Marchisio and Fox [71].
3.3 Numerical Performance
For the DDM computations, which are carried out Humza [68], a hybrid finite volume
method based on the SIMPLER (Semi-Implicit Method for Pressure-Linked Equations
- Revised) algorithm [68, 195] is used to solve the mean conservation equation of the
gas flow, and a Lagrangian stochastic droplet parcel method is used for the spray flow.
The initial and boundary conditions are generated from the experimental data. A
non-equidistant rectangular numerical grid is used, which is finer in the region near
the nozzle exit with a total of 78 × 101 grid nodes. The numerical time step for
the governing gas phase equations is controlled by applying the CFL condition [188].
The solution algorithm and numerical details of the DDM calculation are given by
Humza [68].
The DQMOM simulations are carried out on a PC with two Intel dual core 2.2 GHz
processors having 8 GB RAM. The DDM is simulated on a PC having an AMD quad
Opteron 1.8 GHz processor with 64 GB RAM [68]. The latter PC had several jobs
running simultaneously, so that the available RAM on both the PCs is about identical.
All simulations are run on a single processor. The computations for DQMOM and DDM
take about one hour and three days, respectively. Thus, the DQMOM computations
show a much better performance with respect to the computational cost.