On the Formation of Nitrogen Oxides During the Combustion of ...

4 Numerical Modeling and Simulation
tially pre-vaporized droplets. Either the initial droplet mass m fuel,0 (before prevaporization)
or the reacting fuel mass m fuel,reac (in gas phase) are reasonable
for this purpose. The latter is obtained analogously to Equation (4.45):
∫ tend
∫ R∞
m fuel,reac =
∣
˙ω fuel 4πr 2 dr dt
∣ . (4.46)
t 0
R
Differences between droplet fuel mass and reacting fuel mass arise for a few
reasons. On the one hand, regions with low equivalence ratios are generated
during pre-vaporization, which remain non-flammable throughout the whole
droplet lifetime. On the other hand, as combustion is only considered during
droplet lifetime, unburned fuel remains in the gas phase at the end of the simulation,
even though within the flammable range [297, 298].
4.5 Simulation of Single Droplets
The governing equations introduced above with Equations (4.3) through (4.6)
are simplified for the present numerical studies. Those simplifications are also
implemented in the utilized software package Cosilab ® [362–365].
4.5.1 Single Droplet Model
The final form of the governing equations as well as the motivation behind
those simplifications is presented in the following sections. Still, spherical
symmetry is postulated and detailed chemistry is considered for all computations
[31, 298].
Gas Phase
For the gas phase, a quasi-steady state approach helps in estimating the magnitude
of the radial velocity u r and its derivatives. Since the Soret and Dufour
effects are both hereinafter neglected, the diffusion velocity ∆u m,r changes
in comparison to ∆u m,i in Equation (4.10). According to Williams [461], diffusion
is dominated by concentration gradients. Thus, the thermal diffusion
ratio can be neglected, which would account for the Soret effect. Regarding
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