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

4.3 Modeling of Ignition
Hence, the characteristic time scale of ∆t = 1.0ms utilized for the time between
heat introduction and extraction is reasonable 4 for an investigated
droplet diameter of D 0 = 100µm.
As for the geometrical parameters, the total volume for heat introduction is
optimized for a safe ignition and set to V in = 2.0mm 3 . For heat extraction,
V ex = 20.0 mm 3 is chosen in order to remove heat from a well-spread region of
exhaust gas outside of the burning droplet (cf. Fig. 4.3). The particular profiles
depicted in Figure 4.3 use values of r m,in = 0.8mm and r m,ex = 1.4 mm for the
mean radius. The solid line shows the introduced heat when ˙q v
∣
∣rm
= ˙q v,max (t ),
the dashed line the heat sink when ˙q v
∣
∣rm
= ˙q v,min (t ). Despite the exemplary
r m -values of Figure 4.3, there are two practical approaches to define a reasonable
position for r m,in (and also for r m,ex ). The outcome of both is discussed in
Chapter 5.2 regarding NO x emissions. The first approach uses spatially fixed
values for heat introduction and extraction [298], whereas the second uses
variable positions [297]. In the latter case, the positions are coupled with the
local equivalence ratio φ r . The definition of φ r is derived from the total equivalence
ratio φ. Since fuel and air diffuse into each other in the area around the
droplet, the ratio of nitrogen and oxygen is not constant over the radial coordinate
r . Using mass fractions and considering a differential volume enclosing
a differential mass dm, the local equivalence ratio φ r can be written as:
φ r =
dm fuel
dm O2
(
mfuel
=
dmY fuel
dmY O2
(
mfuel
=
(
mfuel
Y fuel
Y O2
. (4.42)
m O2
)stoich
m O2
)stoich
m O2
)stoich
Figure 4.4 shows the local equivalence ratio φ r in the gas phase of a single
droplet for different pre-vaporization rates Ψ. During vaporization, fuel
(C 10 H 22 ) diffuses away from the droplet, whereas oxidizer (O 2 ) diffuses in the
opposite direction. The local equivalence ratio φ r varies between values ≫ 1
at the droplet surface and zero at the outer gas phase boundary. Consequently,
the spatial position for a constant equivalence ratio also changes for different
pre-vaporization rates Ψ. For low pre-vaporization, this position is close to
the droplet surface but shifts farther away from the droplet for increasing prevaporization
(given that φ r < 1.5).
4 The absolute time between the ignition of neighboring droplets τ ign is much longer in the work of Mikami et al.
[282, 283] and Oyagi et al. [327], as most of the experiments are conducted with an initial droplet diameter of
D 0 = 1mm.
133