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

5 Results
uses constant spatial positions for heat introduction and extraction. It is the
basis for the results of Chapter 5.2.1. The second approach enforces ignition
at a position r m,in of constant local equivalence ratio φ r (cf. Eq. (4.42)). Here,
the effective local equivalence ratio is fixed to φ r = 0.5. The advantage is to
ensure a safe ignition for all degrees of vaporization Ψ without receiving overlapping
volumes of droplet and heat source, which can become crucial particularly
at low values of Ψ. Since the heat source is distributed over a finite
volume around r m,in , heat is also introduced at positions where φ r > 0.5. This
ensures a safe ignition. The results of this second approach are discussed in
the present chapter and compared with the previous results of Chapter 5.2.1.
For the heat extraction following a successful ignition, two different cases are
considered: In the first case, a spatially fixed position of the heat extraction
(i.e. the heat sink in Fig. 5.4) of r m,ex = 1.4×10 −3 m is used in combination with
all ignition positions r m,in (i.e. the heat source in Fig. 5.4). This case of heat
extraction is termed “heat sink at fixed position”. In the second case, the position
of the heat extraction r m,ex is coupled to the (variable) ignition position
r m,in by the fixed distance of 0.6×10 −3 m. This latter case of heat extraction is
termed “heat sink at fixed distance” [297].
Differences of the Approaches Applied for Ignition Modeling
The resulting mean positions of heat introduction, r m,in , and heat extraction,
r m,ex , are depicted in Figure 5.4 as a function of pre-vaporization rate
Ψ. The positions of r m,in are marked with squares for heat sources, whereas
the positions of r m,ex are marked with circles for heat sinks. As introduced
in Chapter 4.3 and Figure 4.3, the expansion of the respective volumes V in/ex
(Eq. (4.36)) is indicated in Figure 5.4 by vertical bars, ranging from r min
(Eq. (4.37)) to r max (Eq. (4.38)). Even though these limiting minimum and
maximum positions vary with Ψ, depending on the mean positions r m,in/ex ,
the effective volumes of heat introduction and extraction are of constant size.
Starting from a low pre-vaporization rate of Ψ= 0.15, Figure 5.4 unveils a shift
of the ignition position away from the droplet until a maximum is reached
around Ψ= 0.7. Furthermore, an overlap can be identified between heat introduction
and extraction volume in the range of Ψ= 0.15 to 0.4. For comparison,
the fixed positions as employed in Chapter 5.2.1 are shown on the right
hand side of Figure 5.4.
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