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

4 Numerical Modeling and Simulation
differently in this work. Here, its location is specified by a mean radius r m
and a volume V . The introduced heat is given as an integral quantity Q in J
(cf. Eqs. (4.34) and (4.35)). Subsequent to ignition, the same amount of heat
can be taken out of the computational domain by a heat sink. This process
conserves the energy of the system and allows a reasonable interpretation
of, for instance, NO x generation. Furthermore, NO x formation is enhanced
at temperatures above 1800 K, and thus it is permissible to promote droplet
vaporization and combustion within a moderate temperature range by using
a well-balanced heat source. 2 Dietrich et al. [102], on the other hand, provide
only a heat source, and the droplet’s energy balance is not kept. This approach
might be sufficient to model droplet ignition and burning rate k, but the results
of a detailed chemistry would possibly become questionable.
Q =
˙q v = ˙q v,max sin
∫ tmax
∫ rmax
t min
r min
˙q v 4πr 2 dr dt (4.34)
( ) ( )
t− tmin
r − rmin
π sin
π
t max − t min r max − r min
(4.35)
The model uses half-sine profiles in space and time for the distribution of the
volumetric heat source ˙q v . The corresponding profiles of heat source and heat
sink are illustrated in Figures 4.2 and 4.3.
The volumetric heat source ˙q v is defined within the spatial limits r min and r max
and the temporal limits t min and t max (Eq. (4.35)). In the first instance, the time
interval t min to t max is arbitrary, with time t = 0s referring to the end of droplet
pre-vaporization. Besides, instead of directly specifying the remaining model
constants r min , r max , and ˙q v,max , more applicable constants are used: r min and
r max are determined from the total volume
V = 4 3 π( r 3 max − r 3 min
)
(4.36)
in conjunction with the mean radius r m , where
r min = r m − ∆r, (4.37)
r max = r m + ∆r. (4.38)
2 This study also includes the auto-ignition of droplets at an elevated temperature level due to the heat transfer
from hot exhaust gas (Chaps. 5.1 and 5.3). This particular droplet burning regime resembles a single droplet
entering the flame front within spray combustion.
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