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

4 Numerical Modeling and Simulation
Equations (4.36) through (4.38) yield a transcendental equation for the determination
of the offset radius ∆r . Finally, the total amount of heat Q introduced
in or extracted from the system needs to be defined by the user. It delivers
the maximum volumetric heat source ˙q v,max .
Two different parameter sets realize the heat introduction and extraction process
within the numerical model, indicated by “in” and “ex”, respectively.
Finding appropriate values for the ignition parameters is crucial regarding
NO x emissions. The parameters must be set to ensure a safe ignition, but NO x
emissions are expected to depend on the amount of introduced heat Q in , position
of the heat source r m,in , and ignition volume V in [297, 298].
The amount of heat introduced by the heat source is Q in = 3×10 −3 J, which
is about 20% of the lower heating value times the mass of the investigated
fuel droplet 3 (see Chap. 5.2). The same amount of heat is nominally extracted
from the computational domain with Q ex = −3.0×10 −3 J during droplet burning
(cf. Fig. 4.2). To avoid negative effects, the absolute amount of heat |Q|
is optimized to be as low as possible but still high enough to ensure ignition.
The associated times are t in,min = 0.0ms and t in,max = 0.5ms as well as
t ex,min = 1.0 ms and t ex,max = 1.5 ms, with a calculative time shift between heat
introduction and extraction of ∆t = 1.0 ms. These time scales are chosen according
to correlations given in Mikami et al. [282, 283] and Oyagi et al. [327]
for the flame spread rate along droplet arrays. The experiments of Mikami
et al. [283] show that the flame spread rate is
v f D 0 = 50mm 2 s −1 (4.39)
for an inter-droplet distance of S ≈ 10D 0 and a preheating temperature of
T = 600K. v f denotes the mean velocity of the flame front and can be expressed
as
v f = S
τ ign
, (4.40)
where τ ign is the time between the ignition of two neighboring droplets. Solving
for time yields
τ ign = S D 1 s
0·
50 mm ≈ D 0· 2 1 s
2 5 mm = 2 2.0×10−3 s. (4.41)
3 The droplet itself has an initial diameter of D 0 = 100µm before the start of pre-vaporization.
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