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

4.5 Simulation of Single Droplets
the Dufour effect, Annamalai and Puri [19] state that heat flux is dominated
by ordinary heat conduction. Consequently, the diffusion velocity in the gas
phase g simplifies to
∆u m,r =− D m ∂X g ,m
. (4.47)
X g ,m ∂r
This is the best first-order approximation to the exact formulation 5 [147, 180,
341] and avoids evaluating a linear system of equations [325]. According to
Poinsot and Veynante [341] and Annamalai and Puri [19] the diffusion coefficient
of a single species m in a multi-component gas can be approximated
by Equation (4.48). In this case, the coefficient D m is not a binary diffusion
coefficient but an equivalent diffusion coefficient of species m into the rest of
the mixture.
D m = 1−Y m
∑n;n≠m X n
D mn
(4.48)
Besides, as the mass flow of the vaporized liquid ṁ(t ) is equal to the mass flow
through a spherical shell with radius r , the radial velocity u r is proportional to
1/r 2 [297, 298].
The mass balance (Eq. (4.3)) and species conservation equation (Eq. (4.4)) remain
unaffected by those simplifications for the gas phase, apart from the outlined
changes in the diffusion velocity ∆u m,r . The viscous terms in the radial
momentum equation (Eq. (4.5)) cancel each other out [363, 364]. This is a
meaningful fact, as Euler flow (neglecting viscous effects), in the strict sense,
is only valid here for constant vaporization mass flow ṁ, constant density ρ,
and constant viscosity η g . Furthermore, a low Mach-Number is presumed.
This finally reduces the momentum equation (Eq. (4.5)) to a single term, still
accounting for Stefan flow [19, 443]. In the energy equation (Eq. (4.6)), viscous
terms are also neglected and Euler flow is presumed at a constant pressure
p g in space. The heat flux ˙q g ,r is evaluated neglecting the so-called Dufour
term, the second term on the right hand side of Equation (4.13). Altogether,
the governing equations for the gas phase as applied in the model are given
below.
5 In-depth analyses of both second-order effects (Soret and Dufour) can be found in literature [328, 335, 361].
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