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

4.2 Basics for Numerical Modeling
4.2.2 Transport Mechanisms in the Gas Phase
The species velocity difference ∆u m,r (i.e. the diffusion velocity) and the radial
heat flux ˙q r can be derived from molecular gas kinetics [169, 180, 213], which
is based on population dynamics. The distribution function
f m (x, v m , t ) (4.7)
is the number of molecules of species m per volume at position x with the
velocity vector v at time t . The bulk velocity
∑ ∫
n m n fn (x, v n , t )v n d v n
u(x, t )= ∫ (4.8)
∑n m n fn (x, v n , t ) d v n
is calculated from the distribution function f n , with m n denoting the mass of
one molecule of species n. f m and f n are used similarly; the only difference is
the name of the species – namely m or n. Thus, the species velocity is given by
∫
fm (x, v m , t )v m d v m
u m (x, t )= ∫ . (4.9)
fm (x, v m , t ) d v m
Note that v m is the coordinate in the space-velocity frame, not the mean velocity
of species m.
After some rearranging [180], the diffusion velocity ∆u m,i can be calculated
from Equation (4.9), with the bulk velocity u i in m s −1 , the velocity coordinate
v m,i in m s −1 , the species density n m in m −3 , and the relation ρ = ∑ m n m m m .
Employing ∆u m,i of Equation (4.10) to the energy (or heat flux) conservation
equation and to the species conservation equation, it accounts for the Dufour
and Soret effect, respectively. Both second-order effects are included in the
general derivations of this section but will be neglected in the final model, as
outlined in Chapter 4.5.
∆u m,i = 1 ∫ (vm,i )
− u i fm dv m,i = 1 ∫
∆v m,i f m d∆v m,i
n m n m
( n
2 ∑
=
m n ˜D mn d m,i −
n m ρ) D T (4.10)
m ∂lnT
n
n m m m ∂x i
The gradients in molar fraction X m = n m /n and in pressure p are summarized
as vector d m,i according to Hirschfelder et al. [180] (see Eq. (4.11)). If the sys-
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