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

4.2 Basics for Numerical Modeling
number L with S m and ν ml representing the species name and the stoichiometric
coefficient, respectively, of the m-th species in the l-th reaction equation.
The stoichiometric coefficients indicate the number of moles reacting,
and they are marked with one stroke ( ′ ) for reactants and two strokes ( ′′ ) for
products.
The net production rate ˙ω m of a species is the sum of its forward and backward
reaction rate, which in turn depend on the species molar densities N m = N X m
and the rate coefficients k + l
and k − l :
˙ω m = ˙ω + m + ˙ω− m . (4.15)
The forward and backward reaction rates are given for species S 1 in
Equation (4.14), expressed on a mass basis in kg m −3 s −1 owing to the molar
mass M S1 :
˙ω + S 1
=−M S1 k + 1
˙ω − S 1
=+M S1 k − 1
(
ν
′′
11 − ) ν ′ ( )
11)( ν′ N 11 ν ′
XS1 N
21
XS2 ... (4.16)
(
ν
′′
11 − )( ) ν ′′ ( ) ν′ 11 N 31 ν ′′
XS3 N
41
XS4 ... (4.17)
For elementary reactions, the exponents ν ′ ml and ν′′ in Equations (4.16) and
ml
(4.17) can be related to the elementary molecule collisions, and thus correspond
to the coefficients in Equation (4.14). For global reactions, however,
reaction rates are often determined empirically, and the exponents may differ
significantly from the stoichiometric coefficients. This is the case, for
instance, with the global kinetics of Duterque et al. [106], Hautman et al.
[172], Jones and Lindstedt [199], and Westbrook and Dryer [459].
The rate coefficient
(
k + 1 = A T b exp − E )
a
R T
(4.18)
for the forward reaction of Equation (4.14) is given as a modified Arrhenius
equation. The empirical law is a measure for the likeliness of collision and
reaction. The pre-exponential factor A describes the likelihood of a collision
of gas molecules, and the pre-exponential contribution to collision frequency
due to temperature is represented by the power of T . The backward rate coefficient
k1
− could be determined similarly to Equation (4.18) but is deduced
more easily and precisely from the constant of thermodynamic equilibrium as
a minimum of the Gibbs function [149, 297, 298, 443, 461].
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