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

4 Numerical Modeling and Simulation
are neglected. Here, the species in multi-component mixtures are quantified
as mole fractions X m or as mass fractions Y m . Specific heat at constant pressure
in J kg −1 K −1 is referenced by c p , standard enthalpy of formation in J kg −1
by h 0 m (at reference temperature T 0), and dynamic viscosity in Pa s by η. ˙ω m denotes
the net production rate of species m and serves as a source term in the
species conservation equation, whereas ˙q is the specific heat flux in W m −2 .
Hence, mass conservation can be written as
∂ρ
∂t + ∂ρu r
∂r
+ 2 ρu r
r
= 0, (4.3)
conservation of species is given by
ρ ∂Y m
∂t
+ ρu r
∂Y m
∂r
+ ∂ρY m∆u m,r
∂r
+ 2 ρY m∆u m,r
r
= ˙ω m , (4.4)
the momentum equation yields
ρ ∂u r
∂t + ρu ∂u r
r
∂r
=− ∂p
∂r + ∂η ( 4 ∂u r
∂r 3 ∂r − 4 ) (
u r 4 ∂ 2 u r
+ η
3 r 3 ∂r + 8 2 3
1
r
∂u r
∂r − 8 )
u r
,
3 r 2
(4.5)
and the energy equation is
ρ ∂∑ ∫ T
m Y m T 0
c p,m d T ∂ ∑ ∫ T
m Y m T
+ ρu
0
c p,m d T
r
∂t
∂r
=− ∂∑ ∫ T ∑ ∫ T
m ρY m T 0
c p,m d T ∆u m,r m ρY m T
− 2
0
c p,m d T ∆u m,r
∂r
r
− ∑ ˙ω m h 0 m + ∂p [ ( )
m ∂t + u ∂p 4 2
r
∂r + η ∂ur
− 8 u r ∂u r
3 ∂r 3 r ∂r + 4 ( ur
) ] 2
3 r
(4.6)
− ∂ ˙q r
∂r − 2 r ˙q r .
This set of equations is not closed. The species velocity difference ∆u m,r , net
production rate ˙ω m , and heat flux ˙q r are unknown. Pressure p is related to
temperature T and density ρ in the ideal gas law.
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