INAUGURAL–DISSERTATION zur Erlangung der Doktorwürde der ...

18 2. Mathematical Modeling
where δ is the tensorial Kronecker delta given by
{
1 : i = j
δ ij =
0 : i ≠ j.
(2.4)
Neglecting the processes of radiation, friction heating, Dufour effect, and the viscous
heating, the conservation equation of total stagnant enthalpy can be written as
∂(ρh)
∂t
+ ∂(ρu jh)
∂x j
= ∂p
∂t − ∂J d q,j
∂x j
− ∂J c q,j
∂x j
+ S l,h , (2.5)
where h is the enthalpy of the gas flow and the terms on the right hand side (R.H.S)
are the change rate of the pressure, the heat diffusion term, the heat conduction term
and the source term due to spray evaporation, S l,h , respectively. The heat conduction
term is expressed by the Fourier’s Law
(
)
Jq,j c = −λ ∂T = λ¯Cp ∂h ∑N s
∂Y α
− h α , (2.6)
∂x j ∂x j ∂x j
where λ, T , ¯Cp are thermal conductivity, gas temperature and specific heat capacity,
respectively.
N s refers to the number of chemical species while h α and Y α are the
enthalpy and mass fraction of species α. The heat diffusion term J d q,j is written as
α=1
∑N s
Jq,j d = h α Jα
m
α=1
∑N s
Y α
= − ρh s,α D α,M , (2.7)
∂x j
α=1
where h s,α and D α,M are the specific sensible enthalpy of species α and diffusion coefficient
of species α, respectively. Assuming a unity Lewis number, which is defined as
the ratio of thermal diffusion to mass diffusion, (Le = k/(ρC p D), and equal diffusibility
of all species, the total heat flux is
J q = J c q,j + J d q,j = − λ¯Cp (
∂h
∂x j
−
∂(ρh)
∂t
+ ∂(ρu jh)
∂x j
∑N s
α=1
)
∂Y α
h α −
∂x j
∑N s
α=1
The conservation equation of species mass can be written as
∂(ρY α )
∂t
ρh s,α D α,M
Y α
∂x j
. (2.8)
= ∂p
∂t + ∂ ( )
∂h
Γ h + S l,h . (2.9)
∂x j ∂x j
+ ∂(ρu jY α )
− ∂ ( )
∂Y α
ρD α = S α + δ L,α S l,Yα , (2.10)
∂x j ∂x j ∂x j
where D α is the diffusion coefficient of species α while S α and S l,α are the source terms
due to chemical reactions and spray evaporation, respectively. The mass fraction may
be used to formulate mixture fraction.
The advantage of an appropriately defined