Untitled - Aerobib - Universidad Politécnica de Madrid

12.3. BOUNDARY CONDITIONS ON THE FLAME 291
The previous system together with the adequate boundary conditions determine
the values for p, ρ, T , ¯v and for the mass fractions Y i at each point as well as the shape
of the flame which so far is unknown.
12.3 Boundary conditions on the flame
Chemical species A i forming the mixture can be classified into three different groups:
a) Reacting species A r i , which diffuses towards the flame where they burn. These
species are unable to cross Σ f .
b) Reaction products A p i , which produce at the flame and diffuse from it towards
both sides.
c) Inert species A d i , which dilute reacting species. These species can cross Σ f .
According to Burke and Schumann’s model mass fractions Yi
r
species must be zero on the flame surface, that is
of the reacting
Y r
1 = Y r
2 = · · · = Y r
l r
= 0 (12.10)
on Σ f , where l r is the number of reacting species. 5
position of the flame surface.
These conditions determine the
Let m r i be the mass of species Ar i that reaches Σ f per unit surface and per unit
time. m r i is given by m r i = −ρY i¯v i · ¯n i , (12.11)
where ¯n i is the unit vector normal to Σ f at A r i side. mr i is consumed by the chemical
reactions taking place at Σ f . Let
∑
ν ijA ′ r i → ∑ ν ijA ′′ p i , (j = 1, 2, . . . , r), (12.12)
i
i
be one of these reactions, which transforms reacting species A r i into reaction products
A p i . Since the thickness of the reaction zone is assumed to be zero, it becomes necessary
to define a surface reaction rate for each reaction. Such rate gives the masses of
the species consumed or produced by the reaction per unit surface and per unit time.
Be r j the value of such surface reaction rate for reaction j. The fraction m r ij of mr i
consumed by this reaction is
where M r i is the molar mass of species Ar i .
m r ij = M r i ν ′ ijr j , (j = 1, 2, . . . , r), (12.13)
5 Actually, these mass fractions are very small but not strictly zero. Their values are practically those of
the equilibrium of species under the prevailing conditions at the flame.