Untitled - Aerobib - Universidad Politécnica de Madrid

168 CHAPTER 6. LAMINAR FLAMES
and
q j =
h j
c p T f
. (6.134)
In comparison with the case of only two chemical species, we observe here the appearing
of a set of parameters Λ ij . However, all these parameters have only one unknown
quantity, the value of m, this is to say the flame velocity, therefore the set Λ ij may be
expressed as a function of just one of the parameters.
Diffusion equations
l∑
( )
Mi
L ij X i ε j − X j ε i
dX
M
i
dθ = j=1
j
, (i = 1, 2, . . . , l), (6.135)
l∑
θ − 1 + q j (ε j − ε jf )
j=1
where
L ij =
λRT
M i c p pD ij
, (6.136)
Boundary conditions
The boundary conditions reduce to the following
θ → θ 0 : ε i → ε i0 , X i → X i0 , (6.137)
θ → 1 : ε i → ε if , X i → X if . (6.138)
Like in the case of two chemical species the difficulty at the cold boundary may be
eliminated by introducing an ignition temperature θ i whose value does not influence
the result when the reduced activation energies θ aj , or at least some of them, are large
as normally happens in practical cases for the same reasons stated in detail in the two
species case.
A recount of the number of boundary conditions when compared to the number
of equations proves that the conditions are superabundant and hence, an eigenvalue of
m, this is, one of the values of Λ ij , must exist which makes compatible the said
conditions and therefore the problem is essentially the same than in the case of two
species only.