Untitled - Aerobib - Universidad Politécnica de Madrid

294 CHAPTER 12. DIFFUSION FLAME
In particular, if condition
D 12 = D 23 = D (12.28)
is satisfied, Y as well as its derivatives are continuous even at the flame. Continuity
Eq. (12.1) for the mixture and Eq. (12.3) for motion still hold unchanged.
As for the energy equation we shall assume that the specific heats at constant
pressure of the three species are constant and equal
c p1 = c p2 = c p3 = c p . (12.29)
In such case, one can immediately check that Eq. (12.7) reduces to the following
which hold throughout space.
ρc p¯v · ∇T − ∇ · (λ∇T ) = 0, (12.30)
Let us assume, that besides condition (12.28) the following is satisfied
λ
ρDc p
= 1, (12.31)
in accordance with the result of the elementary Kinetic Theory of Gases. Then, comparison
between (12.25) and (12.30) suggests the existence of solutions of the form
T = aY + b, (12.32)
where a and b are constant but can be different for the interior and exterior regions.
These solutions are available for the study of diffusion flames if boundary conditions
can be satisfy through them. In particular, since on the flame Y = 0, temperature T b
of the flame will be constant for the cases where Eq. (12.32) is valid. The study of
diffusion flames reduces, hence, to a computation of the values for ρ , T , ¯v and Y . For
this purpose, system of Eqs. (12.1) and (12.3) is available which must be completed
with the boundary conditions adequate for each case. These refer in general to the
conditions at the discharge sections of fuel and oxidizer and at great distance from the
flame. Let us assume, for example, that in the fuel’s discharge section
and in the oxidizer’s discharge section
Y = Y 10 , T = T 0 , (12.33)
Y = Y 30 , T = T 0 , (12.34)
Then Eq. (12.32) holds and one obtains in the interior region,
T = T b − (T b − T 0 ) Y
Y 10
= T b − (T b − T 0 ) Y 1
Y 10
, (12.35)