Untitled - Aerobib - Universidad Politécnica de Madrid

296 CHAPTER 12. DIFFUSION FLAME
Formula (12.42) shows that T b is the temperature that would correspond to a
premixed flame where fuel and its diluent were mixed with oxidizer and its diluent in
the stoichiometric ratio. In fact, in such mixture, mass fraction of fuel is
Y 10 Y 30
Y 30 + νY 10
and the corresponding temperature of the flame is given by (12.42).
12.5 Solutions of the simplified system
Some approximate solutions have been obtained for the case of two-dimensional and
axis symmetrical flames either confined or open. A detailed study of such flames is
available in the works listed in Refs. [1], [3], [6], [10], [15] and [17]. The following
chapter studies in detail the diffusion flame that forms surrounding a fuel droplet
which evaporates and burns in an oxidizing atmosphere. The problem has great importance
in jet propulsion.
The following simplifying assumptions have generally been adopted:
1) Stream velocity is throughout space parallel to the flame axis and uniform throughout
space or at least at each cross-section of the same.
2) All transport coefficients are independent from the composition and temperature
of the mixture.
3) Diffusion produces only in cross direction to the flame axis.
These assumptions reduce the problem to the integration of the one-dimensional
diffusion equation for the computation of the fuel, oxidizer and temperature distributions.
For example, in the case of an axis symmetrical flame, Fig. 12.1, the preceding
assumptions reduce Eq. (12.25) to
v ∂Y
∂x = D 1 (
∂
r ∂Y )
. (12.44)
r ∂r ∂r
Here, D and v are constant or at least independent from r. In the first case
the integration of Eq. (12.44) is straightforward. This also occurs in the second case
through the previous change of variable
x =
∫ τ
0
v
dτ, (12.45)
D
which to be determined would require knowing v/D as a function of distance x to
the flame base, which has been introduced by Hottel [3], Wohl [10] and Barr [5] as