Untitled - Aerobib - Universidad Politécnica de Madrid

12.5. SOLUTIONS OF THE SIMPLIFIED SYSTEM 297
an unknown function empirically determined. The aforementioned simplifying assumptions
only represent a poor approximation to reality. In fact, with respect to the
velocity field it can easily be verified that it is only constant in open flames where fuel
and oxidizer move at equal velocity within the discharge section and, furthermore,
where pressure is uniform throughout space. In such case the gas expansion produces
entirely in cross direction to the flame. As for transport coefficients they can change
in the ratio ten to one between flame surface and unburnt gases. While the assumption
that diffusion produces only in cross direction is well justified, except for special cases
like that of meniscus flames which so far have not been theoretically analyzed.
Before ending this brief review on diffusion flames we shall develop a simple
argument by W. Jost [19] which enables an easy establishment of the influence on the
length of a diffusion flame of some of the fundamental variables of the process. In
view of the rudimentary approximations needed in Burke-Schumann’s calculations in
order to obtain usable expressions, Jost assumes the same validity for his argument.
R
z
L
vt
z
v
fuel
v
oxigen
burner axis
burner wall
Figure 12.9: Schematic diagram of a overventilated diffusion showing the elements of the
Jost’s criterion.
Let us consider, for example, a confined flame with an excess of air or an
open flame, Fig. 12.9. According to Jost, the length of the flame is determined by the
condition that the oxygen of the atmosphere surrounding the jet can reach the jet’s axis
through diffusion. Let D be the oxygen diffusion coefficient and z its mean quadratic
displacement due to diffusion at instant t. Diffusion theory gives for z as function of t
z 2 = 2Dt. (12.46)
But, at the point where the flame closes
z = R, (12.47)