Untitled - Aerobib - Universidad Politécnica de Madrid

304 CHAPTER 13. COMBUSTION OF LIQUID FUELS
or else
R =
γ
(6/δ, (d/d m ) δ)
Γ (6/δ)
(13.5)
Here the parameters have the same meaning that in the Rosin–Rammler formula and γ
is the incomplete gamma function. 2 The Sauter diameter is related with d m by means
of
(
ds
d m
)
= Γ (6/δ)
Γ (5/δ) . (13.6)
13.3 Mixing
Once the fuel is atomized it mixes with the surrounding atmosphere due to the action
of turbulence. Lately, Longwell and Weiss [17] have studied the problem for the case
where the fuel atomizes in a turbulent gas stream flowing with uniform velocity. Let
f be the ratio of the fuel mass to the air mass. Assuming that:
1) Turbulent diffusion produces only transversely to the main flow.
2) Mean motion is stationary.
3) The process has axial symmetry.
The following approximate equation for f is obtained
∂f
∂x = E ( )
1 ∂f
v r ∂r + ∂2 f
∂r 2 . (13.7)
Here v is the motion velocity, E is the coefficient of turbulent diffusion of the
fuel and x and r are the cylindrical coordinates of the system. For the derivation of this
equation E is assumed to be constant. Equation (13.7) is identical to the molecular
diffusion equation under similar conditions.
If the fuel is vaporized, E is the coefficient of turbulent diffusion defined by
Taylor [18]. In this case, E equals the product of intensity by scale of turbulence.
If the fuel is in the liquid state, E is appreciably smaller due to the inertia of the
droplets which are unable to follow the air fluctuations. In a typical case calculated by
Longwell and Weiss, E was only 35% of the value corresponding to the diffusion of
the vapour.
If the boundary conditions in the atomizing section are known, equation (13.7)
can be integrated. For example, if the mixing starts from the origin of coordinates,
which acts as a source of fuel of strength G c , and if the duct radius is large, the
2 Defined as γ(a, x) = R x
0 ta−1 e −t dt, Ed.