Untitled - Aerobib - Universidad Politécnica de Madrid

13.11. NUMERICAL APPLICATION 319
where r si is the initial radius. This formula is the fundamental result of the present
theory.
The combustion time t c of a droplet of radius r si is, obviously,
t c = r2 si
k . (13.75)
The position r l /r s of the flame front is obtained by writing T = T l in Eq. (13.62).
Furthermore if m/r s is eliminated from the result by means of Eq. (13.73), one obtains
r l
r s
=
1 − T l
T ∞
+
kρ e c p
2λ ∞
q ln q − c . (13.76)
pT ∞
c p T ∞ q − c p T l
Therefore, the radius of the flame and the droplet are proportional.
Formula (13.73) enables the study of the influence of the physical constants of
the fuel and the state of the surrounding atmosphere on the burning rate of the droplet. 8
In particular, the said formula shows that the heat transmitted from the flame to the
droplet surface is the determining factor for its burning velocity. This is due to the
fact that the burning rate of the droplet is determined by the heat received by the same.
Therefore, the essential factor in the combustion process is not the volatility of the fuel
but its heat of evaporation. Experiments results confirm this conclusion. In fuels at the
high temperature reached when combustion occurs, the evaporation mechanism differ
essentially from the evaporation when the temperature of the surrounding atmosphere
is normal. In the last case the evaporation is maintained by the diffusion of the vapors
from the droplet towards the exterior. The evaporation velocity is determined by the
difference between the vapour pressures on the droplet surface and at infinity. 9
13.11 Numerical application
The above formulas are applied here to the study of the combustion of a gasoline
droplet in the air. The following typical values are adopted
T ∞ = 300 K, T s = 355 K, ρ c = 0.85 gr/cm 3 ,
λ ∞ = λ s = 55 × 10 −6 cal/cm s K, q l = 95 cal/gr, q = 10 000 cal/gr,
c p1 = 0.60 cal/gr K, c p2 = 0.35 cal/gr K, c p3 = 0.26 cal/gr K,
Y 3∞ = 0.23, ν = 3, δ 1 = δ 2 = 0.9.
8 Information on the subject can be found in Godsave’s works [19].
9 See §15, Eq. (13.103).