Untitled - Aerobib - Universidad Politécnica de Madrid

326 CHAPTER 13. COMBUSTION OF LIQUID FUELS
to the study of combustion. The extrapolation of the obtained values to the case of a
spray is difficult. In fact, it is necessary to know the size distribution of the droplets
and the composition of the surrounding atmosphere. Furthermore, in order to account
for the convection effects, which are very important, the motion of the droplets relative
to the atmosphere must also be known. The available information is not sufficient to
show the way in which this extrapolation can be performed [34].
The evaporation velocity of a droplet when convection is absent can be easily
obtained from the formulas deduced in the preceding paragraphs. In fact, assuming
that the droplet is isothermal, it is enough to make use of the equations corresponding
to the interior region, enlarged to infinity. Therein the following boundary conditions
must be satisfied which determine the integration constants
r → ∞ : T → T ∞ , Y 1 → Y 1∞ , (13.87)
where Y 1∞ is the mass fraction of the fuel vapours at great distance from the droplet.
Moreover, if one keeps the assumptions previously established with respect to the law
of variation of the values of the transport coefficients as functions of temperature, the
two equations for T and Y 1 are Eqs. (13.53) and (13.54), that is
The last equation when applied to the droplet surface r = r s gives the following
relation
4πr 2 λ ∞
T
T ∞
dT
dr − mc p1T = m(q l − c p1 T s ), (13.88)
4πr 2 (ρD 12 ) ∞
T
T ∞
dY 1
dr = −m(1 − Y 1). (13.89)
The solution of this system that satisfies conditions (13.87) is
1
r = 4πλ (
∞
1 − T + q l − c p1 T s
ln c )
p1(T − T s ) + q l
, (13.90)
mc p1 T ∞ c p1 T ∞ c p1 (T ∞ − T s ) + q l
( ) δ
1 − Y 1 cp1 (T − T s ) + q l
=
. (13.91)
1 − Y 1∞ c p1 (T ∞ − T s ) + q l
(
) δ
1 − Y 1s
q l
=
. (13.92)
1 − Y 1∞ c p1 (T ∞ − T s ) + q l
Let p 1s be the partial pressure of the fuel vapour on the droplet surface. Thermodynamics
shows 13 that p 1s is determined by the temperature T s on the droplet surface.
13 See Prigogine, I. and Defay, R.: Chemical Thermodynamics. Longmans Green & Co., 1954, pp. 332
and f.