Untitled - Aerobib - Universidad Politécnica de Madrid

314 CHAPTER 13. COMBUSTION OF LIQUID FUELS
The solution of Eq. (13.44) must satisfy another additional condition. In fact,
the mass fraction of oxygen must tend to the value Y 3∞ corresponding to the composition
of the atmosphere surrounding the droplet at great distance from the same
r → ∞ : Y 3 → Y 3∞ . (13.46)
This equation will be used in determining the burning velocity m of the droplet.
13.9 Combustion velocity of the droplet. Temperature
and position of the flame
From the preceding paragraph it results that the solution of the combustion problem
of a droplet reduces to the following:
1) The integration of the system of differential equations
4πr 2 λ dT ∫ (
T
∫ )
Ts
dr − m c p1 dT =m q l − c p1 dT , (13.32)
T 0
T 0
4πr 2 ρD 12
dY 1
dr = − m(1 − Y 1), (13.42)
for the determination of temperature T and mass fraction Y 1 of the fuel vapours
within the interior region r s ≤ r ≤ r l , with the two boundary conditions
r = r s : T = T s , (13.33)
r = r − l
: Y 1 = 0. (13.43)
2) The integration of the differential equations
4πr 2 λ dT ∫ (
T
dr − m c p dT =m q l − q r −
T 0
with boundary conditions
∫ Ts
T 0
c p1 dT
)
, (13.36)
4πr 2 ρD 23
dY 3
dr =m(ν + Y 3), (13.44)
r →∞ : T →T ∞ , (13.38)
r =r + l
: Y 3 =0, (13.45)
for the determination of temperature and mass fraction Y 3 of oxygen within the
exterior region r ≥ r l .