Untitled - Aerobib - Universidad Politécnica de Madrid

310 CHAPTER 13. COMBUSTION OF LIQUID FUELS
Exterior region r ≥ r l
In this region only oxygen and inert gases exist. Equation (13.11) applied to each of
them gives
Inert gases: 4πr 2 ρY 2 v 2 =m 2 . (13.18)
Oxygen: 4πr 2 ρY 3 v 3 =m 3 . (13.19)
Similarly, Eqs. (13.12) and (13.14) give
Y 2 v 2 + Y 3 v 3 = v, (13.20)
m 2 + m 3 = m. (13.21)
Let ν be the stoichiometric ratio of oxygen mass to fuel mass. Since, the fuel
mass reaching the flame per unit time is m, the oxygen mass that reaches the flame
per unit time must be νm. And since the oxygen moves towards the flame, v 3 must be
negative, that is
m 3 = −νm. (13.22)
From here and (13.21) there results for m 2
m 2 = (1 + ν)m. (13.23)
Equations (13.22) and (13.23) determine the constants for the continuity equations
(13.18) and (13.19) as functions of m, thus obtaining
4πr 2 ρY 2 v 2 = (1 + ν)m, (13.24)
4πr 2 ρY 3 v 3 = −νm. (13.25)
Therefore, within the exterior region the flame acts as a sink of strength νm for the
oxygen and as a source of strength (1 + ν)m for the combustion products.
13.7 Energy equation
The process is stationary and it takes place at constant pressure. Moreover the kinetic
energy of the motion and the work of the viscous forces can be neglected. Therefore
the summation of the heat and enthalpy fluxes through any spherical surface concentric
to the droplet must be constant. Due to the spherical symmetry, this condition can be
expressed in the form
∑
i
m i h i − 4πr 2 λ dT
dr
= const. (13.26)