Untitled - Aerobib - Universidad Politécnica de Madrid

306 CHAPTER 13. COMBUSTION OF LIQUID FUELS
present approximate study its influence is neglected. Otherwise, the analysis of the
problem becomes far more complicated. By doing so the results obtained do not
substantially varies.
The heat received by the droplet is partially used for its heating and partially for
the evaporation of the fuel. The first fraction increases the temperature of the droplet
up to the boiling point T s of the fuel. Thereon it remains constant and equal to T s . All
the heat received is used up in evaporating the fuel. Hereinafter, the transient initial
period will be neglected by assuming that the droplet temperature is initially T s and
that no thermal gradients exist therein.
Strictly the process is non-stationary since the size of the droplet decreases as
combustion progresses. However, the recession velocity of the droplet surface is very
small compared to the velocity of the burnt gases. Thereby, an excellent approximation
is obtained by assuming that the phenomenon is stationary. That is to say by
neglecting the local variations with time of temperature and mass fractions produced
by the droplet reduction in size. This is an important simplification of the problem
since it eliminates an independent variable from the equations.
The existence of free and forced convection introduces a privileged direction
destroying the spherical symmetry of the phenomenon. Even in the case of free convection
with approximately round droplets experimental results show that the shape
of the flame differs appreciably from the spherical within the upper region. A study of
the problem, taking into account these convection effects is very arduous and has not
yet been achieved. Nevertheless, when assuming that both the droplet and the flame
front are spherical, that is when neglecting these effects, the results obtained show a
good agreement with the experimental results as for the overall magnitudes such as the
droplet burning velocity and its law of variation with size. Thereby, neglect of convection
effects is justified. With this assumption and the assumption of quasi-stationary
state previously stated, there is only one independent variable, this is the distance r to
the center of the droplet.
The composition of the fuel vapours, oxidizing atmosphere and burnt gases is
very complex. However, as done in the study of diffusion flames, we shall assume for
simplicity that only three different chemical species exist, namely: fuel, oxidizer and
inert gases, which include these in the atmosphere surrounding the droplet as well as
the combustion products.
Summarizing, the following assumptions are adopted:
1) The phenomenon has spherical symmetry.