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7.2 Liquid Fuel Combustion 197

When the fuel droplets enter a combustion system, elevated surrounding tem-
perature enables the evaporation of the droplets and combustion takes place
between the oxidizer and the fuel vapors.

7.2.1 Droplet Vaporization

There have been many models for fuel droplet evaporation in spray combustion.
Most of them describe fuel droplets as spherical; the only relative motion between
the droplets and gas involves radial convection due to vaporization. A critical
review was given by Sirignano [42] for droplet vaporization in a high-temperature
environment. However, it would deviate too much from the scope of this book if we
continued on the combustion theories. For this reason, only a simple analysis is
introduced as follows.
The energy required to vaporize a fuel droplet can be calculated using

0
ð
q ¼ c p T v T 0 Þ þ h fg ð7:2Þ
where c p is the specific heat of the fuel, T v is the vaporization temperature for the
liquid fuel (K), T 0 is the initial temperature when the droplets are sprayed into
the combustion system, and h fg is the latent heat of vaporization of the fuel (J/mol).
The energy needed for the vaporization of the liquid droplets is taken from the
surrounding gases by thermal radiative and conductive heat transfer, which can be
found in many advanced heat and mass transfer books.
The droplet size decreases as the evaporation proceeds. Assuming vapor velocity
profile is symmetric around the center of the droplet and the rate of vaporization is
described as
2 2
_ m v ¼ q u4pr ¼ q u s 4pr ð7:3Þ
v v s
where _ m v is the vaporization rate (kg/s); it is also the loss rate with respective to the
liquid droplet. q is vapor density, u s is the vapor speed leaving the surface and r s is
v
2
the radius of the droplet. 4pr is the surface area of the droplet sphere.
s
Assuming that only conductive heat transfer dominates and that the radial pro-
files of temperature and compositions are quasi-steady, the change of the droplet
diameter can be related to the heat transfer from the surrounding to the droplet
surface. Conservation of energy for the droplet leads to
dT
2
_ m v c p T T v Þ þ _ m v h fg ¼ 4pr k ð7:4Þ
ð
dr
where T is the temperature at r, k is the thermal conductivity of the liquid droplet at
the surface. LHS is the energy for liquid evaporation and RHS is for the heat
transfer by conduction.