Environmental Engineering Reference
In-Depth Information
The question arises whether the droplet can be assumed to be at a uniform temper-
ature or not. If the thermal conductivity of the fuel is sufficiently large, the heat
received at the droplet surface will spread so rapidly to the interior that at any time
the droplet can be assumed to have uniform temperature T
d
. Models using this
assumption are called infinite conductivity or rapid-mixing models. In the presence
of heat transfer from or to the surroundings, T
d
is time dependent. In the frame of
the rapid-mixing model, it is straightforward to determine the evolution in time of
the droplet diameter d
d
=2r
d
, with r
d
being the radius. During the evaporation process,
the droplet diameter decreases with time, and so does its surface area A
d
=4
r
d
, vol-
π
r
d
, and mass m
d
=
ume V
d
=4
ρ
d
V
d
.
For the classical derivation of the droplet evaporation rate, we here follow Jenny
et al. (2012). In the absence of gravity, buoyancy effects do not play a role, and the
configuration of the evaporating droplet remains spherically symmetric. The problem
becomes mathematically more tractable if it is assumed that the droplet evaporates in a
very large (infinitely large) domain, where the conditions at infinity are not influenced
by the presence of the droplet. To solve the problem, both the continuity equations
expressing mass conservation and energy conservation have to be solved.
The flow around the droplet is caused by the vapor moving away from the droplet
surface. Due to the density difference between liquid and vapor, the vapor takes up
more volume and is pushed away from the droplet surface, causing the so-called
Stefan flux
ð
π
=
3
is the density of the gas mixture at the droplet surface and
u
r
the
mixture velocity at the droplet surface). Diffusive mixing between vapor and the sur-
rounding gas takes place. The surrounding gas diffuses toward the droplet surface, and
the evaporated fuel diffuses into the surroundings.
The net motion of fuel vapor is the sum of convection at the mixture velocity and
the diffusive velocity. The rate of change of the total droplet mass can then be
expressed as
ρ
u
r
(
ρ
dm
d
dt
D
∂
Y
∂
=
m
d
=
_
−
A
d
ρ
u
r
=
−
A
d
ρ
u
r
Y+
ρ
ð
Eq
:
4
:
31
Þ
r
Here, Y is the fuel vapor mass fraction, and
D
is the diffusion coefficient of the fuel
in air. If quasi-steady state is assumed, the fuel vapor conservation equation takes
the form
= 0 for
∂
∂
D
∂
Y
∂
r
2
_
m
d
Y+4
π
ρ
r
≥
r
d
ð
Eq
:
4
:
32
Þ
r
r
with boundary conditions
YrðÞ
=Y
s
and Y r
ð
!
∞
Þ
=Y
b
ð
Eq
:
4
:
33
Þ
Here, quasi-steady state means that the rate of regression of the surface is assumed
to be so slow compared to the change in fuel vapor concentration away from
the droplet that the vapor concentration profile can be found by solving the
Search WWH ::
Custom Search