Environmental Engineering Reference
In-Depth Information
2. When the assumption of infinite conductivity cannot be made, a finite conduc-
tivity model is needed. Then the temperature and consequently also the temper-
ature-dependent thermodynamic properties are spatially varying instead of
being uniform throughout the droplet. In the simplest case, there is no convec-
tive flow pattern inside the droplet, and the temperature only depends on the radial
coordinate. Using a spherical coordinate system, the conservation of energy is
expressed as
!
2
T
d
∂
∂
T
d
∂
=
λ
d
ρ
d
c
p
,
d
∂
+
2
r
∂
T
d
∂
ð
Eq
:
4
:
47
Þ
t
r
2
r
This equation is to be solved starting from an initial condition (the initial droplet
temperature) and using boundary conditions at the droplet center and at the
droplet surface. At the center, the flux in radial direction vanishes:
r=0
∂
T
d
∂
=0
ð
Eq
:
4
:
48
Þ
r
At the droplet surface, the inward heat flux just below the surface equals the sum
of the inward heat flux just above the surface and the heat used for evaporation
at the surface (h
fg
being the heat of evaporation):
inside
outside
−
m
:
d
h
fg
4
λ
d
∂
T
d
∂
λ
g
∂
T
∂
=
ð
Eq
:
4
:
49
Þ
r
d
r
r
π
3. In the case of a slip velocity between droplet and surroundings, also internal
convection can be present, and then heat transfer in large liquid droplets is usu-
ally not only governed by conduction but also by internal convection. The
importance of this effect increases with droplet size. To take convective effects
into account, the thermal conductivity can be replaced by an effective conduc-
tivity larger than the conductivity of the liquid (see Jenny et al. (2012) for details
and references).
4. Usually, the temperature of the droplets is much lower than the gas temperature,
and the contribution of radiation from the droplets is negligible compared to that
of the hot surroundings. The effect of radiation on heating and evaporation of
the droplet can be important for relatively large droplets in hot surroundings
such as spray flames. This has been investigated in detail by Sazhin et al.
(2006). In the derivation of a small-scale model for the net radiative heat source
of droplets, droplets are considered as semitransparent with a uniform but time-
dependent temperature, and the thermal radiation from the surroundings is
assumed to be that of a blackbody at temperature T
ext
. The effect of radiative
heating on droplets can be taken into account by an extra source term Q
r
in the
droplet temperature equation, which is of the form
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