Environmental Engineering Reference
In-Depth Information
A key characteristic of the evaporation process at constant droplet temperature and in
a steady environment is that the rate of change of the droplet diameter squared is a
constant, which is known as the d
2
-law:
d
dt
d
d
=
8
ρ
ρ
d
−
D
ln B
M
+1
ð
Þ
≡
−
K
ð
Eq
:
4
:
40
Þ
From this equation, it follows that the droplet diameter is given by d
d
=d
d
0
ðÞ
t
ðÞ−
Kt
and the droplet lifetime by
t
d
=
d
d
t=0
ð
Þ
ð
Eq
:
4
:
41
Þ
K
In realistic process conditions, the environment is not stagnant, and the evaporating
droplet can have a relative velocity compared to the surroundings, causing enhance-
ment of heat and mass transfer and increase of the evaporation rate by forced convec-
tion effects. To take this effect into account, it is necessary to multiply the rate with the
relative increase of the Nusselt number from the value 2 for stagnant conditions to the
value Nu
d
for flow around a sphere, leading to
d
dt
d
d
=
8
Nu
d
2
ρ
ρ
d
−
D
ln B
M
+1
ð
Þ
ð
Eq
:
4
:
42
Þ
As a first approximation, the Nusselt number Nu
d
for flow around a solid sphere given
by the Ranz
Marshall correlation Equation (4.26) can be used. The Prandtl number of
the surrounding gas appearing in the Nusselt number correlation is given by
-
Pr =
η
c
p
λ
=
ν
α
ð
Eq
:
4
:
43
Þ
In the preceding paragraphs, it was assumed that mass transfer from droplet to
surroundings is taking place while also assuming constant physical properties at the
droplet surface. In reality, the droplet temperature may change with time. This process
is described by the droplet temperature evolution equation. Assuming a homogeneous
droplet temperature, the droplet temperature and the surface temperature are the same
(T
d
=T
s
), and it follows from energy conservation that the droplet temperature
evolves as
dT
d
dt
=
T
b
−
T
d
τ
d
,
T
h
fg
m
d
c
p
,
F
_
+
m
d
ð
Eq
:
4
:
44
Þ
with c
p,F
being the specific heat capacity of the fuel and
τ
d,T
the droplet thermal relax-
ation time, given by
τ
d
,
T
=
ρ
d
c
p
,
F
d
d
6Nu
d
λ
m
ð
Eq
:
4
:
45
Þ
Here, the thermal conductivity and the material properties entering the Nusselt num-
ber are evaluated as an average using the 1/3 rule. Equation (4.44) expresses that the
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