Environmental Engineering Reference
In-Depth Information
V
m
¼ V
m
;
air
V
m
;
water
ð
1
Þ
flow-rate fraction is used to create relationships for the mean density and
viscosity of the two-phase system as de
A mass
fl
ned in Eq. (
2
).
V
a
A
a
q
a
V
a
A
a
q
a
þ
V
w
A
w
q
w
x ¼
ð
2
Þ
Using the mass
fl
flow-rate fraction presented in Eq. (
2
), the two-phase density is
de
ned as shown in Eq. (
3
).
1
x
q
a
þ
1
x
q
w
q
m
¼
ð
3
Þ
The two-phase viscosity is de
ned for the system in Eq. (
4
)
1
x
l
a
þ
1
x
l
w
l
m
¼
ð
4
Þ
The speci
c Reynolds number equation for the system using Eqs. (
1
)
(
4
)is
-
given in Eq. (
5
).
q
m
V
m
D
h
l
m
Re
¼
ð
5
Þ
The speci
c Weber number is written using velocity and density de
nitions from
Eqs. (
1
) and (
3
) and given as:
2
q
m
V
m
D
h
r
We
¼
ð
6
Þ
where
is the surface tension of water with an air interface. Since it is desired to
know the extent of breakup occurring over the experimental section, the mass
conservation principle is applied to liquid water volume beneath the time-averaged
boundary layer between the two-phase and liquid water
˃
flows. With the time-
averaged condition, the conservation law is limited to the liquid water mass entering
and leaving the control volume as given in Eq. (
7
) and (
5
)
fl
m
m
;
w
;
i
¼ m
m
;
w
;
b
þ
m
m
;
w
;
e
ð
7
Þ
where m is the mass
fl
flow rate, and the indices m, w, i, b, and e denote; the mean,
liquid water, inlet
fl
flow, water breakup in the two-phase outlet
fl
flow, and liquid exit
fl
flow, respectively. With constant density, Eq. (
7
) is reduced to volumetric balance
as shown in Eq. (
8
). The constant density condition is valid unless the static
pressure drops to or below the vapor pressure of water. In such cases, the mass
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