two-phase flow across sudden expansion/contractions · 1 richardson (1958) romie (1958) schmidt and...
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![Page 1: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/1.jpg)
Chi-Chuan Wang*, PhD ([email protected])
Fellow ASME, Fellow ASHRAE
*Currently a professor @ Department of Mechanical Engineering, National Chiao Tung University, Hsinchu, Taiwan. The work was presented as a Keynote lecture in HEAT 2008 (the Fifth International Conference on Transport Phenomena in Multiphase Systems, June 30 - July 3, 2008, Bialystok, Poland) when the author was a senior lead researcher in Industrial Technology Research Institute.
Two-phase Flow Across Sudden
Expansion/Contractions
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Introduction
Correlation Review
Two-phase pressure change across sudden
enlargement
Two-phase pressure change across sudden
contraction
Experimental setup
Results and Observations
Correlation Assessment
Conclusions
Outline
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Introduction
Fluid flow across singularity such as sudden expansion/contraction are is quite common in typical heat exchanging devices/piping connections. Irreversible pressure loss accompanies with these singularities.
However, the pressure loss is recognized as minor loss when comparing to major frictional loss. This is often valid in macro tube systems but is questionable in mini/micro tube systems.
For smaller heat exchanging device (e.g. micro heat exchangers). This loss becomes more pronounced and may be as high as 5-20% depending on the contraction ratio and flow rate. The loss becomes even worse for two-phase system (exceeds 35~40% in extreme cases).
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Two-phase flow across contractions/expansions
contraction
expansion
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Motivations of this study
Comparatively rare data subject to sudden area
change in mini/micro channels.
Therefore the objectives of this study includes:
To provide the new data in this range. (mini
channels)
Assessment of the existing correlations.
To develop the a modified predictive method that
is capable of predicting the present database and
existing data.
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Estimation of Pressure drop
- Single phase flow
By combining the pressure balance equation, the static
pressure difference subject to the sudden expansion is
related to the kinetic energy of the flow:
A: contraction ratio
G: mass flux
Sudden contraction is normally in terms of the
contraction loss coefficient (K) and multiplication of
kinetic energy of the fluid flow:
2 2 1
2
eΑ Α
Pσ σ
G
2
2
cPK
G
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Some correlations/models for 2-phase flow
◆Sudden Contraction
※Schmidt and Friedel-1997
※Homogeneous Model
※Chisholm-1967
※Abdelall-2005
◆Sudden Expansion
※Schmidt and Friedel-1996
※Homogeneous Model
※Chisholm and Sutherland- 1969
※Wadle-1989
※Abdelall-2005
※Richardson-1958
※Lottes-1960
※Attou and Bolle-1997
※Romie-1958
![Page 8: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/8.jpg)
Correlations for sudden expansion
Attou and Bolle (1997)
Abdelall et al. (2005)
Chisholm and Sutherland (1969)
Homogeneous Model
Lottes (1961)
Richardson (1958)
Romie (1958)
Schmidt and Friedel (1996)
Wadle (1989)
2
21 1
1
rA Ar
e A AL
GP G
2 2
2
2 2
0.5 12 1 10.5
h AL A A h L A
e eI eR L
GP P P G
2
221 / 1/
1 1h
e A AL
C X XP G x
2 1
1e A AL G
x xP G
2
2
1
1
A Ae
L
GP
2
2 21
0.5 11
A
e AL
xP G
22 21 1 /
1
A A L Ge
L
G x xP
2 22 1
11
1 1
A Ae eff A
eff eff G L
ee A
xxG f
P
2 22 2 11
12
e AL G
x xP m K
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Correlations applicable for sudden contraction
Abdelall et al. (2005)
Chisholm (1983)
Homogeneous Model
Schmidt and Friedel (1997)
2
22
2
1
1
2
h AC C
c
C CP G
2 1 -1 ( 1- )Lc cL
G
P P Bx x x
2 2
1 21 1 1 12
Lc C A
L G
GP C x
1/ 22 2 21 1[ ( ) (1 ) ]
(1 )
11 ( 1)
Acon eff A
eff eff G L
c
con
A
x xG f
p
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Sudden Contraction -Schmidt and Friedel
Area Ratio Inner Pipe diameter
Inlet (mm) Outlet (mm)
0.057 17.2 72.2
0.115 19.0 56.0
0.167 29.5 72.2
0.180 24.0 56.5
0.276 29.5 56.2
0.375 44.2 72.2
0.445 29.5 44.2
0.619 44.2 56.2
•Sudden Contraction
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Sudden Expansion-Schmidt and Friedel
22 1/ 2 21[ ( )(1 ) ]
(1 )
1 (1 )
A Ae eff A
eff eff G L
e
e A
x xG f
p
•Sudden Expansion
Area Ratio Inner Pipe diameter
Inlet (mm) Outlet (mm)
0.057 17.2 72.2
0.115 19.0 56.0
0.167 29.5 72.2
0.180 24.0 56.5
0.276 29.5 56.2
0.375 44.2 72.2
0.445 29.5 44.2
0.619 44.2 56.2
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Sudden Contraction-Abdelall
Micro tubes
•Sudden Contraction
water air
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Sudden Expansion-Abdelall
2
, 2 2
1
2
Ae R
GP
2
,
2 11
2
L A A Le I h A
L
GP
ΔPe = ΔPeR - ΔPeI
•Sudden Expansion
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Some comments
Correlations developed by previous researchers
are generally applicable for their own database
only. Extending the correlation outside their
database are not recommended.
A preliminary screening indicates the deviation
for expansion amid different sources is much
larger in association with data for contraction.
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Schematic for Experimental setup for expansion
Inlet
Outlet
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Some database collected for comparisons
- contractions
Researchers Geiger(1964)
3 tubes
McGee(1966)
2 tubes
Schmidt & Friedel(1997)
1 tube
Abdellal (2005)
1 tube
Present Data
6 tubes
Mass flux
(kg/m2s)
2313-8116
4013-12682
6941-21448
3251-6471
1778-4421 500-4000 2700-6200 100-700
Quality
0.0117-0.265
0.0001-0.245
0.0001-0.159
0-0.981
0-0.323 0.01-0.90 1.9x10-4-1.6x10-3 0.001-0.8
Working fluid Steam-Water
at 194-241 C
Steam-Water
at 141-195 C Air-Water at 25 C Air-Water at 25 C Air-Water at 25 C
Hydraulic
Diameter (mm)
9.70-25.53
12.88-25.53
16.10-25.53
8.64-11.68
11.68-14.99 17.2-72.2 0.84 -1.6
2-2.67 2-4 2-3
2-4.8 3-4 3-4.5
Contraction
ratio
0.144
0.253
0.398
0.546
0.608 0.057 0.26
0.39 0.19 0.26
0.13 0.39 0.26
Bond number 7-19 15-55 9.988 0.024 0.13-0.29
Test Points 210 44 77 26 240
Mean Deviation
(%) by
Homogenous
Model
19.90
24.34
15.56
20.43
19.43 43.64 468.79
33.82 30.39 41.42
47.61 54.27 60.84
![Page 17: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/17.jpg)
Pressure drops for sudden contraction
63 rectangular channel into 3 mm tube
10-2
10-1
100
101
102
PS
ud
den
co
ntr
acti
on
(kP
a)
10-3
10-2
10-1
100
x
G=100kg/m2·s
10-2
10-1
100
101
102
PS
ud
den
co
ntr
acti
on
(kP
a)
10-3
10-2
10-1
100
x
G=300kg/m2·s
10-2
10-1
100
101
102
PS
ud
den
co
ntr
acti
on
(kP
a)
10-3
10-2
10-1
100
x
G=500kg/m2·s
10-2
10-1
100
101
102
PS
ud
den
co
ntr
acti
on
(kP
a)
10-3
10-2
10-1
100
x
G=700kg/m2·s
6 × 3 mm rectangular channel to 3 mm tube
6x3 G=100 kg/m2s G=500 kg/m2
s
0.001 (Plug) (Plug)
0.005 (Plug) (Stratified)
0.01 (Stratified) (Annular)
0.05 (Wavy) (Annular)
0.1 (Annular) (Annular)
![Page 18: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/18.jpg)
10-2
10-1
100
101
102
PS
ud
den
co
ntr
acti
on
(kP
a)
10-3
10-2
10-1
100
x
G=100kg/m2·s
10-2
10-1
100
101
102
PS
ud
den
co
ntr
acti
on
(kP
a)
10-3
10-2
10-1
100
x
G=300kg/m2·s
10-2
10-1
100
101
102
PS
ud
den
co
ntr
acti
on
(kP
a)
10-3
10-2
10-1
100
x
G=500kg/m2·s
10-2
10-1
100
101
102
PS
ud
den
co
ntr
acti
on
(kP
a)
10-3
10-2
10-1
100
x
G=700kg/m2·s
9 × 3 mm rectangular channel to 3 mm tube
9x3 G=100 kg/m2s G=300 kg/m2
s
0.001 (Plug) (Plug)
0.005 (Plug) (Plug)
0.01 (Stratified) (Annular)
0.05 (Wavy) (Annular)
0.1 (Annular) (Annular)
Pressure drops for sudden contraction
93 rectangular channel into 3 mm tube
![Page 19: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/19.jpg)
10-2
10-1
100
101
102
PS
ud
den
co
ntr
acti
on
(kP
a)
10-3
10-2
10-1
100
x
G=100kg/m2·s
10-2
10-1
100
101
102
PS
ud
den
co
ntr
acti
on
(kP
a)
10-3
10-2
10-1
100
x
G=300kg/m2·s
10-2
10-1
100
101
102
PS
ud
den
co
ntr
acti
on
(kP
a)
10-3
10-2
10-1
100
x
G=500kg/m2·s
10-2
10-1
100
101
102
PS
ud
den
co
ntr
acti
on
(kP
a)
10-3
10-2
10-1
100
x
G=700kg/m2·s
4 × 2 mm rectangular channel to 2 mm tube
4x2 G=100 kg/m2s G=500 kg/m2
s
0.001 (Plug) (Plug)
0.005 (Plug) (Plug)
0.01 (Annular) (Annular)
0.05 (Annular) (Annular)
0.1 (Annular) (Annular)
Pressure drops for sudden contraction
42 rectangular channel into 2 mm tube
![Page 20: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/20.jpg)
Contraction: 42 rectangular tube into
a 2mm tube
G = 300 kg/m2s, x =0.05
![Page 21: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/21.jpg)
10-2
10-1
100
101
102
PS
ud
den
co
ntr
acti
on
(kP
a)
10-3
10-2
10-1
100
x
G=100kg/m2·s
10-2
10-1
100
101
102
PS
ud
den
co
ntr
acti
on
(kP
a)
10-3
10-2
10-1
100
x
G=300kg/m2·s
10-2
10-1
100
101
102
PS
ud
den
co
ntr
acti
on
(kP
a)
10-3
10-2
10-1
100
x
G=500kg/m2·s
10-2
10-1
100
101
102
PS
ud
den
co
ntr
acti
on
(kP
a)
10-3
10-2
10-1
100
x
G=700kg/m2·s
4 × 4 mm rectangular channel to 2 mm tube
4x4 G=100 kg/m2s G=500 kg/m2
s
0.001 (Plug) (Plug)
0.005 (Plug) (Plug)
0.01 (Plug) (Annular)
0.05 (Annular) (Annular)
0.1 (Annular) (Annular)
Pressure drops for sudden contraction
44 rectangular channel into 2 mm tube
![Page 22: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/22.jpg)
Contraction: 44 rectangular tube into
a 2mm tube
G = 300 kg/m2s, x =0.01
![Page 23: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/23.jpg)
10-2
10-1
100
101
102
PS
ud
den
co
ntr
acti
on
(kP
a)
10-3
10-2
10-1
100
x
G=100kg/m2·s
10-2
10-1
100
101
102
PS
ud
den
co
ntr
acti
on
(kP
a)
10-3
10-2
10-1
100
x
G=300kg/m2·s
10-2
10-1
100
101
102
PS
ud
den
co
ntr
acti
on
(kP
a)
10-3
10-2
10-1
100
x
G=500kg/m2·s
10-2
10-1
100
101
102
PS
ud
den
co
ntr
acti
on
(kP
a)
10-3
10-2
10-1
100
x
G=700kg/m2·s
6 × 2 mm rectangular channel to 2 mm tube
6x2 G=300 kg/m2s G=700 kg/m2
s
0.001 (Plug) (Plug)
0.005 (Plug) (Annular)
0.01 (Annular)
(Annular)
0.05 (Annular) (Annular)
0.1 (Annular) (Annular)
Pressure drops for sudden contraction
62 rectangular channel into 2 mm tube
![Page 24: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/24.jpg)
Contraction: 62 rectangular tube into
a 2mm tube
G = 300 kg/m2s, x =0.05
![Page 25: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/25.jpg)
10-2
10-1
100
101
102
PS
ud
den
co
ntr
acti
on
(kP
a)
10-3
10-2
10-1
100
x
G=100kg/m2·s
10-2
10-1
100
101
102
PS
ud
den
co
ntr
acti
on
(kP
a)
10-3
10-2
10-1
100
x
G=300kg/m2·s
10-2
10-1
100
101
102
PS
ud
den
co
ntr
acti
on
(kP
a)
10-3
10-2
10-1
100
x
G=500kg/m2·s
10-2
10-1
100
101
102
PS
ud
den
co
ntr
acti
on
(kP
a)
10-3
10-2
10-1
100
x
G=700kg/m2·s
6 × 4 mm rectangular channel to 2 mm tube
6x4 G=300 kg/m2s G=700 kg/m2
s
0.001 (Plug) (Plug)
0.005 (Plug) (Annular)
0.01 (Plug) (Annular)
0.05 (Annular) (Annular)
0.1 (Annular) (Annular)
Pressure drops for sudden contraction
64 rectangular channel into 2 mm tube
![Page 26: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/26.jpg)
Contraction: 64 rectangular tube into a
2mm tube
G = 300 kg/m2s, x =0.005
![Page 27: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/27.jpg)
Correlation Assessment
10-2
10-1
100
101
102
103
P(C
his
ho
lm, K
pa
)
10-2
10-1
100
101
102
103
P(Experiment Data, Kpa)
Schmidt and Friedel (77 Points)
10-2
10-1
100
101
102
103
P(C
his
ho
lm, K
pa
)
10-2
10-1
100
101
102
103
P(Experiment Data, Kpa)
Abdelall (26 Points)
10-2
10-1
100
101
102
103
P(C
his
ho
lm, K
pa
)
10-2
10-1
100
101
102
103
P(Experiment Data, Kpa)
McGee (44 Points)
10-2
10-1
100
101
102
103
P(C
his
ho
lm, K
pa
)
10-2
10-1
100
101
102
103
P(Experiment Data, Kpa)
Geiger (210 Points)
10-2
10-1
100
101
102
103
P(C
his
ho
lm, K
pa
)
10-2
10-1
100
101
102
103
P(Experiment Data, Kpa)
Present Data (240 Points)
Mean Deviation = 43.62%
Mean Deviation = 225.74%
Mean Deviation = 60.82%
Mean Deviation = 48.73%
Mean Deviation = 50.62%
Total Mean Deviation = 50.62%
Total Points = 597
+50%
-50%
10-2
10-1
100
101
102
103
P(S
ch
mid
t a
nd
Frie
del, K
pa
)
10-2
10-1
100
101
102
103
P(Experiment Data, Kpa)
Schmidt and Friedel (77 Points)
10-2
10-1
100
101
102
103
P(S
ch
mid
t a
nd
Frie
del, K
pa
)
10-2
10-1
100
101
102
103
P(Experiment Data, Kpa)
Abdelall (26 Points)
10-2
10-1
100
101
102
103
P(S
ch
mid
t a
nd
Frie
del, K
pa
)
10-2
10-1
100
101
102
103
P(Experiment Data, Kpa)
McGee (44 Points)
10-2
10-1
100
101
102
103
P(S
ch
mid
t a
nd
Frie
del, K
pa
)
10-2
10-1
100
101
102
103
P(Experiment Data, Kpa)
Geiger (210 Points)
10-2
10-1
100
101
102
103
P(S
ch
mid
t a
nd
Frie
del, K
pa
)
10-2
10-1
100
101
102
103
P(Experiment Data, Kpa)
Present Data (240 Points)
Mean Deviation = 48.35%
Mean Deviation = 191.46%
Mean Deviation = 65.25%
Mean Deviation = 52.58%
Mean Deviation = 55.28%
Total Mean Deviation = 59.52%
Total Points = 597
+50%
-50%
![Page 28: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/28.jpg)
Correlation Assessment
10-2
10-1
100
101
102
103
P(A
bd
ela
ll, K
pa)
10-2
10-1
100
101
102
103
P(Experiment Data, Kpa)
Schmidt and Friedel (77 Points)
10-2
10-1
100
101
102
103
P(A
bd
ela
ll, K
pa)
10-2
10-1
100
101
102
103
P(Experiment Data, Kpa)
Abdelall (26 Points)
10-2
10-1
100
101
102
103
P(A
bd
ela
ll, K
pa)
10-2
10-1
100
101
102
103
P(Experiment Data, Kpa)
McGee (44 Points)
10-2
10-1
100
101
102
103
P(A
bd
ela
ll, K
pa)
10-2
10-1
100
101
102
103
P(Experiment Data, Kpa)
Geiger (210 Points)
10-2
10-1
100
101
102
103
P(A
bd
ela
ll, K
pa)
10-2
10-1
100
101
102
103
P(Experiment Data, Kpa)
Present Data (240 Points)
Mean Deviation = 31.89%
Mean Deviation = 16.61%
Mean Deviation = 75.09%
Mean Deviation = 93.11%
Mean Deviation = 63.75%
Total Mean Deviation = 68.84%
Total Points = 597
+50%
-50%
10-2
10-1
100
101
102
103
P(H
om
og
en
eo
us
Mo
del, K
pa
)
10-2
10-1
100
101
102
103
P(Experiment Data, Kpa)
Schmidt and Friedel (77 Points)
10-2
10-1
100
101
102
103
P(H
om
og
en
eo
us
Mo
del, K
pa
)
10-2
10-1
100
101
102
103
P(Experiment Data, Kpa)
Abdelall (26 Points)
10-2
10-1
100
101
102
103
P(H
om
og
en
eo
us
Mo
del, K
pa
)
10-2
10-1
100
101
102
103
P(Experiment Data, Kpa)
McGee (44 Points)
10-2
10-1
100
101
102
103
P(H
om
og
en
eo
us
Mo
del, K
pa
)
10-2
10-1
100
101
102
103
P(Experiment Data, Kpa)
Geiger (210 Points)
10-2
10-1
100
101
102
103
P(H
om
og
en
eo
us
Mo
del, K
pa
)
10-2
10-1
100
101
102
103
P(Experiment Data, Kpa)
Present Data (240 Points)
Mean Deviation = 50.50%
Mean Deviation = 468.79%
Mean Deviation = 19.93%
Mean Deviation = 20.69%
Mean Deviation = 48.50%
Total Mean Deviation = 55.88%
Total Points = 597
+50%
-50%
![Page 29: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/29.jpg)
Further examination of the existing Data and
Some critical parameters
Researchers Geiger(1964)
3 tubes
McGee(1966)
2 tubes
Schmidt & Friedel(1997)
1 tube
Abdellal (2005)
1 tube
Present Data
6 tubes
Mass flux
(kg/m2s)
2313-8116
4013-12682
6941-21448
3251-6471
1778-4421 500-4000 2700-6200 100-700
Quality
0.0117-0.265
0.0001-0.245
0.0001-0.159
0-0.981
0-0.323 0.01-0.90 1.9x10-4-1.6x10-3 0.001-0.8
Working fluid Steam-Water
at 194-241 C
Steam-Water
at 141-195 C Air-Water at 25 C Air-Water at 25 C Air-Water at 25 C
Hydraulic
Diameter (mm)
9.70-25.53
12.88-25.53
16.10-25.53
8.64-11.68
11.68-14.99 17.2-72.2 0.84 -1.6
2-2.67 2-4 2-3
2-4.8 3-4 3-4.5
Contraction
ratio
0.144
0.253
0.398
0.546
0.608 0.057 0.26
0.39 0.19 0.26
0.13 0.39 0.26
Bond number 7-19 15-55 9.988 0.024 0.13-0.29
Test Points 210 44 77 26 240
Mean Deviation
(%) by
Homogenous
Model
19.90
24.34
15.56
20.43
19.43 43.64 468.79
33.82 30.39 41.42
47.61 54.27 60.84
![Page 30: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/30.jpg)
A rational modification to the Homogeneous
Model
0.1
10.072 101 0.92 2.13 1 0.28 1.1
Bo
BoC
Modified HP P e Bo e
2 2 L G
G G
dWe G x
2( )( / 2)L Gg DBo
1 100 AC e
ABo We Bo
10-2
10-1
100
101
102
103
P(R
evis
ed
Hom
ogen
eou
s M
od
el, K
pa)
10-2
10-1
100
101
102
103
P(Experiment Data, Kpa)
Schmidt and Friedel (77 Points)
10-2
10-1
100
101
102
103
P(R
evis
ed
Hom
ogen
eou
s M
od
el, K
pa)
10-2
10-1
100
101
102
103
P(Experiment Data, Kpa)
Abdelall (26 Points)
10-2
10-1
100
101
102
103
P(R
evis
ed
Hom
ogen
eou
s M
od
el, K
pa)
10-2
10-1
100
101
102
103
P(Experiment Data, Kpa)
McGee (44 Points)
10-2
10-1
100
101
102
103
P(R
evis
ed
Hom
ogen
eou
s M
od
el, K
pa)
10-2
10-1
100
101
102
103
P(Experiment Data, Kpa)
Geiger (210 Points)
10-2
10-1
100
101
102
103
P(R
evis
ed
Hom
ogen
eou
s M
od
el, K
pa)
10-2
10-1
100
101
102
103
P(Experiment Data, Kpa)
Present Data (240 Points)
Mean Deviation = 22.89%
Mean Deviation = 29.89%
Mean Deviation = 22.62%
Mean Deviation = 23.96%
Mean Deviation = 37.19%
Total Mean Deviation = 29.30%
Total Points = 597
+50%
-50%
![Page 31: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/31.jpg)
Short summary – Contraction
At a rather low mass flux where the upstream/downstream flow patterns are wavy flow or a “Liquid like vena contracta” occurring at a certain mass flux/quality may lead to a fall-off contraction pressure drop.
The “Liquid like vena contracta” seems to occur in front of the contraction.
Correlation/model assessment indicates that the homogeneous model gives the best predictive ability with a mean deviation of 57.45%.
By introducing a correction factor, a modified homogeneous correlation is proposed. The correction factor takes into account the influence of Bond number with a mean deviation of 29.30%.
![Page 32: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/32.jpg)
Expansion
![Page 33: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/33.jpg)
Test Results for sudden expansion
3 mm tube into 63 rectangular channel
10-2
10-1
100
101
102
PS
ud
den
ex
pa
nsi
on
(k
Pa
)
10-3
10-2
10-1
100
x
G=100kg/m2·s
10-2
10-1
100
101
102
PS
ud
den
ex
pa
nsi
on
(k
Pa
)
10-3
10-2
10-1
100
x
G=300kg/m2·s
10-2
10-1
100
101
102
PS
ud
den
ex
pa
nsi
on
(k
Pa
)
10-3
10-2
10-1
100
x
G=500kg/m2·s
10-2
10-1
100
101
102
PS
ud
den
ex
pa
nsi
on
(k
Pa
)
10-3
10-2
10-1
100
x
G=700kg/m2·s
3 mm tube to 6 × 3 mm rectangular channel
6x3 G=100 kg/m2s G=500 kg/m2
s
0.001 (Plug) (Slug)
0.005 (Plug) (Plug)
0.01 (Annular) (Annular)
0.05 (Annular)
(Annular)
0.1 (Annular) (Annular)
![Page 34: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/34.jpg)
10-2
10-1
100
101
102
PS
ud
den
ex
pa
nsi
on
(k
Pa
)
10-3
10-2
10-1
100
x
G=100kg/m2·s
10-2
10-1
100
101
102
PS
ud
den
ex
pa
nsi
on
(k
Pa
)
10-3
10-2
10-1
100
x
G=300kg/m2·s
10-2
10-1
100
101
102
PS
ud
den
ex
pa
nsi
on
(k
Pa
)
10-3
10-2
10-1
100
x
G=500kg/m2·s
10-2
10-1
100
101
102
PS
ud
den
ex
pa
nsi
on
(k
Pa
)
10-3
10-2
10-1
100
x
G=700kg/m2·s
3 mm tube to 9 × 3 mm rectangular channel
9x3 G=100 kg/m2s G=500 kg/m2
s
0.001 (Plug) (Slug)
0.005 (Plug) (Annular)
0.01 (Plug) (Annular)
0.05 (Annular)
(Annular)
0.1 (Annular) (Annular)
Test Results for sudden expansion
3 mm tube into 93 rectangular channel
![Page 35: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/35.jpg)
10-2
10-1
100
101
102
PS
ud
den
ex
pa
nsi
on
(k
Pa)
10-3
10-2
10-1
100
x
G=100kg/m2·s
10-2
10-1
100
101
102
PS
ud
den
ex
pa
nsi
on
(k
Pa)
10-3
10-2
10-1
100
x
G=300kg/m2·s
10-2
10-1
100
101
102
PS
ud
den
ex
pa
nsi
on
(k
Pa)
10-3
10-2
10-1
100
x
G=500kg/m2·s
10-2
10-1
100
101
102
PS
ud
den
ex
pa
nsi
on
(k
Pa)
10-3
10-2
10-1
100
x
G=700kg/m2·s
4x2 G=100 kg/m2s G=500 kg/m2
s
0.001 (Slug) (Slug)
0.005 (Slug) (Plug)
0.01 (Plug)
(Annular)
0.05 (Annular) (Annular)
0.1 (Annular) (Annular)
2 mm tube to 4 × 2 mm rectangular channel
Test Results for sudden expansion
2 mm tube into 42 rectangular channel
![Page 36: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/36.jpg)
Expansion: 2 mm tube into 42
rectangular cross section
G = 300 kg/m2s, x =0.005 G = 100 kg/m2s, x =0.01
![Page 37: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/37.jpg)
10-2
10-1
100
101
102
PS
ud
den
ex
pa
nsi
on
(k
Pa
)
10-3
10-2
10-1
100
x
G=100kg/m2·s
10-2
10-1
100
101
102
PS
ud
den
ex
pa
nsi
on
(k
Pa
)
10-3
10-2
10-1
100
x
G=300kg/m2·s
10-2
10-1
100
101
102
PS
ud
den
ex
pa
nsi
on
(k
Pa
)
10-3
10-2
10-1
100
x
G=500kg/m2·s
10-2
10-1
100
101
102
PS
ud
den
ex
pa
nsi
on
(k
Pa
)
10-3
10-2
10-1
100
x
G=700kg/m2·s
4x4 G=100 kg/m2s G=500 kg/m2
s
0.001 (Slug) (Plug)
0.005 (Slug) (Plug)
0.01 (Plug) (Annular)
0.05 (Plug) (Annular)
0.1 (Annular) (Annular)
2 mm tube to 4 × 4 mm rectangular channel
Test Results for sudden expansion
2 mm tube into 44 rectangular channel
![Page 38: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/38.jpg)
G = 100 kg/m2s, x =0.01
Expansion: 2 mm tube
into 44 rectangular cross section
![Page 39: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/39.jpg)
10-2
10-1
100
101
102
PS
ud
den
ex
pa
nsi
on
(k
Pa
)
10-3
10-2
10-1
100
x
G=100kg/m2·s
10-2
10-1
100
101
102
PS
ud
den
ex
pa
nsi
on
(k
Pa
)
10-3
10-2
10-1
100
x
G=300kg/m2·s
10-2
10-1
100
101
102
PS
ud
den
ex
pa
nsi
on
(k
Pa
)
10-3
10-2
10-1
100
x
G=500kg/m2·s
10-2
10-1
100
101
102
PS
ud
den
ex
pa
nsi
on
(k
Pa
)
10-3
10-2
10-1
100
x
G=700kg/m2·s
6x2 G=300 kg/m2s G=700 kg/m2
s
0.001 (Slug) (Plug)
0.005 (Plug) (Plug)
0.01 (Plug) (Annular)
0.05 (Annular) (Annular)
0.1 (Annular) (Annular)
2 mm tube to 6 × 2 mm rectangular channel
Test Results for sudden expansion
2 mm tube into 62 rectangular channel
![Page 40: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/40.jpg)
Expansion: 2 mm tube into 62
rectangular cross section
G = 300 kg/m2s, x =0.005
![Page 41: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/41.jpg)
10-2
10-1
100
101
102
PS
ud
den
ex
pa
nsi
on
(k
Pa
)
10-3
10-2
10-1
100
x
G=100kg/m2·s
10-2
10-1
100
101
102
PS
ud
den
ex
pa
nsi
on
(k
Pa
)
10-3
10-2
10-1
100
x
G=300kg/m2·s
10-2
10-1
100
101
102
PS
ud
den
ex
pa
nsi
on
(k
Pa
)
10-3
10-2
10-1
100
x
G=500kg/m2·s
10-2
10-1
100
101
102
PS
ud
den
ex
pa
nsi
on
(k
Pa
)
10-3
10-2
10-1
100
x
G=700kg/m2·s
2 mm tube to 6 × 4 mm rectangular channel
Test Results for sudden expansion
2 mm tube into 64 rectangular channel
![Page 42: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/42.jpg)
Collected dData for comparison/development
Researchers Abdellal Mendler McGee Schmidt & Friedel Present study
Mass flux (kg/m2s) 3000-5000
8928-18480
4528-11280
2304-4576
1778-4416
3248-6464
3248-8096
1000
1000-3000 100-700
Quality 0.0022-0.006
0.032-0.139
0.040-0.181
0.037-0.185
0-0.304
0-0.249
0-0.229
0.01-0.99
0.01-0.99 0.001-0.8
Working fluid Air-Water at 25 C Steam-Water
at 194-252 C
Steam-Water
at 141-195 C Air-Water at 25 C Air-Water at 25 C
Hydraulic
Diameter (mm) 0.84 -1.6
9.55-25.09
12.90-25.09
17.63-25.09
11.43-14.99
8.636-14.99
8.636-11.68
17.2-72.2
19.0-62.1
2-2.67 2-4 2-3
2-4.8 3-4 3-4.5
Contraction ratio 0.26
0.145
0.264
0.493
0.608
0.332
0.546
0.0568
0.0937
0.39 0.19 0.26
0.13 0.39 0.26
Bond number 0.024 7-30 7-20 9.988
12.188 0.13-0.29
Total Points 14 90 65 90 240
![Page 43: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/43.jpg)
Correlation assessment
10-2
10-1
100
101
102
103
P(S
ch
mid
t an
d F
rie
del,k
Pa)
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Schmidt and Friedel (90 Points)
10-2
10-1
100
101
102
103
P(S
ch
mid
t an
d F
rie
del,k
Pa)
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Abdelall (14 Points)
10-2
10-1
100
101
102
103
P(S
ch
mid
t an
d F
rie
del,k
Pa)
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Present Data (217 Points)
10-2
10-1
100
101
102
103
P(S
ch
mid
t an
d F
rie
del,k
Pa)
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
McGee (65 Points)
10-2
10-1
100
101
102
103
P(S
ch
mid
t an
d F
rie
del,k
Pa)
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Mendler (90 Points)
+50%
-50%Mean Deviation = 28.11%
Mean Deviation = 260.85%
Mean Deviation = 74.32%
Mean Deviation =199.34%
Mean Deviation =98.84%
Total Data= 476 Points
Total Mean Deviation = 92.78%
10-2
10-1
100
101
102
103
P(D
elh
aye,k
Pa)
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Schmidt and Friedel (90 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Abdelall (14 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Present Data (217 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
McGee (65 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Mendler (90 Points)
+50%
-50%Mean Deviation = 66.31%
Mean Deviation = 55.96%
Mean Deviation = 96.74%
Mean Deviation = 256.04%
Mean Deviation = 180.84%
Total Data= 476 Points
Total Mean Deviation = 127.44%
![Page 44: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/44.jpg)
Correlation assessment
10-2
10-1
100
101
102
103
P(L
ott
es,
kP
a)
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Schmidt and Friedel (90 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Abdelall (14 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Present Data (217 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
McGee (65 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Mendler (90 Points)
+50%
-50%
Mean Deviation = 6335.55%
Mean Deviation = 80.34%
Mean Deviation = 203.69%
Mean Deviation = 337.20%
Mean Deviation = 322.92%
Total Data= 476 Points
Total Mean Deviation = 1400.22%
10-2
10-1
100
101
102
103
P(A
bd
ela
ll,k
Pa
)
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Schmidt and Friedel (90 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Abdelall (14 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Present Data (217 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
McGee (65 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Mendler (90 Points)
Mean Deviation = 45.30%
Mean Deviation = 96.58%
Mean Deviation = 78.17%
Mean Deviation = 469.15%
Mean Deviation = 300.12%
Total Data= 476 Points
Total Mean Deviation = 167.85%
+50%
-50%
![Page 45: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/45.jpg)
Correlation assessment
10-2
10-1
100
101
102
103
P(C
his
holm
an
d S
uth
erla
nd
,kP
a)
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Schmidt and Friedel (90 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Abdelall (14 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Present Data (217 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
McGee (65 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Mendler (90 Points)
+50%
-50%
Mean Deviation = 29.28%
Mean Deviation = 643.17%
Mean Deviation = 61.33%
Mean Deviation = 1548.46%
Mean Deviation = 732.16%
Total Mean Deviation = 402.30%
Total Data = 497 Points
10-2
10-1
100
101
102
103
P(A
ttou
an
d B
oll
e,k
Pa)
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Schmidt and Friedel (90 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Abdelall (14 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Present Data (217 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
McGee (65 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Mendler (90 Points)
+50%
-50%
Mean Deviation = 89.89%
Mean Deviation = 34.01%
Mean Deviation = 90.76%
Mean Deviation = 249.27%
Mean Deviation = 234.56%
Total Data= 476 Points
Total Mean Deviation = 137.76%
![Page 46: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/46.jpg)
Correlation assessment
10-2
10-1
100
101
102
103
P(W
ad
le,k
Pa
)
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Schmidt and Friedel (90 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Abdelall (14 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Present Data (217 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
McGee (65 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Mendler (90 Points)
+50%
-50%Mean Deviation = 103.18%
Mean Deviation = 140.19%
Mean Deviation = 80.44%
Mean Deviation = 201.33%
Mean Deviation = 277.68%
Total Data= 476 Points
Total Mean Deviation = 140.30%
10-2
10-1
100
101
102
103
P(R
om
ie,k
Pa
)
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Schmidt and Friedel (90 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Abdelall (14 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Present Data (217 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
McGee (65 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Mendler (90 Points)
+50%
-50%Mean Deviation = 66.31%
Mean Deviation = 55.96%
Mean Deviation = 87.85%
Mean Deviation = 256.04%
Mean Deviation = 168.35%
Total Data= 476 Points
Total Mean Deviation = 121.03%
![Page 47: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/47.jpg)
Correlation assessment
10-2
10-1
100
101
102
103
P(M
en
dle
r,k
Pa)
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Schmidt and Friedel (90 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Abdelall (14 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Present Data (217 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
McGee (65 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Mendler (90 Points)
+50%
-50%
Mean Deviation = 93.85%
Mean Deviation = 44.94%
Mean Deviation = 96.58%
Mean Deviation = 67.71%
Mean Deviation = 52.12%
Total Data= 476 Points
Total Mean Deviation = 82.20%
10-2
10-1
100
101
102
103
P(H
om
ogen
eou
s M
od
el,k
Pa)
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Schmidt and Friedel (90 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Abdelall (14 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Present Data (217 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
McGee (65 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Mendler (90 Points)
Mean Deviation = 58.63%
Mean Deviation = 366.12%
Mean Deviation = 74.32%
Mean Deviation = 481.69%
Mean Deviation = 202.08%
Total Mean Deviation = 159.72%
Total Data = 476Points
+50%
-50%
![Page 48: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/48.jpg)
Correlation assessment
22 2
A A
e
L
G 1- 1-xP
2 1-
10-2
10-1
100
101
102
103
P(R
om
ie,k
Pa)
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Schmidt and Friedel (90 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Abdelall (14 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Present Data (217 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
McGee (65 Points)
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
P(Experiment Data,kPa)
Mendler (90 Points)
+50%
-50%Mean Deviation = 66.31%
Mean Deviation = 55.96%
Mean Deviation = 87.85%
Mean Deviation = 256.04%
Mean Deviation = 168.35%
Total Data= 476 Points
Total Mean Deviation = 121.03%
![Page 49: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/49.jpg)
Mean Deviation
ref
Present Data Schmidt and
Friedel
Abdelall McGee Mendler
Schmidt and
Friedel 92.78% 28.11% 260.85% 199.34% 98.84%
Homogeneous
Model 74.32% 58.63% 366.12% 481.69% 202.08%
Abdelall 78.17% 45.30% 96.58% 469.15% 300.12%
Richardson 83.72% 95.54% 29.55% 58.10% 35.81%
Romie 87.85% 66.31% 55.96% 256.04% 168.35%
Chisholm and
Sutherland 61.33% 29.28% 643.17% 1548.47% 732.15%
Lottes 203.69% 6335.54% 80.34% 337.20% 322.92%
Wadle 80.44% 103.18% 140.19% 201.33% 277.68%
Attou and Bolle 90.76% 89.89% 34.01% 249.27% 234.56%
Mendler 96.58% 93.85% 44.94% 67.71% 52.12%
Delhaye 96.74% 66.31% 55.96% 256.04% 180.84%
![Page 50: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/50.jpg)
Conclusions-Sudden expansion
‘‘Liquid jet-like flow pattern’’ at a very low quality and mass flux region may lead to a fall-off of pressure drop.
Elongated bubble becomes the dominate flow pattern when vapor quality or the mass flux is small.
The total mean deviations of the relevant predictions to all the data are 92.78%, 159.72%, 1400.52%, 82.20%, 71.80%, 137.76%, 140.30%, 121.03%, 167.85%, 127.44%, 402.30% by Schmidt and Friedel correlation , Homogeneous model, Lottes, Mendler, Richardson, Attou and Bolle, Wadle, Romie,Abdelall et al. correlation , Delhaye and Chisholm and Sutherland.
Development of suitable correlation applicable for expansion is still in progress.
![Page 51: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/51.jpg)
Acknowledgements
Financial supports provided by the Energy
Bureau, and Department of Industrial
Technology, from the Ministry of Economic
Affairs, Taiwan are higly appreciated.
Collaborating professor: Chen I. Y. From
National Yunlin University of Science and
Technology, Yunlin, Taiwan.
Graduate Students: Chu, M. C. & Lin Y. S.
![Page 52: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/52.jpg)
Thanks for your attention
ITRI (Hsinchu, Taiwan)
(Industrial Technology Research Institute)
A non-profit organization dedicated to
industrial/applied research.
Approx. 5700 employees
Approx. 1000 PhDs
![Page 53: Two-phase Flow Across Sudden Expansion/Contractions · 1 Richardson (1958) Romie (1958) Schmidt and Friedel (1996) Wadle (1989) 2 V 11 2 1 r r AA L G PG V T V V V V T U ' ) 2 220.5](https://reader034.vdocuments.us/reader034/viewer/2022042711/5f7b808b583f910f8105a429/html5/thumbnails/53.jpg)
Experimental setup - expansion
T
T
P
Tank
Air-Dryer
Pump Filter Magnetic Flow Meter
T
Water themostatTank
Mixer
Qu
ick-closin
g
valves
DP
P1 P2 P3 P4 P5 P6 P7 P8 P9
DP
60 60 60 60 60606090 90120
L=30Rectangular Channel with sudden contraction
Width : W=6 and 9 , Gap=3
Round Tube
D=3
Air Flow Meter
unit:mmLL
Air compressor