The movement and distribution of each phase in annular flow can be considered as random events at a microscopic level. Hence, a probability analysis is appropriate to estimate the morphological features and mechanical characteristics of annular flow from a macroscopic scale. In the present work, three characteristic parameters including the film thickness, interfacial shear stress, and characteristic droplet size are predicted by a probability model as the statistical results of abundant samples. The film thickness can be directly calculated as one of the solutions to the basic equations of annular flow. The interfacial shear stress is estimated as a combination of the frictional and dragging components. The droplet size distribution is obtained using a method of undetermined coefficients. These characteristic parameters are well verified by comparing with the experimental data available in the literature. It is demonstrated that the probability model can accurately calculate the film thickness and maximum droplet size, but the predictions of the interfacial shear stress and mean droplet size are relatively coarse. Furthermore, the effects on the film thickness and Sauter mean diameter of other parameters are discussed in detail. Finally, some important phenomena observed in experiments are interpreted by the probability model.

References

1.
Zhang
,
R.
,
Liu
,
H. X.
, and
Liu
,
M. Y.
,
2015
, “
A Probability Model for Fully Developed Annular Flow in Vertical Pipes: Prediction of the Droplet Entrainment
,”
Int. J. Heat Mass Transfer
,
84
, pp.
225
236
.
2.
Pogson
,
J. T.
,
Roberts
,
J. H.
, and
Waibler
,
P. J.
,
1970
, “
Investigation of the Liquid Distribution in Annular-Mist Flow
,”
ASME J. Heat Transfer
,
92
(
4
), pp.
651
658
.
3.
Henstock
,
W. H.
, and
Hanratty
,
T. J.
,
1976
, “
The Interfacial Drag and the Height of the Wall Layer in Annular Flows
,”
AIChE J.
,
22
(
6
), pp.
990
1000
.
4.
Fukano
,
T.
, and
Furukawa
,
T.
,
1998
, “
Prediction of the Effects of Liquid Viscosity on Interfacial Shear Stress and Frictional Pressure Drop in Vertical Upward Gas-Liquid Annular Flow
,”
Int. J. Multiphase Flow
,
24
(
4
), pp.
587
603
.
5.
Hewitt
,
G. F.
, and
Taylor
,
H. N. S.
,
1970
,
Annular Two-Phase Flow
,
Pergamon Press
,
Oxford, UK
.
6.
Wallis
,
G. B.
,
1969
,
One-Dimensional Two-Phase Flow
,
McGraw-Hill
,
New York
.
7.
Kosky
,
P. G.
,
1971
, “
Thin Liquid Films Under Simultaneous Shear and Gravity Forces
,”
Int. J. Heat Mass Transfer
,
14
(
8
), pp.
1220
1224
.
8.
Hughmark
,
G. A.
,
1973
, “
Film Thickness, Entrainment, and Pressure Drop in Upward Annular and Dispersed Flow
,”
AIChE J.
,
19
(
5
), pp.
1062
1065
.
9.
Asali
,
J. C.
, and
Hanratty
,
T. J.
,
1985
, “
Interfacial Drag and Film Height for Vertical Annular Flow
,”
AIChE J.
,
31
(
6
), pp.
895
902
.
10.
Schubring
,
D.
, and
Shedd
,
T. A.
,
2009
, “
Critical Friction Factor Modeling of Horizontal Annular Base Film Thickness
,”
Int. J. Multiphase Flow
,
35
(
4
), pp.
389
397
.
11.
Aliyu
,
A. M.
,
Lao
,
L.
,
Almabrok
,
A. A.
, and
Yeung
,
H.
,
2016
, “
Interfacial Shear in Adiabatic Downward Gas/Liquid Co-Current Annular Flow in Pipes
,”
Exp. Therm. Fluid Sci.
,
72
, pp.
75
87
.
12.
Whalley
,
P. B.
,
1987
,
Boiling, Condensation, and Gas-Liquid Flow
,
Clarendon Press
,
Oxford, UK
.
13.
Brauner
,
N.
, and
Maron
,
D. M.
,
1993
, “
The Role of Interfacial Shear Modeling in Prediction the Stability of Stratified Two-Phase Flow
,”
Chem. Eng. Sci.
,
48
(
16
), pp.
2867
2879
.
14.
Fore
,
L. B.
,
Beus
,
S. G.
, and
Bauer
,
R. C.
,
2000
, “
Interfacial Friction in Gas-Liquid Annular Flow: Analogies to Full and Transition Roughness
,”
Int. J. Multiphase Flow
,
26
(
11
), pp.
1755
1769
.
15.
Pan
,
L.
,
He
,
H.
,
Ju
,
P.
,
Hibiki
,
T.
, and
Ishii
,
M.
,
2015
, “
The Influence of Gas-Liquid Interfacial Properties on Interfacial Shear Stress for Vertical Annular Flow
,”
Int. J. Heat Mass Transfer
,
89
, pp.
1172
1183
.
16.
Tatterson
,
D. F.
,
Dallman
,
J. C.
, and
Hanratty
,
T. J.
,
1977
, “
Drop Sizes in Annular Gas-Liquid Flows
,”
AIChE J.
,
23
(
1
), pp.
68
76
.
17.
Azzopardi
,
B. J.
,
Freeman
,
G.
, and
King
,
D. J.
,
1980
, “
Drop Sizes and Deposition in Annular Two-Phase Flow
,” UKAEA Report, Report No. AERE R9634.
18.
Kataoka
,
I.
,
Ishii
,
M.
, and
Mishima
,
K.
,
1983
, “
Generation and Size Distribution of Droplet in Annular Two-Phase Flow
,”
ASME J. Fluids Eng.
,
105
(
2
), pp.
230
238
.
19.
Ambrosini
,
W.
,
Andreussi
,
P.
, and
Azzopardi
,
B. J.
,
1991
, “
A Physical Based Correlation for Drop Size in Annular Flow
,”
Int. J. Multiphase Flow
,
17
(
4
), pp.
497
507
.
20.
Kocamustafaogullari
,
G.
,
Smits
,
S. R.
, and
Razi
,
J.
,
1994
, “
Maximum and Mean Droplet Sizes in Annular Two-Phase Flow
,”
Int. J. Heat Mass Transfer
,
37
(
6
), pp.
955
965
.
21.
Al-Sarkhi
,
A.
, and
Hanratty
,
T. J.
,
2002
, “
Effect of Pipe Diameter on the Drop Size in a Horizontal Annular Gas-Liquid Flow
,”
Int. J. Multiphase Flow
,
28
(
10
), pp.
1617
1629
.
22.
Patruno
,
L. E.
,
Ystad
,
P. A. M.
,
Jenssen
,
C. B.
,
Marchetti
,
J. M.
,
Dorao
,
C. A.
,
Svendsen
,
H. F.
, and
Jakobsen
,
H. A.
,
2010
, “
Liquid Entrainment-Droplet Size Distribution for a Low Surface Tension Mixture
,”
Chem. Eng. Sci.
,
65
(
18
), pp.
5272
5284
.
23.
Ueda
,
T.
,
1979
, “
Entrainment Rate and Size of Entrained Droplets in Annular Two-Phase Flow
,”
Bull. Jpn. Soc. Mech. Eng.
,
22
(
171
), pp.
1258
1265
.
24.
Jepson
,
D. M.
,
Azzopardi
,
B. J.
, and
Whalley
,
P. B.
,
1989
, “
The Effects of Gas Properties on Drops in Annular Flow
,”
Int. J. Multiphase Flow
,
15
(
3
), pp.
327
339
.
25.
Azzopardi
,
B. J.
,
Piearcey
,
A.
, and
Jepson
,
D. M.
,
1991
, “
Drop Size Measurements for Annular Two-Phase Flow in a 20 mm Diameter Vertical Tube
,”
Exp. Fluids
,
11
(2), pp.
191
197
.
26.
Hay
,
K. J.
,
Liu
,
Z. C.
, and
Hanratty
,
T. J.
,
1996
, “
Relation of Deposition to Drop Size When the Rate Law is Nonlinear
,”
Int. J. Multiphase Flow
,
22
(
5
), pp.
829
848
.
27.
Davies
,
J. T.
,
1972
,
Turbulence Phenomena
,
Academic Press
,
New York
.
28.
Norberg
,
C.
, and
Sundern
,
B.
,
1984
, “
Influence of Stream Turbulence Intensity and Eddy Size on the Fluctuating Pressure Forces on a Single Tube
,”
Symposium on Flow-Induced Vibrations
, New Orleans, LA, Vol.
6
, pp.
43
56
.
29.
Baumert
,
H. Z.
,
2013
, “
Universal Equations and Constants of Turbulent Motion
,”
Phys. Scr., T
,
155
, p.
014001
.
30.
Asali
,
J. C.
,
1984
, “
Entrainment in Vertical Gas-Liquid Annular Flows
,” Ph.D. thesis, University of Illinois at Urbana-Champaign, Urbana, IL.
31.
Jong
,
P. D.
, and
Gabriel
,
K. S.
,
2003
, “
A Preliminary Study of Two-Phase Annular Flow at Microgravity: Experimental Data of Film Thickness
,”
Int. J. Multiphase Flow
,
29
(
8
), pp.
1203
1220
.
32.
Alamu
,
M. B.
, and
Azzopardi
,
B. J.
,
2011
, “
Simultaneous Investigation of Entrained Liquid Fraction, Liquid Film Thickness and Pressure Drop in Vertical Annular Flow
,”
ASME J. Energy Resour. Technol.
,
133
(
2
), p.
023103
.
33.
Pan
,
L.
, and
Hanratty
,
T. J.
,
2002
, “
Correlation of Entrainment for Annular Flow in Vertical Pipes
,”
Int. J. Multiphase Flow
,
28
(
3
), pp.
363
384
.
34.
Belt
,
R. J.
,
Van't Westende
,
J. M. C.
, and
Porela
,
L. M.
,
2009
, “
Prediction of the Interfacial Shear-Stress in Vertical Annular Flow
,”
Int. J. Multiphase Flow
,
35
(
7
), pp.
689
697
.
35.
Fore
,
L. B.
, and
Dukler
,
A. E.
,
1995
, “
Droplet Deposition and Momentum Transfer in Annular Flow
,”
AIChE J.
,
41
(
9
), pp.
2040
2046
.
36.
Fore
,
L. B.
, and
Dukler
,
A. E.
,
1995
, “
The Distribution of Drop Size and Velocity in Gas-Liquid Annular Flow
,”
Int. J. Multiphase Flow
,
21
(
2
), pp.
137
149
.
37.
Azzopardi
,
B. J.
,
1997
, “
Drops in Annular Two-Phase Flow
,”
Int. J. Multiphase Flow
,
23
(
7
), pp.
1
53
.
38.
Simmons
,
M. J. H.
, and
Hanratty
,
T. J.
,
2001
, “
Droplet Size Measurements in Horizontal Annular Gas-Liquid Flow
,”
Int. J. Multiphase Flow
,
27
(
5
), pp.
861
883
.
39.
Fore
,
L. B.
,
Ibrahim
,
B. B.
, and
Beus
,
S. G.
,
2002
, “
Visual Measurements of Droplet Size in Gas-Liquid Annular Flow
,”
Int. J. Multiphase Flow
,
28
(
12
), pp.
1895
1910
.
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