COMPARISON OF VOID FRACTION
CORRELATIONS FOR TWO-PHASE
FLOW IN HORIZONTAL AND
UPWARD INCLINED FLOWS
By
MELKAMU AKLILE WOLDESEMAYAT
Bachelor of Science
Addis Ababa University
Addis Ababa, Ethiopia
1998
Submitted to the Faculty of the
Graduate College of
Oklahoma State University
in partial fulfillment of
the requirements for
the degree of
MASTER OF SCIENCE
May 2006
COMPARISON OF VOID FRACTION
CORRELATIONS FOR TWO-PHASE
FLOW IN HORIZONTAL AND
UPWARD INCLINED FLOWS
Thesis approved:
___________Dr. Afshin J. Ghajar_________
Thesis Advisor
_______ ___Dr. Frank W. Chambers______
_____ _
Dr. P.M. Moretti_____________
_________ Dr. A.Gordon Emslie_________
Dean of the Graduate College
ii
ACKNOWLEDGEMENTS
I would like to express my gratitude to my advisor Dr. A.J. Ghajar for his untiring
support and guidance not only in this work and preparation of the manuscript but also
throughout my entire graduate study.
My sincere thanks go to Dr. F.W. Chambers and P.M. Moretti for being in my
committee and also their kind advice during my masters program.
No words would be good enough to express my gratitude to my parents and all my
family who have done everything to make my graduate studies possible. I will forever
cherish their endless love and support.
iii
TABLE OF CONTENTS
CHAPTER
PAGE
1 INTRODUCTION ......................................................................................................1
2 LITERATURE REVIEW ...........................................................................................5
2.1 VOID FRACTION CORRELATIONS ...........................................................................7
2.1.1 Slip Ratio Correlations .................................................................................9
2.1.2 KαH Correlations .......................................................................................12
2.1.3 Drift Flux Correlations ...............................................................................16
2.1.4 General Void Fraction Correlations ..........................................................20
2.2 PREVIOUS COMPARISON WORK ............................................................................28
2.2.1 Dukler et al. (1964) Horizontal Pipe Comparison .....................................28
2.2.2 Marcano (1973) Horizontal Pipe Comparison...........................................28
2.2.3 Palmer (1975) Inclined Pipe Comparison..................................................29
2.2.4 Mandhane et al. (1975) Horizontal Pipe Comparison ...............................30
2.2.5 Papathanassiou (1983) Horizontal Pipe Comparison................................31
2.2.6 Spedding et al. (1990) Inclined (2.75o) Pipe Comparison..........................32
2.2.7 Abdulmajeed (1996) Horizontal Pipe Comparison ....................................32
2.2.8 Spedding (1997) General (-90o to +90o) Comparison................................33
2.2.9 Friedel and Diener (1998) Horizontal/Vertical Upward Comparison.......33
3 EXPERIMENTAL DATA BASE AND FLUID PROPERTIES..............................35
3.1 EXPERIMENTAL DATABASE .................................................................................35
3.2 FLUID PHYSICAL PROPERTIES ............................................................................38
3.2.1 Properties of Air .........................................................................................38
3.2.2 Properties of Kerosene ...............................................................................39
3.2.3 Properties of Natural Gas...........................................................................40
3.2.4 Properties of Water.....................................................................................42
3.2.5 Measurement Techniques............................................................................42
4 EVALUATION OF VOID FRACTION CORRELATIONS ...................................47
4.1 HORIZONTAL FLOW COMPARISON .......................................................................50
4.2 INCLINED FLOW COMPARISON .............................................................................61
4.2.1 Inclined flow comparison with the data of Beggs(1972) ............................62
4.2.2 Inclined flow comparison with the data of Spedding and Nguyen (1976)..68
4.2.3 Inclined flow comparison with the data of Mukherjee (1979)....................81
4.3 VERTICAL FLOW COMPARISON ..........................................................................104
4.4 SUMMARIZED COMPARISON AND DISCUSSION FOR ALL DATA SETS....................116
5 CONCLUSIONS AND RECOMMENDATIONS .................................................135
5.1 CONCLUSIONS ...................................................................................................135
5.2 RECOMMENDATIONS .........................................................................................136
iv
REFERENCES ..........................................................................................................138
APPENDIX A SUMMARY OF EXPERIMENTAL DATA BASE .......................147
v
LIST OF TABLES
TABLE
PAGE
Table 1. Slip ratio correlations for Butterworth's(1975) correlation ...........................14
Table 2. Constants for the Maier and Coddington (1997) correlation.........................22
Table 3. Coefficients for Beggs (1972) Correlation ....................................................24
Table 4. Coefficients for Mukherjee (1979) correlation..............................................24
Table 5. Characteristics of Data Base sources .............................................................37
Table 6. Optimized constants for the Lee et al. correlation .........................................43
Table 7. Number of data points correctly predicted for horizontal flow by data sets..54
Table 8. Percentage of data points correctly predicted for horizontal flow by data sets
........................................................................................................................57
Table 9. Total number and percentage of data points correctly predicted for horizontal
flow database (all sources).............................................................................63
Table 10. Number of data points correctly predicted for Beggs (1972) inclined data.69
Table 11. Percentage of data points correctly predicted for Beggs (1972) inclined data
......................................................................................................................72
Table 12. Total number and percentage of data points correctly predicted for the
inclined experimental data of Beggs(1972) .................................................75
Table 13. Number of data points correctly predicted for Spedding & Nguyen (1976)
inclined data .................................................................................................82
Table 14. Percentage of data points correctly predicted for Spedding & Nguyen
(1976) inclined data .....................................................................................84
Table 15. Total number and percentage of data points correctly predicted for the
inclined experimental data of Spedding & Nguyen (1976) .........................86
Table 16. Number of data points correctly predicted for Mukherjee (1979) inclined
data...............................................................................................................93
Table 17. Percentage of data points correctly predicted for Mukherjee (1979) inclined
data...............................................................................................................96
vi
Table 18. Total number and percentage of data points correctly predicted for the
inclined experimental data of Mukherjee (1977).........................................99
Table 19. Total number and percentage of data points correctly predicted for the
whole inclined data base ............................................................................106
Table 20. Number of data points correctly predicted for vertical flow by data set ...112
Table 21. Percentage of data points correctly predicted for vertical flow by data set
....................................................................................................................114
Table 22. Total number and percentage of data points correctly predicted for vertical
flow database (all sources).........................................................................117
Table 23. Total number and percentage of data points correctly predicted for the
whole database (all sources) ......................................................................122
Table 24. Total number and percentage of data points correctly predicted by the top
six correlations for the whole database (all sources) .................................124
Table 25. Total number and percentage of data points correctly predicted by the top
six and modified correlation for the whole database (all sources).............129
vii
LIST OF FIGURES
FIGURE
PAGE
Figure 1. Comparison of Rouhani I (1970) correlation with measured horizontal
experimental data ..........................................................................................65
Figure 2. Comparison of Hughmark (1962) correlation with measured horizontal
experimental data ..........................................................................................65
Figure 3. Comparison of Minami and Brill (1987) correlation with measured
horizontal experimetal data...........................................................................66
Figure 4. Comparison of Armand-Massina correlation (Leung 2005) with measured
horizontal experimental data.........................................................................66
Figure 5. Comparison of Mukherjee (1979) correlation with measured horizontal
experimental data ..........................................................................................67
Figure 6. Comparison of Chisholm (1983) – Armand (1946) with measured horizontal
experimental data ..........................................................................................67
Figure 7. Comparison of Armand-Massina (Leung 2005) correlation with measured
inclined experimental data of Beggs(1972) ..................................................77
Figure 8. Comparison of Hughmark (1962) correlation with measured
inclined experimental data of Beggs(1972) ..................................................77
Figure 9. Comparison of Premoli et al. (1970) correlation with measured
inclined experimental data of Beggs(1972) ..................................................78
Figure 10. Comparison of Rouhani I (1970) correlation with measured
inclined experimental data of Beggs(1972) ................................................78
Figure 11. Comparison of Filimonov et al. (1957) correlation with measured
inclined experimental data of Beggs(1972) ................................................79
Figure 12. Comparison of Toshiba (1989) correlation with measured
inclined experimental data of Beggs(1972) ................................................79
Figure 13. Comparison of Dix (1971) correlation with measured inclined
experimental data of Beggs(1972) ..............................................................80
Figure 14. Comparison of Toshiba (1989) correlation with measured inclined
experimental data of Spedding & Nguyen(1976) .......................................88
Figure 15. Comparison of Dix (1971) correlation with measured inclined
viii
experimental data of Spedding & Nguyen(1976) .......................................88
Figure 16. Comparison of Filimonov et al. (1957) correlation with measured inclined
experimental data of Spedding & Nguyen(1976) .......................................89
Figure 17. Comparison of Rouhani I (1970) correlation with measured inclined
experimental data of Spedding & Nguyen(1976) .......................................89
Figure 18. Comparison of Premoli et al. (1970) correlation with measured inclined
experimental data of Spedding & Nguyen(1976) .......................................90
Figure 19. Comparison of Hughmark (1962) correlation with measured inclined
experimental data of Spedding & Nguyen(1976) .......................................90
Figure 20. Comparison of Armand-Massina (Leung 2005) with measured inclined
experimental data of Spedding & Nguyen(1976) .......................................91
Figure 21. Comparison of Armand-Massina (Leung 2005) with measured inclined
experimental data of Mukherjee (1979)....................................................101
Figure 22. Comparison of Rouhani I (1970) correlation with measured inclined
experimental data of Mukherjee (1979)....................................................101
Figure 23. Comparison of Hughmark (1962) correlation with measured inclined
experimental data of Mukherjee (1979)....................................................102
Figure 24. Comparison of Premoli et al. (1970) correlation with measured inclined
experimental data of Mukherjee (1979)....................................................102
Figure 25. Comparison of Toshiba (1989) correlation with measured inclined
experimental data of Mukherjee (1979)....................................................103
Figure 26. Comparison of Dix (1971) correlation with measured inclined
experimental data of Mukherjee (1979)....................................................103
Figure 27. Comparison of Filimonov et al. (1957) correlation with measured inclined
experimental data of Mukherjee (1979)....................................................104
Figure 28. Comparison of Toshiba (1989) correlation with measured combined
inclined experimental data ........................................................................108
Figure 29. Comparison of Dix (1971) correlation with measured combined
inclined experimental data ........................................................................108
Figure 30. Comparison of Rouhani I (1970) correlation with measured combined
inclined experimental data ........................................................................109
Figure 31. Comparison of Premoli et al. (1970) correlation with measured combined
inclined experimental data ........................................................................109
Figure 32. Comparison of Filimonov et al. (1957) correlation with measured
combined inclined experimental data .......................................................110
ix
Figure 33. Comparison of Hughmark (1962) correlation with measured combined
inclined experimental data ........................................................................110
Figure 34. Comparison of Armand-Massina (leung 2005) with measured combined
inclined experimental data ........................................................................111
Figure 35. Comparison of Toshiba (1989) correlation with measured combined
vertical experimental data .........................................................................119
Figure 36. Comparison of Rouhani I (1970) correlation with measured combined
vertical experimental data .........................................................................119
Figure 37. Comparison of Filimonov et al. (1957) correlation with measured
combined vertical experimental data .......................................................120
Figure 38. Comparison of Hughmark (1962) correlation with measured combined
vertical experimental data .........................................................................120
Figure 39. Comparison of Dix (1971) correlation with measured combined
vertical experimental data .........................................................................121
Figure 40. Comparison of Premoli et al. (1970) correlation with measured combined
vertical experimental data .........................................................................121
Figure 41. Comparison of Toshiba (1989) correlation with measured combined
total experimental data..............................................................................131
Figure 42. Comparison of Rouhani I (1989) correlation with measured combined
total experimental data..............................................................................131
Figure 43. Comparison of Dix (1971) correlation with measured combined
total experimental data..............................................................................132
Figure 44. Comparison of Hughmark (1962) correlation with measured combined
total experimental data..............................................................................132
Figure 45. Comparison of Premoli et al. (1970) correlation with measured combined
total experimental data..............................................................................133
Figure 46. Comparison of Filimonov et al. (1957) correlation with measured
combined total experimental data .............................................................133
Figure 47. Comparison of present study correlation with measured combined
total experimental data..............................................................................134
Figure 48. Comparison of present study correlation with measured "refined"
combined total experimental data .............................................................134
x
NOMENCLATURE
A
Constant in Butterworth (1975) correlation
APRM
Premoli et al. (1970) correlation constant
ASM
Constant in Smith (1969) correlation
b
Constant in Butterworth (1975) correlation
B gas
Gas formation volume factor, ft3/scf
c
Constant in Butterworth (1975) correlation
C
Inclination factor constant in Beggs (1972) correlation
C ABD
Abdulmajeed (1996) correction factor
C0
Distribution parameter
C MC1
Constant in Maier and Coddington (1997) correlation
C MC2
Constant in Maier and Coddington (1997) correlation
d
Constant in Butterworth (1975) correlation
d1
Factor in Moussali correlation as given by Isbin and Biddle (1979)
D
Diameter of pipe
DH
Hydraulic diameter of vessel
D*H
Dimensionless hydraulic diameter,
Dv
Wilson et al. (1961) correlation dimensionless number,
EL
Liquid-holdup
xi
DH
⎛ σ
⎞
⎜⎜
⎟⎟
⎝ ρ L − ρG ⎠
0.5
ρG
ρ L − ρG
0.25
F
⎡ ρ − ρG ⎤
Wilson et al. (1961) Froude number, USG ⎢ L
⎥
⎣ gσ ⎦
FD
Constant in Gardner (1980) correlation, FD =
F1
Constant in Premoli et al. (1970) correlation
F2
Constant in Premoli et al. (1970) correlation
Fr
Froude number,
Ft
Froude rate parameter
F (Xtt )
⎛ 1
2.85 ⎞
Constant in Tandon et al. (1985) correlation, 0.15⎜⎜
+ 0.476 ⎟⎟
⎝ Xtt Xtt ⎠
g
Acceleration of gravity
G
Mass flux
j*g
Dimentiev et al. correlation gas flux as reported by Kataoka and Ishii
(1987),
j*L
USG ρ 1L / 2
[(ρ L − ρG )gσ ]1 / 4
2
UM
gD
ρG0.5USG
[gD(ρ L − ρG )]0.5
Kawaji et al. (1987) correlation dimensionless liquid flux,
ρ L0.5USL
[gD(ρ L − ρG )]0.5
K
General coefficient
KB
Bankoff (1960) parameter
K FJ
Constant in Fujie (1964) correlation
KGR
Constant in Gardner (1980) correlation (Either 1.7 or 11.2)
K HO
Hoogendoorn (1959) correlation constant
K HU
Hughmark (1962) parameter
xii
K MO
Moussali constant as given by Isbin and Biddle (1979)
l
Constant in Abdulmajeed (1996) correlation
L
Laplace length scale defined in Gardner (1980) correlation,
⎡
⎤
σ
⎢
⎥
⎣ (ρ L − ρG )g ⎦
m
Mass
N FR
Froude number in Beggs (1972) correlation, Fr
NGV
⎡ρ ⎤
Gas velocity number, USG ⎢ L ⎥
⎣ gσ ⎦
NL
Liquid viscosity number,
0.25
μL
[ρ σ ]
3 0.25
L
0.25
N LV
⎡ρ ⎤
Liquid velocity number, USL ⎢ L ⎥
⎣ gσ ⎦
P
System pressure
PC
Critical pressure (thermodynamic)
R
Parameter in Abdulmajeed (1996) correlation
Re
Reynolds number
Re L
Reynolds number, liquid,
RL
Liquid holdup
Re M
Mixture Reynolds number,
Re SL
Superficial liquid Reynolds number,
S
Slip ratio
UG
Velocity of gas
GD(1 − x )
μL
ρ LU M D
μL
xiii
ρ LUSL D
μL
UGM
Drift velocity, UG − U M
UL
Velocity of liquid
UM
Mixture velocity, USL + USG
USG
Superficial gas velocity
USL
Superficial liquid velocity
v1
Constant in Maier and Coddington (1997) correlation
v2
Constant in Maier and Coddington (1997) correlation
v3
Constant in Maier and Coddington (1997) correlation
v4
Constant in Maier and Coddington (1997) correlation
v5
Constant in Maier and Coddington (1997) correlation
v6
Constant in Maier and Coddington (1997) correlation
We L
Weber number, liquid
x
quality, mass of vapor/total mass
Xtt
⎛ μ ⎞ ⎛1- x ⎞
Lockhart - Martinelli (1949) Parameter, Xtt = ⎜⎜ L ⎟⎟ ⎜
⎟
⎝ μG ⎠ ⎝ x ⎠
y
Constant in Premoli et al. (1970) correlation
Z
Parameter in Hughmark (1962) correlation
Z1
Parameter in Minami and Brill (1987) correlation
K
Parameter in Lee et al. correlation (Jeje and Mattar 2004)
K1
Constant in Lee et al. correlation (Jeje and Mattar 2004)
K2
Constant in Lee et al. correlation (Jeje and Mattar 2004)
K3
Constant in Lee et al. correlation (Jeje and Mattar 2004)
K4
Constant in Lee et al. correlation (Jeje and Mattar 2004)
0.1
xiv
0. 9
⎛ ρG ⎞
⎜⎜ ⎟⎟
⎝ ρL ⎠
0.5
K5
Constant in Lee et al. correlation (Jeje and Mattar 2004)
T
Temperature
P
Pressure
R
The universal gas constant (8314.34 J/ kmol.K)
R2
Correlation coefficient
M
Molecular weight
USG
Superficial gas velocity
UM
Mixture velocity, USL + USG
X
Parameter in Lee et al. correlation (Jeje and Mattar 2004)
x1
Constant in Lee et al. correlation (Jeje and Mattar 2004)
x2
Constant in Lee et al. correlation (Jeje and Mattar 2004)
x3
Constant in Lee et al. correlation (Jeje and Mattar 2004)
Y
Parameter in Lee et al. correlation (Jeje and Mattar 2004)
y1
Constant in Lee et al. correlation (Jeje and Mattar 2004)
y2
Constant in Lee et al. correlation (Jeje and Mattar 2004)
Z
Compressibility factor
Greek Letters
α
Void fraction, average
αH
No slip (homogeneous) void fraction
β
Volumetric quality, α H
λ
Input liquid content,
μ
Viscosity
USL
USL + USG
xv
ρ
Density
σ
Surface tension
θ
Pipe inclination angle
Subscripts
kero
Kerosene
pc
Pseudo-critical (pressure, temperature)
r
Reduced (pressure, temperature)
G
Gas phase
L
Liquid phase
M
Mixture
Ngas
Natural gas
S
Superficial
xvi
CHAPTER 1
INTRODUCTION
Two phase flow, theoretically, is the simultaneous flow of two of any of the three
discrete phases (solid, liquid or gas) of any substance or combination of substances.
Practical applications of a gas-liquid flow, of a single substance or two different
components, are commonly encountered in the petroleum, nuclear and process
industries. The two phases may be of different components and/or there could be a
phase change due to evaporation and condensation of a single fluid.
Considerable effort has been devoted to understanding the underlying physics of two
phase flow in the past. Theoretical and empirical correlations have been developed for
predictions of the various parameters of practical interest such as flow pattern
transitions, pressure drop and void fraction.
In the industrial applications where two phase flow exists, the task of sizing the
equipment for gathering, pumping, transporting and storing such a two phase mixture
requires the formidable task of predicting the phase distribution in the system from
given operating conditions. Once this phase distribution is known, the problem may
be simplified in such a way that it could be approached and tackled in a similar
fashion analogous to single phase flow.
One of the critical unknown parameters involved in predicting the pressure loss and
heat transfer (see for example Kim and Ghajar,2006) in any gas-liquid system is the
void fraction or liquid holdup which is the volume of the space occupied by the gas or
1
liquid respectively. The seemingly benign issue of determining the phase distribution
from input conditions in a given pipe is complicated as a result of the slippage
between the gas and the liquid phases.
Void fraction data was measured by different researchers in the effort for
development of the numerous empirical correlations and also to validate some of the
theoretical work pursued in analyzing two phase flow behavior.
To accomplish the task of measuring void fraction, an appreciable number of
experimental facilities have been built and tests run in order to measure void fraction
under different operating conditions. The experimental set ups have been designed
and constructed ranging in size from very short, small diameters to those which
simulate big petroleum field conditions. As expected, these setups are operated with
different fluids under a broad range of temperatures, pressures and pipe inclinations
spanning from vertically upwards to vertically downwards.
A huge number of studies have been carried out and are still on going in an effort to
accurately predict the void fraction from the known operating conditions. These
analyses range in scope from a fully theoretical one for the simple cases to the fully
empirical methods when the interaction between the phases becomes complex. Owing
to the complexity and lack of understanding of the basic underlining physics of the
problem, the majority of the analyses are more inclined towards empirical
correlations. For the sake of simplicity both the theoretical models as well as the
empirical correlations that are collected from the literature and discussed here would
be simply referred to as void fraction correlations.
The design engineer is faced with the difficult task of choosing the “right” correlation
among the plethora of correlations available. The fact that there are plenty of
correlations available would not be a concern had it not been for the fact that most of
2
the correlations have some form of restrictions attached to them. The lack of
understanding of the exact behavior of two phase flow had led many investigators to
resort to developing a correlation which only works for a given range of flow rates,
holdup values or flow patterns.
The idea of combining different correlations which claim to do well for the specific
flow characteristics for which they are developed in trying to take care of all the
design need would entail its own difficulty. For instance, one of the most common
restrictions to the correlations, flow pattern dependency, is sometimes a purely
subjective judgment of the investigator especially for those points on or near flow
pattern boundaries. Moreover, one is bound to run into some discontinuity when
switching from one correlation to the other for different practical operating
conditions.
The other pitfall is that the majority of the void fraction data and also the correlations
are primarily developed for mostly horizontal orientation though some vertical flows
and inclined cases have been investigated.
The purpose of this work is to find a void fraction correlation that could acceptably
predict most of the experimental data collected for all inclination angles as well as the
different flow patterns without resorting back to too much complex expressions that
require iterative schemes with multiple solutions.
A search is made in the open literature to collect all the available void fraction
correlations as well as the measured void fraction data from various sources.
Comparisons of predictive performance and promise of the capability of the collected
void fraction correlations was made using the collected data from the literature.
Based on the analysis, the best performing correlations were selected and
recommendations were drawn on their strengths and weaknesses. A detailed study of
3
the performance of several of the top performing correlations together with the data
points which they could not handle well was made. Introducing correction factors
which take into account the physical nature of the data sets, an improved version to
one of these correlations was suggested that could handle data points which failed to
be captured by the original correlation.
This work has been organized in such a way that, first, the void fraction correlations
which are collected from the literature would be presented briefly in Chapter 2.
Moreover, most of the void fraction comparison work carried out by different
researchers at different times is also presented.
Chapter 3 will be concerned with the presentation of the experimental data base
collected from the literature along with the associated fluid physical properties of the
working fluids used in the tests.
The detail predictive capability of each of the void fraction correlations reported in
Chapter 2 in capturing the experimental data base compiled (Chapter 3) will be
investigated and results of the comparisons between the correlations would be
discussed in Chapter 4. Steps on how we have developed an improved version of one
of the best performing correlations would be given as an extension in Chapter 4.
Finally, concise conclusions and recommendation based on the overall undertaking
are forwarded in Chapter 5.
4
CHAPTER 2
LITERATURE REVIEW
The extensive literature review presented here had two major parts that were
undertaken in parallel. The primary task was to find most if not all of the void fraction
correlations that are available in the open literature while the other one was gathering
a sufficiently large number of experimental void fraction data that covered a wide
range of fluids and operating conditions such as pipe size, flow rates and the
associated flow patterns, temperature, pressure and different inclination angles.
This chapter would focus on the discussion of the collected void fraction correlations
and some of the previous comparison work done. The experimental data base and
characteristics related to it would be discussed in the next chapter.
In facilitating better understanding of this manuscript, it is worth while to highlight
some of the most common terminologies and definitions of parameters that would be
encountered throughout this work.
Void fraction is defined as the volume of space the gas phase occupies in a given two
phase flow in a pipe, hence for a total pipe cross sectional area of A; the void fraction
is given by
α=
AG
A
Liquid holdup is the complement of the void fraction in the pipe, i.e., it is the
remaining volume of space occupied by the liquid phase. Thus, liquid holdup is
RL = 1 − α =
AL
A
5
The quality of the mixture, x , in the isothermal flow case we are considering here is
taken as the input mass of the gaseous phase to that of the total mixture mass of m,
hence
x=
mG
mM
The slip ratio, S, is defined as the ratio of the actual velocities between the phases. A
slip ratio of unity for a mixture being the homogeneous case where it is assumed that
both phases travel at the same velocity. The slip ratio is defined as
S=
UG
UL
The superficial gas , USG ,and liquid , USL , velocities are defined as the velocities of
the gas or liquid phase in the pipe assuming the flow is a single phase in either gas or
liquid respectively.
From the definitions given above and writing conservation of mass for each phase and
total flow, we can define the relationships,
UG =
USG
UL =
USL
1−α
α
and
⎛ ⎞
x
⎜ ρ ⎟ ⎛U ⎞
= ⎜ .G ⎟ ⎜⎜ SG ⎟⎟
1 − x ⎜ ρ ⎟ ⎝ USL ⎠
⎝ L⎠
Having given the basic definitions of the most important and frequently used
parameters in two phase flow in relation to void fraction, we now move on to
presenting the correlations.
6
2.1 Void Fraction Correlations
Different void fraction (liquid holdup) correlations which appeared from the early
1940s to date collected from the open literature are presented here. The total number
of correlations collected is more than 80. We have presented correlations for which
we have complete information and those with little or no information at all but which
we found to perform well in predicting a good portion of the data points in our
database. These correlations were developed from theoretical and mostly
experimental investigations under various operating conditions.
In presenting the correlations, different criteria were sought into which the developed
correlations might conveniently fall. One advantage of it is that it would shed some
light on the common features or lack thereof between individual and groups of
correlations. There are a number of simple criteria that could easily be used to
categorize these correlations. However, their shortfalls would heavily outweigh the
practical advantage one would get out of them. We have briefly discussed some of
these criteria and their associated shortfalls below.
The majority of the correlations were developed from horizontal test section
experiments with only a few for other inclination angles, the common one being
upward 900. Categorizing the correlations along their applicability with regard to
angle of inclination would not serve any purpose as most of them would fall under the
horizontal case.
As the complex nature of two phase flow has not yielded to any theoretical
formulation of a particular flow pattern analysis let al.one a general one, it can be
observed from a literature search that the trend in correlation development is
becoming more geared towards a specific area of particular interest which in effect
has resulted in correlations for a specific flow regime. Hence, division along the type
7
of flow pattern dependency is a logical step. Due to the highly subjective and tricky
nature of assigning a specific flow pattern to a given flow phenomenon, we have
avoided lumping the correlations to a specific flow pattern even when the
applicability of the correlations has been stated so by the investigator. It is assumed
that the correlation is a general type if no specific restriction is implied by the original
author or no discussion thereof is provided in the reference in which it was reported.
However, a flow pattern specific correlation would therefore be reported so in order to
avoid unfair and misleading judgment during the predictive performance assessment.
Apart from the two criteria discussed above, types of fluid considered, pipe diameters
and mass flow rates are the other areas where the correlations would share some form
of commonality. Most of the data from which the correlations have been developed
was from small diameter, short length pipes in a laboratory setting with controlled,
and relatively small mass flow rates while mixtures of air-water dominate with regard
to the fluids considered. Even though these factors have significant influence in the
predictive capability of a correlation, it was decided not to group them along these
lines as the majority of the correlations would aggregate into a single category.
Finally, it is imperative to mention here that by nature all the empirical correlations
have some sort of limitations, even though not explicitly reported by their authors.
This is due to the very fact that they were fitted to given data sets which in turn
depend on the inherent physical limitations of the experiment under which the data
was collected. Therefore, rather than going after physical parameters(inclination
angles, mass flow rates etc) which are too narrow or subjective criteria like flow
pattern upon which there is no firm consensus to this date, we have decided to follow
the work of Vijayan et al.(2000) to classify the correlations into four categories. These
are:
8
1. Slip ratio correlations
2. Kα H correlations
3. Drift flux correlations
4. General void fraction correlations
We shall now give a brief description of these categories and the correlations that fall
into each of them.
2.1.1 Slip Ratio Correlations
These correlations are of the form
α=
1
c
b
⎛ 1 − x ⎞ ⎛⎜ ρG ⎞⎟
1 + S⎜
⎟
⎝ x ⎠ ⎜⎝ ρ L ⎟⎠
Where the slip ratio S and the general expression has been shown by Butterworth
(1975) to take the form
α=
1
c
d
b
⎛ 1 − x ⎞ ⎛⎜ ρG ⎞⎟ ⎛⎜ μ L ⎞⎟
1 + A⎜
⎟
⎝ x ⎠ ⎜⎝ ρ L ⎟⎠ ⎜⎝ μG ⎟⎠
Discussion of some of the notable void fraction correlations that fall in this category
follows. The constants (A, b, c, d) in the above equation for the different correlations
are given in Table 1.
The most simple of all the correlations with a theoretical background and the
assumption that the gas and liquid velocities are equal or there is no slip between
them is the Homogeneous model or the no-slip correlation.
9
It is almost impossible to find any literature on void fraction correlations which would
not refer to the pioneer work and the void fraction correlation of Lockhart and
Martinelli (1949). The correlation was in a plot form where the void fraction was
correlated with what is now commonly known as the Lockhart-Martinelli Parameter,
X tt . The empirical approximation of this graphical representation given by
Butterworth (1975) is used in this analysis.
The correlation by Fauske (1961) can also be shown to fall into a similar expression
form. A complex expression between steam quality and void fraction was given by
Fujie (1964) for horizontal and vertical flows with and without heating. The
isothermal correlation for the horizontal case is considered here which can be reduced
and cast into the slip ratio model form. It must be noted here that the constant term A
in the Butterworth (1975) expression is a function of the void fraction and hence the
whole process requires an iterative solution.
The work of Butterworth (1975) showed the striking similarity in variables between
six of the correlations he considered. He made approximate transformations of the
graphical representations of the original authors and expressed it with an appropriate
equation. The correlations reported in Butterworth (1975) in addition to the
Homogeneous and Lockhart-Martinelli correlations are that of Thom (1964) for
which the “slip factor” plot was approximated by a parameter which is a function of
the density and viscosity ratios of the two phases and
Zivi (1964) which was
developed from the principle of minimum entropy production for steam void fraction.
The correlation by Moody (1965) which in the limit for the maximum flow rate, is
exactly identical to that of Zivi (1964) has been taken out of the analysis. The Turner
and Wallis (1965) separate cylinders model for turbulent-turbulent flow has also been
cast in the slip ratio from. The Baroczy (1966) correlation which is in graphical form
10
has also been approximated to fit into the general type of slip ratio correlation
presented to show the similarity between the correlations.
Smith (1969) developed a correlation based on equal velocity heads of the
homogeneous mixture core and the annulus liquid phase. The expression for the
correlation is
α=
1
⎡
⎧
⎢
⎪
⎢1 + ⎛⎜ ρG ⎞⎟⎛⎜ 1 − x ⎞⎟ * ⎪0.4 + 0.6
⎢ ⎜⎝ ρ L ⎟⎠⎝ x ⎠ ⎨
⎪
⎢
⎪⎩
⎣⎢
⎫⎤
(ρ L / ρG ) + 0.4⎛⎜ 1 − x ⎞⎟ ⎪⎥
⎝ x ⎠ ⎪⎥
⎬⎥
⎛1− x ⎞
⎪⎥
1 + 0.4⎜
⎟
⎪⎭⎥
x
⎠
⎝
⎦
where the constant in Table 1 is given as A SM
⎛
⎜
⎜
= ⎜ 0.4 + 0.6
⎜
⎜
⎝
⎛ ρL ⎞
1 − x ⎞ ⎞⎟
⎜⎜
⎟⎟ + 0.4⎛⎜
⎟
⎝ x ⎠⎟
⎝ ρG ⎠
⎟
⎛1 − x ⎞ ⎟
1 + 0.4⎜
⎟
⎝ x ⎠ ⎟
⎠
Premoli et al. (1970) correlation or sometimes referred to as the CISE correlation is
developed from a similar analysis as the homogeneous model introducing the Weber
and Reynolds numbers for different conditions of two phase flow in vertical adiabatic
channels. The correlation is given by
1/ 2
⎫
⎧ y
⎛ 1 − α ⎞⎛ x ⎞⎜⎛ ρ L ⎟⎞
= 1 + F1 ⎨
− yF2 ⎬
⎟⎜
⎟⎜
⎜
⎟
⎝ α ⎠⎝ 1 − x ⎠⎝ ρG ⎠
⎭
⎩1 + yF2
Where, the different coefficients in the expression can be calculated via
F1 = 1.578 Re
− 0.19
L
⎛ ρL ⎞
⎟⎟
⎜⎜
⎝ ρG ⎠
F2 = 0.0273WeL Re
y=
β
1− β
− 0.51
L
Re L =
0.22
⎛ ρL ⎞
⎟⎟
⎜⎜
⎝ ρG ⎠
GD
μL
−0.08
We L =
G2D
σρ L g
11
β=
1
⎛ 1 − x ⎞⎛ ρG ⎞
⎟⎟
1+ ⎜
⎟⎜⎜
⎝ x ⎠⎝ ρ L ⎠
1/ 2
where the constant in Table 1 is given as
APRM
⎧ y
⎫
= 1 + F1 ⎨
− yF2 ⎬
⎩1 + yF2
⎭
The Chisholm (1973) correlation also falls into this category. Madsen (1975)
developed a void fraction correlation for the bulk boiling phenomenon where many
flow regimes were encountered. Spedding and Chen (1984) gave a correlation for
void fraction values of 0.8 and greater for horizontal air-water flow data. A further
extension of the Spedding and Chen correlation by Chen (1986) is also included.
The correlation by Hamersma and Hart (1987) which can be cast in the same
expression and with coefficients given for their experimental air-water data is
considered. This correlation has been developed for liquid holdups less than 0.04.The
Petalaz and Aziz (1997) correlation as reported in Ouyang and Aziz (2002) is
primarily for annular mist flow and also can be cast in the same form. Zhao et al.
(2000) correlation was developed using 189 (refined from 255) sets of geothermal
experimental data points (steam and water) and using the seventh power law velocity
distribution.
2.1.2 Kα H Correlations
The other category of void fractions are those which are a constant multiple or some
function (Isbin and Biddle, 1979) of the no slip(homogeneous) void fraction
correlation.
In an effort to predict the void fraction taking into consideration the non
homogeneous nature of the two phase flow, Armand (1946) gave a correlation in the
early days of two phase flow research and is one of the first few correlations to be
developed and is given as
α = 0.833α H
12
The modifications by Chisholm (1983) -Armand(1946) is given by
α=
1
αH
α H + (1 − α H )0.5
Bankoff (1960) correlation is developed from analysis on a single fluid with variable
density and velocity profile for vertical flow. The original form of the equation is
1
ρ ⎛ K ⎞
= 1 − L ⎜1 − B ⎟
x
ρG ⎝
α ⎠
where, K B = 0.71 + (1.45 × 10−2 ) P , P in MPa
Upon rearrangement of which becomes,
α = K Bα H
Hughmark (1962) correlation is a modification of the Bankoff (1960) correlation
primarily targeted at improving the applicability of the correlation to two phase flows
other than steam water. With a new flow parameter Z defined in terms of the
Reynolds and Froude numbers and a no slip liquid volume fraction, it was claimed
that the correlation works for both vertical as well as horizontal flows. Hughmark
modified the flow parameter to be characterized as,
K HU = f (Z )
where
Z=
Re1 / 6 Fr1 / 8
λ
1/ 4
Re =
GD
(1 − α )μ L + αμG
Fr =
2
UM
gD
λ=
1
x ρL
1+
1 − x ρG
The correlation by Nishino and Yamazaki(1963) is given as
⎡1 − x ρG ⎤
α =1− ⎢
⎥
⎣ x ρL ⎦
0.5
(α H )0.5
The correlation presented in 1964 by Kowalczewski that was reported by Isbin and
Biddle(1979) is given as
13
Table 1 Slip Ratio Correlations Coefficients for Butterworth’s (1975) Correlation
Correlation
Homogeneous
Lockhart & Martinelli(1949)
Fauske(1961)
Fujie(1964)
A
b
c
d
1
1
1
0
0.28
0.64
0.36
0.07
1
1
0.5
0
1
1
0
K FJ α + 1
Thom(1964)
1
1
0.89
0.18
Zivi (1964)
1
1
0.67
0
Turner & Wallis(1965)
1
0.72
0.4
0.08
Baroczy(1966)
1
0.74
0.65
0.13
Smith(1969)
ASM
1
1
0
Premoli et al.(1970)
APRM
1
1
0
⎛ ρ ⎞
1 − x⎜⎜1 − L ⎟⎟
⎝ ρG ⎠
1
1
0
-0.5
0
Chisholm(1973)
Madsen(1975)
⎛ρ ⎞
log⎜⎜ L ⎟⎟
⎝ ρG ⎠
1+
⎛1− x ⎞
log⎜
⎟
⎝ x ⎠
1
Spedding & Chen(1984)
2.22
0.65
0.65
0
Chen(1986)
0.18
0.6
0.33
0.07
Hamersma & Hart(1987)
0.26
0.67
0.33
0
-0.2
-0.126
0
0.875
0.875
0.875
Petalaz & Aziz(1997)
Zhao et al.(2000)
⎛ μ 2U 2 ⎞
0.735⎜⎜ L 2SG ⎟⎟
⎝ σ ⎠
α −0.125
14
0.074
⎛
α = α H − 0.71(1 − α H )0.5 Fr − 0.045 ⎜⎜1 −
⎝
P⎞
⎟
Pc ⎟⎠
The correlation of Guzhov et al. (1967) which can handle the plug and stratified flow
regimes in pipes with small inclination angles to the horizontal (±9o) is also
considered here. The correlation is a function of the homogeneous void fraction and
the mixture Froude number.
α = 0.81α H (1 − exp(−2.2 Fr )
The correlation Loscher and Reinhardt presented in 1973 as reported by
Friedel(1977) is
⎛P⎞
α = α H − ⎜⎜ ⎟⎟
⎝ Pc ⎠
−0.22
α
1.39
H
(1 − α H ) Fr
0.8
− 0.25
⎛
P⎞
⎜⎜1 − ⎟⎟
⎝ Pc ⎠
3. 4
Greskovich and Cooper (1975) developed a correlation from air water data for
inclined flows. It was noted that the data showed little diameter dependency above
2.54cm but was considerably dependent on inclination angles.
α=
1
⎡
⎛ (sin θ )0.263
⎞⎤
⎢1 + 0.671⎜⎜
0.5 ⎟
Fr ⎟⎠⎥⎦
⎝
⎣
αH
Moussali correlation reported by Isbin and Biddle (1979) is given by
α = K MOα H
K MO = 1 −
(30.4 / d1 ) + 11
60(1 + 1.6 / d1 )(1 + 3.2 / d1 )
Where d1 =
1 − x ρG
x ρL
Isbin and Biddle (1979) also gave the Kutucuglu correlation as
15
⎛
P⎞
α = α H − (1 − α H ) Fr ⎜⎜1 − ⎟⎟
⎝ Pc ⎠
0. 5
2
0. 2
Czop et al. (1994) correlation was developed based on experimental measurements of
void fraction in a helical tube of internal diameter 19.8mm,coil diameter of 1.17m and
helix angle of 7.45o with water-SF6 mixture. It was cautioned that it may not work for
situations other than for which it was developed, especially small volume flux. The
correlation is given as
α = −0.285 + 1.097α H
A slightly modified form of the original Armand (1946) correlation which takes into
consideration the quality of the two phase mixture is given as the Armand and
Massina correlation reported by Leung (2005). No limitations of any kind are
reported for this correlation. The expression for the correlation is given by
α = (0.833 + 0.167 x )α H
2.1.3 Drift Flux Correlations
This type of correlations are based on the work of Zuber and Findlay (1965) where
the void fraction can be predicted taking into consideration the non uniformity in
flows and the difference in velocity between the two phases. This model is good for
any flow regime. It has the general expression given by
α=
USG
C0U M + UGM
where C0 is the distribution parameter and UGM = UG − U M is the drift velocity.
Filimonov et al. (1957) correlated steam-water data by the drift flux relation with the
distribution parameter C0 and the drift velocity UGM given as
16
0.25
UGM
⎛ D ⎞
= (0.65 − 0.0385P)⎜ H ⎟
⎝ 0.063 ⎠
UGM
⎛ D ⎞
= (0.33 − 0.00133P)⎜ H ⎟
⎝ 0.063 ⎠
For P < 12.7 MPa
0.25
For P ≥ 12.7 – 18.2 MPa
with C0 = 1
Similarly, The correlation in 1959 by Dimentiev et al. as reported by Kataoka and
Ishii(1987) is given as
⎛
⎞
⎟⎟
⎝ ρ L − ρG ⎠
ρG
α = 1.07 j*g0.8 D*H−0.25 ⎜⎜
and
α = 1.9 j
*0.34
g
D
*−0.25
H
⎛ ρG ⎞
⎜⎜
⎟⎟
⎝ ρ L − ρG ⎠
−0.23
0.09
⎛ ρG ⎞
⎟⎟
For j*g ⎜⎜
⎝ ρ L − ρG ⎠
−0.5
⎛ ρG ⎞
⎟⎟
For j ⎜⎜
⎝ ρ L − ρG ⎠
−0.5
*
g
≤ 3.7
> 3.7
Wilson et al. (1961) correlation as given by Gardner (1980) is
α = 0.56157 F
0.62086
0.0917
v
D
⎛L⎞
⎜ ⎟
⎝D⎠
0.11033
For
F<2
For
F≥2
and
⎛L⎞
⎟
⎝D⎠
0.11033
α = 0.68728F 0.41541Dv0.10737 ⎜
In an experiment done in 25.4mm (1 in) diameter vertical tube, Nicklin et al. (1962)
produced an expression for the prediction of bubble velocity from which the void
fraction could also be backed out. The constant 1.2 in the expression is said to be
accurate for Reynolds numbers greater than 8000 and approximate for lesser values.
The expression for the correlation is
α=
U SG
1.2U M + 0.35 gD
Hughmark (1965) developed another correlation for slug flow, the simple one for
horizontal flow is given as
17
α=
U SG
1.2(U SL + U SG )
The correlation by Gregory and Scott (1969) in line with the work of Nicklin et al.
(1962) is
α=
U SG
1.19U M
Rouhani and Axelsson (1970) developed two correlations for the different regions of
flow boiling using the drift flux analysis of Zuber and Findlay (1965) that was
compared with test data that covers a wide range of pressures, heat fluxes, sub cooling
and mass velocities. The correlation is given by
.
x
α=
ρG
⎡ ⎛ x 1 − x ⎞ UGM ⎤
⎟+
+
⎢C0 ⎜⎜
⎥
ρ L ⎟⎠ G ⎦⎥
⎣⎢ ⎝ ρG
⎛ 1.18 ⎞
⎟(gσ (ρ L − ρG ))0.25
where , UGM = ⎜
⎜ ρ ⎟
L ⎠
⎝
Rouhani I : C0 = 1 + 0.2(1 − x )
Rouhani II : C0 = 1 + 0.2(1 − x )(gD )
0.25
⎛ ρL ⎞
⎜ ⎟
⎝G⎠
0.5
A variation of the Nicklin et al. (1962) correlation is given by Bonnecaze et al.
(1971) as
α=
U SG
⎛ ρ ⎞
1.2U M + 0.35 gD ⎜⎜1 − G ⎟⎟
ρL ⎠
⎝
Mattar and Gregory (1974) correlation is expressed as
α=
U SG
1.3U M + 0.7
18
Kokal and Stanislav (1989) correlated their air-oil experimental data in horizontal
and near horizontal (±9o) pipe using the drift flux relation and recommended their
correlation for all flow regimes. It is given as
α=
U SG
⎡ gD(ρ L − ρG )⎤
1.2U M + 0.345⎢
⎥
ρL
⎣
⎦
1/ 2
Coddington and Macian (2002) reported and made a comparison of thirteen void
fraction correlations that were developed based on the drift flux analysis of Zuber and
Findlay (1965).
α=
USG
C0U M + UGM
Those reported as derived from rod bundle data with large distribution parameters and
those with good predictions over the whole range of void fractions are reported here.
One of the oldest drift flux correlations for rod bundle systems was given by Dix in
1971. The parameters for this correlation as reported by Coddington and Macian
(2002) are
⎛ ρG ⎞
⎛
⎜
⎟
USG ⎜ ⎛ USL ⎞⎜⎝ ρ L ⎟⎠
⎜1 + ⎜
⎟
C0 =
U M ⎜ ⎜⎝ USG ⎟⎠
⎜
⎝
0.1
⎞
⎟
⎟ and
⎟⎟
⎠
UGM
⎛ gσ (ρ L − ρG ) ⎞
⎟⎟
= 2.9⎜⎜
ρ L2
⎝
⎠
0.25
The distribution factor and the slip velocity for the correlation of Sun et al. (1980)
developed from tube as well as rod bundle data are given by
C0 =
1
P
0.82 + 0.18
PC
and
U GM
⎛ gσ (ρ L − ρ G ) ⎞
⎟
= 1.41⎜⎜
⎟
ρ L2
⎝
⎠
0.25
Jowitt developed a void fraction correlation in 1981 from rod bundle data.
The parameters for the drift flux expression as reported by Coddington and Macian
(2002) are expressed as
19
⎛
ρ ⎞
C0 = 1 + 0.796 exp⎜⎜ − 0.061 L ⎟⎟
ρG ⎠
⎝
⎛ ρL
⎞
− 1⎟⎟
UGM = 0.034⎜⎜
⎝ ρG
⎠
and
Coddington and Macian (2002) also gave the factors of the Bestion correlation which
was presented in 1985 as
C0 = 1 and
UGM
⎛ gD(ρ L − ρG ) ⎞
⎟⎟
= 0.188⎜⎜
ρG
⎝
⎠
0.5
Similarly, for the Toshiba correlation presented in 1989, constant values for these
variables are given as
C0 = 1.08
and
UGM = 0.45
Inoue et al. (1993) introduced pressure dependence for the distribution factor as well
as the drift velocity terms in the drift flux expression. These are given as
C0 = 6.76 × 10−3 P + 1.026 and
.
⎛
⎞
UGM = ⎜⎜ 5.10 × 10− 3 m + 6.91 × 10 − 2 ⎟⎟ × 9.42 × 10− 2 P 2 − 1.99 P + 12.6
⎝
⎠
(
)
Maier and Coddington (1997) undertook a least square fit to their experimental data
and presented the distribution parameter and drift velocity with empirical constants as
C0 = C MC1P + C MC2
and
(
) (
UGM = v1P 2 + v2 P + v3 G + v4 P 2 + v5 P + v6
)
The values for the constants are given in Table 2.
2.1.4 General Void Fraction Correlations
Void fraction correlations which fall into this category are mostly empirical in nature
though the basic underlying principles have been used to develop the correlations.
20
A general type correlation given in a plot format by Flanigan (1958) put into equation
form by the AGA (American Gas Association) is considered. The correlation assumes
that pipe inclination has no effect on void fraction and is only a function of the
superficial gas velocity.
α=
1
−1.006
1 + 3.0637U SG
Chisholm and Laird (1958) correlation is given as
⎡
⎤
⎢
⎥
0.8
⎥
α = 1+ ⎢
⎢1 + 21 + 1 ⎥
2
⎢ X
X tt ⎥⎦
tt
⎣
1.75
A correlation was adapted by Hoogendoorn (1959) from correlations initially
developed for vertical two-phase flows. It was reported that it is successful in
correlating his experimental data for air-water and air-oil mixtures for horizontal
smooth (ID 24 mm – 140mm) and rough pipes (ID 50mm) considered. Moreover, he
stated that for the different velocity ranges considered, the effect of pipe diameter or
liquid viscosity is negligible. The correlation has taken care of the slip velocity
between the two phases and is given by
⎡
⎛
α
α USL ⎞⎤
⎟⎟⎥
= K HO ⎢USG ⎜⎜1 −
1−α
1
α
U
−
⎢⎣
SG ⎠ ⎥
⎝
⎦
0.85
Wallis (1969) developed an expression that best fits the data of Lockhart - Martinelli.
The correlation is a function of the Lockhart-Martinelli variable and is given by
α = [1 + Xtt0.8 ]
−0.38
Neal and Bankoff (1965) reported their correlation which was determined from
experimental data collected varying the mass flow rates and the relative volumes of
21
Table 2 constants for the Maier and Coddington (1997) correlation
constant
Value
C MC1
2.57 × 10−3
C MC2
1.0062
v1
6.73 × 10−7
v2
− 8.81 × 10−5
v3
1.05 × 10−3
v4
5.63 × 10−3
v5
− 1.23 × 10−1
v6
8.00 × 10 −1
the phases in a vertical mercury-nitrogen flow. The simplified form of the correlation
looks like
1.88
⎛ USG ⎞
⎟⎟
⎝ UM ⎠
α = 1.25⎜⎜
⎛ U2 ⎞
⎜ SL ⎟
⎜ gD ⎟
⎝
⎠
0.2
Beggs (1972) developed correlations from experimental air water data in 1in and
1.5in diameter pipes for all inclination angles. The flow pattern is first predicted and
different constants selected for the different flow regimes in horizontal flow. For all
other inclination angles, the horizontal correlation is first calculated and then properly
adjusted by an inclination correction factor. The horizontal void fraction correlations
for the different flow regimes are given in Table 3. The void fraction at any angle of
inclination for upward flow is given by
1
α (θ )
⎡
⎤
= 1 + C ⎢sin(1.8θ ) − sin 3 (1.8θ )⎥
3
α (0)
⎣
⎦
The different values of C are shown in Table 3.
Mukherjee (1979) developed a correlation from experimental data obtained using
kerosene-air and light lube oil-air mixtures in a 1.5 in ID pipe which could be inclined
22
up or down. Flow regime coordinates and angle of inclination were considered as the
independent variables.
α = 1 − exp(C1 + C 2 sin θ + C 3 sin 2 θ + C 4 N L )
C5
NGV
N CLV6
It was reported that the regression coefficients ( C1 to C6 ) which are given in Table 4
were obtained using a non linear regression programs.
Gardner (1980) came up with an improved correlation from pool void fraction data
collected from the literature and making a comparison of five such correlations. The
proposed correlation has two empirical constants one of which could take any of the
proposed two values depending on the method of void fraction measurement for the
pool void fraction. Both cases are considered in our analysis. The correlation is of the
form
α
= KGR [ FD P m ]n
1/ 2
(1 − α )
With n being constant (2/3) while m could take the value of 0.16 or 0.3.
Of the five pool void fraction correlations considered in Gardner (1980) comparison,
Wilson et al. (1961) and Sterman (1956) are considered here as the other correlations
that predicted void fractions outside the physically realistic range are discarded.
Tandon et al. (1985) correlation is based on the Lockhart-Martinelli model for
pressure drop and von Karman’s velocity profile to represent the velocity distribution.
It is developed for two phase annular flow.
{
}
50 < Re L < 1125
{
}
Re L > 1125
α = 1 − 1.928 Re −L0.315 [F (X tt )]−1 + 0.9293 Re −L0.63 [F (X tt )]−2
α = 1 − 0.38 Re −L0.088 [F (X tt )]−1 + 0.0361 Re −L0.176 [F (X tt )]−2
23
24
-1.330282
-0.516644
All
Stratified
Other
Uphill flow
Downhill
Flow
-0.380113
Flow pattern
C1
0
⎡ 0.011N LV 3.539 ⎤
⎢ 3.768 1.614 ⎥
C = (1 − λ )ln ⎢ λ N FR ⎥
⎥⎦
⎢⎣
⎡ 2.96λ0.305 N FR 0.0978 ⎤
⎥
⎢
0.4473
C = (1 − λ )ln ⎢
N LV
⎥
⎥⎦
⎢⎣
Inclination factor
0.789805
4.808139
0.129875
C2
0.551627
4.171584
-0.119788
C3
15.519214
56.262268
2.343227
C4
Values of Coefficients
0.371771
0.079951
0.475686
C5
Table 4 Coefficients for Mukherjee (1979) correlation
1.065λ0.5824
0.609
N FR
0.845λ0.5351
0.0173
N FR
α (0) = 1 −
α (0) = 1 −
0.98λ0.4846
0.0868
N FR
α (0) = 1 −
Horizontal Void fraction
Flow
Direction
Distributed
Intermittent
Segregated
Flow pattern
Table 3 Coefficients for Beggs (1972) correlation
0.393952
0.504887
0.288657
C6
The correlation by El-Boher et al. (1988) was reported to be developed from 3400
data points collected from three different facilities i.e., steam/lead-bismuth, air/ water
and steam/ mercury. The expression looks like
α=
Fr =
1
0.378
⎡
⎞ ⎛
⎛
⎢1 + 0.27 β − 0.69 (Fr )− 0.177 ⎜ μ L ⎟ ⎜ Re
⎜ μ ⎟ ⎜ We
⎢
L
⎝ G⎠ ⎝
⎣
2
U SL
gD
Re =
ρGU SL D
⎞
⎟
⎟
⎠
0.067
⎤
⎥
⎥
⎦
We =
μL
ρ LU SL D
σ
Minami and Brill (1987) developed a purely empirical general correlation from
experimental air-water and air kerosene data and the data of Beggs (1972). The
general correlation is given by
⎧⎪ ⎡ (ln Z1 + 9.21)⎤ 4.3374 ⎫⎪
⎬
⎥
⎪⎩ ⎣ 8.7115 ⎦
⎪⎭
α = exp⎨− ⎢
Kawaji et al. (1987) correlation is proposed from experimental investigation on a
large diameter high pressure steam water data. This correlation is in terms of
dimensionless flow rates for small mass velocities. It was shown that the Spedding
and Chen (1984) correlation has the same trend as their data which has been
incorporated to cater for the data range with mixture velocity greater than 1.5 m/s.
Their equation is
α = 1.05( jL* )
1/ 2
U SL + U SG < 1.5 m / s
Spedding and Spence (1989) correlation is reported as
⎛U ⎞
α
2
= [0.45 + 0.08 exp(−100(0.25 − U SL
))] ⎜⎜ SG ⎟⎟
1−α
⎝ U SL ⎠
25
0.65
Hart et al. (1989) developed a correlation for prediction of pressure drop and holdup
in two phase horizontal flow for the very high void fraction range of more than 0.94.
The explicit equation is reported as,
U
α = 1 − SL
USG
0. 5
⎧⎪ ⎛
ρ L − 0.726 ⎞ ⎫⎪
Re SL ⎟⎟ ⎬
⎨1 + ⎜⎜108
ρG
⎪⎩ ⎝
⎠ ⎪⎭
Huq and Loth (1992) correlation is given by
α = 1−
2(1 − x) 2
1 − 2 x + (1 + 4 x(1 − x)(
ρL
− 1))0.5
ρG
With the aim of simplifying the iterative mechanistic model of Taitel and Dukler
(1976), Abdulmajeed (1996) proposed an empirical correlation for the experimental
data points collected. It was reported that this correlation was tested against his data
(89 points) and 111 data points from the literature.
l
⎡U ρ μ ⎤ ρ U 2
Xtt = ⎢ SG G L ⎥ L SL
2
⎣ USL ρ L μG ⎦ ρGUSG
Turbulent m=0.2, Laminar m=1
Defining R = ln(Xtt )
For turbulent flow
(EL )theoretical
(
= exp − 0.9304919 + 0.5285852 R − 9.219634 * 10−2 R 2 + 9.02418 * 10−4 R 4
)
For laminar flow
⎛ − 1.099924 + 0.6788495 R − 0.1232191 * 10−2 R 2 − 1.778653 * 10−3 R 3 ⎞
⎟
(EL )theoretical = exp⎜⎜
−3 4
⎟
+
1
.
626819
*
10
R
⎝
⎠
(EL )True = C ABD (EL )theoretical
Where C ABD = 0.528(USGUSL )
−0.216121
26
is a correction factor
from which finally the void fraction could be easily calculated as
α = 1 − (EL )True
Based on experimental data from six sources consisting of 283 points, Gomez et al.
(2000) developed a correlation for predicting liquid holdup for slug flow for
horizontal, inclined and vertical orientations. The data covers pipe diameters between
5.1 – 20.3 in and fluids considered were air, nitrogen, Freon, water and kerosene. The
liquid slug is seen to be dependent on the inclination angle, mixture velocity and
viscosity of the liquid phase. Surface tension has no significant effect on the holdup in
comparison to the viscosity of the liquid. The equation is of the form
α = 1 − e − (0.45θ + 2.48×10
−6
Re M
)
Introducing the Froude rate parameter which is the ratio of the vapor kinetic energy
relative to the energy required to pump liquid from the bottom of a tube to the top of a
tube and following Wallis’(1969) model, Graham et al. (2001) developed a
correlation of the form
⎛
1
1 ⎞
−0.321
⎟⎟
α = ⎜⎜1 + +
⎝ Ft X tt ⎠
⎛
⎞
G 2 x3
⎟⎟
Ft = ⎜⎜
2
⎝ (1 − x )ρG gD ⎠
0.5
After listing the void fraction correlations collected from the literature into their
respective categories where it is thought to give some insight into the similarity
between the correlations, the next logical step one would look into is if any
comparison work has been done in order to identify the “best” or a promising
correlation for all or any particular area of interest. Hence, a search was made to study
most if not all of the previous comparison works done along this line which we will
briefly present in the next section.
27
2.2 Previous comparison work
This section will look into void fraction correlation performance comparisons done
before. It is to be noted that different investigators in the different industries in which
the void fraction is an important factor to be determined have done various
comparisons with selected void fraction correlations commonly used in that specific
industry while some other have tried to consider a broader range of correlations that
are used in different applications.
2.2.1 Dukler et al. (1964) Horizontal Pipe Comparison
The first void fraction correlation comparison work was done by Dukler et al.(1964).
It consists of 706 refined void fraction data points of Hoogendoorn (1959) obtained
from tests run in 1,2,3 ½ and 5 ½ in diameter horizontal pipes with liquid viscosities
of 3 and 20cp. The void fraction correlations considered were Hoogendoorn (1959),
Hughmark (1962) and Lockhart and Martinelli (1949).
Using statistical tools such as arithmetic mean deviation, standard deviation and
defining a new variable to account for the fractional deviation which includes 68% of
the population to measure the spread of data, it was shown that the Hughmark (1962)
correlation was able to perform better that the other two.
2.2.2 Marcano (1973) Horizontal Pipe Comparison
Marcano (1973) compared five correlations; Lockhart and Martinelli(1949),
Hughmark(1962), Dukler et al. (1969), Eaton et al. (1967),Guzhov et al. (1967) and
Beggs(1972) correlations using the data of Eaton (1966) and Beggs(1972).
The total data points consisted of 238 natural gas - water (Eaton) and 58 air - water
(Beggs). The statistical parameters considered in the Dukler et al. (1969) comparison
were calculated here to determine the relative accuracy of the correlations. It was
shown that the Eaton et al. (1967) and Beggs (1972) correlations performed well
28
owing to the fact that the data used for comparison is the data from which these
correlations were developed. The correlations of Dukler et al. (1969) and Lockhart
and Martinelli (1949) are reported to have acceptable results while the rest were
totally unsatisfactory.
Dropping out the “unreliable” low liquid holdup data (below 0.1), the correlations
were further compared. It was noted that the performance of the correlations was
better. The performance of the correlations for specific liquid holdup(void fraction)
ranges were also analyzed where Eaton et al. (1967), Guzhov et al. (1967) and
Beggs(1972) correlations were found to do better for void fraction less than 0.65(or
liquid holdup above 0.35) while on the lower range of holdup only the Eaton et al.
(1967) correlation gave acceptable results. For liquid holdup below 0.1 none of the
correlations gave reasonable accuracy though the Dukler et al. (1969) correlation and
the no slip model gave best estimates. One of the many reasons for this could be the
unreliability of the measurement of the data for this range of holdup.
2.2.3 Palmer (1975) Inclined Pipe Comparison
Using 174 liquid holdup(void fraction) water-natural gas experimental data from a 2
in diameter pipe line with three uphill(at 4.2,7.1 and 7.5 degrees from the horizontal)
and three downhill(at 4.3,3.8 and 6.3 degrees from the horizontal) test sections,
Palmer(1975) compared the correlations of Beggs(1972), Flanigan (1958) and
Guzhov et al. (1967). The percent error, average percent error and the standard
deviation were calculated to make the comparison. It was concluded that the
Beggs(1972) correlation gave good predictions of the void fraction for uphill flow.
The Flanigan (1958) correlation was the least accurate which may be expected as it
didn’t consider downhill flow and also the fact that it was given as an approximate
estimate by Flanigan.
29
2.2.4 Mandhane et al. (1975) Horizontal Pipe Comparison
Mandhane et al. used 2700 void fraction (liquid holdup) data contained in the
University of Calgary Multiphase Pipe Flow Data Bank. A two step procedure for
calculating the void fraction was used in that the flow pattern is first predicted and
then a correlation developed for that flow regime is used to calculate the liquid
holdup. The flow pattern map of Mandhane et al. (1974) was used for the analysis.
Twelve void fraction correlations were considered. They were, Lockhart and
Martinelli(1949), Hoogendoorn(1959), Eaton
et al. (1967), Hughmark(1962),
Guzhov et al. (1967), Chawla(1969), Beggs(1972), Dukler et al. (1969),
Scott(1962),Agrawal et al.(1973), Hughmark(1965) and Levy(1960).
Five different measures of error, namely the root mean-square error, mean absolute
error, simple mean error, mean-percentage absolute error and the mean-percentage
error were used. Arbitrary designation of the void fraction ranges to the flow pattern
was made to see the predictive capability of the correlations within each range of
holdup (void fraction) values or flow regime.
Using the flow pattern map of Mandhane et al.(1974), the Hughmark(1962)
correlation is recommended for the bubble, elongated bubble and slug flow regimes,
the flow pattern specific correlation of Agrawal et al.(1973) for the stratified,
Lockhart and Martinelli(1949) correlation for annular, annular-mist and the
Beggs(1972) correlation for the dispersed-bubble flow regimes. No correlation was
recommended for the total data points even though the recommended error values for
all the parameters were given.
Comparing the correlation using the flow maps of Hoogendoorn(1959), Govier and
Aziz(1972) and Baker(1954), it was shown that the Hughmark(1962) correlation
30
predicts the bubble, elongated bubble regimes in all the four maps and the slug regime
in all but the Baker (1954) map where the Chawla(1969) is seen to outperform it.
The stratified regime is predicted by the Agrawal et al. (1973) correlation in the
Mandhane et al. (1974) and Baker (1954) maps while the Dukler et al. (1964)
correlation predicts that of Hoogendoorn (1959) and Govier and Aziz (1972). This is
easily explainable as it has an advantage over the others due to the bias it has towards
these two data sets from which it was developed.
The annular, annular-mist regime is handled by the Lockhart and Martinelli (1949) in
all the flow pattern maps.
In the dispersed-bubble regime the Beggs (1972) correlation has predicted the data in
Mandhane et al. (1975) while Hughmark (1962) was able to handle that of
Hoogendoorn (1959) and Govier and Aziz (1972). The Hoogendoorn (1959)
correlation is shown to predict the data in the Baker (1954) map quite well.
All in all it was concluded that none of the correlations gave satisfactory results in the
annular and annular mist flow regimes.
2.2.5 Papathanassiou (1983) Horizontal Pipe Comparison
Deducing the physically realistic range within which two phase flow exists for a set of
operating conditions and specified fluid, a void fraction spectrum graph is constructed
which is used to compare the correlation of Lockhart and Martinelli (1949),
Hoogendoorn (1959), Bankoff (1960) and Hughmark (1962). All but the Bankoff
(1960) correlation have similar trend in predicting void fraction. The Bankoff (1960)
correlation is seen to under predict the void fraction mainly due to the fact that it was
developed for vertical flow. It was shown that the Lockhart and Martinelli (1949)
correlation falls out of the physically realistic range at a void fraction of about 0.4. At
very high void fraction ranges of 0.7 and above the Bankoff (1960) correlation is seen
31
to fall in the unrealistic region. The disagreement between the void fraction
correlation predictions in the lower and upper extreme values of void fraction ranges
is explained by the fact that few experimental results at this range exist when the
correlations were developed.
2.2.6 Spedding et al. (1990) Inclined (2.75o) Pipe Comparison
Spedding et al. (1990) considered 60 correlations and the data of Spedding and
Nguyen (1976) to make a comparison of the void fraction correlation at an upward
angle of 2.75o form the horizontal. A recommendation for the 12 flow regimes in this
flow was made. Criteria that the average predicted value to be within ±15% with a
30% spread was used to consider the predictive performance of a correlation as
satisfactory.
2.2.7 Abdulmajeed (1996) Horizontal Pipe Comparison
In an effort to simplify the mechanistic model of Taitel and Dukler (1976) and
developing a new correlation, Abdulmajeed (1996) collected 88 air-kerosene void
fraction data in a horizontal 2in diameter pipe and compared 12 correlations. Namely;
Armand (1946),Hughmark and Pressburg(1961),Hughmark (1962),Eaton et al.
(1967), Guzhov
et al.(1967),
Beggs (1972),Gregory
et
al.(1978),Brill et
al.(1981),Chen and Spedding(1981,1983),Mukherjee and Brill(1983),Minami and
Brill(1987) and Abdulmajeed(1996).The comparison parameters used were average
percent error, absolute average percent error and the standard deviation. It was shown
that the correlation of Abdulmajeed (1996) is able to predict void fraction in stratified,
slug and annular flow regime which covers a range of flow regimes in contrast to the
implicit Taitel and Dukler(1976) model which was specifically developed for
stratified flow.
32
2.2.8 Spedding (1997) General (-90o to +90o) Comparison
Spedding undertook an extensive comparison on more than 100 void fraction
correlations using the air-water data of Spedding and his coworkers (1976, 1979,
1989, 1991, 1993). The pipe diameters ranged from 26mm to 95.3 mm. Prediction of
a correlation was considered satisfactory for average predicted values that fall within
±15% of the data with spread of individual values within ±30%. Eighteen flow
regimes were identified for which the predictive capability of the correlations
assessed. The comparison ranges from -90o to + 90o with fair representation of major
inclination angles.
It was noted that no single model could handle all flow regimes and angle of
inclination satisfactorily. Different void fraction correlations are recommended for
different flow regimes for horizontal/upward inclined and downward flow separately.
2.2.9 Friedel and Diener (1998) Horizontal/Vertical Upward Comparison
Based on 24000 experimental data bank in single component water and R12, and two
component air-water, a comparison of 13 (refined selection of the originally 26
correlations) void fraction correlations was undertaken. Some of the correlations are
proprietary of which no information was given while the others are collected from the
open literature. Some of the correlations considered in the analysis that passed the
limiting void fraction criteria, for x = 0 ,α = 0 and x = 1, α = 1 , and were considered
in our comparison are Huq and Loth (1992), Kowalczewski (1964), Smith (1969), the
Loscher and Reinhardt correlation reported by Friedel (1977), Rouhani and Axelsson
(1970) and Premoli et al. (1970).
The average predictive accuracy of the correlations was established by calculating the
statistical parameters; the scatter of logarithmic ratios, scatter of absolute deviations
and average of logarithmic ratios for the mean void fraction and mean density.
33
The Rouhani and Axelsson’s (1970) first correlation (Rouhani I) has been
recommended as having an accurate predictive capability.
Coming to the conclusion of this chapter, the challenge of the design engineer in
finding a suitable correlation that would be applicable to the task at hand from the
numerous correlations available is unquestionable. Moreover, the few numbers of
void fraction correlation comparisons made and the recommendations put forward by
the investigators are very narrow in nature complicating the task of reaching at
objective conclusions very difficult. This is due to the fact that, the correlations
selected for comparison are those that are popular to a particular industry (like the
petroleum, nuclear industries and so on), limited to a certain physical restriction
which may not be always practical(horizontal pipe, air –water mixture) or biased
towards one experimental set up for which complete and detailed description of
and/or the data that was obtained thereby are not clearly indicated.
Hence, an objective assessment of all the available correlations in the open literature
with all the available data from different sources covering wide range of operating
characteristics including any recent data which was not included in the previous
comparisons is justified. The next section of this work is devoted to the presentation
of the systematically gathered and compiled, acceptably good quality void fraction
database.
34
CHAPTER 3
EXPERIMENTAL DATA BASE AND FLUID PROPERTIES
As pointed out in the previous chapter, the main emphasis of this section would be on
the characteristics of the experimental data base that was collected from the literature
and the various correlations used in determining fluid physical properties of the test
mixtures. A brief highlight to the different void fraction measurement techniques used
by different investigators would be given towards the end of this chapter.
In the collection process, effort has been made to make the data base as un-biased,
wide range covering and having an acceptable quality as was practically possible.
3.1 Experimental Database
After the initial screening process that primarily considered the completeness of the
data set, the data of Eaton (1966), Beggs (1972), Spedding and Nguyen (1976) ,
Mukherjee (1979),Minami and Brill (1987),Franca and Lahey (1992), Abdulmajeed
(1996), and Sujumnong (1997) were selected with which the performance of the void
fraction correlations are to be compared. The range of diameters, inclination angles
and fluids covered by the data is summarized in Table 5. A more detailed description
of the each data set is provided in Appendix A.
The pressure, temperature, diameter of the pipe and the measured and calculated
values for the void fraction from each source is taken as it is, without changing any of
the units involved. One exception is that of the Spedding and Nguyen (1976) data,
where a value for the ratio between the void fraction to the liquid holdup was reported
from which the void fraction is explicitly calculated.
35
Each set of data was tested against simple but essential requirements to determine if
all the data points are within the realistic range of values.
One screening criterion was to identify the measured void fraction in any two phase
flow that exceeded the value calculated by the homogeneous model. This yielded 13
of Beggs (1972), 20 of Spedding and Nguyen (1976), 61 of Mukherjee (1979), 1 of
Eaton (1966), 5 of Abdulmajeed (1996), 3 of Franca and Lahey (1992) and 3 of
Sujumnong (1997) data to be out of the realistic range. These data points were
therefore excluded from the data bank.
In addition, following the work of Zuber and Findlay (1965), plotting any two phase
flow experimental data on the weighted mean velocity ( USG / α ) versus average
volumetric flux density ( U M ) , a few abnormally isolated data points that don’t fall
into any data clusters (accounting for flow pattern change) were suspect to be
unreliable and hence had been taken out. As pointed out by Zuber and Findlay (1965),
deviation of points from a certain cluster may be attributed to a change in flow
regime.
When a group of data points stray far from the rest, verification with the reported flow
pattern by the researcher has been done to check if indeed those data points belong to
a different flow regime. If not, these data points are suspect to be unreliable and hence
discarded.
Data points which are identical in all aspects(repeated runs) but reported as separate
points in some of the reports were also taken out. This is to make the analysis fair
especially for those correlations that could not predict that data point as calculating
the total percentage of data points within the acceptable percentage band is sensitive
to the number of data points and counting the same data point more than once is not
necessary.
36
37
101
Quick-Closing
Valves
Note: For details on this data set, please refer to Appendix A.
Vertical, ID 12.7mm
Sujumnong (1997)
83
Quick-Closing
Valves
Horizontal, ID 50.8mm
Abdulmajeed(1996)
81
Air-Water
Air-Kerosene
Air-Water
Quick-Closing
Valves
Horizontal, ID 19mm
Franca and Lahey(1992)
111
Quick-Closing
Valves
Air-water and
Air-Kerosene
Horizontal, ID 77.93mm
Air-Kerosene
Horizontal, Uphill and Vertical
ID 38.1 mm
Mukherjee(1979)
Minami and Brill(1987)
Capacitance Probes
Air-Water
Horizontal, Uphill and Vertical
ID 45.5mm
Quick-Closing
Valves
558
1383
291
Spedding and
Nguyen(1976)
Quick-Closing
Valves
Air-Water
Horizontal, Uphill and Vertical
ID 25.4mm and 38. 1mm
Beggs(1972)
237
Quick-Closing
Valves
Natural Gas-Water
Horizontal, ID 52.5mm and 102.26mm
No. of Data
points
Measurement
Technique
Eaton(1966)
Mixture considered
Physical flow configuration /
Characteristics
Source
Table 5 Characteristics of Data Base Sources
The next step taken was to collect well known correlations and equations that could
accurately predict the fluid physical properties of the fluids under consideration. We
shall separately discuss each of the fluids considered as the types of correlations
developed differ from one type of fluid to the other.
3.2 Fluid Physical Properties
The experimental data base compiled from the different sources consists of mixtures
of four fluids. These are air, kerosene, natural gas and water. Properties for these
fluids were calculated with correlations either explicitly given by the researcher of the
specific data or from well known correlations and equations of state taken from the
literature. We will briefly give the correlations used for each fluid type in this section.
3.2.1 Properties of Air
Air being the most economical fluid available for any experiment, it has been used as
the gas phase in all data sets except that of Eaton(1966). Neglecting ,Z,
compressibility effects, the density of air is calculated from the ideal gas equation of
state as
ρ air =
P
T (R/M air )
Where T
in (kg/m3)
Temperature in K
P
Absolute pressure in Pa
R
The universal gas constant (8314.34 J/kmol. K)
M air
Molecular weight of air (28.966 kg/kmol)
The viscosity of air is calculated from the correlations by Kays and Crawford(1993)
which is reported to be good for the temperature ranges between -10 to 120 oC with
R 2 = 0.99994. It is given as
μ air = 1.7211 × 10 −5 + 4.8837 × 10 −8 T − 2.9967 × 10 −11 T 2
38
in (Pa.s)
where T is in oC.
3.2.2 Properties of Kerosene
Mukherjee (1979), Minami and Brill (1987) and Abdulmajeed (1996) have used
kerosene as the liquid phase in their experiments. Properties such as density, dynamic
viscosity and surface tension for kerosene were given by approximate equations based
on laboratory measurements. The respective equations put forward by each researcher
have been used in determining the physical properties of kerosene for the respective
data set.
The fitted equations by Mukherjee (1979) are given as
σ ker o = 29.198 − 0.05T
in (dynes/cm)
ρ ker o = 52.8858 − 0.0289T
in (lb/ft3)
μ ker o = Exp(1.4344 − 0.0115T )
in (cp)
where T is in oF.
The fitted equations by Minami and Brill (1987) are
σ ker o = 33.62 − 0.06452T
in (dynes/cm)
ρ ker o = 62.4(0.83756 − 0.000395T )
in (lb/ft3)
μ ker o = 4.955Exp(− 0.0111T )
in (cp)
where T is in oF.
The fitted equations by Abdulmajeed (1996) are given as
σ ker o = 29.9776 − 0.13176T
in (dynes/cm)
ρ ker o = 62.4(0.8252 − 0.00072T )
in (lb/ft3)
μ ker o = 3.473004Exp(− 0.02016T )
in (cp)
where T is in oC.
39
3.2.3 Properties of Natural Gas
The ideal gas equation of state accounting for compressibility effects has been used
in calculating the density of natural gas. Hence, the density of natural gas is calculated
from
ρ Ngas =
P
ZT R/M Ngas
(
in (kg/m3)
)
The molecular weight of the natural gas used in the Eaton (1966) experiment is
calculated from the specific gravity of the natural gas given and the known molecular
weight of air.
The compressibility factor, Z, is calculated by a FORTRAN subroutine of Dranchuk
and Abou-Kassem (1975) which is the approximation of the Standing and Katz
(1942) correlation. As inputs to the subroutine, the pseudo-critical properties, pressure
and temperature have to be determined first.
Since the composition of the natural gas used in Eaton (1966) experiment is not given,
the pseudo-critical properties could only be determined from the Standing (1977)
equation which is a function of the specific gravity of the natural gas only. The
equations for the natural gas systems are given as,
2
T pc = 168 + 325γ Ngas − 12.5γ Ngas
2
Ppc = 677 + 15γ Ngas − 37.5γ Ngas
where γ Ngas
is specific gravity of natural gas.
Then the reduced pressure and temperature for each data point are calculated from the
expressions,
Tr =
T
Tpc
40
Pr =
P
Ppc
These reduced pressures and temperatures are then read from file to calculate the
compressibility factor for each data point. The other thing worth mentioning here
would be, Eaton(1966) reported the volume flow rate of natural gas in standard units
(scf/d). Therefore, to convert this into the common volume flow rate units in m3/sec or
ft3/s, the gas formation volume factor has to be considered in addition to the basic unit
conversions.
The gas formation volume factor is defined as the ratio of the actual volume of gas at
given pressure and temperature to the volume of the same gas at standard conditions
(60oF and 14.7 Psia). Ahmad (2001) gives the gas formation volume factor as,
B gas = 0.02827
ZT
P
where B gas is the gas formation volume factor, ft3/scf and T is Temperature in o R.
The optimized Lee et al. correlation reported by Jeje and Mattar (2004) is used to
calculate the viscosity of natural gas. The assumption made here is that the natural gas
used is a sweet gas (containing no or only traces of non-hydrocarbon gases) due to its
low specific gravity. The Lee et al. correlation is given by,
μ Ngas = 10 −4 K Exp(Xρ Y )
where the different variables are defined as,
X = x1 +
x2
+ x3 M
T
Y = y1 − y 2 X
K=
(k1 + k2 M )T k
3
k4 + k5 M + T
41
The optimized constants for this equation as given by Jeje and Mattar (2004) are
given in Table 6.
3.2.4 Properties of Water
As in the case of air, being the cheapest and very easy to handle, most experiments are
done with water as the liquid phase. There are quite a number of correlations to
determine the thermo physical properties of water in the literature. The equations by
Linstrom and Mallard (2003) reported to be applicable to temperature ranges
between 0 and 100 0C with R2=0.99997are used to calculate the density and viscosity
of water. The equations are given as
ρ water = 999.96 + 1.7158 × 10 −2 T − 5.8699 × 10 −3 T 2 + 1.5487 × 10 −5 T 3
in
(kg/m3)
where, T is in oC.
μ water = 1.7888 × 10 −3 − 5.9458 × 10 −5 T + 1.3096 × 10 −6 T 2 − 1.8035 × 10 −8 T 3
+ 1.3446 × 10 −10 T 4 − 4.0698 × 10 −13 T 5
in(Pa.s)
where, T is in oC.
Popiel and Wojtkowiak (1998) gave a formula for calculating the surface tension of
water that approximates the data of Vargaftik et al. (1983,1996) for the temperature
ranges between 0 and 150 o C as,
σ water = 0.075652711 − 0.00013936956T − 3.0842103 × 10 −7 T 2 + 2.7588435 × 10 −10 T 3
where, T is in oC and σ water in (N/m).
3.2.5 Measurement Techniques
As pointed out in the beginning of this chapter, a brief description of some of the most
commonly used experimental void fraction methods would be discussed. It must be
pointed out that the purpose of this short section here is to bring to the attention of the
42
Table 6 Optimized constants for the Lee et al. correlation
Variable
Optimized Value
k1
16.7175
k2
0.0419188
k3
1.40256
k4
212.209
k5
18.1349
x1
2.12574
x2
2063.71
x3
0.00119260
y1
1.09809
y2
-0.0392851
interested reader what the possibilities are in terms of measuring void fraction and
some of the pros and cons associated with some of the commonly used methods. It
also presents some of the very good references which should be looked into if in
depth understanding in this area is sought.
A considerable amount time and investment has gone into researching and devising a
proper measurement technique that could capture void fraction data accurately. As the
applications and areas of interest to the different researchers were diverse, the number
of void fraction measurement devices or techniques also differ in various aspects
ranging from but not limited to cost, nature of data of interest (local and/or average)
and simplicity in construction as well as operation.
Hewitt (1978) discussed the various measurement techniques available to measure
void fraction classifying them into different categories. This section would be
predominantly a take off from his discussion with emphasis given only to those which
could be commonly encountered in the literature.
43
Three of the most common of the available experimental techniques given by Hewitt
(1978) are the radioactive absorption and scattering method, impedance method and
the direct and indirect volume measurement. A brief discussion and the relative
advantages and disadvantages of these methods will be presented below.
Radioactive Absorption and Scattering Technique
As the name implies this method relies on the property of the two phase fluid mixture
to attenuate or scatter a given beam of radioactive ray. β (beta), Gamma and X rays
have been used by different investigators while the most commonly employed one
being the Gamma ray attenuation technique. This technique will give the
instantaneous local void fraction which may also be integrated over time to calculate
an average while the collected data could simultaneously give prediction on the flow
pattern. One of the major advantages with this method is that it doesn’t disrupt the
flow field while the safety consideration, high cost, complexity of construction and
inaccuracy in measurement are some of the draw backs associated with this method.
A traversing mechanism that scans the length of the test section is a peculiar feature
of this technique.
Impedance Technique
Incorporating the appealing nature of being non intrusive, continuous measurement,
less expensive and simple construction, this technique has been used more commonly
to measure void fraction. This method takes advantage of the change in electrical
impedance between two electrodes separated by a dielectric material, for this case the
dielectric material is the two phase mixture and the change in impedance would be
triggered as a result of the variation in void fraction. Designs for the electrodes can be
flush mounted to the inner wall of the test section or mounted in a different pattern
around the external body.
44
A number of devices based on either the conductance or capacitance or both have
been developed. Some of these are the works by Gregory and Mattar (1973), Rosehart
et al. (1975), Mukherjee (1979), Elkow and Rezkallah (1996) and Song et al. (1998).
The draw backs of this method are the temperature dependency of the dielectric
constant, requirement of high excitation frequency for large conductivity fluid and
also calibration issue associated with purity of the mixture.
Direct Measurement Technique
This is the most commonly used experimental technique where the two phase flow is
trapped between two quick closing valves mounted along the test section after which
the trapped volume of liquid is drained and measured. Since the total volume of the
test section between the valves is predetermined by design, the ratio of the measured
amount to that of the total would then give the required fraction of the liquid and
hence indirectly determining void fraction.
The most noticeable advantages of this method are simple construction and low cost.
It is also used to calibrate and/or validate void fraction measurements from other
methods. The unique feature of this method is that a bypass line would be required not
to damage the system due to disruption to the flow. The major set backs in using this
system are that it is time consuming, would not capture any transient characteristics of
flow and not practicable to high temperature and pressure applications.
For someone thinking of building up or putting together a system for the purpose of
measuring void fraction, it is recommended to start with Hewitt (1978) where he has
discussed some other methods on top of the ones presented above while presenting an
appreciable number of references which could be of practical interest.
To wrap up this chapter, we have briefly presented the different data sets we included
in our database. Applying some basic screening tools to make the data sets more
45
reliable we have excluded data that do not fall within the physically realistic band
within which two phase flow should exist. Well known and widely used equations and
correlations for calculating the physical properties of the fluids under consideration
have been gathered from primary and secondary sources and reported in this section.
Having highlighted the shortcomings of the previous performance comparison work
that were focused on a few selected correlations or limited data sets, we have tried to
gather a good number of void fraction correlations (Chapter 2) and reasonably wide
ranging experimental void fraction data (this chapter). We are now ready to do a
comprehensive comparison of all the void fraction correlations with the experimental
data base that was complied.
The next chapter will be focused on reporting the methodology and results of the
comparison work that was undertaken.
46
CHAPTER 4
EVALUATION OF VOID FRACTION CORRELATIONS
This chapter is mainly concerned with the evaluation of the predictive capabilities of
the different void fraction correlations that were reported in Chapter 2 with the data
base discussed in Chapter 3.
For ease of reference and quick read-through between some of the similarities and/or
disparity of the conclusions drawn from this work and the previous assessments, the
current comparison work is divided into three sections namely horizontal, upward
inclined and vertical. These sections will be separately analyzed and a
recommendation drawn for each section before the final summary is presented.
In line with our ambitious goal of recommending a single or a combination of
correlations that could handle all angles regardless of flow regimes, inclination
angles, fluid types and so on, we have removed all the restrictions on all the
correlations in trying to see their promise to handle most if not all of the diverse data
points in the database.
The fact that this is an over stretch on all of the correlations should be well noted.
Hence, all subsequent analyses and comparisons made which may dictate unfavorable
conclusions on some or all correlations should be taken from this perspective only and
should not be an undermining factor against the correlations in any way.
The degree of accuracy expected from a correlation for a particular application is a
very subjective issue recognizing the fact that none of the correlations would handle
all data points in any data set. Hence, the practicing engineer/researcher could set his
47
own criteria of choosing the “best” performing correlation considering the task at
hand.
As given in the literature review part, different parameters have been used by the
previous comparison analyses to show the relative accuracy of one correlation over
the others. The most commonly used of these include widely known parameters like
percent error and standard deviation while some investigators have defined other
parameters. The fractional deviation above and below the mean used by Dukler et al.
(1964),the scatter and average of logarithmic ratios by Friedel and Diener (1998) and
the relative performance factor reported by Abdulmajeed and Al-Mashat (2000) are
noteworthy.
All of the methods have inherent strengths and weaknesses. Preference of one
technique over the other is again a subjective matter. We have considered the simple
percentage error index in this work for two reasons. The first one is that we would get
hold of some very “stray” data points that would have gone undetected due to the
cancellation effect introduced in some of the error measurement techniques.
The other rationale is that it would give us an insight as to where the weakness of a
correlation lies which could facilitate the investigation to improve a given correlation
or to learn from the pitfalls when developing a new one.
The major drawback of this technique is that there is no general consensus on what a
“good” bandwidth is for a given correlation to be taken as acceptably accurate.
Different researchers have considered different ranges for their comparisons. Madsen
(1975) took ±10% for checking his correlation against measured data; Chexal et al.
(1991) considered ±10% when assessing eight void fraction correlations that are
common in the nuclear industry. Smith (1969) also took ±10% for verification of his
correlation against experimental data and also assessing the relative importance with
48
other correlations. Spedding et al. (1990) and Spedding (1997) have considered the
±15% index together with a maximum spread of 30%. Friedel and Diener (1998) have
presented prediction of two correlations on ±30% index as an example even though
the final recommendations were not based solely on that plot.
The other major area that should be looked into when comparing a given correlation
with a particular data set is the total number (percentage) of the data points that have
to be correctly predicted within the specified index to deem the correlation good or
not. We would refer to this percentage as the cutoff percentage not to confuse with the
percentage error index discussed above.
In order to be able to cater for different (at least three) scenarios to make comparisons
between the correlations that one could pick from; a decision was made to go for a
progressively increasing cutoff percentage as the percentage error index is relaxed
out. By this we mean that a lower cut off percentage of 75% was chosen for the more
restrictive 5% error index while cutoff percentages of 80 and 85% were used for the
10 and 15% indices respectively.
Each of the data sets, even those from the same researcher but with different
inclination angle or fluid, has been considered separately. This would aid in making
sound judgment about the data, the capability of the correlations to predict that
specific data set and other pertinent facts that could have been easily obscured if two
or more data sets were bundled together.
Some approximations and modifications were done on some correlations without
compromising their capability so that the calculations could be done more efficiently.
A brief explanation of the two major such adjustments are given below.
The correlation by Greskovich & Cooper (1975) was developed for inclined flows
only and hence could not directly be applied to horizontal flow case. Moreover,
49
setting a value of zero for the inclination angle in the correlation for horizontal flow,
the different data sets tested were not predicted well. However, the results were seen
to significantly improve when a very small number near zero (10-6) is considered and
this has been integrated in the calculations.
The Hughmark (1962) correlation flow factor K HU which was given in tabular form in
the original report in terms of the variables discussed in Chapter 2, has been
approximated by a function fitted to the values given with a correlation coefficient
of R 2 = 0.999 .
For all correlations requiring iteration, the simple but accurate bisection method has
been used to calculate the void fraction that has lower and upper limits of 0 and 1
respectively.
We shall now present the different comparisons starting with the horizontal flow first.
4.1 Horizontal flow comparison
A total of 900 data points from eight sources was used for the comparison in this
section. The analysis is first focused on the predictive capability of the correlations on
each of the data sets separately and would be consolidated into a general one towards
the end of the section.
The total number of data points correctly predicted by each of the correlations for the
respective individual data sets are reported in Table 7. This table is the base on which
all subsequent percentage tables and analyses there of are made. It gives the reader an
objective assessment ground of the capability of each correlation between the
different data sets as percentage values could sometimes give a misleading impression
at first glance.
To make the relative comparison between the correlations, the numbers of correctly
predicted points are expressed as a percentage for the different error percentage
50
indices. These have been presented in Table 8 with a highlight on those results which
have met our criteria set out earlier and are worth noting.
The data set of Eaton (1966) could not be satisfactorily predicted by any of the
correlations for the more restrictive 5% index. The closest best prediction is given by
Premoli et al. (1970) that could only predict 68.4% of the total data points.
The same trend is observed for the data of Abdulmajeed (1996) and Mukherjee
(1979). The only exception are the correlations of Filimonov et al. (1957) and Premoli
et al.(1970) that were able to predict 77.1% and 74.7 % of Abdulmajeed (1996) data.
The next best prediction comes from Mukherjee (1979) and Minami and Brill (1987)
with 72.3% and 73.5%, respectively of the data points correctly predicted for the
Abdulmajeed (1996) data. The performance of all the rest of the correlations was far
from being satisfactory. Even those of Mukherjee (1979) and Minami and Brill (1987)
failed to give acceptable prediction of Mukherjee (1979) data.
This clearly shows the lack of capability of all of the correlations to satisfactorily
predict void fraction for fluids different from air-water mixtures from which most of
them were originally developed. A point worth mentioning here is that the
correlations of Mukherjee (1979) and Minami and Brill (1987) have an edge over the
others as they are developed from the same data set that was used here for the
comparison.
The other notable observation here is that even the correlations developed from the
respective data sets by Mukherjee (1979) and Abdulmajeed (1996) could only capture
51.6% and 57.8% of their own data within the 5% percentage error index. The
exception in this regard is the correlation by Minami and Brill (1987) that was able to
predict 84.2% of their own air kerosene data within the 5% index. However, it
captured a fair 73.5% of Abdulmajeed (1996), with only 53.2% of Mukherjee (1979)
51
and 55.3% of Eaton (1966) data. Hence, even for the same or similar fluid types,
correlations developed from specific experimental data sets with fitted constants fail
to adequately predict data sets from other test setups and operating conditions.
The trend in predictive capability for the air-water data is almost similar for the 5%
index with a few exceptions. The Beggs (1972) correlation was able to predict 82.1 %
of his own data while Lockhart and Martinelli (1949), Wallis (1969) and Spedding
and Chen (1984) predicted 74.1, 74.4 and 75.6% of Spedding and Nguyen (1976)
data, respectively.
The performance of all correlations improves as the percentage error index is relaxed
out to 10 and 15%.With the 10% error percentage index 9, 7 and 9 correlations were
able to predict the data of Abdulmajeed(1996), Eaton(1966) and the air kerosene data
of Minami and Brill(1987),respectively. It must be noted here that
only the
correlation of Premoli et al. (1970),Mukherjee (1979), Minami and Brill(1987) and
Graham et al. (2001) were able to predict two data sets within this index whereas all
the rest only predicted a single data set.
Considering all the data sets, the correlations of Armand-Massina (Leung 2005),
Premoli et al. (1970) and Minami and Brill (1987) are the ones which came close to
the set criteria and worth a general recommendation for the 10% percentage error
index.
The performance of most of the correlations further improves with an extended error
percentage index of 15%. Here a fair number of correlations were seen to perform
rather well as compared to the lower error percentage indices.
The ones worth consideration here are due to Armand-Massina as reported by Leung
(2005), Hughmark (1962), Smith (1969), Premoli et al.(1970), Rouhani and Axelsson
(1970)first Correlation (Rouhani I), Dix(1971), Beggs(1972), Chisholm(1973),
52
Mukherjee(1979),Chisholm(1983)-Armand(1946),Toshiba(1989),Huq & Loth(1992),
Minami & Brill(1987) and Graham et al.(2001).
Consolidating the results of the individual data sets together, a comparison table was
constructed that shows the total number and percentage of data points correctly
predicted for the whole horizontal database (900 points). This has been presented in
Table 9.
As this is a cumulative analysis, all the issues we raised when discussing the
individual data sets would have their own effect on the overall performance of each
correlation. This is reflected in the table as the performance of all the correlations
have deteriorated heavily especially within the 5% error index. The only correlation
that was able to give a fair result is that of Premoli et al. (1970) with 65.2% which is
much lower than the “acceptable” cutoff index we set out earlier.
The case is seen to improve for the wider error indices where the correlation of
Armand-Massina (Leung 2005), Minami and Brill (1987), Premoli et al. (1970) and
Rouhani I (1970) were able to predict 79.1, 78.3, 78.2 and 77 % of the whole data set
within 10% error index, respectively. On the other hand correlations that gave an
acceptably good prediction within the 15% index are Rouhani I(1970), Hughmark
(1962), Armand Massina(Leung 2005), Mukherjee(1979), Minami and Brill(1987),
Chisholm(1983)-Armand(1946) and Chisholm(1973) in order of decreasing accuracy.
Another set of correlations worth a general recommendation with very close
performance to the ones listed above are due to Smith (1969), Premoli et al. (1970),
Dix (1971),Beggs (1972),Toshiba(1989), Huq and Loth (1992) and Graham et al.
(2001).
53
54
78
43
26
Zivi(1964)
Hughmark(1965)
Neal & Bankoff(1965)
115
113
60
Gregory & Scott(1969)
53
Wallis(1969)
3
Smith(1969)
Kutucuglu
0
81
Thom(1964)
Guzhov et al(1967)
52
Kowalczewski(1964)
9
68
Fujie(1964)
52
23
31
Nicklin et al(1962)
Nishino & Yamazaki(1963)
Baroczy(1966)
35
34
67
Fauske(1961)
Wilson et al(1961,1965)
Hughmark(1962)
Turner & Wallis(1965)
0
19
34
Hoogendroon(1959)
Bankoff(1960)
2
Armand - Massina
Lockhart & Martinelli(1949)
Sterman(1956)
Filimonov et al(1957)
Chisholm & Laird(1958)
Flanigan(1958)
Dimentiev et al
53
118
124
27
147
22
12
Armand(1946)
1
Homogeneous
116
122
171
171
82
0
89
17
102
51
117
126
104
119
79
85
56
59
180
22
50
78
187
179
63
189
52
43
185
197
188
207
104
0
129
40
171
76
143
150
144
159
156
129
82
85
216
86
89
119
205
193
95
208
80
75
28
27
36
16
0
16
4
18
10
17
20
14
22
4
14
12
6
14
2
0
16
39
27
10
26
12
3
23
±5%
14
38
49
46
22
0
22
8
34
12
22
27
23
32
20
22
14
10
41
4
2
22
50
43
19
42
14
6
36
±10%
22
54
50
55
26
0
26
12
50
14
22
29
26
54
42
32
14
18
52
16
10
26
54
50
22
48
16
20
50
±15%
35
Total Points 56
±15%
146
Total Points 237
±10%
104
Beggs
(1972)
(Air -Water)
Eaton
(1966)
(Natural Gas -Water)
Percentage within ±5%
58
Correlation / Data Source
39
201
167
156
0
183
84
41
13
168
131
175
123
45
195
131
21
152
90
24
93
135
200
103
125
107
79
41
±5%
108
94
227
209
190
15
210
115
85
24
190
186
208
167
87
227
148
42
208
143
53
119
221
224
128
170
137
131
90
±10%
143
185
241
235
210
33
220
134
142
39
202
211
218
194
139
236
157
54
234
172
84
135
236
240
143
198
157
155
161
±15%
166
Total Points 270
Spedding & Nguyen
(1976)
(Air -Water)
10
27
33
28
0
21
4
8
9
23
27
28
20
4
22
13
6
7
4
2
9
21
28
10
30
10
4
10
±5%
19
29
38
43
35
0
33
8
19
10
32
34
35
42
14
33
21
13
36
13
5
18
44
39
16
40
23
10
23
±10%
39
57
48
50
40
0
37
14
44
13
35
37
39
53
36
41
26
20
49
23
10
27
55
44
23
47
30
19
52
±15%
47
Total Points 62
Mukherjee
(1979)
(Air -Kerosene)
18
29
29
24
0
28
17
16
0
24
22
29
12
2
28
22
8
30
0
6
5
33
29
12
26
12
0
15
±5%
12
29
37
36
30
0
36
28
25
3
31
32
39
23
20
39
32
14
47
4
11
12
42
38
18
37
20
0
27
±10%
21
45
44
41
37
0
43
33
37
7
40
39
47
31
33
44
40
17
52
18
18
14
45
42
20
43
23
9
42
±15%
28
Total Points 54
Minami & Brill
(1987)
(Air -Water)
14
31
30
25
0
30
16
14
0
29
26
32
14
4
33
24
0
21
0
4
9
32
31
10
31
14
0
14
±5%
14
32
40
41
34
0
37
27
25
5
37
37
42
28
18
41
36
3
47
4
10
14
44
41
15
38
21
0
30
±10%
26
50
48
45
40
0
48
36
40
8
42
43
48
35
31
49
42
10
56
17
18
15
50
46
19
45
29
11
47
±15%
33
Total Points 57
Minami & Brill
(1987)
(Air -Kerosene)
Table 7. Number of data points correctly predicted for horizontal flow by data sets
32
27
22
17
0
22
12
20
6
16
3
22
9
14
25
16
3
37
12
6
9
38
27
1
25
0
9
29
±5%
8
60
41
46
25
0
29
18
47
9
26
21
30
28
40
30
28
9
51
18
8
21
62
33
10
46
10
15
56
±10%
27
73
46
63
30
0
37
26
71
13
34
30
37
56
65
37
30
16
65
26
20
30
72
43
19
55
22
18
73
±15%
49
Total Points 81
Franca & Lahey
(1992)
(Air -Water)
6
40
51
44
0
36
9
4
11
46
47
52
39
6
34
27
9
14
9
5
30
24
42
27
64
28
10
5
±5%
34
27
62
58
54
0
55
24
16
19
55
55
58
62
17
56
39
22
66
31
7
49
62
61
39
71
41
25
23
63
66
67
57
0
60
32
44
23
61
60
63
63
50
60
44
28
73
46
13
61
68
68
48
73
51
40
54
±10% ±15%
52
62
Total Points 83
Abdulmajeed
(1996)
(Air -Kerosene)
55
4
4
0
68
101
2
128
47
82
38
Madsen(1975)
Mukherjee(1979)
Gardner-1(1980)
Gardner-2(1980)
Sun et al(1980)
37
50
131
16
134
26
34
Kawaji et al(1987)
Minami & Brill(1987)
El-Boher et al(1988)
Hart et al(1989)
Kokal & Stanislav(1989)
Spedding et al(1989)
130
103
Tandon et al(1985)
Chen(1986)
Hamersma & Hart(1987)
33
89
Bestion(1985)
Chisholm(1983) - Armand(1946)
Spedding & Chen(1984)
1
72
Jowitt(1981)
5
Moussali
Greskovich & Cooper(1975)
53
25
112
127
105
112
84
Mattar & Gregory(1974)
Rouhani II(1970)
Bonnecaze et al(1971)
Dix(1971)
Beggs(1972)
Chisholm(1973)
Loscher & Reinhardt(1973)
Rouhani I(1970)
86
80
184
51
206
117
70
192
139
126
73
151
8
131
107
6
198
81
118
158
1
164
80
186
188
173
145
184
127
157
200
145
225
174
99
215
161
147
107
203
15
153
202
12
222
115
160
199
5
202
157
210
219
207
154
221
14
4
17
15
33
16
12
14
22
14
14
37
0
13
3
0
35
7
17
30
0
13
4
26
46
38
14
10
±5%
27
18
20
20
32
44
24
14
20
42
20
17
54
0
20
16
0
44
18
26
45
0
26
20
36
50
54
14
38
±10%
39
24
43
22
44
53
32
16
26
46
28
24
56
3
24
32
0
46
26
41
51
1
40
43
44
52
56
19
50
±15%
50
Total Points 56
±15%
208
Total Points 237
±10%
196
Beggs
(1972)
(Air -Water)
Eaton
(1966)
(Natural Gas -Water)
Percentage within ±5%
Premoli et al(1970)
162
Correlation / Data Source
185
45
180
162
149
133
107
172
166
58
204
172
31
97
48
24
138
83
109
128
11
61
45
171
149
165
138
111
±5%
199
210
88
192
205
185
174
138
198
206
118
233
207
70
132
82
37
182
131
149
163
29
101
87
203
185
206
160
217
±10%
236
216
141
197
227
215
186
157
205
230
138
245
229
175
161
128
47
218
151
173
190
51
136
140
220
209
229
173
240
±15%
251
Total Points 270
Spedding & Nguyen
(1976)
(Air -Water)
21
4
17
10
33
17
10
14
39
12
23
27
1
17
7
0
32
9
22
31
2
5
4
26
34
33
26
12
±5%
26
30
14
26
33
44
28
23
27
46
22
33
44
3
29
20
0
44
18
42
43
3
20
14
39
43
45
38
34
±10%
40
37
36
32
40
48
36
31
36
53
31
39
55
7
31
44
0
47
23
49
48
6
31
36
47
52
55
45
48
±15%
47
Total Points 62
Mukherjee
(1979)
(Air -Kerosene)
28
2
32
14
36
26
12
28
27
0
27
31
0
2
5
0
33
0
12
24
0
0
2
24
32
32
12
27
±5%
33
38
20
43
42
46
37
20
40
33
3
37
40
0
7
23
0
51
1
22
33
0
4
20
40
42
38
23
53
±10%
40
44
33
51
54
54
41
24
47
40
14
44
44
0
14
35
1
54
7
28
50
0
12
33
46
46
42
29
54
±15%
45
Total Points 54
Minami & Brill
(1987)
(Air -Water)
31
5
35
1
48
29
14
37
31
0
31
35
0
19
15
0
39
1
14
28
0
1
5
32
37
32
33
29
±5%
40
40
18
52
22
56
36
23
45
37
5
40
43
0
28
34
2
56
7
26
41
0
6
18
48
45
42
36
56
±10%
47
49
32
55
46
57
41
32
54
43
16
49
49
0
34
54
5
57
15
35
50
0
18
32
51
52
46
45
57
±15%
53
Total Points 57
Minami & Brill
(1987)
(Air -Kerosene)
Table 7. Number of data points correctly predicted for horizontal flow by data sets (contd.)
27
14
27
45
27
16
0
26
28
14
28
30
7
12
9
0
8
8
17
24
0
12
14
41
28
30
36
18
±5%
38
33
40
38
57
54
37
11
38
48
18
36
56
24
16
29
0
35
14
39
46
2
28
40
49
49
56
40
49
±10%
40
40
65
40
66
71
39
23
41
62
21
44
69
36
20
54
0
55
18
58
59
11
35
65
66
61
69
41
61
±15%
41
Total Points 81
Franca & Lahey
(1992)
(Air -Water)
37
5
50
18
61
27
29
34
54
22
34
40
2
44
9
0
60
22
34
43
0
8
6
58
53
51
56
15
±5%
62
55
17
63
63
70
44
45
58
60
39
51
63
5
60
28
1
68
37
57
64
0
27
17
75
66
64
63
62
63
50
68
77
74
58
54
66
64
53
61
67
10
64
61
3
74
50
63
70
0
59
50
78
69
67
66
72
±10% ±15%
66
69
Total Points 83
Abdulmajeed
(1996)
(Air -Kerosene)
56
89
20
70
59
15
52
0
137
Inoue et al(1993)
Czop et al(1994)
AbdulMajeed(1996)
Maier and Coddington(1997)
Petalaz & Aziz(1997)
Gomez et al(2000)
Zhao et al(2000)
Graham et al(2001)
203
1
165
45
165
136
30
88
168
223
3
191
107
187
162
46
119
206
20
0
15
0
26
12
6
2
28
32
3
26
17
31
21
6
2
41
±10%
34
44
6
36
24
34
26
10
3
51
±15%
44
181
85
124
43
104
131
10
96
176
±5%
60
197
111
164
78
173
166
17
124
215
±10%
182
213
121
190
126
192
195
30
150
239
±15%
221
Total Points 270
Spedding & Nguyen
(1976)
(Air -Water)
22
0
16
4
23
19
4
0
30
±5%
16
33
2
31
9
40
31
8
0
42
±10%
37
42
4
37
30
43
36
14
1
47
±15%
45
Total Points 62
Mukherjee
(1979)
(Air -Kerosene)
36
10
14
10
19
15
3
14
28
±5%
17
43
18
27
22
32
29
6
21
37
±10%
38
52
22
34
38
40
36
13
26
41
±15%
47
Total Points 54
Minami & Brill
(1987)
(Air -Water)
40
2
16
12
22
20
2
0
31
±5%
22
49
6
30
21
39
34
3
0
40
±10%
39
55
10
36
41
46
38
5
0
45
±15%
48
Total Points 57
Minami & Brill
(1987)
(Air -Kerosene)
29
7
26
10
2
27
0
0
22
±5%
28
41
11
38
23
34
41
2
0
37
±10%
48
43
19
49
40
45
42
2
0
60
±15%
59
Total Points 81
Franca & Lahey
(1992)
(Air -Water)
55
1
43
4
48
51
3
8
51
±5%
31
69
7
56
6
61
56
7
12
56
77
11
65
28
67
59
11
13
65
±10% ±15%
73
75
Total Points 83
Abdulmajeed
(1996)
(Air -Kerosene)
correlation as reported by Leung (2005), 2,5 correlations as reported by Kataoka and Ishii (1987), 3 correlation as reported by Isbin and Biddle (1979) , 4 Rouhani
and Axelsson(1970) correlations I and II.
1
106
Huq & Loth(1992)
Toshiba(1989)
±5%
15
Total Points 56
±15%
209
Total Points 237
±10%
179
Beggs
(1972)
(Air -Water)
Eaton
(1966)
(Natural Gas -Water)
Percentage within ±5%
97
Correlation / Data Source
Table 7. Number of data points correctly predicted for horizontal flow by data sets (contd.)
57
9.3
5.1
Chisholm & Laird(1958)
Flanigan(1958)
0.0
72.2
48.5
47.7
25.3
Smith(1969)
Wallis(1969)
Gregory & Scott(1969)
51.5
72.2
34.6
0.0
7.2
37.6
21.5
43.0
49.4
53.2
22.4
3
Guzhov et al(1967)
Kutucuglu
3.8
21.9
32.9
Zivi(1964)
Turner & Wallis(1965)
Baroczy(1966)
34.2
Thom(1964)
18.1
21.9
11.0
50.2
28.7
Fujie(1964)
Kowalczewski(1964)
Neal & Bankoff(1965)
33.3
35.9
9.7
13.1
Hughmark(1965)
75.9
28.3
Hughmark(1962)
Nicklin et al(1962)
Nishino & Yamazaki(1963)
43.9
23.6
24.9
8.0
14.8
14.3
21.1
9.3
32.9
18.1
21.9
79.7
26.6
75.5
Fauske(1961)
Wilson et al(1961,1965)
0.0
48.9
78.9
Bankoff(1960)
Hoogendroon(1959)
14.3
62.0
Filimonov et al(1957)
2
11.4
Sterman(1956)
Dimentiev et al
52.3
Lockhart & Martinelli(1949)
49.8
22.4
1
Armand(1946)
Armand - Massina
Homogeneous
78.1
83.1
79.3
87.3
43.9
0.0
16.9
54.4
32.1
72.2
60.3
63.3
60.8
67.1
65.8
54.4
91.1
34.6
35.9
37.6
36.3
50.2
31.6
33.8
87.8
40.1
81.4
86.5
50.0
48.2
64.3
28.6
0.0
7.1
28.6
17.9
32.1
30.4
35.7
25.0
39.3
7.1
25.0
25.0
21.4
10.7
0.0
3.6
28.6
5.4
21.4
46.4
17.9
48.2
69.6
41.1
±5%
25.0
67.9
87.5
82.1
39.3
0.0
14.3
39.3
21.4
60.7
39.3
48.2
41.1
57.1
35.7
39.3
73.2
25.0
17.9
3.6
7.1
39.3
10.7
25.0
75.0
33.9
76.8
89.3
64.3
±10%
39.3
96.4
89.3
98.2
46.4
0.0
21.4
46.4
25.0
89.3
39.3
51.8
46.4
96.4
75.0
57.1
92.9
25.0
32.1
17.9
28.6
46.4
35.7
28.6
85.7
39.3
89.3
96.4
89.3
±15%
62.5
Total Points 56
±15%
61.6
Total Points 237
±10%
43.9
Beggs
(1972)
(Air -Water)
Eaton
(1966)
(Natural Gas -Water)
Percentage within ±5%
24.5
Correlation / Data Source
14.4
74.4
61.9
57.8
0.0
31.1
67.8
4.8
15.2
62.2
48.5
64.8
45.6
16.7
72.2
56.3
48.5
7.8
8.9
33.3
34.4
29.3
39.6
46.3
38.1
74.1
50.0
15.2
±5%
40.0
34.8
84.1
77.4
70.4
5.6
42.6
77.8
8.9
31.5
70.4
68.9
77.0
61.9
32.2
84.1
77.0
54.8
15.6
19.6
53.0
44.1
48.5
50.7
63.0
47.4
83.0
81.9
33.3
±10%
53.0
68.5
89.3
87.0
77.8
12.2
49.6
81.5
14.4
52.6
74.8
78.1
80.7
71.9
51.5
87.4
86.7
58.1
20.0
31.1
63.7
50.0
57.4
58.1
73.3
53.0
88.9
87.4
59.6
±15%
61.5
Total Points 270
Spedding & Nguyen
(1976)
(Air -Water)
16.1
43.5
53.2
45.2
0.0
6.5
33.9
14.5
12.9
37.1
43.5
45.2
32.3
6.5
35.5
11.3
21.0
9.7
3.2
6.5
14.5
6.5
16.1
48.4
16.1
45.2
33.9
16.1
±5%
30.6
46.8
61.3
69.4
56.5
0.0
12.9
53.2
16.1
30.6
51.6
54.8
56.5
67.7
22.6
53.2
58.1
33.9
21.0
8.1
21.0
29.0
16.1
37.1
64.5
25.8
62.9
71.0
37.1
±10%
62.9
91.9
77.4
80.6
64.5
0.0
22.6
59.7
21.0
71.0
56.5
59.7
62.9
85.5
58.1
66.1
79.0
41.9
32.3
16.1
37.1
43.5
30.6
48.4
75.8
37.1
71.0
88.7
83.9
±15%
75.8
Total Points 62
Mukherjee
(1979)
(Air -Kerosene)
33.3
53.7
53.7
44.4
0.0
31.5
51.9
0.0
29.6
44.4
40.7
53.7
22.2
3.7
51.9
55.6
40.7
14.8
11.1
0.0
9.3
0.0
22.2
48.1
22.2
53.7
61.1
27.8
±5%
22.2
53.7
68.5
66.7
55.6
0.0
51.9
66.7
5.6
46.3
57.4
59.3
72.2
42.6
37.0
72.2
87.0
59.3
25.9
20.4
7.4
22.2
0.0
37.0
68.5
33.3
70.4
77.8
50.0
±10%
38.9
83.3
81.5
75.9
68.5
0.0
61.1
79.6
13.0
68.5
74.1
72.2
87.0
57.4
61.1
81.5
96.3
74.1
31.5
33.3
33.3
25.9
16.7
42.6
79.6
37.0
77.8
83.3
77.8
±15%
51.9
Total Points 54
Minami & Brill
(1987)
(Air -Water)
24.6
54.4
52.6
43.9
0.0
28.1
52.6
0.0
24.6
50.9
45.6
56.1
24.6
7.0
57.9
36.8
42.1
0.0
7.0
0.0
15.8
0.0
24.6
54.4
17.5
54.4
56.1
24.6
±5%
24.6
56.1
70.2
71.9
59.6
0.0
47.4
64.9
8.8
43.9
64.9
64.9
73.7
49.1
31.6
71.9
82.5
63.2
5.3
17.5
7.0
24.6
0.0
36.8
66.7
26.3
71.9
77.2
52.6
±10%
45.6
87.7
84.2
78.9
70.2
0.0
63.2
84.2
14.0
70.2
73.7
75.4
84.2
61.4
54.4
86.0
98.2
73.7
17.5
31.6
29.8
26.3
19.3
50.9
78.9
33.3
80.7
87.7
82.5
±15%
57.9
Total Points 57
Minami & Brill
(1987)
(Air -Kerosene)
Table 8. Percentage of data points correctly predicted for horizontal flow by data sets
39.5
33.3
27.2
21.0
0.0
14.8
27.2
7.4
24.7
19.8
3.7
27.2
11.1
17.3
30.9
45.7
19.8
3.7
7.4
14.8
11.1
11.1
0.0
30.9
1.2
33.3
46.9
35.8
±5%
9.9
74.1
50.6
56.8
30.9
0.0
22.2
35.8
11.1
58.0
32.1
25.9
37.0
34.6
49.4
37.0
63.0
34.6
11.1
9.9
22.2
25.9
18.5
12.3
56.8
12.3
40.7
76.5
69.1
±10%
33.3
90.1
56.8
77.8
37.0
0.0
32.1
45.7
16.0
87.7
42.0
37.0
45.7
69.1
80.2
45.7
80.2
37.0
19.8
24.7
32.1
37.0
22.2
27.2
67.9
23.5
53.1
88.9
90.1
±15%
60.5
Total Points 81
Franca & Lahey
(1992)
(Air -Water)
7.2
48.2
61.4
53.0
0.0
10.8
43.4
13.3
4.8
55.4
56.6
62.7
47.0
7.2
41.0
16.9
32.5
10.8
6.0
10.8
36.1
12.0
33.7
77.1
32.5
50.6
28.9
6.0
±5%
41.0
32.5
74.7
69.9
65.1
0.0
28.9
66.3
22.9
19.3
66.3
66.3
69.9
74.7
20.5
67.5
79.5
47.0
26.5
8.4
37.3
59.0
30.1
49.4
85.5
47.0
73.5
74.7
27.7
75.9
79.5
80.7
68.7
0.0
38.6
72.3
27.7
53.0
73.5
72.3
75.9
75.9
60.2
72.3
88.0
53.0
33.7
15.7
55.4
73.5
48.2
61.4
88.0
57.8
81.9
81.9
65.1
±10% ±15%
62.7 74.7
Total Points 83
Abdulmajeed
(1996)
(Air -Kerosene)
58
4
4
0.4
49.8
53.6
44.3
47.3
0.0
28.7
42.6
0.8
54.0
19.8
34.6
16.0
Chisholm(1973)
Loscher & Reinhardt(1973)
Mattar & Gregory(1974)
5
Moussali
Greskovich & Cooper(1975)
Madsen(1975)
Mukherjee(1979)
Gardner-1(1980)
Gardner-2(1980)
Sun et al(1980)
53.2
81.0
58.6
13.9
37.6
54.9
43.5
15.6
21.1
55.3
6.8
56.5
11.0
14.3
Spedding & Chen(1984)
Bestion(1985)
Tandon et al(1985)
Chen(1986)
Hamersma & Hart(1987)
Kawaji et al(1987)
Minami & Brill(1987)
El-Boher et al(1988)
Hart et al(1989)
Kokal & Stanislav(1989)
Spedding et al(1989)
3.4
Chisholm(1983) - Armand(1946)
36.3
33.8
77.6
21.5
86.9
29.5
49.4
30.8
63.7
0.4
30.4
Jowitt(1981)
55.3
45.1
34.2
2.5
83.5
66.7
61.2
73.0
79.3
78.5
47.3
Dix(1971)
Beggs(1972)
69.2
77.6
33.8
22.4
35.4
10.5
Bonnecaze et al(1971)
Rouhani II(1970)
Rouhani I(1970)
53.6
66.2
84.4
61.2
94.9
41.8
73.4
67.9
62.0
90.7
45.1
85.7
6.3
64.6
85.2
48.5
5.1
93.7
84.0
67.5
2.1
65.0
87.3
92.4
88.6
66.2
85.2
93.2
25.0
7.1
30.4
26.8
58.9
21.4
28.6
39.3
25.0
25.0
25.0
66.1
0.0
23.2
5.4
12.5
0.0
62.5
53.6
30.4
0.0
25.0
67.9
82.1
46.4
7.1
23.2
17.9
±5%
48.2
32.1
35.7
35.7
57.1
78.6
25.0
42.9
75.0
35.7
35.7
30.4
96.4
0.0
35.7
28.6
32.1
0.0
78.6
80.4
46.4
0.0
25.0
96.4
89.3
64.3
35.7
46.4
67.9
±10%
69.6
42.9
76.8
39.3
78.6
94.6
28.6
57.1
82.1
50.0
46.4
42.9
100.0
5.4
42.9
57.1
46.4
0.0
82.1
91.1
73.2
1.8
33.9
100.0
92.9
78.6
76.8
71.4
89.3
±15%
89.3
Total Points 56
±15%
87.8
Total Points 237
±10%
82.7
Beggs
(1972)
(Air -Water)
Eaton
(1966)
(Natural Gas -Water)
Percentage within ±5%
Premoli et al(1970)
68.4
Correlation / Data Source
68.5
16.7
66.7
60.0
55.2
39.6
49.3
61.5
21.5
63.7
75.6
63.7
11.5
35.9
17.8
30.7
8.9
51.1
47.4
40.4
4.1
51.1
61.1
55.2
63.3
16.7
22.6
41.1
±5%
73.7
77.8
32.6
71.1
75.9
68.5
51.1
64.4
76.3
43.7
73.3
86.3
76.7
25.9
48.9
30.4
48.5
13.7
67.4
60.4
55.2
10.7
59.3
76.3
68.5
75.2
32.2
37.4
80.4
±10%
87.4
80.0
52.2
73.0
84.1
79.6
58.1
68.9
85.2
51.1
75.9
90.7
84.8
64.8
59.6
47.4
55.9
17.4
80.7
70.4
64.1
18.9
64.1
84.8
77.4
81.5
51.9
50.4
88.9
±15%
93.0
Total Points 270
Spedding & Nguyen
(1976)
(Air -Water)
33.9
6.5
27.4
16.1
53.2
16.1
27.4
62.9
19.4
22.6
37.1
43.5
1.6
27.4
11.3
14.5
0.0
51.6
50.0
35.5
3.2
41.9
53.2
54.8
41.9
6.5
8.1
19.4
±5%
41.9
48.4
22.6
41.9
53.2
71.0
37.1
45.2
74.2
35.5
43.5
53.2
71.0
4.8
46.8
32.3
29.0
0.0
71.0
69.4
67.7
4.8
61.3
72.6
69.4
62.9
22.6
32.3
54.8
±10%
64.5
59.7
58.1
51.6
64.5
77.4
50.0
58.1
85.5
50.0
58.1
62.9
88.7
11.3
50.0
71.0
37.1
0.0
75.8
77.4
79.0
9.7
72.6
88.7
83.9
75.8
58.1
50.0
77.4
±15%
75.8
Total Points 62
Mukherjee
(1979)
(Air -Kerosene)
51.9
3.7
59.3
25.9
66.7
22.2
48.1
50.0
0.0
51.9
50.0
57.4
0.0
3.7
9.3
0.0
0.0
61.1
44.4
22.2
0.0
22.2
59.3
59.3
44.4
3.7
0.0
50.0
±5%
61.1
70.4
37.0
79.6
77.8
85.2
37.0
68.5
61.1
5.6
74.1
68.5
74.1
0.0
13.0
42.6
1.9
0.0
94.4
61.1
40.7
0.0
42.6
70.4
77.8
74.1
37.0
7.4
98.1
±10%
74.1
81.5
61.1
94.4
100.0
100.0
44.4
75.9
74.1
25.9
87.0
81.5
81.5
0.0
25.9
64.8
13.0
1.9
100.0
92.6
51.9
0.0
53.7
77.8
85.2
85.2
61.1
22.2
100.0
±15%
83.3
Total Points 54
Minami & Brill
(1987)
(Air -Water)
54.4
8.8
61.4
1.8
84.2
24.6
50.9
54.4
0.0
64.9
54.4
61.4
0.0
33.3
26.3
1.8
0.0
68.4
49.1
24.6
0.0
57.9
56.1
64.9
56.1
8.8
1.8
50.9
±5%
70.2
70.2
31.6
91.2
38.6
98.2
40.4
63.2
64.9
8.8
78.9
70.2
75.4
0.0
49.1
59.6
12.3
3.5
98.2
71.9
45.6
0.0
63.2
73.7
78.9
84.2
31.6
10.5
98.2
±10%
82.5
86.0
56.1
96.5
80.7
100.0
56.1
71.9
75.4
28.1
94.7
86.0
86.0
0.0
59.6
94.7
26.3
8.8
100.0
87.7
61.4
0.0
78.9
80.7
91.2
89.5
56.1
31.6
100.0
±15%
93.0
Total Points 57
Minami & Brill
(1987)
(Air -Kerosene)
Table 8. Percentage of data points correctly predicted for horizontal flow by data sets (contd.)
33.3
17.3
33.3
55.6
33.3
0.0
19.8
34.6
17.3
32.1
34.6
37.0
8.6
14.8
11.1
9.9
0.0
9.9
29.6
21.0
0.0
44.4
37.0
34.6
50.6
17.3
14.8
22.2
±5%
46.9
40.7
49.4
46.9
70.4
66.7
13.6
45.7
59.3
22.2
46.9
44.4
69.1
29.6
19.8
35.8
17.3
0.0
43.2
56.8
48.1
2.5
49.4
69.1
60.5
60.5
49.4
34.6
60.5
±10%
49.4
49.4
80.2
49.4
81.5
87.7
28.4
48.1
76.5
25.9
50.6
54.3
85.2
44.4
24.7
66.7
22.2
0.0
67.9
72.8
71.6
13.6
50.6
85.2
75.3
81.5
80.2
43.2
75.3
±15%
50.6
Total Points 81
Franca & Lahey
(1992)
(Air -Water)
44.6
6.0
60.2
21.7
73.5
34.9
32.5
65.1
26.5
41.0
41.0
48.2
2.4
53.0
10.8
26.5
0.0
72.3
51.8
41.0
0.0
67.5
61.4
63.9
69.9
7.2
9.6
18.1
±5%
74.7
66.3
20.5
75.9
75.9
84.3
54.2
53.0
72.3
47.0
69.9
61.4
75.9
6.0
72.3
33.7
44.6
1.2
81.9
77.1
68.7
0.0
75.9
77.1
79.5
90.4
20.5
32.5
74.7
75.9
60.2
81.9
92.8
89.2
65.1
69.9
77.1
63.9
79.5
73.5
80.7
12.0
77.1
73.5
60.2
3.6
89.2
84.3
75.9
0.0
79.5
80.7
83.1
94.0
60.2
71.1
86.7
±10% ±15%
79.5 83.1
Total Points 83
Abdulmajeed
(1996)
(Air -Kerosene)
59
Graham et al(2001)
85.7
0.4
37.1
12.7
57.4
69.6
19.0
69.6
70.9
94.1
1.3
50.2
19.4
68.4
78.9
45.1
80.6
86.9
35.7
0.0
3.6
10.7
21.4
46.4
0.0
26.8
50.0
57.1
5.4
3.6
10.7
37.5
55.4
30.4
46.4
73.2
±10%
60.7
78.6
10.7
5.4
17.9
46.4
60.7
42.9
64.3
91.1
±15%
78.6
67.0
31.5
35.6
3.7
48.5
38.5
15.9
45.9
65.2
±5%
22.2
73.0
41.1
45.9
6.3
61.5
64.1
28.9
60.7
79.6
±10%
67.4
78.9
44.8
55.6
11.1
72.2
71.1
46.7
70.4
88.5
±15%
81.9
Total Points 270
Spedding & Nguyen
(1976)
(Air -Water)
35.5
0.0
0.0
6.5
30.6
37.1
6.5
25.8
48.4
±5%
25.8
53.2
3.2
0.0
12.9
50.0
64.5
14.5
50.0
67.7
±10%
59.7
67.7
6.5
1.6
22.6
58.1
69.4
48.4
59.7
75.8
±15%
72.6
Total Points 62
Mukherjee
(1979)
(Air -Kerosene)
66.7
18.5
25.9
5.6
27.8
35.2
18.5
25.9
51.9
±5%
31.5
79.6
33.3
38.9
11.1
53.7
59.3
40.7
50.0
68.5
±10%
70.4
96.3
40.7
48.1
24.1
66.7
74.1
70.4
63.0
75.9
±15%
87.0
Total Points 54
Minami & Brill
(1987)
(Air -Water)
70.2
3.5
0.0
3.5
35.1
38.6
21.1
28.1
54.4
±5%
38.6
86.0
10.5
0.0
5.3
59.6
68.4
36.8
52.6
70.2
±10%
68.4
96.5
17.5
0.0
8.8
66.7
80.7
71.9
63.2
78.9
±15%
84.2
Total Points 57
Minami & Brill
(1987)
(Air -Kerosene)
35.8
8.6
0.0
0.0
33.3
2.5
12.3
32.1
27.2
±5%
34.6
50.6
13.6
0.0
2.5
50.6
42.0
28.4
46.9
45.7
±10%
59.3
53.1
23.5
0.0
2.5
51.9
55.6
49.4
60.5
74.1
±15%
72.8
Total Points 81
Franca & Lahey
(1992)
(Air -Water)
66.3
1.2
9.6
3.6
61.4
57.8
4.8
51.8
61.4
±5%
37.3
correlations I and II.
83.1
8.4
14.5
8.4
67.5
73.5
7.2
67.5
67.5
92.8
13.3
15.7
13.3
71.1
80.7
33.7
78.3
78.3
±10% ±15%
88.0 90.4
Total Points 83
Abdulmajeed
(1996)
(Air -Kerosene)
correlation as reported by Leung (2005), 2,5 correlations as reported by Kataoka and Ishii (1987), 3 correlation as reported by Isbin and Biddle (1979) , 4 Rouhani and Axelsson(1970)
0.0
57.8
Zhao et al(2000)
1
6.3
21.9
Gomez et al(2000)
24.9
Maier and Coddington(1997)
Petalaz & Aziz(1997)
8.4
29.5
AbdulMajeed(1996)
37.6
Czop et al(1994)
44.7
Huq & Loth(1992)
Inoue et al(1993)
Toshiba(1989)
±5%
26.8
Total Points 56
±15%
88.2
Total Points 237
±10%
75.5
Beggs
(1972)
(Air -Water)
Eaton
(1966)
(Natural Gas -Water)
Percentage within ±5%
40.9
Correlation / Data Source
Table 8. Percentage of data points correctly predicted for horizontal flow by data sets (contd.)
A simple scatter plot of the measured versus calculated void fractions for the top six
correlations that predicted 85% of the data points in the horizontal flow data base
within the 15% error percentage index is presented. These are Rouhani I(1970),
Hughmark
(1962),
Minami
and
Brill
(1987),
Armand–Massina
(Leung
2005),Mukherjee(1979),Chisholm(1973) and Chisholm(1983) – Armand (1946) that
are given in Figures 1 through 6, respectively. The purpose of these plots is to show
where the rest of the “stray” data points that could not be predicted by the correlations
lie. This would reveal how much consistent the prediction by the correlation is and
also the general tendency to under or over predict a given data set through the range
of void fraction values.
We can see that the correlation of Hughmark (1962), which is the second best
correlation in terms of the total percentage of data points predicted, shows a superior
consistent performance over the others. Rouhani I( 1970) correlation is seen to under
predict the data most of the time while all the other correlations tend to over predict
the data on the whole range of values.
A final note on this analysis is that one could opt for a correlation with a fairly
consistent performance across the whole void fraction range even when the
correlation may not give accurate prediction of most of the data. On the other hand, a
specific correlation which shows good performance in a specific range could be a
viable choice if the range of voidage encountered for the particular application of
interest has been indirectly determined or expected to be in that area.
Considering predicting void fraction for air –water mixture for example, sixteen
correlations are observed to give a prediction of 85% and above for the 15% error
index with the best predictions coming from Armand Massina (Leung 2005), Rouhani
I(1970) and Hughmark(1962) with predictions of 87% and above. A very close
60
performance to these correlations were also given by Chisholm(1973) and
Chisholm(1983)–Armand(1946) which predicted 86% while Smith(1969),Minami &
Brill(1987), El-Boher et al.(1988) and Huq & Loth(1992) captured 85% of the data
within the 15% error index.
On another hand, for the non air-water data, the correlations of Beggs (1972) and
Graham et al. (2001) gave the best prediction of 89 and 88%, respectively. This is
quite unexpected as these correlations were developed from air water and refrigerant
data sets that have quite different physical properties from kerosene and natural gas.
Rouhani I (1970) and Toshiba (1989) predicted 86% of the data set for the 15% index
with close predictions of 85% coming from Armand Massina (Leung 2005),
Filimonov et al. (1957), Chisholm (1973), Premoli et al. (1970) and Chisholm (1983)Armand (1946).
We would now move on to the comparison of the correlations with the inclined flow
database compiled.
4.2 Inclined flow comparison
In this section we would use data sets from three different sources that reported void
fraction data for different upward inclination angles. As the inclination angles
investigated by the different researchers are not the same, the results from one data set
would not be directly comparable to the other except on some inclination angles
which are common to all. Hence, the presentation of the comparison results here
would be a bit different from the previous section in the sense that it would be
restricted to a single data source at a time.
We would therefore subdivide this section into the three different data sources we
have, namely, Beggs (1972), Spedding and Nguyen (1976) and Mukherjee (1979)
61
inclined flow comparisons. All the inclination angles within the data set would then
be considered separately first before making a consolidated summary for all.
A few of the correlations have inclination angle dependent terms or constant in their
expression which has to be adjusted for the individual inclinations angles under
consideration to predict the void fraction. These are the correlation by Beggs (1972),
Greskovich and Cooper (1975) and Gomez et al. (2000).
4.2.1 Inclined flow comparison with the data of Beggs(1972)
Beggs (1972) reported 209 void fraction measurements for seven different pipe
inclination angles between 5 – 75 degrees from the horizontal. The mixtures used
were air and water in 1 and 1.5 in diameter pipes. The total number of data points
correctly predicted for each pipe inclination angle is given in Table 10 while the
percentage comparisons are reported in Table 11.
Owing to the fact that there are only a small number of data points and the inclination
is very close to the horizontal, four correlations were seen to perform acceptably with
the tightest 5% error index for the 5 degree inclination data set.
These are the simpler models by Armand Massina, Chisholm (1973), Chisholm
(1983) – Armand (1946) and the correlation by Beggs (1972). As was observed in the
horizontal comparison, the performance of the correlations improve for the wider
error bands of 10 and 15 % where 12 and 21 correlations were found to predict at
least 80 and 85% of the data sets respectively. The same trend is observed for all
inclination angles except for a few exceptions.
62
Table 9. Total number and percentage of data points correctly predicted for horizontal
flow database (all sources)
Correlation / Data Source
Total data points correctly
Percentage of data points correctly
predicted for the whole database predicted for the whole database
Total Points 900
Total Points 900
±5%
267
190
±10%
434
401
±15%
566
664
±5%
29.7
21.1
±10%
48.2
44.6
±15%
62.9
73.8
Armand - Massina
Lockhart & Martinelli(1949)
Sterman(1956)
440
508
200
712
658
308
785
726
389
48.9
56.4
22.2
79.1
73.1
34.2
87.2
80.7
43.2
Filimonov et al(1957)
Chisholm & Laird(1958)
Flanigan(1958)
474
205
117
633
318
230
717
408
347
52.7
22.8
13.0
70.3
35.3
25.6
79.7
45.3
38.6
Dimentiev et al
Hoogendroon(1959)
Bankoff(1960)
Fauske(1961)
205
117
66
280
333
239
146
374
427
404
262
435
22.8
13.0
7.3
31.1
37.0
26.6
16.2
41.6
47.4
44.9
29.1
48.3
Wilson et al(1961,1965)
Hughmark(1962)
Nicklin et al(1962)
Nishino & Yamazaki(1963)
Fujie(1964)
87
342
102
382
307
172
676
295
533
501
248
797
552
628
645
9.7
38.0
11.3
42.4
34.1
19.1
75.1
32.8
59.2
55.7
27.6
88.6
61.3
69.8
71.7
Kowalczewski(1964)
404
539
622
44.9
59.9
69.1
Thom(1964)
Zivi(1964)
Hughmark(1965)
Neal & Bankoff(1965)
Turner & Wallis(1965)
Baroczy(1966)
Guzhov et al(1967)
3
Kutucuglu
357
401
164
75
155
388
0
518
510
353
133
245
511
15
599
579
599
193
327
600
33
39.7
44.6
18.2
8.3
17.2
43.1
0.0
57.6
56.7
39.2
14.8
27.2
56.8
1.7
66.6
64.3
66.6
21.4
36.3
66.7
3.7
363
472
544
40.3
52.4
60.4
Smith(1969)
Wallis(1969)
Gregory & Scott(1969)
Premoli et al(1970)
4
Rouhani I(1970)
483
495
207
587
650
665
431
704
763
731
724
764
53.7
55.0
23.0
65.2
72.2
73.9
47.9
78.2
84.8
81.2
80.4
84.9
306
693
803
34.0
77.0
89.2
Rouhani II(1970)
Bonnecaze et al(1971)
Dix(1971)
Beggs(1972)
153
105
490
506
376
296
676
668
533
556
762
760
17.0
11.7
54.4
56.2
41.8
32.9
75.1
74.2
59.2
61.8
84.7
84.4
Chisholm(1973)
486
678
771
54.0
75.3
85.7
Loscher & Reinhardt(1973)
Mattar & Gregory(1974)
427
13
519
35
572
74
47.4
1.4
57.7
3.9
63.6
8.2
293
409
26
473
479
593
46
678
607
717
68
773
32.6
45.4
2.9
52.6
53.2
65.9
5.1
75.3
67.4
79.7
7.6
85.9
Percentage within
Homogeneous
Armand(1946)
1
2
4
5
Moussali
Greskovich & Cooper(1975)
Madsen(1975)
Mukherjee(1979)
63
Table 9. Total number and percentage of data points correctly predicted for horizontal
flow database (all sources) (contd.)
Correlation / Data Source
Total data points correctly
Percentage of data points correctly
predicted for the whole database predicted for the whole database
Total Points 900
Total Points 900
Gardner-1(1980)
Gardner-2(1980)
±5%
177
286
±10%
307
423
±15%
405
501
±5%
19.7
31.8
±10%
34.1
47.0
±15%
45.0
55.7
Sun et al(1980)
Jowitt(1981)
Chisholm(1983) - Armand(1946)
Spedding & Chen(1984)
Bestion(1985)
Tandon et al(1985)
134
42
444
394
209
455
339
110
658
520
351
618
610
246
772
613
448
690
14.9
4.7
49.3
43.8
23.2
50.6
37.7
12.2
73.1
57.8
39.0
68.7
67.8
27.3
85.8
68.1
49.8
76.7
Chen(1986)
Hamersma & Hart(1987)
Kawaji et al(1987)
Minami & Brill(1987)
El-Boher et al(1988)
Hart et al(1989)
Kokal & Stanislav(1989)
Spedding et al(1989)
Toshiba(1989)
Huq & Loth(1992)
Inoue et al(1993)
Czop et al(1994)
AbdulMajeed(1996)
Maier and Coddington(1997)
Petalaz & Aziz(1997)
Gomez et al(2000)
Zhao et al(2000)
470
221
314
518
281
492
105
377
286
472
343
103
314
334
43
172
105
611
344
497
705
505
618
297
510
630
636
537
221
575
514
79
247
159
699
436
607
797
699
665
557
600
748
754
638
434
654
594
131
312
196
52.2
24.6
34.9
57.6
31.2
54.7
11.7
41.9
31.8
52.4
38.1
11.4
34.9
37.1
4.8
19.1
11.7
67.9
38.2
55.2
78.3
56.1
68.7
33.0
56.7
70.0
70.7
59.7
24.6
63.9
57.1
8.8
27.4
17.7
77.7
48.4
67.4
88.6
77.7
73.9
61.9
66.7
83.1
83.8
70.9
48.2
72.7
66.0
14.6
34.7
21.8
Graham et al(2001)
520
667
749
57.8
74.1
83.2
Percentage within
1
correlation as reported by Leung (2005), 2,5 correlations as reported by Kataoka and Ishii (1987), 3
correlation as reported by Isbin and Biddle (1979) , 4 Rouhani and Axelsson(1970) correlations I and II.
The Minami and Brill (1987) correlation fairly predicted the 10 and 20 degree data
within the 5% index while Chisholm (1973), Chisholm (1983) Armand (1946) and
Smith (1969) also gave similar results for the 15 degree data. A consistent acceptable
prediction for the high inclination angle ranges (35 -75) comes from Dix (1971)
correlation which predicted the data for all error indices quite well.
64
1.0
E
B
S
0.8
M
O
Calculated void fraction
K
F
A
0.6
0.4
0.2
S
F
S
S
S
S
S
S
S
K
E
A
S
E
S
K
S
E
K
O
S
A
S
S
K
O
B
S
K
B
S
O
A
E
S
S
A
E
E
E
E
E
S
S
M
M
E
K
K
O
S
K
A
A
S
M
S
M
AA
Eaton(1966)
K
A
S
S
K
E
O
S
S
M EE
S
O
S
A
A
O
M
K
K
A
S
M
S
A
EA
K
A
M
E
S
K
EB
EE
O
A
S
M
S
E
O
B
S
O
E
S
S
O
B
B
S
S
ES
A
E
S
M
EFKE
K
S
A
Beggs(1972)
EEE
A
S
O
M
E
M
M
O
O
E
FOK
A
S
KK
S
E
M
A
AA
S
SE
F
M
A
SSA
S
A
A
S
E
K
AS
E
S
E
K FEK
F
KFO
E
E
O
S
E
O
S
S
AA
E
S
SE
F
E
AAM
F
ASF
FKEEF
S
S
FEE
FM
S
O
E
F
S
S
A
S
M
Spedding & Nguyen(1976)
S
F
A
E
E
O
K
A
E
M
E
F
A
S
F
K
B
E
S
O
O
O
F
S
M SFFSSF
E
KFSSS
F
A
EA
FS
E
F
E
EFK
O
S
A
SS
M
F
EBMS
ASS
A
S
FFF
FK
SS
OM
E
EAK
S
O
A
EM
E
M
F
EE
K
O
E
S
AS
B
FA
E
ESAE E M M
K
OKEE
EE
E
E
E
FF SF SSSSSS
EE
K
M
E
A
Mukherjee(1979)
M
M
S FOOK
EM BE
EAOKAEE
SS
E
F
S
SSO
K
S KS
ME S S
OE
F S A
SS S
EOKEO BE
EK
KS EE
E
S
S
M
E
E
E
K
A
K
OBE
S
Minami & Brill(1987)(air-water)
EE O
ES E EEE
EK
K AEO
BEB
SS SS
E
K
KE
SS
EE
KEE
O
E E
S
SE E
E
EO
E
OE
K
S S
B S
S KA
S
E
KEOEBO
K EAE
Minami & Brill(1987)(air-kero)
SE EEB
S
AKSE E OSSE O
EE E
E
S
A MS O
K EEM EE E
E A BB EE
Franca & Lahey(1992)
SE B E
S KS
OEKE
M
SE
A
EEEOEBEMO
E E
M
BE
ESA
AE
A
S E
A
A S
A M E
Abdulmajeed(1996)
BB S EO
A
A
M
S A
O S A
S
EM
E
A E
M
F
O E
S
BB
M
F E
A SS
S
A
M
AE EF EE E
E
M
E
E
EM A
S E BB
S
M
F
B
M
B
E FE
M
SEEE
M
ES
E B
E EFB
E
F
E
E
F
S E ES
E ESB A
FF
M
E ESS
A
F
M
S
E E EE E B M FF
S
F F
E EEEE E
E
S
FA
SE EEEE
FB
S
EE
E E SM
BSB
S
S S
S S
EFEM F
F
F
MS
S
S
S
S
E E FE
S
S
FB
E
S
F
S
S
S
F
SF F
SS
SFM
FF
B
B
S SS F
MFFBF
SB
M
S
S
+15%
-15%
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Measured void fraction
Figure 1. Comparison of Rouhani I (1970) correlation with
measured horizontal experimental data
1.0
E
B
S
0.8
M
O
Calculated void fraction
K
F
A
0.6
0.4
0.2
M
S
S
S
S
SS
S
S
S
S
SS
S
S
S
S
S
S
S
SAS
SS
S
SS
SSS
A
SSS
S
K
O
E
S
A
E
S
K
O
S
A
S
S SA
O
AE
S
S
S
B
E
SSSSS SS
A
B
A
A
S
O
Eaton(1966)
K
K
S
E
S
SA
A
S
SS
O
OA
S S SAS
S
S
M
K
S S
E
EA
AM
O
S
AA
O
SSS F S
E
SO
S
S
A
E
AA
FB
E
ME
K
A
E
S S SF
K
M
A
B
O
E
M
O
FFFS
S
S
AE
B
F
K
M
O
EF
A
Beggs(1972)
K
OSE
EA
S
S SS SSFF
M
M
E
M
A
OE
BA
FO
KK
S
FFSFS
S
FO
M
EE
E
S
E
SSS F SFSO
B
EK
F
A
A
AEA
K
S
SS
M
FO
B
S
ME
EKE
E
E
K
S S
E
KS
B
O
E
F
S
S
F
S
O
F
S
O
M
E
F
F
A
E
A
M
K
E
F
E
S
A
E
S
A
E
E
K
A
A
SFS
SS
ES
A
AS
M
A
M
K
OOOFK
SF
E
E
FF
E
FFF
Spedding & Nguyen(1976)
A
F
S
M
E
S
F
A
O
S
K
S
S
O
M
A
F
E
E
K
E
E
E
A
K
E
A
E
KMS S M
EE
FFS S SS S
O OKOK
E
KES
S
E
A
E
E
FF
E
M M
A
M
OO O
KE
E
K
S
E
S
EE
B
EE
M
M
KEM
SE E
S
S
EE
EA
AK
EK
O
S
O FO
E
K
KA
A
K
EE
OSEB
SSS O
E
O
SSK
Mukherjee(1979)
EKM
K MEE E M
AE
E
E
E
S A
EE
K SE
KK
EE
S
OS KO S
EEEBE
AEE
E
S
KKEE BOM
SS KE EOKE
SAS O O
BE
E
E
E
E
B
E
E
E
E
SE EO
Minami & Brill(1987)(air-water)
E
EES
SS O
OK
AEK E AK ESO A
S
E
KE
BE E E
S
E
K
E
S OE
O
S
OAKK M SEEE EEBB K A E
Minami & Brill(1987)(air-kero)
EE
E BB
SS S
A K A M AE
O
EM E M
E
E
E
S
E
E
E E
E BESE EE
B E
ES S E
A EA
Franca & Lahey(1992)
E E
S
E E EMA
E
A
S E E
BE
A
A S S
BS
S
A
E
B
M
A
M
Abdulmajeed(1996)
A
S
M
EE
M
EA E
S
F
S
A S
M
BB
S
A
M
F E
S
S
M
F E
S
E
E
E
A
B
EE B EE
EM E M
ES SEEE
F
M
M S
B
FE
B
E E
EE
B M
E ES
EE
S
M
F
EFB
S S
F
ES
A
E
S
FB
F
F
E
E
M
S
F
F
E
E E E A
S
S
E BF FM
E
E EE E E E
E
FA
S EE E SES F
E
B
EE
S
S S
B
E
E B
S
M
FF F M F
S
M
S
S
EB F E
F S
S
E
S F F
S
SS
F
F
S
FF FM
B
S
B
SS F
M
SFFBBF
+15%
-15%
S
S
F
0.0
0.0
0.2
0.4
0.6
0.8
Measured void fraction
Figure 2. Comparison of Hughmark (1962) correlation with
measured horizontal experimental data
65
1.0
1.0
E
B
S
0.8
M
O
Calculated void fraction
K
F
A
0.6
S
S
S
S
S
S
S
S
S
A
SS
S
A
O
K
S
S
S
K
S
O
S
A
SS
O
S
A
A
K
M
SSS
K
O
A
S
B
S
A
B
M
SSSS
A
S
K
A
S
K
M
S
K
O
F
O
EE
A
S
M
S
O
A
A
S
SF
F
M
S
K
M
SS
SS
F
O
K
K
E
FS
O
S
SFS
F
A
A
M
O
M
E
FS
SA
S
S
SS
E
OO
A
A
A
FS
KE
M
EEE
FFS F
SF
SSF
O
K
AK
B
E
M
A
B
KAB
SFSS
S
OS
SF
O
B
FM
SS
S
AO
K
AK
M
AB
A
SFS
FS
S
E
S
F
O
F
E
S
S
K
E
F
S
K
M
M
A
S
F F F OO SKKOA
BA K EE
EME
E
SSAA
M
A EM
SF SSS
F
F
E
OOA
SS
F
B
EEEMEE
F
O
EAEEE
SF
F
SSS
M
OOK KA
MAAKAKE
E M
SS MFO
AAAS EEEEEME
S
K KSAKM
S
E
O
EEE
OS
OK O
AM
E
F
S S SS
OK
EE
E
E
E
K
K
E
A
E
E
SS
E
E
E
F
M
E
M
E
B
EEEEEM
O K AM S M KE
S S SSO
K K O EKEM
EEEEE
E
E
E
AEEM
K O
OSA E B
E
S S
A
E EE
E
E
O K S
O
EKE EE
SS
E
E
EE
K KE
EE
S
E EEEB
KO
SA
E
B
E
O
E
E
E
S
O BEE MEE
O K S O
S SA A
K
K
E
OS
E KEAEB
E EES
S
S
A M OAE
EE EE
BBEE
SA
S EEE
E
KOE
E
B
S O K M SK ME EE E
E
E
O B
EEO
E
S
E
S
E K
S
S
M K EB E E EM
E
E
AK
E
S
S
S
O EE ES E E E A E
MAO
S
BB M
E
E
A
S
EE EES
A KS
A
S
M
EE E E
O M BB B F
B
A
E
EMBS
A
M
E
F EE
E S
A
E
AE
F AE
A
B
AE SS
M
S S
E E B E ME
S
E BE F
S
S
S EF BB
EE
S A EF B
M
S
A
E FF F A
M
EE
E
E
E
B
B
F
M
SE E EA E FS AE
M
S ESB
E FF
M
S
S
E
F
E E
EEM
M
S
EE E
M
FF F
E E EM
EBE E
B
EF S
E
A
S
S EE
EF E
S
E
EF S F
S ES
S E E F EE
MS E F
EE
S
F
SS
B
S
S
SF F
+15%
Eaton(1966)
Beggs(1972)
Spedding & Nguyen(1976)
Mukherjee(1979)
Minami & Brill(1987)(air-water)
Minami & Brill(1987)(air-kero)
Franca & Lahey(1992)
Abdulmajeed(1996)
0.4
SSM
0.2
M
FS
S
S
B SF
M F B
FF
S
SF
BB
S
FF
S
-15%
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Measured void fraction
Figure 3. Comparison of Minami and Brill (1987) correlation with
measured horizontal experimental data
1.0
S
S
E
S
K
S
S
A
E
E
K
K
O
S
S
S
A
S
K
O
EB
K
B
E
E
EK
S
O
S
A
O K
S
O
A
A
M
E
AOS
M
E
S
K
KS
K M
O
S
K
M
A
S
O
S
S
E
M
S
E
A
O KE
ASS
E
E
B
E
M
K A E
SE
BE
S
KM
K
A
K MO
O
S
A
S
A
E
S
E
F
A
S
A
A
E
S
S
O
SS
S
ESS
AO
S
FK
S
M
O
S
S
B
B
M
A
S
KSB
A
KE
K E
K E OK OE
E
ES
S
A
K
F
S
EEE
SE
E FM S
O
A
S
A
SE
ESA
AO
M
O
M
A
A
A
EK
E
O
A
FA
E
O K KO KKO
ES
E
AM
EE
S
E
O
S
K
SS
K
AM
F
SSE
S
A
S
S
S
OFM
F FKFEE
BEA
E
S
F
AM
SSSE
FA
AF
FO
E
AM
M
S
SSFE
FF O
F
E
S
S
S
O
E
SE
O FK OE E
F
F
K
A
A
K
F
S
M
S
F
E
E
S
M
S
E
O
S
F
F
S
S
S
O
S
F
E
S
M
S
A
F
A
K
EE BF A
K A FE K E SEE
EEE
FMS
KM
FSA
B
SS
SSS
E
M
SEE
F
S
EE
F
SFK
S
A
S
SEF
OA
AEAFEM
K
O AFO
OS
M
FB
S
EE
M
S
O
S SSO
E
FESS
SSS
ES
ES
AM
FE
ES E A
A
BE
ME
K
SK K M S SSSK SSSKSSFSS
EAOK
ESE A
S OO
O SS O
EKE
AEM
SES
BE
OEEK
O E
EE
EK M
S
A
E E SESESE
E E M
S
S
E
E
S
A KS
E
E
K
E
E
E
K
A
S
K
E E SEEA
B
BBE
OSS EO
OA SS S
O
O
A
M
E
E E
EK
S
BB EE E
S
E
AE
S EM EEBBOK A E E
S
E
EA
AE E E A S SE EM E
E B E E EE SEE
E
S
S
E
E
E
A
A
S
S E E SEMEEEEMA E E
A S E EBEE E
S
M
BB
B A A M
M
SS
M MM
SE
S
S
M
F
A
S
M
BB
F
E
M
A
E
S
E EE
E
B F E
EE E E
EEE B E EE
S SEE S E
E E
F
F
EE
EE
BB M M
M
S S
ESB
S
F FA
F
E B
S
S
M
FB
F
E
ES
A F FF FF
S
E
E
EE EES E EE E B M
E EE
E
E
A
Eaton(1966)
S
S
F
E S
S
EE E
E F
B
S
E
B
BF
Beggs(1972)
FB
S
M
M
FF M
S
S
F
Spedding & Nguyen(1976)
BF
SSS
E
E
F
M
Mukherjee(1979)
E
S
F
O
Minami & Brill(1987)(air-water)
FF
+15%
0.8
Calculated void fraction
-15%
0.6
0.4
0.2
S
F
MS
S
SS
FF S
F
M
S
B
F B
SS
SFF F
M
BB
K
F
A
Minami & Brill(1987)(air-kero)
Franca & Lahey(1992)
Abdulmajeed(1996)
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Measured void fraction
Figure 4. Comparison of Armand Massina correlation (Leung 2005)
with measured horizontal experimental data
66
1.0
S
S
S
S
S
S
S
S
A
K
A
S
K
O
S
S
A
O
S
S
K
O
A
SSS
SSS
M
SS
S
SS
E
B
A
B
SS
S
A
S
FS
K
K
K
MM
SSSS
A
E
M F F SSS
SS
O
O
M
A
F
S
O
M
F
K
K
S
S
A
K
S
S
F
M
A
O
E
M
O
A
K
E
E
O
F
S
K
M
S
OS
SS
A
SS
EEE
A
A
S
A
FS
S
A
E
E
FS F
K
M
AA
FF
KS
B
B
O
S
O
F
S
K
B
F
F SFSSF
O
AO
S
M
A
FSSF
AK
AB
E E
SOS
SS
E
E
Eaton(1966)
Beggs(1972)
Spedding & Nguyen(1976)
Mukherjee(1979)
Minami & Brill(1987)(air-water)
Minami & Brill(1987)(air-kero)
Franca & Lahey(1992)
Abdulmajeed(1996)
B
S
0.8
M
O
Calculated void fraction
K
F
A
0.6
0.4
M
0.2
FS
S
S
SSM
B SF
M F B
FF
B
SB
F
S SF F
S
F
F
FS SFOSFF M
KM
E
A
KBA M
O
EME
FFS
SKAAO
KEEE
SSFO S
SS
+15%
EM
S
F FS
M
E
AA
K K
EE
F
SS
F F
E MEE
O
K AS
AKBKE
FSFS O
EAEEEE
M
SSS
O
M
KOK
O
SA A
EEEE
MFO
SKA AAA
E M
ME
SSS S
S
EE
MA E
S OK O O K
EEE
EE
E
AM
S S
E
EEE
S OK M
KE
EEEEE
E
B
OKE
M
SS
E
E
S S OKF K AM S MA
E
E
M
E EE
SS
K
O
EE E
K
A
EE
KE
E
EM
E
A
SS OS SS KOS O B
EE
EE E
EM
EAE
EKEEEE
E
S A O K
K OEE
K S
EE E
EEBE
A
E
E
O
E
K
EEM
SA S S
E EE
B
S
O
E
AOS
B
K K EE E
O
S
EB
O
AEEE
E E
EKEAE
A KS
S
O
S
EE EE E S
ME EO
E
K
BB
SK M
E
S
E
O
O
E
S
E
B
E
S
MEKEE
S OS
K B S EEE O
M AK E E E EM S
S
E
E
EE A E
S
EE B
AO
S
E
E
E
O
K
S
BB E
M
E
S A M
S
EE SEEA
E EAE
SA
B
BB
MA
A ES
E O SM E
E
EF
A
M
MBS EE
A
A
E A BS
F
M
E
E
S
E
B
F EE
SS
SS
EB
EE E
S S A EE BEAF BB M EE
A
M
M
E E BS F
S
EE
S
E
F
F
F FAE B
S
M
E E EA B E
M
F
ES
E
F
S
M
EE
B
M
E
S ES
FF
EEM
M
SE E
EE
FSE
E E EM
FF F
B
S
A
E EEE F B
S
E E
E
S
E E ESS EE F E S S
SF
EFE F
S MS E
S
F
F B
SS
S
S
F
SF F
F
F
-15%
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Measured void fraction
Figure 5. Comparison of Mukherjee (1979) correlation with
measured horizontal experimental data
1.0
F
F
KB
SS
F SF
M
F
FSM
SS
BAM
S
A
O
S
SKS
EAE
AB
A
E
EAO
S
S
K
FO F OSK FOFK SS O
EFSM
KKM
O
A
E
E
E
F
S
K
O
S
S SSS
AE
SSF
AE
M
M
FAFAM
EEAE
F A
F
SA M
SK K
S S
E
K
KFKM
E
SE
M
O
O
B
E
SS
K
E AE
O KS
EA
O S OS
FS FSO
A
E
S
S
E ME E
S
B
E
F
A
S
E
S
K
E
O
S
M
E
S
AS
FK OK M EM
AE
B EA EE E
S S M
K
EEE
E
KS
OM
MA
SA
S O AMO KS A SSKO SS SO
E
AA
EEEE E E
F
S E E ESEE
BE M
E
E
E
E
O M
S
E EA
E
K
S
K
EE BAEEA
KS S
K
EKAE
E EE
E
E
OO EK O E
EKEM
O S O K S SS S
E
E
E
K
E
O
E
BE
EE E
S
S S
E
M
S
S
E
E
A
E E S EAE E E
E
S
E KS
AK
E
EEE
KE E
E
E
KE
EE E
EB
OS A
MBB
OA SSS
O
O
A
O
E
KEE E
E E E EE
E
S
E E EE
B EEK E
E
AE
S M BBBO
E AE
S
E
S
S
S
E
EB EE E
EE
ES
EA
AE E S E A
E
E
EM EE
E EE
SE EMA E
A
S
A
S
EE EMEEE
E E
SE B
EE
E
AB
S
M
E M
E E
B
S
B
A
M
S
MAM
S
S SE
M
F
AM
S
M
BB
F
M
E
A
S
E EES
BB F EE E
E S E
EE
EE E E
E
E
EF E EE
S
FE
E
E
BB M M
S S
M E E SEB
S
FF
F
E BA
S
S
B
M
FF
E A
ES
F FF FF
S
E E
E
M
E EEE EEE
E
S E E E B FA
Eaton(1966)
E SS S
E
EE S
E F
B
S
E
B
B
Beggs(1972)
FB
S
M
M F
F
F
S
M
F
S
BF
Spedding & Nguyen(1976)
SSS
E
F E
E
M
S
Mukherjee(1979)
F
FF
O
A AF
0.8
S
S
S
S
S
S
S
S
K
S
A
S
K
O
S
K
A
S
S
S
S
S
S
K
O
A
KO
E
S
A
S
SSS
SS
M
S
A
O
S
SS
A
O
B
B
SFF
K
K
+15% OFFF KOF MOAFOFFFKSSSSSKFAFSSFFSSFSOSMMASFSOOKSSMMMKOSSKFSSSSSSSBABAKOSMMAAOEOSKKAKSMMAKAAOAEAEEKEAAAEAEE
Calculated void fraction
-15%
0.6
0.4
0.2
S
F
MS
S
SS
FF S
F
M
S
B
F B
SS
SFF F
M
BB
K
F
A
Minami & Brill(1987)(air-water)
Minami & Brill(1987)(air-kero)
Franca & Lahey(1992)
Abdulmajeed(1996)
0.0
0.0
0.2
0.4
0.6
0.8
Measured void fraction
Figure 6. Comparison of Chisholm (1983) – Armand (1946)
correlation with measured horizontal experimental data
67
1.0
A combined assessment for all the inclination angles was performed and the results of
which are given in Table 12. Excellent prediction of the data was given by Chisholm
(1983)–Armand (1946), Chisholm (1973), Gregory & Scott (1969), Hughmark (1962)
and Armand Massina (Leung 2005). The correlations by Filimonov et al. (1957),
Premoli et al. (1970), Rouhani I (1970), Dix (1971) and Toshiba (1989) gave
comparable results to the other set of correlations mentioned above. Among these
correlations the Hughmark (1962) correlation gave the best prediction of 96.7% of the
data within the 15% index while Premoli et al. (1970), Rouhani I (1970), Filimonov et
al. (1957), Toshiba (1989) and Dix (1971) gave 96.2, 95.7, 95.2, 94.3 and 91.4
percent respectively. A scatter plot of the performance of each of the correlations in
the latter group is shown for each inclination angle in Figures 7 to 13. The decision to
consider these correlations is driven primarily due to the fact that these correlations
would be shown to predict inclined data from other sources very well. On the other
hand those correlations that did perform superior for the Beggs (1972) inclined data
did not do as much on the other data sets.
4.2.2 Inclined flow comparison with the data of Spedding and Nguyen (1976)
An extensive void fraction data restricted to only four inclination angles was given by
Spedding and Nguyen (1976). The angles considered were 2.75, 20.75,45 and 75
degrees. A total of 889 void fraction measurements were reported for which the
number and percentage of data points of predictions by the correlations are reported in
Table 13 and Table 14 respectively.
This data set is where almost all the predictions by the correlations fall short of being
acceptable in all the error indices across all inclination angles.
68
69
±15%
14
29
31
28
12
30
8
9
16
9
10
9
9
30
27
18
26
17
19
15
29
8
6
17
0
15
31
27
±5%
4
13
16
16
6
18
4
3
8
0
0
6
1
14
7
10
9
9
10
7
14
3
3
9
0
9
17
16
±10%
11
21
28
25
10
26
7
6
11
5
4
7
6
25
17
14
16
13
13
11
18
7
4
12
0
10
26
24
±15%
13
27
29
26
11
28
8
10
15
11
10
8
7
28
26
16
21
15
17
15
27
8
6
15
0
14
27
25
Total Points 29
Total Points 31
±10%
10
23
31
27
11
26
7
4
12
4
1
7
8
28
18
15
16
14
16
13
23
7
4
13
0
15
29
25
10 Degree data
5 Degree data
Percentage within ±5%
Homogeneous
4
Armand(1946)
18
1
Armand - Massina
23
Lockhart & Martinelli(1949)
18
Sterman(1956)
7
Filimonov et al(1957)
17
Chisholm & Laird(1958)
4
Flanigan(1958)
2
2
Dimentiev et al
7
Hoogendroon(1959)
0
Bankoff(1960)
0
Fauske(1961)
6
Wilson et al(1961,1965)
4
Hughmark(1962)
14
Nicklin et al(1962)
2
Nishino & Yamazaki(1963)
10
Fujie(1964)
10
Kowalczewski(1964)
12
Thom(1964)
11
Zivi(1964)
11
Hughmark(1965)
13
Neal & Bankoff(1965)
4
Turner & Wallis(1965)
2
Baroczy(1966)
10
Guzhov et al(1967)
0
3
Kutucuglu
9
Smith(1969)
18
Wallis(1969)
18
Correlation / Inclination Angle
±5%
4
18
20
17
5
15
4
2
8
1
0
6
2
18
10
9
8
9
9
7
19
3
3
9
0
10
23
21
±10%
8
24
30
25
11
24
6
6
11
5
4
7
7
28
21
16
14
15
14
12
23
7
5
14
0
13
28
25
±15%
11
29
31
29
12
30
7
11
15
11
16
8
8
31
29
18
20
16
15
15
29
9
6
16
0
15
31
26
Total Points 31
15 Degree data
±5%
4
14
16
15
6
20
3
1
7
0
4
4
3
15
15
12
7
12
8
7
15
3
4
9
0
7
14
14
±10%
6
22
26
24
11
26
5
4
13
4
6
9
9
27
21
17
13
15
14
13
22
5
5
16
0
12
27
23
±15%
10
28
30
29
11
28
6
9
15
11
17
11
9
30
28
21
16
18
17
16
28
8
6
18
0
16
29
26
Total Points 30
20 Degree data
±5%
4
14
17
12
5
15
3
0
8
0
4
4
3
13
15
10
6
8
8
7
14
3
4
8
0
8
12
13
±10%
7
19
21
22
10
25
4
4
12
3
7
9
10
26
20
16
12
15
13
9
19
6
5
13
0
12
23
20
±15%
10
27
28
27
12
28
6
10
14
13
17
10
12
29
27
20
16
17
15
15
27
7
6
17
0
13
26
26
Total Points 30
35 Degree data
±10%
8
16
20
22
12
25
4
8
11
4
7
8
8
25
20
16
10
15
12
12
17
6
5
14
0
12
20
20
±15%
12
24
25
25
13
27
7
12
15
12
15
9
12
27
24
19
17
17
16
15
25
9
6
16
0
13
25
25
Total Points 29
55 Degree data
±5%
6
12
14
11
6
16
3
2
8
0
5
3
1
12
13
11
7
8
9
6
11
3
4
8
0
8
15
12
Table 10. Number of data points correctly predicted for Beggs (1972) Inclined data
±5%
5
12
16
13
8
17
4
3
8
0
3
5
2
12
12
12
9
11
10
7
12
2
4
11
0
8
10
14
±10%
9
19
23
22
11
25
5
5
10
4
5
8
8
25
18
14
14
15
13
12
18
4
5
14
0
12
23
22
±15%
12
26
28
27
12
28
7
10
15
10
12
9
11
27
25
19
17
17
16
15
25
6
7
16
0
13
26
27
Total Points 29
75 Degree data
70
±15%
31
30
29
23
27
26
28
31
13
1
21
30
0
28
15
14
23
0
31
18
17
18
25
8
16
28
25
14
±5%
12
17
14
9
7
16
19
20
7
0
8
14
0
19
6
7
2
0
18
9
9
9
12
4
9
21
8
9
±10%
22
26
25
18
17
24
26
27
9
0
14
21
0
25
11
11
16
0
27
11
12
13
16
7
12
25
21
12
±15%
29
27
28
25
26
26
27
29
11
0
17
28
0
28
15
13
22
0
29
17
18
16
22
8
16
27
25
15
Total Points 29
Total Points 31
±10%
23
26
26
18
18
24
27
29
8
0
14
25
0
24
9
12
12
0
29
13
13
13
19
8
14
26
21
12
10 Degree data
5 Degree data
Percentage within ±5%
Gregory & Scott(1969)
17
Premoli et al(1970)
19
4
Rouhani I(1970)
9
4
Rouhani II(1970)
9
Bonnecaze et al(1971)
2
Dix(1971)
14
Beggs(1972)
26
Chisholm(1973)
25
Loscher & Reinhardt(1973)
7
Mattar & Gregory(1974)
0
5
Moussali
9
Greskovich & Cooper(1975)
19
Madsen(1975)
0
Mukherjee(1979)
22
Gardner-1(1980)
5
Gardner-2(1980)
7
Sun et al(1980)
2
Jowitt(1981)
0
Chisholm(1983) - Armand(1946)
24
Spedding & Chen(1984)
10
Bestion(1985)
8
Tandon et al(1985)
9
Chen(1986)
11
Hamersma & Hart(1987)
4
Kawaji et al(1987)
10
Minami & Brill(1987)
20
El-Boher et al(1988)
4
Hart et al(1989)
10
Correlation / Inclination Angle
±5%
16
19
18
8
10
20
21
23
7
0
7
14
0
19
5
4
7
0
23
7
9
9
7
4
7
20
12
9
±10%
25
29
27
23
21
26
24
29
9
0
12
20
0
26
8
10
18
0
31
13
14
14
13
7
12
25
21
12
±15%
31
31
31
27
29
27
27
31
10
2
15
30
0
28
14
13
24
0
31
18
18
18
22
7
17
28
25
14
Total Points 31
15 Degree data
±5%
15
20
20
10
15
20
14
17
6
0
7
9
0
18
3
6
10
0
19
9
10
12
6
3
9
21
14
11
±10%
22
27
26
19
21
25
20
28
9
0
9
16
0
28
6
11
20
0
28
16
13
16
13
5
16
27
26
14
±15%
30
30
30
28
28
28
27
30
15
2
13
28
0
28
13
15
26
1
30
18
19
19
18
6
20
28
27
17
Total Points 30
20 Degree data
±5%
13
17
20
12
15
23
13
12
8
0
5
7
0
21
3
4
13
0
15
8
11
10
8
3
8
20
16
8
±10%
18
25
26
21
20
28
20
21
9
0
9
13
0
28
9
11
20
0
22
14
15
15
13
4
15
28
28
12
±15%
27
28
29
27
27
29
27
30
13
2
13
27
0
29
14
16
24
4
30
19
19
19
18
6
20
29
28
15
Total Points 30
35 Degree data
±5%
11
17
17
8
13
22
12
15
7
0
5
6
0
20
6
5
12
0
16
6
10
7
7
4
6
19
13
8
±10%
18
22
26
21
20
28
19
20
9
0
10
16
0
26
11
11
18
0
20
14
15
15
14
5
17
26
25
11
±15%
25
27
26
28
24
28
25
27
13
2
13
25
0
28
16
16
23
4
29
17
20
19
18
7
20
28
26
16
Total Points 29
55 Degree data
Table 10. Number of data points correctly predicted for Beggs (1972) Inclined data (contd.)
±5%
12
18
18
10
12
22
16
14
7
0
5
8
0
17
6
3
9
0
16
8
9
9
6
4
8
19
15
7
±10%
20
22
25
19
18
26
21
25
9
0
12
16
0
27
9
13
16
0
25
15
13
15
13
5
14
26
22
12
±15%
29
28
27
27
25
27
28
29
13
1
15
25
0
28
15
16
22
4
28
16
18
18
20
7
17
28
25
16
Total Points 29
75 Degree data
71
1
±5%
7
9
13
12
11
4
9
7
2
0
0
13
±10%
17
12
26
25
18
7
17
13
6
1
2
19
±15%
26
16
28
27
24
17
19
16
7
1
3
27
±5%
10
7
16
18
11
5
8
7
3
0
0
13
±10%
21
12
25
27
20
10
16
13
5
1
2
25
±15%
29
16
29
31
24
17
19
16
6
2
3
29
Total Points 31
15 Degree data
±5%
15
9
17
14
12
9
8
9
1
0
0
14
±10%
21
16
28
26
23
13
17
17
4
0
2
27
±15%
28
17
29
29
26
19
21
19
5
0
3
28
Total Points 30
20 Degree data
±5%
15
8
21
12
13
6
10
12
2
0
0
15
±10%
20
14
27
23
23
15
16
15
5
1
2
25
±15%
27
16
28
26
26
20
20
18
8
1
4
28
Total Points 30
35 Degree data
±5%
13
6
18
11
13
6
11
10
1
0
0
14
±10%
20
14
27
23
23
14
15
16
6
1
2
25
±15%
24
16
27
24
26
18
19
17
7
1
3
26
Total Points 29
55 Degree data
±5%
12
8
16
10
11
7
11
10
2
1
0
12
±10%
18
15
25
21
22
12
18
16
6
1
2
24
±15%
25
16
27
25
25
16
20
16
7
1
3
27
Total Points 29
75 Degree data
correlation as reported by Leung (2005), 2,5 correlations as reported by Kataoka and Ishii (1987), 3 correlation as reported by Isbin and Biddle (1979) , 4 Rouhani and Axelsson(1970)
correlations I and II.
±15%
27
16
29
30
23
16
20
19
6
2
3
26
Total Points 29
Total Points 31
±10%
18
12
26
26
17
9
18
12
3
2
2
21
10 Degree data
5 Degree data
Percentage within ±5%
Kokal & Stanislav(1989)
2
Spedding et al(1989)
9
Toshiba(1989)
11
Huq & Loth(1992)
17
Inoue et al(1993)
10
Czop et al(1994)
2
AbdulMajeed(1996)
11
Maier and Coddington(1997)
7
Petalaz & Aziz(1997)
2
Gomez et al(2000)
2
Zhao et al(2000)
0
Graham et al(2001)
14
Correlation / Inclination Angle
Table 10. Number of data points correctly predicted for Beggs (1972) Inclined data (contd.)
72
22.6
54.8
12.9
6.5
22.6
0.0
0.0
19.4
12.9
45.2
6.5
32.3
32.3
38.7
35.5
35.5
41.9
12.9
6.5
32.3
0.0
29.0
58.1
58.1
Sterman(1956)
Filimonov et al(1957)
Chisholm & Laird(1958)
Flanigan(1958)
Dimentiev et al
Hoogendroon(1959)
Bankoff(1960)
Fauske(1961)
Wilson et al(1961,1965)
Hughmark(1962)
Nicklin et al(1962)
Nishino & Yamazaki(1963)
Fujie(1964)
Kowalczewski(1964)
Thom(1964)
Zivi(1964)
Hughmark(1965)
Neal & Bankoff(1965)
Turner & Wallis(1965)
Baroczy(1966)
Guzhov et al(1967)
Kutucuglu
Smith(1969)
Wallis(1969)
3
58.1
Lockhart & Martinelli(1949)
2
74.2
48.4
93.5
80.6
48.4
51.6
45.2
51.6
41.9
74.2
22.6
12.9
41.9
0.0
38.7
12.9
3.2
22.6
25.8
90.3
58.1
22.6
12.9
35.5
83.9
87.1
74.2
100.0
58.1
Armand(1946)
1
Armand - Massina
Homogeneous
48.4
100.0
87.1
58.1
83.9
54.8
61.3
48.4
93.5
25.8
19.4
54.8
0.0
51.6
29.0
32.3
29.0
29.0
96.8
87.1
25.8
29.0
38.7
96.8
90.3
100.0
93.5
±15%
45.2
31.0
58.6
55.2
34.5
31.0
31.0
34.5
24.1
48.3
10.3
10.3
31.0
0.0
27.6
0.0
0.0
20.7
3.4
48.3
24.1
13.8
10.3
20.7
62.1
55.2
55.2
44.8
±5%
13.8
34.5
89.7
82.8
48.3
55.2
44.8
44.8
37.9
62.1
24.1
13.8
41.4
0.0
37.9
17.2
13.8
24.1
20.7
86.2
58.6
24.1
20.7
34.5
89.7
86.2
96.6
72.4
±10%
37.9
48.3
93.1
86.2
55.2
72.4
51.7
58.6
51.7
93.1
27.6
20.7
51.7
0.0
51.7
37.9
34.5
27.6
24.1
96.6
89.7
27.6
34.5
37.9
96.6
89.7
100.0
93.1
±15%
44.8
Total Points 29
Total Points 31
±10%
32.3
10 Degree data
5 Degree data
Percentage within ±5%
12.9
Correlation / Inclination Angle
32.3
74.2
67.7
29.0
25.8
29.0
29.0
22.6
61.3
9.7
9.7
29.0
0.0
25.8
3.2
0.0
19.4
6.5
58.1
32.3
12.9
6.5
16.1
48.4
54.8
64.5
58.1
±5%
12.9
41.9
90.3
80.6
51.6
45.2
48.4
45.2
38.7
74.2
22.6
16.1
45.2
0.0
35.5
16.1
12.9
22.6
22.6
90.3
67.7
19.4
19.4
35.5
77.4
80.6
96.8
77.4
±10%
25.8
48.4
100.0
83.9
58.1
64.5
51.6
48.4
48.4
93.5
29.0
19.4
51.6
0.0
48.4
35.5
51.6
25.8
25.8
100.0
93.5
22.6
35.5
38.7
96.8
93.5
100.0
93.5
±15%
35.5
Total Points 31
15 Degree data
23.3
46.7
46.7
40.0
23.3
40.0
26.7
23.3
50.0
10.0
13.3
30.0
0.0
23.3
0.0
13.3
13.3
10.0
50.0
50.0
10.0
3.3
20.0
66.7
50.0
53.3
46.7
±5%
13.3
40.0
90.0
76.7
56.7
43.3
50.0
46.7
43.3
73.3
16.7
16.7
53.3
0.0
43.3
13.3
20.0
30.0
30.0
90.0
70.0
16.7
13.3
36.7
86.7
80.0
86.7
73.3
±10%
20.0
53.3
96.7
86.7
70.0
53.3
60.0
56.7
53.3
93.3
26.7
20.0
60.0
0.0
50.0
36.7
56.7
36.7
30.0
100.0
93.3
20.0
30.0
36.7
93.3
96.7
100.0
93.3
±15%
33.3
Total Points 30
20 Degree data
26.7
40.0
43.3
33.3
20.0
26.7
26.7
23.3
46.7
10.0
13.3
26.7
0.0
26.7
0.0
13.3
13.3
10.0
43.3
50.0
10.0
0.0
16.7
50.0
40.0
56.7
46.7
±5%
13.3
40.0
76.7
66.7
53.3
40.0
50.0
43.3
30.0
63.3
20.0
16.7
43.3
0.0
40.0
10.0
23.3
30.0
33.3
86.7
66.7
13.3
13.3
33.3
83.3
73.3
70.0
63.3
±10%
23.3
43.3
86.7
86.7
66.7
53.3
56.7
50.0
50.0
90.0
23.3
20.0
56.7
0.0
46.7
43.3
56.7
33.3
40.0
96.7
90.0
20.0
33.3
40.0
93.3
90.0
93.3
90.0
±15%
33.3
Total Points 30
35 Degree data
27.6
51.7
41.4
37.9
24.1
27.6
31.0
20.7
37.9
10.3
13.8
27.6
0.0
27.6
0.0
17.2
10.3
3.4
41.4
44.8
10.3
6.9
20.7
55.2
37.9
48.3
41.4
±5%
20.7
41.4
69.0
69.0
55.2
34.5
51.7
41.4
41.4
58.6
20.7
17.2
48.3
0.0
37.9
13.8
24.1
27.6
27.6
86.2
69.0
13.8
27.6
41.4
86.2
75.9
69.0
55.2
±10%
27.6
44.8
86.2
86.2
65.5
58.6
58.6
55.2
51.7
86.2
31.0
20.7
55.2
0.0
51.7
41.4
51.7
31.0
41.4
93.1
82.8
24.1
41.4
44.8
93.1
86.2
86.2
82.8
±15%
41.4
Total Points 29
55 Degree data
Table 11. Percentage of data points correctly predicted for Beggs (1972) Inclined data
27.6
34.5
48.3
41.4
31.0
37.9
34.5
24.1
41.4
6.9
13.8
37.9
0.0
27.6
0.0
10.3
17.2
6.9
41.4
41.4
13.8
10.3
27.6
58.6
44.8
55.2
41.4
±5%
17.2
41.4
79.3
75.9
48.3
48.3
51.7
44.8
41.4
62.1
13.8
17.2
48.3
0.0
34.5
13.8
17.2
27.6
27.6
86.2
62.1
17.2
17.2
37.9
86.2
75.9
79.3
65.5
±10%
31.0
44.8
89.7
93.1
65.5
58.6
58.6
55.2
51.7
86.2
20.7
24.1
55.2
0.0
51.7
34.5
41.4
31.0
37.9
93.1
86.2
24.1
34.5
41.4
96.6
93.1
96.6
89.7
±15%
41.4
Total Points 29
75 Degree data
73
83.9
80.6
22.6
0.0
Beggs(1972)
Chisholm(1973)
Loscher & Reinhardt(1973)
Mattar & Gregory(1974)
29.0
61.3
0.0
71.0
16.1
22.6
6.5
0.0
77.4
32.3
25.8
29.0
35.5
12.9
32.3
64.5
12.9
32.3
Moussali
Greskovich & Cooper(1975)
Madsen(1975)
Mukherjee(1979)
Gardner-1(1980)
Gardner-2(1980)
Sun et al(1980)
Jowitt(1981)
Chisholm(1983) - Armand(1946)
Spedding & Chen(1984)
Bestion(1985)
Tandon et al(1985)
Chen(1986)
Hamersma & Hart(1987)
Kawaji et al(1987)
Minami & Brill(1987)
El-Boher et al(1988)
Hart et al(1989)
5
6.5
45.2
29.0
Bonnecaze et al(1971)
Dix(1971)
Rouhani II(1970)
4
83.9
67.7
38.7
38.7
38.7
0.0
93.5
41.9
41.9
41.9
61.3
25.8
45.2
45.2
80.6
0.0
77.4
29.0
25.8
0.0
87.1
93.5
58.1
77.4
58.1
83.9
83.9
61.3
29.0
Premoli et al(1970)
4
Rouhani I(1970)
90.3
80.6
45.2
45.2
74.2
0.0
100.0
58.1
54.8
58.1
80.6
25.8
51.6
67.7
96.8
0.0
90.3
48.4
41.9
3.2
90.3
100.0
87.1
83.9
74.2
93.5
96.8
±15%
100.0
72.4
27.6
31.0
24.1
6.9
0.0
62.1
31.0
31.0
31.0
41.4
13.8
31.0
27.6
48.3
0.0
65.5
20.7
24.1
0.0
65.5
69.0
24.1
55.2
31.0
48.3
58.6
±5%
41.4
86.2
72.4
41.4
37.9
55.2
0.0
93.1
37.9
41.4
44.8
55.2
24.1
41.4
48.3
72.4
0.0
86.2
37.9
31.0
0.0
89.7
93.1
58.6
82.8
62.1
86.2
89.7
±10%
75.9
93.1
86.2
51.7
44.8
75.9
0.0
100.0
58.6
62.1
55.2
75.9
27.6
55.2
58.6
96.6
0.0
96.6
51.7
37.9
0.0
93.1
100.0
89.7
89.7
86.2
96.6
93.1
±15%
100.0
Total Points 29
Total Points 31
±10%
74.2
10 Degree data
5 Degree data
Percentage within ±5%
Gregory & Scott(1969)
54.8
Correlation / Inclination Angle
64.5
38.7
29.0
12.9
22.6
0.0
74.2
22.6
29.0
29.0
22.6
12.9
22.6
22.6
45.2
0.0
61.3
16.1
22.6
0.0
67.7
74.2
32.3
64.5
25.8
58.1
61.3
±5%
51.6
80.6
67.7
38.7
32.3
58.1
0.0
100.0
41.9
45.2
45.2
41.9
22.6
38.7
38.7
64.5
0.0
83.9
25.8
29.0
0.0
77.4
93.5
67.7
83.9
74.2
87.1
93.5
±10%
80.6
90.3
80.6
45.2
41.9
77.4
0.0
100.0
58.1
58.1
58.1
71.0
22.6
54.8
48.4
96.8
0.0
90.3
45.2
32.3
6.5
87.1
100.0
93.5
87.1
87.1
100.0
100.0
±15%
100.0
Total Points 31
15 Degree data
70.0
46.7
36.7
20.0
33.3
0.0
63.3
30.0
33.3
40.0
20.0
10.0
30.0
23.3
30.0
0.0
60.0
10.0
20.0
0.0
46.7
56.7
50.0
66.7
33.3
66.7
66.7
±5%
50.0
90.0
86.7
46.7
36.7
66.7
0.0
93.3
53.3
43.3
53.3
43.3
16.7
53.3
30.0
53.3
0.0
93.3
20.0
30.0
0.0
66.7
93.3
70.0
83.3
63.3
86.7
90.0
±10%
73.3
93.3
90.0
56.7
50.0
86.7
3.3
100.0
60.0
63.3
63.3
60.0
20.0
66.7
43.3
93.3
0.0
93.3
43.3
50.0
6.7
90.0
100.0
93.3
93.3
93.3
100.0
100.0
±15%
100.0
Total Points 30
20 Degree data
66.7
53.3
26.7
13.3
43.3
0.0
50.0
26.7
36.7
33.3
26.7
10.0
26.7
16.7
23.3
0.0
70.0
10.0
26.7
0.0
43.3
40.0
50.0
76.7
40.0
66.7
56.7
±5%
43.3
93.3
93.3
40.0
36.7
66.7
0.0
73.3
46.7
50.0
50.0
43.3
13.3
50.0
30.0
43.3
0.0
93.3
30.0
30.0
0.0
66.7
70.0
66.7
93.3
70.0
86.7
83.3
±10%
60.0
96.7
93.3
50.0
53.3
80.0
13.3
100.0
63.3
63.3
63.3
60.0
20.0
66.7
43.3
90.0
0.0
96.7
46.7
43.3
6.7
90.0
100.0
90.0
96.7
90.0
96.7
93.3
±15%
90.0
Total Points 30
35 Degree data
65.5
44.8
27.6
17.2
41.4
0.0
55.2
20.7
34.5
24.1
24.1
13.8
20.7
17.2
20.7
0.0
69.0
20.7
24.1
0.0
41.4
51.7
44.8
75.9
27.6
58.6
58.6
±5%
37.9
89.7
86.2
37.9
37.9
62.1
0.0
69.0
48.3
51.7
51.7
48.3
17.2
58.6
34.5
55.2
0.0
89.7
37.9
31.0
0.0
65.5
69.0
69.0
96.6
72.4
89.7
75.9
±10%
62.1
96.6
89.7
55.2
55.2
79.3
13.8
100.0
58.6
69.0
65.5
62.1
24.1
69.0
44.8
86.2
0.0
96.6
55.2
44.8
6.9
86.2
93.1
82.8
96.6
96.6
89.7
93.1
±15%
86.2
Total Points 29
55 Degree data
Table 11. Percentage of data points correctly predicted for Beggs (1972) Inclined data (contd.)
65.5
51.7
24.1
10.3
31.0
0.0
55.2
27.6
31.0
31.0
20.7
13.8
27.6
17.2
27.6
0.0
58.6
20.7
24.1
0.0
55.2
48.3
41.4
75.9
34.5
62.1
62.1
±5%
41.4
89.7
75.9
41.4
44.8
55.2
0.0
86.2
51.7
44.8
51.7
44.8
17.2
48.3
41.4
55.2
0.0
93.1
31.0
31.0
0.0
72.4
86.2
62.1
89.7
65.5
86.2
75.9
±10%
69.0
96.6
86.2
55.2
55.2
75.9
13.8
96.6
55.2
62.1
62.1
69.0
24.1
58.6
51.7
86.2
0.0
96.6
51.7
44.8
3.4
96.6
100.0
86.2
93.1
93.1
93.1
96.6
±15%
100.0
Total Points 29
75 Degree data
74
1
6.5
67.7
9.7
6.5
58.1
38.7
54.8
29.0
83.9
51.6
9.7
83.9
19.4
6.5
64.5
61.3
74.2
51.6
96.8
93.5
0.0
44.8
6.9
0.0
31.0
24.1
37.9
13.8
41.4
44.8
31.0
±5%
24.1
6.9
65.5
20.7
3.4
58.6
44.8
62.1
24.1
86.2
89.7
41.4
±10%
58.6
10.3
93.1
24.1
3.4
65.5
55.2
82.8
58.6
93.1
96.6
55.2
±15%
89.7
0.0
41.9
9.7
0.0
25.8
22.6
35.5
16.1
58.1
51.6
22.6
±5%
32.3
6.5
80.6
16.1
3.2
51.6
41.9
64.5
32.3
87.1
80.6
38.7
±10%
67.7
9.7
93.5
19.4
6.5
61.3
51.6
77.4
54.8
100.0
93.5
51.6
±15%
93.5
Total Points 31
15 Degree data
0.0
46.7
3.3
0.0
26.7
30.0
40.0
30.0
46.7
56.7
30.0
±5%
50.0
6.7
90.0
10.0
93.3
16.7
0.0
70.0
63.3
56.7
56.7
13.3
0.0
86.7
63.3
96.7
96.7
56.7
±15%
93.3
76.7
43.3
86.7
93.3
53.3
±10%
70.0
Total Points 30
20 Degree data
0.0
50.0
6.7
0.0
33.3
40.0
43.3
20.0
40.0
70.0
26.7
±5%
50.0
6.7
83.3
16.7
3.3
53.3
50.0
76.7
50.0
76.7
90.0
46.7
±10%
66.7
13.3
93.3
26.7
3.3
66.7
60.0
86.7
66.7
86.7
93.3
53.3
±15%
90.0
Total Points 30
35 Degree data
0.0
48.3
3.4
0.0
37.9
34.5
44.8
20.7
37.9
62.1
20.7
±5%
44.8
6.9
86.2
20.7
3.4
51.7
55.2
79.3
48.3
79.3
93.1
48.3
±10%
69.0
10.3
89.7
24.1
3.4
65.5
58.6
89.7
62.1
82.8
93.1
55.2
±15%
82.8
Total Points 29
55 Degree data
0.0
41.4
6.9
3.4
37.9
34.5
37.9
24.1
34.5
55.2
27.6
±5%
41.4
6.9
82.8
20.7
3.4
62.1
55.2
75.9
41.4
72.4
86.2
51.7
±10%
62.1
10.3
93.1
24.1
3.4
69.0
55.2
86.2
55.2
86.2
93.1
55.2
±15%
86.2
Total Points 29
75 Degree data
correlation as reported by Leung (2005), 2,5 correlations as reported by Kataoka and Ishii (1987), 3 correlation as reported by Isbin and Biddle (1979) , 4 Rouhani and Axelsson(1970)
correlations I and II.
0.0
45.2
Zhao et al(2000)
Graham et al(2001)
35.5
22.6
AbdulMajeed(1996)
Maier and Coddington(1997)
6.5
6.5
32.3
6.5
Inoue et al(1993)
Czop et al(1994)
Petalaz & Aziz(1997)
Gomez et al(2000)
54.8
Huq & Loth(1992)
83.9
38.7
29.0
35.5
Spedding et al(1989)
Toshiba(1989)
±15%
87.1
Total Points 29
Total Points 31
±10%
58.1
10 Degree data
5 Degree data
Percentage within ±5%
Kokal & Stanislav(1989)
6.5
Correlation / Inclination Angle
Table 11. Percentage of data points correctly predicted for Beggs (1972) Inclined data (contd.)
Table 12. Total Number and percentage of data points correctly predicted for the
inclined experimental data of Beggs(1972)
Correlation / Inclination Angle
Total data points correctly
Percentage of data points correctly
predicted for the whole database predicted for the whole database
Total Points 209
Percentage within
Homogeneous
Armand(1946)
1
Armand - Massina
Lockhart & Martinelli(1949)
Sterman(1956)
Filimonov et al(1957)
Chisholm & Laird(1958)
Flanigan(1958)
2
Dimentiev et al
Hoogendroon(1959)
Bankoff(1960)
Fauske(1961)
Wilson et al(1961,1965)
Hughmark(1962)
Nicklin et al(1962)
Nishino & Yamazaki(1963)
Fujie(1964)
Kowalczewski(1964)
Thom(1964)
Zivi(1964)
Hughmark(1965)
Neal & Bankoff(1965)
Turner & Wallis(1965)
Baroczy(1966)
Guzhov et al(1967)
3
Kutucuglu
Smith(1969)
Wallis(1969)
Gregory & Scott(1969)
Premoli et al(1970)
4
Rouhani I(1970)
Total Points 209
±5%
31
101
±10%
59
144
±15%
82
190
±5%
14.8
48.3
±10%
28.2
68.9
±15%
39.2
90.9
122
102
43
118
25
13
179
167
76
177
38
37
202
191
83
199
49
71
58.4
48.8
20.6
56.5
12.0
6.2
85.6
79.9
36.4
84.7
18.2
17.7
96.7
91.4
39.7
95.2
23.4
34.0
54
1
16
34
16
98
74
74
56
69
65
52
98
21
24
64
0
80
29
34
55
56
184
135
108
95
102
95
82
140
42
33
96
0
105
77
97
64
68
202
186
131
133
117
115
106
190
55
43
115
0
25.8
0.5
7.7
16.3
7.7
46.9
35.4
35.4
26.8
33.0
31.1
24.9
46.9
10.0
11.5
30.6
0.0
38.3
13.9
16.3
26.3
26.8
88.0
64.6
51.7
45.5
48.8
45.5
39.2
67.0
20.1
15.8
45.9
0.0
50.2
36.8
46.4
30.6
32.5
96.7
89.0
62.7
63.6
56.0
55.0
50.7
90.9
26.3
20.6
55.0
0.0
59
109
108
96
127
86
176
159
148
177
99
195
182
202
201
28.2
52.2
51.7
45.9
60.8
41.1
84.2
76.1
70.8
84.7
47.4
93.3
87.1
96.7
96.2
116
181
200
55.5
86.6
95.7
Rouhani II(1970)
Bonnecaze et al(1971)
Dix(1971)
Beggs(1972)
66
74
137
121
139
135
181
157
185
186
191
189
31.6
35.4
65.6
57.9
66.5
64.6
86.6
75.1
88.5
89.0
91.4
90.4
Chisholm(1973)
126
179
207
60.3
85.6
99.0
Loscher & Reinhardt(1973)
Mattar & Gregory(1974)
49
0
62
0
88
10
23.4
0.0
29.7
0.0
42.1
4.8
Moussali
Greskovich & Cooper(1975)
Madsen(1975)
Mukherjee(1979)
46
77
0
136
80
127
0
184
107
193
0
197
22.0
36.8
0.0
65.1
38.3
60.8
0.0
88.0
51.2
92.3
0.0
94.3
Gardner-1(1980)
34
63
102
16.3
30.1
48.8
4
5
75
Table 12. Total Number and percentage of data points correctly predicted for the
inclined experimental data of Beggs(1972)(contd.)
Correlation / Inclination Angle
Total data points correctly
Percentage of data points correctly
predicted for the whole database predicted for the whole database
Total Points 209
Total Points 209
Gardner-2(1980)
Sun et al(1980)
±5%
36
55
±10%
79
120
±15%
103
164
±5%
17.2
26.3
±10%
37.8
57.4
±15%
49.3
78.5
Jowitt(1981)
Chisholm(1983) - Armand(1946)
Spedding & Chen(1984)
Bestion(1985)
Tandon et al(1985)
Chen(1986)
0
131
57
66
65
57
0
182
96
95
101
101
13
208
123
129
127
143
0.0
62.7
27.3
31.6
31.1
27.3
0.0
87.1
45.9
45.5
48.3
48.3
6.2
99.5
58.9
61.7
60.8
68.4
Hamersma & Hart(1987)
Kawaji et al(1987)
Minami & Brill(1987)
El-Boher et al(1988)
Hart et al(1989)
Kokal & Stanislav(1989)
Spedding et al(1989)
Toshiba(1989)
Huq & Loth(1992)
Inoue et al(1993)
Czop et al(1994)
AbdulMajeed(1996)
Maier and Coddington(1997)
Petalaz & Aziz(1997)
Gomez et al(2000)
Zhao et al(2000)
Graham et al(2001)
26
57
140
82
62
74
56
112
94
81
39
68
62
13
3
0
95
41
100
183
164
85
135
95
184
171
146
80
117
102
35
7
14
166
49
126
196
181
107
186
113
197
192
174
123
138
121
46
8
22
191
12.4
27.3
67.0
39.2
29.7
35.4
26.8
53.6
45.0
38.8
18.7
32.5
29.7
6.2
1.4
0.0
45.5
19.6
47.8
87.6
78.5
40.7
64.6
45.5
88.0
81.8
69.9
38.3
56.0
48.8
16.7
3.3
6.7
79.4
23.4
60.3
93.8
86.6
51.2
89.0
54.1
94.3
91.9
83.3
58.9
66.0
57.9
22.0
3.8
10.5
91.4
Percentage within
1
correlation as reported by Leung (2005), 2,5 correlations as reported by Kataoka and Ishii (1987), 3
correlation as reported by Isbin and Biddle (1979) , 4 Rouhani and Axelsson(1970) correlations I and II.
For near horizontal data (2.75 degree) the simple correlation of Lockhart and
Martinelli (1949) and Wallis (1969) were the only ones to give a fairly acceptable
prediction for the 5% error index. Surprisingly the Lockhart and Martinelli (1949)
correlation could not predict the wider error indices acceptably which was not
logically expected.
Wallis(1969) and Rouhani I(1970) are two correlations which gave a close to
acceptable results for the 10% error index for the near horizontal inclination and
76
1.0
+15%
0.8
Calculated void fraction
-15%
0.6
0.4
5 Degree
10 Degree
15 Degree
20 Degree
35 Degree
75 Degree
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Measured void fraction
Figure 7. Comparison of Armand Massina (Leung 2005) correlation
with measured inclined experimental data of Beggs(1972)
1.0
+15%
0.8
Calculated void fraction
-15%
0.6
5 Degree
10 Degree
15 Degree
20 Degree
35 Degree
75 Degree
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
Measured void fraction
Figure 8. Comparison of Hughmark (1962) correlation with
measured inclined experimental data of Beggs(1972)
77
1.0
1.0
+15%
0.8
Calculated void fraction
-15%
0.6
0.4
5 Degree
10 Degree
15 Degree
20 Degree
35 Degree
75 Degree
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Measured void fraction
Figure 9. Comparison of Premoli et al. (1970) correlation with
measured inclined experimental data of Beggs(1972)
1.0
+15%
0.8
Calculated void fraction
-15%
0.6
0.4
5 Degree
10 Degree
15 Degree
20 Degree
35 Degree
75 Degree
0.2
0.0
0.0
0.2
0.4
0.6
0.8
Measured void fraction
Figure 10. Comparison of Rouhani I (1970) correlation with
measured inclined experimental data of Beggs(1972)
78
1.0
1.0
+15%
0.8
Calculated void fraction
-15%
0.6
0.4
5 Degree
10 Degree
15 Degree
20 Degree
35 Degree
75 Degree
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Measured void fraction
Figure 11. Comparison of Filimonov et al. (1957) correlation with
measured inclined experimental data of Beggs(1972)
1.0
+15%
0.8
Calculated void fraction
-15%
0.6
0.4
5 Degree
10 Degree
15 Degree
20 Degree
35 Degree
75 Degree
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Measured void fraction
Figure 12. Comparison of Toshiba (1989) correlation with measured
inclined experimental data of Beggs(1972)
79
1.0
+15%
0.8
Calculated void fraction
-15%
0.6
0.4
5 Degree
10 Degree
15 Degree
20 Degree
35 Degree
75 Degree
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Measured void fraction
Figure 13. Comparison of Dix (1971) correlation with measured inclined
experimental data of Beggs(1972)
Toshiba(1989) for 20.75 degree with none of the correlations acceptably predicting
any of the data set at all.
Filimonov et al. (1957), Rouhani I (1970) and Dix (1971) were able to predict the
15% error index for 45 and 75, 2.75 and 20.75 & 75 inclination angles respectively.
The performance of the correlations for all inclination angles for this data set is
summarized in Table 15.
The correlation of Toshiba (1989) was the only one that fairly predicted the data set
across all inclination angles for error indices of 10 and 15%. The next best
performance comes from Dix (1971) followed by Filimonov et al. (1957) and
Rouhani I (1970). Though the predictions by Hughmark (1962) and Premoli et al.
(1970) fall short of the expected cutoff percentage, we have included a scatter plot to
show the void fraction range where they underperformed.
80
The graphical representation of the capability of the selected correlations with the
Spedding and Nguyen(1976) data are given in Figures 15 through 20.
4.2.3 Inclined flow comparison with the data of Mukherjee (1979)
Mukherjee (1979) reported air - kerosene void fraction data for six pipe inclination
angles. The pipe inclinations researched were 5, 20,30,50,70 and 80 Degrees. A total
of 444 data points were considered here for making the comparison between the void
fraction correlations. The number of data points correctly predicted for each
inclination angle by correlations is given in Table 16 while percentage figures of
those points are tabulated in Table 17.
Here again the capability of the correlations to predict the data sets within the 5%
error index could only average 57.3% of the data points across all inclination angles
which is far below what we have set out as an acceptable cut off percentage and far
from being applicable to any use by any standard.
The prediction performance of the correlations within the 10% error index has seen a
general improvement with the only exception of the 20 degree set for which none of
the correlations were able to predict the data on and above the specified 80% cutoff
percentage.
81
Table 13. Number of data points correctly predicted for Spedding & Nguyen (1976)
Inclined data
Correlation / Inclination Angle
2.75 Degree data
20.75 Degree data
45 Degree data
Total Points 226
Total Points 188
Total Points 196
Percentage within ±5%
88
32
Homogeneous
Armand(1946)
1
70 Degree data
Total Points 279
±10%
115
70
±15%
133
124
±5%
42
45
±10%
55
74
±15%
67
106
±5%
40
30
±10%
53
52
±15%
68
74
±5%
38
47
±10%
53
86
±15%
72
135
123
119
55
101
53
92
136
133
63
134
61
107
74
79
42
95
40
52
97
110
51
139
50
83
103
122
71
170
65
92
82
99
39
105
37
68
130
141
57
176
50
98
151
172
80
240
67
109
Armand - Massina
Lockhart & Martinelli(1949)
Sterman(1956)
Filimonov et al(1957)
Chisholm & Laird(1958)
Flanigan(1958)
2
Dimentiev et al
95
163
89
103
85
83
171
167
114
132
111
128
184
175
128
152
123
153
82
97
44
71
42
56
78
114
137
61
80
98
47
71
99
54
92
117
Hoogendroon(1959)
Bankoff(1960)
88
16
135
38
158
57
59
18
100
41
109
67
58
11
91
24
103
45
83
35
112
72
130
115
Fauske(1961)
Wilson et al(1961,1965)
Hughmark(1962)
96
20
146
112
33
174
124
43
188
49
16
98
60
28
134
65
46
149
49
18
90
63
29
101
69
44
109
51
18
117
74
43
143
83
54
172
Nicklin et al(1962)
Nishino & Yamazaki(1963)
37
148
80
170
132
180
58
104
95
125
124
129
32
95
67
120
94
134
78
106
136
157
186
179
Fujie(1964)
101
132
157
51
86
106
40
57
82
41
65
102
Kowalczewski(1964)
140
163
169
85
107
108
83
98
107
92
127
143
Thom(1964)
98
150
168
60
90
109
57
91
110
61
105
137
Zivi(1964)
147
157
163
74
88
95
64
81
90
76
98
116
Hughmark(1965)
Neal & Bankoff(1965)
31
20
63
31
109
45
49
17
73
33
101
49
23
20
49
38
65
55
46
36
89
61
127
84
Turner & Wallis(1965)
63
87
104
34
48
54
38
54
61
34
55
68
Baroczy(1966)
153
166
171
85
117
127
75
102
115
93
132
159
Guzhov et al(1967)
3
22
50
1
20
45
1
7
23
0
7
32
139
161
167
69
91
98
72
89
97
74
98
117
Smith(1969)
143
171
185
87
126
135
78
102
106
97
131
156
Wallis(1969)
163
182
189
93
119
130
75
102
114
104
139
169
Kutucuglu
3
Gregory & Scott(1969)
29
73
146
47
75
113
33
53
84
49
82
140
Premoli et al(1970)
153
174
189
100
136
146
101
121
125
132
176
194
4
Rouhani I(1970)
95
184
197
97
136
153
71
107
120
104
169
194
Rouhani II(1970)
47
92
132
71
106
141
44
76
105
64
104
153
Bonnecaze et al(1971)
38
80
132
58
95
124
32
67
94
78
136
186
Dix(1971)
150
168
175
114
148
160
112
139
154
165
228
243
4
Beggs(1972)
120
158
175
57
96
121
54
92
102
60
109
140
Chisholm(1973)
146
172
182
84
121
130
81
97
105
90
124
151
Loscher & Reinhardt(1973)
Mattar & Gregory(1974)
142
10
154
23
156
44
78
3
88
23
99
53
66
5
75
21
81
59
78
4
90
22
102
85
Moussali
Greskovich & Cooper(1975)
89
110
119
147
137
173
46
51
63
79
79
94
40
41
53
57
68
78
38
41
53
56
73
90
Madsen(1975)
Mukherjee(1979)
20
99
32
144
40
172
14
57
21
88
24
131
12
58
21
93
25
113
8
73
13
127
14
173
Gardner-1(1980)
Gardner-2(1980)
Sun et al(1980)
86
92
35
127
131
73
152
154
120
64
60
55
97
94
88
110
120
117
55
56
27
81
90
62
98
103
90
70
79
80
103
114
139
117
129
184
Jowitt(1981)
Chisholm(1983) - Armand(1946)
32
148
65
171
157
182
43
89
66
121
111
130
26
82
65
98
120
106
48
93
100
130
165
152
Spedding & Chen(1984)
155
170
181
94
122
130
88
110
125
104
145
174
5
82
Table 13. Number of data points correctly predicted for Spedding & Nguyen (1976)
Inclined data (contd.)
Correlation / Inclination Angle
2.75 Degree data
20.75 Degree data
45 Degree data
Total Points 226
Total Points 188
Total Points 196
Percentage within ±5%
Bestion(1985)
69
Tandon et al(1985)
147
Chen(1986)
Hamersma & Hart(1987)
Kawaji et al(1987)
Minami & Brill(1987)
El-Boher et al(1988)
Hart et al(1989)
143
87
104
117
145
150
70 Degree data
Total Points 279
±10%
121
164
±15%
144
171
±5%
64
76
±10%
89
92
±15%
98
105
±5%
34
67
±10%
66
86
±15%
84
101
±5%
46
83
±10%
81
110
±15%
97
130
168
113
151
150
169
158
182
124
167
177
184
161
84
42
51
51
92
85
119
55
81
83
130
97
129
63
97
119
149
105
80
40
56
48
105
66
99
52
79
86
120
83
104
65
92
104
124
91
96
37
63
57
144
84
135
51
101
110
191
104
154
68
118
150
206
117
Kokal & Stanislav(1989)
36
78
132
58
94
124
32
66
93
77
136
183
Spedding et al(1989)
Toshiba(1989)
141
56
154
154
160
174
77
60
87
148
93
174
80
63
96
132
102
155
93
123
122
229
138
260
Huq & Loth(1992)
Inoue et al(1993)
Czop et al(1994)
152
105
25
173
145
55
188
156
95
99
56
35
131
82
56
138
113
79
84
77
23
102
108
51
110
133
75
107
91
55
144
150
104
160
192
135
AbdulMajeed(1996)
Maier and Coddington(1997)
81
110
130
143
156
158
50
59
92
86
115
109
51
67
85
101
95
111
59
80
114
136
136
156
Petalaz & Aziz(1997)
3
8
18
1
4
9
4
7
11
11
23
33
Gomez et al(2000)
80
118
144
57
82
101
45
70
83
56
84
97
Zhao et al(2000)
63
80
92
33
43
51
35
44
51
27
42
53
Graham et al(2001)
160
169
173
85
108
125
74
99
120
101
139
167
1
correlation as reported by Leung (2005), 2,5 correlations as reported by Kataoka and Ishii (1987), 3
correlation as reported by Isbin and Biddle (1979) , 4 Rouhani and Axelsson(1970) correlations I and II.
Referring to the combined inclined data set of Mukherjee (1979) presented in Table
18, the correlation of Gregory & Scott (1969), Armand – Massina (Leung
2005),Rouhani I (1970) and Hughmark (1962) have performed well on the 15% index
and in aggregate across the whole range of inclination angles considered and are
worth a recommendation for this data set.
The correlations that fall into the next best category are that of Beggs (1972),
Chisholm (1973), Greskovich & Cooper (1975), Chisholm (1983) – Armand (1946),
Sun et al. (1980) and Minami & Brill (1987).
The correlations of Premoli et al. (1970), Toshiba (1989), Dix (1971) and Filimonov
et al. (1957) which will be shown to have an acceptable performance for the
combined
83
Table 14. Percentage of data points correctly predicted for Spedding & Nguyen
(1976) Inclined data
Correlation / Inclination Angle
2.75 Degree data
20.75 Degree data
45 Degree data
Total Points 226
Total Points 188
Total Points 196
Percentage within ±5%
38.9
Homogeneous
Armand(1946)
±10%
50.9
±15%
58.8
±5%
22.3
±10%
29.3
±15%
35.6
±5%
20.4
±10%
27.0
±15%
34.7
70 Degree data
Total Points 279
±5%
13.6
±10%
19.0
±15%
25.8
14.2
31.0
54.9
23.9
39.4
56.4
15.3
26.5
37.8
16.8
30.8
48.4
Armand - Massina
42.0
75.7
81.4
43.6
65.4
72.3
37.8
49.5
52.6
29.4
46.6
54.1
Lockhart & Martinelli(1949)
72.1
73.9
77.4
51.6
63.3
70.7
40.3
56.1
62.2
35.5
50.5
61.6
Sterman(1956)
Filimonov et al(1957)
39.4
45.6
50.4
58.4
56.6
67.3
23.4
37.8
29.3
53.7
33.5
71.3
21.4
48.5
26.0
70.9
36.2
86.7
14.0
37.6
20.4
63.1
28.7
86.0
Chisholm & Laird(1958)
Flanigan(1958)
37.6
36.7
49.1
56.6
54.4
67.7
22.3
29.8
28.2
48.9
32.4
56.9
20.4
26.5
25.5
42.3
33.2
46.9
13.3
24.4
17.9
35.1
24.0
39.1
Dimentiev et al
Hoogendroon(1959)
Bankoff(1960)
34.5
38.9
7.1
50.4
59.7
16.8
60.6
69.9
25.2
32.4
31.4
9.6
42.6
53.2
21.8
52.1
58.0
35.6
24.0
29.6
5.6
36.2
46.4
12.2
50.5
52.6
23.0
19.4
29.7
12.5
33.0
40.1
25.8
41.9
46.6
41.2
Fauske(1961)
Wilson et al(1961,1965)
42.5
8.8
49.6
14.6
54.9
19.0
26.1
8.5
31.9
14.9
34.6
24.5
25.0
9.2
32.1
14.8
35.2
22.4
18.3
6.5
26.5
15.4
29.7
19.4
1
2
Hughmark(1962)
64.6
77.0
83.2
52.1
71.3
79.3
45.9
51.5
55.6
41.9
51.3
61.6
Nicklin et al(1962)
16.4
35.4
58.4
30.9
50.5
66.0
16.3
34.2
48.0
28.0
48.7
66.7
Nishino & Yamazaki(1963)
65.5
75.2
79.6
55.3
66.5
68.6
48.5
61.2
68.4
38.0
56.3
64.2
Fujie(1964)
Kowalczewski(1964)
Thom(1964)
Zivi(1964)
44.7
61.9
43.4
65.0
58.4
72.1
66.4
69.5
69.5
74.8
74.3
72.1
27.1
45.2
31.9
39.4
45.7
56.9
47.9
46.8
56.4
57.4
58.0
50.5
20.4
42.3
29.1
32.7
29.1
50.0
46.4
41.3
41.8
54.6
56.1
45.9
14.7
33.0
21.9
27.2
23.3
45.5
37.6
35.1
36.6
51.3
49.1
41.6
Hughmark(1965)
13.7
27.9
48.2
26.1
38.8
53.7
11.7
25.0
33.2
16.5
31.9
45.5
Neal & Bankoff(1965)
8.8
13.7
19.9
9.0
17.6
26.1
10.2
19.4
28.1
12.9
21.9
30.1
Turner & Wallis(1965)
Baroczy(1966)
27.9
67.7
38.5
73.5
46.0
75.7
18.1
45.2
25.5
62.2
28.7
67.6
19.4
38.3
27.6
52.0
31.1
58.7
12.2
33.3
19.7
47.3
24.4
57.0
Guzhov et al(1967)
3
Kutucuglu
Smith(1969)
Wallis(1969)
Gregory & Scott(1969)
Premoli et al(1970)
1.3
9.7
22.1
0.5
10.6
23.9
0.5
3.6
11.7
0.0
2.5
11.5
61.5
63.3
72.1
12.8
71.2
75.7
80.5
32.3
73.9
81.9
83.6
64.6
36.7
46.3
49.5
25.0
48.4
67.0
63.3
39.9
52.1
71.8
69.1
60.1
36.7
39.8
38.3
16.8
45.4
52.0
52.0
27.0
49.5
54.1
58.2
42.9
26.5
34.8
37.3
17.6
35.1
47.0
49.8
29.4
41.9
55.9
60.6
50.2
67.7
77.0
83.6
53.2
72.3
77.7
51.5
61.7
63.8
47.3
63.1
69.5
Rouhani I(1970)
4
Rouhani II(1970)
42.0
81.4
87.2
51.6
72.3
81.4
36.2
54.6
61.2
37.3
60.6
69.5
20.8
40.7
58.4
37.8
56.4
75.0
22.4
38.8
53.6
22.9
37.3
54.8
Bonnecaze et al(1971)
Dix(1971)
16.8
66.4
35.4
74.3
58.4
77.4
30.9
60.6
50.5
78.7
66.0
85.1
16.3
57.1
34.2
70.9
48.0
78.6
28.0
59.1
48.7
81.7
66.7
87.1
4
Beggs(1972)
53.1
69.9
77.4
30.3
51.1
64.4
27.6
46.9
52.0
21.5
39.1
50.2
Chisholm(1973)
64.6
76.1
80.5
44.7
64.4
69.1
41.3
49.5
53.6
32.3
44.4
54.1
Loscher & Reinhardt(1973)
Mattar & Gregory(1974)
62.8
4.4
68.1
10.2
69.0
19.5
41.5
1.6
46.8
12.2
52.7
28.2
33.7
2.6
38.3
10.7
41.3
30.1
28.0
1.4
32.3
7.9
36.6
30.5
Moussali
Greskovich & Cooper(1975)
39.4
48.7
52.7
65.0
60.6
76.5
24.5
27.1
33.5
42.0
42.0
50.0
20.4
20.9
27.0
29.1
34.7
39.8
13.6
14.7
19.0
20.1
26.2
32.3
Madsen(1975)
Mukherjee(1979)
8.8
43.8
14.2
63.7
17.7
76.1
7.4
30.3
11.2
46.8
12.8
69.7
6.1
29.6
10.7
47.4
12.8
57.7
2.9
26.2
4.7
45.5
5.0
62.0
Gardner-1(1980)
Gardner-2(1980)
Sun et al(1980)
38.1
40.7
15.5
56.2
58.0
32.3
67.3
68.1
53.1
34.0
31.9
29.3
51.6
50.0
46.8
58.5
63.8
62.2
28.1
28.6
13.8
41.3
45.9
31.6
50.0
52.6
45.9
25.1
28.3
28.7
36.9
40.9
49.8
41.9
46.2
65.9
Jowitt(1981)
Chisholm(1983) - Armand(1946)
14.2
65.5
28.8
75.7
69.5
80.5
22.9
47.3
35.1
64.4
59.0
69.1
13.3
41.8
33.2
50.0
61.2
54.1
17.2
33.3
35.8
46.6
59.1
54.5
Spedding & Chen(1984)
68.6
75.2
80.1
50.0
64.9
69.1
44.9
56.1
63.8
37.3
52.0
62.4
5
84
Table14. Percentage of data points correctly predicted for Spedding & Nguyen (1976)
Inclined data (contd.)
Correlation / Inclination Angle
Bestion(1985)
2.75 Degree data
20.75 Degree data
45 Degree data
Total Points 226
Total Points 188
Total Points 196
Percentage within ±5%
30.5
±10%
53.5
±15%
63.7
±5%
34.0
±10%
47.3
±15%
52.1
±5%
17.3
±10%
33.7
±15%
42.9
70 Degree data
Total Points 279
±5%
16.5
±10%
29.0
±15%
34.8
Tandon et al(1985)
65.0
72.6
75.7
40.4
48.9
55.9
34.2
43.9
51.5
29.7
39.4
46.6
Chen(1986)
63.3
74.3
80.5
44.7
63.3
68.6
40.8
50.5
53.1
34.4
48.4
55.2
Hamersma & Hart(1987)
38.5
50.0
54.9
22.3
29.3
33.5
20.4
26.5
33.2
13.3
18.3
24.4
Kawaji et al(1987)
Minami & Brill(1987)
46.0
51.8
66.8
66.4
73.9
78.3
27.1
27.1
43.1
44.1
51.6
63.3
28.6
24.5
40.3
43.9
46.9
53.1
22.6
20.4
36.2
39.4
42.3
53.8
El-Boher et al(1988)
Hart et al(1989)
64.2
66.4
74.8
69.9
81.4
71.2
48.9
45.2
69.1
51.6
79.3
55.9
53.6
33.7
61.2
42.3
63.3
46.4
51.6
30.1
68.5
37.3
73.8
41.9
Kokal & Stanislav(1989)
Spedding et al(1989)
Toshiba(1989)
15.9
62.4
24.8
34.5
68.1
68.1
58.4
70.8
77.0
30.9
41.0
31.9
50.0
46.3
78.7
66.0
49.5
92.6
16.3
40.8
32.1
33.7
49.0
67.3
47.4
52.0
79.1
27.6
33.3
44.1
48.7
43.7
82.1
65.6
49.5
93.2
Huq & Loth(1992)
Inoue et al(1993)
67.3
46.5
76.5
64.2
83.2
69.0
52.7
29.8
69.7
43.6
73.4
60.1
42.9
39.3
52.0
55.1
56.1
67.9
38.4
32.6
51.6
53.8
57.3
68.8
Czop et al(1994)
11.1
24.3
42.0
18.6
29.8
42.0
11.7
26.0
38.3
19.7
37.3
48.4
AbdulMajeed(1996)
35.8
57.5
69.0
26.6
48.9
61.2
26.0
43.4
48.5
21.1
40.9
48.7
Maier and Coddington(1997)
48.7
63.3
69.9
31.4
45.7
58.0
34.2
51.5
56.6
28.7
48.7
55.9
Petalaz & Aziz(1997)
Gomez et al(2000)
Zhao et al(2000)
Graham et al(2001)
1.3
35.4
27.9
70.8
3.5
52.2
35.4
74.8
8.0
63.7
40.7
76.5
0.5
30.3
17.6
45.2
2.1
43.6
22.9
57.4
4.8
53.7
27.1
66.5
2.0
23.0
17.9
37.8
3.6
35.7
22.4
50.5
5.6
42.3
26.0
61.2
3.9
20.1
9.7
36.2
8.2
30.1
15.1
49.8
11.8
34.8
19.0
59.9
1
correlation as reported by Leung (2005), 2,5 correlations as reported by Kataoka and Ishii (1987), 3
correlation as reported by Isbin and Biddle (1979) , 4 Rouhani and Axelsson(1970) correlations I and II.
inclined database only gave a reasonable average prediction of 75% on the 15% error
index for this data set . Once again the scatter plot is used to show where the
weaknesses of the top performing correlations lie rather than presenting the
correlations that have performed well for this dataset only. These are given in Figures
21 to 27.
Having separately considered each data set for each inclination angle, we have
consolidated all the results from all the sources by error index so that we may infer
general conclusions on the performance of the correlations. The total number and
percentage of the total data points correctly predicted for the combined inclined
database for each error indices is reported in Table 19.
85
Table 15. Total Number and percentage of data points correctly predicted for the
inclined experimental data of Spedding & Nguyen (1976)
Correlation / Inclination Angle
Total data points correctly
Percentage of data points correctly
predicted for the whole database predicted for the whole database
Total Points 889
Total Points 889
±5%
208
154
±10%
276
282
±15%
340
439
±5%
23.4
17.3
±10%
31.0
31.7
±15%
38.2
49.4
333
438
214
374
204
259
521
537
277
548
264
401
574
602
342
696
316
461
37.5
49.3
24.1
42.1
22.9
29.1
58.6
60.4
31.2
61.6
29.7
45.1
64.6
67.7
38.5
78.3
35.5
51.9
240
288
80
245
72
451
205
453
233
400
276
361
149
93
169
406
5
357
438
175
309
133
552
378
572
340
495
436
424
274
163
244
517
56
451
500
284
341
187
618
536
622
447
527
524
464
402
233
287
572
150
27.0
32.4
9.0
27.6
8.1
50.7
23.1
51.0
26.2
45.0
31.0
40.6
16.8
10.5
19.0
45.7
0.6
40.2
49.3
19.7
34.8
15.0
62.1
42.5
64.3
38.2
55.7
49.0
47.7
30.8
18.3
27.4
58.2
6.3
50.7
56.2
31.9
38.4
21.0
69.5
60.3
70.0
50.3
59.3
58.9
52.2
45.2
26.2
32.3
64.3
16.9
354
405
435
158
486
439
530
542
283
607
479
582
602
483
654
39.8
45.6
48.9
17.8
54.7
49.4
59.6
61.0
31.8
68.3
53.9
65.5
67.7
54.3
73.6
367
596
664
41.3
67.0
74.7
Rouhani II(1970)
Bonnecaze et al(1971)
Dix(1971)
Beggs(1972)
226
206
541
291
378
378
683
455
531
536
732
538
25.4
23.2
60.9
32.7
42.5
42.5
76.8
51.2
59.7
60.3
82.3
60.5
Chisholm(1973)
401
514
568
45.1
57.8
63.9
Loscher & Reinhardt(1973)
Mattar & Gregory(1974)
364
22
407
89
438
241
40.9
2.5
45.8
10.0
49.3
27.1
Moussali
Greskovich & Cooper(1975)
Madsen(1975)
Mukherjee(1979)
213
243
54
287
288
339
87
452
357
435
103
589
24.0
27.3
6.1
32.3
32.4
38.1
9.8
50.8
40.2
48.9
11.6
66.3
Gardner-1(1980)
275
408
477
30.9
45.9
53.7
Percentage within
Homogeneous
Armand(1946)
1
Armand - Massina
Lockhart & Martinelli(1949)
Sterman(1956)
Filimonov et al(1957)
Chisholm & Laird(1958)
Flanigan(1958)
2
Dimentiev et al
Hoogendroon(1959)
Bankoff(1960)
Fauske(1961)
Wilson et al(1961,1965)
Hughmark(1962)
Nicklin et al(1962)
Nishino & Yamazaki(1963)
Fujie(1964)
Kowalczewski(1964)
Thom(1964)
Zivi(1964)
Hughmark(1965)
Neal & Bankoff(1965)
Turner & Wallis(1965)
Baroczy(1966)
Guzhov et al(1967)
3
Kutucuglu
Smith(1969)
Wallis(1969)
Gregory & Scott(1969)
Premoli et al(1970)
4
Rouhani I(1970)
4
5
86
Table 15. Total Number and percentage of data points correctly predicted for the
inclined experimental data of Spedding & Nguyen (1976) (contd.)
Correlation / Inclination Angle
Total data points correctly
Percentage of data points correctly
predicted for the whole database predicted for the whole database
Total Points 889
Percentage within
Gardner-2(1980)
Sun et al(1980)
Jowitt(1981)
Chisholm(1983) - Armand(1946)
Spedding & Chen(1984)
Bestion(1985)
Tandon et al(1985)
Chen(1986)
Hamersma & Hart(1987)
Kawaji et al(1987)
Minami & Brill(1987)
El-Boher et al(1988)
Hart et al(1989)
Kokal & Stanislav(1989)
Spedding et al(1989)
Toshiba(1989)
Huq & Loth(1992)
Inoue et al(1993)
Czop et al(1994)
AbdulMajeed(1996)
Maier and Coddington(1997)
Petalaz & Aziz(1997)
Gomez et al(2000)
Zhao et al(2000)
Graham et al(2001)
±5%
287
197
149
412
441
213
373
403
206
274
273
486
385
203
391
302
442
329
138
241
316
19
238
158
420
±10%
429
362
296
520
547
357
452
521
271
412
429
610
442
374
459
663
550
485
266
421
466
42
354
209
515
Total Points 889
±15%
506
511
553
570
610
423
507
569
320
474
550
663
474
532
493
763
596
594
384
502
534
71
425
247
585
±5%
32.3
22.2
16.8
46.3
49.6
24.0
42.0
45.3
23.2
30.8
30.7
54.7
43.3
22.8
44.0
34.0
49.7
37.0
15.5
27.1
35.5
2.1
26.8
17.8
47.2
±10%
48.3
40.7
33.3
58.5
61.5
40.2
50.8
58.6
30.5
46.3
48.3
68.6
49.7
42.1
51.6
74.6
61.9
54.6
29.9
47.4
52.4
4.7
39.8
23.5
57.9
±15%
56.9
57.5
62.2
64.1
68.6
47.6
57.0
64.0
36.0
53.3
61.9
74.6
53.3
59.8
55.5
85.8
67.0
66.8
43.2
56.5
60.1
8.0
47.8
27.8
65.8
1
correlation as reported by Leung (2005), 2,5 correlations as reported by Kataoka and Ishii (1987), 3
correlation as reported by Isbin and Biddle (1979) , 4 Rouhani and Axelsson(1970) correlations I and II.
For the range of inclination angles considered from all sources, none of the
correlations gave acceptable results within the 5% error index. The only correlation
worth consideration being that of Minami and Brill (1987) which predicted 67% of
the inclined data set of Beggs (1972).
The results of the wider error bands are a little bit encouraging where quite a number
of correlations were able to predict the data of Beggs (1972) and Mukherjee (1979).
The data of Spedding and Nguyen (1976) yielded only to the correlation of Toshiba
87
1.0
+15%
2.75 Degree
20.75 Degree
45 Degree
70 Degree
0.8
Calculated void fraction
-15%
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Measured void fraction
Figure 14. Comparison of Toshiba (1989) correlation with measured
inclined experimental data of Spedding & Nguyen (1976)
1.0
+15%
2.75 Degree
20.75 Degree
45 Degree
70 Degree
0.8
Calculated void fraction
-15%
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Measured void fraction
Figure 15. Comparison of Dix (1971) correlation with measured
inclined experimental data of Spedding & Nguyen (1976)
88
1.0
+15%
2.75 Degree
20.75 Degree
45 Degree
70 Degree
0.8
Calculated void fraction
-15%
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Measured void fraction
Figure 16. Comparison of Filimonov et al. (1957) correlation with
measured inclined experimental data of Spedding & Nguyen (1976)
1.0
+15%
0.8
Calculated void fraction
-15%
0.6
0.4
2.75 Degree
20.75 Degree
45 Degree
70 Degree
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Measured void fraction
Figure 17. Comparison of Rouhani I (1970) correlation with
measured inclined experimental data of Spedding & Nguyen (1976)
89
1.0
+15%
0.8
Calculated void fraction
-15%
0.6
0.4
2.75 Degree
20.75 Degree
45 Degree
70 Degree
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Measured void fraction
Figure 18. Comparison of Premoli et al. (1970) correlation with
measured inclined experimental data of Spedding & Nguyen (1976)
1.0
+15%
0.8
Calculated void fraction
-15%
0.6
0.4
2.75 Degree
20.75 Degree
45 Degree
70 Degree
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Measured void fraction
Figure 19. Comparison of Hughmark (1962) correlation with
measured inclined experimental data of Spedding & Nguyen (1976)
90
1.0
+15%
0.8
Calculated void fraction
-15%
0.6
0.4
2.75 Degree
20.75 Degree
45 Degree
70 Degree
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Measured void fraction
Figure 20. Comparison of Armand Massina (Leung 2005)
correlation with measured inclined experimental data of Spedding
& Nguyen (1976)
(1989) and Dix (1971) where the latter correlation falls short of the set cutoff
percentage by a slight margin.
It is our conclusion that the poor performance of the correlations within the narrow
error index (5%) may not be due to the weaknesses of the correlations alone but the
accuracy of the data sets which for most of the cases was not explicitly reported and
should also be questioned. This observation is not confined to the current data set
al.one but equally applies to all the data sets used in the horizontal, inclined and the
subsequent vertical comparisons.
As was in the horizontal comparison, a simple scatter plot of the measured versus
calculated void fraction for six correlations that fairly predicted the whole inclined set
is given in Figures 28 through 34. The correlations worth recommendation for this
91
data set are that of Toshiba(1989), Dix(1971), Rouhani I(1970), Filimonov et al.
(1957) ,Premoli et al. (1970) and Hughmark(1962) in order of decreasing accuracy.
The correlation of Toshiba (1989) and Dix (1971) were seen not only to give a very
good prediction but also the most consistent performance across the whole void
fraction range. The slight under prediction by Dix (1971) and Toshiba (1989) for void
fractions less than 0.8 is quite acceptable. Rouhani I (1970) has a moderate over
prediction for void fraction less than 0.6 while the correlations of Premoli et al. (1970)
and Hughmark (1962) have an excessive over prediction for void fractions less than
0.8. The range involved is this case is very wide making a general recommendation of
these two correlations a bit controversial despite their high percentage of data points
correctly predicted within the 15% error index especially for the correlation of
Premoli et al. (1970).
The fifth ranked Filimonov et al. (1957) correlation has a fairly acceptable scatter
across the whole range which is much better than the performance by Premoli et al.
(1970) which is ranked fourth in terms of the aggregate data points correctly predicted
for the inclined data set.
Splitting the consolidated results by mixture type, we have analyzed the performance
of the correlations for the air-water and air-kerosene mixtures separately. For the
wider 15% error percentage index, the correlation of Toshiba (1989) predicted 87.4%
of the combined data (1098 points) of Beggs (1972) and Spedding and Nguyen (1976)
while Dix (1971) and Filimonov et al. (1957) gave 84.1 and 81.5%, respectively.
92
93
10
7
17
4
21
22
23
21
21
16
11
Fauske(1961)
Wilson et al(1961,1965)
Hughmark(1962)
Nicklin et al(1962)
Nishino & Yamazaki(1963)
Fujie(1964)
Kowalczewski(1964)
Thom(1964)
Zivi(1964)
Hughmark(1965)
Neal & Bankoff(1965)
16
Guzhov et al(1967)
Kutucuglu
0
Baroczy(1966)
3
1
20
Turner & Wallis(1965)
28
0
27
9
24
14
30
34
36
40
21
27
13
15
39
18
2
19
8
6
0
42
33
20
46
18
16
23
22
6
31
6
7
Hoogendroon(1959)
Bankoff(1960)
1
34
0
37
12
44
16
35
36
42
47
38
39
17
21
53
25
12
29
54
43
29
47
27
23
±15%
37
52
18
0
20
6
12
10
23
25
27
30
11
19
10
10
21
6
3
11
20
23
10
37
10
12
±5%
25
15
29
0
28
12
28
16
30
35
36
50
25
32
19
25
45
27
10
26
45
40
26
58
24
26
±10%
42
30
40
1
37
16
51
22
32
39
40
61
50
40
22
32
60
41
20
37
65
47
35
64
30
37
±15%
50
60
Total Points 74
Total Points 57
±10%
32
31
20 Degree data
5 Degree data
±5%
16
16
Armand - Massina
Lockhart & Martinelli(1949)
Sterman(1956)
Filimonov et al(1957)
Chisholm & Laird(1958)
Flanigan(1958)
2
Dimentiev et al
Homogeneous
Armand(1946)
Percentage within
Correlation / Inclination Angle
14
0
14
3
16
7
14
17
17
30
6
15
8
11
23
5
3
12
28
17
8
45
4
15
±5%
16
20
22
0
20
7
32
9
17
28
31
51
28
24
10
27
46
29
5
25
47
31
21
58
13
32
±10%
37
34
25
0
29
8
65
11
22
32
36
68
58
32
12
39
66
46
11
34
68
49
31
60
20
41
±15%
50
67
Total Points 74
30 Degree data
13
0
13
1
12
4
12
17
15
18
11
15
11
10
19
5
4
11
17
16
6
31
5
3
±5%
10
13
23
0
22
8
27
5
21
29
30
32
28
26
16
22
39
19
10
22
43
32
12
44
10
18
±10%
27
36
32
4
31
13
46
10
29
33
34
45
53
32
20
27
54
31
15
33
51
42
22
47
19
30
±15%
37
50
Total Points 67
50 Degree data
30
0
17
2
15
12
21
27
28
16
9
14
11
10
19
4
5
16
27
28
9
42
6
4
±5%
21
16
40
0
30
6
36
24
32
38
41
52
26
33
17
16
51
11
10
37
59
56
20
54
19
12
±10%
37
40
45
1
40
14
61
25
38
41
43
66
53
45
20
25
62
32
19
46
70
67
30
55
29
25
±15%
52
69
Total Points 80
70 Degree data
Table 16. Number of data points correctly predicted for Mukherjee (1979) Inclined data
26
0
20
6
22
5
29
31
28
23
21
22
22
12
32
14
4
15
39
34
15
50
13
2
±5%
18
22
36
0
37
15
46
10
36
40
50
52
43
41
28
18
69
29
12
35
72
63
36
64
25
15
±10%
34
49
46
0
50
25
66
12
49
48
56
71
66
56
31
20
81
42
20
42
79
78
42
71
35
27
±15%
50
72
Total Points 92
80 Degree data
94
Percentage within
11
18
8
Gardner-1(1980)
Gardner-2(1980)
Sun et al(1980)
11
10
32
10
Bestion(1985)
Tandon et al(1985)
Chen(1986)
Hamersma & Hart(1987)
0
0
24
Madsen(1975)
Mukherjee(1979)
25
20
0
21
23
Mattar & Gregory(1974)
5
Moussali
Greskovich & Cooper(1975)
Chisholm(1983) - Armand(1946)
Spedding & Chen(1984)
23
30
Chisholm(1973)
Loscher & Reinhardt(1973)
Jowitt(1981)
32
Beggs(1972)
13
5
4
26
4
16
31
19
47
23
22
43
24
0
26
29
20
0
42
0
34
43
43
41
47
17
21
38
35
33
42
27
50
32
27
52
36
7
48
35
24
0
51
0
41
51
52
42
54
29
38
44
52
55
47
±15%
48
43
14
35
12
13
24
21
0
13
23
16
0
35
3
31
40
26
36
34
10
11
35
19
13
42
±5%
29
24
25
54
24
30
45
33
4
30
39
31
0
51
4
47
56
46
48
51
29
25
50
40
30
51
±10%
44
36
30
56
34
42
60
44
14
58
44
41
0
59
8
57
64
60
53
59
45
50
63
61
67
60
±15%
54
48
Total Points 74
Total Points 57
±10%
42
29
20 Degree data
5 Degree data
±5%
24
22
Rouhani II(1970)
Bonnecaze et al(1971)
Dix(1971)
Rouhani I(1970)
4
Gregory & Scott(1969)
Premoli et al(1970)
Smith(1969)
Wallis(1969)
Correlation / Inclination Angle
4
40
8
22
30
14
0
14
21
15
0
38
0
22
29
32
41
33
23
6
34
20
21
43
±5%
25
18
13
51
17
37
46
22
0
39
30
29
0
54
1
46
53
48
53
53
42
28
56
41
38
56
±10%
49
31
21
55
25
45
61
31
9
63
44
38
0
62
3
55
67
64
56
62
58
58
63
66
66
61
±15%
57
52
Total Points 74
30 Degree data
7
29
9
17
18
15
4
16
22
11
0
32
2
14
30
16
30
26
19
11
21
16
13
20
±5%
17
16
11
41
20
29
36
22
6
42
31
27
0
40
4
33
49
37
38
47
37
28
44
38
36
42
±10%
36
31
19
47
34
36
51
32
6
57
37
33
2
49
7
40
57
51
51
52
49
53
51
56
51
54
±15%
49
43
Total Points 67
50 Degree data
9
43
14
15
28
17
0
16
30
7
0
41
0
18
27
31
44
46
11
9
30
19
16
42
±5%
35
25
21
49
24
32
61
26
0
41
41
19
0
57
0
39
53
60
57
59
25
26
52
49
41
57
±10%
58
53
30
53
42
40
70
43
1
64
48
32
0
63
0
60
63
69
60
68
53
53
56
61
70
62
±15%
68
64
Total Points 80
70 Degree data
Table 16. Number of data points correctly predicted for Mukherjee (1979) Inclined data (contd.)
14
44
19
21
34
21
0
19
32
10
0
46
0
19
27
41
39
54
7
21
44
34
26
43
±5%
40
25
27
56
35
42
70
39
2
55
49
16
0
61
4
37
53
71
55
68
36
43
63
73
52
58
±10%
68
52
36
62
50
58
80
54
11
78
56
34
0
76
10
61
72
78
77
74
56
66
72
83
80
72
±15%
79
70
Total Points 92
80 Degree data
95
20
23
23
23
Spedding et al(1989)
Toshiba(1989)
Huq & Loth(1992)
Inoue et al(1993)
37
28
45
40
21
36
27
43
37
47
47
38
41
31
31
20
26
27
11
30
20
±5%
13
41
46
31
50
44
25
53
28
±10%
25
53
54
36
64
53
50
59
34
±15%
36
62
36
14
37
23
7
32
13
±5%
10
48
54
22
58
43
28
50
17
±10%
15
57
57
29
61
53
59
57
27
±15%
19
62
Total Points 74
30 Degree data
22
13
26
20
11
14
12
±5%
9
27
38
22
43
33
28
28
22
±10%
19
46
47
31
50
41
53
43
29
±15%
28
51
Total Points 67
50 Degree data
29
16
31
30
9
18
16
±5%
10
45
42
25
52
54
26
39
30
±10%
19
54
50
43
55
63
53
58
39
±15%
38
62
Total Points 80
70 Degree data
34
21
38
33
21
17
31
±5%
14
43
61
45
70
53
43
54
38
±10%
34
67
46
49
57
10
6
3
70
69
51
76
80
66
66
47
±15%
46
75
Total Points 92
80 Degree data
7
13
30
7
11
26
0
9
30
7
16
27
6
11
38
5
14
21
33
36
32
43
46
26
36
41
19
25
37
28
40
46
26
40
28
35
39
31
40
47
31
35
38
15
24
30
30
37
43
37
51
2
5
8
2
5
7
2
6
10
2
4
7
7
9
11
4
8
1
1
1
0
1
2
2
2
4
0
2
5
0
0
1
2
4
0
1
4
2
3
6
0
1
3
0
0
0
0
0
0
0
1
19
31
36
24
39
49
22
35
50
19
34
42
18
43
51
37
56
1
correlation as reported by Leung (2005), 2,5 correlations as reported by Kataoka and Ishii (1987), 3 correlation as reported by Isbin and Biddle (1979) , 4 Rouhani and Axelsson(1970)
correlations I and II.
4
Kokal & Stanislav(1989)
Czop et al(1994)
AbdulMajeed(1996)
Maier and Coddington(1997)
Petalaz & Aziz(1997)
Gomez et al(2000)
Zhao et al(2000)
Graham et al(2001)
21
18
El-Boher et al(1988)
Hart et al(1989)
±15%
28
52
Total Points 74
Total Points 57
±10%
21
44
20 Degree data
5 Degree data
Percentage within ±5%
Kawaji et al(1987)
10
Minami & Brill(1987)
33
Correlation / Inclination Angle
Table 16. Number of data points correctly predicted for Mukherjee (1979) Inclined data (contd.)
96
1
29.8
7.0
36.8
38.6
40.4
36.8
36.8
28.1
19.3
1.8
35.1
0.0
Hughmark(1962)
Nicklin et al(1962)
Nishino & Yamazaki(1963)
Fujie(1964)
Kowalczewski(1964)
Thom(1964)
Zivi(1964)
Hughmark(1965)
Neal & Bankoff(1965)
Turner & Wallis(1965)
Baroczy(1966)
Guzhov et al(1967)
28.1
14.0
10.5
0.0
17.5
12.3
Dimentiev et al
Hoogendroon(1959)
Bankoff(1960)
Fauske(1961)
Wilson et al(1961,1965)
3
10.5
12.3
Chisholm & Laird(1958)
Flanigan(1958)
Kutucuglu
10.5
54.4
Sterman(1956)
Filimonov et al(1957)
2
38.6
40.4
28.1
49.1
42.1
24.6
15.8
47.4
0.0
47.4
70.2
63.2
59.6
52.6
68.4
36.8
33.3
31.6
3.5
22.8
26.3
31.6
28.1
35.1
80.7
57.9
73.7
54.4
±10%
56.1
59.6
77.2
28.1
21.1
64.9
0.0
68.4
82.5
73.7
63.2
61.4
93.0
66.7
50.9
43.9
21.1
29.8
36.8
47.4
40.4
50.9
82.5
75.4
94.7
91.2
±15%
64.9
24.3
16.2
13.5
8.1
27.0
0.0
25.7
40.5
36.5
33.8
31.1
28.4
14.9
14.9
8.1
4.1
13.5
13.5
13.5
16.2
13.5
50.0
31.1
27.0
20.3
±5%
33.8
39.2
37.8
21.6
16.2
37.8
0.0
43.2
67.6
48.6
47.3
40.5
60.8
33.8
35.1
36.5
13.5
25.7
33.8
32.4
35.1
35.1
78.4
54.1
60.8
40.5
±10%
56.8
54.1
68.9
29.7
21.6
50.0
1.4
54.1
82.4
54.1
52.7
43.2
81.1
67.6
50.0
55.4
27.0
29.7
43.2
40.5
50.0
47.3
86.5
63.5
87.8
81.1
±15%
67.6
Total Points 74
Total Points 57
±5%
28.1
20 Degree data
5 Degree data
Lockhart & Martinelli(1949)
Armand - Massina
Armand(1946)
Homogeneous
Percentage within
Correlation / Inclination Angle
18.9
21.6
9.5
4.1
18.9
0.0
20.3
40.5
23.0
23.0
18.9
31.1
8.1
16.2
6.8
4.1
10.8
14.9
5.4
20.3
10.8
60.8
23.0
37.8
27.0
±5%
21.6
29.7
43.2
12.2
9.5
27.0
0.0
32.4
68.9
41.9
37.8
23.0
62.2
37.8
33.8
39.2
6.8
13.5
36.5
17.6
43.2
28.4
78.4
41.9
63.5
45.9
±10%
50.0
33.8
87.8
14.9
10.8
39.2
0.0
43.2
91.9
48.6
43.2
29.7
89.2
78.4
45.9
62.2
14.9
16.2
52.7
27.0
55.4
41.9
81.1
66.2
91.9
90.5
±15%
67.6
Total Points 74
30 Degree data
19.4
17.9
6.0
1.5
19.4
0.0
22.4
26.9
22.4
25.4
17.9
28.4
16.4
16.4
7.5
6.0
16.4
14.9
7.5
4.5
9.0
46.3
23.9
25.4
19.4
±5%
14.9
34.3
40.3
7.5
11.9
32.8
0.0
38.8
47.8
44.8
43.3
31.3
58.2
41.8
32.8
28.4
14.9
23.9
32.8
14.9
26.9
17.9
65.7
47.8
64.2
53.7
±10%
40.3
47.8
68.7
14.9
19.4
46.3
6.0
47.8
67.2
50.7
49.3
43.3
80.6
79.1
49.3
46.3
22.4
29.9
40.3
28.4
44.8
32.8
70.1
62.7
76.1
74.6
±15%
55.2
Total Points 67
50 Degree data
37.5
18.8
15.0
2.5
21.3
0.0
17.5
20.0
35.0
33.8
26.3
23.8
11.3
20.0
5.0
6.3
13.8
12.5
7.5
5.0
11.3
52.5
35.0
33.8
20.0
±5%
26.3
50.0
45.0
30.0
7.5
37.5
0.0
41.3
65.0
51.3
47.5
40.0
63.8
32.5
46.3
13.8
12.5
21.3
20.0
23.8
15.0
25.0
67.5
70.0
73.8
50.0
±10%
46.3
56.3
76.3
31.3
17.5
50.0
1.3
56.3
82.5
53.8
51.3
47.5
77.5
66.3
57.5
40.0
23.8
25.0
31.3
36.3
31.3
37.5
68.8
83.8
87.5
86.3
±15%
65.0
Total Points 80
70 Degree data
Table 17. Percentage of data points correctly predicted for Mukherjee (1979) Inclined data
28.3
23.9
5.4
6.5
21.7
0.0
23.9
25.0
30.4
33.7
31.5
34.8
22.8
16.3
15.2
4.3
23.9
13.0
14.1
2.2
16.3
54.3
37.0
42.4
23.9
±5%
19.6
39.1
50.0
10.9
16.3
40.2
0.0
44.6
56.5
54.3
43.5
39.1
75.0
46.7
38.0
31.5
13.0
30.4
19.6
27.2
16.3
39.1
69.6
68.5
78.3
53.3
±10%
37.0
50.0
71.7
13.0
27.2
54.3
0.0
60.9
77.2
60.9
52.2
53.3
88.0
71.7
45.7
45.7
21.7
33.7
21.7
38.0
29.3
45.7
77.2
84.8
85.9
78.3
±15%
54.3
Total Points 92
80 Degree data
97
28.1
54.4
22.8
8.8
7.0
45.6
56.1
40.4
52.6
0.0
36.8
40.4
0.0
42.1
19.3
31.6
14.0
0.0
43.9
35.1
19.3
17.5
56.1
17.5
Gregory & Scott(1969)
Premoli et al(1970)
Rouhani I(1970)4
Rouhani II(1970)4
Bonnecaze et al(1971)
Dix(1971)
Beggs(1972)
Chisholm(1973)
Loscher & Reinhardt(1973)
Mattar & Gregory(1974)
Moussali5
Greskovich & Cooper(1975)
Madsen(1975)
Mukherjee(1979)
Gardner-1(1980)
Gardner-2(1980)
Sun et al(1980)
Jowitt(1981)
Chisholm(1983) - Armand(1946)
Spedding & Chen(1984)
Bestion(1985)
Tandon et al(1985)
Chen(1986)
Hamersma & Hart(1987)
38.6
33.3
75.4
42.1
38.6
40.4
82.5
73.7
35.1
50.9
45.6
0.0
75.4
0.0
82.5
75.4
71.9
0.0
59.6
36.8
66.7
61.4
29.8
73.7
57.9
50.9
±10%
73.7
47.4
91.2
63.2
47.4
56.1
87.7
89.5
42.1
61.4
84.2
12.3
89.5
0.0
94.7
91.2
73.7
0.0
71.9
66.7
77.2
91.2
50.9
82.5
96.5
75.4
±15%
84.2
18.9
32.4
28.4
17.6
16.2
47.3
47.3
21.6
31.1
17.6
0.0
54.1
0.0
45.9
35.1
48.6
4.1
41.9
14.9
47.3
25.7
13.5
56.8
17.6
32.4
±5%
39.2
33.8
60.8
44.6
40.5
32.4
73.0
68.9
41.9
52.7
40.5
5.4
75.7
0.0
68.9
62.2
64.9
5.4
63.5
33.8
67.6
54.1
39.2
68.9
40.5
48.6
±10%
59.5
40.5
81.1
59.5
56.8
45.9
75.7
79.7
55.4
59.5
78.4
18.9
86.5
0.0
79.7
81.1
71.6
10.8
77.0
67.6
85.1
82.4
60.8
81.1
90.5
64.9
±15%
73.0
Total Points 74
Total Points 57
±5%
42.1
20 Degree data
5 Degree data
Wallis(1969)
Smith(1969)
Percentage within
Correlation / Inclination Angle
5.4
40.5
18.9
29.7
10.8
54.1
51.4
20.3
28.4
18.9
0.0
39.2
0.0
44.6
43.2
55.4
0.0
29.7
8.1
45.9
27.0
31.1
58.1
28.4
24.3
±5%
33.8
17.6
62.2
29.7
50.0
23.0
68.9
73.0
39.2
40.5
52.7
0.0
71.6
0.0
71.6
64.9
71.6
1.4
62.2
37.8
75.7
55.4
56.8
75.7
51.4
41.9
±10%
66.2
28.4
82.4
41.9
60.8
33.8
74.3
83.8
51.4
59.5
85.1
12.2
90.5
0.0
83.8
86.5
75.7
4.1
74.3
78.4
85.1
89.2
78.4
82.4
89.2
70.3
±15%
77.0
Total Points 74
30 Degree data
10.4
26.9
22.4
25.4
13.4
43.3
47.8
16.4
32.8
23.9
6.0
44.8
0.0
38.8
23.9
44.8
3.0
20.9
16.4
31.3
23.9
28.4
29.9
19.4
23.9
±5%
25.4
16.4
53.7
32.8
43.3
29.9
61.2
59.7
40.3
46.3
62.7
9.0
73.1
0.0
70.1
55.2
56.7
6.0
49.3
41.8
65.7
56.7
55.2
62.7
53.7
46.3
±10%
53.7
28.4
76.1
47.8
53.7
50.7
70.1
73.1
49.3
55.2
85.1
9.0
85.1
3.0
77.6
76.1
76.1
10.4
59.7
79.1
76.1
83.6
73.1
80.6
76.1
64.2
±15%
73.1
Total Points 67
50 Degree data
11.3
35.0
21.3
18.8
17.5
53.8
51.3
8.8
37.5
20.0
0.0
33.8
0.0
57.5
38.8
55.0
0.0
22.5
11.3
37.5
23.8
13.8
52.5
20.0
31.3
±5%
43.8
26.3
76.3
32.5
40.0
30.0
61.3
71.3
23.8
51.3
51.3
0.0
66.3
0.0
73.8
75.0
71.3
0.0
48.8
32.5
65.0
61.3
31.3
71.3
51.3
66.3
±10%
72.5
37.5
87.5
53.8
50.0
52.5
66.3
78.8
40.0
60.0
80.0
1.3
78.8
0.0
85.0
86.3
75.0
0.0
75.0
66.3
70.0
76.3
66.3
77.5
87.5
80.0
±15%
85.0
Total Points 80
70 Degree data
Table 17. Percentage of data points correctly predicted for Mukherjee (1979) Inclined data (contd.)
15.2
37.0
22.8
22.8
20.7
47.8
50.0
10.9
34.8
20.7
0.0
29.3
0.0
58.7
44.6
42.4
0.0
20.7
22.8
47.8
37.0
7.6
46.7
28.3
27.2
±5%
43.5
29.3
76.1
42.4
45.7
38.0
60.9
66.3
17.4
53.3
59.8
2.2
57.6
0.0
73.9
77.2
59.8
4.3
40.2
46.7
68.5
79.3
39.1
63.0
56.5
56.5
±10%
73.9
39.1
87.0
58.7
63.0
54.3
67.4
82.6
37.0
60.9
84.8
12.0
78.3
0.0
80.4
84.8
83.7
10.9
66.3
71.7
78.3
90.2
60.9
78.3
87.0
76.1
±15%
85.9
Total Points 92
80 Degree data
98
1.8
0.0
Inoue et al(1993)
Czop et al(1994)
AbdulMajeed(1996)
Maier and Coddington(1997)
Petalaz & Aziz(1997)
Gomez et al(2000)
Zhao et al(2000)
1
1.8
1.8
64.9
22.8
57.9
61.4
8.8
78.9
70.2
1.8
7.0
75.4
52.6
63.2
68.4
14.0
82.5
82.5
66.7
64.9
40.5
0.0
2.7
41.9
9.5
43.2
41.9
2.7
35.1
36.5
14.9
27.0
27.0
71.6
1.4
4.1
62.2
14.9
58.1
54.1
6.8
67.6
59.5
33.8
41.9
37.8
2.7
8.1
73.0
35.1
62.2
63.5
9.5
86.5
71.6
67.6
48.6
45.9
79.7
83.8
2.7
0.0
48.6
0.0
35.1
41.9
2.7
50.0
31.1
9.5
18.9
17.6
43.2
64.9
±5%
13.5
2.7
1.4
73.0
12.2
48.6
47.3
8.1
78.4
58.1
37.8
29.7
23.0
67.6
77.0
±10%
20.3
5.4
4.1
77.0
40.5
55.4
51.4
13.5
82.4
71.6
79.7
39.2
36.5
77.0
83.8
±15%
25.7
0.0
0.0
32.8
10.4
28.4
22.4
3.0
38.8
29.9
16.4
19.4
17.9
20.9
40.3
±5%
13.4
3.0
0.0
56.7
23.9
37.3
35.8
6.0
64.2
49.3
41.8
32.8
32.8
41.8
68.7
±10%
28.4
7.5
0.0
70.1
40.3
55.2
44.8
10.4
74.6
61.2
79.1
46.3
43.3
64.2
76.1
±15%
41.8
0.0
0.0
36.3
7.5
35.0
37.5
8.8
38.8
37.5
11.3
20.0
20.0
22.5
56.3
±5%
12.5
0.0
0.0
52.5
13.8
50.0
46.3
11.3
65.0
67.5
32.5
31.3
37.5
48.8
67.5
±10%
23.8
1.3
0.0
62.5
47.5
57.5
53.8
13.8
68.8
78.8
66.3
53.8
48.8
72.5
77.5
±15%
47.5
2.2
0.0
37.0
5.4
28.3
40.2
4.3
41.3
35.9
22.8
22.8
33.7
18.5
46.7
±5%
15.2
66.3
15.2
43.5
55.4
8.7
76.1
57.6
46.7
48.9
41.3
58.7
72.8
±10%
37.0
75.0
50.0
53.3
62.0
10.9
82.6
87.0
71.7
55.4
51.1
71.7
81.5
±15%
50.0
Total Points 92
80 Degree data
76.1
40.4
12.3
36.8
49.1
3.5
Toshiba(1989)
Huq & Loth(1992)
36.8
49.1
71.9
54.4
71.6
±15%
48.6
Total Points 80
70 Degree data
Graham et al(2001)
33.3
54.4
63.2
32.4
52.7
66.2
29.7
47.3
67.6
28.4
50.7
62.7
22.5
53.8
63.8
40.2
60.9
correlation as reported by Leung (2005), 2,5 correlations as reported by Kataoka and Ishii (1987), 3 correlation as reported by Isbin and Biddle (1979) , 4 Rouhani and Axelsson(1970)
correlations I and II.
40.4
40.4
Kokal & Stanislav(1989)
Spedding et al(1989)
63.2
47.4
55.4
±10%
33.8
Total Points 67
50 Degree data
6.5
3.3
7.0
35.1
Hart et al(1989)
91.2
±5%
17.6
Total Points 74
30 Degree data
4.3
1.1
36.8
31.6
El-Boher et al(1988)
77.2
57.9
Minami & Brill(1987)
±15%
49.1
Total Points 74
Total Points 57
±10%
36.8
20 Degree data
5 Degree data
Percentage within ±5%
Kawaji et al(1987)
17.5
Correlation / Inclination Angle
Table 17. Percentage of data points correctly predicted for Mukherjee (1979) Inclined data (contd.)
Table 18. Total Number and percentage of data points correctly predicted for the
inclined experimental data of Mukherjee (1979)
Correlation / Inclination Angle
Total data points correctly
Percentage of data points correctly
predicted for the whole database predicted for the whole database
Total Points 444
Percentage within
Homogeneous
Armand(1946)
1
Armand - Massina
Lockhart & Martinelli(1949)
Sterman(1956)
Filimonov et al(1957)
Chisholm & Laird(1958)
Flanigan(1958)
2
Dimentiev et al
Hoogendroon(1959)
Bankoff(1960)
Fauske(1961)
Wilson et al(1961,1965)
Hughmark(1962)
Nicklin et al(1962)
Nishino & Yamazaki(1963)
Fujie(1964)
Kowalczewski(1964)
Thom(1964)
Zivi(1964)
Hughmark(1965)
Neal & Bankoff(1965)
Turner & Wallis(1965)
Baroczy(1966)
Guzhov et al(1967)
3
Kutucuglu
Smith(1969)
Wallis(1969)
Gregory & Scott(1969)
Premoli et al(1970)
4
Rouhani I(1970)
4
Rouhani II(1970)
Bonnecaze et al(1971)
Dix(1971)
Beggs(1972)
Total Points 444
±5%
106
102
±10%
209
220
±15%
276
370
±5%
23.9
23.0
±10%
47.1
49.5
±15%
62.2
83.3
154
140
54
236
44
43
308
255
135
324
109
119
387
326
189
344
160
183
34.7
31.5
12.2
53.2
9.9
9.7
69.4
57.4
30.4
73.0
24.5
26.8
87.2
73.4
42.6
77.5
36.0
41.2
73
40
19
72
60
131
62
106
139
138
138
120
93
49
19
104
0
164
133
49
103
123
289
171
183
277
224
204
166
193
78
57
164
0
221
217
97
122
164
376
318
244
358
251
229
205
333
96
88
224
6
16.4
9.0
4.3
16.2
13.5
29.5
14.0
23.9
31.3
31.1
31.1
27.0
20.9
11.0
4.3
23.4
0.0
36.9
30.0
11.0
23.2
27.7
65.1
38.5
41.2
62.4
50.5
45.9
37.4
43.5
17.6
12.8
36.9
0.0
49.8
48.9
21.8
27.5
36.9
84.7
71.6
55.0
80.6
56.5
51.6
46.2
75.0
21.6
19.8
50.5
1.4
117
170
130
105
221
178
297
232
230
306
222
355
320
389
356
26.4
38.3
29.3
23.6
49.8
40.1
66.9
52.3
51.8
68.9
50.0
80.0
72.1
87.6
80.2
121
276
379
27.3
62.2
85.4
75
62
190
225
186
171
303
325
290
318
349
369
16.9
14.0
42.8
50.7
41.9
38.5
68.2
73.2
65.3
71.6
78.6
83.1
Chisholm(1973)
169
305
374
38.1
68.7
84.2
Loscher & Reinhardt(1973)
Mattar & Gregory(1974)
220
5
292
13
339
28
49.5
1.1
65.8
2.9
76.4
6.3
125
176
0
216
236
307
0
305
314
374
2
360
28.2
39.6
0.0
48.6
53.2
69.1
0.0
68.7
70.7
84.2
0.5
81.1
5
Moussali
Greskovich & Cooper(1975)
Madsen(1975)
Mukherjee(1979)
99
Table 18. Total Number and percentage of data points correctly predicted for the
inclined experimental data of Mukherjee (1979)(contd.)
Correlation / Inclination Angle
Total data points correctly
Percentage of data points correctly
predicted for the whole database predicted for the whole database
Total Points 444
Total Points 444
Gardner-1(1980)
Gardner-2(1980)
±5%
70
146
±10%
142
219
±15%
202
264
±5%
15.8
32.9
±10%
32.0
49.3
±15%
45.5
59.5
Sun et al(1980)
Jowitt(1981)
Chisholm(1983) - Armand(1946)
Spedding & Chen(1984)
Bestion(1985)
Tandon et al(1985)
86
4
159
108
99
72
233
12
301
166
192
143
368
48
374
240
248
217
19.4
0.9
35.8
24.3
22.3
16.2
52.5
2.7
67.8
37.4
43.2
32.2
82.9
10.8
84.2
54.1
55.9
48.9
Chen(1986)
Hamersma & Hart(1987)
Kawaji et al(1987)
Minami & Brill(1987)
El-Boher et al(1988)
Hart et al(1989)
Kokal & Stanislav(1989)
Spedding et al(1989)
Toshiba(1989)
Huq & Loth(1992)
Inoue et al(1993)
Czop et al(1994)
AbdulMajeed(1996)
Maier and Coddington(1997)
Petalaz & Aziz(1997)
Gomez et al(2000)
Zhao et al(2000)
223
58
66
237
132
110
63
104
181
156
175
32
152
172
19
5
2
298
116
133
321
260
162
171
173
318
267
278
74
217
222
37
10
6
323
163
195
364
324
207
319
227
353
337
320
197
255
254
53
19
16
50.2
13.1
14.9
53.4
29.7
24.8
14.2
23.4
40.8
35.1
39.4
7.2
34.2
38.7
4.3
1.1
0.5
67.1
26.1
30.0
72.3
58.6
36.5
38.5
39.0
71.6
60.1
62.6
16.7
48.9
50.0
8.3
2.3
1.4
72.7
36.7
43.9
82.0
73.0
46.6
71.8
51.1
79.5
75.9
72.1
44.4
57.4
57.2
11.9
4.3
3.6
Graham et al(2001)
139
238
298
31.3
53.6
67.1
Percentage within
1
correlation as reported by Leung (2005), 2,5 correlations as reported by Kataoka and Ishii (1987), 3
correlation as reported by Isbin and Biddle (1979) , 4 Rouhani and Axelsson(1970) correlations I and II.
On the other hand, for the total inclined air-kerosene data of Mukherjee(1979), the
best prediction was given by Gregory and Scott(1969), Armand Massina (Leung
2005), Rouhani I(1970) and Hughmark(1962) predicting 87.6,87.2,85.4 and 84.7% of
the data within 15% error margin respectively. The excellent prediction by the simple
drift flux correlation of Gregory and Scott (1969) that was developed for horizontal
slug flow could only be explained by the fact that most of the data in the set
considered is dominated by slug flow regime. However, the simple deduction that the
100
1.0
+15%
0.8
Calculated void fraction
-15%
0.6
0.4
5 Degree
20 Degree
30 Degree
50 Degree
70 Degree
80 Degree
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Measured void fraction
Figure 21. Comparison of Armand Massina (Leung 2005)
correlation with measured inclined experimental data of Mukherjee
(1979)
1.0
+15%
0.8
Calculated void fraction
-15%
0.6
0.4
5 Degree
20 Degree
30 Degree
50 Degree
70 Degree
80 Degree
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Measured void fraction
Figure 22. Comparison of Rouhani I (1970) correlation with
measured inclined experimental data of Mukherjee (1979)
101
1.0
+15%
0.8
Calculated void fraction
-15%
0.6
0.4
5 Degree
20 Degree
30 Degree
50 Degree
70 Degree
80 Degree
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Measured void fraction
Figure 23. Comparison of Hughmark (1962) correlation with measured
inclined experimental data of Mukherjee (1979)
1.0
+15%
0.8
Calculated void fraction
-15%
0.6
0.4
5 Degree
20 Degree
30 Degree
50 Degree
70 Degree
80 Degree
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Measured void fraction
Figure 24. Comparison of Premoli et al. (1970) correlation with
measured inclined experimental data of Mukherjee (1979)
102
1.0
+15%
0.8
Calculated void fraction
-15%
0.6
0.4
5 Degree
20 Degree
30 Degree
50 Degree
70 Degree
80 Degree
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Measured void fraction
Figure 25. Comparison of Toshiba (1989) correlation with measured
inclined experimental data of Mukherjee (1979)
1.0
+15%
0.8
Calculated void fraction
-15%
0.6
0.4
5 Degree
20 Degree
30 Degree
50 Degree
70 Degree
80 Degree
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Measured void fraction
Figure 26. Comparison of Dix (1971) correlation with measured
inclined experimental data of Mukherjee (1979)
103
1.0
+15%
0.8
Calculated void fraction
-15%
0.6
0.4
5 Degree
20 Degree
30 Degree
50 Degree
70 Degree
80 Degree
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Measured void fraction
Figure 27. Comparison of Filimonov et al. (1957) correlation with
measured inclined experimental data of Mukherjee (1979)
drift velocity between the phases is zero which could be taken from the drift flux
correlation expression would be quite erroneous. We have presented the horizontal
and inclined comparisons so far and what remains to be discussed is the vertical flow
comparison which will follow next.
4.3 Vertical flow comparison
The presentation of the comparison in this section would be very similar to that of the
horizontal comparison. A total of 403 data points are included for these comparisons
from four sources namely, Beggs (1972), Spedding and Nguyen (1976), Mukherjee
(1979) and Sujumnong (1997).
Over half of the data points in this data set are from Spedding and Nguyen (1976)
which would influence most of the final conclusions and recommendations.
104
Most of the general observations and comments on the horizontal and inclined
comparison would also hold for this section.
The total number of data points correctly predicted within each error index for each
data set is given in Table 20. The respective percentage figures are presented in Table
21.
The correlation by Beggs (1972) and Minami and Brill (1987) were the only ones that
made an acceptably close prediction within the 5% index for the data of Beggs
(1972). None of the correlations were able to predict a sufficient percentage of the
data points in any of the data sets within this range. The performance shows
improvement for the wider error bands for all the data sets except the data of
Spedding and Nguyen (1976).
This has been the case for all the previous comparisons. The only new correlation that
came into the picture is that of El-Boher et al. (1988) which gave an 83.5% prediction
of the Spedding and Nguyen (1976) data within the 15% error index even though it
was not able to satisfactorily predict the data of Sujumnong (1997).
Considering their general performance across the three other data sets, the correlation
by Armand – Massina (Leung 2005), Filimonov et al. (1957), Hughmark (1962),
Smith (1969), Gregory & Scott (1969) ,Premoli et al. (1970), Rouhani I(1970),
Chisholm (1973), Chisholm (1983) Armand (1946),Toshiba (1989) and Huq and Loth
(1992) have acceptable prediction and are worth consideration.
Consolidating our observation of the separate data sets into one, we have presented
the predictive capability of the correlations for the whole vertical data base in Table
22. The drift flux correlations by Toshiba (1989), Rouhani I (1970) and Filimonov et
al. (1957) once again have shown an excellent performance in predicting the vertical
data base within the 15% index. Six other correlations have given a fairly good result
105
Table 19. Total Number and percentage of data points correctly predicted for the
whole inclined data base
Correlation / Data Source
Total data points correctly
Percentage of data points correctly
predicted for the whole database predicted for the whole database
Total Points 1542
Percentage within
Homogeneous
Armand(1946)
1
Armand - Massina
Lockhart & Martinelli(1949)
Sterman(1956)
Filimonov et al(1957)
Chisholm & Laird(1958)
Flanigan(1958)
2
Dimentiev et al
Hoogendroon(1959)
Bankoff(1960)
Fauske(1961)
Wilson et al(1961,1965)
Hughmark(1962)
Nicklin et al(1962)
Nishino & Yamazaki(1963)
Fujie(1964)
Kowalczewski(1964)
Thom(1964)
Zivi(1964)
Hughmark(1965)
Neal & Bankoff(1965)
Turner & Wallis(1965)
Baroczy(1966)
Guzhov et al(1967)
3
Kutucuglu
Smith(1969)
Wallis(1969)
Gregory & Scott(1969)
Premoli et al(1970)
4
Rouhani I(1970)
4
Rouhani II(1970)
Bonnecaze et al(1971)
Dix(1971)
Beggs(1972)
Chisholm(1973)
Loscher & Reinhardt(1973)
Mattar & Gregory(1974)
5
Moussali
Greskovich & Cooper(1975)
Madsen(1975)
Mukherjee(1979)
Total Points 1542
±5%
345
357
±10%
544
646
±15%
698
999
±5%
22.4
23.2
±10%
35.3
41.9
±15%
45.3
64.8
609
680
311
728
273
315
1008
959
488
1049
411
557
1163
1119
614
1239
525
715
39.5
44.1
20.2
47.2
17.7
20.4
65.4
62.2
31.6
68.0
26.7
36.1
75.4
72.6
39.8
80.4
34.0
46.4
367
329
115
351
148
680
341
633
428
607
479
533
340
163
212
574
5
601
600
258
467
312
1025
684
863
712
821
735
672
607
283
334
777
56
777
794
478
527
419
1196
1040
997
938
895
868
775
925
384
418
911
156
23.8
21.3
7.5
22.8
9.6
44.1
22.1
41.1
27.8
39.4
31.1
34.6
22.0
10.6
13.7
37.2
0.3
39.0
38.9
16.7
30.3
20.2
66.5
44.4
56.0
46.2
53.2
47.7
43.6
39.4
18.4
21.7
50.4
3.6
50.4
51.5
31.0
34.2
27.2
77.6
67.4
64.7
60.8
58.0
56.3
50.3
60.0
24.9
27.1
59.1
10.1
530
684
673
359
834
703
1003
933
661
1090
800
1132
1104
1074
1211
34.4
44.4
43.6
23.3
54.1
45.6
65.0
60.5
42.9
70.7
51.9
73.4
71.6
69.6
78.5
604
1053
1243
39.2
68.3
80.6
367
342
868
637
696
633
27
703
684
1167
937
998
761
102
1006
1040
1272
1096
1149
865
279
23.8
22.2
56.3
41.3
45.1
41.1
1.8
45.6
44.4
75.7
60.8
64.7
49.4
6.6
65.2
67.4
82.5
71.1
74.5
56.1
18.1
384
496
54
639
604
773
87
941
778
1002
105
1146
24.9
32.2
3.5
41.4
39.2
50.1
5.6
61.0
50.5
65.0
6.8
74.3
106
Table 19. Total Number and percentage of data points correctly predicted for the
whole inclined data base(contd.)
Correlation / Data Source
Total data points correctly
Percentage of data points correctly
predicted for the whole database predicted for the whole database
Total Points 1542
Total Points 1542
Gardner-1(1980)
Gardner-2(1980)
±5%
379
469
±10%
613
727
±15%
781
873
±5%
24.6
30.4
±10%
39.8
47.1
±15%
50.6
56.6
Sun et al(1980)
Jowitt(1981)
Chisholm(1983) - Armand(1946)
Spedding & Chen(1984)
Bestion(1985)
Tandon et al(1985)
338
153
702
606
378
510
715
308
1003
809
644
696
1043
614
1152
973
800
851
21.9
9.9
45.5
39.3
24.5
33.1
46.4
20.0
65.0
52.5
41.8
45.1
67.6
39.8
74.7
63.1
51.9
55.2
Chen(1986)
Hamersma & Hart(1987)
Kawaji et al(1987)
Minami & Brill(1987)
El-Boher et al(1988)
Hart et al(1989)
Kokal & Stanislav(1989)
Spedding et al(1989)
Toshiba(1989)
Huq & Loth(1992)
Inoue et al(1993)
Czop et al(1994)
AbdulMajeed(1996)
Maier and Coddington(1997)
Petalaz & Aziz(1997)
Gomez et al(2000)
Zhao et al(2000)
683
290
397
650
700
557
340
551
595
692
585
209
461
550
51
246
160
920
428
645
933
1034
689
680
727
1165
988
909
420
755
790
114
371
229
1035
532
795
1110
1168
788
1037
833
1313
1125
1088
704
895
909
170
452
285
44.3
18.8
25.7
42.2
45.4
36.1
22.0
35.7
38.6
44.9
37.9
13.6
29.9
35.7
3.3
16.0
10.4
59.7
27.8
41.8
60.5
67.1
44.7
44.1
47.1
75.6
64.1
58.9
27.2
49.0
51.2
7.4
24.1
14.9
67.1
34.5
51.6
72.0
75.7
51.1
67.3
54.0
85.1
73.0
70.6
45.7
58.0
58.9
11.0
29.3
18.5
Graham et al(2001)
654
919
1074
42.4
59.6
69.6
Percentage within
1
correlation as reported by Leung (2005), 2,5 correlations as reported by Kataoka and Ishii (1987), 3
correlation as reported by Isbin and Biddle (1979) , 4 Rouhani and Axelsson(1970) correlations I and II.
within this range. These are, Hughmark(1962), Nicklin et al.(1962), Premoli et
al.(1970), Bonnecaze et al.(1971), Dix(1971) and Kokal and Stanislav(1989). The
correlations of Hughmark (1962), Premoli et al. (1970) and Dix (1971) are worth a
general recommendation while the other three being discarded as unsatisfactory as
could be seen from their poor performance within the 10% error index.
We have again implemented our simple scatter check to see the consistency of the top
performing six correlations presented in Figures 35 to 40. Toshiba (1989) has once
again showed a very good consistency with only a few “stray” under predictions.
107
1.0
+15%
0.8
S
S
S
S
S
S
S
S
S
S
SS
S
S
S
S
SS
S
S
S
S
S
S
SS
SS
S
SS
SS
SS
S
S
SS
S
S
S
S
S
S
S
SS
S
S
S
S
S
S
M
SS
S
M
S
S
S
S
S
S
S
SS
S
SS
MM
SS
SM
S
S
S
S
SS
S
S
S
M
S
S
S
M
S
M
S
S
SS
S
S
M
M
S
S
B
MM
M
S
S
S
SS
S
S
S
S
S
S
S
B
S
SS
B
BS
S
S
S
S
S
SSS
B
M
S
S
SS
MS
M
M
M
B
M
MM
S
S
S
SS
S
S
B
SM
BM
B
MSM
S
M
SS
M
M
M
M
S
SBS
S
SM
BM
M
S
S
M
S
SM
S
SM
M
M
SS
SM
B
M
M
SS
SM
SS
M
SS
MMM
SSS
MS
SS
S
S
MM
SS
M
S
MM
S
M
S SS S
MM
S
SS
MMM
M
S
M
M
B
S
M
M
SS
M
M
SM
S
SS
BM
M
B
S
M
M
S
M
M
MMS
S
M
S
M
S
S
S
S
M
M
M
M
S
M
M
B
M
S
S
S
B
M
S SS
MBBB
S S SS SSM MMMM
BBMBBB
S S
BBM
M
MM M
MM
S S S S SMM
SSS SSS
SM
M
MM
S
M
MM
M
S S S SS
SS S SSSSBMB
BSM
BB
MM
BBM
B
S
BB
SB
M
MB
SS S S
M
M
M
S
SM
M
SB
M
S S SS
M
BM
MM
S SS S
S SMM SBBBS
M
M
MB B M
M
B B
BM
S
B
B
S
S
S
B
M
M
M
S
S
B
B
M
S
B B B MBM
B
S M S BB
MB
B
M M
BBBBM
MBMM
SB
BMSS
M
BB
SB MM
S
S SS
S
M M
S B
M MB
B
BBM
M
S
S M MM
BSB BS
S SS SS S SSSB
BM
SM MMM
M
SS M M
M
SS
S M
S S S S
B M M M
M
SSS S S B
S S MMB M MM
S
B
M
BM
S
M
S
S S SMS S
S
BBBBBMBMB
M
S S B S S SB
S BM B M M S B
SMSBB
SBM
S
B
BSS
B BB
M
BMMB M B MB M
S
B
S
B
M
M
SS
S
SS S
S MSS
M
S
S S SS
MS
S MM M
SS
SBBB
BB B
S
S MMM
B BS
S S
SSB
SM S M
S
B BS
SS
S
M
B
M
B
M
B
B
S
B
S S
MSS
S
SS
S
B
S BMSBM BM S
M
S SM B S
B
S S SSS
M
S BSSM
S
M
SB
S
M
SS SSBS SSM
SS
M
M
MS S
S SS B
S
M
S M SMSS
S S
B S
SS BM
M SSSSSM
M
B M S M
SS
S
S
SS
M
S
S
SS SMS
M
S
SS M
S
SM S
S M
M
SB M M
M
BS
S M
S S
M
S M
B
SSSS SBBSM
SM S M
S S MSS SSSM S
M
M
SSS
SSSSS SS B
S
S
S
S
B
S S
M
M
S
MMMS
S SS B
S
S
M
S S
S
S
S
S
M SM M
S SSS S S
M
S SS SSM M S
S
M
S
SM B
S
M
MSS M MB
SS
M S MM
SSM
S
M
S SS
SS S
SS S B B S
M
S SSS
SS
M
M
S
S
S M
S SS S S B SB M
M
SS
S
M
S MMS B
SSSSSSSSSSSS
B
SM
Beggs(1972) data
SSS MS M M
M M M M M
S
M
SSSSM
S
S
S
SSM
S
M
SS
SS SS M
SM
S
SSS
SSM
SS
S
S
SS
BS S
S
M
S
S
B
M
S
Spedding & Nguyen(1976) data
SS S B
SS
B
MSM
S
S
S
M
S
S
S
S
SMSS SSBMS
MMM
S
SSSS SS
MM MMMM
M
BM
Mukherjee(1979) data
S
M
M
S
M
SSS
S
SM
S M SM
SSS
SSS
SS
SS
MS
SSSS
SS
SS
S
S
SS
Calculated void fraction
-15%
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Measured void fraction
Figure 28. Comparison of Toshiba (1989) correlation with
measured combined inclined experimental data
1.0
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
SS
S
S
S
S
S
S
M
S
S
S
S
M
S
S
SSS S MS
S
S
S
S
S
S
M
S
S
S
B
S
B
MS
S SMS S
S
M
S
S
S
S
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M
MS
S
S
S
M
S
S
M
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S
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M
M SS S
S
M
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M
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B
S
M
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M
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S
M
M
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M
B
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SSS
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SM
M
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BB M M
M
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S
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M
S
M
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SSS
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SS
SS
S
M
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M
M
M
S
M
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S
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M
M
MM
MM
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B BM
M
S
S
M
M
SSM
S
M
SSSS
SB
M
M
B
S
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SSS
BBM
M
M
BM
S
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B
BB
S
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SS
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BBM
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MMM
M
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SS
S
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SS
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SS
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B
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M
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M
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B
M
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SM
M
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BMMB MM
S
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SMSS S SS
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SS M M
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BB
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BS
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B
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S S
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M
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M
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S SB
SSS SS
S
M
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SSBB
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B
BB B
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MS
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M
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S M
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M SMM
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BBS
S SBMS
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M
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BBB
S
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S S
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SM M
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M
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B MM MMM
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SSS S
BB MM
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B
M
S SS B
M
MM
S SSS SMM
S MB M
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SS SS
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MB
MM
B
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B
B
M
B
S
B
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S
B
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SMSMBS
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M S
MSB
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M
B
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B
M
S
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M
BMB
SSBS SBS
M BMS
B MBS
M M
SS SMS
SSB
M
M
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B
S
S S B
BBS B
M SS
SSM M
B
S
M
B S BM
SBSS S
M
MS
SB
M
B
M
B
S
S
BS B
SM B B M
S
BSSB SSSSS
SM
SS
MM
S M M SS
S
S
S
S
B
M
B
B
S
M
S
M
SMS BSM B M
S
S
M
M
M
SSS SS B
M S
BB
M SS
SS S
S
S
S
S
S
M M MS
M
S MS S
SM
S
S
M S S SS M MSM
SSB
S SS
M MSMS BSB
S
BS
S S
BM
M
S SS S M
M
M
S
M
SS SSS SM S M S
S
SSS
S
M
B
B
S
S
S
S
S
S
S
BS
S S
S S M
M
S MS
SS
SBSSS
S MMSM
SS
SS B
M
S SS M SS S MM SM S
M
S SSSM
M
SS SM
SS
S
S
SS SS
M S
SM
S BM
M
S
S S
S
M
M
S
S
S
MS
S S SS
M
S
M SS S
M
S
S
M
S
S S SB M
S
S
SB M
M
MSSMM
MSSS S
S
S
SS
M
M
S
SS S
S SS
B MB S
S
M
S M
S
SS SSSS
S
S
B SM M
S SMSS
M
M M
M
M
SSSBSSS SS
SSB
B
SS
S
MSM
M
SS
SS
SBSS S
S SS
M M
S S
S MSS
S
B
Beggs(1972) data
S S SBM
SSSSSS SSS SS
S
S
S S SSSSS
M
M
SSM S
SMMS
S SM
S MSM
MSS
MM
S
S SS
S
S
Spedding & Nguyen(1976) data
S M SM
MSSS MM
S
S
S
S
S
SSS SS
S
MS
S
S
S
SSS
M SM
SSS
M
S
SS
Mukherjee(1979) data
S
S
SS
+15%
0.8
M
Calculated void fraction
-15%
0.6
0.4
0.2
S
S
SSS
0.0
0.0
0.2
0.4
0.6
0.8
Measured void fraction
Figure 29. Comparison of Dix (1971) correlation with
measured combined inclined experimental data
108
1.0
1.0
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
SS
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
B
SM
S
S
B
S
S
S
MS
M
S
S SS S
S
S
S
S
M
M
M
S
M
S
S
M
SMS
S M
MM
SS
S
SSM
S
M
M
S
S
S
S
S
M
M
S
S
S
MMS
M
M
S
S
S
S
BBM
B M
S
BBB
SS
B
B
M
M
S
S
B
BM
SS
M
S
B
S
BS
S
M
M
SMBBB BMM
M
S
M
S
S
S
M
M
M
S
SS
B
M
S
SS
M
S
B
M
B
B
S S MMM
M
S
M
S
S
S
M
SS
S
M
S
S
S
S
SMS
S
M
SS
B
S
SM
SS
S
M
M
S
S
S
SS
M
S
S
SSS
MM
M
SM
SSMSMM
SS
S
M
S
SS
S
M
M
S
S
M
M
S
M
S
M
S
M
S
S
M
S
B
S
S
M
M
S
M
S
S
S
S
S
M
M
MBMMMMM
S
MMSMMSS
SS
SS
M
S
S
SM
SS
SM
S
M
S
M
M
M
S
S
S
M
S
M
S
S
S
S M SSSSM
S
S
S
B
M
S
S
S
S
M
M
S
M
M
S
M
S
B
SSSM
MMM
S
S
SS
S
BM
M
S
S
M
S
MB MM
M
B SS
BS
M
SSSS
SBM
BSMM
S
BM
SB
S
SS
S
S
M
M
SSS
SM
S
M
B
MMM
S
S
S
B
SBSS
BS
M
S
S
S S
M
B
BS
B
BM
B
S
S S SS
M
B M
S
M
B
M
B
MM
S
S SS
SBM
M
SS
B
M
B
B
SB
M
S
M
S
B
MM
BMM
B
S
BB
M
BSSSB
SSM
BBS
B
BM
SM
S SSS SS
SBS
S
M
M
BSB MB M
M
M
M
S
SSS
S SS SS SSSSMS
M
M
S
M
M
M
S BM M M
S S
SS
MM
S
S S SSS M
B BBB
M
S
BS SSB
M M
S
B
B MM
S
M
BB
BBBM
M
S
S S M SSB MM
M
S
SS SS SS SSSSBSMMM M
M M
M
B
S
S
B
B
SBBB B BM
M
M
S
S
M S SSS S SSSSS
BBM M M
S M
M
SSS
M
M
S SS
S SS S B BS
MBM
B M
MM
MBBM SB
S SS B
S SS S
S
B M
S
S
SS SMMB M M
S SS B
S
S
S
S B B
SSBB
MM
S
SSB SM
S
S
BBM
SB B
BSM
S S S SS
S SS
SB S
MB M
M
S
S
M
M
S S S BS S
B
M
M MSM
S
S S
SSSSS B S S
MSM
S
S
MS M MM
B BB M
M
S
S
S
S
B
S
B
S
BMM
S
SS
S
SB M
BB
BSSSB BSBB
S S S SSS SMS SS
SM
M M
SS S
SSB S BM B B M
S SS S
S
SM M M
S S BS BM
M
SBMS M
S S S
S S
MB
S
S
S
S
S
S
S
SS
S S S S B MM
S
S
SS S
M
S
M
SS S S M
M
S SS
M
S S S S S SSMSS M BMM M
S
S
SS MMS SS S
B M M
S SS
M
BS
M
M
SM S
S S SS
M
B
M
S
S
S
S
M
S
M
M
M
S S S MM
SBS
S
M
B
S
S
S
S
S
S
M
S
S
B
B B MM
S
S
S S SSM S
S M M
S
M
S M S M M
S SS
S
M
SM S
S
S
SSM B
S SSS SSS S
M M
BMM
S
S S SS
M M
S
S S SS M S
MS
MMSB
SM
S
SS
S
M
SS S S S SS S
M
B
SS SS SS S BMBB SM
S
M
SSS
S SS
S
MM M
S M
S
M
SSM S
S S SS S
SB B M
M
S SS
MSSSSS MS BSMM MM
S
S S
S
S
B
M M
Beggs(1972) data
SS M
SS
S
S
M SSM
SM
M
MS
MM
M
S S SMS S MMM M
BM
S
S S SM M
Spedding & Nguyen(1976) data
M
BMB
SS S MB
S
S
S
M
S
S
S
S M S SB
M
M
M
S S
SSM
Mukherjee(1979) data
BM
S
S
S
S
S
SS
SSS
SS
SS
MS
SSSS
SS
SS
S
S
SS
+15%
0.8
Calculated void fraction
-15%
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Measured void fraction
Figure 30. Comparison of Rouhani I (1970) correlation with
measured combined inclined experimental data
1.0
S
S
S
S
S
S
S
S
S
S
SS
S
S
S
S
S
S
S
S
S
M
S
S
SS
S
S
SS M S
SM
M
B
S
B
S
M
S
S
S
S
SS
S
S
MS
M
S
M S MS SSSM
M
M
M
S
S
S
S
S
M
S
S
M
S
S
SS MMSS
S
M
S
S
S
SSS
M
MBBB
S
MM
MM
S
B
S
BBMSS
S
BS
B
S
B
M
S
S
S
S
B
S
B
S
S
M
M
M
M
B
B
S
B
M
B
B
S
M
S
S
M
S
S
M
S
SS
M
SS
SMSSBB
S
SM
B
M M
S
S
M
M
BM
M
S
M
M
S
S
S
M
M
S
MS
SS
S SM
M
SS
M
MM
SS
S
S
S
M
M
BMM
SS
M
S
M
M
SS
M
S
SSM
S
SS
M
MM
M
SS
MM
S
SSSM
S
MM
S
BMM
S
M
SS
SSM
M S SS
M
S
SMMM
M
M
S
SS
SS
MBM
MM M
SSS
S SM
S
MSBB
S
SMM
M
S
S
M
BSMB
M
BS
MSM
BBMM
M
SS
M S
S
SSS
M
S
S
SS
B
M
M
S
S
M
M
S
S
S
M
M
S
S
SMM
M
M
S S
S
SMS
S
S M
S
S SSS M
M
S SS
S
M
MB M
M
M
SM
M
S
SS
SB
S
B
BSB
B
B
BMMMM
S
SBBSS
B
B
MS
BS
S
B BS
B
S
B
MM
B
SBB
B
B
SB
B
S
SBS
M
S
S
SSM
M M
B
S S
MM
S
B
M
S
M
MB
M
S S
MB
S
SS
S
MMBSM
S
SB
MS
B M
SSM
SS
SSMSM
S
M
M
S
S
B
M
S
S
B
SB
M
S
S
S
S
SM
S
S
S
B
S
S
S
M
S
S S
MM
M
SS M
MSS S M
MM
S S SS S
SS
S
B
M
B
M
S
S
S
B
S
BB
B
S M S S
S
M
S
MB S SB
S M
MBS B S
S
S
S S S
S
S
S
SMM S BBBBBS MM M
S
M
S
BS
S S
S
S
M
B M
S
B M
S M SS BB
B SSS M
SS
MSS
M B
S
M
M
SS MS M M
S
S
S
M
M
S
S
B
S
B
S
S
M
M
SSS BM
S
S S
S
S BS
BM
SB SS SB SBBM
SS
S
S
M M
M
S
M
S
S
M
M
S
S
M
S
S
S
S
MM
S
B
S
S S SS B S
M
M
S
M
M SSM
S
S
S
MS S SSB M S MMSB
S
SS B
M
S
S SB BB M M
SSBBB
M
SS M
S
S M S SS S S
B SB B
SS
S
SM
S SB
M
M
S
S
S
M
B
M
S
S
M
S
M
M
S
B
S
B
S B SSM
S M M
S
M
M
S
BM
S
SS
SS SBS
S
B
MBSB B
MS S S S
S
SS
S
SBS MM M
S S
SS BBB
M
B
S M
BS S
SS
S
S M
M
S S
S B
S
S B SBSS B B M M
SM
S S MS S S
SM
S
S
S S B
S BBMM
S B MS
S
M
S S
B SMSSSSM BM BM S
M
SS
SS
S
S B
M
S S S S S S SS
S
MS
M
M
SMMS SMSMS
S
M
M
M
S
S
M
S
B
S
M S
M
M
SMS
S
S S
S
S
M
S
B
S
M
M
S
M
S
BS
SM M S
S SS
S S
B
S
S
B
S
M
M
MS M S
S
S
S
B
S
S S
BS
B MM
S S S BS
S
S S SS M M
S S
SS
SS
SM SSM SMMS
MM
S
B
MSSM
SMMSB
B
S S S S M SMM
SS
S
S
S
S
S
S
SS
S S S SM SSBSB M S M
S
M SBM M
B
M
S
SS
M
S
M
S SS SS B B M
M
S SSS
S S
S
M SM S
S
SS M S M B M
S M
S
S SMS
MSM M M
S
M
S S
S
S
S S S MS
SSSMM
S
MM
B
Beggs(1972) data
S
B
M
MMM
S S
SSSSMMB
BM
B
SS SS
M
SS
S
S SSMS
Spedding & Nguyen(1976) data
S SB M
SM SS
M
B
M
S
SS
SS
M
S S
Mukherjee(1979) data
SSS
SS
SS
MS
SSSS
S SS
S SS
S
S
+15%S
0.8
M
Calculated void fraction
-15%
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Measured void fraction
Figure 31. Comparison of Premoli et al. (1970) correlation with
measured combined inclined experimental data
109
1.0
S
S
S
S
S
S
S
S
S
SS
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
SS
S
SS
S
S
SS
S
SS
S
S
S
S
S
SS
S
S
S
SS
S
S
S
S
M
S
M
M
S
S
S
S
S
S
S
S
MM
SS
SS
S
S
S
S
S
S
SS
S
S
SS
S
SM
B
SS
M
M
S
S
SS
S
M
S
S
S
SS
SS
M
BS
B
M
B
S
S
S
M
S
M
S
S
M
S
S
S
S
M
S
B
B
S
B
S
S
S
M
S
SM
S
S
M
M
SS
S
M
M
BS
M
S
SS
M
M
S
SS
S
M
B
B
S
S
MS
SS
SS
SSS
M
MSS
BSM
S
S
BS
M
M
M
S
M
SS
M
SM
S
SSS
S
S
SS
S
M
M
S
M
M
S
SS
SBS
M
S
S
SSS
S
S
M
SS
S
SSS
M
M
SM
M
M
MS
MM
M
BBM
SS
M
MM
M
MMMM
M
SS
M
SSS
B
M
M
S
M
M
S SSSSSS
S
S
S
M
M
S
BMM
S MMM
S
S SSM
M
SS
SS
MSSS
M
SS
MM S
S SS
SS
MBM
MBMM
M
B
S
MM
B BB M
MMMM
SMM
S
B B
BM
S SSSS SM
MMBBB
SSS SSSS
MM
MM
MM
M
MM MM
S
M
SS
S S S SS
BBM
SM
MM
M
BSSB
M
BM
B
B MM
M
SS S SS
M
BB
BSM
BB
SS
S
S
B
M
M
M
SS S
M
SS S
M
SS B BSM
M
M
SM
MMMB
M
SS
S
M
MM
M SB BS
MM
S SB
M
B
BB
S SB
B
B
M
M
B
S S M SMS
S
B
S
B
B
B
B
B
B
M
B
B
B
M
B BMB M
B BBB
BS
MB M
BMB
S
S
BM MM
SB BM MS
M MM
S S
B
BBB
BS BBM
M
M
S B SB
S MM
S SS SS SS S MMM
MM MM
B SM M M
B
M
S S S B SS M
M
M
M
MSSM M
SSMM
M
SSS
SMB M
S
S SS
B
S
S
M
S BS S S S B BBBS MM
S B
M B SM MM
BS BB
SBMM
BB
M SBBB B B B
SS
M BM
B SM
B S
B MBS
S
M
SM
S B MB
MS
S
S
SS S
M
M SM
M
S
S
M
BS BSSSM
S BSBBB
S
B
MSM M S
B
S
S
M
S
B
M
B BBBBBSS
M
BSB SBSM
SS SS
S
S
S
SBSSB M
MMSS
BSS
BS M
S S
M
BS S
SSM S
SS
S BS B
S
M
M
M
S
SS SSB
S
M
S
S
S
S
M
S
S
M
SS
M S MS
SSSBS
M
M
B
M
M
S
M
SS M
S
MS
S M
S
S
MSSB
S S SM S S M MM
SS
M M
S SS
S
S M
S S SS
S
B BSSS M
SBB
S
M SS
SM M SM S
S SM
M
S
M
S
SSS S S SS M S M
M
B
B
S
S
S
M
S SSB SS
M S
M S
SMS
MSS S
SSS SS
SS
SS
SS
B SSSMS S S MS
M SM
M
S BS S
S MS MSSM
M
S
SMS M
M
S
SS M
S
M
M
SSMS
S
M
SSBSSS SB BS
M
M
S
SSS S S
MS S
MM
S
SS
M
M
S S MS
SS SSSS
SSB BM
S SS S
M
M
M
S
M
SS S
S
SS
SS
S
S
M
MM
S
S
S
S
B
M
S
M
S
Beggs(1972) data
M
M
SS
M
S SSS M M S
M M
SSSM
SSS
S
SSSM
SM
SSS
M
BBSS
M M
S
B SSS
SSSSB
M
S
S
SSSSS
Spedding & Nguyen(1976) data
S
SS SMS
M
BSS
S
S
MS
S
S
S
S
S SS
SB
SSS
M
S
SS
S
SSSS SS
MMM
SMM M
M
S MSMMM
M
SS SM
Mukherjee(1979) data
M
MS
S SM
SS M S
S
S
S
S
S
S
M
S
S
S
SS
SS
M
SSSS
SS
S
S
S
SSS
+15%
0.8
Calculated void fraction
-15%
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Measured void fraction
Figure 32. Comparison of Filimonov et al. (1957) correlation with
measured combined inclined experimental data
1.0
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
SS
S
S
S
SS
SS
S
S
S
S
S
SS
S
S
S
S
S
S
S
S
S
S
SS
S
S
S
S
S
S
S
S S S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
SSSS
S
S
M
SS
S SSSSM
S
S
B
S
SSSS
BM
S
SS
S
S
S
M
S
S
S
SB
S
SM
MS
S
S
S
S
S
SSS
S
S
SM
M
SSS
S
S
S
S
S
S
S
SS
S
SMM
SSS
S
M
S
S
B
SM
B
M
S
SS
S
M
SSS
SM
SS
MM
M
S
S
M
S
S
MBBS
BM
S
S
B
SSSS
S
B
MM
SM
M
S
SM
S
S
SS
SS
M
SS
SM
M
S
SS
SS
M
M
S
MM
SM
M
S
M
SS
MM
MS
S
S
S
S
MM
B
S
M
MM
S
M
S
M
B
S
M
S
B
M
S
S
M
B
S
S
B
S
B
B
M
B
B
M
S
S
S
M
M
S
S
S
B
M
S
B
M
M
M
S
S
S
B
S
B
M
SSSS M
MMM
MM
BM
M
S
M S
SSSM
SM
M
MM
SSSSS
S S SS S
MMM
MM
MM
B
S M
BM
M
S
M
S
M
M
BM
M
S
S
SSM
MM
SSSS
S
MM
M
M
M
M S
S
M
M
B
S
S
S
M
S
S
S
S
M
M
M
M
M
S
S
M
M
S
M
S
M
M
S S S S SS
SS SBS
MB M
B SM
B
M
B
M
M
M
M
M
S
S
M
M
S
S
M
M
M
B
SS S
B
B
S
S
M
SM
S B
M M
B
BBB
SM
BMBMM
BMB
S
M
M
SSBBSB
B
SS S
B
S
S
SBSSB
BBBS
BBB
B BM
BM
M
S
S
MM
M
S
SSBSS S
SS SSSS SS S
M
MSMBM
B MMB
M
S
S
SMS S
S
MM
S SS S
M MBM M
M
S
S
SB
S
S SS S
S
SBBBBB MM M MM
SS
S
B
S
B
M
S
S
S
B
M
S
M
M
M
S
S
M
BBB M
S
S S SB BM
S
M M
S
S S SS
SS
SS B MM BBMM M
MS SSBBB
S
S
SB
B
S
SMM
SS
M
S
S S
M
S S S SS S S SS
B M MM M
S SS
S SSB B S BS BS
SS S
M
S B BM
BSS
B M M
S
S
S SM S M
SS
S
M
M
M M
S S S
S SS
S
S
M
S M
SB
S SS
S
M
SS
S SS
SSS S BB
S S
M
M M
S
SS M
S
B MMMB
S
B M
S MS
S SS
M
S
S
S
M
S
SSS
S
S
SS S S
SS SBB
BBBSMB
B
MMM
SS S
SS
S
B
S S
B
S
B
B
M
S
S
S BS S B
S
B MMMM MSM M
SSSS
MS S S SB SS
M
S
M
S
S
MM
BS
SS SS
B M
S
S S S SB BB M
S
S
BM
M
S S
S
SS
M
BS BS
BB
BB
B
B
S
S
M
S
S
M
S
S
S
SBSS BB
S SS
M S SS
M MM
S
B
M
M
S
MBMB M M M
B S
SS S S S
S
S
S
M
M M
B
S
SS
S S S S MM
S
S
S S
B
S M
S
SS S MSS
S
MS M S M M
M
S
SM
BM
S
M
SSS
M
S S S MS M
S
S
MS
S
B
S
S
M
S
M
S
S
S
S MS SM S MM
BM
M
B
S S
SM M M
SS
M
SS S
S S
S
B BB
M
S
B
S
MSBM
SS
S S
M
S
MM M
S MM
SS S
SS
S
M
M
SS
S
BM
M SSM B
M M
M
S SS
S
S S M SB
S
MM
S MS MS
B SM
S S SSS
M
S SSM SS B B
B
M
S
M
S
S SSS SMS M
M
S
S
M
M
M SB B
S S S
SS
M SS
M M
SM M
M SMSSS S MS
S B M MM
SS S
S M
S
SS
M M
MS
M
B
Beggs(1972) data
MM
S S S MS M
MM
M
S
S S
MBM
M B
S SS
SS
BMB M M
S
S S SS M
Spedding & Nguyen(1976) data
SSSSMSSS B M M
M
M
S
SS M BM
SS
M
Mukherjee(1979) data
S S
SSS
SS
SS
MS
SSSS
S
S
S
S
SS S
SS
+15%
0.8
Calculated void fraction
-15%
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
Measured void fraction
Figure 33. Comparison of Hughmark (1962) correlation with
measured combined inclined experimental data
110
1.0
1.0
S
S
S
S
S
S
S
S
SS
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
B
M
SS
B
S
SS
S
S
S
S
SM
S
S
M S S SS M
S
S
S
S
M
SM
S
S
M
S
MS
S
S
M
BBB SB
MBBB
S
M
S
B
BBBM
B
M
B
S
M
S
S
SM
SM
S
M
M
S
S
S
S
MMS
SSMM
S
M S
M
S
BSS
S
MM
S
B
BS
S
S
S
S
M
M
M S
S
M
M
M
M
S
M
S
S S MM
M
S
S
S
B
S
S
S
B
SS
B
S
M S M
MS
S
M
S
S
M
S
S MM
SBS
M
S
S
SM
SS
M BMB M
SS
SM
S
S
M
S
S
SS
M
S
S
BS S SM
S
M
S
M
M BBMM
S
S S
S SM
SS
S
M
M
M
S
SS
S
S
S
M
S
S
M
S
S
S
S
S
M
M
M
M
M
S
S
S
M
S
S
S
S
BMM M
SS
SS
S
S
M
SS
SB MM
M
S
MBS SM
S
M
SM
S
SS
M
SS SS S S S
S MM SBM
M
S
S
SS
S
S
S
MM
MM
M
S
S
S
SS
S
S
S
S
S
SM
SB
SS
S
MBM
SS
S
S
SSS
BM
M
S
S
MM
S
S
S
S
S
S
S
M
B
M
SBSSB
SBBB BB BB
B
S
S
B
BS
S
M
MM
M
S
S
S
S
S
M
S
S
M
MM
B
S
BB
BM
M
MBM
S
B
M
S M
S
S
M
S
S
S
S S S S
M
SM
M
S
SS S S
BM
B
S
S
S
BS M
S
B
BS
S
B
SSSBM
S
S
M
B
S
M
S
SB
S
S
S
MS
S SS S S
M
SS
SS
M
S
S
S
S
MMM MMM
SM
SS S
S
S
SS
M
S
S
SM
S
M
S M
S
M
M
SS
SM
BM
SS
MMM
S
SSS
M
S
MSS S
S S SS S
SSS M
S SS
SM
SB
M
S
SS S
S
M
M
B
S
S
M
M
S
B
S
S
S
S
M
M
M
S
S S
M
SS
SS
S S SMB M
S
SB SB
S
S
S
S
B
S SS SSS SM
S
S
M
B
S
M
M
B
M
S
S
B
B
M
B
B
B
S S
SM
S
M
SSS
SS
S
SS
M
S SBMS
S
M
M
SBBB
M
S
B B
S MM M
S S
M SS
SS
MM
SS M
S
S
S S
BB
MM
S SSS MSSB S
BB
MBM
S S
S
BS
S
SB
M M
S S S SS
SB S B
BS
S
SS M
S
SM
MM MM MM
B M
S
S
S
B
S
M
M
M
S
SSS
S
BM
S
S
SS M
M
S
B
M
M
S
S
S
M
B
M
M
S
M M
SS
M SS S S
SSM S
S SSS SS
S S
M M
S S
S S
S SSS SS BBM
S
B
S MB
M
M B M
S S BS
S SS
S
SS
S BB S B S MM
S
SS
S SS S S SS
S
B M B
BSB M BB
M
S
B
M
S
S
B M MB B
SB
S
S
S
S
M
MM M M M
S SB BB SB BSB
B
S M
S
S S
B
S S
S
B SM S MM
S
S
SS SSSB S
M
M
M
S S
S
S
MM
BB
BS S B BBB
B
S
M
M
MS
M
S B
S
BM
MM M
MS
SS M
S
S SM
S
M S S BMB M M
S S SS
SMS S
B
SM
S
S
S
S
SMSSS M
S
B MM M
S S
M
SM
M
S
S
MB
S SS S S S
S SS
S
S S
S
M
M
B M
S
S M
BM
S MS M
S
S S
M
B
SS
B
M
M
S SMM B M
S S
S
M
M
S
B
B
B
M
S
S
S
SS S
M
M
SS
M SSM SSMS
S
M
S MSS S
MM S
B
B
S SS
MBM
SBMM
S
M
SMS
S
S M MM
M
B
S
S
M
SS
M
S B M
S S
M
S
S M
B
S BB
S
MM
SS S SS M
SMM
S
M MM M
S
MM
S
S SMS
MB M
S S
S
S
M
S
S
S
S
S
S
M M MM
S S
SMMM
SS SSS
SSSMM M BM
B
M MB
Beggs(1972) data
S SS S S S B
B
SSS SB M
S
M
S
Spedding & Nguyen(1976) data
S
S
S MM BM
S
S
S
SSSS
SS
SS
M
Mukherjee(1979) data
S
S
M
SS S
S SS
S SS
SS
+15%
0.8
M
Calculated void fraction
-15%
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Measured void fraction
Figure 34. Comparison of Armand Massina (Leung 2005)
correlation with measured combined inclined experimental data
111
Table 20. Number of data points correctly predicted for vertical flow by data set
Beggs
(1972)
(Air -Water)
Correlation / Data Source
Spedding & Nguyen
(1976)
(Air -Water)
Total Points 26
Percentage within ±5%
5
12
Homogeneous
Armand(1946)
1
Armand - Massina
Lockhart & Martinelli(1949)
Sterman(1956)
Filimonov et al(1957)
Chisholm & Laird(1958)
Flanigan(1958)
2
Dimentiev et al
Hoogendroon(1959)
Bankoff(1960)
Fauske(1961)
Wilson et al(1961,1965)
Hughmark(1962)
Nicklin et al(1962)
Nishino & Yamazaki(1963)
Fujie(1964)
Kowalczewski(1964)
Thom(1964)
Zivi(1964)
Hughmark(1965)
Neal & Bankoff(1965)
Turner & Wallis(1965)
Baroczy(1966)
Guzhov et al(1967)
3
Kutucuglu
Smith(1969)
Wallis(1969)
Gregory & Scott(1969)
Premoli et al(1970)
4
Rouhani I(1970)
Mukherjee
(1979)
(Air -Kerosene)
Total Points 224
Total Points 52
Sujumnong
(1997)
(Air -Water)
Total Points 101
±10%
9
18
±15%
14
23
±5%
34
47
±10%
55
84
±15%
79
135
±5%
13
16
±10%
27
25
±15%
33
46
±5%
24
35
±10%
40
60
±15%
59
85
17
13
6
19
4
2
24
20
10
24
6
5
25
25
11
24
7
9
70
95
47
77
33
56
131
126
73
136
53
81
147
151
90
180
68
101
25
22
6
33
4
1
40
33
22
42
19
9
47
44
26
45
26
19
44
46
15
38
13
26
77
56
26
69
21
35
92
63
33
89
26
40
7
0
0
7
3
10
3
9
7
10
9
9
9
3
3
9
0
9
4
2
7
7
24
14
15
15
13
13
12
17
5
5
14
0
13
9
5
8
10
25
23
17
20
15
16
14
23
6
7
15
0
45
63
32
41
17
127
90
108
39
99
62
82
39
24
24
96
0
79
102
54
61
37
160
140
141
70
118
110
102
82
45
39
124
0
99
117
81
71
60
169
181
167
111
137
127
118
124
67
60
150
21
11
4
0
10
6
16
9
23
16
22
19
23
14
6
5
25
0
31
9
0
24
12
39
18
31
31
34
28
31
22
8
6
31
0
33
25
12
26
18
46
40
38
38
37
35
37
38
9
13
37
0
21
44
2
19
3
69
33
39
32
37
20
28
35
6
7
37
0
30
64
7
24
5
79
60
47
58
49
37
40
56
11
13
48
0
37
75
31
26
13
89
85
52
79
52
47
42
79
20
22
49
4
6
13
14
12
15
12
22
22
18
21
15
25
24
26
24
79
106
99
49
129
99
129
128
90
164
118
153
156
140
177
20
21
25
16
30
27
38
39
27
43
36
44
46
46
45
24
48
43
36
48
32
83
55
62
62
41
96
69
94
83
13
22
25
107
180
189
17
36
43
35
75
92
Rouhani II(1970)
Bonnecaze et al(1971)
Dix(1971)
Beggs(1972)
9
3
16
19
14
14
23
25
21
23
25
25
43
90
141
79
92
139
167
116
129
181
188
146
4
9
26
26
15
18
41
40
33
40
43
43
34
33
56
37
73
60
65
57
88
85
73
71
Chisholm(1973)
17
25
26
101
126
146
24
41
46
58
91
96
Loscher & Reinhardt(1973)
Mattar & Gregory(1974)
7
0
8
0
10
0
59
0
90
3
99
43
26
0
35
0
42
2
36
0
41
1
50
19
Moussali
Greskovich & Cooper(1975)
6
11
11
20
18
26
35
45
56
84
80
139
16
24
27
35
35
41
24
42
47
70
65
89
Madsen(1975)
Mukherjee(1979)
0
16
0
24
0
25
0
74
4
137
5
162
0
28
0
37
0
43
0
32
0
55
0
68
Gardner-1(1980)
Gardner-2(1980)
Sun et al(1980)
4
3
2
7
12
13
13
14
19
52
66
84
77
95
132
95
114
168
7
30
12
14
36
27
23
37
45
23
26
20
30
31
50
36
36
75
Jowitt(1981)
Chisholm(1983) - Armand(1946)
0
16
0
24
0
26
30
105
75
126
134
146
1
24
1
39
3
45
9
58
28
90
60
96
Spedding & Chen(1984)
8
12
15
101
125
159
23
28
33
40
55
62
Bestion(1985)
6
10
15
30
65
84
9
28
37
42
57
71
4
5
Tandon et al(1985)
8
15
16
69
107
121
21
29
34
22
35
36
Chen(1986)
10
18
22
103
129
146
26
36
45
47
66
75
Hamersma & Hart(1987)
4
7
7
33
53
68
9
20
27
14
21
27
112
Table 20. Number of data points correctly predicted for vertical flow by data set
(contd.)
Beggs
(1972)
(Air -Water)
Correlation / Data Source
Spedding & Nguyen
(1976)
(Air -Water)
Total Points 26
Percentage within ±5%
Kawaji et al(1987)
7
Minami & Brill(1987)
19
Total Points 224
Mukherjee
(1979)
(Air -Kerosene)
Sujumnong
(1997)
(Air -Water)
Total Points 52
Total Points 101
±10%
12
24
±15%
14
26
±5%
52
71
±10%
93
131
±15%
115
163
±5%
18
29
±10%
29
41
±15%
36
44
±5%
27
28
±10%
39
53
±15%
48
68
El-Boher et al(1988)
Hart et al(1989)
Kokal & Stanislav(1989)
Spedding et al(1989)
Toshiba(1989)
Huq & Loth(1992)
9
9
3
8
12
12
20
12
14
12
24
17
23
15
23
15
24
25
140
84
89
91
105
112
166
102
139
109
186
139
187
113
179
133
201
157
11
22
9
23
19
20
34
28
18
32
43
36
42
34
40
36
45
43
35
27
33
33
48
43
49
30
60
43
89
72
58
34
85
46
96
91
Inoue et al(1993)
Czop et al(1994)
AbdulMajeed(1996)
Maier and Coddington(1997)
Petalaz & Aziz(1997)
Gomez et al(2000)
Zhao et al(2000)
Graham et al(2001)
Spedding et al(1989)
Toshiba(1989)
Huq & Loth(1992)
Inoue et al(1993)
Czop et al(1994)
AbdulMajeed(1996)
Maier and Coddington(1997)
Petalaz & Aziz(1997)
Gomez et al(2000)
9
3
11
7
1
0
0
14
8
12
12
9
3
11
7
1
0
17
8
16
13
3
0
3
16
12
24
17
17
8
16
13
3
0
23
14
19
15
6
1
3
24
15
24
25
23
14
19
15
6
1
95
39
63
90
9
49
26
96
91
105
112
95
39
63
90
9
49
132
84
110
125
18
77
37
124
109
186
139
132
84
110
125
18
77
142
122
124
134
24
87
49
146
133
201
157
142
122
124
134
24
87
26
9
17
26
0
0
0
23
23
19
20
26
9
17
26
0
0
37
13
29
35
2
1
0
34
32
43
36
37
13
29
35
2
1
43
31
35
39
2
2
0
40
36
45
43
43
31
35
39
2
2
51
16
21
33
4
14
6
40
33
48
43
51
16
21
33
4
14
77
24
41
50
8
19
13
50
43
89
72
77
24
41
50
8
19
85
42
49
53
13
25
17
59
46
96
91
85
42
49
53
13
25
Zhao et al(2000)
Graham et al(2001)
0
14
3
16
3
24
26
96
37
124
49
146
0
23
0
34
0
40
6
40
13
50
17
59
1
correlation as reported by Leung (2005), 2,5 correlations as reported by Kataoka and Ishii (1987), 3
correlation as reported by Isbin and Biddle (1979) , 4 Rouhani and Axelsson(1970) correlations I and II.
The Rouhani I(1970) correlation gave an excellent prediction all in all and has a
noticeable improvement from its performance for the other inclined data sets where it
had excessive over predictions in the lower void fraction range. Filimonov et al.
(1957) also has good uniformity in its prediction with slight under predictions for void
fractions less than 0.7. The Hughmark (1962) correlation over predicted the data
while Dix (1971) gave a slight random scatter for the lower void fraction range. The
excellent performance by Dix (1971) for the very high void fraction range should be
well noted where measurement of experimental data is usually very difficult.
Premoli et al. (1970) repeated its tendency to over predict the data as observed for the
other inclination angles though there has been a significant improvement for the
113
Table 21. Percentage of data points correctly predicted for vertical flow by data set
Beggs
(1972)
(Air -Water)
Correlation / Data Source
Spedding & Nguyen
(1976)
(Air -Water)
Total Points 26
Percentage within ±5%
19.2
Total Points 224
Sujumnong
(1997)
(Air -Water)
Mukherjee
(1979)
(Air -Kerosene)
Total Points 52
Total Points 101
±10%
34.6
±15%
53.8
±5%
15.2
±10%
24.6
±15%
35.3
±5%
25.0
±10%
51.9
±15%
63.5
±5%
23.8
±10%
39.6
±15%
58.4
46.2
69.2
88.5
21.0
37.5
60.3
30.8
48.1
88.5
34.7
59.4
84.2
65.4
92.3
96.2
31.3
58.5
65.6
48.1
76.9
90.4
43.6
76.2
91.1
Lockhart & Martinelli(1949)
Sterman(1956)
Filimonov et al(1957)
50.0
23.1
73.1
76.9
38.5
92.3
96.2
42.3
92.3
42.4
21.0
34.4
56.3
32.6
60.7
67.4
40.2
80.4
42.3
11.5
63.5
63.5
42.3
80.8
84.6
50.0
86.5
45.5
14.9
37.6
55.4
25.7
68.3
62.4
32.7
88.1
Chisholm & Laird(1958)
Flanigan(1958)
15.4
7.7
23.1
19.2
26.9
34.6
14.7
25.0
23.7
36.2
30.4
45.1
7.7
1.9
36.5
17.3
50.0
36.5
12.9
25.7
20.8
34.7
25.7
39.6
Dimentiev et al
Hoogendroon(1959)
Bankoff(1960)
26.9
0.0
0.0
34.6
15.4
7.7
50.0
34.6
19.2
20.1
28.1
14.3
35.3
45.5
24.1
44.2
52.2
36.2
21.2
7.7
0.0
59.6
17.3
0.0
63.5
48.1
23.1
20.8
43.6
2.0
29.7
63.4
6.9
36.6
74.3
30.7
Fauske(1961)
Wilson et al(1961,1965)
26.9
11.5
26.9
26.9
30.8
38.5
18.3
7.6
27.2
16.5
31.7
26.8
19.2
11.5
46.2
23.1
50.0
34.6
18.8
3.0
23.8
5.0
25.7
12.9
Hughmark(1962)
38.5
92.3
96.2
56.7
71.4
75.4
30.8
75.0
88.5
68.3
78.2
88.1
Nicklin et al(1962)
11.5
53.8
88.5
40.2
62.5
80.8
17.3
34.6
76.9
32.7
59.4
84.2
Nishino & Yamazaki(1963)
34.6
57.7
65.4
48.2
62.9
74.6
44.2
59.6
73.1
38.6
46.5
51.5
Fujie(1964)
Kowalczewski(1964)
26.9
38.5
57.7
50.0
76.9
57.7
17.4
44.2
31.3
52.7
49.6
61.2
30.8
42.3
59.6
65.4
73.1
71.2
31.7
36.6
57.4
48.5
78.2
51.5
Thom(1964)
Zivi(1964)
34.6
34.6
50.0
46.2
61.5
53.8
27.7
36.6
49.1
45.5
56.7
52.7
36.5
44.2
53.8
59.6
67.3
71.2
19.8
27.7
36.6
39.6
46.5
41.6
Hughmark(1965)
34.6
65.4
88.5
17.4
36.6
55.4
26.9
42.3
73.1
34.7
55.4
78.2
Neal & Bankoff(1965)
11.5
19.2
23.1
10.7
20.1
29.9
11.5
15.4
17.3
5.9
10.9
19.8
Turner & Wallis(1965)
Baroczy(1966)
11.5
34.6
19.2
53.8
26.9
57.7
10.7
42.9
17.4
55.4
26.8
67.0
9.6
48.1
11.5
59.6
25.0
71.2
6.9
36.6
12.9
47.5
21.8
48.5
Homogeneous
Armand(1946)
Armand - Massina
1
2
Guzhov et al(1967)
3
Kutucuglu
Smith(1969)
Wallis(1969)
0.0
0.0
0.0
0.0
0.0
9.4
0.0
0.0
0.0
0.0
0.0
4.0
23.1
50.0
53.8
46.2
84.6
84.6
57.7
96.2
92.3
35.3
47.3
44.2
44.2
57.6
57.1
52.7
68.3
69.6
38.5
40.4
48.1
51.9
73.1
75.0
69.2
84.6
88.5
23.8
47.5
42.6
31.7
82.2
54.5
40.6
95.0
68.3
Gregory & Scott(1969)
46.2
69.2
100.0
21.9
40.2
62.5
30.8
51.9
88.5
35.6
61.4
93.1
Premoli et al(1970)
57.7
80.8
92.3
57.6
73.2
79.0
57.7
82.7
86.5
47.5
61.4
82.2
4
Rouhani I(1970)
50.0
84.6
96.2
47.8
80.4
84.4
32.7
69.2
82.7
34.7
74.3
91.1
Rouhani II(1970)
Bonnecaze et al(1971)
Dix(1971)
34.6
11.5
61.5
53.8
53.8
88.5
80.8
88.5
96.2
19.2
40.2
62.9
41.1
62.1
74.6
57.6
80.8
83.9
7.7
17.3
50.0
28.8
34.6
78.8
63.5
76.9
82.7
33.7
32.7
55.4
72.3
59.4
64.4
87.1
84.2
72.3
Beggs(1972)
73.1
96.2
96.2
35.3
51.8
65.2
50.0
76.9
82.7
36.6
56.4
70.3
Chisholm(1973)
65.4
96.2
100.0
45.1
56.3
65.2
46.2
78.8
88.5
57.4
90.1
95.0
Loscher & Reinhardt(1973)
Mattar & Gregory(1974)
26.9
0.0
30.8
0.0
38.5
0.0
26.3
0.0
40.2
1.3
44.2
19.2
50.0
0.0
67.3
0.0
80.8
3.8
35.6
0.0
40.6
1.0
49.5
18.8
Moussali
Greskovich & Cooper(1975)
23.1
42.3
42.3
76.9
69.2
100.0
15.6
20.1
25.0
37.5
35.7
62.1
30.8
46.2
51.9
67.3
67.3
78.8
23.8
41.6
46.5
69.3
64.4
88.1
Madsen(1975)
Mukherjee(1979)
0.0
61.5
0.0
92.3
0.0
96.2
0.0
33.0
1.8
61.2
2.2
72.3
0.0
53.8
0.0
71.2
0.0
82.7
0.0
31.7
0.0
54.5
0.0
67.3
Gardner-1(1980)
Gardner-2(1980)
Sun et al(1980)
15.4
11.5
7.7
26.9
46.2
50.0
50.0
53.8
73.1
23.2
29.5
37.5
34.4
42.4
58.9
42.4
50.9
75.0
13.5
57.7
23.1
26.9
69.2
51.9
44.2
71.2
86.5
22.8
25.7
19.8
29.7
30.7
49.5
35.6
35.6
74.3
Jowitt(1981)
Chisholm(1983) - Armand(1946)
0.0
61.5
0.0
92.3
0.0
100.0
13.4
46.9
33.5
56.3
59.8
65.2
1.9
46.2
1.9
75.0
5.8
86.5
8.9
57.4
27.7
89.1
59.4
95.0
4
5
Spedding & Chen(1984)
30.8
46.2
57.7
45.1
55.8
71.0
44.2
53.8
63.5
39.6
54.5
61.4
Bestion(1985)
23.1
38.5
57.7
13.4
29.0
37.5
17.3
53.8
71.2
41.6
56.4
70.3
Tandon et al(1985)
30.8
57.7
61.5
30.8
47.8
54.0
40.4
55.8
65.4
21.8
34.7
35.6
Chen(1986)
38.5
69.2
84.6
46.0
57.6
65.2
50.0
69.2
86.5
46.5
65.3
74.3
Hamersma & Hart(1987)
15.4
26.9
26.9
14.7
23.7
30.4
17.3
38.5
51.9
13.9
20.8
26.7
114
Table 21. Percentage of data points correctly predicted for vertical flow by data set
(contd.)
Beggs
(1972)
(Air -Water)
Correlation / Data Source
Spedding & Nguyen
(1976)
(Air -Water)
Total Points 26
Percentage within ±5%
Kawaji et al(1987)
26.9
±10%
46.2
±15%
53.8
Total Points 224
±5%
23.2
±10%
41.5
±15%
51.3
Mukherjee
(1979)
(Air -Kerosene)
Total Points 52
±5%
34.6
±10%
55.8
±15%
69.2
Sujumnong
(1997)
(Air -Water)
Total Points 101
±5%
26.7
±10%
38.6
±15%
47.5
Minami & Brill(1987)
73.1
92.3
100.0
31.7
58.5
72.8
55.8
78.8
84.6
27.7
52.5
67.3
El-Boher et al(1988)
34.6
76.9
88.5
62.5
74.1
83.5
21.2
65.4
80.8
34.7
48.5
57.4
Hart et al(1989)
Kokal & Stanislav(1989)
Spedding et al(1989)
34.6
11.5
30.8
46.2
53.8
46.2
57.7
88.5
57.7
37.5
39.7
40.6
45.5
62.1
48.7
50.4
79.9
59.4
42.3
17.3
44.2
53.8
34.6
61.5
65.4
76.9
69.2
26.7
32.7
32.7
29.7
59.4
42.6
33.7
84.2
45.5
Toshiba(1989)
Huq & Loth(1992)
46.2
46.2
92.3
65.4
92.3
96.2
46.9
50.0
83.0
62.1
89.7
70.1
36.5
38.5
82.7
69.2
86.5
82.7
47.5
42.6
88.1
71.3
95.0
90.1
Inoue et al(1993)
Czop et al(1994)
AbdulMajeed(1996)
34.6
11.5
42.3
65.4
30.8
61.5
88.5
53.8
73.1
42.4
17.4
28.1
58.9
37.5
49.1
63.4
54.5
55.4
50.0
17.3
32.7
71.2
25.0
55.8
82.7
59.6
67.3
50.5
15.8
20.8
76.2
23.8
40.6
84.2
41.6
48.5
Maier and Coddington(1997)
Petalaz & Aziz(1997)
26.9
3.8
50.0
11.5
57.7
23.1
40.2
4.0
55.8
8.0
59.8
10.7
50.0
0.0
67.3
3.8
75.0
3.8
32.7
4.0
49.5
7.9
52.5
12.9
Gomez et al(2000)
0.0
0.0
3.8
21.9
34.4
38.8
0.0
1.9
3.8
13.9
18.8
24.8
Zhao et al(2000)
0.0
11.5
11.5
11.6
16.5
21.9
0.0
0.0
0.0
5.9
12.9
16.8
Graham et al(2001)
53.8
61.5
92.3
42.9
55.4
65.2
44.2
65.4
76.9
39.6
49.5
58.4
1
correlation as reported by Leung (2005), 2,5 correlations as reported by Kataoka and Ishii (1987), 3
correlation as reported by Isbin and Biddle (1979) , 4 Rouhani and Axelsson(1970) correlations I and II.
vertical case. It also gave an excellent result for the higher void fraction range
comparable to Dix (1971).
The combined air water data of Beggs (1972), Spedding and Nguyen (1976) and
Sujumnong (1997) is well captured by Toshiba (1989) that predicted 81.5% of the
data within the 15% index. The correlations of Rouhani I (1970) and Filimonov et al.
(1957) gave comparably close predictions to that of Toshiba (1989). The next best
predictions are from Dix (1971), Premoli et al. (1970) and Hughmark (1962) which
gave results of 81.5, 80.9 and 80.6% for the same error index.
Owing to the small number of data in the air kerosene vertical data set of Mukherjee
(1979), fifteen correlations were able to predict 85% and above for the 15% error
percentage. Armand Massina (Leung 2005) gave 90.4% while Chisholm (1973) and
Armand (1946) predicted 88.5% each, respectively. The correlations by Filimonov et
al. (1957), Premoli et al. (1970), Sun et al. (1980) Chisholm (1983) – Armand (1946),
115
Chen (1986) and Toshiba (1989) all predicted 86.5% of the data sets within the wider
error band of 15%.
4.4 Summarized comparison and discussion for all data sets
We have presented an extensive comparison of the predictive capability of the
different void fraction correlations with experimental data for horizontal, inclined and
vertical flow cases. Recommendations and conclusions were drawn for each section
with regard to the performance of the top performing correlations. The top six
correlations which predicted about 85% of the whole data base within the 15% error
index were selected and their consistency across the void fraction spectrum presented.
The total number and percentage of data points correctly predicted for the whole data
base by each correlation is given in Table 23.
Based on the analysis for the three scenarios considered, six correlations are
recommended for acceptably predicting void fraction for horizontal and upward
inclined pipes regardless of flow regimes.
Basing our judgment of the capability of the correlations only on the total number of
data points correctly predicted, we can have two groups of correlations where the
three correlations within each group have very close performance to one another.
The first group of correlations consists of Toshiba (1989), Rouhani I(1970) and
Dix(1971) with 85.3,84.2 and 83.1 %,respectively, prediction of the total points
within the 15% error index. The second group is comprised of Hughmark (1962),
Premoli et al. (1970) and Filimonov et al. (1957) with 81.6,81 and 80.6 %,
respectively, of the total points correctly captured within the 15% error index.
Hence, the correlation by Toshiba (1989) has shown best prediction capability in the
inclined and vertical flow cases with a very good agreement with the horizontal
experimental data set making it the top correlation to be worth a general
116
Table 22. Total number and percentage of data points correctly predicted for Vertical
flow database (all sources)
Correlation / Data Source
Total data points correctly
Percentage of data points correctly
predicted for the whole database predicted for the whole database
Total Points 403
Percentage within
Homogeneous
Armand(1946)
1
Armand - Massina
Lockhart & Martinelli(1949)
Sterman(1956)
Filimonov et al(1957)
Chisholm & Laird(1958)
Flanigan(1958)
2
Dimentiev et al
Hoogendroon(1959)
Bankoff(1960)
Fauske(1961)
Wilson et al(1961,1965)
Hughmark(1962)
Nicklin et al(1962)
Nishino & Yamazaki(1963)
Fujie(1964)
Kowalczewski(1964)
Thom(1964)
Zivi(1964)
Hughmark(1965)
Neal & Bankoff(1965)
Turner & Wallis(1965)
Baroczy(1966)
Guzhov et al(1967)
3
Kutucuglu
Smith(1969)
Wallis(1969)
Gregory & Scott(1969)
Premoli et al(1970)
4
Rouhani I(1970)
Total Points 403
±5%
76
110
±10%
131
187
±15%
185
289
±5%
18.9
27.3
±10%
32.5
46.4
±15%
45.9
71.7
156
176
74
167
54
85
272
235
131
271
99
130
311
283
160
338
127
169
38.7
43.7
18.4
41.4
13.4
21.1
67.5
58.3
32.5
67.2
24.6
32.3
77.2
70.2
39.7
83.9
31.5
41.9
84
111
34
77
29
222
135
179
94
168
110
142
97
39
39
167
0
149
179
63
116
61
302
232
234
174
214
188
185
177
69
63
217
0
182
226
129
131
101
329
329
274
248
241
225
211
264
102
102
251
25
20.8
27.5
8.4
19.1
7.2
55.1
33.5
44.4
23.3
41.7
27.3
35.2
24.1
9.7
9.7
41.4
0.0
37.0
44.4
15.6
28.8
15.1
74.9
57.6
58.1
43.2
53.1
46.7
45.9
43.9
17.1
15.6
53.8
0.0
45.2
56.1
32.0
32.5
25.1
81.6
81.6
68.0
61.5
59.8
55.8
52.4
65.5
25.3
25.3
62.3
6.2
129
188
181
113
222
170
272
244
197
290
210
318
295
306
329
32.0
46.7
44.9
28.0
55.1
42.2
67.5
60.5
48.9
72.0
52.1
78.9
73.2
75.9
81.6
172
313
349
42.7
77.7
86.6
Rouhani II(1970)
Bonnecaze et al(1971)
Dix(1971)
Beggs(1972)
90
135
239
161
194
231
296
238
271
329
329
285
22.3
33.5
59.3
40.0
48.1
57.3
73.4
59.1
67.2
81.6
81.6
70.7
Chisholm(1973)
200
283
314
49.6
70.2
77.9
Loscher & Reinhardt(1973)
Mattar & Gregory(1974)
128
0
174
4
201
64
31.8
0.0
43.2
1.0
49.9
15.9
Moussali
Greskovich & Cooper(1975)
Madsen(1975)
Mukherjee(1979)
81
122
0
150
141
209
4
253
198
295
5
298
20.1
30.3
0.0
37.2
35.0
51.9
1.0
62.8
49.1
73.2
1.2
73.9
Gardner-1(1980)
Gardner-2(1980)
86
125
128
174
167
201
21.3
31.0
31.8
43.2
41.4
49.9
4
5
117
Table 22. Total number and percentage of data points correctly predicted for Vertical
flow database (all sources) (contd.)
Correlation / Data Source
Total data points correctly
Percentage of data points correctly
predicted for the whole database predicted for the whole database
Total Points 403
Total Points 403
Sun et al(1980)
Jowitt(1981)
Chisholm(1983) - Armand(1946)
Spedding & Chen(1984)
Bestion(1985)
Tandon et al(1985)
Chen(1986)
Hamersma & Hart(1987)
±5%
118
40
203
172
87
120
186
60
±10%
222
104
279
220
160
186
249
101
±15%
307
197
313
269
207
207
288
129
±5%
29.3
9.9
50.4
42.7
21.6
29.8
46.2
14.9
±10%
55.1
25.8
69.2
54.6
39.7
46.2
61.8
25.1
±15%
76.2
48.9
77.7
66.7
51.4
51.4
71.5
32.0
Kawaji et al(1987)
Minami & Brill(1987)
El-Boher et al(1988)
Hart et al(1989)
Kokal & Stanislav(1989)
Spedding et al(1989)
Toshiba(1989)
Huq & Loth(1992)
Inoue et al(1993)
Czop et al(1994)
AbdulMajeed(1996)
Maier and Coddington(1997)
Petalaz & Aziz(1997)
Gomez et al(2000)
Zhao et al(2000)
Graham et al(2001)
104
147
195
142
134
155
184
187
181
67
112
156
14
63
32
173
173
249
269
172
231
196
342
264
263
129
196
223
31
97
53
224
213
301
310
196
327
230
366
316
293
209
227
241
45
115
69
269
25.8
36.5
48.4
35.2
33.3
38.5
45.7
46.4
44.9
16.6
27.8
38.7
3.5
15.6
7.9
42.9
42.9
61.8
66.7
42.7
57.3
48.6
84.9
65.5
65.3
32.0
48.6
55.3
7.7
24.1
13.2
55.6
52.9
74.7
76.9
48.6
81.1
57.1
90.8
78.4
72.7
51.9
56.3
59.8
11.2
28.5
17.1
66.7
Percentage within
1
correlation as reported by Leung (2005), 2,5 correlations as reported by Kataoka and Ishii (1987), 3
correlation as reported by Isbin and Biddle (1979) , 4 Rouhani and Axelsson(1970) correlations I and II.
recommendation. The next on the list would be that of Rouhani I (1970) with a very
close performance to that of Toshiba (1989). On top of the slight numerical percent
advantage, its relatively good consistency across the whole void fraction range for the
total data set has been given weight over the excellent consistency shown by Dix
(1971) for the combined inclined experimental data set with a wide scatter for the
horizontal data which is placed third. However, it should be pointed out that the Dix
(1971) correlation has the best prediction with the tighter error indices of 5 and 10%
above all of the correlations considered in this analysis.
118
1.0
+15%
U
U
UU
UU
SS
S
U
S
S
SS
SS
S
S
U
S
U
S
U
SUSS
U
S
U
S
S
S
SS
SSU
S
US
U
U
S
U
S
S
USS
S
S
S
M
SM
B
S SM
BM
USSS
USS
MS B
M
SM
SSS
SS
SSUM
US
S
S
S
MM
US
US
U
MM
MM
S
SS
MM
U S
SSSSM SM B M
S
S
S
M
BB
S
SS
USMSSS M
MM
S SUSSU B B
M
SS S
U
U
M
SM
MM
S S U
M
UM
U S
USS S S S
M
U MBB
U
S B MB
S
U
US
SB
U
BM
S US
U
U
U U SSS S M
UU S S
B
S
U
S
S
U
USS
U
SU B
B
S
M
U BB
UU
SS S
S
S S UU
S
S
SS
M S
BB
S
S U
U
U U
U
S
S
S
U US SMS
U S B
S S
S
UU S
SS
U
S
SU SS
S S
M
S
S S
B
MS
U B S
S SSU
S
U
S U
S SS
SU
S
S
SUS
M SS
SS
S U S S
M
S SS SU S
USS
S
U SS
S
S
U
SU
S
S
S
SS
SB
S
M
SU U S
M
U
M
UU U
S SUSS
M
S
U
U
S
S
MU
M
US SU
UU
B
0.8
Beggs(1972)
Spedding & Nguyen(1976)
Mukherjee(1979)
Sujumnong(1997)
Calculated void fraction
-15%
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Measured void fraction
Figure 35. Comparison of Toshiba (1989) correlation with
measured combined vertical experimental data
1.0
S
S
SS
S
SU
SSSB
US
S
S MU
SU
U
S
S
MMSUSS
S
SMSSSSU
U
S M BSB M
BMU
U
B
MS
UB
SS
Beggs(1972)
M
SU
SS
M
M
U
M
USS
M
MU
U
S
M
S
SS M UM
S SS
M
U
S
U
SS
SU
S
U
S
M
S
S
U
S
M
S
S
U
SM
M
S M
S
S
Spedding & Nguyen(1976)
US
SS
US S
S
U
S
S
S
S
S
S
U
S
S
U
MM
SSS
SS
MB
S SS B MM
M
US
SUS
M
USMS
M
S
S
SS
USS
M
M
Mukherjee(1979)
SS UU B
MB B
SS
S SB
U S S
U
S
S SUSU
U
M
U
S
U
Sujumnong(1997)
S SS S SU
UM B
S
M
S USS B
S
B UU
USS
U
U
S B
U
SSS S
S
S M
U
B
S
U
S U
S
U
S S US
S SU MU
U US
S
S
BBB
S
S S U MU
S
S SS B S U
U
S S S
U
B
M
SS U S
S
S
S
S SS S
U UU
S
S
U
S
S
S
U U B M
S
S
S S S U
U
S SS
SS
S
BU
S
SS S M
M
B
S SS
SS
SS
S UU
S
U U
S
S U
SS
S
S S SS UM
M
S
M
U
SS
U
S SSS MU
M
S
S S S UU U
B
M
S
U
M S SS U
U
S
U
UUU
S SU
U
S U
US U
U
U
+15%
0.8
S
Calculated void fraction
-15%
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
Measured void fraction
Figure 36. Comparison of Rouhani I (1970) correlation with
measured combined vertical experimental data
119
1.0
1.0
U
U
UU
UU
U
S
S
S
S
S
S
SS
U
S
U
SSS
U
SU
S
U
S
U
SUS
S
US
U
S
U
SB
S
SM
USUSS
SB
SSS
S
SS
S
SS
S
US
USS
S
S
S
SSUSM
BM
M
SM
M B
SSSSS
M
S SM
UUUS
SS
US
M
SBM
S
SM
S SM
S
S
M
S
M
MM M
S S S SM M
S
USU MS S
SSS S SS
M BB
SSU M
M
B
Beggs(1972)
MS
U SS
U S
M
SS S
U B B
S
S
U
U
S
U
MM
Spedding & Nguyen(1976)
UM
M
SM
SS S
MB
M
U
B
S BU
U
M
M
Mukherjee(1979)
B SS S
SB
U
U
BU
U
US U S S
Sujumnong(1997)
S M M
UU
S
U
B
S U UB S S S
U
US
S
S
U
S
U US S
B
M
B
B
UU
S
S
SS S
B
S
B
S
UM
UU
S
SU
U SSS
U U
S
S
M
S
U
B
S
UU
S S
SS
U
US
SS
S
S
S
S
S
S
B
S
S
M
S
U B
M
U
US
S S
S
US S
S
S
US S
U
S
S
M S
SS
U SS SSS
S
S
U
U
S
S S
S
M
S
SS
UU SU SS
SS
S
S
S
B
U
S
S
S
S
S
U SSS S S
U
M
US
USU U S
M
M
SS
U
U
SS S
M
U
S
M
M
U
US S
UU
+15%U
0.8
Calculated void fraction
-15%
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Measured void fraction
Figure 37. Comparison of Filimonov et al. (1957) correlation with
measured combined vertical experimental data
1.0
U
U
S
S
S
S
US
US
UUS
U
S
UU
SS
USS
S
S
UU
S SS
SB
B
US S S
US S
US
US SS
S
SS
US
SUSBM
SS
S
UU
SS
S
BM
SSUSSS M
SUS
SS
S
USS
M
M
M
M
U S
SSSS
B
M
S
S
M
S
US MM BM BMM
SS
US
S
U
S
M
S S
SS
M M
S
S
MMM
SUS M
SS
UMS
SS
MS
S
UM BB B
U
S SUSSU
MM
M
S
SS
M
S
S
M
U
M
B
S
U
U
SBSU U B
U
S S
S
S
M
S SS
U
S
MUU
S S US S B
M
U
M
S BS
S S
US
S
U B
U
S B
U SS
S M
S
U U
S
S
US S
S U
SU
S
UM
S
S
UB
U BB
S S SU U
B
SS
UM
S
M
U
B
U
SU
U
B M
+15%
0.8
Calculated void fraction
-15%
S
SS
S
0.6
S
0.4
SS
S
0.2
0.0
0.0
0.2
S
S
S
S
U
SSS
U
S U U
SS
UU
S
BU
S
S
S
SS
S
S S
S
S S S
S
S
S
UM
S
M
S
U
S
U
U
S
S
S M
M
SS S U U
UB
SS
M SS S U U
UU
U
U
S SU U
U
S U
US U
U
U
SS
S
S
S
S
S
S
S
S
S
S
S
SS
UU
U
S U
U
S
M
BM
M
B
M
S
M
U
0.4
0.6
Beggs(1972)
Spedding & Nguyen(1976)
Mukherjee(1979)
Sujumnong(1997)
0.8
Measured void fraction
Figure 38. Comparison of Hughmark (1962) correlation with
measured combined vertical experimental data
120
1.0
1.0
S
S
S
U
S
S
S
U
S
SU
S
SU
UU
U
B
U
SSS
S
S
SS
MS
SS
U
M
SS
S USUU
M
MS
US
SB
S
U
M UUM
S
B
S
M
S
S
M
S
S
UUSS
S
S
SS MU
SSM M
SU
S
SM
B MB M
U
B
MS
SS
Beggs(1972)
M USS S
M
SU M M
S
S
SUSS S SB MMM
SS U
SSS
MM SS
S
USSS
M
U
Spedding & Nguyen(1976)
S
S S
S
S
M
SM
USM S
M
S S
S
M S
M
SU M B B
Mukherjee(1979)
S
B
S
S
S
M
U
U
S S SU
UB
U
US
Sujumnong(1997)
SM
SUBS
M
SS U M
S B MU
S
U
U
S
S
S
BS BU
S
S
U
U
SS
B
M
S UUSS
U
U
S S
UU
U B S
S
SU SU
U
M
US
B
S BB
U U U SSS
U
U
S BS S MS S
B
U
S
S SS
U UU U
SM
S
UU S
B
U
S
U
U
U
S S
S
S
S
SS S
S
M S SS
B
U
U U
M
U
S
B
S
SS
S
U
U
S
S
S
S
US
U M
S
S
S
S
S
S
S
SS
U
S
S
S
UU
U S SS
U
U
S
S
S
S
U
M
U
S S
SS
SS
UU
S
BS
SS
UU
SS
S
U
U
SS S
SS
SS
US
M
S S S
S
U
M
S
M
U S S
S
S
M
S
SS M
M
UU
+15%
0.8
Calculated void fraction
-15%
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Measured void fraction
Figure 39. Comparison of Dix (1971) correlation with
measured combined vertical experimental data
1.0
S
S
S
S
S
S
U
SU
S
S
SSU
B
S
U
SS
S M
U
S
S
SU
U
SSU
SS
SM
U
M
S S
M
S
BSB S
U
M
MSS
S
SM
B
S
M
S
S
B
S
U
U
M
S
S
SUM
SSMM SSS
SS
S
U
S
S
U
S
S
MM
SM
S
USM
M
M
S
S
U
U
SU U
M
U
S SS
MS
M
BSS S
S
S
MSS
S
U B
US
SMM S SS
M
S
SS S
SMSM S
UUS U MM
MM SBS
S
SU BS
S
B
S BS
U
S MB
S SUSU
U
SS S S U
MUS
S S
BSU UM
BS U U
S SS SU
U
U
S
BS S
SB
S
U
U
SU U M
+15%
S
0.8
S
S
-15%
Calculated void fraction
S
S
S
S
S
0.6
S
S
S
0.2
0.0
M
S
S
S
S
S
S
S
0.2
S
SU S
S
UU
S
M
S
S S
S B
S SS
S
M
U
S
S
U U
S
S M
S
U
S
M
S
S
S
S U U
S SS
U
S
S
U
S
U
M
SS
S UU BU
SS
U
S S
S
U
U
S S UU U
U
U
S U
US U
U
U
0.0
S
S
S
0.4
S
S
S
USS
U
S
S
S SU
SB
U
S S
S
U SU
SU
S
S SU
SUSUU SM
B
U
SS BU
B
S
US
M S B S
U
M UM
S BS U
S
U
SS
S
U
U
U
S
U
B
M
UM
M
B
S
M
U
0.4
0.6
S
Beggs(1972)
Spedding & Nguyen(1976)
Mukherjee(1979)
Sujumnong(1997)
0.8
Measured void fraction
Figure 40. Comparison of Premoli et al. (1970) correlation with
measured combined vertical experimental data
121
1.0
Table 23. Total number and percentage of data points correctly predicted for the
whole database (all sources)
Correlation / Data Source
Total data points correctly
Percentage of data points correctly
predicted for the whole database predicted for the whole database
Total Points 2845
Percentage within
Homogeneous
Armand(1946)
1
Armand - Massina
Lockhart & Martinelli(1949)
Sterman(1956)
Filimonov et al(1957)
Chisholm & Laird(1958)
Flanigan(1958)
2
Dimentiev et al
Hoogendroon(1959)
Bankoff(1960)
Fauske(1961)
Wilson et al(1961,1965)
Hughmark(1962)
Nicklin et al(1962)
Nishino & Yamazaki(1963)
Fujie(1964)
Kowalczewski(1964)
Thom(1964)
Zivi(1964)
Hughmark(1965)
Neal & Bankoff(1965)
Turner & Wallis(1965)
Baroczy(1966)
Guzhov et al(1967)
3
Kutucuglu
Smith(1969)
Wallis(1969)
Gregory & Scott(1969)
Premoli et al(1970)
4
Rouhani I(1970)
4
Rouhani II(1970)
Bonnecaze et al(1971)
Dix(1971)
Beggs(1972)
Chisholm(1973)
Loscher & Reinhardt(1973)
Mattar & Gregory(1974)
5
Moussali
Greskovich & Cooper(1975)
Madsen(1975)
Mukherjee(1979)
Gardner-1(1980)
Gardner-2(1980)
Total Points 2845
±5%
688
657
1205
1364
585
1369
532
517
±10%
1109
1234
1992
1852
927
1953
828
917
±15%
1449
1952
2259
2128
1163
2294
1060
1231
±5%
24.2
23.1
42.4
47.9
20.6
48.1
18.7
18.2
±10%
39.0
43.4
70.0
65.1
32.6
68.6
29.1
32.2
±15%
50.9
68.6
79.4
74.8
40.9
80.6
37.3
43.3
656
557
215
708
264
1244
578
1194
829
1179
946
1076
601
277
406
1129
5
1083
1018
467
957
545
2003
1211
1630
1387
1574
1441
1367
1137
485
642
1505
71
1386
1424
869
1093
768
2322
1921
1899
1831
1758
1692
1565
1788
679
847
1762
214
23.1
19.6
7.6
24.9
9.3
43.7
20.3
42.0
29.1
41.4
33.3
37.8
21.1
9.7
14.3
39.7
0.2
38.1
35.8
16.4
33.6
19.2
70.4
42.6
57.3
48.8
55.3
50.7
48.0
40.0
17.0
22.6
52.9
2.5
48.7
50.1
30.5
38.4
27.0
81.6
67.5
66.7
64.4
61.8
59.5
55.0
62.8
23.9
29.8
61.9
7.5
1022
1355
1349
679
1643
1345
1925
1842
1289
2084
1554
2213
2130
2104
2304
35.9
47.6
47.4
23.9
57.8
47.3
67.7
64.7
45.3
73.3
54.6
77.8
74.9
74.0
81.0
1082
2059
2395
38.0
72.4
84.2
610
582
1597
1304
1382
1188
40
1273
1211
2139
1843
1959
1454
141
1810
1925
2363
2141
2234
1638
417
21.4
20.5
56.1
45.8
48.6
41.8
1.4
44.7
42.6
75.2
64.8
68.9
51.1
5.0
63.6
67.7
83.1
75.3
78.5
57.6
14.7
758
1027
80
1262
642
880
1224
1575
137
1872
1048
1324
1583
2014
178
2217
1353
1575
26.6
36.1
2.8
44.4
22.6
30.9
43.0
55.4
4.8
65.8
36.8
46.5
55.6
70.8
6.3
77.9
47.6
55.4
122
Table 23. Total number and percentage of data points correctly predicted for the
whole database (all sources)(contd.)
Correlation / Data Source
Total data points correctly
Percentage of data points correctly
predicted for the whole database predicted for the whole database
Total Points 2845
Total Points 2845
Sun et al(1980)
Jowitt(1981)
Chisholm(1983) - Armand(1946)
Spedding & Chen(1984)
Bestion(1985)
Tandon et al(1985)
Chen(1986)
Hamersma & Hart(1987)
±5%
590
235
1349
1172
674
1085
1339
571
±10%
1276
522
1940
1549
1155
1500
1780
873
±15%
1960
1057
2237
1855
1455
1748
2022
1097
±5%
20.7
8.3
47.4
41.2
23.7
38.1
47.1
20.1
±10%
44.9
18.3
68.2
54.4
40.6
52.7
62.6
30.7
±15%
68.9
37.2
78.6
65.2
51.1
61.4
71.1
38.6
Kawaji et al(1987)
Minami & Brill(1987)
El-Boher et al(1988)
Hart et al(1989)
Kokal & Stanislav(1989)
Spedding et al(1989)
Toshiba(1989)
Huq & Loth(1992)
Inoue et al(1993)
Czop et al(1994)
AbdulMajeed(1996)
Maier and Coddington(1997)
Petalaz & Aziz(1997)
Gomez et al(2000)
Zhao et al(2000)
Graham et al(2001)
815
1315
1176
1191
579
1083
1065
1351
1109
379
887
1040
108
481
297
1347
1315
1887
1808
1479
1208
1433
2137
1888
1709
770
1526
1527
224
715
441
1810
1615
2208
2177
1649
1921
1663
2427
2195
2019
1347
1776
1744
346
879
550
2092
28.6
46.2
41.3
41.9
20.4
38.1
37.4
47.5
39.0
13.3
31.2
36.6
3.8
16.9
10.4
47.3
46.2
66.3
63.6
52.0
42.5
50.4
75.1
66.4
60.1
27.1
53.6
53.7
7.9
25.1
15.5
63.6
56.8
77.6
76.5
58.0
67.5
58.5
85.3
77.2
71.0
47.3
62.4
61.3
12.2
30.9
19.3
73.5
Percentage within
1
correlation as reported by Leung (2005), 2,5 correlations as reported by Kataoka and Ishii (1987), 3
correlation as reported by Isbin and Biddle (1979) , 4 Rouhani and Axelsson(1970) correlations I and II.
On the second group of correlations, though ranked lower than that Hughmark (1962)
in Table 24 based on the 15 % error index for the whole data set, Premoli et al.(1970)
correlation has the best capability in the tighter indices which is very comparable to
that of Dix(1971). Moreover, the Premoli et al. (1970) correlation has predicted all the
data bases with very good accuracy and is the second only correlation to do so behind
Rouhani I (1970). Hence it would deserve a general recommendation above that of
Hughmark (1962) and Filimonov(1957) within this group.
The dilemma to chose between the correlations of Rouhani I(1970) and Premoli et
al.(1970) over Dix(1971) and Filimonov et al.(1957) respectively could be resolved if
123
Table 24. Total number and percentage of data points correctly predicted by the top
six correlations for the whole database (all sources)
Total data points
Percentage of data points
correctly predicted for
correctly predicted for the
Correlation
the whole database
whole database
Total Points 2845
Total Points 2845
Percentage within ±5%
±10% ±15% ±5%
±10%
±15%
Toshiba(1989)
1065
2137
2427
37.4
75.1
85.3
1
Rouhani I(1970)1
Dix(1971)
Hughmark(1962)
Premoli et al.(1970)
Filimonov et al.(1957)
1082
1597
1244
1643
1369
2059
2139
2003
2084
1953
2395
2363
2322
2304
2294
38.0
56.1
43.7
57.8
48.1
72.4
75.2
70.4
73.3
68.6
84.2
83.1
81.6
81.0
80.6
Rouhani and Axelsson(1970) correlations I.
one is interested to predict void fraction in inclined flow only. For this case the
superior performances of the latter two correlations would out weigh the general
applicability of the former ones.
In addition to the correlations discussed above, the correlations of Chisholm (1973),
Chisholm (1983) – Armand (1946) and Armand-Massina (Leung 2005) have also
given reasonable prediction for the whole data base with quite a very good
performance for horizontal data sets. Their performances are very much close to one
another which is to be expected as they are combinations and/or derivations of each
other.
The scatter plots for the correlations of Toshiba (1989),Rouhani I (1970),Dix (1971),
Premoli et al. (1970), Hughmark (1962) and Filimonov et al. (1957) with the whole
data set are given in Figures 41 to 46.
We have mentioned the general failure of all the correlations in predicting any of the
experimental data within the 5% index. The accuracy of the data set within this range
is also one issue which we have raised as open to question and may not at all be the
problem of the correlations.
124
To make a conclusive statement on this and also in trying to predict the compiled
experimental data base with greater accuracy than what we have achieved so far, an
effort is made to modify one of the best performing correlations so as to remove some
of its weaknesses.
The next section would discuss the details of how this has been undertaken.
MODIFIED CORRELATION
The drift flux analysis given by Zuber and Findlay (1965) has been proven to be quite
a powerful tool to correlate experimental data as well as develop void fraction
correlations. Of the six best performing void fraction correlations we have
recommended all except the correlation of Hughmark (1962) are developed based on
that model. It is to be recalled that this types of correlations have the general
expression of the form
α=
USG
C0U M + UGM
where C0 is the distribution parameter and UGM = UG − U M is the drift velocity.
A decision was made to make improvements on the top performing correlations so as
to overcome some of the weaknesses observed in the analysis. It is worth mentioning
here that the correlations of Rouhani I (1970) and Dix (1971) share similar parameters
in their correlations. They differ in their expression for the distribution parameter
while they share exactly the same expression for their drift velocity parameter with a
very minor difference in the constant term.
Considering the fact that the Dix (1971) correlation has an outstanding consistency in
all its predictions over the entire void fraction range and the fact that it has the highest
prediction of data points within the tighter 5% error index makes it a prime candidate
for further improvement.
125
The correlation of Dix (1971) and all the other correlations have significantly
underperformed for the data of Spedding and Nguyen (1976). A close look at the data
reveals that the system pressure is quite different from that of all the other data sets
considered here. The prediction of the correlations further deteriorates with
inclination angle for this data set. The other variable with this data set is the slightly
“off standard” pipe diameter used in their experiment. Using these inputs, a damping
correction factor is suggested which takes into account some of the observations made
in previous void fraction developments. These are:
The drift velocity in the two phase slug flow which is predominantly encountered in
horizontal as well as upward inclined two phase flow applications has been shown by
Zuber and Findlay(1965) and many others to have the general form
UGM
⎡ g(ρ L − ρG )D ⎤
= C⎢
⎥
ρ Ln
⎢⎣
⎥⎦
m
where C , m and n are constants which have been shown to be flow pattern
dependent.
An observation of the Dix (1971) correlation would reveal that the correlation is more
geared to the “churn-turbulent bubbly flow” correlation suggested by Zuber and
Findlay (1965). Considering the fact that slug flow is common in horizontal and
upward inclined flows, the diameter term is introduced to take care of that as could be
found in many of the correlations developed for slug flow.
The inclination angle and the system pressure effect on the measured void fraction
data were closely analyzed. As outlined by Zuber and Findlay (1965) the drift velocity
is a function of the concentration profiles and also depends on the momentum transfer
between the two phases. Bankoff (1960) has shown that the non uniformity in phase
distribution is a function of pressure while Zuber and Findlay (1965) has stated that
126
the drift velocity is a function of the concentration profiles and also depends on the
momentum transfer between the phases. The concentration profiles across the pipe for
a given inclination are generally assumed constant for flows without mass transfer
along the pipe length. Integrating these observations and noting that this effect is
normal to the pipe inclination, a correction factor of the form
[1 + cosθ ]0.25
is introduced.
The functional dependency of drift velocity on the momentum transfer between the
phases assuming one dimensional force interaction would be a maximum for pipes
with very high inclination angles operating near atmospheric pressure. This effect
could conveniently be captured by the factor
Patm
[1 + sin θ ]P
system
where Patm and Psystem are the atmospheric and system pressures respectively. These
two correction factors have been introduced into the drift velocity expression of Dix
(1971) correlation.
Hence the combined drift velocity expression for the modified correlation is,
⎡ gDσ (1 + cosθ )(ρ L − ρG )⎤
U GM = 2.9 ⎢
⎥
ρ L2
⎣
⎦
0.25
Patm
(1.22 + 1.22 sin θ )P
system
while the distribution coefficient, C0 , being the same as given by Dix(1971). This
correlation would be referred to as the present study correlation and is given as
α=
USG
⎛ ρG ⎞
⎛
⎜ ⎛ U ⎞⎜⎜⎝ ρ L ⎟⎟⎠
USG ⎜1 + ⎜⎜ SL ⎟⎟
⎜⎜ ⎝ USG ⎠
⎝
0.1
⎞
0.25
⎟
Patm
⎡ gDσ (1 + cosθ )(ρ L − ρG )⎤
⎟ + 2.9⎢
Psystem
(
)
1
.
22
+
1
.
22
sin
θ
⎥
ρ L2
⎟⎟
⎣
⎦
⎠
127
The performance of the top ranked Toshiba(1989) correlation could also be improved
by replacing the constant drift velocity value with our modified expression given
above while retaining the near uniform constant distribution coefficient, C0 = 1.08 ,
given by Toshiba(1989) which is seen to better capture the inclined data at lower void
fraction quite well. The detail discussion of this correlation has been omitted here as
the inherent consistency problem associated with the fixed value taken for the
distribution parameter has been considered a weak point as compared to the present
study correlation we have suggested here.
The performance of the present study correlation on the whole database in comparison
to the best performing correlations is presented in Table 25 while the scatter plot for
the whole data set is presented in Figure 47. It can be seen that the under prediction by
the original Dix (1971) correlation on a substantial number of horizontal, inclined and
vertical data points has been improved.
The excessively under predicted twelve horizontal data points which are also common
to the prediction of Toshiba(1989), Rouhani I(1970), Dix(1971) and Filimonov et
al.(1957) were further scrutinized individually.
These data points coincidentally come from the horizontal data of Spedding and
Nguyen (1976) with common characteristics like,
1. All of them are the first data points to be measured within their group
2. Their flow regime was reported to be stratified
3. They have excessively low air mass flow rates in comparison to their next
neighbors with jumps ranging from 237% to 853% while the water mass flow
rate is literally unchanged
4. The void fraction to liquid hold up ratio reported was the least within their
group which is quite expected.
128
Table 25. Total number and percentage of data points correctly predicted by the top
six and modified correlation for the whole database (all sources)
Total data points
Percentage of data points
correctly predicted for
correctly predicted for the
Correlation
the whole database
whole database
Total Points 2845
Total Points 2845
Percentage within ±5%
±10% ±15% ±5%
±10%
±15%
Toshiba(1989)
1065
2137
2427
37.4
75.1
85.3
1
Rouhani I(1970)1
Dix(1971)
Hughmark(1962)
Premoli et al.(1970)
Filimonov et al.(1957)
Present Study
1082
1597
1244
1643
1369
1718
2059
2139
2003
2084
1953
2234
2395
2363
2322
2304
2294
2436
38.0
56.1
43.7
57.8
48.1
60.4
72.4
75.2
70.4
73.3
68.6
78.5
84.2
83.1
81.6
81.0
80.6
85.6
Rouhani and Axelsson(1970) correlations I.
In addition to the fact that measuring the liquid holdup for such a situation where the
liquid phase is very small in comparison to the test section volume can prove difficult,
the inaccuracy of the presented data for these points may have also arisen when the
erroneous assumption that whatever volume is left of the drained liquid is taken as the
gas volume fraction. This would not be valid as the flow being considered here is
nearly a single phase(liquid) flow with very small volume flow rate which would
translate into a small volume of liquid being trapped between the sections. The
remaining portion of the test section already occupied with air even without starting
the test run may have been wrongly interpreted as the void fraction in the final
computations which is not reflective of the input conditions. The case is seen to
improve as the mass flow rate of air is increased hence minimizing the difference
between the actual volumes occupied by the “test” air with that of the preexisting one.
Therefore, without additional information as to how the gas fraction was calculated, it
was concluded that the data for all cases where the air mass flow rate is almost
negligible is very questionable. Hence, discarding these data sets is justifiable and the
predictions of all the correlations should be seen from this perspective. The present
129
study correlation after removing the 12 “irregular” data points is presented in Figure
48.
130
1.0
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Figure 41. Comparison of Toshiba (1989) correlation with
measured combined total experimental data
1.0
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IV
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V
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V
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Inclined data
I
IH
VVH
IH
II
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II
H
I
I
V
V
V
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H
I
H
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H
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V
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V
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Vertical data
IIIH
H
IIH
IV
V
VHI I II
IV
IH
H
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V
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IIH
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+15%
0.8
I
Calculated void fraction
-15%
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
Measured void fraction
Figure 42. Comparison of Rouhani I (1970) correlation with
measured combined total experimental data
131
1.0
1.0
III
IH
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Horizontal data
H
V
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Inclined data
V
V
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Vertical data
IVIIIIIV
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V
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III
+15%
0.8
I
Calculated void fraction
-15%
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Measured void fraction
Figure 43. Comparison of Dix (1971) correlation with
measured combined total experimental data
1.0
III
H
V
IIII
IIIH
H
H
IV
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H
IV
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H
VV
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Horizontal data
H
HH
H
IH
IH
IIV
IIIIIIIV
H
H
V
III V
V
I
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V
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V
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Inclined data
IIIVV
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H
V
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H
H
H
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H
H
H
V
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IV
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Vertical data
I
I
I
H
I
H
H
H HV I IHIH
H
IH
H H
H
H
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HV
VV
IIHV
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132
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Measured void fraction
Figure 46. Comparison of Filimonov et al. (1957) correlation with
measured combined total experimental data
133
1.0
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Measured void fraction
Figure 47. Comparison of present study correlation with
measured combined total experimental data
1.0
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0.6
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0.6
0.8
Measured void fraction
Figure 48. Comparison of present study correlation with
measured “refined”combined total experimental data
134
1.0
CHAPTER 5
CONCLUSIONS AND RECOMMENDATIONS
5.1 Conclusions
A very extensive comparison of most of the void fraction correlations available in the
open literature was made with data collected from different sources. The total number
of data points considered was 2845 of which 900 were horizontal, 1542 inclined and
403 for vertical pipe setups. The best performing correlations within each of the three
pipe orientations considered were given with detailed tables and plots showing their
strengths and weakness with respect to inclination angle, fluid type and dataset.
A combined comparison for all the pipe inclinations from horizontal all the way up to
vertical flow was done. The best performing correlations considering total number of
data points predicted as well as their relative consistency in performance was
highlighted. Appropriate recommendations were also given.
It was confirmed that the drift flux analysis method is a powerful tool in developing
void fraction correlations as well as analyzing experimental data. Out of the six best
performing correlations for the whole void fraction database, all except one were
developed based on the drift flux model.
Detail study of the top performing correlations with each of the data points for which
their performance deteriorated was carried out. An improvement to the correlation of
Dix (1971) was made by systematically introducing appropriate physical parameters.
It was shown that a total of 121 data points were additionally captured within the
restrictive 5% index than that was predicted by the original correlation.
135
This improvement made the correlation to be the top correlation in predicting the
whole database within each of the three error indices. The strong part of the
modification done for the original correlation is that it is not a regression fit to the
data with some very strange looking “lucky” constants bur rather plausible physical
arguments and parameters were forwarded and included. This would make it much
more appealing to other correlations with fitted constants that would have to be
changed if another data set were to be included in the analysis.
5.2 Recommendations
The comparative analysis done on the collected void fractions is extensive both in the
number of void fraction correlations considered and in the effort to include datasets
from different sources. However, additional data is still required to make the
conclusions more sound and valid. Some of the directions to which additional data
collection should be geared towards and their justifications are:
1. The data base is more biased towards air water flow. Hence data with working
fluids other than air-water would not only widen our data base but also would
give definitive insight to the dependency of void fraction on physical
properties like viscosity and surface tension.
2. The accuracy of some of the experimental data may be questionable especially
for error percentage of ±5%. Systematically controlled and accurate data
within this range or even the confirmation of the fact that this is physically
practicable with the methods currently used would be useful in “relieving” the
burden from the correlations to predict the data within this range.
3. Experimental data from setups which operate close to atmospheric pressure
and also with diameters less than 1 in. would be useful in validating some of
136
the dataset used in this analysis. It would also be used to validate and/or
correct the correction factors introduced in the modified correlation.
4. Additional inclined experimental data with frequent angle intervals (5 degrees)
would prove useful in filling some of the gaps in the database considered here.
This would especially have a greater importance if the working fluids are
different from air - water as was suggested in 1 above.
5. A similar comparative analysis of the correlations for downward flow is
another area which may be further explored.
6. Taking into consideration the fact that there are some “gaps” in the inclined
air-water experimental data set (refer to 2, 3 and 4 above), most of the best
performing correlations didn’t predict one of the vertical datasets with a small
diameter and the fact that there is no sufficient data for downward flow, an
experimental test setup with a diameter of 0.375in which can cover inclination
angles of -900 all the way up to +900 is built. This test setup is a multi purpose
facility where heat transfer, pressure drop and void fraction measurements for
single and two phase flows could be undertaken aided with flow visualization.
137
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146
APPENDIX A
SUMMARY OF EXPERIMENTAL DATA BASE
In this section a concise summary of the characteristics of each of the data set used in
this work is given. This would include source of data set, pipe diameter, inclination
angle, total data points in the set, the types of fluids used for the study and range of
void fraction covered. Moreover, minimum and maximum values for parameters like
pressure and temperature which are usually average values for both phases and mass
or volumetric flow rate of each phase is also given. It is worth mentioning here that
the calculation procedure in the comparison takes in the superficial velocities of the
two phases as an input. Therefore, all mass or volumetric flow quantities would be
converted to superficial velocities at the beginning of the calculations when
evaluating the void fraction correlations. However, for reporting the data set here, no
change is made in form to the original data set and everything would be reported as
presented by the original researcher(s).
The complete and detailed data set is available from:
Dr. Afshin J. Ghajar
School of Mechanical and Aerospace Engineering
Oklahoma State University
218 Engineering North
Stillwater, Oklahoma 74078
147
Data Source: Eaton (1966)
Inclination angle: Horizontal
Diameter of pipe: 2.067 & 4.026 in (0.0525 & 0.10226m respectively)
Total data points: 237
Fluids used: natural gas – water
Pressure range: 305.30 – 868.70 Psia
Temperature range: 57.0 – 112.0 0 F
Range of liquid volume flow rate: 46.00– 5620.00 bbl / d
Range of gas volume flow rate: 36609.00 – 9126789.00 scf / d
Range of void fraction covered: 0.27 – 0.99
As mentioned in Chapter 3, Eaton (1966) reported the volume flow rate of natural gas
in standard units (scf/d). Therefore, to convert this into the common volume flow rate
units in m3/sec or ft3/s, the gas formation volume factor and the compressibility
coefficient have to be calculated in addition to the basic unit conversions. The ranges
of the calculated parameters and the converted volume flow rates are given below.
Calculated input parameter ranges
Range of calculated compressibility factor: 0.88– 0.96
Range of calculated gas formation factor: 0.56 – 1.78
Range of converted liquid volume flow rate: 0.0001– 0.01 m 3 / s
Range of converted gas volume flow rate: 0.0005– 0.09 m 3 / s
148
Data Source: Beggs (1972)
Inclination angle: Horizontal
Diameter of pipe: 1 & 1.5 in (0.0254 & 0.0381m respectively)
Total data points: 56
Fluids used: air – water
Pressure range: 51.92 – 98.6 Psia
Temperature range: 38 - 85 0 F
Range of liquid phase flow rate: 4.63 - 535.6 lbm / s. ft 2
Range of gas phase flow rate: 0.48 – 25.41 lbm / s. ft 2
Range of void fraction covered: 0.14 – 0.98
Data Source: Beggs (1972)
Inclination angle: 5 Degrees
Diameter of pipe: 1 & 1.5 in (0.0254 & 0.0381m respectively)
Total data points: 31
Fluids used: air – water
Pressure range: 65.70 – 98.97 Psia
Temperature range: 49 - 89 0 F
Range of liquid phase flow rate: 4.63 – 624.73 lbm / s. ft 2
Range of gas phase flow rate: 0.49 – 25.53 lbm / s. ft 2
Range of void fraction covered: 0.12 – 0.98
Data Source: Beggs (1972)
Inclination angle: 10 Degrees
Diameter of pipe: 1 & 1.5 in (0.0254 & 0.0381m respectively)
Total data points: 29
Fluids used: air – water
Pressure range: 65.45 – 98.51 Psia
Temperature range: 51 - 90 0 F
Range of liquid phase flow rate: 4.63 – 1065.09 lbm / s. ft 2
Range of gas phase flow rate: 0.49 – 25.75 lbm / s. ft 2
Range of void fraction covered: 0.14 – 0.98
149
Data Source: Beggs (1972)
Inclination angle: 15 Degrees
Diameter of pipe: 1 & 1.5 in (0.0254 & 0.0381m respectively)
Total data points: 31
Fluids used: air – water
Pressure range: 75.70 – 98.47 Psia
Temperature range: 53 - 90 0 F
Range of liquid phase flow rate: 4.63 – 636.96 lbm / s. ft 2
Range of gas phase flow rate: 0.53 – 29.75 lbm / s. ft 2
Range of void fraction covered: 0.15 – 0.98
Data Source: Beggs (1972)
Inclination angle: 20 Degrees
Diameter of pipe: 1 & 1.5 in (0.0254 & 0.0381m respectively)
Total data points: 30
Fluids used: air – water
Pressure range: 64.49 – 98.75 Psia
Temperature range: 53 - 90 0 F
Range of liquid phase flow rate: 4.63 – 1134.40 lbm / s. ft 2
Range of gas phase flow rate: 0.49 – 24.97 lbm / s. ft 2
Range of void fraction covered: 0.17 – 0.98
Data Source: Beggs (1972)
Inclination angle: 35 Degrees
Diameter of pipe: 1 & 1.5 in (0.0254 & 0.0381m respectively)
Total data points: 30
Fluids used: air – water
Pressure range: 81.67 – 97.32 Psia
Temperature range: 39 - 89 0 F
Range of liquid phase flow rate: 4.63 – 517.21 lbm / s. ft 2
Range of gas phase flow rate: 0.49 – 24.81 lbm / s. ft 2
Range of void fraction covered: 0.19 – 0.98
150
Data Source: Beggs (1972)
Inclination angle: 55 Degrees
Diameter of pipe: 1 & 1.5 in (0.0254 & 0.0381m respectively)
Total data points: 29
Fluids used: air – water
Pressure range: 74.81 – 96.96 Psia
Temperature range: 46 - 92 0 F
Range of liquid phase flow rate: 4.63 – 632.12 lbm / s. ft 2
Range of gas phase flow rate: 0.53 – 25.37 lbm / s. ft 2
Range of void fraction covered: 0.20 – 0.98
Data Source: Beggs (1972)
Inclination angle: 75 Degrees
Diameter of pipe: 1 & 1.5 in (0.0254 & 0.0381m respectively)
Total data points: 29
Fluids used: air – water
Pressure range: 76.76 – 96.84 Psia
Temperature range: 51 - 95 0 F
Range of liquid phase flow rate: 4.62 – 624.43 lbm / s. ft 2
Range of gas phase flow rate: 0.53 – 25.21 lbm / s. ft 2
Range of void fraction covered: 0.15 – 0.98
Data Source: Beggs (1972)
Inclination angle: 90 Degrees
Diameter of pipe: 1 & 1.5 in (0.0254 & 0.0381m respectively)
Total data points: 26
Fluids used: air – water
Pressure range: 76.40 – 98.17 Psia
Temperature range: 67 - 96 0 F
Range of liquid phase flow rate: 4.62 – 516.78 lbm / s. ft 2
Range of gas phase flow rate: 0.31 – 24.43 lbm / s. ft 2
Range of void fraction covered: 0.15 – 0.98
151
Data Source: Spedding & Nguyen (1976)
Inclination angle: Horizontal
Diameter of pipe: 4.55 cm (0.0455m)
Total data points: 270
Fluids used: air – water
Pressure range: 747.84 – 860.52 mm Hg (absolute pressure)
Temperature range: 15.9 – 27.6 0C
Range of liquid mass flow rate: 6.01– 6093.40 Kg / h
Range of gas mass flow rate: 0.64 – 474.03 Kg / h
Range of void fraction covered: 0.06 – 1.00
Data Source: Spedding & Nguyen (1976)
Inclination angle: 2.75 degrees
Diameter of pipe: 4.55 cm (0.0455m)
Total data points: 226
Fluids used: air – water
Pressure range: 744.78 – 834.50 mm Hg (absolute pressure)
Temperature range: 18.20 – 23.70 0C
Range of liquid mass flow rate: 5.95– 8097.94 Kg / h
Range of gas mass flow rate: 0.06 – 523.71 Kg / h
Range of void fraction covered: 0.04 – 1.00
Data Source: Spedding & Nguyen (1976)
Inclination angle: 20.75 degrees
Diameter of pipe: 4.55 cm (0.0455m)
Total data points: 188
Fluids used: air – water
Pressure range: 711.94 – 876.58 mm Hg (absolute pressure)
Temperature range: 16.70 – 22.72 0C
Range of liquid mass flow rate: 5.85– 6089.19 Kg / h
Range of gas mass flow rate: 0.35 – 515.78 Kg / h
Range of void fraction covered: 0.03 – 1.00
152
Data Source: Spedding & Nguyen (1976)
Inclination angle: 45 degrees
Diameter of pipe: 4.55 cm (0.0455m)
Total data points: 196
Fluids used: air – water
Pressure range: 763.00 – 900.98 mm Hg (absolute pressure)
Temperature range: 16.13 – 24.33 0C
Range of liquid mass flow rate: 5.75– 6093.91 Kg / h
Range of gas mass flow rate: 0.69 – 482.99 Kg / h
Range of void fraction covered: 0.04 – 1.00
Data Source: Spedding & Nguyen (1976)
Inclination angle: 70 degrees
Diameter of pipe: 4.55 cm (0.0455m)
Total data points: 279
Fluids used: air – water
Pressure range: 741.54 – 939.94 mm Hg (absolute pressure)
Temperature range: 17.60 – 23.60 0C
Range of liquid mass flow rate: 5.88– 6098.97 Kg / h
Range of gas mass flow rate: 0.14 – 440.04 Kg / h
Range of void fraction covered: 0.01 – 0.99
Data Source: Spedding & Nguyen (1976)
Inclination angle: 90 degrees
Diameter of pipe: 4.55 cm (0.0455m)
Total data points: 224
Fluids used: air – water
Pressure range: 799.21 – 937.72 mm Hg (absolute pressure)
Temperature range: 16.35 – 23.73 0C
Range of liquid mass flow rate: 32.41– 6095.97 Kg / h
Range of gas mass flow rate: 0.59 – 436.30 Kg / h
Range of void fraction covered: 0.04 – 0.99
153
Data Source: Mukherjee (1979)
Inclination angle: Horizontal
Diameter of pipe: 1.5 in (0.0381m)
Total data points: 62
Fluids used: air – kerosene
Pressure range: 28.20 – 91.90 Psia
Temperature range: 62 - 132 0 F
Range of liquid superficial velocity: 0.05 – 13.07 ft / s
Range of gas superficial velocity: 0.75 – 78.93 ft / s
Range of void fraction covered: 0.08 – 1.00
Data Source: Mukherjee (1979)
Inclination angle: 5 degrees
Diameter of pipe: 1.5 in (0.0381m)
Total data points: 57
Fluids used: air – kerosene
Pressure range: 26.70 – 68.70 Psia
Temperature range: 56 - 118 0 F
Range of liquid superficial velocity: 0.09 – 13.07 ft / s
Range of gas superficial velocity: 0.31 – 77.24 ft / s
Range of void fraction covered: 0.01 – 1.00
Data Source: Mukherjee (1979)
Inclination angle: 20 degrees
Diameter of pipe: 1.5 in (0.0381m)
Total data points: 74
Fluids used: air – kerosene
Pressure range: 26.90 – 72.90 Psia
Temperature range: 55 – 165.5 0 F
Range of liquid superficial velocity: 0.09 – 14.31 ft / s
Range of gas superficial velocity: 0.31 – 101.03 ft / s
Range of void fraction covered: 0.06 – 0.99
154
Data Source: Mukherjee (1979)
Inclination angle: 30 degrees
Diameter of pipe: 1.5 in (0.0381m)
Total data points: 74
Fluids used: air – kerosene
Pressure range: 27.70 – 88.70 Psia
Temperature range: 57 – 117.0 0 F
Range of liquid superficial velocity: 0.09 – 12.98 ft / s
Range of gas superficial velocity: 0.29 – 76.78 ft / s
Range of void fraction covered: 0.09 – 0.99
Data Source: Mukherjee (1979)
Inclination angle: 50 degrees
Diameter of pipe: 1.5 in (0.0381m)
Total data points: 67
Fluids used: air – kerosene
Pressure range: 23.90 – 93.70 Psia
Temperature range: 40 – 127.0 0 F
Range of liquid superficial velocity: 0.05 – 11.33 ft / s
Range of gas superficial velocity: 0.15 – 79.34 ft / s
Range of void fraction covered: 0.03 – 1.00
Data Source: Mukherjee (1979)
Inclination angle: 70 degrees
Diameter of pipe: 1.5 in (0.0381m)
Total data points: 80
Fluids used: air – kerosene
Pressure range: 40.20 – 83.70 Psia
Temperature range: 44.5 – 122.0 0 F
Range of liquid superficial velocity: 0.06 – 11.09 ft / s
Range of gas superficial velocity: 0.14 – 89.98 ft / s
Range of void fraction covered: 0.14 – 0.99
155
Data Source: Mukherjee (1979)
Inclination angle: 80 degrees
Diameter of pipe: 1.5 in (0.0381m)
Total data points: 92
Fluids used: air – kerosene
Pressure range: 40.20 – 87.20 Psia
Temperature range: 60.0 – 113.0 0 F
Range of liquid superficial velocity: 0.06 – 11.27 ft / s
Range of gas superficial velocity: 0.19 – 66.43 ft / s
Range of void fraction covered: 0.17 – 0.99
Data Source: Mukherjee (1979)
Inclination angle: 90 degrees
Diameter of pipe: 1.5 in (0.0381m)
Total data points: 52
Fluids used: air – kerosene
Pressure range: 38.70 – 88.40 Psia
Temperature range: 54.0 – 118.0 0 F
Range of liquid superficial velocity: 0.09 – 10.97 ft / s
Range of gas superficial velocity: 0.12 – 80.08 ft / s
Range of void fraction covered: 0.02 – 0.98
156
Data Source: Minami & Brill (1987)
Inclination angle: Horizontal
Diameter of pipe: 3.068 in (0.07793m)
Total data points: 54
Fluids used: air – water
Pressure range: 46.40 – 85.40 Psia
Temperature range: 76 - 117 0 F
Range of liquid superficial velocity: 0.02 – 2.96 ft / s
Range of gas superficial velocity: 1.56 – 49.13 ft / s
Range of void fraction covered: 0.55 – 0.99
Data Source: Minami & Brill (1987)
Inclination angle: Horizontal
Diameter of pipe: 3.068 in (0.07793m)
Total data points: 57
Fluids used: air – kerosene
Pressure range: 43.70 – 96.70 Psia
Temperature range: 82 - 118 0 F
Range of liquid superficial velocity: 0.02 – 3.12 ft / s
Range of gas superficial velocity: 1.78 – 54.43 ft / s
Range of void fraction covered: 0.56 – 0.99
157
Data Source: Franca & Lahey (1992)
Inclination angle: Horizontal
Diameter of pipe: 0.019m
Total data points: 81
Fluids used: air – water
Pressure range: 0.0 – 1.47 m H 2O (Gauge Pressure)
Temperature range: 22 - 22 0C
Range of liquid superficial velocity: 0.01 – 1.49 m / s
Range of gas superficial velocity: 0.13 – 23.76 m / s
Range of void fraction covered: 0.06 – 0.94
158
Data Source: Abdulmajeed (1996)
Inclination angle: Horizontal
Diameter of pipe: 2 in (0.0508m)
Total data points: 83
Fluids used: air – kerosene
Pressure range: 197.20 – 919.10 KPa
Temperature range: 27.8 – 48.9 0C
Range of liquid superficial velocity: 0.002 – 1.83 m / s
Range of gas superficial velocity: 0.20 – 48.91 m / s
Range of void fraction covered: 0.39 – 0.99
159
Data Source: Sujumnong (1997)
Inclination angle: 90 degrees
Diameter of pipe: 0.5 in (0.0127m)
Total data points: 101
Fluids used: air – water
Pressure range: 101351.35 – 342664.09 Pa
Temperature range: 18.97 – 32.13 0C
Range of liquid superficial velocity: 0.05– 8.46 m / s
Range of gas superficial velocity: 0.04 – 118.36 m / s
Range of void fraction covered: 0.02 – 0.99
160
VITA
Melkamu Aklile Woldesemayat
Candidate for the Degree of
Master of Science
Thesis: COMPARISON OF VOID FRACTION CORRELATIONS FOR TWOPHASE FLOW IN HORIZONTAL AND UPWARD INCLINED FLOWS
Major Field: Mechanical Engineering
Biographical:
Personal Data: Born in Addis Ababa (Ethiopia) on February 1, 1976.
Education:
Received Bachelor of Science degree in Mechanical
Engineering, Addis Ababa University, Faculty of Technology
in July 1998.
Completed the requirements for the Master of Science degree at
Oklahoma State University, Stillwater (USA) in May 2006.
Professional Experience:
1998-2000 Design and Drafting engineer
Radel Foundry
2000-2001 Associate Engineer
Ethiopian Airlines
2001-2004 Aviation Sales Manager
Shell Ethiopia
2004-2006 Teaching Assistant
Oklahoma State Univ.
Professional Membership:
American Society of Mechanical Engineers
(ASME)
Name: Melkamu Woldesemayat
Date of Degree: May 2006
Institution: Oklahoma State University
Location: Stillwater, Oklahoma
Title of Study: COMPARISON OF VOID FRACTION CORRELATIONS FOR
TWO-PHASE FLOW IN HORIZONTAL AND UPWARD
INCLINED PIPES
Pages in Study: 160
Candidate for the Degree of Master of Science
Major Field: Mechanical Engineering
Scope and Method of Study: An extensive literature search was made for the available
void fraction correlations and experimental void fraction data. After
systematically refining the data, the performance of the correlations in
correctly predicting the diverse data sets was evaluated. Comparisons
between the correlations were made and appropriate recommendations
drawn. An improved version of one of the top performing correlations was
also suggested by introducing physical parameters backed by plausible
arguments that capture the physics of the problem.
Findings and Conclusions: The analysis showed that most of the correlations
developed are very restricted in terms of handling a wide variety of data sets.
Based on the recommendations made, an improved void fraction correlation
which could acceptably handle all data sets regardless of flow patterns for
horizontal and upward inclination angles was suggested. It was shown that
this correlation has the best predictive capability than all the correlations
considered in this study.
Advisor’s Approval:
Dr. Afshin J. Ghajar
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