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Pressure drop in three-phase oilЦwaterЦgas horizontal co-current flow experimental data and development of prediction models.

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ASIA-PACIFIC JOURNAL OF CHEMICAL ENGINEERING
Asia-Pac. J. Chem. Eng. 2008; 3: 531?543
Published online 15 September 2008 in Wiley InterScience
(www.interscience.wiley.com) DOI:10.1002/apj.165
Research Article
Pressure drop in three-phase oil?water?gas horizontal
co-current flow: experimental data and development
of prediction models
P. L. Spedding,1 Adrian Murphy,1 * G. F. Donnelly,1 E. Benard1 and A. P. Doherty2
1
2
School of Mechanical and Aerospace Engineering, Queen?s University, Belfast, UK
School of Chemistry and Chemical Engineering, Queen?s University, Belfast, UK
Received 3 March 2008; Revised 6 June 2008; Accepted 8 June 2008
ABSTRACT: Pressure-loss data are reported for three-phase oil?water?gas flow in horizontal pipes of 0.0259 and
0.0501 m i.d. The pressure profiles exhibited a maximum value above a critical gas rate, which progressively moved to
higher oil ratios as the gas rate was increased due to increased turbulence within the system. The actual pressure-drop
profiles obtained depended mainly on the flow regimes present within the system. As the pipe diameter was reduced,
the pressure drop was observed to increase for the same flow conditions. Above a critical gas rate the magnitude of
the pressure drop was higher if approached for the oil-dominated (OD) region compared to the opposite direction.
A momentum-balance model was proposed for prediction of the pressure drop, which gave improved performance
over other existing methods. ? 2008 Curtin University of Technology and John Wiley & Sons, Ltd.
KEYWORDS: oil?water?air flow; pressure drop; predictive models
INTRODUCTION
Pressure drop in multi-phase flow is given by
dP
dP
dP
dP
=
+
+
d T
d H
d A
d F
(1)
For horizontal pipe at low gas-flow rates without
change in area or phase, the latter friction component
in Eqn (1) is the only element of concern. Frictional
pressure loss is caused by both the shearing action of the
fluids on the inner pipe wall and at the fluid interfaces.
Sobocinski and Huntington[1] conducted a limited
study of pressure drop with an air?water?oil (祇 =
0.00383 kg m?1 s?1 at 38 ? C) system in a 0.0796 m
i.d. horizontal pipe for the stratified, stratified plus wave
and semi-annular regimes. The pressure drop exhibited
a maximum at an oil to liquid volumetric ratio of fo =
0.23 which was attributed to the transition from waterdominated (WD) to oil-dominated (OD) dispersion. A
series of studies were conducted on three-phase pressure
drop in production systems[2 ? 5] but the results proved to
be of limited application.[6] However, Guzhov et al .[5]
showed that the pressure drop possessed a maximum
*Correspondence to: Adrian Murphy, School of Mechanical and
Aerospace Engineering, Queen?s University, Belfast, UK.
E-mail: a.murphy@qub.ac.uk
? 2008 Curtin University of Technology and John Wiley & Sons, Ltd.
value at the transition point around an oil to liquid
volumetric ratio of fo = 0.5.
Both Malinowski,[7] and Laflin and Oglesby[8] conducted experiments on air?water?diesel (祇
= 0.00492 kg m?1 s?1 at 15.6 ? C) flow in a 0.0381 m
i.d. horizontal pipe. The main focus was to calculate
an effective liquid viscosity using correlations developed for two-phase systems.[9,10] The apparent liquid
viscosity obtained dramatically increased at an oil to liquid volumetric ratio of fo = 0.49. Fayed and Otten[11]
extended the approach and showed that pressure-loss
prediction for the slug-froth and bubble regimes was
respectively higher and lower than measurement.
Stapelberg and Mewes[12] determined three-phase
pressure drop in two horizontal tubes of 0.0238 and
0.0590 m i.d. using an oil of 祇 = 0.031 kg m?1 s?1 .
The flow was mainly in the slug regime pattern with
a few results in the stratified regime. Their work
demonstrated the inadequacy of current methods for
prediction of pressure drop in three-phase systems.
Pleshko and Sharma[13] showed that, in general, adapted
two-phase models tended to underestimate three-phase
pressure gradients, an observation already made by
Gregory and Forgassi.[14] Hall[6] and Pan et al .[15]
studied three-phase slug flow in a 0.078 m i.d. horizontal pipe using oils in the viscosity range 祇 =
0.04?0.06 kg m?1 s?1 . Attempts were made to model
532
P. L. SPEDDING ET AL.
Asia-Pacific Journal of Chemical Engineering
the data using an extension of two-phase momentumbalance relations. Taitel et al .[16] adapted the two-phase
momentum-balance model to the prediction of threephase systems. Data for 0.05 m i.d. horizontal pipe
using oils of various viscosities were used to attempt
validation of the model.
This work extends previous horizontal three-phase
studies reported on flow regimes[17] and hold-up[18] to
include the pressure-drop data for the same apparatus.
A momentum-balance model is developed in order
to attempt to predict three-phase flow pressure-drop
phenomena.
EXPERIMENTAL
The frictional pressure loss was obtained for horizontal co-current three-phase flow in both 0.0259 and
0.0501 m i.d. pipes which were also used to determine
flow regimes[17] and hold-up.[18] Pressure readings for
the smaller diameter were taken over a 2-m pipe length
while for the larger diameter pipe data were obtained
over a 4.25 m length. The fluids were measured in calibrated rotameters. The air flow was first mixed with
the water phase and then with the oil phase. The liquids were injected through a series of fine holes into
the air stream. The exiting fluids were separated in
a cyclone and the liquids passed to storage through
separation tanks. Preliminary experiments were performed to ensure that adequate mixing of the phases was
achieved and that the inlet and outlet lengths around the
measurement zones were sufficient at 4 and 1.7 m for
the 0.0259 m i.d. pipe; and 6 and 2 m for the 0.0501 m
i.d. pipe.
The oil properties were, for the 0.0259 m i.d. facility,
?o = 828.5 kg m?3 , 祇 = 0.0122 kg m?1 s?1 and for
the 0.0501 m i.d. apparatus ?o = 854.2 kg m?3 , 祇 =
0.0395 kg m?1 s?1 , all at 24 ? C
Tapping points of 2.5-mm diameter were set in the
top of the pipe and led to small separation chambers to ensure that no liquid entrainment was in
the neoprene lines leading to the pressure measurement equipment. A Neotronics Zephyr differential pressure transducer (�1 Pa) was used for the 0.0259 m
i.d. facility and a Foxboro differential pressure transducer (�1%) for the 0.0501 m i.d. rig. Preliminary
experiments with single-phase and air?water two-phase
flow were conducted to ensure the validity of the
experiments. Some of this initial work is detailed in
previous publications.[17,18] Further details are given
elsewhere.[19]
EXPERIMENTAL RESULTS
Pressure-loss profiles of some data across the entire
range of the oil to liquid volumetric ratio fo values are
presented in Figs 1?5 for a number of set gas rates each
at set liquid rates. The 0.0501 m i.d. pipe data are given
in Figs 1?4 while some of the 0.0259 m i.d. data are
shown in Figs 3 and 5. In general, the two-phase oil
only condition at fo = 1.0 gave a higher pressure loss
than the corresponding two-phase water only system at
fo = 0.
In Fig. 1 at the lower-gas rate of VSG = 7 m s?1 and
a diameter of 0.0501 m, the stratified ripple regime
was observed at fo = 0 and there was a slight drop
in pressure loss as oil was admitted. This was due
Figure 1. Pressure drop, oil?liquid volume profiles for VSLT = 0.0173 m s?1
and i.d. = 0.0501 m.
? 2008 Curtin University of Technology and John Wiley & Sons, Ltd.
Asia-Pac. J. Chem. Eng. 2008; 3: 531?543
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
PRESSURE DROP IN THREE-PHASE HORIZONTAL CO-CURRENT FLOW
Figure 2. Pressure drop, oil?liquid volume profiles for VSLT = 0.0329 m s?1
and i.d. = 0.0501 m.
to the oil dampening the wave structure and thus
reducing coupling between the gas and liquid flows.
Further addition of oil resulted in a consistent rise in
pressure loss up to the air?oil two-phase pressure drop
at fo = 1.0, with a slight elevation at an oil to liquid
volumetric ratio of about fo ? 0.4 where the ripples
showed a measure of height increase. As the gas rate
was raised in three stages, definite maxima appeared
in the pressure profiles at steadily increasing values of
oil to liquid volumetric ratio fo . For example at VSG =
10 m s?1 , fo = 0.45; at VSG = 14 m s?1 , fo = 0.725
and at VSG = 19 m s?1 , fo = 0.825. These pressure-loss
maxima were not necessarily at the onset or completion
of the stripping and pasting mechanism, described by
Woods et al .[20] and were illustrated previously as a
flow regime,[17] around the transition from water to OD
flow. Nor were the pressure-loss maxima necessarily
at the maxima observed in the hold-up profiles which
were usually at an oil to liquid volumetric ratio of fo =
0.75,[18] but were associated with observable changes
within the flow pattern structure. For example, with
the VSG = 10 m s?1 pressure profile, the stratified rollwave regime was noted at the two-phase air?water
point at fo = 0. There was little observable change in the
wave structure until beyond an oil to liquid volumetric
ratio of about fo > 0.15 where a steady increase in wave
height was observed as oil was added until the stratified
? 2008 Curtin University of Technology and John Wiley & Sons, Ltd.
roll-wave plus droplet regime was initiated at an oil to
liquid volumetric ratio of fo = 0.4 in the region of the
pressure maximum.
There are three relevant characteristics of these pressure fo profiles. Firstly, Monson[21] and Woelfin[22] both
showed that the measured viscosity exhibited a sharp
maximum value at an intermediate value of the oil
to liquid volumetric ratio. This was observed to be
caused by a reduction in the size distribution of the dispersed phase[23] culminating in a close-packed structure
at transition.[24] Secondly, the observed pipe-pressure
drop sometimes possessed a maximum, which has often
been assumed to be at the transition point.[25] Consequently, the transition point between the liquid-phase
dominated flows has often been assumed to be at the
maxima of both the measured viscosity and pressure
profiles. However, the reality is much more complex.
Marsden and Raghaven[26] have demonstrated that the
point of maximum measured viscosity and the actual
transition region varied widely with the latter depending
on factors such as the type of oil, dispersion size, size
distribution, degree of mixing etc. In fact, there was a
marked difference between the profiles found from rheological analysis and for flowing pipe systems.[8,27,28]
Arirachakaran[29] suggested a correlation between oil
viscosity 祇 and transition point foi but in this work the
relation was sometimes shown to fail.[19] Pan et al .[15]
Asia-Pac. J. Chem. Eng. 2008; 3: 531?543
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P. L. SPEDDING ET AL.
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Figure 3. Pressure drop, oil?liquid volume profiles for VSLT = 0.0501 m s?1
and i.d. = 0.0501 m.
have demonstrated that the phase transition did not necessarily occur at the point of maximum pressure loss but
depended on the degree of mixing actually present in
the pipe, with the emulsion viscosity in situ being different from rheological measurements. Lunde et al .[30]
reported no unusual pressure characteristics for tranquil stratified flow and pressure maxima at an oil to
liquid volumetric ratio of about fo ? 0.1 for more agitated pipe-flow regimes. By contrast, for well-mixed
conditions of slug[8] and annular type flows[20,30] significant pressure maxima were reported at an oil to
liquid volumetric ratio fo of 0.775. Thus, the observed
pressure-profile maxima appeared at increasing values
of the oil to liquid volumetric ratio fo as the turbulence intensity of the liquid phases increased due to the
resident flow regimes present in the pipe.
The pressure profiles for the highest three gas rates
of VSL = 10, 14 and 19 m s?1 show an influence of
the direction that the transition approached, particularly
around the maximum point. Tracing the pressure profile
from the WD condition gave a smaller pressure drop to
? 2008 Curtin University of Technology and John Wiley & Sons, Ltd.
that with the converse OD direction, although the position of maximum pressure loss did not alter. This excess
pressure loss observed when commencing from an OD
condition was not a time-dependent phenomenon due to
preferential wall wetting detailed by Woods et al .[20] If
it were, it would be transitory in nature and occur for
all gas rates including the lowest VSG = 7 m s?1 value.
Other data not reported in detail in this work obtained
by holding both the gas rate and one liquid phase-rate
constant while altering the second liquid-phase rate also
indicated that the pressure drop depended on the choice
of the initial continuous phase condition. The effect may
be associated with the wave/droplet conditions present.
Dishing of the transpipe liquid?gas interface has been
shown to have a pronounced effect on two-phase parameters, particularly the hold-up.[31] It can be assumed that
similar effects can be expected with the three-phase system. Others have reported that the form of the emulsion
had an effect on the pressure drop. Chen et al .[32] found
an increase in frictional pressure drop for OD emulsion over that for the converse WD system. Zhou and
Asia-Pac. J. Chem. Eng. 2008; 3: 531?543
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Asia-Pacific Journal of Chemical Engineering
PRESSURE DROP IN THREE-PHASE HORIZONTAL CO-CURRENT FLOW
Figure 4. Pressure drop, oil?liquid volume profiles for VSLT = 0.066 m s?1
and i.d. = 0.0501 m.
Jepson[33] showed that an OD system gave a reduction
in turbulence intensity but an increased shear rate over
that in the WD system. Pacek and Nienow[34] reported
finding oil drops in water droplets for the OD initiated
system, which were absent from the converse WD condition. This was an observation also made by Pal and
Rhodes.[35] It is possible, therefore, that differences in
the fine structure of the liquid dispersion within the system that arise at initiation conditions result in a variation
of the observed pressure loss for the same volumetric
composition. The whole subject needs further investigation to ascertain the importance of these three possible
mechanisms.
The data in Fig. 2 at VSL = 0.0329 m s?1 exhibited
an increase in pressure drop with oil to liquid volumetric
ratio fo for all gas flows. There was an initial fall in
pressure drop at the lowest gas rate of VSG = 7 m s?1
as the addition of oil was seen to dampen the surface
ripples to give essentially stratified flow. At an oil
? 2008 Curtin University of Technology and John Wiley & Sons, Ltd.
to liquid volumetric ratio of fo = 0.25 ripples again
appeared on the surface, while at an oil to liquid
volumetric ratio of fo = 0.875 the stratified roll-wave
regime appeared briefly. Here, the roll-wave crests were
observed to be non-breaking. As the gas rate was
increased, not only did the pressure-drop rise but there
was an increase in the rate of droplet deposition. At
the highest gas rate of VSG = 19 m s?1 the film plus
droplet regime was present and the position of the
maximum pressure drop coincided with that observed
for the corresponding hold-up profile[18] and where
transition from water to OD flow was completed. For
Fig. 3 at VSL = 0.0501 m s?1 and the lowest gas rate
of VSG = 7 m s?1 ,the stratified ripple to stratified rollwave transition occurred at an oil to liquid volumetric
ratio of fo = 0.5, while at the highest gas rate of
VSG = 19 m s?1 the film plus droplet regime changed
into the broken film pattern at the same point. The
data for VSL = 0.066 m s?1 in Fig. 4 had a dramatic
Asia-Pac. J. Chem. Eng. 2008; 3: 531?543
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Figure 5. Pressure drop, oil?liquid volume profiles for i.d. 0.0259 m.
initial drop of 35% in the pressure gradient as oil
was added at the lowest gas rate of VSG = 7 m s?1 .
Initially, at the two-phase air?water condition at fo = 0,
the stratified blow through slug regime was present
and with oil addition the flow pattern changed into
the stratified condition. Thereafter, the stratified rollwave regime prevailed until the stratified roll-wave plus
droplet regime appeared at an oil to liquid volumetric
ratio of fo = 0.77. Mewes et al .[12,36 ? 39] also observed
a dramatic pressure-drop reduction at low gas-flow
rates for the initial slug flow pattern. Observing the
data across Figs 1?4 at constant gas rates the point of
maximum pressure loss tended to occur at increasing
values of the oil to liquid volumetric ratio fo as the total
liquid rate was raised for the lowest gas rate of VSG =
7 m s?1 . However, the trend reversed at the highest
gas rate of VSG = 19 m s?1 . These results indicated that
the point of maximum pressure drop depended on the
turbulence level within the liquid flow regime with a
tendency to move to higher oil to liquid volumetric rate
fo values as the turbulence was raised. This observation
aligned with conclusions by others noted above.[15,30]
? 2008 Curtin University of Technology and John Wiley & Sons, Ltd.
Figure 5 and part of Fig. 3 present the pressure-loss
profiles for the 0.0259 m i.d. pipe. The predominant
flow regimes present were in the stratified roll-wave and
the stratified roll-wave plus droplet patterns. In general
the observed pressure drops were greater than those
found for the larger 0.0501 m i.d. pipe. This is well
illustrated in Fig. 3 where data for the two diameters,
admittedly with different oil viscosities are shown but
the observed effect of diameter is in agreement with the
results of other workers.[12,40] In addition, the pressure
profiles did not exhibit the dramatic changes with oil
to liquid volumetric ratio values that were apparent
at the larger diameter but did possess maxima at
approximately the same positions.
Illustrations with descriptions of the flow regimes are
given by Spedding et al .[17] in Part I of this series.
EVALUATION OF PREDICTION MODELS
Most methods used to predict three-phase pressure loss
(and hold-up) are extensions of two-phase momentumAsia-Pac. J. Chem. Eng. 2008; 3: 531?543
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Asia-Pacific Journal of Chemical Engineering
PRESSURE DROP IN THREE-PHASE HORIZONTAL CO-CURRENT FLOW
balance models. Stapelberg[39] adapted the two-phase
model of Taitel and Dukler[40] by grouping both liquid phases. Predictions were poor. Hall[6] similarly
extended the two-phase model by defining the oil-wall
stress and phase heights which were derived from holdup measurements. Figures 6?8 show predictions were
mostly on the low side and somewhat different in performance for the three groups of data. Figure 6 exhibits
a deterioration in prediction as the gas rate was lowered.
At higher rates the prediction agreed substantially with
the corresponding performance of the 0.0259 m i.d. data
of this work presented in Fig. 7. Also, in this figure, the
two stratified roll-wave plus droplet data sets were for
the short and long wave regimes but gave similar result
trends in both Figs 6 and 7. The predictions at low liquid flows shown in Fig. 8 showed a good performance
at low gas flows but under estimation of pressure drop
as the gas rate was increased in a similar manner to the
data in Figs 6 and 7.
Taitel et al .[16] developed a three-phase force balance
relation for separated stratified flow. The model is
illustrated in Fig. 11 and Eqns (4)?(8). Initially, the
wall-shear stresses were evaluated for both the aqueous
and oil phases, then grouped to give a combined pseudoliquid-wall stress.
?L SL = ?w Sw + ?o So
(2)
The density of the combined liquid was also defined.
?L =
?w Aw + ?o Ao
AL
(3)
The viscosity was treated in a similar fashion with no
consideration of emulsion effects. Using this approach,
a simplified two-phase model was then applied from
which an overall liquid height hL was derived. The
resultant solution for the height of the pseudo liquid, hL ,
may then be re-substituted to determine the level of the
aqueous phase hW . On the premise that the momentum
given by Eqn (8) is satisfied for hW , the expression in
Eqn (7) is tested. If it is not satisfied, another value of hL
is chosen and the procedure repeated until convergence
Figure 6. Measured and predicted pressure gradient for the data for
Lunde[30] using the method of Hall.[6] .
Figure 7. Measured and predicted pressure gradient using the method
of Hall[6] and i.d. = 0.0259 m data.
? 2008 Curtin University of Technology and John Wiley & Sons, Ltd.
Asia-Pac. J. Chem. Eng. 2008; 3: 531?543
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P. L. SPEDDING ET AL.
Asia-Pacific Journal of Chemical Engineering
Figure 8. Measured and predicted pressure gradient for the data of
Sobocinski[1] using the method of Hall.[6] .
Figure 9. Measured and predicted pressure gradient against fo using the
Taitel et al.[16] model and i.d. = 0.0508 m data.
is attained. Figure 9 indicates that the model grossly
underpredicts the pressure drop (and also the holdup[18] ) giving overall under performance except for the
stratified regime at water cuts above 70%. This is to be
expected since the model assumed that geometry prediction progressively deteriorated as the gas rate increased
and the resulting turbulence steadily rose with changes
in flow regime until maximum divergence occurred
with the annular roll-wave pattern. In addition, the
poor underperformance was magnified in the central fo
region in the phase-transition area. Figure 10 gives the
same data and model, but with the Brinkman[41] viscosity relation. Overall prediction performance improved
particularly for stratified type flows around the phase
transition. The low and high water-cut regions were left
largely unaltered. The implication of Figs 6?10 is that
the three-phase model of Taitel et al .[16] does not give
satisfactory prediction of pressure-loss data. The work
of Hall[6] shows that the model gives better performance
if attention was paid to the stresses, particularly the
? 2008 Curtin University of Technology and John Wiley & Sons, Ltd.
liquid?gas interfacial stress. This view is further supported by the advances made in two-phase momentumbalance models by attention to the gas?liquid interface.
Further improvements to the prediction will occur if
attention was directed to the liquid emulsion viscosity.
In this work, the three-phase momentum balance
of Taitel et al .[16] is extended to include these two
additional elements identified above.
DEVELOPMENT OF PREDICTION MODEL
A schematic representation of the geometry involved is
presented in Fig. 11. The force balances are:
?Aw
dP
d
? ?w Sw + ?iow Siow
w
? ?w Aw g sin ? = 0
(4)
Asia-Pac. J. Chem. Eng. 2008; 3: 531?543
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PRESSURE DROP IN THREE-PHASE HORIZONTAL CO-CURRENT FLOW
Figure 10. Measured and predicted pressure gradient against fo using the
Taitel et al.[16] model and incorporating the Brinkman[41] viscosity relation
with i.d. = 0.0508 m data.
?Ao
dP
d
overall combined liquid-phase momentum balance.
? ?o So ? ?iow Siow + ?iog Siog
o
? ?o Aog g sin ? = 0
dP
? ?g Sg ? ?iog Siog
?Ag
d g
? ?g Ag g sin ? = 0
(5)
??L SL ?g Sg
+
+ ?go Sgo
AL
Ag
1
1
+
AL Ag
? (?L ? ?g ) g sin ? = 0
(6)
Since the liquids are incompressible and neglecting
any inertial liquid level gradient along the pipe, the
three-phase pressure loss is equal in all phases. The
methodology involved the evaluation of the wall-shear
stresses for both water and oil liquids into a homogeneous expression. A two-phase model was then applied
to give an overall liquid height, which satisfied the balance relations. The height of the pseudo liquid, hL , was
assumed and then used to determine the level of the
aqueous phase, hW , using Eqn (8) on the premise that
Eqn (7) was satisfied. The determined phase heights
were then validated through re-substitution into the
??w Sw ?o So ?go Sgo
+
?
+ ?ow Sow
Aw
Ao
Ao
? (?w ? ?o ) g sin ? = 0
1
1
+
Aw Ao
(7)
(8)
The procedure is repeated until a balance is achieved.
Relevant equations are detailed in the Appendix.
The pressure gradient may then be inferred from
Eqns (4)?(6).
In this work modifications are proposed to both
the mechanisms by which the gas interfacial friction
factor was determined and the actual effective mixture
viscosity. The interfacial friction factor was estimated
from the Colebrook?White[42] expression
1
2?
9.35
+
= 3.84 ? 4 log
?
d
f
ReG f
(9)
where the interfacial roughness to diameter ratio, ?/d,
was taken as the film height to equivalent diameter
using the methodology of Baker and Gravestock.[43] The
effective viscosity was determined using the Brinkman
type model[41] validated by rheological measurements.
PREDICTION RESULTS
Figure 11. Three-phase stratified flow geometry.
? 2008 Curtin University of Technology and John Wiley & Sons, Ltd.
The predictive results of the model are illustrated in
Figs 12?14 and show an enhanced level of performance
over other three-phase models when judged against a
diverse range of data. In general, hold-up prediction
was inferior to that obtained for pressure drop.
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P. L. SPEDDING ET AL.
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Figure 12. Measured and predicted pressure gradient against fo using the
model of this work and i.d. = 0.0508 m data.
Figure 13. Measured and predicted pressure gradient against fo using the
model of this work and i.d. = 0.0259 m data.
Figure 14. Measured and predicted pressure gradient for the data of
Sobocinski[1] using the method of this work.
Despite the observed advances in performance, the
model still has certain deficiencies. Notably, the deterioration in performance with annular type flows where
underprediction prevailed and for smooth stratified
flows where overprediction was in evidence. The
selection of the interfacial wave height introduced as an
effective surface roughness into the Colebrook?White
? 2008 Curtin University of Technology and John Wiley & Sons, Ltd.
relation for the interfacial friction factor is not without
its limitations. Andritsos and Hanratty[44] showed that
the interfacial friction factor did not always qualitatively
follow the amplitude of the interfacial wave height to
amplitude ratio. Secondly, the modelling assumed a
constant stratified wetted-wall fraction and did not consider the effects of either wave spreading or surface
Asia-Pac. J. Chem. Eng. 2008; 3: 531?543
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Asia-Pacific Journal of Chemical Engineering
PRESSURE DROP IN THREE-PHASE HORIZONTAL CO-CURRENT FLOW
disturbance both of which have been shown to be
significant in two-phase momentum-balance studies.[45]
While the Brinkman[41] or the Pal and Rhodes[35] viscosity correlations improve prediction performance of
momentum-balance models they are not always reliable,
showing wide deviations depending on the assumed
inversion point. When actual rheological data were used
to check the predictions, an improvement in predictive
performance was in evidence. However, they are not
the actual emulsion viscosity conditions experienced in
the pipe, and their inclusion in the model should lead
to improved prediction performance.
Ag =
CONCLUSIONS
Aw =
The pressure-drop profiles obtained showed a dependence on gas and liquid flow rates and pipe diameter.
Above a critical total liquid and gas rate, a maximum
appeared in the pressure drop vs the oil to liquid volume ratio fo profile. The position of the maximum
pressure loss altered to higher fo values as the gas
rate was raised, due mainly to the formation of flow
regimes that possessed more turbulent motion in the
liquid phases. Most of the particular features observed
in the pressure profiles could be explained in terms of
the flow regimes and the turbulence present within the
system. The relationship between the rheological and
pressure-drop profile and the inversion point are discussed. The differences between the measured emulsion
viscosity and the conditions in the pipe are emphasised. Under similar velocity conditions the pressure
drop was observed to rise as the pipe diameter was
reduced.
Most of the predictive models for three-phase flow
were developed as an extension of the two-phase
momentum-balance condition and gave poor performance when tested against data. Existing three-phase
models also gave poor predictive performance against
the data of this work.
The Taitel et al .[16] three-phase momentum-balance
model was extended to predict the pressure-drop data
for a wide range of data. The methodology involved
using the actual rheological data for the emulsion and
the Colebrook?White relation to estimate interfacial
roughness. The model gave improved predictions over
existing methods.
Ag
d
2
= 0.25 cos?1 (2 h L ? 1)
2
? (2 h L ? 1) 1 ? (2 h L ? 1)
AL
AL =
d2
= 0.25 ? ? cos?1 (2 h L ? 1)
2
+ (2 h L ? 1) 1 ? (2 h L ? 1)
Aw
d
2
= 0.25 ? ? cos?1 (2 h w ? 1)
+ (2 h w ? 1) 1 ? (2 h w ? 1)2
Ao =
Ao
= AL ? Aw
d2
Sg =
Sg
= cos?1 (2 h L ? 1)
d
SL =
SL
= ? ? cos?1 (2 h L ? 1)
d
Sw =
Sw
= ? ? cos?1 (2 h w ? 1)
d
So =
So
= SL ? Sw
d
S go
Sgo
=
=
d
S wo =
Swo
=
d
1 ? (2 h L ? 1)2
1 ? (2 h w ? 1)2
Total holdup = 1 ?
1 ?1
cos (2 h L ? 1)
?
2
? (2 h L ? 1) 1 ? (2 h L ? 1)
APPENDIX
Three-phase geometries in terms of dimensionless
heights.
hL =
ho + hw
hL
=
= ho + hw
d
d
? 2008 Curtin University of Technology and John Wiley & Sons, Ltd.
Reg =
4 Vg Ag g
(Sg + Sgo ) 礸
Rew =
4 Vw Aw w
Sw 祑
Asia-Pac. J. Chem. Eng. 2008; 3: 531?543
DOI: 10.1002/apj
541
542
P. L. SPEDDING ET AL.
Reo =
Asia-Pacific Journal of Chemical Engineering
4 Vo Ao o
So 祇
�
?
?
= Viscosity, kg m?1 s?1
= Density, kg m?3
= Shear, kg m?1 s?2
1
?g = fg Vg 2 g
2
SUBSCRIPTS
fg = Cg Reg ?n
?w , ?o
laminar
Cg
= 16
n = 1.0
turbulent
Cg = 0.046
n = 0.2
follow on
BRINKMAN VISCOSITY MODEL
礒 =
祇
(1 ? ? )?
? = 2.5
NOMENCLATURE
A =
A =
B =
BF =
BTS=
C =
d =
D =
f =
fo =
F =
Fro =
g =
h =
i.d. =
=
n =
OD =
P =
P =
R =
Re =
RW =
S =
SL =
ST =
V =
W =
WD =
? =
? =
? =
? =
Annular regime
Area, m2
Bubble regime
Broken film regime
Blow through slug regime
Constant
Diameter, m
Droplet regime
Friction factor
Oil/liquid vol/vol
Film regime
Froth regime
Gravity, m s?2
Liquid height, m
Internal diameter, m
Length, m
Index
Oil dominated
Pressure, kg m?1 s?2
Pressure gradient, kg m?2 s?2
Ripple regime
Reynolds number
Roll-wave regime
Surface length, m
Slug regime
Stratified regime
Velocity, ms?1
Wave regime
Water dominated
Angle, degrees
Roughness, m
Index
Volume fraction dispersed phase
? 2008 Curtin University of Technology and John Wiley & Sons, Ltd.
A =
E =
F =
g =
go =
H =
i =
iog =
iow =
L =
o =
S =
T =
W =
Acceleration
Emulsion
Friction
Gas
Gas, oil
Head
Transition
Interface oil gas
Interface oil water
Liquid
Oil
Superficial
Total
Water
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