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Momentum Balance for Two-Phase Horizontal Pipe Flow Part 1 Friction Factors.

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Dev. Chem. Eng. Mineral Process. 12(1/2), pp. 179-188, 2004.
Momentum Balance for Two-Phase
Horizontal Pipe Flow
Part 1: Friction Factors
P.L. Spedding", J.S. Cole and G.F. Donnelly
School of Aeronautical Engineering, The Queen 's University of Belfast,
Stranmillis Road, Belfast BT9 SAG, Northern Ireland, UK
Estimations of gas wall, liquid wall and interfacial fiiction factors for two-phase
horizontal co-current pipe flow are discussed critically after being checked against
reliable data obtained under a wide range of conditions. The use of equivalent
diameters and the Blasius relation were shown to be valid for estimation of the gas
wall fiiction. Prediction of liquid wall and interfacial friction factors proved to be
more drfficcult but estimation improved if consideration was given to the efects of
liquid holdup and interfacial liquid shape.
Introduction
The one dimensional momentum balance model has been proposed for the prediction
of pressure loss, holdup and multiphase flow characteristics by considering the
transfer of momentum between the gas and liquid phases. This mechanistic approach
is not entirely theoretical since the development of representative wall and interfacial
friction factors are necessary for closure of the derived equations. The approach also
neglects any variations in the flow geometry. The model was originally developed by
Johannessen [ 13 for turbulent flows and extended by others [2, 31. Taitel and Dukler
[4] suggested a dimensionless form of the momentum balance which has been shown
*Authorfor correspondence.
I79
P.L. Spedding, J.S. CoIe and G.F. Donnelly
to have a multivalued solution under certain circumstances [5-71.
Successive
modifications have been proposed for the relationships governing the interfacial and
wall friction factors [3, 6, 8-24]. It is the purpose of this study to examine the friction
factors to ascertain which are the most appropriate for use with the momentum
balance two-phase model.
Friction Factors
(0 Gas Wall Friction Factor
Andritsos [ 171 showed experimentally that the use of equivalent diameters was valid
when estimating the gas shear stress while others have experimentally verified that
the Blasius relation correctly predicted the gas wall friction [19, 20, 23, 241. There is
perhaps a slight under prediction at higher gas Reynolds numbers in the annular
regimes where surface waves and droplet formation were in evidence.
(ii) Liquid Wall Friction Factor
Figure 1 gives an example of the results of two-phase studies performed over a
diameter range of 0.0259 to 0.0935 m [25-271 where both the Poiseuille and Blasius
relations are shown to underpredict the liquid wall friction, except for certain of the
stratified data. These findings are in agreement with other detailed work [3, 17, 19,
20, 23, 241. The deviations between experiment and theory increased steadily with
gas velocity as evidenced by the St+R and A+D data.
The failure to predict
accurately arises because effects of the interfacial shear stress exerted on the liquid by
the gas, and the shape of the gadliquid interface, have been ignored. In order to
improve prediction performance a number of workers have proposed the use of liquid
height [17, 191, holdup [20, 241 or interfacial friction [23, 281 in the relations for
predicting liquid wall friction. In addition, some workers have given consideration to
the degree of surface wetting by the liquid [23, 281 thus challenging the conventional
smooth stratified flow assumption.
Attempts to model data using two of the correlations, notably Equation (1) by
Kowalski [20] and Equation (2) by Hand [26], were an improvement on the Blasius
relation but were not entirely successful, as illustrated by Figure 2.
I80
Momentum Balance for Two Phase Horizontal Pipe Flow: Friction Factors
Figurel. Liquid wall friction against liquid Reynolds group compared to the
Poiseuille and Blasius relations using the 0.0454 m diameter data of
Nguyen [25].
.-8
5
.-
-r;
5J
..-s
,J
0.1 -:
--._
-...
ST+R
ST+RW+D
0
a - _
.P
0
A
i.;-,rt"
*$GH-.-$.
x
U
0.01 -:
e
a
......- Kowelski
Q
A A
J
-Hand
ST+RW
A+BTS
.d-s...+---x.+*+ *.
*x
X
x
- - X.. ..- .._
...--- .
.---..
5
0.001 9
181
P.L. Spedding, JS. Cole and G.F. Donneliy
600
500
400
300
200
100
0
0
100
200
300
400
500
Measured h
Figure 3. Prediction of h; against measurement using the 0.0953 m diameter data
of Andritsos [17/.
Figure 4. Intevacial to superficial gas friction against superficial gas Reynolds
number using the 0.0454 m diameter data of Nguyen [25].
182
Momentum Balancefor Two Phase Horizontal Pipe Flow: Friction Factors
f, = 0.263(gLRe,)-'"
f, = 0.0262(gLResL)-o'139
The main reasons for poor performance of these models was that no consideration
was given to the influence of interfacial friction and the actual shape of the wetted
surface. Secondly the relations depend on accurate prediction of liquid holdup.
Andritsos and Hanratty [18] proposed a relationship based on the work of
Cheremisinoff and Davis [13J to describe the wall stress in terms of a dimensionless
film height,
hl ,as given by:
d, = 4A, I S ,
(5)
Figure 3 shows as an example that the model performed favourably for a large
proportion of the data but was inadequate at lugher gas rates in the annular type
regimes. Tests of other models against data gave similar trends with neglect of
interfacial shape and friction contributing to a deterioration in prediction performance.
(iii)
Interfacial Friction Factor
The interfacial friction is usually inferred from the measured axial flowing pressure
gradient and liquid holdup, although other measurements have been used [8-101 such
as velocity profiles, etc. The magnitude of the interfacial shear stress is influenced by
both the surface roughness of the liquid and droplet generation. Johannessen [I]
neglected
A.
altogether. Others assigned a constant value to interfacial friction
[9, lo], sometimes only for lower gas rates [13, 151. Either way, such methods led to
gross under predictions of two-phase parameters [29]. Taitel and Dukler [4]used the
gas wall friction while others employed it only at low gas flows [13, 171. Generally
more complex relations have been proposed [8, 12, 20, 22, 23, 281. Hagiwara et al.
183
P.L. Spedding, J.S.Cole and G.F. Donnelly
0.06
0.05
0.04
I
,2
0.03
z
0.02
0.01
0
0
0.01
0.02
0.03
M e a s d fi
0.04
0.05
0.06
Figure 5. Measured and predicted interfacial friction factor using the method of
Hand [26] tested against data of Andritsos [17], d = 0.02515 m.
-
I":
0.01
0.1
Increased liquid loading
' """4
'
1
""":'
10
"'UA
loo
Measured
Figure 6. Measured and predicted interfacial fitiction factor using the method of
Grolman [28] tested against data of Hand [26], d = 0.0935 m.
I84
Momentum Balancefor Two Phase Horizontal Pipe Flow: Friction Factors
[30] demonstrated experimentally that the gas wall shear stress increased with both
wave amplitude at the liquid interface and associated turbulence, thus justifjmg an
emphasis on relations using a function of interfacial to gas wall friction for prediction
purposes [ 16, 17, 19,261. The profile given in Figure 4 highlights that as the gas rate
was increased so as to generate roll waves, there was a corresponding sharp rise in the
magnitude of the
f,. / fsc
ratio. In modelling
f,.
solely in terms of gas wall friction
factor, Crowley and Rothe [6] recognised the variation in the magnitude of the
interfacial shear stress associated with scale-up and suggested that data may be
bounded within a
f, I f s G ratio of
1 to 10. The data of Donnelly [27] was also
located within these limits. Hand [26] indicated that a sizable selection of 0.0935 m
diameter data was located on a
f, / fsc
loci of 4. For the larger diameter of 0.18 m
data, Kawaji et al. [21] found a loci of 3 provided the most appropriate data fit. The
interfacial friction factor proposed by Hand [26] is tested against data in Figure 5, and
gave a generally poor performance but certainly the best fit for this general type of
f, prediction method.
Both Hart et al. [23] and Grolman [28] advanced the accuracy of prediction, as
shown in Figures 6 to 8, by considering the effect of interfacial liquid shape when
developing methods of interfacial shear estimation.
Discussion
The prediction of shear stress deteriorated in accuracy when proceeding progressively
from
?wG
, the gas wall value, to ,T
for the liquid wall, through to
‘ti
for the
gadliquid interface. The value of the latter shear stress t i has a pronounced effect on
the momentum balance and its accurate estimation presents the greatest contemporary
challenge to successful phenomenological modelling of multiphase flow. It is clear
that neither the liquid wall friction nor the interfacial shear can be accurately
accounted for without proper consideration being given to both the liquid holdup
value and interfacial shape present within the system. Accurate prediction of liquid
holdup has been detailed by Spedding et al. [3 1-33] while both the interfacial shape
185
P.L.Spedding, J.S.Cole and G.F. Donnelly
100
10
B
1
B 0.1
L4
0.01
0.01
0.1
1
10
100
Measuxed
Figure 7. Measured and predicted interfacial frication factor using the method of
Grolman [28J tested against data of Nguyen [25], d = 0.0454m.
Increased liquid loading
0.01
0.01
0.1
1
10
100
Measured
Figure 8. Measured and predicted inte~acialpiction factor using the method of
Grolman [28J tested against data of Donnelly [27]. d = 0.0259m.
186
Momentum Balance for Two Phase Horizontal Pipe Flow: Friction Factors
and shear stress have been explored for low liquid holdup values [23, 291. A detailed
examination of these factors and their relationship to the momentum balance
modelling will be the subject of a future study.
Conclusions
The gas wall shear stress in two-phase flow can be accurately predicted using the
Blasius relation in association with the equivalent diameter. Prediction of both liquid
wall and interfacial shear exhibited scattered results particularly with the latter.
Attention to the effects of liquid holdup and interfacial shape led to an improvement
in prediction.
Nomenclature
A
AL
BTS
d
dL
D
f
F
hL
h'
IW
LRW
R
R
Annular regime
Cross sectional liquid area (m')
Blow through Slug regime
diameter (m)
Equivalent liquid diameter (m)
Droplet regime
Friction factor
Film regime
Centre line liquid height (m)
Dimensionless film height (Eqn 3)
Inertial wave regime
Long roll wave regime
Ripple regime
Holdup
RW
Re
SL
St
V
P
P
'5
Roll wave regime
Reynolds number, dVp / p
Liquid wetted length (m)
Stratified regime
Velocity (m s-'
Viscosity (kg m-' s-' )
Density (kg m-3)
Shear stress (kg m" s-' )
Subscripts
G
Gas
i
L
S
W
Interface
Liquid
Superficial
Wall
References
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Received: 16 July 2002; Accepted afer revision: 1 May 2003.
188
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