International Journal of Multiphase Flow 91 (2017) 120–129

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International Journal of Multiphase Flow

journal homepage: www.elsevier.com/locate/ijmulflow

Investigation of pressure drop in horizontal pipes with different

diameters
F.A. Hamad a,∗, F. Faraji a, C.G.S. Santim b, N. Basha a, Z. Ali a
a
School of Science & Engineering, Teesside University, Middlesbrough, TS1 3BA, UK
b
Faculty of Mechanical Engineering, University of Campinas, CEP: 13083-860, Campinas, Brazil

a r t i c l e i n f o a b s t r a c t

Article history: The pressure drop has a significant importance in multiphase flow systems. In this paper, the effect of
Received 27 August 2016 the volumetric quality and mixture velocity on pressure drop of gas-liquid flow in horizontal pipes of
Revised 8 January 2017
different diameters are investigated experimentally and numerically. The experimental facility was de-
Accepted 23 January 2017
signed and built to measure the pressure drop in three pipes of 12.70, 19.05 and 25.40 mm. The water
Available online 27 January 2017
and air flow rates can be adjusted to control the mixture velocity and void fraction. The measurements
Keywords: are performed under constant water flow rate (CWF) by adding air to the water and constant total flow
Air-water flow rate (CTF) in which the flow rates for both phases are changed to give same CTF. The drift-flux model is
Pressure drop also used to predict the pressure drop for same cases. The present data is also compared with a number
Horizontal pipes of empirical models from the literature. The results show that: i) the pressure drop increases with higher
Experimental measurement volumetric qualities for the cases of constant water flow rate but decreases for higher volumetric qualities
Drift-flux model
of constant total flow rate due to the change in flow pattern. ii) The drift-flux model and homogenous
model are the most suitable models for pressure drop prediction.
© 2017 Elsevier Ltd. All rights reserved.

1. Introduction In horizontal flow, the phases tend to separate due to the dif-
ference in densities and the effect of fluid gravity, thereby causing
The application of single and multiphase flow has been fre- a form of stratification. The heavier fluid tends to concentrate at
quently observed in many diverse fields of science and engineer- the bottom of the pipe whereas the lighter fluid concentrates at
ing such as agricultural, biomedical, chemical, food science and the top. Several flow patterns can be observed during the flow of
petroleum engineering. It is necessary to predict design parame- mixed phases as flow rates of water and air are varied. These flow
ters such as friction factor, pressure drop, bubble size, void fraction, patterns also depend on the physical properties of the fluids such
heat and mass transfer coefficient in order to determine the de- as the density and viscosity, surface tension and the flow system
sired operating conditions and the size of the equipment required geometry.
for the specific application. The pressure drop in horizontal pipes According to Awad (2012) the formation of specific flow pattern
is the parameter to be investigated in this paper is governed by competition of different forces in the system such
The pressure drop in horizontal pipes has been studied by a as momentum, viscous, gravitational, and surface tension. When
number of researchers to develop empirical models to use in the the momentum force in two-phase flow is dominant, the bubbles
design of new equipment. However, there is no general model tend to disperse uniformly into the pipe. This usually occurs at
available to predict the pressure drop within acceptable accuracy high mixture flow rate, which leads to a bubbly flow.
(Michaelides, 2006). This is attributed to the complexities inher- The pressure drop of a fluid is due to the variation of kinetic
ited from the single-phase flow like the non-linearity, transition to and potential energy of the fluid and that is due to friction on
turbulence and instabilities plus additional two-phase characteris- the walls of the flow channel. Therefore, the total pressure drop
tics like motion and deformation of the interface, non-equilibrium is represented by the sum of the static pressure drop (elevation
effects and interactions between the phases (Ghajar, 2005). head), the momentum pressure drop (acceleration) and the fric-
tional pressure drop. Here the most problematic and important
term is the frictional pressure drop, which can be expressed as a
∗
Corresponding author. function of the two-phase friction factor.
E-mail addresses: f.hamad@tees.ac.uk (F.A. Hamad), foadfaraji5@gmail.com Two distinct approaches are available from engineering point
(F. Faraji), christianosantim@gmail.com (C.G.S. Santim), n.mehboobbasha@tees.ac.uk of view in accounting for the behaviour of multiphase flow sys-
(N. Basha), z.ali@tees.ac.uk (Z. Ali).

http://dx.doi.org/10.1016/j.ijmultiphaseflow.2017.01.007
0301-9322/© 2017 Elsevier Ltd. All rights reserved.
 F.A. Hamad et al. / International Journal of Multiphase Flow 91 (2017) 120–129 121

tem. The first is a global approach that relies on the practical Table 1
The values of coefficient C
method in developing simplified models that contain parameters,
(Holland and Bragg, 1999).
which are evaluated from the experimental data (Lockhart and
Martinelli, 1949; Friedel, 1979; Mishima and Hibiki, 1996; Chen Liquid Gas C
et al., 2001). The second is a continuum approach in which more Turbulent Turbulent 20
complex physically-based models are used to describe the flow Viscous Turbulent 12
phenomena (Ivor, et al., 2004; Beattie and Whalley, 1982; Awad Turbulent Viscous 10
Viscous Viscous 5
and Muzychka, 2014a, b). The two-phase frictional pressure drop in
gas-liquid flow is determined by either finding of two-phase fric-
tion factor (homogeneous flow model) or a two phase friction mul-
2.2. Separated flow models
tiplier (separated flow model). A summary of the empirical models
is given hereafter.
2.2.1. Lockhart-Martinelli (1949)
2. Review of two-phase frictional pressure drop correlation The Lockhart-Martinelli pressure drop correlation is the most
typical form of separated flow model. A majority of correlation that
2.1. Homogeneous flow model has been proposed by many researchers such as (Friedel, 1979;
Mishima and Hibiki, 1996; Chen et al., 2001) were proposed on the
In Homogenous model, it is assumed that there is no slip be- basis of two-phase friction multiplier suggested by Lockhart and
tween the two phase flow at similar velocities. However, with Martinelli (1949) and the fitting correlation of the multipliers from
an exception for very small values of void fraction (bubbly flow Chisholm (1983).
region) there exists a significant slip between the two phases Lockhart and Martinelli (1949) performed the most represen-
(Bhagwat and Ghajar (2014)). tative investigation that developed the theory of separated flow
The frictional pressure drop equation is the Darcy equation model. Their work based on experimental analysis of a circular
which uses Blasius relation to calculate the friction factor from pipe with the diameter ranging from 1.48 to 25.83 mm using two
the average mixture properties. The Blasius equation for two-phase phase mixture of air with benzene, kerosene, water and several
flow is represented by: oils. Their work was based on two hypotheses, the first assumption
states that the static pressure drop for both liquid and gas phases
fT P = 0.079/Re−0
m
.25
, (1) are the same regardless of the flow pattern as long as the changes
where Rem = Gm d/μm is the mixture Reynolds number, fTP is the in radial direction are not significant and the second assumption
two-phase flow friction factor, Gm : mass flus (kg/m2 s), d is the pipe states that the total volume of the pipe is equal to sum of the vol-
diameter (m) ume occupied by gas and liquid at any instant (continuity equa-
The mixture viscosity (μm ) is represented as in Awad (2014), in tion). Based on these assumptions and their experimental analysis,
terms of the mass quality (x): Lockhart and Martinelli (1949) developed the concept of two-phase
μm = x μg + ( 1 − x ) μl , (2a) flow multipliers which can be used to calculate the two-phase flow
pressure drop (dp/dx)TP as ((Holland and Bragg, 1995):
Where x is the mass quality, μg is the gas viscosity μl is
(N.s/m2 ),    
the liquid viscosity (N.s/m2 ) dp dp
= φl2 . (5)
Rodrigo et al., (2016) studied experimentally the pressure drop dx dx
TP l
of multicomponent zeotropic mixtures boiling in small channels Chisholm (1983) developed the theoretical basis to calculate the
over temperatures ranging from 100 K to room temperature along liquid phase multiplier φl2 from the following simplified correla-
with the sensitivity of frictional pressure drop to parameters such tion,
as mass flux, pressure, tube diameter, and mixture composition.
c 1
The measured data were compared to several pressure drop cor- φl2 = 1 + + 2, (6)
relations available in the literature. They found that Awad and X X
Muzychka (definition 1) correlation (Awad and Muzychka, 2008) where

for multiphase flow viscosity was able to predict the pressure drop X= (d p/dx )l /(d p/dx )g . (7)
over the range of experimental data considered, with an Absolute
The values of C are into the range of 5 ≤ C ≤ 20 for different
Average Deviation (AAD) of 17%. Awad and Muzychka (2008) (defi-
flow regimes, as given in Table 1.
nition 1) is also used in this study to predict the pressure drop for
all the tests in present work. The viscosity is given as: 2.2.2. The Friedel correlation (1979)
2 μl + μa − 2 ( μl − μa ) x Friedel model is one of the most accurate methods to deter-
μ2 ph = μl (2b)
2 μl + μa + ( μl − μa ) x mine the pressure drop in two-phase flow (Quiben (2005)). Friedel
Whereas, the mixture density (ρ m ) can be evaluated as (Awad and developed a correlation based on 16.0 0 0 measured data points
Muzychka, 2008; Awad 2015): and for wide range of pipe diameters. The model includes the
gravity effect through the Froude number (Fr), the effects of sur-
ρm = αρg + (1 − α )ρl (3) face tension and total mass flux using the Weber number (We)
Where α is void fraction, ρ g is the gas density ρ l is the
(kg/m3 ), (Suwankamnerd and Wongwises, 2014).
liquid density, (kg/m3 ) The two-phase flow pressure drop, ( ddxp )T P can be obtained as:
Based on the above average properties, two phase frictional    
pressure drop for horizontal tube of internal diameter, d is calcu- dp dp
=φ 2
lo . (8)
lated as: dx dx
TP lo
2 fT P G2m l
p = (4) Where φlo2 is the two-phase friction multiplier based on pressure
ρm d gradient for total flow assumed liquid. It can be calculated from
Where l is the length of the pipe. The homogenous model becomes the following equation:
more accurate for density ratio lower than 10 and mass flux lower 3.24F H
than 20 0 0 kg/m2 s (Crowe, 20 06). φlo2 = E + , (9)
Fr0.45 We0.035
122 F.A. Hamad et al. / International Journal of Multiphase Flow 91 (2017) 120–129

The Froude and weber numbers are given respectively as F r = 3. Experimental facility
G2m G2m d
and W e = ρm σ , where σ is the surface tension. The dimen-
gdρm2
The experimental facility shown in Fig. 1 is designed and built
sionless parameters F, H and E (Thome, (1990)) are defined as fol-
at Teesside University, to investigate the pressure drop for single
lows:
and two-phase flows pressure drop. The test rig has three PVC
ρl fgo transparent pipes of 1 m in length and inner diameters of 0.0127 m,
E = ( 1 − x )2 + x2 ,
ρg flo 0.01905 m and 0.0254 m. The main components of the test rig are
the water centrifugal pump, air compressor, water and air flow me-
F = x0.78 (1 − x )0.224 , ters, water tank, and differential pressure transducer.
To generate air-water mixture, the water is pumped from the
 0.91  tank to the test section using a centrifugal pump. Then, the air is
ρl μg 0.19  μ g 0 . 7 supplied from the main compressor in the building via a filter and
H= 1− .
ρg μl μl pressure regulator, this is done to minimise the fluctuation in air
flow rate. The air and water flow rates are measured by the flow
The flo and fgo can be calculated from the single phase fric- meters at the upstream of the mixing point.
tion correlation (Blasius equation) based on liquid Reynolds num- In this investigation, the measurements are performed under
ber, (Relo = Gd/μl ) and gas Reynolds number (Rego = Gd/μg ). From constant water flow rate (CWF) by adding air to the water and con-
Blasius equation: flo = 0.079/Re0lo.25 and fgo = 0.079/Re0go.25 . stant total flow rate (CTF) in which the flow rates for both phases
The pressure drop for assuming the total flow is liquid, (dp/dx)lo are changed to give same CTF. The water flow rate up to 40 l/min
can be calculated as, was measured by the digiflow 6710 M meter. The air flow rate was
  measured by Platon air flow meter with accuracy of ± 1.25%. The
dp 2 floG2 vl
= (10) differential pressure transducer (C9553 COMARK) is connected to
dx d
lo the test section by two flexible plastic tubes via two taps at inlet
Where vl is the liquid specific volume (m3 /kg) is Thus, the two- and exit of the pipe. The images of flow patterns are obtained by a
phase flow pressure gradient can be obtained by substituting the high-speed digital camera (NiKon 1J1). Due to the limited length of
Eqs. (9) and (10) in Eq. (8). the pipes, a perforated pate was used a flow conditioned eliminate
velocity profile distortion and uneven void fraction distribution.

2.2.3. Muller-Steinhagen and heck

4. Numerical solution
Muller-Steinhagen and Heck (1986) proposed the following
two-phase frictional pressure drop correlation based on all liquid
A Roe-type Riemann solver based on drift-flux model proposed
flow and all gas flow,
by Santim and Rosa (2016) was used to calculate the pressure drop
 
dp numerically. The model is assumed to be isothermal with no mass
= GMS (1 − x )1/3 + Bx (11) transfer between the phases. The system of equations for the con-
dx
TF
servation laws is given by Eqs. (15)–(17). The first two equations
where the factor GMS is defined as, consist of mass formulations for each phase i.e, liquid and gas and
the last equation for the mixture momentum conservation.
GMS = A + 2(A − B )x. (12)
∂ ∂
Assuming the total flow is liquid, the pressure drop (dp/dx)lo [(1 − α )ρl ] + [(1 − α )ρl ul ] = 0, (15)
∂t ∂x
can be calculated as:
  ∂ ∂
dp 2 floG2 vl (αρg ) + (αρg ug ) = 0, (16)
A= = . (13) ∂t ∂x
dx d
lo

Similarly, the pressure drop for assuming the total flow is gas
∂ ∂  
((dp/dx)go ): [(1 − α )ρl ul + αρg ug ] + (1 − α )ρl u2l + αρg u2g + P = FW ,
  ∂t ∂x
dp 2 fgoG2 vg (17)
B= = . (14)
dx d where P is the pressure, α represents the void fraction, ρ is the
go
density, u is the velocity, with the subscripts l and g refer to the
The literature review presented above shows that in spite a
liquid and gas phases. The last term on Eq. (17), FW , is a momen-
number of empirical models developed to predict the pressure
tum source term that represents the wall friction force and is given
drop of multiphase flow, there is still a dearth of research work
as:
needed in this area as there are no reliable models that can be
used for different geometries and flow patterns. Hence, the pur- ρm |um |um
FW = f (18)
pose of this paper is twofold. The first is to collect new experi- 2d
mental data on pressure drop for various pipe diameters with dif- In which um = (1 − α )ul + α ug represents the mixture velocity,
ferent flow patterns (Teesside University) at Constant Water Flow ρm = αρg + (1 − α )ρl is the density of mixture in terms of the
rate (CWF) where the air is added to the water. In addition to Con- void fraction, f represents the friction factor and d is the ID pipe.
stant Total Flow rate (CTF) where the flow rates for both phases The friction factor (f) depends on Reynolds number of the mix-
are changed to give CTF. The second is to examine the possibility ture (Rem ), which is defined as:
of using the drift-flux model (utilizing the approximate Riemann
ρm |um |d
solver proposed by Santim and Rosa (2016) to predict the pressure Rem = , (19)
drop for two-phase flows by comparing the experimental data with
μm
predictions from the model. In addition, the present experimental where μm is the mixture viscosity. The relation proposed by
measurements are also compared with predictions from empirical Beattie and Whalley (1982) is used: μm = (1 − α )μl (1 + 2.5α ) +
models in the literature. αμg in the range 0 < α < 1.
 F.A. Hamad et al. / International Journal of Multiphase Flow 91 (2017) 120–129 123

Fig. 1. Schematic diagram of the experimental rig.

For laminar flows, the friction factor is defined as f = 64/Rem . length is too short or flow to fully develop and achieve a define
The implicit relation proposed by Colebrook, Eq. (21), is utilized defined flow pattern.
to calculate f for turbulent flows since the Eq. (20), proposed by  
ρg −18α
Haaland, is assumed as an initial guess for Colebrookś equation. 1.2 − 0.2 ρl 1 − e
2
  C0 =  Rem 2 +  1000 2 , (24)
1 ε 1.11 6.9 1+ 1+
 = −1.8 log + , (20) 10 0 0 Rem
f 3.7D Rem  1/4

gσ ρ
ud = C cos (θ ) + D sin(θ ). (25)
1 ε /D 2.51 ρl2
 = −2 log +  , (21)
f 3.7 Rem f The discretization scheme used in the simulations is an upwind
discretization as demonstrated by Leveque (2002), in which the
where ε represents the equivalent roughness of the pipe, consid- vector U of conservative variables has its components Ui evaluated
ered as 10−9 m. using an explicit numerical procedure depicted below:
The thermodynamic state equations for the liquid and gas den-  _ +

sities are expressed in terms of the sound velocities, cl and cg , as t m m
Uin+1 = Uin − (λ ) wi+1/2 +
p p
(λ ) wi−1/2 ,
p p
(26)
presented below x p=1 p=1
P − Pl,0 P
ρl = ρl,0 + and ρg = 2 , in which,
cl2 cg
βi−1/2 = R−1 (Ui − Ui−1 ) and wip−1/2 = βip−1/2 r p , (27)
where ρ l,0 and Pl,0 are given as constants.
The system of the conservation laws, given by Eqs. (15)–(17), where w represents the waves crossing the cells’ interface, and λ−
can be written in the conservative form, as: λ+ are the characteristic velocities (superscript ’-’ means left going
waves). The matrix R represents the right eigenvector matrix, and
∂U ∂F
+ = S, (22) p is the counter of eigenvalues (m is the total number).
∂t ∂ x This explicit scheme must satisfy a CFL (Courant-Friedrichs-
where U, F and S are the vectors of the conservative variables, Lewy) condition as stability criterion:
fluxes and source terms written as follows: t



|λmax | < 1. (28)
U1 (1 − α )ρl F1 x
U= ≡
U2 αρg F = F2 The wall friction force source term is treated using the
U3 (1 − α )ρl ul + αρg ug F3 Fractional-Step method studied by Leveque (2002). The hyperbolic

( 1 − α ) ρl u l S1 0 system is split into two sub-problems which are solved indepen-

≡ αρg ug S = S2 ≡ 0 . (23) dently. The first consists of a homogeneous system using the up-
(1 − α )ρl u2l + αρg u2g + P S3 FW wind scheme previously presented in Eq. (26).
 _ +

Since the system has three equations and four unknowns, we t m m
Ui∗ = Uin − (λ ) wi+1/2 +
p p
(λ ) wi−1/2
p p
(29)
need to obtain the system closure by using of a drift-flux rela- x p=1 p=1
tion. The relation chosen was ug = C0 um + ud , presented by Zuber
and Findlay (1965). The parameter C0 and the drift velocity ud are The ODE must be solved in a second step, as
defined primarily considering the fluid transport properties and
Uin+1 = Ui∗ − t S3 . (30)
sometimes by the flow pattern regime. The drift parameters imple-
mented on the solver were proposed by Choi et al. (2012) and are To obtain a numerical solution on pressure drop along a pipe,
pattern independent. This correlation was chosen since the pipe a spatial mesh with 100 nodal points is chosen after a mesh test,
124 F.A. Hamad et al. / International Journal of Multiphase Flow 91 (2017) 120–129

The results show that the pressure drop increased significantly

for small pipe diameter. To have a better understanding of this
effect, the variation of pressure drop with volume fraction for
the same water Reynolds number (Re = 26,0 0 0) considering three
pipes is given in Fig. 3d. The results show that reducing pipe di-
ameter lead to nonlinear increase in friction pressure drop similar
to single phase flow which can be approximated by the follow-
ing formula: p = 4.733d−1.45 , this formula is given by Bhagwat et
al. (2012). A similar finding was reported by Kaji and Azzopardi
(2010) for air-water flow in vertical pipes of diameters in a range
of 0.010 m – 0.050 m.

5.3.1. Pressure drop and flow pattern

From the visual observation of the flow pattern during the pres-
sure drop tests, it is possible to verify that the flow pattern in
Fig. 2. Comparision of measured single phase flow friction factor with prediction the pipe is changed. To investigate this in further detail, the wa-
from Blasius correlation. ter single phase flow of 40 l/min in 0.0254 m diameter is selected
to observe and the flow pattern for both CWF and CTF at different
volumetric qualities. Fig. 4a shows the pressure variation for the
and a temporal mesh with 20,0 0 0 points is used. Finally, the tran- vomumetric qualities under investigation.
sient simulation is halted to record the results when it reaches the The corresponding flow pattern at points 1, 2, 3, 4, 5 and 6 are
steady state. given in Fig. 4b and c. The flow patterns are observed visually and
recorded photographically using High Speed Camera for water flow
5. Results and discussion rate of 40 l/min in 0.0254 m ID pipe. For CWF cases (Fig. 4b), the
photos show that two different zones of bubbly flow and water can
5.1. Single phase pressure gradient be observed. For lower air flow rate of 5 l/min (β = 11.1%), bubbly
flow of 1 cm height is formed in the upper part of the pipe while
The single-phase flow measurements are carried out to validate water zone of 1.5 cm height flow in the rest of the pipe separated
the experimental facility along with the accuracy of instrumenta- by wavy interface. The thickness of the bubbly flow zone expand
tion and thereby set the reference velocities to be used for the further down to become 1.5 cm for air flow of 15 l/min (β = 27.3%)
comparison with two-phase flow measurements. The pressure drop and 2 cm for air flow rate of 25 l/min (β = 38.5%). The large long
data for different water flow rates is recorded and this is used to bubbles appears near the upper pipe wall at the upper region of
calculate the friction factor by applying Darcy equation for smooth the bubbly flow zone for β = 38.5%.
pipe as: For the CTF, the photos show that by replacing 5 l/min (12.5%)
 pd of water with air, generates two layers of the flow similar to
fexp = . (31) flow pattern at point 1 for CWF. Increasing the air to 15 l/min
2Lρv2
(β = 37.5%) leads to an appearance of air of about 0.6 cm thickness
Fig. 2 presents the comparison of the experimental friction at the top, water layer at the bottom of the pipe and this bottom
factor fexp with predicted values from Blasius correlation ( f = layer is separated by a middle layer of bubbly flow. The bubbly
0.079/Re0.25 ) for each flow rate (Re). The error associated with the flow lies in the central region of the pipe. The flow in the middle
pressure drop measurements proves to be quite reasonable (± 10%). layer is very chaotic and has unstable wavy interface with air and
The discrepancy in the data compared to well-known correlation water. A similar flow pattern is observed for higher air flow rate
developed by Blasius may attributed to the inaccuracy of flow me- of 25 l/min (β = 62.5%), but the layer of air becomes thicker and
ters and pressure transducer and the change in temperature due water layer becomes thinner.
the heat added by the pump. The temperature affects the density The changes in volumetric quality (β ), flow structure and pres-
and viscosity of the fluid used in calculation of Renolds number sure drop in photos of Fig. 4b & c indicate the strong correlation
used in calculation of friction factor. between the pressure drop and the flow structure. The pressure
drop is a result of energy dissipation, eddy viscosity and bubble
5.2. Two-phase flow measurements induced turbulence. Energy dissipation occurs due to the fluid vis-
cosity and eddy viscosity occurs due the wall generated turbu-
The frictional pressure drop in this study was investigated lence. In case of two-phase flow, the turbulence becomes more
with two different methods of two-phase generation. In the first complicated due to: i) the interaction between the drop-induced
method, air is injected while the water flow rate kept is constant, turbulence (bubble motion) and shear–induced turbulence (the ex-
hence that increases the total mass flow rate and β of mixture. ternal force due to main flow). ii) The modification of water ra-
This method is called Constant Water Flow (CWF). In the second dial velocity distribution due to the radial distribution of bubble
method, the total flow rate of the mixture is kept constant and the slip velocity, bubble diameter and volume fraction. To add to this,
flow rate is changed for both the phases, this leads to a two-phase the introduction of bubble can either enhance or attenuate the liq-
flow. This method is called Constant Total Flow rate (CTF). For CWF, uid turbulence (Serizawa and Kataoka (1990); Hamad and Ganesan
the results show that the increase in β lead to an increase in the (2015).
mixture mass flux and the pressure drop. This is a similar trend The modification in energy dissipation (pressure drop) may be
that was observed by Muller-Steinhagen and Heck (1986), Shannak attributed to (Hamad and Ganesan (2015): i) the breakup of large
(2008) and Hamayun et al. (2010). In contrast to the CWF cases, scale eddies containing higher energy by the bubble moving faster
the results show that an increase in β lead to a reduction in mass than water. ii) The drop wake turbulence due the slip velocity be-
velocity and pressure drop. The measured pressure drop for CWF tween the drops and the continuous phase in bubbly layers gen-
and CTF are presented in Fig. 3a, b and c, relative to the 0.0254 m, erate additional turbulence as described by Risso et al. (2008). iii)
0.01905 m and 0.0127 m ID, respectively. The unstable fluctuating interface zone between the layers can be
 F.A. Hamad et al. / International Journal of Multiphase Flow 91 (2017) 120–129 125

a b

c d

Fig. 3. (a) Variation of measured pressure drop for the CWF and CTF rates in 0.0254 m pipe with volumetric quality. (b) Variation of measured pressure drop for the CWF
and CTF rates in 0.01905 m pipe with volumetric quality. (c) Variation of measured pressure drop for the CWF and CTF rates in 0.0127 m pipe with volumetric quality. (d)
Effect of pipe diameter on pressure drop (Re = 26,0 0 0).

considered as an addition wall due to the velocity difference be- to predict the pressure drop. For all simulations the system shows
tween the two layers which lead to generation of addition eddies. to be hyperbolic.
Fig. 5 represents a comparison between the experimental pres-
5.3.2. Experimental uncertainties sure drop and the numerical prediction as a function of the mix-
The experimental uncertainty is a combined effect of pipe di- ture Reynolds number. The results show that there are different
mensions (length and diameter and surface roughness), the accu- representative zones: i) for 0 < Re < 75,0 0 0, the data is not scat-
racy of instrumentation (flow meters and pressure transducer), the tered, ii) for 75,0 0 0 < Re < 125,0 0 0, there is a high scatter and
fluid properties (density, viscosity and surface tension) and the op- iii) Re> 125,0 0 0 the pressure drop increases smoothly. These three
erating conditions (flow rate and temperature). The effect of these zones are similar to laminar, transition and turbulent flow for sin-
variables will lead to some error in pressure drop measurements. gle phase flow. The comparison proves that the drift-flux model
The error can be estimated using the available statistical correla- can be used for pressure drop prediction of two-phase flows.
tion and incorporated into the plotted graph as error bars. Fig. 4d
present the experimental data for two-phase flow with error bar.
5.5. Comparison of present pressure drop data with empirical models
The height of the bars reflects the level of uncertainty at the dif-
ferent Reynolds numbers. It can be observed that the uncertainty
The investigation was carried out for three horizontal pipes of
is peaked for Re in the rage of 75,0 0 0 −125,0 0 0 reflecting the high
different diameters using air-water mixture. Table 2 provides the
instability of the flow this
range of various parameters used in the experiments.
The measured pressure drop values have been compared with
5.4. The drift-flux model the predictions from the most common existing empirical models,
the models selected for this purpose are Lockhart and Martinelli
The model takes into account the effects of non-uniform veloc- (1949), Friedel (1979), Müller-Steinhagen and Heck (1986), Awad
ity and void fraction profiles as well as effect of local relative ve- and Muzychka, (2008) and the homogeneous model. The most
locity between the phases (Shen et al., 2014). The relative motion of the above-mentioned models are applicable for smooth pipes.
between the phases is governed by a subset of the parameters in- Therefore, the test conditions of the present experimental data in
herent to the flow. The model comes from the Two-Fluid model transparent acrylic pipes are applied to the above-mentioned mod-
(TFM) through neglecting the static head terms and assuming the els. The comparison between the measurements and the predic-
momentum conservation of the mixture. Therefore, a third bound- tions are presented in Figs. 6–9.
ary condition is not necessary at the inlet region and the interfacial The accuracy of the predictions can be measured by calculating
friction term is cancelled out. Other advantage is that the equa- the average percent error (APE) and average absolute percent error
tions can be put in a conservative form, facilitating to discretize (AAPE) of each data source.
by finite volume methods. The system of the conservation laws is The percentage error at each point (PE) can be calculated as:
generally hyperbolic depending on the slip law used.  
The approximate Roe-type Riemann solver proposed by Santim (d p/dx ) pred − (d p/dx )exp
PE = 100. (32)
and Rosa (2016), which is based on the Drift-Flux model, is applied (d p/dx )exp
126 F.A. Hamad et al. / International Journal of Multiphase Flow 91 (2017) 120–129

a b

Fig. 4. (a) Variation of the pressure drop with CWF and CTF in the pipe of 0.0254 m. (Water flow rate = 40 l/min). (b) Photographs of CWF cases (d = 0.0254 m, 40l/min). (c)
Photographs of CWF (d = 0.0254 m, 40 l/min). (d) Experimental data for pressure drop with error bars.

Table 2
Experimental measurements.

Pipe ID (m) No of tests Flow Type Usw (m/s) Usa (m/s) β (%) p (Pa) Rem

min max min max min max min max min max

0.0254 19 CWF 0.66 1.32 0 0.822 0 55 260 1300 15,0 0 0 65,0 0 0

0.0254 14 CTF 0.16 1.32 0 0.82 0 75 160 750 15,0 0 0 46,0 0 0
0.0191 24 CWT 0.585 0.234 0 1.46 0 72 10 0 0 4800 130,005 95,0 0 0
0.0191 20 CTF 0.29 2.33 0 1.4 0 83 240 2800 13,300 77,0 0 0
0.0127 24 CWF 1.21 5.26 0 3.28 0 71 50 0 0 33,0 0 0 33,0 0 0 190,0 0 0
0.0127 19 CTF 0.65 5.26 0 3.28 0 83 1200 21,0 0 0 24,0 0 0 125,0 0 0

The average percentage error is defined as follows: the absolute errors are considered so that the positive and nega-
   tive errors are taken into account (the positive and negative errors
1  (d p/dx ) pred − (d p/dx )exp
n
are not cancelled out). The equation is given as:
AP E = 100. (33)
n (d p/dx )exp
k=1

Eqs. (32) and (33) are used to estimate the error of individual ⎡ ⎤
points and average error of the data. The average abolute percent-   2 1 / 2
1 (d p/dx ) pred − (d p/dx )exp
n
age error (AAPE) is calculated to evaluate the prediction capability AAP E = ⎣ ⎦100. (34)
of the emprical correlation. Unlike the average percent error (APE), n (d p/dx )exp
k=1
 F.A. Hamad et al. / International Journal of Multiphase Flow 91 (2017) 120–129 127

Fig. 5. Comparision between the experimental data and prediction from drifit flux
model.
Fig. 7. Comparison of experimental data with Lockhart-Martinelli model.

a
perimental data within acceptable error and can be recommended
as a guide for pressure drop prediction in multiphase flow systems.
The Friedel model gives the highest percentage of errors and not
recommendable for pressure drop prediction of multiphase flow in
horizontal pipes.
Through analysis of Fig. 6a & b, it can be observed that the
assumption of multiphase flow as homogeneous flow is reason-
able as the trend lines of experimental and predictions are very
close. Ghajar (2005) considers that the homogenous model is more
suitable for predicting pressure drop in bubbly flow patterns. The
present results also confirm this behaviour once the homogenous
model gives more accurate results for the bubbly flow cases of
CWF and the low air flow rate (5 l/min) for CTF. The error increases
for the cases of CTF with high volumetric quality when wavy strat-
ified flow is observed. The discrepancy can be related to the orig-
inal assumption, which is made in the equation of the homoge-
b neous model, that the flow is homogenous and the velocities of
the gas and liquid are the same.
By analysis of Fig. 7, it can be verified that the Lockhart-
Martinelliḿodel is suitable for CWF cases as the error is lower
compared to CTF. The higher error for CTF may be attributed to the
change in the flow pattern to wavy stratified flow similar to ho-
mogenous model. Spedding et al. (2006) found that the Lockhart-
Martinelli over/under predicts the data within ± 40% error espe-
cially in higher mass velocity values which are much higher than
present study. Awad and Muzychka (2014a) and Quiben (2005),
recommended that Lockhart-Martinelli as one of the best methods
that can be used for predicting pressure drop in two-phase flow as
it can be used for any flow pattern. The results in Fig. 7 demon-
strate that at higher mass flux (Reynolds number) of the mixture,
this method under predicts the data. This may be attributed to the
assumptions considered in the development of the model such as:
i) interaction between the two-phases is ignored, ii) the accelera-
Fig. 6. (a) Comparison of experimental data with Homogenous model. (b) Compar- tions and static heads for the phases are neglected, therefore the
ison of experimental data with Awad and Muzychka, 2008. pressure drop in gas and liquid phases is assumed to be the same.
From Fig. 8, it can be seen that the Friedel model gives a high
discrepancy compared to the other correlations which is reflected
A summary of the error percentage calculated from Eqs. in average error of 56% as given in Table 3. As it can be verified
(33) and (34) for all the models (empiricals and numerical) is given from the comparison, this model over predicts the experimental
in Table 3. data. The present finding is supported by even higher error of 66%
The error values in the table show that the drift-flux model given by Xu et al (2012) and 83% by Awad and Muzychka (2014b).
and homogenous model and Awad and Muzychka, (2008) model In contrast, some authors (Quiben (2005) and Ghajar (2005)) rec-
are the most accurate models for pressure prediction as the values ommended the Friedel correlation is capable of providing the most
of AAPE, APE, average negative error and average positive error are accurate results for pressure drop analysis in two phase flows. The
the lowest compared to the other models. The Lockhart-Martinelli high discrepancy may be attributed to the difference in test oper-
and Muller-Steinhagen & Heck models can predict most of the ex- ating condition, pipe diameter and using fluids of different densi-
128 F.A. Hamad et al. / International Journal of Multiphase Flow 91 (2017) 120–129

Table 3
Estimated error for each model.

Model APE (%) APE (-Ve%) APE (+Ve%) AAPE (%)

Drift-flux model 3 −15.2 17.2 16.2

Homogenous –1.9 −17.3 15.4 16.2
Awad and Muzychka (2008) 2.8 −16.2 19.7 18
Lockhart-Martinelli (1949) −10 −21 16.2 10
The Friedel Correlation (1979) 56 −9.5 67 60.7
Muller-Steinhagen and Heck (1986) −12.2 −22.5 14.2 18.6

various empirical correlations and by using of the drift-flux model.

The superficial water velocity in the range of 0.16 - 5.263 m/s and
superficial air velocity in the range of 0.16 – 3.289 m/s were used
to give the different values of volumetric qualities. The pressure
drop measurements were performed under constant water flow
rate (CWF) and constant total flow rate (CTF). Through compari-
son between the pressure drop obtained by the models against the
experimental acquisitions, the main findings can be summarised as
follows:

- Single phase flow tests were performed and the results con-
firmed the accuracy of the instrumentation and the suitabil-
ity of the test facility which can be used for two-phase flow
investigation.
- The friction pressure drop enhanced with the increasing of
Fig. 8. Comparison of experimental data with Friedel model. gas flow rate for CWF. On the other hand, it decreased with
the increasing of gas flow rate for CTF. This behaviour is at-
tributed to the flow patterns transition in pipes.
- Drift-flux model predicts the experimental data with good ac-
curacy. The average error is of around 0.8% which is the low-
est compared to other models.
- The prediction from Homogenous and Awad and Muzychka
(2008) models is concluded as the most accurate one com-
pared to other empirical models in the literature to measure
the friction pressure drop with an average percentage error
less than 3%.
- The Lockhart-Martinelli and Muller-Steinhagen and Heck
model are considered as the second best empirical models
from the literature to predict the experimental data with
satisfactory average percentage is less than - 12%.
- The Friedel model can be used as a guide to predict the pres-
sure drop but it is not quantitatively reliable as the average
percentage error is around 56%.

Fig. 9. Comparison of experimental data with Muller-Stainhagen model.

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