Download Governing Equation

A Novel Explicit Equation for the Friction Factor Prediction in the Annular
Flow with Drag-Reducing Polymer
Esmail Lakzian1*, Amir Masoudifar2, Hassan Saghi3
1Assistant
professor, Department of Mechanical Engineering, Hakim Sabzevari University, Sabzevar, Iran
student, Department of Mechanical Engineering, Hakim Sabzevari University, Sabzevar, Iran
3Assistant professor, Department of Civil Engineering, Hakim Sabzevari University, Sabzevar, Iran
Corresponding author: [email protected]
2Msc
Abstract
In this paper, a novel explicit equation is presented for the friction factor prediction in the
annular flow with drag reducing polymer (DRP). By using dimensional analyses and curve
fitting on the published experimental data, the suggested equation is derived based on the
logarithmic velocity profiles and power law in boundary layers. In the next step, a least squares
method is used to calibrate the presented equation. Then, the equation is used to friction factor
prediction of the gas–liquid mixture with DRP and the results are compared with the
experimental data and Al-Sarkhi ones. Finally, drag reduction (DR) is applied as the ratio of
the friction factor reduction using DRP to the friction factor without DRP. The DR results show
that the suggested equation has a better agreement with the experimental data in comparison
with the pervious equations. The results also show that DR prediction decreases with the
increase of the gas superficial velocity.
Keywords: Annular Flow, Friction Factor, Drag Reducing Polymer, Logarithmic Velocity
Profiles, Power Law.
Introduction
A proper description of the annular flow with DRP and the calculation of friction losses are
of specific interest to the natural gas and petroleum researchers and engineers. Calculations of
friction factor are used to determine the parameters such as pressure drop in the piping system.
For laminar flow, friction factor calculation is simple as it is a function of the Reynolds number
(Re). But in turbulent flow, friction factor is a complex function of relative surface roughness
and Re. Blasius [1] presented the first correlation for friction factor as a curve fit to smooth wall
data collected for pipes. Of course, his equation has a limited range of applicability for
Re  1105 . Therefore, Prandtl [2] derived a better formula from the logarithmic velocity profile
and experimental data on smooth pipes. His formula is valid for Re>4000 and it is implicit and
needs an iterative solution. Although, the recent studies show that the constants of Prandtl’s
friction factor relationship are unsuitable for extrapolation to high Re [3-4]. So, McKeon et al.
[4] proposed the new formula for friction factor. Nikuradse [5] investigated turbulent pipe flow
and developed an approximate equation for the calculation of friction factor in rough pipes.
Some researchers presented several formulas for the transitional roughness region [6-12].
Recently, numerous researches were presented explicitly [13-21] and some researches were
presented implicit equations [22-23] to calculate the friction factor. For example, Avci and
Karagoz [13] proposed a new explicit equation for friction factor for both smooth and rough
wall turbulent flows in pipes and channels. The form of the proposed equation was based on a
new logarithmic velocity profile and the model constants were determined by using the
published experimental data. Sablani et al. [14] used an artificial neural network approach to
develop an explicit procedure for calculation of friction factor of laminar and turbulent flow of
plastic fluids. Taler [21] reviewed the most popular explicit correlations for the friction factor
in smooth tubes.
1
The added polymers in fluids are called drag- reducing polymers (DRP). The reduction of
frictional factor caused by the injection of small amounts of polymers has been studied by some
researchers [24-25]. For example, Manfield et al. [24] gave a comprehensive review of drag
reduction with additives in multiphase flows. Oliver and Young Hoon [26] reported the first
experiments on drag reduction in gas– liquid flows. Greskovich and Shrier [27] used the term
DRP in multi-phase flow. They found that drag reduction could reach 40% during slug air–
water flow. Some researchers investigated drag reduction in a variety of systems with different
results [28-30]. Al-Sarkhi and Hanratty [31-32] investigated the influence of a new polymer on
annular air–water flow in pipes for reduction of interfacial waves.
In the natural gas and petroleum industries, two-phase gas-liquid flow in pipes often occurs.
The gas velocity is commonly large enough that an annular flow exists. In contrast to the
numerous studies of the friction factor calculation in single phase flow, the friction factor
calculation in annular flow has been done in few types of researches. Therefore, in this study,
by using dimensional analyses and curve fitting on published experimental data, the suggested
equation is derived based on the logarithmic velocity profiles and power law in boundary layers.
Governing Equation
The pressure loss due to viscous effects in a cylindrical pipe of uniform diameter D, flowing
full, is proportional to length L and can be characterized by the Darcy–Weisbach equation as
follows [33]:
p L  f
 um2
(1)
2 D
where p L is the pressure loss per unit length (Pa/m),  is the fluid density (kg/m3), D is the
hydraulic diameter of the pipe (m), um is the mean flow velocity (m/s) and f is the friction
factor.
The friction factor is not a constant and it depends on some parameters such as the
characteristics of the pipe (diameter, D and roughness height, ε), the characteristics of the fluid
(kinematic viscosity, ν), and the velocity of the fluid flow, um. Friction factor can be measured
to high accuracy within certain flow regimes and may be evaluated by using the various
empirical relations, or it may be read from published charts. These charts are often referred to
as Moody diagrams. Friction factor equations are applied for different types of flow, consisting
of laminar, transition, fully turbulent flow in smooth and rough pipes and free surface flows.
For example, the friction factor equation for laminar flow in a circular pipe is defined as follows
[33]:
64
f 
(2)
Re
where Re is the Reynolds number.
DRP have been applied to reduce Reynolds shear stresses and velocity fluctuations on gas–
liquid flows in a direction normal to the wall. The friction factors for the gas–liquid mixture
without and with DRP are defined as Eqs. 3 and 4, respectively:
fm 
p L D
1 2mum2
f MD 
(3)
p L D D
1 2 mum2
(4)
Some researchers presented formulas for friction factor prediction of two- phase flows. For
instance, Al-Sarkhi [33] presented the new formula as:
Vsg 0.5 0.595
D
f MD  3.36  10 7 0 (Re m (
) )
(5)
D
Vsl
This study aims to suggest a new equation for friction factor prediction in annular flow with
2
DRP. Therefore, mathematical modeling of the present study is introduced in the next section.
Mathematical Modeling
Logarithm velocity profiles and power law formulations in boundary layers are the basis of
the friction factor equation. The velocity varied in the overlap region, logarithmically and it is
called the logarithmic-overlap layer as follow [13]:
1
y u
u
 ln(
) B
(6)
u 
l
where u    l ,  w ,  ,  l , and B are the friction velocity, the wall shear stress, the von Karman
w
constant, the kinematics viscosity of the liquid, and a constant for turbulent flow past smooth
impermeable walls, respectively. Assuming the velocity profile as a combination of logarithmic
and power law, we have [13]:
u
 K (ln(
u
( R  r ) u
l
)  p) N
(7)
where y=R-r (See Fig. 1) and K , N should be determined experimentally.
u max
Fig. 1: Typical velocity profile in the pipe
By setting r=0 in Eq. (7), the maximum velocity (umax) can be calculated as:
Ru
u max
 K (ln(  )  p) N
u
l
(8)
The velocity profile is used for the calculation of friction factor. For this aim, Eq. (7) should
be integrated in order to get the mean velocity in a pipe. Since integration of this equation is
not easy, it is assumed that the mean velocity um is a fraction of the maximum velocity and
therefore, the mean velocity has the same form as in Eq. (8) with different constants.
Consequently, the mean velocity can be written as:
(9)
um
Re
 a (ln( m )) b
u

where, u m  Vsl  Vsg , Re m  um D , and
l

is introduced in the next paragraph. The term
Ru
l
in Eq.
8 is approximated as a logarithmic function of Re.
The friction factor in the Darcy–Weisbach equation is defined by using dimensional
analysis for the gas–liquid mixture with DRP as follow [13]:

f MD  8 w 2
(10)
 l um
by inserting  w   l u2 , Eq. (10) is rewritten as follows:
f MD
u2
8 2
um
(11)
3
combining Eqs. 9 and 11, leads to:
f MD 
8
  Re  
a 2  ln  m  
   
2b
(12)
This equation covers turbulent internal flows in pipes and channels with Re in the range of
2.4 105 < Re m < 4106 . It also recovers Prandtl’s law of friction for smooth pipes. D0 , D , V sg , Vsl
, u m and  l are the important parameters in the gas–liquid mixture with DRP. Therefore, some
dimensionless parameters have been derived by using these parameters and dimensional
analysis. These dimensionless parameters are D0 D , Vsg Vsl 0.5 and Re m . On the other hand, with
compare between Eq. (5) and Eq. (12), we propose that a and  parameters are the function of
0.5
D0 D and Vsg Vsl  , respectively. Substituting these proposed parameters in the Eq. (12), leads
to:
f MD 
8
1
Re m
 D0   
a1 
 ln a 2
 0.5


 D    V sg V sl 
2b

(13)



where, the coefficients a1 , a2 and b were derived by using the experimental data and least
squares method. In this method, the parameter S is defined as follows:



n
8

S    f MDEi 

1
i 1

Re m
D   
a1  0   ln  a 2

 0.5


 D    V sg V sl 





2b









2
(14)
where, n and f MDE is the number of experimental data and experimental friction factors for the
gas–liquid mixture with DRP, respectively. In the least square method, the parameter S is
derivative relative to parameters a1 , a2 and b as follows:






n
16a1 2
S
8



f

0
2 b   MDEi
2b 
1 
1 
a1
i 1








Re m
Re m
 D0   
D 

 
a1  0   ln  a 2

 ln a 2

 0.5  
 0.5  




D
D


   V sg V sl   

   V sg V sl    





 2 b 1


n
 


Re
S
8
16
b

m
 ln  a

 2  f MDEi 
0
2b 
1

  2 Vsg Vsl 0.5  
1 
a2
D
i 1





0




Re m
 D0   


a1a2  

a1   ln a2
 D

D    Vsg Vsl 0.5   


 





 
Re



n
 
  2b ln ln a V V 
Re
S
8
16


m

 ln  a
e

 2 f MDE 
ln
2b 
1

  2 Vsg Vsl 0.5  
1
b
i 1
   a  D0 

 
Re m
 D0   



1
a1   ln a2
0.5 
D






D


V
V
   
sg
sl
 

m
2
sg
i
4
sl
 0.5
(15)
(16)




0
(17)
by solving Eqs. 15 to 17, simultaneously, parameters a1 , a2 and b are derived as a1  1.319 1020
, a2  158 and b  6.4 . Substituting a1 , a2 and b in the Eq. (13) leads to:
12.8
f MDp  6.02 10
20
Vsg 0.5 
 D0  
  ln 158  Re m ( ) 
Vsl 
D 
(18)
Model Validation
In this step, friction factors for the gas–liquid mixture with DRP  f MDp  is calculated by
using Eq. 16 for different pipe diameters (D) and the results are validated by using the
experimental data [33]  f MDE  in Fig. 2. Furthermore, a new parameter  f MD D0 D1  is
introduced and compared with the experimental data (See Fig. 3). The good agreement between
the present results and the experimental data is achieved.
0.008
0.007
Exprimental data
Present study
0.007
Exprimental data
Present study
0.006
0.006
f MDp
f MDp 0.005
0.005
0.004
0.004
0.003
0.003
0.002
0.002
0.001
0.001
0
0
1E+07
2E+07
Re m Vsg VSL 
0
3E+07
0
1E+07
Re m Vsg VSL 
2E+07
0.5
0.5
(a)
(b)
0.0045
0.007
Exprimental data
Present study
0.006
0.004
Exprimental data
Present study
0.0035
f MDp
f MDp 0.005
0.003
0.0025
0.004
0.002
0.003
0.0015
0.002
0.001
0.001
0
0.0005
0
2E+07
4E+07
0
6E+07
Re m Vsg VSL 
(c)
Fig. 2: Comparison between
0
5E+07
1E+08
1.5E+08
Re m Vsg VSL 
(d)
0.5
0.5
f MDp and f MDE [33] for different pipe diameters (D): a) D= 0.0125 m, b)
D=0.019 m, c) D=0.0250 m, d) D=0.0953 m
5
f MD D0 D
1
Re m Vsg VSL 
0.5
Fig. 3: Comparison between f MD D0 D1 predicted and the experimental data [33]
Results
In this step, f MDp and friction factor for the gas–liquid mixture with DRP was calculated by
Al-Sarkhi  f MDAL  [33] are compared with f MDE [33] and the results were shown in the Fig. 4.
The better agreement between f MDp and f MDE [33] than between f MDAL and f MDE is achieved.
f MDp
f MDAL
f MDE
f MDE
Fig. 4: Comparison between
f MDp and f MDAL with f MDE [2]
In this step, mean absolute percentage deviation (MAPE) and standard deviation   of the
results are estimated and summarized in Table. 1 and they show the accuracy of the using Eq.
16.
Table 1: Comparison between
f MDp and f MDAL by using mean absolute percentage deviation (MAPE)
and standard deviation
  criteria
MAPE

f MDAL
0.103
0.09
f MDp
0.062
0.089
Drag reduction (DR) is defined as the ratio of reduction in the friction factors using DRA
 f MDp  to the friction factors without DRA  f m  as follow:
DR % 
f m  f MDP
100
fm
(19)
6
where f m and f MDP are estimated by using the Eqs. 3 and 18, respectively. In this step, the
variation of DR versus to superficial gas velocity Vsg  and superficial liquid velocity Vsl  are
compared with the experimental data [34-35] in the Figs. 5 and 6, respectively. The results show
a good agreement between the results of the present study and the experimental data in annular
flows (high Vsg ).
90
80
80
Exprimental data
Present study
Exprimental data
Present study
70
DR (%)
60
60
DR (%) 50
40
40
30
20
20
25
30
Vsg (m / s )
35
10
40
10
20
(a)
40
50
(b)
90
90
80
70
Exprimental data
80
Present study
70
60
60
DR (%) 50
DR (%) 50
40
40
30
30
20
20
10
10
0
30
Vsg (m / s )
20
30
40
50
0
60
Vsg (m / s )
Exprimental data
Present study
20
30
40
50
60
Vsg (m / s )
(c)
(d)
Fig. 5: Comparison between the estimated DR and the experimental data [34-35] for different superficial
gas velocities Vsg  and diameters (D), a) Vsl =0.1 m/s, D=0.0127 m, b) Vsl =0.4 m/s, D=0.0127 m, c) Vsl
=0.104 m/s, D=0.0254 m, d) Vsl =0.125m/s, D=0.0254 m
7
DR (%)
Vsl (m / s)
Fig. 6: Comparison between the estimated DR and the experimental data [34-35] for different superficial
liquid velocities Vsl  at Vsg =38 m/s and D=0.0127 m
Conclusion
In this paper, a novel explicit equation is presented for the friction factor prediction in the
annular flow with the drag-reducing polymer (DRP). By using dimensional analyses and curve
fitting on published experimental data, the suggested equation is derived based on the
logarithmic velocity profiles and power law in boundary layers. In the next step, f MDp results
are validated by using the experimental data. The good agreement between the present results
and the experimental data is achieved. After validation, f MDp and f MDAL are compared with f MDE
. The better agreement between f MDp and f MDE than between f MDAL and f MDE is achieved.
Finally, the variation of DR versus to Vsg is compared with the experimental. A good agreement
between the results of the present study and the experimental data in annular flows (high
velocity) is achieved. The results also show that DR prediction decreases with the increase of
Vsg .
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[32] A. Al-Sarkhi and T.J. Hanratty, Effect of Pipe Diameter on the Performance of Drag Reducing Polymer in
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[33] A. Al-Sarkhi, M. El Nakla and W. Ahmed, Friction Factor Correlations for Gas–Liquid/Liquid–Liquid Flows
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9
Nomenclature
a, a1, a2 , b, B, K , N , 
D
D0
DR
DRP
f
fm
constant
hydraulic diameter of pipe
reference pipe diameter
drag reduction
drag reduction polymer
friction factor
friction factor for the gas–liquid mixture
f MDE
friction factor for the gas–liquid mixture with DRA
experimental friction factors for the gas–liquid mixture with DRP
f MDp
friction factors for the gas–liquid mixture with DRP by using Eq. 16
f MDAL
friction factors for the gas–liquid mixture with DRP by Al-Sarkhi [33]
f MD
n
Re
Re m
S
um
u m ax
u
V sg
V sl
Greek symbols
number of experimental data
Reynolds number
mixture Reynolds number
The parameter in the least squares method
mean flow velocity
maximum velocity
friction velocity
gas superficial velocity
liquid superficial velocity
p
pressure loss

von Karman constant
kinematics viscosity
kinematics viscosity of the liquid

l

l

w
Subscripts
0
g
l
m
MD
MDE
sl
sg
fluid density
liquid density
standard deviation
wall shear stress
reference
gas
liquid
mixture or mean
mixture with DRP
experimental mixture with DRP
liquid superficial
gas superficial
10