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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 1105 . 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 2mum2 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 u2 , Eq. (10) is rewritten as follows: f MD u2 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 < 4106 . 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 D1 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 D1 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 . References [1] P.H.R. Blasius, Das Aehnlichkeitsgesetz Bei Reibungsvorgangen in Flüs Sigk Eaten, Forschungsheft, 131, 141 (1913). [2] L. 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Multiphase Flow, 32, 926–934 (2006). 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