See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/346503366 Friction Factor Equations Accuracy for Single and Two-Phase Flows Conference Paper · August 2020 DOI: 10.1115/OMAE2020-18682 CITATIONS READS 2 965 5 authors, including: Germano S. C. Assuncao Dykenlove Marcelin Petróleo Brasileiro S.A. Centro Universitário Univel (UNIVEL) – Cascavel/Paraná - Brasil 5 PUBLICATIONS 11 CITATIONS 2 PUBLICATIONS 2 CITATIONS SEE PROFILE SEE PROFILE João Carlos von Hohendorff Filho Denis José Schiozer University of Campinas University of Campinas 40 PUBLICATIONS 273 CITATIONS 535 PUBLICATIONS 3,553 CITATIONS SEE PROFILE Some of the authors of this publication are also working on these related projects: Use of Quality Maps and Flow Unit Definition in Reservoir Management View project Integration of Reservoir Simulation and Production Systems View project All content following this page was uploaded by Germano S. C. Assuncao on 30 November 2020. The user has requested enhancement of the downloaded file. SEE PROFILE Proceedings of the ASME 2020 39th International Conference on Ocean, Offshore and Arctic Engineering OMAE2020 August 3-7, 2020, Virtual, Online OMAE2020-18682 FRICTION FACTOR EQUATIONS ACCURACY FOR SINGLE AND TWO-PHASE FLOWS Germano Scarabeli Custódio Assunção1,2,*, Dykenlove Marcelin2, João Carlos Von Hohendorff Filho1, Denis José Schiozer1, Marcelo Souza De Castro1 1 University of Campinas, Campinas, Brazil Univel University Center, Cascavel, Brazil 2 ABSTRACT Estimate pressure drop throughout petroleum production and transport system has an important role to properly sizing the various parameters involved in those complex facilities. One of the most challenging variables used to calculate the pressure drop is the friction factor, also known as Darcy–Weisbach’s friction factor. In this context, Colebrook’s equation is recognized by many engineers and scientists as the most accurate equation to estimate it. However, due to its computational cost, since it is an implicit equation, several explicit equations have been developed over the decades to accurately estimate friction factor in a straightforward way. This paper aims to investigate accuracy of 46 of those explicit equations and Colebrook implicit equation against 2397 experimental points from single-phase and two-phase flows, with Reynolds number between 3000 and 735000 and relative roughness between 0 and 1.40x10-3. Applying three different statistical metrics, we concluded that the best explicit equation, proposed by Achour et al. (2002), presented better accuracy to estimate friction factor than Colebrook’s equation. On the other hand, we also showed that equations developed by Wood (1966), Rao and Kumar (2007) and BrkiΔ (2016) must be used in specifics conditions which were developed, otherwise can produce highly inaccurate results. The remaining equations presented good accuracy and can be applied, however, presented similar or lower accuracy than Colebrook’s equation. 1. INTRODUCTION Methodologies to solve coupling of petroleum reservoir and production systems have generated great interest in industry, because such approach can improve production forecasts and optimize projects. Discoveries in the Brazilian pre-salt intensified this interest since old and new fields can share the same production system. In these projects, estimation of pressure drop has a key role to correctly sizing the production system and correct analyze production forecasts. For accurately calculate pressure drop throughout production system, it is well known that, at least, seven parameters may be known: (1) internal pipe diameter; (2) pipe length and fittings; (3) fluids velocities, (4) fluids properties; (5) flow regimes, (6) void fraction and (7) stress at the pipe wall, which determines the so-called friction factor. The first four parameters can be easily obtained by direct measurement, differently of the last three. Evaluation of flow regime and void fraction are totally related to the development of multiphase flow models from the mid-20th century onward, because both parameters only make sense to be studied when more than a single-phase exists, which usually happens in petroleum production system, due to natural or artificial phenomena. However, although the evaluation of both parameters is difficult, it can still be carried out, using, for example, cameras and sensors to observe flow regimes or to obtain void fraction in laboratory applications and then scale it up to real production system through models. The seventh and last parameter, friction factor, is the most complex to be evaluated, because it is a variable and not a “measurable” parameter. It should always be estimated, considering Reynolds number, wall roughness and pipe cross section shape. Its analysis has been considered since the Keywords: friction factor, Colebrook’s equation, explicit equations, petroleum production system. * E-mail addresses: germano@univel.br (G.S.C. Assunção), marcelin.dykens@gmail.com (D. Marcelin), hohendorff@cepetro.unicamp.br (J.C.v. Hohendorff Filho), denis@unicamp.br (D.J. Schiozer), mcastro@fem.unicamp.br (M.S.d. Castro). 1 Copyright © 2020 ASME development of the first models to calculate pressure drop in hydraulic studies, back to 18th century. One of the first and notable works addressing this issue was developed by Prony (1804 apud Rennels and Hudson, 2012), which expressed water pressure drop in internal conduits using two empirical coefficients. Later, Weisbach (1845) proposed the use of a dimensionless group called “friction factor (π)” in a pressure drop equation. Then, Darcy (1857) developed another equation using three empirical coefficients. Although he had not used the dimensionless group proposed by Weisbach (1845), Darcy (1857) had an important role, identifying that pressure drop depends on type and condition of the boundary material. For these reasons, it is traditional to call friction factor as Darcy friction factor or Darcy–Weisbach’s friction factor, although Darcy (1857) had not proposed it, neither Weisbach (1845) had a theorical basis for friction factor meaning. As suggests Brown (2002), the first to put together concepts developed by Weisbach (1845) and Darcy (1857) was Fanning (1877), who studied a large compilation of π values as a function of pipe material, pipe dimension and velocity. However, instead of diameter, he used radius in his analysis, which returned π values equal to 1/4th of Darcy–Weisbach’s π. To give credit to his work, we can find frequently in literature (mainly American works) applications with the so-called Fanning friction factor. Those pioneering works were the basis for several subsequent publications in beginning of 20th century. At this time, friction factor for laminar flows was already known, due to works of Hagen (1839) and Poiseuille (1841). Thus, friction factor estimations remained challenging for turbulent flows. Aiming to solve this challenge, several authors proposed equations as the ones developed by Blasius (1913) and Von Kármán (1930). Blasius (1913) proposed an explicit equation to be used in smooth pipes under fully turbulent flow, which is still used today. Von Kármán (1930) proposed an explicit equation for applications in smooth and rough pipes, also for fully turbulent flows. Nikuradse (1933) confirmed the equation developed by Von Kármán (1930) testing against several experimental points. He concluded that that equation could be used in smooth and rough pipes, in laminar, turbulent and transition flows. However, in his work, he used “artificial roughened surfaces” — uniform layers of sand grains, with known size, glued on the inner surfaces of the pipes. In this context, Colebrook and White (1937) used experimental data from air flow to show that variation in size and pattern of individual protuberances (non-uniform layers of sand grains), generates different friction factor responses, mainly for flows in transition zone (laminar to turbulent). From that conclusion, Colebrook (1939)† performed a comprehensive study of friction factors in laminar, turbulent and transition flow conditions using commercial pipes (galvanized, cast and wrought iron pipes). He proposed a theorical transition equation from laminar to turbulent, which is an implicit equation and since then, has been largely used as response for friction factor under different flow conditions: π = (−2 log ( 2.51 π π√π + π )) 3.7π· −2 (1) where π is the friction factor (or Darcy–Weisbach’s friction factor), π π is Reynolds number, π is the surface roughness and π· is the internal diameter. One of the main causes of the popularity of Colebrook’s equation is due to the work of Moody (1944), who summarized the results of the equation 1 in a diagram. Equation and diagram have been tested against several data over the decades, presenting outstanding accuracy. It is common both to be studied in introductory courses of fluid mechanics. However, since the development of Colebrook’s equation an effort has been made to develop an explicit equation that provides such accurate results, because to find friction factor from equation 1, numerical algorithms are required. Some examples of those explicit equations were developed by Moody (1947), Atshul (1952), Wood (1966), Churchill (1973), Eck (1973)‡, Jain (1976), Swamee and Jain (1976) and several others, until the most recent ones, proposed by Azizi et al. (2018) and BrkiΔ and Praks (2019). Thus, the objective of this work is comparing friction factor estimates from several explicit equations and from Colebrook’s equation against experimental data. We can find in literature several review papers that already investigated the performance of some of those explicit equations against Colebrook equation. We highlight recent works of Génic et al. (2011), Asker et al. (2014), Pimenta et al. (2018) and Zeghadnia et al. (2019), whose studies compared several explicit equations to Colebrook equation. Génic et al. (2011) concluded that Zigrang and Sylvester (1982) presented the most accurate explicit equation. Asker et al. (2014) obtained that Sonnard and Goudar (2006) and Serghides (1984) are the most accurate explicit equations. Zeghadnia et al. (2019) showed that Vatankhah and Kpuchakzadeh (2009) is the better one and Pimenta et al. (2018) concluded that all 29 explicit equations analyzed in their work presented accurate results, however, recommend application of equation developed by Offor and Alabi (2016). Although these works are extremely relevant, a drawback is the fact that accuracy of explicit equations was compared to estimates from Colebrook’s equation, in other words, estimates from Colebrook equation were used as real response (“measured” data). Nonetheless, Colebrook’s equation is also an approximation for the variable π, therefore, such analysis can generate some bias in the results and consequently in conclusions † The equation proposed by Colebrook (1939) is also well known as Colebrook-White’s equation, due to the collaboration of Dr. White in the development of formula, as mentioned by Colebrook in this original paper. ‡ Although extensively cited in the literature as an equation developed by Eck, in fact, the equation presented by Eck (1973) was developed by the Yugoslav engineer Pecornik, in 1963, on the paper entitled “Determination of the coefficient of friction in pipes at stationary uniform turbulent flow in a transitional region”. A note was presented in Eck (1973) about the work of Pecornik. 2 Copyright © 2020 ASME taken. A proof of this claim is that none of the four works had the same conclusion about the most accurate explicit equation. Therefore, we propose here a comprehensive comparison between estimated friction factors (ππΈπ π‘ππππ‘ππ ) and “measured”§ friction factors (πππππ π’πππ ). To perform this analysis, we used real observed data acquired from literature and from an experimental facility. This paper is organized as follows: in section 2, a description of experimental data from literature is presented, as well as the facility to obtain experimental data acquired in this work. Section 3 describes methodologies used to obtain experimental data and comparison criterions used to test performance of explicit equations and Colebrook’s equation against experimental data. Section 4 shows the results, including a scrutiny analysis of the most accurate explicit equation found and finally section 5 the main conclusions obtained. applying an amplitude parameter Ra — the most common parameter. Pressure drop between P1 and P2 (Figure 1) was measured using a digital differential pressure manometer with range of operation among 0 to 250 mm H20 — resolution 0.25 mm H20 — coupled to a software and data acquisition interface. This software/interface is able to record measured data for every 5 millisecond or more, thus, for every set velocity, several data were recorded and then statistically analyzed, in order to obtain more accurate data. Details about those steps are shown in section 3.1. Note that the manometer was 1000 mm way from both ends of the test section, to decrease the inlet and outlet effects. To obtain this value, we used three literature equations, as shows Table 2. Since in our experiment Reynolds number ranged between 12600 to 50600, taking into account the internal diameter of 27.8 mm and water flow rate of 4000 liters/hour — which give us the longer entrance length — we conclude, from results presented in Table 2, that 1000 mm would be a safe value for our analyses. Water temperature was also measured by thermometer to assure values within 23 ± 3 β during experiment. 2. DATA DESCRIPTION Table 1 presents a summary of experimental data used in this work. A total of 47 datasets under different conditions were used, which gives 2397 experimental points. We divided our data in four main conditions: (1) horizontal single-phase flow, (2) horizontal two-phase flow, (3) upward two-phase flow and (4) downward two-phase flow. Since the focus here is applications in petroleum industry, most of data used are from two-phase flows, which is most common to observe in production pipelines than single-phase flows. Most of data obtained uses water-air and kerosene-air mixtures, but some oil-air mixtures were also analyzed. Internal diameter varies among 21.6 mm and 149.6 mm, while pipe inclination varies between -90º and +90º. For single-phase flow analysis, an experimental facility was built to obtain datasets 1 and 2 shown in Table 1 — section 2.1 presented more details about this facility. Other experimental data were acquired from work developed by Silva (2018). 3. METHODOLOGY In this section we present methodologies used in this work. In section 3.1, we present the steps followed to collect experimental pressure drop of datasets 1 and 2 (Table 1), while section 3.2 regards the means to measure the agreement between ππΈπ π‘ππππ‘ππ and πππππ π’πππ . 3.1 Methodology to measure pressure drop of datasets1 and 2 The follow steps were applied for both pipe diameters setups: 1. For each diameter, we set the flow rates in liters/hour (lph): a. For D = 21.6 mm: (1) 1000 lph, (2) 1500 lph, (3) 2000 lph and (4) 2500 lph; b. For D = 27.6 mm: (1) 1000 lph; (2) 1500 lph; (3) 2000 lph, (4) 2500 lph, (5) 3000 lph, (6) 3500 lph and (7) 4000 lph. 2. After set each of the flow rates from step 1, we wait flow stabilization for 60 seconds. Then, we started to record pressure drop between P1 and P2 (Figure 1) for each second, during 200 seconds. After that, we stopped the line. So, in the end, we had 200 pressure drop points measured. 2.1 Experimental facility to acquirer datasets 1 and 2 The experimental setup consisted of a water reservoir, a centrifugal pump, a flow meter, a test section with a digital differential pressure manometer and a return line of PVC pipe. The flow meter is a rotameter able to accurately measure water flow rates from 0 to 5000 liters/hour. Water flows in a closed loop following the layout presented in Figure 1 and the flow rate is set using a variable-frequency drive in the centrifugal pump. Two PVC pipes of internal diameter 21.6 mm and 27.8 mm were used in test section, therefore, two experimental pipelines between Valve 1 and Valve 2 (Figure 1) were used. Roughness of both pipes were measured using a portable roughness tester, § In fact, it is not measured, but obtained from measured data, using, for instance, equation 4. 3 Copyright © 2020 ASME Figure 1. Schematic of experimental setup for pressure drop measurement. 3. To give more accuracy in our measurements, we repeated step 2. five times. Thus, for every flow rate, we obtained 1000 points. 4. Using these 1000 points, we identified outliers using Tukey’s method. In such method, one point can be identified as outlier if it is outside lower and upper quartiles. More details about Tukey’s method and other methods to identify outliers can be found in Seo (2002) and Wilcox (2012). 5. Subsequently, error (e) of measurements were calculated using the following equation: π = √(10 π √π 2 ) + ππππ π‘ππ’ππππ‘ . π calculate frictional pressure drop per unit length, which is related to the wall shear stress. In inclined flows, it is necessary to remove gravitational pressure gradient, as presents equation 3. In horizontal flows, it is not necessary, since sin θ = 0. βπ ( ) βπ βπΏ πΉππππ‘πππππ = ( ) βπΏ ππππ π’πππ βπ where ( ) βπΏ ππππ π’πππ − ππ . π. π πππ (3) is the pressure drop experimentally measured, ππ is mixture density for two-phase flow used in homogenous non-slip model; π is gravitational acceleration and π is pipe inclination. (2) After that, πππππ π’πππ can be obtained using the following equation: where π is the sample standard deviation, π is the sample size and π is the sample average. Note that we used “10” in equation 2 due to Chebyshev's inequality with confidence of 99% (values must lie within 10 standard deviation of the mean), while ππππ π‘ππ’ππππ‘ is due to pressure sensor error, which is, according to manufacturer, 2% of measured value. 6. Finally, pressure drop estimates for every flow rate were obtained: π ± π. Values of viscosity and density of water were assumed from literature (taking into account measured temperature), while roughness was obtained using a roughness tester. In both pipes, we measured roughness in 4 points 90° spaced. The cross-section area used to measure roughness was located in the half of the pipe used in test section (Figure 1). We measured the roughness for both pipes, thus, we obtained 8 values, which were used to calculate the mean value. To calculate error of measured roughness, we applied equation 2, using ππππ π‘ππ’ππππ‘ = 10%, as recommend by the manufacturer. βπ πππππ π’πππ = 2( βπΏ ) πΉππππ‘πππππ 2 ππ ππ π· (4) where π· is the pipe internal diameter and ππ is the mixture velocity (equation 5). Note that we applied homogenous no-slip model for twophase flow analysis. As explained by Shoham (2006), in the homogeneous no-slip model, two-phase mixture is considered a pseudo single-phase flow with fluids proprieties based on the noslip void fraction (λ). Thus, flow velocity and fluid proprieties are calculated as follows: ππ = πππΏ + πππΊ (5) where ππ is the mixture velocity, πππΏ and πππΊ are the liquid and gas superficial velocities, respectively. In single-phase flows, mixture velocity is equal to liquid velocity. These superficial velocities represent the phase volumetric flow rate per unit area, it is the velocity which would be observed if only that specific phase flows throughout the pipe. 3.2 Methodology to compare ππ¬ππππππππ and ππ΄πππππππ To perform the comparison between ππΈπ π‘ππππ‘ππ and πππππ π’πππ , for every point of 2397 available, first step was to 4 Copyright © 2020 ASME Table 1: Summary of experimental data used for performance evaluation of friction factor equations. Nº of π½ Reynolds range Data Reference Fluids D (mm) points (x103) (deg.) Horizontal single-phase flow Data 1 Authors Water 0 21.6 4 12 – 50 Data 2 Authors Water 0 27.8 7 24 – 41 Data 3 Silva (2018) Water 0 53.55 64 7 – 126 Data 4 Silva (2018) Water 0 82.25 186 4 – 70 Data 5 Silva (2018) Water 0 106.47 24 34 – 400 Horizontal two-phase flow Data 6 Silva (2018) Water-Air 0 53.55 70 14 – 163 Data 7 Silva (2018) Water-Air 0 82.25 129 13 – 104 Data 8 Silva (2018) Water-Air 0 106.47 78 21 – 451 Data 9 Mukherjee (1979) Kerose-Air 0 38.1 59 4 – 124 Data 10 Magrini (2009) Water-Air 0 76.2 20 238 – 604 Data 11 Johnson (2005) Water-Air 0 100 206 65 – 735 Data 12 Cheremisinoff (1977) Water-Air 0 63.5 174 57 – 232 Data 13 Fan (2005) Water-Air 0 50.8 51 34 – 232 Data 14 Fan (2005) Water-Air 0 149.6 87 84 – 545 Data 15 Brito (2012) Oil-Air 0 50.8 8 4–7 Data 16 Brill et al. (1995) Kerose-Air 0 77.9 48 28 – 151 Data 17 Kouba (1986) Kerose-Air 0 76.2 18 35 – 287 Data 18 Meng (2001) Oil-Air 0 50.8 31 26 – 141 Data 19 Zheng (1989) Kerose-Air 0 76.2 6 65 – 141 Upward two-phase flow Data 20 Mukherjee (1979) Kerose-Air 90º 38.1 39 7 – 128 Data 21 Yuan (2011) Kerose-Air 30º 76.2 22 9 – 521 Data 22 Yuan (2011) Kerose-Air 60º 76.2 43 68 – 482 Data 23 Yuan (2011) Kerose-Air 75º 76.2 44 70 – 482 Data 24 Yuan (2011) Kerose-Air 90º 76.2 44 39 – 482 28 70 – 241 Data 25 Fan (2005) Water-Air 1º 50.8 Data 26 Fan (2005) Water-Air 2º 50.8 28 71 – 245 Data 27 Fan (2005) Water-Air 2º 149.6 33 204 – 538 Data 28 Meng (2001) Oil-Air 1º 50.8 45 32 – 143 Data 29 Meng (2001) Oil-Air 2º 50.8 36 32 – 143 Data 30 Zheng (1989) Kerose-Air 1º 76.2 3 67 – 217 Data 31 Alsaadi (2013) Water-Air 2º 76.2 56 44 – 525 Data 32 Alsaadi (2013) Water-Air 5º 76.2 56 42 – 525 Data 33 Alsaadi (2013) Water-Air 10º 76.2 56 40 – 524 Data 34 Alsaadi (2013) Water-Air 20º 76.2 56 42 – 526 Data 35 Alsaadi (2013) Water-Air 30º 76.2 64 40 – 526 Downward two-phase flow Data 36 Mukherjee (1979) Kerose-Air -5º 38.1 33 38 – 482 Data 37 Mukherjee (1979) Kerose-Air -20º 38.1 49 3 – 435 Data 38 Mukherjee (1979) Kerose-Air -30º 38.1 43 10 – 377 Data 39 Mukherjee (1979) Kerose-Air -50º 38.1 56 3 – 332 Data 40 Mukherjee (1979) Kerose-Air -70º 38.1 50 3 – 338 Data 41 Mukherjee (1979) Kerose-Air -80º 38.1 43 3 – 305 Data 42 Mukherjee (1979) Kerose-Air -90º 38.1 38 4 – 334 Data 43 Fan (2005) Water-Air -1º 50.8 35 34 – 233 Data 44 Fan (2005) Water-Air -2º 50.8 25 34 – 218 Data 45 Fan (2005) Water-Air -2º 149.6 34 116 – 538 Data 46 Meng (2001) Oil-Air -1º 50.8 32 23 – 146 Data 47 Meng (2001) Oil-Air -2º 50.8 36 29 – 143 5 Copyright © 2020 ASME Table 2: Definition of entrance length Equation Entrance length for D = 27.8 mm and Re ~ 50600 548 mm 1.359π·π π 1/4 Author Bhatti and Shah (1987) White (2011) 4.4π·π π 1/6 744 mm Çengel and Boles (2013) 10π· 278 mm Fluids proprieties such as density and viscosity of the pseudo single-phase flow are calculated as follows: apply this statistical metric was observing whether equations were overestimating or underestimating πππππ π’πππ . (6) ππ = ππΏ (1 − λ) + ππΊ λ ππ ± (%) = and where ππ is the mixture density, ππΏ is liquid density and ππΊ is gas density, while ππ , ππΏ and ππΊ are mixture, liquid and gas viscosity, respectively. More details about homogenous non-slip model can be found in Shoham (2006). For two-phase flow analysis, application of homogeneous no-slip model increases absolute errors, since it is well known that such approach is an approximation of real multiphase flow. However, using such model we can have an overall picture of friction factor accuracy equations in a standard way, which is the focus of the present work. A next step would be to observe how can accuracy of friction factor affect accuracy of other multiphase flow models such as correlations, two-fluid model, drift-flux and others. Then, absolute percentage error (ππ ) between πππππ π’πππ and ππΈπ π‘ππππ‘ππ for every equation studied was calculated by equation 8. Table 3 presents all explicit equations used to calculate ππΈπ π‘ππππ‘ππ . Beyond these equations, Colebrook equation was used to estimate friction factors in this study. ππ (%) = 1 ππ π π (∑π=1 ( |πππππ π’πππ −ππΈπ π‘ππππ‘ππ | πππππ π’πππ ) . 100) πππππ π’πππ −ππΈπ π‘ππππ‘ππ πππππ π’πππ ± ππΆ ± (%) = ∑47 π=1|ππ | ) 100) (10) (11) To observe average percentage errors, we used the average percentage error with sign (ππ΄ ± ), calculated by equation 12: ππ΄ ± (%) = ± ∑47 π=1|ππ | π± (12) where π± is the size of the entire sample (from Data 1 to Data 47) with positive or negative error. We highlight that equations from (8) to (12) were applied for each explicit equation presented in Table 3 and for Colebrook’s equation, as well. In literature there are several others statistical metrics that can be used to perform such comparison. It is commonly found: (1) percentage error, (2) percentage standard deviation, (3) average error, (4) absolute average error or (5) standard deviation error, as presents Shoham (2006) for example. In general, theses metrics present the same trend and for this reason only two (ππΆ and ππ΄ ± ) were applied here. In addition, other statistical index that is not usually applied in those analyses was also used, the refined Willmott index (ππ ). This metric was proposed by Willmott et al. (2012) and evaluate index of agreement for meteorological and agricultural studies, called Willmott index (π), proposed by Willmott (1981). The refined Willmott index is a dimensionless number, bounded by -1.0 and 1.0. Values of ππ near to 1.0 means that both data are in agreement and near to -1.0 the converse. To calculate ππ , equation 13 must be used when ∑ππ=1|ππ − ππ | ≤ 2 ∑ππ=1|ππ − πΜ |: (8) where index π represents each data studied, presented in column “Data” in Table 1, while ππ is the size of each of these data, presented in column “Nº of points” in Table 1. The cumulative absolute percentage error (ππΆ ) can be calculated by equation 9. The aim in calculating ππΆ was to measure performance of each equation against the entire databank. ππΆ (%) = ∑47 π=1 ππ π π (∑π=1 ( where ππ ± is the size of each data, presented in column “Nº of points” in Table 1. We used ππ + for the sum of points with positive errors, while ππ − for the sum of points with negative errors. Cumulative percentage error with sign (ππΆ ± ) was also calculated, as follows: (7) ππ = ππΏ (1 − λ) + ππΊ λ 1 ππ ± (9) where 47 represents Data 47 from Table 1. Similarly, percentage error with sign (ππ ± ) between πππππ π’πππ and ππΈπ π‘ππππ‘ππ was also calculated. The reason to 6 Copyright © 2020 ASME ππ = 1 − ∑ππ=1|ππ − ππ | 2 ∑ππ=1|ππ − πΜ | Colebrook’s equation in predicting friction factor under different “error intervals”. (13) 4. RESULTS AND DISCUSSION However, when ∑ππ=1|ππ − ππ | > 2 ∑ππ=1|ππ − πΜ | equation 14 must be applied: 2 ∑ππ=1|ππ − πΜ | ππ = π −1 ∑π=1|ππ − ππ | 4.1 Pressure drop from datasets 1 and 2 Using experimental facility described in section 2.1 and applying methodology from section 3.1, we obtained average values and uncertainties presented in Table 4. As a whole, 11,000 points were collected. Roughness values measured for 21.6 mm and 27.8 mm pipes are presented in Table 5. The average value found following methodology proposed in section 3.1 was (0.9 ± 0.3) μm. (14) where ππ is the estimates or predicted values and ππ the pairwise-matched observations. We can interpret refined Willmott index as modified Pearson correlation coefficient, widely used in regression analyses. More details about ππ and other statistical metrics related to Willmott indices can be found in Willmott et al. (2012) and Pereira et al. (2018). Table 4: Pressure drop measured for Data 1 and Data 2. βπ· π·π Data D (mm) ( ) π³ π (37.5 ± 0.9)π₯101 (7.4 ± 0.2)π₯ 102 Data 1 21.6 (13.6 ± 0.3)π₯102 (19.9 ± 0.4)π₯102 (10.8 ± 0.6)π₯101 (24.5 ± 0.7)π₯101 (37.1 ± 0.8)π₯101 Data 2 27.8 (6.2 ± 0.2)π₯102 (6.6 ± 0.2) π₯102 (7.6 ± 0.2)π₯102 (10.3 ± 0.3) π₯102 3.2.1 Methodology to compare the best explicit equation to Colebrook’s equation After definition of the best explicit equation among all studied, we performed an additional comparison using scatter plots, which is a two-dimensional data graph that uses dots to represent the values obtained for two different sources — one plotted along the x-axis (estimated data) and the other plotted along the y-axis (measured data). Examples of studies that used scatter plots in this context are the works developed by Ouyang and Aziz (1996), Shoham (2006), Babajimopoulos and Terzidis (2013) and Azizi et al. (2018). Instead of the simple visualization of those scatter plots, we defined three intervals (±5%, ±25% and ±50%) to quantify percentage of estimated data within each of those intervals. Thus, we measured the accuracy of the chosen explicit equation and Table 3: Friction factor explicit equations. Equation Eq. number Reference 15 Moody (1947) π = 0.005 (1 + ((20000 16 Altshul (1952) π = 0.11 ( 17 Wood (1966) 18 Churchill (1973) π = (−2 log ( 19 Pecornik (1963) apud Eck (1973) π = (−2 log ( 20 Jain (1976) 21 Swamee and Jain (1976) 22 Chen (1979) 1 π π 0.225 π· π· 68 + π π π = 0.53 ( ) + (0.094 ( ) π π· π π· ) π = (−2 log ( π 3.7065π· π ) 3.7π· − π π π 0.44 + ( 88 ( ) π· 0.25 log( + π π·×3.71 + π π·×3.715 ( 7 π π 0.9 15 + π π π π )) 21.25 π· π π 0.9 3.7π· 5.0452 )) π π + log ( ) ) 0.25 π = (1.14 − 2 log ( ) + ( π= ( 106 3 5.74 π π 2 π π π 0.134 ) π· (−1.62 ( ) ) −2 −2 −2 )) −2 0.9 ) ) π 1.1098 2.8257π· + 5.8506 π π −2 0.8981 ))) Continues on the next page 7 Copyright © 2020 ASME Continued from Table 3 8 12 π = (8 ( ) π π 23 Churchill (1977) 25 Shacham (1979) −1.5 1⁄12 ) 1 (( Round (1980) ) where: π΄= 24 16 + ((2.457πππ (π΄)) 7 0.9 0.27π 37530 ) +( )) π π π· π π π 6.5 π· π π π = (1.8 πππ (0.135 + −2 )) −2 π = (−2πππ ( π 3.7π· π = (−2 log ( 26 Barr (1981) 5.02 − πππ ( π π π 3.7π· π 14.5 + 3.7π· π π 4.518 log(π π/7) + ))) )) π π (π΄) −2 where: π΄= 1+( π π 0.52 π 0.7 π· ) 29( ) 27 Zigrang and Sylvester I (1982) 28 Zigrang and Sylvester II (1982) 29 Haaland (1983) −2 π = (−2πππ ( π 3.7π· − 5.02 π π πππ (( π 3.π· 13 + π π )))) −2 π = (−2πππ ( π 3.7π· − 5.02 π π πππ ( π 3.π· 6.9 π = (−1.8log ( π π + +( π = (0.25 (π΄ − 5.02 π π πππ ( 1.11 π π·×3.7 ) π 3.π· + 13 π π )))) −2 )) −2 (π΅−π΄)2 )) (πΆ−2π΅+π΄) where: 30 Serghides I (1984) π΄ = −2log ( π 3.7π· + 12 π π ) π 2.51π΄ π΅ = −2log ( + ) 3.7π· π π π 2.51π΅ πΆ = −2log ( + ) 3.7π· π π π = 0.25 (4.781 − (π΄−4.781)2 −2 ) (π΅−2π΄+4.781 ) where: 31 Serghides II (1984) π΄ = −2log ( π 3.7π· + 12 π π ) π 2.51π΄ π΅ = −2log ( + ) 3.7π· π π 32 Robaina (1992) π = (−2 log ( 33 Manadilli (1997) π = ((−2 log 10 (( 34 Sousa, Cunha and Marques (1999) π = (−2πππ10 ( π 0.27π π· + 5.62 π π 0.9 )) 95 3.7×π· −2 −2 )) + ((π π 0.983))) − ( 96.82 π π )) −2 π 3.7×π· )− 5.16 π 5.09 )+( 0.87 )) 3.7π· π π π π×πππ10(( ) Continues on the next page 8 Copyright © 2020 ASME Continued from Table 3 −2 π π = (−2πππ10 ( 3.7065π· 35 Romeo, Royo and Monzon (2002) 5.0272 − πππ(π΄))) π π where: π΄= π 3.827 − 4.567 π π πππ (( π 7.79π· ) 0.9924 +( 5.3326 208.82+π π ) 0.9345 ) −2 π = (−2 × log 10 (( 36 Achour et al. (2002) π ) + (π΄)) ) 3.7 × π· where: 4.5 4.5 π π π΄ = ( ) log ( ) log ( ) π π π π 6,97 −2 π = ((0.8686 × ππ (( 0.4587 × π π (π΄ − 37 Sonnard and Goudar (2006) π΄ ))) 0.31)π΄+1 ) where: π π΄ = 0.124 ( ) π π + ln(0.4587π π) π· π −1 38 Rao and Kumar (2007) π = 2πππ ( ((2(π·) −2 ) 0.444+0.135π π )πΉ(π π) π π ( ) where: πΉ(π π) = (1 − 0.55)π −0.33(ππ( π π 2 ) ) 6.5 −2 π΅ π = (π΅1 − ( π΅1 +(2 log π π2 ) 1+ 2.18 π΅2 )) where: 39 Buzzelli (2008) π΅1 = ( (0.774 ln(π π))−1.41 π΅2 = (( π π· ) (1+1.32√( )) π 3.7π· ) π π) + 2.51π΄ −2 π = (0.8686ln ( 40 Vantankah and Kouchakzadeh (2008) 0.4587π π π ( ) (π−0.31) π+0.9633 )) where: π π = 0.124π π ( ) + ln (0.4587π π) π· Continues on the next page 9 Copyright © 2020 ASME Continued from Table 3 π π πΌ π π 64 6.8 2(1−πΌ)π½ π = (( ) (1.8πππ ( )) (2.0πππ10 ( 3.7π· 2(1−πΌ)(1−π½) ) π −1 )) where: 41 Cheng (2008) 1 πΌ= 1+ ( 1 π½= 1+ ( 42 Avci and Karagoz (2009) 43 Papaevangelou et al. (2010) π π 9 ) 2720 2 π π π·) 160 π 2.4 π=( 6.4 log(Re) π π π· π· − log (1 + 0.01π π ( ) (1 + 10√ ))) 4 π= ( 0.2479−0.0000947×(7−πππ10(π π)) (πππ10( π 7.366 )) + 3.615π· π π0.9142 π = (−2πππ (10−0.4343π½ + 44 BrkiΔ I (2011) Fang et al. (2011) 47 Ghanbari et al. (2011) 3.71 )) π π 1.1π π ) ln(1+1.1π π) 2.18π½ π π + −2 π 3.71 )) where: π½ = ln 46 −2 π 1.816 ln( π = (−2πππ ( BrkiΔ II (2011) ) where: π½ = ln 45 2 π π 1.1π π ) ln(1+1.1π π) 1.816 ln( π 1.1007 60.525 56.291 −2 π = 1.613 (ln (0.234 ( ) − 1.1105 + 1.0712 )) π· π π π π π = (−1.52πππ (( π 7.21π· 1.042 ) π = (S1 − + ( 2.731 0.9152 (π2 −π1 )2 π3 −2π2 +π1 π π ) −2.169 )) −2 ) where: 48 ΔojbašiΔ and BrkiΔ I (2013) π1 = −2πππ ( π 3.71π· )+( π π2 = −2πππ ( 3.71π· π3 = −2πππ ( 3.71π· π 12.585 )+( )+( π π 2.51π1 π π 2.51π2 π π ) ) ) Continues on the next page 10 Copyright © 2020 ASME Continued from Table 3 49 π = (−2πππ ( ΔojbašiΔ and BrkiΔ II (2013) π 3.7106π· )− ( 5 π π πππ ( Achour and Bedjaoui (2012) 3.8597π· 4.9755 206.2795+π π π = (−2πππ10 ( 50 π ) )− 0.8795 4.795 π π πππ (( π 7.646π· ) 0.9685 + −2 )) π 10.04 )+ ( )) 3.7π· π Μ −2 where: π Μ = 2π π (−2πππ ( π 3.7π· + −1 5.5 π π 0.9 )) −2 51 BrkiΔ (2016) π 2.51 (1.14 − 2πππ ( )) π π· −2πππ ( )+( ) π π 3.71π· π= ( ( )) −2 52 Offor and Alabi (2016) 53 Azizi, Hojjati and Homayoon (2018) 1.092 π 1.975 π 7.627 π = (−2πππ ( )− (πππ ( ) + ( ))) 3.71π· π π 3.93π· π π + 395.6 π = (1.805πππ ( π 1.108 π· ( ) 4.267 −2 )+( 5.164 π π 0.966 )) −2 1.038ln(π΅+π΄) π = (0.8686 (π΅ + ( (0.332+π΅+π΄) ) − ln(π΅ + π΄))) 54 BrkiΔ and Praks (2019) where: π π΄= ( π π(π·) 8.0878 π΅ = ln ( ) π π 2.18 ) π = (0.8686 (π΅ + ( 55 BrkiΔ and Praks II (2019) 1.0119ln(π΅+π΄) (π΅+π΄) ) − ln(π΅ + π΄) + −2 ln(π΅+π΄)−2.3849 )) (π΅+π΄)2 where: π π΄= ( π π(π·) 8.0878 π΅ = ln ( ) π π 2.18 ) 56 Blasius (1913) π = 0.316(π π −0.25 ) 57 Ward-Smith (1980) π = 0.3052(π π −0.25 ) 58 Ward-Smith (1980) π = 0.2252(π π −0.222 ) 59 Knudsen and Katz (1958) π = 0.1748(π π −0.20 ) 60 Ward-Smith (1980) π = 0.139(π π −0.182 ) 11 Copyright © 2020 ASME (2014), Zeghadnia et al. (2019) and Pimenta et al. (2018) considered Cheng’s equation in their studies. Figure 3 presents results from analysis of average percentage error with sign (ππ΄ ± ). We note that Achour’s equation can estimate friction factor within the lowest error interval, between [-27%; +34%]. Cheng’s equation presented the second lowest error, with errors between [-26%; +36%]. From this perspective, most of explicit equation (33 equations out of 47 studied) presented similar trend, with interval between [-30%; +37%], including Colebrook’s equation. Equations from 56 to 60, called Blasius derivative equations, had presented good performance in both analyses, as we can observe in Figures 2 and 3. We measured their accuracy in this work because several handbooks and articles recommend them and we concluded that they present better accuracy than most equations — including Colebrook’s equation. Nonetheless, we believe that those equations will not be that accurate whether pipes with greater roughness would be applied, since they do not include roughness to estimate π. Thus, we can recommend Equations 56 to 60 only for flows under low roughness pipes, which is the case here studied, once relative roughness in this work varies from 0 to 1.40x10-3. On the other hand, equations 17, 38 and 51 presented the worst results, with values of cumulate absolute percentage error greater than 3000%. Cumulative percentage errors observed in equations 17 and 51 can be explained by the fact that they were applied in intervals that were not recommended by authors: Wood (1966) recommend applications for relative roughness from 10-5 to 4x10-2, while BrkiΔ (2016) recommended applications of his equation for relative roughness from 10 -2 to 5x10-2. Despite that, we used both equations to access their accuracy, since we are looking for an equation that can be employed under different conditions. Asker et al. (2014) compared 27 explicit equation to Colebrook’s equation and also concluded that Wood (1966) was one of the worst equations to predict friction factor. Same conclusions were shown by Genic et al. (2011) and Zeghadnia et al. (2019). The equation developed by BrkiΔ (2016) was only systematically analyzed by Pimenta et al. (2018), because it is a relatively recent equation. Pimenta et al. (2018) concluded that BrkiΔ equation have low accuracy, as shown here. In their work, they compared explicit equations to Colebrook’s. Differently of equations 17 and 51, equation 38 developed by Rao and Kumar (2007) do not have an applicable range. They proposed this equation to cover the whole turbulent range flow based on experimental data from Nikuradse (1933) and did not specify any limitations concerning relative roughness. Similar results found in our work were also observed by BrkiΔ (2011) and Pimenta et al. (2018), whose conclusions showed that errors from equation 34 were the highest in relation to others explicit equations analyzed. From Figure 3 we can also note a trend to underestimate π values, which means that usually πππππ π’πππ > ππΈπ π‘ππππ‘ππ . This trend is predominant in equations that showed greater cumulative absolute percentage errors in Figure 2, namely, Table 5: Measured roughness of Data 1 and Data 2. D (mm) πΊ (ππ) ± 10% 0.80 1.01 21.6 0.98 0.97 0.89 0.90 27.8 0.89 0.96 Mean 0.9 ± 0.3 4.2 Comparison between ππ¬ππππππππ and ππ΄πππππππ Figure 2 presents stacked bar plot for overall cumulate absolute percentage error. Each color represents absolute percentage error (ππ ) for each data shown in Table 1. We used stacked bar instead of simple bar plot because we were interested in observing if one of the datasets affected cumulative error more than others. Nonetheless, as shown in Figure 2, there is not a notable percentage error for a specific data. We did not plot legend in Figure 2 for each data once we were interested in overall performance. Observing Figure 2, equations 36, 41, 56, 57, 58, 59 and 60 produced better results than others equations studied. Equation 36, developed by Achour et al. (2002), exhibited the lowest cumulate error, 1951%. The lowest percentage error between ππΈπ π‘ππππ‘ππ (using equation 36) and πππππ π’πππ was observed for Data 2, with π2 = 5.83 %, while the greatest one was observed for Data 28, π28 = 79.63 %. Zeghadnia et al. (2019) has already shown that equation developed by Achour et al. (2002) brings accurate results, however, they highlighted that this equation is not very known and it is not, indeed — it can be partially explained by the fact that Achour’s original paper was written in French. In their work, Zeghadnia et al. (2019) tested Achour’s equation against Colebrook’s equation. Here, we tested this equation against real data, and we showed its good accuracy in comparison to other equations. Note that we were interested in relative error, that is, we considered Achour’s equation more accurate in comparison to others 46 equations studied. With a cumulative percentage error of 1951%, we can considerer that it is not that accurate, but it is important to take into account that most of data studied in this work is from two-phase flows, thus, errors related to homogenous no-slip model must be considered as main source of error in friction factor estimates. However, this error is equal for all 47 equations studied, therefore, from a relative perspective, Achour’s equation can be considered accurate. Equation 41 developed by Cheng (2008) also had good accuracy, with the second lowest cumulative percentage error, 1976 % (Figure 2). In his work, he compared equation 41 against Nikuradse (1933) experimental data and obtained accurate results. This paper reinforces the accuracy of the equation developed by Cheng (2008). However, from literature review, we concluded that Cheng’s equation has not been used frequently, as also observed for Achour’s equation. As example, none of the recent works developed by Genic et al. (2011), Asker et al. 12 Copyright © 2020 ASME equations 17, 38 and 51. The best results were again observed for equations 36, 41, 56, 57, 58, 59 and 60. We also used the refined Willmott index (ππ ) to measure the agreement between ππΈπ π‘ππππ‘ππ and πππππ π’πππ . Table 6 presents results from this statistical metric. Equations 17, 38 and 51 presented the worst results, with dr of 0.11, 0.01 and 0.11, respectively. Therefore, from three different statistical metrics, equations 17, 38 and 51 shown to be inaccurate. This conclusion was already observed in previous works, but here we proved their inaccuracy using real data. On the other hand, Achour’s equation presented the most accurate result again, with dr = 0.40, followed by Cheng’s equation with dr = 0.39, the same conclusion observed from Figures 3 to 5. However, it is interesting to note that using refined Willmott index, equations 57, 58, 59 and 60 did not presented greater accuracy than others equations, following the trend of d r between 0.36 and 0.37. This result complement previous observations about applicability of Blasius derivative equations. Thus, although they presented a good accuracy, Achour’s equation can generate better results with the same mathematical cost, once all are straightforward equations. Note that taking into account only data from single-phase flows, errors decreased significantly (Figure 4), since errors in modelling multiphase flows are not included. However, the same trend from two-phase flow can be observed for this one: the best explicit equation was the proposed by Achor et al. (2002), with an average confidence interval of [-12 %; +10 %], while equations 17, 38 and 51 presented the worst results. mean), it is possible note that Achour’s equation gives slightly better results than Colebrook’s — the same conclusion taken from section 4.2. We must highlight that our study covers relative roughness from 0 to 1.40x10-3, thus we cannot guarantee that accuracy in rough pipes. Colebrook’s equation is frequently presented as the most accurate equation to predict friction factor, usually followed by discussions about computational cost involving its applications and subsequent presentation of a “less accurate” explicit equation that can be applied. We showed here that, under some conditions, Colebrook’s equation can be exchanged by an explicit, low computational cost, equation without losing any accuracy in estimates of π. 5. CONCLUSION Comparing 46 explicit equations and Colebrook’s equation for friction factor against 2897 experimental points, we concluded that estimates from Achour’s equation was the most accurate, stood out from the others. We highlight that, under conditions here considered, Achour (2002) performed better than Colebrook’s equation. An innovation of the present work in comparison to recent similar works was the application of experimental data to measure accuracy of explicit equations instead of use Colebrook’s estimates as a benchmark. This approach allowed us to observe that, under certain conditions, Colebrook’s equation is not the most accurate one. The equation developed by Cheng (2008) also presented outstanding performance, while equations developed by Wood (1966), Rao and Kumar (2007) and BrkiΔ (2016) must be used in specifics conditions that were developed, otherwise can produce highly inaccurate results. The remaining 41 equations presented good accuracy and can be applied. The next step is to study how uncertainty of the friction factor equation can affect application of the most accurate friction factor equation can affect mechanistic and empirical correlations for two-phase flows to predict pressure drop in oil production systems and how these results can affect the coupled simulation with petroleum reservoirs. 4.3 Comparing Achour’s and Colebrook’s equations In previous section we showed that Achour’s equation presented the most accurate estimates from all 47 equations studied, including Colebrook’s equation. Here, we perform a comparative study between both equations. Figure 5 presents scatter plot for dataset 1. As we can see, both equations presented roughly the same results. Using the seven points analyzed, two were within an error interval ±5%, which gives the value of 28.5% points within this interval as observed in Table 7. Within interval ±25%, we can observe six points in Figure 5, that is, 85.7% and 100% of the points are within ±50% interval. Figure 6 shows scatter plot for dataset 2 and again the same trend was observed for both equations: 50% of both estimates are within an error interval of ±5% and 100 % of estimates within an error interval of ±25%. Both figures show error intervals from data measured in this study — datasets 1 and 2. Table 7 summarizes this analysis for the entire data bank. Observing overall performance (weighted 13 Copyright © 2020 ASME Table 6: Refined Willmott index (dr) Equation dr Equation dr Equation dr Equation dr Equation dr 1 11 12 13 14 15 16 17 18 19 0.37 0.37 0.36 0.11 0.37 0.35 0.37 0.37 0.37 0.37 20 21 22 23 24 25 26 27 28 29 0.36 0.37 0.37 0.37 0.37 0.37 0.37 0.37 0.36 0.37 30 31 32 33 34 35 36 37 38 39 0.37 0.37 0.40 0.37 0.01 0.37 0.37 0.39 0.38 0.37 40 41 42 43 44 45 46 47 48 49 0.36 0.37 0.37 0.37 0.37 0.37 0.37 0.11 0.37 0.37 50 51 52 53 54 55 56 0.37 0.37 0.38 0.37 0.37 0.37 0.36 Figure 2. Cumulative absolute percentage error (ππͺ ) between observed and estimated friction factor for the entire data bank. 14 Copyright © 2020 ASME Figure 3. Average percentage error with sign (ππ¨ ± ) between “measured” and estimated friction factor for the entire data bank. Figure 4. Average percentage error with sign (ππ¨ ± ) between “measured” and estimated friction factor using data from singlephase flows. 15 Copyright © 2020 ASME Figure 5. “Measured” friction factor vs. estimated friction factor for Data 1. Figure 6. “Measured” friction factor vs. estimated friction factor for Data 2. 16 Copyright © 2020 ASME Table 7: “Measured” vs. estimated friction factors within intervals of ±5%, ±25% and ±50%. Achour et al. (2002) Colebrook (1939) Data ± 5% ± 25% ± 50% ± 5% ± 25% ± 50% Horizontal single – phase flow Data 1 28,6 % 85,7 % 100 % 28,6 % 85,7 % 100 % Data 2 50,0 % 100 % 100 % 50,0 % 100 % 100 % Data 3 51,5 % 100 % 100 % 10,9 % 62,2 % 100 % Data 4 29,0 % 87,6 % 96,2 % 32,3 % 86,0 % 96,2 % Data 5 83,3 % 100 % 100 % 0% 75 % 100 % Horizontal two-phase flow Data 6 21,4 % 75,7 % 85,7 % 8,6 % 28,6 % 72,9 % Data 7 7,8 % 43,4 % 78,3 % 10,0 % 49,6 % 79,8 % Data 8 30,8 % 69,2 % 89,7 % 5,1 % 50,0 % 82,0 % Data 9 8,5 % 37,3 % 67,8 % 8,5 % 39,0 % 57,6 % 0% 0% 0% 0% Data 10 90 % 95 % Data 11 15,0 % 55,8 % 99,5 % 18,9 % 92,7 % 99,0 % Data 12 9,8 % 51,7 % 77,0 % 9,8 % 52,3 % 75,9 % Data 13 9,8 % 52,9 % 84,3 % 9,8 % 52,9 % 84,3 % Data 14 19,5 % 66,7 % 88,5 % 19,5 % 66,7 % 88,5 % Data 15 0% 50,0 % 75,0 % 0% 50,0 % 75,0 % Data 16 8,3 % 32,25 % 85,4 % 8,3 % 31,2 % 85,4 % Data 17 5,6 % 5,6 % 11,1 % 5,6 % 5,6 % 11,1 % Data 18 6,5 % 25,8 % 77,4 % 6,5 % 25,8 % 77,4 % Data 19 0% 0% 50 % 0% 0% 50 % Upward two-phase flow Data 20 7,7 % 33,3 % 59,0 % 0% 43,6 % 69,2 % Data 21 0% 0% 27,3 % 0% 0% 27,3 % Data 22 0% 2,3 % 4,7 % 0% 2,3 % 4,7 % Data 23 0% 0% 0% 0% 0% 0% Data 24 0% 0% 0% 0% 0% 0% 3,6 % 28,6 % 3,6 % 28,6 % Data 25 100 % 100 % Data 26 3,6 % 32,1 % 100 % 3,6 % 32,1 % 100 % Data 27 21,2 % 100 % 100 % 21,2 % 100 % 100 % Data 28 0% 11,1 % 60,0 % 0% 11,1 % 60,0 % Data 29 0% 16,7 % 52,8 % 0% 16,7 % 52,8 % Data 30 0% 0% 60,0 % 0% 0% 60,0 % Data 31 0% 3,6 % 35,7 % 0% 3,6 % 35,7 % Data 32 0% 0% 28,6 % 0% 0% 28,6 % Data 33 0% 0% 17,9 % 0% 0% 17,9 % Data 34 0% 0% 3,6 % 0% 0% 3,6 % Data 35 0% 0% 0% 0% 0% 0% Downward two-phase flow Data 36 24,2 % 54,5 % 63,6 % 3,0 % 27,3 % 63,6 % Data 37 22,4 % 71,4 % 71,4 % 4,1 % 42,9 % 71,4 % Data 38 25,6 % 55,9 % 69,8 % 0% 32,6 % 62,8 % Data 39 12,5 % 44,7 % 51,8 % 7,1 % 25,0 % 51,8 % Data 40 14,0 % 50,0 % 62,0 % 0% 30,0 % 60,0 % Data 41 14,0 % 51,2 % 65,1 % 7,0 % 30,2 % 67,4 % Data 42 13,2 % 44,8 % 60,5 % 7,9 % 39,5 % 60,5 % Data 43 11,4 % 28,6 % 80,0 % 11,4 % 28,6 % 80,0 % Data 44 8,0 % 28,0 % 92,0 % 8,0 % 28,0 % 92,0 % Data 45 17,7 % 64,7 % 91,2 % 17,7 % 64,7 % 91,2 % Data 46 0% 21,9 % 78,1 % 0% 21,9 % 78,1 % Data 47 2,8 % 16,7 % 58,3 % 2,8 % 16,7 % 58,3 % Weighted mean 13,5 % 44,1 % 68,7 % 9,2 % 41,4 % 67,8 % 17 Copyright © 2020 ASME ACKNOWLEDGEMENTS The authors would like to thank the following institutions for supporting this work: UNISIM Research Group, School of Mechanical Engineering and Center for Petroleum Studies (CEPETRO) both at the University of Campinas — UNICAMP, Brazil. Acknowledgements are extended to UNIVEL University Center (Brazil) and ALGETEC. The authors also acknowledge Petrobras for the Financial Support and Energi Simulation. Finally, thanks to Profs. Cem Sarica and Eduardo Pereyra from University of Tulsa (TUFFP) for providing part of the datasets used in this work. Blasius, H. (1913). Das Ähnlichkeitsgesetz bei Reibungsvorgängen in Flüssigkeiten. Arbeit des Ingenieur-Wesens. Brill, J. P., Chen, X. T., Flores, J. G., & Marcano, R. (1995). Transportation of liquids in multiphase pipelines under low liquid loading conditions. Pennsylvania: Final Report, The Pennsylvania State University. Brito, R. (2012). Effect of medium oil viscosity on two-phase oil-gas flow behavior in horizontal pipes. Tulsa: MSc Thesis, The University of Tulsa. BrkiΔ , D., & Praks, P. (2019). Accurate and efficient explicit approximations of the colebrook flow friction equation based on the wright ω-function. Mathematics. BrkiΔ, D. (2011). 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