Friction Factor Correlations Research Articles

Three-Dimensional Thermo-Hydraulic Analysis of Solar Air Heater With Equilateral Prism-Shaped Rib Roughness

Abstract Thermal transport and flow friction characteristics due to roughness on the absorber plate of solar air heater are evaluated by applying three-dimensional finite volume based code. Renormalization group (RNG) k–ɛ model is employed to capture the turbulent nature of the flow. The effect of equilateral prism-shaped rib roughness geometrical parameters in terms of relative roughness height (e/D) and relative roughness pitch (p/e) on heat transfer and flow friction is analyzed. Further, the effect of flow parameter, Re in the range of 4000–18,000 is also explored. Results are elucidated in terms of average Nusselt number, friction factor, turbulent kinetic energy, and eddy dissipation. Results are compared with a smooth absorber plate solar air heater. Thermo-hydraulic performance of the roughened solar air heater is analyzed. Noteworthy augmentation in heat transport is obtained. The thermal enhancement factor is calculated for optimal performance and found to vary from 1.7 to 3.5. However, friction factor and pressure loss for roughened plate is significantly higher than its smooth counterpart. The pressure drop across the test section increases with the rise in roughness height due to flow obstruction. A minimum value of the friction factor enhancement ratio worth 2.13 is obtained. Enhancement in thermal transport and pressure losses are combined by introducing a thermo-hydraulic performance factor (THHP). For the range of parameters investigated, the optimum value of the thermo-hydraulic performance factor is found to be 3.41. Correlations for average Nusselt number and friction factor are offered at the end.

Assessment of thermo-hydraulic performance of inward dimpled tubes with variation in angular orientations

This paper presents a numerical investigation and assessment of thermal and hydraulic performance of dimpled tubes of varying topologies at constant heat flux of 10kW/m2 and Reynolds numbers ranging from 2300 to 15,000. The performance of the tubes consisting of conical, spherical and ellipsoidal dimples with equivalent flow volumes were compared using steady state Reynolds Averaged Navier Stokes simulations. The ellipsoidal dimples, in comparison to other dimple shapes, demonstrated large increment in heat transfer rate. The variation in the orientation of the ellipsoidal dimples was examined to further improve thermal and hydraulic performances of the tube. A 45° inclination angle of ellipsoidal dimple, from its major axis, increased the thermo-hydraulic performance by 58.1% and 20.2% in comparison to smooth tube and 0° ellipsoidal dimpled tube, respectively. Furthermore, Large Eddy Simulations (LES) were carried out to investigate the role geometrical assistance to fluid flow and heat transfer enhancement for the 45° and 90° ellipsoidal dimpled tubes. LES results revealed a flow channel of connected zones of wakes which maximized fluid-surface contact and therefore enhanced the thermal performance of the tube. In addition, correlations for Nusselt number and friction factor for all angular topologies of ellipsoidal dimpled tube have been proposed.

Frictional Factor Correlation for Laminar High-Viscosity Oil/Gas Flow in Horizontal Pipes

Summary Pressure drop prediction of high-viscosity oil/gas flow is an important issue in several industries, and it has a great interest for many researchers. This paper presents a simplified correlation for two-phase high-viscosity oil/gas flow friction factor in horizontal pipes. The developed friction factor is a sole function of Reynolds number on the basis of the mixture velocity and liquid properties. The recently developed friction factor correlation for laminar and turbulent gas/liquid flow is not suitable for the high-viscosity liquid/gas flow for which Reynolds number is less than 2,000. The validation of the proposed correlation against experimental data and the newly developed correlation showed accurate results for the two-phase friction factor for the cases matching gas/heavy oil flow with liquid holdup greater than 0.5 and mixture Reynolds number on the basis of liquid properties less than 2,000. The proposed correlation accuracy is better than the generalized correlation developed for all viscosities and Reynolds numbers and liquid holdup greater than 0.5. The less the Reynolds number, the more accurate the results.

Detailed heat transfer measurements of impinging swirling and non-swirling jet arrays emitted from grooved orifice plate

An innovative type of heat exchanger with heat transfer benefits from jet impingements is proposed. The increase of surface area to volume ratio for the heat exchanger requires the minimization of the separation distance for the impinging jets, which motivates the present study to comparatively examine the heat transfer performances between the non-swirling and swirling impinging jet arrays emitted from the orifice plate with vertical and horizontal grooves at small separation distances less than one orifice diameter. As demonstrated by the selective experimental results, either the grooves on the orifice plate or the swirling jets can effectively alleviate the unfavorable crossflow effect to boost the heat trasnfer rates at small separation distances. After analyzing the isolated and coupled Rej and separation-distance effects on the local and area-averaged heat transfer properties for the non-swirling and swirling impinging jet arrays, two sets of physically consistent heat transfer and friction factor correlations are devised for practical applications.

Thermal performance of air to water heat exchanger with plain surface disc turbulators: experimental and numerical study

The influence of plain surface disc turbulators on thermo-hydraulic characteristics of water to air double pipe heat exchanger is studied. Tests are conducted for three different diameter ratios, three different pitch ratios and Reynolds number varies betweess 3200 and 10,500. Water (hot fluid) and air (cold fluid) flow along inner tube and through annulus, respectively. Heat exchanger involving smallest diameter ratio (0.420) and pitch ratio (8.420) offers highest augmentation in heat transfer with larger pressure drop. In addition to this, finite volume method has been employed to numerically evaluate the hydrothermal characteristics of heat exchanger using disc turbulators for diameter ratio = 0.420 and pitch ratio = 8.420. Correlations for Nusselt number, friction factor, and thermal enhancement factor are presented as a function of experimental parameters.

Numerical study of heat transfer and friction factor in a tube with groove and rib on the wall

AbstractHeat transfer and friction factor in a tube with turbulator internally grooved and rib are studied using numerical simulation. Two different turbulator geometries, including only groove and both groove and rib, on the tube wall are investigated. The study is developed for different parameters of turbulator including pitch ratio (PR), groove and rib height (DR), and Reynolds number in the range of 10 000 to 38 000. The results revealed that the best Nusselt number is achieved for GR turbulator and for the tube wall with PR = 2 and DR = 1. Furthermore, for a specific turbulator parameter, Nusselt number is improved by 46% compared with the experimental result. The results show that the friction factor decreases by increase of both step and height ratios. For example, based on Reynolds number, 21% to 69% reduction in the friction factor was observed for PR = 7 and DR = 3. In addition, the correlations for Nusselt number and friction factor with acceptable deviation are presented for the studied turbulators. Finally, thermal performance enhancement factor is defined to be a good criterion for selection of the optimum turbulator and their parameters.

Heat transfer and friction of molten salt and supercritical CO2 flowing in an airfoil channel of a printed circuit heat exchanger

Airfoil printed circuit heat exchanger (PCHE) is considered as one of the competitive candidates in the 3rd generation of concentrating solar power (CSP) plant, where the molten salt and supercritical carbon dioxide (S-CO2) are adopted as the heat transfer fluids (HTFs). To study the flow features and heat transfer performance of the two HTFs in the airfoil channel of PCHE, a three-dimensional numerical model was firstly developed and validated by experiment. Then, the friction features and heat transfer of the two HTFs under different mass flow rate and inlet temperature conditions were numerically investigated. Then, the performance of distributed airfoil channels was compared with that of straight channels and zigzag channels. Finally, the heat transfer and friction factor correlations were fitted for the two HTFs in the airfoil channel of PCHE, which can be used in relatively wide ranges of Reynolds number and temperature. The results show that the larger inlet temperature leads to higher heat transfer performance for molten salt, but causes lower heat transfer performance for S-CO2. The airfoil channels have the best comprehensive heat transfer performance among these three channels at the given pumping power. Moreover, the maximum deviations between the simulation results and the proposed heat transfer correlations are within ±6% for both the molten salt and S-CO2. Finally, the maximum deviations between the proposed friction correlations and the calculated results are within ±4% and ±8% for the salt and S-CO2, respectively. The correlations and results given in current study can contribute to the design and application of airfoil PCHEs in the 3rd generation of CSP plant.

Development of heat transfer and pressure drop correlations in twisted tube heat exchangers

Heat transfer in concentric twisted tubes exchangers has several applications in the chemical and food industry mainly due to its compact structure and high transport coefficients, however, its use has been limited as a consequence of drop pressure. This study reports Nusselt number and friction factor correlations as a function of the dimensionless Dean number as well as Prandtl number, and a torsional step parameter. Simulations were carried out using ANSYS CFX® V16.0 software at workstations belonging to Grupo de Investigación en Materiales Avanzados y Energía of Instituto Tecnológico Metropolitano. Six geometries were developed with SolidEdge® ST9 academic license. Geometry modification, cleanup, and repair were made although DesignModeler® 2019 academic license. Finally Meshing® for discretization. The mesh size independence, and incidence of the turbulence model, and twisting step validated the computational tools use. Correlation numerically showed an increase of 60% average respect to smooth bending tube.

Heat transfer and flow characteristics of offset fin and flat tube heat exchangers under low pressure environment

Due to the performance degradation of the vehicle cooling system heat exchanger in the low pressure environment, the problem of overheating of the cooling system occurs when the vehicle is running on the plateau. This seriously affects the use of vehicles on the plateau environment. Therefore, it is necessary to study the influence of the plateau low pressure environment on the heat transfer and flow characteristics of the vehicle cooling system heat exchanger. In this paper, an experimental study of an offset fin and flat tube heat exchanger was conducted on a low pressure wind tunnel test bench, and the heat transfer and flow characteristics of the heat exchanger under different gauge pressure conditions were analyzed. Based on this, numerical simulations were carried out to study the heat transfer and flow characteristics of the heat exchanger with different structural parameters under low pressure environment. The empirical correlation of Colburn factor, j, and friction factor, f, were fitted using multiple linear regression. The research results can provide help for the engineering calculation of the offset fin and flat tube heat exchanger.

Thermal hydraulic performance of printed circuit heat exchanger with various channel configurations and arc ribs for SCO2 Brayton cycle

Printed circuit heat exchanger (PCHE) is an efficient and high-compactness heat exchanger which can withstand extreme conditions of high temperature and high pressure, thus is a promising candidate in the nuclear, marine propulsion and power applications. In this paper, eight multi flow paths (MFP) PCHEs with different channel configurations are proposed, and seven kinds of arc ribs, including continuous and discontinuous 1/3 ribs, 2/3 ribs and semi-circular ribs, are innovatively introduced into MFP channels. Numerical investigations are carried out to explore the local and overall thermal hydraulic performances of various MFP-PCHEs in supercritical carbon dioxide (SCO2) Brayton cycle, and to figure out the effects of channel configuration and arc rib arrangement systematically. Additionally, the improvement in system performances of SCO2 Brayton cycle is further evaluated when the MFP-PCHEs are applied as recuperator. The results show that the 90° turning bend and arc rib effectively enhance the fluid disturbance by the impingement and acceleration effects, contributing to the augmentation of turbulence kinetic energy (TKE) and heat transfer level. The increase of flow path is also helpful to improve the overall thermal performance. Among all the MFP-PCHEs, the C3 (with 14 turning bends) produces the largest Nusselt number, with 16.6%–30.6% improvement compared with the straight channel (C0 case), at the same time its friction is 3.41–4.90 times of C0 case. The maximum heat exchanger effectiveness is obtained in C7 case (with 12 turning bends) at the Re of 30,000, which is 75.5%. According to the simulation results, the correlations of Nusselt number and friction factor for hot and cold channels of eight MFP-PCHEs are developed. Moreover, the adopt of semi-circular rib (R5) improves the Nusselt number by 7.3%. For a SCO2 Brayton cycle, the adoption of MFP-PCHEs can effectively enhance the system energy efficiency, and the improvement effect will reach 41.2% when the C3 configuration is applied. This research has provided critical data support for the development of MFP-PCHE in SCO2 Bryton cycle.

Experimental and numerical investigation of thermal-hydraulic performance of supercritical nitrogen in airfoil fin printed circuit heat exchanger

Printed circuit heat exchangers (PCHE) are a new kind of high efficiency and compact plate heat exchangers, which can be used in extreme conditions, such as high pressure, high heat flux and volumetric limitation. Supercritical fluids have various favorable characteristics when used as working fluids in heat exchangers. In this paper, the thermal-hydraulic performance of supercritical nitrogen was studied both experimentally and numerically in an airfoil fin PCHE. The mass flow rate varied from 203 kg·m−2·s−1 to 228 kg·m−2·s−1 while the operating pressure ranged from 5 MPa to 8 MPa. The cold-side outlet temperature and pressure drop in the PCHE were investigated experimentally. The outlet temperature increased as pressure increased and the average outlet temperature at 8MPa was 7.81 K higher than that at 5 MPa for the same mass flow rate. The gradient of pressure drop increased as the mass flow rate increased. Moreover, a numerical analysis of nitrogen in the channel of airfoil fins PCHE was conducted. Nusselt number and Fanning friction factor correlations were developed for Reynolds number values from 10000 to 14500. The accuracy of the proposed correlations was validated using the experimental and numerical data obtained, and results showed that they have a good accuracy for predicting the thermal hydraulic performance of supercritical nitrogen in the channel of airfoil fin PCHE.

Compressible Fanno flows in micro-channels: An enhanced quasi-2D numerical model for turbulent flows

Fanno theory provides an analytical model for one-dimensional confined viscous compressible flows. The model holds under the assumptions of adiabatic flow and constant cross-section channel. From theory, the differential of every flow-related quantity is expressed as a function of Mach number and friction factor. One-dimensional flow numerical models can be derived by discretizing Fanno equations. However, theory does not assess how to evaluate friction, while the model works properly only if friction is estimated correctly. Compressibility and turbulence act by deforming the velocity profile making it flatter. Assuming the friction factor function of the Reynolds number alone, in line with incompressible flow theory, is thus not correct. Better correlations should include the Mach number to address compressibility effects. Here, the impact of turbulence and compressibility on the velocity profiles in a micro-channel is analysed by means of CFD simulations. Friction factor correlations are deduced for turbulent micro-flows. The impact of the velocity profile on other quantities, such as dynamic pressure and bulk temperature, needed for the numerical model operation, is also evaluated. Additional correlations for these quantities overcome the instrinsic limits of the one-dimensional model, necessarily unaware of local velocity profiles, in a quasi-2D fashion significantly improving its predicting capabilities.

Thermohydraulic characteristics of inline and staggered angular cut baffle inserts in the turbulent flow regime

In the present work, heat transfer and pressure drop characteristics in flow through a tube with inline and staggered baffles having angular cut at the edge are reported for various operating conditions. An experimental test rig is designed and developed to investigate heat transfer and pressure drop behavior for different conditions. Effects of different geometrical parameters, i.e., pitch ratios, baffle arrangement and cutting angle of baffles on heat transfer rate and pressure drop characteristics, have been investigated for turbulent flow regime. Reynolds number ranging from 10,000 to 52,000 has been considered in the present study. The maximum heat transfer rate has been observed for staggered arrangement with pitch ratio of 0.1 and cutting angle of 60°, while minimum heat transfer rate has been observed for inline arrangement with pitch ratio of 0.2 and cutting angle of 30°. Empirical correlations for Nusselt number and friction factor have been developed as a function of geometrical and flow parameters. The deviations between experimental and predicted values of Nusselt number and friction factor for staggered arrangements have been observed as ± 10%, ± 4%, respectively, whereas for inline arrangement the deviation has been observed as ± 12%, ± 5%, respectively. Results from empirical correlations are well agreed with the experimental data.

Influence of perforated triple twisted tape on thermal performance characteristics of a tube heat exchanger

One of the challenging tasks in the development of heat exchangers is to select an appropriate geometry that will consume minimum energy. This could be achieved by designing a suitable enhancing device which will boost the thermal performance characteristics of heat exchangers. Twisted tape swirl generator, a heat transfer enhancement device equipped in a tube heat exchanger would induce turbulence and superimposed vortex motion causing a reduction in the hydrodynamic or thermal boundary layer thickness which results in the improvement of convective heat transfer. In this study, a new design of perforated triple twisted tape (PTTT) insert was applied as a swirl flow device with four different porosities (Rp) of 1.2, 4.6, 10.4 and 18.6% to enhance the convective heat transfer of a tube heat exchanger. The effects of porosity (Rp) of PTTT swirl generators on heat transfer, fluid friction and thermal performance characteristics of a tube heat exchanger were experimentally investigated. The experiments were conducted under constant heat flux condition using air as the working fluid in turbulent flow regime for variation in Reynolds number (Re) from 7250 to 49,800. The Nusselt number of the tube installed with PTTT inserts for porosity ranging from 1.2 to 18.6% was found to be 88–320% higher, whereas, the friction factor was achieved to be 112–355% higher in comparison to the plain tube. The highest heat transfer result of 320% was obtained at a porosity of 4.6% with PTTT insert compared to the smooth tube. Despite the significant enhancement in heat transfer, the friction factor was obtained to be 355% higher in comparison to the plain tube at a porosity of 4.6% with PTTT insert. The experimental results demonstrated that Nusselt number, friction factor, and thermal enhancement efficiency (TEE) were increased with decreasing the porosity of the tape inserts except for 1.2%. The heat transfer performance was evaluated based on constant blower power and the values were found to be 1.13–1.5 relative to the plain tube. The maximum TEE of 1.5 was achieved using PTTT insert with a porosity of 4.6%. Based on the experimental data, the empirical correlations of heat transfer, friction factor and TEE were developed in terms of Rp, Re and Prandtl number (Pr) to predict the heat transfer, friction factor and TEE. The optimal design of PTTTs based on the evaluation of thermal performance with suitable porosity could be an excellent heat transfer enhancement device for many industrial applications.

Hydraulic characterization of conductor prototypes for fusion magnets

Forced flow HTS conductors designed for the Winding Demonstrator (WD) or EU DEMO Toroidal Field (TF) and Central Solenoid coils consist of several CroCo or twisted stack monolithic strands, embedded in a stainless steel jacket. Such type of conductors have never been tested for pressure drop. Cable-in-Conduit conductors for the DEMO TF coils designed by ENEA have two low-impedance cooling channels, separated from the cable bundle by flat spirals, with the inner/outer diameter of about 5/7 mm. Experimental pressure drop data for such small spiral ducts are also unavailable yet. To fill this gap, four dedicated short samples, namely dummy conductors with the layout similar to the DEMO HTS or WD conductors and a small spiral-walled pipe have been prepared and tested for pressure drop using demineralised water at 2–3 different temperatures. The tests using water have been performed covering the same range of Reynolds number experienced by the supercritical He at operating conditions typical for DEMO coils. The experimental data have been used to develop experimental friction factor correlations for the considered ducts, which could be utilized in future thermal-hydraulic studies of the DEMO coils or WD.

A Novel Bulk-Flow Model for Pocket Damper Seals

Abstract In this paper, a novel bulk-flow model for pocket damper seals (PDS) is introduced. The model is based on two control volumes (CVs) for each circumferential pocket of the seal. The continuity, circumferential momentum, and energy equations are considered for each control volume. The circumferential recirculating flow within the pocket is modeled for the first time. The boundary layer theory is used to estimate the recirculating flow area, and the Swamee–Jain friction factor correlation allows for defining the dissipation of the circumferential velocity. The perturbation method is used to solve the partial derivative governing equations in the zeroth- and first-order system of equations. The rotordynamic coefficients are evaluated by integrating the dynamic pressure and rotor shear stresses along the circumferential direction. The predictions are compared to the experimental data, which refer to test conditions representative of high-pressure centrifugal compressors. Numerical predictions are accurate for both high positive–negative inlet preswirl ratios. Leakage predictions are also aligned with measurements. Finally, sealing selection approach is introduced in the paper for comparing the dynamic behavior of two different sealing technologies and identifying stable regions as a function of the rotor natural frequency and preswirl ratio.

Optimized Condition for Pairing of Different Friction Factor and Viscosity Equations for the Frictional Pressure Drop of R22 and R290

Accurate prediction of the friction factor and consequently the pressure drop of two-phase flow in small channels is still an issue. Many correlations exist for the determination of the viscosity and the friction factor that appear in the frictional pressure drop and their combination often determined the degree of disagreements between the experimental data and predicted outcomes. Demands for environmentally friendly refrigerants have further posed a challenge to find compatible alternatives with as good a performance as the current coolants. Despite the many available correlations developed to date, many more are studied in effort to reduce the discrepancies. This paper presents the outcomes of a study comparing the optimized conditions when three different viscosity equations are paired with eight different friction factor correlations to minimize the frictional pressure drop. The approach used multi-objective genetic algorithm (MOGA) to assist in selecting the best pairing. Comparison is then completed with available experimental data. The study showed that the Blasius friction factor paired with the Dukler viscosity produced the least percentage difference for R22, while when paired with the McAdams viscosity produced a lower difference for R290, an environmentally friendly refrigerant being considered to replace R22.

Thermal performance characteristics of a triple spiral tube heat exchanger

In the current paper, the thermo fluid characteristics of a new design of curved tube called a triple spiral tube heat exchanger (TSTHE) was achieved with a brave comparative with a double spiral tube heat exchanger (DSTHE) as particular reference. A three-dimensional computational fluid dynamic model was developed using Ansys 14.5 software package. The Realize k–ε model had been applied with enhanced wall treatment to simulate the thermo-fluid characteristics. The governing equations were solved by a finite volume discretization method. The new design was created by adding a third tube to a double concentric spiral tube heat exchanger. The study carried out under consideration of turbulent fluid to fluid heat transfer condition. The influence of different key design parameter such as; inlet hot fluid temperatures, flow arrangements, concentration of Alumina (Al2O3) nanoparticles, and Dean number were the main point of interest. The effect of Alumina-water nanofluid was considered in three different concentrations of 0.5%, 1% and 2% by volume. The results carried out at Dean number ranged from 1100 to 10,500 in the turbulent region corresponding to Reynolds number from 6400 to 57,400. The results presented that the Nusselt number obtained from a TSTHE was higher than that value of DSTHE at the same flow conditions. The results also presented that a significant effect of hot water inlet temperature on the Nusselt number while the increase in pressure drop can be approximately considered neglect. Also, the counter flow arrangement gives better effect on hot fluid Nusselt number compared with the three other flow arrangements. A notice augmentation of heat transfer enhancement occurs due to use Alumina–water nanofluid in TSTHE by 20.8 %. New correlations for inner annulus hot fluid Nusselt number, effectiveness, and friction factor of heat exchanger were presented.

Thermal performance analysis of CuO/water nanofluid flow in a pipe with single and double strip helical screw tape

This paper presents the experimental analysis to evaluate thermo-hydraulic characteristics of single and double strip helical screw tape inserts in the copper tube with water and CuO/water nanofluid at constant heat flux conditions at twist ratio of 1.5, 2.5 and 3. Results show that Nusselt number and friction factor with double strip helical screw tape inserts achieves substantial enhancement as compared to single strip helical screw tape insert. The Nusselts number with nano-fluid increases and achieves the increment by 182% and 170% for double strip and single strip helical screw tape inserts respectively, at a twist ratio of 1.5. Whereas at twist ratio of 2.5 and 3, thermal performance factor shows excellent performance for double strip helical screw tape inserts, that makes it suitable for heat exchanger to reduce the size of it for thermal applications. Correlations for Nusselt number and friction factor are also developed within the range of Reynolds number, twist ratio and Number of strips.

Heat Transfer and Friction Factor Correlations for Solar Air Heater With Gap in V-Rib With Symmetrical Gap and Staggered Ribs

Abstract The present experimental study is concerned with heat transfer analysis of air flowing in solar air heater duct with a gap in V-rib with symmetrical gap and staggered ribs geometry. The investigated parameters are Reynolds number (Re) of 4000–14,000, relative roughness pitch (p/e) of 12, relative roughness height (e/Dh) of 0.043, angle of attack (α) of 60 deg, relative staggered rib pitch (p′/p) of 0.65, relative gap size (g/e) of 4, relative staggered rib size (r/e) of 4, relative gap position of additional gap in each symmetrical rib elements (d/w) of 0.65, relative gap size of additional gap in each symmetrical rib elements (g′/e) of 1, number of main gaps (Ng) of 1, 2, 3, 4, and number of additional gap (ng) varying from 1 to 5. Fourteen roughened absorber plates were tested. The maximum enhancement in Nusselt number (Nu) and friction factor (f) was 2.34 and 2.79 times that of smooth surface corresponding to the number of main gaps (Ng) of 4 with the number of additional gaps (ng) of 4. The performance of the gap in V-rib with symmetrical gap and staggered rib geometry has been compared with the existing latest V-rib geometry and smooth surface. The proposed gap in V-rib with symmetrical gap and staggered ribs geometry has a better performance than the existing latest V-rib geometry. The following correlations have been developed for heat transfer and friction factor in terms of roughness and operating parameters. Heat transfer:Nur=0.0073(Re)0.9788(Ng)0.2790(ng)0.0184exp[−0.1678(ln(Ng))2]exp[−0.0129(ln(ng))2] Friction factor:fr=0.0477(Re)−0.0678(Ng)0.5919(ng)−0.0562exp[−0.4922(ln(Ng))2]exp[−0.0487(ln(ng))2]