Flow Friction Factor Research Articles

Testing a theoretical resistance law for overland flow under simulated rainfall with different types of vegetation

In this paper a recently theoretically deduced flow resistance equation, based on a power-velocity profile, was tested using data collected for overland flow under simulated rainfall carried out in plots with vegetation. The available data were obtained exploring a wide range of rainfall intensities (from 60 to 181 mm h−1) and slopes (from 3.6 to 39.6%), and with four different types of vegetation. The database, including measurements of flow velocity, water depth, cross sectional flow area, wetted perimeter and bed slope, was divided in four datasets (one for each vegetation type), which allowed the calibration of the relationship between the velocity profile parameter Γ, the slope steepness, the flow Froude number, and the rainfall Reynolds number. The effect of rainfall intensity and different types of vegetation on flow resistance was investigated.The results showed that the theoretically deduced flow resistance equation allowed an accurate estimate of the Darcy-Weisbach friction factor for overland flow under simulated rainfall and in presence of vegetation. The developed analysis also suggested that flow resistance increases with rainfall intensity for laminar overland flow. The available data demonstrated that a quasi-independence between slope and mean velocity occurred. Finally, a single flow resistance equation resulted applicable to all investigated vegetation types and this equation was affected by flow regime represented by flow Reynolds number.

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.

Experimental thermal analysis of a turbulent nano enriched water flow in a circular tube

In the present study the effect of adding MgO nanoparticles to the water on the turbulent flow thermal properties in a counter-flow heat exchanger is investigated experimentally. Experiments have been done in Reynolds number of 6,000 to 32,000 and nanoparticles volume fraction of 0.005 to 0.02. In this investigation convective heat transfer coefficient, pressure drop, friction factor and Nusselt number are measured and discussed. According to experimental data convective heat transfer coefficient, average Nusselt number, and pressure drop have direct mathematical relation with nanoparticles volume fraction and Reynolds number (with maximum enhancement of 31.96% in nanoparticles volume fraction of 2%). However, at certain nanoparticle volume fractions and Reynolds numbers, increasing nanoparticle volume fraction led to a reduction of the flow friction factor (1.63% when φ=0.5%). Also, irrespective of the nanoparticle volume fraction, increasing the Reynolds numbers caused decreases of the flow friction factor. Considering high cost of experimental studies as a problem, a new mathematical and statistical based correlation is presented to predict the Nusselt number with independent variables of φ, Pr and Re.

Externally Pressurized Thrust Spherical Bearing Performance Under Variable Lubricant Viscosity (Un-recessed Clearance Type)

In a previous study handling this subject, a series of six papers has been published covering the fitted type (equal radii of this sort of bearings) with its configurations in details. The present study handles the same subject investigating the clearance type of this sort of bearings starting with the un-recessed one. The clearance type of bearings, where the sphere radius is smaller than that of the seat, provides plenty of complications and great divergence in its behavior compared with the fitted one. The study investigates theoretically the bearing behavior under the lubricant viscosity variation in the presence of the centripetal inertia due to the shaft rotation and the surface roughness. Solutions have been derived for the un-recessed clearance type of bearings with hemispherical and partial hemispherical seats in addition to its two special cases using capillary tube and orifice restrictors. Unlike other studies, using the conventional integration (i.e., without using the Sommerfeld substitution), only one equation with one form for the pressure gradient is derived to cover the positive and negative eccentricity ratios. Expressions for the pressure distribution, temperature distribution in turn the temperature rise, load carrying capacity; volume flow rate, frictional torque, friction factor, power losses and stiffness factor are obtained. The study shows the combined effects of the viscosity variation, the centripetal inertia and the surface roughness on the bearing performance. The optimum design (with constant viscosity) based on the minimum power losses, minimum flow rate and the optimal restrictor dimensions, in a previous study is checked where it is found out that designing this type of bearings on such basis only is not sufficient in despite of the bearing consistency. A natural dynamic phenomenon ignored in the previous study is revealed and briefly touched.

Mechanism Study and Geometric Parameter Optimization of the Vortex Downhole Tool

In the present study, the mechanism of the vortex downhole tool (VDT) to improve gas well production efficiency is investigated and the optimal geometric parameter combination of VDT is obtained by an orthogonal experiment. Results show that VDT has the potential to reduce the critical gas velocity only when its helix angle is less than 63°, and the theoretical reduction will not exceed 21.5%, compared to the Turner model. Under the action of VDT, the friction factor of the gas-liquid two-phase flow is reduced, as well as the total flow pressure drop. The field testing results show that the proposed structure optimization method of VDT is feasible and the optimal VDT has greatly improved the production conditions of well A.

Rational Approximation for Solving an Implicitly Given Colebrook Flow Friction Equation

The empirical logarithmic Colebrook equation for hydraulic resistance in pipes implicitly considers the unknown flow friction factor. Its explicit approximations, used to avoid iterative computations, should be accurate but also computationally efficient. We present a rational approximate procedure that completely avoids the use of transcendental functions, such as logarithm or non-integer power, which require execution of the additional number of floating-point operations in computer processor units. Instead of these, we use only rational expressions that are executed directly in the processor unit. The rational approximation was found using a combination of a Padé approximant and artificial intelligence (symbolic regression). Numerical experiments in Matlab using 2 million quasi-Monte Carlo samples indicate that the relative error of this new rational approximation does not exceed 0.866%. Moreover, these numerical experiments show that the novel rational approximation is approximately two times faster than the exact solution given by the Wright omega function.

Experimental investigation of flow and heat transfer in a channel with dimpled plate

This study presents the experimental investigation on the effect of dimpled arrangements on flow and heat transfer characteristics. Three plate surfaces were prepared (smooth, evenly distributed spherical dimples and unevenly distributed spherical dimples) and were placed successively in a channel. The unevenly distributed dimpled plate had the same dimple density with the evenly distributed dimpled plate but had varying transverse pitches to concentrate the dimples at mid-plate in flow direction. Data obtained from the experiment were analysed to determine the performance of each dimpled plate channel. It was observed that the average Nusselt number due to the heat interaction with the air-flow increases with the Reynolds number. The evenly and unevenly dimple plate channels had respectively, 75.7% and 91.8% increase in Nusselt number over the smooth channel. The flow friction factors of the evenly and unevenly dimple plate channels were merely more than that of smooth plate channel by 0.59% and 0.67%, respectively. Thus, the unevenly dimple plate channel had the highest overall thermal-hydraulic performance, followed by the evenly dimple plate channel.

Estimation of Equivalent Sand–Grain Roughness for Coated Water Supply Pipes

AbstractThe Colebrook equation gives the Darcy-Weisbach friction factor for fluid flow in pipes as a function of sand-grain roughness and the Reynolds number. Values of equivalent sand-grain roughn...

Flow Resistance and Structures in Viscoelastic Channel Flows at Low Re.

The flow of viscoelastic fluids in channels and pipes remains poorly understood, particularly at low Reynolds numbers. Here, we investigate the flow of polymeric solutions in straight channels using pressure measurements and particle tracking. The flow friction factor f_{η} versus flow rate exhibits two regimes: a transitional regime marked by rapid increase in drag, and a turbulentlike regime characterized by a sudden decrease in drag and a weak dependence on flow rate. Lagrangian trajectories show finite transverse modulations not seen in Newtonian fluids. These curvature perturbations far downstream can generate sufficient hoop stresses to sustain the flow instabilities in the parallel shear flow.

Generalized correlations for predicting heat transfer and friction factor of turbulent flow in tubes with inner helical ribs

Generalized correlations for predicting heat transfer and pressure drop for turbulent flow of Newtonian fluid through tubes with inner helical ribs are presented. A comparison with existing correlations in the literature shows that revealed correlations allow decreasing the data deviation and cover the wide ranges of working fluids, operational conditions, and different dimensions of helical ribs.

Convective heat transfer performance of hybrid nanofluid in a horizontal pipe considering nanoparticles shapes effect

In this paper, the problem of steady forced convection heat transfer and fluid flow characteristics of a hybrid nanofluid flowing through an isothermally heated horizontal tube considering various nanoparticle shapes has been investigated numerically. The three dimensionless cylindrical coordinate equations are discretized using the finite volume method and solved via a FORTRAN program. A numerical parametric investigation is carried out for a tube filled with regular water, (TiO2/water) nanofluid and (Ag–TiO2/water) hybrid nanofluid. Four different types of nanoparticle shapes are considered in this study, spherical, cylindrical, platelets and blades, with different volume fractions ranging from 0 to 8% using water as a base liquid. The influence of nanoparticle shape, nanoparticle concentration and Reynolds number on the local Nusselt number and the friction factor is essentially examined. The results showed that the friction factor of both nanofluid and hybrid nanofluid flow was increased as the nanoparticle volume fraction increased for all kinds of nanoparticle shapes, whereas it decreased as the Reynolds number increased. Nusselt number increased with increase in the nanoparticle concentration and Reynolds number. The highest heat transfer rate was acquired for the maximum nanoparticle volume concentration by using blade nanoparticle shape followed by platelet shape, cylindrical shape and lastly the sphere shape. It was found that the maximum values of the friction factor were registered for platelet-shape nanoparticles.

Colebrook’s Flow Friction Explicit Approximations Based on Fixed-Point Iterative Cycles and Symbolic Regression

The logarithmic Colebrook flow friction equation is implicitly given in respect to an unknown flow friction factor. Traditionally, an explicit approximation of the Colebrook equation requires evaluation of computationally demanding transcendental functions, such as logarithmic, exponential, non-integer power, Lambert W and Wright Ω functions. Conversely, we herein present several computationally cheap explicit approximations of the Colebrook equation that require only one logarithmic function in the initial stage, whilst for the remaining iterations the cheap Padé approximant of the first order is used instead. Moreover, symbolic regression was used for the development of a novel starting point, which significantly reduces the error of internal iterations compared with the fixed value staring point. Despite the starting point using a simple rational function, it reduces the relative error of the approximation with one internal cycle from 1.81% to 0.156% (i.e., by a factor of 11.6), whereas the relative error of the approximation with two internal cycles is reduced from 0.317% to 0.0259% (i.e., by a factor of 12.24). This error analysis uses a sample with 2 million quasi-Monte Carlo points and the Sobol sequence.

(Invited) Thoughts on the Viscous Sublayer

Years ago people postulated that there is a laminar sublayer near a solid wall in turbulent flow. Murphree (1932) and Levich (1942, 1944) postulated that the turbulence eddies decay only gradually as the solid wall is approached. There are two schools of thought: one says that the eddy diffusivity is proportional to y 3 near the wall; the other says y 4. Which is right? Perhaps there is some truth in both views. It may be that y 4 prevails near the outer limit of the viscous sublayer, adjacent to the outer turbulent flow, but that y3 prevails very close to the wall. It has been thought that this was important only for mass transfer in turbulent flow at high Schmidt numbers, that is, for diffusion coefficients much smaller than the kinematic viscosity. However, it is shown that Nikuradse’s data (1932) for pressure drops (or friction factors) in turbulent pipe flow do not agree with his very careful measurements of profiles of the eddy viscosity. In other words, the viscous sublayer cannot be ignored even in fluid flow. The profile of predicted eddy diffusivity of Wasan, Tien, and Wilke (1963) show regions of y 3 dependence very close to the wall and y dependence in the outer turbulent flow with perhaps a y 4 dependence in the intermediate region. On this diagram, one can identify the thickness of the diffusion layer as a function of the Schmidt number. As is well known, the diffusion layer gets very thin at high Schmidt numbers. Using this profile, one can show that the y 4 dependence will not show up clearly in mass-transfer data at any Schmidt number. Any region of y 4 dependence of the eddy diffusivity is outside the diffusion layer where the concentration does not change (at high Schmidt numbers). These thoughts inspire a method of calculating the profiles of eddy diffusivity and eddy viscosity from consideration of fluctuations themselves. Hopefully this can be a subject of future work.

Upward interfacial friction factor in gas and high-viscosity liquid flows in vertical pipes

In this study, experiments were carried out in a vertical 60-mm internal diameter pipe with air and oil (viscosities 100–330 mPa s) constituting the gas and liquid phases. Superficial air and oil velocity ranges used were 9.81–59.06 m/s and 0.024–0.165 m/s, respectively. Visual observations and change in slope of pressure drop– plot were used to identify flow pattern transition to annular flow. Using the experimental data as well as other reported data, a new correlation to predict interfacial friction factor in upward gas–viscous liquid annular flow regime was developed. Compared to the performance of 16 existing correlations using higher viscosity liquids, that of the new correlation was better. The performance of another correlation we derived for predictions at both low and higher low viscous showed good agreement with measurements. In addition, a neural network model to predict the interfacial friction factor involving both low and high viscous liquids was developed and it excellently described the experimental data.

Friction factor correlation for hexagonal bundles of bare tubes/rods and with flat and scalloped walls

Accurately calculating the pressure losses in hexagonal bundles of bare tubes or rods and with flat and scalloped walls is important for the design and safety analyses of small modular reactors, cooled by forced or natural circulation during nominal operation and after reactor shutdown, and to medium and large, liquid metal cooled nuclear reactors. Extensive Computational Fluid Dynamics (CFD) analyses are performed to calculate the friction factor for laminar, transition and turbulent flows in hexagonal bundles of bare tubes/rods and with flat and scalloped walls. The results for the bundles with flat walls are in good agreement with previously reported experimental data for different number of tubes (7, 19, 37, and 61) in a triangular lattice with 1.07 ≤ P/D ≤ 2.416. The compiled CFD results and the reported experimental data are used to develop a continuous friction factor correlation that spans all three flow regions (102 < Rein <106). The developed correlation is in excellent agreement with the compiled numerical and experimental database, and is also applicable to hexagonal bundles of bare tubes and with scalloped walls.

Experimental investigation on the packed bed of rodlike particles

Rodlike particles have been usually found in industrial applications, such as the straw and needle catalyst in energy and chemical engineering. Compared to spherical particles, rodlike particles exhibit different behaviour in the packing structure due to their rotational movement. In this work, we have experimentally explored the packing structure and its friction factor for fluid flow. The porosity of packing structure generated by two packing methods is measured for four kinds of rodlike particles. The experimental results show that the porosity of bed of rodlike particles in the poured packing is not a monotonic function of the aspect ratio of particles. This is due to the competition between the “self-fitting” effect and excluded effect. The porosity of bed of rodlike particles is more sensitive to the packing method than that of spherical particles. To describe the pressure drop of fluid flow through the packing structure, the Ergun equation is further modified by introducing the modified Reynolds number and Galileo number. By combing the experimental data for packed bed generated by the fluidised packing method, and other experimental work in current literature, a new empirical equation is proposed to predict the friction factor of the packing structure of rodlike particles, in which the effects of the particle orientation and particle shape are both considered by the equivalent sphericity. These experimental results would be of interest from applied standpoints as well as revealing fundamental effects of the aspect ratio of rodlike particles on the packing structure.

Laminar flow friction factor in highly curved helical pipes: Numerical investigation, predictive correlation and experimental validation using a 3D-printed model

Highly curved helical pipes offer attracting potentialities for intensified mass/heat transfer performances as they generate intense Dean-type vortices. The evaluation of friction factor in such geometries is necessary for assessing the trade-offs between the increase of transfer efficiency and the associated specific energy requirement. Unfortunately, such data are lacking for highly curved helixes, probably due to the difficulty to manufacture these geometries through traditional manufacturing techniques.In this paper, CFD simulations are carried out for determining the laminar flow friction factor in helical pipes, particularly highly curved ones. For an experimental validation of the numerical results, a highly curved helix was built by 3D-printing. Existing correlations are shown to fail for the accurate prediction of the friction factor in highly curved helixes. A new correlation is thus proposed. An excellent agreement is obtained between the experimental pressure drop measurements and the proposed correlation predictions.

An experimental investigation in forced convective heat transfer and friction factor of air flow over aligned round and flattened tube banks

AbstractHeat transfer coefficients and pressure drop are studied experimentally for airflow over aligned round and flattened tube configurations. The Reynolds number is based on the outer diameter of the round tube or the outside transverse diameter of the flattened tube, which is used for various flows, ranging from 133 to 800 with a constant input heat flux. In the present work, a total of 30 samples of round and flattened tubes heat exchangers with three transverse pitches, 2.0, 3.0, and 4.5, were studied to investigate their thermal performance. The results indicate that the relative gain in the overall Nusselt number is about 32.5 to 60.6% in flattened tubes, while the reduction range in the friction factor is about 11 to 30.6%. Correlations are proposed for the overall Nusselt number, friction factor, and Colburn j–factor for both round and flattened tube banks. A higher value means that a deviation error of 9.9% in the round tube banks and 10.1% in the flattened tube banks are expected. In addition, the best value for thermal performance for the flattened tube bundle was found to be coincident with a smaller Reynolds number.

What Can Students Learn While Solving Colebrook’s Flow Friction Equation?

Even a relatively simple equation such as Colebrook’s offers a lot of possibilities to students to increase their computational skills. The Colebrook’s equation is implicit in the flow friction factor and, therefore, it needs to be solved iteratively or using explicit approximations, which need to be developed using different approaches. Various procedures can be used for iterative methods, such as single the fixed-point iterative method, Newton–Raphson, and other types of multi-point iterative methods, iterative methods in a combination with Padé polynomials, special functions such as Lambert W, artificial intelligence such as neural networks, etc. In addition, to develop explicit approximations or to improve their accuracy, regression analysis, genetic algorithms, and curve fitting techniques can be used too. In this learning numerical exercise, a few numerical examples will be shown along with the explanation of the estimated pedagogical impact for university students. Students can see what the difference is between the classical vs. floating-point algebra used in computers.

Correlations for friction factor of turbulent pipe flow under supercritical pressure: Review and a new correlation

This paper presents a comprehensive review of correlations of friction factor of turbulent pipe flow under supercritical pressure to provide a guide to choosing a proper correlation for a given application and proposes a high accuracy correlation of friction factor for adiabatic supercritical turbulent pipe flow. Two databases containing 1279 experimental data points of friction factor of turbulent pipe flow under supercritical pressure were compiled from 12 available sources. One has 820 data points at adiabatic conditions, and the other has 459 data points at non-isothermal conditions. Based on the two databases, six adiabatic and 13 non-isothermal correlations are evaluated, and a new model for adiabatic supercritical fluids is developed. The new model has a mean absolute deviation (MAD) of 8.2%, while the best existing one only has an MAD of 20.4%, indicating that the new model improves the prediction accuracy of friction pressure drop of adiabatic supercritical pipe flow remarkably.