Friction Factor For Turbulent Flow Research Articles

Computer-based analysis of explicit approximations to the implicit Colebrook–White equation in turbulent flow friction factor calculation

The implicit Colebrook-White equation has been widely used to estimate the friction factor for turbulent fluid-flow in rough-pipes. In this paper, the state-of-the-art review for the most currently available explicit alternatives to the Colebrook-White equation, is presented. An extensive comparison test was established on the 20x500 grid, for a wide range of relative roughness (@e/D) and Reynolds number (R) values (1x10^-^6=<@e/D=<5x10^-^2; 4x10^3=

Friction factor for turbulent flow in rough pipes from Heisenberg’s closure hypothesis

We show that the main results of the analysis of the friction factor for turbulent pipe flow reported by Gioia and Chakraborty [Phys. Rev. Lett. 96, 044502 (2006)] can be recovered by assuming the Heisenberg closure hypothesis for the turbulent spectrum. This highlights the structural features of the turbulent spectrum underlying the analysis of Gioia and Chakraborty.

Discussion of “Turbulent Flow Friction Factor Calculation Using a Mathematically Exact Alternative to the Colebrook–White Equation” by Jagadeesh R. Sonnad and Chetan T. Goudar

Discussion of “Turbulent Flow Friction Factor Calculation Using a Mathematically Exact Alternative to the Colebrook–White Equation” by Jagadeesh R. Sonnad and Chetan T. Goudar

Closure to “Turbulent Flow Friction Factor Calculation Using a Mathematically Exact Alternative to the Colebrook–White Equation” by Jagadeesh R. Sonnad and Chetan T. Goudar

Closure to “Turbulent Flow Friction Factor Calculation Using a Mathematically Exact Alternative to the Colebrook–White Equation” by Jagadeesh R. Sonnad and Chetan T. Goudar

Discussion of “Turbulent Flow Friction Factor Calculation Using a Mathematically Exact Alternative to the Colebrook–White Equation” by Jagadeesh R. Sonnad and Chetan T. Goudar

Discussion of “Turbulent Flow Friction Factor Calculation Using a Mathematically Exact Alternative to the Colebrook–White Equation” by Jagadeesh R. Sonnad and Chetan T. Goudar

A power-law approximation of the turbulent flow friction factor useful for the design and simulation of urban water networks

An approximation of the friction factor of the Colebrook–White equation is proposed, which is expressed as a power-law function of the pipe diameter and the energy gradient and is combined with the Darcy–Weisbach equation, thus yielding an overall power-law equation for turbulent pressurised pipe flow. This is a generalised Manning equation, whose exponents are not unique but depend on the pipe roughness. The parameters of this equation are determined by minimising the approximation error and are given either in tabulated form or as mathematical expressions of roughness. The maximum approximation errors are much smaller than other errors resulting from uncertainty and misspecification of design and simulation quantities and also much smaller than the errors in the original Manning and the Hazen–Willians equations. Both these can be obtained as special cases of the proposed generalised equation by setting the exponent parameters constant. However, for large roughness the original Manning equation improves in performance and becomes practically equivalent with the proposed generalised equation. Thus its use, particularly when the networks operate with free surface flow is absolutely justified. In pressurised conditions the proposed generalised Manning equation can be a valid alternative to the combination of the Colebrook–White and Darcy–Weisbach equations, having the advantage of simplicity and speed of calculation both in manual and computer mode.

Explicit Reformulation of the Colebrook−White Equation for Turbulent Flow Friction Factor Calculation

In this paper, we present an improvement of a mathematically equivalent representation of the Colebrook−White (CW) equation to compute friction factors for turbulent flow in rough pipes. This new form is simple and very well-suited for accurately estimating the friction factor, because no iterative calculations are necessary. Specifically, the friction factor is expressed as the sum of known simple functions and an unknown correction term. This correction term satisfies an auxiliary equation that can be accurately and easily solved with predictable error bounds over the complete range of pipe roughness and Reynolds number values encountered in practice. The simplest case, with the unknown correction term set to zero, resulted in friction factor estimates with errors of <1%. A simple linear approximation of the correction term resulted in a maximum error of 3.64 × 10-4%, whereas friction factor estimates from a continued-fractions-based approximation had a maximum error of 1.04 × 10-10%. These maximum erro...

Turbulent Flow Friction Factor Calculation Using a Mathematically Exact Alternative to the Colebrook–White Equation

We present a novel, mathematically equivalent representation of the Colebrook–White equation to compute friction factor for turbulent flow in rough pipes. This new form is simple, no iterative calculations are necessary, and is well suited for accurate friction factor estimation. A limiting case of this equation provided friction factor estimates with a maximum absolute error of 0.029 and a maximum percentage error of 1% over a 20×500 grid of e∕D and R values ( 10−6 ⩽e∕D⩽5× 10−2 ; 4× 103 <R< 108 ). This was more accurate than the best currently available noniterative approximation of the Colebrook–White equation (maximum absolute error of 0.058; maximum percentage error of 1.42%). The superior accuracy, however, was obtained at the expense of a 30% increase in computational effort over the noniterative approximation. The novel equation presented in this study is theoretical and eliminates the need for best fit parameters or complicated initial guesses that are a characteristic of various empirical approxi...

Constraints for Using Lambert W Function-Based Explicit Colebrook–White Equation

We analyze the general applicability of a recent explicit expression of the Colebrook.White equation for turbulent flow friction factor calculation. This explicit expression, which is based on the Lambert \IW\N function, is characterized by an exponential term which imposes restrictions on its use. These constraints have been expressed in terms of pipe roughness (ϵ/\ID\N) and the Reynolds number R that are required for friction factor calculation. These constraints were determined as 8.0666 ln(R) + (ϵ/\ID\N)R<721.97 and 8.0666 ln(R) + (ϵ/\ID\N)R<5731.83, respectively, for machines using single precision and double precision computations. Using the Lambert W function, an explicit equation relating Rand ϵ/\ID\N was derived at the limiting case which allowed for a graphical representation of the applicability of the explicit form of the Colebrook.White equation in the Rversus ϵ/\ID\N space. Before computing friction factors using the explicit Colebrook-White equation, a quick check must be performed to see if the desired combination of Rand ϵ/\ID\N values satisfies the applicable constraint mentioned above.

Explicit Friction Factor Correlation for Turbulent Flow in Smooth Pipes

We present a truly explicit representation of the Nikuradse, Prandtl, von Kármán (NPK) friction factor equation for turbulent fluid flow in smooth pipes. Specifically, the Lambert W function is used to relate the friction factor explicitly to the Reynolds number. In addition to the elegance of an explicit expression, this new representation allows for the computation of friction factor values that are accurate to machine precision. As this is an exact explicit representation, it is far more accurate than current methods such as graphical estimation of friction factors from friction factor versus Reynolds number charts or several other explicit approximations of the NPK equation. We believe that this is the first truly explicit representation of the NPK equation and the elegance and high accuracy of this representation should make it the method of choice for estimating turbulent flow friction factors in smooth pipes.

Turbulent forced convection in microtubes

The anomalous and opposing trends in the published data dealing with turbulent flow friction factor and heat transfer coefficient in microchannels, and their apparent disagreement with macroscale correlations, are discussed. It is shown that the modification of turbulent eddy diffusivities, consistent with the way suspended particles may modify turbulence, can explain the observed higher-than-expected heat transfer coefficients in some data. It is thus suggested that suspended microscopic particles may be a major contributor to the aforementioned inconsistencies and disagreements in some of the published data.

A Generalized and Consistent Pressure Drop and Flow Regime Transition Model for Drilling Hydraulics

A Generalized and Consistent Pressure Drop and Flow Regime Transition Model for Drilling Hydraulics

Turbulent heat transfer and pressure drop in pinned annular regions

Experiments were performed to determine average heat transfer coefficients and friction factors for turbulent flow through annular ducts with pin fins. The measurements were carried out by means of a double-pipe heat exchanger. The total number of pins attached to the inner wall of the annular region was 560. The working fluids were air, flowing in the annular channel, and water through the inner circular tube. The average heat transfer coefficients of the pinned air-side were obtained from the experimental determination of the overall heat transfer coefficients of the heat exchanger and from the knowledge of the average heat transfer coefficients of the circular pipe (water-side), which could be found in the pertinent literature. To attain fully developed conditions, the heat exchanger was built with additional lengths before and after the test section. The inner circular duct of the heat exchanger and the pin fins were made of brass. Due to the high thermal conductivity of the brass, the small tube thickness and water temperature variation, the surface of the internal tube was practically isothermal. The external tube was made of an industrial plastic which was insulated from the environment by means of a glass wool batt. In this manner, the outer surface of the annular channel can be considered adiabatic. The results are presented in dimensionless forms, in terms of average Nusselt numbers and friction factors as functions of the flow Reynolds number, ranging from 13,000 to 80,000. The pin fin efficiency, which depends on the heat transfer coefficient, is also determined as a function of dimensionless parameters. A comparison of the present results with those for smooth sections (without pins) is also presented. The purpose of such a comparison is to study the influence of the presence of the pins on the pressure drop and heat transfer rate.

Turbulent heat transfer, pressure drop and fin efficiency in annular regions with continuous longitudinal rectangular fins

Abstract Experiments were performed to determine average heat transfer coefficients and friction factors for turbulent flow through annular ducts with continuous longitudinal rectangular fins. In addition, the fin efficiency was also determined by means of a numerical two-dimensional heat transfer analysis. The total number of fins attached to the inner wall of the annular region was 20. The measurements were made by using a double-pipe heat exchanger. The fluids were, air flowing in the annular section, and water through the inner tube. The average heat transfer coefficients were obtained from the experimental determination of the overall heat transfer coefficients of the heat exchanger. To attain fully developed conditions at the entrance and exit, the heat exchanger was built with additional lengths before and after the test section (30 hydraulic diameters). The double-pipe heat exchanger and fins were made of brass. Due to the high thermal conductivity of the brass and the small water temperature variation, the surface of the inner tube was practically isothermal. The external tube was well insulated and can be considered adiabatic. The results are presented in dimensionless forms, in terms of the average Nusselt number, friction factor and fin efficiency, as functions of the flow Reynolds number. A comparison of the present results with those for smooth sections (without fins) is also presented. The purpose of such comparison is to study the influence of the presence of the fins on the pressure drop and heat transfer rate.

Turbulent heat transfer and pressure drop in an internally finned equilateral triangular duct

Experiments were carried out to determine average heat transfer coefficients and friction factors for turbulent flow through equilateral triangular ducts with pin fins. The measurements were made by using a triangular duct heat exchanger. The fluids in the heat exchanger were air and water, and the average heat transfer coefficients were determined by measuring the overall heat transfer coefficients of the heat exchanger. To attain fully developed conditions at the entrance and exit, the heat exchanger was built with additional lengths before and after the test section (30 hydraulic diameters). The triangular ducts of the heat exchanger consisted of two metal walls (brass) and a wall of a less conductive material. The metal walls of the air side were finned with an array of short pin fins. Due to their high thermal conductivity and the small temperature variation of the water, these two walls were practically isothermal. The third wall of the duct was well insulated and can be considered adiabatic. The results are presented in dimensionless form, in terms of Nusselt numbers and friction factors as functions of the Reynolds number. The pin fin efficiency, which depends on the heat transfer coefficient, is also determined as a function of dimensionless parameters. It was found that the fins increase both the heat transfer and the pressure drop compared with an unfinned (smooth) duct. A better comparison between finned and smooth ducts was made for the same pumping power and heat transfer area. The results show that the Nusselt number for the finned duct, in the present work, is always greater than the one for the smooth duct.

Analytic Prediction of the Friction Factor for Turbulent Flow in Internally Finned Channels

This work develops an analytical model for the friction factor with turbulent flow in internally finned channels. Such channels are an important class of enhanced heat transfer surfaces. Until this work, no analytical models for the turbulent friction factor have been proposed. The present model assumes the validity of the Law of the Wall and applies the logarithmic velocity distribution to the interfin and core regions of the flow. Theoretically based friction factor equations are developed for internally finned circular tubes and rectangular channels. The model predicts Carnavos data for 21 internally finned tubes within ± 10 percent. Friction factor data were taken for five internally finned, rectangular channels. The analytical model predicts these data within ± 10 percent, except for the case of a very high fin.

Effect of Wetting on Friction Factors for Turbulent Flow of Mercury in Annuli

Friction factors were determined for fully developed turbulent flow of mercury in smooth concentric annuli under conditions where either both walls were unwetted, or both were wetted, or the inner wall was wetted and the outer one unwetted. Three radius ratios (r2/r1) were used, i.e., 2.09, 2.78, and 4.00. Unwetted walls gave the lowest friction factors, which were practically independent of the r2/r1 ratio over the limited range tested. The factors were 10 ± 1 percent higher than the commonly accepted values for smooth pipes (at the same Reynolds number). The highest friction factors were obtained with the inner wall wetted and the outer wall unwetted, and the greater the r2/r1 ratio the greater was the effect. For example, at r2/r1 = 4.00, the friction factors were 9.9% greater than for the situation when both walls were unwetted. The wetting conditions affected the location of the radius of maximum velocity (rm); and it was found that the nearer rm approached r2, the higher was the friction factor.

Simple method of predicting friction factors of turbulent flow in non-circular channels

On the basis of friction factor relationships of a simple model a method is developed which allows the prediction of friction factors for turbulent flow in non-circular channels if only the geometry factor of the pressure drop relationship for laminar flow is known. The proposed method of calculation is tested with numerous experimental results from the literature with respect to non-circular channels such as triangular shaped ducts, eccentric annuli and rod bundles in hexagonal and square arrays in circular tubes, hexagonal and square channels. It turns out that the proposed method provides an excellent description of all the experimental data at hand.

Experimental friction factors for turbulent flow with suction in a porous tube

Measurements of pressure gradient have been obtained with air flowing in a porous tube of circular cross-section with fully-developed turbulent profile at the entrance and with uniform mass extraction through the wall; an empirical correlation for the axial gradients with suction is presented. A slight but insignificant gradient in the radial direction was detected. The experiments covered an inlet Reynolds number range of 11000–101000 with a ratio of the transverse velocity at the wall to the mean axial velocity at inlet from zero to about 0.027. The radial distributions of both the temporal-mean axial velocity and the absolute turbulent velocity fluctuation were measured at the tube exit and the velocity profile parameter has been correlated with a suction parameter. The form of the temporal-mean velocity profile is found to depend critically upon the suction rate, becoming more flat at modest rates of suction but more peaked at high rates. The relative turbulence level is found to increase with suction at all radii, save for some reduction in the region of the wall at very low rates of suction. Local and average values of the effective friction factor with suction have been computed and are presented in graphical form. At a fixed inlet Reynolds number average values increased markedly with suction but decreased along the tube. For a given local Reynolds number local values displayed a similar trend.

Non‐newtonian flow through open channels

AbstractA general method is proposed for prediction of the flow rate and maximum velocity in the isothermal, steady, uniform, laminar flow of any incompressible, time‐independent non‐Newtonian fluid in straight open channels of arbitrary cross section. The method requires only a knowledge of two geometric coefficients and a function of shear stress, used to characterize the behavior of the fluid model. The slip effect at the solid boundary has been taken into consideration. Numerical values of the geometric parameters have been determined for flow through an inclined plane of infinite width, semi‐circular, semi‐elliptical, rectangular, and 90° and 60° symmetric triangular open channels. Applications have been made to various non‐Newtonian fluid models such as the power‐law, Bingham, Ellis, Meter and the Reiner‐Rivlin general model. Numerical examples are presented.A generalization of the Fanning friction factor — Reynolds number is also presented. The problem of determining the point of transition from laminar to turbulent flow in the general case is examined, as is the problem of prediction of the friction factor in turbulent flow.