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• Evolutionary optimization of Colebrook’s turbulent flow friction approximations

  D. Brkić, Ž. Ćojbašić

  Fluids (2017). 2 (2), article 15

  
  

  Abstract

  This paper presents evolutionary optimization of explicit approximations of the empirical Colebrook’s equation that is used for the calculation of the turbulent friction factor (λ), i.e., for the calculation of turbulent hydraulic resistance in hydraulically smooth and rough pipes including the transient zone between them. The empirical Colebrook’s equation relates the unknown flow friction factor (λ) with the known Reynolds number (R) and the known relative roughness of the inner pipe surface (ε/D). It is implicit in the unknown friction factor (λ). The implicit Colebrook’s equation cannot be rearranged to derive the friction factor (λ) directly, and therefore, it can be solved only iteratively [λ = f(λ, R, ε/D)] or using its explicit approximations [λ≈f(R, ε/D)], which introduce certain error compared with the iterative solution. The optimization of explicit approximations of Colebrook’s equation is performed with the aim to improve their accuracy, and the proposed optimization strategy is demonstrated on a large number of explicit approximations published up to date where numerical values of the parameters in various existing approximations are changed (optimized) using genetic algorithms to reduce maximal relative error. After that improvement, the computational burden stays unchanged while the accuracy of approximations increases in some of the cases very significantly.

  Abstract
  This paper presents evolutionary optimization of explicit approximations of the empirical Colebrook’s equation that is used for the calculation of the turbulent friction [...]

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• Solution of the implicit Colebrook equation for flow friction using Excel

  D. Brkić

  Spreadsheets in Education (eJSiE) (2017). 10 (2), article 4663

  
  

  Abstract

  Empirical Colebrook equation implicit in unknown flow friction factor (λ) is an accepted standard forcalculation of hydraulic resistance in hydraulically smooth and rough pipes. The Colebrook equation givesfriction factor (λ) implicitly as a function of the Reynolds number (Re) and relative roughness (ε/D) of innerpipe surface; i.e. λ0=f(λ0, Re, ε/D). The paper presents a problem that requires iterative methods for thesolution. In particular, the implicit method used for calculating the friction factor λ0is an application of fixed-point iterations. The type of problem discussed in this "in the classroom paper" is commonly encountered influid dynamics, and this paper provides readers with the tools necessary to solve similar problems. Students’task is to solve the equation using Excel where the procedure for that is explained in this “in the classroom”paper. Also, up to date numerous explicit approximations of the Colebrook equation are available where as an additional task for students can be evaluation of the error introduced by these explicit approximations λ≈f(Re,ε/D) compared with the iterative solution of implicit equation which can be treated as accurate.

  Abstract
  Empirical Colebrook equation implicit in unknown flow friction factor (λ) is an accepted standard forcalculation of hydraulic resistance in hydraulically smooth and [...]

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• Unified friction formulation from laminar to fully rough turbulent flow

  D. Brkić, P. Praks

  Applied Sciences (2018). 8 (11), article 2036

  
  

  Abstract

  This paper provides a new unified formula for Newtonian fluids valid for all pipe flow regimes from laminar to fully rough turbulent flow.This includes laminar flow; the unstable sharp jump from laminar to turbulent flow; and all types of turbulent regimes, including the smooth turbulent regime, the partial non-fully developed turbulent regime, and the fully developed rough turbulent regime. The new unified formula follows the inflectional form of curves suggested in Nikuradse’s experiment rather than the monotonic shape proposed by Colebrook and White. The composition of the proposed unified formula uses switching functions and interchangeable formulas for the laminar, smooth turbulent, and fully rough turbulent flow regimes. Thus, the formulation presented below represents a coherent hydraulic model suitable for engineering use. This new flow friction model is more flexible than existing literature models and provides smooth and computationally cheap transitions between hydraulic regimes.

  Abstract
  This paper provides a new unified formula for Newtonian fluids valid for all pipe flow regimes from laminar to fully rough turbulent flow.This includes laminar flow; the unstable [...]

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• Advanced iterative procedures for solving the implicit Colebrook equation for fluid flow friction

  P. Praks, D. Brkić

  Advances in Civil Engineering (2018). 2018, article 5451034

  
  

  Abstract

  The empirical Colebrook equation from 1939 is still accepted as an informal standard way to calculate the friction factor of turbulent flows (4000 < Re < 108) through pipes with roughness between negligible relative roughness (ε/D ⟶ 0) to very rough (up to ε/D = 0.05). The Colebrook equation includes the flow friction factor λ in an implicit logarithmic form, λ being a function of the Reynolds number Re and the relative roughness of inner pipe surface ε/D: λ = f(λ, Re, ε/D). To evaluate the error introduced by the many available explicit approximations to the Colebrook equation, λ ≈ f(Re, ε/D), it is necessary to determinate the value of the friction factor λ from the Colebrook equation as accurately as possible. The most accurate way to achieve that is by using some kind of the iterative method. The most used iterative approach is the simple fixed-point method, which requires up to 10 iterations to achieve a good level of accuracy. The simple fixed-point method does not require derivatives of the Colebrook function, while the most of the other presented methods in this paper do require. The methods based on the accelerated Householder’s approach (3rd order, 2nd order: Halley’s and Schröder’s method, and 1st order: Newton–Raphson) require few iterations less, while the three-point iterative methods require only 1 to 3 iterations to achieve the same level of accuracy. The paper also discusses strategies for finding the derivatives of the Colebrook function in symbolic form, for avoiding the use of the derivatives (secant method), and for choosing an optimal starting point for the iterative procedure. The Householder approach to the Colebrook’ equations expressed through the Lambert W-function is also analyzed. Finally, it is presented one approximation to the Colebrook equation with an error of no more than 0.0617%.

  Abstract
  The empirical Colebrook equation from 1939 is still accepted as an informal standard way to calculate the friction factor of turbulent flows (4000 < Re < 108) through [...]

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• One-log call iterative solution of the Colebrook equation for flow friction based on Padé polynomials

  P. Praks, D. Brkić

  Energies (2018). 11 (7), article 1825

  
  

  Abstract

  The 80 year-old empirical Colebrook function ξ, widely used as an informal standard for hydraulic resistance, relates implicitly the unknown flow friction factor λ, with the known Reynolds number R e and the known relative roughness of a pipe inner surface ε*; λ= ξ (R e, ε*, λ). It is based on logarithmic law in the form that captures the unknown flow friction factor λ in a way that it cannot be extracted analytically. As an alternative to the explicit approximations or to the iterative procedures that require at least a few evaluations of computationally expensive logarithmic function or non-integer powers, this paper offers an accurate and computationally cheap iterative algorithm based on Padé polynomials with only one l o g-call in total for the whole procedure (expensive l o g-calls are substituted with Padé polynomials in each iteration with the exception of the first). The proposed modification is computationally less demanding compared with the standard approaches of engineering practice, but does not influence the accuracy or the number of iterations required to reach the final balanced solution

  Abstract
  The 80 year-old empirical Colebrook function ξ, widely used as an informal standard for hydraulic resistance, relates implicitly the unknown flow friction factor λ, [...]

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• Choosing the optimal multi-point iterative method for the Colebrook flow friction equation

  P. Praks, D. Brkić

  Processes (2018). 6 (8), article 130

  
  

  Abstract

  The Colebrook equation is implicitly given in respect to the unknown flow friction factor λ; λ=ζ(Re, ε∗, λ)which cannot be expressed explicitly in exact way without simplifications and use of approximate calculus. A common approach to solve it is through the Newton–Raphson iterative procedure or through the fixed-point iterative procedure. Both require in some cases, up to seven iterations. On the other hand, numerous more powerful iterative methods such as threeor two-point methods, etc. are available. The purpose is to choose optimal iterative method in order to solve the implicit Colebrook equation for flow friction accurately using the least possible number of iterations. The methods are thoroughly tested and those which require the least possible number of iterations to reach the accurate solution are identified. The most powerful three-point methods require, in the worst case, only two iterations to reach the final solution. The recommended representatives are Sharma–Guha–Gupta, Sharma–Sharma, Sharma–Arora, Džuni´c–Petkovi´c–Petkovi´c; Bi–Ren–Wu, Chun–Neta based on Kung–Traub, Neta, and the Jain method based on the Steffensen scheme. The recommended iterative methods can reach the final accurate solution with the least possible number of iterations. The approach is hybrid between the iterative procedure and one-step explicit approximations and can be used in engineering design for initial rough, but also for final fine calculations.

  Abstract
  The Colebrook equation is implicitly given in respect to the unknown flow friction factor λ; λ=ζ(Re, ε∗, λ)which cannot be expressed [...]

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• Symbolic regression-based genetic approximations of the Colebrook equation for flow friction

  P. Praks, D. Brkić

  Water (2018). 10 (9), article 1175

  
  

  Abstract

  Widely used in hydraulics, the Colebrook equation for flow friction relates implicitly to the input parameters; the Reynolds number, Re and the relative roughness of an inner pipe surface, ε/D with an unknown output parameter; the flow friction factor, λ; λ = f (λ, Re, ε/D). In this paper, a few explicit approximations to the Colebrook equation; λ≈f (Re, ε/D), are generated using the ability of artificial intelligence to make inner patterns to connect input and output parameters in an explicit way not knowing their nature or the physical law that connects them, but only knowing raw numbers, {Re, ε/D}→{λ}. The fact that the used genetic programming tool does not know the structure of the Colebrook equation, which is based on computationally expensive logarithmic law, is used to obtain a better structure of the approximations, which is less demanding for calculation but also enough accurate. All generated approximations have low computational cost because they contain a limited number of logarithmic forms used for normalization of input parameters or for acceleration, but they are also sufficiently accurate. The relative error regarding the friction factor λ, in in the best case is up to 0.13% with only two logarithmic forms used. As the second logarithm can be accurately approximated by the Padé approximation, practically the same error is obtained also using only one logarithm.

  Abstract
  Widely used in hydraulics, the Colebrook equation for flow friction relates implicitly to the input parameters; the Reynolds number, Re and the relative roughness of an inner [...]

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• Accurate and efficient explicit approximations of the Colebrook flow friction equation based on the Wright ω-function: Reply to Discussion

  D. Brkić, P. Praks

  Mathematics (2019). 7 (5), article 410

  
  

  Abstract

  This reply gives two corrections of typographical errors in respect to the commented article, and then provides few comments in respect to the discussion and one improved version of the approximation of the Colebrook equation for flow friction, based on the Wright ω-function. Finally, this reply gives an exact explicit version of the Colebrook equation expressed through the Wright ω-function, which does not introduce any additional errors in respect to the original equation. All mentioned approximations are computationally efficient and also very accurate. Results are verified using more than 2 million of Quasi Monte-Carlo samples.

  Abstract
  This reply gives two corrections of typographical errors in respect to the commented article, and then provides few comments in respect to the discussion and one improved [...]

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• Accurate and efficient explicit approximations of the Colebrook flow friction equation based on the Wright ω-function

  D. Brkić, P. Praks

  Mathematics (2019). 7 (1), article 34

  
  

  Abstract

  The Colebrook equation is a popular model for estimating friction loss coefficients in water and gas pipes. The model is implicit in the unknown flow friction factor, f . To date, the captured flow friction factor, f , can be extracted from the logarithmic form analytically only in the term of the Lambert W-function. The purpose of this study is to find an accurate and computationally efficient solution based on the shifted Lambert W-function also known as the Wright ω-function. The Wright ω-function is more suitable because it overcomes the problem with the overflow error by switching the fast growing term, y=W(ex), of the Lambert W-function to series expansions that further can be easily evaluated in computers without causing overflow run-time errors. Although the Colebrook equation transformed through the Lambert W-function is identical to the original expression in terms of accuracy, a further evaluation of the Lambert W-function can be only approximate. Very accurate explicit approximations of the Colebrook equation that contain only one or two logarithms are shown. The final result is an accurate explicit approximation of the Colebrook equation with a relative error of no more than 0.0096%. The presented approximations are in a form suitable for everyday engineering use, and are both accurate and computationally efficient.

  Abstract
  The Colebrook equation is a popular model for estimating friction loss coefficients in water and gas pipes. The model is implicit in the unknown flow friction factor, f . [...]

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• Short overview of early developments of the Hardy Cross type methods for computation of flow distribution in pipe networks

  D. Brkić, P. Praks

  Applied Sciences (2019). 9 (10), article 2019

  
  

  Abstract

  Hardy Cross originally proposed a method for analysis of flow in networks of conduits or conductors in 1936. His method was the first really useful engineering method in the field of pipe network calculation. Only electrical analogs of hydraulic networks were used before the Hardy Cross method. A problem with flow resistance versus electrical resistance makes these electrical analog methods obsolete. The method by Hardy Cross is taught extensively at faculties, and it remains an important tool for the analysis of looped pipe systems. Engineers today mostly use a modified Hardy Cross method that considers the whole looped network of pipes simultaneously (use of these methods without computers is practically impossible). A method from a Russian practice published during the 1930s, which is similar to the Hardy Cross method, is described, too. Some notes from the work of Hardy Cross are also presented. Finally, an improved version of the Hardy Cross method, which significantly reduces the number of iterations, is presented and discussed. We also tested multi-point iterative methods, which can be used as a substitution for the Newton–Raphson approach used by Hardy Cross, but in this case this approach did not reduce the number of iterations. Although many new models have been developed since the time of Hardy Cross, the main purpose of this paper is to illustrate the very beginning of modeling of gas and water pipe networks and ventilation systems. As a novelty, a new multi-point iterative solver is introduced and compared with the standard Newton–Raphson iterative method.

  Abstract
  Hardy Cross originally proposed a method for analysis of flow in networks of conduits or conductors in 1936. His method was the first really useful engineering method in the [...]

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Articles (Documents not available in Scipedia)

• Discussion of “Friction modeling of flood flow simulations” by Vasilis Bellos, Ioannis Nalbantis, and George Tsakiris

  D Brkić, P Praks

  Journal of Hydraulic Engineering 146 (5), article 07020007, 2020

• Discussion of “Approximate analytical solutions for the Colebrook equation” by Ali R. Vatankhah

  D Brkić, P Praks

  Journal of Hydraulic Engineering 146 (2), article 07019011, 2020

• A possible publishing model profitable for authors

  D Brkić

  GRIN Verlag, 2019

• Correction: Brkić, D., and Praks, P. Accurate and efficient explicit approximations of the Colebrook flow friction equation based on the Wright ω-function. Mathematics 2019, 7, 34

  D Brkić, P Praks

  Mathematics 7 (10), article 951, 2019

• Discussion of “Economics and statistical evaluations of using Microsoft Excel Solver in pipe network analysis”

  D Brkić

  Journal of Pipeline Systems Engineering and Practice 9 (3), article 07018002, 2018

• Discussion of “Exact analytical solutions of the Colebrook-White equation”

  D Brkić

  Journal of Hydraulic Engineering ASCE 143 (9), article 07017007, 2017

• Discussion of “Gene expression programming analysis of implicit Colebrook–White equation in turbulent flow friction factor calculation”

  D Brkić

  Journal of Petroleum Science and Engineering 124, 399-401, 2014

• Discussion of “Method to cope with zero flows in Newton solvers for water distribution systems”

  D Brkić

  Journal of Hydraulic Engineering 140 (4), 456-459, 2014

• Very accurate explicit approximations for calculation of the Colebrook friction factor

  Ž Ćojbašić, D Brkić

  International Journal of Mechanical Sciences 67, 10-13, 2013

• Discussion of “Sustainable electric power system: Is it possible? Case study: Croatia”

  D Brkić

  Journal of Energy Engineering 138 (3), 163-164, 2012

• Discussion of “Jacobian matrix for solving water distribution system equations with the Darcy-Weisbach head-loss model”

  D Brkić

  Journal of Hydraulic Engineering 138 (11), 1000-1001, 2012

• Discussion of “Water distribution system analysis: Newton-Raphson method revisited”

  D Brkić

  Journal of Hydraulic Engineering 138 (9), 822-824, 2012

• Can pipes be actually really that smooth?

  D Brkić

  International Journal of Refrigeration 35 (1), 209-215, 2012

• Comparison of the Lambert W‐function based solutions to the Colebrook equation

  D Brkić

  Engineering Computations, 2012

• Determining friction factors in turbulent pipe flow

  D Brkić

  Chemical Engineering N.Y. 119 (3), 34-39, 2012

• Comments on "Experimental study of the pressure drop in the cathode side of air-forced open-cathode proton exchange membrane fuel cells"

  D Brkić

  International Journal of Hydrogen Energy 37 (14), 10963-10964, 2012

• Discussion of “Classroom activities to illustrate concepts of Darcy’s Law and hydraulic conductivity”

  D Brkić, V Mitrović

  Journal of Professional Issues in Engineering Education and Practice 137 (1 …, 2011

• Iterative methods for looped network pipeline calculation

  D Brkić

  Water Resources Management 25 (12), 2951-2987, 2011

• An explicit approximation of Colebrook's equation for fluid flow friction factor

  D Brkić

  Petroleum Science and Technology 29 (15), 1596-1602, 2011

• New explicit correlations for turbulent flow friction factor

  D Brkić

  Nuclear Engineering and Design 241 (9), 4055-4059, 2011

• Review of explicit approximations to the Colebrook relation for flow friction

  D Brkić

  Journal of Petroleum Science and Engineering 77 (1), 34-48, 2011

• Comments on “Settling velocities of particulate systems 15: Velocities in turbulent Newtonian flows”

  D Brkić

  International Journal of Mineral Processing 92 (3-4), 201-202, 2009

• Serbian gas sector in the spotlight of oil and gas agreement with Russia

  D Brkić

  Energy Policy 37 (5), 1925-1938, 2009

• An improvement of Hardy Cross method applied on looped spatial natural gas distribution networks

  D Brkić

  Applied Energy 86 (7-8), 1290-1300, 2009

• Serbian, Russian pipeline accord enhances European gas security

  D Brkić

  Oil & Gas Journal 106 (48), 52-54, 2008

• Systematic approach to natural gas usage for domestic heating in urban areas

  D Brkić, TI Tanasković

  Energy 33 (12), 1738-1753, 2008

• Prirodni gas kao gorivo za grejanje

  D Brkić

  Zadužbina Andrejević, 2006

• Određivanje graničnih parametara upotrebe prirodnog gasa u Beogradu

  D Brkić

  University of Belgrade, Faculty of Mining and Geological Engineering, 2005