@article{Brkić2014,
abstract = {Maximal relative error of the explicit approximation to the Colebrook equation for flow friction presented in the discussed paper by Saeed Samadianfard [J. Pet. Sci. Eng. 92-93 (2012), 48-55; doi. 10.1016/j.petrol.2012.06.005] is investigated. Samadianfard claims that his approximation is very accurate with the maximal relative error of no more than 0.08152%. Here is shown that this error is about 7%. Related comments about the paper are also enclosed. ; JRC.F.3-Energy Security, Systems and Market},
author = {Brkić, Dejan and Dejan, Brkic},
doi = {10.1016/j.petrol.2014.06.007},
journal = {Journal of Petroleum Science and Engineering},
month = {jun},
pages = {399-401},
title = {Discussion of “Gene expression programming analysis of implicit Colebrook–White equation in turbulent flow friction factor calculation” by Saeed Samadianfard [J. Pet. Sci. Eng. 92-93 (2012), 48-55]},
url = {https://zenodo.org/record/889996/files/article.pdf},
volume = {124},
year = {2014}
}
@article{Brkić2016,
abstract = {Nowadays, the Colebrook equation is used as a mostly accepted relation for the calculation of fluid flow friction factor. However, the Colebrook equation is implicit with respect to the friction factor (λ). In the present study, a noniterative approach using Artificial Neural Network (ANN) was developed to calculate the friction factor. To configure the ANN model, the input parameters of the Reynolds Number (Re) and the relative roughness of pipe (ε/D) were transformed to logarithmic scales. The 90,000 sets of data were fed to the ANN model involving three layers: input, hidden, and output layers with, 2, 50, and 1 neurons, respectively. This configuration was capable of predicting the values of friction factor in the Colebrook equation for any given values of the Reynolds number (Re) and the relative roughness (ε/D) ranging between 5000 and 108 and between 10−7 and 0.1, respectively. The proposed ANN demonstrates the relative error up to 0.07% which had the high accuracy compared with the vast majority of the precise explicit approximations of the Colebrook equation.},
author = {Brkić, Dejan and Ćojbašić, Žarko},
doi = {10.1155/2016/5242596},
month = {jan},
title = {Intelligent Flow Friction Estimation},
url = {https://zenodo.org/record/889687/files/article.pdf},
year = {2016}
}
@article{Brkić2016_2,
author = {Brkić, Dejan},
month = {jan},
title = {Study on the impacts of possible amendments to the ATEX, the Machinery, and the Pressure Equipment Directives with respect to equipment intended for use in the offshore oil and gas industry},
url = {https://zenodo.org/record/3274260/files/article.pdf},
year = {2016}
}
@article{Brkić2016_3,
author = {Brkić, Dejan},
doi = {10.1016/j.ijheatmasstransfer.2015.08.109},
journal = {International Journal of Heat and Mass Transfer},
month = {feb},
pages = {513-515},
title = {A note on explicit approximations to Colebrook’s friction factor in rough pipes under highly turbulent cases},
url = {https://doi.org/10.1016/j.ijheatmasstransfer.2015.08.109},
volume = {93},
year = {2016}
}
@article{Brkić2016_4,
abstract = {An example of hydraulic design project for teaching purpose is presented. Students’ task is to develop a looped distribution network for water (i.e. to determinate node consumptions, disposal of pipes, and finally to calculate flow rates in the network’s pipes and their optimal diameters). This can be accomplished by using the original Hardy Cross method, the improved Hardy Cross method, the node-loop method, etc. For the improved Hardy Cross method and the node-loop method, use of matrix calculation is mandatory. Because the analysis of water distribution networks is an essential component of civil engineering water resources curricula, the adequate technique better than the hand-oriented one is desired in order to increase students’ understanding of this kind of engineering systems and of relevant design issues in more concise and effective way. The described use of spreadsheet solvers is more than suitable for the purpose, especially knowing that spreadsheet solvers are much more matrix friendly compared with the hand-orientated calculation. Although matrix calculation is not mandatory for the original Hardy Cross method, even in that case it is preferred for better understanding of the problem. The application of commonly available spreadsheet software (Microsoft Excel) including two real classroom tasks is presented. ; JRC.C.3-Energy Security, Distribution and Markets},
author = {Brkić, Dejan and Dejan, Brkic},
month = {aug},
title = {Spreadsheet-Based Pipe Networks Analysis for Teaching and Learning Purpose},
url = {https://zenodo.org/record/889609/files/article.pdf},
year = {2016}
}
@article{Brkić2017,
author = {Brkić, Dejan and Ćojbašić, Žarko},
doi = {10.3390/fluids2020015},
journal = {Fluids},
month = {mar},
pages = {15},
title = {Evolutionary Optimization of Colebrook’s Turbulent Flow Friction Approximations},
url = {https://doi.org/10.3390/fluids2020015},
volume = {2},
year = {2017}
}
@article{Brkić2017_2,
author = {Brkić, Dejan},
month = {aug},
title = {Solution of the Implicit Colebrook Equation for Flow Friction Using Excel},
url = {https://zenodo.org/record/889581/files/article.pdf},
year = {2017}
}
@article{Brkić2017_3,
author = {Brkić, Dejan},
doi = {10.1061/(asce)hy.1943-7900.0001341},
journal = {Journal of Hydraulic Engineering},
month = {sep},
pages = {07017007},
title = {Discussion of “Exact Analytical Solutions of the Colebrook-White Equation” by Yozo Mikata and Walter S. Walczak},
url = {https://zenodo.org/record/889703/files/article.pdf},
volume = {143},
year = {2017}
}
@article{Brkić2018,
author = {Brkić, Dejan},
doi = {10.1061/(asce)ps.1949-1204.0000319},
journal = {Journal of Pipeline Systems Engineering and Practice},
month = {aug},
pages = {07018002},
title = {Discussion of “Economics and Statistical Evaluations of Using Microsoft Excel Solver in Pipe Network Analysis” by I. A. Oke, A. Ismail, S. Lukman, S. O. Ojo, O. O. Adeosun, and M. O. Nwude},
url = {https://zenodo.org/record/1310963/files/article.pdf},
volume = {9},
year = {2018}
}
@article{Brkić2018_2,
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.},
author = {Brkić, Dejan and Praks, Pavel},
doi = {10.3390/app8112036},
journal = {Applied Sciences},
month = {oct},
pages = {2036},
title = {Unified Friction Formulation from Laminar to Fully Rough Turbulent Flow},
url = {https://www.mdpi.com/2076-3417/8/11/2036/pdf},
volume = {8},
year = {2018}
}
@article{Brkić2018_3,
author = {Brkić, Dejan and Praks, Pavel},
doi = {10.3390/math7010034},
journal = {Mathematics},
month = {dec},
pages = {34},
title = {Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function},
url = {https://doi.org/10.3390/math7010034},
volume = {7},
year = {2018}
}
@article{Brkić2019,
abstract = {The original and improved versions of the Hardy Cross iterative method with related modifications are today widely used for the calculation of fluid flow through conduits in loop-like distribution networks of pipes with known node fluid consumptions. Fluid in these networks is usually natural gas for distribution in municipalities, water in waterworks or hot water in district heating systems, air in ventilation systems in buildings and mines, etc. Since the resistances in these networks depend on flow, the problem is not linear like in electrical circuits, and an iterative procedure must be used. In both versions of the Hardy Cross method, in the original and in the improved one, the initial result of calculations in the iteration procedure is not flow, but rather a correction of flow. Unfortunately, these corrections should be added to or subtracted from flow calculated in the previous iteration according to complicated algebraic rules. Unlike the Hardy Cross method, which requires complicated formulas for flow corrections, the new Node-loop method does not need these corrections, as flow is computed directly. This is the main advantage of the new Node-loop method, as the number of iterations is the same as in the modified Hardy Cross method. Consequently, a complex algebraic scheme for the sign of the flow correction is avoided, while the final results remain accurate.},
author = {Brkić, Dejan and Praks, Pavel},
doi = {10.3390/fluids4020073},
journal = {Fluids},
month = {apr},
pages = {73},
title = {An Efficient Iterative Method for Looped Pipe Network Hydraulics Free of Flow-Corrections},
url = {https://doi.org/10.3390/fluids4020073},
volume = {4},
year = {2019}
}
@article{Brkić2019_2,
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.},
author = {Brkić, Dejan and Praks, Pavel},
doi = {10.3390/math7050410},
journal = {Mathematics},
month = {may},
pages = {410},
title = {Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function: Reply to Discussion},
url = {https://doi.org/10.3390/math7050410},
volume = {7},
year = {2019}
}
@article{Brkić2019_3,
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.},
author = {Brkić, Dejan and Praks, Pavel},
doi = {10.3390/app9102019},
journal = {Applied Sciences},
month = {may},
pages = {2019},
title = {Short Overview of Early Developments of the Hardy Cross Type Methods for Computation of Flow Distribution in Pipe Networks},
url = {https://doi.org/10.3390/app9102019},
volume = {9},
year = {2019}
}
@article{Brkić2019_4,
abstract = {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.},
author = {Brkić, Dejan and Praks, Pavel},
doi = {10.3390/fluids4030114},
journal = {Fluids},
month = {jun},
pages = {114},
title = {What Can Students Learn While Solving Colebrook’s Flow Friction Equation?},
url = {https://doi.org/10.3390/fluids4030114},
volume = {4},
year = {2019}
}
@article{Brkić2019_5,
author = {Brkić, Dejan},
month = {sep},
title = {Colebrook’s Flow Friction Explicit Approximations Based on Fixed-Point Iterative Cycles and Symbolic Regression},
url = {https://zenodo.org/record/3384700/files/article.pdf},
year = {2019}
}
@article{Praks2018,
author = {Praks, Pavel and Brkić, Dejan},
doi = {10.3390/en11071825},
journal = {Energies},
month = {jul},
pages = {1825},
title = {One-Log Call Iterative Solution of the Colebrook Equation for Flow Friction Based on Padé Polynomials},
url = {https://doi.org/10.3390/en11071825},
volume = {11},
year = {2018}
}
@article{Praks2018_2,
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%.},
author = {Praks, Pavel and Brkić, Dejan},
doi = {10.1155/2018/5451034},
journal = {Advances in Civil Engineering},
month = {jul},
pages = {1-18},
title = {Advanced Iterative Procedures for Solving the Implicit Colebrook Equation for Fluid Flow Friction},
url = {https://doi.org/10.1155/2018/5451034},
volume = {2018},
year = {2018}
}
@article{Praks2018_3,
author = {Praks, Pavel and Brkić, Dejan},
doi = {10.3390/pr6080130},
journal = {Processes},
month = {aug},
pages = {130},
title = {Choosing the Optimal Multi-Point Iterative Method for the Colebrook Flow Friction Equation},
url = {https://doi.org/10.3390/pr6080130},
volume = {6},
year = {2018}
}
@article{Praks2018_4,
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.},
author = {Praks, Pavel and Brkić, Dejan},
doi = {10.3390/w10091175},
journal = {Water},
month = {sep},
pages = {1175},
title = {Symbolic Regression-Based Genetic Approximations of the Colebrook Equation for Flow Friction},
url = {https://www.mdpi.com/2073-4441/10/9/1175/pdf},
volume = {10},
year = {2018}
}