Dejan Brkić's personal collection : Scipedia

Symbolic Regression Approaches for the Direct Calculation of Pipe Diameter
D. Brkić, P. Praks, R. Praksová, T. Kozubek

Axioms (2023). Brkić, D.; Praks, P.; Praksová, R.; Kozubek, T. Symbolic Regression Approaches for the Direct Calculation of Pipe Diameter. Axioms 2023, 12, 850. https://doi.org/10.3390/axioms12090850

Abstract
This study provides novel and accurate symbolic regression-based solutions for the calculation of pipe diameter when flow rate and pressure drop (head loss) are known, together with the length of the pipe, absolute inner roughness of the pipe, and kinematic viscosity of the fluid. PySR and Eureqa, free and open-source symbolic regression tools, are used for discovering simple and accurate approximate formulas. Three approaches are used: (1) brute force of computing power, which provides results based on raw input data; (2) an improved method where input parameters are transformed through the Lambert W-function; (3) a method where the results are based on inputs and the Colebrook equation transformed through new suitable dimensionless groups. The discovered models were simplified by the WolframAlpha simplify tool and/or the equivalent Matlab Symbolic toolbox. Novel models make iterative calculus redundant; they are simple for computer coding while the relative error remains lower compared with the solution through nomograms. The symbolic-regression solutions discovered by brute force computing power discard the kinematic viscosity of the fluid as an input parameter, implying that it has the least influence.
Abstract
This study provides novel and accurate symbolic regression-based solutions for the calculation of pipe diameter when flow rate and pressure drop (head loss) are known, together [...]
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Serbian Natural Gas Sector in a Gap Between East and West: ChatGPT Decision Support
D. Brkić

Dejan Brkić's personal collection (2023). 2

Abstract
This article should examine ideas of how a chatbot can support decision making processes by collecting general opinion about the certain topics. The ChatGPT is tested in this article in respect of natural gas security of supply of the Republic of Serbia in the context of the current political situation in the world. The author first describes its own stance on the topic and then the ChatGPT is asked for its opinion (which most probably collects and compiles phrases from English speaking sources to build a text in a smart way). The topic is ort only about security of gas supply of Serbia but also political and highly sensitive, so it is examined if the ChatGPT can be useful tool to reach a balance between less on more favorable options in the process of policy making.
Abstract
This article should examine ideas of how a chatbot can support decision making processes by collecting general opinion about the certain topics. The ChatGPT is tested in this [...]
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Discussion of “Explicit Solution for Pipe Diameter Problem Using Lambert W-Function”
D. Brkić, P. Praks

Journal of Irrigation and Drainage Engineering (2023). Brkić, D. and Praks, P., 2023. Discussion of “Explicit Solution for Pipe Diameter Problem Using Lambert W-Function”. Journal of Irrigation and Drainage Engineering, 149(7), p.07023016. https://doi.org/10.1061/JIDEDH.IRENG-10071
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Selection of Appropriate Symbolic Regression Models Using Statistical and Dynamic System Criteria: Example of Waste Gasification
P. Praks, D. Brkić

Axioms (2022). Praks, P.; Lampart, M.; Praksová, R.; Brkić, D.; Kozubek, T.; Najser, J. Selection of Appropriate Symbolic Regression Models Using Statistical and Dynamic System Criteria: Example of Waste Gasification. Axioms 2022, 11, 463.

Abstract
In this paper, we analyze the interpretable models from real gasification datasets of the project “Centre for Energy and Environmental Technologies” (CEET) discovered by symbolic regression. To evaluate CEET models based on input data, two different statistical metrics to quantify their accuracy are usually used: Mean Square Error (MSE) and the Pearson Correlation Coefficient (PCC). However, if the testing points and the points used to construct the models are not chosen randomly from the continuum of the input variable, but instead from the limited number of discrete input points, the behavior of the model between such points very possibly will not fit well the physical essence of the modelled phenomenon. For example, the developed model can have unexpected oscillatory tendencies between the used points, while the usually used statistical metrics cannot detect these anomalies. However, using dynamic system criteria in addition to statistical metrics, such suspicious models that do fit well-expected behavior can be automatically detected and abandoned. This communication will show the universal method based on dynamic system criteria which can detect suitable models among all those which have good properties following statistical metrics. The dynamic system criteria measure the complexity of the candidate models using approximate and sample entropy. The examples are given for waste gasification where the output data (percentage of each particular gas in the produced mixture) is given only for six values of the input data (temperature in the chamber in which the process takes place). In such cases instead, to produce expected simple spline-like curves, artificial intelligence tools can produce inappropriate oscillatory curves with sharp picks due to the known tendency of symbolic regression to produce overfitted and relatively more complex models if the nature of the physical model is simple.
Abstract
In this paper, we analyze the interpretable models from real gasification datasets of the project “Centre for Energy and Environmental Technologies” (CEET) discovered [...]
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Reliability-Based Criterion for Evaluating Explicit Approximations of Colebrook Equation
S. Easa, A. Lamri, D. Brkić

Journal of Marine Science and Engineering (2022). Easa SM, Lamri AA, Brkić D. Reliability-Based Criterion for Evaluating Explicit Approximations of Colebrook Equation. Journal of Marine Science and Engineering. 2022; 10(6):803. https://doi.org/10.3390/jmse10060803

Abstract
Numerous explicit approximations of the Colebrook equation have been developed and evaluated based on two criteria: prediction accuracy and computational efficiency. This paper introduces a new evaluation criterion based on the reliability of each equation. The reliability is defined by the coefficient of variation (CV) of the explicit friction factor that is a function of the variabilities of component random variables (roughness height of the internal pipe surface and kinematic viscosity of the fluid). The coefficient of variation of the friction factor depends on its first derivative for roughness height of the inner pipe surface and kinematic viscosity of the fluid and their correlation. Seven explicit approximations were evaluated using the new reliability-based criterion. The results show that all explicit approximations are very reliable, but variations exist regarding the reliability level. The reliabilities of the seven approximations is very close for the rough-flow regime and when the CV of the viscosity is minimal. However, for the smooth-flow regime, and when the CV of the roughness is minimal, various approximations showed substantially different reliabilities. The novelty of the proposed criterion is that it addresses an evaluation dimension that complements the accuracy and efficiency criteria.
Abstract
Numerous explicit approximations of the Colebrook equation have been developed and evaluated based on two criteria: prediction accuracy and computational efficiency. This [...]
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Hydraulic Losses in Systems of Conduits with Flow from Laminar to Fully Turbulent: A New Symbolic Regression Formulation
M. Milošević, D. Brkić, P. Praks, D. Litričin, Z. Stajić

Axioms (2022). Milošević, Marko, Dejan Brkić, Pavel Praks, Dragan Litričin, and Zoran Stajić. 2022. "Hydraulic Losses in Systems of Conduits with Flow from Laminar to Fully Turbulent: A New Symbolic Regression Formulation" Axioms 11, no. 5: 198. https://doi.org/10.3390/axioms11050198

Abstract
Separate flow friction formulations for laminar and turbulent regimes of flow through pipes are in common use in engineering practice. However, variation of different parameters in a system of conduits during conveying of fluids can cause changes in flow pattern from laminar to fully turbulent and vice versa. Because of that, it is useful to unify formulations for laminar and turbulent hydraulic regimes in one single coherent equation. In addition to a physical interpretation of hydraulic friction, this communication gives a short overview of already available Darcy’s flow friction formulations for both laminar and turbulent flow and additionally includes two simple completely new approximations based on symbolic regression.
Abstract
Separate flow friction formulations for laminar and turbulent regimes of flow through pipes are in common use in engineering practice. However, variation of different parameters [...]
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Steady-State Analysis of Electrical Networks in Pandapower Software: Computational Performances of Newton–Raphson, Newton–Raphson with Iwamoto Multiplier, and Gauss– Seidel Methods
J. Vysocký, L. Foltyn, D. Brkić, R. Praksová, P. Praks

Sustainability (2022). Vysocký, J.; Foltyn, L.; Brkić, D.; Praksová, R.; Praks, P. Steady-State Analysis of Electrical Networks in Pandapower Software: Computational Performances of Newton–Raphson, Newton–Raphson with Iwamoto Multiplier, and Gauss–Seidel Methods. Sustainability 2022, 14, 2002. https://doi.org/10.3390/su14042002

Abstract
At the core of every system for the efficient control of the network steady-state operation is the AC-power-flow problem solver. For local distribution networks to continue to operate effectively, it is necessary to use the most powerful and numerically stable AC-power-flow problem solvers within the software that controls the power flows in these networks. This communication presents the results of analyses of the computational performance and stability of three methods for solving the AC-power-flow problem. Specifically, this communication compares the robustness and speed of execution of the Gauss–Seidel (G–S), Newton–Raphson (N–R), and Newton–Raphson method with Iwamoto multipliers (N–R–I), which were tested in open-source pandapower software using a meshed electrical network model of various topologies. The test results show that the pandapower implementations of the N–R method and the N–R–I method are significantly more robust and faster than the G–S method, regardless of the network topology. In addition, a generalized Python interface between the pandapower and the SciPy package was implemented and tested, and results show that the hybrid Powell, Levenberg–Marquardt, and Krylov methods, a quasilinearization algorithm, and the continuous Newton method can sometimes achieve better results than the classical N–R method.
Abstract
At the core of every system for the efficient control of the network steady-state operation is the AC-power-flow problem solver. For local distribution networks to continue [...]
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Approximate flow friction factor: Estimation of the accuracy using Sobol’s Quasi-Random sampling
P. Praks, D. Brkić

Axioms (2022). Praks, P.; Brkić, D. Approximate Flow Friction Factor: Estimation of the Accuracy Using Sobol’s Quasi-Random Sampling. Axioms 2022, 11, 36. https://doi.org/10.3390/axioms11020036

Abstract
The unknown friction factor from the implicit Colebrook equation cannot be expressed explicitly in an analytical way, and therefore to simplify the calculation, many explicit approximations can be used instead. The accuracy of such approximations should be evaluated only throughout the domain of interest in engineering practice where the number of test points can be chosen in many different ways, using uniform, quasi-uniform, random, and quasi-random patterns. To avoid picking points with undetected errors, a sufficient minimal number of such points should be chosen, and they should be distributed using proper patterns. A properly chosen pattern can minimize the required number of testing points that are sufficient to detect maximums of the error. The ability of the Sobol quasi-random vs. random distribution of testing points to capture the maximal relative error using a sufficiently small number of samples is evaluated. Sobol testing points that are quasi-randomly distributed can cover the domain of interest more evenly, avoiding large gaps. Sobol sequences are quasi-random and are always the same, which allows the exact repetition of scientific results.
Abstract
The unknown friction factor from the implicit Colebrook equation cannot be expressed explicitly in an analytical way, and therefore to simplify the calculation, many explicit [...]
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Probability Analysis and Prevention of Offshore Oil and Gas Accidents: Fire as a Cause and a Consequence
D. Brkić, P. Praks

Fire (2021). Brkić, Dejan, and Pavel Praks. 2021. "Probability Analysis and Prevention of Offshore Oil and Gas Accidents: Fire as a Cause and a Consequence" Fire 4, no. 4: 71. https://doi.org/10.3390/fire4040071

Abstract
ailures during the drilling and exploitation of hydrocarbons that result in catastrophic offshore oil and gas accidents are relatively rare but if they occur the consequences can be catastrophic in terms of loss of life and environmental damage. Therefore, to gain insight into their prevention, the largest major offshore oil and gas accidents, those with more than 10 fatalities or with a large environmental impact, are analyzed in this article. Special attention is placed on fire as a cause and a consequence. Relevant technological and legislative changes and updates regarding safety that have followed such accidents and that can prevent potential future similar misfortunes are evaluated. Two main approaches to safety are compared: (1) the American prescriptive vs. (2) the European goal-oriented approach. The main causes of accidents are tested statistically in respect of failure probability, where the exact confidence limits for the estimated probabilities are computed. The results of the statistical test based on exact confidence intervals show that there is no significant difference between the analysed factors, which describe the main causes of offshore oil and gas accidents. Based on the small but carefully chosen group of 24 of the largest accidents, it can be concluded that there is no evidence of a difference between the categories of the main causes of accidents.
Abstract
ailures during the drilling and exploitation of hydrocarbons that result in catastrophic offshore oil and gas accidents are relatively rare but if they occur the consequences [...]
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Excel VBA-based user defined functions for highly precise Colebrook’s pipe flow friction approximations: A comparative overview
D. Brkić, Z. Stajić

Facta Universitatis, series: Mechanical Engineering (FU Mech Eng) (2021). Brkić, D. and Stajić, Z., 2021. Excel VBA-Based User Defined Functions for Highly Precise Colebrook’s Pipe Flow Friction Approximations: A Comparative Overview. Facta Universitatis, Series: Mechanical Engineering 9(2):7267. Available from: http://casopisi.junis.ni.ac.rs/index.php/FUMechEng/article/view/7267

Abstract
This review paper gives Excel functions for highly precise Colebrook’s pipe flow friction approximations developed by users. All shown codes are implemented as User Defined Functions – UDFs written in Visual Basic for Applications – VBA, a common programming language for MS Excel spreadsheet solver. Accuracy of the friction factor computed using nine to date the most accurate explicit approximations is compared with the sufficiently accurate solution obtained through an iterative scheme which gives satisfying results after sufficient number of iterations. The codes are given for the presented approximations, for the used iterative scheme and for the Colebrook equation expressed through the Lambert W-function (including its cognate Wright ωfunction). The developed code for the principal branch of the Lambert W-function has additional and more general application for solving different problems from variety branches of engineering and physics. The approach from this review paper automates computational processes and speeds up manual tasks.
Abstract
This review paper gives Excel functions for highly precise Colebrook’s pipe flow friction approximations developed by users. All shown codes are implemented as User [...]
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Offshore Oil and Gas Safety: Protection against Explosions
D. Brkić, Z. Stajić

Journal of Marine Science and Engineering (2021). 9(3), article 331.

Abstract
Offshore oil and gas operations carry a high risk of explosions, which can be efficiently prevented in many cases. The two most used approaches for prevention are: (1) the “International Electrotechnical Commission System for Certification to Standards Relating to Equipment for Use in Explosive Atmospheres” (IECEx) and (2) European “Atmosphere Explosible” (ATEX) schemes. The main shortcoming for the IECEx scheme is in the fact that it does not cover nonelectrical equipment, while for the ATEX scheme, it is due to the allowed self-certification for a certain category of equipment in areas with a low probability of explosions, as well as the fact that it explicitly excludes mobile offshore drilling units from its scope. An advantage of the IECEx scheme is that it is prescribed by the US Coast Guard for protection against explosions on foreign mobile offshore drilling units, which intend to work on the US continental shelf but have never operated there before, with an additional requirement that the certificates should be obtained through a US-based Certified Body (ExCB). Therefore, to avoid bureaucratic obstacles and to be allowed to operate with minimized additional costs both in the US and the EU/EEA’s offshore jurisdictions (and very possibly worldwide), all mobile offshore drilling units should be certified preferably as required by the US Coast Guard.
Abstract
Offshore oil and gas operations carry a high risk of explosions, which can be efficiently prevented in many cases. The two most used approaches for prevention are: (1) the [...]
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Simplified Indirect Estimation of Pump Flow Discharge: An Example from Serbia
M. Amon, M. Radić, Z. Stajić, D. Brkić, P. Praks

Water (2021). 13(6), article 796.

Abstract
In the absence of a flowmeter or due to its inaccuracy, the flow rate at the discharge section of the pipeline following the observed pump can be roughly estimated if the pressure can be measured instead. To use the proposed procedure two main conditions should be achieved: (1) a manometer should be installed at the discharge pipeline between the pump and the flow regulation valve, and (2) the actual curve that relates pressure and flow for the observed pump unit should be known in advance. The described example is from Serbia, but it is of interest for any water pumping station with a submersible pump (installed in wells or tanks) where a limited number of adequate places for the measuring of flow are available (if any are available at all), but where the pressure at the discharge pipe of the observed pump can be measured. This simplified method can find applicability in installations in remote rural regions where limited resources are available. The results show that the calculated values of the flow obtained by the presented method deviate greatly in relation to the measured values provided by the portable ultrasonic flowmeter, up to 60% at one of the measuring points. However, in relation to the measured values provided by the permanently installed flowmeter the discrepancy is significantly lower (0.6–6.8%).
Abstract
In the absence of a flowmeter or due to its inaccuracy, the flow rate at the discharge section of the pipeline following the observed pump can be roughly estimated if the [...]
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Suitability for coding of the Colebrook’s flow friction relation expressed by symbolic regression approximations of the Wright-ω function
P. Praks, D. Brkić

Reports in Mechanical Engineering (2020). Vol. 1, No. 1, 2020, pp. 174-179. ISSN: 2683-5894

Abstract
This article analyses a form of the empirical Colebrook’s pipe flow friction equation given originally by the Lambert W-function and recently also by the Wright ω-function. These special functions are used to explicitly express the unknown flow friction factor of the Colebrook equation, which is in its classical formulation given implicitly. Explicit approximations of the Colebrook equation based on approximations of the Wright ω-function given by an asymptotic expansion and symbolic regression were analyzed in respect of speed and accuracy. Numerical experiments on 8 million Sobol’s quasi-Monte points clearly show that also both approaches lead to the approximately same complexity in terms of speed of execution in computers. However, the relative error of the developed symbolic regression-based approximations is reduced significantly, in comparison with the classical basic asymptotic expansion. These numerical results indicate promising results of artificial intelligence (symbolic regression) for developing fast and accurate explicit approximations.
Abstract
This article analyses a form of the empirical Colebrook’s pipe flow friction equation given originally by the Lambert W-function and recently also by the Wright ω-function. [...]
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List of Equations in https://doi.org/10.23967/j.rimni.2020.09.001
P. Praks, D. Brkić

Dejan Brkić's personal collection (2020). 59
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Proper Use of Technical Standards in Offshore Petroleum Industry
D. Brkić, P. Praks

Journal of Marine Science and Engineering (2020). 8(8), article 555

Abstract
Ships for drilling need to operate in the territorial waters of many different countries which can have different technical standards and procedures. For example, the European Union and European Economic Area EU/EEA product safety directives exclude from their scope drilling ships and related equipment onboard. On the other hand, the EU/EEA offshore safety directive requires the application of all the best technical standards that are used worldwide in the oil and gas industry. Consequently, it is not easy to select the most appropriate technical standards that increase the overall level of safety and environmental protection whilst avoiding the costs of additional certifications. We will show how some technical standards and procedures, which are recognized worldwide by the petroleum industry, can be accepted by various standardization bodies, and how they can fulfil the essential health and safety requirements of certain directives. Emphasis will be placed on the prevention of fire and explosion, on the safe use of equipment under pressure, and on the protection of personnel who work with machinery. Additionally considered is how the proper use of adequate procedures available at the time would have prevented three large scale offshore petroleum accidents: the Macondo Deepwater Horizon in the Gulf of Mexico in 2010; the Montara in the Timor Sea in 2009; the Piper Alpha in the North Sea in 1988.
Abstract
Ships for drilling need to operate in the territorial waters of many different countries which can have different technical standards and procedures. For example, the European [...]
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Air-Forced Flow in Proton Exchange Membrane Fuel Cells: Calculation of Fan-Induced Friction in Open-Cathode Conduits with Virtual Roughness
D. Brkić, P. Praks

Processes (2020). 8(6), article 686

Abstract
Measurements of pressure drop during experiments with fan-induced air flow in the open-cathode proton exchange membrane fuel cells (PEMFCs) show that flow friction in its opencathode side follows logarithmic law similar to Colebrook’s model for flow through pipes. The stable symbolic regression model for both laminar and turbulent flow presented in this article correlates air flow and pressure drop as a function of the variable flow friction factor which further depends on the Reynolds number and the virtual roughness. To follow the measured data, virtual inner roughness related to the mesh of conduits of fuel cell used in the mentioned experiment is 0.03086, whereas for pipes, real physical roughness of their inner pipe surface goes practically from 0 to 0.05. Numerical experiments indicate that the novel approximation of the Wright-ω function reduced the computational time from half of a minute to fragments of a second. The relative error of the estimated friction flow factor is less than 0.5%.
Abstract
Measurements of pressure drop during experiments with fan-induced air flow in the open-cathode proton exchange membrane fuel cells (PEMFCs) show that flow friction in its [...]
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Вештачка неуронска мрежа као алат за процену хидрауличког отпора
Ž. Ćojbašić, D. Brkić

Tehnika (2011). 66 (6), 947-952,
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Прилог пројектовања гасних дистрибуционих мрежа
D. Brkić

Istraživanja i projektovanja za privredu (2008). 6 (22), 7-18,
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Варијанте индиректне и директне употребе природног гаса за грејање станова
D. Brkić

Tehnička dijagnostika (2007). 6 (3), 41-48,
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Sedbuk: Британска методологија за одређивање сезонске ефикасности гасних котлова
T. Tanasković, D. Brkić

Gas (2007). 12 (1), 17-22,
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Примена Ламберт-W функције за прорачун отпора трења у цевима
D. Brkić

Vodoprivreda (2009). 41 (1-3), 69-78,
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Пример прорачуна гасоводне дистрибутивне мреже прстенастог типа обједињеном методом чворова и прстенова
D. Brkić, T. Tanasković

Gas (2010). 15 (4), 17-22,
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Цене природног гаса у Србији и у свету
D. Brkić

7:419-425 - MAREN 2006, Beograd, Srbija
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Природни гас као еколошки најприхватљивије фосилно гориво
D. Brkić, T. Tanasković

7:225-230 - MAREN 2006, Beograd, Srbija
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Котлови кондензационе технологије
T. Tanasković, D. Ivezić, M. Živković, D. Brkić

MAREN 2006, Beograd, Srbija
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Провера модела условних површина грејања на гас у насељима
D. Brkić

Tehnička dijagnostika (2007). 6 (4), 39-44,

Abstract
In this paper is shown short particular review of the book by the same author “Natural Gas as Heating Fuel”. First criterions for decision of appending on distribution gas network or district heating system for particular part of town are determinate on conditional urban area base. These conclusions are verified on specific settlement in Belgrade and in this paper are shown results on Karaburma settlement.
Abstract
In this paper is shown short particular review of the book by the same author “Natural Gas as Heating Fuel”. First criterions for decision of appending on distribution [...]
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Пројектовање посебне класе гасних дистрибутивних мрежа
D. Brkić

Istraživanja i projektovanja za privredu (2005). 3, 49-56,
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Оптимизација пречника цеви у прстенастој цевоводној мрежи модификованом методом петљи
T. Tanasković, D. Brkić

XXXVI Sym-Op-Is 2009, Ivanjica, Serbia

Abstract
After the calculation of flows in a looped pipeline distribution network is done, very often velocity of fluid in some of the pipeline branches are undersized while in other are oversized. Hence, it is necessary to perform optimization of the pipes’ diameter to fit in range of suggested values. Suggested velocities for water flow in waterworks systems are 0.75-1.5 m/s, while in the case of natural gas flow in gas-pipelines this velocity limit is increased up to20 m/s. Problem of velocity in a gas pipeline is more complex in comparison to water velocity. This issue is caused by compressibility of gas. In this paper is shown procedure of diameters’ optimization after the modified Hardy Cross method. Difference in comparison to classical problem in which flow distribution is unknown is that on the contrary, in optimization problem flows are now constants while the diameters are treated as variables.
Abstract
After the calculation of flows in a looped pipeline distribution network is done, very often velocity of fluid in some of the pipeline branches are undersized while in other [...]
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Прорачун расподеле протока у прстенастој цевоводној мрежи обједињеном методом чворова и прстенова
D. Brkić, T. Tanasković

XXXVI Sym-Op-Is 2009, Ivanjica, Serbia

Abstract
Calculation of gas i.e. water distribution network is much more complex compared with branch-like distribution pipelines. In this paper is shown one more suitable method in comparison to modified Hardy Cross. Here shown method as well as modified Hardy Cross method are iterative methods. For balancing of network, approximately equal number of iteration are to be calculated after the here presented node-loop method as for the improved Hardy Cross. Main advantage of the node-loop method in comparison to improved Hardy Cross method is that the final result in each calculation is flow which is used directly as input parameter in the next iteration cycle. Disadvantage of the improved Hardy Cross method is that this flows are calculated indirectly using flow corrections which they need to be added after complicated algebraic scheme to the flows calculated in previous iteration.
Abstract
Calculation of gas i.e. water distribution network is much more complex compared with branch-like distribution pipelines. In this paper is shown one more suitable method in [...]
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Класификација кућних гасних котлова на основу ефикасности
T. Tanasković, D. Brkić

Termotehnika (2008). 34 (1), 83-92

Abstract
Methodology for calculation of Seasonal Efficiency of Comestic Gas Boilers is set by the UK Government's Standard Assessment Procedure for Energy Rating of Dwellings. This methodology is general applicable, and can be applied in our country. This methodology is compatible with EU Council Directive 92/42/EEC on efficiency requirements for new hot-water boilers fired with liquid or gaseous fuels. Thus calculated Seasonal Efficiency is relevant parameter for comparisons of different models of boilers. Energy Efficiency Bands are as signed, in a first place based on verified values of seasonal efficiency. In correlation with seasonal efficiency stands annual fuel consumption of certain boiler, and consequently annual fuel consumption of certain boiler, and consequently annual fuel costs are comparable.
Abstract
Methodology for calculation of Seasonal Efficiency of Comestic Gas Boilers is set by the UK Government's Standard Assessment Procedure for Energy Rating of Dwellings. [...]
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Анализа карактеристика гасификације и топлификације на моделу условне грађевинске површине
T. Tanasković, N. Đajić, M. Tanasijević, D. Brkić

69-72, XXXI Sym-Op-Is 2004, Iriški Venac, Srbija
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Критеријуми за прекид итеративног поступка при прорачуну гасне дистрбутивне мреже са прстеновима
D. Brkić

Tehnička dijagnostika (2005). 4 (3-4), 71-75
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Дијагностиковање проблема насталих при прорачуну прстенасте гасне дистрибутивне мреже ниског притиска
D. Brkić

Tehnička dijagnostika (2005). 4 (2), 11-16
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Оптимизација методе контура прилагођене за прорачун гасних дистрибутивних мрежа
D. Brkić, T. Tanasković

XXXV Sym-Op-Is 2008, Soko Banja, Srbija

Abstract
Gas distributive network in settlements are usually with contours, which assure that gas is available for consumers from few different directions. By application of concept of looped network least possible number of consumers can remain without gas in the case of malfunction in some branch. But, calculation of these types of networks is much more complicated compared with branch type of network. Here is shown example of one possible way of improvement of iterative procedure during the calculation of network according to Hardy Cross method.
Abstract
Gas distributive network in settlements are usually with contours, which assure that gas is available for consumers from few different directions. By application of concept [...]
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Сезонска ефикасност кућних гасних котлова
T. Tanasković, D. Brkić

171-174, XXXIV Sym-Op-Is 2007, Zlatibor, Srbija

Abstract
Methodology for calculating Seasonal Efficiency of Domestic Gas Boilers is set by The UK Government’s Standard Assessment Procedure for Energy Rating of Dwellings. This methodology is compatible with EU Council Directive 92/42/EEC on efficiency requirements for new hot-water boilers fired with liquid or gaseous fuels. Thus calculated Seasonal Efficiency is relevant parameter for comparisons of different models of boilers.
Abstract
Methodology for calculating Seasonal Efficiency of Domestic Gas Boilers is set by The UK Government’s Standard Assessment Procedure for Energy Rating of Dwellings. This [...]
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Поређење гасних котлова у домаћинствима на основу сезонске ефикасности
D. Brkić, T. Tanasković

167-170, XXXIV Sym-Op-Is 2007, Zlatibor, Srbija

Abstract
This paper presents a review of the boilers according to performed seasonal efficiency. Energy Efficiency Bands are assigned, primarily based on the verified values of seasonal efficiency. In correlation with seasonal efficiency stands annual fuel consumption of a certain boiler, and consequently annual fuel costs are comparable.
Abstract
This paper presents a review of the boilers according to performed seasonal efficiency. Energy Efficiency Bands are assigned, primarily based on the verified values of seasonal [...]
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Утицајни фактори на формирање цене природног гаса
D. Brkić

181-184, XXXIII Sym-Op-Is 2006 Banja Koviljača, Srbija
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Ефикасност кондензационе технологије
T. Tanasković, D. Ivezić, M. Živković, D. Brkić

165-168, XXXIII Sym-Op-Is 2006 Banja Koviljača, Srbija
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Energy situation in the Republic of Serbia
D. Brkić

4th Sustainable Energy and Environmental Protection SEEP, Bari, Italy, June 29–July 2 2010

Abstract
Paper presents the energy policy of the Republic of Serbia with special attention to the energy situation on the government controlled territory. South Serbian autonomous province of Kosovo and Metohija is under UN jurisdiction since the 1999 according to UNSC Resolution 1244. Renewable energy sources are rarely used in Serbia with exception of energy from hydropower plants, but in this sector priorities in geothermal and energy coming from biomass recently increased. In natural gas sector, Serbia has the deal with Russia for construction of South Stream gas-line through Serbia and for construction of the first underground storage in depleted gas reservoir in Banatski Dvor. In 2008, Serbia also sold 51% of the government founded petroleum industry – NIS which has exclusive monopoly for exploitation of crude oil. Serbian government has complete monopoly in electric power sector. Electric power infrastructure became technologically obsolete, and operative efficiency is at very low level. Serbia has not yet decided whether Serbian Electric Power Industry – EPS will be privatized. District heating sector mostly natural gas fuelled is highly inefficient and it is in jurisdiction of local municipalities but also has social component dictated by central government.
Abstract
Paper presents the energy policy of the Republic of Serbia with special attention to the energy situation on the government controlled territory. South Serbian autonomous [...]
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Similarities and differences in optimization of water-and gas-distribution pipeline networks
D. Brkić

5th Dubrovnik Conference on Sustainable Development of Energy, Water and Environment Systems- SDEWES, Dubrovnik, Croatia, September 29-October 4 2009
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Two efficient methods for gas distributive network calculation
D. Brkić

24th International Conference on Efficiency, Cost, Optimization, Simulation and Environmental Impact of Energy Systems ECOS, Lausanne, Switzerland, June 13–17 2010

Abstract
Today, two very efficient methods for calculation of flow distribution per branches of a looped gas pipeline are available. Most common is improved Hardy Cross method, while the second one is so-called unified node-loop method. For gas pipeline, gas flow rate through a pipe can be determined using Colebrook equation modified by AGA (American Gas Association) for calculation of friction factor accompanied with Darcy-Weisbach equation for pressure drop and second approach is using Renouard equation adopted for gas pipeline calculation. For the development of Renouard equation for gas pipelines some additional thermodynamic properties are involved in comparisons with Colebrook and Darcy-Weisbach model. These differences will be explained. Both equations, the Colebrook’s (accompanied with Darcy-Weisbach scheme) and Renouard’s will be used for calculation of flow through the pipes of one gas pipeline with eight closed loops which are formed by pipes. Consequently four different cases will be examined because the network is calculated using improved Hardy Cross method and unified node-loop method. Some remarks on optimization in this area of engineering also will be mentioned.
Abstract
Today, two very efficient methods for calculation of flow distribution per branches of a looped gas pipeline are available. Most common is improved Hardy Cross method, while [...]
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Вишедимензионе гасне дистрибутивне мреже
D. Brkić

Tehnička dijagnostika (2008). 7 (2), 33-40,
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Повећање тачности при прорачуну гасне дистрибутивне мреже Харди Крос методом
D. Brkić, N. Đajić

175-178, XXXII Sym-Op-Is 2005, Vrnjačka Banja, Srbija
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Избор централизваног система снабдевања енергијом демо насеља
M. Živković, D. Brkić

187-190, XXXII Sym-Op-Is 2005, Vrnjačka Banja, Srbija
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Прорачун расподеле протока у прстенастој цевоводној мрежи обједињеном методом чворова и прстенова
D. Brkić, T. Tanasković

39-42, XXXVI Sym-Op-Is 2009, Ivanjica, Srbija
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Унапређење методе контура прилагођене за прорачун гасних дистрибутивних мрежа
D. Brkić, T. Tanasković

97-100, XXXV Sym-Op-Is 2008, Soko Banja, Srbija
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Оптимизација пречника цеви у прстенастој цевоводној мрежи модификованом методом петљи
43-46, XXXVI Sym-Op-Is 2009, Ivanjica, Srbija
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Програмски пакет за пројектовање бушотина
D. Tomić, N. Banjac, D. Brkić

Istraživanje i Projektovanje za Privredu (2005). Journal of Applied Engineering Science 3 (10), 21-27, 2005
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Are fossil fuels the main cause of today's global warming?
D. Brkić

Facta Universitatis-series: Working and Living Environmental Protection (2009). Facta Universitatis-series: Working and Living Environmental Protection 6 (1 …, 2009

Abstract
Gas will increasingly be seen as the fossil fuel of choice, especially when considering environmental impacts. Natural gas is the chance for Serbia for sustainable development and with its intensive consumption in the XXI century to conciliate the 4Es (Energy, Economy, Efficiency and Environment). In this paper we will compare the impact of different fossil fuels used for domestic heating with a special emphasis on natural gas. Some other causes of climate changes will be also discussed such as the Milanković astronomical cycles. These factors will be compared with climate changes caused by the consumption of fossil fuels.
Abstract
Gas will increasingly be seen as the fossil fuel of choice, especially when considering environmental impacts. Natural gas is the chance for Serbia for sustainable development [...]
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Natural gas heating in Serbian settlements according to urbanity parameters
D. Brkić

Facta universitatis-series: Architecture and Civil Engineering (2008). Facta universitatis-series: Architecture and Civil Engineering 6 (1), 139-153, 2008

Abstract
Natural gas can be directly used for heating of flats by gas distribution system. Indirectly, heating power plant can disburse natural gas and deliver hot water or steam for heating of flats. Decision of optimal way for gas heating usage is done based on spatial disposal of building, number and size of buildings in settlement, etc. Optimal solution, between gas distribution and district heating system (local or district heating by natural gas), can be done according to methodology (model approach) shown in this paper. According to variety of Serbian settlements (in density, size and layout of buildings) model which has ability to represent their different characteristics is formed. This model could be simple and useful tool for initial decision about energy supply system.
Abstract
Natural gas can be directly used for heating of flats by gas distribution system. Indirectly, heating power plant can disburse natural gas and deliver hot water or steam for [...]
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Ефикасност дистрибуције и коришћења природног гаса у домаћинствима
D. Brkić

(2010). University of Begrade, Department of Petroleum and Natural Gas Engineering, Belgrade, Serbia

Abstract
http://nardus.mpn.gov.rs/123456789/2654
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Gas distribution network topology problem
D. Brkić

Mathematica Balkanica (2012). 26 (3-4), 277-284

Abstract
Problem of gas fow in looped network is nonlinear problem and these kind of problems have to be solved using some kind of iterative procedure. For the pipeline network, two topology matrices can be written; i.e. loop matrix and node matrix. The node matrix is related to the first Kirchho 's law while the loop matrix is related to second Kirchho 's law. Here will be shown effcient method in which both matrices, i.e. the node and the loop matrix are united in one coherent procedure for solution of looped gas pipeline problem.
Abstract
Problem of gas fow in looped network is nonlinear problem and these kind of problems have to be solved using some kind of iterative procedure. For the pipeline network, two [...]
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Lambert W function in hydraulic problems
D. Brkić

Mathematica Balkanica (2012). 26 (3-4), 285-292

Abstract
Darcy’s flow friction factor is expressed in implicit form in some of the relations such as Colebrook’s and have to be solved by iteration procedure because the unknown friction factor appears on both sides of the equation. Lambert W function is implicitly elementary but is not, itself, an elementary function. Implicit form of the Lambert W function allows us to transform other implicit functions in explicit form without any kind of approximations or simplifications involved. But unfortunately, the Lambert W function itself cannot be solved easily without approximation. Two original transformations in explicit form of Colebrook’s relation using Lambert W function will also be shown. Here will be shown efficient procedure for approximate solutions of the transformed relations.
Abstract
Darcy’s flow friction factor is expressed in implicit form in some of the relations such as Colebrook’s and have to be solved by iteration procedure because the [...]
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Intelligent flow friction estimation
D. Brkić, Ž. Ćojbašić

Computational Intelligence and Neuroscience (2016). 2016, article 5242596

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 (). 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 () 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 () 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.
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 [...]
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A note on explicit approximations to Colebrook’s friction factor in rough pipes under highly turbulent cases
D. Brkić

International Journal of Heat and Mass Transfer (2016). 93, 513-515

Abstract
New explicit approximations to the implicitly given Colebrook equation for flow friction factor are given. They are with improved accuracy compared with the one recently published [Shaikh et al.: Int. J. Heat Mass Tran. 88 (2015) 538–543]. The new approximations are highly accurate only in rough pipes under fully developed turbulent cases of flow.
Abstract
New explicit approximations to the implicitly given Colebrook equation for flow friction factor are given. They are with improved accuracy compared with the one recently published [...]
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Spreadsheet-based pipe networks analysis for teaching and learning purpose
D. Brkić

Spreadsheets in Education (eJSiE) (2016). 9 (2), article 4646

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.
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 [...]
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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 …
A. Aznárez, D. Brkić, S. Contini, M. Christou, S. Tarantola

(2016). Publications Office of the European Union

Abstract
The purpose of this study is to provide the Commission with the information necessary for the assessment of the economic, social, and environmental impacts of a possible extension of the scope of the EU product safety legislation (ATEX Directive 2014/34/EU, Pressure Equipment Directive 2014/68/EU and Machinery Directive 2006/42/EC) with respect to equipment intended for the use in the offshore oil and gas industry. Whereas equipment on fixed units is in the scope of these three Directives, mobile offshore units and equipment installed on them are currently in general excluded. In addition the Pressure Equipment Directive excludes also well-control equipment. The study investigates whether there are safety issues which could be addressed by extending the scope of the Directives and what would be the impacts of such an extension. Title Study on the impacts of possible amendments to the ATEX, the Machinery, and the Pressure Equipment Directive with respect to equipment intended for use in the offshore oil and gas industry.
Abstract
The purpose of this study is to provide the Commission with the information necessary for the assessment of the economic, social, and environmental impacts of a possible extension [...]
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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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An efficient iterative method for looped pipe network hydraulics free of flow-corrections
D. Brkić, P. Praks

Fluids (2019). 4 (2), article 73

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.
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 [...]
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Colebrook’s flow friction explicit approximations based on fixed-point iterative cycles and symbolic regression
D. Brkić, P. Praks

Computation (2019). 7 (3), article 48

Abstract
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 WrightΩ 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.
Abstract
The logarithmic Colebrook flow friction equation is implicitly given in respect to an unknown flow friction factor. Traditionally, an explicit approximation of the Colebrook [...]
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What can students learn while solving Colebrook’s flow friction equation?
D. Brkić, P. Praks

Fluids (2019). 4 (3), article 114

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.
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 [...]
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Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function: Reply to the Discussion by Majid Niazkar
P. Praks, D. Brkić

Mathematics (2020). 8(5), article 796

Abstract
In this reply, we present updated approximations to the Colebrook equation for flow friction. The equations are equally computational simple, but with increased accuracy thanks to the optimization procedure, which was proposed by the discusser, Dr. Majid Niazkar. Our large-scale quasi-Monte Carlo verifications confirm that the here presented novel optimized numerical parameters further significantly increase accuracy of the estimated flow friction factor.
Abstract
In this reply, we present updated approximations to the Colebrook equation for flow friction. The equations are equally computational simple, but with increased accuracy thanks [...]
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Rational approximation for solving an implicitly given Colebrook flow friction equation
P. Praks, D. Brkić

Mathematics (2020). 8 (1), artice 26

Abstract
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.
Abstract
The empirical logarithmic Colebrook equation for hydraulic resistance in pipes implicitly considers the unknown flow friction factor. Its explicit approximations, used to [...]
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Brkić, Dejan 15566

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editors

Brkić, Dejan 15566

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