Intelligent flow friction estimation

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

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

The 80 year-old empirical Colebrook function zeta, widely used as an informal standard for hydraulic resistance, relates implicitly the unknown flow friction factor lambda, with the known Reynolds number Re and the known relative roughness of a pipe inner surface epsilon* ; lambda = zeta(Re, epsilon* ,lambda). It is based on logarithmic law in the form that captures the unknown flow friction factor l 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 Pade polynomials with only one log-call in total for the whole procedure (expensive log-calls are substituted with Pade 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.Web of Science117art. no. 182

Unified Friction Formulation from Laminar to Fully Rough Turbulent Flow

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.Web of Science811art. no. 203

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

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, epsilon/D with an unknown output parameter; the flow friction factor, ; = f (, Re, epsilon/D). In this paper, a few explicit approximations to the Colebrook equation; approximate to f (Re, epsilon/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, epsilon/D}{}. 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 , 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 Pade approximation, practically the same error is obtained also using only one logarithm.Web of Science109art. no. 117

Evolutionary optimization of Colebrook’s turbulent flow friction approximations

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

Accurate solutions of Colebrook-White’s friction factor formulae

Estimations of friction factor (Ff) in pipeline systems and fluid transport are essential ingredients in engineering fields and processes. In this paper explicit friction factor formulae (Fff) were proposed and evaluated with an aim of developing error free Fff. General Fff that relate Ff, Reynolds number (Re) and relative roughness (Rr) were proposed. Colebrook – White’s formula was used to compute different Ff for Re between 4 x 103 and 1.704 x 108, and Rr between 1.0 x 10-7 and 0.052 using Microsoft Excel Solver to fix the Fff. The fixed Fff were used to compute Ff for Re between 4 x 103 and 1.704 x 108 and Rr between 1.0 x 10-7 and 0.052. Accuracy of the fixed Fff was evaluated using relative error; model of selection (MSC) and Akaike Information Criterion (AIC) and compared with the previous Fff using Colebrook–White’s Ff as the reference. The study revealed that Ff estimated using the fixed Fff were the same as Ff estimated using Colebrook – White’s Fff. The fixed Fff provided the lowest relative error of (0.02 %; 0.06 % and 0.04 % ), the highest MSC (14.03; 12.42 and 13.07); and the lowest AIC (-73006; -64580 and -67982). The study concluded that modeling of Fff using numerical methods and Microsoft Excel Solver are better tools for estimating Ff in pipeline flow problems.Keywords: Friction factor, MSC; AIC; Reynolds number; Engineering Field; pipe flow, statistical method

Colebrook’s flow friction explicit approximations based on fixed-point iterative cycles and symbolic regression

The logarithmic Colebrook flow friction equation is implicitly given in respect to an unknown flow friction factor. Traditionally, an explicit approximation of the Colebrook equation requires evaluation of computationally demanding transcendental functions, such as logarithmic, exponential, non-integer power, Lambert W and 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

Solution of the implicit Colebrook equation for flow friction using Excel

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

What can students learn while solving Colebrook's flow friction equation?

Even a relatively simple equation such as Colebrook's offers a lot of possibilities to students to increase their computational skills. The Colebrook's equation is implicit in the flow friction factor and, therefore, it needs to be solved iteratively or using explicit approximations, which need to be developed using different approaches. Various procedures can be used for iterative methods, such as single the fixed-point iterative method, Newton-Raphson, and other types of multi-point iterative methods, iterative methods in a combination with Pade 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.Web of Science43art. no. 11

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

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 solutio