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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.
Highlights
The Colebrook equation for flow friction is an empirical formula developed by C
Keeping in mind that the Moody diagram represents a graphical interpretation of the Colebrook equation, students should make a sketch of the Moody diagram [18,46], which will add an additional value both from the engineering and mathematical points of view (Figure 1 of this article can be used for this purpose)
Some iterative methods belong to the group of multi-point methods [8] that are very powerful and that can reach the accurate solution of the Colebrook equation in the first iteration
Summary
The Colebrook equation for flow friction is an empirical formula developed by C. The Colebrook equation follows logarithmic law and, at first sight, it seems to be very simple It is given implicitly, with respect to the unknown flow friction factor f in a way that it can be expressed explicitly only in terms of the Lambert W function, with further difficulties in its evaluation [4,5,6] (it can be rearranged for gas flow [7]). A first simple explicit approximate formula with inner iterative cycles can be introduced using such approaches [24] Transcendental functions, such as logarithmic and exponential functions, on computers require the execution of additional floating point operations [25,26,27]. Keeping in mind that the Moody diagram represents a graphical interpretation of the Colebrook equation, students should make a sketch of the Moody diagram [18,46], which will add an additional value both from the engineering and mathematical points of view (Figure 1 of this article can be used for this purpose)
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