What Can Students Learn While Solving Colebrook’s Flow Friction Equation?

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.

Excel Vba-Based User Defined Functions for Highly Precise Colebrook’s Pipe Flow Friction Approximations: A Comparative Overview

“…Computational effort for the mathematical operations was determined by performing 100 million calculations for each mathematical operation using random input each repeated five times, with the average computational time recorded. The results are [44]: Addition-23.40sec, Subtraction-27.50sec, Multiplication-36.20sec, Division-31.70sec, Squared-51.10sec, Square root-53.70sec, Fractional exponential-77.60sec, Napierian natural logarithm-63.00sec, and Briggsian decimal logarithm to base 10-78.80sec. Accuracy is checked using 2 Million quasi-random and as well 90 thousand and 740 uniformly distributed samples, as in [9,23,35,36], which covers the whole domain of the Reynolds number, Re and of the relative roughness of inner pipe surface, ε, which are commonly used in engineering practice; 2320<Re<10 8 and 0<ε<0.05.…”

“…Algorithms and VBA codes are given here only for extremely accurate approximations while coding of the further approximations is not shown [44].…”

Excel VBA-Based User Defined Functions for Highly Precise Colebrook’s Pipe Flow Friction Approximations: A Comparative Overview

“…Computational effort for the mathematical operations was determined by performing 100 million calculations for each mathematical operation using random input each repeated five times, with the average computational time recorded. The results are [44]: Addition-23.40sec, Subtraction-27.50sec, Multiplication-36.20sec, Division-31.70sec, Squared-51.10sec, Square root-53.70sec, Fractional exponential-77.60sec, Napierian natural logarithm-63.00sec, and Briggsian decimal logarithm to base 10-78.80sec. Accuracy is checked using 2 Million quasi-random and as well 90 thousand and 740 uniformly distributed samples, as in [9,23,35,36], which covers the whole domain of the Reynolds number, Re and of the relative roughness of inner pipe surface, ε, which are commonly used in engineering practice; 2320<Re<10 8 and 0<ε<0.05.…”

Hybrid Models for Solving the Colebrook–White Equation Using Artificial Neural Networks

“…Moreover, the study and development of the approximation of the Colebrook-White equation are still going on today. Some of the latest studies on this can be seen in several kinds of literature [18][19][20][21][22][23].…”