One-Log Call Iterative Solution of the Colebrook Equation for Flow Friction Based on Padé Polynomials

Abstract

The eighty years old empirical Colebrook function widely used as an informal standard for hydraulic resistance relates implicitly the unknown flow friction factor , with the known Reynolds number and the known relative roughness of a pipe inner surface ;. It is based on logarithmic law in the form that captures the unknown flow friction factor in a way from which 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 -call in total for the whole procedure (expensive -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.

Choosing the Optimal Multi-Point Iterative Method for the Colebrook Flow Friction Equation – Numerical Validation

“…Choosing the different starting point compared with the here selected for the shown numerical validation does not alter the number of required iterations significantly [29,66].…”

“…Calculation of flow through complex networks of pipes such as for water or gas distribution requires multiple evaluations of flow friction factor [18][19][20][21][22][23]. In general the less number of iterations required, the solution is more efficient with decreased burden for computers [24][25][26][27][28] (recent approach based on Padé polynomials shows how the computational burden can be minimized with the same number of used iterations [29]).…”

Colebrook’s Flow Friction Explicit Approximations Based on Fixed-Point Iterative Cycles and Symbolic Regression

“…Praks and Brkić [24] recently showed a Newton-Raphson iterative solution of the Colebrook equation based on Padé approximants [25][26][27][28][29]. Based on their solution, one simplified approach and a novel starting point that significantly reduces numerical error will be offered herein.…”

Symbolic Regression-Based Genetic Approximations of the Colebrook Equation for Flow Friction

“…In our case this is not of interest, knowing that for y=Re; y>>1. Of course, to use only numbers between 1 and 10, we can use the rule ln(Re)=ln(z·10 n )=ln(z)+n·ln (10) where n=len(int(z)) and z=Re/10 n ; len is a function which calculates number of digits in a number while int is function which gives a number down to the nearest integer; ln(10)=2.30258509. Similar can be done for the normalized parameter b=-log10(ε/D) where ε/D is between 0 and 0.05.…”

“…A fast but still reliable Padé approximation useful for the Colebrook equation has been recently introduced in [10]. The logarithm term , where argument z~1 is approximated by the rational function:…”