An Improved Expression for a Classical Type of Explicit Approximation of the Colebrook White Equation with Only One Internal Iteration

The solution of the Colebrook White equation continues to attract the attention of engineers and mathematicians. Some solutions proposed in recent years have two or more internal iterations and present relative complexity. At the same time, micro-computers and calculators have shown increased calculation capacity, virtually eliminating approximations that have a certain accuracy but that are too complex. A direct approximation of the equation allowing for quick and accurate solution is still useful. This technical note brings together some explicit approximations with only one internal iteration presented in the 1970s and 1980s into a single category and proposes a single model with five parameters to be determined. Moreover, this technical note also presents the values for these five parameters that minimize the mean square error related to the outcome of the Colebrook White equation, considered as a reference. The proposed model allows to obtain results with mean square error equal to 1.14 x 10-8, with maximum relative error of 4.44% and 96.78% of results below 1.00% of difference relative to Colebrook White equation, which can be considered quite satisfactory.


Introduction
The Colebrook White equation was proposed [1] in 1937 and its non-algebraic characteristic, hindering its solution, still attracts the attention of engineers and mathematicians. Moody [2] proposed a diagram containing all the information summarized in the equation and serving as an abacus for determining the friction factor f.
During the 70s and 80s, the first approximations to the equation of Colebrook White, basically with only one internal iteration have been proposed. The first personal computers became popular, but still showing inability to solve the Colebrook White equation in problems involving the calculation of a large number of pipes.
During the following decades, several other approximations have been proposed, with increasing complexity, some with two or three [3] or even more internal iterations and others with different shapes or different methods [4] of the structure proposed by Colebrook and White. Currently, the Colebrook White equation can be solved by personal computers and even an exact although complex solution was proposed [5,6]. Even with great accuracy [7,8], the evolution of computers reduces the usefulness of approximations with high computational costs.
Currently, there is a large number of papers proposing alternative equations to the expression proposed by Colebrook White, with different shapes and several of them with better results for a particular type of flow [9,10] or for a certain type of fluid [11]. This paper presents an equation with scope and applicability equivalent to the original expression, therefore suitable for Newtonian fluids and turbulent flows, for a wide range of roughness values.
An approximation of the Colebrook White equation that represents a computational cost equivalent to only one iteration and displays reasonable accuracy could still be applied to calculations of large pipe networks.
This technical note includes six classical approximations of the Colebrook White equation with a similar shape in a same category. These approximations have been proposed in the 1970s and 1980s and have computational cost equivalent to one internal iteration.
This technical note also proposes a generic model for this category, with five parameters to be determined so as to minimize the difference of their results with the Colebrook White equation, considered as reference in this study.
Finally, this technical note provides values for these five parameters so that the model proposed to provide better results than those obtained with the six models taken as the starting point.

The Friction Factor and the Colebrook White Equation
The friction factor f appears in the Darcy equation for energy loss in pipes. The factor f is dimensionless and depends on the pipe diameter, roughness of the material of the pipeline and the Reynolds number of the flow. Darcy Eq. (3) directly indicates the value of f after each iteration. The initial value used in the iterations can reduce the amount of calculations, but the process usually converges to a reasonable accuracy after a few iterations. 2 Re 51 .
Good accuracy in result is already obtained with few iterations. For simple calculations, it is not interesting to adopt approaches to this equation. The use of approximations that directly provide the final value, even with a loss of precision, will be interesting for the design of networks with large numbers of tubes.

Some Classical Basic Approximations of the Colebrook White Equation
Several approximations for the Colebrook White equation have been proposed in last decades and they are still being proposed. The trend that can be expected from new approaches is that they will present lower computational costs. Among them, one interesting class can be established for the approximations with only one internal iteration and similar constitution to the Colebrook White equation.
These equations can be generalized to a model with five parameters, aiming to optimize these parameters to reduce the error in relation to the Colebrook White equation. The next section discusses a model developed in this way.

A Model for an Explicit Approximation with Only One Internal Iteration
It is possible to build a generic model for this class of explicit approximations containing five variables to be optimized, thereby reducing the error in relation to the Colebrook White equation. Eq. (10) shows the proposed model. Comparing the expressions (4) to (9), changes were observed in five locations on the basic structure of the Colebrook White equation.
The five variables together must compensate for the withdrawal of the factor f from the inside of the logarithm. Z is a factor external to the logarithm function. A and B respectively integrate the terms including roughness and Reynolds number. The exponents a and b do not appear in the original structure proposed by Colebrook and White. (10) Table 1 shows the values of the five parameters of the model of Equation (10) corresponding to the approaches presented in previous section.
The determination of the values of the five parameters of the model of equation (10) to minimize the RE and MSE values can be performed with the Matlab software. The ER and MSE values were obtained for the values of Z between 1 and 2, a between 1 and 2, A between 1 and 2, b between 1 and 2 and B between 1 and 2.
The next section presents and discusses the results.

Results and Discussion
The MSE minimization of the five parameters resulted in the following values: Z=0.30091, A=1.09540, B=0.969491, a=3.73150 and b=5.98017. These values may be rounded to the fourth decimal with no significant change in accuracy of the results, as shown in Eq. (14).
This equation is the best result of this work, with a maximum relative error of 4.44% and MSE equal to 1.14 x10 -8 . Among the values of relative roughness and Reynolds numbers used in the calculation, 92.60% are lower than the percentage relative error of 0.08% and 96.78% are lower than PRE of 1.00%. Figure 1 shows the relative error distribution to the ranges of relative roughness and Reynolds number considered. Table 2 shows a comparison between the MSE for the classical equations and the MSE obtained for the equation proposed in this paper. This table also shows maximum values of RE, difference between negative and positive maximum values of RE and percentage of values of RE lower than 0.08% and lower than 1.00%. The equation of this paper is the best approximation when compared with the classical equations discussed in previous section. Table 2 shows that the proposed equation has the lowest MSE and the greatest number of results with relative error less than 0.8% and less than 1.0%. The low MSE value makes the surface appearing in Figure 1 "flatter" than the surfaces corresponding to other approximations and allows a greater amount of values with less difference with respect to the Colebrook White equation.

Conclusions
This technical note included six classical approximations of the Colebrook White equation in a same category and also proposed a generic model for this category, with five parameters to be determined. These five parameters were determined, presenting a new alternative equation with only a single iteration with RMSE equal to 1.125x10 -8 .