Analysis of the Colebrook-White Equation and further approaches to solve Fluid Loss Coefficient Definition Problems

— With the advancement of fluid mechanics in engineering, the need to estimate the pressure drop coefficient, becomes necessary for flow loss calculations in order to be able to measure pipe diameter or stipulate flow regimes that are required for a given situation. This coefficient appears in the Darcy-Weisbach formula in equality with the Poiseuille equation and is now measured by the Colebrook-White equation. However, because this equation presents a different characteristic, where the coefficient appears on both sides of the same equation, scholars of the area over time modeled approximations derived from this previous knowledge. In this work we will approach the Colebrook-White principles and their subsequent approaches. The aim of this paper is to analyze the correlations cited, as well as their authors, also analyzing the relative errors between the approximations and the Colebrook-White equation at specific intervals for the relative roughness and the Reynolds number and, from this, to determine which ones. have the lowest relative error.

This article has as general objectives to analyze the explicit equations and verify which ones have the lowest and highest average relative error and analyze the relative errors and organize them from the lowest to the highest percentage, since the lowest percentage will have results of coefficient closest to those of Colebrook-White and, by definition, the ideal approximation will be considered. The highest percentage will demonstrate the opposite.

II. THE COLEBROOK-WHITE EQUATION AND ITS APPROXIMATIONS
Below are briefly presented the equations used for study, followed by the calculations to make comparisons.

Colebrook-White equation
To Baqer (2015), the Colebrook equation is an implicit equation that combines experimental results from studies of turbulent flow in rough tubes. The equation is used to iteratively solve the Darcy-Weisbach friction factor "λ".

Churchill approach
According to Brkić

Relative error
To Asker et al (2014), the calculation that will be the basis for the analysis of the approximations in relation to the Colebrook-White equation will be that of the relative error. The relative error can demonstrate how close the result of the coefficient of the explicit equation will be when compared to the coefficient of the equation. Such an equation of relative error can be expressed in equation 2.14.

III. RESULTS AND DISCUSSION
The present work obtained friction factor data considering that the turbulent flow, with Reynolds number greater than four thousand. To obtain the data, sixteen values of relative roughness were used, which correspond from the smooth surface to a rougher surface.
After that, there will be a discussion of relative errors between Colebrook-White and their approximations, to observe the best explicit equations regarding relative errors.

Colebrook-White
The Colebrook-White equation will be used as a reference for comparison with the other explicit equations.For the friction factor calculations, the following parameters were used: 4x10 3 ≤ Re ≤ 10 8 and 10 -6 ≤ ε/D ≤ 5x10 -2 . The graph 1 shows the calculated values of the friction factor for Colebrook -White equation. It can be seen from Graph 1 that the greater the Reynolds number, the lower the value of the coefficient "λ" for the relative roughness intervals. It is also noticed that there is a tendency for values of "λ" very close to the Reynolds intervals, especially in the periods of 5x10 4 ≤ Re ≤ 10 8 , where the results approach the equality as the value of the relative roughness grows.

Graph 1 -Friction factor with Colebrook-White equation
This can be explained by the fact that Equation (2.1), together with the explicit equations, presents a sum of the relative roughness (ε/D) with the Reynoldsnumber (1/Re), since the rest will be just a relation of mathematical operations with constants. This sum, as the relative roughness increases and goes through the Reynolds number intervals, it tends to have a common result. For example, for a relative roughness = 5x10 -2 , in the Reynolds number range between 5x10 4 ≤ Re ≤ 10 8 , the sum will tend to 5x10 -2 , as the term "1 / Re" will tend to zero. Bandeira (2015) reports that the viscous sublayer presents a thickness which is capable of covering the rough elements, it will not have a significant loss, in this condition it can be said that the flow is in a hydraulically smooth regime. However, the thickness of the viscous sublayer is influenced by the Reynolds number, as the Reynolds number increases, the thickness of the viscous sublayer decreases and for a given high Reynolds number some rough elements emerge significantly, at that moment the friction becomes a function of Reynolds number and roughness as well. For even higher Reynolds values, all the rough elements emerge through the viscous sublayer and the loss of pressure depends on the size of the rough elements, in this condition the flow is in a rough regime.
According to Schlichting (1979), the friction factor varies up to a certain Reynolds number, this is due to the ratio between the protrusions of the surfaces and the height of the boundary layer, however, after a certain point the friction factor stops varying , that is, the friction factor no longer depends on the Reynolds number, this is because the flow has reached a completely rough regime, being possible to visualize in the graph 1 the friction factor remains constantfor each line that represents each relative roughness.

Moody
Calculations will be performed at the intervals above for Equation 2.2. Graph 2 shows the values of λ according to the relative roughness and Reynolds number.

Graph 2 -Friction factor for the Moody equation
It can be seen from Graph 2 that there is a difference with Graph 1 in values for "λ", such discrepancies will be addressed in the error percentage, using Equation 2.14. The behavior and explanation for it are similar to Graph 1, but there are differences in values due to the approximation of the model equations.

Wood
The wood approximation, equation (2.3) was used to obtain data that are shown in Graph 3.
Graph 3 shows the data in which the Wood equation was used, with the behavior of the lines slightly different from the previous graphs, it being possible to observe that for low values of the Reynolds number the results are more different than the Colebrook -white data.

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Graph 4 -Friction factor with Churchill's equation
Graph 4 shows the behavior of the lines and margins of values remarkably similar for the coefficient when compared with the Colebrook -White data.

Eck
The Eck model was also simulated with the same conditions as the simulation of the other models.
The Graph 5 contemplates the results of the "λ" coefficient for equation (2.8), for the "Re" intervals and the relative roughness.
Graph 5-Friction factor with Eck equation It can be seen from Graph 5 that there is the same behavioral similarity of the graphs of coefficient values previously mentioned, and with values of "λ" awfully close to the results of Colebrook-White.

Haaland
For equation 2.9, the following graph 6 is made to demonstrate the values of the friction factor.Graph 6 shows the results of "λ" for the pre-determined "Re" and "ε/D" intervals.
It can be seen from Graph 6 that there is the same behavioral similarity and with coefficient values tending to equality when compared to Graph 1.

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Graph 7-Friction factor with Tsal equation
According to the above data, there is a similarity in behavior but there is a considerable discrepancy, it is possible to verify that for low Reynolds numbers the relative roughness present nearest friction factor values than the other methods, it becomes clearer when it is held a comparison with other graphics.

Buzzelli
Buzelli propose the equation (2.11) for determination friction factor. The graph 8 shown the coefficient values for the "Re" and "ε/D".
It can be seen in Graph 8 that it is most similar to Graph 1 in the values of "λ", with low discrepancies will be addressed in the percentage of error, using Equation 2.14.
And the low discrepancy makes this approximation method has good results compared to Colebrook -White.

Relative Error
Considering that all the models presented above are an approximation of the Colebrook -White equation, the relative error will occur when comparing the results of each model with Colebrook-White.
The comparison made in the present work lists the results of all models for each relative roughness.

Error for relative roughness of 0.000001
The graph 9 shows the result of all models, including Colebrook -White, for the relative roughness of 0.000001.

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[ Graph 9-Comparison of the friction factor models for relative roughness of 0.000001 The data shown in graph 9, it is possible to observe that for the relative roughness of 0.000001, Wood's method presented a more discrepant result when compared to the Colebrook-White data. Table 2 shows percentage values of the relative error between all the models. From the analysis of the graph 9 and the table2, it is possible to verify that the Wood, Moodyand Eck models generate results with greater errors in relation to the Colebrook -White equation, while the Haaland and Buzzelli models present good approximations.

Error for relative roughness of 0.000005
The graph 10 shows the result of all models, including Colebrook -White, for the relative roughness of 0.000005.

Graph 10-Comparison of the friction factor models for relative roughness of 0.000005
The data shown in graph 10, it is possible to observe that for the relative roughness of 0.000005,in general method all have a tendency to next Colebrook-White data. Table 3 shows percentage values of the relative error between all the models.

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[  Error for relative roughness of 0.00001 The graph 11 shows the result of all models, including Colebrook -White, for the relative roughness of 0.00001.

Graph 11-Comparison of the friction factor models for relative roughness of 0.00001
The data shown in graph 11, it is possible to observe that for the relative roughness of 0.00001,in general method all have a tendency to next Colebrook-White data. Table 4 shows percentage values of the relative error between the models. Error for relative roughness of 0.00005 The graph 12 shows the result of all models, including Colebrook -White, for the relative roughness of 0.00005.
Graph 12-Comparison of the friction factor models for relative roughness of 0.00005 The data shown in graph 12, it is possible to observe that for the relative roughness of 0.00005,in general method all have a tendency to next Colebrook-White data. Table 5 shows percentage values of the relative error between the models. Error for relative roughness of 0.0001 The graph 13 shows the result of all models, including Colebrook -White, for the relative roughness of 0.0001.
The data shown in graph 13, it is possible to observe that for the relative roughness of 0.0001,in general method all have a tendency to next Colebrook-White data.
Graph 13-Comparison of the friction factor models for relative roughness of 0.0001. Table 6 shows percentage values of the relative error between the models.

Error for relative roughness of 0.0002
The graph 14 shows the result of all models, including Colebrook -White, for the relative roughness of 0.0002.
Graph 14-Comparison of the friction factor models for relative roughness of 0.0002.

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[  Table 7 shows percentage values of the relative error between the models.

Error for relative roughness of 0.0005
The graph 15 shows the result of all models, including Colebrook -White, for the relative roughness of 0.0005.
Graph 15-Comparison of the friction factor models for relative roughness of 0.0005.
The data shown in graph 15, it is possible to observe that for the relative roughness of 0.0005,in general method all have a tendency to next Colebrook-White data. Table 8 shows percentage values of the relative error between the models.

Error for relative roughness of 0.001
The graph 16 shows the result of all models, including Colebrook -White, for the relative roughness of 0.001.
The data shown in graph 16, it is possible to observe that for the relative roughness of 0.001,in general method all have a tendency to next Colebrook-White data.
Graph 16-Comparison of the friction factor models for relative roughness of 0.001. Table 9 shows percentage values of the relative error between the models.

Error for relative roughness of 0.002
The graph 17 shows the result of all models, including Colebrook -White, for the relative roughness of 0.002.

Error for relative roughness of 0.005
The graph 18 shows the result of all models, including Colebrook -White, for the relative roughness of 0.005.
Graph 18-Comparison of the friction factor models for relative roughness of 0.005.
The data shown in graph 18, it is possible to observe that for the relative roughness of 0.005, Moody's, Wood's and Tsal's method presented a more discrepant result when compared to the Colebrook-White data. Table 11 shows percentage values of the relative error between the models.

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[  Error for relative roughness of 0.01 The graph 19 shows the result of all models, including Colebrook -White, for the relative roughness of 0.01.
Graph 19-Comparison of the friction factor models for relative roughness of 0.01.
The data shown in graph 19, it is possible to observe that for the relative roughness of 0.01, Wood's and Tsal's method presented a more discrepant result when compared to the Colebrook-White data. Table 12 shows percentage values of the relative error between the models.

Error for relative roughness of 0.015
The graph 20 shows the result of all models, including Colebrook -White, for the relative roughness of 0.015.
Graph 20-Comparison of the friction factor models for relative roughness of 0.015.
The data shown in graph 20, it is possible to observe that for the relative roughness of 0.015, Moody's and Tsal's method presented a more discrepant result when compared to the Colebrook-White data. Table 13 shows percentage values of the relative error between the models. Tsal's method presented a more discrepant result when compared to the Colebrook-White data.
Graph 21-Comparison of the friction factor models for relative roughness of 0.02. Table 14 shows percentage values of the relative error between the models. Error for relative roughness of 0.03 The graph 22 shows the result of all models, including Colebrook -White, for the relative roughness of 0.03.
Graph 22-Comparison of the friction factor models for relative roughness of 0.03. Table 15 shows percentage values of the relative error between the models.

Error for relative roughness of 0.05
The graph 24 shows the result of all models, including Colebrook -White, for the relative roughness of 0.05.
The data shown in graph 24, it is possible to observe that for the relative roughness of 0.05, Moody's, Wood's and Tsal's method presented a more discrepant result when compared to the Colebrook-White data.
Graph 24-Comparison of the friction factor models for relative roughness of 0.05. Table 17 shows percentage values of the relative error between the models. There are several other correlations, statistics, and values for relative roughness (absolute roughness and pipe diameter) and Reynolds number (turbulent fluid) that can be determined.
As future work it is possible to estimate such approximations, statistical calculations and Reynolds values, absolute roughness, and diameters, for a more statistically concrete analysis and / or a more specific analysis depending on the values adopted for relative roughness and Reynolds.