Scientific Paper / Artículo Científico 



https://doi.org/10.17163/ings.n30.2023.05 


pISSN: 1390650X / eISSN: 1390860X 

CORRELATION FOR THE CALCULATION OF TURBULENT FRICTION IN PIPES 

CORRELACIÓN PARA EL CÁLCULO DE LA FRICCIÓN TURBULENTA EN TUBERÍAS 
Received: 25012023, Received after review: 21042023, Accepted: 18052023, Published: 01072023 
Abstract 
Resumen 
One of the essential parameters in hydraulic systems of pipe networks is the friction factor λ. The friction factor is determined using the implicit ColebrookWhite equation through iterative methods, whichmakes its application challenging. In this work, a correlation based on the recursive method is developed to calculate the friction factor using the ColebrookWhite equation. Two empirical relationships are proposed to finalise the correlation, with coefficients and exponents calibrated in Excel 2019. The results of the two proposed relationships were compared with the SwameeJain and Haaland relationships for recursive increments. For the λ_{8} correlation, the maximum percentage error of the friction factor was 0,0000017%, for a relative roughness of 0,00001 and a Reynolds number of 4000. Additionally, the calculations yielded seven exact decimal digits for the friction factor. For Reynolds numbers greater than 4000, the percentage error decreases. As a result, it is concluded that the correlation based on the proposed explicit relationships satisfies the solution of the implicit ColebrookWhite equation. 
En los sistemas hidráulicos de redes de tuberías, uno de los parámetros fundamentales es el factor de fricción λ. El factor de fricción se determina con la ecuación implícita de ColebrookWhite por medios iterativos, lo cual dificulta su aplicación. En el presente trabajo se construye una correlación basada en el método recursivo para el cálculo del factor de fricción, para lo cual se empleó la ecuación de ColebrookWhite. Para el cierre de la correlación se proponen dos relaciones empíricas, donde sus coeficientes y exponentes fueron calibrados en Excel 2019. Se compararon los resultados de las dos relaciones que se proponen con las relaciones de SwameeJain y Haaland, para incrementos recursivos, donde para la correlación λ_{8} se obtuvo el error porcentual máximo del factor de fricción de 0,0000017 %, para la rugosidad relativa de 0,00001 y número de Reynolds 4000; así como, los decimales arrojaron siete dígitos decimales exactos para el factor de fricción. Para Reynolds mayores de 4000, el error porcentual disminuye. Se concluye que la correlación en función de las relaciones explícitas que se proponen satisface a la solución de la ecuación implícita de ColebrookWhite. 
Keywords: Correlation, ColebrookWhite equation, Percentage error, Friction factor, Recursive method 
Palabras clave: correlación, ecuación de ColebrookWhite, error porcentual, factor de fricción, método recursivo 
^{1,*}Group of Mathematical Modeling and Numerical Simulation (GMMNS), Universidad Nacional de Ingeniería, Lima, Perú. Corresponding author ✉: sanluist@gmail.com. ^{2}Universidad Nacional Hermilio Valdizán, Huánuco, Perú.
Suggested citation: Tolentino, S. L. and González, O. “Correlation for the calculation of turbulent friction in pipes ,” Ingenius, Revista de Ciencia y Tecnología, N.◦ 30, pp. 5463, 2023, doi: https://doi.org/10.17163/ings.n30.2023.05. 
1. Introduction
In pipe network systems in industrial processes, internal flow exhibits fluctuations in velocity, pressure, temperature, and other parameters. The flow is driven by pressure differentials, and flow friction is always present. For a pipeline or measuring instrument, a gradient of pressure, velocity, and temperature occurs. The flow velocity is maximum in the central region, while at the wall, it is zero due to the noslip condition. Therefore, a pressure decrease caused by viscous stresses represents an irreversible pressure loss known as pressure drop [1, 2]. There are abrupt pressure drops in the throat section in experimental flow measurement devices, such as the Venturi tube. In contrast, the pressure drops on the walls of the junctions between the throat section and the converging and diverging sections are lower compared to the central region of the flow [3]. The flow regime is classified into laminar, transitional, and turbulent. Smooth and parallel streamlines with orderly motion characterize the laminar flow. This flow is expected in fluids with high viscosity and lowspeed motion. On the other hand, turbulent flow is characterized by random fluctuations of eddies at different scales. These eddies transport mass, amount of motion, and energy to other flow regions, increasing the amount of motion, mass, and heat transfer in the affected areas. As a result, turbulent flow is associated with significant variations in the values of friction coefficients, mass transfer, and heat transfer [1, 2]. Osborne Reynolds [4] conducted experiments on pipe sections to study the flow and discovered that the flow regime is related to the ratio of inertial forces to viscous forces in the fluid. This ratio is known as the Reynolds number, R_{e}, and is calculated using the formula: R_{e} = V d/ν, where V is the average velocity, d is the internal diameter of the pipe, ν = μ/ρ is the kinematic viscosity, μ is the dynamic viscosity, and ρ is the density of the fluid [1, 2]. The transition from laminar flow to turbulent flow is determined by the viscosity and velocity of the flow, as well as the pipe geometry, internal wall roughness, wall temperature, and other factors. In most practical conditions, flow is classified as follows: laminar flow for R_{e} ≤ 2300, turbulent flow for R_{e} ≥ 4000, and transitional flow in the range of 2300 ≤ R_{e} ≤ 4000 [1, 2]. Colebrook and White [5, 6] proposed an implicit equation to calculate the friction factor λ, in turbulent flow in pipes, based on the results of their experimental research. This equation is known as the ColebrookWhite equation (1).

The ColebrookWhite equation combines data for transitional and turbulent flow in smooth and rough pipes. The parameter ε represents the relative roughness and is defined as ε = k/d, where k is the average height of the material’s roughness and d is the pipe’s internal diameter. The parameter R_{e} is the Reynolds number. The parameters λ, ε and R_{e} are dimensionless. The friction factor in the ColebrookWhite equation cannot be solved accurately and explicitly using algebraic procedures. Therefore, the friction factor is determined using numerical methods through iterative procedures in computational codes, such as the NewtonRaphson method, bisection, fixedpoint iteration, etc. This makes it challenging to obtain the friction factor in the design of pipeline networks due to the extensive and laborious calculations required by these iterative methods. As an alternative solution to the implicit ColebrookWhite equation, Moody [7] proposed using a graph that represents this equation. This graph is used in engineering to determine the friction factor. However, a numerical error is generated when determining the friction factor, which implies obtaining an approximate result. The literature describes empirical correlations that provide an approximate solution for calculating the friction factor. These correlations are based on the ColebrookWhite equation. Among the most wellknown and widely used empirical correlations are the SwameeJain equation (2) [8], with a maximum estimated error of 3.2%, and the Haaland equation (3) [9], with a maximum estimated error of 2.1%.
Several authors have proposed various empirical and explicit relationships to decrease the numerical error associated with the friction factor in relation to ε and R_{e}. They have employed different methods to obtain a more accurate solution to achieve this. Some authors, such as Mikata and Walczack [10], Rollmann and Spindler [11] and Biberg [12] apply the Lambert ω function. Serghides [13], Vatankhah [14], and Azizi et al. [15] obtain correlations through combinations of algebraic procedures. Chen [16], Schorle et al. [17], Zigrang and Sylvester [18], Sousa et al. [19], Romeo et al. [20] and Offor and Alabi [21] obtain correlations using the recursive method with modifications of constants and exponents. Santos et al. [22] and Alfaro et al. [23] conduct experimental evaluations to calculate the friction factor.



In addition to the aforementioned authors, Pérez et al. [24] present a list of fortynine (49) explicit relationships for calculating the friction factor. This list starts with the Moody equation [7] and ends with the one proposed by Azizi et al. [15].
It is worth mentioning that some recent studies have conducted reviews of the errors associated with the friction factor in correlations reported in the literature [25]. These studies reveal that the relationship proposed by Praks and Brkić [26] has a maximum percentage error of 0,001204%, the relationship proposed by Serghides [13] yields an error of 0,00256%, the relationship proposed by Vantakhak [14] has an error of 0,005952%, and the relationship proposed by Romeo et al. [20] presents an error of 0,007468%. Lamri and Easa [27] apply the Lagrange inversion theorem and obtain an error of 0,002% for four terms.
Based on the results obtained by the mentioned authors, it can be inferred that the error produced by each empirical equation is due to the structure of the equation with the algebraic terms it comprises and the coefficients and exponents used.
The numerical precision in the number of decimal digits of the friction factor is related to the percentage relative error. Therefore, it is essential to calibrate the coefficients and exponents to create a new empirical correlation that has a simple structure as a mathematical model.
This work develops an explicit correlation using the recursive method to calculate the friction factor for turbulent flow in pipes. Additionally, this correlation is evaluated for four explicit relationships that calculate the friction factor for the initial approximation. The methodology is described in Section 2, the results obtained for the friction factor and percentage errors are presented in Section 3, and finally, the conclusions of the analysis are discussed in Section 4.
2. Materials and methods
2.1.Graphical representation of the correlation curve fit
Figure 1 displays a generic scheme with three curve trajectories. It illustrates the curve of an implicit analytic function y = f(x, y). Additionally, it shows the curve of an explicit empirical function y = h(x), which is offset from the curve of the analytic function. The segmented curve corresponds to the correlation, which is the recursive function y_{n+1} = f(x, y_{n}). This function approaches the curve of the analytic function. 
At a local reference point (x_{o}, y_{o}) for the empirical function y_{o} = h(x_{o}), the data x_{o} is within the range from x_{a} to x_{b} (xaxis), and the output data y_{o} is within the range from y_{a} to y_{b} ( yaxis). For the analytic function, when xo, the output data is y_{m}, establishing the reference point (x_{o}, y_{m}). Similarly, for recursion, when x_{o}, is the input data, the result is y_{n+1}, defining the reference position (x_{o}, y_{n+1}). For a fixed point x_{o}, as the algebraic terms of the recursion y_{n+1} increase, starting from y_{o}, the dependent variable y_{n+1} approaches the fixed value y_{m} This implies that the numerical error gradually decreases until achieving numerical convergence y_{m} = y_{n+1}. Therefore, for different input values x_{o}, the recursion curve would overlap the analytic curve in the range x_{a} ≤ x_{o} ≤ x_{b}, and the output data would fall within the range y_{a} ≤ y_{n+1} ≤ y_{b}.
Figure 1. Basic schematic representation of the curves for the analytic, empirical, and recursive functions
In Figure 1 the steps of the recursive method shown in equation (4).
Where the first approximation is y_{1}, the second is y_{2}, the third is y_{3}, and the last one y_{n+1}. The succession for y_{n+1}, increases progressively, starting from n = 0. The function y_{o} = h(x_{o}) is an explicit mathematical expression that provides a representation of a closed initial calculation, where y_{o} is the first approximate solution.




2.2.Explicit relationship
To calculate the initial first approximation of the friction factor λ_{o}, it is necessary to establish a mathematical expression as an explicit relationship for this approximate solution. To obtain the explicit relationship λ_{o}, equation (1) of ColebrookWhite [5, 6] was considered. was removed from the argument of the ColebrookWhite equation, and the positions of the coefficients a_{i} and exponents n_{i} were adjusted. As a result, the explicit relationship λ_{o} for the initial friction factor calculation was structured as equation (5).
The input data consists of the independent parameters: the relative roughness ε and the Reynolds number R_{e}. These parameters are set in the range of 1×10^{−06} ≤ ε ≤ 0, 05 and 4000 ≤ R_{e} ≤ 1×10+08. The output data is λ_{o}. The coefficients a_{1}, a_{2}, a_{3} and the exponents n_{1} and n_{2} of equation (5) were iteratively calibrated in an Excel 2019 spreadsheet using equation (1) of ColebrookWhite as a reference. The best results regarding the magnitudes of the coefficients and exponents were selected to establish two explicit relationships. Table 1 displays the magnitudes of the coefficients and exponents for the two proposed explicit relationships, presented as equations (6) and (7).
Table 1. Calibrated values of coefficients and exponents
2.3.Correlation adjustment
Based on the recursive method, equation (1) of ColebrookWhite [5, 6] was adjusted to calculate the friction factor using λn+1, as the output data and λ_{n} as the input data. The modified equation (8) is expressed as follows.

Where a = −2, b = ε/3, 7 y c = 2, 51/R_{e}. Equation (8) is based on the base 10 logarithm. This equation is used to perform the calculations in this work. It can also be expressed in terms of the natural logarithm as follows: = a_{1}In[b+c/], where a_{1} = a/ln (10). Using equation (8), the correlation expressed as equation (9) was established to calculate the increment of the succession λ_{n+1} = λ_{2}, λ_{4}, λ_{6}, λ_{8}.
Where D= , C= , B= y A= . Equation (9) was extended by adding terms to evaluate up to λ_{n+1} = λ20, although the mathematical expression for λ_{20} is not presented because the procedure is similar. The same principle as in equation (8) is applied. The purpose of evaluating the correlation in a segmented manner was to determine the decrease in the percentage error of the friction factor for different values of the Reynolds number and relative roughness. As input value of λ_{o }for A = 1/ en in equation (9), four explicit relationships were considered: the SwameeJain relationship [8] (equation (2)), the Haaland relationship [9] (equation (3)), and the equations (6) and (7) proposed in this work. Each relationship was evaluated separately in equation (9). The percentage error of the friction factor λ(%) was calculated using the following the equation (10).
Where λ_{m} represents the friction factor of the exact solution of the ColebrookWhite equation, and λ_{n+1} represents the friction factor obtained from equation (9). It is worth mentioning that all numerical calculations and graphs were performed in an Excel 2019 spreadsheet.
3. Results and discussion
3.1.Correlation and explicit relationships for friction factor calculation
The correlation created using the recursive method is expressed by equation (9), where the parameter values are a = −2, b = ε/3, 7, c = 2, 51/R_{e}. This correlation is a mathematical expression that models the trajectory of the friction factor curve, λ, in relation to the relative 

roughness, ε, and the Reynolds number, R_{e}. Considering the term A = 1/, in which the explicit relationships of 1/are substituted, acting as the correlation’s finalizing factor. The two explicit relationships proposed in this work to close equation (9) are equations (6) and (7), and they are expressed as follows. It is worth mentioning that equation (9) has a simple structure for direct calculations, which improves the precision in reducing the error of the friction factor with each additional term in the recursion. Therefore, the exact decimal digits of the friction factor increase. Below, we present the percentage errors of the friction factor yielded by equation (9) for the explicit relationships that serve as a closure for A = 1/ , equation (2) by SwameeJain, equation (3) by Haaland, and the proposed equations (6) and (7).
3.2.Percentage errors of the friction factor
Figure 2 shows the plots of the curve trajectories of the percentage errors of the friction factor λ_{o}, calculated by the initial first approximation for equations (2), (3), (6) and (7). These curve trajectories are essential to understand the effect of the coefficients and exponents on variable values of the relative roughness, ε, and the Reynolds number, R_{e}.
For the Reynolds number range of 4000 ≤ R_{e} ≤1×10^{08} and the relative roughness range of 1×10^{−06} ≤ ε ≤ 0, 05, equation (6) exhibits an estimated maximum percentage error of 2.1%, while equation (7) has an error of 3.1%. SwameeJain’s equation (2) shows an estimated maximum error of 3.2%, and Haaland’s equation (3) has an error of 2.1%. In certain areas, equations (2), (3), (6), and (7) exhibit errors of around 0.1% (Figure 2). It is worth mentioning that the figures only depict curve trajectories for the relative roughness range of 0, 00001 ≤ ε ≤ 0, 05.
For ε = 0, 05 (Figure 2a), equations (3) and (6) exhibit trajectories with a horizontal trend starting from the local position R_{e} = 1 × 10^{05}. The curve of equation (7) overlaps with the curve of equation (2) and shows a straightline trend with a negative slope. For ε = 0, 00001 (Figure 2d), the fluctuations in the friction factor curves are greater compared to the other curves illustrated in Figure 2.
For hydraulically rough pipes (ε = 0, 05), the trends differ much more from each other than in hydraulically smooth pipes (ε = 0, 00001), especially in the case of equations (2) and (7) for the curve in Figure 2a. 
The curves show that the coefficients and exponents of each relationship have a dominant effect that defines their own trajectory behavior. It is worth mentioning that, in Figure 2 and other figures shown below, there are inflection points in the curves during the descending peaks for specific Reynolds numbers, which are not visible because the error output data (yaxis) are absolute values according to equation (10). Additionally, it is essential to note that the vertical axis is on a logarithmic scale base 10 to facilitate the analysis of the curve trajectories.
Figure 2. Percentage errors of the friction factor for λ_{o}, as the calculation of the first approximation of Equations (2), (3), (6) and (7)
Figures 3, 4, 5 and 6 show the trajectories of the curves for the percentage errors of the friction factor for λ_{2}, λ_{4}, λ_{6} and λ_{8}. As the terms increase, equations (2), (3), (6) and (7) define their own curve trajectories. The curve trajectories exhibit the highest percentage relative error of the friction factor for each value of ε, at the position R_{e} = 4000, for λ_{2}, λ_{4}, λ_{6} and λ_{8} The regions with the highest errors of the friction factor are found at the local positions ε = 0, 00001 and R_{e} = 4000, as illustrated in Figures 3d, 4d, 5d and 6d. The equations (2), (3), (6) and (7) for λ_{8} (Figure 6d) have errors less than 0,000002%. For λ_{6} (Figure 5d), the maximum errors are 0,00006%. Forλ_{4} (Figure 4d), the errors are 0,002%, and for λ_{2} (Figure 3d), the errors are 0,061%. 
The magnitudes of the coefficients and exponents in each explicit relationship exhibit a specific behavior, which influences the evolution of the curve trajectories as the relative roughness increases within the same range of Reynolds numbers, as illustrated in Figures 2 to 6. For future work, it is recommended to perform comparisons with other explicit relationships by substituting them into equation (9) to determine which one yields the lowest percentage errors. It is worth mentioning that something similar to what is observed in Figure 2a also occurs in Figure 3a. For λ_{2} and ε = 0, 05, equations (2) and (7) show a more significant difference in hydraulically rough pipes, with a trend of straight lines with a negative slope. Similarly, equations (3) and (6) also exhibit straight trajectories and intersect in the region around R_{e} = 1 × 10^{05}. This phenomenon is also observed in Figures 4a, 5a, and 6a, where the trajectories show a trend of straight lines for rough pipes (ε = 0, 05), respectively. Table 2 displays the maximum errors of the friction factor for ε = 0, 00001 and the local Reynolds numbers 4 × 10^{03}, 1 × 10^{04}, 1 × 10^{05}, 1 × 10^{06}, 1 × 10^{07} and 1 × 10^{08}. These values correspond to λ_{o}, λ_{2}, λ_{4}, λ_{6} and λ_{8}, which are related to Figures 2, 3, 4, 5 and 6. For equation (6) for λ_{8}, considering the conditions ε = 0, 00001 and R_{e} = 4×10^{03}, a maximum percentage error of the friction factor of 1, 7×10^{−06}% is obtained. For equations (2), (3) and (7), the errors are below 1, 7 × 10^{−06} %. It is worth mentioning that as the recursion increases, the percentage errors decrease. Consequently, the friction factors for λ_{8} exhibit seven exact decimal digits for ε = 0, 001, eight exact decimal digits for ε = 0, 00001 and nine exact decimal digits for ε = 0, 05 and ε = 0, 0001, compared to the friction factor of ColebrookWhite equation (1), as shown in Table 3. For recursion values lower than λ_{8}, the exact decimal digits decrease. For λ_{6} there are six exact decimal digits, for λ_{4} there are five exact decimal digits, and for λ_{2} there are four exact decimal digits. In equation (9) for λ_{8}, the maximum percentage error of the friction factor was obtained, with a value of 0.0000017%, which is significantly lower than other reported percentage errors. Brkić and Stajić [25], obtained errors around 0.001204%, Praks and Brkić [26] obtained an error of 0.002560%, and Serghides [13] obtained an error of 0.002560%. It is worth mentioning that the percentage errors reported by Brkić and Stajić [25] have not been verified by the authors of this study through numerical calculations. Therefore, they are presented solely for comparative purposes. 
Figure 3. Percentage errors of the friction factor for λ_{2} _{ }
Figure 4. Percentage errors of the friction factor for λ4 
Figure 5. Percentage errors of the friction factor for λ_{6} _{ }
Figure 6. Percentage errors of the friction factor for λ_{8} 
Table 2. Percentage errors, λ (%), of the friction factor for the recursions, for ε = 0, 00001 and local Re
Figure 7 displays the decrease in the percentage error of the friction factor as recursion increases for λ_{2}, λ_{4}, λ_{6}, λ_{8}, up to λ_{20}. For equations (2), (3) (6) and (7), with, ε = 0, 00001 and a local position of R_{e} = 4E + 03, the following percentage errors are obtained: λ_{9}, 2, 85E − 07%, λ_{10}, 4, 95E − 08 %, λ_{11}, 8, 55E − 09 %; λ_{12}, 1, 5E − 09 %, λ_{14}, 4, 5E − 11 %, λ_{16}, 1, 5E − 12%, λ_{18}, 5, 5E − 14%, and λ_{20} 0,0%. For other values of relative roughness and Reynolds numbers, the percentage errors of the friction factor are lower. This indicates that as the Reynolds number increases, fewer terms in the recursion are required to achieve numerical convergence. 
Table 3. Comparison of numerical values of friction factors for the recursions considering the ColebrookWhite equation R_{e} = 4E + 03 and local ε
Equation (9) is applicable for hydraulic pipes with values lower than ε = 0, 00001, and even for ε = 0. When ε = 0, equation (9) simplifies, and numerically, the error curves of the friction factor resemble the trajectories shown in Figure 7c. The results from the graphs and tables are not presented due to their similarity. For λ_{8}, and the range 4000 ≤ R_{e} ≤ 1E + 08, the percentage error is lower than 1,5E06%.
The SwameeJain and Haaland equations were evaluated in equation (9), starting from λ_{2}, and yielded similar results to the evaluations of the proposed equations (6) and (7).
Any equation that has structures different from equations (6) and (7) and that is used in equation (9) can reduce the percentage error of the friction factor.
The advantage of equation (9) lies in its simple structure and ease of use for directly calculating the friction factor as an approximate solution. 
Figure 7. Percentage errors of the friction factor for equations (2), (3), (6) and (7)
4. Conclusions
The correlation expressed in equation (9) for λ_{8}, and the explicit relationships represented by equations (6) and (7) for calculating λ_{o} as an initial approximation provide an approximate solution for the implicit ColebrookWhite equation (1). Equation (9) is applicable for turbulent flows within the Reynolds number range of 4000 ≤ R_{e} ≤ 1E + 08 and the relative roughness range of 0, 05 ≥ ε ≥ 0, 00001. It can also be applied to smooth pipes within the same Reynolds number range. However, equation (9) is not applicable for R_{e} < 4000. 
Within the relative roughness range of, 0, 05 ≥ ε ≥ 0, 00001, and Reynolds number range of 4 × 10^{03} ≤ R_{e} ≤ 1 × 10^{08}, the estimated value of the maximum percentage error of the friction factor is 1, 7 × 10^{−06} %, for ε = 0, 00001 and R_{e} = 4 × 10^{03}. In this case, the friction factor has seven exact decimal digits. For other values of relative roughness and Reynolds numbers, the numerical magnitudes of the friction factor exhibit more than seven exact decimal digits. For recursions beyond λ_{8}, the exact decimal digits increase. For ε = 0, 00001 and R_{e }= 4 × 10^{03}, λ_{10} exhibits an error of 4, 95×10^{−08} %, λ_{14} yields an error of 4, 5 × 10−11 %, and λ_{20} exhibits an error of 0,0%.
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