In general, this new equation is significant for designing and operating a pipeline to predict flow discharge. In order to predict the flow discharge, accurate determination of the flow loss due to pipe friction is very important. However, existing pipe friction coefficient equations have difficulties in obtaining key variables or those only applicable to pipes with specific conditions. Thus, this study develops a new equation for predicting pipe friction coefficients using statistically based entropy concepts, which are currently being used in various fields. The parameters in the proposed equation can be easily obtained and are easy to estimate. Existing formulas for calculating pipe friction coefficient requires the friction head loss and Reynolds number. Unlike existing formulas, the proposed equation only requires pipe specifications, entropy value and average velocity. The developed equation can predict the friction coefficient by using the wellknown entropy, the mean velocity and the pipe specifications. The comparison results with the Nikuradse’s experimental data show that the R2 and RMSE values were 0.998 and 0.000366 in smooth pipe, and 0.979 to 0.994 or 0.000399 to 0.000436 in rough pipe, and the discrepancy ratio analysis results show that the accuracy of both results in smooth and rough pipes is very close to zero. The proposed equation will enable the easier estimation of flow rates.
In order to accurately predict the flow discharges in pipe flow, not only the diameter and flow velocity, but also the discharge loss due to pipe friction, is very important. However, the most important factor in calculating flow loss in pipes is the friction coefficients. Existing equations are limited in accurately predicting friction factors due to material properties.
Studies with various experiments have been performed for calculating the friction coefficient. Jones [
Studies of numerical analysis have been undertaken by many researchers. Romeo et al. [
Studies with computer methods have been developed and combined. Tonye [
Other papers, such as that by Liakopoulos et al. [
Although various computer programs and methods have been developed for estimating pipe friction loss, research has been found to be insufficient to calculate the exact pipe friction coefficient. For every material condition, there is friction value ranged for its own value. However, sometimes these friction values are out of date or miscalculated. Previous equations on friction coefficient has limits to certain roughness of pipe or flow. Therefore, in order to estimate the pipe friction coefficient in any type of roughness and in laminar and turbulence flow, as mentioned in this study, we would like to propose an equation using entropy for estimating friction loss.
Chiu [
The mean velocity equation (Equation (3)) using K(M) and water level is proposed and used, as shown in Equation (2).
In fluid dynamics, the Darcy–Weisbach equation is used for estimating friction coefficient which is shown in Equation (4).
The friction coefficient was estimated by using the equation regarding the frictional velocity, as shown in Equation (5).
In this part, an equation for calculating the friction head loss using Chiu’s velocity distribution equation and friction head loss equations is purposed.
If Equation (3) is differentiated with respect to the velocity gradient, and
As for the velocity (u is zero at the boundary layer), Equation (6) can be modified as Equation (7).
Equation (7) can be represented with shear stress equation (Equation (8)), as follows:
Additionally, Equation (9) can be represented with shear velocity equation (Equation (10)), as follows:
The friction coefficient equation can be determined by submitting Equation (5) into Equation (12) following as Equation (13).
The proposed friction coefficient equation is as follows:
F(M) is defined as Equation (15).
In this paper, the pipe friction coefficient was proposed. The proposed equation was verified using Nikuradse’s experimental data [
The measured data are classified into five diameters of 1, 2, 3, 5, and 10 listed in
The friction coefficient results determined from the developed equation with the data measured by Nikuradse from the smooth pipes experiments were compared. Then, entropy M values were determined for 125 cases, considering the flow rate and diameter of the smooth pipe. From these entropy M, F(M) was calculated by using Equation (14). Then, the friction coefficient was estimated from Equation (15).
The estimated friction coefficients were compared with the measured data from Nikuradse’s experiments as shown in
According to
In the case of the rough pipe experiments performed by Nikuradse, the roughness was represented with sand attached to the pipe wall. Nikuradse measured from total of 362 cases experiments and the classified into six relative roughness conditions: 507, 252, 126, 60, 30.6, 15. Each type of relative roughness was classified into three diameters of 9.94 cm, 4.94 cm and 2.474 cm. In the data, relative roughness (
The entropy M values were determined for 362 cases investigated by Nikuradse considering the flow rate and diameter of the smooth pipe. We also calculated F(M) using Equation (14) and determined the friction coefficient from Equation (15).
The determined friction coefficients were compared with the measured friction coefficients from Nikuradse’s experiments as shown in
For the visibility of the graph,
The RMSE is a measure of the residual, which is the difference between the values predicted by the model and actual observed values. The RMSE enables predictive power to be integrated into a single unit of measurement. The RMSE of the model’s prediction for the estimated variable
The determined friction coefficients of the pipe were compared with those measured by Nikuradse, as shown
The more quantitative validation, this study used discrepancy ratio method which method is a statistical analysis method for calculating ratio between measured and determined coefficients Equation (17).
Each calculated constraint is sorted in ascending order and then expressed as a percentage of certain divided section. If the value is greater than 0, it means overdetermination, and if it is less than 0, it means underdetermination.
Determined friction coefficients were compared with Nikuradse’s results, as shown in
This study proposed a new equation using the entropybased mean velocity equation for estimating the pipe friction coefficient, which is a very important factor for the determination of pipe friction loss. The proposed equation used the F(M) factor derived by combining Darcy–Weisbach’s formula for friction head loss, shear velocity equation and Chiu’s average velocity equation. Additionally, the proposed equation can be simply and easily calculated using only the pipe specifications, entropy values and average velocity without knowing the friction head loss and Reynolds number.
The evaluation results show that the proposed equation well represented the Nikuradse’s measured data and also the R^{2} value were 0.998 for smooth pipes and 0.979 to ~0.994 for rough pipes. In addition, the RMSE were determined to be 0.00036 for smooth pipes and from 0.00039 to 0.00044 for rough pipes. The discrepancy ratio had range from 0.055 to −0.029 for smooth pipes and a range from −0.011 to −0.003 in the case of rough pipes. Through these evaluation studies, the accuracy of the proposed equation was high and simple in applications. This shows that proposed equation can be applied to various pipe roughness values and laminar and turbulence flow.
In the future, it is expected that the pipe friction coefficient equation proposed in this study will enable the convenient estimation of flow rates to a greater extent than the existing equations. In addition, it will be possible to develop a new method for calculating roughness coefficients through continuous research on the proposed equation. Additionally, it will also be possible to respond to the Reynolds number through the study of the application method for F(M) used in the proposed equation.
Y.M.C. and Y.W.C. carried out the survey of previous studies. Y.M.C. wrote the manuscript. Y.W.C. conducted all simulations. Y.M.C., S.B.S., Y.W.C. conceived the original idea of the proposed method. All authors have read and agreed to the published version of the manuscript.
This research was funded by Research Program to Solve Regional Convergence Issues of the National Research Foundation of Korea (NRF) funded by the Korean government (Ministry of Science and ICT(MSIT)), grant number 2020M3F8A1080153.
This research was supported by Research Program to Solve Regional Convergence Issues of the National Research Foundation of Korea (NRF) funded by the Korean government (Ministry of Science and ICT(MSIT)) (No. 2020M3F8A1080153).
The authors declare no conflict of interest.
Comparison of friction factor (smooth pipe).
Comparison of friction factor (rough pipe).
Comparison and verification of friction factor (smooth pipe).
Comparison and verification of friction factor (rough pipe, r/k = 507).
Comparison and verification of friction factor (rough pipe, r/k = 252).
Comparison and verification of friction factor (rough pipe, r/k = 126).
Comparison and verification of friction factor (rough pipe, r/k = 60).
Comparison and verification of friction factor (rough pipe, r/k = 30.6).
Comparison and verification of friction factor (rough pipe, r/k = 15).
Discrepancy ratio for velocity results in smooth pipe.
Discrepancy ratio for velocity results in rough pipe.
Smooth pipe.
D 




F(M) 



1  0.25  3.124 
42.8 
0.014 
4.072 
3.07 
0.016692 
2  0.5  6.711 
114.6 
0.0114 
14.555 
17 
0.015797 
3  0.75  4.895 
91.4 
0.0114 
15.948 
37 
0.01444 
5  1.25  3.857 
71.2 
0.0134 
21.517 
29.3 
0.0124344 
10  2.5  11.234 
259 
0.0122 
112.076 
238.8 
0.0095988 
Rough pipe.
r/k  D 




F(M) 



507  9.94  2.485  0.923 
15.45 
0.009 
11.622 
13.0017 
0.0171 
252  9.94  2.485  3.753 
72.3 
0.0089 
37.815 
55.9758 
0.0209 
4.94  1.235  2.578 
43.4 
0.0086 
14.325 
16.2181 
0.0209 

126  9.94  2.485  7.018 
121 
0.0081 
85.427 
88.1251 
0.026 
2.474  0.6185  1.597 
22.8 
0.128 
0.524 
2.4917 
0.0246 

60  9.8  2.434  6.611 
101 
0.0092 
80.32 
74.9894 
0.0342 
2.434  0.6085  1.665 
23.8 
0.0114 
5.54 
4.4978 
0.0303 

30.6  9.64  2.41  7.449 
99 
0.009 
121.686 
85.9014 
0.0447 
4.87  1.2175  5.153 
70 
0.0105 
36.08 
26.6073 
0.0426 

2.434  0.6085  1.704 
24.9 
0.0107 
5.738 
4.69894 
0.0378 

15  4.82  1.205  6.515 
75.5 
0.0072 
51.318 
27.5423 
0.0596 
2.412  0.603  2.427 
30.81 
0.0098 
9.153 
5.88844 
0.0497 
Prediction results R^{2} and RMSE.




0.9977  0.000366  

507  0.9923  0.000436 
252  0.9796  0.000434  
126  0.9886  0.000423  
60  0.9875  0.000399  
30.6  0.9941  0.000433  
15  0.9846  0.000420 