# Application of AI-Based Techniques on Moody’s Diagram for Predicting Friction Factor in Pipe Flow

^{*}

## Abstract

**:**

^{2}), and Nash–Sutcliffe efficiency (NSE)), it was revealed that the predictions made by the Random Forest model were the most reliable when compared to other AI tools. The main objective of this study was to highlight the limitations of artificial intelligence (AI) techniques when attempting to effectively capture the characteristics and patterns of the friction curve in certain regions of turbulent flow. To further substantiate this behavior, the conventional algebraic equation was used as a benchmark to test how well the current AI tools work. The friction factor estimates using the algebraic equation were found to be even more accurate than the Random Forest model, within a relative error of ≤±1%, in those regions where the AI models failed to capture the nature and variation in the friction factor.

## 1. Introduction

_{e}) and relative roughness of the pipe $\left(\frac{\epsilon}{D}\right)$ [3,4,5,6,7,8,9], where $\epsilon $ is the roughness height of the pipe in meters, and D is the diameter of the pipe in meters. In the past, the equation of the friction factor given by [3,4] was used for gas and oil transporting pipelines in the petroleum and chemical industries. The expression given by [3,4] has been extensively used to estimate the friction factor for turbulent flow (R

_{e}≥ 4000). One of the major disadvantages of this expression is that it requires a number of iterations to obtain the final solution. As the formula has to be solved iteratively, it is a time-consuming solution. Research by [5] reanalyzed the work of [3,4] and rearranged all of the datasets systematically in a log-log graph, where f is a function of relative roughness $\left(\frac{\epsilon}{D}\right)$ and the Reynolds number (R

_{e}). The explicit equation proposed by [6] is valid for 0 ≤ $\left(\frac{\epsilon}{D}\right)$ ≤ 10

^{−2}and 4 × 10

^{3}≤ R

_{e}≤ 10

^{7}. Likewise, the expression proposed by [7] is applicable for 10

^{−5}≤ $\left(\frac{\epsilon}{D}\right)$ ≤ 4 × 10

^{−2}and 4 × 10

^{3}≤ R

_{e}≤ 5 × 10

^{7}. However, the expression given by [7] depends on three parameters, a, b, and c, as given in Table 1. This expression gives an accurate result with an error of ±5%, but the major disadvantage of this expression is that it cannot be applied to smooth turbulent flow.

^{−6}≤ $\left(\frac{\epsilon}{D}\right)$ ≤ 10

^{−2}and 5 × 10

^{3}≤ R

_{e}≤ 10

^{8}, with an error of ±1%. This expression is an explicit solution of the friction factor. The explicit expression given by [8] is used for calculating the friction factor and also can be used for finding a few more important factors in pipe design. Research by [9] contributed a universal expression that is effective for all flow regions, such as laminar flow, turbulent flow, and transition flow. The expression given by [9] predicts the friction factor more accurately than the values obtained from [3,4]. The expression given by [9] can also be simplified for the turbulent flow region. The expression for the turbulent flow region gives an accurate result with an error of ≤ 1.5%; however, in the near transition flow range, the error increases to 3%.

_{f}, $\epsilon ,$ and $\vartheta $, and stated the expression for determining the discharge in the pipe flow in a turbulent region, as given in Table 2. Here, Q is the discharge in m

^{3}/s, $\vartheta $ is the kinematic viscosity in m

^{2}/s, $L$ is the pipe length in meters, and ${h}_{f}$ is the head loss due to friction in meters. Later research conducted by [10] gave an alternative expression that was valid for all flow conditions. This is also an exact solution with a maximum error of 0.1%.

_{f}, $\epsilon $, Q, and $\vartheta $, and stated an expression for determining the diameter of the pipe in a turbulent region, which is given in Table 3. Later, research by [10] gave an alternative expression that was valid for all flow conditions. The diameter is found within an error of 2.75%, while the inaccuracy is only around 4% in the transition range. Further details on this are available in [11], including use of the friction factor in network synthesis, water transition lines, water distribution systems, etc. Likewise, many more expressions were given by [12] for finding the drag coefficient of a particle and the fall velocity of a sediment particle, which could be useful along with the friction factor to describe slurry transport in pipelines. Table 2 and Table 3 provide valuable linkages between the expressions of discharge, diameter, and the friction factor.

## 2. AI-Based Models

_{e}) and relative pipe roughness $\left(\frac{\epsilon}{D}\right)$. All of these models are described in this section.

## 3. Collection of Data and Model Selection

^{−2}≤ $\left(\frac{\epsilon}{D}\right)$≤ 10

^{−6}and 5 × 10

^{3}≤ R

_{e}≤ 10

^{8}. The purpose was to test the efficiency of the AI tools in capturing the friction factor from Moody’s diagram. Hence, 1052 data points were collected. In approach-1, the data were randomly collected without taking the nature of the curve into consideration. Out the whole dataset, 789 (75% of data) data points were taken for training and 263 were taken for testing of the AI-based models, as shown in Figure 1a.

^{−2}≤ $\left(\frac{\epsilon}{D}\right)$ ≤ 10

^{−6}and 3 × 10

^{3}≤ R

_{e}≤ 10

^{8}. For training and testing, 3111 data points were collected. In the 2nd approach, 2332 (75% of data) data points were taken for training and 779 were taken for testing the AI-based models. Here, in approach-2, the data were selected at a particular interval. The graphical presentation of data selection is shown in Figure 1b. In approach-3, different curves with different $\left(\frac{\epsilon}{D}\right)$ values were chosen for training and testing of the AI-based models. The purpose of this approach was to make the AI techniques understand the trends of the friction factor with variations in $\left(\frac{\epsilon}{D}\right)$ and R

_{e}. In approach-3, 2418 (78% of data) data points were taken for training and 693 were taken for testing of the AI-based models. Here, in approach-3, the data were selected at a particular interval with respect to $\left(\frac{\epsilon}{D}\right)$. The graphical presentation of data selection is shown in Figure 1c.

^{2}), root-mean-square error (RMSE), mean absolute error (MAE), and Nash–Sutcliffe efficiency (NSE) as described by McCuen et al. [25] were chosen. The representation of R

^{2}, MAE, NSE, and RMSE is as follows:

_{obs}is the observed data of the friction factor, and y is the simulated friction factor. For better accuracy, the values of R

^{2}and NSE must be close to 1 and those of MAE and RMSE should be close to zero. In modeling the AI tools, the primary work was to collect the data and divide it into training and testing datasets. This was followed by applying different models and choosing the optimal parameters of the models using the hit and trial method. Subsequently, the statistical parameters were evaluated, as mentioned above. In the final stage, the models were ranked as per their performance in predicting accurate results. The process involved in organizing the data, selecting the different models, and choosing the best-fit model as per the statistical parameters is explained in the following flow diagram (Figure 2) in which all three approaches are adopted.

## 4. Results and Analysis

^{2}, MAE, and NSE. For each training and testing dataset, the statistical parametric evaluation was performed separately. For the SVM model, the kernel parameters to obtain the optimum predicted values are shown in Table 4. The model performances of the training and testing datasets for approach-1, -2, and -3 are shown in Table 5, Table 6 and Table 7. As per the statistical analysis, the RF model was the most accurate model and ranked 1st among all of the AI-based models, with high R

^{2}and NSE values, as shown in Table 5, Table 6 and Table 7. The RT and REPTree models were ranked as the 2nd and 3rd most accurate models in predicting the friction factor. SVM_POLY was the least accurate model, ranked 8th, with a lower value of R

^{2}during training and testing. For all three approaches, the RF model showed better results as compared to the other AI models based on the statistical analysis.

^{6}≤ R

_{e}≤ 4 × 10

^{7}. From Figure 9, it can be seen that during training as well as testing, the predicted values were less accurate in the datasets of the turbulent flow region in the range of 2 × 10

^{5}≤ R

_{e}≤ 10

^{8}for approach-2. Likewise, in testing, most of the datasets that failed to be captured by the AI tools were in the range of 3 × 10

^{4}≤ R

_{e}≤ 10

^{7}. But, in other sections, the AI tools were good enough to capture accurate results. Equally, for approach-3, the friction factor was plotted in two different segments, such as relative error ≤ 5% and 5% ≤ relative error ≤ 12% for training and relative error ≤ 5% and 5% ≤ relative error ≤ 20% for the testing datasets. The approach that exhibited the highest degree of error in data capture during the testing phase was approach-3.

^{5}≤ R

_{e}≤ 10

^{8}, both in the training and testing datasets. But, in the testing phase, most of the datasets failed to be captured by the AI tools for almost all regions. The testing datasets were unable to capture the variation as well as the trend in the curve and showed a maximum error of 20%. Thus, it could be inferred that the Al tools were not successful in capturing the trends in the friction factor in turbulent regions with higher Reynolds numbers.

_{e}) and relative roughness $\left(\frac{\epsilon}{D}\right)$ in the current context.

## 5. Conclusions

^{−6}≤ $\left(\frac{\epsilon}{D}\right)$ ≤ 10

^{−2}and 5 × 10

^{3}≤ R

_{e}≤ 10

^{8}. It is worth emphasizing that the algebraic expression is a powerful method and has the potential to predict the friction factor in pipe flow with greater accuracy than current AI-based methods in turbulent flow regions.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Wang, H.; Xu, Y.; Shi, B.; Zhu, C.; Wang, Z. Optimization and intelligent control for operation parameters of multiphase mixture transportation pipeline in oilfield: A case study. J. Pipeline Sci. Eng.
**2021**, 1, 367–378. [Google Scholar] [CrossRef] - Xu, Y.; Wang, Z.; Hong, J.; Zhou, B.; Pu, H. An Insight into Wax Precipitation, Deposition, and Prevention Stratagem of Gas-Condensate Flow in Wellbore Region. J. Energy Resour. Technol.
**2023**, 145, 093101. [Google Scholar] [CrossRef] - Colebrook, C.F.; White, C.M. Experiments with Fluid Friction Factor in Roughened Pipes. Proc. R. Soc. Lond. Ser. A
**1937**, 161, 367–381. [Google Scholar] [CrossRef] - Colebrook, C.F. Turbulent Flow in Pipes, with Particular Reference to the Transition Region between the Smooth and Rough Pipe Laws. J. Inst. Civ. Eng.
**1939**, 11, 133–156. [Google Scholar] [CrossRef] - Moody, L.F. Friction Factors for Pipe Flow. Trans. Am. Soc. Mech. Eng.
**1944**, 66, 671–681. [Google Scholar] [CrossRef] - Moody, L.F. An approximate formula for pipe friction factors. Trans. Am. Soc. Mech. Eng.
**1947**, 69, 1005–1011. [Google Scholar] - Wood, D.J. An explicit friction factor relationship. Civil. Eng.
**1966**, 36, 60–61. [Google Scholar] - Swamee, D.K.; Jain, A.K. Explicit Equations for Pipe Flow Problems. J. Hydraul. Div.
**1976**, 102, 657–664. [Google Scholar] [CrossRef] - Swamee, P.K. Design of a submarine oil pipeline. J. Transp. Eng.
**1993**, 119, 159–170. [Google Scholar] [CrossRef] - Swamee, P.K.; Swamee, N. Full range pipe-flow equations. J. Hydraul. Res.
**2007**, 45, 841–843. [Google Scholar] [CrossRef] - Swamee, P.K.; Sharma, A.K. Design of Water Supply Pipe Networks; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2008; pp. 11–41. [Google Scholar]
- Swamee, P.K.; Ojha, C.S.P. Drag coefficient and fall velocity of nonspherical particles. J. Hydraul. Eng.
**1991**, 117, 660–667. [Google Scholar] [CrossRef] - Kumar, S.; Ojha, C.S.P.; Tiwari, N.K.; Ranjan, S. Exploring the potential of artificial intelligence techniques in prediction of the removal efficiency of vortex tube silt eject. Int. J. Sediment Res.
**2023**, 38, 615–627. Available online: https://www.sciencedirect.com/science/article/abs/pii/S100162792300015X?via%3Dihub (accessed on 31 March 2023). [CrossRef] - Breiman, L. Bagging Predictors. Mach. Learn.
**1996**, 24, 123–140. Available online: https://link.springer.com/content/pdf/10.1023/A:1018054314350.pdf (accessed on 22 September 2023). [CrossRef] - Breiman, L.; Friedman, J.H.; Olshen, R.A.; Stone, C.J. Classification and Regression Trees; Chapman & Hall: New York, NY, USA, 1984. [Google Scholar] [CrossRef]
- Breiman, L. Using Adaptive Bagging to Debias Regression; Report No. 547; Statistics Department, University of California at Berkeley: Berkeley, CA, USA, 1999. [Google Scholar]
- Vapnik, V. The support vector method of function estimation. In Nonlinear Modeling; Springer: Boston, MA, USA, 1998; pp. 55–85. Available online: https://link.springer.com/chapter/10.1007/978-1-4615-5703-6_3 (accessed on 22 September 2023).
- Han, S.; Qubo, C.; Meng, H. Parameter selection in SVM with RBF kernel, function. In Proceedings of the World Automation Congress, Puerto Vallarta, Mexico, 24–28 June 2012; IEEE: Piscataway, NJ, USA, 2012; pp. 1–4. [Google Scholar]
- Sihag, P.; Jain, P.; Kumar, M. Modelling of impact of water quality on recharging rate of stormwater filter system using various kernel function-based regression. Model. Earth Syst. Environ.
**2018**, 4, 61–68. Available online: https://link.springer.com/article/10.1007/s40808-017-0410-0 (accessed on 22 September 2023). [CrossRef] - Smola, A.J.; Sch€olkopf, B. A tutorial on support vector regression. Stat. Comput.
**2004**, 14, 199–222. Available online: https://link.springer.com/article/10.1023/B:STCO.0000035301.49549.88 (accessed on 22 September 2023). [CrossRef] - Quinlan, J.R. Learning with continuous classes. In Proceedings of the 5th Australian Joint Conference on Artificial Intelligence, Hobart, Tasmania, 16–18 November 1992; Volume 92, pp. 343–348. [Google Scholar]
- Holmes, G.; Hall, M.; Prank, E. Generating rule sets from model trees. In Lecture Notes in Computer Science; Springer: Berlin/Heidelberg, Germany, 1999. [Google Scholar] [CrossRef]
- Ayaz, Y.; Kocamaz, A.F.; Karakoç, M.B. Modeling of compressive strength and UPV of high-volume mineral-admixtured concrete using rule-based M5 rule and tree model M5P classifiers. Constr. Build. Mater.
**2015**, 94, 235–240. [Google Scholar] [CrossRef] - Rajesh, P.; Karthikeyan, M. A Comparative study of data mining algorithms for decision tree approaches using WEKA tool. Am. Eurasian Netw. Sci. Inf.
**2017**, 11, 230–241. [Google Scholar] - McCuen, R.H.; Knight, Z.; Cutter, A.G. Evaluation of the nashi-sutcliffe efficiency index. J. Hydrol. Eng.
**2006**, 11, 597–602. [Google Scholar] [CrossRef]

**Figure 1.**Selection of training and testing datasets of Moody’s diagram for the friction factor in pipe flow. (

**a**) Approach-1; (

**b**) Approach-2; (

**c**) Approach-3.

**Figure 3.**Agreement diagram of observed and predicted friction factors of training and testing datasets for approach-1. (

**a**) M5P model; (

**b**) RF model; (

**c**) M5Rules model; (

**d**) REPTree model; (

**e**) RT model.

**Figure 4.**Agreement diagram of observed and predicted friction factors of training and testing datasets for approach-2. (

**a**) M5P model; (

**b**) RF model; (

**c**) M5Rules model; (

**d**) REPTree model; (

**e**) RT model.

**Figure 5.**Agreement diagram of observed and predicted friction factors of training and testing datasets for approach-3. (

**a**) RF model; (

**b**) RT model; (

**c**) REPTree model; (

**d**) M5rules model; (

**e**) M5P model.

**Figure 6.**Taylor diagrams of AI models RF, RT, REPTree, M5P, and M5Rules. (

**a**) Training datasets for approach-1; (

**b**) Testing datasets for approach-1; (

**c**) Training datasets for approach-2; (

**d**) Testing datasets for approach-2; (

**e**) Training datasets for approach-3; (

**f**) Testing datasets for approach-3.

**Figure 7.**Violin error box diagrams. (

**a**) Training datasets of all models for approach-1; (

**b**) Testing datasets of all models for approach-1; (

**c**) Training datasets of all models for approach-2; (

**d**) Testing datasets of all models for approach-2; (

**e**) Training datasets of all models for approach-3; (

**f**) Testing datasets of all models for approach-3.

**Figure 11.**Taylor diagram of training and testing datasets for (

**a**) approach-1; (

**b**) approach-2; (

**c**) approach-3 [8].

**Figure 12.**Violin error box diagrams of training and testing datasets for (

**a**) approach-1; (

**b**) approach-2; (

**c**) approach-3.

Reference | Expression | Condition | Flow Type |
---|---|---|---|

[3,4] | $\frac{1}{\sqrt{f}}=-2log\left(\frac{\epsilon}{3.7D}+\frac{2.51}{{R}_{e}\sqrt{f}}\right)$ | R_{e} ≥ 4000 | Turbulent flow |

[6] | $f=0.0055\left[1+{\left(2\times {10}^{4}\frac{\epsilon}{D}+\frac{{10}^{6}}{{R}_{e}}\right)}^{1/3}\right]$ | $0\text{\u2264}\left(\frac{\epsilon}{D}\right)$≤ 10^{−2}4 × 10 ^{3} ≤ R_{e} ≤ 10^{7} | Turbulent flow |

[7] | $f=a+b{R}_{e}^{-c}$$a=0.094{\left(\frac{\epsilon}{D}\right)}^{0.225}+0.53\left(\frac{\epsilon}{D}\right);b=88{\left(\frac{\epsilon}{D}\right)}^{0.44};$$c=1.62{\left(\frac{\epsilon}{D}\right)}^{0.134}$ | ${10}^{-5}\text{\u2264}\left(\frac{\epsilon}{D}\right)$≤ 4 × 10^{−2}4 × 10 ^{3} ≤ R_{e} ≤ 5 × 10^{7} | Turbulent flow |

[8] | $f=\frac{0.25}{{\left[log\left(\frac{\epsilon}{3.7D}\right)\right]}^{2}}$ | Smooth turbulent flow | |

[8] | $f=\frac{0.25}{{\left[log\left(\frac{5.74}{{{R}_{e}}^{0.9}}\right)\right]}^{2}}$ | Rough turbulent flow | |

[8] | $f=\frac{0.25}{{\left[log\left(\frac{\epsilon}{3.7D}+\frac{5.74}{{{R}_{e}}^{0.9}}\right)\right]}^{2}}$ | ${10}^{-6}\text{\u2264}\left(\frac{\epsilon}{D}\right)$≤ 10^{−2}5 × 10 ^{3} ≤ R_{e} ≤ 10^{8} | Transition zone of turbulent flow |

[9] | $f={\left\{{\left(\frac{64}{{R}_{e}}\right)}^{8}+9.5{\left[ln\left(\frac{\epsilon}{3.7D}+\frac{5.74}{{{R}_{e}}^{0.9}}\right)-{\left(\frac{2500}{{R}_{e}}\right)}^{6}\right]}^{-16}\right\}}^{0.125}$ | All regions | |

[9] | $f=1.325{\left[ln\left(\frac{\epsilon}{3.7D}+\frac{5.74}{{{R}_{e}}^{0.9}}\right)\right]}^{-2}$ | Turbulent flow |

Reference | Expression (m^{3}/s) | Condition |
---|---|---|

[8] | $Q=-0.965{D}^{2}\sqrt{\frac{gD{h}_{f}}{L}}$$\mathrm{ln}\left(\frac{\epsilon}{3.7D}+\frac{1.78\vartheta}{D\sqrt{\frac{gD{h}_{f}}{L}}}\right)$ | Turbulent region |

[10] | $Q={D}^{2}\sqrt{\frac{gD{h}_{f}}{L}}{\left\{{\left(\frac{128\vartheta}{\pi D\sqrt{\frac{gD{h}_{f}}{L}}}\right)}^{4}+1.153{\left[{\left(\frac{415\vartheta}{D\sqrt{\frac{gD{h}_{f}}{L}}}\right)}^{8}-ln\left(\frac{\epsilon}{3.7D}+\frac{1.775\vartheta}{D\sqrt{\frac{gD{h}_{f}}{L}}}\right)\right]}^{-4}\right\}}^{-0.25}$ | All regions |

Reference | Expression (m) | Condition |
---|---|---|

[8] | $D=0.66$${\left[{\epsilon}^{1.25}{\left(\frac{L{Q}^{2}}{g{h}_{f}}\right)}^{4.75}+\vartheta {Q}^{9.4}{\left(\frac{L}{g{h}_{f}}\right)}^{5.2}\right]}^{0.04}$ | Turbulent region |

[10] | $D=0.66{\left[{{\left(214.75\frac{Q\vartheta L}{g{h}_{f}}\right)}^{6.25}+\epsilon}^{1.25}{\left(\frac{L{Q}^{2}}{g{h}_{f}}\right)}^{4.75}+\vartheta {Q}^{9.4}{\left(\frac{L}{g{h}_{f}}\right)}^{5.2}\right]}^{0.04}$ | All regions |

Method | User-Defined Parameter Training and Testing |
---|---|

SVM-POLY | C = 10, d = 3.6 |

SVM-RBF | C = 10, d = 4.8 |

SVM-PUK | C = 10, ω = 1.7, σ = 1 |

Method | R^{2}Training | RMSE | MAE | NSE | Ranking | R^{2}Testing | RMSE | MAE | NSE |
---|---|---|---|---|---|---|---|---|---|

RF | 0.9998 | 0.0001 | 0.0001 | 0.9989 | 1 | 0.9990 | 0.0004 | 0.0003 | 0.9971 |

RT | 0.9992 | 0.0003 | 0.0002 | 0.9983 | 2 | 0.9972 | 0.0005 | 0.0004 | 0.9967 |

REPTree | 0.9982 | 0.0004 | 0.0003 | 0.9973 | 3 | 0.9970 | 0.0005 | 0.0004 | 0.9952 |

M5Rules | 0.9926 | 0.001 | 0.0008 | 0.9892 | 4 | 0.9892 | 0.0014 | 0.0012 | 0.9793 |

M5P | 0.9854 | 0.0013 | 0.0009 | 0.9822 | 5 | 0.9706 | 0.0017 | 0.0015 | 0.9641 |

SVM-PUK | 0.7793 | 0.0049 | 0.0029 | 0.7522 | 6 | 0.6798 | 0.0062 | 0.0039 | 0.5616 |

SVM-RBF | 0.6266 | 0.0061 | 0.0047 | 0.6161 | 7 | 0.4881 | 0.0068 | 0.0051 | 0.4751 |

SVM-POLY | 0.6255 | 0.0061 | 0.0046 | 0.6098 | 8 | 0.4881 | 0.007 | 0.0053 | 0.4397 |

Method | R^{2}Training | RMSE | MAE | NSE | Ranking | R^{2}Testing | RMSE | MAE | NSE |
---|---|---|---|---|---|---|---|---|---|

RF | 0.9998 | 0.0003 | 0.0003 | 0.9995 | 1 | 0.9996 | 0.0004 | 0.0003 | 0.9994 |

RT | 0.9992 | 0.0006 | 0.0004 | 0.9989 | 2 | 0.9990 | 0.0006 | 0.0004 | 0.9988 |

REPTree | 0.9992 | 0.0006 | 0.0004 | 0.9989 | 3 | 0.9990 | 0.0006 | 0.0004 | 0.9988 |

M5Rules | 0.9984 | 0.0008 | 0.0006 | 0.9979 | 4 | 0.9984 | 0.0008 | 0.0006 | 0.9979 |

M5P | 0.9978 | 0.0009 | 0.0006 | 0.9974 | 5 | 0.9978 | 0.0009 | 0.0006 | 0.9974 |

SVM-PUK | 0.9731 | 0.0033 | 0.0017 | 0.9713 | 6 | 0.9735 | 0.0032 | 0.0018 | 0.9732 |

SVM-RBF | 0.9204 | 0.0056 | 0.0041 | 0.9152 | 7 | 0.9191 | 0.0056 | 0.0041 | 0.9192 |

SVM-POLY | 0.8877 | 0.0065 | 0.0053 | 0.8855 | 8 | 0.8862 | 0.0065 | 0.0053 | 0.8894 |

Method | R^{2}Training | RMSE | MAE | NSE | Ranking | R^{2}Testing | RMSE | MAE | NSE |
---|---|---|---|---|---|---|---|---|---|

RF | 0.9998 | 0.0003 | 0.0003 | 0.9995 | 1 | 0.9972 | 0.0024 | 0.0021 | 0.9816 |

RT | 0.9996 | 0.0004 | 0.0003 | 0.9993 | 2 | 0.9968 | 0.0024 | 0.0021 | 0.9814 |

REPTree | 0.9994 | 0.0005 | 0.0004 | 0.9991 | 3 | 0.9968 | 0.0024 | 0.0022 | 0.9810 |

M5Rules | 0.9986 | 0.0008 | 0.0006 | 0.9983 | 4 | 0.9952 | 0.0015 | 0.0011 | 0.9807 |

M5P | 0.9978 | 0.0009 | 0.0006 | 0.9975 | 5 | 0.9950 | 0.0019 | 0.0014 | 0.9806 |

SVM-PUK | 0.9753 | 0.0032 | 0.0018 | 0.9732 | 6 | 0.9692 | 0.0032 | 0.0017 | 0.9518 |

SVM-RBF | 0.9250 | 0.0056 | 0.0041 | 0.9195 | 7 | 0.9101 | 0.0054 | 0.0039 | 0.9033 |

SVM-POLY | 0.8903 | 0.0066 | 0.0054 | 0.8887 | 8 | 0.8847 | 0.0061 | 0.0048 | 0.8746 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Mishra, R.; Ojha, C.S.P.
Application of AI-Based Techniques on Moody’s Diagram for Predicting Friction Factor in Pipe Flow. *J* **2023**, *6*, 544-563.
https://doi.org/10.3390/j6040036

**AMA Style**

Mishra R, Ojha CSP.
Application of AI-Based Techniques on Moody’s Diagram for Predicting Friction Factor in Pipe Flow. *J*. 2023; 6(4):544-563.
https://doi.org/10.3390/j6040036

**Chicago/Turabian Style**

Mishra, Ritusnata, and Chandra Shekhar Prasad Ojha.
2023. "Application of AI-Based Techniques on Moody’s Diagram for Predicting Friction Factor in Pipe Flow" *J* 6, no. 4: 544-563.
https://doi.org/10.3390/j6040036