# Variation of Coefficient of Friction and Friction Head Losses Along a Pipe with Multiple Outlets

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## Abstract

**:**

_{1}) and last segment (f

_{n}) of the multiple outlet pipe was noted to be minimal.

## 1. Introduction

_{m}and that without outlets (hf)

_{p}:

## 2. The Physical Model

## 3. Results and Discussion

_{m}is the friction head loss in the multiple outlet pipe and (hf)

_{p}is the friction head loss in the pipe without outlets. For a pipe with multiple outlets, the discharge is decreasing towards the end of the pipe and this leads to a smaller head loss compared to a pipe without outlets carrying the same discharge. Figure 6 shows the variation between the normalized head loss ((hf)

_{m}/H) with the area ratio. The normalized head loss is the ratio between total head loss along a pipe with multiple outlets and the inlet head while area ratio is the ratio between the area of outlet and the area of the main pipe. Table 1 summarizes the head losses for both cases (pipes having multiple outlets and without outlets). The piezometric head at each pipe outlet was measured, and the difference between the measurements yields the head loss along a certain pipe segment while the difference between the piezometric head at the first and last outlets yields the total head loss, (hf)

_{m}. The same principal was utilized in determining the head losses in a pipe without an outlet, (hf)

_{p}, in this case, the piezometers head was read at pipe inlet and outlet. The G factor for different flow conditions cases was calculated using Equation (13) which shows that there is no definitive relationship between the G factor for differing pipe diameter, area ratio, outlet spacing and inlet pressure as illustrated in Figure 7 and Figure 8.

_{n}/f

_{1}) is a dimensionless ratio between the coefficients of friction at the first and last segments of the pipe while the horizontal axis (S/d) is dimensionless ratio between outlet spacing and the main pipe diameter. For an inlet head of 2.2 m, the friction ratio (f

_{n}/f

_{1}) varied between 1.18 and 1.82, the friction ratio (f

_{n}/f

_{1}) decreased when S/d ratio increased. The values of the coefficient of friction in each segment of the multiple outlet pipe (f

_{1}, f

_{2}, f

_{3}, …, f

_{n}) is affected by the decreasing discharge towards the dead end (q

_{1}, q

_{2}, q

_{3}, …, q

_{n}) and the ratio (q

_{n}/q

_{1}) which is called uniformity of flow. Table 2 shows the effect of various q

_{1}and q

_{n}values on discharge. For Spacing of 1.5 m, outlet diameter of 6 mm, area ratio of 0.24, pipe diameter of 25.4 mm and inlet head of 2.2 m, the values of q

_{1}, q

_{n}and total discharge were found to be 0.154, 0.1 and 1.4 L/s respectively.

_{max}is the value of the maximum experimental G factor; G

_{min}is the value of the minimum experimental G factor; and G

_{a}= is the mean experimental G factor. In statistics, RMSD is used to compare a calculated value with a measured value (Arbat et al. [29]) and also it give an indication about the accuracy of the model prediction or model efficiency (Legates and McCabe [30]). A low RMSD or NRMSD indicate an accurate prediction. A value of 1.0 for the model efficiency (ME) indicates a perfect agreement between the experimental and calculated G factors, however, ME can be a positive or negative value. The OIMP combines the NRMSD and ME indicators to verify the performance of a selected formula, an OIMP value of 1.0 indicates a perfect agreement between the experimental and calculated G factors. The CRM parameter represents the difference between experimental and calculated G factors in relation to the experimental G factor value. For any tested formula, a zero value for CRM indicates a perfect agreement while a positive and a negative value indicate overestimation and underestimation respectively. Table 3 summarizes the values of the statistical indices for all the tested formulae.

## 4. Conclusions

_{m}along a PVC pipe with multiple outlets. The findings of this study show that the friction loss in a pipe with multiple outlets was affected mainly by the outlet spacing (S), inlet head (H) and area ratio (AR). For a given pipe diameter and an inlet head, the smaller outlet spacing led to a greater number of outlets, greater discharge and hence a greater head loss. In addition, there was a proportional relationship between friction head loss and area ratio.

_{n}to the friction in the first pipe segment, f

_{1}) and S/d. Equation (7) which was proposed by Scaloppi [16] yielded the most satisfactory estimation for the G factor among the eight tested formulae in this study. The performance of the tested formulae was assessed by using statistical indices which were RMSD, NRMSD, ME, OIMP and CRM. The vales of these indices for Equation (7) were found to be 0.026, 0.102, 0.93, 0.91 and 0.00 respectively.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Variation of pressure head data with different outlet spacing (inlet head = 1.7 m, AR = 0.24).

**Figure 4.**Relationship between the head loss and the outlet spacing for two inlet heads (AR = 0.24).

**Figure 5.**Variation of the pressure head for various area ratios (inlet head = 1.7 m, spacing = 6.0 m).

Pipe Diameter (mm) | Inlet Head (m) | Spacing (m) | (hf)_{m} (Pipe with Outlets) (mm) | (hf)_{p} (Pipe without Outlets) (mm) | Experimental G Factor |
---|---|---|---|---|---|

25.4 | 1.7 | 1.5 | 403.5 | 974.0 | 0.4143 |

25.4 | 1.7 | 3.0 | 216.0 | 660.0 | 0.3273 |

25.4 | 1.7 | 4.5 | 120.0 | 485.0 | 0.2474 |

25.4 | 1.7 | 6.0 | 65.0 | 235.0 | 0.2766 |

25.4 | 1.7 | 7.5 | 32.0 | 235.0 | 0.1362 |

25.4 | 2.2 | 1.5 | 479.0 | 1314.0 | 0.3645 |

25.4 | 2.2 | 3.0 | 271.0 | 512.0 | 0.5293 |

25.4 | 2.2 | 4.5 | 125.0 | 412.0 | 0.3034 |

25.4 | 2.2 | 6.0 | 92.0 | 339.0 | 0.2714 |

25.4 | 2.2 | 7.5 | 57.5 | 308.0 | 0.1867 |

38.1 | 1.7 | 1.5 | 165.5 | 1010.0 | 0.1639 |

38.1 | 1.7 | 3.0 | 50.0 | 254.0 | 0.1969 |

38.1 | 1.7 | 4.5 | 33.0 | 189.0 | 0.1746 |

38.1 | 1.7 | 6.0 | 14.5 | 196.0 | 0.0740 |

38.1 | 1.7 | 7.5 | 12.0 | 145.0 | 0.0828 |

38.1 | 2.2 | 1.5 | 121.0 | 1180.0 | 0.1025 |

38.1 | 2.2 | 3.0 | 57.5 | 195.0 | 0.2949 |

38.1 | 2.2 | 4.5 | 35.0 | 132.0 | 0.2652 |

38.1 | 2.2 | 6.0 | 12.0 | 145.0 | 0.0828 |

38.1 | 2.2 | 7.5 | 5.0 | 100.0 | 0.0500 |

50.8 | 1.7 | 1.5 | 15.6 | 765.0 | 0.0204 |

50.8 | 1.7 | 3.0 | 408.0 | 747.0 | 0.5462 |

50.8 | 1.7 | 4.5 | 313.0 | 682.0 | 0.4589 |

50.8 | 1.7 | 6.0 | 224.0 | 655.0 | 0.3420 |

50.8 | 1.7 | 7.5 | 301.0 | 667.0 | 0.4513 |

50.8 | 2.2 | 1.5 | 25.5 | 1127.0 | 0.0226 |

50.8 | 2.2 | 3.0 | 590.0 | 1091.0 | 0.5408 |

50.8 | 2.2 | 4.5 | 400.0 | 1112.0 | 0.3597 |

50.8 | 2.2 | 6.0 | 675.0 | 1050.0 | 0.6429 |

50.8 | 2.2 | 7.5 | 364.0 | 940.0 | 0.3872 |

76.2 | 1.7 | 3.0 | 19.0 | 488.0 | 0.0389 |

76.2 | 1.7 | 4.5 | 85.5 | 481.5 | 0.1776 |

76.2 | 1.7 | 6.0 | 22.0 | 498.0 | 0.0442 |

76.2 | 1.7 | 7.5 | 25.0 | 495.5 | 0.0505 |

76.2 | 2.2 | 1.5 | 16.4 | 665.0 | 0.0247 |

76.2 | 2.2 | 3.0 | 30.5 | 644.0 | 0.0474 |

76.2 | 2.2 | 4.5 | 39.0 | 685.5 | 0.0569 |

76.2 | 2.2 | 6.0 | 31.0 | 553.0 | 0.0561 |

76.2 | 2.2 | 7.5 | 27.0 | 553.0 | 0.0488 |

Experiment | Spacing (m) | Pipe Diameter (mm) | S/d | q_{1} (L/s) | q_{n} (L/s) | q_{n}/q_{1} |
---|---|---|---|---|---|---|

1.5 | 25.4 | 59.1 | 0.154 | 0.100 | 0.651 | |

D = 25.4 mm | 3.0 | 25.4 | 118.1 | 0.150 | 0.133 | 0.882 |

d = 6.0 mm | 4.5 | 25.4 | 177.2 | 0.163 | 0.111 | 0.684 |

6.0 | 25.4 | 236.2 | 0.162 | 0.141 | 0.872 | |

7.5 | 25.4 | 295.3 | 0.158 | 0.149 | 0.946 | |

1.5 | 38.1 | 39.4 | 0.193 | 0.132 | 0.681 | |

D = 38.1 mm | 3.0 | 38.1 | 78.7 | 0.183 | 0.152 | 0.828 |

d = 6.0 mm | 4.5 | 38.1 | 118.1 | 0.191 | 0.145 | 0.758 |

6.0 | 38.1 | 157.5 | 0.178 | 0.159 | 0.894 | |

7.5 | 38.1 | 196.9 | 0.188 | 0.140 | 0.746 | |

3.0 | 50.8 | 59.1 | 1.859 | 1.174 | 0.631 | |

D = 50.8 mm | 4.5 | 50.8 | 88.6 | 2.247 | 1.534 | 0.683 |

d = 25.4 mm | 6.0 | 50.8 | 118.1 | 2.370 | 1.757 | 0.742 |

7.5 | 50.8 | 147.6 | 2.358 | 1.543 | 0.654 | |

1.5 | 76.2 | 19.7 | 1.379 | 1.135 | 0.823 | |

D = 76.2 mm | 3.0 | 76.2 | 39.4 | 1.898 | 1.808 | 0.953 |

d = 25.4 mm | 4.5 | 76.2 | 59.1 | 2.299 | 1.883 | 0.819 |

6.0 | 76.2 | 78.7 | 2.421 | 2.242 | 0.926 | |

7.5 | 76.2 | 98.4 | 2.984 | 1.988 | 0.666 |

Formula for G Factor | RMSD | NRMSD | ME | OIMP | CRM |
---|---|---|---|---|---|

Oran and Walker [15] | 0.073 | 0.282 | 0.464 | 0.591 | −0.013 |

Christianson [9] | 0.065 | 0.253 | 0.568 | 0.658 | 0.109 |

Alazba [18] | 0.244 | 0.946 | −5.023 | −2.485 | 0.592 |

Alberston et al. [14] | 0.122 | 0.471 | −0.493 | 0.018 | −0.173 |

Alazba et al. [19] | 0.237 | 0.918 | −4.663 | −2.290 | 0.569 |

Valiantzas [8] | 0.092 | 0.357 | 0.143 | 0.393 | −0.088 |

Scaloppi [16] | 0.026 | 0.102 | 0.930 | 0.914 | 0.000 |

Mostafa [17] | 0.220 | 0.853 | −3.893 | −1.873 | 0.477 |

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**MDPI and ACS Style**

Alawee, W.H.; Almolhem, Y.A.; Yusuf, B.; Mohammad, T.A.; Dhahad, H.A. Variation of Coefficient of Friction and Friction Head Losses Along a Pipe with Multiple Outlets. *Water* **2020**, *12*, 844.
https://doi.org/10.3390/w12030844

**AMA Style**

Alawee WH, Almolhem YA, Yusuf B, Mohammad TA, Dhahad HA. Variation of Coefficient of Friction and Friction Head Losses Along a Pipe with Multiple Outlets. *Water*. 2020; 12(3):844.
https://doi.org/10.3390/w12030844

**Chicago/Turabian Style**

Alawee, Wissam H., Yousef A. Almolhem, Badronnisa Yusuf, Thamer A. Mohammad, and Hayder A. Dhahad. 2020. "Variation of Coefficient of Friction and Friction Head Losses Along a Pipe with Multiple Outlets" *Water* 12, no. 3: 844.
https://doi.org/10.3390/w12030844