# Discharge Coefficients of Standard Spillways at High Altitudes

^{1}

^{2}

^{*}

## Abstract

**:**

_{d}) ranging from 0.5 to 2. The experimental results indicated that, the P/H

_{d}ratio influences the discharge coefficients in Condoroma, and P/H

_{d}≥ 1 values are recommended for the design of the spillway profile. In addition, for all the P/H

_{d}ratios studied, the discharge coefficients adjusted to the Condoroma altitude were lower than those reported by classical formulations used in conventional spillway designs. Finally, a generalized equation is proposed to estimate the discharge coefficient for standard spillways located in dams at similar elevations above sea level.

## 1. Introduction and Objective

#### Previous Studies

## 2. Materials and Methods

#### 2.1. Experimental Facility

#### 2.2. Instrumentation

#### 2.3. Operating Conditions

#### 2.4. Methodology

## 3. Results

#### 3.1. Data Processing

**Figure 6.**(

**a**) Water depth (in m) and flow rates measured for the case P/H

_{d}= 2 , test duration > 5 min; (

**b**) relation between discharge coefficients and the dimensionless head H/H

_{d}obtained from the measured data in (

**a**).

**Figure 7.**Boxplot graphs for the tests on P/H

_{d}= 2 related to (

**a**) water depth and (

**b**) flow rate measurements.

**Figure 8.**(

**a**) Water depth (in m) and flow rates after the detection of outliers; (

**b**) relation between discharge coefficients (m) and the dimensionless head (H/H

_{d}) after the detection of outliers. The thick black dots correspond to the average values.

#### 3.2. Spillway $P/{H}_{d}=2.0;1.5;1;0.5;0.25$

#### 3.3. P/H_{d} Aggregated Analysis

## 4. Discussion

## 5. Conclusions

- -
- The experimental results were compared with previous research. At altitudes above 4000 m a.s.l., the discharge coefficients show substantial differences, with consistently lower values than those obtained to date in previous works conducted at lower altitudes above sea level.
- -
- Particularly, the discharge coefficient C
_{o}of Equation (5) when H/H_{d}≈ 1 at Condoroma seems to tend towards a value slightly lower than 2.1, lower than the 2.17 proposed by USBR [4]. - -
- The P/H
_{d}ratio influences the discharge coefficients in Condoroma, and P/H_{d}≥ 1 values are recommended for the design of the spillway profile. The authors observed more stable and predictable flow behavior for higher P/H_{d}ratios. - -
- Although the results obtained for P/H
_{d}≥ 1 at Condoroma show significantly lower values than those obtained in previous experience at much lower altitudes, it can be seen that the tests for P/H_{d}< 1 the discharge coefficients, although still significantly lower, are more similar to the results obtained by other authors. - -
- The equations to determine the discharge coefficients (Equations (10) and (11)) for Condoroma could be used in areas at similar altitudes in the absence of experimental data.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**General view of the 21 m long experimental channel located close to Condoroma dam at 4075 m a.s.l.

**Figure 4.**View of two of the ogee-tested profiles: (

**a**) P/H

_{d}= 2; (

**b**) P/H

_{d}=0.5. Additionally, P/H

_{d}= 1.5, 1 and 0.25 were tested.

**Figure 9.**P/H

_{d}= 2. (

**a**) Graph of residuals of the fitted regression; (

**b**) fitted regression curve.

**Figure 11.**(

**a**) P/H

_{d}= 1.5; (

**b**) P/H

_{d}= 1.0; (

**c**) P/H

_{d}= 0.5; (

**d**) P/H

_{d}= 0.25. Adjustment of the dimensionless discharge coefficient for the case P/H

_{d}= 1; 0.5; 0.25 and comparison with the results obtained by Maynord [28].

**Figure 12.**(

**a**) Comparison of the values of the dimensionless discharge coefficient obtained from experimental results at Condoroma with those obtained by Brudenell [15], Randolph [18], Senturk [27] and Erpicum [31]; (

**b**) Generalized equations for the dimensionless discharge coefficient obtained by Hager [29], Dillman [12], Rouse and Reid [13] and Montes [60].

**Figure 14.**C

_{o}coefficient of Equation (5) for the Condoroma design head compared with that proposed by USBR [4].

Author | Width (m) | H_{d}(m) | P/H_{d} | Q (L/s) | Elevation m a.s.l. |
---|---|---|---|---|---|

Dillman [12] | - | 0.05 | - | - | 520 |

Cassidy [24] | - | - | 2; 2.5; 3.7; 6.6 | - | 210 |

Rouse [48] | 0.500 | - | - | 62 | 115 |

Murphy [26] | 0.732 | 0.305 | 3.5; 7.0 | 560 | 38 |

Maynord [28] | 0.762 | 0.249 | 0.25; 0.5; 1.0; 2.0 | 385 | 38 |

Hager [29] | 0.500 | 0.20/0.1 | 3.5/7.0 | 375 | 495 |

Erpicum [31] | 0.200 | 0.10/0.15 | - | 358 | 240 |

Condoroma dam | 0.915 | 0.20/0.175 | 0.25; 0.5; 1; 1.5/2 | 415 | 4075 |

P (m) | P/H_{d} | Q (L/s) | Stage (m) | Fr | y_{b}(m) | Fr_{b} |
---|---|---|---|---|---|---|

(1) | (2) | (3) | (4) | (5) | (6) | (7) |

0.35 | 2.0 | 53–363 | 0.45–0.634 | 0.062–0.25 | 0.073–0.21 | 0.96–1.28 |

0.30 | 1.5 | 49–400 | 0.40–0.607 | 0.068–0.31 | 0.074–0.24 | 0.85–1.18 |

0.20 | 1.0 | 52–415 | 0.30–0.492 | 0.11–0.42 | 0.074–0.23 | 0.89–1.29 |

0.10 | 0.50 | 56–391 | 0.20–0.368 | 0.22–0.62 | 0.086–0.21 | 0.78–1.40 |

0.05 | 0.25 | 61–345 | 0.15–0.280 | 0.37–0.93 | 0.100–0.22 | 0.63–1.15 |

P/H_{d} | Data | m_{0} | n | R^{2} |
---|---|---|---|---|

2.0 | Condoroma | 0.467 | 0.184 | 0.962 |

2.0 | Maynord [28] | 0.494 | 0.157 | 0.984 |

2.0 | Cassidy [24] | 0.518 | 0.186 | 0.993 |

2.5 | Murphy [26] | 0.503 | 0.139 | 0.974 |

3.5 | Hager [29] | 0.493 | 0.122 | 0.988 |

1.5 | Condoroma | 0.470 | 0.172 | 0.969 |

1 | Condoroma | 0.472 | 0.158 | 0.974 |

1 | Maynord [28] | 0.490 | 0.129 | 0.989 |

0.5 | Condoroma | 0.480 | 0.098 | 0.82 |

0.5 | Maynord [28] | 0.487 | 0.099 | 0.849 |

0.25 | Condoroma | 0.465 | 0.085 | 0.807 |

0.25 | Maynord [28] | 0.468 | 0.063 | 0.489 |

**Table 4.**Fitting factors corresponding to Equation (4) for the discharge coefficient m proposed by different authors.

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© 2024 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/).

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**MDPI and ACS Style**

Rendón, V.; Sánchez-Juny, M.; Estrella, S.; Sanz-Ramos, M.; Rucano, P.; Huarca Pulcha, A.
Discharge Coefficients of Standard Spillways at High Altitudes. *Designs* **2024**, *8*, 22.
https://doi.org/10.3390/designs8020022

**AMA Style**

Rendón V, Sánchez-Juny M, Estrella S, Sanz-Ramos M, Rucano P, Huarca Pulcha A.
Discharge Coefficients of Standard Spillways at High Altitudes. *Designs*. 2024; 8(2):22.
https://doi.org/10.3390/designs8020022

**Chicago/Turabian Style**

Rendón, Víctor, Martí Sánchez-Juny, Soledad Estrella, Marcos Sanz-Ramos, Percy Rucano, and Alan Huarca Pulcha.
2024. "Discharge Coefficients of Standard Spillways at High Altitudes" *Designs* 8, no. 2: 22.
https://doi.org/10.3390/designs8020022