# Discharge Coefficients of a Specific Vertical Slot Fishway Geometry—New Fitting Parameters

^{*}

## Abstract

**:**

## 1. Introduction

## 2. State-of-the-Art

**(a)**,

**(b)**] are commonly used to determine the discharge Q in vertical slot passes [2,6].

**(a)**The first approach is based on the flow through a slot as a channel constriction without flow transition according to Venturi or Poleni [18,19]:

**(b)**The second approach is based on the Torricelli equation, with a maximum flow velocity ${v}_{\mathrm{max}}$ (see Equation (1)):

## 3. Experimental Model Data

## 4. Results and Discussion

#### 4.1. General

#### 4.2. Discharge Coefficients in Accordance with Poleni

#### 4.3. Discharge Coefficients in Accordance with Torricelli

## 5. Summary and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Notations

$A,B$ | coefficients depending on the fishway geometry (-) |

${A}_{d}$ | discharge area (m^{2}) |

${\alpha}_{1},{\alpha}_{2},\beta $ | empirically determined coefficients (-) |

${b}_{0}$ | slot width (m) |

${C}_{d}$ | discharge coefficient (-) |

${C}_{d,1}$ | discharge coefficient according to Poleni (-) |

${C}_{d,2}$ | discharge coefficient according to Torricelli (-) |

${C}_{d,1,\mathrm{calc}}$ | calculated discharge coefficient (-) |

${C}_{d,2,\mathrm{calc}}$ | calculated discharge coefficient (-) |

$\Delta h$ | water level difference (m) |

$\Delta {h}_{m}$ | mean water level difference (m) |

$\Delta z$ | geodetic height difference (m) |

g | acceleration due to gravity (ms^{−2}) |

h | water depth (m) |

${h}_{1}$ | headwater depth (m) |

${h}_{2}$ | tailwater depth (m) |

${h}_{1,m}$ | calculated mean headwater depth (m) |

${h}_{2,m}$ | calculated mean tailwater depth (m) |

${{h}_{2}{h}_{1}}^{-1}$ | dimensionless calculated water depth (-) |

${{h}_{2,m}{h}_{1,m}}^{-1}$ | dimensionless calculated mean water depth (-) |

${h}_{m}$ | area-averaged mean water depth in the basin (m) |

${l}_{g}$ | guide element length (m) |

${l}_{g}{b}_{0}^{-1}$ | ratio of guide element length to slot width (-) |

L | basin length (m) |

$L{b}_{0}^{-1}$ | ratio of basin length to slot width (-) |

Q | discharge (m^{3}s^{−1}) |

${Q}^{*}$ | dimensionless discharge (-) |

${R}^{2}$ | coefficient of determination (-) |

S | slope (-) |

${v}_{\mathrm{max}}$ | maximum flow velocity (ms^{−1}) |

W | basin width (m) |

$W{L}^{-1}$ | ratio of basin width to basin length (-) |

$W{b}_{0}^{-1}$ | ratio of basin width to slot width (-) |

${y}_{0}$ | average flow depth (m) |

${y}_{0}{b}_{0}^{-1}$ | ratio of average flow depth to slot width (-) |

## References

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**Figure 1.**Photograph of an examplary vertical slot pass at a weir in North-Rhine Westphalia, Germany.

Design | W | L | ${\mathit{b}}_{0}$ | ${\mathit{WL}}^{-1}$ | ${\mathit{Wb}}_{0}^{-1}$ | ${\mathit{Lb}}_{0}^{-1}$ | S | Q | ${\mathit{l}}_{\mathit{g}}$ | ${\mathit{l}}_{\mathit{g}}{\mathit{b}}_{0}^{-1}$ | Typ of Guide Element | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Mohlfeld and Oertel (2021) [3] | 3 | 0.8 | 1.021 | 0.119 | 0.78 | 6.72 | 8.58 | 0.025; 0.050 | 0.016; 0.024; 0.032 | 0.190 | 1.60 | linear |

4 | 0.8 | 1.021 | 0.119 | 0.78 | 6.72 | 8.58 | 0.178 | 1.50 | linear | |||

5 | 0.8 | 1.021 | 0.119 | 0.78 | 6.72 | 8.58 | 0.178 | 1.50 | bevelled | |||

6 | 0.8 | 1.021 | 0.119 | 0.78 | 6.72 | 8.58 | 0.178 | 1.50 | spline | |||

7 | 0.8 | 1.021 | 0.119 | 0.78 | 6.72 | 8.58 | 0.178 | 1.50 | quadrant | |||

BAW (2019) [23] | 3 | 0.8 | 1.01 | 0.120 | 0.79 | 6.67 | 8.42 | 0.028; 0.039; | 0.010 to 0.035 | linear | ||

3 | 0.79 | 1.02 | 0.122 | 0.77 | 6.43 | 8.36 | 0.050; 0.028 | linear | ||||

Klein and Oertel (2018) [25] | 3 | 0.8 | 1.021 | 0.119 | 0.78 | 6.72 | 8.58 | 0.190 | 1.60 | linear | ||

3.67 | 0.8 | 1.021 | 0.119 | 0.78 | 6.72 | 8.58 | 0.190 | 1.60 | linear | |||

4 | 0.8 | 1.021 | 0.119 | 0.78 | 6.72 | 8.58 | 0.025; 0.050 | 0.008 to 0.044 | 0.178 | 1.50 | linear | |

5 | 0.8 | 1.021 | 0.119 | 0.78 | 6.72 | 8.58 | 0.178 | 1.50 | bevelled | |||

6 | 0.8 | 1.021 | 0.119 | 0.78 | 6.72 | 8.58 | 0.178 | 1.50 | spline | |||

7 | 0.8 | 1.021 | 0.119 | 0.78 | 6.72 | 8.58 | 0.178 | 1.50 | quadrant | |||

Klein and Oertel (2017) [6] | 37 | 0.8 | 1.021 | 0.119 | 0.78 | 6.72 | 8.58 | 0.005 to 0.056 | 0.190 | 1.60 | linear | |

52 | 0.8 | 1.021 | 0.119 | 0.78 | 6.72 | 8.58 | 0.050 | 0.190 | 1.60 | linear | ||

67 | 0.8 | 1.021 | 0.119 | 0.78 | 6.72 | 8.58 | 0.190 | 1.60 | linear | |||

Krüger et al. (2010) [20] | 1 | 3.3 | 4.5 | 0.45 | 0.73 | 7.33 | 10.00 | 0.056 | 0.769; 0.933; 1.101 | none | ||

Bermúdez et al. (2010) [7] | 2 | 0.3 | 0.38 | 0.168; 0.084; 0.043 | 0.80 | 1.79; 3.58; 7.16 | 2.24; 4.47; 8.94 | 0.050 | 0.0009 to 0.0270 | none | ||

2 | 0.3 | 0.75 | 0.168; 0.084 | 0.40 | 1.79; 3.58 | 4.47; 8.94 | none | |||||

2 | 0.3 | 1.50 | 0.168 | 0.20 | 1.79 | 8.94 | none | |||||

2 | 0.3 | 2.25 | 0.168 | 0.13 | 1.79 | 13.42 | none | |||||

2 | 0.3 | 0.28 | 0.126 | 1.07 | 2.39 | 2.24 | none | |||||

2 | 0.3 | 0.57 | 0.126; 0.043 | 0.53 | 2.39; 7.16 | 4.47; 13.42 | none | |||||

2 | 0.3 | 1.13 | 0.126; 0.084 | 0.27 | 2.39; 3.58 | 8.94; 13.42 | none | |||||

2 | 0.3 | 1.70 | 0.126 | 0.18 | 2.39 | 13.34 | none | |||||

2 | 0.3 | 0.19 | 0.084; 0.043 | 1.60 | 3.58; 7.16 | 2.24; 4.47 | none | |||||

2 | 0.3 | 0.095 | 0.043 | 3.20 | 7.16 | 2.24 | none | |||||

Puertas et al. (2004) [16] | 1 | 0.99 | 1.21 | 0.16 | 0.82 | 6.19 | 7.58 | 0.0570; 0.1005 | 0.0159 to 0.1250 | 0.243 | 1.52 | linear |

2 | 0.99 | 1.21 | 0.15 | 0.82 | 6.60 | 8.09 | none | |||||

Rajaratnam et al. (1992) [17] | 1 | 0.458 | 0.572 | 0.057 | 0.80 | 8.00 | 10.00 | 0.10 | 0.0009 to 0.026 | 0.086 | 1.50 | bevelled |

6 | 0.305 | 0.381 | 0.043 | 0.80 | 7.16 | 8.94 | 0.057; 0.10 | none | ||||

8 | 0.305 | 0.191 | 0.043 | 1.60 | 7.16 | 4.47 | 0.10; 0.149 | none | ||||

9 | 0.153 | 0.191 | 0.043 | 0.80 | 3.58 | 4.47 | 0.10; 0.149 | none | ||||

11 | 0.305 | 0.572 | 0.043 | 0.53 | 7.16 | 13.42 | 0.050; 0.10 | none | ||||

14 | 0.305 | 0.381 | 0.043 | 0.80 | 7.16 | 8.94 | 0.050; 0.10; 0.148 | 0.086 | 2.00 | linear | ||

15 | 0.305 | 0.381 | 0.043 | 0.80 | 7.16 | 8.94 | 0.050; 0.10; 0.148 | 0.086 | 2.00 | linear | ||

16 | 0.305 | 0.381 | 0.046 | 0.80 | 6.65 | 8.31 | 0.050; 0.10 | 0.092 | 2.00 | linear | ||

17 | 0.305 | 0.381 | 0.038 | 0.80 | 8.00 | 10.00 | 0.10; 0.150 | 0.057 | 1.50 | round | ||

18 | 0.305 | 0.381 | 0.038 | 0.80 | 8.00 | 10.00 | 0.10; 0.150 | 0.057 | 1.50 | spline |

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## Share and Cite

**MDPI and ACS Style**

Kasischke, K.; Oertel, M.
Discharge Coefficients of a Specific Vertical Slot Fishway Geometry—New Fitting Parameters. *Water* **2023**, *15*, 1193.
https://doi.org/10.3390/w15061193

**AMA Style**

Kasischke K, Oertel M.
Discharge Coefficients of a Specific Vertical Slot Fishway Geometry—New Fitting Parameters. *Water*. 2023; 15(6):1193.
https://doi.org/10.3390/w15061193

**Chicago/Turabian Style**

Kasischke, Kimberley, and Mario Oertel.
2023. "Discharge Coefficients of a Specific Vertical Slot Fishway Geometry—New Fitting Parameters" *Water* 15, no. 6: 1193.
https://doi.org/10.3390/w15061193