# Contribution of Different Elements of Inclined Trash Racks to Head Losses Modeling

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Setup

#### 2.2. Measurements

#### 2.3. Calculation

#### 2.4. Methodology of Modeling

#### 2.4.1. Bar Shape

#### 2.4.2. Support

- If profiled shape, h is constant.
- If U-shaped, $h=AB\times sin\left(\beta \right)+OB\times cos\left(\beta \right)$.

## 3. Results

#### 3.1. Effect of the Bar Profiles on Head Losses

#### 3.1.1. Experimental Head Loss Coefficients

#### 3.1.2. Modeled Head Loss Coefficients

#### 3.2. Effect of the Support on Head Losses

#### 3.2.1. Experimental Comparison with and without U-Shaped Supports

#### 3.2.2. Experimental Comparison with and without the Profiled Support

#### 3.2.3. Modeled Head Loss Coefficients

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

${A}_{i}$ | Bar shape coefficient (-) |

b | Bar thickness (m) |

p | Bar depth (m) |

B | Channel width (m) |

e | Bar spacing (m) |

h | Projection of the support diagonal (m) |

g | Gravitation acceleration (m${\mathrm{s}}^{-2}$) |

${H}_{1}$, ${H}_{2}$ | Upstream and downstream head water (m) |

${O}_{g}$ | Blockage ratio due to the bars (-) |

${O}_{sp}$ | Blockage ratio due to the spacing bars (-) |

${O}_{support}$ | Blockage ratio due to the supports (-) |

${V}_{1}$, ${V}_{2}$ | Upstream and downstream velocities (m${\mathrm{s}}^{-1}$) |

$\beta $ | Angle of inclination (${}^{\xb0}$) |

${K}_{b}$ | Ratio of the bars (-) |

${K}_{\beta}$ | Ratio of the angle of inclination (-) |

${K}_{U\text{-}support}$ | U-shaped support coefficient (-) |

${K}_{Profiled\text{-}support}$ | Profiled support coefficient (-) |

${\xi}_{bars}$ | Head loss coefficient due to the bars (-) |

${\xi}_{spacers}$ | Head loss coefficient due to the spacers (-) |

${\xi}_{support}$ | Head loss coefficient due to the supports (-) |

${\xi}_{total}$ | Total head loss coefficient (-) |

## Appendix A

**Table A1.**Measured head loss coefficient (${\xi}_{measured}$) as a function of the bar shapes, the bar spacing (e), and the angle of inclination ($\beta $).

Bar Shape | Droplet | Plétina | Tadpole 8 | Tadpole 10 | ||||||
---|---|---|---|---|---|---|---|---|---|---|

e (mm) | 20.2 | 30.8 | 17.2 | 27.8 | 18.2 | 28.8 | 18.2 | 28.8 | ||

$\mathbf{\beta}$(${}^{\xb0}$) | ||||||||||

${\xi}_{measured}$ | 15 | 0.44 | 0.46 | 0.37 | 0.34 | 0.54 | 0.44 | 0.36 | 0.44 | |

25 | 0.43 | 0.30 | 0.43 | 0.29 | 0.28 | 0.10 | 0.41 | 0.29 | ||

35 | 0.51 | 0.38 | 0.56 | 0.37 | 0.37 | 0.20 | 0.46 | 0.32 | ||

45 | 0.71 | 0.52 | 0.75 | 0.52 | 0.48 | 0.32 | 0.62 | 0.46 | ||

60 | 0.88 | 0.60 | 0.98 | 0.64 | 0.53 | 0.43 | 0.77 | 0.54 | ||

90 | 1.21 | 0.81 | 1.32 | 0.85 | 0.66 | 0.47 | 1.04 | 0.67 |

**Table A2.**Measured head loss coefficient (${\xi}_{measured}$) with (1 or 2) the U-shaped or profiled shape support or without support as a function of the bar shape (PR or PH), the bar spacing (e), and the angle of inclination ($\beta $).

$\mathbf{\beta}$(${}^{\xb0}$) | Support Shape | Without | With 1 U-Shaped | With 2 U-Shaped | With Profiled Shape | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Bar Profile | PR | PH | PR | PH | PR | PH | PR | PH | |||

e (mm) | |||||||||||

${\xi}_{measured}$ | 15 | 10 | 0.71 | 0.58 | 2.43 | 2.24 | 3.08 | 2.90 | 0.82 | 0.70 | |

15 | 0.56 | 0.55 | 1.75 | 1.44 | 2.75 | 2.87 | 0.69 | 0.64 | |||

25 | 10 | 0.56 | 0.50 | 1.51 | 1.36 | 2.20 | 2.02 | 0.66 | 0.55 | ||

15 | 0.45 | 0.63 | 1.23 | 1.37 | 1.93 | 1.81 | 0.53 | 0.66 | |||

35 | 10 | 0.75 | 0.44 | 1.49 | 1.28 | - | - | 0.79 | 0.45 | ||

15 | 0.53 | 0.43 | 1.37 | 1.18 | - | - | 0.58 | 0.50 | |||

45 | 10 | 0.93 | 0.54 | 1.61 | 1.16 | - | - | 0.96 | 0.64 | ||

15 | 0.61 | 0.36 | 1.21 | 0.98 | - | - | 0.64 | 0.44 | |||

60 | 10 | 1.17 | 0.63 | 1.63 | 0.99 | - | - | 1.22 | 0.71 | ||

15 | 0.74 | 0.47 | 1.14 | 0.82 | - | - | 0.75 | 0.50 | |||

90 | 10 | 1.53 | 0.75 | 2.00 | 1.06 | - | - | 1.70 | 0.89 | ||

15 | 0.88 | 0.54 | 1.27 | 0.78 | - | - | 0.97 | 0.63 |

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**Figure 2.**Trash rack components (bars, spacers, transversal supports, and longitudinal supports) and tested bar shapes.

**Figure 4.**Transversal support dimensions: U-shaped on the left and profiled shapes on the right, placed in front of the spacers.

**Figure 7.**Comparison of measured head loss coefficients $\xi $ for the four bar shapes ((

**a**) for Droplet, (

**b**) for Plétina, (

**c**) for Tadpole 8, (

**d**) for Tadpole 10) and the two bar spacings for each one, as a function of the angle of inclination $\beta $.

**Figure 8.**Linear regression of the ${\xi}_{bars}$ for the four bar shapes ((

**a**) for Droplet, (

**b**) for Plétina, (

**c**) for Tadpole 8, (

**d**) for Tadpole 10) as a function of the ${K}_{b}\times {K}_{\beta}$.

**Figure 9.**Comparison of measured head loss coefficients $\xi $ without, with one or two U-shaped supports (US) for (

**a**) PR for a bar spacing of 10, (

**b**) PR for a bar spacing of 15, (

**c**) PH for a bar spacing of 10, and (

**d**) PH for a bar spacing of 15, as a function of the angle of inclination $\beta $.

**Figure 10.**Comparison of measured head loss coefficients $\xi $ with one and without the profiled shape (PS) support for (

**a**) PR for a bar spacing of 10, (

**b**) PR for a bar spacing of 15, (

**c**) PH for a bar spacing of 10, and (

**d**) PH for a bar spacing of 15, as a function of the angle of inclination $\beta $.

**Figure 11.**Linear regression between the measured ${\xi}_{measured}$ and modeled ${\xi}_{modeled}$ head loss coefficients for (

**a**) the U-shaped support and (

**b**) the profiled shape support.

Parameters | Values | Units |
---|---|---|

Bar spacing e | 17.2/18.2/20.2/27.8/28.8/30.8 | (mm) |

Angle of inclination $\beta $ | 15/25/35/45/60/90 | (${}^{\xb0}$) |

Discharge Q | 0.29/0.48/0.5 | (m${}^{3}$s${}^{-1}$) |

Upstream water depth ${H}_{1}$ | 0.42/0.67/0.7 | (m) |

Approach velocity ${V}_{1}$ | 0.72 | (ms${}^{-1}$) |

Reynolds number | 720,000 | (-) |

Bar-Reynolds number | 3600/5760/7200/8640 | (-) |

Froude number | 0.27/0.28/0.35 | (-) |

**Table 2.**Bar parameters (b,p) for the different bar shapes (Droplet, Plétina, Tadpole 8, Tadpole 10, hydrodynamic, rectangular).

Bar Shape | Maximum Thickness b (mm) | Depth p (mm) | e/b (-) |
---|---|---|---|

Droplet | 10 | 80 | 2/3.1 |

Plétina | 12 | 60 | 1.4/2.3 |

Tadpole 8 | 8 | 60 | 2.3/3.6 |

Tadpole 10 | 10 | 80 | 1.8/2.9 |

Hydrodynamic | 5 | 40 | 1/2/3/4 |

Rectangular | 5 | 40 | 1/2/3/4 |

Bar Shape | Droplet | Plétina | Tadpole 8 | Tadpole 10 | Hydrodynamic | Rectangular |
---|---|---|---|---|---|---|

Bar coefficient ${A}_{i}$ | 2.47 | 1.75 | 1.27 | 1.79 | 2.10 | 3.85 |

ratio (%) | 64.2 | 45.5 | 33 | 46.5 | 54.5 | 100 |

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**MDPI and ACS Style**

Lemkecher, F.; Chatellier, L.; Courret, D.; David, L.
Contribution of Different Elements of Inclined Trash Racks to Head Losses Modeling. *Water* **2020**, *12*, 966.
https://doi.org/10.3390/w12040966

**AMA Style**

Lemkecher F, Chatellier L, Courret D, David L.
Contribution of Different Elements of Inclined Trash Racks to Head Losses Modeling. *Water*. 2020; 12(4):966.
https://doi.org/10.3390/w12040966

**Chicago/Turabian Style**

Lemkecher, Fatma, Ludovic Chatellier, Dominique Courret, and Laurent David.
2020. "Contribution of Different Elements of Inclined Trash Racks to Head Losses Modeling" *Water* 12, no. 4: 966.
https://doi.org/10.3390/w12040966