# Experimental and Numerical Modelling of Bottom Intake Racks with Circular Bars

^{*}

## Abstract

**:**

^{2}> 95%. Expressions to calculate the discharge coefficient and the collected water through the rack in the clear water case have been proposed and show a good agreement with the laboratory data.

## 1. Introduction

#### 1.1. Wetted Rack Length

_{1}. The required length may differ by up to double in some cases, depending on the author (García [10]). This is due to the variation of the experimental conditions used to adjust the discharge coefficient, such as the shape of the bars, their separation and width, the void ratio, the approximation flow conditions, the initial flow depth h

_{1}, or the longitudinal rack slope θ (Castillo et al. [24]).

#### 1.2. Discharge Coefficient

_{d}= dq/dx (m

^{3}/s/m). As the collected flow through the rack plane is influenced by the velocity distribution close to it, the velocity coefficient, C

_{v}, considers the deviation from the uniform distribution. In a similar way, as there is a change in the available section, which involves the flow contraction, a contraction coefficient, C

_{c}, may be considered. Both coefficients require experimental measurement and depend on the shape of the bars and the spacing between them. The orifice equation as a function of the total energy available referred to the plane of the rack has been considered by several authors (De Marchi [34]; Motskow [15]; Nakagawa [35]; Krochin [17]; Ahmad & Mittal [36]; Ghosh & Ahmad [37]; Righetti & Lanzoni [9]; Kumar et al. [38]). The general expression may be written as:

_{qH}the discharge coefficient depending on the energy head (C

_{qH}= C

_{c}C

_{v}).

_{qh}is the discharge coefficient as a function of the water depth normal to the rack plane.

_{0}is the specific energy at the beginning of the rack, and Δz the vertical difference between the initial rack section and the analyzed section.

#### 1.3. Discharge Coefficient

## 2. Physical Device

_{1}flow was measured with an electromagnetic flowmeter Endress Häuser Promag 53W of 125 mm with an accuracy of 0.5%, the specific discharge flow, q

_{2}, was obtained with a calibrated v-notch, while the specific discharge flow collected in the intake system, q

_{d}, was obtained as the difference between the inlet and discharged flows. Further details of the model may be obtained in García [10] and in Castillo et al. [3].

## 3. Numerical Modelling

_{i}represents the coordinates directions (i = 1 to 3 for x, y, z directions, respectively), ρ the flow density, t the time, U the velocity vector, p the pressure, ${u}_{i}^{\prime}$ presents the turbulent velocity in each direction (i = 1 to 3 for x, y, z directions, respectively), μ is the molecular viscosity, ${S}_{ij}$ is the mean strain-rate tensor and $-\rho \overline{{u}_{i}^{\prime}{u}_{j}^{\prime}}$ is the Reynolds stress.

#### 3.1. Details of the Numerical Model

^{−4}for all the variables. A fix time step of 0.05 s was considered. Around 7–8 iterations were required to reach the convergence criteria in each time step. The transient statistics were obtained once the steady state was reached.

#### 3.2. Free Surface Modelling

_{α}) is the unit in each control volume. It may be assumed that the free surface is on the 0.5 air volume fraction.

_{P}the number of phases.

#### 3.3. Boundary Conditions

#### 3.4. Mesh Size Independence

_{s}= 1.25 (ASCE [43]). The analysis was based on the water depths obtained at the beginning of the spacing between bars (cross section X = 0.00 m), giving GCI values of less than 1.5% for the test carried out. The GCI is not a direct measurement of the mesh accuracy, however, it does ensure with a level of confidence that the solution is approaching to the mesh convergence solution. The comparison with the laboratory measurements also shows good agreement. Table 3 shows the GCI calculations for an inlet specific flow of 114.6 l/s/m and a rack slope of 20%.

#### 3.5. Turbulence Models

_{t}being the eddy viscosity or turbulent viscosity, $k=1/2\overline{{u}_{i}^{\prime}{u}_{j}^{\prime}}$ the turbulent kinetic energy and δ the Kronecker delta function.

^{+}value was smaller than 200 in all the simulations. More details are given in the ANSYS CFX Manual [40].

#### 3.6. Turbulence Model Comparison

## 4. Results and Discussion

#### 4.1. Flow Profile over ther Rack

_{c}, the void ratio m, and the discharge coefficient measured in static conditions C

_{q0}(Castillo et al. [24,49]). Considering the 30 configurations carried out, Figure 10 and Figure 11 show the non-dimensional flow profiles over the bar and over the spacing between bars.

_{c}· m · C

_{q}

_{0}< 0.50. For values greater than 0.50, the behavior over the bars and over the spacing between bars seems to follow different fit curves. Figure 10 and Figure 11 also show polynomial curve fits with r

^{2}> 0.95:

#### 4.2. Wetted Rack Length

_{1}as the distance from the beginning of the rack to the section where the nappe enters directly through the racks (measured between the bars) (Figure 12).

_{1}values have been obtained in the laboratory device and in the numerical simulations considering different specific flows and slopes. Table 5 and Table 6 show the comparison of measured and simulated length values. Despite the good behavior shown in the flow profiles (with a relative error of around 1%), the spacing between bars is a challenge region for the free surface modelling. Maximum differences were around 4% of the laboratory value for all the cases considered.

#### 4.3. Discharge Coefficient

_{0}= H

_{min}for the incoming flow, the discharge coefficient C

_{qH}along the rack has been calculated from Equation (3) using the measurements of the rejected flow at each cross-section.

_{qH}variation in a dimensionless way for several flows and rack slopes. Results are similar than those obtained from CFD simulations. Values obtained with T-shaped bars have also been considered for the same void ratio (Castillo et al. [24]). There are remarkable differences among the behavior of the racks made with circular and T-shape bars. Racks with circular bars have higher values of discharge coefficients. That drives to a smaller wetted rack length to derive the same flow. However, these discharge coefficients correspond to clear water tests.

_{0}= H

_{min}for the incoming flow, and horizontal energy level along the rack. The accuracy of Equation (13) will be discussed in the following section.

_{0}, and the minimum energy H

_{min}. To obtain the empirical discharge coefficient related to the real energy head, Equation (13) has been multiplied by the square root of the ratio H

_{min}/H

_{0}.

_{qH}≈ sinα. While the different flows obtain similar maximum C

_{qH}values, the rack slope seems to influence the maximum. In this way, smaller maximum C

_{qH}values are obtained with bigger rack slopes. After the maximum, the discharge coefficient tends to reduce with the decreasing of the water depth over the rack. Differences with the empirical formula proposed in Equation (13) are related with the consideration that the energy remains constant along the rack.

#### 4.4. Water Collected

## 5. Conclusions

^{2}> 95%.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Detail of the fluid domain: (

**a**) boundary conditions in a longitudinal plane view; (

**b**) spacing between bars in cross section A-A’; (

**c**) boundary conditions in cross section A-A’.

**Figure 5.**Flow profiles over the center of the bar with a rack slope of 20% for different mesh sizes.

**Figure 6.**Discretization of the volume fraction over the spacing between bars and over the center of a bar for an inlet flow of 198 l/s/m, a rack slope of 20% and a 0.004 m mesh size.

**Figure 7.**Flow profiles over the center of the bar with a rack slope of 20% for different turbulence models.

**Figure 15.**Variation of the discharge coefficient along the rack for circular and T-shaped bars with void ratios m = 0.28.

Horizontal Energy Level | Energy Level Parallel to the Rack Plane |
---|---|

Bouvard & Kunztmann [4] Frank [5] Free overfall (Henderson [6]) Vargas [7] Drobir et al. [8] Brunella et al. [2] Righetti & Lanzoni [9] García [10] | Chaguinov [11] Noseda [12,13] Gherardelli [14] Motskow [15] Dagan [16] Krochin [17] |

Mesh Size (m) | Number of Elements | Mean Required Time |
---|---|---|

0.006 | 108,072 | 0.05 h |

0.005 | 154,017 | 0.5 h |

0.004 | 245,068 | 6 h |

Mesh Size (m) | Water Depth (m) | Relative Error (%) | Laboratory Value (m) | Relative Error with Laboratory Value (%) | GCI (%) |
---|---|---|---|---|---|

0.006 | 0.0595 | - | 0.0611 | 1.63 | - |

0.005 | 0.0601 | 1.09 | 0.0611 | 0.98 | 3.08 |

0.004 | 0.0604 | 0.46 | 0.0611 | 0.71 | 1.02 |

**Table 4.**Coefficients of the polynomial curve fits of flow profiles over bar and over the spacing between bars.

x/h_{c} · m · C_{q0} | Flow Profile | a | b | c | d | e |
---|---|---|---|---|---|---|

<0.50 | - | −0.0041 | −0.0975 | −0.3946 | −0.6253 | +0.6807 |

≥0.50 | Over bar | – | −0.3105 | +1.3284 | −1.9036 | +0.9225 |

Over spacing | – | – | −0.7695 | −1.703 | +0.9249 |

Rack Slope = 10% | |||
---|---|---|---|

q_{1} (l/s/m) | L_{1_lab} (m) | L_{1_CFD} (m) | Error (%) |

198.0 | 0.512 | 0.518 | 1.17 |

155.4 | 0.449 | 0.452 | 0.67 |

114.6 | 0.357 | 0.365 | 2.24 |

77.00 | 0.284 | 0.276 | 2.82 |

53.8 | 0.209 | 0.215 | 2.87 |

Rack Slope = 30% | |||
---|---|---|---|

q_{1} (l/s/m) | L_{1_lab} (m) | L_{1_CFD} (m) | Error (%) |

198.0 | 0.541 | 0.522 | 3.51 |

155.4 | 0.451 | 0.433 | 3.99 |

114.6 | 0.352 | 0.351 | 0.28 |

77.00 | 0.279 | 0.268 | 3.94 |

53.8 | 0.207 | 0.212 | 2.42 |

tanθ | H_{min}/H_{0} |
---|---|

0.10 | 0.89 |

0.30 | 0.82 |

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**MDPI and ACS Style**

Carrillo, J.M.; García, J.T.; Castillo, L.G.
Experimental and Numerical Modelling of Bottom Intake Racks with Circular Bars. *Water* **2018**, *10*, 605.
https://doi.org/10.3390/w10050605

**AMA Style**

Carrillo JM, García JT, Castillo LG.
Experimental and Numerical Modelling of Bottom Intake Racks with Circular Bars. *Water*. 2018; 10(5):605.
https://doi.org/10.3390/w10050605

**Chicago/Turabian Style**

Carrillo, José M., Juan T. García, and Luis G. Castillo.
2018. "Experimental and Numerical Modelling of Bottom Intake Racks with Circular Bars" *Water* 10, no. 5: 605.
https://doi.org/10.3390/w10050605