# Multi-Parametrical Tool for the Design of Bottom Racks DIMRACK—Application to Small Hydropower Plants in Ecuador

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{3}/s/m. In all the cases, an occlusion factor f = 0 was considered. The parameter f multiplies the wetted rack length calculated with a clear water hypothesis.

^{3}/s/m); C

_{q}

_{0}the discharge coefficient measured in racks with m = 0.60 in static conditions, i.e., with still flow dissipating the kinetic energy [7,8,10,16] (-); L is the wetted rack length to be calculated (m); h

_{0}the flow depth at the beginning of the rack (m); k

_{c}a coefficient to correct the flow depth at the beginning of the rack depending on the longitudinal slope [14,15] (-); h

_{c}the critical depth calculated as ${h}_{c}={\left(q/\sqrt{g}\right)}^{2/3}$ (m); and g the gravity acceleration (m/s

^{2}). Figure 5 shows the values of the static discharge coefficients proposed by Frank.

^{®}software and available online, DIMRACK, gathers together all the conclusions and serves as a Tool for the Design of Bottom Racks with the possibility of taking into account multiple different parameters such as different void ratios, bar types, longitudinal slope, and flowrates.

## 2. Experimental Setting

_{0}, and the wetted rack length L were measured with a vertical point gauge (accuracy ± 0.5 mm in vertical and ± 1.0 mm in horizontal) as shown in Figure 7. The point gauge measures only vertical distances. To define the water depth at the beginning of the rack, it was necessary to measure several points to trace the water surface profile. From this, the water depth was defined by geometric projection [16,21,22,23,24,25,26,27,28,29,30].

^{3}/s/m. The inlet total flow was measured with an electromagnetic flowmeter Endress Häuser Promag 53W of 125 mm with an accuracy of 0.50% of the flow. Tests were performed with five different longitudinal rack slopes. Further details of the model are available in [16,21,22,23,24,25,26,27,28,29,30]. The approaching flow is subcritical in all the cases at the beginning of the inlet channel. The flow reaches supercritical conditions at the beginning of the rack. The experimental data from these authors carried out from 2010 were analyzed as a whole in the present work. Table 4 summarizes the experimental work that is analyzed through the present work.

## 3. Results and Discussion

#### 3.1. Generalized Nomogram for the Rack Length Calculation

^{3}/s/m. Details of that work can be found in [25]. The resulting equations are presented as follows:

_{0}is the mean velocity at the beginning of the rack, calculated as the relation between the incoming flow and the flow depth at the beginning of the rack, $q/{h}_{0}$, μ the kinematic viscosity; ρ the density; σ the surface tension of water; tanθ expresses the longitudinal slope of the rack; m the void ratio, C

_{q}

_{0}the discharge coefficient measured in static conditions; g the gravitational acceleration; $\overline{{C}_{qH}}$ the mean discharge coefficient for each wetted rack length, and R

_{e}

_{0}, W

_{e}

_{0}, and F

_{r}

_{0}are the Reynolds, Weber, and Froude numbers, respectively. The sub-index 0 indicates that those variables are calculated at the beginning of the rack. For both inspectional and empirical analyses in the bottom racks, it is stated that ${h}_{0}=f\left({q}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)$, allowing us to rewrite the previous equation, $\overline{{C}_{qH}}=f\left({q}^{power},tan\theta ,m,{C}_{q0}\right)$. This assumption fits with the experimental measurements presented in [25] and it is consistent with the Equation (4) of the current work, $\overline{{C}_{qH}}=\frac{am{C}_{q0}}{\left(1+tan\theta \right)}{q}^{b},$ where both “a” and “b” variables are functions of the void ratio, m, maintaining in this way the consistency with previous equations that arose from dimensional analyses. Besides this, from experimental measurements, these variables are adjusted as a function of the void ratio that is included as a dependent variable of the mean discharge coefficient, $\overline{{C}_{qH}}$.

_{q}

_{0}variables that appear in Equation (4), beyond the values presented in Table 5 were sought. The wetted rack length is defined as the maximum length reached by the flow over a rack to completely derive the incoming flow [20]. The total wetted length, L, can be observed in Figure 8, as the photos clearly show. These rack lengths must usually be measured experimentally in the laboratory.

_{min}= 1.50h

_{c}, is the minimum energy of the flow and $\overline{{C}_{qH}}$ the mean discharge coefficient according to Equation (4). Equation (11), comes from the classical orifice equation $\frac{dq}{dx}={C}_{qH}\left(x\right)m\sqrt{2g{H}_{min}}$. In this equation, x is the rack length longitudinal coordinate, H

_{min}the energy height at the beginning of the rack that is the energy associated to the critical conditions, i.e., minimum energy, θ the angle between the bottom rack and the horizontal, m the void ratio, and g the gravitational acceleration. Once integrated along the wetted rack length considering a mean discharge coefficient, $\overline{{C}_{qH},}$ the Equation (11), is obtained. In equation (11), L is the wetted rack length. In the lab, L was measured for more than 376 different cases. Other cases proposed by [20,31] from field and lab experiments were also included to support the proposed Equations. Equation (10), presents the static discharge coefficient, C

_{q}

_{0}, adjusted from experimental measurements. This was obtained in the lab disregarding the approaching velocity, placing a vertical wall at the end of the rack. Therefore, there is only potential energy and the Froude number of the approaching flow, F

_{r0}, tends to 0. In the lab, the flow depth for different flowrates were measured and the orifice equation was solved, obtaining the static discharge coefficient. Results were in agreement with those proposed by several authors [7,10,11,14,15,17]. The Equation (10), was originally published in [16,22]. In the present work, new adaptations have been done to include the static discharge coefficient associated to other profiles like the fish body bars. Regarding Equations (8) and (9), from dimensional analysis, it was previously stated that $\overline{{C}_{qH}}=f\left({q}^{power},tan\theta ,m,{C}_{q0}\right)$ [25]. In agreement with this, it was proposed that Equation (4),$\overline{{C}_{qH}}=\frac{am{C}_{q0}}{\left(1+tan\theta \right)}{q}^{b}$, which was also derived from the analysis of the experimental data. Equation (4) includes two variables, “a” and “b”. These parameters were proposed for the generalization of Equation (4) to any void ratio and are a novelty of the current work. The variable “a” takes into account the influence of the void ratio (in Figure 9, it follows a potential law); and the variable “b” accomplishes the potential law shown in the dimensional analysis, which relates $\overline{{C}_{qH}}$ and the specific discharge, q, for any void ratio. Both variables together with the static discharge coefficient allow for the extension of Equation (4) to any void ratio and bar profile.

#### 3.2. Occlusion Factor from Experiment Data

_{1}with d

_{50}= 8.3 mm, g

_{2}with d

_{50}= 14.8 mm, and g

_{3}with d

_{50}= 22.0 mm. The d

_{50}of the gravels is similar to the space between the bars, which favors the embedding of the gravels in the slits between the bars. The mean values of the occlusion factor f, obtained in the experimental campaign, can be observed in Figure 20. These values were in the range of 1.65–1.37 for longitudinal slopes of 0–33%, and the minimum mean occlusion factor achieved in the 30% slope was 1.35. The mean of the occlusion factors of T-shaped bars was calculated, omitting the higher value of f in each figure, under the consideration that this value would excessively penalize their design. From this, the deviation of each of the figures in terms of the occlusion factor was between 0.12 and 0.18.

_{3}and d

_{50}= 22.0 mm. The results are presented in Figure 21.

#### 3.3. Application Case

#### 3.4. Application to Design Cases in Ecuador

_{0}, included in Figure 4, which was only for the void ratio of m = 0.60. Figure 24 shows that the rack lengths concur with those proposed by the present work.

#### 3.5. Multi-Parametrical Tool for Designing Purposes

^{®}, includes the methodology proposed in the current work and serves as a tool for the design of bottom racks with the possibility of taking into account multiple different parameters such as: clear water or with sediment dragging, different void ratios, bar types, longitudinal slope, and flowrates as presented below.

- (a)
- Project Information: This section is designed to enter information related to the design conditions of the bottom grid, such as the width of the section of the river where the intake will be located, B
_{r}, and the design flow to derive, Q_{d}. Schemes of the components of the bottom intake are also included, as presented in Figure 27. This part also includes the bottom rack information, which allows the information chosen for the racks, such as rack width, B, longitudinal slope, tanθ, and the selection of bottom racks profile (fish body, T-shaped, and circular), to be entered. It also includes schemes of the selected rack profile and its components. The user also needs to provide the bar width, b_{w}, and bar spacing, b_{1}. - (b)
- Hydraulic Design Parameters: This section presents the results of the calculations made by the program and the recommended rack length for the design. In the case of the design considering sediment clogging, the rack length is expressed as L
_{f}, where f is the occlusion factor. The clear water design length is expressed by L.

## 4. Conclusions

^{3}/s/m and longitudinal slopes from 0 to 33% [25].

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**View of the Chalpi A project: (

**a**) construction of the first bottom intake system and sand trap; (

**b**) detail of the bottom rack bar profile [6]; and (

**c**) project location in Ecuador (without scale).

**Figure 6.**Rack length depending on the flowrate to derive, q, in the case of m = 0.60 and longitudinal rack slope = 20% and 30%.

**Figure 8.**Wetted rack length for different flows and longitudinal slopes with racks made of circular bars and void ratio m = 0.28.

**Figure 12.**Factor of wetted rack length in Equation (8), L, in the case of T-shaped bars for different void ratios, m, and flowrates, q.

**Figure 13.**Factor of mean discharge coefficient in Equation (4), $\overline{{C}_{qH}}$, for T-shaped bars depending on the void ratio, m, and the flowrate, q.

**Figure 14.**Factor of wetted rack length in Equation (8), L, in the case of circular bars for different void ratios, m, and flowrates, q.

**Figure 15.**Factor of mean discharge coefficient in Equation (4), $\overline{{C}_{qH}}$, for circular bars depending on the void ratio, m, and the flowrate, q.

**Figure 16.**Factor of wetted rack length in Equation (8), L, in the case of prismatic rounded bars for different void ratios, m, and flowrates, q.

**Figure 17.**Factor of mean discharge coefficient in Equation (4), $\overline{{C}_{qH}}$, for prismatic rounded bars depending on the void ratio, m, and the flowrate, q.

**Figure 18.**Factor of wetted rack length in Equation (8), L, in the case of fish body bars for different void ratios, m, and flowrates, q.

**Figure 19.**Factor of mean discharge coefficient in Equation (4), $\overline{{C}_{qH}}$, for fish body bars depending on the void ratio, m, and the flowrate, q.

**Figure 23.**Rack length depending on the flowrate to derive, q, in the case of m = 0.60 and longitudinal rack slope = 20%.

**Figure 24.**Rack lengths calculated for SHP bottom intakes in Ecuador compared with the values proposed in the present work (clear water case).

**Figure 25.**Rack length depending on the flowrate to derive, q, in the case of m = 0.60 and longitudinal rack slope = 20% according to Ecuador SHP designs, including the results of the current work.

**Figure 26.**Rack lengths calculated for SHP bottom intakes in Ecuador compared with the values proposed in the present work, considering the occlusion factor f = 1.40.

Name | Head (m) | Installed Capacity (MW) | Estimated Annual Production (GWh/year) | Turbines |
---|---|---|---|---|

Dudas [3] | 294.00 | 7.40 | 41.35 | 1 Pelton unit |

San Antonio [3] | 195.00 | 7.19 | 44.87 | 1 Pelton unit |

Jondachi-Sardinas [4] | 98.77 | 6.71 | 44.06 | 2 Francis units |

Nanegal [5] | 110.00 | 5.30 | 37.16 | 2 Francis units |

Alambi [5] * | 110.00 | 5.50 | 29.50 | 2 Francis units |

Tulipe [5] | ||||

Chuquiraguas [5] | 300.00 | 2.35 | 13.15 | 2 Pelton units |

Chanchán [5] | 220.00 | 7.25 | 38.78 | 3 Pelton units |

Chalpi A [6] ** | 398.62 | 7.67 | 24.70–35.96 (depending on urban water supply requirements) | 2 Pelton units |

Encantado [6] | ||||

Chalpi B [6] | ||||

Chalpi C [6] |

**Table 2.**Different setups of experiments to define the length of the bottom racks for q

_{1}= 1.25 m

^{3}/s/m.

Autor | Required Length (m) | Shape of the Bars | Setting of the Rack, Void Ratio | Slope of the Experiments |
---|---|---|---|---|

Righetti & Lanzoni [7] | 0.91 | Prismatic with rounded edge | m = 0.20; b_{1} = 0.50 cm; b_{w} = 2.00 cm | <3.5% |

Mostkow [8] | 1.01 | Prismatic | Not specified | |

Vargas [9] | 1.17 | Circular | Two setups: (a) m = 0.33; b _{1} = 0.50 cm; b_{w} = 1 cm;(b) m = 0.5; b _{1} = 1 cm; b_{w} = 1 cm | slope: 0°–20° |

Brunella et al. [10] | 1.56 | Circular | Two setups: (a) b _{1} = 0.60 or 0.30 cm; b_{w} = 1.20 or 0.60 cm; m = 0.352(b) b _{1} = 1.80 or 0.90 cm; b_{w} = 1.20 or 0.60 cm; m = 0.664 | 0°–51° |

Noseda [11] | 1.96 | T- shaped | 0.16 < m < 0.28 0.57 < b _{1} < 1.17 cm | 0–20% |

Krochin [12] f = 0% | 2.04 | Prismatic | Not specified |

_{1}is the space between bars; b

_{w}the width of a bar; and $m={b}_{1}/\left({b}_{1}+{b}_{w}\right)$ the void ratio.

Length (m) | Width b_{w} (m) | Bar Type (mm) | Width of the Bars (mm) | Direction of the Bars | Spacing between Bars, b_{1} (mm) | Void Ratio $\mathrm{m}=\frac{{\mathrm{b}}_{1}}{{\mathrm{b}}_{1}+{\mathrm{b}}_{\mathrm{w}}}$ |
---|---|---|---|---|---|---|

0.90 | 0.50 | T30/25/2 | 30 | Longitudinal | 5.7 | 0.16 |

T30/25/2 | 8.5 | 0.22 | ||||

T30/25/2 | 11.7 | 0.28 | ||||

O30/30 | 11.7 | 0.28 | ||||

O30/30 | 45.0 | 0.60 |

Experimental Work Description | Flow Rates (L/s/m) | Void Ratios, m | Longitudinal Slope (%) | References |
---|---|---|---|---|

T-shaped bars with clear water. Flow profile, rack length, and discharge coefficients | 53.8, 77.0, 114.6, 138.8, 155.5 | m = 0.16, 0.22 and 0.28 | 0, 10, 20, 30, 33 | [22,23,24,25] |

T-shaped bars in a flow with gravels. Occlusion factor with three different gravels: d_{50} = 8.3, 14.8, and 22.0 mm | 114.6, 138.8, 155.5 | m = 0.16, 0.22 and 0.28 | 0, 10, 20, 30, 33 | [26,27] |

Circular bars with clear water. Flow profile, rack length, and discharge coefficients | 53.8, 77.0, 114.6, 138.8, 155.5, 198.0 | m = 0.28, 0.60 | 0, 10, 20, 30, 33 | [24,25,28] |

Circular bars in a flow with gravels. Occlusion factor with three different gravels: d_{50} = 22.0 and 58 mm | 114.6, 138.8, 155.5, 198.0, 250.0 | m = 0.28, 0.60 | 0, 10, 20, 30, 33 | [29,30] |

**Table 5.**Constants of Equation (10) for the adjustment of the mean discharge coefficient $\overline{{C}_{qH}}$ [25].

Bar Type | m | a | b |
---|---|---|---|

T | 0.16 | 3.3 | 0.05 |

T | 0.22 | 2.1 | 0.05 |

T | 0.28 | 1.5 | 0.05 |

O | 0.28 | 1.45 | 0.05 |

O | 0.60 | 0.70 | 0.20 |

Intake | Width of the River, B_{r} (m) | Width of the Intake, B (m) | Design Flowrate, Q_{D} (m^{3}/s) | Bar Type | Longitudinal Slope, tanθ (%) | b_{1} (m) | b_{w} (m) | Void Ratio, m (-) | Rack Length, L |
---|---|---|---|---|---|---|---|---|---|

Chalpi A | 35.00 | 6.00 | 2.20 | T | 20 | 0.050 | 0.030 | 0.625 | 1.20 |

Encantado | 16.00 | 5.00 | 1.14 | T | 20 | 0.050 | 0.030 | 0.625 | 0.70 |

Chalpi B | 12.00 | 3.50 | 0.47 | T | 20 | 0.050 | 0.030 | 0.625 | 0.50 |

Chalpi C | 5.00 | 1.00 | 0.12 | T | 20 | 0.0254 | 0.020 | 0.559 | 0.60 |

Jondachi Sardinas | 40.00 | 9.00 | 8.80 | Prismatic | 21 | 0.050 | 0.030 | 0.625 | 2.50 |

Nanegal | 48.00 | 10.00 | 7.00 | T | 20 | 0.050 | 0.030 | 0.625 | 3.00 |

Alambi | 17.00 | 8.00 | 11.50 | T | 20 | 0.050 | 0.030 | 0.625 | 3.00 |

Tulipe | 6.00 | 6.00 | 1.40 | T | 20 | 0.050 | 0.030 | 0.625 | 1.20 |

Chuquiraguas | 11.00 | 6.00 | 1.40 | T | 20 | 0.050 | 0.030 | 0.625 | 1.20 |

Chanchán | 10.00 | 5.00 | 4.83 | T | 20 | 0.050 | 0.030 | 0.625 | 2.30 |

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**MDPI and ACS Style**

García, J.T.; Castillo, L.G.; Haro, P.L.; Carrillo, J.M. Multi-Parametrical Tool for the Design of Bottom Racks DIMRACK—Application to Small Hydropower Plants in Ecuador. *Water* **2019**, *11*, 2056.
https://doi.org/10.3390/w11102056

**AMA Style**

García JT, Castillo LG, Haro PL, Carrillo JM. Multi-Parametrical Tool for the Design of Bottom Racks DIMRACK—Application to Small Hydropower Plants in Ecuador. *Water*. 2019; 11(10):2056.
https://doi.org/10.3390/w11102056

**Chicago/Turabian Style**

García, Juan T., Luis G. Castillo, Patricia L. Haro, and José M. Carrillo. 2019. "Multi-Parametrical Tool for the Design of Bottom Racks DIMRACK—Application to Small Hydropower Plants in Ecuador" *Water* 11, no. 10: 2056.
https://doi.org/10.3390/w11102056