# Energy Dissipation of Type a Piano Key Weirs

^{*}

## Abstract

**:**

_{i}/W

_{o}) were studied, with 255 tests comprising this new dataset, along with detailed observations of the flow field. Results were compared to existing published data regarding energy dissipation downstream of trapezoidal and rectangular labyrinth weirs. To support design efforts, two equations, both functions of head-water ratio (H/P) and W

_{i}/W

_{o}, are proposed to predict the relative residual energy downstream of PK weirs. The energy dissipation of PK weirs is largest at low flows and decreases in a logarithmic-like manner as flow increases. PK weirs with increased hydraulic efficiency, caused by an increase in W

_{i}/W

_{o}, resulted in slightly smaller energy dissipation values within the range 0.2 ≤ H/P ≤ 0.8. The energy dissipation of PK weirs was found to be relatively constant, independent of W

_{i}/W

_{o}, and in the ranges 0.07 ≤ H/P ≤ 0.2 and 0.8 ≤ H/P ≤ 0.95.

## 1. Introduction

#### 1.1. PK Weir Overview

#### 1.2. Energy Dissipation

_{1}/H

_{0}) downstream of trapezoidal labyrinth weirs, where H

_{0}is the upstream specific energy (H

_{0}= H + P, P = weir height) and H

_{1}is the downstream specific energy. An empirical equation was fit to the data to predict H

_{1}/H

_{0}, which was also compared with energy dissipation downstream of vertical drops [7,8]. They concluded that H

_{1}/H

_{0}at the base of labyrinth weirs increases nonlinearly as H

_{0}increases. Additionally, for a given H

_{0}, labyrinth weirs were shown to dissipate more energy than vertical drops, in part due to the colliding nappes in the downstream cycles. Lopes et al. [19,20] estimated ±10% or better agreement with [17] and also documented the characteristic depths, air concentrations, and described flow patterns downstream of trapezoidal labyrinth weirs. They observed a three-dimensional flow field directly downstream of labyrinth weirs, noting shockwaves and areas of air entrainment. It was also noted that as the headwater ratio (H/P) increases, the horizontal downstream distance to a normalized two-dimensional flow regime also increases. Finally, within the two-dimensional flow regime, characteristic depths (which increase with increasing H/P) and air concentration profiles become similar regardless of transverse location.

_{i}/W

_{o}= 1.0 to assist with the optimization of stilling basin configurations. They noted that energy dissipation provided by rectangular labyrinth weirs is somewhat similar to that of trapezoidal labyrinth weirs. Al-Shukur and Al-Khafaji [22] studied the role of PK weir slopes on energy dissipation and concluded that dissipation decreases as the slope decreases. The magnitude of the energy dissipation was determined by the distance of the hydraulic jump downstream from the toe of the PK weir. They assumed that the dissipation of energy decreases as the distance to the hydraulic jump increases.

_{i}/W

_{o}) of 1.0, 1.25, 1.28, and 1.5 and two size scales for a wide range of hydraulic conditions 0.05 ≤ H/P ≤ 1.0 to estimate dissipation of energy and corresponding flow features.

## 2. Experimental Setup

_{i}/W

_{o}≤ 1.5 with four datasets of 0.05 ≤ H/P ≤ 1.0, comprising 255 tests in total. The experimental test matrix is summarized in Table 1, with an overview of the experimental setup in each flume presented in Figure 3.

_{i}/W

_{o}PK weir was directly mounted to the floor of Flume 1. In Flume 2, the remaining three PK weirs were mounted to a thin plate (P

_{p}= 6.4 mm thick) that was necessary for installing in the flume. P

_{p}was accounted for when estimating specific energy. Flows were gravity fed from the reservoir adjacent to the laboratory and were measured using calibrated orifice meters (±0.25% accuracy) connected to pressure transducers (0.1% accuracy). Flows were averaged over 5 min sample periods, with 10% of tests repeated as part of experimental uncertainty and quality control.

_{0}) and downstream flow depths, h

_{1}, were carefully measured with a point gage (±0.1 mm) and stilling well setup with taps at locations that considered the flow fields in the flume selected at 4P upstream [34] and at 10P downstream of each PK weir. Piezometric head for PK1.28 in Flume 1 was also measured at 8P. The downstream measurement locations to estimate H

_{1}were selected based upon the complex flow field exiting immediately downstream of the PK weirs and the downstream distance needed for the flow field to return to gradually varied with a quasi-constant flow depth in a given cross section, and the assumption of a uniform unit discharges (q) is reasonable. Specific energy upstream and downstream of the PK weirs (H

_{0}and H

_{1}, respectively) was calculated as h

_{0}+ V

_{0}

^{2}/2g or h

_{1}+ V

_{1}

^{2}/2g with the flume bottom as reference, where h

_{0}and h

_{1}are the piezometric head and V

_{0}and V

_{1}are the average cross-sectional velocity. P

_{p}was also considered for tests in Flume 2. Values of H

_{1}and H

_{0}were then used to compute relative energy dissipation [(H

_{0}− H

_{1})/H

_{0}] and relative residual energy (H

_{1}/H

_{0}). This analysis also looked specifically at local velocities, U, since the appropriate estimation of V is necessary. According to [34], U measurements upstream of labyrinth weirs were uniform at approximately 1–3P, depending on flow. With this guidance, U measurements (sample period of 2 min) in this study therefore focused on the downstream flow field, where U profiles were systematically taken at 8P and 10P at 0.25, 0.5, and 0.75W across the width W of the flume using a Sontek Acoustic Doppler Velocimeter [35]. Velocity analysis results found agreement between these two approaches, resulting in a difference of less than 5% in H

_{1}for the range of Q.

_{1}/h

_{j}= 0.5[(1 + 8F

_{j}

^{2}) − 1], where h

_{j}is the inflow depth to the jump and F

_{j}is the inflow or approach Froude number. Such dissipation estimates were small when juxtaposed to energy dissipated by the PK weirs. The authors point out that there is some uncertainty in this indirect approach since F

_{j}and Belanger’s equation assume uniform inflow conditions but are being applied to a three-dimensional rapidly varied flow field at the base of the PK weirs. However, since the focus of this analysis is to provide design information to industry where safety factors are typical and there may be much greater uncertainties in hydrologic studies, the authors believe that this estimation is reasonable, certainly as a first-order approximation. Nonetheless, the authors would encourage additional research focused on this complex region to expand and build upon the results presented herein.

_{0}and H

_{1}to the datasets by [21] of a rectangular labyrinth weir and [17] (also presented in [18]). The second comparison is specific to PK weirs, where the Q vs. H data collected herein were compared to a commonly used head-discharge method [36] and a scale effects study [37], where [36] tested Type A PK weirs with P = 0.197 m and W

_{i}/W

_{o}= 1.50, 1.25, and 1.0 (H was computed at 1P and 2P), while [37] included data for P = 0.42 m and W

_{i}/W

_{o}= 1.28 (H

_{0}was computed at ~4.7P). As shown in Table 2, the comparison of Q for a given H was made by computing the mean absolute percentage error (MAPE) and the root mean square error (RMSE) between the data from this study and that from [36,37].

## 3. Results

#### 3.1. Type A PK weir Flow Features

_{o}+ B

_{b}streamwise distance is locally submerged [39]. Local submergence is a contributor to reduced discharge capacity and is also noteworthy when considering the reduction in energy dissipation for higher H/P ratios, as discussed in the subsequent section.

#### 3.2. Dissipation of Energy

_{0}− H

_{1})/H

_{0,}in Figure 7. The published data for laboratory-scale trapezoidal labyrinth weirs [17] and rectangular labyrinth weirs [21] is included for comparison and reference. A second quantification of energy dissipation is relative residual energy, H

_{1}/H

_{0}(see Figure 8). (H

_{0}− H

_{1})/H

_{0}of PK and labyrinth weirs are directly inverse to the trends for H

_{1}/H

_{0}.

_{0}− H

_{1})/H

_{0}is inversely proportional to H/P with a nonlinear trend. Energy dissipation is greatest for lowest flow depths, and a PK weir is less effective at dissipating energy at higher heads. Also observed is for H/P < 0.3, a small change in H/P results in a relatively large change in relative energy dissipation. As H/P increases, the rate of change in (H

_{0}− H

_{1})/H

_{0}gradually decreases. This is in part due to the progressive change in the portion of the nappe affected by local submergence [39]. The labyrinth or PK weir nappe conditions transition from aerated to partially aerated and then drown [40], energy dissipation provided by jet collision and downstream impact is altered and consequently reduced. The trends discussed above are also true when comparing relative energy dissipation to q of dissimilar structures such as vertical drops and overflow weirs.

_{0}− H

_{1})/H

_{0}across PK weirs is relativity uniform at the dataset extremities (low and high ends of H/P) and independent of W

_{i}/W

_{o}values. At 0.2 ≤ H/P ≤ 0.9, the relative energy dissipation of PK weirs exhibits a dependence upon key geometries, and thus values of W

_{i}/W

_{o}are a convenient parameter. Furthermore, the results of [21] for rectangular labyrinth weirs compared with the PK 1.00 tested herein do not clarify the quantitative effects of energy dissipation by the inlet and outlet ramps, as it would be anticipated that less energy would be dissipated by the presence of inlet and perhaps also outlet ramps.

#### 3.3. Residual Energy Estimation for Design

_{i}/W

_{o}.

^{2}, mean absolute percentage error (MAPE), and root mean square error (RMSE) for each set of coefficients.

_{1}/H

_{0}using Equation (1) and the corresponding coefficients from Table 3 are plotted against the observed data for each weir in Figure 9. It can be seen that the predicted values of relative residual energy for PK1.00 and PK1.25 correspond well to the observed data. Larger differences are noted between the predicted and observed values of relative residual energy of PK1.28 and PK1.50. The authors recommend that this tool be considered a first-order approximation that may be acceptable for conceptual designs and alternative analyses. Hydraulic modeling of site-specific conditions using physical and, when appropriate, computational fluid dynamics is recommended for final design.

^{2}, MAPE, and RMSE values of Equation (2) relative to each Type A PK weir geometry.

_{i}/W

_{o}and the change in H

_{1}/H

_{0}, causing a slight overestimation of the H

_{1}/H

_{0}for PK1.5. Additionally, for PK1.28, the predicted values of H

_{1}/H

_{0}at 0.5 ≤ H/P are slightly less than the observed values.

## 4. Discussion and Conclusions

- Similar to labyrinth weirs, PK weirs provide some energy dissipation that may be desirable in rehabilitation and new projects. The rate of energy dissipation is not linear and is greatest at low heads.
- Energy dissipation is in part provided by flow entering the structure with the front perpendicular faces of the PK weir beneath the upstream overhangs. Some vortex shedding is noted along with capillary waves. The nappes are considered as opposing planar jets producing turbulence, mixing, increasing aeration, and forming jets directed downstream at a trajectory mimicking the outlet cycle ramps. These jets expand and interact at the toe of the PK weir and with the tailwater.
- The parameter Wi/Wo appears to affect energy dissipation of a PK weir in the following range: 0.2 ≤ H/P ≤ 0.8. As Wi/Wo increases, the hydraulic efficiency also increases, resulting in decreased energy dissipations. At values of H/P ≤ 0.2 and H/P ≥ 0.8, energy dissipation appears to remain relatively constant, independent of the parameter Wi/Wo.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Overview of the new Piano Key weir (PK weir) under construction at West Fork of Eno Reservoir Dam (North Carolina, USA). (

**a**) Looking upstream; (

**b**) across weir towards right wingwall, and (

**c**) looking downstream. Photos courtesy Schnabel Engineering.

**Figure 4.**Type A PK weir (W

_{i}/W

_{o}= 1.28) with flow field modification due to structure geometry and energy dissipated in part by outlet keys including upstream face and jet interaction.

**Figure 7.**Relative energy dissipation [(H

_{0}− H

_{1})/H

_{0}] with respect to (

**a**) the head water ratio (H/P) and (

**b**) unit discharge (q) for Type A PK weirs.

**Figure 8.**Relative residual energy (H

_{1}/H

_{0}) with respect to (

**a**) the head water ratio (H/P) and (

**b**) unit discharge (q) for Type A PK weirs.

**Figure 9.**Observed and predicted values of relative residual energy using Equation (1) and Table 3.

Model | Range of H | W_{i}/W_{o} | W_{i} | P | L | B | N | Flume |
---|---|---|---|---|---|---|---|---|

(m) | (mm) | (m) | (m) | (m) | ||||

PK 1.50 | 0.014–0.186 | 1.50 | 124.7 | 0.20 | 4.82 | 0.50 | 4 | 2 |

PK 1.28 | 0.020–0.211 | 1.28 | 248.4 | 0.42 | 5.15 | 1.04 | 2 | 1 |

PK 1.25 | 0.014–0.188 | 1.25 | 115.6 | 0.20 | 4.75 | 0.50 | 4 | 2 |

PK 1.00 | 0.014–0.187 | 1.00 | 103.9 | 0.20 | 4.81 | 0.50 | 4 | 2 |

**Table 2.**Q comparison for corresponding H between the present study and published data for Type A PK weir.

Geometry | Published Data | 0.2 ≤ H/P | 0.2 ≤ H/P ≤ 0.9 | ||
---|---|---|---|---|---|

MAPE ^{1} | RMSE ^{2} | MAPE ^{1} | RMSE ^{2} | ||

PK 1.50 | Anderson and Tullis (2013) [36] | 3.12% | 0.0113 | 4.97% | 0.0327 |

PK 1.28 | Young (2018) [37] | 1.79% | 0.0061 | 3.39% | 0.0226 |

PK 1.25 | Anderson and Tullis (2013) [36] | 1.94% | 0.0086 | 3.01% | 0.0179 |

PK 1.00 | Anderson and Tullis (2013) [36] | 1.15% | 0.0045 | 1.86% | 0.0103 |

^{1}MAPE = Mean Absolute Percentage Error;

^{2}RMSE = Root Mean Square Error.

Geometry | A | B | C | R^{2 1} | MAPE ^{2} | RMSE ^{3} |
---|---|---|---|---|---|---|

PK 1.50 | 0.7699 | 0.2107 | 0.01 | 0.9931 | 2.83% | 0.0154 |

PK 1.28 | 0.7835 | 0.2108 | 0.01 | 0.9954 | 2.27% | 0.0113 |

PK 1.25 | 0.7528 | 0.2088 | 0.02 | 0.9994 | 1.29% | 0.0062 |

PK 1.00 | 0.7478 | 0.2075 | 0.10 | 0.9993 | 0.75% | 0.0046 |

^{1}R

^{2}= coefficient of determination;

^{2}MAPE = Mean Absolute Percentage Error;

^{3}RMSE = Root Mean Square Error.

Geometry | R^{2 1} | MAPE ^{2} | RMSE ^{3} |
---|---|---|---|

PK 1.50 | 0.9981 | 6.84% | 0.0313 |

PK 1.28 | 0.9943 | 2.73% | 0.0146 |

PK 1.25 | 0.9990 | 6.93% | 0.0320 |

PK 1.00 | 0.9994 | 3.10% | 0.0150 |

^{1}R

^{2}= coefficient of determination;

^{2}MAPE = Mean Absolute Percentage Error;

^{3}RMSE = Root Mean Square Error.

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**MDPI and ACS Style**

R. Eslinger, K.; Crookston, B.M.
Energy Dissipation of Type a Piano Key Weirs. *Water* **2020**, *12*, 1253.
https://doi.org/10.3390/w12051253

**AMA Style**

R. Eslinger K, Crookston BM.
Energy Dissipation of Type a Piano Key Weirs. *Water*. 2020; 12(5):1253.
https://doi.org/10.3390/w12051253

**Chicago/Turabian Style**

R. Eslinger, Kam, and Brian M. Crookston.
2020. "Energy Dissipation of Type a Piano Key Weirs" *Water* 12, no. 5: 1253.
https://doi.org/10.3390/w12051253