Experimental study of the effect of the input and output key angles on the discharge coefficient and efficiency of the PKWs Open Access

The piano key weir (PKW) is a type of hydraulic infrastructure that aims to increase the passing flow discharge and discharge capacity, improve weir performance, and simultaneously decrease construction costs (Bhukya et al. 2022). PKWs are used with gravity dams and natural channels (Anderson & Tullis 2013; Erpicum et al. 2016; Crookston et al. 2019; Tullis et al. 2020) and can replace any affected gated weirs to increase operational performance and maintenance (Laugier 2007; Leite Ribeiro et al. 2012). The discharge coefficient of PKWs is very important, and it has been addressed in Machiels et al. (2009), Ghanbari & Heidarnejad (2020) and Roushangar et al. (2021). Kabiri-Samani & Javaheri (2012) studied the effect of weir geometry including weir length and height, upstream key and downstream key width, as well as upstream and downstream apex overhangs on the discharge coefficient of PKWs in free and submerged flow conditions. Machiels et al. (2014) performed a parametric study of the flow over PKWs and presented equations to determine the flow discharge passing over a cycle of PKWs. Safarzadeh & Noroozi (2017) used the Flow-3D numerical model to study the effects of the inlet key area and the angle of the side walls on the discharge coefficient of PKWs considering the 3D flow condition. Crookston et al. (2018) studied two approaches: (1) empirical prediction methods ranging from simple to sophisticated (including five experimental design methods) and (2) computational fluid dynamics (two different turbulence models) in PKWs. Kumar et al. (2019) considered the importance of the discharge coefficient in PKWs and compared the validity of four equations presented by different researchers under different experimental conditions and data to calculate the discharge coefficient. Akbari et al. (2019) studied the hydraulic performance of PKWs by adding a gate to the input keys. Abhash & Pandey (2021) experimentally and numerically studied the discharge capacity and sediment-carrying capacity of different geometries of PKWs. Behroozi & Vaghefi (2022) studied the discharge capacity and hydraulic behavior of type-A PKWs (with symmetrical consoles) considering different thresholds and geometries. In the laboratory, Mero et al. (2022) investigated the effect of the inlet-to-outlet width ratio (W_i/W_o) on the hydraulic performance of PKWs. Li et al. (2023) performed a numerical simulation to study the hydraulic characteristics of PKWs including the flow pattern. Kadia et al. (2023) used extensive experimental data to present a comprehensive equation with high efficiency and appropriate accuracy to calculate the flow coefficient of type-A PKWs.

In the present study, using experimental data and taking into account the importance of weirs in flood control, solutions were presented to increase the discharge coefficient (C_d) and efficiency (R_a) of rectangular PKWs. In other words, the effect of different input and output key inclination angles including 5°, 30°, 45°, 60°, 75°, and 90° on the discharge coefficient and efficiency of PKWs was studied.

The reservoir was located under the main flume, and water was pumped to the flume using a pump and a pipe with an external diameter of 145 mm. Upon passing the flume and spillways, water was redirected to the reservoir. A hydraulic valve was used to control five different flow discharges.

Equation (2) is in the metric system in which Q_N is the passing discharge over the Ogee linear weir, L is the length of the weir threshold, H_T is the height of the water nappe over the weir threshold, and C_d is the weir discharge coefficient, which was considered as 2.1804.

It is worth noting that since the P/H_T ratio was greater than 3 for all tests, the value of C_d was considered the highest value for the USBR threshold.

In order to study the changes in the weir discharge coefficient, efficiency, passing flow discharge and flow nappe height over the weir, a distance of 9.5 cm from the weir was considered to be studied. In other words, five different flow discharges were passed over each weir at different weir installation angles. Then, for each flow discharge, the mentioned parameters were measured and calculated at different distances such as 9.5 cm from the weir. Different parts of Figure 4 were then plotted. Similar figures can be plotted and presented at all distances.

According to Figure 4(a), C_d and H_T/P are inversely related. Therefore, the highest discharge coefficient occurred at the lowest H_T/P, resulting in the lowest flow discharge. This can be attributed to the low interference of the water nappes on the weir. The reduction in C_d occurred in the 15° PKW with a lower intensity and, in other weirs, with a higher intensity. In addition, at constant H_T/P values, weirs with lower angles had lower C_d. In other words, a 90° PKW had the highest C_d among all cases.

According to Figure 4(b), R_a is inversely related to the H_T/P ratio. Due to the low interference of the water nappes on the weir, the highest efficiency occurred at the lowest discharge, and consequently, at the lowest H_T/P ratio. By increasing the flow nappe height on the weir, the PKW was out of its desired efficiency and acted like a linear weir. This reduction in PKW efficiency occurred with a lower intensity for a 15° threshold angle and a higher intensity for other threshold angles. The PKW with a 90° threshold angle also had the highest efficiency.

According to Figure 4(c), there was a direct relationship between Q_L and H_T/P. At constant H_T/P values, weirs with a higher angle had a higher passing flow discharge. Among all the PKWs, the highest discharge occurred for the 90° threshold angle.

Figure 4(d) shows the changes in C_d versus the angle of the input and output keys of the weir for five different values of H_T/P (from 0.05 to 0.25). For all constant values of H_T/P, C_d was directly related to the angle of the input and output keys. The increase in C_d occurred with more intensity at H_T/P = 0.05 and less intensity in other cases. In addition, at a constant angle, weirs with higher H_T/P had lower C_d. Among all cases, the highest C_d was related to a 90° PKW and H_T/P = 0.05.

Figure 4(e) shows the changes in R_a versus the angle of the input and output keys of the weir for five different values of H_T/P (from 0.05 to 0.25). For all constant values of H_T/P, R_a was directly related to the angle of the input and output keys. The increase in efficiency occurred with more intensity at H_T/P = 0.05 and less intensity in other cases. In addition, at a constant angle, weirs with higher H_T/P had lower R_a. Among all weirs, the highest efficiency was related to a 90° PKW and H_T/P = 0.05.

It is worth noting that the fitted binomial relations between the discharge coefficient (C_d) and the efficiency (R_a) considering the angle of the input and output keys for different H_T/P ratios are also shown in Figure 4.

According to Tables 1–6 and Figures 4 and 5, the discharge coefficient, efficiency, and passing flow discharge are directly related to the inclination angle of the input and output keys. In low H_T/P ratios, these types of weirs do not need aeration. In other words, according to the equation of the flow discharge passing over the PKWs (Equation (1)), the hydraulic characteristics such as the discharge coefficient and the efficiency of the PKWs generally improve with the inclination angle of the input and output keys. Consequently, the efficiency of this type of weir in passing flow discharge increases.

Weirs are one of the most important structures in the design of dams. They are responsible for transferring water over the dam's capacity during floods. The PKWs have high efficiency, and as a result, have a higher passing discharge capacity in flood conditions in comparison to the linear weirs. In the present study, the effect of the angle of the input and output keys on the discharge coefficient (C_d) and efficiency (R_a) (the ratio of the passing flow over a PKW to that of a linear weir) of the rectangular PKWs with identical indentations was studied experimentally for six different angles 15°, 30°, 45°, 60°, 75°, and 90°.

In general, the results of the present study include the following:

• - The discharge coefficient, efficiency and passing discharge for constant values of the H_T/P ratio are directly related to the angle of the input and output keys. A decrease in these mentioned parameters occurs at an angle of 15° with less intensity than other angles.

• - In the PKWs with different input and output key angles, the dropping nappe bonds at low H_T/P ratios and needs aeration. By increasing the H_T/P ratio, the flow nappes drop in the aerated condition and do not need aeration. The increase in the ratio continues until the submergence of the weir with its performance converting to that of a linear weir. After submergence, the performance of the PKW decreases.

• - Due to the lack of interference of the dropping nappes, the highest discharge coefficient and efficiency occur at low H_T/P ratios. By increasing the ratio, the dropping nappes in the output keys start to interfere with each other, causing the flow to rise and the weir to become submerged. As a result, the discharge coefficient and efficiency of the weir decrease.

In general, the highest discharge coefficient of the rectangular PKWs occurs at a 90° angle of the input and output keys with the best efficiency.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.