*3.1. Results of Thermodynamic Method*

3.1.1. High-Pressure Section

For each of the different conditions and working sections, the temperature, velocity and pressure in the manifolds were obtained as required by IEC 60041. Similarly, the amount of volumetric flow that exits the manifolds located on the penstock and draft pipe was tested. As it is a stationary type of simulation, the value of temperature, pressure and velocity is obtained by exporting a series of values provided by the software in each of the locations of interest at the end of the numerical calculation (high- and low-pressure section). This series of values is averaged and shown below.

Table 5 contains the average temperature values in the four manifolds; Table 6 contains the average velocity and pressure values of the manifolds. Section 14.3.1 "General"; of the IEC-60041 standard establishes that the thermodynamic method for the average yield is based on the laws of thermodynamics, using the thermodynamic temperature ϑ in Kelvin (K). In case of temperature differences, the temperature can be directly expressed in Celsius ( ◦C) degrees, as ϑ<sup>1</sup> − ϑ<sup>2</sup> = *θ*<sup>1</sup> − *θ*<sup>2</sup> [2].


**Table 5.** Manifold´s temperature, high-pressure section.

**Table 6.** Manifold's velocity and pressure, high-pressure section.


#### 3.1.2. Low-Pressure Section

The analysis of results in the low-pressure section (runner and draft tube) involves a comparison of the mechanical power and torque generated by the turbine for each flow condition (Table 7) in the software.


**Table 7.** Comparison between mechanical power and torque, reported vs. simulated.

By demonstrating the same mechanical power and torque conditions, the results in the draft tube can be analyzed. The manifolds attached to the draft tube acquired samples of the main flow (water) to obtain the energy distribution at different points. Variables such as temperature, velocity and pressure, obtained in each of the containers, are shown in Tables 8 and 9.

**Table 8.** Manifold´s temperature, low-pressure section.


**Table 9.** Manifold´s velocity and pressure, low-pressure section.


## **4. Discussion**

The results obtained in the low-pressure section (draft tube) show that the direction of runner rotation (clockwise) and the geometry of the draft tube discharges water from a turbine, in addition to acting as an energy-recovery device, helping to improve the overall performance of the unit. It can also allow the downstream water level to be lower or higher than the equatorial plane of the turbine, depending on the needs of the facility. The draft tube, due to its divergent shape, causes a deceleration in the velocity of the water leaving the turbine, converting the kinetic energy of the fluid into pressure energy (Figure 15) [18].

**Figure 15.** Velocity streamlines on the complete turbine, (**a**) upper view, (**b**) lateral View.

By coupling the manifolds in the draft tube, the flow distribution is affected, causing recirculation or vorticity in the area in which manifolds are located. The location of the manifolds is suggested by IEC-60041. Depending on the dimensions of probes, vorticity can be created behind the probes and then dissipated. The flow disturbance will be downstream once velocity, pressure, and temperature variables have been measured, so they cannot influence efficiency calculations. Therefore, the average temperatures in the manifolds *T*<sup>22</sup> and *T*<sup>23</sup> are slightly higher than the average temperature of *T*<sup>21</sup> and *T*24, as derived from the flow distribution behavior in the turbine (Figure 16).

**Figure 16.** Manifold´s in the draft tube, (**a**) Recirculation flow (normalized symbols), (**b**) Recirculation flow in manifolds, left section "zoom" (normalized symbols) (**c**) Temperature contour.

The summary of results obtained from the temperature differences *T*1–*T*<sup>2</sup> (Δ*T*), *Em*, *Eh*, *Pm*, *Ph* and *ηh*, for different cases is presented in Table 10. Figures 17 and 18 show the main comparison of the results, between what was reported in [18,24] and the current case study.


**Table 10.** Summary of results, application of Thermodynamic Method.

<sup>1</sup> Δ*T*: Temperature difference between measured sections (*T*1–*T*2).

**Figure 17.** Comparison, Reported hydraulic efficiency (Gibson method) vs. Simulated hydraulic efficiency.

**Figure 18.** Comparison of mechanical power generated, reported (Gibson method) vs. simulated.

CFD simulations are a proven tool to investigate hydraulic turbine performance, while measurements of some parameters, such as flow or pressure, are common in calculations of their efficiency. In the present study, the design of manifolds and CFD applications contribute to the assay, with sampling system (manifolds) and experimental measurement times in the power plant, complying with the criteria established to apply the TM to low-load turbines.

Experimental studies report that the water temperature at the turbomachine outlet must be higher than that at the inlet. With a lower temperature difference between the measurement sections, the maximum hydraulic efficiency is presented. According to those mentioned above [3], the difference between the efficiency curves is around 0.5%; however, for the present study, the maximum and minimum differences in efficiency are 15.12% and 1.09%, respectively, for the Gibson method (reported). As one of the most important variables for the study is the temperature on surfaces of principal components, such as the runner, penstock, draft tube, etc., and these are unknown, the domain was specified as adiabatic. As a result, there is a low-temperature increase in the water between the high- and low-pressure sections. These cause a low-energy exchange and higher efficiency than expected. If the temperature in these components was known, the boundary conditions could be set differently, and a lower efficiency would be expected in different cases. Likewise, the efficiency would present results closer to those reported. The hydraulic efficiency of the turbine is susceptible to temperature changes between one section and another. This sensitivity is presented with values up to 0.0001 K; the assumed temperature, or a change in temperature in any of the components, has a direct effect on efficiency.

The simulated TM presented differences in the mechanical power and efficiency; however, the behavior of the generated curve shows the same tendency as the curve in the experimental data obtained using the Gibson method (reported), presenting a gradual increase in efficiency until a maximum point is reached. This subsequently decreases. The results obtained for each operating condition are similar to those reported by the Gibson method, meaning an adequate comparison for the study of the proposed manifolds design, considering the head limits (less than 100 m), the amount of maximum volumetric input flow (89.67 m3/s), the type of turbine (Francis slow) and the specific speed of the turbomachine (less than 110). In future studies, the authors recommend developing transitory simulations for other operating conditions, as well as using the experimental test to measure temperature in the main components, and set different variables in the numerical simulations.

According to [3,12], the present study used a hybrid vertical detraction system and a mixing chamber for each tube, reducing the number of sensors that are required to facilitate installation in the low-pressure section. In addition, the manifolds proposed in the low-pressure section are compatible at different outlet heights for the draft tube, as it is only necessary to adjust the tube length.

#### **5. Conclusions**

Based on the location of the manifolds in the input and output sections, the proposed design of manifolds to measure properties of the main flow of a Francis-type low-head hydraulic turbine meet with the requirements suggested by the IEC—60041 Standard to carry out the Thermodynamic Method (TM) employing Computational Fluid Dynamics (CFD).

The distance from the turbine center to the measuring section is essential. The minimum distance set in the standard [2] is five times its maximum diameter, and the measurements show that it should be the absolute minimum. According to Figure 3, a shorter distance could improve energy distribution.

Using a mixing chamber inside the draft tube allows for a direct measurement of temperature in the principal flow at the outlet. In addition, inside the mixing chamber, there is a water flow concentrator, which helps to direct the flow into the temperature sensor, obtaining a direct measurement. The IEC-60041 establishes that the minimum number of tubes consists of two units that collect partial flows. However, increasing the number of tubes and manifolds at the outlet makes it possible to improve the temperature measurements. In this case, four manifolds were used in the low-pressure section. In both the left and right section, two manifolds were installed after the division to avoid a high recirculation or vorticity in the area in which manifolds are located.

On the other hand, the results obtained from the mechanical power and torque in the turbine runner were identical to those reported by the Gibson method (GM); however, the efficiency between the above methods is similar. To obtain results that are closer to reality, the numerical simulations used in CFD must be supplied from as many boundary conditions as possible (actual conditions). It is necessary to set the temperature on the surface of principal components so that the main flow of water makes contact via its passage through the turbomachine to the efficiency results, with the application of TM.

The efficiency calculation is higher under particular volumetric flow conditions (35.68 m3/s and 68.73 m3/s) compared to the efficiency reported when applying the GM. The maximum efficiency generated by the turbine applying the TM was 92.10%, corresponding to a flow of 68.73 m3/s. After the maximum efficiency point, the TM's efficiency is lower than the GM's.

**Author Contributions:** Conceptualization: L.L.C.G.; Investigation: E.O.C.M. and L.L.C.G.; Methodology: L.L.C.G.; Project administration: G.U.B.; Resources: G.U.B.; Software: L.L.C.G.; Supervision: G.U.B.; Writing—original draft preparation: E.O.C.M.; Writing—review & editing: L.L.C.G. and J.C.G.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work is partly supported by the National Council for Science and Technology [Conacyt], CVU Number: 707755.

**Acknowledgments:** To Arturo Nava Torres, for his unconditional collaboration in the presented project. To the "Centro de Investigación en Ingeniería y Ciencias Aplicadas (CIICAp)", for all facilities provided during my stay.

**Conflicts of Interest:** The authors declare no conflict of interest.
