*2.1. Measurement System Design*

2.1.1. Manifolds Design for the High-Pressure Section (Inlet)

According to Castro [18] and Urquiza [19], the principal parameters were obtained to design the manifolds used in TM on the turbine´s inlet section. The values shown in Table 1 are the final results of the Gibson method, applied on a 52.54 m head turbine under different working conditions. (*QT*) it is the net volumetric flow, (*Q*0) is leakage flow when wicked gates are closed, (*P*1) is inlet pressure in the flow of water, (*Pm*) is the mechanical power energy generated by the runner, (*Pe*) is the electrical power measurement in the generator, (*Torque*) is the torque generated by the runner, (*ηh*) is the hydraulic efficiency of the turbine and (*ηg*) is the efficiency measured in the generator. The number of manifolds and their positioning is shown in Figure 2. The proposed design is shown in Figure 3 [20].


**Table 1.** Parameters of the turbine on study [18,19].

**Figure 2.** Measurement system, high-pressure section: (**a**) general view, (**b**) upper-right probe and manifold, zoom.

**Figure 3.** Manifold proposed and instrumentation.

According to [19], for each volumetric flow, the rotational velocity is 180 RPM (18.84 rad/s), and the total deviation of measurements was ±1.6%. It is possible to define the total deviation of measurements of the flow in a systematic way, with Equation (2):

$$\delta\_Q = \pm \sqrt{\delta\_{\Lambda\rho}^2 + \delta\_{\Lambda A}^2 + \delta\_C^2 + \delta\_\delta^2 + \delta\_{D\rho}^2 + \delta\_{\Lambda pf}^2 + \delta\_t^2 + \delta\_{Ql}^2 + \delta\_{rp}^2} \tag{2}$$

where:

*δ*Δ*<sup>ρ</sup>* —Uncertainty regarding the change in water density due to subsequent pressure change.

*δ*Δ*A*—Uncertainty regarding the change of pipe section due to the change in pressure.

*δC*—Uncertainty regarding the determination of the C-value (C = L/A).

*δρ*—Uncertainty regarding the value of water density.

*δ*Δ*p*—Uncertainty regarding errors in measuring pressure differences between sections of the pressure pipe.

*δ*Δ*pf*—Uncertainty regarding the decrease in pressure in the section of the pipe that generates hydraulic losses.

*δt*—Error relating to measurement over time.

*δQl*—Relative uncertainty of measurement under final conditions by assessing flow intensification (leakage intensification).

*δrp*—Error regarding the pressure change log.

The probe intrusion depth in the pressure tube for the extracted water samples is 170 mm, placed diametrically opposite to, or at 90◦ from, each other. According to Côté [9], the increase in the intrusion length does not represent significant changes between the results obtained with a longer probe (50 mm minimum). The differences between the results obtained with probes of different length were small, and no greater than those obtained with probes of the same length. On the other hand, the intrusion depth of the probe is at an optimum point where the main velocity produces a velocity equal to the average falling velocity of the turbine at the probe inlet. The optimal penetration where this condition is fulfilled is reported for different flow velocity profiles within the penstock [8].

However, the power of the turbine shaft (*Pm*) or mechanical power has been calculated with Equation (3):

$$P\_m = \left(P\_\mathbf{e} / \eta\_\mathbf{g}\right) - P\_f \tag{3}$$

where *Pe* is the generator active power (measured on site), *η*<sup>g</sup> is the efficiency of the generator (obtained from the manufacturer), and *Pf* = (*PtB* + *PgB*) are the losses in the load-bearing block (*PtB*) and the guide-bearing (*PgB*). The losses have been calculated in accordance with the IEC 60041 standard.

#### 2.1.2. Manifolds Design for the Low-Pressure Section (Outlet)

For the study of energy transfer in the low-pressure section, the geometry and design parameters were obtained by Castro [18]. The low-pressure section is made up of a rotating domain and a stationary one. The first is made up of the runner, hub and shroud of the turbine; the second is made up of the draft tube, divider and outlet of the section.

According to the standard, the distance of the traction intakes in this section must be located at a distance from the runner of at least five times its maximum diameter; for the turbine in question, the tip diameter of the runner is 3.5 m and the minimum distance required is 17.5 m. However, the manifolds were located farther away than the minimum distanced required to avoid turbulence generated in the walls, close to the division of the draft tube (see Figure 4).

**Figure 4.** General geometry low section pressure (isometrical view).

Hulaas establishes that, under favorable conditions, the application of TM can be extended to falls of less than 100 m; on the other hand, since it is an inaccessible, closed measurement selection, the only possibility of exploring the temperature is through an intake device located inside the tube. This device consists of at least two tubes that collect partial flows [1,2].

Based on Figure 4, four fluid withdrawal intakes were coupled to perform temperature, flow rate and pressure measurements at the outlet of the draft tube; the proposed design is shown in Figure 5.

**Figure 5.** Manifold vessels coupling, outlet section: (**a**) manifolds T21, T22, T23 and T24, (**b**) view outlet section left, (**c**) isometric view of manifold vessel, (**d**) mixing chamber (inside).

#### *2.2. Numerical Simulation (CFD)*

The computational fluid dynamics (CFD) analysis for the high- and low-pressure sections was performed in commercial software (ANSYS CFX). The domain discretization was performed by ICEM for both domains, and both the numerical calculation, and the post-process were performed by ANSYS CFX.

The discretization of the high-pressure section was of the non-structured tetrahedral type, presenting a total of 1,273,913 elements. In both the high- and low-pressure section, the element unit is millimeters (mm).

For the high-pressure section, the minimum size of the element is 1 mm, and the maximum size is 480 mm. This section includes the temperature sensors, probes, manifolds, inlet, outlet, and penstock.

The discretization for the low-pressure section is also that of the non-structured tetrahedral type, presenting a total of 6,297,796 elements. On the other hand, united with the elements, smaller bodies such as collector tubes (manifolds), mixing chambers, RTD's, and the flow inlet and outlet locations are added. For the low-pressure section, the minimum size of the element is 1 mm, and the maximum size is 600 mm. This section includes the temperature sensors, manifolds, runner, inlet and outlet of turbine, and draft tube, respectively.

For each of the numerical simulations, mass flow conditions calculated from the inlet volumetric flow were established.

According to [21], some turbulence models, such as k−Epsilon, are only valid for fully developed turbulence, and do not perform well in the area close to the wall. Two ways of dealing with the near-wall region are usually proposed.

One way is to integrate the turbulence with the wall, where turbulence models are modified to enable the viscosity-affected region to be resolved with all the mesh down to the wall, including the viscous sublayer. When using a modified low-Reynolds turbulence model to solve the near-wall region, the first cell center must be placed in the viscous sublayer (preferably y+ = 1), leading to the requirement of abundant mesh cells. Thus, substantial computational resources are required.

Another way is to use the so-called wall functions, which can model the near-wall region. When using the wall functions approach, there is no need to resolve the boundary layer, causing a significant reduction in the mesh size and the computational domain. Then:


For the present case, the absolute distance from the wall in temperature sensors (walls of greater interest) is 0.97 mm (*y*), the *Re* number is 3998.2, the skin friction (*Cf*) is 0.013, the Wall shear stress (*τw*) is 2.44 Pa, the friction velocity (*u\**) is 0.049 m/s and the *y+* value is 47. As the *y+* value is in the range 30 < *y+* < 300, both the turbulence model k-Epsilon and mesh are applicable for the study.
