2.2.1. High Pressure Section

The high-pressure domain (penstock, Figure 6) was established as a stationary numerical analysis, with a k-Epsilon turbulence model and the Total Energy model to obtain temperature changes at strategic points in the domain. The fluid temperature at the inlet was 25 ◦C, and the walls of the study domain were defined as adiabatic.

**Figure 6.** CFD, Post-processing. High-pressure section: Isometric view.

The boundary condition at the input was established as a mass flow rate and the outlet was established as a pressure outlet. Both the inlet and outlet conditions are presented in Table 1; for example, the first simulation is a development to 89,418.9 kg/s (89.67 m3/s) and 390 kPa values, respectively. A total of 2000 iterations were established, with a convergence criterion of residual type "RMS", with a value of 1 × <sup>10</sup>−<sup>6</sup> and, for energy, a value of <sup>1</sup> × <sup>10</sup><sup>−</sup>4.

The post-processing of the interest variable in the software shows the water temperature inside the manifolds (Figure 7), and the temperature on the surface of the RTD instrument through color contours (Figure 8), in which the higher value corresponds to the red color and the minor to the blue. The RTD sensor, a simulated surface within the study domain, directly obtains the necessary resolution for temperature measurement. The dimensions of the simulated sensor are 4 mm in diameter and 152 mm long [20]. Proper mixing of the fluid is confirmed by means of the temperature contours inside the manifolds, and a constant temperature is ensured. The maximum temperature of the fluid inside the manifolds is 25.1 ◦C, and the maximum temperature on the surface of the RTD sensor is 25.09 ◦C.

**Figure 7.** Internal temperature vessel, high-pressure section: (**a**) Longitudinal view, (**b**) Cross view.

**Figure 8.** RTD temperature, high-pressure section: (**a**) Isometric view, (**b**) Longitudinal view (zoom).

According to the standard, at the manifold outlet, the volumetric flow must be between 0.1 × <sup>10</sup>−<sup>3</sup> and 0.5 × <sup>10</sup>−<sup>3</sup> m3/s; therefore, the expected velocity range will be between 0.29 m/s and 1.46 m/s, respectively, since the outlet diameter of the manifolds is 0.02 m. Figure 9 shows the outlet velocity of the manifolds using colored contours. The obtained results confirm the values that are allowed by the standard.

**Figure 9.** Velocity outlet, high-pressure section: (**a**) Isometric view, (**b**) Front view, (**c**) location velocity outlet (zoom).

On the other hand, Figure 10 shows the pressure contours at a location where a relevant sensor is physically attached.

**Figure 10.** Pressure location, high-pressure section: (**a**) Isometric view, (**b**) Front view, (**c**) location pressure outlet (zoom).

#### 2.2.2. Low Pressure Section

The CFD in the low-pressure section, as well as in the high-pressure one, used different inlet mass flows (presented in Table 1); however, the pressure at the outlet of the turbine (draft tube) was established as a pressure static outlet or open to the atmosphere. The numerical simulation was of the "turbo-machinery" type, defining a rotating domain (runner) and a stationary domain (draft tube and manifolds). When using two types of domains, it is necessary to establish a new boundary condition, defined as an interface. This configures itself as a "stage" type, since it adapts the results of a domain with movement to a stationary one, in which it is determined to be a "fluid–fluid" interface with corresponding 360◦ angles. A volumetric flow inlet with a direction based on cylindrical components was defined, a rotational velocity of the runner at 180 rpm and the temperature of the inlet fluid was that obtained at the outlet of the penstock for each of the different cases. The k-Epsilon turbulence model and the Total Energy equation were enabled; similarly, the domain walls were adiabatic, as in the penstock. In both the low- and high-pressure section, one of the most prominent turbulence models, the (k-Epsilon) model, was used. This is implemented in most general purpose CFD codes and is considered the industry standard model. It has proven to be stable and numerically robust and has a well-established regime of predictive capability. Therefore, for general-purpose simulations, the model offers a good compromise in terms of accuracy and robustness.

Within CFX, the turbulence model uses the scalable wall-function approach to improve robustness and accuracy when the near-wall mesh is refined. The scalable wall functions enable solutions to arbitrarily fine near-wall grids, significantly improving standard wall functions. Defined thus, a total of 10,000 iterations were established with a convergence criterion of residual type "RMS" with a value of 1 × <sup>10</sup>−<sup>6</sup> and, for energy, a value of <sup>1</sup> × <sup>10</sup><sup>−</sup>4.

The processing of variables of interest in the software shows the temperature measured by the RTD sensor fitted inside the manifold (Figure 11) at the outlet of the draft tube. The dimensions of the simulated sensor are 4 mm in diameter and 50 mm long.

**Figure 11.** Temperature, low-pressure section: (**a**) Isometric view (**b**) RTD Sensor, zoom.

Figure 12 shows the velocity and pressure at the outlet of the manifold.

**Figure 12.** Outlet location, (**a**) Velocity outlet, (**b**) Pressure Outlet.

A view of the flow inlet through velocity vectors to the mixing chamber is shown in Figure 13. The total length of the collecting tubes is 4.06 m, equivalent to the outlet height of the draft tube for correct sampling in the zone, the diameter of the tubes is 30.8 mm or 1 1/2 in., 10 inlet holes to the collection tube with a diameter of 10 mm satisfy the minimum dimensions required by the standard [2].

**Figure 13.** Internal flow (velocity vectors), low-pressure section.

#### *2.3. Application of Grid Convergence Index (GCI)*

According to [22], the computer code used for CFD applications must be fully referenced, and previous code verification studies must be briefly described or cited. Appropriate methods could be selected to validate that CFD results do not depend on the quality or size of the grid. For the present study, the Grid Convergence Index (*GCI*) method was used.

The recommended procedure to calculate the fine-grid convergence index (*GCI*) is based on Equation (4)

$$\text{GCI}^{21} = (1.25c\_a^{21})/(r\_{21}^p - 1)\tag{4}$$

where *ea* <sup>21</sup> is approximated relative error, calculated by Equation (5). *φ* are the values of critical variables. For the present case, *φ* is the temperature (*T*<sup>11</sup> or *T*21) at specific points in specific domains.

$$
\varepsilon\_a{}^{21} = |\left(\Phi\_1 - \Phi\_2\right)/\Phi\_1|\tag{5}
$$

*r*21*<sup>p</sup>* is the grid refinement factor *r* = *hcoarse*/*hfine*. It is desirable that this is greater than 1.3. The 21 subscripts correspond to the relationship between grid 1 (fine) and grid 2 (coarse); see Equation (6)

$$r\_{21}{}^p = h\_2/h\_1 \tag{6}$$

where "*p*" is the apparent order of the method used. For estimation of discretization error, it is necessary to define a representative cell, mesh or grid size "*h*" (mm). For example, Equation (7) is employed for three-dimensional calculations.

$$h = \left[\frac{1}{N} \ast \sum\_{i=1}^{N} (\Delta V\_i)^{\left(\frac{1}{N}\right)}\right] \tag{7}$$

Δ*Vi* is the volume and *N* is the total number of cells used for the computations. Another method to obtain the size of the grid (*h*) is analyzing the grid in the software used. This analysis can be conducted according to volume, the maximum/minimum length or the maximum/minimum side or the density of the grid.

In comparison with Equation (4), Roache [23] establishes that the grid convergence index (*GCI*) is based on Equation (8)

$$GCI\_{Ro} = 3 \mid \varepsilon \mid / (r^p - 1) \tag{8}$$

where *ε* is equivalent to *ea* 21, and *rp* is equivalent to *r*21*p*. A summary and comparison of results for two grids are shown in Tables 2 and 3.

**Table 2.** Summary of results, high-pressure section.


**Table 3.** Summary of results, low-pressure section.


The grid convergence index (*GCI*) is adequate when the result is less than 1%, according to Roache. Despite the *CGI* differences between the authors, a value of less than 1% was obtained for both cases. Due to the presented results, it is possible to carry out the current study with the first generated grid.
