*2.4. Thermodynamic Method Application*

The calculation of Hydraulic Efficiency (*ηh*) is defined by the ratio of the mechanical power (*Pm*) and the hydraulic power (*Ph*) of the turbine, respectively, as in Equation (9).

$$
\eta\_h = P\_m / P\_h \tag{9}
$$

The mechanical power (*Pm*) of the turbine is calculated by the specific mechanical energy (*Em*), density (*ρ*) and the volumetric flow (*QT*) that passes through the turbine, as in Equation (10).

$$P\_m = E\_m \* (Q\_T \* \rho) \tag{10}$$

The hydraulic power (*Ph*), in contrast with the *Pm*, is obtained by means of the Specific Hydraulic Energy (*Eh*), as in Equation (11). The correction factor (Δ*Ph*) is neglected since Urquiza [8] considered this factor in the presented results.

$$P\_{\rm h} = E\_{\rm h} \ast (Q\_T \ast \rho) \pm \Delta P\_{\rm h} \tag{11}$$

The *Em* was calculated with the variables measured in the manifolds, such as pressure (*p*), temperature (*T*) and velocity (*v*), (see Equation (12)). The reference heights (*z*) are assigned for each manifold and the isothermal factor (*a˙*), as well as the specific heat (*Cp*), are obtained from the annexes of IEC 60041, Appendix E physical data, Table EV and EVI [2] (Table 4), and an interpolation of the temperature and average pressure for each of the case studies.


**Table 4.** Properties of water [2].

Finally, gravity (*g*) was obtained from Reference [8]. The subscripts 11 and 21 correspond to the manifolds in the inlet and outlet section, respectively. Similarly, *T*<sup>1</sup> and *T*<sup>2</sup> belong to the corresponding sections.

$$E\_m = \left[\dot{a} \ast (p\_{11} - p\_{21})\right] + \left[\mathbb{C}\_p \ast (T\_1 - T\_2)\right] + \left[(\upsilon\_{11}^2 - \upsilon\_{21}^2)\right]/2\left[ + \left[\mathbb{g} \ast (z\_{11} - z\_{21})\right] \right] \tag{12}$$

The *Eh* is obtained by the properties measured in the main water flow (subscripts 1 and 2), Equation (13). Pressure (*p*), velocity (*v*) and height (*z*) are geodetic sampling points or reference points with respect to the height of the sea level at which the turbine is located. *ρ*, as well as *a˙* and *Cp*, are obtained by interpolation.

$$E\_h = \left[ \left( \left( p\_1 - p\_2 \right) \right) / \rho \right] + \left[ \left( \left( \upsilon\_1 \right)^2 - \upsilon\_2 \right) \right] / 2 \right] + \left[ \mathbf{g} \* \left( z\_1 - z\_2 \right) \right] \tag{13}$$

The sampling points are observed in Figure 14, which is a general diagram of the turbine in question (original C.H. Temascal plane), as well as the areas in which the fluid properties are measured.

**Figure 14.** Longitudinal view, measurement points [24].

According to [6], the mechanical energy (*Em*) is calculated by Equation (14). In this equation, *a˙* is an isothermal factor of the water, *p*11, the inlet pressure in the diffuser, *p*21, the outlet pressure of the suction tube, *T*11, the inlet temperature of the suction tube, *T*20, the outlet temperature of the suction tube aspiration, *z*11, is a reference point for temperature measurement, and *z*1*<sup>m</sup>* is the reference point for measuring *p*11.

$$E\_m = \left[\dot{a} \ast (p\_{11} - p\_{12})\right]$$

$$E\_m = \left[C\_p \ast (T\_{11} - T\_{20})\right] + \left[(\upsilon\_1^2 - \upsilon\_2^2)\right)/2\left[ + \left[\mathcal{g} \ast (z\_{1m} - z\_{11})\right] \right] \tag{14}$$

However, the variables for the present study were adapted to the previously established conditions, defining *Em* as Equation (15).

$$E\_m = \left[\mathbb{C}p\*(T\_1 - T\_2)\right] + \left[\left(\upsilon\_{11}\,^2 - \upsilon\_{21}\,^2\right)/2\right] + \left[\mathbb{g}\*(z\_{11} - z\_{21})\right] \tag{15}$$
