**5. Case Study and Discussion**

In this section, we compare the off-design performance of a group of radial turbines with different design values of guide vane outlet flow angle. To ensure the comparability of different radial turbine performances, the preliminary design results of radial turbine obtained under the same design conditions were employed for the case study. Moreover, the performance evaluation of the radial turbine was conducted using the mean-line model method.

The mean-line model method previously developed in our group is a proven performance evaluation method for radial turbines. Since the model evaluation of radial turbines is not the focus of this study, please refer to one of our previous publications for more details [7]. The model uses limited turbine geometry information to predict the turbine performance under variable operating conditions, which is especially valuable in the preliminary design process. The total-to-static efficiency of the radial turbine was calculated by the specific work output, isentropic enthalpy drops, and turbine losses, as shown in Equation (8). According to the Euler equation of turbomachines [32], the specific work output can be calculated with Equation (9). Table 2 gives the loss model combination suitable for the turbine loss evaluation under variable guide vane opening, which involves four types of main turbine losses: stator loss, incidence loss, rotor loss, and exit loss.

$$
\eta\_{t-s} = \frac{\mathcal{W}\_t}{\Delta h\_{1-6s}} = \frac{\Delta h\_{1-6s} - \sum h\_{\text{loss}}}{\Delta h\_{1-6s}},\tag{8}
$$

$$\mathcal{W}\_l = \mathfrak{u}\_4 \mathfrak{c}\_{\mathfrak{u}4} - \mathfrak{u}\_6 \mathfrak{c}\_{\mathfrak{u}6} \tag{9}$$

where *Wt* denotes the specific work output of the radial turbine, Δ*h*1−6*<sup>s</sup>* denotes the isentropic enthalpy drop between the guide vane inlet and rotor outlet, and *hloss* denotes the turbine loss.


**Table 2.** The loss model combination of the radial turbine.

The design conditions of the radial turbines in this case study are shown in Table 3. Based on the conclusions of the matching analysis in Section 4, three typical design values of guide vane outlet flow angle (<sup>α</sup>4,*d*= [70◦; 75◦; 80◦) were employed for the case study. The main parameters of the radial turbines were obtained by a preliminary design algorithm proposed by Ventura et al. [17] (Table 4).

**Parameter Value** Inlet Temperature, K 430 Inlet Total Pressure, MPa 10.00 Outlet Pressure, MPa 3.33 Mass Flow Rate, kg/s 24.80

**Table 3.** The design conditions of the radial turbines for the case study.

**Table 4.** The preliminary design results of the radial turbines for the case study.


Figure 10 presents the performance curves of radial turbines with different design values of guide vane outlet flow angle operating under pressure ratio change. It can be seen that the larger the design value of guide vane outlet flow angle, the higher the design efficiency of the radial turbine, and the higher the efficiency operating under variable pressure ratio in a certain range (0.5 < β*<sup>t</sup>*,*<sup>i</sup>*/β*<sup>t</sup>*,*<sup>d</sup>* < 1.5). These are consistent with the findings of the rotor loss characteristic analysis under pressure ratio change (Section 3.1). It should also be noted that the turbine efficiency deteriorated significantly as the pressure ratio decreased to less than about 60% of the design value. However, simply optimizing the design value of the guide vane outlet angle did not seem to be effective in improving this deterioration.

Figure 11a,b, gives the performance curves of two typical cases of guide vane opening change: the constant pressure ratio case and the constant flow coefficient case. First, it can be seen that the radial turbines with different design values of guide vane outlet flow angle had significant differences in the distribution of the efficient operating range under guide vane opening changes, which verifies the finding of the rotor loss characteristic analysis in Section 3.2. Specifically, the radial turbine with a larger design value of guide vane outlet flow angle (e.g., <sup>α</sup>4,*d* = 80◦) had a broader efficient operating range (e.g., turbine efficiency > 0.8) and higher efficiency for the up-regulation of the guide vane opening, while a smaller design value of guide vane outlet flow angle resulted in broader efficient operating range and higher efficiency for the down-regulation of the guide vane opening. Furthermore, comparing Figure 11a,b, it can be found that the deterioration of the turbine efficiency under the down-regulation of the guide vane opening could be alleviated by reducing the reverse increase in the pressure ratio versus the guide vane opening change. Similarly, for the up-regulation of the guide vane opening, when the design value of the guide vane outlet flow angle was large, diminishing the inverse reduction of the pressure ratio could improve the turbine efficiency. However, for a turbine with a smaller design value of guide vane outlet flow angle (e.g., <sup>α</sup>4,*d* = 70◦), this could worsen efficiency. These are also consistent with the findings in Section 3.2.

Based on the above, for a multistage radial turbine that simultaneously needs up- and down-regulation of the guide vane opening, the design value of guide vane outlet flow angle for the high-pressure turbine stage (or other turbine stages with variable geometry guide vane) is recommended to be about 80◦. In this case, higher design efficiency and wide efficient operating range for the up-regulation can be obtained, while the turbine efficiency deterioration under down-regulation can be improved with combined control of the guide vane openings of the multistage radial turbine. The combined control of the guide vane openings of the multistage radial turbine has been proved to be able to alleviate the reverse change in pressure ratio versus the guide vane opening [6]. This could be an effective way to achieve optimum performance for a multistage radial turbine operating under variable working conditions.

**Figure 10.** Turbine performance under pressure ratio change.

**Figure 11.** Turbine performances when the guide vane opening changes and: (**a**) the pressure ratio is constant; (**b**) the pressure ratio changes inversely to the guide vane opening.
