*3.1. Pressure Ratio Change*

For radial turbines with fixed geometry guide vane, the rotor loss under off-design operating conditions is dominated by viscous loss with a nearly constant loss coefficient [24]. Therefore, the value of *Wr* is used to predict the rotor loss characteristics of radial turbines with different <sup>α</sup>4,*d* indirectly in the case of pressure ratio change.

In the case where only the change of pressure ratio is considered, the off-design flow coefficient can be estimated using the improved Flügel formula [28] and the definition of flow coefficient (Equation (5)). It can be seen that the off-design flow coefficient is approximately proportional to the pressure ratio change.

$$\begin{cases} \frac{\dot{m}\_t}{\dot{m}\_{t,d}} = \frac{P\_{\text{in}}}{P\_{\text{in},d}} \sqrt{\frac{1 - 1/\beta\_t^2}{1 - 1/\beta\_{t,d}^2}}\\ \frac{\dot{m}\_t}{\dot{m}\_{t,d}} = \frac{c\_{\text{in}} \beta\_6 \rho\_6}{c\_{\text{in}} \rho\_4 P\_{\text{\theta},d}} \approx \frac{\left(\frac{c\_{\text{in}}}{\dot{m}\_4}\right) \cdot P\_{\text{\theta}}}{\left(\frac{c\_{\text{in}}}{\dot{m}\_4}\right) \cdot P\_{\text{\theta},d}} \rightarrow \frac{\phi \cdot P\_6}{\phi\_d \cdot P\_{\text{\theta},d}} \approx \sqrt{\frac{\beta\_t^2 - 1}{\beta\_{t,d}^2 - 1}} \end{cases} \tag{5}$$

where .*mt* denotes the mass flow, subscript *d* denotes the design value, *Pin* denotes the inlet pressure of turbine, β*t* denotes the pressure ratio, *A*6 denotes the rotor outlet area, ρ6 denotes the density of gas at the rotor outlet, and *P*6 denotes the rotor outlet pressure.

Substituting the off-design flow coefficient into Equation (3), the change characteristics of *Wr* in the case of pressure ratio change could be obtained as shown in Figure 5. The figure shows that for a radial turbine with a small design value of guide vane outlet flow angle (e.g., <sup>α</sup>4,*d* = 60◦), the corresponding *Wr* had a characteristic of decreasing first and then increasing when the pressure ratio was reduced relative to the design point. However, for a radial turbine with a large design value of guide vane outlet flow angle (e.g., <sup>α</sup>4,*d* = 80◦), an increase or decrease in the pressure ratio relative to the design point resulted in an increase in *Wr*. The rotor losses also exhibited the characteristics described above due to the change in *Wr*. The above results can be explained as follows. For turbines with a small

<sup>α</sup>4,*d*, the flow coefficient corresponding to minimum rotor loss is less than the design flow coefficient (Figure 4). Therefore, in the process of decreasing the pressure ratio, the off-design flow coefficient (which is approximately proportional to the pressure ratio) is close to the minimum rotor-loss-based flow coefficient first and then gradually deviates from it, resulting in rotor loss exhibiting similar change characteristics. Similarly, for turbines with a large <sup>α</sup>4,*d*, the flow coefficient corresponding to minimum rotor loss is coincident with the design flow coefficient. An increase or decrease in the pressure ratio will cause the off-design flow coefficient to deviate from the minimum rotor-loss-based flow coefficient, resulting in an increase in rotor loss.

Moreover, within a wide range of pressure ratio change (0.5 < β*<sup>t</sup>*,*<sup>i</sup>*/β*<sup>t</sup>*,*<sup>d</sup>* < 1.5), the larger the guide vane outlet flow angle, the smaller *Wr* is. It can be concluded that the turbine efficiency under the same range of pressure ratio change increases proportionally to the design value of guide vane outlet flow angle.

**Figure 5.** Effect of design guide vane outlet flow angle on the change characteristics of *Wr* under pressure ratio change (β<sup>6</sup>*m* = <sup>−</sup>52.5◦, β4 = <sup>−</sup>30◦, β*<sup>t</sup>*,*<sup>d</sup>* = 3).
