*3.2. Guide Vane Opening Change*

For radial turbines with variable geometry guide vane, the rotor loss under off-design operating conditions becomes more complex due to the change in the internal flow of the rotor with guide vane opening. Meitner et al. [29] show a relationship between the loss coefficient of rotor kinetic energy and guide vane opening where the loss coefficient increased significantly at small guide vane opening. This occurs because the rotor loss mechanism is not only viscous loss but also increased secondary flow loss at small guide vane opening, according to Otsuka et al. [30]. Therefore, in the case of decreasing guide vane opening (<100% design value), the relative velocity *Wr*/*Wr*,*<sup>d</sup>* is proposed to indirectly and qualitatively estimate rotor loss for comparison between radial turbines with different design values of guide vane outlet flow angle (<sup>α</sup>4,*d*). On the other hand, the value of *Wr* is used for increasing guide vane opening (>100% design value).

According to Spence et al. [26,27], the mass flow rate of the radial turbine is approximately proportional to the guide vane opening. As a result, the existing Flügel formula [28] can be updated by multiplying a guide vane opening ratio (*Or*) to estimate the mass flow rate under guide vane opening change. From the updated Flügel formula and the definition of flow coefficient, the off-design flow coefficient can be obtained as shown in Equation (6).

$$\begin{cases} \frac{\dot{m}\_t}{\dot{m}\_{t,d}} \approx O\_r \frac{P\_{\text{in}}}{P\_{\text{in},d}} \sqrt{\frac{1 - 1/\beta\_t^2}{1 - 1/\beta\_{t,d}^2}}\\ \frac{\dot{m}\_t}{\dot{m}\_{t,d}} = \frac{c\_{\text{on},d} \rho\_{\text{off}}}{c\_{\text{on},d} \rho\_{\text{off}} \rho\_{\text{d},d}} \approx \frac{\left(\frac{c\_{\text{on},d}}{\dot{u}\_4}\right) \cdot P\_{\text{\dot{\theta}}}}{\left(\frac{c\_{\text{on},d}}{\dot{u}\_4}\right) \cdot P\_{\text{\dot{\theta}},d}} \rightarrow \frac{\phi \cdot P\_{\text{\dot{\theta}}}}{\phi\_d \cdot P\_{\text{\dot{\theta}}}} \approx O\_r \sqrt{\frac{\beta\_t^2 - 1}{\beta\_{t,d}^2 - 1}}, \tag{6}$$

 where *Or* is defined as the ratio of the off-design guide vane opening to the design value. *Energies* **2019**, *12*, 2550

Moreover, from the guide vane geometric (Figure 6), a sinus rule can be employed to estimate the updated α4 caused by the guide vane opening change as shown in Equation (7) [23]. Finally, by substituting the off-design flow coefficient and the updated α4 into Equation (3), the off-design *Wr* the corresponding *Wr*/*Wr*,*<sup>d</sup>* can be obtained.

$$a\_4 = \cos^{-1}(\frac{O}{S}) = \cos^{-1}(\frac{O}{O\_d} \times \frac{O\_d}{S}) = \cos^{-1}(O\_r \cdot \cos(\alpha\_{4,d})),\tag{7}$$

where *O* denotes the guide vane opening and *S* denotes the span of the guide vane at the outlet.

**Figure 6.** Typical variable-geometry guide vanes for radial turbines.

From Equation (6), the variation in flow coefficient was closely related to the pressure ratio in the case of guide vane opening change. Thus, the rotor loss analysis of two representative cases of guide vane opening change was carried out respectively; these were the constant pressure ratio case and a more complicated case where the pressure ratio changed inversely to the guide vane opening.

Figure 7a shows that in the case where the pressure ratio remained constant, *Wr*/*Wr*,*<sup>d</sup>* decreased significantly as the guide vane opening decreased. Although the difference between different <sup>α</sup>4,*d* was not notable, it can be seen that the *Wr*/*Wr*,*<sup>d</sup>* of the radial turbine with a smaller <sup>α</sup>4,*d* was smaller, which means a smaller increase in the rotor loss with decreased guide vane opening. In the case where the guide vane opening was increased, the larger the <sup>α</sup>4,*d*, the smaller the *Wr*. This means a smaller increase in the rotor loss with the increased guide vane opening. The above results can be explained as follows. In the case of a constant pressure ratio, the flow coefficient is proportional to the guide vane opening ratio (Equation (6)). Since smaller <sup>α</sup>4,*d* leads to a larger corresponding design flow coefficient (Figure 4), it can be inferred that the relative velocity of air flow at the rotor inlet (*w*4) is also larger (Figure 3). Thus, within the same regulation range of the guide vane opening, the changes in *w*4 caused by the flow coefficient change are more pronounced in both the increasing and decreasing directions. This directly leads to a more significant change in the rotor loss.

**Figure 7.** Effect of design guide vane outlet flow angle on the change characteristics of *Wr* with variable guide vane opening: (**a**) constant pressure ratio; (**b**) constant flow coefficient (β<sup>6</sup>*m* = <sup>−</sup>52.5◦, β4 = −30◦).

Figure 7b shows that in the case where the pressure ratio changed inversely with the guide vane opening and maintained a constant flow coefficient, *Wr*/*Wr*,*<sup>d</sup>* increased significantly as the guide vane opening decreased. In contrast, the radial turbine with smaller <sup>α</sup>4,*d* had a smaller increase in *Wr*/*Wr*,*d*, which means a relatively smaller increase in the rotor loss with decreased guide vane opening. In the case where the guide vane opening was increased, the larger the <sup>α</sup>4,*d*, the smaller the *Wr*. This means a smaller increase in the rotor loss with the increased guide vane opening. The above results can be explained as follows. According to Figure 4, the larger α4, the more obvious the change (decreased) of the flow coefficient corresponding to the minimum rotor loss. Thus, for the radial turbine with a smaller <sup>α</sup>4,*d*, the deviation of the off-design flow coefficient (approximately equal to the design value) from the flow coefficient (decreased) corresponding to the minimum rotor loss caused by the decrease of the guide vane opening is smaller. It can also be inferred that the increase in the rotor loss would be smaller. In contrast, in the case where the guide vane opening was increased, the increased flow coefficient corresponding to the minimum rotor loss tended to be gentle with the increase of α4. The deviation of the off-design flow coefficient (approximately equal to the design value) from the flow coefficient (increased) corresponding to the minimum rotor loss caused by the increase of the guide vane opening can be ignored for radial turbines with a different <sup>α</sup>4,*d*. Therefore, due to the smaller *Wr* at the design point, the larger the <sup>α</sup>4,*d*, the smaller the off-design rotor loss.

Furthermore, by comparing Figure 7a,b, we can find that the inverse change of pressure ratio with the change in the guide vane opening will increase the rotor loss in the case of reduced guide vane opening. On the contrary, it can mitigate the increase of the rotor loss in the case of increased guide vane opening.

In conclusion, the difference in the rotor loss characteristics under guide vane opening change for radial turbines with different <sup>α</sup>4,*d* is determined by the directionality of the guide vane regulation. It can be concluded that the optimum design of the guide vane outlet flow angle is closely related to the directionality of the guide vane regulation.

### **4. Optimal Design of the Guide Vane Outlet Flow Angle for Multistage Radial Turbines**

This section presents a discussion of the optimal design of the guide vane outlet flow angle (<sup>α</sup>4,*d*) for a multistage radial turbine from the perspective of matching the rotor loss characteristics with variable operating conditions. In addition, the multistage radial turbine of a CAES pilot plant named "TICC-500" [31] was taken as an example for analysis (Table 1).


**Table 1.** Parameters of the multistage turbine in a "TICC-500" CAES pilot system [31].

To obtain the representative variable operating conditions of the multistage radial turbine based on guide vane control, only the case of regulating the guide vane opening of the high-pressure turbine stage was considered here. Since the rotational speed of the multistage radial turbine in a CAES system is generally constant, the mass flow rate of each turbine stage under guide vane control can be estimated by the updated Flügel formula in Equation (6), which includes a guide vane opening ratio (*Or*). On this basis, to solve the operating conditions of each turbine stage in the multistage radial turbine at a certain inlet pressure and guide vane opening, the pressure ratios were updated to iteratively calculate the mass flow for each turbine stage until the continuity of mass flow was met. Furthermore, by substituting off-design the operating conditions (guide vane opening and pressure ratio) into Equation (6), the corresponding the flow coefficient could be obtained.

Depending on the directivity of the guide vane opening regulation, the case where the guide vane opening is reduced and increased with respect to the design value is defined as a down-regulation and an up-regulation, respectively. Figure 8a,b presents the variable operating conditions of the multistage radial turbine in the case of down-regulation and up-regulation, respectively. We assumed steady-state operation conditions with different guide vane openings, and the hysteresis during continuous downand up-regulation was not considered in this study.

### (1) Down-Regulation of the Guide Vane Opening

When the inlet pressure of the multistage radial turbine is high while the load demand is low, the down-regulation of the guide vane opening is conducted to reduce the mass flow rate. Figure 8a shows that the pressure ratio of the low-pressure (LP) turbine decreased significantly with the decrease of the guide vane opening of the high-pressure (HP) turbine and relatively small change in the expansion ratio of the medium-pressure (MP) turbine. In contrast, the pressure ratio of the HP turbine changed inversely to the guide vane opening, which significantly increased with the decrease of the guide vane opening.

### (2) Up-Regulation of the Guide Vane Opening

When the inlet pressure of the multistage radial turbine is relatively low while the load demand is high, the up-regulation of the guide vane opening is conducted to increase the mass flow rate. Figure 8b shows that the pressure ratio of the LP turbine increased significantly with the increase of the guide vane opening of the HP turbine, while the expansion ratio of the MP turbine remained almost constant. Like the down-regulation of the guide vane opening, the pressure ratio of HP turbine changed inversely to the guide vane opening, which reduced significantly with the increase of the guide vane opening.

**Figure 8.** Variable operating conditions for a multistage radial turbine under guide vane opening change: (**a**) down-regulation at rated inlet pressure; (**b**) up-regulation at 60% the rated inlet pressure.

Figure 9a,b presents the flow coefficient variation characteristics of the multistage radial turbine under the variable operating conditions shown in Figure 8a,b, respectively. Combined with the findings regarding the relationship of off-design performance versus the design value of the guide vane outlet flow angle (<sup>α</sup>4,*d*) for the radial turbine shown in Section 3, the recommendations for the optimum <sup>α</sup>4,*d* in the preliminary design of the multistage radial turbine are as follows.


**Figure 9.** Flow coefficient variation for the multistage turbine under guide vane opening change: (**a**) down-regulation at rated inlet pressure; (**b**) up-regulation at 60% of the rated inlet pressure.
