**6. Conclusions**

This study improves the classical loading-to-flow diagram method to meet the design needs of radial turbines with variable operating conditions. To optimize the off-design performance of radial turbines in the early design phase, we proposed a hypothesis that uses the ratio of the mean velocity of the fluid relative to the rotor passage with respect to the circumferential velocity of the rotor as an indicator to indirectly and qualitatively estimate the rotor loss, as it plays a key role in the off-design efficiency. This hypothesis is based on the findings from existing studies indicating that rotor loss is a function of the flow velocity within the rotor.

The findings of o ff-design rotor loss analysis for radial turbines with a di fferent design value of guide vane outlet flow angle are as follows:

(1) For a radial turbine with a smaller design value of guide vane outlet flow angle, the rotor loss first decreased and then increased with the decrease of mass flow. This means better o ff-design performance in the case of reducing the mass flow. This applies both under conditions of pressure ratio change and of guide vane opening change. However, due to a higher rotor loss at design conditions, the turbine e fficiency under pressure ratio changes may be lower than for radial turbines with a larger guide vane outlet flow angle.

(2) A radial turbine with a larger design value of guide vane outlet flow angle not only had higher efficiency at design conditions but also had better o ff-design performance in the case of increased mass flow. This held for changes in both pressure ratio and guide vane opening.

The above findings were validated with the mean-line model method [7]. Furthermore, based on the findings, this study discusses the optimization of the design value of guide vane outlet flow angle based on the matching of rotor loss characteristics with specified variable operating conditions for a multistage radial turbine in a compressed air energy storage system. It provides important guidance for the design optimization of multistage radial turbines operating under variable working conditions.

**Author Contributions:** Conceptualization, Z.C., Z.T., and S.T.; Methodology, Z.C.; Formal analysis, Z.C.; Validation, Z.C.; Writing, Z.C. and Z.T.; Supervision, S.T. and Z.T.

**Funding:** We would like to acknowledge the National Key R&D Program of China (2018YFB0606105), National Natural Science Foundation of China (51708493), Zhejiang Provincial Natural Science Foundation (LR19E050002), Zhejiang Province Key Science and Technology Project (2018C01020, 2018C01060, 2019C01057), and the Youth Funds of State Key Laboratory of Fluid Power & Mechatronic Systems (SKLoFP\_QN\_1804).

**Conflicts of Interest:** The authors declare no conflict of interest.
