## 4 Conclusions and Remarks  
To investigate the effects of impeller-diffuser interaction on the performance of high-pressure ratio centrifugal compressors and establishing a design criteria of the radial gap ratio in a centrifugal compressor, an extensive numerical simulation series of centrifugal compressor stage with six various settings of a radial gap ratio for five rotational speeds have been conducted and some conclusion are as follows:  
1. Effects of radial gap ratio on the stage performance:  
Reducing the radial gap size between blade rows contributes to a considerable increase in the Mach number value (reaching to Ma = 1) at the diffuser leading edge. This is because of a reduction in diffuser throat area size when the diffuser vane gets closer to the impeller blades. Consequently,  
Figure 29: Stage efficiency vs. normalized mass flow. the choke mass flow rate reduces with a reduction in the radial gap ratio. However, there is an exception at M\({}_{u}\) = 1.16, as the choke mass flow rate remains constant for a radial gap ratio of 1.10 or more. This is because the choke occurs in the impeller. Therefore, because the choke occurs earlier in smaller radial gaps, the stable flow range increases with an increase in the radial gap. The effect of changing radial gap on the stage efficiency is noticeable at high rotational speeds, while this effect is not notable for a tip-speed Mach number of less than 1.0. 2. Effects of radial gap ratio on the impeller Performance:  
Recirculation is expected to be responsible for a higher working input when the diffuser leading edge gets closer to the impeller trailing edge, contributing to a growth in the total pressure ratio. Furthermore, at the same mass flow rate, decreasing the radial gap ratio reduces the impeller enthalpy slightly. These lead to a reduction in impeller efficiency by increasing the radial gap ratio. The deviation in impeller efficiency is less than 0.3%. Unlike the stage efficiency, the maximum efficiency of an impeller for a pressure ratio of less than 6.7 increases significantly by increasing the radial gap ratio. However, for a pressure ratio of more than 6.7, the maximum efficiency decreases slightly. 3. Effects of radial gap ratio on the diffuser performance:  
Due to a reduction of mixing of non-uniform flow at diffuser leading edge area in smaller radial gaps, the loss in a smaller radial gap ratio should be less than that in a larger radial gap size. This phenomenon truly occurs when the tip-speed Mach number is less than 1.0. However, when the tip-speed Mach number exceeds 1.0, the shock wave occurs in the diffuser leading edge of the small radial gaps. Therefore, at the tip-speed Mach number more than 1.0, the loss for small radial gap increases significantly. Furthermore, for the tip-speed Mach number less than 1.0, the pressure recovery is larger and the diffuser performs better in smaller radial gaps. However, for the tip-speed Mach number more than 1.0, the diffuser with a larger radial gap have better performance than that with smaller radial gaps. 4. Considering the whole stage efficiency as well as the diffuser performance, when the tip-speed Mach is smaller than one, the radial gap could be within a wide range. When it is larger than 1.0, the radial gap should be about 1.08 to achieve a better compressor performance. This research was supported by the National Natural Science Foundation of China (Grant No. 51176087). All authors contributed directly to write this paper. [PERSON] did the numerical simulations, acquired the data, analyzed the data and wrote the paper. [PERSON] had collaboration in data analyzing. [PERSON] revised the paper. [PERSON] supervised the research work, proposed helpful comments to write the paper and did the final revision. The authors declare no conflict of interest. The following nomenclature is used in this manuscript: