## 3 Results and Discussion
### Effects of Radial Gap Ratio on the Compressor Performance  
Performance map is the most important characteristic of a compressor. Figure 12 shows the performance map of the centrifugal compressor for six various radial gap ratios at five tip-speed Mach numbers. As the curves on the map shows, the peak pressure ratio reaches 8.12 at the maximum rotational speed and the radial gap ratio of 1.10. The maximum corrected mass flow is 2.6982 kg/s.  
Figure 12 also offers more information about how compressor performance is affected by the radial gap ratio. The discrepancy in total to total pressure ratio is 0.66% for M\({}_{\rm u}\) = 0.69 (radial gap ratio 1.10 is compared to the radial gap ratio of 1.14). As the rotational speed increases, the discrepancy is higher. This difference reaches 2.21% and 3.63% for M\({}_{\rm u}\) = 1.11 and 1.16, respectively.  
Efficiency can be affected by many factors. Figure 13 shows the compressor adiabatic efficiencies for different tip-speed Mach numbers and radial gap sizes. Adiabatic efficiency for the whole stage of the compressor is dependent on the radial gap size. The maximum efficiency for the tip-speed Mach number of 1.16 is achieved at the radial gap ratio of 1.08. Interestingly, maximum efficiency does not change considerably with the radial gap ratio when M\({}_{\rm u}\) is smaller than 1.0. The correlation between the peak efficiency of each tip-speed Mach number (refer to rotational speed) and the various radial gap ratios is shown clearly in Figure 14. At a high tip-speed Mach number (M\({}_{\rm u}\geq 1\)), enlarging the distance between the impeller trailing edge and the diffuser leading edge has a significant effect on the compressor efficiency. However, this effect is not considerable for a low tip-speed Mach number (M\({}_{\rm u}<1\)).  
Figure 12: Performance map for five tip-speed Mach numbers and six radial gap ratios.  
As discussed later, closing the distance between the impeller trailing edge and diffuser leading edge decreases the diffuser throat area. Thus, the Mach number at the diffuser leading edges increases. This is a significant and major cause for diminishing the value of the choke mass flow rate, which influences the matching of the impeller and diffuser [35]. At the smaller radial gap ratio, the Mach number exceeds unit value sooner. Therefore, the choke in the lower radial gap ratio occurs earlier (Figure 15). Comparing a radial gap ratio of 1.04 with a radial gap ratio of 1.14, the choke mass flows for M\({}_{\text{u}}\) = 1.16, 1.11, 1.05, 0.93 and 0.69 are smaller by 1.15%, 5.66%, 7.25%, 8.55%, and 5.9%, respectively. However, for the cases with the radial gap ratios of 1.10, 1.12 and 1.14 at M\({}_{\text{u}}\) = 1.16, the choke mass flow rate is constant. This is because the choke occurs in the impeller instead of the diffuser. The stable flow range (SFR) can also be obtained by Figure 12, which is defined by the following equation:  
Figure 14: Maximum adiabatic efficiency map for five tip-speed Mach numbers and six radial gap ratios.  
Figure 13: Compressor adiabatic efficiencies map.  
\[\mathbf{SFR}=\ \left(\frac{\hat{\mathbf{m}}_{\mathbf{choke}}-\hat{\mathbf{m}}_{ \mathbf{surge}}}{\hat{\mathbf{m}}_{\mathbf{choke}}}\right)_{\mathbf{N}=\text{ constant}}\times\mathbf{100} \tag{1}\]  
Varying the radial gap ratio also affects the SFR. A smaller radial gap size has a higher Mach number so that the choke happens earlier. Additionally, higher turbulence due to separation and recirculation leads to higher entropy and thus, an earlier surge/stall. Therefore, as the diffuser leading edge recedes from the impeller trailing edge, the SFR increases (Figure 15). When M\({}_{\text{u}}\) = 0.93, the SFR for a radial gap ratio of 1.14 and 1.04 reaches 21.08% and 11.18%, respectively. For all investigated radial gap sizes, low tip-speed Mach numbers have higher values of stable flow range compared to the high tip-speed Mach numbers.