## 7 Models for Centrifugal Compressor  
Incidence loss [29]  
For impeller:  
\[\Delta h_{inc}=0.4[W_{1}-C_{m1}/\cos\beta_{b1}]^{2}+[W_{1}Z_{FB}t_{1}/(\pi D _{1}\cos\beta_{b1})]^{2}/2 \tag{20}\]  
For the diffuser:  
\[\Delta h_{inc}=0.4[C_{3}-C_{m3}/\cos\beta_{3}]^{2}+[C_{3}Z_{v}t_{3}/(\pi D_{ 3}\cos\beta_{3})]^{2}/2 \tag{21}\]  
Blade loading loss [29]  
\[\begin{array}{c}\Delta h_{bl}=\frac{\Delta W^{2}}{\Delta W}\\ \Delta W=\frac{2\pi D_{2}^{2}U_{2}I_{b}}{ZL_{b}}\end{array} \tag{22}\]  
Blade velocity difference  
Impeller skin friction [29]  
\[\Delta h_{sf}=2C_{f1-2}\frac{L_{b}}{d_{HB}}\overline{W}^{2} \tag{23}\]  
Figure 9: Variation of each stage’s efficiency.  
Length of blade mean camber line:  
\[L_{b}=\frac{\pi}{8}(D_{2}-\frac{D_{s1}+D_{h1}}{2}-b_{2}+2L_{ax})\frac{2}{0.5(\cos \beta_{bs1}+\cos\beta_{bh1})+\cos\beta_{b2}} \tag{24}\]  
\[d_{HB}=\frac{D_{2}\cos\beta_{b2}}{\frac{Z}{\pi}+\frac{D_{2}\cos\beta_{b2}}{b_{2 }}}+\frac{0.5(D_{s1}+D_{h1})(\cos\beta_{bs1}+\cos\beta_{bh1})}{\frac{Z_{2}}{ \pi}+\frac{D_{4}+D_{h1}}{D_{s1}-D_{h1}}\frac{\cos(\beta_{bh1}+\cos\beta_{bh1}) }{2}} \tag{25}\]  
Here,  
\[C_{f1-2}=\frac{1}{4}\bigg{(}0.0032+\frac{0.221}{\text{Re}_{1-2}\text{0.237}} \bigg{)}\bigg{[}1+0.075\cdot\text{Re}_{1-2}^{0.25}\cdot\sqrt{\frac{d_{HB}}{2 \cdot r_{c}}}\bigg{]} \tag{26}\]  
\[r_{c}=0.5(r_{2}-\frac{r_{s1}+r_{h1}}{2}-\frac{b_{2}}{2}+L_{ax}) \tag{27}\]  
\[\overline{W}=\frac{W_{s1}+W_{h1}+2W_{2}}{4} \tag{28}\]  
\[\text{Re}_{1-2}=0.5(\rho_{1}+\rho_{2})(W_{1}+W_{2})d_{HB}/(\mu_{1}+\mu_{2}) \tag{29}\]  
Due to the path curvature of a centrifugal compressor, the modified equation of [PERSON] (1932) [30] was used for the friction coefficient. But, it needs to be noted that the friction coefficient is also influenced by adverse pressure gradient, which is usually ignored by 1-D calculation.  
Hub-to-shroud loading loss [29]  
\[\Delta h_{h-s}=(\overline{k}_{m}\overline{b}\overline{W})^{2}/12 \tag{30}\]  
Mean streamline curvature  
\[\overline{k}_{m}=\Delta\alpha/L_{m} \tag{31}\]  
\(L_{m}\) represents the length of mean streamline.  
Abrupt expansion loss [29]  
\[\Delta h_{\text{exp}}=(\lambda_{2}-1)^{2}\frac{C_{m2}{}^{2}}{2} \tag{32}\]  
\[\lambda_{2}=1/(1-B_{2}) \tag{33}\]  
Mixing loss [29]  
\[\Delta h_{mix}=\frac{(C_{m2}-C_{m2,mix})^{2}}{2} \tag{34}\]  
where \(C_{m2,mix}=\dot{m}/(\pi D_{2}b_{2}\rho_{2})\).  
Clearance loss [29]  
\[\Delta h_{cl}=\frac{\dot{m}_{cl}\Delta P_{cl}}{\dot{m}\overline{\rho}} \tag{35}\]  
\[\Delta P_{cl}=\frac{\dot{m}(r_{2}C_{2u}-r_{1s}C_{u1})}{ZL\frac{r_{1s}+r_{2}}{2 }\frac{b_{1}+b_{2}}{2}} \tag{36}\]  
Velocity of clearance gap leakage flow  
\[U_{cl}=0.816\sqrt{\frac{2\Delta P_{cl}}{\rho_{2}}} \tag{37}\]  
\[\dot{m}_{cl}=\rho_{2}ZLU_{cl}\delta_{c} \tag{38}\]Vaneless diffuser loss [29]  
\[\Delta h_{vaneless}=4C_{f2-3}\frac{r_{3}-r_{2}}{2b_{av}\cos{a_{av}}}\frac{\overline {C}^{2}}{2}=\frac{C_{f2-3}}{2\cos{a_{av}}}(r_{3}-r_{2})(\frac{{C_{2}}^{2}}{b_{2 }}+\frac{{C_{3}}^{2}}{b_{3}}) \tag{39}\]  
In the vaneless diffuser, the friction coefficient is distinctly different of that inside a pipe. Thus, the experimental equation of [PERSON] (1982) was adopted [31].  
\[C_{f2-3}=0.015(1.8\times 10^{5}/{\rm Re}_{2-3})^{0.2} \tag{40}\]  
\[{\rm Re}_{2-3}=0.5(\rho_{2}+\rho_{3})(C_{2}+C_{3})(b_{2}+b_{3})/(\mu_{2}+\mu_{3}) \tag{41}\]  
Vaned diffuser loss [32]  
\[\Delta h_{vanes}=\frac{1}{2}\bigg{[}0.07+0.0038\Big{(}\frac{C_{\rm max,3}}{C_ {5}}-1\Big{)}^{3}+0.0038\Big{(}\frac{C_{5}}{C_{5}}-1\Big{)}^{3}\bigg{]}C_{3}^{ 2}\bigg{\{}1+4.4\Big{(}1-\frac{W_{2}}{W_{2}}\Big{)}\bigg{[}1-\Big{(}\frac{D_{ 1}}{D_{2}}-1\Big{)}^{0.33}\bigg{]}\bigg{\}} \tag{42}\]  
\[C_{\rm max,3}=C_{3}(1+A_{v}|i|) \tag{43}\]  
Diffuser outlet loss  
\[\Delta h_{tex}=K_{ex}\frac{C_{m4}}{2} \tag{44}\]  
where \(K_{ex}\) is the loss coefficient.  
Slip factor [33]  
\[\mu=1-\frac{\sqrt{\cos{\beta_{2b}}}}{Z_{2}0^{.7}} \tag{45}\]