## 4 Results and Discussion
### Correlations for the Volumetric and Isentropic Efficiency  
The capacity and power consumption of the Roots hydrogen pump are influenced by the operation parameters and working fluids properties. In order to realize the fast and accurate matching between the hydrogen recycling content of stack and the capacity of the pump, a performance equation is needed for engineering application. Based on a great amount of the simulated data mentioned above, equations can be correlated for the volumetric efficiency and isentropic efficiency in function of the pressure difference, suction temperature, ration speed and nitrogen content. The basic form would be:  
\[\eta=f(\Delta P,n,T_{s},n_{N_{2}}) \tag{10}\]  
Figure 8: Comparison of the performance with various working gases: (**a**) flow rate; (**b**) power consumption.  
As shown in Figure 8, the flow rate of the pump presents an obvious linear relationship with the speed. This means that the absolute leakage is almost the same under different speed. Thus, the relative leakage is low and volumetric efficiency is high when the speed is high. Hence, the impact of rotation speed on volumetric efficiency can be considered as:  
\[\eta_{v}=1-a/(V_{r}\cdot n) \tag{11}\]  
Considering the pressure difference and composition separately, the correlation equation could be given as:  
\[\eta_{v}=1-f\Big{(}T_{s},n_{N_{2}}\Big{)}/(V_{r}\cdot n)\cdot g(\Delta P) \tag{12}\]  
Applying the least square method, a specific correlation is obtained as:  
\[\eta_{v}=1-\frac{(12T_{s}+2667)n_{N_{2}}^{2}+(19.6T_{s}-2780)n_{N_{2}}-5.1T_{s }+1063}{V_{r}\cdot n}\sqrt{\frac{\Delta P-2.1}{25-2.1}} \tag{13}\]  
The most important value of this correlation equation is to supply a simple adjusting method to meet the quick response of the pump capacity with the stack requirement. By combining Equations (5) and (13), the rotation speed needed corresponding to the required flow rate of pump can be expressed as below:  
\[n=V_{Target}/V_{r}-\Big{[}(12T_{s}+2667)n_{N_{2}}^{2}+(19.6T_{s}-2780)n_{N_{2} }-5.1T_{s}+1063\Big{]}\sqrt{\frac{\Delta P-2.1}{25-2.1}}/V_{r} \tag{14}\]  
Based on the derived volumetric efficiency, the correlation formula of isentropic efficiency can be obtained:  
\[\eta_{is}=\frac{0.4\cdot V_{r}\cdot n\cdot\eta_{V}}{\Big{[}(1.6546-0.05\Delta P )n_{N_{2}}+0.1559T_{s}+50.182\Big{]}n/1000-0.8363T_{s}-(916-28\Delta P)n_{N_{2 }}-25.94} \tag{15}\]  
Furthermore, the power consumption of the pump for compressing the gas in real conditions can be expressed as below. This can be used to show how much parasitic power is consumed by the hydrogen pump:  
\[N=N_{is}/\eta_{is} \tag{16}\]  
As shown in Figure 9, these two correlation equations can achieve high accuracy within the range of pressure difference of 15-25 kPa, a temperature of 70-80 \({}^{\circ}\)C, a rotation speed of 5000-9000 rpm and a nitrogen mole fraction of 5-25%. In most cases, the absolute error of the predicted value of volumetric efficiency did not exceed 1%, which could precisely reflect the influence of the four factors on volumetric efficiency. Although most of coefficients in the correlation equation are fitted at a speed of 7000 and a pressure difference of 25 kPa, we can see from Figure 9 that high accuracy can still be ensured at different speeds and pressure differences, which suggests that the assumption about correlation equation is feasible. Through these two correlations, the volumetric efficiency and isentropic efficiency of the Roots blower can be quickly estimated as the four factors change.