## 3 Simulation Model Development
### Performance Parameters Definition  
As a positive displacement machine, the performance of a Roots pump is mainly indicated by the volume flow rate and power consumption. In addition, volumetric efficiency and isentropic efficiency are introduced in order to compare the pumps with different sizes and various operating conditions.  
The volume flow rate is defined as the volume of the gas delivered by the pump per unit time with the state of the gas being converted to the parameters at the inlet of the pump. It is the capability of the Roots pump to deliver the fluids in the anode recirculate loop of the fuel cell system. Moreover, the volume flow rate is also an indicator of the initial cost of the pump because the geometrical size of a Roots pump is dependent on its volume flow rate as well as rotation speed. In actuality, for a positive displacement pump or compressor, there is always an optimum speed for the best performance and minimum size. High speed can reduce the size and thus reduce the cost, but the performance will become worse for higher flow losses. That is to say, a pump or compressor with a certain volume flowrate implies the relation between size and speed. With the actual average mass flow rate calculated by the CFD simulation, the volume flow rate of the Roots pump can be obtained by the state equation of the ideal gas:  
\[V=60 mR_{g}T_{s}/p_{s} \tag{4}\]  
where \(R_{g}\) is the gas constant calculated by Equation (1).  
The volumetric efficiency is defined as the ratio of the volume flow rate to the theoretical capacity of the Roots pump, and a higher volumetric efficiency implies better sealing within the Roots pump:  
\[\eta_{V}=\frac{V}{V_{t}} \tag{5}\]  
The theoretical capacity \(V_{t}\) of the Roots pump is the geometric function of the rotor profile and is defined as the swept volume per revolution multiplied by the rotation speed of the pump:  
\[V_{t}=V_{r}n=2 ZA_{0}Ln \tag{6}\]  
where \(V_{r}\) is the displaced volume per rotation, \(Z\) is the lobe number, \(L\) is the rotor length, \(n\) is the rotation speed and \(A_{0}\) is the area between two lobes of the rotor and the casing.  
The power consumption of the Roots pump is the electric power input to the controller of the pump. Most of the power is used to compress and deliver the fluid from the inlet to the outlet of the Roots pump, while a small part of it is used to compensate the losses in the controller and in the motor as well as the friction power consumed within the Roots pump by the timing-gears, bearing, and shaft seals. In the following sections of the present study, these losses are neglected in order to focus on the compression process of the Roots pump. Then, the power consumption of the pump can be calculated with the average torque M, which are obtained based on the CFD simulation results:  
\[N=2\pi Mn/60 \tag{7}\]  
The isentropic efficiency is defined as the ratio of the isentropic compression power consumption to the power consumption of the Roots pump. A higher isentropic efficiency means a more efficient compression process and less power consumption of the pump. It can be calculated by the following equation:  
\[\eta_{\text{is}}=N_{\text{is}}/N \tag{8}\]  
where the isentropic compression power consumption \(N_{\text{is}}\) can be expressed as:  
\[N_{\text{is}}=\frac{\kappa}{\kappa-1}p_{s}V((\frac{p_{d}}{p_{s}})^{\frac{\kappa- 1}{\kappa}}-1) \tag{9}\]  
where \(\kappa\) is the isentropic index calculated by Equation (2).