## 1 Introduction  
The proton exchange membrane (PEM) fuel cell system has been used increasingly more often in the applications of various vehicles due to its advantages, such as high efficiency, compact structure and because it is easy to start [1]. In order to decrease hydrogen consumption and increase reliability, the configurations with hydrogen recirculation are widely adopted for the hydrogen supply to the stack of the fuel cell system in these applications [2]. A hydrogen pump is one of the most important components in the hydrogen supply loop and the behavior of the hydrogen pump has a significant impact on the performance the fuel cell system. The accurate prediction of the hydrogen pump performance is an essential task in the design and operation of the PEM fuel cell system. On the one hand, a sub-model for describing the performance of the hydrogen pump is a critical part in the system-level simulation in the design of a PEM fuel system [3; 4]. It is also very important for the realization of the control strategies during the operation of the fuel cell system [5; 6]. On the other hand, the evaluation of the hydrogen pump performance is very challenging, both through experimental testing and theoretical modeling. The challenge comes mainly from the complex composition of the fluid recirculated by the pump. Despite that the hydrogen is excessively supplied to the stack of the fuel cell, there are usually some impurities being recirculated with the hydrogen, and these impurities include water vapor and nitrogen [7; 8; 9; 10]. Regarding the experimental investigation, the performance measured with dry and pure hydrogen on a test rig differs greatly from the performance of the pump integrated in the PEM fuel cell system. It would be very expensive and time-consuming to evaluate the performance of the pump if all these impurities are presented and controlled in the experimental investigation. Therefore, the theoretical modeling and simulation of the performance of the hydrogen pump becomes a feasible and effective solution to meet this need from the fuel cell system [11; 12]. The simulation of the hydrogen pump performance is of course dependent on the specific types of the pump. Up to now, the hydrogen pumps which have been widely used for hydrogen recirculation in the PEM fuel cell systems mainly include an ejector and mechanical pump. The ejector utilizes the pressure difference between the hydrogen tank and the fuel cell stack for providing energy. This kind of pump has advantages, such as having no moving parts, low maintenance cost and relatively high efficiency [13; 14; 15; 16]. 'Mechanical pump' is actually the general name of a group of different pumps. According to the structures of the key components, the mechanical pump can be divided into several types such as the regenerative pump [17; 18], claw pump [19], scroll pump [20; 21] and Roots pump. The Roots pump belongs to the positive displacement machine and its advantages include high reliability, low cost and permission with reversible rotation. The hydrogen recirculation pump used in the fuel cell system of the Toyota Mirai is a Roots pump with two straight lobes [22]. Although the Roots pump has been used for hydrogen recirculation in the fuel cell system in the past, it is hard to find published papers related to the Roots pump for this application, except one primary simulation work by the current authors [23]. In our previous study, a three-dimensional unsteady flow simulation model of a three-lobes Roots pump in fuel cell system was established and solved based on the SST k-w turbulence model. The structured dynamic meshes of the deforming fluid domain was generated by Twinmesh software. The results revealed that the flow field is influenced by the pressure field and the mass flow fluctuation is affected greatly by the pressure ratio. The type of Roots pump studied previously has three straight lobes, instead of helical rotors in this study; moreover, the working gas is just pure hydrogen, and water vapor and nitrogen are not considered.