# Potential Evaluation of Twin-Screw Air Expanders with Dual-Lead Rotors Used in PEMFC Systems
## 2 Methods
### Mathematical Model of the TSE Working Process  
The mathematical model describing the TSE working process was developed on the basis of the principles of mass and energy conservation, with one chamber of the TSE selected as the control volume, as depicted in Figure 3. In order to simplify the calculation, several assumptions were made as follows:  
1. The changes in the kinetic and gravitational energy of the gas were neglected;
2. The thermophysical properties of a gas were assumed to be uniform within the control volume;
3. The heat transfer between a gas and solid, such as the rotor surface and casing wall, was disregarded. \begin{table}
\begin{tabular}{c c c} \hline
**Category** & **Symbol** & **Meaning** \\ \hline \multirow{8}{*}{physical quantity} & \(A\) & area, m\({}^{2}\) \\  & \(C\) & flow coefficient \\  & \(cv\) & specific heat at a constant volume, \\  & \(h\) & specific enthalpy, J/kg \\  & \(L\) & length, m \\  & \(m\) & mass, kg \\  & \(\dot{m}\) & mass flowrate, kg/s \\  & \(n\) & rotating speed of a male rotor, rpm \\  & \(p\) & pressure, Pa \\  & \(Q\) & heat exchange flux, J/s \\  & \(T\) & temperature, Klead, mm \\  & \(V\) & volume, m\({}^{3}\) \\  & \(W\) & compression power, kW \\  & \(v\) & specific volume, m\({}^{3}\)/kg \\  & \(z_{1}\) & tooth number of a male rotor \\ \hline \multirow{8}{*}{Greek symbols} & \(\delta\) & clearance height, m \\  & \(\theta\) & rotating angle of a male rotor, \({}^{\circ}\) \\  & \(\kappa\) & isentropic exponent \\  & \(\tau\) & wrap angle, \({}^{\circ}\) \\  & \(\mu\) & dynamic viscosity, Pa-s \\  & \(\rho\) & density, kg/m\({}^{3}\) \\  & \(v\) & velocity, m/s \\ \hline \multirow{8}{*}{subscripts} & a & dry air \\  & dis & discharge \\  & HP & high pressure \\  & in & inflow of control volume \\  & ind & indicated \\  & is & isentropic \\  & LP & low pressure \\  & out & outflow of control volume \\  & r & whole rotor \\  & suc & suction \\ \hline \end{tabular}
\end{table} 
Table 2: Nomenclature. Based on the above assumptions, differential equations representing the temperature and pressure of air in the control volume can be expressed in the form of Equations (2) and (3) [41]. \[\frac{\mathrm{d}T_{\mathrm{a}}}{\mathrm{d}\theta}=\frac{1}{m_{\mathrm{a}}c_{ \mathrm{p,a}}}\biggl{[}-T_{\mathrm{a}}\biggl{(}\frac{\partial p}{\partial T} \biggr{)}_{v}\biggl{(}\frac{\mathrm{d}V_{\mathrm{a}}}{\mathrm{d}\theta}-v_{ \mathrm{a}}\frac{\mathrm{d}m_{\mathrm{a}}}{\mathrm{d}\theta}\biggr{)}-h_{ \mathrm{a}}\frac{\mathrm{d}m_{\mathrm{a}}}{\mathrm{d}\theta}+\frac{1}{6n}\bigl{(} h_{\mathrm{a,in}}\dot{m}_{\mathrm{a,in}}-h_{\mathrm{a,out}}\dot{m}_{ \mathrm{a,out}}-Q\bigr{)}\biggr{]} \tag{2}\]  
\[\frac{\mathrm{d}p_{\mathrm{a}}}{\mathrm{d}\theta}=\biggl{(}\frac{\partial p}{ \partial T}\biggr{)}_{v}\frac{\mathrm{d}T_{\mathrm{a}}}{\mathrm{d}\theta}+ \frac{1}{m_{\mathrm{a}}}\biggl{(}\frac{\partial p}{\partial v}\biggr{)}_{T} \biggl{(}\frac{\mathrm{d}V_{\mathrm{a}}}{\mathrm{d}\theta}-v_{\mathrm{a}}\frac {\mathrm{d}m_{\mathrm{a}}}{\mathrm{d}\theta}\biggr{)} \tag{3}\]  
The leakage model presented herein takes into account the flow state of the working fluid and the geometry of the leakage paths, and hence, various leakage models are employed for various types of leakage paths. A Couette-Poiseuille model was employed to simulate the leakage flow via end-face clearance and rotor-tip clearance. Consequently, it became feasible to calculate the mass flow rate by applying Equation (4). \[\dot{m}=\rho L\int_{0}^{\delta}v\mathrm{d}y=\rho L\biggl{[}-\frac{U\delta}{2}+ \frac{\delta^{3}}{12\mu}\frac{p_{\mathrm{high}}-p_{\mathrm{low}}}{w}\biggr{]} \tag{4}\]  
where \(\delta\) is the clearance height, \(U\) is the wall velocity, \(L\) is the clearance length, \(w\) is the clearance width and \(\mu\) is the dynamic viscosity. The flow via other leakage paths and ports was simplified as the flow through an isentropic nozzle, for independent determination of the rotating speed, as Equation (5) shows. The relationships for both the subsonic and chocked regimes are provided.