# Matching and Control Optimisation of Variable-Geometry Turbochargers for Hydrogen Fuel Cell Systems
## 2 Materials and Methods
### Fuel Cell Air Path Matching Tool (FCMT)  
The Fuel Cell Matching Tool (FCMT) is used to select the best matching compressor and turbine for a given fuel cell stack and target operating conditions (stack inlet pressure and oxygen excess ratio) across a range of current densities. For this study, the compressor and fuel cell stack were pre-defined, the VGT was selected by the FCMT, and the FGT turbine map was represented by the VGT operating with a \"flush gap\". This allows for a fair comparison of the FGT and VGT architectures by ensuring they have the same efficiency characteristics and swallowing capacity. The initial operating conditions were estimated based on previous work [24] and consisted of a set of desired cathode inlet pressures and oxygen excess ratios (OERs) for a range of 18 current densities from 0.05 A/cm\({}^{2}\) to 1.7 A/cm\({}^{2}\). These are used in conjunction with a set of parameters that describe the environment and the design of the air path system (e.g., pressure drop across the fuel cell) to determine the boundary conditions for both the compressor and turbine as follows. The boundary conditions for the compressor and turbine are defined by their pressure/expansion ratios and corrected air mass flow rates, which can be extracted directly from the 1D model results or calculated within the FCMT from the desired fuel cell operating conditions. The FCMT models the pressure drop of the air inlet, intercooler, fuel cell stack, and exhaust (illustrated in Figure 2) as fixed orifices based on Equation (1). The orifice area and discharge coefficient are summarised by a parameter (\(k\)) that is automatically generated based on experimental data, as shown in Figure 3.  
\[\Delta p=k\frac{\dot{m}^{2}}{\rho} \tag{1}\]  
Figure 2: Fuel cell system diagram including definition of air system. The mass flow through the compressor (\(\dot{m}_{cmp}\)) is calculated as the product of the inverse of the mass fraction of oxygen in air (100/23.3), the OER (\(R_{EO}\)), and the required oxygen flow (\(\dot{m}_{\lambda,O2}\)) as described by Equation (2). The required oxygen flow is calculated from the current density (\(i\)) using Equation (3), where \(n_{c}\) is the number of cells, \(A_{c}\) is the active area of the cell, \(M_{O2}\) is the molar mass of molecular oxygen and \(F\) is Faraday's constant. \[\dot{m}_{cmp}=\left(\frac{100}{23.3}\right)R_{EO}\dot{m}_{\lambda,O2} \tag{2}\]  
\[\dot{m}_{\lambda,O2}=\frac{in_{c}A_{c}M_{O2}}{4F} \tag{3}\]  
It is assumed that the back-diffusion of water generated at the cathode through the membrane is negligible under steady-state conditions, and therefore, the mass flow through the turbine is calculated as shown in Equations (4) and (5). \[\dot{m}_{trb}=\dot{m}_{cmp}-\dot{m}_{\lambda,O2}+\dot{m}_{\lambda,H2O}=\dot{m} _{cmp}+\dot{m}_{\lambda,H2} \tag{4}\]  
\[\dot{m}_{\lambda,H2}=\frac{in_{c}A_{c}M_{H2}}{2F} \tag{5}\]  
Once the boundary conditions have been determined, the speed and efficiency of the compressor (\(\dot{m}_{mp}\)) and turbine (\(\dot{m}_{trb}\)) can be calculated using a map-based approach as described by SAE J922 [41] and J1826 [42]. This, in turn, enables the calculation of mechanical work as shown in Equations (6)-(8) for the compressor and Equations (9)-(11) for the turbine. Equations (6)-(11) use temperature (\(T\)), the ratio of specific heats (\(\gamma\)), power (\(P\)), specific heat capacity (\(c_{p}\)), efficiency (\(\eta\)), and the pressure ratio (\(\pi\)), while subscripts '_isu_' and '_mch_' denote 'isentropic' and'mechanical', respectively. \[T_{cmp,out,isn}=T_{cmp,in}\pi_{cmp}\frac{\gamma-1}{\gamma} \tag{6}\]  
\[P_{cmp,isn}=\dot{m}_{cmp}*c_{p}*\left(T_{cmp,out,isn}-T_{cmp,in,isn}\right) \tag{7}\]  
\[P_{cmp,mch}=\frac{P_{cmp,isn}}{\eta\left(\dot{m}_{cmp},\pi_{cmp}\right)} \tag{8}\]  
\[\Delta p=k\frac{\dot{m}^{2}}{\rho} \tag{1}\]0  
\[\Delta p=k\frac{\dot{m}^{2}}{\rho} \tag{1}\]1 Orifice model of pressure drop.