# Analysis of Hybrid Ship Machinery System with Proton Exchange Membrane Fuel Cells and Battery Pack
## 3 Numerical Background of Fuel Cell Model  
The fuel cell model utilized in this article was designed and extensively examined in [24]. The model is a 0-dimensional electrochemical model that is thermodynamically consistent. It is used to manage the operation of proton exchange membrane fuel cells (PEMFC). Furthermore, the model was also expanded to include the transportation of gases through channels and the gas diffusion layer (GDL), making it quasi-dimensional. The primary characteristic of the model is the simplified form of the Butler-Volmer equation, which clarifies the correlation between voltage and current in an electrochemical cell. The main assumption is a charging transfer coefficient value of 0.5, enabling the utilization of the hyperbolic sine function. Therefore, Equation (2) is derived, which is analogous to the Butler-Volmer equation but considerably easier to handle. The equations are presented below:  
\[I_{c}=I_{0}^{c}\cdot\mathrm{e}^{-\frac{E_{0}^{c}}{k_{B}^{2}}}\Big{(}\widetilde{ C}_{O_{2}}\Big{)}^{0.5}\Big{(}\widetilde{C}_{H_{2}O}\Big{)}2\mathrm{sinh} \bigg{(}-\frac{e_{0}Z_{c}\eta_{c}}{k_{B}T}\bigg{)} \tag{1}\]  
\[I_{0}^{c}=Z\mathrm{Fe}^{-\frac{a_{C}\alpha_{C}(T-T^{0})}{k_{B}T}}\mathrm{e}^{ -a_{\kappa}\cdot\mathrm{ln}(\frac{k_{B}^{c}}{P_{O}CC})}k_{RDC}^{*} \tag{2}\]  
where \(I_{c}\) is the cathode current, \(F\) is the Faraday constant, \(\widetilde{C}_{O_{2}}\) and \(\widetilde{C}_{H_{2}O}\) are the normalized concentrations of oxygen and water, \(k_{B}\) is the Boltzmann constant, \(Z\) is the number of electrons in the electrochemical reaction, \(T\) is the temperature, \(k_{RDC}^{*}\) and \(k_{OXC}\) are the reaction rate constants, \(e_{0}\) is the elementary charge, and \(\eta_{C}\) is the reaction kinetics overvoltage on the cathode. Based on Equation (1), it is possible to derive the expression for the overpotential of reaction kinetics, \(\eta_{C}\), for the cathode side. However, analogous derivations can also apply to the anode side. This adaptability allows the model to be efficiently implemented in both fuel cell operations and electrolyzer mode. The single-cell voltage \(U_{cell}\) is defined as given:  
\[U_{cell}=U_{th}-\eta_{c}-\eta_{a}-U_{R} \tag{3}\]  
where \(U_{th}\) is the thermodynamic potential, and \(U_{R}\) is voltage ohmic loss. Ohmic losses are caused by the flow of protons across the membrane, and they have a substantial impact on ionic conductivity. This conductivity is computed using data collected from sources in the literature. Assuming constant ionic conductivity, the ohmic loss increases linearly as the current density increases, which determines the slope of the polarization curve for medium current. Transport losses refer to the phenomenon of species diffusing through the channels and gas diffusion layer and into the catalyst layer. These losses are quantified using [PERSON]'s law of diffusion. These processes define the maximum current densities, namely the current density at which the concentration of the reactant at the catalyst layer begins to decrease. The transport model was modeled with the idea of estimating the concentration of reactants at the catalyst layer (\(C_{r_{CL}}\) ) by multiplying the reactant concentration in the channel (\(C_{r_{chem}}\) ) by the ratio of the limiting current (\(I_{L}\)) to the current density (\(I\)), as expressed by the following equation:  
\[C_{r_{CL}}=C_{r_{\mathrm{chan}}}\left(1-\frac{I}{I_{L}}\right) \tag{4}\]  
\[I_{L}=ZFSD_{rr}\frac{C_{r_{\mathrm{chan}}}}{\delta_{GDL}} \tag{5}\]  
where \(D_{rr}\) is diffusion coefficient, and \(\delta_{GDL}\) is \(GDL\) width. The electrochemical model incorporates five calibration parameters, each corresponding to a distinct loss mechanism. Due to the greater significance of activation and transport losses on the cathode side, the parameters for the anode are typically believed to be irrelevant.