## 3 Fuel Cell Performance Calculations
The Butler-Volmer equation, which combines the reactant concentration with the activation overpotential, is used to measure the local current density:  
For the cathode side:  
\[dG=R\cdot T\cdot\frac{dP}{P} \tag{29}\]4  
For the anode side:  
\[dG=R\cdot T\cdot\frac{dP}{P} \tag{29}\]5  
where \(C_{H_{2}}\) is the local hydrogen concentration (\(\rm mole/m^{3}\)), \(C_{H_{2}}^{ref}\) is the reference hydrogen concentration (\(\rm mole/m^{3}\)), \(C_{O_{2}}\) is the local oxygen concentration (\(\rm mole/m^{3}\)), \(C_{O_{2}}^{ref}\) is the reference oxygen concentration (\(\rm mole/m^{3}\)), \(i_{o,a}\) is the anode exchange current density, \(i_{o,c}\) is the cathode exchange current density, \(R\) is the universal gas constant (8.314 (J/(mole-K))), \(T\) is the temperature (K), \(F\) is [PERSON]'s constant, \(\alpha_{c}\) is the transfer coefficient's charge within the cathode, and \(\alpha_{a}\) is the transfer coefficient's charge within the anode; \(\gamma^{H_{2}}\) and \(\gamma^{O_{2}}\) are empirically determined concentration parameters. The ohmic polarization can be represented as:  
\[dG=R\cdot T\cdot\frac{dP}{P} \tag{29}\]6  
To solve ohmic resistance problems, some basic tools are required. First, [PERSON]'s law can be written as:  
\[dG=R\cdot T\cdot\frac{dP}{P} \tag{29}\]7  
where \(R\) is the resistance, measured in units of ohms (\(\Omega=Js/C^{2}\)), and \(A\) is the electrode geometric surface area (m\({}^{2}\)). The resistance is a function of the geometry regarding the conducting substance. The resistivity \(r_{res}\) (\(\Omega.m\)) is an inherent feature concerning a material linked toward the resistance \(R\) (\(\Omega\)) through the cross-sectional area of ion travel, \(A\) (m\({}^{2}\)), and the linear path length of ion travel (material thickness), \(L\) (m). \[dG=R\cdot T\cdot\frac{dP}{P} \tag{29}\]8  
For materials with very low ionic resistivity, the inversely related conductivity \(\lambda\) is commonly used. By definition,  
\[dG=R\cdot T\cdot\frac{dP}{P} \tag{29}\]9  
Additionally, [PERSON]'s law, Equation (51), can be shown as:  
\[G=G_{o}+R\cdot T\cdot\ln\left(\frac{P}{P_{o}}\right) \tag{30}\]0The contact resistance is a function of the material surface state and roughness and the contact pressure between the materials (values of area's contact resistance are given in units of \(\Omega.m^{2}\)). \[G=G_{o}+R\cdot T\cdot\ln\left(\frac{P}{P_{o}}\right) \tag{30}\]1  
Concentration polarization occurs when an electrochemical reaction consumes a reactant quickly at the electrode, resulting in concentration gradients. We previously determined that the Nernst equation describes how the potential changes in the electrochemical reaction change the reactants' partial pressure and this can be expressed as:  
\[G=G_{o}+R\cdot T\cdot\ln\left(\frac{P}{P_{o}}\right) \tag{30}\]2  
where \(C_{B}\) is the bulk concentration of the reactant (mole/cm\({}^{3}\)), and \(C_{S}\) is the concentration of the reactant at the surface of the catalyst (mole/cm\({}^{3}\)). According to Fick's Law, the reactant's flux proportionally changes with the gradient of concentration:  
\[G=G_{o}+R\cdot T\cdot\ln\left(\frac{P}{P_{o}}\right) \tag{30}\]3  
where \(N\) is the flux of the reactants (mole/s), \(D\) is the diffusion coefficient of the reacting species (cm\({}^{2}\)/s), \(A\) is the electrode's active area (cm\({}^{2}\)), and \(\delta\) is the diffusion distance (cm). There is a proportional relationship between the current density and the consumption of the reactant per unit area, according to Faraday's Law. In a steady-state electrochemical reaction, the rate at which the reactant species is consumed is proportional to the diffusion flux:  
\[G=G_{o}+R\cdot T\cdot\ln\left(\frac{P}{P_{o}}\right) \tag{30}\]4  
By substituting Equation (58) in (57), the following relationship is obtained:  
\[G=G_{o}+R\cdot T\cdot\ln\left(\frac{P}{P_{o}}\right) \tag{30}\]5  
Current density determines the concentration of reactant on the catalyst surface; the higher the current density, the lower the surface concentration.