# Green Hydrogen Production--Fidelity in Simulation Models for Technical-Economic Analysis
## 2 System Component Modeling
### Application of Hydrogen as a Fuel Cell
#### 2.4.2 The Detailed Fuel Cell Model  
The detailed model accurately simulates a specific fuel cell stack by adjusting for changes in parameters such as pressure, temperature, composition, and flow rates of the fuel and air.  
Figure 3: Simplified model representation of fuel cell.  
These variations influence the Tafel slope (\(A\)), the exchange current (\(i_{0}\)), and the open-circuit voltage (\(E_{oc}\)). The equivalent circuit of the detailed model (illustrated in Figure 4) is identical to that of the simplified model, except that the parameters (\(E_{oc},i_{0},A\)) need to be updated based on the input pressures, flow rates, stack temperature, and gas compositions [31].  
According to the detailed model, the next set of equations has been modified:  
\[E_{oc}=K_{C}\times E_{n} \tag{15}\]  
\[i_{o}=\frac{zFk(P_{H_{2}}+P_{O_{2}})}{Rh}\times\exp(\frac{-\Delta G}{RT}) \tag{16}\]  
\[A=\frac{RT}{z\times\alpha\times F} \tag{17}\]  
where \(R\) represents the unit of molar entropy, \(F\) the Faraday constant, \(z\) the number of moving electrons (two in this case), \(E_{n}\) the Nernst voltage, which is the thermodynamics voltage of the cells and depends on the temperatures and partial pressures of reactants and products inside the stack, \(\alpha\) the charge transfer coefficient, \(P_{H_{2}}\) the partial pressure of hydrogen inside the stack, \(P_{O_{2}}\) the partial pressure of oxygen inside the stack, \(k\) [PERSON]'s constant, \(h\) [PERSON]'s constant, \(\Delta G\) the activation energy barrier, \(T\) the temperature of operation, and \(K_{C}\) the voltage constant at nominal conditions.  
As illustrated in Figure 4, Blocks A, B, and C are used to calculate new values for \(E_{oc}\), \(i_{0}\), and \(A\). Initially, Block A determines the conversion rates (utilization) of hydrogen (\(U_{fH_{2}}\)) and oxygen (\(U_{fO_{2}}\)) as follows:  
\[U_{fH_{2}}=\frac{60000R\times Ti_{fc}}{z\times F\times P_{fud}\times V_{fuel} \times\times\%} \tag{18}\]  
\[U_{fO_{2}}=\frac{60000R\times Ti_{fc}}{2\times z\times F\times P_{air}\times V_ {air}\times y\%} \tag{19}\]  
In Equation (18), \(P_{fuel}\) represents the absolute supply pressure of fuel, \(P_{air}\) the absolute supply pressure of air, \(V_{fuel}\) the fuel flow rate, \(V_{air}\) the air flow rate, \(x\%\) the percentage of hydrogen in the fuel, and \(y\%\) the percentage of hydrogen in the oxidant.  
Block B, shown in Figure 4, determines the partial pressures and the Nernst voltage as follows:  
\[P_{H_{2}}=(1-U_{fH_{2}})\times x\%P_{fuel} \tag{20}\]  
Figure 4: Detailed fuel cell stack model.  
\[P_{O_{2}}=(1-U_{fO_{2}})\times y^{\infty}P_{air} \tag{21}\]  
\[P_{H_{2}O}=(w+2y\%\times U_{fO_{2}})\times P_{air} \tag{22}\]  
\[E_{n}=1.299+(T-298)\times\frac{-44.43}{zF}+\frac{RT}{zF}\ln\left(P_{H_{2}}\times P _{O_{2}}^{1/2}\right) \tag{23}\]  
for \(T>373\) K:  
\[E_{n}=1.299+(T-298)\times\frac{-44.43}{zF}+\frac{RT}{zF}\ln\left(\frac{P_{H_{2} \times P_{O_{2}}^{1/2}}}{P_{H_{2}O}}\right) \tag{24}\]  
In Equation (22), \(P_{H_{2}O}\) represents the partial pressure of water vapor and \(w\) the percentage of water vapor in the oxidant. The nominal value and Nernst voltage \(E_{n}\) can also be calculated from the Equation (24), where \(T\) represents temperature operation, \(R\) refers to the amount of energy needed to raise the temperature of one mole of a substance by one kelvin, and finally \(z\) is the number of moving electrons.  
By knowing the partial pressures of gases and the Nernst voltage, the new values for the open-circuit voltage and the exchange current can be calculated using Equation (13) and (16), respectively. Block C then calculates the new Tafel slope value using Equation (17).  
To model the effect of oxygen depletion (due to air compressor delay) on the stack voltage, parameters for flow dynamics such as the peak utilization (\(U_{fO_{2}}\)(peak)) and the corresponding voltage undershoot (Vu) are needed. Oxygen depletion within the cell increases its utilization above the nominal value, and the Nernst voltage is subsequently adjusted as follows.