gaspype/examples/sofc_methane.md

# SOFC with Methane

This example shows a 1D isothermal SOFC (Solid oxide fuel cell) model.

The operating parameters chosen here are not necessarily realistic due to
constraints not included in this model. Under the example condition the
formation of solid carbon is for example very likely. A pre-reforming step
is typically applied when operating on methane.

```python
import gaspype as gp
from gaspype.constants import R, F
import numpy as np
import matplotlib.pyplot as plt
```

Calculate equilibrium compositions for fuel and air sides
in counter flow along the fuel flow direction:
```python
fuel_utilization = 0.90
air_utilization = 0.5
t = 800 + 273.15  # K
p = 1e5  # Pa

fs = gp.fluid_system('H2, H2O, O2, CH4, CO, CO2')
feed_fuel = gp.fluid({'CH4': 1, 'H2O': 0.1}, fs)

o2_full_conv = np.sum(gp.elements(feed_fuel)[['H', 'C' ,'O']] * [1/4, 1, -1/2])

feed_air = gp.fluid({'O2': 1, 'N2': 4}) * o2_full_conv * fuel_utilization / air_utilization

conversion = np.linspace(0, fuel_utilization, 32)
perm_oxygen = o2_full_conv * conversion * gp.fluid({'O2': 1})

fuel_side = gp.equilibrium(feed_fuel + perm_oxygen, t, p)
air_side  = gp.equilibrium(feed_air  - perm_oxygen, t, p)
```

Plot compositions of the fuel and air side:
```python
fig, ax = plt.subplots()
ax.set_xlabel("Conversion")
ax.set_ylabel("Molar fraction")
ax.plot(conversion, fuel_side.get_x(), '-')
ax.legend(fuel_side.species)

fig, ax = plt.subplots()
ax.set_xlabel("Conversion")
ax.set_ylabel("Molar fraction")
ax.plot(conversion, air_side.get_x(), '-')
ax.legend(air_side.species)
```

Calculation of the oxygen partial pressures:
```python
o2_fuel_side = gp.oxygen_partial_pressure(fuel_side, t, p)
o2_air_side = air_side.get_x('O2') * p
```

Plot oxygen partial pressures:
```python
fig, ax = plt.subplots()
ax.set_xlabel("Conversion")
ax.set_ylabel("Oxygen partial pressure / Pa")
ax.set_yscale('log')
ax.plot(conversion, np.stack([o2_fuel_side, o2_air_side], axis=1), '-')
ax.legend(['o2_fuel_side', 'o2_air_side'])
```

Calculation of the local nernst potential between fuel and air side:
```python
z_O2 = 4  # Number of electrons transferred per O2 molecule
nernst_voltage = R*t / (z_O2*F) * np.log(o2_air_side/o2_fuel_side)
```

Plot nernst potential:
```python
fig, ax = plt.subplots()
ax.set_xlabel("Conversion")
ax.set_ylabel("Voltage / V")
ax.plot(conversion, nernst_voltage, '-')
print(np.min(nernst_voltage))
```

The model uses between each node a constant conversion. Because
current density depends strongly on the position along the cell
the constant conversion does not relate to a constant distance.

![Alt text](../../media/soc_inverted.svg)

To calculate the local current density (**node_current**) as well
as the total cell current (**terminal_current**) the (relative)
physical distance between the nodes (**dz**) must be calculated:
```python
cell_voltage = 0.77  # V
ASR = 0.2  # Ohm*cm²

node_current = (nernst_voltage - cell_voltage) / ASR  # A/cm² (Current density at each node)

current = (node_current[1:] + node_current[:-1]) / 2  # A/cm² (Average current density between the nodes)

dz = 1/current / np.sum(1/current)  # Relative distance between each node

terminal_current = np.sum(current * dz) # A/cm² (Total cell current per cell area)

print(f'Terminal current: {terminal_current:.2f} A/cm²')
```

Plot the local current density:
```python
z_position = np.concatenate([[0], np.cumsum(dz)])  # Relative position of each node

fig, ax = plt.subplots()
ax.set_xlabel("Relative cell position")
ax.set_ylabel("Current density / A/cm²")
ax.plot(z_position, node_current, '-')
```