# SOEC with Methane

This example shows a 1D isothermal SOEC (Solid oxide electrolyzer cell) model.

The operating parameters chosen here are not necessary realistic

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

Calculation of the local equilibrium compositions on the fuel and air
side 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 / 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
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  # mA/cm² (Current density at each node)

current = (node_current[1:] + node_current[:-1]) / 2  # mA/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) # mA/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, '-')
```