gaspype/examples/soec_syngas.md

# SOEC Co-Electrolysis

This example shows a 1D isothermal SOEC (Solid oxide electrolyzer cell) model for
converting carbon dioxide and steam into syngas.

The operating parameters chosen here are not necessarily realistic. For example,
a utilization of 0.95 causes issues with the formation of solid carbon.

```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
utilization = 0.95
air_dilution = 0.2
t = 700 + 273.15  # K
p = 1e5  # Pa

fs = gp.fluid_system('H2, H2O, O2, CH4, CO, CO2')
feed_fuel = gp.fluid({'H2O': 2, 'CO2': 1}, fs)

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

feed_air = gp.fluid({'O2': 1, 'N2': 4}) * o2_full_conv * utilization * air_dilution

conversion = np.linspace(0, utilization, 128)
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'])
```

The high oxygen partial pressure at the inlet is in reality lower.
The assumption that gas inter-diffusion in the flow direction is slower
than the gas velocity does not hold at this very high gradient. However
often the oxygen partial pressure is still to high to prevent oxidation of the
cell/electrode. This can be effectively prevented by recycling small amounts of
the hydrogen riche output gas.

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 = 1.3  # 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, '-')
```