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Electrochemical Modelling of Na-MCl_{2} Battery Cells Based on an Expanded Approximation Method

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## Abstract

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## 1. Introduction

## 2. Model Development

- 1.
- Ohmic drop due to electronic and ionic conducting structures, e.g., separator and current collector.
- 2.
- Electrode losses/electrode overpotential: Combines electrode processes such as transport processes and charge transfer kinetics.

- ${R}_{\kappa ,n}$ represents the ionic resistance of the secondary electrolyte in the cathode segment n, leading to an Ohmic potential loss.
- ${R}_{\sigma ,n}$ represents the electrical resistance of the nickel and iron matrix in the cathode segment n, leading to an Ohmic potential loss.

- Cylindrical cell geometry with the cathode space divided into 100 segments;
- $FeC{l}_{2}$ as an additional active compound besides $NiC{l}_{2}$;
- Electron transfer in the metal matrix of the cathode;
- A constant current charging and discharging cycle;
- Heat generation.

- In this case, only $NiC{l}_{2}$ is converted, because in this voltage range the iron reduction is thermodynamically not preferred.
- The electrochemical reaction follows Equation (1).

- In this case, both $NiC{l}_{2}$ and $FeC{l}_{2}$ are converted, because in this voltage range iron reduction is also thermodynamically preferred.

#### Modelling Heat Generation

## 3. Results

#### 3.1. Cell Voltage for Discharge and Charge Cycle

#### 3.2. Impact of C-Rate on the Discharge Voltage

#### 3.3. Voltages Losses during Cell Cycling

#### 3.4. Hysteresis Effect

#### 3.5. Material and Volume Distribution

#### 3.6. Heat Generation

Symbol | Value | Additional |
---|---|---|

T | $573.15$ K | |

${j}_{0}$ | 0.14 A cm${}^{-3}$ | [15] |

${j}_{0,A}$ | 5 A cm${}^{-3}$ | [15] |

${h}_{K}$ | 21 cm | Estimated value |

${r}_{K}$ | 1.8 cm | Estimated value |

${d}_{\beta}$ | 0.15 cm | Estimated value |

${\sigma}_{Ni}$ | $1/(6.24\xb7{10}^{-6}(1+0.0069\xb7(T-(20+273.15))))$ | [34] |

${\sigma}_{Fe}$ | $1/(9.71\xb7{10}^{-6}(1+0.0065\xb7(T-(20+273.15))))$ | [34] |

$\kappa $ | $0.145-1.827{m}_{B}+(-0.5715+6.358{m}_{B})\xb7{10}^{-3}\xb7T)$ | [14] |

${m}_{B}$ | $0.5436-1.972\xb7{10}^{-4}\xb7T+2.346\xb7{10}^{-7}\xb7{T}^{2}$ | Molar ratio in the electrolyte [14] |

$me{t}_{uti}$ | $0.25776$ | Metal utilization |

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Reaction scheme during $NiC{l}_{2}$ and $FeC{l}_{2}$ reduction inside the cathode space, RF (reaction front).

**Figure 2.**Equivalent circuit representing the segmented cathode space based on Orchard and Weaving [15].

**Figure 4.**Model validation with 1/8C ($I=5.125\phantom{\rule{3.33333pt}{0ex}}A$): (

**a**) discharge (dch) and (

**b**) charge (chg). Highlighted are three different segments:

**1 and can be removed. The following highlights are the same.**refers to the voltage plateau according to Equation (1);

**2**refers to the transition towards the next plateau; and

**3**is related to the voltage plateau according to Equation (2).

**Figure 5.**(

**a**) Simulated and measured discharge voltages for four different C-rates over DoD. (

**b**) Modelled charge and discharged voltage for 1/8C showing a hysteresis.

**Figure 6.**(

**a**) Resistance due to ion transport over DoD. (

**b**) Electrical resistance in the solid metal matrix over segment number, for $DoD=0.5$. Both graphs have been modelled for discharging with $1/8C$.

**Figure 7.**(

**a**) Volume fraction distribution of $NiC{l}_{2}$ over the segments 0−100 for different DoDs. (

**b**) Volume fraction distribution of $FeC{l}_{2}$ over the segments 0−100 for different DoDs. (

**c**) Calculated charge transfer current for the corresponding DoDs shown in (

**a**,

**b**).

**Figure 8.**Results of the thermal model. (

**a**) Overall heat generation of the reaction volume in one discharging cycle. (

**b**) Contribution of the different heat generation terms. (

**c**) Heat generation in a single segment (#50) over one discharging cycle with 1/8C.

Features | 1990 Sudoh et al. [11] | 1990 Boom et al. [19] | 1993 Orchard et al. [15] | 2008 Vallance et al. [20] | 2010 Rexed et al. [21] | 2012 Eroglu et al. [13] | 2015 Christin [14] | 2016 Zhu et al. [22] | 2018 Bracco et al. [23] | |
---|---|---|---|---|---|---|---|---|---|---|

Operation mode | Discharge | Yes | Yes | Yes | Yes | Yes | Yes | Yes | Yes | Yes |

Charge | Yes | No | No | No | Yes | Yes | No | Yes | No | |

Cell chemistry | Fe | Yes | No | Yes | Yes | No | Yes | Yes | Yes | Yes |

Ni | No | Yes | No | No | Yes | No | Yes | Yes | No | |

Cell geometry | Planar | No | No | Yes | No | No | No | No | No | No |

Radial | Yes | Yes | No | No | No | Yes | No | Yes | Yes | |

Cloverleaf | No | No | No | Yes | No | No | Yes | No | No | |

Processes | Porosity | Yes | No | Yes | Yes | No | Yes | Yes | Yes | Yes |

Na${}^{+}$ Transport | Yes | Yes | Yes | Yes | Yes | Yes | Yes | Yes | Yes | |

MCl${}_{2}$ lattice | Yes | No | No | Yes | Yes | Yes | Yes | No | No | |

Heat formation | No | No | No | No | No | No | Yes | No | No | |

Validation | No | Yes | Yes | Yes | No | No | Yes | No | Yes |

Symbol | Unit | Description |
---|---|---|

$\Delta G$ | J | Gibbs free energy change |

F | A s mol${}^{-1}$ | Faraday constant |

R | J K${}^{-1}$ mol${}^{-1}$ | Universal gas constant |

T | K | Temperature |

z | Number of electrons transferred | |

${E}_{OCV}$ | V | Open circuit v t V oltage |

E | V | Cell voltage |

$\eta $ | V | Overpotential |

${\eta}_{C}$ | V | Cathode overpotential |

${\eta}_{A}$ | V | Anode overpotential |

${\eta}_{n}$ | V | Overpotential in segment n |

I | A | Current |

${i}_{\kappa ,n}$ | A | Ionic current in segment n |

${i}_{\sigma ,n}$ | A | Electronic current in segment n |

${j}_{a,0}$ | A cm${}^{-3}$ | Anodic standard exchange current density |

${j}_{a}$ | A cm${}^{-3}$ | Anodic exchange current density |

${j}_{n,j,M}$ | A cm${}^{-3}$ | Charge exchange current density of active material M |

${j}_{n,j}$ | A cm${}^{-3}$ | Sum of exchange current densities |

${j}_{0}$ | A cm${}^{-3}$ | Standard exchange current density |

${j}_{0,n,j,M}$ | A cm${}^{-3}$ | Standard exchange current density of active material M |

${d}_{n,j,M}$ | Local depth of discharge | |

${R}_{Ohmic}$ | $\mathsf{\Omega}$ | Separator resistance |

${R}_{\sigma ,n}$ | $\mathsf{\Omega}$ | Ionic resistance |

${R}_{\kappa ,n}$ | $\mathsf{\Omega}$ | Electronic resistance |

$\kappa $ | ${\mathsf{\Omega}}^{-1}$ cm${}^{-1}$ | Standard conductivity |

${\kappa}_{n,j}$ | ${\mathsf{\Omega}}^{-1}$ cm${}^{-1}$ | Ionic conductivity in segment n at timestep j |

${\sigma}_{n,j}$ | ${\mathsf{\Omega}}^{-1}$ cm${}^{-1}$ | Standard conductivity |

${\sigma}_{e,n}$ | ${\mathsf{\Omega}}^{-1}$ cm${}^{-1}$ | Electronic conductivity in segment n |

${\u03f5}_{n,j}$ | Cathode porosity | |

${\u03f5}_{M,n}$ | Metal matrix porosity | |

$\tau $ | Totuosity | |

$\alpha $ | Charge transfer coefficient | |

${l}_{n}$ | cm | Length of each segment |

${A}_{n}$ | cm${}^{2}$ | Exchange area |

q | W cm${}^{-3}$ | Volumetric heat generation rate |

${\mathsf{\Phi}}_{1}$ | V | Potential in solid phase |

${\mathsf{\Phi}}_{2}$ | V | Potential in the electrolyte |

${v}_{MCl,n,j}$ | mol cm${}^{-3}$ | Molar volume of active material M in segment n at timestep j |

Subscripts | ||

j | Time step index | |

n | Segment number | |

N | Total number of segments | |

M | Metal indexing ($Ni$,$Fe$) |

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**MDPI and ACS Style**

Büttner, N.; Purr, F.; Sangrós Giménez, C.; Richter, M.; Nousch, L.; Zellmer, S.; Michaelis, A. Electrochemical Modelling of *Na*-*MCl*_{2} Battery Cells Based on an Expanded Approximation Method. *Batteries* **2023**, *9*, 200.
https://doi.org/10.3390/batteries9040200

**AMA Style**

Büttner N, Purr F, Sangrós Giménez C, Richter M, Nousch L, Zellmer S, Michaelis A. Electrochemical Modelling of *Na*-*MCl*_{2} Battery Cells Based on an Expanded Approximation Method. *Batteries*. 2023; 9(4):200.
https://doi.org/10.3390/batteries9040200

**Chicago/Turabian Style**

Büttner, Nils, Foelke Purr, Clara Sangrós Giménez, Maria Richter, Laura Nousch, Sabrina Zellmer, and Alexander Michaelis. 2023. "Electrochemical Modelling of *Na*-*MCl*_{2} Battery Cells Based on an Expanded Approximation Method" *Batteries* 9, no. 4: 200.
https://doi.org/10.3390/batteries9040200