Cylindrical Cell Electrode Estimation: *An alternative geometric approach*.

Here we present a simple method for estimating electrode length in a cylindrical cell. The method is equally applicable to other formats since we make an estimation of the total active electrode area. Results require knowledge of one electrode Active Material (AM) chemistry, electrode porosity and thickness and cell capacity. We assume that 100% of the active material is utilised and contributes to the cell capacity. This is optimistic and will likely lead to an underestimate of the length.

Taking the Telsa 4680 cell as an example, and using the benchmark data from here, an estimate of the electrode length for the cathode can be calculated.

Starting data:

- Cell capacity = 23.35 Ah
- Cathode material – NMC811 – Specific practical capacity
^{1}= 200 mA/g - Crystallographic density
^{1}– ρ_{NMC}= 4.87 g/cm^{3} - Electrode thickness = 85 micron
- Estimated porosity = 0.35

Assumptions:

- Electrode composition = 96:2:2 wt% AM:CA:B
- Cell capacity represents specific practical capacity for cathode active material
- Electrode density is dominated by AM and pore space only

First, calculate the total mass of cathode active material = 23.35 Ah/0.200 Ah/g = 116.7 g

Total electrode mass: 116.7 g/0.96 = 122 g

Estimated electrode density (based on AM and estimated porosity) = 4.87 g/cm3 * 0.65 = 3.17 g/cm^{3}

Total active electrode area can be calculated geometrically from electrode thickness and density: A = 122 g / (3.17 g/cm^{3} * 0.0085 cm) = 4513 cm^{2}.

We estimate (based on 4680 teardown and 1865 data) that active electrode width is 85% of can height 80 x 0.85 = 68mm. Therefore the total length can be calculated 4513 cm^{2}/6.8 cm = 664 cm (of single side equivalent) = 332cm of double sided coating. The Battery Design spiral calculator estimates 345 cm.

##### Electrode Estimation

Knowing the outer and inner diameter of the spiral along with it’s thickness we can calculate the length of the material to create it.

D is the inner diameter of the cylindrical can.

The inner diameter is that of the mandrel around which we wind the spiral.

Again, this is a simple approach to cylindrical cell electrode estimation that makes several assumptions (in this case sensitive to active material utilisation) but the results are in reasonable agreement with the alternative spiral calculation method.

Denis Cumming – a research-focused academic with interests in energy storage, material development and electrode manufacturing technology. Project Leader of the Faraday Institution Nextrode project, with 15 years academic and industrial research expertise in colloidal processing and electrochemistry.

## References

- Hasa, I., et al., (2021). Table S4 in Challenges of today for Na-based batteries of the future: From materials to cell metrics.
*Journal of Power Sources*,*482*, 228872.