Heat generation in a cell can be defined quite simple for the case where the cell is operating within it’s normal limits. The following expression gives the heat flow [W]:

Where: *I* = current [A],* V _{oc}* = open circuit voltage [V],

*T*= reference temperature [K],

_{ref}*T*= cell temperature [K]

The first part of this equation is the irreversible Joule heating term:

The second part is the reversible entropy term or Reaction heat terms:

A comparison of these two terms is shown based on the experimental analysis [1] of a power type

prismatic LMO-G (lithium manganese oxide/graphite) with nominal capacity 8 Ah. The battery has a maximum discharge current rate of 20C and maximum charge current rate of 10C.

From these graphs you see that the entropy term can be endothermic under certain conditions. When charging this cell the entropy term becomes very significant, especially at low SoC. Charging the fully discharged cell shows that it will cool down even further until it reaches around 20% SoC.

This shows how important it is to fully characterise the thermal behaviour of a cell in order to properly model and then design a battery pack to optimise charging.

At high discharge rates the joule heating term dominates, as expected. It is under high discharge rates that often the heating is approximated as just I^{2}R_{int}. Where R_{int} is the internal resistance of the cell.

#### References

- Guangming Liu, Minggao Ouyang, Languang Lu, Jianqiu Li, Xuebing Han, Analysis of the heat generation of lithium-ion battery during charging and discharging considering different influencing factors, J Therm Anal Calorim (2014) 116:1001–1010
- Kotub Uddin et al, “An Acausal Li-Ion Battery Pack Model for Automotive Applications“, Energies 2014, 7, 5675-5700
- NREL, Energy Storage Thermal Performance
- Widanalage Dhammika Widanage, Ryan Prosser, Sabine Paarmann, Oliver Queisser and Mohammad Shahjalal, Estimation of the Entropy Coefficient of Lithium-Ion Batteries Via a Temperature and Open-Circuit Voltage Kernel Function, ECS Meeting Abstracts, Volume MA2020-01, A01: Battery and Energy Technology Joint General Session

#### Thermal Conductivity of the Active Layers

If we look at the active layers of a cell the thermal conductivity in the plane of the layers is approximately 10x to 100x that through the planes.

This should not be unexpected as the electrodes are made from sheets of aluminium and copper. Two of the best materials for thermal conductivity.

These values though have a large range [Ref 1]:

- 15 to 160 W/mK In-Plane
- 0.2 to 8 W/mK Through-Plane

#### Specific Heat Capacity of Lithium Ion Cells

The specific heat capacity of lithium ion cells is a key parameter to understanding the thermal behaviour. From literature we see the specific heat capacity ranges between 800 and 1100 J/kg.K

Heat capacity is a measurable physical quantity equal to the ratio of the heat added to an object to the resulting temperature change. Specific heat is the amount of heat per unit mass required to raise the temperature by kelvin (one degree Celsius).