# Radial Thermal Conductivity Measurements of Cylindrical Lithium-Ion Batteries—An Uncertainty Study of the Pipe Method

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**Table 1.**Values found in literature for the thermal conductivity of cylindrical cells and their individual layers.

Material/Layer | ${\mathit{\lambda}}_{\mathbf{radial}}$ | ${\mathit{\lambda}}_{\mathbf{axial}}$ | Cross-Plane | In-Plane | Geometry | Capacity | Refs. |
---|---|---|---|---|---|---|---|

[Wm^{−1}·K^{−1}] | [Wm^{−1}·K^{−1}] | [Wm^{−1}·K^{−1}] | [Wm^{−1}·K^{−1}] | [Ah] | |||

NMC, LCO, NCA | 0.15–2.59 | 1.78–32 | 18650 | 2.14–3.5 | [8,10,11,12,13] | ||

3 | 21,700 | [11] | |||||

0.2 | 30.4 | 26,650 | [8,10] | ||||

Single Layer | |||||||

LiCoO2 (LCO) (Cathode) | 1.58–3.5 | 21.6–28 | [5,9,14,15] | ||||

Graphite (Anode) | 0.89–1.2 | 8.72–139 | [9,14,15,16] | ||||

Separator | 0.1–1 | [9,14,16,17] |

^{−1}·K

^{−1}. Using the data found in the references [8,10,11,12,13], for the radial thermal conductivity for 18650 cells, a standard deviation of 1.61 Wm

^{−1}·K

^{−1}can be calculated, which can be considered very high, especially compared to the actual measured values. Therefore, this paper’s objective is to analyze the experimental process and to identify reasons why these deviations in the radial thermal conductivity occur when the pipe method is used.

^{−1}·K

^{−1}) was used to reduce the influence of the contact resistance. Even with these improvements, the deviation of the radial thermal conductivity could not be adequately reduced. Therefore, a second independent method, which was intended to further limit the possible range of results, was introduced. This second method calculates the effective thermal conductivity from the results of the LFA and DSC measurements that are described in Section 1.2. However, this second method is not intended to serve as a benchmark, since the cell must be completely disassembled and therefore influences like the missing electrolyte cannot be considered. Using this second method, the thermal conductivity values of each of the materials of a LIB were established, which made it possible to create a simplified FEM-model that depicts the measurement setup of the pipe-method for a cylindrical cell. With the help of this thermal simulation, the authors were able to show that not only the slightest changes in the measurement setup can lead to major variations. Furthermore, a minor deviation in the structure of the cell can lead to a major variation in the measured thermal conductivity. For the first time, uncertainties in the pipe method used for LIB-based thermal conductivity measurements were analyzed in detail. In addition to this discovery, the authors also investigated the influence of the separator on the thermal conductivity and analyzed the state-of-charge (SOC) dependency of the thermal conductivity on functional cells. Figure 2 shows the flowchart of the methods used, including the process to measure or calculate the radial thermal conductivity.

#### 1.1. Steady-State Methods

^{−1}·K

^{−1}] can be calculated after the system reaches a steady state, which is the case when the heat flux through the specimen is constant. This method uses two calibrated heat flux sensors to measure the heat flux density $\dot{q}$ [W/m

^{2}]. With known sensor area, the heat flux $\dot{Q}$ [W] through the specimen can be calculated. Therefore, the specimen is put between a heated and a cooled plate. With a defined temperature gradient and the measured heat flux, the thermal conductivity of the specimen can be calculated according to Equation (1).

#### 1.2. Transient Methods

## 2. Experimental Method

#### 2.1. Thermocouple Based Setup

#### 2.2. FBG Sensor-Based Setup

#### 2.3. Reference Material Measurements

^{−1}·K

^{−1}), which represents the lower range of thermal conductivity for this setup, and one specimen of fused quartz ($\lambda =1.38$ Wm

^{−1}·K

^{−1}), which represents the upper bound of this measurement setup, were selected. The dimensions of the reference materials were chosen to be comparable to those of the cylindrical cell, to ensure similar conditions as for the measurements conducted later. According to Equation (2) the expected $\mathsf{\Delta}T$ for Polymethylmethacrylate (PMMA) is 32 ${}^{\circ}$C and for fused quartz it is 10 ${}^{\circ}$C. Therefore, the influence in temperature measurement uncertainty increases the greater the thermal conductivity of the reference material is.

^{−1}·K

^{−1}for the PMMA cylinder. For the fused quartz cylinder, a 0.75 W higher heat flow compared to the steady-state power dissipation was measured. For these measurements the insulation material, which was put on the poles to mitigate axial heat flow, partially covered some surface area of interest, leading to a reduced surface area for heat dissipation. If the fused quartz cylinder area is reduced by the area of the insulation material, the calculated heat flow is 0.25 W smaller than the measured power dissipation. Using the heat flow based on the reduced cylinder area leads to $\lambda =1.33$ Wm

^{−1}·K

^{−1}for the fused quartz cylinder.

#### 2.4. 18650 Cell Measurements

#### 2.4.1. Pipe Method Inactive Cell Measurements

^{−1}·K

^{−1}and $\lambda =0.73$ Wm

^{−1}·K

^{−1}. Due to safety reasons the maximum inner temperature of the cell was limited to 55 ${}^{\circ}$C, which also limits the maximum power dissipation and therefore the maximum temperature difference between the inner and outer surface of the cell. The temperature difference achieved in the measurements varies between 6.8 ${}^{\circ}$C and 12 ${}^{\circ}$C depending on the power dissipation. Except for cell number 4, two temperature sensors were mounted on the outer surface of each cell. For the thermocouple-based setup each temperature sensor is mounted with thermal adhesive on the middle of the cell on different sides. Between these outer temperature sensors, a maximum difference of 1 ${}^{\circ}$C occurred. This can happen due to the tolerance of each temperature sensor, but also because of uneven distributed heat. Due to the high fluctuations in the individual measurement results, the influence in heating wire and sensor position will be analyzed more accurately in the simulation chapter.

#### 2.4.2. Measurements at Different State of Charge (SOC) Levels Using the Pipe Method

#### 2.4.3. LFA and DSC Based Cell Measurements

^{−1}·K

^{−1}[9,14,16,17]. Therefore, they have a great impact on heat transfer in the cell [9]. By varying the value of the thermal conductivity of the separator within the previously mentioned range and calculating the thermal resistance with Equation (10) together with the obtained values from Table 6, the influence of the separator can be seen in Figure 10.

^{−1}·K

^{−1}depending on the separator material. In comparison to the pipe method measurements, where the results vary between 0.51 and 0.73 Wm

^{−1}·K

^{−1}, the LFA and DSC based measurement results are much higher. A possible explanation is given in the simulation chapter where the temperature sensor position inside the cell is varied and an additional layer of air, indicating the possibility of missing thermal paste, is simulated.

## 3. Simulation

^{−1}·K

^{−1}was assumed for the separator layer. Therefore, the effective thermal conductivity of this simplified battery model is $\lambda =1.17$ Wm

^{−1}·K

^{−1}, which can be calculated using Equation (2). The primary purpose of the simulation model is to determine the effects of an incorrectly assumed temperature sensor position. Furthermore, the effects of missing thermal paste between the temperature sensor and the bulk material of the cell are shown, as well as the influence of different heating wire positions on the calculation result. For this simplified model, the thickness of the anode, cathode and separator layer is calculated by summing up all the individual layer thicknesses in radial direction. Due to this simplification, no statement about the temperature distribution in the bulk material can be made. However, the core and the surface temperature of the cell can be simulated. Figure 11 shows the simulation model and the cross-section showing possible uncertainties. The simulation model contains a heat source in the center of the circle that represents the heating wire. A heat flux boundary condition on the outer surface of the cylinder is added to create similar conditions as in the measurements. With this boundary condition, a constant heat flow of 1.38 W is created. Different virtual measurement points (T1–T9) next to the heating wire are placed to represent the possible locations of the introduced temperature sensor as shown in Figure 11c.

^{−1}·K

^{−1}to create the same conditions as in the pipe method. As shown in Figure 12 the closer the virtual sensor moves to the heating wire, the smaller the calculated effective thermal conductivity becomes. At a sensor distance of 1.7 mm to the inner surface the calculated thermal conductivity of the cell is 0.87 Wm

^{−1}·K

^{−1}, which is already a reduction of 25% compared to the given effective thermal conductivity of 1.17 Wm

^{−1}·K

^{−1}. For the second simulation an additional air gap was inserted between the sensor and the bulk material. This gap simulates the absence of thermal paste between the inner surface and the temperature sensor. The gap is varied between 10 µm to 100 µm, which is in the range of the thickness of a single separator layer and a negative electrode. Figure 12 also shows that a 50 µm air gap reduces the thermal conductivity to 0.56 Wm

^{−1}·K

^{−1}.

^{−1}·K

^{−1}, which is the conductivity of the used thermal paste, leads to an even higher deviation in the simulation results, as shown in Figure 13d.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BMS | Battery management system |

CT | Computed tomography |

DSC | Differential scanning calorimetry |

FBG | Fiber Bragg grating |

FEM | Finite element method |

HFM | Heat flow meter |

LFA | Laser flash method |

LIB | Lithium-ion secondary battery |

NCA | Nickel cobalt aluminium oxide |

PMMA | Polymethylmethacrylate |

SOC | State of charge |

$\lambda $ | Thermal conductivity (Wm^{−1}·K^{−1}) |

$\dot{Q}$ | Heat flow (W) |

$\dot{q}$ | Heat flux density (W/m^{2}) |

$\mathsf{\Delta}T$ | Temperature difference (${}^{\circ}$C) |

A | Surface area of the specimen (m^{2}) |

a | Thermal diffusivity (m^{2}/s) |

$\rho $ | Density (kg·m^{−3}) |

${c}_{p}$ | Heat capacity (J/K) |

${R}_{th}$ | Thermal resistance (K/W) |

${r}_{o}$ | Outer radius (m) |

${r}_{i}$ | Inner radius (m) |

${r}_{s}$ | Sensor radius (m) |

${A}_{cell}$ | Surface area of cell without top and bottom area (m^{2}) |

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**Figure 6.**Reference material measurements carried out with the thermocouple-based measurement setup: measured temperatures and dissipation power from the PMMA specimen (

**left**side) and from the Quartz-specimen (

**right**side).

**Figure 9.**CT scan of the cell showing the different layers and the diameter of the inner hole (

**Left**: A horizontal scan of an 18650 cell.

**Right**: Vertical scan showing the diameter of the inner hole).

**Figure 10.**Influence in varying thermal conductivity values of separator material on the effective thermal conductivity of a 18650 cylindrical cell.

**Figure 11.**Cross section of the cell (

**a**) showing possible uncertainties that can influence the measurement results of the radial thermal conductivity. The simulation model (

**b**) with virtual temperature sensors (T1–T9) and different positions (Pos.0–Pos.4) of the heating wire (

**c**).

**Figure 12.**Simulation results of different temperature sensor positions and missing thermal paste leading to wrong calculations.

**Figure 13.**Influence of different heating wire positions on the calculated thermal conductivity when the sensor position is unknown. The

**left**side (

**a**,

**c**) shows the results when the thermal paste is not considered by using Equation (2). The

**right**side (

**b**,

**d**) shows the results when the thermal paste is considered by using Equation (3).

Sensitivity | FBG1 pm/${}^{\circ}$C | FBG2 pm/${}^{\circ}$C |
---|---|---|

Sensor 1 | 9.98 ± 0.08 | 10.01 ± 0.09 |

Sensor 2 | 10.37 ± 0.07 | 10.38 ± 0.14 |

Sensor 3 | 9.96 ± 0.11 | 10.03 ± 0.11 |

Sensor 4 | 10.06 ± 0.12 | 10.16 ± 0.12 |

Sensor 5 | 10.29 ± 0.10 | 10.34 ± 0.10 |

Material | Dimensions (${\mathit{r}}_{\mathit{o}}$, ${\mathit{r}}_{\mathit{i}}$, Length) | $\mathbf{\Delta}$T | Q | ${\mathit{\lambda}}_{\mathbf{Literature}}$ | $\mathit{\lambda}\mathbf{Measured}$ |
---|---|---|---|---|---|

[mm] | [${}^{\circ}$C] | [W] | [Wm^{−1}·K^{−1}] | [Wm^{−1}·K^{−1}] | |

Acrylic glass (PMMA) | 10, 2, 61 | 31.8 | 1.46 | 0.19 | 0.197 |

Quartz | 12.5, 1.35, 100 | 13–17.3 | 4.28–5.3 | 1.38 | 1.31–1.42 |

Quartz (FBG) | 12.5, 1.35, 100 | 13–16.5 | 4.45 | 1.38 | 1.21–1.36 |

Cell Number | Dimensions (${\mathit{r}}_{\mathit{o}}$, ${\mathit{r}}_{\mathit{i}}$, Length) | $\mathbf{\Delta}$T | Q | ${\mathit{\lambda}}_{\mathbf{Measured}}$ |
---|---|---|---|---|

[mm] | [${}^{\circ}$C] | [W] | [Wm^{−1}·K^{−1}] | |

1 | 9, 1.9, 65 | 9–10 | 1.19 | 0.51–0.58 |

2 | 9, 1.9, 65 | 8.5–9.3 | 1.38 | 0.66–0.73 |

3 | 9, 1.9, 65 | 10.1–12 | 1.46 | 0.52–0.64 |

4 | 9, 1.9, 65 | 6.8 | 0.86 | 0.67 |

5 (FBG) | 9, 1.9, 65 | 10.1–10.7 | 1.46 | 0.56–0.61 |

Layer | Thickness | Quantity | Thickness in Total |
---|---|---|---|

[µm] | [µm] | ||

Separator | 18 | 60 | 1080 |

Positive electrode | 90 | 29 | 2610 |

Negative electrode | 110 | 30 | 3300 |

Case | 152 | 1 | 152 |

Inner hole | 1900 | 1 | 1900 |

${\mathit{\lambda}}_{\mathbf{Calculated}}$ | a | ${\mathit{c}}_{\mathit{p}}$ | $\mathit{\rho}$ | |
---|---|---|---|---|

[Wm^{−1}·K^{−1}] | $\times {10}^{6}$ [m^{2}/s] | [J/g·K] | [g/cm^{3}] | |

Anode | 3.4 | 1.57 | 0.97 | 2.23 |

Cathode | 1.8 | 0.52 | 1.02 | 3.45 |

Case | 136 | 54.1 | 0.524 | 4.8 |

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**MDPI and ACS Style**

Koller, M.; Unterkofler, J.; Glanz, G.; Lager, D.; Bergmann, A.; Popp, H.
Radial Thermal Conductivity Measurements of Cylindrical Lithium-Ion Batteries—An Uncertainty Study of the Pipe Method. *Batteries* **2022**, *8*, 16.
https://doi.org/10.3390/batteries8020016

**AMA Style**

Koller M, Unterkofler J, Glanz G, Lager D, Bergmann A, Popp H.
Radial Thermal Conductivity Measurements of Cylindrical Lithium-Ion Batteries—An Uncertainty Study of the Pipe Method. *Batteries*. 2022; 8(2):16.
https://doi.org/10.3390/batteries8020016

**Chicago/Turabian Style**

Koller, Markus, Johanna Unterkofler, Gregor Glanz, Daniel Lager, Alexander Bergmann, and Hartmut Popp.
2022. "Radial Thermal Conductivity Measurements of Cylindrical Lithium-Ion Batteries—An Uncertainty Study of the Pipe Method" *Batteries* 8, no. 2: 16.
https://doi.org/10.3390/batteries8020016