# A Novel Quick Temperature Prediction Algorithm for Battery Thermal Management Systems Based on a Flat Heat Pipe

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. FHP-Based BTMS Electro-Thermal Coupled Model

#### 2.1. FHP-Based BTMS

#### 2.2. Battery Heat Generation Model

_{ocv}are the terminal voltage and open-circuit voltage of the battery, respectively. Since U

_{ocv}is related to State of Charge (SOC), battery temperature and entropy change, a resistance-based heat generation model could be adopted to calculate the irreversible heat in order to simplify calculations. Thus, Equation (1) could be changed into:

_{p}and θ

_{o}are, respectively, the polarization resistance and ohmic resistance.

_{1}to U

_{2}, and then slowly decreasing to U

_{3}. After the end of pulse loading, the voltage starts with a jump rebound and then slowly increases. Correspondingly, the sudden change of voltage from U

_{1}to U

_{2}and the decrease from U

_{2}to U

_{3}are caused by θ

_{o}and θ

_{p}, respectively. According to ohmic law, the θ

_{o}and θ

_{p}under the tested conditions can be obtained, and the total resistance (θ

_{o}+ θ

_{p}) of the battery can be obtained. Then, the polynomial fitting relationship of (θ

_{p}+ θ

_{o}) and dU

_{ocv}/dT with respect to temperature, SOC, I based on experimental results is shown in Equations (3) and (4).

#### 2.3. Dynamic Heat Transfer Model

_{cxi}and R

_{cyi}are the thermal resistance of cell structure in the x and y axis directions, respectively, and R

_{ei}is the external thermal resistance due to natural convection or physical contact. The heat transfer process occurs in the following steps. The heat generated by two inner nodes enters the peripheral nodes through thermal conduction, then the heat is divided into four directions with three modes: (1) dissipates to the environment, (2) moves into other cells and (3) moves into the FHP. The last mode is the main basis for battery thermal management. After heat transfers through the solid shell and porous wick structure of the FHP, it enters the vapor chamber after the phase change of acetone, and finally is carried away at the condensation section.

_{c}configured and the heat is transferred into the vapor chamber by the phase change process. For the proposed FHP, the corresponding thermal network model is also described in Figure 3, where R

_{si}and R

_{wi}, respectively, represent the thermal resistances of the shell and the wick structure, R

_{pci}is the phase change thermal resistance and R

_{vi}is the resistance generated due to the pressure drop along the vapor chamber.

_{w}is obtained based on the assumption of a porous sintered material filled with liquid medium, as shown in the following equation [36]:

_{l}represents the thermal conductivity of the liquid-phase working fluid, λ

_{s}represents the thermal conductivity of the solid-phase material and φ denotes the porosity of the porous wick.

_{f}represents the thermal conductivity of the cooling fluid. d

_{e}represents the hydraulic diameter of the flow channel. Re

_{f}and Pr

_{f}, respectively, represent the Reynolds number and Prandtl number of the cooling fluid. s

_{fin}, w

_{fin}and t

_{fin}are, respectively, the space, width and thickness structure parameters of the fin, whose values have been shown in Table 2.

_{v}and T

_{v}represent the pressure and temperature at the evaporation interface, respectively. R

_{gas}is the gas constant, and h

_{fgi}represents the phase change latent heat of the working fluid.

_{FHP}represents the total length of the flat heat pipe along the heat transfer direction. Μ denotes the dynamic viscosity of the vapor, and t

_{v}represents the thickness of the vapor chamber in the FHP.

#### 2.4. Electro-Thermal Coupled Modeling Approach

#### 2.5. Verification of FHP-Based BTMS Thermo-Electric Coupled Model

^{–1}(natural convection) and 5 m·s

^{–1}are adopted, and the results are shown in Figure 6a and Figure 6b, respectively. The discharging capacity is set to 80% SOC, so that the experimental times are 6400 s, 2880 s and 1920 s for 0.5 C, 1 C and 1.5 C. After every discharging process, 2–4 h is applied to the battery for relaxation. When the measuring temperature has dropped to be equal to the environment, the charging process is started up. The upper and lower cut-off voltages are set to 4.2 V and 2.7 V per cell, in case over-charging/discharging occur. Due to the random variation of the environmental temperature in different cases, temperature rise is utilized as the comparison indicator. All experiments are started with the environmental temperature around 20 °C; thus, the initial temperature of the simulation is also set to 20 °C.

## 3. Implementation of Thermal Convolution Method

#### 3.1. Temperature Response of a Particular Node inside a Battery Cell to Impulse Excitation

**h**(n) and h(t) to represent the response of a linear system to the unit impulse signal

**δ**(n) and δ(t) under continuous and discrete conditions, respectively. Its response to any signal is a combination of

**h**(n) after weighting and translation, which means the system response

**y**(n) to a discrete time signal

**u**(n) can be expressed through linear combination of a series of delayed unit impulse responses by Equation (15). When it involves continuous time systems, y(t) can be rewritten into the integral form as shown in Equation (16), which is very similar to the formation of convolution.

_{i}(t).

#### 3.2. Correction of the Heat Flux throughout the FHP

_{FHP}term in Equation (17) to account for the dynamic thermal characteristics of the FHP. Mathematical models for the heating and cooling dynamic processes of flat plate heat pipes have been established in the literature and experimentally validated [42,43].

#### 3.3. Implementation Procedure of Online Temperature Prediction Based on the TCM

**q**and the inverse response function

**h**in the time domain yields the temperature rise at the current time step. By summing the results obtained from the inner loop, the temperature at the current time can be determined. Once the outer loop concludes, the temperature variations of the nodes from time t to t + k can be obtained.

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematic of an FHP-based BTMS of 12 cells and working principle of the FHP [32].

**Figure 6.**Comparison of experiments and simulation under constant discharging current: (

**a**) FHP with natural convection, (

**b**) FHP with air cooling.

**Figure 10.**Temperature response h

_{i}(t) of node inside cell no. 6 to different unit pulse heat sources.

**Figure 11.**Heat-up and cool-down time constants variation with (

**a**) power input and (

**b**) airflow rate.

**Figure 12.**Comparison between the TCRN and the FHP-DM under different heat input and shut-down conditions.

**Figure 13.**Comparison between the FHP-TCRN model and the FHP-DM. (

**a**) Continuous power input 12 W(200 s)–8 W, (

**b**) Continuous power input 12 W(50 s)–0 W.

**Figure 14.**Flow chart of quick temperature prediction algorithm based on the thermal convolution method.

**Figure 15.**Results comparison of the lumped model, the TCM, battery surface temperature, battery core temperature: (

**a**) 1 C-break condition; (

**b**) 2 C-break condition; (

**c**) 3 C-break condition; (

**d**) 4 C-break condition; (

**e**) 5 C-break condition.

**Figure 17.**Temperature prediction comparison between the lumped model and the TCM under (

**a**) 1C, (

**b**) 2C, (

**c**) 3C, (

**d**) 4C, (

**e**) 5C conditions, when the fan starts at 120 s and the temperature of cooling air is set to 10 °C.

**Figure 18.**Temperature prediction comparison between the lumped model and the TCM under (

**a**) 1C, (

**b**) 2C, (

**c**) 3C, (

**d**) 4C, (

**e**) 5C condition, when the temperature of cooling air changes from 20 °C to 10 °C at 120 s.

**Figure 19.**Results of the lumped model, the TCM, battery surface temperature and core temperature (under VTOL mission profile).

Parameters | Value |
---|---|

Dimensions | 148 mm × 26.7 mm × 98 mm |

Nominal capacity | 50 Ah |

Energy | 182.5 Wh |

Nominal voltage | 3.65 V |

Anode material | NCM |

Electrolyte material | LiPF_{6} |

Cathode material | Graphite |

Mass | 895 g |

Parameters | Value |
---|---|

Shell Material of FHP | Aluminum |

Wick Material of FHP | (Porous sintered) Aluminum particles |

Working Fluid of FHP | Acetone |

Evaporator length (each cell) | 0.026 m |

Condenser length of FHP | 0.1 m |

FHP width | 0.148 m |

FHP length | 0.44 m |

Total thickness of FHP | 0.005 m |

Thickness of shell | 0.001 m |

Thickness of wick | 0.0015 m |

Thickness of vapor channel | 0.0015 m |

Space of fin | 0.01 m |

Width of fin | 0.08 m |

Thickness of fin | 0.0005 m |

Wick porosity | 0.48 |

Module overall size | 440 mm × 150 mm × 103 mm |

Cooling method | Axial fan |

Fan size | 120 mm × 120 mm × 50 mm |

Max airflow rate | 0.12 kg/m^{3} |

Type | Symbol | Expression | |
---|---|---|---|

Conduction thermal resistance | R_{c} | ${R}_{i}=\frac{{t}_{i}}{{\lambda}_{i}{A}_{i}}$ | (5) |

R_{w} | |||

R_{s} | |||

Convection thermal resistance | R_{ec} | ${R}_{eci}=\frac{1}{{h}_{\mathrm{si}}{A}_{i}}$ | (6) |

Phase change thermal resistance | R_{pc} | ${R}_{\mathrm{pc}i}=\frac{2-\sigma}{2\sigma}\frac{{\left(2\pi {R}_{\mathrm{gas}}{T}_{\mathrm{v}i}\right)}^{0.5}{R}_{\mathrm{gas}}{T}_{\mathrm{v}i}^{2}}{{A}_{\mathrm{e}i}{p}_{\mathrm{v}i}{h}_{\mathrm{fg}i}^{2}}$ | (7) |

Vapor flow thermal resistance | R_{v} | ${R}_{\mathrm{vi}}=\frac{{R}_{\mathrm{gas}}{T}_{\mathrm{v}i}^{2}}{{p}_{\mathrm{v}i}{h}_{\mathrm{fg}i}}\frac{12{\mu}_{\mathrm{v}i}}{{t}_{\mathrm{v}}^{3}{\rho}_{\mathrm{v}i}{w}_{\mathrm{FHP}}{h}_{\mathrm{fg}i}}{l}_{\mathrm{e}i}$ | (8) |

Battery Cell | FHP Shell | FHP Wick | Sources | |
---|---|---|---|---|

^{#} Density(kg/m^{3}) | 2.519 × 10^{3} | 2.7 × 10^{3} | 1.520 × 10^{3} | [32,40] |

^{#} Thermal Capacity(J/kg·K) | 1.023 × 10^{3} | 920.9 | 1.059 × 10^{3} | [32,40] |

Thermal Conductivity (W·m^{–1}·K^{–1}) | ^{&} x axis: 1.096 | 200 | 9.965 | [32,40] |

^{&} y axis: 22.446 |

^{#}Based on mass average method: $\rho ={\displaystyle \sum _{i}{\rho}_{i}{m}_{i}}/{\displaystyle \sum _{i}{m}_{i}},{c}_{\mathrm{p}}={\displaystyle \sum _{i}{c}_{\mathrm{p}i}{m}_{i}}/{\displaystyle \sum _{i}{m}_{i}}$.

^{&}Based on linear average method: $\lambda ={\displaystyle \sum _{i}{l}_{i}{\lambda}_{i}}/{\displaystyle \sum _{i}{l}_{i}}$.

Operating Condition | 0.5 C | 1 C | 1.5 C | WLTC | |||
---|---|---|---|---|---|---|---|

Relative RSMEs | Fan off 8.14% | Fan on 8.74% | Fan off 3.10% | Fan on 1.82% | Fan off 2.93% | Fan on 1.98% | 12.38% |

TCM | Lumped Model | TCRN | |
---|---|---|---|

CPU-time | 47 ms | 5 ms | 26.81 s |

**Table 7.**Relative error (RE) and maximum absolute error (MAE) of the TCM and the lumped model under different dynamic conditions.

TCM | Lumped Model | ||||
---|---|---|---|---|---|

RE | MAE | RE | MAE | ||

Dynamic current | 1 C | 2.20% | 0.01 °C | 12.70% | 0.07 °C |

2 C | 2.20% | 0.06 °C | 12.69% | 0.28 °C | |

3 C | 2.19% | 0.12 °C | 12.67% | 0.62 °C | |

4 C | 2.18% | 0.21 °C | 12.64% | 1.04 °C | |

5 C | 2.18% | 0.31 °C | 12.61% | 1.55 °C | |

Step | 6.24% | 1.27 °C | 45.06% | 10.39 °C | |

Dynamic air velocity | 1 C | 26.12% | 0.13 °C | 141.45% | 0.84 °C |

2 C | 7.37% | 0.33 °C | 92.29% | 1.50 °C | |

3 C | 6.54% | 0.60 °C | 25.02% | 2.32 °C | |

4 C | 6.23% | 0.98 °C | 23.39% | 3.46 °C | |

5 C | 6.10% | 1.43 °C | 22.64% | 4.80 °C | |

Dynamic air temperature | 1 C | 34.24% | 0.14 °C | 175.90% | 0.83 °C |

2 C | 7.66% | 0.31 °C | 28.04% | 1.37 °C | |

3 C | 6.40% | 0.61 °C | 22.56% | 2.19 °C | |

4 C | 6.02% | 0.99 °C | 20.82% | 3.23 °C | |

5 C | 5.85% | 1.44 °C | 20.02% | 4.46 °C |

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## Share and Cite

**MDPI and ACS Style**

Li, W.; Xie, Y.; Li, W.; Wang, Y.; Dan, D.; Qian, Y.; Zhang, Y.
A Novel Quick Temperature Prediction Algorithm for Battery Thermal Management Systems Based on a Flat Heat Pipe. *Batteries* **2024**, *10*, 19.
https://doi.org/10.3390/batteries10010019

**AMA Style**

Li W, Xie Y, Li W, Wang Y, Dan D, Qian Y, Zhang Y.
A Novel Quick Temperature Prediction Algorithm for Battery Thermal Management Systems Based on a Flat Heat Pipe. *Batteries*. 2024; 10(1):19.
https://doi.org/10.3390/batteries10010019

**Chicago/Turabian Style**

Li, Weifeng, Yi Xie, Wei Li, Yueqi Wang, Dan Dan, Yuping Qian, and Yangjun Zhang.
2024. "A Novel Quick Temperature Prediction Algorithm for Battery Thermal Management Systems Based on a Flat Heat Pipe" *Batteries* 10, no. 1: 19.
https://doi.org/10.3390/batteries10010019