# Arrhenius Equation-Based Cell-Health Assessment: Application to Thermal Energy Management Design of a HEV NiMH Battery Pack

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Effect of Temperature Gradient on Cell Degradation Variation

#### 2.1. Model Structure Based on Arrhenius Equation

^{.}K)]. ΔE and T are activation energy in the unit of J/mol and absolute temperature in the unit of K, respectively [21,22]. Here, we view Equation (1) as an empirical formula such that ⋀ and λ = ΔE/R are interpreted as two unknown parameters to be calibrated. Integrating both sides of Equation (1) over the cell lifespan yields:

_{r}is the capacity reduction threshold indicative of the cell failure, and n

_{c}is the cycle life. Assign two different temperatures, T

_{1}and T

_{2}(T

_{1}> T

_{2}), to Equation (2) to constitute:

_{c}is the lifetime deviation. It is evident that Equation (3) is a quantitative description of the effect of the temperature gradient on cell life.

#### 2.2. Model Parameter Calibration

#### 2.2.1. Linear Identifiable Form

**ϕ**is the regressor, and

**θ**is the parameters to be identified. For electrified vehicle applications, C

_{r}is often specified as 20%. In order to extract the unknown parameters, the cell cycle life n

_{c}under different temperatures should be determined or estimated.

#### 2.2.2. HEV NiMH Cell Aging Test

_{d}= C

_{r}= 20%, the solutions are shown in Table 2.

**Table 1.**Quadratic functions depicting the relationship between capacity reduction and the number of aging cycles.

Temperature/°C | Quadratic function/% |
---|---|

25 | C_{d} = 3.2 × 10^{−5} n^{2} − 9.8 × 10^{−4} n + 0.21 |

40 | C_{d} = 2.6 × 10^{−5} n^{2} + 6 × 10^{−3} n + 0.15 |

50 | C_{d} = 2.3 × 10^{−5} n^{2} + 9 × 10^{−3} n + 0.13 |

Temperature/°C | 25 | 40 | 50 |
---|---|---|---|

Cycle life | 801.87 | 765.96 | 754.19 |

#### 2.2.3. Least-Squares Algorithm Based Parameterization

_{1}, T

_{2}and T

_{3}are the three operating temperatures. After obtaining , we can determine the model parameters:

^{−5}. Note that thanks to the nonlinear Equation (8), the standard error of ⋀ is derived from Monte Carlo simulation, based on statistics (mean and standard deviation) of .It is clear from error analysis that the fitting equation has good accuracy.

Result | ⋀ | λ |
---|---|---|

Estimate | −5.61 × 10^{−4} | 240.74 |

Standard error (1 Sigma) | 6.20 × 10^{−5} | 34.29 |

#### 2.3. Assessment Criterion

## 3. Feasibility Assessment of Primitive NiMH Battery Pack Design with its Cooling System

#### 3.1. Experimental Analysis

**Figure 3.**Configuration of the primitive HEV NiMH battery pack: 36 cells (3 rows and 12 columns) in the first (top) layer; 42 cells (3 rows and 14 columns) in each of the other layers.

Thickness | 23.5 mm |

Width | 60 mm |

Height | 84 mm |

Weight | 180 g |

Nominal Voltage | 1.2 V |

Nominal Capacity | 6.5 Ah |

Power Density | ≥ 1400 W/kg |

Energy Density | ≥ 44 Wh/kg |

Ambient Temperature | −40 ~ 80 °C |

Working Temperature | −30 ~ 60 °C |

Usable SOC range | 30% ~ 80% |

#### 3.2. Analysis of the Shortcomings of the Preliminary Design

Layer No. | Flow/ Kg/s | Flow percent |
---|---|---|

1 | 0.0130 | 29.23% |

2 | 0.0170 | 38.11% |

3 | 0.0145 | 32.66% |

## 4. Improved Thermal Energy Management

#### 4.1. Enhanced Structure Design

^{3}/h. The air enters the pack from the intake duct under the suction role of the centrifugal fun, and then the convective heat transfer occurring at air ducts on the cell body and the internal gaps of the pack structure is able to cool the cells. After that, the air continues to dissipate the heat from the DC/DC converter and the controller and is finally extracted by the fan.

Width | 444 mm |

Length | 542 mm |

Height | 198 mm |

Weight | 35 kg |

Nominal Capacity | 6.5 Ah |

Nominal Voltage | 144 V |

Number of cells | 120 |

#### 4.2. Simulation Validation

^{3}/h, and its temperature was assumed to be 35 °C. Since the average heating power of the pack was measured to be 494 W in the practical HEV filed test at 40 °C, the heat flux of 718.5 W/m

^{3}was specified for the battery pack system in the simulation. Please refer to [27] for more details on the boundary conditions applied. The simulation result of the first row of cells is shown in Figure 15. We can see similar temperature change of air in the ducts and gaps for eight cells, indicating relatively uniform heat dissipation. The maximum temperature of cell surface is 49.3 °C; the minimum is 45.5 °C; the maximum deviation is therefore 3.8 °C. It meets the performance index of thermal energy control depicted in Section 2. The simulation results of the other rows are similar.

#### 4.3. Experimental Validation

## 5. Conclusions

## Acknowledgments

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**MDPI and ACS Style**

Yang, Y.; Hu, X.; Qing, D.; Chen, F.
Arrhenius Equation-Based Cell-Health Assessment: Application to Thermal Energy Management Design of a HEV NiMH Battery Pack. *Energies* **2013**, *6*, 2709-2725.
https://doi.org/10.3390/en6052709

**AMA Style**

Yang Y, Hu X, Qing D, Chen F.
Arrhenius Equation-Based Cell-Health Assessment: Application to Thermal Energy Management Design of a HEV NiMH Battery Pack. *Energies*. 2013; 6(5):2709-2725.
https://doi.org/10.3390/en6052709

**Chicago/Turabian Style**

Yang, Yalian, Xiaosong Hu, Datong Qing, and Fangyuan Chen.
2013. "Arrhenius Equation-Based Cell-Health Assessment: Application to Thermal Energy Management Design of a HEV NiMH Battery Pack" *Energies* 6, no. 5: 2709-2725.
https://doi.org/10.3390/en6052709