### 1. Introduction

Air cooling [1], liquid cooling [2], and PCM cooling [3] are extensively applied to thermal safety design for lithium-ion energy storage batteries (LFPs). They are highly effective in reducing the working temperature of LFPs. Therefore, the study of heat dissipation during operation is a significant topic [4–8].

Yuan [9] and Golubkov [10] experimentally studied the main gas composition of lithium batteries after the thermal runaway. Jin et al. [11] proposed a detection method of micro-scale Li dendrite precipitation based on H

_{2}detection, applied it to the safety warning of lithium-ion batteries and carried out experimental verification in a real storage tank. Ye et al. [8] used Fluent to simulate and analyze the temperature field variation of LFP. And The structure design of the lithium iron phosphate battery was optimized based on this model. Mei et al. [12] used the COMSOL to establish an electrochemical-thermal coupling model for an 18.5 Ah lithium-ion battery. Then the thermal behavior and temperature field distribution of lithium-ion battery was obtained. Chiew et al. [13] established an electrochemical-thermal coupling model for a 26650 cylindrical lithium-ion battery. The validity of the numerical simulation results is verified by experiments. The results demonstrated that the battery cell performs differently at a lower ambient temperature and lower discharge rate where the exothermic reactions are milder. Park [14] optimized the air-cooling structure of lithium-ion battery packs. The result shows that the cooling effect of the optimized structure is better. Nieto et al. [15] designed a new lithium-ion battery management system. The maximum difference in peak temperature between the simulation and experiment is less than 2°C. Zhao et al. [16] established thermal model of 75 18650 lithium-ion batteries. Simulation results show that increasing liquid flow can significantly reduce the temperature of the battery module, and improves the temperature uniformity in the battery module. Zhao et al. [17] studied the effect of phase change material cooling on the temperature of lithium batteries. The maximum temperature of the lithium battery was reduced by 11.22% compared with air cooling.However, previous studies have considered the heat generation rate of LFP as a constant. For example, Gang Zhao et al. [1] set the heat generation rate of LFP at a specific C-rate as a constant parameter and performed thermal management simulations for cylindrical LFP with air cooling. Similarly, Lu Ling [18] and Hu Shuntao [19] set the heat generation heat as a constant parameter in their studies on PCM-enhanced finned heat transfer thermal management and serpentine channel liquid cooling for batteries, respectively. Likewise, the studies by Xiong Huimin [20] and Ma Jing [21] also adopted this approach. This is not accurate since the heat generation rate of LFP varies with discharge time. In this study, we assume that LFP is a transient source and utilize Fluent software to simulate the temperature field variation with discharge time for a 100 Ah LFP. We investigate the heat dissipation processes of the LFP using both air cooling and air coupled with PCM cooling methods. The numerical simulation method provides insights into the temperature distribution and evolution within the battery during discharge, allowing us to evaluate the security of LFP and the effectiveness of different cooling methods.

### 2. Theoretical model and Computational method

### 2.1 Theoretical model of LFP

LFP cells are composed of a variety of stacked materials, such as cathode, diaphragm, and anode. To simplify the calculation process while maintaining generality, it is assumed that: 1) The heat generated inside the cell is uniform; 2) Neglect radiation heat transfer with the cell; 3) The electrolyte inside the LFP does not experience relative displacement, and convection heat transfer is neglected; 4) The thermal physical parameters of the LFP remain constant with variations in its operating temperature [9].

The simulation was conducted on a 100Ah LFP cell provided by China Aviation Lithium Battery Company. Technical parameters of the LFP are presented in Table 1.

Fig. 1 illustrates the simplified components of the LFP, namely the positive pole, negative pole, core, and shell, and a hexahedral mesh with a cell size of 0.002 m is utilized for the simulation. The mesh consists of 27032 grid cells and 259150 grid nodes, meeting the requirements for grid quality with a minimum orthogonal quality inspection value of 0.89163 and a maximum value of 1.

### 2.2 Thermophysical parameter

The thermophysical parameters of the LFP, including density, specific heat capacity, and thermal conductivity, are considered. The parameters of each part of LFP materials are obtained using a weighted average based on the data from Ref. [22]. The obtained results are presented in Table 2.

The composite PCM of Graphite-expanded paraffin exhibits favorable characteristics such as high latent heat, high specific heat capacity, and high thermal conductivity. The Phase-transition temperature of the PCM is 44.63°C and just within the 0–55°C range of the operating temperature of LFP [4,5]. Hence, the composite PCM of graphite-expanded paraffin is chosen as the cooling material in this study. The thermophysical parameters are presented in Table 3 [5].

### 2.3 Heat generation rate of LFP

In 1985 Bernardi et al. [23] proposed the following formula to calculate the heat generation rate,

*q*, of lithium batteries:Where

*V*is the volume of the battery;*R*represents the internal resistance of the battery,*E*and*U*represent the open-circuit voltage and operating voltage of the battery, and*T*and*I*represent the temperature and current of the battery.*dE/dT*represents the entropy heat coefficient.The

*dE/dT*is typically not constant [24], especially under low-temperature conditions [25]. To optimize the model and simplify calculations, we set this value to a reasonable constant for the following reasons: Firstly, the lithium iron phosphate energy storage battery studied in this paper typically operates in a constant temperature environment equipped with temperature control devices such as air conditioning, avoiding the effects of low temperatures. Secondly, according to Eq. 1, the entropy heat coefficient affects the electrochemical reaction heat (the second term on the right side of Eq. 1). However, compared to the main heat source during normal operation—ohmic heat (the first term on the right side of Eq. 1), the electrochemical reaction heat is relatively small and can usually be neglected [26]. Finally, based on Ref. [21,27], the range of*dE/dT*is 0.1–0.28 mV/K. We selected three scenarios: high (0.28 mV/K), low (0.1 mV/K), and medium (0.19 mV/K), and compared them with experimental results. It was found that 0.19 mV/K had the best fit with the experimental results. Through further trials, we determined that 0.148 mV/K is the most accurate value.The operating current and internal resistance of the LFP can be obtained by solving Eq. 1. Under normal working conditions, the current of the LFP can be controlled, and thus only one variable of the internal resistance

*R*needs to be calculated as a function of discharge time. We consider the relationship between SOC and the discharge time of the LFP [22]:In Eq. 2,

*I*represents the discharge current,*t*represents the discharge time, and*C**is the rated capacity of the battery. In this study,*_{n}*C**is taken as 100 Ah. Furthermore, Ye et al. [8] experimentally investigated the correlation between the internal resistance of the LFP and SOC under various discharge rates. By incorporating the experimental results and Eq. 1 and 2, we can derive the heat generation rate,*_{n}*q*, as a polynomial function of discharge time,*t*, of the LFP at various discharge rates. Table 4 presents the coefficients of the polynomials for different discharge rates.The relationship between the internal heat source of the LFP,

*q*, and the discharge time,*t*, is implemented in Fluent through a user-defined function (UDF) that utilizes the polynomial functions specified in Table 4.### 3. Results and Discussion

### 3.1 Validation of numerical simulation

In the simulation, the ambient temperature was set at 25°C, and the convective heat transfer coefficient was assumed to be 5 W/(m

^{2}·K) [28]. The variation of heat generation (*q*) with time for the core heat source was modeled as a polynomial function, as defined in Table 4. The thermophysical parameters for the core, shell, positive, and negative electrodes of the LFP, as provided in Table 2, were utilized.
Fig. 2(a) displays the numerical results of peak temperature versus discharge capacity (DC), along with corresponding experimental data [8].

As observed in Fig. 2(a), the curve representing the peak temperature of the LFP obtained through numerical simulation aligns well with the experimental results for discharge rates of 1C and 1.5C, respectively. However, upon a careful comparison between theoretical data and experimental findings, it becomes apparent that, during the initial stages of discharge, the theoretical peak temperature slightly exceeds the experimental values. This disparity can be attributed to the fact that the experimental temperature is measured at the surface of the LFP using thermocouples, while the theoretical temperature represents the internal temperature of the LFP. It takes some time in the experiment to establish a balance between the surface and internal temperatures of the LFP, resulting in the theoretical peak temperature consistently surpassing the experimental one before the equilibrium stage is reached, as depicted in Fig. 2(a). At the end of discharge, the peak temperatures for the LFP, as determined through theoretical and experimental approaches, were recorded as 40.76°C/47.1°C and 41.3°C/47.6°C, for discharge rates of 1C/1.5C, respectively. The relative errors were all below 1.3%. We also calculated the peak temperature as a function of discharge time under the discharge rates of 0.2C, 0.5C, 2C, 3C, and 5C for the LFP, and the results were shown in Fig. 2(a). From the figure it is seen that as the discharge rate increases, the increase of peak temperature with the discharge time is faster and faster. This phenomenon is in agreement with the experimental results for the LFP under the discharge rate of 1C and 1.5C [8], and also it does for a 205 Ah LFP under the discharge rate of 0.5C and 1C [21].

To further validate the universality of the proposed numerical model, we also conducted numerical simulations of the variation of peak temperature vs. discharge time for a 205 Ah LFP. The numerical simulation results, together with experimental data are shown in Fig. 2(b). From the figure, the agreement between the theoretical and the experimental is also found. The findings further confirm the effectiveness and applicability of the proposed numerical model.

### 3.2 Temperature field of LFP

We conducted temperature field analysis for the LFP under transient conditions. Fig. 3 presents the temperature distribution as a cloud diagram for the LFP discharged at 1C. The temperature fields of the entire LFP are shown in Fig. 3(a) and the internal distribution of the LFP is in Fig. 3(b). As depicted in Fig. 3, at the end of the discharge, the peak temperature of 40.757°C is observed at the center of the LFP, while the lowest temperature of 38.151°C is recorded at the eight outer corners of the LFP. The temperature gradually decreases from the center towards the outer regions of the LFP. It is worth noting that the thermal conductivity in the Y direction is the lowest among the three directions, as presented in Table 2. This indicates that heat transfer along the thickness direction (Y direction) is slowest compared to the X direction (length) and the Z direction (height). At the end of the 1C discharge, the temperature difference between the maximum and minimum temperatures of the LFP was found to be 2.606°C.

Fig. 4 illustrates the temperature curves of the maximum and minimum temperatures of LFP at different discharge rates. Analysis of Fig. 4 and Table 4 reveals that, when the discharge capacity is below 80 Ah, the temperature increase exhibits an approximately linear relationship with the DC. This behavior can be attributed to a relatively constant heat generation rate. However, when the DC exceeds 80 Ah, the temperature increase becomes nonlinear. This non-linearity arises from significant changes in the heat generation rate during discharge time.

Fig. 4(a) presents the maximum temperature of the LFP at the end of discharge for various discharge rates of 0.2C, 0.5C, 1C, C, 3C, and 5C. The corresponding temperatures are recorded as 26.01°C, 30.81°C, 40.757°C, 62.062°C, 83.461°C, and 126.615°C, respectively. Under natural convection conditions, when the discharge rate of the LFP is 1C or below, the maximum temperature remains below 41°C. This temperature range falls within the normal operating temperature range of the LFP, as indicated in Table 1, and therefore does not necessitate any additional cooling treatment. However, when the discharge rate of the LFP exceeds 1C, the maximum temperature of the LFP exceeds 55°C at the end of discharge. This temperature exceeds the normal operating temperature range of 55°C for the LFP. Similar results are also obtained for the variation of minimum temperatures of the LFP with DC at different discharge rates (see Fig. 4(b)). The reasons for this are as follows: When LFP are ch/discharged at high C-rates, the operating current of LFP increases significantly. For example, under a 5C discharge condition, the battery current reaches 500 A, which is five times the current of a 1C discharge. According to Eq. 1, the larger the current, the greater the heat generation, leading to a significant increase in the internal temperature of LFP. Therefore, high C-rate charging and discharging result in a noticeable increase in the temperature of LFP. To ensure the safe operation of the LFP at discharge rates higher than 1C, heat dissipation treatments such as air cooling or air coupled with PCM cooling are required.

### 3.3 Air cooling

Air cooling is commonly used as a cooling method for lithium batteries, where air is utilized as the cooling medium. When air flows over the surfaces of a lithium battery, it absorbs the heat generated by the battery, primarily through convection heat transfer. Enhancing the convective heat transfer coefficient of the LFP surface can lead to improving heat dissipation for the battery.

The convective heat transfer coefficient of air can vary depending on the conditions. The natural convection coefficient ranges from 0 to 20 W/(m

^{2}·K), while the forced convection coefficient ranges from 20 to 100 W/(m^{2}·K) [28]. For simulating the air cooling of the LFP at different airflow rates, different convective heat transfer coefficients were selected as 5, 20, 50, and 100 W/(m^{2}·K) respectively. Fig. 5 illustrates the impact of these coefficients on the maximum temperature of the LFP for different LFP discharge rates.As depicted in Fig. 5, when LFP DC is below 20 Ah, 40 Ah, 50 Ah, and 70 Ah for discharge rates of 1C, 2C, 3C, and 5C respectively, the convective heat transfer coefficient has almost little influence on the maximum temperature of the LFP. This is because the internal heat of the LFP has not been fully conducted to its surface, and thus convective heat transfer does not play a significant role in heat dissipation. However, when LFP DC exceeds 70 Ah, particularly for the batteries with discharge rates of 1C, 2C, and 3C, the effect of convective heat transfer on the peak temperature becomes evident. In this scenario, the heat generation efficiency within the LFP remains relatively constant, and the conducted heat is eventually transferred to its surface. Thus, convective heat transfer becomes crucial for effective heat dissipation. From Fig. 5(a–d) it is also demonstrated that for the LFP with a discharge rate of 1C and 2C if the air heat transfer coefficient is not smaller than 5 and 50 W/(m

^{2}·K), air cooling can lower the maximum temperature of the LFP to the smaller temperature than 55°C. The temperature falls within the normal operating temperature range of the LFP. However, when the discharge rate of LFP is larger than 2C, even air heat transfer coefficient is 100 W/(m^{2}·K), air cooling cannot lower the maximum temperature of the LFP to 55°C at the end of discharge time, as depicted in Fig. 5(c,d). In this case, a more efficient cooling method is required.### 3.4 Air-PCM coupling cooling

PCM is a passive cooling material that stores energy through phase transformation. When the temperature reaches the phase transition temperature the phase transition occurs, and the PCM absorbs most of the heat through latent heat. In this study, the solidification and melting model in Fluent was used to simulate the cooling of LFP by air-PCM coupling.

In general, the thickness of the PCM significantly affects the heat efficiency conducted by the LFP [29]. It should be stressed that on the one hand when the PCM thickness is too thin, it fails to fully exploit the heat dissipation advantages of the PCM material, resulting in insufficient heat dissipation; on the other hand, when the PCM thickness is too thick, although it increases the latent heat, the rapid heat generation rate of the battery at high rates limits the PCM’s ability to complete phase change heat dissipation within a short period. Additionally, the obstructive nature of the PCM inhibits effective convective heat dissipation. Therefore, there exists a critical thickness at which both convective and phase change heat dissipation can be optimized. To investigate this, we conducted a study on the critical thickness of PCM. Fig. 6(a) shows the effect of PCM thicknesses on the peak temperature of the LFP. From the figure, it can be seen that under the discharge rate of 3C and 5C, the best effective thicknesses of the PCM are all to be 8 cm. In this thickness, the LFP possesses the best effect to lower peak temperature.

The effects of cooling way on the peak temperature of the LFP are shown in Fig. 6(b–d). The LFP coated by PCM can increase the convective heat transferring area and reduce the temperature of the LFP. From Fig. 6(b–d) it is seen that in these cooling ways, the best way is the air-PCM coupling cooling, the middling the air cooling, and the worst the natural cooling. It is also seen that the cooling of air-PCM coupling plays a key role in enforcing the peak temperature of the LFP below 50°C, 70°C, and 70°C, where the LFP with a discharge rate of 2C, 3C, and 5C at the DC of 100 Ah, 100 Ah, 70 Ah, respectively. The results indicate that the air-PCM coupling cooling method reduces the peak temperature of the LFP by 12.04°C, 15.213°C, and 18.554°C for discharge rates of 2C, 3C, and 5C, respectively, compared to natural cooling. It is worth noting that, for the LFP with a discharge rate of 5C and a DC larger than 70Ah, even with coupled cooling, the peak temperature of LFP exceeds 75°C, eventually reaching 108.06°C. Therefore, to prevent high temperatures and ensure safe operation, the DC should not exceed 80 Ah and 40 Ah for discharge rates of 3C and 5C, respectively. This implies that the discharge time should not exceed 960 s and 288 s for these respective discharge rates.

### 4. Conclusions and Prospect

### 4.1 Conclusions

Using experimental data, a transient heat source for LFP was established, and Fluent software was employed to simulate the impact of air cooling and air coupled with PCM cooling on the temperature distribution of the LFP. Firstly, the validity of the transient heat source model is verified. Subsequently, it was observed that the highest temperature within the LFP occurred at its center, while the lowest temperature was recorded at the corners. Air cooling is only effective for the LFP with a discharge rate of 2C or lower. The results indicated that by employing an 8 cm-thick PCM layer, the maximum operating temperature of LFP could be decreased by 12.04°C, 15.213°C, and 18.55°C under discharge rates of 2C, 3C, and 5C, respectively, in comparison to natural cooling. Furthermore, for the LFP with discharge rates of 3C and 5C, it is recommended to limit the discharge capacity to 80 Ah and 40 Ah, respectively, which corresponds to a maximum discharge time of 960 s and 288 s.

### 4.2 Prospect

The following are potential directions for further research in this study:

(1) By simulating the voltage profile of the lithium battery, obtaining the power loss, and coupling it with the heat transfer model, we can calculate the heat generation power of the lithium battery.

(2) Based on the transient source model, complete the thermal behavior simulation of LFP at the pack level and cabinet level.