Cooling characteristics of a lithium‐ion battery module based on flat aluminum heat pipe by considering thermal radiation

Aiming at the thermal safety and inconsistency caused by the high temperature of lithium‐ion (Li‐ion) battery, a cooling structure embedded with a flat aluminum heat pipe (FAHP) for a Li‐ion battery module is proposed. The three‐dimensional thermal model of the FAHP module is established by considering regionalized thermal radiation. The thermal characteristics of the module are compared with four progressive cooling schemes, and the temperature performance affected by different convection (hconv) and radiation (hrad) heat transfer coefficients are analyzed. Results show that, the thermal model with the thermal radiation is more precise than that without the thermal radiation under the natural convection. Adding FAHPs can effectively reduce the maximum temperature (Tmax) and the maximum temperature difference (ΔTmax,pack) of the FAHP module. Especially adding FAHPs with fins, even at 3C discharging, the average cooling performance can be improved by 33.3%, 25.0%, and 14.4% than that of natural convection, aluminum plates sandwiched between cells and FAHPs with no fins, respectively. Meanwhile, the decrease in rates of Tmax and ΔTmax,pack are gradually increasing with the increasing of hrad, but decreased with the increasing of hconv. When 5 W·m−2·K−1 ≤ hconv ≤ 35 W·m−2·K−1, the average ratio of radiation heat dissipation to total heat dissipation (η) is 35.7%, but when hconv > 55 W·m−2·K−1, η is less than 1.5%.


| INTRODUCTION
With the rapid growth of motor vehicles, the energy consumption and the exhaust pollution caused by traditional vehicles with the internal combustion engines as the main power source are seriously increasing. As the leader of traditional automobile industry turning to new energy market, electric vehicle (EV) plays an important role in the field of new energy vehicles. Lithium-ion (Liion) battery, as one of the core components of EVs, has become the preferred power source because of its advantages of high specific energy, no memory effect, no pollution, low maintenance, and long cycle life. [1][2][3] Under the service conditions, the performance of Li-ion battery is very sensitive to the change of temperature. At high temperature (usually higher than 45°C), the capacity can be reduced due to the battery aging, which seriously affect its safety and service life. 4,5 At low temperature (usually below 0°C), the charge/discharge capacity of the battery is decreased due to the increasing of the electrode polarization and the internal resistance. 6 Thus, the normal working temperature range of Li-ion battery is narrow. Its optimal working temperature range is 20-45°C, and the maximum temperature difference should be controlled within 5°C. 7,8 Therefore, a reasonable and efficient battery thermal management system (BTMS) is crucial, so as to ensure that the battery can work in a suitable temperature environment and meet the requirements of safety, reliability and environmental adaptability of EVs.
At present, the research about BTMS is mainly focused on the selection of cooling mediums and the optimization of heat dissipation structure. 9 The cooling mediums of BTMS include air, liquid, phase change material (PCM) and heat pipe. 10,11 Besides, other solutions are also effective for BTMS, such as nanofluid cooling, 12 thermoelectric air-cooling, 13 thermoelectric ferrofluid cooling 14 and pulsating water/nanofluid flowing in thermoelectric cooling. 15 Air cooling is the most traditional heat dissipation method, which is widely used in BTMS of EVs, such as Zhidou, Toyota Prius, Nissan Leaf and BYD E6. Air cooling is simple in structure, low in cost, easy to maintain and excellent in stability. 16 However, the low thermal conductivity of air limits its cooling performance, which makes it difficult to ensure the thermal equilibrium of the battery module/pack. And the contradiction between the nonuniform temperature field and the system energy loss will be unavoidablely caused by fans or air pumps even using forced convection. 17,18 Compared with air, liquid has higher heat transfer coefficient. The liquid cooling plate can be inserted between battery cells, or the cells can be immersed in the insulating liquid medium to provide higher cooling capacity. 19 At present, there are some market applications of EVs with liquid cooling, such as JAC iEV7s, BYD Yuan EV360, Southeast DX3 EV400, Beiqi New Energy EX360, and so on. Because of the complex structure of the liquid cooling system, it is difficult to be compatible with lightweight of EVs. Furthermore, it has high requirements for the tightness of the cooling system and needs a hydraulic pump which will lead to additional energy consumption. 20,21 PCM cooling is to use PCM to absorb a large amount of latent heat in the process of solid-liquid transformation to dissipate heat for battery. 22 It has advantages of good heat dissipation performance, no extra energy consumed from the battery and use for both heat cooling and heating. However, the low thermal conductivity and high thermal instability of PCM seriously restricted its application on the service conditions for EVs. 23,24 Heat pipe cooling can rapidly transfer heat from the heat source due to its high-efficiency phase change heat transfer characteristic. Heat pipes possess the advantages of high thermal conductivity (more than 10-100 times than that of ordinary metals), compact structure, flexible geometry, bidirectional heat transfer and convenient maintenance, which has been garnering more attention in BTMSs. 25,26 Zhao et al. 27 experimentally studied a BTMS with pulsating heat pipes. The battery can be better controlled at low temperature and stable state in the proposed system compared with that in the PCM system. Liu et al. 28 investigated the heat dissipation of a Li-ion battery module (3.2V50Ah) with microcopper heat pipes. The comparison of the heat pipe BTMS with natural and forced convection at different discharge rates by the simulation and test validation, showed that heat pipe had obvious heat dissipation advantages even discharging at high rates. Gan et al. 29 explored a thermal equivalent circuit model of heat pipes for thermal management of a battery module with cylindrical cells. Compared with natural convection, the temperature of the battery module with heat pipe cooling could be reduced by 14°C at 5C discharge rate. Behi et al. 30 demonstrated that the flat heat pipe can improve the heat dissipation performance of batteries by 29.1% at 8C discharge rate through simulation and experiment. Compared with the natural air, the forced liquid and heat pipe used in the battery module could reduce the battery temperature by 29.9% and 32.6%. Yuan et al. 31 adopted the heat pipe-cooper plate structure consisting of 4 L-shaped heat pipes and two copper plates, and coupled with liquid cooling to dissipate a 3.2V50Ah battery cell. The proposed structure could maintain the maximum temperature and the maximum temperature difference at 38.6°C and 1°C, respectively during three charging/ discharging cycles even when the ambient temperature is 35°C. Hamidreza et al. 32 designed the cooling structure of Li-ion batteries sandwiched with copper heat pipes, the temperature of the battery was 37.8°C at 8C discharge rate, which was 33.4% lower than that of forced convection. Li et al. 33 analyzed the thermal characteristics of a LiFePO 4 battery pack at different discharge rates by using copper heat pipe. A three-dimensional (3D) numerical model was established and verified for a 50 Ah battery module. Compared with the liquid cooling plate, the maximum temperature and the temperature difference of the heat pipe BTMS could be reduced by 6.95% and 11.08%. Behi et al. 34 put forward a sandwiched configuration of the heat pipes cooling system (SHCS) for Li-ion battery at 8C high discharge rate. Through the test validation, the battery temperature could be reduced by 13.7%, 31.6%, and 33.4%, respectively, by the SHCS with natural convection, forced convection and forced convection without heat pipes. Xie et al. 35 proposed a BTMS based on the embedded heat pipe system, which can reduce the maximum temperature of battery by 19.93°C compared with natural convection at the condition of 100 A discharge current.
Among the above research, the traditional copper heat pipe is mainly used as the heat dissipation element. Due to its high thermal conductivity, heat pipe BTMS has become a preferred candidate for efficient heat dissipation. But the expensive manufacturing cost can not be beneficial for BTMS. Meanwhile, considering the limited layout space and the lightweight of EVs, aluminum heat pipe (AHP) has become a potential advantage for BTMS due to its good heat transfer performance and low density. Research 36-39 on the feasibility of using AHP reveals that it will be an effective method to reduce the cost and weight of heat pipes. Wang et al. 40 experimentally investigated a compact cooling system with combination of AHP and PCM for thermal management of cylindrical Li-ion battery packs, and found that AHPs had advantages in reducing the maximum temperature, which is 47.7°C at 2C discharge rate. Furthermore, many investigations show that the discharge rate is the main factor affecting the temperature rise of Li-ion batteries. When establishing the 3D thermal simulation model, the influence of thermal radiation on the temperature field and thermal distribution of the battery is often ignored. However, ignoring the influence of thermal radiation on the battery is not feasible under any conditions. Chen et al. 41 put forward an accurate thermal model of the battery. The simulation results show that thermal radiation accounts for 43%-63% of the total heat dissipation when adopting passive cooling. Liu et al. 42 conducted transient thermal simulation research on a cylindrical Li-ion power battery. The effects of different discharge rate, ambient temperature and convective heat transfer coefficient on thermal radiation are analyzed. However, the convection and radiation boundaries are simply superimposed when establishing the 3D thermal model. For thermal analysis of BTMS, accurate thermal model is crucial. However, there also exists a challenge to establish the thermal model more in line with the actual battery working conditions.
Aiming at the problem of efficient heat dissipation and lightweight of BTMS, taking a Li-ion battery module consisting of seven prismatic cells (3.2V15Ah) in series as the research object, the heat dissipation structure of the battery module based on flat aluminum heat pipe (FAHP) is proposed. Due to the thermal radiation effect caused by the nonuniformity of the battery surface temperature, the 3D thermal model of the FAHP module is established by considering regionalized thermal radiation. The thermal characteristics of the module are compared with or without thermal radiation. On the basis of model verification, the temperature performance of the battery module is analyzed with four progressive cooling schemes. Furthermore, the interaction influence of different radiation and convection heat transfer coefficients on the FAHP module is investigated. The research results can provide some reference for the application of AHPs in BTMS for EVs.

| DESCRIPTION OF BTMS BASED ON FAHP COOLING
Considering the effective heat dissipation and thermal balance of Li-ion battery at high temperature, and taking into account the lightweight requirements of EVs, the heat dissipation structure of a battery module consisting of seven prismatic Li-ion cells (3.2V15Ah) in series based on FAHPs is designed. The FAHP battery cooling system is shown in Figure 1. Each cell of the module is numbered from Cell 1 to Cell 7 along the z direction. The aluminum plate is sandwiched between every two adjacent battery cells in the module. The two surfaces of each aluminum plate in contact with cells are respectively provided with a fillister along the positive direction of the z axis and another one along the negative direction of the z axis (i.e., each aluminum plate has two fillisters). Each FAHP is inserted into each fillister. FAHPs are arranged in two layers along the x direction. One surface of the evaporator section of FAHP (l eva ) along the y direction is contacted with the cell through the thermal conductive silicone, the opposite surface is in contact with the fillister. The condenser section of FAHP (l con ) extends to the outside of the module and is contacted with the air directly. Six FAHPs and nine aluminum fins in each layer along the z direction are combined into a FAHP group. The initial structure parameters of the FAHP battery cooling system are shown in Table 1. Among them, FAHP is manufactured by inorganic high-efficiency flat porous heat pipe technology. That is, the upper and lower substrates are respectively processed first, the bonding of the upper and lower substrates is realized by metal welding, the capillary structure is manufactured by groove method, and then the working medium is injected into the plate-shaped cavity for sealing and molding. The working medium in the heat pipe is an inorganic liquid mixture mainly composed of acetone, and its service temperature ranges from −120°C to 200°C. It has the advantages of quick start-up, high thermal conductivity, good temperature uniformity, and long service life.

| Grid configuration
For the FAHP battery cooling system (shown in Figure 1), because the fins are thin and the length of the heat pipe is long, the hexahedral grid is adopted in this grid division. Considering the running speed of the computer, two boundary layers are set in this grid model. The grid model of the FAHP battery cooling system shown in Figure 2 can be obtained by spatially dispersing the computational domain in Figure 1.

| Thermal model of the battery cell
Due to the complex electrochemical reactions of Liion cells involved in the process of charging/ discharging, to simplify the model, the assumptions are as follows: (1) Each component material of the cell is uniform, and the density, thermal conductivity, specific heat capacity, and current density are all regarded as constants. (2) The influence of thermal convection and radiation inside the cell is ignored.
A 3D thermal model of the Li-ion cell in a Cartesian coordinate system has been developed based on the heat generation-transfer mechanism of the cell. The equation is as follows 28 : where ρ is the density; c is the specific heat capacity; λ x , λ y , λ z represents the thermal conductivity in the x, y, and z directions, respectively; T is the temperature; q is the heat generation rate. Subscript cell stands for the battery cell.

| Thermo-physical parameters
Based on the geometrical characteristics and thermophysical properties of the cell multilayer structure, the thermal parameters can be calculated as follows 28 : where m is the mass; l, b, and δ represent the length, width, and height respectively; Subscript i represents the different materials of the cell, including positive electrode, negative electrode, separator sheet and shell. The thermo-physical parameters of the mentioned battery cell in this paper can be obtained from Equation (2), that is ρ cell = 1971 kg·m −3 , c cell = 910 J·kg −1 ·K −1 , λ x = λ y = 19.32 W·m −1 ·K −1 , and λ z = 1.44 W·m −1 ·K −1 .

| Initial conditions
Considering the temperature distribution of the cell and FAHPs is a function of time and space, the initial boundary can be expressed as: where T 0 is the initial temperature of the cells and FAHPs;  T is the ambient temperature.

| Boundary conditions of cells
Thermal conduction There exists thermal conduction between the cells and the aluminum plates due to their temperature difference T A B L E 1 Initial structure parameters of the FAHP module (unit: mm).

Size Value
Thermal convection There exists convective heat transfer between the cell and the external environment due to their temperature difference. Thermal convection can be expressed as follows: For the laminar flow between the prismatic cell and air 43 : For the turbulence flow between the prismatic cell and air 43 : where h conv is the convective heat transfer coefficient; Re is the Reynolds number; Pr is the Prandtl number; n represents the x, y, or z direction; l n is the characteristic length of heat transfer along the n direction.

Regionalized thermal radiation
Due to the nonuniformity of the surface temperature distribution of the cells in the battery module, this results in the regionalization of the external radiation energy on the cell surface. Each cell is evenly divided into n regions from the maximum temperature T max,cell to the minimum temperature T min,cell . T cell(i) is set as the average temperature of each region. The radiation heat transfer coefficients of each region (h rad(i) ) and the radiation heat transfer coefficient of the cell (h rad ) to the external environment are expressed as follows: where where C 0 is the blackbody radiation coefficient, the value is 5.67 W·m −2 ·K −4 ; A cell is the radiation surface area of the cell; ε cell is the surface emissivity of the cell, the value is 0.95.

| Boundary conditions between aluminum plates and FAHPs
Because the cell is in direct contact with the aluminum plate and the FAHP respectively, there exist thermal conduction boundaries between the cell and the aluminum plate, and between the cell and the FAHP, which can be described as follows: where subscript plate and hp represent the aluminum plate and the FAHP, respectively; superscript a and h represent the contact surface between the cell and the aluminum plate, and the surface between the cell and the FAHP, respectively; the equivalent thermal conductivity of the FAHP (λ hp ) is adopted as its thermal conductivity during simulation, and the calculation formulas are as follows 35 : where R is thermal resistance; subscript eva, wall, wick, con, and p represent the evaporator, wall, wick, condenser and powder of the FAHP, respectively; Q is the transferred heat of each powder; N is the number of powder; superscript e, v and c represent evaporator (powder), vapor and condenser (powder), respectively; superscript V and H represent vertical and horizontal of the FAHP, respectively.

| Boundary conditions of FAHPs
The heat transfer of the fins fixed on the condenser section of the FAHP is mainly determined by the convective heat transfer between the fins and the surrounding air. The convective heat transfer coefficient between the fin and the surrounding air (h con ) is related to the geometry of the fin, which can be described as 28 : where s fin is the fin spacing; h fin is the fin height; δ fin is the fin thickness; λ f is the fluid thermal conductivity.

| Numerical procedure
For the computational domain, the interfaces between the battery, aluminum plate, fins, and air are all no-slip walls. The solver used in the numerical simulation is a double-precision pressure-based separation solver. Among them, the pressure-velocity coupling scheme is SIMPLE, and the discrete format of pressure is Standard format. The gradient scheme is least squares cell based. The transient formulation is first order implicit. And the energy and momentum are in the first-order upwind scheme. When the residual error of continuity equation is lower than 1e−6 and the residual error of energy equation is lower than 1e−8, the calculation is considered to be convergent.

| Heat generation rate of the cell
According to the heat generation rate model of the battery proposed by Bernardi, 44 the heat generation rate model is as follows: where T and SOC is obtained when the ambient temperature (  T ) is set at the range of −20 to 50°C, as shown in Figure 3. From Figure 3, the curves of E ocv varying with SOC at different temperatures are close, the values are almost coincident especially when 0.2 < SOC < 0.9. When SOC < 0.2, E ocv is decreased with the increasing of  T . But when SOC > 0.9, E ocv is increased with the increasing of  T . While, E ocv is 3.36 V when  T = −20°C, E ocv is 3.45 V when  T = 50°C. Thus，the change of E ocv does not exceed ±0.32 mV if  T changes by 1°C. The internal resistance (R t ) of the cell at discharging can be obtained by the hybrid pules power characteristic (HPPC) test. When the ambient temperature (  T ) is set at 25°C, an HPPC test is carried out by every 10% decrease from SOC = 1.0 to SOC = 0.0. The test R t varying with SOC is obtained, and can be described as follows: The variation of R t and q cell of the cell varying with SOC are shown in Figure 4. It can be seen from Figure 4, R t is gradually increased with the discharging time. Especially at the end of discharging, the increasing trend is more obvious. Therefore, q cell is increased with the increase of discharging time and discharge rate. Especially when SOC < 0.3, it is increased obviously. q cell can reach 210152.17 W·m −3 at the end of 3C discharging, which is 58.97% higher than that at the beginning of discharging.

| EXPERIMENTAL SET-UP
The battery module based on FAHP cooling consisting of seven cells (3.2V15Ah) in series is taken as the test object. The test module has the same structure as the initial module described in Figure 1. The test rig is shown in Figure 5. A high-performance battery test system (CE-6002n-30V100A-H) with a maximum range of 30V100A and an accuracy of 0.1%RD ± 0.1%FS is used to charge/ discharge battery at different rates. A constant temperature and humidity test chamber (SC-80-CB-2) ranged from −20°C to 150°C with an accuracy of ±0.2°C is used to adjust the actual working (environmental) temperature during the experiment. A computer is used to control the charging/discharging program of the battery and collect data such as voltage, current, capacity, and temperature of the battery. 14K-type thermocouples ranged from −200°C to 350°C with an accuracy of ±1.5°C are used to obtain the surface temperature of each measuring point on the cells and the FAHPs. The tests of the cell and the module are shown in Figure 5.
For the test of the cell, two thermocouples are arranged to test the surface temperature at the center and the temperature at the edge of the cell respectively. Also, the cell can be equally divided into n regions. A K-type thermocouple is arranged on the center of each region to test the temperature at different discharge rates. Considering the symmetry of the temperature field distribution of the battery module consisting of seven cells in series, 14 thermocouples are arranged on the surface of cells 1-4 and the corresponding FAHPs, as shown in Figure 6. Among them, T b1 -T b8 are used to test the temperature of the module; T b1 -T b2 , T b3 -T b4 , T b5 -T b6 , and T b7 -T b8 are used to test the temperature uniformity of the Cells 1-4, respectively; T p1 -T p2 are used to test the temperature of the evaporator section of the FAHPs. By comparing and analyzing the measured temperatures, the maximum temperature and temperature difference of the cells and the module can be obtained.
The module was first charged at a constant current rate of 0.5C to SOC = 1.0. The charging cut-off voltage was 3.65 ± 0.03 V. Stay for more than 30 min. Then the module was discharged at constant current rate of 3C to SOC = 0.0. The discharging cut-off voltage was 2.5 V. The temperature value of each measuring point of the FAHP module at different discharge rates was recorded by the data acquisition unit. For the cooling performance test of the FAHP module, each condition was tested for three times. It could be found that there is good repeatability for each test. For all the tests, the temperature deviation measured at the same location of the FAHP module was lower than 1.0°C. Hence, the mean measured temperature was reported in this paper.

| Model verification
From the test of the cell at 3C discharge rate, for the natural convection, the experimental maximum temperature (T max,cell ) (i.e., the surface temperature at the center) is 51.31°C, the experimental minimum temperature (T min,cell ) (i.e., the surface temperature at the edge) is 46.13°C, the temperature difference is 5.18°C. Then, the cell is equally divided into nine regions. According to the test temperature of nine regions and the Equation (8), the radiation heat transfer coefficient of each region (h rad(i) ) on is shown in Table 2.
Based on the 3D thermal model and thermal boundary conditions of the FAHP module, According to Equation (9), the radiation heat transfer coefficient of the cell (h rad ) is calculated for the 3D simulation, which is 7.58 W·m −2 ·K −1 . When h conv = 5.0 W·m −2 ·K −1 and  T = 25°C, the temperature performance of the FAHP module with and without thermal radiation is simulated. When at 3C discharge rate, the temperature contour of the FAHP module is shown in Figure 7. When  considering thermal radiation, the central section (x-y plane) temperature contour of cells 1-4 is shown in Figure 8. It can be seen from Figures 7 and 8, the temperature field distribution of the FAHP module is consistent, which presents symmetrical, whether or not the thermal radiation is considered. Among the module, the temperature on the edge of cell 4 (T b8 ) is the minimum. The temperature of the cell gradually rises from the center of the module to the outside. The outer surface temperature of cell 1 (T b1 ) and cell 7 is the maximum. This is because only one surface of cell 1 (or cell 7) is dissipated by the FAHP, but there are two surfaces of other cells in contact with FAHPs. Although there is heat accumulation at the center of cell 4 due to the increasing heat generation of the cell, the heat dissipation is great there because of the large temperature difference between the evaporator and condenser section of the FAHP. The temperature near the evaporator section is higher than that near the condenser section due to the heat transfer of the FAHP. Without considering the thermal radiation, the maximum temperature of the module (T max ) can reach 48.20°C and the maximum temperature difference of the module (ΔT max,pack ) is 5.05°C. But considering the thermal radiation, T max is only 44.68°C and ΔT max,pack is 4.51°C.
Compared with the test under the same conditions, the temperature rising of the simulation and test is shown in Figure 9. As can be seen from Figure 9, the changing trends of T b1 , T b8 , T p1 , and T p6 are consistent. At the end of discharging, T max (T b1 ) is 44.25°C. The T A B L E 2 Temperature and radiation heat transfer coefficient of the cell on each region at 3C discharge rate. | 1867 difference between the simulated and experimental temperature of the module without considering thermal radiation is assumed as ΔT 1 , the difference is assumed as ΔT 2 when considering thermal radiation. For T b1 , ΔT max = max |ΔT 1 − ΔT 2 | appears at 900 s, which is 2.5°C. For T b8 , ΔT max appears at 750 s, which is 2.6°C. For T p1 , ΔT max appears at 900 s, which is 2.8°C. For T p6 , ΔT max appears at 1050 s, which is 2.9°C. During the whole discharging process, the maximum error of all the test points between the simulated and experimental temperature of the module is 1.7% when considering thermal radiation, but the maximum error is 3.9% without considering thermal radiation. It indicates that the simulated temperature with thermal radiation is closer to the experimental temperature, so as to improve the accuracy of the thermal model of the FAHP module. Furthermore, it reveals that the thermal radiation of the cell can not be ignored under the condition of natural convection when analyzing the heat dissipation performance of BTMS. From the variation of T p1 and T p6 , it can be seen that FAHPs start to work as soon as there is a little temperature difference between the evaporator section and the condenser section, which demonstrates that FAHPs can be used for effective heat dissipation of BTMS.  Figure 1A. The temperature contour of the battery module at 3C discharge rate under different schemes is shown in Figure 10. As can be seen from Figure 10, the center temperature of cell 4 with Scheme 1 is the maximum, which can reach 60.28°C. The minimum temperature is located on the edge of the module, and ΔT max,pack is 7.24°C. T max of Scheme 2 appears on the center of cell 4, which is 57.40°C, which is 2.88°C lower than that of Scheme 1. The minimum temperature is also presented on the edge of the module, and ΔT max,pack is 6.37°C. It indicates that the aluminum plates embedded into the module not only can reduce the maximum temperature but also improve the temperature uniformity of the module. Adding AHPs with Scheme 3 can effectively reduce the temperature on the center of the module and transfer the maximum temperature to the outer surface of cell 1 and cell 7. T max of Scheme 3 is 51.34°C, which is 14.8% and 10.5% lower than that of Scheme 1 and Scheme 2, respectively. And ΔT max,pack of Scheme 3 is 5.68°C, which is 21.5% and 10.8% lower than that of Scheme 1 and Scheme 2, respectively. The temperature distribution trend of Scheme 4 is similar to that of Scheme 3. Due to the high thermal conductivity of FAHPs and the enhanced heat transfer of the fins with Scheme 4, the temperature of the outer surface of cell 1 and cell 7 near the evaporation section of the FAHP is the maximum, which can reach 44.68°C. The minimum temperature region is near the condenser section of cell 4, and ΔT max,pack of Scheme 4 is 4.51°C.

| Different cooling schemes
The simulated temperature values of different cooling schemes at the end of 1C, 2C, and 3C discharging are shown in Table 3. From Table 3, under different discharge rates, adding FAHPs to the module can effectively reduce T max and ΔT max,pack . Especially, the heat transfer can be strengthened by adding fins on the condenser section of FAHPs. The maximum temperature difference of the cell (ΔT max,cell ) in the module is 4.03°C even when discharging at 3C rate. which is 2.29°C lower than that of Scheme 1, 1.24°C lower than that of Scheme 2 and 0.43°C lower than that of Scheme 3. By comparing T max , ΔT max,pack and ΔT max,cell , the average heat dissipation performance of Scheme 4 is improved by 33.3%, 25.0%, and 14.4% than that of Scheme 1, Scheme 2, and Scheme 3, respectively. It indicates that FAHPs with fins on the condenser section can be used for highefficient heat dissipation of BTMS.

| Radiation heat transfer coefficients
When the FAHP module of Scheme 4 is discharged at 3 C rate,  T = 25°C, h conv = 5.0 W·m −2 ·K −1 , h rad is set as 6.0, 6.5, 7.0, 7.5, and 8.0 W·m −2 ·K −1 , the temperature performance of the FAHP module with different h rad is shown in Figure 11. As can be seen from Figure 11, T max with different h rad is increased with the increasing of discharging time. At the early stage of discharging (t < 200 s), the influence of different h rad on the temperature of the module is little because the heat generation rate of the cell (q cell ) is changed little. The maximum temperature difference with different h rad is 1.60°C. When t = 600 s, the difference becomes 4.41°C. When t > 800 s, q cell is increased gradually due to the increasing internal resistance, which leads to the greater external radiation energy of the cell. The higher h rad is, the lower T max will be. And with the increase of h rad , the decrease rate of T max gradually increases. When h rad = 8.0 W·m −2 ·K −1 , T max is 44.09°C at the end of discharging, which is decreased by 6.74°C, 5.41°C, 3.81°C, and 2.01°C than that when h rad is 6.0, 6.5, 7.0, and 7.5 W·m −2 ·K −1 , respectively. When h rad is increased from 6.0 to 6.5 W·m −2 ·K −1 , 7.0, 7.5, and 8.0 W·m −2 ·K −1 , the ratio of the heat generated by the thermal radiation to the total heat generated by the cell (φ) is 2.6%, 5.7%, 9.5%, 13.8%, and 18.5%, respectively under the condition of natural convection. With the increasing of the thermal radiation of the cell, ΔT max,pack is decreased from 8.11°C to 7.62°C, 6.72°C, 5.67°C, and 4.49°C, respectively. ΔT max,cell is decreased from 6.33°C to 5.91°C, 5.40°C, 4.73°C, and 3.98°C, respectively. It indicates that the influence of different thermal radiation coefficients on the temperature of the FAHP module is gradually increased with the increasing of discharge time.

| Convection heat transfer coefficients
When h rad is 6.0, 7.0, and 8.0 W·m −2 ·K −1 , h conv is set as 5, 15, 25, 35, 45, 55, and 65 W·m −2 ·K −1 , respectively, the temperature performance of the FAHP module with different h conv and h rad is shown in Figure 12. From Figure 12A, T max is decreased with the increasing of h conv , and finally tends to be stable. Under the same h conv and  T , the higher h rad is, the lower T max will be. And the temperature difference between different h rad is decreased with the increasing of h conv . For h rad = 8.0 W·m −2 ·K −1 , when h conv = 5, 15, 25, 35, 45, 55, and 65 W·m −2 ·K −1 , the temperature difference between T max with thermal radiation and T max without thermal radiation is 6.74°C, 6.24°C, 5.52°C, 1.88°C, 0.42°C, and 0.08°C respectively. While, the ratio of the radiation heat dissipation to the total heat dissipation(η) is 35.31%, 36.68%, 37.81%, 33.33%, 18.76%, 4.29%, and 1.52%, respectively. It indicates that, the average η(η avg ) is 35.7% when 5 W·m −2 ·K −1 ≤ h conv ≤ 35 W·m −2 ·K −1 . The radiation heat transfer of the FAHP module can not be ignored, and the accuracy of the thermal model can be improved by considering the thermal radiation. η avg is 18.8% when 35 W·m −2 ·K −1 < h conv < 55 W·m −2 ·K −1 . With the weakening of the radiation heat transfer, it can be comprehensively determined whether to consider the influence of thermal radiation according to the actual demand. η avg is lower than 1.5% when h conv ＞ 55 W·m −2 ·K −1 . In this situation, the thermal radiation has little influence on the temperature rise of the module, which can be ignored.

| CONCLUSIONS
(1) Based on the thermal model of the FAHP module considering the thermal radiation, the maximum error between the simulated and experimental temperature of the module is 1.7%, but the maximum error is 3.9% without considering thermal radiation.
Even at 3 C discharge rate, T max is only 44.68°C and ΔT max,pack is 4.51°C. Therefore, the thermal radiation effect of the cell cannot be ignored under the condition of natural convection, so as to improve the accuracy of the thermal model for the FAHP module. maximum temperature to the outer surface of cell 1 and cell 7. Due to the high thermal conductivity of FAHPs and the enhanced heat transfer of the fins with Scheme 4, the average heat dissipation performance of Scheme 4 is improved by 33.3%, 25.0%, and 14.4% than that of Scheme 1 (i.e., natural convection), Scheme 2 (i.e., aluminum plates sandwiched between cells) and Scheme 3, respectively. (3) At the same  T , the decreasing rate of T max and ΔT max,pack is gradually increased with the increasing of h rad , but decreased with the increasing of h conv . η avg is 35.7% when 5 W·m −2 ·K −1 ≤ h conv ≤ 35 W·m −2 ·K −1 , and the accuracy of the thermal model can be improved by considering the thermal radiation. η avg is 18.8% when 35 W·m −2 ·K −1 < h conv < 55 W·m −2 ·K −1 , and the model can be comprehensively determined whether to consider the influence of thermal radiation or not. η avg is lower than 1.5% when h conv > 55 W·m −2 ·K −1 and the thermal radiation has little influence on the temperature rise of the FAHP module.