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Shortcut computation for the thermal management of a large air-cooled battery pack Zhongming Liu a, Yuxin Wang a, *, Jun Zhang b, Zhibin Liu b a School of Chemical Engineering and Technology, State Key Lab of Chemical Engineering, Co-Innovation Centre of Chemistry and Chemical Engineering of Tianjin, Tianjin University, Tianjin 300072, China b Pylon Technologies Co., Ltd., No. 887-73 Zuchongzhi RD, Zhangjiang HiTech Park, Pudong District, Shanghai 201203, China

h i g h l i g h t s Shortcut computation for the thermal management of a large battery pack is developed. The effects of non-uniform airﬂow on battery temperature uniformity are considered. A ﬂow resistance network model is built to rapidly estimate the non-uniform airﬂow. Structural parameters are analyzed to improve the battery temperature uniformity.

a r t i c l e i n f o

a b s t r a c t

Article history: Received 18 September 2013 Accepted 17 February 2014 Available online 25 February 2014

Thermal management is crucial to maintain the performance of large battery packs in electric vehicles. To this end, we present herein a shortcut computational method to rapidly estimate the ﬂow and temperature proﬁles in a parallel airﬂow-cooled large battery pack with wedge-shaped plenums for airﬂow distribution. The method couples a ﬂow resistance network model with a transient heat transfer model to calculate the temperature distribution in a battery pack as inﬂuenced by the airﬂows within and among battery modules in the pack. The effects of the plate angle of the plenums, the minimal plenum width and the battery unit spacing on the airﬂow and temperature distributions are presented. Additionally, an example of collective parameter adjustment for acceptable temperature uniformity of a battery pack subjected to total volume constraint is given. Ó 2014 Elsevier Ltd. All rights reserved.

Keywords: Battery pack Thermal management Air-cooling Flow resistance network

1. Introduction The development of electric vehicles (EVs) and hybrid electric vehicles (HEVs) under the pressures of oil shortages and environmental protection has stimulated a high demand for high energy rechargeable battery packs, e.g., lithium-ion battery packs, in recent years. The thermal management of a battery pack is important to ensure a safe and sustained power supply from the battery pack. A number of thermal management systems have been proposed to date, and these systems can be divided into active cooling and passive cooling generally. For the former, the heat generated within the battery is actively removed by forced air [1e3] or liquids [4] via convective heat transfer; for the latter, the heat is passively absorbed by phase change materials [5e7]. Rao et al. [8] has published a detailed comparison of these methods. It is shown that the forced

* Corresponding author. Tel./fax: þ86 22 27890515. E-mail address: [email protected] (Y. Wang). http://dx.doi.org/10.1016/j.applthermaleng.2014.02.040 1359-4311/Ó 2014 Elsevier Ltd. All rights reserved.

parallel airﬂow cooling is advantageous over other active cooling methods in maintaining the uniform battery temperature distribution, because the cooling air distributed into different cooling channels or modules can theoretically be of the same ﬂow rate and initial temperature. The thermal characteristics of parallel airﬂow-cooled battery packs have been numerically simulated using Computational Fluid Dynamics (CFD) in recent years [9e14]. In some of these works [9e 12], the computational domain was reduced from a whole battery pack to one of its repeating units to save computation time because the CFD method is demanding on high-performance computer hardware. This treatment was justiﬁed by the theoretically uniform airﬂow distribution of forced parallel airﬂow cooling. However, the airﬂow distribution is affected by many factors and its uniformity cannot be always guaranteed. When the airﬂow is non-uniform, the uniformity of temperature distribution in battery packs will become worse. In this case, the entire battery pack should be included in the computational domain and the simulation can be

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very time consuming because of a huge number of mesh elements required. For example, in Fan’s simulation [13], the computational domain had about 100,000 elements, and in Park’s work [14], the total number of meshes was about 2,400,000. Alternative computation methods other than CFD, consequently, are necessary when quick analysis of the pack’s thermal behaviour is required. However, the factor of airﬂow maldistribution was not considered in previously reported methods [15e17], new method that takes account of uneven airﬂow should be developed. In this paper, we report a shortcut method to estimate the ﬂow and temperature proﬁles in a parallel airﬂow-cooled large battery pack (as shown in Fig. 1). The method couples a ﬂow resistance network model and a transient heat transfer model. The former simulates the ﬂow ﬁeld of the air-cooling system and provides convective heat transfer coefﬁcients on battery uniteair interfaces for subsequent use; the latter calculates the temperature rise of each battery unit without considering its internal details. As such, the non-uniform airﬂow distribution and its effect on the battery temperature uniformity are considered in the simulation. The method is then applied to discuss structure parameters to improve the temperature uniformity in the battery pack. 2. Model development 2.1. Illustrations of the parallel airﬂow conﬁguration The parallel airﬂow-cooled large battery pack simulated in this paper is shown in Fig. 1. It consisted of eight battery modules, and each module contained eight battery units with ﬁve battery cells per unit. The cylindrical 18650 lithium-ion battery cells were used. A nested parallel airﬂow cooling system was built to manage the thermal behaviour of the battery pack, which could be divided into two subsystems, one between the battery units inside each module and the other between the modules. The structural similarity of these subsystems led to the same computational principle, so the former was chosen to expound the ﬂow resistance network model in detail. In the subsystem shown in the dashed area of Fig. 1(a), air is distributed and converged by wedge-shaped plenums called the distribution plenum (DP) and the convergence plenum (CP). Heat transfer takes place in the cooling channels (CCs) between adjacent battery units. The plenums’ plate angle, q, the minimal plenum width, wmin, and the battery unit spacing, lsp, determine the structure of the subsystem and thus affect the airﬂow distribution. The cooling channels, whose cross sections are the same as the ﬁlled area in Fig. 1(b), should be separated from each other in the design of the air-cooling system to ensure parallel airflow.

Fig. 2. The ﬂow resistance network model for the air-cooling system in a battery module. Calculation segments are partitioned by the distribution/convergence points of the airﬂow, and the blocks indicate the ﬂow resistances. The static pressure, P, is read on the main duct side at each distribution/convergence point of the airﬂow.

2.2. The ﬂow resistance network model When airflows inside the air-cooling system, its static pressure is simultaneously varied and labelled by segment in Fig. 2. This phenomenon is primarily caused by (1) the energy transformation of air between its kinetic energy and static pressure when the airﬂow velocity changes; and (2) the energy losses, including the irreversible loss due to the friction between the air and the rough channel walls and the local loss due to the air distribution and convergence at speciﬁc sites. The air is assumed to be an incompressible Newtonian ﬂuid here. For a continuous ﬂow process,

Fig. 1. (a) The schematic of a nested air-cooling system in a battery pack, in which the arrows indicate the airﬂow directions. (b) The arrangement of battery cells in a battery module.

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according to Bernoulli’s equation [18], the static pressures at the two ends of the ith cooling channel are deﬁned as follows: i X 1 DPloss;DP;k PDP;i ¼ PDP;1 rair v2DP;i v2DP;1 2

(1)

i1 X 1 DPloss;CP;k PCP;i ¼ PCP;1 rair v2CP;i v2CP;1 2

(2)

k¼1

k¼1

Furthermore, the static pressure drops in the 1st and ith cooling channels are deﬁned as follows:

PDP;1 PCP;1

1 ¼ rair v2CP;1 v2DP;1 þ DPloss;CC;1 2

(3)

1 rair v2CP;i v2DP;i þ DPloss;CC;i 2

(4)

PDP;i PCP;i ¼

Thus, combining Eqs. (1)e(4) yields the following:

DPloss;DP;i þ DPloss;CC;i DPloss;CP;i1 DPloss;CC;i1 ¼ 0

(5)

where v is the ﬂow velocity and DPloss is the corresponding static pressure loss. Eq. (5) is similar to the Kirchhoff Voltage Law; in other words, it implies that the sum of static pressure losses equals zero in each directional circuit shown in Fig. 2. It is the main governing equation that forms the ﬂow resistance network model. The static pressure loss can be related to the corresponding ﬂow velocity, v. Taking the ith segment of the distribution plenum as an example, the irreversible loss due to the friction with the surrounding walls can be expressed by the Fanning equation [18]:

DPfriction;DP;i ¼ lDP;i

lDP;i r v2 2DDP;i air DP;i

(6)

where l is the dimensionless friction factor, rair is the air density and l and D are the length and the equivalent diameter of the ith segment, respectively. This equation can also be used to evaluate the irreversible losses in the convergence plenum and the cooling channels because they are in the same form. In addition, the mean value of the equivalent diameter in each segment can be adopted to reduce the model size because the plenums are wedge-shaped. The irreversible loss is relatively small in this model because the length of the segments is short, whereas the local loss caused by the distribution of airﬂow at the T-junction area is more signiﬁcant and given by the following [18]:

DPlocal;DP;i ¼

xDP;i 2

rair v2DP;i1

(7)

where x is the local loss coefﬁcient. This coefﬁcient is always related to the ﬂow velocity in the main duct of T-junctions, i.e., the ﬂow velocity before the air distribution in the distribution plenum or after the air convergence in the convergence plenum. In addition, the local loss in each cooling channel is the sum of the pressure losses caused by air distribution and convergence at both ends of the cooling channel. Therefore, the local losses induce several changes in the convergence plenum and the cooling channels, which can be written as follows:

DPlocal;CP;i ¼ DPlocal;CC;i ¼

xCP;i 2

rair v2CP;iþ1

xDP/CC;i 2

rair v2DP;i þ

(8)

xCC/CP;i 2

rair v2CP;i

(9)

447

Consequently, the total pressure loss in the ith segment is the sum of the corresponding irreversible loss and local loss. In particular, the friction factor, l, and the local loss coefﬁcient, x, are the most important parameters in these equations. The friction factor is related to the state of airﬂow, or more explicitly, the Reynolds number Re. In laminar ﬂow and transition regions where Re is below 3000, this factor can be calculated by Ref. [19],

l ¼ F

64 Re

(10)

Re ¼

rair Dv mair

(11)

where F is the shape factor for non-circular channels, which equals 0.89 for rectangular cross-sections; rair and mair are the air density and the air viscosity, respectively. When 3 103 < Re < 105, the Blasius equation is given by the following [19]:

l ¼

0:3164

(12)

Re0:25

The local loss coefﬁcient is related to the system geometry and ﬂow status in the air-cooling system. M. Bassett et al. [20] developed a series of analytical expressions to calculate the steady ﬂow pressure loss coefﬁcients for a T-junction. The conditions were restricted as follows: the main duct and the branch in a straight line must have the same cross sectional area and the intersection of the channels at the junction should be sharp. According to the study by M. Basset et al., deﬁning j as the cross sectional area ratio of the plenums to the cooling channels and q the ratio of airﬂow in the branch to that in the main duct in this paper allows the local loss coefﬁcient to be expressed as follows:

3 2

xDP;i ¼ q2DP;i qDP;i þ

1 2

(13)

xDP/CC;i ¼ q2DP/CC;i j2DP;i 2 cos

3 p qDP/CC;i jDP;i þ 1 8

(14)

xCP;i ¼ 1 q2CP;i

(15)

xCC/CP;i ¼ q2CC/CP;i j2CP;i þ 2q2CC/CP;i 1

(16)

The correlations between airﬂow velocities in each independent circuit depicted in Fig. 2 have been clearly described with Eqs. (5)e (16). They provide 8 independent equations with a total of 27 unknown velocities to be solved. Because the cooling air is assumed to be incompressible in this model, the sum of airﬂow rates (Q ¼ v A) at each distribution/convergence point naturally satisﬁes the following relationship:

vDP;i ADP;i vDP;iþ1 ADP;iþ1 vCC;i ACC;i ¼ 0

(17)

vCP;iþ1 ACP;iþ1 vCP;i ACP;i vCC;iþ1 ACC;iþ1 ¼ 0

(18)

where A is the cross-sectional area in each segment. Similar to the equivalent diameter, A is the mean value in each segment of the plenums. Eqs. (17) and (18) provide 18 independent equations; thus, only one variable remains left unknown. Once the inlet airﬂow rate, QDP,1, is given, the above ﬂow resistance network model can be solved, and the airﬂow distribution within the module is consequently obtained. Although the above model was designed for the subsystem in a module, it can be simply modiﬁed

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to investigate the non-uniform distribution of inlet airﬂow rates between modules if each module in the pack is treated as a cooling channel.

2.3. The transient heat transfer model The uniformity of airﬂow distribution in the parallel cooling channels somewhat reﬂects the uniformity of the battery unit temperatures in a module. Nevertheless, a transient heat transfer model was built to obtain temperature proﬁles for a unit, module and pack to provide more intuitive results. To reduce the model size, the internal details of the battery cells and cooling air are neglected, which results in volume average values for the calculated temperatures. Heat exchange is also assumed to only occur on the battery uniteair interface in the cooling channels; thus, the cooling air in the plenums does not exchange heat with the battery units. The heating power of a battery unit was considered ﬁrst. The heat generated when a battery cell’s discharges primarily originates from the Joule heat and the heat of reaction [3]. The former is caused by the cell internal resistance, while the latter is the result of cell internal reactions. Previous studies [21] have suggested that the heat of reaction can be neglected at high current discharge rates because the Joule heat becomes dominant. Because each battery unit consists of ﬁve cells, the heating power can be evaluated by the following:

H ¼ 5 I2 R

Fig. 3. The schematic of heat transfer on the surface of a battery unit.

equation that links the two models. The temperatures of the cooling air in the battery unit differ between the two sides, as shown in Fig. 3, thus the temperature difference between the battery unit and either side of the air was calculated individually. A logarithmic averaged temperature difference was adopted in the model:

Tunit;i T0 Tunit;i Tair;iþ1

(23)

where T0 is the initial temperature of both the air and battery cells and Tair is the outlet temperature of the air in the cooling channels. According to the law of the conservation of energy, the transient heat transfer equations can be written as follows:

where Tunit is the battery unit temperature and SOC is calculated by the coulomb accumulation method [12]. The generated heat is partly removed by the surrounding air, and the convective heat transfer coefﬁcient is given by Ref. [18]:

YRey Pr1=3

unit

rCp QCC;i

(20)

kair

(22)

DTup;i ¼ Tair;iþ1 T0 ln

rCp V

223:71 R ¼ 27:54 27:68 expð 1:91=Tunit Þ þ 1 þ 21:61 SOC expð 1:91=Tunit Þ 225:06 1 þ 21:61 SOC

dcell

Tunit;i T0 Tunit;i Tair;i

(19)

where I and R are the discharge current and the internal resistance in a battery cell, respectively. The internal resistance R, which varies with battery’s state-of-charge (SOC) and temperature, can be obtained experimentally. Our experiments on a 2.2 Ah LiFePO4 battery yield:

h ¼

DTdown;i ¼ Tair;i T0 ln

(21)

where dcell is the diameter of a battery cell and the characteristic length for Re; kair and Pr are the thermal conductivity and the Prandtl Number of air, respectively; Y and y are empirical parameters that vary with Re and are given in Table 1. Eq. (21) is the key

Table 1 Values of Y and y under different Reynolds numbers in Eq. (21). Re

Y

y

1e4 4e40 40e4000 4000e40,000 40,000e250,000

0.989 0.911 0.683 0.193 0.0266

0.330 0.385 0.466 0.618 0.805

vTunit;i ¼ H hi SDTdown;i hiþ1 SDTup;i vt

vTair;i ¼ hi S DTup;i1 þ DTdown;i air vt

(24)

(25)

where S is the area of the battery uniteair interface on either side of each cooling channel and Q is the airﬂow rate in each cooling channel. The speciﬁc heat at constant pressure, Cp, is approximated as a constant in the above equations. A RungeeKutta algorithm was applied to solve the above differential equations, and the necessary constants are provided in Table 2. The ﬂow resistance network model and the transient heat transfer model were both built in the MatlabÒ software, and the block diagram of the computing process is shown in Fig. 4.

Table 2 Constants used in the shortcut computation and the CFD model. Air properties at 300 K & 1 bar [18]

Parameters relative to the battery cell

rair [kg/m3]

rcell [kg/m3] Cp,cell [J/(kg K)] T0 [ C] kcell [W/(m K)]

Cp,air [J/(kg K)] mair [Pa s] kair [W/(m K)]

1.1614 1007 1.85 105 0.0263

Pr

0.707

a b c

Denotes average properties of a battery cell. Denotes the property in the radial direction. Denotes the property in the height direction.

5400a 502.35a 25 1.373a,b 17.33a,c

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Fig. 4. The block diagram of the solving process for a battery module.

3. Results and discussion 3.1. Flow and temperature proﬁles in a module For the battery module in Fig. 1(b), the inlet airﬂow rate, QDP,1, was set to be 0.04 m3/s, and the structure parameters, including the plenums’ plate angle q and the battery unit spacing, lsp, were ﬁxed to 15 and 1 mm, respectively. In addition, the minimal plenum width (wmin) was set at 5 mm to prevent sharp changes in the airﬂow velocities at the terminals of the two plenums. A constant current discharge process from 95% SOC to 5% SOC was considered with 5C discharge rate, thus the duration was set to be 11 min. As a comparison, a three-dimensional CFD model was also developed by using commercial software COMSOL Multiphysics, which is based on the ﬁnite element method. Fig. 5 shows the rise in the temperature of each battery unit in the battery module. The battery units closer to the air inlet of the module were found to maintain a higher temperature over the entire discharge time and take longer to reach steady state than those near the rear. As a result, the temperature distribution in the module becomes less uniform over the entire discharge process, and reaches a maximal temperature difference at the end of discharge. This result can be related to the airﬂow maldistribution in the parallel cooling channels, as depicted in Fig. 6(a), where the airﬂow velocities gradually increase from the ﬁrst channel to the ninth. According to Eq. (21), a low airﬂow velocity leads to low convective heat transfer coefﬁcient on the battery uniteair interface and thus a high temperature of the corresponding battery unit. The result in Fig. 5 highlights the necessity of considering the airﬂow maldistribution in thermal simulations. Fig. 6(a) suggests that the non-uniform airﬂow distribution in the cooling channels is unavoidable, even if the air is distributed by wedge-shaped plenums. According to the ﬂow resistance network model, the airﬂow distribution depends on the pressure loss distribution in the air-cooling system, and the latter is furthered affected by pressure loss coefﬁcients, which depend on the system structure and ﬂow status. Because the pressure loss coefﬁcient of air distribution is smaller than that of air convergence, the pressure loss in the distribution plenum grows much slower than that in the convergence plenum, as shown in Fig. 6(b). Thus, the pressure differences and the airﬂow velocities within the cooling channels increase from the ﬁrst channel to the ninth. Therefore, the growth of the pressure losses in the plenums,

Fig. 5. The temperature rise of each battery unit and the growth of the maximal temperature difference in the module calculated by (a) the shortcut method (b) the CFD method (q ¼ 15 , wmin ¼ 5 mm and lsp ¼ 1 mm).

especially the growth in the convergence plenum, should be slowed to improve the uniformity of airﬂow distribution. This reduction can be achieved by adjusting the structural parameters of the parallel air-cooling system.

Fig. 6. (a) The distribution of the airﬂow velocities in the cooling channels. (b) The pressure loss from the air inlet to each distribution/convergence point in the module (q ¼ 15 , wmin ¼ 5 mm and lsp ¼ 1 mm).

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Figs. 5 and 6(a) show that the results obtained by the shortcut method are closely approximate to those obtained by the CFD method. The maximal temperature differences calculated by the two methods only differ by 0.5 C. Compared to the CFD method, the shortcut method saves considerable amount of computation time. In the CFD model, the total number of meshes was about 1,720,000 considering the accuracy of the results and computation time. Consequently, it took about 10 h to get a solution. However, by the shortcut method, the computation time was reduced to 2e 3 s using the same desktop computer. 3.2. Effects of structural parameters on battery temperature uniformity According to the analysis above, three parameters, the plenums’ plate angle, q, the minimal plenum width, wmin, and the battery unit spacing, lsp, are discussed in this section. The effect of the plate angle, q, is depicted in Fig. 7, in which the maximal differences in the airﬂow velocities and the battery unit temperatures are used to represent the non-uniformity of the airﬂow distribution and the temperature distribution. A larger plate angle was shown to increase the uniformity of the airﬂow and temperature distributions. This angle should be no less than 16.5 for 5C-rate and 13.2 for 4Crate discharge process to maintain a maximal temperature difference lower than 5 C. Analogously, the effect of the minimal plenum width on the uniformity of both distributions is similar to that of the plate angle, as shown in Fig. 8(a). With a 10 plate angle, this width should be 16.4 mm or 10.1 mm for 5C-rate or 4C-rate discharge process to ensure that the maximal temperature difference does not exceed 5 C. These values arise from the expansion of the two parameters, which mitigates not only the variations but also the average values of the airﬂow velocities in the two plenums to dampen the pressure losses in the plenums, especially in the convergence plenum. However, the expansion of the plate angle and the minimal plenum width will notably enlarge the volume of a battery module and thus decrease its volumetric power density. Only the minimal width of the convergence plenum is adjusted to reduce the volume of the air-cooling system because the growth of the pressure loss is more evident in the convergence plenum than in the distribution plenum. The result is shown in Fig. 8(b), which suggests that the required width to maintain the maximal temperature difference in the module lower than 5 C is 14.1 mm for

Fig. 7. The effect of plenums’ plate angle, q, on the uniformity of airﬂow velocities in the cooling channels and battery unit temperatures in the module (wmin ¼ 5 mm and lsp ¼ 1 mm).

Fig. 8. The effect of the minimal plenum width on the uniformity of airﬂow velocities in the cooling channels and battery unit temperatures in the module: (a) the minimal width of both plenums is adjusted; (b) only the minimal plenum width of the convergence plenum is adjusted (q ¼ 10 and lsp ¼ 1 mm).

5C-rate and 8.3 mm for 4C-rate discharge process. It also indicates that it’s better to only adjust the minimal width of the convergence plenum than to adjust that of both plenums. The reason is that the pressure loss curve in the distribution plenum is slightly concave, which is beneﬁcial to the uniform airﬂow distribution in the cooling channels. Generally, this adjustment for 5C-rate discharge process will decrease the module spacing by 17.3% or 23.4% compared to only adjusting the plate angle or only adjusting the minimal width of both plenums, respectively. The battery unit spacing determines the cross-sectional area of the cooling channels in the battery module. This spacing should be tight to ensure the compactness of the battery module and enhance the convective heat transfer coefﬁcients on the battery celleair interfaces [12]. In the above discussion, this spacing was to 1 mm in each channel. However, the airﬂow rate was increased in the ﬁrst cooling channel to improve the uniformity of battery unit temperatures because the battery unit near the air inlet showed the highest temperature rise, as depicted in Fig. 5. This objective was met by broadening the battery unit spacing in the ﬁrst channel to 9 mm while narrowing it to zero in the other channels while maintaining the module volume the same. The result and its comparison to the result obtained for uniform battery unit spacing is shown in Fig. 9. In this scenario, more uniform temperatures and a 0.39 C/0.24 C improvement in the temperature difference for a

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5C-rate/4C-rate discharge process was resulted. However, the airﬂow was less uniform in this conﬁguration. In this case, the uniform airﬂow rates, rather than the airﬂow velocities, primarily inﬂuence the uniformity of the battery temperature. All the above parameter adjustments are generally designed to optimise the temperature uniformity in a battery pack and minimise the total volume. The results in Fig. 9 suggest an acceptable adjustment for the battery module, i.e., a plate angle of 10 , a minimal width of the convergence plenum of 14 mm/9 mm, and the battery unit spacing (9 mm, 0, ., 0) for a 5C-rate/4C-rate discharge process. 3.3. Flow and temperature proﬁles in a battery pack For the battery pack depicted in Fig. 1, the airﬂow rates distributed to different battery modules cannot be uniform. A modiﬁed ﬂow resistance network model was used to compute this non-uniform inlet airﬂow distribution. In this model, the role of previous irreversible pressure loss was replaced by the total pressure loss in each module. When the correct inlet airﬂow rate was determined for each module, the ﬂow and temperature proﬁles inside each module could be individually calculated. The total computing process was completed in less than half a minute.

Fig. 10. The non-uniform distribution of the inlet airﬂow rates for different modules in the battery pack, and the temperature differences in the modules (The total inlet airﬂow rate is 0.32 m3/s; q ¼ 30 , wCP,min ¼ 14 mm for the pack plenums; q ¼ 10 , wCP,min ¼ 14 mm and lsp ¼ (9, 0, 0, 0, 0, 0, 0, 0, 0) mm for each module).

A 5C-rate discharge process was considered for the battery pack. As shown in Fig. 10, the amount of air each module receives increased from the ﬁrst module to the last, with a maximal difference of 0.0074 m3/s. Thus, the volumetric average temperatures of each module could not be equal and showed an opposite trend compared with the curve of the inlet airﬂow rates. Consequently, the maximal temperature difference, i.e., the temperature uniformity, increased to 6.1 C over the entire battery pack. The temperature proﬁles in different modules were also affected by the non-uniform inlet airﬂow rates. The temperature difference inside each module gradually reduced with the increasing inlet airﬂow rate. The inlet airﬂow rates positively correlated with the airﬂow velocities in the cooling channels but did not signiﬁcantly affect the differences in the velocity values. However, the differences in the enlarged heat transfer coefﬁcients on the battery unite air interfaces are diminished due to the enhanced airﬂow velocities, according to Eq. (21). Therefore, a higher inlet airﬂow rate increases the temperature uniformity in the battery module. Overall, the above battery pack thermal simulation conﬁrms the speed of the proposed shortcut computation method. 4. Conclusion This paper presents a shortcut computational method to obtain the ﬂow and temperature proﬁles in a parallel airﬂow-cooled large battery pack. The airﬂow distribution in the parallel cooling channels was found to be non-uniform, even though the air was distributed by wedge-shaped plenums. This pattern decreased the temperature uniformity among the battery units and modules in the pack. The structural parameters, including the plate angle, the minimal plenum width and the battery unit spacing, were all related to the airﬂow and temperature uniformity. A collective adjustment of these parameters was provided to maintain the maximal temperature difference below 5 C. Generally, this method can quickly estimate the ﬂow and temperature proﬁles in a battery pack and be used to conduct a preliminary analysis or design of a battery pack. Acknowledgements Fig. 9. The effect of the non-uniform distribution of the battery unit spacing on the uniformity of the airﬂow velocities and battery unit temperatures in the module. (a) 5C-rate discharge process, wCP,min ¼ 14 mm; (b) 4C-rate discharge process, wCP,min ¼ 9 mm (q ¼ 10 , the spacing in formation A is equal with a value of 1 mm, while formation B is arranged as (9, 0, 0, 0, 0, 0, 0, 0, 0) mm.).

The authors would like to thank Shenzhen ZTE Holdings Co., LTD for funding this work. This work was also supported by Tianjin Natural Science Foundation (Grant No. 11CYBCJC07800).

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Nomenclature A cross-sectional area [m2] Cp speciﬁc heat at constant pressure [J/(kg K)] D equivalent diameter [m] F shape factor for non-circular channel H heat power [Watt] I current [A] P static pressure [Pa] Pr Prandtl number Q airflow rate [m3/s] R electric resistance [U] Re Reynolds number S heat transfer area [m2] T temperature [K] V volume [m3] d diameter [m] h convective heat transfer coefﬁcient [W/(m K)] l length [m] q ﬂow ratio [m3/s] v velocity [m/s] w width [m]

Greek

l r m x k q J

friction factor density [kg/m3] viscosity [Pa s] local loss coefﬁcient thermal conductivity [W/(m K)] plate angle [ ] cross sectional area ratio

Subscript air cell CP CC DP friction local loss min sp

air properties battery properties convergence plenum cooling channel distribution plenum friction loss local loss total pressure loss the minimal value spacing

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