A statistical analysis of the effect of design factors of the cylindrical battery module on the thermal behavior by C-rate

Abstract

In real-world driving scenarios of electric vehicles, the C-rate fluctuates with changes in speed, and a Battery Thermal Management System (BTMS) design that does not account for a range of C-rates may fail to ensure thermal safety of the battery. In this study, we statistically analyzed the effects of cooling plate geometry and coolant velocity on battery thermal behavior using Design of Experiments (DoE) and Analysis of Variance (ANOVA) across various C-rates. The ANOVA results by C-rate demonstrate that the influence of design factors on BTMS thermal behavior varies with the C-rate. Specifically, at 0.5C, the percentage contribution of the cooling plate outer width to the temperature difference is 41.56 %, whereas at 3C, it is reduced to 39.41 %. Conversely, the percentage contribution of the cooling plate outer height to the temperature difference increases from 6.88 % at 0.5C to 9.13 % at 3C. These findings suggest that to ensure the thermal safety of the BTMS, it is crucial to consider the thermal behavior under various C-rate conditions during the design process.

Nomenclature

T	temperature (°C)	Subscripts
$c_p$	specific heat (J/kg $\bullet$ K)	a	axial
m	Mass (kg)	r	radial
V	volume (${\mathrm{m}}^3$)	b	battery
k	thermal conductivity (W/m $\bullet$ K)	cp	cooling plate
t	time (s)	c	coolant
A	area (${\mathrm{m}}^2$)	ir	irreversible
L	length (m)	re	reversible
v	velocity (m/s)	ca	cathode
P	pressure (Pa)	an	anode
Q	heat generation (W)	Abbreviations
U	voltage (V)	BTMS	Battery Thermal Management System
I	current (A)	C-rate	Current rate
R	resistance (Ω)	DoE	Design of Experiment
Re	Renolds number	ANOVA	Analysis of Variance
$D_0$	hydraulic diameter (m)	FFD	Full Factorial Design
l	wetted perimeter (m)	NMC	Nickel-Manganese-Cobalt
Greek symbols		CFD	Computational Fluid Dynamics
ρ	density (kg/ ${\mathrm{m}}^3$)	DoD	Depth of Discharge
μ	viscosity (kg/m $\bullet$ s)	SS	Sum of Square
		MS	Mean Square
		DoF	Degree of Freedom

1. Introduction

Internal combustion engine vehicles using fossil fuels as energy sources are contributing to climate change through global warming. To address the issue of climate change, many countries are encouraging the development of zero-emission electric vehicles as replacements for internal combustion engine cars. Li-ion batteries, known for their low self-discharge rate, high capacity, long lifespan, and high energy density are predominantly utilized as the power supply for electric vehicles [[1], [2], [3]] (see Table 8).

Table 8.

The effect of $h_{in}$ in the cooling plate on the variations of BTMS mass and pressure drop.

$h_{in}$ [$mm$]	BTMS mass [g]	Pressure drop [Pa]
0.3	2516	469.64
0.5	2549	880.71
0.7	2582	2406.37

During the charging/discharging process, Li-ion batteries generate considerable heat due to internal resistance and reaction heat [[4], [5], [6]]. This thermal generation can lead to battery overheating and result in uneven temperature distribution within the battery pack. If the battery temperature surpasses the range of 20 $°C$ to 40 $°C$ or if the temperature variance between battery cells exceeds 5 $°C$, it significantly degrades battery performance and escalates the risk of thermal runaway [7,8]. Hence, an effective Battery Thermal Management System (BTMS) is indispensable for regulating battery thermal dynamics. Several studies incorporating experimental trials and computational simulations are presently underway to improve the cooling efficiency of diverse cooling methodologies [[9], [10], [11], [12]].

Air-cooled BTMS offers the advantage of not requiring additional equipment for cooling but suffers from lower cooling efficiency. Conversely, liquid-cooled BTMS boasts superior cooling efficiency but necessitates separate equipment such as a coolant circulation pump. Dafen et al. compared air-cooled and liquid-cooled BTMS using pouch type battery in terms of cooling performance, power consumption, additional weight, etc [13]. The results showed that air-cooled BTMS added little weight to the cooling system but consumed over 2.5 times more power to achieve similar cooling performance as liquid cooling systems. Air-cooled BTMS, known for its simplicity and cost-effectiveness, has been widely applied in early hybrid vehicles or electric vehicles (e.g., Nissan Leaf, Toyota Prius, Honda Insight). However, with advancements in electric vehicle technology leading to increased power density and capacity of batteries, generating more heat, there is a growing need for more efficient cooling. Hence, in recent times, liquid-cooled BTMS employing coolant with high thermal conductivity and specific heat capacity has been adopted in most electric vehicles (e.g., Tesla, GM Volt, French Peugeot Citroen, BMW i3) instead of air cooling.

In recent years, parametric studies on liquid-cooled BTMS have been conducted to further improve the cooling efficiency of batteries [[14], [15], [16]]. Haitao et al. studied the impact of battery module flow rate, cooling mode (serial/parallel cooling), and liquid flow direction on the thermal performance of cylindrical batteries discharged at 3C [14]. Yuzhang et al. investigated the thermal characteristics of pouch-type batteries discharged at 1C by setting design parameters such as channel count, channel aspect ratio, and channel inlet layout [17]. Jiaqiang et al. investigated the effects of the channel height, channel width, channel number, and coolant flow rate on the cooling effectiveness of liquid-cooled BTMS when prismatic batteries were discharged at 4C [18]. They utilized an orthogonal array to comprehensively study the combined impact of multiple variables on the cooling efficiency of battery thermal management systems. Wu et al. developed a BTMS design based on variable heat transfer path (VHTP). They parameterized the geometry of the VHPH layer of the VHTP cold plate and investigated the optimal VHTP-based BTMS when the battery is discharged to 3C [19]. Zhao et al. studied the thermal behavior of a battery thermal management system based on a liquid cold plate with honeycomb flow channels as a function of the width of the cooling channel, the thickness of the cold plate, and the coolant inflow rate when the battery is discharged to 5C [20].

One of the primary limitations of previous BTMS parametric studies is the lack of consideration for various C-rates when investigating battery temperature variations in response to changes in design factors. The C-rate, short for current rate, indicates the speed at which a battery is charged or discharged, defined as the multiple of the battery's capacity in ampere-hours ($Ah$). In actual driving scenarios of electric vehicles, the C-rate fluctuates depending on speed changes, and design that does not take this variability into account can be a factor limiting the accuracy and performance of the system. Recently, there has been a growing body of research analyzing the thermal behavior of batteries for various C-rate. Wang et al. studied hybrid cooling BTMS using PCM and wave microchannel cold plates. They investigated the effects of C-rate, number of microchannels, flow direction, PCM thickness, mass flow rate, etc. on the thermal behavior of the battery [21]. Li et al. studied the effects of cooling surface, inlet water, coolant flow direction, mass flow rate, and C-rate on the thermal behavior of prismatic battery modules [22]. Subhedar et al. investigated the performance of liquid-cooled cylindrical battery packs using nano coolant. They observed the thermal behavior of the battery as a function of the volume fraction of nano coolant, the velocity of the coolant, and the C-rate [23]. However, even papers that consider these different C-rates only investigate the thermal behavior of the battery by C-rate, and do not analyze the impact of C-rate-specific design factors on the thermal behavior of the battery.

In this study, the design factors of the BTMS commonly used in existing studies were statistically analyzed to determine the impact of the design factors on the thermal behavior of the battery by C-rate. To analyze the thermal behaviors based on each design factor, we computed the average temperature of the 71 cells within the Li-ion battery module, identifying the cell with the highest average temperature ($T_{\mathit{max}}$), the one with the lowest average temperature ($T_{\mathit{min}}$), and the temperature difference between these two cells ($T_{diff}$). Employing Design of Experiment (DoE), specifically the Full Factorial Design (FFD) sampling method, we conducted statistical analysis to investigate the relationship between design variables and responses. A total of 972 cases were analyzed, comprising 243 cases for each of the four C-rates, to confirm the influence of design factors on responses. The effect of each design factor on thermal properties across all C-rates was analyzed using analysis of variance (ANOVA). The effect of each design variable on BTMS Mass and pressure drop was also analyzed.

2. Model and method

2.1. Physical structure and properties of the battery module

In this study, a commercial nickel-manganese-cobalt (NMC) 18650 type cylindrical Li-ion battery cell was used. The specifications of the cylindrical battery cell are shown in Table 1.

Table 1.

Specification of the 18650-type battery [[24], [25], [26]].

Items	Value
Type	18650-type
Diameter ($mm$)	$18$
Height ($mm$)	$65$
Mass ($g$)	$45$
Volume ($c{m}^3$)	$16.54$
Nominal capacity ($Ah$)	$1.5$
Nominal voltage ($V$)	$3.6$
Maximum charge voltage ($V$)	$4.2$
Discharge cut-off voltage ($V$)	$3$

The battery module used in this study is the battery model utilized in the Tesla Model S (see Table 2). The battery pack contains 7104 separate cells and is cooled by coolant. The battery pack consists of 16 sheets linked in series, each of which contains six battery modules also connected in series [16]. In this study, design a battery module with 71 battery cells connected in parallel, as shown in Fig. 1.

Table 2.

thermodynamic properties of materials [[24], [25], [26]].

Material	$\rho \left(kg/m^3\right)$	$c_p\left(J/kg\bullet K\right)$	$k\left(W/m\bullet K\right)$	$\mu \left(kg/m\bullet s\right)$
$Electrolyte$	1290	133.9	0.45	–
$Anodecopperfoil$	8933	385	398	–
$Anodeelectrode$	2660	1437.4	5	–
$Separator$	1200	700	1	–
$Cathode$	1500	700	5	–
$Cathodealuminumfoil$	2702	903	238	–
$CoolingPlate$	$2,719$	$896.0$	$201$	–
$Coolant$	$998.2$	$4,182$	$0.6$	$0.001003$

Fig. 1.

(A) Cylindrical Li-ion battery module (b) Schematic of the cylindrical Li-ion battery module (c) Schematic of a cooling plate with coolant.

A battery cell consists of an anode, cathode, current collector, and separator. Each battery component is made of a very thin layer, and each component is heavily overlapped. Therefore, the detailed model considering battery components increases the computational cost significantly. To reduce the computational cost, the method of using a lumped-model instead of a detailed model is mainly used and lumped-model also used in this study. The equivalent physical properties of the lumped-model are calculated by Ref. [27]:

$${\rho }_b=\frac{m_b}{V_b}$$ (1)

$${\rho }_b{c}_{pb}=\frac{{\sum }_i{\rho }_i{c}_{p_i}{V}_i}{{\sum }_i{V}_i}$$ (2)

where $\rho$ is the density, $m$ is the mass, $V$ is the volume, and $c_p$ is the specific heat of the battery cell. The subscripts $b$ and $i$ represent the battery and each component of the battery, respectively.

A cylindrical battery cell has four major components rolled up in a jelly-roll form. Therefore, the axial thermal conductivity and radial thermal conductivity are different. Axial thermal conductivity and radial thermal conductivity are calculated by Eqs. (3), (4) respectively [28].

$$k_a=\frac{{\sum }_i{A}_i{k}_i}{{\sum }_i{A}_i}$$ (3)

$$k_r=\frac{{\sum }_i{L}_i}{{\sum }_i{k}_i/L_i}$$ (4)

Here, $k_a$ is the axial thermal conductivity, $k_r$ is the radial thermal conductivity $L$ is the length, and $A$ is the area.

2.2. Numerical analysis

2.2.1. Governing equations

This study was performed using ANSYS Fluent. Computational fluid dynamics (CFD) method was used to calculate the temperature distribution of the battery. The simulated system comprises three domains: battery, cooling plate, and coolant. The governing equation for the battery domain and cooling plate domain is the energy conservation equation, which is expressed as [29]:

$$\frac{\partial }{\partial t}\left({\rho }_b{c}_{pb}{T}_b\right)=\nabla \bullet \left(k_b\nabla T_b\right)+\frac{Q_{gen}}{V_b}$$ (5)

$$\frac{\partial }{\partial t}\left({\rho }_{cp}{c}_{pcp}{T}_{cp}\right)=\nabla \bullet \left(k_{cp}\nabla T_{cp}\right)$$ (6)

The governing equations for the coolant domain are the mass continuity equation, momentum conservation equation, and energy conservation equation, which are expressed as [30]:

$$\frac{\partial {\rho }_c}{\partial t}+\nabla \bullet \left({\rho }_c\overrightarrow{v}\right)=0$$ (7)

$$\frac{\partial }{\partial t}\left({\rho }_c\overrightarrow{v}\right)+\nabla \bullet \left({\rho }_c\overrightarrow{v}\overrightarrow{v}\right)=-\nabla \mathrm{P}+\nabla \bullet \left({\mu }_c\nabla \overrightarrow{v}\right)$$ (8)

$$\frac{\partial }{\partial t}\left({\rho }_c{c}_{pc}{T}_c\right)+\nabla \bullet \left({\rho }_c{c}_{pc}\overrightarrow{v}{T}_c\right)=\nabla \bullet \left(k_c\nabla {\mathrm{T}}_c\right)$$ (9)

In Eqs. (5), (6), (7), (8), (9), $k$ and $T$ are thermal conductivity and temperature, respectively. The subscripts $b,cp$, and $c$ represent the battery, the cooling plate, and the coolant, respectively. $\overrightarrow{v},P$, $\mu$ and $Q_{gen}$ are the velocity, the static pressure, the dynamic viscosity, and the heat generation, respectively.

2.2.2. Heat generation of the battery cell

When the battery discharges, the temperature of the battery increases due to irreversible and reversible heat. Bernardi et al. presented the heat generation of Li-ion Battery can be expressed as [31]:

$$Q_{gen}=Q_{ir}+Q_{re}=I\left(U_{ca}-U_{an}-U\right)-IT\frac{d\left(U_{ca}-U_{an}\right)}{dT}$$ (10)

Where $Q_{gen}$ is the total heat generation rate of battery cell, $Q_{ir}$ and $Q_{re}$ are the irreversible heat and the reversible heat, $I$ is the current, $U_{ca}$ and $U_{an}$ are the open-circuit potential voltage of the cathode and anode electrode, respectively. $U$ is the battery voltage, $T$ is the battery temperature and $\frac{d\left(U_{ca}-U_{an}\right)}{dT}$ is derivative of the open-circuit potential with respect to the temperature.

During the discharge process, $U$ obtained by Eq. (11) as follows [32]:

$$U=U_{ca}-U_{an}-IR$$ (11)

Where $R$ represents the total internal resistance, encompassing both the polarization equivalent resistance and the ohmic resistance.

We adopted the heat generation of 18650 NMC battery cell derived experimentally by Zhao et al. [16]. Zhao et al. performed discharge experiments of battery cell at four C-rates at 0.5 C, 1 C, 3 C and 5 C. In these experiments, the single battery cell was placed in a temperature-controlled chamber, and during the discharge process at the specified C-rate, $U$ and $I$ were recorded. The corresponding $U_{ca}$ and $U_{an}$ as a function of depth of discharge (DoD) were taken from Refs. [33,34]. The correlation of $U_{ca}$ and $U_{an}$ on temperature was derived from Ref. [35]. The source term is determined by

$$\dot{Q}=\frac{Q_{gen}}{V_b}$$ (12)

Here, $\dot{Q}$ denotes the heat generation rate, $V_b$ is the volume of battery. The heat generation rate was calculated based on the voltage $U$ and current $I$ presented. As the C-rate increases, chemical reactions inside the battery occur more rapidly, which contributes to heat generation, and with increased current passing through the battery, more heat is generated due to internal resistance, leading to an increase in heat output. The calculated heat generation rate is shown in Fig. 2.

Fig. 2.

Heat generation rate with DOD at 25 °C ambient temperature.

2.2.3. Calculation of Reynolds number for determining coolant flow

To determine the flow of the coolant, the Reynolds number was calculated using Eq. (13) [36]. The calculated Reynolds number is 898.38. Therefore, flow of the coolant used laminar flow.

$$Re=\frac{{\rho }_c\bullet v\bullet D_0}{{\mu }_c}$$ (13)

Where $Re$ is Reynolds number, $v,D_0,{\mu }_w$ are the velocity, a hydraulic diameter, dynamic viscosity respectively. $D_0$ is formulated as follows [37]:

$$D_0=\frac{4A_{cross}}{l}$$ (14)

Here $A_{cross}$ is the cross-sectional area of the channel, and $l$ is the wetted perimeter.

2.2.4. Initial and boundary conditions

The initial temperature of the entire system and the coolant inlet temperature were set to $298.15K$. The coolant inlet is set to velocity-inlet and the coolant inlet velocity ($v_{in})$ is set to $0.1m/s$. The outlet of the coolant was set to pressure-outlet and set to $0Pa$, assuming atmospheric pressure. Considering natural convection, convection conditions were set with an ambient temperature of $298.15K$, and a convection heat transfer coefficient of $5W/m^2\bullet K$ was applied to all external walls of the BTMS and Li-ion battery cells. Both the outer wall of the system and the inner wall of the cooling plate were designated as non-slip walls. The dimensions of the outer wall of the cooling plate ($h_{out}$) are $65mm$ in height and $2mm$ in width ($w_{out}$), while the inner wall dimensions are $0.5mm$ in height ($h_{in}$) and width ($w_{in}$).

2.2.5. Mesh independence test

To ensure the accuracy of numerical simulations, it is crucial to investigate the mesh independence of the model. Generally, a larger mesh size can reduce the accuracy of the analysis because it may fail to capture fine details of physical phenomena. Conversely, while a smaller mesh size can enhance the accuracy of the results, it can also lead to a significant increase in computational cost and time. Therefore, to efficiently manage computational cost and time, it is essential to run a mesh independence test. In this paper, the $T_{\mathit{max}}$ and $T_{\mathit{min}}$ at each C-rate according to the number of elements are presented in Table 3. The meanings of $T_{\mathit{max}}$ and $T_{\mathit{min}}$ are as follows:

$$T_{\mathit{max}}=\mathit{Max}\left(T_{avg,1},T_{avg,2},T_{avg,3},\cdots T_{avg,69},T_{avg,70},T_{avg,71}\right)$$ (15)

$$T_{\mathit{min}}=\mathit{Min}\left(T_{avg,1},T_{avg,2},T_{avg,3},\cdots T_{avg,69},T_{avg,70},T_{avg,71}\right)$$ (16)

Where $T_{avg,n}$ is the average temperature of the nth cell. The results of the mesh independence test indicate that when the number of elements exceeds 4,109,224, there is almost no difference in the changes in $T_{\mathit{max}}$ and $T_{\mathit{min}}$. Therefore, this study uses a mesh with 4,109,224 elements.

Table 3.

Mesh independence test

Number of Element	$T_{\mathit{max}}$ [$°C]$				$T_{\mathit{min}}$ [$°C]$
	0.5C	1C	3C	5C	0.5C	1C	3C	5C
1,929,419	28.42	30.13	34.08	49.05	27.91	29.01	31.05	38.82
2,357,648	27.83	29.50	33.37	48.03	27.14	28.20	30.17	37.80
4,109,224	26.97	28.59	32.34	46.55	26.18	27.20	29.10	36.45
5,528,531	26.97	28.59	32.35	46.55	26.18	27.20	29.10	36.46

2.3. Numerical analysis result

In this paper, The average temperature difference of the battery cells was defined to calculate the temperature uniformity of battery modules as follows.

$$T_{diff}=T_{\mathit{max}}-T_{\mathit{min}}$$ (17)

The cylindrical battery module with cooling plate BTMS was fully discharged from a full charge to various C-rate, with $T_{\mathit{max}}$, $T_{\mathit{min}}$, and $T_{diff}$ shown in Fig. 3. The battery module exhibits similar temperature behavior across all C-rates during discharge. Initially, the temperature of the battery module rises from the start to the mid-discharge phase due to continuous heat accumulation. From mid-discharge to mid-to-late discharge, the temperature decreases because of the battery's low heat generation rate. At the end of the discharge, the battery's heat generation rate rapidly increases, causing the temperature of the battery module to rise quickly. The rapid increase in battery heat rate at the end of discharge, which causes the temperature of the battery module to rise rapidly, is due to the increased internal resistance of the battery, which results in higher heat generation [38]. As the C-rate increases, so does the heat generation rate, resulting in higher overall temperatures ($T_{\mathit{max}},T_{mim}\text{,}$ and $T_{diff}$) during discharge. Table 4 presents the values of $T_{\mathit{max}}$, $T_{\mathit{min}}$, and $T_{diff}$ at each C-rate at the end of the discharge. The increase in $T_{\mathit{max}}$ is more pronounced than the increase in $T_{\mathit{min}}$ as the C-rate rises. Specifically, when the C-rate changes from 0.5C to 5C, $T_{\mathit{min}}$ increases by $10.27°C$, while $T_{\mathit{max}}$ increases by $19.58°C$, indicating that the heat generation rate has a greater impact on $T_{\mathit{max}}$ than on $T_{\mathit{min}}$. Fig. 4 showed that the temperature distribution of the battery module upon completing the discharge at each C-rate. In Fig. 4, $T_{\mathit{max}}$ is located at the inlet side of the cooling plate, and the cell with $T_{\mathit{min}}$ is at the outlet side. This occurs because the coolant temperature increases as it flows through the cooling channel, leading to higher temperatures at the outlet than at the inlet, thus reducing cooling efficiency. At the battery cell level, the center of the cell has a higher temperature than the top and bottom, indicating greater heat accumulation in the center. As a result, in a battery module, the maximum temperature is seen in the center of the cell near the outlet, and the minimum temperature is found at the top and bottom of the cell near the inlet.

Fig. 3.

Thermal fluid analysis results of the battery module for (a) $T_{\mathit{max}}$, (b) $T_{\mathit{min}}$, and (c) $T_{diff}$.

Table 4.

Average temperature of cells after discharge for different C-rate.

C-rate	$T_{\mathit{max}}(°C)$	$T_{\mathit{min}}(°C)$	$T_{diff}(°C)$
$0.5C$	$26.97$	$26.18$	$0.80$
$1C$	$28.59$	$27.2$ 0	$1.39$
$3C$	$32.34$	$29.1$ 0	$3.24$
$5C$	$46.55$	$36.45$	$10.11$

Fig. 4.

Temperature distribution of a battery module and cooling plate after discharge for different C-rate (a) $0.5C$ (b) $1C$ (c) $3C$ (d) $5C$.

3. Statistical analysis for thermal behaviors of an Li-ion battery

3.1. Design factors and responses of design of experiment (DoE)

DoE is a statistical technique used to analyze the correlation between responses and design factors to obtain information about the impact of design factors on responses with minimal experiments. This method is highly efficient, enabling researchers to derive meaningful insights while conserving resources. FFD is a statistical method used in experimental design to systematically explore the influence of multiple factors on response variables. This method allows for an exhaustive evaluation of all possible combinations of factors and their levels, which is crucial for identifying both main effects and interactions among factors. By keeping other factors constant and varying each factor at all levels, we can obtain a comprehensive understanding of how each design factor impacts the responses. In this study, the geometry of the cooling plate and the coolant inlet velocity ($h_{out}$, $w_{out}$, $h_{in}$, $w_{in}$, $v_{in}$) were selected as design factors to investigate their influence on the temperature distribution of the BTMS, the mass of the BTMS, and the pressure drop at four different C-rate (0.5C, 1C, 3C, and 5C). Table 5 lists the range and levels of five design factors. In this design, all possible combinations of factors and their levels are tested. For each C-rate, there are $3^5$ sampling points, resulting in a total of 972 sampling points being analyzed across the four C-rates. This extensive sampling ensures a robust dataset, which enhances the reliability of the conclusions drawn from the study.

Table 5.

Range and levels of design factors.

Design factors	Analysis range	Interval	Level
$h_{out}$	$50-80\left[mm\right]$	$15\left[mm\right]$	$3$
$w_{out}$	$2-4\left[mm\right]$	$1\left[mm\right]$	$3$
$h_{in}$	$0.3-0.7\left[mm\right]$	$0.2\left[mm\right]$	$3$
$w_{in}$	$0.3-0.7\left[mm\right]$	$0.2\left[mm\right]$	$3$
$v_{in}$	$0.075-0.125\left[m/s\right]$	$0.025\left[m/s\right]$	$3$

3.2. Analysis of variance (ANOVA)

ANOVA is a statistical method utilized to determine whether there are considerable differences among the means of three or more independent groups. This method compares the variance within each group to the variance between the groups. ANOVA is employed to evaluate the theory that the means of multiple groups are equivalent. It helps identify whether any of the differences between the means are statistically significant. ANOVA is a powerful tool in experimental design and analysis, allowing researchers to make informed decisions about the significance of their results [39]. The basic definition of ANOVA is as follow:

$$S{S}_{total}=S{S}_{between}+S{S}_{within}$$ (18)

In the context of ANOVA, the total sum of squares ($S{S}_{total}$) is decomposed into two components: the variability between groups ($S{S}_{between}$) and the variability within groups ($S{S}_{within}$).

$$S{S}_{total}=\sum _{i=1}^a\sum _{j=1}^n{\left(X_{ij}-\overline{X_{..}}\right)}^2$$ (19)

$$S{S}_{between}=n\sum _{i=1}^a{\left(\overline{X_{i.}}-\overline{X_{..}}\right)}^2$$ (20)

$$S{S}_{within}=\sum _{i=1}^a\sum _{j=1}^n{\left(X_{ij}-{\bar{X}}_{i.}\right)}^2$$ (21)

$S{S}_{total}$ is computed by summing the squared differences between each individual observation ($X_{ij})$ and the overall mean ($\overline{X_{..}}$), where $a$ represents the number of groups and $n$ represents the number of observations per group. $S{S}_{between}$ is calculated by summing the squared differences between the group means ($\overline{X_i})$ and $\bar{X}$, multiplied by $n$. Lastly, $S{S}_{within}$ is determined by summing the squared differences between $X_{ij}$ and the mean of its respective group ($\overline{X_{i.}}$). These components help in assessing the significance of the differences among group means, thus allowing researchers to determine whether any observed differences are statistically significant [5,40].

$$\sum _{i=1}^a\sum _{j=1}^n{\left(X_{ij}-\overline{X_{..}}\right)}^2=n\sum _{i=1}^a{\left(\overline{X_{i.}}-\overline{X_{..}}\right)}^2+\sum _{i=1}^a\sum _{j=1}^n{\left(X_{ij}-{\bar{X}}_{i.}\right)}^2$$ (22)

Using Eq. (18) through Eq. (21), we can derive Eq. (22), which encapsulates the overall decomposition of the sum of squares in ANOVA.

$$\overline{X_{..}}=\frac{1}{n}\sum _{i=1}^a\sum _{j=1}^n\left(X_{ij}\right)$$ (23)

$$\overline{X_{ij}}=\frac{1}{n}\sum _{j=1}^n\left(X_{ij}\right)$$ (24)

$$Do{F}_{between}=a-1$$ (25)

$$Do{F}_{whithin}=an-a$$ (26)

Eq. (23) ∼ Eq. (26) provide a detailed breakdown of the sum of squares in ANOVA, along with the definitions of the overall mean and group means, as well as the degrees of freedom ($DoF$) associated with the analysis. By calculating the group means (${\bar{X}}_{i.}$) and the overall mean (${\bar{X}}_{..}$), we measure the variation between and within groups.

$$F-value=\frac{M{S}_{between}}{M{S}_{within}}=\frac{\frac{S{S}_{between}}{Do{F}_{between}}}{\frac{S{S}_{within}}{Do{F}_{within}}}$$ (27)

$$P-value=f\left(F-value,Do{F}_{between},Do{F}_{within}\right)$$ (28)

$$Percentcontribution(\%)=\frac{S{S}_{Factor}}{S{S}_{Total}}\times 100$$ (29)

The F-value, which is the ratio of the mean square between treatments ($M{S}_{between}$) to the mean square within treatments ($M{S}_{within}$), indicates the likelihood that the differences between the means are significant, with a higher F-value suggesting a greater significance. The P-value is derived from the F-value and the degrees of freedom, indicating the probability that the observed differences occurred by chance. The statistical significance of the design factors was established based on the P-value [41]. Finally, the percentage contribution of individual design factor to the total variation is evaluated by the ratio of the $S{S}_{Factor}$ to $S{S}_{Total}$.

4. Result and discussion

We analyzed the changes in $T_{\mathit{max}}$, $T_{\mathit{min}}$, and $T_{diff}$ of the battery module according to the shape of the cooling plate and the inlet velocity of the coolant when the battery module was fully discharged at four different C-rates (0.5C, 1C, 3C, 5C). Particularly, a higher rate of change in $T_{\mathit{max}}$, $T_{\mathit{min}}$, and $T_{diff}$ was observed at higher C-rates due to increased heat generation. Additionally, we analyzed the changes in BTMS mass and pressure drop according to the shape of the cooling plate and the inlet velocity of the coolant. We used ANOVA to analyze the contribution of each design factor to the responses for the five design factors. As a result, we confirmed that $w_{out}$ had the most significant impact on $T_{diff}$ and pressure drop, while $h_{out}$ influenced BTMS mass the most.

4.1. Effect of design factors on the responses

Fig. 5 showed that the variations in $T_{\mathit{max}},T_{\mathit{min}}$, and $T_{diff}$ with respect to the height of the outer wall of the cooling plate ($h_{out}$). Increasing $h_{out}$ results in an increased contact area between the cooling plate and the battery cell, thereby increasing the heat transfer surface area and enhancing the cooling capacity. Accordingly, Fig. 5, $T_{\mathit{max}}$ and $T_{\mathit{min}}$ decrease as $h_{out}$ increases in all C-rates. Furthermore, it is evident that the decrease in $T_{\mathit{max}}$ surpasses that of $T_{\mathit{min}}$, leading to a significant reduction in $T_{diff}$. When $h_{out}$ was 50 mm, $T_{diff}$ was observed to be $0.95°C$, $1.66°C$, $4.09°C$, and $12.65°C$ for each C-rate, respectively. When $h_{out}$ increased to 65 mm, $T_{diff}$ decreased by 15.79 %, 16.27 %, 20.78 %, and 20.08 % to $0.8°C$, $1.89°C$, $3.24°C$, and $10.11°C$, respectively, for each C-rate. Upon further increase of $h_{out}$ from 65 mm to 80 mm, the reduction rate slightly decreased. Consequently, $T_{diff}$ for each C-rate decreased by 15 %, 15.11 %, 16.67 %, and 16.82 %, amounting to $0.68°C$, $1.18°C$, $2.70°C$, and $8.41°C$, respectively. Table 6showed that the variation in the mass and pressure drop of the BTMS with changing $h_{out}$. As $h_{out}$ increased, the size of the cooling plate expanded, resulting in a slight increase in the mass of the BTMS. Conversely, the pressure drop decreased slightly, attributed to the increase in hydraulic diameter, which reduced the frictional resistance between the fluid and the internal walls of the cooling plate (see Table 7).

Fig. 5.

The effect of $h_{out}$ on the variation of temperature at different C-rate (a) $T_{\mathit{max}}$ (b) $T_{\mathit{min}}$ (c) $T_{diff}$.

Table 6.

The effect of $h_{out}$ in the cooling plate on the variations of BTMS mass and pressure drop.

$h_{out}$ [$mm$]	BTMS mass [g]	Pressure drop [Pa]
50	2508	883.47
65	2549	880.71
80	2590	878.77

Table 7.

The effect of $w_{out}$ in the cooling plate on the variations of BTMS mass and pressure drop.

$w_{out}$ [$mm$]	BTMS mass [g]	Pressure drop [Pa]
2	2549	880.71
3	2598	255.14
4	2647	134.33

Fig. 6 showed that the variations in $T_{\mathit{max}}$, $T_{\mathit{min}}$, and $T_{diff}$ with respect to the width of the outer wall of the cooling plate ($w_{out}$). As $w_{out}$ increases, the flow within the cooling plate capable of cooling the battery also increases. Consequently, when $w_{out}$ increases from $3mm$ to $5mm$, $T_{\mathit{max}}$ and $T_{\mathit{min}}$ notably decrease. However, when $w_{out}$ increases from 5 mm to 7 mm, the reduction in $T_{\mathit{max}}$ and $T_{diff}$ diminishes significantly. When $w_{out}$ is 3 mm, $T_{diff}$ is observed to be $0.8°C$, $1.39°C$, $3.24°C$, and $10.11°C$ for each C-rate, respectively. Upon increasing $w_{out}$ by $5mm$, $T_{diff}$ decreases by 25.00 %, 23.74 %, 28.40 %, and 29.38 %, resulting in values of $0.6°C$, $1.06°C$, $2.32°C$, and $7.14°C$ for each C-rate, respectively. Similarly, when $w_{out}$ increases from 5 mm to 7 mm, the reduction rate decreases slightly, and $T_{diff}$ for each C-rate decreases by 3.33 %, 4.72 %, 5.60 %, and 6.02 %, resulting in values of $0.58°C$, $1.01°C$, $2.19°C$, and $6.71°C$, respectively. Like the increase in $h_{out}$, the increase in $w_{out}$ leads to a slight increase in the size of the cooling plate, resulting in a slight increase in the mass of the BTMS. The pressure drop also decreases more significantly as $w_{out}$ increases compared to the increase in $h_{out}$. This is attributed to the higher impact of $w_{out}$ on the hydraulic diameter than $h_{out}$, leading to a greater reduction in frictional resistance between the fluid and the internal walls of the cooling plate.

Fig. 6.

The effect of $w_{out}$ on the variation of temperature at different C-rate (a) $T_{\mathit{max}}$ (b) $T_{\mathit{min}}$ (c) $T_{diff}$.

Fig. 7 showed that the variations in $T_{\mathit{max}}$, $T_{\mathit{min}}$, and $T_{diff}$ with respect to the height of the inner wall of the cooling plate ($h_{in}$) and the width of the inner wall of the cooling plate ($w_{in}$), respectively. It can be observed that $h_{in}$ does not exert a significant influence on $T_{\mathit{max}}$, $T_{\mathit{min}}$, and $T_{diff}$. Conversely, $w_{in}$ appears to have a substantial impact on $T_{\mathit{max}}$, $T_{\mathit{min}}$, and $T_{diff}$. As $w_{in}$ increases, the gap between the cooling fluid and the battery cells increases, which increases the conduction thermal resistance. However, $h_{in}$ does not affect the conduction thermal resistance between the fluid and the battery cell. Therefore, $h_{in}$ does not have a significant impact on $T_{diff}$, but $w_{in}$ has a greater impact on $T_{diff}$ as it increases. When $h_{in}$ is $0.3mm$, $T_{diff}$ is calculated to be $0.79°C$, $1.38°C$, $3.22°C$, and $10.03°C$ for each C-rate, respectively. When $h_{in}$ increases to 5 mm, there is little change in $T_{diff}$, which is calculated at $0.80°C$, $1.39°C$, $3.24°C$, and $10.11°C$ at each C-rate. Similarly, as $h_{in}$ increases from 5 mm to 7 mm, $T_{diff}$ remains almost unchanged at each C-rate, with values of $0.80°C$, $1.39°C$, $3.25°C$, and $10.14°C$. In contrast, when $w_{in}$ increases from $0.3mm$ to $0.5mm$, $T_{diff}$ increases significantly from $0.65°C$, $1.14°C$, $2.56°C$, and $7.91°C$ to $0.80°C$, $1.39°C$, $3.24°C$, and $10.11°C$ at each C-rate. When $w_{in}$ increases from 5 mm to 7 mm, the $T_{diff}$ for each C-rate increases even more, to $1.21°C$, $2.10°C$, $5.11°C$, and $15.92°C$, respectively. As the values of both design factors increase, the volume of the cooling plate increases, leading to a rise in the mass of the BTMS. Since $h_{in}$ dominates the volume of the cooling plate more than $w_{in}$, it can be observed that $h_{in}$ also significantly influences the change in the mass of the BTMS. Additionally, while $h_{in}$ has a significant impact on the pressure drop, $w_{in}$ does not appear to significantly affect the pressure drop (see Table 9).

Fig. 7.

The effect of $h_{in}$ and $w_{in}$ on the variation of temperature at different C-rate (a) $T_{\mathit{max}}$ (b) $T_{\mathit{min}}$ (c) $T_{diff}$ (d) $T_{\mathit{max}}$ (e) $T_{\mathit{min}}$ (f) $T_{diff}$.

Table 9.

The effect of $w_{in}$ in the cooling plate on the variations of BTMS mass and pressure drop.

$w_{in}$ [$mm$]	BTMS Mass [g]	Pressure Drop [Pa]
0.3	2549	880.65
0.5	2549	880.71
0.7	2550	880.79

Fig. 8 showed that the variation in $T_{\mathit{max}}$, $T_{\mathit{min}}$, and $T_{diff}$ with respect to the coolant inlet velocity ($v_{in}$), while Table 10 represents the changes in the mass of the BTMS and pressure drop corresponding to $v_{in}$. $v_{in}$ significantly influences the $T_{diff}$ and the pressure drop. The heat transfer coefficient is typically determined by the velocity of the flowing fluid, with an increase in fluid velocity generally causing an increase in the heat transfer coefficient. This is because higher fluid velocity disrupts the boundary layer around the flowing fluid more efficiently and increases the heat transfer area. Therefore, as $v_{in}$ increases, $T_{\mathit{max}}$ decreases linearly, ultimately leading to a decrease in $T_{diff}$. When $v_{in}$ is $0.075m/s$, $T_{diff}$ is observed to be $1.00°C$, $1.74°C$, $4.19°C$, and $13.05°C$ for each C-rate, respectively. Upon increasing $v_{in}$ to $0.1m/s$, $T_{diff}$ decreases to $0.80°C$, $1.39°C$, $3.24°C$, and $10.11°C$ for each C-rate, respectively. When $v_{in}$ is $0.125m/s$, $T_{diff}$ is observed to be $0.68°C$, $1.18°C$, $2.67°C$, and $8.30°C$ for each C-rate, respectively. Furthermore, as $v_{in}$ increases, the pressure drop increases due to energy losses resulting from fluid friction and resistance.

Fig. 8.

The effect of $v_{in}$ on the variation of temperature at different C-rate (a) $T_{\mathit{max}}$ (b) $T_{\mathit{min}}$ (c) $T_{diff}$.

Table 10.

The effect of $v_{in}$ in the cooling plate on the variations of BTMS mass and pressure drop.

$v_{in}$ [$m/s$]	BTMS Mass [g]	Pressure Drop [Pa]
0.075	2549	658.15
0.1	2549	880.71
0.125	2549	1104.97

4.2. ANOVA results for responses

For each design factor, the BTMS mass, pressure drop, and $T_{diff}$ at the four C-rates were statistically analyzed for each design factor using ANOVA. The higher the percent contribution, the more it is a design factor that affects the response. A design factor was considered to have a significant effect on the response if its P-value was below the significance level of 0.05 (5 %).

According to Table 11, the design factor that has the greatest impact on $T_{diff}$ is $w_{out}$, and all design factors except $h_{in}$ appear to have a significant impact on the response. Although it is slightly different for each C-rate, $w_{out}$'s contribution to $T_{diff}$ was about 40 %, $w_{in}$ was about 14 %, $h_{in}$ was about 8 %, and $v_{in}$ was about 7 %. As C-rate increases from 0.5C to 3C, $h_{out}$'s influence on $T_{diff}$ increases little by little, and then decreases again when it reaches 5C. Conversely, the impact of $w_{out}$ and $h_{in}$ on $T_{diff}$ gradually decreases as the C-rate increases from 0.5C to 3C, and then increases again when it reaches 5C. And $v_{in}$'s influence on $T_{diff}$ was found to steadily increase as C-rate increased. The ANOVA results for each C-rate show that the impact of design factors on the response varies depending on the C-rate. Table 12 shows the ANOVA results of design factors for BTMS mass and pressure drop. It was confirmed that the three design elements in the order $h_{out}>w_{out}>w_{in}$ had a significant effect on the BTMS mass, and the three design elements in the order $w_{out}>v_{in}>w_{in}$ had a significant effect on the pressure drop. As a result, it was confirmed that all four design factors that affect cooling performance also affect BTMS mass and pressure drop. Since cooling performance, BTMS mass, and pressure drop have a trade-off relationship, when designing a BTMS, the impact of each design element on cooling performance, BTMS mass, and pressure drop must be considered.

Table 11.

ANOVA results for $T_{diff}$ in various C-rates.

Design factors		SS	DoF	MS	F-value	Percent contribution (%)	P-value
0.5C	$h_{out}$	0.914395	2	0.457197	26.78223	6.88	$\approx$ 0.001
$T_{diff}$	$w_{out}$	5.522692	2	2.761346	161.7571	41.56	$\approx$ 0
	$h_{in}$	0.000618	2	0.000309	0.018104	4.65 $\times {10}^{-3}$	$\approx$ 0.982
	$w_{in}$	1.980988	2	0.990494	58.02223	14.91	$\approx$ 0
	$v_{in}$	0.910499	2	0.455249	26.66811	6.85	$\approx$ 0.001
	$Error$	3.960458	232	0.017070	–	–	–
1C	$h_{out}$	2.876479	2	1.438239	26.94356	7.07	$\approx$ 0.001
$T_{diff}$	$w_{out}$	16.58119	2	8.290595	155.3135	40.73	$\approx$ 0
	$h_{in}$	0.003027	2	0.001513	0.028362	7.44 $\times {10}^{-3}$	$\approx$ 0.972
	$w_{in}$	6.031294	2	3.015647	56.49424	14.82	$\approx$ 0
	$v_{in}$	2.833092	2	1.416546	26.53715	6.96	$\approx$ 0.001
	$Error$	12.38409	232	0.053379	–	–	–
3C	$h_{out}$	29.71925	2	14.85962	35.319979	9.13	$\approx$ 0
$T_{diff}$	$w_{out}$	128.3163	2	64.15817	152.49815	39.41	$\approx$ 0
	$h_{in}$	0.022284	2	0.011142	0.0264844	6.84 $\times {10}^{-3}$	$\approx$ 0.974
	$w_{in}$	45.33518	2	22.66759	53.878810	13.93	$\approx$ 0
	$v_{in}$	24.55749	2	12.27874	29.185464	7.54	$\approx$ 0
	$Error$	97.60574	232	0.420714	–	–	–
5C	$h_{out}$	272.6903	2	136.3451	35.917095	8.90	$\approx$ 0
$T_{diff}$	$w_{out}$	1247.605	2	623.8029	164.32698	40.73	$\approx$ 0
	$h_{in}$	0.223037	2	0.111518	0.0293771	3.36 $\times {10}^{-3}$	$\approx$ 0.975
	$w_{in}$	428.7984	2	214.3992	56.478694	13.9976	$\approx$ 0
	$v_{in}$	233.3529	2	116.6764	30.735817	7.62	$\approx$ 0
	$Error$	880.6969	232	3.796107	-	-	–

Table 12.

ANOVA results for BTMS mass and pressure drop.

Design factors		SS	DoF	MS	F-value	Percent contribution (%)	P-value
BTMS	$h_{out}$	0.438797	2	0.2193985	2644.8255	43.25	$\approx$ 0
Mass	$w_{out}$	0.386056	2	0.1930284	2326.9365	38.05	$\approx$ 0
	$h_{in}$	0.000165	2	8.250 $\times {10}^{-5}$	0.9946132	0.02	$\approx$ 0.73
	$w_{in}$	0.170216	2	0.0851083	1025.9716	16.78	$\approx$ 0
	$v_{in}$	0	2	0	0	0	$\approx$ 0.5
	$Error$	0.019245	232	8.295 $\times {10}^{-5}$	–	–	–
Pressure	$h_{out}$	968.5805	2	484.29029	0.00293730	1.94 $\times {10}^{-3}$	$\approx$ 0.995
Drop	$w_{out}$	6006058.6	2	30030293	182.139068	12.02	$\approx$ 0
	$h_{in}$	52.22036	2	26.110180	0.00015836	1.05 $\times {10}^{-4}$	$\approx$ 0.99
	$w_{in}$	2323768.2	2	11618843	70.4703523	4.65	$\approx$ 0
	$v_{in}$	3387429	2	1693714.5	10.2726796	6.78	$\approx$ 0
	$Error$	38251145	232	164875.62	–	–	–

5. Conclusion

In this study, the effects of cold plate geometry and coolant velocity on battery cooling performance, BTMS mass, and pressure drop at various C-rates (0.5C, 1C, 3C, and 5C) were statistically analyzed. The results revealed that across all C-rates, the order of impact on BTMS $T_{diff}$ was $w_{out}>w_{in}>v_{in}>h_{out}$. However, the magnitude of this influence varied with the C-rate. Specifically, as the C-rate increased from 0.5C to 1C, the impact of $w_{out}$ on $T_{diff}$ decreased from 41.56 % to 40.73 %, and further decreased to 39.41 % as the C-rate increased from 1C to 3C. When the C-rate increased from 3C to 5C, the influence of $w_{out}$ increased again to 40.73 %. In contrast, the effect of $h_{out}$ on $T_{diff}$ exhibited an increasing trend, rising from 6.88 % to 7.07 % as the C-rate increased from 0.5C to 1C, and then significantly increasing to 9.13 % when the C-rate further increased from 1C to 3C.

This study also confirmed that the BTMS mass is significantly influenced in the order of $h_{out}>w_{out}>w_{in}$. The percent contributions of each design variable to the BTMS mass were 43.25 %, 38.05 %, and 16.78 %, respectively. Additionally, it was found that pressure drop is most significantly affected in the order of $w_{out}>v_{in}>w_{in}$. The percent contributions of each design variable to the pressure drop were 12.02 %, 6.78 %, and 4.65 %, respectively. It was confirmed that there is a trade-off relationship between BTMS cooling performance, mass, and pressure drop, highlighting the necessity of considering these trade-offs when designing BTMS.

The results of this study suggest that BTMS design should consider thermal behaviors under various C-rate conditions. Since the influence of design factors varies depending on the discharge rate, it is important to consider not only a single C-rate but also various C-rate conditions when designing a BTMS to ensure safety. Furthermore, the design of BTMS should consider the effects of each design factor on cooling performance, BTMS mass, and pressure drop.

CRediT authorship contribution statement

Seokjun Park: Writing – review & editing, Writing – original draft, Validation, Software, Methodology, Investigation, Conceptualization. Hamin Lee: Writing – review & editing, Investigation, Data curation. Cheonha Park: Writing – review & editing, Validation, Investigation. Chang-Wan Kim: Writing – review & editing, Supervision, Project administration, Funding acquisition.

Availability of data and materials

The data that support the findings of this study are available from the authors, upon reasonable request.

Funding

This research was supported by National Research Foundation of Korea (NRF) grant funded by the Korea government the Ministry of Science, ICT & Future Planning (RS-2024-00352401), and the Konkuk University Researcher Fund in 2024.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.