Electrochemical–Thermal Fluid Coupled Analysis and Statistical Analysis of Cooling System for Large Pouch Cells

Abstract

In large-format pouch cells for electric vehicles, the issues of elevated and non-uniform temperatures resulting from heat generation are intensified, necessitating the use of liquid cooling systems. The design factors of the liquid cooling system influence the maximum temperature and temperature differences in the module as well as the pumping power of the cooling system. Although it is known that these design factors interact, research on these interactions and their effects is currently lacking. In this study, the individual as well as interaction effects of design factors on the performance of the liquid cooling system for a large-format pouch cell module were investigated using design of experiment and analyzed through statistical methods. Electrochemical–thermal fluid coupled analysis was used to calculate the performance according to the design factors of the liquid cooling system. The wall and channel widths are factors that directly determine the coolant flow velocity and cooling plate heat capacity, and they exhibited major effects on all three responses. Moreover, the influence of each design factor tended to change in response to variations in the other design factors. Thus, the effects of each factor individually and of interactions between factors were quantitatively compared and evaluated for significance. The width of the walls was found to contribute the most to the maximum temperature (36.00%) and pumping power (57.56%), while the width of the channels contributed the most to the temperature difference (38.24%), indicating that they are the main influencing factors.

1. Introduction

Lithium-ion batteries (LIBs) have recently become the most widely used energy source for electric vehicles (EVs) [1,2]. LIBs are suitable for use in EVs because they have a high energy density and long life cycle compared to other batteries. Among the various types of LIB cells, pouch-type cells have been applied to many vehicles due to their high energy density resulting from their stacked structure. In addition, the size of pouch-type cells has been increased to ensure their capacity is sufficient for EV battery systems [3]. The performance of LIBs is critically affected by temperature [4,5]. During the charging and discharging of LIBs, electrochemical heat is generated, which can exacerbate the problem of temperature increases and differences due to heat generation, especially in large-format pouch cells. In EVs, battery cooling systems are applied to prevent the deterioration and failure of the battery system due to these temperature fluctuations. Liquid cooling can more effectively remove heat compared to air cooling, so it is currently widely used in battery cooling systems for EVs [6].

Research has been conducted on reducing power consumption while effectively cooling battery systems containing large-format pouch cells via liquid-cooling systems. Using numerical methods, Zang et al. analyzed the effects of coolant flow rate, cooling start time, and glycol solution concentration on cooling performance in an indirect liquid-cooling plate installed between large-format pouch cells of 55 Ah [7]. Koorata et al. analyzed the effect of the coolant flow rate and temperature on the maximum temperature and temperature differences in battery cells and the pressure drop in an indirect liquid-cooling plate with a U-shaped channel enclosed in a 20 Ah pouch cell [8]. Li et al. investigated the temperature distribution and power dissipation in relation to the position of the cooling plate for a 20 Ah pouch cell stack [9]. Li et al. compared the cooling efficiency of different flow types in a liquid cooling system for 20 Ah pouch cells [10]. While these studies are significant in that they analyzed the impact of design factors to effectively cool large-format pouch cells, they are limited in that they only analyzed the influence of each factor independently. To effectively improve the performance of battery liquid cooling systems, a comprehensive method is needed to analyze the interactions between various factors and their influence on performance.

Design of experiments (DOE) is a systematic methodology used to plan, conduct, and analyze experiments to understand the effects of multiple variables on a particular outcome. In studies conducted at the level of LIB cells, the DOE method was adopted to comprehensively analyze the influence of design factors on performance. Li et al. used a Box–Behnken experimental design based on a 3D electrochemical–thermal model that allows for the consideration of various design factors in prismatic cells [11]. The influence of design factors was subsequently analyzed using analysis of variance (ANOVA). Hosseinzadeh et al. utilized a three-level full-factorial design (FFD) based on a 1D electrochemical–thermal model of a pouch cell [12]. ANOVA was performed using the DOE data to analyze which design factors have a valid effect on the specific energy and specific power. Lee et al. applied FFD to a 3D electrochemical–thermal model of a pouch cell to analyze the effect of design factors on the temperature distribution of the cell [13]. They used ANOVA to evaluate the significance of the design factors. These studies are notable because they demonstrate that DOE can be used to comprehensively analyze the effects of multiple design factors on LIB performance and assess their significance, especially by applying statistical analysis such as ANOVA. However, these studies are limited as they only studied single cells and did not consider cooling systems.

Research has been conducted to extend the application of DOE to battery cooling systems. Ling et al. used thermal fluid analysis and central composite design (CCD) to analyze the performance of a phase change material (PCM)-liquid hybrid cooling system for cylindrical cell modules [14]. Using the DOE results, they applied sensitivity analysis and RSM to comprehensively analyze the influence of design factors on the maximum temperature, temperature differences, mass, and volume of the module. Ye et al. performed thermal fluid analysis and an orthogonal experimental design to analyze the performance of a liquid cooling system in a module composed of prismatic cells [15]. As a result, the influence of the design factors on the temperature difference and pressure drop of the battery module was determined. Rangappa et al. used FFD to comprehensively analyze the thermal behavior, overall volume, and power dissipation depending on design factors of a PCM–liquid hybrid cooling system for pouch-type cells [16]. The authors performed ANOVA using the data collected through FFD to analyze the effect of design factors on the response and used RSM to improve performance. Wang et al. used orthogonal analysis to analyze the design factors of a bionic spider-web channel cold plate and their effect on the temperature of a 7 Ah small pouch cell and pressure drop [17]. Orthogonal analysis was also used by Guo and Li to examine the design factors of parallel-spiral serpentine channel liquid cooling plates for a prismatic cell module and their effects on the thermal behavior of the module and pressure drop [18]. In both studies, they selected the best design of those obtained from the orthogonal analysis method for improving the performance of the cooling system. However, despite the demand for liquid cooling systems for large-scale pouch cells, there are few studies in which DOE was applied to analyze the effects of design factors in such systems. Moreover, of these studies, only a few have applied statistical analysis.

In our study, we propose a methodology based on DOE for statistically analyzing the effects of multiple design factors, both individually and resulting from interactions, on the performance of the liquid cooling system for a large-format pouch cell module. This performance was determined using electrochemical–thermal fluid coupled analysis. We defined the number of walls, channel width, wall width, and cooling plate thickness as design factors and the maximum temperature, temperature difference, and pumping power as responses. A five-level FFD sampling method was used to select sampling points, and 625 data points were obtained. Based on the DOE data, ANOVA was used to analyze the contribution and significance of each design factor on the performance. This study addresses the lack of research considering the effects of multiple design factors on the performance of large-format pouch cell module cooling systems by conducting a comprehensive analysis and quantitative comparison, thereby providing a foundation for future design improvements.

2. Electrochemical–Thermal Fluid Coupled Model of Large-Format Pouch Cell Module Cooling System

2.1. 2P6S Pouch Type Large-Format Pouch Cell Module

The battery module consists of 12 large-format 65 Ah pouch cells and a liquid-cooling plate, as shown in Figure 1. The 12 large-format pouch cells are connected in 2P6S. For the 65 Ah pouch cells that compose the module, the specifications for cells used in production EVs are adopted. A single pouch cell cannot achieve the required voltage and capacity of an electric vehicle. To achieve the targeted performance, the individual cells of electric vehicle battery systems are connected in series or parallel to form modules, which are, in turn, connected to each other to form a pack. Depending on the EV model, the battery module is composed of various connections of individual cells. We adopted a 2P6S (two-parallel six-series) structure, which is the module configuration applied in real commercial vehicles.

Due to their stacked structure, the pouch cells have low thermal conductivity in the out-of-plane direction and high thermal conductivity in the in-plane direction. In many electric vehicles that use large-format pouch cells, the modules are constructed by standing the pouch cells on their sides with one side facing up or down, and the cooling plate is attached to the bottom of the module. In this configuration, the cooling plate can cool the sides of the pouch cell to take advantage of its high in-plane thermal conductivity. Coolant flows inside the cooling plate to facilitate indirect liquid cooling. The internal channels of the cooling plate are equipped with walls that allow the coolant to spread widely and flow in parallel.

In most LIBs, aluminum and copper are used for the positive and negative collectors, respectively, due to their electrical conductivity and chemical reactivity. The application of these materials substantially impacts the thermal behavior in the near-tap region, where the current density is concentrated; in particular, high heating is observed near the positive tap due to aluminum having a lower electrical conductivity than copper. For the electrochemical–thermal fluid coupled model used in this study, the material of each tab was chosen to simulate the behavior of a real pouch cell. Copper is used for the negative tap and busbar, and aluminum for the positive tap and cooling plate.

Compared to water, ethylene glycol–water mixtures have a low freezing point, high boiling point, and high specific viscosity while maintaining good thermal conductivity. They are also chemically stable and can prevent the corrosion of metals. Because of these advantages, ethylene glycol–water mixtures are utilized in a variety of cooling systems, including in automotive applications, and are commonly used in the battery cooling systems of electric vehicles. Therefore, we used 50 ethylene glycol for the coolant. The properties of the used materials are given in

Table 1. Properties of the materials used in this study.

Materials	Density (kg/m3)	Thermal Conductivity (W/m⋅K)	Specific Heat (J/kg⋅K)	Viscosity Coefficient (kg/m⋅s)
Pouch cell [19]	2630.7	In-plane: 28.706 Normal: 1.394	954	-
Copper	8978	381	387.6	-
Aluminum	2719	871	202.4	-
50 Ethylene glycol	1071.11	0.384	3300	0.00339

.

2.2. Electrochemical–Thermal Coupled Model of 65 Ah Pouch Cell

In order to capture the changing heat generation and localized temperature distribution of the battery cell, we used the model proposed by Newman, Tiedemann, Gu, and Kim (NTGK) [20,21,22,23]. In the NTGK model, the volumetric current transfer rate (j_ECH) is calculated as follows [20]:

$$j_{ECH}={\displaystyle \frac{Q_{nominal}}{Q_{refrence}·Vol}}Y(U-({\phi }_{+}-{\phi }_{-}\left)\right)$$

(1)

where Q_nominal and Q_reference are the nominal capacity of the battery and the capacity measured in the discharge test, respectively, and Vol is the total volume of the active area of the cell. φ₊ and φ₋ are the potentials of the positive and negative electrodes, respectively.

Gu expressed Y and U as a function of Depth of Discharge (DoD) [21]. Based on Gu’s proposed equation, consideration of the effect of temperature was added [22]. Y and U are obtained by fitting a cell discharge curve from experiments and expressed as a function of Depth of Discharge (DoD), as follows:

$$U=\left({\sum }_{n=0}^5{a}_n{\left(DoD\right)}^n\right)-C_2\left(T-T_{refrence}\right)$$

(2)

$$Y=\left({\sum }_{n=0}^5{b}_n{\left(DoD\right)}^n\right)\exp \left[-C_1\left({\displaystyle \frac{1}{T}}-{\displaystyle \frac{1}{T_{reference}}}\right)\right]$$

(3)

where a_n and b_n are the model fitting parameters, and C₁ and C₂ are the battery-specific NTGK model constants to reflect the effect of temperature. T and T_reference are the temperature of the cell and the reference temperature, respectively.

The DoD values in Equations (2) and (3) are calculated as follows:

$$DoD={\displaystyle \frac{Vol}{3600Q_{nominal}}}{\int }_0^t{j}_{ECH} dt$$

(4)

In the NTGK model, the heat generated by the electrochemical reaction is calculated as follows:

$${\dot{q}}_{ECH}=j_{ECH}\left[U-\left({\phi }_{+}-{\phi }_{-}\right)-T{\displaystyle \frac{dU}{dT}}\right]$$

(5)

In LIB cells, electrochemical–thermal reactions occur across multiple domains and multiple scales and have been modeled in many studies that distinguish between the particle domain, electrode domain, and cell domain. Accordingly, the Multi-Scale Multi-Domain (MSMD) approach, with coupling of each domain and scale, has been proposed [24]. The NTGK model calculates the electrochemical reaction in the electrode domain. Using the MSMD approach, the temperature and electric field in the cell domain are calculated by coupling the electrode domain with the NTGK model as follows:

$${\displaystyle \frac{\partial {\rho }_b{c}_{p,b}{T}_b}{\partial t}}-{\displaystyle \frac{\partial }{\partial x}}\left(k_{b,x}{\displaystyle \frac{\partial T_b}{\partial x}}\right)-{\displaystyle \frac{\partial }{\partial y}}\left(k_{b,y}{\displaystyle \frac{\partial T_b}{\partial y}}\right)-{\displaystyle \frac{\partial }{\partial z}}\left(k_{b,z}{\displaystyle \frac{\partial T_b}{\partial z}}\right)={\sigma }_{+}{\left|\nabla {\phi }_{+}\right|}^2+{\sigma }_{-}{\left|\nabla {\phi }_{-}\right|}^2+{\dot{q}}_{ECH}$$

(6)

$${\displaystyle \frac{\partial }{\partial x}}\left({\sigma }_{+}{\displaystyle \frac{\partial {\phi }_{+}}{\partial x}}\right)+{\displaystyle \frac{\partial }{\partial y}}\left({\sigma }_{+}{\displaystyle \frac{\partial {\phi }_{+}}{\partial y}}\right)=-j_{ECH}$$

(7)

$${\displaystyle \frac{\partial }{\partial x}}\left({\sigma }_{-}{\displaystyle \frac{\partial {\phi }_{-}}{\partial x}}\right)+{\displaystyle \frac{\partial }{\partial y}}\left({\sigma }_{-}{\displaystyle \frac{\partial {\phi }_{-}}{\partial y}}\right)=j_{ECH}$$

(8)

where σ₊ and σ₋ are the electrical conductivity of the positive and negative electrodes, respectively, and ρ, c_p, and T are the density, specific heat, and temperature, respectively. The subscript b indicates that it refers to a property of a battery cell.

By using the NTGK model, it was able to calculate the voltage over time of a large-format pouch cell during discharge at different discharge rates, as shown in Figure 2. The comparison of the analysis results with the experimental data indicates that our electrochemical model accurately reflects the electrochemical behavior of the actual pouch cell [19].

2.3. Thermal Fluid Model

The transfer of heat generated by the battery module to the cooling plate and out through the coolant can be calculated using thermal fluid analysis. First, the heat transfer in the cooling plate, which is a solid region, is calculated using the energy conservation equation as follows:

$${\displaystyle \frac{\partial }{\partial t}}\left({\rho }_s{c}_{p,s}{T}_s\right)={\displaystyle \frac{\partial }{\partial x}}\left(k_s{\displaystyle \frac{\partial T_s}{\partial x}}\right)+{\displaystyle \frac{\partial }{\partial y}}\left(k_s{\displaystyle \frac{\partial T_s}{\partial y}}\right)+{\displaystyle \frac{\partial }{\partial z}}\left(k_s{\displaystyle \frac{\partial T_s}{\partial z}}\right)$$

(9)

where the subscript s indicates that it is a property of the solid region.

In the fluid domain, flow and heat transfer are calculated using the continuity equation, the momentum equation, and the energy equation, which are presented in order as follows:

$${\displaystyle \frac{\partial {\rho }_f}{\partial t}}+{\displaystyle \frac{\partial \left({\rho }_{f}u\right)}{\partial x}}+{\displaystyle \frac{\partial \left({\rho }_{f}v\right)}{\partial y}}+{\displaystyle \frac{\partial \left({\rho }_{f}w\right)}{\partial z}}=0$$

(10)

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial \left({\rho }_{f}u\right)}{\partial t}}+{\displaystyle \frac{\partial \left({\rho }_{f}uu\right)}{\partial x}}+{\displaystyle \frac{\partial \left({\rho }_{f}uv\right)}{\partial y}}+{\displaystyle \frac{\partial \left({\rho }_{f}uw\right)}{\partial z}}=-{\displaystyle \frac{\partial p}{\partial x}}+{\displaystyle \frac{\partial }{\partial x}}\left({\mu }_f{\displaystyle \frac{\partial u}{\partial x}}\right)+{\displaystyle \frac{\partial }{\partial y}}\left({\mu }_f{\displaystyle \frac{\partial u}{\partial y}}\right)+{\displaystyle \frac{\partial }{\partial z}}\left({\mu }_f{\displaystyle \frac{\partial u}{\partial z}}\right)\\ {\displaystyle \frac{\partial \left({\rho }_{f}v\right)}{\partial t}}+{\displaystyle \frac{\partial \left({\rho }_{f}vu\right)}{\partial x}}+{\displaystyle \frac{\partial \left({\rho }_{f}vv\right)}{\partial y}}+{\displaystyle \frac{\partial \left({\rho }_{f}vw\right)}{\partial z}}=-{\displaystyle \frac{\partial p}{\partial x}}+{\displaystyle \frac{\partial }{\partial x}}\left({\mu }_f{\displaystyle \frac{\partial v}{\partial x}}\right)+{\displaystyle \frac{\partial }{\partial y}}\left({\mu }_f{\displaystyle \frac{\partial v}{\partial y}}\right)+{\displaystyle \frac{\partial }{\partial z}}\left({\mu }_f{\displaystyle \frac{\partial v}{\partial z}}\right)\\ {\displaystyle \frac{\partial \left({\rho }_{f}v\right)}{\partial t}}+{\displaystyle \frac{\partial \left({\rho }_{f}wu\right)}{\partial x}}+{\displaystyle \frac{\partial \left({\rho }_{f}wv\right)}{\partial y}}+{\displaystyle \frac{\partial \left({\rho }_{f}ww\right)}{\partial z}}=-{\displaystyle \frac{\partial p}{\partial x}}+{\displaystyle \frac{\partial }{\partial x}}\left({\mu }_f{\displaystyle \frac{\partial w}{\partial x}}\right)+{\displaystyle \frac{\partial }{\partial y}}\left({\mu }_f{\displaystyle \frac{\partial w}{\partial y}}\right)+{\displaystyle \frac{\partial }{\partial z}}\left({\mu }_f{\displaystyle \frac{\partial w}{\partial z}}\right)\end{array}\right.$$

(11)

$${\displaystyle \frac{\partial }{\partial t}}\left({\rho }_f{c}_{p,f}{T}_f\right)+{\displaystyle \frac{\partial }{\partial x}}\left({\rho }_{f}u{c}_{p,f}{T}_f\right)+{\displaystyle \frac{\partial }{\partial y}}\left({\rho }_{f}v{c}_{pf}{T}_f\right)+{\displaystyle \frac{\partial }{\partial z}}\left({\rho }_{f}w{c}_{p,f}{T}_f\right)={\displaystyle \frac{\partial }{\partial x}}\left(k_f{\displaystyle \frac{\partial T_f}{\partial x}}\right)+{\displaystyle \frac{\partial }{\partial y}}\left(k_f{\displaystyle \frac{\partial T_f}{\partial y}}\right)+{\displaystyle \frac{\partial }{\partial z}}\left(k_f{\displaystyle \frac{\partial T_f}{\partial z}}\right)$$

(12)

where μ is the viscosity modulus, and u, v, and w are the velocities of the fluid in the x, y, and z directions, respectively. The subscript f indicates that it refers to a property of the fluid.

2.4. Conditions of Electrochemical–Thermal Fluid Coupled Analysis

The initial temperature of the battery cell, cooling plate, and coolant was set to 298.15 K. The heat generation of the pouch cell is calculated based on the electrochemical–thermal model, and a discharge condition with a discharge rate of 2 C was applied. In actual EV driving conditions, it is common for lithium-ion batteries to have a discharge rate of 1 C or less [25]. Robustness is achieved by adopting a constant current discharge condition of 2 C, which is a harsh condition of an EV battery system. The coolant flow rate and temperature at the inlet were given as 5 LPM and 298.15 K, respectively. The outlet was set as a pressure outlet with standard atmospheric pressure. Non-slip conditions were applied to the walls of the coolant channel. Convection conditions with an ambient temperature of 298.15 K and a convective heat transfer coefficient of 5 W/m²∙K were applied to all external walls of the battery and cooling plate. The coolant flow was calculated using the realizable k-epsilon model. The turbulence model was selected to consider the complex structure of the channel and changing geometry depending on design factors and to calculate the performance of the indirect liquid-cooling system. The electrochemical–thermal fluid coupled analysis was solved using the finite volume method (FVM) in ANSYS Fluent 2024 R1.

3. Statistical Analysis for the Effects of Design Factors on Performance of Battery Cooling System

3.1. Design Factors and Responses

The design factors were defined as the number of walls (n), channel width (w_channel), wall width (w_wall), and cooling plate thickness (t). Of these, the width of the wall is defined as the ratio of the sum of all wall thicknesses to the total channel width to consider geometry changes due to changes in channel width, as follows:

$$w_{wall}={\displaystyle \frac{n t}{w_{channel}}}$$

(13)

where t is the thickness of each wall. Each wall has the same thickness, and the spacing between walls is the same. The upper and lower bounds for each design factor are set, as shown in

Table 2. Upper and lower bounds of design factors.

Design Factors	Upper Bounds	Lower Bounds
n	2	6
wchannel (mm)	80	140
wwall	0.2	0.6
t (mm)	10	30

. The number of walls does not affect the cross-sectional area of the channel, but the contact area between the coolant and the cooling plate increases as the number of walls increases. Increasing the channel width increases the cross-sectional area of the channel, resulting in a slower flow velocity and an increase in the volume of the channel relative to the cooling plate. There is an opposite effect on wall width when adjusting the ratio of the total thickness of the walls to the channel width. Increasing the thickness of the cooling plate enhances its heat capacity and increases the thermal resistance between the battery cells and the coolant.

The response is defined as the maximum temperature of the battery cells (T_max), the maximum temperature difference (T_diff), and the pumping power (P_pump). T_max is the maximum temperature of all cells in the 2P6S module, calculated as shown in Equation (14). The maximum temperature difference is the difference between the maximum and minimum average temperature of each cell in the module, defined as Equation (15), and is used to evaluate the temperature unevenness within the module. Pumping power is defined as shown in Equation (16), which evaluates the power consumed for cooling.

$$T_{max}=max\left(T_{max,i}\right), \left(1\le i\le 12\right)$$

(14)

$$T_{diff}=max\left(T_{avg, i}\right)-\mathrm{m}\mathrm{i}\mathrm{n}(T_{avg}, j), \left(1\le i,j\le 12\right)$$

(15)

$$P_{pump}=p_{drop}{Q}_{in}$$

(16)

where T_max,i and T_avg,i are the maximum and average temperature of the i-th cell, respectively, and p_drop and Q_in are the pressure drop and inlet flow rate, respectively.

3.2. Five-Level Full Factorial Design

Traditional experimental methods have been commonly used in previous studies to analyze the effects of design factors on the performance of battery cooling systems. However, changes were considered for only one design factor at a time, making it impossible to analyze the effects on performance when simultaneously changing multiple design factors. DOE can allow for the effective analysis of the influence of design factors on battery cooling systems by considering the interaction of multiple design factors [26]. In this study, a full-factorial design (FFD) method was utilized. FFD is a DOE method that takes all possible combinations of levels for all factors, so it is suitable for experimental situations where interactions between factors need to be considered. If we perform l-level FFD for k number of design factors, the total number of sampling points is as follows:

$$number of sampling points=l^k$$

(17)

In this study, five levels of FFD were performed for the four design factors to fully account for the non-linear effects of the factors on each other and on the responses. Each design factor was divided into five equally spaced levels within the upper and lower bounds, as shown in

Table 3. Levels of each design factor.

Design Factors	1st Level	2nd Level	3rd Level	4th Level	5th Level
n	2	3	4	5	6
wchannel (mm)	80	95	110	125	140
wwall	0.2	0.3	0.4	0.5	0.6
t (mm)	10	15	20	25	30

. A total of 625 sampling points were generated, and the responses were obtained through electrochemical–thermal fluid coupled analysis.

3.3. Analysis of Variance (ANOVA)

Analysis of variance (ANOVA) has been used to effectively evaluate the effect of design factors on responses and allows the analysis of multiple factors simultaneously. Thus, the importance of factors can be assessed by considering the individual effects of each factor as well as the interaction effects between factors. Using ANOVA, the total variation is decomposed into between-group variation and within-group variation. The variation is evaluated through sums of squares, and the relationship between the total sum of squares (SST), the between-group sum of squares (SSB), and the within-group sum of squares (SSW) can be expressed as follows:

$$\mathrm{S}\mathrm{S}\mathrm{T}=\mathrm{S}\mathrm{S}\mathrm{B}+\mathrm{S}\mathrm{S}\mathrm{W}$$

(18)

The total sum of squares in Equation (18) represents the total variation in the data, calculated as follows:

$$\mathrm{S}\mathrm{S}\mathrm{T}={\sum }_{i=1}^k{\sum }_{j=1}^{n_i}{\left(Y_{ij}-\overline{Y}\right)}^2$$

(19)

where Y_ij is the j-th observation in the i-th population and $\overline{Y}$ is the overall mean. In this section, i and j denote the indexes of the population and the observation within the population, respectively, and n_i denotes the number of samples in the i-th population.

The between-group sum of squares in Equation (18) represents the variation due to the difference between the mean of each group and the overall mean, and it is calculated as follows:

$$\mathrm{S}\mathrm{S}\mathrm{B}={\sum }_{i=1}^k{n}_i{\left({\overline{Y}}_i-\overline{Y}\right)}^2$$

(20)

where ${\overline{Y}}_i$ means the mean of the i-th population.

The within-group sum of squares in Equation (18) represents the variation due to the difference between the data points within each group and the mean of that group, and it is calculated as follows:

$$\mathrm{S}\mathrm{S}\mathrm{W}={\sum }_{i=1}^k{\sum }_{j=1}^{n_i}(Y_{ij}-{\overline{Y}}_i)$$

(21)

By decomposing total variation into between-group and within-group variation, ANOVA can be used to determine whether the effect of a factor is statistically significant across different combination levels.

The variation can be normalized, and the comparison of different magnitudes facilitated using the root mean square, which is the sum of squares divided by the degrees of freedom. The between-group degrees of freedom (df_between) and within-group degrees of freedom (df_within) are, respectively, as follows:

$$d{f}_{between}=k-1$$

(22)

$$d{f}_{within}=N-k$$

(23)

The between-group mean squares (MSB) and within-group mean squares (MSW) are calculated by dividing the sum of squares by the degrees of freedom, respectively, as follows:

$$\mathrm{M}\mathrm{S}\mathrm{B}={\displaystyle \frac{\mathrm{S}\mathrm{S}\mathrm{B}}{d{f}_{between}}}$$

(24)

$$\mathrm{M}\mathrm{S}\mathrm{W}={\displaystyle \frac{\mathrm{S}\mathrm{S}\mathrm{W}}{d{f}_{within}}}$$

(25)

The effect of each factor can be evaluated in ANOVA by comparing between-group variation to within-group variation using the F-value. The F-statistic is the between-group mean squared divided by the within-group mean squared, defined as follows:

$$\mathrm{F}={\displaystyle \frac{\mathrm{M}\mathrm{S}\mathrm{B}}{\mathrm{M}\mathrm{S}\mathrm{W}}}$$

(26)

A large F-value means that the between-group variation is greater than the within-group variation, suggesting that the factor has a valid effect on the response. To assess whether a factor is statistically significant, the p-value should be used. If the p-value is less than the significance level (0.05), the effect of the design variable is considered statistically significant and not due to chance.

The importance of a factor can also be assessed based on its contribution. The contribution (η²) is the ratio of the sum of squares between groups to the total sum of squares, calculated as follows:

$${\eta }^2={\displaystyle \frac{\mathrm{S}\mathrm{S}\mathrm{B}}{\mathrm{S}\mathrm{S}\mathrm{T}}}$$

(27)

Contribution indicates the percentage of variation in the response accounted for by the influence of the design factor. This provides an intuitive way to understand how much influence a factor has on a response.

4. Result and Discussion

4.1. Five-Level FFD Results

The results from analyzing the effects of design factors on maximum temperature, temperature difference, and pumping power using five-level FFD are illustrated in Figure 3, Figure 4 and Figure 5. Each figure consists of a small 5 × 5 grid within a larger 5 × 5 grid. The local x-axis of the small grid represents the number of walls, and the local y-axis represents the channel width. The global x-axis represents the wall width, and the global y-axis is the cooling plate thickness. The color of each grid cell indicates the magnitude of the response value for the factor values in that grid.

As shown in Figure 3, there is no consistent trend in the effect of each design factor on the maximum temperature when considering the variation of multiple factors simultaneously through FFD. Overall, the maximum temperature decreased as n increased. However, when w_wall and w_channel is small, the maximum value was observed at n = 3 instead of n = 2. Increasing w_channel tends to decrease the maximum temperature, but with n = 2, a trend in decreasing maximum temperature toward the median value of w_channel is observed. w_wall had a relatively consistent effect on maximum temperature, with a decrease in maximum temperature as w_wall increases. The maximum temperature tends to decrease as t increases, but the opposite trend is evident for larger n, w_channel, and w_wall. The design factors are observed to influence each other, and the same design factor can have opposite effects on the maximum temperature depending on the combination of design factors. This suggests that a design considering only the individual effects of design factors may fall short of achieving the target performance.

Figure 3. Five-level FFD results: maximum temperature.

Figure 3. Five-level FFD results: maximum temperature.

The FFD results for temperature difference are shown in Figure 4. As opposed to the maximum temperature, the temperature difference increases as n is increased. However, this trend weakens as w_channel increases, with a minimum value at n = 4 or 5. Increasing w_channel tends to diminish the temperature difference, but for n = 1, the temperature difference decreases toward the median value of w_channel. w_wall has a similar effect on the temperature difference as on the maximum temperature. t tends to be inversely related to temperature difference overall but proportional when n is small and w_channel and w_wall are large. For the temperature difference, each design factor appears to have an interactive effect. As described in Section 3.1, each design factor influences and interacts with the channel cross-sectional area, coolant-cooling plate contact area, flow rate, heat capacity, and thermal resistance. The heat dissipated through the cooling channel is the result of a complex interaction between the channel cross-sectional area, coolant contact area, and flow rate. Therefore, when changing the design factors that determine the cooling plate and channel geometry for improving the thermal performance of the cooling system, the interaction between the design factors must be carefully considered.

Figure 4. Five-level FFD results: temperature difference.

Figure 4. Five-level FFD results: temperature difference.

As seen in Figure 5, the cooling plate thickness has no effect on the pumping power since it does not change the cooling channel geometry. However, n, w_channel, and w_wall, which determine the channel geometry, have significant effects. The pumping power increases as n and w_wall increase and as w_channel decreases. The effects of the three design factors on pumping power exhibit a consistent trend. In addition, it can be seen that the three design factors interact with each other in terms of pumping power. n has a stronger effect on pumping power as w_channel decreases and w_wall increases. w_channel has a stronger effect on pumping power as n and w_wall are increased. The change in pumping power due to w_wall is enhanced as n increases and w_channel decreases. Thus, even though the effects of individual design factors on the response tend to be consistent, better performance can be achieved by considering the interactions between design factors.

We compared these results with those of the study by Qian et al., who investigated the influence of individual design factors on the performance of indirect cooling plates [27]. Their cooling plate is similar to the cooling system used in this study in terms of its mini-channels and structure. First, they analyzed the effect of the mini-channel number on the performance of the battery cooling system, and in our study, the number of channels is determined based on the number of walls. In their study, the maximum temperature and pressure drop decreased as the number of channels was increased, which is consistent with the trend in our study. However, they did not capture that if the walls and channels are small in width, the highest maximum temperature may not occur at the lowest number of channels. Meanwhile, they found that an even number of channels resulted in a lower maximum temperature and higher pressure drop. The performance differences based on whether n is even or odd are also observed in this study, as shown in Figure 3, Figure 4 and Figure 5; however, this was only when w_wall was small, and the even or oddness of n had no effect at higher values of w_wall. They also investigated the effect of channel width. In this case, the number of channels was fixed, and they found that both the maximum temperature of batteries and the pressure drop decreased as the channel width increased. However, in our results, we can see that the maximum temperature does not decrease as w_channel increases when n is 2 but has the lowest maximum temperature in the middle of the range. The DOE method is thus able to successfully capture the interaction between the factors, which is not captured using traditional methods.

Figure 5. Five-level FFD results: pumping power.

Figure 5. Five-level FFD results: pumping power.

By using five-level FFD, it was possible to comprehensively analyze the influence of each design factor and the interactions between design factors when multiple design factors are varied simultaneously. However, it is difficult to quantitatively determine the valid influence of each design factor compared to the remaining design factors based on the DOE results alone. Applying statistical analysis to the DOE results is necessary to determine whether each design factor has significant effects and which have a dominant influence.

4.2. Results of Statistical Analysis Using ANOVA

The effects of each of the three design factors on performance and their interaction effects were analyzed using ANOVA. The influence of a design factor is regarded as stronger the higher the percentage contribution of the factor. The influence of design factors is considered significant when the p-value is less than 0.05, which is the significance level. First,

Table 4. ANOVA results for maximum temperature.

Source	Sum of Squares	Degrees of Freedom	Mean Square	F-Value	Percent Contribution	p-Value
n	0.18988	4	0.04747	1220.06	7.31%	<0.05
wchannel	0.79786	4	0.19947	5126.58	30.72%	<0.05
wwall	0.93507	4	0.23377	6008.19	36.00%	<0.05
t	0.27986	4	0.06996	1798.19	10.77%	<0.05
n:wchannel	0.02689	16	0.00168	43.19	1.04%	<0.05
n:wwall	0.00502	16	0.00031	8.06	0.19%	<0.05
n:t	0.00156	16	0.0001	2.51	0.06%	<0.05
wchannel:wwall	0.02077	16	0.0013	33.37	0.80%	<0.05
wchannel:t	0.20709	16	0.01294	332.66	7.97%	<0.05
wwalll:t	0.11347	16	0.00709	182.28	4.37%	<0.05
Error	0.01992	512	0.00004

shows the ANOVA results for maximum temperature. The influence of individual design factors on the maximum temperature decreases according to the order w_wall, w_channel, t, and n. It is observed that the factors affecting the coolant flow velocity and the heat capacity of the cooling plate have a strong influence on the maximum temperature. The contribution of the interaction of w_channel and t to the maximum temperature was 7.97%, which is higher than the contribution of factor n alone, at 7.31%. The p-values of all design factors and their interactions for maximum temperature were less than significant. This means that the change in the effect of an individual factor when the combination of factors changes is statistically significant and not due to chance.

As shown in

Table 5. ANOVA results for temperature differences.

Source	Sum of Squares	Degrees of Freedom	Mean Square	F-Value	Percent Contribution	p-Value
n	0.05333	4	0.01333	590.38	3.11%	<0.05
wchannel	0.65672	4	0.16418	7270.63	38.24%	<0.05
wwall	0.13284	4	0.03321	1470.65	7.73%	<0.05
t	0.49593	4	0.12398	5490.59	28.88%	<0.05
n:wchannel	0.01263	16	0.00079	34.96	0.74%	<0.05
n:wwall	0.00877	16	0.00055	24.28	0.51%	<0.05
n:t	0.01659	16	0.00104	45.93	0.97%	<0.05
wchannel:wwall	0.00708	16	0.00044	19.58	0.41%	<0.05
wchannel:t	0.27298	16	0.01706	755.56	15.90%	<0.05
wwalll:t	0.04899	16	0.00306	135.59	2.85%	<0.05
Error	0.01156	512	0.00002

, w_channel and t have a dominant influence on the temperature difference, with contributions of 38.24% and 28.88%, respectively. The interaction of w_channel and t contributes 15.90% to the temperature difference, which is higher than the 3.11% of n and 7.73% of w_wall. When designing a liquid cooling system for improved temperature uniformity of large pouch cell modules, it would be advantageous to improve w_channel and t. At that time, the interaction between design factors should be carefully considered. It can be seen from the FFD results in Section 4.1 that the trend in temperature difference depending on each design factor was not absolutely maintained. However, the statistical significance of all design factors and their interactions is indicated by their p-values being below the significance level.

Table 6. ANOVA results for pumping power.

Source	Sum of Squares	Degrees of Freedom	Mean Square	F-Value	Percent Contribution	p-Value
n	673,478.4	4	168,369.6	3420.49	10.77%	<0.05
wchannel	1,276,188	4	319,047.1	6481.57	20.40%	<0.05
wwall	3,600,269	4	900,067.3	18,285.22	57.56%	<0.05
t	0.4	4	0.1	0	0.00%	1.000
n:wchannel	60,502	16	3781.4	76.82	0.97%	<0.05
n:wwall	127,049.5	16	7940.6	161.32	2.03%	<0.05
n:t	1.6	16	0.1	0	0.00%	1.000
wchannel:wwall	492,314.7	16	30,769.7	625.1	7.87%	<0.05
wchannel:t	4.3	16	0.3	0.01	0.00%	1.000
wwalll:t	5.6	16	0.4	0.01	0.00%	1.000
Error	25,202.6	512	49.2

shows the ANOVA results for pumping power. t had no effect on pumping power and is therefore statistically insignificant. On a related point, Animasaun et al. [28] remarked that reducing the plate thickness leads to a significant increase in temperature gradients across the battery module, potentially compromising the thermal management efficiency. t also did not interact with other design factors. The remaining three design factors, w_wall, w_channel, and n, had a significant effect on pumping power. In particular, w_wall had a dominant effect on pumping power with a contribution of more than 50%. The interactions between w_channel and n and between n and w_wall had contributions of 7.87% and 2.03%, respectively. The effects of the three factors that determine the channel geometry and the interactions between them were all significant. These three factors can be considered to reduce the power consumption of liquid cooling systems, and knowledge of the interactions between them can be leveraged to produce more effective designs.

5. Conclusions

In this study, the individual and interaction effects of design factors of a large-format pouch cell module liquid cooling system were statistically analyzed based on DOE. For the maximum temperature, the individual design factors were found to have a dominant influence in the following order: w_wall contributed the most significantly (36.00%), followed by w_channel (30.72%), t (10.77%), and n (7.31%). For the temperature difference, individual design factors contributed significantly, decreasing according to the order w_channel (38.24%), t (28.88%), w_wall (7.73%), and n (3.11%). For pumping power, the individual design factors had major contributions, decreasing in the order of w_wall (57.56%), w_channel (20.40%), n (10.77%), and t, which had no effect. w_wall and w_channel are factors that directly determine the coolant flow velocity and cooling plate heat capacity, and they had major effects on all three responses. In some cases, w_wall and w_channel interacted with the other design factors to produce a larger contribution from each response than from the individual design factors. In the DOE results, it was observed that the influence of each design factor tended to change as the other design factors varied. The change in the effect of the design factors was confirmed by ANOVA to be statistically significant and not due to chance. This suggests that the influence of each design factor, as well as their interactions with each other, should be carefully considered when designing a liquid cooling system for large pouch cell modules. It can also be an important reference point for establishing the initial design by considering local optimum points when performing the design optimization of the battery cooling system in further study.