Combined Experimental, Statistical and CFD Study of the Thermal–Electrical Behavior of a LiFePO₄ Battery Pack Under Varying Load and Cooling Conditions

Abstract

Thermal control represents one of the most important parameters influencing the safety and reliability of lithium-ion batteries, especially at high rates required for modern electric vehicles. The present paper investigates the thermal and electrothermal performance of a lithium iron phosphate (LiFePO₄) battery pack using a combination of experimental, statistical, and numerical methods. The 8S5P module was assembled and examined under load tests of 200, 400, and 600 W with and without active air-based cooling. The findings indicate that cooling reduced cell surface temperature by up to 10 °C and extended discharge time by 7–16% under various load conditions, emphasizing the effect of thermal management on battery performance and safety. In order to more systematically investigate the impact of ambient temperature and load, a RSM study with a central composite design (CCD; 13 runs) was performed, resulting in two very significant quadratic models (R² > 0.98) for peak temperature and discharge duration prediction. The optimum conditions are estimated at a 200 W load and an ambient temperature of 20 °C. Based on experimentally determined parameters, a finite-element simulation model was established, and its predictions agreed well with the measured results, which verified the analysis. Integrating measurements, statistical modeling, and simulation provides a tri-phase methodology to date for determining and optimizing battery performance under the electrothermal dynamics of varied environments.

1. Introduction

The electric-vehicle market has expanded rapidly in recent years, increasing the demand for battery technologies that deliver high performance while maintaining strict safety standards [1,2]. Lithium iron phosphate (LiFePO₄) cells are strong candidates due to their thermal stability, long cycle life, and safer behavior compared with other Li-ion chemistries [3,4]. However, the shift toward high-power applications driven by fast-charging requirements and aggressive load profiles introduces additional thermal and electrochemical challenges [5]. Under such conditions, cells experience increased thermal and mechanical stress, accelerated aging, and wider temperature variations, which can reduce service life and compromise safety if not properly managed [6,7].

Besides the characteristics of the electrode materials, several package-level obstacles also exist, such as thermal-runaway hazards, non-uniform temperature profiles, and accelerated performance degradation, which are more prominent during heavy-duty or high-power operation. In practice, real-world driving introduces additional complications, such as the pack experiencing changing ambient temperatures, varying loads, and non-steady discharge/charge patterns [8,9,10]. Keeping cell temperatures in the range of 20–45 °C, however, is necessary for both efficiency and long life. A variety of thermal-management approaches have been investigated, including passive phase change materials and active air- and liquid-cooling solutions, as well as hybrid concepts; however, only a few have been thoroughly validated under realistic transient operation conditions [11,12].

More essentially, experimental works under various load and cooling conditions are still needed to reveal the thermal response and performance of batteries. Statistical methods, such as Response Surface Methodology (RSM), enable the determination of how factors, including ambient temperature and electrical loads, influence this behavior [13,14,15]. When integrated with numerical simulations (e.g., such as COMSOL Multiphysics), the experiments provide an improved understanding of heat sources, cooling effectiveness, and temperature distribution in the pack [16,17]. The present combination of experimental, statistical, and numerical methodologies provides a sound basis for refining the design of more reliable models, as well as predictive thermal control strategies [18,19].

The thermal management of batteries remains a key challenge in electric vehicle applications, and it is exacerbated when LiFePO₄ systems operate under different load conditions [20,21]. It is also widely known from research that effective temperature regulation is necessary to achieve safe and reliable cell operation under varying environmental and electrical loads [22,23,24,25]. Integrated thermal-management systems (TMS) have therefore become central to improving durability and stability. El Kassar et al. [26] reported that, using a refrigerant-based active cooling approach, cell temperatures can be reduced by as much as 15 °C at high rates, which substantially enhances safety and prolongs life.

Hybrid cooling solutions have been investigated in various studies for fast charging [27,28,29]. Dai and Long [30] designed a liquid-cooled/phase change material system that controlled the temperature of a 95 Ah Li-ion cell at a 6 °C rate charge. They produced designs that reduced surface temperatures to less than 45 °C and shortened charging times to approximately six minutes. COMSOL Multiphysics simulations confirmed these findings, with a cooling rate of approximately 0.33 °C/min, demonstrating the potential of numerical modeling to optimize advanced thermal management systems. Zhou et al. [31] also investigated the effect of temperature on LiFePO₄ cells, concluding that operating above 50 °C can accelerate capacity fade and reduce cycle life. They also observed that temperature inconsistencies of >5 °C within a pack result in uneven aging, meaning this is perhaps more important under variable load levels, such as those considering 200–600 W.

Direct phase change coolers have also been investigated for high discharge rates [32,33,34,35]. Goodarzi et al. [36] experimentally demonstrated that liquid–vapor phase-change cooling on the cell achieves 40% better heat removal than the air-cooling condition, while maintaining a temperature variation of less than 3 °C, even with a discharge C-rate of 5 °C. These outcomes underscore the potential for using alternative methods of cooling when air or passive options are inadequate [37,38], and they are in good agreement with air-based cooling setups typically employed in experiments, including those used here.

For EV applications in practice, Zhu et al. [39] emphasized the benefit of laboratory testing in conjunction with numerical simulations, improving thermal-management effectiveness. Through COMSOL finite-element analysis, the heat distribution was studied under real-world drive cycles, and it was found that temperature control and battery life could be enhanced by 35% when the cooling strategies were tailored to specific scenarios, such as highway cruising or stop-and-go traffic. In addition, design-of-experiments (DOE) methods have been recently adopted for optimizing TMS protocol parameters by others [40,41,42,43,44]. Chavan et al. [45] applied the Box–Behnken experimental design to investigate the effects of coolant flow rate, inlet temperature, and channel geometry, and observed combined optimization for both temperature and uniformity with a prediction accuracy of over 95%. Na [46] also utilized the Central Composite Design to optimize flame-spray synthesis parameters for LiFePO4 and demonstrated that minor modifications in the design can significantly impact both electrochemical and thermal performance.

Several studies couple DOE with electrochemical and thermal modeling to facilitate battery design [47,48]. Kim et al. [49] introduced a novel quadratic response surface approach to improve the specific energy density while maintaining good thermal stability. By tuning parameters such as the thickness and porosity of the electrode, their model achieved 285 Wh/kg with a low prediction error, demonstrating how the integrated modeling can assist the DOE-based optimization approach, as applied here in optimizing discharge time and preventing excessive temperature rise. COMSOL is also a key tool for validating the thermo-electrochemical performance of thermochemical systems [50,51]. Liu [52] formulated a three- dimensional electrochemical–thermal model for accurate prediction characterization of discharge and temperature profiles at various C-rates, with validation within 2 °C and RMSE of 0.02 V; thus proving the ability of COMSOL in handling load-dependent simulations that tally well with the outcome from erratic discharge tests found between 200 W and 600 W, as observed in this study. Jannat [53] went on to apply this analysis methodology in coin-cell geometries, with an emphasis on how electrode dimensions affect temperature and performance, again highlighting the need for accurate geometric modeling of pack configurations at the level tested here (8S5P).

Ebbs-Picken et al. [54] examined optimization strategies for battery thermal management and identified response surface methodology as a practical compromise between predictive accuracy and computational cost. They recommended RSM for parametric and multi-objective problems where trade-offs between cooling effectiveness and energy consumption must be evaluated. These principles underpin the statistical modeling of discharge time and temperature adopted in the present study.

From a sustainability perspective, effective thermal management of battery systems plays a critical role in supporting reliable and energy-efficient electric-vehicle operation by limiting thermal degradation, improving safety margins, and extending service life. By enabling more stable performance and longer usable lifetime of LiFePO₄ battery packs, the proposed framework contributes to more sustainable energy utilization and reduced resource demand in electric-mobility applications.

In this context, the present study investigates the thermal and electrothermal behavior of an 8S5P LiFePO₄ pack under different load levels and cooling configurations. The experiments quantify the effect of active air-based cooling on temperature rise and discharge duration, using no-cooling tests as a baseline for comparison. Response surface methodology (RSM) is then applied to evaluate the combined influence of ambient temperature and electrical load and to identify conditions that lower cell temperature while extending discharge time. To complement the experiments, a finite-element model is developed in COMSOL using parameters obtained from testing, and its predictions are compared with the measured data to confirm agreement between simulation and experiment.

Building on earlier work in thermal control and optimization, this study integrates controlled load testing, active cooling assessment, response surface modeling, and physics-based simulation within a unified framework. Beyond confirming general trends, the proposed approach quantitatively resolves the coupled influence of electrical load and ambient temperature on peak cell temperature and discharge duration, enabling the identification of operating envelopes that simultaneously constrain thermal stress and preserve usable capacity. In addition, the coupled CFD–electrothermal analysis elucidates how pack layout and airflow direction govern temperature non-uniformity and hotspot formation under active air cooling. Together, these results provide predictive and design-relevant insight into the thermal behavior of LiFePO₄ packs in practical operation and support the development of more effective battery thermal management strategies for electric-vehicle applications. In contrast to many prior studies that focus on single cells or C-rate-based testing, the present work considers a full 8S5P LiFePO₄ pack operated under fixed power demand representative of practical electric-vehicle loads. This allows for direct assessment of thermal–electrical coupling at the module level and improves the relevance of the derived models for real-world battery thermal management design.

2. Materials and Methods

2.1. Battery Pack Fabrication

A lithium–iron–phosphate (LiFePO₄)-based pack was constructed for thermal characterization under various cooling conditions. The pack consisted of 40 cylindrical cells (3.2 V and 6000 mAh each), arranged in an 8S5P configuration, which incorporated eight series and five parallel branches, providing a nominal voltage of 24 V and a current capacity of 30 A (Figure 1). The corresponding theoretical power output is 720 W (24 V × 30 A).

Figure 1. The battery cells’ structure.

The specifications of the cells are presented in Table 1. The weight of a single unit was approximately 141 g ± 2 g. Dimensional tolerances were measured according to the manufacturer’s datasheet, with a height of 70.5 mm ± 0.4 mm and diameters ranging from 32.15 mm to 32.5 mm, with varying tolerances. The dimensional parameters are shown in millimeters in Figure 2, with a height of 70.5 mm, a width of 15.92 mm, and diameters of D1, D2, and D3 being 32.15 mm, 32.5 mm, and 32.4 mm, respectively.

Table 1. Cell Properties.

Items	Specification	Remarks
Nominal Capacity	6000 mAh	0.5 C discharge
Nominal Voltage	3.2 V	0.5 C discharge
Discharge Cut-off Voltage	2.0 V
Charge Limited Voltage	3.65 V	0 °C~60 °C
Standard Charge Current	0.5 C	0 °C~60 °C
Standard Discharge Current	0.5 C	0 °C~60 °C
Rapid Charge Current	1 C	0 °C~45 °C
Rapid Discharge Current	1 C	0 °C~45 °C
Max. Discharge Current	3 C	Limited Voltage 2.0 V
Max. Pulse Discharge Current	5 C	10 ms
Internal Impedance	≤10 mΩ	AC internal resistance

Figure 2. Cell Dimensions.

Battery assembly was performed with the assistance of the R F BOX Company, a subsidiary of the Arab Organization for Industrialization. The assembled unit is presented in Figure 3a, and the pack was subsequently connected to a Battery Management System (BMS) for programming and calibration, as shown in Figure 3b.

Figure 3. Battery Assembly and BMS Programming; (a) LiFePO₄ battery pack assembly; (b) BMS hardware and wiring used for monitoring and programming.

2.2. Charging and Load Circuit Design

The pack was first charged with a 24 V, 2 A slow charger, requiring approximately 14 h. To shorten the process, two identical chargers were later connected in parallel to supply 4 A, reducing the charging time to roughly 7 h. The chargers, provided by RF Box Company, are shown in Figure 4, with the corresponding charging circuit illustrated in Figure 4b. Overcurrent fuses were included for protection, and the BMS handled cell balancing and overvoltage control throughout charging.

Figure 4. Charger and Charging Circuit; (a) Battery charger unit; (b) Charging setup connected to the battery.

To assess battery performance under varying demand, a resistive load was built using fourteen 12 V, 50 W halogen lamps. Each pair of these lamps was connected in series to create 24 volts, with 100 watts per branch, and seven such branches were paralleled. This configuration provided three modes of operation: 600 W with all 7 branches switched on, 400 W with four branches on, and ~200 W with two branches. The full-load configuration is presented in Figure 5.

Figure 5. Experimental Work Diagram.

The air-based cooling configuration employed in this study corresponds to an open-cycle, room-scale setup. The battery pack was placed within a controlled indoor environment where conditioned air supplied by a room air-conditioning system circulated freely around the pack without the use of dedicated ducts or flow-directing elements. Airflow rate was not directly measured; instead, the cooling effectiveness was assessed through the resulting battery surface temperature response under identical operating conditions.

2.3. Experimental Framework

The experiments were performed on a purpose-built test rig to measure the thermal and electrical performance of the 8S5P LiFePO₄ pack with defined loading and cooling. The surface temperature was recorded using an infrared thermometer (±0.2 °C), the voltage with a digital multimeter (±0.01 V), and the discharge time was measured with a stopwatch. All instruments were calibrated as per the manufacturer’s guidelines before testing. As illustrated in Figure 6, the battery pack was placed on an insulated platform behind a barrier to limit unwanted heat transfer from the load elements.

Figure 6. Test Rig Components.

The test program consisted of four cases to investigate the influence of load level and cooling on battery performance. Two tests were conducted at full load (600 W), one without cooling at an ambient temperature of 33 °C, and the other under active air-based cooling at an ambient temperature of 22 °C. The process was repeated for a half-load (400 W) under no-cooling conditions at 28 °C and with air-based cooling at 22 °C. In all cases, the pack was fully charged prior to discharge, and temperature, voltage, and current were sampled every 3 min until cut-off. The measurements were performed in triplicate, and the averaged data were analyzed using MATLAB R2021a and Microsoft Excel 2019 for comparison.

All tests were conducted indoors in a controlled environment with limited natural airflow to maintain stable ambient conditions; humidity was monitored but not actively controlled. For the cooled cases, conditioned air supplied by a room air-conditioning system was directed over the battery pack to enhance convective heat transfer, while the refrigerant cycle remained internal to the air-conditioning unit and did not interact with the battery. This setup maintained the battery surface temperature in the range of approximately 24–27 °C during discharge. The recorded data were post-processed to generate temperature and voltage time histories, demonstrating the direct influence of load level and air-based cooling on thermal stability and discharge performance.

The selected load levels and ambient temperature range were chosen to represent controlled yet practically relevant steady operating conditions, allowing systematic evaluation of thermal–electrical coupling and statistically robust model development. A resistive load was employed to provide a stable and repeatable power demand for controlled thermal analysis; while it does not replicate the dynamic behavior of electric-vehicle traction systems, it enables isolation of battery-internal thermal effects without additional variability introduced by power-electronics switching. The halogen lamps were used solely as external resistive elements and were physically isolated from the battery pack to avoid any direct thermal interaction. Consequently, the battery temperature evolution is governed by internal heat generation and cooling conditions rather than by heat emitted from the load elements. Dynamic load profiles, transient driving cycles, and fast-charging events, which are critical in real electric-vehicle operation, are addressed as future extensions of the proposed framework.

3. Design of Experiments and Statistical Analysis

3.1. RSM Experimental Design

A Response Surface Methodology (RSM) was employed to determine how ambient temperature and electrical load affect the peak battery temperature and discharge time. This method estimates the effects of inputs and their interactions. In this study, two factors were chosen: A (room temperature) and B (applied load). Each factor was investigated at three levels (−1, 0, +1) as presented in Table 2.

Table 2. Experimental factors and their coded levels for RSM design.

Factor	Unit	Code	Levels
			−1	0	+1
Room Temperature	°C	A	20	27.5	35
Load	W	B	200	400	600

A Central Composite Design (CCD) was applied to study the combined effects of these variables and their interactions on the response parameters. CCD was chosen because it enables efficient estimation of a complete quadratic model while providing rotatability and information about curvature through the inclusion of axial (star) points. CCD thus affords improved predictive capability across the experimental region, enabling the localization of the true optimum with a limited number of runs. The axial point levels were carefully selected to avoid unsafe or unrealistic operating conditions, such as combining high ambient temperatures with high loads.

$$n=2^K+2K+C_p$$

(1)

where K is the number of process variables (K = 2), and Cₚ is the number of center points (Cₚ = 5). Accordingly, thirteen experimental runs were performed. All experiments were conducted in a randomized order to minimize potential bias and reduce the influence of uncontrolled external factors.

Figure 7 outlines the RSM procedure used in this study. The workflow begins by defining the input factors, ambient temperature (20–35 °C) and load (200–600 W), and selecting the corresponding response variables. A central composite design (CCD) is then created, and the resulting data are screened before regression analysis. ANOVA is used to test the significance of the model terms; if the criteria are not satisfied, the screening step is revisited.

Figure 7. RSM–CCD optimization workflow for battery thermal and discharge performance.

Once a statistically acceptable model is obtained, the fitted surface is examined to determine the optimal conditions for minimizing battery temperature and maximizing discharge time. The optimized results are subsequently validated against both experimental and simulation data to confirm the model’s adequacy. Overall, the flowchart summarizes the structured sequence used to develop, refine, and validate the RSM-based predictive model.

The general quadratic regression model used for responses (maximum battery temperature and discharge time) is expressed as:

$$\mathrm{Response}=b_0+{\displaystyle \sum _{i=1}^n{b}_i}{x}_i+{\displaystyle \sum _{i=1}^n{b}_{ii}}{x}_i^2+{\displaystyle \sum _{i=1}^{n-1}{\displaystyle \sum _{j=i+1}^n{b}_{ij}}}{x}_i{x}_j+\epsilon$$

(2)

where Y denotes the predicted response, b₀ is the model intercept, bᵢ, bᵢᵢ, and bᵢⱼ represent the linear, quadratic, and interaction coefficients, respectively, and xᵢ, xⱼ are the coded independent variables.

Model adequacy was evaluated using the coefficient of determination (R²), adjusted R² (R²_adj), and predicted R² (R²_pred). The statistical significance of individual model terms and the overall model was assessed by Analysis of Variance (ANOVA) using Fisher’s F-test at a 95% confidence level (p < 0.05).

Numerical optimization was performed using Design Expert Software (Version 13) to determine the optimal combination of room temperature and load that minimizes battery temperature while maximizing discharge time. The predictive model, ANOVA results, and optimized conditions are presented in Section 4.

3.2. Model Validation and ANOVA

The statistical model was validated to determine its applicability for predicting battery behavior. Validation involved verifying the normality of residuals, a key assumption for regression models. If the data shows significant differences from normality, there would be a poor fit or misinformation in the model.

Design Expert Software (Version 13) was used to develop regression equations for the factors and response variables: room temperature (A) and load (B) for maximum battery temperature and battery discharge time. The relationships were described in terms of polynomial equations up to order two, fitted to the experimental data listed in Table 3.

Table 3. RSM-Based Experimental Design Matrix and Observed Responses for Maximum Battery Temperature and Discharge Time.

Std	Run	A: Room_Temp	B: Load	Max_Batt_Temp	Discharge_Time
		°C	W	°C	min
5	1	20	400	24	112
3	2	20	600	27	75
11	3	27.5	400	29	100
13	4	27.5	400	29	101
2	5	35	200	30	118
9	6	27.5	400	28	100
1	7	20	200	22	143
12	8	27.5	400	30	98
8	9	27.5	600	32	70
7	10	27.5	200	25	127
6	11	35	400	34	90
4	12	35	600	38	60
10	13	27.5	400	28	102

The fit summary given in Table 4 indicates that the two-factor interaction (2FI) model is the most suitable for describing the maximum battery temperature response. The model demonstrated a sequential p-value of 0.0473, confirming its statistical significance. In contrast, the lack-of-fit p-value of 0.8923 suggests that the model fits the data well, with minimal error. The adjusted R² value (0.9760) and predicted R² value (0.9605) are in close agreement, confirming the strong predictive capability and model adequacy. Therefore, the 2FI model was selected as the best fit for this response.

Table 4. Statistical evaluation of empirical models for maximum battery temperature and discharge time as functions of room temperature (A) and load (B).

Source	Sequential p-Value	Lack of Fit p-Value	Adjusted R2	Predicted R2	Remarks
(A) Maximum Battery Temperature
Linear	<0.0001	0.6249	0.9657	0.9433
2FI	0.0473	0.8923	0.9760	0.9605	Suggested
Quadratic	0.5143	0.9056	0.9744	0.9662
Cubic	0.8739	0.6141	0.9661	0.8669	Aliased
(B) Battery Discharge Time
Linear	<0.0001	0.1167	0.9892	0.9795
2FI	0.0257	0.2499	0.9933	0.9851	Suggested
Quadratic	0.2903	0.2523	0.9939	0.9763
Cubic	0.1007	0.8748	0.9966	0.9968	Aliased

For discharge time, the fit summary (Table 4) again identified the 2FI model as the most suitable. The sequential p-value (0.0257) shows that the model is statistically significant, while the lack-of-fit p-value (0.2499) indicates an acceptable fit to the data. The adjusted R² (0.9933) and predicted R² (0.9851) are closely aligned, demonstrating strong accuracy and predictive capability. Accordingly, the 2FI model was adopted for the discharge-time response.

For maximum battery temperature, the regression equation is expressed as:

$$\mathrm{Max}\_\mathrm{Batt}\_\mathrm{Temp}=28.92+4.83\mathrm{A}+3.33\mathrm{B}+0.75\mathrm{AB}$$

(3)

For battery discharge time, the regression equation is expressed as:

$$\mathrm{Discharge}\_\mathrm{Time}=99.69-10.33\mathrm{A}-30.50\mathrm{B}+2.50\mathrm{AB}$$

(4)

where A and B represent the coded values of room temperature and load, respectively.

Analysis of Variance (ANOVA) was employed to evaluate the statistical significance and relative importance of the model parameters on the response variables. The contribution of each term was assessed through the F-statistics and corresponding p-values, where a larger F-statistic and smaller p-value indicate a more substantial influence of that factor on the response.

(a)

Maximum Battery Temperature

The ANOVA results for the maximum battery temperature model are summarized in Table 5. The model exhibited a highly significant overall fit with an F-value of 163.36 and a p-value less than 0.05, confirming that the developed regression model is statistically significant. Among the factors studied, both room temperature (A) and load (B) exerted significant effects on the maximum battery temperature, with F-values of 328.54 and 156.26, respectively (p < 0.0001). The interaction term (AB) also showed a statistically significant but less dominant effect (F = 5.27, p = 0.047).

Table 5. Analysis of variance (ANOVA) for the empirical models of maximum battery temperature and discharge time as functions of room temperature (A) and load (B).

Source	Sum of Squares	df	Mean Square	F-Value	p-Value	Remarks
(A) Maximum Battery Temperature
Model	209.08	3	69.69	163.36	<0.0001	significant
A-Room_Temp	140.17	1	140.17	328.54	<0.0001
B-Load	66.67	1	66.67	156.26	<0.0001
AB	2.25	1	2.25	5.27	0.0473
Residual	3.84	9	0.4266
Lack of Fit	1.04	5	0.2079	0.2971	0.8923	not significant
Pure Error	2.80	4	0.7000
Cor Total	212.92	12
(B) Battery Discharge Time
Model	6247.17	3	2082.39	593.04	<0.0001	significant
A-Room_Temp	640.67	1	640.67	182.45	<0.0001
B-Load	5581.50	1	5581.50	1589.54	<0.0001
AB	25.00	1	25.00	7.12	0.0257
Residual	31.60	9	3.51
Lack of Fit	22.80	5	4.56	2.07	0.2499	not significant
Pure Error	8.80	4	2.20
Cor Total	6278.77	12

The lack-of-fit test was found to be insignificant (p = 0.8923), confirming that the developed model accurately represents the experimental data without any systematic deviation. The coefficient of determination (R²) was 0.9820, while the adjusted R² (0.9760) and predicted R² (0.9605) values were in close agreement (see Table 6), indicating strong consistency between the model’s predictions and the observed results. The adequate precision value of 45.08 further signifies a high signal-to-noise ratio, validating the model’s robustness. The standard deviation (0.65) and low coefficient of variation (2.26%) reflect the model’s precision and reproducibility. Moreover, the close correlation between the predicted and experimental maximum battery temperature values (see Figure 8A) confirms that the RSM model can reliably estimate the response behavior across the studied parameter range. Furthermore, Figure 9A illustrates the residual plots for the maximum battery temperature. The random distribution of residuals around the zero line indicates the absence of systematic errors, validating the adequacy of the models and confirming that the developed equations well represent the experimental data.

Table 6. Statistical performance of the empirical models for maximum battery temperature and discharge time.

Parameter	Maximum Battery Temperature	Discharge Time
Standard Deviation (Std. Dev.)	0.6532	1.87
Mean	28.92	99.69
Coefficient of Variation (C.V. %)	2.26	1.88
R2	0.9820	0.9950
Adjusted R2	0.9760	0.9933
Predicted R2	0.9605	0.9851
Adequate Precision	45.0803	78.5682

Figure 8. Comparison between predicted and experimental values of (A) maximum battery temperature and (B) discharge time.

Figure 9. Residual plots for (A) maximum battery temperature and (B) discharge time.

(b)

Battery Discharge Time

The ANOVA results for the battery discharge time model are presented in Table 5. The model was statistically significant, with an overall F-value of 593.04 and p-value < 0.0001, confirming the strong predictive capability of the regression model. Among the factors, room temperature (A) and load (B) exhibited highly significant influences on discharge time, with F-values of 182.45 (p ≤ 0.0001) and 1589.54 (p ≤ 0.0001), respectively. The interaction term (AB) also contributed significantly (F = 7.12, p = 0.0257).

The adopted central composite design provides an efficient and statistically robust basis for second-order modeling with two input variables. As with all response surface models, the predictive capability of the proposed formulations is intended for interpolation within the investigated parameter ranges, and extrapolation beyond these ranges should be treated with caution.

Figure 10 and Figure 11 present the 3D surface and contour plots illustrating how maximum battery temperature and discharge time change with the combined effects of room temperature and load. In each plot, the unused variable was fixed at its midpoint. These visuals help clarify how the two operating parameters interact and identify the ranges that yield favorable thermal and discharge performance.

Figure 10. Influence of (A) room temperature and (B) load on the maximum battery temperature.

Figure 11. Influence of (A) room temperature and (B) load on discharge time.

Tests performed at room temperatures from 20 to 35 °C show a clear linear increase in maximum battery temperature, as seen in Figure 10A. As ambient temperature rose from 20 to 35 °C, peak battery temperature increased from about 24 to 34 °C, demonstrating that higher ambient conditions amplify heating during operation.

Discharge time showed the opposite behavior of temperature with respect to ambient conditions, as seen in Figure 11A. When room temperature increased from 20 °C to 35 °C, discharge time fell from about 112 to 90 min, indicating faster internal reactions and reduced operating duration at higher temperatures.

The effect of load was evaluated by varying it from 200 W to 600 W while holding room temperature at its midpoint. As shown in Figure 10B, maximum battery temperature rose from roughly 25 °C at 200 W to 32 °C at 600 W, reflecting the greater heat generation caused by higher current and resistive losses. Load increases also reduced discharge time, which dropped from about 127 min at 200 W to 70 min at 600 W (see Figure 10B). This confirms that higher power demand accelerates energy consumption and shortens the available operating period.

Unlike conventional parametric studies, the RSM formulation reveals that the interaction term between ambient temperature and load is statistically significant for both peak temperature and discharge time. This indicates that thermal stress cannot be accurately assessed by considering each factor independently. At elevated ambient temperatures, the marginal thermal penalty of increasing load is amplified, whereas at lower temperatures the same load increase produces a comparatively weaker effect. This non-linear coupling is critical for thermal management design, particularly for electric vehicles operating across seasonal and climatic variations.

The optimization aimed to identify the best combination of room temperature and load to minimize maximum battery temperature while maximizing discharge time. Design Expert V13 was used to perform this multi-objective optimization within the RSM framework, applying a desirability function to combine both objectives into a single evaluation metric.

The optimization criteria were set to lower peak temperature and extend discharge time. The resulting plot (see Figure 12) shows the predicted region of optimal performance. According to the RSM results, the most favorable conditions occur at a room temperature of 20.00014 °C and a load of 200.12 W, corresponding to a predicted maximum temperature of 21.51 °C and a discharge time of 143.00 min. These outputs demonstrate the model’s ability to identify operating parameters that enhance battery performance while maintaining thermal safety.

Figure 12. RSM-based optimization results for maximum battery temperature and discharge time.

4. Experimental Results and Discussion

4.1. Temperature Performance

Figure 13 Shows thermal performance of the 8S5P LiFePO₄ pack under three load conditions (200, 400, and 600 W) with and without active cooling. These experiments were used to characterize the influence of the cooling system on temperature decay and discharge period.

Figure 13. Temperature profiles of the LiFePO₄ battery pack under 200, 400, and 600 W loads, with and without active cooling.

At 200 W (see Figure 13a), for the uncooled pack, it was heated from 29.5 to approximately 33 °C and discharged after about 123 min. Upon cooling, however, the temperature dropped from 27.7 °C to approximately 25 °C, where it remained relatively constant, and the discharge time increased to approximately 132 min, representing a 7.3% increase and a decrease in temperature of roughly 8 °C. At 400 W (see Figure 13b), The uncooled pack warmed to 34 °C and discharged for roughly 93 min, and the chilled pack remained at a constant 26 °C while discharging for almost exactly 108 min, a gain of about 16.2% in runtime. This difference was even larger at 600 W (see Figure 13c). The temperature rose to almost 38 °C, and the pack turned off after approximately 63 min without cooling. The temperature with cooling was lower than 28 °C for up to 56 min, and the pack worked for 72 min, achieving a recovery of 14.3%. It is demonstrated that active cooling effectively restrains the temperature rise, enhances heat uniformity, and prolongs the operation duration to a certain extent, which are key factors in ensuring the safety and performance of LiFePO₄ under high power.

The observed thermal benefit of active air-based cooling is consistent with recent experimental and numerical studies reporting that forced convection reduces peak temperature and improves thermal uniformity, particularly as discharge demand increases [55]. CFD investigations have shown that airflow distribution and inter-cell spacing strongly influence hotspot formation in cylindrical modules, and that improving flow access to interior cells can yield measurable reductions in maximum temperature and thermal gradients [55]. Related optimization studies further indicate that geometric and flow-design parameters, such as inlet direction or duct arrangement, can be as influential as increasing flow rate when suppressing hotspots and improving temperature uniformity [56]. Compared with reports on single cells or smaller modules, the absolute temperature reduction observed here is lower, which can be attributed to differences in module scale, load definition (fixed power versus C-rate), and boundary conditions, including ambient control and inlet air temperature or velocity [57]. Importantly, the present results extend these prior findings by providing pack-level LiFePO₄ data (8S5P) under practical power demands (200–600 W) and by linking experimental trends to predictive RSM models and electrothermal CFD validation.

4.2. Voltage Behavior

Figure 14 shows the voltage–time curves of the 8S5P LiFePO₄ pack under 200, 400, and 600 W loads, with and without active cooling. These results complement the thermal measurements by illustrating how temperature control influences voltage stability and discharge duration.

Figure 14. Voltage measurement of the LiFePO₄ battery pack under three load conditions (200, 400, and 600 W) with and without the active cooling system.

At 200 W (Figure 14a), both cases began near 28.5 V. Without cooling, the voltage fell quickly to 25 V within the first 20 min and reached 22 V after about 123 min. With cooling, the voltage declined more gradually and remained above 23 V even after 132 min. The roughly 10 min increase in runtime reflects reduced internal resistance and a slower voltage drop under cooled conditions. At 400 W (Figure 14b), the uncooled pack reached 23 V after about 93 min, while the cooled pack continued operating for roughly 108 min at a slightly higher terminal voltage of around 23.5 V. Lower temperatures limit resistive and polarization losses, enabling the pack to maintain voltage longer and deliver more usable energy before cutoff.

In Figure 14c, at 600 W, the difference between the two cases was most evident. Without cooling, the voltage dropped sharply from 28 V to about 22 V within 63 min, reflecting significant heating and rising internal impedance. With cooling, the decline was more gradual, and the pack operated for about 72 min before reaching the same cutoff voltage. These findings indicate that active cooling limits temperature rise, improves voltage stability, and boosts energy efficiency, extending runtime by roughly 10–20%. Such improvements contribute to better performance and longer service life for LiFePO₄ systems operating under high power demand.

5. Numerical Simulation and Validation

5.1. Simulation Setup in COMSOL

A simulation model of the battery was developed in COMSOL Multiphysics 6.0 using the exact specifications as those used in the experimental setup. The purpose was to ensure that no measurement errors occurred during the practical tests and to verify the computational simulation’s ability to reproduce real-world results accurately.

COMSOL Multiphysics was selected for the present study due to its capability to directly couple electrical conduction, Joule heat generation, and heat transfer with fluid flow within a single computational framework. This integrated multiphysics environment enables consistent representation of load-dependent heat generation and temperature evolution, which is essential for validating electrothermal behavior experimentally. Similar COMSOL-based electrothermal modeling approaches have been widely reported in lithium-ion battery studies, particularly for module- and pack-level analyses [58,59,60].

The numerical model assumes homogeneous material properties within each cell, uniform internal heat generation derived from measured electrical parameters, and steady inlet airflow conditions for the air-based cooling cases. Contact thermal resistance between cells and structural components was simplified, and electrochemical aging effects were not included. These assumptions are appropriate for capturing short-term thermal behavior and comparative cooling performance, while long-term degradation effects are identified as future research directions.

The numerical model is validated at the pack level through comparison with experimentally measured surface temperature evolution and voltage decay under identical operating conditions. While internal airflow fields and internal cell temperature distributions were not directly measured in this study, the simulations are intended to provide comparative insight into cooling configuration effects rather than fully resolved internal flow characterization.

A set of LiFePO₄ (LFP) cylindrical cells (70 mm × 30 mm) was assembled in the configuration shown in Figure 15. The pack consisted of eight series-connected columns, each made up of five cells in parallel. Cell specifications are provided in Table 7. The model was evaluated under two loading conditions, half load and full load, to compare simulated cell-voltage behavior with the experimental results described in the previous section.

Figure 15. Arrangement of cells in Battery Pack.

Table 7. Thermo-physical parameters of Lithium-ion battery pack.

Parameter	Expression	Value
C rate	4	4
Battery cell capacity	4 [A*h]	14,400 C
1C current	Q_cell/1 [h]	4 A
In-layer thermal conductivity	30 [W/m/K]	30 W/(m·K)
Through-layer thermal conductivity	1 [W/m/K]	1 W/(m·K)
Activation energy	24 [kJ/mol]	24,000 J/mol
Activation energy	−59 [kJ/mol]	−59,000 J/mol
Activation energy	24 [kJ/mol]	24,000 J/mol
Reference temperature	20 [degC]	293.15 K
J0 at reference temperature	0.85	0.85
tau at reference temperature	1000 [s]	1000 s
eta_1C at reference temperature	4.5 [mV]	0.0045 V
Battery density	2000 [kg/m3]	2000 kg/m3
Battery heat capacity	1400 [J/(kg*K)]	1400 J/(kg·K)
Heat transfer coefficient	30 [W/m2/K]	30 W/(m2·K)
Initial/external temperature	20 [degC]	293.15 K

The battery pack was tested under half-load and full-load conditions to compare the simulated cell-voltage and thermal behavior with the experimental results presented in the previous chapter. A three-dimensional CFD model was developed to evaluate the cooling performance of a lithium-ion battery module subjected to an air-cooling system. The inlet air temperature was set to 24 °C (297.15 K), equivalent to conditioned air from an HVAC outlet, and the airflow was directed across the battery surfaces to track the evolution of temperature and airflow distribution along the cells. Moreover, the influence of the inlet air velocity (0.1 m/s) on cooling effectiveness was examined to approximate realistic airflow conditions encountered in practical battery-module installations.

The geometry of the model was developed with reference to commercial cylindrical Li-ion cells. Two different packing configurations, configurations A and B were analyzed to investigate the influence of module arrangement on temperature distribution and voltage response. Both layouts and their flow domains are depicted in Figure 16. The thermal and voltage responses at a full-load condition are presented.

Figure 16. (a) Battery Arrangement A with its airflow domain. (b) Battery Arrangement B with an alternative cell layout and airflow domain.

Arrangement A and Arrangement B represent two distinct airflow configurations relative to the battery pack layout. In Arrangement A, the cooling airflow is directed primarily across the outer cell surfaces, resulting in limited penetration toward the interior cells. In contrast, Arrangement B promotes more uniform airflow access to the inner cells, enhancing convective heat removal and reducing peak temperature and thermal gradients across the pack.

Effect of Cell Layout on Thermal Performance: The impact of cell layout on the thermal performance is illustrated in Figure 17. In Arrangement A, the internal cells created obvious hot spots with weak air flow penetration (the uneven and deviant streamlines). This reduced flow resulted in increased peak temperatures and thermal gradients. Conversely, a more homogeneous temperature field was obtained from Arrangement B. The air was more uniformly diverted through the intercell gaps, which decreased dead zones and both the maximum temperature and gradient. It is clear that the cooling of Arrangement B is more efficient and exhibits higher thermal stability when in operation.

Figure 17. Simulated surface temperatures (°C) and airflow streamlines for (a) Arrangement A and (b) Arrangement B under full-load operation.

These results demonstrate that thermal management effectiveness is not solely determined by cooling capacity, but also by geometric configuration and flow orientation. Two modules with identical cooling power can exhibit markedly different peak temperatures and gradients depending on airflow access to inner cells. This provides a practical guideline for pack-level thermal design: improving temperature uniformity through layout optimization can be as important as increasing airflow rate.

The direction of the cooling airflow has a significant impact on the temperature field of the Li-ion battery module. The orientation of the inlet affects how the air enters and interacts with the cell’s surfaces; turbulent levels and the breakup of the stagnant thermal layer will change accordingly. Higher turbulence mitigates dead zones and decreases the thermal resistance, whereas less favorable orientations correspond to longer flow paths or even blocked channels that enhance frictional losses and impair cooling.

Airflow direction also influences whether the cells are cooled uniformly. While some orientations may result in effective cooling of outer cells, others allow insufficient airflow to reach inner cells, and thus, the temperature difference is very noticeable. These gradients lead to accelerated aging and reduced lifetime, which makes uniform cooling a vital design aim.

Figure 18 Illustrates how altering the direction of air flow affects thermal results. In Models A.1 and A.2, the peak temperature and location of the hotspot are influenced by the inlet angle, indicating that there are beneficial directions that allow the least possible heat to be removed. In contrast, others block access to the inner cells. Similar to the results of models B.1 and B.2, the same configuration can yield very different outcomes depending on variations in the direction of airflow. There are inlet directions that result in less temperature uniformity and considerably smaller maximum temperatures. These results underscore the importance of airflow direction in achieving temperature uniformity and improving cooling efficiency for prolonged module operation.

Figure 18. Temperature distribution (°C) for four airflow-direction cases: (a) Model A.1, (b) Model A.2, (c) Model B.1, and (d) Model B.2.

5.2. Temperature Comparison with Simulation

The experimental data are compared to the numerical prediction, and a good agreement is observed in terms of temperature trends and peak values. At full load, the surface temperature of the COMSOL model was estimated to be around 37 °C, which correlates well with the experimentally measured values (37–38 °C), as shown in Figure 19. The higher temperatures in the central cells, where airflow is relatively restricted, were also replicated by the simulated temperature field. This agreement demonstrates that the heat-generation model and material constants used in this simulation accurately describe the thermal behavior of a pack of cells.

Figure 19. Comparison between experimental and simulated temperature at full load condition.

At half load this agreement was of approximately the same quality. The temperature of the peak in experiments is quite well reproduced (34–35 °C) by the model, which gives an estimate of about 34.4 °C as shown in Figure 20. Both sets of data have the same general behavior, an initial rise that eventually becomes a slow increase as heat generation levels off at lower voltage. The multi-cell simulation also included small temperature differences in the pack (typically <1 °C) to reflect the spread seen during testing.

Figure 20. Comparison between experimental and simulated temperature at half load condition.

The close agreement between the experimental and simulated results provides evidence to support the thermal model as well as of the selected parameters including heat generation rates, thermal properties and boundary conditions. This validates model application for temperature evolution prediction across load profiles and indicates the model’s applicability in future optimization, such as investigating alternative cooling methods, studying pack geometry, or simulating high-stressed operating conditions without extensive experimental runs.

5.3. Voltage Comparison with Simulation

The simulation and experiment of the voltage characteristic under full load and half-load were used to examine the correctness of the electrothermal model. At a half load and full load, the average cell voltage stabilized at approximately 3.32 V and 3.20 V in the simulation, respectively. For the 8-series configuration, this corresponds to pack voltages of 26.56 V and 25.6 V, respectively.

These predictions align well with the experimental measurements, which remained close to 25 V during discharge for both load levels, aside from the initial drop during the current surge. The matching trends, particularly the steady voltage decline over time, indicate that the electrothermal model captures the discharge behavior of the LiFePO₄ cells reliably across different loading conditions.

Figure 21 presents the comparative profiles for both the simulated cell voltages and the experimentally measured pack voltages, confirming that the model captures the essential voltage-time behavior under full- and half-load operation.

Figure 21. Comparison between experimental and simulated voltage results at full and half loading.

6. Conclusions

This study presents an integrated experimental, statistical, and numerical framework for analyzing the thermal–electrical behavior of a LiFePO₄-based battery pack under varying load and cooling conditions. The results on discharge temperature and duration were found to be very sensitive to load level and cooling conditions, while air-based cooling provides a lower operating temperature and consistently prolongs the discharge periods.

RSM analysis quantified the coupled effects of ambient temperature and load on battery performance, enabling predictive identification of operating conditions that limit heat accumulation while preserving usable discharge capacity. The good match between experimental data and COMSOL simulation, both in temperature increase and voltage decay, also validates the thermal–electrochemical models employed here.

Beyond confirming the expected influence of load and ambient temperature, this study demonstrates how their coupled effect governs both thermal stress and usable discharge capacity in LiFePO₄ battery packs. Statistically validated response surfaces provide a predictive tool for defining safe and efficient operating regions, while the CFD-based analysis highlights the decisive role of airflow distribution and pack geometry in controlling temperature non-uniformity. Together, these results advance battery thermal management from descriptive assessment toward model-based design and optimization, supporting more reliable implementation of air-cooled LiFePO₄ systems in electric-vehicle applications. In contrast to many prior studies limited to single cells or C-rate-based testing, this work addresses a full 8S5P LiFePO₄ pack under fixed power demand (200–600 W) representative of practical electric-vehicle operation. Moreover, the tight coupling of statistically derived response surfaces with CFD-based electrothermal modeling enables direct evaluation of load-temperature interaction effects and cooling-layout sensitivity at the module level, thereby extending existing experimental-numerical frameworks toward predictive thermal management design.

Future work will build upon the present framework to further extend its applicability and address aspects beyond the scope of this study. The experimental investigation was conducted using a specific air-based cooling configuration and inlet condition; therefore, subsequent studies will examine alternative cooling strategies, including liquid, phase-change, and hybrid systems, to assess their comparative effectiveness. In addition, while the present results provide pack-level insight for an 8S5P LiFePO₄ module, scaling effects in larger battery modules or full vehicle packs merit further investigation, particularly with respect to airflow distribution and temperature uniformity. Future experimental studies will also incorporate controlled and ducted air-cooling configurations with quantified airflow rates, together with internal temperature sensing and airflow characterization, to enable direct validation of simulated internal flow and thermal fields. From a modeling perspective, future developments will incorporate more detailed representations of contact resistance, material heterogeneity, and electrochemical aging, including degradation under repeated thermal cycling, to improve long-term thermal and performance prediction under realistic operating conditions. Validation under dynamic drive cycles and fast-charging conditions representative of real vehicle operation, together with systematic comparison of alternative cooling technologies, will further support assessment of the robustness and scalability of the proposed framework.