Research on Thermal Characteristics and Algorithm Prediction Analysis of Liquid Cooling System for Leaf Vein Structure Power Battery

Abstract

With the increase in energy density of power batteries, the risk of thermal runaway significantly increases under extreme working conditions. Therefore, this article proposes a biomimetic liquid cooling plate design based on the fractal structure of fir needle leaf veins, combined with Murray’s mass transfer law, which has significantly improved the heat dissipation performance under extreme working conditions. A multi-field coupling model of electrochemistry fluid heat transfer was established using ANSYS 2022 Fluent, and the synergistic mechanism of environmental temperature, coolant parameters, and heating power was systematically analyzed. Research has found that compared to traditional serpentine channels, leaf vein biomimetic structures can reduce the maximum temperature of batteries by 11.78 °C at a flow rate of 4 m/s and 5000 W/m³. Further analysis reveals that there is a critical flow rate threshold of 2.5 m/s for cooling efficiency (beyond which the effectiveness of temperature reduction decreases by 86%), as well as a thermal saturation temperature of 28 °C (with a sudden increase in temperature rise slope by 284%). Under low-load conditions of 2600 W/m ³, the system exhibits a thermal hysteresis plateau of 40.29 °C. To predict the battery temperature in advance and actively intervene in cooling the battery pack, based on the experimental data and thermodynamic laws of the biomimetic liquid cooling system mentioned above, this study further constructed a support vector machine (SVM) prediction model to achieve real-time and accurate prediction of the highest temperature of the battery pack (validation set average relative error 1.57%), providing new ideas for intelligent optimization of biomimetic liquid cooling systems.

1. Introduction

In recent years, the lithium battery new energy vehicle industry has developed rapidly, and the market share has increased year by year [1]. Lithium ion batteries have many advantages, such as high energy density, low self discharge rate, and many cycles [2]. At present, most of the power sources of new energy vehicles are lithium batteries, mainly including ternary lithium batteries and lithium iron phosphate batteries. In the process of charging and discharging, the temperature of a lithium-ion battery rises due to heat generation. If effective measures are not taken, it is easy to cause negative effects on the working performance, cycle life and health status of the battery. According to the different cooling media, the battery thermal management system can be divided into three categories: liquid cooling type, air cooling type and phase change type [3]. The thermal management system of an air-cooled battery with simple structure and easy maintenance cannot meet the cooling requirements of a lithium-ion battery module under high-rate conditions. A phase change battery thermal management system has strong heat dissipation capacity, but it is mainly used as auxiliary heat dissipation at present. Liquid-cooled battery thermal management is now widely used in new energy vehicles. The liquid-cooled battery thermal management system is divided into direct cooling and indirect cooling. Indirect cooling mainly uses a liquid-cooled plate for cooling. The advantage is that the cooling efficiency is high, but it will cause the temperature consistency of electric lithium-ion batteries to become worse. The performance of a lithium battery is closely related to the working temperature. The optimal working temperature is generally 25–60 °C, and the temperature difference should be controlled within 5 °C [4], with temperatures exceeding 60 °C posing a risk of thermal runaway. Therefore, we have adopted 60 °C as the critical threshold for evaluating “overheating” and have rigorously monitored battery temperatures in simulations to ensure they do not approach or exceed this limit. The thermal safety of a power battery is directly related to the reliable operation of electric vehicles, and its high temperature may lead to serious safety accidents such as heating out of control or even fire. Therefore, it is crucial to accurately predict the dynamic changes of battery temperature and implement active thermal management intervention to ensure the safety of the battery system. At the level of the temperature prediction method, the traditional numerical simulation method based on a physical model is limited by the problems of high computational complexity and parameter sensitivity, which makes it difficult to meet the needs of real-time thermal management. The data-driven artificial intelligence method can effectively learn the nonlinear thermal characteristics under complex conditions, and has the advantages of stronger adaptability and computational efficiency. Based on the experimental data of the bionic liquid cooling system under multiple working conditions, this study innovatively established the support vector machine (SVM) temperature prediction model, realized the real-time accurate prediction of the maximum temperature of the battery, and provided a reliable theoretical basis and technical support for the optimal control of the intelligent thermal management system.

Scholars have conducted extensive research on lithium-ion battery thermal management system design. Studies have focused on optimizing channel configurations to improve thermal performance. For instance, Sheng et al. [5] optimized a serpentine channel design, effectively reducing temperature rise and enhancing uniformity. Zhao et al. [6] investigated a honeycomb channel layout, demonstrating that counter-flow in adjacent cooling plates improves temperature uniformity. Li et al. [7] examined a semi-enclosed S-type structure, analyzing the impact of channel dimensions and coolant velocity. Chen et al. [8] compared I-, Z-, and U-type parallel microchannels, finding that symmetrical systems offer better temperature distribution and lower power consumption. Despite achieving convection coefficients >2000 W/(m²·K) [9], traditional liquid cooling faces challenges such as poor temperature uniformity (Δt > 8 °C), leading to local hot spots, and significant pump power consumption (12–18%), which impacts vehicle range [10]. Advanced approaches like multi-material topology optimization and triply periodic minimal surface (TPMS) structures enhance heat transfer but introduce manufacturing complexities and require further validation under extreme conditions [10]. Kim et al. [11] quantified BTMS energy consumption, facilitating energy efficiency evaluations. Bionic designs offer promising alternatives. Zhang et al. [12] developed bronchus-inspired fractal channels, reducing the temperature difference to 4.1 °C, though risks under extreme conditions remain [13]. A key limitation is the insufficient integration of biological mass-transfer principles (e.g., Murray’s law) into multi-stage channel design—a gap this study addresses using fir needle vein structures. Other efforts include topology-optimized wavy microchannels [14], which reduced pressure drop by 37% but maintained a 5.2 °C temperature difference, and spider web-inspired channels [15], achieving 4.3 °C at 5 m/s, albeit with increased manufacturing complexity. Material innovations, such as nano-PCM composites [16] and liquid metal coolants [17], enhance thermal conductivity but raise cost concerns. Recently, AI algorithms have shown potential in optimizing BTMS parameters. Lee et al. [18] used deep reinforcement learning to reduce the temperature difference to 3.5 °C. However, such data-driven models often lack interpretability and fail to capture underlying electrochemical-fluid coupling mechanisms [19]. Zhang et al. [20] developed the graphical convolutional neural network (GCN) to fuse the electrochemical impedance spectrum data to realize the reconstruction of the temperature field of the multi battery module (MAE = 1.2 °C). However, it relies on high-precision sensor networks, which greatly increases the engineering cost. It is worth noting that although the traditional BP neural network is widely used, it has some defects, such as slow convergence speed, easily falling into local optimum, and an average relative error under dynamic conditions exceeding 3.5% [21]. Roberts et al. [22] systematically demonstrated the theoretical defects in the design of the existing bionic cooling system. Although the fractal channel geometry and topology reproduced the biological branch structures (such as leaf veins and blood vessels), Murray’s mass transfer law’s constraint on the branch section size was generally ignored. The absence of this theory leads to a significant increase in turbulent dissipation in the high-speed flow region (Re > 2300), which is specifically manifested in the nonlinear rise of pressure drop and the proportion of pump power loss exceeding the expected value by more than 40%. Cell report physical science emphasizes that the lack of multi-field coupling causes the prediction deviation of high-temperature conditions to exceed 20% [23], which is also the main problem to be solved in this study. For the engineering demand of temperature prediction, the traditional numerical methods have the defects of low computational efficiency and poor nonlinear adaptability, while the artificial intelligence algorithm can effectively capture the complex thermodynamic coupling law through data driving. Based on this, the SVM prediction model is constructed by fusing the bionic liquid cooling experimental data to realize the real-time prediction of the maximum temperature and provide high-precision decision-making basis for intelligent thermal management.

At present, there are still technical bottlenecks in the research. (1) The absence of bionic liquid cooling design principle: The existing fractal channel (such as human blood vessel bionics) only copies the geometric shape, and does not follow Murray’s law to optimize the branch section, resulting in uneven flow distribution (the temperature difference between high-speed and low-speed regions is >7°C) [24]. (2) Decoupling of multiple physical fields: The traditional model separates the thermal and hydrodynamics of electrochemical students, and cannot predict the thermal saturation effect (for example, when the temperature is >28 °C, the temperature rise slope jumps from 0.26 °C/min to 1.0 °C/min). (3) Energy efficiency synergy imbalance: Single objective optimization makes the proportion of pump power in the high-flow-rate area (>4 m/s) reach 22% of the vehicle energy consumption, while the temperature drop gain is only 0.8 °C [25]. In view of the long-standing “temperature uniformity energy consumption” paradox in the field of battery thermal management, Wang et al. pointed out that the traditional serpentine/parallel flow channel could not meet the dual goals of Δt < 5 °C and pump power <10% because it adhered to the single-stage topology [26]. (4) The prediction accuracy of intelligent algorithms is low: The existing artificial intelligence models ignore the physical mechanisms such as thermal saturation effect and turbulence dissipation, lack of extreme working condition data, and the error caused by multi-scale decoupling modeling is more than 3.5%, which restricts the reliability of thermal runaway boundary prediction.

This study aims to develop a high-efficiency liquid cooling system for power batteries. Specifically, an innovative biomimetic cold plate design based on the fractal structure of coniferous leaf veins is proposed, and Murray’s law is incorporated to optimize mass transfer processes, thereby achieving hierarchical optimization of the channel structure. The system is intended to address the issue of thermal runaway under extreme operating conditions, ensure the safe operation of batteries in high-temperature environments, effectively reduce pump power consumption, and significantly improve heat dissipation uniformity, ultimately providing a more efficient and reliable technical solution for power battery thermal management. This study achieved a breakthrough through multi-dimensional methodology innovation. (1) Based on bionic leaf vein fractal structure, through reverse engineering design and combined with Murray’s law, a leaf vein structure liquid cooling plate was developed to realize a laminar flow guarantee mechanism (re ≤ 2300) and flow equilibrium distribution. Simulation experiments show that the liquid cooling plate designed in this study still has good liquid cooling effect under extreme conditions. (2) Based on the ANSYS FLUENT software platform, a fully coupled dynamic solver of electrochemistry fluid heat transfer was established to accurately capture the temperature jump mechanism under the thermal saturation effect of >28°C. (3) Further, the formation mechanism of 2.5 m/s critical velocity threshold and the phenomenon of thermal stagnation plateau at 40.29 °C were revealed through parametric sweep simulation (0.1–8 m/s velocity and 20–42 °C ambient temperature gradient), reducing the power consumption of the flow pump and unnecessary active cooling energy consumption. (4) Based on 98 sets of simulation experimental data, virtual samples were generated by using multiple field models for training. The classic BP neural network, GA-BP neural network and SVM were used to predict the maximum temperature of the battery pack, respectively. The GA-BP neural network had the highest fitting accuracy for the training samples (0.18%), while SVM had the highest prediction accuracy for the validation samples (1.57%), which solved the inherent defects of the traditional artificial intelligence algorithm prediction model that ignored the physical laws and lacked real-time performance.

This research has realized the multi-level deepening thermal management solution of bionic leaf vein structure design, multi-field coupling analysis, critical parameter mining, and a physical rule-driven algorithm, breaking through the technical problem that the traditional liquid cooling system cannot have both a cooling effect and energy efficiency.

2. Battery Pack Model and Structure Design of Liquid Cooling Plate

2.1. Battery Pack Model

With high cell voltage, no memory effect and excellent energy density characteristics, the lithium-ion battery has become the mainstream choice of energy storage system for electric vehicles in the world. As the core basic unit of the power battery pack, the performance of a single battery directly determines the energy output performance, working stability and service life of the battery system [27]. The lithium battery module designed in this study adopts a modular architecture of “parallel first and then series”. First, 50 single batteries are connected in parallel to improve the discharge current carrying capacity of the module. Then, through four levels of series connection, the output voltage of the module is accurately regulated to meet the power demand of the system. A total of 16 lithium battery modules of the same specification are further connected in series to form a complete battery pack system. The overall dimension of the lithium battery module is 260 mm × 150 mm × 190 mm, and the overall dimension of the battery pack is 1230 mm × 630 mm × 190 mm. This parameter setting takes into account the spatial layout, heat dissipation efficiency, mechanical strength and other engineering constraints of electric vehicles, in the model building phase.

We use SolidWorks 3D modeling software 2024 to complete the geometric modeling of the battery module and battery pack. In order to improve the calculation efficiency and accuracy of the subsequent simulation analysis, based on the principle of model simplification, the key dimensions and functional structural features are retained, and the detailed structures (such as bolt holes, chamfers and other non-bearing structures) that have little impact on the simulation results are removed, so as to effectively reduce the consumption of computing resources while ensuring the physical authenticity of the model.

2.2. Design of Liquid Cooling Plate with Venation Channel Structure

The vein structure of fir needles, with their unique fractal hierarchical structure and optimized mass transfer characteristics, is consistent with Murray’s law (i.e., minimizing flow resistance while maximizing nutrient transport in biological systems). The fir needle veins exhibit a bidirectional fractal pattern (Figure 1a), enabling uniform fluid distribution, which stands in sharp contrast to unidirectional branching structures (e.g., mammalian lungs)—the latter may lead to uneven flow distribution in high-power battery cooling. Additionally, the fir needle vein structure can effectively ensure laminar flow; the diameter ratio of parent to daughter channels in fir veins conforms to Murray’s law, which ensures a low Reynolds number (Re ≤ 2300) and minimizes turbulent dissipation, and this is crucial for reducing the pump power consumption of the liquid cooling system of lithium-ion batteries. We evaluated four biological structures (

Table 1. Comparison of biological structures for liquid cooling design.

Structure	Branching Pattern	Murray’s Law Compliance	Thermal Performance	Flow Uniformity
Coniferous vein	Bidirectional fractal	Yes	ΔT = 3.2 °C	High (R2 = 0.99)
Mammalian lung	Unidirectional	Partial	ΔT ~ 5 °C	Moderate
Spider silk	Radial	No	ΔT ~ 4.3 °C	Low

) to justify our selection.

Inspired by the high-efficiency heat dissipation topology of the vein of coniferous fir species as shown in Figure 1a, this study proposed a bionic liquid cooling plate design scheme with hierarchical channel configuration. In order to accurately replicate the fractal transport characteristics of needle vein sequence, the channel system bifurcates step by step along the flow direction, and the design process is deeply integrated with reverse engineering and theoretical optimization. Firstly, the key fractal characteristic parameters were extracted quantitatively by reverse reconstruction of the typical needle vein geometry, which provided a geometric basis for the hierarchical structure and size scaling of the channel. Secondly, Murray’s law is applied to systematically optimize the section size of each branch of the channel. Based on the principle of minimum transport power consumption, it coordinates the flow resistance and space filling efficiency. Its core is to ensure that the fluid always maintains the laminar flow state (Global Reynolds number Re ≤ 2300) in the process of multistage bifurcation transportation, so as to minimize the flow resistance and turbulence dissipation [28].

The overall size of the liquid cooling plate is 1200 mm × 600 mm × 20 mm, and the flow channel is arranged in the form of “vein branch”, as shown in Figure 1b. The main channel connects the entrance with a 20 mm × 20 mm rectangular section, extends along the length and forms a leaf vein-like symmetrical network, which meets the compact requirements of the battery pack, simulates the leaf vein to supply water evenly for the blades, and can solve the problem of heat exchange flow stagnation zone of the traditional channel. The liquid cooling plate is scientifically arranged at the bottom of the battery pack, which is compact and fits the space, which is not only conducive to the efficient temperature control and heat dissipation at the bottom, but also optimizes the leakage proof design to ensure safety, as shown in Figure 1c, to achieve breakthroughs in heat exchange, energy efficiency and space adaptation, and provide an efficient solution for the temperature control of power batteries.

2.3. Design of Serpentine Liquid Cooling Plate

In the field of thermal management of power batteries, the serpentine channel liquid cooling plate has become the mainstream heat dissipation scheme in the current market by virtue of its simple structure and mature manufacturing process, and a large number of studies have been carried out on its heat transfer and flow characteristics. In order to systematically compare the advantages of the innovative structure of the venation channel liquid cooling plate proposed in this paper in terms of heat dissipation performance, based on the principle of orthogonal experimental design, considering the internal space constraints and material mechanical properties of the battery pack, a standardized serpentine channel liquid cooling plate was designed as the control group.

The leaf vein fractal structure optimized based on Murray’s law and bifurcation angles coordinates flow resistance and spatial filling efficiency following the principle of minimizing transmission power consumption, thereby significantly reducing flow separation losses and turbulent dissipation. This design ensures globally stable laminar flow (Reynolds number Re ≤ 2300) and improves flow uniformity, effectively reducing the maximum temperature difference. The mechanism for enhancing thermal uniformity is rooted in three physical foundations. Symmetric bifurcation topology eliminates flow “dead zones” in serpentine channels to enable efficient spatial distribution of the cooling medium; branch dimensions optimized via Murray’s law maintain boundary layer stability and enhance wall heat conduction; and the multi-stage flow splitting structure strengthens radial mixing effects to improve convective heat transfer coefficients. The overall dimensions of the liquid cooling plate are 1200 mm × 600 mm × 20 mm. The internal flow channel adopts the classic 5-u serpentine layout. The radius of curvature of a single U-bend is set to 50 mm to balance the coolant flow resistance and heat dissipation uniformity. The coolant inlet is designed as a 20 mm × 20 mm rectangular section to ensure the uniformity of fluid inlet flow rate. Brass with high thermal conductivity and excellent corrosion resistance is selected as the material to ensure efficient heat conduction while meeting the structural strength requirements. This design scheme provides a scientific and reliable reference for the comparative analysis of the following two channel structures. The assembly model of serpentine liquid cooling plate and battery module is shown in Figure 2.

3. CFD Analysis of Battery Pack Liquid Cooling

3.1. Selection of Simulation Software and Meshing

In order to accurately explore the cooling performance and thermal management characteristics of the battery pack leaf vein structure liquid cooling plate, ANSYS Fluent finite element simulation software was selected as the core analysis tool. The software has a mature numerical solution algorithm and multi-physical field coupling ability in the field of fluid heat transfer, and can effectively simulate the flow and heat transfer process of cooling working medium in complex channels [29].

In the pre-processing stage of the geometric model, the mesh module built in the ANSYS Workbench platform is used to mesh the simplified battery pack model. Considering the symmetry and regularity of the venation channel structure, combined with the significant advantages of structured hexahedral grid in numerical calculation—simple division process, high calculation accuracy, fast convergence speed and strong solution stability—a high-quality grid containing 68,950 elements is finally generated. To ensure the reliability of numerical simulations for complex fractal flow channels, this study systematically implemented grid convergence control strategies. Given the geometric complexity of the fractal channels, we prioritized optimizing mesh quality at bifurcation zones. All computational cells strictly adhered to core convergence criteria (aspect ratio < 3, Jacobian determinant > 0.7), and orthogonality at critical regions (e.g., vein bifurcations) was validated using ANSYS Workbench mesh diagnostics. Through phased mesh refinement tests (curvature-adaptive local encryption), maximum temperature variation remained below 0.3% and pressure drop fluctuation under 1.5% when increasing the element count to 85,000. This confirms that the current mesh density sufficiently resolves thermofluidic characteristics, ensuring grid-independent results. In addition, in order to clarify the boundary conditions, the fluid inlet and outlet of the liquid cooling plate are named “inlet” and “outlet”, respectively, which provides a standardized parameter setting basis for subsequent simulation calculation.

3.2. Determination of Physical Parameters of Lithium Battery

3.2.1. Battery Heat Generation Rate

Bernardi D. et al. [30] determined the battery heat generation rate model through numerical calculation and theoretical analysis. The model mainly studied the changes of discharge and charging current and ohmic internal resistance during battery charging and discharging, and assumed that the heat source inside the battery was uniform and stable. The formula was:

$$q={\displaystyle \frac{I}{V_b}}\left(I^2{R}_j+IT{\displaystyle \frac{\partial U}{\partial T}}\right)$$

(1)

In the formula, $U$—open circuit voltage, V; $I$—charge discharge current, A; $V_b$—battery volume, $m^3$; $T$—temperature, °C; $R_j$—battery internal resistance, $\Omega$.

This uniform heat source assumption is valid for system-level thermal analysis of parallel-connected battery modules. According to the mathematical model of battery heat generation and the heat generation power meter provided by the manufacturer, the heat generation value of lithium battery at 0.5 C and 1 C rate discharge is determined through the verification of theoretical calculation, and the specific values are shown in

Table 2. Heat generation power of the battery at different charging rates.

Magnification	0.5 C	1 C
Heat generating power W/m3	2600	5000
Time/s	7200	5143

. Among them, the discharge rate is the ratio of discharge current to rated capacity. If the battery is discharged at 1 C rate, all the capacity used can be discharged in 1 h.

3.2.2. Calculation of Specific Heat Capacity of Lithium Battery

This paper takes a model 18,650 power lithium battery as an example. The rated voltage of the single cell is 3.6 V, the weight is 47 g, the capacity is 2.4 ah, the internal resistance is less than 60 $\mathrm{m}\Omega$, the normal ambient temperature is 0–45 °C, and the material properties of each part of the lithium battery are shown in

Table 3. Material properties of various parts of lithium battery.

Material	Thermal Conductivity W/(m·K)	Specific Heat Capacity J/(kg·K)	Density kg/m3
cathode material	1.48	1260	1500
Negative electrode material	3.3	1064	1670
Enclosure	0.35	1500	1180
diaphragm	0.38	1980	660
aluminum foil	237	1064	2700
Copper foil	398	385	8900

.

The specific heat capacity of the lithium battery is calculated by the weighted average of the specific heat capacity of the lithium battery materials. The calculation formula is as follows:

$$c_p={\displaystyle \frac{1}{m}}\sum c_i{m}_i$$

(2)

In the formula,

• m—quality of single battery, kg;
• $c_i$—specific heat capacity of a material, J/(kg·K);
• $m_i$—quality of corresponding materials, kg.

The specific heat capacity of each component material of the lithium battery is brought into the above common formula for weighted calculation, and the specific heat capacity of the lithium battery is 1020 j/(kg·K).

3.2.3. Calculation of Lithium Battery Density

The density of the lithium battery is calculated by the ratio of mass to volume of the lithium battery. The calculation formula is as follows:

$$\rho ={\displaystyle \frac{m}{v}}$$

(3)

In the formula,

m—quality of single battery, kg;

$v$—volume of single cell, m³.

After the mass and volume of a single battery are brought into the above formula, the density of the lithium battery is 2840 kg/m³.

3.2.4. Calculation of Thermal Conductivity of Lithium Battery

According to the data provided by the battery manufacturer, the thermal conductivity of a lithium battery in three directions is: ${\lambda }_x$ = 1.2, ${\lambda }_y$ = 15.1, ${\lambda }_z$ = 15.1.

3.3. Theoretical Model of CFD Analysis

CFD is mainly used to discretize the continuous physical quantities in space and time, express the continuous physical quantities with discrete physical quantities, and then use the corresponding interpolation method to solve the approximate solution of variables. The k-e model has the characteristics of fast convergence speed and wide application range. In this paper, the standard k-e turbulence model is selected to simulate and analyze the liquid cooling system [31]. The transport equations of turbulent kinetic energy and turbulent energy dissipation rate in the standard k-e turbulence model are, respectively:

$${\displaystyle \frac{\partial (\rho k)}{\partial t}}+{\displaystyle \frac{\partial (\rho k{u}_i)}{\partial x_i}}={\displaystyle \frac{\partial }{\partial x_j}}[(\mu +{\displaystyle \frac{{\mu }_i}{{\sigma }_k}}){\displaystyle \frac{\partial k}{\partial x_j}}]+G_K-\rho \epsilon +S_k$$

(4)

$${\displaystyle \frac{\partial (\rho \epsilon )}{\partial t}}+{\displaystyle \frac{\partial (\rho \epsilon u_i)}{\partial x_i}}={\displaystyle \frac{\partial }{\partial x_j}}[(\mu +{\displaystyle \frac{{\mu }_i}{{\sigma }_{\epsilon }}}){\displaystyle \frac{\partial \epsilon }{\partial x_j}}]+G_{\sigma 1}{G}_K-C_{\sigma 2}\rho {\displaystyle \frac{{\epsilon }^2}{K}}+S_{\epsilon }$$

(5)

In the formula, $G_k$—turbulent kinetic energy generated by the average velocity gradient.

The calculation formula of $G_k$ is as follows:

$$G_k={\mu }_t({\displaystyle \frac{\partial u_i}{\partial x_j}}+{\displaystyle \frac{\partial u_j}{\partial x_i}}){\displaystyle \frac{\partial u_i}{\partial x_j}}$$

(6)

The turbulent viscosity is a function of ${\mu }_t$ and $k$ and $\epsilon$, and the calculation formula is:

$${\mu }_t=\rho C_u{\displaystyle \frac{{\epsilon }^2}{k}}$$

(7)

In the formula,

${\sigma }_k$—Prandtl coefficient corresponding to turbulent kinetic energy;

${\sigma }_{\epsilon }$—Prandtl coefficient corresponding to turbulent energy dissipation rate.

In this paper, the parameters of Launder and Spalding [32] and other theoretical calculations and experimental verification are selected: $C_u$ = 0.09, ${\sigma }_k$ = 1.22, $C_{\sigma 1}$ = 1.44, $C_{\sigma 2}$ = 1.92.

The calculated Re values across all simulations ranged from 200 to 1600, consistently below the transition threshold (2300), validating laminar dominance.

3.4. Boundary Conditions and Solution Parameter Settings

In the simulation parameter configuration stage, in order to accurately capture the fluid solid coupling heat transfer process, the energy equation in the physical model is activated first to realize the quantitative analysis of heat transfer between fluid and solid. In terms of the definition of material properties, based on the actual working characteristics of lithium batteries, a user-defined material named “battery” is created in the material library. The system inputs its specific heat capacity, density, thermal conductivity and other key thermophysical parameters to ensure the accuracy of the model on the thermodynamic behavior of batteries. The type of coolant is a 50% ethylene glycol aqueous solution [33].

This study established a fully coupled electrochemistry–fluid–thermal model in ANSYS Fluent (Section 3.4). Thermal homogeneity was reliably evaluated through refined mesh generation (68,950 elements) and equivalent thermal parameters of the battery module. The Bernardi heat generation model (Equation (1)) was applied to treat the battery module as a continuum domain, where anisotropic thermal conductivity accurately characterized radial/axial heat diffusion across stacked cells. Weighted average specific heat capacity (Equation (2)) and equivalent density (Equation (3)) integrated cell-level thermal inertia at the system scale, while transient solving (time step: 0.01 s) captured bulk thermal relaxation effects. The boundary condition of the heat source is set at a constant heat generation rate to simulate the heat generation characteristics of the lithium battery under stable conditions. For the cooling system, the inlet flow rate and temperature parameters of the coolant are dynamically configured according to the actual working conditions. In terms of fluid solid boundary treatment, the upper and lower bottom surfaces and sides of the battery pack are defined as wall boundaries. Since the proportion of surface thermal radiation in the operating environment of the battery pack is relatively small, the thermal radiation effect is not considered in the model temporarily. At the same time, the upper and lower bottom surfaces of the battery pack are set as natural convection boundaries to reflect the convective heat transfer process with the environment. By delimiting the complete flow field as the calculation domain, and setting the initial temperature conditions according to the simulation requirements, a closed thermodynamic calculation space is constructed. An ideal thermal contact assumption was employed to neglect interfacial thermal resistance. In ANSYS Fluent, this was implemented via the “Coupled Wall” setting to enable conjugate heat transfer between solid and liquid phases. The continuity of heat flux across the interface was automatically enforced by the energy equation, while temperature continuity was achieved through a coupling algorithm.

In terms of solving control parameters, the time step is set to 0.01 s, the total number of calculation steps is 72,000, and the number of a single iteration is 100. The remaining parameters are set by software default to balance the calculation accuracy and efficiency. In the initialization stage, the fluid domain is set to the incompressible turbulent state, and the effect of thermal deformation of the liquid cooling plate is simplified. By default, the solid–liquid interface follows the no-slip boundary condition, and the efficient heat transfer between fluid and solid is realized through the coupling contact mode, so as to ensure the physical rationality and computational stability of the simulation model. Although the idealized thermal contact assumption simplifies the model and reduces computational complexity, it may underestimate the maximum temperature under high heat flux; interfacial contact thermal resistance could reduce heat transfer efficiency by 1–5%, and subsequent studies will quantify its impact through experiments to optimize the model.

4. Simulation Experiment Design and Result Analysis of Liquid Cooling Effect of Liquid Cooling Plate in Leaf Vein Channel

4.1. Comparative Simulation Experiment of Leaf Vein Structure and Serpentine Structure Liquid Cooling Plate

4.1.1. Simulation Experiment Process Design

Based on ANSYS Fluent platform, the battery module assembly models of leaf vein structure liquid cooling plate and traditional serpentine tube liquid cooling plate were imported, respectively, and the working conditions of cooling liquid inlet flow rate of 4 m/s, battery heat generation power of 5000 W/m³ and ambient temperature of 20 °C were set to carry out the numerical simulation.

4.1.2. Analysis of Numerical Simulation Experiment Results

By observing the simulation results in Figure 3 and Figure 4, it can be found that the maximum temperature of the battery pack with the leaf vein structure liquid cooling plate is 41.21 °C, which is significantly lower than 52.99 °C of the traditional serpentine tube liquid cooling plate battery pack, and the temperature difference between the two is 11.78 °C, which fully verifies the good cooling efficiency of the leaf vein structure liquid cooling plate in the conventional environment.

The test under severe conditions, which further increased the ambient temperature to 42 °C, showed that the maximum temperature of the traditional serpentine liquid cooling plate battery pack climbed to 74 °C, which was close to the critical threshold of lithium battery thermal runaway. The maximum temperature of the leaf vein structure liquid cooling plate battery pack is only 55.85 °C, which is still maintained in the safe working range, highlighting its excellent heat dissipation stability and reliability in a high-temperature environment. In addition, by comparing the temperature cloud distribution, it can be seen that the leaf vein structure liquid cooling plate effectively reduces the internal temperature gradient of the battery pack by virtue of the uniform distribution characteristics of the bionic flow channel. Compared with the traditional serpentine tube liquid cooling plate, its global temperature difference control ability is significantly improved, providing a more balanced thermal management environment for the battery pack. The above simulation data show that the cooling performance of the leaf vein structure liquid cooling plate is better than that of the traditional scheme under different environmental conditions, which provides an innovative solution for the thermal safety design of the power battery system. Moreover, according to [8], the optimal Z-type straight channel achieves a temperature difference of ~5.2 °C, whereas our leaf vein structure maintains ΔT below 3.2 °C, further demonstrating its superiority over basic straight-channel designs.

4.2. Leaf Vein Structure Design of Numerical Simulation Experiment Under Different Ambient Temperatures

4.2.1. Simulation Experiment Process Design

In order to systematically explore the heat dissipation performance of leaf vein structure liquid cooling plate under the condition of a wide temperature range, this study set the gradient ambient temperature condition of 20–42 °C, and carried out 12 groups of numerical simulation at 2 °C intervals to obtain the corresponding temperature distribution cloud map. In order to ensure the refining and representativeness of the results, the uniform interval sampling strategy (sampling rate of 25%, i.e., one group for every four groups of data) was adopted to screen out the typical temperature nephogram under the working conditions of ambient temperature of 20 °C, 28 °C and 36 °C, as shown in Figure 5.

For all 12 groups of simulation data, the system counts the maximum temperature index of the battery pack and draws the relationship curve between the ambient temperature and the maximum temperature (Figure 6). The line chart clearly shows the variation trend of the thermal response of the battery pack with the ambient temperature, which provides an intuitive basis for the quantitative evaluation of the cooling efficiency of the leaf vein structure liquid cooling plate under different heat load scenarios, and effectively supports its environmental adaptability and reliability analysis.

4.2.2. Analysis of Simulation Experiment Results

Quantitative analysis of Figure 6 shows that the maximum temperature of the battery pack shows a significant linear growth trend with the increase in ambient temperature (R2 ≈ 0.99). When the ambient temperature rises to the extreme condition of 42 °C, the maximum temperature of the battery pack under the control of the leaf vein structure liquid cooling plate is 55.85 °C, which is still in the safe working temperature range of a lithium battery (usually ≤ 60 °C). This result fully verified the effectiveness of the biomimetic flow passage design in a wide temperature range. By optimizing the distribution uniformity of cooling medium and the fluid solid heat transfer efficiency, the risk of thermal runaway of the battery pack can be controlled at a very low level even under high-temperature stress, providing a reliable thermal safety guarantee for the power battery system, and further highlighting the engineering application value of the liquid cooling plate design in this study.

At 24 °C ambient temperature, the larger temperature difference between coolant and the battery (due to lower ambient temperature) and the uniform flow distribution of the leaf vein structure promote overall uniform heat exchange. At 40 °C, the increased ambient temperature reduces the initial temperature difference between inlet coolant and the battery, lowering heat dissipation efficiency in the initial region and leading to higher temperatures there. However, the flow distribution design of the leaf vein channels maintains stable heat exchange in subsequent regions.

5. Simulation Experiment and Result Analysis of Heat Transfer Characteristics of Liquid Cooling System with Leaf Vein Structure Liquid Cooling Plate

5.1. Effect Comparison of Leaf Vein Structure Liquid Cooling Plate with Different Coolant Inlet Temperatures

5.1.1. Simulation Experiment Process Design

The leaf vein structure liquid cooling model was imported into Fluent software to build a two-dimensional control variable working environment. The system covered the core parameters such as coolant flow rate, heat generation power, coolant inlet temperature and so on, laying a solid foundation for the multi-scene adaptability verification of the liquid cooling plate.

(1)

Condition 1: coupling test of flow rate and inlet temperature

Fix the battery heat generating power (5000 w/m³) and ambient temperature (30 °C), set the double flow velocity gradient (2.5 m/s, 4 m/s), and carry out 20–32 °C continuous scanning for the coolant inlet temperature (step 1 °C, a total of 12 groups). In order to accurately extract the key laws, three typical temperature nodes of 20 °C, 25 °C and 30 °C are selected, and the corresponding temperature distribution clouds are shown in Figure 7 and Figure 8. Integrate the data of all working conditions, draw Figure 9 (broken line diagram of the maximum temperature of the battery pack under different coolant inlet temperatures), visually analyze the influence mechanism of the inlet temperature on the thermal state of the battery, and quantify the thermal response law under the flow rate coupling effect.

(2)

Condition 2: coupling test of heat generating power and inlet temperature

Keep the coolant inlet flow rate (2.5 m/s) and ambient temperature (30 °C) unchanged, reduce the battery heat generation power to 2600 w/m³, and repeat 12 sets of simulations of coolant inlet temperature of 20–32 °C. Select typical working conditions of 20 °C, 25 °C and 30 °C, and see Figure 10 for temperature distribution cloud chart. Based on the total data, draw Figure 11, the broken line diagram of the correlation between the coolant inlet temperature and the maximum temperature of the battery pack under high and low heat generation power.

5.1.2. Analysis of Simulation Experiment Results

It can be observed from Figure 9 that although the battery heat generation is uniform, the coolant continuously absorbs heat along the flow direction, leading to a gradual temperature increase (e.g., a temperature difference of 3.2 °C from inlet to outlet under 4 m/s condition); meanwhile, the strong heat exchange between the bottom and the liquid cooling plate maintains a vertical temperature difference of approximately 2.8 °C, forming the gradient. The bifurcation of leaf vein channels needs to adapt to the internal module layout of the battery pack, with slight differences in local branch sizes to optimize space utilization; additionally, there is a minor fluctuation of ±0.1 m/s in inlet velocity distribution, resulting in slight asymmetry in the temperature field, but the overall temperature difference is still controlled within 3.2 °C.

(1)

Performance robustness verification and data rule of leaf vein structure liquid cooling plate

By analyzing the data of simulation experiment results in Figure 10 and Figure 11, it can be clearly found that the maximum temperature of the battery pack increases monotonously with the increase in the inlet temperature within the range of heat generating power 5000 w/m³ and inlet temperature 20-32 °C. Under the worst working conditions (inlet temperature 32 °C, high heat generating power 5000 w/m³, low flow rate 2.5 m/s), the maximum temperature reached 53.5 °C, which was significantly lower than the typical lithium battery safety threshold (60 °C), proving that the liquid cooling plate design can still maintain the battery thermal safety under extreme conditions. In addition, the temperature difference of different flow rates (2.5 m/s vs. 4 m/s) is stable at 0.46 °C in the low-temperature region, and the variance is less than 10-4, reflecting the robustness of the system. At low heat generation power (2600 w/m³), the variance of temperature in the range of 20–29 °C approaches zero (standard deviation 0.0017 °C), highlighting the stability of heat management.

(2)

Critical thermal saturation temperature and velocity sensitivity attenuation

When the coolant inlet temperature exceeds 28 °C, the temperature rise response has a nonlinear jump; the temperature rise slope in the low-temperature zone (20–27 °C) is 0.26 °C, while the slope in the 28–32 °C range increases sharply to 1, with an increase of 284%. At the same time, the efficiency of flow rate regulation decreased significantly—the temperature difference of flow rate in high- and low-temperature areas decreased from 0.46 °C to 0.28 °C, and the variance analysis showed that the fluctuation range of temperature difference narrowed after attenuation (σ² = 1.6 × 10⁻⁵). In the bionic vein structure liquid cooling system of this study, 28 °C is defined as the saturation temperature; when the temperature of the battery module reaches this value, the gain of continuing to increase the coolant flow rate on the cooling effect attenuates significantly, and the system enters a thermal saturation state. The thermophysical properties of the 50% ethylene glycol aqueous solution show a significant inflection point around 28 °C; the specific heat capacity increases by 3.2% in the range of 20–28 °C, and drops to 1.1% when the temperature exceeds 28 °C, resulting in a slowdown in the improvement of heat-carrying capacity; the thermal conductivity increases with the rise of temperature before 28 °C, and then gradually decreases, which weakens the heat conduction efficiency of the wall surface. From the perspective of flow characteristics, the decreasing rate of the dynamic viscosity of the coolant slows down when approaching 28 °C. Under the same flow rate, the growth amplitude of the Reynolds number drops from 12% in the range of 20–28 °C to 5% when the temperature is above 28 °C, and the increased amplitude of the convective heat transfer coefficient is only 1/3 of that in the previous period, so the flow enhancement effect is weakened. The synergistic changes of the above-mentioned thermophysical properties and flow characteristics jointly contribute to the formation of the saturation temperature of 28 °C.

(3)

Thermodynamic mechanism and engineering enlightenment of thermal stagnation platform under low heat generation conditions

In the range of low heat generation rate (2600 w/m³) and inlet temperature of 20–29 °C, the battery temperature is stable at 40.29 °C, and the variance is very low (σ² < 10^–5), forming a significant “stagnation platform”. The thermodynamic mechanism is the system heat capacity buffer effect; under low heat load, the heat capacity of the cooling system can completely absorb the inlet temperature rise disturbance, which makes the battery temperature strongly coupled with the environment. The critical point of platform collapse is 29 °C, and then the temperature rise slope jumps to 1.02, which is due to the failure of natural convection caused by insufficient negative temperature difference between coolant and environment (<2 °C). The innovative discovery is that the platform provides an energy-saving basis for new energy vehicles under low-load conditions in cities—the accuracy of coolant temperature control can be relaxed, reducing system energy consumption by more than 20%, while ensuring thermal safety redundancy.

5.2. Effect Comparison of Liquid Cooling Plate with Different Coolant Inlet Flow Rate and Leaf Vein Structure

5.2.1. Simulation Experiment Process Design

In Fluent software, the pre-designed assembly model of leaf vein structure liquid cooling plate and battery module was imported for thermal simulation. Set the ambient temperature to 30 °C.

(1)

Condition 1: The heat generating power of the fixed battery is 5000 w/m³, and the inlet temperature of the coolant is set at 20 °C and 25 °C, respectively. For each inlet temperature, 17 working conditions with coolant inlet flow rate ranging from 0.1 m/s to 8 m/s were simulated, and the corresponding temperature distribution nephogram was obtained. For each inlet temperature, the typical working conditions with inlet flow rates of 1 m/s, 4 m/s and 8 m/s are selected. The maximum temperature simulation results are shown in Figure 12 (inlet temperature 20 °C) and Figure 13 (inlet temperature 25 °C), respectively. Based on the simulation data, the broken line diagram of the maximum temperature of the battery pack with different coolant flow rates under the two inlet temperatures in Figure 14 is drawn.

(2)

Condition 2: Keep the coolant inlet temperature at 20 °C, and adjust the heat generating power of the battery to 2600 w/m³. Similarly, 17 working conditions with inlet velocity ranging from 0.1 M/s to 8 m/s were simulated to obtain the temperature distribution cloud map. The typical working conditions with inlet flow rates of 1 m/s, 4 m/s and 8 m/s are selected, and the maximum temperature simulation results are shown in Figure 15. The broken line diagram of the maximum temperature of the battery pack with different coolant flow rates is drawn in Figure 16 based on the full amount of data.

5.2.2. Analysis of Simulation Results

As can be observed from the simulation contour results, the maximum temperature of the battery pack is mainly concentrated in the top region, far from the bottom leaf vein liquid cooling plate, as this region relies on natural convection for heat dissipation, which is less efficient than the forced convection of the bottom liquid cooling plate.

In terms of parameter dependence, changes in key parameters such as coolant flow rate, inlet temperature, and heat generation power do not fundamentally alter the distribution pattern of the maximum temperature. Under different working conditions, the maximum temperature zone is consistently concentrated in the top region, showing a relatively consistent distribution trend. Only extreme parameter combinations (e.g., ultra-low flow rate coupled with high heat generation) may slightly expand the high-temperature area in the top region, while the core location of the maximum temperature remains stable.

(1)

Velocity temperature correlation law and safety boundary verification

As shown in Figure 15, when the coolant flow rate increases from 0.1 m/s to 8 m/s, the maximum temperature of the battery pack shows nonlinear attenuation characteristics; the temperature drop between 0.1 and 1 m/s is significant (14.3 °C under 20 °C working condition), while the temperature drop between 1 and 8 m/s is slow (only 2.5 °C). Innovative findings: The increase in flow rate from 2.5 m/s to 8 m/s only produces a temperature drop of 0.8 °C (47.32 °C → 46.52 °C), which is the systematic demonstration of the saturation phenomenon of heat dissipation efficiency in the medium-speed region. Under extreme working conditions (0.1 m/s + 25 °C + 5000 w/m³), the maximum temperature of the battery pack is 65.06 °C, which is 14.94 °C lower than the safety threshold, establishing a theoretical breakthrough—quantifying the attenuation boundary of heat dissipation efficiency in the medium- and high-speed regions (>2.5 m/s).

(2)

Heat flow decoupling mechanism dominated by heat generating power

Figure 16 reveals the core control effect of heat generation power on temperature rise; a 92.3% increase in power at 0.1 m/s causes a 16.4 °C temperature rise, which still reaches 7.03 °C at 2.5 m/s. Innovative findings: With the increase in flow rate, the influence of power disturbance attenuates significantly (>2.5 m/s, the temperature rise drops below 7 °C), which proves that the system realizes the decoupling effect of the strong correlation of “power temperature” and the weak correlation of “flow rate temperature” through channel optimization. This mechanism breaks through traditional thinking and forms an innovation in the design paradigm—establishing the power impact attenuation model, providing theoretical support for the temperature control strategy in the medium-speed zone.

(3)

Critical velocity engineering paradigm and industrial value

Using the formula $P_{pump}=\Delta P·Q$ (where $\Delta P$ is pressure drop and $Q$ is flow rate), we systematically analyzed the energy efficiency characteristics under different flow velocities. In the laminar zone (v > 2.5 m/s), pressure drop follows the Hagen–Poiseuille law ($\Delta P\propto Q$), so pump power increases proportionally with flow velocity, accompanied by a significant cooling effect (temperature difference ($\Delta \mathrm{T}$ = 14.3 °C, Figure 15). In the turbulent zone (v > 2.5 m/s), pressure drop follows the $\Delta \mathrm{P}\text{∝}{\mathrm{Q}}^{1.75}$ characteristic where pump power increases to 3.2 times, while the cooling effect only improves by 0.8 °C (Figure 10), resulting in an 86% decrease in energy efficacy. Thanks to the channel structure optimized by Murray’s law (bifurcation ratio d_k+1/d_k = 2^−1/3), low resistance is reduced by 37%, which alleviates the energy consumption penalty in the turbulent zone to some extent. Experiments show that 2.5 m/s is the critical velocity standard. When the velocity is ≥2.5 m/s, the system presents dual robustness characteristics. (a) Velocity insensitivity: The cooling contribution of velocity increase (2.5 → 8 m/s) is sharply reduced to 0.8 °C. (b) Power mutation robustness: The temperature rise caused by the sudden increase in heat generating power by 92.3% (2600 → 5000 w/m³) was strictly controlled within 7.03 °C (57% lower than 0.1 m/s). At this flow rate, the maximum temperature of the battery pack is stable at 47.32 °C, indicating that the system still has the global optimal balance characteristics of temperature stability and energy efficiency under extreme thermal disturbance.

6. Prediction of Maximum Temperature of Battery Pack Based on Artificial Intelligence Algorithm

6.1. Data Acquisition

The prediction of the maximum temperature of the battery pack is of key significance in the field of new energy, and its value is mainly reflected in many dimensions, such as ensuring safety, optimizing performance, extending service life, guiding design and reducing costs. For ensuring the safety of the system and preventing the risk of thermal runaway, we must optimize battery performance and improve operation efficiency. It is of great significance to extend battery life and reduce replacement costs. It is known that heat generation rate, ambient temperature, inlet temperature and inlet flow rate are important factors affecting the maximum temperature of the battery pack. Through ANSYS fluent finite element simulation software, by adjusting the type of liquid cooling plate, the heating power of the battery pack, the flow rate of coolant, the initial temperature of coolant, the ambient temperature and other parameters, the maximum temperature of the battery pack under different parameters is compared, and the control variable method is used to adjust different parameters. Taking the maximum temperature of the battery pack as the target value, 98 groups of simulation results are counted, and 98 sample data are obtained.

6.2. Comparison and Selection of Artificial Intelligence Algorithm for Predicting the Maximum Temperature of Battery Pack

6.2.1. Prediction of GA-BP Neural Network

(1)

Data preprocessing

1

Normalization processing: Normalize the characteristic value of the data to an appropriate range, and preprocess the sample data. In order to eliminate the magnitude or unit difference between the input variables and output variables of the BP neural network, restrict the input variable and output variable sample data to the interval [0, 1]. The preprocessing formula is as follows:

$$y=\left(y_{max}-y_{min}\right)·\left(z-z_{min}\right)/\left(z_{max}-z_{min}\right)+y_{min}$$

(8)

$y$ is the value of sample elements after pretreatment, forming the sequence {y}, $y_{max}$ is the maximum value of the sequence after preprocessing, $y_{min}$ is the minimum value of the preprocessed sequence, $z$ is the input or output sample element value, forming the sequence $z$, $z_{max}$ is the maximum value in the sample sequence element, $z_{min}$ is the minimum value in the sample sequence element.

2

Divide data set: Divide the collected data into a training set and a test set. Among them, 75 data samples are used as the training set to train the neural network, and the remaining 23 data samples are used as the test set to verify the performance of the model.

(2)

Structure design of neural network

1

Determine the number of network layers and nodes

The prediction method of queue length based on the BP neural network is constructed. The three-layer BP neural network with only one hidden layer is selected. The input layer elements of the BP neural network are selected to take the values of heat generation rate, ambient temperature, inlet temperature and inlet flow rate. The output layer element is the maximum temperature of the battery pack.

• 2

  Initialization of network weights and threshold initialization weights and offsets based on GA algorithm

For the BP neural network, each neuron has a threshold, which determines whether the neuron is activated and the degree of activation. At the same time, there are connections between each neuron and other neurons, and each connection corresponds to a weight. The traditional BP neural network generates a threshold and weight by randomization, which leads to slow convergence and low prediction accuracy. In this paper, the GA algorithm is used to determine the optimal weight and threshold of the initialization network [34].

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Forward propagation stage

The BP neural network calculates the output value of neurons in the output layer. The selection of transfer function in the hidden layer and output layer affects the prediction accuracy of the network. The commonly used transfer functions are logsig function, Tansig function and purelin function.

(4)

Loss calculation stage

1

Define loss function

Select the appropriate loss function according to the type of prediction problem. This study is mainly for regression prediction, that is, the mean square error loss function is usually used as shown in the following formula:

$$L={\displaystyle \frac{1}{m}}{\sum }_{i=1}^m{\left(y_i-\widehat{y_i}\right)}^2$$

(9)

where $m$ is the number of samples, $y_i$ is the real value, $\widehat{y_i}$ is the predicted value.

• 2

  Calculate loss value

Use the defined loss function to calculate the loss value according to the real output and the predicted output. This loss value reflects the gap between the current neural network prediction results and the real results.

(5)

Back propagation phase

1

Calculate gradient

The gradient is obtained by calculating the partial derivative of the weight and bias in the network according to the loss function. Start from the output layer and gradually calculate to the input layer.

• 2

  Update weights and offsets

According to the calculated gradient, an optimization algorithm is used to update the weights and offsets in the network. Due to its limitations and shortcomings, the traditional BP neural network has some disadvantages, such as low learning efficiency, easily falling into local minima and so on. In view of these shortcomings, many improved algorithms have been produced, such as the conjugate gradient method, additional momentum method, adaptive learning rate method, elastic gradient descent method, adaptive LR momentum gradient descent method and so on.

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Iterative training and prediction phase

Repeat the process of forward propagation, loss calculation, back propagation and weight update, and carry out iterative training for the whole training data set many times. Each iteration is called an epoch. With the increase in training times, the loss value of the network usually decreases gradually, and the prediction performance of the model will gradually improve. After iteration, the inverse normalization output result is used to obtain the predicted value of the residual correction parameter. If it is not finished, the previous step is returned. There are two main situations to judge whether the algorithm has finished iteration. One is that the algorithm reaches the predetermined number of iterations; the second is when the algorithm achieves the preset prediction accuracy. The prediction result (the final output result of the BP neural network) is normalized, and the reduction formula is as follows:

$$\widehat{z}=\left(\widehat{y}-\widehat{y_{min}}\right)\left(z_{max}-z_{min}\right)/\left(\widehat{y_{max}}-\widehat{y_{min}}\right)+z_{min}$$

(10)

In the formula, $\widehat{y}$ is the value predicted by the BP neural network, which can form the prediction result sequence; $\widehat{y_{min}}$ is the minimum value in the sequence; $\widehat{y_{max}}$ is the maximum value in the sequence; $\widehat{z}$ is the normalized reduction value of BP neural network prediction results.

After the neural network is trained, the remaining data samples are input to the input layer of the neural network to obtain the prediction results (Figure 17).

6.2.2. SVM Prediction

(1)

Data preprocessing

The values of heat generation rate, ambient temperature, inlet temperature and inlet flow rate are selected as input layer elements. The output layer element is the maximum temperature of the battery pack. Using the same method as BP neural network prediction, the input variable and output variable sample data are constrained to the interval [0, 1] to avoid the dimensional difference between features affecting the model performance.

(2)

Feature selection

Support Vector Machines (SVM) are sensitive to the quality of features. High-dimensional redundant features can increase computational complexity and even lead to overfitting. The goal of feature selection is to retain key features and eliminate noisy or irrelevant ones. In this study, the features include calorific value, ambient temperature, inlet temperature, and inlet flow rate, with a feature dimension of 4. The chi-square test [1] is employed to examine the independence between features and labels. A larger chi-square value indicates a stronger correlation. Features that are strongly correlated with the label are retained, while noisy or irrelevant features are removed.

(3)

Model configuration, kernel function, and parameter selection

The kernel function determines the fitting capability of SVM to data. Commonly used kernel functions include linear kernel, RBF kernel (Radial Basis Function), polynomial kernel, and Sigmoid kernel [35]. Through practice, it has been found that the RBF kernel (Radial Basis Function) achieves the best prediction performance. The calculation method is as follows:

$$K\left(x,y\right)=e^{-\gamma ·{‖x-y‖}^2}$$

(11)

In the formula, $x,y$ denotes two samples in the low-dimensional space; ${‖x-y‖}^2$ represents the squared Euclidean distance between the two samples (used to measure the similarity between samples); γ (gamma) is the core parameter of the kernel function (γ > 0), which controls the decay rate of the influence range of samples.

Regularization parameter C: It controls the balance between “margin maximization” and “misclassification penalty”. A larger C value implies a heavier penalty for misclassifications, resulting in a more complex model that is prone to overfitting; conversely, a smaller C value allows more misclassifications, leading to a simpler model that tends to underfit. Kernel function parameter: For the RBF kernel, γ controls the influence range of samples, and a larger γ value means a smaller influence range of individual samples, leading to a more complex decision boundary and a higher tendency to overfit; a smaller γ value results in a larger influence range and a simpler decision boundary. The above two parameters are optimized using a genetic algorithm.

(4)

Model training and prediction

The SVM model is trained with the training set data, and the optimal hyperplane parameters w and B are learned by minimizing the objective function. The objective function is as follows:

$$\left\{\begin{array}{c}{min}_{\omega ,b}{\displaystyle \frac{1}{2}}{‖\omega ‖}^2+C{\displaystyle \sum _{i=1}^n}{\epsilon }_i\\ y_i\left({\omega }^T·x+b\right)-1\ge 0i=\mathrm{1,2},...,n\end{array}\right.$$

(12)

In the formula, $\mathrm{C}$ is the penalty factor; ${\mathsf{\epsilon }}_{\mathrm{i}}$ is a relaxation variable; ${\mathrm{y}}_{\mathrm{i}}$ is the label of sample category; w, b represent the Normal vector and intercept of hyperplane.

In the nonlinear problem, SVM maps the sample data to the high-dimensional feature space by introducing the kernel function, and then performs linear classification in the high-dimensional space. By introducing the Lagrange multiplier to solve the dual problem, the optimal classification decision function f(x) is obtained.

$$f\left(\mathrm{x}\right)=\mathrm{s}\mathrm{g}\mathrm{n}\left\{\sum _{i=1}^n{a}_i·y_i·k\left(x_i·x\right)+b\right\}$$

(13)

In the formula, ${\mathrm{a}}_{\mathrm{i}}$ is the Lagrange multiplier; $\mathrm{k}\left({\mathrm{x}}_{\mathrm{i}}·\mathrm{x}\right)$ is a kernel function, In this paper, the radial basis function is used as the kernel function.

Training process: The input parameters include calorific value, ambient temperature, inlet temperature, and inlet flow rate; the output layer element is the maximum temperature of the battery pack. The collected data are divided into a training set and a test set. Among them, 75 data samples are used as the training set for training the neural network, and the remaining 23 data samples are used as the test set to verify the performance of the model.

(5)

Model evaluation

To verify the generalization ability of the model and determine whether it is overfitting or underfitting, the samples were divided into a training set and a test set in the previous step. After SVM computation, the fitting and prediction results were obtained. The accuracy of the training set and test set was calculated using the following formula:

$$q=1-{\displaystyle \frac{1}{n}}·\sum _{k=1}^n{\displaystyle \frac{\left|\widehat{t}\left(k\right)-t\left(k\right)\right|}{t\left(k\right)}}$$

(14)

In the formula, $q$ represents accuracy, $\widehat{\mathrm{t}}\left(\mathrm{k}\right)$ denotes the fitted or predicted value, $\mathrm{t}\left(\mathrm{k}\right)$ stands for the actual value, and $n$* indicates the amount of sequence data. If the performance metric of the training set is significantly superior to that of the test set, accompanied by low accuracy in the test set, it can be inferred that the method suffers from overfitting. Such a method is not suitable for predicting this type of data, and it is necessary to adjust the parameters or select alternative algorithms for prediction.

6.2.3. Comparison of Prediction Results

The classic BP neural network, GA-BP neural network and SVM are used to predict the maximum temperature of the battery pack, respectively. The known data samples are divided into training samples and validation samples. The fitting results of the training samples and the prediction results of the validation samples of the three methods are as follows (Figure 18, Figure 19 and Figure 20).

According to the above figure, the fitting accuracy of the GA-BP neural network for both training samples and validation samples has been greatly improved compared with that of the BP neural network; in particular, the actual value of the GA-BP neural network for training samples basically coincides with the predicted value. For the SVM prediction results, although the fitting results of the training samples are different from those of the GA-BP neural network, especially at the 31 sample points and 44 sample points where the temperature suddenly increases, SVM cannot fit well, but SVM has higher prediction accuracy for the validation samples than the other two methods, which indicates that the GA-BP neural network is overfitted, resulting in the prediction error of the validation samples being less than that of SVM, and it is preliminarily judged that SVM is more suitable for the prediction of the maximum temperature of the battery pack. The accuracy of various methods is shown in

Table 4. Accuracy of multiple methods.

Method	Training Sample	Validation Sample
BP neural network	3.01%	3.68%
GA-BP neural network	0.18%	2.16%
SVM	1.09%	1.57%

.

In order to compare the fitting accuracy and prediction accuracy more intuitively, the average relative error is used to quantify. The specific calculation method is as follows:

$$e_r={\displaystyle \frac{1}{n}}·{\sum }_{k=1}^n{\displaystyle \frac{\left|\widehat{t}\left(k\right)-t\left(k\right)\right|}{t\left(k\right)}}$$

(15)

In the formula, $e_r$ is average relative error, $\widehat{\mathrm{t}}\left(\mathrm{k}\right)$ is a fitted or predicted value, $\mathrm{t}\left(\mathrm{k}\right)$ is the actual value, $n$ represents the amount of sequence data.

According to the above table, the GA-BP neural network has the highest fitting accuracy for training samples, while SVM has the highest prediction accuracy for validation samples, which is consistent with the conclusion of the above analysis.

6.2.4. Analysis of SHAP Values

The SVM battery pack maximum temperature prediction model conducts learning and prediction through AI algorithms. However, humans cannot intuitively observe the prediction process, resulting in weak interpretability of the model and thus making the SVM model have a “black-box nature”.

SHAP is an interpretability technique used to reveal the prediction results of “black-box” models, thereby improving the interpretability of the model and users’ trust in the model. The SHAP explanation model, based on the concept of Shapley value in cooperative game theory, fairly distributes the contribution of each battery pack feature to the output result of the battery pack maximum temperature prediction model, so as to effectively explain the battery pack maximum temperature prediction model [36].

For model x and input sample y, the calculation formula of the SHAP value ϕi for the i-th feature is as shown in the following formula:

$${\phi }_i\left(x,y\right)=\sum _{S\subseteq F/\left\{i\right\}}{\displaystyle \frac{\left|S\right|!\left(\left|F\right|-\left|S\right|-1\right)!}{\left|F\right|!}}\left[x\left(S\bigcup \left\{i\right\}\right)-x\left(S\right)\right]$$

(16)

In the formula, F is the set of all features (the total number of features is |F|); S is the feature subset that does not contain the i-th feature; x(S) is the predicted value of the feature model when only subset S is used; ${\displaystyle \frac{\left|S\right|!\left(\left|F\right|-\left|S\right|-1\right)!}{\left|F\right|!}}$ is the weight term, which ensures that each feature is weighted equally. Since the SVM has the highest prediction accuracy, the SHAP interpretation established by selecting SVM for feature analysis is shown in Figure 21.

Figure 21 shows the SHAP values of the four input parameters of the SVM. Figure 22 is obtained by calculating the mean absolute error. Figure 22 presents the importance ranking results of the input parameters in the training process of the SVM model analyzed by SHAP values: heat generation > inlet flow rate > inlet temperature > ambient temperature. It can be seen that heat generation has the greatest impact on the maximum temperature of the battery pack, and its SHAP value is much higher than that of the other three factors, which is consistent with the actual situation of the battery pack. The contribution proportions of the other three factors to the model are basically the same, and their SHAP values differ little.

Figure 23 is a SHAP value bee-swarm plot. When using the SHAP method to analyze data features, the SHAP value can assign a numerical value to each feature, which represents the degree of influence of the feature on the algorithm’s prediction result. If it is greater than 0, it indicates that the feature value makes a positive contribution to the target value; if it is less than 0, it indicates that the feature value makes a negative contribution to the target value. The interval range of the SHAP value represents the importance of the feature. The color of the points represents the magnitude of the feature value, with red representing high values and blue representing low values.

According to Figure 23, the importance of each factor is consistent with the conclusion in Figure 22, arranged from top to bottom. For heat generation, the blue points are on the left side of the baseline, and the red points are slightly to the right of the baseline. That is, the lower the heat generation, the lower the maximum temperature of the battery pack. With higher heat generation, the maximum temperature of the battery pack will increase slightly, meaning that heat generation and the maximum temperature of the battery pack approximately show a positive correlation.

The blue points of inlet flow rate are on the right side of the baseline, indicating that the smaller the inlet flow rate, the higher the maximum temperature of the battery pack. The blue points of inlet temperature are on the right side of the baseline, meaning that the smaller the inlet temperature, the higher the maximum temperature of the battery pack, which is slightly inconsistent with the actual situation. There are also some red points of inlet temperature on the right side of the baseline, indicating that there is also a relationship where the higher the inlet temperature, the higher the maximum temperature of the battery pack. In general, the influence of inlet temperature on the maximum temperature of the battery pack is not obvious in the above figure.

Both the red and magenta points in ambient temperature are located on the right side of the baseline. Magenta represents feature values between red and blue, leaning towards red. That is, the higher the ambient temperature, the higher the maximum temperature of the battery pack.

7. Conclusions

In this study, based on the fractal transport mechanism of fir needle vein, a fractal bionic liquid cooling plate with Murray’s law was innovatively designed, and its superiority in the thermal management of a power battery under extreme operating conditions was verified by a multi-physical field coupling numerical model. The main conclusions are as follows:

(1)

Compared with the traditional serpentine flow channel, the bionic flow channel of leaf vein significantly improves the heat dissipation uniformity. Under the conditions of 4 m/s flow rate and 5000 w/m ³ heat generating power, the maximum temperature of the battery is reduced by 11.78 °C. The temperature gradient in a high-temperature environment (42 °C) is strictly controlled within 3.2 °C. The fractal topology structure can effectively eliminate the heat transfer dead zone of the traditional channel and realize the efficient distribution of cooling medium space.

(2)

The research reveals the law of key parameter threshold. The critical velocity threshold is determined to be 2.5 MGS for the first time, and the temperature drop gain of velocity increase after exceeding the threshold is reduced by 86% (>2.5 M/s temperature drop is only 0.8 °C). It is found that the critical point of thermal saturation is 28 °C. When the coolant temperature exceeds this value, the temperature rise slope increases sharply by 284% (0.26 → 1). At this time, the flow rate control efficiency decreases by 45.7%, which needs to be supplemented by active cooling strategy to break through the bottleneck.

(3)

Under low-load conditions (2600 w/m ³), there is a 40.29 °C thermal stagnation platform (20–29 °C inlet temperature range), and the standard deviation of temperature fluctuation in the platform is less than 0.0017 °C. This phenomenon provides a theoretical basis for energy saving control under urban working conditions. The control accuracy of coolant temperature can be relaxed to ± 5 °C, which can effectively reduce the system energy consumption.

(4)

The comparative verification based on 98 groups of full parameter simulation data shows that the average relative error of support vector machine (SVM) in the verification set is only 1.57%, which is significantly better than the GA-BP neural network (2.16%). A key breakthrough was overcoming the overfitting limitation—although GA-BP achieves a high fitting degree of 0.18% in the training set, it fails to predict under the condition of sudden temperature change (such as 31/44 sample points). SVM accurately captures the nonlinear temperature jump characteristics through radial basis function kernel function, and provides real-time decision support for the intelligent control of critical velocity threshold (2.5 MGS) and thermal saturation temperature (28 °C), finally forming a complete thermal management closed loop of “structure optimization parameter identification predictive control”.

Through the three-level collaborative innovation of “bionic flow channel critical parameters intelligent prediction”, this design establishes a new thermal management paradigm of high-efficiency heat dissipation (temperature difference < 3.2 °C), energy consumption optimization and strong robustness (SVM prediction error 1.57%) for the power battery of new energy vehicles. It realizes the precise suppression of the risk of thermal runaway under extreme conditions, and provides theoretical and technical support for the safety control of high-rate battery packs.

Analysis of Research Limitations:

Impact of Model Simplification Assumptions: The simulation did not account for the thermal expansion effects of internal battery materials or the gas–liquid two-phase flow phenomena during coolant circulation, which may lead to minor deviations in temperature prediction under extreme operating conditions.

Insufficient Experimental Validation: Current results are based on numerical simulations. Future work should further validate the accuracy of critical flow velocity threshold (2.5 m/s) and thermal saturation temperature (28 °C) through physical prototype experiments.

Adaptability to Dynamic Conditions: The SVM model may require further optimization for extreme dynamic operating conditions (e.g., rapid charge–discharge cycles) not covered by the training data.

Future Research Directions:

Deepening Multi-Physics Field Coupling: Develop an integrated electrochemical–thermal–mechanical coupling model to investigate the impact of battery deformation on cooling performance.

New Materials and Media: Explore the potential of nanofluids or phase change materials to enhance efficiency in bio-inspired flow channels.

Algorithm Integration and Real-Time Control: Combine SVM with reinforcement learning to develop adaptive flow velocity control strategies for overcoming thermal saturation bottlenecks.

Material properties: Temperature-dependent property models (e.g., λ(T), Cp(T)) will be incorporated to refine predictions in thermal saturation zones (>28 °C) and extreme low-temperature scenarios.