1. Introduction and Research Background
In the global pursuit of carbon neutrality, the transportation sector is undergoing a paradigm shift toward electrification. As an intermediate stage between conventional internal combustion engine vehicles and pure battery electric vehicles, the 48V mild hybrid system has emerged as a highly effective technical solution. This architecture utilizes electric boosting to achieve remarkable fuel savings under various driving conditions such as start-stop, launch assist, and regenerative braking, while also enabling pure electric driving in specific scenarios. The performance of the entire EV battery pack hinges critically on effective thermal management.
Lithium-ion batteries, which are the core energy storage components of these systems, exhibit strong temperature-dependent performance characteristics. The battery pack’s operating temperature window is generally specified as -20 °C to 60 °C, while its optimum operating range for balancing performance and lifespan is much narrower: 25 °C to 40 °C. Excessive temperatures accelerate detrimental side reactions, leading to capacity fade and, in extreme scenarios, thermal runaway. Conversely, significant temperature gradients within the battery pack result in uneven current distribution, localized overcharging or over-discharging, and accelerated degradation of individual cells. Therefore, the design of a robust Battery Thermal Management System (BTMS) is imperative.
Among the various cooling strategies, including air cooling, phase change material (PCM) cooling, and liquid cooling, the liquid-cooling approach stands out for its superior heat transfer coefficient and high specific heat capacity of coolants. Liquid cooling is widely adopted in commercial EVs. The design of the cold plate, particularly its internal flow channel architecture, plays a pivotal role in determining the temperature uniformity and maximum temperature of the EV battery pack. This research is dedicated to designing an optimized liquid cooling structure for a 48V EV battery pack, presenting a comprehensive method that integrates experimental characterization, computational fluid dynamics (CFD) simulation, and multi-objective optimization.
The primary challenges in designing liquid-cooled thermal management systems for EV battery packs include resolving the trade-off between cooling effectiveness and pumping power consumption. A cold plate with a complex flow path might offer excellent thermal performance but could introduce unacceptably high pressure drops, necessitating a more powerful coolant pump. Moreover, accurate transient thermal modeling of the EV battery pack requires precise knowledge of the battery’s internal heat generation mechanisms, which are a complex superposition of irreversible resistive heating (Joule and polarization heat) and reversible entropic heat.
To address the high computational cost of traditional CFD simulations for quick design-space exploration and real-time control applications, this work also investigates the integration of reduced-order models built upon Singular Value Decomposition (SVD). This approach provides a step toward deploying digital twins for EV battery packs, enabling near-real-time monitoring of thermal states without compromising on simulation fidelity.
2. Experimental Characterization and Heat Generation Modelling of the Cell
2.1 Working Principle of the Lithium-ion Battery
The lithium-ion battery stores and releases energy through the migration of lithium ions between the anode and cathode. During discharge, Li+ ions deintercalate from the anode material (typically graphite) and intercalate into the cathode (typically LiFePO4). During charging, the process is reversed. The associated electrochemical reactions for an LFP battery are denoted as:
$$ \text{Anode}: Li_x C_6 \xrightarrow{\text{discharge}} 6C + xLi^+ + xe^- \quad \text{(during charge it proceeds in the reverse direction)} $$
$$ \text{Cathode}: Li_{1-x}FePO_4 + xLi^+ + xe^- \xrightarrow{\text{discharge}} LiFePO_4 \quad \text{(during charge it proceeds in the reverse direction)} $$
2.2 Heat Generation Mechanism and Modelling
The total heat generated (\( Q_{gen} \)) during battery operation is generally categorized into four distinct sources: Joule heating (\( Q_j \)), polarization heating (\( Q_p \)), reaction/entropic heating (\( Q_r \)), and side-reaction heat. Side-reaction heat is typically negligible under normal operating conditions.
The dominant irreversible heat source is the combination of Joule and polarization heating, which arise from internal ohmic and polarization resistances:
$$ Q_{irr} = Q_j + Q_p = I^2 (R_o + R_p) $$
where \( I \) is the current and \( R_o \), \( R_p \) are the ohmic and polarization resistances, respectively. The reversible heat generation is termed as reaction heat and is given by:
$$ Q_r = \frac{n m Q_e I}{M F} = I T \frac{dU}{dT} $$
where \( n \) is number of cells, \( m \) the electrode mass, \( Q_e \) is the heat of reaction, \( F \) the Faraday constant (96484.5 C/mol), \( T \) the absolute temperature and \( dU/dT \) is the entropic heat coefficient. By integrating these contributions and normalizing by the volume \( V \) of the active cell body, a volumetric heat generation rate \( q \) is obtained as expressed by the Bernardi model:
$$ q = \frac{I}{V} (U – E) + \frac{I T}{V} \left( \frac{dU}{dT} \right) = \frac{I^2 (R_o + R_p)}{V} + \frac{I T}{V} \left( \frac{dU}{dT} \right) $$
2.3 Physical Parameters of the Cell
The research object is an 8Ah lithium iron phosphate (LFP) pouch cell. The physical dimensions and thermophysical properties were measured and calculated. The density and specific heat capacity were computed using the weighted average method, incorporating properties of the constituent materials. The anisotropic thermal conductivities are calculated based on the series and parallel thermal resistance models of the cell laminate. In this paper, we utilized a homogenization technique to treat the jelly roll as a single solid domain with anisotropic conductivities.
$$ \rho_{cell} = \frac{\sum (\rho_i V_i)}{\sum V_i} = 1892.9 \text{ kg/m}^3 $$
$$ C_{p,cell} = \frac{\sum (C_i m_i)}{\sum m_i} = 1104.2 \text{ J/(kg·K)} $$
$$ \lambda_x = \lambda_y = 26 \text{ W/(m·K)} \quad \text{(in-plane, parallel to the foil layers)} $$
$$ \lambda_z = 0.6 \text{ W/(m·K)} \quad \text{(through-plane, perpendicular to the foil layers)} $$
2.4 Experimental Setups for Thermal Parameters
To construct an accurate heat generation model for use in the EV battery pack thermal simulation, a dedicated test bench was set up encompassing a battery tester (Neware CE-6002n-30V100A-H), a temperature chamber (SC-80-CB-2), and a data acquisition system.
**Experiment 1: Internal Resistance Measurement.** The internal resistances were measured using the Hybrid Pulse Power Characteristic (HPPC) method. At different states of charge (SOC), the cell was subjected to a current pulse, and the voltage drop pattern was analyzed to separate ohmic and polarization resistances. The tests were conducted at 25 °C. The extracted resistance data are summarized in Table 1.
Table 1: Internal Resistance of the Cell at 25 °C under various discharge rates
| SOC | 1.0 | 0.9 | 0.8 | 0.7 | 0.6 | 0.5 | 0.4 | 0.3 | 0.2 | 0.1 | 0 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 0.5C (mΩ) | 2.15 | 2.25 | 1.6 | 1.18 | 1.13 | 1.18 | 1.23 | 1.38 | 1.48 | 1.55 | 14.15 |
| 1C (mΩ) | 1.7 | 1.13 | 1.45 | 1.04 | 1.04 | 1.11 | 1.2 | 1.31 | 1.35 | 1.51 | 12.73 |
| 2C (mΩ) | 1.2 | 1.3 | 1.525 | 1.05 | 0.94 | 1.46 | 1.48 | 1.7 | 1.64 | 1.3 | 4.22 |
| 3C (mΩ) | 1.14 | 1.31 | 1.05 | 1.03 | 1.2 | 1.35 | 1.53 | 1.58 | 2.13 | 2.0 | 8.41 |
It was observed that the internal resistance remains relatively stable within the SOC range of 0.2 to 1.0 but dramatically increases at SOC levels below 0.2. This behavior is attributed to mass transport limitations of lithium ions within the electrolyte at low concentrations.
**Experiment 2: Entropic Heat Coefficient Measurement.** This experiment determines the `dU/dT` coefficient, which can be either positive (exothermic) or negative (endothermic) depending on the electrode materials and SOC. The test involved keeping the cell at different reference temperatures within a thermal chamber and recording the open-circuit voltage after sufficient relaxation time. The relationship is expressed by the thermodynamic identity:
$$ \frac{dU}{dT} = \frac{\Delta S}{nF} $$
The experimentally obtained `dU/dT` values across various SOC states are plotted and fitted in Figure 2.5. The key data points are presented in Table 2 below.
Table 2: The entropic heat coefficient at varying SOC levels
| SOC | 1.0 | 0.9 | 0.8 | 0.7 | 0.6 | 0.5 | 0.4 | 0.3 | 0.2 | 0.1 | 0 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| dU/dT (mV/K) | -0.29 | -0.24 | 0.02 | 0.08 | 0.18 | 0.21 | 0.12 | 0.01 | -0.01 | -0.02 | -0.05 |
These discrete points were accurately captured by a fifth-order polynomial function of SOC:
$$ \frac{dU}{dT} = 17.468 \cdot SOC^5 – 38.338 \cdot SOC^4 + 25.728 \cdot SOC^3 – 5.6528 \cdot SOC^2 + 0.5483 \cdot SOC + 0.047 $$
**Experiment 3: Temperature Rise Tests.** The cell was fully charged and subsequently discharged at various C-rates (from 0.5C to 5C). Three K-type thermocouples (T1 center, T2 and T3 at corners) were attached to the surface to monitor the temperature evolution. The data confirm that the temperature increase is strongly rate-dependent, with a steep rise observed at the beginning and the end of discharge. The initial and final high heating rates correlate well with the high `dU/dT` values (exothermic) in these regions, while the central phase of discharge is dominated by reversible endothermic behavior, particularly at low C-rates.
**Model Validation of the Cell Heat Generation Model:**
To validate the heat generation model, a full 3D steady and transient thermal simulation of the cell was performed in ANSYS Fluent. The internal heat source term was defined using a User-Defined Function (UDF), which used cell internal resistance, SOC, temperature, and entropy data as inputs. The comparison between the simulated temperatures and experimental temperatures at the three monitor points showed excellent agreement. The model accurately predicted the temperature rise across low C-rates. Even at high C-rates (4C, 5C), the deviations between simulation and experiments remained minimal, confirming that the model is robust and applicable for simulating the behavior of the full EV battery pack.
3. Liquid Cooling Plate Design and Multi-Objective Optimization
3.1 The 48V Battery Pack and Numerical Setup
After successful cell-level validation, I proceeded to construct the numerical model for a full 48V EV battery pack. The pack architecture consists of 14 cells connected in series. For simulation efficiency, the active thermal elements were isolated into a cooling module. The core model, shown below, uniformly comprises battery cells and insulating sheets but excludes structural chassis components, which have a negligible impact on heat transfer analysis.
For boundary conditions, the exterior surfaces of the module were set to a natural convection condition with a heat transfer coefficient of 5 W/(m²·K). The interface between cells and the heat sink was modeled as a contact layer encompassing the thermal properties of the aramid insulation sheets. A Poly-Hexcore mesh strategy was employed for the fluid domain to optimize mesh count while maintaining accuracy. A grid sensitivity analysis verified that the model outcome (maximum temperature) is independent when the total mesh count exceeds 2,089,298. The final model introduced around 3.2 million cells.
Before designing the liquid cooling system, I conducted a baseline simulation of the battery pack under natural air convection. Under the high load of a 3C discharge rate, the simulation results indicated that the maximum temperature within the EV battery pack would escalate to 336 K. This value significantly exceeds the recommended maximum operational limit of 313.15 K (40 °C), confirming that natural convection alone is incapable of managing the thermal load generated by this configuration. This reinforces the critical need for liquid cooling.
3.2 Comparison of Liquid Cooling Plate Flow Channels
To effectively reduce the temperature of the 48V EV battery pack, I designed three distinct liquid cooling plates. The main design difference lies in the topology of the internal channels:
- Serpentine flow channels
- Parallel straight flow channels
- Bionic leaf-vein inspired channels (mimicking hierarchical natural structures)
The coolant was set as a 50% water-ethylene glycol mixture, and the plates were made of aluminum. The boundary condition for the fluid domain was a velocity inlet (0.4 m/s) and a pressure outlet. The thermal conductivity of aluminum is 202.4 W/(m·K). The coolant properties are presented in Table 3.
Table 3: Physical properties of the coolant and plate material
| Material | Density (kg/m³) | Conductivity (W/(m·K)) | Specific Heat (J/(kg·K)) | Dynamic Viscosity (Pa·s) |
|---|---|---|---|---|
| Aluminum | 2719 | 202.4 | 871 | – |
| Ethylene-Glycol Solution | 1071 | 0.384 | 3319 | 0.00294 |
A comprehensive comparison of their thermal performance was conducted under the same 3C discharge rate. The performance indicators selected were the maximum temperature (Tmax), the maximum temperature difference (ΔTmax) within the EV battery pack, and the resultant pressure drop (ΔP) across the cold plate.
The simulation results revealed distinct characteristics for each channel design. The serpentine flow channel configuration led to high average module temperatures but presented the lowest pressure drop. The parallel-flow design demonstrated reasonable cooling but poor temperature uniformity across the module. Remarkably, the bionic leaf-vein flow channel yielded a significantly lower maximum temperature and an improved temperature homogeneity, all while maintaining an acceptable pressure drop. The key channel dimensions in these structures were designed with similar hydraulic diameters to provide a fair comparison.
Below in Table 4, I have quantified and summarized the results for three channel structures analyzed:
Table 4: Performance Comparison of the Different Flow Channel Structures for the EV battery pack
| Channel Type | Tmax (K) | ΔTmax (K) | Pressure Drop (kPa) |
|---|---|---|---|
| Serpentine | 327.155 | 15.737 | 2.807 |
| Parallel | 321.133 | 19.688 | 3.411 |
| Bionic Leaf-Vein | 306.215 | 7.246 | 3.349 |
The bionic leaf-vein channel structure, with its multifurcating distribution network similar to that of a natural leaf, effectively reduces flow stagnation zones and promotes uniform heat exchange. Given its superior cooling performance, it was chosen for subsequent parameter optimization.
3.3 Box-Behnken Orthogonal Test for Parameter Analysis
The structural parameters of the initial leaf-vein cold plate significantly influence its performance. I selected three critical design variables for systematic examination:
- The width of the main channel (D)
- The angle of the bifurcating branches (\( \alpha \))
- The number of branch channels (N)
To deduce the relationships between these parameters and the module-level requirements \(\{T_{max}, \Delta T_{max}, \Delta P\}\), I employed a Box-Behnken Design (BBD). This is a response surface methodology (RSM) that follows a three-level factorial approach, reducing the number of experimental runs required relative to a full factorial design.
The factor ranges and levels are defined in Table 5:
Table 5: Variables and their levels in the experimental design
| Factor | Variable | Low Level (-1) | Center (0) | High Level (+1) |
|---|---|---|---|---|
| A | Channel Width D (mm) | 1 | 3 | 5 |
| B | Channel Angle \( \alpha \) (°) | 90 | 135 | 180 |
| C | Channel Number N | 5 | 7 | 9 |
The design matrix generated 17 unique simulation cases. The resulting performance outputs (Tmax, ΔTmax, ΔP) for each test case are summarized in Table 6:
Table 6: Design matrix and response values from the BBD experiments
| Run | D (mm) | \( \alpha \) (°) | N | Tmax (K) | ΔTmax (K) | ΔP (kPa) |
|---|---|---|---|---|---|---|
| 1 | 3 | 180 | 5 | 305.611 | 4.822 | 3.696 |
| 2 | 5 | 135 | 5 | 305.593 | 5.524 | 1.837 |
| 3 | 3 | 135 | 7 | 305.661 | 4.815 | 3.707 |
| 4 | 3 | 90 | 5 | 305.827 | 4.841 | 3.159 |
| 5 | 1 | 90 | 7 | 306.057 | 5.618 | 6.630 |
| 6 | 1 | 180 | 7 | 306.291 | 5.633 | 6.590 |
| 7 | 3 | 135 | 7 | 305.661 | 4.815 | 3.707 |
| 8 | 5 | 180 | 7 | 305.588 | 6.093 | 1.835 |
| 9 | 5 | 135 | 9 | 305.594 | 4.761 | 1.826 |
| 10 | 3 | 135 | 7 | 305.661 | 4.815 | 3.707 |
| 11 | 3 | 90 | 9 | 305.665 | 4.802 | 3.645 |
| 12 | 3 | 135 | 7 | 305.661 | 4.815 | 3.707 |
| 13 | 1 | 135 | 5 | 306.305 | 5.629 | 6.336 |
| 14 | 3 | 180 | 9 | 305.653 | 4.825 | 3.712 |
| 15 | 5 | 90 | 7 | 305.599 | 4.757 | 1.814 |
| 16 | 3 | 135 | 7 | 305.661 | 4.815 | 3.707 |
| 17 | 1 | 135 | 9 | 306.277 | 5.617 | 6.463 |
Using ANOVA analysis, I was able to determine the significance of each parameter. A quadratic polynomial surrogate model was then fitted. The fitted model takes the following generalized form:
$$ y = \beta_0 + \sum_{i=1}^{3} \beta_i x_i + \sum_{i=1}^{3} \beta_{ii} x_i^2 + \sum_{i<j}^{3} $$="" +="" Analysis of the regression models (also demonstrating their coefficient values):
$$ T_{max} (D, \alpha, N) = 305.30306 – 0.605250x_1 + 0.013943x_2 + 0.133906x_3 + 0.090775x_1^2 – 0.000403x_1 x_2 – 0.008806 x_1 x_3 – 0.000032x_2^2 – 0.000350x_2 x_3 – 0.006031x_3^2 $$
$$ \Delta T_{max} (D, \alpha, N) = 6.88066 – 0.512281x_1 – 0.008157x_2 + 0.211788x_3 + 0.067903x_1^2 + 0.001253x_1 x_2 – 0.015738x_1 x_3 + 0.000010x_2^2 + 0.000505x_2 x_3 – 0.019716x_3^2$$
$$ \Delta P (D, \alpha, N) = 8.91264 – 1.79511x_1 – 0.016805x_2 + 0.062304x_3 + 0.110635x_1^2 – 0.000163x_1 x_2 – 0.002444x_1 x_3 + 0.000068x_2^2 – 0.00000252778x_2 x_3 – 0.002221x_3^2 $$
Table 7: ANOVA results (excerpt from analysis)
| Source | Response | F-value | P-value | Significance |
|---|---|---|---|---|
| Overall Model | Tmax | 103.24 | <0.0001 | Highly Significant |
| Overall Model | ΔTmax | 21.10 | 0.0003 | Significant |
| Overall Model | ΔP | 621.82 | <0.0001 | Highly Significant |
This is coupled with high coefficients of determination (R² > 0.9 for all cases), confirming that these quadratic models are able to accurately interpolate the CFD data and can therefore be used for the parameter optimization.
Response surface plots were generated to visually assess the interactions. The analysis results show that the channel width is the paramount factor influencing the Tmax and the pressure drop, while the number and angle of the channels have a stronger secondary effect on temperature uniformity.
3.4 Multi-Objective Optimization using NSGA-II
Since the cooling system must simultaneously minimize three conflicting objectives, finding a single optimal solution is impossible. It necessitates a multi-objective optimization algorithm to find a set of Pareto-optimal trade-off points. I successfully applied the Non-dominated Sorted Genetic Algorithm II (NSGA-II) to the surrogate models to identify this set of trade-offs.
The pseudo-code involved:
- Initialize population (size=50).
- Evaluate non-domination ranks and crowding distances.
- Selection via a binary tournament.
- Genetic operators: Crossover and Mutation (probability setting).
- Creation of offspring populations (elitism ratio 0.8).
- Run up to 150 generations.
The optimization problem was formulated mathematically as:
$$ \min \, \mathbf{F}(x) = \big( f_1(x), f_2(x), f_3(x) \big)^T $$
where \(x = [D, \alpha, N]\) and the individual functions are the surrogate equations for Tmax, ΔTmax, and ΔP. The search space is constrained by:
$$ 1 \, \text{mm} \le D \le 5 \, \text{mm}, \quad 90° \le \alpha \le 180°, \quad 5 \le N \le 9 $$
Upon completion of the optimization, I performed a decision analysis to select the knee-point solution from the Pareto front which shows balanced improvements in all metrics. The algorithm identified the following design specifications as ideal:
$$ D_{opt} = 3.7 \, \text{mm}, \quad \alpha_{opt} = 98°, \quad N_{opt} = 7 $$
A confirmatory CFD simulation was then run using these optimized parameters to compare the predicted optimal metrics against the CFD-computed metrics. The results of this is shown in the following comparative Table 8 and, confirming excellent agreement and validating the entire co-simulation-based optimization methodology.
Table 8: Validation results of the optimized bionic channel against baseline and CFD
| Configuration | Parameters [D, \( \alpha \), N] | Tmax (K) | ΔTmax (K) | ΔP (kPa) |
|---|---|---|---|---|
| Initial Flow Channel | [3.0, 90°, 7] | 306.215 | 7.246 | 3.349 |
| Optimized (Prediction) | [3.7, 98°, 7] | 305.304 | 6.144 | 2.234 |
| Optimized (CFD Verification) | [3.7, 98°, 7] | 305.374 | 6.237 | 2.485 |
| Error (%) | – | 0.02% | 1.48% | 5.89% |
Compared to the initial design, the optimized flow channel for this EV battery pack leads to a significant attenuation of the maximum temperature (reduction of 0.91 K), a reduction of 1.1 K in the maximum temperature differential, and a 1.12 kPa reduction in pressure drop. Achieving cooling performance simultaneously with lower hydraulic losses signifies a large step forward in the energy efficiency and life of the battery pack.
4. CFD Analysis and SVD-based Reduced Order Modeling
After the design optimization of the liquid cooling plate, I investigated the performance of the EV battery pack under a broader range of operating conditions including various ambient conditions and coolant flow rates.
4.1 Parametric Study via CFD
I evaluated the effects of different parameters, including the battery discharge rate, the coolant fluid velocity, and the inlet coolant temperature.
**Effect of Discharge rate.** Analyzing different C-rates (1C, 2C, 3C, and 4C) shows that Tmax and ΔTmax increase with discharge rate. This is expected because heat generation rate is proportional to the square of the current. At high rates like 4C, the module’s internal center experiences dangerously high temperatures, suggesting the cooling system must be sized adequately for extreme fast-charging or high power demands.
Table 9: Effect of different discharge rates on EV battery pack temperature
| Discharge Rate | Tmax (K) | ΔTmax (K) |
|---|---|---|
| 1C | 301.377 | 3.322 |
| 2C | 303.402 | 5.334 |
| 3C | 305.914 | 6.237 |
| 4C | 309.200 | 9.180 |
**Effect of Inlet Velocity.** The coolant flow velocity strongly influences the convective heat transfer coefficient. Simulations show that increasing the velocity from 0.2 m/s to 0.6 m/s results in a cooler battery. However, the returns on temperature reduction diminish as velocity increases, while the pressure drop increases super-linearly due to increased friction. Therefore, the optimal flow velocity represents a balance between heat transfer and pump power requirements.
Table 10: Effect of coolant velocity on EV battery pack metrics (3C rate)
| Inlet Velocity (m/s) | Tmax (K) | ΔTmax (K) | Pressure Drop (kPa) |
|---|---|---|---|
| 0.2 | 307.948 | 8.340 | 1.010 |
| 0.3 | 306.897 | 7.030 | 1.700 |
| 0.4 | 306.386 | 6.552 | 2.485 |
| 0.5 | 305.914 | 6.237 | 2.485 |
| 0.6 | 305.850 | 5.812 | 4.310 |
**Effect of Inlet Temperature.** The coolant inlet temperature defines the baseline cooling potential. Lower inlet temperatures will reduce the maximum temperature, but this comes at the expense of higher energy consumption from the cooling unit. A trade-off is also observed: an excessively low coolant temperature can increase the thermal gradient within the EV battery pack.
Table 11: Effect of coolant inlet temperature on EV battery pack performance
| Inlet Temperature (K) | Tmax (K) | ΔTmax (K) |
|---|---|---|
| 289.15 | 299.134 | 7.846 |
| 292.15 | 301.694 | 7.608 |
| 295.15 | 304.293 | 7.414 |
| 298.15 | 305.914 | 6.237 |
The conclusion is drawn that coolant at 298.15 K is suitable to manage the heat. Overall, it can be observed that a lower inlet temperature is not always ideal as it hurts the uniformity when the cold plate is cooling the cells unevenly.
4.2 Needs for Reduced order Modeling
The full-scale CFD simulations of the thermal management system, while accurate, can be computationally heavy, especially when considering the 3D transient nature of the simulation and the high mesh counts. For engineers to perform real-time monitoring (through a digital twin), conduct thousands of “what-if” scenarios for safety assessment, or embed in system-level controllers, a set of reduced-order models (ROMs) is required. The computational time to run these individual cases is shown in Table 12.
Table 12: Comparison of solve times between traditional CFD and SVD reduced-order models
| Discharge rate | CFD Simulation time (s) | ROM Simulation time (s) |
|---|---|---|
| 1C | 3514 | 0.813 |
| 2C | 1821 | 0.591 |
| 3C | 1307 | 0.192 |
| 4C | 932 | 0.393 |
4.3 Digital Twin and Data-driven ROM Generation
I employed a physics-based data-driven method to construct a reduced-order model using Singular Value Decomposition (SVD). The process starts by carefully defining a set of input parameters (like mass flow rate, temperature, and heat generation rate) and output parameters (like the temperature distribution across the battery pack and pressure fields). A parametric sweep was generated based on a Latin Hypercube sampling method to create a training matrix. The resulting high-fidelity simulation data is then assembled into a snapshot matrix, M.
This matrix M (M × N) is comprised of N snapshots. The SVD algorithm decomposes this matrix into three matrices, as described by:
$$ M = U \Sigma V^{*} $$
Where \( U \) and \( V \) are orthogonal matrices and \( \Sigma \) is a diagonal matrix function of the singular values \( \sigma_i \). The dimensions of these matrices are such that:
$$ (M \times N) = (M \times M) \times (M \times N) \times (N \times N). $$
Key to dimensionality reduction is the understanding that the singular values decay rapidly. Thus, one can form a reduced-rank approximation \( M_r \) of the system, using only a number of “dominant” modes \( r \). The approximation error is defined by:
$$ \text{relativeError} = \sqrt{\frac{\sum_{i=r+1}^{n} \sigma_i^2}{\sum_{i=1}^{n} \sigma_i^2}} $$
A small relative error means that the truncated modes contain only a tiny fraction of the system’s energy. The solution field for any new parameter set is generated via interpolation of the modal coefficients \( \alpha_i \), which forms the response surface for the physical variable:
$$ X \approx \sum_{i=1}^{r} \alpha_i U_i $$
By projecting the 3D fields onto a lower-dimensional space and applying an interpolation scheme, the model predicts the grid point values (temperatures and pressures) almost instantly.
4.4 Accuracy and Efficiency of the SVD Reduced-order Model
The effectiveness of the SVD reduced-order model was assessed by comparing its predictions against validation CFD cases that were not part of the training dataset. The relative error was computed using:
$$ \epsilon_{rel} = \frac{|X_{ref} – X_{rom}|}{X_{ref}} \times 100\% $$
The validation considered the effect of training data volume, revealing that using 93% of the 15 samples for training results in a temperature error near zero (0.0055% relative error). The analysis conclusively shows that the ROM and CFD results for The Tmax exhibit a maximum relative error of merely 0.2% when varying the discharge rate. When looking at Pressure drop, the relative error is less than 5%. This confirms the ability of the ROM to serve as a robust stand-in for detailed simulations.
**The data from the ROM vs. CFD for other conditions is plotted in the main text.**
Table 13: Relative Errors of the SVD model across different operating parameters
| Operating Parameter | Metrics | Max Relative Error (%) |
|---|---|---|
| Discharge Rate (1C~4C) | Tmax | 0.198 |
| Discharge Rate (1C~4C) | ΔTmax | 4.788 |
| Coolant Inlet Velocity (0.2~0.6 m/s) | Tmax | 1.0 |
| Coolant Inlet Velocity (0.2~0.6 m/s) | ΔTmax / ΔP | < 5.0 |
| Coolant Inlet Temperature (289~298 K) | Tmax | 0.182 |
| Coolant Inlet Temperature (289~298 K) | ΔTmax | 4.422 |
In practical terms, while performing a CFD calculation takes one computational core around 20-50 minutes to simulate 1200 seconds of operation, the SVD-ROM yields these results in milliseconds when integrated into the Ansys Twin Builder environment. Based on the analysis of the table, the computation efficiency of the ROM is nearly 2000 times higher than traditional CFD simulations.
5. Conclusions
In this work, I conducted a systematic study to design and optimize a liquid cooling system for a 48V lithium-ion EV battery pack. The main findings of this research are summarized as follows:
1. **Battery Thermal Model**: From an accurate UDF heat source model, built from experimental pressure measurements (HPPC test) and entropic heat coefficients, I validated the simulation against experimental temperature rise tests. The model proved highly accurate, even at high C-rates, indicating its suitability for pack-level simulations.
2. **Cooling Structure Selection and Optimization**: The baseline natural convection analysis for the 48V EV battery pack showed it would exceed its temperature limit under a 3C discharge. To solve this, I developed a bionic leaf-vein cooling plate. It efficiently reduced the EV battery pack’s maximum temperature by more than 30 K compared to natural convection. By creating surrogate models with a Box-Behnken design, I identified that the channel width is the primary factor to reduce both Tmax and pressure drop. By utilizing the NSGA-II algorithm, I obtained the Pareto optimal set. The optimized cold plate parameters (D=3.7 mm, α=98°, N=7) decreased the Tmax by almost 1 K and ΔTmax by 1.1 K compared to a baseline, while also lowering pressure drop by 1.12 kPa. This offers a dual benefit: enhanced battery lifespan and lower energy consumption for the cooling loop.
3. **Real-Time Thermal State Estimation via ROM**: I demonstrated that the detailed CFD model can be converted into a reduced-order model (ROM) using SVD. The resulting simulation model predicts the temperature and pressure fields of the battery pack with minimal fidelity loss—its error is below 1% for peak temperatures and below 5% for pressure drops. This model’s accuracy holds across varying C-rates, coolant inlet temperatures, and flow velocities. This SVD technique enables a step-change in computational efficiency, performing up to 2000 times faster than conventional CFD. This seamless integration of accurate ROMs into digital twin platforms is the cornerstone of the next generation of real-time battery management systems.
Acknowledgements
The authors would like to express their gratitude for various supports during this project, especially the Provincial Key R&D Program focused on key technologies and applications for energy storage system integration.