In the rapid evolution of electric vehicles, the thermal management system has become a critical factor that governs the safety, durability, and efficiency of the electric vehicle battery pack. As a core energy storage device, the electric vehicle battery must operate within a narrow optimal temperature window to ensure high charging/discharging efficiency and long cycle life. My research focuses on a prismatic lithium iron phosphate battery, which is widely adopted in modern electric vehicles due to its excellent thermal stability and cost-effectiveness. The central aim of this work is to design and optimize a hybrid cooling structure that combines heat pipes with a liquid cooling system. I established a comprehensive numerical model using the NTGK electrochemical approach to simulate the heat generation behavior of the electric vehicle battery module. I then coupled this thermal model with a fluid dynamics model of the cooling channels. Through extensive simulation, I evaluated the cooling performance of traditional parallel and serpentine flow channels, and I proposed an innovative series-parallel flow channel design. To further enhance the thermal performance, I applied the Taguchi experimental design and grey relational analysis for multi-objective optimization, which allowed me to determine the optimal structural parameters. My results demonstrate that the optimized hybrid cooling system can significantly reduce the maximum temperature of the electric vehicle battery pack while balancing the pressure drop across the flow channels. This study provides valuable insights for the development of efficient and reliable battery thermal management systems in electric vehicles.
The motivation for this study stems from the increasing energy density and fast-charging capabilities of modern electric vehicle batteries. When an electric vehicle battery operates under high current rates, the internal heat generation rate rises sharply. If this heat is not dissipated rapidly, the local temperature may exceed the safe threshold, leading to accelerated degradation and even thermal runaway. Therefore, the battery thermal management system must be designed to extract heat efficiently while maintaining a uniform temperature distribution. Among various cooling techniques, liquid cooling offers high heat transfer coefficients, while heat pipes provide passive and ultra-high effective thermal conductivity. The combination of these two technologies can overcome the limitations of each individual method. In my research, I placed heat pipes between the battery cells to conduct heat from the core to the bottom liquid cooling plate. This configuration reduces the risk of coolant leakage coming into direct contact with the cells, which is a key safety advantage for electric vehicle battery packs.
1. Introduction and Background
The global automotive industry is shifting toward electrification to reduce carbon emissions. According to recent statistics, the sales of electric vehicles have grown exponentially over the past decade. The electric vehicle battery is the most expensive and safety-critical component in an electric vehicle. Lithium-ion batteries, especially lithium iron phosphate batteries, are favored for their long cycle life and inherent thermal stability. However, even lithium iron phosphate batteries generate significant heat during high-rate discharge. The optimal operating temperature range for most lithium-ion batteries is between 20°C and 40°C. Temperatures above this range accelerate side reactions, increase internal resistance, and reduce capacity. Conversely, temperatures below the optimal range increase the risk of lithium plating. Therefore, the battery thermal management system must not only cool the battery but also maintain temperature uniformity. A temperature gradient of more than 5 K within a module can lead to uneven aging and unbalanced state of charge among cells. This imbalance reduces the overall capacity of the electric vehicle battery pack and poses a safety risk.
In this work, I address the challenge of designing an effective cooling system for an electric vehicle battery module with 24 prismatic cells arranged in a 3×8 configuration. Each cell has a nominal capacity of 180 Ah and a nominal voltage of 3.2 V. The battery pack is connected as 1 parallel and 24 series. The total heat generation under 1C discharge is substantial. I selected the NTGK model for the battery heat source because it provides a computationally efficient yet accurate representation of the electrochemical heat generation. The NTGK model relates the volumetric current transfer rate to the potential difference and depth of discharge. The model parameters were calibrated from experimental data reported in literature. I validated my numerical methodology against experimental results from independent researchers, achieving an error of less than 2%.
The cooling system I propose consists of an aluminum liquid cooling plate at the bottom of the battery module and L-shaped heat pipes inserted between the cells. The heat pipes have a thickness of 5 mm and a width of 20 mm. The long leg of the L-shaped heat pipe extends along the height of the cell (207 mm), and the short leg extends along the length (174 mm). The evaporator section of the heat pipe contacts the battery surface, while the condenser section is embedded in the liquid cooling plate. This arrangement allows heat to be efficiently transported from the entire cell surface to the cooling plate. The coolant used is a 50% ethylene glycol-water solution, which has a freezing point of -33.8°C and a boiling point of 107.2°C. The flow regime is laminar for all operating conditions considered, as verified by the Reynolds number. I used the commercial software Ansys Fluent for all simulations.
2. Battery Heat Generation Model
I established the geometric model of the electric vehicle battery pack using SpaceClaim. The battery cell dimensions are 207 mm in height, 174 mm in length, and 72 mm in width. To reduce computational cost, I simplified the model by neglecting the rounded corners and terminal connectors. The main components include the cell body, positive and negative tabs, and the busbars connecting the cells in series. The thermophysical properties of the cell are listed in Table 1.
| Parameter | Value |
|---|---|
| Cell size (H×L×W) | 207 × 174 × 72 mm |
| Nominal capacity | 180 Ah |
| Density | 2248 kg/m³ |
| Specific heat capacity | 850 J/(kg·K) |
| Thermal conductivity (x, y, z) | 15 / 15 / 1 W/(m·K) |
| Nominal voltage | 3.22 V |
| Maximum discharge rate | 3C |
The NTGK model assumes that the internal heat generation is spatially uniform within the cell. The governing energy equation is given by:
$$\rho \frac{\partial h_e}{\partial t} + \nabla \cdot (\mathbf{V} \rho h_e) = \nabla \cdot (k \nabla T) + S_h$$
where ρ is the density, he is the specific enthalpy, V is the velocity vector, k is the thermal conductivity, T is the temperature, and Sh is the volumetric heat source. The heat source term is composed of the electrochemical reaction heat, ohmic heat, and polarization heat. The volumetric current transfer rate is expressed by:
$$J_{ech} = Y \left[ U – (\phi_+ – \phi_-) \right]$$
Here, U and Y are model parameters that depend on the depth of discharge (DOD) and temperature. The DOD is defined as:
$$DOD = \frac{\int_0^t I \, dt}{Q_{ah}} \times 100\%$$
where I is the current and Qah is the total capacity. The parameters U and Y are fitted as polynomial functions of DOD:
$$U = \sum_{n=0}^{5} a_n (DOD)^n$$
$$Y = \sum_{n=0}^{5} b_n (DOD)^n$$
The coefficients an and bn are listed in Table 2.
| Parameter | a0 | a1 | a2 | a3 | a4 | a5 |
|---|---|---|---|---|---|---|
| U | 4.12 | -0.804 | 1.075 | -1.177 | 0 | 0 |
| Y | 1168.59 | -8928 | 52504.6 | -136231 | 158531.7 | -67578.5 |
The electrochemical heat generation rate is calculated as:
$$S_{ech} = J_{ech} \left[ (\phi_+ – \phi_-) – T \frac{dU}{dT} \right]$$
I set the initial battery temperature to 298 K and the ambient temperature to 298 K. The convective heat transfer coefficient on the external surfaces was set to 5 W/(m²·K). The discharge rate was 1C, corresponding to a discharge time of 3600 s. The simulation was transient, with a time step of 20 s and a maximum of 20 iterations per step.
The mesh independence study was performed by varying the mesh count from 5.14×10⁵ to 1.21×10⁸. The maximum temperature and pressure drop were monitored. The results, shown in Table 3, indicate that a mesh count of 2.108×10⁷ provides a good balance between accuracy and computational cost. Further increasing the mesh count changes the maximum temperature by only 0.08% and the pressure drop by 0.009%.
| Mesh count (×10⁵) | Max temperature (K) | Pressure drop (Pa) |
|---|---|---|
| 5.14 | 330.595 | 394.508 |
| 7.06 | 337.194 | 405.862 |
| 14.4 | 327.464 | 416.453 |
| 21.5 | 327.493 | 406.741 |
| 38.1 | 326.331 | 412.331 |
| 76.7 | 323.605 | 411.164 |
| 210.8 | 321.717 | 400.406 |
| 1215.1 | 321.458 | 400.370 |
To validate my numerical model, I compared my simulation results with experimental data from a published study that used a similar battery module with liquid cooling. The boundary conditions were matched exactly. The experimental results and my simulation results are compared in Table 4. The maximum error is 2.01%, which confirms the reliability of my model.
| Condition | Experimental data (°C) | Simulation data (°C) | Error |
|---|---|---|---|
| 1C discharge, 0.5 L/min | 36.31 | 36.72 | 1.13% |
| 1C discharge, 0.5 L/min | 32.37 | 33.02 | 2.01% |
3. Cooling System Numerical Model
My cooling system combines a liquid cooling plate with heat pipes. The heat pipes are modeled as solid conductors with an effective thermal conductivity of 5000 W/(m·K), which is a common assumption in battery thermal management simulations because the internal phase-change heat transfer results in very low thermal resistance. The liquid cooling plate is made of aluminum, and the coolant is a 50% ethylene glycol-water solution. The thermophysical properties of the coolant and aluminum are listed in Table 5.
| Material | Density (kg/m³) | Specific heat (J/(kg·K)) | Thermal conductivity (W/(m·K)) | Dynamic viscosity (Pa·s) |
|---|---|---|---|---|
| 50% ethylene glycol-water | 1069 | 3494 | 0.419 | 0.00315 |
| Aluminum | 2702 | 903 | 237 | – |
The flow state of the coolant is determined by the Reynolds number. For a rectangular channel with cross-section dimensions a and b, the hydraulic diameter is:
$$D_h = \frac{2ab}{a+b}$$
At the maximum velocity of 0.6 m/s, the Reynolds number is:
$$Re = \frac{\rho v D_h}{\mu} = \frac{1069 \times 0.6 \times 2 \times 0.005 \times 0.02}{0.00315 \times (0.005+0.02)} \approx 1364$$
Since this value is below 2300, the flow is laminar. Therefore, the governing equations are the laminar continuity, momentum, and energy equations:
$$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0$$
$$\frac{\partial (\rho \mathbf{u})}{\partial t} + \nabla \cdot (\rho \mathbf{u} \mathbf{u}) = – \nabla p + \nabla \cdot \tau + \rho \mathbf{F}$$
$$\frac{\partial (\rho e)}{\partial t} + \nabla \cdot \left[ \mathbf{u} \left( \rho e + \frac{1}{2} \rho u^2 \right) \right] = – \nabla \cdot \mathbf{q} + \tau : \nabla \mathbf{u} + Q$$
The boundary conditions for the cooling system are listed in Table 6.
| Boundary | Condition |
|---|---|
| Coolant inlet | Velocity inlet, uniform velocity and temperature |
| Coolant outlet | Pressure outlet, gauge pressure = 0 Pa |
| External walls | Adiabatic, no-slip |
| Interfaces | Coupled heat transfer |
| Initial temperature | 298 K |
I analyzed two traditional flow channel designs: a parallel flow channel and an S-shaped flow channel. Both channels have a width of 20 mm and a thickness of 5 mm to match the heat pipe condenser dimensions. The parallel channel offers low pressure drop but moderate heat transfer, while the S-shaped channel provides larger contact area and higher heat transfer but at the cost of significant pressure drop. I simulated both with and without heat pipes to quantify the contribution of heat pipes.
4. Performance of Traditional Flow Channels with Heat Pipes
4.1 Parametric Study on Coolant Velocity
I first investigated the influence of coolant velocity on the cooling performance. The coolant velocity was varied from 0.1 m/s to 0.6 m/s with an increment of 0.1 m/s. The inlet temperature was held constant at 298 K. Table 7 lists the results for the parallel flow channel without and with heat pipes.
| Velocity (m/s) | Max temp without HP (K) | Temp difference without HP (K) | Pressure drop (Pa) |
|---|---|---|---|
| 0.1 | 323.30 | 3.68 | 20.15 |
| 0.2 | 322.33 | 3.77 | 71.31 |
| 0.3 | 321.91 | 3.93 | 149.08 |
| 0.4 | 321.63 | 4.09 | 262.60 |
| 0.5 | 321.41 | 4.12 | 418.50 |
| 0.6 | 321.25 | 4.19 | 585.01 |
| Velocity (m/s) | Max temp with HP (K) | Temp difference with HP (K) | Pressure drop (Pa) |
|---|---|---|---|
| 0.1 | 317.14 | 1.87 | 20.15 |
| 0.2 | 315.04 | 1.63 | 71.31 |
| 0.3 | 314.98 | 3.77 | 149.08 |
| 0.4 | 314.28 | 3.67 | 262.60 |
| 0.5 | 312.13 | 2.38 | 418.50 |
| 0.6 | 313.49 | 3.64 | 585.01 |
The results demonstrate that adding heat pipes reduces the maximum temperature of the electric vehicle battery pack by 4 to 8 K across all velocity cases. In particular, the temperature difference is significantly improved at low velocities. At 0.2 m/s, the temperature difference drops from 3.77 K to 1.63 K. The pressure drop remains identical because the heat pipes do not alter the flow field.
For the S-shaped channel, the results are shown in Tables 9 and 10.
| Velocity (m/s) | Max temp without HP (K) | Temp difference without HP (K) | Pressure drop (Pa) |
|---|---|---|---|
| 0.1 | 323.29 | 4.82 | 223.75 |
| 0.2 | 322.09 | 4.98 | 942.64 |
| 0.3 | 321.57 | 5.06 | 2174.33 |
| 0.4 | 321.27 | 5.11 | 3945.49 |
| 0.5 | 321.07 | 5.15 | 6054.63 |
| 0.6 | 320.93 | 5.17 | 8874.06 |
| Velocity (m/s) | Max temp with HP (K) | Temp difference with HP (K) | Pressure drop (Pa) |
|---|---|---|---|
| 0.1 | 316.04 | 2.10 | 223.75 |
| 0.2 | 314.03 | 3.78 | 942.64 |
| 0.3 | 313.08 | 3.70 | 2174.33 |
| 0.4 | 312.54 | 3.65 | 3945.49 |
| 0.5 | 312.20 | 3.64 | 6054.63 |
| 0.6 | 311.96 | 3.62 | 8874.06 |
The S-shaped channel with heat pipes achieves an average reduction of about 8.5 K in maximum temperature compared to the channel without heat pipes. However, the pressure drop of the S-shaped channel is extremely high, reaching 8874 Pa at 0.6 m/s, which would lead to excessive pump power consumption. This highlights the need for a new channel design that can provide high heat transfer while maintaining a moderate pressure drop.
4.2 Parametric Study on Coolant Inlet Temperature
I also studied the effect of coolant inlet temperature, varying it from 298 K down to 278 K in steps of 5 K, while keeping the velocity at 0.5 m/s. The results for the parallel channel are given in Tables 11 and 12.
| Inlet temperature (K) | Max temperature (K) | Temperature difference (K) |
|---|---|---|
| 298 | 321.41 | 4.12 |
| 293 | 320.40 | 7.53 |
| 288 | 318.26 | 7.77 |
| 283 | 316.00 | 7.81 |
| 278 | 313.71 | 7.25 |
| Inlet temperature (K) | Max temperature (K) | Temperature difference (K) |
|---|---|---|
| 298 | 313.49 | 3.64 |
| 293 | 308.95 | 2.48 |
| 288 | 305.80 | 2.53 |
| 283 | 304.10 | 3.58 |
| 278 | 300.88 | 3.38 |
Lowering the inlet temperature significantly reduces the maximum temperature. When heat pipes are added, the temperature difference is dramatically reduced, especially at moderate inlet temperatures. For instance, at 293 K, the temperature difference drops from 7.53 K to 2.48 K. The same trend is observed for the S-shaped channel, as shown in Tables 13 and 14.
| Inlet temperature (K) | Max temperature (K) | Temperature difference (K) |
|---|---|---|
| 298 | 320.93 | 5.17 |
| 293 | 320.06 | 8.44 |
| 288 | 317.53 | 8.48 |
| 283 | 314.96 | 8.36 |
| 278 | 312.36 | 8.03 |
| Inlet temperature (K) | Max temperature (K) | Temperature difference (K) |
|---|---|---|
| 298 | 311.96 | 3.62 |
| 293 | 308.62 | 3.59 |
| 288 | 305.05 | 3.48 |
| 283 | 301.49 | 3.28 |
| 278 | 298.47 | 4.44 |
From these parametric studies, I concluded that the combination of heat pipes and liquid cooling is highly effective. The parallel channel has a low pressure drop but a relatively high temperature difference, while the S-shaped channel has a lower temperature difference but an excessive pressure drop. Therefore, I designed a new series-parallel flow channel that combines the advantages of both geometries.
5. Design and Optimization of the Series-Parallel Flow Channel
The new series-parallel flow channel is shown conceptually by merging the parallel and S-shaped features. It consists of a main inlet manifold that distributes coolant into several branch channels (parallel section). These branch channels then converge into a series section that runs along the heat pipe condenser region, after which the coolant exits through an outlet manifold. The key structural parameters are the number of branch channels (N), the branch channel width (D1), the spacing between branches (D2), and the channel thickness (D3). Figure 5.2 in my original thesis shows the geometry; here I describe the parameter definitions in Table 15.
| Parameter | Symbol | Level 1 | Level 2 | Level 3 | Level 4 |
|---|---|---|---|---|---|
| Number of branch channels | N | 2 | 3 | – | – |
| Branch width (mm) | D1 | 15 | 20 | 25 | 30 |
| Spacing (mm) | D2 | 30 | 36 | 44 | 50 |
| Thickness (mm) | D3 | 2 | 3 | 4 | 5 |
Before optimizing the structure, I determined the optimal coolant velocity and inlet temperature using a control variable method. The results are summarized in Tables 16 and 17.
| Velocity (m/s) | Max temperature (K) | Temperature difference (K) | Pressure drop (Pa) |
|---|---|---|---|
| 0.1 | 321.36 | 4.89 | 111.78 |
| 0.2 | 319.61 | 6.93 | 481.07 |
| 0.3 | 318.53 | 6.86 | 1106.89 |
| 0.4 | 317.90 | 6.81 | 1995.40 |
| 0.5 | 317.21 | 6.78 | 3165.14 |
| 0.6 | 317.50 | 6.77 | 4582.90 |
The minimum maximum temperature is achieved at 0.5 m/s. Beyond that, the maximum temperature slightly increases due to insufficient heat exchange time. Therefore, I selected 0.5 m/s as the optimal velocity.
| Inlet temperature (K) | Max temperature (K) | Temperature difference (K) |
|---|---|---|
| 298 | 317.50 | 6.78 |
| 293 | 313.85 | 6.68 |
| 288 | 310.21 | 6.24 |
| 283 | 306.58 | 6.51 |
| 278 | 303.65 | 7.52 |
The temperature difference reaches a minimum at 288 K. At lower temperatures, the temperature difference increases because the bottom of the battery becomes too cold relative to the top. Thus, I chose 288 K as the optimal inlet temperature.
5.1 Taguchi Experimental Design
I employed the Taguchi method to design the experiments for structural optimization. The optimization objectives are the battery temperature difference (ΔT) and the pressure drop (ΔP) across the flow channel. Both are smaller-is-better characteristics. The signal-to-noise (S/N) ratio for a smaller-is-better characteristic is calculated as:
$$SN = -10 \log_{10} \left( \frac{1}{n} \sum_{i=1}^{n} y_i^2 \right)$$
where yi is the measured response and n is the number of experiments. I constructed an L32(21 × 43) orthogonal array, resulting in 32 different structural configurations. For each configuration, I built the geometric model, meshed it, and performed the CFD simulation. The results are shown in Table 18.
| No. | N | D1 (mm) | D2 (mm) | D3 (mm) | Tmax (K) | ΔT (K) | ΔP (Pa) |
|---|---|---|---|---|---|---|---|
| 1 | 2 | 15 | 30 | 2 | 308.04 | 4.42 | 3150.66 |
| 2 | 2 | 15 | 36 | 3 | 306.25 | 3.57 | 3159.78 |
| 3 | 2 | 15 | 44 | 4 | 305.47 | 3.31 | 3719.57 |
| 4 | 2 | 15 | 50 | 5 | 305.25 | 3.23 | 3603.73 |
| 5 | 2 | 20 | 30 | 2 | 309.48 | 5.28 | 2933.53 |
| 6 | 2 | 20 | 36 | 3 | 306.09 | 3.48 | 3152.66 |
| 7 | 2 | 20 | 44 | 4 | 306.30 | 3.93 | 3105.48 |
| 8 | 2 | 20 | 50 | 5 | 305.69 | 3.69 | 2746.01 |
| 9 | 2 | 25 | 30 | 3 | 307.81 | 4.28 | 2548.24 |
| 10 | 2 | 25 | 36 | 2 | 312.28 | 6.93 | 2179.57 |
| 11 | 2 | 25 | 44 | 5 | 310.08 | 4.20 | 2678.60 |
| 12 | 2 | 25 | 50 | 4 | 305.33 | 3.23 | 1962.44 |
| 13 | 2 | 30 | 30 | 3 | 306.08 | 3.32 | 2661.06 |
| 14 | 2 | 30 | 36 | 2 | 305.95 | 3.36 | 2390.11 |
| 15 | 2 | 30 | 44 | 5 | 307.86 | 4.11 | 2376.79 |
| 16 | 2 | 30 | 50 | 4 | 306.06 | 3.77 | 927.75 |
| 17 | 3 | 15 | 30 | 5 | 307.21 | 3.76 | 2912.72 |
| 18 | 3 | 15 | 36 | 4 | 306.04 | 4.33 | 2786.14 |
| 19 | 3 | 15 | 44 | 3 | 305.42 | 3.48 | 2390.25 |
| 20 | 3 | 15 | 50 | 2 | 306.46 | 3.90 | 1665.56 |
| 21 | 3 | 20 | 30 | 5 | 308.55 | 4.43 | 2910.49 |
| 22 | 3 | 20 | 36 | 4 | 305.57 | 3.85 | 2633.19 |
| 23 | 3 | 20 | 44 | 3 | 305.69 | 3.58 | 1968.11 |
| 24 | 3 | 20 | 50 | 2 | 307.32 | 4.02 | 939.55 |
| 25 | 3 | 25 | 30 | 4 | 309.21 | 4.09 | 2795.90 |
| 26 | 3 | 25 | 36 | 5 | 305.40 | 3.13 | 2440.17 |
| 27 | 3 | 25 | 44 | 2 | 306.70 | 4.73 | 854.02 |
| 28 | 3 | 25 | 50 | 3 | 309.07 | 4.00 | 606.99 |
| 29 | 3 | 30 | 30 | 4 | 308.51 | 4.06 | 2309.17 |
| 30 | 3 | 30 | 36 | 5 | 305.71 | 3.37 | 828.45 |
| 31 | 3 | 30 | 44 | 2 | 308.09 | 4.77 | 694.73 |
| 32 | 3 | 30 | 50 | 3 | 310.08 | 4.60 | 590.94 |
The S/N ratio analysis for ΔT and ΔP is presented in Table 19. For ΔT, the thickness D3 has the largest range (1.20), indicating it has the greatest influence on temperature uniformity. For ΔP, the branch width D1 has the largest range (6.48), meaning it most strongly affects the pressure drop.
| Objective | Level | N | D1 | D2 | D3 |
|---|---|---|---|---|---|
| ΔT | 1 | -16.26 | -16.29 | -16.49 | -17.03 |
| 2 | -16.50 | -16.49 | -16.50 | -16.22 | |
| 3 | – | -16.34 | -16.32 | -15.83 | |
| 4 | – | -16.38 | -16.20 | -16.43 | |
| Range | – | 0.24 | 0.20 | 0.30 | 1.20 |
| ΔP | 1 | -68.29 | -69.09 | -68.84 | -64.13 |
| 2 | -63.88 | -67.62 | -67.22 | -65.14 | |
| 3 | – | -65.03 | -65.78 | -67.50 | |
| 4 | – | -62.06 | -62.51 | -67.58 | |
| Range | – | 4.41 | 6.48 | 6.33 | 3.46 |
The single-objective optimal configurations are:
– For minimum ΔT: N=2, D1=15 mm, D2=50 mm, D3=4 mm.
– For minimum ΔP: N=3, D1=30 mm, D2=50 mm, D3=2 mm.
However, these two objectives conflict. The configuration that minimizes ΔT increases ΔP by 11.56%, while the one that minimizes ΔP increases ΔT by 26.32%. Therefore, a multi-objective optimization is necessary.
5.2 Grey Relational Analysis for Multi-Objective Optimization
Grey relational analysis (GRA) is suitable for solving multi-objective optimization problems with limited and uncertain information. The procedure involves normalizing the response data, calculating the grey relational coefficients, and then the grey relational grade. Since both ΔT and ΔP are smaller-is-better, the normalization is performed using:
$$x_i^*(k) = \frac{\max x_i(k) – x_i(k)}{\max x_i(k) – \min x_i(k)}$$
where xi(k) is the k-th response in the i-th experiment. The grey relational coefficient is given by:
$$\gamma_i(k) = \frac{\Delta_{\min} + \zeta \Delta_{\max}}{\Delta_i(k) + \zeta \Delta_{\max}}$$
where Δi(k) = |x0(k) – xi*(k)|, ζ is the distinguishing coefficient (taken as 0.5), and Δmin and Δmax are the minimum and maximum differences. The overall grey relational grade is:
$$\gamma_i = \sqrt{\frac{\gamma_i(\Delta T)^2 + \gamma_i(\Delta P)^2}{2}}$$
I calculated the grey relational coefficients for all 32 experiments. The results for the overall grade γ are listed in Table 20.
| No. | γ | No. | γ | No. | γ | No. | γ |
|---|---|---|---|---|---|---|---|
| 1 | 0.52 | 9 | 0.71 | 17 | 0.66 | 25 | 0.84 |
| 2 | 0.57 | 10 | 0.59 | 18 | 0.69 | 26 | 0.72 |
| 3 | 0.56 | 11 | 0.59 | 19 | 0.70 | 27 | 0.71 |
| 4 | 0.51 | 12 | 0.68 | 20 | 0.64 | 28 | 0.82 |
| 5 | 0.56 | 13 | 0.70 | 21 | 0.69 | 29 | 0.82 |
| 6 | 0.60 | 14 | 0.58 | 22 | 0.72 | 30 | 0.71 |
| 7 | 0.59 | 15 | 0.58 | 23 | 0.73 | 31 | 0.71 |
| 8 | 0.54 | 16 | 0.68 | 24 | 0.67 | 32 | 0.83 |
By averaging the grey relational grade for each factor at each level, I obtained the response table shown in Table 21.
| Level | N | D1 | D2 | D3 |
|---|---|---|---|---|
| 1 | 0.5000 | 0.4281 | 0.7767 | 0.4528 |
| 2 | 1.0000 | 0.5660 | 0.5857 | 0.8144 |
| 3 | – | 0.8498 | 0.5976 | 0.7906 |
| 4 | – | 0.8339 | 0.7453 | 0.4678 |
The maximum grey relational grade corresponds to N=3, D1=25 mm, D2=30 mm, D3=3 mm. This combination yields the best trade-off between temperature uniformity and pressure drop. The influence ranking of the factors is: N > D1 > D3 > D2.
6. Performance of the Optimized Series-Parallel Channel
Using the optimal structural parameters, I constructed the final geometric model. The coolant velocity and inlet temperature were set to 0.5 m/s and 288 K, respectively. The simulation was performed at 1C discharge for 3600 s. The maximum temperature of the electric vehicle battery pack reached 303.22 K, which is significantly lower than the baseline. Table 22 compares the optimized configuration with the initial series-parallel configuration and the traditional channels.
| Configuration | Max temperature (K) | Temperature difference (K) | Pressure drop (Pa) |
|---|---|---|---|
| No cooling (initial battery model) | 325.00 | – | – |
| Parallel channel with heat pipes | 312.13 | 2.38 | 418.50 |
| S-shaped channel with heat pipes | 312.20 | 3.64 | 6054.63 |
| Initial series-parallel (N=2, D1=20, D2=36, D3=3) | 305.56 | 3.45 | 3165.14 |
| Optimized series-parallel | 303.22 | 3.24 | 1956.16 |
The optimized series-parallel channel reduces the maximum temperature by 2.33 K compared to the initial series-parallel design, reduces the temperature difference by 6.10%, and reduces the pressure drop by 38.20%. Furthermore, the optimized channel achieves a maximum temperature reduction of 21.78 K compared to the battery pack without any cooling system. The temperature uniformity is far superior to the S-shaped channel, and the pressure drop is much lower than that of the S-shaped channel. Table 23 summarizes the normalized performance improvements.
| Comparison baseline | Max temperature reduction (K) | Temperature difference reduction | Pressure drop reduction |
|---|---|---|---|
| vs. parallel channel with HP | 2.58 | – | – |
| vs. S-shaped channel with HP | – | 6.94% | 67.69% |
| vs. initial series-parallel | 2.33 | 6.10% | 38.20% |
| vs. no cooling | 21.78 | – | – |
These results validate the effectiveness of the proposed multi-objective optimization methodology. The optimized hybrid cooling system achieves an excellent balance between thermal control and energy consumption. The temperature of the electric vehicle battery pack is maintained well within the optimal range, while the pressure drop is kept at an acceptable level for the coolant pump.
7. Discussion
The role of the heat pipe in my cooling system is crucial. By transporting heat from the battery cell surfaces to the liquid cooling plate, the heat pipe effectively reduces the maximum temperature and improves the temperature uniformity. Without heat pipes, the liquid cooling plate only cools the bottom surface of the battery pack, leading to a large vertical temperature gradient. With heat pipes, the heat is extracted along the full height of the cell, which drastically reduces the temperature difference. This is particularly evident in the parallel channel results: at a coolant velocity of 0.2 m/s, the temperature difference dropped from 3.77 K to 1.63 K when heat pipes were added. The heat pipes also reduce the sensitivity of the thermal performance to the coolant flow rate. In the S-shaped channel, the pressure drop is extremely high, but with heat pipes, the same level of cooling can be achieved at a much lower flow rate, thereby saving pump energy.
I also observed that the coolant inlet temperature plays a significant role. Lowering the inlet temperature reduces the maximum temperature but may increase the temperature difference. With heat pipes, the temperature difference is less sensitive to the inlet temperature, which is beneficial for practical applications where the ambient temperature varies. The optimized series-parallel channel further mitigates this issue by providing more uniform coolant distribution.
The grey relational analysis proved to be an effective tool for balancing two conflicting objectives. By combining the Taguchi method with GRA, I was able to identify the optimal structural parameters with only 32 simulations, which is computationally efficient. The ranking of factors from the GRA (N > D1 > D3 > D2) indicates that the number of branch channels and the branch width are the most influential parameters for the overall performance. The optimized structure increases the number of branches from 2 to 3, which improves the coolant distribution and heat transfer area without causing excessive pressure drop. The width of 25 mm is a compromise between the wider channels that reduce pressure drop and the narrower channels that increase velocity and heat transfer. The thickness of 3 mm provides sufficient structural strength while maintaining a low thermal resistance.
8. Conclusions
In this research, I conducted a comprehensive numerical investigation of a hybrid heat pipe and liquid cooling system for an electric vehicle battery pack. The following conclusions can be drawn from my work:
1. The NTGK electrochemical model accurately reproduced the heat generation characteristics of the prismatic lithium iron phosphate battery. The simulated maximum temperature agreed with experimental data within 2% error, validating the numerical methodology.
2. Coupling heat pipes with conventional liquid cooling channels significantly improved the thermal performance. For the parallel channel, the maximum temperature was reduced by an average of 7.5 K across different coolant velocities. For the S-shaped channel, the reduction was about 8.5 K. When varying the coolant inlet temperature, the average reductions were 11.5 K and 12 K for the parallel and S-shaped channels, respectively.
3. The S-shaped channel provided better temperature control than the parallel channel but suffered from a very high pressure drop. To overcome this, I designed a novel series-parallel flow channel that combines the benefits of both geometries while reducing pressure drop.
4. Using the Taguchi experimental design and grey relational analysis, I optimized the structural parameters of the series-parallel channel. The optimal parameters were determined as: branch channel number N = 3, branch width D1 = 25 mm, spacing D2 = 30 mm, and thickness D3 = 3 mm. The grey relational grade ranking was N > D1 > D3 > D2.
5. The optimized cooling system achieved a maximum battery temperature of 303.22 K under 1C discharge, which is 21.78 K lower than the uncooled battery pack. Compared to the initial series-parallel design, the temperature difference was reduced by 6.10% and the pressure drop was reduced by 38.20%. The optimized structure also outperformed the traditional channels in both temperature uniformity and energy consumption, demonstrating that the proposed multi-objective optimization method is effective for designing high-performance battery thermal management systems for electric vehicle batteries.
Future work will extend this study to higher discharge rates (2C and 3C), explore the effect of different coolants, and incorporate the thermal management system into a complete battery pack design with structural components. Experimental validation of the optimized cooling system is also planned to further confirm the simulation results.