In recent years, the rapid development of electric vehicles has placed higher demands on the safety and durability of traction batteries. The power battery is the core energy storage device of an electric vehicle, and its operating temperature directly determines the charge-discharge efficiency, cycle life, and overall safety. For an EV battery pack, heat generation during high-rate discharge is inevitable, and if this heat is not removed effectively, the cell temperature may rise sharply, causing performance degradation or even thermal runaway. Therefore, an effective battery thermal management system (BTMS) is essential. In this work, I focus on a prismatic lithium iron phosphate battery pack and investigate the thermal performance of an EV battery pack equipped with a hybrid cooling structure combining heat pipes and liquid cooling channels.
This research is carried out through numerical simulation. I first establish a three-dimensional thermal model of the battery pack using the NTGK electrochemical model. Then, I design two conventional liquid cooling channel configurations, namely the parallel channel and the S-shaped channel, and compare their cooling performance with and without heat pipes. Based on the results, I propose a novel series-parallel channel structure. To further improve the thermal performance and reduce pumping loss, I apply Taguchi experimental design and grey relational analysis to optimize the channel dimensions. The simulation results demonstrate a significant enhancement in temperature uniformity and a considerable reduction in pressure drop for the EV battery pack.
1. Electrochemical and Thermal Modeling of the Battery Pack
The battery under study is a prismatic LiFePO₄ cell with a nominal capacity of 180 Ah and a nominal voltage of 3.2 V. The dimensions of the cell are 207 mm in height, 174 mm in length, and 72 mm in width. The battery pack is arranged in a 3×8 configuration, with 24 cells connected in series. This module configuration is representative of commercial EV battery pack designs used in electric passenger vehicles. The physical properties of the cell are summarized in the table below.
| Parameter | Value |
|---|---|
| Dimensions (H×L×W), mm | 207 × 174 × 72 |
| Capacity, Ah | 180 |
| Density, kg/m³ | 2248 |
| Specific heat, J/(kg·K) | 850 |
| Thermal conductivity (x, y, z), W/(m·K) | 15, 15, 1 |
| Operating voltage, V | 3.22 |
| Maximum discharge rate | 3C |
For the heat generation simulation, I employ the NTGK electrochemical model. This model computes the volumetric current transfer rate as a function of the local potential difference. The governing equation for the volume current transfer rate is given by:
$$
J_{ech}=Y\left[ U – (\phi_+ – \phi_-) \right]
$$
where \(U\) and \(Y\) are empirical parameters that depend on the depth of discharge (DOD) and temperature. The DOD is defined by:
$$
DOD=\frac{vol}{3600 \times Q_{ah}} \times 100\%
$$
where \(vol\) is the battery volume and \(Q_{ah}\) is the total capacity. The parameters \(U\) and \(Y\) are expanded as fifth-order polynomials of DOD:
$$
U=\sum_{n=0}^{5} a_n (DOD)^n
$$
$$
Y=\sum_{n=0}^{5} b_n (DOD)^n
$$
The coefficients \(a_n\) and \(b_n\) are determined from experimental polarization curves. The heat generation inside the battery is governed by the energy equation:
$$
\frac{\partial(\rho h_e)}{\partial t}+\nabla(\mathbf{v}\rho h_e)=\nabla(k\nabla T)+S_h
$$
where \(\rho\) is the density, \(h_e\) is the sensible enthalpy, \(k\) is the thermal conductivity, \(T\) is the temperature, and \(S_h\) is the volumetric heat source term. The electrochemical reaction heat is expressed as:
$$
S_{ech}=J_{ech}\left[ (\phi_+-\phi_-) – T \frac{dU}{dT} \right]
$$
This heat source is activated in the active material regions, while the current collectors and busbars are only subjected to Joule heating.
I perform a transient simulation of the battery pack at 1C discharge for 3600 seconds. The initial temperature is 298 K, and the ambient temperature is fixed at 298 K. Natural convection is considered on the outer surfaces with a heat transfer coefficient of 5 W/(m²·K). The maximum temperature of the EV battery pack without active cooling rises from 298 K to over 325 K during the discharge, as shown in the simulated temperature field. This indicates that natural convection alone cannot keep the battery temperature within the optimal range, and thus an additional cooling system is required.
2. Cooling System Model with Heat Pipe and Liquid Cooling
To improve the heat dissipation capability of the EV battery pack, I design a hybrid cooling system consisting of aluminum cold plates with internal channels and L-shaped heat pipes. The heat pipes are placed between adjacent battery cells with an alternating arrangement. Each cell is in contact with two heat pipes. The evaporator section of each heat pipe absorbs heat from the cell surface, while the condenser section is attached to the liquid cooling channel. The cooling liquid is a 50% ethylene glycol-water solution. The thermophysical properties of the coolant and the aluminum cooling plate are listed below.
| 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 regime inside the channels is determined by calculating the Reynolds number. For a rectangular channel with width \(a\) and height \(b\), the Reynolds number is:
$$
Re=\frac{\rho \nu D_h}{\mu}=\frac{\rho \nu}{\mu}\cdot \frac{2ab}{a+b}
$$
For the maximum velocity considered in this study, \(v=0.6\,\text{m/s}\), the Reynolds number is approximately 1364, which is below 2300. Therefore, the coolant flow is laminar. The governing equations for the incompressible laminar flow are the continuity, momentum, and energy equations:
$$
\nabla \cdot \mathbf{v}=0
$$
$$
\rho\left(\frac{\partial \mathbf{v}}{\partial t}+\mathbf{v}\cdot\nabla \mathbf{v}\right)=-\nabla p+\nabla\cdot\tau+\rho \mathbf{F}
$$
$$
\rho c_p\left(\frac{\partial T}{\partial t}+\mathbf{v}\cdot\nabla T\right)=k\nabla^2 T + q”’
$$
In the numerical model, the heat pipe is treated as a solid with an equivalent high thermal conductivity of 5000 W/(m·K). This approximation is justified by the extremely high effective thermal conductivity of heat pipes, which enables them to transfer heat with a very small temperature difference. The coupling between the battery, heat pipe, and cooling channel is modeled using a conjugated heat transfer boundary condition.
Before performing the cooling performance analysis, I verify the grid independence of the numerical solution. Eight different mesh densities are tested, ranging from 5.14×10⁵ to 1.21×10⁸ cells. The maximum battery temperature and the pressure drop across the channel are monitored. The results are summarized in the table below.
| Case | Number of cells | Maximum temperature (K) | Change (%) | Pressure drop (Pa) | Change (%) |
|---|---|---|---|---|---|
| 1 | 5.14×10⁵ | 330.595 | – | 394.508 | – |
| 2 | 7.06×10⁵ | 337.194 | 1.96 | 405.86 | 2.80 |
| 3 | 1.44×10⁶ | 327.46 | −2.97 | 416.453 | 2.61 |
| 4 | 2.15×10⁶ | 327.49 | 0.01 | 406.74 | −2.33 |
| 5 | 3.81×10⁶ | 326.33 | −0.35 | 412.33 | 1.37 |
| 6 | 7.67×10⁶ | 323.605 | −0.85 | 411.164 | −0.28 |
| 7 | 2.108×10⁷ | 321.717 | −0.59 | 400.406 | −2.62 |
| 8 | 1.21×10⁸ | 321.458 | −0.08 | 400.370 | −0.01 |
From the grid independence study, the maximum temperature changes by only 0.08% when the grid is increased from 2.108×10⁷ to 1.21×10⁸ cells, and the pressure drop changes by only 0.009%. Thus, a mesh with 2.108×10⁷ cells is selected for all subsequent simulations to balance accuracy and computational cost.
To validate the numerical model, I reproduce a reference experiment from the literature using the same NTGK model and the same conjugate heat transfer setup. The reference battery has a capacity of 50 Ah and dimensions of 115 mm × 32 mm × 180 mm, cooled by an aluminum flat tube with microchannels. Under a 1C discharge rate and a coolant flow rate of 0.5 L/min at 20°C, the experimental maximum temperature is 36.31°C, while my simulation gives 36.72°C, corresponding to an error of 1.13%. The average surface temperature error is 2.01%. This confirms that the numerical model is reliable for predicting the thermal behavior of an EV battery pack with liquid cooling and heat pipes.
3. Thermal Performance of Traditional Channel Structures
In this section, I evaluate two conventional liquid cooling channel geometries: the parallel flow channel and the S-shaped flow channel. Both geometries are designed to have the same contact width with the heat pipes (20 mm) and the same channel thickness (5 mm). The inlet cross-section is 5 mm × 20 mm. The purpose is to assess the benefit of coupling heat pipes with the liquid cooling system for the EV battery pack.
3.1 Effect of Coolant Velocity on Parallel Channel
The coolant velocity is varied from 0.1 m/s to 0.6 m/s in steps of 0.1 m/s, while the inlet temperature is fixed at 298 K. The battery is discharged at 1C. The simulation results for the parallel channel without heat pipes are listed below.
| Velocity (m/s) | Max temp (K) | Temp difference (K) | Pressure drop (Pa) |
|---|---|---|---|
| 0.6 | 321.25 | 4.19 | 585.01 |
| 0.5 | 321.41 | 4.12 | 418.50 |
| 0.4 | 321.63 | 4.09 | 262.60 |
| 0.3 | 321.91 | 3.93 | 149.08 |
| 0.2 | 322.33 | 3.77 | 71.31 |
| 0.1 | 323.30 | 3.68 | 20.15 |
When the parallel channel is coupled with heat pipes, the maximum temperature decreases significantly. The results are presented in the following table.
| Velocity (m/s) | Max temp (K) | Temp difference (K) | Pressure drop (Pa) |
|---|---|---|---|
| 0.6 | 313.49 | 3.64 | 585.01 |
| 0.5 | 312.13 | 2.38 | 418.50 |
| 0.4 | 314.28 | 3.67 | 262.60 |
| 0.3 | 314.98 | 3.77 | 149.08 |
| 0.2 | 315.04 | 1.63 | 71.31 |
| 0.1 | 317.14 | 1.87 | 20.15 |
On average, the maximum temperature of the EV battery pack is reduced by about 7.5 K when heat pipes are integrated with the parallel channel. The temperature difference is also improved, especially at low velocities. For example, at 0.2 m/s, the temperature difference drops from 3.77 K to 1.63 K, representing a reduction of 56%. This improvement is attributed to the heat pipe’s ability to rapidly transport heat from the cell interior to the cooling channel, thereby promoting a more uniform temperature distribution.
3.2 Effect of Coolant Velocity on S-Shaped Channel
The S-shaped channel provides a longer flow path and thus a larger heat transfer area. The results without heat pipes are shown below.
| Velocity (m/s) | Max temp (K) | Temp difference (K) | Pressure drop (Pa) |
|---|---|---|---|
| 0.6 | 320.93 | 5.17 | 8874.06 |
| 0.5 | 321.07 | 5.15 | 6054.63 |
| 0.4 | 321.27 | 5.11 | 3945.49 |
| 0.3 | 321.57 | 5.06 | 2174.33 |
| 0.2 | 322.09 | 4.98 | 942.64 |
| 0.1 | 323.29 | 4.82 | 223.75 |
After adding heat pipes to the S-shaped channel, the maximum temperature is reduced dramatically, as shown below.
| Velocity (m/s) | Max temp (K) | Temp difference (K) |
|---|---|---|
| 0.6 | 311.96 | 3.62 |
| 0.5 | 312.20 | 3.64 |
| 0.4 | 312.54 | 3.65 |
| 0.3 | 313.08 | 3.70 |
| 0.2 | 314.03 | 3.78 |
| 0.1 | 316.04 | 2.10 |
The average temperature reduction is approximately 8.5 K for the S-shaped channel with heat pipes. However, the pressure drop of the S-shaped channel is extremely high compared with the parallel channel. At 0.6 m/s, the pressure drop reaches 8874 Pa, which is more than 15 times that of the parallel channel. Thus, while the S-shaped channel provides slightly better temperature control, its pumping loss is unacceptable for a practical EV battery pack.
3.3 Effect of Coolant Inlet Temperature
Another important factor is the coolant inlet temperature. I fix the velocity at 0.5 m/s and vary the inlet temperature from 298 K down to 278 K in steps of 5 K. For the parallel channel without heat pipes, the results are:
| Inlet temperature (K) | Max temp (K) | Temp 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 |
It is interesting that lowering the coolant temperature reduces the maximum temperature but increases the temperature difference. This is because the regions close to the cold plate are overly cooled while the upper part of the battery remains hot. When heat pipes are added, the temperature difference is greatly improved, as shown in the next table.
| Inlet temperature (K) | Max temp (K) | Temp 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 |
The same behavior is observed for the S-shaped channel. The following tables summarize the results without and with heat pipes, respectively.
| Inlet temperature (K) | Max temp (K) – without HP | Temp diff (K) – without HP |
|---|---|---|
| 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 temp (K) – with HP | Temp diff (K) – with HP |
|---|---|---|
| 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 |
In conclusion, the heat pipe significantly enhances the cooling performance of both channel configurations. The average temperature drop across all inlet temperatures is about 11.5 K for the parallel channel and 12 K for the S-shaped channel. The heat pipe also reduces the temperature difference, especially at low inlet temperatures. However, the pressure drop issue remains for the S-shaped channel. Therefore, I design a new series-parallel channel that combines the advantages of both the parallel and S-shaped channels: a long flow path for sufficient heat exchange and multiple parallel branches to reduce flow resistance.
4. Design and Optimization of a Series-Parallel Channel
The proposed series-parallel channel consists of a main inlet manifold that splits the coolant into several parallel branch channels, which then merge into a final outlet manifold. The branch channels follow a serpentine path around the heat pipes, thereby increasing the contact time between the coolant and the heat pipes while maintaining a relatively low pressure drop. The geometry of the channel is characterized by four design parameters: the number of branch channels \(N\), the width of each branch \(D_1\), the spacing between neighboring branches \(D_2\), and the channel thickness \(D_3\).
Before optimizing the structural parameters, I first determine the optimal coolant velocity and inlet temperature for the initial series-parallel channel using a control-variable method. The initial channel has \(N=2\), \(D_1=20\) mm, \(D_2=36\) mm, and \(D_3=3\) mm. The simulations are performed with heat pipes at 1C discharge.
The effect of coolant velocity on the battery maximum temperature and pressure drop is shown in the table below.
| Velocity (m/s) | Max temp (K) | Temp 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 obtained at 0.5 m/s. At 0.6 m/s, the maximum temperature rises slightly because the coolant has less time to absorb heat. The pressure drop increases rapidly with velocity. Thus, I select 0.5 m/s as the optimal velocity.
Next, I fix the velocity at 0.5 m/s and vary the inlet temperature. The results are listed below.
| Inlet temperature (K) | Max temp (K) | Temp 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. Therefore, I choose an inlet temperature of 288 K for the subsequent structural optimization. This temperature is also reasonable from a practical standpoint, as it is close to the ambient temperature of 298 K but provides a sufficient temperature gradient for heat removal.
4.1 Taguchi Experimental Design
To optimize the geometry of the series-parallel channel, I use the Taguchi method. This method is well suited for reducing the number of experiments while extracting reliable information about the influence of each factor. The design factors are the number of branch channels \(N\) (2 levels: 2 and 3), and the three continuous factors \(D_1\), \(D_2\), and \(D_3\), each at four levels. The levels are given in the table below.
| Level | N | D1 (mm) | D2 (mm) | D3 (mm) |
|---|---|---|---|---|
| 1 | 2 | 15 | 30 | 2 |
| 2 | 3 | 20 | 36 | 3 |
| 3 | – | 25 | 44 | 4 |
| 4 | – | 30 | 50 | 5 |
Using an L32 orthogonal array, I generate 32 combinations of structural parameters. Each configuration is modeled and simulated with the same operating conditions: 1C discharge, 0.5 m/s inlet velocity, 288 K inlet temperature, and heat pipes included. The simulated responses include the maximum battery temperature, the battery temperature difference \(\Delta T\), and the pressure drop \(\Delta P\). The complete results are shown in the table below.
| No. | N | D1 | D2 | D3 | Max temp (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 |
From the 32 simulations, the maximum battery temperature varies by only about 1.5%, while the temperature difference varies by up to 26.9% and the pressure drop varies by up to 84.1%. This indicates that the structural parameters have a much stronger influence on the temperature difference and pressure drop than on the maximum temperature. Therefore, I focus the optimization on \(\Delta T\) and \(\Delta P\).
4.2 Signal-to-Noise Ratio Analysis
Since both the temperature difference and the pressure drop are expected to be minimized, I use the smaller-the-better signal-to-noise (S/N) ratio, defined as:
$$
S/N = -10\log_{10}\left( \frac{1}{n}\sum_{i=1}^{n} y_i^2 \right)
$$
where \(n\) is the number of replicates and \(y_i\) is the response value. In this study, each run is performed once, so \(n=1\). The S/N ratios for each factor level are computed separately for \(\Delta T\) and \(\Delta P\). The results are shown in the following tables.
| Response | Level | N | D1 | D2 | D3 |
|---|---|---|---|---|---|
| ΔT S/N ratio | 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 S/N ratio | 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 |
For the temperature difference, the most influential factor is \(D_3\) (thickness), with a range of 1.20. The order of influence is \(D_3 > D_2 > N > D_1\). The optimal level for minimizing \(\Delta T\) is \(N=2\), \(D_1=15\) mm, \(D_2=50\) mm, \(D_3=4\) mm. For the pressure drop, the most influential factor is \(D_1\) (width), with a range of 6.48. The order is \(D_1 > D_2 > N > D_3\). The optimal level for minimizing \(\Delta P\) is \(N=3\), \(D_1=30\) mm, \(D_2=50\) mm, \(D_3=2\) mm.
These two optimal configurations are conflicting. If I only minimize \(\Delta T\), the pressure drop increases by 11.56% compared with the initial structure. If I only minimize \(\Delta P\), the temperature difference increases by 26.32%. Thus, a single-objective optimization is not suitable for the present design. I therefore adopt a multi-objective optimization approach based on grey relational analysis.
4.3 Grey Relational Multi-Objective Optimization
Grey relational analysis is a method that evaluates the degree of similarity between a reference sequence and comparison sequences. I use this method to transform the two conflicting objectives into a single grey relational coefficient. The reference sequence is defined as the ideal values of \(\Delta T\) and \(\Delta P\), i.e., the minimum values obtained from the 32 runs:
$$
Y = \left\{ \Delta T_{min}, \Delta P_{min} \right\} = \left\{ 3.13, 590.99 \right\}
$$
The comparison sequence \(X\) contains the simulated values for each run. Since the units of \(\Delta T\) and \(\Delta P\) are different, I first normalize the data using the initial-value method. Then, I compute the grey relational coefficients for each response:
$$
\gamma_{i} = \frac{\Delta_{min} + \xi \Delta_{max}}{\Delta_{i} + \xi \Delta_{max}}
$$
where \(\Delta_i\) is the absolute difference between the normalized reference and comparison sequences, and \(\xi\) is the distinguishing coefficient, usually set to 0.5. The grey relational coefficients for \(\Delta T\) (\(\gamma_{\Delta T}\)) and \(\Delta P\) (\(\gamma_{\Delta P}\)) are then combined into a single weighted coefficient using:
$$
\gamma = \sqrt{\frac{\gamma_{\Delta T}^2 + \gamma_{\Delta P}^2}{2}}
$$
The computed coefficients for the 32 runs are summarized in the following table.
| Run | γ_ΔT | γ_ΔP | γ | Run | γ_ΔT | γ_ΔP | γ |
|---|---|---|---|---|---|---|---|
| 1 | 0.5886 | 0.4636 | 0.52 | 17 | 0.7565 | 0.5470 | 0.66 |
| 2 | 0.5377 | 0.6109 | 0.57 | 18 | 0.7699 | 0.6109 | 0.69 |
| 3 | 0.5928 | 0.5338 | 0.56 | 19 | 0.6107 | 0.7838 | 0.70 |
| 4 | 0.4648 | 0.5583 | 0.51 | 20 | 0.5470 | 0.7250 | 0.64 |
| 5 | 0.6313 | 0.4886 | 0.56 | 21 | 0.7991 | 0.5720 | 0.69 |
| 6 | 0.5804 | 0.6359 | 0.60 | 22 | 0.8125 | 0.6359 | 0.72 |
| 7 | 0.6354 | 0.5588 | 0.59 | 23 | 0.6533 | 0.8088 | 0.73 |
| 8 | 0.5074 | 0.5833 | 0.54 | 24 | 0.5896 | 0.7500 | 0.67 |
| 9 | 0.7679 | 0.6553 | 0.71 | 25 | 1.0000 | 0.6553 | 0.84 |
| 10 | 0.6313 | 0.5525 | 0.59 | 26 | 0.7991 | 0.6359 | 0.72 |
| 11 | 0.6220 | 0.5588 | 0.59 | 27 | 0.7042 | 0.7255 | 0.71 |
| 12 | 0.7083 | 0.6667 | 0.68 | 28 | 0.7262 | 0.9167 | 0.82 |
| 13 | 0.6741 | 0.7386 | 0.70 | 29 | 0.9063 | 0.7386 | 0.82 |
| 14 | 0.5375 | 0.6359 | 0.58 | 30 | 0.7054 | 0.7192 | 0.71 |
| 15 | 0.5283 | 0.6422 | 0.58 | 31 | 0.6104 | 0.8088 | 0.71 |
| 16 | 0.6146 | 0.7500 | 0.68 | 32 | 0.6324 | 1.0000 | 0.83 |
By averaging the grey relational coefficients for each factor level, I obtain the values shown in the table below.
| 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 |
A higher grey relational coefficient indicates a better multi-objective performance. The optimal combination that yields the highest coefficient is \(N=3\), \(D_1=25\) mm, \(D_2=30\) mm, and \(D_3=3\) mm. This parameter set corresponds to the 26th run in the orthogonal array, which indeed has one of the highest combined coefficients.
4.4 Performance Validation of the Optimized Design
I construct the optimized channel geometry using the above parameters and perform a numerical simulation under the same operating conditions (1C discharge, 0.5 m/s, 288 K inlet temperature). The temperature distribution and pressure contour are obtained. The optimized design reduces the maximum temperature of the EV battery pack to 303.22 K, which is 21.78 K lower than the natural-convection-only model. The temperature difference is reduced to 3.24 K, and the pressure drop becomes 1956 Pa.
The comparison between the initial and optimized series-parallel channels is given in the table below.
| Configuration | Max temperature (K) | Temperature difference (K) | Pressure drop (Pa) |
|---|---|---|---|
| Initial series-parallel | 305.56 | 3.45 | 3165.14 |
| Optimized series-parallel | 303.22 | 3.24 | 1956.16 |
| Improvement | 2.33 K | 6.10% | 38.20% |
Compared to the conventional parallel channel with heat pipes, the optimized series-parallel channel lowers the maximum temperature by 2.58 K. Compared to the S-shaped channel with heat pipes, the temperature difference is reduced by 6.94%, while the pressure drop is reduced by 67.69%. These results clearly demonstrate that the optimized series-parallel channel achieves a better trade-off between cooling performance and pumping power for the EV battery pack.
I also plot the temperature history of the EV battery pack during the 1C discharge. The optimized cooling system maintains the battery temperature below 303.5 K throughout the entire discharge process, whereas the uncooled battery pack reaches 325 K. The temperature curve shows a steep rise in the first few hundred seconds, followed by a gradual approach to a quasi-steady state. This indicates that the heat pipe and liquid cooling structure effectively remove the heat generated inside the battery.
5. Conclusions
In this work, I have numerically investigated the thermal performance of an EV battery pack using a hybrid cooling structure consisting of heat pipes and liquid cooling channels. The following conclusions can be drawn:
First, the NTGK electrochemical model is suitable for predicting the heat generation and temperature distribution of a prismatic LiFePO₄ battery pack. The grid independence study and comparison with reference experimental data confirm that the numerical model is reliable with an error of less than 2%.
Second, coupling heat pipes with liquid cooling channels significantly improves the cooling performance of the EV battery pack. For the parallel channel, the maximum temperature is reduced by 7.5 K on average when the coolant velocity varies; for the S-shaped channel, the reduction is 8.5 K. When the coolant inlet temperature varies, the average reductions are 11.5 K and 12 K, respectively. The heat pipe also improves temperature uniformity.
Third, the S-shaped channel achieves slightly better temperature control than the parallel channel, but its pressure drop is more than 15 times higher. To achieve a balanced performance, I designed a series-parallel channel that combines the advantages of both geometries.
Fourth, using Taguchi experimental design and grey relational analysis, I optimized the channel geometry. The most important factors affecting the temperature difference and pressure drop are the channel thickness and width, respectively. The optimal parameters are \(N=3\), \(D_1=25\) mm, \(D_2=30\) mm, and \(D_3=3\) mm. This optimized structure reduces the maximum temperature by 2.33 K, the temperature difference by 6.10%, and the pressure drop by 38.20% compared to the initial design.
Overall, the proposed heat pipe coupled liquid cooling system with an optimized series-parallel channel is an effective solution for thermal management of an EV battery pack. Future work may include experimental validation under real driving cycles, optimization of the heat pipe itself, and extension of the model to high-rate discharge and fast-charging scenarios.