In this work, I focus on the thermal performance of a high-voltage battery module equipped with a hybrid cooling structure combining heat pipes and a liquid-cooling system. The high-voltage battery is the core component of modern electric vehicles, and its temperature distribution directly affects charging efficiency, cycle life, and operational safety. As the energy density of the high-voltage battery continues to rise, the heat generated during discharge has become increasingly difficult to remove using conventional air cooling. Therefore, I have conducted a systematic numerical investigation to understand how heat pipes coupled with liquid cooling channels can enhance the thermal management of a high-voltage battery pack. The entire study is based on the NTGK electrochemical model and computational fluid dynamics simulations. I also designed a novel series-parallel flow channel and optimized its structural parameters using Taguchi experimental design and grey relational analysis.
Background and Significance
The rapid development of low-carbon transportation and electric mobility has led to a dramatic growth in the global market share of electric vehicles. In this context, the high-voltage battery is responsible for storing and delivering electric energy to the drivetrain. The operating temperature of the high-voltage battery is one of the most important factors governing its performance. If the temperature is too high, internal side reactions are accelerated, causing capacity fade and potentially leading to thermal runaway. If the temperature distribution is non-uniform, the local degradation rate differs among cells, which reduces the usable capacity of the entire high-voltage battery pack. In order to design an effective thermal management system, I first classify the heat sources of a lithium-ion cell. There are three major heat-generation mechanisms: electrochemical reaction heat, ohmic heat, and polarization heat. These are expressed in the following equations:
Q_r = – n F T \frac{dE}{dT}
$$
Q_\Omega = I^2 R_o
$$
Q_p = I \eta
$$
where \(Q_r\) is the electrochemical reaction heat, \(n\) is the number of transferred electrons, \(F\) is Faraday’s constant, \(T\) is temperature, \(\frac{dE}{dT}\) is the temperature coefficient of the open-circuit potential, \(Q_\Omega\) is ohmic heat generated by the internal resistance, \(R_o\) is the equivalent internal resistance, \(I\) is the applied current, \(Q_p\) is polarization heat, and \(\eta\) is the polarization overpotential. The total heat source inside the high-voltage battery is approximately equal to the sum of these three contributions. In practical numerical simulations, I omit the side-reaction heat because its magnitude is much smaller than the above terms.
In addition to heat generation, heat transfer within the high-voltage battery pack includes conduction through the cell layers, convection from the external surfaces to the surrounding air, and radiation to the ambient environment. The governing equation for heat conduction in solid regions is
\psi = – \lambda \nabla T
$$
where \(\psi\) is the heat flux and \(\lambda\) is the thermal conductivity. The convective heat flux at the boundary is described by Newton’s law of cooling:
q = h A \left( T_s – T_f \right)
$$
where \(h\) is the convective heat transfer coefficient, \(A\) is the heat transfer area, \(T_s\) is the surface temperature, and \(T_f\) is the fluid temperature. Radiative heat transfer follows the Stefan-Boltzmann law:
q = \varepsilon \sigma A T^4
$$
where \(\varepsilon\) is the surface emissivity and \(\sigma\) is the Stefan-Boltzmann constant. These fundamental equations form the basis of the thermal model used for the high-voltage battery studied in this paper.
Selection of the High-Voltage Battery Type
There are several types of automotive power batteries, including lithium-ion, nickel-metal hydride, fuel cells, and solid-state batteries. A comparison of these technologies is summarized in Table 1.
| Type | Advantages | Disadvantages | Applications |
|---|---|---|---|
| Lithium-ion | High energy density, long cycle life | Thermal management required | Most EVs |
| Nickel-metal hydride | Safe, low cost | Low energy density | Hybrid vehicles |
| Fuel cell | Zero emission, high specific energy | High hydrogen cost | Commercial vehicles |
| Solid-state | Very high energy density | Technology not mature | Demonstration vehicles |
Among these candidates, the lithium-ion high-voltage battery is the most widely used because it provides a reasonable balance of specific energy, power capability, cost, and life. Within lithium-ion chemistry, lithium iron phosphate (LiFePO4) has become increasingly popular in Chinese electric vehicles because of its high safety, long cycle life, and lower material cost. According to my review of market data, the high-voltage battery market now sees a strong share of LiFePO4 cells for both passenger vehicles and energy-storage applications. Therefore, I selected a prismatic LiFePO4 high-voltage battery as the research object in this study. The cells are rated at 3.2 V and 180 Ah. The dimensions of one cell are 207 mm by 174 mm by 72 mm in height, length, and width respectively.
Battery Pack Configuration and Heat-Pipe Placement
A high-voltage battery pack usually consists of several cells connected in series or parallel. In this research, I built a module with 24 cells arranged in a three-row by eight-column layout. The connection type is 1 parallel and 24 series. The cell geometry was simplified by omitting the external casing details and busbar connections, but the electrical tabs and interconnecting busbars were preserved in the model because they act as heat-conducting paths.
To reduce the maximum temperature and improve the temperature uniformity of the high-voltage battery pack, I placed L-shaped heat pipes between the cells. A heat pipe works by means of internal evaporation and condensation of a working fluid. It is capable of transferring large amounts of heat with a very low temperature drop. The effective thermal conductivity of a heat pipe is several orders of magnitude higher than that of copper or aluminum. In the numerical model, I treated the heat pipe as a solid conductor with an equivalent thermal conductivity of 5000 W/(m·K), which is widely used in the literature. The L-shaped heat pipe has its long side in contact with the surface of the high-voltage battery cells and its short side inserted into a liquid-cooled aluminum plate at the bottom of the module. Two heat pipes are assigned to each cell, and they are placed alternately to maximize the heat-extraction path from the inner region of the pack. For a clearer understanding, a typical high-voltage battery pack with this arrangement is shown in the following relevant image:
This configuration ensures that the heat pipe evaporator section is close to the hottest internal zone of the high-voltage battery, while the condenser section is in direct contact with the coolant flow channels. The liquid coolant carries the heat away to the external radiator. Thus, the thermal path from the cell core to the ambient environment is very efficient.
Numerical Modeling of the High-Voltage Battery
My numerical simulation of the high-voltage battery was carried out using the ANSYS Fluent software. For the electrochemical heat source, I used the NTGK model available through the MSMD (Multi-Scale Multi-Domain) solution method. The NTGK model links the heat generation to the depth of discharge (DOD) through two polynomial functions \(U\) and \(Y\). The governing equation for energy conservation in the battery and in the cooling system is written as:
\frac{\partial (\rho h_e)}{\partial t} + \nabla \cdot (\mathbf{V} \rho h_e) = \nabla \cdot (k \nabla T) + S_h
$$
where \(\rho\) is density, \(h_e\) is the effective enthalpy per unit mass, \(\mathbf{V}\) is the velocity vector, \(k\) is thermal conductivity, and \(S_h\) is the volumetric source term. The volume-specific current transfer rate is obtained from:
J_{ech} = a \left[ U \left( \phi^+ – \phi^- \right) – Y \right]
$$
where \(Y\), \(U\) are model parameters that depend on the DOD and temperature, \(a\) is the specific electrode area, and \(\phi^+\), \(\phi^-\) are the positive and negative phase potentials. The DOD is defined as:
DOD = \frac{\text{vol}}{3600 \, Q_{ah}} \cdot \int_0^t I \, dt
$$
where \(\text{vol}\) is the cell volume and \(Q_{ah}\) is the capacity in ampere-hours. The NTGK parameters \(U\) and \(Y\) are fitted by sixth-order polynomials:
U = \sum_{n=0}^{5} a_n (DOD)^n
$$
Y = \sum_{n=0}^{5} b_n (DOD)^n
$$
The polynomial coefficients are summarized in Table 2.
| Parameter | Coefficient | Value |
|---|---|---|
| U | a0 | 4.12 |
| a1 | -0.804 | |
| a2 | 1.075 | |
| a3 | -1.177 | |
| a4 | 0 | |
| a5 | 0 | |
| Y | b0 | 1168.59 |
| b1 | -8928 | |
| b2 | 52504.6 | |
| b3 | -136231 | |
| b4 | 158531.7 | |
| b5 | -67578.5 |
My high-voltage battery model used the following properties: density 2248 kg/m3, specific heat 850 J/(kg·K), and anisotropic thermal conductivities of 15 W/(m·K) in the in-plane directions and 1 W/(m·K) in the through-plane direction. The operating voltage was set to 3.22 V and the maximum discharge rate was 3C. The initial temperature was 298 K, and the external walls of the battery pack exchanged heat with a 298 K environment with a natural-convection coefficient of 5 W/(m²·K).
For the fluid domain, the coolant was a 50% ethylene glycol-water solution. Its thermophysical properties are listed in Table 3 together with the solid aluminum plate properties.
| Material | Density (kg/m³) | Specific heat (J/(kg·K)) | Thermal conductivity (W/(m·K)) | Dynamic viscosity (Pa·s) |
|---|---|---|---|---|
| 50% glycol solution | 1069 | 3494 | 0.419 | 0.00315 |
| Aluminum | 2702 | 903 | 237 | — |
I considered the cooling liquid as an incompressible Newtonian fluid with constant physical properties. Since the highest inlet velocity in the study is 0.6 m/s, I calculated the maximum Reynolds number based on the hydraulic diameter of the channel:
Re = \frac{\rho v D_h}{\mu}, \quad D_h = \frac{2ab}{a+b}
$$
where \(a\) and \(b\) denote the cross-sectional width and height. For a typical 20 mm wide and 5 mm high channel, the maximum Reynolds number is about 1364, well below 2300. Therefore, the flow is laminar. The fluid domain thus satisfies the mass, momentum, and energy conservation equations:
\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{V}) = 0
$$
\frac{\partial (\rho \mathbf{V})}{\partial t} + \nabla \cdot (\rho \mathbf{V}\mathbf{V}) = – \nabla p + \nabla \cdot \tau + \rho \mathbf{F}
$$
\frac{\partial}{\partial t}(\rho e) + \nabla \cdot \left[ \mathbf{V} \left( \rho e + \frac{1}{2} \rho V^2 \right) \right] = -\nabla \cdot q + \tau : \nabla \mathbf{V} + Q
$$
where \(p\) is pressure, \(\tau\) is the viscous stress tensor, \(\mathbf{F}\) is the body force, \(e\) is internal energy, and \(q\) is heat flux. These equations are solved by the finite-volume method with the SIMPLE scheme.
Validation and Grid-Independence Check
Any numerical simulation must be verified against independent experimental data. I compared my modeling method with the results from an experimental benchmark described in the literature. The benchmark was a large lithium-ion high-voltage battery with a capacity of 50 Ah, using a flat microchannel cooling plate. I reproduced the same geometry and boundary conditions and simulated the 1C discharge process with a coolant flow rate of 0.5 L/min and an inlet temperature of 20°C. The comparison is shown in Table 4.
| Operating condition | Experiment (°C) | Simulation (°C) | Error |
|---|---|---|---|
| Battery maximum temperature | 36.31 | 36.72 | 1.13% |
| Cooling plate surface temperature | 32.37 | 33.02 | 2.01% |
The errors were below 2%, which proves that the NTGK model and the fluid-solid coupling model can accurately predict the thermal performance of a high-voltage battery with a liquid-cooling system.
To ensure that my grid resolution did not change the solution, I conducted a grid-independence study by varying the number of elements from 5.14 × 105 to 1.21 × 108. The monitored results were the maximum temperature and the pressure drop in the flow channel. The data are shown in Table 5.
| Number of cells (×105) | Maximum temperature (K) | Temperature change (%) | Pressure drop (Pa) | Pressure change (%) |
|---|---|---|---|---|
| 5.14 | 330.595 | — | 394.508 | — |
| 7.06 | 337.194 | 1.96 | 405.860 | 2.80 |
| 14.4 | 327.460 | -2.97 | 416.453 | 2.61 |
| 21.5 | 327.490 | 0.01 | 406.740 | -2.33 |
| 38.1 | 326.330 | -0.35 | 412.330 | 1.37 |
| 76.7 | 323.605 | -0.85 | 411.164 | -0.28 |
| 210.8 | 321.717 | -0.59 | 400.406 | -2.62 |
| 1215.1 | 321.458 | -0.08 | 400.370 | -0.01 |
At a grid count of 2.108 × 107, the temperature deviation from the finest grid is only 0.08%, and the pressure drop deviation is 0.01%. Therefore, I selected about 21 million cells for all subsequent simulations of the high-voltage battery module with the cooling system.
Performance of Conventional Flow Channels with Heat Pipes
I first evaluated two conventional liquid-cooling channel configurations, namely the parallel flow channel and the serpentine flow channel. For each configuration, I compared the baseline liquid cooling only and the hybrid case with heat pipes coupled between the high-voltage battery cells and the cooling plate. The operating conditions were 1C discharge, environment temperature 298 K, and the coolant was the same 50% glycol solution. I varied the inlet velocity between 0.1 m/s and 0.6 m/s. Table 6 lists the results for the parallel flow channel without heat pipes.
| Velocity (m/s) | Maximum temperature (K) | Temperature 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 heat pipes were added, the improved thermal path between the high-voltage battery cores and the liquid-cooling plate reduced the maximum temperature considerably. The results for the parallel channel with heat pipes are shown in Table 7.
| Velocity (m/s) | Maximum temperature (K) | Temperature 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 |
The heat pipe addition gave an average reduction of about 7.5 K in maximum temperature over the velocity range. At the same time, the temperature difference was greatly reduced at moderate velocities. The pressure drop in the parallel channel remained low due to the direct simple flow paths.
For the serpentine flow channel, the coolant travels a longer winding path, which improves heat transfer but substantially increases the pressure drop. The results without heat pipes are listed in Table 8.
| Velocity (m/s) | Maximum temperature (K) | Temperature 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 serpentine channel, the maximum temperature of the high-voltage battery pack dropped substantially, as indicated in Table 9.
| Velocity (m/s) | Maximum temperature (K) | Temperature difference (K) | Pressure drop (Pa) |
|---|---|---|---|
| 0.6 | 311.96 | 3.62 | 8874.06 |
| 0.5 | 312.20 | 3.64 | 6054.63 |
| 0.4 | 312.54 | 3.65 | 3945.49 |
| 0.3 | 313.08 | 3.70 | 2174.33 |
| 0.2 | 314.03 | 3.78 | 942.64 |
| 0.1 | 316.04 | 2.10 | 223.75 |
The average maximum temperature reduction was about 8.5 K in the serpentine channel. The heat pipe also produced a lower temperature difference at lower velocities. Nevertheless, the pressure drop of the serpentine channel remained extraordinarily high compared to the parallel channel. This is an important constraint when designing an efficient thermal management system for a high-voltage battery.
I also tested the influence of the coolant inlet temperature for both conventional channel types. The inlet velocity was fixed at 0.5 m/s, while the inlet temperature varied from 298 K down to 278 K. The high-voltage battery results are summarized in Table 10.
| Inlet temperature (K) | Parallel channel (without/with heat pipe) | Serpentine channel (without/with heat pipe) | ||
|---|---|---|---|---|
| Tmax no-HP | Tmax HP | Tmax no-HP | Tmax HP | |
| 298 | 321.41 / 313.49 | 320.93 / 311.96 | 313.49 / 311.96 | 313.49 / 311.96 |
| 293 | 320.40 / 308.95 | 320.06 / 308.62 | 308.95 / 308.62 | 308.95 / 308.62 |
| 288 | 318.26 / 305.80 | 317.53 / 305.05 | 305.80 / 305.05 | 305.80 / 305.05 |
| 283 | 316.00 / 304.10 | 314.96 / 301.49 | 304.10 / 301.49 | 304.10 / 301.49 |
| 278 | 313.71 / 300.88 | 312.36 / 298.47 | 300.88 / 298.47 | 300.88 / 298.47 |
When heat pipes were included, the parallel and serpentine channels respectively achieved an average maximum-temperature reduction of 11.5 K and 12 K across the tested inlet temperatures. The heat pipes also attenuated the large temperature differences that appeared when only the inlet temperature was lowered. In all cases, the high-voltage battery pack was more uniformly cooled when heat pipes were present.
Design of a Series-Parallel Flow Channel
The previous results demonstrated that the serpentine flow channel has a strong cooling capability but a very high pressure drop, whereas the parallel flow channel has a low pressure drop but slightly worse temperature uniformity. I therefore designed a new series-parallel flow channel that combines the advantages of both structures. In the series-parallel channel, the main flow is split into several branch channels which then recombine at the outlet region. The branch passages travel along the length of the condenser section of the heat pipes, allowing the coolant to interact with the same effective heat-transfer area as in the serpentine channel while keeping the flow resistance close to that of a parallel channel. This design is especially suitable for an integrated cooling plate under a high-voltage battery pack.
Before optimizing the structural parameters, I fixed the coolant inlet velocity and inlet temperature using a control variable approach. The selected velocity was 0.5 m/s, because at this velocity the maximum temperature of the high-voltage battery reached the lowest value and the temperature difference was relatively acceptable. The corresponding pressure drop was 3165 Pa, which was still lower than that of the long serpentine channel. In Table 11 I present the effect of velocity on the series-parallel flow channel coupled with heat pipes.
| Velocity (m/s) | Tmax (K) | ΔT (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 |
Although increasing the velocity from 0.5 to 0.6 m/s substantially raised the pressure drop, the maximum temperature increased slightly due to insufficient residence time for the coolant to extract heat from the heat pipe. Therefore, an inlet velocity of 0.5 m/s was chosen. Next, with the velocity fixed at 0.5 m/s, I varied the inlet temperature from 298 K to 278 K. Table 12 shows the resulting maximum temperatures and differences.
| Inlet temperature (K) | Tmax (K) | ΔT (K) |
|---|---|---|
| 298 | 317.21 | 6.78 |
| 293 | 313.85 | 6.68 |
| 288 | 310.21 | 6.24 |
| 283 | 306.58 | 6.51 |
| 278 | 303.65 | 7.52 |
The first four rows show a decreasing trend in temperature difference as the inlet temperature decreases from 298 K to 288 K, after which further cooling causes the difference to increase. Thus, I selected an inlet temperature of 288 K as the optimum coolant temperature for this high-voltage battery application. These inlet conditions were then used for the structural optimization of the series-parallel flow channel.
Taguchi Experimental Design
The Taguchi method is a robust design methodology that investigates how different design parameters affect the performance of a product under various noise factors. I applied it to determine the optimal geometrical parameters of the series-parallel flow channel for the high-voltage battery thermal management system. I selected four structural factors: the number of branch channels \(N\), the branch width \(D_1\), the spacing between branches \(D_2\), and the channel thickness \(D_3\). The first factor \(N\) has two levels, while the others have four levels. The design parameters are listed in Table 13.
| Factor | A: N | B: D1 (mm) | C: D2 (mm) | D: D3 (mm) |
|---|---|---|---|---|
| Level 1 | 2 | 15 | 30 | 2 |
| Level 2 | 3 | 20 | 36 | 3 |
| Level 3 | — | 25 | 44 | 4 |
| Level 4 | — | 30 | 50 | 5 |
Using the Taguchi orthogonal array \(L_{32}(2^1 \times 4^3)\), I obtained 32 structural combinations. For each one, I created a corresponding three-dimensional model of the high-voltage battery module with heat pipes and the series-parallel cooling plate. I then ran a full CFD simulation under the fixed operating condition of 1C discharge, inlet velocity 0.5 m/s, inlet temperature 288 K, and ambient temperature 298 K. Table 14 shows the first 16 of the 32 simulation results.
| No. | N | D1 | D2 | D3 | 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 |
The remaining 16 runs are shown in Table 15.
| No. | N | D1 | D2 | D3 | Tmax (K) | ΔT (K) | ΔP (Pa) |
|---|---|---|---|---|---|---|---|
| 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 runs, the largest and smallest maximum temperatures differ by only 1.5 percent. Such a result indicates that the peak temperature of the high-voltage battery is not strongly affected by the channel geometry under these fixed coolant conditions. In contrast, the temperature difference changes by about 26.9 percent and the pressure drop changes by as much as 84.1 percent. Thus, I focused the optimization on two key criteria: the temperature difference \(\Delta T\) and the pressure drop \(\Delta P\). For a smaller-is-better characteristic, the signal-to-noise ratio is calculated from:
S/N = -10 \log_{10} \left( \frac{1}{n} \sum_{i=1}^{n} y_i^2 \right)
$$
where \(y_i\) is the measured output for the \(i\)-th repetition and \(n\) is the total number of repetitions. Since each combination was simulated only once, \(n=1\) and the formula reduces to \(-20 \log_{10}(y)\). Table 16 gives the signal-to-noise ratio response tables obtained by averaging the values at each level for \(\Delta T\) and \(\Delta P\).
| 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 response for \(\Delta T\) indicates that the channel thickness \(D_3\) has the largest effect, followed by the spacing \(D_2\), the number of branches \(N\), and finally the width \(D_1\). The best level for minimizing \(\Delta T\) is \(N=2\), \(D_1=15\) mm, \(D_2=50\) mm, and \(D_3=4\) mm. In contrast, the response for \(\Delta P\) shows that \(D_1\) is the most influential factor, with a range of 6.48, followed by \(D_2\), \(N\), and \(D_3\). The best level for minimizing \(\Delta P\) is \(N=3\), \(D_1=30\) mm, \(D_2=50\) mm, and \(D_3=2\) mm.
These two single-objective optimizations lead to conflicting parameter choices. Since a flow channel that minimizes \(\Delta T\) may increase \(\Delta P\), and a channel that minimizes \(\Delta P\) may give a larger \(\Delta T\), I performed a multi-objective optimization based on grey relational analysis. This method normalizes the two objectives into a single score, allowing me to identify a compromise configuration that is favorable for both performance criteria.
Grey Relational Multi-Objective Optimization
Grey relational analysis is a useful technique when the system has multiple uncertain or non-linear relationships. In my study, I considered the two optimization targets: \(\Delta T\) and \(\Delta P\). For each group in the 32 Taguchi experiments, I calculated the grey relational coefficient for temperature difference and pressure drop. The necessary equations are as follows. First, I define the reference sequence as the ideal minimum values:
Y = \left[ y_1^*, y_2^* \right] = \left[ \min(\Delta T), \min(\Delta P) \right]
$$
These values are 3.13 K and 590.9948 Pa. Then, I construct the comparison sequence \(X\) from the measured outputs of each experiment \(j\), i.e. \(x_j = \left[ \Delta T_j, \Delta P_j \right]\). After normalizing the sequences, the grey relational coefficient is:
\gamma_i(k) = \frac{\Delta_{\min} + \zeta \Delta_{\max}}{\Delta_{i}(k) + \zeta \Delta_{\max}}
$$
where \(\Delta_i(k)\) is the absolute difference between the normalized reference and comparison values at the \(k\)-th objective, \(\Delta_{\min}\) and \(\Delta_{\max}\) are the minimum and maximum absolute differences across all experiments, and \(\zeta\) is the distinguishing coefficient, usually taken as 0.5. Once I compute the coefficients for \(\Delta T\) and \(\Delta P\), I combine them into an overall grey relational grade using the equal-weight average:
\Gamma_i = \sqrt{\frac{\gamma_{\Delta T}^2 + \gamma_{\Delta P}^2}{2}}
$$
I calculated the values of \(\gamma_{\Delta T}\), \(\gamma_{\Delta P}\), and \(\Gamma\) for all 32 experimental combinations. The full set of grey coefficients is shown in Table 17.
| No. | γΔT | γΔP | Γ | No. | γΔT | γΔP | 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 |
I then averaged the overall grey grade at each level for each factor to evaluate the influence ranking. Table 18 shows the resulting grey relational grade for each factor at all levels.
| Factor | Level 1 | Level 2 | Level 3 | Level 4 |
|---|---|---|---|---|
| N | 0.5000 | 1.0000 | — | — |
| D1 | 0.4281 | 0.5660 | 0.8498 | 0.8339 |
| D2 | 0.7767 | 0.5857 | 0.5976 | 0.7453 |
| D3 | 0.4528 | 0.8144 | 0.7906 | 0.4678 |
The maximum grey grade is obtained when the number of branch channels is three, the branch width is 25 mm, the spacing is 30 mm, and the channel thickness is 3 mm. Therefore, the optimum structural parameters of the series-parallel flow channel are:
N^* = 3, \quad D_1^* = 25~mm, \quad D_2^* = 30~mm, \quad D_3^* = 3~mm
$$
The factor ranking based on the grey grade is \(N > D_1 > D_3 > D_2\). This indicates that the number of branch channels has the highest combined influence on the temperature uniformity and pressure drop of the high-voltage battery cooling system.
Performance of the Optimized Channel
After completing the optimization, I rebuilt the series-parallel flow channel model with the optimum dimensions and conducted a simulation under the same boundary conditions. The optimized channel was also compared with the original series-parallel design that had a velocity of 0.5 m/s and an inlet temperature of 288 K. The resulting flow structure shows that the coolant is distributed evenly among the three branches, leading to a relatively uniform temperature profile across the bottom of the high-voltage battery pack. Table 19 presents a comparison between the initial series-parallel design and the optimized design.
| Configuration | Tmax (K) | ΔT (K) | ΔP (Pa) |
|---|---|---|---|
| Initial series-parallel | 305.56 | 3.45 | 3165.14 |
| Optimized series-parallel | 303.22 | 3.24 | 1956.16 |
| Performance improvement | 2.33 K drop | 6.10% drop | 38.20% drop |
The optimized design reduced the maximum temperature of the high-voltage battery pack from 305.56 K to 303.22 K, lowered the temperature difference by 6.1 percent, and decreased the coolant pressure drop by 38.2 percent. This result is particularly important because the optimized structure improves thermal performance while simultaneously reducing pumping power, thereby balancing temperature control and energy consumption.
I also compared the optimized series-parallel channel with the best conventional channels discussed earlier. Table 20 provides a comprehensive comparison of the different channel configurations with heat pipes under the same operating condition of 1C discharge and coolant at 0.5 m/s and 288 K.
| Channel type | Tmax (K) | ΔT (K) | ΔP (Pa) |
|---|---|---|---|
| Parallel + heat pipe | 305.80 | 2.53 | 418.50 |
| Serpentine + heat pipe | 305.05 | 3.48 | 6054.63 |
| Optimized series-parallel + heat pipe | 303.22 | 3.24 | 1956.16 |
The optimized series-parallel channel yields a maximum temperature that is 2.58 K lower than the parallel channel with heat pipes, while its pressure drop is only about one-third of the serpentine channel. Although the best possible temperature uniformity is still achieved by the parallel channel with heat pipes, the optimized series-parallel design gives a much lower maximum temperature and a much lower pressure drop than the serpentine design. In practical high-voltage battery packages, both the pressure drop and the maximum temperature are important because the coolant pump consumes auxiliary power and the lifecycle of the high-voltage battery is shortened at elevated temperatures.
To quantify the overall benefit, I calculated the maximum-temperature reduction compared with the original standalone battery model without any active cooling. In the no-cooling case, the high-voltage battery reached a maximum temperature of about 325 K at the end of 1C discharge. With the optimized series-parallel channel and heat pipes, the final maximum temperature was 303.22 K. This means that my proposed cooling system reduced the maximum temperature of the high-voltage battery pack by roughly 21.78 K. Such a remarkable temperature reduction is highly favorable for extending the cycle life and preventing thermal runaway.
Heat Transfer and Flow Field Discussion
The superior behavior of the optimized series-parallel channel can be explained by examining the local flow distribution and its interaction with the heat-pipe condensers. In a conventional parallel channel, the flow resistance is small but not all branch paths receive equal flow. In a serpentine channel, the coolant is forced to pass all heat pipes in sequence, which ensures good heat exchange but generates a very large pressure gradient. The optimized series-parallel design splits the coolant into three streams, each of which travels along several heat-pipe condensers. Because the branch width is 25 mm and the spacing is 30 mm, the fluid residence time is sufficient to absorb heat without drastically raising the pressure loss. The thickness of 3 mm provides an appropriate cross-section for heat exchange without creating excessive flow velocity or pumping loss.
One physical quantity used to evaluate the overall heat-transfer enhancement is the dimensionless Nusselt number for the laminar internal flow:
Nu = \frac{h D_h}{k}
$$
where \(h\) is the convective heat transfer coefficient between the coolant and the channel wall, \(D_h\) is the hydraulic diameter, and \(k\) is the thermal conductivity of the coolant. For a fixed geometry and coolant properties, an increase in flow velocity raises the Reynolds number and consequently the Nusselt number. However, high velocity also raises the friction factor and pressure drop. The friction factor for fully developed laminar flow is:
f = \frac{64}{Re}
$$
Thus, at the optimum velocity of 0.5 m/s, the modular series-parallel channel avoids the high friction loss that appears in the serpentine channel while maintaining a relatively high Nusselt number on the heat-pipe condenser surface. The heat pipe itself transports the internal heat from the high-voltage battery cells to the condenser at an extremely high equivalent conductance. The combination of a mild cross-flow and a high-conductivity heat path is the key reason for the excellent performance.
In addition, the heat pipes placed on both sides of each cell provide an equalized heat-transport path to the bottom cooling plate. Without heat pipes, heat generated in the center of the high-voltage battery module has to travel through the entire battery width to reach the liquid-cooled surface at the bottom. Since the through-plane thermal conductivity of the battery is only about 1 W/(m·K), the cell core temperature becomes much higher than the surface temperature. The heat pipe effectively short-circuits the poor thermal path by collecting heat along its evaporator section and delivering it to the cold plate. This is why the maximum temperature of the high-voltage battery pack drops by more than 10 K when heat pipes are added.
Table 21 summarizes the effect of heat-pipe addition on the average maximum temperature decrease for the traditional flow channels, under both velocity sweeps and inlet-temperature sweeps.
| Variable sweep | Parallel channel | Serpentine channel |
|---|---|---|
| Velocity 0.1-0.6 m/s, ΔTmax reduction | 7.5 K | 8.5 K |
| Inlet temperature 278-298 K, ΔTmax reduction | 11.5 K | 12 K |
These results are consistent with the known behavior of heat pipes. The heat pipe works by evaporation and condensation of a working fluid inside a sealed container. The heat pipe begins to operate once the evaporator temperature exceeds the saturation temperature of the internal fluid. In this application, the high-voltage battery surface temperature is well above 30°C during operation, so the heat pipe is always active. The effective thermal conductivity of the selected heat pipe model was fixed at 5000 W/(m·K). Figure 13 in the original paper showed the pressure distribution in the two conventional channels; the parallel channel had a uniform pressure distribution whereas the serpentine channel had a sharp pressure gradient at the entrance. That observation motivated the development of the series-parallel channel, which inherently has a more moderate pressure field.
From a view of the temperature field in the optimized model, the obtained temperature slices indicated that the battery cells had a small vertical temperature gradient because the heat was efficiently extracted from both the side surfaces via heat pipes and the bottom via the liquid cooling plate. The maximum temperature occurred near the top center of the cells, which is expected because the heat pipes are mainly located on the lower side and the bus bars also generate ohmic heat. Nonetheless, the temperature difference in the optimized high-voltage battery module was only 3.24 K, which is well below the usual recommended limit of 5 K for a high-voltage battery pack.
Discussion of Parametric Optimization
The combination of Taguchi design and grey relational analysis proved to be a very effective method for reducing the number of numerical simulations while still finding the optimal structural combination. Without using this method, a full factorial design would require \(2 \times 4^3 = 128\) experiments, whereas the Taguchi method required only 32 simulations. The grey relational analysis then transformed the two-objective problem into a single-objective grade, avoiding the need to manually trade off between \(\Delta T\) and \(\Delta P\). Table 22 summarizes the optimum factor levels obtained from each analysis.
| Method | N | D1 (mm) | D2 (mm) | D3 (mm) | Optimization criterion |
|---|---|---|---|---|---|
| Taguchi S/N for ΔT | 2 | 15 | 50 | 4 | Minimize ΔT |
| Taguchi S/N for ΔP | 3 | 30 | 50 | 2 | Minimize ΔP |
| Grey relational grade | 3 | 25 | 30 | 3 | Multi-objective |
It is interesting to see that the multi-objective optimum is not the same as either single-objective optimum. Instead, it balances the requirements by using three branch channels rather than two, because the third branch reduces the pressure drop while still performing enough heat exchange. A branch width of 25 mm is a compromise between the wide 30 mm channel that reduces pressure and the narrow 15 mm channel that increases coolant velocity and heat transfer. The spacing of 30 mm is relatively small, allowing more heat-pipe condensers to be covered within the available footprint. The channel thickness of 3 mm provides a cross-sectional area that gives a moderate coolant flow rate and enough wetted surface.
From the manufacturing and integration perspective, the optimized flow channel remains simple enough to be produced by conventional aluminum stamping or extruded-plate bonding processes. Furthermore, the high-voltage battery module can be assembled layer by layer: first the heat pipes are inserted between cells, then the bottom cold plate is positioned so that the condenser sections of all heat pipes lie inside the coolant flow passages. A thermal interface material may be added to eliminate contact resistance. Such a modular design is attractive for high-voltage battery packs because it allows independent replacement of cooling components when necessary.
Design Robustness and Limitations
I also investigated the sensitivity of the optimized design to changes in the coolant inlet temperature and the discharge rate. In the previous sections all simulations were performed at 1C discharge, but modern electric vehicles often use high-voltage battery packs at 2C or 3C fast-charging conditions. Although I have not included a full analysis of higher discharge rates in the present manuscript, the NTGK model is inherently capable of simulating these conditions by changing the applied current. The heat source strength increases quadratically with the current because the ohmic term is \(I^2R\). Therefore, the maximum temperature of the high-voltage battery at 2C is expected to rise more rapidly. The combination of heat pipes and liquid cooling will still be effective, but the coolant flow rate and inlet temperature may need to be adjusted in accordance with the actual heat load.
One limitation is that the equivalent thermal conductivity of the heat pipe was assumed constant. In reality, heat pipes have a maximum heat-transport capacity known as the capillary limit. If the heat flux exceeds this limit, the evaporation section dries out and the effective thermal conductivity decreases sharply. In the present high-voltage battery cooling design, the heat flux per heat pipe is relatively moderate because the heat pipes are distributed along every cell. Thus, the capillary limit is not likely to be reached at 1C rate. At higher C-rates or in the case of abusive thermal loading, the heat-pipe model should be updated with a more detailed two-phase model. I suggest future work in this direction.
Another practical issue is the contact thermal resistance between the heat pipe and the cell surface. In the numerical model, I assumed that the contacting surfaces are perfectly joined with zero contact resistance. In an actual manufacturing process, the surfaces may be coated with a thermal interface material or held with mechanical compression. The small gap between the cell and the heat pipe will produce a thermal resistance that could be represented by an equivalent thin boundary layer. If the contact pressure is insufficient, the thermal performance of the high-voltage battery cooling system would degrade. However, prior experimental work with battery modules using heat pipes has shown that a tightly assembled module with a thin gap pad still behaves acceptably if the surfaces are flat.
Table 23 lists some important design guidelines for heat-pipe based liquid cooling systems of high-voltage battery packs, which I derived from the numerical simulations.
| Design guideline | Recommended value or strategy |
|---|---|
| Heat pipe arrangement | Place heat pipes alternately between adjacent cells to increase surface coverage. |
| Heat pipe length | Longer condenser sections improve heat rejection but increase heat-pipe mass; optimize with cold-plate area. |
| Coolant flow velocity | For a plate thickness of 3-5 mm, an inlet velocity near 0.5 m/s provides good thermal performance without excessive pressure loss. |
| Coolant inlet temperature | For 1C operation, 288 K yielded the lowest temperature difference; lower temperatures may reduce liquid-crystal film issues but increase energy consumption. |
| Channel geometry | Series-parallel branches moderate pressure and improve surface coverage; three branch channels gave the best overall grade. |
| Multi-objective optimization | Taguchi plus grey relational analysis is computationally efficient and can balance conflicting design criteria. |
The optimum structural parameters obtained here were determined for a given fixed heat-pipe thermal conductivity, cell dimensions, and cooling plate footprint. When the high-voltage battery size changes, the corresponding channel geometry may need to be rescaled. The general conclusion, that the series-parallel branch channel can reduce pumping power while maintaining good heat extraction, is expected to hold regardless of scale.
Conclusion and Future Work
Through the course of this thesis, I have presented a complete numerical study of a thermal management system for a prismatic LiFePO4 high-voltage battery module. A coupled strategy consisting of L-shaped heat pipes inserted between battery cells and a liquid-cooling plate at the bottom was considered. The NTGK electrochemical model together with a conjugate heat-transfer simulation was validated against an experimental benchmark. I then simulated two common flow-channel shapes, the parallel and serpentine, with and without heat pipes. The addition of heat pipes caused the maximum temperature of the high-voltage battery to drop by approximately 7.5 K in the parallel channel and 8.5 K in the serpentine channel when the coolant velocity was varied. When the inlet temperature was varied, the reductions were 11.5 K and 12 K respectively. These findings clearly indicate that heat pipes can significantly improve the cooling capability of a liquid-cooling system for high-voltage battery applications.
To overcome the disadvantageous high pressure drop of the serpentine channel while retaining its large effective heat-transfer region, I individually designed a new series-parallel channel. In this design, the coolant enters an inlet manifold, is distributed into several parallel branches, and then recombines before exiting. The branches are shaped so that they pass over the condenser sections of all heat pipes. I used a control variable method to first fix the coolant velocity of 0.5 m/s and inlet temperature of 288 K for the structural optimization. Then I applied the Taguchi method to generate 32 different structural combinations from four factors: branch number \(N\), branch width \(D_1\), branch spacing \(D_2\), and branch thickness \(D_3\). For each combination, I carried out a CFD simulation and collected the maximum temperature, temperature difference, and pressure drop.
The single-objective Taguchi analysis showed that minimizing \(\Delta T\) and minimizing \(\Delta P\) lead to different structural choices. To combine them into a unique design, I implemented a grey relational analysis. The resulting optimum geometry was \(N=3\), \(D_1=25\) mm, \(D_2=30\) mm, and \(D_3=3\) mm. The grey relational grades indicated that the most influential factor for the combined objective was the number of branch channels, followed by the branch width, the thickness, and the branch spacing. The optimized model was then simulated and compared with the original series-parallel design. The maximum temperature decreased by 2.33 K, the temperature difference decreased by 6.10 percent, and the pressure drop decreased by 38.20 percent. Also, when compared with the parallel channel and the serpentine channel with heat pipes, the optimized design achieved a maximum temperature 2.58 K lower than the parallel channel and a pressure drop about one-fifth that of the serpentine channel.
In summary, the newly designed series-parallel flow channel in combination with heat pipes provides a balanced solution for high-voltage battery thermal management. By using the Taguchi-grey method, I achieved a meaningful improvement in both thermal uniformity and pumping efficiency without needing 128 full-factorial simulations. This work demonstrates that the coupling of a high-conductivity passive heat pipe with a carefully optimized liquid-cooling flow structure is a promising way to manage the increasingly high heat fluxes of advanced high-voltage batteries in modern electric vehicles.
For future research, I plan to examine the performance of the proposed cooling system at higher discharge rates, especially at 3C and under fast-charging cycles. In addition, the heat pipe model should be replaced by a more detailed porous-media model that can capture the capillary limit and the local temperature saturation behavior of the working fluid. Experimental validation of the optimized series-parallel flow channel should also be performed using thermal-imaging techniques and pressure transducers to measure the temperature distribution and pumping power in a real high-voltage battery module. Finally, the control strategy of the coolant pump can be optimized with feedback from the battery temperature sensor to reduce auxiliary losses under partial load.
As high-voltage battery technology evolves toward even higher specific energy and faster charging, innovative thermal management solutions will become ever more critical. The heat-pipe/liquid-cooling hybrid architecture presented in this study offers a practical and efficient path to keep high-voltage battery cells within their narrow optimal temperature window. My results contribute not only to the development of robust thermal design methods, but also to the safe deployment of high-voltage battery packs in next-generation electric vehicles.
The combination of Taguchi experimental design, CFD simulation, and grey relational analysis proved to be a powerful framework for thermal design of high-voltage battery systems. The same framework can be extended to other heat-transfer enhancing features such as wavy channels, pin fins, or composite phase-change materials. By applying this systematic workflow, the thermal performance of any high-voltage battery pack can be optimized with limited computational resources.
I hope that this thesis provides a valuable reference for the design of liquid-cooled high-voltage battery thermal management systems using heat pipes and offers new insights into the trade-off between cooling performance and hydraulic losses. The proposed optimized series-parallel channel is a simple but effective change to conventional flow fields, and it may be directly used in industrial high-voltage battery packs that utilize flat cooling plates. With further testing and refinement, high-voltage battery systems can achieve an even higher level of safety and performance.