1. Introduction
As an energy storage device, batteries have been widely used in fields such as mobile devices, household appliances, and electric vehicles due to the advantages of high energy density, long life cycle, and stable performance [
1,
2,
3]. However, heat production problems arise during the charge and discharge of the battery, which may lead to a capacity reduction, inefficiency, and the risk of spontaneous combustion with temperature rise in the battery [
4,
5]. Therefore, to further contribute to this research topic, a topology optimization pipeline for the cooling plate is presented to improve the heat dissipation efficiency and ensure the working performance and stability of the battery in this paper.
When compared with air and phase change-based cooling systems, the liquid-based cooling system is of good performance regarding the trade-off between cooling efficiency, cost, and reliability, and it has been widely used in battery thermal management [
6,
7,
8]. The liquid-based cooling system usually works using a cooling plate, and a coolant takes away the heat through flowing in the internal fluid channels [
9,
10]. Current researche projects on cooling plate optimization mainly focus on the size and shape parameters. Huo et al. [
11] designed a straight mini-channel cold plate-based battery thermal management system and numerically investigated the effects of the number of channels, flow direction, inlet mass flow rate and ambient temperature on temperature rise and distribution. To further develop the performance of the cooling plate, snake, U-shape, and bionic wave-based runners have been investigated. Jarrett et al. [
12] presented a serpentine shape-based cooling plate to account for the average temperature, temperature uniformity, and pressure drop of the cooling plate and investigated the relationship between the objectives, distribution of the input heat flux, and the coolant flow rate. Li et al. [
13] presented an optimization pipeline that included the Gaussian process and NSGA-II methods for U-shaped cooling plate design and claimed that the U-shape channel can significantly decrease the pressure drop loss when compared with a serpentine channel cooling plate. Li et al. [
14] proposed a novel and efficient liquid cooling scheme for battery thermal management by using a bionic wave-based channel, and they thoroughly investigated the effects of the discharge rate, channel shape, coolant mass flow rate, coolant inlet direction, and dynamic operation strategy on the battery maximum temperature and temperature variation. Even the size and shape optimization of the cooling plate can develop the performance regarding heat dissipation, while the prior knowledge-based method is troublesome in the process of the structure design of the cooling channel.
Topology optimization [
15,
16], a structure design method, offers an effective approach for structure design that satisfies prescribed constraints and boundary conditions, which makes it suitable for the optimal design of the cooling plate [
17,
18]. Dong et al. [
19] proposed a density-based topology optimization framework for thermal cooling device optimization considering the performance of heat resistance, energy dissipation, and pressure drop. Mo et al. [
20] presented a novel cooling plate design for battery thermal management, which was of good performance regarding the trade-off between heat exchange efficiency and pressure drop. Numerical experiments demonstrated that the new design significantly developed the pressure drop and maximum temperature when compared against a traditional cooling plate. Wang et al. [
21] comprehensively studied the effect of different optimization objectives, including heat transfer terms such as heat transfer, outlet fluid enthalpy, and a solid domain temperature multi-objective function consisting of the kinetic energy difference and the heat transfer and their effects on the topology optimization flow channel shape and cooling performance. Liu et al. [
22] presented a novel iterative topology optimization method for a three-dimensional cooling plate design to achieve good performance regarding the temperature difference of lithium-ion batteries. The topology optimization method can effectively optimize the structure for heat dissipation. However, the current study mainly focused on the balance of multi-objectives, wherein the nonuniform distribution of thermal load may lead to unsatisfying results of the structure design by the conventional SIMP-based topology optimization, which lacks attention to deal with most traditional cooling plate designs. Therefore, to decease the maximum temperature and improve the thermal homogeneity of the battery, an appropriate method for structural design to improve the temperature distribution of the battery should be further studied.
To further contribute to this research topic, a density-based topology optimization pipeline is proposed for a cooling plate design with a functionally graded structure to decrease the maximum temperature and improve the thermal homogeneity of the battery. We first employ a Helmholtz-type Partial Differential Equation (PDE)-based filter and smoothed Heaviside projection operator to produce the discrete solution, and another variable-radius Helmholtz PDE filter with a local volume constraint is utilized for the functionally graded structure design. Several examples are conducted to demonstrate the development of the proposed method for cooling plate optimization.
The rest of this paper is organized as follows:
Section 2 presents the theory of topology optimization of the heat–fluid coupling problem and formulates the optimization objective of a cooling plate with functionally graded structure for thermal load management. Then, numerical examples are carried out to demonstrate the effectiveness of the proposed method in
Section 3. Finally, conclusion and future works are summarized in
Section 4.
3. Evaluation and Analysis
In this section, the effectiveness of the proposed method for topology optimization of cooling plate is evaluated, and uniform and nonuniform thermal load cases are utilized to demonstrate the advancement of the functionally graded structure for cooling performance.
Figure 2 shows the layout of the battery and cooling plate. Here, COMSOL Multiphysics 6.0 has been utilized for numerical simulation. The interpolation parameters
were set to 0.005, respectively. The maximum penalty force was set to
, the volume constraint for the fluid domain was set to 0.5, the mesh size and global filter radius were set to 1 mm, respectively, and the projection parameters
. The Method of Moving Asymptotes was introduced as the solver for the numerical simulation. The material property for topology optimization of the cooling plate is organized in
Table 1, where
and
are the density, conduction coefficient, heat capacity at constant pressure, and dynamic viscosity, respectively.
3.1. Numerical Example 1
The first case is to evaluate the proposed method for uniform thermal load management, the design domain and boundary condition are given in
Figure 3, where the length and height of the cooling plate are 120 and 60 mm, respectively, and the inlet and outlet of the rectangular area define the width of the inlet and outlet at 10 mm. The local filter radius
R was set to 2 mm. The velocity of the flow for inlet was set to 0.1 m/s.
To demonstrate the advancement of the optimized structure, the classic straight micro-channel cooling plate is introduced for a comparison, which can be found in
Figure 4. We can observe that the right part of the structure suffered from high temperature, and the temperature of the middle part of the design was comparatively lower due to the different velocities of flow. The maximum temperature approached 365.38 K.
Figure 4b shows the layout of the traditional SIMP-based topology optimization. It was constructed with several macro-runners and numerous micro-channels, and the channel distribution ended up being more complicated and novel when compared with the classical straight channel one. As the topology optimization technology introduced more freedom for the structure design, and the material distribution of the design was comparatively random, the maximum temperature of the optimal structure was lower than the classical straight channel cooling plate, which was 357.85 K.
Figure 4c shows the optimal cooling plate with its lattice structure constructed using the proposed method. It can be found that the optimal design has been constructed by channels with similar widths, the distribution of the channel is even, and fewer slender runners exist when compared with the result of the traditional SIMP-based topology optimization. Thus, the velocity of flow in the structure constructed using the proposed method is consistent, and the maximum temperature of the optimal design came out to 324.12 K.
Table 2 records the average temperature, the maximum temperature of the cooling plate designs, and shows the standard error of the temperature distribution in the cooling plate. It can be observed that the proposed lattice structure performed best among the cooling plate designs, which not only significantly decreased the maximum temperature of the cooling plate but also ensured the thermal homogeneity of the design. The standard error of the temperature distribution decreased from 21.80 to 7.88, which demonstrates that the proposed method is of good performance in the control of the temperature rise and homogeneity of the cooling plate.
3.2. Numerical Example 2
In practical application, a battery may work under nonuniform thermal load conditions. Thus, the second case is introduced to demonstrate the advancement of the functionally graded structure for nonuniform thermal load management. The boundary condition is shown in
Figure 5, where the thermal load linearly increased from the left edge to the right one, and the thermal power of the left and right edges were
W/m
2 and
W/m
2, respectively. The local filter radius varied from 1 mm to 3 mm.
Figure 6a shows the result of the traditional density-based method, as the thermal source is high at the right part of the design domain; thus, the material gathered at the right part for better heat dissipation. However, the slender channels still existed, which caused the velocity of flow to be comparatively slow at these channels; thus, the performance of heat dissipation on this part is not desirable, in which the maximum temperature approached 333.85 K.
Figure 6b shows the result of the proposed method with a constant filter radius. Similarly, the introduction of the local volume constraint avoided the creation of the slender channel. Nevertheless, the local volume constraint prevented the gathering of the material, which caused the thermal load at the right part to not be taken away. The maximum temperature was 327.04 K.
Figure 6c shows the result of the proposed method with variable filter radius values, in which the lower and upper boundaries were set to
r and 3
r, respectively. It can be observed that the width of the channel gradually widened from the left to the right side, which acted consistently with the thermal load in the design domain. The maximum temperature of the optimal design further decreased to 324.20 K.
To quantitatively measure the radiation performance of the structures, the average and maximum temperature and the standard error were taken as the indicators for evaluation, which are given in
Table 3. We can find that the result of the functionally graded structure had good performance regarding heat dissipation.
3.3. Discussion
Different ranges of local filter radius will affect the material distribution of the solid phase and fluid flow. Therefore, two cases have been introduced to evaluate the effect of the range on the structural design. The local volume constraint was set to 0.5. The design domain and boundary condition are the same as
Figure 5.
Figure 7a shows that the shape and width of the flow channels were similar when the local filter range was [
r, 2
r], while subplot (b) indicates that the slender runners arose when the influence radius was the range
. When compared with subplot (c) in
Figure 6, we can observe that enlarging the search range of the filter radius enabled the slender flow channel of the cooling plate, and a relatively small range of the filter radius ensured structural homogeneity. Therefore, the utilization of the filtering range should follow the demand of the practical application.
Secondly, a laminar flow model requires a Reynolds number Re = less than the critical value, where , u, are the density, velocity, and viscosity, respectively, and parameter L is feather size of the passage way. Note that the proposed method assumes the stable laminar flow in the channel as the velocity field, and the size of the runner are both small. Turbulence may occur if the boundary conditions or the feature size of the runner for the numerical simulation violates the standard of the laminar flow model.
4. Conclusions and Future Works
This paper proposed a topology optimization of the cooling plate with functionally graded structure for thermal load management to reduce the average temperature and temperature difference of the overall structure. We first utilized the Brinkman penalty force to distinguish the solid and fluid phases, and then the local volume constraint with a variable filtering radius was introduced to obtain functionally graded structure for thermal load management. The measurement indicators, including the maximum temperature and the standard error of the temperature distribution, were introduced to evaluate the radiation performance of the cooling plate. Numerical experiments, including uniform and nonuniform thermal loads, were carried out to demonstrate the effectiveness of the proposed method for thermal management, which show that the designed cooling plate with a functionally graded lattice structure significantly improved the measurement indicators when compared with a conventional straight mini-channel cooling plate.
The proposed method is not limited to cooling plate design at a small scale for battery thermal management, but it can be extended to structural design in a large scale such as thermal management of electronic vehicles. Further works will consider manufacturing-constrained topology optimization for cooling plate design, especially the length scale and manufacturing error.