Topology Optimization of Functionally Graded Structure for Thermal Management of Cooling Plate

Abstract

The fast charge and discharge of a battery will significantly increase the overall temperature and thermal difference of the battery, which will further affect the working performance and safety of the battery. Therefore, a heat–fluid coupling topology optimization pipeline for developing radiation performance of the cooling plate is presented to ensure the thermal homogeneity of the battery in this paper. First, the Brinkman penalty model is utilized to construct the solid and fluid structures. Then, a local volume constraint is introduced to create the lattice structure to reduce the temperature difference of the cooling plate. Furthermore, a functionally graded lattice structure via a variable influence radius is presented to improve the radiation performance of the cooling plate when the thermal load is uneven. Numerical experiments are carried out to evaluate the performance of the presented methods on the optimization of the cooling plate, which indicates that the designed cooling plate by the proposed method improves the radiation performance when compared against a traditional straight channel and a SIMP-based optimal design.

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.

2. Methodology

2.1. Heat–Fluid Coupling

The internal heat transfer process in the cooling plate includes heat conduction in the solid region and convective heat transfer in the fluid region. The convective heat transfer part is further influenced by the fluid flow. Therefore, it is necessary to analyze the conjugate heat transfer process involving both heat conduction and convective heat transfer, as well as the fluid flow process simultaneously.

2.1.1. Fluid Flow

Generally, the coolant flowing within the cooling plate exhibits low speed and a stable laminar flow. Consequently, the incompressible laminar Navier–Stokes equations are employed to describe the flow. To distinguish between the fluid and solid regions, the Brinkman penalty force is added to the Navier–Stokes equations:

$${\rho }_f(u·\nabla )u=\nabla (-pI+\mu (\nabla u+{(\nabla u)}^T))-\alpha u$$

(1)

$$\nabla ·u=0$$

(2)

Equations (1) and (2) describe the law of the conservation of momentum and mass, respectively, where ${\rho }_f,u,p,\mu ,$ and $\alpha$ define the fluid density, velocity, pressure, dynamic viscosity, and Brinkman penalty factor, respectively. The Brinkman penalty strategy distinguishes a solid and a fluid through the introduction of a virtual body force $\alpha u$, where the fluid velocity approaches 0 when the body force is at a large value, and the corresponding part is the solid phase.

2.1.2. Heat Transfer

The heat transfer process in the cooling plate involves thermal conductivity in the solid structure and fluid flow, respectively. According to Fourier transform theory, the governing equation for the thermal conductivity in a solid structure can be constructed as follows:

$${\mathrm{k}}_s{\nabla }^{2}T+Q=0$$

(3)

where $k_s$ is the thermal conductivity in the solid phase, T is the temperature, and Q is the internal thermal source.

The governing equation for the thermal conductivity in the fluid phase is given as follows:

$${\rho }_f{C}_{Pf}u·\nabla T=k_f{\nabla }^{2}T+Q$$

(4)

where $C_{pf}$ and $k_f$ are the fluid-specific heat capacity and the conduction coefficient in fluid. Combining Equations (3) and (4), we can obtain a unified governing equation for the thermal conductivity, which can be formulated as follows:

$$\overline{\varphi }{\rho }_f{C}_p\left(\overline{\varphi }\right)u·\nabla T=k\left(\overline{\varphi }\right){\nabla }^{2}T+Q$$

(5)

where $\overline{\varphi }$ is the physical density of the cooling plate, in which 0 and 1 indicate the solid and fluid phases, respectively.

2.2. Material Interpolation

A conventional interpolation function for solid structural topology optimization is numerically stable in fluid problems; thus, the interpolation model for a heat–fluid coupling optimization problem is introduced as follows:

$$k=k_s+(k_f-k_s){\displaystyle \frac{\overline{\varphi }(1+q_k)}{(\overline{\varphi }+q_k)}}$$

(6)

$$\rho ={\rho }_s+({\rho }_f-{\rho }_s){\displaystyle \frac{\overline{\varphi }(1+q_{\rho })}{(\overline{\varphi }+q_{\rho })}}$$

(7)

$$C_p={C_p}_s+({C_p}_f-{C_p}_s){\displaystyle \frac{\overline{\varphi }(1+q_c)}{(\overline{\varphi }+q_c)}}$$

(8)

where $q_k,q_{\rho },q_c$ are the penalty factors for the conduction coefficient, density, and heat capacity at a constant pressure. Furthermore, the interpolation function for the Brinkman penalty force is given in Equation (9):

$$\alpha ={\alpha }_{max}+(0-{\alpha }_{max}){\displaystyle \frac{\overline{\varphi }(1+q_{\alpha })}{(\overline{\varphi }+q_{\alpha })}}$$

(9)

where ${\alpha }_{max}$ is an upper limit for the Brinkman penalty factor.

2.3. Filtering and Projection

To avoid the phenomenon of mesh dependency and check-broad problems and to limit the minimum length scale of the optimized structure, the Helmholtz filtering operator is utilized to deal with a such situation, which is constructed as Equation (10):

$$-r^2{\nabla }^2\tilde{\varphi }+\tilde{\varphi }=\varphi$$

(10)

where r is the influence factor that can adjust the feature size of the structure, and $phi,\tilde{\varphi }$ are the original and filtered design variables, respectively.

Although the filtering operation can relieve the phenomenon of check-broad problems, it introduces numerous gray elements during optimization. Thus, the projection function constructed in Equation (11) is utilized to clear the layout of the topology structure, where $\overline{\varphi }$ is the projected design variable, namely, physical density. The parameter $\beta$ controls the sharpness of the projection function, and $\eta$ is the projection threshold, which is usually set to 0.5.

$$\overline{\varphi }={\displaystyle \frac{tanh\left(\beta (\tilde{\varphi }-\eta )\right)+tanh\left(\beta \eta \right)}{tanh\left(\beta \right(1-\eta \left)\right)+tanh\left(\beta \eta \right)}}$$

(11)

2.4. Objective for Lattice Structure

A reasonable layout of the runner can reduce the temperature difference of the cooling plate and develop the radiation performance. Therefore, a type of lattice structure using local volume constraint is utilized to decrease the temperature difference of each local area for the plate structure. Similarly, Helmholtz filtering is used to smooth the element density within the influence circle with a radius R:

$$-R^2{\nabla }^2\widehat{\varphi }+\widehat{\varphi }=\overline{\varphi }$$

(12)

A local volume constraint is defined by Equation (13) to create the lattice structure:

$$\underset{\forall e}{max}\left(\widehat{\varphi }\right)\le V_l$$

(13)

Equation (13) is computationally expensive for the optimization. Thus, a p-norm strategy is utilized to approximate Equation (13), which can be constructed as follows:

$$\underset{\forall e}{max}\left(\widehat{\varphi }\right)\approx {\widehat{\varphi }}_p={\left(\sum _e{\widehat{\varphi }}_e^p\right)}^{{\displaystyle \frac{1}{p}}}$$

(14)

where p is the penalty factor, and the approximation error approaches to zero when p goes to infinity. However, a large value of p leads to numerical instability. Therefore, Equation (13) can be reformulated as follows:

$${\left({\displaystyle \frac{1}{n}}\sum _e{\widehat{\varphi }}_e^p\right)}^{{\displaystyle \frac{1}{p}}}\le V_l$$

(15)

2.5. Optimization Objective

Based on the stated heat–fluid coupling governing equation, interpolation model, and constraint on the lattice structure, the formulation of the topology optimization for the cooling plate can be constructed as follows:

$$\begin{array}{c}\underset{\varphi R}{minimize}c={\displaystyle \frac{\int Td\Omega }{S}}\hfill \\ s.t.\left\{\begin{array}{c}\rho (u·\nabla )u=\nabla (-pI+\mu (\nabla u+{(\nabla u)}^T))-\alpha u\hfill \\ \nabla ·u=0\hfill \\ \overline{\varphi }{\rho }_f{C}_p\left(\overline{\varphi }\right)u·\nabla T=k\left(\overline{\varphi }\right){\nabla }^{2}T+Q\hfill \\ -R^2{\nabla }^2\widehat{\varphi }+\widehat{\varphi }=\overline{\varphi }\hfill \\ {\mathrm{g}}_k\left({\widehat{\varphi }}_e\right)={\left({\displaystyle \frac{1}{n}}\sum {\widehat{\varphi }}_e^p\right)}^{{\displaystyle \frac{1}{p}}}\le V_l\hfill \\ V\left(\overline{\varphi }\right)\le V\hfill \\ R_{min}\le R\le R_{max}\hfill \\ 0<\varphi <1\hfill \end{array}\right.\hfill \end{array}$$

(16)

where R is set as an optimization variable, of which the lower and upper boundaries are $R_{min}$ and $R_{max}$, respectively. A constant influence radius leads to a uniform lattice structure, which is suitable for the boundary condition with an even thermal load. However, nonuniform thermal load commonly appears in practice, previous methods cannot cope with such situations well. Therefore, a changeable control variable R is introduced to create a functionally graded lattice structure to improve the radiation performance of the optimal structure.

Figure 1 shows the proposed topology optimization pipeline for the cooling plate design. It starts to initialize the design variable to the design domain and filters the design variable using Equation (10), and it projects the smoothed density to 0–1 using Equation (11). Then, we interpolate the material parameters via Equations (6)–(9) and solve the objective function shown in Equation (16) and the constraint in Equation (15). Finally, we conduct the sensitivity analysis and update the design variables via a numerical solver.

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 $q_k,q_{\rho },q_{\alpha },q_c$ were set to 0.005, respectively. The maximum penalty force was set to ${10}^7$, 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 $\eta =0.5,\beta =4$. 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. Material property for numerical simulation.

	$\mathit{\rho }$ (kg/m3)	k (W/(m·K))	${\mathit{C}}_{\mathit{p}}$ (J/(kg·K))	$\mathit{\nu }$ (Pa·s)
solid	2700	217	880	-
fluid	1000	0.6	4200	${10}^{-3}$

, where $\rho ,k,C_p,$ and $\nu$ 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. Comparison of temperature responses of various cooling plate designs (unit: K).

	Average Temperature	Maximum Temperature	Standard Error
Straight channel	326.81	365.38	21.80
SIMP	311.38	357.85	14.16
Proposed method	307.13	324.12	7.88

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 $5\times {10}^6$ W/m² and $1.1\times {10}^7$ W/m², 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 3r, 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. Comparison of temperature response of various cooling plate designs (unit: K).

	Average Temperature	Maximum Temperature	Standard Error
SIMP	305.9	333.85	8.82
Lattice structure	304.49	327.04	8.45
Proposed method	303.18	324.07	7.13

. 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, 2r], while subplot (b) indicates that the slender runners arose when the influence radius was the range $[r,4r]$. 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 = $\rho uL/\nu$ less than the critical value, where $\rho$, u, $\nu$ 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.