Thermal Performance and Geometric Optimization of Fractal T-Shaped Highly-Conductive Material for Cooling of a Rectangular Chip

Abstract

To improve the thermal performance of inserted highly-conductive material (HCM) for the cooling of a chip, the present work numerically investigates the effects of various geometric and structural parameters of a fractal T-shaped branched HCM on the maximum temperature of the chip. These parameters include the length ratios of branches at two consecutive branching levels α, the width ratio of branches at two consecutive branching levels β, the maximum branching level m, the length of the branch at the initial level L₀, the thickness of the HCM H, and the total volume of the HCM V. The results indicate that the maximum temperature of the chip first drops and then rises with the increase of β, which means the existence of the optimal geometric structure of the branched HCM for the cooling of the chip. In addition, the maximum temperature of the chip decreases with the increase of m and V, decreases with the decrease of H, while first drops and then rises with the increase of α and L₀. Further, the present work investigates the effects of the thermal conductivity ratio of HCM and chip γ on the optimal width ratio β_m of the branched HCM with a different length ratio α, maximum branching level m, length of the branch at the initial level L₀, thickness H, total volume V, and thermal conductivity of the rectangular chip K_c. It was found that β_m increases with the increase of γ and V, and decreases with the increase of α, L₀, and H. The present finding is beneficial to the improvement of the thermal performance of the inserted HCM via geometric optimization.

1. Introduction

Due to the increasing degree of integration of electronic products, electronic devices are suffering from more challenges involving the dramatic increase in heat flux. Searching for a wide spectrum of optimal architectures of cooling routes in heat-generating systems is of paramount interest nowadays, such as electronic components, where heat is generated within the body or induced into the outer surfaces of the piece. Thus, various optimization works have been carried out to improve the heat dissipation of electronic products to increase the lifetime of electronic devices [1,2]. Considering the dependence of the liquid cooling technique on the external driving source and consequent system complexity, the heat conduction cooling system can become an essential and straightforward alternative for the cooling of the electronic component, for which the heat generation can be released by inserting a suitable highly-conductive material (HCM) with explicit structural design into the electronic chip.

The challenge that comes with designing the highly-conductive inserts is to optimize the architecture of the paths for heat removal from a low-conductive domain (electronic component). To improve the thermal performance of the inserted HCM, Bejan [3] proposed to insert a fixed volume of tree-like HCM into the low thermal conductivity material with fixed geometric parameters to achieve the purpose of collecting heat and optimized the HCM to achieve the optimal distribution pathway. Following Bejan’s research, many scholars have conducted extensive studies and optimizations of heat conduction models and further obtained considerable model structure designs with better heat dissipation performances [4,5,6,7,8,9,10,11,12,13,14,15]. For example, Hajmohammadi et al. [12] optimized the geometric parameters of two types of fork-shaped highly-conductive pathways under a fixed volume and found that the peak temperatures of the heat generator with two types of fork-shaped pathways could be reduced by 41% and 46%. Hajmohammadi et al. [13] embedded ‘phi’ and ‘psi’-shaped highly-conductive pathways into the heat generator and found that the peak temperature of the heat generator inserted in these highly-conductive pathways was 50% below the heat generator with x-shaped pathways. Feng et al. [14] optimized the structure of the highly-conductive pathways with single and multilevel ‘+’ shapes in the square heat generator and found that the peak temperature of the square heat generator with the highly-conductive pathway with multilevel ‘+’ shape was 75.79% lower than the pathway with single ‘+’ shape. Mazloomi et al. [15] studied the conductive cooling of a rectangular chip heated from the bottom surface. They found that using side branches and increasing the thickness of the main channel could decrease the maximum temperature of the rectangular chip considerably.

Natural branches, such as leaf vein branches, tree branches, and river veins, play an essential role in transporting matter and energy. People are inspired by the branched structures of nature and apply the branch of nature to the cooling system of electronic products [16,17,18,19,20,21,22,23]. Due to the lack of reliable design standards for the design of heat conduction networks, Liu et al. [19] innovatively put forward a criterion to simulate the process of leaf vein generation and establish a generating design by combining the standard with a flexible optimization framework. Li et al. [20] presented a growth simulation method named the ‘conductivity spreading approach’ to optimize the cooling geometry inside the heat transfer system. Hajmohammadi et al. [21] investigated the optimal distribution configuration of the branched HCM with unequal branches by changing the length of the branch and found that the heat dissipation performance of the tree-like HCM with unequal branches was better than the HCM with equal branches under the fixed volume of the HCM and the number of branches. Xu et al. [22] investigated the thermal conduction law of the branched network with a circular cross-section. They obtained the result that the optimal diameter ratio of the branch makes the network achieve the highest heat transfer rate. Liu and Jing [23] numerically analyzed the influences of geometrical and structural parameters on the total thermal resistance of tree-like networks with three kinds of cross-sectional shapes (circular, rectangle, and triangle) under a fixed volume, and verified the law that the value of total thermal resistance of the tree-like heat conduction network would reach the smallest when the ratio of the cross-section area of the daughter branch and parent branch reaches the inverse of the branching number.

To further understand the thermal performance of inserted fractal tree-like HCM for the cooling of electronic chips and find the optimal structure of fractal tree-like HCM with the best cooling capability, this paper investigates the influences of various geometric and structural parameters of the T-shaped branched HCM on the maximum temperature of the chip and the influences of diverse parameters of the T-shaped branched HCM on its optimal geometric structure. The object of the work is to improve the heat dissipation performance of the HCM through geometric optimization.

2. Numerical Methods

2.1. Description of the Simulation Model

The schematic of a simulation model applied in the present work is displayed in Figure 1, which can be considered a simplified model of a heat-conduction-based cooling plate inserted with T-shaped branched HCM. The branching level of the branched HCM is 3 and the branching number at each level is 2 for the example given in Figure 1. In this simulation model, a T-shaped branched HCM with thermal conductivity of K_h is inserted into a rectangular chip with thermal conductivity of K_c. The rectangular chip can be considered a simplified model of the thermal spreader. The dimensions of the rectangular chip are L_c × W_c × H_c = 18 × 22 × 1.5 mm³. At the same time, a small heat sink with the same material as the inserted HCM is added to the outside of the chip to collect the heat from the chip through the transfer of the branched HCM. The dimensions of the heat sink are fixed at L_s × W_s × H_s = 7.5 × 0.8 × 1 mm³. The cross-section of each branch at any branching level is a rectangle with uniform dimensions. The length ratio and the width ratio of two branches at any two consecutive branching levels of the T-shaped branched HCM keep constant. The thickness of the whole branched HCM also keeps uniform. The main target of the present work is investigating the effects of various geometric and structural parameters of the T-shaped branched HCM and the thermal conductivity ratio of the HCM to the chip γ = K_h/K_c on the maximum temperature of the chip and finding the optimal geometric parameter of the T-shaped branched material with the best thermal performance for the cooling of the chip. The geometric and structural parameters of the HCM considered in the present work are as follows:

(1)

α = L_i+1/L_i (i = 0,1,2,3,…). The α is the length ratio of branches at two consecutive branching levels.

(2)

β = W_i+1/W_i (i = 0,1,2,3,…). The β is the width ratio of branches at two consecutive branching levels.

(3)

m. The m is the maximum branching level.

(4)

L₀. The L₀ is the length of the branch at the initial level.

(5)

H. The H is the thickness of the HCM.

(6)

V. The V is the total volume of the HCM.

In this study, the geometric and structural parameters α, β, m, L₀, and H of the highly-conductive material will be adjusted under several fixed volume V to study their effects on the maximum temperature of the chip. The detailed geometric and structural parameters employed in the present work are listed in

Table 1. Geometric parameters of branched HCM with different α.

m = 3, H = 0.8 mm, V = 50 mm3
α	β	L0 (mm)	L1 (mm)	L2 (mm)	L3 (mm)
0.65	0.2–0.6	9	5.85	3.80	2.47
0.70		9	6.30	4.41	3.09
0.75		9	6.75	5.06	3.79

,

Table 2. Geometric parameters of branched HCM with different m.

α = 0.65, H = 0.8 mm, V = 50 mm3
m	β	L0 (mm)	L1 (mm)	L2 (mm)	L3 (mm)	L4 (mm)
2	0.2–0.6	9	5.85	3.80	/	/
3		9	5.85	3.80	2.47	/
4		9	5.85	3.80	2.47	1.61

,

Table 3. Geometric parameters of branched HCM with different L₀.

α = 0.65, m = 3, H = 0.8 mm, V = 50 mm3
L0 (mm)	β	L1 (mm)	L2 (mm)	L3 (mm)
8	0.2–1.2	5.20	3.38	2.19
9		5.85	3.80	2.47
10		6.50	4.22	2.74

,

Table 4. Geometric parameters of branched HCM with different H.

α = 0.65, m = 3, V = 50 mm3
H (mm)	β	L0 (mm)	L1 (mm)	L2 (mm)	L3 (mm)
0.6	0.2–0.6	9	5.85	3.80	2.47
0.8
1

and

Table 5. Geometric parameters of branched HCM with different V.

α = 0.65, m = 3, H = 0.8 mm
V (mm3)	β	L0 (mm)	L1 (mm)	L2 (mm)	L3 (mm)
40	0.2–0.6	9	5.85	3.80	2.47
50
60

. Utilizing the parameters listed in these Tables, the width of each branch at different branching levels can be given as,

$$W_i=\frac{V(1-2\alpha \beta )}{[1-{\left(2\alpha \beta \right)}^{m+1}{]\mathit{HL}}_0}{\beta }^i$$

(1)

2.2. Numerical Method

Using the above geometric model, a uniform and constant heat flux density q = 20,000 W/m² is applied to the upper surface of the chip, and the outer surface of the heat sink is set to be the minimum temperature of T_min = 0 °C, as shown in Figure 1. Furthermore, the other surfaces are set to be adiabatic. Since the thermal conductivity of the HCM is larger than that of the rectangular chip, the HCM will collect the heat applied on the chip and transfer it to the outer surface of the heat sink with a minimum temperature.

To simplify the numerical analysis in the study, the following assumptions are put forward for analysis:

(1)

The heat conduction is steady state.

(2)

The physical properties of the solid materials used in the study are not affected by temperature, and the heat exchange and radiation between the chip and the environment are ignored.

Based on the above assumptions, the heat transfer problem in the present work can be solved by the following governing equation,

$$K_i{\nabla }^2{T}_i=0 i=h,c$$

(2)

where K_i is the thermal conductivity of the solid and T_i is the temperature of the solid. In the present work, γ = K_h/K_c under constant K_c = 10 W/(m·K) will be adjusted to study its effect on the dimensional optimization of branched material with the best thermal performance for the cooling of the chip.

Based on the governing equation and boundary conditions listed above, the Multiphysics simulation software COMSOL is used to solve the problem.

2.3. Mesh Test and Data Validation

To ensure the accuracy of the numerical study, the mesh sensitivity analysis is carried out. In this study, the model with α = 0.65, β = 0.5, L₀ = 9 mm, H = 0.8 mm, V = 50 mm³, m = 3, N = 2, and γ = 100 was selected for mesh sensitivity analysis, and

Table 6. The mesh independence test.

Test Number	Mesh Number	Tmax [°C]	$\frac{\left|{\mathit{T}}_{\mathit{max}}^{\mathit{i}+1}-{\mathit{T}}_{\mathit{max}}^{\mathit{i}}\right|}{{\mathit{T}}_{\mathit{max}}^{\mathit{i}}}$
1	3809	55.7806
2	10,351	56.0943	5.6238 × 10−3
3	16,959	56.1542	1.0678 × 10−3
4	25,931	56.2084	9.6520 × 10−4
5	45,639	56.2407	5.7465 × 10−4

presents the results of the mesh sensitivity analysis. It can be found that the result of the fourth group of grid settings is accurate enough for the numerical simulation, thus, the fourth group of grids is used for the grid setting of all the simulations with different parameter setups.

In addition to the mesh sensitivity test to ensure the accuracy of the numerical calculation, the data comparison in literature [24] is also carried out to keep the correctness of the present numerical method. In the literature, a highly-conductive rectangular material is inserted in a rectangular chip to numerically study its effect on the maximum temperature of the chip. Using the model setup in the literature, the data comparison of the maximum temperature calculated by the present method with that in the literature is given in Figure 2. It can be found that the results obtained by the method used in the present work are consistent with the results in the literature. This shows that the method used in the present work is feasible.

3. Results and Discussion

3.1. The Thermal Performances of the Branched HCM

After testing the feasibility of the method and the accuracy of the numerical results, the effects of the length ratio of branches at two consecutive branching levels α, the width ratio of branches at two consecutive branching levels β, the maximum branching level m, the length of the branch at the initial level L₀, the thickness of the HCM H and the total volume of the HCM V on the maximum temperature T_max of the chip are illustrated and discussed in this section. As the results show in Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7, the maximum temperature of the chip always first drops and then rises with the increase of β. It can be inferred that the HCM has an optimal width ratio β_m to minimize the maximum temperature of the chip. This result is in agreement with the previous studies on the heat conduction of tree-like networks [22,23,25,26,27].

Furthermore, from Figure 3a, it can be found that the effect of α on the maximum temperature of the chip is a nonmonotonic change, and the maximum temperature of the chip shows the transition from drop to rise with the increase of α, which indicates there might be an optimal length ratio to obtain the minimum value of the maximum temperature of the chip. To prove this, Figure 3b gives the effects of α of the T-shaped branched HCM with different β on the maximum temperature of the chip. It can be found that there is an optimal length ratio to obtain the minimum value of the maximum temperature of the chip.

In addition, it can be found from Figure 4a that the effect of L₀ on the maximum temperature of the chip is also a nonmonotonic change, and the maximum temperature of the chip first decreases and then increases with L₀, which means there might be an optimal length of the branch at the initial level to obtain the minimum value of the maximum temperature of the chip, which can be proved by the results shown in Figure 4b.

Regarding the effect of m on the maximum temperature of the chip, it can be found from Figure 5 that the maximum temperature declines gradually with the increase of m = 2, 3, and 4. In this paper, the maximum branching level is chosen to be 4, this is because the further increase of m may lead to the crossing of the branches, as discussed in the previous study [28], and destroy the structure of the fractal branched HCM. When m increases from 2 to 4, the minimum value of the maximum temperature of the chip reduces from 332.75 K to 324.15 K, which indicates that the branched HCM with a large branching level displays better heat dissipation performance. The main reason is that the heat exchange area between the HCM and the chip will become larger with the increase of m, thereby promoting heat dissipation performance and further reducing the maximum temperature of the chip.

In Figure 6, the effect of β on the maximum temperature of the chip was studied by adjusting the fixed volume V of the HCM under fixed parameters of α, m, L₀, and H. It is found that the increase of V is beneficial to enhance the heat dissipation performance of the branched HCM. The curve of the maximum temperature gradually declines with the increase of V. The main reason is that the heat exchange area will also increase with the increase of V, which further promotes heat dissipation performance and reduces the maximum temperature of the chip. This result is consistent with previous studies [26].

In Figure 7, the effect of β on the maximum temperature of the chip was studied by adjusting H at fixed parameters of α, m, L₀, and V. It can be found that the maximum temperature of the chip will gradually decline with the decrease of the thickness of the HCM, which means that reducing the thickness of the HCM can improve the heat dissipation performance of the HCM, which is also due to the increasing heat exchange area with the decrease of the thickness of the HCM.

3.2. The Optimal Width Ratio of the Branched HCM

According to the above analysis, there exists an optimal width ratio β_m to make the thermal performance of the branched HCM obtain the best; thus, it might be interesting and significant to investigate the optimal width ratio. Correspondingly, Figure 8 shows the effects of the thermal conductivity ratio γ on the optimal width ratio of the branched HCM with a different length ratio α, the maximum branching level m, the length of the branch at the initial level L₀, the thickness H, the total volume V and the thermal conductivity of the rectangular chip K_c. It shows that the optimal width ratio of the branched HCM always rises with the increase of γ, which means with the increased conductivity of the inserted HCM, the branched HCM with wider terminal branches is preferred. In addition, the results are apparent in Figure 8 that the optimal width ratio decreases with the increase of α, L₀, and H while increasing with the increase of V. Further, it can be found that the effects of m and K_c on the optimal width ratio are nonmonotonic change, which still needs further studies. At the same time, it is found that the optimal width ratio of the branched HCM obtained in the present work is not consistent with the findings of the previous literature [22,23] regarding the optimal structure of a fractal heat-conduction network with a single material. For the fractal heat-conduction network with a single material, its optimal width ratio is always equal to the reciprocal of the branching number at each level when the cross-section shape is a rectangle with uniform height. However, when the branched HCM is inserted in the rectangular chip with lower heat conductivity, the dual-materials system increases the complexity of optimization design and leads to the discrepancy with the single-material system.

4. Conclusions

This paper proposed a rectangular chip inserted with branched HCM, and numerically investigated the influences of different geometric and structural parameters of the branched HCM, including the length ratio of branches at two consecutive branching levels α, width ratio of branches at two consecutive branching levels β, maximum branching level m, length of the branch at the initial level L₀ and thickness of the HCM H on the maximum temperature of the chip under several fixed volume V. The effects of the thermal conductivity ratio γ on the optimal width ratio β_m of the branched HCM with a different length ratio α, the maximum branching level m, the length of the branch at the initial level L₀, the thickness H, the total volume V, and the thermal conductivity of the rectangular chip K_c are also studied.

The present work found that the maximum temperature of the chip first drops and then rises with the increase of β, which means there is an optimal geometric structure for the branched HCM to achieve the best thermal performance. In addition, the maximum temperature of the chip decreases with the increase of m and V and decreases with the decrease of H, while first drops and then rises with the increase of α and L₀. For the optimal geometric structure of HCM, it obtains the result that the optimal width ratio β_m of the branched HCM always increases with the increase of γ and V, while decreasing with the increase of α, L₀, and H. However, the effects of m and K_c on the optimal width ratio still need further study due to the nonmonotonic variation of β_m. The present finding provides a potential method to improve the cooling performance of the inserted HCM via explicit geometric optimization, which can be applied to the cooling of the microelectronic chip, LED lamp, and lithium battery. In addition, this work focus on the thermal performance of symmetric branched highly-conductive material inserted in the low-conductive material for cooling of uniform heat distribution. However, an asymmetrical branched highly-conductive material structure may be better for the cooling of the non-uniform heat and hotspot distribution, and its thermal performance and optimization design still need deep study.