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
Tmax 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
βm 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
βm 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
Kc. 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
Kc 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.