Investigation of Flow and Heat Transfer Performance of Double-Layer Pin-Fin Manifold Microchannel Heat Sinks

: The manifold microchannel (MMC) heat sink is characterized by high heat transfer efﬁ-ciency, high compactness, and low ﬂow resistance. It can be an effective method for the high-ﬂux removal of high-power electronic components. To further enhance the performance of the MMC, a double-layer pin–ﬁn MMC structure was designed. The thermodynamic properties, including the ﬂow and heat transfer characteristics, were numerically investigated using ANSYS Fluent with deionized water as the working liquid. Compared with the single-layer MMC, the temperature uniformity is better, the pressure drop is lower, and the comprehensive performance is improved at the cost of slightly larger thermal resistance for the double-layer MMC. The geometric effects on the thermodynamic performance were also analyzed. The results show that among the pin–ﬁn structures with round, diamond-shaped, and rectangular cross-sections, the round pin–ﬁns demonstrate the best comprehensive performance and the minimal thermal resistance. Under the same inlet velocity, the thermal resistance is decreased, and the comprehensive performance is ﬁrst increased and then decreased as the pin–ﬁn size increases. In addition, it is recommended to adopt a larger height ratio for low inlet velocity and a smaller height ratio for high inlet velocity.


Introduction
With the rapid development of micro-electro-mechanical systems, microchannel structures have been widely applied in chemical and biological engineering, microelectronics, aerospace, and especially high-power chip cooling [1][2][3][4]. According to Moore's Law, the transistors accommodated on an integrated circuit double about every 18 months [5], and the chip's performance also doubles. Meanwhile, the heat flux of electronic chips also increased sharply. The traditional cooling technology cannot meet the requirements for heat dissipation, which limits the further improvement of the chip's performance [6]. Harpole and Eninger et al. [7] proposed the manifold microchannel (MMC), which has excellent thermal performance. It is expected to become an effective high-power chip cooling method.
Traditional microchannel heat sinks have several disadvantages, including poor temperature uniformity and high pressure drop. Therefore, the study of novel structures and patterns of flow channels has been conducted to enhance the thermal performance of traditional microchannel heat sinks. Li et al. [8] conducted an experimental survey of zigzag microchannels. They found that the zigzag channel has the minimum pressure drop and the best thermal performance at an attack angle of 30 • . Zhai et al. [9] proposed a fan-shaped microchannel and concluded that its thermal performance is better than that of rectangular microchannels. Xu et al. [10] performed a simulation study on a tree-shaped microchannel heat sink. It was found that the tree-shaped structure has better temperature uniformity, and the cooling effect of the seven-level tree-shaped microchannel has the best minimal pressure drop. Li et al. [11] conducted an experimental research study on a bidirectional counter-flow microchannel heat sink. The results display that its heat transfer capacity was better and the pressure drop was reduced compared to the rectangular microchannel.
Some other researchers proposed a pin-fin structure to replace the traditional straight microchannel. The heat transfer capacity can be improved due to the larger heat transfer area and enhanced flow turbulence. Current research studies have focused on the arrangement and geometric parameters of pin-fin as they are critical parameters affecting the flow and thermal performance. Alam et al. [12] proposed a triangular micro-pin-fin structure, which significantly improved the heat sink's heat dissipation capacity. Yang et al. [13] compared the heat transfer characteristics of heat sinks with five micro-pin-fin structures through a combined simulation and experimental study. It showed that the cylindrical micro-pin-fin structure demonstrates a minimum pressure drop. Xia et al. [14] proposed a rhomboid-shaped micro-pin-fin design. The experimental results show that the thermal performance of this structure is better compared with the cylindrical structure at high Re. Ambreen et al. [15] investigated the effect of pin-fin shapes on heat transfer performance and concluded that the round pin-fins have the best thermal performance.
Double-layer microchannels proposed by Vafai et al. [16] have attracted the interest of many researchers. Many studies have been engaged in improved structures for heat transfer enhancement. The thermal performance of single-layer and double-layer heat sinks was compared by Hu et al. [17], and the authors claimed that the double-layer structure shows better temperature uniformity. Lin et al. [18] investigated the thermal transfer capacity of a double-layer microchannel structure under a heat flux of 100 W/cm 2 and a pump power of 0.2 W. Compared with a single-layer structure, the thermal resistance decreased by about 11% and the temperature difference was lowered from 4.6 K to 0.5 K. Shen et al. [19] proposed a novel X-structured double-layer microchannel heat sink. It was found that this structure displays better temperature uniformity compared with the traditional double-layer microchannel heat sinks. Chen et al. [20] conducted a simulation study on a double-layer trapezoidal microchannel heat sink with a pyramidal perturbation structure. They found that the novel design demonstrates the best thermal performance when the bottom-to-height ratio of the perturbation structure is 0.6 and the spacing is 300 µm. Zhang et al. [21] proposed a double-layer microchannel with fins of different thicknesses in the upper and lower layers. The results found that the heat transfer performance is improved and the pressure drop is reduced for thicker fins in the lower layer.
Recently, the MMC heat sink has become a research hotspot due to its excellent thermal performance. The MMC structure comprises a manifold divider layer and a microchannel layer, as illustrated in Figure 1. The upper manifold divider layer provides numerous inlet and outlet channels, which are alternately distributed in the length direction of the bottom microchannel. After flowing through the inlet of the manifold structure, the liquid ejects downwards to the bottom surface. Then, the liquid turns direction and flows in the parallel microchannels to dissipate the heat. Finally, the liquid flows upward at the outlet of the manifold divider layer. The manifold structure of the MMC divides the entire long microchannels into many short microchannel units, which reduces the effective length of the flow channel, thereby reducing the pressure drop of the overall structure. Meanwhile, the thermal boundary layer is difficult to fully develop due to a reduction in the effective flow length and the multi-inlet jet effect, leading to a better heat thermal performance. Most of the studies on the MMC focus on structural assembly and optimization. Pan et al. [22] numerically studied the geometric structure effect on the MMC performance. It was found that there is an optimal aspect ratio, which demonstrates the most efficient heat transfer performance. Gan et al. [23] performed a simulation investigation on a novel orifice MMC sink. They found that the temperature uniformity of the novel orifice MMC structure has been dramatically improved. Yang et al. [24] proposed a MMC structure with secondary oblique channels. The optimal structure was obtained through a genetic algorithm, and the numerical simulation results show that this structure has better thermal performance than the traditional MMC. Gilmore et al. [25] proposed an open MMC structure. A smaller pressure drop was obtained at the cost of lower heat transfer performance compared to the traditional MMC. Erp et al. [26] directly etched the MMC structure on the silicon-based chips, which decreased the thermal resistance since the contact layer between the heat source and the heat sink was removed. This result showed that the MMC structure has excellent application prospects in chip cooling.
Water 2022, 14, 3140 3 of 16 structure with secondary oblique channels. The optimal structure was obtained through a genetic algorithm, and the numerical simulation results show that this structure has better thermal performance than the traditional MMC. Gilmore et al. [25] proposed an open MMC structure. A smaller pressure drop was obtained at the cost of lower heat transfer performance compared to the traditional MMC. Erp et al. [26] directly etched the MMC structure on the silicon-based chips, which decreased the thermal resistance since the contact layer between the heat source and the heat sink was removed. This result showed that the MMC structure has excellent application prospects in chip cooling. As discussed above, although the pin-fin structure can enhance heat transfer, it also increases the pressure drop. The double-layer heat sinks can reduce the pressure loss at the cost of higher thermal resistance. The MMC heat sink can improve temperature uniformity with a lower pump power. In the current studies, the heat transfer improvement of the MMC is generally achieved at a higher pressure drop penalty. To solve this problem, this work designs a double-layer pin-fin MMC structure by combining the advantages of double-layer and pin-fin structures to enhance heat transfer and improve the temperature uniformity at a lower pressure drop. On this basis, the flow and thermal characteristics of the double-layer pin-fin MMC structure are numerically investigated and the influences of the pin-fin shape, pin-fin size, and height ratio of the upper and lower microchannel layer are discussed. Figure 2 illustrates the geometric scheme of the double-layer pin-fin MMC. The difference from the traditional MMC structure is that it has two microchannel layers, and the discontinuous microchannels in the microchannel layers are formed with a pin-fin structure. As shown in Figure 2, the cold liquid is injected downwards onto the upper and lower microchannel layers after flowing through the manifold inlet. Then, the liquid spreads along the microchannels among the regularly arranged pin-fins for heat exchange. The liquid flow direction in the microchannel layer is shown in Figure 3. Finally, the hot liquid flows upwards and exits from the outlet layer. As discussed above, although the pin-fin structure can enhance heat transfer, it also increases the pressure drop. The double-layer heat sinks can reduce the pressure loss at the cost of higher thermal resistance. The MMC heat sink can improve temperature uniformity with a lower pump power. In the current studies, the heat transfer improvement of the MMC is generally achieved at a higher pressure drop penalty. To solve this problem, this work designs a double-layer pin-fin MMC structure by combining the advantages of double-layer and pin-fin structures to enhance heat transfer and improve the temperature uniformity at a lower pressure drop. On this basis, the flow and thermal characteristics of the double-layer pin-fin MMC structure are numerically investigated and the influences of the pin-fin shape, pin-fin size, and height ratio of the upper and lower microchannel layer are discussed. Figure 2 illustrates the geometric scheme of the double-layer pin-fin MMC. The difference from the traditional MMC structure is that it has two microchannel layers, and the discontinuous microchannels in the microchannel layers are formed with a pin-fin structure. As shown in Figure 2, the cold liquid is injected downwards onto the upper and lower microchannel layers after flowing through the manifold inlet. Then, the liquid spreads along the microchannels among the regularly arranged pin-fins for heat exchange. The liquid flow direction in the microchannel layer is shown in Figure 3. Finally, the hot liquid flows upwards and exits from the outlet layer.

Governing Equations and Boundary Conditions
The governing equations for the liquid region are shown below. Continuity equation: Momentum equation: Energy equation: where cp, μ, ρ, and λ are the specific heat capacity at a constant pressure, the viscosity, the density, and the thermal conductivity of the liquid, respectively. i and j are the tensor index symbols, which are equal to 1, 2, and 3 in Cartesian coordinates.
To conduct numerical simulation, the commercial software Ansys Fluent 2020R2 is adopted, and deionized water is used as the working liquid. The inlet temperature and pressure are set at 293.15 K and 0.1 MPa, respectively. The inlet velocity covers 1.2 m/s to 3.6 m/s. Due to the turbulence caused by numerous pin-fins and the multi-inlet jet effect, there is a possibility of local turbulent flow despite the corresponding Re values calculated for microchannels being lower than the transitional range of 2500 [27]. The reliable k-e

Governing Equations and Boundary Conditions
The governing equations for the liquid region are shown below. Continuity equation: Energy equation: where c p , µ, ρ, and λ are the specific heat capacity at a constant pressure, the viscosity, the density, and the thermal conductivity of the liquid, respectively. i and j are the tensor index symbols, which are equal to 1, 2, and 3 in Cartesian coordinates.
To conduct numerical simulation, the commercial software Ansys Fluent 2020R2 is adopted, and deionized water is used as the working liquid. The inlet temperature and pressure are set at 293.15 K and 0.1 MPa, respectively. The inlet velocity covers 1.2 m/s to 3.6 m/s. Due to the turbulence caused by numerous pin-fins and the multi-inlet jet effect, there is a possibility of local turbulent flow despite the corresponding Re values calculated for microchannels being lower than the transitional range of 2500 [27]. The reliable k-e turbulence model is adopted, and the boundary conditions are set as a velocity inlet and a pressure outlet. A uniform heat source of 100 W/cm 2 is maintained at the bottom boundary. All simulation works are assumed to be performed under steady-state conditions, and the liquid flow is incompressible.

Data Reduction
The effective thermal resistance of the MMC is calculated as: where T s is the average bottom surface temperature of the MMC, T in is the inlet temperature, A is the effective heating area, and q is the heat flux.
The temperature uniformity is assessed by calculating the temperature difference between the maximum and the minimum temperature at the bottom surface of the MMC: where T max and T min are the maximum and the minimum temperatures, respectively. The comprehensive performance index of the MMC can be described by the factor ζ [28]: where j and f are the Colburn factor and the Fanning friction factor, which can be obtained using the following formula: where u, ρ, L, and ∆P are the inlet velocity, the deionized water density, the overall length of the flow channel, and the pressure drop, respectively. The hydraulic diameter of the flow channel D is obtained by the following equation.
where V, p, and S are the volume of the liquid domain, the wetted perimeter, and the wetted area of the flow channel, respectively. The Reynolds number and the Prandtl number can be calculated as follows: where µ, C p , and λ are the dynamic viscosity coefficient, the specific heat capacity at constant pressure, and the thermal conductivity of the deionized water, respectively. The Nusselt number is: where h is the convective heat transfer coefficient: where T av is the average temperature where T in and T out are the inlet and outlet liquid temperatures, respectively.

Grid Independence Verification and Simulation Method Validation
FLUENT meshing is used to generate polyhedral meshes to ensure the accuracy of numerical calculations. The advantage of the polyhedral mesh is that it can capture delicate geometric features, although the number of meshes is significantly reduced compared to the tetrahedral mesh. A higher mesh generation rate compared with ICEM makes FLUENT meshing more suitable for liquid simulation and preprocessing complex geometric shapes. As shown in Figure 4, the fluid domain meshes near the walls are set to five refined boundary layers. Figure 5 shows the grid independence verification results. It can be seen that the Nu and pressure drop barely change as the grid number exceeds 594,758. To save computing resources, the following simulations are performed with the number of grids set at 594,758.
to the tetrahedral mesh. A higher mesh generation rate compared with ICEM makes FLU-ENT meshing more suitable for liquid simulation and preprocessing complex geometric shapes. As shown in Figure 4, the fluid domain meshes near the walls are set to five refined boundary layers. Figure 5 shows the grid independence verification results. It can be seen that the Nu and pressure drop barely change as the grid number exceeds 594,758. To save computing resources, the following simulations are performed with the number of grids set at 594,758.  The numerical simulation method is validated by comparing the calculation results with the experimental data from Drummond [29]. The geometric parameters of the singlelayer MMC and the experimental conditions in Drummond's experiment are displayed in Table 1 and Figure 6, respectively. In the verification experiment, the heat sink material is silicon and the working liquid is HFE7100. The set convergence condition is that the residual error is less than 10 −12 and the convergence time is 1000 steps. The calculation results are compared with the experimental data in Figure 7. Apparently, the calculated average temperature of the heat sink shows the same trend as the experimental data in a singlephase heat transfer, and the maximum deviation is lower than 1%. Therefore, the reliability of the simulation method in this study is validated. to the tetrahedral mesh. A higher mesh generation rate compared with ICEM makes FLU-ENT meshing more suitable for liquid simulation and preprocessing complex geometric shapes. As shown in Figure 4, the fluid domain meshes near the walls are set to five refined boundary layers. Figure 5 shows the grid independence verification results. It can be seen that the Nu and pressure drop barely change as the grid number exceeds 594,758. To save computing resources, the following simulations are performed with the number of grids set at 594,758.  The numerical simulation method is validated by comparing the calculation results with the experimental data from Drummond [29]. The geometric parameters of the singlelayer MMC and the experimental conditions in Drummond's experiment are displayed in Table 1 and Figure 6, respectively. In the verification experiment, the heat sink material is silicon and the working liquid is HFE7100. The set convergence condition is that the residual error is less than 10 −12 and the convergence time is 1000 steps. The calculation results are compared with the experimental data in Figure 7. Apparently, the calculated average temperature of the heat sink shows the same trend as the experimental data in a singlephase heat transfer, and the maximum deviation is lower than 1%. Therefore, the reliability of the simulation method in this study is validated. The numerical simulation method is validated by comparing the calculation results with the experimental data from Drummond [29]. The geometric parameters of the singlelayer MMC and the experimental conditions in Drummond's experiment are displayed in Table 1 and Figure 6, respectively. In the verification experiment, the heat sink material is silicon and the working liquid is HFE7100. The set convergence condition is that the residual error is less than 10 −12 and the convergence time is 1000 steps. The calculation results are compared with the experimental data in Figure 7. Apparently, the calculated average temperature of the heat sink shows the same trend as the experimental data in a single-phase heat transfer, and the maximum deviation is lower than 1%. Therefore, the reliability of the simulation method in this study is validated.  Figure 6. A schematic diagram of single-layer structure from Drummond [29].  inlet width win 500 μm outlet width wout 250 μm divider width wdiv 250 μm mass flow rate G 1300 kg/(m 2 ·s) heat flux q 0-75 W/cm 2 inlet temperature Tin 332 K Figure 6. A schematic diagram of single-layer structure from Drummond [29].

Comparison of the Flow and Heat Transfer Performance of the Double-Layer and Single-Layer Pin-Fin MMC
The flow and thermal performances of the double-layer and single-layer pin-fin MMC are obtained and compared in this section. For both, the height of the microchannel layer is 0.3 mm, and the pin-fin has a round cross-section and a diameter of 0.8 mm. The difference is that the double-layer MMC owns two microchannel layers while the single-layer structure lacks the upper microchannel layer, as depicted in Figure 2. The other detailed working conditions, such as the inlet velocity and temperature, are referred to in Section 2.2. Figure 8 exhibits the thermal resistance, the maximum temperature difference, the pressure drops, and the comprehensive performance of both the double-layer and singlelayer pin-fin MMC. As depicted in Figure 8a, the thermal resistance of the double-layer MMC is 2.1% to 5.7% higher compared with the single-layer heat sink. Since the flow area in the microchannel layer of the double-layer MMC is doubled compared with the single-layer heat sink, the latter has a much higher flow velocity in the microchannel layer under the same total inlet velocity. A higher flow velocity in the single-layer MMC contributes to the multi-inlet jet effect and flow turbulence, which destroy the thermal boundary and enhance the convective heat transfer. Therefore, the single-layer heat sink demonstrates a slightly lower thermal resistance. Figure 8b indicates that the maximum temperature difference of the double-layer heat sink is lower compared with the single-layer structure. This is because the lower microchannel layer can be further cooled by the working liquid in the upper layer for the double-layer MMC, thus achieving better temperature uniformity. In the meantime, the maximum temperature differences for both structures decrease with an increase in the inlet velocity. This is because the higher velocity can improve the convection  Figure 8c shows that the pressure loss of the double-layer heat sink is lower compared with the single-layer structure. Specifically, the pressure drop can be reduced by up to 35.2%. This is because under the same total inlet velocity, the doubled flow area in the double-layer MMC causes lower velocities in each microchannel layer, resulting in a lower pressure drop. Figure 8d shows that the double-layer MMC demonstrates better comprehensive performance compared with the single-layer MMC. This result is synonymous with the fact that the double-layer MMC can sharply reduce the pressure loss at the cost of slightly higher thermal resistance. enhance the convective heat transfer. Therefore, the single-layer heat sink demonstrates a slightly lower thermal resistance. Figure 8b indicates that the maximum temperature difference of the double-layer heat sink is lower compared with the single-layer structure. This is because the lower microchannel layer can be further cooled by the working liquid in the upper layer for the double-layer MMC, thus achieving better temperature uniformity. In the meantime, the maximum temperature differences for both structures decrease with an increase in the inlet velocity. This is because the higher velocity can improve the convection heat transfer in the flow dead zone and decrease the temperature difference of the MMC. Figure 8c shows that the pressure loss of the double-layer heat sink is lower compared with the single-layer structure. Specifically, the pressure drop can be reduced by up to 35.2%. This is because under the same total inlet velocity, the doubled flow area in the double-layer MMC causes lower velocities in each microchannel layer, resulting in a lower pressure drop. Figure 8d shows that the double-layer MMC demonstrates better comprehensive performance compared with the single-layer MMC. This result is synonymous with the fact that the double-layer MMC can sharply reduce the pressure loss at the cost of slightly higher thermal resistance.

Pin-Fin Shape
The flow and thermal characteristics of the double-layer MMC with various pin-fin shapes are analyzed in this section. Figure 9 illustrates the pin-fins with round, diamondshaped, and rectangular cross-sections. For comparing different pin-fin shapes, the circumscribed circle diameters of the diamond and rectangle are set to be the same as that of the round shape.
The flow and thermal characteristics of the double-layer MMC w shapes are analyzed in this section. Figure 9 illustrates the pin-fins with shaped, and rectangular cross-sections. For comparing different pincumscribed circle diameters of the diamond and rectangle are set to be the round shape.  Figure 10 shows the thermal resistance, maximum temperature d drops, and comprehensive performance of the double-layer MMC w shapes mentioned above. As shown in Figure 10a, the thermal resistanc round, diamond-shaped, and rectangular cross-sections decreases with inlet velocity. The convective heat transfer is enhanced as the inlet veloc ing to a decreased average temperature. Therefore, the thermal resis also decreases. Moreover, the thermal resistance of the MMC with the r  Figure 10 shows the thermal resistance, maximum temperature difference, pressure drops, and comprehensive performance of the double-layer MMC with three pin-fin shapes mentioned above. As shown in Figure 10a, the thermal resistance for pin-fins with round, diamond-shaped, and rectangular cross-sections decreases with an increase in the inlet velocity. The convective heat transfer is enhanced as the inlet velocity increases, leading to a decreased average temperature. Therefore, the thermal resistance of the MMC also decreases. Moreover, the thermal resistance of the MMC with the round-shaped pin-fin is lower compared to the other pin-fin shapes. This is because the effective flow channel area among the round pin-fins is smaller, and the flow velocity is faster, leading to a higher convective heat transfer coefficient and a lower thermal resistance for the round pin-fin under the same inlet velocity. Figure 10b shows that the maximum temperature difference for the diamond-shaped pin-fin is higher compared with the others. The velocity contour shown in Figure 11 proves that the local velocity around the diamond-shaped pin-fin is lower, resulting in a poor local heat transfer. Thus, the temperature uniformity for the diamond-shaped pin-fin deteriorates. Figure 10c shows that the pressure drops increase with an increase in the flow velocity for three pin-fin shapes. This is because increasing the flow velocity enhances the jet effect and flow turbulence, causing a higher frictional resistance. However, the pressure drops for different pin-fin shapes hardly change under the inlet velocity. This is because the average flow velocity for three pin-fin shapes is similar. The pin-fin shape almost has no effect on the pressure drops. Figure 10d indicates that the comprehensive performance for all pin-fin shapes increases as the inlet velocity increases. The pin-fin with a round cross-section demonstrates the best comprehensive performance. This is because the convective heat transfer for round pin-fins is enhanced due to a higher local velocity in smaller flow passages formed by the round pin-fins. Meanwhile, the pressure drops caused by different pin-fin shapes are similar, which means that the comprehensive performance of the pin-fin with a round cross-section is the best.
indicates that the comprehensive performance for all pin-fin shapes increases as the inlet velocity increases. The pin-fin with a round cross-section demonstrates the best comprehensive performance. This is because the convective heat transfer for round pin-fins is enhanced due to a higher local velocity in smaller flow passages formed by the round pinfins. Meanwhile, the pressure drops caused by different pin-fin shapes are similar, which means that the comprehensive performance of the pin-fin with a round cross-section is the best.

Pin-Fin Size
This section analyzes the flow and thermal characteristics of the double-layer MMC with various pin-fin sizes. Since the pin-fin with a round cross-section demonstrates the best comprehensive performance, it is employed in this section. The diameter of the round cross-section ranges from 0.4 to 1.1 mm, while the other geometric parameters stay constant. Figure 12 displays the thermal resistance, the maximum temperature difference, the pressure drops, and the comprehensive performance of the double-layer MMC with various pin-fin sizes. Figure 12a shows that despite different pin-fin sizes, the thermal resistance of the MMC decreases as the inlet velocity rises. Meanwhile, the larger the pin-fin size, the lower the thermal resistance. As the pin-fin size increases, the space among the pinfins becomes smaller, which enlarges the flow velocity and enhances the convective heat transfer. Therefore, the thermal resistance of the MMC reduces as the inlet velocity and the pin-fin size both increase. Figure 12b shows that the maximum temperature difference for all pin-fin sizes slightly decreases with an increase in the inlet velocity. This outcome agrees with a previous study. The maximum temperature difference also decreases with an increase in the pin-fin size. This is because a larger pin-fin size leads to smaller flow passages and a higher flow velocity in the microchannel layer, which enhances the local heat transfer for the flow dead zone and improves the temperature uniformity of the MMC heat sinks. Figure 12c shows that for the pin-fin diameter ranging from 0.4 to 1.0 mm, the pressure drops of the heat sink hardly change. However, once the pin-fin size exceeds 1 mm, the pressure drops increase sharply. This is because when the pin-fin size exceeds 1 mm, the flow area among the pin-fins is too small, resulting in a sudden increase in pressure drops. Figure 12d shows that the comprehensive performance of the double-layer MMC with different pin-fin sizes is significantly enhanced with an increase in the inlet velocity. Meanwhile, the comprehensive performance increases with an increase in the pin-fin size. This suggests that the heat transfer enhancement could overcome the pressure drop penalty caused by increasing pin-fin sizes. However, when the pin-fin size reaches 1.1 mm, the comprehensive performance of the pin-fin MMC significantly decreases due to a sharply increased pressure drop. This indicates that properly increasing the pin-fin size could enhance the comprehensive performance of the pin-fin MMC. However, there is a critical diameter at which comprehensive performance begins to decrease.

Height Ratio
The effects of the height ratio are discussed in this section. Here, the height ratio, α, refers to the ratio of the upper layer height to the lower layer height (α = H 2 /H 1 , as illustrated in Figure 2), which is adjusted by changing the upper layer height and keeping the lower layer height constant. α changes from 0.4 to 1.2. Figure 13 shows the flow and thermal characteristics for the double-layer MMC with different height ratios. Figure 13a,b show that the thermal resistance and the maximum temperature difference for all height ratios decrease with an increase in the inlet velocity, which agrees with the previous study. However, the maximum temperature difference and thermal resistance increase with an increase in the height ratio. Since the liquid flow in the upper and lower microchannel layers works in parallel in this work, the mass flow rate in the upper microchannel layer increases, and that in the lower microchannel layer decreases with an increase in the upper layer height. Theoretically, more heat is removed by the liquid in the lower layer as it directly contacts the heat source [30]. A reduced flow rate in the lower layer results in a higher thermal resistance and maximum temperature difference. Figure 13c shows that the pressure drops decrease with an increase in height ratios under the same inlet velocity. This is a product of the fact when the flow area increases as the upper layer height increases, it reduces the average flow velocity and the pressure drops. Figure 13d shows that the comprehensive performance for all height ratios increases with an increase in the inlet velocity. However, the comprehensive performance increases with an increase in the height ratios at lower velocities, while the situation is opposite at higher velocities. With the decreased height ratio, the liquid velocity in each microchannel layer increases, causing a higher pressure drop but a larger heat transfer coefficient. It is observed that under a higher inlet velocity, the heat transfer enhancement could overcome the cost of pressure drop caused by a decreased height ratio. Therefore, the comprehensive performance improves with a decreased height ratio at a higher inlet velocity. On the contrary, the comprehensive performance decreases with a decrease in the height ratio at a lower inlet velocity. Thus, a larger height ratio for a low inlet velocity and a smaller height ratio for a high inlet velocity are recommended.

Conclusions
In this study, a double-layer pin-fin MMC structure is designed, and its flow and thermal performance are studied using ANSYS Fluent. By taking deionized water as the working liquid, the present work comes to the following conclusions.
(1) Compared with the single-layer MMC, the double-layer heat sink demonstrates lower pressure drop, better temperature uniformity, and improved comprehensive performance at the cost of slightly larger thermal resistance.
(2) Among the pin-fin structures with round, diamond-shaped, and rectangular crosssections, the round pin-fin structure demonstrates minimal thermal resistance and the best comprehensive performance, although its temperature uniformity is relatively poor.
(3) Under the same inlet velocity, the thermal resistance is decreased, and the comprehensive performance is first increased and then decreased as the pin-fin size increases.
(4) To achieve better flow and heat transfer performance, the employment of a larger height ratio at a low inlet velocity and a smaller height ratio at a high inlet velocity is recommended.
(5) Further study on the effect of manifold arrangements on the double-layer MMC is advised. Moreover, the performance of the proposed two-layer MMC should be experimentally validated in the future.