Thermal–Hydrodynamic Behavior and Design of a Microchannel Pin-Fin Hybrid Heat Sink

A three-dimensional convective heat transfer model of a microchannel pin-fin hybrid heat sink was established. Considering the non-uniform heat generation of 3D stacked chips, the splitting distance of pin-fins was optimized by minimizing the maximum heat sink temperature under different heat fluxes in the hotspot, the Reynolds numbers at the entrance of the microchannel, and the proportions of the pin-fin volume. The average pressure drop and the performance evaluation criteria were considered to be the performance indexes to analyze the influence of each parameter on the flow performance and comprehensive performance, respectively. The results showed that the maximum temperature of the hybrid heat sink attained a minimum value with an increase in the splitting distance. The average pressure drop in the center passage of the microchannel first increased and then decreased. Furthermore, the optimal value could not be simultaneously obtained with the maximum temperature. Therefore, it should be comprehensively considered in the optimization design. The heat flux in the hotspot was positively correlated with the maximum heat sink temperature. However, it had no effect on the flow pressure drop. When the Reynolds number and the pin-fin diameter increased, the maximum heat sink temperature decreased and the average pressure drop of the microchannel increased. The comprehensive performance of the hybrid heat sink was not good at small Reynolds numbers, but it significantly improved as the Reynolds number gradually increased. Choosing a bigger pin-fin diameter and the corresponding optimal value of the splitting distance in a given Reynolds number would further improve the comprehensive performance of a hybrid heat sink.


Introduction
Electronic devices are developing toward miniaturization, being lightweight, and having a high integration, promoted by the developments in science and technology. A high-level integration leads to a high-power consumption density per unit volume. The performance of a chip is substantially affected if the large amount of heat generated by the integrated circuit cannot be removed in time. Furthermore, thermal failure may occur in severe cases. The proposed microchannel provides a solution to this problem [1].
The heat generation of chips in practical engineering applications is occasionally nonuniform. This leads to the formation of a "hotspot" in the local areas of chips. Certain studies have shown that the heat flux in a hotspot area can be over eight times the average heat flux in the background area [2]. This easily leads to a great non-uniform temperature distribution in an electronic device, reducing the service life of the system. Accordingly, it is of high practical significance to optimize the design of a hybrid heat sink that integrates multiple areas with different cooling capacities (hereinafter referred to as a "hybrid heat sink"); e.g., the combination of a microchannel heat sink in the background area and a pin-fin heat sink in the hot area. In addition, an in-chip cooling structure that is embedded in a semiconductor can reduce the parasitic thermal resistance and improve the overall heat dissipation effect [3,4]. Therefore, an interlayer microchannel pin-fin hybrid heat sink has a broad application prospect for meeting the challenges from high-power microchip-level thermal management problems.
Scholars worldwide have carried out a substantial amount of research on microchiplevel cooling technology [5][6][7][8][9][10][11][12][13]. Ansari et al. [14] proposed a hybrid micro-heat sink for cooling microprocessors with non-uniform heat generation. The temperature inhomogeneity and maximum temperature were chosen as the performance indexes as well as the thermal resistance and pump power. Comparative analyses were performed between these with those of a conventional microchannel heat sink. Wang et al. [15] employed a large eddy simulation to investigate the heat transfer performance and flow characteristics of a rectangular channel with three-row miniature square column vortex generators when the Reynolds number varied from 3745 to 11,235. Ling et al. [16] designed a novel interlaced microchannel. Two sidewalls of the microchannel functioned as the main heat transfer surface. The thermal and hydraulic performance of the interlaced and parallel microchannels was numerically investigated using a full-sized conjugate heat transfer model and then compared with the experimental results. The optimized geometric dimensions of the interlaced microchannel, including the depth, width, and spacing, were obtained. Feng et al. [17] investigated the effects of several fluid flow parameters and double-sided heating power on the performance of a heat sink with embedded microfins by experimental and numerical simulation methods, taking the temperature uniformity as the optimization objective. Cheng et al. [18] investigated the hydrodynamic and thermal characteristics of a slot array in a microchannel based on a three-dimensional numerical simulation, and a comparison with a seamless array was conducted. Ding et al. [19] analyzed the temperature distribution at the bottom of a processor and the flow field distribution inside a microchannel for a 3D-IC sandwich microchannel structure. Xie et al. [20] studied the effects of an inclined angle and arrangement method on the hydraulic and thermal performance of microchannel heat sinks. Chen et al. [21] presented a novel cross-rib microchannel heat sink to induce a fluid to self-rotate and then studied the effects of the aspect ratio of the microchannel. Omri et al. [22] presented a microchannel heat exchanger equipped with triangular fins and applied a CNT-water nanofluid as a coolant. They found that the triangular fins in the heat exchanger could significantly improve the thermal performance of the microchannel heat exchanger and the geometrical parameters of the triangular fins had various effects on the comprehensive performance of the microchannel heat exchanger.
Based on the literature [14,17,18], this study applied split pin-fins to a hybrid heat sink; considered the thermal-hydrodynamic coupling effect of, and the non-uniform heat generation by, the chip; optimized the splitting distance of the pin-fins; and analyzed the influence of the heat flux on the hotspot, the Reynolds number at the entrance of the microchannel, and the volume ratio of the pin-fins on the thermal-hydrodynamic performance of the chip. The results may provide theoretical guidance for the design of a high-power microchip-level thermal management scheme. Figure 1 shows a 3D stacked chip with a microchannel interlayer cooling system. As shown in the figure, each chip layer had an independent microchannel interlayer cooling the heat sink. The microchannels and pin-fins arrays were embedded to enhance the heat dissipation effect. Each chip was connected by thermal interface material (TIM) and a micropump. The TSV integrated in the microchannel sidewall and pin-fin arrays enabled an electrical and communication interconnection between the different chips. The silicon substrate was the base that supported the entire 3D stacked chip and acted as a fluid inlet and outlet. The liquid coolant was pumped through a micropump into the fluid inlet. It then flowed through the hybrid heat sink and finally into the fluid outlet. As shown in Figure 1 [11], each chip layer had a device layer that generated heat and transferred it to the embedded hybrid heat sink. Thus, an individual hybrid heat sink (see Figure 2a [9]) was heated on both sides. To simplify the model and reduce the computational cost, an individual hybrid heat sink was selected for an optimization design and the boundary conditions of the double-sided heat flow were specified. In addition, the heat flux on both sides of the hybrid heat sink was non-uniformly distributed to account for the nonuniform heat generation of the chip. The heated area was divided into the hot area and the background area. The heat flux in the hot area was significantly higher than that in the background area.  The heat sink was integrated with 20 microchannels and 25 pin-fins, which were mainly divided into two areas. The part around the heat sink was the background area of the low heat flux cooled by the microchannels. The red part in the center was the hot area of the high heat flux cooled by the pin-fin array. The values of the geometric parameters of the model are presented in Table 1. A heat sink was used for the interlayer cooling of the chip. The upper and lower surfaces were heated equally and the remaining outer surface was insulated. In this example, the heat flux in the background area was q bg = 50 W·cm −2 , the heat flux in the hot area was q hs = 400-600 W·cm −2 , and the Reynolds number at the entrance of the microchannel was Re = 100-300. The proportion of the pin-fin volume was controlled by the pin-fin diameter, which was D fin = 100-260 µm in this example. As the structure and boundary conditions of the model were horizontally symmetrical, only half of the model was developed and calculated for simplification (see Figure 2b). Figures 3 and 4 show a hotspot model diagram and a cross-sectional diagram of the split pin-fins used in this study, respectively.

Physical Model
The material of the microchannel pin-fin hybrid heat sink was silicon (constant pressure specific heat capacity c p,s = 700 J·kg −1 ·k −1 , density ρ s = 2329 kg·m −3 , and thermal conductivity k s = 130 W·m −1 ·k −1 ). The cooling medium was deionized water with constant physical parameters. The cooling medium flowed in from one end of the heat sink and flowed out from the other end after a convective heat transfer with the hightemperature wall and high-temperature fins. The Reynolds number at the inlet was set to Re = U in D h /v = 100~300, in which U in (m·s −1 ) was the inlet velocity of the deionized water, D h = 2W ch · H ch /(W ch + H ch ) (m) was the hydraulic diameter of the microchannel, and ν (m 2 ·s −1 ) was the kinematic viscosity of the deionized water. The pressure boundary condition at the exit was p out = 1 atm pressure. The upper and lower surfaces of the rectangular heat sink were heated equally and the remaining outer surface was insulated. In this example, the heat flux from the background area was q bg = 50 W·cm −2 and the heat flux from the hotspot area was q hs = 400~600 W·cm −2 . At the fluid-solid interface, there was no slip in the fluid velocity (i.e., U = 0) and the temperature was seen as continuous (i.e., T s = T w and q" = λ s ∂T s /∂n = λ f ∂T f /∂n). The other external surfaces were adiabatic surfaces (i.e., ∂T/∂n = 0).
The heat flux in the background area and hotspot area was uniform: The continuity equation, momentum equation, and energy equation of the fluid laminar flow were: is the heat source term (including the viscous dissipation and pressure work), and T denotes the matrix transpose operation.

Numerical Methods and Grid Tests
In this study, the finite element calculation software COMSOL Multiphysics was used to solve the thermal-hydrodynamic-coupled model; the flow viscous dissipation was considered whereas the gravitational force and radiation were neglected. The unstructured tetrahedral mesh was divided into fluid and solid areas; the denser boundary layer mesh was divided into areas near the fluid-solid interfaces. The independence of the grid was tested to ensure the calculation accuracy. Considering q hs = 500 W·cm −2 , Re = 200, D fin = 180 µm, and L fin = 100 µm as an example, the number of grids under the three partitioning strategies of coarsening, conventional, and refinement were 276,205, 873,442, and 2,965,967, respectively. The corresponding T max was 365.93 K, 368.96 K, and 371.31 K, and the relative errors were 0.828% and 0.637%, respectively. A second mesh division strategy was adopted in this study to consider the calculation accuracy and efficiency. Figure 5 shows the schematic diagram of the meshes generated in the microchannel pin-fin hybrid heat sink, where the hotspot area was partially enlarged. The common default convergence criterion was used for the continuity, momentum, and energy equations. To further verify the accuracy of the algorithm in this study, the modeling method in this study was used to establish the microchannel pin-fin hybrid heat sink and heat dissipation model developed in [14] and its calculation was carried out. When the other conditions were fixed, the parameter relationship between the maximum pressure drop of the cooling channels and the Reynolds number was obtained, as shown in Figure 6. As is evident from the numerical comparison, the deviations between the maximum pressure drop of the cooling channel in [14] and the simulation model in this study were negligible at 9 Reynolds numbers.

Thermal Behavior and Optimization
(1) With different heat fluxes at the hotspot Figure 7 shows the influence of the heat flux q hs in the hotspot on the relationship between the maximum temperature T max and the splitting distance L fin when the inlet Reynolds number was Re = 200 and the pin-fin diameter was D fin = 180 µm.  Figure 7 shows that when q hs was specified and as L fin increased, T max first marginally increased, then substantially decreased, and finally increased again. The fluid flow rate in the interlayer of the split pin-fins was highly marginal when L fin was marginal. This was due to the influence of the boundary layer. At this time, the main heat transfer mode was heat conduction and the convective heat transfer made a limited contribution to the heat dissipation effect. Simultaneously, the vertical streamline length of the pin-fins increased, the flow stagnation area of the back flow side of the pin-fins increased, and the heat dissipation performance decreased. Thus, T max increased with the increase in L fin . As L fin continued to increase, the flow rate of the fluid flowing through the interlayer of the pin-fins gradually increased and the main heat transfer mode became a convective heat transfer. The heat transfer performance of the pin-fins was enhanced and T max decreased accordingly. When L fin > 100 µm, the fluid flow on both sides of the pin-fins and the heat dissipation performance on both sides of the surface continued to decrease with the increase in L fin . This caused T max to gradually increase.
A comparison of the relationship curves between T max and L fin at different q hs revealed that the curves for q hs = 400 W•cm −2 , 500 W•cm −2 , and 600 W•cm −2 had a similar variation trend. Here, the curves first marginally increased, then substantially decreased, and finally increased again. T max minima were obtained at (L fin ) opt = 100 µm for the three curves under different q hs conditions. This indicated that q hs had no effect on the thermal behavior of the element body. The extreme values of T max for the three curves increased with the increase in q hs (i.e., 356.85 K, 368.96 K, and 381.16 K, respectively) with increments of 12.11 K and 12.20 K, respectively. It was observed that when q hs increased, T max proportionally increased; these were positively correlated. This was because the increase in q hs increased the heat output of the chip and the heat dissipation load of the heat sink. If the generated heat was not emitted in time, T max accordingly increased until a thermal balance was attained. (T max ) min under different q hs was reduced by 9.84 K, 12.04 K, and 14.16 K, respectively, compared with T max = 366.69 K, 381.00 K, and 395.32 K under L fin = 0 µm (reduction of 2.68%, 3.16%, and 3.58%, respectively). That is, the larger the value of q hs , the more significant the effect of L fin on T max . This implied that the higher the heat flux of the 3D stacked chip, the more significant the optimization effect of varying the needle rib splitting distance on the maximum temperature.
(2) With different Reynolds numbers at the microchannel entrance Figure 8 shows the influence of the inlet Reynolds number (Re) on the relationship between the maximum temperature T max and the splitting distance L fin when the heat flux in the hotspot was q hs = 500 W·cm −2 and the pin-fin diameter was D fin = 180 µm.  Figure 8 shows that when the Re was specified and as L fin increased, T max marginally increased, then decreased, and finally increased again. A comparison between the relationship curves of T max and L fin for different Re revealed that the curves for Re = 100, 200, and 300 had a similar trend. When Re differed, the three curves achieved a T max minima at (L fin ) opt = 100 µm with extreme values of 400.46 K, 368.96 K, and 355.93 K, respectively, and incremental values of −31.50 K and −13.03 K, respectively. The overall heat transfer coefficient h avg corresponding with these T max minima were 30,290 W·m −2 ·K −1 , 35,077 W·m −2 ·K −1 , and 38,432 W·m −2 ·K −1 , respectively. It was observed that for the same model, the fluid flow rate increased and the convective heat transfer effect was enhanced when Re increased. This, in turn, improves the heat dissipation performance of the heat sink and decreased T max . The larger the value of Re, the smaller the effect of varying Re on T max .
That is, an appropriate increase in Re when it was small could reduce T max more effectively. (T max ) min under different Re was reduced by 11.19 K, 12.04 K, and 11.19 K compared with T max = 411.65 K, 381.00 K, and 367.12 K, respectively, for L fin = 0 µm (reduction of 2.72%, 3.16%, and 3.05%, respectively). Therefore, when T max was used as a performance metric, variations in T max were less affected by Re as L fin varied.
(3) With different volume ratios (pin-fin diameters) Figure 9 shows the influence of the pin-fin diameter D fin on the relationship between the maximum temperature T max and the splitting distance L fin when the heat flux in the hotspot was q hs = 500 W·cm −2 and the inlet Reynolds number was Re = 200. Corresponding with Figure 9, Figure ?? shows the temperature color maps of the hotspot area of the hybrid heat sink when D fin = 180 µm and L fin = 10, 100, and 140 µm, respectively. As can be observed from Figure 9, when D fin = 180 µm and 260 µm and as L fin increased, T max marginally increased, then decreased, and finally increased. When D fin = 100 µm and as L fin increased, T max first decreased and then increased. When D fin = 180 µm and 260 µm, T max passed through three stages as L fin increased. The first stage was where L fin was small. Here, the main heat transfer mode in the interlayer of the split pin-fins was heat conduction owing to the influence of the small fluid flow rate in the boundary layer. T max marginally increased with the increase in L fin . When D fin = 180 µm, the T max maximum in the first stage was 381.96 K and the corresponding overall heat transfer coefficient h avg was 33,612 W·m −2 ·K −1 . In the second stage, L fin continued to increase, the flow-around effect increased, the disturbance of the fluid flow increased, the convective heat transfer effect increased and became the dominant heat transfer mode, and T max accordingly decreased. When D fin = 180 µm, the T max minimum in the second stage was 368.60 K and the corresponding overall heat transfer coefficient h avg was 35,067 W·m −2 ·K −1 . The third stage was when L fin increased to a certain value. With the increase in L fin , the fluid mainly flowed through the interlayer of the pin-fins and the convective heat transfer effect on the semicircular surface of the pin-fins weakened. This resulted in an increase in T max . When D fin = 180 µm and L fin = 140 µm, T max was 373.23 K and the corresponding overall heat transfer coefficient h avg was 34,527 W·m −2 ·K −1 . When D fin = 100 µm, the obtained data did not reflect the first stage of T max because the numerical simulation calculation step (10 µm) was large. These showed a decreasing trend followed by an increasing trend with the increase in L fin .
When D fin was 100 µm, 180 µm, and 260 µm, T max attained a minimum value when (L fin ) opt = 110 µm, 100 µm, and 90 µm, respectively. The extreme values were 382.57 K, 368.96 K, and 368.13 K, respectively, and the increments were −13.61 K and −0.83 K, respectively. T max was reduced by 12.48 K, 12.04 K, and 5.52 K compared with T max = 395.05 K, 381.00 K, and 373.65 K, respectively, for L fin = 0 µm (reduction of 3.15%, 3.16%, and 1.48%, respectively). It was observed that with the increase in D fin , the surface area of the pin-fin heat transfer and the heat transfer increased. The smaller the value of (L fin ) opt , the less optimal the T max that was obtained by varying L fin . Furthermore, the smaller the value of D fin , the more significant the effect of varying D fin and L fin was on T max . This indicated that D fin and L fin had optimal matching issues when T max was considered to be the performance index. Therefore, in practical engineering applications, the pin-fin diameter can be determined by comprehensively considering the cost, process, and other factors. (L fin ) opt can then be determined according to the relationship curve between L fin and T max so that the heat sink T max is minimized and the performance is optimal.

Hydrodynamic Behavior Analysis
(1) Influence of the heat flux at the hotspot Figure 11 shows the influence of the heat flux q hs at the hotspot on the relationship between the average pressure drop ∆p avg of the center passage and the splitting distance L fin when the inlet Reynolds number was Re = 200 and the pin-fin diameter D fin = 180 µm. Figure 11. Effect of q hs on ∆p avg versus L fin .
As can be observed from Figure 11, when q hs was given and as L fin increased, ∆p avg marginally increased, then substantially increased, and finally continuously decreased. When L fin was small, the fluid flow in the interlayer of the split pin-fins was gradual and the disturbance was weak owing to the influence of the boundary layer. However, the increase in the thickness of the split pin-fins in the vertical streamline direction obstructed the fluid flow. This marginally increased the flow resistance and ∆p avg increased with it. When L fin > 10 µm and continued to increase, the fluid flow in the split pin-fin interlayer gradually increased, so that part of the fluid flowed through the split pin-fin interlayer, the disturbance was enhanced, the flow resistance increased, and ∆p avg greatly increased. When L fin > 50 µm and with the increase in L fin , the fluid flow on both sides of the pin-fins gradually decreased, the fluid mainly flowed through the interlayer of the split pin-fins, the disturbance weakened, and ∆p avg continued to decrease.
A comparison of the relationship curves between ∆p avg and L fin for different q hs revealed that the curves for q hs = 400 W·cm −2 , 500 W·cm −2 , and 600 W·cm −2 essentially overlapped. Moreover, ∆p avg attained a maximum when (L fin ) opt = 50 µm. A minimum value of 2079.9 Pa was obtained at (L fin ) opt = 140 µm. This was 84.8 Pa lower than ∆p avg = 2164.7 Pa for the cylindrical pin-fins at L fin = 0 µm (a reduction of 3.92%). It was observed that the variation in q hs essentially had no effect on ∆p avg . This was because the effect of the variation in q hs only altered the temperature field. However, the microchannel retained a single-phase incompressible laminar flow (Re = 200; the fluid was liquid and the physical properties did not vary with the temperature). Thus, the variation in the temperature field at a certain range had no effect on the fluid flow and the flow field and ∆p avg were not affected by q hs . In addition, from the perspective of the fluid flow, increasing the pin-fin splitting distance to the maximum extent within the range permitted by the manufacturing process could be considered. This would reduce the pressure drop in the flow passage and the pump power consumption.
(2) Influence of the Reynolds number at the microchannel entrance Figure 12 shows the influence of the Reynolds number (Re) at the entrance of the microchannel on the relationship between the average pressure drop ∆p avg in the center passage and the splitting distance L fin when the heat flux in the hotspot was q hs = 500 W·cm −2 and the pin-fin diameter was D fin = 180 µm. As can be observed from Figure 12, when Re was specified and as L fin increased, ∆p avg marginally increased, then substantially increased, and finally continuously decreased. A comparison of the relationship curves of ∆p avg and L fin for different Re revealed that the curves for Re = 100, 200, and 300 had a similar variation trend. ∆p avg attained the maximum when (L fin ) opt = 60 µm, 50 µm, and 40 µm and the respective minimums were 964.7 Pa, 2079.9 Pa, and 3298.0 Pa when (L fin ) opt = 140 µm, respectively. The increments were 1115.2 Pa and 1218.1 Pa, which were 9.8 Pa, 84.8 Pa, and 191.8 Pa lower than ∆p avg = 974.5 Pa, 2164.7 Pa, and 3489.8 Pa when L fin = 0 µm. The reduction percentages were 1.00%, 3.92%, and 5.50%, respectively. It was observed that the flow velocity and flow resistance increased as Re increased and ∆p avg required by the flow accordingly increased. The higher the value of Re, the higher the impact of ∆p avg on the Re variations. That is, the effect of reducing the heat sink ∆p avg by reducing Re was better when Re was small. In addition, when Re was large, the fluid flow rate was large, the disturbance was strong, the fluid was substantially affected by the variation in the pin-fin geometric parameters, L fin had a more significant effect on ∆p avg , and the optimization space was larger. Therefore, in engineering applications, the higher the value of Re, the higher the significance of the optimal design related to the pin-fin splitting distance.
(3) Influence of the volume ratio (pin-fin diameter) Figure 13 shows the influence of the pin-fin diameter D fin on the relationship between the average pressure drop ∆p avg in the center passage and the splitting distance L fin when the heat flux in the hotspot was q hs = 500 W·cm −2 and the inlet Reynolds number was Re = 200. Corresponding with Figure 13, Figure 14 shows the velocity color maps of the hotspot area of the hybrid heat sink when ∆p avg reached the maxima and minima at D fin = 100, 180, and 260 µm, respectively.  As can be observed from Figure 13, as L fin increased from 0 µm to 140 µm, the fluid flow in the microchannel underwent a process wherein it mainly flowed through both sides of the pin-fins to the interlayer of the split pin-fins. The fluid disturbance first increased and then weakened. ∆p avg varied with it; increasing first and then continuously decreasing. A comparison between the relationship curves of ∆p avg and L fin with different D fin revealed that for D fin = 100 µm, 180 µm, and 260 µm, ∆p avg attained the maximum value when L fin = 80 µm, 50 µm, and 40 µm, respectively. Meanwhile, the minimum values were obtained when L fin = 0 µm, 140 µm, and 140 µm (namely, 1946.1 Pa, 2079.9 Pa, and 2322.7 Pa, respectively, with increments of 133.8 Pa and 242.8 Pa, respectively). Compared with ∆p avg = 1946.1 Pa, 2164.7 Pa, and 2612.4 Pa for L fin = 0 µm, it was reduced by 0 Pa, 84.8 Pa, and 289.7 Pa, respectively (i.e., by 0%, 3.92%, and 11.09%, respectively). It was observed that the larger the D fin , the smaller the minimum value of ∆p avg . Furthermore, the larger the decrease with the increase in D fin , the larger the variation in the influence of L fin . That is, the larger the D fin , the higher its influence on ∆p avg . Compared with the cylindrical pin-fin hybrid heat sink when L fin =0 µm and when ∆p avg was taken as the performance index, the larger the D fin , the better the optimal performance that could be achieved by tweaking L fin for the heat sink. In addition, when D fin = 100 µm, the minimum value of ∆p avg was L fin = 0 µm. The cylindrical pin-fins then formed the optimal configuration.

Comprehensive Performance of the Microchannel Pin-Fin Hybrid Heat Sink
From the above thermal and hydrodynamic behavior analysis of the microchannel pinfin hybrid heat sink, the influences of the flow and structure parameters on the maximum temperature T max and the overall heat transfer coefficient h avg of the hybrid heat sink were exactly contrary to the average pressure drop ∆p avg of the center passage of the hybrid heat sink. This indicated that it was necessary to further confirm if the comprehensive performance of the microchannel pin-fin hybrid heat sink was better than the conventional microchannel heat sink.
The performance evaluation criteria PEC = (Nu a / Nu 0 ) / ( f a / f 0 ) 1/3 is a wellknown and widely used indicator to describe the comprehensive performance of heat sinks [23]. Nu is the overall Nusselt number, f is the apparent flow friction coefficient, the subscript "0" indicates the conventional microchannel heat sink, and the subscript "a" indicates the microchannel pin-fin hybrid heat sink.
When the heat flux in the hotspot q hs = 500 W·cm −2 and the Reynolds number Re = 100~300, the apparent flow friction coefficients f 0 of the conventional microchannel heat sink with the exact dimensions as Table 1 were 0.1652, 0.0871, and 0.0591, respectively; the overall Nusselt numbers Nu 0 of the conventional microchannel heat sink were 13.958, 15.674, and 16.842, respectively. The above parameters were used as the benchmark to compare the comprehensive performance of the microchannel pin-fin hybrid heat sink.
Considering the effect of the Reynolds number (Re) and the splitting distance L fin on the comprehensive performance of the microchannel pin-fin hybrid heat sink (D fin = 180 µm), PEC = 0.8595 when (L fin ) opt = 100 µm at Re = 100, PEC = 1.2248 when (L fin ) opt = 100 µm at Re = 200, and PEC = 1.5131 when (L fin ) opt = 100 µm at Re = 300, respectively. This indicated that the comprehensive performance of the microchannel pin-fin hybrid heat sink was not good at small Reynolds numbers, but could be significantly improved as the Reynolds number gradually increased.
Considering the effect of the pin-fin diameter D fin and the splitting distance L fin on the comprehensive performance of the microchannel pin-fin hybrid heat sink (q hs = 500 W·cm −2 ; Re = 200), PEC = 1.1607 when (L fin ) opt = 110 µm at D fin = 100 µm, PEC = 1.2217 when (L fin ) opt = 100 µm at D fin = 180 µm, and PEC = 1.2312 when (L fin ) opt = 100 µm at D fin = 180 µm, respectively. This indicated that, at a given Reynolds number, the thermal performance and the comprehensive performance of the microchannel pin-fin hybrid heat sink could be further improved with a bigger pin-fin diameter D fin and corresponding optimal value of the splitting distance (L fin ) opt .

Conclusions
In this study, split pin-fins were applied to a hybrid heat sink and the thermalhydrodynamic behavior and optimal design of a split pin-fin array at the hotspot was obtained. The influence of the heat flux at the hotspot, the Reynolds number at the entrance of the microchannel, and volume proportion of the pin-fins were analyzed.
The results showed that the maximum temperature of the hybrid heat sink had a minimum value with the increase in the splitting distance of the pin-fins. The average pressure drop in the center passage of the microchannel first increased and then decreased. An optimal value could not be simultaneously obtained with the maximum temperature. Therefore, it should be comprehensively considered in the optimization design. The heat flux in the hotspot was positively correlated with the maximum heat sink temperature. However, it had no effect on the flow pressure drop. The effects of the Reynolds number at the entrance of the microchannel and the volume proportion of the pin-fins were similar. That is, when the Reynolds number and the volume proportion increased, the maximum heat sink temperature decreased whereas the average pressure drop of the microchannel increased and the comprehensive performance of the microchannel pin-fin hybrid heat sink was improved. The larger the value of the Reynolds number and the volume proportion, the smaller the effect.
Based on the concept of a thermal-hydrodynamic coupling optimization design, this study provides a new method and example for the optimization design of a hybrid heat sink with a microchannel and pin-fins. Considering the different processes and structural forms of microchannels and pin-fins, further research to carry out a multiobjective optimization design, including non-uniform heat production and its dynamic characteristics, is worthwhile.
Author Contributions: X.G., conceptualization, data curation, formal analysis, methodology, derivation and calculation, validation, writing-original draft, and software; Z.X., conceptualization, methodology, project administration, supervision, writing-reviewing and editing, and funding acquisition; G.N. and K.X., methodology and visualization; Z.L., visualization and writing-original draft; Y.G., validation and visualization. All authors have read and agreed to the published version of the manuscript.
Funding: This research was funded by the National Natural Science Foundation of China (grant numbers 51979278 and 51579244).

Informed Consent Statement: Not applicable.
Data Availability Statement: Data will be made available on request.