A Hybrid Microchannel Heat Sink Featuring Micro Pin-Fins with Secondary Channels for Hotspot Suppression

Abstract

Non-uniform heat flux in high-power electronic devices can induce local hotspots. To address hotspot thermal management, a hybrid microchannel heat sink featuring micro pin-fins with secondary channels (MPSCs) is proposed. Straight microchannels are used in the low-heat-flux region, while staggered pin-fins are arranged in the central hotspot region. Five models are considered: solid pin-fins without secondary channels (M1) and pin-fins with one-, three-, four-, and six-directional secondary channels (M2–M5). Over Re = 200–800, the thermal resistance of M3 is reduced by up to 43.3% compared with the SPMS, and the maximum PEC reaches 1.178. MPSC structures achieve better cooling at lower pumping power. At similar pumping power, the thermal resistance of M3 is 36.8% lower than the minimum value attainable by the SPMS. However, the marginal reduction in thermal resistance gradually weakens as the pumping power increases. The effects of hotspot heat flux and relative pin-fin height are investigated. MPSCs can withstand higher heat loads. Under a hotspot heat flux of 800 W/cm², the maximum temperature of M3 remains below the safety threshold. Relative pin-fin height has a coupled effect on the thermal–hydraulic performance. At Re = 800, the PEC of M3 at RPH = 0.9 is about 2.8% higher than that at RPH = 1.0, and a PEC fitting equation is developed. The results show that the hybrid micro pin-fin heat sink with secondary channels disrupts the thermal boundary layer through internal flow diversion and local disturbance, thereby balancing hotspot suppression and pressure drop control. This design provides a reference for hotspot thermal management and thermal–hydraulic optimization of microchannel heat sinks.

1. Introduction

Increasing integration density in microprocessors, power electronics, and three-dimensional integrated chips raises local power density and imposes stricter thermal constraints on device reliability [1,2]. Current advanced high-performance processors can have thermal design power (TDP) of up to 400 W [3]. During actual chip operation, heat is not generated uniformly. Core regions and switching units often form local hotspots with heat flux much higher than that in the background region [4,5]. These hotspots can raise the peak temperature and thermal gradients, degrading performance and device lifetime [5]. Conventional air cooling is generally limited to heat fluxes on the order of 100 W/cm² and is inadequate for high-heat-flux devices [6]. Liquid cooling with localized hotspot control therefore offers a more suitable approach [7,8].

To meet the increasing thermal management demand of electronic devices, various cooling technologies have been studied, including thermoelectric cooling, immersion cooling, and heat pipe cooling [9,10,11,12]. Among them, microchannel heat sinks (MHS) have become an important route for electronic cooling because of high specific surface area, short heat transfer path, and strong convective heat transfer capability [13]. Tuckerman and Pease [14] first demonstrated high-heat-flux cooling with a straight parallel microchannel heat sink (SPMS) in 1981. Subsequent studies have examined cross-section, inlet or outlet arrangement, coolant type, and other key factors [15,16,17]. Kose et al. [18] showed that heat transfer enhancement generally increases pumping power demand. In straight microchannels, limited disturbance allows the thermal boundary layer to grow downstream, reducing local heat-transfer capability. Passive enhancements such as ribs [19,20], grooves [21,22], manifolds [23,24], multilayer channels [25,26], and micro pin-fins [27,28] are therefore used to interrupt this development and promote mixing.

Micro pin-fins have strong fluid disturbance capability and have become an important structure for heat transfer enhancement in MHS. Yang et al. [29] summarized the heat transfer enhancement mechanisms of pin-fins and reported that pin-fins can induce accelerated flow, thereby disrupting the thermal boundary layer. They also noted that the shape and size of pin-fins have significant effects on heat transfer performance. Ismail et al. [30] optimized the geometry of pin-fins and showed that pin-fin design should balance heat transfer enhancement and pressure drop control. In addition to shape, the arrangement of micro pin-fins also affects the cooling performance. Chiu et al. [31] studied a variable-density arrangement of micro pin-fins and found that it provided better temperature uniformity than the staggered arrangement, with a minimum effective thermal resistance of 0.258 K/W. In recent years, micro pin-fin structures have increasingly been developed as composite designs. Li et al. [28] designed an open microchannel with diamond pin-fins. At a pumping power of 0.1 W, its total thermal resistance was 23.77% lower than that of a conventional microchannel. Shen et al. [32] experimentally studied a silicon-based thermal test chip cooled by a backside embedded-manifold staggered pin-fin microchannel heat sink. At a mass flow rate of 5 mL/s and a heat flux of 700 W/cm², this design reduced the maximum chip temperature by 8 K compared with a rectangular microchannel. Haque et al. [33] investigated perforated, twisted, and grooved pin-fins. At Re = 21,367, the hydrothermal performance factor (HTPF) of the grooved–perforated heat sink was about 27% higher than that of the baseline case, but the pressure increased by about 80%. Complex pin-fin structures can enhance fluid mixing and improve heat transfer efficiency, but they also increase hydraulic loss and manufacturing complexity.

However, most MHS studies assume uniform heat flux, although practical chips exhibit spatially non-uniform power dissipation. Taking microprocessors as an example, the cores, caches, interconnects, and peripheral logic units have different power levels. Therefore, hotspot thermal management has gradually become an important direction in MHS research. Local heat-transfer enhancement has been achieved by embedding pin-fin arrays in the hotspot region of microchannels. Feng et al. [34] proposed gradient annular-cavity micro pin-fins for non-uniform hotspots in 3D chips. Under a hotspot heat flux of 700 W/cm², the thermal resistance was reduced by 34.5%. Fallahtafti et al. [35] optimized the shape of hotspot-targeted micro pin-fins. Using a multi-objective optimization algorithm, they achieved a hotspot thermal resistance of 0.208 K·cm²/W under a pressure drop constraint of 10 kPa, and improved the temperature uniformity by about 50%. These studies show that local cooling enhancement for non-uniform heat sources is more consistent with the practical thermal management demand of modern multi-core chips. However, although micro pin-fins can effectively enhance local heat transfer, they often cause large hydraulic losses. Composite designs that balance thermal and hydraulic performance are still limited.

Secondary channels offer another passive approach for balancing heat transfer and flow resistance. By redirecting coolant between adjacent regions, they promote mixing and disturb boundary-layer growth with a relatively small power penalty [36]. Sadique et al. [36] systematically reviewed the enhancement mechanisms of secondary channels in microchannels and reported that secondary flow can enhance heat transfer level with a limited power penalty. Wen et al. [37] reported up to a 34% improvement in overall performance for a composite secondary-channel MHS at a relative channel width of 0.6. However, most existing secondary channels are arranged between adjacent straight microchannels or near the channel walls. Studies on secondary-channel structures for micro pin-fin heat sinks are still limited. Representative MHS architectures and their reported thermal–hydraulic performance are compared in Table 1.

Table 1. Comparison of representative microchannel heat sink structures, reported performance, and thermal–hydraulic characteristics.

Enhanced Structure			Reported Performance	Structural Characteristics and Trade-Offs	References
Straight microchannel			Heat flux up to about 790 W/cm2.	Simple structure and continuous downstream boundary-layer growth limits hotspot cooling.	Tuckerman and Pease [14]
Cross-section-optimized microchannel			Pumping power ↓ 17% relative to trapezoidal channels.	Cross-section optimization balances heat transfer and flow resistance, but provides limited mixing and boundary-layer disturbance.	Kose et al. [18]
Periodic merging chamber with diamond ribs			Enhancement ↑ 3.59–13.24%.	Merging chambers and ribs induce local acceleration and secondary flow, with additional form drag.	Lu et al. [19]
Rib–groove composite microchannel			Nu ↑ 131%, Rth ↓ 66.6%, PEC ↑ 17.9%.	Ribs and grooves enhance flow diversion and mixing.	Li et al. [21]
Manifold microchannel			Maximum temperature ↓ by up to 4.1 K.	Manifold feeding shortens flow paths, but complicates flow distribution and structural design.	Kang et al. [23]
Double-layer microchannel			Thermal performance factor ↑ 38%.	They enlarge the heat-transfer area, but complicate interlayer flow and pressure distribution.	Maheswari et al. [25]
Micro pin-fins	Open microchannel with diamond pin-fins		Rth ↓ 23.77% at 0.1 W pumping power.	Tip clearance provides bypass flow, while pin-fins enhance mixing.	Li et al. [28]
	Geometrically optimized micro pin-fins		Minimum thermal resistance ↓ 35%.	Fin dimensions and clearance balance heat transfer enhancement against flow blockage.	Ismail et al. [30]
	Variable-density micro pin-fins		Minimum effective Rth of 0.258 K/W.	Variable fin density matches cooling capacity to local heat load, but dense regions may restrict flow.	Chiu et al. [31]
	Embedded-manifold staggered pin-fins		Chip temperature ↓ 8 K at 700 W/cm2.	Manifold feeding and staggered fins enhance cooling, but increase pressure drop and fabrication complexity.	Shen et al. [32]
	Hotspot thermal management	Hotspot-targeted gradient micro pin-fins	Rth ↓ 34.5%.	Gradient layouts concentrate cooling in hotspot regions, improving coolant utilization but increasing design complexity.	Feng et al. [34]
		Hotspot-targeted optimized micro pin-fins	Hotspot thermal resistance reached 0.208 K·cm2/W.	Regional optimization matches cooling capacity with heat load, but depends on hotspot location and parameter optimization.	Fallahtafti et al. [35]
Composite secondary-channel microchannel			Overall performance ↑ 34%.	Secondary channels promote flow diversion and transverse exchange, while ribs further disturb the boundary layer; complexity increases.	Wen et al. [37]

↑ and ↓ represent the magnitude of performance improvement and reduction, respectively.

Table 1 shows that conventional straight channels offer structural simplicity but limited hotspot control. Ribs, grooves, manifolds, multilayer channels, and pin-fins strengthen heat transfer at the expense of additional hydraulic or fabrication penalties. Secondary channels can relieve local blockage, but their direct integration within hotspot-targeted micro pin-fins has received limited attention.

Based on these characteristics, this study proposes a hybrid microchannel heat sink featuring micro pin-fins with secondary channels. Straight microchannels are retained in the low-heat-flux background region, while staggered micro pin-fins are arranged only in the central hotspot region. One-, three-, four-, and six-directional secondary channels are embedded inside the pin-fins to promote local flow diversion and reduce the blockage associated with solid pin-fins. The proposed structure therefore combines hotspot-targeted enhancement with internal secondary flow paths, aiming to improve hotspot suppression while controlling the hydraulic penalty. Numerical simulations are conducted to compare several pin-fin structures with embedded secondary channels against the baseline SPMS. The influence of Re on the thermal–hydraulic performance is examined, and the energy benefit is evaluated using the relationship between thermal resistance and pumping power. The hotspot load capacity of the hybrid heat sink is analyzed under different hotspot heat fluxes, and the effect of pin-fin height is also investigated. This study provides a hotspot thermal management solution that balances thermal and hydraulic performance for high-power electronic devices.

2. Numerical Method

2.1. Geometric Model

A hybrid microchannel heat sink featuring micro pin-fins with secondary channels (MPSCs) is proposed to enhance local cooling of the central hotspot under non-uniform heat flux. Its geometric structure is shown in Figure 1. The heat sink consists of straight microchannels in the background region and micro pin-fins in the central hotspot region. In this hybrid design, the heat transfer enhancement structures are concentrated in the high-heat-flux region, which improves local cooling capacity while controlling the overall flow resistance.

Figure 1. Geometric model of MHS. (a) Overall structure of the MPSC and five pin-fin models; (b) the SPMS model; (c) top view of M3 and enlarged staggered pin-fin array; (d) geometrical parameters of microchannels; (e) hotspot and background regions of MHS.

Figure 1a shows the geometric structure of the MPSC heat sink. The cover plate is hidden in this view. A circular pin-fin array is arranged in the central hotspot region, and different types of secondary channels are embedded inside the pin-fins. According to the number and direction of the internal channels, five geometric models are constructed and denoted as Model 1–Model 5 (M1–M5). M1 is a solid circular pin-fin without internal secondary channels and is used as the pin-fin baseline model. M2–M5 introduce one-, three-, four-, and six-directional secondary channels inside the circular pin-fins, respectively. All models have the same external pin-fin envelope size. The secondary channels are formed only by removing solid regions inside the circular pin-fins, and their characteristic width is W_sch. The SPMS is also established as the control model, as shown in Figure 1b. Except for the central enhancement structure, the SPMS and M1–M5 have the same geometric dimensions and boundary conditions. Figure 1c presents the top view of the hotspot region in M3. The pin-fin array is located in the central hotspot region and arranged in a staggered pattern. The staggered arrangement promotes flow diversion in the gaps between the pin-fins. The secondary channels inside the pin-fins further connect the surrounding flow regions, which provides additional internal flow paths and promotes local coolant exchange. Figure 1d describes the local geometric parameters of the straight microchannel region. Figure 1e illustrates the bottom heat source regions and the corresponding geometric dimensions. The central pin-fin enhancement region is aligned with the hotspot region in the planar direction. The dimensional parameters of all models are listed in Table 2.

Table 2. Size parameters of all models.

Parameter Description	Symbol	Value (mm)
Heat sink length	Lsink	12
Heat sink width	Wsink	12
Hotspot length	Lhs	4
Hotspot width	Whs	4
Substrate thickness	Hb	0.20
Channel height	Hch	0.80
Channel width	Wch	0.25
Wall thickness	Wwall	0.25
Pin-fin height	Hpf	0.80
Pin-fin radius	rpf	0.27
Secondary-channel width	Wsch	0.08
Transverse spacing of pin-fins	WT	0.66
Longitudinal spacing of pin-fins	WL	0.58

2.2. Governing Equations and Boundary Conditions

Numerical simulations are conducted using computational fluid dynamics (CFD) with a steady fluid–solid conjugate heat transfer model. The computational domain includes the complete heat sink model, as shown in Figure 1. To reduce unnecessary computational complexity, the following assumptions are adopted:

• The Reynolds number of deionized water ranges from 200 to 800. Therefore, the flow is assumed to remain laminar in all models. The applicability of this treatment at Re = 800 is further assessed using local Reynolds numbers and a transient stability test.
• The thermal–hydraulic analysis is performed under steady-state conditions. The stability of the steady-state solution at Re = 800 is further examined in Section 2.7.
• Deionized water is treated as a single-phase incompressible fluid. The calculated coolant temperature remained below 340 K in all cases, so phase change was not considered.
• Gravity, thermal radiation, and natural convection between the outer walls and the surrounding environment are neglected. Conjugate heat transfer conditions are applied at the solid–fluid interfaces, and the no-slip boundary condition is used.

Silicon is selected as the heat-sink material because of its compatibility with microfabrication and integrated devices. Its properties are listed in Table 3, with a constant thermal conductivity of 148 W/(m·K). Although the thermal conductivity of silicon decreases with increasing temperature, the present study mainly focuses on the thermal–hydraulic differences between geometric configurations. The same material property and boundary conditions are applied to all models, providing a consistent basis for evaluating the effects of pin-fins and secondary channels. Therefore, the constant k_si is retained as a reasonable simplification for the comparative analysis in this study.

Table 3. The material parameter of silicon (Si).

Parameter	Symbol	Value
Density	ρsi (kg/m3)	2330
Specific heat capacity	Cp,si (J/(kg·K))	712
Thermal conductivity	ksi (W/(m·K))	148

Deionized water is used as the coolant in the heat sink, and its physical properties are expressed as polynomial functions of temperature [38,39,40]. The density (ρ_f (T), kg/m³), specific heat capacity (C_p,f (T), J/(kg·K)), thermal conductivity (k_f (T), W/(m·K)), and dynamic viscosity (μ_f (T), Pa·s) are calculated using the correlations listed in Equations (1)–(4), respectively.

$${\rho }_f\left(T\right)=765.33+1.8142T-0.0035T^2$$

(1)

$$c_{p,f}\left(T\right)=28070-281.7T+1.25T^2-\left(2.48\times {10}^{-3}\right)T^3+\left(1.857\times {10}^{-6}\right)T^4$$

(2)

$$k_f\left(T\right)=-0.5752+\left(6.397\times {10}^{-3}\right)T-\left(8.151\times {10}^{-6}\right)T^2$$

(3)

$${\mu }_f(T)=\left\{\begin{array}{ll}0.06119-0.0003845T+6.111\times {10}^{-7}{T}^2,& 273\hspace{0.17em}\mathrm{K}\le T<308\hspace{0.17em}\mathrm{K}\\ 0.01611-8.688\times {10}^{-5}T+1.199\times {10}^{-7}{T}^2,& 308\hspace{0.17em}\mathrm{K}\le T<343\hspace{0.17em}\mathrm{K}\\ 0.07097-3.371\times {10}^{-5}T+4.141\times {10}^{-8}{T}^2,& 343\hspace{0.17em}\mathrm{K}\le T<373\hspace{0.17em}\mathrm{K}\end{array}\right.$$

(4)

Since the model is solved under steady-state conditions, three-dimensional steady governing equations are adopted.

The continuity equation is

$$\nabla \cdot \left({\rho }_f\overrightarrow{V}\right)=0$$

(5)

The momentum equation is

$$\nabla \cdot \left({\rho }_f\overrightarrow{V}\cdot \overrightarrow{V}\right)=-\nabla p+\nabla \cdot \tau$$

(6)

where τ is defined as

$$\tau ={\mu }_f\left[\left(\nabla \overrightarrow{V}+\nabla {\overrightarrow{V}}^T\right)-{\displaystyle \frac{2}{3}}\nabla \cdot \overrightarrow{V}I\right]$$

(7)

The energy equations are expressed as follows:

The energy equation for the fluid is

$$\overrightarrow{V}\cdot \nabla \left({\rho }_f{C}_{p,f}{T}_f\right)=\nabla \cdot \left(k_f\nabla T_f\right)$$

(8)

The energy equation for the solid is

$$\nabla \cdot \left(k_s\nabla T_s\right)=0$$

(9)

where $\overrightarrow{V}$ is the velocity vector (m/s), p is the static pressure (Pa), τ is the stress tensor (Pa), and I is the identity tensor (dimensionless). k_f and k_s are the thermal conductivities of the fluid and solid (W/(m·K)). T_f and T_s are the fluid temperature and solid temperature (K), respectively.

Three-dimensional numerical simulations are performed using Ansys Fluent 2026 R1 based on the finite volume method. A pressure-based steady solver is employed, and pressure–velocity coupling is handled using the Coupled algorithm. The gradients are evaluated using the least-squares cell-based method. A second-order scheme is used for pressure interpolation, while the momentum and energy equations are discretized using the second-order upwind scheme. These settings are applied consistently to all steady-state models and operating conditions.

The convergence of the numerical solution is evaluated using both residuals and the monitored thermal–hydraulic parameters. The convergence criterion for the scaled residuals of the continuity, momentum, and energy equations is set to 10⁻⁸. The iterations are continued until the residual criteria are satisfied and T_max and ΔP no longer show appreciable changes. The converged solution is used for subsequent performance evaluation and flow-field analysis. All computations are performed on a hardware platform equipped with an Intel^® Core™ Ultra 7 251 HX processor (5.1 GHz) and 32 GB DDR5 memory.

The boundary conditions are simplified based on the physical assumptions of this study. The inlet is set as a uniform velocity inlet. The average inlet velocity (u_avg) is determined by the Reynolds number, Re. The Reynolds number ranges from 200 to 800, corresponding to an inlet velocity range of 0.45–1.81 m/s. The inlet temperature (T_in) is fixed at 293.15 K. The outlet is defined as a pressure boundary with zero gauge pressure. A partitioned heat flux boundary is applied at the bottom heating surface. The heat sink footprint is 12 mm × 12 mm. The central 4 mm × 4 mm region is defined as the hotspot region, while the remaining bottom surface is defined as the background region. The initial heat flux is set to 3 × 10⁶ W/m² in the hotspot region (q_hs) and 5 × 10⁵ W/m² in the background region (q_bg). Except for the inlet, outlet, and bottom heating surface, all other external solid surfaces are treated as adiabatic. After comparing the thermal–hydraulic performance of different structures at the same Reynolds number, the hotspot heat flux is gradually increased at Re = 400 to evaluate the maximum hotspot heat flux that the heat sink structure can withstand.

Accordingly, the applicability of the present numerical model is limited to the operating conditions and geometric configurations investigated in this study. The model considers steady, single-phase laminar cooling of a silicon heat sink using deionized water at T_in = 293.15 K. Therefore, the quantitative results should not be directly extrapolated to boiling or turbulent flow, other coolants or substrate materials, or substantially different hotspot layouts and geometries without additional validation.

2.3. Practical Manufacturability

The proposed MPSC structures are compatible with established silicon microfabrication processes based on photolithography and deep reactive-ion etching (DRIE). For the present design, the nominal etching aspect ratios are approximately 3.2 for the main microchannels (H_ch/W_ch = 0.80/0.25) and 10 for the secondary channels (H_pf/W_sch = 0.80/0.08), which are within the demonstrated capability of optimized DRIE processes [41,42]. In the present models, M5 has the largest number of intersecting secondary channels, and its manufacturing requirements are mainly associated with dimensional accuracy at the channel intersections. In practical fabrication, mask compensation and optimized DRIE parameters can be used to control etch bias and junction dimensions, thereby preserving the designed flow passages.

Surface roughness produced during DRIE can increase wall friction and consequently raise the pressure drop. Its influence is more pronounced in the secondary channels because of their larger relative roughness. Sidewall roughness can be substantially reduced through optimized etching and post-etch smoothing treatments [43]. Therefore, practical fabrication should focus on controlling the intersection geometry and sidewall quality. These manufacturing considerations provide a feasible route for fabricating the proposed MPSC structures while limiting deviations in their thermal–hydraulic performance. Accordingly, the smooth-wall numerical results can be regarded as an idealized benchmark for the proposed structures. In fabricated devices, residual surface roughness and small dimensional deviations may lead to slightly higher hydraulic resistance and some variation in thermal–hydraulic performance, while these discrepancies can be limited through appropriate fabrication control.

2.4. Performance Parameters

The following performance parameters are selected as the main evaluation indices. The Reynolds number was used to define the inlet flow condition:

$$Re={\displaystyle \frac{{\rho }_f{u}_{avg}{D}_h}{{\mu }_f}}$$

(10)

where ρ_f, u_avg, D_h, and μ_f are the coolant density (kg/m³), mean inlet velocity (m/s), hydraulic diameter (m), and dynamic viscosity (Pa·s), respectively [44].

For the rectangular inlet channel, the hydraulic diameter of the rectangular inlet microchannel is defined as

$$D_h={\displaystyle \frac{4A_{ch}}{P_{ch}}}={\displaystyle \frac{2W_{ch}{H}_{ch}}{W_{ch}+H_{ch}}}$$

(11)

where A_ch and P_ch are the inlet cross-sectional area (m²) and wetted perimeter of a single microchannel (m), respectively. W_ch and H_ch are the width and height of the inlet microchannel [44].

The maximum substrate temperature is defined as the larger value between the peak temperature in the hotspot region and that in the background region:

$$T_{max}=\mathrm{m}\mathrm{a}\mathrm{x}\left(T_{max,hs},T_{max,bg}\right)$$

(12)

where T_max,hs is the maximum temperature in the hotspot region (K), and T_max,bg is the maximum temperature in the background region (K).

The temperature uniformity of the hotspot is characterized by the mean absolute temperature difference (MATD) [5]:

$$\delta T_{hs}={\displaystyle \frac{\left|T_{max,hs}-T_{avg,hs}\right|}{2}}+{\displaystyle \frac{\left|T_{min,hs}-T_{avg,hs}\right|}{2}}$$

(13)

where δT_hs is the mean absolute temperature difference in the hotspot region (K), and T_max,hs, T_min,hs, and T_avg,hs are the maximum, minimum, and average temperatures in the hotspot region (K), respectively. A smaller value of this index indicates a more uniform surface temperature distribution and weaker hotspot concentration [5].

The convective heat transfer intensity is expressed by the Nusselt number:

$$Nu={\displaystyle \frac{Q_{tot}{D}_h}{k_f{A}_h\left(T_c-T_f\right)}}$$

(14)

where Q_tot is the total heat input (W), A_h is the solid–fluid contact area involved in heat transfer (m²), T_c is the average temperature of the heated wall (K), and T_f is the average fluid temperature (K).

The total thermal resistance is used to describe the ability of the heat sink to limit the maximum temperature under a given heat input:

$$R_{th}={\displaystyle \frac{T_{max}-T_{in}}{q_{bg}{A}_{bg}+q_{hs}{A}_{hs}}}$$

(15)

where R_th is the total thermal resistance (K/W); T_max and T_in are the maximum substrate temperature and inlet fluid temperature (K), respectively; q_bg and q_hs are the background and hotspot heat fluxes (W/m²), respectively; and A_bg and A_hs are the corresponding heating areas (m²).

The average pressure drop is defined as

$$\Delta P={\displaystyle \frac{1}{n_{ch}}}{\displaystyle \sum _{i=1}^{n_{ch}}}\Delta p_i$$

(16)

where ΔP is the average pressure drop (Pa). n_ch is the number of channels, and Δp_i is obtained from the inlet–outlet pressure difference of channel i (Pa) [44].

The surface friction factor reflects the flow resistance:

$$f={\displaystyle \frac{2D_h\Delta P}{L{\rho }_f{u}_{avg}^2}}$$

(17)

where f is the friction factor (dimensionless), and L is the effective channel length (m).

The pumping power is used to evaluate the hydraulic power required to drive the coolant:

$$P_{pump}=n_{ch}\Delta P{u}_{avg}{A}_{in}$$

(18)

where P_pump is the pumping power (W). A_in is the inlet cross-sectional area of a single channel (m²). This parameter can be used to evaluate the flow penalty caused by heat transfer enhancement.

The performance evaluation criterion is used to compare the overall benefit of the enhanced structures relative to the baseline MHS:

$$PEC={\displaystyle \frac{Nu/N{u}_0}{{\left(f/f_0\right)}^{1/3}}}$$

(19)

where Nu₀ and f₀ are the average Nu and f of the baseline MHS at the same Re, respectively. When PEC > 1, the heat transfer gain of the new structure can offset its flow resistance penalty. When PEC is clearly lower than 1, the structure can reduce the maximum temperature, but its overall thermal–hydraulic performance is insufficient. For M1, M2, M3, M4, and M5 in this study, the SPMS is used as the common baseline.

2.5. Grid Independence Test

To ensure that the numerical results are not affected by the mesh number, a grid independence test is performed for M5 at Re = 400 under q_bg = 50 W/cm² and q_hs = 300 W/cm², as M5 has relatively complex geometric features. The results are shown in Figure 2. An unstructured polyhedral mesh is used in the calculation, with local refinement near the pin-fins, secondary channels, and solid–fluid heat transfer interfaces. T_max and ΔP are used as the evaluation indices for grid independence. Meshes with different densities are established and compared. As the mesh number increases from 2.65 × 10⁶ to 12.33 × 10⁶, both T_max and ΔP gradually become stable. When the mesh number is further increased to 20.33 × 10⁶, T_max changes only from 320.241 K to 320.418 K, with a relative variation of 0.055%. ΔP changes from 5.00225 kPa to 4.98389 kPa, with a relative variation of 0.363%. The variations in both indices are lower than the convergence criterion of 1%, indicating that further mesh refinement has little effect on the calculation results. Considering both computational accuracy and cost, a mesh with approximately 12.33 × 10⁶ cells is finally selected for the following numerical simulations.

Figure 2. (a) Mesh diagram of M5. The colors are used only for visualization of different mesh regions; (b) grid independence validation.

2.6. Numerical Validation

To validate the accuracy of the numerical method used in this study, the simulation results are compared with experimental data reported in the literature. The experimental study on tree-like fractal microchannel heat sinks by Ji et al. [45] is selected as the validation reference, and the FM-30 structure with better overall performance is used for numerical validation. As shown in Figure 3, the global heat transfer coefficient (h_eff) and pressure drop (ΔP) are selected as the validation indices. As the flow rate increases from 35.5 mL/min to 121.0 mL/min, the maximum relative error of h_eff is 2.36%, and the average relative error is 1.73%. For ΔP, the maximum relative error is 3.53%, and the average relative error is 2.79%. Overall, the numerical results are consistent with the experimental data in both h_eff and ΔP. The relative errors remain low, confirming that the present model can provide dependable predictions for the subsequent simulations.

Figure 3. Validation of the present numerical model against the experimental data reported by Ji et al. [45].

2.7. Validation of Laminar and Steady-State Assumptions

2.7.1. Validation of the Laminar-Flow Assumption

To assess the laminar-flow assumption at the highest inlet Reynolds number, the local flow characteristics of M3 are examined at Re = 800, q_hs = 300 W/cm², and q_bg = 50 W/cm². The maximum local velocity in the hotspot fluid region is 4.2621 m/s, indicating local acceleration around the pin-fins and secondary-channel junctions. Since this maximum velocity is a local point value, mass-averaged velocities at representative cross-sections are used to calculate the local Reynolds numbers. At the pin-fin region, the characteristic Reynolds number is defined as

$${Re}_{\mathrm{p}\mathrm{f}}={\displaystyle \frac{{\rho }_{\mathrm{f}}{u}_{\mathrm{g}\mathrm{a}\mathrm{p}}{d}_{\mathrm{p}\mathrm{f}}}{{\mu }_{\mathrm{f}}}}$$

(20)

where d_pf = 0.54 mm is the pin-fin diameter and u_gap is the mass-averaged velocity at the minimum gap between adjacent pin-fins. Representative sections are selected at the upstream, middle, and downstream locations of the hotspot region. As shown in Table 4, the corresponding Re_pf values are 1547.7, 1050.1, and 1899.7, respectively.

Table 4. Local flow parameters and characteristic Reynolds numbers of M3 at Re = 800.

Location	ugap (m/s)	Repf	usch (m/s)	Resch
Upstream	2.4724	1547.7	3.0137	501.7
Middle	1.7634	1050.1	3.5398	563.3
Downstream	3.2981	1899.7	2.3087	362.1

The flow inside the secondary channels is evaluated using

$${Re}_{\mathrm{s}\mathrm{c}\mathrm{h}}={\displaystyle \frac{{\rho }_{\mathrm{f}}{u}_{\mathrm{s}\mathrm{c}\mathrm{h}}{D}_{\mathrm{h},\mathrm{s}\mathrm{c}\mathrm{h}}}{{\mu }_{\mathrm{f}}}}$$

(21)

where D_h,sch = 0.145 mm is the hydraulic diameter of the secondary channel and u_sch is the corresponding mass-averaged velocity. The calculated Re_sch values are 501.7, 563.3, and 362.1 at the upstream, middle, and downstream sections, respectively. A recent study on water flow in rectangular microchannels reported transition-related changes at Reynolds numbers above approximately 2200 [46]. The present Re_sch values are much lower than this level, indicating that the flow inside the secondary channels remains in the laminar regime.

2.7.2. Transient Stability Test

To further assess the applicability of the steady-state assumption and the stability of the numerical results at the upper limit of the investigated Reynolds number, a transient laminar calculation is performed for M3 at Re = 800, q_hs = 300 W/cm², and q_bg = 50 W/cm². The fully converged steady-state solution is used as the initial field. The physical models, material properties, and boundary conditions are kept unchanged. This treatment avoids the startup process caused by reinitialization and allows the stability of the converged solution to be examined directly under transient calculation. A time step of 1 × 10⁻⁶ s is used, and the total flow time is 0.02 s.

Figure 4 shows the transient histories of T_max and ΔP. Both parameters remain within narrow ranges and gradually approach stable levels during the calculation. Table 5 compares the steady-state results with the time-averaged transient results. The time-averaged T_max and ΔP are 313.1463 K and 21.8014 kPa, while the steady-state values are 313.1434 K and 21.7818 kPa, respectively. The relative deviations are only 0.00092% and 0.09023%. The transient results are very close to the steady-state predictions. This confirms that no significant time-dependent variation develops in the monitored thermal–hydraulic parameters at Re = 800, supporting the use of the steady-state model for the present thermal–hydraulic analysis.

Figure 4. Transient histories of the thermal–hydraulic responses of M3 at Re = 800.

Table 5. Comparison of steady-state and time-averaged transient results for M3 at Re = 800.

Parameter	Steady-State Result	Time-Averaged Transient Result	Relative Deviation (%)
Tmax (K)	313.1434	313.1463	0.00092
ΔP (kPa)	21.78182	21.80148	0.09023

3. Results and Discussion

This study evaluates the thermal–hydraulic performance of MPSC microchannel heat sinks (M1–M5) under several thermal and flow conditions. First, the performance of M1–M5 and the SPMS is compared under the same heat flux, and the effect of Reynolds number is examined. Then the hotspot heat flux is gradually increased at Re = 400 to evaluate the hotspot load capacity of all designs. The effect of the pin-fin/channel height ratio on the thermal–hydraulic performance of the heat sink is analyzed under the same Reynolds number and hotspot heat flux. The detailed conditions are listed in Table 6.

Table 6. Thermal and flow conditions used in this study.

	qbg (W/cm2)	qhs (W/cm2)	Re	Pin-Fin/Channel Height Ratio (RPH)
Case 1	50	300	200–800	—
Case 2	50	300–800	400	—
Case 3	50	300	200–800	0.7–1.0

3.1. Effect of Different Micro Pin-Fin Structures

The heat transfer and flow performance of the SPMS and M1–M5 are compared at the q_bg = 50 W/cm² and q_hs = 300 W/cm².

Figure 5 shows the bottom surface temperature distributions of the SPMS and M1–M5 at Re = 400. All results use the same temperature scale. For the SPMS, heat is concentrated in the central hotspot, and the peak temperature reaches 333.2 K. This indicates that straight channels alone cannot suppress the central local hotspot. After the pin-fin structures are introduced, the bottom surface temperature is clearly reduced. The surface temperatures in the hotspot region of M2–M5 are all lower than that of M1. Among them, M3 shows the lowest temperature, with a peak value of 319.5 K, which is 13.7 K lower than that of the SPMS. In addition, along the transverse direction of the flow, M1–M5 show lower surface temperatures. This indicates that the secondary channels enhance transverse fluid disturbance and can effectively suppress the central hotspot. Figure 6 depicts temperature variation of each structure along the flow direction at the bottom centerline, y = 6 mm. When the fluid enters the hotspot region, the temperature increases rapidly, and the differences between different structures become more pronounced. The temperatures of M1–M5 are lower than that of the SPMS along the entire centerline. M2–M5 also show a smoother temperature rise, indicating weaker local heat accumulation and improved surface temperature uniformity. The associated near-wall heat transfer behavior is further examined below using the flow and thermal fields.

Figure 5. Bottom surface temperature distributions of the SPMS and MPSC structures (M1–M5). The maximum temperature is marked in the lower right corner. All cases are presented using the same temperature range and color scale to ensure direct comparison among different configurations.

Figure 6. Bottom surface temperature profiles of different structures along the flow direction at y = 6 mm at Re = 400.

To explain the differences in heat transfer performance, flow behavior in the SPMS and M1–M5 is analyzed. Figure 7 shows the velocity distributions in the central hotspot region at Re = 400. The central eight channels passing through the hotspot are selected, and the section is located at z = 0.6 mm, parallel to the x–y plane. In the SPMS, the fluid flows uniformly along the straight channels, and the thermal boundary layer continues to develop along the flow direction. After the central pin-fins are introduced, the fluid in the hotspot region of M1–M5 is clearly accelerated. The high-velocity regions are mainly located in the main channels and the constricted regions around the pin-fins. The local velocity approaches 2.0 m/s, which is much higher than the overall velocity in the SPMS. The secondary channels change the flow distribution. The average flow velocity in M2–M5 is lower than that in M1 because through or crossed secondary flow paths are developed inside the pin-fins. Part of the coolant is diverted through the interior of the pin-fin array, which reduces the flow velocity and pressure drop.

Figure 7. Velocity field distributions in the x–y section of the central hotspot region for the SPMS and M1–M5 at Re = 400.

Figure 8 further shows the velocity streamlines around the pin-fin array. In M1, the fluid separates at the front edge of the pin-fins, accelerates in the gaps, and forms low-velocity wake regions behind the pin-fins. Continuous high-velocity regions are observed in M2 and M4, indicating that the secondary channels provide through-flow paths for the fluid and reduce the local blockage caused by the pin-fins. In M4, the fluid also shows clear transverse low-velocity flow with small vortices. M3 has three-directional secondary channels, and its streamline distribution is more balanced. It maintains strong local impingement while avoiding excessive transverse flow dispersion. M5 has the largest number of internal connected paths. The fluid is distributed into more small channels, and the local high-velocity core becomes less concentrated. The results show that M1–M5 have higher flow velocities and larger pressure drops than the SPMS. For M2–M5, the embedded secondary channels allow the fluid to enter the pin-fins. This reduces the dependence on external bypass flow around the pin-fins and promotes transverse exchange between adjacent main channels. The secondary channels introduce pressure-relief paths in the pin-fin array. Part of the flow is shared by the internal channels, local acceleration occurs in the internal channels and junction regions, and small-scale vortices are generated downstream of the pin-fins. These effects enhance the mixing of hot and cold fluids.

Figure 8. Local velocity streamlines in the pin-fin region of SPMS and M1–M5 at Re = 400.

Compared with M1, M2–M5 not only increase the solid–fluid contact area but also reduce the pressure penalty caused by solid pin-fins. These channels create additional coolant paths in the hotspot region. They refresh the near-wall fluid and weaken thermal boundary-layer growth, thereby improving heat transfer around the pin-fins. The corresponding changes in the near-wall thermal field are examined directly in Figure 9.

Figure 9. Cross-sectional temperature and near-wall temperature-gradient distributions of the SPMS, M1, and M3 at Re = 400. (a–c) Fluid temperature contours of the SPMS, M1, and M3; (d–f) temperature-gradient magnitude $\left|\nabla T\right|$ of the SPMS, M1, and M3.

Figure 9 compares the thermal fields of the SPMS, M1, and M3 at Re = 400. The temperature contours are plotted on longitudinal x–z sections at y = 6.5 mm for the SPMS and y = 6.0 mm for M1 and M3. The near-wall temperature-gradient distributions are shown over z = 0.20–0.25 mm on the corresponding sections.

Figure 9a–c show that the SPMS maintains a continuous heated-fluid layer adjacent to the bottom wall, which expands downstream. This reflects the continuous development of the thermal boundary layer in the straight microchannel. In M1, the solid pin-fins interrupt this pattern. The near-wall isotherms are redistributed between successive pin-fins and redevelop downstream of the local disturbances. This behavior agrees with the flow separation and local acceleration observed in Figure 8. For M3, the longitudinal section passes through a streamwise secondary channel inside the pin-fins. The internal passage allows coolant to pass through the pin-fin region and changes the local isotherm pattern. Relative to M1, the near-wall temperature field shows repeated local redistribution along the secondary-channel path. Together with the streamlines, this indicates that the secondary channels promote local coolant exchange and modify the local redevelopment of the thermal boundary layer.

To characterize the local spatial variation of the thermal field near the heated wall, the temperature-gradient magnitude is defined as

$$\left|\nabla T\right|=\sqrt{{\left({\displaystyle \frac{\partial T}{\partial x}}\right)}^2+{\left({\displaystyle \frac{\partial T}{\partial y}}\right)}^2+{\left({\displaystyle \frac{\partial T}{\partial z}}\right)}^2}$$

(22)

where T is temperature (K), and x, y, and z are the Cartesian coordinates (m). $\left|\nabla T\right|$ has a unit of K/m and reflects the local spatial variation in temperature.

Figure 9d–f further compare the temperature-gradient magnitude $\left|\nabla T\right|$ within z = 0.20–0.25 mm. In the SPMS, the near-wall gradient band remains relatively continuous along the flow direction. In M1, the pin-fin disturbance causes clear local changes in the gradient distribution. M3 shows further local redistribution of the gradient field along the secondary-channel path. The comparison indicates that the secondary channels redistribute the near-wall temperature gradient. Combined with the temperature contours and velocity streamlines, these results support that the internal channels promote coolant exchange and redevelopment of the thermal boundary layer.

3.2. Effect of Reynolds Number on the Thermal–Hydraulic Performance of Different Structures

The thermal–hydraulic performance of each structure is investigated under different Re values at the same hotspot heat flux of 300 W/cm² and background heat flux of 50 W/cm².

Figure 10 shows the variations in the performance parameters of different structures over Re = 200–800. Figure 10a shows the maximum temperature (T_max) of the SPMS and M1–M5. As Re increases, T_max decreases continuously for all six structures, but the marginal benefit of increasing the flow rate gradually weakens. M1–M5 show much lower T_max values than the SPMS over the entire Re range, and M3 has the lowest T_max. At Re = 200 and 800, the T_max of M3 is about 12.0 K and 15.4 K lower than that of the SPMS, respectively. Compared with M1, the reductions are 5.4 K and 3.1 K, respectively. This is because the secondary channels promote flow diversion and recombination, which disrupts the thermal boundary layer. Figure 10b shows the variation in MATD (δT_hs) over Re = 200–800. As Re increases, δT_hs decreases continuously for all six structures. The SPMS always has the largest δT_hs, indicating that straight microchannels have limited ability to reduce temperature accumulation caused by the central hotspot. The secondary-channel structures in M2–M5 further improve fluid redistribution in the pin-fin region, making the temperature field in the hotspot region more uniform. Among them, M5 has the lowest δT_hs. At Re = 200 and 800, the δT_hs of M5 is 4.0 K (27.9%) and 5.8 K (53.7%) lower than that of the SPMS, respectively. This is attributed to the strong flow diversion effect of the six-directional secondary channels in M5. As the flow velocity increases, fluid mixing is enhanced, and the advantage of the secondary channels in improving temperature uniformity becomes more pronounced.

Figure 10. Variations in the performance parameters of SPMS and M1–M5 at Re = 200–800. (a) T_max; (b) δT_hs; (c) ΔP; (d) R_th.

Figure 10c shows the variation in channel pressure drop (ΔP). All structures show a clear nonlinear increase in ΔP. Compared with the SPMS, M1–M5 show a faster increase in pressure drop. M1 has the strongest blockage effect, and its pressure drop is 125.1% higher than that of the SPMS at Re = 800. Compared with M1, M2–M5 show lower pressure drops overall. Among them, the pressure drop of M5 is 13.22 kPa at Re = 800, which is close to that of the SPMS. This is because the embedded secondary channels provide additional through-flow paths for the coolant. Part of the flow is transferred into the internal channels of the pin-fins, which reduces the flow velocity and lowers part of the blockage loss while maintaining flow disturbance in the hotspot region. Figure 10d shows the variation in total thermal resistance (R_th). At Re = 200 and 800, the R_th of M3 is 24.9% and 43.3% lower than that of the SPMS, respectively. The internal secondary channels promote fluid exchange and allow heat to be transferred more effectively to the coolant. This advantage becomes more evident as Re increases.

Figure 11 depicts PEC for M1–M5 over Re = 200–800. As Re increases, the PEC values of M1–M5 all show an upward trend. At higher flow velocities, the heat transfer gains of M1–M5 gradually exceed their additional flow resistance. The PEC values of M2–M5 are clearly higher than that of M1 at Re = 200–800. Among them, M3 shows the best performance, with a maximum PEC of 1.178. This indicates that the embedded secondary channels maintain heat transfer enhancement and reduce pressure drop penalty through internal flow diversion, leading to better overall thermal–hydraulic performance.

Figure 11. Variation in PEC for M1–M5 at Re = 200–800.

Since different structures have different flow resistances, comparison at the same Re alone cannot directly reflect the energy consumption level. Therefore, Figure 12a further compares R_th among different structures using pumping power (P_pump) as the reference. M1–M5 all show lower R_th values at similar P_pump levels. At 5.8 mW, the R_th of M3 is about 0.32 K/W, which is about 25.0% lower than that of the SPMS. This indicates that the MPSC structures already have a clear thermal resistance advantage under low-energy-consumption conditions. To reach a similar thermal resistance level, the SPMS requires a pumping power of about 92 mW. At a comparable pumping power, the R_th of M3 decreases to 0.20 K/W, corresponding to a reduction of 36.8%. This shows that the MPSC structures can improve heat transfer while maintaining good flow performance, which is of practical engineering value. Figure 12b shows a similar nonlinear trend in δT_hs. With secondary channels, M2–M5 achieve better temperature uniformity at lower pumping power. For the SPMS, δT_hs is 10.81 K at 91.9 mW, while M3 reduces δT_hs by 45.1% at a similar P_pump level.

Figure 12. Variations in (a) total thermal resistance (R_th) and (b) MATD (δT_hs) with pumping power (P_pump) over Re = 200–800. (c) Thermal resistance reduction ratio (ΔR_th) and pumping power increase ratio (ΔP_pump) of the SPMS; (d) ΔR_th and ΔP_pump of M3.

Figure 12a,b show diminishing thermal returns with increasing P_pump, which is not cost-effective in engineering applications. Figure 12c,d further quantify the relationship between R_th benefit and P_pump consumption. Taking Re = 200 as the baseline condition and Re = 800 as the normalized endpoint, the thermal resistance reduction ratio and pumping power increase ratio are defined as

$$\Delta R_{th}\left(R{e}_i\right)={\displaystyle \frac{R_{th}\left(200\right)-R_{th}\left(R{e}_i\right)}{R_{th}\left(200\right)-R_{th}\left(800\right)}}\times 100\%$$

(23)

$$\Delta P_{pump}\left(R{e}_i\right)={\displaystyle \frac{P_{pump}\left(R{e}_i\right)-P_{pump}\left(200\right)}{P_{pump}\left(800\right)-P_{pump}\left(200\right)}}\times 100\%$$

(24)

where ΔR_th (%) represents the cumulative thermal resistance reduction ratio obtained when Re increases from 200 to Re_i, and ΔP_pump (%) represents the cumulative pumping power increase ratio required to reach this condition. For the SPMS, raising Re from 200 to 300 consumes only 6.6% of the total pumping-power increase but delivers 42.1% of the total R_th reduction. In contrast, the final increase from Re = 600 to 800 requires another 51.7% of pumping power for only 12.8% additional R_th reduction. M3 shows a similar trend, but its absolute performance improvement is better than that of the SPMS. Practical operation should therefore avoid unnecessarily high flow rates.

3.3. Heat Transfer Performance Analysis Under Different Hotspot Heat Fluxes

The safe operating temperature of a microprocessor is 358.15 K. To evaluate the load capacity of different structures, the hotspot heat flux is gradually increased at Re = 400, and the heat transfer performance of different structures is systematically compared. It should be noted that, in actual chip packaging, the heat flux that the heat sink can safely withstand may be lower because of the thermal interface layer, the inherent thermal resistance inside the chip, and the decrease in silicon thermal conductivity with increasing temperature.

Figure 13 depicts the surface temperature distributions of the SPMS, M1, and M3 under different q_hs at Re = 400. The SPMS shows the largest temperature rise and approaches the safe temperature limit at 500 W/cm². In contrast, the T_max of M1 is clearly lower than that of the SPMS under the same heat flux. M3 shows stronger hotspot suppression capability. As q_hs increases from 300 to 800 W/cm², the T_max of M3 rises from 319.5 K to 357.5 K. Its heat load capacity is 60% higher than that of the SPMS, and its temperature distribution is also smoother.

Figure 13. Bottom surface temperature distributions of the SPMS, M1, and M3 under different hotspot heat fluxes at Re = 400.

Figure 14a shows the variation in T_max under different q_hs values. As q_hs increases from 300 to 800 W/cm², M1–M5 all reduce T_max significantly compared with the SPMS, and their advantage becomes more evident in the high-heat-flux range. M3 has the lowest T_max. Its temperature reduction relative to the SPMS increases from 13.7 K at 300 W/cm² to 27.0 K at 600 W/cm². This is because the secondary channels improve the heat transfer capacity in the pin-fin region, and their effect becomes stronger as the hotspot load increases. The results show that a reasonable secondary-channel design can significantly improve the hotspot load capacity of the heat sink. Figure 14b shows the variation in δT_hs under different q_hs values. When q_hs increases from 300 to 600 W/cm², the reduction in δT_hs of M5 relative to the SPMS increases from 4.9 K to 9.9 K, with a relative reduction of 39.0% at 600 W/cm². This indicates that the improvement in temperature uniformity by the secondary channels becomes more evident as the hotspot load increases. Although M5 gives the lowest δT_hs, M3 maintains the lowest T_max over the investigated heat-flux range. This difference indicates that broader flow redistribution does not necessarily provide stronger local heat removal. The flow mechanism responsible for this difference is further examined in Figure 15.

Figure 14. (a) T_max and (b) δT_hs under different hotspot heat fluxes.

Figure 15. Local momentum-flux indicator and bottom-wall shear-stress distributions of M3–M5 at Re = 400 and q_hs = 800 W/cm². (a–c) Momentum-flux indicator; (d–f) wall shear stress.

Figure 15 compares the local flow momentum and bottom-wall shear stress of M3–M5 under the high-heat-flux condition. To characterize the local inertial momentum carried by the coolant, a local momentum-flux indicator is defined as

$$I_m={\rho }_f{u}^2$$

(25)

where ρ_f is the coolant density and u is the local velocity magnitude. I_m has a unit of Pa and represents the strength of local inertial momentum.

For the laminar Newtonian flow considered here, the wall shear stress is expressed as

$${\tau }_w={\mu }_f{\displaystyle \frac{\partial u_t}{\partial n}}$$

(26)

where τ_w is the wall shear stress, μ_f is the dynamic viscosity of the coolant, u_t is the wall-tangential velocity, and n is the wall-normal direction. The momentum distributions are plotted on the x–y plane at z = 0.6 mm, while the wall shear stress is evaluated on the heated bottom wall at z = 0.2 mm.

Figure 15a–c show that the number of secondary-channel branches has a clear effect on the spatial distribution of momentum. In M3, high-I_m regions of approximately 2.9 × 10³–4.2 × 10³ Pa are continuously maintained in the main flow passages and streamwise secondary channels, while the inclined branches provide additional fluid exchange. With additional internal branches in M4, the high-momentum regions are distributed among more flow paths and their continuity begins to decrease. This trend becomes more evident in M5. Most of its inclined internal branches have I_m values below approximately 1.3 × 10³ Pa, while the main flow passages are mostly within approximately 1.7 × 10³–2.5 × 10³ Pa. The six-directional secondary channels distribute the coolant among more internal flow paths, but the high-momentum cores become less concentrated. Therefore, increasing the number of secondary-channel branches broadens the flow redistribution but also reduces the local momentum retained in individual effective flow paths.

The wall shear stress in Figure 15d–f shows a corresponding change. M3 maintains relatively continuous moderate-to-high-shear-stress regions along the main flow paths and near the channel junctions, with some local values approaching approximately 140–160 Pa. In M4, these regions become weaker and more dispersed. M5 shows the most dispersed shear-stress distribution, and the wall shear stress over most of the internal-branch region remains below approximately 60 Pa. The spatial correspondence between the momentum and wall-shear distributions indicates that the additional internal branches in M4 and M5 do not produce the same level of wall-adjacent flow action. As the coolant is distributed among more flow paths, the local near-wall shear is reduced in part of the hotspot region.

Under high thermal loads, the more dispersed flow momentum and weaker near-wall shear in M4, especially M5, reduce local coolant renewal near the heated surface and limit further disturbance of the thermal boundary layer. On the other hand, M5 still provides broader flow redistribution, which is consistent with its lower δT_hs and better temperature uniformity. These results indicate that M3 achieves a better balance between flow redistribution and local momentum retention.

Figure 16 shows the variation in R_th for different structures under different hotspot heat fluxes. M1 shows a clear improvement compared with the SPMS, but its R_th is always higher than those of M2–M5. M3 has the lowest R_th among all designs. Under q_hs = 300–600 W/cm², its R_th is about 35.0% lower than that of the SPMS, indicating that the MPSC structures can maintain a stable thermal resistance reduction under different heat load levels.

Figure 16. Variation in R_th under different hotspot heat fluxes.

3.4. Effect of Relative Pin-Fin Height on Thermal–Hydraulic Performance

Pin-fin height influences both heat-transfer area and the fraction of coolant entering the secondary channels, thereby affecting thermal performance and pressure drop. Its effect is evaluated using the relative pin-fin height (RPH):

$$RPH={\displaystyle \frac{H_{pf}}{H_{ch}}}$$

(27)

where H_pf and H_ch are the pin-fin height and channel height (m), respectively.

In this section, M3, which has the highest PEC, is selected as the representative structure. Figure 17 shows the performance variation of M3 under RPH = 0.7–1.0 over Re = 200–800. Figure 17a shows that ΔP increases with RPH. The ΔP reaches 21.78 kPa at RPH = 1.0, while it is only 11.20 kPa at RPH = 0.7, corresponding to a reduction of 48.6%. This is because increasing the pin-fin height compresses the upper flow space and increases the form resistance. When RPH increases from 0.9 to 1.0, ΔP rises more sharply. This indicates that when the pin-fins approach the channel top wall, the flow resistance does not increase linearly. Instead, it worsens rapidly because the top clearance is greatly reduced. Therefore, an appropriate reduction in pin-fin height can significantly reduce flow blockage and save pumping power, especially at high flow velocities.

Figure 17. Variations in the performance parameters of M3 under different RPH values over Re = 200–800. (a) ΔP; (b) R_th; (c) R_th with P_pump; (d) PEC.

Figure 17b shows that at Re = 800, when RPH = 1.0, R_th is 0.1790 K/W, while it is 0.2433 K/W at RPH = 0.7, corresponding to a reduction of about 26.4%. Increasing the pin-fin height enlarges the heat transfer area and allows more coolant to participate in the heat transfer process through the pin-fins and secondary channels. Figure 17c further shows the need for RPH optimization from the perspective of pumping power. At Re = 600, the pumping power of RPH = 0.9 is 74.39 mW, which is 60.5% lower than that of RPH = 1.0 at Re = 800, 188.41 mW. However, its R_th increases by only about 19.5%. Therefore, M3 at RPH = 0.9 can achieve a large saving in pumping power with only a small loss in thermal performance. Figure 17d shows M3 has the lowest PEC at RPH = 0.7, indicating that the flow disturbance is insufficient when the pin-fin height is too low. At Re = 200–400, RPH = 1.0 gives the highest PEC because the pressure drop penalty is not significant at low flow velocities. When Re approaches 500, RPH = 0.9 shows higher PEC. At Re = 800, the PEC reaches 1.2106, which is about 2.8% higher than that at RPH = 1.0. This indicates that at higher flow velocities, the PEC at RPH = 1.0 is weakened by the sharp increase in flow resistance. In contrast, RPH = 0.9 can avoid excessive channel blockage and lead to better overall performance.

The above analysis shows that the effect of RPH on PEC is regulated by Re, indicating a coupled relationship between them. To further analyze the effects of Re and RPH on PEC, a quadratic response surface model is used to fit PEC in this section.

Re is normalized as follows:

$$R_n={\displaystyle \frac{Re-200}{600}},200\le Re\le 800$$

(28)

Based on the normalized Re, a quadratic response surface model including linear terms, quadratic terms, and an interaction term is adopted:

$$PEC=a_0+a_1{R}_n+a_{2}RPH+a_3{R}_n^2+a_{4}RP{H}^2+a_5{R}_{n}RPH$$

(29)

where a₀, a₁, a₂, a₃, a₄, and a₅ are constant coefficients.

The fitting process is performed using the least-squares method, with the PEC data of M3 under different combinations of Re and RPH used as sample points. The fitted quadratic response surface equation is obtained as

$$PEC=0.33628+0.08630R_n+1.36828RPH-0.07045R_n^2-0.64804RP{H}^2+0.13088R_{n}RPH$$

(30)

where 200 ≤ Re ≤ 800 and 0.7 ≤ RPH ≤ 1.0.

Figure 18 shows the fitting results of PEC. Figure 18a presents the fitting error curve. All sample points fall within an error range of ±4%, which is acceptable. Figure 18b further shows the variation trend of PEC in the continuous Re-RPH parameter space. The surface shows clear curvature along the RPH direction, indicating that the effect of relative pin-fin height on PEC is constrained by the pressure drop. Combined with the results in Figure 17d, the maximum RPH does not necessarily correspond to the optimal PEC. At low flow velocities, increasing RPH is more favorable for using the heat transfer benefits from the larger heat transfer area and secondary channels. At higher flow velocities, the pumping power penalty caused by an excessively high RPH begins to weaken the overall benefit. Therefore, RPH optimization should be based on the balance between heat transfer gain and flow penalty, rather than simply pursuing the maximum pin-fin height. This provides a reference for selecting a suitable pin-fin height and flow condition according to practical cooling requirements.

Figure 18. Quadratic response surface fitting of PEC for the M3 structure with respect to Re and RPH: (a) Fitting error; (b) response surface.

4. Conclusions

This study proposes a hybrid microchannel heat sink featuring micro pin-fins with secondary channels for hotspot thermal management in high-heat-flux electronic devices. Straight microchannels are used in the background region, while staggered pin-fins are arranged in the hotspot region. Five models are considered: solid pin-fins without secondary channels (M1) and pin-fins with one-, three-, four-, and six-directional secondary channels (M2–M5). The effects of pin-fin structure, Reynolds number, hotspot heat flux, and relative pin-fin height on the thermal–hydraulic performance of the heat sink are investigated. The main conclusions are as follows:

• Compared with the SPMS, M1–M5 significantly improve hotspot suppression. The secondary channels provide internal flow diversion and transverse exchange paths inside the pin-fins. They reduce the local blockage caused by solid pin-fins and promote redevelopment of the thermal boundary layer.
• Over Re = 200–800, M3 shows the best overall performance. Compared with the SPMS, the R_th of M3 is reduced by 24.9% and 43.3% at Re = 200 and 800, respectively, and the maximum PEC reaches 1.178.
• The MPSC structures can achieve stronger hotspot cooling at lower pumping power. At similar P_pump, the R_th of M3 is 36.8% lower than the minimum R_th attainable by the SPMS. Meanwhile, the marginal reduction in R_th gradually weakens as P_pump increases. This indicates that the coolant flow velocity should be reasonably selected in practical engineering.
• The MPSC structures have a higher hotspot load capacity. M3 can control the maximum temperature at 357.5 K under q_hs = 800 W/cm². Compared with the SPMS, M3 increases the allowable hotspot heat flux from 500 W/cm² to 800 W/cm², corresponding to a 60% improvement in hotspot load capacity.
• The relative pin-fin height, RPH, has a coupled effect on the thermal–hydraulic performance. At Re = 800, the PEC of RPH = 0.9 reaches 1.2106, which is about 2.8% higher than that of RPH = 1.0. The response surface fitting further shows that the pin-fin height should be designed according to the flow condition and pumping power constraint.

Overall, the hybrid micro pin-fin structure with embedded secondary channels can balance hotspot suppression and pressure drop control. Among the tested structures, M3 shows the best overall thermal–hydraulic performance. This study provides a reference for hotspot suppression in high-power electronic devices and for the thermal–hydraulic optimization of microchannel heat sinks.

Future work will focus on experimental evaluation of the MPSC heat sink using both controlled constant-power heat sources and representative high-power electronic devices. These studies will be used to verify its thermal–hydraulic performance under practical operating conditions and further evaluate the engineering applicability of the proposed design.

The main limitations of the present study and the corresponding future work are summarized as follows:

• The present study is mainly based on CFD analysis. The immediate next step is experimental validation. Future work will fabricate silicon-based MPSC prototypes and compare the measured thermal resistance and pressure drop with the corresponding CFD predictions.
• The present model uses a constant thermal conductivity for silicon and does not account for its temperature dependence. Future simulations will incorporate the temperature-dependent thermal conductivity k_si(T), particularly for high-heat-flux conditions.
• The present design has not yet been integrated with other advanced microchannel architectures. Future work will explore combinations of MPSCs with manifold-fed or double-layer microchannels to further improve coolant distribution and shorten the effective flow path while retaining the local heat-transfer enhancement of MPSCs.