Introduction

Technology and the rapid updating of technological devices bring with them heating problems. As a result, the issue of cooling electronic devices has emerged. While researchers have introduced many innovations, the main ones are passive methods to improve heat transfer and provide processor cooling.

Research in the field of heat sinks has yielded valuable insights into their thermal performance and cooling capabilities. Chiu et al. [1, 2] conducted experimental and numerical studies, highlighting the importance of pin diameter and density on heat transfer and temperature uniformity. Additionally, the study examined the hydrothermal properties of micro-pin configurations with a circular diameter, revealing the advantages of a specific inlet and double outlet design in terms of heat transfer coefficient [3]. Fin sizes [4], shapes [5], number of fins, gap areas, fin arrangement, heat surface location, and the coolant type used are just a few of the variables that affect how well heat sinks perform. Among the different designs, the V-type heat sink stands out because of its unique angle and distance configuration, which helps to keep temperatures evenly distributed and create vortices, which improves heat transfer performance [3]. Further investigations have explored the relationship between pin height and heat transfer performance in micro-pin-type heat sinks, revealing an inverse correlation [6]. Studies also examined the heat transfer and flow characteristics of various fin shapes in micro-channel heat sinks, indicating a lower Nusselt number with equal fin height [7]. Efforts to optimize heat sink designs continue, with topology optimizations leading to reductions in pressure drop and increases in the Nusselt number [8]. Given the paramount role of surface area in cooling processes, the exploration of heat sinks with diverse geometric structures remains of utmost importance to researchers. In this regard, perforated structures, such as triangular, rectangular, and square fins, have significantly improved heat transfer performance, demonstrating the potential for 51–64% higher heat transfer [9,10,11]. RSM and CFD studies have further refined our understanding, revealing the advantages of square perforated fins with larger opening spacing and circular perforated fins with smaller spacing [12]. Towsyfyan et al. [13] investigated the thermal performance of hollow circular pins in heat sinks, reporting a remarkable 20% improvement. The use of non-Newtonian nanofluids with CuO nanoparticles in offset ribbon-finned micro-channel heat sinks displayed a 2.29-fold increase in heat transfer relative to pressure drop when compared to base fluids [14]. The proliferation of 3D printing has introduced innovative possibilities in heat sink design, leading to the creation of compact heat sinks with unconventional geometries and high PEC values [15]. A comparison of different heat sink types revealed that perforated structures consistently outperformed their non-perforated counterparts, particularly in terms of heat transfer [16]. Adding longer surfaces like ribs, pin fins, and bifurcation plates to micro-channel heat sinks has made them work better. In terms of heat transfer and pressure drop [17], bifurcated divergent micro-channel heat sinks are better than parallel micro-channel heat sinks [18]. Advancements in heat sink design have also been realized through techniques such as rib cavities, alongside arc, S, and wave-type geometries. The ribbed and hollow wave channel structures proved to be the most effective, achieving a performance evaluation criterion of 1.68 [19]. When phase change material (PCM) was added to heat sinks with double-sided spiral structures, the critical temperatures went down as the fin diameter got bigger [20]. Moreover, increasing the channel height within double-layer mini-channels improved heat transfer performance by up to 52.8% [21]. Researchers looked at how heat sinks with fin-type pin structures behaved in water and found that staggered arrangements always worked better than inline ones [22]. Numerical studies of mini-channel heat sinks with different secondary channel shapes showed better heat transfer efficiency and lower thermal resistance, leading to a maximum PEC of 1.69 [23]. When rectangular micro-channels and pin heat sinks were combined to fix the problem of hotspots in electronics, they worked well, cutting the amount of power needed for pumping by 98.4% [24]. In addition, using both active and passive cooling mechanisms together in heat exchangers with synthetic jets and phase change material (PCM) has made cooling much better [25]. Many researchers have actively conducted important studies on the cooling effects and entropy investigations of the geometrical structure of heat sinks [26,27,28,29,30].

The use of nanofluids with geometry changes is also quite common. Nanofluids are increasingly used in heat transfer enhancement due to their thermal conductivity and density factor. Studies have shown that Al2O3/water nanofluid can improve heat transfer by 68% in fin-type heat sinks. The sinusoidal winding fin design is more efficient than the serpentine design. Adding straight fins or extending fin lengths can improve heat sink box performance. Nanofluid-cooled finned structures show maximum Nu increases at high pressure drops [27,28,29,30,31,32,33,34,35,36,37,38], The performance of nanofluids such as wavy structures, twisted structures, and mono-hybrids in cooling heat sinks is also investigated [39,40,41,42].

These extensive studies and developments contribute to a deeper understanding of heat sinks, shedding light on their critical role in maintaining the thermal stability of electronic devices. Researchers continue to explore innovative designs and optimization techniques to further enhance heat sink performance and advance the field of thermal management. Central processing unit (CPU) cooling and methods have been extensively studied through experimental and numerical investigations, as mentioned previously. Notably, the circular cross-section micro-pin structure is the most widely used geometry type [1,2,3, 6, 7, 24, 43]. This study examines the cooling effects and thermal performance improvements of circular, triangular, square, and hexagonal heat sinks, as well as their perforated versions, under four different pressure boundary conditions. Unlike the previous literature, each geometry is examined separately. Previous investigations have only examined circular or triangular structures alone, or their perforated structures in isolation. This numerical study analyzes all geometries under the same boundary condition and emphasizes cooling effects. Additionally, the study examines in detail the effect of temperature distributions on cooling using flow imaging techniques with using FLUENT. To ensure efficient parametric comparison, the perforated structures were placed in the same geometrical area and at the same gap distance.

Geometry and boundary conditions

The current study investigates the cooling performances of eight different geometries of heat sinks under the same boundary conditions. With the use of an experimental investigation from the literature, all geometry is compared and validated [1]. The geometries studied are circular, square, triangular, hexagonal, and their perforated structure. All the examined geometries have an edge length or diameter of 0.6 mm. In addition, the distance between the geometries is 0.6 mm for all geometries. In the experimental study of [1], the heat flux applied to the heat sink’s base (10 × 10 mm2) as a boundary condition is 300 kW m−2. In addition, water is used as a cooling fluid in the heat sink. The inlet temperature of the water in the system is 300 K. Another boundary condition was compared under four pressures (689, 1370, 2040, and 2750 Pa). The height of the heat sink is 4 mm, and the height of the structures, such as circles, squares, and triangles in the pin structure, is 2.5 mm. The base height of the heat sink is 1.5 mm. The aspect distances of the heat sink are 13*21 mm. Figure 1 shows the whole geometry, while Fig. 2 shows the detailed geometric parameters.

Fig. 1
figure 1

The geometries studied, the geometries in the left column are circular, square, triangular and hexagonal, respectively, and the ones on the right are their perforated versions

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Fig. 2
figure 2

a Geometric details of pin–fin heat sinks, b boundary conditions, c. flow chart

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Numerical analysis

In this study, where processor cooling performances are compared, the CFD module of the ANSYS program was used. All geometries were drawn in ANSYS, and the mesh structure was created here. Meshes with three different element numbers were created for mesh independence in all eight geometries. Approximately 1.5 million, 3.2 million, and 5.5 million tetrahedral mesh structures were created. In order to obtain efficient results from the obtained meshes, a tighter mesh structure was created in the flow domain. According to the results obtained, it was seen that mesh structures with approximately 3.2 million elements in each geometry were sufficient. Figure 3 shows the case, where the tetrahedron-type mesh structure is transformed into a polyhedron type. The reason for using a polyhedral-type mesh structure is that it reduces the number of mesh elements and shortens the solution time. The range of polyhedron-type mesh element numbers used for all geometries is in the range of 400000–800000. The number of elements indicated shows that the average reduces the number of elements to approximately one quarter.

Fig. 3
figure 3

Polyhedron mesh types for square and triangular-type pin–fin heat sinks

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A SIMPLE solver was chosen. The boundary conditions used are inlet temperature, pressure, and constant heat flux, which are 689, 1370, 2040, 2750 Pa, 300 K, and 300 kW, respectively. Constant heat flux 300 kW m−2 was applied to the bottom side of the heat sink. The adiabatic boundary condition was applied to all walls of the heat sink. Convergence requirements for continuity, momentum, and the energy equation were established at 10–5, 10–5, and 10–8, respectively, to track the calculation’s residual.

The equations used in the numerical method (CFD) are as follows [3]:

Mass conservation:

$$\frac{{\partial v}_{{\text{x}}}}{\partial x}+\frac{{\partial v}_{{\text{y}}}}{\partial y}+\frac{{\partial v}_{{\text{z}}}}{\partial z}=0$$
(1)
$$\rho \left({v}_{{\text{x}}}\frac{{\partial v}_{{\text{x}}}}{\partial x}+{v}_{{\text{y}}}\frac{{\partial v}_{{\text{y}}}}{\partial y}+{v}_{{\text{z}}}\frac{{\partial v}_{{\text{z}}}}{\partial z}\right)=-\frac{\partial P}{\partial x}+\mu \left(\frac{{\partial }^{2}{v}_{{\text{x}}}}{\partial {x}^{2}}+\frac{{\partial }^{2}{v}_{{\text{x}}}}{\partial {y}^{2}}+\frac{{\partial }^{2}{v}_{{\text{x}}}}{\partial {z}^{2}}\right)$$
(2)
$$\rho \left({v}_{{\text{x}}}\frac{{\partial v}_{{\text{y}}}}{\partial x}+{v}_{{\text{y}}}\frac{{\partial v}_{{\text{y}}}}{\partial y}+{v}_{{\text{z}}}\frac{{\partial v}_{{\text{y}}}}{\partial z}\right)=-\frac{\partial P}{\partial y}+\mu \left(\frac{{\partial }^{2}{v}_{{\text{y}}}}{\partial {x}^{2}}+\frac{{\partial }^{2}{v}_{{\text{y}}}}{\partial {y}^{2}}+\frac{{\partial }^{2}{v}_{{\text{y}}}}{\partial {z}^{2}}\right)$$
(3)
$$\rho \left({v}_{{\text{x}}}\frac{{\partial v}_{{\text{x}}}}{\partial x}+{v}_{{\text{y}}}\frac{{\partial v}_{{\text{y}}}}{\partial y}+{v}_{{\text{z}}}\frac{{\partial v}_{{\text{z}}}}{\partial z}\right)=-\frac{\partial P}{\partial x}+\mu \left(\frac{{\partial }^{2}{v}_{{\text{z}}}}{\partial {x}^{2}}+\frac{{\partial }^{2}{v}_{{\text{z}}}}{\partial {y}^{2}}+\frac{{\partial }^{2}{v}_{{\text{z}}}}{\partial {z}^{2}}\right)$$
(4)

Energy conservation:

$$\rho {C}_{{\text{p}}}\left({v}_{{\text{x}}}\frac{\partial T}{\partial x}+{v}_{{\text{y}}}\frac{\partial T}{\partial y}+{v}_{{\text{z}}}\frac{\partial T}{\partial z}\right)=k\left(\frac{{\partial }^{2}T}{\partial {x}^{2}}+\frac{{\partial }^{2}T}{\partial {y}^{2}}+\frac{{\partial }^{2}T}{\partial {z}^{2}}\right)$$
(5)

The most critical parameter that shows the system’s efficiency and how effective the cooling is in heat sinks is the effective thermal resistance (Rth) [1, 2, 34]. The lower this value is, the more efficient the system is. The calculation of Rth and the heat transfer coefficient used in heat transfer studies are shown in Eqs. 6 and 7. Equation 8 defines the Reynolds number, while Eq. (9) computes the friction factor. Equation 10 indicates the performance evaluation criterion (PEC). The circular cross-section pin structure is used in calculating the value of parameter a in Eq. (10) [3].

$${R}_{{\text{th}}}=\frac{{T}_{{\text{CPU}},{\text{m}}}-{T}_{{\text{in}}}}{{q}^{{\prime}{\prime}}}$$
(6)
$$h=\frac{{q}^{{\prime}{\prime}}}{{T}_{{\text{CPU}},{\text{m}}}-{T}_{{\text{bf}}}}$$
(7)
$${\text{Re}} = \frac{{\rho \cdot V \cdot D_{{\text{h}}} }}{\mu }$$
(8)
$$f = \frac{{2 \cdot \Delta P \cdot D_{{\text{h}}} }}{{\rho \cdot L \cdot V^{2} }}$$
(9)
$${\text{PEC}} = \frac{{h_{{\text{a}}} /h_{0} }}{{\left( {f_{{\text{a}}} /f_{0} } \right)^{1/3} }}{ }$$
(10)

‘a’ and ‘0’ sub-indices have been defined in the study for the purpose of comparing performance with the literature study. The terms ‘a’ and ‘0’ used in this context refer to the specified geometry and Chiu’s study [1] in circular geometry, respectively.

Results and discussion

The presented numerical study investigates the cooling performance of different types of heat sink geometries. Before starting the investigations, the numerical study is validated with an experimental study [1], an essential contribution to the literature. The diameter of the circular pin used in the experimental study is 0.66 mm, while the diameter and edge length in the presented study are 0.6 mm. The geometries investigated are circular, square, triangular, hexagonal, circular perforated, square perforated, and triangular perforated. The cross-sections in the perforated structures have a side length, diameter, and distance between each other of 0.6 mm, just as in the case of circles, squares, and triangles. The numerical results were compared with the volumetric flow rate parameters found in the experimental study. Four different pressure boundary conditions and constant heat fluxes were applied to all geometries. The fluid used is water. According to the numerical results, the maximum deviation for Rth is 4.69% at the highest flow rate, while the maximum deviation is 6.42% at P = 1370 Pa. These values are pretty reasonable, as they are below 10%. Other authors in the literature [3, 6], confirming the experimental study [1], also obtained deviations within the specified ranges. Figure 4 shows the validation of the experimental study against the volumetric flow rate.

Fig. 4
figure 4

Comparison of experimental [1] and numerical results

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Figure 5 shows the volumetric flow rate versus pressure for all geometries. The common observation for all geometries is that the volumetric flow rate increases as the pressure increases. An expected result is that the increase in pressure difference will increase the volumetric flow rate. If the geometries are compared within themselves, the highest volumetric flow rate is obtained in the circular pin fin geometry for all pressure values. The highest value is 213.62 mL min−1 at P = 2750 Pa. The lowest volumetric flow rate was V̇ = 34.13 mL min−1 for the triangular pin fin geometry at P = 689 Pa. The study’s volumetric flow rate ranged from 34.13 to 213.62 mL min−1. The order of volumetric flow rate for circular, square, triangular, and hexagonal pin fin geometries is circle > hexagon > triangle > square. The volumetric flow rate ranking of perforated structures is square perforated, circular perforated, hexagonal perforated, and triangular perforated. The results of cross-section and perforated cross-section are different because of the resistance to flow.

Fig. 5
figure 5

Pressure vs. volumetric flow rate of all geometries

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Thermal resistance is a parameter used in the cooling of electronic equipment, such as heat sinks, and is used as a parameter to indicate efficiency. Figure 6 shows the thermal resistance to pressure of all the geometries studied. As can be clearly seen from the figure, the lowest thermal resistance Rth = 0.29 K W−1 was obtained at P = 2750 Pa and a circular cross-section pin-type heatsink, while the highest thermal resistance Rth = 0.51 K W−1 was obtained at P = 689 Pa on triangular perforated structure. In both perforated and non-perforated structures, the triangular cross-section exhibited the highest Rth, while the circular cross-section exhibited the lowest. This similarity is important for the compared geometries because it shows that perforated and non-perforated geometric structures have the same effect on thermal resistance. This finding makes a valuable contribution to the literature on this topic. The thermal resistance range of perforated structures is Rth = 0.306–0.51, while that of non-perforated structures is 0.29–0.448. If the role of low thermal resistance for cooling efficiency is compared with the data in the literature, it can be clearly seen that a very efficient result is obtained. In fact, in the study [2], which is particularly noteworthy for its similarity to the present study from a geometrical point of view, a value of 0.25 K W−1 was obtained, but this value was obtained at P = 5000 Pa. From this point of view, the results obtained in this study represent an important contribution to the literature in this field. As a result, more efficient cooling is obtained by using less pumping power. Among eight different heatsinks, the most efficient value was obtained for the circular heatsink geometry, similar to other studies in the literature [1,2,3, 6, 12, 33] (Fig. 7).

Fig. 6
figure 6

Compare of pressure vs. thermal resistance all geometries

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Fig. 7
figure 7

Mean temperature vs. pressure for all geometries

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Upon analyzing eight distinct geometries, it was found that the temperature range of objects that were pinned and perforated fell between 308.73 and 315.27 K. This suggests a tightly constrained and consistent temperature distribution. The temperature distribution of the pinned structure in circular shape is represented by a lower and smaller band (311–308 K), as shown by the literature sources [1,2,3, 6, 7, 24, 43]. Nevertheless, it is crucial to acknowledge that temperature distribution alone cannot function as a reliable indication. Hence, the assessment of PEC will also serve as a vital factor in choosing the most advantageous geometrical configuration. The circular perforated structure exhibits a variety of bands at low temperatures, but the square perforated geometric structure follows. The triangular perforated structure has the most elevated temperature distribution band, ranging from 315.27 to 310.70 K. Temperature distribution contours provide the most insightful viewpoint for analyzing temperature bands. The imaging approach is essential for revealing the flow and temperature features, as shown in Fig. 8, which presents the previously indicated temperature distributions. The sequence of presentation in Fig. 8 and the other figures adheres to that of Fig. 1. In order to maintain consistency, identical temperature ranges are used when examining the temperature bands. The structures located at the uppermost position of the left column and the second position of the right column provide a distinct depiction of the flow-temperature representation, showcasing the effectiveness of the temperature distribution. These structures have circular and square perforations. The investigation revealed that the circular section exhibited the lowest Tm across all pressure boundary conditions, but the triangular sections had the greatest Tm at all pressures except P = 689 Pa, among the triangular, square, hexagonal, and circular sections. In addition, the square cross-section perforated structure exhibited the lowest temperature (Tm) compared to the other perforated structures, but the triangle cross-section perforated structure had the highest Tm. It is noteworthy that the Tm values of the triangular cross-section perforated structure exhibited a substantial increase compared to the other constructions. Nevertheless, the temperature distribution in the triangle cross-sectional heat sinks or perforated version was not improved to the same extent as in other cross-sections. The temperature range for the triangle cross-sectional constructions is 310.704 K to 315.273 K, while for the square perforated structure, it is 309.22 K to 312.15 K. The temperature differential of around 3 degrees is substantial and plays a vital role in effectively cooling the heat sink.

Fig. 8
figure 8

a Top view temperature contour of square-type heat sink, b bottom view temperature contour of square-type heat sink, c temperature contours at P = 2750 Pa of all geometries

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The performance evaluation criterion is the best way to find out how well the structure made for temperature distribution and heat transfer analyses works. Figure 9 displays the PECs of the structures obtained thus far. The range of Reynolds numbers obtained is between 410 and 2747. However, after verifying the literature, the PEC graph was plotted against pressure to ensure an objective evaluation. This decision was made in pursuit of assessing the data in a more comprehensive manner. In contrast to the temperature distribution, the square-section structure demonstrated the greatest efficiency in terms of PEC. The highest PEC value achieved across all pressures and geometrical structures was 1.1872 at 689 Pa on square cross-sectioned geometry, with the subsequent highest value attained from the triangular perforated structure (PEC = 1.1811). The triangular cross-sectioned structure resulted in the lowest PEC value, 0.67 at 2750 Pa. This structure exhibited a greater increase in the friction factor compared to the other structures, resulting in a significant outcome. The evaluation of both Tm and PEC indicates that an excessive increase in temperature has a positive effect on heat transfer, while a high temperature distribution has a negative effect on the heat sink. Therefore, it is clear that evaluating Tm or PEC alone is insufficient. The temperature difference has an impact on the heat transfer coefficient, which in turn affects thermal resistance and temperature distribution. Figures 10 and 11 show the pressure contours and streamlines of the newly designed geometries. Comparisons are made for the case, where P = 2750 Pa under the same pressure. According to the numerical results obtained, the pressure contours in the fluid domain are lower and more uniformly distributed for heat sinks with circular pins, hexagonal pins, and square and triangular pins. These two figures can be examples of figures that can well express the efficiency of the system; both the pressure distribution and the flow images express very well the reflection of the effect of the pressure distribution on the thermal resistance parameter. The streamlines shown in Fig. 11 are the regions with the lowest flow velocity. Vortex flow characteristics, flow steering, and flow direction determination and shaping are seen in eight different geometries. While vortex and secondary flow are seen in almost all geometries, the pin geometry with a circular cross-section is the least common. Increasing the flow time triggers an increase in the heat transfer coefficient.

Fig. 9
figure 9

PEC vs. pressure drop

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Fig. 10
figure 10

Pressure drop contours at P = 2750 Pa of all geometries

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Fig. 11
figure 11

Streamline at P = 2750 Pa of all geometries

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Conclusions

The presented numerical study aims to provide a more uniform temperature distribution and a more efficient heat transfer improvement for heat sinks designed in eight different types. The geometries studied are circular, square, triangular, hexagonal, and their perforated structure. The geometries specified were examined under a constant surface heat flux and four different pressure inlet boundary conditions. The findings can be summarized as follows:

  1. 1.

    According to the results obtained, it was observed that as the pressure increases, the volumetric flow rate increases while the PEC, Tm, and thermal resistance decreases.

  2. 2.

    The lowest thermal resistance was obtained from the geometry designed in the circular cross-section pin structure. This value was obtained as Rth ≅ 0.29 K W−1 when P = 2750 Pa. The thermal resistance range of all geometries studied was obtained between 0.29 and 0.51 K W−1.

  3. 3.

    The highest pressure resulted in the lowest thermal resistance. When comparing the most efficient structures with and without holes at this pressure, there was a difference in Rth of 0.017.

  4. 4.

    The average temperature range of the whole study is between 308.732 and 315.273 K. In this range, the most efficient range in terms of temperature homogeneity is seen in the circular cross-section and is between 308.732 and 311.306 K. The temperature distribution in perforated structures was found to be most efficient in square sections, ranging from 309.22 to 312.15 K.

  5. 5.

    In general, the results for thermal resistance and average temperature distribution are parallel to each other.

  6. 6.

    The pin-type structure with a square cross-section achieved the highest PEC of 1.1872 at a pressure of 689 Pa, while the pin-type structure with a triangular cross-section achieved the lowest PEC of 0.67 at a pressure of 2750 Pa.

  7. 7.

    The study found that, in terms of PEC, perforated structures generally outperformed other pin structures, with the exception of the square-section pin structure. This is a significant finding. The study also highlights that evaluating Rth, Tm, or PEC alone is insufficient.

The study evaluated the impact of geometric effects on heat sinks across various architectures. Future research will explore the cooling impact of innovative geometric designs in conjunction with novel mono or hybrid nanofluids.