Comparison of Forced Convective Heat-Transfer Enhancement of Conventional and Thin Plate-Fin Heat Sinks under Sinusoidal Vibration

Abstract

Applying sinusoidal vibration to heat sinks has proven to be a promising technique for improving heat transfer by disrupting the thermal boundary layer. However, applying sinusoidal vibration to the base of thin plate-fin heat sinks can cause a flapping motion within the fins, further enhancing heat transfer. Therefore, the current study numerically investigates and compares the effects of sinusoidal vibrations on the thermal performance of conventional and thin plate-fin heat sinks. The study concludes that increased vibrational amplitude and frequency (f ˃ 30 Hz) increases the vibration-assisted thermal performance. It was found that the thin plate-fin heat sink provides higher thermal performance compared to the conventional heat sink at every level of vibrational characteristics. The study found that the application of vibration enhances the Nusselt number up to a maximum of 20% and 15% in thin plate-fin and conventional heat sinks, respectively. Furthermore, the Reynolds number is reduced by 33.3% and 28% with thin plate-fin and conventional heat sinks compared with non-vibrating heat sinks, indicating a potential reduction of the size of the cooling system or fin size.

1. Introduction

Packed electronic components have increased heat flux generation, prompting the investigation of various cooling strategies [1]. The application of conventional heat sinks, extended surfaces used to increase the heat-transfer area in electronic components, is a popular cooling mechanism among electronic cooling systems [2]. Two types of conventional heat sinks are available: plate-fin and pin-fin heat sinks. The application of conventional plate-fin heat sink is popular due to their advantages of being simple to design and fabricate [3]. Recent studies have proposed conventional plate-fin designs, including cross-fin arrangements [4], wavy [5,6] or Y-shaped plate fins [7], radial plate fins [8], combinations of plate and pin fins [9,10], plate fins with fillets [11,12], slots [13,14,15], dimples [16], and rods [17].

The current trend in electronic product design is focused on making them lighter, thinner, shorter, and smaller [18]. Thus, the challenging tasks lie not only in improving heat dissipation rates but also in reducing the size and weight of the cooling system. Consequently, numerous studies have been conducted on different conventional plate-fin heat sink modules to achieve high-performance heat removal while maintaining compactness. The utilization of thin plate fins in a heat sink allows for a higher number of fins within a given substrate size compared to conventional plate fins. This increase in the number of fins results in a larger heat-transfer area, ensuring improved heat removal. Furthermore, the application of thin plate-fin heat sinks enables the reduction of weight and material usage in the cooling system as compared to conventional plate-fin heat sinks. As a result, researchers have become increasingly interested in enhancing heat transfer in thin plate-fin heat sinks.

Heat-transfer enhancement through modifications in heat sink geometry has its limitations due to the development of thermal boundary layers. To achieve further improved thermal performance, heat sinks have incorporated passive and active cooling methods. Researchers have explored roughened heat sinks [19,20,21], nanoscale structuration techniques [22], and the application of reverse or swirl flow devices [23] as passive cooling methods to obtain higher thermal performance in heat sinks. However, the increasing demand for heat transfer necessitates the use of active techniques that utilize external power. The implementation of active cooling methods such as fan-based air cooling systems, air jets [24,25,26], electro-hydrodynamics (EHDs), and magneto-hydrodynamics (MHDs) in heat sinks is common among researchers [27]. The combination of passive and active cooling technologies, such as the implementation of a jet system on heat sinks with micro-scale surface treatment [28] or heat-sink cooling with nanofluids, which is subjected to a magnetic field [29], has also been investigated by researchers. Additionally, vibration-assisted convection cooling mechanisms have shown significant heat-transfer enhancement. Nevertheless, the application of vibrations in the cooling system can lead to reliability and stability issues within the cooling system due to the formation of fatigue stress on the vibrating surface [30].

Vibrational energy provided to the heated surfaces is propagated through the interface of the solid and fluid, inducing secondary flows within the flow regime, disrupting thermal boundary layers, thus enhancing the heat transfer [31]. However, this induced disturbance depends on the vibration characteristics, flow parameters, and geometrical parameters of the heated body. Experimental and numerical investigations have been conducted to prove the concept of enhancing heat transfer via vibration using gauge wire [32], spheres [33], and cylinders [34,35,36].

The results of these studies have shown significant improvements in heat transfer due to induced secondary flows within the flow field. Moreover, past studies investigated the potential of mechanical vibration for enhancing the thermal efficiency of fins or heat sinks. For instance, the vibration-assisted heat transfer of 2D fins has been numerically investigated by Gunaratna and Li [37], Fu and Wang [38] and Rahman et al. [30]. These studies show that vibration induces the secondary flows within the flow regime, which disrupts the thermal boundary layer, resulting in enhanced heat transfer. Najim et al. [39] investigated the effect of sinusoidal vertical vibration on heat sinks. Their findings revealed that vibrational frequency enhances heat transfer, and the Nu value is increased by up to a maximum of 6.5% compared with the static conditions due to the fluid flow circulation caused by the vibration. Moreover, Rasangika et al. [40] investigated the thermal performance of horizontally vibrating conventional heat sinks under sinusoidal and square wave-shaped vibrations, and they observed a maximum of 25% enhancement in the Nusselt number.

There is a growing interest among researchers in enhancing the heat transfer of fins or heat sinks through the utilization of piezoelectric actuators to induce movement in heated solid bodies. Piezoelectric technology is widely suitable for electronic cooling purposes due to its suitability in terms of power densities, operating frequencies, and compactness [41]. Hussain et al. [42] conducted an experimental study on heat-transfer augmentation using piezo-actuated single and triple thin fins, which are oscillated by piezoelectric actuator patches connected to the base of the fin. The study revealed that vibration enhances heat transfer and achieves a maximum temperature reduction of 35% at the fin tip. In their numerical investigation, Ma et al. [43] examined the thermal performance of six piezo-actuated fins and observed an 11.6% enhancement in thermal effectiveness. Another numerical study by Dey and Chakraborty [44] focused on the impact of piezo-actuated fin motion with a static substrate on heat-transfer enhancement. They reported a 98.7% increase in the Nusselt value by utilizing higher vibrational characteristics. They attributed this enhancement to the formation of two counter-rotating vortices at the tip of the fin, and it tends to enhance heat transfer.

The application of vibration on a thin plate-fin heat sink can cause a flapping motion within the fins, which may positively contribute to the formation of excessive secondary flows and additional mixing effects within the flow field. The induced secondary flows and the mixing effect may result in disturbing the thermal boundary layer, enhancing heat transfer further. As a result, the size of the cooling system or heat sink can be reduced. Moreover, thin plate fins have a small thickness compared to conventional heat sinks, resulting in material saving, weight reduction and more compactness. In fact, applying vibration to a thin plate-fin heat sink provides a more feasible way to achieve actuator motion within the thin fins. However, to the authors’ best knowledge, no research has been conducted to demonstrate the heat-transfer enhancement and potential cooling system size reduction of thin plate-fin heat sinks under sinusoidal vibration or to compare their thermal performance with vibrating conventional heat sinks to determine which heat sink is more superior in terms of thermal performance.

This study aims to analyze and compare the heat-transfer enhancement of conventional and thin plate-fin heat sinks subjected to sinusoidal vibrations using ANSYS/FLUENT V 2021 R2 CFD software. The vibration of a conventional heat sink causes the heat sink to vibrate within the vibration wave frequency and amplitude range; however, for thin fins, the vibration causes a flapping motion. Commercial CFD software does not model by default vibration. Therefore, User-Defined Function (UDF) codes using C++ need to be developed to simulate each vibration and then incorporated into the CFD model. The developed incorporated codes were validated against published research. Different vibration levels and Reynolds numbers are applied to the heat sink structures, and the study examines and compares the heat-transfer enhancement depending on the frequency and amplitude of the vibrations in both conventional and thin plate-fin heat sinks, which results in reducing the size of the heat sink. Moreover, the potential reduction in both the Reynolds number and the size of the cooling system due to the application of both heat sinks under vibrating conditions has also been investigated.

2. Methodology

2.1. Problem Description

The current study consists of two types of heat sinks: conventional and thin aluminum plate-fin heat sinks, which have similar heat-transfer surface areas of 0.016 m². Each heat sink has two fins connected to the base with the dimension of 0.095 × 0.016 × 0.005 m, and the average fin spacing is maintained at a constant of 0.01 m. The thickness of the thin plate fins is selected as 0.00025 m based on the fin thickness of the study of Dey and Chakrborty [44], maintaining the same heat-transfer areas of both heat sinks. In both simulation models, a uniform velocity is employed in the X direction (as shown in Figure 1) with a Reynolds number of 1000 at an inlet temperature of 25 °C. The heat sinks are subjected to a heat flux of 6250 W/m², evenly distributed at the bottom surface, which is typically generated in data centers [18,40]. After performing the comparative analysis of the vibration-assisted thermal performance of conventional and thin plate-fin heat sinks, the Reynolds number of the thin plate-fin heat sink model is varied between 1000 and 2000, which is related to the velocity of the convectional cooling fans in electronic systems [45].

2.2. Governing Equations

In this study, the flow characteristics near the heat sink are influenced by the vibration of the heat sink, resulting in turbulent flow despite the main flow having a relatively low Reynolds number of 1000. To accurately simulate this turbulent flow, various simulation models, including lamina, k-ω SST, and large eddy simulation (LES), are used, and the results of those models are compared with experimental and previously published results. Among the different turbulence models, the LES with the dynamic Smagorinsky–Lilly sub-grid model demonstrated good agreement with published experimental measurements. Thus, the LES turbulence model was chosen for the numerical simulation and subsequent validation in this study.

The flow was assumed to be incompressible ideal gas with constant thermo-physical properties. Also, the no-slip condition is employed in the solid–fluid interface of the heat sink models. Thus, the continuity, momentum and energy equations for turbulent flow can be written as below [46]:

Continuity equation:

$$\frac{\partial }{\partial x_i}\left(\rho U_i\right)=0$$

(1)

Momentum equation:

$$\frac{\partial }{\partial \mathrm{t}}\left(\rho {\overline{U}}_i\right)+\frac{\partial }{\partial x_j}\left(\rho {\overline{U}}_i{\overline{U}}_j\right)=-\frac{\partial \overline{P}}{\partial x_i}+\frac{\partial }{\partial x_j}\left(\mu \frac{\partial {\overline{U}}_i}{\partial x_j}\right)-\frac{\partial {\tau }_{ij}}{\partial x_j}$$

(2)

where ρ represents the density of the working fluid, P represents the pressure, U_i represents the velocity in ith direction, and τ_ij represents the sub-grid scale stress which can be written below:

$${\tau }_{ij}=\rho \overline{U_i{U}_j}-\rho U_i{\overline{U}}_j$$

(3)

Energy equation:

$$\frac{\partial \left(\rho C_{\mathrm{p}}T\right)}{\partial t}+\frac{\partial \left(\rho C_{\mathrm{p}}{U}_{\mathrm{j}}T\right)}{\partial x_{\mathrm{j}}}=\frac{\partial }{\partial x_{\mathrm{j}}}\left(k\frac{\partial T}{\partial x_{\mathrm{j}}}\right)$$

(4)

where T represents the temperature, k represents the thermal conductivity of the fluid, and C_p represents the specific heat capacity of the fluid.

2.3. Computational Details

This study solves the governing Equations (1)–(4) using the ANSYS/FLUENT CFD software. The finite-volume method is utilized, and the SIMPLE algorithm is employed to ensure accurate pressure-velocity coupling. A second-order upwind scheme is utilized for the spatial discretization of momentum and energy equations. The transient formulation was obtained using a second-order implicit scheme. To achieve convergence of the iterative solution, continuity, momentum, and energy convergence criteria are predefined as 10⁻⁶.

ANSYS/FLUENT software is unable to simulate the vibration and hence requires the development of User-Defined Function (UDF) codes, which consist of displacement equations of the heat sinks, to enable the software to simulate the vibration of the heat sinks and their interaction with the air domain. The horizontal sinusoidal vibration is applied to the conventional and thin plate-fin heat sink in the direction of Y (Figure 1). The conventional heat sink and the base of the thin plate-fin heat sink vibrate as a rigid body with displacement Equation (5), while the thin fins flap with the displacement Equation (6) [47,48].

$$Y\left(t\right)=A_{b}sin\left(2\pi ft\right)$$

(5)

$$\begin{gathered} W_{rel}\left(Z,t\right)=0.2345A_b\left( \frac{sin\left(2\pi ft\right)-\frac{f}{f_n}sin\left(2\pi f_{n}t\right)}{{\left(\frac{f_n}{f}\right)}^2-1} \right)(\mathit{cos}\left(\frac{\lambda }{H}Z\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} -\mathit{cosh}\left(\frac{\lambda }{H}Z\right)+0.734 (\mathit{sin}\left(\frac{\lambda }{H}Z\right)-\mathit{sinh}(\frac{\lambda }{H}Z)) \end{gathered}$$

(6)

Here, Y(t) represents the displacement at a given time, A_b represents the base amplitude, t represents the time, H represents the height of the fin, Z represents the height between any point on the fin to the base of the fin, and f and fn represent the supplied frequency and natural frequency of the fin, respectively. λ represents the solutions of the equation cos λ cosh λ + 1 = 0. The induced flapping motion of the thin fins is based on the first mode of the vibration; thus, λ = 1.8751.

A mesh sensitivity analysis is performed to ensure that the numerical results are independent of the number of cells that are utilized in each simulation model. Figure 2 illustrates the variation in Nusselt number of different grid sizes in each model. The Nusselt number initially decreases significantly as the number of cells increases. However, beyond the number of cells of 522,764 and 558,771, a further increase in the number of cells results in a deviation in the Nusselt number of less than 1% in conventional and thin plate-fin heat sink models, respectively. Therefore, the meshes with 522,764 and 558,771 cells are chosen as the optimal mesh for all simulation scenarios in the conventional and thin plate-fin heat sink models, respectively.

Time-step sensitivity analysis is carried out using three different time steps—Δt = τ/8, τ/16 and τ/32—at a frequency of 100 Hz for both simulation models. Figure 3 shows the area average Nusselt number variations in each heat sink model with the cycle. It can be seen that reducing the time step (Δt) from τ/16 to τ/32 s leads to a peak-to-peak variation in the Nusselt number of less than 1% in each simulation model. Here, τ represents the cycle time of vibration. Hence, the time step of τ/16 s is utilized as a time step for both simulation models.

2.4. Validation

The developed CFD models incorporated the UDF codes and were validated against a published experimental measurement. The CFD modeling of conventional heat sinks was validated with the experimental study of Najim et al. [39]. For validation purposes, the numerical investigations were conducted under a range of vibrational frequencies with the amplitude, supplied power, and air velocity of 0.005 m, 17 W, and 0.6 m/s, respectively. compared with the experimental measurements of Najim et al. [39], the validation results show good agreement, as depicted in Figure 4a, with a maximum deviation of 3%.

Park et al. [41] experimentally investigated the heat-transfer augmentation of flapping thin plate fins due to the sinusoidal excitation of the base under natural convection phenomena. Therefore, the CFD modeling of thin plate-fin heat sink was validated using the experimental measurements of Park et al. [41]. Numerical computations were performed using a thin plate fin with the dimension of 0.1 × 0.025 × 0.0004 m. In simulations, the vibration amplitude was varied, while the frequency and supplied temperature were kept constant at 59 Hz and 328.15 K, respectively. Figure 4b shows a strong agreement between numerical and experimental data, with a maximum percentage error of 2.6%. Thus, it can be concluded that the CFD modelings used in the present study can be confidently employed to numerically analyze the vibration-assisted thermal performance of conventional and thin plate-fin heat sinks.

3. Results

After validating each simulation model, the CFD modeling was initially performed with heat sinks in static conditions at a Reynolds number of 1000. Then, conventional and thin plate-fin heat sinks were subjected to sinusoidal vibration with a range of vibrational frequency and amplitude of 0–100 Hz and 0–0.003 m, respectively. After performing the comparative analysis of the vibration-assisted thermal performance of conventional and thin plate-fin heat sinks, the Reynolds number of the thin plate-fin heat sink model was varied between 1000 and 2000 and we determined the effect of vibration on the heat-transfer enhancement of thin plate-fin heat sinks under different Reynolds numbers.

Vibration induces secondary flows within the flow regime, resulting in a mixing effect and consequently enhancing heat transfer. Thus, a brief discussion is presented on the effect of vibration on the flow profiles shed from an oscillating conventional and thin plate fin. The end-view schematic of the conventional and thin plate-fin heat sink is presented in Figure 5.

Figure 6 shows the velocity profiles of non-vibrating and vibrating heat sinks at a vibrational frequency of 100 Hz and an amplitude of 0.0025 m. When a conventional or thin plate-fin heat sink is not subjected to vibration, the flow moves smoothly from the inlet to the outlet, and no secondary flows are induced within the flow field, as shown in Figure 6a,b.

When the conventional heat sink is at the start of the first quarter of the sinusoidal vibration, the velocity vector at the tip of the heat sink shows that the flow recirculation appears near the trailing edges of both fins (Figure 6c). Furthermore, the velocity vector at the cross-section of the heat sink shows that the clockwise two-flow recirculation zones near each fin tip and airflow within the channel area follow the movement direction of the heat sinks and hit the inner surface of fin 1. However, when vibration is subjected to a thin plate-fin heat sink, as shown in Figure 6g, as compared to the conventional heat sink, stronger recirculation zones are formed near the trailing edges of both fins and a stronger counterclockwise recirculation zone is formed near the tip of the fins due to the flapping motion of the thin fins. Moreover, flow within the channel area hits the inner surface of fin 1. At the start of the second quarter, the heat sink moves to the maximum positive amplitude, inducing flow recirculation near the trailing edge of the heat sinks (as shown in Figure 6d). Moreover, two counterclockwise zones and one clockwise recirculation zone tend to appear near the tip of the conventional heat sink. However, the vibration of the thin plate-fin heat sink induces stronger recirculation zones near the trailing edge of the fins, and two stronger counterclockwise and clockwise recirculation zones are formed near the tip of the fins (Figure 6h).

At the start of the quarter of the sinusoidal vibration (Figure 6e,i), the motion of both heat sinks induces the recirculation zones near the trailing edge of the fins, and one counterclockwise recirculation zone is formed near the tip of both heat sink fins. Moreover, the flow between the channel area follows the movement direction of the heat sinks and hits the inner surface of fin 2 in both heat sinks. However, as expected, induced recirculation zones are found to be stronger with thin plate fins due to the flapping motion of the fins. At the start of the fourth quarter, it can be seen in Figure 6f,j that the recirculation zones formed near the tip of both heat sinks become opposite to those shown in Figure 6d and Figure 6h, respectively.

Figure 7 shows the temperature profile passing through the tip of stationary conventional and thin plate-fin heat sinks compared to increasing vibrational frequency and amplitude. The temperature profiles (Figure 7c–h) are captured at the end of the second quarter of the vibration cycle. As shown in Figure 7a,b, the stationary heat sinks have strong thermal wake regions around the fin surfaces, exhibiting a higher temperature gradient near the vicinity of the heat sink, resulting in a limited cooling effect. This is due to the streamlines flowing from the inlet to the outlet without disturbance under no vibration.

However, when heat sinks vibrate under higher vibrational frequencies and amplitudes, the continuous formation of recirculation zones within the flow field disrupts the thermal boundary layer and sets the fluid in motion, thus enhancing the cooling effect. As seen in Figure 7c–f, with an increase in vibrational frequency from 50 Hz to 100 Hz at an amplitude of 0.0015 m, the thermal wakes formulated on either side of the fins are spread over the fluid domain, resulting in higher heat dissemination form heated fins to fluid and consequently reducing the fins temperature. However, as seen in Figure 7d,f, the implementation of vibration on the thin plate-fin heat sink provides a wider spread in the thermal wake behind the fins while reducing the overall thermal boundary layer thickness of the fins. Moreover, it can be seen that air temperature near the thin plate-fin heat sink is significantly reduced compared to the conventional heat sink, indicating a higher cooling effect at similar vibrational frequency conditions. In Figure 7e–h, with the increase in vibrational amplitude from 0.0015 m to 0.003 m at a frequency of 100 Hz, the thermal boundary layer around the fins in both heat sink models becomes thinner, and air temperature in the vicinity of the heat sinks is found to be significantly less as compared to that at a lower vibrational amplitude. Moreover, a thin plate-fin heat sink with higher amplitudes leads to the thinnest thermal boundary layer, owing to the induced flapping motion, leading to higher heat transfer.

The heat sink base is attached to the electronic component, which emits the heat. Therefore, it is necessary to provide a comprehensive analysis of the vibration effect on the temperature profiles of the heat sink base. Thus, the temperature profiles of each heat sink baseare illustrated in Figure 8 under different vibrational conditions. The temperature profiles (Figure 8c–h) are captured at the end of the second quarter of the vibration cycle.

When heat sinks are not subjected to vibration, the thermal boundary layer hinders the heat transfer between the heat sink and air. Therefore, as shown in Figure 8a,b, the base temperature of each heat sink is found to be higher with no vibration conditions, limiting the cooling effect. However, as shown in Figure 8c–f, the base temperature of each heat sink reduced with the increase in vibrational frequency for both conventional and thin plate-fin heat sinks. Induced flow recirculation zones around heat sink fins enhance the heat transfer of fin surfaces. This leads to a lower temperature in the heat sink base, indicating a higher cooling effect. Moreover, Figure 8d,f shows that the application vibration with higher frequency leads to a higher reduction in the heat-sink base temperature of the thin plate-fin heat sink as compared to the conventional heat sink (Figure 8c,e) owing to the flapping motion of thin plate fins which induced strong flow recirculation zones.

As shown in Figure 8e–h, an increase in vibrational amplitude leads to a further reduction in base temperature in both heat sinks. As expected, as seen in Figure 8h, a higher decrease in heat-sink base temperature is recorded with the thin plate-fin heat sink compared to the conventional heat sink, indicating a higher cooling effect.

As seen in Figure 9a, with low vibrational frequencies (f < 30 Hz), the increase in Nusselt number is insignificant. This might be due to the small frequency values, which are not adequate to formulate the recirculation zones within the flow field, resulting in limited heat transfer. Moreover, this threshold frequency is common for both conventional and thin plate-fin heat sinks. When heat sinks vibrate with higher frequencies (f ˃ 30 Hz), an increase in frequency leads to a higher Nusselt number. It is noteworthy that an enhancement in Nusselt number is prominent with the thin plate-fin heat sink in comparison with the conventional heat sink. The Nusselt numbers are increased up to a maximum of 16% and 12% with the conventional heat sink and thin plate-fin heat sinks, respectively.

The variation in the Nusselt value with an increase in vibrational amplitude is shown in Figure 9b. As amplitude increases, the Nusselt value increases due to the flow disturbance and recirculation zones caused by the higher vibrational amplitude. As expected, the vibration of the thin plate-fin heat sink provides higher Nu enhancement than the conventional heat sink, due to the stronger recirculation zones induced by the translation and flapping motion of the heat sink fins. The thin plate-fin heat sink achieved a maximum of 20% enhancement in Nusselt value, while the conventional heat sink recorded a maximum 15% increase in Nusselt number compared to its static conditions.

Since the application of vibration provides a higher heat-transfer enhancement in the thin plate-fin heat sink as compared to the conventional heat sink, the thin plate-fin heat sink is selected to investigate the effect of varying Reynolds numbers on the vibration-assisted thermal performance. Figure 9c shows the variation in the Nusselt value with the vibrational frequency at different Reynold numbers under a constant amplitude of 0.0015 m. It can be seen that the enhancement of the Nusselt value is insignificant when the vibrational frequency is less than 30 Hz. However, when the frequency exceeds 30 Hz, the Nusselt value increases exponentially with the increase in vibrational frequency. It can be seen in Figure 9c that Nusselt number values increase at higher Reynolds values. The results indicate that at a frequency and amplitude of 100 Hz and 0.0015 m, the Nusselt number of the thin plate-fin heat sink increased to a maximum of 16%, 9% and 6% at Reynolds numbers of 1000, 1500 and 2000, respectively.

In order to estimate the possible reduction of the size of the fin through the utilization of vibration, it is necessary to calculate the enhancement of the Nusselt number in terms of the Nusselt ratio, which is calculated based on Equation (7).

$$Nusselt Ratio=\frac{Nusselt number under vibrating conditions}{Nusselt number under non-vibrating conditions}$$

(7)

Figure 10 shows the variation in the Nusselt ratio with respect to the vibrational characteristics and Reynolds numbers. Since an enhancement in the Nusselt number is found to be insignificant with a vibrational frequency below 30 Hz, the enhancement of the Nusselt ratio is also found to be insignificant at low vibrational frequencies (f < 30 Hz). As shown in Figure 10a, at the vibrational frequency of 100 Hz, the Nusselt ratio is increased up to a maximum of 16% and 12% with the conventional heat sink and thin plate-fin heat sink, respectively. Moreover, as shown in Figure 10b, at the vibrational amplitude of 0.003 m, the thin plate-fin heat sink achieved a maximum of 20% enhancement in the Nusselt ratio, while the conventional heat sink achieved a 15% maximum increase in the Nusselt ratio in comparison to static conditions.

Figure 10c shows the variation in the Nusselt ratio of the thin plate-fin heat sink for frequencies at different Reynolds numbers. Below the critical frequency of 30 Hz, the effect of vibration on the thermal performance of the thin plate-fin heat sink is insignificant for all considered Reynolds values, resulting in an insignificant increase in the Nusselt ratio. However, beyond the 30 Hz critical frequency, an increase in vibrational frequency leads to an increase in the Nusselt ratio for each Reynolds value. It can be seen that higher Nusselt number ratio values were recorded at lower Reynolds numbers. This shows that vibration is less effective in increasing the Nusselt number (compared to Nusselt at 0 frequency) when the Reynolds number increases. At a heat-sink base frequency and amplitude of 100 Hz and 0.0015 m, the maximum enhancement in the Nusselt ratio is found to be 16%, 9% and 6% at Reynolds numbers of 1000, 1500 and 2000, respectively.

The findings of this study demonstrate that an increase in vibrational frequency and amplitude enhances the thermal performance of both heat sink models. It is also demonstrated that at lower Reynolds number values, the effect of vibration on increasing the Nusselt number values from no-vibration values is more significant at lower Reynolds numbers. This heat-transfer enhancement under sinusoidal vibration can be utilized to reduce the size of the cooling system or fin, eventually leading to a compact cooling system. Thus, to estimate the possible reduction of the cooling system, it is necessary to present the percentage reduction in Reynolds number due to the application of vibration. Since the thermal performance of the thin plate-fin heat sink assisted by vibration diminishes with higher Reynolds numbers, the percentage reduction in Reynolds value is calculated and presented at a lower Reynolds value of 1000 for both heat sinks. Based on the vibration-assisted Nusselt value, Reynolds values corresponding to each vibrating condition are calculated using Equation (8) [49], and the percentage reduction in Reynolds number is calculated using Equation (9).

$${Nu}_{without}=0.664 \sqrt{\frac{Re.b}{L}}{. Pr}^{1/3}\sqrt{1+\frac{3.65}{\sqrt{\frac{Re.b}{L}}}}$$

(8)

$${PR}_{Re}=\left(1-\frac{Reynolds number under the vibrating condition}{Reynolds number under the non-vibrating condition}\right)\times 100$$

(9)

Here, Nu_without is the Nusselt number in static conditions, Re represents the Reynolds number, Pr represents the Prandtl number, b and L represent the fin spacing and fin length, respectively.

The variation in the percentage reduction of the Reynolds number with respect to vibrational frequency is presented in Figure 11a. Since the enhancement of the Nusselt value is found to be insignificant below the threshold frequency, the increase in percentage enhancement in Reynolds number below 30 Hz is insignificant. As vibration frequency increases above the threshold value, the percentage enhancement in Reynolds increases with an increase in frequency, and a higher enhancement is recorded with a thin plate-fin heat sink. It was found that the percentage reduction in Reynolds number increased up to a maximum of 23% and 27% with conventional and thin plate-fin heat sinks, respectively.

Since the increase in amplitude resulted in an enhancement of Nusselt value, as seen in Figure 11b, an increase in vibrational amplitude leads to an increase in percentage reduction in Reynolds number. As expected, the enhancement of PR_Re is found to be prominent with a thin plate-fin heat sink. In comparison with non-vibrating heat sinks, a 28% and 33.3% maximum enhancement in PR_Re is recorded with conventional and thin plate-fin heat sinks, respectively.

These results revealed that the application of vibration to heat sinks provides a possible reduction in the size of the cooling system or fin size, while a thin plate-fin heat sink provides higher thermal benefit and a reduction in the size of the cooling system in comparison with the conventional heat sink module.

Thus, a vibrating thin plate-fin heat sink that is subjected to sinusoidal vibration will be more suitable for applications in electronic cooling systems such as data centers for heat-transfer enhancement, due to the higher possible reduction in the cooling system size or heat sink size.

This research focused on investigating the effect of vibration frequencies, amplitudes and Reynolds number values on heat sinks’ thermal performance. Future research could be conducted by investigating more variables such as fin height on heat sink heat-transfer enhancement.

4. Conclusions

A numerical study is performed to value and compare the effect of sinusoidal vibration on the thermal performance of conventional and thin plate-fin heat sinks. It was found that at a vibration frequency below 30 Hz, an enhancement in the Nusselt value is insignificant with both heat sinks. However, an increase in frequency beyond the threshold value and amplitude increases the Nusselt number. Moreover, the effect of the vibration on the heat-transfer enhancement was proved to be stronger with the thin plate-fin heat sink than the conventional heat sinks due to the induced flapping motion of the fins, which positively contributes to the formation of higher recirculation zones within the flow field. Compared to the non-vibrating heat sink, the thin plate-fin heat sink achieved a maximum increase in 20% in Nusselt values, while a 15% maximum enhancement in the Nusselt number is recorded with the conventional heat sink. Moreover, the effect of the Reynolds number on the vibration-assisted thermal performance of the thin plate-fin heat sink is investigated, and it was found that an increase in the Reynolds number leads to a reduction in the thermal performance of the heat sink at vibrating conditions. At a base frequency and amplitude of 100 Hz and 0.0015 m, the maximum enhancement in the Nusselt value of the vibrating thin plate-fin heat sink is recorded as 16%, 9% and 6% at Reynolds numbers of 1000, 1500 and 2000, respectively. Furthermore, a possible reduction of the size of the cooling system is estimated using a percentage reduction of the Reynolds number, and a 28% and 33.3% maximum enhancement in PR_Re is recorded with conventional and thin plate-fin heat sinks, respectively, in comparison with non-vibrating heat sinks at a Reynolds value of 1000.