Numerical Investigation of the Effect of Square and Sinusoidal Waves Vibration Parameters on Heat Sink Forced Convective Heat Transfer Enhancement

Abstract

Among numerous electronic cooling methods, a vibrating heat sink using sinusoidal wave vibration effectively enhances the heat transfer by disturbing the thermal boundary layer. However, sinusoidal wave vibration has reached its limits in enhancing heat transfer. The present study utilizes a new square wave-shaped vibration and numerically investigates the thermal performance of a heat sink subjected to sinusoidal and square waves vibration. It is found that using the square wave vibration is more beneficial to the thermal performance of the heat sink than the sinusoidal wave. The sudden impulsive motion of square wave vibration induces a higher randomness of the airflow profile and recirculation zones than the sinusoidal wave, causing the air flow to impinge directly into the fin surfaces, and further enhances the heat transfer. Furthermore, increasing the frequency and amplitude leads to a higher heat transfer enhancement. Moreover, square wave vibration achieves a 25% increase in Nusselt values compared to the nonvibrating fins and it is 11% higher than the Nusselt number recorded by the sinusoidal vibration. Consequently, Reynolds number values can be reduced by 42.2% to achieve the Nusselt number values of nonvibrating fins, potentially reducing the cooling system or fin size. This reduction may contribute to solving the challenges of electronic systems compactness.

1. Introduction

Continued development in the electronic industry has led to a significant increase in energy consumption [1,2]. A higher percentage of this energy is converted into heat [3]. In addition, reducing the size of electronic components is on the rise, making the cooling of these components smaller whilst maintaining the compactness of the system a challenge. Among numerous cooling enhancements, one mechanism is the fan-based cooling system, a well-known method due to its high performance-to-cost ratio [4]. However, the thermal performance of fan-based electronic cooling systems is saturated by the formation of the thermal boundary layer, which is found to hinder the heat transfer between the cooling fluid and heated surface [5]. In addition, increasing cooling requires increasing fan size, which compromises the compactness of electronic components.

Consequently, researchers have implemented different mechanisms in electronic cooling systems to overcome the limited thermal performance. In the literature, the most popular passive thermal enhancement techniques in electronic cooling have been investigated, such as utilizing foam-based heat sinks [6,7], and optimizing the heat sink shape, size, arrangement, and artificial roughness [8,9,10]. Various approaches by enhancing the mixing process of working fluid with vortex generators [4,11,12], vortex tubes [13], mixing promoters [14], or flow-induced vibration [15,16] have also been attempted by many researchers. However, the benefits of electronic cooling by utilizing passive technologies have been found to be inefficient for high-power dissipation electronic systems.

Therefore, active cooling methods have been utilized to enhance the electronic cooling, including the projection of air at high speed toward a heated surface using air jets [17,18,19,20] or spray [14,21], investigated to enhance the thermal performance. Furthermore, using heat pipes [22] or adding oscillating heat pipes [23] and utilizing nanofluids in conjunction with a heat sink [24,25] as well as electronic cooling systems [26,27] have shown to be promising techniques for the thermal management of electronic systems.

However, another effective technique has been used; applying external vibrations to the heated object has also been studied as a heat transfer intensification technique. To prove the concept of vibrational heat transfer augmentation, experimental and numerical studies have been performed with different geometries, including gauge wire [28], spheres [29], and cylinders [30,31,32,33]. These studies noted that forced convective heat transfer is significantly influenced by external vibration. Due to the need to reduce the size of electronic components and cooling systems, applying vibration on fins and heat sinks could be a viable option to achieve a reduction in the electronic cooling system. Thus, several studies have considered the influence of vibrational frequency and amplitude variation on heat transfer augmentation. Gururatana and Li [34] investigated the heat transfer of a vibrating 2D pin fin numerically and observed an increase in heat transfer coefficient with frequency. An analysis detailing the influence of a vibrating 2D plate-fin on the flow field was reported by Rahman and Tafti [35]. The maximum heat transfer was obtained at the highest vibrational characteristics.

Dey and Chakrborty [5] numerically investigated the heat transfer augmentation of a vibrating actuated 3D fin. The authors obtained limited enhancement with lower vibrational characteristics. However, they noted an enhanced thermal performance with an increase in vibrational characteristics. Hussain, et al. [36] experimentally studied the thermal performance of piezo-actuated cooling fins and observed a higher heat transfer with the increase in vibrational frequency. Najim, et al. [37] conducted an experimental study indicating a potential increase in the thermal performance of the vibrating heat sink at higher frequencies.

Given that the application of external vibration is beneficial to thermal performance, it may also cause an adverse effect on the reliability of the system. There are studies that have investigated the reliability of the electronic systems under the vibration that occurs during shipping, handling, and the service life of the system. The reliability assessment of electronic assemblies under shock, harmonic, and random vibration has been performed experimentally [38] and analytically [39,40] to investigate the strains and stresses of printed circuit boards (PCBs) and the solder interconnect. Moreover, finite element modeling analysis (FEA) has been performed by several studies [41,42,43] to investigate the fatigue performance and failure modes of electronic products under vibrating conditions. However, no research has been performed to investigate the reliability of electronic systems due to the application of external vibration. Thus, a special consideration on the relatability of the system may be required in designing a vibrating system.

As mentioned, vibration has been a viable method to enhance heat transfer, and further investigation has been warranted to increase the enhancement. Although previous research has focused on the effect of sinusoidal vibration on forced convective heat transfer in heated geometries, new vibration methods by introducing different vibrational wave shapes on thermal systems are needed.

The vibrating heated surface enhances heat transfer mainly by setting the fluid in motion and consequently disrupting the thermal boundary layer [14]. However, the motion of the fluid particles depends not only on the vibrational characteristics but also on the type of vibrational wave shape. The different vibrational wave shapes may have a distinct effect on the flow profile, potentially affecting the thermal performance of the heat sink. This effect has not been considered in previous studies and is worth investigating.

In fact, Zhang, et al. [44] conducted an experimental study to investigate the effect of vibration modes on the detachment of low-rank coal particles from oscillating bubbles. The authors found that different vibration modes of oscillating bubbles have a distinct effect on path changes, and applying square wave excitations leads to the highest dramatic movement path changes due to its impulsive motion. In addition, the authors claimed that the square wave-shaped vibration induces the greatest acceleration within the particles compared to other vibration modes at similar vibrational frequencies and amplitudes. On the other hand, several experimental studies have been performed by utilizing a square wave-shaped vibration using a piezo-shaker [45], vibrators [44], and piezoelectrical actuators [46]. As applying square wave-shaped vibration has resulted in the highest movement path change as compared with the sinusoidal wave, utilizing square wave vibrations in heat sinks may result in a higher heat transfer enhancement, which has not been previously utilized in electronic systems and could be considered a new implementation to further enhance heat transfer.

As seen in the vibrational wave plot (Figure 1), square wave-shaped vibration causes a sudden impulsive motion of the heated geometry, potentially causing rapid velocity changes. Therefore, introducing the new square wave-shaped vibration in comparison with sinusoidal vibration may produce a higher heat transfer enhancement and consequently further reduce the size of the cooling system. To the best of the author’s knowledge, the application of square wave-shaped vibration on a heat sink to improve the thermal performance of the heat sink is new and has not yet been investigated by previous researchers.

The present study investigates the heat transfer enhancement resulting from a vibrating heat sink under the influence of the newly introduced square wave shape and sinusoidal wave shape using ANSYS/FLUENT CFD software. The examined heat sink structure is subjected to different vibration levels. Therefore, the dependence of heat transfer enhancement on the vibration frequency and amplitude of sinusoidal and square wave shapes is investigated. The possible reduction in both Reynolds number and size of the cooling system is discussed. Furthermore, the parameterization of the Nusselt number with frequency and amplitude under sinusoidal and square wave shapes is established to identify the operating envelope of vibration characteristics for enhanced thermal performance. Outcomes from the current study are expected to enable identification of the appropriate envelope of the vibrational characteristics, thereby enhancing the heat transfer of the heat sink and consequent reduction in the electronic cooling system and heat sink size.

2. Simulation Model and Methods

2.1. Model Development

The aluminum heat sink considered in the current study is presented in Figure 2 and the size of the rectangular fluid domain is 0.305 × 0.310 × 0.037 m³. A uniformly distributed power input of 9.5 W is supplied to the bottom of the heat sink, which corresponds to 6250 W/m², which is usually generated in data centers [47,48]. The uniform velocity is employed at the air inlet with a temperature of 25 °C, and the airflow is laminar at a Reynolds number of 1000, which corresponds to an averaged air inlet velocity of 1.46 m/s.

Mass, momentum, and energy conservation laws form the governing equations for the current study. The flow is assumed to be a laminar incompressible ideal gas with constant thermo-physical properties. The body forces are neglected, and a no-slip condition is applied on the interface between the heat sink and fluid. Thus, the governing equations can be presented as:

Continuity equation:

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0,$$

(1)

Momentum equations:

$$\frac{\partial u}{\partial t}+\rho u\frac{\partial u}{\partial x}+\rho v\frac{\partial u}{\partial y}+\rho w\frac{\partial w}{\partial z}=\mu \left({\nabla }^{2}u\right)-\frac{\partial P}{\partial x}$$

(2)

$$\rho \frac{\partial v}{\partial t}+\rho u\frac{\partial v}{\partial x}+\rho v\frac{\partial v}{\partial y}+\rho w\frac{\partial w}{\partial z}=\mu \left({\nabla }^{2}v\right)-\frac{\partial P}{\partial y}$$

(3)

$$\rho \frac{\partial w}{\partial t}+\rho u\frac{\partial w}{\partial x}+\rho v\frac{\partial w}{\partial y}+\rho w\frac{\partial w}{\partial z}=\mu \left({\nabla }^{2}w\right)-\frac{\partial P}{\partial z}$$

(4)

Energy conservation equation:

$$\rho c_p\frac{\partial T}{\partial t}+\rho c_p\frac{\partial }{\partial x}\left(uT\right)+\rho c_p\frac{\partial }{\partial y}\left(vT\right)+\rho c_p\frac{\partial }{\partial z}\left(wT\right)=k_{fluid}\left({\nabla }^{2}T\right)$$

(5)

Energy equation for heat sink:

$${\nabla }^{2}T=\frac{1}{\alpha } \frac{\partial T}{\partial \mathrm{t}}$$

(6)

where u, v, and w are the velocities along the x, y, and z directions, respectively, ρ is the fluid density, P is the hydrostatic pressure, and μ is the fluid viscosity, which is assumed to be constant. k_fluid is the thermal conductivity of the fluid, c_p is the constant specific heat, T is the temperature, and α is the thermal diffusivity of the fin material (k_fin/ρc_p). Past research has found that a higher heat transfer enhancement is obtained with horizontally vibrating heated objects [49,50]. Hence, the sinusoidal and square wave-shaped vibration is applied to the heat sink in the Y direction (Figure 2). To simulate vibration in FLUENT, User Define Function codes (UDFs) are developed, which include the displacement equations of sinusoidal and square wave-shaped vibration, given in Equations (7) and (8), respectively.

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

(7)

$$Y=\frac{4A_v}{\pi } {\displaystyle \sum }_{n=1,3,5..}^N\frac{1}{n}sin\left(2\pi nft\right).$$

(8)

where Y is the displacement of the heat sink at any time, f is the vibrational frequency, A_v is the vibrational amplitude, and t is time.

The investigation is conducted over a range of vibrating frequencies 0–100 Hz with peak-to-peak amplitudes ranging from 0 m to 0.005 m. A Reynolds number of 1000 is used for the vibrating case. Theoretical Reynolds numbers under nonvibrating conditions have been calculated using Equations (9) and (10) [51,52], which are used for comparative analyses with the vibrating heat sink.

$$Nu=\frac{\mathit{tanh}\sqrt{2N{u}_{ideal} \frac{k_{fluid}}{K_{fin}} \frac{h}{b} \frac{h}{t_{fin}} \left(\frac{t_{fin}}{L_{fin}}+1\right)}}{\sqrt{2N{u}_{ideal}\frac{k_{fluid}}{K_{fin}} \frac{h}{b} \frac{h}{t_{fin}} \left(\frac{t_{fin}}{L_{fin}}+1\right)}}\cdot N{u}_{ideal}$$

(9)

$$N{u}_{ideal}=\left[{\left(\frac{\frac{Re.b}{L_{fin}}Pr}{2}\right)}^{-3}+{\left.{\left(0.664\sqrt{\frac{Re.b}{L_{fin}}}. P{r}^{1/3}\sqrt{1+\frac{3.65}{\sqrt{\frac{Re.b}{L_{fin}}}}}\right)}^{-3}\right]}^{-1/3}\right.$$

(10)

where Nu is the Nusselt number under the normal condition with the fin effect, Nu_ideal is the ideal Nusselt number, Re is the Reynolds number, b is the fin spacing, and L_fin is the length of the fins. k_fin is the thermal conductivity of the fin material and Pr is the Prandtl number. h is the height of the fins and t_fin is the thickness of the fin.

2.2. Computational Details

The governing equations solved by ANSYS/FLUENT CFD software, discretized by the finite-volume method and the SIMPLE algorithm, is used to obtain the pressure–velocity coupling. The spatial discretization of momentum and energy equations is obtained by using a second-order upwind scheme. The transient formulation is obtained by using a second-order implicit scheme. The criteria for convergence of the iterative solution are specified as 10⁻⁶ for the residual of continuity, momentum, and energy.

Figure 3 shows the computational domain of the current study. While the heat sink is moving with sinusoidal and square wave-shaped vibration, the mesh of the fluid domain must be rearranged. A dynamic mesh setup is used to obtain the mesh motion, and a layering scheme is used to regenerate the mesh.

A mesh independency analysis is performed to ensure that the numerical results are independent of the number of cells. As shown in Figure 4, eight grids are considered in the current study with numbers of cells of 234,556, 324,568, 435,892, 498,460, 522,764, 580,356, 784,322, and 985,662. When the number of cells is increasing from 234,556 to 522,764, the Nusselt number is found to significantly decrease. However, a further increment in the number of cells leads to less than 1% of deviation in the Nusselt number. Thus, the grid with 522764 cells is used as the optimum mesh for all developed models.

The quality of a mesh used in the simulation is indicated by skewness and aspect ratio values. The maximum skewness and aspect ratio of the mesh for the present study are 0.62 and 6.12, respectively, meeting the requirements of acceptable mesh quality.

In addition, a time step independency investigation is carried out using three different time steps Δt = τ/8, τ/16, and τ/32 at a frequency of 100 Hz. The variations in the area-averaged Nusselt number of the heat sink with cycle time are shown in Figure 5. Reducing the time step (Δt) from τ/16 to τ/32 s gives less than 1% of variation in the peak-to-peak variation in the area-averaged Nusselt number. Here, τ is the cycle time of each vibration case. Thus, the τ/16 s time step is used in all simulated cases.

Validation of the CFD modeling is performed in terms of the sinusoidal vibrating heat sink, which is simulated under similar experimental conditions to Najim, et al. [37]. Numerical computations are performed under a range of vibrational frequencies, while the amplitude, supplied power, and air velocity are kept constant at 0.005 m, 17 W, and 0.6 m/s, respectively. As shown in Figure 6, a reasonable agreement is achieved in the validation against Najim, et al.’s [37] experimental measurements with a deviation of up to 8.3%. Therefore, CFD modeling in this study can be utilized with confidence to numerically analyze the heat transfer enhancement of the vibrating heat sink.

However, it can be seen that the average Nusselt number of the validation study is consistently lower than that of the experimental finding for the entire range of vibrational frequency. Hussain, et al. [36] performed numerical and experimental investigations and validated their numerical results against the experiments they performed. Their validation results were presented in terms of temperature of the fin, and they showed that their experimental temperature graph was consistently lower than the numerically obtained temperature. In other words, when the temperature graph corresponds to a higher Nusselt number, it can be concluded that the results obtained by the [36] is in agreement in terms of the trend with our validation results.

3. Results

CFD modelling is initially performed for the static heat sink (shown in Figure 2) at a Reynolds number of 1000. Then, Re = 1000 is used in the investigation of the vibration and amplitude variation effect on heat transfer. The airflow and vibrating directions are shown in the black arrows in the top view of the heat sink (Figure 7).

The vibration of the heat sink improves heat transfer by inducing randomness of the airflow profile and recirculation in the flow field. Thus, investigating the velocity profile shed from the vibrating heat sink is important. The effect of using the sinusoidal and square wave-shaped vibrations on the flow profile is expected to be different. Investigating the transient development of the velocity profile under each type of vibration compared with the velocity profile of the stationary heat sink is therefore important. The velocity profile of the nonvibrating heat sink and the transient development of air velocity vectors on a plane passing through the top section of the heat sink at different vibration cycle stages for sinusoidal and square wave shapes are presented. The air velocity vectors shown in Figure 8b–i are obtained with sinusoidal and square wave-shaped vibrations at frequencies of 100 Hz at a constant peak-to-peak amplitude of 0.003 m. In each figure, the two fins of the vibrating heat sink are shown as red surfaces.

After performing the validation of the model, the heat sink shown in Figure 2 is modeled under normal and vibration conditions under the range of vibrational frequencies and amplitudes for both sinusoidal and square wave shapes.

At the beginning of the sinusoidal motion, the heat sink moves to a positive Y direction. Figure 8b shows that, at the end of the first quarter of the cycle, the vibrating fins induce airflow recirculation near the trailing edges of the fins, whereas the flow between the two fins moves upward toward fin 1 following the vibration displacement direction and hitting the surface of fin 1.

At the end of the second and third quarter of the vibration cycle, Figure 8c,d show that the recirculation of air at the trailing edges of the fins continues. However, the airflow in the middle channel between the two fins changes its direction and hits the interior surfaces, especially the interior surface of fin 2. This induced flow change in direction becomes more noticeable at the end of the third quarter of the cycle. When the vibration continues at the end of the cycle, it can be seen in Figure 8e that the flow direction and recirculation become similar to Figure 8c.

At the beginning of the square wave-shaped motion, the heat sink reaches its maximum positive amplitude almost instantly. As a result, Figure 8f shows that the square wave excitation of the fins induces strong air recirculation near the trailing edges of fins compared to the sinusoidal motion at the end of the first quarter of the vibration cycle.

With increasing time, the heat sink moves to the end of the second quarter of the cycle (Figure 8g). The impulsive motion of the heat sink gives a higher air velocity magnitude to the air particles. Thus, the fluid adjacent to it gains noticeable movement path changes. This leads to the formation of strong clockwise and counter-clockwise recirculation zones near the leading and trailing edges of both fins.

Afterward, the heat sink moves in the negative y-direction continuously at the start of the third quarter of the vibration cycle. As indicated in Figure 8h, the flow circulation zones are found to be similar to those mentioned earlier at the end of the second quarter. However, the strength of the circulation zones near the leading and trailing edges of the fins decreases gradually at the end of the third quarter of the vibration cycle.

As vibration continuous, the heat sink moves to the positive y-direction at the end of the vibration cycle (Figure 8i), and the flow direction and recirculation zones are found to be opposite to those, as shown in Figure 8g.

Compared with the nonvibrating heat sink shown in Figure 8a, air velocity vectors under the vibration effect are not straight streamlines. When the heat sink vibrates with some frequency and amplitude, the airflow direction continuously changes toward the fins' surface, which is expected to improve the fins’ thermal performance. When the heat sink vibrates under a square wave shape, strong recirculation zones and secondary flows are formed around both fins at each quadrant of the vibration cycle owing to the sudden movement of the heat sink, causing a noticeable velocity increase in the flow field. By comparing air velocity vectors between the square shape and sinusoidal vibration, the velocity magnitude and strength of the recirculation zones are higher under square-shaped vibration. This phenomenon induces more disruption to the developed thermal boundary layer, thereby further enhancing the thermal performance of the heat sink.

Figure 9 shows the temperature contours of the nonvibrating heat sink compared with other cases of increasing frequency and amplitude at each type of vibrational wave shape. To avoid repetition, only one representing case for vibration frequency and amplitude under each wave shape is selected to show the effect of vibration on the thermal boundary layer. As seen in Figure 8a, the nonvibrating heat sink has a uniform airflow distribution in the direction of the flow field. Thus, the temperature gradient is high between the air stream and near the fin surfaces (Figure 9a), showing that the effect of the heat resistance of the thermal boundary layer causes and limits the cooling effect.

However, when vibration is applied, it causes a rapid change in the air velocity distribution and randomness and recirculation, causing the fluid to move directly toward the heated fin surfaces, as shown in Figure 8. This phenomenon consequently ruptures the thermal boundary layer. As seen in Figure 9b,c, applying vibration on the heat sink induces a disturbance to the continuous thermal wake. It leads to a widespread decrease in air temperature near the fin surfaces, which tends to have a higher cooling effect than the static fin. This randomness and recirculation are more noticeable with a higher air velocity magnitude when a square-shaped vibration is used. Figure 8f–i show that using a square wave-shaped vibration induces stronger recirculation zones, leading to a higher mixing effect within the flow field. Thus, the overall spread of thermal disturbance and lower air temperature becomes more noticeable when square wave excitation is applied compared with the sinusoidal wave shape.

The cooling effect of the square shape vibration can be seen in Figure 9c, where it causes a widespread low-temperature region at both fin surfaces, indicating a further cooling effect.

When the peak-to-peak amplitude is increased from 0.003 to 0.005 m at a constant frequency of 100 Hz, the recirculation zones and randomness of the airflow profile increase as the peak-to-peak amplitude increases. As a result, the thermal boundary layer is disturbed further, resulting in increased heat transfer. As seen in Figure 9b–e, increasing the amplitude leads to a widespread decrease in air temperature near the fin surfaces, tending to have a stronger cooling effect. However, as seen in Figure 9e, increasing the amplitude with square wave-shaped vibration leads to a higher decrease in air temperature near the fin surfaces due to stronger flow circulations, leading to a higher mixing effect within the flow field.

The discussion shows that vibrational frequency, amplitude, and wave shape are three important parameters that induce air mixing and rupture the thermal boundary layer. To evaluate the thermal effectiveness and consequent cooling enhancement of the system by varying those parameters, time-averaged Nusselt numbers are used as an indicator. As a result, the parameterization of time-averaged Nusselt number with vibration frequency and amplitude for each wave shape is performed and presented in Figure 10. The parametrization helps to more simply estimate the heat transfer enhancement and consequent reduction in the cooling system based on the vibrational parameters under each wave shape. The parametrization also aids in determining the optimum scenario of parameter combinations that provide the best thermal performance of the system.

Moreover, according to the Nusselt number value obtained from the modeling, Reynolds numbers are calculated from Equations (9) and (10), which gives the value of the Reynolds number under no vibrating conditions, and those values are presented at each data point in Figure 10.

As shown in Figure 10a below, when the vibrational frequency is below 30 Hz, the Nusselt number almost remains unchanged with each type of wave shape compared with the static heat sink. When the frequency is lower than the threshold value, the energy of vibration propagates through the solid interface, and the fluid may not be adequate to induce the strong recirculation zones or secondary flows within the flow field. Thus, applied vibration is insufficient to disturb the developed thermal resistance caused by the thermal boundary layer. Notably, this threshold is observed to be independent of the vibration wave shape. Thus, using sinusoidal or square wave-shaped vibration on the heat sink with a lower frequency (f < 30 Hz) does not influence the thermal performance of the system. However, after exceeding the threshold frequency, Nusselt values show a steady rise with increasing frequency at each vibrational wave shape.

This result can be attributed to the vibration effect, where a lower vibration frequency may not induce sufficient air perturbation, which cannot disrupt the thermal boundary layer. However, when the frequency increases beyond the threshold, it causes a higher agitation and air perturbation, causing disruption to the thermal boundary layer and consequently enhancing heat transfer.

As expected, the square wave excitation of the heat sink causes the Nusselt number to shift to an elevated value compared with the sinusoidal motion because square wave excitation of the heat sink attains the maximum amplitude instantly, which induces a strong impulsive motion, leading to a significant velocity change and stronger recirculation zones, thereby enhancing the cooling effect. Moreover, the relative Nusselt number enhancement difference in square and sinusoidal wave motion is improved with increased frequency. Compared with the static heat sink, the Nusselt number is improved up to 20% with square wave-shaped vibration, whereas a 12% increase in Nusselt number values is recorded when the sinusoidal frequency is used.

To investigate the effect of varying the amplitude at a constant frequency on heat transfer enhancement, a 100 Hz frequency is selected to show the effect on the Nusselt number. As seen in Figure 10b above, an increase in peak-to-peak amplitude at a frequency of 100 Hz leads to an increase in time-averaged Nusselt number in both types of excitations. The increase in the peak-to-peak amplitude at higher frequency imparts a higher fin displacement, resulting in a greater recirculation within the flow field and improved heat transfer.

However, the heat transfer enhancement is found to be higher with square wave excitations. Once the amplitude of the square-shaped vibration is increased, the impulse motion of the vibrating fins combined with the higher amplitude causes a higher air perturbation due to the impulsive nature of this vibration, resulting in a rapid change in airflow field direction near the fin surface, further enhancing the heat transfer. When the amplitude is increased from 0 to 0.005 m at 100 Hz, a 14.2% increase in Nusselt number values is recorded when the sinusoidal vibration is used. However, at 100 Hz frequency and 0.005 m amplitude, the Nusselt number magnitude increases by up to 25% with square wave-shaped vibration, providing the system with the highest thermal performance over the evaluated parameter range.

From Figure 10, the results show that increasing amplitude and frequency lead to a higher thermal performance of the vibrating heat sink. Therefore, it is also necessary to present the heat transfer enhancement in terms of the ratio of the Nusselt number of vibrating and nonvibrating conditions at each vibration level, which could be used as an indicator to perform the comparative analysis of the increase in heat transfer enhancement under vibrating conditions:

$$Nusselt Ratio=\left(\frac{Nusselt number with vibrating conditions }{Nusselt number with non-vibrating conditions}\right)$$

(11)

Figure 11a,b show the variation in Nusselt ratio with vibrational frequency and amplitude under sinusoidal and square wave shapes. Due to the insignificant heat transfer enhancement with a lower frequency (f < 30 Hz), the Nusselt ratio is found to be one at lower vibrational frequencies. However, after exceeding the critical frequency, the Nusselt ratio is found to steadily increase with increasing frequency. As expected, a higher enhancement of the Nusselt ratio is obtained with square wave-shaped excitation as compared to the sinusoidal vibration at each vibration frequency. As compared to the nonvibrating heat sink, 12% and 20% enhancements in Nusselt number ratios are obtainable by applying sinusoidal and square wave-shaped excitation, respectively, at f = 100 Hz and Amp = 0.003 m.

With the increase in peak-to-peak amplitude at constant frequency of 100 Hz, the Nusselt ratio also increases. As expected, an increase in Nusselt ratio is found to be higher with square wave-shaped vibration. As compared to the nonvibrating heat sink, 14.2 and 25% enhancements in Nusselt number ratios are obtained by utilizing the sinusoidal and square wave-shaped vibration, respectively, with the conventional heat sink at f = 100 Hz and Amp = 0.005 m.

Given that the parametric study shows the possibility of heat transfer intensification of the vibrating heat sink under sinusoidal and square wave shapes, this method can be used to reduce the size of the conventional cooling system. The Reynolds number values required under normal conditions (no vibration) to achieve the Nusselt number value under vibration conditions are presented in Figure 10 at each frequency and amplitude. Using the presented Reynolds number values, the percentage reduction in Reynolds number (PR_Re) is calculated. It is used as an indicator to perform the comparative analysis of the reduction in Reynolds number under vibrating conditions. In this regard, investigating PR_Re eventually estimates the possible reduction in the size of the cooling system based on the vibrational characteristics at each wave shape.

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

(12)

Figure 12a,b show the variations in PR_Re with vibrational frequency and amplitude at both sinusoidal and square wave shapes. Given that no heat transfer enhancement with a lower frequency range (f < 30 Hz) occurs, no opportunity for reductions in Reynolds number arises in this frequency range. However, after exceeding the critical frequency (f = 30 Hz), PR_Re shows a steady rise with an increase in frequency for each type of wave shape. Given that the square wave-shaped vibration shows a higher heat transfer enhancement than the sinusoidal vibration at each frequency level, a higher percentage reduction in Reynolds number is obtained with increasing frequency under square wave-shaped excitation. As observed from Figure 12a, 25% and 36.5% reductions in Reynolds numbers are obtainable by applying sinusoidal and square wave-shaped excitation, respectively, at f = 100 Hz and Amp = 0.003 m compared to the conventional cooling system.

When the peak-to-peak amplitude increases at a constant frequency of 100 Hz, PR_Re also increases. As expected, an increase in PR_Re is found to be higher with square wave-shaped vibration. Figure 12b shows that PR_Re rises to 28.8% with sinusoidal wave-shaped and 42.2% with square wave-shaped vibration at f = 100 Hz and Amp = 0.005 m. This result indicates that using vibration on heat sink structures can achieve a similar thermal performance at lower Reynolds numbers than can the conventional Reynolds numbers required to achieve the same thermal performance under nonvibrating conditions. Moreover, applying square wave-shaped vibration helps to achieve a further reduction in the size of the cooling system compared to sinusoidal vibration at every investigated frequency (f > 30 Hz) and amplitude level. Thus, the square-shaped vibration can achieve similar thermal performance with a smaller cooling system than a static heat sink with a much larger cooling system. Thus, this enhancement method might be well suited to the electronic industry because it is coherent with the challenges of compactness experienced in most modern electronic systems.

4. Discussion

Conclusively, the parameterization of Nusselt number with frequency and amplitude under sinusoidal and square wave shapes is established in the current study, helping to identify the appropriate operating envelope of vibration characteristics for enhanced thermal performance. Increasing the frequency or amplitude within the appropriate operating envelope region increases the heat sink thermal performance at each vibrational wave shape. However, square wave-shaped vibration is found to be more effective in the enhancement of heat transfer than sinusoidal vibration. Thus, applying square wave-shaped vibration on the forced convective cooling system is more beneficial to its thermal performance and further reduces the cooling system size compared with sinusoidal vibration at similar vibrational frequencies and amplitudes. By using square wave-shaped vibration at f = 100 Hz and 0.005 m amplitudes, the maximum enhancement of the Nusselt number is 25% compared with the nonvibrating heat sink, which is 11% higher than the Nusselt number recorded by the sinusoidal vibrating heat sink. This finding results in a 42.2% reduction in Reynolds number compared with the Reynolds number of nonvibrating cooling systems, which is 13.4% higher than the reduction in Reynolds number by the sinusoidal vibrating heat sink at similar vibrational frequencies and amplitudes.

Researchers have been continuously investigating new methods to enhance the cooling of electronic systems. Zhuang, et al. [53] used a fan-integrated microchannel heat sink with rhombus fractal-like units. They achieved an 18% increase in Nusselt number compared with a conventional parallel microchannel heat sink. Dey and Chakrborty [5] applied sinusoidal vibrations on a piezo-actuated fin. They achieved up to a 25.5% improvement in Nusselt number compared with a nonvibrating fin. Freegah, et al. [54] used plate-fin heat sinks with corrugated half-round pins in a vertical arrangement subject to parallel flow and recorded an approximately 34.5% increase in Nusselt number values compared with the traditional plate-fin heat sink. Masip, et al. [55] used an impinging air jet with a crossflow configuration. They recorded an up-to-40% increase in the average Nusselt number compared with the conventional cross-flow configuration. Moon, et al. [56] applied a pulsating flow through heated blocks. They recorded increases of up to approximately 23% in Nusselt number values compared with a nonpulsating flow.

As mentioned, using square-shaped wave vibration achieved an up-to-25% increase in Nusselt number and a 42.2% reduction in Reynolds number compared with nonvibrating fins. This result shows that square shape-waved vibration resulted in a higher cooling effect compared with other methods, making it achieve further reductions in the cooling system size of electronic components.

It is worth mentioning that the results presented in this research could be assumed applicable to heat sinks with more fins under the same heat flux and air velocity, because past research reported by Adhikari, et al. [57] found that the Nusselt number does not change with the increase in the number of static fins. They also reported that when they varied the number of fins, the flow and thermal contours indicated very small differences between the heat sink fin channels. Since the finding of [57] that the Nusselt number remains unchanged when the number of static fins increases, and in this research, the application of vibration on fins is equal, it is expected that the results presented in this research will be the same when number of fins increases under the same heat flux and air velocity. However, further research is required to confirm this assumption.

It is also important to mention that special consideration may be required in designing a vibrating system. Therefore, future research could be performed to investigate the reliability of the system demonstrating the mechanical stress factors and fatigue life due to the supplied vibration.

5. Conclusions

Numerical simulations are conducted to investigate the effect of vibration on the heat transfer performance of a heat sink. A parametric study is performed by using vibrational frequency, amplitude, and wave shapes to quantify the dependence of thermal performance on each parameter. When the vibrational frequency is lower than the critical frequency (f < 30 Hz), the thermal performance of the heat sink is not influenced by the vibration. However, after exceeding the critical frequency, an increase in amplitude and frequency leads to a higher enhancement in thermal performance at each vibrational wave shape. This result enables the identification of the appropriate operating envelope of the vibrational characteristics, thereby enhancing the heat transfer of the heat sink. Using square wave-shaped excitations provides more benefit to heat transfer than does sinusoidal vibration at every frequency and amplitude level. A maximum increase of 14.2% in Nusselt number values is recorded with sinusoidal vibration, while a maximum enhancement of 25% is obtained with square wave-shaped vibration. Thus, using a square wave-shaped vibration on the heat sink at the frequency of 100 Hz and amplitude of 0.005 m is found to achieve the highest thermal performance of the system over the range of the evaluated parameters. The process results in a 42.2% reduction in Reynolds number, making using the square wave-shaped vibration beneficial in reducing the conventional cooling system and achieve the compactness needed in modern electronic systems.

However, the current study is performed at a single Reynolds number and power input. Future research could be performed on a range of Reynolds numbers and power inputs. Moreover, this current study could be extended with experimental work and investigating the energy used to vibrate the fins as compared with heat transfer enhancement. This research could also be extended to investigate the reliability of the system due to the vibration of the heat sink.