Simulation of Alumina/Water Nanofluid Flow in a Micro-Heatsink With Wavy Microchannels: Impact of Two-Phase and Single-Phase Nanofluid Models

In this article, alumina/water nanofluid (NF) flow in a heatsink (H-S) with wavy microchannels (W-MCs) is simulated. The H-S is made of aluminum containing four similar parts. Each part has an inlet and outlet. Constant heat flux is applied on the bottom of the H-S. The study is based on two-phase (T-P) mixture and single-phase (S-P) models to determine the difference between these two types of simulations. FLUENT software and the control volume method were used for simulations. The volume control method is employed to solve equations. The effective variables include the volume fraction 0 < φ < 5% of alumina and Reynolds number (Re) 300 < Re < 1800. The maximum H-S bottom temperature, the required amount of pumping power (PP), the temperature uniformity, and the heat resistance of the H-S are the outputs studied to simulate the S-P and T-P models. The results show that the use of the T-P model has less error in comparison with the experimental data than the S-P model. An increment in the Re and φ reduces the maximum temperature (M-T) of the H-S. The S-P model, especially at a higher value of φ, leads to a lower M-T value than the T-P model. The S-P model shows a 0.5% greater decrease than the T-P model at the Reynolds number of 300 by enhancing the volume percentage of nanoparticles (NPs) from 1 to 5%. Temperature uniformity is improved with Re and φ. The reduction of H-S thermal resistance with Re and φ is the result of this study. Adding NPs to water, especially at higher amounts of φ, enhances the required PP. The T-P model predicts higher PP than the S-P one, especially at a high value of φ. The T-P model shows 4% more PP than the S-P model at Re = 30 and a volume fraction of 4%.

In this article, alumina/water nanofluid (NF) flow in a heatsink (H-S) with wavy microchannels (W-MCs) is simulated. The H-S is made of aluminum containing four similar parts. Each part has an inlet and outlet. Constant heat flux is applied on the bottom of the H-S. The study is based on two-phase (T-P) mixture and single-phase (S-P) models to determine the difference between these two types of simulations. FLUENT software and the control volume method were used for simulations. The volume control method is employed to solve equations. The effective variables include the volume fraction 0 < φ < 5% of alumina and Reynolds number (Re) 300 < Re < 1800. The maximum H-S bottom temperature, the required amount of pumping power (PP), the temperature uniformity, and the heat resistance of the H-S are the outputs studied to simulate the S-P and T-P models. The results show that the use of the T-P model has less error in comparison with the experimental data than the S-P model. An increment in the Re and φ reduces the maximum temperature (M-T) of the H-S. The S-P model, especially at a higher value of φ, leads to a lower M-T value than the T-P model. The S-P model shows a 0.5% greater decrease than the T-P model at the Reynolds number of 300 by enhancing the volume percentage of nanoparticles (NPs) from 1 to 5%. Temperature uniformity is improved with Re and φ. The reduction of H-S thermal resistance with Re and φ is the result of this study. Adding NPs to water, especially at higher amounts of φ, enhances the required PP. The T-P model predicts higher PP than the S-P one, especially at a high value of φ.
The T-P model shows 4% more PP than the S-P model at Re 30 and a volume fraction of 4%.

INTRODUCTION
Heatsinks (H-Ss) are widely used in various industries such as electronics and electrical industries, solar industries, military industries, aerospace, etc. H-Ss are employed to prevent the temperature enhancement of electronic components in different devices (Nakayama, 1986). Electronic components need to be cooled so that the components are not damaged. They are heated during the operation and power consumption and therefore need to be cooled (Alihosseini et al., 2020). These parts are used in many devices. The H-Ss play a pivotal role, and many researchers have studied their performance (SohelMurshed and Nieto de Castro, 2017;Zhao et al., 2019;Qi et al., 2020a;Cheng et al., 2021;Tian et al., 2021). The purpose of these studies has been to enhance the thermal efficiency of the H-S and thus reduce the temperature of the electronic components on which the H-S is installed. Researchers have used different base fluids for their studies (Choi et al., 2012;Zhang et al., 2020a). Air has gained a lot of attention due to its availability and cheapness, but due to its limited cooling capacity, it is not suitable for parts with very high processing power. However, various researchers have used air as a H-S working fluid (Khattak and Ali, 2019;Elsayed et al., 2020). Kalbasi (2021) introduced a new H-S using phase-change material (PCM) and air-adopted with electrical equipment. He used the characteristics of PCM and air to remove heat and keep the temperature of the electronic device low.
Single-phase (S-P) and two-phase (T-P) mixture models can be used to simulate the cooling performance of NFs in different equipment SafdariShadloo, 2021). The possible difference in the results of using these methods encouraged a group of researchers to simulate various problems using these models and compare the results with experimental data (MokhtariMoghari et al., 2011;Göktepe et al., 2014). In the above-mentioned articles, considering NF flow in various heat exchangers, the S-P model has been used to model the NF flow; however, the researchers have rarely employed the T-P method to simulate the performance of NF.
In one of these articles, Akbari et al. (2012) analyzed the cooling performance of water-Al 2 O 3 NF in a tube using these two schemes. It was reported that the outcomes of the T-P method are more consistent with the experimental data. In another study, Moraveji and Ardehali (2013) compared the results of two methods and revealed the superior performance of the T-P method compared to the S-P one.
Due to the wide range of applications of H-Ss in the applied industry, a large number of studies have been conducted in this field. The reason for researchers' attention to H-Ss is their widespread use and importance in industries. This study aims to enhance the thermal efficiency of H-Ss by changing their geometry. On the other hand, the use of NFs in recent decades has been very much considered by researchers due to the challenge of using S-P or T-P models in recent years (Peng et al., 2020;Ahmadi et al., 2020;Ahmadi et al., 2020;Giwa et al., 2021;Bagherzadeh et al., 2019). In the present article, the effect of using alumina/water NF flow in a H-S with wavy walls is numerically investigated. One of the innovations of this work is the comparison of S-P and T-P schemes in the H-S. H-S temperature, temperature uniformity, pumping power (PP), and heat resistance of H-S at different values of Re and φ are studied. The effective variables are Reynolds number (Re) and φ when S-P and T-P models are employed. In summary, the effect of S-P and T-P modeling of NFs in a H-S with wavy walls is the innovation of the present work.

PROBLEM DESCRIPTION
The H-S is a rectangle structure of four similar parts made of aluminum, as shown in Figure 1. The dimensions of the H-S, including its thickness, wall thickness, and the dimensions of the inlets and outlets can be seen in Figure 1. A constant heat flux of 1 MW/m 2 is applied to the bottom of the H-S. The heat flux is applied to the enclosure with a surface area of 111.6 mm 2 , which is located under the microchannels. The wavy microchannel (W-MC) walls are designed to be corrugated to enhance heat transfer. Alumina/water NF flows in the middle of the W-MCs. By changing φ from 0 to 5%, the thermal efficiency of the H-S is evaluated using the T-P and S-P models.

S-P Model Equations
The equations governing fluid flow within the H-S, as S-P for the laminar and continuous flow of incompressible Newtonian fluid, are as follows. These equations include mass, momentum, and energy conservation (Akbari et al., 2011): where, vis the velocity; T is the temperature, P is the pressure, ρ is the density, k is thermal conductivity, c p is the specific heat capacity, and μ is the viscosity. These properties correspond to NFs, which are calculated as follows: where, the indices p and f refer to nanoparticles (NPs) and the base fluid, respectively, and φ is the volume fraction of NP. The viscosity of the NF is calculated according to the following equation, and this equation is specific to alumina NF (Khanafer and Vafai, 2011): The equation of thermal conductivity, which depends on the diameter of NPs, is expressed as follows (Teng et al., 2010): where d is the diameter of NPs equal to 40 nm, ω is the mass percentage of NPs, and T is the temperature. Other properties of water and alumina NPs are given in Table 1.

T-P Mixture Model Equations
The T-P model solves mass, momentum, and energy conservation equations for the mixture as a volume percentage equation for the second phase (Akbari et al., 2012;Moraveji and Ardehali, 2013). Then, the relative velocities are calculated using the equations. Relative equations are defined as follows: Conservation of mass:  Conservation of momentum: where the index m represents the mixture. In the above equations, the mean velocity values of the mixture and density can be expressed as follows: Fi k ρ k .
Conservation of energy: Volumetric percentage: It should be pointed out that the S-P model equations (Eq. 6 and Eq. 7) are used for the thermal conductivity and viscosity models in the T-P model. Drift velocity expressed for the second phase and can be calculated using the following equation for the k th phase: Slip velocity, or relative velocity, is defined as the secondary phase velocity depending on the primary phase. v The gravitational acceleration is also defined as follows: where the heat transfer is by natural convection, and the gravitational acceleration is considered in the equations.

Boundary Conditions
It should be pointed out that the no-slip boundary condition is applied to the walls. In the H-S, which is made of aluminum (Al) with thermal conductivity of 202.4 W/mK and a specific heat capacity of 871 J/kg.K, conductive heat transfer occurs. A constant heat flux of 1 MW/m 2 is applied on the H-S bottom FIGURE 3 | Comparison of the average Nusselt number between the present study, an experimental work (Ho and Chen, 2013), and a numerical one (Moraveji and Ardehali, 2013). with an area of 111.6 mm 2 . The top wall of the H-S is assumed to be insulated. The fluid enters the H-S at a temperature of 293 K and exits at a p 101.325 kPa boundary condition.

Numerical Method and Validation
To simulate the problem, geometry is first generated and meshed by mesh software. The generated mesh type is tetrahedral. The control volume method is employed for numerical solution, and the SIMPLE algorithm is used to couple the velocity and pressure fields. The second-order upwind model is used to solve the equations of momentum and energy. The convergence criterion used for all equations is set to 10 −5 . A PC with CPU i7 and 8-GB DDR4 RAM had been used to solve the equations. The approximate time of the runs varied from 150 to 200 min depending on the Reynolds number. The mesh generated on the geometry is evaluated for S-P and T-P models. Different results are examined for different numbers of the grid point, and eventually, the grid with 1,552,680 elements is selected as the optimal one. Figure 2 demonstrates the average temperature of the H-S for various numbers of elements at two Re, when φ 5% for T-P and S-P models. For the validation, the results are compared with two other articles to investigate the accuracy of the S-P and T-P models using the thermal conduction model and the selected viscosity. The verification is performed using numerical simulations of Moraveji and Ardehali (2013) and experimental data of Ho and Chen (2013). The results of the comparison are provided in Figure 3. As can be seen, the differences between the results are small, particularly with experimental results, and the results are more promising. The maximum error for the S-P model was observed for Re 655 compared to the experimental data of Ho and Chen (2013). The maximum error of the S-P model was 4.1% and that of the T-P one was 1.9% at Re 1,300. Besides, the value of the local heat transfer coefficient is compared between the present work and that of Kim et al. (2009) for two different channel lengths ( Table 2). The comparison table shows that the simulation results are reasonable due to a maximum error of 5.5% between the present work and the experimental data.

RESULTS AND DISCUSSION
It is necessary to measure the error of both S-P and T-P models with a valid reference to determine which method is more accurate with respect to the experimental data. Table 3 shows the error values of T-P and S-P models compared with the experimental model of Ho and Chen (2013). It can be seen that the amount of error between the T-P model and the experimental work is less than the S-P one. It is generally seen that the amount of simulation error at lower amounts of Re is lower for both models than for high values of Re.   Finally, it is found that the T-P model leads to better results in the W-MCs than the S-P one. Figure 4 presents contours on the middle plane of the H-S for various amounts of Re for water/Al 2 O 3 , with φ 5%. As can be seen, the enhancement in Re raises the velocity in the microchannels. An increment in the Re means an enhancement in the fluid velocity in the inlet of the H-S. It is observed that the maximum velocity occurs in the internal microchannels of the H-S. In such a H-S, the fluid moves faster due to shorter distance. The use of NFs slightly reduces the maximum velocity in the H-S. Figure 5 demonstrates the velocity vector for the NF flow at Re 1800. This figure clearly displays the motion direction of the fluid. It can be observed that the fluid is directed toward the W-MCs after entering, and the velocity in the internal W-MC is higher than in other regions. This channel has a higher velocity because of its shorter passage to the outlet and less energy loss in the fluid due to its viscosity. In sharp corners, the velocity variations are more noticeable. In the W-MC, the velocity is enhanced in its spiral parts. Figure 6 shows the maximum temperature (M-T) of the H-S at different values of Re when φ 2% for T-P and S-P simulation models. It can be seen that an increment in the Re reduces the M-T for both types of NF models. Also, an enhancement in the φ has the same effect. For T-P simulation, the mixture model is used, and for the S-P model, the expressed relationships for viscosity and temperature-dependent thermal conductivity are used. It can be seen that at low amounts of φ, the difference between the S-P and T-P models is low, but the amount of temperature difference is enhanced with enhancing φ. The temperature value obtained by the T-P method is slightly lower than that achieved from the S-P one, which is more accurate due to the validation of the T-P model. Figure 7 shows the temperature uniformity at the H-S bottom at different values of Re when φ 2% for the T-P and S-P simulation models. The amount of temperature uniformity depends on the M-T and minimum temperature of H-S. As the Re is enhanced, the amount of Teta is reduced, which means better temperature uniformity. Enhancing the φ has the same result. The reduction in the M-T is the most important reason for the increase in temperature uniformity. It can be seen that the difference between the simulations using the T-P and S-P models when φ 0.05 is more noticeable. In the S-P model, the results are dependent on the viscosity and thermal conductivity models, and changing the viscosity and thermal conductivity models can change the results. Figure 8 shows the H-S thermal resistance at different amounts of Re and two values of φ for the T-P and S-P simulation models. It can be seen that the enhancement in Re reduces the amount of heat resistance of the H-S. The addition of more NPs has the same effect. A reduction in the average temperature of the H-S with Re and φ reduces the heat resistance of the H-S. It can be observed that there is a difference in the simulation results of the S-P and T-P models, which is higher at larger amounts of φ. Of course, the difference between the T-P and S-P models is more noticeable at lower magnitudes of Re. At lower flow rates, the effect of NF on heat transfer is higher, and as a result, the effect of NF simulation is more noticeable. Figure 9 shows PP at different values of Re when φ 2% for T-P and S-P simulation models. An increment in the Re means an enhancement in the volume flow rate of the fluid, leading to intensification in the amount of power required by the pump. The addition of NPs in both NF models enhances the viscosity, so the shear stress in the fluid is increased, and more power is required  for the NF pumping. Since the effect of adding NPs to the fluid is much less than the variations of Re on the PP, the effect of the S-P or T-P model on the PP is low. However, at high fluid velocities and at values of Re where the shear stress has a more significant effect, the effect of the S-P and T-P models is more visible. At high values of Re, for example, 1800, the role of the type of viscosity is more important. In this respect, at Re 1800, the difference between the results is more remarkable. The value expressed by the T-P model for PP is more significant than the S-P one.

CONCLUSION
This article simulated the water/alumina NF flow in a H-S with W-MC using S-P and T-P models. By changing the Re and φ, the M-T, H-S temperature uniformity, thermal resistance, and PP were studied. The results of numerical analysis and comparison of two models demonstrated the following: 1) In general, the error percentage of using the T-P mixer model is lower than the S-P method, but the error of both methods is enhanced compared to the experimental data by increasing φ. 2) An increment in the Re and φ reduces the M-T in the H-S. The S-P method predicts a lower M-T than the T-P one. At Re of 300, an increment in the volume percentage of NPs from 1 to 5% reduces the maximum H-S temperature by 3.9 and 3.4% for the S-P and T-P models, respectively. 3) An enhancement in the flow rate and the use of thicker NF provide better temperature uniformity in the H-S. The S-P method predicts a better amount of temperature uniformity than the T-P model. 4) An increment in the φ and the Re reduces the thermal resistance of the H-S. The T-P mixer model shows higher thermal resistance than the S-P one. 5) At low NF velocities, an enhancement in the φ and the type of phase model do not have a considerable effect on the PP, while at higher velocities, the addition of NPs enhances the PP, and the T-P model has a higher prediction than the S-P one.
Enhancing the volume percentage of NPs from 1 to 5% increases the amount of PP by 24 and 28% for the S-P and T-P models, respectively, when Re 300.

DATA AVAILABILITY STATEMENT
The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.