Thermophysical properties of the base fluid and Cu nanoparticles .
Nanofluids are liquid/solid suspensions with higher thermal conductivity, compared to common working fluids. In recent years, the application of these fluids in electronic cooling systems seems prospective. In the present study, the laminar mixed convection heat transfer of different water–copper nanofluids through an inclined ribbed microchannel––as a common electronic cooling system in industry––was investigated numerically, using a finite volume method. The middle section of microchannel’s right wall was ribbed, and at a higher temperature compared to entrance fluid. The modeling was carried out for Reynolds number of 50, Richardson numbers from 0.1 to 10, inclination angles ranging from 0° to 90°, and nanoparticles’ volume fractions of 0.0–0.04. The influences of nanoparticle volume concentration, inclination angle, buoyancy and shear forces, and rib’s shape on the hydraulics and thermal behavior of nanofluid flow were studied. The results were portrayed in terms of pressure, temperature, coefficient of friction, and Nusselt number profiles as well as streamlines and isotherm contours. The model validation was found to be in excellent accords with experimental and numerical results from other previous studies.
- mixed convection heat transfer
- inclined ribbed microchannel
- finite volume method
- friction factor
Electronics have turned smaller, quicker, and more powerful due to the current development in computing technology during the past few decades, resulting in a dramatic rise in the rate of heat generation from electronic appliances. One way to keep the heat generated by different parts of electronic devices within safety zone is to cool the chips via forced air flow. However, standard cooling procedures seem insufficient to deal with the parts which are comprised of billions of transistors functioning at high frequency, considering the fact that the temperature can rise up to a critical point. Thus, microscale cooling appliances like microchannel heat sinks have vital roles in heat removal applications in appliances including high-energy mirrors and laser diode arrays . In 1981, Tuckerman and Pease brought up the concept of a microchannel heat exchanger first . The major advantage of a microchannel heat sink is the fact that its heat transfer coefficient is much higher than the traditional heat exchangers . This causes microchannels to become useful for being employed in semiconductor power devices, very large-scale integrated (VLSI) circuits, etc. . The first idea was to utilize water as a coolant in microchannels as cooling systems [5, 6]. Nevertheless, water is subjected to weak thermophysical properties. The convective heat transfer rate of these types of working fluids can be enhanced by improving their thermophysical properties. Nanofluids prepared through dispersing nanosized particles into the base fluid for the sake of enhancing the thermophysical properties of the working fluid are considered to support higher heat transfer compared to conventional fluids, such as water , ethylene glycol , kerosene , etc.
Recently, heat transfer and nanofluid flow in microchannels have drawn enormous interests by researchers. However, most of the researches are concerned with the forced convection heat transfer in smooth microchannels. The laminar forced convection heat transfer of γ-Al2O3/deionized water nanofluid through a rectangular microchannel heat sink was studied by Kalteh et al. , using a finite volume method. Moreover, they carried out experimental study to make comparison between the outcomes with numerical results. Their findings demonstrated that average Nusselt number rises with a growth in Re and vol. % of nanofluid besides a reduction in the nanoparticle size.
The theoretical study of laminar forced convection heat transfer of Al2O3/H2O nanofluid inside a circular microchannel accompanied by a uniform magnetic field was carried out by Malvandi and Ganji . Due to the nonadherence of the fluid–solid interface accompanied by nanoparticle migration, considered as a slip condition, and also the microscopic roughness in circular microchannels, the Navier’s slip boundary condition was applied to the walls. The results of this research showed that the near-wall velocity gradients rise by applying the magnetic field, improving the slip velocity, and therefore, the pressure drop and heat transfer rate rise.
The heat transfer and fluid flow of MWCNT/water-based nanofluids in a microchannel, with frequent change of heat flux and slip boundary condition, were studied by Nikkhah et al. . Based on their results, local Nusselt number, along the length of the microchannel, changes periodically and enhances with the rising of Reynold’s number. Furthermore, it was pointed out that an increase in the weight percentage of nanoparticles and slip coefficient results in the rise of Nusselt number, which is higher in upper Reynolds numbers.
The experimental investigation of forced convection of various nanofluids in a 500 μm width, 800 μm height, and 40 mm length microchannel was conducted by Nitiapiruk et al. . Pure water and TiO2-water with 0.5–2 vol.% were studied in this research. According to the outcomes of this research, the use of nanofluid with a volume fraction of 2 vol.% and minimum rate of heat flux and Reynold’s number is more beneficial than other conditions.
An analytical approach was taken to study the entropy generation of alumina–water nanofluid inside circular microchannels and minichannels by Hassan et al. . In their research, the Reynolds number was maintained constant at 1500, while the nanoparticle volume fraction and the diameter of channels differed from 0 to 0.14, and 3 mm (minichannel) to 0.05 mm (microchannel), respectively. They realized that water/Al2O3 nanofluid is an excellent coolant in minichannels under laminar flow regime. Nonetheless, employing high-viscous H2O/Al2O3 nanofluid for laminar flow in microchannels is undesirable. Therefore, it is necessary to develop low-viscous Al2O3/water nanofluids in order to apply in microchannels under laminar flow condition.
Rimbault et al.  investigated convection heat transfer of nanofluids in a rectangular microchannel heat sink. The nanofluids were comprised of CuO nanoparticles combined with water as the base fluid in 0.24–4.5 volume fractions. The findings reveal that employment of copper oxide/water nanofluid for microchannel under the examined conditions does not offer great heat transfer improvement, compared to water. Such results were inconsistent with the experimental results reported by Zhang et al.  for alumina–water nanofluid flow through a circular microchannel, who indicated a significant increase in heat transfer rate and Nusselt number while using 0.25–0.75 vol.% nanofluid. While employing non-Newtonian Al2O3 nanofluid of up to 4% in a rectangular microchannel, the findings of Esmaeilnejad et al.  were consistent with those of Zhang et al. .
The literature survey reveals that combined use of nanofluids with microchannels gives higher heat transfer performance compared to the use of common, traditional fluids in conventional systems [18–20]. However, fulfilling the requirements from other applications of the microchannels needs additional advancement. A particular still uncomprehending case is the natural and mixed convection heat transfer of nanofluids in vertical and inclined ribbed microchannel heat sinks. In this work, dilute mixture of Cu nanoparticles and water has been analyzed in an inclined microchannel with four rectangular shaped ribs. Laminar mixed convection heat transfer was studied by the use of FLUENT software. Properties of nanofluids have been extracted from the available formulations in literature and introduced in the software. Model validation has been performed by the comparison of the simulation results and the existing literature. The focus was on the heat transfer of water-based nanofluids with variable volume fractions of solid nanoparticles in microchannels with different angles of inclination. Results of this study may be applied in the use of coolants in various electronic devices such as high-power light-emitting diodes (LED), VLSI circuits, and micro-electro mechanical system (MEMS) .
2. Governing equations for laminar nanofluids
Dimensionless governing equations comprised of continuity, momentum, and energy equations, which are solved for laminar, steady-state flow in Cartesian coordinate system, are as follows [22, 23]:
In the above equations, the following dimensionless parameters are used [24, 22]:
To calculate the local Nusselt number along the lower wall, the following relation is used :
The local Nusselt number across the ribs is given as
The local Nusselt number along each horizontal and vertical part of the lower wall can be expressed as follows:
To calculate the local friction factor along the lower wall, the following relation is used:
Substituting dimensionless parameters of Eq. (5) in Eq. (11), relation (12) and (13) are obtained as follows:
The average friction factor along each horizontal part of the lower wall can be calculated
The average friction factor across each rib is defined as
Total friction factor:
2.1. Nanofluid properties
Table 1 shows the thermophysical properties of copper (as nanoparticles) and water (as base fluid). The thermophysical properties of the nanofluid can be acquired from the nanoparticles’ characteristics as well as that of the base fluid.
Density and heat capacity of nanofluids can be computed through the recommended expressions by Goodarzi et al. , Togun et al.  and Safaei et al. :
For nanofluid thermal conductivity, Chon et al.  suggested a model for Al2 O3–water which includes the influences of Brownian motion, viscous sublayer thickness, and temperature :
where and are the Brownian Reynolds and Prandtl numbers, is the mean free path of base fluid (0.17 nm for water), and is the temperature-dependent viscosity of the base fluid, represented as
Dynamic nanofluid viscosity is evaluated based on the recommendations of Brinkman :
The thermal expansion coefficient can be obtained from the suggested formula by Khanafer et al.  and Abouali and Ahmadi :
3. Boundary conditions
A 2-D microchannel with four same rectangular ribs was selected for the analysis. Investigation of heat transfer and fluid dynamics, including the study of velocity, thermal field, and friction effects, was performed in different angles of inclination. The schematic of the investigated microchannel is illustrated in Figure 1. The microchannel is 1350 μm long and 90 μm high. The length of the lower wall of the microchannel was divided into three parts. The temperature of 290.5 K was set at the inlet. The temperature of 305.5 K was considered in the middle part of the microchannel with the length of 450 μm, consisting of four ribs. The channel was insulated on the total length of the upper wall (
In this investigation, flow is considered to be incompressible, Newtonian, laminar, and single-phase. Thermophysical properties of nanofluid are assumed to remain unchanged with temperature.
4. Numerical method
The FLUENT commercial code was used to solve the partial differential equations that govern to the flow. The software applies the finite volume method, which is a particular case of the residual weighting method. This procedure is based on dividing the computational domain into finite control volumes, each node of which surrounds with a control volume. The partial differential equation is afterward integrated over each finite volume .
The QUICK scheme  was applied for the discretization of all convective terms, while the SIMPLEC algorithm  was employed for pressure/velocity coupling. At one point, when the residuals for all equations fell under 10-7, the calculation reached the convergence . Heat transfer and fluid dynamics parameters can be assessed, after solving the governing equations.
5. Numerical procedure validation
5.1. Comparison with numerical and experimental study of water
Results of this study were compared with those of Salman et al.  for validation purposes. Validation has been performed with numerical and experimental data, considering fluid flow of water in a smooth microtube with Reynolds number equal to 90. Figure 2 shows an excellent agreement between the simulation results of this work with both experimental and numerical results.
5.2. Comparison with numerical study of nanofluid
Aminossadati et al.  numerically investigated forced convection of water/Al2O3 in a horizontal microchannel. Middle part of the microchannel was exposed to a constant magnetic field and heated by a constant heat flux. The effect of parameters such as Reynolds number, volume fraction of solid nanoparticles, and Hartmann number on the flow field and thermal performance of the microchannel was studied.
Figure 3 demonstrates excellent agreement between the present model’s predictions and the numerical results of Aminossadati et al.  in different Reynolds numbers and volume fractions, in terms of average Nusselt number. This comparison shows that the present numerical method is reliable and is useful in predicting forced convection heat transfer inside a microchannel for nanofluids.
5.3. Grid independence
A structured, nonuniform grid has been chosen for the discretization of the computational domain. A more refined grid was applied near the walls, where temperature and velocity gradients are sensitive. Grid independency of the computational domain was tested by using various grid distributions. Average Nusselt number and dimensionless pressure drop for each number of grids are shown in Figure 4(A, B) , from which a grid of 60 × 900 was chosen for all the simulation cases.
6. Results and discussion
Inside an inclined microchannel with four rectangular ribs, mixed convection heat transfer of water–copper nanofluid is studied, utilizing finite volume method (Figure 1). The distances between the ribs and their lengths and widths are supposed to be constant. The simulation results are plotted in the form of contours and diagrams.
The isotherm contours and streamlines for
The dimensionless velocity contours for Ri = 10, 2% volume fraction, and 30°, 45°, 60°, and 90° inclination angles, in line with (A) and perpendicular (B) to the flow are demonstrated in Figure 6 (A, B). The increasing
The average Nusselt number for a ribbed microchannel with various volume fractions, different Ri, and inclination angles is depicted in Figure 7. The results endorse that the average Nusselt number increased by increasing the Richardson number, inclination angle, and nanofluid volume fraction. Nevertheless, a significant increase in the average Nusselt number is seen compared to others for Ri = 10. Also, in all volume fractions, the average Nusselt number is higher when Ri = 1 compared with Ri = 0.1. This can be attributed to the fact that as the Richardson number increases, the effective terms in natural convection heat transfer are strengthened. Moreover, increased volume fraction of nanoparticles significantly affects fluids’ thermal conductivity, which enhances the rate of nanofluid heat transfer. Even though when Ri = 0.1, the increment of average Nusselt number is nearly independent of
The local Nusselt number for the distillated water and nanofluids on the lower wall of the microchannel,
The pressure drop values for different Richardson numbers and volume fractions are shown in Figure 9. It was observed that in all studied cases, pressure drop augments as the volume fractions of nanoparticles, Richardson number, or inclination angle increase. In the cooling fluid, solid nanoparticles cause a significant drop in pressure owing to the flow of denser, high-viscous fluid, compared with the fluid with lower density and viscosity. More vortexes are created as a result of increased inclination angle of the microchannel and the flow is reversed, which necessitate more energy to increase the pressure drop. Likewise, the pressure drop increases by transition from forced convection domination to free convection one, as a result of high variations of gravity components in higher Richardson numbers.
The average friction factor for different Richardson numbers and volume fractions on the upper wall of the microchannel are shown in Figure 10. The average friction factor drops as the nanoparticle volume fraction decreases, and Richardson number and inclination angle increase. The density and dynamic viscosity of the fluid are intensified as the nanoparticle volume fraction increases, which leads to an increment in the average friction factor. Also, collision of particles with the microchannel’s surface increases in higher nanoparticle volume fraction, which raises the friction factor. The friction factor is more or less independent from
The dimensionless temperature profiles in different inclination angles and microchannel cross sections for
Higher inclination angles result in development of intensive vortexes and better fluid mixing, which significantly decrease the dimensionless temperature, particularly in near-inlet cross sections. Thus, the dimensionless temperature for the vertical microchannel has the least value in all cross sections, meaning that this microchannel angle has the maximum rate of heat transfer.
At sections near to the entry, the dimensionless temperature profile drops, because the thermal boundary layer has not been developed yet. The thermal boundary layer becomes fully developed as the entry length is increased, which increases the dimensionless temperature.
In this work, the fluid flow and heat transfer of laminar Cu–water nanofluid in a 2D rectangular ribbed microchannel with different inclination angles and Richardson numbers were investigated. Simulation of the problem was performed by the use of finite volume method. Reynolds number of 50 and Richardson numbers between 0.1 and 10 were applied to the simulation. Solid nanoparticles were chosen to have a volume fraction of 0.0–4.0%.
The results of this research revealed that increasing the inclination angle of microchannel or volume fraction of solid particles enhances the heat transfer rate. Existence of ribs through the flow path results in velocity gradient and increases the fluid contact with the surfaces of the microchannel, which in turn enhances heat transfer, while increasing the average friction factor. Addition of nanoparticles to the base fluid does not majorly affect the hydrodynamic parameters of the flow such as fluid velocity. Of all the studied cases, maximum heat transfer can be seen in a vertical microchannel, dominated by natural convection, because of the significant effect of gravity on the fluid structure and enhanced mixing of the fluid layers; and the lowest Nusselt number belongs to the horizontal microchannel dominated by forced convection.
The authors gratefully acknowledge High Impact Research Grant UM.C/HIR/MOHE/ENG/23 and Faculty of Engineering, University of Malaya, Malaysia for support in conducting this research work.
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