Numerical investigation of fluid flow and heat transfer within multilayer wavy microchannels

Abstract

Keywords: Friction Factor; Microchannel; Nusselt Number; Thermal Performance

Introduction

The subject of transferring the heat was the objective of many works in the literature such as investigation of natural convection [1], addition of nanoparticles to a base fluid [2], role of magnetic field [3] and also other fields such as

aerospace applications, automobile sectors, biomedical engineering and chemical engineering areas [4]. In this context, the microchannel heat sinks have received attentions by researchers as these devices could be used as an effective means for cooling process of the electronic devices. In this field, the capacity of the channel in removing the heat and

*Corresponding author. *E-mail address: kharati@sutech.ac.ir This paper was recommended for publication in revised form by Editorin-Chief Ahmet Selim Dalkılıç Published by Yıldız Technical University Press, İstanbul, Turkey Copyright 2021, Yıldız Technical University. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

also the pumping power requirement are essential factors which must be taken into account. Various efforts have been made in order to enhance the capacity of channels in removing the heat such as using fins [5]. One of the main techniques is to employ wavy surface for the channel wall. Mohammed et al. [6] investigated the effect of different wave amplitude on the heat transfer enhancement. They found that the heat transfer phenomenon was improved compared to the straight channel and heat transfer enhancement achievement was superior to the pressure drop penalty. Liu et al. [7] studied the effect of longitudinal and transversal-wavy structures on the thermal performance of the microchannels. Their findings revealed that the thermal performance of longitudinal wavy structure was inferior to that of straight channel while the thermal performance of transversal wavy was superior. Solehati et al. [8] performed a study to investigate mixing performance in microchannel T-junction using a wavy structure. In their study, they realized that employing wavy structure could improve the mixing quality and also performance index especially at high Reynolds numbers. Rostami and Abbassi [9] and Bazdar et al. [10] conducted a study to assess the role of wavy structure in heat transfer enhancement of microchannels using nanofluid as working fluid. They indicated that the use of wavy structure could improve the rate of heat transfer. Eltaweel et al. [11] focused on the heat transfer characteristics of wavy-tapered microchannels. They found that tapering of the wavy channel could improve the channel performance in terms of thermal resistance in comparison with un-tapered wavy channel. Gong et al. [12] performed a research to evaluate the role of dimples in wavy microchannels. In their research, they exhibited that the existence of dimples caused the boundary layer to break up and thus improvement in the channel thermal performance. Lin et al. [13] analyzed heat transfer phenomenon in wavy microchannels while the wavelength and amplitude of the wavy units were variable. They realized that wavy microchannels employing variable wavelength or amplitude wavy units could enhance the thermal behavior of the channel than those using constant wavelength and amplitude wavy units. Kumar et al. [14] compared performance of circular wavy microchannel with sinusoidal wavy microchannel. In their study, they declared that heat transfer performance of circular wavy microchannel was higher than that of sinusoidal one. Zhu et al. [15] compared performance of wavy microchannels with wavy bottom rib and side rib designs. Their results showed that the up-down wavy design led to a better performance. In the context of fluid flow through microchannels, some studies were devoted to characterize fluid flow through straight multi-layer channels. Chong et al. [16] investigated the fluid flow within single and double layer counter flow in microchannels. Their research revealed that single and double layer counter flow microchannels have better performance in laminar flow than turbulent flow. Some studies focused on the optimization of double layer microchannels [17-20]. In

these works, they found the best geometric sizes to boost the channel performance. Effat et al. [21] studied the effect of number of layers on the thermal performance of microchannels. Their results showed that employing multilayer microchannel caused the maximum base surface temperature and pressure drop to decrease. Wong and Ang [22] and Kumar et al. [23] characterized fluid flow through double layer microchannels with channel contraction. Their research indicated that tapered channel could lead to the increase in the thermal performance but caused a higher pumping power. Shen et al. [24] characterize fluid flow through the parallel- and counter-flow staggered double layer microchannel. They mentioned that the staggered configuration could lead to a better heat transfer rate and thermal performance than the conventional double layer microchannel and counter flow case had better performance than the parallel flow case. Khodabandeh et al. [25] conducted a research to evaluate the effect of sinusoidal cavities and rectangular ribs on the thermal performance of double layer microchannels. They stated that the proposed design was capable of increase in the Nusselt number but was recommended for low Reynolds number due to pressure drop penalty. Shen et al. [26] studied the effect of multiple staggered flow alternation structures in a double layer microchannel. They showed that the staggered flow alteration structure had a favorable effect on the thermal behavior with a minor increase in the pressure drop. In the other work by Shen et al. [27], the effect of staggered alteration location was studied. This work exhibited that the change in the location of staggered alteration had a favorable effect on the thermal performance while an insignificant increase in the pressure loss was observed. Some researches were devoted to analyze fluid flow through wavy multilayer microchannels. Xie et al. [28] assessed thermal performance of single- and double-layer wavy microchannels. Their study declared that for a same flow rate, the double-layer caused lower pressure losses and overall thermal resistance than the single-layer microchannel. In another study by Xie et al. [29], the flow through parallel and counter flow double layer wavy microchannels was compared. In this study, it was found that at large flow rate, the counter-flow double-layer wavy microchannel had a good performance while for a slightly larger flow rate, the parallel configuration resulted in a better behavior. Shen et al. [30] performed a research to examine convective heat transfer of counter-flow in staggered wavy double layer microchannels. They found that in these microchannels, the ones with opposite amplitude could enhance the heat transfer rate. Wang et al. [31] carried out a research to study fluid flow and heat transfer phenomena through a new design of a double layer microchannel with combination of wavy structure and porous vertical ribs. They found that the proposed design had a good cooling performance over a wide range of pumping power. Referring to the work mentioned above, one could see that the fluid flow and heat transfer phenomena within the wavy multilayer microchannels with variable wavelength

and amplitude has not been investigated yet. In the present work, the role of both increase in the number of channel layer as well as the different wavy patterns with variable wavelength and amplitude on the channel walls on the friction factor, Nusselt number and also overall performance of the microchannels are characterized. In the present research, laminar flow is concerned and results are obtained at different Reynolds number.

Problem Description

In the research undertaken, the fluid flow and heat transfer phenomena through one-, two- and three layer channels are investigated. The side walls of the channels are wavy and the channel top and down walls are flat. In this work, five different wavelengths and amplitudes are considered for the side walls. The first wavy channel, which is named as the base channel, has a constant amplitude and wavelength of A = 30 μm and λ=2000 μm, respectively. For the base channel, the wall profile is obtained using the following expression: (1) where A and λ are amplitude and wavelength, respectively, and x denotes the channel length.

Figure 1. The one-layer base microchannel. Figure 1 represents the single base microchannel along with geometrical parameters as well as the direction of flow entering the channel. The single microchannels are known as one-layer microchannels. The four other wavy microchannels are obtained by varying the amplitude and wavelength of the base channel. For the two wavy channels, the amplitude is kept constant to that of the base channel (i.e., A=30 μm) and wavelength decreases and increases along the channel in such away that the difference between wave length of the two adjacent waves is ∆λ=300 μm. For the two other wavy channels, the wavelength is fixed to that

Figure 2. Inner wall profiles for the wavy microchannels on the x-y cross section at the distance of 250 μm from the channel bottom. Channels are presented with 50 magnification in y-direction.

of the base channel (i.e., λ=2000 μm) and amplitude decreases and increases such that the difference between amplitude of the two adjacent waves is ∆A=4 μm. In the present work, seven wave units are considered for numerical calculation and so, the lengths of wavy microchannels would be 14000 μm. Figure 2 depicts the inner wall profiles for the considered wavy microchannels on the x-y cross section at the distance of 250 μm from the channel bottom. The multilayer microchannel is obtained by placing the same single microchannels on each other. In this work, two- and three-layer microchannels are considered to study. The considered microchannels are subjected to a constant heat flux at the bottom. Figure 3 represents front views for the one-, two- and three-layer microchannels along with geometrical parameters and also the heated wall of the microchannels. In this study, conjugate heat transfer is concerned. Working fluid is water and material of solid wall is silicon. Fluid and channel wall properties, except fluid viscosity, are kept constant to those in the inlet temperate (i.e., Tin=300 K). Table 1 represents fluid and channel wall properties at the fluid inlet temperature. Since water viscosity is more influenced by temperature than other properties, the water viscosity at fluid mean temperature is used in numerical calculation. At first, numerical

calculation is performed using water viscosity at the inlet fluid temperature. Then, mass average fluid temperature within the channel is computed and water viscosity at the average fluid temperature is used for numerical recalculation. Repeating this procedure, one could obtain flow field within the channel accounting temperature dependency of water viscosity. In the present work, a constant heat flux of q"=1000 kw/m2 is applied at the bottom of the microchannels (see figure 3). To examine fluid flow and heat transfer phenomena within the considered channels, friction factor (f ), Nusselt number (Nu) and overall performance of the microchannels (φ) are computed. Friction factor and Nusselt number are defined as the followings: (2)

(3) In equation (2), dp/dx stands for pressure drop through the channel and Dh, ρ and U represent channel hydraulic diameter, fluid density and velocity, respectively. In equation (3), hydraulic diameter is defined as Dh = 4S/Pw, where S and Pw are cross sectional area and wetter perimeter of the channel. In equation (3), hm also denotes average heat transfer coefficient within the channel and kf is thermal conductivity of the fluid. For multilayer channels, the average values of friction factors and Nusselt numbers of the layers has been considered as the overall friction factor and Nusselt number of the microchannel. Heat transfer coefficient is obtained from the following: (4) where Tw is average wall temperature and Tm is the mass average of the fluid temperature within the channel. Overall performance of the channel is obtained via the following relation [32]:

Figure 3. Front views for the (a) one-, (b) two- and (c) three-layer microchannels.

In equation (5), f0 and Nu0 are friction factor and Nusselt number of the one-layer straight channel, respectively. Reynolds number is also defined as Re=ρUDh/µ, where µ is fluid viscosity at the fluid inlet temperature. For better understanding of fluid flow and heat transfer phenomena within the microcahnnels, results are obtained at Reynolds numbers of Re=300 to 1050. This flow Reynolds number range is lower than the critical Reynolds number of Recr ≈ 2300 and so the assumption of laminar flow is appropriate.

Numerical Procedure

In the current research, an incompressible flow through the channels is considered. Consequently, for the fluid flow, governing equations would be the incompressible form of the continuity, momentum and energy equations as the followings: (6)

(8) Governing equation within the channel wall is heat conduction equation and is of the following form: (9)

In equations (6) to (9), ui and p are denoted for the fluid velocity components and pressure, respectively, and T represents temperature. In these equations, the subscripts f and s indicate fluid and solid properties, respectively. Based on the considered Reynolds numbers, velocity is computed and sets for the fluid entering the channel. Pressure outlet is also set for the channel outlet. As mentioned earlier, it is assumed that the fluid with constant temperature of Tin=300 K enters the channel and a constant heat flux of q"=1000 kw/m2 is imposed on the channel lower wall. Other channel walls are considered to be adiabatic. No slip boundary condition is also considered for the channel inner walls. In this research, the generation of grid and numerical computation are performed using the commercial flow solver software, ANSYS FLUENT 16.2. In the current numerical model, the second order upwind scheme is used for discretization of momentum and energy equations and SIMPLE algorithm is utilized for the pressure-velocity coupling. The convergence criterion is based on the residuals and iteration is stopped as residuals reach less than about 10-8. Grid Study and Validation A comprehensive grid study is performed to obtain grid independent results. To do this, for each channel, four grid levels are considered and at Reynolds number of Re=900, friction factor and Nusselt number obtained from the considered grid levels are compared. Grid independent study reveals that for the wavy and straight channels, the grids with around 600,000 (40×750×20), 1,150,000 (77×750×20) and 1,700,000 (112×750×20) computational cells have adequate resolution for the one-, two- and three-layer channels, respectively, and maximum errors for the friction factor and

Table 2. Friction factor and Nusselt number for one-layer channel at different gird levels Number of computational cells

Table 3. Friction factor and Nusselt number for two-layer channel at different gird levels Number of computational cells

Table 4. Friction factor and Nusselt number for three-layer channel at different gird levels Number of computational cells

Nusselt number are 2.2% and 0.5%, respectively. For more illustration, tables 2 to 4 show friction factor and Nusselt number for one-, two- and three-layer microchannel at different grid resolutions for the wavy channel in which the wavelength and amplitude vary along the channel.

For more illustration concerning the topology of the used grids, figure 4 depicts the grid associated to the base channel for the one-layer microchannel. The dark and white regions represent the solid and fluid portions of the computational domain.

Figure 4. The grid used for numerical calculations for one-layer microchannel.

Figure 5. Friction factor (a) and Nusselt number (b) associated to the present work and experiment for the straight channel.

Figure 6. Friction factor (a) and Nusselt number (b) associated to the present work and experiment for the wavy channel. The appropriateness of grids in Reynolds number of Re=900 ensures the adequacy of the considered grids for numerical calculation at other considered Reynolds numbers. To validate the used numerical model, friction factor and Nusselt number obtained from the present work are compared with those of experimental work of Sui et al. [33]. In the work of Sui et al. [33], the one-layer straight and wavy channel with amplitude and wavelength of 139 μm and 2500 μm, respectively, were studied. In experiment, channel wall was made of copper and a constant heat flux of q"=500 kw/m2 was applied on the channel lower wall. Figures 5 and 6 compare friction factor and Nusselt number. Exploring figures 5 and 6 indicates that the numerical model used for numerical calculation has adequate accuracy in prediction of friction factor and Nusselt number for the flow through the considered microchannels.

Results And Discussion

For better illustration regarding the effects of employing different wavy walls for the channels or the increase in the number of channel layers, the values of friction factor and Nusselt number relative to those of the one-layer straight channel (i.e., (f-f0)/f0×100 and (Nu-Nu0)/Nu0×100, where f and Nu are friction factor and Nusselt number of the considered channel, respectively, and f0 and Nu0 denote friction factor and Nusselt number of one-layer straight channel at the same Reynolds number, respectively) are presented. Effect on Friction Factor Figures 7 represents variation of relative friction factor for the one-, two- and three-layer microchannels for the straight and wavy channels at different Reynolds numbers. This figure shows that for one-, two- and three-layer channels, as expected, the wavy channels result in a higher friction factor than the straight channel.

Figure 7 shows that the relative friction factor increases as the channel layers increase. To illustrate this, temperature contour at the distance of 12000 μm from the channel inlet for one-, two- and three-layer channel at Reynolds number of Re=600 is depicted in figure 8. This figure is provided for the base channel. Exploring figure 8 shows that as the number of channel layer increases, the heat absorbed by the channel is divided between the channel layers and so a lower temperature rise is achieved for the fluid. Thus, as the water viscosity increases with decreasing the fluid temperature, one could conclude that an increase in the channel layer leads to a higher fluid viscosity and so higher pressure loss and friction factor. Figure 7 reveals that for one-, two- and three-layer channels, the lowest and highest relative friction factors of the wavy channels correspond to the cases of increasing amplitude and increasing wavelength, respectively. Numerical findings show that for one-, two- and three-layer channels, the minimum relative friction factors are 0.1%, 3.7% and 4.7%, respectively. Figure 7 indicates that for one-, two- and three-layer channels, the relative friction factor increases as the channel configuration changes from decreasing wavelength to constant wavelength and amplitude, and finally to decreasing amplitude. Exploring numerical findings show that the change in the fluid viscosity governs the change in the friction factor. The fluid viscosity decreases as the fluid absorbs heat. Consequently, the more the fluid temperature increases, the lower the friction factor would be. To examine how the fluid temperature rises within the considered channels, the increase percentages in the mass averages of the fluid temperature relative to fluid inlet temperature for the one-layer channels at the distance of 12000 μm from the channel inlet at Reynolds number of Re=600 is shown in figure 9. Referring to this figure, one could see that the increase percentage in the fluid temperature is the highest for the increasing amplitude cases and is the lowest for the increasing wavelength cases. This figure also indicates that

Figure 7. Variation of relative friction factor for (a) one-, (b) two- and (c) three-layer microchannels for the straight and wavy channels at different Reynolds numbers.

Figure 8. Temperature contour for the base channel at the distance of 12000 μm from the channel inlet for (a) one-, (b) two- and (c) three-layer base channel at Reynolds number of Re=600.

Figure 9. Increase percentages in the mass averages of the fluid temperature relative to fluid inlet temperature for the one-layer channels at the distance of 12000 μm from the channel inlet at Reynolds number of Re=600. the increase percentage in the fluid temperature diminishes as the channel structure changes from decreasing wavelength to constant wavelength and amplitude (base case), and then to decreasing amplitude. Figure 7(a) reveals that for one-layer channel, an increase in Reynolds number for the wavy channel causes the relative friction factor to increase. As expected, the wavy channel results in a higher pressure loss than the straight channel as Reynolds number increases. Although the wavy channels absorb more heat than the straight channels, however, as Reynolds number increases, the effect of decrease in the fluid viscosity (due to heat absorption by the fluid) in decreasing the pressure loss is inferior to the impact of increase in the Reynolds number in increasing the pressure loss. Consequently, the relative friction factor of one-layer channel increases with increasing the Reynolds number. Exploring figure 7(b) indicates that for the twolayer channel, the relative friction factor decreases as the Reynolds number increases from Re=300 to Re=600. As mentioned earlier, the two-layer channel causes higher pressure loss than the one-layer channel due to higher fluid viscosity. With increasing the Reynolds number from Re=300 to Re=1050, the fluid absorbs more heat which results in a lower fluid viscosity. In this range of Reynolds number, the effect of decrease in the pressure loss due to decrease in the fluid viscosity is superior to the effect of increase in the pressure loss due to increase in the Reynolds number. Thus, as the Reynolds number increases from Re=300 to Re=600, a lower friction factor compared to that of the one-layer straight channel could be obtained. Figure 7(b) shows that as the Reynolds number increases from Re=600 to Re=1050, the relative friction factor increases. Although the heat

absorption capability of the fluid increases as the Reynolds number increases from Re=600 to Re=1050 (which leads to a lower fluid viscosity), however, the effect of decrease in the fluid viscosity in decreasing the pressure loss is not significant. Thus, the relative friction factor increases as the Reynolds number varies from Re=600 to Re=1050. Figure 7(c) reveals that for the three-layer channel, an increase in Reynolds number causes the relative friction factor to decrease. As explained earlier, the three-layer channel leads to the highest pressure loss than the one- and twolayer channels due to lowest temperature rise and so higher viscosity for the fluid. As the Reynolds number increases from Re=300 to Re=1050, the impact of decease in the pressure loss due to decrease in the fluid viscosity is superior to the effect of increase in the pressure loss due to increase in the Reynolds number. Accordingly, for the three-layer microchannel, an increase in the Reynolds number in the considered range of Reynolds numbers causes the relative friction factor to decreases. Effect on Nusselt Number Figures 10 represents variations of relative Nusselt number for the one-, two- and three-layer microchannels for the straight and wavy channels at different considered Reynolds numbers. As presented in this figure, the one-, two- and three-layer wavy channels lead to a higher value for the Nusselt number than the one-layer straight channel. Figure 10 shows that for the wavy channels, the relative Nusselt number increases as the channel layer increases from one to two. As this figure exhibits, the increase in the channel layer from two to three also causes the relative Nusselt number to increase but the increase in relative Nusselt number as

Figure 10. Variation of relative Nusselt number for (a) one-, (b) two- and (c) three-layer microchannels for the straight and wavy channels at different Reynolds numbers. the channel layer increases from one to two is more remarkable than the channel layer increases from two to three. To justify this, one could refer to figure 8 that represent temperature contours at specified values of Reynolds number and distance from the channel inlet for one-, two- and threelayer base channel. This figure shows that an increase in the number of layers leads to the lower temperature difference between the wall and fluid temperatures especially for the upper layers of the microchannels. Since the Nusselt number has an inverse relationship with the temperature difference between the wall and fluid mean temperatures (see equations (3) and (4)), one could deduce that an increase in the number of the channel layers results in the increase in Nusselt number. Figure 8 also shows that as the number of layers increase from one to two, the difference between the wall and fluid temperatures becomes higher than that the number of layer increases from two to three. This confirms that as the number of layers increases from one to two, the increase in Nusselt number is more significant than that the number of layers increase from two to three. To explain the effect of change in the number of layers on the fluid temperature rise, figure 11 presents mass

average fluid temperature at the distance of 12000 μm from the channel inlet for the one-, two- and three-layer channels. This figure is provided for the base channels at Reynolds number of Re=600. Exploring this figure reveals that the increase in the channel layer causes the fluid mean temperature to decrease. Figure 11 also shows the decrease in the fluid temperature is higher as the channel layers increase from one to two than that the channel layers increase from two to three. Figure 8 reveals that for one-, two- and three-layer wavy channels, the highest relative Nusselt number corresponds to the decreasing wavelength and increasing amplitude cases. This due to the fact that for the decreasing wavelength channels, a higher mixing for the fluid could be attained than the other wavy channels. For the increasing amplitude channels, higher mixing as well as the increase in the contact surface area between the channel and fluid are the main causes for the increase in the Nusselt number. Referring to figure 8, one could see that the maximum relative Nusselt numbers for one-, two- and three-layer wavy channels are found to be 18.2%, 45% and 45.2%, respectively. Referring to figure 8, one could observe that for

Figure 11. Average fluid temperature at the distance of 12000 μm from the channel inlet for the base channel at Reynolds number of Re=600. one-, two- and three-layer wavy channels, the increasing wavelength cases yield the lower relative Nusselt number as these cases result in a lower mixing for the fluid and also lower surface area than the other wavy channels. Figure 8 shows that, as expected, an increase in the Reynolds number results in the increase in the relative Nusselt number for the considered channels. This figure reveals that the increase in the relative Nusselt number as the Reynolds number increases from Re=300 to Re=600 is remarkable than the Reynolds number increases from Re=600 to Re=1050. To explain this, temperature contours

at the distance of 12000 μm from the channel inlet for the base channel at Retynolds numbers of Re=300, 600 and 900 for the one-, two- and three-layer channels are shown in figures 12 to 14. Referring to these figures, one could realize that as the Reynolds number increases from Re=300 to Re=600, the decrease in the temperature of the solid portion of the domain is higher than that the Reynolds number increases from Re=600 to Re=900. These figures also represent that temperature distributions within the solid part at Reynolds numbers of Re=600 and Re=900 are closer to each other than those at Reynolds numbers of Re=300 and

Figure 12. Temperature contours for the base channel at Reynolds number of (a) Re=300, (b) Re=600 and (c) Re=900 for one-layer channel.

Figure 13. Temperature contours for the base channel at Reynolds number of (a) Re=300, (b) Re=600 and (c) Re=900 for two-layer channel.

Figure 14. Temperature contours for the base channel at Reynolds number of (a) Re=300, (b) Re=600 and (c) Re=900 for three-layer channel. Re=600. These imply that as the Reynolds number increases from Re=300 to Re=600, the channel capability in transferring the heat is higher than that the Reynolds number increases from Re=600 to Re=1050.

Effect on Overall Performance Figures 15 shows variations of overall performance for one-, two- and three-layer channels at different values of Reynolds numbers. Overall performance is a significant

Figure 15. Variations of microchannel overall performance for (a) one-, (b) two- and (c) three-layer channel at different Reynolds numbers. issue which must be taken into account in the design process of heat sink microchannel. As discussed in the work conducted by Kharati-Koopaee and Zare [32], overall performance compare somehow the capability of microchannel in transferring the heat with the penalty imposed by pressure loss. Figure 15 shows that the overall performance increases with increasing the channel layers and overall performance enhancement is higher as the number of layers increases from one to two than that the number of layers increases from two to three. As presented in this figure, an increase in the Reynolds number causes the channel overall performance to increase. As presented in figure 15, for one-, two- and three-layer wavy channels, the lowest overall performances is associated to the case of increasing wavelength and the highest overall performance corresponds to increasing amplitude, although the case of decreasing wavelength has the same performance as the increasing amplitude at high Reynolds numbers (i.e., Re=600 to 1050). Results also reveals that for one-, two- and three-layer channels, the change in channel structure form increasing

wavelength to decreasing amplitude, and then to constant wavelength and amplitude, and finally to decreasing wavelength causes the channel overall performance to increase. Figure 15 shows that for one-, two- and three-layer wavy channels, the maximum overall performances are 1.16, 1.43 and 1.43, respectively.

Conclusion

In this work, the effect of change in the number of layers for the variable wavelength and amplitude wavy microchannels on the fluid flow and heat transfer phenomena at different Reynolds number are examined. To perform this, the changes in the friction factor and Nusselt number relative to those of the one-layer straight microchannel are studied. In the considered range of parameters, following results could be drawn: • The relative friction factor increases as the channel layers increase due to increase in the fluid viscosity. • The lowest and highest relative friction factors of the wavy microchannels correspond to the cases of

increasing amplitude and increasing wavelength, respectively. For one-, two- and three-layer microchannels, the relative friction factor increases as the channel configuration changes from decreasing wavelength to constant wavelength and amplitude, and finally to decreasing amplitude. For one-, two- and three-layer channels, the minimum relative friction factors are 0.1%, 3.7% and 4.7%, respectively. For the one-layer wavy microchannel, an increase in Reynolds number causes the relative friction factor to increase. This is due to the fact that for the one-layer microchannels, as Reynolds number increases, the impact of decrease in the fluid viscosity in decreasing the pressure loss is inferior to the effect of increase in the Reynolds number in increasing the pressure loss. For the two-layer channel, the relative friction factor decreases as the Reynolds number increases from Re=300 to Re=600 whereas the relative friction factor increases as the Reynolds number increases from Re=600 to Re=1050. The reason for that is due to superiority or inferiority of decrease in the fluid viscosity in reducing the pressure loss. For the three-layer channel, an increase in Reynolds number causes the relative friction factor to decrease. This is due to superiority of decrease in the fluid viscosity in decreasing the pressure loss than the increase in the Reynolds number in increasing the pressure loss. The relative Nusselt number increases as the channel layer increases and this increase is remarkable as the number of layers increases from one to two than the number of layers increases from two to three. For one-, two- and three-layer wavy channels, the highest relative Nusselt number corresponds to the decreasing wavelength and increasing amplitude cases and the lowest relative Nusselt number corresponds to the increasing wavelength case. Results also reveal that an increase in the Reynolds number results in the increase in the relative Nusselt number and this increase is remarkable as the Reynolds number increases from Re=300 to Re=600 than that the Reynolds number increases from Re=600 to Re=1050. The maximum relative Nusselt numbers for one-, two- and three-layer wavy channels are found to be 18.2%, 45% and 45.2%, respectively. The overall performance enhances with increasing the channel layers and overall performance enhancement is higher as the number of layers increases from one to two than that the number of layers increases from two to three. Results also show that an increase in the Reynolds number results in the in the increase in the channel overall performance. Numerical findings indicate that the lowest overall performances is associated to the case of increasing wavelength and the highest overall performance corresponds to increasing amplitude case. It is also shown that for one-, two- and three-layer channels, the change in the channel configuration from

increasing wavelength to decreasing amplitude, and then to constant wavelength and amplitude, and finally to decreasing wavelength causes the channel overall performance to increase. For one-, two- and three-layer wavy channels, the maximum overall performance are 1.16, 1.43 and 1.43, respectively.

Nomenclature

Wave amplitude Specific heat, J/ kgK Hydraulic diameter, m Friction factor Average heat transfer coefficient Thermal conductivity, W / m K Nusselt number Pressure, N/m2 Wetted perimeter, m Heat flux, W/m2 Reynolds number Critical Reynolds number Area, m2 Temperature, K Mass average of fluid, K Average wall temperature, K Fluid velocity, m/s Fluid velocity components, m/s Spatial direction

Greek symbols ρ Density, kg/m3 λ Wavelenght, m µ Viscosity, Ns/m2 φ Overall performance Subscripts f Refers to fluid s Refers to solid 0 Refers to one-layer straight channel in Refers to inlet property

Data Availability Statement

The authors confirm that the data that supports the findings of this study are available within the article. Raw data that support the finding of this study are available from the corresponding author, upon reasonable request.

Conflict Of Interest

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Ethics

There are no ethical issues with the publication of this manuscript.

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