Introduction

Microchannel heat sinks are one of the most efficient solutions to heating and cooling the equipment. Nanofluids, which are fluids containing suspensions of nanoparticles, have been reported to possess substantially higher thermal conductivity than anticipated from the effective medium theories. Recently, nanotechnology gains interest to explore the microchannel cooling benefits of nanofluids as working fluid [1,2,3,4,5,6,7,8,9]. Lin et al. [10] optimized the geometry and flow rate distribution for double-layer microchannel heat sink. The optimal design variables were obtained at fixed pumping powers, coolant volumetric flow rates and pressure drops through the MCHS, respectively. Sakanova et al. [11] optimized and compared double-layer and double-layer microchannel heat sinks with nanofluid for power electronics cooling. Their results indicated the nanofluids at a higher concentration yield a better cooling performance by about 17.3% at 5% concentration and 10.6% at 1% concentration. Mohebbi et al. [12] studied convection heat transfer of Al2O3–water nanofluid turbulent flow through internally ribbed tubes with different rib shapes. They observed that the ribbed tubes with Al2O3–water nanofluid flow are thermodynamically advantageous. Akbari et al. [13] studied the impact of ribs on flow parameters and laminar heat transfer of water–aluminum oxide nanofluid with different nanoparticle volume fractions in a three-dimensional rectangular microchannel. They observed that an increase in nanoparticle volume fractions causes nanofluid heat transfer properties to have a higher heat transfer and friction factor compared with the base fluid used in cooling due to an increase in viscosity. Leng et al. [14] presented an improved design of double-layer microchannel heat sink with truncated top channels. They concluded that the simulations confirm the effectiveness of the improved design which reduces not only the overall thermal resistance of the heat sink but also the maximum temperature difference on the bottom wall as compared to the original design. Wang et al. [15] investigated geometric parameters on flow and heat transfer performance of microchannel heat sinks. They found that the increase in channel number reduces the thermal resistance, but at the expense of high-pressure drop. Osanloo et al. [16] studied performance enhancement of the double-layer microchannel heat sink by the use of tapered channels. They concluded that thermal performance was improved through increasing the channels convergence angle, although more pressure drop occurred and therefore increased pumping power was required. Arabpour et al. [17] investigated the effects of slip boundary condition on nanofluid flow in a double-layer microchannel. Their results showed that by enhancing the volume fraction of nanoparticles, slip velocity coefficient, Reynolds number and significant reduction in thermal resistance of solid wall, Nusselt number enhances. Behnampour et al. [18] analyzed heat transfer and nanofluid fluid flow in microchannels with trapezoidal-, rectangular- and triangular-shaped ribs. They found that for all studied Reynolds numbers, heat transfer values are least for rectangular rib. Therefore, trapezoidal-shaped ribs are recommended in high Reynolds numbers. Arabpour et al. [19] studied the heat transfer and laminar flow of kerosene/multi-walled carbon nanotubes (MWCNTs) nanofluid in the microchannel heat sink with slip boundary condition. They found that by increasing the slip velocity coefficient on the solid surfaces, the amount of minimum temperatures reduces significantly which behavior remarkably entails the heat transfer enhancement.

In this paper, laminar flow of water nanofluid/GNP–SDBS (graphene nanoplatelet–sodium dedecylbenzene sulfonate) for 0–0.1% solid nanoparticles mass fraction was studied for Reynolds numbers of 50–1000 in 3D space via finite volume method. In the new proposed microchannel design, the cooling fluid is moving in countercurrent in the upper and lower layers of the microchannels, and there are cavities and sinusoidal routes on the solid walls of the microchannel, and the presence of rectangular ribs on the flow centerline along the fluid path enhances mixing for cooling fluid and creates better heat transfer for warm surfaces.

Problem definition and mathematical method

Geometry

Using single-layer microchannels for heat transfer devices has been the topic of research by many researchers, and in most of the papers, the heat transfer behavior is fully investigated. In designing single-layer microchannel due to direct contact of microchannel wall with heat flux and the fact that the surfaces confining the fluid (inner walls of the microchannel) act as heat transfer surfaces, a nonuniform temperature distribution can occur. In order to fix this problem, increasing heat transfer surface is considered in this study and the heat transfer and fluid flow behavior in the new microchannel for the double-layer microchannel configuration is considered. Figure 1 shows the new design of double-layer microchannel considered in this study. In this special design other than the structure of the double-layer microchannel, the sinusoidal cavities in the microchannel walls are used for disturbing thermal and velocity boundary layers at the walls. Also rectangular rips are used at microchannel center for improving fluid mixing in those regions. Based on Fig. 1, the fluid flow in the upper and lower layers passes in opposing directions. Also the microchannel in the upper layer and lower layers is separated by a silicon substrate with a thickness of Ws = 100 μm. The lower-layer microchannel is under the influence of constant heat flux of q” = 50 W cm−2, and the cool fluid in the upper and lower layers enters at Tin = 293.15 K. For improving heat transfer performance, GNP–SDBS (graphene nanoplatelet–sodium dodecylbenzene sulfonate) is used with mass fraction of 0, 0.02, 0.06 and 0.1% solid nanoparticles in water as base fluid. This investigation is done or Reynolds numbers of 50, 300, 700 and 1000 in 3D space and simulations are done with finite volume method.

Fig. 1
figure 1

New design of two-layer microchannel considered in this study

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This study is done on the 3D space in Cartesian coordinates. Based on Fig. 2-a, the coordinate system is placed at the bottom of silicon substrate and in the lower layer and in the center. In Fig. 2, parameters Wf and Hf are width and height of inlet and outlet sections for upper and lower layers of the microchannel which are the same for both layers. Parameters Lt and Hs are overall length and height of silicon substrate of the microchannel. Dimensions La and Ha are length and depth of sinusoidal cavities placed at microchannel walls. The considered microchannel has a thickness of silicon substrate equal to Ws = 100 μm, and this thickness is constant for microchannel walls. The dimensions Lr and Wp are length and height of ribs placed in the microchannel center, respectively.

Fig. 2
figure 2

Introducing parametric dimensions for the considered geometry

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Also, the upper layer of the microchannel is considered without thickness and insulated. The precise dimensions introduced for microchannel geometry are given in Table 1. The precise dimensions in Fig. 2 are according to values in Table 1. Also, all of the dimensions have micrometer units.

Table 1 Precise dimensions of the considered geometry in this study
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In this numerical study, nanofluid flow and heat transfer are simulations in laminar flow and 3D space in Cartesian coordinates. The nanofluid properties are considered to be constant and independent of temperature. The solid–liquid suspension is simulation in mass fractions of 0–0.1% as single phase and Newtonian.

A constant heat flux is imposed on the microchannel walls. The no-slip boundary condition is applied on microchannel walls. The nanofluid behavior in the considered mass fractions is considered to be Newtonian. The effects of dispersion and thermophoresis are neglected. The effect of gravity acceleration is negligible, and the fluid flow is completely forced.

Numerical approach

Continuity equation

$$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0.$$
(1)

Momentum equation

$$\begin{aligned} u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial x} + w\frac{\partial u}{\partial x} = - \frac{1}{{\rho_{\text{nf}} }}\frac{\partial p}{\partial x} \hfill \\ \quad +\, \nu_{\text{nf}} \left( {\frac{\partial }{\partial x}\left( {\frac{\partial u}{\partial x}} \right) + \frac{\partial }{\partial y}\left( {\frac{\partial u}{\partial y}} \right) + \frac{\partial }{\partial z}\left( {\frac{\partial u}{\partial z}} \right)} \right). \hfill \\ \end{aligned}$$
(2)
$$\begin{aligned} u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} + w\frac{\partial v}{\partial z} = - \frac{1}{{\rho_{\text{nf}} }}\frac{\partial p}{\partial y} \hfill \\ \quad +\, \nu_{\text{nf}} \left( {\frac{\partial }{\partial x}\left( {\frac{\partial v}{\partial x}} \right) + \frac{\partial }{\partial y}\left( {\frac{\partial v}{\partial y}} \right) + \frac{\partial }{\partial z}\left( {\frac{\partial v}{\partial z}} \right)} \right). \hfill \\ \end{aligned}$$
(3)
$$\begin{aligned} u\frac{\partial w}{\partial x} + v\frac{\partial w}{\partial y} + w\frac{\partial w}{\partial z} = - \frac{1}{{\rho_{\text{nf}} }}\frac{\partial p}{\partial z} \hfill \\ \quad +\, \nu_{\text{nf}} \left( {\frac{\partial }{\partial x}\left( {\frac{\partial w}{\partial x}} \right) + \frac{\partial }{\partial y}\left( {\frac{\partial w}{\partial y}} \right) + \frac{\partial }{\partial z}\left( {\frac{\partial w}{\partial z}} \right)} \right). \hfill \\ \end{aligned}$$
(4)

Energy equation

$$u\frac{\partial T}{\partial x} + v\frac{\partial T}{\partial y} + w\frac{\partial T}{\partial z} = \frac{1}{{\rho_{\text{nf}} C_{\text{p,nf}} }}\frac{\partial }{\partial x}\left( {\left( {K_{\text{nf}} \frac{\partial T}{\partial x}} \right) + \frac{\partial }{\partial y}\left( {K_{\text{nf}} \frac{\partial T}{\partial y}} \right) + \frac{\partial }{\partial z}\left( {K_{\text{nf}} \frac{\partial T}{\partial z}} \right)} \right).$$
(5)

Boundary conditions

Normal uniform distribution of temperature and velocity is applied for fluid at the channel. In addition, zero relative pressure is employed at the channel outlets. A no-slip condition is utilized at the interface of walls and fluid. Moreover, a constant heat flux (50 W cm−2) is applied on the bottom of down channel, and the temperature of the inlet is considered 293.15 K for the outer solid walls, and the symmetry condition is considered. Moreover, the temperature of nanofluid is equal wall temperature at the interface of wall and base fluid.

Table 2 summarizes the thermophysical properties of water nanofluid/GNP–SDBS which is calculated by Arzani et al. [20], employed in the current research.

Table 2 Thermophysical properties of nanofluid
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CFD simulation, mesh study and validation

Although the relative error of numerical prediction in comparison with experimental results is not low, due to high cost of experimental platform for any case, recently, the tendency of researchers toward using numerical method instead of experimental approach is elevated [21,22,23,24,25].

Discretization of the nonlinear differential equations is carried out by the finite volume approach using the second-order upwind method [26,27,28]. To couple the velocity and pressure, the SIMPLE [29, 30] scheme is employed. Moreover, the convergence criterion is considered 10−6 for the relative residuals related to all parameters [31, 32]. The structured meshes with uniform grid are applied for both domains (fluid and solid) according to Fig. 3. The assumptions used during simulation are as follows: single-phase approach for simulation of fluid, steady and Newtonian fluid flow, temperature independent for nanofluid characteristic, no-slip condition on interface of fluid and solid domain and symmetry condition on the outer solid wall.

Fig. 3
figure 3

The grid type used in this study

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For grid independency, the average Nusselt number is calculated with different cell numbers. The results of mesh study are presented in Table 3. As can be observed, by increasing the number of grids, the difference percentage of Nusselt numbers decreases and no significant change is noticed in the Nusselt number for the grids with the cell number higher than 428,600.

Table 3 Mesh study results for Re = 50, ϕ = 0.00% (water)
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Hence, these grids are selected to investigate the nanofluid flow in these geometries.

For verification of simulated results, the numerical predictions are compared with experimental results reported by Wei et al. [33] which showed wall temperature of double-layer microchannel with rectangular cross section, 83 mL min−1 mass flow rate for each Layer, water inlet temperature at 20 °C and 71 w cm−2 for uniform constant heat flux at the bottom on microchannel. The experimental platform was involved two channels with 1.8 mm and 1 mm as length and 284 µm × 56 µm and 243 µm × 56 µm for upper and lower layers, respectively. According to Table 4, the numerical predictions are in a good agreement with the considered experimental work. (Maximum relative error is higher than 1 °C.) Therefore, it can be noted that the numerical approach is valid.

Table 4 Comparisons between numerical and experimental results
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Data reduction

The Nusselt number (Nu) is calculated through the following equation [34]:

$$Nu = \frac{hD}{k},$$
(6)

where h and D denote the convective heart transfer coefficient and hydraulic diameter, which are obtained as follows [35]:

$$D = \frac{4A}{P},$$
(7)
$$h = \frac{{q^{\prime\prime}}}{{\bar{T}_{\text{s}} - T_{\text{m}} }},$$
(8)

where \(\bar{T}_{\text{s}}\) is the average wall temperature of the bottom wall, q” denotes the wall heat flux, Tm indicates the mean temperature of the fluid and A and Pare cross-sectional area and wetted perimeter of the cross section, respectively. In order to calculate the friction factor (f), Darcy friction factor is used, according to Eq. (9) [36].

$$f = \frac{2D\Delta P}{{L\rho U_{\text{m}}^{2} }},$$
(9)

where L, D and Um represent the channel length, the hydraulic channel diameter and the cross-sectional area-weighted average of the flow velocity, respectively. Also, ∆P indicates the average of bottom and upper channels of pressure drop. The main purpose of applying nanofluid is to improve the Nusselt number. However, the use of nanofluids also increases the friction factor and pumping power. Hence, a parameter called performance evaluation criteria (PEC) is defined here to examine overall thermo-hydraulic performance. This parameter shows the enhancement of Nusselt number in comparison with that of pressure drop, when nanoparticles are injected into a tube. It is defined as follows [37]:

$$PEC = \frac{{{{Nu} \mathord{\left/ {\vphantom {{Nu} {Nu_{0} }}} \right. \kern-0pt} {Nu_{0} }}}}{{{{(f} \mathord{\left/ {\vphantom {{(f} {f_{0} )^{1/3} }}} \right. \kern-0pt} {f_{0} )^{1/3} }}}},$$
(10)

where Nu0 and ∆P0 are the Nusselt number and pressure drop of the base fluid, respectively.

The amount of thermal resistance at the bottom wall of microchannel is calculated by [38]:

$$R = \frac{{T_{\hbox{max} } - T_{\text{in}} }}{{q^{\prime\prime} \times A_{\text{bottom}} }},$$
(11)

where Tmax, Tin, A and \(q^{\prime\prime}\) wall are, respectively, the maximum temperature of bottom wall, minimum temperature which is the inlet temperature of fluid, cross-sectional area of applied heat flux and applied heat flux on the bottom wall.

Results and discussion

In Fig. 4, the graphs of variations of average Nusselt number are shown for Reynolds numbers of 50, 300, 700 and 1000 for different solid nanoparticles mass fractions which are dissolved in water. Based on this figure, the increase in fluid velocity (Reynolds number) has a significant effect on increasing convective heat transfer ratio of cooling fluid in the considered geometry, so that by increasing Reynolds number of the fluid, the Nusselt number is increased significantly. Increasing solid nanoparticles in the base fluid improves heat conduction of the cooling fluid and improves temperature distribution between different fluid layers, and this behavior reduces temperature gradients in the cooling fluid. For different considered nanoparticle mass fractions in the base fluid, it can be said that the variations of increasing Nusselt number for Reynolds number of 50 have a lower distinction compared with other Reynolds numbers. The use of solid nanoparticles for mass fractions of 0.02, 0.04 and 0.1 can have a clear effect on increasing Nusselt number for Reynolds numbers of 300 and 700. For Reynolds number of 1000, increasing mass fraction of solid nanoparticles in the base fluid does not make any changes in increasing Nusselt number and this behavior can be due to negative impacts of adding solid nanoparticles in a special mass fraction or higher mass fractions or the selection of Newtonian fluid model for approximating flow field and heat transfer in the given mass fraction might not be a good model. Curves in Fig. 5 show the Darcy friction factor for Reynolds number of 50–1000 for mass fractions of 0, 0.02, 0.04 and 0.1% solid nanoparticles. This factor is dependent on increasing fluid pressure drop in the microchannel path and value of average inlet velocity for each of the different Reynolds number parameters and mass fractions. By increasing fluid velocity due to placement of ribs in the center and cavities on the microchannel walls, strong velocity gradients are formed and the fluid in the longitudinal path and in some areas due to surface features will have secondary flows, especially in high Reynolds numbers. These features reduce kinetic energy of the fluid in motion and change them to pressure drop.

Fig. 4
figure 4

Changes in average Nusselt number for different Reynolds numbers and different solid nanoparticles mass fractions

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Fig. 5
figure 5

Darcy friction factor for Reynolds number in the range 50–1000 for different solid nanoparticles mass fractions

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By increasing fluid momentum (higher Reynolds number), the formation of secondary flows and changes in direction of fluid components is increased. Therefore, by increasing Reynolds number the level of friction factor curves is significantly increased. Increasing solid nanoparticles mass fraction based on the given properties in this study causes an increase in density and viscosity of the cooling fluid. By increasing viscosity of the cooling fluid, the effect of shear stress on the fluid motion, especially in regions close to wall, becomes significant and these effects have a more prominent effect by further movement of fluid toward outlet section, while moving to central regions of the flow and because of an increase in friction factor and momentum dissipation increase, their effect becomes significant.

Also by increasing cooling fluid density, the fluid motion and convection in the microchannel, especially in sinusoidal cavities, will require higher energy; as a result, this behavior results in higher momentum dissipation and fluid kinetic energy loss in higher solid nanoparticles mass fraction and increasing the level of friction factor curves are seen consequently. In general, employing the new microchannel design with sinusoidal cavities in solid walls and placement of ribs in the central flow regions results in better flow mixing and increases Nusselt number; on the other hand, it increases Darcy friction factor and this behavior is more prominent by increasing Reynolds number.

In Fig. 6, the effect of Reynolds number variations and using nanofluid with different mass fractions on the thermal resistance behavior of the solid wall for the bottom of the lower layer of the microchannel is considered. Thermal resistance is an important factor which considers the effects of flow and geometric parameters on the cooling performance of solid walls of the microchannel (decrease in maximum temperature in the lower microchannel layer). Parameters such as increasing Reynolds number and solid nanoparticles mass fraction result in increased Nusselt number which reduces maximum temperature in the lower layer of the microchannel and reduces thermal resistance. On the other hand, the meaning of reducing thermal resistance is that any agent that creates uniform temperature distribution in the microchannel can eliminate temperature gradients and hot zones and reduces maximum temperature of the microchannel solid walls. Also decreasing thermal resistance of the lower microchannel wall in lower Reynolds number is dependent of increasing solid nanoparticles mass fraction and the dependence is strong. The distinction in behavior related to decreasing thermal resistance of the warm wall of the microchannel for Reynolds number of 1000 has a smaller effect for different mass fractions so that decreasing maximum temperature of the bottom wall of the microchannel tends to a constant value by increasing Reynolds number and solid nanoparticles mass fraction. Based on the behavior of thermal resistance graphs, the lowest value of this factor happens for Reynolds number of 1000 and solid nanoparticles mass fraction of 0.1%.

Fig. 6
figure 6

Changes in thermal resistance of the solid wall for the bottom of the lower layer of the microchannel

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In Fig. 7, the thermal efficiency curves for different Reynolds numbers and solid nanoparticles mass fractions are compared. The thermal efficiency factor is the ratio of an increase in Nusselt number divided by an increase in friction coefficient for each mass fraction compared with the base fluid. This investigation is done for each Reynolds number compared with the similar cases. The trends of thermal efficiency are influenced by heat transfer, and changes in friction factor show different behaviors for Reynolds numbers of 50, 300, 700 and 1000. Adding the solid nanoparticles mass fraction, presence of sinusoidal cavities and counterflow in the upper and lower layers of the channel has an effect on temperature field and flow structure and creates complete mixing for all of the regions of the microchannel along fluid route. Based on the considered factors for increasing heat transfer, and the magnitude of its effect on increasing friction factor, it can have different behaviors. In curves of Fig. 7, adding solid nanoparticles mass fraction results in increased Nusselt number and improved thermal efficiency factor and the level of the curves for this factor has shown a good trend for all solid nanoparticles mass fractions. The presence of sinusoidal cavities on the walls of the microchannel and rectangular ribs in the microchannel center has a good effect on increasing Nusselt number, but according to friction factor curves, it causes a significant increase in friction factor values, and this will result in decreasing trend of thermal efficiency curves by increasing Reynolds number.

Fig. 7
figure 7

Thermal efficiency curves for different Reynolds numbers and solid nanoparticles mass fractions

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In general, the best compromise between Nusselt number performance and friction factor happens for Reynolds number of 50 and solid nanoparticles mass fraction of 0.1%. By increasing Reynolds number, the effect of increasing friction factor compared with Nusselt number will become dominant and result in decreasing trend of thermal efficiency curves.

In bar charts of Fig. 8, the average static temperature in the outlet sections of the upper and lower layers of the microchannel is compared for Reynolds number of 50–1000 and mass fraction of 0–0.1% solid nanoparticles. During entrance of cooling fluid with lower temperature, heat exchange happens between fluid and walls and the warm temperature of the walls penetrates into the fluid. By further movement of fluid into the microchannel, the effects of heat penetration in the central areas of the flow become pronounced. Due to an increase in temperature of the fluid, the value of heat transfer is reduced. The reason for this behavior is an increase in fluid temperature and a decrease in temperature difference between cooling fluid and warm surfaces. Also factors such as sinusoidal cavities and ribs placed in the channel improve flow mixing and elimination of hot fluid zones at the neighborhood of solid walls. Also increasing solid nanoparticles mass fraction results in more uniform temperature distribution between fluid layers. By slow movement of fluid in Reynolds number of 50, the penetration of heat from warm walls to different cooling fluid zones is increased and heat transfer between fluid and warm surface is reduced. By increasing fluid velocity, the value of convective heat transfer is elevated and the temperature of the inlet fluid compared with the penetrated heat into the microchannel becomes dominant. As a result, by increasing Reynolds number the average temperature of the outlet section is reduced. Also among the considered cases the highest average outlet temperature was observed for pure water fluid, and by increasing solid nanoparticles mass fraction, the value of fluid temperature in the outlet section is significantly reduced which can improve heat transfer. In general factor such as increased fluid velocity, an increase in solid nanoparticles mass fraction causes an increase in convective heat transfer in the microchannel and results in domination of inlet fluid temperature in the microchannel and a reduction in heat penetration into fluid layers.

Fig. 8
figure 8

Average static temperature in the outlet section for the upper and lower layers of the microchannel

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In bar charts of Fig. 9, the average static pressure drop for outlet sections in the upper and lower layers of the microchannel for different Reynolds numbers and solid nanoparticles mass fractions is compared. By movement of fluid in the microchannel and contact of fluid with solid walls, the effects of surface shear stress on the fluid layers are increased and some part of the fluid kinetic energy is lost.

Fig. 9
figure 9

Average static pressure drop for outlet sections for upper and lower layers of the microchannel

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Also factors such as presence of sinusoidal cavities in the walls of the considered geometry and presence of ribs at flow center cause changes in the components of fluid velocity and result in formation of secondary flows in different flow regions, and this behavior has a high effect on dissipating kinetic energy of the cooling fluid and a large portion of the kinetic energy of the fluid is converted to pressure drop. By increasing fluid velocity, the above-mentioned effects about decreasing fluid momentum become stronger and the pressure drop becomes more significant. For low Reynolds number (Reynolds of 50 and 300) due to lower fluid velocity, the fluid can adapt to surfaces better and the presence of surface features for flow mixing will have a smaller effect on increasing pressure drop. Therefore, the level of pressure drop curves for these Reynolds numbers is lower compared with Reynolds numbers of 700 and 1000. Increasing pressure drop for Reynolds numbers of 700 and 1000 is mainly because of presence of fluid mixing features on the surface such as presences of cavities and ribs, and by increasing solid nanoparticles mass fraction, then the level of pressure drop will increase significantly. Based on the behavior of the graph, the implementation of the mentioned geometry for high Reynolds numbers is good for mass fraction lower than 0.06% and for mass fractions of 0.06 and above the value of pressure drop in the microchannel is significant.

Contours of static temperature distribution are shown in Fig. 10 for section Y = 0 for solid nanoparticles mass fraction of 0.02% for Reynolds numbers of 50, 300, 700 and 1000. In these contours, the distribution of static temperature in the longitudinal flow plane and in different regions such as regions behind and in front of ribs for upper and lower layers is clearly exhibited. The presence of surface features which result in flow mixing has a high effect on uniform distribution of heat in the microchannel. But according to the flow direction in the upper and lower layers in the regions behind the ribs along flow route due to presence of separation zones and formation of vortices, the value of heat transfer in these areas is not uniform similar to other parts of the microchannel and in these areas zones with lower heat transfer (hot zones) area created. Also the bottom of the lower layer of the microchannel has direct contact with the constant heat flux and therefore has the highest temperature and temperature gradients which are obvious in the figure. In the lower layer of the microchannel and the beginning of the inlet section due to temperature difference between fluid and wall, the value of heat transfer is elevated and the temperature of the inlet fluid is dominant and these areas are colder. By further progression of fluid into the microchannel, the value of heat transfer is reduced and hot zones are formed in the lower layer. By increasing Reynolds number, the temperature distribution in the lower layer is reduced because of an increase in heat transfer and the position of maximum temperature moves toward the outlet section.

Fig. 10
figure 10

Static temperature/K distribution contours for Y = 0 section for solid nanoparticles mass fraction of 0.02%

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Contours of static temperature for the lower layer (y = 150 μm plane) and the upper microchannel layer (y = 450 μm plane) longitudinally along the flow for different Reynolds numbers and solid nanoparticles mass fraction of 0.02% are investigated in Fig. 11. By movement of fluid in the microchannel, the effect of heat penetration from warm walls to central areas of the flow will become significant by further progression of fluid toward outlet section for the lower and upper layers of the microchannel. The presence of sinusoidal cavities on the solid walls and presence of rectangular ribs in the flow centerline (ribs) improve mixing and enhance temperature distribution of the fluid in different longitudinal sections of the microchannel. Due to behavior of temperature contours in Fig. 11, because the lower layer is in direct contact with the heat flux, the highest effect due to presence of temperature gradients on solid walls and cooling fluid happens in the lower layer. In the upper layer due to weaker and indirect contact with the warm wall of the microchannel for all of the considered Reynolds number, a more uniform temperature distribution is obtained. In general, for geometries for fluid mixing features such as sinusoidal cavity and ribs along fluid route, because of better mixing and disturbance of thermal and velocity boundary layers, in higher velocities the temperature distribution is more uniform and hot zones are largely eliminated for high Reynolds numbers such as 700 and 1000.

Fig. 11
figure 11

Static temperature/K contours for plane y = 150 μm and y = 450 μm along the length of the microchannel for different Reynolds numbers and solid nanoparticles mass fraction of 0.02%. aRe = 50, bRe = 300, cRe = 700, dRe = 1000

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Conclusions

In this numerical study, the simulation of water nanofluid–GNP–SDBS for mass fraction of 0–0.1% for a new microchannel heat sink design with two layers in the presence of sinusoidal cavities was carried out. The numerical study was done for laminar Reynolds numbers for range of 50, 300, 700 and 1000, and the numerical simulation was conducted with finite volume method in 3D space. In this study, the use of fluid mixing factors in the new microchannel design by application of nanofluid as a suitable and highly efficient tool was studied. The application of nanofluid, according to previous studies [39,40,41,42,43,44,45,46,47,48,49], has had a significant contribution toward improving heat transfer of the new design of the microchannel by increasing heat conduction of the nanofluid and micron-scale mechanisms of heat transfer, and on the other hand, by increasing Reynolds number, because of reinforcement of convective heat transfer coefficient, this behavior is amplified. Among the considered Reynolds numbers, the Reynolds numbers of 1000, 700, 300 and 50 have the highest and lowest Nusselt number values, respectively. The presence of ribs in the microchannel improves heat transfer and causes a sharp rise in friction factor and pressure drop in the microchannel. The above behavior shows a significant increase for Reynolds numbers of 1000 and 700 which is because of intense variations of velocity gradients and fluid momentum dissipation in high Reynolds numbers because of application of these devises in the microchannel. Also by increasing viscosity of the cooling fluid, the effects of shear stress due to movement of fluid on the solid walls, especially in areas close to wall, become significant and these effects are transferred to central areas of the flow upon further movement of fluid toward exit section and have a contribution in raising friction factor and increasing momentum dissipation. By increasing density of cooling fluid (solid nanoparticles mass fraction), the motion of the fluid and convection of fluid in the microchannel especially in the sinusoidal cavities will require higher energy; as a result, this behavior results in higher fluid momentum and kinetic energy dissipation in higher solid nanoparticles mass flow rates and will result in elevation of level of curves related to friction factor. Any feature that can cause a uniform distribution of heat in the microchannel can eliminate temperature gradients and warm zones and will reduce maximum temperature in the solid wall of the microchannel. Because of direct contact of lower layer with the heat flux, the highest effect of presence of temperature gradients on solid walls and cooling fluid happens in the lower layer. In the upper layer because of weaker and indirect contact with the warm wall of the microchannel, for all of the considered Reynolds numbers, a more uniform temperature distribution is observed. In general, the new investigation design can improve heat transfer efficiency significantly and reduction in effects such as pressure drop and friction factor can be achieved by using different rib designs which provide less blockage against the flow.