1 Introduction

The microchannel heat sink, also known as the MCHS, is an innovative new cooling system that was developed to meet the growing demand for cooling in electronic equipment. It is a component that increases the heat flow away from the electronic devices by increasing the surface area and the amount of fluid that moves across it. Compared to conventional heat dissipation devices, the MCHS has been found to have various advantages, including a minimal size, the capacity to produce a high heat transfer coefficient, and reduced coolant requirements (Hussien et al. 2016).

The fluid involved in the heat transfer mechanism also plays an essential role in improving the cooling performance. The solid particles suspended in the base fluid, which together create a colloidal solution of nanofluid, contribute to the reduction of the temperature difference and the thermal resistance between the flowing fluid and the heated heat sink wall. Nanofluids typically contain metals, carbides, oxides, or carbon nanotubes as the type of nanoparticles that are dispersed in the fluid, while the typical base fluids consist of water, oil, and ethylene glycol. (Hung et al. 2012). A previous study has confirmed that nanoparticles help in enhance thermal performance in the microchannel because of greater thermal conductivity. The base fluid containing dispersed nanoparticles causes an increase in the rate of heat transfer of 14–20% points higher than that of water by itself (Snoussi et al. 2017). Aside from that, the particles with a size of 20 and 47 nm were tested at 1.5 vol.% and 2 vol.%, respectively, and their thermal conductivity increased by 40.1% and 12.1%, respectively (Teng et al. 2010).

When evaluating the effectiveness of heat transfer and fluid flow, it is necessary to take into account several criteria, including the concentration and diameter of nanoparticles. Thermal conductivity improvement was discovered to be 18% higher after adding alumina nanoparticles with a diameter of 11 nm and a concentration of 4 vol.% (Pantzali et al. 2009). The microchannel heat sink that uses water-based Al2O3 has a better heat transfer coefficient, particularly when the volume percentage is increased to 4% or when the nanoparticles’ diameter is reduced from 70 to 25 nm (Alfaryjat et al. 2018). An increment in the volume fraction and a decrease in the size of the nanoparticles are both significant for the nanofluid's increased thermal conductivity. Besides that, nanofluid improves the overall microchannel heat sink performance by 19%, making it superior to pure water concerning heat transfer coefficient (Heidarshenas et al. 2021). They explained that when the average size of the Al2O3 nanoparticles is increased from 29 to 40 nm, the heat transfer of the fluid is decreased. This is because the total surface area between the surrounding liquid and the particles decreases as nanoparticle size increases (Naranjani et al. 2021).

Apart from thermal performance, it is essential to consider the fluid flow in the microchannel to predict nanofluids’ behavior since incorrect viscosity might result in excessively difficult pumping. The pressure drop of nanofluids would be greater than water (Alfaryjat et al. 2018). This is because the suspended solid particles in some fluids enhance the base fluid's dynamic viscosity. An experiment was conducted to investigate the effects of different particles size, and it was discovered that as nanoparticle size increases, the pressure drop would also increase (Heidarshenas et al. 2021). This is because the particles tend to agglomerate together, and then sediment over time that occurred in the base fluid, resulting in channel blocking and clogging. Besides, it also revealed that the pressure drops enhanced to 7.7%, 11.5%, 15.3%, and 25% for particle diameters of 20, 50, 80, and 135 nm, respectively (Heidarshenas et al. 2021). MCHS with a hexagonal cross-section was subjected to numerical analysis to determine the effect of various nanofluids, volume concentrations, and nanoparticle sizes on thermal and hydraulic performance (Alfaryjat et al. 2018). According to the findings, when the volume percentage was doubled to 4% and the nanoparticles were reduced from 70 to 25 nm in diameter, the friction factor increased.

As the particle diffusion coefficient is directly related to fluid temperature, particles dispersed in a base fluid have greater Brownian motion during the heating cycle. When the diffusion coefficient is high, there is a greater likelihood of particle aggregation due to increased collisions between particles. The formation of bigger aggregates or clusters can result in sedimentation and a decrease in the thermal conductivity of nanofluid (Timofeeva et al. 2007). It has been discovered that as the temperature rises, nanoparticles tend to cluster together and increase in the size of the particles on average (Chang et al. 2005). Moreover, at a temperature of 197 °C, accelerated Brownian motion and increased particle-to-particle collision cause agglomeration to occur, which eventually drop the thermal performance (Tavares and Coulombe 2011).

The nanoparticles' dispersion condition may be affected by time over time. The thermal conductivities of fresh nanofluids were found to be somewhat greater than those of nanofluids that had been kept for up to two months (Eastman et al. 2001). As time goes on, the stability of nanoparticle dispersal may decrease, which might explain the phenomenon. When nanoparticles are stored for extended periods, they may begin to agglomerate together. On the other hand, there is a concern that needs to be properly addressed, particularly with the thermal performance of nanofluids in a turbulent flow. Although there was an 8% improvement in laminar flow, there was no discernible difference in the convective heat transfer performance of amorphous carbonic nanofluids in turbulent flow (Kim et al. 2007). Another study also found that, as opposed to turbulent flow, replacing traditional coolants with nanofluids appeared to be advantageous for laminar flow (Panzali et al. 2009). Another nanofluids' drawback is their thermal performance in fully developed regions is mediocre. The convective heat transfer coefficient of nanofluids with a low Reynold number is maximum at tube entry length, decreases with axial distances, and approaches constant value in a completely formed region (Ding et al. 2009).

In 2011, new material of nanoparticles was introduced in the thermal management system called MXene. From the conducted experiment (Wang et al. 2021), the materials needed in the synthesis of MXene (Ti3C2) without any further purification are MAX phase (Ti3AlC2, 400 mesh, 98%) powders, Graphene nanosheets, sodium citrate, hydrochloric acid (HCl), and Lithium fluoride (99.9%). As for the characterization of MXene material, transmission electron microscopy (TEM) and atomic force microscopy (AFM) was used to further characterize the microstructure of MXene. TEM results showed that MXene is a graphene-like 2D nanostructure material with a wrinkled structure, the particle size of MXene is about 5 nm based on AFM, and HRTEM indicates the cross-section of the etched single-layer nanosheets is 1 nm (Wang et al. 2021). Most research suggests that the increase in the concentration of MXene nanofluids will increase its thermal conductivity, which results in a higher heat transfer coefficient.

A study into the enhancement of the energy and thermal performance of hybrid PV/T systems with MXene nanoparticles in olein palm oil has been carried out (Samlingam et al. 2020). They found that when MXene particles with 0.01 wt.% are added to the base fluid, the improvement percentage is 5.2% at 25°. At the same temperature, with concentrations of nanoparticles ranging from 0.03 wt.% to 0.2 wt.%, thermal conductivity improves by 21%, 31.5%, 42.1%, 57.9%, and 68.5%, respectively. On the other hand, the results obtained from the experiment revealed that thermal conductivity was enhanced by 53.1% and 64.9% in nanofluids containing 5% Ti3C2Tx MXene multilayers and single layer, respectively (Bao et al. 2021). In terms of hydraulic performance, adding 3 wt.% of MXene nanoparticles to water resulted in a viscosity rise of up to 1.8% (Samlingam et al. 2020). The viscosity of nanofluids is decreased when the MXene content in the base fluid is less than 1% volume fraction (Wang et al. 2021). Compared to single-layer and multi-walled graphene and carbon nanotubes with a 0.1% volume concentration respectively, MXene nanofluids with a 1% volume concentration have a significantly lower viscosity than those materials. MXene nanoparticles have been used in many applications, which include solar collectors (Bakthavatchalam et al. 2021), Photovoltaic/Thermal systems (Samlingam et al. 2020), supercapacitors (Zang et al. 2020), battery technologies (Dong et al. 2020), and sensors (Xin et al. 2020), but yet never been applied in a microchannel heat sink. Plus, there was a lack of studies investigating the impact of various nanoparticle sizes on the heat transfer and fluid flow performance of microchannel heat sinks. Therefore, this present study aims to close the gap by investigating the performance of different MXene nanoparticle diameters in a microchannel heat sink.

In this present research, numerical analysis was conducted to explore the flow pattern of MXene nanofluids along with the microchannel heat sink as well as to determine the effects of different nanoparticles’ diameters on the thermophysical properties of nanofluids. In addition, the main concern is to analyze the heat transfer and fluid flow performance of nanoparticles concerning the Reynold number (Re) by comparing their sizes to that of water. The considered diameter of nanoparticles is 20, 50, 80, and 135 nm, and the range of the Re is set between 400 and 1000, where the values are intentionally similar to the previous paper (Heidarshenas et al. 2021) to relate the results.

The following sections will include the MXene nanoparticle's thermophysical properties and a detailed geometry with the dimension of the microchannel heat sink. The equation of thermophysical properties of nanofluids, the governing equation, and data formulation is shown in Sects. 2.3, 2.4, and 2.5, respectively. The boundary condition for fluid and solid domains were listed in Sect. 2.8. The mesh sensitivity and validation study were conducted and discussed in Sects. 2.9 and 2.10, respectively. Finally, Sect. 3 includes the results and discussion while Sect. 4 is the conclusion.

2 Methodology

2.1 Nanoparticles variable and properties

The thermal and hydraulic performance in the MCHS will be investigated with different nanoparticle sizes. In this study, the material of nanoparticles that is mixed with the base fluid to form nanofluids is MXene nanoparticles. Since MXene is newly introduced in two-dimensional materials, there was least study on its nanoparticle’s diameter. However, the effects of different sizes of MXene nanoparticles will be studied by using a numerical solutions approach. The investigation performs at different particle sizes ranging from 20, 50, 80, and 135 nm, while the fixed parameter of the nanoparticles is volume concentration which is 0.05 vol.%. The variables in sizes and constant parameters were referred from past research in which they investigated the heat transfer coefficient of \({Al}_{2}{O}_{3}\) nanofluid (Heidarshenas et al. 2021). From this, the results obtained will be compared to the other nanofluids in terms of their performances in heat transfer and fluid flow. Table 1 lists the thermophysical properties of water and MXene nanoparticles at a temperature of 30 °C. The property values of MXene nanoparticles were obtained from the research on solar collectors’ performance using MXene nanoparticles (Li et al. 2020).

Table 1 Thermophysical properties of water and MXene nanoparticles
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2.2 Model description

The geometrical shape and specifications of the MCHS are illustrated in Fig. 1 and Table 2 accordingly, which were obtained from the previous paper (Bezaatpour and Goharkhah 2019). As for this research, the shape of the MCHS investigated is conventional rectangular, and the type of material is silicone, the same material at the base of the MCHS. The top layer, which acts as an adiabatic cover, is attached to the MCHS to block vertical heat loss and improve system effectiveness. All microchannels are symmetrical; therefore, only one channel with half of that unit is used for further simulation which can be seen in the figure. The dimension of the half-symmetrical channel is 1 mm × 5.283 mm × 40 mm (width,\({w}_{c}\) x height, \({H}_{c}\) x length, L). The working fluids are discharged through each channel, and the heat sink’s base is supplied with constant heat flux.

Fig. 1
figure 1

The geometry of the MCHS

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Table 2 The dimensions of the MCHS.(mm)
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2.3 Thermophysical properties of the nanofluid

The equations below can be used to implement the thermophysical properties of the nanofluid (Alfaryjat et al. 2018) in which the density of the nanofluid can be obtained as follows:

$${\uprho }_{nf}=\left(1-\varnothing \right){\uprho }_{bf}+ \varnothing {\uprho }_{np,}$$
(1)

where the nf, bf, and np mean nanofluid, base fluid, and nanoparticles, respectively, while \(\varnothing\) representing the volume concentration of the nanoparticles. The specific heat capacity of the nanofluid can be calculated as follows:

$${{(\uprho c}_{p)}}_{nf}=\left(1-\varnothing \right){{(\uprho c}_{p})}_{bf}+ \varnothing {{(\uprho c}_{p})}_{np}$$
(2)

As for thermal conductivity of the nanofluid, it comprises static and dynamic relations which can be expressed as follows:

$${k}_{eff}={k}_{static}+ {k}_{Brownian}$$
(3)
$${k}_{static}= {k}_{bf}\left[\frac{{(k}_{np}+2{k}_{bf})+2\varnothing ({k}_{np}-{k}_{bf})}{\left({k}_{np}+2{k}_{bf}\right)-\varnothing \left({k}_{np}-{k}_{bf}\right)}\right]$$
(4)
$${k}_{Brownian}= 5 X {10}^{4}\beta \varnothing {{c}_{p}}_{bf} {\rho }_{bf}\sqrt{\frac{k T}{ {d}_{np} {\rho }_{np}}} f\left(T,\varnothing \right),$$
(5)

where \({d}_{np}\) is the diameter of the nanoparticles, \(\beta\) is the liquid volume fraction, k is the Boltzmann constant with the value of \(1.3087\mathrm{x}{10}^{-23}\frac{J}{K}\) and the empirical function, \(f\) is defined as follows:

$$\begin{gathered} f\left( {T,\emptyset } \right) = (2.8217 x 10^{ - 2} \emptyset + 3.917 x 10^{ - 3} \left( {\frac{T}{{T_{o} }}} \right) + \hfill \\ \left( { - 3.0699 x 10^{ - 2 } \emptyset - 3.91123 x 10^{ - 3} } \right) \hfill \\ \end{gathered}$$
(6)

The effective viscosity can be obtained by the addition of the static and dynamic viscosity which can be calculated as follows:

$${\mu }_{eff}={\mu }_{static} + {\mu }_{Brownian}$$
(7)
$${\mu }_{static}= \frac{{\mu }_{bf}}{{\left(1-\varnothing \right)}^{2.5}}$$
(8)
$${\mu }_{Brownian}=5 x {10}^{4}\beta \varnothing {\rho }_{bf}\bullet \sqrt{\frac{k T}{{d}_{np }{\rho }_{np}}} f(T,\varnothing )$$
(9)

2.4 Governing equation

For analytical simplification, the following assumptions are made: (i) the steady-state heat transfer and fluid flow are both three-dimensional fluxes; (ii) fluid flow is laminar, single-phase, and incompressible Newtonian fluid for water and multiphase for nanofluids. All solid and liquid property characteristics are constant, including viscosity's temperature dependence; (iii) the effects of thermal radiation, gravity, surface tension, and volume forces are neglected. The continuity, momentum, and governing equations can be derived from the assumptions as follows:

Continuity:

$$\frac{\partial \mathrm{u}}{\partial \mathrm{x}}+\frac{\partial \mathrm{v}}{\partial \mathrm{y}}+\frac{\partial \mathrm{w}}{\partial \mathrm{z}}=0$$
(10)

X Momentum:

$$\frac{{\upmu }_{\mathrm{f}}}{{\uprho }_{\mathrm{f}}} \left(\frac{{\partial }^{2}\mathrm{u}}{\partial {\mathrm{x}}^{2}}+\frac{{\partial }^{2}\mathrm{u}}{\partial {\mathrm{y}}^{2}}+\frac{{\partial }^{2}\mathrm{u}}{\partial {\mathrm{z}}^{2}}\right)+\frac{1}{{\uprho }_{\mathrm{f}}}\frac{\partial \mathrm{p}}{\partial \mathrm{x }}=\mathrm{ u}\frac{\partial \mathrm{u}}{\partial \mathrm{x}}+\mathrm{v}\frac{\partial \mathrm{u}}{\partial \mathrm{y}}+\mathrm{w}\frac{\partial \mathrm{u}}{\partial \mathrm{z}}$$
(11)

Y Momentum:

$$\frac{{\upmu }_{\mathrm{f}}}{{\uprho }_{\mathrm{f}}} \left(\frac{{\partial }^{2}\mathrm{v}}{\partial {\mathrm{x}}^{2}}+\frac{{\partial }^{2}\mathrm{v}}{\partial {\mathrm{y}}^{2}}+\frac{{\partial }^{2}\mathrm{v}}{\partial {\mathrm{z}}^{2}}\right)+\frac{1}{{\uprho }_{\mathrm{f}}}\frac{\partial \mathrm{p}}{\partial \mathrm{y }}=\mathrm{ u}\frac{\partial \mathrm{v}}{\partial \mathrm{x}}+\mathrm{v}\frac{\partial \mathrm{v}}{\partial \mathrm{y}}+\mathrm{w}\frac{\partial \mathrm{v}}{\partial \mathrm{z}}$$
(12)

Z Momentum:

$$\frac{{\upmu }_{\mathrm{f}}}{{\uprho }_{\mathrm{f}}} \left(\frac{{\partial }^{2}\mathrm{w}}{\partial {\mathrm{x}}^{2}}+\frac{{\partial }^{2}\mathrm{w}}{\partial {\mathrm{y}}^{2}}+\frac{{\partial }^{2}\mathrm{w}}{\partial {\mathrm{z}}^{2}}\right)+\frac{1}{{\uprho }_{\mathrm{f}}}\frac{\partial \mathrm{p}}{\partial \mathrm{z }}=\mathrm{ u}\frac{\partial \mathrm{w}}{\partial \mathrm{x}}+\mathrm{v}\frac{\partial \mathrm{w}}{\partial \mathrm{y}}+\mathrm{w}\frac{\partial \mathrm{w}}{\partial \mathrm{z}}$$
(13)

Energy (fluid):

$$\frac{{\upmu }_{\mathrm{f}}}{{\uprho }_{\mathrm{f}}} \left(\frac{{\partial }^{2}{\mathrm{T}}_{\mathrm{f}}}{\partial {\mathrm{x}}^{2}}+\frac{{\partial }^{2}{\mathrm{T}}_{\mathrm{f}}}{\partial {\mathrm{y}}^{2}}+\frac{{\partial }^{2}{\mathrm{T}}_{\mathrm{f}}}{\partial {\mathrm{z}}^{2}}\right)+\frac{1}{{\uprho }_{\mathrm{f}}}\frac{\partial \mathrm{p}}{{\mathrm{C}}_{\mathrm{pf}} }=\mathrm{ u}\frac{\partial {\mathrm{T}}_{\mathrm{f}}}{\partial \mathrm{x}}+\mathrm{v}\frac{\partial {\mathrm{T}}_{\mathrm{f}}}{\partial \mathrm{y}}+\mathrm{w}\frac{\partial {\mathrm{T}}_{\mathrm{f}}}{\partial \mathrm{z}}$$
(14)

Energy (substrate condition):

$${\mathrm{k}}_{\mathrm{s}}\left(\frac{{\partial }^{2}{\mathrm{T}}_{\mathrm{s}}}{\partial {\mathrm{x}}^{2}}+\frac{{\partial }^{2}{\mathrm{T}}_{\mathrm{s}}}{\partial {\mathrm{y}}^{2}}+\frac{{\partial }^{2}{\mathrm{T}}_{\mathrm{s}}}{\partial {\mathrm{z}}^{2}}\right)=0,$$
(15)

where u, v, and w represent the x, y, and z velocity components, respectively, while P, \({T}_{\mathrm{f}}\) and \({T}_{\mathrm{s}}\) are the pressure and temperature of the liquid and solid, respectively, while dynamic viscosity, density, the thermal conductivity of liquid and solid, and specific heat capacity are represented by \({\upmu }_{\mathrm{f}}\), \({\uprho }_{\mathrm{f}}\), \({\mathrm{k}}_{\mathrm{f}}\), \({\mathrm{k}}_{\mathrm{s}}\), and \({\mathrm{C}}_{\mathrm{pf}}\) ,respectively. Using these governing equations with specified boundary conditions, the fluid pressure drop and temperature are solved. The acquired data are then analyzed to determine the thermal and fluid flow performances along the channel.

2.5 Data formulation

As to obtain the results for this research in terms of thermal and hydraulic performance, the mathematical model has been determined. The mathematical formulation is shown below, where the hydraulic diameter, \({D}_{h}\), is computed by using the formula given as follows:

$${D}_{h}=\frac{{2H}_{c}{w}_{c}}{{H}_{c}+{w}_{c}},$$
(16)

where \({H}_{c}\) and \({w}_{c}\) is the height and width of the microchannel, respectively. The Re can be expressed as follows:

$$Re=\frac{\rho {{v}_{in}D}_{h}}{\mu },$$
(17)

where \({v}_{in}\) is the inlet velocity, while \(\mu\) and \(\rho\) are the dynamic viscosity and density of the working fluids, respectively. The pressure drop of the working fluids can be calculated as follows:

$$\Delta P={P}_{in}-{P}_{out},$$
(18)

and the pumping power can be obtained as follows:

$${P}_{p}=\Delta P.Q,$$
(19)

where Q represents the volumetric flow rate of the working fluids. The apparent friction coefficient can be calculated as follows:

$${f}_{app,ave}=\frac{{2\Delta pD}_{h}}{\rho L{v}^{2}},$$
(20)

where L is the length of the microchannel and \(v\) is the average velocity. The formula for thermal resistance is as follows:

$${R}_{th}=\frac{\Delta T}{q{A}_{b}},$$
(21)

where \(q\) and \({A}_{b}\) is the heat flux and the area of channel at the base, respectively, and the temperature difference, \(\Delta T\), can be obtained as follows:

$$\Delta T={T}_{b}-{T}_{f,m},$$
(22)

where \({T}_{b}\) is the average temperature of the base while \({T}_{f,m}\) is the average fluid temperature. The equation below is the formula to calculate the average convective heat transfer which is written as follows:

$${h}_{ave}=\frac{{qA}_{b}}{\Delta T}$$
(23)

and the Nusselt number is shown as follows:

$$Nu=\frac{{{h}_{ave}D}_{h}}{{k}_{f}},$$
(24)

where \({k}_{f}\) is the fluid thermal conductivity.

2.6 Boundary condition

2.6.1 Inlet channel

The working fluid enters the heat sink at a fixed velocity and temperature of 25 °C, normal to the inlet boundary. For thermal and hydraulic performance analysis, the inlet velocity depends on the Re which ranges from 400 to 1000. Flow condition into the inlet is under-developed laminar flow.

2.6.2 Outlet channel

The gauge pressure at the outlet is set to 0 pa (atmospheric relative pressure).

2.6.3 MCHS walls

The top surface is adiabatic so that there is no heat loss in the vertical direction. The thermal conductivity of the microchannel walls is 153.04 W/m K which is made from silicone. A boundary condition that prevents slip is applied to the microchannel walls, and a symmetry region is generated along the sidewalls of the heat sink. A fixed uniform heat flux of q = 66 kW/\({m}^{2}\) is applied to the rectangular base of the microchannel.

2.7 Domain creation

A 3-D microchannel heat sink with fluid and solid interface was modeled to analyze the thermal deformation and heat transfer of the model, which is shown in Fig. 2. The single half channel was developed as geometry in the simulation which was set as a symmetry region on both sides. The MCHS walls were set as the solid domain where the material of the heat sink is made from silicon, while the working fluid was set as the fluid domain in which the type of fluid is water and nanofluids. The height and the length of the channel are 5.283 mm and 40 mm, respectively, which represents the fluid domain. The flow direction of the working fluid is depicted in the figure by a blue arrow, which denotes the direction of the inflow, and a red arrow, which indicates the direction of the outflow. The bottom surface wall of the channel was applied to a heat flux of 66 kW/m2, whereas the top surface wall was considered as an adiabatic wall.

Fig. 2
figure 2

MCHS model in ANSYS

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Since the type of flow in the heat sink is under-developed laminar flow, the condition of the behavior is viscous laminar sublayer where it is a very thin layer near a no-slip wall in which viscous factors dominate. In this case, the interface mesh between the solid and fluid domain needs to be defined with an inflation layer to catch the viscous sublayer. The size of the mesh should be reduced and made to be sufficiently dense toward the wall to effectively catch the entirety of the impacts caused by the sublayer. The important parameter for this condition is y + , where the targeted range is below 5 which can be seen in Fig. 3. In terms of meshing, the value of y + indicates the non-dimensional distance between the wall and the first mesh node located between the MCHS wall and the water. The mesh size near the wall is determined by considering the y+ values. This value is needed to be below 5 in order to capture the viscous effect accurately during computation. Hence, the first layer thickness was calculated to be 0.8 mm. After that, the region close to the wall is meshed using the computed value for the first cell height, and the gradual growth of the mesh is 1.2.

Fig. 3
figure 3

Inflation layer structure over a flat plate

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2.8 Boundary conditions in CFD

Table 3 shows the setup of boundary conditions in CFD. The heat and flow conditions were set as steady-state and laminar flow. The material of the microchannel is made of silicon, and the thermal conductivity is 153.04 W/mK. Inlet temperature and velocity were set to be 25 °C and 0.225 m/s, respectively, while outlet pressure is relative atmospheric pressure, 0 Pa. A heat flux of 66 kW/\({m}^{2}\) is applied on the bottom of the microchannel. The CFD Simulation programmed by ANSYS Fluent is implemented to solve the governing equations of the numerical model. Couple pressure and flow rate were solved using the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) algorithm, pressure interpolation was done using a second-order scheme, the momentum and energy equation was discretized using a second-order upwind scheme. This iterative solution process starts with a fluid field pressure prediction where the continuity equation is a useful tool for calculating pressure drop. For the continuity, momentum, and energy equations, the convergence requirement is to achieve scaled residuals of less than \({10}^{-6}\), \({10}^{-6}\), and \({10}^{-7}\), respectively. Iterations were repeated until the total of residuals was minimal and the temperature was obtained by calculating the energy equation while the velocity component remained.

Table 3 Boundary condition in CFD
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2.9 Mesh sensitivity study

A mesh sensitivity study is conducted to achieve the optimum mesh before performing further simulations. This study begins with a mesh size that is coarse and then refined. After that, the simulation was run again but with increasing mesh size to approach the exact value. The obvious solution is meshing with a super fine mesh which definitely can capture every cell perfectly and compute the highest accuracy. However, the finest mesh will consume more time and require much computational running. In this study, five sets of mesh were produced to test the precision of the response where the number of elements ranges from 10,000 to 800,000 and the response is thermal resistance which is shown in Fig. 4. While decreasing the mesh size, the representative cell length was also considered in which the values depend on the number of elements. The difference between the mesh generated was computed based on the error where it must exceed 30% to obtain the new mesh size. To aid the mesh sensitivity study, Richardson Extrapolation iterations were applied to determine the extrapolated error. The representative cell length with a 0 value means that the response is the exact value which obtains from the iteration process. The meshing was stopped at the cell length with the extrapolated error below 10%. From the figure, the representative cell length for coarse, medium, and fine meshes is 22.31 mm, 15.47 mm, and 10.79 mm, respectively. Hence, the representative cell length selected for further study is 10.79 mm which will compute the best accuracy results.

Fig. 4
figure 4

Mesh sensitivity study

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2.10 Model validation

After determining the optimum representative cell length, a validation of the computational analysis has to be compared with experimental results to verify the validity and reliability of the simulation conducted for MCHS. As for this study, the numerical results of the friction coefficient for the microchannel heat sinks are compared with the experimental solutions (Shah and London 1978). Figure 5 displays the validation of the present data with experimental results. The friction coefficient obtained from the current data indicates a decreasing trend as the Re number increases, which correlates with the experimental data trend. In terms of accuracy, the relative error decreases when Re increases, and the numerical results obtained were below the experimental results. At Re 150, the relative error shows the highest value at 8.83%. Thus, there is a good accord with the experimental results and the numerical analysis is verified for further study.

Fig. 5
figure 5

Friction coefficient validation

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3 Results and discussion

The effects of different nanoparticle diameters have been analyzed numerically on the heat transfer and fluid flow performances in the microchannel heat sink. The considered particle's diameters are 20, 50, 80, and 135 nm, and the Re was kept in the range between 400 and 1000. The chosen values were intended to compare the present study with the past literature review. The flow pattern of MXene nanoparticles was investigated to observe the behavior of the nanofluids along with the heat sink. Both thermal and hydraulic performance of nanofluids were also studied with different Re to select the best overall performance.

3.1 Flow pattern of MXene nanoparticles

3.1.1 Velocity contour plot

The velocity contour of water and nanofluids with different nanoparticle sizes is depicted in Fig. 6. The figure demonstrates that the water velocity is the highest at 55% of the channel length. It is essential to emphasize the fact that the addition of nanoparticles alters the physical properties of the working fluids. From Eq. (7), the effective viscosity relies on static and dynamic viscosity, where the diameter of the nanoparticles affects the dynamic viscosity. It can be expected that the presence of nanoparticles increases viscosity. If the viscosity increases, the velocity of the fluid flow will be reduced which can be referred to as Eq. (17). In other words, the suspended particles in the base fluid contribute to increase particle contact, which can lead to particle aggregation and an increase in fluid viscosity. Besides, nanofluids with constant volume concentration but different particles diameter shows almost identical velocity which can be seen in the figure. In this case, a critical analysis is essential to investigate the impact of different nanoparticle's diameters on the fluid flow along the microchannel.

Fig. 6
figure 6

Velocity distribution along the entire channel length

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3.1.2 Pressure contour plot

The pressure distribution along the length of the microchannel for water and nanofluids with four different diameters is shown in Fig. 7. Generally, the pressure of all working fluids along the channel decreases to 0 Pa at the outlet. In comparison between water and nanofluids, the contour plot shows almost no difference in pressure along the channel. The addition of 0.05 vol.% nanofluids with various particles’ diameter reveals that the pressure increases by less than 2%. It can be said that at constant Re, the effects of nanoparticles’ diameter do not affect much the pressure in the microchannel. The factor that influences the slight increment of pressure for nanofluids is because of the higher viscosity and velocity. From this, it can be judged that nanofluids will have a higher pumping power than water which is defined in Eq. (19). However, water cannot be considered to have the lowest pumping power because its velocity is greater compared to nanofluids.

Fig. 7
figure 7

Pressure distribution along the entire channel length

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3.1.3 Temperature Contour Plot

Figure 8 depicts the temperature contours of water and four different nanoparticle sizes along the microchannel. The bottom wall of the microchannel shows the highest temperature since the base of the heat sink is attached to the heated electronic chip. In terms of fluid flow, the temperature of both water and nanofluids rises as they move through the channel, and the highest temperature is found at the outlet end of the base surface of the channel. The heat transfer region for nanofluids is more significant than water. The thermal conductivity of nanoparticles may be the factor that influences the performance of the nanofluids. Equations (3) and (5) describe that the thermal conductivity of working fluids increase as the solid particles are introduced into the base fluid. Thus, it increases the heat transfer coefficient which causes an effective heat distribution along the microchannel. Apart from that, the temperature distribution of nanofluids with different particle diameter shows almost identical results. Therefore, a deeper investigation is crucial to obtain significant findings.

Fig. 8
figure 8

Temperature distribution along the entire channel length

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3.2 Thermophysical properties of MXene nanoparticles

3.2.1 Temperature difference

The temperature difference between water and MXene nanofluid along the relative length in the microchannel is shown in Fig. 9. Generally, both water and nanofluids show an increasing trend until at the relative length of 0.7 and decrease to the end of the microchannel. It can be seen that the lowest temperature difference for all nanofluids occurred at the inlet of the microchannel. Based on the graph, it clearly shows that water has a higher temperature difference than nanofluids. The addition of nanoparticles into the base fluid contributes to a lower temperature difference in the microchannel since the fluid temperature is increased. This is due to the fact that the highly conductive particles in the base fluid improve the coolant's thermal conductivity, allowing more heat to be drawn from the heat sink. The percentage difference between the water and the nanofluids with nanoparticles diameter of 20, 50, 80, and 135 nm is 37.2%, 37.1%, 36.9%, and 36.9%, respectively. The result has shown that when the nanoparticle's diameter reduced, the temperature difference in the nanofluid decreased. This finding alligned with the past researcher (Alfaryjat et al. 2018).

Fig. 9
figure 9

The temperature difference between water and nanofluids along the microchannel

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3.2.2 Thermal resistance

Figure 10 illustrates the thermal resistance of water and MXene nanofluids with different diameters along the relative length in the microchannel. The lines show an increasing trend until at the relative length of 0.7 and decrease to the outlet of the channel. According to the figure, nanofluids at all diameters have much lower thermal resistance than water. This occurs as a result of the mixing of nanoparticles into the base fluid, which in turn aids the working fluid in enhancing its thermal conductivity, hence, reducing the thermal resistance. The particles promote thermal diffusion more efficiently in the base fluid which enhances the heat transfer in the heat sink. In addition to the significance of thermal diffusion, the effect of Brownian motion also plays a role in the improvement of heat transfer performance. Given that the working fluid is discharged from the inlet, this demonstrates that MXene nanoparticle travels across the channel via forced convection, which is related to kinetic energy. The particles move in random dynamic motion which ultimately leads to collisions between the particles. Therefore, heat transfer will take place because of the heat energy being transferred from one particle to another as well as from one particle to the base fluid. Moreover, it is observed that the decrease in thermal resistance of nanofluids with diameters of 20, 50, 80, and 135 nm is 37.25%, 37.17%, 36.91%, and 36.91%, respectively, at 70% of the channel length. This indicates that 20 nm nanofluid has the lowest thermal resistance since the percentage reduction is the highest. In addition, the findings also reveal that the reduction in thermal resistance is because of the decrease in particle diameter, where this finding was in good agreement with a recent study (Alfaryjat et al. 2018). From Eq. (21), the thermal resistance relies strongly on the temperature difference. As the temperature difference between the fluid and the bottom wall increases, the thermal resistance will also increase.

Fig. 10
figure 10

The thermal resistance of water and nanofluids along the microchannel

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3.2.3 Heat transfer coefficient

The heat transfer coefficient of water and various nanoparticle diameters along the relative length in the microchannel is shown in Fig. 11. The trend of the lines for both water and nanofluids show decreasing trend starting from the entrance of the microchannel until at the relative length of 0.7 and increasing to the outlet of the channel. The downward trends shown were in good agreement with a recent literature review (Alfaryjat et al. 2018). In general, nanoparticles of all diameters have a greater heat transfer coefficient than water. At 70% relative length of the microchannel, both water and nanofluids show the lowest heat transfer coefficient. According to the zoomed figure, it is observed that nanofluid with a 20-nm diameter has the highest heat transfer coefficient of all nanofluids. The experimental results (Heidarshenas et al. 2021) were aligned with the present data where 20 nm nanofluids contribute to a higher heat transfer performance. The percentage difference of 20, 50, 80, and 135 nm at 0.7 of relative length is 59.36%, 59.15%, 58.49% and 58.49%, respectively. These results reveal that the heat transfer coefficient increases as the diameter of the nanoparticles reduce. This can be related to the Brownian motion in which the smaller particles improve the nano-convection in the base fluid. This is because the particles move faster with a smaller diameter and hence increase the rate of energy transportation. Besides, the enhancement in heat conductivity of the coolant also can be attributed to an increase in the relative surface area that occurs when the size of the particles reduces. Due to their substantially bigger relative surface area, nanoparticles will greatly enhance heat conductivity capabilities and increase mixture stability. This phenomenon is caused by the fact that a larger relative surface area provides a greater heat interaction between solid particles and base fluid.

Fig. 11
figure 11

Heat transfer coefficient of water and nanofluids along the microchannel

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3.2.4 Nusselt number

The Nusselt number (Nu) of water and nanofluids at different diameters are shown in Fig. 12. The result indicates that the Nu decreases to the relative length of 0.7 and increases to the outlet of the microchannel for both water and nanofluids. The values of Nu for both water and nanofluids show the highest at the entrance while the lowest at 70% of the relative length. It is observed that nanofluids of all sizes have a higher Nu than water. The presence of nanoparticles enhances the thermal conductivity of the nanofluids and, therefore, increases the heat transfer coefficient. Nu is derived from convective heat transfer and thermal conductivity, which can be referred to as Eq. (24). It is expected that as the heat transfer coefficient increase, the Nu will also increase due to the relationship being directly proportional. Moreover, the interaction between MXene nanoparticles and the surrounding fluid results in the formation of a liquid layering on the particles. This interfacial liquid exhibits the same thermal conductivity as the MXene particles, resulting in a particle–liquid interface structure with an immense effective volume. Therefore, the coolant's heat transfer diffusion is enhanced. The results of the nanofluids with different particles diameter were zoomed in and are shown in the figure. It is obvious that the smaller the particles, the higher the Nu. The enhancement of Nu for nanofluids with a diameter of 20, 50, 80, and 135 nm is 59.4%, 59.2%, 58.5%, and 58.5%, respectively. These present results were in line with the experimental results (Heidarshenas et al. 2021). From the calculated percentage, 20 nm nanofluid has the most significant effect on the Nu. The reason for this is that smaller particles contribute a better particle dispersion which then gives a better heat distribution, thus increasing the heat transfer energy. On the other hand, the effects of solid–liquid interfacial layers also have a significant effect in accordance with the size of nanoparticles. The impact of liquid layering intensified as nanoparticle size reduced. Nanofluid with 20 nm particles has the highest effective volume and strongest interaction between nanoparticle surfaces and the base fluid. As a result, it exhibits the optimum rates of energy transportation and heat absorption.

Fig. 12
figure 12

Nusselt number of water and nanofluids along the microchannel

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3.2.5 Pressure

Figure 13 displays the pressure of water and nanofluids with different nanoparticle diameters along the microchannel. It is shown that the pressure of water and nanofluids reduce at the outlet of the channel. The highest pressure occurred at the entrance and dropped to zero at the outlet. These findings are comparable with the pattern that was shown by the results of the previous studies (Alfaryjat et al. 2018). From the zoomed results, nanofluids of all diameters have a greater pressure drop compared to water. This is because the presence of solid particles in the base fluid increases the viscosity of the working fluid. This occurrence is correlated with the immobilization of fluid layers around the MXene nanoparticle surfaces, which travel with the particles in the flow. Because of the large volume of fluid that is immobilized, there is a tendency for the viscosity to rise due to the higher adhesive and cohesive forces. At constant Re, it can be seen that nanofluids with a diameter of 20, 50, and 80 nm have almost the same pressure, while 135 nm nanofluids display the highest pressure. The difference in pressure between 135 nm nanofluids with water and the other nanofluids is 2.5% and 1.4%, respectively. This indicates that the pressure of the working fluid increases when the diameter of solid particles increases. Larger particles’ diameter grows the sedimentation rate and forms big clusters along the microchannel. Such behavior would obstruct the flow and lead to a rise in pressure.

Fig. 13
figure 13

The pressure of water and nanofluids along the microchannel

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3.2.6 Reynold number (Re)

The result of Re for both water and nanoparticles with different diameters is presented in Fig. 14. Generally, the Re increases as the relative length increase for both water and nanofluids in the microchannel. According to the figure, it shows that water has the highest Re along the microchannel. This is because of the lower viscosity of water which makes the working fluid flow easily with less friction in the channel. Due to the fact that temperature is a measure of the total kinetic energy of the particles, a rise in temperature will contribute to an increase in the movement of particles. The increment of the temperature along the channel promotes the MXene nanoparticles to have a higher Brownian motion which causes the particles to move faster due to the enhancement of the kinetic energy. In addition, nanofluid of 20 nm diameter shows the greatest Re of all nanofluids with the lowest percentage reduction, which is 1.2%. It can be said that the lower the diameter of the particle, the higher the Re. This can be associated with the effect of the viscosity of each decrement particle's diameter. The equation in Brownian motion viscosity comprises nanoparticles diameter which can be referred to in Eq. (9). It is expected that dynamic viscosity will decrease as the particle's diameter decreases. Equation (17) describes that the relationship between Re with viscosity is inversely proportional. When the viscosity of nanofluids decreases, the Re will increase.

Fig. 14
figure 14

Re of water and nanofluids along the microchannel

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3.2.7 Friction coefficient

The friction coefficient that occurs over the length of the microchannel for water and nanofluids of four different sizes is shown in Fig. 15. From the graph, it can be said that the friction coefficient of the working fluids shows downward trends where these results were aligned with recent research (Alfaryjat et al. 2018). The highest friction coefficient of water and nanofluids in the channel occurs at the entrance, while the lowest is at the outlet. Base fluid has the lowest friction coefficient as compared to nanofluids along the microchannel. The dispersion of nanoparticles into the base fluid will create a higher viscosity due to the particle's agglomeration and sedimentation. The higher the viscosity, the lower the velocity, hence an increase in friction coefficient. The friction coefficient of the nanofluids increased by 4.7%, 5.3%, 5.1%, and 6.3%, respectively, for particles diameter of 20, 50, 80, and 135 nm. These results indicate that as the diameter is reduced, the friction coefficient will also reduce. When smaller particles are suspended in the base fluid, the collisions and sedimentation rate are lower due to the ease of particle movement. Thus, the viscosity is lower than that of nanofluids with larger particles diameter. In addition, the lower friction coefficient of 20 nm nanofluids reduces the force that keeps solid surfaces and fluid layers from moving relative to each other.

Fig. 15
figure 15

The friction coefficient of water and nanofluids along the microchannel

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3.2.8 Pumping power

The pumping power along the microchannel for water and nanofluids is presented in Fig. 16. It is discovered that the pumping power decreases in the direction of flow for both types of fluids. At constant Re, the pumping power of water at the inlet of the channel is the lowest and becomes the highest when it reaches the midway of the channel. This is because the volumetric flow rate of water developed the highest at a relative length of 0.3. From Eq. (19), pumping power depends on the volumetric flow rate. The greater the flow rate, the greater the pumping power. Moreover, nanofluids with 50 nm and 80 nm diameters require the lowest pumping power of all nanofluids. At 50% of the relative length, the reduction in pumping power for both nanofluids is 3.2% which is greater than that of 20 nm nanofluids. This finding may be related to the movement and collision of the particles in the base fluid. Nanofluid of 20 nm diameter has a smaller size where the particles will move faster and freely which contributes to a higher flow rate, while 50 nm and 80 nm nanofluids give a lower flow rate due to the possibility of many collisions. From this, it can be said that the larger the particles, the lower the required pumping power at constant Re.

Fig. 16
figure 16

Pumping power of water and nanofluids along the microchannel

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3.3 Thermal and Hydraulic performance

3.3.1 Temperature difference

The temperature difference between water and nanofluids at various Re is given in Fig. 17. According to this figure, when Re is increased, the temperature difference between water and nanofluids decreases along the microchannel where the result obtained for water is much higher than nanofluids. Re of nanofluids has a significant relation with an increase in heat transfer rate, with an increase in flow rate, it increases the dispersion of particles in the fluid and collision between particles. Therefore, it boosts the rate at which heat is transferred between the nanoparticles and the base fluid. Nanoparticles with a diameter of 20, 50, 80, and 135 nm reduce the temperature difference from the base fluid by 64.75%, 64.62%, 64.61%, and 64.61%, respectively. From the figure, the temperature difference of 20 nm nanoparticles diameter has the lowest value of all working fluids. This is aligned with the past study (Alfaryjat et al. 2018) which stated that as the diameter of nanoparticles decreases, the thermal conductivity of nanofluids increases, hence the temperature difference decreases. Nanofluid of 20 nm diameter gives better thermal conductivity in which it increases the fluid temperature in the channel.

Fig. 17
figure 17

The temperature difference between water and nanofluids with various Re

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3.3.2 Thermal resistance

Figure 18 indicates the thermal resistance of water and nanofluids with different nanoparticles diameters versus Re. From the graph, it can be said that the trends of the water and nanofluids decrease with increasing Re. It was discovered that the thermal transport performances and the heat transfer rate were enhanced with an increase in the flow rate of the nanofluid as a result of better mixing of the particles. The results also show that the thermal resistance of the water in the microchannel is more significant compared to nanofluids. Among the four nanofluids, the nanofluid with a 20-nm diameter has the lowest thermal resistance at all Re which reduced the thermal resistance by 35.68% on average. This is proven by a recent study (Alfaryjat et al. 2018) where decreasing the diameter of nanoparticles will decrease the thermal resistance because of the increase in thermal conductivity. One possible explanation is that smaller particles have a greater percentage of their total surface area exposed. Solid particles in the base fluid form liquid layers around the particles which can increase the thermal conductivity. When the diameter of the particle is reduced, the specific area of the particles increases. Thus enhance its thermal conductivity due to the nanolayers of smaller particles. Despite the same concentration, nanofluids with smaller-sized nanoparticles may contain a greater number of particles than those with bigger-sized nanoparticles. More smaller particles can achieve greater interaction with the base fluid, higher movement velocity, and higher transport energy, resulting in an increase in heat transfer rate. Moreover, thermal resistance relies on the temperature difference which can be referred to in Eq. (21). Since the relationship is directly proportional, it is expected that the thermal resistance will increase as the temperature difference increases.

Fig. 18
figure 18

The thermal resistance of water and nanofluids with various Re

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3.3.3 Heat transfer coefficient

Figure 19 depicts the heat transfer coefficient in microchannels containing water and nanofluids. It reveals that an increase in Re increases the convective heat transfer of both water and nanofluids. The increase in flow rate mainly causes a greater in the collision between particles as well as an increase in the collision between the wall and particles which can result in a higher heat transfer rate. On the other hand, it has been shown that the heat transfer coefficient of nanofluids is significantly higher than that of water. The enhancement in heat transfer performance of the nanofluids is increased by 45.57%, 45.25%, 45.24%, and 45.24% for nanoparticles of diameter 20, 50, 80, and 135 nm, respectively. This shows that nanofluid with 20 nm has the greatest heat transfer coefficient. The finding is supported by the previous literature review (Naranjani et al. 2021) in which a smaller nanoparticle diameter will result in an improvement in the overall surface area for heat transfer between nanoparticles and the fluid. Thus, an increase in nano-mixing in the base fluid leads to an effective thermal conductivity of the nanofluids. In addition, smaller particles contribute to a stronger Brownian motion. This is because smaller particles move faster and enhance the rate of energy transfer in the base fluid. The nano-convection of particles increases and hence increase the heat transfer. Besides, Eq. (23) describes that the relationship between heat transfer coefficients is inversely proportional to temperature difference. It is noted that smaller particles have lower temperature differences. Hence, the lower the temperature difference the higher the heat transfer coefficient.

Fig. 19
figure 19

Heat transfer coefficient of the working fluids at various Re

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3.3.4 Nusselt number

Figure 20 presents the Nu of water and MXene nanofluids at various Re. As shown in the graph, the Nu of the water and nanofluids increased with Re which correlated with the trends in past research (Heidarshenas et al. 2021). The results also show that the Nu of the nanofluids at all diameters in the microchannel is higher than water. This is because the nanoparticles that are dispersed into the base fluid increase the heat distribution rate. According to the zoomed figure, 20 nm diameter has the greatest Nu compared to other particle sizes where the enhancement is about 45.57%. A smaller nanoparticle diameter contributes to a better particle dispersion in the base fluid and hence gives a greater heat distribution. From this, the nanoparticles increase the heat transfer which leads to an increase in the Nu. In addition, the results obtained also can be compared with the equation which can predict the Nu. The equation is as follows (Heidarshenas et al. 2021):

$$\frac{Nu}{Pr}=0.184\left({Re}^{0.629}{a}^{0.067}\right),$$
(25)

where Pr is the Prandtl number and ‘a’ is a dimensionless number in which the hydraulic diameter is divided by nanoparticles size. From the equation, it can be predicted that as the diameter of the nanoparticles decreases, the value of ‘a’ will increase, and hence, the Nu increases.

Fig. 20
figure 20

Nusselt number of the flowing fluids at different Re

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3.3.5 Pressure drop

Figure 21 shows the pressure drop for water and nanofluids with four nanoparticles diameter at different Re. The result indicates that the pressure drop for both water and nanofluids increases when Re increases. These increasing patterns aligned with the results shown by Heidarshenas et al. (2021). It can be said that, at all Re, water shows the lowest pressure drop compared to nanofluids. This is because the nanoparticles in the based fluid collide and agglomerate with each other which results in a higher viscosity. As the particles get more entangled, they grow in size and density at the points of agglomeration, leading ultimately to an increase in pressure drop. Moreover, at Re below 700, the pressure drops increased to 1.032%, 1.036%, 1.033%, and 1.035% for particles diameter of 20, 50, 80, and 135 nm, respectively. The pressure drop increases as the particle's diameter increases which due to the bigger diameter affects the increment of the viscosity in the nanofluids. However, when the Re exceeds by 700, the 20-nm nanoparticles show the highest pressure drop. This is because smaller particles have a higher tendency to move faster with random motion, contributing to particle clustering and eventually, causing a blockage along the channel. The pressure drop of 20 nm nanofluid increased by only 1.032% at Re 676, which is considered to be the optimum flow rate for this particle’s size. Among all the sizes, the smaller diameter of particles gives a better dispersion and reduces the possibility of agglomeration and sedimentation in the microchannel. Thus, it is best to describe that 20 nm nanofluid gives a better pressure drop performance in the microchannel when the Re is less than 700.

Fig. 21
figure 21

Pressure drops of the working fluids at different Re

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3.3.6 Friction coefficient

The friction coefficient of water and nanoparticles with four different diameters is presented in Fig. 22. In general, the results show that when Re is applied increased, the friction coefficient will drop for both water and nanofluids where the present results are in line with the previous findings (Alfaryjat et al. 2018). Based on the figure, it is clearly shown that water has lower values of friction coefficient at all Re. Nanofluids tend to have a greater friction factor because the particles agglomerate and sediment. This means that there is a stronger transfer of momentum between the nanoparticles and the molecules of the base fluid that occurred in the microchannel. This will result in channel blocking and clogging, thus increasing the friction coefficient. Moreover, at Re less than 700, the increment of the friction coefficient for 20 nm nanofluid is only 0.797% which is the lowest of all nanofluids. It can be said that a smaller particle's diameter gives a lower friction coefficient since a better dispersion of particles occurred in the base fluid. Decreasing nanoparticle diameter is attributed to the easier movements where there may be less probability of particles collision. Besides, the decrease in friction coefficient with increasing Re is due to the increase in pressure drop. This is because the relationship between the friction coefficient and the Re is inversely proportional.

Fig. 22
figure 22

The friction coefficient of water and nanofluids with various Re

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3.3.7 Pumping power

The relationship between pumping power with various Re is shown in Fig. 23. As can be seen from the graph, the pumping power for water and nanofluids increases as the Re increases. This trend is aligned with the previous study (Naranjani et al. 2021) in which base fluid always has a lower pumping power than nanofluids. Plus, the pumping power is also affected by the pressure drop where both are directly proportional. When the pressure increases, the pumping power will also increase. From the figure, it reveals that nanoparticle of size 20 nm provides the lowest pumping powers by 1.95% as compared with others nanofluid's diameter. This demonstrates that a smaller diameter of nanoparticles requires a lesser rate of pumping force to drive the coolant through the microchannel. This is due to the smaller diameter of nanoparticles reducing the hydrodynamic interactions between particles. Moreover, Eq. (19) defines that pumping power is directly proportional to the pressure drop. From this, it can be said that the pumping power of 20 nm nanofluid is the lowest since the pressure drop is also the lowest among all nanofluids.

Fig. 23
figure 23

Pumping power of water and nanofluids with various Re

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4 Conclusion

This numerical study is implemented to study the effects of water-based nanofluids containing different MXene nanoparticles diameter on the performance of thermal and hydraulic in a microchannel heat sink. The following results were obtained following the three objectives:

  1. 1.

    The flow distribution of velocity, pressure and temperature showed almost no difference between water and nanofluids. At 55% of the channel length, the velocity of water shows the highest. Adding of MXene nanoparticles into the base fluid indicates an increment of less than 2% for pressure. However, there was a significant difference which can be seen from temperature distribution where the temperature of the bottom wall for nanofluids is lower than water.

  2. 2.

    The thermal resistance for water and nanofluids shows the lowest at the outlet of the channel while in the case of heat transfer coefficient, it shows the highest. Nanofluids with a 20-nm diameter reduced the thermal resistance by 37.25% and enhanced the heat transfer coefficient by 59.36% which is the lowest and the highest, respectively, among all nanofluids. Besides, it was revealed that the pressure drop and pumping power for the working fluids decreased throughout the channel. At constant Re, nanofluids of 50 and 80 nm diameter require the lowest pumping power since the velocity along the channel is slightly lower than nanofluids with 20 nm diameter.

  3. 3.

    The best heat transfer performance for water and nanofluids occurred at 1000 of Re. It was discovered that 20 nm nanofluid enhanced the heat transfer coefficient by 45.57% which is the highest of all nanofluids. This is because the smaller particle's diameter contributes to better heat distribution and increases thermal conductivity. In addition, the hydraulic performance for nanofluids is the best when the flow applied is less than 700 of Re. Of all sizes of nanoparticles, it was discovered that 20 nm nanofluids increase the pressure drop by 1.95% which is the lowest.