Mehdi Bahiraei *,Ali Monavari

1 Institute of Research and Development,Duy Tan University,Da Nang 550000,Vietnam

2 Department of Mechanical Engineering,Razi University,Kermanshah,Iran

Keywords:Nanofluid Thermohydraulic performance Nanoparticle shape Microchannel heat sink Boehmite nanoparticles

ABSTRACT In this article,the thermal-hydraulic efficacy of a boehmite nanofluid with various particle shapes is evaluated inside a microchannel heat sink.The study is done for particle shapes of platelet,cylinder,blade,brick,and oblate spheroid at Reynolds numbers (Re) of 300,800,1300,and 1800.The particle volume fraction is assumed invariant for all of the nanoparticle shapes.The heat transfer coefficient (h),flow irregularities,pressure loss,and pumping power heighten by the elevation of the Re for all of the nanoparticle shapes.Also,the nanofluid having the platelet-shaped nanoparticles leads to the greatest h,and the nanofluid having the oblate spheroid particles has the lowest h and smallest pressure loss.In contrast,the nanofluid having the platelet-shaped nanoparticles leads to the highest pressure loss.The mean temperature of the bottom surface,thermal resistance,and temperature distribution uniformity decrease by the rise in the Reynolds number for all of the particle shapes.Also,the best distribution of the temperature and the lowest thermal resistance are observed for the suspension containing the platelet particles.Thereby,the thermal resistance of the nanofluid with the platelet particles shows a 9.5%decrement compared to that with the oblate spheroid particles at Re=300.For all the nanoparticle shapes,the figure of merit (FoM) uplifts by elevating the Re,while the nanofluids containing the brickand oblate spheroid-shaped nanoparticles demonstrate the highest FoM values.

1.Introduction

After the production of electronic devices in micro dimensions,heat generation rate of electronic devices has been increased,which causes destructive effects on their processors.So,different ways are considered for their heat disposal,and one of these methods is use of microchannel heat sinks in electronic components.Thus,microchannel heat sinks are dramatically utilized for cooling purposes.Kandlikar and Grande [1] reported that hydraulic diameters of microchannels are between 10 μm and 200 μm.

Several research surveys have been performed on the thermofluidic features of heat sinks.Such analyses can clarify the values of heat exchange and power consumption in these devices.Kumar and Singh[2]evaluated the effects of the stream inlet angle on the heat exchange attributes of the H2O within a mini-channel liquid block.Their results showed that the pressure loss heightens with the rise of the stream rate for all the flow inlet angles.Liet al.[3] evaluated the thermal efficacy of the water in the micro heat sinks equipped with the dimple and pin-fin.The authors concluded that the Nusselt number(Nu)increased through increasing the Reynolds number.Mushtaqet al.[4]researched the thermohydraulic performance of the water in a heat sink equipped with the pin fin.They indicated that the parameter of the pressure drop rises through the elevation of the Reynolds number.Also,by the increase of theRe,the uniformity of the temperature profile reduced.

In the recent decades,varied nanoparticles have been employed in different aspects of technology[5-14].Since conventional fluids such as water and ethylene glycol possess relatively unsatisfactory thermophysical properties for use in thermal engineering,many research studies have been performed to uplift the thermal features of working fluids.It is well-known that nanofluids are producedviadispersing nano-scale solid particles inside common liquids,which have shown superb thermal attributes in various systems[15-25].Muhammad Ali[26]researched the heat transfer of a SiO2-water nanofluid in a copper tube.It was found that when the parameters of the Reynolds number and concentration increased,the Nusselt number was increased.Ahmadlouydarabet al.[27]analyzed the performance of the flat plate solar collector with using a TiO2-water nanofluid.The authors concluded that by rising the concentration of the nanofluid,the absorbed heat energy was increased.Ghaneifaret al.[28]evaluated the heat transfer of a horizontal channel subjected with two heat sources with using an Al2O3-water nanofluid.The findings unveiled that the Nusselt number increased by rising the Reynolds number.

According to the fact that the thermophysical properties of working fluids dramatically affect the thermohydraulic performance of heat sinks,several researchers have utilized diverse nanofluids in different heat sinks.In an empirical study,Hoet al.[29] evaluated the cooling efficiency of a H2O-based alumina nanofluid inside a mini-channel liquid block with a phase change material (PCM) layer embedded in its ceiling.The empirical outcomes indicated that the average of the Nusselt number and figure of merit(FoM)values increase by elevating theRe.Tariqet al.[30]studied the thermohydraulic characteristics of a TiO2-water nanofluid in a normal channel facile heat sink.The results unveiled that for all the concentrations of the nanofluid,the parameters of the Nusselt number and pressure drop increased when the volumetric flow increased.Ghasemiet al.[31]explored the hydrothermal performance of the circular heat sinks with using a nanoparticle dispersion.They concluded that the parameters of theNuand pumping consumption increase by rising the volumetric flow rate.Also,by increasing the concentration of the nanofluids,theNuand pumping consumption increased.Ambreenet al.[32] assessed the heat exchange features of a nanoparticle dispersion within a water block having micro pin-fin using the Eulerian-Lagrangian method.The results indicated that thehenhances by the elevation of the Reynolds number and concentration of the nanofluid.Zargartalebi and Azaiez[33]explored the thermal efficacy of a nanoparticle suspension inside a microchannel heat sink.Based on the results,the meanNuimproved with elevating the Reynolds number.Ambreen and Kim[34]investigated the impacts of the fin shape on the thermal efficiency in the water blocks having the micro pin-fins.They indicated that the Nu improves with the increment in theRe.Furthermore,the fin with the circular shape led to the best thermal performance.Bahiraei and Heshmatian [35] analyzed the heat transfer efficacy and entropy production characteristics of a hybrid nanoparticle suspension in the two microchannel water blocks.It was concluded that for all the flow velocities,the thermal resistance reduced with the rise of the nanoparticle concentration.Besides,by rising the flow velocity,the pumping power increased for all the concentrations.Hassaniet al.[36]evaluated the thermal features of a nanoparticle suspension within a liquid block by using different fins.The findings revealed that the performance index is increased then it is decreased with the increment of the Reynolds number.Ghasemiet al.[37] analyzed the forced heat transfer of a nanoparticle dispersion in a water block.The authors stated that the average of the CPU temperature declined with theReelevation.

One of the key parameters that influences hydrothermal attributes of nanofluids in different equipment is nanoparticle shape.Several surveys have been done on the impacts of particle shape in problems related to nanofluids.Most of them have reported significant influence of this parameter on behavior of nanofluids in various thermal devices.Kimet al.[38]investigated the influences of a H2O-Al2O3nanofluid with diverse particle shapes on the thermal resistance of a flat-plate water block.These scholars analyzed the influences of nanoparticle shapes of sphere,cylinder,and brick.Their numerical outputs revealed that the suspension with the cylindrical particles leads to the lowest thermal resistance.Sheikholeslami [39] researched the influences of a copper oxide-H2O nanoparticle dispersion with diverse particle shapes within a porous duct.It was concluded that the platelet nanoparticles result in the highestNu.Abbasian Araniet al.[40] evaluated the impacts of various particle shapes on thermofluidic features inside a wavy mini-conduit.They stated that the Nusselt number and pressure drop intensified with the rise of the Reynolds number for all of the particle shapes.In addition,the suspension having the brick nanoparticles demonstrated the largest Nu,while that containing the blade nanoparticles showed the smallest pressure drop.Khan[41] provided the influences of the MoS2particles with various shapes on the magnetohydrodynamic slip stream of a molybdenum disulphide nanoparticle dispersion.It was found that by increasing the concentration of the nanofluid,the enhancement of the heat transfer increased.Also,the nanofluid having the blade particles led to the greatest increment of the heat transfer.Akbaret al.[42] explored the influences of different particle shapes in a tiny-length non-flat conduit with a nanofluid as the coolant.The findings unveiled that the thermal conductivity of the dispersion uplifted by rising the volume fraction.Meanwhile,the dispersion containing the platelet nanoparticles resulted in the maximum thermal conductivity.

In this paper,the hydrothermal performance of a boehmite nanofluid with diverse particle shapes in an improved microchannel heat sink is analyzed at various Reynolds numbers.Theh,pressure drop(ΔP),pumping power,thermal resistance(R),uniformity of temperature distribution (θ),and FoM are evaluated.Based on the corresponding literature,very few studies have been done about impacts of nanoparticle shape on the thermohydraulic performance in heat sinks.In fact,the novelty of the current study is to analyze the effects of nanoparticle shape on the hydrothermal performance of a boehmite nanofluid in a modified microchannel heat sink.

2.Defining the Structure and Nanofluid

Fig.1 displays the geometry under evaluation where the material of the solid part is silicon (k=148 W·m-1·K-1),while the dimensions of the heat sink are micro-scale.Fig.2 depicts the main sizes of the geometry under study.As is seen,only one quadrant of the geometry is considered because of the symmetry.The current study is done for the boehmite nanofluid with five nanoparticle shapes of the Os(Oblate spheroid),platelet,brick,blade,and cylinder at four Reynolds numbers of 300,800,1300,and 1800,and the base fluid of water-ethylene glycol (50:50).The nanofluid concentration is regarded to be constant and its magnitude equals 1%.Table 1 lists the thermophysical properties of the base liquid and particles.In addition,Fig.3 exhibits the particle shapes under examination.

Fig.1. Schematic of the microchannel heat sink.

Fig.2. Main dimensions for one quadrant of the geometry.

3.Governing Equations

To properly evaluate the cooling attributes of the boehmite alumina nanofluid in the micro heat sink under laminar flow regime,the continuity,energy,and momentum equations are solved in the present work.Also,the governing equations have been assumed under the incompressible,Newtonian and steady state.

Conservation of mass:

Conservation of momentum:

Conservation of energy:

whereinT,P,and v respectively indicate the temperature,pressure,and velocity.Also,kdenotes the thermal conductivity,cpindicates the specific heat.In addition,ρ and μ indicate the density and dynamic viscosity of the coolant,respectively.

3.1.Nanofluid properties

Eqs.(4) and (5) are respectively used to calculate the density and specific heat of the nanofluids having various nanoparticle shapes:

In which φ represents the concentration of the nanofluid.Besides,subscripts np,nf,and f respectively show the nanoparticle,nanofluid,and base fluid.

Eq.(6)is adopted for calculating the thermal conductivity of the nanofluid with the Os nanoparticles [44].

in whichnindicates the shape factor(n=3/ψ)in which the magnitude of ψ for the particle shape of the Os is stated in Table 2.

The viscosity of the nanofluid having the Os nanoparticles is calculated as below:

Muelleret al.[45]evaluated the amount of the φmfor the nanofluid with the Os nanoparticles in an empirical study,and its value is shown in Table 2.

The δ magnitude,meaning the aspect ratio of the Os nanoparticles,is calculatedviaEq.(8) [40]:

regarding Os particles,meaning ofcandais illustrated in Fig.3.Furthermore,the δ value is shown in Table 2.

The thermal conductivity for the nanofluids having the platelet-,cylinder-,brick-,and blade-shaped nanoparticles is computed as follows [47]:

in whichCkis shown in Table 3.

Eq.(10) is used to compute the nanofluid viscosity with the platelet-,cylinder-,brick-,and blade-shaped nanoparticles [47].

in which coefficientsA1andA2existing in Eq.(10)are mentioned in Table 4.

3.2.Boundary conditions

Fig.4 demonstrates the boundary conditions in the microchannel heat sink for the current investigation.As can be seen,this configuration is symmetrical,so only one quadrant of that is investigated.According to the figure,the heat load of the bottom wall of the Microchannel Heat Sink (MCHS) is considered as 100 W·cm-2,and all the planes are adiabatic.The temperature at the entrance of the nanofluid is considered to be uniform and its value equals 300 K,while the velocity profile at the inlet is also selected uniform.Besides,the gage pressure in the MCHS output is equal to the atmospheric pressure.In addition,the no-slip criterion is considered to all the solid walls,and the regime of the flow is laminar.

4.Numerical Method and Validation

The finite volume approach is selected to discretize the governing equations for the flow in the microchannel heat sink.The second-order upwind approach is implemented for solution of the mass,momentum,and energy equations,and also,the SIMPLE method is considered for the pressure-velocity coupling.Convergence condition to finish the computational iterations is assumed 10-6for all the parameters of the governing equations.

Table 5 summarizes the results of the investigation of the mesh for the current contribution.The mesh study is considered for the pure water at entrance velocity of 1 m·s-1.Also,the heat load considered on the bottom surface is 100 W·cm-2.The parameters of the pressure drop andhare employed to the mesh study in this investigation.As can be noticed in this table,there is not any noticeable change in the pressure drop andhwhen the number of cells becomes higher than 1,600,000.Hence,this grid is selected for the further simulations.

For validation of the numerical approach,Fig.5 shows the mean Nusselt number against the pressure drop for the results of Chein and Chen [48] compared to the current investigation for the pure water.The geometry used in the validation is a straight microchannel heat sink,whose length,width,and height are 18 mm,6.2 mm,and 0.5 mm,respectively.Also,the flow velocities of 1.35,1.73,and 2.15 m·s-1are selected for the validation,which these velocitiescause 25 kPa,35 kPa,and 50 kPa pressure drops,respectively.As can be seen in Fig.5,the data of the current numerical approach show suitable consistency with those of Chein and Chen[48](maximum error is about 1.5%) that certifies the strength of the employed computational technique.

Table 1 The thermophysical properties of the boehmite alumina particles and base fluid [43]

Fig.3. The nanoparticle shapes under study:(a) platelet,(b) cylinder,(c) blade,(d) brick,(e) Os.

Table 2 The characteristics for the Os nanoparticles considered in this contribution [46]

5.Data Processing

To evaluate the thermal performance of the nanofluid with various shapes of nanoparticles,thehis evaluated as follows:

whereinq′′andTbwrepresent respectively the heat load and average temperature of the bottom wall.Also,Tmis defined as below:

whereTinandToutdenote the average temperature at the entrance and outlet,respectively.

To assess the efficiency of the microchannel heat sink,the parameter of the pumping power is evaluated as below:

For evaluating the performance of MCHSs,the uniformity of the temperature distribution(θ)on the plane of electronic components is calculated as below:

whererepresent the maximum temperature and minimum temperature of the heating wall,respectively.

Eq.(15) is also implemented for evaluating the thermal resistance.

To attain the ratio of heat transfer increment to pressure loss intensification,the FoM criterion is calculated as below:

wherehnfand ΔPnfrespectively indicate thehand pressure loss of the nanofluid.Also,hfand ΔPfindicate thehand pressure loss of the base fluid,respectively.

The Reynolds number is calculated as below:

Table 3 Aspect ratio of the various nanoparticle shapes as well as the factors utilized in Eq.(9) [47]

Table 4 The parameters existing in Eq.(10)to calculate the viscosity of the nanofluids having the various nanoparticles [47]

Fig.4. Boundary conditions in the MCHS for the current investigation.

Table 5 Mesh sensitivity evaluation regarding the pure water

Fig.5. The comparison between the data of the current research and those presented by Chein and Chen [48].

where v indicates the average velocity of nanofluid andDhis the hydraulic diameter.

6.Results and Discussion

In this research,the influences of five nanoparticle shapes on the thermohydraulic performance in the MCHS are evaluated.This investigation is done at φ=1%for all the modes and fourRevalues(i.e.,Re=300,800,1300,and 1800).

Fig.6 exhibits thehagainst theRefor various particle shapes.It is observed that the inertia force rises with the rise in theRe,which intensifies the heat transfer coefficient (h) for all of the particle shapes.Besides,the dispersion with the platelet particles leads to the greatest heat transfer coefficient,pursued by those containing the cylinder-shaped,blade-shaped,brick-shaped,and Os-shaped nanoparticles,respectively.To clarify the reason of this outcome,Table 6 summarizes the mean velocity of the dispersions for different particle shapes atRe=1300.As can be perceived,the plateletbased nanofluid has the highest velocity,and the average velocity sequence is the same as thehorder,because by increasing the average velocity,the flow mixing increases,which causes the improvement of the thermal performance and the increase of theh.It is worth mentioning that although the thermal conductivity of the Os-based nanofluid (i.e.,knf/kf=1.06) is greater than that of the platelet-based nanofluid (i.e.,knf/kf=1.04),which can uplift the heat transfer,the Os-based nanofluid has smallest velocity among all the nanofluids.Consequently,the mixing of the flow is not strong for the nanofluid containing this particle shape,and therefore,it demonstrates the lowesth.

The amount of the velocity for the varied nanofluids at an invariantRedepends on the amount of the viscosity (see Eq.(17)).Indeed,cylindrical particles lead to more significant viscosity at a constant concentration compared to the quasi-sphere particles due to physical limitations in rotational and transitional Brownian motions.As a result,the platelet-and cylinder-based nanofluids possess larger viscosities,which induce higher velocities compared to the brick-and Os-based nanofluids (see Table 6).As per the research of Muelleret al.[45],sphere-shaped particles slide past each other rapidly,but cylindrical particles may be in contact with each other for lengthier period.This fact can be a main cause for the more intense viscosity of the dispersions containing platelet and cylinder particles in the current study.Indeed,cylindershaped particles (i.e.,platelet and cylinder) experience the higher degree of interaction between themselves compared to the other particle shapes.

Fig.7 shows the velocity contours at the mid plane of the MCHS for the nanofluid with platelet-shaped nanoparticles at differentRevalues.Note that the flow direction is from the bottom to the top.As can be observed,the growth of the boundary layer reduces with rising theRe,which results in the intensification of the mixing of the stream,and so the increment of theh.As stated before in Fig.6,thehenhances by increasing the Reynolds number.

Fig.8 displays the temperature contours at the bottom wall of the MCHS for the nanofluid with the cylinder-shaped nanoparticles at differentRevalues.Clearly,the temperature of the bottom surface decreases by elevating theRe,which shows the enhancement in the cooling process and causes the increment of thehby rising theRe.

Fig.6. The h against the Reynolds number for various particle shapes.

Table 6 Mean velocity of nanofluids with various particle shapes at Re=1300

Fig.9 illustrates the temperature contours at the mid plane of the MCHS for diverse particle shapes atRe=300.As can be perceived,the suspension with the Os particles causes the maximum temperature,pursued by the dispersions with the brick-shaped,blade-shaped,cylinder-shaped,and platelet-shaped nanoparticles.This result certifies the outcome observed in Fig.6 because the lower temperature indicates that the nanofluid has improved heat excretion in the constant heat flux,and has augmented the heat transfer coefficient.As mentioned,although the platelet-based nanofluid has not a relatively good thermal conductivity,its velocity is high,which positively affects the heat transfer features.About the thermal conductivity,based on the study of Timofeevaet al.[47] the thermal conductivity of nanofluids is dramatically dependent on the total area of the liquid/solid interface.It can be said that in nanofluids with non-spherical particles,the thermal conductivity increase predicted by the Hamilton-Crosser model diminishes because of the negative contribution of the interfacial thermal resistance as the sphericity of nanoparticles decreases.Thus,the nanofluids containing cylindrical nanoparticles including cylinder-,blade-,and platelet-shaped particles possess smaller thermal conductivities,since these nanofluids have greater surfaces,which heightens the adverse contribution of the interfacial thermal resistance.Between different cylindrical particles,a long cylindrical particle possesses further energetic atoms compared with conventional cylindrical particles.Ghosh and Pabi [49] indicated that increasing contact area with increment of aspect ratio of nanoparticles causes larger heat transport during the collisions,and thus,a long cylinder-shaped nanoparticle results in a greater thermal conductivity in comparison with a platelet-shaped nanoparticle.The reason is that a lengthy cylindrical particle has a larger specific surface area,and it is known that atoms on the surface have more potential energy than internal atoms.Such indication has also been stated by Mirmohammadiet al.[50] through a molecular dynamics simulation.All in all,in this contribution,the platelet-based nanofluid shows better heat transfer characteristics due to its higher velocity at a constant Reynolds number.

Fig.10 displays the velocity vectors for the suspension with the platelet particles atRevalues of 300 and 1800.It is perceived that the mixing of the flow increases with the rise of theRe,which improves thehas Fig.6.Also,the flow in the inlet is uniform but it is mixed at the outlet because of the sudden route change,which this is more evident at the higher Reynolds numbers.

Fig.7. Velocity contours at the mid plane of the MCHS for the nanofluid with the platelet-shaped nanoparticles at Re values of:(a) 300,(b) 800,(c) 1300,(d) 1800.

Fig.11 exhibits the mean temperature of the MCHS bottom surface against theRefor various particle shapes.It is seen in this figure that by the increase of theRe,the mean temperature of the bottom surface decreases due to the intensification in the disturbance of the boundary layer,which is also evident in Fig.8.

Fig.12 depicts the temperature contours for the solid part of the MCHS for the particle shape of the cylinder at diverseRemagnitudes.It is noticed that with the elevation of theRe,the cooling process is improved,which tends to decrease the temperature of the MCHS and increase theh.Also,this is evident in Figs.8 and 11 in which the temperature of the MCHS bottom surface declines with theReelevation.

The pathlines of the dispersion flow with the platelet particles are exhibited in Fig.13 atRe=300,Re=1300,andRe=1800.It is noteworthy that the color of the pathlines is based on the temperature,and as can be noticed in this figure,since the residence time of the nanofluid in the MCHS is smaller at largerRe,the temperature of the nanofluid does not increase significantly,and therefore,the performance of the cooling is improved.

Fig.14 illustrates the temperature contours at the mid surface of the MCHS atRe=1800 for particle shapes of the Os and platelet.As stated in Fig.11,the suspension having the platelet nanoparticles causes the lowest temperature and the best cooling,pursued by those having the cylinder-shaped,blade-shaped,brick-shaped,and Os-shaped nanoparticles,respectively.Therefore,the suspensions with the platelet-shaped and Os-shaped particles respectively have the best and worst cooling performances,which is displayed in Fig.14 as well.

Fig.8. Temperature contours at the bottom wall of the MCHS for the nanofluid with the cylinder-shaped nanoparticles at Re values of:(a) 300,(b) 800,(c) 1300,(d) 1800.

Fig.9. Temperature contours at the mid plane of the MCHS at Re=300 for particle shapes of:(a) Os,(b) brick,(c) blade,(d) cylinder,(e) platelet.

Fig.10. Velocity vectors for the suspension with the platelet particles at Re values of:(a) 300,(b) 1800.

Fig.11. Mean temperature of the MCHS bottom surface against the Reynolds number for various particle shapes.

Fig.12. Temperature contours for the solid part of the MCHS for the particle shape of cylinder at Re values of:(a) 300,(b) 800,(c) 1300,(d) 1800.

Fig.13. Pathlines of the dispersion with the platelet particles at Re values of:(a) 300,(b) 1300,(c) 1800.

Fig.14. Temperature contours at the mid surface of the MCHS at Re of 1800 for particle shapes of:(a) Os,(b) platelet.

Fig.15 shows the pressure dropversustheRefor diverse particle shapes.It is observed that the pressure drop heightens with the increase of theRefor all the nanoparticle shapes.The reason is that the mean velocity intensifies with theRerise,which intensifies the velocity gradient on the walls.Furthermore,the suspension containing the platelet-shaped nanoparticles causes the most intense ΔP,pursued by those with the cylinder-shaped,blade-shaped,brick-shaped,and Os-shaped nanoparticles,respectively.Table 7 summarizes the relative viscosity of the nanofluids for diverse particle shapes.The suspension containing the platelet-shaped particles has the highest viscosity,and it is seen that the order of the viscosity for the particle shapes is the same as their order in the ΔP,because by rising the viscosity of nanofluid at a constant Reynolds number,the average velocity heightens (see Eq.(17)),and hence,the ΔPincreases by increasing the average velocity.

The cylindrical nanoparticles (i.e.,platelet-,cylinder-,and blade-shaped nanoparticles) have a larger surface area,which heightens the viscosity of the nanofluid as per Table 7.This may be due to the intensified interactions with the fluid layer,within themselves,and also with the solid surfaces.The theoretic studies[51] have also unveiled that nanoparticles having higher surface area result in a more unsatisfactory stability,which negatively affects the viscosity.It can also be said that rotation of the cylindrical particles is difficult,which can rise the flow resistance and increase the viscosity.Therefore,as seen in Table 7,the platelet nanoparticles cause a high viscosity,which intensifies the pressure drop.

Fig.16 shows the pressure drop contours at the mid plane of the MCHS for the nanofluid with the cylindrical nanoparticles at differentRevalues.It is perceived that the ΔPintensifies with theRerise,which proves the conclusion of Fig.15.

Fig.15. Pressure drop against the Reynolds number for diverse particle shapes.

Fig.17 depicts the pumping power due to theRechange for various particle shapes.Clearly,the pumping power heightens by the increase in theRefor all of the particle shapes.Fig.18 shows the pathlines for the dispersion with the Os particles atRe=300 andRe=1800.As can be noticed in this figure,the pathlines of the flow demonstrate more significant irregularities at the greater Reynolds number,which causes the increase of the pressure drop and hence pumping power.In addition,as per Fig.17,the particle shape of platelet indicates the highest pumping power,pursued by the suspensions having the cylinder-shaped,blade-shaped,brick-shaped,and Os-shaped particles,respectively.The pumping power directly depends on the ΔP(see Eq.(13)),and so by increasing the pressure drop,the pumping power augments as well.As mentioned before in Table 7,the order of the viscosity for the particle shapes is the same as their pumping power order,such that the pumping power heightens by increasing the viscosity of the nanofluid.

Table 7 The relative viscosity of the dispersions with different nanoparticles

Fig.16. Pressure drop contours at the mid plane of the MCHS for the nanofluid with the cylindrical nanoparticles at Re values of:(a) 300,(b) 800,(c) 1300,(d) 1800.

Fig.17. Pumping power against the Re for various particle shapes.

Fig.19 illustrates the thermal resistance versus theRefor various particle shapes.Obviously,the thermal resistance reduces with theReincrement for all of the particle shapes.As stated before in Fig.6,thehenhances with the rise of theRe.The lower values of theRdemonstrate the lower thermal resistance (see Eq.(15)),which shows a better cooling performance.Also,among the different particle shapes,the dispersion having the platelet particles is the best case,because its thermal resistance is smaller than the other nanoparticle shapes,which causes the greaterhcompared to the other shapes of the nanoparticles.For example,the thermal resistance of the dispersion with the platelet particles shows a 9.5%decrement compared to the dispersion with the Os nanoparticles forRe=300.

Table 8 indicates the uniformity of the temperature distribution(θ)versustheRefor various particle shapes.It is seen that by elevating theRe,the uniformity of the temperature distribution is improved due to the increase of the average velocity.In fact,the smaller θ indicates that the temperature distribution is more uniform(observe Eq.(14)).Furthermore,the suspension with the platelet nanoparticles results in the best temperature distribution,while that having the Os particles demonstrates the worst temperature distribution.It is noteworthy that the uniformity of temperature distribution is a very important parameter since in the case of more uniform temperature profile,the possibility of formation of hot spots reduces.

The FoM criterion illustrates the ratio of heat transfer augmentation to pressure loss intensification(see Eq.(16)).In fact,increase of heat transfer is a desirable phenomenon and increase of pressure drop is an undesirable phenomenon that causes pumping power intensification.It is noteworthy that FoM magnitudes larger than 1 reveal that adding the nanoparticles rises heat transfer more than pressure loss,and so the addition of the nanoparticles to base liquid is optimal.Fig.20 presents the FoM versus theRefor various particle shapes.As is evident,the values of the FoM increase by the increase of theRefor all of the particle shapes,which shows higher increase in the heat transfer compared to the pressure loss at greater Reynolds numbers.Though thehvalues of the dispersions having the brick-and Os-shaped nanoparticles were smaller than the nanofluids with the other nanoparticles,they have the higher FoM values,because adding them to the base fluid leads to a smaller intensification in the pressure loss compared to adding the other nanoparticles.

Fig.18. The pathlines for the dispersion with the Os particles at Re values of:(a) 300,(b) 1800.

Fig.19. The R due to the Re variation for various shapes of particles.

Fig.20. FoM against the Re for diverse shapes of nanoparticles.

7.Conclusions

In this article,the influences of five shapes of particles(i.e.,platelet,brick,cylinder,blade,and Os)on the thermofluidic efficacy in the MCHS are explored at four Reynolds numbers.The most important results are defined as below:

-Thehenhances by theRerise for all the particle shapes.

-The dispersion containing the platelet nanoparticles shows the greatesth,while the particle shape of Os demonstrates the lowest heat transfer coefficient.

-By elevating theRe,the average temperature of the bottom surface reduces for all of the particle shapes,and so the thermal efficiency is improved.

-The particle shapes of platelet and Os lead to the minimum and maximum temperatures,respectively.

-At all of the particle shapes,the pumping power rises with the rise of the Reynolds number.

-The dispersion with the platelet particles indicates the highest pressure drop,pursued by those with the cylinder-,blade-,brick-,and Os-shaped nanoparticles,respectively.

-The parametersRand θ decrease by increasing theRefor all the particle shapes.

-The dispersion containing the platelet nanoparticles presents the lowest thermal resistance and uniformity of temperature distribution.

-The FoM values enhance with theRerise.Furthermore,the nanoparticle shapes of the Os and brick have the maximum FoM values.

Table 8 The uniformity of the temperature distribution (θ) versus the Re for various shapes of nanoparticles

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Nomenclature

cpspecific heat,J·kg-1·K-1

Ddiameter,m

hheat transfer coefficient,W·m-2·K-1

kthermal conductivity,W·m-1·K-1

NuNusselt number

nshape factor

Ppressure,Pa

q′′heat flux,W·m-2

Rthermal resistance,m2·K·W-1

ReReynolds number

Ttemperature,K

vvelocity,m·s-1

θ temperature uniformity,m2·K·W-1

μ dynamic viscosity,Pa·s

ρ density,kg·m-3

? nanoparticle volume fraction

Subscripts

bw bottom wall

f fluid

in inlet

m mean

nf nanofluid

np nanoparticle

out outlet