international communications in heat and mass...
TRANSCRIPT
International Communications in Heat and Mass Transfer 68 (2015) 69–77
Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer
j ourna l homepage: www.e lsev ie r .com/ locate / ichmt
Forced convective heat transfer of water/functionalized multi-walledcarbon nanotube nanofluids in a microchannel with oscillating heat fluxand slip boundary condition☆
Zahra Nikkhah a, Arash Karimipour a, Mohammad Reza Safaei b,⁎, Pezhman Forghani-Tehrani a,Marjan Goodarzi b, Mahidzal Dahari c, Somchai Wongwises d,⁎a Department of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Isfahan, Iranb Young Researchers and Elite Club, Mashhad Branch, Islamic Azad University, Mashhad, Iranc Department of Mechanical Engineering, Faculty of Engineering, University of Malaya, Kuala Lumpur, Malaysiad FluidMechanics, Thermal Engineering andMultiphase FlowResearch Lab. (FUTURE), Department ofMechanical Engineering, Faculty of Engineering, KingMongkut's University of Technology Thonburi,Bangmod, Bangkok 10140, Thailand
☆ Communicated by W.J. Minkowycz.⁎ Corresponding authors.
E-mail addresses: [email protected] (M.R. Safaei)(S. Wongwises).
http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.08.000735-1933/© 2015 Elsevier Ltd. All rights reserved.
a b s t r a c t
a r t i c l e i n f oAvailable online 4 September 2015
Keywords:Convective heat transferNanofluidMicrochannelOscillating heat fluxSlip velocity
In the present work, forced convective heat transfer of water/functionalized multi-walled carbon nanotube(FMWCNT) nanofluid in a two-dimensional microchannel is investigated. To solve the governing Navier–Stokesequations and discritization of the solution domain, thenumericalmethod offinite volumeand SIMPLE algorithmhave been employed. Walls of the microchannel are under a periodic heat flux, and slip boundary conditionsalong the walls have been considered. Effect of different values of shear forces, solid nanoparticles concentration,slip coefficient, and periodic heat flux on the flow and temperature fields as well as heat transfer rate has beenevaluated. In this study, changes of the variables considered to be from 1 to 100 for Reynolds number, 0–25%for weight percentage of solid nanoparticles, and 0.001–0.1 for velocity slip coefficient. Results of the currentwork showed good agreement with the numerical and experimental studies of other researchers. Data arepresented in the form of velocity and temperature profiles, streamlines, and temperature contours as well asamounts of slip velocity and Nusselt number. Results show that local Nusselt number along the length ofmicrochannel changes in a periodic manner and increases with the increase in Reynold number. It is alsonoted that rise in slip coefficient and weight percentage of nanoparticles leads to increase in Nusselt number,which is greater in higher Reynolds numbers.
© 2015 Elsevier Ltd. All rights reserved.
1. Introduction
Recently, flow in small sizes has been widely considered and effortshave been made to make smaller and yet more efficient devices. One ofthe applications of microchannels is in the cooling of electronic chips,where heat transfer rate is of great importance [1–3]. Choi [4] was thefirst who gave the name of nanofluid to the suspension of nanoparticlesin a host fluid and showed considerable increase in their heat transfercoefficient. In recent years, CNTs have attracted many attentions dueto their significant mechanical, electrical, and thermal properties.These particles are cylindrical with a high length-to-width ratio andsuspend well in water [5,6]. On the other hand, there has been a lotof research regarding energy transfer through microdevices likemicrochannels as well as thermal properties of nanofluids.
8
Lately, microchannels with different cross-sections such as circular,rectangular, and trapezoidal have been used to evaluate the heat trans-fer and characteristics of nanofluids flow. Researchers found thatNusselt number increases by the increase in Reynolds number andthat solid nanoparticles have significant effect on the fully developedthermal boundary conditions. They also showed that when thefluid temperature difference between the inlet and outlet of themicrochannel is substantially high, temperature-related propertiesaffect the augmentation of heat transfer [7–9]. Many investigatorsassessed the nanofluids thermal performance taking into account slipboundary condition and found that thermal performance of the fluidenhances by the rise in slip coefficient [10,11].
To better understand the role of liquid and solid phases on the heattransfer process, researchers have adopted a two-phase model for thenanofluidflow. They evaluated the temperature and velocity differencesbetween the liquid and solid phases and observed that the relativevelocity and temperature of phases are very small and therefore negli-gible [12–15]. Reviewing published articles, one can understand thesignificance of utilizing nanofluids in cooling applications [16,17].
Nomenclature
FMWCNT Functionalized multi-walled carbon nanotubeK Thermal conductivity, W/mkCp Specific heat, J/kgkdp Diameter of nanoparticle, mh Microchannel height, mH non-dimensional microchannel height, H = h/h = 1l Microchannel length, mL non-dimensional microchannel length, mB non-dimensional slip coefficient, B = β/hRe Reynold number, Re = uch/υfPr Prandtl number, pr = υf/αf
q′′ Heat flux, W/m2
q0′′ Amplitud heat flux, W/m2
x, y Cartesian coordinates, mX, Y non-dimensional coordinates, X = x/h, Y = y/hT Temperature, KTc Cold temperature, KP Fluid pressure, pap Average pressure, p ¼ pþ ρcgyP Non-dimensional fluid pressure, P ¼ p=ðρn fu2
c Þu, v Velocity components in x, y directions, ms−1
U,V Non-dimensional velocity components, U = u/uc, V =v/uc
us Slip velocity, ms−1
uc Nanofluid inlet velocity, ms−1
Us Non-dimensional slip velocityNux Local Nusselt numberNum Averaged Nusselt number
Greek symbolsα Thermal diffusivity, m2s−1
β Slip coefficient, mφ Weight percentage of nanoparticlesμ Dynamic viscosity, pasρ Density, kg m−3
λ Convection heat transfer coefficient, W/m2kθ Non-dimensional temperature, θ = (T− Tc)/ΔTυ Kinematic viscosity, m2s−1
Subscriptsf Fluidnf Nanofluidm Averaged value
70 Z. Nikkhah et al. / International Communications in Heat and Mass Transfer 68 (2015) 69–77
However, nanofluids require more extensive research [18,19]. Besides,evaluation of the thermal performance of nanofluids in devices ofmicro and nano sizes needs more attention [20].
Fig. 1. Schematic of anal
In the present research, laminar forced convective heat transfer ofwater/FMWCNT nanofluid in a microchannel under periodic heat fluxwas numerically investigated. Slip boundary condition was taken intoaccount. Thermal performance of nanofluid under the effect of shearforce, nanoparticle concentration, velocity slip coefficient, and periodicheat flux has been studied. Obtained results from numerical solutionof the problem have been presented by temperature lines, streamlines,and local and average Nusselt numbers. Data have been validatedthrough comparison with the available results.
2. Problem statement
The investigated problem is a two-dimensional microchannel asillustrated in Fig. 1. Nanofluid consists of water and FMWCNT solidnanoparticles. FMWCNT nanoparticles are all spherical with the samediameter of dp = 30 ± 5nm. Cold nanofluid with the temperature ofTc = 306K and uniform velocity of uc enters the microchannel andexits from the other side after cooling the microchannels walls. Thefluid flow inside themicrochannel is considered as Newtonian, laminar,steady and incompressible. Thermophysical properties of water andFMWCNT at constant temperature of T = 33∘C are tabulated inTable 1. A periodic heat flux of q″ is imposed on the microchannel'swalls. The amplitude of heat flux (q0″) is calculated using non-dimensional parameters of Eq. (5). Slip boundary conditions are alsoconsidered for the velocity at the walls. Reynolds number for thenanofluid at the inlet are considered equal to Re = 1, Re = 10, andRe = 100. Variation of slip coefficient is considered as B = 0.001,B = 0.01, and B= 0.1. The weight percentages of solid nanoparticlesare also taken as ϕ = 0 %, ϕ = 0.12 %, and ϕ = 0.25 %.
3. Governing equations
Governing 2-D Navier–Stokes equations in the Cartesian coordinateare shown below [21]:
Continuity equation:
∂u∂x
þ ∂v∂y
¼ 0 ð1Þ
Momentum equation in X direction:
u∂u∂x
þ v∂u∂y
¼ −1ρn f
∂p∂x
þ υn f∂2u∂x2
þ ∂2u∂y2
!ð2Þ
Momentum equation in Y direction: Energy equation:
u∂v∂x
þ v∂v∂y
¼ −1ρn f
∂p∂y
þ υnf∂2v∂x2
þ ∂2v∂y2
!ð3Þ
yzed configuration.
Table 1Experimental data for physical properties of FMWCNT nanofluids at T = 33 °C [5,16].
Wt.%(FMWCNT/water) ρ (kg m−3) μ (Pas) K (W/mk) Cp (J/kgk)
0 995.8 7.65 × 10−4 0.62 41780.12 1003 7.80 × 10−4 0.68 41780.25 1008 7.95 × 10−4 0.75 4178
Fig. 3. Variation of heat transfer coefficient with Re from present work versus of experi-mental study [5] for different values of bulk temperature.
71Z. Nikkhah et al. / International Communications in Heat and Mass Transfer 68 (2015) 69–77
u∂T∂x
þ v∂T∂y
¼ αn f∂2T∂x2
þ ∂2T∂y2
!ð4Þ
Non-dimensional used parameters in Eqs. (1) to (4) are as below [22]:
H ¼ h=h ¼ 1; L ¼ l=h ¼ 32Y ¼ y=h; X ¼ x=hV ¼ v=uc; U ¼ u=uc
θ ¼ T‐TcΔT
; ΔT ¼ q000hkf
P ¼ Pρn fu2
c; Re ¼ uch
ν f; Pr ¼ ν f =α f
ð5Þ
Therefore, the non-dimensional governing equations can beexpressed as: Non-dimensional continuity equation:
∂U∂X
þ ∂V∂Y
¼ 0 ð6Þ
Non-dimensional momentum equation in X direction:
U∂U∂X
þ V∂U∂Y
¼ −∂P∂X
þ 1PrRe
υnf
α f
∂2U∂X2 þ
∂2U∂Y2
!ð7Þ
Non-dimensional momentum equation in Y direction:
U∂V∂X
þ V∂V∂Y
¼ −∂P∂Y
þ 1PrRe
υn f
α f
∂2V∂X2 þ
∂2V∂Y2
!ð8Þ
Fig. 2. Comparing fully developed dimensionless velocity profiles with those of Raisi et al.[31].
Non-dimensional energy equation:
U∂θ∂X
þ V∂θ∂Y
¼ αn f
α f
1PrRe
∂2θ∂X2 þ
∂2θ∂Y2
!ð9Þ
4. Boundary Conditions
Nanofluid slip velocity on the moving wall is defined by [23]:
us ¼ �β∂u∂y
����y¼0;h
ð10Þ
or in non-dimensional form:
Us ¼ �B∂U∂Y
����y¼0;1
ð11Þ
Generally, governing dimensionless boundary conditions are:
U ¼ 1;V ¼ 0 and θ ¼ 0 for X ¼ 0 and 0≤Y≤1
V ¼ 0 and∂U∂X
¼ ∂θ∂X
¼ 0 for X ¼ 32 and 0≤Y≤1
V ¼ 0;Us ¼ B∂U∂Y
and∂θ∂Y
¼ 2q ′′0 þ q ′′
0 sinπX4
� �for Y ¼ 0 and 0≤X≤32
V ¼ 0;Us ¼ −B∂U∂Y
and∂θ∂Y
¼ 2q ′′0 þ q ′′
0 sinπX4
� �for Y ¼ 1 and 0≤X≤32
ð12Þ
Table 2Grid independence tests for Re = 1 and Re = 10 at ϕ = 0.12% and B = 0.001.
Grid points 400 × 40 500 × 50 600 × 60
Re = 1Num 0.183 0.183 0.184Uout(Y = H/2) 1.490 1.491 1.491
Re = 10Num 0.836 0.835 0.835Uout(Y = H/2) 1.489 1.490 1.490
72 Z. Nikkhah et al. / International Communications in Heat and Mass Transfer 68 (2015) 69–77
Oscillation heat flux applies on the whole length of top and bottommicrochannel walls are given as:
q ′′ Xð Þ ¼ 2q ′′0 þ q ′′
0 sinπX4
� �ð13Þ
Local Nusselt number along the microchannel walls can beexpressed as [22]:
Nux ¼ λhkf
ð14Þ
λ ¼ q ′′0
Ts−Tcð15Þ
Fig. 4. Streamlines (top) and isotherms (bottom) at ϕ
Therefore, local Nusselt number can be achieved by using dimen-sionless parameters in the Eq. (5) as follows:
Nux Xð Þ ¼ 1θs Xð Þ ð16Þ
By integrate local Nusselt number along heated surfaces, the averageNusselt number is calculated as follows:
Num ¼ 1L
Z L
0Nux Xð ÞdX ð17Þ
5. Numerical solution
To solve the governing Navier–Stokes equations, the finite volumemethod has been utilized [24,25]. In this method, conservation
= 0.12 %, B = 0.1 and for different values of Re.
73Z. Nikkhah et al. / International Communications in Heat and Mass Transfer 68 (2015) 69–77
equations are integrated over the grids, while these equations are validin the entire computational domain. This is applicable in any number ofnodes, evenwhen the number of nodes is small [26–28]. To discretize allthe equations terms, second-order upwind has been used and to couplethe pressure and velocity, SIMPLE algorithmhas been employed [29,30].When the residuals of all parameters become lower than 10−8, the so-lution converges and the results can be obtained.
6. Validation of numerical solution
6.1. Comparison with the work of Raisi et al. [31]
In order to validate our results, developed velocity profiles of water–copper nanofluid in a microchannel were compared with the resultspresented by Raisi et al. [31]. Comparison is shown in Fig. 2 for differentvalues of slip coefficient (B). Raisi et al. [31] numerically investigatedlaminar forced convective fluid flow of water–copper nanofluid in amicrochannel, considering slip and no slip boundary conditions. Theyevaluated the cooling capacity of pure water and nanofluid and alsostudied the effect of Reynolds number, volume percentage of nanopar-ticles, and slip coefficient on the flow domain and heat transfer.
6.2. Comparison with the experimental study of Amrollahi et al. [5]
Forced convective heat transfer coefficient at the inlet of a horizontalpipe for water/FMWCNTnanofluid in φ = 0.12 % and different bulktemperatures and Reynolds numbers were compared with the work ofAmrollahi et al. [5] in Fig. 3. Amrollahi et al. [5] experimentally mea-sured forced convective heat transfer coefficient of water/FMWCNTnanofluid in the inlet section of a horizontal pipe under constant heatflux in laminar and turbulent flow regimes. Reynolds number wasconsidered to be in the range of 1592 ≤ Re ≤ 4778 and the experimentalarea was a pipe of 1 m length and 11.42 mm diameter. They were thefirst who examined the effect of Reynolds number, mass fraction, andtemperature on the convective heat transfer coefficient at the inletsection. Their results revealed that compared to water, heat transfercoefficient of the nanofluid in laminar and turbulent regimes increaseswith the rise in nanoparticles concentration.
Fig. 5. Dimensionless velocity profile of FMWCNT at central vertical cross-section ofmicrochannel (X = L/2), at ϕ= 0.12 %, Re = 10 and for different values of B.
7. Mesh independent results
Table 2 presents average Nusselt number of nanofluid inside themicrochannel for different grids of 400 × 40, 500 × 50, and 600 × 60atReynolds numbers of Re = 1, Re = 10 in φ = 0.12 %, B = 0.001. It isobserved that the difference between the two grids of 500 × 50,600 × 60 is small; therefore the 500 × 50 grid was selected for thecalculation.
8. Results and discussion
Forced convective heat transfer of water/FMWCNT nanofluid in a2-D horizontal microchannel was numerically investigated. As shownin Fig. 1, walls of microchannel were under sinusoidal heat flux q″.Thermophysical properties of water/FMWCNT nanofluid are presentedin Table 1. Moreover, slip boundary conditions (Us) was consideredalong the microchannel's walls for different values of slip coefficientequal to B = 0.001, B = 0.01, and B = 0.1.
Fig. 6. Dimensionless temperature profiles of FMWCNT at different cross-sections of themicrochannel, at ϕ= 0.12 %, B = 0.1 and for different values of Re.
Fig. 7. Variations of dimensionless velocity along the horizontal microchannel centerline(Y = H/2), at ϕ = 0.12 %, B = 0.1 and for different values of Re.
74 Z. Nikkhah et al. / International Communications in Heat and Mass Transfer 68 (2015) 69–77
Effect of different values of Reynolds number on the streamlines andtemperature lines of the nanofluid inφ=0.12 %, B=0.1 is displayed inFig. 4. Based on the streamlines, it can be observed that for all the threeReynolds numbers, flow becomes fully developed after passing a smalllength further from the microchannel inlet. It is also found that thelower the Reynolds number, the flow would quicker become fullydeveloped. Temperature lines show that nanofluid temperatureincreases along the length of the microchannel due to the heat fromheat flux, which is higher in lower Reynolds numbers. It is also notedthat as the Reynolds number becomes greater, it takes longer time forthe fluid flow to become thermally fully developed. This is because ofthe higher fluid velocity and less heat transfer between the nanofluidand microchannel's walls.
Fig. 5 demonstrates the effect of different values of B on the velocityprofile (U) along the vertical centerline of microchannel (X = L/2) inφ = 0.12 %. As indicated in Fig. 5, due to the slip boundary conditionon the walls of microchannel, there is a velocity for the nanofluid onthe walls and as slip coefficient increases, this velocity gets higher. The
Fig. 8. Variations of dimensionless velocity along the horizontal microchannel centerline(Y = H/2), at ϕ = 0.12 %, Re = 10 and for different values of B.
maximum velocity across the vertical centerline of microchannel isalso increased by the slip coefficient.
Fig. 6 displays non-dimensional temperature profiles, θ, at differentcross-sections ofmicrochannel inφ=0.12 %, B=0.1 for various valuesof Reynolds number. Temperature profiles at Re=1 are almost uniformfor different cross-sections due to the low flow velocity and higher heattransfer between the nanofluid and microchannel's walls. As Reynoldsnumber increases, considerable changes can be observed in tempera-ture profiles. Nanofluid temperature in the fully developed regionincreases across the length of microchannel with the increase in cross-section. In X = 0.9 L, where flow is fully affected by the walls
Fig. 9.Variations of dimensionless temperature along the horizontalmicrochannel center-line (Y = H/2), at B = 0.1 for various Re and ϕ.
75Z. Nikkhah et al. / International Communications in Heat and Mass Transfer 68 (2015) 69–77
temperature, maximum temperature with a parabolic profile is ob-served. Temperature profiles show that nanofluid temperature is in-creased near the warm walls of microchannels.
Figs. 7 and 8 illustrate the effects of different values of Re and B onthe non-dimensional velocity (U) across the horizontal centerline ofmicrochannel (Y = H/2), at φ = 0.12 %. Velocity profiles show that forall the considered Reynolds numbers, velocity increases with X in ashort length from the microchannel inlet and then becomes constantwhen flow becomes hydro-dynamically fully developed. It is alsonoted that in lower Reynolds number, transition to fully developedflow occurs in a shorter time.Non-dimensional velocity across the hori-zontal centerline of microchannel decreases with the slip coefficient.
Fig. 10. Variations of dimensionless temperature along the horizontal microchannelcenterline (Y = H/2), at ϕ = 0.12 % for various Re and B.
This is because the velocity grows near the microchannel's walls asthe slip coefficient increases and based on the conservation of mass,velocity decreases across the horizontal centerline of microchannel.
Figs. 9 and 10 show the effect of different values of Re, B, and ϕ onthe non-dimensional temperature (θ) across the horizontal centerlineof microchannel (Y=H/2). As it can be seen from temperature profiles,inlet temperature of the nanofluid periodically increases with the rise inX, because of the heat transfer with the microchannel's walls warmedby the periodic heat flux. The rate of this increment is higher in lowerReynolds numbers. The reason is that when the velocity is low, there
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8 B = 0.001B = 0.01B = 0.1
Re = 1
Us
X
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8 B = 0.001B = 0.01B = 0.1
Re = 10
Us
X
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8 B = 0.001B = 0.01B = 0.1
Re = 100
Us
X
Fig. 11. Dimensionless slip velocity along the microchannel wall at ϕ=0.12 % for variousRe and B.
Fig. 12. Variations of local Nusselt number along the microchannel wall at B = 0.1 forvarious Re and ϕ.
Fig. 14. Variation of averaged Nusselt number with ϕ for various Re and B.
76 Z. Nikkhah et al. / International Communications in Heat and Mass Transfer 68 (2015) 69–77
is enough time for the heat to be transferred between the nanofluid andmicrochannel's walls. It is clear in Fig. 9 that increasing the nanoparti-cles weight percentages results in the elevation in the non-dimensional temperature of the nanofluid due to the rise in the thermalconductivity of nanofluid. This augmentation is more evident in higherReynolds numbers. One can observe from Fig. 10 that with the decreasein slip coefficient, non-dimensional temperature of Nanofluid along thehorizontal centerline of the channel increases. When Reynolds numbergrows, this temperature rise is more visible.
Shown in Fig. 11 is the effect of different values of B on the slipvelocity Us across the length of microchannel for various values ofReynolds number at ϕ = 0.12 %, B = 0.1. It is observed that with therise in the slip coefficient, flow become closer to the slip flow and slipvelocity across the walls increases. It is also seen that the slip velocityhas the maximum value at the inlet of microchannel and decreases as
Fig. 13. Variations of local Nusselt number along the microchannel wall at ϕ = 0.12 % forvarious Re and B.
X increases within a short length from the inlet. Then it becomesconstant and the flow begins to be fully developed. As the Reynoldsnumber increases, it takes more time for the slip velocity to get fullydeveloped.
To better study the mechanism of heat transfer in microchannel,values of local Nusselt number Nux, across the length of microchannelfor various values of Re, B, and φ are presented in Figs. 12 and 13.Nusselt number has the highest value at the inlet of channel becauseof the maximum temperature difference between the nanofluid andmicrochannel's walls. This value periodically decreases by X along thelength of channel due to the increase in the nanofluid temperature.Nusselt number reduces more quickly in lower Reynolds numbers. Atthe outlet of channel, where temperatures of nanofluid and wallsapproach each other, Nusselt number turns to zero. As Reynoldsnumber gets higher, nanofluid velocity near the walls increases andtemperature difference between the nanofluid and walls becomesgreater and therefore Nusselt number increases.
Fig. 12 shows that in different Reynolds numbers, local Nusseltnumber grows as the nanoparticle weight fraction increases. The reasonis that by increasing the weight percentage, nanofluid thermal perfor-mance enhances due to the high thermal conductivity of nanoparticles.
Fig. 13 reveals the effect of various values of B on the local Nusseltnumber across the length of microchannel at φ = 0.12 % for differentReynolds numbers. At Re = 1, local Nusselt number is not a functionof slip coefficient and therefore when the slip coefficient increases, itremains unchanged along the length of the microchannel's wall. Withthe growth in Reynolds number, local Nusselt number increases withthe slip coefficient, because the temperature gradient along the lengthof the warmed wall of the microchannel gets higher.
Values of Num on the channel wall for different values of B, ϕ, and Reare given in Fig. 14. For all the Reynolds numbers, average Nusseltnumber increases with the rise in ϕ. Because increasing the weightpercentage of nanoparticle intensifies the thermal conductivity of thenanofluid. By the rise in Reynolds number and therefore nanofluidvelocity, temperature difference of the nanofluid and channel wallincreases. Then, averageNusselt number along the length of the channelwall enhances. It is observed that at Re = 1, Nusselt number remainsunchanged by the slip coefficient. However, as Reynolds numberincreases, the average Nusselt number along the length of channelenhances with the growth in slip coefficient.
77Z. Nikkhah et al. / International Communications in Heat and Mass Transfer 68 (2015) 69–77
9. Conclusion
In the current research, forced convective heat transfer of water/FMWCNT in a microchannel under periodic heat flux has been numeri-cally evaluated. Slip boundary condition across the length of the channelwalls for different slip coefficients equal to B = 0.001, B = 0.01, andB = 0.1 has been considered. Effect of weight percentage ofnanoparticles, shear force, velocity slip coefficient, and periodicheat flux on the flow field, temperature, and heat transfer rate hasbeen investigated.
Key observations are summarized as follows:
1- Streamlines and temperature lines vary along the length ofmicrochannel by the Reynolds number.
2- Velocity profiles on the vertical centerline of channel are constantfor all Reynolds numbers and remain the same with the change inReynolds number, notable changes are observed in temperatureprofiles.
3- Nanofluid velocity on the channel walls increases with the rise inslip coefficient.
4- Nanofluid temperature across the length ofmicrochannel increasesby X, and the rate of this increment is higher in lower Reynoldsnumbers.
5- Increasing the slip coefficient and weight percentage of nanoparti-cles, results in higher nanofluid temperature across the horizontalcenterline of microchannel.
6- Slip velocity begins at the channel entrance with the maximumvalue and gradually decreases by X to become finally constantalong the length of walls, where flow becomes fully developed.
7- Augmentation of slip coefficient significantly increases the slipvelocity.
8- The imposed periodic heat flux on the walls causes a periodicchange for both non-dimensional temperature along the horizontalcenterline and local Nusselt number along the length ofmicrochannel walls.
9- Nusselt number is maximum at the inlet and decreases with Xalong the length of microchannel.
10- Nusselt number elevates with the rise in Reynolds number, weightpercentage of nanoparticles, and slip coefficient, and themaximumvalue occurs at φ = 0.25 %, B = 0.1 and Re = 100.
11- In lowReynolds numbers, Nusselt number is not affected by the slipcoefficient and remains constant.
Acknowledgements
The authors gratefully acknowledge High Impact Research GrantUM.C/HIR/MOHE/ENG/23 and Faculty of Engineering, University ofMalaya, Malaysia, for support in conducting this research work. Theseventh authorwould like to thank theNational Science andTechnologyDevelopment Agency (NSTDA 2013), the Thailand Research Fund(IRG5780005) and National Research University Project (NRU 2014)for the support.
References
[1] J. Driker, W. Liu, D. VanWyk, J.P. Meyer, A.G. Malan, Embedded solid state heatextraction in integrated power electronic modules, IEEE Trans. Power Electron.(2005) 694–703.
[2] M.A. Alavi, Mohammad Reza Safaie, Omid Mahian, Marjan Goodarzi, M. Dahari,Hooman Yarmand, Somchai Wongwises, A hybrid finite element/finite differencescheme for solving the 3-D energy equation in transient non-isothermal fluid flowover a staggered tube bank, Numer. Heat Transfer, Part B 68 (2) (2015) (accepted).
[3] G. Hestroni, A. Mosyak, E. Pogrebnyak, L.P. Yarin, Heat transfer in microchannels:comparison of experiments with theory and numerical esults, Int. J. Heat Transf.48 (2005).
[4] S.U.S. Choi, J.A. Eastman, Enhancing thermal conductivity of fluid with nanoparticles,in: D.A. Siginer, H.P. Wangeds (Eds.),Developments and Applications of Non-Newtonian flows, 1995.
[5] A. Amrollahi, A.M. Rashidi, R. Lotfi, M. EmamiMeibodi, K. Kashefi, Convection heattransfer of functionalized MWNT in aqueous fluids in laminar and turbulent flowat the entrance region, Int. Commun. Heat Mass Transfer 37 (2010) 717–723.
[6] M. NuimLabib, Md.J. Nine, H. Afrianto, H. Chung, H. Jeong, Numerical investigationon effect of base fluids and hybrid nanofluid in forced convective heat transfer,Int. J. Therm. Sci. 71 (2013) 163–171.
[7] A. Karimipour, A. Hossein Nezhad, A. D'Orazio, E. Shirani, Investigation of the gravityeffects on the mixed convection heat transfer in a microchannel using latticeBoltzmann method, Int. J. Therm. Sci. 54 (2012) 142–152.
[8] J. Jung, H. Oh, H. Kwak, Forced convective heat transfer of nanofluids inmicrochannels, Int. J. Heat Mass Transf. 52 (2009) 466–472.
[9] P. Rosa, T.G. Karayiannis, M.W. Collins, Single-phase heat transfer in microchannels:the importance of scaling effects, Appl. Therm. Eng. 29 (2009) 3447–3468.
[10] A. Karimipour, New correlation for Nusselt number of nanofluid with Ag/Al2O3/Cunanoparticles in a microchannel considering slip velocity and temperature jumpby using lattice Boltzmann method, Int. J. Therm. Sci. 91 (2015) 146–156.
[11] A. Karimipour, A.H. Nezhad, A. D'Orazio, M.H. Esfe, M.R. Safaei, E. Shirani, Simulationof copper-water nanofluid in a microchannel in slip flow regime using the latticeBoltzmann method, Eur. J. Mech. B. Fluids 49 (2015) 89–99.
[12] A. Behzadmehr,M. Saffar-Avval, N. Galanis, Prediction of turbulent forced convectionof a nanofluid in a tube with uniform heat flux using a two phase approach, Int. J.Heat Fluid Flow 28 (2007) 211–219.
[13] S.A. Moshizi, A. Malvandi, D.D. Ganji, I. Pop, A two-phase theoretical study of Al2O3-water nanofluid flow inside a concentric pipe with heat generation/absorption, Int.J. Therm. Sci. 84 (2014) 347–357.
[14] M.M. Rashidi, A. Hosseini, I. Pop, S. Kumar, N. Freidoonimehr, Comparative Numericalstudy of single and two phase models of nanofluid heat transfer in a wavy channel,Appl. Math. Mech. 35 (7) (2014) 831–848.
[15] M. Kalteh, A. Abbassi, M. Saffar-Avval, J. Harting, Eulerian-Eulerian two-phasenumerical simulation of nanofluid laminar forced convection in a microchannel,Int. J. Heat Fluid Flow 32 (2011) 107–116.
[16] M.R. Safaei, H. Togun, K. Vafai, S.N. Kazi, A. Badarudin, investigation of heat transferenchantment in a forward-facing contracting channel using FMWCNT nanofluids,Numer. Heat Transfer - Part A 66 (2014) 1321–1340.
[17] M. Goodarzi, M.R. Safaei, G. Ahmadi, K. Vafai, M. Dahari, N. Jomhari, S.N. Kazi,Investigation of nanofluid mixed convection in a shallow cavity using a two-phasemixture model, Int. J. Therm. Sci. 75 (2014) 204–220.
[18] A. Karimipour, M.H. Esfe, M.R. Safaei, D.T. Semiromi, S. Jafari, S.N. Kazi, Mixedconvection of copper-water nanofluid in a shallow inclined lid driven cavity usingthe lattice Boltzmann method, Physica A 402 (2014) 150–168.
[19] H. Togun, M.R. Safaei, R. Sadri, S.N. Kazi, A. Badarudin, K. Hooman, E. Sadeghinezhad,Heat transfer to turbulent and laminar Cu/water flow over a backward-facing step,Appl. Math. Comput. 239 (2014) 153–170.
[20] M.R. Safaei, O. Mahian, F. Garoosi, K. Hooman, A. Karimipour, S.N. Kazi, S.Gharehkhani, Investigation of micro- and nanosized particle erosion in a 90° pipebend using a two-phase discrete phase model, Sci. World J. 2014 (2014), ArticleID 740578, http://dx.doi.org/10.1155/2014/740578 (12 pages).
[21] A.H. Mahmoudi, I. Pop, M. Shahi, Effect of magnetic field on natural convection in atriangular enclosure filled with nanofluid, Int. J. Therm. Sci. 59 (2012) 126–140.
[22] S.M. Aminossadati, A. Raisi, B. Ghasemi, Effects of magnetic field on nanofluid forcedconvection in a partially heated microchannel, Int. J. Non Linear Mech. 46 (2011)1373–1382.
[23] G.D. Ngoma, F. Erchiqui, Heat flux and slip effects on liquid flow in a microchannel,Int. J. Therm. Sci. 46 (2007) 1076–1083.
[24] M. Goodarzi, M.R. Safaei, A. Karimipour, K. Hooman, M. Dahari, S.N. Kazi, E.Sadeghinezhad, Comparison of the finite volume and lattice Boltzmann methodsfor solving natural convection heat transfer problems inside cavities and enclosures,Abstr. Appl. Anal. 2014 (2014), Article ID 762184, http://dx.doi.org/10.1155/2014/762184 (15 pages).
[25] M.R. Safaei, B. Rahmanian, M. Goodarzi, Numerical study of laminar mixed convec-tion heat transfer of power-law non-Newtonian fluids in square enclosures by finitevolume method, Int. J. Phys. Sci. 6 (33) (2011) 7456–7470.
[26] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington,1980.
[27] M.R. Safaei, H.R. Goshayshi, B. Saeedi Razavi, M. Goodarzi, Numerical investigation oflaminar and turbulent mixed convection in a shallow water-filled enclosure byvarious turbulence methods, Sci. Res. Essays 6 (22) (2011) 4826–4838.
[28] H.R. Goshayshi, M.R. Safaei, Y. Maghmoumi, Numerical Simulation of UnsteadyTurbulent and Laminar Mixed Convection in Rectangular Enclosure with HotUpper Moving Wall by Finite Volume Method, The 6th International ChemicalEngineering Congress & Exhibition (IChEC 2009), Kish Island, Iran, 2009.
[29] A. Karimipour, M. Afrand, M. Akbari, M.R. Safaei, Simulation of fluid flow and heattransfer in the inclined enclosure, Int. J. Mech. Aerosp. Eng. 6 (2012) 86–91.
[30] M. Goodarzi, M.R. Safaei, Hakan F. Oztop, E. Sadeghinezhad, M. Dahari, S.N. Kazi,Numerical study of entropy generation due to coupled laminar and turbulentmixed convection heat transfer and thermal radiation in a square enclosure filledwith a semitransparent medium, Sci. World J. 2014 (2014) http://dx.doi.org/10.1155/2014/761745 (8 pages).
[31] A. Raisi, B. Ghasemi, S.M. Aminossadati, A numerical study on the forced convectionof laminar nanofluid in a microchannel with both slip and no-slip conditions,Numer. Heat Transfer, Part A 59 (2012) 114–129.