Physica E 84 (2016) 474–481

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Physica E

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The variations of heat transfer and slip velocity of FMWNT-waternano-fluid along the micro-channel in the lack and presence of amagnetic field

Masoud Afrand a, Arash Karimipour a,n, Afshin Ahmadi Nadooshan b, Mohammad Akbari a

a Department of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iranb Shahrekord University, Faculty of Engineering, P.O. Box 115, Shahrekord, Iran

H I G H L I G H T S

� The effects of Ha, Re, slip coefficient and nano-particles volume fraction were investigated.

� The stronger magnetic field leaded to less maximum of U at horizontal centerline and greater fluid velocity near the walls.� Magnetic field affected severely the slip velocity along the walls of micro-channel.

a r t i c l e i n f o

Article history:Received 1 June 2016Received in revised form20 June 2016Accepted 12 July 2016Available online 16 July 2016

Keywords:FMWNT-water nano-fluidMagneto hydrodynamicsMicro-channelSlip velocityNusselt number

x.doi.org/10.1016/j.physe.2016.07.01377/& 2016 Elsevier B.V. All rights reserved.

esponding author.ail address: ArashKarimipour@gmail.com (A. K

a b s t r a c t

Simulation of forced convection of FMWNT-water (functionalized multi-walled carbon nano-tubes)nano-fluid in a micro-channel under a magnetic field in slip flow regime is performed. The micro-channel wall is divided into two portions. The micro-channel entrance is insulated while the rest oflength of the micro-channel has constant temperature (TC). Moreover, the micro-channel domain isexposed to a magnetic field with constant strength of B0. High temperature nano-fluid (TH) enters themicro-channel and exposed to its cold walls. Slip velocity boundary condition along the walls of themicro-channel is considered. Governing equations are numerically solved using FORTRAN computer codebased on the SIMPLE algorithm. Results are presented as the velocity, temperature, and Nusselt numberprofiles. Greater Reynolds number, Hartmann number, and volume fraction related to more heat transferrate; however, the effects of Ha and ϕ are more noteworthy at higher Re.

& 2016 Elsevier B.V. All rights reserved.

1. Introduction

In recent decades, study on different methods to enhance theheat transfer of fluids has attracted the attention of researchers. Inthis way, a new generation of fluids, called nano-fluids, with greatpotential in industrial application was considered as a result of thedistribution of nanoparticles in a common liquid [1–5].

Common heat transfer fluids including water, oil and ethyleneglycol generally have low thermal conductivity, while solid na-noparticles have high thermal conductivity. Hence, dispersion ofnanoparticles in the base fluid can lead to increase its thermalconductivity. Therefore, nano-fluids could be considered an optionfor improving the heat transfer rate [6–10]. Enhancement of heattransfer coefficient of nano-fluids is subject to many factors. Forexample, heat transfer coefficient can be increased by changing

arimipour).

the flow geometry, boundary conditions or enhancement of ther-mal conductivity of the fluid [11]. Numerous studies on heattransfer in different geometric shapes and the effects of externalfactors on the heat transfer have been performed. In this regard,Tahir and Mital [12] investigated the heat transfer in channels witha circular cross section. Heat transfer in channels with differentgeometry has been reported in the literature [13,14].

Cooling systems are the main concerns of thermal and electronicsystems. In these circumstances, the use of improved and optimalcooling systems is inevitable. Tullius et al. [15] examined the effect ofmicro-channel geometry and fluid type on the cooling rate from mi-cro-channels. The most common fluids used in micro-channels wereair and water, which have low thermal conductivity.

Low heat transfer rate in micro-channels containing water orair led to use of various methods such as increasing the heattransfer surface, while this method increase the size of systems.Therefore, to overcome this problem, a new and effective cooling isrequired and nano-fluids were introduced as a new approach inthis field [16].

Fig. 1. The schematic configuration of the micro-channel.

Table 1Empirical data for thermo-physical properties of the FMWNT nanofluid [50].

wt% FMWNT/water ρ (kg/m3) k (W/mK) μ (Pas)

0 995.8 0.62 7.65�10�4

0.12% 1003 0.68 7.80�10�4

0.25% 1008 0.75 7.95�10�4

M. Afrand et al. / Physica E 84 (2016) 474–481 475

In order to study the heat transfer in micro-channels, differentmodels were studied [17–19]. The effects of micro-channel geo-metry, Reynolds number and nanoparticles volume fraction on thethermal performance of micro-channels have also been in-vestigated. In general, it seems that using new nano-fluid shows abetter performance against the heat load generated by smallelectronic devices [20–26].

Surface effects in micro-scale have dramatic impact than that inmacro-scale. For example, no-slip condition commonly used forthe macro-scale, in micro-channels is not true; consequently, slipcondition should be used along the walls. Moreover, for micro andnano scales, specific methods such as Lattice-Boltzmann methodor molecular dynamics, which are based on particles, should beused [27–31]. Raisi et al. [32] investigated numerically the heattransfer in micro-channels by assuming existence or lack of slipvelocity. They also reported the effect of nano-fluids on heattransfer rate in micro-channel in the presence of slip velocity. Tothis day, much researches assuming slip velocity have been pre-sented in the field of heat transfer in micro-channel containingnano-fluids with different boundary conditions such as constanttemperature or constant flux, which quickly slip boundary condi-tion have also been studied [33–37].

In recent years, the flow and heat transfer in different sections ofmacro-scales exposed to a magnetic field have attracted the attentionof the researchers [38–44]. In this case, the magnetic field leads toproduce a force, called the Lorentz force, which affects the fluid flow[45–48]. Aminossadati et al. [49] studied the effects of magnetic fieldon a micro-channel under constant heat flux, while the slip velocityalong the walls of the micro-channel was neglected.

However, a review of previous researches [53,54] showed thatregarding the study of flow and heat transfer in micro-channels underthe magnetic field, thes lip velocity as a boundary condition has beenignored. Up to now, there is no comprehensive research on the Si-multaneous effects of the magnetic field, slip velocity boundary con-dition, forced convection on the flow and heat transfer in micro-channels. In this study, all of these conditions are considered si-multaneously and efforts were also made to simulate the effects ofmagnetic field on the slip velocity of molecules adjacent wall. More-over, in this study water-based nano-fluid containing functionalizedmulti-walled carbon nanotubes (FMWNT) is used as the working fluid.

2. Problem statement

Simulation of forced convection of FMWNT-water (functiona-lized multi-walled carbon nano-tubes suspended in water) nano-fluid in a two dimensional micro-channel under a magnetic fieldfor slip flow regime is performed.

Single-walled carbon nanotubes (SWNTs) and multi-walled carbonnanotubes (MWNTs) are similar in certain respects but they also havestriking differences. SWNTs structure is a cylindrical tube includingsix-membered carbon rings similar to graphite. They consist of ahollow cylinder of carbon �1 nm (in present work for one cylinder) indiameter, up to 1000 times as long as it is wide. Analogously MWNTsinclude several tubes layers (concentric tubes) of graphene in con-centric cylinders. The interlayer distance in multi-walled nanotubes isclose to the distance between graphene layers in graphite, approxi-mately 3.4 Å. The number of these concentric walls may vary from 6 to25 or more. The diameter of MWNTs may be 30 nm (used at presentarticle) when compared to 0.7–2.0 nm for typical SWNTs. The uniqueproperties of carbon nanotubes enable a wide range of novel appli-cations and improvements in the performance of existing ones.However, one can functionalize the nanotubes to enhance both thestrength and dispersibility of composites.

The supposed micro-channel aspect ratio is 30; hence the fullydeveloped condition is achieved at outlet. As shown in Fig. 1, the

micro-channel wall is divided into two portions. The micro-channel entrance (X¼0.3L) is insulated while the rest of length ofthe micro-channel has constant temperature (TC). This configura-tion leads to approach the hydrodynamic fully developed condi-tion. Moreover, the micro-channel domain is exposed to a mag-netic field with constant strength of B0. Slip velocity boundarycondition along the walls of the micro-channel is considered. Hightemperature nano-fluid (TH) enters the micro-channel and ex-posed to its cold walls. Finally, it leaves the micro-channel fromthe other side. The empirical data for thermo-physical propertiesof the FMWNT-water nano-fluid are presented in Table 1.

With regard to consider thevery low weight fraction of nano-tubes, nano-fluid is assumed as a Newtonian fluid. Reynoldsnumber (Re) commonly is low in a micro-channel duo to express areal physical condition (Re¼20 and 200). Moreover, in order toinvestigate the slip velocity boundary condition different values ofslip coefficient such as B¼0.005 and B¼0.05 are assumed. Basedon Table 1, three different values of weight fractions of FMWNTsare applied. Earlier investigations showed that the Hartmannumbers greater than 40 have no significant effect on the flow andheat transfer; thus, a reasonable range of Hartmann number isfound lower than 40. It should be noted that the nanofluid is in-compressible and homogeneous mixture.

3. Mathematical formulation

The two-dimensional Navier–Stokes equations (continuity,momentum and energy) with regard to the effect of a magneticfield strength (B0) are as follows:

∂∂

+ ∂∂

=( )

ux

vy

01

ρσ

ρ∂∂

+ ∂∂

=− ∂∂

+ϑ ∂∂

+ ∂∂

−( )

⎛⎝⎜

⎞⎠⎟u

ux

vuy

px

ux

uy

B1

2

2

2

2

202

ρϑ∂

∂+ ∂

∂=− ∂

∂+ ∂

∂+ ∂

∂ ( )

⎛⎝⎜

⎞⎠⎟u

vx

vvy

py

vx

vy

1

3

2

2

2

2

α∂∂

+ ∂∂

= ∂∂

+ ∂∂ ( )

⎛⎝⎜

⎞⎠⎟u

Tx

vTy

Tx

Ty 4

2

2

2

2

s indicates the electrical conductivity of the nano-fluid and isequal to 4.99�10�2 (S/cm).

M. Afrand et al. / Physica E 84 (2016) 474–481476

It is necessary that Eqs. (1)–(4) are written in the form of di-mensionless as follows:

∂∂

+ ∂∂

= ( )UX

VY

0 5

∂∂

+ ∂∂

=− ∂∂

+ ∂∂

+ ∂∂

−( )

⎛⎝⎜

⎞⎠⎟U

UX

VUY

PX

UX

UY

U1

ReHaRe 6

2

2

2

2

2

∂∂

+ ∂∂

=−∂∂

+ ∂∂

+ ∂∂ ( )

⎛⎝⎜

⎞⎠⎟U

VX

VVY

PY

VX

VY

1Re 7

2

2

2

2

θ θ θ θ∂∂

+ ∂∂

= ∂∂

+ ∂∂ ( )

⎛⎝⎜

⎞⎠⎟U

XV

Y X Y1

Pr. Re 8

2

2

2

2

In the above equations, dimensionless numbers have been

defined in terms of nano-fluid properties: =ρ

μRe

u hnf i

nf، =

υ

αPr nf

nf

and =σ

μ⎜ ⎟⎛⎝

⎞⎠B hHa 0

0.5nf

nf.

Moreover, in Eqs. (5)–(8) the following dimensionless para-meters have been used:

ρθ

= = = = = = = =

= =−

( )−

Hhh

Llh

Xxh

Yyh

Uuu

Vvu

P

pu

T TT T

1, 30, , , , ,

,9

i i

nf i

c

H c2

U

0.0 0.5 1.0 1.5

Y

0.00

0.25

0.50

0.75

1.00

Lines: present work

Symbols: Hooman and Ejlali [51]

Kn = 0.0

Kn = 0.1

Fig. 2. Horizontal velocity profiles at fully developed condition from present workversus [51].

Table 2Grid independency study for the values of U and θ at the point of X¼L/2 and Y¼H/2for ϕ¼0.0, Re¼20, Ha¼0 and B¼0.005.

375�25 450�30 525�35

U 1.480 1.481 1.481θ 0.766 0.768 0.769

4. Boundary conditions and solution algorithm

It is obvious that at the macro-scale no-slip boundary conditionshould be used, while in micro-channels the slip flow regime isestablished. The slip velocity can be calculated by the followingequation:

β= ± ∂∂ ( )=

⎛

⎝⎜⎜

⎞

⎠⎟⎟u

uy 10

sy h0,

where β represents the slip coefficient.Dimensionless form of Eq. (10) on the wall is expressed as

follows:

= ± ∂∂ ( )=

⎛⎝⎜⎜

⎞⎠⎟⎟U B

UY 11

sY 0,1

in which B is called dimensionless slip coefficient.Other dimensionless boundary conditions are as follows:

( )θ= = = = ≤ ≤U V X Y1, 0, 1 at 0, 0 1

( )θ∂∂

= ∂∂

= = ≤ ≤UX X

X Y0 at 30, 0 1

Fig. 3. Num from present work versus [52] (Symbols: [52], Lines: Present work).

Re=100

Ha10 20 30

Nu m

0.0

2.5

5.0

7.5

10.0

Present workAminossadati et al.

Re=10

Fig. 4. Num from present work versus [49] at ϕ¼0.02.

M. Afrand et al. / Physica E 84 (2016) 474–481 477

θ= = = ( = < ≤ )

( = < ≤ ) ( )

V U U Y X

Y X

0, , 0 at 0, 9 30 and

1, 9 30 12s

θ= = ∂∂

= ( = ≤ ≤ ) ( = ≤ ≤ )V U UY

Y X Y X0, , 0 at 0, 0 9 and 1, 0 9s

Based on above boundary conditions, Eqs. (5)–(8) are numeri-cally solved using FORTRAN computer code. The finite volumemethod based on the SIMPLE algorithmwas employed to solve thegoverning equations. Discretization of the nonlinear equations wasperformed by the power law scheme.

Local Nusselt number is:

] θ= − ∂∂ ( )

==

⎤⎦⎥⎥

k

k yNu

13X Y

nf

f Y0

0

] θ= − ∂∂ ( )

==

⎤⎦⎥⎥

k

k yNu

14X Y

nf

f Y1

1

Nusselt number with integration over the wall to be expressedas follows

∫ )= ( ( )LNu X dXNu

10.7 15m

L

L

X0.3

Fig. 5. U profiles at ϕ¼0.0, Re¼20, Ha¼40, B¼0.005 and B¼0.05 for differentcross sections of the micro-channel.

5. Results and discussion

Forced convective heat transfer of nano-fluid composed ofwater and functionalized multi-walled carbon nanotubes(FMWNT) in a two-dimensional micro-channel has been numeri-cally investigated. The results were classified in the following sub-sections.

5.1. Grid independency and code verification

Table 2 presents the values of U and θ at the central point of themicro-channel in the absence of magnetic field for various grids.

It can be found from Table 2 that difference between those of450�30 and 525�35 is negligible; consequently, grid nodes of450�30 is selected for the next computations.

In order to ensure the accuracy of the results obtained by thedeveloped FORTRAN code, results of present work versus those ofHooman and Ejlali [51] are presented in Fig. 2. In this figure,horizontal velocity profiles at fully developed condition through amicro-channel for different values of Knudsen number (Kn) arecompared. Moreover, Fig. 3 shows a comparison between theaveraged Nusselt number obtained by the code and that presentedby Santra et al. [52] for Reynolds numbers of 100 and 200. The lastselected case for confirmation would be a work of Aminossadatiet al. [49] which is displayed in Fig. 4 and pertains to a forcedconvection flow of nanofluid in a micro-channel exposed to a

Fig. 6. Fully developed horizontal velocity profiles at ϕ¼0.0, Re¼20, B¼0.005 andB¼0.05 for different values of Ha.

M. Afrand et al. / Physica E 84 (2016) 474–481478

magnetic field and in the absence of slip velocity. Good agree-ments can be observed in Figs. 2–4.

5.2. Effects of dimensionless slip coefficient

Fig. 5 shows the dimensionless velocity profiles (U) at differentsections of micro-channel for various values of B. This figure showsthat with increasing X, velocity profile becomes smoother; as aresult, the shear stress increases near the walls. It is also foundfrom Fig. 5 that with increasing slip coefficient the slip velocityincreases. For the case of B¼0.05, with increasing X, the slip ve-locity enhances that leads to a smoother velocity profile. For ex-ample, slip velocity values are approximately 0.25, 0.40 and 0.60 atX¼0.1, 0.3 and 0.6, respectively.

5.3. Effects of Hartmann number

Fig. 6 shows U profiles along the vertical line in developedregion of micro-channel for various Hartman numbers and slipcoefficients of 0.005 and 0.05. We know that applying the mag-netic field leads to the generation of the Lorentz force in the re-verse direction of movement of nano-fluid. Hence, the strongermagnetic field can lead to less maximum of U at horizontal cen-terline and greater fluid velocity near to the walls. Consequently,the fully developed velocity profile would vary with Ha. This be-havior can be found well in Fig. 6 at B¼0.005; thus, stronger

Fig. 7. Temperature profiles of θ at ϕ¼0.0, Re¼20, B¼0.005, Ha¼0 and Ha¼40 fordifferent cross sections of the micro-channel.

magnetic field leads to thinner boundary layer along the walls.Furthermore, the growth of thickness of the boundary layer issmall at greater Ha values.

It can be also observed in Fig. 6 that the slip velocity is changedby Hartmann number at B¼0.05. As shown in this figure, magneticfield affectsthe slip velocity on the walls of micro-channels suchthat slip velocity increases with increasing Hartmann number. Forexample,slip velocity is equal to 0.25 for in absence of magneticfield, while it would be 0.5 and 0.65 for Ha¼20 and Ha¼40,respectively.

In order to demonstrate the effects of Ha on θ profiles, thetemperature profiles of θ at different sections of micro-channelunder various values of Ha for B¼0.005 and B¼0.05 are displayedin Figs. 7 and 8, respectively. In both figures, at X¼0.3L, where theflow is not significantly affected by the cold walls, the temperatureand its gradient are great. The high temperature gradient indicatesthat the heat transfer rate in this zone is considerable. In themiddle of the micro-channel (0.5LrXr0.6L), where nano-fluidexchanges heat with the cold walls, the slope of the temperatureprofile is reduced. In the area near the outlet (XZ0.7L), nano-fluidtends to the temperature of the walls, and the temperature gra-dient is reduced. Comparison of temperature profiles in the ab-sence of magnetic field (Ha¼0) with those at Ha¼40 indicatesthat the nanofluid temperature in the various sections of the mi-cro-channel decreases and tends to the wall's one in the presenceof magnetic field Ha¼40. This fact denotes more heat transfer

Fig. 8. Temperature profiles of θ at ϕ¼0.0, Re¼20, B¼0.05, Ha¼0 and Ha¼40 fordifferent cross sections of the micro-channel.

M. Afrand et al. / Physica E 84 (2016) 474–481 479

existences due to more nanofluid velocity near the walls for thecase of H¼40.

Comparison between Figs. 7 and 8 shows that temperatureprofiles for the case of B¼0.005 are almost similar to those for thecase of B¼0.05, implying the inconsiderable effects of B on tem-perature profiles. Fig. 9 displays the isotherms in the micro-channel at Reynolds numbers of 20 and 200. This figure denotes agood visual aspect of nanofluid flow through the micro-channel.

5.4. Effects of Carbon nano-tubes concentration

Fig. 10 depicts the of Us along the micro-channel wall at variousvalues of B for Ha¼0, Ha¼20 and Ha¼40. Us starts from its

Fig. 10. Slip velocity profiles along the micro-channel wall at B¼0.005 and B¼0.05for different values of Ha.

Fig. 9. Isotherms at ϕ¼0.0, B¼0.005, Ha¼0 for Re¼20 (top) and Re¼200(bottom).

maximum value at inlet and then it diminishes with increasing Xand, eventually, becomes constant along the wall. Furthermore,slip coefficient significantly affects the slip velocity such that itincreases with enhancing slip coefficient. However, this eventcould be severely invigorated for more Ha.

Moreover, Fig. 11 illustrates NuX along the micro-channel wallat Re¼20, B¼0.005, Ha¼0 and Ha¼40 for different values of ϕ.Nusselt number starts from its maximum value at X¼0.3L andthen decreases mildly along the micro-channel length. More ϕcorresponds to more Nu however the effects of Ha can be ignoredat low values of Reynolds like Re¼20. In spite of that, the positiveeffect of Re on Nu can be observed well in Fig. 12. It is well knownthat the effects of slip coefficient are significant especially at en-trance length. On the other hand, less Re leads to lower entrancelength; so that the effects of slip coefficient on Nu would benegligible at Re¼20 in comparison with those of Re¼200.

More focus on heat transfer rate, the influences of all Ha, ϕ, Band Re on Num are presented in Fig. 13. It is seen that higher valuesof Nu can be achieved at larger amounts of Hartman number andvolume fraction at higher Reynolds number (Re¼200). There isinconsistency in this figure which means the slop of line is notconstant and decreases with Ha. This event obviously shows thathigher Hartmann number corresponds to more Nusselt number;however this increasing trend will not achieve continuously. As aresult more Ha (Ha450) might not be useful for this purpose.

Fig. 11. NuX along the micro-channel wall at Re¼20, B¼0.005, Ha¼0 and Ha¼40for different values of ϕ.

Fig. 12. NuX along the micro-channel wall at ϕ¼0.0025, Ha¼20, Re¼20 andRe¼200 for B¼0.005 and B¼0.05.

Fig. 13. Num on the micro-channel upper wall at different values of Ha, ϕ and B forRe¼20 and Re¼200.

M. Afrand et al. / Physica E 84 (2016) 474–481480

6. Conclusion

Simulation of forced convection of FMWNT-water nano-fluid ina two dimensional micro-channel under a magnetic field for slipflow regime is performed. Slip velocity and constant wall tem-perature were considered as boundary conditions. The effects ofHartmann number, Reynolds number, slip coefficient and solidvolume fraction on velocity, temperature and Nusselt numberwere investigated. The following results were obtained:

� With increasing slip coefficient the slip velocity increased.� The stronger magnetic field leaded to less maximum of U at

horizontal centerline and greater fluid velocity near to the walls.Consequently, the fully developed velocity profile would varywith Ha. Moreover, stronger magnetic field leaded to thinnerboundary layer along the walls. Furthermore, the growth ofthickness of the boundary layer was small at greater Ha values.

� Magnetic field affected the slip velocity on the walls of micro-channels such that slip velocity increased with increasingHartmann number.

� There was more heat transfer rate due to more nanofluid ve-locity near the walls in the presence of magnetic field.

� At higher Hartmann numbers, the effects of slip coefficient onthe slip velocity were more than that at lower Hartmannnumbers.

� More solid volume fractions corresponded to more Nu, whilethe effects of Ha were negligible at low values of Reynolds.

� Higher values of Nu were achieved at larger amounts of Hart-man number and volume fraction at higher Reynolds number(Re¼200).

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