International Journal of Heat and Fluid Flow 32 (2011) 107–116

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International Journal of Heat and Fluid Flow

journal homepage: www.elsevier .com/ locate/ i jhf f

Eulerian–Eulerian two-phase numerical simulation of nanofluid laminarforced convection in a microchannel

Mohammad Kalteh a,b, Abbas Abbassi a,⇑, Majid Saffar-Avval a, Jens Harting b,c

a Department of Mechanical Engineering, Amirkabir University of Technology, Hafez Ave., P.O. Box 15916-34311, Tehran, Iranb Department of Applied Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlandsc Institute for Computational Physics, University of Stuttgart, Pfaffenwaldring 27, 70569 Stuttgart, Germany

a r t i c l e i n f o a b s t r a c t

Article history:Received 4 March 2010Received in revised form 1 August 2010Accepted 5 August 2010

Keywords:NanofluidMicrochannelTwo-phaseLaminarHeat transfer

0142-727X/$ - see front matter � 2010 Elsevier Inc. Adoi:10.1016/j.ijheatfluidflow.2010.08.001

⇑ Corresponding author. Tel.: +98 2164543425; faxE-mail address: abbassi@aut.ac.ir (A. Abbassi).

In this paper, laminar forced convection heat transfer of a copper–water nanofluid inside an isothermallyheated microchannel is studied numerically. An Eulerian two-fluid model is considered to simulate thenanofluid flow inside the microchannel and the governing mass, momentum and energy equations forboth phases are solved using the finite volume method. For the first time, the detailed study of the rel-ative velocity and temperature of the phases are presented and it has been observed that the relativevelocity and temperature between the phases is very small and negligible and the nanoparticle concen-tration distribution is uniform. However, the two-phase modeling results show higher heat transferenhancement in comparison to the homogeneous single-phase model. Also, the heat transfer enhance-ment increases with increase in Reynolds number and nanoparticle volume concentration as well as withdecrease in the nanoparticle diameter, while the pressure drop increases only slightly.

� 2010 Elsevier Inc. All rights reserved.

1. Introduction

Emerging developments in MEMS (Micro-Electro-Mechanical-Systems) make it possible to fabricate very small scale devices.On the other hand, these small scale devices can generate a highamount of heat flux that should be taken away by a cooling systemto guarantee their appropriate performance. One possible way tocool these devises can be the use of so-called nanofluids. A nano-fluid is a suspension of nano-sized (10–100 nm) metallic andnon-metallic solid particles in a conventional cooling liquid suchas water, ethylene glycol, or oil. The term nanofluid for the firsttime was used by Choi (1995) for such a suspension. After that,many researchers focused on studying the thermophysical and alsoheat and fluid flow properties of nanofluids. Most of the studiesconcentrated on the modeling of the effective thermal conductivityof nanofluids (e.g. Xuan et al., 2003; Koo and Kleinstreuer, 2004;Feng et al., 2007). Recently, researchers have focused on the nano-fluid heat and fluid flow behavior.

There are many experimental studies for nanofluids on macroand micro-scales (e.g. Wen and Ding, 2004; Heris et al., 2006; Junget al., 2009; Wu et al., 2009). Wen and Ding (2004) investigated theheat transfer of an Al2O3–water nanofluid in the entrance region ofa 4.5 mm diameter copper tube under the constant heat flux con-ditions. Their measurements showed enhancement in heat transfer

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especially in the entrance region of the tube. They described thisbehavior as the particle migration effect (non-uniform nanoparti-cle volume concentration) that reduces the thermal boundary layerthickness. For an annular tube with a 6 mm inner diameter coppertube and a 32 mm outer diameter stainless steel tube, Heris et al.(2006) studied CuO and alumina nanoparticles in water. They com-pared the experimental results with homogeneous model results(single-phase correlations with nanofluid effective properties)and reported that the homogeneous modeling under-estimatesthe heat transfer enhancement, especially in higher volumeconcentrations.

Jung et al. (2009) did experiments for Al2O3–water nanofluids inrectangular microchannels. The particle diameter in their experi-ments was 170 nm. With only 1.8% of volume concentration theyreported a 32% increase of the heat transfer coefficient in compar-ison to single distilled water. Also, experiments on nanofluid heattransfer in trapezoidal silicon microchannels have been performedby Wu et al. (2009). For channels with a hydraulic diameter of194.5 lm and an Al2O3–water nanofluid, they reported an increasein the Nusselt number with increasing particle concentration, Rey-nolds and Prandtl numbers, while the pressure drop increasedslightly when compared to pure water.

For the theoretical study of the pressure-driven nanofluid heatand fluid flow commonly homogenous (single-phase) and two-phase models are used. In homogeneous modeling it is assumedthat the particles and the base fluid have the same temperatureand velocity and thus, the single-phase equations along with the

Nomenclature

A defined in Eq. (25)B defined in Eq. (24)Cp specific heat at constant pressure, J kg�1 K�1

Cd drag coefficientdp nanoparticle diameter, mDh hydraulic diameter, mFcol particle–particle interaction force, Pa m�1

Fd drag force, Pa m�1

Fvm virtual mass force, Pa m�1

G particle–particle interaction modulus, Pah heat transfer coefficient based on the mean tempera-

ture, W m�2 K�1

hv volumetric heat transfer coefficient, W m�3 K�1

hp liquid-particle heat transfer coefficient, W m�2 K�1

H channel height, mk thermal conductivity, W m�1 K�1

L channel length, mNu Nusselt number (hDh/kl)Nu average Nusselt numberNup Particle Nusselt numberp pressure, PaP non-dimensional pressurePr liquid Prandtl numberq00 wall convective heat flux, W m�2

Re Reynolds number (qluinDh/ll)ReH Reynolds number (qluinH/ll)

Rep particle Reynolds number ulql j~Vl�~Vp jdp

ll

� �u, v velocity components in the x- and y-directions respec-

tively, m s�1

U, V non-dimensional velocity in the x- and y-directionsrespectively

t time, sT temperature, Kx, y axial and vertical coordinates respectively, mx* non-dimensional axial length (xD�1

h Re�1Pr�1)X, Y non-dimensional axial and vertical coordinates respec-

tively

Greek symbolsb friction coefficient, kg m�3 s�1

C defined in Eq. (23)h non-dimensional temperaturel viscosity, Pa sq density, kg m�3

u volume concentrationx defined in Eq. (25)

Subscriptsb bulkeff effectivei phase index (=l, p)in inletl liquid phasem meannf nanofluidp particle phasepw pure waterw wall

108 M. Kalteh et al. / International Journal of Heat and Fluid Flow 32 (2011) 107–116

appropriate effective thermophysical properties (thermal conduc-tivity, viscosity, specific heat and density) for the nanofluid aresolved. In this method, the accuracy of the models used as effectivethermophysical properties is very important. Most of the theoreti-cal studies in this field are based on the homogeneous approach(e.g. Koo and Kleinstreuer, 2005; Li and Kleinstreuer, 2008; Santraet al., 2009). In addition to the pressure-driven nanofluid flows, it ispossible to use electroosmotic transport for nanofluids especiallyin the micro-scale. In this case, the thickness of the electrical dou-ble layer and effective electrical conductivity of the nanofluid canaffect the nanofluid heat transfer behavior (Chakraborty and Pad-hy, 2008; Chakraborty and Roy, 2008).

In despite of the homogeneous modeling, in the two-phasemodeling, the nanoparticle and the base fluid are considered astwo different phases with different velocities and temperatures.In this method, the interactions between the phases are taken intoaccount in the governing equations. There are a few studies thatused two-phase approach to study nanofluids. Behzadmehr et al.(2007) used a two-phase mixture model to study the turbulentnanofluid convection inside a circular tube. Comparing with anexperimental study they reported that the two-phase results aremore precise than the homogeneous modeling results. However,they considered thermal equilibrium conditions (the same temper-ature) for the phases. Mirmasoumi and Behzadmehr (2008a) usedthe same method as in Behzadmehr et al. (2007) to study themixed convection of the nanofluid in a tube. Also, Mirmasoumiand Behzadmehr (2008b) and Akbarinia and Laur (2009) investi-gated the nanoparticles size effect on the mixed convective heattransfer of a nanofluid using the two-phase mixture method. Inboth studies an increase in heat transfer with a decrease in thenanoparticles size was reported. Bianco et al. (2009) modeled thenanofluid flow and heat transfer inside a tube. They used both sin-

gle-phase and two-phase methods. For the two-phase method,they implemented Lagrangian approach to model the particle mo-tion. They reported a maximum difference of 11% between the sin-gle and two-phase results. Kurowski et al. (2009) used threedifferent homogeneous, Eulerian–Lagrangian and mixture methodsto simulate nanofluid flow inside a minichannel. Their resultsshowed almost the same behavior for all the methods. Fard et al.(2010) studied the nanofluid heat transfer inside a tube consider-ing both the single and two-phase methods. For a 0.2% copper–water nanofluid, they reported that the average relative error be-tween the experimental data and single-phase model was 16%while for the two-phase method it was 8%. On the other hand, Lotfiet al. (2010) used homogeneous, two-phase Eulerian and mixturemodels for nanofluid flow inside a tube. They reported that amongthese methods, the two-phase mixture method is more precisethan the others.

According to the literature, there is a non-uniform nanoparticlevolume concentration distribution in the entrance region (Wenand Ding, 2004) and the homogeneous model under-estimatesthe observed heat transfer enhancement in the experiments (Heriset al., 2006; Behzadmehr et al., 2007; Bianco et al., 2009; Fard et al.,2010; Lotfi et al., 2010). Thus, the two-phase modeling can be analternative method. On the other hand, the existing studies forthe two-phase method do not consider the temperature differencebetween the phases (Behzadmehr et al., 2007; Mirmasoumi andBehzadmehr, 2008a,b; Akbarinia and Laur, 2009) or do not presentdetailed results on the relative velocity and temperature betweenthe phases and the volume concentration distribution (Biancoet al., 2009; Fard et al., 2010). The amount of the relative velocityand temperature between the phases along with the nanoparticleconcentration distribution can provide an estimation of the accu-racy of the assuming nanofluid as a homogeneous solution. On

M. Kalteh et al. / International Journal of Heat and Fluid Flow 32 (2011) 107–116 109

the other hand, all the above mentioned two-phase studies are formacro-sized circular tubes and to the best of the knowledge of theauthors there is no such study for microchannels. So, this paperaims to study the nanofluid laminar forced convection in a deeprectangular (parallel plate) microchannel with isothermally heatedwalls, using the Eulerian–Eulerian two-phase model. To do this,mass, momentum and energy conservation equations for bothphases are solved with the iterative numerical methods. Thetwo-phase results are compared with the single-phase results fromthe literature and then the nanoparticle size, nanoparticle concen-tration and Reynolds number effects on the nanofluid heat transferbehavior are studied. Also, the relative velocity and temperaturebetween the phases and the particle volume concentration distri-bution in the field are investigated. To the best knowledge of theauthors, this is the first paper reporting the detailed two-phasenanofluid modeling in a microchannel that considers differentvelocity and temperatures for the phases.

2. Governing equations

The geometry of the present problem is shown in Fig. 1. It con-sists of a parallel plate microchannel with height 200 lm and thelength L is 100 times larger than the height (L/H = 100). The originof the Cartesian coordinate system is considered to be at the platesymmetry axis and only the top half of the channel is used fornumerical simulation. In this problem laminar nanofluid flow thatis a mixture of water and copper nanoparticles enters the channelwith a uniform velocity and temperature and exchanges heat withthe isothermal microchannel walls.

Considering the laminar, steady state and two-dimensionalEulerian–Eulerian two-phase model for the nanofluid, the govern-ing mass, momentum and energy equations for the particle andbase liquid phases can be written as follows (Fluent user’s guide,2006; Hao and Tao, 2004).

2.1. Continuity equations

@ðulqlulÞ@x

þ @ðulqlv lÞ@y

¼ 0 ð1Þ

@ðupqpupÞ@x

þ@ðupqpvpÞ

@y¼ 0 ð2Þ

where x, y, u, v, q and u are axial and vertical direction, axial andvertical velocity, density and volume concentration, respectively.

Fig. 1. Geometry of the isothermally-heated parallel plate microchannel with length L anleaves the channel while it is fully developed.

Also, subscripts l and p stand for liquid and particle phases,respectively. According to the volume concentration definition wehave

ul þup ¼ 1 ð3Þ

2.2. Momentum equations

Momentum equations in the x-direction are

@ðulqlululÞ@x

þ @ðulqlv lulÞ@y

¼ �ul@p@xþ @

@xulll

@ul

@x

� �

þ @

@yulll

@ul

@y

� �þ ðFdÞx þ ðFvmÞx ð4Þ

and

@ðupqpupupÞ@x

þ@ðupqpvpupÞ

@y¼ �up

@p@xþ @

@xuplp

@up

@x

� �

þ @

@yuplp

@up

@y

� �þ ðFcolÞx

� ðFdÞx � ðFvmÞx ð5Þ

Here, p, l, (Fd)x, (Fvm)x and (Fcol)x are the pressure, viscosity, drag,virtual mass (added mass) and particle–particle interaction forcesin the x-direction, respectively. Due to the very small size of thenanoparticles, the lift force between the phases is neglected inthe present study.

Momentum equations in the y-direction can be written asfollows:

@ðulqlulv lÞ@x

þ @ðulqlv lv lÞ@y

¼ �ul@p@yþ @

@xulll

@v l

@x

� �

þ @

@yulll

@v l

@y

� �þ ðFdÞy þ ðFvmÞy ð6Þ

@ðupqpupvpÞ@x

þ@ðupqpvpvpÞ

@y¼ �up

@p@yþ @

@xuplp

@vp

@x

� �

þ @

@yuplp

@vp

@y

� �þ ðFcolÞy

� ðFdÞy � ðFvmÞy ð7Þ

where (Fd)y, (Fvm)y and (Fcol)y are the drag, virtual mass (added mass)and particle–particle interaction forces in the y-direction, respec-tively. Due to the small size of the channel, the gravitational forceis neglected in the present study.

d height H. Nanofluid enters the channel with uniform velocity and temperature and

110 M. Kalteh et al. / International Journal of Heat and Fluid Flow 32 (2011) 107–116

The drag force between the phases is calculated as

Fd ¼ �b ~Vl � ~Vp

� �ð8Þ

where ~V is the velocity vector.The friction coefficient b is calculated according to the particle

volume concentration range. For very dilute two-phase flows withparticle diameter dp, the friction coefficient is (Syamlal and Gidas-pow, 1985)

b ¼ 34

Cdulð1�ulÞ

dpj~Vl � ~Vpju�2:65

l ð9Þ

Eq. (9) is valid for two-phase flows with ul > 0.8 and Cd is thedrag coefficient and its magnitude depends on the particle Rey-nolds number:

Cd ¼24Repð1þ 0:15Re0:697

p Þ Rep < 1000

0:44 Rep P 1000

(ð10Þ

where

Rep ¼ulqlj~Vl � ~Vpjdp

llð11Þ

The virtual mass force is proportional to the relative accelera-tion of the two phases and is written as (Drew and Lahey, 1993)

Fvm ¼ 0:5upqlDDtð~Vl � ~VpÞ ð12Þ

In Eq. (12) D is the material derivative, such that, for the steadystate case, the convective terms of the material derivative areretained.

The particle–particle interaction force is calculated as (Bouillardet al., 1989)

Fcol ¼ GðulÞ~rul ð13Þ

Here, G is the particle–particle interaction modulus and it is cal-culated as

G ¼ 1:0 expð�600½ul � 0:376�Þ ð14Þ

Eqs. (8)–(14) are not correlations developed for nano-sizedparticles. But, since there are no such correlations for nano-sizedparticles, it is assumed that it is reasonable to use them for nano-particles. Also, the importance of drag, virtual mass and particle–particle interaction forces in nanofluid will be discussed inSection 6.1.

2.3. Energy equations

Considering the base fluid and the particle phase as incompress-ible fluids, and neglecting the viscous dissipation and radiation, theenergy equation is written as

@

@xðulqlulcpl

TlÞ þ@

@yðulqlmlcpl

TlÞ

¼ @

@xulkeff ;l

@Tl

@x

� �þ @

@yulkeff ;l

@Tl

@y

� �� hvðTl � TpÞ ð15Þ

@

@xðupqpupcpp

TpÞ þ@

@yðupqpmpcpp

TpÞ

¼ @

@xupkeff ;p

@Tp

@x

� �þ @

@yupkeff ;p

@Tp

@y

� �þ hvðTl � TpÞ ð16Þ

where cp, T, keff and hv are the heat capacity at constant pressure,temperature, effective thermal conductivity and volumetric inter-phase heat transfer coefficient, respectively. For mono-dispersedspherical particles hv can be calculated from

hv ¼6ð1�ulÞ

dphp ð17Þ

where hp is the fluid–particle heat transfer coefficient that shouldbe calculated from empirical correlations. In the present study thefluid–particle heat transfer coefficient is calculated based on theWakao and Kaguei (1982):

Nup ¼hpdp

k1¼ 2þ 1:1Re0:6

p Pr13 ð18Þ

Here Pr is the base liquid Prandtl number.The effective thermal conductivities for liquid and particle

phases are estimated as (Kuipers et al., 1992)

keff ;1 ¼kb;l

ul; ð19Þ

keff ;p ¼kb;p

upð20Þ

where

kb;l ¼ 1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1�ulÞ

q� �kl ð21Þ

kb;p ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1�ulÞ

qðxAþ ½1�x�CÞkl ð22Þ

and

C ¼ 21� B

A

� � BðA� 1ÞA 1� B

A

� �2 lnAB

� �� ðB� 1Þ

1� BA

� � � Bþ 12

( )ð23Þ

with

B ¼ 1:25½1�ul�

ul

� �109

ð24Þ

For spherical particles one can use

A ¼ kp

kland x ¼ 7:26� 10�3 ð25Þ

Eqs. (18)–(25) are not correlations developed for nano-sizedparticles. But, due to the lack of such correlations for nano-sizedparticles, they are used in the present study.

2.4. Nusselt number definition

The Nusselt number is defined based on the temperature differ-ence between the microchannel wall and the nanofluid mean(bulk) temperature:

Nu ¼ ðhDhÞkl¼ q00Dh=klðTw � TmÞ ð26Þ

where h, Dh, q00 are the convective heat transfer coefficient, micro-channel hydraulic diameter and the wall convective heat transferflux, respectively. Also, subscripts w and m stand for wall and mean,respectively. The mean temperature for a two-phase fluid can becalculated from (Boulet and Moissette, 2002)

Tm ¼Pp

i¼1ðRR

qiuicpiTidAÞPpi¼1ðRR

qiuicpidAÞð27Þ

where the integration is performed on the channel cross section. Fortwo-phase flow according to Eqs. (15) and (16), wall convectiveheat transfer flux can be calculated as

q00 ¼ ulkeff ;l@Tl

@y

w

þupkeff ;p@Tp

@y

w

ð28Þ

M. Kalteh et al. / International Journal of Heat and Fluid Flow 32 (2011) 107–116 111

According to the local Nusselt number, the average Nusseltnumber is

Nu ¼ 1L

Z L

0Nudx ð29Þ

2.5. Non-dimensionalization

Before the solution, all the governing equations are converted tonon-dimensional form, using the following non-dimensionalparameters:

X ¼ xDh

; Y ¼ yDh

; Ui ¼ui

uin; Vi ¼

v i

uin; P ¼ p� pin

qlu2in

;

hi ¼Ti � Tin

Tw � Tinð30Þ

where i = l, p stands for the liquid and the particle phases.

2.6. Boundary conditions

Both phases enter the channel at the inlet with the same uni-form axial velocity that is specified according to the flow Reynoldsnumber. At the channel outlet, outflow velocity boundary condi-tion is assumed for both phases. For liquid molecules the mean freepath is 0.1–1 nm, so that the channel hydraulic diameter should besmaller than 1 lm to be in the slip region (Morini, 2005). There-fore, for both of the phases the no-slip boundary condition at thewalls is appropriate for the present study.

For thermal boundary conditions, it is assumed that the nano-fluid enters the channel with 293 K and the isothermal walls havea temperature of 303 K. For the channel outlet, the outflow bound-ary condition is considered for both phases.

Table 1Grid-independency study results for pure water flow (up = 0.0) and Re = 1500.

No. of grid points in x-direction

No. of grid points in y-direction

Average Nusselt numberat the wall

150 10 12.252300 20 12.155600 40 12.122250 15 12.182500 30 12.130

1000 60 12.116

Table 2Comparison between average Nusselt number results from numerical simulationsand Ebadian and Dong (1998) results for pure water flow at different Reynoldsnumbers.

Re Present study Ebadian and Dong (1998) Deviation (%)

20 7.66 7.621 0.5150 7.756 7.739 0.22

100 7.924 7.934 �0.13500 9.299 9.288 0.12

1000 10.843 10.638 1.931500 12.13 11.775 3.012000 13.238 12.775 3.62

3. Numerical method

The non-dimensional form of the mass, momentum and energyconservation equations for liquid and particle phases along withthe interphase correlations and boundary conditions are discret-ized using the finite volume method on the upper half of thechannel (Patankar, 1980; Versteeg and Malalasekera, 1995). Anon-uniform grid is employed in the computational domain. Thegrids are finer close to the wall and also in the channel entranceregion using the cosine weighting function for control volumelength and height. The power-law scheme (Patankar, 1980;Versteeg and Malalasekera, 1995) is used for the convection–diffusion term discretization. The set of the discretized equationsis solved iteratively using the line by line method (Patankar, 1980;Versteeg and Malalasekera, 1995) and for solving the pressure–velocity coupling the well-known SIMPLE (Semi-Implicit Methodfor Pressure-Linked Equations) algorithm of Patankar (1980) isused. To use the SIMPLE algorithm, the pressure-correction equa-tion is derived by combining the mass conservation equations forparticle and liquid phases. Also, for accelerating the convergenceof the SIMPLE algorithm, the under-relaxation for velocity andpressure is used. For convergence criteria, the sum of the scaledabsolute residual for every parameter (mass, velocity and temper-ature) in all control volumes is calculated. The sum of the scaledabsolute residuals for every parameter is restricted to be smallerthan 10�6. For convergence criteria up to 10�9, the calculatedaverage Nusselt number remains unchanged up to the third digitof the decimal. Thus, the convergence criterion is selected to be10�6 in the present study.

4. Grid-independence study

To check for the independency of the results from the numberof grid points used, a grid independency study is done by consider-ing the amount of the calculated average Nusselt number. To dothis, different numbers of grid points are used in the x- and y-direc-tions. The results are shown in Table 1, where the flow Reynoldsnumber is 1500. According to this study, the number of the gridpoints in x- and y-directions are considered 500 and 30 respec-tively in the present study.

5. Code validation

Due to the lack of experimental data for nanofluid flow in a par-allel plate microchannel, the calculated average Nusselt numbersfor the special case of pure water flow (up = 0.0) at different Rey-nolds numbers are compared to corresponding available data inthe literature in order to check the accuracy of the written com-puter code. For a single-phase fluid flow in an isothermally-heatedparallel plate channel, the average Nusselt number is calculated as(Ebadian and Dong, 1998)

Nu ¼ 7:55þ 0:024x��1:14

1þ 0:0358Pr0:17x��0:64ð31Þ

where x� ¼ xD�1h Re�1Pr�1 is the non-dimensional axial length.

Table 2 shows the calculated average Nusselt number in thenumerical simulation, the corresponding results from Eq. (31)and the difference between the two results for a wide range of Rey-nolds number. The results show that the agreement between thepresent numerical solution results and the existing solution fromEq. (31) is very good especially for smaller Reynolds numbers.According to Table 2 for lower Reynolds numbers the deviation isless than 1% and it is less than 4% for Re = 2000. After checkingfor the accuracy of the computer code, in the following section heattransfer and pressure drop results for different Reynolds numbers,nanoparticle diameters and concentrations are presented.

6. Results and discussion

Since in the present study the particle phase is considered as acontinuum, its viscosity lp has to be obtained. In fact, due to the

Table 3Sensitivity study of the average Nusselt num-ber on the particle phase viscosity for Re = 100,up = 0.01 and dp = 100 nm.

Particle viscosity(Pa s)

Average Nusseltnumber

0.01 11.6740.005 11.6750.002 11.6580.00138 11.6590.001 11.6480.0008 11.660.0002 11.6550.00005 11.6510.00001 11.663

Nanoparticle volume concentration

Ave

rage

Nus

selt

num

ber

enha

ncem

ent (

%)

0 0.01 0.02 0.03 0.04 0.050

5

10

15

20

25

30

35

40

Present StudyHomogeneous Model of Santra et al. (2009)

ReH=100, dp=100 nm

Fig. 2. Comparison of percentage enhancement in average Nusselt number withrespect to pure water flow for homogeneous and two-phase models at ReH = 100and different nanoparticle volume concentrations. Two-phase results show higherheat transfer enhancement in comparison to homogeneous results.

112 M. Kalteh et al. / International Journal of Heat and Fluid Flow 32 (2011) 107–116

lack of experimental data, the solid viscosity for a liquid–solid two-phase mixture is not available. So, for the first degree of approxi-mation the following method is adopted in the present study:The corresponding pressure drop and the average Nusselt numberof a highly dilute nanofluid with volumetric concentration of0.00001 (which is quite close to pure water), is compared with thatof pure water, for Re = 1500. Using the trial and error method, thevalue of viscosity in the solid phase of the highly dilute nanofluid ischanged up to a point where the pressure drop and the averageNusselt number of the highly dilute nanofluid and the pure waterare matched. In doing so, the viscosity of the solid phase reaches avalue of 1.38 � 10�3 Pa s. Under such conditions, the difference be-tween the pressure and the average Nusselt number of the highlydilute nanofluid and that of pure water are 0% and 1.1%, respec-tively. In order to investigate the effect of Reynolds number ofthe flow, the same method of comparison between the pressuredrop and the average Nusselt number of the highly dilute nanofluidand the pure water is made, under the condition of very low Rey-nolds number (equal to 100). Under these circumstances, the dif-ference between the pressure drop and the average Nusseltnumber correspond to 0.04% and 1.4%, respectively. It can be con-cluded that the viscosity of the solid phase is independent of theReynolds number. Also, Table 3 shows the sensitivity study of aver-age Nusselt number on the solid viscosity magnitude for Re = 100,up = 0.01 and dp = 100 nm. As can be seen from Table 3 changingthe solid viscosity, the amount of the average Nusselt numberchanges slightly. i.e., when the solid viscosity changes three ordersof magnitude (from 0.01 to 0.00001) the average Nusselt numberremains unchanged up to the first digit of the decimal. So, it canbe concluded that the results are not very sensitive to the solid vis-cosity and it is not critical to know the exact magnitude for particleviscosity. With these discussions, the particle viscosity is consid-ered to be 1.38 � 10�3 Pa s in the present study.

6.1. Importance of the terms in the momentum equation

There are three interphase forces in the momentum equation(i.e. drag, virtual mass and particle–particle interaction forces).Table 4 shows the average Nusselt number for Re = 300,

Table 4Effect of the different terms in the momentum equation on the average Nusselt num

Volume concentration(%)

Considering allterms

Neglecting the dragterm

Negterm

1 12.343 12.385 12.32 14.068 14.151 14.03 15.507 15.637 15.54 16.818 16.998 16.85 18.051 18.292 18.0

dp = 100 nm and different term conditions. According to Table 4virtual mass and particle–particle interaction forces do not haveany effect on the average Nusselt number. But, the drag force af-fects the average Nusselt number slightly. Also, the drag force ef-fect on the average Nusselt number increases with an increase inthe nanoparticle volume concentration. So, it is possible to neglectthe virtual mass and particle–particle interaction forces for nano-fluid in the mathematical modeling.

6.2. Comparison with homogeneous model results

In this section, the two-phase modeling results are compared tothe homogeneous modeling results of Santra et al. (2009). They re-ported average Nusselt numbers in different conditions for 100 nmcopper–water nanofluid flowing inside a parallel plate channel.They used the channel height as a characteristic length scale to de-fine Reynolds number (i.e., ReH) and Nusselt numbers. Also, theyused the temperature difference between the wall and the inletfluid to define the heat transfer coefficient (in despite of Eq.(26)). So, to be able to compare the present two-phase results withthe homogeneous modeling results of Santra et al. (2009), all thepresent results in this section are based to their definitions. Fig. 2depicts the two-phase modeling results in comparison with thehomogeneous modeling results for ReH = 100 and 100 nm particles.It can be seen from Fig. 2 that the two-phase modeling resultsshow higher heat transfer enhancement in comparison to homoge-neous modeling results. Also, the heat transfer enhancement in-creases non-linearly with increase in nanoparticle volume

ber for Re = 300, dp = 100 nm and different volume concentrations.

lecting the particle–particle interaction Neglecting the virtual massterm

43 12.34368 14.06807 15.50718 16.81851 18.051

Table 5Percentage increase in average Nusselt number with respect to pure water flow for homogeneous model simulations of Santra et al. (2009) and two-phase models at differentReynolds numbers and nanoparticle volume concentrations. Two-phase results show higher heat transfer enhancement in comparison to the homogeneous modeling results.

up (%) ReH = 500 ReH = 1000 ReH = 1500

Present study Santra et al. (2009) Present study Santra et al. (2009) Present study Santra et al. (2009)

0.0 0.0 0.0 0.0 0.0 0.0 0.00.5 20.75917 3.21628 20.90864 3.26732 20.94575 3.303071.0 29.31589 6.40564 29.5805 6.51014 29.62227 6.58251.5 35.96397 9.56952 36.36213 9.72955 36.42431 9.838942.0 41.64701 12.70968 42.21522 12.9274 42.30332 13.074152.5 46.75102 15.82767 47.52114 16.10541 47.62851 16.289763.0 51.49046 18.92487 52.42912 19.26517 52.58449 19.487273.5 55.84388 22.00248 57.05521 22.40813 57.25646 22.668084.0 60.02573 25.06157 61.49892 25.53565 61.70122 25.833474.5 64.07892 28.10305 65.7105 28.64899 66.01818 28.984675.0 67.76753 31.12774 69.88891 31.74931 70.19313 32.12279

M. Kalteh et al. / International Journal of Heat and Fluid Flow 32 (2011) 107–116 113

concentration for two-phase modeling results, while it is almostlinear for homogeneous modeling results. For instance, for 0.05nanoparticle volume concentration, two-phase modeling shows32.6% heat transfer enhancement, while it is 19% for homogeneousmodeling results. For quantitative comparisons between the two-phase and homogeneous modeling results for different Reynoldsnumbers the data are collected in Table 5. The same behavior asFig. 2 can be seen for other Reynolds numbers in Table 5. Thisobservation also has been reported by previous two-phase model-ing studies (Behzadmehr et al., 2007; Bianco et al., 2009; Fard et al.,2010; Lotfi et al., 2010). One reason for this behavior can be relatedto the accuracy of the nanofluid thermophysical property modelsthat are used in the homogeneous modeling.

6.3. Effect of the nanoparticles on the velocity and temperature field

Two-phase simulation results show that the relative velocityand temperature between the phases is very small and negligible.For instance, for Re = 100 and 0.01 nanoparticle volume concentra-tion, the relative velocity in almost the full computation field is ofthe order of 10�6 or smaller and in a very small region close to theentrance it is of the order of 10�4. For the temperature difference,in almost the entire flow field it is of order 10�4 and in the verysmall region close to the channel entrance it is of order 10�2. Also,the results show that the nanoparticle volume concentration dis-tribution is uniform in the entire flow field. Such a result for nano-

0.50.9

1.2

.4

Y

0 10 200

0.25nanopa

0.5

1.21.3

Y

0 10 200

0.25base liq

0.50.9

1.2 .3

Y

0 10 200

0.25pure

Fig. 3. Velocity contour lines for pure water (bottom plot) and nanofluid (base liquidNanoparticles affect the velocity profile slightly and the relative velocity between the n

particle distribution also has been reported by Akbarinia and Laur(2009) for a curved tube. Thus, considering the nanofluid as ahomogeneous solution seems to be reasonable. Fig. 3 shows the ef-fect of nanoparticles on the velocity field for the Re = 1500 andup = 0.01 case. In Fig. 3 the bottom contour lines show the velocityfield for pure water flow in the computational domain, while themiddle and the top ones show the velocity field for base liquidand nanoparticle phases in the nanofluid. From Fig. 3 it can be seenthat the effect of nanoparticle on the velocity profile is very smalland it slightly increases the hydrodynamic entrance length. Thisbehavior is expectable due to very small size and concentrationof the nanoparticles. This is the reason for a small increase in pres-sure drop for a nanofluid in comparison to the pure water. Accord-ing to the velocity contour lines for base liquid and nanoparticlephases in the nanofluid in Fig. 3, it is clear that the relative velocitybetween the phases is very small and negligible.

Fig. 4 shows the effect of nanoparticles on the temperature fieldfor the Re = 1500 and up = 0.01 case. In this figure the temperaturecontour line for pure water (the bottom plot) is compared to tem-perature contour lines for base liquid (middle plot) and nanoparti-cle (top plot) phases in the nanofluid. According to Fig. 4nanoparticles increase the thermal boundary layer developmentconsiderably. This can be interpreted as increase in the thermalconductivity of the fluid due to the presence of the nanoparticlesand as a result an increase in the heat transfer rate betweenthe nanofluid and the microchannel wall. Also, the very small

1.3

X30 40 50

rticle phase

0.9

X30 40 50

uid phase

1.2

X30 40 50

water

phase in middle plot and nanoparticle phase in top plot) at Re = 1500, up = 0.01.anoparticles and base liquid is very small and negligible.

0.010.1

0.30.5

X

Y

0 10 20 30 40 500

0.25particle phase

0.01

0.01

0.1

10.3

X

Y

0 10 20 30 40 500

0.25base liquid phase

0.010.1 0.3

0.5

X

Y

0 10 20 30 40 500

0.25pure water

Fig. 4. Temperature contour lines for pure water (bottom plot) and nanofluid (base liquid phase in middle plot and nanoparticle phase in top plot) at Re = 1500, up = 0.01.Nanoparticles cause the temperature boundary layer to develop faster while the relative temperature between the nanoparticle and the base liquid phases is very small andnegligible.

240%1%2%

114 M. Kalteh et al. / International Journal of Heat and Fluid Flow 32 (2011) 107–116

difference between the temperature profiles for base fluid and thenanoparticle phases is clear in Fig. 4.

Ave

rage

Nus

selt

num

ber

12

14

16

18

20

223%4%5%

6.4. Effect of particle volume concentration on heat and fluid flow

Fig. 5 depicts the effect of varying the particle volume concen-tration on the pressure drop for 100 nm particles. As it can be seen,the pressure drop increases slightly with increase of the nanopar-ticle volume concentration for all Reynolds numbers. This corre-sponds to the results reported by previous studies (e.g. Junget al., 2009; Wu et al., 2009). This behavior is expectable sincethe nanoparticles have a very small effect on the velocity field aspresented in the previous section. For instance, for Re = 1000 andup = 0.01, the percentage increase in pressure drop in comparison

Re

Non

-dim

ensi

onal

pre

ssur

e dr

op

400 800 1200 1600

5

10

15

20

25

30

0%2%3%5%

Fig. 5. Non-dimensional pressure drop versus Reynolds number for differentnanoparticle volume concentrations (dp = 100 nm). The pressure drop increasesslightly with an increase of the nanoparticle volume concentration.

Re200 400 600 800 1000 1200 1400 16008

10

Fig. 6. Average Nusselt number ðNuÞ versus Reynolds number for differentnanoparticle volume concentrations (dp = 100 nm). The average Nusselt numberincreases with an increase of the Reynolds number and the nanoparticle volumeconcentration.

to pure water is 1.99%. This is one of the advantages of using nano-particles rather than millimeter or micrometer sized particles.

Fig. 6 shows the particle volume concentration and flow Rey-nolds number effect on the average Nusselt number for 100 nmparticles. It can be seen that the average Nusselt number increaseswith an increase in the nanoparticle volume concentration as wellas an increase in the flow Reynolds number. By comparing Figs. 5and 6 it is obvious that the increase in the Nusselt number is muchhigher than the increase in the pressure drop. So, this shows thatnanofluids can be used as efficient working fluids in cooling sys-tems. According to Fig. 6, increasing the Reynolds number andthe nanoparticle volume concentration increases the heat transferamount due to higher convection effects and higher nanoparticle

Re

Nu nf

/Nu pw

200 400 600 800 1000 1200 1400 16001.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

2.1

2.21%2%3%4%5%

Fig. 7. Nanofluid average Nusselt number normalized by the pure water averageNusselt number versus Reynolds number for different particle concentrations anddp = 100 nm. The normalized average Nusselt number decreases slightly with anincrease in the Reynolds number.

M. Kalteh et al. / International Journal of Heat and Fluid Flow 32 (2011) 107–116 115

participation in increasing the effective thermal conductivity of thenanofluids, respectively. This behavior is in agreement with theexperimental study of Wu et al. (2009) and the numerical studyof Li and Kleinstreuer (2008) for microchannels.

Fig. 7 shows the nanofluid average Nusselt number normalizedby the corresponding Nusselt number for pure water flow. Fromthe figure it can be seen that for every Reynolds number, an in-crease in nanoparticle volume concentration increases the averageNusselt number ratio. This behavior is expected since the highernanoparticle concentration increases the thermal conductivity ofthe nanofluid and thus causes an increase in the heat transfer rate.Also, Fig. 7 shows that for the same nanoparticle volume concen-tration, with an increase in Reynolds number, the average Nusseltnumber ratio decreases slightly. In other words, the effect of a

Nanoparticle volume concentration

Ave

rage

Nus

selt

num

ber

enha

ncem

ent (

%)

0.01 0.02 0.03 0.04 0.0540

50

60

70

80

90

100

110

120

100 nm70 nm50 nm30 nm

Fig. 8. Average Nusselt number enhancement in comparison to pure water flowversus nanoparticle volume concentration for different nanoparticle diameters andRe = 500. The effect of the size of the nanoparticles is more pronounced for highernanoparticle volume concentrations.

specified amount of nanoparticles (constant nanoparticle volumeconcentration) on enhancing the heat transfer ratio is larger forthe lower Reynolds numbers. For instance, at up = 0.01, the averageNusselt number ratio is 1.45 and 1.34 for Re = 200 and Re = 1600,respectively.

6.5. Effect of the nanoparticle diameter on heat transfer

Fig. 8 depicts the nanoparticle size effect on the amount of theheat transfer enhancement in comparison to pure water flow, atdifferent volume concentrations and for Re = 500. It can be seenthat an increase of the volume concentration and a decrease ofthe nanoparticle diameter increases the heat transfer enhance-ment. According to Fig. 8, the increase in heat transfer enhance-ment with a decrease in the nanoparticle size is not verypronounced especially for lower volume concentrations, since theparticle sizes considered here are at the same order of magnitude.For instance, for a nanofluid with 0.01 nanoparticle volume con-centration, for 100 nm and 30 nm particle sizes, the enhancementin heat transfer is 40.43% and 42.47%, respectively. The increase inheat transfer rate with a decrease in particle size was also reportedby other researchers (Heris et al., 2007; Mirmasoumi and Behzad-mehr, 2008b; Anoop et al., 2009; Akbarinia and Laur, 2009). Anoopet al. (2009) reported an increase in heat transfer enhancementwith a decrease in particle size for experimental study of the alu-mina-water nanofluid flow in the developing region of a tube withnanoparticle diameters 45 nm and 150 nm. But, He et al. (2007)experimentally studied the TiO2 nanoparticle size effect on theheat transfer rate inside a tube. They performed the experimentsfor 95, 145 and 210 nm particles with 0.006 particle volume con-centrations. Their heat transfer rate results did not show sensitivityto the nanoparticle size. They suggested a particle migration effectcausing a depletion layer that is created due to migration of the lar-ger particles to the channel center to explain this behavior. How-ever, the continuum method used in the present study is notable to resolve a very small depletion layer of the order of the par-ticle diameter close to the wall and the results show a uniformnanoparticle volume concentration distribution in the computa-tional field.

7. Conclusions

Pressure drop and heat transfer due to copper–water nanofluidflow inside an isothermally-heated parallel plate microchannel isstudied numerically for a wide range of Reynolds numbers, nano-particle volume concentrations and nanoparticle diameters. To dothis, the nanofluid flow is modeled using the Eulerian two-fluidmodel. In this method, the difference between the velocity andtemperature for liquid and nanoparticle phases are consideredand the governing equations for both phases are solved numeri-cally using the finite volume method. It is observed that the rela-tive velocity and temperature for base liquid and nanoparticlephases are very small and negligible. Thus, the liquid and the nano-particles have almost the same velocity and temperature. Also, thenanoparticle volume concentration distribution is uniform in thecomputational domain. Therefore we conclude that consideringthe nanofluid as a homogeneous solution is reasonable.

The heat transfer enhancement results for two-phase modelingshow higher magnitudes in comparison to the homogeneous mod-eling results. Such an observation also reported by other researches(Behzadmehr et al., 2007; Bianco et al., 2009; Fard et al., 2010; Lotfiet al., 2010). Thus, under-estimation of the heat transfer enhance-ment by homogeneous modeling seems to be related to the insuf-ficient accuracy of the nanofluid thermophysical property modelsthat are used in the homogeneous modeling. Also, the pressure

116 M. Kalteh et al. / International Journal of Heat and Fluid Flow 32 (2011) 107–116

drop for nanofluids is slightly higher than the pressure drop for thepure water flow, while the average Nusselt number increases withincrease in the Reynolds number and particle volume concentra-tion. For the same nanoparticle volume concentration, the averageNusselt number ratio is higher for lower Reynolds numbers. Keep-ing all the other parameters constant, heat transfer enhancement ishigher for the nanofluids with smaller nanoparticle sizes. However,this effect is not very pronounced at low nanoparticle volume con-centrations. Also, the most important advantage of this method incomparison to homogenous modeling is that there is no need foreffective thermophysical models for the nanofluid.

Acknowledgements

The first author would like to acknowledge the Ministry of Sci-ence, Research and Technology of the Islamic Republic of Iran forfinancial support to perform the research at the Eindhoven Univer-sity of Technology. The authors also like to thank Anton Darhuberfor fruitful discussions.

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