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International Journal of Thermal Sciences 84 (2014) 347e357

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International Journal of Thermal Sciences

journal homepage: www.elsevier .com/locate/ i j ts

A two-phase theoretical study of Al2O3ewater nanofluid flow inside aconcentric pipe with heat generation/absorption

S.A. Moshizi a, A. Malvandi d, D.D. Ganji b, I. Pop c, *

a Young Researchers and Elite Club, Neyshabur Branch, Islamic Azad University, Neyshabur, Iranb Department of Mechanical Engineering, Babol University of Technology, Babol, Iranc Department of Mathematics, Babes-Bolyai University, 400084 Cluj-Napoca, Romaniad SuneAir Research Institute, Ferdowsi University of Mashhad, Mashhad, Iran

a r t i c l e i n f o

Article history:Received 13 December 2013Received in revised form23 April 2014Accepted 6 June 2014Available online

Keywords:NanofluidTwo phase mixtureConcentric tubeHeat generation/absorptionSlip condition

* Corresponding author.E-mail addresses: s.a.moshizi@aut.ac.ir (S.A. Moshi

malvandi@staff.um.ac.ir (A. Malvandi), popm.ioan@yacom (I. Pop).

http://dx.doi.org/10.1016/j.ijthermalsci.2014.06.0121290-0729/© 2014 Elsevier Masson SAS. All rights res

a b s t r a c t

Convective heat transfer and pressure drop characteristics of Al2O3ewater nanofluid inside a concentricpipe with constant heat flux boundary conditions at the both walls is investigated theoretically. Theemployed model for nanofluid includes the two-phase modified Buongiorno model that fully accounts forthe effects of nanoparticle volume fraction distribution. Due to the nanoparticlesmigration in the fluid, theno-slip condition of the fluidesolid interface at the pipe walls is abandoned in favor of a slip conditionwhich appropriately represents the non-equilibrium region near the interface. Governing equations weretransformed into a system of ordinary ones via the similarity variables and solved numerically. The effectsof heat generation/absorption s, slip parameter l, and heat flux ratio ε on nanoparticle volume fraction,velocity, temperature, heat transfer coefficient at bothwalls, and the dimensionless pressure gradient havebeen investigated in detail. The results obtained indicated that the nanoparticles move from the wall withhigher heating energy towards thewall with lower heating energy (along the temperature gradient) due tothe thermophoretic force. This non-uniform distribution of nanoparticles at the cross section of the pipe,pushes the peak of the axial velocity from thewall with lower heating energy towards thewall with higherheating energy. In addition, slip velocity at the pipe walls enhances heat transfer coefficient and increasethe dimensionless pressure gradient ratio. Moreover, the changes of the heat transfer coefficientenhancement in the case of heat generation ismuchmore that in the case of heat absorption, for lowvaluesof ratio of Brownian diffusivity to thermophoretic diffusivities NBT.

© 2014 Elsevier Masson SAS. All rights reserved.

1. Introduction

The issue of heat transfer in the engineering applications hasdrawn considerable attention from numerous researchers over thepast few years. However, the heat transfer capacity of conventionalfluids such as water, oil, ethylene glycol mixture, which have lowthermal conductivity, is not suitable for many engineering pro-cesses. In 1873, Maxwell [1] investigated the effects of grafting solidmicro-particles onto base fluid and found a thermal conductivityenhancement of the obtained mixture. Later, many researchersstudied the influence of solideliquid mixture on the possible heattransfer enhancement [2,3]. Nevertheless, the volume fraction of

zi), amirmalvandi@aut.ac.ir, a.hoo.co.uk, pop.ioan@hotmail.

erved.

micro-particle in the base fluid had disadvantages such as abrasion,fouling and further pressure drop. Consequently, they were not asuitable choice for heat transfer enhancement and the investigationcontinued for new heat transfer fluids. Hence, Masuda et al. [4]dispersed solid nanoparticles in pure fluid and studied thermody-namic properties of the mixture. They claimed that using the nano-scale particles could overcome the majority of micro-particles re-strictions. After that, Choi [5] investigated thermal conductivity offluids with nanoparticle and proposed the term “nanofluid” for themixture of nanoparticles suspended in liquid.

Convective heat transfer in nanofluids can be modeled on twotheories. The first assumes that the base fluid and nanoparticlemixture homogeneous. According to this viewpoint, there is no slipvelocity between the base fluid and nanoparticles. This assumptionwas used by many researchers; however, it was not able to foreseethe behavior of the Nusselt number with the added volume fractionof nanofluid accurately [6,7]. The second approach is to considernon-homogeneous nanofluid mixture where there is slip velocity

Nomenclature

cp specific heat of nanofluid (m2/s2 K)Dh ¼ 2(Ro � Ri) hydraulic diameter (m)DB Brownian diffusion coefficientDT thermophoresis diffusion coefficienth heat transfer coefficient (W/(m2 K))k thermal conductivity of nanofluid (W/(m K))kBO Boltzmann constant (¼1.3806488 � 10�23 m2 kg/

(s2 K))Nu Nusselt numberNBT ratio of Brownian and thermophoretic diffusivitiesp pressure (Pa)q00 surface heat flux (W/m2)Q0 dimensional heat generation or absorption coefficient

(W/(m3 K))r radial coordinate (m)R radius (m)T temperature (K)u axial velocity (m/s)

x, r cylindrical coordinate system (m)

Greek symbolsε heat flux ratio (q

00i =q

00o)

ɸ nanoparticle volume fractiong ratio of wall and fluid temperature difference to

absolute temperatureh similarity variablel slip parameterm dynamic viscosity of nanofluid (kg/m s)r density of nanofluid (kg/m3)s heat generation/absorption parameterz ratio of the inner to outer diameter

SubscriptsB bulk meanbf base fluidi inner wallo outer wall

S.A. Moshizi et al. / International Journal of Thermal Sciences 84 (2014) 347e357348

between nanoparticles and the base fluid. Buongiorno [8] showedthat homogeneous models tend to under predict the heat transfercoefficient. This is because the nanofluid mixture is considered asingle-phase fluid and the dispersion effect is assumed to beentirely trivial. He then classified slip mechanisms that can producea relative velocity between the nanoparticles and the base fluid, inaccordance with the following seven mechanisms: inertia, Brow-nian diffusion, thermophoresis, diffusiophoresis, Magnus effect,fluid drainage, and gravity. He claimed that only Brownian diffusionand thermophoresis are the important slip mechanisms in nano-fluids. Accordingly, he proposed a two-component four-equationnon-homogeneous equilibrium model for explaining convectivetransport in nanofluids. Many researchers have studied and thereported results on convective heat transfer in nanofluids aftertaking Buongiorno's model into consideration in different geome-tries. For example, Kuznetsov and Nield [9], Malvandi et al. [10],Grosan and Pop [11], Sheikholeslami et al. [12e19], Xu et al. [20]and Aziz et al. [21]. Many excellent reviews on this concept havebeen carried out by Trisaksri and Wongwises [22], Sarkar [23], andSundar and Singh [24] in recent years. Recently, Yang et al. [25]studied the fully developed forced convective heat transfer ofnanofluids in a concentric annulus by modifying the Buongiornomodel to fully explain the effects of nanoparticle volume fractiondistributions on continuity, momentum, and energy equations.However, their investigation employed a no-slip condition at theboundary. Later, Malvandi and Ganji [26,27] used the modifiedBuongiorno's model in order to investigate the aluminaewaternanofluid in a channel.

In view of several physical problems such as those dealing withchemical reactions and those concerned with dissociating fluids,the investigation of heat generation/absorption in moving fluidsplays an important role on the fluid flow and heat transfer rate.Possible heat generation/absorption effects may alter the temper-ature distribution and therefore, the particle deposition rate thatmay occur in applications related to semi-conductor wafers, elec-tronic chips, and nuclear reactors.

In the majority of the above-mentioned studies, the analyseswere restricted to flow field and heat transfer with no slip condi-tion. The standard no slip boundary condition of classical fluidmechanics is not suitable for fluid flows at the micro and nano scale

and must be replaced by a boundary condition that allows somedegree of tangential slip. For the first time, Navier [28] describedthe phenomenon of the slip condition in which the component ofthe fluid velocity tangential to the surface is assumed proportionalto the tangential stress and the constant of proportionality is calledthe slip length [29]. Later, some studies [30e32] extended Navier'svelocity slip condition, so that this condition is well known [33,34]nowadays.

The aim of this paper is to study the fully developed forcedconvective heat transfer of nanofluids in a concentric annulus in thepresence of heat generation or absorption by modifying Buon-giorno's model and using that of Yang et al. [25]. Due to thenanoparticle migrations in the fluid, the no-slip condition of thefluidesolid interface at the walls of the pipe is ignored in favor ofthe velocity slip condition, which appropriately represents the non-equilibrium region near the interface. In addition, using the simi-larity variables, the governing partial differential equations weretransformed into a system of ordinary ones with a constraintparameter, and a solution was obtained using a reciprocal numer-ical algorithm. The effects of heat generation/absorption s and slipparameter l on nanoparticle volume fraction, velocity, tempera-ture, heat transfer coefficient, and pressure gradient have beeninvestigated in detail.

2. Governing equations

Consider the hydrodynamically and thermally fully developedAl2O3ewater nanofluid flow inside a straight annulus with heatgeneration/absorption, in which both walls are subjected to aconstant wall heat flux. The geometry and coordinate systems ofthe annulus is shown in Fig. 1, in which the inner and outer piperadii correspond to Ri and Ro, respectively. Here, the flow is drivenby the pressure gradient and the gravity is negligible. The two-dimensional coordinate frame was selected in which the x-axis isaligned horizontally and the r-axis is normal to the pipe walls.Buongiorno [8] considered the following assumptions: incom-pressible flow with negligible external forces, small viscous dissi-pation, negligible radiative heat transfer, local thermal equilibriumbetween nanoparticles and base fluid, no chemical reactions anddilute mixture. The two-phase modified Buongiorno's model [25]

Fig. 1. Physical model and coordinate system.

S.A. Moshizi et al. / International Journal of Thermal Sciences 84 (2014) 347e357 349

was used for the nanofluid, which is treated as a two-componentnon-homogeneous mixture, including the base fluid and nano-particles. The modified version includes the nanofluid density r inthe mass, momentum, and energy conservation equations. Since r

strongly depends on the nanoparticle volume fraction ɸ, themodified model entirely considers the effects of nanoparticle con-centration distribution. Hence, the basic incompressible conserva-tion equations of the mass, momentum, thermal energy andnanoparticle fraction can be expressed as

dpdx

¼ 1r

ddr

�mr

dudr

�(1)

rcpuvTvx

¼ 1r

v

vr

�kr

vTvr

�þ Q0ðT � TBÞ (2)

v

vr

�DB

vf

vrþ DT

TvTvr

�¼ 0 (3)

where u, T and P represent the axial velocity, local temperature andpressure respectively. Furthermore, the Brownian diffusion coeffi-cient DB and thermophoretic diffusion coefficient DT are defined by

DB ¼ kBOT3pmbfdp

(4)

and

DT ¼ 0:26kbf

2kbf þ kp

mbfrbf

f (5)

respectively. kBO is the Boltzmann constant and dp is the nano-particle diameter which varies between 1 and 10 nm. Also, m, r, kand c describe the dynamic viscosity, density, thermal conductivityand specific heat capacity of nanofluid which depend on thenanoparticle volume fraction as follows:

m ¼ mbf

�1þ 39:11 fþ 533:9 f2

�(6a)

r ¼ frp þ ð1� fÞrbf (6b)

cp ¼ frpcpp þ ð1� fÞrbfcpbf

r(6c)

k ¼ kbf ð1þ 7:47fÞ (6d)

Eqs. (6a) and (6b) are the empirical equations correlated byBuongiorno [8] using the data of Pak and Cho [35], and the ther-mophysical properties of Al2O3 nanoparticle and base fluid (water)are also provided as follows [35]:

cpbf ¼ 4182 J.

kg Kð Þ (7a)

rbf ¼ 998:2 kg.m3 (7b)

kbf ¼ 0:597 W=ðm KÞ (7c)

mbf ¼ 9:93� 10�4 kg.

m sð Þ (7d)

cpp ¼ 773 J=ðkg KÞ (7e)

rp ¼ 3880 kg.m3 (7f)

kp ¼ 36 W�m Kð Þ (7g)

where the subscript bf and p denote the base fluid and nanoparticleproperties, respectively. To solve Eqs. (1)e(3) the followingboundary conditions must be considered

u ¼ Nm

r

dudr

�kvTvr

¼ q00i

DBvf

vrþ DT

TvTvr

¼ 0

9>>>>>>>=>>>>>>>;at r ¼ Ri ;

u ¼ �Nm

r

dudr

kvTvr

¼ q00o

DBvf

vrþ DT

TvTvr

¼ 0

9>>>>>>>=>>>>>>>;at r ¼ Ro

(8)

whereN is the slip velocity factor. Furthermore, the average value ofparameters required should be calculated over the annular cross-section by

4≡1A

ZA

4dA ¼ 1

p�R2o � R2i

� ZRo

Ri

2pr4dr (9)

and the bulk mean temperature is defined as

TB≡rcuTrcu

(10)

Due to the constant diffusionmass flux of nanofluid (Eq. (3)) andthe impermeable wall condition, everywhere in the annulus, theBrownian diffusion flux and thermophoretic diffusion flux iscanceled out (DBvɸ/vr ¼ (�DT/T) (vT/vr)). Then, by averaging energyequation (Eq. (2)) from r ¼ Ri to Ro the following equation can beobtained

rcpuvTvx

¼ 1r

v

vr

�rk

vTvr

�þ Q0ðT � TBÞ (11)

After substituting Eqs. (8)e(10) into Eq. (11), it can be expressedas

Fig. 2. Comparison of fully developed velocity inside a concentric annulus for ɸB ¼ 0,l ¼ 0 and s ¼ 0 with analytical results of Kays et al. [36].

S.A. Moshizi et al. / International Journal of Thermal Sciences 84 (2014) 347e357350

rcpudTBdx

¼ 2�R2o � R2i

��Roq00o þ Riq

00i

�þ Q0ðT � TBÞ (12)

which is implemented to simplify the energy equation (Eq. (2))according to the thermally fully developed condition for the uni-form wall heat flux as follows

dTdx

¼ dTBdx

(13)

Introducing the following non-dimensional parameters

u* ¼ u

ð�dp=dxÞR2o.mo

; T* ¼ T � TB

q00oRo.ko

; g ¼ q00oRokoTB

z ¼ RiRo

; h ¼ 1� rRo

; NBT ¼DBokoTofo

DToRoq00o

; s ¼ Q0R2o

kbf; ε ¼ q

00i

q00o

(14)

Fig. 3. Comparison of the nanoparticle volume fraction, velocity and temperature profiles forN

Eq. (12) can be rewritten as00 00

rcpudTBdx

¼ 2qoRo�1� z2

� ð1þ zεÞ þ RoQ0qoko

T* (15)

Finally, Eqs. (1)e(3) can be reduced as

d2u*

dh2¼�

11� h

� 1m

dmdf

dfdh

�du*

dh� mo

m(16)

d2T*

dh2¼� ko

k

24� rcpu*

rcpu*

0@ 21� z2

ð1þ zεÞ þ sT*

1þ 7:47fo

1A

þ T*s1þ 7:47fo

þ�

7:471þ 7:47fo

dfdh

� k=ko1� h

�dT*

dh

� (17)

vf

vh¼ � f

NBTð1þ gT*Þ2�1þ gT*o

�2vT*vh

(18)

with the boundary conditions

u* ¼ l

forp

.rbf þ ð1� foÞ

du*

dh

vT*

vh¼ �1

f ¼ fo

9>>>>>>=>>>>>>;

at h ¼ 0 ðouter wallÞ

(19a)

u* ¼ � l

4irp

.rbf þ 1� 4ið Þ

du*

dh

vT*

vh¼ ε

koki

9>>>=>>>;

at h ¼ 1� z inner wallð Þ

(19b)

where the slip parameter l, the bulk mean dimensionless tem-perature T*B and the bulk mean nanoparticle volume fraction ɸB canbe obtained as

T*B ¼ rcu*T*

rcu*; fB ¼ u*f

u*; l ¼ N

mbfRorbf

(20)

BT¼ 0.2, ɸB¼ 0.02, s¼ 0, l¼ 0, ε¼ 0, and g¼ 0with the reported results of Yang et al. [25].

Fig. 4. Comparison of bulk Nusselt number inside a concentric annulus for ε ¼ 0,ɸB ¼ 0.02, l ¼ 0.01 and s ¼ 0 with numerical results of Malvandi et al. [10].

Table 1Comparison of the fully developed Nusselt number at the inner and outer wall withprevious results presented by Kays et al. [36].

ε ¼ q00i

q00oKays et al. [36] Present work

Nui Nuo Nui Nuo

0 0 5.036 0 5.0361 13.111 6.417 13.117 6.4232 8.401 8.842 8.403 8.8663 7.503 14.215 7.504 14.3074 7.122 36.223 7.122 37.029

S.A. Moshizi et al. / International Journal of Thermal Sciences 84 (2014) 347e357 351

Note that the average value 4 over the annular cross-section (Eq.(9)) can be simplified as

4 ¼ 2�1� z2

� Z1�z

0

ð1� hÞ 4dh (21)

The Nusselt number at the inner wall is defined as

Fig. 5. Effects of bulk nanofluid volume fraction ɸB on the profiles of flow characteristics in afraction; (b) temperature; (c) velocity profiles.

Nui ¼hiDh

ki¼ qiDh

ðTi � TBÞki¼ 2ð1� zÞ ε

T*ikiko

(22)

and, at the outer wall

Nuo ¼ hoDhko

¼ qoDhðTo � TBÞko

¼ 2ð1� zÞ 1T*o

(23)

The Nusselt number based on the thermal conductivity of basefluid can be defined as the non-dimensional heat transfer coeffi-cient by

HTC ¼ hDhkbf

¼ Nukkbf

¼ Nuð1þ 7:47fÞ (24)

So, the heat transfer coefficient enhancement may be evaluatedas

hhbf

¼ HTCHTCbf

(25)

The engineering quantity of interest is the dimensionless pres-sure gradient that can be defined as

Ndp ¼ �dpdx

, mbf

uBð2RoÞ2

!¼ 4rB

ru*�mbf

.mo

� (26)

3. Results and discussions

The system of equations (16)e(18) with the boundary condi-tions (19) represent a system of nonlinear ordinary differentialequations which have been solved numerically via the Run-geeKuttaeFehlberg scheme. The solutions were obtained bydifferent initial guesses for the missing values of du*/dh (0), T* andrcu*. In practice, the bulk mean particle volume fraction ɸB is pre-scribed rather than that at the wall ɸo. As a consequence, a recip-rocal algorithm was needed as well to obtain the appropriate ɸo toreach a required ɸB. The method is to treat these terms as certainvalues that should be determined in advance, and then an extraiterative loop in the program has been applied to find these certainvalues in a way that satisfies the Farfield conditions. The Run-geeKuttaeFehlberg scheme as a standard integration scheme wasused to determine the distributions of the velocity and temperatureprofiles. In order to avoid the grid dependency, the integration stephas been altered from 10�5 to 10�6 and no dependency was

concentric annulus with NBT ¼ 0.3, s ¼ �5, ε ¼ 1, and l ¼ 0.01: (a) nanoparticle volume

Fig. 6. Effects of heat generation/absorption s on the profiles of flow characteristics in a concentric annulus with NBT ¼ 0.3, ɸB ¼ 0.02, l ¼ 0.01, and ε ¼ 1: (a) nanoparticle volumefraction; (b) temperature; (c) velocity profiles.

Fig. 7. Effects of slip parameter l on the velocity, temperature, and concentration profiles in a concentric annulus with NBT ¼ 0.3, ɸB ¼ 0.02, s ¼ �5, and ε ¼ 1 for aluminaewater: (a)nanoparticle volume fraction; (b) temperature; (c) velocity profiles.

S.A. Moshizi et al. / International Journal of Thermal Sciences 84 (2014) 347e357352

observed. Furthermore, the ratio of Brownian diffusivity to ther-mophoretic NBT with dpy10 nm and fBy0:1 can be varied over awide range of 0.1e10. Therefore, the results have been carried outfor different values of NBT, s, l, ε, ɸB and with z ¼ 0.5 and g ¼ 0.01.

Fig. 2 shows the comparison of the results obtained fordimensionless fully developed velocity profile with the

Fig. 8. Effects of heat flux ratio ε on the velocity, temperature, and concentration profiles in a(a) nanoparticle volume fraction; (b) temperature; (c) velocity profiles.

corresponding analytical results of Kays et al. [36]. As it is shownthe predicted velocity profile for pure fluid is in good agreementwith the analytical results. In addition, the accuracy of the presentresults for the nanoparticle volume fraction, velocity and temper-ature profiles with the study of Yang et al. [25] have been shown inFig. 3. Further, to validate the slip condition part of the present

concentric annulus with NBT ¼ 0.3, ɸB ¼ 0.02, s ¼ �5, and l ¼ 0.01 for aluminaewater:

Fig. 9. Effects of NBT on the nanoparticle volume fraction profile in a concentricannulus with ɸB ¼ 0.02, s ¼ �5, l ¼ 0.01, and ε ¼ 1.

Fig. 10. Effects of heat flux ratio ε on the heat transfer coefficient ratio at both wall and dil ¼ 0.01: (a) hi/hi(bf); (b) ho/ho(bf); (c) Ndp/Ndp(bf).

S.A. Moshizi et al. / International Journal of Thermal Sciences 84 (2014) 347e357 353

results, NuB ¼ 2/(To*kB/ko) for different values of NBT was comparedwith the reported data of Malvandi et al. [10] as indicated in Fig. 4.Evidently, the numerical results are in the best accuracy. Further-more, in order to confirm the precision of the thermal boundaryconditions, fully developed values of the Nusselt numbers at innerand outer walls for single phase flow are compared with the resultspresented by Kays et al. [36] at different values of ratio betweenheat flux at the inner and outer wall (ε ¼ q

00i =q

00o), as presented in

Table 1.Fig. 5aec illustrates the influence of bulk mean volume fraction

of nanoparticle for a range of ɸB¼ 0, 0.02, 0.04, 0.06, 0.08 and 0.1 onnanoparticle volume fraction ɸ/ɸB, temperatureT*=T*

B ¼ ðTw � TÞ=ðTw � TBÞ, and velocity profilesu=uB ¼ u*=ðru*=rBÞ and at which heat absorption s ¼ �5 existswith l¼ 0, ε¼1 and NBT ¼ 0.3. In this case, h¼ 0 corresponds to theouter wall, whereas the inner wall is placed in h ¼ 0.5. In Fig. 5a, asɸB increases, a downward trend for the volume fraction of nano-particles in the middle region of the flow is observable; however,this trend is vice versa at the walls. This rise in the nanoparticlevolume fraction near the outer wall is higher than that at the innerwall. Because, the ratio of inner wall heat flux to the outer wall heatflux is unit, the total heating energy at the outer wall is higher thanthat of at the inner wall. This rise in ɸB near the outer wall leads toan increase in the viscosity of the nanofluid in that region. Hence,the shear stress intensifies on the outer wall which results inshifting the peak of the axial velocity towards the higher heatedsurface (outer wall), as supported in Fig. 5c. This increase in the

mensionless nanofluid pressure gradient in a range of NBT for ɸB ¼ 0.02, s ¼ �5, and

Fig. 11. Effects of heat generation/absorption s on the heat transfer coefficient ratio at both wall and dimensionless nanofluid pressure gradient in a range of NBT for ɸB ¼ 0.02, ε ¼ 1,and l ¼ 0.01: (a) hi/hi(bf); (b) ho/ho(bf); (c) Ndp/Ndp(bf).

S.A. Moshizi et al. / International Journal of Thermal Sciences 84 (2014) 347e357354

dimensionless velocity near the walls forces the dimensionlesstemperature there to decrease, as shown in Fig. 5b.

Fig. 6aec shows the effects of the heat generation/absorptionparameter s for a range of s ¼ �10, �5, 0, 5 and 10 on the nano-particle volume fraction, temperature and velocity profiles whenl¼ 0, ɸB¼ 0.02, ε¼1 andNBT¼ 0.3. For a given heat flux ratio, whenthe heat absorption (s < 0) intensifies, more energy will be absor-bed by the fluid. Consequently, the dimensionless temperatureprofile becomes to be more uniform as supported by Fig. 6b. Losingheat from the system, the concentration of nanoparticles in thefluid becomes more uniform which is a result of thermophoreticforces, as shown by Fig. 6a. In case of heat generation, the oppositebehavior is experienced. Due to indirect effects of heat generation/absorption on the momentum equation (Eq. (16)), variations on thevelocity profiles are insignificant in a particular range of s (seeFig. 6c).

Because of the nanoparticle migrations in the fluid, the no-slipcondition of the fluidesolid interface is abandoned in favor of aslip condition which appropriately represents the non-equilibriumregion near the interface. The effects of slip parameter l for a rangeof l ¼ 0, 0.01, 0.02, 0.04, and 0.08 on nanoparticle volume fraction,temperature and velocity profiles with ɸB ¼ 0.02, s ¼ �5, ε ¼ 1 andNBT¼ 0.3are presented in Fig. 7aec. The slip parameter signifies theamount of slip velocity at the surface. It is obvious that the velocitynear the pipe walls goes up due to increasing slip velocity, and alsothe velocity profile peak declines, as shown by Fig. 7c. An increasein the velocity close to the surfaces leads to an upward trend in the

nanoparticle volume fraction. In fact, the momentum in the coreregion moves away toward the pipe walls and leads to a highernanoparticle concentration there and a decrease in temperatureprofiles as fully supported by Fig. 7a and b.

In order to present the effect of different heat fluxes ratio at theinner and outer walls, assuming l ¼ 0.01, ɸB ¼ 0.02, s ¼ �5, andNBT ¼ 0.3, nanoparticle volume fraction, velocity and temperatureprofiles are presented in Fig. 8aec, respectively. As expected, havingincreased the ratio of heat flux at the inner wall to the outer wall,temperature gradients near the inner wall become steeper, asshown in Fig. 8b. This result leads to the conclusion that velocitynear the inner wall takes a decreasing trend with increased thermalenergy at the inner wall (see Fig. 8c). It is known that as ε reachesunity, the values of temperature at the inner and outer walls are notequal since the imposed heating energy at these two walls aredifferent. It is worth noting that nanoparticle volume fractiongradient is proportional to the temperature gradient with oppositesign (Eq. (18)), as supported in Fig. 8a.

It is interesting to note that the Brownian motion, which is thepresumably randommoving of particles suspended in a fluid, is onethe key heat transfer mechanisms in nanofluids. On the other hand,thermophoresis involves nanoparticle migration due to kinetictheory inwhich high energy molecules in awarmer region of liquidimpinge on the molecules with greater momentum than moleculesfrom a cold region [37]. So, there is a migration of particles from theouter wall (at h ¼ 0) to the inner wall (at h ¼ 0.5) due to thermo-phoretic force. Needless to say that at a constant heat flux on two

S.A. Moshizi et al. / International Journal of Thermal Sciences 84 (2014) 347e357 355

surfaces (ε ¼ 1), the one with higher area receives more thermalenergy. Hence, in this case, as Figs. 5a, 6a, 7a and 8a confirm, thenanoparticle concentration rises along the inner wall and declinesclose to the outer wall when NBT is fixed.

Increasing NBT enhances Brownian diffusion flux versus ther-mophoretic diffusion flux in the whole nanofluid domain, whichleads nanoparticles to scatter in the fluid and tends to have a moreuniform nanoparticle distribution (see Fig. 9). Consequently,increasing NBT causes heat transfer coefficient enhancement h/hbfand dimensionless pressure gradient parameter ratio Ndp=Ndp ðbfÞto reach asymptotically to a constant level, as shown in Figs. 10e12.Fig. 10aec shows the effect of heat fluxes ratio ε on the heat transfercoefficient enhancement at the inner wall (hi/hi (bf)), the outer wall(ho=ho ðbfÞ) and the dimensionless pressure gradient ratio(Ndp=Ndp ðbfÞ) in a wide range of NBT respectively. Obviously, thedimensionless temperature has an upward trend close to the innerwall and a downward trend near the outer wall when ε increases.This effect on the dimensionless temperature could have reverseeffect on the heat transfer coefficient enhancement at bothwalls fora particular NBT, (see Fig. 10a and b). This arises from the definitionof convective heat transfer coefficient that inversely depends on thedimensionless temperature. It is interesting to note that addingnanoparticles to the pure fluid increases Ndp which is a result of thefact that the pressure drops of nanofluids for given uB are higherthan those of the pure fluid due to the viscosity increase near thewalls, as adding the particles (see Fig. 5a). Moreover, increasing theheat flux ratio decreases the overall nanoparticle volume fraction at

Fig. 12. Effects of heat generation/absorption s on the heat transfer coefficient ratio at both wNBT ¼ 0.3, ε ¼ 1, and ɸB ¼ 0.02: (a) hi/hi(bf); (b) ho/ho(bf); (c) Ndp/Ndp(bf).

both walls as well as the overall viscosity there which is shown inFig. 8a; therefore, Ndp=Ndpðbf Þ has a downward trend with ε.

Fig.11aec studies hi=hi ðbfÞ, ho=ho ðbfÞ andNdp=Ndp ðbfÞ at differentvalues of heat generation/absorption s versus NBT. Clearly, for anidentical imposed heat flux at two walls, intensifying heat absorp-tion decreases the dependence of h=hðbfÞ at walls on the size ofnanoparticle (NBTf1=dp). As illustrated in Fig. 6c, increasing theabsorption of heat results in an increase in the dimensionless tem-perature at inner wall and a decrease at outer wall; thus, the heattransfer coefficient of nanofluid andpurefluiddecreases at the innerwall and increases at the outerwall simultaneously (see Tables 2e4),whereas the ratio between the two mentioned terms experiences areverse trend based on Fig.11a and b. It ismore apparent that for thelow values of NBT, the variations of h/h(bf)in the case of heat ab-sorption are smaller than that of in the case of heat generation.Furthermore, regarding Fig. 6a, increasing the net accumulation ofthe nanoparticles at the two walls causes a slight rise in total vis-cosity near thewalls, as heat absorption grows (s<0). Consequently,heat absorption climbs up and the values of the pressure drop innanofluid declines a trifle, as presented inTables 3 and 4 and Fig.11c.In the case of heat generation, the opposite behavior is observed (seeTables 2 and 3 and Fig. 11c). For the case of pure fluid, ɸB ¼ 0, themomentum equation (Eq. (16)) is mathematically decoupled to thethermal energy and mass equations (Eqs. (17) and (18)) and so, thehydrodynamical part of the equations (i.e. Eq. (16)) is not dependenton the temperature; Consequently, Ndp is independent of the valueof s, as presented in Tables 2e4.

all and dimensionless nanofluid pressure gradient in a range of the slip parameter l for

Table 2Effects of slip parameter l on HTC and Ndp for nanofluid and base fluid in the case ofheat generation.

l s ¼ 5

HTCi HTCo Ndp

Nanofluid Base fluid Nanofluid Base fluid Nanofluid Base fluid

0 15.486 13.827 7.041 6.038 336.770 190.5200.01 16.466 14.748 7.286 6.235 300.722 170.0140.02 17.299 15.543 7.492 6.402 271.675 153.5090.04 18.632 16.827 7.818 6.674 227.740 128.5760.08 20.417 18.558 8.266 7.059 172.160 97.099

Table 3Effects of slip parameter l on HTC and Ndp for nanofluid and base fluid without theheat source.

l s ¼ 0

HTCi HTCo Ndp

Nanofluid Base fluid Nanofluid Base fluid Nanofluid Base fluid

0 14.871 13.111 7.402 6.417 337.873 190.5200.01 15.745 13.902 7.653 6.621 301.673 170.0140.02 16.487 14.581 7.864 6.795 272.514 153.5090.04 17.667 15.674 8.199 7.077 228.426 128.5760.08 19.247 17.147 8.662 7.477 172.666 97.099

S.A. Moshizi et al. / International Journal of Thermal Sciences 84 (2014) 347e357356

Finally, the effects of the slip condition on the heat transfercoefficient enhancement and the dimensionless pressure gradientratio in the existence of considering heat generation/absorption arepresented in Fig. 12a to c, respectively. Increasing l, for a prescribeds, increases the heat transfer coefficient of nanofluid and pure fluidat the walls simultaneously, because of a downward trend in thedimensional temperature near two walls, as indicated inTables 2e4. This is due to the momentum intensification close tothe pipe walls. However, the ratio of heat transfer coefficient be-tween nanofluid and pure fluid have different trend, as shown inFig. 12a and b. Additionally, Fig. 12a shows that in the case of heatgeneration (s ¼ 10), hi/hi(bf) experiences a swift decline-around 3%-as l increases from 0 to 0.08, whereas this parameter goes down byabout 0.3% for heat absorption (s ¼ �10). Therefore, it can beconcluded that a decrease in hi=hi ðbfÞ for the case of heat genera-tion is more than the case of heat absorption. Considering the nextfigure, it is clear that ho=hoðbfÞ grows monotonously as l increasesfor a prescribed s. Also, Fig. 12c indicates that at a constant heatgeneration/absorption parameter, Ndp=Ndp ðbfÞ increases slightly byalmost 0.3%, as l rises from 0 to 0.08.

4. Summary and conclusions

In this paper, forced convective heat transfer of Al2O3ewaternanofluid inside a horizontal annulus is investigated. The two-

Table 4Effects of slip parameter l on HTC and Ndp for nanofluid and base fluid in the case ofheat absorption.

l s ¼ �5

HTCi HTCo Ndp

Nanofluid Base fluid Nanofluid Base fluid Nanofluid Base fluid

0 14.488 12.699 7.742 6.767 338.934 190.5200.01 15.288 13.406 7.998 6.977 302.591 170.0140.02 15.966 14.012 8.213 7.157 273.324 153.5090.04 17.045 14.988 8.557 7.447 229.083 128.5760.08 18.485 16.302 9.032 7.860 173.156 97.099

phase modified Buongiorno model for nanofluid was used thatfully accounts for the effects of nanoparticle volume fraction dis-tribution. Due to the nanoparticle migrations in the fluid, the no-slip condition of the fluidesolid interface at the pipe walls isabandoned in favor of a slip condition which appropriately repre-sents the non-equilibrium region near the interface. The effects ofheat generation/absorption s, slip parameter l and heat flux ratio ε

on nanoparticle volume fraction, velocity, temperature, heattransfer coefficient enhancement, and dimensionless pressuregradient ratio have been investigated in more detail. The resultsobtained indicated that due to themophoretic force, the nano-particles accumulate in the middle region of the flow where thelower temperature exists. This non-uniform distribution of nano-particles at the cross section of the pipe, push the peak of the axialvelocity towards the pipe wall with higher heating energy. Also,

❖ Heat transfer coefficient enhancement at the inner wall andouter wall experiences a downward and an upward trend, asincreasing the heat flux ratio ε. This arises from definition ofconvective heat transfer coefficient that inversely depends onthe dimensionless temperature. Furthermore, a rise in ε leads toan increase in the dimensional pressure gradient ratioNdp=Ndp ðbfÞ.

❖ In the case of heat absorption, by imposing heat flux at the bothwalls, the dimensional temperature profile become to be moreuniform. The variations on the heat transfer coefficientenhancement in the case of heat absorption are smaller that incase of heat generation, for a moderate range of NBT. Further-more, the heat absorption boosts the pressure drops ofnanofluid.

❖ Slip parameter plays an important role in the heat transferaugmentation. The results show that despite the fact thatgrowing the slip parameter increases the heat transfer coeffi-cient of nanofluid and pure fluid at the both walls simulta-neously, the heat transfer coefficient ratio hi/hi(bf)have anopposite trend. In addition, Ndp=Ndp ðbfÞ goes up as the slipparameter increases.

❖ Increasing the ratio of Brownian and thermophoretic diffusiv-ities NBT, leads to a scatter distribution of nanoparticles in thefluid e more uniform distribution e ; therefore, h/h(bf)andNdp=Ndp ðbfÞ reach asymptotically to a constant level.

❖ The variations of heat transfer coefficient enhancement at theinner wall for the case of heat generation are more than the caseof heat absorption, whereas that at the outer wall is monoto-nously, as rising l. Also, for a prescribed s, slip condition causesNdp=Ndp ðbfÞ to increase.

Overall, these results demonstrate that the addition of the Al2O3nanoparticles to the water leads to higher heat transfer coefficientand dimensionless pressure gradient than the water itself.

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