Thermal performance analysis of hydromagnetic Al2O3-water nanofluid flows inside a concentric microannulus considering nanoparticle migration and asymmetric heating

International Journal of Thermal Sciences 109 (2016) 10e22

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Thermal performance analysis of hydromagnetic Al2O3-water nanofluid flows inside a concentric microannulus considering nanoparticle migration and asymmetric heating A. Malvandi a, *, Amirmahdi Ghasemi b, D.D. Ganji c a b c

Department of Mechanical Engineering, Neyshabur Branch, Islamic Azad University, Neyshabur, Iran Mechanical Engineering Department, Worcester Polytechnic Institute, Worcester, MA, USA Mechanical Engineering Department, Babol Noshirvani University of Technology, Babol, Iran

a r t i c l e i n f o

a b s t r a c t

Article history: Received 29 October 2015 Received in revised form 18 May 2016 Accepted 23 May 2016

Direction and intensity of nanoparticle migration are able to tune thermophysical properties of nanofluids to improve thermal performance of heat exchange equipment. In this paper, thermal performance of hydromagnetic alumina/water nanofluid inside a vertical microannular tube is investigated numerically considering different modes of nanoparticle migration. The model used for the nanofluid is able to consider nanoparticle migration originating from the thermophoresis and Brownian motion. In order to study the thermal performance, the figures of merit for different range of parameters including the ratio of Brownian motion to thermophoresis (NBT), slip parameter (l), mixed convective parameter (Nr), Hartmann number (Ha), bulk mean nanoparticle volume fraction (fB), radii (z) and heat flux (ε) ratios are investigated in detail. It is revealed that increasing the slip velocity and magnetic field strength intensify the thermal performance, whereas increasing the ratio of inner wall to the outer wall radius, nanoparticle volume fraction, and heat flux ratio decrease it. © 2016 Elsevier Masson SAS. All rights reserved.

Keywords: Nanofluid MHD Microannulus Scale analysis Nanoparticles migration Thermophoresis

1. Introduction Nanofluids are basically used as thermal enhancing agent and they are used extensively for cooling, such as heat exchangers, cooling of solar PV, microchannel, and multi-phase flow. Nanofluids flow in channels has many purposes in chemical processes, Tribological applications, medical applications - drug delivery, pharmaceutical applications, surfactant and coating. In 1873, the act of adding particles to the regular fluids such as water, oil, and ethylene-glycol to improve the thermal conductivity of mixture with solid particle was emerged by Maxwell [1]. Several researchers then studied the impact of fluid additives on possible heat transfer enhancement. But, there were some problems such as abrasion, fouling, clogging, and additional pressure loss of the system, which were unsuitable for heat transfer systems. In 1995, the term “nanofluid” was established by Choi [2] to introduce dilute suspensions formed by functionalized nanoparticles less than 100 nm in diameter created before by Masuda et al. [3] as alumina-

* Corresponding author. E-mail address: [email protected] (A. Malvandi). http://dx.doi.org/10.1016/j.ijthermalsci.2016.05.023 1290-0729/© 2016 Elsevier Masson SAS. All rights reserved.

water. These nanoparticles are quite close in size to the molecules of the base fluid and, therefore, can provide extremely stable suspensions with only minor gravitational settling over long periods. Thus, theoretical studies emerged to model the nanofluid behaviors. Up to now, various models are proposed for the determination of flow and thermal characteristics of nanofluids. The models, however, depend on certain inputs from experimentation. Whichever model gives close agreement with the experimental data is referred as the “Best model” by those authors. The determination of heat transfer coefficients, for instance, one have to model the nanofluid as whether single or two phase flow [4,5]. In 2006, Buongiorno [6] declared that the homogeneous flow models are in disagreement with the experimental observations and has a tendency to underestimate the nanofluid heat transfer coefficient, whereas the dispersion effect can be neglected because of the nanoparticle size. Therefore, Buongiorno developed a nanofluid model to explain the abnormal convective heat transfer enhancement in nanofluids and so eliminate the limitations of the homogeneous and dispersion models. He asserted that the anomalous heat transfer occurs due to particle migration in the fluid and proposed a two-component four-equation non-homogeneous equilibrium model for convective transport in nanofluids.

A. Malvandi et al. / International Journal of Thermal Sciences 109 (2016) 10e22

Nomenclature B cp d

DB DT h Ha HTC

k kBO Nr

NBT p q’’ R T u x,r

uniform magnetic field strength (T) specific heat (m2/s2K) nanoparticle diameter (m) Brownian diffusion coefficient thermophoresis diffusion coefficient heat transfer coefficient (W/m2.K) Hartmann number dimensionless heat transfer coefficient thermal conductivity (W/m.K) Boltzmann constant (¼ 1:3806488  1023 m2 kg=s2 K) mixed convective parameter due to nanoparticle distribution ratio of the Brownian to thermophoretic diffusivities pressure (Pa) surface heat flux (W/m2) radius (m) temperature (K) axial velocity (m/s) coordinate system

11

Greek symbols proportionality factor volumetric thermal expansion coefficient non-dimensional temperature nanoparticle volume fraction ratio of wall and fluid temperature difference to absolute temperature h transverse direction m dynamic viscosity (kg/m.s) r density (kg/m3) s electric conductivity (1/U.m) l slip parameter

a b q f g

Subscripts B bulk mean bf base fluid condition at the inner wall i condition at the outer wall o p nanoparticle Superscripts dimensionless variable

*

Buongiorno’s model then was modified by Yang et al. [7,8] to fully account for the effects of the nanoparticle volume fraction distribution. Malvandi et al. [9], then, extended their study to consider the mutual effects of buoyancy and nanoparticle migration for mixed convection of nanofluids in a vertical annular tubes. In another study, Malvandi and Ganji [10] studied the impacts of nanoparticle migration on alumina/water nanofluids in a parallelplate channel. They indicated that the motion of nanoparticles is from the adiabatic wall (nanoparticle depletion) to the cold wall (nanoparticle accumulation) which it leads to construction a nonuniform nanoparticle distribution. Hedayati and Domairry [11,12] investigated the effects of nanoparticle migration on titania/water nanofluids in horizontal and vertical channels. The modified Buongiorno’s model were applied to different heat transfer problems including forced convection [13,14], mixed convection [15], and free convection [16]. The results revealed that the modified model is completely appropriate for considering the effects of nanoparticle migration in nanofluids. Investigation on the effects of magnetic field has important applications in medicine, physics and engineering. Many industrial equipments, like pumps, MHD generators, bearings and boundary layer control are influenced by the interaction between the electrically conducting fluid and a uniform magnetic field. The flow behavior strongly relies on the intensity and orientation of the exerted magnetic field. The applied magnetic field manipulates suspended nanoparticles and rearranges their volume fraction in the fluid which completely changes heat transfer characteristics of
n the flow. The basic study on MHD flows was investigated by Alfve who won the Nobel Prize for his works. Then, a unique investigation on MHD flow in a channel was studied by Hartman. Later, several researchers have emphasized this and their details are exist in literature such as [17e32]. In addition, slip condition at the surface is a suitable assumption for fluid flow and heat transfer microchannels. In fact, because of surface roughness in microchannels, there is no longer adhesion of working fluid at the fluidsolid interface which results in slip velocity at the surface. Slip velocity, as a result, is considered by many authors in different heat transfer system, some of which are [33,34]. A complete review on

heat and mass transfer of nanofluids in microchannel is conducted by Salman et al. [35]. The current investigation is motivated by the need to evaluate the detailed behavior of nanoparticle migration inside a vertical microannulus in the presence of a uniform magnetic field. The nanoparticle volume fraction distribution is obtained considering the nanoparticle fluxes originating from the Brownian diffusion and thermophoresis. As the thermophoresis is the key mechanism of the nanoparticle migration, different temperature gradient is imposed by different wall heat fluxes; q;;i for the inner wall and q;;o for the outer wall. A physical quantity of interest, figure of merit, is considered to examine the thermal performance of Al2O3 nanofluid inside a concentric microannular tube. In addition, the investigation is performed for the different modes of nanoparticle migration induced by asymmetric heating. The impact of a uniform magnetic field and different modes of nanoparticle migration on the thermal performance is of particular interest.

2. Nanoparticle transport equation In nanofluids, it is recognized that nanoparticles do not follow the fluid streamlines passively. In fact, there are some reasons that induce a slip velocity between the nanoparticles and the base fluid. Movements of nanoparticles has significant impact on rheological and thermophysical properties of the nanofluids. Therefore, investigating the nanoparticles motion is critical for evaluating the performance of nanoparticles inclusion to the base fluid as a heat transfer medium. Since the nanoparticles are very small (<100 nm), Brownian and thermophoretic diffusivities are the main slip mechanisms in nanofluids, as Buongiorno [6] declared. Brownian diffusion is due to random drifting of suspended nanoparticles in the base fluid which originates from continuous collisions among the nanoparticles and liquid molecules. Regarding Einstein-Stokes’s equation, Brownian diffusion is proportional to the nanoparticle volume fraction gradient and expressed by the Brownian diffusion coefficient, DB, given by Ref. [6].

12

DB ¼

A. Malvandi et al. / International Journal of Thermal Sciences 109 (2016) 10e22

kB T ; 3pmbf dp

(1)

where kB is Boltzmann’s constant, mbf dynamic viscosity of the base fluid, T temperature and dp nanoparticle diameter. The nanoparticle flux because of Brownian diffusion (J) can be written as

Jp;B ¼ r p DB Vf:

(2)

Thermophoresis induces nanoparticle migration from warmer to colder region (in opposite direction of the temperature gradient), making a non-uniform nanoparticle volume fraction distribution. The thermophoresis is expressed by the thermal diffusion, DT, given by Ref. [6].

DT ¼ a

mbf f; rbf

(3) k

where a ¼ 0:26 2k bfþk . The nanoparticle flux due to thermophop bf resis (J) can be given by

Jp;T ¼ r p DT

VT : T

(4)

Thus, the total nanoparticle flux consists of two parts as described above


VT : Jp ¼ Jp;B þ Jp;T ¼ r p DB Vf þ DT T

(5)

Since DB~T and DT~f (depend on the flow field), it is beneficial to re-write Eq. (5) as [36].


VT ; Jp ¼ r p CB TVf þ CT f T

(6)

where CB ¼ DB/T and CT ¼ DT/f are independent to temperature and nanoparticle volume fraction. As a result, nanoparticle volume fraction distribution can be obtained via [6].

vt ðfÞ þ V$ðufÞ ¼ 


1   VT : V Jp ¼ V$ CB TVf þ CT f T rp

(7)

Eq. (7) represents the nanoparticle volume fraction equations and will be coupled with the governing equations in section 3.

Fig. 1. The geometry of physical model and coordinate system.

vt ðrÞ þ V$ðruÞ ¼ 0;

(8)

h vt ðruÞ þ V$ðruuÞ ¼ Vp þ V$t  sB2 u þ ð1  fB Þrbf bðT  TB Þ  i   rp  rbf ðf  fB Þ g;

3. Problem formulation and governing equations Fig. 1 illustrates the schematic of laminar two-dimensional flow of Al2O3-water nanofluid inside vertical microannuli, subjected to 00 00 different heat fluxes at inner (qi ) and outer (qo ) walls. A twodimensional Cartesian coordinate frame has been chosen in which x-axis is aligned vertically and r-axis is normal to walls. Due to non-adherence of fluid-solid interface because of microscopic roughness inside microchannels, flow will have a slip velocity at surface walls. To consider nanoparticle migration, the modified Buongiorno’s two-component heterogeneous mixture model [9] is used for the nanofluid, as described before (in section 2). From the numerical outcomes of Koo and Kleinstreuer [37], for the most regular nanofluids inside a channel about 50 mm, the viscous dissipation can be neglected and ohmic heating and Hall effects can be assumed to be small. Therefore, the basic conservation equations of the mass, momentum, and thermal energy can be written as [6,7]:

(9) vt ðrcTÞ þ V$ðrcuTÞ ¼ V$q þ hp V$Jp ;

(10)

where hp is the specific enthalpy of nanoparticles, t ¼ mðVu þ ðVuÞt Þ is the shear stress, b is the nanofluid thermal expansion, g is the gravity, and q is the energy flux with respect to the nanofluid velocity, which can be written as the sum of the conduction and diffusion heat flux as below:

q¼

kVT |fflfflffl{zfflfflffl} conduction heat flux

þ

hp Jp |ffl{zffl}

:

(11)

nanoparticle diffusion heat flux

In addition, r, m, k, c are the density, dynamic viscosity, thermal conductivity, and specific heat capacity of Al2O3-water nanofluid respectively, which depends on nanoparticle volume fraction:

A. Malvandi et al. / International Journal of Thermal Sciences 109 (2016) 10e22





13

m ¼ mbf 1 þ 39:11f þ 533:9f2 ;

evaluate coefficients of the governing equations. With regard to Eq. (13), the scale values can be written as

r ¼ frp þ ð1  fÞrbf ;

Dp

cp ¼

frp cpp þ ð1  fÞrbf cpbf

r

Lref

;

∝

(12)

mDU L2ref |ffl{zffl}

;



ð1  fB Þrbf g bDT ; |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}

72

6:11104



rp  rbf gDf ; 8:94103

sB2 U : |fflffl{zfflffl} 2104

k ¼ kbf ð1 þ 7:47fÞ;

b¼

frp bp þ ð1  fÞrbf bbf

r

(16) ;

where subscripts bf and p stand for base fluid and particle, respectively. Because the rheological and thermophysical properties (r, m, k and c) are dependent to the nanoparticle concentration, the nanoparticle distribution equation, Eq. (7) (see section 2), should be coupled with Eqs. (8)e(10). Therefore, substituting Eq. (11) into Eq. (10) and equating Vhp ¼ cpVT, the governing equations for steady, incompressible, hydrodynamically and thermally fully developed flow can be written as:


 1 d du dp h rm  þ ð1  fB Þrbf bðT  TB Þ r dr dr dx   i  rp  rbf ðf  fB Þ g  sB2 u

2:0861010

(14)

(15)

Equations (13)e(15) are able to shorten further using scale analysis. Our physical conditions is the ones that basically occur in the flow of Al2O3/water nanofluids in microannular tubes. Table 1 illustrates material and device properties of regular water based nanofluids with alumina nanoparticles, from which one can

Table 1 Device and material properties of alumina/water nanofluid. Variable name

Variable symbol

Value

Particle diameter Heat Flux Conductivity

dp (nm) q (kW) kp (W/(m K)) kbf (W/(m K)) rp (kg/m3) rbf (kg/m3) (cp)p (J/(kg K)) (cp)bf (J/(kg K)) mbf (Ns/m2) kB (J/K) g (m/s2)

20 5000 36 0.597 3880 998.2 773 4182 9.93  104 1.38  1023 9.82 0.1 8.4  106 2.07  104

Reference values Length scale Velocity Temperature Velocity difference Axial dimension difference Temperature difference Concentration difference

0

1

2:654103

2:77102

fB bp (1/K) bbf (1/K) L (mm) U (cm/s) T (K) DU (cm/s) Dx (cm) DT (K)

Df

(17) The scale analysis reveals that the LHS term has the same order of magnitude with the first term of RHS which both of them are about 1000 times higher than the second RHS term. In essence, heat transfer originates from nanoparticle diffusion (second RHS term) can be neglected relative to the other terms. Thus, governing equations (Eqs. (13)e(15)) can be written as:


  1 d du dp  r mðfÞ ¼ þ rp  rbf ðf  fB Þg þ sB2 u; r dr dr dx

3.1. Scale analysis of governing equations

Viscosity Boltzmann’s constant Gravity Bulk concentration Thermal expansion

2:0285108

(13)


 1 v vf CT f vT CB T þ ¼ 0: r vr vr T vr

Specific heat capacity

 U DT  ref fB rp cp þð1 fB Þrbf cbf ; Dx |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

B C B T Df DT C ref B C RHS : kbf ð1þ7:47fB Þ 2 ; rp cp B CB þCT fB C DT : Tref Lref C |{z} Lref Lref |ffl{zffl} B @ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 3106 |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} A 50


 
dT 1 d dT vf CT f vT vT ¼ rk þ rp cpp CB T þ ; dx r dr dr vr T vr vr

Density

LHS :

DT

¼ 0;

rcp u

With regard to Eq. (16) it can be stated that buoyancy effects because of temperature gradient (second RHS term) can be neglected relative to buoyancy effects because of nanoparticle distribution (third RHS term) and shear stress terms (first RHS term). Therefore, this term can be neglected from Eq. (13). But, the scale analysis of Eq. (14) yields to

50 20 300 15 1 50 0.3

rcp u


 dT 1 d dT ¼ rkðfÞ ; dx r dr dr

(18)

(19)


 1 v vf CT f vT CB T þ ¼ 0: r vr vr T vr

(20)

Due to non-adherence of the fluid-solid interface because of microscopic roughness in microannular tubes, the flow has a slip velocity at surface walls. Different heat fluxes have been considered at the walls, q;;i for the inner wall and q;;o at the outer one. Thus, appropriate boundary conditions for this problem can be expressed as [7].

r ¼ Ri : r ¼ Ro :

du vT ; ki ¼ q;;i ; dr vr du vT u ¼ N ; ko ¼ q;;o ; dr vr u¼N

vf CT f vT þ ¼ 0; vr T vr vf CT f vT þ ¼ 0: CB T vr T vr

CB T

(21) where N represents slip velocity factor. It should be stated that the boundary condition for nanoaprticle volume fraction (CBTvf/ v rþCTf/TvT/v r ¼ 0) comes from no flux boundary condition for nanoparticle volume fraction at the walls [38]. Eq. (19) revealed that the energy equation for nanofluid is similar to the regular fluid. Consequently, the heat transfer in nanofluids only influenced by the rheological and thermophysical properties depending on nanoparticle distribution. With regard to the nanoparticle distribution equation, Eq. (20), and no flux for

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A. Malvandi et al. / International Journal of Thermal Sciences 109 (2016) 10e22

nanoparticle volume fraction at the walls (Eq. (21)), it is clear that the Brownian diffusion and thermophoretic diffusion are canceled out (CB/fvf/vr ¼ CT/T2vT/vr) everywhere (see also Ref. [7]), therefore, by averaging Eq. (19) from r ¼ Ri to Ro and considering the thermally fully developed condition for the uniform heat flux at the walls, dT/dx ¼ dTB/dx, and introducing the following nondimensional parameters:

h¼

2r ; u* ¼ Dh

D2h




u ddpx

mbf

T T

q ¼ ðqo þqi ÞDi ;

;

h

2


 f gðqw  qÞ : ¼ Ln fw NBT gð1 þ gqÞð1 þ gqw Þ

gðqw qÞ f ¼ eNBT gð1þgqÞð1þgqw Þ ; fw

q;; ε ¼ i;; ; qo

CB kbf TB fB ; CT Dh ðqo þ qi Þ

NBT ¼

Before solving the governing equations, it is beneficial to analyze them. From equation (25), it is clear that the differential equation has the solution which can be written as

Nr

z¼

(30)

and the bulk mean nanoparticle volume fraction fB can be obtained via

Ri : Ro

(22)

Z

1 1z

z

fB ¼ fw

Eqs. (18)e(20) can be simplified as


i d u 1 1 dm df du* mbf h 1  Nrðf  fB Þ  Ha2 u* ; ¼  þ  2 h m df dh dh 4m dh 2 *

1z

gðqw qÞ

u* eNBT gð1þgqÞð1þgqw Þ hdh Z

1 z

;

1z

where subscript w stands for both right and left walls. Equation (30)

. 2 3
* k kbf  dq kbf d2 q r cu ð1 þ z εÞ df 4 5; þ 7:47 ¼ þ dh k h dh dh2 rcu* ð1 þ zÞð1 þ εÞ (24) vf f vq ¼ ; 2 vh vh NBT ½1 þ gq

(25)

where the average value of parameters can be calculated over the cross-section by

Z

1

fdA ¼ 

p R2o  R2i

A



ZRo Ri

1

1z fð2prÞdr ¼ 2 1þz

Z1z

fhdh:

z

1z

(26) Hence, the bulk mean dimensionless temperature qB, and the bulk mean nanoparticle volume fraction fB can be obtained by

fB ¼

u* f u*

;

qB ≡

rcp uq : rcp u

(27)

Substituting Eq. (22) into Eq. (21), the reduced boundary conditions can be expressed as

h¼

z 1z

: u*  l

h¼

¼ fw ¼ fi ; ¼

du* vq ε ; ¼ 0; ¼ 2ð1 þ 7:47fi Þð1 þ εÞ dh vh 1 : 1z

u* þ l

du* ¼ 0; dh

f

vq vh

1 : 2ð1 þ 7:47fo Þð1 þ εÞ (28) 2m

where l ¼ N Dh rbf is the slip parameter. bf

(31)

u* hdh

(23)

1 f≡ A

(29)

This can be reduced for the nanoparticle volume fraction as

g

kbf

sB2 D2h Ha ¼ ; mbf

ðqo þ qi ÞDh ¼ ; TB kbf   rp  rbf g ; ¼
dp=dx

4. Theoretical analysis of governing equations

Fig. 2. Algorithm of the numerical method.

A. Malvandi et al. / International Journal of Thermal Sciences 109 (2016) 10e22 Table 2 Validation of the results with the ones reported by Kays and Crawford [40] when.Nr ¼ l ¼ Ha ¼ fB ¼ 0

z

ε

0.2

0.6

1

a

a

0 0.5 1 1.5 0 0.5 1 1.5 0 0.5 1 1.5

Kays and Crawford [40]

Present study

HTCi

HTCo

HTCi

HTCo

0 10.493 89.463 21.426 0 109.481 11.218 8.635 0 17.484 8.234 7.000

4.883 5.151 5.450 5.787 5.099 5.812 6.758 8.071 5.385 6.511 8.234 11.195

0 10.513 88.712 21.397 0 115.490 11.248 8.646 0 17.500 8.235 7.000

4.883 5.150 5.449 5.784 5.099 5.813 6.759 8.072 5.385 6.512 8.235 11.200

In order to avoid singularity at z ¼ 1, the results are obtained at z ¼ 0.99999.

signifies that the nanoparticle concentration distribution is very sensitive to the temperature gradient depending on the values of g and NBT. The value of g, is usually smaller than 1 and regarding Refs. [7], its influence on the solution are insignificant, as g varies from 0 to 0.2. So, the impact of NBT on nanoparticle volume fraction distribution is considered here.

15

4.1. Limiting cases of NBT (NBT/0 and NBT/∞) Because of the significance of NBT on nanoparticle concentration, it is beneficial to consider the limiting cases of NBT. With regard to Eq. (30), for NBT/0 we can write

fi;o /0 or qw /qi;o ;

For NBT /0 :

(32)

or in a more general way, from Eq. (24) it can be stated that

For NBT /0 :

vq  f/0: vh

(33)

Eqs. (32) and (33) shows that for the lower values of NBT, the nanoparticle volume fraction decreases significantly at the walls (fi,o / 0) except for the adiabatic wall (vq/vh|i,o ¼ 0). Therefore, for the lower values of NBT, it can be stated that heat flux at the walls reduces the nanoparticle concentration and leads to a nanoparticle depleted region. Thus, thermal conductivity and viscosity fall at the walls, which considerably influenced the heat transfer rate and shear stresses. But, from Eqs. (30) and (31) for NBT / ∞ we may have

fzfB zfi zfo :

(34)

Eq. (34) indicated that for the higher values of NBT, the

Fig. 3. a. The effects of NBT on nanoparticle distribution (f/fB), velocity (u/uB) and temperature (q/qB) profiles when ε ¼ 0.5, Nr ¼ 50, fB ¼ 0.06, Ha ¼ 5, l ¼ 0.1 and z ¼ 0.6. b. The effects of l on nanoparticle distribution (f/fB), velocity (u/uB) and temperature (q/qB) profiles when ε ¼ 0.5, Nr ¼ 50, fB ¼ 0.06, NBT ¼ 0.5, Ha ¼ 5, and z ¼ 0.6.

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A. Malvandi et al. / International Journal of Thermal Sciences 109 (2016) 10e22

nanoparticle volume fraction is completely uniform and there is no dependency to heat flux at the walls. In fact, increasing NBT reduces the nanoparticle migration from the heated walls and prevents the nanoparticle depletion region there. More discussion on the effects of NBT can be found in Ref. [39].

different values of ε and z are compared with the outcomes of Kays and Crawford [40] in Table 2. As is clear, the results are in a desirable agreement.

5. Numerical method and accuracy

Nanoparticle volume fraction distribution is determined by the mutual impact of the Brownian and the thermophoretic diffusivities. In the present study, this impact is considered with the ratio of Brownian diffusion to the thermophoresis, NBT. With dp y 20nm and fB y 0.1, this value can be changed over a wide range of 0.2e10 for alumina/water nanofluid (NBT ∝ 1/dp). Furthermore, the results have been carried out for g y TwTB/Tw ¼ 0.1, since its impact on the solution is insignificant, see Refs. [7,8].

The Eqs. (23) and (24) with boundary conditions of Eq. (28) present a system of ordinary differential equations which should be solved numerically. In each step, Eq. (30) has to be employed to find the distribution of nanoparticles, which is required for calculating the local values of thermophysical variables. Convergence criterion was considered to be 106 for the relative errors of the velocity, temperature, and nanoparticle concentration. The numerical algorithm considers a reciprocal procedure where fw, rcu* , and qB are utilized to calculate the values of fB, rcu* , and qB. The process is repeated till a prescribed value of fB is reached, and the relative errors among the assumed values of rcu* and qB with the calculated ones after solving equations (23) and (24) are less than 106. To clarify, numerical algorithm is shown in Fig. 2. Furthermore, to check the accuracy of the numerical code, the outcomes obtained for a horizontal annulus with Nr ¼ fB ¼ Ha ¼ l ¼ 0 and

6. Results and discussion

6.1. Velocity, temperature and concentration profiles The variations in nanoparticle concentration (f/fB), velocity (u/ uB), and temperature (q/qB) profiles for different values of NBT are depicted in Fig. 3a. A non-uniform nanoparticle volume fraction distribution can be seen for the lower values of NBT because of the thermophoresis. On the contrary, at greater values of NBT, the Brownian motion induces the nanoparticle concentration, and

Fig. 4. a. The effects of Ha on nanoparticle distribution (f/fB), velocity (u/uB) and temperature (q/qB) profiles when ε ¼ 0.5, f B¼ 0.06, Nr ¼ 50, l ¼ 0.1, NBT ¼ 0.5, and z ¼ 0.6. b. The effects of Nr on nanoparticle distribution (f/fB), velocity (u/uB) and temperature (q/qB) profiles when ε ¼ 0.5, f B¼ 0.06, Ha ¼ 5, l ¼ 0.1, NBT ¼ 0.5, and z ¼ 0.6.

A. Malvandi et al. / International Journal of Thermal Sciences 109 (2016) 10e22

reduces the formation of concentration gradients (described completely in Section 4). It can be observed that the nanoparticle volume fraction takes its lowest value at the inner wall, a rise to the maximum value in the region away from the wall, and then a drop toward the outer wall. This is because the heat flux at the inner wall is greater than at the outer wall, so this provides a greater temperature gradient and thermophoretic force. The movements of the nanoparticles from the heated walls to the core region at low values of NBT constructs a nanoparticle poor regions near the walls. This decreases the viscosity and the shear stress of the nanofluid at the walls. Therefore, the momentum in the vicinity of the walls grows, especially at the inner wall, having a lower nanoparticle volume fraction. Fig. 3b illustrates the impact of slip parameter on the flow fields. As l increases, it can be seen that peak of the velocity profile disappears from the core region and moves toward the surface walls. As is evident from Eq. (28), the slip parameter l reveals the amount of slip velocity at the surface walls. A rise in l results in an increase in the slip velocity near the wall and due to constant mass flow rate inside the microannulus, the momentum in the core region reduces. This results in a grow in the temperature gradients and nanoparticles concentration at the walls. Hartmann number (Ha) signifies the significance of magnetic field, which its effect on the flow fields is depicted in Fig. 4a.

17

Evidently, Ha has a similar effect on the velocities as the slip parameter does; the peak of velocity profile at the core region is reduced, while near wall velocity gradients enhance. In essence, the momentum at the core region migrates toward the walls. As a result, the velocities closed to the walls grow with rising Ha, increasing the slip velocity. Fig. 4b depicts a gentle increase of the fluid velocities near the inner wall, followed by a fall at the core region without any considerable change of the velocities at the outer wall. In essence, the fluid flow at the core region moves toward the inner wall. This is because rising Nr, increases the buoyancy, which accelerates the velocities at the walls and because of a constant mass flow rate in the microchannel, the momentum in the core region slightly decreases. The impact of radii (z) and heat fluxes (ε) ratios on the flow fields is depicted in Fig. 5a and b, respectively. As z increases, the effects of the inner wall grow, which improve the effects of viscous forces; so, the velocities eject themselves from the inner wall. Furthermore, it can be seen that the nanoparticle concentration becomes more non-uniform, when z rises. From Fig. 5b, for ε ¼ 1 the nanoparticle rich region is formed at the inner wall, having a lower heat transfer area. Growing ε pushes the peak of the nanoparticle concentration toward the outer wall. The effects of ε on nanoparticles distribution have significant impact on the velocity and temperature profiles.

Fig. 5. a. The effects of z on nanoparticle distribution (f/fB), velocity (u/uB) and temperature (q/qB) profiles when fB ¼ 0.06, Nr ¼ 50, ε ¼ 2, l ¼ 0.1, Ng ¼ 50, and NBT ¼ 0.5. b. The effects of ε on nanoparticle distribution (f/fB), velocity (u/uB) and temperature (q/qB) profiles when fB ¼ 0.06, Nr ¼ 50, z ¼ 0.5, l ¼ 0.1, Ng ¼ 50, and NBT ¼ 0.5.

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Obviously, the peak of the velocity profile moves toward the inner wall, when ε grows. On the contrary, the dip point of the temperature profile reduces and shifts toward the outer wall, where the nanoparticles accumulated.

HTCR ¼ ¼

6.2. Heat transfer rate and pressure drop

HTCi  Ri þ HTCo  Ro HTCi½base fluid  Ri þ HTCo½base fluid  Ro HTCi  z þ HTCo : HTCi½base fluid  z þ HTCo½base fluid

Besides, the pressure drop ratio can be defined as

hi Dh qi Dh ε ¼ ¼ ; ð1 þ εÞqB kbf ðTi  TB Þkbf

ho Dh q;;o Dh 1 : ¼ ¼ HTCo ¼ kbf ðTo  TB Þko ð1 þ εÞðqo  qB Þ

(35)

(36)

,




The dimensionless heat transfer coefficient (HTC) at the inner and outer walls can be defined respectively as

HTCi ¼

(37)

dp dx Nrp ¼


 dp dx

, ½base fluid

mbf uB

!

D2h

mbf uB½base fluid D2h

!¼0

rB ru*

1

:

@ rB A ru*

½base fluid

Figs. 6 e 10 reveal the effects of different parameters including

fB, z, ε, Ha, l, and Nr on total heat transfer ratio (a), and pressure

The total heat transfer ratio can be defined as the nondimensional by

drop ratio (b), respectively. From Fig. 6a and b, rising NBT leads to a grow in Nrp, while HTCR decreases. This is because as NBT increases, the momentum at the walls moves toward the core region; so the

Fig. 6. a. The effects of fB in terms of NBT on the total heat transfer enhancement when ε ¼ 2, Nr ¼ 50, l ¼ 0.1, Ha ¼ 5, and z ¼ 0.5. b. The effects of fB in terms of NBT on the pressure drop ratio when ε ¼ 2, Nr ¼ 50, l ¼ 0.1, Ha ¼ 5, and z ¼ 0.5.

Fig. 7. a. The effects of z in terms of NBT on total heat transfer enhancement when fB ¼ 0.06, Nr ¼ 50, l ¼ 0.1, Ha ¼ 5, and ε ¼ 2. b. The effects of z in terms of NBT the pressure drop ratio when fB ¼ 0.06, Nr ¼ 50, l ¼ 0.1, Ha ¼ 5, and ε ¼ 2.

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Fig. 8. a. The effects of ε in terms of NBT on the total heat transfer enhancement when fB ¼ 0.06, Nr ¼ 50, l ¼ 0.1, Ha ¼ 5, and z ¼ 0.5. b. The effects of ε in terms of NBT on the pressure drop ratio when fB ¼ 0.06, Nr ¼ 50, l ¼ 0.1, Ha ¼ 5, and z ¼ 0.5.

Fig. 9. a. The effects of l in terms of NBT on the total heat transfer enhancement when fB ¼ 0.06, Nr ¼ 50, Ha ¼ 5, ε ¼ 2, and z ¼ 0.5. b. The effects of l in terms of NBT on the pressure drop ratio when fB ¼ 0.06, Nr ¼ 50, Ha ¼ 5, ε ¼ 2, and z ¼ 0.5.

heat transfer rate reduces (see Fig. 3). In contrast, increment of the pressure loss originates from a rise in nanoparticle concentration at the walls, increasing the local viscosity of the nanofluid and the shear stresses. Fig. 6a and b also show that the heat transfer and pressure drop ratios increase with rising fB.These are due to increasing the viscosity and thermal conductivity of nanofluids at the walls. On the contrary, the heat transfer rate at greater values of NBT increases with fB since the momentum reduces at the walls. The impact of radii ratio z is demonstrated in Fig. 7a and b. It is clear that the heat transfer rate decreases, as z increases. However, the pressure drop has increases with z. As a result, it can be concluded that z has a negative influence on the performance of the system. Fig. 8a and b depict that despite the pressure drop, the heat transfer ratio reduces for higher values of ε. This is because when ε increases, impact of heat flux at inner wall, having a lower area, increases. For a constant heat flux at both walls, this decreases the total heat transfer rate. From Fig. 9a and b, total heat transfer ratio reduces as l increases, though a rise in the heat transfer rate may be expected [39]. This reveals that the heat transfer enhancement of

the nanofluid is significantly lower than that of the base fluid, which reduces the heat transfer ratio. Similar trend can be seen for the pressure drop ratio (Nrp). Reducing the pressure drop is a useful attribute, commonly in microtubes where the head loss extremely pronounced, because it allows a considerable drop in necessity pumping power to drive the nanofluid flow. Fig. 10a and b depict the effects of Hartmann number Ha on HTCR and Nrp for a range of Nr. Clearly, the heat transfer and the pressure drop ratios reduced for the greater values of Ha. It is worth noting that the total heat transfer ratio is below the unity, indicating that nanoparticle inclusion in the presence of high magnetic field results in a lower heat transfer rate. On the contrary, the pressure loss for the nanofluid is higher than that of the base fluid, signifying the negative performance of adding nanoparticles. Furthermore, Nr has a positive influence on the total heat transfer ratio and almost no impact on the pressure drop ratio. Augmenting the heat transfer rate is because of a rapid improvement of the velocities at the walls which intensifies the heat transfer rate there.

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6.3. Cooling performance analysis With regard to the heat transfer performance and pressure drop required in the system, figure of merit (h) is introduced as a measure of comparison of the heat transfer enhancement with respect to the penalty in pumping power. For the nanofluid, h can be defined as [41,42]:

. HTCtðnanofluidÞ HTCtðbasefluidÞ h¼ . 1=3 : NpðnanofluidÞ NpðbasefluidÞ

(39)

Fig. 11 shows the computed figures of merit for the various values of the parameters investigated in this study. At first, it can be observed that the value of h for all the figures is less than unity. This indicates that the performance of the system is reduced for the nanofluid and changing the variables can only lessen or exacerbate the negative performance of the system. According to

Fig. 11a, increasing the nanoparticle volume fraction reduces the performance, which can be easily understood from Fig. 6a and b. Fig. 11b shows that the performance of the system is increased when the ratio of the inner wall to the outer wall is reduced (decreasing z), particularly for the lower values of NBT. Accordingly, it can be deduced that the performance of a microannulus is always greater than a parallel-plate microchannel (z- > 1). From Fig. 11c, it can be seen that the slip velocity at the walls enhances the performance of the system. Accordingly, slippage at the fluidsolid interface is a useful attribute since it intensifies the performance of the system which is followed with a reduction in the pressure drop (see Fig. 9b). Generally, slippage at the surfaces can be created using liquid-repellent surface technologies, e.g. hydrophobic, superhydrophobic, and hydrophilic surfaces. Fig. 11d indicates that the thermal performance of heating is decreased when the contribution of heat transfer on the inner wall increases. In addition, it can be observed that using smaller nanoparticles reduces the thermal performance of the system, except for the higher values of ε in which a gentle rise on the heat transfer rate can be observed. Thus, it can be stated that using larger nanoparticles has more advantage than the smaller ones. Finally, Fig. 11e shows that the mixed convection and magnetic field enhances the thermal performance of the system. Similar to the effects of the slip velocity, the presence of a uniform magnetic field decreases the pressure loss, while it improves the thermal performance. It should be mentioned that the fall in the pressure loss is a key feature in microchannels in which the pressure loss is very high. Accordingly, the slip velocity and magnetic field have more advantageous than the other parameters.

7. Summary and conclusions

Fig. 10. a. The effects of Ha and Nr on the total heat transfer enhancement when ε ¼ 2, fB ¼ 0.06, l ¼ 0.1, NBT ¼ 0.5, and z ¼ 0.5. b. The effects of Ha and Nr on the pressure drop ratio when ε ¼ 2, fB ¼ 0.06, l ¼ 0.1, NBT ¼ 0.5, and z ¼ 0.5.

The impact of nanoparticle migration on thermal performance of hydromagnetic Al2O3-water nanofluid inside a vertical microannulus is investigated numerically. To consider different modes of nanoparticle migration, non-uniform temperature gradient is induced in the flow field by means of different heat fluxes at the walls; q;;i for the inner wall and q;;o for the outer wall. Assuming hydrodynamically and thermally fully developed flow, the partial differential equations including continuity, momentum, energy, and nanoparticle volume fraction equations are simplified to ordinary differential equations and solved numerically. Having examined the scale analysis for different terms of the governing equations, it is revealed that both the heat transport by nanoparticle flux and buoyancy force because of the temperature gradient can be neglected. To analyze the performance, the figures of merit for various range of parameters including the ratio of Brownian motion to thermophoresis NBT, slip parameter l, mixed convective parameter Nr, Hartmann number Ha, nanoparticle concentration fB, radii (z) and heat flux (ε) ratios have been studied in detail. It is found that the performance improves as the inner to outer wall radius ratio decreases. As a result, it can be concluded that the performance of nanoparticle inclusion in a microannular tubes is completely higher than that of a parallelplate microchannel. Furthermore, the thermal performance of heating reduces as the contribution of heat transfer rate at the inner wall enhances. What is more, it can be seen that smaller nanoparticles decreases the thermal performance of the system, except for the greater values of heat flux ratio (ε), where there is a gentle increase of the heat transfer rate. Therefore, it can be deduced that larger nanoparticles has more advantage than the smaller ones.

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Fig. 11. a. Figures of merit for different values of NBT and fB when ε ¼ 2, Nr ¼ 50, l ¼ 0.1, Ha ¼ 5, and z ¼ 0.5. b. Figures of merit for different values of NBT and z when fB ¼ 0.06, Nr ¼ 50, l ¼ 0.1, Ha ¼ 5, and ε ¼ 2. c. Figures of merit for different values of NBT and l when fB ¼ 0.06, Nr ¼ 50, ε ¼ 2, Ha ¼ 5, and z ¼ 0.5. d. Figures of merit for different values of NBT and ε when fB ¼ 0.06, Nr ¼ 50, Ha ¼ 5, l ¼ 0.1, and z ¼ 0.5. e. Figures of merit for different values of Nr and Ha when ε ¼ 2, fB ¼0 .06, l ¼ 0.1, NBT ¼ 0.5, and z ¼ 0.5.

Acknowledgments The authors wish to thank the respectful reviewers for their helpful and constructive comments which improve the paper. References [1] Maxwell JC. A treatise on electricity and magnetism. second ed. Oxford, UK: Clarendon press; 1873. [2] Choi SUS. Enhancing thermal conductivity of fluids with nanoparticles. In: Siginer DA, Wang HP, editors. Developments and applications of nonnewtonian flows, vol. 66. ASME; 1995. p. 99e105. [3] Masuda H, Ebata A, Teramae K, Hishinuma N. Alteration of thermalconductivity and viscosity of liquid by dispersing ultra-fine particles. Netsu Bussei 1993;7.

[4] Heidary H, Kermani MJ. Effect of nano-particles on forced convection in sinusoidal-wall channel. Int Commun Heat Mass Transf 2010;37:1520e7.
g OA. A study on peristaltic flow of nanofluids: application in [5] Tripathi D, Be drug delivery systems. Int J Heat Mass Transf 2014;70:61e70. [6] Buongiorno J. Convective transport in nanofluids. J Heat Transf 2006;128: 240e50. [7] Yang C, Li W, Nakayama A. Convective heat transfer of nanofluids in a concentric annulus. Int J Therm Sci 2013;71:249e57. [8] Yang C, Li W, Sano Y, Mochizuki M, Nakayama A. On the anomalous convective heat transfer enhancement in nanofluids: a theoretical answer to the nanofluids controversy. J Heat Transf 2013;135. 054504e054504. [9] Malvandi A, Moshizi SA, Soltani EG, Ganji DD. Modified Buongiorno’s model for fully developed mixed convection flow of nanofluids in a vertical annular pipe. Comput Fluids 2014;89:124e32. [10] Malvandi A, Ganji DD. Effects of nanoparticle migration on force convection of alumina/water nanofluid in a cooled parallel-plate channel. Adv Powder Technol 2014;25:1369e75.

22

A. Malvandi et al. / International Journal of Thermal Sciences 109 (2016) 10e22

[11] Hedayati F, Domairry G. Effects of nanoparticle migration and asymmetric heating on mixed convection of TiO2eH2O nanofluid inside a vertical microchannel. Powder Technol 2015;272:250e9. [12] Hedayati F, Domairry G. Nanoparticle migration effects on fully developed forced convection of TiO2ewater nanofluid in a parallel plate microchannel. Particuology February 2016;24:96e107. [13] Malvandi A, Ganji DD. Brownian motion and thermophoresis effects on slip flow of alumina/water nanofluid inside a circular microchannel in the presence of a magnetic field. Int J Therm Sci 2014;84:196e206. [14] Moshizi SA, Malvandi A, Ganji DD, Pop I. A two-phase theoretical study of Al2O3ewater nanofluid flow inside a concentric pipe with heat generation/ absorption. Int J Therm Sci 2014;84:347e57. [15] Malvandi A, Kaffash MH, Ganji DD. Nanoparticles migration effects on magnetohydrodynamic (MHD) laminar mixed convection of alumina/water nanofluid inside microchannels. J Taiwan Inst Chem Eng 2015;52:40e56. [16] Malvandi A, Ganji DD. Magnetic field and slip effects on free convection inside a vertical enclosure filled with alumina/water nanofluid. Chem Eng Res Des 2015;94:355e64. [17] Rashidi MM, Abelman S, Freidooni Mehr N. Entropy generation in steady MHD flow due to a rotating porous disk in a nanofluid. Int J Heat Mass Transf 2013;62:515e25. [18] Mushtaq A, Mustafa M, Hayat T, Alsaedi A. Nonlinear radiative heat transfer in the flow of nanofluid due to solar energy: a numerical study. J Taiwan Inst Chem Eng 2014;45:1176e83. [19] Yadav D, Nam D, Lee J. The onset of transient Soret-driven MHD convection confined within a Hele-Shaw cell with nanoparticles suspension. J Taiwan Inst Chem Eng January 2016;58:235e44. [20] Sheikholeslami M, Ganji DD, Gorji-Bandpy M, Soleimani S. Magnetic field effect on nanofluid flow and heat transfer using KKL model. J Taiwan Inst Chem Eng 2014;45:795e807. [21] Yadav D, Bhargava R, Agrawal GS. Thermal instability in a nanofluid layer with a vertical magnetic field. J Eng Math 2013;80:147e64. [22] Sarkar S, Ganguly S. Fully developed thermal transport in combined pressure and electroosmotically driven flow of nanofluid in a microchannel under the effect of a magnetic field. 2014. [23] Malvandi A, Ganji DD. Effects of nanoparticle migration on hydromagnetic mixed convection of alumina/water nanofluid in vertical channels with asymmetric heating. Phys E Low-dimensional Syst Nanostructures 2015;66: 181e96. [24] Yadav D. Hydrodynamic and hydromagnetic instability in nanofluids. VDM Publishing; 2014. [25] Bondareva NS, Sheremet MA, Ioan P. Magnetic field effect on the unsteady natural convection in a right-angle trapezoidal cavity filled with a nanofluid: Buongiorno’s mathematical modelnull. Int J Numer Methods Heat Fluid Flow 2015;25(8):1924e46. [26] Nadeem S, Haq RU, Khan ZH. Numerical study of MHD boundary layer flow of a Maxwell fluid past a stretching sheet in the presence of nanoparticles. J Taiwan Inst Chem Eng 2014;45:121e6.

[27] Malvandi A, Ganji DD. Effects of nanoparticle migration and asymmetric heating on magnetohydrodynamic forced convection of alumina/water nanofluid in microchannels. Eur J Mech e B/Fluids 2015;52:169e84. [28] Yadav D, Kim M. The onset of transient soret-driven buoyancy convection in nanoparticle suspensions with particle-concentration-dependent viscosity in a porous medium. J Porous Media 2015;18:369e78. [29] Rashidi MM, Ali M, Freidoonimehr N, Nazari F. Parametric analysis and optimization of entropy generation in unsteady MHD flow over a stretching rotating disk using artificial neural network and particle swarm optimization algorithm. Energy 2013;55:497e510. [30] Yadav D, Kim C, Lee J, Cho HH. Influence of magnetic field on the onset of nanofluid convection induced by purely internal heating. Comput Fluids 2015;121:26e36. [31] Sheikholeslami M, Ganji DD, Rashidi MM. Ferrofluid flow and heat transfer in a semi annulus enclosure in the presence of magnetic source considering thermal radiation. J Taiwan Inst Chem Eng 2015;47:6e17. [32] Yadav D, Lee J. The onset of MHD nanofluid convection with Hall current effect. Eur Phys J Plus 2015;130:1e22. [33] Karimipour A, Hossein Nezhad A, D’Orazio A, Hemmat Esfe M, Safaei MR, Shirani E. Simulation of copperewater nanofluid in a microchannel in slip flow regime using the lattice Boltzmann method. Eur J Mech e B/Fluids 2015;49:89e99. Part A. [34] Nikkhah Z, Karimipour A, Safaei MR, Forghani-Tehrani P, Goodarzi M, Dahari M, et al. Forced convective heat transfer of water/functionalized multiwalled carbon nanotube nanofluids in a microchannel with oscillating heat flux and slip boundary condition. Int Commun Heat Mass Transf 2015;68: 69e77. [35] Salman BH, Mohammed HA, Munisamy KM, Kherbeet AS. Characteristics of heat transfer and fluid flow in microtube and microchannel using conventional fluids and nanofluids: a review. Renew Sustain Energy Rev 2013;28: 848e80. [36] MacDevette MM, Myers TG, Wetton B. Boundary layer analysis and heat transfer of a nanofluid. Microfluid Nanofluid 2014:1e12. [37] Koo J, Kleinstreuer C. Laminar nanofluid flow in microheat-sinks. Int J Heat Mass Transf 2005;48:2652e61. [38] Pakravan HA, Yaghoubi M. Analysis of nanoparticles migration on natural convective heat transfer of nanofluids. Int J Therm Sci 2013;68:79e93. [39] Ganji DD, Malvandi A. Natural convection of nanofluids inside a vertical enclosure in the presence of a uniform magnetic field. Powder Technol 2014;263:50e7. [40] Kays WM, Crawford ME. Convective heat and mass transfer. McGraw-Hill Ryerson, Limited; 1980. [41] Manca O, Nardini S, Ricci D. A numerical study of nanofluid forced convection in ribbed channels. Appl Therm Eng 2012;37:280e92. [42] Bahiraei M, Hosseinalipour SM, Hangi M. Laminar forced convection of a water-TiO2 nanofluid in annuli considering mass conservation for particles. Chem Eng Technol 2013;36:2057e64.