water nanofluid in microchannels

Accepted Manuscript Effects of nanoparticle migration and asymmetric heating on magnetohydrodynamic forced convection of alumina/water nanofluid in microchannels A. Malvandi, D.D. Ganji PII: DOI: Reference:

S0997-7546(15)00039-4 http://dx.doi.org/10.1016/j.euromechflu.2015.03.004 EJMFLU 2875

To appear in:

European Journal of Mechanics B/Fluids

Received date: 28 July 2014 Revised date: 23 February 2015 Accepted date: 16 March 2015 Please cite this article as: A. Malvandi, D.D. Ganji, Effects of nanoparticle migration and asymmetric heating on magnetohydrodynamic forced convection of alumina/water nanofluid in microchannels, European Journal of Mechanics B/Fluids (2015), http://dx.doi.org/10.1016/j.euromechflu.2015.03.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

1

Effects of nanoparticle migration and asymmetric heating on

2

magnetohydrodynamic forced convection of alumina/water nanofluid in

3

microchannels

4

A. Malvandi a,*, D. D. Ganji b a

5

Young Researchers and Elite Club, Qazvin Branch, Islamic Azad University, Qazvin, Iran b

6

Mechanical Engineering Department, Babol University of Technology, Babol, Iran

7

Abstract

8

The present paper is a theoretical investigation on effects of nanoparticle migration and

9

asymmetric heating on forced convective heat transfer of alumina/water nanofluid in

10

microchannels in presence of a uniform magnetic field. Walls are subjected to different heat

11

,, ,, fluxes; qt for top wall and qb for bottom wall, and because of non-adherence of the fluid-solid

12

interface due to the microscopic roughness in microchannels, Navier's slip boundary condition is

13

considered at the surfaces. A two-component hetregeneous mixture model is used for nanofluid

14

with the hypothesis that Brownian motion and thermophoretic diffusivities are the only

15

significant slip mechanisms between solid and liquid phases. Assuming a fully developed flow

16

and heat transfer, the basic partial differential equations including continuity, momentum, and

17

energy equations have been reduced to two-point ordinary boundary value differential equations

18

and solved numerically. It is revealed that nanoparticles eject themselves from heated walls,

19

construct a depleted region, and accumulate in the core region, but more likely to accumulate

20

near the wall with lower heat flux. Also, the non-uniform distribution of nanoparticles causes

21

velocities to move toward the wall with a higher heat flux and enhances heat transfer rate there.

22

In addition, inclusion of nanoparticles in a very strong magnetic field and slip velocity at the

23

walls has a negative effect on performance.

24 25

Key Words: Nanofluid; microchannel; nanoparticles migration; magnetic field; slip velocity; modified Buongiorno’s model.

*

Corresponding author: [email protected] , Tel.: +98 26 34323570; Fax: +98 21 65436660

1

Nomenclature B

uniform magnetic field strength

cp

specific heat (m2/s2K)

d

Nanoparticle diameter (m)

DB

Brownian diffusion coefficient

DT

thermophoresis diffusion coefficient

h

heat transfer coefficient (W/m2.K)

hp

specific enthalpy of nanoparticles

H

channel height (m)

Ha

Hartmann number

HTC

dimensionless heat transfer coefficient

Jp

nanoparticle flux

k

thermal conductivity (W/m.K)

kBO

Boltzmann constant ( = 1.3806488 × 10

N BT

ratio of the Brownian to thermophoretic diffusivities

Np

non-dimensional pressure drop

Ndp

pressure drop ratio

p

pressure (Pa)

q''

surface heat flux (W/m2)

T

temperature (K)

u

axial velocity (m/s)

x, y

coordinate system

−23

m2kg s2 K )

Greek symbols φ

nanoparticle volume fraction

γ

ratio of wall and fluid temperature difference to absolute temperature

η

transverse direction

μ

dynamic viscosity (kg/m.s) 2

ρ

density (kg/m3)

σ

electric conductivity

λ

slip parameter

Subscripts B

bulk mean

bf

base fluid

nf

nanofluid

t

condition at the top wall

p

nanoparticle

b

condition at the bottom wall

Superscripts *

dimensionless variable

26 27 28 29

1. Introduction

30

Economic incentives, energy saving and space considerations have increased efforts to construct

31

more efficient heat exchange equipment. Many techniques have been presented by researchers to

32

improve heat transfer performance, which is referred to as heat transfer enhancement,

33

augmentation, or intensification. Bergles [1] was the first to classify heat transfer enhancement

34

techniques to a) active techniques which require external forces to maintain the enhancement

35

mechanism such as an electrical field or vibrating the surface and b) passive techniques which do

36

not require external forces, including geometry refinement (e.g., micro/nano channels), special

37

surface geometries [2], or fluid additives (e.g., micro/nano particles).

3

38

The idea of adding particles to heat transfer fluids as an effective method of passive techniques,

39

emerged in 1873 [3]. The motivation was to improve thermal conductivity of the most common

40

fluids such as water, oil, and ethylene-glycol mixture, with solid particles which have

41

intentionally higher thermal conductivity. Then, many researchers studied the influence of solid-

42

liquid mixtures on potential heat transfer enhancement. However, they were confronted with

43

problems such as abrasion, clogging, fouling and additional pressure loss of the system, which

44

makes these unsuitable for heat transfer systems. In 1995, the word “nanofluid” was proposed by

45

Choi [4] to indicate dilute suspensions formed by functionalized nanoparticles smaller than

46

100nm in diameter which had already been created by Masuda et al. [5] as Al2O3-water. These

47

nanoparticles are fairly close in size to the molecules of the base fluid and, thus, can enable

48

extremely stable suspensions with only slight gravitational settling over long periods. Likewise,

49

in 1999, Lee et al. [6] measured the thermal conductivity of Al2O3 and CuO nanoparticle

50

suspensions in water and ethylene glycol. In 2001, Eastman et al. [7] and Choi et al. [8] found an

51

anomalous thermal conductivity enhancement of Cu and nanotube dispersions in ethylene glycol

52

and oil, respectively. In the light of these pioneering works, numerous experimental

53

investigations on the behaviors of nanofluids has been carried out which can be found in

54

literature such as Fan and Wang [9]. Meanwhile, theoretical studies emerged to model the

55

nanofluid behaviors. At the outset, the proposed models were twofold: homogeneous flow

56

models and dispersion models. In 2006, Buongiorno [10] demonstrated that the homogeneous

57

flow models are in conflict with the experimental observations and tend to underpredict the

58

nanofluid heat transfer coefficient, whereas the dispersion effect is completely negligible due to

59

the nanoparticle size. Hence, Buongiorno developed an alternative model to explain the

60

anomalous convective heat transfer in nanofluids and so eliminate the shortcomings of the

4

61

homogeneous and dispersion models. He asserted that the anomalous heat transfer occurs due to

62

particle migration in the fluid. Investigating the nanoparticle migration, he considered seven slip

63

mechanisms — the inertia, Brownian diffusion, thermophoresis, diffusiophoresis, magnus forces,

64

fluid drainage, and gravity — and maintained that, of these seven, only Brownian diffusion and

65

thermophoresis are important slip mechanisms in nanofluids. Taking this finding as a basis, he

66

proposed a two-component four-equation non-homogeneous equilibrium model for convective

67

transport in nanofluids. Then, Savino and Patterson [11] presented a similar model accounting

68

for the gravitational effects. These models have been used by Kuznetsov and Nield [12, 13] to

69

study the influence of nanoparticles on the natural convection boundary-layer flow past a vertical

70

plate, Tzou [14] for the analysis of nanofluid Bernard convection, Hwang et al. [15] for the

71

analysis of laminar forced convection. Then, the model considered by many researchers, for

72

example [16-18]. Recently, Buongiorno’s model has been modified by Yang et al. [19, 20] to

73

fully account for the effects of the nanoparticle volume fraction. Then, Li and Nakayama [21]

74

obtained an exact solution for fully developed flow of nanofluid in a tube subject to constant heat

75

flux, with and without accounting for the temperature dependency of thermophysical properties.

76

The comparison reveals that the effects of temperature-dependent thermophysical properties on

77

both dimensionless velocity and temperature profiles are not as sensitive as those of the

78

nanoparticle volume fraction. Furthermore, they indicated that the temperature dependency on

79

thermophysical properties only alters the level of the Nusselt number. Accordingly, the behavior

80

of heat transfer rate can easily be understood with considering the nanoparticle volume fraction

81

distribution. The modified Buongiorno’s model has been applied to different heat transfer

82

concepts including forced [22-24], mixed [25-30], and natural convection [31, 32]. The results

5

83

indicated that the modified model is suitable for considering the effects of nanoparticle migration

84

in nanofluids.

85

Recent progress in microfabrication-the process of fabrication of miniature structures of

86

micrometer scales- has resulted in the development of a variety of micro-devices involving heat

87

and fluid flows. Such devices found their application in various industries, such as

88

microelectronics, biotechnology, and microelectromechanical systems (MEMS). Several

89

research initiatives have been conducted to improve our understanding of the fluid flow and heat

90

transfer at the micro level; these initiatives which thoroughly reviewed by Adham et al. [33] and

91

Salman et al. [34] have resulted in an increased interest in the possibility of a slip boundary

92

condition. Adherence of fluid to solid at the boundaries, known as “no-slip” boundary condition,

93

is one of the commonplace assumptions of the Navier-Stokes theory which is not valid at

94

microscale channels. Slippage of liquids near the walls of microscale channels has encountered

95

as a result of the interaction between a coated solid wall (hydrophobic, hydrophilic or

96

superhydrophobic materials) and the adjacent fluid particle. In fact, because of the repellent

97

nature of the hydrophobic and superhydrophobic surfaces, the fluid molecules closed to the

98

surface do not follow the solid boundary, resulting in an overall velocity slip. More discussion on

99

the slip effects can be found in open literature, e.g. [35-38].

100

On the other hand, active techniques commonly present higher augmentation thought they need

101

additional power that increases initial capital and operational costs of the system. In this class,

102

the study of the magnetic field has important applications in medicine, physics and engineering.

103

Many industrial types of equipment, such as MHD generators, pumps, bearings and boundary

104

layer control are affected by the interaction between the electrically conducting fluid and a

105

magnetic field. The behavior of the flow strongly depends on the orientation and intensity of the 6

106

applied magnetic field. The exerted magnetic field manipulates the suspended particles and

107

rearranges their concentration in the fluid which strongly changes heat transfer characteristics of

108

the flow. The seminal study about MHD flows was conducted by Alfvén who won the Nobel

109

Prize for his works. Later, Hartmann did a unique investigation on this kind of flow in a channel.

110

Afterwards, many researchers have emphasized this concept and the details can be found in

111

literature such as [39-48].

112

The current progress on theoretical modeling of nanofluids has resulted in an increased interest

113

in explaining the thermophysical characteristics of nanofluids. Lately, it has been shown that

114

nanoparticle migration has considerable effects on the flow and heat transfer characteristics of

115

nanofluids, and it is responsible for the abnormal heat transfer characteristics of nanofluids [10,

116

19, 20]. Up to now, very few studies have been investigated on theoretical modeling of

117

nanofluids in microchannels, most of which used homogeneous models for nanofluids [49-51],

118

while the effects of nanoparticle migration have commonly been ignored. In the current research,

119

the distribution of the nanoparticle volume fraction is obtained considering the nanoparticle

120

fluxes due to the Brownian diffusion and thermophoresis in order to take into account the effects

121

of nanoparticle migration on fully developed forced convective heat transfer of alumina/water

122

nanofluid in microchannels in the presence of a uniform magnetic field. Walls are subjected to

123

,, ,, different heat flux; qt for the top wall and qb for the bottom wall and because of the

124

microscopic roughness at the wall of the microchannel, instead of a conventional no-slip

125

condition, the Navier's slip condition has been employed at the walls. The modified

126

Buongiorno’s model [26] has been used for nanofluids that fully account for the effects of

127

nanoparticle volume fraction distribution. The motivation behind this study is associated with the

128

migration of nanoparticle owing to the mutual effects of thermophoresis and Brownian

129

diffusivities, which is not only greatly influenced with the ratio of heat flux at the walls, but also

130

is very sensitive to the nanoparticle size. One of the most significant aspects of the present study 7

131

is that the researcher for the study combines different methods of heat transfer enhancement,

132

namely, passive (microchannel and nanofluid concepts) and active (presence of a magnetic field)

133

techniques.

134

2. Nanoparticle migration

135

In many studies, a nanofluid was considered to be a homogeneous liquid, and its material

136

properties were assumed to be constant in all positions of the system. These assumptions are not

137

realistic, and might cause misunderstandings in the heat transfer mechanism of a nanofluid.

138

Therefore, an examination of the motion of nanoparticles in a nanofluid is essential for

139

examining nanofluids as a heat transfer medium in heat exchanger or heat transfer equipment.

140

Microscopically, because of the very small dimension of the nanoparticles ( < 100nm), there are

141

two possible reasons for the in-homogeneity; Brownian and thermophoretic diffusivities [10].

142

Brownian diffusion can be observed due to random drifting of suspended nanoparticles within

143

the base fluid which comes from continuous collisions between nanoparticles and liquid

144

molecules. It is proportional to the concentration gradient and described by the Brownian

145

diffusion coefficient, DB, which is given by the Einstein-Stokes’s equation [10] DB =

k BT 3πμbf d p

(1)

146

where kB is the Boltzmann’s constant, μ b f is the dynamic viscosity of the base fluid , T is the

147

local temperature and dp is the nanoparticle diameter. The nanoparticle flux due to Brownian

148

diffusion ( J p , B ) can be given as

J p , B = − ρ p DB ∇φ

8

(2)

149

On the other hand, the thermophoresis (“particle” equivalent of the Soret effect), tries to induce

150

the nanoparticles migration in the opposite direction of the temperature gradient (warmer to

151

colder region), causing a non-uniform nanoparticle distribution. The thermophoresis is described

152

by the thermal diffusion, DT, which is given by [10]

DT = β

153

where β = 0.26

154

as

kbf 2kbf + k p

μbf φ ρbf

. The nanoparticle flux due to thermophoresis ( J p ,T ) can be calculated

J p ,T = − ρ p DT

155

(3)

∇T T

(4)

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

∇T ⎞ ⎛ J p = J p,B + J p,T = −ρ p ⎜ DB∇φ + DT ⎟ T ⎠ ⎝

(5)

156

Since DB ∼ T and DT ∼ φ (depend on the flow field), it is advantageous to re-write Eq. (5) as

157

follows [52]

∇T ⎞ ⎛ J p = −ρ p ⎜ CBT ∇φ + CTφ ⎟ T ⎠ ⎝

(6)

DB D and CT = T do not depend on the flow field. As a result, distribution of T φ

158

where C B =

159

nanoparticle can be obtained via

160

Nanoparticle distribution equation

∂ t (φ ) + ∇. ( uφ ) = −

∇T ⎞ ⎛ ∇ ( J p ) = ∇. ⎜ CBT ∇φ + CT φ ⎟ ρp T ⎠ ⎝ 1

9

(7)

161

3. Problem Formulation and Governing Equations

162

Consider an MHD, laminar and two-dimensional flow of the alumina/water nanofluid inside a

163

microchannel, which is subjected to different heat fluxes at the top ( qt" ) and bottom ( qb" ) walls

164

such that qt" > qb" . The geometry of the problem is shown in Figure 1. A two-dimensional

165

coordinate frame has been selected where the x-axis is aligned horizontally and the y-axis is

166

normal to the walls. Because of non-adherence of the fluid-solid interface due to microscopic

167

roughness in microchannels, the flow has a slip velocity at the surface walls. In view of

168

considering the nanoparticle migration, the modified Buongiorno’s two-component mixture

169

model [26] is applied to the nanofluid which was described before (in section 2). From the

170

numerical solutions provided by Koo and Kleinstreuer [53], for the most standard nanofluid

171

flows inside a channel of around 50μm, the viscous dissipation is negligible and ohmic heating

172

and Hall effects are also assumed to be small. Consequently, the basic incompressible

173

conservation equations of the mass, momentum, thermal energy, and nanoparticle fraction can be

174

expressed in the following manner [10, 52]:

175

Continuity equation

176

177

∂ t ( ρ ) + ∇. ( ρ u ) = 0

(8)

∂ t ( ρ u ) + ∇. ( ρ uu ) = −∇p + ∇.τ − σ B 2u

(9)

∂ t ( ρ cT ) + ∇. ( ρ cu T ) = −∇.q + h p ∇.J p

(10)

Momentum equation

Energy equation

10

(

) is the shear stress, and q

178

where h p is the specific enthalpy of nanoparticles, τ = μ ∇u + ( ∇u )

179

is the energy flux relative to the nanofluid velocity, which can be expressed as the sum of the

180

conduction and diffusion heat flux as below:

q=

−k ∇T conduction heat flux

+

t

hp J p nanoparticle diffusion heat flux

(11)

181

Further, ρ , μ , k , c are the density, dynamic viscosity, thermal conductivity, and specific heat

182

capacity of alumina/water nanofluid respectively, depending on the nanoparticle volume fraction

183

as follows:

μ = μ bf ( 1 + 39 .11φ + 533 .9φ 2 ), ρ = φρ p + ( 1 − φ ) ρ bf c=

φρ p c p + (1 − φ ) ρbf cbf ρ

(12) , k = kbf (1 + 7.47φ ), β = φρ p β p + (1 − φ ) ρ f β f

184

where bf represents base fluid and p for particle. Since the rheological and thermophysical

185

properties ( ρ , μ , k and c ) are dependent on nanoparticle concentration, the nanoparticle

186

distribution equation, Eq. (7), should be coupled with Eq. (8)-(10). Thus, substituting Eq. (11)

187

into Eq. (10) and equating ∇ h p = c p ∇ T , one may simply obtain governing equations for steady,

188

incompressible, hydrodynamically and thermally fully developed flow as follows: dp d ⎛ du ⎞ 2 = ⎜μ ⎟ −σ B u dx dy ⎝ dy ⎠

ρ cu

⎛ φ ∂T ⎞ ∂T ∂T ∂ ⎛ ∂T ⎞ ∂φ + CT = ⎟ ⎜k ⎟ + ρ p c p ⎜ C BT ∂y ∂x ∂ y ⎝ ∂ y ⎠ T ∂y⎠∂y ⎝

∂ ⎛ ∂φ CT φ ∂T ⎞ + ⎜ CBT ⎟=0 ∂y ⎝ ∂y T ∂y ⎠ 189 190

3-1

Scale analysis of governing equations

The scale analysis for Eq. (14) can be expressed as 11

(13) (14)

(15)

LHS :

(φ ρ c B

p

p

+ (1 − φB ) ρ bf cbf

)

U ref ΔT Δx

2.0285×108 RHS :

k bf (1 + 7.47φB )

ΔT 2

Lref

2.086×1010

,

⎛ ⎞ ⎜ ⎟ Tref Δφ ΔT ⎟ ⎜ C ρ pcp + C φB ΔT ⎜ Lref Tref Lref ⎟ 50 ⎟ 3×106 ⎜ ⎝ 2.654×10−3 2.77×10−2 ⎠ B

(16)

T

191

Scale analysis indicates that the LHS term is of the same order of magnitude as the first and third

192

RHS terms and these are about 1000 times more than the second RHS term. In fact, heat transfer

193

associated with nanoparticle diffusion (second RHS term) can be neglected in comparison with

194

the other terms. Hence, the governing equations (Eqs. (13)-(15)) can be given as: d ⎛ du ⎞ dp − σ B 2u = 0 ⎜μ ⎟− dy ⎝ dy ⎠ dx

ρ cu

∂T ∂ ⎛ ∂T ⎞ − ⎜k ⎟=0 ∂x ∂ y ⎝ ∂ y ⎠

∂ ⎛ ∂φ CT φ ∂T ⎞ + ⎜ CBT ⎟=0 ∂y ⎝ ∂y T ∂y ⎠

(17) (18)

(19)

195

Eq. (18) indicates that the energy equation for a nanofluid is of the same form of a regular fluid.

196

As a result, the heat transfer in nanofluids is only affected via the rheological and thermophysical

197

properties which depend on nanoparticle distribution. Regarding the nanoparticle distribution

198

equation, Eq. (19), it is obvious that the Brownian diffusion flux and thermophoretic diffusion

199

flux cancel out (

200

and according to the thermally fully developed condition for the uniform wall heat flux,

201

dT dTB , and introducing the following non-dimensional parameters: = dx dx

CB ∂φ C ∂T = − T2 ) everywhere, thus, by averaging Eq. (18) from y = 0 to H T ∂y φ ∂y

12

η=

T − Tt y u σ B2 H 2 , u* = 2 , T * = ,, , Ha 2 = , H qt H kt μbf H ⎛ dp ⎞ − μbf ⎜⎝ d x ⎟⎠

(20)

CB ktTt 2 qt,, H qb,, γ= , ε = ,, , N BT = CT Hqt ktTt qt 202

Eqs. (18)-(20) can be reduced to (see Appendix)

d 2u* 1 d μ dφ du* μbf + + 1 − Ha2u* = 0 dη 2 μ dφ dη dη μ

(21)

d 2T * kt ⎡ ρ cu* 1 dk dφ dT * ⎤ − + − 1 ε ) ⎥=0 ⎢( dη 2 k ⎣ < ρ cu* > kt dφ dη dη ⎦

(22)

∂φ φ ∂T * + =0 ∂η N BT (1 + γ T * ) 2 ∂η

(23)

(

)

ε is the heat flux ratio and the average value of parameters can be calculated over the

203

where

204

cross-section by Γ ≡

1 A

∫ ΓdA = A

H

1 Γdy H ∫0

(24)

205

Hence, the bulk mean dimensionless temperature T B* , and the bulk mean nanoparticle volume

206

fraction φB can be obtained by TB* =

207

3-2

ρcu*T * ρcu*

, φB =

u*φ u*

(25)

Boundary conditions

208

Because of non-adherence of the fluid-solid interface due to microscopic roughness in

209

microchannels, the flow has a slip velocity at the surface walls. Different heat fluxes are taken

13

210

,, ,, ,, into account at the walls, qt for the top wall and the other one is considered to be qb < qt . As a

211

result, appropriate boundary conditions for this problem can be expressed as

⎧ ⎪y = 0 : ⎪ ⎨ ⎪y = H : ⎪⎩

du ∂T ∂φ C φ ∂T , − kt = qt,, , CBT + T = 0. dy ∂y ∂y T ∂y du ∂T ∂φ C φ ∂T , kb u = −N = qb,, , CBT + T = 0. dy ∂y ∂y T ∂y u=N

(26)

212

where N represents slip velocity factor. Substuting Eq. (21) into Eq. (27), the boundary

213

conditions can be expressed as

du* ∂T * η = 0 (Top wall) : u − λ = 0, = −1, T * = 0, φ = φt dη ∂η *

du* η = 1 (Bottom wall) : u + λ = 0. dη

(27)

*

214

215

where λ =

N is the slip parameter. H

4. Theoretical Analysis of governing equations

216

Before we go on to the numerical simulations of the governing equations, it would be

217

advantageous to analyze the governing equations. Regarding equations (23), it is evident that the

218

differential equation is separable and has solution in the following form

γ (Tw* − T * ) ⎛φ ⎞ Ln ⎜ ⎟ = * * ⎝ φw ⎠ N BT γ (1 + γ T )(1 + γ Tw ) 219

which can be simplified for the nanoparticle volume fraction as

(

γ Tw* −T *

φ = e N BT γ (1+γ T φw 220

(28)

*

)

)(1+γ Tw* )

and the bulk mean nanoparticle volume fraction φB can be obtained as 14

(29)

(

γ Tw* −T * 1

φB = φw

)

* N BT γ (1+γ T * )(1+γ Tw* )

∫ue 0

1

dη

(30)

∫ u dη *

0

221

where the subscript w stands for both top and bottom walls. Eq. (30) indicates that the

222

nanoparticle volume fraction distribution is very sensitive to the temperature profile (so does to

223

the thermal boundary conditions) and its sensitivity depends on the values of γ and NBT. The

224

value of γ , is usually smaller than 1 and according to Refs. [19], its effects on the solution are

225

insignificant, as γ varies from 0 to 0.2. So, the effects of NBT on nanoparticle distribution have

226

been considered here.

227

4-1

Limiting Cases of NBT (NBTÆ0 and NBTÆ∞)

228

Due to the importance of NBT on nanoparticle volume fraction distribution, it is beneficial to

229

consider the limiting case of NBT. Regarding Eq. (29), for NBTÆ0 we may have *

*

For N BT → 0 : φ w → 0 or T → Tw 230

(31)

Or in a more general way, from Eq. (23) it can be concluded that

For N BT → 0 :

∂T

*

∂η

φ →0

(32)

231

Eqs. (31) and (32) indicate that for the lower values of NBT, the nanoparticle volume fraction is

232

reduced significantly at the walls ( φw → 0 ) except when the wall is adiabatic (

∂T * ∂η

= 0 ). Thus, w

233

for the lower values of NBT, it can be concluded that heat flux at the surface walls eject the

234

nanoparticles at the walls and leads to a nanoparticle deplete region at the walls. As a result, 15

235

thermal conductivity and viscosity decrease at the walls (significantly affects the heat transfer

236

rate and shear stresses). In addition, for a prescribed φB , it can be concluded that the nanoparticle

237

volume fraction far from the walls takes an increasing trend, leading to a nanoparticle rich

238

region. Thus, it can be concluded the nanoparticle distribution becomes non-uniform at the lower

239

values of NBT. On the other hand, regarding Eqs. (29) and (30) for NBTÆ∞ we may have

φ ≈ φB ≈ φlw ≈ φrw

(33)

240

Eq. (33) indicates that for higher values of NBT, the nanoaprticle volume fraction is completely

241

uniform and there is no dependency on heat flux at the walls. In other words, increasing NBT

242

decreases the nanoparticle migration from the heated walls and prevents the nanoparticle

243

depletion region there.

244

To explain these characteristics in a physical manner, one should bear in mind that NBT is the

245

ratio of Brownian motion to thermophoretic diffusivities. Brownian motion, tries to push

246

nanoparticles in the opposite direction of the concentration gradient and make the nanofluid

247

more homogeneous. Conversely, the thermophoresis, tries to induce the nanoparticle migration

248

in the opposite direction of the temperature gradient, causing a non-uniform nanoparticle

249

distribution. In other words, once the nanoparticles concentration gradient is developed by the

250

thermophoretic force, Brownian force tends to counterbalance the former effect. Thus, at the

251

lower values of NBT the thermophoretic force is predominant and results in a non-uniform

252

nanoparticle concentration; however, at the higher values of NBT, the Brownian motion takes

253

control of the nanoparticle concentration and prevents the formation of concentration gradients.

254

Consequently, the nanoparticle concentration becomes more uniform ( φ ≅ φw ≅ φB ) at the higher

16

255

values of NBT. More graphical results and discussions on the effects of NBT and nanoparticle

256

migration on flow and thermal fields will be shown later in section 6.

257

5. Numerical Method and Accuracy

258

A system of equations (21) and (22) with boundary conditions of equation (27) represents a

259

system of nonlinear ordinary differential equations which are solved numerically via the Runge-

260

Kutta-Fehlberg method. In each iteration, Eq. (29) is used to determine the distribution of

261

nanoparticles, which is necessary for calculating the local values of thermophysical variables.

262

Convergence criterion is considered to be 10-6 for relative errors of the velocity, temperature, and

263

nanoparticle volume fraction. Practically, the bulk mean nanoparticle volume fraction

264

prescribed rather than that at the wall

265

to obtain the appropriate

266

term as a certain value which should be determined in advance, and then an extra iterative loop

267

in the program is applied to find this certain value. The same algorithm is also required to find

268

the value of ρ cu * . To clarify, the algorithm of the numerical solution is shown in Fig. 2. In

269

view of helping others to regenerate the results as well as providing material for possible future

270

references, the corrected values of

271

To check the accuracy of the numerical code, the results obtained for a parallel-plate channel

272

with Ha=λ=0, and ε = 1 are compared with the reported results of Yang et al. [20] in Figure 3.

273

As shown in the figure, the results are in desirable agreement. In addition, the numerical code

274

developed here is run with three different integration steps (dη) of 10-5, 5×10-5, and 10-6 to verify

275

the results are independent of the grid size. The obtained numerical results are presented in Table

φB is

φw . As a consequence, a reciprocal algorithm was required

φw in order to reach the required φB . The method used is to treat this

φw and ρ cu* for a special case are presented in Table 2.

17

276

3. The results clearly indicate good performance of the grid independence of the code. It must be

277

mentioned that all the numeric results presented here are carried out using the integration step

278

dη=10-6.

279

6. Results and Discussions

280

Nanoparticle distribution is determined by the mutual effects of the Brownian diffusion and the

281

thermophoresis. Here, these effects are considered by means of NBT, which is the ratio of the

282

Brownian diffusion to the thermoporesis. With d p ≅ 20nm and

283

motion to thermophoretic forces N BT ∝ 1/ d p can be changed over a wide range of 0.2 to 10 for

284

alumina/water nanofluid. In addition, the results are presented for γ ≅

285

effect on the solution is insignificant, see Ref. [19, 20].

286

6-1

φB ≅ 0.1, the ratio of Brownian

Tw − TB = 0.1 , since its Tw

Velocity, temperature and concentration profiles

287

The variations in the nanoparticle volume fraction ( φ / φB ), velocity (u/uB), and temperature

288

(T*/T*B) profiles for different values of NBT are shown in Figure 4, where η=0 corresponds to the

289

top wall, and the bottom wall to η=1. A non-uniform nanoparticle distribution can be observed

290

for the lower values of NBT because of the thermophoresis. In contrast, at higher values of NBT,

291

the Brownian motion takes control of the nanoparticle concentration, and prevents formation of

292

concentration gradients (i.e. the nanoparticle distribution becomes uniform). This is in complete

293

consonance with theoretical outcomes of Section 4. Physically, as the size of a nanoparticle is

294

reduced, its Brownian diffusion is increased. This increase leads to a higher value of NBT

295

(NBT~DB~1/dp). Accordingly, the use of smaller nanoparticles leads to a more uniform

296

nanoparticle distribution. Again, with the enlargement of nanoparticles, the effect of the 18

297

Brownian force is reduced. Hence, thermophoresis effects become significant, which results in a

298

greater concentration of nanoparticles in the core regions of the microchannel away from the

299

heated walls. The migration of the nanoparticles from the heated walls toward the core region at

300

lower values of NBT constructs nanoparticle-depleted regions at the walls. This reduces the

301

viscosity, and also the shear stress of the nanofluid on the walls. However, the viscosity and

302

shear stress increase in the core region, where a nanoparticle-accumulated region has been

303

constructed. Hence, the velocity shifts toward the top wall which has the lowest viscosity. In

304

addition, because of the strong dependence of the thermal conductivities on the volume fraction

305

of the nanoparticles, nanoparticle migration has a significant impact on the thermal

306

characteristics. Considering Eq. (12), for the lower values of NBT, the thermal conductivity is

307

higher in the nanoparticle accumulation region, where the nanoparticles concentration is higher;

308

whereas, the thermal conductivity is reduced at the heated walls (a negative behavior for the heat

309

transfer rate). Further discussion of the effects of NBT on the heat transfer rate will be carried out

310

later. Moreover, it can be easily observed that the nanoparticle concentration has its lowest value

311

at the top wall (higher wall heat flux), a slight increase to the maximum in the region away from

312

the wall, but decreasing rapidly towards the bottom wall (lower wall heat flux). This is purely

313

because the heat flux at the top wall is higher than at the bottom wall, so this produces a

314

temperature gradient and thermophoresis. It is not surprising that the peak of the nanoparticle

315

concentration is closer to the bottom wall, having a lower heat flux. This phenomenon can be

316

observed in more details in Figure 5 where the variations of the nanoparticle volume fraction (

317

φ / φB ), velocity (u/uB), and temperature (T*/T*B) profiles for different values of

318

shown. For the case ε = 0 , where the bottom wall is kept adiabatic, nanoparticle concentration is

319

higher near the bottom (adiabatic) wall. In fact, the nanoparticle accumulation region formed

19

ε have been

320

near the adiabatic wall. Increasing ε , shifts the peak of nanoparticle concentration toward the

321

middle of the microchannel in such a way that for ε = 1 (symmetric heat flux at the walls), the

322

nanoparticle distribution becomes symmetric and the nanoparticle accumulation region is formed

323

in the middle of the microchannel. Further increase in ε leads the peak of the nanoparticle

324

concentration toward the top wall (not shown). As an explanation, it should be noted that the

325

thermophoresis, which is related to the temperature gradient, is the mechanism of the

326

nanoparticle migration from the heated walls. Any change in ε causes the temperature gradient

327

at the walls to change, as well as the thermophoresis. For ε < 1 , the temperature gradient at the

328

top wall is more than that at the bottom wall; therefore, the nanoparticle migration from the top

329

wall is higher, which leads the nanoparticle rich region to move toward the bottom wall. This

330

phenomenon continues until ε = 0 , in which there is no temperature gradient at the bottom wall;

331

so a nanoparticle accumulation region is formed on the bottom wall. Moreover, the effects of ε

332

on nanoparticles concentration have considerable influence on the velocity and temperature

333

profiles. Obviously, the velocity and temperature profiles are symmetric for ε = 1 , in which the

334

nanoparticle volume fraction is symmetric. For ε < 1 a regular symmetry in velocity profile

335

disappears and the peak of the velocity profiles shifts toward the top wall where the nanoparticle

336

depleted region is located (lower viscosity). However, the dip point of the temperature profile

337

moves toward the bottom wall in which the nanoparticles accumulated (higher thermal

338

conductivity). In fact, the velocity profile has a tendency to shift toward the nanoparticle

339

depleted region. However, it is vice versa for the temperature profile. Figure 6 depicts the

340

variations of the nanoparticle volume fraction ( φ / φB ), velocity (u/uB), and temperature (T*/T*B)

341

profiles for different values of φB . It can be observed that φB does not have any significant

342

change in nanoparticle distribution, velocity and temperature profiles. However, increasing 20

φB

343

strengthen the effects of the nanoparticle volume fraction on the viscosity and thermal

344

conductivity (will be shown later in Figure 10). Hartmann number (Ha) represents the

345

importance of magnetic field on the flow, which its effects on the profiles have been illustrated

346

in Figure 7. As it is obvious, Ha has a strong influence on the velocity profile; the peak of the

347

velocity profile in the core region decreases, however, the velocity near the walls increases. In

348

fact, the momentum in the core region moves toward the walls (so does the nanoparticle

349

concentration). Hence, the near wall velocity gradients increase with increasing Ha, enhancing

350

the slip velocity (slip velocity is proportional to the velocity gradients, see Eq. (26)). This is

351

because in the presence of transverse magnetic field sets in a resistive type force (Lorentz force),

352

which is a retarding force on the velocity field. Finally, an examination of Figure 8 reveals an

353

increase in the velocities of the fluid closed to the walls, followed by a decrease in the core

354

region, as λ increases. This is because the slip parameter λ signifies the amount of slip velocity at

355

the surface. An increase in λ leads to a rise in the slip velocity near the walls and because of the

356

constant mass flow rate inside the microchannel, the velocity in the core region reduces. Thus,

357

the momentum in the core region moves away toward the walls and leads to a more uniform

358

velocity profile, as λ increases. The momentum enrichments near the walls increase the heat

359

removal ability of the flow and reduces the walls' temperature. As a result, the nanoparticle

360

migration from the heated walls is reduced. This leads to a rise in the nanoparticles concentration

361

and temperature gradients at the walls.

362 363

6-2

Nanoparticle migration effects on the viscosity and thermal conductivity

364

As described before, nanoparticle migration has a strong effect on the viscosity and thermal

365

conductivity of nanofluids. Therefore, the effects of different parameters including 21

φB , ε, Ha,

366

and λ on the ratio of the viscosity and thermal conductivity of nanofluids to that of the base fluid

367

at the heated walls, in terms of NBT, have been shown in Figures 9-12, respectively. As it is

368

obvious, increasing NBT (smaller nanoparticles) leads to an increase in the viscosity and thermal

369

conductivity ratio except for the case where the bottom wall is adiabatic, i.e. ε=0. This is

370

because as NBT increases, the nanoparticle migration from the walls reduces and there is no

371

nanoparticle depletion region at the walls (nanoparticle concentration at the walls increase).

372

However, for ε=0, nanoparticles accumulate at the bottom wall and reducing the nanoparticle

373

migration results in a decrease in the accumulation of nanoparticles at the bottom walls. Hence,

374

the viscosity and thermal conductivity at the bottom wall take a decreasing trend for ε=0. In

375

addition, it can be observed that the viscosity and thermal conductivity at the bottom wall is

376

always greater than that at the top wall which is due to a higher value of nanoparticle

377

concentration there. Regarding Figure 9, it can be seen that increasing

378

viscosity and thermal conductivity of nanofluids. Furthermore, it can be observed that for higher

379

values of

380

increases. Figure 10 indicated that as ε gets smaller, the viscosity and thermal conductivity at the

381

bottom wall (lower heat flux) increases, leading to a region with a very large viscosity and

382

thermal conductivity, whereas a reversed trend can be observed at the top wall. This explains

383

why the symmetry of the profiles which is shown in Figure 5, rapidly disappear. Moreover, the

384

range of variations in the viscosity is much higher than the thermal conductivity which explains

385

the higher sensitivity of the velocity profile compared to the temperature one. Figure 11 shows

386

that increasing Ha leads to a slight rise in the viscosity and thermal conductivity at the bottom

387

wall. Also, an increase in λ leads to a rise in the viscosity and thermal conductivity at both walls

388

which has been shown in Figure 12.

φB intensifies the

φB , the difference between the viscosity and thermal conductivity of the heated walls

22

389

Up to now, we describe how the different parameters influence the nanoparticle migration and

390

accordingly how the profiles of the velocity, temperature and nanoparticle volume fraction

391

profiles are changed with the non-uniform distribution of the viscosity and thermal conductivity.

392

In the following section, we study these effects on the physical quantities of interest, including

393

the heat transfer rate and the pressure drop.

394

6-3

Heat transfer rate and pressure drop

395

The dimensionless heat transfer coefficient (HTC) at the top and the bottom walls can be defined

396

respectively as

HTClw =

ht H qt H 1 k 1 = = − * t = − * (1 + akφ( 0) + bkφ(20) ) kbf (Tt − TB ) kbf TB kbf TB

hb H qb'' H k ε ε HTCrw = = = * * t = (1 + akφ( 0) + bkφ(20) ) * * kbf (Tb − TB ) kbf (Tb − TB ) kbf T (1) − TB

(

397

)

(34)

(35)

Besides, the pressure drop ratio can be defined as

Np =

ρB −dp ⎛ μbf uB ⎞ ⎜ ⎟= dx ⎝ H ⎠ ρu*

(36)

398

For finding the advantage of the nanofluids, it is better to define the heat transfer ratio at the

399

top and bottom walls as

HTCRt =

HTCRrw = 400

ht[ nf ] ht[bf ]

hb[nf ] hb[bf ]

ht[ nf ] H ht[bf ] H HTCt = / kbf kbf HTCt[bf ]

(37)

=

hb[nf ] H hb[bf ] H HTCb = / kbf kbf HTCb[bf ]

(38)

the total heat transfer ratio can be expressed as

HTCRtotal = 401

=

ht[ nf ] + hb[ nf ] ht [bf ] + hb[bf ]

=

HTCt [ nf ] + HTCb[ nf ] HTCt[bf ] + HTCb[bf ]

and the pressure drop ratio can be defined as 23

(39)

N dp = 402

N p[ nf ] N p[bf ]

Figures 13 to 16 show the effects of different parameters including

(40)

φB , ε, Ha, and λ on heat

403

transfer ratio at (a) the top wall, (b) the bottom wall, (c) the total heat transfer ratio, and (d) the

404

pressure drop ratio, respectively. It should be mentioned that the heat transfer ratio does not

405

indicate the heat transfer rate. In fact, it shows the heat transfer enhancement (the advantage of

406

the nanofluid). For example, if the heat transfer ratio (HTCR) is decreased by increasing NBT, it

407

means that it is better to use the nanofluid at the lower values of NBT since the rate of the heat

408

transfer enhancement is greater. Regarding the figures, the heat transfer rate and the pressure

409

drop ratios do not have a unique behavior with increasing NBT, indicating the existence of a

410

complex behavior of the rheological and thermophysical properties of nanofluids with the

411

parameters. However, it was shown that an increase in NBT intensifies the nanoparticle

412

concentration at the walls (Figure 4), so does the thermal conductivity. Thus, it is expected that

413

the heat transfer rate at the walls would increase, but this does not always happen. The reason is

414

that two mechanisms affect the heat transfer rate. The first one is the thermal conductivity

415

enhancement and the second one is the hydrodynamic boundary layer, which is affected by

416

the viscosity of the nanofluid. These two mechanisms run against each other such that the

417

dominant effect determines the behavior of the heat transfer rate. This is the main reason that the

418

behavior of the viscosity and thermal conductivity for different parameters have been explained

419

before in Figures 9-12. In what follows, the effects of each mechanism are described separately

420

for each parameter.

421

Figure 13 depicts the enhancement of both heat transfer rate ratios (HTCR) and pressure drop

422

ratio (Ndp) with increasing

φB which is due to enhancement of the viscosity and thermal 24

423

conductivity of nanofluids at the walls (see Figure 10). Figure 14 shows an identical heat transfer

424

ratio for the symmetric heating at the walls (ε=1). But, it can be observed that the heat transfer

425

rate ratio at the top wall is the lowest, whereas, the heat transfer ratio at the bottom wall is the

426

highest. This means that reducing ε enhance the advantage of nanofluid at the top wall, however,

427

at the bottom wall it is vice versa. For the case where the bottom wall is adiabatic (ε=0), the heat

428

transfer ratio at the top wall is the greatest due to the momentum enrichments there and it is

429

evident that the heat transfer rate at the bottom wall is zero (not shown since it is adiabatic).

430

Total heat transfer ratio (total advantage of nanofluid) is shown in Figure 16c, indicating that one

431

directional heating (one of the walls kept adiabatic) leading to the best heat transfer

432

enhancement. For finding the reason it should bear in mind that for ε=0, due to very much

433

viscosity at the bottom wall (Figure 10a), momentum enrichments at the top wall (second

434

mechanism) is significant and responsible for the greatest heat transfer enhancement. It should be

435

noted that for ε=1, the thermal conductivity enhancement (first mechanism) increases the heat

436

transfer rate. Thus, it can be concluded that the heat transfer enhancement due to momentum

437

enrichments at the walls due to the nanoparticle depletion (second mechanism) is much stronger

438

than the thermal conductivity enhancement (first mechanism). In addition, for ε=0.5, two heat

439

fluxes at the walls eject the nanoaprticles from the walls and prevent the high increment of

440

thermal conductivity (first mechanism) and the viscosity. In this case, depletion at the walls is

441

not strong enough and momentum does not shift very much toward the walls to enhance the heat

442

transfer rate. As a result, lowest heat transfer enhancement is obtained for asymmetric heating

443

ε=0.5. In addition, inducing the temperature gradient at the walls with increasing ε, the viscosity

444

at the walls decreases and leads to a fall in the pressure drop ratio Ndp. Figure 15 shows the

445

effects of Hartmann number Ha on HTCR and Ndp for a range of NBT. As it is obvious, the heat

25

446

transfer ratio at both the walls is decreased for the higher values of Ha. In other words, the

447

advantage of the nanofluid is decreased with increasing Ha. It is interesting to note that for the

448

higher values of Ha, the total heat transfer ratio is lower than unity, meaning that using

449

nanofluids in the presence of high magnetic field can lead to a lower heat transfer rate. To show

450

this, the values of the total heat transfer coefficient (HTC) and pressure drop (Ndp) for the

451

nanofluid and base fluid is tabulated in Table 4. As it is obvious, for Ha=10, heat transfer

452

coefficient of the base fluid (37.183) is higher than the nanofluid (32). However, the pressure

453

drop for the nanofluid is always greater than the base fluid, suggesting the negative performance

454

of the nanoaprticles inclusion. Regarding Figure 16, the total heat transfer ratio is decreased by

455

increasing the slip parameter λ, although an increase of the heat transfer ratio at the top wall is

456

observed. This indicates that the heat transfer enhancement of the nanofluid is much lower than

457

that of base fluid, and decreases the heat transfer ratio. Similar trend can be observed for the

458

pressure drop (Ndp). Decreasing the pressure drop (Ndp) is a useful attribute, particularly in

459

microchannels where the head loss is extremely pronounced, since it allows a significant

460

reduction in the pumping power required to drive the nanofluid flow. However, it should be

461

noted that slip effects reduce the advantage of using the nanoparticles such that in the higher

462

values of slip parameter λ, heat transfer rate of nanofluid is lower than the base fluid.

463

7. Summary and Conclusions

464

The current study deals with the nanoparticle migration in magnetohydrodynamic laminar forced

465

convection of alumina/water nanofluid in microchannels. Walls are subjected to different heat

466

,, ,, fluxes; qwl for the top wall and qwr for the bottom wall, and nanoparticles are assumed to have a

467

slip velocity relative to the base fluids induced by Brownian motion and thermophoresis. The

468

effects of different parameters including the ratio of Brownian motion to thermophoretic

469

diffusivities NBT, the ratio of heat fluxes at the walls ε, Hartmann number Ha, slip parameter λ, 26

φB on the velocity, temperature, and nanoparticle

470

and bulk mean nanoparticle volume fraction

471

concentration profiles as well as heat transfer rate and pressure drop were investigated in detail.

472

The major findings of this paper can be expressed in the following ways:

473

•

Due to the thermophoresis, nanoparticles migrate from the heated walls (making a

474

depleted region) and accumulate at the central regions of the microchannel, more likely to

475

accumulate toward the wall with a lower heat flux. As a result, an inhomogeneous

476

concentration distribution of nanoparticles develops, which leads to a non-uniform

477

distribution of the viscosity and thermal conductivity of nanofluids.

478

•

Nanoparticle flux at the heated walls is increased with the use of larger nanoparticles (for

479

lower values of NBT), which results in more depletion of nanoparticles near the heated

480

walls. Thus, the local viscosity and thermal conductivity of nanofluids at the heated walls

481

is reduced for larger nanoparticles. Reduction of thermal conductivity has negative

482

effects on heat transfer rate; however, a decrease in the viscosity leads to momentum

483

enrichment near the walls, which enhances the heat removal from the surface walls. So,

484

two mechanisms have been observed for the heat transfer enhancement in nanofluids.

485

The first is the case where the Brownian motion is prominent and nanoparticle

486

distribution is uniform (thermal conductivity enhancement). This is the case where there

487

are no abnormal variations in heat transfer rate (expected heat transfer enhancement). The

488

second is the case where the thermophoresis has been overwhelming the Brownian

489

motion and leads to a reduction in momentum enrichments near the walls followed by

490

thermal conductivity reductions at the walls. If the contribution of momentum

491

enrichments near the walls is lower than the effects of thermal conductivity reductions,

492

the heat transfer rate does not augment as it was expected. This explains why the pure

493

fluid correlations are not successful in modelling the heat transfer and shows the

494

importance of considering the nanoparticle migration in nanofluids.

27

495

•

One direction heating (one of the walls is kept adiabatic) intensifies heat transfer

496

enhancement. However, asymmetric heating at the walls, prevents the enrichments of the

497

velocities at the walls, which reduces the heat transfer enhancement.

498

•

In the presence of a strong magnetic field, for the higher values of NBT, the total heat

499

transfer ratio is lower than unity, meaning that using nanofluids leads to a lower heat

500

transfer rate. In other words, heat transfer coefficient of the base fluid is higher than the

501

nanofluid. However, the pressure drop for the nanofluid is always greater than the base

502

fluid, suggesting the negative performance of the nanoaprticles inclusion in the presence

503

of a strong magnetic field.

504

•

Heat transfer rate increases with λ and due to the smoother velocity gradients at the walls,

505

the pressure drop Np has a decreasing trend. However, these trends for the base fluid and

506

the nanofluid are different such that for the higher values of λ, inclusion of nanoparticles

507

has negative effects on the heat transfer rate and the pressure drop. Hence, combinations

508

of the heat transfer enhancement techniques (slip velocity due to the surface roughness

509

and nanoparticles inclusion) should be carefully considered.

510

Appendix

511

A- Momentum Equation

d ⎛ du ⎞ dp 2 ⎜ μ ⎟ − −σ B u = 0 dy ⎝ dy ⎠ dx

(A-1)

2 ⎛ ⎛ d μ dφ dη ⎞⎛ du dη ⎞ d 2u ⎛ dη ⎞ ⎞ dp 2 ⎜⎜ ⎟ + μ 2 ⎜ ⎟ ⎟⎟ − − σ B u = 0 ⎜ ⎝ dφ dη dy ⎟⎜ dη ⎝ dy ⎠ dx ⎠⎝ dη dy ⎠ ⎝ ⎠

(A-2)

y = Hη 28

(A-3)

dη 1 = dy H

(A-4)

(A-3) and (A-4) in (A-2): ⎛ 1 d μ dφ ⎞ du μ d 2u dp 2 ⎜ 2 ⎟ + 2 2 − −σ B u = 0 ⎝ H dφ dη ⎠ dη H dη dx μbf μ

× (A-5):

μbf d 2u ⎛ μbf 1 d μ dφ ⎞ du μbf dp μbf σ B2u = −⎜ 2 + ⎟ + 2 2 H dη ⎝ H μ dφ dη ⎠ dη μ dx μ u* =

(A-5)

(A-6)

u H ⎛ dp ⎞ − μbf ⎜⎝ d x ⎟⎠ 2

(A-7)

d 2u* 1 d μ dφ du* μbf H 2 =− − + σ B2u* 2 dη μ dφ dη dη μ μ Ha 2 =

σ B2 H 2 μbf

d 2u * 1 d μ dφ du * μbf = − − 1 − Ha 2u * dη 2 μ dφ dη dη μ

(

512

)

B- Thermally fully developed relations

ρcu

dT d ⎛ dT ⎞ = ⎜k ⎟ dx d y⎝ d y⎠ TB ≡

(B-1)

ρ cuT ρ cu

(B-2)

y = Hη H

For each parameter (Γ):

Γ ≡

29

1 1 ΓdA = ∫ Γdy ∫ AA H0

(B-3)

H dTB 1 ∂ ⎛ ∂T ⎞ = ρcu ⎜ k ⎟dy dx H ∫0 ∂y ⎝ ∂y ⎠

(B-4)

H

dT 1 ⎛ ∂T ⎞ ρcu B = ⎜ k ⎟ dx H ⎝ dy ⎠0 − kt

∂T ⎞ ∂T ⎞ = qt,, , − kb = − qb,, ⎟ ⎟ ∂y ⎠ y =0 ∂y ⎠ y = H

ρ cu

dTB 1 ,, = qt + qb,, dx H

(

ρcu

514

dTB qt,, = (ε +1) dx H ε=

513

)

(B-5)

(B-6) (B-7) (B-8)

qb,, qt,,

C- Energy equation

∂T ∂ ⎛ ∂T ⎞ = ⎜k ⎟ ∂x ∂ y ⎝ ∂ y ⎠

(C-1)

ρ cu

dT d ⎡ ⎛ Hqt dT * ⎞⎤ = ⎢k ⎜ ⎟⎥ dx dy ⎢⎣ ⎝ kt dy ⎠⎥⎦

(C-2)

ρ cu

q d ⎡ ⎛ dT * ⎞ ⎤ dT = t ⎢k ⎜ ⎟⎥ dx Hkt dη ⎣ ⎝ dη ⎠ ⎦

(C-3)

q ⎛ dk dφ dT * dT d 2T * ⎞ = t ⎜ +k 2 ⎟ dx Hkt ⎝ dφ dη dη dη ⎠

(C-4)

qt,, dT dTB = = (ε + 1) dx dx H < ρ cu >

(C-6)

qt,, qt,, ⎛ dk dφ dT * d 2T * ⎞ ρcu* k ε 1 + = + ( ) ⎜ ⎟ H d 2η ⎠ < ρcu* > Hkt ⎝ dφ dη dη

(C-7)

ρ cu

ρ cu

ρcu* 1 dk dφ dT * k d 2T * = + (ε +1) < ρcu* > kt dφ dη dη kt d 2η 30

k = k bf (1 + ak φ + bk φ 2 )

d 2T * kt ⎡ ρcu* 1 dk dφ dT * ⎤ = ⎢ ( ε +1) − ⎥ d 2η k ⎣ < ρcu* > kt dφ dη dη ⎦

(C-8) (C-9)

515 516

D- Nanoparticle fraction conservation equation

∂ ⎛ ∂φ CTφ ∂T ⎞ + ⎜ CBT ⎟=0 ∂y⎝ ∂y T ∂y⎠

(D-1)

∂ ⎛ ∂φ CTφ ∂T ⎞ ∂η + =0 ⎜ CBT ⎟ ∂η ⎝ ∂ y T ∂ y⎠∂ y

(D-2)

qt H * T kt

(D-3)

dT qt H dT * = dη kt dη

(D-4)

qt H ∂T * CT φ ∂φ =− (D-3) and (D-4) in (D-2): qH ∂η (1 + t T * )2 Tt 2CB kt ∂η

(D-5)

T −Tt =

Tt kt

qt H Tk t t

(D-6)

CB ktTt 2 = CT Hqt

(D-7)

γ= N BT

∂φ φ ∂T * =− ∂η NBT (1+ γ T * )2 ∂η

(D-8)

517 518

References

519 520

[1] A.E. Bergles, Heat Transfer Augmentation, in: S. Kakaç, A. Bergles, E.O. Fernandes (Eds.) TwoPhase Flow Heat Exchangers, Springer Netherlands, 1988, pp. 343-373.

521 522

[2] M.R.H. Nobari, A. Malvandi, Torsion and curvature effects on fluid flow in a helical annulus, International Journal of Non-Linear Mechanics, 57(0) (2013) 90-101.

31

523

[3] J.C. Maxwell, A Treatise on Electricity and Magnetism, 2nd ed., Clarendon press, Oxford, Uk, 1873.

524 525

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35

652

*Tables

653

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

variable symbol

value

Particle diameter

dp (nm)

20

Heat Flux

q (kW)

5000

kp (W/(m.K))

36

kbf (W/(m.K))

0.597

Conductivity

3

3880

3

ρbf (kg/m )

998.2

(cp)p (J/(kg.K))

773

(cp)bf (J/(kg.K))

4182

Viscosity

μbf (Ns/m2)

9.93×10-4

Boltzmann’s constant

kB (J/K)

1.38×10-23

Density Specific heat capacity

Bulk concentration

ρp (kg/m )

0.1

B

Reference values Length scale

L (μm)

50

Velocity

U(cm/s)

20

Temperature

T(K)

300

Velocity difference

ΔU (cm/s)

15

Axial dimension difference

Δx (cm)

1

Temperature difference

ΔT (K)

50

Concentration difference

Δ

0.3

654 655 656

36

657 Table 2 Results of the numerical solution when λ = 0.1 , ε=0.5, and Ha=5. B=0.02 B =0.06 NBT * * -4 < ρ cu >×10 < ρ cu >×10-4 w w 0.011494 10.40014 0.035796 7.018205 0.5 1

0.015102

10.20633

0.045757

6.64008

5

0.018888

10.01938

0.056664

6.283874

10

0.019435

9.993556

0.058295

6.235048

658

Table 3 Grid independence test for different values of dη when NBT=1, λ = 0.1 , ε=0.5, and

Ha=5. dη

B=0.02

B=0.06

Np

HTClw

HTCrw

Np

HTClw

HTCrw

10-5

40.6638

4.1354

19.6801

61.7802

5.1582

22.1184

5×10-6

40.6705

4.1355

19.7001

61.8059

5.1609

22.1215

10-6

40.6824

4.1386

19.7084

61.8179

5.1670

22.1274

659

Table 4 HTCtotal and Np for different values of Ha when NBT=1, λ = 0.1 , and ε=0.5. Np

HTCtotal

Fluid Ha=0

Ha=5

Ha=10

Ha=0

Ha=5

Ha=10

Nanofluid

25.6358

27.2823

31.9969

35.138239

61.80586

140.808

Water

18.2092

23.4685

37.1825

33.980

111.111

660 661

37

7.5

*

Figures

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

Fig. 2. Algorithm of the numerical method

Fig. 3. Comparison of numerical results for NuB Yang et al. [20] when

HTC kbf / k B with the ones reported by

1 and Nr

Ha

0.

Fig. 4. The effects of NBT on nanoparticle distribution ( / *

*

temperature (T /T B) profiles when ε=0.5, Ha=5,

Fig. 5. The effects of ε on nanoparticle distribution ( / *

B

*

(T /T B) profiles when NBT =1, Ha=5,

B

B

B

), velocity (u/uB) and

=0.06 and λ=0.1.

), velocity (u/uB) and temperature

=0.06 and λ=0.1.

Fig. 6. The effects of B on nanoparticle distribution ( / B ), velocity (u/uB) and temperature (T*/T*B) profiles when ε=0.5, Ha=5, NBT =1 and λ=0.1.

Fig. 7. The effects of Ha on nanoparticle distribution ( / temperature (T*/T*B) profiles when ε=0.5, NBT =1,

B B

), velocity (u/uB) and

=0.06 and λ=0.1.

Fig. 8. The effects of λ on nanoparticle distribution ( / (T*/T*B) profiles when ε=0.5, Ha=5,

B

B

), velocity (u/uB) and temperature

=0.06 and NBT =1.

(a)

(b)

Fig. 9. The effects of B on the dimensionless heat transfer coefficient viscosity ratio (a) and thermal conductivity ratio (b) when ε=0.5, Ha=5, and λ=0.1.

(a)

(b)

Fig. 10. The effects of ε on the dimensionless heat transfer coefficient viscosity ratio (a) and thermal conductivity ratio (b) when Ha=5, B =0.06 and λ=0.1.

(a)

(b)

Fig. 11. The effects of Ha on the dimensionless heat transfer coefficient viscosity ratio (a) and thermal conductivity ratio (b) when ε=0.5, B =0.06 and λ=0.1.

(a)

(b)

Fig. 12. The effects of λ on the dimensionless heat transfer coefficient viscosity ratio (a) and thermal conductivity ratio (b) when ε=0.5, Ha=5, and B =0.06.

(a)

(b)

(c)

(d)

Fig. 13. The effects of B on the dimensionless heat transfer coefficient at the top wall (HTCRt) (a) , at the bottom wall (HTCRb) (b), total dimensionless heat transfer coefficient (HTCRtotal) (c), and dimensionless pressure drop (Ndp) (d) when ε=0.5, Ha=5 and λ=0.1.

(a)

(b)

(c)

(d)

Fig. 14. The effects of ε on the dimensionless heat transfer coefficient at the top wall (HTCRt) (a) , at the bottom wall (HTCRb) (b), total dimensionless heat transfer coefficient (HTCRtotal) (c), and dimensionless pressure drop (Ndp) (d) when Ha=5, B =0.06 and λ=0.1.

(a)

(b)

(c)

(d)

Fig. 15. The effects of Ha on the dimensionless heat transfer coefficient at the top wall (HTCRt) (a) , at the bottom wall (HTCRb) (b), total dimensionless heat transfer coefficient (HTCRtotal) (c), and dimensionless pressure drop (Ndp) (d) when ε=0.5, B =0.06 and λ=0.1.

(a)

(b)

(c)

(d)

Fig. 16. The effects of λ on the dimensionless heat transfer coefficient at the top wall (HTCRt) (a) , at the bottom wall (HTCRb) (b), total dimensionless heat transfer coefficient (HTCRtotal) (c), and dimensionless pressure drop (Ndp) (d) when ε=0.5, Ha=5 and B =0.06.

Highlights (for review)

Highlights

1. Magnetohydrodynamic forced convection of alumina/water nanofluid in microchannels 2. Nanoparticles

migration

effects

on

rheological

and

thermophysical

characteristics 3. Brownian motion and thermophoresis effects on nanoparticles migration 4. Effects of asymmetric heating on the heat transfer enhancement 5. Describing the anomalous heat transfer enhancement in nanofluids

Graphical Abstract (for review)