The Cheng–Minkowycz problem for the double-diffusive natural convective boundary layer flow in a porous medium saturated by a nanofluid

International Journal of Heat and Mass Transfer 54 (2011) 374–378

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The Cheng–Minkowycz problem for the double-diffusive natural convective boundary layer flow in a porous medium saturated by a nanofluid D.A. Nield a, A.V. Kuznetsov b,⇑ a b

Department of Engineering Science, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand Department of Mechanical and Aerospace Engineering, North Carolina State University, Campus Box 7910, Raleigh, NC 27695-7910, USA

a r t i c l e

i n f o

Article history: Received 10 June 2010 Received in revised form 18 August 2010 Accepted 18 August 2010 Available online 12 October 2010 Keywords: Nanofluid convection Porous media Brownian motion Thermophoresis Natural convection Vertical plate

a b s t r a c t The paper presents an analytical treatment of double-diffusive nanofluid convection in a porous medium. The problem treated is natural convection past a vertical plate when the base fluid of the nanofluid is itself a binary fluid such as salty water. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis, while the Darcy model is used for the porous medium. In addition the thermal energy equations include regular diffusion and cross-diffusion terms. A similarity solution is presented. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction The term ‘‘nanofluid” (the term was introduced by Choi [1]) refers to a liquid containing a suspension of metallic or non-metallic (for example, metal oxide) nanometer-sized solid particles or fibres (nanoparticles or nanorods). Nanofluids are characterized by enhanced thermal conductivity, a phenomenon observed by Masuda et al. [2]. This makes nanofluids attractive for numerous engineering applications involving cooling and heat exchange, including advanced nuclear systems [3]. Heat transfer in nanofluids has been surveyed in a review article by Das and Choi [4] and a book by Das et al. [5]. Heat transfer characteristics of nanofluids have been reviewed by Wang and Mujumdar [6]. Buongiorno [7] noted that the nanoparticle absolute velocity can be viewed as the sum of the base fluid velocity and a relative velocity (that he calls the slip velocity). He considered in turn seven slip mechanisms: inertia, Brownian diffusion, thermophoresis, diffusiophoresis, Magnus effect, fluid drainage, and gravity settling. He concluded that in the absence of turbulent effects it is the Brownian diffusion and the thermophoresis that will be important. Buongiorno proceeded to write down conservation equations based on these two effects. His model is the basis of the present study. ⇑ Corresponding author. E-mail addresses: [email protected] (D.A. Nield), [email protected] (A.V. Kuznetsov). 0017-9310/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2010.09.034

The problem of natural convection in a porous medium past a vertical plate is a classical problem first studied by Cheng and Minkowycz [8]. The problem is presented as a paradigmatic configuration and solution in the book by Bejan [9]. The extension to the case of heat and mass transfer was made by Bejan and Khair [10]. Further work on this topic is surveyed in Sections 5.1 and 9.2.1 in Nield and Bejan [11]. Of particular interest are the papers by Khaled and Chamkha [12], Chamkha and Quadri [13], and Chamkha and Pop [14]. The extension to the case of a nanofluid was made by Nield and Kuznetsov [15]. The goal of the present paper is to investigate cross-diffusion in nanofluids, with the aim of making a detailed comparison with regular cross diffusion effects and the cross-diffusion effects peculiar to nanofluids, and at the same time investigating the interaction between these effects when the base fluid of the nanofluid is itself a binary fluid such as salty water. The outcome is that we investigating a sort of triple-diffusion problem involving heat, the nanoparticles and the solute. The analysis is an extension of that presented by Nield and Kuznetsov [15]. 2. Analysis It is assumed that nanoparticles are suspended in the nanofluid using either surfactant or surface charge technology. This prevents particles from agglomeration and deposition on the porous matrix. We consider a two-dimensional problem. We select a coordinate frame in which the x-axis is aligned vertically upwards. We

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Nomenclature C DB DT DCT DTC DSm f

solutal concentration Brownian diffusion coefficient thermophoretic diffusion coefficient Soret diffusivity Dufour diffusivity solutal diffusivity of the porous medium dimensionless rescaled nanoparticle volume fraction, defined by Eq. (23) gravitational acceleration vector effective thermal conductivity of the porous medium permeability of the porous medium Dufour-solutal Lewis number, defined by Eq. (34) regular Lewis number, defined by Eq. (33) nanofluid Lewis number, defined by Eq. (35) buoyancy ratio, defined by Eq. (29) Brownian motion parameter, defined by Eq. (30) regular buoyancy ratio, defined by Eq. (28) modified Dufour parameter, defined by Eq. (32) thermophoresis parameter, defined by Eq. (31) Nusselt number, defined by Eq. (45) reduced Nusselt number, Nu/Rax1/2 pressure wall heat flux local Rayleigh number, defined by Eq. (21) dimensionless stream function, defined by Eq. (23) temperature temperature at the vertical plate

g km K Ld Le Ln Nr Nb Nc Nd Nt Nu Nur p q 00 Rax s T Tw

T1 v (x, y)

ambient temperature attained as y tends to infinity Darcy velocity (u, v) Cartesian coordinates (x-axis is aligned vertically upwards, plate is at y = 0)

Greek symbols m am thermal diffusivity of the porous medium, ðqkcÞ f bT volumetric thermal expansion coefficient of the fluid bC volumetric solutal expansion coefficient of the fluid c dimensionless solutal concentration, defined by Eq. (23) e porosity g similarity variable, defined by Eq. (22) h dimensionless temperature, defined by Eq. (23) l viscosity of the fluid qf fluid density qp nanoparticle mass density (qc)f heat capacity of the fluid effective heat capacity of the porous medium (qc)m (qc)p effective heat capacity of the nanoparticle material ðqcÞ r heat capacity ratio, defined by Eq. (15), ðqcÞm f eðqcÞ s nanofluid heat capacity ratio, defined by Eq. (15), ðqcÞ p f / nanoparticle volume fraction /w nanoparticle volume fraction at the vertical plate /1 ambient nanoparticle volume fraction attained as y tends to infinity U dimensionless variable defined by Eq. (40) w stream function, defined by Eq. (16)

consider a vertical plate at y = 0. At this boundary the temperature T, the solute concentration C and the nanoparticle fraction / take constant values Tw, Cw and /w , respectively. The ambient values, attained as y tends to infinity, of T, C and / are denoted by T1, C1 and /1 , respectively. The Oberbeck–Boussinesq approximation is employed. Homogeneity and local thermal equilibrium in the porous medium is assumed. We consider a porous medium whose porosity is denoted by e and permeability by K. The Darcy velocity is denoted by v. The following four field equations embody the conservation of total mass, momentum, thermal energy, and nanoparticles, respectively. The field variables are the Darcy velocity v, the temperature T, the solutal concentration C and the nanoparticle volume fraction /

particles. The gravitational acceleration is denoted by g. We have introduced the effective heat capacity (qc)m, and the effective thermal conductivity km of the porous medium. The coefficients that appear in Eqs. (3) and (4) are the Brownian diffusion coefficient DB and the thermophoretic diffusion coefficient DT and DTC is a diffusivity of Dufour type, DSm is the solutal diffusivity for the porous medium and DCT is a diffusivity of Soret type. Details of the derivation of Eqs. (3) and (5) are given in the papers by Buongiorno [7], and Nield and Kuznetsov [15]. The flow is assumed to be slow so that an advective term and a Forchheimer quadratic drag term do not appear in the momentum equation. It is assumed that the solute does not affect the transport of the nanoparticles. The boundary conditions are taken to be

r  v ¼ 0;

v ¼ 0;

ð1Þ

u ¼ v ¼ 0;

qf @ v l ¼ rp  v þ ½/qp þ ð1  /Þfqf ð1  bT ðT  T 1 Þ e @t K  bC ðC  C 1 ÞÞgg; ðqcÞm

@T þ ðqcÞf v  rT ¼ km r2 T þ eðqcÞp ½DB r/  rT @t þ ðDT =T 1 ÞrT  rT þ ðqcÞm DTC r2 C;

T ¼ T w;

ð2Þ

ð3Þ

@C 1 þ v  rC ¼ DSm r2 C þ DCT r2 T; @t e

ð4Þ

@/ 1 þ v  r/ ¼ DB r2 / þ ðDT =T 1 Þr2 T: @t e

ð5Þ

C ¼ Cw;

T ! T 1;

/ ¼ /w at y ¼ 0;

C ! C1;

/ ! /1 as y ! 1:

ð6Þ ð7Þ

We consider a steady state flow. In keeping with the Oberbeck–Boussinesq approximation and an assumption that the nanoparticle concentration is dilute, and with a suitable choice for the reference pressure, we can linearize the momentum equation and write Eq. (2) as

l

v þ ½ðqp  qf 1 Þð/  /1 Þ K þ ð1  /1 Þqf 1 fbT ðT  T 1 Þ þ bC ðC  C 1 Þgg:

0 ¼ rp 

ð8Þ

We now make the standard boundary-layer approximation, based on a scale analysis, and write the governing equations

We write v = (u, v). Here qf, l and bT are the density, viscosity, and volumetric volume expansion coefficient of the fluid, bC is the analogous solutal coefficient (something that is negative if, as is usual, the solute is more dense that the solvent) while qP is the density of the

@u @ v þ ¼ 0; @x @y @p l ¼  u þ ½ð1  /1 Þqf 1 gfbT ðT  T 1 Þ þ bC ðC  C 1 Þg @x K  ðqp  qf 1 Þgð/  /1 Þ;

ð9Þ

ð10Þ

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@p ¼ 0; @y

ð11Þ

"   2 # @T @T @/ @T DT @T @2C 2 u þv ¼ am r T þ s DB þ þ rDTC 2 ; @x @y @y @y @y @y T1 ð12Þ   @C @C @2C @2T þv ¼ DSm 2 þ DCT 2 ; u @y @y @y e @x

1

    2 1 @/ @/ @2/ DT @ T ¼ DB 2 þ þv u ; @y @y e @x T 1 @y2

km ; ðqcÞf

s¼

eðqcÞp ðqcÞm ; r¼ : ðqcÞf ðqcÞf

v

@w ¼ ; @x

ð17Þ

"   2 # @w @T @w @T @2T @/ @T DT @T @2C  ¼ am 2 þ s DB þ þ rDTC 2 ; @y @x @x @y @y @y @y @y @y T1 ð18Þ

  1 @w @C @w @C @2C @2T  ¼ DSm 2 þ DCT 2 ; @y @y e @y @x @x @y

ð19Þ

    2 1 @w @/ @w @/ @2/ DT @ T ¼ DB 2 þ  : @y e @y @x @x @y T 1 @y2

ð20Þ

We now follow the path blazed by Bejan and Khair [10]. We investigate the situation where heat transfer dominates over mass transfer and the case where the Lewis number is large. Scale analysis (see Section 9.2.1 of Nield and Bejan [11]) then indicates that the following transformation of variables is appropriate. We introduce the local Rayleigh number Rax defined by

lam

ð21Þ

;

and the similarity variable

y x

g ¼ Rax1=2 :

ð22Þ

We also introduce the dimensionless variables s, h, c and f defined by

sðgÞ ¼

cðgÞ ¼

w

am Ra1=2 x

;

C  C1 ; Cw  C1

hðgÞ ¼

T  T1 ; Tw  T1

f ðgÞ ¼

/  /1 : /w  /1

ð23Þ

Then, on substitution in Eqs. (17)–(20), we obtain the ordinary differential equations 00

0

0

where the various parameters are defined by

Nc ¼

bC ðC w  C 1 Þ ; bT ðT w  T 1 Þ

ð28Þ

Nr ¼

ðqp  qf 1 Þð/w  /1 Þ ; qf 1 bT ðT w  T 1 Þð1  /1 Þ

ð29Þ

Nb ¼

eðqcÞp DB ð/w  /1 Þ ; ðqcÞf am

ð30Þ

Nt ¼

sDT ðT w  T 1 Þ ; am T 1

ð31Þ

Nd ¼

rDTC ðC w  C 1 Þ ; am ðT w  T 1 Þ

ð32Þ

ð16Þ

  ðqp  qf 1 ÞgK @/ @ 2 w ð1  /1 Þqf 1 gK @T @C  þ b ; ¼ b T C @y2 @y @y @y l l

Rax ¼

ð27Þ

ð15Þ

so that Eq. (9) is satisfied identically. We are then left with the following four equations:

ð1  /1 Þqf 1 bT gKx

1 Nt 00 0 f 00 þ Lnsf þ h ¼ 0; 2 Nb

ð14Þ

One can eliminate p from Eqs. (10) and (11) by cross-differentiation. At the same time one can introduce a stream function w defined by

@w u¼ ; @y

ð26Þ

ð13Þ

where

am ¼

1 2

c00 þ Lesc0 þ Ldh00 ¼ 0;

0

s  h  Ncc þ Nrf ¼ 0;

Le ¼

am : eDSm

ð33Þ

Ld ¼

DTC : DSm

ð34Þ

Ln ¼

am : eDB

ð35Þ

Here Nc, Nr, Nd, Nb, Nt, denote the regular double-diffusive buoyancy ratio, a nanofluid buoyancy ratio, a modified Dufour parameter, a Brownian motion parameter, a thermophoresis parameter, respectively, while Le is the usual Lewis number, Ln is a nanofluid Lewis number (denoted by Le in [15]), and Ld is a Dufour-solutal Lewis number. Eqs. (24)–(27) are solved subject to the following boundary conditions:

At g ¼ 0 : s ¼ 0; As g ! 1 : s0 ¼ 0;

h ¼ 1; h ¼ 0;

c ¼ 1; f ¼ 1; c ¼ 0; f ¼ 0:

ð36Þ ð37Þ

Integrating Eq. (24) once and using boundary conditions (37) results in

s0  h  Ncc þ Nrf ¼ 0;

ð38Þ

Eqs. (38) and (25)–(27) are solved subject to boundary conditions at g = 0 given by Eq. (36) and the following boundary conditions at g ? 1:

At g ! 1 : h ¼ 0;

c ¼ 0; f ¼ 0:

ð39Þ

When Nc, Nr, Nb, Nt, and Nd are all zero, Eqs. (24)–(27) involve just two dependent variables, namely s and h, and the boundary-value problem for these two variables reduces to the classical problem solved by Cheng and Minkowycz [8]. (The boundary value problems for f and c then becomes ill-posed and are of no physical significance.) In fact, if Nb = Nt = Nd = 0 and also Le = Ln = 1 then the transformation

U¼

h þ Ncc  Nrf 1 þ Nc  Nr

ð40Þ

ð24Þ leads to the differential equation system

1 h þ sh0 þ Nbf 0 h0 þ Nth02 þ Ndc00 ¼ 0; 2 00

ð25Þ

s0  U ¼ 0;

ð41Þ

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1 2

U00 þ sU ¼ 0;

ð42Þ

sð0Þ ¼ 0;

ð43Þ

s0 ð1Þ ¼ 0;

Uð0Þ ¼ 1; Uð1Þ ¼ 0;

ð44Þ

which is precisely the Cheng–Minkowycz system. In the case of a regular fluid we recover the differential equations derived by Bejan and Khair [10], and reproduced as Eqs. (9.49)–(9.51) in [11], when we allow for changes in notation and a change in sign in the streamfunction. In the absence of the regular cross-diffusion terms (letting Nc = Nd = 0) and dropping Eq. (26) (effectively by letting Le tend to infinity) we recover the expressions given in [15]. A quantity of practical interest is the Nusselt number Nu defined by

Nu ¼

q00 x ; km ðT w  T 1 Þ

ð45Þ

where q00 is the wall heat flux and km is the effective thermal conductivity of the porous medium. In the present context Nu/Rax1/2

(a)

2

s, ds/dη, θ, γ, f

s, ds/dη, θ, γ, f

s ds/dη θ γ f

−θ'(0)=0.3343

s ds/dη θ γ f

1

0.5

0.5

0

2

2

4

η

6

8

(d)

−θ'(0)=0.1770

0

2

4

η

6

8

10

−θ'(0)=0.1053

2

s, ds/dη, θ, γ, f

1.5 s ds/dη θ γ f

1

0.5

0

0

10

1.5

s, ds/dη, θ, γ, f

The differential equations become very stiff as the values of the parameters Ln and Le become large and then the amount of computer time needed to produce results becomes prohibitive. Thus our results are limited to relatively small values of Ln and Le. The large number of other relevant parameters also limits the parameter ranges for which we report results. Plots of the similarity variables for a typical case, chosen as that for Le = 10, Ln = 10, Ld = 1, with various values of the other parameters, are shown in Fig. 1. The various cases illustrate in turn typically a regular mono-diffusive fluid, a mono-diffusive nanofluid, a regular double-diffusive fluid and a double-diffusive nanofluid, respectively. In general the boundary layer profiles for the temperature function h(g) and the stream function s(g) have

1.5

1

(c)

3. Results and discussion

(b) 2

−θ(0)=0.4439

1.5

0

is represented by h0 (0). (Likewise the dimensionless solutal mass flux is represented by a regular Sherwood number which is proportional to c0 (0), and the dimensionless particle mass flux is represented by a nanafluid Sherwood number which is proportional to f0 (0) but these are of lesser interest here.)

s ds/dη θ γ f

1

0.5

0

2

4

η

6

8

10

0

0

2

4

η

6

8

10

Fig. 1. Plots of dimensionless similarity functions s(g), s0 (g), h(g), c(g), f(g) for Le = 10, Ln = 10, Ld = 1 for the case (a) Nr = 0, Nb = 0, Nt = 0, Nc = 0, Nd = 0 (these parameter values simulate a mono-diffusive regular fluid, see Cheng and Minkowycz [8]); (b) Nr = 0.2, Nb = 0.2, Nt = 0.2, Nc = 0, Nd = 0 (these parameter values simulate a monodiffusive nanofluid, see Nield and Kuznetsov [15]); (c) Nr = 0, Nb = 0, Nt = 0, Nc = 0.2, Nd = 0.2 (these parameter values simulate a double-diffusive regular fluid, see Bejan and Khair [10]) and (d) Nr = 0.2, Nb = 0.2, Nt = 0. 2, Nc = 0.2, Nd = 0.2 (these parameter values simulate a double-diffusive nanofluid).

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Table 1 Linear regression coefficients and error bound for the reduced Nusselt number. . Here ¼ Cr, Cb, Ct, Cc, Cd, are the coefficients in the linear regression estimate Nuest R1=2 x 0:444 þ C r Nr þ C b Nb þ C t Nt þ C c Nc þ C d Nd, and e is the maximum relative error defined by  ¼ jðNuest  NuÞj=Nu, applicable for Nr, Nb, Nt, Nc, Nd each taking the values [0.05, 0.1, 0.15, 0.2]. Le

Ln

Ld

Cr

Cb

Ct

Cc

Cd

e

1.0 3.0 10 1.0 3.0 10 1.0 3.0 10

1.0 1.0 1.0 3.0 3.0 3.0 10 10 10

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

0.259 0.238 0.182 0.151 0.128 0.073 0.077 0.063 0.018

0.117 0.098 0.073 0.240 0.219 0.186 0.325 0.292 0.239

0.180 0.173 0.162 0.174 0.164 0.153 0.172 0.162 0.155

0.242 0.134 0.029 0.231 0.127 0.024 0.224 0.123 0.020

0.069 0.463 1.334 0.081 0.465 1.326 0.091 0.465 1.305

0.095 0.111 0.328 0.037 0.046 0.119 0.019 0.022 0.115

1.0 3.0 10 1.0 3.0 10 1.0 3.0 10

1.0 1.0 1.0 3.0 3.0 3.0 10 10 10

3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0

0.285 0.222 0.093 0.155 0.104 0.018 0.051 0.024 0.072

0.216 0.150 0.043 0.365 0.288 0.165 0.482 0.384 0.225

0.230 0.178 0.103 0.221 0.169 0.097 0.209 0.162 0.102

0.377 0.229 0.116 0.373 0.227 0.114 0.377 0.229 0.109

0.525 0.233 1.747 0.416 0.294 1.762 0.306 0.350 1.747

0.086 0.521 2.051 0.063 0.554 3.375 0.108 0.126 3.253

essentially the same form as in the case of a regular mono-diffusive fluid. The thickness of the boundary layer for the mass fraction function f(g) is smaller than the thermal boundary layer thickness when Le > 1, and this feature is seen in all cases shown. It is well known that, in the case of a regular fluid, the profile for ds/dg (something that represents the longitudinal component of the velocity, u) is identical with that for the temperature h, and indeed this feature is seen in Fig. 1(a). Further, in the case of a mono-diffusive nanofluid the two profiles diverge within a layer whose thickness is comparable with that of the mass fraction. For the parameters illustrated (Le = Ln = 10), the plots for c(g) and f(g) coincide. At first sight the plots for Fig. 1(a) and (d) appear to differ little from each other, but a closer examination reveals that in Fig. 1(d) the thermal boundary layer thickness is larger, and this is consistent with the smaller value of the reduced Nusselt number Nu=Ra1=2 (something that we denote by Nur) in that case. x For the case Le = 1, Ln = 1, Ld = 1, the value of Nur was calculated for sets of values of Nr, Nb, Nt, Nc, Nd in the range [0.05, 0.10, 0.15, 0.20] and a linear regression was performed on the results. This yielded the correlation

Nurest ¼ 0:444  0:259Nr  0:117Nb  0:180Nt þ 0:242Nc  0:069Nd;

ð46Þ

valid for Nr, Nb, Nt, Nc, Nd each taking values in the range [0, 0.20], with a maximum error of about 10%. Clearly an increase in any of the buoyancy-ratio number Nr, the Brownian motion parameter Nb, and the thermophoresis parameter Nt leads to a decrease in the value of the reduced Nusselt number (corresponding to an increase in the thermal boundary-layer thickness), in agreement with the results for a mono-diffusive nanofluid reported in [15]. An increase in the modified Dufour parameter Nd also leads to a decrease in the value of the reduced Nusselt number, but an increase in the regular double-diffusive buoyancy ratio Nc causes a change in the opposite direction. This exercise was repeated for other values of Le, Ln and Ld, with the results shown in Table 1. These results show that so long as Le stays moderately small the regression formula gives consistent results, but as Le increases to the value 10 the formula becomes unreliable. The maximum error becomes unacceptably

large. In fact the formula leads to negative values of Nur (which are physically meaningless) when the value of Nd (in particular) becomes relatively large. 4. Conclusions We have examined the influence of nanoparticles on natural convection boundary layer flow in a porous medium past a vertical plate, using a model in which Brownian motion and thermophoresis are accounted for. We have employed the Darcy model for the momentum equation and we have assumed the simplest possible boundary conditions, namely those in which both the temperature and the nanoparticle fraction are constant along the wall. This permits a simple similarity solution which depends on eight dimensionless parameters, namely a regular Lewis number Le, a nanofluid Lewis number Ln, a Dufour-solutal Lewis number Ld, a nanofluid buoyancy-ratio parameter Nr, a Brownian motion parameter Nb, a thermophoresis parameter Nt, a regular doublediffusive buoyancy Nc, and a modified Dufour parameter Nd. We have explored the way in which the wall heat flux, represented by a reduced Nusselt number Nur, depends on these eight parameters. In general a decrease in the reduced Nusselt number, associated with an increase in the thickness of the thermal boundary layer, is found with increase of Nt, Nb, Nt, Nd and decrease of Nc. In each case the amount of change depends on the values of Le, Ln and Ld. In this paper we have confined our attention to the case of mutually assisting buoyancy forces (Nr > 0). The case of opposing buoyancy forces is of considerable interest because of the possible existence of multiple solutions but this case is left for further investigation. References [1] S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, in: D.A. Siginer, H.P. Wang (Eds.), Developments and Applications of Non-Newtonian Flows, ASME FED, vol. 231/MD-vol. 66, 1995, pp. 99–105. [2] H. Masuda, A. Ebata, K. Teramae, N. Hishinuma, Alteration of thermal conductivity and viscosity of liquid by dispersing ultra-fine particles, Netsu Bussei 7 (1993) 227–233. [3] J. Buongiorno, W. Hu, Nanofluid coolants for advanced nuclear power plants, in: Proceedings of ICAPP ’05, Seoul, May 15–19, 2005, Paper No. 5705. [4] S.K. Das, S.U.S. Choi, A review of heat transfer in nanofluids, Adv. Heat Transfer 41 (2009) 81–197. [5] S.K. Das, S.U.S. Choi, W. Yu, T. Pradeep, Nanofluids: Science and Technology, Wiley, Hoboken, NY, 2008. [6] X.Q. Wang, A.S. Mujumdar, Heat transfer characteristics of nanofluids: a review, Int. J. Therm. Sci. 46 (2007) 1–19. [7] J. Buongiorno, Convective transport in nanofluids, ASME J. Heat Transfer 128 (2006) 240–250. [8] P. Cheng, W.J. Minkowycz, Free convection about a vertical flat plate embedded in a porous medium with application to heat transfer from a dike, J. Geophys. Res. 82 (1977) 2040–2044. [9] A. Bejan, Convection Heat Transfer, third ed., Wiley, Hoboken, 2004. pp. 586– 591. [10] A. Bejan, K.R. Khair, Heat and mass transfer by natural convection in a porous medium, Int. J. Heat Mass Transfer 28 (1985) 909–919. [11] D.A. Nield, A. Bejan, Convection in Porous Media, third ed., Springer, New York, 2006. [12] A.R.A. Khaled, A.J. Chamkha, Variable porosity and thermal dispersion effects on coupled heat and mass transfer by natural convection from a surface embedded in a non-metallic porous medium, Int. J. Numer. Methods Heat Fluid Flow 11 (2001) 413–429. [13] A.J. Chamkha, M.M.A. Quadri, Simultaneous heat and mass transfer by natural convection from a plate embedded in a porous medium with thermal dispersion effects, Heat Mass Transfer 39 (2003) 561–569. [14] A.J. Chamkha, I. Pop, Effect of thermophoresis particle deposition in free convection boundary layer from a vertical flat plate embedded in a porous medium, Int. Commun. Heat Mass Transfer 31 (2004) 421–430. [15] D.A. Nield, A.V. Kuznetsov, The Cheng–Minkowycz problem for natural convective boundary layer flow in a porous medium saturated by a nanofluid, Int. J. Heat Mass Transfer 52 (2009) 5792–5795.