Combined thermophoresis and Brownian motion effects on nanofluid free convection heat transfer in an L-shaped enclosure

Accepted Manuscript

Combined thermophoresis and Brownian motion effects on nanofluid free convection heat transfer in an L-shaped enclosure M. Sheikholeslami , Ali J. Chamkha , P. Rana , R. Moradi PII: DOI: Reference:

S0577-9073(17)30513-0 10.1016/j.cjph.2017.09.011 CJPH 352

To appear in:

Chinese Journal of Physics

Received date: Revised date: Accepted date:

29 April 2017 26 September 2017 26 September 2017

Please cite this article as: M. Sheikholeslami , Ali J. Chamkha , P. Rana , R. Moradi , Combined thermophoresis and Brownian motion effects on nanofluid free convection heat transfer in an L-shaped enclosure, Chinese Journal of Physics (2017), doi: 10.1016/j.cjph.2017.09.011

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ACCEPTED MANUSCRIPT

Highlights  Nanofluid free convection heat transfer is studied  Two phase model is employed for nanofluid.  CVFEM is used as simulation tool.

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 Nusselt number decreases with increase of aspect ratio.

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 Nusselt number enhances with increase of Lewis number.

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Combined thermophoresis and Brownian motion effects on nanofluid free convection heat transfer in an L-shaped enclosure

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M. Sheikholeslami a, Ali J. Chamkhab1 , P. Rana c, R. Moradi d Department of Mechanical Engineering, Babol Noshirvani University of Technology, Iran b

Mechanical Engineering Department, Prince Mohammad Bin Fahd University P.O. Box 1664, Al-Khobar 31952, Kingdom of Saudi Arabia

Department of Mathematics, Indian Institute of Technology, Roorkee 247667, India d

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c

Department of Chemical Engineering, School of Engineering and Applied Science, Khazar University, Baku, Azerbaijan

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Abstract

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In this study, natural convection heat transfer in an L-shaped enclosure filled with a nanofluid is studied. The numerical investigation is carried out using Control Volume based Finite Element Method (CVFEM). The important effects of Brownian motion and

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thermophoresis have been included in the model of the nanofluid. Comparisons with previously published works are conducted and the results are found to be in good agreement. The influence

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of pertinent parameters such as the Lewis number, aspect ratio, thermal Rayleigh number and the concentration Rayleigh number on the flow, heat transfer and Nusselt number are illustrated

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graphically and discussed. Results show that the Nusselt number increases with increases in either of the thermal Rayleigh number and the Lewis number but it decreases with increases in either of the aspect ratio and the concentration Rayleigh number.

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Corresponding Author: Emails: [email protected] (Ali J. Chamkha) , [email protected], [email protected] (M. Sheikholeslami)

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Keywords: Brownian motion; Thermophoresis; Nanofluid; Natural convection; CVFEM; Lshaped enclosure. Nomenclature Cp

Specific heat at constant pressure

Greek symbols

DB

Brownian diffusion coefficient



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Angle measured from the insulated right plane

Thermophoretic diffusion coefficient



Thermal diffusivity

g

Gravitational acceleration vector



Volume fraction

k

Thermal conductivity



Dynamic viscosity

L

Gap between inner and outer boundary of

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DT

the enclosure L  rout  rin

Le

Lewis number (  ) DB

Brownian motion parameter ( (  c) p DB (h  c ) )

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(  c)

Nt

Thermophoretic

Kinematic viscosity

 &

stream function & dimensionless stream function

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Nb





dimensionless temperature



Fluid density

parameter ( (  c) p DT (Th  Tc ) )

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(  c) Tc

Nusselt number



Thermal expansion coefficient

Pr

Prandtl number (  /  )



Aspect ratio   rin / L 

r

Non-dimensional radial distance

Subscripts

Ra

thermal Rayleigh number

c

Cold

h

Hot

loc

Local

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Nu

(

Rn

1  c  0 g  L3 Th  Tc  ) 

The concentration Rayleigh number (   p  0  h  c  gL ) 3



T

Fluid temperature 3

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u ,v

Velocity components in the x-direction

ave

Average

in

Inner

out

Outer

and y-direction

U ,V

Dimensionless velocity components in the X-direction and Y-direction Space coordinates

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x ,y

1. Introduction

The study of free convection in horizontal annuli is of importance in many industrial and

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geophysical problems. This topic is of useful interest in several sciences such as in solar collector-receiver, underground electric transmission cables, vapor condenser for water distillation and food process. Kuehn et al. [1] presented experimental and numerical studies of steady-state free convection heat transfer in horizontal concentric annuli, in which the effects of Rayleigh and Prandtl numbers and aspect ratio were parametrically explored and correlating

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equations, were proposed as well. Karki and Patankar [2] presented a numerical study for the combined convection in the entrance region of a horizontal annulus with a fixed radius ratio of 2.

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The results were presented for developing velocity and temperature fields, local pressure gradient, and the circumferentially averaged Nusselt number. Xu et al. [3] have studied laminar

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convection heat transfer from a horizontal triangular cylinder to its concentric cylindrical enclosure. They indicated that at constant aspect ratio, both the inclination angle and cross-

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section geometry have insignificant effects on the overall heat transfer rates. MHD effect on natural convection heat transfer in an enclosure filled with nanofluid was reported by Sheikholeslami et al. [4]. Their results indicated that Nusselt number is an increasing function of

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buoyancy ratio number but it is a decreasing function of Lewis number and Hartmann number. Control Volume based Finite Element Method (CVFEM) is a scheme that uses the

advantages of both finite volume and finite element methods for simulation of multi-physics problems in complex geometries [5] and [6]. Sheikholeslami and Chamkha [7] studied the magnetohydrodynamic nanofluid flow in a double-sided lid-driven wavy cavity. Sheikholeslami and Chamkha [8] investigated the effect of electric field on nanofluid natural convection. 4

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Sheikholeslami et al. [9] presented single phase modeling of nanofluid flow in an inclined enclosure. Sheikholeslami and Rokni [10] utilized the two phase model for nanofluid flow in presence of magnetic field considering melting heat transfer. Sheikholeslami and Rokni [11] employed Buongiorno Model for the effect of melting heat transfer on nanofluid thermal behavior. Sheikholeslami and Chamkha [12] investigated Marangoni convection effect on

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nanofluid forced convection over a plate. Sheikholeslami [13] presented water based nanofluid flow in a porous enclosure considering various shapes of nanofluid. Sheikholeslami Kandelousi [14] utilized KKL model for nanofluid flow in a porous channel. Chamkha et al. [15] reported a review on MHD convection of nanofluids. In addition, Chamkha and co-workers [16 - 24] have analyzed the use of various types and models of nanofluids in different types of geometries

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including single and double lid-driven cavities, triangular wavy cavities, annulus, odd-shaped cavities and C-shaped cavities.

All the above studies assumed that no slip velocities exist between the nanoparticles and the fluid molecules and assumed that the nanoparticle concentration is uniform. It is believed that in natural convection of nanofluids, the nanoparticles could not accompany fluid molecules due

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to some slip mechanisms such as Brownian motion and thermophoresis, so the volume fraction of nanofluids may not be uniform anymore and there would be a variable concentration of

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nanoparticles in a mixture. Buongiorno [25] investigated different slip mechanisms between nanoparticles and base fluid. He indicated that there are many slip mechanisms such as inertia,

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Brownian diffusion, thermophoresis, diffusiophoresis, magnus effect, fluid drainage, and gravity. He concluded that only Brownian diffusion and thermophoresis are important slip mechanisms in

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the absence of turbulent effects. Khan and Pop [26] published a paper on boundary-layer flow of a nanofluid past a stretching sheet as a first paper in that field. Their model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. They have taken into

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account the Prandtl number, Lewis number, Brownian motion numbers and thermophores number. They indicated that the reduced Nusselt number is a decreasing function of each dimensionless number. Yadav et al. [27] studied the effect of internal heat source on the onset of Darcy-Brinkman convection in a porous layer saturated by nanofluid. They found that the internal heat source, nanoparticle Rayleigh number, modified diffusivity ratio and Lewis number have a destabilizing effect while Darcy number and the porosity show stabilizing effects on the system. Steady, laminar boundary fluid flow which results from the non-linear stretching of a flat 5

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surface in a nanofluid has been investigated numerically by Rana and Bhargava [28]. They reported that the dimensionless mass transfer rates increases with the increase in the Brownian motion parameter and thermophoresis parameter. Yadav et al. [29] considered thermal instability of rotating nanofluids heated from below. They showed that the rotation has a stabilizing effect depending upon the values of various nanofluid parameters. Recently, several authors utilized

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various models for nanofluid [30-50]. The main goal of the present work is to conduct a numerical investigation of natural convection heat transfer in an L-shape enclosure filled with a nanofluid using the Control Volume based Finite Element method. The combined effects of Brownian motion and thermophoresis are considered to get the gradient of nanoparticles’ volume fraction. The

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numerical investigation is carried out for different governing parameters such as the Lewis number, aspect ratio, thermal Rayleigh number and the concentration Rayleigh number.

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2. Geometry definition and boundary conditions:

The physical model along with the important geometrical parameters and the mesh of the

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enclosure used in the present CVFEM program are shown in Fig 1. The right and top wall of the enclosure are maintained at constant cold temperatures Tc whereas the inner circular hot wall is

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maintained at constant hot temperature Th and the two bottom and left walls with the length of

H / 2 are thermally insulated. Under all cases T h > T c condition is maintained.

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To assess the shape of inner circular and outer rectangular boundary which consists of the right and top walls, a supper elliptic function can be used as follows; 2n

(1)

2n

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X Y      a b

1

When a  b and n  1 the geometry becomes a circle. As n increases from 1 the geometry would approach a rectangle for a  b and square for a  b .

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3. Mathematical modeling and numerical procedure: 3.1. Problem formulation The nanofluid’s density,  is    p  (1   ) f

(2)





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  p  (1   ) f 0 1   (T T c ) 

where  f , is the base fluid’s density, Tc , is a reference temperature,  f0 is the base fluid’s density at the reference temperature ,  is the volumetric coefficient of expansion. Taking the density of base fluid as that of the nanofluid, as adopted by Yadav et al. [29], The density  in

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Eq. (2), Thus becomes    p  (1   )0 1   (T Tc ) 

(3)

 0 is the nanofluid’s density at the reference temperature.

The continuity, momentum under Boussinesq approximation and energy equations for the

dimensional form as follows:

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u v  0 x y

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laminar and steady state natural convection in a two-dimensional enclosure can be written in

(4)

  2u  2u   u u  P v    2  2  y  x y   x  x

(5)

  2v  2v   v v  P v     2  2    p  (1   ) 0 1   T  Tc    g y  y y   x  x

(6)

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0 u

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0 u

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   T  T    .   DB  .     2T  2T    c  p   x x y y  T T u v   2  2   2  2  T   T   x y y    c    x  ( DT / Tc )        x  y      

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  2  2   D    2T  2T    v  DB  2  2    T   2  2  x y y   Tc   x y   x

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(7)

(8)

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The physical boundary conditions are: on the inner cylinder

T  Tc ,   c

on the outer cylinder

T   0, 0 n n

on other walls

  0,

on all solid boundaries

(9)

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T  T h ,   h

The stream function and vorticity are defined as follows: u

 v u  , v ,   y x y x

(10)

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The stream function satisfies the continuity Eq. (4). The vorticity equation is obtained by eliminating the pressure between the two momentum equations, i.e. by taking y-derivative of Eq. (5) and subtracting from it the x-derivative of Eq. (6). This gives:

  2  2         T   g 0 1         2  2   g  p  0  0  T  Tc  x x y   y x x y   x



0 



(12)

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   T  T    .   DB  .     2T  2T  (  c) p   x x y y   T  T    2  2   2  2  T   T   y x x y y  (  c)   x  ( DT / Tc )       x   y     

(11)

  2  2   D    2T  2T        DB  2  2    T   2  2  y x x y y   Tc   x y   x

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 2  2    x 2 y 2

(13) (14)

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By introducing the following non-dimensional variables:

X

T  Tc   c x y  L2  , Y  , ,  ,  , L L   Th  Tc h  c

where in Eq. (15) L  rout  rin  rin , and using the dimensionless parameters, the equations now become:

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(15)

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(16)

  2  2        2   2            2  2   Nb     Nt  2  2  Y X X Y  X Y  Y   X X Y Y   X

(17)

    1   2  2  Nt   2  2           Y X X Y Le  X 2 Y 2  NbLe  X 2 Y 2 

(18)

2 2    X 2 Y 2

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1         2  2           2  2   Ra    Rn     Pr  Y X X Y   X Y   X   X 

(19)

1  c  0 g  L3 Th  Tc   where Ra  is the thermal Rayleigh number for the nanofluid,

 Rn 

p



 0 h  c  gL3



is the concentration Rayleigh number and Pr   /  is the Prandtl

number for the nanofluid, Nb  (  c) p DT (Th  Tc ) (  c) Tc

(  c) p DB (h  c ) (  c)

is the Brownian motion parameter of nanofluids.

is the thermophoretic parameter of nanofluids, Le 

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Nt 

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

 DB

is Lewis number.

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It should be noted that Eq. (13) has been obtained using small temperature gradient in a dilute suspension of nanoparticles.

(20)

on the inner cylinder

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  1.0,   1.0

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The dimensionless boundary conditions as shown in Fig. 1 are:

on the outer cylinder

   0, 0 n n

on other walls

  0.0

on all solid boundaries

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  0.0,   0.0

The values of vorticity on the boundary of the enclosure can be obtained using the stream function formulation and the known velocity conditions during the iterative solution procedure.

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The local Nusselt number on the hot circular wall can be expressed as: Nuloc  

Tc   n Th  Tc

     Nb n  Nt n   

(21)

where n is the direction normal to the inner cylinder surface. The average number on the hot circular wall is evaluated as: 1 0.5

0.5



(22)

Nu loc   d 

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Nu ave 

0

3.2. Numerical procedure

Triangular elements are considered as the building block of the discretization using CVEM. The values of variables are approximated with linear interpolation within the elements.

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A control volume is created by joining the center of each element in the support to the mid points of the element sides that pass through the central node i, which creates a close polygonal control volume (see fig. 1 (b)). To illustrate the solution procedure using the CVFEM, one can consider the general form of advection-diffusion equation for node i in integral form:







V

A

A

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or point form

(24)

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  (k )    (v )  Q  0

(23)

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 Q dV  k   n dA  (v  n) dA  0

which can be represented by the system of CVFEM discrete equations as: ni

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 ai  Qci  Bci i   ai , jS j 1

i, j

(25)

 QBi  BBi

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In the above equation, the a ’s are the coefficients, the index ( i, j ) indicates the j

th

node in the

th

support of node i , the index Si , j provides the node number of the j node in the support, the B ’s account for boundary conditions, and the Q ‘s for source terms. For the selected triangular element which is shown in Fig. 2. This approximation without considering the source term leads to

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











 a1k  a1u i  a2k  a2u Si ,3  a2k  a2u Si ,4  0

(26)

Using upwinding the advective coefficients, identified with the superscripts ( )u , are given by a1u  max  q f 1 ,0   max  q f 2 ,0  a2u  max  q f 1 ,0  a3u  max  q f 2 ,0 

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(27)

and the diffusion coefficients, identified with the superscripts ( )k , are given by a1k   k f 1 N1x  y f 1  k f 1 N1 y  x f 1  k f 2 N1x  y f 2  k f 2 N1 y  x f 2 a2k   k f 1 N 2 x  y f 1  k f 1 N 2 y  x f 1  k f 2 N 2 x  y f 2  k f 2 N 2 y  x f 2

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a2k   k f 1 N 3 x  y f 1  k f 1 N 3 y  x f 1  k f 2 N 3 x  y f 2  k f 2 N 3 y  x f 2

(28)

In Eq. (21) the volume flow across face 1 and 2 in the direction of the outward normal, is

q f 2  v  n A f 2  vxf 2  y f 2  v yf 2  y f 2

(29)

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q f 1  v  n A f 1  vxf 1 y f 1  v yf 1 y f 1

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The value of the diffusivity at the mid-point of face 1 can be obtained as 5 5 2 k1  k2  k3 12 12 12

(24)

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k f 1   N1k1  N2 k2  N3k3  f 1 

and at the mid-point of face 2

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k f 2   N1k1  N2 k2  N3k3  f 2 

5 2 5 k1  k2  k3 12 12 12

(30)

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The velocity components at the midpoint of face 1 are: 5 5 2 vx1  vx2  vx3 12 12 12 5 5 2 v yf 1  v y1  v y2  v y3 12 12 12 vxf 1 

(31)

and on face 2:

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5 2 5 vx1  vx2  vx3 12 12 12 5 2 5  v y1  v y2  v y3 12 12 12

vxf 2  v yf 2

(32)

These values can be used to update the ith support coefficients through the following equation:

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ai  ai  a1k ai ,3  ai ,3  a2k

(33)

ai ,4  ai ,4  a3k

x3 x2 x1 x x x   , x f 2   2  3  1 3 6 6 3 6 6 y y y y y y y f1  3  2  1 , y f 2   2  3  1 3 6 6 3 6 6 x f 1 

the derivatives of the shape functions are: N1  y2  y3  N  x  x   , N1 y  1  3 ele2 ele x y 2V 2V

N2 x 

N 2  y3  y1  N  x  x   , N 2 y  1  1 ele3 ele x y 2V 2V

N3 x 

N x  x  N 2  y1  y2   , N 3 y  3  2 ele1 ele x y 2V 2V

(34)

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N1x 

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In Eq. (28), moving counter-clockwise around node i , the signed distances are:

PT

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(35)

and the volume of the element is

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 x2 y3  x3 y2   x1  y2  y3   y1  x3  x2 

(36)

2

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V ele 

The obtained algebraic equations from the discretization procedure using CVFEM are solved by the Gauss-Seidel Method. 3.3. Implementation of source terms and boundary conditions: The boundary conditions for the present problem can be enforced using BBi and BCi as follows: 12

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Insulated boundary: BBi  0 and BCi  0

(37)

Fixed Value Boundary: BBi  value 1016 and BCi  1016

(38)

where value is the prescribed value on the boundary and Ak is the length of the control volume

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surface on the boundary segment. The volume source terms can be applied to Eq. (25) as: (39)

elements

  Q dV  Q V

i i

j 1

Vj

or after linearizing the as source term

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QV i i  QCi i  QBi

4. Grid testing and code validation

(40)

A mesh testing procedure was studied to guarantee the grid-independency of the present solution.

Various

mesh

combinations

were

explored

for

the

case

at

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Ra  Rn  105 , Nt  Nb  108 ,   0.8 and Pr  5 as shown in Table 1. The present code was tested for grid independence by calculating the average Nusselt number on the inner circular

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wall. In harmony with this, it was found that a grid size of 81  241 ensures a grid-independent solution. The convergence criterion for the termination of all computations is:

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max grid  n 1   n  107

(41)

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where n is the iteration number and  stands for the independent variables ( , ,  ).The present FORTRAN code is validated by comparing the obtained results for Pr=0.7 with other

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works reported in literature [9] and [51]. As shown in Table 2, they are in a very good agreement.

5. Results and discussion In this study, natural convection heat transfer in an L-shape enclosure filled with nanofluid is investigated numerically using the Control Volume based Finite Element Method (CVFEM).

Calculations

are

made

for 13

various

values

of

Lewis

number

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( Le  100,500,5000 and 10 4 ), aspect ratio (   rin / L  0.2, 0.4, 0.6 and 0.8 ), thermal Rayleigh number( Ra  10 3 ,10 4 and 10 5 ) and concentration Rayleigh number( Rn  10 3 ,10 4 and 10 5 ) at constant Brownian motion parameter of nanofluids ( Nb  108 ) , thermophoretic parameter of nanofluids ( Nt  108 ) and Prandtl number ( Pr  5 ). The effects of the thermal Rayleigh number, concentration Rayleigh number, Lewis

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number and the aspect ratio on isotherms, streamlines and isoconcentration contours are shown in Figs. 3-7. As thermal Rayleigh number increases, the buoyancy forces enhance so that the thermal boundary layer thickness reduces near the hot wall. As the Lewis number increases, the temperature gradient increases. As the aspect ratio rises, the distance between the hot and cold walls reduces. Thus, the conduction heat transfer mechanism becomes dominant and in turn, the

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temperature gradient reduces. Figs. 8 and 9 show that the effects of the thermal Rayleigh number, concentration Rayleigh number, Lewis number and the aspect ratio on the local and average Nusselt number, respectively. The temperature gradient is an increasing function with the thermal Rayleigh number and the Lewis number but it is a decreasing function with concentration Rayleigh number and aspect ratio. Nusselt number increases with the rise of

Rayleigh number and aspect ratio.

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6. Conclusion

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thermal Rayleigh number and Lewis number while it reduces with the rise of concentration

In this study, nanofluid free convection heat transfer in an inclined L-shape enclosure is

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studied numerically using the Control Volume based Finite Element Method. Two-phase model has been applied for the nanofluid. The effects of the Lewis number, aspect ratio, thermal

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Rayleigh number and the concentration Rayleigh number on the flow, heat and mass transfer are studied. From the numerical investigation, it can be concluded that the aspect ratio can be a

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control parameter for heat and fluid flow. In addition, the rate of heat transfer enhances with rise of the thermal Rayleigh number and Lewis number while the opposite trend is observed for aspect ratio and concentration Rayleigh number.

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of Mechanics - B/Fluids 36 (2012) 82-96. [19] N. Ben Cheikh, A. J. Chamkha, B. Ben Beya and T. Lili, Natural convection of water-based nanofluids in a square enclosure with non-uniform heating of the bottom wall, Journal of Modern Physics 4 (2013) 147-159.

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[36] A.J. Chamkha, M.A. Ismael, Magnetic field effect on mixed convection in lid-driven trapezoidal cavities filled with a Cu–water nanofluid with an aiding or opposing side wall, J.

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Thermal Sci. Eng. Appl. 2016; 8(3):031009-031009-12. [37] M. Sheikholeslami, Lattice Boltzmann Method simulation of MHD non-Darcy nanofluid

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free convection, Physica B 516 (2017) 55–71 [38] M. Sheikholeslami, Influence of magnetic field on nanofluid free convection in an open

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tilted elliptic inner cylinder, J. Mol. Liquids 229 (2017) 137–147

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considering Darcy model, J. Mol. Liquids 225 (2017) 903–912

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convection in an enclosure with elliptical cylinders, Int. J. Heat Mass Transfer 115 (2017) 981– 991

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[47] E. Magyari, A. J. Chamkha, Exact analytical results for the thermosolutal MHD Marangoni

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boundary layers, International Journal of Thermal Sciences, 47(7) (2008) 848-857 [48] A. Al-Mudhaf, A. J. Chamkha, Similarity solutions for MHD thermosolutal Marangoni convection over a flat surface in the presence of heat generation or absorption effects, Heat and

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Mass Transfer, 42 (2005) 112–121 [49] T. Tayebi, A. J. Chamkha, M. Djezzar, A. Bouzerzour , Natural convective nanofluid flow in an annular space between confocal elliptic cylinders , J. Thermal Sci. Eng. Appl. 2016; doi: 10.1115/1.4034599 [50] M. Sheikholeslami, M.K. Sadoughi, Numerical modeling for Fe3O4 -water nanofluid flow in porous medium considering MFD viscosity, J. Mol. Liquids 242 (2017) 255-264 19

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[51] G. De Vahl Davis, Natural convection of air in a square cavity, a benchmark numerical

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solution, Int. J. Numer. Methods Fluids 3 (1962) 249–264.

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(a)

(b)

Fig. 1. (a)Geometry and the boundary conditions with (b) the mesh of enclosure considered

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in this work.

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Fig. 2. A sample triangular element and its corresponding control volume.

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Streamlines

Isoconcentration

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λ= 0.2

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Isotherms

 max  0.001,  min  0.691

λ= 0.6

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λ= 0.4

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 max  0.015,  min  0.680

22

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λ= 0.8

 max  0.0008,  min  0.380

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 max  0.0004,  min  0.158

Fig. 3. Comparison of the isotherms, streamlines and isoconcentration contours for different values of 8

aspect ratio when Ra  Rn  10 , Le  10 , Nt  Nb  10 4

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3

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and Pr  5 .

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 max  0.670,  min  0.026

λ= 0.6

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λ= 0.4

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 max  0.457,  min  0.079

Isoconcentration

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Streamlines

λ= 0.2

Isotherms

 max  0.583,  min  0.012

24

 max  0.172,  min  0.007

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λ= 0.8

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Fig. 4. Comparison of the isotherms, streamlines and isoconcentration contours for different values of 8

aspect ratio when Ra  10 , Rn  10 , Le  10 , Nt  Nb  10 5

4

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3

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and Pr  5 .

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λ= 0.4

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 max  0.005,  min  22.01

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 max  0.003,  min  20.05

λ= 0.6

Isoconcentration

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Streamlines

λ= 0.2

Isotherms

 max  0.003,  min  17.24

26

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λ= 0.8

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 max  0.487,  min  11.73

Fig. 5. Comparison of the isotherms, streamlines and isoconcentration contours for different values of 8

aspect ratio when Ra  10 , Rn  10 , Le  10 , Nt  Nb  10 3

4

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5

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and Pr  5 .

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λ= 0.4

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 max  0.558,  min  20.08

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 max  0.438,  min  19

λ= 0.6

Isoconcentration

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Streamlines

λ= 0.2

Isotherms

 max  4.082,  min  16.71

28

 max  0.205,  min  11.45

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λ= 0.8

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Fig. 6. Comparison of the isotherms, streamlines and isoconcentration contours for different values of 8

aspect ratio when Ra  10 , Rn  10 , Le  10 , Nt  Nb  10 5

4

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5

29

and Pr  5 .

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Streamlines

Isoconcentration

Ra=Rn= 103

λ= 0.2

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Isotherms

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 max  3.1E  5,  min  8.1E  5

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Ra=Rn= 105 λ= 0.2

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λ= 0.8

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 max  0.019,  min  0.032

 max  0.682,  min  19.59

30

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λ= 0.8

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 max  0.047,  min  10.78

Fig. 7. Comparison of the isotherms, streamlines and isoconcentration contours for different values of 8

aspect ratio and Lewis number when Le  10 , Nt  Nb  10

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2

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and Pr  5 .

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(a) Ra  Rn  10 , Le  10

(b) Ra  Rn  10 , Le  10

2

5

4

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5

(c) Rn  10 , Le  10 ,   0.2 4

(d) Rn  10 , Le  10 ,   0.8 5

4

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PT

ED

5

(e) Ra  10 , Le  10 ,   0.2 5

(f) Ra  10 , Le  10 ,   0.2

4

5

4

Fig. 8. Effects of Lewis number, aspect ratio thermal Rayleigh number and concentration Rayleigh 8 number on Local Nusselt number at Nt  Nb  10 and Pr  5 .

32

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(b)

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PT

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M

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(a)

(c)

Fig. 9. Effects of thermal Rayleigh number, concentration Rayleigh number, Lewis number and aspect

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ratio on average Nusselt number (a)

  0.2, Le  104 ; (b)   0.8, Le  104 ;(c) Ra  Rn  105 at Nt  Nb  108 and Pr  5 .

33

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Table 1. Comparison of the average Nusselt number Nuave for different grid resolution at Ra  Rn  10 , Nt  Nb  10 ,   0.8 and Pr  5 . 8

5

31  91

41  121

51  151

61  181

71  211

81  241

91  271

101  301

111  331

Nuave

5.895654

5.937205

5.956422

5.960397

5.960149

5.95846

5.956164

5.954207

5.951092

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Mesh size

Table 2. Comparison of the present results with previous works for different Rayleigh numbers when

Present

103

1.1432

104

2.2749

105

4.5199

Khanafer et al. [9]

De Vahl Davis [51]

1.118

1.118

2.245

2.243

4.522

4.519

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ED

M

Ra

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Pr=0.7.

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