Investigation of Laminar Heat Transfer of Binary Nanofluid on Horizontal Stretching Plate

Intl. j. Basic. Sci. Appl. Res. Vol., 2 (10), 852-861, 2013

852

International Journal of Basic Sciences & Applied Research. Vol., 2 (10), 852-861, 2013 Available online at http://www.isicenter.org ISSN 2147-3749 ©2013 VictorQuest Publications

Investigation of Laminar Heat Transfer of Binary Nanofluid on Horizontal Stretching Plate

Hojatollah Gholami1,2*, Mahdi Shehnehpour Borazjani 2,3

1Department of Mechanical Engineering, Islamic Azad University, Dashtestan Branch, Dashtestan, Iran

1Member of Young Research Club 3Department of Mechanical Engineering, Science and Research Branch, Islamic Azad University, Bushehr, Iran

*Corresponding Author email: [email protected]

Abstract

In this paper, laminar heat transfer of binary nanofluid on horizontal stretching plate investigated numerically. This paper focuses effect of the nanofluid's (Brownian diffusion, thermophoresis effect) and the binary fluid parameters on a convective heat transfer flux. The coupled government equations were solved by the similarity solution. To investigate thermal parameters of the nanofluid and the binary fluid, dimensionless parameters such as Prandtl number, Lewis number, Brownian diffusion parameter, Thermophoresis parameter and Dufour number defined and effect of them on heat transfer rate investigated. Results show nanofluid in solutal fluid has a slightly lower heat transfer rate; rising nanofluid's physical parameters such as Brownian diffusion, Thermophoresis parameter and Physical parameters of diffusive fluid (Dufour Number) lead to decrease thermal performance.

Keywords: Nanofluid, Forced heat transfer, Similarity solution.

Introduction

Fluid flow over the stretching plates is popular in engineering applications, for example, the stretching polymer plates in

chemical factories (Fisher, 1986; Tadmor and Klein, 1970) and process on the metals in metallurgy (Altan et al., 1979). The

boundary layer of flow was studied by Sakiadis (1961a, 1961b). After that researchers such as Crane (1970), Banks (1983), Ali

(1994, 1995), Chaudhary et al (1995), Erickson et al (1966) and Magyari and Keller (1999, 2000) have considered parts of the

boundary layer of the stretching plate. Regular fluids which were considered by them, usually regard as theoretical and laboratory

applications, but their presents in environment are insignificant. Actually, for most cases a binary fluid is in attendance such as river

water, chemical settlement, and melting and freezing process (Nithyadevi & Yang, 2009). In other hands, today in the industrial

usages, binary fluid prefers than regular fluid in cooling applications because of prevention of local corrosion in pipes and

sedimentation. A binary fluid in front of a heated surface, encounters with thermal and concentration's gradients that led to double-

diffusive convection's phenomenon.

This phenomenon has the different mass and thermal diffusivity, and makes a mass and heat transfer, simultaneity (Shi

and Lu., 2006). In more problems, the mass diffusivity due to a thermal's gradient (Soret effect) and the thermal gradient due to a

concentration's gradient (Dufour effect) were ignored, whiles in the high, mass and thermal's gradients, they have significant

influence. Emphasis of these effects considered in several papers (Joly et al., 2000; Bahloul et al., 2003; Mansour et al., 2006; Patha

et al., 2006; Platten, 2006). In this study, the Soret and the Dufour effect also are considered. Nanofluid is a fluid with the

nanometers nanoparticles suspension (Choi, 1995). Nanofluid, which has main specifications to improve thermal conduction than

fluid (Masuda et al., 1993) was interested by researchers. In literature review, few copper's nanoparticles (0.3%) and multi-walled

carbon nanotubes (4%), in ethylene glycol and oil, rise 40% and 150% the conductive heat transfer, respectively (Eastman et al.,

2001; Choi et al., 2001). For same enhancement, more than 10% of the common particles' suspensions are required. Special

nanofluid's properties include the conductive heat transfer more than conventional suspensions, better stability (Godson et al., 2010),


853

simple preparation's methods, and acceptable viscosity, makes it as a suitable choice in coolant fluid such as nuclear's powerhouse

application (Buongiorno, 2005).

The comprehensive study by Buongiorno (2006) was done to investigate the phenomenon of nanofluid's convective heat

transfer. He presented proper description of reasons of extraordinary heat transfer enhancement and viscosity. He pointed the

distributing suspension particles in fluid-which more previous researchers accounted as major reason– has slightly the effect on the

heat-transfer enhancement. He described turbulence's phenomenon is independence of suspension particles, and it is not the

reason of the conductive heat transfer enhancement. A particle' swirl as another cause discussed and low influence was reported.

Buongiorno proposed a new model base on a mechanic's relative velocity of particle respect to regular fluid. He considered seven

mechanisms of particles' corresponding motion than fluid: inertia; Brownian diffusion; thermophoresis; diffusiophoresis; Magnus

affect; fluid drainage, and gravity. Kuznetsov and Nield in 2010 investigated the boundary layer of nanofluid in a natural convection

near a constant temperature's vertical plate. They used a mathematical model with Brownian diffusion and thermophoresis' effect to

evaluate the influence of Prandtl number and the others nanofluid's parameters on boundary layer. Khan and Aziz in 2011 studied

same problem with the constant heat flux boundary condition.

Boundary layer of secondary fluid with nanofluid in 2011 was considered by Kuznetsov and Nield. They discussed the

effect of binary fluid's parameters on boundary layer and also derived a reduced Nusselt number with dimensionless parameters.

Nanofluid's behavior next to stretching plate was explored analytically (Khan and Pop, 2010; Bachok et al., 2010). Base of solution

was similarity solution and reduced Nusselt number as well as Sherwood’s number derived. The Magnetic field also was the other

parameter, which considered (Hamad et al., 2011a; Hamad, 2011b). Aim of this study is investigating nanofluid's behavior near the

stretching plate with binary fluid. For this purpose, government equations derived as dimensionless parameters and then results was

obtained by numerical methods.

Methodology

Government equations

The problem is a steady flow two-dimensional laminar boundary layer of incompressible binary nanofluid over stretching

plate with the linear velocity as wu ax, which a is constant and x is the coordinate measured from some point along the plate

(Figure 1). The flow takes over the plate in 0 y , where y is a coordinate measured normal to the stretching plate. A steady

uniform stress leading to equal and opposite forces is applied along the x-axis so that the sheet is stretched keeping the origin fixed.

At this boundary the temperatureT , the solutal concentration C, and the nanoparticle fraction

take constant values wT, wC

,

and w , respectively. The ambient values, attained as y tends to infinity, of T, C, and

are denoted byT ,C , and ,

respectively.

Figure 1. Physical model and coordinate system.

Principal equations of problem in x, y coordinate are as follow:

0u v

x y

(1)

2 2

2 2

1

f

u u p u uu v

x y x x y

(2)

2 2

2 2

1

f

v v p v vu v

x y y x y

(3)


854

22 2

2 2

TB TC

DT T T T T Cu v D D

x y y y y T y y

(4)

2 2

2 2S CT

C C C Tu v D D

x y y y

(5)

2 2

2 2

TB

D Tu v D

x y y T y

(6)

Above equations have these boundary conditions:

In 0y : ( ) , 0, , ,w w w wu u x ax v T T C C

(7)

For y : 0, 0, , ,u v T T C C

(8)

Where u, and v are flow velocity in x, and y direction, respectively, and f is dynamic viscosity.

Now, we introduce the dimensionless variables defined by:

1/2( ) ( ), ( ) , ( ) , ( )w w w

T T C Ca xf

T T C C

(9)

is a stream function, and were defined ,u v

y x

.

By substitution

in Eqs. (1-6) and using Eq. (9), the ordinary differential equations obtain.

"' " '2 0f ff f (10)

" ' ' ' '2 "10

Prf Nb Nt Nd

(11)

" ' '' 0NT

Le fNb

(12)

" ' '' 0Ln f Ld

(13)

Where the various parameters are defined by:

Pr

(14)

( )

wC

T w

C CNc

T T

(15)

f

f 1 ( )

P w

T w

NrT T

(16)

Bw

DNb

(17)

Tw

DNt T T

T

(18)

wTC

w

C CDNd

T T

(19)

B

LeD

(20)

CT w

S w

D T TLd

D C C

(21)


855

S

LnD

(22)

Here Nc, Nr, Nd, Nb, Nt, denote the typical double-diffusive buoyancy ratio, a nanofluid buoyancy ratio, a modified Dufour

parameter, a Brownian diffusion parameter, a thermophoresis parameter, respectively. Furthermore, Le is the regular Lewis

number; Ld is a Dufour-solutal Lewis number, while Ln is the nanofluid Lewis number.

Equations (10)-(13) are solved subject to the following boundary conditions:

At '(0) 0, (0) 1, (0) 1,f f (23)

At ( ) 0, ( ) 0,f (24)

In this paper, Nusselt number and Sherwood’s number considered and defined as:

, ,

w m m

c B

w S w B w

xq xq xqNu Sh Sh

k T T D D

(25)

wq and

mq are thermal flux and mass flux, respectively.

In this problem, total mass flux consists of both nanoparticles and solutal diffusivities in their base fluid. Two Sherwood’s

numbers introduced, B

Shand

CSh , that

BSh

is Sherwood’s number of Brownian diffusion and C

Sh is solutal fluid due to the

concentration's gradient. Kuznetsov and Nield (2010) showed that reduced Nusselt number and reduced Sherwood’s number have

been relative with negative thermal and concentration's derivatives, respectively. The reduced Sherwood’s number of solutal fluid

also can be defined with negative solutal concentration's derivative. It so can be writing:

(26) 1/2 'Re (0)

xNur Nu

(27) 1/2 '

1Re (0)

B xShr Sh

(28) 1/2 'Re (0)

c x cShr Sh

Where:

(29) Re ( )

x w

xu x

(30)

( )

w

w

xqNu

k T T

(31)

( )

m

B

B w

xqSh

D

(32)

( )

m

c

s w

xqSh

D

Rex

is local Reynolds number that use the velocity at the plate ( )w

u x .

Results and Discussion

Equations (10-13) with boundary conditions (23-24) were solved with numerical methods for various dimensionless

parameters. Results were calculated with Maple 13. Figure (2) illustrates general result for Pr=1, Nt=0.2, Nb=0.2, Ln=1, Le=10,

Nd=0.2, Ld=0.2.


856

f q F g

h

0 1 2 3 4 5

f, q, F , g

0

0.2

0.4

0.6

0.8

1

Figure 2. General result for Pr=1, Nt=0.2, Nb=0.2, Ln=1, Le=10, Nd=0.2, Ld=0.2.

To validate results, solutal parameters neglected (Ln, Ld, Nd=0.00002) and reduced Nusselt number compared with results

of Khan and Pop (2010) (Table 1). Only small difference reports among results.

Table 1. Comparison reduced Nusselt number for Pr=Le=10.

Effect of the thermophoresis parameter Nt and the Brownian diffusion Nb on the reduced Nusselt number is shown in

figure (3). As it seem, Nur decrease by rising Nb and Nt.

Pr=1 , Ln=5 , Le=10 , Ld=0.2 , Nd=0.2.

Figure 3. Effect of the thermophoresis parameter Nt and the Brownian diffusion Nb on the reduced Nusselt number.

Figure (4) represents solutal parameter influences Nd and Ld on reduced Nusselt number.

Pr=1, Nb=0.1, Nt=0.1, Ln=1, Le=10

Nt 0.1 0.2 0.3 0.4 0.5

Nb=0.1 Khan & Pop 0.9524 0.6932 0.5201 0.4026 0.3211

This study 0.9522 0.6933 0.5203 0.4028 0.3213

Nb=0.3 Khan & Pop 0.2522 0.1816 0.1355 0.1046 0.0833

This study 0.2518 0.1821 0.1359 0.1041 0.0835

Nb=0.5 Khan & Pop 0.0543 0.0390 0.0291 0.0225 0.0179

This study 0.0541 0.0393 0.0295 0.0230 0.0182


857

Figure 4. Solutal parameter influences Nd, Ld on reduced Nusselt number.

Figure (4) shows the Nur has direct relation with the Ld, but for the Nd, it is opposite. It was expected because of Nd and

Ld definitions; two dimensionless parameters have reverse influences with thermal and concentration's gradients and raising the Nd

lead to decreasing the Ld effect. Nusselt number is the ratio of convective heat transfer on conductive heat transfer. Each of the Nd,

Nb, and Nt parameters increase convective heat transfer (Figures 7-8), but due to increasing thermal gradient and physical

nanofluid's properties, enhancement is lower than conductive heat transfer.

In Figure (5), variations of the reduced Sherwood’s number illustrate versus Nb and Nt. Raising the Nb increase reduced

Sherwood’s number as well as the Nt. As Eq. (31), Sherwood’s number is total mass flux on top of the mass flux of the Brownian

diffusion. Since the total mass flux comprises the mass fluxes of Brownian diffusion, thermophoresis effect, and solutal fluid, so the

Nb directly should affect the mass fluxes of the thermophoresis or solutal fluid.

The Nt increment due to thermal gradient ( wT T ) or physical properties of fluid ( TD

) (Buongiorno, 2006) also

increase reduced Sherwood’s number. Since the mass flux of thermophoresis effect is ,p T P T

TJ D

T

(Buongiorno, 2006)

and Tw

DNt T T

T

, it can declare the effect of Nt on the total mass flux is because of increment of thermophoresis

effect.

Pr=1, Nd=0.3, Ln=1, Le=10, Ld=1

Figure 5. The reduced Sherwood’s number versus Nt and Nb.

Figure (6) shows an effect of Ld and Nd on the reduced Sherwood’s number of the solutal fluid. Raising the Nd and Ln

increase reduced Sherwood’s number, but for the Nd, it is slightly.

Pr=1, Nt=0.3, Nb=0.2, Le=10, Ld=1


858

Figure 6. Effect of Ln and Nd on the reduced Sherwood’s number of the solutal fluid.

By consider the Nd from Eq. (19), it is obvious, increasing the solutal fluid gradient, decrease the thermal gradient.

Therefore, the mass fluxes of solutal fluid and thermophoresis effect are against together.

Figure (7) shows temperature's profile with influence of Nb and Nt. Two parameters develop the temperature's profile.

Pr=3, Nd=0.2, Ld=0.2, Ln=1, Le=10

h

0 1 2 3 4 5 6 7 8 9

q h

0

0.2

0.4

0.6

0.8

1

Figure 7. Nanoparticles' parameters effect on temperature's profile.

In figure (8), effect of Ld and Nd on temperature's profile illustrate. As it expects, increasing Ld and Nd expand the domain of

temperature's profile.

Pr=3, Nt=0.2, Nb=0.2, Ln=1, Le=10

Nb, Nt= 0.1, 0.3, 0.5, 0.7, 0.9


859

h

0 1 2 3 4 5 6 7 8 9

q h

0

0.2

0.4

0.6

0.8

1

Figure 8. Effect of the Nd and Ld on the temperature's profile.

Conclusion

In this paper, the laminar heat transfer of binary nanofluid on horizontal stretching plate investigated numerically. This

paper focuses the effect of the nanofluid's (Brownian diffusion, thermophoresis effect) and the binary fluid parameters on a

convective heat transfer flux. The coupled government equations were solved by the similarity solution. The dimensionless

parameters, which defined for solution, have indirect relation with the reduced Nusselt number. The defined reduced Sherwood’s

number rises with Nt and Nb, as well as Nd and Ld.

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Appendix

Nomenclature

C solutal concentration c

Shr

Reduced Sherwood’s number of solutal fluid

wC solutal concentration at the plate B

Shr

Reduced Sherwood’s number

C ambient solutal concentration attained as y tends

to infinity

f dimensionless stream function

DB

Brownian diffusive coefficient T Temperature

DT thermophoretic diffusive coefficient wT temperature at the plate

DCT Soret diffusivity T ambient temperature attained as y tends to infinity

DTC Dufour diffusivity v velocity, (u, v)

DS solutal diffusivity of the porous medium (x,

y)

Cartesian coordinates (x-axis is aligned horizontally

to right, plate is at y = 0)

g gravitational acceleration vector Greek symbols

,p TJ

nanoparticle mass flux to thermophoresis (kg/m2

s)

α thermal diffusivity

k thermal conductivity of the nanofluid T volumetric thermal expansion coefficient of the fluid

Ld Dufour-solutal Lewis number C volumetric solutal expansion coefficient of the fluid

Le regular Lewis number dimensionless solutal concentration

Ln nanofluid Lewis number similarity variable

Nr buoyancy ratio θ dimensionless temperature

Nb Brownian diffusion parameter μ dynamic viscosity of the nanofluid

Nd modified Dufour parameter kinematic viscosity, µ/f

Nt thermophoresis parameter f density of the base fluid

Nu Nusselt number p nanoparticle mass density

Nur reduced Nusselt number, 1/4/ xNu Ra ( )fC

heat capacity of the fluid

Pr Prandtl number ( )pC

effective heat capacity of the nanoparticle material

p Pressure parameter defined by ( )pC / ( )fC

mq Mass flux at the wall (Kg/m2) nanoparticle volume fraction

wq heat flux at the wall (W/m2) w nanoparticle volume fraction at the plate

Rax local Rayleigh number ambient nanoparticle volume fraction attained as y

tends to infinity

CSh

Sherwood’s number of solutal fluid stream function

BSh

Sherwood’s number

Investigation of Laminar Heat Transfer of Binary Nanofluid on Horizontal Stretching Plate

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