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Soret effect on mixed convection flow in a nanofluid under convective boundary condition
International Journal of Heat and Mass Transfer 64 (2013) 384–392
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Soret effect on mixed convection flow in a nanofluid under convective boundary condition Ch. RamReddy a,⇑, P.V.S.N. Murthy b, Ali J. Chamkha c, A.M. Rashad d,e a
Department of Mathematics, National Institute of Technology, Rourkela 769008, Odisha, India Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, India c Manufacturing Engineering Department, Public Authority for Applied Education and Training, Shuweikh 70654, Kuwait d Department of Mathematics, Aswan University, Faculty of Science, Aswan 81528, Egypt e Department of Mathematics, Salman Bin Abdul Aziz University, College of Science and Humanity Studies, AL-Kharj, Saudi Arabia b
a r t i c l e
i n f o
Article history: Received 30 November 2012 Received in revised form 13 April 2013 Accepted 14 April 2013
Keywords: Mixed convection Nanofluid Soret effect Convective boundary condition Numerical solution
a b s t r a c t In this investigation, we intend to present the influence of the prominent Soret effect on mixed convection heat and mass transfer in the boundary layer region of a semi-infinite vertical flat plate in a nanofluid under the convective boundary conditions. The transformed boundary layer ordinary differential equations are solved numerically using the implicit iterative finite difference method. Consideration of the nanofluid and the convective boundary conditions enhanced the number of non-dimensional parameters considerably thereby increasing the complexity of the present problem. A wide range of parameter values is chosen to bring out the effect of Soret parameter on the mixed convection process with the convective boundary condition. The effect of mixed convection, Soret and Biot parameters on the flow, heat and mass transfer coefficients is analyzed. The numerical results obtained for the velocity, temperature, volume fraction, and concentration profiles, as well as the local skin-friction coefficient, local wall temperature, local nanoparticle concentration and local wall concentration reveal interesting phenomenon, some of these qualitative results are presented through plots and tables. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction The process of heat and mass transfer caused by the simultaneous effect of free and forced convection is known as mixed convection flow. Considerable attention has been paid to the theoretical and numerical study of mixed convection boundary layer flow along a vertical plate in the recent past as it plays a crucial role in diverse applications, such as electronic devices cooled by fans, nuclear reactors cooled during an emergency shutdown, a heat exchanger placed in a low-velocity environment, solar collectors and so on. In the study of fluid flow over heated surfaces, the buoyancy force is neglected when the flow is horizontal. However, for vertical or inclined surfaces, the buoyancy force exert strong influence on the flow field. Extensive studies on mixed convection heat and mass transfer from isothermal and non-isothermal vertical surface under usual boundary layer approximation for viscous fluids have been undertaken by several researchers. Somers [1] analyzed theoretical results for combined thermal and mass transfer from a flat plate. The theoretical solution of heat transfer by mixed convection about a vertical flat plate has been obtained ⇑ Corresponding author. E-mail addresses: (Ch. RamReddy).
[email protected],
[email protected]
0017-9310/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.04.032
by Kliegel [2]. Merkin [3] investigated the mixed convection boundary layer flow on a semi-infinite vertical flat plate when the buoyancy forces aid and oppose the development of the boundary layer. Lloyd and Sparrow [4] used a local similarity method to solve the mixed convection flow on a vertical surface and showed that the numerical solutions ranged from pure forced convection to mixed convection. Kafoussias [5] presented analysis of the effects of buoyancy forces in a laminar uniform forced-convective flow with mass transfer along a semi-infinite vertical plate. An analysis is carried out by Chamkha et al. [6] to study the effects of localized heating (cooling), suction (injection), buoyancy forces and magnetic field for the mixed convection flow on a heated vertical plate. A detailed review of mixed convective heat and mass transfer can be found in the book by Bejan [7]. Recently, Subhashini et al. [8] discussed the simultaneous effects of thermal and concentration diffusions on a mixed convection boundary layer flow over a permeable surface under convective surface boundary condition. The diffusion of mass due to temperature gradient is called Soret or thermo-diffusion effects and this effect might become significant when large density differences exist in the flow regime. For example, Soret effect can be significant when species are introduced at a surface in fluid domain, with a density lower than the surrounding fluid. The Soret parameter has been utilized for isotope separation and in a mixture between gases with very light molecular weight
Ch. RamReddy et al. / International Journal of Heat and Mass Transfer 64 (2013) 384–392
385
Nomenclature Bi C Cw Cf C1 c DB DS DT DCT f g Grx hf J k Le Nb Nc Nr Nt Nux Pr qm qn qw Rex S Sc Shx NShx ST T Tf
Biot number solutal concentration solutal concentration at the wall skin friction coefficient ambient solutal concentration constant Brownian diffusion coefficient solutal diffusivity thermophoretic diffusion coefficient soret diffusivity dimensionless stream function gravitational acceleration local Grashof number convective heat transfer coefficient ratio between the effective heat capacity of the nanoparticle material and heat capacity of the fluid thermal conductivity of the nanofluid Lewis number Brownian motion parameter regular buoyancy ratio nanoparticle buoyancy ratio thermophoresis parameter local Nusselt number Prandtl number regular mass flux at the wall nanoparticle mass flux at the wall heat flux at the wall local Reynolds number dimensionless concentration Schmidt number local Sherwood number local nanoparticle Sherwood number Soret number temperature temperature of the hot fluid
(H2,He) and of medium molecular weight (N2,air). Dursunkaya and Worek [9] studied diffusion-thermo and thermal-diffusion effects in transient and steady natural convection from a vertical surface, whereas Kafoussias and Williams [10] presented the same effects on mixed convective and mass transfer steady laminar boundary layer flow over a vertical flat plate with temperature dependent viscosity. The linear stability analysis of Soret-driven thermosolutal convection in a shallow horizontal layer of a porous medium subjected to inclined thermal and solutal gradients of finite magnitude has been investigated theoretically by Narayana et al. [11]. Recently, the effect of melting and/or thermodiffusion on convective transport in a non-Newtonian fluid saturated non-Darcy porous medium are presented by Kairi and Murthy [12] and Srinivasacharya and RamReddy [13]. In recent times, the flow analysis of nanofluids has been the topic of extensive research due to its characteristic in increasing thermal conductivity in heat transfer process. Several ordinary fluids including water, toluene, ethylene glycol and mineral oils etc. in heat transfer processes have rather low thermal conductivity. The nanofluid (initially introduced by Choi [14]) is an advanced type of fluid containing nanometer sized particles (diameter less than 100 nm) or fibers suspended in the ordinary fluid. Undoubtedly, the nanofluids are advantageous in the sense that they are more stable and have an acceptable viscosity and better wetting, spreading and dispersion properties on a solid surface. Nanofluids are used in different engineering applications such as microelectronics,
T1 u1 u,v x,y
am bT,bC
g c k h / /w /1
l m q qf1 qp (qc)f (qc)p sw w
ambient temperature characteristic velocity velocity components in x and y directions coordinates along and normal to the plate thermal diffusivity volumetric thermal and solutal expansion coefficients of the base fluid similarity variable dimensionless volume fraction mixed convection parameter dimensionless temperature nanoparticle volume fraction nanoparticle volume fraction at the wall nanoparticle volume fraction at large values of y (ambient) dynamic viscosity of the base fluid kinematic viscosity density of the fluid density of the base fluid nanoparticle mass density heat capacity of the fluid effective heat capacity of the nanoparticle material wall shear stress stream function
Subscripts w wall condition 1 ambient condition C concentration T temperature Superscript 0 differentiation with respect to g
microfluidics, transportation, biomedical, solid-state lighting and manufacturing. The research on heat and mass transfer in nanofluids has been receiving increased attention worldwide. Many researchers have found unexpected thermal properties of nanofluids, and have proposed new mechanisms behind the enhanced thermal properties of nanofluids. Excellent reviews on convective transport in nanofluids have been made by Buongiorno [15] and Kakac and Pramuanjaroenkij [16]. Kuznetsov and Nield [17] studied analytically the natural convective boundary-layer flow of a nanofluid past a vertical plate. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. Also, it is interesting to note that the Brownian motion of nanoparticles at molecular and nanoscale levels is a key nanoscale mechanism governing their thermal behaviors. In nanofluid systems, due to the size of the nanoparticles, the Brownian motion takes place, which can affect the heat transfer properties. As the particle size scale approaches to the nanometer scale, the particle Brownian motion and its effect on the surrounding liquids play an important role in the heat transfer. The steady boundary-layer flow of a nanofluid past a moving semi-infinite flat plate in a uniform free stream is analyzed by Bachok et al. [18]. Recently, the double-diffusive natural convective boundary-layer flow of a nanofluid past a vertical plate has been studied analytically by Kuznetsov and Nield [19]. Gorla et al. [20] presented a boundary layer analysis for the mixed convection past a vertical wedge in a porous medium saturated with a nanofluid. But, very little attention has been paid to study
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the significance of buoyancy forces on mixed convective in a nanofluid under convective boundary condition. Makinde and Aziz [21] studied numerically the boundary layer flow induced in a nanofluid due to a linearly stretching sheet in the presence of a convective heating boundary condition. However, a clear picture about the nanofluid boundary layer flows is still to emerge. Motivated by the above referenced work and the vast possible industrial applications, it is of paramount interest to consider the effect of Soret parameter on mixed convective flow along a vertical plate in a nanofluid under convective boundary condition. The presence of convective boundary conditions make the mathematical model of the present physical system a little more complicated leading to the complex interactions of the flow, heat and mass transfer mechanism. Analytical solution is ruled out in the current set up and hence a numerical solution is obtained for the present problem. Consideration of the nanofluid and the convective boundary conditions enhanced the number of non-dimensional parameters considerably. The effects of mixed convection, Soret and Biot parameters on the physical quantities of the flow, heat and mass transfer coefficients are analyzed. To examine the convergence of the numerical code written to solve the present problem, (i.e., for code validation) we compare the present results for the clear fluid mixed convection results with previously published works with convective boundary conditions and the comparison shows that the results are in very good agreement. 2. Mathematical formulation Choose the coordinate system such that the x-axis is along the vertical plate and y-axis normal to the plate. The physical model and coordinate system are shown in Fig. 1. Consider the steady laminar two-dimensional mixed convection heat and mass transfer along a flat vertical surface embedded in a nanofluid having T1, C1, and /1 as the temperature, concentration and nanoparticle volume fraction respectively in the ambient medium. Also assume that a free stream with uniform velocity u1 goes past the flat plate. The plate is either heated or cooled from left by convection from a fluid of temperature Tf with Tf > T1 corresponding to a heated surface (assisting flow) and Tf < T1 corresponding to a cooled surface (opposing flow) respectively. On the wall the solutal concentration and the nanoparticle volume fraction are taken to be constant and are given by Cw, and /w, respectively. By employing Oberbeck–Boussinesq approximation, making use of the standard boundary layer approximations and eliminating pressure, the governing equations for the nanofluid are given by
@u @ v þ ¼0 @x @y
qf 1 u
ð1Þ
@u @u þv @x @y
¼l
@2u þ g qf 1 ð1 /1 Þ½bT ðT T 1 Þ @y2
þ bC ðC C 1 Þ ðqp qf 1 Þgð/ /1 Þ " 2 # @T @T @2T @/ @T DT @T u þv ¼ am 2 þ J DB þ @x @y @y @y @y T 1 @y
ð2Þ
ð3Þ
u
@/ @/ @ 2 / DT @ 2 T þv ¼ DB 2 þ @x @y @y T 1 @y2
ð4Þ
u
@C @C @2C @2T þv ¼ Ds 2 þ DCT 2 @x @y @y @y
ð5Þ
where u and v are the velocity components along the x and y axes, respectively, T is the temperature, / is the nanoparticle volume fraction, C is the solutal concentration, g is the gravitational acceleration, am = k/(qc)f is the thermal diffusivity of the fluid, m = l/qf1 is the kinematic viscosity coefficient and J ¼ ðqcÞp =ðqcÞf . Further, qf1 is the density of the base fluid and q, l, k, bT, and bC are the density, viscosity, thermal conductivity, volumetric thermal expansion coefficient and volumetric solutal expansion coefficient of the nanofluid, while qp is the density of the nanoparticles, (qc)f is the heat capacity of the fluid and (qc)p is the effective heat capacity of the nanoparticle material. The coefficients that appear in Eqs. (3)–(5) are the Brownian diffusion coefficient DB, the thermophoretic diffusion coefficient DT, the solutal diffusivity DS and the Soret-type diffusivity DCT. For, details of the derivation of Eqs. (1)–(5), one can refer the papers by Buongiorno [15] and Nield and Kuznetsov ([17];[19]). The boundary conditions are
@T ¼ hf ðT f TÞ; / ¼ /w ; C ¼ C w at y ¼ 0 ð6aÞ @y u ! u1 ; T ! T 1 ; / ! /1 ; C ! C 1 as y ! 1; ð6bÞ
u ¼ 0;
v ¼ 0;
k
here, hf is the convective heat transfer coefficient and the subscripts w and 1 indicate the conditions at the surface and at the outer edge of the boundary layer respectively. In view of the continuity equation (1), we introduce the stream function w by
u¼
@w ; @y
v ¼
@w @x
ð7Þ
Substituting (7) in Eqs. (2)–(5) and then using the following nondimensional transformation 1=2 p1ffiffi g ¼ pyffiffi2x Re1=2 wðgÞ; x ; f ðgÞ ¼ 2m Rex
9 =
1 1 1 ; cðgÞ ¼ /// ; SðgÞ ¼ CCC ;; hðgÞ ¼ TTT T 1 w C 1 w /1
ð8Þ
f
with the local Raynold’s number Rex ¼ u1m x, we get the transformed system of ordinary differential equations as 00
f 000 þ ff þ 2k½h þ Nc S Nr c ¼ 0 2 1 00 h þ f h0 þ Nb h0 c0 þ Nt h0 ¼ 0 Pr
ð10Þ
Nt 00 h ¼0 Nb
ð11Þ
1 00 S þ ST h00 þ f S0 ¼ 0 Sc
ð12Þ
c00 þ Le f c0 þ
Fig. 1. Physical model and coordinate system.
ð9Þ
where the primes indicate differentiation with respect to g. In usual notations, Pr ¼ amm is the Prandtl number, Sc ¼ DmS is the Schmidt
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Ch. RamReddy et al. / International Journal of Heat and Mass Transfer 64 (2013) 384–392 Table 1 Comparison of f00 (0) and h0 (0) for mixed convection along a vertical flat plate in Newtonian fluids ([23,8]). Bi
Pr
k
Makinde and Olanrewaju [23] 00
0.1 1.0 0.1 0.1 0.1
0.1 0.1 1.0 0.1 0.1
0.72 0.72 0.72 3.00 7.10
Subhashini et al. [8]
0
00
Present
f (0)
h (0)
f (0)
h (0)
f00 (0)
h0 (0)
0.36881 0.44036 0.63200 0.34939 0.34270
0.07507 0.23750 0.07704 0.08304 0.08672
0.36875 0.44032 0.63198 0.34937 0.34270
0.07505 0.23746 0.07700 0.08301 0.08670
0.36898 0.44049 0.63197 0.34957 0.34289
0.07509 0.23765 0.07705 0.08308 0.08674
1.6
0
0.8 ST=0.2 ST=0.4
ST=0.2 ST=0.4 ST=1.0
f'( η )
1.2
λ=1.0
Nr=1.0 Nb=0.5 Nt=0.5 Nc=1.0 Bi=1.0 Pr=1.0 Sc=0.6
0.4
0.0
θ (η )
Le=1.0,10,100
0.8
0
1
η
(a)
2
3
0.4
λ=1.0
Nr=1.0 Nb=0.5 Nt=0.5 Nc=1.0 Bi=1.0 Pr=1.0 Sc=0.6
Le=1.0,10,100
0.2
0.0
4
0
1
(b)
1.0
2
η
3
4
1.0 ST=0.2 ST=0.4
0.8
ST=0.2 ST=0.4
0.8
ST=1.0
ST=1.0
Le=1.0,10,100
γ (η)
0.6
S ( η)
0.6 λ=1.0
Nr=1.0 Nb=0.5 Nt=0.5 Nc=1.0 Bi=1.0 Pr=1.0 Sc=0.6
0.4
0.2
0.0
ST=1.0
0.6
0
1
2
3
(c)
4
Le=1.0,10,100
λ=1.0
Nr=1.0 Nb=0.5 Nt=0.5 Nc=1.0 Bi=1.0 Pr=1.0 Sc=0.6
0.4
0.2
5
0.0
0
(d)
η
1
2
η
3
4
Fig. 2. Effects of ST and Le on (a) velocity, (b) temperature, (c) concentration, and (d) volume fraction profiles. ðqp qf 1 Þð/w /1 Þ f 1 bT ðT f T 1 Þð1/1 Þ
number and Le ¼ DmB is the Lewis number. Nr ¼ q
is the
w C 1 Þ is the regular buoyancy nanofluid buoyancy ratio and Nc ¼ bbCTðC ðT T 1 Þ f
ratio. Further, Nb ¼ Nt ¼
ðqcÞp DT ðT f T 1 Þ ðqcÞf mT 1
gbT ð1/1 ÞðT f T 1 Þx3
m2
ðqcÞp DB ð/w /1 Þ ðqcÞf m
is
the
is the Brownian motion parameter,
thermophoresis
parameter,
is the local Grashof number and k ¼
convection parameter [8]. Finally, ST ¼
DCT ðT f T 1 Þ mðC w C 1 Þ
Grx Re2x
Grx ¼
is the mixed
is the Soret number.
Boundary conditions (6) in terms of f, h, c, S become
g ¼0:f ð0Þ¼0; f 0 ð0Þ¼0; h0 ð0Þ¼Bi½1hð0Þ; cð0Þ¼1; Sð0Þ¼1 ð13aÞ g !1:f 0 ð1Þ!1; hð1Þ!0; cð1Þ!0; Sð1Þ!0 ð13bÞ
qffiffiffiffiffi where Bi ¼ kc u21m is the Biot number. This boundary conditions will be free from the local variable x when we choose hf = cx1/2. The Biot number Bi is a ratio of the internal thermal resistance of the plate to the boundary layer thermal resistance of the hot fluid at the bottom of the surface. It is important to note that this boundary value problem reduces to the classical problem of flow and heat and mass transfer due to a the Blasius problem when Nb and Nt are zero. Most nanofluids examined to date have large values for the Lewis number Le > 1 [17]. For water nanofluids at room temperature with nanoparticles of 1–100 nm diameters, the Brownian diffusion coefficient DB ranges from 4 104 to 4 1012 m2/s. Furthermore, the ratio of the Brownian diffusivity coefficient to the thermopho-
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Ch. RamReddy et al. / International Journal of Heat and Mass Transfer 64 (2013) 384–392
resis coefficient for particles with diameters of 1–100 nm can be varied in the ranges of 2–0.02 for alumina, and from 2 to 20 for copper nanoparticles (see Buongiorno [15] for details). Hence, the variation of the non-dimensional parameters of nanofluids in the present study is considered in the mentioned range.
C f ð2Rex Þ1=2 ¼ f 00 ð0Þ; Nux ðRex =2Þ1=2 ¼ h0 ð0Þ; NShx ðRex =2Þ1=2 ¼ c0 ð0Þ; Shx ðRex =2Þ1=2 ¼ S0 ð0Þ:
Effect of the various parameters involved in the investigation on these coefficients is discussed in the following section.
3. Skin friction, heat and mass transfer coefficients The primary objective of this study is to estimate the skin friction coefficient Cf, the Nusselt number Nu, the nanoparticle Sherwood number NSh, and the regular Sherwood number Sh. These parameters characterize the surface drag, the wall heat, nanoparticle mass transfer and regular mass transfer rates, respectively. The shearing stress, local heat, local nanoparticle mass and local regular mass fluxes from the vertical plate can be obtained from
@u @T ; qw ¼ k ; @y y¼0 @y y¼0 @/ @C qn ¼ DB and qm ¼ DS : @y y¼0 @y y¼0
sw ¼ l
ð14Þ
The non-dimensional shear stress C f ¼ q swu2 , the Nusselt number f1 1 x wx Nux ¼ kðTqT , the nanoparticle Sherwood number NShx ¼ DB ð/qwn/ 1Þ 1Þ f x and the regular Sherwood number Shx ¼ DS ðCqwmC , are given by 1Þ
4. Results and discussion The resulting transport Eqs. (9)–(12) are non-linear, coupled, ordinary differential equations, which possess no closed-form solution. Therefore, these are solved numerically subject to the boundary conditions given by Eq. (13). The implicit, iterative finite-difference method discussed by Blottner [22] has proven to be adequate for the solution of this type of equations. For this reason, this method is employed in the present work. Eqs. (9)–(12) are discretized using three-point central difference quotients. This converts the differential equations into linear sets of algebraic equations, which can be readily solved by the well-known Thomas algorithm [22]. On the other hand, Eqs. (9)–(12) is discretized and solved subject to the appropriate boundary conditions by the trapezoidal rule. The computational domain in the g-direction was made up of 196 non-uniform grid points. It is expected that most
1.6
1.0 Nr=0.1 Nr=0.5 Nr=0.9
1.2
0.6 λ=1.0
Nb=0.5 Nt=0.5 Nc=1.0 ST=1.0 Le=10 Pr=1.0 Sc=0.6
0.4
1
2
3
0.2
0.0
4
0
1
η
(b)
1.0
2
3
4
1.0 Nr=0.1 Nr=0.5 Nr=0.9
0.8
Nr=0.1 Nr=0.5 Nr=0.9
0.8
0.6 γ (η )
S( η )
0.6 λ=1.0
Nb=0.5 Nt=0.5 Nc=1.0 ST=1.0 Le=10 Pr=1.0 Sc=0.6
0.4
0.2 Bi=0.1,10,20
0.0
λ=1.0 Nb=0.5 Nt=0.5 Nc=1.0 ST=1.0 Le=10 Pr=1.0 Sc=0.6
0.4
η
(a)
Bi=0.1,10,20
θ (η )
f'( η )
0.8
0
Nr=0.1 Nr=0.5 Nr=0.9
0.8
Bi=0.1,10,20
0.0
ð15Þ
0
(c)
1
η
2
3
λ=1.0 Nb=0.5 Nt=0.5 Nc=1.0 =1.0 S T Le=10 Pr=1.0 Sc=0.6
0.4 Bi=0.1,10,20
0.2
4
0.0
0
(d)
1
η
2
3
Fig. 3. Effects of Bi and Nr on (a) velocity, (b) temperature, (c) concentration, and (d) volume fraction profiles.
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Ch. RamReddy et al. / International Journal of Heat and Mass Transfer 64 (2013) 384–392
changes in the dependent variables occur in the region close to the plate where viscous effects dominate. However, small changes in the dependent variables are expected far away from the plate surface. For these reasons, variable step sizes in the g-direction are employed. The initial step size Mg1 and the growth factor K⁄ employed such that Mgi+1 = K⁄Mgi (where the subscript i indicates the grid location) were 103 and 1.0375, respectively. These values were found (by performing many numerical experimentations) to give accurate and grid-independent solutions. The solution convergence criterion employed in the present work was based on the difference between the values of the dependent variables at the current and the previous iterations. When this difference reached 105, the solution was assumed converged and the iteration process was terminated. To have a better understanding of the flow characteristics, numerical results for the velocity, temperature, volume fraction and concentration are calculated for different sets of values of the parameters ST, Le, k, Nc, Nb, Nt, Nr, Bi. Also, the effect of these parameters on skin friction, non-dimensional heat, nanoparticle mass and regular mass transfer coefficients is discussed. The mixed convection flow of Newtonian fluids along a vertical flat plate studied by Makinde and Olanrewaju [23] and Subhashini et al. [8] can be obtained here as a special case by taking Nc = 0, Nr = 0, Nb = 0, Nt = 0, Le = 0, ST = 0 and c(g) ? 0 (i.e., the Newtonian
(non-nanofluid) fluid formulation of the present investigation). Blottner [22] method is extended here to study the mixed convective transport problem with the convective boundary conditions. It is more relevant and significant to compare a special case result of the present problem with the convective boundary condition with the existing solutions in the literature. Therefore, a comparison is made with the solution presented in Makinde and Olanrewaju [23] and Subhashini et al. [8] and it is found to be in good agreement, as seen from the Table 1. Fig. 2(a)–(d) show the non-dimensional velocity, temperature, nanoparticle volume fraction and concentration profiles for various values of Soret number ST along with varying values of the Lewis number Le for a given k = 1.0, Nr = 1.0, Nc = 1.0, Bi = 1.0, Nb = 0.5, Nt = 0.5, Pr = 1.0, Sc = 0.6. The Soret number ST accounts for the additional mass diffusion due to the temperature gradients. It is noticed from Fig. 2 that an increase in the Soret number ST resulted in an increase in the velocity and concentration fields while a decrease in the temperature and the nanoparticle volume fraction is noted within the boundary layer. The present analysis shows that the flow field is appreciably influenced by the Soret parameter. It is clear from these figures that an increase in Lewis number Le increased the momentum boundary layer thickness, while a reduction in the thermal, nanoparticle nanoparticle volume fraction and solutal boundary layer thickness is noted.
1.6 Nb=0.1 Nb=0.5 Nb=0.9
1.2
0.6
0.8
θ (η )
f'( η )
Nt=0.1,0.5,0.9 λ=1.0 Nr=1.0 Nc=1.0 Bi=1.0 ST=1.0 Le=10 Pr=1.0 Sc=0.6
0.4
0.0
0
1
η
(a)
2
3
λ=1.0
0.4
Nr=1.0 Nc=1.0 Bi=1.0 ST=1.0 Le=10 Pr=1.0 Sc=0.6
0.2 Nt=0.1,0.5,0.9
4
0.0
0
1
(b)
1.0
η
2
3
1.0 Nb=0.1 Nb=0.5 Nb=0.9
Nt=0.1,0.5,0.9
0.8
Nb=0.1 Nb=0.5 Nb=0.9
0.8
γ (η )
0.6
S( η )
0.6 λ=1.0
Nr=1.0 Nc=1.0 Bi=1.0 ST=1.0 Le=10 Pr=1.0 Sc=0.6
0.4
0.2
0.0
Nb=0.1 Nb=0.5 Nb=0.9
0.8
0
(c)
1
2
η
3
4
λ=1.0
0.4
Nt=0.1,0.5,0.9
0.2
5
0.0
0
(d)
1
η
2
Nr=1.0 Nc=1.0 Bi=1.0 ST=1.0 Le=10 Pr=1.0 Sc=0.6
3
Fig. 4. Effects of Nb and Nt on (a) velocity, (b) temperature, (c) concentration, and (d) volume fraction profiles.
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Ch. RamReddy et al. / International Journal of Heat and Mass Transfer 64 (2013) 384–392
The non-dimensional velocity for different values of Biot number Bi with fixed values of other parameters is plotted in Fig. 3(a). Increased convective heating associated with an increase in Bi is seen to thicken the momentum boundary layer. A reverse trend is seen in the case of nanofluid buoyancy ratio Nr. Given that convective heating increases with Biot number, Bi ? 1 simulates the isothermal surface, which is clearly seen from Fig. 3(b), where h(0) = 1 as Bi ? 1. In fact, a high Biot number indicates higher internal thermal resistance of the plate than the boundary layer thermal resistance. As a result, an increase in the Biot number leads to increase of fluid temperature efficiently, these figures confirm this fact also. A similar nature is observed in the case of Nr. Fig. 3(c) brings out the effect on the nanoparticle volume fraction with different values of Biot number Bi and Nr for fixed values of other parameters. As the parameter values Bi and Nr increase, the volume fraction increased for the fixed values of the other parameters. Fig. 3(d) indicates the variation of dimensionless concentration for different values of Bi and Nr. A reduction in concentration boundary layer thickness is seen with increasing values of the Biot number Bi whereas the concentration boundary layer thickens with increasing values of Nr. Fig. 4 present the effect of the Brownian motion Nb and thermophoresis Nt parameters on the velocity, temperature, volume fraction and concentration distributions. When Nb = 0, there is no
additional thermal transport due to buoyancy effects created as a result of nanoparticle concentration gradients. It is observed that the momentum boundary layer thickness increases with the increase in values of Nb but it decreases with increasing values of Nt. With the rise in the values of parameters Nt and Nb, an increase in the temperature is noted whereas a reduction in the concentration field is noticed. The nanoparticle volume fraction decreased with increase in Nb and it increased with increasing values of Nt. It is also noticed that the nanoparticle volume fraction increased with an increase in Nb in the case of forced convection flow. Nt > 0 indicates a cold surface while negative Nt < 0 corresponds a hot surface, in case of a hot surface, thermophoresis tends to blow the nanoparticle volume fraction away from the surface since a hot surface repels the sub-micron sized particles from it, thereby forming a relative particle-free layer near the surface. Variation of non-dimensional velocity, temperature, volume fraction and concentration distribution against the similarity variable g, is shown respectively in Fig. 5, for a few sets of values of k and Nc with fixed values of other parameters. As the parameters k and Nc increase, the velocity increased whereas temperature, volume fraction and concentration of the nanofluid decreased. This is due to the fact that the buoyancy force act like pressure gradient which accelerates/decelerates the fluid within the boundary layer. The effect of buoyancy on the temperature, volume fraction and
0.8 Nc=1.0 Nc=2.0 Nc=3.0
3.0
Nc=1.0 Nc=2.0 Nc=3.0
0.6
2.4
f'( η )
θ (η )
1.8
λ=1.0,2.0,3.0
0.4
1.2 λ=1.0,2.0,3.0
0.6
0.0
0.2
Nr=1.0,Nb=0.5,Nt=0.5,Bi=1.0 ST=1.0,Le=10,Pr=1.0,Sc=0.6
0
(a)
1
2
η
3
4
1.0
0
1
(b)
2
η
3
1.0 Nc=1.0 Nc=2.0 Nc=3.0
0.6 S( η )
λ=1.0,2.0,3.0
0.4
0.2
0
(c)
1
2
η
3
Nc=1.0 Nc=2.0 Nc=3.0
0.8
0.6 Nr=1.0 Nb=0.5 Nt=0.5 Bi=1.0 ST=1.0 Le=10 Pr=1.0 Sc=0.6
Nr=1.0 Nb=0.5 Nt=0.5 Bi=1.0 ST=1.0 Le=10 Pr=1.0 Sc=0.6
γ (η )
0.8
0.0
0.0
Nr=1.0 Nb=0.5 Nt=0.5 Bi=1.0 ST=1.0 Le=10 Pr=1.0 Sc=0.6
0.4
λ=1.0,2.0,3.0
0.2
4
0.0
0.0
(d)
0.5
1.0
η
1.5
2.0
2.5
Fig. 5. Effects of k and Nc on (a) velocity, (b) temperature, (c) concentration, and (d) volume fraction profiles.
Ch. RamReddy et al. / International Journal of Heat and Mass Transfer 64 (2013) 384–392 Table 2 Effects of skin friction coefficient, and heat, nanoparticle mass and regular mass transfer rates for various values of ST, Le, Bi, and Nr when Nb = Nt = 0.5, Nc = Pr = 1.0, k = 1.0 and Sc = 0.6. ST
Le
Bi
Nr
f00 (0)
h0 (0)
S0 (0)
c0 (0)
0.2 0.4 0.6 0.2 0.4 0.6 0.2 0.4 0.6 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
1.0 1.0 1.0 10 10 10 100 100 100 10 10 10 10 10 10 10 10 10
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.1 0.1 0.1 10 10 10 20 20 20
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.1 0.5 0.9 0.1 0.5 0.9 0.1 0.5 0.9
1.660426 1.713282 1.764715 2.331555 2.378589 2.425132 2.620651 2.669386 2.717777 2.264569 2.030422 1.786345 3.232459 3.019211 2.800324 3.256729 3.043529 2.825050
0.2648041 0.2673099 0.2699203 0.2574143 0.2591855 0.2606758 0.2500289 0.2517529 0.2531556 0.06391265 0.06375668 0.06362097 0.3909954 0.3866136 0.3827407 0.4015757 0.3980230 0.3931959
0.4990276 0.5023820 0.5051266 0.5738778 0.5769784 0.5797088 0.5887031 0.5962021 0.6038435 0.5650807 0.5578854 0.5495672 0.5883316 0.5835845 0.5789469 0.5873385 0.5823732 0.5781532
0.6064302 0.6167969 0.6259490 1.777230 1.791779 1.805668 4.007293 4.034007 4.060444 1.686413 1.646561 1.603247 1.937524 1.908359 1.876994 1.941364 1.911234 1.880945
Table 3 Effects of skin friction coefficient, and heat, nanoparticle mass and regular mass transfer rates for various values of Nb, Nt, k, and Nc when Bi = Nr = ST = Pr = 1.0, Le = 10 and Sc = 0.6. Nb
Nt
k
Nc
f00 (0)
h0 (0)
S0 (0)
c0 (0)
0.1 0.5 0.9 0.1 0.5 0.9 0.1 0.5 0.9 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
0.1 0.1 0.1 0.5 0.5 0.5 0.9 0.9 0.9 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 2.0 2.0 2.0 3.0 3.0 3.0
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 2.0 3.0 1.0 2.0 3.0 1.0 2.0 3.0
2.231877 2.368428 2.477065 2.202382 2.425132 2.538059 2.199755 2.484781 2.599391 2.425132 3.611774 4.676075 3.800878 5.801891 7.591507 4.987329 7.700205 10.12352
0.3779059 0.3122588 0.2496606 0.3130586 0.2606758 0.2090975 0.2608640 0.2195165 0.1762226 0.2606758 0.2802926 0.2939260 0.2848226 0.3070663 0.3219541 0.3003111 0.3237350 0.3394796
0.4822507 0.5418518 0.5957121 0.5212756 0.5797088 0.6256282 0.5517263 0.6099822 0.6494735 0.5797088 0.6412663 0.6880689 0.6569492 0.7360554 0.7947425 0.7114092 0.8018535 0.8682382
1.643036 1.707198 1.733590 1.7813728 1.805668 1.862133 1.909306 1.953483 2.432659 1.805668 2.036074 2.205986 2.086694 2.373713 2.582202 2.279164 2.602796 2.836214
concentration profiles are less significant for the chosen values of the parameters, and those profiles are not shown for the sake of brevity. Variation of skin friction coefficient, heat, nanoparticle mass and regular mass transfer rates against the Soret number ST together with the Lewis number Le is presented in Table 2 with fixed values of other parameters. It is observed that all these non-dimensional rate coefficients increase with the increasing value of Soret number ST. Similar behavior is seen with increasing values of Lewis number Le. Further, it can be noted that the skin friction coefficient, heat, nanoparticle mass and regular mass transfer rates decrease with an increase in both Bi and Nr for fixed values of other parameters. The effect of the Brownian motion parameter Nb together with the thermophoresis parameter Nt on these non-dimensional rate coefficients is tabulated for fixed values of other parameters, in Table 3. It can be observed from this table that skin friction coefficient, nanoparticle mass and regular mass transfer rates increase but heat transfer rate decreases with the increasing value of the Brownian motion parameter Nb. Similar nature can be seen for increasing values of thermophoresis parameter Nt. The behavior of these non-dimensional rate coefficients against k and Nc is also
391
seen from this Table 3, it is clear from this data that all these coefficients increases with an increase in both k and Nc. 5. Conclusions In this paper, the effect of Soret parameter on mixed convection flow along a vertical plate in a nanofluid is analysed under the convective boundary conditions. Using the similarity variables, the governing equations are transformed into a set of non-dimensional parabolic equations. These equations are solved numerically using the implicit, iterative finite-difference method discussed by Blottner [22] by modifying the scheme to suite to the convective boundary conditions. The numerical results are obtained for a wide range of values of the physical parameters. To ascertain the convergence of the numerical method adopted, comparison is made for the case of the mixed convection with convective boundary conditions with those results available in the literature, the comparison is found to be in very good agreement. Both the nanoparticle and regular buoyancies are considered in the analysis. The skin friction, heat, nanoparticle mass and regular mass transfer coefficients are obtained for a physically realistic values of governing parameters. The results are analyzed thoroughly for different values of ST, Bi and k on the flow, thermal and solutal fields. The major conclusion is that the Soret effect enhanced the skin friction, heat, nanoparticle mass and regular mass transfer rates in the medium. Further, these quantities show a reverse trend with the increasing value of Bi. On the other hand, the skin friction, heat, nanoparticle mass and regular mass transfer rates are enhanced with increasing values of mixed convection parameter k. Acknowledgements The authors are thankful to the reviewers for their valuable suggestions and comments for improving the clarity of the manuscript. References [1] E.V. Somers, Theoretical considerations of combined thermal and mass transfer from a flat plate, ASME J. Appl. Mech. 23 (1956) 295–301. [2] J.R. Kliegel, Laminar free and forced convection heat and mass transfer from a vertical flat plate, Ph.D. thesis, University of California, 1959. [3] J.H. Merkin, The effects of buoyancy forces on the boundary layer flow over a semi-infinite vertical flat plate in a uniform stream, J. Fluid Mech. 35 (1969) 439–450. [4] J.R. Lloyd, E.M. Sparrow, Combined free and forced convective flow on vertical surfaces, Int. J. Heat Mass Transfer 13 (1970) 434–438. [5] N.G. Kafoussias, Local similarity solution for combined free-forced convective and mass transfer flow past a semi-infinite vertical plate, Int. J. Energy Res. 14 (1990) 305–309. [6] A.J. Chamkha, H.S. Takhar, G. Nath, Mixed convection flow over a vertical plate with localized heating (cooling), magnetic field and suction (injection), Heat Mass Transfer 40 (2004) 835–841. [7] A. Bejan, Convection Heat Transfer, John Wiley, New York, 2004. [8] S.V. Subhashini, Samuel Nancy, I. Pop, Double-diffusive convection from a permeable vertical surface under convective boundary condition, Int. Commun. Heat Mass Transfer 38 (2011) 1183–1188. [9] Z. Dursunkaya, W.M. Worek, Diffusion-thermo and thermal diffusion effects in transient and steady natural convection from a vertical surface, Int. J. Heat Mass Transfer 35 (1992) 2060–2065. [10] N.G. Kafoussias, N.G. Williams, Thermal-diffusion and diffusion-thermo effects on mixed free-forced convective and mass transfer boundary layer flow with temperature dependent viscosity, Int. J. Engng. Sci. 33 (1995) 1369–1384. [11] P.A.L. Narayana, P.V.S.N. Murthy, R.S.R. Gorla, Soret-driven thermo-solutal convection induced by inclined thermal and solutal gradients in a shallow horizontal layer of a porous medium, J. Fluid Mech. 612 (2008) 1–19. [12] R.R. Kairi, P.V.S.N. Murthy, The effect of melting and thermo-diffusion on natural convection heat mass transfer in a non-Newtonian fluid saturated nonDarcy porous medium, Open Transp. Phenom. J. 1 (2009) 7–14. [13] D. Srinivasacharya, Ch. RamReddy, Mixed convection heat and mass transfer in a non-Darcy micropolar fluid with Soret and Dufour effects, Nonlinear Anal. Model. Contr. 16 (2011) 100–115. [14] S.U.S. Choi, Enhancing thermal conductivity of fluid with nanoparticles, in: D.A. Siginer, H.P. Wang (Eds.), Developments and Applications of NonNewtonian Flows, FED-V. 231/MD-V. 66, ASME, New York, 1995, pp. 99–105.
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[15] J. Buongiorno, Convective transport in nanofluids, ASME J. Heat Transfer 128 (2006) 240–250. [16] S. Kakac, A. Pramuanjaroenkij, Review of convective heat transfer enhancement with nanofluids, Int. J. Heat Mass Transfer 52 (2009) 3187–3196. [17] A.V. Kuznetsov, D.A. Nield, Natural convective boundary-layer flow of a nanofluid past a vertical plate, Int. J. Therm. Sci. 49 (2010) 243–247. [18] N. Bachok, A. Ishak, I. Pop, Boundary-layer flow of nanofluids over a moving surface in a flowing fluid, Int. J. Therm. Sci. 49 (2010) 1663–1668. [19] A.V. Kuznetsov, D.A. Nield, Double-diffusive natural convective boundarylayer flow of a nanofluid past a vertical plate, Int. J. Therm. Sci. 50 (2011) 712– 717.
[20] R.S.R. Gorla, Ali J. Chamkha, A.M. Rashad, Mixed convective boundary layer flow over a vertical wedge embedded in a porous medium saturated with a nanofluid: natural convection dominated regime, Nanoscale Res. Lett. 6 (2011) 207–214. [21] O.D. Makinde, A. Aziz, Boundary layer flow of a nanofluid past a stretching sheet with a convective boundary condition, Int. J. Therm. Sci. 50 (2011) 1326–1332. [22] F.G. Blottner, Finite-difference methods of solution of the boundary layer equations, AIAA J. 8 (1970) 193–205. [23] O.D. Makinde, P.O. Olanrewaju, Buoyancy effects on thermal boundary layer over a vertical plate with a convective surface boundary condition, J. Fluids Eng. 132 (2010). 044502-1.
Contents lists available at SciVerse ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Soret effect on mixed convection flow in a nanofluid under convective boundary condition Ch. RamReddy a,⇑, P.V.S.N. Murthy b, Ali J. Chamkha c, A.M. Rashad d,e a
Department of Mathematics, National Institute of Technology, Rourkela 769008, Odisha, India Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, India c Manufacturing Engineering Department, Public Authority for Applied Education and Training, Shuweikh 70654, Kuwait d Department of Mathematics, Aswan University, Faculty of Science, Aswan 81528, Egypt e Department of Mathematics, Salman Bin Abdul Aziz University, College of Science and Humanity Studies, AL-Kharj, Saudi Arabia b
a r t i c l e
i n f o
Article history: Received 30 November 2012 Received in revised form 13 April 2013 Accepted 14 April 2013
Keywords: Mixed convection Nanofluid Soret effect Convective boundary condition Numerical solution
a b s t r a c t In this investigation, we intend to present the influence of the prominent Soret effect on mixed convection heat and mass transfer in the boundary layer region of a semi-infinite vertical flat plate in a nanofluid under the convective boundary conditions. The transformed boundary layer ordinary differential equations are solved numerically using the implicit iterative finite difference method. Consideration of the nanofluid and the convective boundary conditions enhanced the number of non-dimensional parameters considerably thereby increasing the complexity of the present problem. A wide range of parameter values is chosen to bring out the effect of Soret parameter on the mixed convection process with the convective boundary condition. The effect of mixed convection, Soret and Biot parameters on the flow, heat and mass transfer coefficients is analyzed. The numerical results obtained for the velocity, temperature, volume fraction, and concentration profiles, as well as the local skin-friction coefficient, local wall temperature, local nanoparticle concentration and local wall concentration reveal interesting phenomenon, some of these qualitative results are presented through plots and tables. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction The process of heat and mass transfer caused by the simultaneous effect of free and forced convection is known as mixed convection flow. Considerable attention has been paid to the theoretical and numerical study of mixed convection boundary layer flow along a vertical plate in the recent past as it plays a crucial role in diverse applications, such as electronic devices cooled by fans, nuclear reactors cooled during an emergency shutdown, a heat exchanger placed in a low-velocity environment, solar collectors and so on. In the study of fluid flow over heated surfaces, the buoyancy force is neglected when the flow is horizontal. However, for vertical or inclined surfaces, the buoyancy force exert strong influence on the flow field. Extensive studies on mixed convection heat and mass transfer from isothermal and non-isothermal vertical surface under usual boundary layer approximation for viscous fluids have been undertaken by several researchers. Somers [1] analyzed theoretical results for combined thermal and mass transfer from a flat plate. The theoretical solution of heat transfer by mixed convection about a vertical flat plate has been obtained ⇑ Corresponding author. E-mail addresses: (Ch. RamReddy).
[email protected],
[email protected]
0017-9310/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.04.032
by Kliegel [2]. Merkin [3] investigated the mixed convection boundary layer flow on a semi-infinite vertical flat plate when the buoyancy forces aid and oppose the development of the boundary layer. Lloyd and Sparrow [4] used a local similarity method to solve the mixed convection flow on a vertical surface and showed that the numerical solutions ranged from pure forced convection to mixed convection. Kafoussias [5] presented analysis of the effects of buoyancy forces in a laminar uniform forced-convective flow with mass transfer along a semi-infinite vertical plate. An analysis is carried out by Chamkha et al. [6] to study the effects of localized heating (cooling), suction (injection), buoyancy forces and magnetic field for the mixed convection flow on a heated vertical plate. A detailed review of mixed convective heat and mass transfer can be found in the book by Bejan [7]. Recently, Subhashini et al. [8] discussed the simultaneous effects of thermal and concentration diffusions on a mixed convection boundary layer flow over a permeable surface under convective surface boundary condition. The diffusion of mass due to temperature gradient is called Soret or thermo-diffusion effects and this effect might become significant when large density differences exist in the flow regime. For example, Soret effect can be significant when species are introduced at a surface in fluid domain, with a density lower than the surrounding fluid. The Soret parameter has been utilized for isotope separation and in a mixture between gases with very light molecular weight
Ch. RamReddy et al. / International Journal of Heat and Mass Transfer 64 (2013) 384–392
385
Nomenclature Bi C Cw Cf C1 c DB DS DT DCT f g Grx hf J k Le Nb Nc Nr Nt Nux Pr qm qn qw Rex S Sc Shx NShx ST T Tf
Biot number solutal concentration solutal concentration at the wall skin friction coefficient ambient solutal concentration constant Brownian diffusion coefficient solutal diffusivity thermophoretic diffusion coefficient soret diffusivity dimensionless stream function gravitational acceleration local Grashof number convective heat transfer coefficient ratio between the effective heat capacity of the nanoparticle material and heat capacity of the fluid thermal conductivity of the nanofluid Lewis number Brownian motion parameter regular buoyancy ratio nanoparticle buoyancy ratio thermophoresis parameter local Nusselt number Prandtl number regular mass flux at the wall nanoparticle mass flux at the wall heat flux at the wall local Reynolds number dimensionless concentration Schmidt number local Sherwood number local nanoparticle Sherwood number Soret number temperature temperature of the hot fluid
(H2,He) and of medium molecular weight (N2,air). Dursunkaya and Worek [9] studied diffusion-thermo and thermal-diffusion effects in transient and steady natural convection from a vertical surface, whereas Kafoussias and Williams [10] presented the same effects on mixed convective and mass transfer steady laminar boundary layer flow over a vertical flat plate with temperature dependent viscosity. The linear stability analysis of Soret-driven thermosolutal convection in a shallow horizontal layer of a porous medium subjected to inclined thermal and solutal gradients of finite magnitude has been investigated theoretically by Narayana et al. [11]. Recently, the effect of melting and/or thermodiffusion on convective transport in a non-Newtonian fluid saturated non-Darcy porous medium are presented by Kairi and Murthy [12] and Srinivasacharya and RamReddy [13]. In recent times, the flow analysis of nanofluids has been the topic of extensive research due to its characteristic in increasing thermal conductivity in heat transfer process. Several ordinary fluids including water, toluene, ethylene glycol and mineral oils etc. in heat transfer processes have rather low thermal conductivity. The nanofluid (initially introduced by Choi [14]) is an advanced type of fluid containing nanometer sized particles (diameter less than 100 nm) or fibers suspended in the ordinary fluid. Undoubtedly, the nanofluids are advantageous in the sense that they are more stable and have an acceptable viscosity and better wetting, spreading and dispersion properties on a solid surface. Nanofluids are used in different engineering applications such as microelectronics,
T1 u1 u,v x,y
am bT,bC
g c k h / /w /1
l m q qf1 qp (qc)f (qc)p sw w
ambient temperature characteristic velocity velocity components in x and y directions coordinates along and normal to the plate thermal diffusivity volumetric thermal and solutal expansion coefficients of the base fluid similarity variable dimensionless volume fraction mixed convection parameter dimensionless temperature nanoparticle volume fraction nanoparticle volume fraction at the wall nanoparticle volume fraction at large values of y (ambient) dynamic viscosity of the base fluid kinematic viscosity density of the fluid density of the base fluid nanoparticle mass density heat capacity of the fluid effective heat capacity of the nanoparticle material wall shear stress stream function
Subscripts w wall condition 1 ambient condition C concentration T temperature Superscript 0 differentiation with respect to g
microfluidics, transportation, biomedical, solid-state lighting and manufacturing. The research on heat and mass transfer in nanofluids has been receiving increased attention worldwide. Many researchers have found unexpected thermal properties of nanofluids, and have proposed new mechanisms behind the enhanced thermal properties of nanofluids. Excellent reviews on convective transport in nanofluids have been made by Buongiorno [15] and Kakac and Pramuanjaroenkij [16]. Kuznetsov and Nield [17] studied analytically the natural convective boundary-layer flow of a nanofluid past a vertical plate. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. Also, it is interesting to note that the Brownian motion of nanoparticles at molecular and nanoscale levels is a key nanoscale mechanism governing their thermal behaviors. In nanofluid systems, due to the size of the nanoparticles, the Brownian motion takes place, which can affect the heat transfer properties. As the particle size scale approaches to the nanometer scale, the particle Brownian motion and its effect on the surrounding liquids play an important role in the heat transfer. The steady boundary-layer flow of a nanofluid past a moving semi-infinite flat plate in a uniform free stream is analyzed by Bachok et al. [18]. Recently, the double-diffusive natural convective boundary-layer flow of a nanofluid past a vertical plate has been studied analytically by Kuznetsov and Nield [19]. Gorla et al. [20] presented a boundary layer analysis for the mixed convection past a vertical wedge in a porous medium saturated with a nanofluid. But, very little attention has been paid to study
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the significance of buoyancy forces on mixed convective in a nanofluid under convective boundary condition. Makinde and Aziz [21] studied numerically the boundary layer flow induced in a nanofluid due to a linearly stretching sheet in the presence of a convective heating boundary condition. However, a clear picture about the nanofluid boundary layer flows is still to emerge. Motivated by the above referenced work and the vast possible industrial applications, it is of paramount interest to consider the effect of Soret parameter on mixed convective flow along a vertical plate in a nanofluid under convective boundary condition. The presence of convective boundary conditions make the mathematical model of the present physical system a little more complicated leading to the complex interactions of the flow, heat and mass transfer mechanism. Analytical solution is ruled out in the current set up and hence a numerical solution is obtained for the present problem. Consideration of the nanofluid and the convective boundary conditions enhanced the number of non-dimensional parameters considerably. The effects of mixed convection, Soret and Biot parameters on the physical quantities of the flow, heat and mass transfer coefficients are analyzed. To examine the convergence of the numerical code written to solve the present problem, (i.e., for code validation) we compare the present results for the clear fluid mixed convection results with previously published works with convective boundary conditions and the comparison shows that the results are in very good agreement. 2. Mathematical formulation Choose the coordinate system such that the x-axis is along the vertical plate and y-axis normal to the plate. The physical model and coordinate system are shown in Fig. 1. Consider the steady laminar two-dimensional mixed convection heat and mass transfer along a flat vertical surface embedded in a nanofluid having T1, C1, and /1 as the temperature, concentration and nanoparticle volume fraction respectively in the ambient medium. Also assume that a free stream with uniform velocity u1 goes past the flat plate. The plate is either heated or cooled from left by convection from a fluid of temperature Tf with Tf > T1 corresponding to a heated surface (assisting flow) and Tf < T1 corresponding to a cooled surface (opposing flow) respectively. On the wall the solutal concentration and the nanoparticle volume fraction are taken to be constant and are given by Cw, and /w, respectively. By employing Oberbeck–Boussinesq approximation, making use of the standard boundary layer approximations and eliminating pressure, the governing equations for the nanofluid are given by
@u @ v þ ¼0 @x @y
qf 1 u
ð1Þ
@u @u þv @x @y
¼l
@2u þ g qf 1 ð1 /1 Þ½bT ðT T 1 Þ @y2
þ bC ðC C 1 Þ ðqp qf 1 Þgð/ /1 Þ " 2 # @T @T @2T @/ @T DT @T u þv ¼ am 2 þ J DB þ @x @y @y @y @y T 1 @y
ð2Þ
ð3Þ
u
@/ @/ @ 2 / DT @ 2 T þv ¼ DB 2 þ @x @y @y T 1 @y2
ð4Þ
u
@C @C @2C @2T þv ¼ Ds 2 þ DCT 2 @x @y @y @y
ð5Þ
where u and v are the velocity components along the x and y axes, respectively, T is the temperature, / is the nanoparticle volume fraction, C is the solutal concentration, g is the gravitational acceleration, am = k/(qc)f is the thermal diffusivity of the fluid, m = l/qf1 is the kinematic viscosity coefficient and J ¼ ðqcÞp =ðqcÞf . Further, qf1 is the density of the base fluid and q, l, k, bT, and bC are the density, viscosity, thermal conductivity, volumetric thermal expansion coefficient and volumetric solutal expansion coefficient of the nanofluid, while qp is the density of the nanoparticles, (qc)f is the heat capacity of the fluid and (qc)p is the effective heat capacity of the nanoparticle material. The coefficients that appear in Eqs. (3)–(5) are the Brownian diffusion coefficient DB, the thermophoretic diffusion coefficient DT, the solutal diffusivity DS and the Soret-type diffusivity DCT. For, details of the derivation of Eqs. (1)–(5), one can refer the papers by Buongiorno [15] and Nield and Kuznetsov ([17];[19]). The boundary conditions are
@T ¼ hf ðT f TÞ; / ¼ /w ; C ¼ C w at y ¼ 0 ð6aÞ @y u ! u1 ; T ! T 1 ; / ! /1 ; C ! C 1 as y ! 1; ð6bÞ
u ¼ 0;
v ¼ 0;
k
here, hf is the convective heat transfer coefficient and the subscripts w and 1 indicate the conditions at the surface and at the outer edge of the boundary layer respectively. In view of the continuity equation (1), we introduce the stream function w by
u¼
@w ; @y
v ¼
@w @x
ð7Þ
Substituting (7) in Eqs. (2)–(5) and then using the following nondimensional transformation 1=2 p1ffiffi g ¼ pyffiffi2x Re1=2 wðgÞ; x ; f ðgÞ ¼ 2m Rex
9 =
1 1 1 ; cðgÞ ¼ /// ; SðgÞ ¼ CCC ;; hðgÞ ¼ TTT T 1 w C 1 w /1
ð8Þ
f
with the local Raynold’s number Rex ¼ u1m x, we get the transformed system of ordinary differential equations as 00
f 000 þ ff þ 2k½h þ Nc S Nr c ¼ 0 2 1 00 h þ f h0 þ Nb h0 c0 þ Nt h0 ¼ 0 Pr
ð10Þ
Nt 00 h ¼0 Nb
ð11Þ
1 00 S þ ST h00 þ f S0 ¼ 0 Sc
ð12Þ
c00 þ Le f c0 þ
Fig. 1. Physical model and coordinate system.
ð9Þ
where the primes indicate differentiation with respect to g. In usual notations, Pr ¼ amm is the Prandtl number, Sc ¼ DmS is the Schmidt
387
Ch. RamReddy et al. / International Journal of Heat and Mass Transfer 64 (2013) 384–392 Table 1 Comparison of f00 (0) and h0 (0) for mixed convection along a vertical flat plate in Newtonian fluids ([23,8]). Bi
Pr
k
Makinde and Olanrewaju [23] 00
0.1 1.0 0.1 0.1 0.1
0.1 0.1 1.0 0.1 0.1
0.72 0.72 0.72 3.00 7.10
Subhashini et al. [8]
0
00
Present
f (0)
h (0)
f (0)
h (0)
f00 (0)
h0 (0)
0.36881 0.44036 0.63200 0.34939 0.34270
0.07507 0.23750 0.07704 0.08304 0.08672
0.36875 0.44032 0.63198 0.34937 0.34270
0.07505 0.23746 0.07700 0.08301 0.08670
0.36898 0.44049 0.63197 0.34957 0.34289
0.07509 0.23765 0.07705 0.08308 0.08674
1.6
0
0.8 ST=0.2 ST=0.4
ST=0.2 ST=0.4 ST=1.0
f'( η )
1.2
λ=1.0
Nr=1.0 Nb=0.5 Nt=0.5 Nc=1.0 Bi=1.0 Pr=1.0 Sc=0.6
0.4
0.0
θ (η )
Le=1.0,10,100
0.8
0
1
η
(a)
2
3
0.4
λ=1.0
Nr=1.0 Nb=0.5 Nt=0.5 Nc=1.0 Bi=1.0 Pr=1.0 Sc=0.6
Le=1.0,10,100
0.2
0.0
4
0
1
(b)
1.0
2
η
3
4
1.0 ST=0.2 ST=0.4
0.8
ST=0.2 ST=0.4
0.8
ST=1.0
ST=1.0
Le=1.0,10,100
γ (η)
0.6
S ( η)
0.6 λ=1.0
Nr=1.0 Nb=0.5 Nt=0.5 Nc=1.0 Bi=1.0 Pr=1.0 Sc=0.6
0.4
0.2
0.0
ST=1.0
0.6
0
1
2
3
(c)
4
Le=1.0,10,100
λ=1.0
Nr=1.0 Nb=0.5 Nt=0.5 Nc=1.0 Bi=1.0 Pr=1.0 Sc=0.6
0.4
0.2
5
0.0
0
(d)
η
1
2
η
3
4
Fig. 2. Effects of ST and Le on (a) velocity, (b) temperature, (c) concentration, and (d) volume fraction profiles. ðqp qf 1 Þð/w /1 Þ f 1 bT ðT f T 1 Þð1/1 Þ
number and Le ¼ DmB is the Lewis number. Nr ¼ q
is the
w C 1 Þ is the regular buoyancy nanofluid buoyancy ratio and Nc ¼ bbCTðC ðT T 1 Þ f
ratio. Further, Nb ¼ Nt ¼
ðqcÞp DT ðT f T 1 Þ ðqcÞf mT 1
gbT ð1/1 ÞðT f T 1 Þx3
m2
ðqcÞp DB ð/w /1 Þ ðqcÞf m
is
the
is the Brownian motion parameter,
thermophoresis
parameter,
is the local Grashof number and k ¼
convection parameter [8]. Finally, ST ¼
DCT ðT f T 1 Þ mðC w C 1 Þ
Grx Re2x
Grx ¼
is the mixed
is the Soret number.
Boundary conditions (6) in terms of f, h, c, S become
g ¼0:f ð0Þ¼0; f 0 ð0Þ¼0; h0 ð0Þ¼Bi½1hð0Þ; cð0Þ¼1; Sð0Þ¼1 ð13aÞ g !1:f 0 ð1Þ!1; hð1Þ!0; cð1Þ!0; Sð1Þ!0 ð13bÞ
qffiffiffiffiffi where Bi ¼ kc u21m is the Biot number. This boundary conditions will be free from the local variable x when we choose hf = cx1/2. The Biot number Bi is a ratio of the internal thermal resistance of the plate to the boundary layer thermal resistance of the hot fluid at the bottom of the surface. It is important to note that this boundary value problem reduces to the classical problem of flow and heat and mass transfer due to a the Blasius problem when Nb and Nt are zero. Most nanofluids examined to date have large values for the Lewis number Le > 1 [17]. For water nanofluids at room temperature with nanoparticles of 1–100 nm diameters, the Brownian diffusion coefficient DB ranges from 4 104 to 4 1012 m2/s. Furthermore, the ratio of the Brownian diffusivity coefficient to the thermopho-
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resis coefficient for particles with diameters of 1–100 nm can be varied in the ranges of 2–0.02 for alumina, and from 2 to 20 for copper nanoparticles (see Buongiorno [15] for details). Hence, the variation of the non-dimensional parameters of nanofluids in the present study is considered in the mentioned range.
C f ð2Rex Þ1=2 ¼ f 00 ð0Þ; Nux ðRex =2Þ1=2 ¼ h0 ð0Þ; NShx ðRex =2Þ1=2 ¼ c0 ð0Þ; Shx ðRex =2Þ1=2 ¼ S0 ð0Þ:
Effect of the various parameters involved in the investigation on these coefficients is discussed in the following section.
3. Skin friction, heat and mass transfer coefficients The primary objective of this study is to estimate the skin friction coefficient Cf, the Nusselt number Nu, the nanoparticle Sherwood number NSh, and the regular Sherwood number Sh. These parameters characterize the surface drag, the wall heat, nanoparticle mass transfer and regular mass transfer rates, respectively. The shearing stress, local heat, local nanoparticle mass and local regular mass fluxes from the vertical plate can be obtained from
@u @T ; qw ¼ k ; @y y¼0 @y y¼0 @/ @C qn ¼ DB and qm ¼ DS : @y y¼0 @y y¼0
sw ¼ l
ð14Þ
The non-dimensional shear stress C f ¼ q swu2 , the Nusselt number f1 1 x wx Nux ¼ kðTqT , the nanoparticle Sherwood number NShx ¼ DB ð/qwn/ 1Þ 1Þ f x and the regular Sherwood number Shx ¼ DS ðCqwmC , are given by 1Þ
4. Results and discussion The resulting transport Eqs. (9)–(12) are non-linear, coupled, ordinary differential equations, which possess no closed-form solution. Therefore, these are solved numerically subject to the boundary conditions given by Eq. (13). The implicit, iterative finite-difference method discussed by Blottner [22] has proven to be adequate for the solution of this type of equations. For this reason, this method is employed in the present work. Eqs. (9)–(12) are discretized using three-point central difference quotients. This converts the differential equations into linear sets of algebraic equations, which can be readily solved by the well-known Thomas algorithm [22]. On the other hand, Eqs. (9)–(12) is discretized and solved subject to the appropriate boundary conditions by the trapezoidal rule. The computational domain in the g-direction was made up of 196 non-uniform grid points. It is expected that most
1.6
1.0 Nr=0.1 Nr=0.5 Nr=0.9
1.2
0.6 λ=1.0
Nb=0.5 Nt=0.5 Nc=1.0 ST=1.0 Le=10 Pr=1.0 Sc=0.6
0.4
1
2
3
0.2
0.0
4
0
1
η
(b)
1.0
2
3
4
1.0 Nr=0.1 Nr=0.5 Nr=0.9
0.8
Nr=0.1 Nr=0.5 Nr=0.9
0.8
0.6 γ (η )
S( η )
0.6 λ=1.0
Nb=0.5 Nt=0.5 Nc=1.0 ST=1.0 Le=10 Pr=1.0 Sc=0.6
0.4
0.2 Bi=0.1,10,20
0.0
λ=1.0 Nb=0.5 Nt=0.5 Nc=1.0 ST=1.0 Le=10 Pr=1.0 Sc=0.6
0.4
η
(a)
Bi=0.1,10,20
θ (η )
f'( η )
0.8
0
Nr=0.1 Nr=0.5 Nr=0.9
0.8
Bi=0.1,10,20
0.0
ð15Þ
0
(c)
1
η
2
3
λ=1.0 Nb=0.5 Nt=0.5 Nc=1.0 =1.0 S T Le=10 Pr=1.0 Sc=0.6
0.4 Bi=0.1,10,20
0.2
4
0.0
0
(d)
1
η
2
3
Fig. 3. Effects of Bi and Nr on (a) velocity, (b) temperature, (c) concentration, and (d) volume fraction profiles.
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Ch. RamReddy et al. / International Journal of Heat and Mass Transfer 64 (2013) 384–392
changes in the dependent variables occur in the region close to the plate where viscous effects dominate. However, small changes in the dependent variables are expected far away from the plate surface. For these reasons, variable step sizes in the g-direction are employed. The initial step size Mg1 and the growth factor K⁄ employed such that Mgi+1 = K⁄Mgi (where the subscript i indicates the grid location) were 103 and 1.0375, respectively. These values were found (by performing many numerical experimentations) to give accurate and grid-independent solutions. The solution convergence criterion employed in the present work was based on the difference between the values of the dependent variables at the current and the previous iterations. When this difference reached 105, the solution was assumed converged and the iteration process was terminated. To have a better understanding of the flow characteristics, numerical results for the velocity, temperature, volume fraction and concentration are calculated for different sets of values of the parameters ST, Le, k, Nc, Nb, Nt, Nr, Bi. Also, the effect of these parameters on skin friction, non-dimensional heat, nanoparticle mass and regular mass transfer coefficients is discussed. The mixed convection flow of Newtonian fluids along a vertical flat plate studied by Makinde and Olanrewaju [23] and Subhashini et al. [8] can be obtained here as a special case by taking Nc = 0, Nr = 0, Nb = 0, Nt = 0, Le = 0, ST = 0 and c(g) ? 0 (i.e., the Newtonian
(non-nanofluid) fluid formulation of the present investigation). Blottner [22] method is extended here to study the mixed convective transport problem with the convective boundary conditions. It is more relevant and significant to compare a special case result of the present problem with the convective boundary condition with the existing solutions in the literature. Therefore, a comparison is made with the solution presented in Makinde and Olanrewaju [23] and Subhashini et al. [8] and it is found to be in good agreement, as seen from the Table 1. Fig. 2(a)–(d) show the non-dimensional velocity, temperature, nanoparticle volume fraction and concentration profiles for various values of Soret number ST along with varying values of the Lewis number Le for a given k = 1.0, Nr = 1.0, Nc = 1.0, Bi = 1.0, Nb = 0.5, Nt = 0.5, Pr = 1.0, Sc = 0.6. The Soret number ST accounts for the additional mass diffusion due to the temperature gradients. It is noticed from Fig. 2 that an increase in the Soret number ST resulted in an increase in the velocity and concentration fields while a decrease in the temperature and the nanoparticle volume fraction is noted within the boundary layer. The present analysis shows that the flow field is appreciably influenced by the Soret parameter. It is clear from these figures that an increase in Lewis number Le increased the momentum boundary layer thickness, while a reduction in the thermal, nanoparticle nanoparticle volume fraction and solutal boundary layer thickness is noted.
1.6 Nb=0.1 Nb=0.5 Nb=0.9
1.2
0.6
0.8
θ (η )
f'( η )
Nt=0.1,0.5,0.9 λ=1.0 Nr=1.0 Nc=1.0 Bi=1.0 ST=1.0 Le=10 Pr=1.0 Sc=0.6
0.4
0.0
0
1
η
(a)
2
3
λ=1.0
0.4
Nr=1.0 Nc=1.0 Bi=1.0 ST=1.0 Le=10 Pr=1.0 Sc=0.6
0.2 Nt=0.1,0.5,0.9
4
0.0
0
1
(b)
1.0
η
2
3
1.0 Nb=0.1 Nb=0.5 Nb=0.9
Nt=0.1,0.5,0.9
0.8
Nb=0.1 Nb=0.5 Nb=0.9
0.8
γ (η )
0.6
S( η )
0.6 λ=1.0
Nr=1.0 Nc=1.0 Bi=1.0 ST=1.0 Le=10 Pr=1.0 Sc=0.6
0.4
0.2
0.0
Nb=0.1 Nb=0.5 Nb=0.9
0.8
0
(c)
1
2
η
3
4
λ=1.0
0.4
Nt=0.1,0.5,0.9
0.2
5
0.0
0
(d)
1
η
2
Nr=1.0 Nc=1.0 Bi=1.0 ST=1.0 Le=10 Pr=1.0 Sc=0.6
3
Fig. 4. Effects of Nb and Nt on (a) velocity, (b) temperature, (c) concentration, and (d) volume fraction profiles.
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Ch. RamReddy et al. / International Journal of Heat and Mass Transfer 64 (2013) 384–392
The non-dimensional velocity for different values of Biot number Bi with fixed values of other parameters is plotted in Fig. 3(a). Increased convective heating associated with an increase in Bi is seen to thicken the momentum boundary layer. A reverse trend is seen in the case of nanofluid buoyancy ratio Nr. Given that convective heating increases with Biot number, Bi ? 1 simulates the isothermal surface, which is clearly seen from Fig. 3(b), where h(0) = 1 as Bi ? 1. In fact, a high Biot number indicates higher internal thermal resistance of the plate than the boundary layer thermal resistance. As a result, an increase in the Biot number leads to increase of fluid temperature efficiently, these figures confirm this fact also. A similar nature is observed in the case of Nr. Fig. 3(c) brings out the effect on the nanoparticle volume fraction with different values of Biot number Bi and Nr for fixed values of other parameters. As the parameter values Bi and Nr increase, the volume fraction increased for the fixed values of the other parameters. Fig. 3(d) indicates the variation of dimensionless concentration for different values of Bi and Nr. A reduction in concentration boundary layer thickness is seen with increasing values of the Biot number Bi whereas the concentration boundary layer thickens with increasing values of Nr. Fig. 4 present the effect of the Brownian motion Nb and thermophoresis Nt parameters on the velocity, temperature, volume fraction and concentration distributions. When Nb = 0, there is no
additional thermal transport due to buoyancy effects created as a result of nanoparticle concentration gradients. It is observed that the momentum boundary layer thickness increases with the increase in values of Nb but it decreases with increasing values of Nt. With the rise in the values of parameters Nt and Nb, an increase in the temperature is noted whereas a reduction in the concentration field is noticed. The nanoparticle volume fraction decreased with increase in Nb and it increased with increasing values of Nt. It is also noticed that the nanoparticle volume fraction increased with an increase in Nb in the case of forced convection flow. Nt > 0 indicates a cold surface while negative Nt < 0 corresponds a hot surface, in case of a hot surface, thermophoresis tends to blow the nanoparticle volume fraction away from the surface since a hot surface repels the sub-micron sized particles from it, thereby forming a relative particle-free layer near the surface. Variation of non-dimensional velocity, temperature, volume fraction and concentration distribution against the similarity variable g, is shown respectively in Fig. 5, for a few sets of values of k and Nc with fixed values of other parameters. As the parameters k and Nc increase, the velocity increased whereas temperature, volume fraction and concentration of the nanofluid decreased. This is due to the fact that the buoyancy force act like pressure gradient which accelerates/decelerates the fluid within the boundary layer. The effect of buoyancy on the temperature, volume fraction and
0.8 Nc=1.0 Nc=2.0 Nc=3.0
3.0
Nc=1.0 Nc=2.0 Nc=3.0
0.6
2.4
f'( η )
θ (η )
1.8
λ=1.0,2.0,3.0
0.4
1.2 λ=1.0,2.0,3.0
0.6
0.0
0.2
Nr=1.0,Nb=0.5,Nt=0.5,Bi=1.0 ST=1.0,Le=10,Pr=1.0,Sc=0.6
0
(a)
1
2
η
3
4
1.0
0
1
(b)
2
η
3
1.0 Nc=1.0 Nc=2.0 Nc=3.0
0.6 S( η )
λ=1.0,2.0,3.0
0.4
0.2
0
(c)
1
2
η
3
Nc=1.0 Nc=2.0 Nc=3.0
0.8
0.6 Nr=1.0 Nb=0.5 Nt=0.5 Bi=1.0 ST=1.0 Le=10 Pr=1.0 Sc=0.6
Nr=1.0 Nb=0.5 Nt=0.5 Bi=1.0 ST=1.0 Le=10 Pr=1.0 Sc=0.6
γ (η )
0.8
0.0
0.0
Nr=1.0 Nb=0.5 Nt=0.5 Bi=1.0 ST=1.0 Le=10 Pr=1.0 Sc=0.6
0.4
λ=1.0,2.0,3.0
0.2
4
0.0
0.0
(d)
0.5
1.0
η
1.5
2.0
2.5
Fig. 5. Effects of k and Nc on (a) velocity, (b) temperature, (c) concentration, and (d) volume fraction profiles.
Ch. RamReddy et al. / International Journal of Heat and Mass Transfer 64 (2013) 384–392 Table 2 Effects of skin friction coefficient, and heat, nanoparticle mass and regular mass transfer rates for various values of ST, Le, Bi, and Nr when Nb = Nt = 0.5, Nc = Pr = 1.0, k = 1.0 and Sc = 0.6. ST
Le
Bi
Nr
f00 (0)
h0 (0)
S0 (0)
c0 (0)
0.2 0.4 0.6 0.2 0.4 0.6 0.2 0.4 0.6 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
1.0 1.0 1.0 10 10 10 100 100 100 10 10 10 10 10 10 10 10 10
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.1 0.1 0.1 10 10 10 20 20 20
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.1 0.5 0.9 0.1 0.5 0.9 0.1 0.5 0.9
1.660426 1.713282 1.764715 2.331555 2.378589 2.425132 2.620651 2.669386 2.717777 2.264569 2.030422 1.786345 3.232459 3.019211 2.800324 3.256729 3.043529 2.825050
0.2648041 0.2673099 0.2699203 0.2574143 0.2591855 0.2606758 0.2500289 0.2517529 0.2531556 0.06391265 0.06375668 0.06362097 0.3909954 0.3866136 0.3827407 0.4015757 0.3980230 0.3931959
0.4990276 0.5023820 0.5051266 0.5738778 0.5769784 0.5797088 0.5887031 0.5962021 0.6038435 0.5650807 0.5578854 0.5495672 0.5883316 0.5835845 0.5789469 0.5873385 0.5823732 0.5781532
0.6064302 0.6167969 0.6259490 1.777230 1.791779 1.805668 4.007293 4.034007 4.060444 1.686413 1.646561 1.603247 1.937524 1.908359 1.876994 1.941364 1.911234 1.880945
Table 3 Effects of skin friction coefficient, and heat, nanoparticle mass and regular mass transfer rates for various values of Nb, Nt, k, and Nc when Bi = Nr = ST = Pr = 1.0, Le = 10 and Sc = 0.6. Nb
Nt
k
Nc
f00 (0)
h0 (0)
S0 (0)
c0 (0)
0.1 0.5 0.9 0.1 0.5 0.9 0.1 0.5 0.9 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
0.1 0.1 0.1 0.5 0.5 0.5 0.9 0.9 0.9 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 2.0 2.0 2.0 3.0 3.0 3.0
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 2.0 3.0 1.0 2.0 3.0 1.0 2.0 3.0
2.231877 2.368428 2.477065 2.202382 2.425132 2.538059 2.199755 2.484781 2.599391 2.425132 3.611774 4.676075 3.800878 5.801891 7.591507 4.987329 7.700205 10.12352
0.3779059 0.3122588 0.2496606 0.3130586 0.2606758 0.2090975 0.2608640 0.2195165 0.1762226 0.2606758 0.2802926 0.2939260 0.2848226 0.3070663 0.3219541 0.3003111 0.3237350 0.3394796
0.4822507 0.5418518 0.5957121 0.5212756 0.5797088 0.6256282 0.5517263 0.6099822 0.6494735 0.5797088 0.6412663 0.6880689 0.6569492 0.7360554 0.7947425 0.7114092 0.8018535 0.8682382
1.643036 1.707198 1.733590 1.7813728 1.805668 1.862133 1.909306 1.953483 2.432659 1.805668 2.036074 2.205986 2.086694 2.373713 2.582202 2.279164 2.602796 2.836214
concentration profiles are less significant for the chosen values of the parameters, and those profiles are not shown for the sake of brevity. Variation of skin friction coefficient, heat, nanoparticle mass and regular mass transfer rates against the Soret number ST together with the Lewis number Le is presented in Table 2 with fixed values of other parameters. It is observed that all these non-dimensional rate coefficients increase with the increasing value of Soret number ST. Similar behavior is seen with increasing values of Lewis number Le. Further, it can be noted that the skin friction coefficient, heat, nanoparticle mass and regular mass transfer rates decrease with an increase in both Bi and Nr for fixed values of other parameters. The effect of the Brownian motion parameter Nb together with the thermophoresis parameter Nt on these non-dimensional rate coefficients is tabulated for fixed values of other parameters, in Table 3. It can be observed from this table that skin friction coefficient, nanoparticle mass and regular mass transfer rates increase but heat transfer rate decreases with the increasing value of the Brownian motion parameter Nb. Similar nature can be seen for increasing values of thermophoresis parameter Nt. The behavior of these non-dimensional rate coefficients against k and Nc is also
391
seen from this Table 3, it is clear from this data that all these coefficients increases with an increase in both k and Nc. 5. Conclusions In this paper, the effect of Soret parameter on mixed convection flow along a vertical plate in a nanofluid is analysed under the convective boundary conditions. Using the similarity variables, the governing equations are transformed into a set of non-dimensional parabolic equations. These equations are solved numerically using the implicit, iterative finite-difference method discussed by Blottner [22] by modifying the scheme to suite to the convective boundary conditions. The numerical results are obtained for a wide range of values of the physical parameters. To ascertain the convergence of the numerical method adopted, comparison is made for the case of the mixed convection with convective boundary conditions with those results available in the literature, the comparison is found to be in very good agreement. Both the nanoparticle and regular buoyancies are considered in the analysis. The skin friction, heat, nanoparticle mass and regular mass transfer coefficients are obtained for a physically realistic values of governing parameters. The results are analyzed thoroughly for different values of ST, Bi and k on the flow, thermal and solutal fields. The major conclusion is that the Soret effect enhanced the skin friction, heat, nanoparticle mass and regular mass transfer rates in the medium. Further, these quantities show a reverse trend with the increasing value of Bi. On the other hand, the skin friction, heat, nanoparticle mass and regular mass transfer rates are enhanced with increasing values of mixed convection parameter k. Acknowledgements The authors are thankful to the reviewers for their valuable suggestions and comments for improving the clarity of the manuscript. References [1] E.V. Somers, Theoretical considerations of combined thermal and mass transfer from a flat plate, ASME J. Appl. Mech. 23 (1956) 295–301. [2] J.R. Kliegel, Laminar free and forced convection heat and mass transfer from a vertical flat plate, Ph.D. thesis, University of California, 1959. [3] J.H. Merkin, The effects of buoyancy forces on the boundary layer flow over a semi-infinite vertical flat plate in a uniform stream, J. Fluid Mech. 35 (1969) 439–450. [4] J.R. Lloyd, E.M. Sparrow, Combined free and forced convective flow on vertical surfaces, Int. J. Heat Mass Transfer 13 (1970) 434–438. [5] N.G. Kafoussias, Local similarity solution for combined free-forced convective and mass transfer flow past a semi-infinite vertical plate, Int. J. Energy Res. 14 (1990) 305–309. [6] A.J. Chamkha, H.S. Takhar, G. Nath, Mixed convection flow over a vertical plate with localized heating (cooling), magnetic field and suction (injection), Heat Mass Transfer 40 (2004) 835–841. [7] A. Bejan, Convection Heat Transfer, John Wiley, New York, 2004. [8] S.V. Subhashini, Samuel Nancy, I. Pop, Double-diffusive convection from a permeable vertical surface under convective boundary condition, Int. Commun. Heat Mass Transfer 38 (2011) 1183–1188. [9] Z. Dursunkaya, W.M. Worek, Diffusion-thermo and thermal diffusion effects in transient and steady natural convection from a vertical surface, Int. J. Heat Mass Transfer 35 (1992) 2060–2065. [10] N.G. Kafoussias, N.G. Williams, Thermal-diffusion and diffusion-thermo effects on mixed free-forced convective and mass transfer boundary layer flow with temperature dependent viscosity, Int. J. Engng. Sci. 33 (1995) 1369–1384. [11] P.A.L. Narayana, P.V.S.N. Murthy, R.S.R. Gorla, Soret-driven thermo-solutal convection induced by inclined thermal and solutal gradients in a shallow horizontal layer of a porous medium, J. Fluid Mech. 612 (2008) 1–19. [12] R.R. Kairi, P.V.S.N. Murthy, The effect of melting and thermo-diffusion on natural convection heat mass transfer in a non-Newtonian fluid saturated nonDarcy porous medium, Open Transp. Phenom. J. 1 (2009) 7–14. [13] D. Srinivasacharya, Ch. RamReddy, Mixed convection heat and mass transfer in a non-Darcy micropolar fluid with Soret and Dufour effects, Nonlinear Anal. Model. Contr. 16 (2011) 100–115. [14] S.U.S. Choi, Enhancing thermal conductivity of fluid with nanoparticles, in: D.A. Siginer, H.P. Wang (Eds.), Developments and Applications of NonNewtonian Flows, FED-V. 231/MD-V. 66, ASME, New York, 1995, pp. 99–105.
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