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Soret and Dufour effects on natural convection boundary layer flow over a vertical cone in a porous medium with constant wall heat and mass fluxes
International Communications in Heat and Mass Transfer 38 (2011) 44–48
Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i c h m t
Soret and Dufour effects on natural convection boundary layer flow over a vertical cone in a porous medium with constant wall heat and mass fluxes☆ Ching-Yang Cheng Department of Mechanical Engineering, Southern Taiwan University, Yungkang 71005, Taiwan
a r t i c l e
i n f o
Available online 19 November 2010 Keywords: Natural convection Heat and mass transfer Boundary layer Vertical cone Porous medium Dufour effect Soret effect
a b s t r a c t This work studies the Soret and Dufour effects on the boundary layer flow due to natural convection heat and mass transfer over a vertical cone in a fluid-saturated porous medium with constant wall heat and mass fluxes. A similarity analysis is performed, and the obtained similar equations are solved by the cubic spline collocation method. The effects of the Dufour parameter, Soret parameter, Lewis number, and buoyancy ratio on the heat and mass transfer characteristics have been studied. The local surface temperature tends to increase as the Dufour parameter is increased. The effect of the Dufour parameter on the local surface temperature becomes more significant as the Lewis number is increased. Moreover, an increase in the Soret parameter leads to an increase in the local surface concentration and a decrease in the local surface temperature. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Heat and mass transfer problems on natural convection flow in fluid-saturated porous media have received considerable attention during the last several decades because of numerous applications in engineering problems, such as the design of building components for energy consideration, control of pollutant spread in groundwater, compact heat exchangers, solar power collectors and food industries. Bejan and Khair [1] examined the heat and mass transfer by natural convection in a porous medium. Lai [2] studied the heat and mass transfer by natural convection from a horizontal line source in a saturated porous medium. Nakayama and Hossain [3] examined the heat and mass transfer by natural convection in a porous medium by integral methods. Cheng [4] studied the effect of a magnetic field on heat and mass transfer by natural convection from vertical surfaces in porous media by an integral approach. Yih [5] examined the coupled heat and mass transfer by free convection over a truncated cone in porous media for variable wall temperature and concentration or variable heat and mass fluxes. Chamkha and Khaled [6] studied the hydromagnetic heat and mass transfer by mixed convection from a vertical plate embedded in a uniform porous medium. Yih [7] examined the uniform transpiration effect on coupled heat and mass transfer in mixed convection about inclined surfaces in porous media for the entire regime. Khanafer and Vafai [8] studied the double-diffusive mixed convection in a lid-driven enclosure filled with a fluid-saturated porous medium. Rathish Kumar et al. [9] studied the effect of thermal stratification on double-diffusive natural convection in a vertical porous enclosure. Cheng [10] examined the double-diffusive natural convection ☆ Communicated by W.J. Minkowycz. E-mail address: [email protected]. 0735-1933/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2010.09.011
along a vertical wavy truncated cone in non-Newtonian fluid-saturated porous media with thermal and mass stratification. Moreover, Cheng [11] studied the combined heat and mass transfer in natural convection flow from a vertical wavy surface in a power-law fluid-saturated porous medium with thermal and mass stratification. The Soret effect referred to species differentiation developing in an initial homogeneous mixture submitted to a thermal gradient. The Dufour effect referred to heat flux produced by a concentration gradient. Postelnicu [12] examined the heat and mass characteristics of natural convection about a vertical surface embedded in a saturated porous medium subjected to a magnetic field by considering the Dufour and Soret effects. Partha et al. [13] studied the Soret and Dufour effects in a non-Darcy porous medium. Mansour et al. [14] examined the multiplicity of solutions induced by thermosolutal convection in a square porous cavity heated from below and submitted to horizontal concentration gradient in the presence of Soret effect. Lakshmi Narayana and Murthy [15] examined the Soret and Dufour effects on free convection heat and mass transfer in a doubly stratified Darcy porous medium. Lakshmi Narayana and Murthy [16] examined the Soret and Dufour effects on free convection heat and mass transfer from a horizontal flat plate in a Darcy porous medium. Cheng [17] studied the Soret and Dufour effects on natural convection heat and mass transfer froma vertical cone in a porous medium with constant wall temperature and concentration. Moreover, Cheng [18] examined the Soret and Dufour effects on free convection boundary layer flow over a vertical cylinder in a porous medium with constant wall temperature and concentration. The objective of this paper is to study simultaneous heat and mass transfer by natural convection from a vertical cone in a fluid-saturated porous medium with constant wall heat and mass fluxes, including
C.-Y. Cheng / International Communications in Heat and Mass Transfer 38 (2011) 44–48
45
Nomenclature A C D D DM f g K Le mw N Nu qw r S S Ra Sh T u, v x, y
half angle of the cone concentration Dufour parameter Dufour coefficient mass diffusivity of the porous medium dimensionless stream function acceleration due to gravity permeability of the porous medium Lewis number mass flux at the wall buoyancy ratio local Nusselt number heat flux at the wall local radius of the cone Soret parameter Soret coefficient Rayleigh number Sherwood number temperature dimensional velocity components along x andy axes dimensional Cartesian coordinates along and normal to the cone
Greek symbols α thermal diffusivity of the porous medium βC coefficient of concentration expansion βT coefficient of thermal expansion η similarity variable θ dimensionless temperature ν kinematic viscosity ϕ dimensionless concentration ψ stream function
Fig. 1. Physical model and coordinates.
boundary layers are assumed to be sufficiently thin compared with the local radius. Under the Boussinesq and the boundary layer approximations, the governing equations for the flow, heat and mass transfer within the boundary layer near the vertical cone can be written in two-dimensional Cartesian coordinates (x,y) as [5,19] ∂ðruÞ ∂ðrvÞ + =0 ∂x ∂y u=
2. Analysis Consider the boundary layer flow due to natural convection heat and mass transfer from a vertical cone of half angle A embedded in a fluid-saturated porous medium with Soret and Dufour effects. The origin of the coordinate system is placed at the vertex of the full cone, with x being the coordinate along the surface of the cone measured from the origin and y being the coordinate perpendicular to the conical surface, as shown in Fig. 1. The surface of the cone is held at a constant heat flux qw while the porous medium temperature sufficiently far from the surface of the cone is T∞. The mass flux of a certain constituent in the solution that saturated the porous medium is held at mw near the surface of the cone while the concentration of this constituent in the solution that saturated the porous medium sufficiently far from the surface of the cone is maintained at C∞. The fluid properties are assumed to be constant except for density variations in the buoyancy force term. The thermal and concentration
gK cos A ½βT ðT−T∞ Þ + βC ðC−C∞ Þ ν
u
∂T ∂T ∂2 T ∂2 C +v =α 2 +D 2 ∂x ∂y ∂y ∂y
u
∂C ∂C ∂ C ∂ T +v = DM 2 + S 2 ∂x ∂y ∂y ∂y
Subcripts w condition at wall
Soret and Dufour effects. A similarity analysis is performed, and the obtained similar equations are solved by the cubic spline collocation method. The effects of the Dufour parameter, Soret parameter, Lewis number, and buoyancy ratio on the heat and mass transfer characteristics have been studied.
ð1Þ
2
ð2Þ
ð3Þ
2
ð4Þ
Here u and v are the volume-averaged velocity components in the x- and y-directions, respectively. T and C are the volume-averaged temperature and concentration, respectively. Property ν is the kinematic viscosity of the fluid, and K is the permeability of the porous medium. Furthermore, βT and βC are the coefficient of thermal expansion and the coefficient of concentration expansion, respectively. α and DM are the thermal diffusivity and mass diffusivity of the porous medium, respectively. D and S are the Dufour coefficient and Soret coefficient of the porous medium, respectively. g is the gravitational acceleration. The boundary conditions are given by y=0:v=0 y→∞ : u→0
∂T −k ∂y y = 0 T→T∞
C→C∞
= qw
−DM
∂C ∂y y = 0
= mw ð5Þ ð6Þ
Because the boundary layer thickness is small, the local radius to a point in the boundary layer r can be represented by the local radius of the vertical cone,
r = x sin A
ð7Þ
46
C.-Y. Cheng / International Communications in Heat and Mass Transfer 38 (2011) 44–48
We introduce the similarity variables η=
Table 1 Comparison of values of θ(0) and ϕ(0) for D = 0 and S = 0.
y 1=3 ψ kðT−T∞ ÞRa1 = 3 D ðC−C∞ ÞRa1 = 3 ;θ = Ra ; f ðηÞ = ;ϕ = M 1 = 3 x qw x mw x α rRa
ð8Þ where ψ is the stream function defined as: u=
1 ∂ψ 1 ∂ψ ;v = − r ∂y r ∂x
ð9Þ
and the Rayleigh number is given by Ra =
gKβT x2 qw cos A ναk
f ′ = θ + Nϕ
ð11Þ
5 1 f θ′− f ′θ + Dϕ″ = 0 3 3
ϕ″ 5 1 + f ϕ′− f ′ϕ + Sθ″ = 0 Le 3 3
ð12Þ ð13Þ
where primes denote differentiation with respect to η. Moreover the Lewis number, buoyancy ratio, Dufour parameter, and Soret parameter are respectively defined as Le = D=
θ(0)
Le
4 4 4 1 1 1 0 0 0
1 10 100 1 10 100 1 10 100
ϕ(0)
Yih [5]
Present
Yih [5]
Present
0.6178 0.9490 1.0416 0.8385 1.0211 1.0523 1.0564 1.0564 1.0564
0.6178 0.9488 1.0414 0.8384 1.0210 1.0521 1.0563 1.0563 1.0563
0.6178 0.2273 0.0781 0.8385 0.2618 0.0829 1.0564 0.2804 0.0849
0.6178 0.2274 0.0784 0.8384 0.2619 0.0832 1.0563 0.2805 0.0853
ð10Þ
Upon using these variables, the boundary layer governing equations can be written in a non-dimensional form as
θ″ +
N
for variable grids is used to calculate the value of f at every position from the boundary conditions, Eqs. (15) and (16). At every position, the iteration process continues until the convergence criterion for all the variables, 10− 6, is achieved. Variable grids with 300 grid points are used in the η-direction. The optimum value of boundary layer thickness is used. To assess the accuracy of the solution, the present results are compared with the results obtained by other researchers. Table 1 shows the numerical values of θ(0) and ϕ(0) for S = 0 and D = 0, the conditions for natural convection heat and mass transfer of an vertical
a
0.9 S=0.03 S=0.3 S=0.03 S=0.3
0.8 0.7
α β m k ;N = C w ; DM βT qw DM
0.6
ð14Þ
Dmw k Sq D ;S = w M αqw DM αmw k
D=0.3 D=0.3 D=0.03 D=0.03
N=3 Le=2
0.5 0.4
The corresponding boundary conditions are given by η=0:f =0
θ′ = −1
ð15Þ
0.2
ð16Þ
0.1
In terms of the new variables, the Darcian velocities in the x- and ydirections can be expressed as
0.0
η→∞ : f ′→0 θ→0
u=
ϕ′ = −1
0.3
ϕ→0
αRa2 = 3 f′ x 2=3
αRa v=− x
ð17Þ 5 1 f− ηf 3 3
ð18Þ
The local Nusselt number can be given by Nu 1 = θð0Þ Ra1 = 3
b
0.5
1.0
1.5
2.0
ð19Þ
2.5
3.0
0.6 S=0.3 S=0.3 S=0.03 S=0.03
0.5
D=0.03 D=0.3 D=0.03 D=0.3
N=3 Le=2
0.4
The local Sherwood number can be given by Sh 1 = ϕð0Þ Ra1 = 3
0.0
0.3 0.2
ð20Þ
3. Results and discussion The transformed governing equations, Eqs. (12) and (13), and the associated boundary conditions, Eqs. (15) and (16), can be solved by the cubic spline collocation method [20]. The velocity f′ is calculated from the momentum equation, Eq. (11). Moreover, the Simpson's rule
0.1 0.0 0.0
0.5
1.0
1.5
2.0
2.5
Fig. 2. The effect of the Dufour parameter and Soret parameter on (a) the temperature profile and (b) the concentration profile for N = 3 and Le = 2.
C.-Y. Cheng / International Communications in Heat and Mass Transfer 38 (2011) 44–48
cone of Newtonian fluids in porous media with constant wall heat and mass fluxes. It is shown that the present results are in excellent agreement with the results reported by Yih [5]. Fig. 2 shows the effects of the Dufour parameter D and the Soret parameter S on the temperature profile θ(η) and the concentration profile ϕ(η) near a vertical cone for N = 3 and Le = 2. Increasing the Dufour parameter tends to increase the local surface temperature, thus decreasing the local Nusselt number, as shown in Fig. 2(a). Moreover, Fig. 2(b) shows that, as the Soret parameter is increased, the local surface concentration increases, thus decreasing the local Sherwood number. Fig. 3 depicts the variation of the local surface temperature θ(0) and the local surface concentration ϕ(0) with the Dufour parameter D for various values of buoyancy ratio N and Lewis number Le with S = 0.3. It is clearly seen that the local surface temperature and the local surface concentration tend to decrease as the buoyancy ratio increases. As the Lewis number is increased, the local surface temperature increases while the local surface concentration decreases. Fig. 3(a) shows that the local surface temperature increases as the Dufour parameter is increased. Moreover, an increase in the Dufour parameter tends to decrease the surface concentration, as shown in Fig. 3(b). Comparing the curves in Fig. 3(a), we can deduce that the effect of the
a
Dufour parameter on the local surface temperature becomes more significant as the Lewis number is increased. Fig. 4 shows the effect of Soret parameter S on the local surface temperature θ(0) and the local surface concentration ϕ(0) for various values of buoyancy ratio N and Lewis number Le with D = 0.3. Fig. 4(a) shows that an increase in the Soret parameter tends to decrease the local surface temperature. Moreover, the local surface concentration increases as the Soret number is increased, as shown in Fig. 4(b). Comparing the curves in Fig. 4(b), we can deduce that the effect of the Soret parameter on the local surface concentration is more pronounced as the Lewis number is increased. 4. Conclusions This work studies the Dufour and Soret effects on the steady boundary layer flow due to natural convection heat and mass transfer over a vertical cone embedded in a porous medium saturated with Newtonian fluids with constant wall temperature and concentration. A similarity analysis is performed, and the obtained similar equations are solved by the cubic spline collocation method. The effects of the Dufour parameter, Soret parameter, Lewis number, and buoyancy ratio on the heat and mass transfer characteristics have been studied. As the Dufour parameter is increased, the local surface temperature
a
1.2 N=1 Le=5 N=1 Le=2
1.1
47
1.4 N=1 N=1 N=3 N=3
S=0.3
1.3
N=3 Le=5 N=3 Le=2
1.2
Le=5 Le=2 Le=5 Le=2
D=0.3
1.0 1.1 0.9
1.0 0.9
0.8
0.8 0.7 0.7 0.6
0.6 0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.00
0.05
0.10
0.15
D
b
b
1.0 N=1 N=3 N=1 N=3
0.9
0.20
0.25
0.30
S
S=0.3
Le=2 Le=2 Le=5 Le=5
1.0 N=1 N=3 N=1 N=3
0.9 0.8
0.8
D=0.3
Le=2 Le=2 Le=5 Le=5
0.7 0.6
0.7
0.5 0.6
0.4 0.3
0.5
0.2 0.4
0.1 0.0
0.3 0.00
0.05
0.10
0.15
0.20
0.25
0.30
D Fig. 3. The effect of Dufour parameter on (a) the local surface temperature and (b) the surface concentration for various values of buoyancy ratio and Lewis number with S = 0.3.
0.00
0.05
0.10
0.15
0.20
0.25
0.30
S Fig. 4. The effect of Soret parameter on (a) the local surface temperature and (b) the local surface concentration for various values of buoyancy ratio and Lewis number with D = 0.3.
48
C.-Y. Cheng / International Communications in Heat and Mass Transfer 38 (2011) 44–48
increases while the local surface concentration decreases. The effect of the Dufour parameter on the local surface temperature becomes more significant as the Lewis number is increased. An increase in the Soret parameter tends to increase the local surface concentration and to decrease the local surface temperature. Moreover, the effect of the Soret parameter on the local surface concentration is more pronounced as the Lewis number is increased.
References [1] A. Bejan, K.R. Khair, Heat and mass transfer by natural convection in a porous medium, International Journal of Heat and Mass Transfer 28 (1985) 909–918. [2] F.C. Lai, Coupled heat and mass transfer by natural convection from a horizontal line source in saturated porous medium, International Communications in Heat and Mass Transfer 17 (1990) 489–499. [3] A. Nakayama, M.A. Hossain, An integral treatment for combined heat and mass transfer by natural convection in a porous medium, International Journal of Heat and Mass Transfer 38 (1995) 761–765. [4] C.Y. Cheng, Effect of a magnetic field on heat and mass transfer by natural convection from vertical surfaces in porous media—an integral approach, International Communications in Heat and Mass Transfer 26 (1999) 935–943. [5] K.A. Yih, Coupled heat and mass transfer by free convection over a truncated cone in porous media: VWT/VWC or VHF/VMF, Acta Mechanica 137 (1999) 83–97. [6] A.J. Chamkha, A.A. Khaled, Nonsimilar hydromagnetic simultaneous heat and mass transfer by mixed convection from a vertical plate embedded in a uniform porous medium, Numerical Heat Transfer; Part A: Applications 36 (1999) 327–344. [7] K.A. Yih, Uniform transpiration effect on coupled heat and mass transfer in mixed convection about inclined surfaces in porous media: the entire regime, Acta Mechanica 132 (1999) 229–240. [8] K. Khanafer, K. Vafai, Double-diffusive mixed convection in a lid-driven enclosure filled with a fluid-saturated porous medium, Numerical Heat Transfer; Part A: Applications 42 (2002) 465–486.
[9] B.V. Rathish Kumar, P. Singh, V.J. Bansod, Effect of thermal stratification on double-diffusive natural convection in a vertical porous enclosure, Numerical Heat Transfer; Part A: Applications 41 (2002) 421–447. [10] C.Y. Cheng, Double-diffusive natural convection along a vertical wavy truncated cone in non-Newtonian fluid saturated porous media with thermal and mass stratification, International Communications in Heat and Mass Transfer 35 (2008) 985–990. [11] C.Y. Cheng, Combined heat and mass transfer in natural convection flow from a vertical wavy surface in a power-law fluid saturated porous medium with thermal and mass stratification, International Communications in Heat and Mass Transfer 36 (2009) 351–356. [12] A. Postelnicu, Influence of a magnetic field on heat and mass transfer by natural convecion from vertical surfaces in porous media considering Soret and Dufour effects, International Journal of Heat and Mass Transfer 47 (2004) 1467–1472. [13] M.K. Partha, P.V.S.N. Murthy, G.P. Raja Sekhar, Soret and Dufour effects in a nonDarcy porous medium, Journal of Heat Tansfer 128 (2006) 605–610. [14] A. Mansour, A. Amahmid, M. Hasnaoui, M. Bourich, Multiplicity of solutions induced by thermosolutal convection in a square porous cavity heated from below and submitted to horizontal concentration gradient in the presence of Soret effect, Numerical Heat Transfer; Part A: Applications 49 (2006) 69–94. [15] P.A. Lakshmi Narayana, P.V.S.N. Murthy, Soret and Dufour effects on free convection heat and mass transfer in a doubly stratified Darcy porous medium, Journal of Porous Media 10 (2007) 613–624. [16] P.A. Lakshmi Narayana, P.V.S.N. Murthy, Soret and Dufour effects on free convection heat and mass transfer from a horizontal flat plate in a Darcy porous medium, Journal of Heat Tansfer 130 (2008)8 104504-1-104504-5. [17] C.Y. Cheng, Soret and Dufour effects on natural convection heat and mass transfer from a vertical cone in a porous medium, International Communications in Heat and Mass Transfer 36 (2009) 1020–1024. [18] C.Y. Cheng, Soret and Dufour effects on free convection boundary layer flow over a vertical cylinder in a saturated porous medium, International Communications in Heat and Mass Transfer 37 (2010) 796–800. [19] D.A. Nield, A. Bejan, Convection in Porous Media, Springer-Verlag, New York, 1992, p. 34. [20] P. Wang, R. Kahawita, Numerical integration of a partial differential equations using cubic spline, International Journal of Computer Mathematics 13 (1983) 271–286.
Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i c h m t
Soret and Dufour effects on natural convection boundary layer flow over a vertical cone in a porous medium with constant wall heat and mass fluxes☆ Ching-Yang Cheng Department of Mechanical Engineering, Southern Taiwan University, Yungkang 71005, Taiwan
a r t i c l e
i n f o
Available online 19 November 2010 Keywords: Natural convection Heat and mass transfer Boundary layer Vertical cone Porous medium Dufour effect Soret effect
a b s t r a c t This work studies the Soret and Dufour effects on the boundary layer flow due to natural convection heat and mass transfer over a vertical cone in a fluid-saturated porous medium with constant wall heat and mass fluxes. A similarity analysis is performed, and the obtained similar equations are solved by the cubic spline collocation method. The effects of the Dufour parameter, Soret parameter, Lewis number, and buoyancy ratio on the heat and mass transfer characteristics have been studied. The local surface temperature tends to increase as the Dufour parameter is increased. The effect of the Dufour parameter on the local surface temperature becomes more significant as the Lewis number is increased. Moreover, an increase in the Soret parameter leads to an increase in the local surface concentration and a decrease in the local surface temperature. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Heat and mass transfer problems on natural convection flow in fluid-saturated porous media have received considerable attention during the last several decades because of numerous applications in engineering problems, such as the design of building components for energy consideration, control of pollutant spread in groundwater, compact heat exchangers, solar power collectors and food industries. Bejan and Khair [1] examined the heat and mass transfer by natural convection in a porous medium. Lai [2] studied the heat and mass transfer by natural convection from a horizontal line source in a saturated porous medium. Nakayama and Hossain [3] examined the heat and mass transfer by natural convection in a porous medium by integral methods. Cheng [4] studied the effect of a magnetic field on heat and mass transfer by natural convection from vertical surfaces in porous media by an integral approach. Yih [5] examined the coupled heat and mass transfer by free convection over a truncated cone in porous media for variable wall temperature and concentration or variable heat and mass fluxes. Chamkha and Khaled [6] studied the hydromagnetic heat and mass transfer by mixed convection from a vertical plate embedded in a uniform porous medium. Yih [7] examined the uniform transpiration effect on coupled heat and mass transfer in mixed convection about inclined surfaces in porous media for the entire regime. Khanafer and Vafai [8] studied the double-diffusive mixed convection in a lid-driven enclosure filled with a fluid-saturated porous medium. Rathish Kumar et al. [9] studied the effect of thermal stratification on double-diffusive natural convection in a vertical porous enclosure. Cheng [10] examined the double-diffusive natural convection ☆ Communicated by W.J. Minkowycz. E-mail address: [email protected]. 0735-1933/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2010.09.011
along a vertical wavy truncated cone in non-Newtonian fluid-saturated porous media with thermal and mass stratification. Moreover, Cheng [11] studied the combined heat and mass transfer in natural convection flow from a vertical wavy surface in a power-law fluid-saturated porous medium with thermal and mass stratification. The Soret effect referred to species differentiation developing in an initial homogeneous mixture submitted to a thermal gradient. The Dufour effect referred to heat flux produced by a concentration gradient. Postelnicu [12] examined the heat and mass characteristics of natural convection about a vertical surface embedded in a saturated porous medium subjected to a magnetic field by considering the Dufour and Soret effects. Partha et al. [13] studied the Soret and Dufour effects in a non-Darcy porous medium. Mansour et al. [14] examined the multiplicity of solutions induced by thermosolutal convection in a square porous cavity heated from below and submitted to horizontal concentration gradient in the presence of Soret effect. Lakshmi Narayana and Murthy [15] examined the Soret and Dufour effects on free convection heat and mass transfer in a doubly stratified Darcy porous medium. Lakshmi Narayana and Murthy [16] examined the Soret and Dufour effects on free convection heat and mass transfer from a horizontal flat plate in a Darcy porous medium. Cheng [17] studied the Soret and Dufour effects on natural convection heat and mass transfer froma vertical cone in a porous medium with constant wall temperature and concentration. Moreover, Cheng [18] examined the Soret and Dufour effects on free convection boundary layer flow over a vertical cylinder in a porous medium with constant wall temperature and concentration. The objective of this paper is to study simultaneous heat and mass transfer by natural convection from a vertical cone in a fluid-saturated porous medium with constant wall heat and mass fluxes, including
C.-Y. Cheng / International Communications in Heat and Mass Transfer 38 (2011) 44–48
45
Nomenclature A C D D DM f g K Le mw N Nu qw r S S Ra Sh T u, v x, y
half angle of the cone concentration Dufour parameter Dufour coefficient mass diffusivity of the porous medium dimensionless stream function acceleration due to gravity permeability of the porous medium Lewis number mass flux at the wall buoyancy ratio local Nusselt number heat flux at the wall local radius of the cone Soret parameter Soret coefficient Rayleigh number Sherwood number temperature dimensional velocity components along x andy axes dimensional Cartesian coordinates along and normal to the cone
Greek symbols α thermal diffusivity of the porous medium βC coefficient of concentration expansion βT coefficient of thermal expansion η similarity variable θ dimensionless temperature ν kinematic viscosity ϕ dimensionless concentration ψ stream function
Fig. 1. Physical model and coordinates.
boundary layers are assumed to be sufficiently thin compared with the local radius. Under the Boussinesq and the boundary layer approximations, the governing equations for the flow, heat and mass transfer within the boundary layer near the vertical cone can be written in two-dimensional Cartesian coordinates (x,y) as [5,19] ∂ðruÞ ∂ðrvÞ + =0 ∂x ∂y u=
2. Analysis Consider the boundary layer flow due to natural convection heat and mass transfer from a vertical cone of half angle A embedded in a fluid-saturated porous medium with Soret and Dufour effects. The origin of the coordinate system is placed at the vertex of the full cone, with x being the coordinate along the surface of the cone measured from the origin and y being the coordinate perpendicular to the conical surface, as shown in Fig. 1. The surface of the cone is held at a constant heat flux qw while the porous medium temperature sufficiently far from the surface of the cone is T∞. The mass flux of a certain constituent in the solution that saturated the porous medium is held at mw near the surface of the cone while the concentration of this constituent in the solution that saturated the porous medium sufficiently far from the surface of the cone is maintained at C∞. The fluid properties are assumed to be constant except for density variations in the buoyancy force term. The thermal and concentration
gK cos A ½βT ðT−T∞ Þ + βC ðC−C∞ Þ ν
u
∂T ∂T ∂2 T ∂2 C +v =α 2 +D 2 ∂x ∂y ∂y ∂y
u
∂C ∂C ∂ C ∂ T +v = DM 2 + S 2 ∂x ∂y ∂y ∂y
Subcripts w condition at wall
Soret and Dufour effects. A similarity analysis is performed, and the obtained similar equations are solved by the cubic spline collocation method. The effects of the Dufour parameter, Soret parameter, Lewis number, and buoyancy ratio on the heat and mass transfer characteristics have been studied.
ð1Þ
2
ð2Þ
ð3Þ
2
ð4Þ
Here u and v are the volume-averaged velocity components in the x- and y-directions, respectively. T and C are the volume-averaged temperature and concentration, respectively. Property ν is the kinematic viscosity of the fluid, and K is the permeability of the porous medium. Furthermore, βT and βC are the coefficient of thermal expansion and the coefficient of concentration expansion, respectively. α and DM are the thermal diffusivity and mass diffusivity of the porous medium, respectively. D and S are the Dufour coefficient and Soret coefficient of the porous medium, respectively. g is the gravitational acceleration. The boundary conditions are given by y=0:v=0 y→∞ : u→0
∂T −k ∂y y = 0 T→T∞
C→C∞
= qw
−DM
∂C ∂y y = 0
= mw ð5Þ ð6Þ
Because the boundary layer thickness is small, the local radius to a point in the boundary layer r can be represented by the local radius of the vertical cone,
r = x sin A
ð7Þ
46
C.-Y. Cheng / International Communications in Heat and Mass Transfer 38 (2011) 44–48
We introduce the similarity variables η=
Table 1 Comparison of values of θ(0) and ϕ(0) for D = 0 and S = 0.
y 1=3 ψ kðT−T∞ ÞRa1 = 3 D ðC−C∞ ÞRa1 = 3 ;θ = Ra ; f ðηÞ = ;ϕ = M 1 = 3 x qw x mw x α rRa
ð8Þ where ψ is the stream function defined as: u=
1 ∂ψ 1 ∂ψ ;v = − r ∂y r ∂x
ð9Þ
and the Rayleigh number is given by Ra =
gKβT x2 qw cos A ναk
f ′ = θ + Nϕ
ð11Þ
5 1 f θ′− f ′θ + Dϕ″ = 0 3 3
ϕ″ 5 1 + f ϕ′− f ′ϕ + Sθ″ = 0 Le 3 3
ð12Þ ð13Þ
where primes denote differentiation with respect to η. Moreover the Lewis number, buoyancy ratio, Dufour parameter, and Soret parameter are respectively defined as Le = D=
θ(0)
Le
4 4 4 1 1 1 0 0 0
1 10 100 1 10 100 1 10 100
ϕ(0)
Yih [5]
Present
Yih [5]
Present
0.6178 0.9490 1.0416 0.8385 1.0211 1.0523 1.0564 1.0564 1.0564
0.6178 0.9488 1.0414 0.8384 1.0210 1.0521 1.0563 1.0563 1.0563
0.6178 0.2273 0.0781 0.8385 0.2618 0.0829 1.0564 0.2804 0.0849
0.6178 0.2274 0.0784 0.8384 0.2619 0.0832 1.0563 0.2805 0.0853
ð10Þ
Upon using these variables, the boundary layer governing equations can be written in a non-dimensional form as
θ″ +
N
for variable grids is used to calculate the value of f at every position from the boundary conditions, Eqs. (15) and (16). At every position, the iteration process continues until the convergence criterion for all the variables, 10− 6, is achieved. Variable grids with 300 grid points are used in the η-direction. The optimum value of boundary layer thickness is used. To assess the accuracy of the solution, the present results are compared with the results obtained by other researchers. Table 1 shows the numerical values of θ(0) and ϕ(0) for S = 0 and D = 0, the conditions for natural convection heat and mass transfer of an vertical
a
0.9 S=0.03 S=0.3 S=0.03 S=0.3
0.8 0.7
α β m k ;N = C w ; DM βT qw DM
0.6
ð14Þ
Dmw k Sq D ;S = w M αqw DM αmw k
D=0.3 D=0.3 D=0.03 D=0.03
N=3 Le=2
0.5 0.4
The corresponding boundary conditions are given by η=0:f =0
θ′ = −1
ð15Þ
0.2
ð16Þ
0.1
In terms of the new variables, the Darcian velocities in the x- and ydirections can be expressed as
0.0
η→∞ : f ′→0 θ→0
u=
ϕ′ = −1
0.3
ϕ→0
αRa2 = 3 f′ x 2=3
αRa v=− x
ð17Þ 5 1 f− ηf 3 3
ð18Þ
The local Nusselt number can be given by Nu 1 = θð0Þ Ra1 = 3
b
0.5
1.0
1.5
2.0
ð19Þ
2.5
3.0
0.6 S=0.3 S=0.3 S=0.03 S=0.03
0.5
D=0.03 D=0.3 D=0.03 D=0.3
N=3 Le=2
0.4
The local Sherwood number can be given by Sh 1 = ϕð0Þ Ra1 = 3
0.0
0.3 0.2
ð20Þ
3. Results and discussion The transformed governing equations, Eqs. (12) and (13), and the associated boundary conditions, Eqs. (15) and (16), can be solved by the cubic spline collocation method [20]. The velocity f′ is calculated from the momentum equation, Eq. (11). Moreover, the Simpson's rule
0.1 0.0 0.0
0.5
1.0
1.5
2.0
2.5
Fig. 2. The effect of the Dufour parameter and Soret parameter on (a) the temperature profile and (b) the concentration profile for N = 3 and Le = 2.
C.-Y. Cheng / International Communications in Heat and Mass Transfer 38 (2011) 44–48
cone of Newtonian fluids in porous media with constant wall heat and mass fluxes. It is shown that the present results are in excellent agreement with the results reported by Yih [5]. Fig. 2 shows the effects of the Dufour parameter D and the Soret parameter S on the temperature profile θ(η) and the concentration profile ϕ(η) near a vertical cone for N = 3 and Le = 2. Increasing the Dufour parameter tends to increase the local surface temperature, thus decreasing the local Nusselt number, as shown in Fig. 2(a). Moreover, Fig. 2(b) shows that, as the Soret parameter is increased, the local surface concentration increases, thus decreasing the local Sherwood number. Fig. 3 depicts the variation of the local surface temperature θ(0) and the local surface concentration ϕ(0) with the Dufour parameter D for various values of buoyancy ratio N and Lewis number Le with S = 0.3. It is clearly seen that the local surface temperature and the local surface concentration tend to decrease as the buoyancy ratio increases. As the Lewis number is increased, the local surface temperature increases while the local surface concentration decreases. Fig. 3(a) shows that the local surface temperature increases as the Dufour parameter is increased. Moreover, an increase in the Dufour parameter tends to decrease the surface concentration, as shown in Fig. 3(b). Comparing the curves in Fig. 3(a), we can deduce that the effect of the
a
Dufour parameter on the local surface temperature becomes more significant as the Lewis number is increased. Fig. 4 shows the effect of Soret parameter S on the local surface temperature θ(0) and the local surface concentration ϕ(0) for various values of buoyancy ratio N and Lewis number Le with D = 0.3. Fig. 4(a) shows that an increase in the Soret parameter tends to decrease the local surface temperature. Moreover, the local surface concentration increases as the Soret number is increased, as shown in Fig. 4(b). Comparing the curves in Fig. 4(b), we can deduce that the effect of the Soret parameter on the local surface concentration is more pronounced as the Lewis number is increased. 4. Conclusions This work studies the Dufour and Soret effects on the steady boundary layer flow due to natural convection heat and mass transfer over a vertical cone embedded in a porous medium saturated with Newtonian fluids with constant wall temperature and concentration. A similarity analysis is performed, and the obtained similar equations are solved by the cubic spline collocation method. The effects of the Dufour parameter, Soret parameter, Lewis number, and buoyancy ratio on the heat and mass transfer characteristics have been studied. As the Dufour parameter is increased, the local surface temperature
a
1.2 N=1 Le=5 N=1 Le=2
1.1
47
1.4 N=1 N=1 N=3 N=3
S=0.3
1.3
N=3 Le=5 N=3 Le=2
1.2
Le=5 Le=2 Le=5 Le=2
D=0.3
1.0 1.1 0.9
1.0 0.9
0.8
0.8 0.7 0.7 0.6
0.6 0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.00
0.05
0.10
0.15
D
b
b
1.0 N=1 N=3 N=1 N=3
0.9
0.20
0.25
0.30
S
S=0.3
Le=2 Le=2 Le=5 Le=5
1.0 N=1 N=3 N=1 N=3
0.9 0.8
0.8
D=0.3
Le=2 Le=2 Le=5 Le=5
0.7 0.6
0.7
0.5 0.6
0.4 0.3
0.5
0.2 0.4
0.1 0.0
0.3 0.00
0.05
0.10
0.15
0.20
0.25
0.30
D Fig. 3. The effect of Dufour parameter on (a) the local surface temperature and (b) the surface concentration for various values of buoyancy ratio and Lewis number with S = 0.3.
0.00
0.05
0.10
0.15
0.20
0.25
0.30
S Fig. 4. The effect of Soret parameter on (a) the local surface temperature and (b) the local surface concentration for various values of buoyancy ratio and Lewis number with D = 0.3.
48
C.-Y. Cheng / International Communications in Heat and Mass Transfer 38 (2011) 44–48
increases while the local surface concentration decreases. The effect of the Dufour parameter on the local surface temperature becomes more significant as the Lewis number is increased. An increase in the Soret parameter tends to increase the local surface concentration and to decrease the local surface temperature. Moreover, the effect of the Soret parameter on the local surface concentration is more pronounced as the Lewis number is increased.
References [1] A. Bejan, K.R. Khair, Heat and mass transfer by natural convection in a porous medium, International Journal of Heat and Mass Transfer 28 (1985) 909–918. [2] F.C. Lai, Coupled heat and mass transfer by natural convection from a horizontal line source in saturated porous medium, International Communications in Heat and Mass Transfer 17 (1990) 489–499. [3] A. Nakayama, M.A. Hossain, An integral treatment for combined heat and mass transfer by natural convection in a porous medium, International Journal of Heat and Mass Transfer 38 (1995) 761–765. [4] C.Y. Cheng, Effect of a magnetic field on heat and mass transfer by natural convection from vertical surfaces in porous media—an integral approach, International Communications in Heat and Mass Transfer 26 (1999) 935–943. [5] K.A. Yih, Coupled heat and mass transfer by free convection over a truncated cone in porous media: VWT/VWC or VHF/VMF, Acta Mechanica 137 (1999) 83–97. [6] A.J. Chamkha, A.A. Khaled, Nonsimilar hydromagnetic simultaneous heat and mass transfer by mixed convection from a vertical plate embedded in a uniform porous medium, Numerical Heat Transfer; Part A: Applications 36 (1999) 327–344. [7] K.A. Yih, Uniform transpiration effect on coupled heat and mass transfer in mixed convection about inclined surfaces in porous media: the entire regime, Acta Mechanica 132 (1999) 229–240. [8] K. Khanafer, K. Vafai, Double-diffusive mixed convection in a lid-driven enclosure filled with a fluid-saturated porous medium, Numerical Heat Transfer; Part A: Applications 42 (2002) 465–486.
[9] B.V. Rathish Kumar, P. Singh, V.J. Bansod, Effect of thermal stratification on double-diffusive natural convection in a vertical porous enclosure, Numerical Heat Transfer; Part A: Applications 41 (2002) 421–447. [10] C.Y. Cheng, Double-diffusive natural convection along a vertical wavy truncated cone in non-Newtonian fluid saturated porous media with thermal and mass stratification, International Communications in Heat and Mass Transfer 35 (2008) 985–990. [11] C.Y. Cheng, Combined heat and mass transfer in natural convection flow from a vertical wavy surface in a power-law fluid saturated porous medium with thermal and mass stratification, International Communications in Heat and Mass Transfer 36 (2009) 351–356. [12] A. Postelnicu, Influence of a magnetic field on heat and mass transfer by natural convecion from vertical surfaces in porous media considering Soret and Dufour effects, International Journal of Heat and Mass Transfer 47 (2004) 1467–1472. [13] M.K. Partha, P.V.S.N. Murthy, G.P. Raja Sekhar, Soret and Dufour effects in a nonDarcy porous medium, Journal of Heat Tansfer 128 (2006) 605–610. [14] A. Mansour, A. Amahmid, M. Hasnaoui, M. Bourich, Multiplicity of solutions induced by thermosolutal convection in a square porous cavity heated from below and submitted to horizontal concentration gradient in the presence of Soret effect, Numerical Heat Transfer; Part A: Applications 49 (2006) 69–94. [15] P.A. Lakshmi Narayana, P.V.S.N. Murthy, Soret and Dufour effects on free convection heat and mass transfer in a doubly stratified Darcy porous medium, Journal of Porous Media 10 (2007) 613–624. [16] P.A. Lakshmi Narayana, P.V.S.N. Murthy, Soret and Dufour effects on free convection heat and mass transfer from a horizontal flat plate in a Darcy porous medium, Journal of Heat Tansfer 130 (2008)8 104504-1-104504-5. [17] C.Y. Cheng, Soret and Dufour effects on natural convection heat and mass transfer from a vertical cone in a porous medium, International Communications in Heat and Mass Transfer 36 (2009) 1020–1024. [18] C.Y. Cheng, Soret and Dufour effects on free convection boundary layer flow over a vertical cylinder in a saturated porous medium, International Communications in Heat and Mass Transfer 37 (2010) 796–800. [19] D.A. Nield, A. Bejan, Convection in Porous Media, Springer-Verlag, New York, 1992, p. 34. [20] P. Wang, R. Kahawita, Numerical integration of a partial differential equations using cubic spline, International Journal of Computer Mathematics 13 (1983) 271–286.