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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
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 heat and mass transfer from a vertical cone in a porous medium☆ 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 11 August 2009 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 downward-pointing vertical cone 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 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 Nusselt number tends to decrease as the Dufour parameter is increased. The effect of the Dufour parameter on the local Nusselt number becomes more significant as the Lewis number is increased. Moreover, an increase in the Soret number leads to a decrease in the local Sherwood number and an increase in the local Nusselt number. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction Coupled heat and mass transfer in fluid-saturated porous media has been of growing interest during the last several decades because of its great practical applications in modern industry, 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] studied the heat and mass transfer by natural convection in a porous medium. Lai [2] examined the heat and mass transfer by natural convection from a horizontal line source in saturated porous medium. Nakayama and Hossain [3] studied the combined 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. Cheng [8] uses an integral approach to study the heat and mass transfer by natural convection from truncated cones in porous media with variable wall temperature and concentration. Khanafer and Vafai [9] studied the
☆ Communicated by W.J. Minkowycz. E-mail address: [email protected]. 0735-1933/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2009.07.003
double-diffusive mixed convection in a lid-driven enclosure filled with a fluid-saturated porous medium. Rathish Kumar et al. [10] examined the effect of thermal stratification on double-diffusive natural convection in a vertical porous enclosure. Moreover, Cheng [11] studied the doublediffusive natural convection along a vertical wavy truncated cone in nonNewtonian fluid-saturated porous media with thermal and mass stratification. Cheng [12] examined 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. There are a few studies about the Soret and Dufour effects in a Darcy or non-Darcy porous medium. Postelnicu [13] studied 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. [14] examined the Soret and Dufour effects in a non-Darcy porous medium. Mansour et al. [15] studied 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 et al. [16] studied the Soret and Dufour effects in a doubly stratified Darcy porous medium. Lakshmi Narayana and Murthy [17] examined the Soret and Dufour effects on free convection heat and mass transfer from a horizontal flat plate in a Darcy porous medium. The previous studies are concerned about the Soret and Dufour effects on the steady boundary layer flow by natural convection heat and mass transfer over a horizontal and vertical surfaces embedded in a Darcy or non-Darcy porous medium. The present study, however, considers the natural convection heat and mass transfer over a downward-pointing vertical cone in a Darcy porous medium saturated with Newtonian fluid
C.-Y. Cheng / International Communications in Heat and Mass Transfer 36 (2009) 1020–1024
1021
Nomenclature A C D D̅ DM f g K Le N Nu r S S̅ 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 buoyancy ratio local Nusselt number local radius of the cone Soret parameter Soret coefficient Rayleigh number Sherwood number temperature dimensional velocity components along x and y axes dimensional Cartesian coordinates along and normal to the cone Fig. 1. Physical model and coordinates.
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 v kinematic viscosity ϕ dimensionless concentration ψ stream function
Subcripts w condition at wall ∞ condition at infinity
considering Soret and Dufour effects. The similarity solutions are derived and then solved by the cubic spline collocation method to study the effects of the Dufour parameter, Soret parameter, Lewis number, and buoyancy ratio on the heat and mass transfer characteristics.
2. Basic equations and similarity analysis Consider the boundary layer flow due to natural convection heat and mass transfer from a downward-pointing vertical cone of half angle A embedded in a porous medium saturated with a Newtonian fluid 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 maintained at a constant temperature Tw, which is different from the porous medium temperature sufficiently far from the surface of the cone. The concentration of a certain constituent in the solution that saturated the porous medium varies from Cw on the fluid side of the surface of the cone to C∞ sufficiently far from the surface of the cone. The fluid properties are assumed to be constant except for density variations in the buoyancy force term. Assuming that the thermal and concentration boundary layers are sufficiently thin compared with the local radius, 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 [5,18] AðruÞ AðrvÞ + =0 Ax Ay u=
gK cos A ½βT ðT − T∞ Þ + βC ðC − C∞ Þ m
ð1Þ
ð2Þ
u
AT AT A2 T A2 C +v =α 2 +D 2 Ax Ay Ay Ay
ð3Þ
u
AC AC A2 C A2 T +v = DM 2 + S 2 Ax Ay Ay Ay
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 v is 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 T = Tw C = Cw
ð5Þ
yY∞ : u→0 T→T∞ C→C∞
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
We introduce the similarity variables η=
y 1=2 ψ T − T∞ C − C∞ Ra ; f ðηÞ = ; θ= ; /= x Tw − T∞ Cw − C∞ αrRa1 = 2
ð8Þ
1022
C.-Y. Cheng / International Communications in Heat and Mass Transfer 36 (2009) 1020–1024
where ψ is the stream function defined as: u=
1 Aψ 1 Aψ ; v= − r Ay r Ax
ð9Þ
and the Rayleigh number is given by Ra =
gKβT xðTw − T∞ Þ cos A rα
ð10Þ
Upon using these variables, the basic equations of the boundary layer for the problem under consideration can be written in nondimensional form as f V= θ + N/ θW +
ð11Þ
3 f θ V + D/ W = 0 2
ð12Þ
/W 3 + f /V+ SθW = 0 Le 2
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 a vertical cone of Newtonian fluids in porous media with constant wall temperature and concentration. 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 Le = 2 and N = 1. Increasing the Dufour parameter tends to thicken thermal boundary layer, thus decreasing the heat transfer rate at the wall, as shown in Fig. 2(a). Moreover, Fig. 2(b) shows that, as the Soret parameter is increased,
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 =
α β ðC − C∞ Þ DðCw − C∞ Þ SðTw − T∞ Þ ; D= ; S= ; N= C w ð14Þ DM βT ðTw − T∞ Þ α ðTw − T∞ Þ α ðCw − C∞ Þ
The boundary conditions are transformed to η=0: f =0 θ=1 /=1
ð15Þ
ηY∞ : f′→0 θ→0 ϕ→0
ð16Þ
The local Nusselt number can be derived as Nu = − θ Vð0Þ Ra1 = 2
ð17Þ
The local Sherwood number can be derived as Sh = − / Vð0Þ Ra1 = 2
ð18Þ
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 [19]. The velocity f′ is calculated from the momentum equation, Eq. (11). Moreover, the Simpson's rule 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 Table 1 Comparison of values of − θ′(0) and − ϕ′(0) for D = 0 and S = 0. N
4 4 4 1 1 1 0 0 0
Le
1 10 100 1 10 100 1 10 100
− θ′(0)
− ϕ′(0)
Yih [5]
Present
Yih [5]
Present
1.7186 1.1795 0.9019 1.0869 0.9030 0.8141 0.7686 0.7686 0.7686
1.7186 1.1794 0.9022 1.0870 0.9032 0.8143 0.7685 0.7685 0.7685
1.7186 5.6977 18.2208 1.0869 3.8139 12.3645 0.7686 2.9102 9.6032
1.7186 5.6949 18.1312 1.0870 3.8134 12.3377 0.7685 2.9103 9.5931
Fig. 2. The effect of the Dufour parameter and Soret parameter on (a) the temperature profile and (b) the concentration profile for N = 1 and Le = 2.
C.-Y. Cheng / International Communications in Heat and Mass Transfer 36 (2009) 1020–1024
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the concentration boundary layer thickness increases, thus decreasing the mass transfer rate at the wall. Fig. 3 depicts the variation of the local Nusselt number NuRa− 0.5 and the Sherwood number ShRa− 0.5 with the Dufour parameter D for various values of buoyancy ratio N and Lewis number Le with S = 0.1. It is clearly seen that the local Nusselt number and the Sherwood number tend to increase as the buoyancy ratio increases. As the Lewis number is increased, the local Nusselt number decreases while the local Sherwood number increases. Moreover, as the Dufour number is increased, the local Nusselt number decreases while the local Sherwood number slightly increases. Comparing the curves in Fig. 3 (a), we can deduce that the effect of the Dufour parameter on the local Nusselt number becomes more significant as the Lewis number is increased. Fig. 4 shows the effect of Soret parameter S on the local Nusselt number NuRa− 0.5 and the Sherwood number ShRa− 0.5 for various values of buoyancy ratio N and Lewis number Le with D = 0.1. In-
creasing the buoyancy ratio accelerates the flow, decreasing the thermal and concentration boundary layer thickness and thus increasing the heat and mass transfer rates between the fluid and the wall. Moreover, as the Soret number is increased, the local Sherwood number decreases while the local Nusselt number increases. Comparing the curves in Fig. 4(a), we can deduce that the effect of the Soret parameter on the local Nusselt number is more pronounced as the Lewis number is increased.
Fig. 3. The effect of Dufour parameter on (a) the local Nusselt number and (b) the Sherwood number for various values of buoyancy ratio and Lewis number with S = 0.1.
Fig. 4. The effect of Soret parameter on (a) the local Nusselt number and (b) the Sherwood number for various values of buoyancy ratio and Lewis number with D = 0.1.
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 downward-pointing 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
1024
C.-Y. Cheng / International Communications in Heat and Mass Transfer 36 (2009) 1020–1024
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 number is increased, the local Nusselt number decreases while the local Sherwood number slightly increases. An increase in the Soret number tends to decrease the local Sherwood number and to increase the local Nusselt number.
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] C.Y. Cheng, An integral approach for heat and mass transfer by natural convection from truncated cones in porous media with variable wall temperature and
[9]
[10]
[11]
[12]
[13]
[14] [15]
[16] [17]
[18] [19]
concentration, International Communications in Heat and Mass Transfer 27 (2000) 437–548. 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. B.V. Rathish Kumar, P. Singh, V.J. Bansod, Effect of thermal stratification on doublediffusive natural convection in a vertical porous enclosure, Numerical Heat Transfer. Part A, Applications 41 (2002) 421–447. 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. 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. 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. M.K. Partha, P.V.S.N. Murthy, G.P. Raja Sekhar, Soret and Dufour effects in a nonDarcy porous medium, Journal of Heat Transfer 128 (2006) 605–610. 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. P.A. Lakshmi Narayana, P.V.S.N. Murthy, Soret and Dufour effects in a doubly stratified Darcy porous medium, Journal of Porous Media 10 (2007) 613–624. 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) 104504-1–104504-5. D.A. Nield, A. Bejan, Convection in Porous Media, Springer-Verlag, New York, 1992, p. 34. 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 heat and mass transfer from a vertical cone in a porous medium☆ 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 11 August 2009 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 downward-pointing vertical cone 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 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 Nusselt number tends to decrease as the Dufour parameter is increased. The effect of the Dufour parameter on the local Nusselt number becomes more significant as the Lewis number is increased. Moreover, an increase in the Soret number leads to a decrease in the local Sherwood number and an increase in the local Nusselt number. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction Coupled heat and mass transfer in fluid-saturated porous media has been of growing interest during the last several decades because of its great practical applications in modern industry, 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] studied the heat and mass transfer by natural convection in a porous medium. Lai [2] examined the heat and mass transfer by natural convection from a horizontal line source in saturated porous medium. Nakayama and Hossain [3] studied the combined 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. Cheng [8] uses an integral approach to study the heat and mass transfer by natural convection from truncated cones in porous media with variable wall temperature and concentration. Khanafer and Vafai [9] studied the
☆ Communicated by W.J. Minkowycz. E-mail address: [email protected]. 0735-1933/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2009.07.003
double-diffusive mixed convection in a lid-driven enclosure filled with a fluid-saturated porous medium. Rathish Kumar et al. [10] examined the effect of thermal stratification on double-diffusive natural convection in a vertical porous enclosure. Moreover, Cheng [11] studied the doublediffusive natural convection along a vertical wavy truncated cone in nonNewtonian fluid-saturated porous media with thermal and mass stratification. Cheng [12] examined 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. There are a few studies about the Soret and Dufour effects in a Darcy or non-Darcy porous medium. Postelnicu [13] studied 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. [14] examined the Soret and Dufour effects in a non-Darcy porous medium. Mansour et al. [15] studied 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 et al. [16] studied the Soret and Dufour effects in a doubly stratified Darcy porous medium. Lakshmi Narayana and Murthy [17] examined the Soret and Dufour effects on free convection heat and mass transfer from a horizontal flat plate in a Darcy porous medium. The previous studies are concerned about the Soret and Dufour effects on the steady boundary layer flow by natural convection heat and mass transfer over a horizontal and vertical surfaces embedded in a Darcy or non-Darcy porous medium. The present study, however, considers the natural convection heat and mass transfer over a downward-pointing vertical cone in a Darcy porous medium saturated with Newtonian fluid
C.-Y. Cheng / International Communications in Heat and Mass Transfer 36 (2009) 1020–1024
1021
Nomenclature A C D D̅ DM f g K Le N Nu r S S̅ 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 buoyancy ratio local Nusselt number local radius of the cone Soret parameter Soret coefficient Rayleigh number Sherwood number temperature dimensional velocity components along x and y axes dimensional Cartesian coordinates along and normal to the cone Fig. 1. Physical model and coordinates.
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 v kinematic viscosity ϕ dimensionless concentration ψ stream function
Subcripts w condition at wall ∞ condition at infinity
considering Soret and Dufour effects. The similarity solutions are derived and then solved by the cubic spline collocation method to study the effects of the Dufour parameter, Soret parameter, Lewis number, and buoyancy ratio on the heat and mass transfer characteristics.
2. Basic equations and similarity analysis Consider the boundary layer flow due to natural convection heat and mass transfer from a downward-pointing vertical cone of half angle A embedded in a porous medium saturated with a Newtonian fluid 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 maintained at a constant temperature Tw, which is different from the porous medium temperature sufficiently far from the surface of the cone. The concentration of a certain constituent in the solution that saturated the porous medium varies from Cw on the fluid side of the surface of the cone to C∞ sufficiently far from the surface of the cone. The fluid properties are assumed to be constant except for density variations in the buoyancy force term. Assuming that the thermal and concentration boundary layers are sufficiently thin compared with the local radius, 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 [5,18] AðruÞ AðrvÞ + =0 Ax Ay u=
gK cos A ½βT ðT − T∞ Þ + βC ðC − C∞ Þ m
ð1Þ
ð2Þ
u
AT AT A2 T A2 C +v =α 2 +D 2 Ax Ay Ay Ay
ð3Þ
u
AC AC A2 C A2 T +v = DM 2 + S 2 Ax Ay Ay Ay
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 v is 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 T = Tw C = Cw
ð5Þ
yY∞ : u→0 T→T∞ C→C∞
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
We introduce the similarity variables η=
y 1=2 ψ T − T∞ C − C∞ Ra ; f ðηÞ = ; θ= ; /= x Tw − T∞ Cw − C∞ αrRa1 = 2
ð8Þ
1022
C.-Y. Cheng / International Communications in Heat and Mass Transfer 36 (2009) 1020–1024
where ψ is the stream function defined as: u=
1 Aψ 1 Aψ ; v= − r Ay r Ax
ð9Þ
and the Rayleigh number is given by Ra =
gKβT xðTw − T∞ Þ cos A rα
ð10Þ
Upon using these variables, the basic equations of the boundary layer for the problem under consideration can be written in nondimensional form as f V= θ + N/ θW +
ð11Þ
3 f θ V + D/ W = 0 2
ð12Þ
/W 3 + f /V+ SθW = 0 Le 2
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 a vertical cone of Newtonian fluids in porous media with constant wall temperature and concentration. 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 Le = 2 and N = 1. Increasing the Dufour parameter tends to thicken thermal boundary layer, thus decreasing the heat transfer rate at the wall, as shown in Fig. 2(a). Moreover, Fig. 2(b) shows that, as the Soret parameter is increased,
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 =
α β ðC − C∞ Þ DðCw − C∞ Þ SðTw − T∞ Þ ; D= ; S= ; N= C w ð14Þ DM βT ðTw − T∞ Þ α ðTw − T∞ Þ α ðCw − C∞ Þ
The boundary conditions are transformed to η=0: f =0 θ=1 /=1
ð15Þ
ηY∞ : f′→0 θ→0 ϕ→0
ð16Þ
The local Nusselt number can be derived as Nu = − θ Vð0Þ Ra1 = 2
ð17Þ
The local Sherwood number can be derived as Sh = − / Vð0Þ Ra1 = 2
ð18Þ
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 [19]. The velocity f′ is calculated from the momentum equation, Eq. (11). Moreover, the Simpson's rule 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 Table 1 Comparison of values of − θ′(0) and − ϕ′(0) for D = 0 and S = 0. N
4 4 4 1 1 1 0 0 0
Le
1 10 100 1 10 100 1 10 100
− θ′(0)
− ϕ′(0)
Yih [5]
Present
Yih [5]
Present
1.7186 1.1795 0.9019 1.0869 0.9030 0.8141 0.7686 0.7686 0.7686
1.7186 1.1794 0.9022 1.0870 0.9032 0.8143 0.7685 0.7685 0.7685
1.7186 5.6977 18.2208 1.0869 3.8139 12.3645 0.7686 2.9102 9.6032
1.7186 5.6949 18.1312 1.0870 3.8134 12.3377 0.7685 2.9103 9.5931
Fig. 2. The effect of the Dufour parameter and Soret parameter on (a) the temperature profile and (b) the concentration profile for N = 1 and Le = 2.
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the concentration boundary layer thickness increases, thus decreasing the mass transfer rate at the wall. Fig. 3 depicts the variation of the local Nusselt number NuRa− 0.5 and the Sherwood number ShRa− 0.5 with the Dufour parameter D for various values of buoyancy ratio N and Lewis number Le with S = 0.1. It is clearly seen that the local Nusselt number and the Sherwood number tend to increase as the buoyancy ratio increases. As the Lewis number is increased, the local Nusselt number decreases while the local Sherwood number increases. Moreover, as the Dufour number is increased, the local Nusselt number decreases while the local Sherwood number slightly increases. Comparing the curves in Fig. 3 (a), we can deduce that the effect of the Dufour parameter on the local Nusselt number becomes more significant as the Lewis number is increased. Fig. 4 shows the effect of Soret parameter S on the local Nusselt number NuRa− 0.5 and the Sherwood number ShRa− 0.5 for various values of buoyancy ratio N and Lewis number Le with D = 0.1. In-
creasing the buoyancy ratio accelerates the flow, decreasing the thermal and concentration boundary layer thickness and thus increasing the heat and mass transfer rates between the fluid and the wall. Moreover, as the Soret number is increased, the local Sherwood number decreases while the local Nusselt number increases. Comparing the curves in Fig. 4(a), we can deduce that the effect of the Soret parameter on the local Nusselt number is more pronounced as the Lewis number is increased.
Fig. 3. The effect of Dufour parameter on (a) the local Nusselt number and (b) the Sherwood number for various values of buoyancy ratio and Lewis number with S = 0.1.
Fig. 4. The effect of Soret parameter on (a) the local Nusselt number and (b) the Sherwood number for various values of buoyancy ratio and Lewis number with D = 0.1.
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 downward-pointing 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
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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 number is increased, the local Nusselt number decreases while the local Sherwood number slightly increases. An increase in the Soret number tends to decrease the local Sherwood number and to increase the local Nusselt number.
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