Soret and Dufour effects on free convection heat and mass transfer from an arbitrarily inclined plate in a porous medium with constant wall temperature and concentration

International Communications in Heat and Mass Transfer 39 (2012) 72–77

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Soret and Dufour effects on free convection heat and mass transfer from an arbitrarily inclined plate in a porous medium with constant wall temperature and concentration ☆ 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 23 September 2011 Keywords: Arbitrarily inclined plate Porous medium Free convection Soret effect Dufour effect

a b s t r a c t The free convection boundary layer flow over an arbitrarily inclined heated plate in a porous medium with Soret and Dufour effects is studied by transforming the governing equations into a universal form. The generalized equations can be used to derive the similarity solutions for limiting cases of horizontal and vertical plates and to calculate the heat and mass transfer characteristics between these two limiting cases. The heat and mass transfer characteristics are presented as functions of Soret parameter, Dufour parameter, inclination variable, Lewis number, and buoyancy ratio. Results show that an increase in the Dufour parameter tends to decrease the local heat transfer rate, and an increase in the Soret parameter tends to decrease the local mass transfer rate. As the inclination variable increases, the local Nusselt number and the local Sherwood number decrease from their respective values for horizontal plates, reach their respective minima, and then increase to their respective values for vertical plates. The minima are where the tangential and normal components of buoyancy force are comparable. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction Natural convection heat and mass transfer in a porous medium saturated with Newtonian fluids may be met in geophysical, geothermal and industrial applications, such as the migration of moisture through air contained in fibrous insulations and the underground spreading of chemical contaminants through water-saturated soil. Cheng and Chang [1] studied the natural convection heat transfer from impermeable horizontal surfaces in a saturated porous medium. Bejan and Khair [2] examined the natural convection boundary layer flow driven by both temperature and concentration gradients. Lai and Kulacki [3] studied the natural convection boundary layer along a vertical surface with constant heat and mass flux. Nakayama and Hossain [4] presented an integral treatment for combined heat and mass transfer by natural convection in a porous medium. Pop and Na [5] studied the natural convection heat transfer from an arbitrarily inclined plate in a porous medium. Li and Lai [6] studied the double diffusive natural convection from horizontal surfaces in porous media. Cheng [7] examined 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 [8] studied the uniform transpiration effect on coupled heat and mass transfer in mixed convection about inclined surfaces in porous media for the entire regime. Yih

☆ Communicated by W.J. Minkowycz. E-mail address: [email protected]. 0735-1933/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2011.09.003

[9] studied the heat and mass transfer driven by natural convection from a truncated cone embedded in a porous medium with variable wall temperature and concentration or variable heat and mass flux. Chamkha and Khaled [10] examined the hydromagnetic simultaneous heat and mass transfer by mixed convection from a vertical plate embedded in a uniform porous medium. Murthy and Singh [11] studied the heat and mass transfer by natural convection near a vertical surface embedded in a non-Darcy porous medium. Cheng [12] presented an integral approach for heat and mass transfer by natural convection from truncated cones in porous media with variable wall temperature and concentration. Cheng [13] examined the double diffusive natural convection along an inclined wavy surface in a porous medium. 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 [14] examined the heat and mass characteristics of free 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. [15] studied the Soret and Dufour effects in a non-Darcy porous medium. Lakshmi Narayana et al. [16] examined the Soret and Dufour effects in a doubly stratified Darcy porous medium. Lakshmi Narayana and Murthy [17] studied the Soret and Dufour effects on free convection heat and mass transfer from a horizontal flat plate in a Darcy porous medium. Mahdy [18] examined the problem of MHD non-Darcian free convection from a vertical wavy surface embedded in porous media in the presence of

C.-Y. Cheng / International Communications in Heat and Mass Transfer 39 (2012) 72–77

Nomenclature A C D ¯ D DM g f K Le N Nu Ra S S¯ Sh T ¯ u; ¯ v ¯ x; ¯y

inclination angle concentration Dufour parameter Dufour coefficient mass diffusivity of the porous medium gravitational acceleration dimensionless stream function permeability of porous medium Lewis number buoyancy ratio local Nusselt number modified Rayleigh number Soret parameter Soret coefficient local Sherwood number temperature velocity components Cartesian coordinates

Greek symbols α thermal diffusivity of the porous medium βT coefficient of concentration expansion βC coefficient of thermal expansion η, ξ dimensionless coordinates θ dimensionless temperature ν kinematic viscosity ϕ dimensionless concentration ψ stream function

Subscripts w condition at wall ∞ condition at infinity

Soret and Dufour effect. Cheng [19] studied the Soret and Dufour effects on free convection boundary layer over a vertical cylinder in a saturated porous medium. Cheng [20] examined the Soret and Dufour effects on heat and mass transfer by natural convection from a vertical truncated cone in a fluid-saturated porous medium with variable wall temperature and concentration. This work aims to study the Soret and Dufour effects on free convection heat and mass transfer above an arbitrarily inclined plate in a porous medium. The surface of the inclined plate is kept at constant temperature and concentration. The governing equations are transformed into a set of coupled differential equations, and the obtained boundary layer equations are solved by the cubic spline collocation method [21]. The effects of Soret parameter, Dufour parameter, inclination, Lewis number, and buoyancy ratio on the heat and mass transfer characteristics over an arbitrarily inclined plate in a porous medium saturated with a Newtonian fluid are carefully examined. 2. Analysis Consider the boundary layer flow driven by natural convection with temperature and concentration gradients above a semi-infinite plate embedded in a porous medium saturated with a Newtonian fluid in the presence of Soret and Dufour effects. This plate is above the horizontal and is inclined at an angle A (0 ∘ ≤ A ≤ 90 ∘) to the horizontal. The surface of the inclined plate is maintained at a constant temperature Tw greater than the porous medium temperature T∞

73

sufficiently far from the inclined plate. The concentration of a certain constituent in the solution that saturates the porous medium varies from a higher concentration Cw on the fluid side of the surface of the inclined plate to a lower concentration C∞ sufficiently far from the inclined plate. Based on the boundary layer and Boussinesq approximations, we can write the governing equations for boundary layer Darcy flow by free convection of a Newtonian fluid embedded in a porous medium near an arbitrarily inclined plate with Soret and Dufour effects in two-dimensional Cartesian coordinates ( x¯; ¯yÞ as [5, 22] ∂ ¯u ∂ ¯v þ ¼0 ∂ ¯x ∂ ¯y

ð1Þ


 ∂u¯ ∂ ¯v gK ∂T ∂C ∂T ∂C βT − ¼ sinA þ βC sinA−βT cosA−βC cosA ð2Þ ν ∂ ¯y ∂ x¯ ∂ ¯y ∂ ¯y ∂ ¯x ∂ ¯x ¯u

∂T ∂T ∂2 T ∂2 C ¯ ¯ 2 þv ¼α 2þ D ∂ ¯x ∂ ¯y ∂ ¯y ∂ ¯y

ð3Þ

¯u

∂C ∂C ∂2 C ∂2 T ¯ þv ¼ DM 2 þ S¯ 2 : ∂ x¯ ∂ ¯y ∂ ¯y ∂ ¯y

ð4Þ

The boundary conditions are written as T ¼ Tw ; C ¼ Cw ;

¯v ¼ 0

on

¯y ¼ 0

ð5Þ

C→C∞ ; T→T∞ ;

¯u →0

as

¯y→∞

ð6Þ

Here ū 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. βT and βC are the coefficients for thermal expansion and for concentration expansion of the saturated porous medium, respectively. ν and K are the kinematic viscosity of the fluid and the permeability of the porous medium, 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. Here we introduce a nondimensional parameter R¼

ðRa sin AÞ1=2 ðRa cos AÞ1=3

ð7Þ

where Ra ¼ gKβT ðTw −T∞ Þ x¯=ðανÞ is the Rayleigh number. This parameter represents the relative strength of the longitudinal to normal components of the buoyancy force within the boundary layer. Here we define the nondimensional variables: θ ¼ ðT−T∞ Þ=ðTw −T∞ Þ ;

ϕ ¼ ðC−C∞ Þ=ðCw −C∞ Þ

h i η¼ð ¯ y= ¯xÞ ðRa cosAÞ1=3 þ ðRa sinAÞ1=2 ;

f ¼



¯ =α ψ

h

ξ ¼ R=ð1 þ RÞ ;

ðRa cosAÞ1=3 þ ðRa sinAÞ1=2

i−1

:

ð8Þ ¯ to satisfy the relations: Here we introduce the stream function ψ ¯ =∂ ¯y; ¯v ¼ −∂ ψ ¯ =∂ ¯x: ¯u ¼ ∂ ψ

ð9Þ

Substituting Eqs. (7)–(9) into Eqs. (1)–(4), we obtain the following equations:   1 ″ 3 ′ ′ f − ð4−ξÞð1−ξÞ η θ þ Nϕ 6     1 ∂ϕ 2 ′ ′ 4 ∂θ ¼ ξ θ þ Nϕ − ξð1−ξÞ þN 6 ∂ξ ∂ξ

ð10Þ

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C.-Y. Cheng / International Communications in Heat and Mass Transfer 39 (2012) 72–77

  1 1 ″ ′ ″ ′ ∂θ ′ ∂f θ þ ð2 þ ξÞf θ þ Dϕ ¼ ξð1−ξÞ f −θ 6 6 ∂ξ ∂ξ   ″ ϕ 1 1 ′ ″ ′ ∂ϕ ′ ∂f þ ð2 þ ξÞf ϕ þ Sθ ¼ ξð1−ξÞ f −ϕ 6 Le 6 ∂ξ ∂ξ

ð11Þ

N

ð12Þ

where prime denotes differentiation with respect to the variable η. Moreover, the Lewis number, the buoyancy ratio, the Dufour parameter, and the Soret parameter are respectively defined as Le ¼

α ; DM

N¼

βC ðCw −C∞ Þ ; βT ðTw −T∞ Þ

¯ ðC −C∞ Þ ¯ðT −T∞ Þ D¼ D w ; S¼ S w : αðTw −T∞ Þ αðCw −C∞ Þ

Table 1 Comparison of values of − θ′(ξ, 0) and − ϕ′(ξ, 0) for ξ = 1, D = 0 and S = 0.

ð13Þ

1 1 1 4 4 4

Le

1 2 4 1 2 4

− θ′(ξ, 0)

− ϕ′(ξ, 0)

Bejan and Khair [2]

Present

Bejan and Khair [2]

Present

0.628 0.593 0.559 0.992 0.899 0.798

0.6276 0.5927 0.5585 0.9923 0.8986 0.7976

0.628 0.930 1.358 0.992 1.431 2.055

0.6276 0.9295 1.3575 0.9923 1.4311 2.0550

Moreover, the local Sherwood number is Sh ′ ¼ −ð1 þ RÞ ϕ ðξ; 0Þ: ðRa cos AÞ1=3

ð25Þ

The boundary conditions are f ¼ 0; θ ¼ 1; ϕ ¼ 1 f ′ →0;

θ→0;

on η ¼ 0

ϕ→0 as

η→∞ :

ð14Þ

3. Results and discussion

ð15Þ

The transformed governing equations, Eqs. (10)–(12), and the associated boundary conditions, Eqs. (14)–(15), can be solved by the cubic spline collocation method [21]. At every position, the iteration process continues until the convergence criterion for all the variables, 10 − 6, is achieved. Variable grids 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 values of − θ′(ξ, 0) and − ϕ′(ξ, 0) for ξ = 1, D = 0 and S = 0, the conditions for heat and mass transfer from a vertical surface of Newtonian fluids in porous media, obtained by the present method and by Bejan and Khair [2], respectively. It is shown that these two results are in excellent agreement. Fig. 1 shows the effects of the Dufour parameter D and the inclination variable ξ on the temperature profile θ(ξ, η) and the concentration profile ϕ(ξ, η) near an inclined plate for Le = 3, N = 1 and S = 0.2. Increasing the Dufour parameter tends to thicken thermal boundary layer, thus decreasing the heat transfer rate at the wall, as shown in Fig. 1. The thermal boundary layer thickness at ξ = 1 is lower than that at ξ = 0. That means that the local Nusselt number at ξ = 0, which is the condition of horizontal orientation (A = 0), is lower than those at ξ = 1, which is the condition of vertical orientation (A = π/2). Fig. 2 shows the effects of the Soret parameter S and the inclination variable ξ on the concentration profile ϕ(ξ, η) near an inclined plate for Le = 3, N = 1 and D = 0.2. An increase in the Soret parameter tends to thicken concentration boundary layer, thus decreasing the

For N = 0 and D = 0, Eqs. (10)–(11) become the equations for the Darcy natural convection heat transfer near an arbitrarily inclined plate in a porous medium saturated with Newtonian fluids with constant wall temperature presented by Pop and Na [5]. 1 1 ″ 3 ′ 2 ′ 4 ∂θ f − ð4−ξÞð1−ξÞ ηθ ¼ ξ θ − ξð1−ξÞ 6 6 ∂ξ

ð16Þ

  1 1 ″ ′ ′ ∂θ ′ ∂f −θ θ þ ð2 þ ξÞf θ ¼ ξð1−ξÞ f 6 6 ∂ξ ∂ξ

ð17Þ

For ξ = 1, D = 0, and S = 0, Eqs. (10)–(12) become the equations for the Darcy natural convection heat and mass transfer from vertical surfaces in porous media with constant wall temperature and concentration presented by Bejan and Khair [2]. ″

′

′

f ¼ θ þ Nϕ

ð18Þ

1 ′ ″ θ þ fθ ¼ 0 2

ð19Þ

ϕ″ 1 ′ þ fϕ ¼ 0 Le 2

ð20Þ

1.0

For ξ = 0, D = 0, and S = 0, Eqs. (10)–(12) become the equations for the Darcy natural convection heat and mass transfer from horizontal surfaces in porous media with Newtonian fluids with constant wall temperature and concentration presented by Li and Lai [6]. ð21Þ

1 ′ ″ θ þ fθ ¼ 0 3

ð22Þ

ϕ″ 1 ′ þ fϕ ¼ 0 Le 3

ð23Þ

0.8

0.6

θ(ξ,η)

 2  ′ ″ ′ f − η θ þ Nθ ¼ 0 3

D=0.2 ξ=0 D=0.2 ξ=1 D=0.05 ξ=0 D=0.05 ξ=1 Le=3 N=1 S=0.2

0.4

0.2

0.0 0

The local Nusselt number is Nu ′ ¼ −ð1 þ RÞ θ ðξ; 0Þ: ðRa cos AÞ1=3

ð24Þ

1

2

3

η

4

5

6

7

Fig. 1. Effect of Dufour parameter and inclination variable on the temperature profile for Le = 3, N = 1 and S = 0.2.

C.-Y. Cheng / International Communications in Heat and Mass Transfer 39 (2012) 72–77

1.0 S=0.2 ξ=0 S=0.2 ξ=1 S=0.02 ξ=0 S=0.02 ξ=1 Le=3 N=1 D=0.2

0.8

φ(ξ,η)

0.6

0.4

0.2

0.0 0

1

2

3

η

4

5

Fig. 2. Effect of Soret parameter and inclination variable on the concentration profile for Le = 3, N = 1 and D = 0.2.

mass transfer rate at the wall, as shown in Fig. 2. Moreover, the concentration boundary layer thickness at ξ = 1 is lower than that at ξ = 0. That means that the local Sherwood number at ξ = 0, which is the condition of horizontal orientation (A = 0), is lower than those at ξ = 1, which is the condition of vertical orientation (A = π/2). Fig. 3 shows the variation of the local Nusselt number Na(1 + R) − 1 (Ra cos A) − 1/3 and the local Sherwood number Sh(1 + R) − 1

a

(Ra cos A) − 1/3 with the inclination variable ξ for various values of Dufour parameter D with Le=3, N=1 and S=0.2. Results show that an increase in the Dufour parameter leads to a decrease in the local Nusselt number and an increase in the local Sherwood number. As the inclination variable increases from 0 to 1, the local Nusselt number and the local Sherwood number decrease, reach their respective minima, and then increase. Note that the local Nusselt number and the local Sherwood number at ξ=0, which is the condition of horizontal orientation (A=0) are respectively lower than those at ξ=1, which is the condition of vertical orientation (A=π/2). The minima are where the tangential and normal components of buoyancy force are comparable. Fig. 4 shows the variation of the local Nusselt number Na(1 + R) − 1 (Ra cos A) − 1/3 and the local Sherwood number Sh(1 + R) − 1 (Ra cos A) − 1/3 with the inclination variable ξ for various values of Soret parameter S with Le = 3, N = 1 and D = 0.2. It is shown in Fig. 4 that an increase in the Soret parameter decreases the local Sherwood number and slightly increases the local Sherwood number. Fig. 5 plots the variation of the local Nusselt number Nu(1 + R) − 1 (Ra cos A) − 1/3 and the local Sherwood number Sh(1 + R) − 1 (Ra cos A) − 1/3 with inclination variable ξ for various values of buoyancy ratio N with Le = 3, D = 0.2 and S = 0.2. Results show that higher buoyancy ratio leads to higher flow velocities, thinning the thermal and concentration boundary layers and thus increasing the heat and mass transfer rates between the fluid and the wall. Fig. 6 plots the variation of the local Nusselt number Nu(1 + R) − 1 (Ra cos A) − 1/3 and the local Sherwood number Sh(1 + R) − 1 (Ra cos A) − 1/3 with inclination variable ξ for various values of Lewis number Le for N = 1, D = 0.2 and S = 0.2. As the Lewis number

a

0.7 D=0.1 D=0.2 D=0.3

0.6

0.6 0.5

Le=3 N=1 D=0.2

(Ra cos A)1 3

0.4

Nu (1 + R )−1

(Ra cos A)1 3

S=0.3 S=0.2 S=0.1

Le=3 N=1 S=0.2

0.5

Nu (1 + R )−1

75

0.4 0.3

0.3 0.2

0.2 0.1

0.1

0.0

0.0 0.0

0.2

0.4

0.6

0.8

0.0

1.0

0.2

0.4

b

1.5

1.0

0.9

(Ra cos A)

0.6

Le=3 N=1 D=0.2

S=0.1 S=0.2 S=0.3

1.2

Sh(1 + R )−1

(Ra cos A)

13

Sh(1 + R )−1

1.2

0.8

1.5

Le=3 N=1 S=0.2

D=0.3 D=0.2 D=0.1

13

b

0.6

ξ

ξ

0.9

0.6

0.3

0.3

0.0

0.0 0.0

0.2

0.4

0.6

0.8

1.0

ξ Fig. 3. Effects of Dufour parameter and inclination variable on (a) the local Nusselt number and (b) the local Sherwood number for Le = 3, N = 1 and S = 0.2.

0.0

0.2

0.4

0.6

0.8

1.0

ξ Fig. 4. Effects of Soret parameter and inclination variable on (a) the local Nusselt number and (b) the local Sherwood number for Le = 3, N = 1 and D = 0.2.

76

C.-Y. Cheng / International Communications in Heat and Mass Transfer 39 (2012) 72–77

a

0.8

0.6

0.6

0.5

0.5

0.4 0.3

0.4 0.3

0.2

0.2

0.1

0.1

0.0

N=1 D=0.2 S=0.2

Le=2 Le=3 Le=5

0.7

Nu (1 + R )−1

(Ra cos A)1 3

Nu (1 + R )−1

0.7

0.8

Le=3 D=0.2 S=0.2

N=3 N=2 N=1

(Ra cos A)1 3

a

0.0 0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

ξ

b

1.8

1.0

1.4

1.4

1.2

1.2

0.8 0.6

(Ra cos A)

1.0

N=1 D=0.2 S=0.2

Le=5 Le=3 Le=2

1.6

Sh(1 + R )−1

(Ra cos A)

13

Sh(1 + R )−1

1.6

0.8

1.8

Le=3 D=0.2 S=0.2

N=3 N=2 N=1

13

b

0.6

ξ

1.0 0.8 0.6

0.4

0.4

0.2

0.2 0.0

0.0 0.0

0.2

0.4

0.6

0.8

0.0

1.0

0.2

0.4

0.6

0.8

1.0

ξ

ξ Fig. 5. Effects of buoyancy ratio and inclination variable on (a) the local Nusselt number and (b) the local Sherwood number for Le = 3, D = 0.2 and S = 0.2.

Fig. 6. Effects of Lewis number and inclination variable on (a) the local Nusselt number and (b) the local Sherwood number for N = 1, D = 0.2 and S = 0.2.

is increased, the local Nusselt number decreases while the local Sherwood number increases, as shown in Fig. 6.

thermal and concentration boundary layers and thus increasing the heat and mass transfer rates between the fluid and the wall. As the Lewis number is increased, the local Nusselt number decreases while the local Sherwood number increases.

4. Conclusions The Soret and Dufour effects on free convection heat and mass transfer near an arbitrarily inclined plate in a porous medium saturated with Newtonian fluids have been studied. The inclined plate is kept at constant temperature and concentration. The problem is studied by transforming the governing equations into a universal form. The generalized equations can be used to derive the similarity solutions for limiting cases of horizontal and vertical plates and to calculate the heat and mass transfer characteristics between these two limiting cases. The heat and mass transfer characteristics are presented as functions of Soret parameter, Dufour parameter, inclination variable, Lewis number, and buoyancy ratio. Results show an increase in the Dufour parameter tends to decrease the local Nusselt number and to increase the local Sherwood number. Moreover, increasing the Soret parameter leads to a decrease in the local Sherwood number and an increase in the local Nusselt number. As the inclination variable increases, the local Nusselt number and the local Sherwood number decrease from their respective values for horizontal plates, reach their respective minima, and then increase to their respective values for vertical plates. The minima are where the tangential and normal components of buoyancy force are comparable. Moreover, higher buoyancy ratio leads to higher flow velocities, thinning the

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