Transp Porous Med (2015) 109:279–295 DOI 10.1007/s11242-015-0517-6
Mixed Convection Heat and Mass Transfer from a Vertical Surface Embedded in a Porous Medium Alin V. Ro¸sca1 · Natalia C. Ro¸sca2 · Ioan Pop2
Received: 13 February 2015 / Accepted: 15 May 2015 / Published online: 2 June 2015 © Springer Science+Business Media Dordrecht 2015
Abstract An analysis is performed to study the heat and mass transfer characteristics of mixed convection flow along a vertical flat plate embedded in a fluid-saturated porous medium under the combined buoyancy effects of thermal and mass diffusion. The plate is impermeable and is maintained at a uniform temperature/concentration. The boundary-layer partial differential equations are transformed into a set of nonlinear ordinary differential equations using a similarity transformation and then are solved numerically using the function bvp4c from MATLAB for different values of the governing parameters. Results show that dual solutions exist for a certain range of these parameters. Very interesting analytical solutions have also been included in the paper. The results have been compared with the ones from the open literature. Keywords transfer
Mixed convection · Boundary layer · Vertical surface · Porous medium · Mass
1 Introduction Convective heat transfer from surfaces embedded in porous media has been the topic of many studies in recent years. The interest for such studies is motivated by a wide range of thermal engineering applications such as geothermal systems, oil extractions, groundwater pollution, thermal insulation, solid matrix heat exchangers, cooling of nuclear reactors, grain storage installations, the storage of nuclear wastes, volcanic eruption, electronic circuits, see, for example, the books by Nakayama (1995); Nield and Bejan (2013); Pop and Ingham (2001); Ingham and Pop (2005); Vafai (2005, 2010) and Vadasz (2008). There are many
B
Ioan Pop
[email protected]
1
Department of Statistics–Forecasts–Mathematics, Faculty of Economics and Business Administration, Babe¸s-Bolyai University, 400084 Cluj-Napoca, Romania
2
Department of Mathematics, Faculty of Mathematics and Computer Science, Babe¸s-Bolyai University, 400084 Cluj-Napoca, Romania
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transport processes in industry and in the environment in which buoyancy forces arise from both thermal and mass diffusion as a result of the coexistence of temperature gradients and concentration differences of dissimilar chemical species. The problems of combined buoyancy modes of thermal and mass diffusion have been studied rather extensively for laminar free convection flow along a vertical flat plate embedded in a fluid-saturated porous medium (see, for example, the book by Nield and Bejan 2013). The analysis, in general, has been based on mass diffusion processes for which very low concentration level exists, such that the diffusion-thermo and thermo-diffusion effects, along with the interfacial velocities from mass diffusion at the surface, are neglected. In their analysis, Bejan and Khair (1985) have reported a fundamental study of the phenomenon of natural convection heat and mass transfer near a vertical surface embedded in a fluid-saturated porous medium. The buoyancy effect is due to the variation of temperature and concentration across the boundary layer. A survey of the literature reveals that the mixed convection flow of the combined effects of buoyancy forces from thermal and mass diffusion near a vertical surface embedded in a fluidsaturated porous medium has not been studied. Many such problems exist in engineering and environmental processes. On the other hand, it is worth mentioning to this end that Merkin (1980) has studied in a pioneering paper the problem of steady mixed convection flow past an impermeable vertical flat plate embedded in a porous medium, where the plate is assumed to be at a constant heat flux or at a constant temperature, respectively. Dual (upper and lower branch) solutions are found for the case when the flow is opposing (cooled plate). These dual solutions have been discussed by Merkin (1985) in his second paper, where he showed that the dual solutions occur in a determined range of the mixed convection parameter. The case of the steady mixed convection flow past a vertical flat plate embedded in a porous medium has been also studied by Aly et al. (2003) for the case when the temperature of the plate is variable. Ahmad and Pop (2010) studied the steady mixed convection boundarylayer flow past a vertical flat plate embedded in a porous medium filled with nanofluids using different types of nanoparticles such as Cu (copper), Al2 O3 (aluminum) and TiO2 (titanium). Ahmad and Pop (2014) have also very recently studied the effect of melting phenomenon on the steady mixed convection boundary-layer flow about a vertical surface embedded in a fluid-saturated porous medium. Therefore, the aim of this paper is to study the heat and mass transfer characteristics of mixed convection flow along a vertical flat plate embedded in a porous medium under the combined buoyancy effects of thermal and mass diffusion. The boundary-layer partial differential equations are transformed into a set of nonlinear ordinary differential equations using similarity transformations and then are solved numerically using the function bvp4c from MATLAB for different values of the governing parameters. In our opinion, the problem is original and the results are new. It should be, however, said that the present paper is an extension of the classical work of Bejan and Khair (1985) from natural to the mixed convection resulting in the occurence of a main velocity component U∞ = 0 (mixed convection flow) instead of U∞ = 0 (free convection flow).
2 Mathematical Model We consider the steady mixed convection flow near a vertical impermeable surface embedded in a porous medium saturated with fluid over which is flowing a uniform free stream U∞ as shown in Fig. 1, where x and y are the Cartesian coordinates measured along the plate and normal to it, respectively. It is assumed that the surface is maintained at a constant temperature Tw , while the temperature of the ambient (inviscid) fluid is T∞ , where Tw > T∞
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Mixed Convection Heat and Mass Transfer from a Vertical...
x
Tw
T
281
x
g
u
Tw
u
T
T T
u
u
v
y
0
0
y
U
U
(a)
(b)
Fig. 1 Physical model and coordinate system. a Assisting flow, b opposing flow
(heated plate) corresponds to the assisting flow and Tw < T∞ (cooled plate) corresponds to the opposing flow. In addition, the concentration of a certain constituent C in the solution that saturates the porous medium varies from Cw on the fluid side of the vertical surface to C∞ , where we assume that Cw > C∞ . Under these assumptions along with the Boussinesq and boundary-layer approximations, the basic equations are (see Bejan and Khair 1985) ∂u ∂v + = 0, ∂x ∂y u = U∞ +
gKρ [βT (T − T∞ ) + βC (C − C∞ )] , μ ∂2T ∂T ∂T +v = αm 2 , u ∂x ∂y ∂y ∂C ∂C ∂ 2C u +v =D 2, ∂x ∂y ∂y
(1) (2) (3) (4)
subject to the boundary conditions v = 0, T = Tw , C = Cw at y = 0, u → U∞ , T → T∞ , C → C∞ as y → ∞,
(5)
where u and v are the velocity components along x and y axes, T is the fluid temperature, g is the acceleration due to gravity, αm is the thermal diffusivity of the porous medium, μ is the dynamic viscosity, ρ is the fluid density, βT and βC are the volumetric coefficients of thermal and concentration expansion, D is the coefficient of diffusion in the mixture, and K is the permeability of the porous medium.
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3 Solution Following Merkin (1980), we look for a similarity solution of Eqs. (1)–(4) of the form U∞ T − T∞ C − C∞ 1/2 , , φ(η) = , η=y ψ = (2αm U∞ x) f (η) , θ (η) = Tw − T∞ Cw − C∞ 2αm x (6) where ψ is the stream function, defined by u = ∂ψ/∂ y and v = −∂ψ/∂ x. As has been suggested by a very competent reviewer of this paper, these similarity variables imply two mixed convection parameters λ ≡ λT and N λ ≡ λC , respectively, defined as λT =
gKβT T gKβC C , λC = , νU∞ νU∞
(7)
where T = Tw − T∞ , C = Cw − C∞ , and the buoyancy ratio parameter N , which describes the competition between thermal and concentration buoyancy forces, is defined as N=
βC C . βT T
(8)
This parameter can take positive or negative values because βC can be positive or negative (see Bejan and Khair 1985). However, the case λC = N λ excludes the buoyancy ratio parameter N from the governing similarity equations. Therefore, we will further study the present problem by considering only the parameters λT , N and Le. Thus, substituting (6) into Eqs. (2)–(4), we get the following ordinary (similarity) differential equations f = 1 + (θ + N φ)λT ,
(9)
θ + f θ = 0,
(10)
φ + Le f φ = 0,
(11)
together with the boundary conditions f (0) = 0, θ (0) = 1, φ(0) = 1, θ (η) → 0, φ(η) → 0 as η → ∞.
(12)
Here, prime denotes differentiation with respect to η and Le = αm /D is the Lewis number. We notice that λT > 0 corresponds to an assisting flow (heated plate), λT < 0 corresponds to an opposing flow (cooled plate), and λT = 0 corresponds to a forced convection flow. Quantities of physical interest in this problem are the skin friction coefficient C f , the local Nusselt N u x , and the local Sherwood Sh x numbers, which are defined as ∂T ∂u μ x Cf = − , N u = , x 2 ρU∞ ∂ y y=0 (Tw − T∞ ) ∂ y y=0 x ∂C Sh x = . (13) − (Cw − C∞ ) ∂ y y=0 Using (6) and (13), we get 1 1/2 Pex / Pr C f = √ f (0), 2
−1/2
Pex
1 N u x = − √ θ (0), 2
−1/2
Pex
1 Sh x = − √ φ (0), 2 (14)
where Pr = ν/αm is the Prandtl number for a porous medium. Further, we are grateful to an anonymous reviewer for suggesting that there are several specific cases to be considered.
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3.1 The Forced Convection-Governed Regime |λT | << 1 It is formally obtained by letting in the governing equations λT → 0, i.e., U∞ → ∞. Thus, Eq. (9) reduces to f (η) = 1 which admits the solution f (η) = η. Equations (10) and (11) along with the boundary conditions (12) have then the closed analytical solution 1 Le θ (η) = erfc η , φ(η) = erfc η , (15) 2 2 and thus
2 2Le , φ (0) = − , θ (0) = − π π
(16)
where erfc(ξ ) is the complementary error function.
3.2 Case Le = 1 with N = −1 In this case, the two diffusivities are equal, αm = D, and Eqs. (10) and (11) along with the boundary conditions (12) imply that θ (η) = φ(η). Thus, f (η) is obtained as solution of the boundary-value problem f + f f = 0, f (0) = 0,
f (0) = 1 + (1 + N )λT
f (∞) = 1.
(17)
Once this problem is solved, the corresponding temperature and concentration fields θ (η) and φ(η) are obtained (for N = −1) as θ (η) = φ(η) =
f (η) − 1 . λ(1 + N )
(18)
This problem coincides exactly with the classical boundary-value problem (8, 9) solved by Merkin (1980) for f (0) = 1 + ε, which in the present notation means ε ≡ (1 + N )λT . The same problem has been discussed by Merkin (1985) in more detail in terms of a parameter α which here corresponds to notation α ≡ −1 − (1+ N )λT . Therefore, in case Le = 1 (equal diffusivities), all the results of Merkin (1980, 1985) can easily be transcribed for the present case by replacing Merkin’s ε and α by (1 + N )λT and −1 − (1 + N )λT , respectively.
3.3 Case Le = 1, λ = −1/(N + 1) with N = −1 In this case, the boundary-value problem (17) reduces to f + f f = 0, f (0) = 0,
f (0) = 0,
f (∞) = 1,
(19)
which is precisely the classical Blasius (1908) problem. As it is well known, the solution of f = f (η) of this boundary-value problem corresponds to f (0) = 0.4696 and is unique. The corresponding temperature and concentration fields θ (η) and φ(η) are given by
and thus
θ (η) = φ(η) = 1 − f (η),
(20)
θ (0) = φ (0) = − f (0) = −0.4696.
(21)
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3.4 Case Le = 1 with N = −1 When, in addition to the equality of diffusivities, αm = D (i.e., Le = 1), which implies θ (η) = φ(η), the thermal and concentration buoyancy forces are equal and opposite, i.e., N = −1; the boundary-value problem admits the exact solution f (η) = η. In this case, the above Eqs. (10) and (11) can also be solved exactly, yielding 1 1 θ (η) = φ(η) = erfc (22) η , θ (0) = φ (0) = − √ . 2 π Further, we notice that when N = 0 (concentration is absent), Eqs. (9) and (10) can be combined into the equation f + f f = 0, (23) subject to the boundary conditions f (0) = 1 + λ,
f (0) = 0,
f (η) → 1
η → ∞.
(24)
Equations (23) and (24) are the same with Eqs. (8) and (9) from the paper by Merkin (1980).
4 Results and Discussion Equations (9)–(11) subject to boundary conditions (12) were solved numerically using the function bvp4c from MATLAB for selected values of the mixed convection parameter λT and the buoyancy parameter N , when the values of the Lewis number Le are Le = 0.1 (strong concentration diffusivity), Le = 1 (equal diffusivities) and Le = 10 (strong thermal diffusivity). The bvp4c is a finite difference method code that implements the three-stage Lobatto IIIa formula, which is a collocation method with fourth-order accuracy. The differential equations are first reduced to a system of first-order equations by introducing new variables. The mesh selection and error control are based on the residual of the continuous solution. We set here the relative error tolerance to 10−10 . In this method, we have chosen a suitable finite value of η → ∞, namely η = η∞ = 20 for the upper-branch solution and η = η∞ = 180 for the lower-branch solution. The numerical method used is very well described in the papers by Ro¸sca and Pop (2013a) and Ro¸sca and Pop (2013b). It is found that dual (upper and lower branch) solutions exist for Eqs. (9)–(11) with the boundary conditions (12) when the flow is opposing (λT < 0). By comparing the present results with those reported by Merkin (1980) when N = 0 (concentration is absent), it follows that our results are in very good agreement. Thus, we are deeply confident that the present results are correct. Further, we convert Eqs. (7)–(9) to the form presented in the paper by Bejan and Khair (1985) by considering the following transformations √ 1 f (η) = − √ F(z), θ (η) = (z), φ(η) = (z), z = 2η. 2
(25)
Thus, Eqs. (9)–(11) can be written as F˙ + 1 + ( + N )λT = 0, 1 ˙ = 0, ¨ − F 2 Le ¨ − ˙ = 0, F 2
123
(26) (27) (28)
Mixed Convection Heat and Mass Transfer from a Vertical...
285
subject to the boundary conditions F(0) = 0, (0) = 1, (0) = 1, (z) → 0, (z) → 0 as z → ∞,
(29)
where dots denote differentiation with respect to z. Thus, from (14), we have −1/2
Pex
−1/2
˙ N u x = −(0),
Pex
˙ Sh x = −(0).
(30)
We notice that for Le = 1, it results in from Eqs. (27) and (28) that = , so that for N = −1, the solution of the boundary-value problem (26–29) is given by z F(z) = −z, (z) = (z) = erfc . (31) 2 Thus, using (30) and (31), we get −1/2
Pex
−1/2
N u x = Pex
1 Sh x = √ , π
(32)
for Le = 1 and N = −1. It should be mentioned that the rescaled equations (26)–(29) have only been used for the data of Table 1, but all the profiles of Figs. 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 and 16 are based on the solutions of the original form of Eqs. (9)–(12) of the boundary-value problem. The next results are given for the variations of the reduced skin friction f (0), reduced local Nusselt −θ (0) and local Sherwood −φ (0) numbers with the mixed convection parameter λT or buoyancy parameter N , which are presented in Figs. 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 and 13. All these figures show that dual (upper and lower branch) solutions exist for certain chosen parameters. The solid lines show the upper-branch solutions, while the dot lines represent the lower-branch solutions, respectively. The two branches solutions merge with one another at a critical point λT = λT c < 0 and N = Nc > 0, respectively. It should be stated that for λT < λT c < 0 and N > Nc > 0, the full Darcy, energy and concentration equations have to be solved. We can see from Fig. 2 for f (0) that the value of λT c is λT c = −1.3541 when N = 0, which is in perfect agreement with that reported by Merkin (1980). It suggests again that the present numerical method works very efficient also for this problem. It can be seen from Figs. 3 and 4 that for the upper-branch solution, −θ (0) and −φ (0) increase with the buoyancy ratio parameter N > 0 when λT > 0 (assisting flow), while for λT < 0 (opposing flow), they decrease with N > 0. Further, we notice from Fig. 6 that when N = 0.1 (heat transfer-governed flow), −θ (0) decreases with the above-considered values of Le when λT > 0 and increases with Le when λT < 0. On the other hand, we observe from Fig. 7 that −φ (0) increases with Le for both λT > 0 and λT < 0. The same behavior happens in Figs. 9 and 10 when N = 10 (mass transfer-governed flow). One can also see from Figs. 11, 12 and 13 that the critical values of Nc decrease with |λT |. As an anonymous reviewer pointed out, the forced convection case λT = 0 gives f (η) = η and it corresponds in Figs. 2, 5 and 8 to the intersection of in
all the curves
the √ point λT , f (0) = (0, 0) and in Figs. 3, 6 and 9 in the point λT , −θ (0) = 0, 2/π , for all the corresponding parameters N and Le. The point of the curves in
√intersection Fig. 4 is, for Le = 10, at point λT , −φ (0) = 0, 20/π ≈ (0, 2.52). The situation √ is different in Figs. 7 and 10 because the value of −φ (0) = 2Le/π depends on Le, so that the curves associated with different values of Le do not intersect each other for λT = 0. In these Figs. 7 and 10,√the values of −φ (0) at λT = 0 for the considered values Le = 0.1, 1, 10 are −φ (0) = 2Le/π ≈ 0.25, 0.8, 2.52. However, it should be noticed
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A. V. Ro¸sca et al. −1/2
Table 1 Values of Pex λT
N
Le
−1/2
N u x and Pex
Mixed convection present results −1/2
Pex −2
−5
−2
−1
−2
0.1
1
−1/2
Pex
Sh x
−1/2
Pex
N ux
−1/2
Pex
1.380081
1.380081
0.888
0.888 –
4
1.022619
2.656240
–
0.731963
4.121138
–
–
100
0.183460
12.546231
–
–
1
0.847579
0.847579
0.444
0.444
4
0.637410
1.537162
–
–
10
0.447024
2.288210
–
–
100
0.183387
6.398758
–
–
1
0.720584
0.720584
0.444
0.444
4
0.613955
1.358488
–
–
10
0.540017
2.086637
–
–
100
0.410032
6.365588
–
1
0.296373
0.296373
0.465
0.465
4
0.303687
0.456476
–
–
10
0.308319
0.592387
–
–
0.959017
–
–
−1
1
0.317025 √ 1/ π = 0.564189
0.1
1
0.734310
0.734310
0.465
0.465
4
0.729655
1.541995
–
–
10
0.726978
2.484766
–
–
100
0.722883
8.028849
–
–
1
0.847579
0.847579
0.628
0.628 1.358
1
4
1.5
N ux
Free convection results by Bejan and Khair (1985)
10
100 1
Sh x for several values of λT , N and Le
−1.1
0.1
1
123
No flow
4
0.804524
1.762062
0.559
10
0.779439
2.828165
0.521
2.202
100
0.741478
9.092659
0.470
7.139
1
1.145597
1.145597
0.992
0.992
4
1.002205
2.345433
0.798
2.055
10
0.915882
3.742201
0.681
3.290
100
0.787910
11.944965
0.521
10.521
1
0.536518
0.536518
0.144
0.144
4
0.623487
1.202633
–
–
10
0.671124
1.984829
–
–
100
0.744479
6.603549
–
–
1
0.805485
0.805485
0.465
0.465
4
0.798993
1.712704
–
–
10
0.795324
2.771032
–
–
100
0.789798
8.989954
–
–
1
0.957475
0.957475
0.628
0.628
4
0.898563
2.005777
0.559
1.358
Sh x
Mixed Convection Heat and Mass Transfer from a Vertical...
287
Table 1 continued λT
N
Le
Mixed convection present results −1/2
Pex
4
N ux
−1/2
Pex
Free convection results by Bejan and Khair (1985)
Sh x
−1/2
Pex
−1/2
N ux
Pex
10
0.864647
3.227220
0.521
2.202
100
0.814136
10.398663
0.470
7.139 0.992
1
1.343866
1.343866
0.992
4
1.153858
2.758947
0.798
2.055
10
1.039647
4.405886
0.681
3.290
100
0.872874
14.072240
0.521
10.521
Sh x
1 λ
Tc
Upper solution branch
λ 0.5
= −1.1099
Tc
= − 1.2130 Lower solution branch (0, 0) N = 0, 0.1, 0.15, 0.19
f ’’ (0)
0
N = 0, 0.1, 0.15, 0.19
λ
−0.5
Tc
−1
−1.5 −1.5
λ
Tc
= − 1.1533
= − 1.3541Merkin (1980) Present study
−1
−0.5
0
0.5
1
λ
T
Fig. 2 Variation of f (0) with λT for several values of N when Le = 10
√ that λT , −φ (0) = 0, 2Le/π holds regardless the values of N (compare Figs. 7, 8, 9 and 10). By performing a stability analysis, one can show that the upper-branch solutions are stable and physically realizable, while the lower-branch solutions are unstable and, therefore, not physically realizable, but this is beyond the scope of this paper. It is worth mentioning that Merkin (1980) has performed the corresponding stability analysis of these dual solutions for the case when N = 0 (concentration is absent), so that we will not present it here. The velocity f (η), temperature θ (η) and concentration φ(η) profiles are shown in Figs. 14, 15 and 16 for several values of the parameter N when λT = −1.1 (opposing flow). Dual solutions of these profiles are also observed in these figures, and they satisfy the far-field boundary conditions (12) asymptotically, which support the validity of the numerical results obtained. On the other hand, we observe that for a given value of N , the boundary-
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A. V. Ro¸sca et al. 1.4
1.2
1 λ − θ ’ (0)
Tc
0.8
= − 1.3541 Merkin (1980) Present study
N = 0, 0.1, 0.15, 0.19
N = 0, 0.1, 0.15, 0.19 λ
0.6
Tc
λ
Tc
= − 1.1533
= − 1.2130
0.4 Upper solution branch 0.2
Lower solution branch
λ
Tc
0 −1.5
= −1.1099
−1
−0.5
0
0.5
1
λ
T
Fig. 3 Variation of −θ (0) with λT for several values of N when Le = 10 4 3.5 3
φ ’ (0)
2.5
N = 0.1, 0.15, 0.19
N = 0.1, 0.15, 0.19
2 λ
1.5 1
Tc
λ
Tc
= − 1.1533
= − 1.2130
0.5 Upper solution branch 0 λ
Tc
−0.5 −1.5
−1
Lower solution branch
= −1.1099 −0.5
λ
0
0.5
1
T
Fig. 4 Variation of −φ (0) with λT for several values of N when Le = 10
layer thicknesses for the lower-branch solutions are larger than those of the upper-branch solutions. In Table 1, we present several values of the local Nusselt and Sherwood numbers −1/2 −1/2 Pex N u x and Pex Sh x given by (30) for some values of the parameters λT , N and Le. Comparing the present results for the mixed convection flow with those by Bejan and
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289
1 λ
Tc
λ
Tc
0.5
= − 1.1533
= − 1.1774
Upper solution branch Lower solution branch (0, 0)
0
f ’’ (0)
Le = 0.1, 1, 10
Le = 0.1, 1, 10
−0.5 λ
Tc
= − 1.1626
−1
−1.5
−2 −1.5
−1
−0.5
λ
0
0.5
1
T
Fig. 5 Variation of f (0) with λT for several values of Le when N = 0.15 (small) 1.4
1.2
Le = 0.1, 1, 10
− θ ’ (0)
1
0.8
0.6 λ
Tc
= − 1.1774 Le = 0.1, 1, 10
0.4
Upper solution branch λ
0.2
Tc
λ
0 Tc −1.5
= − 1.1533
Lower solution branch
= − 1.1626
−1
−0.5
λ
0
0.5
1
T
Fig. 6 Variation of −θ (0) with λT for several values of Le when N = 0.15 (small)
Khair (1985) for the case of free convection, it can be said that the physical effect of the −1/2 −1/2 driving external stream on Pex N u x and Pex Sh x increases for λT < 0 (oppos−1/2 −1/2 N u x and Pex Sh x increase when ing flow) and |N | > 1. On the other hand, Pex λT > 0 for any values of N comparing with those of Bejan and Khair (1985). However, −1/2 −1/2 N u x = Pex Sh x for both the mixed and free convection flows when Le = 1 Pex
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A. V. Ro¸sca et al. 4 Upper solution branch
3.5
Lower solution branch 3 2.5
λ
− φ ’ (0)
Tc
= − 1.1626
2 1.5
λ
Tc
= − 1.1774
1 0.5 0 λ = − 1.1533 Tc −0.5 −1.5
Le = 0.1, 1, 10 −1
−0.5
0
λ
0.5
1
T
Fig. 7 Variation of −φ (0) with λT for several values of Le when N = 0.15 (small) 5 4
λ
Tc
= − 0.1219
Upper solution branch
3
Lower solution branch
λ
2
Tc
= − 0.1231
f ’’ (0)
1
Le = 0.1, 1, 10
(0, 0)
0 −1
Le = 0.1, 1, 10
−2 −3 −4
λ
Tc
−5 −0.2
= − 0.1213
−0.1
0
0.1
0.2
λ
0.3
0.4
0.5
0.6
0.7
T
Fig. 8 Variation of f (0) with λT for several values of Le when N = 10 (large)
(equal diffusivities). Further, it should be noticed that the case Le = 1 implies that = . This, along with N = −1 and λT = −1 (which holds also for λT = −1), gives the relation √ −1/2 −1/2 (32), that is Pex N u x = Pex Sh x = 1/ π = 0.564189. This implies in turn that the thermal and concentration buoyancy forces are equal and opposite, so that they compensate each other. Therefore, there remains active only the driving effect of the external
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291
2 Le = 0.1, 1, 10
λ
1.5
Tc
λ
Tc
= − 0.1219
= − 0.1231
− θ ’ (0)
1
0.5 Le = 0.1, 1, 10 Upper solution branch
0
Lower solution branch λ
Tc
−0.5 −0.2
= − 0.1213 0
0.2
0.4
0.6
0.8
1
λ
T
Fig. 9 Variation of −θ (0) with λT for several values of Le when N = 10 (large) 6 5 λ
Tc
= − 0.1219
4
− φ ’ (0)
3 λ
2
Tc
= − 0.1231
1 0 Upper solution branch Le = 0.1, 1, 10
−1 λ
Tc
−2 −0.2
Lower solution branch
= − 0.1213 0
0.2
0.4
0.6
0.8
1
λ
T
Fig. 10 Variation of −φ (0) with λT for several values of Le when N = 10 (large)
stream, which gives rise to a “quasi-forced convection flow.” The corresponding solution of the boundary-value problem (26–29) is given by (32), and it proves explicitly that in the present mixed convection case, there exists a quasi-forced convection flow, while in the case of the free convection flow studied by Bejan and Khair (1985), where no external stream exists, the statement “no flow” is correct.
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A. V. Ro¸sca et al. 1 N = 0.0357
N = 0.1104
c
c
0.5
f ’’ (0)
0 λ = −1.1, −1.2, −1.3 T
N = 0.1997
−0.5
c
λ = −1.1, −1.2, −1.3 T
−1 Upper solution branch −1.5
−2 −1
Lower solution branch
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
N
Fig. 11 Variation of f (0) with N for several values of λT (< 0) opposing flow when Le = 10 1 Upper solution branch 0.8
Lower solution branch λ = −1.1, −1.2, −1.3 T
− θ ’ (0)
0.6 N = 0.1104 c
0.4
N = 0.1997 c
0.2 N = 0.0357 c
0 −1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
N
Fig. 12 Variation of −θ (0) with N for several values of λT (< 0) opposing flow when Le = 10
5 Conclusion In this paper, the well-known classical work of Bejan and Khair (1985) on free convection boundary-layer flow and mass transfer from a vertical flat plate embedded in a fluid-saturated porous medium has been extended to the case of a mixed convection flow. The extension consists of imposing the driving effect of a uniform stream of velocity U∞ along the plate. By
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293
2.5
λ = −1.1, −1.2, −1.3
2
T
− φ ’ (0)
1.5
N = 0.1104
1
c
N = 0.1997 c
0.5
Upper solution branch
0
Lower solution branch
N = 0.0357 c
−0.5 −1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
N Fig. 13 Variation of −φ (0) with N for several values of λT (< 0) opposing flow when Le = 10 1 N = 0.1, 0.15, 0.19
0.8 Upper solution branch Lower solution branch
f ’ ( η)
0.6
0.4 N = 0.1, 0.15, 0.19
0.2
0
−0.2
−0.4 0
2
4
6
8
10
12
14
16
18
20
η Fig. 14 Velocity profiles f (η) for different values of N when λT = −1.1 and Le = 10
means of appropriate similarity transformations, the partial differential equations describing the problem are transformed into a system of ordinary (similarity) differential equations. For solving numerically the similarity Eqs. (9)–(11) subject to the boundary conditions (12), we used the bvp4c function from MATLAB, which is very well described in the papers by Ro¸sca and Pop (2013a) and Ro¸sca and Pop (2013b). Some closed-form analytical solutions of these similarity equations are obtained for several particular values of the governing parameters.
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A. V. Ro¸sca et al. 1
Upper solution branch
0.8 N = 0.1, 0.15, 0.19
Lower solution branch
θ (η)
0.6
0.4
0.2
N = 0.1, 0.15, 0.19
0 0
2
4
6
8
10
12
14
16
18
20
η
Fig. 15 Temperature profiles θ (η) for different values of N when λT = −1.1 and Le = 10 1 Upper solution branch 0.8
Lower solution branch
N = 0.1, 0.15, 0.19
φ (η )
0.6
0.4
0.2
N = 0.1, 0.15, 0.19
0 0
2
4
6
8
10
12
14
η
Fig. 16 Concentration profiles φ(η) for different values of N when λT = −1.1 and Le = 10
Results show that dual solutions exist for a certain range of the parameters λT and N , which had not been reported before. The presented results show that the driving mechanism of the uniform stream has a significant effect on the flow, heat and mass transfer characteristics. Acknowledgments The authors would like to express their very sincere thanks to the very competent two reviewers for the very good comments and suggestions, which led to a substantial improvement of the paper.
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References Ahmad, S., Pop, I.: Mixed convection boundary layer flow from a vertical flat plate embedded in a porous medium filled with nanofluids. Int. Comm. Heat Mass Transf. 37, 987–991 (2010) Ahmad, S., Pop, I.: Melting effect on mixed convection boundary layer flow about a vertical surface embedded in a porous medium: opposing flows case. Transp. Porous Media 102, 317–323 (2014) Aly, E.H., Elliott, L., Ingham, D.B.: Mixed convection boundary-layer flow over a vertical surface embedded in a porous medium. Eur. J. Mech. B -Fluids. 22, 529–543 (2003) Bejan, A., Khair, R.K.: Heat and mass transfer by natural convection in a porous medium. Int. J. Heat Mass Transf. 28, 909–918 (1985) Blasius, H.: Grenzschichten in Flüssigkeiten mit kleiner Reibung. Z. Math. Phys. 56, 1–37 (1908) Ingham, D.B., Pop, I. (eds.): Transport Phenomena in Porous Media, vol. III. Elsevier, Oxford (2005) Merkin, J.H.: Mixed convection boundary layer flow on a vertical surface in a saturated porous medium. J. Eng. Math. 14, 301–313 (1980) Merkin, J.H.: On dual solutions occurring in mixed convection in a porous medium. J. Eng. Math. 20, 171–179 (1985) Nakayama, A.: PC-Aided Numerical Heat Transfer and Convective Flow. CRC Press, Tokyo (1995) Nield, D.A., Bejan, A.: Convection in Porous Media, 4th edn. Springer, New York (2013) Pop, I., Ingham, D.B.: Convective Heat Transfer: Mathematical and Computational Modeling of Viscous Fluids and Porous Media. Pergamon, Oxford (2001) Ro¸sca, A.V., Pop, I.: Flow and heat transfer over a vertical permeable stretching/shrinking sheet with a second order slip. Int. J. Heat Mass Transf. 60, 355–364 (2013a) Ro¸sca, N.C., Pop, I.: Mixed convection stagnation point flow past a vertical flat plate with a second order slip: heat flux case. Int. J. Heat Mass Transf. 65, 102–109 (2013b) Vadasz, P.: Emerging Topics in Heat and Mass Transfer in Porous Media. Springer, New York (2008) Vafai, K. (ed.): Handbook of Porous Media. Taylor and Francis, New York (2005) Vafai, K.: Porous Media: Applications in Biological Systems and Biotechnology. CRC Press, Tokyo (2010)
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