Transp Porous Med (2012) 93:223–240 DOI 10.1007/s11242-012-9954-7
Entropy Generation in Double-Diffusive Convection in a Square Porous Cavity using Darcy–Brinkman Formulation Ali Mchirgui · Nejib Hidouri · Mourad Magherbi · Ammar Ben Brahim
Received: 14 October 2011 / Accepted: 30 January 2012 / Published online: 16 February 2012 © Springer Science+Business Media B.V. 2012
Abstract The article reports a numerical study of entropy generation in doublediffusive convection through a square porous cavity saturated with a binary perfect gas mixture and submitted to horizontal thermal and concentration gradients. The analysis is performed using Darcy–Brinkman formulation with the Boussinesq approximation. The set of coupled equations of mass, momentum, energy and species conservation are solved using the control volume finite-element method. Effects of the Darcy number, the porosity and the thermal porous Rayleigh number on entropy generation are studied. It was found that entropy generation considerably depends on the Darcy number. Porosity induces the increase of entropy generation, especially at higher values of thermal porous Rayleigh number. Keywords Entropy generation · Heat and mass transfer · Porous medium · Numerical method
A. Mchirgui (B) · N. Hidouri · A. B. Brahim Chemical and Process Engineering Department, Engineers National School of Gabès, Applied Thermodynamics Unit, Gabès University, Omar Ibn El Khattab Street, 6029, Gabès, Tunisia e-mail:
[email protected] N. Hidouri e-mail:
[email protected] A. B. Brahim e-mail:
[email protected] M. Magherbi Civil Engineering Department, High Institute of Applied Sciences and Technology, Gabès University, Omar Ibn El Khattab Street, 6029, Gabès, Tunisia e-mail:
[email protected]
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List of Symbols Variables C Concentration (mol m−3 ) D Molecular diffusivity (m2 s−1 ) DA Darcy number g Gravitational acceleration (m s−2 ) GrT Thermal Grashof number GrS Solutal Grashof number k Thermal conductivity (W m−1 K−1 ) K Permeability of the porous medium (m2 ) Le Lewis number N Buoyancy ratio (GrS /GrrT ) Nu Average Nusselt number P Pressure (kg m−1 s−2 ) p Dimensionless pressure R Universal gas constant (J kg−1 K−1 ) Pr Prandtl number Ra Rayleigh number Ra* Thermal porous Rayleigh number RK Thermal conductivity ratio (km /kf ) ST Dimensionless total entropy generation Sc Schmidt number Sh Sherwood number T Temperature (K) t Time (s) u, v Dimensionless velocity components U, V Velocity components along X, Y directions (m s−1 ) x, y Dimensionless coordinates X, Y Cartesian coordinates (m) W Characteristic velocity (m s−1 )
Greek Symbols α Thermal diffusivity (m2 s−1 ) αe Effective thermal diffusivity (m2 s−1 ) βT Thermal volumetric expansion coefficients (K−1 ) Solutal volumetric expansion coefficients (m3 mol−1 ) βC Viscosity ratio (μeff /μ) ε Porosity of the medium μ Fluid dynamic viscosity (kg m−1 s−1 ) μeff Viscosity in the Brinkman model (kg m−1 s−1 ) ν Kinematic viscosity (m2 s−1 ) φ Dimensionless concentration ρ Fluid density (kg m−3 ) σ Specific heat ratio [(ρc)m /(ρc) f ] θ Dimensionless temperature τ Dimensionless time
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Subscripts 0 Reference c Cold side C Solutal f Fluid h Hot side T Thermal m Porous medium
1 Introduction Thermosolutal convection, also called Double-diffusive convection occurs when a fluid is subjected to density gradients caused by local variations of temperature and concentration. Double-diffusive convection is frequently encountered in seawater flow and mantle flow, in the earth’s crust, in geophysics, in geology, as well as in many engineering applications. During the last years, convection in porous media becomes the subject of many studies. The most part of these studies have been documented by Nield and Bejan (1999). Goyeau et al. (1996) carried out a numerical study about double-diffusive convection in a porous cavity using Darcy–Brinkman formulation. They showed the influence of the Darcy number in the transport phenomena. Lauriat and Prasad (1987) investigated the buoyancy effects on natural convection in a vertical enclosure using Brinkman-extended Darcy formulation. Bennacer et al. (2001) numerically studied double-diffusive natural convection in a rectangular cavity filled with a saturated anisotropic porous medium, where the side walls are maintained at constant temperatures and concentrations, while the horizontal walls are adiabatic and impermeable. They found that the evolution of heat and mass transfers with permeability ratio can be divided into three regimes: a diffusive regime, an intermediate regime and a convective regime. Chen et al. (2010) investigated double-diffusive convection in vertical annuluses with opposing temperature and concentration gradients. They studied the influence of the buoyancy, the aspect and the radius ratios on the convectional patterns. Kramer et al. (2007) used the boundary domain integral method to study double-diffusive natural convection in porous media. Khadiri et al. (2011) numerically studied thermosolutal natural convection through homogeneous and isotropic porous media, saturated with a binary fluid in two- and three-dimensional approximations. The critical Rayleigh number at the onset of the natural convection in anisotropic horizontal porous layers with high porosity was determined by Shiina and Hishida (2010). They showed that the critical Rayleigh number decreases with the increase of the Darcy number and inversely, with the decrease of the effective thermal diffusivity ratio. Barletta and Nield (2011) studied the linear stability analysis of the basic horizontal flow in a porous layer taking into account the effects of both viscous dissipation and mass diffusion. Hooman and Gurgenci (2007) numerically studied forced convection with viscous dissipation in a parallel plate channel filled with a saturated porous medium. Three different viscous dissipation models are examined. Cheng (2011) studied Soret and Dufour effects on the boundary layer flow due to natural convection over a vertical cone in a fluid-saturated porous medium. He investigated the effects of the Dufour parameter, the Soret parameter, the Lewis number and the buoyancy ratio on heat and mass transfer. The study of nanofluid double-diffusive convection was carried out by Nield and Kuznetsov (2011) and by Kuznetsov and Nield (2008, 2010, 2011).
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Recently, many researchers are attempted to study an aspect of great importance in order to predict the performance of engineering processes: the second law of thermodynamics. In fact, the second law is applied to investigate the irreversibilities in terms of entropy generation. Entropy generation is a measure of the degraded energy of any given industrial system, it is the basis of most formulations of both equilibrium and non-equilibrium thermodynamics. Convection includes phenomena that induce a non-equilibrium state, which means that a given evolution is described as irreversible. Many researches studied entropy generation and its localization. Baytas (1997, 2000) carried out a numerical study about the minimization of entropy generation in an inclined enclosure (1997) and an inclined porous cavity (2000). He showed that minimum entropy generation depends on the Rayleigh number and the enclosure inclination angle. Hidouri et al. (2006) investigated the influence of Soret effect on entropy generation in steady state of thermosolutal convection. They found that entropy generation increases with the increase of thermodiffusion ratio and takes a minimum value at a specific buoyancy ratio value. Magherbi et al. (2007a,b) numerically studied entropy generation in double-diffusive convection in presence of Soret and Dufour effects. First, they investigated the influence of Dufour effect on entropy generation in thermosolutal convection through a square cavity, filled with a binary perfect gas mixture (2007a). They showed that Dufour effect tends to decrease thermal irreversibility and to increase viscous and diffusive irreversibilities. Second, they indicated that entropy generation takes a minimum value when thermal and concentration gradients are equal in intensity but act in opposite way in presence of Soret effect, for the case of an inclined square cavity, filled with a binary perfect gas mixture, and submitted to thermal and concentration gradients (2007b). El Jery et al. (2010) numerically investigated the influence of an oriented magnetic field on entropy generation in natural convection flow for air and liquid gallium. For air, they found that transient entropy generation exhibits an oscillatory behaviour for small values of Hartmann number, when thermal Grashof number GrT ≥ 104 . For liquid gallium, transient irreversibility always exhibits an asymptotic behaviour. Further, for air, increasing the inclination angle of the magnetic field, tends to decrease the critical Hartmann number, consequently, the transient oscillatory behaviour of entropy generation decreases. Bouabid et al. (2011) analysed the magnetic field effect on entropy generation in a square cavity submitted to thermosolutal convection. Their results showed that, magnetic effect is more pronounced than friction one. The magnetic field suppresses the flow in the cavity and this leads to the decrease of entropy generation. Entropy generation caused by double-diffusive convection in presence of rotation was investigated by Chen (2011). He found that only fast rotation has significant influence on entropy generation. Famouri and Hooman (2008) numerically studied entropy generation in free convection in a partitioned cavity. They investigated the effect of the Rayleigh number and the position of the heated partition on entropy generation. The effect of non-uniform heat flux on entropy generation of a heating fluid in a developing laminar pipe flow was investigated by Esfahani and Shahabi (2010). They showed that in all cases, entropy generation owing to the heat flux with decreasing tendency, is more pronounced than that owing to the uniform heat flux distribution. The first and the second laws characteristics of a fully developed forced convection inside a saturated porous rectangular duct, were analytically investigated by Hooman et al. (2008) using the Darcy–Brinkman model. This article is a complementary study of those established by Hidouri et al. (2006) and Magherbi et al. (2007a,b) concerning entropy generation determination in double-diffusive convection, for the case of a saturated square porous cavity, filled with a binary perfect gas mixture. The Darcy–Brinkman model is employed and a numerical study is carried out by the variation of the Darcy number, the porosity and the thermal porous Rayleigh number. Both
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∂T ∂C = 0, =0 ∂Y ∂Y
Fig. 1 Mathematical Darcy–Brinkman model
Ch
Porous Medium
Cc
g
Th
Tc ∂T ∂C = 0, =0 ∂Y ∂Y
cases of cooperative and opposite buoyancy forces are considered, which is in our knowledge, have not been encountered in the previous studies.
2 Mathematical Formulation The considered geometry in this study is shown in Fig. 1. It consists of a square cavity filled with a fluid composed by a binary perfect gas mixture (air and a pollutant species), saturating a porous medium. Left and right walls are submitted to different but uniform temperatures and concentrations (Th , Ch ) and (Tc , Cc ), respectively. The two horizontal walls are insulated and adiabatic. The porous medium is isotropic, homogeneous and in thermodynamic equilibrium with the fluid. The flow in the cavity is laminar and two dimensional, and Soret and Dufour effects are neglected. All physical properties of the fluid are assumed to be constant, except its density which satisfies the Boussinesq approximation such that ρ(C, T ) = ρ0 [1 − βT (T − T0 ) − βC (C − C0 )]
(1)
ρ0 , T0 and C0 are the reference mass density, the reference temperature and the reference concentration of the pollutant species, respectively. βT and βC are the thermal and the solutal expansion coefficients, respectively. They are given by 1 ρ0 1 βC = − ρ0 βT = −
∂ρ ∂T ∂ρ ∂C
(2)
The Darcy–Brinkman formulation is adopted in the analysis. The conservation equations of mass, momentum, energy and chemical species describing the phenomenon inside the cavity are given as follows: ∂U ∂V + (3) ∂X ∂Y 2 1 ∂U ∂ U ∂ 2U 1 ∂U μ ∂P ∂U + ρ0 . + 2 U +V = − ·U − + μeff (4) ε ∂t ε ∂X ∂Y K ∂X ∂ X2 ∂Y 2 2 1 ∂V ∂ V ∂2V 1 ∂V μ ∂P ∂V + · + 2 U +V =− ·V − + μeff − ρg ρ0 2 ε ∂t ε ∂X ∂Y K ∂Y ∂X ∂Y 2 (5)
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∂2T ∂2T + ∂ X2 ∂Y 2 2 ∂C ∂C ∂ C ∂ 2C ∂C +U · + V. = De · + ε ∂t ∂X ∂Y ∂ X2 ∂Y 2 σ
∂T ∂T ∂T +U +V = αe · ∂t ∂X ∂Y
(6) (7)
K , μ, μeff , αe , De , σ and ε are the medium permeability, the fluid dynamic viscosity, the effective viscosity (i.e. the viscosity in the Brinkman model), the effective thermal diffusivity, the species diffusivity through the fluid-saturated porous matrix, the specific heat capacities ratio and the medium porosity, respectively. αe , De and σ are given by αe =
km (ρc)m , De = ε · D, σ = (ρc)f (ρc)f
(8)
The effective thermal diffusivity is the ratio of the saturated porous medium thermal conductivity (km ) by the specific heat capacity of the fluid ((ρc)f ). c is the specific heat. The effective diffusivity is given as the product of the medium porosity by the species molecular diffusivity through the fluid mixture. The thermal conductivity of the porous medium is given by km = ε · kf + (1 − ε) · ks
(9)
Subscript f refers to fluid properties, while subscript m refers to the fluid–solid mixture and s to the solid matrix. The dimensionless macroscopic conservation equations of mass, momentum, energy and chemical species (pollutant species) can therefore be written as follows:
1 ∂u · ε ∂τ 1 ∂v · ε ∂τ
∂u ∂v + ∂x ∂y 2 ∂ u ∂p ∂ 2u 1 ∂u Pr ∂u u− + 2 + 2 u +v =− + Λ · Pr · ε ∂x ∂y DA ∂x ∂x2 ∂y 2 ∂ v ∂p ∂ 2v 1 ∂v Pr ∂v v− + + 2 u +v =− + Λ · Pr · ε ∂x ∂y DA ∂y ∂x2 ∂ y2 + GrT (·θ + N φ) 2 ∂θ ∂θ ∂θ ∂ θ ∂ 2θ σ +u +v = Rk · + 2 ∂τ ∂y ∂y ∂x2 ∂y ε
∂ 2φ ∂φ ∂φ ∂φ ε ∂ 2φ +u· +v· = ·( 2 + 2) ∂τ ∂x ∂y Le ∂ x ∂y
(10) (11)
(12) (13) (14)
The governing equations are established using the following dimensionless variables: U T − T0 V X Y t·W P − P0 C − C0 ;θ = ; u= ; x= ; y= ; τ= ; p= ; φ= W W a a a ρ.W 2 ΔT ΔC GrS αf km ν μe ΔT = Th − Tc ; C = Ch − Cc ; N = ; Le = ; Pr = ; Λ= ; RK = GrT D W ·a μ kf
u=
(15) ΔT a 3
and GrS = gβCυΔCa a and W are scales of length and viscosity, respectively. GrT = gβT υ 2 2 are the thermal and the solutal Grashof numbers, respectively, and N is the buoyancy ratio. D A = aK2 is the Darcy number. Λ is the ratio of the viscosity in the Brinkman term to the fluid viscosity and RK is the ratio of the thermophysical properties of the porous medium to kf the fluid for the thermal conductivity. αf = (ρc) is the thermal conductivity of the fluid. f
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The source term in the momentum equation is written in terms of Grashof numbers but the thermal porous Rayleigh number Ra ∗ = Pr D A GrT will be used in the analysis. The appropriate initial and boundary conditions of the problem are For the hole space, at τ = 0 : u = v = 0, p = 0, θ = 0.5 − x and φ = 0.5 − x.
(16)
At x = 0; φ = θ = 0.5
(17)
At x = 1; φ = θ = −0.5 ∂θ ∂φ At y = 0 and y = 1; = =0 ∂y ∂y
(18) (19)
3 Entropy Generation In general, the study of the irreversibility requires the determination of the rate of local entropy generation which is given by the sum of products of conjugate fluxes and forces. For a given n-components in a thermodynamic system, local entropy generation rate can be written as follows: S=
n
Ji · Fi
(20)
i=1
Ji and Fi are the fluxes and the driving thermodynamic forces, respectively. In the present problem, entropy is generated through heat, fluid flow and species diffusion. The volumetric entropy generation is therefore the sum of irreversibilities due to thermal gradients, viscous dissipation and concentration gradients. In such problem, the Darcy dissipation term should not be neglected compared to the clear fluid term in the contribution of fluid friction irreversibility. Entropy generation due to fluid friction denotes both the wall and the fluid layer shear stress and the momentum exchange at the solid boundaries (Hooman et al. 2008). Under the previous assumptions, the expression of the volumetric entropy generation in double-diffusive convection through a porous medium for a single diffusing species of a binary perfect gas mixture in 2D approximation is given by (Hidouri et al. 2006; Magherbi et al. 2007a; Hooman et al. 2007, 2008; Tamayol et al. 2010): km ∂T 2 ∂T 2 μ S= 2 + (U 2 + V 2 ) + ∂x ∂y T0 · K T0 ∂V 2 μ ∂V 2 ∂U ∂U 2 + + +2 + 2 T0 ∂X ∂Y ∂Y ∂X 2 2 ∂C ∂C ∂C ∂T ∂C ∂T R De R De + + + · + · (21) C0 ∂X ∂Y T0 ∂X ∂X ∂Y ∂Y The first term of the right hand side of Eq. 21 is the entropy generation due to thermal gradients, the second is due to viscous dissipation composed by the Darcy dissipation term (the velocity square term) and the clear fluid dissipation term. The third term denotes diffusive entropy generation composed by irreversibilities due to pure concentration gradients and a mixed product of temperature and concentration gradients. The dimensionless form of local entropy generation is obtained by using the dimensionless variables previously listed and takes the following form: slT = slθ + slv + sld
(22)
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A. Mchirgui et al.
2 ∂θ 2 ∂θ + slθ = ∂x ∂y
2 2 ∂u ∂v ∂u ∂v 2 ∗ 2 2 +2 + slv = Br u + v + Da · 2 + ∂x ∂y ∂y ∂x ∂φ 2 ∂φ ∂θ ∂φ ∂θ ∂φ 2 · + · + sld = ϕ1 + ϕ2 ∂x ∂y ∂x ∂x ∂y ∂y
(23)
(24)
(25)
The expression of the dimensionless local entropy generation is the result of regrouping the three irreversibilities due to thermal gradients (slθ ), fluid friction (slv ) and diffusion (sld ). Equation 25 is the local entropy generation due to diffusion, the right hand side of this equation denotes irreversibility due to pure concentration gradient (sdd ) and a crossed term of both thermal and concentration gradients (sdθ ), respectively. ϕ1 and ϕ2 are dimensionless coefficients, called irreversibility distribution ratios and are related to diffusive irreversibility. They are given by De R Ω ϕ1 = ΔC (26) k·Ω Ω De R · C (27) ϕ2 = k·Ω Br* is the modified Darcy–Brinkman number, it is given by Br ∗ =
Br Ω
(28)
Br, Ω and Ω are the Brinkman number, the temperature and the concentration ratios, respectively. They are given by Br =
μ · W 2 · a2 C T , Ω = ,Ω = km · T · K T0 C0
(29)
It is important to notice that the Brinkman number depends on the Darcy number. Consequently, the modified Brinkman number can be written as follows: 30 Ra = 10
6
25 20
ST
15 10
Ra = 10
5
Ra = 10 Ra = 10
0 21x21
5
31x31
4
3
41x41
51x51
61x61
Grid size Fig. 2 Total entropy generation versus gird size (Pr = 0.71, D A = 10−2 , Le = 1.2, ε = 0.5)
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Br ∗ =
ϕ3 DA ;
231
where ϕ3 is the third irreversibility distribution ratio defined by ϕ3 =
μT0 km
W (T )
2 (30)
The total dimensionless entropy generation is obtained by numerical integration, over the cavity volume A, of the dimensionless local entropy generation. It is given by sT = slT · dA. (31) A
The average heat and mass transfer through the heated wall are given in dimensionless terms by Nusselt and Sherwood numbers, respectively. They are defined as follows: 1 Nu =
−
∂θ ∂x
−
∂φ ∂x
0
1 Sh = 0
dy
(32)
dy
(33)
4 Numerical Procedure A modified version of the control volume finite-element method (CVFEM) of Saabas and Baliga (1994) is adapted to standard-staggered girds, in which pressure and velocity components are calculated and stored at different points. The SIMPLE algorithm of Patankar (1980) is applied to resolve the coupled pressure–velocity equations in conjunction with an alternating direction implicit (ADI) scheme for performing the time evolution. From the known temperature, concentration and velocity fields, calculated at any time τ by solving Eqs. 11–14, local entropy generation SlT is then calculated at any nodal point of the cavity. The total entropy generation for the entire cavity ST is then obtained by numerical integration. The shape function describing the variation of the dependent variable ( = u, v, θ, φ) is needed to calculate the flux across the control volume faces. We have followed Saabas and Baliga (1994) in assuming linear and exponential variations, respectively, when the dependent variable is calculated in the diffusive and in the convective terms of the conservation equations. The used numerical code written in FORTRAN language was described and validated in detail in Abassi et al. (2001a,b). More details and discussion about CVFEM are available in Prakash (1986), Hookey (1989), El Kaim et al. (1991) and Saabas and Baliga (1994). As mentioned, entropy generation is a function of thermal gradients, viscous dissipation and concentration gradients. To confirm the validity of the established numerical code concerning the grid independence analysis, Fig. 2 illustrates the variation of total entropy generation values versus gird sizes in steady state, for different values of Rayleigh number. Results show that, for a given value of the Darcy number (in this case Da = 10−2 ), the relative error given −ST(n,n) | |S × 100 (where ST(n,n) is the calculated entropy generation at a by Er = T(n+10,n+10) ST(n,n) grid of size n × n), is the adopted criterion for choosing the grid size. As an example, for Ra = 105 , the relative error is equal to 2.9% when we pass from 31 × 31 to 41 × 41 nodal points, and it becomes 1.33% when we pass from 41 × 41 to 51 × 51 nodal points. Thus, the grid of size 41 × 41 nodal points is sufficiently enough for the corresponding Rayleigh
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number value. In this study, and for all the considered Rayleigh number values, grids are chosen for a relative error: Er < 2%. The accuracy of the present numerical study has been performed over a large range of parameters. For purely thermal natural convection, average values of Nusselt number are compared with those given by Lauriat and Prasad (1987) and Kramer et al. (2007) (Table 1) and with those found by Younsi et al. (2002) and Nithiarasu et al. (1996) (Table 2). For double-diffusive convection, the average values of Nusselt and Sherwood numbers are compared with those given by Kramer et al. (2007) (Table 3). As it can be seen, the results are in good agreement with those given by the literature.
5 Results and Discussion As mentioned above, this article is a complementary study of Hidouri et al. (2006) and Magherbi et al. (2007a,b). In this case, the considered medium is a square porous cavity filled with a binary perfect gas mixture characterized by Pr = 0.71 and Le = 1.2. The operating parameters are in the following ranges: 10−6 ≤ D A ≤ 10, 50 ≤ Ra ∗ ≤ 500. For the present binary mixture, it was found that ϕ1 ranges between 10−1 and 0.5; ϕ2 between 10−5 and 10−2 and ϕ3 between 10−5 and 10−7 . In order to see the contributions of difTable 1 Average Nusselt number for N = 0, Pr = 0.71, Ra ∗ = 100 10−1
DA
10−2
10−3
10−4
10−5
10−6
This study
1.067
1.72
2.42
2.84
3.04
3.15
Kramer et al. (2007)
1.08
1.70
2.43
2.83
2.99
3.12
Lauriat and Prasad (1987)
–
1.70
2.41
2.84
3.02
3.06
Table 2 Average Nusselt number for N = 0, Pr = 1 Ra ∗
DA
10
10−2
100
10−2
1.72
1.71
1.68
1000
10−2
4.26
4.26
4.24
10
10−6
1.1
1.08
1.06
100
10−6
3.02
3.00
2.98
1000
10−6
12.19
12.25
12.11
This study
Younsi et al. (2002)
1.009
1.02
Nithiarasu et al. (1996) 0.99
Table 3 Average Nusselt and Sherwood numbers for Le = 10, D A = 10−1 , Ra ∗ = 100 −1
N This study Kramer et al. (2007)
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0
1
2
Nu
0.98
0.99
1.07
1.08
Sh
0.98
1.09
2.70
3.01
Nu
1.0
1.0
1.07
1.09
Sh
1.0
1.08
2.66
2.95
Entropy Generation in Double-Diffusive Convection in a Square Porous Cavity
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ferent irreversibilities, the following values of irreversibility coefficients are considered: ϕ1 = 0.5, ϕ2 = 10−2 , ϕ3 = 10−6 . All the above parameters are chosen after several numerical computations. Due to large number of parameters, the porous medium properties are kept constant, they are given by Λ = 1, σ = 1, Rk = 1. The buoyancy ratio N ranges between −6 and 6. Entropy generation determination for the case of double-diffusive convection in a porous medium has not been encountered. This article aims to give the influence of different operating parameters mentioned above on irreversibility inside the considered medium. The imposed global and local convergence criteria are given, respectively, by
∂u ∂v + ∂x ∂y
≤ 10
−5
τ +τ − τ ≤ 10−5 , max τ +τ
(34)
is the dependent variable, = (u, v, θ, φ). This means that the continuity equation should verify the first convergence criterion at each step time of calculation, and the dependent variable should verify the second criterion at each point of the cavity and at each step time. 5.1 Influence of the Darcy Number on Entropy Generation In this section, the influence of Darcy number (10−6 ≤ D A ≤ 10), consequently the permeability effect on entropy generation is studied by the variation of the buoyancy ratio (N = −6, −2, 2, 6). Thermal porous Rayleigh number and porosity are fixed (Ra ∗ = 100 and ε = 0.5). Figure 3 illustrates the dependence of total entropy generation on Darcy number for different values of buoyancy ratio. Because of the important difference between the Darcy number and the entropy generation values, a logarithmic scale of the axis is used. As it can be seen, for a fixed value of Darcy number, total entropy generation increases with the absolute value of the buoyancy ratio. This augmentation can be justified by the increase of the solutal Grashof number and consequently, the importance of the convection diffusion phenomena which induces the increase of entropy generation. As the Darcy number increases, entropy generation amplitude decreases and tend towards a minimum constant value for all the studied buoyancy ratio values. This is due to the fact that, increasing Darcy number induces the increase of the medium permeability, which causes the decrease of friction effects and then a decrease of entropy generation. The decrease of Darcy number causes an augmentation of total entropy generation for both cases of cooperative and opposite buoyancy forces. This augmentation is more pronounced for higher values of the buoyancy ratio. In fact, the inverse of the Darcy number directly affects the convective motion by the Darcy term in the momentum equation and indirectly, in the temperature and the concentration fields which depend on the velocity. The influence of Darcy number on irreversibilities is plotted in Figs. 4 and 5. Two values of the buoyancy ratio are considered (N = 6, N = −6). As it can be seen, viscous entropy generation predominates both thermal and diffusive irreversibilities for 10−6 ≤ D A ≤ 10−3 . As Darcy number increases D A > 10−3 , thermal irreversibility becomes slightly higher than diffusive and viscous irreversibilities. The three irreversibilities tend towards a constant minimum value when D A ≥ 1, regardless the buoyancy ratio value. It is important to notice that the amplitude of all the irreversibilities is more important in the case of cooperative buoyancy forces then that in opposite case, since thermal and concentration gradients act in the same direction which induces an augmentation of entropy generation.
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N = -2 N=2 N = -6 N=6
10
4
10
3
10
2
ST
10
1 10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
1
10
DA Fig. 3 Total entropy generation versus Darcy number (Pr = 0.71, Le = 1.2, Ra ∗ = 100, ε = 0.5)
4
10
sd sθ sv
S θ ,S d ,S v
3
10
2
10 10 1 10
-1
10 10
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
-2
-3
-4
1
DA Fig. 4 Entropy generation due to thermal, diffusive and viscous effects versus Darcy number (N = −6, Pr = 0.71, Le = 1.2, Ra ∗ = 100, ε = 0.5)
To explain the increase of entropy generation with the decrease of Darcy number, Fig. 6 depicts heat and mass transfer which are expressed by Nusselt and Sherwood numbers, versus Darcy number. As it can be seen, for both cases of cooperative and opposite buoyancy forces, heat and mass exchanged between the heated wall and the porous medium increase with the decrease of the medium permeability. Thus, the decrease of the medium permeability induces a considerable increase of fluid friction irreversibility (Figs. 4, 5), and an increase in heat and mass transfer leading to the increase of thermal and diffusive irreversibilities. As mentioned above, at higher values of Darcy number (D A > 10−3 ), entropy generation is mainly due to heat and mass transfer. Entropy generation is mainly due to viscous effects at lower values of Darcy number (D A ≤ 10−3 ). In fact, at low Darcy numbers, the Darcy term increases which indicates that the balance between the viscous force and the buoyancy
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Entropy Generation in Double-Diffusive Convection in a Square Porous Cavity
235 10
4
10
3
10
2
S θ ,S d ,S v
sd sθ sv
10 1 10
-1
10 10 10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
-2
-3
1
DA Fig. 5 Entropy generation due to thermal, diffusive and viscous effects versus Darcy number (N = 6, Pr = 0.71, Le = 1.2, Ra ∗ = 100, ε = 0.5) 12
Nu, Sh
N=6
10
Nu Sh Nu Sh
N = -6
8 6
4 2 0
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
DA Fig. 6 Nusselt and Sherwood numbers versus Darcy number (Pr = 0.71, Le = 1.2, Ra ∗ = 100, ε = 0.5)
force in the boundary layer is progressively replaced by a Darcy term versus buoyancy term balance, causing the increase of the velocity of convective motion. The effect of Darcy number on the flow velocity is illustrated in Fig. 7, showing the midsection y-velocity component at x = 0.5 for different values of Darcy number at N = 6 as an illustrative example. It can be seen that the magnitude of the velocity, tends to increase as the Darcy number decreases. An inversion of the convective motion is seen when the Darcy number reaches 10−3 . This can be explained by the competition between the Darcy term and the buoyancy term. In fact, the Darcy term becomes dominant when D A ≤ 10−3 . 5.2 Influence of the Porosity on Entropy Generation The effect of the porosity on the total entropy generation is investigated by the variation of the thermal porous Rayleigh number ranging between 50 and 500, when the Darcy term is
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2
v
a
0 0
0,2
0,4
-2
0,6
0,8
1
c -4 a : DA = 10-1 d : DA = 10 -5 b : DA = 10-2 e : DA = 10 -6 c : DA = 10-3 f : DA = 10
d e
-4 -6
f
-8
y Fig. 7 y-component velocity at x = 0.5 for different values of Darcy number (N = 6, Pr = 0.71, Le = 1.2, Ra ∗ = 100, ε = 0.5)
12 ε = 0,3 ε = 0,5 ε = 0,7 ε = 0,9
10 8
ST
6 4 2 0 0
50
100
150
200
250
300
350
400
450
500
Ra* Fig. 8 Total entropy generation versus thermal porous Rayleigh number (N = −5, Pr = 0.71, Le = 1.2, D A = 10−2 )
dominant (i.e. D A ≤ 10−3 ), and when the buoyancy term is dominant (i.e. D A > 10−3 ). Figures 8 and 9 illustrate entropy generation versus thermal porous Rayleigh number for the two above mentioned cases at N = −5. As it can be seen from Figs. 8 and 9, entropy generation increases with thermal porous Rayleigh number for any fixed porosity value. Amplitude of entropy generation is more important for the case of Darcy term predominance (Fig. 9) as compared to buoyancy term predominance (Fig. 8). This was explained by the fact that, when D A > 10−3 , fluid velocity decreases (Fig. 7), heat and mass transfer also decrease (Fig. 6) inducing the decrease of entropy generation magnitude. The reversed case is obtained when D A ≤ 10−3 . Since porosity measures the ratio of the volume pores occupied by the fluid mixture to the total medium volume, entropy generation increases with porosity for any value of thermal porous Rayleigh number. As a consequence, the medium porosity plays an important role concerning entropy generation minimization. As shown in Fig. 10, the average values of Nusselt and Sherwood
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ε = 0,3 ε = 0,5 ε = 0,7 ε = 0,9
150
ST
100
50
0 0
50
100
150
200
250
300
350
400
450
500
Ra* Fig. 9 Total entropy generation versus thermal porous Rayleigh number (N = −5, Pr = 0.71, Le = 1.2, D A = 10−4 )
6,5 Nu, Sh 6 5,5 5
Nu Sh
4,5 4 3,5 3 0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
ε Fig. 10 Average Nusselt and Sherwood numbers versus porosity (N = −5, Pr = 0.71, Le = 1.2, Ra ∗ = 100, D A = 10−4 )
numbers increase with the increase of the porosity, which implies that entropy generation due to thermal and diffusion effects increase with porosity. Figure 11 shows that magnitude of the net velocity tends to increase as the porosity increases. This result allows us to conclude that entropy generation due to viscous effects increases for higher values of porosity. It is important to notice that the porosity clearly affects the flow, the heat and the mass transfer and consequently, the entropy generation in thermosolutal convection. The porosity should be considered as an independent parameter in such cases, this result is consistent with the findings of Nithiarasu et al. (1998).
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a : ε = 0.3 b : ε = 0.5 c : ε = 0.9
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0 -1 -2
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0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
a
-3 -4
b c
-5
y Fig. 11 y-Component velocity at x = 0.5 for different values of the porosity (N = −5, Pr = 0.71, Le = 1.2, Ra ∗ = 100, D A = 10−4 )
6 Conclusion Entropy generation in double-diffusive convection through a square porous cavity filled with a binary perfect gas mixture is numerically studied using the CVFEM. The obtained results show that, for both cases of cooperative and opposite buoyancy forces, entropy generation increases with the decrease of the Darcy number. This augmentation is more important for higher values of the buoyancy ratio. It was found that entropy generation is mainly due to viscous irreversibility at lower values of the Darcy number D A ≤ 10−3 . The porosity considerably affects entropy generation in thermosolutal convection, especially at higher values of thermal porous Rayleigh numbers. At lower thermal porous Rayleigh number values, the porosity effect is smaller but observed.
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