Combined Heat and Mass Transfer in Two Sided Lid-Driven ...

21ème Congrès Français de Mécanique Bordeaux, 26 au 30 août 2013

Combined Heat and Mass Transfer in Two Sided Lid-DrivenPorous Cavity

TOUFIK BENMALEK*, FERHAT SOUIDI

Physics department, laboratory of applied fluid mechanics, USTHB University. Algiers, AlgeriaEmail: [email protected]

Abstract:In this study, a numerical investigation was conducted to analyze the steady state two-dimensional mixedconvection in two sided lid-driven porous cavity under the combined buoyancy effects of thermal and massdiffusion. The left and right vertical walls are submitted to horizontal thermal and concentration gradientsand they are moving in opposite directions. The horizontal walls are impermeable and thermally insulated.The finite volume method is used to resolve the governing equations of the flow using the formulation ofDarcy-Brinkman. The heat and mass transfer rates were examined using several operational dimensionlessparameters, such as the Richardson number Ri, Schmidt number Sc, Darcy number Da and buoyancy ratioparameter N.

Keywords: lid-driven flows, heat and mass transfer, porous media

Introduction:The problem of lid-driven flows in cavities has been major topic for research studies due to its fundamentalnature and owing to the wide spectrum of engineering applications such as electronic device cooling, crystalgrowth, high-performance building insulations, multi shield structures used for nuclear reactors, foodprocessing, float glass production, solar power collectors, furnace, drying technologies, etc. Numerousinvestigations on lid-driven cavity flow and heat transfer involving various combinations of cavityconfigurations, various fluids, and imposed temperature gradients have been published in the literaturefrequently.Ouertatani et al [1] have simulated three dimensional mixed convection in a double lid driven cubic cavity atvarious Reynolds and Richardson numbers. The fluid flow and heat transfer induced by the combined effectsof the driven lid and the buoyancy force within rectangular enclosures were investigated by Waheed [2].Investigation for various Prandtl number, Richardson number, and aspect ratio at Reynolds number of 100was performed by this author. Sivakumar et al. [3] performed analysis of the mixed convection heat transferin lid-driven cavities with different heating portion lengths at different locations. Cheng [4] studied the flowand heat transfer in a two-dimensional square cavity where the flow is induced by a shear force resultingfrom the motion of the upper lid combined with buoyancy force due to bottom heating. Lioua et al [5]examined numerically laminar mixed convection and entropy generation in a cubic lid-driven cavity. Basaket al [7] analyzed four different thermal boundary conditions, and heat flow patterns in mixed convectionusing heatlines concept of Bejan [8] for wide ranges of parameters. Numerical simulation of unsteady mixedconvection in a driven cavity using an externally excited sliding lid has been studied by Khanafer et al. [9].A summary of double diffusive convection in porous media made in the past is presented in the book ofNield and Bejan [10]. Goyeau et al [11] studied natural convection in porous media confined. They proposedcorrelations for heat and mass transfer in the cooperating case. Bennacer et al. [12] studied the diffusiveconvection in a porous square cavity with cooperating imposed horizontal temperature and concentrationgradients. The authors demonstrated that the anisotropic properties of the porous medium significantly affectthe rate of heat transfer and mass transfer in the cavity. A numerical study of the flow Double oppositediffusive in a porous cavity was performed by Younsi et al. [13]. The temperature and concentrationgradients were imposed horizontally. The model Darcy-Brinkman Forchheimer was used. The results showthat the effect of porous medium has been to reduce the vertical velocity in the boundary layer. Heat andmass transfer and the flow field are also deeply affected.


The objective of the present study is to analyze heat and mass transfer through a square cavity filled with aporous media and to examine the effects of various parameters, such as the Darcy, Richardson and Schmidtnumbers and the buoyancy ratio.

Problem description:

The physical model is shown schematically in Fig.1. A two-dimensional squarecavity of height H is filled with fluid-saturated porous medium. The left wall ismoving up and the right wall is moving down. They are maintained at high andlow temperatures and concentrations respectively. The upper and the bottom fixedwalls are impermeable and are thermally insulated.The flow in the cavity is considered laminar, two dimensional,and incompressible. The thermo physical properties are considered to beconstant except for the body force term in the momentum equation, according to the Boussinesqapproximation given by:ρ= ρ0[1– βT (T–T0) – βC(C–C0)]Where ρ is the density, βT and βC are the thermal and concentration expansion coefficients respectively.Radiation heat transfer is considered negligible with respect to other modes of heat transfer.

Mathematical modelling:

With the assumptions mentioned above, the nondimensional form of the governing equations reads:∂u∂x + ∂v∂y = 0 (1)u ∂u∂x + v ∂u∂y = −dpdx + ε 1Re ∂ u∂x + ∂ u∂y − ε 1ReDau (2)u ∂v∂x + v∂v∂y = −dpdy + ε 1Re ∂ v∂x + ∂ v∂y − ε 1ReDa v + Ri(θ + NC) (3)u ∂θ∂x + v ∂θ∂y = 1RePr ∂ θ∂x + ∂ θ∂y (4)u ∂C∂x + v ∂C∂y = 1ReSc ∂ C∂x + ∂ C∂y (5)The initial and boundary conditions corresponding to the problem are :

0,0,0,0,10,1

0,0,0,0,10,0

0,0,1,0,10,1

1,1,1,0,10,0

yC

yvuxy

yC

yvuxy

Cvuyx

Cvuyx

The following non-dimensional variables are definedx = x′L , y = y′L , u = u′U , v = v′U , p = p′ρU , θ = T − TT −T , C = C′ − C′C′ −C′The dimensionless parameters as buoyancy ratio “N”, Darcy number “Da”, Prandtl number “Pr”, Schmidtnumber “Sc” and Richardson number “Ri” are then defined as:N = β (C − C )β (θ − θ ) = GrGr , Da = KL , Pr = αϑ , Sc = ϑD , Ri = Gr /ReWhere Re = U Hϑ , Gr = gβ (T − T )Lϑ , Gr = gβ (C − C )Lϑ

Fig.1 physical model







0,0,0,0,10,1

0,0,0,0,10,0

0,0,1,0,10,1

1,1,1,0,10,0

yC

yvuxy

yC

yvuxy

Cvuyx

Cvuyx









0,0,0,0,10,1

0,0,0,0,10,0

0,0,1,0,10,1

1,1,1,0,10,0

yC

yvuxy

yC

yvuxy

Cvuyx

Cvuyx




The Nusselt and Sherwood numbers are defined respectively as:Nu = ∫ ϴ dy Sh = ∫ dyNumerical solution:Numerical solutions of the governing dimensionless equations subject to their corresponding initial andboundary conditions were obtained using the control-volume procedure described in detail by Patankar [14].The system of linear equations is solved by a tridiagonal matrix algorithm (TDMA). The pressure-velocitycoupling is ensured by the SIMPLER algorithm (Patankar [14]).

The code is validated by the mixed convection in two-sided lid driven differentially heated square cavity byHakanF. Oztop [6].

Fig.2. streamline and isotherm plots of the present work and of Hakan Oztop for case.II

Results and discussion

The numerical code has been carry out a number of simulations for a wide range of controlling parameterssuch as buoyancy ratio, Richardson number, Darcy number, and Schmidt number.

The buoyancy number N for this investigation was in the range 1≤N≤10. The Richardson number used in thisstudy was 0.01≤ Ri ≤ 100 by varying the Reynolds number while the Grashof number was fixed at 104,Darcy number was in the range10-5 ≤ Da ≤1, and Schmidt number was 0.01 ≤ Sc ≤ 100.

1. Richardson number EffectTo vary Richardson number from 0.01 to100, Grashof number is fixed at Gr=104 and changing Reynoldsnumber which represent the plate velocity Vp. The Darcy and the Schmidt numbers are fixed at Da=1.E-2and Sc=1.The effect of Richardson number on heat and mass transfer is represented in fig.3 and fig.4respectively for two Darcy numbers. We can see that the heat and mass flux is higher for low Richardsonnumber where the forced convection is dominates. As the Richardson number increases, the Nusselt and theSherwood numbers decrease. This is expected since for increasing Richardson number, natural convectionbecomes dominant and the flow motion is generally subdued. As a result the heat and mass transfer in theinterior are dominated by conduction mode.


5.2. Darcy number Effect

The effect of the porous medium on the streamlines and isotherms is shown in Figure.5 for various Darcynumbers. Consider a case of equal thermal and solutal Grashof numbers (GrT = GrC = 10-4) and Schmidtnumber of unity. It can be seen from this figure that as the Darcy number decreases, the flow become flatter(fig.6) as seen for Da=1.E-5 where the flow is formed by three recirculation cells , the midal one is due to thebuoyancy forces, and the two others due to the lids motion. As results, the microscopic drag force increases,which suppresses the buoyancy forces and consequently decreases the heat and mass transfer rates.

In fig.9 which represents velocity profile for different Darcy number; we can see that the velocity magnitudedecreases when Da number is increased.

Da=1

Da=1.E-3

Da=1.E-5C T ψ

Fig. 3. Sherwood number for different Ri number Fig. 4. Nusselt number for different Ri number

Fig.5. isoconcentrations, isotherms and streamlines for different Da numbers at Ri=1 , Sc=1 and N=1

0,01 0,1 1 10 100

2

4

6

8

10

12

14

16

18

Da=1E-2

Nu

Ri

Da=1

0,01 0,1 1 10 100

2

4

6

8

10

12

14

16

18

20

Da=1E-2

Sh

Ri

Da=1


The effect of the buoyancy ratio on the streamlines, isotherms and isoconcentrations contours is shown inFigure 7. It can be seen from this figure that as the buoyancy ratio increases the solutal effect dominates andthe mass transfer enhances due to the decrease in the thermal boundary-layer thickness as the Schmidtnumber increases, while the heat transfer decreases as seen in figure 8.

N=1

N=6

N=10

Figure 8 presents the heat transfer for different Schmidt numbers and buoyancy ratios. It can be seen fromthis figure that the heat transfer decreases with the increase of Schmidt number and buoyancy ratio due to thedecreases of the thermal boundary layer as sees in figure7, while it decreases for high Richardson numbers.

The effect of Schmidt numbers and buoyancy ratios on mass transfer is presented in figure 9. It can see thatthe solutal rate is enhanced with the increase of Schmidt numbers and buoyancy ratios where the solutal

Fig.6. velocity profile for different Darcy number at Ri=1 and Sc=1

Fig.7. isoconcentrations, isotherms and streamlines for different buoyancy numbers at Ri=10, Sc=10

0,0 0,2 0,4 0,6 0,8 1,0-0,6

-0,4

-0,2

0,0

0,2

0,4

0,6

Da=1E-2

Da=1E-2

Da=1

u

x

Da=1E-5

Ri=1


0,01 0,1 1 10 1000

10

20

30

40

50

60

Ri=10

N=1

Ri=0.1

N=1

N=6Sh

Sc

N=10

boundary layer decreases which product a high solutal gradient. It can see two that the variation of Sherwoodnumber is higher for higher values of Schmidt numbers comparing with low ones.

References

[1] N. Ouertatani, N.B. Cheikh, B.B. Beya, T. Lili, A. Campo, Mixed convection in a lid-driven cubiccavity, Int. J. Therm. Sci. 48(2009) 1265–1272.[2] M.A. Waheed, Mixed convective heat transfer in rectangular enclosures driven bya continuously moving horizontal plate, Int. J. Heat Mass Transf. 52(2009) 5055–5063.[3] V. Sivakumar, S. Sivasankaran, P. Prakash, J. Lee, Effect of heating location and size on mixedconvection in lid-driven cavities, Comput. Math. Appl. 59(2010) 3053–3065.[4] T.S. Cheng, Characteristic of mixed convection heat transfer in a lid-driven square cavity with variousRichardson and Prandtl numbers, Int. J. Therm. Sci. 50(2011) 197–205.[5] K. Lioua, H.F. Oztop, M.N. Borjini, K. Al-Salem, Second law analysis in a three dimensional lid-drivencavity, Int. Commun. Heat Mass Transfer38(2011) 1376–1383.[6] Hakan F. Oztop , Ihsan Dagtekin , Mixed convection in two-sided lid-driven differentiallyheated square cavity , International Journal of Heat and Mass Transfer 47 (2004) 1761–1769[7] T. Basak, P.V.K. Pradeep, S. Roy, A complete heating analysis on visualization of heat flow and thermalmixing during mixed convection in a square cavity with various wall heating, Ind. Eng. Chem. Res.50(2011) 7608–7630.[8] S. Kimura, A. Bejan, The heatline visualization of convective heat transfer, J. Heat Transfer 105 (1983)916–919.[9] K.M. Khanafer, A.M. Al-Amiri, I. Pop, Numerical simulation of unsteady mixed convection in a drivencavity using an externally sliding lid, Eur. J. Mech. B/ Fluids26(2007) 669–687.[10] D. A. Nield, A. Bejan, Convection in Porous Media, third ed., Springer-Verlag (2006).[11] B. Goyeau, J.-P. Songbe and D. GOBIN, Numerical study of double-diffusive natural convectionin a porous cavity using the Darcy-Brinkman formulation. Int. J. Heat Mass Transfer. Vol. 39,No. 7, pp. 1363-1378, (1996).[12] R. Younsi, A. Harkati and D. Kalache, Numerical simulation of double-diffusive naturalconvection in porous cavity: opposing flow. The Arabian Journal for Science and Engineering,vol. 27, N° 1C, (2002).[13] R. Bennacer, A. Tobbal and H. Beji, Convection naturelle Thermosolutale dans une CavitéPoreuse Anisotrope: Formulation de Darcy-Brinkman. Rev. Energ. Ren. Vol. 5, pp. 1-21, (2002).[14] S.V. Patankar, Numerical Heat transfer and fluid flow, Hemisphere, Washington, DC, (1980).

0,01 0,1 1 10 100

2

4

6

8

Ri=10

Ri=0.1

N=10

N=6

Nu

Sc

N=1

Fig. 9. Sherwood number for different Sc numberFig. 8. Nusselt number for different Sc number

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