mixed con in porous media

HYDROMAGNETIC MIXED CONVECTION IN A LID-DRIVENCAVITY FILLED WITH A FLUID-SATURATED POROUS MEDIUM

M. Muthtamilselvan1, P. Kandaswamy2, and J. Lee2

1Department of Mathematics, Kalaignar Karunanidhi Institute of Technology, Coimbatore-641046, India

Email: [email protected] of Mechanical Engineering, Yonsei University, Seoul, Republic of Korea

Email: [email protected]

Received 3 February 2009; accepted 3 October 2009

ABSTRACT

A numerical investigation is conducted to study the effect of magnetic field on mixed convectionin a lid-driven cavity filled with a fluid saturated porous medium. The left and right verticalwalls of the cavity are insulated while both the top and bottom horizontal walls are kept atconstant but different temperatures. The top horizontal wall is moving on its own plane ata constant speed while all other walls are fixed. A uniform magnetic field is applied in thevertical direction normal to the moving wall. The governing equations are solved using finite-volume approach along with the SIMPLE algorithm. Numerical solutions are obtained for awide range of parameters. The effects of the controlling parameters involved in the heat transfercharacteristics are studied in detail.

Keywords: Mixed convection, Lid-driven, Porous medium, Hydromagnetic.

Nomenclature

B0 magnetic field strengthDa Darcy number, K/H2g gravitational acceleration, [m/s2]Gr Grashof number, gH3

2

H enclosure length [m]Ha Hartmann numberK effective thermal conductivity of the porous medium [Wm1K1]Nu local Nusslet numberNuavg average Nusslet numberP pressure [Pa]Pr Prandtl number,

e

Re Reynolds number, U0H

Ri Richardson number, Gr/Re2T dimensionless temperature, c

hcU, V dimensionless velocities in X- and Y-directionU0 lid-driven velocity [m/s]

Int. J. of Appl. Math. and Mech. 5(7): 28-44, 2009.

Hydromagnetic mixed convection in a lid-driven cavity filled with a fluid-saturated porous medium 29

Uc dimensionless velocity in x-direction at mid-plane of the cavityu, v velocities in x- and y-directionVc dimensionless velocity in y-direction at mid-plane of the cavityX, Y dimensionless cartesian coordinatesx, y cartesian coordinatesGreek symbols effective thermal diffusivity of porous medium [m2s1] coefficient of thermal expansion of fluid [K1] temperature difference temperature [oC]K permeability of porous medium [m2] effective dynamic viscosity [Pa s1] effective kinematic viscosity, / fluid density [kgm3] porositySubscriptsavg averageC cold wallH hot wall

1 INTRODUCTION

The problem of mixed convection in a lid-driven cavity has been a major topic for research stud-ies due to its frequent occurrence in industrial and technological applications. This includescrystal growth, electronic cooling, oil extraction, solar collectors, etc. Many numerical tech-niques have been proposed to tackle this fundamental problem (Iwatsu et al. 1993; Moallemiand Jang 1992; Aydin and Yang 2000; Singh and Sharif 2003). Prasad and Koseff (1996)performed an experimental investigation of mixed convection flow in a lid-driven cavity. Theyconsidered heated moving bottom wall with high Reynolds number and high Grashof number.Their results indicate that the overall heat transfer rate is a very weak function of the Grashofnumber for the examined range of Reynolds number. They have also analyzed the mean heatflux values over the entire boundary to produce Nusselt numbar and Stanton number correla-tions which are useful for design applications.

Mixed convection flow in a lid-driven porous cavity has recently received considerable attentionbecause of numerous applications in engineering and science. Lai and Kulacki (1991) carriedout an experimental study of free and mixed convection in horizontal porous layers locallyheated from bellow. They considered three different sizes of the heat source and compared theexperimental Nusselt numbers with numerically calculated values. Mixed convection flow in anenclosure filled with a Darcian fluid saturated uniform porous medium in the presence of inter-nal heat generation is numerically investigated by Khanafer and Chamkha (1999). Their resultsindicate that heat transfer mechanisms and the flow characteristics inside the cavity are stronglydependent on the Richardson number. Also they find significant influence on the features ofthe isotherms and slight effects on the streamlines for small values of the Richardson number.A similar trend has also been observed by Al-Amiri (2000). Jue (2002) studied the convectionflow caused by a torsionally-oscillatory lid with thermally stable stratification in an enclosurefilled with porous medium. He found that the oscillatory frequency seriously affects the heattransfer phenomena.


30 M. Muthtamilselvan, P. Kandaswamy, and J. Lee

In the last several decades, there has been considerable interest in studying the influence ofmagnetic fields on the fluid flow dynamics and performance of various processes employingelectrically conducting fluids. Some of these studies considered hydromagnetic flows and heattransfer in many different porous and non porous geometries, for example, (Oreper and Szekely1983; Vajravelu and Hadjinicolaou 1998; Borjini et al. 2006; Chamkha 2002; Al-Nimr andHader 1999). Natural convection of an electrically conducting fluid in a rectangular enclosurein the presence of a magnetic field is studied numerically by Rudraiah et al. (1995). They in-dicate that the average Nusselt number decreases with an increase in the Hartmann number andthe Nusselt number approaches unity for a strong magnetic field. This shows that the convectionin the enclosure is suppressed due to the introduction of the magnetic field. Recently Robillardet al. (2006) investigated numerically as well as analytically the effect of an electromagneticfield on the natural convection in a vertical rectangular porous cavity saturated with an electri-cally conducting binary mixture. They conclude that under the condition of constant fluxes ofheat and mass imposed at the long side walls of the layer, the flow is parallel in the core of thecavity and turns through 180o in regions close to the end boundaries. This flow structure is notaffected by the imposition of a magnetic field. Pangrle et al. (1992) performed an experimentalinvestigation of magnetic resonance imaging an incompressible, laminar fluid flow in poroustube and shell systems flow. They used porous tube module in closed end mode for Reynoldsnumber of 100 to 200 based on the tube radius to study the flow behaviour and heat transfer.Other experimental studies dealing with magnetohydrodynamic flows in porous media were re-ported by McWhirter et al. (1998) and Kuzhir et al. (2003). Khanafer and Chamkha (1998)studied numerically hydromagnetic natural convection heat transfer in an inclined square enclo-sure filled with a fluid-saturated porous medium with heat generation. Their results indicate thatthe effects of magnetic field and the porous medium are found to reduce the heat transfer andfluid circulation within the cavity. Recently Chamkha (2002) investigated combined forced-freeconvection flow in a lid-driven square cavity in the presence of magnetic field. He conducteda parametric study for both aiding and opposing flow conditions. His results show that signifi-cant reductions in the average Nusselt number are produced for both aiding and opposing flowsituations as the strength of the applied magnetic field is increased. Also he concludes that fora fixed value Gr, increasing the Re produces higher values in the average Nusselt number foraiding flow.

The objective of the present numerical investigation is to examine the characteristics of a hydro-magnetic mixed convection in a lid-driven porous enclosure. The mathematical formulations arebased on the Brinkman-extended Darcy equation model. The governing equations are solvedusing the finite-volume formulation. Detailed results are presented in the form of the stream-lines and isotherms. The results of the present work for the particular case (Ha = 0 and allwalls are fixed) is in excellent agreement with those of the already publish work.

2 MATHEMATICAL ANALYSIS

Consider a steady-state two-dimensional square cavity filled with a porous medium of heightH as shown in Fig. 1. It is assumed that the top wall is moving from left to right at a constantspeed U0 and is maintained at a constant temperature h. All other remaining walls are fixed.The bottom wall is maintained at a constant temperature c(h > c). The vertical sidewalls areconsidered to be adiabatic. A uniform magnetic field is applied in the vertical direction normalto the moving wall. The one way magnetic field is coupling to y momentum equation. Thephysical properties are considered to be constant except the density variation in the body forceterm of the momentum equation which is satisfied by the Boussinesqs approximation. The



magnetic Reynolds number is assumed to be very small so that the induced magnetic field andHall effect are negligible (Cramer and Pai 1973). A consequence of small magnetic Reynoldsnumber is the uncoupling of the Navier-Stokes equations from Maxwells equation (Cramer andPai 1973). In the present investigation the porous medium is assumed to be hydrodynamicallyand thermally isotropic and saturated with a fluid that is in local thermal equilibrium (LTE)with the solid matrix. A general Brinkman-extended-Darcy model is used to account for theflow in the porous medium. Using the above assumptions, the governing equations for mass,momentum and energy can be written in the nondimensional form as (Nield and Bejan 2006):U

X+

V

Y= 0 (1)

1

2

(U

U

X+ V

U

Y

)= P

X+

1

Re2 U U

ReDa

1.75150

(U2 + V 2)1/2

Da

U

3/2(2)

1

2

(U

V

X+ V

V

Y

)= P

Y+

1

Re2 V V

ReDa

1.75150

(U2 + V 2)1/2

Da

V

3/2+

Gr

Re2T Ha

2

ReV (3)

UT

X+ V

T

Y=

1

PrRe2 T. (4)

Here U and V are dimensionless velocity components in the X and Y directions respectively,T is the dimensionless temperature, P is the dimensionless pressure and is the porosity. Theterm on the right-hand side of equation (3) includes the Larentz force induced by the interactionof the magnetic field with convective motion (Garandet et al. (1992)).The dimensionless variables are defined as:

X =x

H, Y =

y

H, U =

u

U0, V =

v

U0, T =

ch c , P =

p

U20, Re =

U0H

,

Gr =gH3

2, Ha2 =

B20H2

, Da =

K

H2, P r =

e.

The dimensionless boundary conditions can be written as:

U = 1, V = 0, T = 1 (Y = 1)

U = 0, V = 0, T = 0 (Y = 0)

U = V = 0,T

X= 0 (X = 0, 1).

The local Nusselt number can be expressed as

Nu =T

YdX .

and the average Nusselt number can be expressed as



Nuavg =

10

T

YdX .

3 METHOD OF SOLUTION

The governing equations along with the boundary conditions are solved numerically employingfinite volume method using staggered grid arrangement. The semi-implicit method for pressurelinked equation (SIMPLE) is used to couple momentum and continuity equations as given byPatankar (1980). The third order accurate deferred QUICK scheme of Hayase et al. (1992) isemployed to minimize the numerical diffusion for the convective terms for both the momen-tum equations and energy equation. The solution of the discretized momentum and pressurecorrection equations is obtained by TDMA line-by-line method (Patankar 1980). The pseudo-transient approach is followed for the numerical solution as it is useful for a situation in whichthe governing equations give rice to stability problems, for example, buoyant flows (Versteegand Malalasekera 1995). At each step, the solution is iterated until the normalized residuals ofthe mass, momentum and temperature equation become smaller than 107.

The grid independence test is performed using successively sized grids, 31 31 to 91 91 forPr = 7.0, Ri = 103, Ha = 25, Da = 103 and = 0.4. Uniform grid is used for all thecomputations. The distribution of the v-velocity in the vertical mid-plane is shown in Fig. 2. Itcan be seen from this figure that by using lower than 81 81 grid the solution for the velocityprofile is dependent on the mesh size. Also this result shows that the two grids 81 81 and91 91 give the same result. Therefore considering both the accuracy and the computationaltime, all the computations are performed with a 81 81 grid.

4 CODE VALIDATION

In order to check on the accuracy of the computational code employed for the solution of theproblem considered in the present study, it is validated by performing simulation for the analysisof Benard convection in rectangular cavities filled with a porous medium in the absence of amagnetic field which is reported earlier by Jue (2001). Fig. 3 shows the mid-plane velocityprofile for Da = 102, Pr = 7.0 and different Rayleigh numbers. This comparison showsgood agreement between the two numerical results. In addition, table 1 clearly shows goodagreement of the average Nusselt number between the two numerical results. These effectsprovide credence to the accuracies of the present numerical solutions.

5 RESULTS AND DISCUSSION

The main objective of this investigation is to study the effects of the Hartmann number on theflow and heat transfer in a lid-driven square cavity driven by a combined shear and buoyancyforces. The present computation will be focused on the parameters having the following ranges:The Hartmann number from 0 to 100, the Darcy number from 104 to 101, the porosity from0.2 to 0.8 and the Reynolds and Grashof numbers from 102 to 103. The ratio Gr/Re2 (Richard-son number Ri) provides a measure of the importance of buoyancy-driven natural convectionrelative to lid-driven forced convection. For increasing value of Ri, three different heat trans-port regimes were defined namely, the forced convection, the mixed convection and the naturalconvection (Aydin 1999).



In general fluid circulation is strongly dependent on the Hartmann number as shown in Figs.4-6. Figs. 4-6 illustrate the streamlines (on the left) and isotherms (on the right) for differentvalues of the Hartmann number for a fixed value of the Darcy number. Fig. 4(a) shows that themain circulation fills the entire cavity and the minor cell is visible near the right bottom cornerin the absence of a magnetic field. The isotherms are clustered close to the bottom wall whichpoints to the existence of steep temperature gradients in the vertical direction in this region. Inthe bulk of the cavity except this localized area, however, the temperature gradients are weak.This implies that, due to the vigorous actions of mechanically driven circulations fluids arewell mixed. Consequently the temperature differences in this interior region are very small.When Ha is increased to 50, the clustering of the lines near the sliding lid for the streamlinesand toward the bottom wall for the isotherms (see Fig. 4(b)). In this case weakened flowactivities due to the stratification suggest the formation of a separate cell in the lower half of thecavity. As Ha is increased to 100, streamlines show three rotating cells to exists (see Fig. 4(c)).The vertical temperature stratification is substantially linear in the stagnant bulk of the interiorregions. Only in a relatively small region, where the mechanically induced convective activitiesare appreciable. This reflects the fact that heat transfer is mostly conduction in the middle andbottom parts of the cavity.

Fig 5(a) shows the streamlines to collapse together toward the right top corner where the slidingtop wall impinges on the vertical right wall for Ha = 25 and Ri = 102. In addition the resultsshow that on increasing the Ha values the rotating cell increases to two for Ha = 50 and itincreases to three for Ha = 100. In all the three ranges of the Hartmann number conductionis more dominating in the most portions of the cavity. At the top right portions of the cavityfluids are comparatively well mixed. The reading from the temperature contours point outthat conduction is the dominant mode of heat transfer and that the convection heat transfer isconfined to the top right corner. This portion decreases for increasing values of Ha. Alsothe streamlines are elongated and the bottom region becomes broadly stagnated. For a strongmagnetic field, the axis of the stream line is tilted. This is due to the retarding effect of theLorentz force. Fig. 6 displays similar case of Fig. 5. Comparing Figs. 4-6 for fixed value ofHa the mixed convection portion is decreased for increasing value of Ri.

Fig. 7 illustrates the streamlines and isotherms for Ri = 103, Da = 102 and Ha = 50. In thiscase for increasing values of conduction is more dominating in the entire cavity because thebuoyancy is still too weak to affect the flow pattern. Similarly Fig. 8 shows that the buoyancyeffect is increasing at the right top corner for increasing values of the Darcy numbers. For highDarcy values (Da = 101) mixed convection is more dominating in the entire cavity in theabsence of a magnetic field.

The effect of magnetic field on the velocity profile in a vertical cavity for various values ofthe Richardson number at mid-sections of the cavity is depicted in Fig. 9. The presence of amagnetic field within the cavity results in a force, opposite to the flow direction, which tendsto resist the flow. This causes suppression in the thermal currents of the flow. This is clearlynoticed from the horizontal and vertical velocity profiles at the center of the cavity as depictedin Fig. 9. Fig. 10 shows that the local Nusselt number for Ri = 101, = 0.4 and Da = 102at different Hartmann numbers. As discussed earlier for increasing value of Ha conductionis more dominating mode in the cavity. The average Nusselt number for different Hartmannnumbers is depicted in Fig. 11. In general for a fixed value of the Richardson number, theaverage Nusselt number decreases with increasing values of the Ha. The reason is that themagnetic field affect the flow pattern therefore the buoyancy effect is still too weak. Also theaverage Nusselt number is increasing with decreasing values of the Richardson number. This



implies that substantial contribution of convective heat transfer in the middle and upper portionsof the cavity.

6 CONCLUSION

In this paper, mixed convection in a lid-driven porous cavity in the presence of a magnetic fieldis studied numerically. The finite-volume method is employed for the solution of the presentproblem. The streamlines and the isotherms for various parametric conditions are presented anddiscussed. It is found that the heat transfer is strongly dependent on the strength of the magneticfield and the Darcy number. The effect of the magnetic field is found to reduce the heat transferand fluid circulation within the cavity. In general, for fixed value of Richardson number, theaverage Nusselt number decreases with increasing values of the Hartmann number. Also theaverage Nusselt number increases with increasing values of the Darcy number.

REFERENCES

Al-Amiri AM (2000). Analysis of momentum and energy transfer in a lid-driven cavity filledwith a porous medium. Int. J. Heat Mass Transfer 43, pp. 35133527.Al-Nimr MA and Hader MA (1999). Mhd free convection flow in open-ended vertical porouschannels. Chemical Eng. Sci. 54, pp. 18831889.

Aydin O (1999). Aiding and opposing mechanisms of mixed convection in a shear and buoyancydriven cavity. Int. Comm. Heat Mass Transfer 26, pp. 10191028.Aydin O and Yang WJ (2000). Mixed convection in cavities with a locally heated lower walland moving sidewalls. Numer. Heat Transfer Part A 37, pp. 695710.Borjini MN, Aissia HB, Halouani K, and Zeghmati B (2006). Effect of optical properties onoscillatory hydromagnetic double-diffusive convection within semitransparent fluid. Int. J. HeatMass Transfer 49, pp. 39843996.Chamkha AJ (2002). Hydromagnetic combined convection flow in a vertical lid-driven cavitywith internal heat generation of absorption. Numer. Heat Transfer Part A 41, pp. 529546.Cramer KR and Pai SI (1973). Magnatofluid Dynamics for Engineers and Physicists. McGraw-Hill, New York.

Hayase T, Humphrey JAC, and Grief R (1992). A consistently formulated quick scheme forfast and stable convergence using finite-volume iterative procedures. J. Compt. Phys. 98, pp.108118.

Iwatsu R, Hyun JM, and Kuwahara K (1993). Mixed convection in a driven cavity with a stablevertical temperature gradient. Int. J. Heat Mass Transfer 36, pp. 16011608.Jue TC (2001). Analysis of benard convection in rectangular cavities filled with a porousmedium. Acta Mech. 146, pp. 2129.

Jue TC (2002). Analysis of flows driven by a torsionally-oscillatory lid in a fluid-saturatedporous enclosure with thermal stable statification. Int. J. Therm. Sci. 41, pp. 795804.



Khanafer KM and Chamkha AJ (1998). Hydromagnetic natural convection from an inclinedporous square enclosure with heat generation. Numer. Heat Transfer Part A 33, pp. 891910.Khanafer KM and Chamkha AJ (1999). Mixed convection flow in a lid-driven enclosure filledwith a fluid-saturated porous medium. Int. J. Heat Mass Transfer 42, pp. 24652481.Kuzhir P, Bossis G, Bashtovoi V, and Volkova O (2003). Flow of magnetorheological fluidthrough porous media. European Journal of Mechanics B/Fluids 22, pp. 331343.Lai FC and Kulacki FA (1991). Experimental study of free and mixed convection in horizontalporous layers locally heated from below. Int. J. Heat Mass Transfer 34, pp. 525541.McWhirter JD, Crawford ME, and Klein DE (1998). Magnetohydrodynamic flows porous me-dia ii: Experimental results. Fusion Science and Tech. 34, pp. 187197.

Moallemi MK and Jang KS (1992). Prandtl number effects on laminar mixed convection heattransfer in a lid-driven cavity. Int. J. Heat Mass Transfer 35, pp. 18811892.Nield DA and Bejan A (2006). Convection in Porous Media (third ed). Springer-Verlag, NewYork.

Oreper GM and Szekely J (1983). The effect of an externally imposed magnetic field on buoy-ancy driven flow in a rectangular cavity. J. Crystal Growth 64, pp. 505515.

Pangrle BJ, Walsh EG, Moore SC, and Dibiasioo D (1992). Magnetic resonance imaging oflaminar in porous tube and shell systems flow. Chemical Eng. Sci. 47, pp. 517526.

Patankar SV (1980). Numerical Heat Transfer and Fluid Flow. Hemisphere, Washington, DC.Prasad AK and Koseff JR (1996). Combined forced and natural convection heat transfer in adeep lid-driven cavity flow. Int. J. Heat and Fluid Flow 17, pp. 460467.

Robillard L, Bahloul A, and Vasseur P (2006). Hydromagnetic natural convection of a binaryfluid in a vertical porous enclosure. Chem. Eng. Comm. 193, pp. 14311444.

Rudraiah N, Barron RM, Venkatachalappa M, and Subbaraya CK (1995). Effect of a magneticfield on free convection in a rectangular enclosure. Int. J. Engng. Sci. 33, pp. 10751084.

Singh S and Sharif MAR (2003). Mixed convective cooling of a rectangular cavity with inletand exit opening on differently heated side walls. Numer. Heat Transfer Part A 44, pp. 233253.Vajravelu K and Hadjinicolaou A (1998). Nonlinear hydromagnetic convection at a continuousmoving surface. Nonlinear Analysis, Theory, Methods and Applications 31, pp. 867882.

Versteeg HK and Malalasekera W (1995). An Introduction to Computational Fluid Dynamics:the Finite Volume Method. Longman Group Ltd., Malaysia.



Table 1: Comparison of the average Nusselt number between the present solution and Jue (2001)for Da = 102 at Pr = 7.0

Ra Present Nuavg Jue (2001)

3 104 0.6 2.2008 2.22223 104 0.4 1.9929 2.00125 104 0.6 2.5820 2.5969

105 0.6 3.0893 3.0877105 0.4 2.9004 2.9435

U0

g

x

y

h

c

Adia

batic

HAd

iaba

tic

Porousmedium

B0

Figure 1: Flow configuration and coordinate system.



X

V

0 0.5 1

-0.04

0

0.04

31X3151X5171X7181X8191X91

C

Y=0.5

Figure 2: Horizontal velocity profile for different mesh sizes at Ri = 103, Da = 103, = 0.4, Ha = 0 and Pr = 7.0.

X

V

0 0.5 1-80

-40

0

40

80PresentJue (2001)

c

Ra= 3 104

= 0.4,Ra=

105

, = 0.6Ra=10 5, = 0.6

Figure 3: Comparison of the present horizontal velocity profiles with Jue (2001) for Da = 102and Pr = 7.0.



0.88

0.94

0.06

(a) Ha = 0

0.94

0.31

0.06

(b) Ha = 50

0.06

0.94

0.44

(c) Ha = 100

Figure 4: Streamlines (on the left) and isotherms (on the right) for Ri = 103, = 0.4 andDa = 102.



0.94

0.13

0.56

(a) Ha = 25

0.94

0.44

0.13

(b) Ha = 50

0.94

0.44

0.06

(c) Ha = 100




0.94

0.38

0.13

0.63

(a) Ha = 0

0.94

0.50

0.31

0.06

(b) Ha = 25

0.94

0.50

0.25

0.06

(c) Ha = 100




0.94

0.56

0.25

0.06

(a) = 0.2

0.94

0.38

0.06

(b) = 0.8

Figure 7: Streamlines (on the left) and isotherms (on the right) for Ri = 103, Ha = 50 andDa = 102.



0.06

0.88

0.94

0.81

(a) Da = 104

0.88

0.94

0.31

(b) Da = 101

Figure 8: Streamlines (on the left) and isotherms (on the right) for Ri = 103, Ha = 0 and = 0.4.



U

Y

-0.2 -0.1 0 0.1 0.20

0.5

1

C

Ha= 100, 50, 25, 0

X=0.5

Ri=10-3

XV

0 0.25 0.5 0.75 1-0.3

-0.2

-0.1

0

0.1

0.2

Ha= 100, 50, 25, 0

C

Y=0.5 Ri=10-3

U

Y

-0.3 0 0.3 0.6 0.90

0.5

1

Ha= 100, 50, 25, 0

C

X=0.5Ri=10

-2

Y

V

0 0.5 1-0.2

-0.1

0

0.1

Ha= 100, 50, 25, 0

C

Y=0.5Ri=10-2

U

Y

-0.3 0 0.3 0.6 0.90

0.5

1

C

X=0.5

Ha= 100, 50, 25, 0Ri=10

-1

X

V

0 0.5 1-0.2

-0.1

0

0.1

0.2

Ha= 100, 50, 25, 0

Y=0.5

C

Ri=10-1

Figure 9: Vertical (on the the left) and horizontal (on the right) velocity profiles at mid-plane ofthe cavity for = 0.4 and Da = 102.



X

Nu

0 0.5 10

5

10

15

20

25

loc

Da = 100, 50, 25, 0

Ri= 10 -1 Hot wall

Figure 10: Local Nusselt number for = 0.4 and Da = 102.

Ha

Nu

5

10

15Hot wall

0 25 50 100

avg

Ri=10-3

Ri=10-2

Ri=10-1

Figure 11: Average Nusselt number for = 0.4 and Da = 102.


mixed con in porous media

Documents