Transp Porous Med (2010) 82:337–346 DOI 10.1007/s11242-009-9429-7 TECHNICAL REPORT

Convection in a Lid-Driven Heat-Generating Porous Cavity with Alternative Thermal Boundary Conditions M. Muthtamilselvan · Manab Kumar Das · P. Kandaswamy

Received: 5 March 2009 / Accepted: 5 June 2009 / Published online: 21 June 2009 © Springer Science+Business Media B.V. 2009

Abstract Mixed convection flow in a two-sided lid-driven cavity filled with heatgenerating porous medium is numerically investigated. The top and bottom walls are moving in opposite directions at different temperatures, while the side vertical walls are considered adiabatic. The governing equations are solved using the finite-volume method with the SIMPLE algorithm. The numerical procedure adopted in this study yields a consistent performance over a wide range of parameters that were 10−4 ≤ Da ≤ 10−1 and 0 ≤ Ra I ≤ 104 . The effects of the parameters involved on the heat transfer characteristics are studied in detail. It is found that the variation of the average Nusselt number is non-linear for increasing values of the Darcy number with uniform or non-uniform heating condition. Keywords

Mixed convection · Heat generation · Porous medium · Finite-volume method

List of symbols H Enclosure length (m) Ke Effective thermal conductivity of porous medium (Wm−1 K−1 ) Nu Local Nusselt number N u avg Average Nusselt number q  Volumetric heat generation Ra E External Rayleigh number (Gr·Pr) c T Dimensionless temperature, θθh−θ −θc U0 Lid-driven velocity (m/s) Uc Dimensionless velocity in the x-direction at the mid-plane of the cavity M. Muthtamilselvan (B) · P. Kandaswamy Centre for Fluid Dynamics, Department of Mathematics, Bharathiar University, Coimbatore 641046, India e-mail: [email protected] M. K. Das Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, Kharagpur 721302, India

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u, v Vc x, y

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Velocities in x- and y-directions Dimensionless velocity in the y-direction at the mid-plane of the cavity Cartesian coordinates

Greek symbols αe Effective thermal diffusivity of the porous medium (m2 s−1 ) β Coefficient of thermal expansion of fluid (K−1 ) θ Temperature difference θ Temperature (◦ C) K Permeability of porous medium (m2 ) µ Effective dynamic viscosity (Pa s−1 ) ν Effective kinematic viscosity, µ/ρ ρ Fluid density (kgm−3 ) Subscripts avg Average C Cold wall H Hot wall l Local

1 Introduction Mixed convection flow in a lid-driven enclosure has attracted considerable attention due to its relevance to a wide variety of application area in engineering and science. Some of these include cooling of electronic devices, oil extraction, solar collectors, nuclear reactors, crystal growth, etc. A number of experimental and numerical studies have been conducted to investigate the flow field and heat transfer characteristics of lid-driven cavity flow in the past several decades. Prasad and Koseff (1996) performed an experimental investigation of mixed convection flow in a lid-driven cavity. They considered heated moving bottom wall and high Reynolds number and high Grashof number. Flow through a confined porous medium has received a great deal of attention in the recent years. This is due to a large number of technological and industrial applications such as storage of nuclear waste, separation processes in chemical industries, crude oil production, etc. The presence of internal heat generation provides an additional dynamic in overall convective flow systems. Natural convection in enclosures partially filled with a heat-generating porous medium was numerically solved by Du and Bilgen (1990) and Kim et al. (2001). Recently, Kim and Hyun (2004) investigated buoyant convection in a rectangular cavity filled with heat-generating porous medium. They used a power-law non-Newtonian fluid in enclosure and concluded that as the enclosure aspect ratio increases or power-law model increases, the flow enters the boundary-layer regime at the higher Reynolds number. Khanafer and Chamkha (1999) examined the mixed convection flow in a lid-driven enclosure filled with a Darcian fluid-saturated uniform porous medium and formulated the presence of internal heat generation by using the vorticity–stream function formulation approach. They indicated that the results are strongly dependent on the Richardson number and focused on the internal heat generation for small Richardson number.

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Wan and Kuznetsov (2004) numerically investigated the acoustic streaming in a rectangular cavity induced by the vibration of its lid. Mixed convection in a lid-driven square enclosure filled with water-saturated aluminum foams was investigated numerically by Jang and Tzeng (2008). They found that the higher porosity promotes much more enhancement of convective heat transfer, but the lower porosity is desired for higher total heat transfer due to the higher value of effective thermal conductivity. The similar type of research was carried out by Khanafer and Vafai (2002); Kandaswamy et al. (2008); Vishnuvardhanarao and Das (2008). From a particle point of view, flows at higher Ra E and Ra I are of much interest. An extensive series of experimental investigations has been carried out by Kawara et al. (1990). They performed the range of parameters that were 106 ≤ Ra E ≤ 108 and 108 ≤ Ra I ≤ 1010 and for the aspect ratio A, 0.3 ≤ A ≤ 5.0. Their results show clear identifications and physically insightful descriptions were given on the main structure of the flow field, as the relative strength of internal heat-generation is altered. This study is carried out to examine the characteristics of a lid-driven flow in a two-dimensional square cavity filled with heat-generating porous medium. The obtained numerical results are presented graphically in terms of streamlines, isotherms, and velocity profiles at the mid-section of the cavity to illustrate the interesting features of the solution.

2 Mathematical Analysis Consider a steady-state two-dimensional square cavity filled with a heat-generating porous medium of height H as shown in Fig. 1. It is assumed that the bottom wall is moving from left to right at a constant speed U0 and is maintained at uniform or non-uniform temperatures. The top wall is moving from right to left at a constant speed U0 and is maintained at a constant temperature. The vertical side walls are considered to be adiabatic. The physical properties are considered to be constant except the density variation in the body-force term of

Tc U0

Adiabatic

H

Adiabatic

Porous medium

g

y U0

x

Th Fig. 1 Flow configuration and coordinate system

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the momentum equation, which is satisfied by the Boussinesq’s approximation. In this investigation, the porous medium is assumed to be hydrodynamically and thermally isotropic and saturated with a fluid that is in local thermal equilibrium (LTE) model with the solid matrix. Using the above assumptions, the governing equations for mass, momentum, and energy can be written in the non-dimensional form: ∂U ∂V + =0 ∂X ∂Y

(1)


U ∂U ∂P 1 ∂U +V =− + 2 U − U ∂X ∂Y ∂X  Re ReDa 1/2  U 1.75 U 2 + V 2 −√ √  3/2 Da 150 
∂V ∂P 1 ∂V V 1 + V =− + 2 V − U 2  ∂X ∂Y ∂Y  Re ReDa  2 1/2 2 V Gr 1.75 U + V + T −√ √ 3/2 2  Re Da 150 Ra I 1 ∂T ∂T 1 U +V = 2 T + . ∂X ∂Y Pr Re Ra E Pr Re

1 2

(2)

(3) (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. The dimensionless variables are defined as

Re =

X=

x v θ − θc y u , V = , T = , , Y = , U= H H U0 U0 θ h − θc

U0 H ν

(Reynolds number), Gr =

number), Pr = gβθ H 3 ναe

ν αe

(Prandtl number),

P=

p , ρU02

gβθ H 3 (Grashof number), Da = HK2 (Darcy ν2  H 5 Ra I = gβq ναe K e (Internal Rayleigh number), and

(External Rayleigh number) Ra E = The dimensionless boundary conditions can be written as U = −1, V = 0, T = 0 (Y = 1) U = 1, V = 0, T = 1 or T = sin(π X ) (Y = 0) ∂T = 0 (X = 0, 1). U = V = 0, ∂X The local Nusselt number can be expressed as N u = 1 number can be defined as N u avg = 0 N udX.

∂T ∂Y

− Pr ReV T , and the average Nusselt

3 Method of Solution and Code Validation Numerical solutions to the governing equations are secured by employing the finite-volume computational procedure using staggered grid arrangement with the SIMPLE algorithm as given in Patankar (1980). The convective fluxes in the interior points are discretized by using the deferred QUICK scheme (Hayase et al. 1992), and the central difference scheme was

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used in adjacent to the boundaries. The resulting algebraic equations are solved by using tridiagonal matrix (TDMA) algorithm. The pseudo-transient approach is followed for the numerical solution, as it is useful for a situation in which the governing equations give rise to stability problems, e.g., buoyant flows (Versteeg and Malalasekera 1995). The iteration is carried out until the normalized residuals of the mass, momentum, and temperature equation become less than 10−7 . Considering both the accuracy and the computational time, all the computations are performed with a 81 × 81 uniform grid. The validation of the present computational code has been verified by Khanafer and Chamkha (1999) for the mixed convection flow in a lid-driven enclosure filled with a fluid-saturated porous medium problem. Figure 2 shows the mid-plane velocity profiles for different Darcy Fig. 2 Comparison of the present velocity profiles with Khanafer and Chamkha (1999) for Pr = 0.71 and Ra I = 0

1

X=0.5

Khanafer and Chamkha

Y

Present 0.5

-2

-3

-4

Da= ∝ , 10, 10 and 10

0 -0.4

0

0.4

0.8

UC

Y=0.5 Khanafer and Chamkha

0.2

Vc

Present

0 -2

-3

-4

Da= ∝ , 10, 10 and 10

-0.2

0

0.5

1

X

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numbers. This comparison shows a good agreement between the two numerical results. This effect provided credence to the accuracies of the present numerical solutions.

4 Results and Discussion The present computation focuses on the parameters having the following ranges: the Darcy number from 10−4 to 10−1 , the internal Rayleigh number from 0 to 105 , the Prandtl number from 0.01 to 10, the Reynolds number from 100 to 500, and the Grashof number from 103 to 105 . For all the cases,  = 0.4 is taken. The results are reduced to carry out a parametric study showing influences of various non-dimensional parameters. In general, the fluid circulation is strongly dependent on Darcy numbers as shown in Fig. 3. Figure 3 illustrates the streamlines and isotherms for Ra I = 103 and different values of the Darcy numbers with uniform and non-uniform heating conditions. For Da = 10−4 , the streamlines show a recirculation flow region of small intensity close to the top part of the cold wall and to the bottom part of the hot wall and a spin of the fluid toward the center of the cavity (Fig. 3a). The isotherms which are nearly horizontal indicate conduction heat transfer mode. When the Darcy number is increased to 10−3 , the streamlines indicate an elongation of the recirculation region of the flow(Fig. 3b). It is interesting to note that when Da = 10−2 , the streamlines precipitate in merging two cells into one (Fig. 3c). In this case, isotherms show that the horizontal stratification of isotherms breaks down and becomes vertical in the middle part of the cavity. The non-uniform heating isotherm plots indicate that the maximum temperature occurs at the center of the bottom wall of the cavity due to the non-uniform heating condition. A similar type of heating condition is used by Basak et al. (2006). The crowded thermal boundary layer on top of the cavity for uniform heating condition is reduced for non-uniform heating condition. Darcy number helps in minimizing the conduction heat transfer and in increasing the convection heat transfer. When the Darcy number is increased to 10−1 , streamlines show that the circulation of flow increases and the isotherms illustrate a dominance of mixed convection. When the Darcy number is increased from 10−4 to 10−1 , the isotherm plots indicate that the thermal boundary layers become denser. Similar trend of figures is obtained for Ra I = 0 and slight variations are also obtained for Ra I = 104 (Figures are not shown). The velocity profiles of the mid-plane of the cavity for various Darcy numbers with Ra I = 0 is displayed in Fig. 4. It is clearly observed that for Da ≥ 10−2 , the flow is very strong. Therefore, convection heat transfer is more dominant for high Darcy numbers (Da > 10−3 ). Figure 5 illustrates the effect of Darcy number with uniform and non-uniform heating conditions. For uniform heating condition of the bottom wall, the heat transfer rate is very high at the left edge of the bottom wall and the heat transfer rate reduces toward the right corner of the bottom wall for all Darcy numbers. For non-uniform heating condition of the bottom wall, the heat transfer rate is very high at the edges, X = 0.3 and X = 0.9 of the bottom wall, and it is minimum between X = 0.1 and 0.2 and X = 0.6 to 0.7. The non-uniform heating condition produces a sinusoidal type of local heat transfer rate. The average Nusselt number for different values of the Darcy number and internal Rayleigh number with uniform and non-uniform heating conditions is displayed in Fig. 6. It is observed that when the internal Rayleigh number is 0, the average heat transfer rate increases with increase in Darcy numbers. This figure also shows that the heat transfer rate is increased suddenly from Da = 0.001 to 0.01. For non-uniform heating condition, the average heat transfer is more compared to uniform heating condition with fixed value of Darcy number. It is also observed that when the internal Rayleigh number is increased to 105 , the average

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0.06

0.13

0.44

0.42

0.88

0.71

(a) 0.1

3

0.2

63 0.

0

0.4

9

(b)

0.81

0.20

0.25

0. 64

0.6

0.25

0.13

(c)

9

0.42 0.64

(d) Fig. 3 Streamlines (on the left), isotherms uniform heating (on the middle), and isotherms non-uniform heating (on the right) for Ra I = 103

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1

RaI= 0

RaI= 0

V

Y

0.5

0.5 -4

-3

-2

0

-1

Da= 10, 10, 10 and 10

-4

0

-3

-2

-1

Da= 10, 10, 10 and 10

-0.5

-1 -1

0

1

0

0.5

Uc

1

X

Fig. 4 Velocity profiles at the mid-plane of the cavity for uniform heating case

Fig. 5 Variation of local Nusselt number for uniform heating and non-uniform heating

10

Hot wall

uniform non-uniform RaI= 10

4

-1

Nu l

Da= 10

5

-3

Da= 10

0 0

0.5

1

X

heat transfer decreases for increasing values of the Darcy number. It is found that when the internal heat generation decreases, the external heat increases inside the cavity. The variation of the average Nusselt number for different values of Prandtl number and internal Rayleigh number with uniform and non-uniform heating conditions are displayed in Fig. 7. It is observed that at both Ra I = 103 and 105 , the average heat transfer rate increases with increasing of Prandtl number for uniform and non-uniform heating conditions. Figure 8 shows the average Nusselt number for different values of Darcy number and Grashof number with uniform and non-uniform conditions. This figure shows the fixed value of Grashof number. The average heat transfer rate is non-linear with increasing Darcy number. This figure also shows that the maximum heat transfer rate is observed when Da = 0.1, Gr = 105 and non-uniform heating condition.

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5

RaI=10 6

5

10

Nu avg

4.5

0 3

0 1.5

Uniform Non uniform

0 0.001

0.0001

0.1

0.01

Da Fig. 6 Variation of average Nusselt number (Da vs. Ra I ) for uniform heating and non-uniform heating Fig. 7 Variation of average Nusselt number (Ra I vs. Pr ) for uniform heating and non-uniform heating

35 5

3

RaI=10, 10

30

Uniform

Nu avg

25

Non uniform

20

15 5

10

10

3

10

5

0 0.01

7.0

0.71

10

Pr

5 Conclusion Mixed convection flow and heat transfer are examined numerically in a two-sided lid-driven cavity with internal heat-generating porous medium. Two different variations of the boundary conditions are considered. It is found that the fluid flow switches from conduction heat transfer to mixed convection heat transfer at Da > 10−3 for both the boundary conditions. When the Darcy number is increased from 10−4 to 10−1 , the thermal boundary layer becomes more crowded for both heating conditions. Also, it is found that when Ra I > 102 , the average heat transfer rate decreases with fixed value of the Darcy number.

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Uniform 6

Non uniform

5

Nu avg

4

3

5

Gr=10, 10 3

2

1

0 0.0001

0.01

0.001

0.1

Da Fig. 8 Variation of average Nusselt number (Gr vs. Da) for uniform heating and non-uniform heating

References Basak, T., Roy, S., Paul, T., Pop, I.: Natural convection in a square cavity filled with a porous medium: effects of various thermal boundary conditions. Int. J. Heat Mass Transf. 49, 1430–1441 (2006) Du, Z.G., Bilgen, E.: Natural convection in vertical cavities with partially filled with heat-generating porous media. Numer. Heat Transf. Part A 18, 371–386 (1990) Hayase, T., Humphrey, J.A.C., Grief, R.: A consistently formulated QUICK scheme for fast and stable convergence using finite-volume iterative procedures. J. Comput. Phys. 98, 108–118 (1992) Jang, T.M., Tzeng, S.C.: Heat transfer in a lid-driven enclosure filled with water-saturated aluminum foams. Numer. Heat Transf. Part A 54, 178–196 (2008) Kandaswamy, P., Muthtamilselvan, M., Lee, J.: Prandtl number effects on mixed convection in a lid-driven porous cavity. J. Porous Media 11, 791–801 (2008) Kawara, Z., Kishiguchi, I., Aoki, N., Michiyoshi, I.: Natural convection in a vertical fluid layer with internal heating. Proceedings of 27th national heat transfer symposium Japan, II. pp. 115–117 (1990) Khanafer, K.M., Chamkha, A.J.: Mixed convection flow in a lid-driven enclosure filled with a fluid-saturated porous medium. Int. J. Heat Mass Transf. 42, 2465–2481 (1999) Khanafer, K.M., Vafai, K.: Double diffusive mixed convection in a lid-driven enclosure filled with a fluidsaturated porous medium. Numer. Heat Transf. Part A 42, 465–486 (2002) Kim, G.B., Hyun, J.M.: Buoyant convection of a power-law fluid in an enclosure filled with heat-generating porous media. Numer. Heat Transf. Part A 45, 569–582 (2004) Kim, G.B., Hyun, J.M., Kwak, H.S.: Buoyant convection in a square cavity partially filled with heat-generating porous medium. Numer. Heat Transf. Part A 40, 601–618 (2001) Patankar, S.V.: Numerical Heat Transfer and Fluid Flow. Hemisphere, Washington DC (1980) Prasad, A.K., Koseff, J.R.: Combined forced and natural convection heat transfer in a deep lid-driven cavity flow. Int. J. Heat Fluid Flow 17, 460–467 (1996) Versteeg, H.K., Malalasekera, W.: An Introduction to Computational Fluid Dynamics: The Finite Volume Method. Longman Group Ltd., Malaysia (1995) Vishnuvardhanarao, E., Das, M.K.: Laminar mixed convection in a parallel two-sided lid-driven differentially heated square cavity filled with fluid-saturated porous medium. Numer. Heat Transf. Part A 53, 83–110 (2008) Wan, Q., Kuznetsov, A.V.: Investigation of the acoustic streaming in a rectangular cavity induced by the vibration if its lid. Int. Commun. Heat Mass Transf. 31, 467–476 (2004)

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