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Numerical analysis of natural convection for a porous rectangular enclosure with sinusoidally varying temperature profile on the bottom wall
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International Communications in Heat and Mass Transfer 35 (2008) 56 – 64 www.elsevier.com/locate/ichmt
Numerical analysis of natural convection for a porous rectangular enclosure with sinusoidally varying temperature profile on the bottom wall☆ Yasin Varol a , Hakan F. Oztop b , Ioan Pop c,⁎ a b
Department of Mechanical Education, Firat University, TR-23119, Elazig, Turkey Department of Mechanical Engineering, Firat University, TR-23119, Elazig, Turkey c Faculty of Mathematics, University of Cluj, R-3400 Cluj, CP 253, Romania Available online 3 July 2007
Abstract Numerical investigations of steady natural convection flow through a fluid-saturated porous medium in a rectangular enclosure with a sinusoidal varying temperature profile on the bottom wall were conducted. All the walls of the enclosure are insulated except the bottom wall which is partially heated and cooled. The governing equations were written under the assumption of Darcy-law and then solved numerically using finite difference method. The problem is analyzed for different values of the Rayleigh number Ra in the range 10 ≤ Ra ≤ 1000, aspect ratio parameter AR in the range 0.25 ≤ AR ≤1.0 and amplitude λ of the sinusoidal temperature function in the range 0.25 ≤ λ ≤ 1.0. It was found that heat transfer increases with increasing of amplitude λ and decreases with increasing aspect ratio AR. Multiple cells were observed in the cavity for all values of the parameters considered. © 2007 Elsevier Ltd. All rights reserved. Keywords: Porous medium; Free convection; Rectangular enclosure; Sinusoidal temperature profile
1. Introduction The heat and fluid flow in fluid-saturated porous media is an important problem in engineering and it has gained significant attention over the last three decades. Applications of porous media can be found in grain storage, chemical catalytic reactors, geophysical problems, solar collectors, heat exchangers, etc. These applications are reviewed in recent books by Nield and Bejan [1], Vafai [2], Ingham and Pop [3]. Natural convection in a rectangular/square enclosure filled with a fluid-saturated porous medium under different temperature or heat flux boundary conditions has been extensively analyzed in earlier studies by Bejan [4], Prasad and Kulacki [5], Gross et al. [6], Goyeau et al. [7], Manole and Lage [8], Holzbecher [9], Kumar et al. [10], Khanafer and Chamkha [11], Alchaar et al. [12] etc. In most of these studies, isothermal or isoflux thermal boundary conditions were applied to the side walls of the rectangular/square enclosures. However, non-isothermal thermal boundary conditions, which vary with time can be found in electrical arc furnaces and rotary burners, as given by Gogus et al. [13]. Another ☆
Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (I. Pop).
0735-1933/$ - see front matter © 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2007.05.015
Y. Varol et al. / International Communications in Heat and Mass Transfer 35 (2008) 56–64
57
Nomenclature AR g Gr H K L Nux Nu Pr Ra Tref ΔT u,v X,Y
aspect ratio parameter gravitational acceleration Grashof number height of rectangular permeability of the porous medium length of the bottom wall local Nusselt number mean Nusselt number Prandtl number Rayleigh number reference temperature temperature difference axial and radial velocities non-dimensional coordinates
Greek letters αm thermal diffusivity β thermal expansion coefficient λ amplitude of the sinusoidal temperature distribution θ non-dimensional temperature υ kinematic viscosity ψ non-dimensional stream function
Subscript C cold H hot
application of this boundary condition occurred in case of cylindrical heater, as given by Saeid [14]. Hossain and Wilson [15] performed a numerical study to investigate the natural convection in a porous enclosure with internal heat generation under linearly varying temperature boundary condition. Roy and Basak [16] performed a numerical study to analyze the natural convection in a square cavity filled with a viscous fluid (non-porous media) with non-uniformly heated wall (with a sinusoidal temperature variation) by using Galerkin finite element method. They observed that in case of uniform heating, the heat transfer rate is very high at the right edge of the bottom wall and almost uniform at the rest part of the bottom wall, as well as the hot vertical wall. In contrast, for the case of non-uniform heating, the heat transfer rate is minimum at the edges of the heated walls and the heat transfer rate reaches its maximum value at the center of both heated walls. Other studies of the natural convection over a flat plate or in enclosures with sinusoidal temperature boundary condition are due to Bilgen and Ben Yedder [17], Storesletten and Pop [18], Bradean et al. [19], Basak et al. [20] and Yoo [21]. The main purpose of this study is to analyze the natural convective heat transfer in a square enclosure filled with a fluid-saturated porous medium with non-isothermally heated wall for different Rayleigh numbers. The above literature review shows that the case of non-isothermally heated enclosure under non-isothermal boundary conditions has not been addressed. However, to the authors' best knowledge, sinusoidal thermal boundary conditions applied to a porous media enclosure has been considered only by Saeid [14]. The schematical diagram of the considered physical model is depicted in Fig. 1 with the mentioned boundary conditions and the Cartesian coordinates. It is a square enclosure filled with a fluid saturated porous medium. All walls are insulated except the bottom wall, which has a sinusoidally varying temperature profile.
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Y. Varol et al. / International Communications in Heat and Mass Transfer 35 (2008) 56–64
2. Basic equations Darcy's law is used in the porous medium with the Boussinesq approximation. Both viscous drag and inertia force terms are neglected in the basic equations. Thus, the steady two-dimensional governing equations (continuity, Darcy, and energy equations) are ∂u ∂v þ ¼0 ∂x ∂y
ð1Þ
∂u ∂v gbK ∂T ¼ ∂y ∂x t ∂x
ð2Þ
2 ∂T ∂T ∂ T ∂2 T þv ¼ am u þ ∂x ∂y ∂x2 ∂y2
ð3Þ
where u and v are the velocity components along x and y axes, respectively, T is the fluid temperature, K is the permeability of the porous medium and the physical meaning of the other quantities are mentioned in the Nomenclature. Eqs. (1) and (2) can be written in terms of the stream function ψ defined as u¼
∂w ∂w ;v ¼ ∂y ∂x
ð4Þ
According to above parameters boundary conditions can be written as on all solid walls: w¼0
ð5Þ
on the adiabatic walls: ∂T ¼0 ∂n
ð6Þ
on the bottom wall: T ð xÞ ¼ Tref FDT Sinð 2px=LÞ
ð7Þ
We define now the following non-dimensional variables x y w ðu; vÞH T TC ; ðU ; V Þ ¼ ;h ¼ X ¼ ;Y ¼ ;W ¼ L H am am DT
ð8Þ
so that Eqs. (2) and (3) can be written as AR2
AR
∂2 W ∂2 W ∂h þ ¼ AR Ra ∂X 2 ∂Y 2 ∂X
∂W ∂h ∂W ∂h ∂2 h ∂2 h þ ¼ AR2 ∂Y ∂X ∂X ∂Y ∂X 2 ∂Y 2
ð9Þ
ð10Þ
where AR = H/L is the cavity ratio and Ra = g β K(TH − TC)H/αmυ is the Rayleigh number for the porous medium. The boundary conditions for the considered model are shown in Fig. 1. Thus, on all solid walls: W¼0
ð11Þ
Y. Varol et al. / International Communications in Heat and Mass Transfer 35 (2008) 56–64
59
Fig. 1. Physical model, coordinates and boundary conditions.
on the adiabatic walls: ∂h ¼0 ∂n
ð12Þ
on the bottom wall: h ¼ ksinð 2pX Þ
ð13Þ
The physical quantities of interest in this problem are the local and mean Nusselt numbers, which are given by Z L ∂h Nux ¼ ; Nu ¼ Nux dx ð14a:bÞ ∂Y Y ¼0 0 3. Numerical technique The finite difference method is used to solve Eqs. (9) and (10) with central differences for discretization of these equations. The solution of linear algebraic equations was performed using Successive Under Relaxation (SUR)
Table 1 Comparison of the mean Nusselt number Nu for AR = 1.0 and Ra = 1000 Paper
Nu
Bejan [4] Gross et al. [6] Goyeau et al. [7] Manole and Lage [8] Baytas and Pop [22] Saeid and Pop [23] This study
15.800 13.448 13.470 13.637 14.060 13.726 13.564
60
Y. Varol et al. / International Communications in Heat and Mass Transfer 35 (2008) 56–64
Fig. 2. Streamlines (on the left) and isotherms (on the right) for AR = 1.0, λ = 0.50 and different values of the Rayleigh numbers: a) Ra = 10, b) Ra = 100, c) Ra = 500.
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Fig. 3. Streamlines (on the left) and isotherms (on the right) for λ = 0.50, Ra = 1000, and different values of the ratios: a) AR = 1.0, b) AR = 0.50, c) AR = 0.25.
−4
method. As convergence criteria 10 is chosen for all depended variables and value of 0.1 is taken for under-relaxation parameter. The number of grid points is taken as 61 × 61 with uniform spaced mesh in both X- and Y- directions. In order to verify the accuracy of the results, the numerical algorithm used was tested with the classical natural convection heat transfer problem in a differentially heated square porous enclosure (AR = 1.0). The obtained numerical results for the mean Nusselt number Nu, as given by Eq.(14b), are compared with those given by different authors as tabulated in Table 1. As can be seen from the table, the obtained results show good agreement with the results reported in the literature. Contours of streamline and isotherms are almost the same to those given in the literature for a square porous enclosure but they are not presented here to save space. 4. Results and discussion The temperature distribution and the flow field due to buoyancy-forces in a rectangular enclosure filled with a fluid saturated porous medium with the temperature distribution of the bottom wall varying sinusoidally has been analyzed numerically for different values of the aspect ratio AR, Rayleigh number Ra and amplitude λ of sinusoidal temperature. Results for streamlines, isotherms, local and mean Nusselt numbers for these governing parameters will be presented. Streamlines (left) and isotherms (right) are presented in Fig. 2 for λ = 0.50 and AR = 1.0 to show the effect of the Rayleigh number Ra on the free convection flow. It is seen that the streamline contours exhibit circulation patterns,
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Y. Varol et al. / International Communications in Heat and Mass Transfer 35 (2008) 56–64
Fig. 4. Variation of the local Nusselt number with X: a) different values of Ra when λ = 0.50 and AR = 1.0; b) different values of AR when Ra = 1000 and λ = 0.50; c) different value of λ when Ra = 1000 and AR = 1.0.
which are characterized by the three vortices. The fluid motion, as it is driven by the effect of the buoyancy, is distributed from the heated part of the bottom wall through the inside of the enclosure. Two small cells were formed in the left and right bottom corners in clockwise and counterclockwise rotating direction, respectively. However, a huge main cell was formed with its center located almost in the middle of the enclosure, as shown in Fig. 2 (a). The center of the cell moves towards the left bottom corner with increasing of Ra (Fig. 2 (b) and (c)). The circulation strength is very weak ψmin = − 0.25 due to the domination of conduction mode of heat transfer. But the flow strength increases with increasing of Ra and it can be seen from the values of the stream function on the streamline. The length and locations of the small cells become almost the same when the values of the Rayleigh number are changed. Isotherms show an almost symmetrical distribution for low Rayleigh number due to domination of conduction mode of heat transfer. But temperature gradient at the symmetry plane is very small. Thermal boundary layer becomes thinner with increasing of the Rayleigh number and plume like temperature distribution was observed on the heated part (positive values of sinusoidal function) of the enclosure. Fig. 3 shows the flow field and temperature distribution at different values of AR when Ra = 1000. In Fig. 3 (a) the streamlines (left) and isotherms (right) are shown for AR = 1.0 and Ra = 1000. It can be seen from that the flow strength becomes strong and convection mode of heat transfer is dominant to conduction. As AR becomes smaller, the main cell is depressed, see Fig. 3 (b). Thus, the length of the main and corner cells increase with decreasing AR. On the contrary, the flow strength increases with increasing the values of AR. Results of the local Nusselt number Nux are presented in Fig. 4 (a) to (c). Fig. 4 (a) shows the effect of Ra on the variation of the local Nusselt number with the coordinate X. It is seen that Nux is positive when the heat is transferred into the enclosure and negative when it is transferred from the enclosure to the environment. This result is similar to
Y. Varol et al. / International Communications in Heat and Mass Transfer 35 (2008) 56–64
63
Fig. 5. Variation of the mean Nusselt number with the Rayleigh number for different values of AR when λ = 0.50 and 1.0.
those reported by Bilgen and Yedder [17] for a rectangular cavity filled with air the cavity being heated and cooled from the left vertical side. The maximum value of Nux increases with increasing the Rayleigh number in both positive and negative regions due to the increase of the effect of convection mode of heat transfer. Fig. 4 (b) is plotted to obtain the effect of AR on Nux. The figure shows that the effect of AR is very small, especially at cooling region (right side of the figure). But Nux decreases with increasing AR due to decreasing the volume of the enclosure. Effect of amplitude of sinusoidal function on Nux is given in Fig. 4 (c). Further, the value of the amplitude λ is directly related to the temperature distribution at the bottom wall and it affects the heat transfer inside the enclosure. As expected, heat transfer increases with increasing the values of λ. Finally, Fig. 5 summarizes the variation of the average Nusselt number Nu with Ra for some values of AR and two values of λ. It is observed that Nu increases with increasing of Ra and λ. However AR shows small variation of Nu for low value of λ. On the contrary, Nu increases with decreasing of AR due to low volume of the enclosure. 5. Conclusions A numerical analysis of the steady natural convection flow through a fluid-saturated porous medium in a rectangular enclosure due to sinusoidally varying temperature distribution of the bottom wall has been performed. It was observed that multiple circulation cells were formed for all governing parameters which describe this problem. It was found that heat transfer increases with increasing of amplitude of sinusoidal function and decreases with increasing of aspect ratio. Effects of aspect ratio become significant especially for higher values of amplitude. Both flow strength and heat transfer increases with increasing of the Rayleigh number. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
D.A. Nield, A. Bejan, Convection in Porous Media, 3rd edition, Springer, New York, 2006. K. Vafai, Handbook of Porous Media, Marcel Dekker, New York, 2000. D.B. Ingham, I. Pop, Transport Phenomena in Porous Media, Pergamon, Oxford, 1998. A. Bejan, On the boundary layer regime in a vertical enclosure filled with a porous medium, Lett. Heat Mass Transf. 6 (1979) 93–102. V. Prasad, F.A. Kulacki, Natural convection in a rectangular porous cavity with constant heat flux on one vertical wall, ASME J. Heat Transfer 106 (1984) 152–157. R.J. Gross, M.R. Bear, C.E. Hickox, The application of flux-corrected transport (FCT) to high Rayleigh number natural convection in a porous medium, Proc. 8th Int. Heat Transfer Conf., San Francisco, CA, 1986. B. Goyeau, J.P. Songbe, D. Gobin, Numerical study of double-diffusive natural convection in a porous cavity using the Darcy–Brinkman formulation, Int. J. Heat Mass Transf 39 (1996) 1363–1378. D.M. Manole, J.L. Lage, Numerical benchmark results for natural convection in a porous medium cavity, Heat and Mass Tr. Porous Media, ASME Conf., HTD-216, 1992, pp. 55–60. E. Holzbecher, Free convection in open-top enclosures filled with a porous medium heated from below, Numer. Heat Transf., A Appl. 46 (2004) 241–254.
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[10] B.V.R. Kumar, P. Singh, V.J. Bansod, Effect of thermal stratification on double diffusive natural convection in a vertical porous enclosure, Numer. Heat Transf., A Appl. 41 (2002) 421–447. [11] K.M. Khanafer, A.J. Chamkha, Hydromagnetic natural convection from an inclined porous square enclosure with heat generation, Numer. Heat Transf., A Appl. 34 (1998) 343–356. [12] S. Alchaar, P. Vasseur, E. Bilgen, Hydromagnetic natural convection in a tilted rectangular porous enclosure, Numer. Heat Transf., A Appl. 27 (1995) 107–127. [13] Y.A. Gogus, U. Camdali, M. Kavsaoglu, Energy balance of a general system with variation of environmental conditions and some applications, Energy 27 (2002) 625–646. [14] N.H. Saeid, Natural convection in porous cavity with sinusoidal bottom wall temperature variation, Int. Comm. Heat Mass Transfer 32 (2005) 454–463. [15] M.A. Hossain, M. Wilson, Natural convection flow in a fluid-saturated porous medium enclosed by non-isothermal walls with heat generation, Int. J. Thr. Sci. 41 (2002) 447–454. [16] S. Roy, T. Basak, Finite element analysis of natural convection flows in a square cavity with non-uniformly heated wall(s), Int. J. Eng. Sci. 43 (2005) 668–680. [17] E. Bilgen, R. Ben Yedder, Natural convection in enclosure with heating and cooling by sinusoidal temperature profiles on one side, Int. J. Heat Mass Transfer 50 (2007) 139–150. [18] L. Storesletten, I. Pop, Free convection in a vertical porous layer with walls at non-uniform temperature, Fluid Dyn. Res. 17 (1996) 107–119. [19] R. Bradean, D.B. Ingham, P.J. Heggs, I. Pop, Free convection fluid flow due to a periodically heated and cooled vertical flat plate embedded in a porous media, Int. J. Heat Mass Transf. 39 (1996) 2545–2557. [20] T. Basak, S. Roy, T. Paul, I. Pop, Natural convection in a square cavity filled with a porous medium: Effects of various thermal boundary conditions, Int. J. Heat Mass Transf. 49 (2006) 1430–1441. [21] J.S. Yoo, Thermal convection in a vertical porous slow with spatially periodic boundary conditions: low Ra flow, Int. J. Heat Mass Transf. 46 (2003) 381–384. [22] A.C. Baytas, I. Pop, Free convection in a square porous cavity using a thermal nonequilibrium model, Int. J. Heat Mass Transf. 41 (2002) 861–870. [23] N.H. Saeid, I. Pop, Natural convection from a discrete heater in a square cavity filled with a porous medium, J. Porous Media 8 (2005) 55–63.
International Communications in Heat and Mass Transfer 35 (2008) 56 – 64 www.elsevier.com/locate/ichmt
Numerical analysis of natural convection for a porous rectangular enclosure with sinusoidally varying temperature profile on the bottom wall☆ Yasin Varol a , Hakan F. Oztop b , Ioan Pop c,⁎ a b
Department of Mechanical Education, Firat University, TR-23119, Elazig, Turkey Department of Mechanical Engineering, Firat University, TR-23119, Elazig, Turkey c Faculty of Mathematics, University of Cluj, R-3400 Cluj, CP 253, Romania Available online 3 July 2007
Abstract Numerical investigations of steady natural convection flow through a fluid-saturated porous medium in a rectangular enclosure with a sinusoidal varying temperature profile on the bottom wall were conducted. All the walls of the enclosure are insulated except the bottom wall which is partially heated and cooled. The governing equations were written under the assumption of Darcy-law and then solved numerically using finite difference method. The problem is analyzed for different values of the Rayleigh number Ra in the range 10 ≤ Ra ≤ 1000, aspect ratio parameter AR in the range 0.25 ≤ AR ≤1.0 and amplitude λ of the sinusoidal temperature function in the range 0.25 ≤ λ ≤ 1.0. It was found that heat transfer increases with increasing of amplitude λ and decreases with increasing aspect ratio AR. Multiple cells were observed in the cavity for all values of the parameters considered. © 2007 Elsevier Ltd. All rights reserved. Keywords: Porous medium; Free convection; Rectangular enclosure; Sinusoidal temperature profile
1. Introduction The heat and fluid flow in fluid-saturated porous media is an important problem in engineering and it has gained significant attention over the last three decades. Applications of porous media can be found in grain storage, chemical catalytic reactors, geophysical problems, solar collectors, heat exchangers, etc. These applications are reviewed in recent books by Nield and Bejan [1], Vafai [2], Ingham and Pop [3]. Natural convection in a rectangular/square enclosure filled with a fluid-saturated porous medium under different temperature or heat flux boundary conditions has been extensively analyzed in earlier studies by Bejan [4], Prasad and Kulacki [5], Gross et al. [6], Goyeau et al. [7], Manole and Lage [8], Holzbecher [9], Kumar et al. [10], Khanafer and Chamkha [11], Alchaar et al. [12] etc. In most of these studies, isothermal or isoflux thermal boundary conditions were applied to the side walls of the rectangular/square enclosures. However, non-isothermal thermal boundary conditions, which vary with time can be found in electrical arc furnaces and rotary burners, as given by Gogus et al. [13]. Another ☆
Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (I. Pop).
0735-1933/$ - see front matter © 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2007.05.015
Y. Varol et al. / International Communications in Heat and Mass Transfer 35 (2008) 56–64
57
Nomenclature AR g Gr H K L Nux Nu Pr Ra Tref ΔT u,v X,Y
aspect ratio parameter gravitational acceleration Grashof number height of rectangular permeability of the porous medium length of the bottom wall local Nusselt number mean Nusselt number Prandtl number Rayleigh number reference temperature temperature difference axial and radial velocities non-dimensional coordinates
Greek letters αm thermal diffusivity β thermal expansion coefficient λ amplitude of the sinusoidal temperature distribution θ non-dimensional temperature υ kinematic viscosity ψ non-dimensional stream function
Subscript C cold H hot
application of this boundary condition occurred in case of cylindrical heater, as given by Saeid [14]. Hossain and Wilson [15] performed a numerical study to investigate the natural convection in a porous enclosure with internal heat generation under linearly varying temperature boundary condition. Roy and Basak [16] performed a numerical study to analyze the natural convection in a square cavity filled with a viscous fluid (non-porous media) with non-uniformly heated wall (with a sinusoidal temperature variation) by using Galerkin finite element method. They observed that in case of uniform heating, the heat transfer rate is very high at the right edge of the bottom wall and almost uniform at the rest part of the bottom wall, as well as the hot vertical wall. In contrast, for the case of non-uniform heating, the heat transfer rate is minimum at the edges of the heated walls and the heat transfer rate reaches its maximum value at the center of both heated walls. Other studies of the natural convection over a flat plate or in enclosures with sinusoidal temperature boundary condition are due to Bilgen and Ben Yedder [17], Storesletten and Pop [18], Bradean et al. [19], Basak et al. [20] and Yoo [21]. The main purpose of this study is to analyze the natural convective heat transfer in a square enclosure filled with a fluid-saturated porous medium with non-isothermally heated wall for different Rayleigh numbers. The above literature review shows that the case of non-isothermally heated enclosure under non-isothermal boundary conditions has not been addressed. However, to the authors' best knowledge, sinusoidal thermal boundary conditions applied to a porous media enclosure has been considered only by Saeid [14]. The schematical diagram of the considered physical model is depicted in Fig. 1 with the mentioned boundary conditions and the Cartesian coordinates. It is a square enclosure filled with a fluid saturated porous medium. All walls are insulated except the bottom wall, which has a sinusoidally varying temperature profile.
58
Y. Varol et al. / International Communications in Heat and Mass Transfer 35 (2008) 56–64
2. Basic equations Darcy's law is used in the porous medium with the Boussinesq approximation. Both viscous drag and inertia force terms are neglected in the basic equations. Thus, the steady two-dimensional governing equations (continuity, Darcy, and energy equations) are ∂u ∂v þ ¼0 ∂x ∂y
ð1Þ
∂u ∂v gbK ∂T ¼ ∂y ∂x t ∂x
ð2Þ
2 ∂T ∂T ∂ T ∂2 T þv ¼ am u þ ∂x ∂y ∂x2 ∂y2
ð3Þ
where u and v are the velocity components along x and y axes, respectively, T is the fluid temperature, K is the permeability of the porous medium and the physical meaning of the other quantities are mentioned in the Nomenclature. Eqs. (1) and (2) can be written in terms of the stream function ψ defined as u¼
∂w ∂w ;v ¼ ∂y ∂x
ð4Þ
According to above parameters boundary conditions can be written as on all solid walls: w¼0
ð5Þ
on the adiabatic walls: ∂T ¼0 ∂n
ð6Þ
on the bottom wall: T ð xÞ ¼ Tref FDT Sinð 2px=LÞ
ð7Þ
We define now the following non-dimensional variables x y w ðu; vÞH T TC ; ðU ; V Þ ¼ ;h ¼ X ¼ ;Y ¼ ;W ¼ L H am am DT
ð8Þ
so that Eqs. (2) and (3) can be written as AR2
AR
∂2 W ∂2 W ∂h þ ¼ AR Ra ∂X 2 ∂Y 2 ∂X
∂W ∂h ∂W ∂h ∂2 h ∂2 h þ ¼ AR2 ∂Y ∂X ∂X ∂Y ∂X 2 ∂Y 2
ð9Þ
ð10Þ
where AR = H/L is the cavity ratio and Ra = g β K(TH − TC)H/αmυ is the Rayleigh number for the porous medium. The boundary conditions for the considered model are shown in Fig. 1. Thus, on all solid walls: W¼0
ð11Þ
Y. Varol et al. / International Communications in Heat and Mass Transfer 35 (2008) 56–64
59
Fig. 1. Physical model, coordinates and boundary conditions.
on the adiabatic walls: ∂h ¼0 ∂n
ð12Þ
on the bottom wall: h ¼ ksinð 2pX Þ
ð13Þ
The physical quantities of interest in this problem are the local and mean Nusselt numbers, which are given by Z L ∂h Nux ¼ ; Nu ¼ Nux dx ð14a:bÞ ∂Y Y ¼0 0 3. Numerical technique The finite difference method is used to solve Eqs. (9) and (10) with central differences for discretization of these equations. The solution of linear algebraic equations was performed using Successive Under Relaxation (SUR)
Table 1 Comparison of the mean Nusselt number Nu for AR = 1.0 and Ra = 1000 Paper
Nu
Bejan [4] Gross et al. [6] Goyeau et al. [7] Manole and Lage [8] Baytas and Pop [22] Saeid and Pop [23] This study
15.800 13.448 13.470 13.637 14.060 13.726 13.564
60
Y. Varol et al. / International Communications in Heat and Mass Transfer 35 (2008) 56–64
Fig. 2. Streamlines (on the left) and isotherms (on the right) for AR = 1.0, λ = 0.50 and different values of the Rayleigh numbers: a) Ra = 10, b) Ra = 100, c) Ra = 500.
Y. Varol et al. / International Communications in Heat and Mass Transfer 35 (2008) 56–64
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Fig. 3. Streamlines (on the left) and isotherms (on the right) for λ = 0.50, Ra = 1000, and different values of the ratios: a) AR = 1.0, b) AR = 0.50, c) AR = 0.25.
−4
method. As convergence criteria 10 is chosen for all depended variables and value of 0.1 is taken for under-relaxation parameter. The number of grid points is taken as 61 × 61 with uniform spaced mesh in both X- and Y- directions. In order to verify the accuracy of the results, the numerical algorithm used was tested with the classical natural convection heat transfer problem in a differentially heated square porous enclosure (AR = 1.0). The obtained numerical results for the mean Nusselt number Nu, as given by Eq.(14b), are compared with those given by different authors as tabulated in Table 1. As can be seen from the table, the obtained results show good agreement with the results reported in the literature. Contours of streamline and isotherms are almost the same to those given in the literature for a square porous enclosure but they are not presented here to save space. 4. Results and discussion The temperature distribution and the flow field due to buoyancy-forces in a rectangular enclosure filled with a fluid saturated porous medium with the temperature distribution of the bottom wall varying sinusoidally has been analyzed numerically for different values of the aspect ratio AR, Rayleigh number Ra and amplitude λ of sinusoidal temperature. Results for streamlines, isotherms, local and mean Nusselt numbers for these governing parameters will be presented. Streamlines (left) and isotherms (right) are presented in Fig. 2 for λ = 0.50 and AR = 1.0 to show the effect of the Rayleigh number Ra on the free convection flow. It is seen that the streamline contours exhibit circulation patterns,
62
Y. Varol et al. / International Communications in Heat and Mass Transfer 35 (2008) 56–64
Fig. 4. Variation of the local Nusselt number with X: a) different values of Ra when λ = 0.50 and AR = 1.0; b) different values of AR when Ra = 1000 and λ = 0.50; c) different value of λ when Ra = 1000 and AR = 1.0.
which are characterized by the three vortices. The fluid motion, as it is driven by the effect of the buoyancy, is distributed from the heated part of the bottom wall through the inside of the enclosure. Two small cells were formed in the left and right bottom corners in clockwise and counterclockwise rotating direction, respectively. However, a huge main cell was formed with its center located almost in the middle of the enclosure, as shown in Fig. 2 (a). The center of the cell moves towards the left bottom corner with increasing of Ra (Fig. 2 (b) and (c)). The circulation strength is very weak ψmin = − 0.25 due to the domination of conduction mode of heat transfer. But the flow strength increases with increasing of Ra and it can be seen from the values of the stream function on the streamline. The length and locations of the small cells become almost the same when the values of the Rayleigh number are changed. Isotherms show an almost symmetrical distribution for low Rayleigh number due to domination of conduction mode of heat transfer. But temperature gradient at the symmetry plane is very small. Thermal boundary layer becomes thinner with increasing of the Rayleigh number and plume like temperature distribution was observed on the heated part (positive values of sinusoidal function) of the enclosure. Fig. 3 shows the flow field and temperature distribution at different values of AR when Ra = 1000. In Fig. 3 (a) the streamlines (left) and isotherms (right) are shown for AR = 1.0 and Ra = 1000. It can be seen from that the flow strength becomes strong and convection mode of heat transfer is dominant to conduction. As AR becomes smaller, the main cell is depressed, see Fig. 3 (b). Thus, the length of the main and corner cells increase with decreasing AR. On the contrary, the flow strength increases with increasing the values of AR. Results of the local Nusselt number Nux are presented in Fig. 4 (a) to (c). Fig. 4 (a) shows the effect of Ra on the variation of the local Nusselt number with the coordinate X. It is seen that Nux is positive when the heat is transferred into the enclosure and negative when it is transferred from the enclosure to the environment. This result is similar to
Y. Varol et al. / International Communications in Heat and Mass Transfer 35 (2008) 56–64
63
Fig. 5. Variation of the mean Nusselt number with the Rayleigh number for different values of AR when λ = 0.50 and 1.0.
those reported by Bilgen and Yedder [17] for a rectangular cavity filled with air the cavity being heated and cooled from the left vertical side. The maximum value of Nux increases with increasing the Rayleigh number in both positive and negative regions due to the increase of the effect of convection mode of heat transfer. Fig. 4 (b) is plotted to obtain the effect of AR on Nux. The figure shows that the effect of AR is very small, especially at cooling region (right side of the figure). But Nux decreases with increasing AR due to decreasing the volume of the enclosure. Effect of amplitude of sinusoidal function on Nux is given in Fig. 4 (c). Further, the value of the amplitude λ is directly related to the temperature distribution at the bottom wall and it affects the heat transfer inside the enclosure. As expected, heat transfer increases with increasing the values of λ. Finally, Fig. 5 summarizes the variation of the average Nusselt number Nu with Ra for some values of AR and two values of λ. It is observed that Nu increases with increasing of Ra and λ. However AR shows small variation of Nu for low value of λ. On the contrary, Nu increases with decreasing of AR due to low volume of the enclosure. 5. Conclusions A numerical analysis of the steady natural convection flow through a fluid-saturated porous medium in a rectangular enclosure due to sinusoidally varying temperature distribution of the bottom wall has been performed. It was observed that multiple circulation cells were formed for all governing parameters which describe this problem. It was found that heat transfer increases with increasing of amplitude of sinusoidal function and decreases with increasing of aspect ratio. Effects of aspect ratio become significant especially for higher values of amplitude. Both flow strength and heat transfer increases with increasing of the Rayleigh number. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
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