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Combined convection heat transfer in a porous lid-driven enclosure due to heater with finite length
International Communications in Heat and Mass Transfer 33 (2006) 772 – 779 www.elsevier.com/locate/ichmt
Combined convection heat transfer in a porous lid-driven enclosure due to heater with finite length☆ Hakan F. Oztop Department of Mechanical Engineering, Fırat University, TR-23119, Elazıg, Turkey Available online 29 March 2006
Abstract A numerical work is performed to analyze combined convection heat transfer and fluid flow in a partially heated porous liddriven enclosure. The top wall of enclosure moves from left to right with constant velocity and temperature. Heater with finite length is located on the fixed wall where its center of location changes along the walls. The finite volume-based finite-difference method is applied for numerical experiments. Parameters effective on flow and thermal fields are Richardson number, Darcy number, center of heater and heater length. The results are shown that the best heat transfer is formed when the heater is located on the left vertical wall. © 2006 Elsevier Ltd. All rights reserved. Keywords: Combined convection; Lid-driven cavity; Porous medium
1. Introduction Flows and heat transfer in a lid-driven cavity with buoyancy or without buoyancy effect, for steady or unsteady cases, have been an important topic because of its wide application area in engineering and science. Some of these applications include oil extraction, cooling of electronic devices and heat transfer improvement in heat exchanger devices. Shankar and Deshpande [1] reviewed and showed its applications from a scientific and engineering point of view. Thus, some researchers proposed numerical and experimental technique to solve temperature and flow field in that geometry for non-porous medium [2–9]. Lid-driven cavities with filled saturated porous medium are another important application because of its wide applications in engineering such as heat exchangers, solar power collectors, packed-bed catalytic reactors, nuclear energy systems and so on [10,11]. Al-Amiri [12] made a numerical study to perform laminar flow and heat transfer for square lid-driven cavity heated from a driving wall filled with a porous medium. They wrote the governing equation in the streamline-vorticity form and these equations were solved via the finite-volume method. Khanafer and Chamkha [13] numerically solved a similar problem of boundary conditions with Al-Amiri by taking into account volumetric heat-generating fluid for mixed convection. They concluded that both Ri number and internal Ra number are effective on heat transfer. Jue [14] made a numerical study to investigate mixed convection flow caused by a torsionally oscillatory lid with thermal stable stratification in an enclosure filled with porous medium using semi-implicit projection finite-element method. He indicated that the permeability of the porous medium deeply dominates the flow strength as ☆
Communicated by W.J. Minkowycz. E-mail address: [email protected].
0735-1933/$ - see front matter © 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2006.02.003
H.F. Oztop / International Communications in Heat and Mass Transfer 33 (2006) 772–779
773
Nomenclature CF Da g Gr H K Nu P Pr Re Ri T u,v Ulid x,y
Forcheimer coefficient Darcy number gravity Grashof number height of cavity permeability Nusselt number pressure Prandtl number Reynolds number Richardson number temperature dimensionless velocities in x- and y-directions moving wall velocity dimensionless Cartesian coordinates
Greek letters ε porosity ψ stream function φ any dependent variable β thermal expansion coefficient
the permeability is decreased. Das and Morsi [15], Kim and Hyun [16] studied the natural convection in an enclosure for heat-generating porous media. Khanafer and Vafai [17] analyzed the double-diffusive mixed convection in a lid-driven porous enclosure with horizontal walls kept at constant and different temperatures and concentrations. They indicated that the buoyancy ratio, Da number, Le number and Ri number have important effects on the double-diffusive phenomenon. Asbik et al. [18] solved the equations for mixed convection in a vertical saturated porous enclosure. To the best of the author's knowledge, there is no information on the combined convection heat transfer in a partially heated lid-driven porous enclosure. Therefore, the present study is different from the others that concentrate on length and locations of the heater. It is believed that the study will contribute to academic and engineering research. The main aim of the present study is to determine the combined convection temperature and flow field due to the position of
Fig. 1. Physical model of a square lid-driven cavity (geometry, coordinate system and boundary conditions).
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Table 1 Source term in general equation Equation Continuity x-momentum
S 0
( ) Ap A Au A Av C CF 2 2 1=2 þ C þ C u þ e u þv Ax Ax Ax Ay Ax K=e ðK=eÞ1=2 ( ) A Au A Au C CF 2 2 Gr 2 Ap 2 1=2 eþ þ C þ C þ eþ þ T v þ v u Ay Ax Ay Ay Ay K=e ðK=eÞ1=2 Re2 þ2
y-momentum
heaters partially heated in a square porous enclosure. Parameters for flow and temperature fields are the Richardson number, Darcy number, and the center of location of heater. The studied physical model for a lid-driven partially heated porous enclosure can be shown in Fig. 1. The top wall moves from left to right and has a cold temperature. The figure defines a two-dimensional square enclosure with a side length, H. The enclosure is filled with a porous material that is homogenous and isotropic. The heater with finite length mounted on the fixed wall and its center of location, which shown with C (x,y) on the figure, can be changed along the walls. Except the heater, fixed walls are insulated. 2. Governing equations and numerical method The fluid is assumed as Newtonian and incompressible, and the flow is laminar. Boussinesq approximation is applied. It is assumed that porosity and permeability of the porous medium are uniform. The general equation in Cartesian coordinates for any dependent variables is written as A A/ A A/ u/ C v/ C þ ¼S ð1Þ Ax Ax Ay Ay In this equation, ϕ stands for u, v and T and the term S is the source term which is defined in Table 1 for continuity, momentum and energy equations. Eq. (1) is given in dimensionless form with Ergun relations [21], and dimensionless parameters are given as xV yV uV vV pV T V Tc K x¼ ; y¼ ; u¼ ; v¼ ; p¼ ; T¼ ; Da ¼ 2 ; Gr 2 H H Ulid Ulid H Th Tc qUlid CK 3=2 gbDTH 3 Ulid H Gr m ; Ri ¼ 2 ; Pr ¼ : ð2Þ ; Re ¼ ¼ 2 m a m Re The calculation of mean Nu number is obtained via Eq. (3), which is calculated along the moving wall Z L AT dx ð3Þ Nu ¼ Ay 0 Boundary conditions are depicted on the physical model (Fig. 1) which on the top wall (Ulid = U, T = 0), other walls (u = v = 0) and the heater have constant temperature with T = 1. Porosity (ε) is taken as 0.9. Finite volume-based finitedifference method is used to solve the governing equation using a staggered grid arrangement. A modified version according to the present study of the general purpose of SAINTS (Software for arbitrary integration of Navier–Stokes Equation with Turbulence and Porous Media Simulator) code, proposed by Nakayama [19], is used with SIMPLE algorithm [20]. A hybrid of the central difference and upwind schemes was used for the convective and diffusive terms. Solution of linear algebraic equation is made by the TDMA method. The stream function is calculated from its definition as u = ∂ψ/∂y, v = ∂ψ/∂x. An under-relaxation parameter of 0.3, 0.3, 0.2 and 0.5 were used in order to obtain a stable convergence for the solution of u-velocity, v-velocity, pressure and energy equations. The tolerance for the Table 2 Comparison of mean Nusselt number between this study and literature for Da = 0.1 P Ri Nu (present)
P Nu (Al-Amiri [12])
0.1 1 5
3.92 2.31 1.83
4.196 2.312 1.746
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residuals upon convergence is set to 10 7 in every calculation case. In this study, the Gr number is taken as 104. Thus, a different Re number is chosen to change the Ri number. 48 × 48 grids are chosen for the grid arrangement where fine grids were placed next to boundaries and it is found that this grid solution is enough. To obtain the validation of the code, a study was tested with the literature and the obtained results are listed in Table 2. The general agreement between the present computation and that of Al-Amiri [12] is seen to be very well. The author is not aware of any existing experimental data to compare the numerical results (Table 2). 3. Results and discussion A numerical analysis is performed to obtain a combined convection temperature and flow fields in a porous lid-driven enclosure due to the location of a partial heater on the wall. Pr number is taken as 0.71. The main parameter is Ri number; Ri = Gr/Re2, which determines the importance of buoyancy-driven convection, is varied as 0.1, 1 and 10. Fig. 2a–c shows the streamline and isotherm when the location of the heater C(x = 0, y = 0.5) is on the left vertical wall. In this case, a single cell is formed in a clockwise manner for all Ri numbers but its center moves to the middle of the enclosure as Ri number
Fig. 2. Streamline (on the left) and isotherm (on the right) at Da = 0.01, C (x = 0, y = 0.5), h = 0.5, (a) Ri = 0.1, (b) Ri = 1, (c) Ri = 10.
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Fig. 3. Streamline (on the left) and isotherm (on the right) at Da = 0.01 and h = 0.5, (a) C (x = 1, y = 0.5), Ri = 0.1 and Ri = 10, (b) C (x = 0.5, y = 0), Ri = 0.1 and Ri = 10.
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increases. The egg-shaped cell is obtained for the lowest Ri number, and it turns to a circle with the increase of Ri number. However, the intensity of the temperature is increased with the increasing buoyancy effect. When the heater is located on the middle of right vertical wall C(x = 1, y = 0.5), four different cells are obtained in the cavity, in which three of them settled at the corners as can be seen
a)
1
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0.9 Ri=0.1 Ri=1 Ri=10
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0.3 0.2
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b)
Ri=0.1 Ri=1 Ri=10
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Fig. 4. Velocity (on the left) and temperature (on the right) profiles for different locations of heater as a function of Richardson number, Da = 0.01 and h = 0.5. (a) Variation of temperature profiles for different cases (a) C (x = 0, y = 0.5), (b) C (x = 0.5, y = 0), (c) C (x = 1, y = 0.5).
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from Fig. 3a. When compared with Fig. 2a with the same Ri number, the flow behaves as an impinged jet on to a hot plate. However, when buoyancy force becomes effective on the thermal field, two different cells are obtained wherein the top one takes place near the lid-driven wall clockwise and the other one counterclockwise. For the highest Ri number, the problem turns to natural convection from partially heated cavity, and a plume-like behavior can be seen in isotherms. Thus, the cell which is obtained at the bottom is dominant to the top one. Fig. 3c and d illustrates the streamline and isotherm when the heater is located on the middle of the bottom wall C(x = 0.5, y = 0), which has the longest distance between hot and cold wall. Single cell and double cells are obtained for Ri < 1 and Ri ≥ 1, respectively. The length of the cell which is located at the left bottom corner increases with the increasing of Ri number. Velocity (on the left) and temperature (on the right) profiles at the middle of enclosure are shown in Fig. 4 for different Ri numbers and the location of the heaters. Due to forced convection, the heater does not affect the flow behavior inside the enclosure as indicated Khanafer and Champka [13]. This result can be seen for all different locations of the heater (Fig. 4a–c). But velocity decreases with increasing Ri number (Fig. 4a) due to a natural convection dominant regime. The maximum velocities are obtained when the heater is located on the right vertical wall, whose coordinates are C(x = 1, y = 0.5). However, the maximum value of the temperature is increased with increasing Ri number due to buoyancy effect. Fig. 5a shows the variation of mean Nu number as a function of heater length and heater location. This figure is summarized by the overall heat transfer for the considered system. As indicated in the figure, when the heater is located on left vertical wall, the highest heat transfer is obtained due to the assisting flow; however, the worst heat transfer is formed when the heater is located on the bottom because there is a long distance between hot and cold walls in this case. The mean Nu number is increased with increasing heater length for all cases as expected. The effect of Da number on heat transfer is given for C(x = 0.5, y = 0), namely, the locations of the heater on the bottom wall. As Da number increases, heat transfer is increased, and it is decreased with increasing Ri number because natural convection becomes dominant.
a) 4 C(x=0,y=0.5) C(x=1,y=0.5) C(x=0.5,y=0)
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Nu
3 2.5 2 1.5 1 0.5 0.2
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h
b) 2 Da=0.001 Da=0.01 Da=0.1
Nu
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0
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4
6
8
10
Ri
Fig. 5. Variation of mean Nusselt numbers (a) effects of location for different heater length, (Da = 0.01), (b) effects of Da number at (C(x = 0.5, y = 0)).
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4. Conclusions Combined convection thermal and flow field is analyzed numerically in a partial heater located in the porous enclosure, and the main conclusions that are drawn from the present study are as follows. (a) The location center of the heater is the most effective parameter on combined convection flow and temperature field. (b) The highest heat transfer is obtained when the heater is located on the left vertical wall. (c) Heat transfer is decreased with increasing Ri number and increased with increasing Da number. (d) The study can be extended for higher Ri number values using different models. References [1] P.N. Shankar, M.D. Deshpande, Annu. Rev. Fluid Mech. 32 (2000) 93–136. [2] U. Ghia, K.N. Ghia, C.T. Shin, High Re solution for incompressible flow using Navier–Stokes equations and a multigrid method, J. Comput. Phys. 48 (1982) 387–411; K. Torrance, R. Davis, K. Eike, D. Gill, D. Gutman, A. Hsui, S. Lyons, H. Zien, Cavity flows driven by buoyancy and shear, J. Fluid Mech. 51 (1972) 221–231. [3] H.F. Oztop, I. Dagtekin, Natural convection heat transfer by heated partitions within enclosure, Int. J. Heat Mass Transfer 47 (2004) 1761–1769. [4] O. Aydin, Aiding and opposing mechanisms of mixed convection in a shear- and buoyancy-driven cavity, Int. Commun. Heat Mass Tranf. 26 (1999) 1019–1028. [5] C.-J. Chen, H. Nassari-Neshat, K.S. Ho, Numer. Heat Transf. 4 (1981) 179–197. [6] X. Shi, J.M. Khodadadi, Fluid flow and heat transfer in a lid-driven cavity due to an oscillating thin fin: transient behavior, J. Heat Transfer 126 (2004) 924–930. [7] A.K. Prasad, J.R. Koseff, Combined forced and natural convection heat transfer in a deep lid-driven cavity flow, Int. J. Heat Fluid Flow 17 (1996) 460–467. [8] R. Iwatsu, J.M. Hyun, Three-dimensional driven-cavity flow with a vertical temperature gradient, Int. J. Heat Mass Transfer 38 (1995) 3319–3328. [9] V.S. Arpaci, P.S. Larsen, Convection Heat Transfer, Prentice-Hall, 1984, p. 90. [10] D.A. Nield, A. Bejan, Convection in Porous Media, Springer-Verlag, New York, 1999. [11] K. Vafai, Convective flow and heat transfer in variable-porosity media, J. Fluid Mech. 147 (1984) 233–259. [12] A.M. Al-Amiri, Analysis of momentum and energy transfer in a lid-driven cavity filled with a porous medium, Int. J. Heat Mass Transfer 43 (2000) 3513–3527. [13] K.M. Khanafer, A.J. Chamkha, Mixed convection flow in a lid-driven enclosure filled with a fluid-saturated porous medium, Int. J. Heat Mass Transfer 42 (1999) 2465–2481. [14] T.C. Jue, Analysis of flows driven by a torsionally-oscillatory lid in a fluid-saturated porous enclosure with thermal stable stratification, Int. J. Therm. Sci. 41 (2002) 795–804. [15] S. Das, Y.S. Morsi, A non-Darcian numerical modeling in domed enclosures filled with heat-generating porous media, Numer. Heat Transf., A Appl. 48 (2005) 149–164. [16] G.B. Kim, J.M. Hyun, Buoyant convectýon of a power-law fluid in an enclosure filled with heat-generating porous media, Numer. Heat Transf., A Appl. 45 (2004) 569–582. [17] K. Khanafer, K. Vafai, Double-diffusive mixed convection in a lid-driven enclosure filled with a fluid-saturated porous medium, Numer. Heat Transf., A Appl. 45 (2002) 465–486. [18] M. Asbik, H. Sadki, M. Hajar, B. Zeghmati, A. Khmou, Numerical study of laminar mixed convection in a vertical saturated porous enclosure: the combined effect of double diffusion and evaporation, Numer. Heat Transf. 41 (2002) 403–420. [19] A. Nakayama, PC-Aided Numerical Heat Transfer and Convective Flow, CRC Press, 1995. [20] S.V. Patankar, Numerical Methods for Heat Transfer and Fluid Flow, Hemisphere, New York, 1980. [21] S. Ergun, Fluid flow through packed columns, Chem. Eng. Process. 48 (1952) 89.
Combined convection heat transfer in a porous lid-driven enclosure due to heater with finite length☆ Hakan F. Oztop Department of Mechanical Engineering, Fırat University, TR-23119, Elazıg, Turkey Available online 29 March 2006
Abstract A numerical work is performed to analyze combined convection heat transfer and fluid flow in a partially heated porous liddriven enclosure. The top wall of enclosure moves from left to right with constant velocity and temperature. Heater with finite length is located on the fixed wall where its center of location changes along the walls. The finite volume-based finite-difference method is applied for numerical experiments. Parameters effective on flow and thermal fields are Richardson number, Darcy number, center of heater and heater length. The results are shown that the best heat transfer is formed when the heater is located on the left vertical wall. © 2006 Elsevier Ltd. All rights reserved. Keywords: Combined convection; Lid-driven cavity; Porous medium
1. Introduction Flows and heat transfer in a lid-driven cavity with buoyancy or without buoyancy effect, for steady or unsteady cases, have been an important topic because of its wide application area in engineering and science. Some of these applications include oil extraction, cooling of electronic devices and heat transfer improvement in heat exchanger devices. Shankar and Deshpande [1] reviewed and showed its applications from a scientific and engineering point of view. Thus, some researchers proposed numerical and experimental technique to solve temperature and flow field in that geometry for non-porous medium [2–9]. Lid-driven cavities with filled saturated porous medium are another important application because of its wide applications in engineering such as heat exchangers, solar power collectors, packed-bed catalytic reactors, nuclear energy systems and so on [10,11]. Al-Amiri [12] made a numerical study to perform laminar flow and heat transfer for square lid-driven cavity heated from a driving wall filled with a porous medium. They wrote the governing equation in the streamline-vorticity form and these equations were solved via the finite-volume method. Khanafer and Chamkha [13] numerically solved a similar problem of boundary conditions with Al-Amiri by taking into account volumetric heat-generating fluid for mixed convection. They concluded that both Ri number and internal Ra number are effective on heat transfer. Jue [14] made a numerical study to investigate mixed convection flow caused by a torsionally oscillatory lid with thermal stable stratification in an enclosure filled with porous medium using semi-implicit projection finite-element method. He indicated that the permeability of the porous medium deeply dominates the flow strength as ☆
Communicated by W.J. Minkowycz. E-mail address: [email protected].
0735-1933/$ - see front matter © 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2006.02.003
H.F. Oztop / International Communications in Heat and Mass Transfer 33 (2006) 772–779
773
Nomenclature CF Da g Gr H K Nu P Pr Re Ri T u,v Ulid x,y
Forcheimer coefficient Darcy number gravity Grashof number height of cavity permeability Nusselt number pressure Prandtl number Reynolds number Richardson number temperature dimensionless velocities in x- and y-directions moving wall velocity dimensionless Cartesian coordinates
Greek letters ε porosity ψ stream function φ any dependent variable β thermal expansion coefficient
the permeability is decreased. Das and Morsi [15], Kim and Hyun [16] studied the natural convection in an enclosure for heat-generating porous media. Khanafer and Vafai [17] analyzed the double-diffusive mixed convection in a lid-driven porous enclosure with horizontal walls kept at constant and different temperatures and concentrations. They indicated that the buoyancy ratio, Da number, Le number and Ri number have important effects on the double-diffusive phenomenon. Asbik et al. [18] solved the equations for mixed convection in a vertical saturated porous enclosure. To the best of the author's knowledge, there is no information on the combined convection heat transfer in a partially heated lid-driven porous enclosure. Therefore, the present study is different from the others that concentrate on length and locations of the heater. It is believed that the study will contribute to academic and engineering research. The main aim of the present study is to determine the combined convection temperature and flow field due to the position of
Fig. 1. Physical model of a square lid-driven cavity (geometry, coordinate system and boundary conditions).
774
H.F. Oztop / International Communications in Heat and Mass Transfer 33 (2006) 772–779
Table 1 Source term in general equation Equation Continuity x-momentum
S 0
( ) Ap A Au A Av C CF 2 2 1=2 þ C þ C u þ e u þv Ax Ax Ax Ay Ax K=e ðK=eÞ1=2 ( ) A Au A Au C CF 2 2 Gr 2 Ap 2 1=2 eþ þ C þ C þ eþ þ T v þ v u Ay Ax Ay Ay Ay K=e ðK=eÞ1=2 Re2 þ2
y-momentum
heaters partially heated in a square porous enclosure. Parameters for flow and temperature fields are the Richardson number, Darcy number, and the center of location of heater. The studied physical model for a lid-driven partially heated porous enclosure can be shown in Fig. 1. The top wall moves from left to right and has a cold temperature. The figure defines a two-dimensional square enclosure with a side length, H. The enclosure is filled with a porous material that is homogenous and isotropic. The heater with finite length mounted on the fixed wall and its center of location, which shown with C (x,y) on the figure, can be changed along the walls. Except the heater, fixed walls are insulated. 2. Governing equations and numerical method The fluid is assumed as Newtonian and incompressible, and the flow is laminar. Boussinesq approximation is applied. It is assumed that porosity and permeability of the porous medium are uniform. The general equation in Cartesian coordinates for any dependent variables is written as A A/ A A/ u/ C v/ C þ ¼S ð1Þ Ax Ax Ay Ay In this equation, ϕ stands for u, v and T and the term S is the source term which is defined in Table 1 for continuity, momentum and energy equations. Eq. (1) is given in dimensionless form with Ergun relations [21], and dimensionless parameters are given as xV yV uV vV pV T V Tc K x¼ ; y¼ ; u¼ ; v¼ ; p¼ ; T¼ ; Da ¼ 2 ; Gr 2 H H Ulid Ulid H Th Tc qUlid CK 3=2 gbDTH 3 Ulid H Gr m ; Ri ¼ 2 ; Pr ¼ : ð2Þ ; Re ¼ ¼ 2 m a m Re The calculation of mean Nu number is obtained via Eq. (3), which is calculated along the moving wall Z L AT dx ð3Þ Nu ¼ Ay 0 Boundary conditions are depicted on the physical model (Fig. 1) which on the top wall (Ulid = U, T = 0), other walls (u = v = 0) and the heater have constant temperature with T = 1. Porosity (ε) is taken as 0.9. Finite volume-based finitedifference method is used to solve the governing equation using a staggered grid arrangement. A modified version according to the present study of the general purpose of SAINTS (Software for arbitrary integration of Navier–Stokes Equation with Turbulence and Porous Media Simulator) code, proposed by Nakayama [19], is used with SIMPLE algorithm [20]. A hybrid of the central difference and upwind schemes was used for the convective and diffusive terms. Solution of linear algebraic equation is made by the TDMA method. The stream function is calculated from its definition as u = ∂ψ/∂y, v = ∂ψ/∂x. An under-relaxation parameter of 0.3, 0.3, 0.2 and 0.5 were used in order to obtain a stable convergence for the solution of u-velocity, v-velocity, pressure and energy equations. The tolerance for the Table 2 Comparison of mean Nusselt number between this study and literature for Da = 0.1 P Ri Nu (present)
P Nu (Al-Amiri [12])
0.1 1 5
3.92 2.31 1.83
4.196 2.312 1.746
H.F. Oztop / International Communications in Heat and Mass Transfer 33 (2006) 772–779
775
residuals upon convergence is set to 10 7 in every calculation case. In this study, the Gr number is taken as 104. Thus, a different Re number is chosen to change the Ri number. 48 × 48 grids are chosen for the grid arrangement where fine grids were placed next to boundaries and it is found that this grid solution is enough. To obtain the validation of the code, a study was tested with the literature and the obtained results are listed in Table 2. The general agreement between the present computation and that of Al-Amiri [12] is seen to be very well. The author is not aware of any existing experimental data to compare the numerical results (Table 2). 3. Results and discussion A numerical analysis is performed to obtain a combined convection temperature and flow fields in a porous lid-driven enclosure due to the location of a partial heater on the wall. Pr number is taken as 0.71. The main parameter is Ri number; Ri = Gr/Re2, which determines the importance of buoyancy-driven convection, is varied as 0.1, 1 and 10. Fig. 2a–c shows the streamline and isotherm when the location of the heater C(x = 0, y = 0.5) is on the left vertical wall. In this case, a single cell is formed in a clockwise manner for all Ri numbers but its center moves to the middle of the enclosure as Ri number
Fig. 2. Streamline (on the left) and isotherm (on the right) at Da = 0.01, C (x = 0, y = 0.5), h = 0.5, (a) Ri = 0.1, (b) Ri = 1, (c) Ri = 10.
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Fig. 3. Streamline (on the left) and isotherm (on the right) at Da = 0.01 and h = 0.5, (a) C (x = 1, y = 0.5), Ri = 0.1 and Ri = 10, (b) C (x = 0.5, y = 0), Ri = 0.1 and Ri = 10.
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increases. The egg-shaped cell is obtained for the lowest Ri number, and it turns to a circle with the increase of Ri number. However, the intensity of the temperature is increased with the increasing buoyancy effect. When the heater is located on the middle of right vertical wall C(x = 1, y = 0.5), four different cells are obtained in the cavity, in which three of them settled at the corners as can be seen
a)
1
1
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0
0.25
0.5
0.75
1
T
Fig. 4. Velocity (on the left) and temperature (on the right) profiles for different locations of heater as a function of Richardson number, Da = 0.01 and h = 0.5. (a) Variation of temperature profiles for different cases (a) C (x = 0, y = 0.5), (b) C (x = 0.5, y = 0), (c) C (x = 1, y = 0.5).
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from Fig. 3a. When compared with Fig. 2a with the same Ri number, the flow behaves as an impinged jet on to a hot plate. However, when buoyancy force becomes effective on the thermal field, two different cells are obtained wherein the top one takes place near the lid-driven wall clockwise and the other one counterclockwise. For the highest Ri number, the problem turns to natural convection from partially heated cavity, and a plume-like behavior can be seen in isotherms. Thus, the cell which is obtained at the bottom is dominant to the top one. Fig. 3c and d illustrates the streamline and isotherm when the heater is located on the middle of the bottom wall C(x = 0.5, y = 0), which has the longest distance between hot and cold wall. Single cell and double cells are obtained for Ri < 1 and Ri ≥ 1, respectively. The length of the cell which is located at the left bottom corner increases with the increasing of Ri number. Velocity (on the left) and temperature (on the right) profiles at the middle of enclosure are shown in Fig. 4 for different Ri numbers and the location of the heaters. Due to forced convection, the heater does not affect the flow behavior inside the enclosure as indicated Khanafer and Champka [13]. This result can be seen for all different locations of the heater (Fig. 4a–c). But velocity decreases with increasing Ri number (Fig. 4a) due to a natural convection dominant regime. The maximum velocities are obtained when the heater is located on the right vertical wall, whose coordinates are C(x = 1, y = 0.5). However, the maximum value of the temperature is increased with increasing Ri number due to buoyancy effect. Fig. 5a shows the variation of mean Nu number as a function of heater length and heater location. This figure is summarized by the overall heat transfer for the considered system. As indicated in the figure, when the heater is located on left vertical wall, the highest heat transfer is obtained due to the assisting flow; however, the worst heat transfer is formed when the heater is located on the bottom because there is a long distance between hot and cold walls in this case. The mean Nu number is increased with increasing heater length for all cases as expected. The effect of Da number on heat transfer is given for C(x = 0.5, y = 0), namely, the locations of the heater on the bottom wall. As Da number increases, heat transfer is increased, and it is decreased with increasing Ri number because natural convection becomes dominant.
a) 4 C(x=0,y=0.5) C(x=1,y=0.5) C(x=0.5,y=0)
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b) 2 Da=0.001 Da=0.01 Da=0.1
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0
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4
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Fig. 5. Variation of mean Nusselt numbers (a) effects of location for different heater length, (Da = 0.01), (b) effects of Da number at (C(x = 0.5, y = 0)).
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4. Conclusions Combined convection thermal and flow field is analyzed numerically in a partial heater located in the porous enclosure, and the main conclusions that are drawn from the present study are as follows. (a) The location center of the heater is the most effective parameter on combined convection flow and temperature field. (b) The highest heat transfer is obtained when the heater is located on the left vertical wall. (c) Heat transfer is decreased with increasing Ri number and increased with increasing Da number. (d) The study can be extended for higher Ri number values using different models. References [1] P.N. Shankar, M.D. Deshpande, Annu. Rev. Fluid Mech. 32 (2000) 93–136. [2] U. Ghia, K.N. Ghia, C.T. Shin, High Re solution for incompressible flow using Navier–Stokes equations and a multigrid method, J. Comput. Phys. 48 (1982) 387–411; K. Torrance, R. Davis, K. Eike, D. Gill, D. Gutman, A. Hsui, S. Lyons, H. Zien, Cavity flows driven by buoyancy and shear, J. Fluid Mech. 51 (1972) 221–231. [3] H.F. Oztop, I. Dagtekin, Natural convection heat transfer by heated partitions within enclosure, Int. J. Heat Mass Transfer 47 (2004) 1761–1769. [4] O. Aydin, Aiding and opposing mechanisms of mixed convection in a shear- and buoyancy-driven cavity, Int. Commun. Heat Mass Tranf. 26 (1999) 1019–1028. [5] C.-J. Chen, H. Nassari-Neshat, K.S. Ho, Numer. Heat Transf. 4 (1981) 179–197. [6] X. Shi, J.M. Khodadadi, Fluid flow and heat transfer in a lid-driven cavity due to an oscillating thin fin: transient behavior, J. Heat Transfer 126 (2004) 924–930. [7] A.K. Prasad, J.R. Koseff, Combined forced and natural convection heat transfer in a deep lid-driven cavity flow, Int. J. Heat Fluid Flow 17 (1996) 460–467. [8] R. Iwatsu, J.M. Hyun, Three-dimensional driven-cavity flow with a vertical temperature gradient, Int. J. Heat Mass Transfer 38 (1995) 3319–3328. [9] V.S. Arpaci, P.S. Larsen, Convection Heat Transfer, Prentice-Hall, 1984, p. 90. [10] D.A. Nield, A. Bejan, Convection in Porous Media, Springer-Verlag, New York, 1999. [11] K. Vafai, Convective flow and heat transfer in variable-porosity media, J. Fluid Mech. 147 (1984) 233–259. [12] A.M. Al-Amiri, Analysis of momentum and energy transfer in a lid-driven cavity filled with a porous medium, Int. J. Heat Mass Transfer 43 (2000) 3513–3527. [13] K.M. Khanafer, A.J. Chamkha, Mixed convection flow in a lid-driven enclosure filled with a fluid-saturated porous medium, Int. J. Heat Mass Transfer 42 (1999) 2465–2481. [14] T.C. Jue, Analysis of flows driven by a torsionally-oscillatory lid in a fluid-saturated porous enclosure with thermal stable stratification, Int. J. Therm. Sci. 41 (2002) 795–804. [15] S. Das, Y.S. Morsi, A non-Darcian numerical modeling in domed enclosures filled with heat-generating porous media, Numer. Heat Transf., A Appl. 48 (2005) 149–164. [16] G.B. Kim, J.M. Hyun, Buoyant convectýon of a power-law fluid in an enclosure filled with heat-generating porous media, Numer. Heat Transf., A Appl. 45 (2004) 569–582. [17] K. Khanafer, K. Vafai, Double-diffusive mixed convection in a lid-driven enclosure filled with a fluid-saturated porous medium, Numer. Heat Transf., A Appl. 45 (2002) 465–486. [18] M. Asbik, H. Sadki, M. Hajar, B. Zeghmati, A. Khmou, Numerical study of laminar mixed convection in a vertical saturated porous enclosure: the combined effect of double diffusion and evaporation, Numer. Heat Transf. 41 (2002) 403–420. [19] A. Nakayama, PC-Aided Numerical Heat Transfer and Convective Flow, CRC Press, 1995. [20] S.V. Patankar, Numerical Methods for Heat Transfer and Fluid Flow, Hemisphere, New York, 1980. [21] S. Ergun, Fluid flow through packed columns, Chem. Eng. Process. 48 (1952) 89.