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Natural convection effects on heat and mass transfer in a curvilinear triangular cavity
International Journal of Heat and Mass Transfer 55 (2012) 6250–6259
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Natural convection effects on heat and mass transfer in a curvilinear triangular cavity M.M. Rahman a,⇑, Hakan F. Öztop b, A. Ahsan c, J. Orfi d a
Department of Mathematics, Bangladesh University of Engineering and Technology, Dhaka 1000, Bangladesh Department of Mechanical Engineering, Technology Faculty, Firat University, Elazig, Turkey c Department of Civil Engineering, Faculty of Engineering, (Green Engineering and Sustainable Technology Lab, Institute of Advanced Technology), University Putra, Malaysia, 43400 UPM Serdang, Selangor, Malaysia d Department of Mechanical Engineering, College of Engineering, King Saud University, Riyadh, Saudi Arabia b
a r t i c l e
i n f o
Article history: Received 2 February 2012 Received in revised form 18 June 2012 Accepted 20 June 2012 Available online 9 July 2012 Keywords: Double-diffusive Natural convection Finite element method Triangular cavity Absorber plate Inclined glass covers
a b s t r a c t Finite element method is used in this study to analyze the effects of buoyancy ratio and Lewis number on heat and mass transfer in a triangular cavity with zig-zag shaped bottom wall. Buoyancy ratio is defined as the ratio of Grashof number of solutal and thermal. Inclined walls of the cavity have lower temperature and concentration according to zig-zag shaped bottom wall. Enclosed space consists mostly of an absorber plate and two inclined glass covers that form a cavity. Both high temperature and high concentrations are applied to bottom corrugated wall. Computations were done for different values of buoyancy ratio (10 6 Br 6 10), Lewis number (0.1 6 Le 6 20) and thermal Rayleigh number (104 6 RaT 6 106). Streamlines, isotherms, iso-concentration, average Nusselt and Sherwood numbers are obtained. It is found that average Nusselt and Sherwood numbers increase by 89.18% and 101.91% respectively as Br increases from 10 to 20 at RaT = 106. Also, average Nusselt decreases by 16.22% and Sherwood numbers increases by 144.84% as Le increases from 0.1 to 20 at this Rayleigh number. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Natural convection driven flow due to simultaneous temperature and concentration gradients is generally referred to double diffusive or thermosolutal convection. In such cases, complex flow structures often result in liquids due to differences between the thermal and solutal diffusivities [1]. It may be met in geophysical, geothermal and industrial applications, such as the migration of moisture through air contained in fibrous insulations and the underground spreading of chemical contaminants through water saturated soil and desalination process. Double-diffusive heat transfer problem is studied mostly for square or rectangular geometries at different thermal and solutal boundary conditions as given in literature [2–7]. However, natural convection heat and mass transfer finds an important applications in triangular shaped enclosures such as desalination process in solar stills, attic-shaped space etc., Natural convection heat transfer in triangular enclosures is reviewed by Saha and Khan [8]. Earlier studies of the natural convection studies in triangular enclosure are Flack [9], and Poulikakos and Bejan [10,11]. Asan and Namli [12,13], made a numerical works to compare the summer and win-
⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (M.M. Rahman). 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.06.055
ter conditions for a roof of buildings. They indicated that the conditions are extremely important in flow motion inside the roof of building. Then Salmun [14], Varol et al. [15–18], Kent [19], Ridouane et al. [20] and del Campo et al. [21]. In all of these studies, the bottom wall of the enclosure is chosen as flat at different temperature boundary conditions, and most of these works, mass transfer is not considered. Chamkha et al. [22] made a numerical work on double-diffusive natural convection inclined finned triangular porous enclosures in the presence of heat generation/absorption effects. Double-diffusive natural convection can be seen in some solar collectors. Omri et al. [23] deal with a numerical simulation of natural convection flows in a triangular cavity submitted to a uniform heat flux using the Control Volume Finite Element Method. Their results showed that the flow structure is sensitive to the cover tilt angle. Many recirculation zones can occur in the core cavity and the heat transfer is dependent on the flow structure. A numerical model is presented for the study of natural convective heat and mass transfer in a triangular cavity by Omri [24] for greenhouse solar stills where vertical temperature and concentration gradients between the saline water and transparent cover induce flow in a confined space. They investigated the relationship between the pressure gradient and the other variables. The main aim of this study is to investigate the effects of buoyancy ratio and Lewis number on double diffusive natural
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Nomenclature A Br c ch cL C D g H L Le Nu p P Pr RaT Sh T Th TL
aspect ratio, L/H buoyancy ratio concentration of species high species concentration (source) low species concentration (sink) dimensionless species concentration species diffusivity gravitational acceleration enclosure height enclosure length Lewis number average Nusselt number dimensional pressure non-dimensional pressure Prandtl number thermal Rayleigh number average Sherwood number temperature hot wall temperature (source) cold wall temperature (sink)
convection in triangular types of cavity with zig-zag shaped bottom wall. Based on authors’ knowledge and above literature survey, this work is first step to analyze double diffusive natural convection in mentioned geometry.
2. Physical description and mathematical formulation Fig. 1(a) shows the physical model of a triangular cavity with zig-zag shaped bottom wall in which L and H indicate the length and height of the cavity, respectively. Fig. 1(b) shows grid distribution. The enclosure is filled with a viscous, incompressible and Newtonian fluid. The wave length and height is defined as in the
u U v V x X y Y
horizontal velocity component dimensionless horizontal velocity component vertical velocity component dimensionless vertical velocity component horizontal coordinate dimensionless horizontal coordinate vertical coordinate dimensionless vertical coordinate
Greek symbols thermal diffusivity bT thermal expansion coefficient bc compositional expansion coefficient k wave length m kinematic viscosity h non-dimensional temperature q density w stream function C general dependent variable
a
figure that it is taken as k = 0.2 L and h = 0.05 H, respectively. The inclined and the base walls are kept at constant temperatures Th and TL and concentration ch and cL respectively. Both concentration and temperature are higher on wavy wall than that of inclined walls. The Boussinesq approximation is assumed to be valid and constant fluid properties are used except the density variations. The gravity acts in negative – y direction. Due to the relative values of Rayleigh numbers considered in this study, the flow is assumed to be laminar [25]. With this geometry and boundary conditions, the present study reports the computations for cavity at a fixed Prandtl number of Pr = 0.72, buoyancy ratio (Br) ranging from 10 to 10, Lewis number (Le) ranging from 0.1 to 20, and thermal Rayleigh number (RaT) ranging from 104 to 106, and their effect on the heat and mass transfer process is analyzed. Under the above approximations, the governing equations for steady natural convection flow using conservation of mass, momentum and energy can be written as formulated by Hajri et al. [25]. Both concentration and temperature are higher on bottom zig-zag shaped wall than that of inclined walls. For steady two-dimensional laminar convection the coupled transport
Table 1 Comparison of average Nusselt and Sherwood numbers for various grids, with Le = 2, k = 0.2 L, Br = 1 and Ra = 1.E + 6. No. of elements
Nu
% Difference
Sh
% Difference
928 1554 2238 3610 4830
18.59093 19.0223 19.00917 19.007602 19.014506
– 0.069 0.032 0.016 0.011
21.140654 21.782312 21.731108 21.726773 21.733893
– 0.103 0.045 0.022 0.015
Table 2 Comparison of present values of average Nusselt number for U = 0 with literature. Ri
Fig. 1. (a) Schematic of the problem with the domain and boundary conditions and (b) grid distribution.
0.01 0.1 1 10
Nu Results ghasemi and aminossadati
Present results
Error (%)
33.275 29.356 11.073 11.073
32.706 29.049 11.125 10.623
1.71 1.05 0.47 4.06
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equations for U, V, h and C written in Cartesian coordinates in the dimensionless form as:
@U @V þ ¼0 @X @Y
ð1Þ
@U @U @P @2U @2U þV ¼ þ Pr þ U @X @Y @X @X 2 @Y 2 @V @V @P @2V @2V U þV ¼ þ Pr þ @X @Y @Y @X 2 @Y 2
! ð2Þ
! þ RaT Prðh þ BrCÞ
Fig. 2. Streamlines contours with Le = 2 at selected values of buoyancy ratio, Br and Rayleigh number, RaT.
Fig. 3. Isotherms contours with Le = 2 at selected values of buoyancy ratio, Br and Rayleigh number, RaT.
ð3Þ
M.M. Rahman et al. / International Journal of Heat and Mass Transfer 55 (2012) 6250–6259
U
U
@h @h @ 2 h @ 2 h þ þV ¼ @X @Y @X 2 @Y 2 @C @C 1 @2C @2C þV ¼ þ @X @Y Le @X 2 @Y 2
ð4Þ
ð5Þ
where the dimensionless variables are introduced as:
ð6Þ
m gb ðT h T L ÞH3 a b ðch cL Þ ; RaT ¼ T ; Le ¼ and Br ¼ c bT ðT h T L Þ a am D ð7Þ
The dimensionless boundary conditions corresponding to the considered problem are
on the zig-zag wall : U ¼ V ¼ 0; h ¼ C ¼ 1
Nux ¼
@h @C and Shx ¼ @Y @Y
The average heat and mass transfer rates on the surface of heat and contaminant sources can be evaluated by the average Nusselt and Sherwood numbers, which are defined respectively as
The dimensionless parameters appearing in the above equations are the Prandtl number (Pr), the thermal Rayleigh number (RaT), Lewis number (Le) and buoyancy ratio (Br), which are defined respectively as
Pr ¼
on the inclined walls : U ¼ V ¼ 0; h ¼ C ¼ 0 The local heat and mass transfer rates on the surface of heat and contaminant sources are defined respectively as
!
x Y uH vH X¼ ; Y¼ ; U¼ ; V¼ ; H H m m 2 ðp þ qgyÞH T TL c cL P¼ ; h¼ and C ¼ qm2 Th TL ch cL
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Nu ¼
1 A
Z
A
0
and Sh ¼
@h dX @Y 1 A
Z
A
0
@C dX @Y
ð8Þ
ð9Þ
The stream function is calculated from its definition as
U¼
@w ; @Y
V¼
@w @X
ð10Þ
Above governing equations are solved by using the Galerkin finite element scheme [26]. The convergence norm obligatory variation between the previous and current iterations for all of the dependent variables be 106. Grid independency study has been performed in this work to obtain optimum grid dimension. It is presented in Table 1. The grid
Fig. 4. Isoconcentration contours with Le = 2 at selected values of buoyancy ratio, Br and Rayleigh number, RaT.
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Fig. 6. Local Sherwood number vs. (a) bottom corner and (b) top corner of the v-corrugation wall, for different buoyancy ratio at RaT = 1.E+6 and Le = 2. Fig. 5. Local Nusselt number vs. (a) bottom corner and (b) top corner of the v-corrugation wall, for buoyancy ratio at RaT = 1.E+6 and Le = 2.
tests were performed for k = 0.2 L, Le = 2, Br = 1 and RaT = 106. In this work, five cases corresponding to the following number of nodes of 928, 1554, 2238, 3610 and 4830, were tested. For these grid dimensions, average Nusselt and Sherwood numbers are calculated. Grid distribution is presented in Fig. 1(b). As seen from the figure, finer grid distribution is used near the boundaries. Based on those grid tests, the 3610 grid dimension was chosen for this study, because the difference for both Nusselt and Sherwood numbers are lowest at this value. Thus, the comparison of results in Table 1 is acceptable based on differences. Table 2 shows a comparison between present work and study of Ghasemi and Aminossadati [27] for lid-driven enclosure in case of pure fluid. Same conditions are obtained with literature such as Richardson number. Percentage errors are also presented in last column. As seen from the table, results from the present code and literature show good agreement. 3. Results and discussion Effects of buoyancy ratio on heat and mass transfer have been investigated in a triangular enclosure with a zig-zag shaped bottom wall for thermal different Rayleigh number at k = 0.2 and Pr = 0.72. The study is also focused on effects of Lewis number, which changes between 0.1 and 20, on heat mass and fluid flow. Streamlines contours are presented in Fig. 2 for Le = 2 at selected values of buoyancy ratio, Br and thermal Rayleigh number, RaT. Mainly, two cells are formed inside the cavity which is distributed symmetrically except low buoyancy ratio and higher values of Rayleigh number. Each cell rotates in different directions for all values of Rayleigh number. The cell in left side, which rotates in
clockwise direction, becomes dominant to right one. This is clear from the values of streamfunction as shown inside the figure. Center of cells moves to middle axis of the cavity with increasing of buoyancy ratio. The shape of main cell fits with inclined walls of the enclosure due to insignificant effect of zig-zag walls. Flow strength increases with Rayleigh number for all values of buoyancy ratio. Mini cells are observed on zig-zag shaped wall for higher values of Rayleigh numbers. Those are found at the left and right corners starting from RaT = 105 and Br = 10. These mini cells also obtained at the middle of the bottom wall for some values. It is an interesting result that there is no circulation near the zig-zag walls at Br = 0 which indicates the pure thermal convection. This is valid almost for all Rayleigh number. The reason of this situation is weakness of the concentration. Another observation is that main cell center sits to corners for Br = 10 and RaT = 106 due to high concentration and high temperature differences. Fig. 3 illustrates the isotherms contours with Le = 2 at selected values of buoyancy ratio, Br and thermal Rayleigh number, RaT. As indicated in boundary conditions, the cavity is heated from the bottom zig-zag shaped wall and concentration is higher on this wall than that of inclined walls. As shown from the figure, parabolic distribution is observed even at the highest value of Rayleigh number. This distribution is almost valid for Br = 10 and 5 at all values of Rayleigh numbers. For Br = 0 (pure thermal convection) and RaT = 104, plumelike temperature distribution is started due to moving fluid to the upper side. It becomes stronger with increasing of both buoyancy ratio and Rayleigh number. Hat of plume becomes wider for the higher value of buoyancy ratio and Rayleigh number and second plumes are started from bottom corner and it moves to middle of the cavity. Comparison of negative and positive values of buoyancy value becomes important role on isotherms from the decision of convection and conduction mode of heat transfer point of view.
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Fig. 7. Effect of buoyancy ratio, Br on average (a) Nusselt (b) Sherwood numbers with Le = 2.
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Fig. 4 shows isoconcentration contours with Le = 2 for different buoyancy ratio and Rayleigh number values. Concentration distribution shows similar trend with temperature at lower values of Rayleigh numbers. Trapezoidal shaped isoconcentration distribution is formed for negative and zero values of buoyancy ratio for both RaT = 105 and 106. For higher values of buoyancy ratio, a plumelike distribution is observed and it becomes thinner for the highest value of Rayleigh number and buoyancy ratio. As seen from the figure, values of concentrations are increased with the buoyancy ratio and Rayleigh number due to increasing of convection mode. Variation of local Nusselt numbers at bottom corner of the vcorrugation wall for RaT = 106 in Fig. 5(a). As seen from the figure, local Nusselt numbers are decreased for positive values of buoyancy ratio and Br = 0 up to X = 0.5. Because flow strength is decreased through the middle of the cavity and a stagnation flow is occurred at the mid axis in x-direction. However, local Nusselt numbers are almost equal to each other for X = 0.3, 0.5 and 0.7. Because the mass diffusivity is higher. Nevertheless, variation of negative values of buoyancy ratio becomes insignificant on variation of local Nusselt numbers. General observation shows that symmetric distribution is occurred on local Nusselt numbers. Higher Nusselt number values are obtained for higher values of Rayleigh numbers which is an expected result. Fig. 5(b) displays the local Nusselt number vs. top corner of the v-corrugation wall for different Rayleigh numbers. Similar distribution is observed at top corners and the highest value of Rayleigh number is formed for the highest value of buoyancy ratio. Local Sherwood numbers for bottom corner of the v-corrugation wall for RaT = 106 are shown in Fig. 6(a). As well as known from the literature, the Sherwood number is a measure of mass transfer. Variation of negative value of buoyancy ratio becomes insignificant on variation of Sherwood numbers like heat transfer. It is an interesting result that Sherwood number becomes minimum around
Fig. 8. Streamlines contours with Br = 1 at selected values of Lewis number, Le and Rayleigh number, RaT.
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X = 0.5 for negative and positive values of buoyancy ratio and it shows an increment at that point for positive values of buoyancy ratio. For Br = 0, the values are almost the lowest value around X = 0.4 and 0.6 because of the weakness of concentration. Fig. 6(b) shows variation of local Sherwood number vs. top corner
of the v-corrugation wall at RaT = 106 and Le = 2. Values of local Sherwood number increases with increasing of Rayleigh number but trend becomes almost same. Local Sherwood number becomes almost constant with x direction when buoyancy values are negative.
Fig. 9. Isotherms contours with Br = 1 at selected values of Lewis number, Le and Rayleigh number, RaT.
Fig. 10. Isoconcentration contours with Br = 1 at selected values of Lewis number, Le and Rayleigh number, RaT.
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Variation of average Nusselt and Sherwood numbers with buoyancy ratio is shown in Fig. 7(a) and (b), respectively. For negative values of buoyancy ratio, both average Nusselt and Sherwood numbers become almost constant and they increase with increasing of buoyancy ratio except RaT = 103 due to domination of conduction mode of heat transfer. Difference between values at Br = 10 show more increment according to other values. Values of Sherwood numbers are higher than that of average Nusselt numbers for all values of buoyancy ratio and Rayleigh numbers. However, both values are almost same for lower values of Rayleigh number due to low flow velocity. Fig. 8 is illustrated to see the effects of Lewis number on streamlines at different Rayleigh number for Br = 1. As well known from the literature Lewis number is the ratio of thermal diffusivity to mass diffusivity. Variation of flow strength is changed with Lewis number. Mainly two rotating cells are formed for all values of Lewis number. For all cases, flow strength in the left side is dominant to right side. Streamlines are almost symmetric at RaT = 104 for all values of Lewis numbers. When Le = 5, streamfunction values have highest value at left side for RaT = 104 and 105. For higher values of thermal diffusivity, namely Le = 10 and 20, rotating flow at the right side becomes dominant to left side for RaT = 105 and 106. Due to strong convection mini cells are formed at the top and bottom corners at RaT = 106. Fig. 9 is plotted to show isotherms for Br = 1 at different Lewis numbers and Rayleigh numbers. Isotherms are distributed as parabolic distribution for RaT = 104 and their values are not changed with Lewis numbers. Effects of convection increases with Rayleigh number and plumelike distribution is observed. Isotherms move to upward from the middle of the bottom wall up to Le = 5. However, plume moves to left side of the cavity for higher values of Lewis number as Le = 10 and 20. Mushroom shaped distribution indicates
the strength of convection inside the cavity at RaT = 106. It becomes stronger for higher Lewis number due to increasing of thermal diffusivity. Isoconcentration contours with Br = 1 at selected values of Lewis number and Rayleigh number are presented in Fig. 10. When Lewis number smaller than 1 or equal to 1 isocontcentration is distributed regularly at lower values of Rayleigh number. For higher values of Lewis number the layer near boundaries becomes very thin. The hat of plumelike distribution of concentration move to left inclined wall for cases of Le = 10 and 20. This is due to from the distribution of flow as given in Fig. 8. Fig. 11 illustrates the local Nusselt number for bottom corner (Fig. 11(a)) and top corner (Fig. 11(b)) of the v-corrugation wall at different values of Lewis number and RaT = 106 and Br = 1. A minimum value is formed for local Nusselt number at bottom corner around X = 0.5 for lower values of Lewis number. This point is moved to X = 0.3 due to domination of flow at right side of triangular enclosure as given in Figs. 9–11. Results are almost same at Le = 10 and 20 due to domination of thermal diffusivity to mass diffusivity. Variation of local Nusselt number exhibits different trend on top corner of the v-corrugation wall than that of bottom corner. Again almost same values are formed at the highest value of Lewis number but the higher values are formed near left and right corners at Le = 5. Variation of local Sherwood number is presented at Fig. 12(a) and (b) on bottom and top corner, respectively. This figures are given for RaT = 106 and Br = 1. Local Sherwood numbers have a minimum value at X = 5 for Le = 0.1, 1 and 5 on bottom corner of zigzag shaped wall. Like local Nusselt numbers, Sherwood numbers are minimum around X = 3 for Le = 10 and 20. Local Sherwood number has the highest value at X = 0.2 for top corner of the v-cor-
Fig. 11. Local Nusselt number vs. (a) bottom corner and (b) top corner of the vcorrugation wall, for Lewis number at RaT = 1.E+6 and Br = 1.
Fig. 12. Local Sherwood number vs. (a) bottom corner and (b) top corner of the vcorrugation wall, for different Lewis number at RaT = 1.E+6 and Br = 1.
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tion of negative value of buoyancy ratio on heat and mass transfer become less important according to positive values. Heat and mass transfer increase with positive value of buoyancy ratio and Rayleigh number. Number of circulation cell in fluid flow directly related to buoyancy ratio. Geometry, thermal and concentration boundary conditions are symmetric but flow distribution inside the cavity shows asymmetric distribution depends on buoyancy ratio. Conduction mode of heat transfer becomes dominant onto convection even at higher Rayleigh number for negative value of buoyancy ratio.
Acknowledgement Second and fourth authors like to extend our appreciation to the Deanship of scientific research at King Saud University for funding the work through the Research group project No: RGP-VPP-091.
References
Fig. 13. Effect of Lewis number, Le on average (a) Nusselt (b) Sherwood numbers with Br = 1.
rugation wall at Le = 5 due to weak values of flow strength. On the contrary, local Sherwood number has higher value for Le = 10 and 20 at X = 0.6 and 0.8. Average values of Nusselt and Sherwood numbers for Br = 1 are given in Fig. 13(a) and (b), respectively. Average Nusselt number becomes constant with increasing of Lewis number for all values of Rayleigh number. As given in the Fig. 13(a), average Nusselt number decreases with increasing of Lewis number due to increasing of thermal diffusivity. As an expected result, higher heat transfer is increased with increasing of Rayleigh number. On the contrary, Sherwood number increases with increasing of Lewis number and Rayleigh number. Values are almost equal to each other at Le = 0.1 for all values of Rayleigh number due to lower effect of thermal diffusivity. Due to domination conduction mode of heat transfer Sherwood number becomes constant at RaT = 103. 4. Conclusions A computational work has been performed to show the effects of buoyancy ratio on heat and mass transfer in a triangular enclosure with zig-zag shaped bottom wall for different parameters. Some important findings can be drawn as follows: Lewis number is an effective parameter on flow field and temperature distribution. For higher values of Lewis number symmetric distribution of flow field, temperature field and concentration field are distorted. Negative and positive values of buoyancy ratio play important role on temperature distribution. Negative value of buoyancy ratio decreases the strength of temperature distribution. Varia-
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Contents lists available at SciVerse ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Natural convection effects on heat and mass transfer in a curvilinear triangular cavity M.M. Rahman a,⇑, Hakan F. Öztop b, A. Ahsan c, J. Orfi d a
Department of Mathematics, Bangladesh University of Engineering and Technology, Dhaka 1000, Bangladesh Department of Mechanical Engineering, Technology Faculty, Firat University, Elazig, Turkey c Department of Civil Engineering, Faculty of Engineering, (Green Engineering and Sustainable Technology Lab, Institute of Advanced Technology), University Putra, Malaysia, 43400 UPM Serdang, Selangor, Malaysia d Department of Mechanical Engineering, College of Engineering, King Saud University, Riyadh, Saudi Arabia b
a r t i c l e
i n f o
Article history: Received 2 February 2012 Received in revised form 18 June 2012 Accepted 20 June 2012 Available online 9 July 2012 Keywords: Double-diffusive Natural convection Finite element method Triangular cavity Absorber plate Inclined glass covers
a b s t r a c t Finite element method is used in this study to analyze the effects of buoyancy ratio and Lewis number on heat and mass transfer in a triangular cavity with zig-zag shaped bottom wall. Buoyancy ratio is defined as the ratio of Grashof number of solutal and thermal. Inclined walls of the cavity have lower temperature and concentration according to zig-zag shaped bottom wall. Enclosed space consists mostly of an absorber plate and two inclined glass covers that form a cavity. Both high temperature and high concentrations are applied to bottom corrugated wall. Computations were done for different values of buoyancy ratio (10 6 Br 6 10), Lewis number (0.1 6 Le 6 20) and thermal Rayleigh number (104 6 RaT 6 106). Streamlines, isotherms, iso-concentration, average Nusselt and Sherwood numbers are obtained. It is found that average Nusselt and Sherwood numbers increase by 89.18% and 101.91% respectively as Br increases from 10 to 20 at RaT = 106. Also, average Nusselt decreases by 16.22% and Sherwood numbers increases by 144.84% as Le increases from 0.1 to 20 at this Rayleigh number. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Natural convection driven flow due to simultaneous temperature and concentration gradients is generally referred to double diffusive or thermosolutal convection. In such cases, complex flow structures often result in liquids due to differences between the thermal and solutal diffusivities [1]. It may be met in geophysical, geothermal and industrial applications, such as the migration of moisture through air contained in fibrous insulations and the underground spreading of chemical contaminants through water saturated soil and desalination process. Double-diffusive heat transfer problem is studied mostly for square or rectangular geometries at different thermal and solutal boundary conditions as given in literature [2–7]. However, natural convection heat and mass transfer finds an important applications in triangular shaped enclosures such as desalination process in solar stills, attic-shaped space etc., Natural convection heat transfer in triangular enclosures is reviewed by Saha and Khan [8]. Earlier studies of the natural convection studies in triangular enclosure are Flack [9], and Poulikakos and Bejan [10,11]. Asan and Namli [12,13], made a numerical works to compare the summer and win-
⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (M.M. Rahman). 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.06.055
ter conditions for a roof of buildings. They indicated that the conditions are extremely important in flow motion inside the roof of building. Then Salmun [14], Varol et al. [15–18], Kent [19], Ridouane et al. [20] and del Campo et al. [21]. In all of these studies, the bottom wall of the enclosure is chosen as flat at different temperature boundary conditions, and most of these works, mass transfer is not considered. Chamkha et al. [22] made a numerical work on double-diffusive natural convection inclined finned triangular porous enclosures in the presence of heat generation/absorption effects. Double-diffusive natural convection can be seen in some solar collectors. Omri et al. [23] deal with a numerical simulation of natural convection flows in a triangular cavity submitted to a uniform heat flux using the Control Volume Finite Element Method. Their results showed that the flow structure is sensitive to the cover tilt angle. Many recirculation zones can occur in the core cavity and the heat transfer is dependent on the flow structure. A numerical model is presented for the study of natural convective heat and mass transfer in a triangular cavity by Omri [24] for greenhouse solar stills where vertical temperature and concentration gradients between the saline water and transparent cover induce flow in a confined space. They investigated the relationship between the pressure gradient and the other variables. The main aim of this study is to investigate the effects of buoyancy ratio and Lewis number on double diffusive natural
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Nomenclature A Br c ch cL C D g H L Le Nu p P Pr RaT Sh T Th TL
aspect ratio, L/H buoyancy ratio concentration of species high species concentration (source) low species concentration (sink) dimensionless species concentration species diffusivity gravitational acceleration enclosure height enclosure length Lewis number average Nusselt number dimensional pressure non-dimensional pressure Prandtl number thermal Rayleigh number average Sherwood number temperature hot wall temperature (source) cold wall temperature (sink)
convection in triangular types of cavity with zig-zag shaped bottom wall. Based on authors’ knowledge and above literature survey, this work is first step to analyze double diffusive natural convection in mentioned geometry.
2. Physical description and mathematical formulation Fig. 1(a) shows the physical model of a triangular cavity with zig-zag shaped bottom wall in which L and H indicate the length and height of the cavity, respectively. Fig. 1(b) shows grid distribution. The enclosure is filled with a viscous, incompressible and Newtonian fluid. The wave length and height is defined as in the
u U v V x X y Y
horizontal velocity component dimensionless horizontal velocity component vertical velocity component dimensionless vertical velocity component horizontal coordinate dimensionless horizontal coordinate vertical coordinate dimensionless vertical coordinate
Greek symbols thermal diffusivity bT thermal expansion coefficient bc compositional expansion coefficient k wave length m kinematic viscosity h non-dimensional temperature q density w stream function C general dependent variable
a
figure that it is taken as k = 0.2 L and h = 0.05 H, respectively. The inclined and the base walls are kept at constant temperatures Th and TL and concentration ch and cL respectively. Both concentration and temperature are higher on wavy wall than that of inclined walls. The Boussinesq approximation is assumed to be valid and constant fluid properties are used except the density variations. The gravity acts in negative – y direction. Due to the relative values of Rayleigh numbers considered in this study, the flow is assumed to be laminar [25]. With this geometry and boundary conditions, the present study reports the computations for cavity at a fixed Prandtl number of Pr = 0.72, buoyancy ratio (Br) ranging from 10 to 10, Lewis number (Le) ranging from 0.1 to 20, and thermal Rayleigh number (RaT) ranging from 104 to 106, and their effect on the heat and mass transfer process is analyzed. Under the above approximations, the governing equations for steady natural convection flow using conservation of mass, momentum and energy can be written as formulated by Hajri et al. [25]. Both concentration and temperature are higher on bottom zig-zag shaped wall than that of inclined walls. For steady two-dimensional laminar convection the coupled transport
Table 1 Comparison of average Nusselt and Sherwood numbers for various grids, with Le = 2, k = 0.2 L, Br = 1 and Ra = 1.E + 6. No. of elements
Nu
% Difference
Sh
% Difference
928 1554 2238 3610 4830
18.59093 19.0223 19.00917 19.007602 19.014506
– 0.069 0.032 0.016 0.011
21.140654 21.782312 21.731108 21.726773 21.733893
– 0.103 0.045 0.022 0.015
Table 2 Comparison of present values of average Nusselt number for U = 0 with literature. Ri
Fig. 1. (a) Schematic of the problem with the domain and boundary conditions and (b) grid distribution.
0.01 0.1 1 10
Nu Results ghasemi and aminossadati
Present results
Error (%)
33.275 29.356 11.073 11.073
32.706 29.049 11.125 10.623
1.71 1.05 0.47 4.06
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equations for U, V, h and C written in Cartesian coordinates in the dimensionless form as:
@U @V þ ¼0 @X @Y
ð1Þ
@U @U @P @2U @2U þV ¼ þ Pr þ U @X @Y @X @X 2 @Y 2 @V @V @P @2V @2V U þV ¼ þ Pr þ @X @Y @Y @X 2 @Y 2
! ð2Þ
! þ RaT Prðh þ BrCÞ
Fig. 2. Streamlines contours with Le = 2 at selected values of buoyancy ratio, Br and Rayleigh number, RaT.
Fig. 3. Isotherms contours with Le = 2 at selected values of buoyancy ratio, Br and Rayleigh number, RaT.
ð3Þ
M.M. Rahman et al. / International Journal of Heat and Mass Transfer 55 (2012) 6250–6259
U
U
@h @h @ 2 h @ 2 h þ þV ¼ @X @Y @X 2 @Y 2 @C @C 1 @2C @2C þV ¼ þ @X @Y Le @X 2 @Y 2
ð4Þ
ð5Þ
where the dimensionless variables are introduced as:
ð6Þ
m gb ðT h T L ÞH3 a b ðch cL Þ ; RaT ¼ T ; Le ¼ and Br ¼ c bT ðT h T L Þ a am D ð7Þ
The dimensionless boundary conditions corresponding to the considered problem are
on the zig-zag wall : U ¼ V ¼ 0; h ¼ C ¼ 1
Nux ¼
@h @C and Shx ¼ @Y @Y
The average heat and mass transfer rates on the surface of heat and contaminant sources can be evaluated by the average Nusselt and Sherwood numbers, which are defined respectively as
The dimensionless parameters appearing in the above equations are the Prandtl number (Pr), the thermal Rayleigh number (RaT), Lewis number (Le) and buoyancy ratio (Br), which are defined respectively as
Pr ¼
on the inclined walls : U ¼ V ¼ 0; h ¼ C ¼ 0 The local heat and mass transfer rates on the surface of heat and contaminant sources are defined respectively as
!
x Y uH vH X¼ ; Y¼ ; U¼ ; V¼ ; H H m m 2 ðp þ qgyÞH T TL c cL P¼ ; h¼ and C ¼ qm2 Th TL ch cL
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Nu ¼
1 A
Z
A
0
and Sh ¼
@h dX @Y 1 A
Z
A
0
@C dX @Y
ð8Þ
ð9Þ
The stream function is calculated from its definition as
U¼
@w ; @Y
V¼
@w @X
ð10Þ
Above governing equations are solved by using the Galerkin finite element scheme [26]. The convergence norm obligatory variation between the previous and current iterations for all of the dependent variables be 106. Grid independency study has been performed in this work to obtain optimum grid dimension. It is presented in Table 1. The grid
Fig. 4. Isoconcentration contours with Le = 2 at selected values of buoyancy ratio, Br and Rayleigh number, RaT.
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Fig. 6. Local Sherwood number vs. (a) bottom corner and (b) top corner of the v-corrugation wall, for different buoyancy ratio at RaT = 1.E+6 and Le = 2. Fig. 5. Local Nusselt number vs. (a) bottom corner and (b) top corner of the v-corrugation wall, for buoyancy ratio at RaT = 1.E+6 and Le = 2.
tests were performed for k = 0.2 L, Le = 2, Br = 1 and RaT = 106. In this work, five cases corresponding to the following number of nodes of 928, 1554, 2238, 3610 and 4830, were tested. For these grid dimensions, average Nusselt and Sherwood numbers are calculated. Grid distribution is presented in Fig. 1(b). As seen from the figure, finer grid distribution is used near the boundaries. Based on those grid tests, the 3610 grid dimension was chosen for this study, because the difference for both Nusselt and Sherwood numbers are lowest at this value. Thus, the comparison of results in Table 1 is acceptable based on differences. Table 2 shows a comparison between present work and study of Ghasemi and Aminossadati [27] for lid-driven enclosure in case of pure fluid. Same conditions are obtained with literature such as Richardson number. Percentage errors are also presented in last column. As seen from the table, results from the present code and literature show good agreement. 3. Results and discussion Effects of buoyancy ratio on heat and mass transfer have been investigated in a triangular enclosure with a zig-zag shaped bottom wall for thermal different Rayleigh number at k = 0.2 and Pr = 0.72. The study is also focused on effects of Lewis number, which changes between 0.1 and 20, on heat mass and fluid flow. Streamlines contours are presented in Fig. 2 for Le = 2 at selected values of buoyancy ratio, Br and thermal Rayleigh number, RaT. Mainly, two cells are formed inside the cavity which is distributed symmetrically except low buoyancy ratio and higher values of Rayleigh number. Each cell rotates in different directions for all values of Rayleigh number. The cell in left side, which rotates in
clockwise direction, becomes dominant to right one. This is clear from the values of streamfunction as shown inside the figure. Center of cells moves to middle axis of the cavity with increasing of buoyancy ratio. The shape of main cell fits with inclined walls of the enclosure due to insignificant effect of zig-zag walls. Flow strength increases with Rayleigh number for all values of buoyancy ratio. Mini cells are observed on zig-zag shaped wall for higher values of Rayleigh numbers. Those are found at the left and right corners starting from RaT = 105 and Br = 10. These mini cells also obtained at the middle of the bottom wall for some values. It is an interesting result that there is no circulation near the zig-zag walls at Br = 0 which indicates the pure thermal convection. This is valid almost for all Rayleigh number. The reason of this situation is weakness of the concentration. Another observation is that main cell center sits to corners for Br = 10 and RaT = 106 due to high concentration and high temperature differences. Fig. 3 illustrates the isotherms contours with Le = 2 at selected values of buoyancy ratio, Br and thermal Rayleigh number, RaT. As indicated in boundary conditions, the cavity is heated from the bottom zig-zag shaped wall and concentration is higher on this wall than that of inclined walls. As shown from the figure, parabolic distribution is observed even at the highest value of Rayleigh number. This distribution is almost valid for Br = 10 and 5 at all values of Rayleigh numbers. For Br = 0 (pure thermal convection) and RaT = 104, plumelike temperature distribution is started due to moving fluid to the upper side. It becomes stronger with increasing of both buoyancy ratio and Rayleigh number. Hat of plume becomes wider for the higher value of buoyancy ratio and Rayleigh number and second plumes are started from bottom corner and it moves to middle of the cavity. Comparison of negative and positive values of buoyancy value becomes important role on isotherms from the decision of convection and conduction mode of heat transfer point of view.
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Fig. 7. Effect of buoyancy ratio, Br on average (a) Nusselt (b) Sherwood numbers with Le = 2.
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Fig. 4 shows isoconcentration contours with Le = 2 for different buoyancy ratio and Rayleigh number values. Concentration distribution shows similar trend with temperature at lower values of Rayleigh numbers. Trapezoidal shaped isoconcentration distribution is formed for negative and zero values of buoyancy ratio for both RaT = 105 and 106. For higher values of buoyancy ratio, a plumelike distribution is observed and it becomes thinner for the highest value of Rayleigh number and buoyancy ratio. As seen from the figure, values of concentrations are increased with the buoyancy ratio and Rayleigh number due to increasing of convection mode. Variation of local Nusselt numbers at bottom corner of the vcorrugation wall for RaT = 106 in Fig. 5(a). As seen from the figure, local Nusselt numbers are decreased for positive values of buoyancy ratio and Br = 0 up to X = 0.5. Because flow strength is decreased through the middle of the cavity and a stagnation flow is occurred at the mid axis in x-direction. However, local Nusselt numbers are almost equal to each other for X = 0.3, 0.5 and 0.7. Because the mass diffusivity is higher. Nevertheless, variation of negative values of buoyancy ratio becomes insignificant on variation of local Nusselt numbers. General observation shows that symmetric distribution is occurred on local Nusselt numbers. Higher Nusselt number values are obtained for higher values of Rayleigh numbers which is an expected result. Fig. 5(b) displays the local Nusselt number vs. top corner of the v-corrugation wall for different Rayleigh numbers. Similar distribution is observed at top corners and the highest value of Rayleigh number is formed for the highest value of buoyancy ratio. Local Sherwood numbers for bottom corner of the v-corrugation wall for RaT = 106 are shown in Fig. 6(a). As well as known from the literature, the Sherwood number is a measure of mass transfer. Variation of negative value of buoyancy ratio becomes insignificant on variation of Sherwood numbers like heat transfer. It is an interesting result that Sherwood number becomes minimum around
Fig. 8. Streamlines contours with Br = 1 at selected values of Lewis number, Le and Rayleigh number, RaT.
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X = 0.5 for negative and positive values of buoyancy ratio and it shows an increment at that point for positive values of buoyancy ratio. For Br = 0, the values are almost the lowest value around X = 0.4 and 0.6 because of the weakness of concentration. Fig. 6(b) shows variation of local Sherwood number vs. top corner
of the v-corrugation wall at RaT = 106 and Le = 2. Values of local Sherwood number increases with increasing of Rayleigh number but trend becomes almost same. Local Sherwood number becomes almost constant with x direction when buoyancy values are negative.
Fig. 9. Isotherms contours with Br = 1 at selected values of Lewis number, Le and Rayleigh number, RaT.
Fig. 10. Isoconcentration contours with Br = 1 at selected values of Lewis number, Le and Rayleigh number, RaT.
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Variation of average Nusselt and Sherwood numbers with buoyancy ratio is shown in Fig. 7(a) and (b), respectively. For negative values of buoyancy ratio, both average Nusselt and Sherwood numbers become almost constant and they increase with increasing of buoyancy ratio except RaT = 103 due to domination of conduction mode of heat transfer. Difference between values at Br = 10 show more increment according to other values. Values of Sherwood numbers are higher than that of average Nusselt numbers for all values of buoyancy ratio and Rayleigh numbers. However, both values are almost same for lower values of Rayleigh number due to low flow velocity. Fig. 8 is illustrated to see the effects of Lewis number on streamlines at different Rayleigh number for Br = 1. As well known from the literature Lewis number is the ratio of thermal diffusivity to mass diffusivity. Variation of flow strength is changed with Lewis number. Mainly two rotating cells are formed for all values of Lewis number. For all cases, flow strength in the left side is dominant to right side. Streamlines are almost symmetric at RaT = 104 for all values of Lewis numbers. When Le = 5, streamfunction values have highest value at left side for RaT = 104 and 105. For higher values of thermal diffusivity, namely Le = 10 and 20, rotating flow at the right side becomes dominant to left side for RaT = 105 and 106. Due to strong convection mini cells are formed at the top and bottom corners at RaT = 106. Fig. 9 is plotted to show isotherms for Br = 1 at different Lewis numbers and Rayleigh numbers. Isotherms are distributed as parabolic distribution for RaT = 104 and their values are not changed with Lewis numbers. Effects of convection increases with Rayleigh number and plumelike distribution is observed. Isotherms move to upward from the middle of the bottom wall up to Le = 5. However, plume moves to left side of the cavity for higher values of Lewis number as Le = 10 and 20. Mushroom shaped distribution indicates
the strength of convection inside the cavity at RaT = 106. It becomes stronger for higher Lewis number due to increasing of thermal diffusivity. Isoconcentration contours with Br = 1 at selected values of Lewis number and Rayleigh number are presented in Fig. 10. When Lewis number smaller than 1 or equal to 1 isocontcentration is distributed regularly at lower values of Rayleigh number. For higher values of Lewis number the layer near boundaries becomes very thin. The hat of plumelike distribution of concentration move to left inclined wall for cases of Le = 10 and 20. This is due to from the distribution of flow as given in Fig. 8. Fig. 11 illustrates the local Nusselt number for bottom corner (Fig. 11(a)) and top corner (Fig. 11(b)) of the v-corrugation wall at different values of Lewis number and RaT = 106 and Br = 1. A minimum value is formed for local Nusselt number at bottom corner around X = 0.5 for lower values of Lewis number. This point is moved to X = 0.3 due to domination of flow at right side of triangular enclosure as given in Figs. 9–11. Results are almost same at Le = 10 and 20 due to domination of thermal diffusivity to mass diffusivity. Variation of local Nusselt number exhibits different trend on top corner of the v-corrugation wall than that of bottom corner. Again almost same values are formed at the highest value of Lewis number but the higher values are formed near left and right corners at Le = 5. Variation of local Sherwood number is presented at Fig. 12(a) and (b) on bottom and top corner, respectively. This figures are given for RaT = 106 and Br = 1. Local Sherwood numbers have a minimum value at X = 5 for Le = 0.1, 1 and 5 on bottom corner of zigzag shaped wall. Like local Nusselt numbers, Sherwood numbers are minimum around X = 3 for Le = 10 and 20. Local Sherwood number has the highest value at X = 0.2 for top corner of the v-cor-
Fig. 11. Local Nusselt number vs. (a) bottom corner and (b) top corner of the vcorrugation wall, for Lewis number at RaT = 1.E+6 and Br = 1.
Fig. 12. Local Sherwood number vs. (a) bottom corner and (b) top corner of the vcorrugation wall, for different Lewis number at RaT = 1.E+6 and Br = 1.
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tion of negative value of buoyancy ratio on heat and mass transfer become less important according to positive values. Heat and mass transfer increase with positive value of buoyancy ratio and Rayleigh number. Number of circulation cell in fluid flow directly related to buoyancy ratio. Geometry, thermal and concentration boundary conditions are symmetric but flow distribution inside the cavity shows asymmetric distribution depends on buoyancy ratio. Conduction mode of heat transfer becomes dominant onto convection even at higher Rayleigh number for negative value of buoyancy ratio.
Acknowledgement Second and fourth authors like to extend our appreciation to the Deanship of scientific research at King Saud University for funding the work through the Research group project No: RGP-VPP-091.
References
Fig. 13. Effect of Lewis number, Le on average (a) Nusselt (b) Sherwood numbers with Br = 1.
rugation wall at Le = 5 due to weak values of flow strength. On the contrary, local Sherwood number has higher value for Le = 10 and 20 at X = 0.6 and 0.8. Average values of Nusselt and Sherwood numbers for Br = 1 are given in Fig. 13(a) and (b), respectively. Average Nusselt number becomes constant with increasing of Lewis number for all values of Rayleigh number. As given in the Fig. 13(a), average Nusselt number decreases with increasing of Lewis number due to increasing of thermal diffusivity. As an expected result, higher heat transfer is increased with increasing of Rayleigh number. On the contrary, Sherwood number increases with increasing of Lewis number and Rayleigh number. Values are almost equal to each other at Le = 0.1 for all values of Rayleigh number due to lower effect of thermal diffusivity. Due to domination conduction mode of heat transfer Sherwood number becomes constant at RaT = 103. 4. Conclusions A computational work has been performed to show the effects of buoyancy ratio on heat and mass transfer in a triangular enclosure with zig-zag shaped bottom wall for different parameters. Some important findings can be drawn as follows: Lewis number is an effective parameter on flow field and temperature distribution. For higher values of Lewis number symmetric distribution of flow field, temperature field and concentration field are distorted. Negative and positive values of buoyancy ratio play important role on temperature distribution. Negative value of buoyancy ratio decreases the strength of temperature distribution. Varia-
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