- Email: [email protected]
The effects of Prandtl number on natural convection in triangular enclosures with localized heating from below
International Communications in Heat and Mass Transfer 34 (2007) 511 – 519 www.elsevier.com/locate/ichmt
The effects of Prandtl number on natural convection in triangular enclosures with localized heating from below☆ Ahmet Koca a , Hakan F. Oztop b , Yasin Varol c,⁎ a
Department of Mechanical Education, Firat University, TR-23119, Elazig, Turkey Department of Mechanical Engineering, Firat University, TR-23119, Elazig, Turkey Department of Mechanical Engineering, Vanderbilt University, Nashville, TN 37235, USA b
c
Available online 21 February 2007
Abstract The effect of Prandtl number on natural convection heat transfer and fluid flow in triangular enclosures with localized heating has been analyzed by solving governing equations of natural convection in streamfunction–vorticity form with finite-difference technique. Solution of linear algebraic equations was made by Successive Under Relaxation (SUR) method. Bottom wall of triangle is heated partially while inclined wall is maintained at a lower uniform temperature than heated wall while remaining walls are insulated. Computations were carried out for dimensionless heater locations (0.15 ≤ s ≤ 0.95), dimensionless heater length (0.1 ≤ w ≤ 0.9), Prandtl number (0.01 ≤ Pr ≤ 15) and Rayleigh number (103 ≤ Ra ≤ 106). Aspect ratio of triangle was chosen as unity. It is observed that both flow and temperature fields are affected with the changing of Prandtl number, location of heater and length of heater as well as Rayleigh number. © 2007 Elsevier Ltd. All rights reserved. Keywords: Prandtl number; Natural convection; Partial heating; Triangular enclosure
1. Introduction Analysis of natural convection in enclosures plays important role in many diverse applications including home heating, solar collectors, cryogenic storage, crystal growth, nuclear reactor design, furnace design, and electronical equipments. Cooling of electronic equipment must be considered when designing such systems. Naturally induced convection is an effective scheme as cooling in flush mounted or protruding electronical elements. These heaters are located in rectangular or square enclosures but they can work in corrugated enclosures such as PC monitors or TV. Natural convection in different geometrical enclosures has been extensively investigated, and reviews of the literature are found in Ostrach [1,2] and Catton [3]. Most of these studies related with analysis of natural convection in entirely heated or cooled rectangular or square cavity. Triangular enclosures can be used in the applications for roofs of the buildings or electronical heaters. Asan and Namli [4], Holtzman et al. [5], Akinsete and Coleman [6], Tzeng et al. [7], Salmun [8], and Ridouane and Campo [9] ☆
Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (Y. Varol).
0735-1933/$ - see front matter © 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2007.01.006
512
A. Koca et al. / International Communications in Heat and Mass Transfer 34 (2007) 511–519
Nomenclature AR g Gr H L n Nu Pr Ra s T u, v X, Y w
Aspect ratio, AR = H/L Gravitational acceleration Grashof number Maximum height of triangle Length of bottom wall Coordinate in normal direction Nusselt number Prandtl number Rayleigh number Dimensionless heater locations Temperature Velocities Non-dimensional coordinates Dimensionless heater length
Greek Letters υ Kinematic viscosity Ω Non-dimensional vorticity θ Non-dimensional temperature β Thermal expansion coefficient α Thermal diffusivity ψ Streamfunction ω Vorticity Subscript C Cold H Hot
have studied the natural convection in triangular shaped enclosures in detail. Recently, Varol et al. [10] investigated the flush mounted heater located on the vertical wall of right triangular enclosures. Applications of these boundary conditions can be found in the literature for square or rectangular enclosure (Churchill et al. [11], Farouk and Fusegi [12], Turkoglu and Yucel [13], Ben Nasr et al. [14], Aydin and Yang [15] and Prasad and Kulacki [16]).
Fig. 1. Physical model.
A. Koca et al. / International Communications in Heat and Mass Transfer 34 (2007) 511–519
513
In all of the studies given above, air was chosen as working fluid. However, effects of Prandtl number, which is the most important physical property of the fluids, are analyzed in some studies for rectangular enclosures. For instance, Yoo [17,18] investigated the bifurcation phenomena. The existence of dual solutions in natural convection in a horizontal annulus is numerically investigated for the fluids of 0.3 b Pr b 1. He found that bifurcation points are functions of the Prandtl number. In addition, Poujol et al. [19] analyzed the natural convection problem in a square cavity for high Prandtl number. Ganzarolli and Milanez [20] also investigated the natural convection in a differentially heated from vertical walls of rectangular square cavity by comparing air (Pr = 0.71) and water (Pr = 7) as working fluid. Same fluids are tested for square cavity in unsteady regime by Aydin [21]. Effects of Pr number also tested for different geometries in literature [22–25]. The investigation of the influence of Prandtl number on natural convection heat transfer and flow field is the main contribution for the present study. As observed in the literature review provided above, air was used as working fluid in the past researches for triangular geometries. However, effects of different Prandtl number have not been addressed. Thus, this study will give detailed information to researchers and designers about flush mounted heaters. 2. Physical model The schematical configuration of two-dimensional triangular enclosure with the important geometrical parameters is given in Fig. 1. It is a right triangle with bottom length L, and vertical wall height H. The bottom wall is partially heated with TH and inclined wall is cooled with TC. Vertical wall is insulated. The aspect ratio of the cavity is defined as AR = H/L = 1. The length of heater is depicted by w and its location center is s. 3. Governing equations and numerical method The present flow is considered steady, laminar, incompressible and two-dimensional. The viscous dissipation term in the energy equation is neglected. The variation of fluid properties with temperature has been neglected, with the only exception of the buoyancy term, for which the Boussinesq approximation has been adopted. Thus, natural convection is governed by the differential equations expressing the conservation of mass, momentum, and energy which are written in streamfunction–vorticity mode as follows. The governing equations of natural convection (Eqs. (1)–(3)) are written in streamfunction–vorticity form for laminar regime in two-dimensional form for steady, incompressible, and Newtonian fluid with Boussinesq approximation. It is assumed that radiation heat exchange is negligible according to other modes of heat transfer and the gravity acts in vertical direction. A2 W A2 W þ AX 2 AY 2 A2 X A2 X 1 AW AX AW AX Ah − þ ¼ −Ra AX 2 AY 2 Pr AY AX AX AY AX −X ¼
A2 h A2 h AW Ah AW Ah − þ ¼ AX 2 AY 2 AY AX AX AY
ð1Þ ð2Þ ð3Þ
The employed non-dimensional variables are given as x y wPr xðLÞ2 Pr T −TC ;X ¼ ;h ¼ X ¼ ;Y ¼ ;W ¼ L L υ υ TH −TC Aw Aw Av Au bgðTH −TC ÞL3 Pr υ ;v ¼ − ;x ¼ − u¼ ; Pr ¼ : ; Ra ¼ Ay Ax Ax Ay υ2 a
ð4Þ ð5Þ
Local and mean Nusselt numbers are calculated along the heater wall using Eq. (6) a and b, respectively. Z w Ah Nux ¼ − ; Num ¼ Nux dx ð6a:bÞ AY Y ¼0
514
A. Koca et al. / International Communications in Heat and Mass Transfer 34 (2007) 511–519
Boundary conditions for the considered model are depicted on the physical model (Fig. 1). In this model, u and v velocities are equal to zero for all solid boundaries. On the bottom wall; On the heater, T = TH, On insulated wall, AT An ¼ 0 On the vertical wall, AT ¼ 0 An On the inclined wall, T = TC Governing equations in streamline-vorticity form (Eqs. (1)–(3)) are solved through finite-difference method. Algebraic equations are obtained via Taylor series and they solved via Successive Under Relaxation (SUR) technique, iteratively. The central difference method is used for discretization procedure. The detailed solution technique is well described in the literature [26,27]. The convergence criterion, 10− 4 , is chosen for all depended variables and value of 0.1 is taken for under-relaxation parameter. Some grid tests were created between 35 × 35 and 237 × 237 to obtain optimum grid dimension. The test results showed that 61 × 61 grid dimension is enough for calculations. The computational results are compared with the literature for validation of the present computer code as shown in Fig. 2. Obtained results were compared with the results of Asan and Namli [4], Akinsete and Coleman [6] and Tzeng et al. [7]. 4. Results and discussion Numerical analysis of natural convection heat transfer and fluid flow was performed to obtain effects of Prandtl number in a triangular enclosure which is partially heated from the bottom wall. Temperature of inclined wall is lower than the temperature of the heater while remaining walls are adiabatic. Heat transfer and flow distributions were presented via streamlines and isotherms. The effects of governing parameters on heat transfer were presented by Nusselt numbers. The size and location of the heater, Rayleigh number and Prandtl number were taken as governing parameters on heat transfer and fluid flow. The ratio of height to bottom wall is taken as unity. Effects of Rayleigh number on flow and temperature distribution are presented in Fig. 3 for w = 0.5, s = 0.5, AR = 1.0 and Pr = 0.71. As it can be seen from the Fig. 3 (a) and (b), for Ra = 103 and 104 quasi-conduction heat transfer regime is formed. Heated flow moves from bottom and impinges to the inclined wall and two cells are formed in different directions. While the left cell turns in counterclockwise, the right one turns in clockwise. Intersections of these two cells becomes on the middle of the heater. Extremum absolute values of streamfunctions are increased with the increasing of Ra numbers. If Fig. 3 (a) and (b) are compared with each other, locations of minimum and maximum values of streamfunctions are almost same. When Ra = 105, strength of fluid is increased and a big main cell is formed in counterclockwise direction (Fig. 3 (c)). However, multiple cells are formed with the effect of convection regime at higher Ra numbers. The location of the cell in clockwise direction becomes almost same even at the highest Ra numbers and the third cell forms at the top corner of the triangle. The values of streamfunction and their locations are given on the Fig. 3. Plumelike distribution on isotherms starts for Ra N 105 due to strong convection. The similar distribution can be found in the literature for square cavities [15]. The physical model tested for different Prandtl numbers in the range of 0.01 ≤ Pr ≤ 15. Effects of Pr number on streamlines (on the left) and isotherms (on the right) are presented in Fig. 4 (a) to (c) for the parameters as s = 0.5, w = 0.5, AR = 1.0 and Ra = 105 for Pr = 0.1, 1, 15. It can be seen from the figure that the changing of Prandtl number makes important effect on flow field and temperature distribution due to strong convection. For Pr = 0.1, ψmin = −22.7 (X = 0.5, Y = 0.19) and ψmax = 21.10 (X = 0.23, Y = 0.46), Pr = 1, ψmin = − 14.30 (X = 0.67, Y = 0.15) and ψmax = 37.50 (X = 0.30, Y = 0.33). When Pr = 15, ψmin = −17.20 (X = 0.18, Y = 0.55) and ψmax = 24.6 (X = 0.43, Y = 0.21). Multiple cells are formed for all values of Pr numbers. When
Fig. 2. Comparisons of local Nusselt numbers with literature for Ra = 2772.
A. Koca et al. / International Communications in Heat and Mass Transfer 34 (2007) 511–519
515
Fig. 3. Streamlines (on the left) and isotherms (on the right) at w = 0.5, s = 0.5, AR = 1.0, Pr = 0.71, a) Ra = 103, b) Ra = 104, c) Ra = 105, d) Ra = 106.
Pr = 0.1, four cells are formed (Fig. 4 (a)). On the other hand, three cells are formed in Fig. 4 (b) and (c). Strong plumelike distribution occurs for small values of Pr numbers. However, thicker thermal boundary layer is obtained with high Pr number. When Pr = 15, the main cell elongates through the bottom wall of the enclosure. Fig. 5 (a) to (c) shows the variation of mean Nusselt number with Rayleigh number for different Prandtl number. We also tested the effects of the length and location of the heater on natural convection heat transfer. As seen from the figure, the values of heat transfer is almost constant up to Ra = 104. It increases with the increase of Ra number due to increasing of strength of the regime of convection heat transfer. The values of mean Nusselt number also increases with the increasing of Prandtl number. When these three figures are compared with each other (Fig. 5 (a) to (c)) the highest heat transfer is obtained for w = 0.8 and s = 0.5. For the smallest value of the heater length (w = 0.2), values of mean Nusselt numbers are very close to each other for lower Ra numbers and higher Pr numbers. For the same value of the heater length, heat transfer increases when the heater is moved toward the right
516
A. Koca et al. / International Communications in Heat and Mass Transfer 34 (2007) 511–519
Fig. 4. Streamlines (on the left) and isotherms (on the right) at w = 0.5, s = 0.5, AR = 1.0, Ra = 105, a) Pr = 0.01, b) Pr = 1, c) Pr = 15.
corner of the enclosure. Effects of Ra number on heat transfer is presented in Fig. 6 for different heater length at s = 0.5 and Pr = 0.71. It is observed that heat transfer is enhanced with the increasing of heater length and Rayleigh number, which is an expected result. The obtained mean Nusselt numbers are very close to each other for Ra = 103 and Ra = 104 due to conduction dominant regime. However, heat transfer increases with the increasing of Rayleigh number due to domination of convection mode of heat transfer.
5. Conclusions A numerical study has been performed to investigate the effects of Prandtl number on natural convection heat transfer and fluid flow in a partially heated triangular enclosure. The results are listed below: • Heat transfer increases with the increasing of heater length and Rayleigh number. Values of streamfunction (flow strength) also increase with the increasing of Rayleigh number.
A. Koca et al. / International Communications in Heat and Mass Transfer 34 (2007) 511–519
517
Fig. 5. Variation of mean Nusselt number with Rayleigh number for different Prandtl numbers, AR = 1.0, a) w = 0.2, s = 0.5, b) w = 0.8, s = 0.5, c) w = 0.2, s = 0.2.
• Heat transfer increases with the increasing of Prandtl number for all cases. Both flow and temperature fields are affected with the changing of Prandtl number. • Heat transfer increases when heater is located near the right corner of the triangular enclosure.
518
A. Koca et al. / International Communications in Heat and Mass Transfer 34 (2007) 511–519
Fig. 6. Variation of mean Nusselt number with length of the heater for different Rayleigh numbers, s = 0.5, AR = 1.0, Pr = 0.71.
The study can be extended for turbulent flow, different aspect ratios and an optimization study for a future work. References [1] S. Ostrach, Natural convection in enclosures, Advances in Heat Transfer, 8, Academic, New York, 1972, pp. 161–227. [2] S. Ostrach, Natural convection heat transfer in cavities and cells, Proceedings of the 7th Int. Heat Transfer Conference, vol. 1, 1982, pp. 365–379. [3] I. Catton, Natural convection in enclosures, Proceedings at the 6th Int. Heat Transfer Conference, vol. 6, 1978, pp. 13–31. [4] H. Asan, L. Namli, Laminar natural convection in a pitched roof of triangular cross-section: summer day boundary conditions, Energy and Buildings 33 (2000) 69–73. [5] G.A. Holtzman, R.W. Hill, K.S. Ball, Laminar natural convection in isosceles triangular enclosures heated from below and symmetrically cooled from above, J. Heat Transfer 122 (2000) 485–491. [6] V.A. Akinsete, T.A. Coleman, Heat transfer by steady laminar free convection in triangular enclosures, Int. J. Heat Mass Transfer 25 (1982) 991–998. [7] S.C. Tzeng, J.H. Liou, R.Y. Jou, Numerical simulation-aided parametric analysis of natural convection in a roof of triangular enclosures, Heat Transf. Eng. 26 (2005) 69–79. [8] H. Salmun, Convection patterns in a triangular domain, Int. J. Heat Mass Transfer 38 (1995) 351–362. [9] E.H. Ridouane, A. Campo, Relationship between thermal convection intensity and aspect ratio of two triangular cavities inscribed in horizontal rectangular cavities, Int. J. Numer. Methods Heat Fluid Flow 16 (2006) 338–355. [10] Y. Varol, A. Koca, H.F. Oztop, Natural convection in a triangle enclosure with flush mounted heater on the wall, Int. Commun. Heat Mass Transf. 33 (2006) 951–958. [11] S.W. Churchill, C.V.S. Patterson, H.H.S. Chu, The effect of heater size, location, aspect ratio, and boundary conditions on two dimensional, laminar, natural convection in rectangular enclosures, J. Heat Transfer 98 (1976) 1194–1201. [12] B. Farouk, T. Fusegi, Natural convection of a variable property gas in asymmetrically heated square cavities, J. Thermophysics and Heat Transfer 3 (1989) 85–87. [13] H. Turkoglu, N. Yucel, Effect of heater and cooler locations on natural convection in square cavities, Numerical Heat Transfer Part A 27 (1995) 351–358. [14] K. Ben Nasr, R. Chouikh, C. Kerkeni, A. Guizani, Numerical study of the natural convection in cavity heated from the lower corner and cooled from the ceiling, Applied Thermal Engineering 26 (2006) 772–775. [15] O. Aydin, W.J. Yang, Natural convection in enclosures with localized heating from below and symmetrical cooling from sides, Int. J. Numer. Meth. Heat Fluid Flow 10 (2000) 518–529. [16] V. Prasad, F.A. Kulacki, Convective heat transfer in a rectangular porous cavity-effect of aspect ratio on flow structure and heat transfer, J. Heat Transfer 106 (1984) 158–165. [17] J.S. Yoo, Prandtl number effect on transition of free-convective flows in a wide-gap horizontal annulus, Int. Commun. Heat Mass Transf. 26 (1999) 811–817. [18] J.S. Yoo, Prandtl number effect on bifurcation and dual solutions in natural convection in a horizontal annulus, Int. J. Heat Mass Transfer 42 (1999) 3279–3290. [19] F.T. Poujol, J. Rojas, E. Ramos, Natural convection of a high Prandtl number fluid in a cavity, Int. Commun. Heat Mass Transf. 27 (2000) 109–118. [20] M.M. Ganzarolli, L.F. Milanez, Natural convection in rectangular enclosures heated from below and symmetrically cooled from the sides, Int. J. Heat Mass Transfer 38 (1995) 1063–1073.
A. Koca et al. / International Communications in Heat and Mass Transfer 34 (2007) 511–519
519
[21] O. Aydin, Transient natural convection in rectangular enclosures heated from one side and cooled from above, Int. Commun. Heat Mass Transf. 26 (1999) 135–144. [22] Y.L. He, W.W. Yang, W.Q. Tao, Three-dimensional numerical study of natural convective heat transfer of liquid in a cubic enclosure, Numer. Heat Transf., A Appl. 47 (2005) 917–934. [23] A. Hadjadj, M. El Kyal, Effect of two sinusoidal protuberances on natural convection in a vertical concentric annulus, Numer. Heat Transf., A Appl. 36 (1999) 273–289. [24] H.Y. Pak, K.W. Park, Numerical analysis of natural convective and radiative heat transfer in an arbitrarily shaped enclosure, Numer. Heat Transf., A Appl. 34 (1998) 553–569. [25] H.F. Oztop, I. Dagtekin, Effect of Prandtl number on mixed convection with double sided lid driven cavity, CMES-04, 1st Capadoccia Mechanical Engineering Symposium, Turkey, 2004, pp. 358–363. [26] Y. Varol, H.F. Oztop, A. Koca, Natural convection in a shed roof with eave: comparison of summer and winter days boundary conditions, The 2nd International Green Energy Conference, Oshowa, Canada, 2006, pp. 1440–1450. [27] Y. Varol, A. Koca, H.F. Oztop, Application of central difference scheme to the solution of natural convection equations for irregular shaped enclosures, J. Appl. Sci. 7 (2007) 553–558.
The effects of Prandtl number on natural convection in triangular enclosures with localized heating from below☆ Ahmet Koca a , Hakan F. Oztop b , Yasin Varol c,⁎ a
Department of Mechanical Education, Firat University, TR-23119, Elazig, Turkey Department of Mechanical Engineering, Firat University, TR-23119, Elazig, Turkey Department of Mechanical Engineering, Vanderbilt University, Nashville, TN 37235, USA b
c
Available online 21 February 2007
Abstract The effect of Prandtl number on natural convection heat transfer and fluid flow in triangular enclosures with localized heating has been analyzed by solving governing equations of natural convection in streamfunction–vorticity form with finite-difference technique. Solution of linear algebraic equations was made by Successive Under Relaxation (SUR) method. Bottom wall of triangle is heated partially while inclined wall is maintained at a lower uniform temperature than heated wall while remaining walls are insulated. Computations were carried out for dimensionless heater locations (0.15 ≤ s ≤ 0.95), dimensionless heater length (0.1 ≤ w ≤ 0.9), Prandtl number (0.01 ≤ Pr ≤ 15) and Rayleigh number (103 ≤ Ra ≤ 106). Aspect ratio of triangle was chosen as unity. It is observed that both flow and temperature fields are affected with the changing of Prandtl number, location of heater and length of heater as well as Rayleigh number. © 2007 Elsevier Ltd. All rights reserved. Keywords: Prandtl number; Natural convection; Partial heating; Triangular enclosure
1. Introduction Analysis of natural convection in enclosures plays important role in many diverse applications including home heating, solar collectors, cryogenic storage, crystal growth, nuclear reactor design, furnace design, and electronical equipments. Cooling of electronic equipment must be considered when designing such systems. Naturally induced convection is an effective scheme as cooling in flush mounted or protruding electronical elements. These heaters are located in rectangular or square enclosures but they can work in corrugated enclosures such as PC monitors or TV. Natural convection in different geometrical enclosures has been extensively investigated, and reviews of the literature are found in Ostrach [1,2] and Catton [3]. Most of these studies related with analysis of natural convection in entirely heated or cooled rectangular or square cavity. Triangular enclosures can be used in the applications for roofs of the buildings or electronical heaters. Asan and Namli [4], Holtzman et al. [5], Akinsete and Coleman [6], Tzeng et al. [7], Salmun [8], and Ridouane and Campo [9] ☆
Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (Y. Varol).
0735-1933/$ - see front matter © 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2007.01.006
512
A. Koca et al. / International Communications in Heat and Mass Transfer 34 (2007) 511–519
Nomenclature AR g Gr H L n Nu Pr Ra s T u, v X, Y w
Aspect ratio, AR = H/L Gravitational acceleration Grashof number Maximum height of triangle Length of bottom wall Coordinate in normal direction Nusselt number Prandtl number Rayleigh number Dimensionless heater locations Temperature Velocities Non-dimensional coordinates Dimensionless heater length
Greek Letters υ Kinematic viscosity Ω Non-dimensional vorticity θ Non-dimensional temperature β Thermal expansion coefficient α Thermal diffusivity ψ Streamfunction ω Vorticity Subscript C Cold H Hot
have studied the natural convection in triangular shaped enclosures in detail. Recently, Varol et al. [10] investigated the flush mounted heater located on the vertical wall of right triangular enclosures. Applications of these boundary conditions can be found in the literature for square or rectangular enclosure (Churchill et al. [11], Farouk and Fusegi [12], Turkoglu and Yucel [13], Ben Nasr et al. [14], Aydin and Yang [15] and Prasad and Kulacki [16]).
Fig. 1. Physical model.
A. Koca et al. / International Communications in Heat and Mass Transfer 34 (2007) 511–519
513
In all of the studies given above, air was chosen as working fluid. However, effects of Prandtl number, which is the most important physical property of the fluids, are analyzed in some studies for rectangular enclosures. For instance, Yoo [17,18] investigated the bifurcation phenomena. The existence of dual solutions in natural convection in a horizontal annulus is numerically investigated for the fluids of 0.3 b Pr b 1. He found that bifurcation points are functions of the Prandtl number. In addition, Poujol et al. [19] analyzed the natural convection problem in a square cavity for high Prandtl number. Ganzarolli and Milanez [20] also investigated the natural convection in a differentially heated from vertical walls of rectangular square cavity by comparing air (Pr = 0.71) and water (Pr = 7) as working fluid. Same fluids are tested for square cavity in unsteady regime by Aydin [21]. Effects of Pr number also tested for different geometries in literature [22–25]. The investigation of the influence of Prandtl number on natural convection heat transfer and flow field is the main contribution for the present study. As observed in the literature review provided above, air was used as working fluid in the past researches for triangular geometries. However, effects of different Prandtl number have not been addressed. Thus, this study will give detailed information to researchers and designers about flush mounted heaters. 2. Physical model The schematical configuration of two-dimensional triangular enclosure with the important geometrical parameters is given in Fig. 1. It is a right triangle with bottom length L, and vertical wall height H. The bottom wall is partially heated with TH and inclined wall is cooled with TC. Vertical wall is insulated. The aspect ratio of the cavity is defined as AR = H/L = 1. The length of heater is depicted by w and its location center is s. 3. Governing equations and numerical method The present flow is considered steady, laminar, incompressible and two-dimensional. The viscous dissipation term in the energy equation is neglected. The variation of fluid properties with temperature has been neglected, with the only exception of the buoyancy term, for which the Boussinesq approximation has been adopted. Thus, natural convection is governed by the differential equations expressing the conservation of mass, momentum, and energy which are written in streamfunction–vorticity mode as follows. The governing equations of natural convection (Eqs. (1)–(3)) are written in streamfunction–vorticity form for laminar regime in two-dimensional form for steady, incompressible, and Newtonian fluid with Boussinesq approximation. It is assumed that radiation heat exchange is negligible according to other modes of heat transfer and the gravity acts in vertical direction. A2 W A2 W þ AX 2 AY 2 A2 X A2 X 1 AW AX AW AX Ah − þ ¼ −Ra AX 2 AY 2 Pr AY AX AX AY AX −X ¼
A2 h A2 h AW Ah AW Ah − þ ¼ AX 2 AY 2 AY AX AX AY
ð1Þ ð2Þ ð3Þ
The employed non-dimensional variables are given as x y wPr xðLÞ2 Pr T −TC ;X ¼ ;h ¼ X ¼ ;Y ¼ ;W ¼ L L υ υ TH −TC Aw Aw Av Au bgðTH −TC ÞL3 Pr υ ;v ¼ − ;x ¼ − u¼ ; Pr ¼ : ; Ra ¼ Ay Ax Ax Ay υ2 a
ð4Þ ð5Þ
Local and mean Nusselt numbers are calculated along the heater wall using Eq. (6) a and b, respectively. Z w Ah Nux ¼ − ; Num ¼ Nux dx ð6a:bÞ AY Y ¼0
514
A. Koca et al. / International Communications in Heat and Mass Transfer 34 (2007) 511–519
Boundary conditions for the considered model are depicted on the physical model (Fig. 1). In this model, u and v velocities are equal to zero for all solid boundaries. On the bottom wall; On the heater, T = TH, On insulated wall, AT An ¼ 0 On the vertical wall, AT ¼ 0 An On the inclined wall, T = TC Governing equations in streamline-vorticity form (Eqs. (1)–(3)) are solved through finite-difference method. Algebraic equations are obtained via Taylor series and they solved via Successive Under Relaxation (SUR) technique, iteratively. The central difference method is used for discretization procedure. The detailed solution technique is well described in the literature [26,27]. The convergence criterion, 10− 4 , is chosen for all depended variables and value of 0.1 is taken for under-relaxation parameter. Some grid tests were created between 35 × 35 and 237 × 237 to obtain optimum grid dimension. The test results showed that 61 × 61 grid dimension is enough for calculations. The computational results are compared with the literature for validation of the present computer code as shown in Fig. 2. Obtained results were compared with the results of Asan and Namli [4], Akinsete and Coleman [6] and Tzeng et al. [7]. 4. Results and discussion Numerical analysis of natural convection heat transfer and fluid flow was performed to obtain effects of Prandtl number in a triangular enclosure which is partially heated from the bottom wall. Temperature of inclined wall is lower than the temperature of the heater while remaining walls are adiabatic. Heat transfer and flow distributions were presented via streamlines and isotherms. The effects of governing parameters on heat transfer were presented by Nusselt numbers. The size and location of the heater, Rayleigh number and Prandtl number were taken as governing parameters on heat transfer and fluid flow. The ratio of height to bottom wall is taken as unity. Effects of Rayleigh number on flow and temperature distribution are presented in Fig. 3 for w = 0.5, s = 0.5, AR = 1.0 and Pr = 0.71. As it can be seen from the Fig. 3 (a) and (b), for Ra = 103 and 104 quasi-conduction heat transfer regime is formed. Heated flow moves from bottom and impinges to the inclined wall and two cells are formed in different directions. While the left cell turns in counterclockwise, the right one turns in clockwise. Intersections of these two cells becomes on the middle of the heater. Extremum absolute values of streamfunctions are increased with the increasing of Ra numbers. If Fig. 3 (a) and (b) are compared with each other, locations of minimum and maximum values of streamfunctions are almost same. When Ra = 105, strength of fluid is increased and a big main cell is formed in counterclockwise direction (Fig. 3 (c)). However, multiple cells are formed with the effect of convection regime at higher Ra numbers. The location of the cell in clockwise direction becomes almost same even at the highest Ra numbers and the third cell forms at the top corner of the triangle. The values of streamfunction and their locations are given on the Fig. 3. Plumelike distribution on isotherms starts for Ra N 105 due to strong convection. The similar distribution can be found in the literature for square cavities [15]. The physical model tested for different Prandtl numbers in the range of 0.01 ≤ Pr ≤ 15. Effects of Pr number on streamlines (on the left) and isotherms (on the right) are presented in Fig. 4 (a) to (c) for the parameters as s = 0.5, w = 0.5, AR = 1.0 and Ra = 105 for Pr = 0.1, 1, 15. It can be seen from the figure that the changing of Prandtl number makes important effect on flow field and temperature distribution due to strong convection. For Pr = 0.1, ψmin = −22.7 (X = 0.5, Y = 0.19) and ψmax = 21.10 (X = 0.23, Y = 0.46), Pr = 1, ψmin = − 14.30 (X = 0.67, Y = 0.15) and ψmax = 37.50 (X = 0.30, Y = 0.33). When Pr = 15, ψmin = −17.20 (X = 0.18, Y = 0.55) and ψmax = 24.6 (X = 0.43, Y = 0.21). Multiple cells are formed for all values of Pr numbers. When
Fig. 2. Comparisons of local Nusselt numbers with literature for Ra = 2772.
A. Koca et al. / International Communications in Heat and Mass Transfer 34 (2007) 511–519
515
Fig. 3. Streamlines (on the left) and isotherms (on the right) at w = 0.5, s = 0.5, AR = 1.0, Pr = 0.71, a) Ra = 103, b) Ra = 104, c) Ra = 105, d) Ra = 106.
Pr = 0.1, four cells are formed (Fig. 4 (a)). On the other hand, three cells are formed in Fig. 4 (b) and (c). Strong plumelike distribution occurs for small values of Pr numbers. However, thicker thermal boundary layer is obtained with high Pr number. When Pr = 15, the main cell elongates through the bottom wall of the enclosure. Fig. 5 (a) to (c) shows the variation of mean Nusselt number with Rayleigh number for different Prandtl number. We also tested the effects of the length and location of the heater on natural convection heat transfer. As seen from the figure, the values of heat transfer is almost constant up to Ra = 104. It increases with the increase of Ra number due to increasing of strength of the regime of convection heat transfer. The values of mean Nusselt number also increases with the increasing of Prandtl number. When these three figures are compared with each other (Fig. 5 (a) to (c)) the highest heat transfer is obtained for w = 0.8 and s = 0.5. For the smallest value of the heater length (w = 0.2), values of mean Nusselt numbers are very close to each other for lower Ra numbers and higher Pr numbers. For the same value of the heater length, heat transfer increases when the heater is moved toward the right
516
A. Koca et al. / International Communications in Heat and Mass Transfer 34 (2007) 511–519
Fig. 4. Streamlines (on the left) and isotherms (on the right) at w = 0.5, s = 0.5, AR = 1.0, Ra = 105, a) Pr = 0.01, b) Pr = 1, c) Pr = 15.
corner of the enclosure. Effects of Ra number on heat transfer is presented in Fig. 6 for different heater length at s = 0.5 and Pr = 0.71. It is observed that heat transfer is enhanced with the increasing of heater length and Rayleigh number, which is an expected result. The obtained mean Nusselt numbers are very close to each other for Ra = 103 and Ra = 104 due to conduction dominant regime. However, heat transfer increases with the increasing of Rayleigh number due to domination of convection mode of heat transfer.
5. Conclusions A numerical study has been performed to investigate the effects of Prandtl number on natural convection heat transfer and fluid flow in a partially heated triangular enclosure. The results are listed below: • Heat transfer increases with the increasing of heater length and Rayleigh number. Values of streamfunction (flow strength) also increase with the increasing of Rayleigh number.
A. Koca et al. / International Communications in Heat and Mass Transfer 34 (2007) 511–519
517
Fig. 5. Variation of mean Nusselt number with Rayleigh number for different Prandtl numbers, AR = 1.0, a) w = 0.2, s = 0.5, b) w = 0.8, s = 0.5, c) w = 0.2, s = 0.2.
• Heat transfer increases with the increasing of Prandtl number for all cases. Both flow and temperature fields are affected with the changing of Prandtl number. • Heat transfer increases when heater is located near the right corner of the triangular enclosure.
518
A. Koca et al. / International Communications in Heat and Mass Transfer 34 (2007) 511–519
Fig. 6. Variation of mean Nusselt number with length of the heater for different Rayleigh numbers, s = 0.5, AR = 1.0, Pr = 0.71.
The study can be extended for turbulent flow, different aspect ratios and an optimization study for a future work. References [1] S. Ostrach, Natural convection in enclosures, Advances in Heat Transfer, 8, Academic, New York, 1972, pp. 161–227. [2] S. Ostrach, Natural convection heat transfer in cavities and cells, Proceedings of the 7th Int. Heat Transfer Conference, vol. 1, 1982, pp. 365–379. [3] I. Catton, Natural convection in enclosures, Proceedings at the 6th Int. Heat Transfer Conference, vol. 6, 1978, pp. 13–31. [4] H. Asan, L. Namli, Laminar natural convection in a pitched roof of triangular cross-section: summer day boundary conditions, Energy and Buildings 33 (2000) 69–73. [5] G.A. Holtzman, R.W. Hill, K.S. Ball, Laminar natural convection in isosceles triangular enclosures heated from below and symmetrically cooled from above, J. Heat Transfer 122 (2000) 485–491. [6] V.A. Akinsete, T.A. Coleman, Heat transfer by steady laminar free convection in triangular enclosures, Int. J. Heat Mass Transfer 25 (1982) 991–998. [7] S.C. Tzeng, J.H. Liou, R.Y. Jou, Numerical simulation-aided parametric analysis of natural convection in a roof of triangular enclosures, Heat Transf. Eng. 26 (2005) 69–79. [8] H. Salmun, Convection patterns in a triangular domain, Int. J. Heat Mass Transfer 38 (1995) 351–362. [9] E.H. Ridouane, A. Campo, Relationship between thermal convection intensity and aspect ratio of two triangular cavities inscribed in horizontal rectangular cavities, Int. J. Numer. Methods Heat Fluid Flow 16 (2006) 338–355. [10] Y. Varol, A. Koca, H.F. Oztop, Natural convection in a triangle enclosure with flush mounted heater on the wall, Int. Commun. Heat Mass Transf. 33 (2006) 951–958. [11] S.W. Churchill, C.V.S. Patterson, H.H.S. Chu, The effect of heater size, location, aspect ratio, and boundary conditions on two dimensional, laminar, natural convection in rectangular enclosures, J. Heat Transfer 98 (1976) 1194–1201. [12] B. Farouk, T. Fusegi, Natural convection of a variable property gas in asymmetrically heated square cavities, J. Thermophysics and Heat Transfer 3 (1989) 85–87. [13] H. Turkoglu, N. Yucel, Effect of heater and cooler locations on natural convection in square cavities, Numerical Heat Transfer Part A 27 (1995) 351–358. [14] K. Ben Nasr, R. Chouikh, C. Kerkeni, A. Guizani, Numerical study of the natural convection in cavity heated from the lower corner and cooled from the ceiling, Applied Thermal Engineering 26 (2006) 772–775. [15] O. Aydin, W.J. Yang, Natural convection in enclosures with localized heating from below and symmetrical cooling from sides, Int. J. Numer. Meth. Heat Fluid Flow 10 (2000) 518–529. [16] V. Prasad, F.A. Kulacki, Convective heat transfer in a rectangular porous cavity-effect of aspect ratio on flow structure and heat transfer, J. Heat Transfer 106 (1984) 158–165. [17] J.S. Yoo, Prandtl number effect on transition of free-convective flows in a wide-gap horizontal annulus, Int. Commun. Heat Mass Transf. 26 (1999) 811–817. [18] J.S. Yoo, Prandtl number effect on bifurcation and dual solutions in natural convection in a horizontal annulus, Int. J. Heat Mass Transfer 42 (1999) 3279–3290. [19] F.T. Poujol, J. Rojas, E. Ramos, Natural convection of a high Prandtl number fluid in a cavity, Int. Commun. Heat Mass Transf. 27 (2000) 109–118. [20] M.M. Ganzarolli, L.F. Milanez, Natural convection in rectangular enclosures heated from below and symmetrically cooled from the sides, Int. J. Heat Mass Transfer 38 (1995) 1063–1073.
A. Koca et al. / International Communications in Heat and Mass Transfer 34 (2007) 511–519
519
[21] O. Aydin, Transient natural convection in rectangular enclosures heated from one side and cooled from above, Int. Commun. Heat Mass Transf. 26 (1999) 135–144. [22] Y.L. He, W.W. Yang, W.Q. Tao, Three-dimensional numerical study of natural convective heat transfer of liquid in a cubic enclosure, Numer. Heat Transf., A Appl. 47 (2005) 917–934. [23] A. Hadjadj, M. El Kyal, Effect of two sinusoidal protuberances on natural convection in a vertical concentric annulus, Numer. Heat Transf., A Appl. 36 (1999) 273–289. [24] H.Y. Pak, K.W. Park, Numerical analysis of natural convective and radiative heat transfer in an arbitrarily shaped enclosure, Numer. Heat Transf., A Appl. 34 (1998) 553–569. [25] H.F. Oztop, I. Dagtekin, Effect of Prandtl number on mixed convection with double sided lid driven cavity, CMES-04, 1st Capadoccia Mechanical Engineering Symposium, Turkey, 2004, pp. 358–363. [26] Y. Varol, H.F. Oztop, A. Koca, Natural convection in a shed roof with eave: comparison of summer and winter days boundary conditions, The 2nd International Green Energy Conference, Oshowa, Canada, 2006, pp. 1440–1450. [27] Y. Varol, A. Koca, H.F. Oztop, Application of central difference scheme to the solution of natural convection equations for irregular shaped enclosures, J. Appl. Sci. 7 (2007) 553–558.