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Natural convection in a triangle enclosure with flush mounted heater on the wall
International Communications in Heat and Mass Transfer 33 (2006) 951 – 958 www.elsevier.com/locate/ichmt
Natural convection in a triangle enclosure with flush mounted heater on the wall☆ Yasin Varol a,⁎, Ahmet Koca b , Hakan F. Oztop c a
Department of Mechanical Engineering, Vanderbilt University, Nashville, TN 37235, USA b Department of Mechanical Education, Firat University, TR-23119, Elazig, Turkey c Department of Mechanical Engineering, Firat University, TR-23119, Elazig, Turkey Available online 15 June 2006
Abstract Natural convection heat transfer has been analyzed numerically in a triangle enclosure with flush mounted heater on vertical wall. Finite difference method is used in solution of governing equations in streamfunction-vorticity form and linear algebraic equations were solved via Successive Under Relaxation (SUR). Governing parameters on heat transfer and flow fields are aspect ratio of triangle, location of heater, length of heater and Rayleigh number. It is observed that the most important parameter on heat transfer and flow field is the position of heater which can be a control parameter for the present system. © 2006 Elsevier Ltd. All rights reserved. Keywords: Natural convection; Triangular cavity; Particularly heated wall
1. Introduction It is well known that natural convection heat transfer is occurred due to buoyancy forces and temperature difference in enclosure and this phenomenon can be seen in cooling of electronical devices, building heating, and cooling, solar collectors etc. Most of the studies in related area are focused on differentially heated rectangular or square enclosure. However, the shape of enclosure can be in different configuration in most of the engineering area, including triangle, parallelogram or trapezoidal and a heater can be located on their wall or inside as protruding heater or flush mounted in electronical devices. There are some numerical or experimental studies on heater located enclosure in the literature. Chu et al. [1] investigated the effect of heater size, location, aspect ratio and boundary condition on two-dimensional, laminar, natural convection in rectangular channels both experimentally and numerically. They found that the maximum Nusselt number is obtained almost for all Rayleigh numbers when heater is located on the middle of the wall. In another study, Turkoglu and Yucel [2] made a numerical study using control volume approach for the effect of heater and cooler locations on natural convection in cavities. They indicated that for a given cooler position; mean Nusselt number increases as the heater is moved closer to the bottom horizontal wall. Chu and Hickox [3] investigated the thermal convection with viscosity variation in a cavity with localized heating both experimentally and numerically. Besides these studies sometimes natural convection can be seen in ☆
Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (Y. Varol).
0735-1933/$ - see front matter © 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2006.05.003
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Y. Varol et al. / International Communications in Heat and Mass Transfer 33 (2006) 951–958
Nomenclature AR g Gr H L n Nu Pr P3 P2 P1 Ra T u, v X, Y
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 Heater position (bottom) Heater position (middle) Heater position (top) Rayleigh number Temperature Velocities Non-dimensional coordinates
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
cavities with discrete wall heat sources as indicated in Deng et al. [4,5]. Aydin et al. [6] conducted a numerical study on buoyancy-driven laminar flow in an inclined square enclosure heated from one side and cooled from the adjacent side by using finite difference methods. In all of these studies, solution domain was chosen as square enclosure. Aydin and Yang [7] studied the natural convection in an enclosure partially heated from the bottom wall and symmetrically cooled from the side walls. They observed that symmetrical flow fields are obtained when heater located at the center of the bottom wall. Triangular shaped enclosures were investigated by some authors due to its shape is useful especially in the roof design or some of electronical devices. In the study of Asan and Namli [8], the laminar natural convection heat transfer in triangular shaped roofs with different inclination angle and Rayleigh number in winter day conditions is investigated numerically using the finite volume method. They indicated that both aspect ratio and Rayleigh number affect the temperature and flow field. They also found that heat transfer decreases with the increasing of aspect ratio. Akinsete and Coleman [9] illustrated the natural convection heat transfer in a triangular enclosure in steady-state regime. Moukalled and Acharya [10] solved the governing equations of natural convection heat transfer inside a trapezoidal shaped geometry with baffles for building roofs in the conditions of summerlike and winterlike. They observed that in winterlike conditions, convection starts to dominate at a Rayleigh number much lower than that in summerlike conditions. Recently, Tzeng et al. [11] proposed the Numerical Simulation Aided Parametric Analysis method to solve natural convection equations in streamline-vorticity form. Finally, they developed a correlation for parameters which is effective on flow and heat transfer. Rahman and Sharif [12] studied the effects of aspect ratio in detail for an enclosure. Liu and Thien [13] solved the optimum spacing problem for three heated chips mounted on a conductive substrate in a two-dimensional enclosure filled with air by an operator-
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953
splitting pseudo-time stepping finite element method, which automatically satisfies the continuity of the interfacial temperature and heat flux. Varol et al. [14], Wang [15], Kim et al. [16], Wintruff [17], and Ridouane [18] made studies on natural convection in corrugated enclosures using different numerical techniques. The main purpose of the present study is to numerically examine the effects of natural convection in a partially heated triangle shaped cavity from the vertical walls, while inclined wall has cold temperature and remains are maintained adiabatic. According to the provided literature above, streamlines, isotherm, local and average Nusselt numbers are shown to simulate temperature and flow field. 2. Definition of physical model Physical model is defined in Fig. 1 with dimensions and boundary conditions according to different positions of the heater. In these models, heater length can be changed and its dimension denoted by h. Length of bottom wall and height of vertical wall are shown by L and H, respectively. Inclined wall of triangle has constant cold temperature while heater has constant hot temperature and remains are adiabatic.
Fig. 1. Physical model for heater located enclosure a) Position P1, b) Position P2, c) Position P3.
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3. Governing equations and solution method 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 the 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. −X ¼
∂2 W ∂2 W þ ∂X 2 ∂Y 2
ð1Þ
∂2 X ∂2 X 1 ∂W ∂X ∂W ∂X ∂h − −Ra þ ¼ ∂X 2 ∂Y 2 Pr ∂Y ∂X ∂X ∂Y ∂X
ð2Þ
∂2 h ∂2 h ∂W ∂h ∂W ∂h − þ ¼ ∂X 2 ∂Y 2 ∂Y ∂X ∂X ∂Y
ð3Þ
The employed non-dimensional variables are given as x X ¼ ; L u¼
∂w ; ∂y
y Y ¼ ; L v¼−
W¼
∂w ; ∂x
wPr ; υ
x¼
X¼
xðLÞ2 Pr ; υ
∂v ∂u − ; ∂x ∂y
Ra ¼
h¼
T −Tcold Thot −Tcold
bgðThot −Tcold ÞL3 Pr ; υ2
ð4Þ Pr ¼
υ a
ð5Þ
Local Nusselt number is calculated along the vertical wall as Nuy ¼ −
∂T ∂x
ð6Þ
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 vertical wall; On the heater, T = TH, on unheated wall, On the bottom wall,
∂T ¼0 ∂n
∂T ¼0 ∂n
On the inclined wall, T = TC Governing equations in streamline-vorticity form (Eqs. (1)–(3)) are solved using finite difference method. Algebraic equations are obtained via Taylor series and they solved via Successive Under Relaxation (SUR) technique, iteratively [19,20]. The central difference method is used for discretization procedure. The detailed solution technique is well described in the literature [19]. 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 are made 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. Obtained results were compared with the results of Asan and Namli [8], Akinsete and Coleman [9] and Tzeng et al. [11]. The more detailed information can be found in earlier studies [19,20]. 4. Results and discussion Laminar natural convection heat transfer and fluid flow are analyzed in a triangular enclosure for different parameters, including the aspect ratio of triangle, length and position of heater and Rayleigh number. Position of heater was tested for three cases: top (position P1), middle (position P2) and bottom (position P3) of vertical wall as shown in Fig. 1. Prandtl number was chosen as 0.71 which corresponds to air. Fig. 2 shows the streamlines (on the left) and isotherms (on the right) to obtain thermal and flow field for different Rayleigh numbers of 104, 105 and 106. In this case, heater was located in position P2 with length of h = H / 3 and AR = 1. It can be seen from the
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figure, single cell is obtained at values of all Rayleigh numbers in a clockwise circulation. The cell goes through the top corner of triangle with the increasing Rayleigh number. The flow moves through the top corner of the triangle and impinges to the inclined wall and it is stagnant at the corners of the triangle. Temperature is distributed with this motion of flow and temperature boundary layer starts to develop from hotter to cooler. It shows a semi-circle shaped distribution at the smallest value of Rayleigh number due to small velocity. As Rayleigh number increased, plume-like temperature distribution was obtained as indicated in Asan and Namli [8]. Here, isotherms represent the lines with equal intervals between unity (heater) and zero (inclined wall). At the highest Rayleigh number, natural convection is more effective than that of conduction. Similarly, the individual influence of Rayleigh number is shown in Fig. 3 for the same location and heater length with Fig. 2 but different aspect ratios. In this case, Figs. 2 and 3 are comparable to see the effects of aspect ratio on flow and temperature fields. When aspect ratio becomes smaller, which means smaller volume of enclosure, the flow at the right corner of the triangle becomes stagnant due to long distance (Fig. 3). Meanwhile, temperature of the fluid becomes the same at that part of the enclosure. Both flow fields and temperature distribution are almost the same for smaller Rayleigh number due to domination of conduction effects. Circle shaped main circulation cell is obtained in clockwise (Fig. 3a and b). However, with the increasing of Rayleigh number, natural convection becomes dominant, and flow velocity increases. So, an ellipse shaped cell is obtained and plume-like temperature distribution is observed. Even at this value of Rayleigh number, flow temperature is equal to the inclined wall temperature at the corner of the enclosure (Fig. 3c). Variations of local Nusselt number for heated wall for three different positions and length of heater are presented in Fig. 4 at Ra = 103 (on the left) and Ra = 106 (on the right), which are the smallest and highest Rayleigh number in this study, respectively. In both cases the aspect ratio is chosen as AR = 1. The position of heater is an important parameter on natural convection as indicated by Liu and Thien [13]. Thus, in this study, the position of the heater was tested as a parameter and results are shown in Fig. 4. In this situation, due to small Ra number, conduction heat transfer is superior to convection heat transfer. When the heater is located on the bottom of vertical wall (position P3), the Local Nusselt Number (LNN) becomes constant up to the middle of heater, and after that it increases. In this case the value of LNN is small due to the long distance between hot and cold wall. Similar results are observed in the study of Chu et al. [1]. If the heater is located on the middle of vertical wall (position P2), U-shaped distribution was obtained due
Fig. 2. Streamlines (on the left) and isotherms (on the right) for different Rayleigh numbers at h = H / 3, at AR = 1, at the position of P2, a) Ra = 104, b) Ra = 105, c) Ra = 106.
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Fig. 3. Streamlines (on the left) and isotherms (on the right) for different Rayleigh numbers at h = H / 3, at AR = 0.3, at the position of P2, a) Ra = 104, b) Ra = 105, c) Ra = 106.
to higher temperature difference on the starting and end points of the heater. Similar variations of this trend can be seen in the literature [7]. An interesting result was obtained when heater located at the upper position (position P1). In this position, there is an intersection point between heater and cold inclined wall. Thus, maximum Nu number is obtained due to maximum temperature
Fig. 4. Variation of local Nusselt number along vertical wall at AR = 1, at Ra = 103 (on the left), at Ra = 106 (on the right) a) h = H / 3, b) h = H / 6, c) h = H / 10.
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difference. Both trend and values changed for the value Ra = 106 as shown in Fig. 4 (on the right). In this case, as indicated earlier, natural convection becomes dominant to conduction due to higher Ra number. Boundary layer is distorted with the increasing of flow velocity and heat transfer is increased. Thus, higher LNN values are obtained as expected. When heater located at the bottom or middle of vertical wall, LNN decreased almost linearly. However, a U-shaped trend was observed when heater is located at the top of vertical wall. Their minimum values are increased with the decreasing of heater length. Effects of heater length on mean Nu number as h = H / 6 (on the left) and H / 3 (on the right) are presented in Fig. 5 for different Ra number and position of heater at the value of aspect ratios of 0.3, 0.6 and 1. Fig. 5a shows the variation of mean Nu number values for AR = 0.3. When heater located at the top of the enclosure (position P1), the value of mean Nu numbers become constant at the value of h = H / 6 (on the left) for all Ra numbers due to the small distance between hot and cold wall. In this case, conduction heat transfer is more effective than that of convection. However, for h = H / 3 and position P1 (on the right), mean Nu numbers are constant up to Ra = 105 and after that it was increased. It is observed that for the same parameters as Ra number and heater length, heat transfer was decreased with the increasing of aspect ratio since the volume of enclosure is increased (Fig. 5b and c). It is clear that, conduction effect becomes dominant for lower Ra number for all cases. When left column (h = H / 6) and right column (h = H / 3) are compared, it is seen that when heater length increased the heat transfer increases due to higher heat transfer surfaces as indicated in the literature [7]. When heater is located at the position P3, the smallest heat transfer is obtained due to long distance between hot and cold walls. But the trend is similar for the position P2 and P3.
Fig. 5. Variation of mean Nusselt number for different Rayleigh number for h = H / 6 (on the left) and h = H / 3 (on the right), a) AR = 0.3, b) AR = 0.6, c) AR = 1.
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5. Conclusions According to the obtained results from the present numerical study, important conclusions can be drawn as follows: • Flow and temperature field are affected by the shape of enclosure and Rayleigh numbers play an important role on them. When heater is located at the bottom of the vertical wall, Nusselt number becomes constant due to conduction domination heat transfer regime. • Both position and location of heater affect the flow circulation and heat transfer. Thus, position of heater can be a control parameter for heat transfer. • When convection effects become dominant, heat transfer was increased at the value of Ra > 104. Even at the highest Rayleigh number value, single circulation was obtained inside the enclosure. • There may be an optimum solution for all parameters but it is out of the scope in the present study. References [1] H.H.S. Chu, S.W. Churchill, C.V.S. Patterson, The effect of heater size, location, aspect ratio, and boundary conditions on two-dimensional, laminar, natural convection in rectangular channels, J. Heat Transfer 98 (1976) 1194–1201. [2] H. Turkoglu, N. Yucel, Effect of heater and cooler locations on natural convection in square cavities, Num. Heat Trans. Part A 27 (1995) 351–358. [3] T.Y. Chu, C.E. Hickox, Thermal convection with large viscosity variation in an enclosure with localized heating, J. Heat Transfer 112 (1990) 388–395. [4] Q.H. Deng, G.F. Tang, A combined temperature scale for analyzing natural convection in rectangular enclosures with discrete wall heat sources, Int. J. Heat Mass Transfer 45 (2002) 3437–3446. [5] Q.H. Deng, G.F. Tang, Y. Li, M.Y. Ha, Interaction between discrete heat sources in horizontal natural convection enclosures, Int. J. Heat Mass Transfer 45 (2002) 5117–5132. [6] O. Aydin, A. Unal, T. Ayhan, A numerical study on buoyancy-driven flow in an inclined enclosure heated and cooled on adjacent walls, Num. Heat Trans. Part A 36 (1999) 585–589. [7] O. Aydin, W.J. Yang, Natural convection in enclosures with localized heating from below and symmetrical cooling from sides, Int. J. Num. Methods Heat Fluid Flow 10 (2000) 518–529. [8] H. Asan, L. Namli, Numerical simulation of buoyant flow in a roof of triangular cross section under winter day boundary conditions, Energy Build. 33 (2001) 753–757. [9] 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. [10] F. Moukalled, S. Acharya, Natural convection in trapezoidal enclosure with offset baffles, J. Thermophys. Heat Transf. 15 (2001) 212–218. [11] 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. [12] M. Rahman, M.A.R. Sharif, Numerical study of laminar natural convection in inclined rectangular enclosures of various aspect ratios, Num. Heat Trans. Part A 44 (2003) 355–373. [13] Y. Liu, N.P. Thien, An optimum spacing problem for three chips mounted on a vertical substrate in an enclosure, Num. Heat Trans. Part A 37 (2000) 613–630. [14] Y. Varol, A. Koca, H.F. Oztop, Investigation of natural convection heat transfer in corrugated enclosures, J. Thermodyn. 164 (2006) 104–118 (In Turkish). [15] G. Wang, An efficient equal-order finite-element method for natural convection in complex enclosures, Num. Heat Trans. Part B 42 (2002) 307–324. [16] M.Y. Kim, S.W. Baek, J.H. Park, Unstructured finite-volume method for radiative heat transfer in a complex two-dimensional geometry with obstacles, Num. Heat Trans. Part B 39 (2001) 617–635. [17] I. Wintruff, C. Gunther, A.G. Class, Interface-tracking control—volume finite-element method for melting and solidification problems—Part I: formulation, Num. Heat Trans. Part B 39 (2001) 101–125. [18] E.H. Ridouane, A. Campo, M. Hasnaoui, Benefits derivable from connecting the bottom and top walls of attic enclosures with insulated vertical side walls, Num. Heat Trans. Part A 49 (2006) 175–193. [19] A. Koca, Numerical investigation of heat transfer with laminar natural convection in different roof types, PhD Thesis, Firat University, Elazig, 2005. [20] Y. Varol, A. Koca, H.F. Oztop, Natural convection heat transfer in Gambrel Roofs, Building Environment, in press, doi:10.1016/ j.buildenv.2005.11.013.
Natural convection in a triangle enclosure with flush mounted heater on the wall☆ Yasin Varol a,⁎, Ahmet Koca b , Hakan F. Oztop c a
Department of Mechanical Engineering, Vanderbilt University, Nashville, TN 37235, USA b Department of Mechanical Education, Firat University, TR-23119, Elazig, Turkey c Department of Mechanical Engineering, Firat University, TR-23119, Elazig, Turkey Available online 15 June 2006
Abstract Natural convection heat transfer has been analyzed numerically in a triangle enclosure with flush mounted heater on vertical wall. Finite difference method is used in solution of governing equations in streamfunction-vorticity form and linear algebraic equations were solved via Successive Under Relaxation (SUR). Governing parameters on heat transfer and flow fields are aspect ratio of triangle, location of heater, length of heater and Rayleigh number. It is observed that the most important parameter on heat transfer and flow field is the position of heater which can be a control parameter for the present system. © 2006 Elsevier Ltd. All rights reserved. Keywords: Natural convection; Triangular cavity; Particularly heated wall
1. Introduction It is well known that natural convection heat transfer is occurred due to buoyancy forces and temperature difference in enclosure and this phenomenon can be seen in cooling of electronical devices, building heating, and cooling, solar collectors etc. Most of the studies in related area are focused on differentially heated rectangular or square enclosure. However, the shape of enclosure can be in different configuration in most of the engineering area, including triangle, parallelogram or trapezoidal and a heater can be located on their wall or inside as protruding heater or flush mounted in electronical devices. There are some numerical or experimental studies on heater located enclosure in the literature. Chu et al. [1] investigated the effect of heater size, location, aspect ratio and boundary condition on two-dimensional, laminar, natural convection in rectangular channels both experimentally and numerically. They found that the maximum Nusselt number is obtained almost for all Rayleigh numbers when heater is located on the middle of the wall. In another study, Turkoglu and Yucel [2] made a numerical study using control volume approach for the effect of heater and cooler locations on natural convection in cavities. They indicated that for a given cooler position; mean Nusselt number increases as the heater is moved closer to the bottom horizontal wall. Chu and Hickox [3] investigated the thermal convection with viscosity variation in a cavity with localized heating both experimentally and numerically. Besides these studies sometimes natural convection can be seen in ☆
Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (Y. Varol).
0735-1933/$ - see front matter © 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2006.05.003
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Y. Varol et al. / International Communications in Heat and Mass Transfer 33 (2006) 951–958
Nomenclature AR g Gr H L n Nu Pr P3 P2 P1 Ra T u, v X, Y
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 Heater position (bottom) Heater position (middle) Heater position (top) Rayleigh number Temperature Velocities Non-dimensional coordinates
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
cavities with discrete wall heat sources as indicated in Deng et al. [4,5]. Aydin et al. [6] conducted a numerical study on buoyancy-driven laminar flow in an inclined square enclosure heated from one side and cooled from the adjacent side by using finite difference methods. In all of these studies, solution domain was chosen as square enclosure. Aydin and Yang [7] studied the natural convection in an enclosure partially heated from the bottom wall and symmetrically cooled from the side walls. They observed that symmetrical flow fields are obtained when heater located at the center of the bottom wall. Triangular shaped enclosures were investigated by some authors due to its shape is useful especially in the roof design or some of electronical devices. In the study of Asan and Namli [8], the laminar natural convection heat transfer in triangular shaped roofs with different inclination angle and Rayleigh number in winter day conditions is investigated numerically using the finite volume method. They indicated that both aspect ratio and Rayleigh number affect the temperature and flow field. They also found that heat transfer decreases with the increasing of aspect ratio. Akinsete and Coleman [9] illustrated the natural convection heat transfer in a triangular enclosure in steady-state regime. Moukalled and Acharya [10] solved the governing equations of natural convection heat transfer inside a trapezoidal shaped geometry with baffles for building roofs in the conditions of summerlike and winterlike. They observed that in winterlike conditions, convection starts to dominate at a Rayleigh number much lower than that in summerlike conditions. Recently, Tzeng et al. [11] proposed the Numerical Simulation Aided Parametric Analysis method to solve natural convection equations in streamline-vorticity form. Finally, they developed a correlation for parameters which is effective on flow and heat transfer. Rahman and Sharif [12] studied the effects of aspect ratio in detail for an enclosure. Liu and Thien [13] solved the optimum spacing problem for three heated chips mounted on a conductive substrate in a two-dimensional enclosure filled with air by an operator-
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953
splitting pseudo-time stepping finite element method, which automatically satisfies the continuity of the interfacial temperature and heat flux. Varol et al. [14], Wang [15], Kim et al. [16], Wintruff [17], and Ridouane [18] made studies on natural convection in corrugated enclosures using different numerical techniques. The main purpose of the present study is to numerically examine the effects of natural convection in a partially heated triangle shaped cavity from the vertical walls, while inclined wall has cold temperature and remains are maintained adiabatic. According to the provided literature above, streamlines, isotherm, local and average Nusselt numbers are shown to simulate temperature and flow field. 2. Definition of physical model Physical model is defined in Fig. 1 with dimensions and boundary conditions according to different positions of the heater. In these models, heater length can be changed and its dimension denoted by h. Length of bottom wall and height of vertical wall are shown by L and H, respectively. Inclined wall of triangle has constant cold temperature while heater has constant hot temperature and remains are adiabatic.
Fig. 1. Physical model for heater located enclosure a) Position P1, b) Position P2, c) Position P3.
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3. Governing equations and solution method 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 the 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. −X ¼
∂2 W ∂2 W þ ∂X 2 ∂Y 2
ð1Þ
∂2 X ∂2 X 1 ∂W ∂X ∂W ∂X ∂h − −Ra þ ¼ ∂X 2 ∂Y 2 Pr ∂Y ∂X ∂X ∂Y ∂X
ð2Þ
∂2 h ∂2 h ∂W ∂h ∂W ∂h − þ ¼ ∂X 2 ∂Y 2 ∂Y ∂X ∂X ∂Y
ð3Þ
The employed non-dimensional variables are given as x X ¼ ; L u¼
∂w ; ∂y
y Y ¼ ; L v¼−
W¼
∂w ; ∂x
wPr ; υ
x¼
X¼
xðLÞ2 Pr ; υ
∂v ∂u − ; ∂x ∂y
Ra ¼
h¼
T −Tcold Thot −Tcold
bgðThot −Tcold ÞL3 Pr ; υ2
ð4Þ Pr ¼
υ a
ð5Þ
Local Nusselt number is calculated along the vertical wall as Nuy ¼ −
∂T ∂x
ð6Þ
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 vertical wall; On the heater, T = TH, on unheated wall, On the bottom wall,
∂T ¼0 ∂n
∂T ¼0 ∂n
On the inclined wall, T = TC Governing equations in streamline-vorticity form (Eqs. (1)–(3)) are solved using finite difference method. Algebraic equations are obtained via Taylor series and they solved via Successive Under Relaxation (SUR) technique, iteratively [19,20]. The central difference method is used for discretization procedure. The detailed solution technique is well described in the literature [19]. 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 are made 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. Obtained results were compared with the results of Asan and Namli [8], Akinsete and Coleman [9] and Tzeng et al. [11]. The more detailed information can be found in earlier studies [19,20]. 4. Results and discussion Laminar natural convection heat transfer and fluid flow are analyzed in a triangular enclosure for different parameters, including the aspect ratio of triangle, length and position of heater and Rayleigh number. Position of heater was tested for three cases: top (position P1), middle (position P2) and bottom (position P3) of vertical wall as shown in Fig. 1. Prandtl number was chosen as 0.71 which corresponds to air. Fig. 2 shows the streamlines (on the left) and isotherms (on the right) to obtain thermal and flow field for different Rayleigh numbers of 104, 105 and 106. In this case, heater was located in position P2 with length of h = H / 3 and AR = 1. It can be seen from the
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figure, single cell is obtained at values of all Rayleigh numbers in a clockwise circulation. The cell goes through the top corner of triangle with the increasing Rayleigh number. The flow moves through the top corner of the triangle and impinges to the inclined wall and it is stagnant at the corners of the triangle. Temperature is distributed with this motion of flow and temperature boundary layer starts to develop from hotter to cooler. It shows a semi-circle shaped distribution at the smallest value of Rayleigh number due to small velocity. As Rayleigh number increased, plume-like temperature distribution was obtained as indicated in Asan and Namli [8]. Here, isotherms represent the lines with equal intervals between unity (heater) and zero (inclined wall). At the highest Rayleigh number, natural convection is more effective than that of conduction. Similarly, the individual influence of Rayleigh number is shown in Fig. 3 for the same location and heater length with Fig. 2 but different aspect ratios. In this case, Figs. 2 and 3 are comparable to see the effects of aspect ratio on flow and temperature fields. When aspect ratio becomes smaller, which means smaller volume of enclosure, the flow at the right corner of the triangle becomes stagnant due to long distance (Fig. 3). Meanwhile, temperature of the fluid becomes the same at that part of the enclosure. Both flow fields and temperature distribution are almost the same for smaller Rayleigh number due to domination of conduction effects. Circle shaped main circulation cell is obtained in clockwise (Fig. 3a and b). However, with the increasing of Rayleigh number, natural convection becomes dominant, and flow velocity increases. So, an ellipse shaped cell is obtained and plume-like temperature distribution is observed. Even at this value of Rayleigh number, flow temperature is equal to the inclined wall temperature at the corner of the enclosure (Fig. 3c). Variations of local Nusselt number for heated wall for three different positions and length of heater are presented in Fig. 4 at Ra = 103 (on the left) and Ra = 106 (on the right), which are the smallest and highest Rayleigh number in this study, respectively. In both cases the aspect ratio is chosen as AR = 1. The position of heater is an important parameter on natural convection as indicated by Liu and Thien [13]. Thus, in this study, the position of the heater was tested as a parameter and results are shown in Fig. 4. In this situation, due to small Ra number, conduction heat transfer is superior to convection heat transfer. When the heater is located on the bottom of vertical wall (position P3), the Local Nusselt Number (LNN) becomes constant up to the middle of heater, and after that it increases. In this case the value of LNN is small due to the long distance between hot and cold wall. Similar results are observed in the study of Chu et al. [1]. If the heater is located on the middle of vertical wall (position P2), U-shaped distribution was obtained due
Fig. 2. Streamlines (on the left) and isotherms (on the right) for different Rayleigh numbers at h = H / 3, at AR = 1, at the position of P2, a) Ra = 104, b) Ra = 105, c) Ra = 106.
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Fig. 3. Streamlines (on the left) and isotherms (on the right) for different Rayleigh numbers at h = H / 3, at AR = 0.3, at the position of P2, a) Ra = 104, b) Ra = 105, c) Ra = 106.
to higher temperature difference on the starting and end points of the heater. Similar variations of this trend can be seen in the literature [7]. An interesting result was obtained when heater located at the upper position (position P1). In this position, there is an intersection point between heater and cold inclined wall. Thus, maximum Nu number is obtained due to maximum temperature
Fig. 4. Variation of local Nusselt number along vertical wall at AR = 1, at Ra = 103 (on the left), at Ra = 106 (on the right) a) h = H / 3, b) h = H / 6, c) h = H / 10.
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difference. Both trend and values changed for the value Ra = 106 as shown in Fig. 4 (on the right). In this case, as indicated earlier, natural convection becomes dominant to conduction due to higher Ra number. Boundary layer is distorted with the increasing of flow velocity and heat transfer is increased. Thus, higher LNN values are obtained as expected. When heater located at the bottom or middle of vertical wall, LNN decreased almost linearly. However, a U-shaped trend was observed when heater is located at the top of vertical wall. Their minimum values are increased with the decreasing of heater length. Effects of heater length on mean Nu number as h = H / 6 (on the left) and H / 3 (on the right) are presented in Fig. 5 for different Ra number and position of heater at the value of aspect ratios of 0.3, 0.6 and 1. Fig. 5a shows the variation of mean Nu number values for AR = 0.3. When heater located at the top of the enclosure (position P1), the value of mean Nu numbers become constant at the value of h = H / 6 (on the left) for all Ra numbers due to the small distance between hot and cold wall. In this case, conduction heat transfer is more effective than that of convection. However, for h = H / 3 and position P1 (on the right), mean Nu numbers are constant up to Ra = 105 and after that it was increased. It is observed that for the same parameters as Ra number and heater length, heat transfer was decreased with the increasing of aspect ratio since the volume of enclosure is increased (Fig. 5b and c). It is clear that, conduction effect becomes dominant for lower Ra number for all cases. When left column (h = H / 6) and right column (h = H / 3) are compared, it is seen that when heater length increased the heat transfer increases due to higher heat transfer surfaces as indicated in the literature [7]. When heater is located at the position P3, the smallest heat transfer is obtained due to long distance between hot and cold walls. But the trend is similar for the position P2 and P3.
Fig. 5. Variation of mean Nusselt number for different Rayleigh number for h = H / 6 (on the left) and h = H / 3 (on the right), a) AR = 0.3, b) AR = 0.6, c) AR = 1.
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5. Conclusions According to the obtained results from the present numerical study, important conclusions can be drawn as follows: • Flow and temperature field are affected by the shape of enclosure and Rayleigh numbers play an important role on them. When heater is located at the bottom of the vertical wall, Nusselt number becomes constant due to conduction domination heat transfer regime. • Both position and location of heater affect the flow circulation and heat transfer. Thus, position of heater can be a control parameter for heat transfer. • When convection effects become dominant, heat transfer was increased at the value of Ra > 104. Even at the highest Rayleigh number value, single circulation was obtained inside the enclosure. • There may be an optimum solution for all parameters but it is out of the scope in the present study. References [1] H.H.S. Chu, S.W. Churchill, C.V.S. Patterson, The effect of heater size, location, aspect ratio, and boundary conditions on two-dimensional, laminar, natural convection in rectangular channels, J. Heat Transfer 98 (1976) 1194–1201. [2] H. Turkoglu, N. Yucel, Effect of heater and cooler locations on natural convection in square cavities, Num. Heat Trans. Part A 27 (1995) 351–358. [3] T.Y. Chu, C.E. Hickox, Thermal convection with large viscosity variation in an enclosure with localized heating, J. Heat Transfer 112 (1990) 388–395. [4] Q.H. Deng, G.F. Tang, A combined temperature scale for analyzing natural convection in rectangular enclosures with discrete wall heat sources, Int. J. Heat Mass Transfer 45 (2002) 3437–3446. [5] Q.H. Deng, G.F. Tang, Y. Li, M.Y. Ha, Interaction between discrete heat sources in horizontal natural convection enclosures, Int. J. Heat Mass Transfer 45 (2002) 5117–5132. [6] O. Aydin, A. Unal, T. Ayhan, A numerical study on buoyancy-driven flow in an inclined enclosure heated and cooled on adjacent walls, Num. Heat Trans. Part A 36 (1999) 585–589. [7] O. Aydin, W.J. Yang, Natural convection in enclosures with localized heating from below and symmetrical cooling from sides, Int. J. Num. Methods Heat Fluid Flow 10 (2000) 518–529. [8] H. Asan, L. Namli, Numerical simulation of buoyant flow in a roof of triangular cross section under winter day boundary conditions, Energy Build. 33 (2001) 753–757. [9] 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. [10] F. Moukalled, S. Acharya, Natural convection in trapezoidal enclosure with offset baffles, J. Thermophys. Heat Transf. 15 (2001) 212–218. [11] 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. [12] M. Rahman, M.A.R. Sharif, Numerical study of laminar natural convection in inclined rectangular enclosures of various aspect ratios, Num. Heat Trans. Part A 44 (2003) 355–373. [13] Y. Liu, N.P. Thien, An optimum spacing problem for three chips mounted on a vertical substrate in an enclosure, Num. Heat Trans. Part A 37 (2000) 613–630. [14] Y. Varol, A. Koca, H.F. Oztop, Investigation of natural convection heat transfer in corrugated enclosures, J. Thermodyn. 164 (2006) 104–118 (In Turkish). [15] G. Wang, An efficient equal-order finite-element method for natural convection in complex enclosures, Num. Heat Trans. Part B 42 (2002) 307–324. [16] M.Y. Kim, S.W. Baek, J.H. Park, Unstructured finite-volume method for radiative heat transfer in a complex two-dimensional geometry with obstacles, Num. Heat Trans. Part B 39 (2001) 617–635. [17] I. Wintruff, C. Gunther, A.G. Class, Interface-tracking control—volume finite-element method for melting and solidification problems—Part I: formulation, Num. Heat Trans. Part B 39 (2001) 101–125. [18] E.H. Ridouane, A. Campo, M. Hasnaoui, Benefits derivable from connecting the bottom and top walls of attic enclosures with insulated vertical side walls, Num. Heat Trans. Part A 49 (2006) 175–193. [19] A. Koca, Numerical investigation of heat transfer with laminar natural convection in different roof types, PhD Thesis, Firat University, Elazig, 2005. [20] Y. Varol, A. Koca, H.F. Oztop, Natural convection heat transfer in Gambrel Roofs, Building Environment, in press, doi:10.1016/ j.buildenv.2005.11.013.