5-3-2011.pdf Al-Qadisiya Journal For Engineering Sciences Vol. 4 No. 3 Year 2011 A NUMERICAL STUDY OF NATURAL CONVECTION HEAT TRANSFER INSIDE A HORIZONTAL SQUARE ENCLOSURE WITH A CONCENTRIC HEATED ROD AND A BUNDLE OF TRIANGULAR HEATED CYLINDERS Rana L. Natoosh Mechanical Engineering Department - College of Engineering - Basra University ABSTRACT: In this paper, a numerical investigation of steady laminar natural convection in a square enclosure, contain a concentric heated rod and a bundle of four triangular heated cylinders was carried out. Two cases of enclosure inclination angle were studied: case (I) at =0o and case (II) at =45o. Air is filled the enclosure, and the inner heated cylinders are located at equal distance (E) from the enclosure center. A finite element software package (FLEXPDE) is used in the present study to solve the set of non-linear equations governing the process. Solutions are obtained for aspect ratio h/H=0.29, a values of distance E=0.27-0.44 and a range of Rayleigh number 103 105. The effect of Ra, E and were examined. Results are presented by streamlines, isotherms and Nusselt number and they indicates that the Nusselt number is significantly increases with increasing Ra, E and . A comparison of the streamlines, isotherms and mean Nusselt number was made with that obtained by other authors, which it reveals a good agreement. KEYWORDS: Natural convection; Square Enclosure; Triangular heated cylinders; Uniform Wall Temperature جامعة -- ت ة ".مسخنة ،=0o=45o . (E) . )(FLEXPDE. لنسبة h/H=0.29 ،E=0.27-0.44 103 105 .، Al-Qadisiya Journal For Engineering Sciences Vol. 4 No. 3 Year 2011 . E ، Ra مع" .مع NOMENCLATURE Greek symbols Thermal diffusivity (m2/s) Thermal expansion coefficient (1/K) ɛ The distance between the center of enclosure and the center of inner circular cylinder Dimensionless temperature Dynamic viscosity (N/m2.s) kinematics viscosity (m2/s) Fluid density (kg/m3) The angle of inclination of the enclosure Stream function Subscripts i inner L Local m mean o outer D Diameter of circler rod (m) E The distance between the center of enclosure and triangular base g Gravitational acceleration (m/s2) h Height of triangle (m) H Height of square enclosure (m) NuL Local Nusselt number Num Mean Nusselt number p,P Pressure, dimensional(N/m2) and dimensionless Pr Prandtl number Ra Rayliegh number rr Ratio of L/2ri S Dimensionless local coordinate T Temperature (K) u, U Dimensional and dimensionless horizontal velocity component v, V Dimensional and dimensionless velocity component X,Y Non-dimensional coordinates 1. INTRODUCTION The phenomenon of natural convection in enclosures has received considerable attention in recent years. This attention is mainly because this phenomenon often affects the thermal performance in many engineering and science applications such as boilers, nuclear reactor systems, energy storage and conservation, fire control and chemical, food and metallurgical industries. In engineering applications, the geometries that arise however are more complicated than a simple enclosure filled with a convective fluid. A large number of literature is available which deal with the study of natural convection in enclosures with either vertical (Khaled and Bhattacharyy, 2006), inclined (Venkateswara et al.,2006), (Yasin et al., 2008) or horizontal (Natarajan et al, 2008), (Tanmay et al., 2009) imposed temperature difference. (Tanmay et al., 2009) reported the numerical results of natural convection in trapezoidal enclosures for uniformly heated bottom wall, linearly heated vertical wall and of insulated top wall. Parametric studies for the wide range of Rayleigh numbers Ra (103-105) and Prandtl numbers Pr (0.7 -1000) with va have been performed. It was found that, higher heat transfer rates for =0 and the overall heat transfer rates at the bottom wall were larger for the linearly heated left wall and cooled right wall. (Patrick and Jane, 2005) using an iterative, semi-implicit finite-difference method to solve the governing unsteady, three- dimensional free convective flow equations in a rectangular enclosure. The enclosure considered had rectangular horizontal lower and upper surfaces and vertical side surfaces and there were two square isothermal heated sections on the lower surface, the rest of this surface being adiabatic. (Kim et al.2008) studied the examine how the position of the inner circular cylinder relative to the outer square cylinder affects the natural convection phenomena for different Rayleigh numbers ( Al-Qadisiya Journal For Engineering Sciences Vol. 4 No. 3 Year 2011 103 106 ) when a hot inner circular cylinder was located at different positions along the vertical centerline of the outer square cylinder. (Kumar and Dalal, 2006) considered the problem of natural convection around a square, horizontal, heated cylinder placed inside an enclosure in the range of 103 106. Effects of the enclosure geometry have been assessed using three different aspect ratios placing the square cylinder at different heights from the bottom. (Shu et al., 2001) numerically studied natural convection between an outer square enclosure and an inner circular cylinder according to the eccentricity and angular position of the inner circular cylinder at a Rayleigh number of 3*105. It was found that the global circulation, flow separation and the top space between the square outer enclosure and the circular inner cylinder have significant effects on the plume inclination. Laminar convection heat transfer from a horizontal triangular cylinder to its concentric cylindrical enclosure has been numerically studied by (Zi-Tao et al., 2010), to investigate the Prandtl number effect on flow and heat transfer characteristics. They found that the flow and heat transfer characteristics for a low Prandtl number fluid (Pr = 0.03) were unique and they were almost independent of Prandtl number when Pr 0.7. A numerical investigation of steady-state laminar natural convective heat transfer around a horizontal cylinder to its concentric triangular enclosure was studied by (Xu Xu et al., 2010). The enclosure was filled with air and both the inner and outer cylinders were maintained at uniform temperatures. At the highest Rayleigh number studied, the effects of different inclination angles of the enclosure and various cross-section geometries of the inner cylinder were investigated. The computed results indicated that at constant aspect ratio, both the inclination angle and cross-section geometry have insignificant effects on the overall heat transfer rates though the flow patterns are significantly modified. E. (Fuad, 2009) presented a numerical analysis of laminar natural convection in an enclosure of isosceles triangular cross-section has been performed for cold base and hot inclined walls. Base angles varying from 15 to 75 have been used for different Rayleigh numbers ranging from 103 to 105. It was observed that the enclosures with a low aspect ratio have higher heat transfer rates from the bottom surface of the triangular enclosure. (Tanmay Basak et al., 2008) used a finite element analysis with bi-quadratic elements to investigate the effects of uniform and non-uniform heating of inclined walls on natural convection flows within a isosceles triangular enclosure. The numerical solution of the problem was presented for various Rayleigh numbers (Ra), (103 Ra 106) and Prandtl numbers (Pr), (0.026 Pr 1000. A survey of literature reveals that multi-inner triangular cylinders that are inside air-filled square enclosure has not been investigated. This is the motivation of the present study where a bundle of four triangular heated cylinders with concentric heated rod inside a square enclosure is to be studied. The effect of Rayleigh number, inclination angle of the enclosure and the distance between the center of enclosure and triangular cylinders on the flow and thermal fields for constant aspect ratio are studied in detail 2. GOVERNING EQUATIONS A schematic of the two systems considered in the present study are shown in Fig.1. A case (I) (Fig.1a) consists a square enclosure ( =0o) while case (II) (Fig.1 b) consists a tilted square enclosure with =45o. For two cases a square enclosure contains the concentric rod and four triangular cylinders located at equal distances E from the center of enclosure. The temperature of inner cylinders is higher than that of the square walls and both are uniform. The origin of the Cartesian coordinates (x, y) is positioned at a square center. The aspect ratios h/H=0.29 and D/H=0.24, the range of Rayliegh number 103 105, the locations of the inner cylinder 0.27 0.44 and Pr=0.71 were considered in this study. The fluid is assumed to have constant physical properties but obeys the Bussinesq approximation according to which the compressibility effect every where is neglected except for Buoyancy force term. Viscous dissipation and heat generation are absent. Hence, the two-dimensional, incompressible, steady state dimensionless governing equations using conservation of mass, momentum and energy can be written as (Xu Xu et al., 2010): Al-Qadisiya Journal For Engineering Sciences Vol. 4 No. 3 Year 2011 (1)0 Y V X U (2)2 2 2 2 Y U X U X P Y V V X U U (3) Pr2 2 2 2 Ra Y V X V Y P Y V V X V U (4) 2 2 2 2 Pr 1 YXY V X U Non-dimensional parameters can be given as follows: 2 2 ,,,,, v Hgyp P TT TTvH V uH U H y Y H x X oi o 3HTTg Ra oi, and Rayleigh numberPr Prandtl number 3. NUMERICAL SOLUTION It is well known in the numerical solution field that the set of equations (Eqs. (1) to (3)) may be highly oscillatory or even sometimes undetermined because of inclusion of the pressure term in the momentum equations. In the present study, a finite element software package (Flexpde) (Backstrom, 2005) is relied on in solution of the nonlinear system of equations. In finite element method there is a derived approach with purpose of stabilizing pressure oscillations and allowing standard grids and elements. This approach enforces the continuity equation and the pressure to give the following, what called penalty approach (Langtangen, 2002) Y V X U P2 (5) Where λ is a parameter that should be chosen either from physical knowledge or by other means (Langtangen, 2002). A most convenient value for was attained in this study to be 1E8 /H2 . Hence, the continuity equation (1) is excluded from solution system and replaced by equation (5). The numerical solutions are obtained in terms of velocity components (U,V) and the stream components (U,V) as (Tanmay et al., 2007): X V Y U YX 2 2 2 2 (6) -clockwise circulation and the clock wise The heat transfer coefficient in terms of the local Nusselt number NuL along the inner cylinders surface and the outer wall is defined by (Tanmay et al., 2007): Al-Qadisiya Journal For Engineering Sciences Vol. 4 No. 3 Year 2011 (7) n NuL where n is the normal direction on a plane. At steady- state it is obvious that the mean Nusselt numbers along both the inner and outer walls are exactly identical. Hence, the average Nusselt number can be evaluated along the outer wall as: (8)oL Soo m dSuNS Nu 1 The appropriate boundary conditions are as follows - Isothermal surfaces i.e. =1 on the concentric rod and triangular cylinders walls and =0 on the square walls. -No-slip velocity boundary condition, U=V=0 on all solid walls. -pressure gradient normal to all surfaces is zero, 0 n P where n is a normal unit vector. 4. VALIDATION 4.1 Software Validation To check the validation of software, the grid dependency and the distribution values of ( + / ) over the domain Ra=103 and E=0.36 with accuracy 10-4 are presented in Fig.2, a and b. It is clear that the continuity equation is exactly validated, where it as mentioned in section 3, do not contribute in solving the governing equations. 4.2 Validation of numerical results For the purpose of code validation, the natural convection problem for a low temperature outer square enclosure and high temperature inner circular cylinder (Ra=3*105 , Pr=0.71, (rr = L/2ri) equal to 2.6 (where L: side length of the square outer cylinder, ri: radius of the inner cylinder) and the distance from the center of enclosure to the center of the inner cylinder ɛ =0, 0.25 and 0.5) was tested. The streamlines and isothermal patterns at ɛ =0.25 for present result are compared with the (She et al., 2001), see Fig.3. The results of the present work shows a good agreement with those of Ref. (She et al., 2001). The calculated mean Nusselt number Num along the outer wall between the present work and the work of Ref.(She et al., 2001) is compared for ɛ =0, 0.25 and 0.5 as shown in Table (1) . The Num are in good agreement with the values of (She et al., 2001). 5. RESULTS AND DISCUSSION In the present study the effect of the Rayleigh number and the locations of the inner triangular cylinders were systematically investigated for two different cases of the square enclosure. The basic features of flow and heat transfer are analyzed with the help of isotherm and the streamline patterns. Local and average Nusselt numbers on the isothermal surfaces are plotted to evaluate the local and overall heat transfer process. 5.1 Analysis of Flow and Thermal fields For all Rayleigh numbers at each case considered in this study, the flow and thermal fields are symmetric shape about the vertical centerline through the center of the concentric circular rod. In other words, the governing equations given in Eqs (1) to (4) and the boundary conditions are invariant to the following transformation: Symmetry about x =0 : { u,v,θ,x,y} { -u,v,θ,-x,y} (9) Al-Qadisiya Journal For Engineering Sciences Vol. 4 No. 3 Year 2011 ۲۲۰ 5.1.1 Flow and thermal fields when E=0.36 The results of flow fields and isotherms for air flow in a square with heated inner cylinders at E=0.36 are examined in this section. Fig.4 shows streamlines (on the left) and isotherms (on the right) for different values of Ra in case(I) (φ=0). At low Ra (Ra=103), as shown in the left side in Fig.4a, six circulation cells of streamlines are formed inside the enclosure. Two of them are the strongest and localized in the space between each two triangular cylinders near the vertical walls. The others are smaller and located around a concentric rod. This formula of circulations are appear due to the buoyant effects which are caused by the temperature difference between the inner and outer walls, and this lead to form the recirculation vortices which are clearly demonstrated by the closed streamlines. At Ra=3*104, as shown in the left side of Fig.4b, it can be seen that the upper and lower cells around concentric rod merge together and a new cell is formed near each corner of the enclosure. As Ra increases to 105(Fig.4c on left), the cells in the upper two corners begin to merge with the two strong cells while two small new cells appears near the upper part of concentric rod when the old two cells begin to move down . The effect of buoyancy force become stronger compared to viscous forces and the intensity of fluid motion has been increased as indicated by relatively larger magnitudes of stream functions. The thermal fields are presented in the form of isotherms in the right side of Fig.4. At Ra=103(Fig.4a), the heat transfer in the enclosure is mainly dominated by the conduction mode therefore, we can observe that four equivalent like plumes distribution start to appears from the space between each two inner triangular cylinders towards the center of enclosure. With increasing of Rayliegh number (Ra=3*104) as shown in Fig.4b, the upper plume vanish gradually while the lower plum start to growth toward the center of enclosure. As the Rayleigh number increases up to 105, the role of convection in heat transfer becomes more significant, consequently the upper plum disappear while the lower and both side plumes rise upward , giving rise to a stronger thermal gradient in the upper part of the enclosure. Fig.5 shows streamlines (on the left) and isotherms (on the right) for different values of Ra in case (II) (φ=45). At low Ra (Ra=103) ,as in case(I), there are many circulation cells formed inside the enclosure and the number of cells here is larger than in case(I) , this is because of the inclination of the enclosure that leads to increase of the secondary flow. Two small circulation cells localized in the space between each triangular cylinder side and enclosure corners, another two cells appears at each side of the concentric rod. The last four circulation cells are more in strength and localized in the space between each two inner triangular cylinders. With increasing of Rayleigh number (Ra=3*104and Ra=105),as shown in Fig.5b and c, the size of the small cells are growth and the two cells near the concentric rod start to move upward, also the two cells that are in the upper part of the enclosure slightly becomes smaller in size and weaker in strength compared with those in lower part of it. As in case(I), the strength of cells become more at high Rayleigh numbers compared with the lower values of it, due to increase the buoyancy effect and dominant of free convection heat transfer. The behavior of thermal patterns in this case is similar to that in case (I) especially at Ra=103, as shown the right side in Fig.5. At Ra=3*104 we can observe in Fig.5b that the size of each upper like plum distribution is vanish gradually with increasing of Rayleigh number. For high Rayleigh number (Ra=105 )as shown in Fig.5c(right), the lower plumes start to growth to upward and merge with the upper plumes due to reducing the density of fluid which lead to growth the thermal boundary layer. 5.1.2 Flow and temperature fields as a function of E:- The dependence of the flow and thermal fields on the location of inner heated triangular cylinders can be observed in the plots of the streamlines and isotherms for three different values of E (0.27, 0.36, 0.44) at Ra = 3*104 and φ=0 (case(I)) as shown in Fig.6. It can be seen from the left side of this figure that the number of circulation cells decreases with increasing the value of E while the size of each cells increases with increasing of E. This is because a greater space is then available Al-Qadisiya Journal For Engineering Sciences Vol. 4 No. 3 Year 2011 ۲۲۱ for the cells to be larger when E increases. It is also noticed that the strength of cells increases with increasing of E. The right side of Fig.6 shows that the upper like plum distribution is vanish at lower values of E (E=0.27) while the size of each plum increases with increasing of E due to the increasing the growth of the thermal boundary layer 5.2 Heat transfer rates: local and mean Nusselt numbers 5.2.1 Local Nusselt numbers Figs.7 and 8 display the effects of Ra on the local Nusselt number along the circumference of inner triangular cylinders and outer enclosure wall for E=0.36 in case (I) and case (II) respectively. Variation of the local Nusselt number along the upper left inner triangular cylinder at φ=0 for different Rayleigh numbers is plotted in Fig.7-A. As can be seen from this figure that, at Ra=103 the local Nusselt number (NuL) has a maximum value at S1=0, which represent the point on the upper left corner of the triangular, then the NuL decreases until S1=0.11, and it remain constant to the distance S1=0.22, then the NuL increases until S1=0.29, which represent the point on the lower corner of the triangular cylinder. When we move along the inclined wall of the triangular cylinder, the value of NuL decreases, and reaches a minimum at S1=0.5, then increases toward the upper right corner at S1=0.71. Then the turned of curve of NuL along the horizontal surface of triangle (0.71≤ S1≤1) is the same as that of vertical wall. The shape of the local Nusselt number distribution at Ra =3*104 is generally similar to that at Ra=103. At high Rayleigh number (Ra=105), the distribution is different along the inclined and horizontal walls of the triangular cylinder. The values of NuL increases with increasing of Ra along the portion 0.3≤S1≤0.62 and decreases otherwise. This is due to the like plum distribution of the isotherms through the space between the heated cylinders (which is shown in Fig.4) is greatly spread at high Ra due to the increase the strength of streamlines. The maximum values of NuL appear on the corners of the triangular cylinder, due to the growth of thermal boundary layer near the corners and that's lead to dominant the convection instead of conduction near to these regions. Fig7-B shows the distribution of local Nusselt numbers along the lower left inner triangular cylinder for different Rayleigh numbers. The curves have a similar U- shape at each wall of the triangle. The value of NuL has a relatively small variation with increasing of Ra along the vertical and inclined walls of the triangular cylinder while it increases with increasing of Ra along the horizontal wall. The higher values of NuL appears at the vertical and horizontal walls due to the dominant of convection near to these regions, while the values of NuL are lower at the inclined wall of triangular cylinder due to the dominant of conduction. The distribution of NuL along the concentric rod for the same parameters above is shown in Fig.7-C. At Ra = 103, the distribution of NuL show almost a symmetric shape with respect to the horizontal centerline at y = 0 due to the symmetric distribution of isotherms. The variation of NuL is very small and increases with increasing Ra. With increasing of Rayleigh number (Ra=3*104 and 105) the distribution of NuL is not symmetric any more with respect to horizontal due to the effect of convection. Also we can observe that the maximum value of NuL appears at Si=0.5 while the minimum values at the upper point of cylinder (Si=0 or Si=1) .The maximum value of local Nusselt number increases with increasing Ra because the like plumes distribution growth upward in the presence of convection, meaning that the NuL at the lower surface of the concentric rod is larger than that at the upper surface of it. Variations of the local Nusselt number along the outer enclosure wall are plotted in Fig.7-D for the same above parameters. The curves have a similar sinusoidal distribution with M-shape at the distance which represents the base and each vertical side of the square enclosure. The distribution of NuL at the upper surface of enclosure changes with increasing of Ra. At Ra=103 the NuL on the upper wall have the same behavior while when Ra increases to 105, the behavior of NuL is completely different having sinusoidal distribution with very high values of NuL. This different due to the growth the plumes to upward as shown in Fig.4(on left), and as a result the values of local Nusselt number along the upper part of enclosure wall (0 ≤ So≤ 0.125 and 0.625 ≤ So≤ 1) is larger than that at the lower part of it (So=0.125-0.625). Al-Qadisiya Journal For Engineering Sciences Vol. 4 No. 3 Year 2011 ۲۲۲ Fig. 8-A shows the distribution of local Nusselt numbers along the hot surface of the upper triangular cylinder for different Ra and φ=45o. The distribution of NuL shows symmetrical shape with respect to the vertical centerline due to the symmetric shape distribution of flow and thermal fields. The distribution of NuL in this case is similar to that along the upper left triangular cylinder in case (I) , especially at lower values of Ra. Also it can be seen that the values of NuL decreases with increasing of Ra except along the portion that represents the bottom wall of the triangular cylinder (0.29≤ S1 ≤0.71). The maximum values of local Nusselt numbers appears at the upper corner of the triangle (S1=0 or S1=1) while the minimum values appears at S1=0.5 due to the growth of the thermal boundary layer and dominant convection heat transfer near the upper corner while the conduction heat transfer is the dominant near the bottom wall of the triangular cylinder (as showed in Fig.5). The variation of local Nusselt numbers along the hot surface of the left triangular cylinder for different Ra are shown in Fig.8-B. It can be seen that the NuL is increases with increasing Ra along the lower inclined and the vertical walls of the triangular cylinder (0.29≤ S2 ≤0.96), while vise versa for the upper inclined wall. The lower values of NuL appears at the vertical wall due to dominant the conduction near the center of enclosure, while the values of NuL are higher at the lower inclined wall of triangular cylinder due to dominant of the convection. The distribution of local Nusselt numbers along the hot surface of the lower triangular cylinder for different Ra are shown in Fig.8-C. The distribution of NuL is similar to that along the lower left cylinder in case(I). The value of NuL increases with increasing of Ra along the left and right inclined walls of the triangular cylinder due to increasing the effect of convection, while it has a relatively small variation with increasing of Ra along the horizontal wall. The maximum value of NuL is located on the corner of triangular cylinder that is near the enclosure corner at Ra=103 and Ra=3*104. At Ra=105, the maximum values of NuL are located on each upper corner of triangular cylinder, because the isotherms are symmetric shape with respect to the vertical centerline at x = 0 as shown in Fig.5(left). Variation of the local Nusselt number along the concentric rod for the same above parameters is plotted in Fig.8-D. At Ra=103,as in case(I), the distribution of NuL show almost a symmetric shape with respect to the horizontal centerline at y=0 because the dominant effect is conduction. The other curves show two peaks with equal altitude at the left and right sides of the lower part of cylinder. The local Nusselt number increases with increasing Ra except along the upper part of the cylinder because, as previous mentioned, the lower plumes start to growth to upward in the presence of convection as shown in the left side of Fig.5 .The maximum values of NuL, for Ra=3*104 and Ra=105, appears at Si=0.37 and Si=0.63 while the minimum values of Rayleigh number 103≤Ra≤105, appears at the upper point of concentric rod (Si=0 or Si=1). Fig.8-E shows the distribution of local Nusselt number along the outer enclosure wall for the same above parameters. For Ra=103 and 3*104, the distribution exhibits the same form to that along the outer enclosure wall in case (I). At high Rayleigh number (Ra=105), the distribution of NuL along each the lower inclined walls of the square enclosure have the same behavior, while the behavior of NuL along each the upper inclined walls have sinusoidal distribution with very high values of NuL. This can be demonstrated by the steep temperature gradient along the upper inclined walls of enclosure as in Fig.5c right. 5.2.2 Mean Nusselt numbers The mean Nusselt number is utilized to represent the overall heat transfer rate within the domain interest. Fig.9a and b shows the mean Nusselt number of the enclosure as a function of the distance (E) with different Rayleigh numbers for case (I) and case (II) respectively. It can be seen that the mean Nusslet number ,for each case, depends on the variation of the distance (E) and increases with increasing E, due to increasing the thermal gradient with decreasing the space between the inner cylinders and the wall of the outer enclosure as shown in Fig.6 (right). Also we can observe from this figure that for each value of E the value of mean Nusselt numbers is increased with Ra due to dominant the convection heat transfer, and this value of increasing is greater in case(II) than in case(I) for Ra≥104 as illustrated more in Fig.10. The change of mean Nusselt numbers, at the outer enclosure, versus Ra with different (E) for each case is presents in Fig.10. It can be seen that, Al-Qadisiya Journal For Engineering Sciences Vol. 4 No. 3 Year 2011 ۲۲۳ for E=0.27 and E=0.36 the Num is a strongly effected by high various of Rayliegh number especially in case(II) due to the growth of thermal boundary layer and enhancing the heat transfer coefficient. Also for E=0.44 we can observe that a small variation of Num with increasing of Rayleigh number. 6. CONCLUSIONS In the present paper, a numerical investigation of steady laminar natural convection in a square enclosure contain a concentric heated rod and a bundle of 4-trainguler heated cylinders was studied in two cases depending on various inclined angle of the enclosure (φ=0o and φ=45o). A detailed analysis for the distribution of streamlines, isotherms and Nusselt number was carried out to investigate the effect of the locations of the heated inner cylinders on the fluid flow and heat transfer in the square enclosure for different Rayleigh numbers in the range of 103 ≤ Ra ≤105. The conclusions can be drawn as follows: 1. Flow fields and isotherms are strongly affected by changing Rayleigh number, the location of inner cylinders (E), and the angle of inclination of the enclosure (φ). 2. The values of local Nusselt number along the upper part of outer enclosure wall are larger than that at the lower part of it. 3.The mean Nusselt number and the intensity of stream are an increasing function to Rayliegh number. 4. Location of the inner heated cylinders play an important role to enhance the over all heat transfer significantly. 5. The values of Num in inclined square enclosure at φ=45o are larger than in enclosure at φ=0o. 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Al-Qadisiya Journal For Engineering Sciences Vol. 4 No. 3 Year 2011 ۲۲٥ Fig.1 schematic diagrams of the physical domain a) case(I) b) case(II) -b--a- H SO S2 h S1 Si E SO S1 S2 S3 Si φ=45o E Fig.2 (a) grid distribution over the domain ( b) validation of continuity equation -a- -b- Al-Qadisiya Journal For Engineering Sciences Vol. 4 No. 3 Year 2011 ۲۲٦ Table (1) Comparison of Num between the present work and the work of (Shu et al., 2001) for Ra=3*105, pr=0.71 and rr=2.6 0.5 6.95 6.98 0.43 0.25 6.75 0.9036.689 Num (present) Num (Shu et al., 2001) Error %ɛ 6.526.6 1.20 Fig.3 comparison of streamlines and isotherms with (Shu et al., 2001) for Ra=3*105, Pr=.71, and rr=2.6 Shu et al.(2001) Streamlines isotherms Present work isothermsstreamlines Al-Qadisiya Journal For Engineering Sciences Vol. 4 No. 3 Year 2011 ۲۲۷ Fig.4 Streamlines (left) isotherms (right) for case(I) a)Ra=103 b)Ra=3*104 c) Ra=105 a b c Al-Qadisiya Journal For Engineering Sciences Vol. 4 No. 3 Year 2011 ۲۲۸ Fig.5 Streamlines (left) isotherms(right) for case(II) a)Ra=103 b)Ra=3*104 c) Ra=105 a b c Al-Qadisiya Journal For Engineering Sciences Vol. 4 No. 3 Year 2011 ۲۲۹ a Fig.6 Streamlines(left) and isotherms (right) for case(I) and Ra=3*104 a)E=0.27 b)E=0.36 c)E=0.44 b c Al-Qadisiya Journal For Engineering Sciences Vol. 4 No. 3 Year 2011 ۲۳۰ Fig.7 Variation of the local Nusselt number for case(I) along the A) upper left triangular cylinder B) lower left triangular cylinder C)concentric circular rod D)outer enclosure, at E=0.36 -A- -B- distance,S1 distance,S2 NuL 0 5 10 15 20 25 0 0.2 0.4 0.6 0.8 1 Ra = 1 0 0 0 Ra = 3 0 0 0 0 Ra = 1 0 0 0 0 0 NuL 0 1 0 2 0 3 0 4 0 0 0 .2 0 .4 0 .6 0 .8 1 R a= 1 0 0 0 R a= 3 0 0 0 0 R a= 1 0 0 0 0 0 -C- -D- distance, Si distance, So 0 5 10 15 20 25 0 0.2 0.4 0.6 0.8 1 Ra=1000 Ra=30000 Ra=100000 NuL 0 2 4 6 8 0 0.2 0.4 0.6 0.8 1 Ra=1000 Ra=30000 Ra=100000 NuL Al-Qadisiya Journal For Engineering Sciences Vol. 4 No. 3 Year 2011 ۲۳۱ Fig.8 Variation of the local Nusselt number for case(II) along the A) upper triangular cylinder B) left triangular cylinder C)lower triangular cylinder D) concentric circular rod E) outer enclosure, at E=0.36 -E- NuL distance ,So 0 5 10 15 20 25 0 0.2 0.4 0.6 0.8 1 Ra=1000 Ra=40000 Ra=100000 -A- -B- NuL 0 5 10 15 20 25 0 0.2 0.4 0.6 0.8 1 dist ance Ra=1000 Ra=30000 Ra=100000 0 5 10 15 20 25 30 0 0.2 0.4 0.6 0.8 1 distance Ra=1000 Ra=30000 Ra=100000 NuL distance ,S1 distance ,S2 -C- -D- NuL 0 1 0 2 0 3 0 4 0 0 0 .2 0 .4 0 .6 0 .8 1 dist a n c e R a = 1 0 0 0 R a = 3 0 0 0 0 R a = 1 0 0 0 0 0 0 2 4 6 8 0 0.2 0.4 0.6 0.8 1 distance Ra=1000 Ra=30000 Ra=100000 NuL distance ,S3 distance ,Si Al-Qadisiya Journal For Engineering Sciences Vol. 4 No. 3 Year 2011 ۲۳۲ Fig.9 Variation of the mean Nusselt number with E along the outer wall for different Rayleigh number a) case(I) b) case(II) -a- -b- E E Num 0 4 8 12 16 20 0 .2 5 0 .3 0 .3 5 0 .4 0 .4 5 Ra=1 0 0 0 Ra=3 0 0 0 0 Ra=7 0 0 0 0 Ra=1 0 0 0 0 0 Num 0 4 8 1 2 1 6 2 0 0 .2 5 0 .3 0 .3 5 0 .4 0 .4 5 Ra =1 0 0 0 Ra =3 0 0 0 0 Ra =7 0 0 0 0 Ra =1 0 0 0 0 0 Fig.10 Variation of the mean Nusselt number with Rayleigh number along the outer wall for different values of E Ra Num 0 4 8 1 2 1 6 2 0 1 0 0 0 2 1 0 0 0 4 1 0 0 0 6 1 0 0 0 8 1 0 0 0 1 0 1 0 0 0 1 2 1 0 0 0 E =0 .2 7 E =0 .3 6 E =0 .4 4 E =0 .2 7 E =0 .3 6 E =0 .4 4 case(I) ----- case(II) Case(I) Case(II)