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Entropy generation during natural convection in trapezoidal enclosures with various inclination angles (?), ? = 45�, 60�, and 90� for uniformly heated bottom wall and insulated top wall with linearly heated side walls (case 1) or linearly heated left wall with cold right wall (case 2) have been investigated numerically using penalty finite element method. Parametric studies for the wide range of Rayleigh numbers (Ra = 10 3-10 5) and Prandtl numbers (Pr = 0.015-1000) have been performed. Symmetry in flow pattern is observed for case 1. During the conduction regime at low Ra (Ra = 10 3), the entropy generation in the cavity is dominated by heat transfer irreversibility for all Pr. The strength of fluid flow increases with Ra and that leads to an increase in thermal energy transport due to enhanced convection at Ra = 10 5. Consequently, the entropy generation due to heat transfer (S ?) and fluid friction (S ?) also increases with Ra for all Pr. The comparison of magnitudes of S ? and S ? indicates that maximum entropy generation due to heat transfer (S ?,max) and fluid friction (S ?,max) is lower for case 1 and higher for case 2. It is found that S ?,max occurs at the top portion of side walls in case 1, whereas S ?,max occurs at a hot-cold junction due to a high thermal gradient in case 2. The total entropy generation, S total is found to be smaller in case 1 and larger in case 2 for all Pr at Ra = 10 5. It is observed that S ?,max occurs at the top wall in both heating situations for Pr = 0.015 and Ra = 10 5. It is also found that, S ?,max is observed near side walls in case 1, whereas that is observed near the right cold wall in case 2 for Pr = 0.7 and 1000 at Ra = 10 5. The total entropy generation (S total) is larger for ? = 45� in both heating cases at high Pr (Pr = 1000). The total entropy generation (S total) also increases with Pr due to an increase in S ? with Pr. It is found that, high heat transfer rate (Nu b) and minimum entropy generation (S total) occur for square cavities at Ra = 10 5 for all Pr in case 1. It is observed that S total is smaller for case 1 compared to case 2, even though high Nu b is observed at case 2 for all ?s. The inclination angle (?) has almost identical effect on entropy production rate for 45� ? ? ? 60� within the entire range of Pr at all Ra. � 2012 American Chemical Society.

Finite element simulations were carried out to analyze entropy generation during mixed convection inside square enclosures with an isothermally hot bottom wall, adiabatic top wall, and isothermally cold side walls (case 1) or linearly heated side walls (case 2), or linearly heated left wall with isothermally cold right wall (case 3) for Pr = 0.015-7.2, Re = 1-100, and Gr = 103-105. Local entropy maps are studied in detail, and the dominance of thermal (S?,l) and frictional (S?,l) irreversibility is studied using Bejan number maps. In addition, variation in total entropy generation (Stotal), average Bejan number (Beav), and average Nusselt number at the bottom wall with Gr are analyzed to correlate irreversibility and the overall heat transfer rate of the system or process. It is found that, for Pr = 0.015 and 7.2, Re = 100 may be the optimal level for higher convective heat transport with minimum entropy generation in all the cases for Gr = 103-105. Copyright � 2015 Taylor & Francis Group, LLC.

A wide range of applications involving mixed-convection studies can be found in various engineering processes such as thermal discharge of water bodies, float glass production, heat exchangers, nuclear reactors, and crystallization process. The present study focuses on understanding the thermal mixing scenarios for mixed-convection lid-driven flow in a square cavity using heatlines. Thermal mixing is analyzed for four different thermal boundary conditions, and heat flow patterns in mixed convection are analyzed using Bejan's heatlines concept for wide ranges of parameters (Pr = 0.015-7.2, Re = 1-100, and Gr = 103-105, where Pr, Re, and Gr denote the Prandtl, Reynolds, and Grashof numbers, respectively). The results indicate that, at low Pr values (Pr = 0.015), the transport is conduction-dominant irrespective of the values of Gr and Re. The trends of heatlines and streamlines are identical near the core for high-Re cases. A single circulation cell was observed in the streamlines for any Pr ? 0.7 at high Re and low Gr values for uniform heating of the bottom surface with cold side walls. It was observed that thermal mixing increased significantly with subsequent rises in Gr for high-Pr fluids. Patterns of heatlines and multiple circulation cells of heatlines were found to lead to enhanced thermal mixing, with the thermal boundary layer much compressed toward the walls for linearly heated side walls. The heat-transfer rates along the walls are illustrated by the local Nusselt number distribution based on gradients of heatfunctions for the first time in this work. Nusselt numbers with infinitely large magnitudes were observed at hot-cold junctions, illustrating high heat-transfer rates. An oscillatory distribution in the local heatfunction rate was observed as a result of sinusoidal heating of the bottom surface for high-Pr fluids. Negative heat-transfer rates or local Nusselt numbers were observed along the side walls when side wall(s) was/were linearly heated, as explained based on negative heatfunction gradients. Also, the effect of Gr on the local and average Nusselt numbers in different cases can be adequately explained based on heatlines. Dense heatlines signifying higher overall heat-transfer rates along the bottom surface and side walls were observed for uniform bottom surface heating, whereas lower heat-transfer rates were observed for sinusoidal heating. Nonmonotonic distributions in overall heat-transfer rates along the bottom surface and left wall were observed when both walls were linearly heated, whereas a smooth and exponential increase was observed when the right wall was isothermally cooled. � 2011 American Chemical Society.

Mixed convection flows in a lid-driven square cavity filled with porous medium are studied numerically using penalty finite element analysis for uniformly heated bottom wall, linearly heated side walls or cooled right wall. The top wall is well insulated and set with uniform velocity. The relevant parameters in the present study are Darcy number ( Da = 10 - 5 – 10 - 3 ) , Grashof number ( Gr = 10 3 – 10 5 ) , Prandtl number (Pr = 0.015–10) and Reynolds number (Re = 1–102). The isotherms are generally symmetric at smaller Pr irrespective of Da and Re at Gr = 105 for linearly heated side walls. The isotherms are also almost symmetric at small Re with higher Gr ( Gr = 10 5 ) and Da ( Da = 10 - 3 ) and natural convection is found to be dominant whereas the isotherms are compressed near the left and bottom walls at higher Re for linearly heated side walls. Compression of isotherms and dominance of forced convection is also observed at Re = 102 for linearly heated left wall and cooled right wall. The local Nusselt number of the bottom wall ( Nu b ) for low Da is almost constant whereas Nub for higher Da and Pr shows non-monotonic variation at Re = 10 and an overall decreasing trend is observed at Re = 102 for linearly heated side walls. The local Nusselt numbers of left and right walls ( Nu l and Nu r ) increase for Da = 10−3 and Pr = 0.7 whereas oscillatory trend is observed at Re = 10–102 and Pr = 10. For linearly heated left wall and cooled right wall, Nub increases from the left edge towards the right edge of the bottom wall for both cases Re = 10 and 102 irrespective to Pr and Da. It is also observed that Nur for higher Pr and Da is found to increase monotonically at both Re = 10 and 10 2 and Nu l show non-monotonic trend for higher Pr and Da at Re = 10 whereas for Re = 10 2 , Nul increases monotonically. Average Nusselt numbers Nu b ¯ , Nu l ¯ , Nu r ¯ are found almost invariant with Gr for low Pr with all Da for linearly heated side walls or cooled right wall. On the other hand, Nu b ¯ is found to vary exponentially at higher Pr and Da and oscillations in Nu l ¯ and Nu r ¯ are observed for Re = 10 whereas increasing trend is observed for Nu l ¯ and Nu r ¯ for Re = 102 for both linearly heated side walls or cooled right wall.

Finite element simulations have been performed to investigate the influence of linearly heated side wall(s) or cooled right wall on mixed convection lid-driven flows in a square cavity. It is interesting to note that multiple circulation cells appear inside the cavity with the increase of Pr for Re = 10 and Gr = 10 5 in the case of linearly heated side walls. For Pr = 0.015 , only two circulation cells are formed inside the cavity. As Pr increases to 0.7, three circulation cells are formed inside the cavity. Further increase in Pr to 10, leads to the formation of four circulation cells inside the cavity. On the other hand, only two circulation cells are formed inside the cavity for the case of cooled right wall. A detailed analysis of flow pattern shows that as the value of Re increases from 1 to 10 2 , there occurs a transition from natural convection to forced convection depending on the value of Gr irrespective of Pr. It is observed that the secondary vortex at the top left corner disappears for Re = 10 2 and Gr = 10 5 due to enhanced motion of the upper lid in the case of cooled right wall while a small secondary vortex exist at the bottom right corner in the case of linearly heated side walls. The local Nusselt number ( Nu b ) plot shows that heat transfer rate is equal to 1 at the edges for the case of linearly heated side walls case and that is zero at the left edge and thereafter that increases for the case of cooled right wall. It is interesting to observe that Nu b is large within 0.4 ⩽ X ⩽ 0.6 due to compression of isotherms for Pr = 0.7 and 10 in the case of linearly heated side wall. It is also observed that Nu r or Nu l exhibits oscillations especially for Pr = 10 at higher Gr due to the presence of multiple circulations. It is also observed that Nu r ¯ or Nu l ¯ vs Gr plots show oscillation for two case studies. Average Nusselt numbers at the bottom and right walls are strong functions of Grashof number at larger Prandtl numbers whereas average Nusselt number at the left wall at a specific Pr is a weaker function of Gr.