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A Brinkman extended Darcy model has been used to study the effect of spatial nonuniformity of wall temperatures on the multiplicity of steady convective flows within a square porous enclosure saturated with low Prandtl fluid (Pr = 0.026). The convection is assumed to be driven by the hotter bottom wall in conjunction with colder side walls, where bottom-side wall junctions maintain continuity of temperature to represent a more realistic situation. This configuration enforces nonuniformity on either bottom wall or side walls or both bottom and side wall temperatures. The degree of nonuniformity of wall temperatures are varied parametrically in terms of thermal aspect ratio (?) to simulate various possible scenarios of nonuniform bottom/side wall temperatures and steady solution branches are traced in the parameter space of ? to investigate the variation of multiplicity of the flows. A perturbation technique has been used to initiate the steady solution branches, which are then traced by numerical continuation scheme. A penalty finite element approximation in conjunction with Newton-Raphson method and parameter continuation scheme has been used to construct the bifurcation diagrams of various steady branches. This work reveals three steady symmetric branches and four steady asymmetric branches with exhibition of symmetry breaking bifurcation. This work also shows that the nonuniformity of wall temperatures play a crucial role for the existence of multiple solutions as well as the multiplicity of convective flows. � 2012 Elsevier Ltd. All rights reserved.

Thermal energy conservation is at the heart of energy efficiency in process industries. This allows researchers to go for advances in CFD methods to track the requirement of thermal energy for various thermal processing applications. CFD models need to be incorporated into optimization principles to achieve efficient industrial process designs. In this study, entropy generation due to natural convection in right-angled triangular enclosures (cases 1-4) has been studied numerically for energy efficient processing of various fluids (Pr=0.025, 7 and 1000). It is found that maximum value of entropy generation due to heat transfer (S ?,max) occurs near the vertex of the enclosures for cases 1 and 3, near corner between left wall and bottom wall for case 2 and near lower portion of the right wall for case 4. On the other hand, maximum value of entropy generation due to fluid flow (S ?,max) is observed near middle portions of the side walls for all cases and the location of S ?,max mainly depends on the presence of high velocity gradients. The entropy generation rates and heat transfer rates were also shown for various angles. Triangular cavities with specific angles are recommended for various processing fluids. � 2012 The Institution of Chemical Engineers.

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.

The interplay between internal heat generation and externally driven natural convection inside a porous medium annulus is studied in detail using numerical methods. The axisymmetric domain is bounded with adiabatic top and bottom walls and differentially heated side walls sustaining steady natural convection of a fluid with Prandtl number, Pr=5, through a porous matrix of volumetric porosity, ?=0.4. The generalized momentum equation with Brinkman-Darcy-Forchheimer terms and the local thermal non-equilibrium based two-energy equation model are solved to determine the flow and the temperature distribution. Beyond a critical heat generation value defined using an internal Rayleigh number, RaI,cr *, the convection transits from unicellular to bicellular mode, as the annulus Tmax becomes higher than the fixed hot-wall temperature. The RaI,cr * increases proportionately when the permeability based external Rayleigh number RaE * and the solid-fluid thermal conductivity ratio ? are independently increased. A correlation is proposed to predict the overall annulus Nu as a function of RaE *, RaI *, Da and ?. It predicts the results within �20% accuracy. � 2010 Elsevier Ltd.

Applications involving porous media occur widely in chemical engineering processes, e.g., oil and gas production, groundwater pollution, food processing, nuclear waste repository etc. The present study focuses on enhancing thermal processing of materials via distributed heating. Thermal mixing during laminar natural convection is analyzed in four different discretely heated porous square cavities. Heatline approach of visualizing heat flow is implemented to gain in-depth understanding of complex heat flow patterns for wide range of parameters ( Pr = 0.015 – 1000 , Da = 10 − 6 – 10 − 3 , Ra = 10 3 – 10 6 ). Results indicate that at lower Da ( = 10 − 6 ) heat flow is weak and transport is conduction dominant. At Da = 10 − 3 , the smaller hydraulic resistance of porous medium results in enhanced convection. The role of hot regime at bottom wall or hot regime at side walls to efficiently heat the cavity has been depicted via heatlines. It is illustrated by heatlines that when the heat sources are centrally located on the wall of the cavity, thermal mixing is significantly increased with subsequent rise in maximum temperature in the cavity. Effect of Da on local and average Nusselt numbers in different cases of porous cavities are adequately explained based on heatlines for the first time in this work. Average Nusselt number distributions show that hot regime of bottom wall transfers larger amount of heat whereas cold regimes of side wall receive larger amount of heat, especially at higher Da for all cases (cases 1–4). Further, cup-mixing temperature is evaluated for various cases to establish optimal thermal mixing by distributed heating in comparison to conventional bottom wall heating.

Steady state natural convection of a fluid with Pr ≈ 1 within a square enclosure containing uniformly distributed, conducting square solid blocks is investigated. The side walls are subjected to differential heating, while the top and bottom ones are kept adiabatic. The natural convection flow is predicted employing the nondimensional volumetric flow rate ( $$ Q_{\max }^{*} $$ ) by using a network model and also using numerical simulations. For identical solid and fluid thermal conductivities (i.e. k s = k f ), a parametric study of the effect of number of blocks (N 2), gap size (δ) and enclosure Rayleigh number (Ra) on $$ Q_{\max }^{*} $$ is performed using the two approaches. Network model predictions are observed to agree well with that from the simulations until Raδ3 ~ 12. Considering the enclosure with blocks as a porous medium, for a fixed enclosure Ra number, increasing the number of blocks for a fixed volumetric porosity leads to a decrease in enclosure permeability, which in turn reduces the flow rate. When the number of blocks is fixed, and for a given Ra number, the flow rate increases as the porosity increases by widening the gap between the blocks.

Natural convection in trapezoidal enclosures for uniformly heated bottom wall, linearly heated vertical wall(s) in presence of insulated top wall have been investigated numerically with penalty finite element method. Parametric studies for the wide range of Rayleigh numbers ( Ra = 10 3 - 10 5 ) and Prandtl numbers ( Pr = 0.7 - 1000 ) with various tilt angles of side walls ( φ ) have been performed. For linearly heated side walls, symmetry in flow pattern is observed. In addition, secondary circulations are observed near the bottom wall for φ = 0 ° especially for larger Pr ( Pr ⩾ 0.7 ) . In contrast, for linearly heated left wall and cooled right wall, the secondary circulations are stronger near the top portion of the left wall especially for larger Pr. Streamlines show that the strength of convection is larger for φ = 45 ° and flow intensities are found to be larger for higher Prandtl numbers. Local heat transfer rates are found to be relatively larger for φ = 0 ° . Average Nusselt number plots show higher heat transfer rates for φ = 0 ° and the overall heat transfer rates at the bottom wall is larger for the linearly heated left wall and cooled right wall. A non-monotonic trend in average Nusselt number vs Rayleigh number due to presence of multiple circulations is observed for φ = 0 ° .

A penalty finite element analysis with bi-quadratic elements is carried out to investigate the effects of uniform and non-uniform heating of inclined walls on natural convection flows within a isosceles triangular enclosure. Two cases of thermal boundary conditions are considered; case I: two inclined walls are uniformly heated while the bottom wall is cold isothermal and case II: two inclined walls are non-uniformly heated while the bottom wall is cold isothermal. The numerical solution of the problem is presented for various Rayleigh numbers (Ra), ( 10 3 ⩽ Ra ⩽ 10 6 ) and Prandtl numbers (Pr), ( 0.026 ⩽ Pr ⩽ 1000 ). It has been found that at small Prandtl numbers, geometry does not have much influence on flow structure while at Pr = 1000 , the stream function contours are nearly triangular showing that geometry has considerable effect on the flow pattern. In addition, the presence of multiple circulations are observed for small Pr ( Pr = 0.026 ) which causes wavy distribution of local Nusselt number. It is observed that non-uniform heating produces greater heat transfer rates at the center of the walls than the uniform heating; however, average Nusselt numbers show overall lower heat transfer rates for the non-uniform heating case. Critical Rayleigh numbers for conduction dominant heat transfer cases have been obtained and for convection dominant regimes, power law correlations between average Nusselt number and Rayleigh numbers are presented for specific Prandtl numbers.

The velocity field corresponding to the unsteady motion of a viscous fluid between two side walls perpendicular to a plate is determined by means of the Fourier transforms. The motion of the fluid is produced by the plate which after the time t = 0, applies an oscillating shear stress to the fluid. The solutions that have been obtained, presented as a sum of the steady-state and transient solutions satisfy the governing equation and all imposed initial and boundary conditions. In the absence of the side walls they are reduced to the similar solutions corresponding to the motion over an infinite plate. Finally, the influence of the side walls on the fluid motion, the required time to reach the steady-state, as well as the distance between the walls for which the velocity of the fluid in the middle of the channel is unaffected by their presence, are established by means of graphical illustrations.

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.