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A two-phase model based on the double-diffusive approach is used to perform a numerical study of natural convection of alumina-water nanofluids in differentially heated vertical slender cavities. In the mathematical formulation, Brownian diffusion and thermophoresis are assumed to be the only slip mechanisms by which the solid phase can develop a significant relative velocity with respect to the liquid phase. The system of the governing equations of continuity, momentum and energy for the nanofluid, and continuity for the nanoparticles is solved through a computational code relying on the SIMPLE-C algorithm for the pressure-velocity coupling. The effective thermal conductivity and dynamic viscosity of the nanofluid, and the coefficient of thermophoretic diffusion of the suspended solid phase, are evaluated using three empirical correlations based on a high number of experimental data available from diverse sources, and validated by way of literature data different from those used in generating the...

Mixed convection in a inclined cavity has not been investigated by LBM in case of an imposed non zero heat flux. This type of boundary condition, representing very usual situations in physical world, is not simple to model in lattice Boltzmann schemes. In effect, the only boundary condition able to simulate an imposed temperature and an imposed heat flux at a boundary has been presented by D’Orazio et al. in previous works, where the boundary was at rest. In this work, laminar mixed convective heat transfer in two-dimensional rectangular inclined driven cavity is studied numerically by means of a double population thermal Lattice Boltzmann method. The counter-slip internal energy density boundary condition, able to simulate an imposed heat flux at the wall, is applied. Through the top moving lid the heat flux enters the cavity and it leaves the system through the bottom wall; side walls are adiabatic. Results are analyzed over a range of the Richardson numbers and tilting angles of the enclosure, encompassing the dominating forced convection, mixed convection, and dominating natural convection flow regimes. The results show that, as expected, heat transfer rate increases as increases the inclination angle, but this effect is significant for higher Richardson numbers, when buoyancy forces dominate the problem; for horizontal cavity, average Nusselt number decreases with the increase of Richardson number because of the stratified field configuration. This study shows that the counter-slip internal energy density boundary condition can be effectively used to simulate heat transfer phenomena also in case of moving walls and it makes the Lattice Boltzmann Method able to simulate a wide class of cooling process where a given thermal power must be removed.

One of the important objectives of thermal systems engineering is to analyze the utilization of thermal energy in an efficient manner. Analysis on entropy generation may give an insight about thermal efficiency of various thermal systems. Entropy generation analysis during natural convection in a porous trapezoidal structures of various inclination angles ( φ ) with isothermal (case 1) and non-isothermal (case 2) hot bottom wall has been performed. Simulations have been done to obtain solutions in terms of isotherms ( θ ), streamfunction ( ψ ) and entropy generation maps ( S θ and S ψ ) for various ranges of modified Darcy number ( Da ), Prandtl number ( Pr ) at modified Rayleigh number, Ra = 10 6 . It is observed that S θ , max occurs near bottom corners for case 1 whereas that occurs near middle part of the left and right portions of the bottom wall for case 2. The entropy production due to fluid flow irreversibilities ( S ψ , max ) occurs either at the top wall, or side walls or along the interface between two counter-rotating cells. It is found that, square cavities may be the optimal geometry in comparison to trapezoidal cavities for both cases. Also, trapezoidal cavities with φ = 60° may be the alternative optimal design in comparison with φ = 45° for all Pr fluids.

Analysis of natural convection in porous right angled triangular enclosures with a concave/ convex hypotenuse has been carried out using the Bejan's heatlines approach. A generalized non-Darcy model without Forchheimer term is employed for fluid flow in a porous matrix and the governing equations are solved by the Galerkin finite element method. The cavity is subjected to a thermal boundary condition of an isothermal cold left wall, isothermal hot curved right wall, and adiabatic bottom wall. Due to intense closed loop heatlines, thermal mixing is higher in convex cases compared to the concave case for all parameters. Average heat transfer rate is found to be largest in the concave hypotenuse case. Copyright � Taylor & Francis Group, LLC.

Analysis of ‘entropy generation’ is an important strategy to build, optimize and operate the heat exchange systems within their maximum operating efficiency. Porous rhombic cavities with various inclination angles, φ and various thermal aspect ratios, A, have been considered for the numerical investigation of thermal processing of various fluids (Prandtl number, Pr = 0.015 and 1000) in the range of Darcy number (Da = 10−3–10) due to its extensive energy related applications. The effect of A and φ for various governing parameters during convection are discussed in detail via heat transfer irreversibility (Sθ) and fluid friction irreversibility, Sψ. At lower A, the entropy generation in the cavity is dominated by both Sθ and Sψ for all φs irrespective of Da and Pr. As A increases, Sθ as well as Sψ decreases for all φs which in turn decreases Stotal with A irrespective of Da and Pr. The total entropy generation (Stotal) is found to be lower for φ = 30° and higher for φ = 75° for all Pr and Da. Analysis of variations of Beav with A for higher Da (Da = 10) indicates that, entropy generation is highly fluid friction dominant irrespective of φ and A. Lesser entropy generation (Stotal) with larger heat transfer rate ( Nu b ¯ ) and reasonable heat transfer rate ( Nu b ¯ ) occurs for Pr = 0.015 and Pr = 1000, respectively at φ = 30° cavities with all A irrespective of Da. Current work attempts to analyze energy efficient thermal convection strategies and role of thermal aspect ratio within porous rhombic enclosures based on entropy generation minimization vs heat transfer rates for various fluids.

This paper analyzes probable steady state flow structures and temperature patterns that may evolve during mixed convection within a lid-driven trapezoidal enclosure with cold top wall and hot bottom wall as the speed of moving lid varies with respect to the intensity of imposed temperature gradients. In this regard, a Grashof (Gr)-Reynolds (Re)-Prandtl (Pr) number formulation has been used to induce varying contributions from moving lid compared to that from imposed thermal gradients, where Grashof number has been varied from 10 3 to 105 at Re = 1 and 100 for three different fluids of Pr = 0.015, 0.7 and 10. Simulations have been performed for two different scenarios of isothermal (case 1) and non-isothermal (case 2) bottom wall with inclination angle of the side wall being kept at 45o. It has been found that non-isothermal bottom wall (case 2) leads to multiple steady states in either natural convection dominated regime (Gr/Re2 > 1) or mixed convection regime (Gr/Re2 ? O(1)) in convection dominated heat transport regime (Pr�Re?1). Number of steady states are observed to be more in natural convection dominated regime at Re = 1. The flow structures of various steady states are found to be crucial to achieve higher heat transfer rates for non-isothermal bottom wall. � 2013 Elsevier Ltd. All rights reserved.

Natural convection in isosceles triangular enclosures with various configurations (case 1 — inverted, case 2 — straight and case 3 — tilted) is studied via heatline analysis for linear heating of inclined walls. Detailed analysis and comparison for various base angles (φ = 45°, 60°) of triangular enclosures have been carried out for a range of fluids (Pr = 0.015 − 1000) within Ra = 103 − 105 using Galerkin finite element method. The heat flow distributions indicate conduction dominant heat transfer at low Ra (Ra = 103) for case 1 and case 2 whereas in case 3, convective heat flow is observed due to high buoyancy force. As Ra increases, enhanced thermal mixing is observed at the core of the cavity. Wall to wall heat transfer occurs at walls AB and AC due to linear heating boundary condition in all the cases. Although the distributions of fluid flow and heat flow are qualitatively similar for φ = 45° and 60°, the intensity of fluid flow and heat flow decreases as φ increases. Strength of fluid flow and heat flow circulation cells is found to be higher in case 3 for identical parameters. Results show that upper side wall (AC) for case 3 exhibits higher heat transfer rates whereas heat transfer rates for walls AB and AC are the same for case 1 and case 2. Also NuAB is higher for case 2 followed by case 1 and case 3 at the middle portion of wall AB. Thus to achieve high heat transfer from fluid to wall at the central region, case 2 and case 3 configurations may be recommended at high Ra (Ra = 105) and Pr, irrespective of φ.

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.

A detailed study on natural convection heat transfer within porous trapezoidal enclosures has been carried out for two different cases. The effect of linearly heated side walls on flow pattern is investigated in case 1; whereas, the effect of linearly heated left wall and cold right wall is studied in case 2. In both cases, the bottom wall of the cavity is uniformly heated and the upper wall is adiabatic. The results are analyzed for a wide range of parameters such as Rayleigh number, Ra(103Ra106), Prandtl number, Pr(0.015 Pr988.24), and Darcy number, Da(105Da10 3). � 2013 Taylor and Francis Group, LLC.

Entropy generation analysis has been introduced to investigate the effectiveness in heat transfer during natural convection within porous rhombic enclosures of various inclination angles ? for differential (case 1) and Rayleigh Benard (case 2), heating situations. The results are presented in terms of streamlines (?), isotherms (?), and entropy generation maps (S? and S?). Entropy production on the basis of various heating patterns and geometrical orientations have been reported based on the relation between total entropy generation (Stotal) and average Nusselt number (Nul or Nub) via Bejan Number, which relates the available thermal energy to irreversibilities. � 2013 Copyright Taylor and Francis Group, LLC.

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.

Analysis of ‘entropy generation’ is an important strategy to optimize the natural convection process in order to achieve efficient heat transfer within the system. Rhombic enclosures of various inclination angles ( φ = 30 ° , 45° and 75°) filled with porous media and bounded by adiabatic top wall, cold side walls, isothermally (case 1) and non-isothermally (case 2) heated bottom wall have been considered for the analysis of thermal processing of various fluids ( Pr =0.015, 0.7, 7.2 and 1000) in the range of Darcy number ( Da =10 −5 – 10 −3 ). At Da =10 −5 , the total entropy generation, S total , is found to be significantly high for φ = 30 ° and low for φ = 75 ° and is dominated by heat transfer irreversibility ( S θ ) for all Pr in case 1 whereas the distributions of S total for φ = 45 ° , 75° and 90° closely follow the distributions of φ = 30 ° for all Pr in case 2. Cup mixing temperature ( Θ cup ) is higher for φ = 75 ° and 90° whereas overall heat transfer rate, Nu ¯ , is higher for φ = 30 ° compared to other φ s in both heating cases except at high Pr ( 0.7 − 1000 ) fluids at Da =10 −5 . On increase of Da to 10 − 3 , fluid friction irreversibility, S ψ also increases for all φ s irrespective of Pr in both cases. Increase in S total with Da is small for lower φ ( φ = 30 ° ) due to insignificant S ψ of S total and large for higher φ ( φ = 75 ° ) due to significant S ψ of S total at Da ≥ 10 − 4 irrespective of Pr in both cases. Also, Θ cup slightly decreases at Da ≥ 10 − 4 and further, that reaches a constant value at higher Da as well as S total in both cases irrespective of φ for all Pr except Pr =0.015. Maximum Nu ¯ occurs for φ = 30 ° cavities at Da =10 −3 for all Pr in case 1 due to less available energy loss corresponding to less S ψ . Current work attempts to analyze energy efficient thermal convection strategies within porous rhombic enclosures based on entropy generation minimization vs enhanced thermal mixing or heat transfer rates for various fluids in porous media.

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.

Analysis has been carried out for the energy distribution and thermal mixing in steady laminar natural convective flow through the rhombic enclosures with various inclination angles, φ for various industrial applications. Simulations are carried out for various regimes of Prandtl ( Pr ) and Rayleigh ( Ra ) numbers. Dimensionless streamfunctions and heatfunctions are used to visualize the flow and energy distribution, respectively. Multiple flow circulations are observed at Pr = 0.015 and 0.7 for all φ s at Ra = 10 5 . On the other hand, two asymmetric flow circulation cells are found to occupy the entire cavity for φ = 75° at higher Pr ( Pr = 7.2 and 1000) and Ra ( Ra = 10 5 ). Heatlines are found to be parallel circular arcs connecting the cold and hot walls for the conduction dominant heat transfer at Ra = 10 3 . The enhanced convective heat transfer is explained with dense heatlines and convective loop of heatlines at Ra = 10 5 . Heatlines clearly demonstrate that the left wall receives heat from the bottom wall as heatlines directly connect both the walls whereas the convective heat circulation cells play lead role to distribute the heat along the right wall, especially for smaller φ s. On the other hand, the heat flow is evenly distributed to both side walls at higher φ s via convection as well as direct conductive transport. Significant convective heat transfer from the bottom hot wall to the left cold wall occurs for φ = 30° cavity whereas the heat transfer to the right cold wall is maximum for φ = 75° irrespective of Pr . Average Nusselt number studies also show that φ = 30° cavity gives maximum heat transfer rate from the bottom to left wall irrespective of Pr in isothermal heating case. On the other hand, enhanced thermal mixing occurs at φ = 75° for both isothermal and non-isothermal heating strategies except at Pr = 0.015 in isothermal heating case.

The present numerical study deals with natural convection flow in closed trapezoidal enclosures. The detailed analysis is carried out in two cases: (1) linearly heated side walls; (2) linearly heated left wall and cold right wall. In both the cases bottom wall is uniformly heated and top wall is well insulated. A penalty finite element method with bi-quadratic elements is used to obtain the results in the form of isotherms, streamlines and heatlines and local and average Nusselt numbers. Numerical results are obtained for various values of Rayleigh number Ra ( 10 3 ≤ R a ≤ 10 5 ), Prandtl number Pr ( 0.015 ≤ Pr ≤ 1000 ) and inclination angles (φ = 45°, 60° and 90°). Results signify that, at low Ra ( R a = 10 3 ) heat transfer is conduction dominant. At R a = 10 5 , multiple circulations of streamlines and heatlines results in enhanced convection. For linearly heated side walls (case 1), symmetric pattern in fluid flow and heat flow is observed. Enhanced thermal transport is observed from bottom wall to top portion of side walls via dense heatlines along the vertical center line. It is found that, less intense circulations occurs in square cavity (φ = 90°) compared to other cavities φ = 45°, 60°. In case 2, the cold right wall receives larger amount of heat from bottom wall compared to that of linearly heated left wall. The formation of boundary layer on the walls is explained based on heatlines. The local and average Nusselt numbers are also illustrated using heatlines. It is found that, Nub distribution exhibits sinusoidal variation at P r = 1000 in case 1. It is also found that, Nul and Nur display wavy pattern at higher Ra for all Pr in case 2. Finally, it is concluded that, overall heat transfer rates are larger for square cavity (φ = 90°) compared to other angles (φ = 45°, φ = 60°) irrespective of heating patterns for side walls.

One way of increasing the energy efficiency in thermal processing of materials is to reduce exergy losses due to irreversibilities, measured as "entropy generation". In this study, an analysis of entropy generation during natural convection in rhombic enclosures with various inclination angles (? = 30�, 45�, and 75�) has been carried out for the efficient thermal processing of various fluids (Prandtl numbers of Pr = 0.015, 0.7, 7.2, and 1000) in a range of Rayleigh numbers (Ra = 103-105). The enclosure is bounded by an adiabatic top wall, cold side walls, and an isothermally (case 1) and nonisothermally (case 2) heated bottom wall. Isotherms (?), streamlines (?), and entropy generation contours due to heat transfer (S?) and fluid friction irreversibility (S ?) are analyzed for both cases. At low Rayleigh number (Ra = 103), the entropy generation in the cavity is dominated by S ? for all ?, irrespective of Pr. As Ra increases to 10 5, the fluid flow intensifies and S? also increases for all ?, irrespective of Pr. The total entropy generation (S total), average Bejan number (Beav), and average heat-transfer rate (Nub) are plotted for Rayleigh number ranges of 103 ? Ra ? 105. The total entropy generation (S total) is found to be low for ? = 30� and high for ? = 75� for all Pr values at Ra = 105. Analysis of variations of Beav with Ra for high-Pr fluids indicates that S? contributes significantly for increase in Stotal. It is also found that the largest heat-transfer rate (Nub) corresponds to minimum entropy generation (Stotal) for ? = 30� cavities at Ra = 105 with all Pr in case 1. The nonisothermal heating strategy (case 2) is energy efficient, because of its lower Stotal value, corresponding to low Nub, which is due to a lesser heating effect than case 1 for all ?. Overall, rhombic cavities with ? = 30� may be the optimal geometrical design for thermal processing of all types of fluids (Pr = 0.015, 0.7, 7.2, and 1000), irrespective of the heating strategy. � 2011 American Chemical Society.

In this paper, the numerical investigation of natural convection in a porous trapezoidal enclosures has been performed for uniformly or non-uniformly heated bottom wall. Penalty finite element analysis with bi-quadratic elements is used for solving the Navier–Stokes and energy balance equations. The numerical solutions are studied in terms of streamlines, isotherms, heatlines, local and average Nusselt numbers for a wide range of parameters Da(10−5–10−3), Pr(0.015–1000) and Ra(Ra = 103–106). At low Darcy number (Da = 10−5), heat transfer is primarily due to conduction for all φ’s as seen from the heatlines which are normal to the isotherms. As Da increases to 10−4, convection is initiated and the thermal mixing has been observed at the central regime for all φ’s. Distribution of heatlines illustrate that most of the heat transport for high Darcy number (Da = 10−3) occurs from hot bottom wall to the top portion of cold side walls. It has been found that secondary circulations appear at the top corners of the cavity for φ = 45°, 60° and bottom corners of the cavity for φ = 90° with Pr = 0.015, Da = 10−3 and Ra = 106. The physical interpretation of local and average Nusselt numbers are illustrated using heatlines. For uniformly heated bottom wall with cold side walls, Nub values are maximum near the corners of bottom wall for all φ’s irrespective to Da and Pr. In contrast, for non-uniformly heated bottom wall, the local Nusselt number (Nub) is found to be minimum near the corners of bottom wall and that is also found to be a sinusoidal variation with distance at high Da for all angles (φ). The Nus distribution is similar in both cases along the side wall except near the junction of hot and cold wall for all φ’s. Overall, heat transfer analysis for bottom and side walls is presented in terms of average Nusselt numbers ( Nu b ¯ , Nu s ¯ ) . The critical Ra numbers corresponding to the onset of convection are obtained at Da = 10−3 for Pr(0.015–1000). For Da = 10−3, average Nusselt numbers ( Nu b ¯ and Nu s ¯ ) increase exponentially beyond the critical Ra. Overall, the heat transfer rate is large for square cavity (φ = 90°) compared to other angles (φ) irrespective of heating patterns.

Numerical investigations of combined free convection and surface radiation in enclosures for Rayleigh-Benard configuration of air are carried out using FLUENT 6.3, with a view to determine the onset of convection and to propose correlations for convection and radiation Nusselt numbers based on a detailed parametric study. The onset of Rayleigh-Benard convection is delayed with an increase in the emissivity of the sidewalls. The effect of surface radiation on the onset of convection however diminishes with aspect ratio (AR) and for AR = 8, the effect of surface radiation ceases. Post-onset, the effect of surface radiation on the convection heat transfer becomes insignificant beyond an aspect ratio of 5. � 2010 Elsevier Ltd.

Natural convection in right-angled triangular enclosures with various top angles ( φ =15°, 30°, 45°) is studied in detail via heat flow analysis for various uniform isothermal and linear isothermal heating thermal boundary conditions. Detailed analysis on the effects of aspect-ratio and thermal boundary conditions on the fluid and heat flow inside the triangular enclosures have been carried out for a range of fluids ( Pr = 7.2, 1000, 0.015) within Ra = 10 3 –10 5 . Interesting features of heat flow patterns under various thermal boundary conditions are ‘visualized’ by heatlines. The effect of increase in φ of triangular enclosures is such that the maximum heat flux at the top vertex decreases and the thermal mixing in cavity increases with the increase in φ . It is found that, the fluid in the lower corners is adequately heated in presence of hot right wall compared to that in left wall heating cases. Further, the heat transfer characteristics, in terms of local and average Nusselt numbers, indicate that isothermal heating cases exhibit exponential decrease in Nu l whereas linear heating cases interestingly show local intermediate maxima. Also, various qualitative and quantitative features of Nu and N u ¯ are adequately explained based on heatlines. Finally, the correlations for N u l ¯ and Ra are obtained for various fluid with all heating situations.

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.

A computational model is proposed to simulate sodium pool combustion considering the effect of turbulent natural convection in a vented enclosure of the steam generator building (SGB) of a fast breeder reactor. The model is validated by comparing the simulated results with the experimental results available in literature for sodium pool combustion in a CSTF vessel. After validation, the effects of vents and the location of the pool on the burning rate of sodium and the associated heat transfer to the walls are studied in an enclosure comparable in size to one floor of the steam generator building. In the presence of ventilation, the burning rate of sodium increases, but the total heat transferred to the walls of the enclosure is reduced. It is also found that the burning rate of sodium pool and the heat transfer to the walls of the enclosures vary significantly with the location of sodium pool. � 2009 Elsevier B.V. All rights reserved.

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 ° .

The phenomena of natural convection within a trapezoidal enclosure filled with porous matrix for linearly heated vertical wall(s) with various inclination angles φ has been studied numerically. A penalty finite element analysis with bi-quadratic elements is used for solving the Navier–Stokes and energy balance equations. Wide range of parameters such as Rayleigh number, Ra ( 10 3 ⩽ Ra ⩽ 10 6 ) , Prandtl number, Pr ( 0.026 ⩽ Pr ⩽ 1000 ) and Darcy number, Da ( 10 - 5 ⩽ Da ⩽ 10 - 3 ) have been used. Numerical results are presented in terms of streamlines, isotherms and Nusselt numbers. It has been found that secondary and tertiary circulations appear at the bottom half of the cavity for φ = 30° and φ = 0° with Pr = 0.026 and 0.7 , Da = 10 - 3 and Ra = 10 6 for linearly heated side walls. On the other hand, for linearly heated left wall and cold right wall, multiple circulations occur near the top portion of the cavity. For linearly heated side wall, the local Nusselt number ( Nu b ) shows sinusoidal behavior with distance at high Darcy number for all tilt angles whereas increasing trend in Nu s is observed in the upper half of the side wall for all tilt angles. For linearly heated left wall with cold right wall, increasing trend in Nub is observed irrespective to Da and Pr and Nub is even larger for higher Da. Increasing trend is also observed in Nul for all tilt angles for linearly heated left wall. Due to discontinuity in right corner, Nur first decreases and thereafter that increases for Y ⩾ 0.2 . The average Nusselt number remains constant up to Ra = 10 6 at low Da for Pr = 0.026 whereas for Pr = 1000 and high Ra, that starts to decrease for bottom wall whereas that starts to increase for side walls due to convection dominant effect at high Da. In general, the average Nusselt number increases with the increase of Da and Ra for higher Da.

Turbulent natural convection of liquid metal in a cylindrical enclosure with locally distributed heat source has been investigated for Boussinesq number in the range of 4.5 � 0 11- 6.25 � 10 12. The enclosure considered is an ideal model of the lower plenum of a fast reactor with tray(s) holding damaged core debris, which continuously generate heat. The focus of the study has been to assess the heat dissipation capacity of single and multiple trays in respecting the specific temperature limits on the tray(s). Heat conduction in the metallic trays and in the impervious core debris mixture and turbulent natural convection of the liquid sodium are solved as a conjugate heat transfer problem. The equations that govern the various heat transfer processes in 2-D axi-symmetric cylindrical polar coordinate system have been solved by the finite volume method. Turbulence has been modeled by the k-? turbulence model, without the use of wall functions. The predictions of the numerical model have been validated against benchmark data reported in open literature. Also, experiments have been conducted in an ideal water model towards validation of the computational model. For typical enclosure dimensions representing a 500 MWe fast reactor, it is seen that the critical parameters are heat dissipation area of the source (area of trays) and the thickness of the heat source (debris thickness). With increase in the number of trays, the heat transfer area increases while the debris thickness reduces. Both these effects lead to reduction in the tray as well as source temperatures. The heat dissipation capacity exhibits a non-linear relationship with the number of plates. � 2009 Elsevier Ltd. All rights reserved.

The phenomena of natural convection in a trapezoidal enclosure filled with porous matrix has been studied numerically. A penalty finite element analysis with bi-quadratic elements is performed to investigate the influence of uniform and non-uniform heating of bottom wall while two vertical walls are maintained at constant cold temperature and the top wall is well insulated. Parametric study for the wide range of Rayleigh number Ra ( 10 3 ⩽ Ra ⩽ 10 5 ) , Prandtl number Pr ( 0.015 ⩽ Pr ⩽ 1000 ) and Darcy number ( 10 - 3 ⩽ Da ⩽ 10 - 5 ) shows consistent performance of the present numerical approach to obtain the solutions in terms of stream function and isotherm contours. For parameters studied in the above range, a symmetry is observed for temperature and flow simulations. Non-uniform heating of the bottom wall produces greater heat transfer rate at the center of the bottom wall than uniform heating case for all Rayleigh and Darcy numbers but average Nusselt number shows overall lower heat transfer rate for non-uniform heating case. It is observed that the conduction is dominant irrespective of Ra for Da = 10 - 5 . As Rayleigh number increases, there is a change from conduction dominant region to convection dominant region for Da = 10 - 3 . The correlations between average Nusselt number and three parameters (Rayleigh number ( Ra ), Prandtl number ( Pr ) and Darcy number ( Da )) are also obtained.

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.

Buoyancy-induced flow regimes are investigated numerically for the basic case of a horizontal cylinder centred into a long co-axial square-sectioned cavity. In the frame of the 2D assumption, the threshold for the occurrence of time-dependent behaviour is explored. Stable symmetric and non-symmetric steady-state solutions, as well as unsteady regimes are observed, depending on the Rayleigh number, Ra, and the aspect ratio of the cavity, d. Four d-values are considered (d = 0.2, 0.4, 0.6, and 0.8). Heat transfer results are correlated by a single equation covering the full subcritical region.

The energy efficient convection systems can be designed by reducing exergy losses. In this context, the analysis on the entropy generation during natural convection in the fluid-filled (Prandtl number, P r = 0.015 − 1000 ) rhombic enclosures with various inclination angles ( φ ) has been carried out for the efficient thermal processing in various applications such as the chemical reactor modeling, underground coal gasification, and nuclear reactors. The enclosure is subjected to the differential heating (case 1) and Rayleigh–Benard convection (case 2). The conduction based static solution occurs only for φ = 9 0 ∘ and it is observed that the conduction based static solution disappears with a slight perturbation of φ at Rayleigh number, R a ≥ 2 × 10 3 irrespective of P r in case 2. The active zones of the heat transfer irreversibility ( S θ ) and fluid friction irreversibility ( S ψ ) are found to occur near the isothermal walls for all φ s irrespective of P r in both the cases at R a = 10 5 . In addition, the active zones of S ψ are also found to occur near the adiabatic walls of the cavity for all φ s irrespective of P r in both the cases at R a = 10 5 . Also, the region between the fluid layers of primary circulation cells acts as the strong active zone of S ψ for all φ s in case 2 at lower P r ( P r = 0.015 ) and R a = 10 5 . The total entropy generation ( S t o t a l ) and maximum heat transfer rates ( N u ¯ ) are found to be significantly low for φ = 3 0 ∘ in both the cases at R a = 10 5 irrespective of P r . Analysis of heating patterns and geometrical orientations relates the exergy to irreversibilities which establishes that the rhombic cavity ( φ = 3 0 ∘ ) with the differential heating pattern may be the optimal design based on the energy efficient perspective.

Current work attempts to study the heatline patterns during natural convection for different types of Dirichlet heatfunction boundary conditions. The enclosures with various shapes (square, curved, trapezoidal, tilted square and parallelogrammic) are considered with various thermal boundary conditions such as (a) case 1: hot left wall, cold right wall and adiabatic horizontal walls, (b) case 2: hot bottom wall, cold left and right walls and adiabatic top wall and (c) case 3: hot bottom wall with other cold walls. Traditionally, the reference of heatfunction ( Π = 0 ) is assumed at the adiabatic wall and the implementation of reference ( Π = 0 ) may be non-trivial for the case with zero or multiple adiabatic wall(s). Various heatfunction boundary conditions have been formulated based on locations of Π = 0 for systems with more than one adiabatic walls (case 1) or no adiabatic wall (case 3). As test problems, Π = 0 is considered at the junctions of isothermal walls (cases 2 and 3) or on the isothermal wall (case 3). The governing equations are solved via the Galerkin finite element method at various Rayleigh numbers ( 10 3 and 10 5 ) and Prandtl numbers ( Pr = 0.015 and 7.2). The magnitudes of the heatfunctions change drastically with the location of the datum of Π ( Π = 0 ) whereas, the heat flow patterns remain same irrespective of the heatfunction boundary conditions. The gradients of heatfunctions or the heat flux along the active walls (hot/cold) are invariant of the choice of the reference ( Π = 0 ). The local and average Nusselt numbers are also independent of the choice of Π = 0 and the Nusselt numbers are found to be identical with heatfunction gradients obtained with various locations of Π = 0 . Current work may be useful for heat flow visualization in various thermal systems involving complex thermal boundary conditions.