Natural convection effects on heat and mass transfer in a curvilinear triangular cavity

Finite element method is used in this study to analyze the effects of buoyancy ratio and Lewis number on heat and mass transfer in a triangular cavity with zig-zag shaped bottom wall. Buoyancy ratio is defined as the ratio of Grashof number of solutal and thermal. Inclined walls of the cavity have lower temperature and concentration according to zig-zag shaped bottom wall. Enclosed space consists mostly of an absorber plate and two inclined glass covers that form a cavity. Both high temperature and high concentrations are applied to bottom corrugated wall. Computations were done for different values of buoyancy ratio (?10 ? Br ? 10), Lewis number (0.1 ? Le ? 20) and thermal Rayleigh number (10⁴ ? Ra_T ? 10⁶). Streamlines, isotherms, iso-concentration, average Nusselt and Sherwood numbers are obtained. It is found that average Nusselt and Sherwood numbers increase by 89.18% and 101.91% respectively as Br increases from ?10 to 20 at Ra_T = 10⁶. Also, average Nusselt decreases by 16.22% and Sherwood numbers increases by 144.84% as Le increases from 0.1 to 20 at this Rayleigh number.     

Heatlines based natural convection analysis in tilted isosceles triangular enclosures with linearly heated inclined walls: effect of various orientations

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 Nu_AB 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.     

Heat flow visualization for natural convection in rhombic enclosures due to isothermal and non-isothermal heating at the bottom wall

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 φs at Ra = 10⁵. 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 = 10⁵). Heatlines are found to be parallel circular arcs connecting the cold and hot walls for the conduction dominant heat transfer at Ra = 10³. The enhanced convective heat transfer is explained with dense heatlines and convective loop of heatlines at Ra = 10⁵. 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 φs. On the other hand, the heat flow is evenly distributed to both side walls at higher φs 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.     

Natural convection in a square cavity filled with a porous medium: Effects of various thermal boundary conditions

Natural convection flows in a square cavity filled with a porous matrix has been studied numerically using penalty finite element method for uniformly and non-uniformly heated bottom wall, and adiabatic top wall maintaining constant temperature of cold vertical walls. Darcy–Forchheimer model is used to simulate the momentum transfer in the porous medium. The numerical procedure is adopted in the present study yields consistent performance over a wide range of parameters (Rayleigh number Ra, 10³ ⩽ Ra ⩽ 10⁶, Darcy number Da, 10⁻⁵ ⩽ Da ⩽ 10⁻³, and Prandtl number Pr, 0.71 ⩽ Pr ⩽ 10) with respect to continuous and discontinuous thermal boundary conditions. Numerical results are presented in terms of stream functions, temperature profiles and Nusselt numbers. 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 numbers but average Nusselt number shows overall lower heat transfer rate for non-uniform heating case. It has been found that the heat transfer is primarily due to conduction for Da ⩽ 10⁻⁵ irrespective of Ra and Pr. The conductive heat transfer regime as a function of Ra has also been reported for Da ⩾ 10⁻⁴. Critical Rayleigh numbers for conduction dominant heat transfer cases have been obtained and for convection dominated regimes the power law correlations between average Nusselt number and Rayleigh numbers are presented.     

Effects of thermal boundary conditions on natural convection flows within a square cavity

A numerical study to investigate the steady laminar natural convection flow in a square cavity with uniformly and non-uniformly heated bottom wall, and adiabatic top wall maintaining constant temperature of cold vertical walls has been performed. A penalty finite element method with bi-quadratic rectangular elements has been used to solve the governing mass, momentum and energy equations. The numerical procedure adopted in the present study yields consistent performance over a wide range of parameters (Rayleigh number Ra, 10³ ⩽ Ra ⩽ 10⁵ and Prandtl number Pr, 0.7 ⩽ Pr ⩽ 10) with respect to continuous and discontinuous Dirichlet boundary conditions. Non-uniform heating of the bottom wall produces greater heat transfer rates at the center of the bottom wall than the uniform heating case for all Rayleigh numbers; 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 dominated regimes, power law correlations between average Nusselt number and Rayleigh numbers are presented.     

Analysis of Bejan’s heatlines on visualization of heat flow and thermal mixing in tilted square cavities

This article analyzes the detailed heat transfer phenomena during natural convection within tilted square cavities with isothermally cooled walls (BC and DA) and hot wall AB is parallel to the insulated wall CD. A penalty finite element analysis with bi-quadratic elements has been used to investigate the results in terms of streamlines, isotherms and heatlines. The present numerical procedure is performed over a wide range of parameters (10³ ? Ra ? 10⁵,0.015 ? Pr ? 1000,0° ? φ ? 90°). Secondary circulations cells are observed near corner regions of cavity for all φ’s at Pr = 0.015 with Ra = 10⁵. Two asymmetric flow circulation cells are found to occupy the entire cavity for φ = 15° at Pr = 0.7 and Pr = 1000 with Ra = 10⁵. Heatlines indicate that the cavity with inclination angle φ = 15° corresponds to large convective heat transfer from the wall AB to wall DA whereas the heat transfer to wall BC is maximum for φ = 75°. Heat transfer rates along the walls are obtained in terms of local and average Nusselt numbers and they are explained based on gradients of heatfunctions. Average Nusselt number distributions show that heat transfer rate along wall DA is larger for lower inclination angle (φ = 15°) whereas maximum heat transfer rate along wall BC occur for higher inclination angle (φ = 75°).     

Investigation of natural circulation in cavities with uniform heat generation for different Prandtl number fluids

Natural convection in enclosures with uniform heat generation and isothermal side walls is studied here. For the rectangular enclosure, two-dimensional conservation equations are solved using SIMPLE algorithm. Parametric studies are conducted to examine the effects of orientation of the cavity, fluid properties (Pr number), and aspect ratio for Rayleigh numbers up to 10⁶. For a horizontal square cavity, the flow becomes periodically oscillating at Ra = 5 × 10⁴ and chaotic at Ra = 8 × 10⁵. With a slight increase in the inclination angle, the oscillations die and for inclination angles greater than 15⁰, the flow attain a steady state over a range of Ra. It is found that for tall cavities (aspect ratio > 1), the steady-state solution is obtained for all values of Ra considered here. However, for wide cavities (aspect ratio < 1), an oscillatory flow regime is observed. The maximum temperature within the cavity is calculated for the range of Ra, aspect ratio and Pr number. Correlations for the maximum cavity temperature is presented here. The values of critical Rayleigh number at which the convection sets in the rectangular cavity are also studied and two distinct criteria are determined to evaluate the critical Rayleigh number. Further, a three-dimensional simulation is performed for a cubic cavity. It is found that the steady state solutions are obtained for all Rayleigh number, except at Ra = 10⁶. This is in contrast to the predictions for a two-dimensional square cavity, which has an oscillatory zone from Ra = 5 × 10⁴ onwards.     

Numerical modeling of natural convection in an open cavity with two vertical thin heat sources subjected to a nanofluid

This paper presents a numerical study of natural convection cooling of two heat sources vertically attached to horizontal walls of a cavity. The right opening boundary is subjected to the copper–water nanofluid at constant low temperature and pressure, while the other boundaries are assumed to be adiabatic. The governing equations have been solved using the finite volume approach, using SIMPLE algorithm on the collocated arrangement. The study has been carried out for the Rayleigh number in the range 10⁴ ≤ Ra ≤ 10⁷, and for solid volume fraction 0 ≤ φ ≤ 0.05. In order to investigate the effect of heat source location, three different placement configurations of heat sources are considered. The effects of both Rayleigh numbers and heat source locations on the streamlines, isotherms, Nusselt number are investigated. The results indicate that the flow field and temperature distributions inside the cavity are strongly dependent on the Rayleigh numbers and the position of the heat sources. The results also indicate that the Nusselt number is an increasing function of the Rayleigh number, the distance between two heat sources, and distance from the wall. In addition it is observed that the average Nusselt number increases linearly with the increase in the solid volume fraction of nanoparticles.     

Natural convection heat transfer enhancement using adiabatic block: Optimal block size and Prandtl number effect

Numerical methods are used to solve the finite volume formulation of the two-dimensional mass, momentum and energy equations for steady-state natural convection inside a square enclosure. The enclosure consists of adiabatic horizontal walls and differentially heated vertical walls, but it also contains an adiabatic centrally-placed solid block. The aim of the study is to delineate the effect of such a block on the flow and temperature fields. The parametric study covers the range 10³ ⩽ Ra ⩽ 10⁶ and is done at three Pr namely, 0.071, 0.71 and 7.1. In addition the effect of increasing the size (characterized by the solidity Φ) of the adiabatic block is ascertained. It is found that the wall heat transfer increases, with increase in the Φ, until it reaches a critical value Φ = Φ_OPT, where the wall heat transfer attains its maximum. Further increases in the block size beyond Φ_OPT, reduces the wall heat transfer, for as the block size becomes larger than the conduction dominant core size it reduces the thermal mass of the convecting fluid. A steady-state heat transfer enhancement of 10% is observed for certain Ra and Pr values. Useful correlations predicting this optimum block size and the corresponding maximum heat transfer as a function of Ra and Pr are proposed; these predict within ±3%, the numerical results.     

Unsteady conjugate natural convection in a square enclosure filled with a porous medium

Mathematical simulation of unsteady natural convection modes in a square cavity filled with a porous medium having finite thickness heat-conducting walls with local heat source in conditions of heterogeneous heat exchange with an environment at one of the external boundaries has been carried out. Numerical analysis was based on Darcy–Forchheimer model in dimensionless variables such as a stream function, a vorticity vector and a temperature. The special attention was given to analysis of Rayleigh number effect Ra = 10⁴, 10⁵, 10⁶, of Darcy number effect Da = 10^?5, 10^?4, 10^?3, ∞, of the transient factor effect 0 < τ < 1000 and of the heat conductivity ratio k_2,1 = 3.7 × 10^?2, 5.7 × 10^?4, 6.8 × 10^?5 on the velocity and temperature fields. The influence scales of the defining parameters on the average Nusselt number have been detected.     

Natural convection in an enclosure with a localized nonuniform heat source on the bottom wall

Natural convection in an air filled enclosure with a localized nonuniform heat source mounted centrally on the bottom wall is numerically investigated. The vertical walls are cooled while the top wall and the remaining portions of the bottom wall are insulated. The heat source is assumed to be isothermal with a linearly varying temperature. The governing equations were solved using finite volume method on a uniformly staggered grid system. The computational results are presented in the form of isotherm and streamline plots and Nusselt numbers. The effects of the source nonuniformity parameter, λ and the line source length, ε are investigated for the Grashof numbers Gr = 10⁶ and 10⁷. It is found that for Gr = 10⁶ nonuniform heating of the line source enhances the overall heat transfer rate markedly compared to uniform heating of the heat source whereas for Gr = 10⁷ its effect is marginal.     

The effect of the top and bottom wall temperatures on the laminar natural convection in an air-filled square cavity

The effect of the top and bottom wall temperatures on the natural convection heat transfer characteristics in an air-filled square cavity driven by a difference in the vertical wall temperatures was investigated by measuring the temperature distributions along the heated vertical wall and visualizing the flow patterns in the cavity. The experiments were performed at a horizontal Grashof number of 1.9 × 10⁸. Increasing the top wall temperature resulted in a separated flow region on the top wall, which caused a secondary flow between the separated flow and the boundary layer on the heated vertical wall. This secondary flow had a significant effect on the heat transfer in this region. Changes in the top and bottom wall temperatures changed the temperature gradient and the average temperature of the air outside the thermal boundary layers in the cavity. The local heat transfer along much of the heated vertical wall could be correlated by Nu = C · Ra^0.32, but the constant C increased when the average of the top and bottom wall temperatures increased.     

Flow visualization of natural convection in a tall,air-filled vertical cavity

Natural convection of air in a tall vertical cavity was studied using a smoke patterns and interferometry. Experiments covered Rayleigh numbers of 4850 < Ra < 54,800 and aspect ratio A ≈ 40. Secondary cells were noted at Ra as low as 6228. The flow was stable at Ra < 10⁴. As Ra exceeded 10⁴ the flow became irregular, the core flow became increasingly unsteady and 3-D motion became evident. Interferometry showed that most of the temperature drop exists in boundary layers near the walls. The core is well mixed and of relatively uniform temperature with little or no vertical stratification.     

MHD natural convection in an enclosure from two semi-circular heaters on the bottom wall

A computational study of the thermal and dynamical behavior of fluid in an enclosure with two isothermal semi-circular heaters is presented. The top wall and the flat surfaces on bottom wall are adiabatic while the vertical walls are kept at lower temperature than the semi-circular heaters. The radius of curvature of the semi-circular surfaces is chosen as one tenth of the cavity wall length. The governing equations are solved by the Galerkin weighted residual finite element method. The effect of magnetic field on the flow is another important parameter in this study. Numerical simulations were performed for several values of Rayleigh number (10³ ? Ra ? 10⁶), Hartmann number (0 ? Ha ? 50) and the distance between two semi-circular heaters (0.2 ? D ? 0.8). In all cases the Prandtl number is taken as 7. It is found that the distance between the semi-circular heaters is the most important parameter affecting the heat and fluid flow fields. In addition, Hartmann number was found to have an adverse affect on heat transfer.     

Treatment of turbulent heat fluxes with the elliptic-blending second-moment closure for turbulent natural convection flows

In this paper a comparative study on the treatment of the turbulent heat fluxes with the elliptic-blending second-moment closure for natural convection flows is presented. Three different cases for treating the turbulent heat fluxes are considered. Those are the generalized gradient diffusion hypothesis (GGDH), the algebraic flux model (AFM) and the differential flux model (DFM). These models are implemented in a computer code especially designed for an evaluation of turbulent models. Calculations are performed for turbulent natural convection flows in an 1:5 rectangular cavity (Ra = 4.3 × 10¹⁰) and in a square cavity with conducting top and bottom walls (Ra = 1.58 × 10⁹). The calculated results are compared with the available experimental data. The results show that the GGDH, AFM and DFM models produce sufficiently accurate solutions for the turbulent natural convection in an 1:5 rectangular cavity where the strength of the thermal stratification is weak in a central region of the cavity. However, the GGDH model produces very erroneous solutions for the turbulent natural convection in a square cavity with conducting walls where the Rayleigh number is relatively small and the thermally stratified region is dominant. The AFM and DFM produce very accurate solutions for both cases without invoking any numerical problems.     

Numerical study of heat transfer by natural convection and surface thermal radiation in an open cavity receiver

In this work the numerical results of natural convection and surface thermal radiation in an open cavity receiver considering large temperature differences and variable fluid properties are presented. Numerical calculations were conducted for Rayleigh number (Ra) values in the range of 10⁴–10⁶. The temperature difference between the hot wall and the bulk fluid (ΔT) was varied between 100 and 400 K, and was represented as a dimensionless temperature difference (φ) for the purpose of generalization of the trends observed. Noticeable differences are observed between the streamlines and temperature fields obtained for φ = 1.333 (ΔT = 400 K) and φ = 0.333 (ΔT = 100 K). The total average Nusselt number in the cavity increased by 79.8% (Ra = 10⁶) and 88.0% (Ra = 10⁴) as φ was varied from 0.333 to 1.333. Furthermore the results indicate that for large temperature differences (0.667 ? φ ? 1.333) the radiative heat transfer is more important that convective heat transfer.     

Simulation of turbulent natural convection in a porous cylindrical annulus using a macroscopic two-equation model

This work presents numerical computations for laminar and turbulent natural convection within a horizontal cylindrical annulus filled with a fluid saturated porous medium. Computations covered the range 25 < Ra_m < 500 and 3.2 × 10⁻⁴ > Da > 3.2 × 10⁻⁶ and made use of the finite volume method. The inner and outer walls are maintained at constant but different temperatures. The macroscopic k–ε turbulence model with wall function is used to handle turbulent flows in porous media. First, the turbulence model is switched off and the laminar branch of the solution is found when increasing the Rayleigh number, Ra_m. Subsequently, the turbulence model is included and calculations start at high Ra_m, merging to the laminar branch for a reducing Ra_m. This convergence of results as Ra_m decreases can be seen as an estimate of the so-called laminarization phenomenon. Here, a critical Rayleigh number was not identified and results indicated that when the porosity, Prandtl number, conductivity ratio between the fluid and the solid matrix and Ra_m are kept fixed, the lower the Darcy number, the higher is the difference of the average Nusselt number given by the laminar and turbulent models.     

Role of finite element based grids and simulations on evaluation of Nusselt numbers for heatfunctions within square and triangular cavities involving multiple discrete heaters

Nusselt number is an important non-dimensional parameter which quantifies the heat transfer rate. Local Nusselt number is useful in predicting the heat transfer rate along the various hot and cold sections of the side walls in a discretely heated enclosed cavity. In addition, the overall heat balance in an enclosed cavity (total heat delivered by the hot isothermal walls should be equal to the total heat gained by the cold isothermal walls) can be validated via the average Nusselt numbers. Current finite element based simulations and post-processing have been carried out in order to analyze the influence of the multiple heaters on the Nusselt number along various sections (hot and cold) of the side walls in discretely heated square and triangular (design 1 and design 2) cavities. The working fluid is considered to be air (Pr = 0.7) and the numerical studies have been carried out for a large range of Rayleigh number (Ra = 10³–10⁵) for four different biquadratic elements (24 × 24, 28 × 28, 32 × 32 and 34 × 34). The current work also estimates the fractional error in the heat balance (ϵ) and it is clearly inferred that ϵ is comparatively lower for 34 × 34 biquadratic elements. Current work also reveals that the fractional error (ϵ) is mainly induced due to the sharp variations in the Nusselt number at the cold-hot junctions along the side walls. The present study also involves the detailed evaluation of the heatfunction (Π) expressions along the cold-hot junctions of the side walls. The computations of the heatfunctions are intrinsically related to the Nusselt numbers of the hot-cold junctions.     

Critical condition for Rayleigh-Bénard convection of Bingham fluids in rectangular enclosures

Two-dimensional steady-state numerical simulations have been conducted for laminar Rayleigh-Bénard convection of Bingham fluids in rectangular enclosures to analyse the critical Rayleigh number Ra_crit for which convection ceases to influence the thermal transport and thermal conduction becomes the principal heat transfer mechanism. The influences of Bingham number Bn on the critical Rayleigh number Ra_crit have been investigated for different values of aspect ratio (height: length) AR (ranging from 1/4 to 4) and nominal Prandtl number Pr (ranging from 10 to 500) for both constant wall temperature (CWT) and constant wall heat flux (CWHF) boundary conditions for the horizontal walls. It has been found that Ra_crit increases with increasing values of Bn and AR, regardless of the boundary condition. The values of Ra_crit have been found to be greater in the case of CWT boundary condition than in the CWHF configuration for AR ≤ 1, whereas an opposite trend is obtained for AR > 1 for Bingham fluids. Additionally, Ra_crit has been found be insensitive to the change of Pr for Newtonian fluids (i.e. Bn = 0), whereas Ra_crit increases with increasing Pr for Bingham fluids irrespective of the boundary condition. A detailed scaling analysis has also been performed to elucidate the effects of Bn ,Pr , AR on Ra_crit for Bingham fluids. The results of scaling analysis and numerical findings have been utilised to propose a new correlation for Ra_crit for both Newtonian and Bingham fluids in the case of both CWT and CWHF boundary conditions.