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°).     

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.     

Finite element based heatline approach to study mixed convection in a porous square cavity with various wall thermal boundary conditions

A penalty finite element method based simulation is performed to analyze the influence of various walls thermal boundary conditions on mixed convection lid driven flows in a square cavity filled with porous medium. The relevant parameters in the present study are Darcy number (Da = 10^?5 ? 10^?3), Grashof number (Gr = 10³ ? 10⁵), Prandtl number (Pr = 0.7–7.2), and Reynolds number (Re = 1–10²). Heatline approach of visualizing heat flow is implemented to gain a complete understanding of complex heat flow patterns. Patterns of heatlines and streamlines are qualitatively similar near the core for convection dominant flow for Da = 10^?3. Symmetric distribution in heatlines, similar to streamlines is observed irrespective of Da at higher Gr in natural convection dominant regime corresponding to smaller values of Re. A single circulation cell in heatlines, similar to streamlines is observed at Da = 10^?3 for forced convection dominance and heatlines are found to emanate from a large portion on the bottom wall illustrating enhanced heat flow for Re = 100. Multiple circulation cells in heatlines are observed at higher Da and Gr for Pr = 0.7 and 7.2. The heat transfer rates along the walls are illustrated by the local Nusselt number distribution based on gradients of heatfunctions. Wavy distribution in heat transfer rates is observed with Da ? 10^?4 for non-uniformly heated walls primarily in natural convection dominant regime. In general, exponential variation of average Nusselt numbers with Grashof number is found except the cases where the side walls are linearly heated. Overall, heatlines are found to be a powerful tool to analyze heat transport within the cavity and also a suitable guideline on explaining the Nusselt number variations.     

Heatline analysis of heat recovery and thermal transport in materials confined within triangular cavities

Heat recovery from hot fluids in material processing industries is important for environmental and thermal management. Present work involves numerical visualization of heat flow in entrapped cavities filled with hot materials. The concept of heatline is used to visualize the heat energy trajectory. The system involves entrapped triangular cavities filled with hot fluid (Pr = 0.015, 0.026, 0.7 and 1000). At low Rayleigh number (Ra), it is found that the heatlines are smooth and perfectly normal to the isotherms indicating the dominance of conduction for both the triangles. As Ra increases, flow slowly becomes convection dominant. Multiple heat flow circulations with high intensity are formed within the lower triangular domain especially for low Pr numbers, whereas, less intense convective heat flow circulations are observed for the upper triangle. Multiple circulations are absent for both the triangular domains involving fluids with higher Pr numbers. It is observed that the heat transfer rates are monotonic for the upper triangle whereas a few local maxima in heat transfer rates occur for smaller Pr within lower triangular domain. Overall, fluid with any Pr may be useful for enhanced heat transfer within the upper triangle but fluid with high Pr may be preferred for the lower triangle.     

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.     

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.     

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.     

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.     

Visualization of heat flow due to natural convection within triangular cavities using Bejan’s heatline concept

Natural convection and flow circulation within a cavity has received significant attention in recent times. The wide range of applicability of flow inside a cavity (food processing industries, molten metal industries, etc.) requires thorough understanding for cost efficient processes. This paper is based on comprehensive analysis of heat flow pattern using Bejan’s heatline concept. The key parameters for our study are the Prandtl number, Rayleigh number and Nusselt number. The values of Prandtl number (0.015, 0.026, 0.7 and 1000) have been chosen based on wide range of applicability. The Rayleigh number has been varied from 10² to 10⁵. Interesting results were obtained. For low Rayleigh number, it is found that the heatlines are smooth and perfectly normal to the isotherms indicating the dominance of conduction. But as Ra increases, flow slowly becomes convection dominant. It is also observed that multiple secondary circulations are formed for fluids with low Pr whereas these features are absent in higher Pr fluids. Multiple circulation cells for smaller Pr also correspond multiple cells of heatlines which illustrate less thermal transport from hot wall. On the other hand, the dense heatlines at bottom wall display enhanced heat transport for larger Pr. Further, local heat transfer (Nu_l, Nu_t) are explained based on heatlines. The comprehensive analysis is concluded with the average Nusselt number plots. A correlation for average heat transfer rate and Ra has been developed and the range of Rayleigh number is also found, to depict the conduction dominant heat transfer.     

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.     

Turbulent Rayleigh–Bénard convection of water in cubical cavities: A numerical and experimental study

Experimental measurements and numerical simulations of natural convection in a cubical cavity heated from below and cooled from above are reported at turbulent Rayleigh numbers using water as a convective fluid (Pr = 6.0). Direct numerical simulations were carried out considering the Boussinesq approximation with a second-order finite volume code (10⁷ ⩽ Ra ⩽ 10⁸). The particle image velocimetry technique was used to measure the velocity field at Ra = 10⁷, Ra = 7 × 10⁷ and Ra = 10⁸ and there was general agreement between the predicted time averaged local velocities and those experimentally measured if the heat conduction through the sidewalls was considered in the simulations.     

Vortex formation and heat transfer in turbulent flow past a transverse cavity with inclined frontal and rear walls

The process of vortex formation, distributions of pressure coefficients, and convective heat transfer in a turbulent flow past a cavity with a low aspect ratio and inclined frontal and rear walls were experimentally studied. The angle of wall inclination φ was varied in the interval from 30° to 90°. Visualization techniques were applied to trace the evolution of the flow with the angle φ as the transverse cavity became more open. Pressure fields in the longitudinal and transverse sections on the bottom wall of the cavity, and on its frontal and rear walls, were measured. The measured distributions of temperature in the longitudinal and transverse sections on the three heated walls, and the obtained thermographic fields over the whole heated surface, were used to calculate local and average heat-transfer coefficients. It is found that in the interval of wall inclination angles φ = 60–70° the flow in the cavity becomes unstable, with the primary vortex changing its structure from single-cellular to double-cellular. As a result, the distributions of static pressure and surface temperature across and along the cavity suffer dramatic changes. At smallest angles φ the flow re-attachment point gets displaced into the cavity to cause an abrupt growth of pressure and heat-transfer coefficients on the rear wall, which leads to a slight increase of the surface-mean pressure and heat transfer inside the cavity. At the angle of instability, φ = 60°, the local heat-transfer coefficient decreases markedly over the cavity span from the end faces of the cavity toward its center, and a most pronounced intensification of heat transfer is observed.     

Natural convection and fluid flow in inclined enclosure with a corner heater

The phenomena of natural convection in an inclined square enclosure heated via corner heater have been studied numerically. Finite difference method is used for solving momentum and energy equations in the form of stream function–vorticity. One wall of the enclosure is isothermal but its temperature is colder than that of heaters while the remaining walls are adiabatic. The numerical procedure adopted in this analysis yields consistent performance over a wide range of parameters; Rayleigh number, Ra (10³ ? Ra ? 10⁶); Prandtl number, Pr (0.07 ? Pr ? 70); dimensionless lengths of heater in x and y directions (0.25 ? hx ? 0.75, 0.25 ? hy ? 0.75); and inclination angle, ? (0° ? ? ? 270°). It is observed that heat transfer is maximum or minimum depending on the inclination angle and depending on the length of the corner heaters. The effect of Prandtl number on mean Nusselt number is more significant for Pr < 1.     

Heat transfer enhancement in cubical enclosures with vertical fins

Natural convection of air in a cubical enclosure with a thick partition fitted vertically on the hot wall is numerically investigated for Rayleigh numbers of 10³–10⁶. A three dimensional convective circulation is generated, in which the cold flow sweeps the fin faces and the hot wall, with low flow blockage. The combined contributions of these faces cause heat transfer enhancements over 40% at high Rayleigh numbers and thermal conductivity ratios (R_k). These enhancements significantly exceed the ones obtained with horizontal fins. Even low R_k values cause heat transfer enhancements, except at Ra = 10⁴.     

Effect of optical properties on oscillatory hydromagnetic double-diffusive convection within semitransparent fluid

The effect of radiative heat transfer on the hydromagnetic double-diffusive convection in two-dimensional rectangular enclosure is studied numerically for fixed Prandtl, Rayleigh, and Lewis numbers, Pr = 13.6, Ra = 10⁵, Le = 2. Uniform temperatures and concentrations are imposed along the vertical walls while the horizontal walls are assumed to be adiabatic and impermeable to mass transfer. The influences of the optical thickness and scattering albedo of the semitransparent fluid on heat and mass transfer with and without magnetic damping are depicted. When progressively varying the optical thickness, multiple solutions are obtained which are steady or oscillatory accordingly to the initial conditions. the mechanisms of the transitions between steady compositionally dominated flow and unsteady thermally dominated flow are analyzed.     

Bejan’s heatline analysis of natural convection in right-angled triangular enclosures: Effects of aspect-ratio and thermal boundary conditions

Natural convection in right-angled triangular enclosures with various top angles ($\phi$=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 = 10³–10⁵. 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 $\phi$. 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 $\overline{Nu}$ are adequately explained based on heatlines. Finally, the correlations for $\overline{N{u}_l}$ and Ra are obtained for various fluid with all heating situations.     

Numerical simulation of liquid metals in differentially heated enclosure undergoing orthogonal rotation

In this work a numerical investigation has been performed to examine the characteristics of mixed convective heat transfer in square enclosures undergoing orthogonal rotation i.e. rotation axis and gravity axis are orthogonal to each other. A semi implicit finite difference code on a collocated grid is used to solve the momentum and energy equations subject to Boussinesq approximation. The study is carried out for a wide range of operating parameters such as Rayleigh number (Ra), Taylor number (Ta), Rotational Rayleigh number (Ra_w) for a fixed Prandtl number (Pr). The numerical experiments have been carried out for a fixed Pr = 0.01, Ra varies from 10⁵ to 10⁷ while Ta and Ra_w vary from almost 0 to 10⁹. Results reveal that significant increase or decrease in heat transfer rates can be achieved by the rotational effects, mainly influenced by centrifugal force.     

Heatline based thermal management for natural convection within right-angled porous triangular enclosures with various thermal conditions of walls

Analysis of natural convection in porous triangles have many important energy related applications in geophysical and solar energy fields. A numerical study on heat distribution and thermal mixing during steady laminar natural convective flow inside a right-angled triangular enclosure filled with porous media subjected to various wall boundary conditions is investigated in this study using Bejan’s heatlines approach. Influence of various thermal boundary conditions and inclination angles (φ) on evaluation of complex heat flow patterns are studied as a function of Darcy numbers (Da) for various regimes of Prandtl (Pr) and Rayleigh (Ra) numbers. Studies illustrate that maximum heat transfer occurs at the top vertex for lower top angle (φ=^°15) at higher Da(Da=10⁻³). As φ increases to ^°45, the maximum heat flux at the top vertex decreases and thermal mixing increases irrespective of Da and Pr. The enhanced convection at higher Da significantly affects the heat flow distribution, which is clearly depicted by high local Nusselt numbers at Da=10⁻³. It is also found that isothermal heating of walls enhances the heat distribution and thermal mixing. Overall, it is shown that heatlines provide suitable guideline on thermal management in porous right-angled triangular enclosures with various heating strategies.     

Natural convective flow in an inclined lid-driven enclosure with a heated thin plate in the middle

Natural convection heat transfer from a heated thin plate located in the middle of a lid-driven inclined square enclosure has been analyzed numerically. Left and right of the cavity are adiabatic, the two horizontal walls have constant temperature lower than the plate’s temperature. The study is formulated in terms of the vorticity-stream function procedure and numerical solution was performed using a fully higher-order compact (FHOC) finite difference scheme on the 9-point 2D stencil. Air was chosen as a working fluid (Pr = 0.71). Two cases are considered depending on the position of heated thin plate (Case I, horizontal position; Case II, vertical position). Governing parameters, which are effective on flow field and temperature distribution, are Rayleigh number values (Ra) ranging from 10³ to 10⁵ and inclination angles γ (0° ? γ < 360°). The fluid flow, heat transfer and heat transport characteristics were illustrated by streamlines, isotherms and Nusselt number (Nu). It is found that fluid flow and temperature fields strongly depend on Rayleigh numbers and inclination angles. Further, for the vertical located position of thin plate heat transfer becomes more enhanced with lower γ at various Rayleigh numbers.     

Steady natural convection flows in a square cavity with linearly heated side wall(s)

The present numerical study deals with natural convection flow in a closed square cavity when the bottom wall is uniformly heated and vertical wall(s) are linearly heated whereas the top wall is well insulated. Non-linear coupled PDEs governing the flow have been solved by penalty finite element method with bi-quadratic rectangular elements. Numerical results are obtained for various values of Rayleigh number (Ra) (10³ ⩽ Ra ⩽ 10⁵) and Prandtl number (Pr) (0.7 ⩽ Pr ⩽ 10). Results are presented in the form of streamlines, isotherm contours, local Nusselt number and the average Nusselt as a function of Rayleigh number.