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.     

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.     

Analysis of entropy generation for distributed heating in processing of materials by thermal convection

Analysis of entropy generation has been carried out for square cavities with distributed heated sources filled with various materials involving wide range of Pr(=0.015, 0.7, 10, 1000) during the conduction and convection regime within Ra(=10³ ? 10⁵). Entropy generation terms involving thermal and velocity gradients are evaluated accurately based on elemental basis set via Galerkin finite element method. Local entropy maps are analyzed in detail for various cases and the dominance of thermal and frictional irreversibilities is studied via average Bejan number. The heat transfer irreversibility is found to dominate during conduction regime while the fluid friction irreversibility dominates the entropy generation in the convection regime, except for the low Pr fluid based on the heating configuration of the cavity. Further, the variation of total entropy generation has been observed to be similar for different heating configurations for higher Pr fluids (=10, 1000) whereas, the configuration of cavity has been found to have little effect on total entropy generation for fluids with Pr = 0.7 during both conduction and convection regimes. Thermal mixing and degree of temperature uniformity due to distributed heating in various cases are also reported and optimum cases for processing of various fluids are presented based on minimum entropy generation.     

Numerical visualization of mass and heat transport for conjugate natural convection/heat conduction by streamline and heatline

The method of numerical visualization of mass and heat transport for convective heat transfer by streamlines and heatlines are comprehensively studied. Functions are directly defined in terms of dimensionless governing equations or variables. Some basic characteristics of the functions are illustrated in detail, knowledge of which is essential to perceive the results and the philosophy of heat and fluid flow. The consistency of the formulations is especially addressed when dealing with conjugate convection/conduction problem. The functions/lines are unified for both fluid and solid regions, and the diffusion coefficients of the function equations are invariant. The method has been used to visualize the heat and fluid flow structures for natural convection in an air (Pr=0.71) filled square cavity over a wide range of Ra=10³−10⁶, and those for conjugate natural convection/heat conduction problem where the conduction effect of solid body on heat transfer is investigated. As to exhibiting the nature of convective heat transfer, streamlines and heatlines provide a more practical and efficient means to visualize the results than the customary ways.     

Investigation of buoyancy-driven flow and heat transfer in a trapezoidal cavity filled with water–Cu nanofluid

A numerical study is conducted to investigate the transport mechanism of free convection in a trapezoidal enclosure filled with water–Cu nanofluid. The horizontal walls of the enclosure are insulated while the inclined walls are kept at constant but different temperatures. The numerical approach is based on the finite element technique with Galerkin's weighted residual simulation. Solutions are obtained for a wide range of the aspect ratio (AR) and Prandtl number (Pr) with Rayleigh number (Ra = 10⁵) and solid volume fraction (? = 0.05). The streamlines, isotherm plots and the variation of the average Nusselt number at the left hot wall are presented and discussed. It is found that both AR and Pr affect the fluid flow and heat transfer in the enclosure. A correlation is also developed graphically for the average Nusselt number as a function of the Prandtl number as well as the cavity aspect ratio.     

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.     

Computation of turbulent free convection in left and right tilted porous enclosures using a macroscopic k–ε model

Comparisons of computations for turbulent natural convection within clockwise and counter-clockwise inclined cavities, filled with a fluid saturated porous medium, are presented. The finite volume method in a generalized coordinate system is applied. Oblique walls are maintained at constant but different temperatures, whereas horizontal surfaces are kept insulated. Flow and heat transfer characteristics are investigated for Rayleigh number up to 10⁴ and inclination angles up to 45°, in both directions of rotation. Turbulent is handled using a macroscopic two-equation model with a wall function. In this work, the turbulence model is first switched off and the laminar branch of the solution is obtained. Subsequently, the turbulence model is included and the solution merges to the laminar branch for a reducing value of Ra_m. Present computations are compared with published results and the influence of the inclination angle on Ra_cr is analyzed, for both the left and right rotating directions. For Ra_m greater than around 10⁴, both laminar and turbulent flow solutions deviate, possibly indicating that a critical value for Ra_m was reached. Both left and right rotation of the hot wall reduce Nu, but rotating the hot wall on the counter-clockwise direction decreases Nu at a faster rate than when bending the cavity to the right.     

Role of various moving walls on entropy generation during mixed convection within entrapped porous triangular cavities

The aim of the present investigation is to analyze the effect of the motion of horizontal walls on the entropy generation and heat transfer rates in an entrapped triangular porous cavity during mixed convection. Two different thermal boundary conditions are considered as follows: (i) hot inclined walls and cold horizontal walls and (ii) cold inclined walls and hot horizontal walls. Overall, Re?=?100 may be recommended at Pr_m?=?0.026, 7.2, Gr?=?10⁵, and Da_m?=?10^?4 to 10^?2 within the upper and lower cavities for cases 1 and 2.     

Natural convection in triangular enclosures with protruding isothermal heater

Natural convection heat transfer from a protruding heater located in a triangular enclosure has been analyzed numerically. Temperature of inclined boundary of the triangle is lower than the temperature of the heater, which has constant temperature boundary condition. The remaining walls are insulated. The study is formulated in terms of the vorticity-stream function procedure and numerical solution was performed using the finite difference method. Air was chosen as working fluid with Pr = 0.71. Governing parameters, which are effective on flow field and temperature distribution, are; Rayleigh number, aspect ratio of triangle enclosure, dimensionless height of heater, dimensionless location of heater and dimensionless width of heater. Streamlines, isotherms, velocity profiles, local and mean Nusselt numbers are presented. It is found that all parameters related with geometrical dimensions of the heater are effective on temperature distribution, flow field and heat transfer.     

Prandtl number effect on cooling performance of a heated cylinder in an enclosure filled with nanofluid

Natural convection heat transfer from a heated cylinder contained in a square enclosure filled with water–Cu nanofluid is investigated numerically. The main objective of this study is to explore the influence of pertinent parameters such as Prandtl number (Pr) and diameter (D) of the heated body on the flow and heat transfer performance of nanofluids while Rayleigh number (Ra) and the solid particle volume fraction (?) of nanoparticle are considered fixed. The results obtained from finite element method clearly indicate that heat transfer augmentation is possible using highly viscous nanofluid resulting in the compactness of many industrial devices.     

Natural convection for hot materials confined within two entrapped porous trapezoidal cavities

This paper analyzes the detailed heat transfer and fluid flow within two entrapped porous trapezoidal cavities involving cold inclined walls and hot horizontal walls. Flow patterns and temperature distribution were obtained by solving numerically the governing equations, using Darcy's law. Results are presented for different values of the governing parameters, such as Darcy-modified Rayleigh number, aspect ratio of two entrapped trapezoidal cavities and thermal conductivity ratio between the middle horizontal wall and fluid medium. Heat transfer rates are estimated in terms of local and mean Nusselt numbers. Local Nusselt numbers with spatial distribution exhibit monotonic trend irrespective of all Rayleigh numbers for the upper trapezoidal whereas wavy distribution of local Nusselt number occur for the lower trapezoidal.     

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°).     

The effects of Prandtl number on natural convection in triangular enclosures with localized heating from below

The effect of Prandtl number on natural convection heat transfer and fluid flow in triangular enclosures with localized heating has been analyzed by solving governing equations of natural convection in streamfunction–vorticity form with finite-difference technique. Solution of linear algebraic equations was made by Successive Under Relaxation (SUR) method. Bottom wall of triangle is heated partially while inclined wall is maintained at a lower uniform temperature than heated wall while remaining walls are insulated. Computations were carried out for dimensionless heater locations (0.15 ≤ s ≤ 0.95), dimensionless heater length (0.1 ≤ w ≤ 0.9), Prandtl number (0.01 ≤ Pr ≤ 15) and Rayleigh number (10³ ≤ Ra ≤ 10⁶). Aspect ratio of triangle was chosen as unity. It is observed that both flow and temperature fields are affected with the changing of Prandtl number, location of heater and length of heater as well as Rayleigh number.     

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.     

Role of moving horizontal walls on entropy generation during mixed convection within entrapped triangular cavities

A comprehensive analysis based on the irreversibilities associated with the energy flow and entropy generation is highly essential for the optimization of thermal systems. Entropy generation during mixed convection process has been studied in entrapped triangular cavities for moving horizontal walls involving isothermally hot inclined walls and cold horizontal walls (case 1) or isothermally cold inclined walls and hot horizontal walls (case 2). Overall it is found that, Re = 100 may be preferred over Re → 0, Re = 1 and Re = 10 at Pr = 0.026 and 7.2, Gr = 10³ − 10⁵ within the cavities, irrespective of the cases. In addition to Re = 100, Re = 10 may be optimal for the upper cavity with case 1 and lower cavity with case 2 at Gr ≈ 10⁵ (higher Gr regime) and Pr = 7.2 based on moderate heat transfer rates.     

Fully developed mixed convection flow in a vertical channel filled with nanofluids

In this paper, the fully developed mixed convection flow in a vertical channel filled with nanofluids is investigated. Analytical solutions for both the buoyancy-assisted and -opposed flow are obtained. Further analysis shows that the analytical solution for the opposing flow is only valid for a certain region of the Rayleigh number Ra in physical sense. Besides, the effects of the nanoparticle volume fraction φ on the temperature and the velocity distributions are then exhibited. It is confirmed that the nanoparticle volume fraction φ plays a key role for improving the heat and mass transfer characteristics of the fluids.     

Natural convection in a partially open square cavity with internal heat source: An analysis of the opening mass flow

A steady buoyancy-driven flow of air in a partially open square 2D cavity with internal heat source, adiabatic bottom and top walls, and vertical walls maintained at different constant temperatures is investigated numerically in this work. A heat source with 1% of the cavity volume is present in the center of the bottom wall. The cold right wall contains a partial opening occupying 25%, 50% or 75% of the wall. The influence of the temperature gradient between the verticals walls was analyzed for Ra_e = 10³–10⁵, while the influence of the heat source was evaluated through the relation R = Ra_i/Ra_e, investigated at between 400 and 2000. Interesting results were obtained. For a low Rayleigh number, it is found that the isotherm plots are smooth and follow a parabolic shape indicating the dominance of the heat source. But as the Ra_e increases, the flow slowly becomes dominated by the temperature difference between the walls. It is also observed that multiple strong secondary circulations are formed for fluids with a small Ra_e whereas these features are absent at higher Ra_e. The comprehensive analysis is concluded with horizontal air velocity and temperature plots for the opening. The numerical results show a significant influence of the opening on the heat transfer in the cavity.     

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.