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.

Tanmay Basak

Department Of Chemical Engineering

Pratibha Biswal

Average heat transfers

GALERKIN FINITE ELEMENT METHODS

Governing equations

NON-DARCY MODELS

POROUS MATRIXS

Thermal boundary conditions

THERMAL MIXING

TRIANGULAR ENCLOSURE

flow visualization

Isotherms

Enclosures

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Analysis of Convective Heat Flow Visualization within Porous Right Angled Triangular Enclosures with a Concave/Convex Hypotenuse

Numerical Heat Transfer; Part A: Applications

Purpose: The purpose of this paper is to study thermal (natural) convection in nine different containers involving the same area (area= 1 sq. unit) and identical heat input at the bottom wall (isothermal/sinusoidal heating). Containers are categorized into three classes based on geometric conﬁgurations [Class 1 (square, tilted square and parallelogram), Class 2 (trapezoidal type 1, trapezoidal type 2 and triangle) and Class 3 (convex, concave and triangle with curved hypotenuse)]. Design/methodology/approach: The governing equations are solved by using the Galerkin ﬁnite element method for various processing fluids (Pr = 0.025 and 155) and Rayleigh numbers (103 ≤ Ra ≤ 105) involving nine different containers. Finite element-based heat flow visualization via heatlines has been adopted to study heat distribution at various sections. Average Nusselt number at the bottom wall ((Formula presented.)) and spatially average temperature (θ^) have also been calculated based on ﬁnite element basis functions. Findings: Based on enhanced heating criteria (higher (Formula presented.) and higher θ^), the containers are preferred as follows, Class 1: square and parallelogram, Class 2: trapezoidal type 1 and trapezoidal type 2 and Class 3: convex (higher θ^) and concave (higher (Formula presented.)). Practical implications: The comparison of heat flow distributions and isotherms in nine containers gives a clear perspective for choosing appropriate containers at various process parameters (Pr and Ra). The results for current work may be useful to obtain enhancement of the thermal processing rate in various process industries. Originality/value: Heatlines provide a complete understanding of heat flow path and heat distribution within nine containers. Various cold zones and thermal mixing zones have been highlighted and these zones are found to be altered with various shapes of containers. The importance of containers with curved walls for enhanced thermal processing rate is clearly established. © 2019, Emerald Publishing Limited.

International Journal of Numerical Methods for Heat and Fluid Flow

Numerical heat flow visualization analysis on enhanced thermal processing for various shapes of containers during thermal convection

The present study deals with the finite element based numerical simulations of heat transfer and entropy generation rates during natural convection for fluid saturated porous media in enclosures involving curved walls (case 1: lower curvature and case 2: higher curvature) with various thermal boundary conditions. The differential heating (isothermally hot left wall and cold right wall and adiabatic horizontal walls) and Rayleigh-Bénard heating (isothermally hot bottom wall and cold top wall involving adiabatic left and right walls) are considered. The locations and magnitudes of the entropy generation due to heat transfer (Sθ) and fluid friction (Sψ) are presented and discussed based on the spatial distributions of isotherms and streamlines, respectively. The magnitudes of local entropy generation (Sθ,Sψ), total entropy generation (Stotal) and average heat transfer rates (Nur‾ and Nut‾) are significantly lesser for the Rayleigh-Bénard heating compared to the differential heating for all the cases involving all Dam and Prm. The Rayleigh-Bénard heating is the optimal strategy for all Dam and Prm involving both the concave cases except for 10-3⩽Dam⩽10-2,Prm=10 and case 1 (concave) domain. The Rayleigh-Bénard heating is also the optimal strategy compared to the differential heating involving the convex cases at 10-5⩽Dam⩽10-4 whereas the differential heating is the optimal heating strategy for Dam⩾10-3 involving both Prm for the convex cases. © 2018

International Journal of Heat and Mass Transfer

Role of differential vs Rayleigh-Bénard heating at curved walls for efficient processing via entropy generation approach

Natural convection within enclosures in the presence of isothermal curved walls involving (a) differential heating and (b) Rayleigh–Bénard heating is considered for investigation. The two heating strategies are compared based on the distributions of fluid flow, heat flow and local or average heat transfer rates for various Dam (modified Darcy number) and Prm (modified Prandtl number) at the high Ram (modified Rayleigh number). Unidirectional circulation cells occur for differential heating whereas four (combination of clockwise and anticlockwise) circulation cells occur for Rayleigh–Bénard heating. Consequently, unidirectional heatline cell occurs at the center for differential heating and four heatline circulation cells occur for Rayleigh–Bénard heating specially at the high (Formula presented.). Multiple heatline cells lead to larger thermal mixing for Rayleigh–Bénard heating. The heat transfer rate is larger for differential heating case and the percentage enhancement of Nusselt number is calculated in terms of the gain in heat transfer rate for differential heating (E). The gain in heat transfer rate in terms of E involving concave and convex cavities (cases 1 and 2) is found to be the strong function of Dam and Prm at the high Ram. © 2018, © 2018 Taylor & Francis.

Analysis of differential versus Rayleigh–Bénard heating via heat flow visualization for thermal convection due to heating at enclosures with concave/convex walls

Purpose: This paper is aimed to study natural convection in enclosures with curved (concave and convex) side walls for porous media via the heatline-based heat flow visualization approach. Design/methodology/approach: The numerical scheme involving the Galerkin finite element method is used to solve the governing equations for several Prandtl numbers (Prm) and Darcy numbers (Dam) at Rayleigh number, Ram = 106, involving various wall curvatures. Finite element method is advantageous for curved domain, as the biquadratic basis functions can be used for adaptive automated mesh generation. Findings: Smooth end-to-end heatlines are seen at the low Dam involving all the cases. At the high Dam, the intense heatline cells are seen for the Cases 1-2 (concave) and Cases 1-3 (convex). Overall, the Case 1 (concave) offers the largest average Nusselt number ((Formula presented.)) at the low Dam for all Prm. At the high Dam, (Formula presented.) for the Case 1 (concave) is the largest involving the low Prm, whereas (Formula presented.) is the largest for Case 1 (convex) involving the high Prm. Practical implications: Thermal management for flow systems involving curved surfaces which are encountered in various practical applications may be complicated. The results of the current work may be useful for the material processing, thermal storage and solar heating applications Originality/value: The heatline approach accompanied by energy flux vectors is used for the first time for the efficient heat flow visualization during natural convection involving porous media in the curved walled enclosures involving various wall curvatures. © 2018, Emerald Publishing Limited.

Heatlines visualization of convective heat flow during differential heating of porous enclosures with concave/convex side walls

Natural convection in an enclosure (internal convection) is an important problem due to its significant practical applications. In energy related applications, natural convection plays a dominant role in transport of energy for the proper design of enclosures in order to achieve higher heat transfer rates. This review summarizes the studies on natural convection heat transfer in triangular, trapezoidal, parallelogrammic enclosures and enclosures with curved and wavy walls filled with fluid or porous media. In addition, this review also summarizes the natural convection studies in the nanofluid filled enclosures. Studies have been performed for the enclosures subjected to different thermal boundary conditions. A number of the studies demonstrated that the variation of the aspect ratio and base angle of the triangular and rhombic/parallelogrammic enclosures had a wide influence on the flow distribution pattern. In the trapezoidal enclosure, the aspect ratio of the cavity as well as the presence of the baffles along the walls played a significant role in the temperature and flow distribution. The flow patterns within the complex enclosures were found to be largely dependent on the amplitude-wavelength ratio and number of undulations of the wavy walls. In addition, the researchers have also studied the effect of the various parameters such as the Rayleigh numbers, Prandtl numbers, Darcy numbers, Darcy–Rayleigh number, irreversibility distribution ratios, volume fraction of the nanoparticles, etc. Overall, the current review paper presents an useful insight into the potential strategies for enhancing the convection heat transfer performance. © 2016 Elsevier Ltd

Studies on natural convection within enclosures of various (non-square) shapes – A review

Analysis has been carried out for energy distribution and thermal mixing in steady laminar natural convective flow through the porous rhombic cavities with inclination angle φ for various applications such as geothermal, grain storage, electronic cooling, etc. A generalized non-Darcy model without the Forchheimer inertia term is used to predict the flow in porous media. The effect of the Darcy number (Da) and the role of φ on the energy distribution and thermal mixing within porous rhombic cavities with isothermal (case 1) and nonisothermal (case 2) hot bottom walls are illustrated via " heatlines". Heat transfer is found to be primarily conduction dominant at Da = 10 -5 even at a higher Rayleigh number (Ra = 10 6). The onset of convection occurs at Da = 10 -4, and the distorted heatlines from the hot bottom wall take a longer path to reach the cold side walls of the cavity. Larger heat transfer and thermal mixing occurs for Da = 10 -3 at Ra = 10 6 irrespective of φ and Pr. Multiple flow/convective circulations are observed at Pr = 0.015 for all φ values at Da = 10 -3. On the other hand, two asymmetric flow circulation cells are found to occupy the entire cavity at Pr = 0.7, 7.2, and 1000 for φ = 75° at Da = 10 -3. The cavity with inclination angle φ = 30° enhances the convective heat transfer from the hot wall to the cold wall, and the heat transfer to the right cold wall is a maximum for φ = 75°, as depicted by "heatlines" irrespective of Pr at Da = 10 -3. Average Nusselt number studies based on heatfunction gradients also show that the cavity with φ = 30° gives a maximum heat-transfer rate from the bottom to the left wall irrespective of Pr in case 1 at Da = 10 -3. The cup mixing temperature (θ cup) is higher for case 1 compared to case 2, and it is almost invariant with φ for higher Pr (Pr = 7.2 and 1000) in case 1 at Da = 10 -3. © 2011 American Chemical Society.

Industrial and Engineering Chemistry Research

Heatline analysis for natural convection within porous rhombic cavities with isothermal/nonisothermal hot bottom wall

Bejan's heatlines approach has been introduced to visualize heat flow during natural convection within a tilted square cavity inclined at an angle of ψ=30°. The enclosure is bounded by hot wall AB (case 1: isothermal heating and case 2: nonisothermal heating), isothermally cooled walls DA and BC in the presence of adiabatic wall CD. The results are presented in terms of streamlines, isotherms, heatlines, and local and average Nusselt numbers. The nonisothermal heating case produces the greater heat transfer rate at the center of the wall AB compared to that of the isothermal heating case, whereas the average Nusselt number shows an overall lower heat transfer rate for the nonisothermal heating case. © 2012 Copyright Taylor and Francis Group, LLC.

Visualization of heat transport during natural convection in a tilted square cavity: Effect of isothermal and nonisothermal heating

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. © 2011 Elsevier Ltd.

Energy

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

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 (103≤Ra≤105), Prandtl number Pr (0.015≤Pr≤1000) and inclination angles (φ = 45°, 60° and 90°). Results signify that, at low Ra (Ra=103) heat transfer is conduction dominant. At Ra=105, 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 Pr=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. © 2011 Elsevier Masson SAS. All rights reserved.

International Journal of Thermal Sciences

A comprehensive heatline based approach for natural convection flows in trapezoidal enclosures: Effect of various walls heating

In this article, natural convection in a porous triangular cavity has been analyzed. Bejan's heatlines concept has been used for visualization of heat transfer. Penalty finite-element method with biquadratic elements is used to solve the nondimensional governing equations for the triangular cavity involving hot inclined walls and cold top wall. The numerical solutions are studied in terms of isotherms, streamlines, heatlines, and local and average Nusselt numbers for a wide range of parameters Da (10-5-10-3), Pr (0.015-1000), and Ra (Ra=103-5×105). For low Darcy number (Da=10-5), the heat transfer occurs due to conduction as the heatlines are smooth and orthogonal to the isotherms. As the Rayleigh number increases, conduction dominant mode changes into convection dominant mode for Da=10-3, and the critical Rayleigh number corresponding to the on-set of convection is obtained. Distribution of heatlines illustrate that most of the heat transport for a low Darcy number (Da=10-5) occurs from the top region of hot inclined walls to the cold top wall, whereas heat transfer is more from the bottom region of hot inclined walls to the cold top wall for a high Darcy number (Da=10-3). Interesting features of streamlines and heatlines are discussed for lower and higher Prandtl numbers. Heat transfer analysis is obtained in terms of local and average Nusselt numbers (Nu l, Nul) and the local and average Nusselt numbers are found to be correlated with heatline patterns within the cavity. Copyright © Taylor &amp; Francis Group, LLC.

Journal	Numerical Heat Transfer; Part A: Applications
ISSN	10407782
Open Access	No