An accurate prediction of the flow structure and heat distribution in rhombic configurations are of greater importance due to its significant engineering i.e. cooling of electronics devices as well as natural applications i.e. geothermal extraction. Heatline method is used to analyze natural convection in porous rhombic enclosures with various inclination angles,for differential (case 1) and Rayleigh-Benard heating situations (case 2). Increase in ( = 90°) results in pure conduction dominant heat transfer with stagnant fluid condition for= 90° at Da = 10-5 and a slight perturbation of at higher Da (Da ≥ 4 × 10-5) leads to convection based dynamic solution for = 90° in case 2 irrespective of Pr. At Da = 10 -3, strength of fluid and heat flow increase with due to enhanced convection effect and = 90° shows maximum magnitude of streamfunction (ψmax) and heatfunction (Πmax) values in both cases except convection based solution at = 90° for Pr = 7.2. Both cases are compared based on local (Nu) and average Nusselt numbers (Nu) and those are adequately explained based on heatlines. Also, Nu increases with Da in both cases except convection based solution at = 90° for Pr = 7.2. Overall, Nu is higher for case 2 at ≤ 45° whereas case 1 shows larger Nu for ≥ 45° irrespective of Pr at Da = 10-3. Hence, = 45° is the critical rhombic angle which demarcates the heating strategies of case 1 and case 2 to achieve higher heat transfer rates (Nu) in various applications. © 2012 Elsevier Ltd. All rights reserved.

Tanmay Basak

Department Of Chemical Engineering

DARCY NUMBER

DIFFERENTIAL HEATING

DYNAMIC SOLUTIONS

Heatline

Rayleigh-Benard convection

Electronic cooling

Geothermal energy

Natural convection

Nusselt number

Enclosures

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IIT Madras

Energy Conversion and Management

Visualization of heat flow and trajectories has been employed during buoyancy induced (natural) convection within porous rhombic 2D cavities of different base angles involving various intensities of the thermal state of the bottom wall via tuning thermal aspect ratios (Ar). The thermal mixing effect at the core, heat transport rates at side walls and heat supply rate by the bottom wall have been elucidated at various limits of Pr and Ar for low and high Darcy number regimes. The heatline tool has been found to be indispensable to analyze the convection heat transport at various locations and trajectories of heat propagation from the hot zone to cooler zone of walls involving various scenarios. © 2019, © 2019 Taylor & Francis Group, LLC.

Numerical Heat Transfer; Part A: Applications

Numerical visualization via heatlines for natural convection in porous bodies of rhombic shapes subjected to thermal aspect ratio-based heating of walls

Purpose: The purpose of the paper is to study natural convection within porous square and triangular geometries (design 1: regular isosceles triangle, design 2: inverted isosceles triangle) subjected to discrete heating with various locations of double heaters along the vertical (square) or inclined (triangular) arms. Design/methodology/approach: Galerkin finite element method is used to solve the governing equations for a wide range of modified Darcy number, Dam = 10−5–10−2 with various fluid saturated porous media, Prm = 0.015 and 7.2 at a modified Rayleigh number, Ram = 106 involving the strategic placement of double heaters along the vertical or inclined arms (types 1-3). Adaptive mesh refinement is implemented based on the lengths of discrete heaters. Finite element based heat flow visualization via heatlines has been adopted to study heat distribution at various portions. Findings: The strategic positioning of the double heaters (types 1-3) and the convective heatline vortices depict significant overall temperature elevation at both Dam = 10−4 and 10−2 compared to type 0 (single heater at each vertical or inclined arm). Types 2 and 3 are found to promote higher temperature uniformity and greater overall temperature elevation at Dam = 10−2. Overall, the triangular design 2 geometry is also found to be optimal in achieving greater temperature elevation for the porous media saturated with various fluids (Prm). Practical implications: Multiple heaters (at each side [left or right] wall) result in enhanced temperature elevation compared to the single heater (at each side [left or right] wall). The results of the current work may be useful for the material processing, thermal storage and solar heating applications. Originality/value: The heatline approach is used to visualize the heat flow involving double heaters along the side (left or right) arms (square and triangular geometries) during natural convection involving porous media. The heatlines depict the trajectories of heat flow that are essential for thermal management involving larger thermal elevation. The mixing cup or bulk average temperature values are obtained for all types of heating (types 0-3) involving all geometries, and overall temperature elevation is examined based on higher mixing cup temperature values. © 2019, Emerald Publishing Limited.

International Journal of Numerical Methods for Heat and Fluid Flow

Analysis of efficiency of convection in porous geometries (square vs triangular) with multiple discrete heaters on walls: A heatline perspective

Purpose - This study aims to carry out the analysis of Rayleigh-Bénard convection within enclosures with curved isothermal walls, with the special implication on the heat flow visualization via the heatline approach. Design/methodology/approach - The Galerkin finite element method has been used to obtain the numerical solutions in terms of the streamlines (ψ), heatlines (Π), isotherms (θ), local and average Nusselt number (Nut) for various Rayleigh numbers (103 ≤ Ra ≥ 105), Prandtl numbers (Pr = 0.015 and 7.2) and wall curvatures (concavity/convexity). Findings - The presence of the larger fluid velocity within the curved cavities resulted in the larger heat transfer rates and thermal mixing compared to the square cavity. Case 3 (high concavity) exhibits the largest Nut at the low Ra for all Pr. At the high Ra, Nut is the largest for Case 3 (high concavity) at Pr = 0.015, whereas at Pr = 7.2, Nut is the largest for Case 1 (high concavity and convexity). Practical implications - The results may be useful for the material processing applications. Originality/value - The study of Rayleigh-Bénard convection in cavities with the curved isothermal walls is not carried out till date. The heatline approach is used for the heat flow visualization during Rayleigh- Benard convection within the curved walled enclosures for the first time. Also, the existence of the enhanced fluid and heat circulation cells within the curved walled cavities during Rayleigh-Benard heating is illustrated for the first time. © Emerald Publishing Limited.

Role of heatlines on thermal management during Rayleigh-Bénard heating within enclosures with concave/convex horizontal walls

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, 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 s at Ra = 10 5. On the other hand, two asymmetric flow circulation cells are found to occupy the entire cavity for = 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 s. On the other hand, the heat flow is evenly distributed to both side walls at higher s via convection as well as direct conductive transport. Significant convective heat transfer from the bottom hot wall to the left cold wall occurs for = 30°cavity whereas the heat transfer to the right cold wall is maximum for = 75°irrespective of Pr. Average Nusselt number studies also show that = 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 = 75°for both isothermal and non-isothermal heating strategies except at Pr = 0.015 in isothermal heating case. © 2011 Elsevier Ltd. All rights reserved.

International Journal of Heat and Mass Transfer

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

Heat flow patterns in the presence of natural convection have been analyzed with Bejan's heatlines concept. Momentum and energy transfer are characterized by streamfunctions and heatfunctions, respectively such that streamfunctions and heatfunctions satisfy the dimensionless forms of momentum and energy balance equations, respectively. Finite element method has been used to solve the velocity and thermal fields and the method has also been found robust to obtain the streamfunction and heatfunction accurately. The unique solution of heatfunctions for situations in differential heating is a strong function of Dirichlet boundary condition which has been obtained from average Nusselt numbers for hot or cold regimes. The physical significance of heatlines have been demonstrated for a comprehensive understanding of energy distribution and optimal thermal management via analyzing three cases. Case 1 involves the uniform and non-uniform heating of bottom wall with cooled side walls. The studies illustrate that the heat flow primarily occurs from the central regime of the bottom wall to a very small regime of the top portion of side walls. A large portion of central regime of cold side walls do not receive significant amount of heat. In order to maximize the thermal energy distribution, the distributed heating at the middle portions of the bottom and side walls have been considered in case 2 and heatlines clearly depict the distributions of heat from the hot walls to the large regimes of the cold wall. Further case 3 illustrates the enhanced heat flows in presence of heated bottom and left side walls. Heatline is found as an effective numerical tool to visualize energy distribution in order to establish a suitable heating strategy. © 2007 Elsevier Ltd. All rights reserved.

Role of 'Bejan's heatlines' in heat flow visualization and optimal thermal mixing for differentially heated square enclosures

International Communications in Heat and Mass Transfer

Finite element simulation of MHD combined convection through a triangular wavy channel

Computers & Fluids

Inverse determination of a heat source from natural convection in a porous cavity

Experimental analysis and FEM simulation of finned U-shape multi heat pipe for desktop PC cooling

Heatline based thermal management for natural convection in porous rhombic enclosures with isothermal hot side or bottom wall

Journal	Energy Conversion and Management
ISSN	01968904
Open Access	No