Entropy Generation During Natural Convection in a Porous Cavity: Effect of Thermal Boundary Conditions
02 Aug 2012-Numerical Heat Transfer Part A-applications (Taylor & Francis Group)-Vol. 62, Iss: 4, pp 336-364
TL;DR: In this article, the authors investigated the effect of different boundary conditions on entropy generation, and showed that the entropy generation rates are reduced in sinusoidal heating (case 2) when compared to that for uniform heating with a penalty on thermal mixing, and that there exists an intermediate Da for optimal values of entropy generation.
Abstract: Entropy generation plays a significant role in the overall efficiency of a given system, and a judicious choice of optimal boundary conditions can be made based on a knowledge of entropy generation. Five different boundary conditions are considered and their effect of the permeability of the porous medium, heat transfer regime (conduction and convection) on entropy generation due to heat transfer, and fluid friction irreversibilities are investigated in detail for molten metals (Pr = 0.026) and aqueous solutions (Pr = 10), with Darcy numbers (Da) between 10−5–10−3 and at a representative high Rayleigh number, Ra = 5 × 105. It is observed that the entropy generation rates are reduced in sinusoidal heating (case 2) when compared to that for uniform heating (case 1), with a penalty on thermal mixing. Finally, the analysis of total entropy generation due to variation in Da and thermal mixing and temperature uniformity indicates that, there exists an intermediate Da for optimal values of entropy generation, th...
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TL;DR: In this article, an entropy generation minimization (EGM) approach is implemented for the analysis of optimal thermal mixing and temperature uniformity due to natural convection in quadrantal cavity filled with porous medium for the material processing applications or for cooling of electrical equipments.
Abstract: Industrial processes optimization for higher energy efficiency may be effectively carried out based on the thermodynamic approach of entropy generation minimization (EGM). This approach provides the key insights on how the available energy (exergy) is being destroyed during the process and the ways to minimize its destruction. In this study, EGM approach is implemented for the analysis of optimal thermal mixing and temperature uniformity due to natural convection in quadrantal cavity filled with porous medium for the material processing applications or for cooling of electrical equipments. Effect of the permeability of the porous medium and the role of non-uniform heating in enhancing the thermal mixing, temperature effects and minimization of entropy generation is analyzed. The numerical solutions are obtained using finite element method. Stream lines , Isotherms and Entropy generation results are depicted for Ra=106, 1.7x105, 104 and for Da=10-3, 10-4 and 10-5 at Pr=0.71.It is found that at lower Darcy number (Da), the thermal mixing is low and the heat transfer irreversibility dominates the total entropy generation. In contrast, thermal mixing is improved due to enhanced convection at higher Da. It is observed that fluid friction irreversibility dominates over heat transfer irreversibility for only Da=10-3 at Ra=106. The local entropy generation is maximum at the bottom wall, while at the center and top of the enclosure it becomes minimum. Based on EGM analysis, it is established that total entropy production is not significant with larger thermal mixing at high Darcy number, and can be recommended for material processing process or for cooling of electrical equipments.
13 citations
Cites methods from "Entropy Generation During Natural C..."
...Also important to emphasise here is that the numerical simulations are chosen in order to show the effect of Da which are reported to be well within the regimes of possible operating conditions as reported in the literature by Basak et al.( 2012,2013)....
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TL;DR: In this paper, a comparative study based on the entropy generation analysis has been carried out within porous square and triangular enclosures subjected to discrete heating involving double heater, and the results showed that the double heater is more efficient than the single heater.
Abstract: Comparative study based on the entropy generation analysis has been carried out within porous square and triangular (design 1 and 2) enclosures subjected to discrete heating involving double heater...
10 citations
TL;DR: In this article, a meshless Fourier series (FS) solution of the natural convection problem in a porous enclosure driven by thermal or compositional variations is developed. And the FS solution is used to investigate the effect of the large-scale Rayleigh number and level of heterogeneity on NC processes and energy losses.
Abstract: Three-dimensional (3D) natural convection (NC) processes in heterogeneous porous media and associated energy losses and mixing processes are still poorly understood. Studies are limited to two-dimensional domains because of computational burden, worsened by heterogeneity, which may demand grid refinement at high permeability zones for accurate evaluation of buoyancy forces. We develop a meshless Fourier series (FS) solution of the natural convection problem in a porous enclosure driven by thermal or compositional variations. We derive the vector potential formulation of the governing equations for vertical and horizontal heterogeneity of hydraulic conductivity and implement an efficient method to solve the spectral system with an optimized number of Fourier modes. 3D effects are induced either by heterogeneity or variable boundary conditions. The developed FS solution is verified against a finite element solution obtained using COMSOL Multiphysics. We evaluate entropy generation (viscous dissipation and mixing) indicators using FS expansions and assess how they are affected by heterogeneity. We define a large-scale Rayleigh number to account for heterogeneity by adopting an arithmetic average effective permeability. The FS solution is used to investigate the effect of the large-scale Rayleigh number and level of heterogeneity on NC processes and energy losses. Results show that increasing the Rayleigh number intensifies fluid flow, thus enhancing convective transfer, which causes a dramatic increase in total entropy generation. Both viscous dissipation and mixing (and thus chemical reactions in the solute transport case) increase. The third dimension effect, which also enhances flow and entropy indicators, is more pronounced at high Rayleigh numbers. Surprisingly, entropy variation indicators remain virtually unchanged in response to changes in heterogeneity, for fixed Rayleigh number, which we attribute to the arithmetic average permeability being indeed appropriate for NC in 3D. This study not only explores the effect of Rayleigh number and heterogeneity on natural convection processes and the associated entropy generation and mixing processes, but also provides a highly accurate solution that can be used for codes benchmarking.
10 citations
TL;DR: In this paper, the influence of a spatially non-uniform, sinusoidally varying, wall temperature on the local flow and heat transfer in a simple cubic packing configuration was studied.
Abstract: We report numerical simulations of natural convection and conjugate heat transfer in a differentially heated cubical cavity packed with relatively large hydrogel beads (d/L=0.2) in a Simple Cubic Packing configuration. We study the influence of a spatially non-uniform, sinusoidally varying, wall temperature on the local flow and heat transfer, for a solid-to-fluid conductivity ratio of 1, a fluid Prandtl number of 5.4, and fluid Rayleigh numbers between 105 and 107. We present local and overall flow and heat transfer results for both sphere packed and water-only filled cavities, when subjected to variations of the wall temperature at various combinations of the amplitude and characteristic phase angle of the imposed wall temperature variations. It is found that imposing a sinusoidal spatial variation in the wall temperature may significantly alter the local flow and heat transfer, and consequently the overall heat transfer. At identical average temperature difference, applying a spatial variation in wall temperature at well-chosen phase angle can lead to significant heat transfer enhancement when compared to applying uniform wall temperatures. However, this is achieved at the cost of increased entropy generation.
10 citations
01 Jan 2013
TL;DR: In this article, the most fundamental aspects of the convection heat transfer process when the flow is steady and in the Darcy regime are discussed, as well as special features of flows that deviate from theDarcy regime, flows that are time dependent, and flow that are confined in geometries more complicated than the two-dimensional rectangular space shown in Fig. 7.
Abstract: Enclosures heated from the side are most representative of porous systems that function while oriented vertically, as in the insulations for buildings, industrial cold-storage installations, and cryogenics. As in the earlier chapters, we begin with the most fundamental aspects of the convection heat transfer process when the flow is steady and in the Darcy regime. Later, we examine the special features of flows that deviate from the Darcy regime, flows that are time dependent, and flows that are confined in geometries more complicated than the two-dimensional rectangular space shown in Fig. 7.1.
9 citations
References
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Book•
01 Jan 1992TL;DR: In this paper, an introduction to convection in porous media assumes the reader is familiar with basic fluid mechanics and heat transfer, going on to cover insulation of buildings, energy storage and recovery, geothermal reservoirs, nuclear waste disposal, chemical reactor engineering and the storage of heat-generating materials like grain and coal.
Abstract: This introduction to convection in porous media assumes the reader is familiar with basic fluid mechanics and heat transfer, going on to cover insulation of buildings, energy storage and recovery, geothermal reservoirs, nuclear waste disposal, chemical reactor engineering and the storage of heat-generating materials like grain and coal. Geophysical applications range from the flow of groundwater around hot intrusions to the stability of snow against avalanches. The book is intended to be used as a reference, a tutorial work or a textbook for graduates.
5,570 citations
"Entropy Generation During Natural C..." refers background in this paper
...An extensive review of literature on porous media may be found in earlier works [ 8 ]....
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Book•
01 Jan 1984
TL;DR: Second-order Differential Equations in One Dimension: Finite Element Models (FEM) as discussed by the authors is a generalization of the second-order differential equation in two dimensions.
Abstract: 1 Introduction 2 Mathematical Preliminaries, Integral Formulations, and Variational Methods 3 Second-order Differential Equations in One Dimension: Finite Element Models 4 Second-order Differential Equations in One Dimension: Applications 5 Beams and Frames 6 Eigenvalue and Time-Dependent Problems 7 Computer Implementation 8 Single-Variable Problems in Two Dimensions 9 Interpolation Functions, Numerical Integration, and Modeling Considerations 10 Flows of Viscous Incompressible Fluids 11 Plane Elasticity 12 Bending of Elastic Plates 13 Computer Implementation of Two-Dimensional Problems 14 Prelude to Advanced Topics
3,043 citations
01 Jan 1997
TL;DR: This chapter introduces the finite element method (FEM) as a tool for solution of classical electromagnetic problems and discusses the main points in the application to electromagnetic design, including formulation and implementation.
Abstract: This chapter introduces the finite element method (FEM) as a tool for solution of classical electromagnetic problems. Although we discuss the main points in the application of the finite element method to electromagnetic design, including formulation and implementation, those who seek deeper understanding of the finite element method should consult some of the works listed in the bibliography section.
1,820 citations
"Entropy Generation During Natural C..." refers background or methods in this paper
...(5), (9), and (10)] with boundary conditions is solved by using the Galerkin finite element method [41]....
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...(12) and (13), the second term containing the penalty parameter (c) are evaluated with two point Gaussian quadrature (reduced integration penalty formulation, [41])....
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TL;DR: Entropy generation minimization (finite time thermodynamics, or thermodynamic optimization) is the method that combines into simple models the most basic concepts of heat transfer, fluid mechanics, and thermodynamics as mentioned in this paper.
Abstract: Entropy generation minimization (finite time thermodynamics, or thermodynamic optimization) is the method that combines into simple models the most basic concepts of heat transfer, fluid mechanics, and thermodynamics. These simple models are used in the optimization of real (irreversible) devices and processes, subject to finite‐size and finite‐time constraints. The review traces the development and adoption of the method in several sectors of mainstream thermal engineering and science: cryogenics, heat transfer, education, storage systems, solar power plants, nuclear and fossil power plants, and refrigerators. Emphasis is placed on the fundamental and technological importance of the optimization method and its results, the pedagogical merits of the method, and the chronological development of the field.
1,516 citations
"Entropy Generation During Natural C..." refers background in this paper
...The main idea behind thermodynamic optimization is to relate degree of thermodynamic non-ideality of the design to the physical characteristics of the system, such as finite dimensions, shapes, materials, finite speeds, and finite-time of intervals of operation and vary one or more physical characteristics to optimize the design characterized by minimum entropy generation subject to finite-size and finite-constraints [22, 23]....
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TL;DR: In this article, the effects of a solid boundary and the inertial forces on flow and heat transfer in porous media were analyzed, and a new concept of the momentum boundary layer central to the numerical routine was presented.
Abstract: The present work analyzes the effects of a solid boundary and the inertial forces on flow and heat transfer in porous media. Specific attention is given to flow through a porous medium in the vicinity of an impermeable boundary. The local volume-averaging technique has been utilized to establish the governing equations, along with an indication of physical limitations and assumptions made in the course of this development. A numerical scheme for the governing equations has been developed to investigate the velocity and temperature fields inside a porous medium near an impermeable boundary, and a new concept of the momentum boundary layer central to the numerical routine is presented. The boundary and inertial effects are characterized in terms of three dimensionless groups, and these effects are shown to be more pronounced in highly permeable media, high Prandtl-number fluids, large pressure gradients, and in the region close to the leading edge of the flow boundary layer.
1,427 citations
"Entropy Generation During Natural C..." refers methods in this paper
...Under these assumptions and following Vafai and Tien [37] with Forchheimer inertia term being neglected, the governing equations for steady two-dimensional natural convection flow in a porous square cavity using conservation of mass, momentum, and energy may be written with the following dimensionless variables or numbers:...
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...The momentum transfer in porous medium is based on generalized non-Darcy model proposed by Vafai and Tien [37]....
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...Under these assumptions and following Vafai and Tien [37] with Forchheimer inertia term being neglected, the governing equations for steady two-dimensional natural convection flow in a porous square cavity using conservation of mass, momentum, and energy may be written with the following dimensionless variables or numbers: X ¼ x L ; Y ¼ y L ; U ¼ uL a ; V ¼ vL a ; h ¼ T Tc Th Tc P ¼ pL 2 qa2 ; Pr ¼ n a ; Da ¼ K L2 ; Ra ¼ gbðTh TcÞL 3Pr n2 ð1Þ as qU qX þ qV qY ¼ 0 ð2Þ U qU qX þ V qU qY ¼ qP qX þ Pr q 2U qX 2 þ q 2U qY 2 !...
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