Visualization of Heat Transport during Natural Convection Within Porous Triangular Cavities via Heatline Approach
10 Mar 2010-Numerical Heat Transfer Part A-applications (Taylor & Francis Group)-Vol. 57, Iss: 6, pp 431-452
TL;DR: In this paper, a penalty finite element method with biquadratic elements is used to solve the non-dimensional governing equations for the triangular cavity involving hot inclined walls and cold top wall.
Abstract: 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...
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TL;DR: There is, I think, something ethereal about i —the square root of minus one, which seems an odd beast at that time—an intruder hovering on the edge of reality.
Abstract: There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.
Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …
33,785 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
28 Jan 2005
TL;DR: The Q12-40 density: ρ ((kg/m) specific heat: Cp (J/kg ·K) dynamic viscosity: ν ≡ μ/ρ (m/s) thermal conductivity: k, (W/m ·K), thermal diffusivity: α, ≡ k/(ρ · Cp) (m /s) Prandtl number: Pr, ≡ ν/α (−−) volumetric compressibility: β, (1/K).
Abstract: Geometry: shape, size, aspect ratio and orientation Flow Type: forced, natural, laminar, turbulent, internal, external Boundary: isothermal (Tw = constant) or isoflux (q̇w = constant) Fluid Type: viscous oil, water, gases or liquid metals Properties: all properties determined at film temperature Tf = (Tw + T∞)/2 Note: ρ and ν ∝ 1/Patm ⇒ see Q12-40 density: ρ ((kg/m) specific heat: Cp (J/kg ·K) dynamic viscosity: μ, (N · s/m) kinematic viscosity: ν ≡ μ/ρ (m/s) thermal conductivity: k, (W/m ·K) thermal diffusivity: α, ≡ k/(ρ · Cp) (m/s) Prandtl number: Pr, ≡ ν/α (−−) volumetric compressibility: β, (1/K)
636 citations
TL;DR: In this paper, a review of 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 is presented.
Abstract: 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.
168 citations
TL;DR: In this article, a right-angle triangular enclosure with a flush mounted heater with finite size is placed on the left vertical wall, and the rest of walls are adiabatic.
Abstract: Steady-state free convection heat transfer behavior of nanofluids is investigated numerically inside a right-angle triangular enclosure filled with a porous medium. The flush mounted heater with finite size is placed on the left vertical wall. The temperature of the inclined wall is lower than the heater, and the rest of walls are adiabatic. The governing equations are obtained based on the Darcy’s law and the nanofluid model proposed by Tiwari and Das [1] . The transformed dimensionless governing equations were solved by finite difference method and solution for algebraic equations was obtained through Successive Under Relaxation method. Investigations with three types of nanofluids were made for different values of Rayleigh number Ra of a porous medium with the range of 10 ≤ Ra ≤ 1000, size of heater Ht as 0.1 ≤ Ht ≤ 0.9, position of heater Y p when 0.25 ≤ Y p ≤ 0.75, enclosure aspect ratio AR as 0.5 ≤ AR ≤ 1.5 and solid volume fraction parameter ϕ of nanofluids with the range of 0.0 ≤ ϕ ≤ 0.2. It is found that the maximum value of average Nusselt number is obtained by decreasing the enclosure aspect ratio and lowering the heater position with the highest value of Rayleigh number and the largest size of heater. It is further observed that the heat transfer in the cavity is improved with the increasing of solid volume fraction parameter of nanofluids at low Rayleigh number, but opposite effects appear when the Rayleigh number is high.
142 citations
References
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[...]
TL;DR: There is, I think, something ethereal about i —the square root of minus one, which seems an odd beast at that time—an intruder hovering on the edge of reality.
Abstract: There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.
Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …
33,785 citations
"Visualization of Heat Transport dur..." refers background in this paper
...Address correspondence to I. Pop, Faculty of Mathematics, University of Cluj, R-3400 Cluj, CP 253, Romania....
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...VISUALIZATION OF HEAT TRANSPORT DURING NATURAL CONVECTION WITHIN POROUS TRIANGULAR CAVITIES VIA HEATLINE APPROACH Tanmay Basak1, S. Roy2, D. Ramakrishna2, and I. Pop3 1Department of Chemical Engineering, Indian Institute of TechnologyMadras, Chennai, India 2Department of Mathematics, Indian Institute of Technology Madras, Chennai, India 3Faculty of Mathematics, University of Cluj, Cluj, Romania In this article, natural convection in a porous triangular cavity has been analyzed....
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...Previous studies on the convection patterns in various enclosures filled with a porous media are reported in the literature by Martynenko and Khramtsov [15], Bejan and Poulikakos [16], Ingham and Pop [17, 18], Kaviany [19], and Vafai [20]....
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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
"Visualization of Heat Transport dur..." refers methods in this paper
...Further, it is assumed that the temperature of the fluid phase is equal to the temperature of the solid phase in the porous region, and local thermal equilibrium (LTE) is applicable in the present investigation [ 61 ]....
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Book•
01 Jan 1984
TL;DR: In this paper, the authors describe a transition from Laminar boundary layer flow to Turbulent Boundary Layer flow with change of phase Mass Transfer Convection in Porous Media.
Abstract: Fundamental Principles Laminar Boundary Layer Flow Laminar Duct Flow External Natural Convection Internal Natural Convection Transition to Turbulence Turbulent Boundary Layer Flow Turbulent Duct Flow Free Turbulent Flows Convection with Change of Phase Mass Transfer Convection in Porous Media.
4,067 citations
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
Book•
01 Oct 1991
TL;DR: In this article, the authors identify the principles of transport in porous media and compare the available predicted results, based on theoretical treatments of various transport mechanisms, with the existing experimental results, and the theoretical treatment is based on the volume-averaging of the momentum and energy equations with the closure conditions necessary for obtaining solutions.
Abstract: Although the empirical treatment of fluid flow and heat transfer in porous media is over a century old, only in the last three decades has the transport in these heterogeneous systems been addressed in detail. So far, single-phase flows in porous media have been treated or at least formulated satisfactorily, while the subject of two-phase flow and the related heat-transfer in porous media is still in its infancy. This book identifies the principles of transport in porous media and compares the available predicted results, based on theoretical treatments of various transport mechanisms, with the existing experimental results. The theoretical treatment is based on the volume-averaging of the momentum and energy equations with the closure conditions necessary for obtaining solutions. While emphasizing a basic understanding of heat transfer in porous media, this book does not ignore the need for predictive tools; whenever a rigorous theoretical treatment of a phenomena is not available, semi-empirical and empirical treatments are given.
2,551 citations