# 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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01 Jan 1997TL;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

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TL;DR: In this paper, the authors investigated the entropy generation due to conjugate natural convection-conduction heat transfer in a square domain under steady-state condition, and the results showed that both the average Nusselt number and entropy generation are increasing functions of K ro while they are maxima at some critical values of D.

Abstract: Entropy generation due to conjugate natural convection–conduction heat transfer in a square domain is numerically investigated under steady-state condition. The domain composed of porous cavity heated by a triangular solid wall and saturated with a CuO–water nanofluid. Equations governing the heat transfer in the triangular solid together with the heat and nanofluid flow in the nanofluid-saturated porous medium are solved numerically using the over-successive relaxation finite-difference method. A temperature dependent thermal conductivity and modified expression for the thermal expansion of nanofluid are adopted. A new criterion for assessment of the thermal performance is proposed. The investigated parameters are the nanoparticles volume fraction φ (0–0.05), modified Rayleigh number Ra (10–1000), solid wall to base-fluid saturated porous medium thermal conductivity ratio K ro (0.44, 1, 23.8), and the triangular solid thickness D (0.1–1). The results show that both the average Nusselt number and the entropy generation are increasing functions of K ro , while they are maxima at some critical values of D . It is also found that the addition of nanoparticles increases the entropy generation. According to the new proposed criterion, the results show that the largest solid thickness ( D = 1.0) and the lower wall thermal conductivity ratio manifest better thermal performance.

142 citations

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TL;DR: In this article, the entropy generation in natural convection of nanofluid in a wavy cavity using a single-phase model was analyzed using the finite difference method of the second-order accuracy.

Abstract: Purpose
The main purpose of this numerical study is to study on entropy generation in natural convection of nanofluid in a wavy cavity using a single-phase nanofluid model.
Design/methodology/approach
The cavity is heated non-uniformly from the wavy wall and cooled from the right side while it is insulated from the horizontal walls. The physical domain of the problem is transformed into a rectangular geometry in the computational domain using an algebraic coordinate transformation by introducing new independent variables ξ and η. The governing dimensionless partial differential equations with corresponding initially and boundary conditions were numerically solved by the finite difference method of the second-order accuracy. The governing parameters are Rayleigh number (Ra = 1000-100000), Prandtl number (Pr = 6.82), solid volume fraction parameter of nanoparticles (φ = 0.0-0.05), aspect ratio parameter (A = 1), undulation number (κ = 1-3), wavy contraction ratio (b = 0.1-0.3) and dimensionless time (τ = 0-0.27).
Findings
It is found that the average Bejan number is an increasing function of nanoparticle volume fraction and a decreasing function of the Rayleigh number, undulation number and wavy contraction ratio. Also, an insertion of nanoparticles leads to an attenuation of convective flow and enhancement of heat transfer.
Originality
The originality of this work is to analyze the entropy generation in natural convection within a wavy nanofluid cavity using single-phase nanofluid model. The results would benefit scientists and engineers to become familiar with the flow behaviour of such nanofluids, and will be a way to predict the properties of this flow for the possibility of using nanofluids in advanced nuclear systems, in industrial sectors including transportation, power generation, chemical sectors, ventilation, air-conditioning, etc.

128 citations

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TL;DR: It is found that the applied magnetic field can suppress both the natural convection and the entropy generation rate, and the nanoparticles addition can be useful if a compromised magnetic field value represented by a Hartman number of 30 is applied.

Abstract: This paper investigates the entropy generation and natural convection inside a C-shaped cavity filled with CuO-water nanofluid and subjected to a uniform magnetic field. The Brownian motion effect is considered in predicting the nanofluid properties. The governing equations are solved using the finite volume method with the SIMPLE (Semi-Implicit Method for Pressure Linked Equations) algorithm. The studied parameters are the Rayleigh number (1000 ≤ Ra ≤ 15,000), Hartman number (0 ≤ Ha ≤ 45), nanofluid volume fraction (0 ≤ φ ≤ 0.06), and the cavity aspect ratio (0.1 ≤ AR ≤ 0.7). The results have shown that the nanoparticles volume fraction enhances the natural convection but undesirably increases the entropy generation rate. It is also found that the applied magnetic field can suppress both the natural convection and the entropy generation rate, where for Ra = 1000 and φ = 0.04, the percentage reductions in total entropy generation decreases from 96.27% to 48.17% for Ha = 45 compared to zero magnetic field when the aspect ratio is increased from 0.1 to 0.7. The results of performance criterion have shown that the nanoparticles addition can be useful if a compromised magnetic field value represented by a Hartman number of 30 is applied.

120 citations

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TL;DR: In this paper, a numerical study is made on the mixed convection of copper-water nanofluid inside a differentially heated skew enclosure, where the finite volume based SIMPLEC algorithm is used to solve the transformed equations for fluid flow and heat transfer equations in the computational domain.

Abstract: A numerical study is made on the mixed convection of copper–water nanofluid inside a differentially heated skew enclosure. Co-ordinate transformations are used to transform the physical domain to the computational domain in an orthogonal co-ordinate. The finite volume based SIMPLEC algorithm is used to solve the transformed equations for fluid flow and heat transfer equations in the computational domain. The fluid flow and heat transfer characteristics are studied for a wide range of skew angles ( 30 ° ⩽ λ ⩽ 150 ° ) , nanoparticle volume fraction ( 0.0 ⩽ ϕ ⩽ 0.2 ) and Richardson number ( 0.1 ⩽ Ri ⩽ 5 ) at a fixed value of Reynolds number. The entropy generation and Bejan number are evaluated to demonstrate the thermodynamic optimization of the mixed convection. It is shown that the heat transfer rate increases remarkably by the addition of nanoparticles. The flow field is sensible to the skew angle variation. Our results show that the heat transfer augmentation through nanoparticles with lower rate in entropy generation enhancement can be achieved in a skewed cavity.

76 citations

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TL;DR: In this paper, the authors investigated the heat and mass transfer in an unsaturated wet cylindrical porous bed packed with quartz particles for relatively low convective drying rates, and the results indicated that the drying process could be divided into three periods, the temperature rise period, the constant drying rate period and the decreasing rate period.

Abstract: The heat and mass transfer in an unsaturated wet cylindrical porous bed packed with quartz particles was investigated theoretically for relatively low convective drying rates. Local thermodynamic equilibrium was assumed in the mathematical model describing the multi-phase flow in the unsaturated porous media using the energy and mass conservation equations to describe the heat and mass transfer during the drying. The drying model included convection and capillary transport of the free water, diffusion of bound water, and convection and diffusion of the gas. The numerical results indicated that the drying process could be divided into three periods, the temperature rise period, the constant drying rate period and the decreasing drying rate period. The numerical results agreed well with the experimental data verifying that the mathematical model can evaluate the drying performance of porous media for low drying rates. The effects of drying conditions such as the ambient temperature, the relative humidity, and the velocity of the drying air, on the drying process were evaluated by numerical solution.

29 citations

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TL;DR: In this article, a comprehensive numerical investigation on the natural convection in a hydrodynamically anisotropic porous enclosure is presented, which is due to nonuniformly heated bottom wall and maintenance of constant temperature at cold vertical walls along with adiabatic top wall.

Abstract: A comprehensive numerical investigation on the natural convection in a hydrodynamically anisotropic porous enclosure is presented. The flow is due to nonuniformly heated bottom wall and maintenance of constant temperature at cold vertical walls along with adiabatic top wall. Brinkman-extended non-Darcy model, including material derivative, is considered. The principal direction of the permeability tensor has been taken oblique to the gravity vector. The spectral element method has been adopted to solve numerically the governing conservative equations of mass, momentum, and energy by using a stream-function vorticity formulation. Special attention is given to understand the effect of anisotropic parameters on the heat transfer rate as well as flow configurations. The numerical experiments show that in the case of isotropic porous enclosure, the maximum rates of bottom as well as side heat transfers (Nu b and Nu s ) take place at the aspect ratio, A, of the enclosure equal to 1, which is, in general, not true in the case of anisotropic porous enclosures. The flow in the enclosure is governed by two different types of convective cells: rotating (i) clockwise and (ii) anticlockwise. Based on the value of media permeability as well as orientation angle, in the anisotropic case, one of the cells will dominate the other. In contrast to isotropic porous media, enhancement of flow convection in the anisotropic porous enclosure does not mean increasing the side heat transfer rate always. Furthermore, the results show that anisotropy causes significant changes in the bottom as well as side average Nusselt numbers. In particular, the present analysis shows that permeability orientation angle has a significant effect on the flow dynamics and temperature profile and consequently on the heat transfer rates.

27 citations

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TL;DR: In this article, an integrated micro-macro model of reactive flow of molten silicon in a porous preform consisting of carbon-coated silicon carbide fibers was developed and the overall objective of the research was to arrive at a modified equation of Darcy's law for flow through a porous medium with the help of numerical/computational modeling.

Abstract: Reactive infiltration is a fast and cost-effective technique for manufacturing ceramic-matrix composites (CMCs). CMCs are used in elevated temperature applications like rocket engine casings, jet nozzles, gas turbine blades and nuclear cladding. There is an urgent need for minimizing experimental costs as well as optimizing process parameters during manufacture, so that we have minimized manufacturing costs and reduced infiltration times. Towards this end, the objective of this research was to develop an integrated micro-macro model of reactive flow of molten silicon in a porous preform consisting of carbon-coated silicon carbide fibers and then optimize process parameters computationally. The overall objective of the research was to arrive at a modified equation of Darcy's law for flow through a porous medium with the help of numerical/computational modeling. This paper deals with the flow of silicon through porous carbon at the macro level. The macro flow of silicon was integrated with an available micro model by determining the transient porosity from the micro model and using it in Darcy's law written for the macro flow of silicon. From the results of this study, we recommend suitable process parameters such as initial temperature of the solid reactant and the specific kind of reactants to be used for achieving complete infiltration. These conclusions are drawn after observation of the rate of decrease of permeability with more reaction.

21 citations

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TL;DR: In this article, a finite-volume model was developed to simulate the injection of a pure metal through a fibrous preform placed between two plane walls of a metal mold, based on the simultaneous resolutions of the heat equation, Darcy's law, and an advection equation to follow the evolution of the metal front geometry during the infiltration.

Abstract: A method of metal matrix composites processing consists of injecting a liquid metal through a fibrous preform. The metal flow through the porous preform is associated with phase-change phenomena which imply modifications of the porous medium permeability and disturb the metal flow. A finite-volume model was developed to simulate the injection of a pure metal through a fibrous preform placed between two plane walls of a metal mold. This model is based on the simultaneous resolutions of the heat equation, Darcy's law, and an advection equation to follow the evolution of the metal front geometry during the infiltration.

19 citations

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TL;DR: In this paper, the Darcy momentum equation is selected for momentum transfer modeling by considering a relatively small pore Reynolds number (Re p ), and the equations are solved numerically using the finite volume method.

Abstract: Flow, thermal, energy, and irreversibility characteristics inside wavy enclosures packed with microstructures are reported in this paper It is assumed that the entire enclosure has sufficient and interconnected void spaces; those allow fluid movement inside the cavity. The Darcy momentum equation is selected for momentum transfer modeling by considering a relatively small pore Reynolds number (Re p ). Modeled equations are solved numerically using the finite volume method. Streamlines, isothermal lines, energy streamlines, average Nusselt number, and average entropy generation number are calculated and displayed in order to show their dependency on and variation with Rayleigh number (Ra), surface waviness (λ), and aspect ratio (A R ) of the enclosure. Depending on the wall waviness pattern, the enclosure is divided into three modes (phase-plus, phase-zero, and phase-minus modes). However, for the current calculation, wall waviness is kept symmetric with respect to the vertical and horizontal centerlines of the enclosure.

18 citations

### "Entropy Generation During Natural C..." refers background in this paper

...[34] also studied entropy generation characteristics in wavy enclosures filled with microstructures which are modeled as porous medium....

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