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 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
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
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
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
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
References
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TL;DR: In this paper, the Darcy-Forchheimer model is used to simulate the momentum transfer in the porous medium and numerical results are presented in terms of stream functions, temperature profiles and Nusselt numbers.
Abstract: Natural convection flows in a square cavity filled with a porous matrix has been studied numerically using penalty finite element method for uniformly and non-uniformly heated bottom wall, and adiabatic top wall maintaining constant temperature of cold vertical walls. Darcy–Forchheimer model is used to simulate the momentum transfer in the porous medium. The numerical procedure is adopted in the present study yields consistent performance over a wide range of parameters (Rayleigh number Ra , 10 3 ⩽ Ra ⩽ 10 6 , Darcy number Da , 10 −5 ⩽ Da ⩽ 10 −3 , and Prandtl number Pr , 0.71 ⩽ Pr ⩽ 10) with respect to continuous and discontinuous thermal boundary conditions. Numerical results are presented in terms of stream functions, temperature profiles and Nusselt numbers. Non-uniform heating of the bottom wall produces greater heat transfer rate at the center of the bottom wall than uniform heating case for all Rayleigh numbers but average Nusselt number shows overall lower heat transfer rate for non-uniform heating case. It has been found that the heat transfer is primarily due to conduction for Da ⩽ 10 −5 irrespective of Ra and Pr . The conductive heat transfer regime as a function of Ra has also been reported for Da ⩾ 10 −4 . Critical Rayleigh numbers for conduction dominant heat transfer cases have been obtained and for convection dominated regimes the power law correlations between average Nusselt number and Rayleigh numbers are presented.
275 citations
"Entropy Generation During Natural C..." refers methods in this paper
...The detailed solution procedure is given in an earlier work [42]....
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TL;DR: In this paper, the issue of entropy generation in a tilted saturated porous cavity for laminar natural convection heat transfer is analyzed by solving numerically the mass, momentum and energy balance equations, using Darcy's law and Boussinesq-incompressible approximation.
Abstract: The issue of entropy generation in a tilted saturated porous cavity for laminar natural convection heat transfer is analysed by solving numerically the mass, momentum and energy balance equations, using Darcy’s law and Boussinesq-incompressible approximation. As boundary conditions of cavity, two opposite walls are kept at constant but different temperatures and the other two are thermally insulated. The parameters considered are the angle of inclination and the Darcy–Rayleigh number. When available, present solutions are compared with known results from the previous researches. Excellent agreement was obtained between results that validate the used computer code. The results show that the calculation of local entropy generation maps are feasible and can supply useful information for the selection of a suitable angle of inclination.
253 citations
"Entropy Generation During Natural C..." refers background or result in this paper
...The current approach offers a special advantage over finite difference or finite volume solutions [33, 35], where derivatives are calculated using some interpolation functions which are avoided in the current work, and elemental basis sets (defined by Eq....
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...Baytas [33] studied entropy generation in a differentially heated inclined square porous NOMENCLATURE...
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...Similar studies on entropy generation during natural convection in enclosures where viscous dissipation term is omitted in the energy equation have been reported earlier [33, 35]....
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TL;DR: In this article, the partial differential equations which describe steady flow of fluid in saturated homogeneous permeable solid material under non-isothermal conditions are derived using suitable approximations and making use of empirical laws when necessary.
Abstract: The partial differential equations which describe steady flow of fluid in saturated homogeneous permeable solid material under non-isothermal conditions are stated From these are derived the equations for flow of liquid (in particular, water) using suitable approximations and making use of empirical laws when necessaryIt is then postulated that the only ‘ponderomotive’ (ie massmoving) forces present are those due to thermal expansion effects Free convection results An approximate solution of the equations is attempted for plane flow by means of classical perturbation methods, the temperature and stream-function variables being represented by power series in a convection parameter proportional to the Rayleigh numberA numberical example of the method, with boundary conditions based on a geothermal area at Wairakei, New Zealand, is given The results show features which are in fair agreement with temperature measurements made in the area, and it appears that the convection parameter η is of the order of 10
253 citations
TL;DR: In this paper, a unified mathematical theory for the viscous dissipation term in the governing Brinkman equation is derived, which has the correct asymptotic behaviour in both the fully Darcy and Newtonian fluid flow limits.
Abstract: In this paper a unified mathematical theory for the viscous dissipation term in the governing Brinkman equation is derived. This term has, unlike other models, the correct asymptotic behaviour in both the fully Darcy and Newtonian fluid flow limits.
226 citations
TL;DR: In this paper, the floating constants in Weber's boundary layer solution for free convection in a differentially heated vertical porous slab are reevaluated with a new approach, which uses Weber's solution to calculate the net vertical heat flux which is equated to zero near the top and bottom ends of the enclosure.
Abstract: The floating constants in Weber's boundary layer solution for free convection in a differentially heated vertical porous slab are reevaluated with a new approach. This approach uses Weber's solution to calculate the net vertical heat flux which is equated to zero near the top and bottom ends of the enclosure. It is shown that the Nusselt numbers predicted with the new constants are in excellent agreement with experimental and numerical results.
224 citations
"Entropy Generation During Natural C..." refers methods in this paper
...Further, the relative dominance of Sh and Sw is analyzed in terms of average Bejan number and thermal mixing for temperature uniformity via cup-mixing temperature and root-mean square deviation....
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...The average Bejan number (Beav) indicates the dominance of thermal or frictional irreversibility on total entropy generation....
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...In order to illustrate the dominance of Sh and Sw, the average Bejan number (Beav) is presented in Figure 13b....
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...Further, the present numerical procedure is validated for a porous medium case where a porous square cavity, with air (Pr¼ 0.71) as fluid, is heated isothermally on the left wall while the right wall is maintained cold isothermal and the horizontal walls are maintained adiabatic, similar to the problem reported by Bejan [49], Manole and Lage [50], Saeid and Pop [51], and Baytas and Pop [52]....
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...71) as fluid, is heated isothermally on the left wall while the right wall is maintained cold isothermal and the horizontal walls are maintained adiabatic, similar to the problem reported by Bejan [49], Manole and Lage [50], Saeid and Pop [51], and Baytas and Pop [52]....
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