Analysis of Entropy Generation Minimization during Natural

Article pubs.acs.org/IECR

Analysis of Entropy Generation Minimization during Natural Convection in Trapezoidal Enclosures of Various Angles with Linearly Heated Side Wall(s) Tanmay Basak,*,† Pushpendra Kumar,‡ R. Anandalakshmi,† and S. Roy‡ †

Department of Chemical Engineering, and ‡Department of Mathematics, Indian Institute of Technology Madras, Chennai-600036, India ABSTRACT: Entropy generation during natural convection in trapezoidal enclosures with various inclination angles (φ), φ = 45°, 60°, and 90° for uniformly heated bottom wall and insulated top wall with linearly heated side walls (case 1) or linearly heated left wall with cold right wall (case 2) have been investigated numerically using penalty finite element method. Parametric studies for the wide range of Rayleigh numbers (Ra = 103−105) and Prandtl numbers (Pr = 0.015−1000) have been performed. Symmetry in flow pattern is observed for case 1. During the conduction regime at low Ra (Ra = 103), the entropy generation in the cavity is dominated by heat transfer irreversibility for all Pr. The strength of fluid flow increases with Ra and that leads to an increase in thermal energy transport due to enhanced convection at Ra = 105. Consequently, the entropy generation due to heat transfer (Sθ) and fluid friction (Sψ) also increases with Ra for all Pr. The comparison of magnitudes of Sθ and Sψ indicates that maximum entropy generation due to heat transfer (Sθ,max) and fluid friction (Sψ,max) is lower for case 1 and higher for case 2. It is found that Sθ,max occurs at the top portion of side walls in case 1, whereas Sθ,max occurs at a hot−cold junction due to a high thermal gradient in case 2. The total entropy generation, Stotal is found to be smaller in case 1 and larger in case 2 for all Pr at Ra = 105. It is observed that Sψ,max occurs at the top wall in both heating situations for Pr = 0.015 and Ra = 105. It is also found that, Sψ,max is observed near side walls in case 1, whereas that is observed near the right cold wall in case 2 for Pr = 0.7 and 1000 at Ra = 105. The total entropy generation (Stotal) is larger for φ = 45° in both heating cases at high Pr (Pr = 1000). The total entropy generation (Stotal) also increases with Pr due to an increase in Sψ with Pr. It is found that, high heat transfer rate (Nub) and minimum entropy generation (Stotal) occur for square cavities at Ra = 105 for all Pr in case 1. It is observed that Stotal is smaller for case 1 compared to case 2, even though high Nub is observed at case 2 for all φs. The inclination angle (φ) has almost identical effect on entropy production rate for 45° ≤ φ ≤ 60° within the entire range of Pr at all Ra. cylindrical can containing large food particles. Ghani et al.15 have investigated natural convection within a can of liquid food during sterilization. Verboven et al.16 studied heat and mass transfer in a microwave oven, whereas Mistry et al.17 studied processing for electronic ovens. Frost formation inside food hermetic packages preserved in a domestic freezer was studied by Laguerre and Flick.18 Demirkol et al.19 have investigated mass transfer parameters (changes in mass flux, diffusion coefficient, and mass transfer coefficient) during baking of cookies. Kumar et al.20 studied heating processes due to convection of different types of foods. Various other studies involving sterilization and solidification of foods were reported by earlier researchers.21−25 Fargue et al.26 studied separation of natural oils from food substances for industrial applications. A significant number of applications on thermal processing further requires a comprehensive understanding of heat transfer and flow circulations within cavities. Conduction, natural convection, and forced convection are important means of heat transfer in food applications. A comprehensive review by Bejan27 highlights that internal natural convection flow problems are more complex than external ones. A few studies

1. INTRODUCTION Natural convection due to thermal buoyancy effects occurs in various applications of science and technology. Recent interest in convective heat transfer has been mainly motivated by their importance for many natural and industrial problems, especially those associated with mass transfer problems,1−6 dead end reverse osmosis,7 catalytic and electrochemical reactor design,8 molten material processing,9 extraction of oils and lipids from vegetable substrates,10 phase change energy storage process,11 centrifugal separation,12 and oxidation of carbon particles in supercritical water.13 Natural convection flows are particularly complex involving several parameters among which the geometry concerned and thermophysical characteristics of the fluid are the most important. The proper dimensions of systems in various applications are supported by experimental studies and/or numerical simulation. Convective heating processes are also widely used in food industries.14−26 Various applications depend on the product specification, shape of the container, and heating characteristics. Numerical modeling can offer a way to reduce expensive experimental costs. Most of the modeling studies were carried out with conductive heating because of the simplicity of analytical and numerical solutions. A significant amount of earlier studies involve various applications in thermal processing within materials. Rabiey et al.14 studied transient temperature and fluid flow during natural convection, which involves heating of a © 2012 American Chemical Society

Received: Revised: Accepted: Published: 4069

May 23, 2011 January 20, 2012 January 20, 2012 February 29, 2012 dx.doi.org/10.1021/ie201107f | Ind. Eng. Chem. Res. 2012, 51, 4069−4089

Industrial & Engineering Chemistry Research

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described by Carvalho et al.53 where the power dissipation from the horizontal circuit board is limited by its temperature. The boundary conditions of case 1 and case 2 occur commonly in various situations: (i) Case 1 boundary condition is applied for a hot bottom wall with side walls which are gradually maintained cold. (ii) Case 2 denotes the situation where the right wall is exposed to cold ambiance. Case 1 denotes the symmetric heating situations, whereas case 2 denotes the heating mechanisms with heat source applied in the left portion of the cavity. Other potential applications of these boundary conditions are in the fields of solar collectors, energy efficient buildings, etc. The effect of geometry has been illustrated for various angles of the side wall varying within 45−90°. In addition, it is aimed to study and evaluate the role of linear heating to achieve uniform temperature distribution and minimum entropy generation. In the current study, the Galerkin finite element method54 has been employed to solve the nonlinear equations of fluid flow, energy, and entropy. It may be noted that estimation of entropy generation rate involves accurate evaluation of thermal and velocity gradients or derivatives. The finite element approach offers a special advantage over finite difference or finite volume methods as the elemental basis sets are used for the calculation of gradients or derivatives. In addition, the derivatives at the nodes are evaluated based on all adjacent function values via an elemental basis set as every node is shared by several adjacent elements. Simulations are carried out for a range of parameters, Ra = 103−105 and Pr = 0.015−1000 (0.015 (molten metals), 0.7 (air or gaseous substances), and 1000 (olive/engine oils)). The numerical results are presented in terms of contours of isotherms, streamlines, entropy generation due to heat transfer (Sθ), and entropy generation due to fluid friction (Sψ). Also, the effects of Rayleigh number on the average Nusselt Number, total entropy generation, and average Bejan Number, which indicate the relative importance of thermal and fluid irreversibilities, are presented.

on steady natural convection within cavities have been carried out by earlier researchers. Gebhart28 and Hoogendoorn29 emphasized various aspects of natural convection flows in a square cavity. There are extensive studies available in literature for various material processing applications in square or rectangular cavities.30−37 A few investigations on natural convection within trapezoidal enclosures have been carried out by earlier researchers.38−41 Lyican et al.38 investigated the natural convective flow and heat transfer within a trapezoidal enclosure with parallel cylindrical top and bottom walls at different temperatures and plane adiabatic side walls. Karyakin39 reported two-dimensional laminar natural convection in an isosceles trapezoidal cavity. The heat transfer rate is found to increase with the increase in base angle. Peric40 studied natural convection in trapezoidal cavities using a control volume method and observed the convergence of results for grid-independent solutions. Kuyper and Hoogendoorn41 investigated laminar natural convection flow in trapezoidal enclosures to study the influence of the inclination angle on the flow and also the dependence of the average Nusselt number on the Rayleigh number. Although a few studies of convective heating patterns within trapezoidal containers appear in literature, efficient processing with an energy saving approach has not yet been analyzed. Thus, comprehensive analysis on an energy efficient approach on natural convection flow within a trapezoidal enclosure is important. An energy irreversible process corresponds to exergy loss or loss of available energy which is also known as entropy generation. The minimization of entropy generation is a major challenge to optimize energy saving processing. Entropy generation minimization is the method of thermodynamic optimization, where the classical thermodynamics is combined with transport processes into simple models and optimizations are subjected to “finite-size” and “finite-time” constraints.42 The idea of thermodynamic optimization may be stated as follows: every process is inherently a irreversible process and hence some amount of useful or available energy (called “exergy”) is destroyed during the course of the process due to irreversibilities. This leads to reduction in the maximum achievable efficiency of the process. The “loss” or “destruction” of available energy due to irreversibilities can be quantified in terms of “entropy generation” based on the Guoy-Stodola theorem.43 The minimization of entropy generation results in maximum reduction of irreversibilities associated with the process, and thus the overall system efficiency is enhanced. In general, the irreversibilities during thermal convection are due to heat transfer and fluid friction. Therefore, the strategies to minimize their generation may be arrived at by analyzing the entropy generation due to heat transfer and fluid flow irreversibilities to get optimum design for any thermal process. The entropy generation minimization (EGM) method has been extensively discussed by Bejan.42,43 Many studies on the analysis of various physical systems based on the EGM method have been reported in the literature.44−52 The prime objective of this study is to analyze the heat transfer rate and entropy generation due to heat transfer and fluid friction during natural convection in trapezoidal enclosures where the bottom wall is uniformly heated, vertical wall(s) are linearly heated or cooled, and the top wall is well insulated. The natural convection flow due to the temperature difference between a hot horizontal wall and an adjacent cold vertical wall has an important application in the heat transfer analysis of electronic equipment, specifically the so-called slim rack as

2. MATHEMATICAL FORMULATION AND SIMULATION 2.1. Velocity and Temperature Distributions. The physical model of a trapezoidal cavity with the right wall inclined at an angle φ = 45°, 60°, and 90° with X axis is shown in Figure 1. No-slip boundary conditions are assumed at solid boundaries. Fluid is considered as incompressible and Newtonian and the flow is assumed to be laminar. The Boussinesq approximation is invoked to relate density changes to temperature changes. Under these assumptions, governing equations for steady two-dimensional natural convection flow in the trapezoidal cavity using conservation of mass, momentum, and energy may be written with the following dimensionless variables or numbers. y x uL vL , V= , X= , Y= , U= L L α α T − Tc θ= Th − Tc (1) P=

pL2 ρα

, 2

Pr =

ν , α

Ra =

g β(Th − Tc)L3Pr ν2

(2)

as ∂U ∂V + =0 ∂X ∂Y 4070

(3)

dx.doi.org/10.1021/ie201107f | Ind. Eng. Chem. Res. 2012, 51, 4069−4089


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Figure 1. Schematic diagram of the physical system for case 1, linearly heated side walls, and case 2, linearly heated left wall with cold right wall for inclination angles: (a) φ = 45°, (b) φ = 60°, and (c) φ = 90°.

U

U

⎛ ∂ 2U ∂U ∂U ∂P ∂ 2U ⎞ ⎟⎟ +V =− + Pr ⎜⎜ 2 + ∂Y ∂X ∂X ⎝ ∂X ∂Y 2 ⎠

The dimensionless variables in eqs 1−7 are defined as follows. Note that, X and Y are dimensionless coordinates varying along horizontal and vertical directions, respectively; U and V are dimensionless velocity components along the X and Y directions, respectively; θ is the dimensionless temperature; P is the dimensionless pressure; Ra and Pr are Rayleigh and Prandtl numbers, respectively, and φ is the inclination angle. The momentum and energy balance equations (eqs 4−6) are solved using the Galerkin finite element method. The continuity equation (eq 3) has been used as a constraint due to mass conservation, and this constraint may be used to obtain the pressure distribution. To solve eqs 4−6, the penalty finite element method has been used where the pressure (P) is eliminated by a penalty parameter γ and the incompressibility criteria are given by eq 3 which results in

(4)

⎛ ∂ 2V ∂V ∂V ∂P ∂ 2V ⎞ ⎟⎟ + Ra Pr θ +V =− + Pr ⎜⎜ 2 + ∂X ∂Y ∂Y ⎝ ∂X ∂Y 2 ⎠ (5)

U

∂θ ∂θ ∂ 2θ ∂ 2θ +V = + ∂X ∂Y ∂X 2 ∂Y 2

(6)

The boundary conditions for velocities and temperature are U = 0,

V = 0,

θ = 1,

∀ Y = 0,

U = 0,

V = 0,

θ = 1 − Y,

∀ X sin(φ) + Y cos(φ) = 0, U = 0,

V = 0,

θ = 1 − Y,

0≤Y≤1 or

∀ X sin(φ) − Y cos(φ) = sin(φ), U = 0,

V = 0,

∂θ = 0, ∂Y

−cot(φ) ≤ X ≤ 1 + cot(φ)

0≤X≤1

θ = 0,

⎛ ∂U ∂V ⎞⎟ P = −γ⎜ + ⎝ ∂X ∂Y ⎠

0≤Y≤1

(8)

∀ Y = 1,

The continuity equation (eq 3) is automatically satisfied for large values of γ. Typical values of γ that yield consistent

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solutions are 107. Using (eq 8), the momentum balance equations (eqs 4 and 5) reduce to U

Nul = sin φ

⎛ ∂ 2U ∂U ∂U ∂ ⎛ ∂U ∂V ⎞⎟ ∂ 2U ⎞ ⎟⎟ +V =γ ⎜ + + Pr⎜⎜ 2 + ∂X ∂Y ∂X ⎝ ∂X ∂Y ⎠ ⎝ ∂X ∂Y 2 ⎠

Nu r = sin φ

and

(10)

The system of equations (eqs 6, 9 and 10) with boundary conditions (eq 7) is solved by using the Galerkin finite element method.54 2.2. Streamfunction. The fluid motion is displayed using the streamfunction (ψ) obtained from velocity components U and V. The relationships between streamfunction and velocity components for two-dimensional flows are and

∂ψ V=− ∂X

(11)

which yield a single equation ∂ 2ψ ∂U ∂V + = − 2 2 ∂ Y ∂X ∂X ∂Y

∂θ ∂n

i=1 9

Nul =

∂Φi ∂Y

⎛

∑ θi⎜⎝sin φ

i=1

(14)

∂Φi ∂Φ ⎞ + cos φ i ⎟ ∂X ∂Y ⎠

(15)

Nu r = −

∑ i=1

⎛ ∂Φ ∂Φ ⎞ θi⎜sin φ i − cos φ i ⎟ ⎝ ∂X ∂Y ⎠

1

Nu b =

X |10

=

1

∫0

Nu b dX

μTo ⎛ α ⎞2 ⎜ ⎟ k ⎝ LΔT ⎠

(22)

9

∑ f ke k=1

∂Φek ∂n

(23)

f ke

where, is the value of the function at local node k in the element e. Further, since each node is shared by four elements (in the interior domain) or two elements (along the boundary), the value of the derivative of any function at the global node number (i) is averaged over those shared elements (Ne), that is,

(16)

Note that, nine basis functions, Φi, i = 1−9, have been used for biquadratic elements. The average Nusselt numbers at the bottom, left, and right walls are

∫0 Nu b dX

(21)

∂f e = ∂n

and 9

⎧ 2⎫ ⎛⎛ ∂U ⎞2 ⎛ ∂V ⎞2 ⎞ ⎛ ∂U ⎪ ∂V ⎟⎞ ⎪ ⎜⎜⎜ ⎟ + ⎜ ⎟ ⎟⎟ + ⎜ ⎬ 2 S ψ = ϕ⎨ + ⎪ ⎪ ⎝ ∂Y ⎠ ⎠ ⎝ ∂Y ∂X ⎠ ⎭ ⎩ ⎝⎝ ∂X ⎠

In the current study, ϕ is taken as 10−4.55 A higher value for ϕ was assumed by a few earlier researchers,56,57 and several other researchers have also studied the effect of ϕ on total entropy generation.58,59 Accurate evaluation of derivatives is the key issue for proper estimation of Sθ and Sψ. A small error in the calculation of temperature and velocity gradients would propagate to a much larger error since the derivatives are second order in eqs 20 and 21. As mentioned earlier, the derivatives are evaluated on the basis of a finite element method. The current approach offers a special advantage over finite difference or finite volume solutions,59−61 where derivatives are calculated using some interpolation functions which are avoided in the current work and elemental basis sets are used to estimate Sθ and Sψ. Nine node biquadratic elements are used with each element mapped using iso-parametric mapping54 from X−Y to a unit square ξ−η domain as illustrated in Figure (2). Subsequently, the domain integrals in the residual equations are evaluated using nine node biquadratic basis functions in ξ−η domain. The derivative of any function f over an element e is written as

where n denotes the normal direction on a plane. The local Nusselt numbers at bottom wall (Nub), left wall (Nul) and right wall (Nur) are defined as 9

(19)

(20)

ϕ=

(13)

∑ θi

Nu r dS2

⎛⎛ ∂θ ⎞2 ⎛ ∂θ ⎞2 ⎞ Sθ = ⎜⎜⎜ ⎟ + ⎜ ⎟ ⎟⎟ ⎝ ∂Y ⎠ ⎠ ⎝⎝ ∂X ⎠

(12)

Using the above definition of the streamfunction, the positive sign of ψ denotes anticlockwise circulation and the clockwise circulation is represented by the negative sign of ψ. The no-slip condition is valid at all boundaries and there is no cross-flow; hence, ψ = 0 is used as residual equations at the nodes for the boundaries. 2.3. Nusselt Number. The heat transfer coefficient in terms of the local Nusselt number (Nu) is defined by

Nu b =

1/sin φ

∫0

where Sθ and Sψ are local entropy generation due to heat transfer and fluid friction. In eq 21, ϕ is called the irreversibility distribution ratio, defined as

∂ 2ψ

Nu = −

(18)

where dS1 and dS2 are the small elemental lengths along the left and right walls, respectively. 2.4. Entropy Generation. In a natural convection system, the associated irreversibilities are due to heat transfer and fluid friction. According to local thermodynamic equilibrium of linear transport theory,43 the dimensionless total local entropy generation for a two-dimensional heat and fluid flow in Cartesian coordinates in explicit form is written as

⎛ ∂ 2V ∂V ∂V ∂ ⎛ ∂U ∂V ⎞⎟ ∂ 2V ⎞ ⎟⎟ +V =γ ⎜ + + Pr⎜⎜ 2 + U ∂X ∂Y ∂Y ⎝ ∂X ∂Y ⎠ ⎝ ∂X ∂Y 2 ⎠

∂ψ U= ∂Y

Nul dS1

and

(9)

+ Ra Pr θ

1/sin φ

∫0

∂fi

1 = e ∂n N

(17) 4072

N e ∂f e i

∑

e=1

∂n

(24)



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Figure 2. (a) The mapping of the trapezoidal domain to a square domain in ξ−η coordinate system and (b) the mapping of an individual element to a single element in ξ−η coordinate system.

⎧⎛ ⎞2 ⎛ ⎞2 N N ⎪⎜ ∂ ∂ ⎟ ⎜ Sψ,total = ϕ 2⎨ ( ∑ Uk Φk )⎟ + ⎜ ( ∑ Vk Φk )⎟⎟ ∂ X Ω ⎪⎜ ⎠ ⎝ ∂Y k = 1 ⎠ k=1 ⎩⎝ ⎛ ⎞2 ⎫ N N ⎪ ∂ ∂ ⎜ + ⎜ ( ∑ Uk Φk ) + ( ∑ Vk Φk )⎟⎟ ⎬ dX dY ∂X ⎝ ∂Y k = 1 ⎠ ⎪ k=1 ⎭

Therefore, at each node, local entropy generation for thermal (Sθ,i) and flow fields (Sψ,i) are given by ⎛⎛ ∂θ ⎞2 ⎛ ∂θ ⎞2 ⎞ Sθ, i = ⎜⎜⎜ i ⎟ + ⎜ i ⎟ ⎟⎟ ⎝ ∂Y ⎠ ⎠ ⎝⎝ ∂X ⎠

∫

(25)

(29)

⎧ 2⎫ ⎛⎛ ∂U ⎞2 ⎛ ∂V ⎞2 ⎞ ⎛ ∂U ⎪ ∂Vi ⎞ ⎪ i⎟ i⎟ ⎟ ⎜ i ⎜ ⎨ ⎬ ⎜ ⎜ ⎟ Sψ, i = ϕ 2⎜ + + + ⎪ ⎪ ⎝ ∂Y ⎠ ⎟⎠ ⎝ ∂Y ∂X ⎠ ⎭ ⎩ ⎝⎝ ∂X ⎠

The integrals are evaluated using three-point element-wise Gaussian quadrature integration method. The relative dominance of entropy generation due to heat transfer and fluid friction is given by the Bejan number (Beav), a dimensionless parameter defined as,

(26)

Note that, the derivatives, ∂θi/∂X, ∂θi/∂Y, ∂Ui/∂X, ∂Ui/∂Y, ∂Vi/ ∂X, and ∂Vi/∂Y are evaluated following eq 24. The combined total entropy generation (Stotal) in the cavity is given by the summation of total entropy generation due to heat transfer (Sθ,total) and fluid friction (Sψ,total), which in turn are obtained via integrating the local entropy generation rates (Sθ,i and Sψ,i) over the domain Ω. Stotal = Sθ,total + Sψ,total

Beav =

Sθ,total + Sψ,total

=

Sθ,total Stotal

(30)

Therefore, Beav > 0.5 implies dominance of heat transfer irreversibility and Beav < 0.5 implies dominance of fluid friction irreversibility.

3. RESULTS AND DISCUSSION 3.1. Numerical Procedure and Tests. The computational grid within the trapezoidal cavity is generated via mapping the trapezium into a square domain as shown in Figure 2 and the procedure is outlined in the Appendix. To validate the code, benchmark studies on entropy generation for thermal/fluid friction were carried out for the differentially heated square cavity (φ = 90°) with a hot left wall and cold right wall in the presence of adiabatic top and bottom walls, similar to the case reported by Ilis et al.55 The results in terms of streamlines, isotherms, and entropy generation due to heat transfer and fluid friction are in excellent agreement with the earlier work55 and

(27)

where,

Sθ,total =

Sθ,total

⎧⎛ ⎞2 ⎛ ⎞2 ⎫ N N ⎪⎜ ∂ ⎪ ∂ ⎟ ⎜ ⎨ θ Φ + θ Φ ( ) ( ) ∑ k k⎟ ⎜ ∑ k k ⎟⎟ ⎬ dX dY ∂ ∂ X Y Ω ⎪⎜ ⎠ ⎝ ⎠ ⎪ k=1 k=1 ⎩⎝ ⎭

∫

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at the corner nodes and keeping the adjacent grid-nodes at their respective wall temperatures.63 The Gaussian quadrature-based finite element method has been used in the current investigation, and this method provides smooth solutions in the computational domain including the singular points as evaluation of residuals depends on interior Gauss points. The validation of results on entropy generation in trapezoidal cavities are not discussed owing to the lack in data of earlier works. Various test cases have been discussed in terms of isotherms (θ), streamlines (ψ), entropy generation due to heat transfer (Sθ), and fluid friction (Sψ) in the following section. 3.2. Case 1: Linearly Heated Side Walls. Table 2 illustrates average Nusselt numbers at the bottom wall (Nub) Table 2. Comparison of Average Nusselt Number at the Bottom Wall (Nub) and Bejan Number (Beav) for Various Grid Systems in Case 1 at Ra = 105 and Pr = 0.7 with Various Inclination Angles (φ) Nub

Figure 3. Isotherm (θ), streamfunction (ψ), entropy generation due to heat transfer (Sθ) and entropy generation due to fluid friction (Sψ) contours for cavity with hot left wall and cold right wall with adiabatic top and bottom walls with Pr = 0.7 and Ra = 105 (benchmark problem).

Table 1. Comparisons of Present Results with the Benchmark Resolutions of Davis62 for Natural Convection in Air (Pr = 0.71) Filled Square Cavity with 28 × 28 Biquadratic Elements Ra 3

Davis61

10 104 105

|ψ|max

Nu

1.1746 5.0737 9.6158

1.1179 2.2482 4.5640

|ψ|max

Nu

9.612

1.118 2.243 4.519

φ

24 × 24

26 × 26

28 × 28

24 × 24

26 × 26

28 × 28

45° 60° 90°

1.2408 1.4306 1.6618

1.2407 1.4305 1.6591

1.2407 1.4304 1.6571

0.0878 0.0981 0.1451

0.0878 0.0981 0.1451

0.0878 0.0981 0.1451

and Bejan number (Beav) for various grids, and it is found that 28 × 28 biquadratic elements are adequate to obtain gridindependent results. Figures 4−7 illustrate isotherms (θ), streamlines (ψ), entropy generation maps due to heat transfer (Sθ), and fluid friction (Sψ) for various Pr values (= 0.015, 0.7, and 1000) with Ra = 103−105 corresponding to case 1. In this case, the bottom wall of the cavity is heated uniformly and the side walls are linearly heated. It is observed that the flow and temperature patterns are symmetric about its axis due to its geometric symmetry with respect to a central symmetric line (see Figures 4−7). The entropy generation maps are also symmetric since they evolve due to symmetric velocity and temperature gradients. At Ra = 103 for Pr = 0.015, isotherms with θ = 0.1−0.7 occur symmetrically near the side walls of the enclosure for φ = 45° (Figure 4a). The other isotherms with θ ≥ 0.8 are smooth curves symmetric with respect to a vertical symmetrical line at the center. The thickness of the boundary layer decreases with the height along the side walls due to a high-temperature gradient. The distribution of local entropy due to heat transfer (Sθ) depicts that entropy generation is higher (Sθ,max = 1) at the top corners, where a large thermal gradient occurs between the side wall and top wall. In contrast, Sθ is almost negligible at the central zone of the cavity due to a low temperature gradient as seen from nearly uniform temperature (θ = 0.7−0.9) in the core of the cavity. As expected, because of linearly heated side walls and a uniformly heated bottom wall, fluids rise up from the middle portion of the bottom wall and flow down along the two vertical walls forming two symmetric rolls with clockwise and anticlockwise rotations inside the cavity. At Ra = 103, the magnitudes of streamfunction are considerably smaller and the heat transfer is primarily due to conduction as seen in Figure 4a. The strength of the circulation cells is higher near the center and least at the walls due to no-slip boundary conditions as seen from streamfunction contours of Figure 4a. Weak flow circulation cells are observed at low Ra as depicted by |ψ|max = 0.32, and thus entropy generation due to fluid friction, Sψ, is insignificant relative

the simulation results are shown in Figure 3. Further, Table 1 shows that |ψ|max and Nu agree with the results of an earlier researcher62 for a range of Rayleigh numbers (Ra = 103−105).

present

Beav

The computational domain consists of 28 × 28 biquadratic elements which correspond to 57 × 57 grid points. Detailed computations have been carried out for various fluids of Pr (Pr = 0.015, 0.7, and 1000) within Ra = 103−105 for different cases. The fluid motion and heating patterns are studied for a uniformly heated bottom wall with either linearly heated side walls (case 1) or a linearly heated left wall with a cold right wall (case 2). The top wall is adiabatic in both the cases. It may be noted that the jump discontinuity in the Dirichlettype of wall boundary conditions at the right corner point (see Figure 1) corresponds to computational singularity in case 2. In particular, the singularity at the right corner of the bottom wall needs special attention. The grid-size dependent effect upon the Nusselt numbers (local and average) due to temperature discontinuity at the corner point tends to increase as the mesh spacing at the corner is reduced. This problem is resolved by assuming the average temperature of the two walls 4074



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Figure 4. Isotherm (θ), streamfunction (ψ), entropy generation due to heat transfer (Sθ), and entropy generation due to fluid friction (Sψ) contours for case 1 (lineally heated side walls) for Pr = 0.015 at Ra = 103 with (a) φ = 45°, (b) φ = 60°, and (c) φ = 90°.

to Sθ with Sψ,max being only 0.05, which occurs at the side walls (Figure 4a). The effects of entropy generation due to fluid friction irreversibility (Sψ) are negligible near the central portion of the cavity owing to less velocity gradients at the core. It may also be noted that significant entropy generation due to flow irreversibilities occurs near the top portion due to significant velocity gradients between the cavity wall and adjacent flow circulation cell. As φ increases to 60° (Figure 4b), the boundary layer thickness near top portion of the cavity decreases significantly due to increase in heat flow distribution near top portion of the cavity with φ. Similar to φ = 45°, active zones of entropy generation for heat transfer (Sθ) are higher near the top corners of the cavity (Sθ,max = 2.63) due to high-temperature gradient. At φ = 60°, the heat transfer near top corner increases due to increase in temperature gradient and that leads to high values of entropy generation in that regime. Therefore, at φ = 60°, Sθ near top corner (Sθ,max = 2.63) is comparatively higher than at φ = 45° (Sθ,max = 1). It is observed that streamlines at φ = 60° are qualitatively similar to that of the 45° case except that |ψ|max values are low (|ψ|max = 0.26) for φ = 60° compared to φ = 45° (Figure 4b). Similar to φ = 45°, Sψ is almost negligible for φ = 60° due to weak fluid flow. As φ further increases to 90° (Figure 4c), the thickness of the thermal boundary layer near top portion of side walls are low due to higher amount of heat flow in that regime. As φ increases from 60° to 90°, Sθ near top region increases to nearly 5 times. It is also interesting to note that the maximum local entropy generation due to heat transfer (Sθ,max = 12.59) is nearly 13 times of that of φ = 45° (Sθ,max = 1) case. This is mainly due to high-temperature gradient resulting from higher degree of inclination of linearly heated side walls

with adiabatic top wall. The strength of flow circulation cells is weaker for φ = 90° compared to φ = 45° as indicated by |ψ|max values. Note that, |ψ|max = 0.06 for φ = 90° whereas |ψ|max = 0.32 for φ = 45°. Also, due to no-slip boundary condition along the walls, high-velocity gradients exist near the walls and that contributes to entropy generation due to fluid friction (Sψ). But still, Sψ,max is insignificant relative to Sθ,max with Sψ,max being only 0.01. A few interesting patterns in thermal, flow and respective entropy maps are found to occur for Ra = 105. Isotherms with θ ≤ 0.7 are distorted and compressed along the linearly heated side walls, signifying the dominance of convection at Ra = 105 and Pr = 0.015 for φ = 45° (see Figure 5a). Compression of isotherms near side walls results in larger thermal gradients near that region and hence significant entropy generation rate is observed at that zone as shown in Figure 5a. As a result of enhanced convection at Ra = 105, thermal energy transport to the upper portion of the cavity increases leading to reduction in the thermal boundary layer thickness near the upper portion of the cavity as seen from Figure 5a. Therefore, significant heat transfer irreversibility (Sθ) is found near side walls due to significant thermal gradients in that regime for φ = 45° and Ra = 105 (see Figure 5a). At Ra = 105, the buoyancy forces also become dominant and the strength of the flow circulation cells increases as seen from larger magnitudes of streamfunctions with |ψ|max = 8.66 for φ = 45° (see Figure 5a). Also, multiple fluid circulation cells are found near top corners of the cavity for φ = 45°. Because of intense fluid circulation and as a consequence of no-slip boundary conditions along the wall, highvelocity gradients exist near the top portion of the cavity leading to larger entropy generation (Sψ,max = 229.13) due to 4075



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Figure 5. Isotherm (θ), streamfunction (ψ), entropy generation due to heat transfer (Sθ) and entropy generation due to fluid friction (Sψ) contours for case 1 (lineally heated side walls) for Pr = 0.015 at Ra = 105 with (a) φ = 45°, (b) φ = 60° and (c) φ = 90°.

gradient near the side wall compared to φ = 45°, and that leads to a slight increase in the thermal boundary layer thickness near the top portion of the side walls. Note that Sθ,max = 2.45 and Sψ,max = 229.13 for φ = 45°, whereas Sθ,max = 3.85 and Sψ,max = 271.77 for φ = 60°. Overall, the maximum value of Sψ increases as φ increases from 45° to 60° due to a more intense circulation cell occurring near the top adiabatic wall for φ = 60°. As φ increases further to φ = 90°, the boundary layer thickness near the top portion of the side wall increases significantly due to less heat flow in that regime as seen from the isotherm contours of Figure 5c. Multiple fluid circulations are found near bottom portion of the cavity for φ = 90° and the primary circulation cell fills almost three-fifth part of the cavity due to enhanced convection effect (|ψ|max = 4.45). Also, secondary circulation cells (|ψ|max = 4) are found to occur near the bottom portion of the cavity. As mentioned earlier, Sψ is a maximum at the interface between the circulation cell and the cavity wall and that is comparatively higher than Sψ at the interface between two circulation cells (see Figure 5c). It is interesting to note that Sψ,max is lower for φ = 90° (Sψ,max = 201.77) compared to the φ = 60° case (Sψ,max = 271.77), due to low velocity gradients near the interface between the cavity wall and circulation cell. This is further due to a smaller fluid circulation cell near the top portions of the cavity with |ψ|max = 4 at φ = 90° (see Figure 5c), which is much less than |ψ|max at φ = 60° corresponding to single large circulation cell at the core. As Pr increases to 0.7, the isotherms are strongly compressed toward the top portion of the side walls because of the higher

flow irreversibilities (Figure 5a). The entropy generation due to fluid friction is also observed at the intersection of two fluid circulation cells (Sψ = 26) and that is lower than Sψ,max near top portion of the cavity where a high-velocity gradient is maintained between the wall and adjacent circulation cell. Dense Sψ contours (Sψ = 145.66) in the side walls are also found near portions where the circulation cell contacts the walls of the cavity due to a high-velocity gradient for φ = 45° at Ra = 105 (Figure 5a). In contrast, |ψ| near the bottom corners of the cavity is very less and thus no active zones of Sψ occur near that regime. A detailed observation on distribution of Sθ and Sψ for high Ra (Ra = 105) and low Pr (Pr = 0.015) for φ = 45° shows that Sθ is significant in the upper portion, where the temperature gradient is large due to the intersection of top adiabatic wall with linearly heated side walls, whereas Sψ is significant in the portions where the velocity gradient is large, especially where the solid wall is in contact with the adjacent circulation cells. However, the comparison of magnitudes indicates that both Sθ and Sψ is higher for Ra = 105 due to enhanced convection effect. Note that the maximum entropy generation due to heat transfer, Sθ,max = 1 for Ra = 103, whereas Sθ,max = 2.45 for Ra = 105 and the maximum entropy generation due to fluid friction, Sψ,max is 0.05 for Ra = 103, whereas Sψ,max = 229.13 for Ra = 105 (see Figure 5a and Figure 4a). Isotherms are less compressed toward the side walls for φ = 60° compared to φ = 45°. As φ increases from φ = 45° to 60°, heat flow to the side walls decreases due to less temperature 4076



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Figure 6. Isotherm (θ), streamfunction (ψ), entropy generation due to heat transfer (Sθ) and entropy generation due to fluid friction (Sψ) contours for case 1 (lineally heated side walls) for Pr = 0.7 at Ra = 105 with (a) φ = 45°, (b) φ = 60° and (c) φ = 90°.

friction between two strong counter-current symmetric rolls of the fluid. It is interesting to note that entropy is also generated due to the converging and diverging fluid streams of two symmetric rolls near the central bottom and top portion of the cavity. The maximum value of entropy generation due to fluid friction, Sψ,max = 191.4, 263.58, and 182.5 for φ = 45°, 60°, and 90°, respectively, occurs at the side walls (Figure 6). The values of Sψ,max are comparatively less than those for Pr = 0.015 for all φ. This is due to the absence of multiple circulation cells at Pr = 0.7 in contrast to those at Pr = 0.015. As Pr increases to 1000, isotherms along the walls get further compressed and the thickness of the thermal boundary layer is also reduced. Qualitative features of Sθ are identical with those of Pr = 0.7. The values of Sθ,max are comparatively higher than those at Pr = 0.7 for all φ. The maximum value of entropy generation due to heat transfer, Sθ,max = 16.57, 23.68, and 70.5 for φ = 45°, 60°, and 90°, respectively, for Pr = 1000 at Ra = 105 (Figure 7). Similar to previous results for Pr = 0.015 and 0.7, streamlines indicate stronger circulations for φ = 45° and 60°. The strength of flow circulations is represented by comparatively higher |ψ|max values of |ψ|max = 14.6 at φ = 45°, |ψ|max = 14.63 at φ = 60°, and |ψ|max = 11.2 at φ = 90° for Pr = 1000 compared to |ψ|max = 13.2 at φ = 45°, |ψ|max = 12.93 at φ = 60°, and |ψ|max = 6.63 at φ = 90° for Pr = 0.7 (Figure 7). An increase in |ψ|max values further illustrates that the strength of convection increases with Pr. It may also be noted that the flow circulation cells at higher Pr take the shape of the cavity near the walls and that signifies the enhanced convection effects. It is interesting to note that the strength of the secondary circulation reduces for higher Pr (Pr = 1000) at φ = 90° with an increase in the strength of primary circulations. The maximum

temperature gradient. It is observed that the thickness of the thermal boundary layer especially at the top portion of side walls are reduced for Pr = 0.7 compared to Pr = 0.015 (Figure 6a−c). This is due to a decrease in thermal diffusivity for higher Pr fluids. Thus the entropy generation due to heat transfer is significant along the side walls. Enhanced thermal mixing at the central core of the cavity leads to uniform temperature distribution with θ = 0.6−0.8 for φ = 45°, 60°, and θ = 0.5−0.7 for φ = 90°, and consequently entropy generation due to heat transfer is negligible in the core region. The values of Sθ,max are comparatively higher than that for Pr = 0.015 for all φ. The maximum value of entropy generation due to heat transfer Sθ,max = 12.56, 16.35, and 35.84 for φ = 45°, 60°, and 90°, respectively, for Pr = 0.7 at Ra = 105 (Figure 6). At higher Ra (Ra = 105) with Pr = 0.7, the flow circulation cells fill the entire cavity and attain the shape of the cavity especially for φ = 45° and 60°. At φ = 90°, the strength of secondary circulation cells decreases as depicted by |ψ|max = 3. Active zones of frictional irreversibilities are found near the top, bottom, and side walls due to strong counter-rotating circulation cells and no-slip at the walls. Note that, for higher Pr, all the walls of cavity act as strong active sites of Sψ. The magnitudes of Sψ at the walls are found to be very high, and the dense contours of Sψ over a large regime near the side walls illustrate the zones of high entropy generation due to fluid friction. Note that, the bottom wall of the cavity also contributes significantly to Sψ, in contrast to that in the Pr = 0.015 case. The velocity gradients are found to be small along the eye of the vortices and therefore Sψ at core region is small compared to the Sψ at the walls. Note that a significant amount of entropy generation due to fluid friction is observed along the vertical center line. This is due to the 4077



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Figure 7. Isotherm (θ), streamfunction (ψ), entropy generation due to heat transfer (Sθ) and entropy generation due to fluid friction (Sψ) contours for case 1 (lineally heated side walls) for Pr = 1000 at Ra = 105 with (a) φ = 45°, (b) φ = 60°, and (c) φ = 90°.

value of entropy generation due to fluid friction Sψ,max = 189.01, 266.5, and 348.1 for φ = 45°, 60°, and 90°, respectively, occurs at the side walls (Figure 7). The values of Sψ near the top and bottom walls also show active zones of entropy generation due to fluid friction as seen from the Figure 7a−c. It is interesting to note that significant entropy is also generated due to the converging fluid streams of two symmetric rolls near the central bottom of the vertical center line for all φs in case 1 at higher Pr (Pr = 1000) (see Figure 7a−c). 3.3. Case 2: Linearly Heated Left Wall with Cold Right Wall. Table 3 illustrates the average Nusselt number at the bottom wall (Nub) and Bejan number (Beav) for various grids and it is found that 28 × 28 biquadratic elements are adequate to obtain grid-independent results. Figures 8−11 illustrate isotherms (θ), streamlines (ψ), and entropy generation maps due to heat transfer (Sθ) and fluid friction (Sψ) for various Pr values (= 0.015, 0.7, 1000) with Ra = 103−105 corresponding to case 2. In this case, the bottom wall of the cavity is heated uniformly in the presence of a linearly heated left wall with a cold right wall. Because of a uniformly heated bottom wall and cold right wall, the singularity appears at the right bottom edge of the cavity. Distributions of θ, Sθ, ψ, and Sψ for Pr = 0.015

Table 3. Comparison of Average Nusselt Number at the Bottom Wall (Nub) and Bejan Number (Beav) for Various Grid Systems in Case 2 at Ra = 105 and Pr = 0.7 with Various Inclination Angles (φ) Beav

Nub φ

24 × 24

26 × 26

28 × 28

24 × 24

26 × 26

28 × 28

45° 60° 90°

5.9518 4.9346 5.9518

5.9869 4.9524 5.9869

5.5664 4.9718 6.0219

0.1909 0.1735 0.1909

0.1922 0.1744 0.1922

0.1591 0.1753 0.1934

at Ra = 103 are depicted in Figure 8a−c. Isotherms with θ ≤ 0.6 for φ = 45° are compressed alongside the walls. The formation of the thermal boundary layer along the left wall of the cavity is weaker, whereas isotherms are compressed along the right wall of the cavity forming a strong thermal boundary layer. The thickness of the boundary layer increases with the height along the right wall due to a low temperature gradient. The distribution of local entropy due to heat transfer (Sθ) depicts that entropy generation is higher (Sθ,max = 1764) at the bottom corner of the right cold wall for φ = 45° in contrast to a maximum Sθ (Sθ,max) near the top corners of the cavity in case 1 at identical Pr, φ, and Ra (see Figure 4a and Figure 8a). The 4078



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Figure 8. Isotherm (θ), streamfunction (ψ), and entropy generation due to heat transfer (Sθ) and entropy generation due to fluid friction (Sψ) contours for case 2 (linearly heated left wall with cold right wall) for Pr = 0.015 at Ra = 103 with (a) φ = 45°, (b) φ = 60°, and (c) φ = 90°.

9421.5) in the right bottom corner due to high-thermal gradient. Temperature distribution is uniform in the core with θ = 0.6−0.7. Entropy generation due to heat transfer is negligible in the core due to thermal mixing (see Figure 8b). At φ = 60°, secondary circulation cells (|ψ|max = 0.11) are still observed with a primary circulation cell (|ψ|max = 0.9), and therefore, Sψ,max (Sψ,max = 0.34) occurs near the right wall of the cavity due to high-velocity gradients between a primary circulation cell and the bottom wall (see Figure 8b). As φ increases further (φ = 90°), isotherms are uniformly distributed and that results in uniform smaller thermal gradients throughout the domain except the right corner of the bottom wall where Sθ,max = 7056 is observed. Overall, the entropy generation near the left wall is much smaller for φ = 90° due to linear heating. It is interesting to observe that the weak fluid flow at low Ra (Ra = 103) corresponds to weaker Sψ compared to Sθ at φ = 90° similar to case 1 at identical Pr, φ, and Ra (see Figure 8c). Figure 9a displays isotherms (θ), streamlines (ψ), and entropy generation maps due to heat transfer (Sθ) and fluid friction (Sψ) for Pr = 0.015 at Ra = 105. Isotherms with θ ≤ 0.6 are distorted and compressed along the upper portion of the cold left wall, signifying dominant convection at Ra = 105 and Pr = 0.015 for φ = 45° (see Figure 9a). As a result of enhanced convection at Ra = 105, thermal energy transport to the lower and middle

high entropy generation is due to a high-thermal gradient at the bottom corner zone of the right cold wall. In contrast, Sθ is almost negligible at the bottom corner region of the left linearly heated wall due to a lower temperature gradient. The magnitude of streamfunction is considerably smaller, and heat transfer is primarily due to conduction as seen in Figure 8. Note that the symmetric circulation pattern was observed for case 1 and that it is absent in the present case owing to nonsymmetric thermal boundary conditions. Primary circulation cells occupy a larger right portion of the cavity with |ψ|max = 1, whereas secondary circulation cells occupy a smaller left portion of the cavity with |ψ|max = 0.19 (Figure 8a). This is because the strength of fluid circulation is higher near the left wall compared to the right wall due to imposed boundary conditions along the side walls (linear heating on the left wall and cold isothermal right wall). As a result, strong circulation patterns are formed on the right side of the cavity, whereas weak secondary circulation patterns appear on the left side of the cavity. However, due to weak fluid flow at low Ra (Ra = 103) and Pr (Pr = 0.015), Sψ is negligible for φ = 45° (see Figure 8a). At φ = 60°, distributions of θ, Sθ, ψ, and Sψ are qualitatively similar to that of φ = 45°. As φ increases to 60°, isotherms are largely compressed in the region near corners of the right bottom wall. This further leads to active zones of Sθ (Sθ,max = 4079



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Figure 9. Isotherm (θ), stream-function(ψ), entropy generation due to heat transfer (Sθ) and entropy generation due to fluid friction (Sψ) contours for case 2 (linearly heated left wall with cold right wall) for Pr = 0.015 at Ra = 105 with (a) φ = 45°, (b) φ = 60° and (c) φ = 90°.

found near portions where the circulation cell expands almost near to the right wall of the cavity, and that represents the active zones of high fluid friction entropy generation (Sψ) for φ = 45° (Figure 9a). Note that a comparatively small amount of Sψ occurs at the left wall, there being only 54.97 at the interface between the left circulation cell and left wall due to less flow strength of the left circulation cell. In contrast to that at the side walls, streamfunctions near the left bottom corner of the cavity are very less, and thus no significant active zones of Sψ occur near those regimes. A comparison of Sθ and Sψ for high Ra (Ra = 105) and low Pr (Pr = 0.015) for φ = 45° shows that Sθ is significant in the lower portion of the right wall of the cavity where the temperature gradient is large due to immediate contact of the hot bottom wall with the cold right wall, whereas Sψ is significant in the portions where the velocity gradient is large, especially where the solid wall is in contact with the adjacent circulation cells. It is interesting to note that magnitudes of entropy generation due to heat transfer (Sθ) are identical for both Ra = 103 and 105, whereas maximum entropy generation due to fluid friction (Sψ,max) is 0.36 for Ra = 103 and 642.51 for Ra = 105 (see Figures 8a and 9a). It is interesting to observe that Sψ,max is high (Sψ,max = 642.51) for the present case (see Figure 9a) compared to case 1 (Sψ,max = 229.13) for identical Pr and Ra at φ = 45° (see Figure 5a). This is due to the high-flow strength near

portions of the right wall increases leading to reduction in the thermal boundary layer thickness as seen from Figure 9a. As a consequence, a considerable entropy generation rate due to heat transfer irreversibility is observed near those zones as shown in Figure 9a. It may also be noted that the entropy generation due to heat transfer (Sθ) for Ra = 105 and φ = 45° at the right bottom corners remain identical to that of Ra = 103 (see Figures 9a and 8a). However, Sθ,max is much higher for the present case (Sθ,max = 1764) than that of case 1 with Pr = 0.015, φ = 45°, and Ra = 105 (Sθ,max = 2.45) (see Figure 5a). The buoyancy forces become dominant, and the strength of the circulation cells increases as seen from large magnitudes of streamlines with |ψ|max = 12.5 for primary circulation at higher Ra (Ra = 105) (see Figure 9a). Also, secondary fluid circulation cells are found near the top portion of the side walls for φ = 45°. Owing to high intense primary fluid circulation cells and as a consequence of no-slip boundary conditions along the wall, high-velocity gradients exist near the top portion of the cavity leading to larger fluid friction entropy generation (Sψ,max = 642.51) (Figure 9a). Significantly large entropy generation due to fluid friction is also observed near the right corner of the bottom wall (Sψ = 512.32) and that is lower than Sψ,max near the top portion of the cavity where a high-velocity gradient is maintained between the wall and adjacent circulation cell. It may also be observed that dense Sψ contours in the right wall are 4080



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Figure 10. Isotherm (θ), streamfunction(ψ), entropy generation due to heat transfer (Sθ) and entropy generation due to fluid friction (Sψ) contours for case 2 (linearly heated left wall with cold right wall) for Pr = 0.7 at Ra = 105 with (a) φ = 45°, (b) φ = 60° and (c) φ = 90°.

occur near all corners of the cavity. The primary circulation cell occupies almost the central part of the cavity due to an enhanced convection effect (|ψ|max = 20). As mentioned earlier, Sψ is maximum at the interface between the circulation cell and the right cavity wall, and that is comparatively higher than Sψ at the interface between two circulation cells (see Figure 9c). It is interesting to note that Sψ,max is higher for this case (Sψ,max = 749.3) compared to the φ = 60° case (Sψ,max = 714.3) due to high-velocity gradients near the interface between the cavity wall and the circulation cell (see Figure 9b,c). It is found that, Sθ,max and Sψ,max values are comparatively higher for the present case (case 2) than for case 1 owing to preferential heating and flow intensities at the right portion of the cavity at Ra = 105 and Pr = 0.015. It is found that, Sθ,max = 2.45, 3.85, and 22.83 occur for φ = 45°, 60°, and 90°, respectively, for case 1, whereas Sθ,max = 1764, 9421.5, and 7056 occur for φ = 45°, 60°, and 90°, respectively, for case 2 at Ra = 105 and Pr = 0.015. It may also be noted that, Sψ,max = 229.13, 271.77, and 201.77 occur for φ = 45°,60°, and 90°, respectively, for case 1, whereas Sψ,max = 642.51, 714.3, and 749.3 occur for φ = 45°,60°, and 90°, respectively, for case 2 at Ra = 105 and Pr = 0.015 (see Figure 5 and Figure 9). As Pr increases to 0.7 (Figure 10), the convection effect is further enhanced due to strong buoyancy forces. The values of streamfunction in the primary circulations also increase and

the right portion of the cavity in case 2 as seen, based on the larger magnitudes of streamfunction (|ψ|max = 12.5), compared to case 1 (|ψ|max = 8.66) (see Figure 9a and Figure 5a). Isotherms are more compressed toward the right cold wall for φ = 60° compared to φ = 45° (see Figure 9b). As φ increases, heat flow to the right wall increases, and that leads to a slight decrease in thermal boundary layer thickness near the top portion of the right wall. However, active zones of entropy generation due to heat transfer (Sθ) for φ = 60° and Ra = 105 remain identical with φ = 60° and Ra = 103 near the lower corners of the bottom wall (see Figure 8b and Figure 9b). It is interesting to observe that Sψ,max for case 2 (Sψ,max = 714.3) is higher than that for case 1 (Sψ,max = 271.77) at φ = 60° for Pr = 0.015 and Ra = 105 (see Figure 9b and Figure 5b). Note that the local maxima of Sψ occurring at the left wall is 82.51 for φ = 60°, which is comparatively higher than that for φ = 45°. As φ increases further, the boundary layer thickness near the top portion of the cold right wall increases significantly due to less heat flow in that regime as seen from the isotherm contours of Figure 9c. Maximum entropy generation due to heat transfer still occurs at the right corner of the bottom wall similar to φ = 45° and 60°. It may also be noted that the entropy generation due to heat transfer (Sθ) for Ra = 105 and φ = 90° at the right bottom corner remains identical to that of Ra = 103 (see Figures 9c and 8c). Multiple fluid circulations are found to 4081



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Figure 11. Isotherm (θ), streamfunction (ψ), entropy generation due to heat transfer (Sθ) and entropy generation due to fluid friction (Sψ) contours for case 2 (linearly heated left wall with cold right wall) for Pr = 1000 at Ra = 105 with (a) φ = 45°, (b) φ = 60° and (c) φ = 90°.

rolls of the fluid. It is interesting to note that considerable fluid friction entropy is also generated due to converging and diverging fluid streams of two asymmetric rolls near the upper portion of the left wall and the left portion of top wall at φ = 45° (see Figure 10a). The interface between the circulation cell and the cavity wall act as strong active zones for Sψ (Sψ,max = 507.5 and Sψ = 33.81). Velocity gradients are found to be large along the top adiabatic wall and therefore Sψ in that regime is significant for φ = 45° (Sψ = 104.09) (Figure 10a). Note that the maximum of Sψ is found near the middle portion of the right wall for the present case in contrast to the top adiabatic wall of the cavity for Pr = 0.015 case at φ = 45° (see Figure 10a and Figure 9a). Compression of isotherms is more pronounced near the corners of the right corner of the bottom wall and middle portion of the right wall for φ = 60° (see Figure 10b), and entropy generation due to heat transfer near the right wall is qualitatively similar with the previous case of Pr = 0.015 (see Figure 10b and Figure 9b). At φ = 60°, entropy generation due to fluid friction (Sψ) is enhanced with enhanced fluid flow. The right cold wall of the cavity acts as the strong active sites of Sψ (see Figure 10b). Note that Sψ,max = 638.1 occurs near the middle portion of the right wall and Sψ = 103.4 occurs at the top adiabatic wall. It is also found that Sψ = 87.29 occurs near the lower portion of the left wall.

isotherms are more compressed near the cold right wall. The thickness of thermal boundary layer is thinner compared to that in the previous case of Pr = 0.015 as higher Pr fluids represent smaller thermal diffusivity. Thus the entropy generation due to heat transfer is significant along the bottom and the side walls. Enhanced thermal mixing at the central core of the cavity leads to uniform temperature with θ = 0.5−0.6, and, therefore, entropy generation due to heat transfer is negligible in the core region. At Pr = 0.7 and Ra = 105 for φ = 45°, the frictional irreversibility along the top, bottom, and side walls is further enhanced due to strong counter-rotating circulation cells and no-slip at the walls of the cavity. Note that, for higher Pr, all the walls of the cavity act as strong active sites of Sψ. The magnitudes of Sψ at the walls are found to be very high, and the dense contours of Sψ over a large regime near the side walls illustrate the zones of high entropy generation due to fluid friction. Note that the bottom portion of the left wall contributes a significant amount of Sψ in the Pr = 0.7 case, whereas active zones of Sψ are observed near the middle portion of the left wall for the Pr = 0.015 case. The velocity gradients are found to be small along the eye of the vortices, and therefore Sψ near the core region is small or compared to Sψ at the walls. Note that, a significant amount of entropy generation due to fluid friction is observed along the intersection of the circulation cells. This is due to fluid friction between two counter-current symmetric 4082



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Figure 12. Variation of total entropy generation (Stotal), average Bejan number (Beav), average Nusselt number on the left wall (Nul), and average Nusselt number on the bottom wall (Nub) with Rayleigh number (Ra) are shown for both cases (case 1 and case 2) at Pr = 0.015 (molten metals).

As Pr increases to 1000 (Figure 11), distribution of Sθ remains almost similar to that of the Pr = 0.7 case. However the distribution of Sψ is different from that of Pr = 0.7 case. Interestingly, the lower portion of the left wall also acts as active sites for entropy generation owing to fluid friction, especially at φ = 90°. This is due to high-velocity gradients induced by the change of flow direction of the high-velocity fluid streams upon reaching the bottom wall. An increase of Pr to 1000 (Figure 11a−c) also makes an observable increase in Sψ,max (Sψ,max = 441.7, 588.13, and 735.7 for φ = 45°, 60°, and 90°, respectively) compared to that in the Pr = 0.7 case (Sψ,max = 507.5, 638.1, and 807.5 for φ = 45°, 60°, and 90°, respectively). 3.4. Total Entropy Generation, Bejan Number, Average Nusselt Number on the Left and Bottom Walls. The variations of total entropy generation due to heat transfer and fluid friction irreversibilities (Stotal), average Bejan number (Beav), average Nusselt number on the bottom wall (Nub) and left wall (Nul) for various Pr (0.015, 0.7, and 1000) fluids in both heating cases (cases 1 and 2) are presented in Figure 12 and Figure 13. The qualitative features of Stotal, Beav, Nub and Nul at Pr = 0.7 are found to be similar with the Pr = 1000 case and hence results of Pr = 0.7 are not shown for brevity of the manuscript.

The thickness of the thermal boundary layer near the top portion of the right wall becomes smaller compared to that in case 1 with Pr = 0.015 for φ = 90° due to high heat flow in that regime (see Figure 10c). One more entropy active zone due to heat transfer (Sθ) occurs near the right wall due to the compression of isotherms in that regime, and that is further due to high heat transfer in that regime as seen from the entropy map (Sθ) of Figure 10c. Enhanced thermal mixing at the core of the cavity leads to uniform temperature with θ = 0.1 and therefore, entropy generation due to heat transfer is negligible in the core region as shown in Figure 10c. Note that, all the walls of the cavity act as strong active sites of Sψ for higher Ra (see Figure 10c). It is also found that Sψ value near right wall is found to be maximum (Sψ,max = 807.5), and that is confined to a very small zone near the wall. Note that local maxima of Sψ at the left wall is 277.05. The dense contours of Sψ over a large regime at the interface between fluid circulation cells also illustrate the zones of entropy production due to fluid friction irreversibility. Note that, Sψ at the eye of the vortices is insignificant because of small velocity gradients near that region. It may also be noted that, Sψ,max = 191.4, 263.58, and 182.5 occur for φ = 45°, 60°, and 90°, respectively, for case 1, whereas Sψ,max = 507.5, 638.1, and 807.5 occur for φ = 45°,60°, and 90°, respectively, for case 2 at Ra = 105 and Pr = 0.7 (see Figure 6 and Figure 10). 4083



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Figure 13. Variation of total entropy generation (Stotal), average Bejan number (Beav), average Nusselt number on the left wall (Nul) and average Nusselt number on the bottom wall (Nub) with Rayleigh number (Ra) are shown for both cases (case 1 and case 2) at Pr = 1000 (olive/engine oils).

for φ = 45° due to a significant Sψ,total of Stotal (Sθ,total = 0.906, Sψ,total = 7.6439, and Stotal = 8.5499), followed by φ = 60° (Sθ,total = 1.0419, Sψ,total = 7.1083 and Stotal = 8.1502) and φ = 90° (Sθ,total = 1.2132, Sψ,total = 6.1826 and Stotal = 7.3958) (see bottom panel plots of Figure 12a). It is interesting to note that Stotal increases very rapidly for φ = 90° at Ra ≥ 5 × 104 as the oppositely rotated secondary cells become prominent. This is attributed to the formation of secondary circulation cells, which result in a gradual increment of Sψ due to fluid friction irreversibility at the interface between two circulation cells. Average Bejan number (Beav) indicates the importance of entropy generation due to heat transfer (Sθ) or fluid friction (Sψ) irreversibilities. The dominance of heat transfer or fluid friction irreversibilities is indicated by the average Bejan number (Beav). As mentioned earlier, Beav ≥ 0.5 indicates that entropy generation is heat transfer dominant, whereas Beav ≤ 0.5 indicates fluid friction dominant entropy generation. A common decreasing trend in Beav with Ra is observed for all φs in the middle panel plots of Figure 12a. The maximum value for Beav (Beav = 1) occurs at low Ra (Ra = 103), indicating that entropy generation in the cavity is primarily due to heat transfer irreversibility (Sθ,total) at the conduction dominant mode. As Ra increases to 105, fluid friction irreversibility (Sψ,total) increases, and that dominates over heat transfer irreversibility

Figure 12a represents distributions of Stotal, Beav, Nub and Nul for Pr = 0.015 for case 1. The total entropy generation in the cavity increases gradually until Ra ≤ 6 × 103 for all φs (see bottom panel plots of Figure 12a). This is due to smaller entropy generation due to fluid friction (Sψ,total) compared to entropy generation due to heat transfer (Sθ,total) as seen from Sθ and Sψ maps of Figure 4a−c for all φs. Note that Sθ,total = 0.5322 and Sψ,total = 0.006 for 45°, Sθ,total = 0.6359 and Sψ,total = 0.0044 for 60° and Sθ,total = 0.7299 and Sψ,total = 0.0005 for 90° at Ra = 103. Therefore, total entropy generation (Stotal) at lower Ra (Ra = 103) is a maximum for φ = 90° compared to φ = 45°, due to dominant entropy generation due to heat transfer irreversibility. It is observed that the significant convection is initiated at Ra ≥ 6 × 103 for φ = 45° and 60°, and at Ra = 104 for φ = 90° ( Figure 12a). It may also be noted that, Sθ,total = 0.6528 and Sψ,total = 0.4154 for 45°, Sθ,total = 0.7775 and Sψ,total = 0.3953 for 60°, and Sθ,total = 0.7577 and Sψ,total = 0.0854 for 90° at Ra = 104. It is interesting to note that Stotal increases exponentially for Ra ≥ 104 due to a gradual increase of fluid friction irreversibility (Sψ,total) in addition to the heat transfer irreversibility (Sθ,total) based on high convective motion of the fluid as seen from contour plots of ψ and Sψ for all φs (see Figure 5a−c). Note that |ψ|max = 8.66 for φ = 45°, |ψ|max = 8.37 for φ = 60°, and |ψ|max = 4.45 for φ = 90° at Ra = 105 and Pr = 0.015 (see Figure 5a−c). At higher Ra (Ra = 105), it is observed that Stotal is larger 4084



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creases to 105, convective motion of the fluid is enhanced, and that leads to dominant fluid friction irreversibilities (Sψ,total) at Ra ≥ 104 for all φs. The total entropy generation (Stotal) at higher Ra (Ra = 105) is a maximum for φ = 90°, whereas φ = 45° corresponds to minimum Stotal. Note that Sθ,total = 4.15, Sψ,total = 14.56, and Stotal = 18.71 for φ = 45°, whereas Sθ,total = 5.49, Sψ,total = 18.96, and Stotal = 24.45 for φ = 90° at Ra = 105. Even though the Beav map for Pr = 0.015 and case 2 follows a qualitatively similar trend as that of case 1 at lower Ra, Beav at Ra = 105 for case 2 is higher than that of case 1 for all φs (see middle panel plot of Figure 12a and Figure 12b). This may be due to larger Sθ,total for all φs. Note that Beav = 0.221, 0.231, 0.224 for φ = 45°, 60°, and 90° at Ra = 105. These are clearly indicated by the values of Sθ,total and Sψ,total, where Sθ,total and Sψ,total are 4.15 and 14.56 for φ = 45°, 4.81, and 16.04 for φ = 60° and 5.49 and 18.96 for 90° at Ra = 105. It is also found that Stotal at φ = 45° closely follows φ = 60°, therefore Beav distributions for φ = 45° also closely follows φ = 60°. This is due to approximately constant Sθ,total values and gradually increasing Sψ,total values over the range of Ra = 103−105. It is interesting to note that Sψ,total values at Ra = 105 and Pr = 0.015 for case 2 is comparatively higher than that at case 1 for the same Ra and Pr. Note that, Sψ,total = 14.56 for φ = 45°, Sψ,total = 16.04 for φ = 60°, and Sψ,total = 18.96 for φ = 90° at case 2, whereas Sψ,total = 8.55 for φ = 45°, Sψ,total = 8.15 for φ = 60°, and Sψ,total = 7.39 for φ = 90° at case 1 for Ra = 105. This clearly shows that more amounts of available energy (exergy loss is minimum) are utilized to overcome the irreversibilities due to fluid friction (Sψ,total) in case 2 compared to case 1 at Ra ≈ 105. It is observed that Nul for φ = 45° and 60° in case 2 is quantitatively smaller due to a smaller temperature gradient near the left wall. This is due to a weak left circulation cell which is in contact with the left wall which results in a low heat transport at the onset of convection (Ra ≥ 104). It may be noted that Nul for φ = 90° is larger compared to φ = 45° and 60° due to more heat transfer rate from left wall in case 2 due to a large temperature gradient near the left wall. This is due to very weak secondary circulations and the middle portion of the left wall directly contacts with strong primary circulation which results in more heat transport from the left wall at the onset of convection (Ra ≥ 104). Note that Nul = 0.11 for φ = 45°, Nul = 0.29 for φ = 60° and Nul = 1.32 for φ = 90° at Ra = 105 for case 2. It is found that, distributions for Nub for φ = 45° and 60° in case 2 are qualitatively similar to those of case 1 but Nub is larger in case 2 compared to case 1 due to high heat transfer rate from bottom wall in case 2. This is due to a larger temperature gradient which results in high heat transport due to (Ra ≥ 104), than lost energy due to fluid friction irreversibilities (Sψ,total) along the bottom wall. It may also be noted that Nub = 3.91 for φ = 45°, Nub = 3.62 for φ = 60° and Nub = 5.28 for φ = 90° at Ra = 105 in case 2. It is found that, Nub is quite large for φ = 90° as the temperature gradient near the right portion of the bottom wall is much larger to compensate over entropy generation due to heat transfer irreversibility. The distributions of Stotal, Beav, Nul and Nub for Pr = 1000 and case 1 are shown in Figure 13a. Due to higher momentum diffusivity at higher Pr, Sψ,total for Pr = 1000 is comparatively higher than that for Pr = 0.015 and therefore, Stotal values are

(Sθ,total) due to enhanced fluid circulations within the cavity. Therefore, Beav decreases with Ra for all φs as seen in middle panel plots of Figure 12a. It may be noted that Sθ,max ≤ Sψ,max for all φs (see Figure 5(a−c)) and the dominant fluid friction irreversibility (Sψ,total) leads to an increase in Stotal at a higher Ra (Ra = 105). Consequently, a large amount of available energy (exergy loss is maximum) is utilized to overcome the irreversibilities due to fluid friction at high Ra (Ra = 105). It may also be noted that Beav ≤ 0.5 occurs at all φs due to Sψ,total dominance at Ra = 105. It may be interesting to note that Beav = 0.106, 0.128, 0.164, and Stotal = 8.55, 8.15, 7.39 for φ = 45°, 60°, and 90°, respectively, at Ra = 105. Overall, average bejan number (Beav) is maximum for φ = 90° followed by φ = 60° and 45° due to minimum total entropy generation (Stotal) for φ = 90° followed by φ = 60° and φ = 45° at Ra = 105. The average Nusselt numbers for left and right walls remain same due to symmetric boundary conditions in the present case (case 1: linear heating on both the walls). The values of the average Nusselt numbers along the side walls are less compared to that along the bottom wall. This is due to the fact that the heat transferred to the fluid from the bottom wall is more compared to the side wall. The average Nusselt numbers for both bottom and side walls remain constant up to Ra = 5 × 103 for φ = 45° and 60°, and up to Ra = 104 for φ = 90° (see upper panel plots of Figure 12a) for case 1. The dependence of average Nusselt numbers on the Rayleigh number was found to be significant for all φs. It is interesting to note that average Nusselt number (Nub or Nul) increases very rapidly for φ = 90° at Ra ≥ 5 × 104 due to large temperature gradients near the bottom wall and side walls (see upper panel plots of Figure 12a). This leads to increase in heat flow to fluid from the bottom wall and side walls and that is illustrated by highly compressed isotherms near that region. Heat transfer rate (Nul or Nub) is high for φ = 90° at Ra = 105 and that is illustrated by large Nul and Nub for φ = 90° as seen in top panel plots of Figure 12a. It may be noted that Nul = 0.89, 0.36, and 0.23 for φ = 90°, 60°, and 45°, respectively, at Ra = 105. It may also be observed that Nub = 1.79, 0.88, and 0.71 for φ = 90°, 60°, and φ = 45°, respectively, at Ra = 105. Even though Nub and Nul distributions are qualitatively similar for all φs in case 1, Nul is less than Nub due to less heating effect in the case of linearly heated side walls compared to that of the isothermal hot bottom wall (see Figure 12). It is interesting to note that the maximum heat transfer rate ( Nub or Nul) with minimum entropy generation (Stotal) occurs for φ = 90° cavities at Ra = 105 (see bottom and top panel plots of Figure 12a). It may also be noted that lower heat transfer rates (Nub or Nul), higher Stotal, and lower Beav for φ = 45° and 60° are due to higher irreversibilities, and that further leads to higher exergy losses in trapezoidal cavities compared to square cavities at lower Pr (Pr = 0.015) in case 1. Figure 12b shows Stotal, Beav, Nul and Nub for a linearly heated left wall with a cold right wall (case 2). High values of heat transfer irreversibility (Sθ) near the right bottom corner is observed for case 2 in contrast to case 1 for all Ra between 103 and 105 and Pr = 0.015 (see Figure 9). It is also interesting to note that Stotal at Ra = 105 is higher for case 2 compared to case 1 due to high intense convective motion of the fluid (see bottom panel plots of Figure 12a and Figure 12b). As Ra in4085



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higher for Pr = 1000 at convection dominant region (Ra ≥ 104) (see bottom panel plots of Figure 12a and Figure 13a). It is observed that Stotal increases with Ra for all φs at the onset of convection at Ra ≥ 104. It may also be noted that Stotal at φ = 90° is comparatively lower than those at φ = 45° and φ = 60° due to less fluid friction irreversibility at higher Pr and φ (see bottom panel plots of Figure 13a). Note that Sθ,total and Sψ,total are 2.22 and 23.6 for φ = 45°, Sθ,total = 2.36 and Sψ,total = 22.87 for φ = 60°, and Sθ,total = 2.34 and Sψ,total = 19.49 for φ = 90° at Ra = 105. It is also found that Bejan number decreases rapidly with Ra for all φs, implying that Beav is least for φ = 45° and fluid irreversibility is more prominent at φ = 45° (see bottom and middle panel plots of Figure 13a). Heat transfer rate (Nul or Nub) is comparatively higher for Pr = 1000 than Pr = 0.015 at the onset of convection (Ra ≥ 104) and that is due to a larger temperature gradient which results in high heat transport at the onset of convection (Ra ≥ 104) than lost energy due to fluid friction irreversibilities (Sψ,total) along the bottom wall (see top panel plots of Figure 12a and Figure 13a). Maximum heat transfer rate (Nul or Nub) and minimum entropy generation (Stotal) is observed for φ = 90° similar to that for Pr = 0.015 in case 1. Although an increase in Ra (Ra ≥ 104) results in larger Sψ,total over Sθ,total for φ = 45° and φ = 60°, Nub increases sharply at Ra ≥ 104 because of dominant temperature gradients near the bottom wall (see Figure 13a). A similar explanation follows for Nul. Figure 13b shows the distributions for Pr = 1000 and case 2. It is interesting to note that Stotal is comparatively higher for Pr = 1000 at Ra = 105 due to high fluid friction irreversibility, and that is further due to high momentum diffusivity. It may also be noted that Stotal is larger for smaller values of φ due to dominant Sψ,total values over Sθ,total values at the onset of convection. Note that Sθ,total = 4.41, Sψ,total = 31.28, and Stotal = 35.39 for φ = 45°; Sθ,total = 4.1, Sψ,total = 30.76, and Stotal = 34.86 for φ = 60°; and Sθ,total = 4.21, Sψ,total = 30.59, and Stotal = 34.8 for φ = 90° at Pr = 1000 and Ra = 105. A larger amount of available energy is utilized to overcome fluid friction irreversibilities (Sψ,total) for Pr = 1000 in case 2 compared to that of case 1 at Ra = 105. This is due to a larger Sψ,total in case 2 compared to case 1 for all φs (see bottom panel plots of Figure 13b). It may also be noted that Beav decreases with Ra and the smallest Beav is observed for smaller φ (φ = 45°). This is due to an increase in Stotal with Ra and φ (see Figure 13b). Note that Beav = 0.16, 0.17, and 0.19 for φ = 45°, 60°, and 90°, respectively, for case 2 at Pr = 1000 and Ra = 105, whereas Beav = 0.086, 0.093, and 0.107 for φ = 45°, 60°, and 90°, respectively, for case 1 at Pr = 1000 and Ra = 105. Therefore, Beav is comparatively higher for case 2 than for case 1. It is also interesting to note that even though an increase in Sψ,total is more pronounced throughout the cavity, significant thermal gradients near the left wall result in a relatively profound increase in heat transfer rates for φ = 90° at Ra = 105 for case 2 in contrast to case 1 (see bottom and top panel plots of Figure 13b). Note that, Nub = 5.98, 5.04, and 5.98 and Nul = 0.24, 0.56, and 1.2 for φ = 45°, 60°, and 90°, respectively, for case 2 at Pr = 1000 and Ra = 105. On the other hand, Nub = 1.59, 1.83, and 2.12 and Nul = 0.58, 0.81, and 1.09 for φ = 45°, 60°, and 90°, respectively, for case 1 at Pr = 1000 and Ra = 105. Overall, Nub is larger for case 2 for all φ. It may also be noted that Nul is smaller for φ = 45° and 60° and larger for φ = 90° in case 2 compared to that for case 1. It may also be

noted that total entropy generation is less and heat transfer is more at φ = 90° for both cases at Ra = 105 and Pr = 1000. Hence, linear heating strategy with φ = 90° may be used in order to achieve uniform temperature distribution and minimum entropy generation for higher Pr (Pr = 1000) fluids. It is also important to note that the angle has less influence on entropy generation rate, as Stotal does not vary much for φ = 45° and 60°. Thus, the trapezoidal cavities with φ = 45° or 60° will have almost similar entropy production rate or loss effect.

4. CONCLUSION In the present study, the analysis on the heat transfer rate versus entropy generation due to the heat transfer and fluid friction irreversibilities during natural convection for linearly heated side wall(s) in trapezoidal cavities has been carried out for three different angles φ = 45°, 60°, and 90°. Case 1 involves uniform heating on the bottom wall and linearly heated side walls, whereas case 2 involves uniform heating on the bottom wall and linearly heated left wall with a cold right wall. The flow and temperature distributions are obtained for Pr = 0.015, 0.7, 1000 within a range of Rayleigh numbers (Ra = 103−105). The dimensionless entropy generation due to heat transfer (Sθ) and fluid friction (Sψ) irreversibilities, which are the functions of thermal (θ) and flow fields (ψ), are obtained for both cases. In this work, elemental basis sets are used via the Galerkin finite element method to evaluate the entropy generation terms due to heat transfer and fluid friction. The derivatives at a node are evaluated based on the function values of adjacent elements that share the node through the elemental basis set. This current approach offers accurate estimation of Sθ and Sψ. The important results of this analysis are given below: • At Ra = 103, the heat transfer in the cavity is primarily due to conduction and the total entropy generation (Stotal) is found to be higher in case 2 for all Pr. During the conduction regime at low Ra (Ra = 103), the entropy generation in the cavity is dominated by heat transfer irreversibility, which is indicated by a higher value of Beav for all Pr. At Ra = 103, the influence of φ on flow and heat distribution is negligible except at φ = 90°. It is found that, Sθ,max occurs at the top portion of side walls in case 1, whereas Sθ,max occurs at the hot−cold junction due to high thermal gradient in case 2. • As Ra increases to 105, the strength of fluid flow increases and that leads to increase in thermal energy transport due to enhanced convection. Consequently, the entropy generation due to heat transfer (Sθ) and fluid friction (Sψ) also increase for all Pr. Similar to Ra = 103, Sθ,max occurs at the top portion of the side walls in case 1, whereas Sθ,max occurs at the hot−cold junction due to high thermal gradient in case 2. The total entropy generation Stotal is found to be smaller in case 1 and higher in case 2 for all Pr at high Ra (Ra = 105). It is observed that Sψ is significant in the portions, where the velocity gradient is larger especially where the solid wall is in contact with the adjacent circulation cells irrespective of φ. It is found that significant entropy is also generated due to the converging fluid streams of two symmetric rolls near the middle portion of the bottom wall in a linear heating case at higher Pr (Pr = 1000). • Analysis on the variations of Beav with Ra indicates that the contribution of fluid friction irreversibility is significant for the increase in Stotal at high Pr fluids in both cases. It is observed that the Beav value is maximum 4086



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Φ5 = (4ξ − 4ξ2)(4η − 4η2 )

and total entropy generation is minimum for φ = 90° at higher Ra in both heating cases for high Pr (Pr = 1000). • Average Nusselt number vs Rayleigh number illustrates that overall heat transfer rate at the bottom wall is higher for linearly heated left wall and cold right wall for all Pr and φs. Multiple circulations plays an important role in a variation of average Nusselt numbers versus Rayleigh number for φ = 90° at Pr = 0.015. It is observed that heat transfer rate is maximum and total entropy generation is minimum for φ = 90° at higher Rayleigh number in both heating cases at high Pr (Pr = 1000). Further, a square cavity with linear heating on the side walls may be used in thermal processing of olive/engine oils in order to achieve uniform temperature distribution and minimum entropy generation. • As the inclination angle of side walls toward the X axis changes from φ = 45° to 90°, the convective cells near the top wall of the cavity compressed toward its center, whereas the convective cells near the bottom wall of the cavity elongated from its center. Thus, the strength of the circulation cells in the core decrease with φ in both cases. On the other hand, compression of the isotherms near the side and bottom walls increases with inclination angle (φ). Hence, thermal gradient increases with φ, and that further results in an increase in average Nusselt number in both cases. • The temperature dynamics within the enclosure depends on the circulation patterns and area of cross section of the particular shape based on various φ. It is true that convection velocity decreases as φ increases. In addition, the area of cross section also decreases as φ increases. Thus, the circulation patterns with less value of ψmax may cause effective larger degree of compression of isotherms due to lesser cross sectional area. Thus, larger local temperature gradient or local Nu is observed for larger ψ values. It may also be noted that, based on energy balance,

Φ6 = (4ξ − 4ξ2)( −η + 2η2 ) Φ7 = ( −ξ + 2ξ2)(1 − 3η + 2η2 )

Φ8 = ( −ξ + 2ξ2)(4η − 4η2 )

Φ9 = ( −ξ + 2ξ2)( −η + 2η2 )

The above basis functions are used for mapping the trapezoidal domain into the square domain and the evaluation of integrals of residuals.

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*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Authors would like to thank the anonymous reviewer for critical comments and suggestions which improved the quality of the manuscript.

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l bNu b + llNul + l rNu r = 0

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AUTHOR INFORMATION

Corresponding Author

Therefore, geometry plays critical role for determining overall Nusselt number (Nu). • The inclination angle (φ) has almost identical effect on entropy production rate for 45° ≤ φ ≤ 60° within the entire range of Pr and Ra.

APPENDIX The name ‘isoparametric’ derives from the fact that the same parametric function describing the geometry may be used for interpolating a spatial variable within an element. Figure 2 shows a trapezoidal domain which is mapped to a square domain. The transformation between (x,y) and (ξ,η) coordinates can be defined by X = ∑k9= 1 Φk(ξ,η) xk and Y = ∑k9= 1 Φk(ξ,η) yk where (xk,yk) are the X, Y coordinates of the k nodal points as seen in Figure 2 and Φk(ξ,η) is the basis function. The nine basis functions are Φ1 = (1 − 3ξ + 2ξ2)(1 − 3η + 2η2 )

Φ2 = (1 − 3ξ + 2ξ2)(4η − 4η2 )

NOMENCLATURE Be = Bejan number f e = any function f over an element e g = acceleration due to gravity, m s−2 k = thermal conductivity, W m−1 K−1 L = length of the trapezoidal cavity, m N = total number of nodes n = normal vector to the plane Nu = local Nusselt number Nu = average Nusselt number p = pressure, Pa P = dimensionless pressure Pr = Prandtl number R = residual of weak form Ra = Rayleigh number S = dimensionless entropy generation Sθ = dimensionless entropy generation due to heat transfer Sψ = dimensionless entropy generation due to fluid friction Stotal = dimensionless total entropy generation due to heat transfer and fluid friction T = temperature of the fluid, K To = bulk temperature, K Th = temperature of hot wall, K Tc = temperature of cold wall, K u = x component of velocity, m s−1 U = x component of dimensionless velocity v = y component of velocity, m s−1 V = y component of dimensionless velocity X = dimensionless distance along x coordinate x = distance along x coordinate, m Y = dimensionless distance along y coordinate y = distance along y coordinate, m

Greek symbols

α = thermal diffusivity, m2 s−1 β = volume expansion coefficient, K−1 γ = penalty parameter Γ = boundary of two-dimensional domain

Φ3 = (1 − 3ξ + 2ξ2)( −η + 2η2 ) Φ4 = (4ξ − 4ξ2)(1 − 3η + 2η2 ) 4087



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θ = dimensionless temperature μ = dynamic viscosity, kg m−1 s−1 ν = kinematic viscosity, m2 s−1 ρ = density, kg m−3 ϕ = irreversibility distribution ratio φ = inclination angle with the positive direction of X axis Φ = basis functions ψ = dimensionless streamfunction Ω = two-dimensional domain

(15) Ghani, A. G. A.; Farid, M. M.; Chen, X. D.; Richards, P. Numerical Simulation of Natural Convection Heating of Canned Food by Computational Fluid Dynamics. J. Food Eng. 1999, 41, 55−64. (16) Verboven, P.; Datta, A. K.; Anh, N. T.; Scheerlinck, N.; Nicolai, B. M. Computation of Airflow Effects on Heat and Mass Transfer in a Microwave Oven. J. Food Eng. 2003, 59, 181−190. (17) Mistry, H.; Subbu, G.; Dey, S.; Bishnoi, P.; Castillo, J. L. Modeling of Transient Natural Convection Heat Transfer in Electric Ovens. App. Therm. Eng. 2006, 26, 2448−2456. (18) Laguerre, O.; Flick, D. Frost Formation on Frozen Products Preserved in Domestic Freezers. J. Food Eng. 2007, 79, 124−136. (19) Demirkol, E.; Erdogdu, F.; Palazoglu, T. K. Analysis of Mass Transfer Parameters (Changes in Mass Flux, Diffusion Coefficient and Mass Transfer Coefficient) During Baking of Cookies. J. Food Eng. 2006, 72, 364−371. (20) Kumar, A.; Bhattacharya, M.; Blaylock, J. Numerical Simulation of Natural Convection Heating of Canned Thick Viscous Liquid Food Products. J. Food Sci. 1990, 55, 1403−1411. (21) Zhang, L.; Fryer, P. J. Models for the Electrical Heating of Solid−Liquid Food Mixtures. Chem. Eng. Sci. 1993, 48, 633−642. (22) Jung, A.; Fryer, P. J. Optimising the Quality of Safe Food: Computational Modelling of a Continuous Sterilization Process. Chem. Eng. Sci. 1999, 54, 717−730. (23) Dealwis, A. A. P.; Fryer, P. J. A Finite Element Analysis of Heat Generation and Transfer during Ohmic Heating of Food. Chem. Eng. Sci. 1990, 45, 1547−1559. (24) Ghani, A. G. A.; Farid, M. M.; Zarrouk, S. J. The Effect of Can Rotation on Sterilization of Liquid Food using Computational Fluid Dynamics. J. Food Eng. 2003, 57, 9−16. (25) Moraga, N. O.; Barraza, H. G. Predicting Heat Conduction During Solidification of a Food Inside a Freezer Due to Natural Convection. J. Food Eng. 2003, 56, 17−26. (26) Fargue, D.; Costeseque, P.; Jamet, P.; Girard-Gaillard, S. Separation in Vertical Temperature Gradient Packed Thermodiffusion Cells: An Unexpected Physical Explanation to a Controversial Experimental Problem. Chem. Eng. Sci. 2004, 59, 5847−5852. (27) Bejan, A. Convection Heat Transfer, 3rd ed.; Wiley: Hoboke, NJ, 2004. (28) Gebhart, B. Buoyancy Induced Fluid Motions Characteristic of Applications in Technology-1978 Freeman Scholar Lecture. J. Fluids Eng., Trans. ASME 1979, 101, 5−28. (29) Hoogendoorn, C. J. Natural Convection in Enclosures,. Proceedings of the 8th International Heat Transfer Conference; Hemisphere Publishing Corp.: San Francisco, CA, 1986; Vol. I, pp 111−120. (30) Patterson, J.; Imberger, J. Unsteady Natural Convection in a Rectangular Cavity. J. Fluid Mech. 1980, 100, 65−86. (31) Nicolette, V. F.; Yang, K. T.; Lloyd, J. R. Transient Cooling by Natural Convection in a Two-Dimensional Square Enclosure. Int. J. Heat Mass Transfer 1985, 28, 1721−1732. (32) Hall, J. D.; Bejan, A.; Chaddock, J. B. Transient Natural Convection in a Rectangular Enclosure with One Heated Side Wall. Int. J. Heat Fluid Flow 1988, 9, 396−404. (33) Hyun, J. M.; Lee, J. W. Numerical Solutions of Transient Natural Convection in a Square Cavity with Different Sidewall Temperature. Int. J. Heat Fluid Flow 1989, 10, 146−151. (34) Fusegi, T.; Hyun, J. M.; Kuwahara, K. Natural Convection in a Differentially Heated Square Cavity with Internal Heat Generation. Numer. Heat Transfer A 1992, 21, 215−229. (35) Lage, J. L.; Bejan, A. The Ra−Pr Domain of Laminar Natural Convection in an Enclosure Heated from the Side. Numer. Heat Transfer A 1991, 19, 21−41. (36) Lage, J. L.; Bejan, A. The Resonance of Natural Convection in an Enclosure Heated Periodically from the Side. Int. J. Heat Mass Transfer 1993, 36, 2027−2038. (37) Xia, C. M.; Murthy, J. Y. Buoyancy-Driven Flow Transitions in Deep Cavities Heated from Below. J. Heat Transfer, Trans. ASME 2002, 124, 650−659.

Subscripts

b = bottom wall l = left wall r = right wall s = side wall av = spatial average total = summation over the domain Superscripts

e = element

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REFERENCES

(1) Farajzadeh, R.; Zitha, P. L. J.; Bruining, J. Enhanced Mass Transfer of CO2 into Water: Experiment and Modeling. Ind. Eng. Chem. Res. 2009, 48, 6423−6431. (2) Yang, C. D.; Gu, Y. G. Accelerated Mass Transfer of CO2 in Reservoir Brine Due to Density-Driven Natural Convection at High Pressures and Elevated Temperatures. Ind. Eng. Che. Res. 2006, 45, 2430−2436. (3) Sedahmed, G. H.; Zatout, A. A.; Zewail, T. M. Free Convection Mass-Transfer Behavior of a Fixed Bed of Raschig Rings. Ind. Eng. Chem. Res. 1998, 37, 3481−3484. (4) Sedahmed, G. H.; Zahran, R. R.; Hassan, I. Natural-Convection Mass-Transfer at a Fixed-Bed of Cylinders. Ind. Eng. Chem. Res. 1993, 32, 1235−1238. (5) Lee, T. L.; Donatelli, A. A. Mass-Transfer by Natural-Convection from a Solid Sphere to Power Law Fluids. Ind. Eng. Chem. Res. 1989, 28, 105−107. (6) Sedahmed, G. H.; Ahmed, A. M. M. Natural-Convection MassTransfer in Slender Conical Cavities. Ind. Eng. Chem. Res. 1990, 29, 1728−1731. (7) Fernandez-Sempere, J.; Ruiz-Bevia, F.; Salcedo-Diaz, R.; GarciaAlgado, P. Buoyancy Effects in Dead-End Reverse Osmosis: Visualization by Holographic Interferometry. Ind. Eng. Chem. Res. 2007, 46, 1794−1802. (8) Hassan, I.; Nirdosh, I.; Sedahmed, G. H. Natural Convection Mass Transfer Behavior of Fixed Bed of Spheres in Relation to Catalytic and Electrochemical Reactor Design. Ind. Eng. Chem. Res. 2009, 48, 3692−3695. (9) Basak, T.; Roy, S.; Natarajan, E. Analysis of Steady Convective Heating for Molten Materials Processing within Trapezoidal Enclosures. Ind. Eng. Chem. Res. 2008, 47, 8652−8666. (10) Germain, J. C.; del Valle, J. M.; de la Fuente, J. C. Natural Convection Retards Supercritical CO2 Extraction of Essential Oils and Lipids from Vegetable Substrates. Ind. Eng. Chem. Res. 2005, 44, 2879−2886. (11) Ettouney, H.; El-Dessouky, H.; Al-Kandari, E. Heat transfer characteristics during melting and solidification of phase change energy storage process. Ind. Eng. Chem. Res. 2004, 43, 5350−5357. (12) Golombok, M.; Chewter, L. Centrifugal Separation for Cleaning Well Gas Streams. Ind. Eng. Chem. Res. 2004, 43, 1734−1739. (13) Sugiyama, M.; Kataoka, M.; Ohmura, H.; Fujiwara, H.; Koda, S. Oxidation of Carbon Particles in Supercritical Water: Rate and Mechanism. Ind. Eng. Chem. Res. 2004, 43, 690−699. (14) Rabiey, L.; Flick, D.; Duquenoy, A. 3D Simulations of Heat Transfer and Liquid Flow During Sterilization of Large Particles in a Cylindrical Vertical Can. J. Food Eng. 2007, 82, 409−417. 4088



Article

(38) Lyican, L.; Bayazitoglu, Y.; Witte, L. C. An Analytical Study of Natural Convective HeatTransfer within a Trapezoidal Enclosure. J. Heat Transfer, Trans. ASME 1980, 102, 640−647. (39) Karyakin, Y. E. Transient Natural Convection in Prismatic Enclosures of Arbitrary Cross-Section. Int. J. Heat Mass Transfer 1989, 32, 1095−1103. (40) Peric, M. Natural Convection in Trapezoidal Cavities. Numer. Heat Transfer A 1993, 24, 213−219. (41) Kuyper, R. A.; Hoogendoorn, C. J. Laminar Natural Convection Flow in Trapezoidal Enclosures. Numer. Heat Transfer A 1995, 28, 55−67. (42) Bejan, A. Entropy Generation Minimization: The New Thermodynamics of Finite-Size Devices and Finite-Time Processes. J. Appl. Phys. 1996, 79, 1191−1218. (43) Bejan, A. Entropy Generation Minimization; CRC Press: Boca Raton, FL, 1996. (44) Ko, T. H.; Ting, K. Entropy Generation and Thermodynamic Optimization of Fully Developed Laminar Convection in a Helical Coil. Int. Comm. Heat Mass Transfer 2005, 32, 214−223. (45) Ko, T. H. Numerical Investigation of Laminar Forced Convection and Entropy Generation in a Helical Coil with Constant Wall Heat Flux. Numer. Heat Transfer A 2006, 49, 257−278. (46) Yilbas, B. S. Laser Short-Pulse Heating of Gold-Silver Assembly: Entropy Generation Due to Heat and Electricity Flows in Electron Subsystem. Numer. Heat Transfer A 2006, 49, 873−891. (47) Azoumah, Y.; Neveu, P.; Mazet, N. Constructal Design Combined With Entropy Generation Minimization For Solid-Gas Reactors. Int. J. Therm. Sci. 2006, 45, 716−728. (48) Abbassi, H. Entropy Generation Analysis in a Uniformly Heated Microchannel Heat Sink. Energy 2007, 32, 1932−1947. (49) Betchen, L. J.; Straatman, A. G. The Development of a VolumeAveraged Entropy-Generation Function For Nonequilibrium Heat Transfer in High-Conductivity Porous Foams. Numer. Heat Transfer B 2008, 53, 412−436. (50) Al-Qahtani, H.; Yilbas, B. S. Entropy Generation Rate During Laser Pulse Heating: Effect of Laser Pulse Parameters on Entropy Generation Rate. Opt. Lasers Eng. 2008, 46, 27−33. (51) Biyikoglu, A. Entropy Generation Due to Flow across the Abrupt Contraction of Pipe Joints. Appl. Therm. Eng. 2009, 29, 841− 847. (52) Chu, S. X.; Liu, L. H. Entropy Generation Analysis of TwoDimensional High-Temperature Confined Jet. Int. J. Therm. Sci. 2009, 48, 998−1006. (53) Carvalho, R. D. M.; Goldstein, L Jr.; Milanez, L. F. Heat Transfer Analysis of Digital Transmission Equipment with Horizontally Arranged Printed Circuit Boards. Heat transfer Eng. 1988, 9, 44− 53. (54) Reddy, J.N. An Introduction to the Finite Element Method; McGraw-Hill: New York, . (55) Ilis, G. G.; Mobedi, M.; Sunden, B. Effect of Aspect Ratio on Entropy Generation in a Rectangular Cavity with Differentially Heated Vertical Walls. Int. Comm. Heat Mass Transfer 2008, 35, 696−703. (56) Varol, Y.; Oztop, H. F.; Koca, A. Entropy Generation Due to Conjugate Natural Convection in Enclosures Bounded by Vertical Solid Walls with Different Thicknesses. Int. Comm. Heat Mass Transfer 2008, 35, 648−656. (57) Varol, Y.; Oztop, H. F.; Koca, A. Entropy Production Due to Free Convection in Partially Heated Isosceles Triangular Enclosures. Appl. Therm. Eng. 2008, 28, 1502−1513. (58) Magherbi, M.; Abbassi, H.; Brahim, A. B. Entropy Generation at the Onset of Natural Convection. Int. J. Heat Mass Transfer 2003, 46, 3441−3450. (59) Dagtekin, I.; Oztop, H. F.; Bahloul, A. Entropy Generation for Natural Convection in Γ-Shaped Enclosures. Int. Comm. Heat Mass Transfer 2007, 34, 502−510. (60) Erbay, L. B.; Altac, Z.; Sulus, B. Entropy Generation in a Square Enclosure with Partial Heating from a Vertical Wall. Heat Mass Transfer 2004, 40, 909−918.

(61) Famouri, M.; Hooman, K. Entropy Generation for Natural Convection by Heated Partitions in a Cavity. Int. Comm. Heat Mass Transfer 2008, 35, 492−502. (62) Davis, G. D. Natural Convection of Air in a Square Cavity: A Bench Mark Numerical Solution. Int. J. Numer. Methods Fluids 1983, 3, 249−264. (63) Natarajan, E.; Roy, S.; Basak, T. Effect of Various Thermal Boundary Conditions on Natural Convection in a Trapezoidal Cavity with Linearly Heated Side Wall(s). Numer. Heat Transfer B 2007, 52, 551−568.

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