Analysis of Entropy Generation Due to Natural Convection in Tilted

Article pubs.acs.org/IECR

Analysis of Entropy Generation Due to Natural Convection in Tilted Square Cavities Abhishek Kumar Singh,† S. Roy,† and Tanmay Basak*,‡ †

Department of Mathematics, and ‡Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai-600036, India ABSTRACT: In this article, the numerical investigation of entropy generation due to heat transfer irreversibility and fluid friction irreversibility during natural convection within tilted square cavity with hot wall AB, cold side walls (DA and BC), and top insulated wall (CD) has been performed. The numerical simulation has been carried out for various fluids of industrial importance (Pr = 0.015, 0.7, and 1000), Rayleigh numbers (103 ≤ Ra ≤ 105), and different inclination angles (φ = 15°, 45°, and 75°). The results are presented in terms of isotherms (θ), streamlines (ψ), entropy generation maps due to heat transfer (Sθ), and fluid friction (Sψ). The total entropy generation (Stotal), average Bejan number (Beav), and average heat transfer rate (NuAB) are plotted for Rayleigh number 103 ≤ Ra ≤ 105. The maximum values of Sθ occur near the corner regions of wall AB due to a junction of hot and cold walls. On the other hand, maximum values of Sψ are found near the walls of the cavity due to friction between the circulation cells and walls of the cavity. It is found that minimum entropy generation occurs for φ ≥ 45° at convection dominant mode (Ra = 105) for lower Pr (Pr = 0.015 and 0.7). The inclined cavity with φ = 45° may be an alternative optimal inclination angle in optimal thermal processing of high Pr (Pr = 1000). Murthy.21 Several attempts have been made to acquire a basic understanding of natural convection flow and temperature distribution within the enclosure.22−30 However, the studies of natural convection within inclined cavities have also practical significance in many science and engineering applications due to the presence of both tangential and normal components of buoyancy force, which have a strong effect on the fluid flow and heat transfer characteristics.31−36 Entropy generation minimization is a method for modeling and optimizing energy systems related to thermal power plants, heat exchangers, and cooling devices. The total entropy generation in these systems can be minimized under some physical and geometric arrangements, and an optical configuration with minimum loss of available energy may be obtained. As irreversibility destroys the system energy, its minimization has been considered as the optimal design criteria for thermal systems to utilize its maximum available energy. This destruction of available energy due to irreversibilities can be quantified in terms of entropy generation. Hence, minimization of entropy generation is a major challenge to optimize energy saving processing. The second law of thermodynamics is required to determine entropy generation due to heat transfer and fluid flow in the cavity and consequently minimize the entropy generation. However, a large amount of literature survey is available on the study of natural convection within a closed cavity, which are based only on the first law of thermodynamics.37−44 To improve the efficiency of thermal systems, a new approach based on the simultaneous application of the first law and the second law of thermodynamics is

1. INTRODUCTION The buoyancy-driven flow induced by the density difference in the fluid plays an important role in large industrial heat transfer system. Typical applications of buoyancy-driven flows, termed as natural convection, include chemical processing,1−5 CO2 injection projects,6 chemical reactors,7−9 reservoir,10 food engineering,11−13 cooling of electronic and microelectronic equipment,14 and fuel cells,15 etc. The studies of natural convection flows are particularly complex depending on several parameters, among which the geometry and the thermophysical characteristics of the fluid are the most important. The proper shape and position of the enclosure in different applications is supported by experimental studies or numerical simulation. However, the analysis with numerical methods is important to reduce the experimental costs as high cost is involved in the experimental method. Most of the studies related on natural convection are focused within the different shapes of the horizontal enclosure (square, triangular, trapezoidal, cylindrical, etc.) in which buoyancy force has only a normal component.16−21 The numerical study of natural convection within a long horizontal enclosure of rectangular cross section was carried out by Wilkes and Churchill.16 Holtzman et al.17 performed numerical studies of natural convection within the isosceles triangle enclosures heated from below and symmetric cooled from above. Iyican and Bayazitoglu18 analyzed analytically the natural convection heat transfer within the trapezoidal enclosure. Ganguli et al.19 investigated the flow patterns during natural convection within a vertical slot. The left wall of the vertical slot is maintained at constant hot temperature, while the right wall is maintained at the cold. Yang et al.20 analyzed the rotational effect on the natural convection within a horizontal cylinder. A numerical investigation on the flow transition in deep three-dimensional cavities heated from below has been carried out by Xia and © 2012 American Chemical Society

Received: Revised: Accepted: Published: 13300

May 25, 2012 August 22, 2012 August 29, 2012 September 24, 2012 dx.doi.org/10.1021/ie3013665 | Ind. Eng. Chem. Res. 2012, 51, 13300−13318

Industrial & Engineering Chemistry Research

Article

introduced by Bejan.45−47 This new approach is called as exergy analysis, and its optimization tool is based on entropy generation minimization (EGM). An application on optimized heat transfer processes, which is an indirect outcome of minimization of entropy generation, is also investigated by earlier research.48,49 Bejan and Sciubba48 presented the boardto-board spacing and maximum total heat transfer rate from the stack of parallel boards cooled by laminar forced convection. They concluded that surface thermal conditions have a minor effect on the optimal spacing and the maximum total heat transfer rate. Recently, Lucia49 analyzed the minimum and maximum entropy generation principles and concluded that minimum entropy generation is related to the system while maximum entropy generation is related to the interaction between the system and the environment. Every process related to the energy system is inherently a irreversible process, and hence always some amount of useful or available energy (called exergy) is destroyed during the process due to irreversibilities. This leads to the decrease in 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 Gouy−Stodola theorem, which states that the rate of the available work dissipation is proportional to the rate of internal entropy generation.50 The development of improved thermal designs is enhanced by the ability to identify the source and location of entropy generation. In convection flow, the irreversibilities occur due to the heat transfer and fluid friction. Therefore, the strategies to minimize their generation may be important, and that can be analyzed for entropy generation due to heat transfer and fluid flow irreversibilities to achieve optimum design for any thermal process. Few studies on the analysis of entropy generation for square enclosures based on EGM method have been reported in the literature.51−57 Erbay et al.51 studied transient entropy generation during laminar natural convection in a square enclosure being heated either completely or partially from the left side wall and cooled from the opposite side wall. The active sides are found to be near the left bottom corner of the heated wall and the right top corner of the cooled wall with the same magnitude. The entropy generation due to natural convection in a symmetrically and uniformly heated vertical channel was investigated by Andreozzi et al.52 Magherbi et al.53 analyzed the effect of Rayleigh number and irreversibility distribution ratio on entropy generation due to heat transfer and fluid friction during natural convection within square cavity. Famouri and Hooman54 performed numerical studies on the entropy generation for free convection in a partitioned cavity and presented the new equation for prediction of entropy generation in a square cavity. Demirel and Kahraman55 studied the entropy generation in a rectangular enclosure and found that irreversibility distribution is not continuous through the wall and core regions. Entropy generation for natural convection within Γ-shaped enclosures is investigated by Dagtekin et al.56 They found that geometrical parameters are strongly affected on the entropy generation due to heat transfer and fluid friction. Yilbas et al.57 studied the natural convection and entropy generation in a square cavity with differential top and bottom wall temperatures. They observed that total entropy generation increases with wall temperature, and that becomes almost optimum for a particular Rayleigh number. Recently, Kaluri and Basak58 analyzed the role of entropy generation on thermal management during natural convection in porous square cavities with distributed heat sources.

The above literature review shows that few studies exist on entropy generation during natural convection for square cavities. However, the inclined square cavity requires special attention due to its various energy related engineering applications. The effect of inclination angle on the entropy generation during natural convection within inclined square cavity has also been discussed by a few investigators.59−63 Baytas59 analyzed entropy generation in an inclined enclosure during laminar natural convection heat transfer and found that irreversibilities due to local heat transfer and fluid friction depend on inclination angle. Baytas60 also studied the influence of Rayleigh number, Bejan number, and inclination angle on entropy generation for natural convection in an inclined porous cavity. Mahmud and Islam61 presented flow and heat transfer characteristics and entropy generation analysis within an inclined enclosure bounded by two isothermal wavy walls. They found that the rate of average entropy generation decrease with the inclination angle and the minimum entropy production occurs at 90° angular position for fixed Rayleigh number. Varol et al.62 performed the theoretical study of the buoyancy-driven flow and heat transfer in an inclined porous trapezoidal enclosure heated and cooled from inclined walls. Adjlout et al.63 reported natural convection within an inclined cavity with hot wavy wall and cold flat wall and performed the analysis for different inclination angles, amplitudes, and Rayleigh numbers at a fixed Prandtl number. A few studies on entropy generation for various physical systems based on EGM method have also been reported in the literature.64,65 It is seen from the literature that no attempt has been made to evaluate entropy generation terms within inclined square cavity with hot bottom wall and cold side walls in the presence of adiabatic top wall based on an accurate formulation with elemental basis function via the Galerkin finite element approach. The motivation of study of natural convection within inclined enclosure is to observe the inclination effect on the entropy generation so that one may choose the particular inclination angle that corresponds to the minimum entropy generation. The prime objective of this article is to analyze the effect of the inclination angle on the fluid flow and entropy generation due to heat transfer and fluid friction during natural convection within inclined square cavity with hot wall AB and cold side walls (DA and BC) in the presence of insulated wall (CD). The Galerkin finite element method66 with penalty parameter is used to solve the nonlinear coupled partial differentiation equations governing the fluid flow, temperature, and entropy. It may be noted that estimation of entropy generation rate involves accurate evaluation of thermal and velocity gradients or derivative. The finite element approach offers special advantage over finite difference or finite volume methods as the elemental basis sets are used for the calculation of gradients or derivatives. The current work is the first attempt to evaluate entropy generation terms using elemental basis functions via Galerkin finite element method for tilted cavities. Simulations are carried out for a range of parameters, Ra = 103−105 and Pr = 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 temperature (θ), entropy generation due to heat transfer (Sθ), streamlines (ψ), and entropy generation due to fluid friction (Sψ). Also, the effects of Rayleigh number on entropy generation due to thermal and fluid irreversibilities, and on average Bejan number, which indicates the relative importance of 13301

dx.doi.org/10.1021/ie3013665 | Ind. Eng. Chem. Res. 2012, 51, 13300−13318


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thermal and fluid irreversibilities, are presented. Nonorthogonal grid generation has been done with iso-parametric mapping.30,66

U

2. MATHEMATICAL FORMULATION 2.1. Governing Equations, Boundary Conditions, and Simulation Strategy. The physical domain of tilted square cavity inclined at an angle φ = 15°, 45°, and 75° with X axis is shown in Figure 1a−c, respectively. The wall AB of the cavity is

U

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

(2)

⎛ ∂ 2V ∂V ∂V ∂P ∂ 2V ⎞ +V =− + Pr ⎜ 2 + ⎟ + RaPrθ ∂X ∂Y ∂Y ⎝ ∂X ∂Y 2 ⎠ (3)

U

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

(4)

where X=

x , L

pL2 ρα

U=

uL , α

V=

vL , α

T − Tc , Th − Tc

θ=

P=

y , L

Y=

, 2

Pr =

ν , α

Ra =

gβ(Th − Tc)L3Pr ν2

(5)

No slip conditions are assumed at solid boundaries of tilted square cavity, and the boundary conditions for the velocity components and temperature are as follows: U (X , Y ) = 0,

V (X , Y ) = 0,

θ = 1 along wall AB,

U (X , Y ) = 0,

V (X , Y ) = 0,

θ = 0 along wall BC,

U (X , Y ) = 0, V (X , Y ) = 0, U (X , Y ) = 0,

V (X , Y ) = 0,

n ·∇θ = 0 along wall CD, θ = 0 along wall DA (6)

The continuity equation [eq 1] is used as a constraint due to mass conservation, and this constraint can be used to obtain the pressure distribution. The momentum and energy balance equations [eqs 2−4)] are solved using the Galerkin finite element method. To solve eqs 2,3, the penalty finite element method has been employed to eliminate the pressure (P) with a penalty parameter (γ) and the incompressibility criteria given by eq 1 via the following relationship:

Figure 1. Schematic diagram of the computational domain with the boundary conditions for (a) φ = 15°, (b) φ = 45°, and (c) φ = 75°. The dashed line represents the geometric symmetric line.

⎛ ∂U ∂V ⎞⎟ P = −γ ⎜ + ⎝ ∂X ∂Y ⎠

maintained hot and the walls DA and BC are maintained cold in the presence of adiabatic wall CD [see Figure 1a−c]. The boundary conditions of velocity are considered as no-slip on solid boundaries. The laminar flow of liquid material is considered as incompressible and Newtonian. All of the physical properties such as viscosity, thermal conductivity, specific heats, thermal expansion coefficient, and permeability are constant expect density for body force term where density is linearly varying with temperature following the Boussinesq approximation. Under these assumptions, governing equations for steady two-dimensional natural convection flow in the tilted square enclosure using conservation of mass, momentum, and energy in dimensionless form may be written as ∂U ∂V + =0 ∂X ∂Y

(7)

Typically, γ = 107 yields consistent solutions. Applying eq 7, the momentum balance equations [eqs 2 and 3] are reduced to: U

⎛ ∂ 2U ∂U ∂U ∂ ⎛ ∂U ∂V ⎞⎟ ∂ 2U ⎞ +V =γ ⎜ + + Pr ⎜ 2 + ⎟ ∂X ∂Y ∂X ⎝ ∂X ∂Y ⎠ ⎝ ∂X ∂Y 2 ⎠ (8)

and U

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

(9)

The velocity components (U, V) and temperature (θ) are expanded using the basis set {Φk}Nk = 167 as:

(1) 13302



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

U≈

N

∑ Uk Φk(X , Y ), k=1 N

θ≈

V≈

∑ Vk Φk(X , Y ), k=1

∑ θk Φk(X , Y )

⎛N ⎞ ∂Φ ⎤ + ⎜⎜∑ Vk Φk ⎟⎟ k ⎥Φi dX dY ⎝k=1 ⎠ ∂Y ⎥⎦

(10)

The Galerkin finite element method yields the following nonlinear residual equations for eqs 8, 9, and 4, respectively, at nodes of internal domain:

k=1

∫Ω

N

+ Pr ∑ Uk k=1

∂Φi ∂Φk dX dY + ∂X ∂X

N

∑ Vk ∫ k=1

Ω

(11)

k=1

N

+ Pr ∑ Vk k=1

− RaPr

∂Φ ∂Φk dX dY + ∂X

∫Ω ∂Yi

∑ Vk ∫ k=1

Ω

(13)

∑ Φk(ξ , η)xk k=1

⎤ ∂Φi ∂Φk dX dY ⎥ ⎥⎦ ∂Y ∂Y

and 9

Y=

∂Φ ∂Φk ∂Φi ∂Φk dX dY + ∂X ∂Y ∂Y

∫Ω ∂Xi

⎛N ⎞ ⎜⎜∑ θk Φk ⎟⎟Φi dX dY Ω ⎝k=1 ⎠

∫

N

∂Φi ∂Φk ∂Φi ∂Φk + dX dY ∂X ∂X ∂Y ∂Y

9

X=

∑ Vk ∫

⎡N + γ ⎢∑ Uk ⎢⎣ k = 1

Ω

Biquadratic basis functions with three-point Gaussian quadrature are used to evaluate the integrals in residual equations except for the second term in eqs 11 and 12. In eqs 11 and 12, the second term containing the penalty parameter (γ) is evaluated with the two-point Gaussian quadrature method. The nonlinear residual equations [eqs 11−13] are solved using the Newton−Raphson method to determine the coefficients of the expansions in eq 10. Figure 2 shows the grid generation for (x, y) and (ξ, η) coordinates via the following relationships:

⎤ ∂Φi ∂Φk dX dY ⎥ ∂X ∂Y ⎦⎥

⎡ ∂Φ ∂Φk ∂Φi ∂Φk ⎤ + ⎥ dX dY ∂X ∂Y ∂Y ⎦

∫Ω ⎢⎣ ∂Xi

∑ θk ∫ k=1

⎡⎛ N ⎤ ⎞ ⎛N ⎞ ⎢⎜∑ Uk Φk ⎟ ∂Φk + ⎜∑ Vk Φk ⎟ ∂Φk ⎥Φi dX dY ⎜ ⎟ ⎜ ⎟ Ω⎢ ⎠ ∂X ⎝k=1 ⎠ ∂Y ⎥⎦ ⎣⎝ k = 1

N

=

+

∑ Uk ∫

⎡N + γ ⎢∑ Uk ⎢⎣ k = 1

R i(2)

N

⎤ ⎡⎛ N ⎞ ⎛N ⎞ ⎢⎜∑ Uk Φk ⎟ ∂Φk + ⎜∑ Vk Φk ⎟ ∂Φk ⎥Φi dX dY ⎜ ⎟ ⎜ ⎟ Ω⎢ ⎠ ∂X ⎝k=1 ⎠ ∂Y ⎥⎦ ⎣⎝ k = 1

N

∑ θk ∫ k=1

k=1

R i(1) =

⎡⎛ N ⎞ ⎢⎜∑ Uk Φk ⎟ ∂Φk ⎜ ⎟ ∂X Ω⎢ ⎠ ⎣⎝ k = 1

N

R i(3) =

and

∑ Φk(ξ , η)yk k=1

(14)

where (xk, yk) are the X, Y coordinates of the kth nodal points as seen in Figure 2, and Φk(ξ, η) is the basis function. 2.2. Streamfunction, Nusselt Number, and Entropy Generation. 2.2.1. Streamfunction. The fluid motion is displayed using the streamfunction (ψ) obtained from velocity

(12)

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components (U and V). The relationships between streamfunction (ψ) and velocity components (U and V) for twodimensional flows are U=

∂ψ ∂Y

V=−

and

∂ψ ∂X

entropy generated. The exergy destroyed is equal to temperature of the environment (not the system) multiplied by the total entropy generated in the system and environment. The entropy generated is not the entropy change in a system; it is just the new entropy produced due to strictly internal irreversibilities. The Gouy−Stodola theorem is limited to the special case when the system is only in contact with one environment at a fixed temperature. The Gouy−Stodola theorem states that the rate of the available work dissipation is proportional to the rate of internal entropy generation.50 Mathematically, the formulation of entropy generation can be expressed as

(15)

which yield a single equation: ∂ 2ψ ∂ 2ψ ∂U ∂V + = − ∂Y ∂X ∂X2 ∂Y 2

(16)

The sign convention is as follows: Positive sign of ψ denotes anticlockwise circulation, and clockwise circulation is represented by negative sign of ψ. Expanding the streamfunction (ψ) using the basis set {ϕk}Nk = 1 as

Wlost = T0Sgen

Here, Wlost is lost available work, T0 is the environment absolute temperature, and Sgen is the instantaneous rate of entropy generation. The exact mathematical form of the Gouy−Stodola theorem is

N

ψ≈

∑ ψk Φk(X , Y )

(17)

k=1

Wlost ≈ Sgen

and the relationship for velocity components (U and V) from eq 10, the Galerkin finite-element method yields the following linear residual equation for eq 16. N

R i(ψ ) =

∑ ψk ∫

Ω

k=1

⎛ ∂Φi ∂Φk ∂Φi ∂Φk ⎞ ⎜ ⎟ dX dY − + ⎝ ∂X ∂X ∂Y ∂Y ⎠

N

+

∑ Uk ∫

Ω

k=1

Φi

∂Φk dX dY − ∂Y

N

∑ Vk ∫ k=1

Ω

Φi

∫Γ Φin·∇ψ dΓ

∂Φk dX dY ∂X

(18)

∂θ ∂n

(19)

where n denotes the outward normal direction of the plane. The local Nusselt numbers at wall AB (NuAB), wall BC (NuBC), and wall DA (NuDA) are defined as: 9 ⎛ ∂Φ ∂Φ ⎞ NuAB = −∑ θi⎜sin(φ) i − cos(φ) i ⎟ ⎝ ∂X ∂Y ⎠ i=1

⎛

9

NuDA =

∑ θi⎜⎝cos(φ) i=1

∂Φi ∂Φ ⎞ + sin(φ) i ⎟ ∂X ∂Y ⎠

(20)

(21)

and 9 ⎛ ∂Φ ∂Φ ⎞ NuBC = −∑ θi⎜cos(φ) i + sin(φ) i ⎟ ⎝ ∂X ∂Y ⎠ i=1

(22)

The average Nusselt numbers at any wall (AB, BC, or DA) are given as: Nus =

∫0

1

Nus ds

(25)

The proportionality depends on the specific features of the system interest. The lost energy in the separation systems is due to irreversible processes of heat, mass transfer, and thermal mixing, which are directly related to entropy production according to the Gouy−Stodola principle. Therefore, the efforts to minimize the entropy production have become popular to improve the efficiency of the energy system. This innovative approach on the separation systems is called thermodynamics analysis. Later, exergy analysis was developed to identify the parts of systems with excessive irreversibilities and hence to control the lost energy. On the basis of the developments in nonequilibrium thermodynamics, some recent research has reported the implications of the rate of entropy production on the use of available energy in separation systems. Lucia68 analytically proved the principle of maximum for the variation of the entropy due to irreversibility, which states that, “in a general thermodynamics transformation, the condition of the stability for the open systems’ equilibrium states consists of the maximum for the variation of the entropy due to irreversibility”. Lucia69 also obtained the general approach to the analysis of irreversible systems and developed the application of entropy generation to study the dynamics system. The analysis for entropy generation is also based on the following thermodynamics consideration. Local thermal equilibrium (LTE) is applicable in the present investigation to limit the discussion to a domain in which a linear expansion of the entropy terms can be made.70 The system is well enough behaved that locally (spatially) equilibrium thermodynamics apply. Thermodynamics consideration is also based on the reference state, which is asymptotically stable. Under the assumption of LTE, all thermodynamic properties in a small volume have the thermodynamic equilibrium values at the local values of temperature and pressure. 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,50 the dimensionless total local entropy generation for a twodimensional heat and fluid flow in Cartesian coordinates in explicit form is written as:

The no-slip condition is valid at all boundaries, and there is no cross-flow; hence, ψ = 0 is used for all boundaries. The biquadratic basis function is used to evaluate the integrals in eq 18, and ψ values are obtained by solving the N linear residual equations [eq 18]. 2.2.2. Nusselt Number. The heat transfer coefficient in terms of local Nusselt number (Nu) is defined by Nu = −

(24)

(23)

where ds denotes the small elemental length along sides of the tilted square cavity. Note that NuDA + NuBC = NuAB, whereas NuDA ≠ NuBC due to tilted geometry with respect to gravity. 2.2.3. Entropy Generation. The Gouy−Stodola theorem states the relationship between the exergy destroyed and the

⎡⎛ ∂θ ⎞2 ⎛ ∂θ ⎞2 ⎤ Sθ = ⎢⎜ ⎟ + ⎜ ⎟ ⎥ ⎝ ∂Y ⎠ ⎦ ⎣⎝ ∂X ⎠ 13304

(26)



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2 ⎧ ⎡⎛ ∂U ⎞2 ⎛ ∂V ⎞2 ⎤ ⎛ ∂U ∂V ⎟⎞ ⎫ ⎬ Sψ = ϕ⎨2⎢⎜ ⎟ + ⎜ ⎟ ⎥ + ⎜ + ⎝ ∂Y ⎠ ⎦ ⎝ ∂Y ∂X ⎠ ⎭ ⎩ ⎣⎝ ∂X ⎠ ⎪

⎪

⎪

⎪

Sψ ,total

(27)

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

μTo ⎛ α ⎞2 ⎜ ⎟ k ⎝ LΔT ⎠

9

∑ f ke k=1

∂Φek ∂n

1 = e ∂n N

Ne

∑ e=1

Beav =

(30)

(31)

2 ⎧ ⎡⎛ ∂U ⎞2 ⎛ ∂V ⎞2 ⎤ ⎛ ∂U ⎪ ⎪ ∂V ⎞ ⎫ ⎢⎜ i ⎟ + ⎜ i ⎟ ⎥ + ⎜ i + i ⎟ ⎬ 2 S ψ , i = ϕ⎨ ⎪ ⎝ ∂Y ⎠ ⎥⎦ ⎝ ∂Y ∂X ⎠ ⎪ ⎩ ⎢⎣⎝ ∂X ⎠ ⎭

(32)

Sθ ,total Sθ ,total + Sψ ,total

=

Sθ ,total Stotal

(36)

3. RESULTS AND DISCUSSION 3.1. Numerical Tests and Parameters. The computational grid within the tilted square cavity is generated via mapping the tilted square into horizontal square in the ξ−η coordinate system as shown in Figure 2. The computational domain consists of 28 × 28 biquadratic elements, which correspond to 57 × 57 grid points. The biquadratic elements with lesser number of nodes smoothly capture the nonlinear variations of the field variables, which are in contrast with finite difference/finite volume solutions. Detailed computations have been carried out for various fluids of Pr (Pr = 0.015−1000) within Ra = 103−105. The jump discontinuities in Dirichlettype boundary conditions at the corner points correspond to computational singularity. This problem is resolved by specifying the average temperature of the hot and cold sections/walls at the junction and keeping the adjacent grid-nodes at the respective section/wall temperatures. To validate the code, benchmark studies were carried out for the differentially heated square cavity with 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.71 The results in terms of entropy generation due to heat transfer and fluid friction are in excellent agreement with earlier work71 as shown in Figure 3. Table 1 illustrates average Nusselt numbers at the wall AB (NuAB), Stotal, and Bejan number (Beav) for various grid systems (24 × 24, 26 × 26, and 28 × 28). It is found that 28 × 28 biquadratic elements are adequate to obtain grid independent results. In the current investigation, the Gaussian quadrature based finite element method provides smooth solutions in the computational domain including the singular points as evaluation of residuals depends on interior Gauss points. For the case of horizontal square (φ = 0°), with isothermally hot wall AB and cold walls (DA and BC), fluid rises from the center of the hot wall AB due to buoyancy, directed against the gravity, and rolls down along the cold walls (DA and BC), forming two symmetric rolls with clockwise and anticlockwise rotations inside the cavity.67 In the case of inclined square, the effects of the tangential and normal components of buoyancy force, relative to wall AB, play a critical role for both flow and thermal characteristics. As tilted angle increases, buoyancy force along

(29)

⎡⎛ ∂θ ⎞2 ⎛ ∂θ ⎞2 ⎤ Sθ , i = ⎢⎜ i ⎟ + ⎜ i ⎟ ⎥ ⎝ ∂Y ⎠ ⎥⎦ ⎢⎣⎝ ∂X ⎠

Note that the derivatives, (∂θi/∂X), (∂θi/∂Y), (∂Ui/∂X), (∂Ui/∂Y), (∂Vi/∂X), (∂Vi/∂Y), are evaluated following eq 30. 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. (33)

where Sθ ,total

)

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

Therefore, at each node, local entropy generation for thermal (Sθ,i) and fluid friction (Sψ,i) are given by

Stotal = Sθ ,total + Sψ ,total

(

The integrals are evaluated using the 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

(28)

∂f ie ∂n

⎤2 ⎫ ⎪ ∑k= 1 Vk Φk ⎥⎥ ⎬ dX dY ⎦⎪ ⎭ N

(35)

where f ek is the value of the function at local node k in the element e, and Φek is the value of basis function at local node k in the element e. Further, because 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: ∂fi

∫

⎡ ⎛N ⎞ ∂ ∂ + ⎢ ⎜⎜∑ Uk Φk ⎟⎟ + ⎢⎣ ∂Y ⎝ k = 1 X ∂ ⎠

In the current study, ϕ is taken as 10−4. A similar value for ϕ was considered by Ilis et al.71 A higher value for ϕ was assumed by a few earlier researchers,72,73 and several other researchers have also studied the effect of ϕ on total entropy generation.53,56,71 As mentioned earlier, the derivatives are evaluated based on the finite element method. Nine node biquadratic elements are used with each element mapped using iso-parametric mapping66 from X−Y to a unit square ξ−η domain as illustrated in Figure 2 . The domain integrals in the residual 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:

∂f e = ∂n

2 2 ⎧⎡ ⎛ N ⎡ ⎛N ⎞⎤ ⎞⎤ ⎪ ∂ ∂ 2⎨⎢ ⎜⎜∑ Uk Φk ⎟⎟⎥ + ⎢ ⎜⎜∑ Vk Φk ⎟⎟⎥ =ϕ ⎢⎣ ∂Y ⎝ k = 1 Ω ⎪⎢ ∂X ⎠⎥⎦ ⎠⎥⎦ ⎩⎣ ⎝ k = 1

2 2 ⎧⎡ ⎛ N ⎡ ⎛N ⎞⎤ ⎞⎤ ⎫ ⎪ ∂ ⎪ ∂ ⎨⎢ ⎜⎜∑ θk Φk ⎟⎟⎥ + ⎢ ⎜⎜∑ θk Φk ⎟⎟⎥ ⎬ dX dY = ⎢⎣ ∂Y ⎝ k = 1 ∂X Ω ⎪⎢ ⎠⎥⎦ ⎠⎥⎦ ⎪ ⎩⎣ ⎝ k = 1 ⎭

∫

(34) 13305



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along the side wall due to low temperature gradient in the top portion of the cavity, and therefore Sθ is almost negligible at the upper portion of the cavity [see Figure 4a]. This is further due to less heat transfer in the upper portion of the cavity. The distribution of local entropy due to heat transfer (Sθ) is higher (Sθ,max = 3528) at the edges of the wall AB. This is due to the largest temperature gradient in those regions where the hot wall AB is in direct contact with the cold side walls (DA and BC). The entropy generation due to flow irreversibilities for φ = 15° is displayed in Figure 4a. It is observed that two asymmetric rolls consisting of anticlockwise strong primary circulation cells and clockwise weak secondary circulation cells are formed for small inclination angle (φ = 15°). The left circulation cell spreads out more than the right circulation cell, and that occupies 80% of the cavity [see Figure 4a]. This is due to a dominant tangential component of buoyancy force over normal component along the wall AB. Consequently, right circulation cells become smaller in size, and magnitudes of streamfunction are considerably small as compared to left circulation cell. It may be noted that |ψ|max = 0.4 in the left circulation cells and |ψ|max = 0.01 in the right circulation cells [see Figure 4a]. Because of the no-slip boundary condition along the walls, high velocity gradients exist near the walls and lead to enhanced entropy generation due to fluid friction in region near walls AB and DA. Therefore, active zones of entropy generation due to fluid friction are found near the middle portion of walls AB and DA. The dense contours of Sψ near wall AB and middle portion of wall DA represent the active zones of entropy generation due to fluid friction. However, the magnitudes of Sψ are quite low at the core of the cavity as there is no significant velocity gradient due to weak fluid flow. The flow intensity is weak at low Ra as depicted by |ψ|max = 0.4, and thus entropy generation due to fluid friction, Sψ, is insignificant relative to Sθ with Sψ,max being only 0.12. Similar to φ = 15°, active zones of the entropy generation due to the heat transfer are higher near the lower corners of the cavity (Sθ,max = 3528) based on the direct contact of hot and cold walls for φ = 45° [see Figure 4a,b]. It may be noted that there is no significant change in distribution of Sθ on increase of the inclination angle from φ = 15° to 45° due to a similar pattern of isotherms. It is interesting to observe that left circulation cells tend to become stronger and occupy the whole cavity. Clockwise and anticlockwise asymmetric circulation cells tend to become a single anticlockwise circulation cell with |ψ|max = 0.75 [see Figure 4b]. This is due to an increase in the tangential component of buoyancy force, which is stronger in the anticlockwise direction. Similar to the previous case with φ = 15°, Sψ,max is almost negligible in the center of the cavity and is higher near the left portion of the wall AB (Sψ,max = 0.35). Entropy generation due to fluid friction (Sψ) increases due to enhanced fluid flow. As φ increases from 45° to 75°, the Sθ contours are similar with φ = 45°, and Sθ,max = 3528 is observed at the corner regions of wall AB. Insignificant Sθ is still observed

Figure 3. Local entropy generation due to heat transfer (Sθ) and fluid friction (Sψ) for a square cavity (φ = 0°) with the boundary conditions θ(X,Y) = 1 (wall DA), θ(X,Y) = 0 (wall BC), and n·∇θ = 0 (walls AB and CD) at Pr = 0.71 and (a) Ra = 103 and (b) Ra = 105.71

the wall AB gradually increases, leading to stronger anticlockwise fluid circulations. This is further due to an increase in the tangential component of buoyancy force, which is zero in the case of horizontal square (φ = 0°). Detailed explanations of the results with particular emphasis on the effect of inclination angle on the isotherms, streamlines, entropy generation due to heat transfer, and fluid friction are given in various succeeding sections. 3.2. Flow, Temperature, and Entropy Generation Characteristics. Figures 4−8 show isotherms (θ), entropy generation due to heat transfer (Sθ), streamlines (ψ), and entropy generation due to fluid friction (Sψ) for various fluids (Pr = 0.015, 0.7, and 1000) with Ra = 103−105 within tilted square cavity of different inclination angles (φ = 15°, 45°, and 75°). The wall AB of the cavity is maintained isothermally hot and side walls (DA and BC) are maintained isothermally cooled in the presence of adiabatic wall CD [see Figure 1a−c]. It is observed that flow and temperature patterns are not symmetric along the central vertical line due to geometric asymmetry. The entropy generation maps are also not symmetric as they are based on nonsymmetric velocity and thermal fields. At Ra = 103 for Pr = 0.015, isotherms with θ = 0.1 occur symmetrically near the side walls (DA and BC) of the enclosure, and other isotherms with θ ≥ 0.2 are smooth curves symmetric with respect to geometric symmetric line for φ = 15° [see Figure 4a]. The smooth isotherms and weak fluid flow circulations illustrate the conduction dominant heat transfer. The thickness of the boundary layer increases with the height

Table 1. Comparison of Average Nusselt Number at Wall AB (NuAB), Stotal, and Bejan Number (Beav) for Various Grid Systems at Ra = 105 and Pr = 1000 with Various Inclination Angles (φ) of Tilted Square Cavity Nu AB

Stotal

Beav

φ (deg)

24 × 24

26 × 26

28 × 28

24 × 24

26 × 26

28 × 28

24 × 24

26 × 26

28 × 28

15 45 75

9.56 8.56 8.74

9.64 8.64 8.83

9.72 8.72 8.90

44.77 30.80 32.91

45.88 30.90 33.02

45.97 31.00 33.16

0.20 0.27 0.26

0.21 0.27 0.26

0.21 0.28 0.26

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Figure 4. Isotherms (θ), local entropy generation due to heat transfer (Sθ), streamlines (ψ), and local entropy generation due to fluid friction (Sψ) for a tilted square cavity with inclination angle (a) φ = 15°, (b) φ = 45°, and (c) φ = 75° with boundary conditions, θ(X,Y) = 1 (wall AB), θ(X,Y) = 0 (walls BC and DA) and n·∇θ = 0 (wall CD) for Pr = 0.015 and Ra = 103.

φ = 75° [see Figure 4b,c]. The active zone of entropy generation (Sψ = 0.27) is also observed near the middle portion of wall BC. Temperature profiles are almost invariant with respect to inclination angle. Overall, there is no significant change in the Sθ,max due to an almost invariant temperature gradient near the wall AB irrespective of inclination angle [see Figure 4a−c].

in the top portion of the cavity due to a low temperature gradient in that region. The entropy generation due to fluid friction is higher (Sψ,max = 0.47) near the middle portion of the wall AB as compared to the φ = 45° case (Sψ,max = 0.35) due to a high velocity gradient near the region where the circulation cells are in contact with the walls of cavity for 13307



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Figure 5. Isotherms (θ), local entropy generation due to heat transfer (Sθ), streamlines (ψ), and local entropy generation due to fluid friction (Sψ) for a tilted square cavity with inclination angle (a) φ = 15°, (b) φ = 45°, and (c) φ = 75° with boundary conditions, θ(X,Y) = 1 (wall AB), θ(X,Y) = 0 (walls BC and DA), and n·∇θ = 0 (wall CD) for Pr = 0.015 and Ra = 104.

This is further due to weak fluid flow, and flow strength is invariant as |ψ|max = 0.4−0.9 for φ = 15°−75°. It may also be noted that entropy generation due to fluid friction increases slightly with inclination angle at lower Rayleigh number (Ra = 103) as the magnitude of Sψ,max varies as Sψ,max = 0.12 for

φ = 15°, Sψ,max = 0.35 for φ = 45°, and Sψ,max = 0.47 for φ = 75° [see Figure 4a−c]. As the Rayleigh number increases to 104, isotherms with θ ≤ 0.2 are shifted along the side walls (DA and BC), and other isotherms with θ ≥ 0.3 are distorted due to the onset of 13308



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convection mode for φ = 15° (see Figure 5a). The isotherms with θ ≤ 0.5 are compressed near the lower portion of wall BC for φ = 15°. This is due to enhanced heat transfer along the wall BC, and thus the thermal boundary layer thickness is small in that region. Compression of isotherms near the wall BC

results in larger thermal gradients near that region. It may be noted that the thermal boundary layer thickness along the wall BC is less than that along the wall DA due to enhanced heat transfer along the wall BC. Therefore, large entropy generation due to heat transfer is observed near wall BC in contrast to near 13309



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correspond to significant entropy production with a local maxima of Sθ being 16. Similar to the previous case (Ra = 103), the entropy generation due to heat transfer is maximum (Sθ,max = 3528) at the bottom corners of the cavity due to the contact of hot wall with the cold walls [see Figures 4a and 5a]. It may be noted that significant entropy generation due to heat

the wall DA. On the other hand, the isotherms with less magnitudes (θ = 0.1−0.3) occur at the top portion of the cavity leading to less temperature gradient, which indicate less heat transfer in that region. Note that the top portion of the walls BC and DA corresponds to Sθ = 1.08 and 0.98, respectively. It is interesting to observe that lower portions of the cavity 13310



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Figure 8. Isotherms (θ), local entropy generation due to heat transfer (Sθ), streamlines (ψ), and local entropy generation due to fluid friction (Sψ) for a tilted square cavity with inclination angle (a) φ = 15°, (b) φ = 45°, and (c) φ = 75° with boundary conditions, θ(X,Y) = 1 (wall AB), θ(X,Y) = 0 (walls BC and DA), and n·∇θ = 0 (wall CD) for Pr = 1000 and Ra = 105.

of buoyancy force over normal component. It may be noted that fluid flow patterns via streamlines are completely circular in contrast to Ra = 103 [see Figures 4a and 5a]. Because of intense fluid circulation and as a consequence of no-slip boundary condition along the wall, high velocity gradients exist near cavity walls leading to significant entropy generation due to

transfer occurs near the left portion of the wall AB where isotherms are highly compressed. It is also found that that trend of the fluid flow patterns show single primary circulation cells with less intense multiple circulation cells near the lower and upper corners of the wall BC. The strong anticlockwise circulation cells are due to dominance of tangential component 13311



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velocity gradient near the wall DA at φ = 75°. Overall, Sθ,max = 3528 is the same for all of the inclination angles due to constant temperature gradients at the corner regions. The maximum value of Sψ,max increases with the inclination angle as indicated by the following Sψ,max values: Sψ,max = 14.70 for φ = 15°, Sψ,max = 27.69 for φ = 45°, and Sψ,max = 30.24 for φ = 75° [see Figure 5a−c]. As Ra increases to 105, the thermal boundary layer thickness near the side walls (DA and BC) becomes thinner due to a larger heat flow in that regime for φ = 15° as seen from isotherm contours of Figure 6a. It may be noted that isotherms are highly compressed near the bottom portion of the wall BC, whereas those are dispersed near the top portion. As a result, thermal gradient decreases along the wall from bottom to top. Hence, the entropy generation due to heat transfer varies as 3528 ≥ Sθ ≥ 1.54 along the wall BC. In contrast, the isotherms are dispersed near bottom and top portions of the wall DA, whereas those are compressed near the middle portion of wall DA. Hence, Sθ varies as 120 ≥ Sθ ≥ 1.65 along the 0.05 ≤ Y ≤ 0.32, and that varies as 1.65 ≤ Sθ ≤ 6.63 along 0.32 ≤ Y ≤ 1. It may be observed that local maxima in Sθ near the top edge of side walls DA (Sθ = 6.63) and BC (Sθ = 6.96) are larger for φ = 15° at Ra = 105 as compared to the previous case with the same inclination angle (φ = 15°) at Ra = 104 [see Figures 5a and 6a]. This is due to an increase in temperature gradient near the top edge during convection dominant mode. It is interesting to observe that Sθ is negligible at the core due to enhanced thermal mixing resulting in a uniform temperature distribution within θ = 0.3−0.4 leading to low thermal gradient in that region. Maximum entropy generation due to heat transfer still occurs at the corner regions A and B as Sθ,max = 3528, similar to the lower Ra case. At higher Ra (Ra = 105), high intense fluid circulation and large velocity gradients near the walls result in larger entropy generation throughout the enclosure. The intensity of fluid flow is enhanced as seen from the large magnitudes of the streamfunction (|ψ|max = 20) [see Figure 6a]. Multiple fluid circulations are found near the corner region of the cavity, and that result in active zones of entropy generation due to fluid friction near the corner regions. Because of intense fluid circulation and as a consequence of no-slip boundary conditions along the wall, high velocity gradients exist near the lower portion of the cavity leading to larger entropy generation due to fluid friction (Sψ,max = 577.46) at the middle portion of wall AB. It is found that variation of Sψ with 1 ≤ Sψ ≤ 40 occurring in the interior region is largely due to the velocity gradients induced by primary circulations [see Figure 6a]. Active zones of Sψ are found to be near the walls due to friction between the fluid circulation cells and the walls of the cavity, and hence significant entropy generation due to fluid friction is observed near wall DA (Sψ = 529.08) and BC (Sψ = 237.16). The local maximum of Sψ on adiabatic wall CD is 167.95. It may be noted that that maximum entropy generation due to heat transfer Sθ,max = 3528 is almost similar as compared to the previous case (Ra = 104), whereas maximum entropy generation due to fluid friction increases as Sψ,max = 577.46 for this case, and Sψ,max = 14.70 was observed for Ra = 104 for φ = 15° [see Figures 5a and 6a]. Flow, temperature, and entropy generation characteristics for φ = 45° are qualitatively similar to that at φ = 15° [see Figure 6a,b]. It is interesting to observe that for φ = 45°, local Sθ at the top portion of the side walls is less than that at φ = 15°. Note that Sθ = 6.53 near the top portion of wall DA and Sθ = 6.06

fluid friction irreversibility (Sψ,max = 14.70 near wall AB) in contrast to lower Ra = 103 (Sψ,max = 0.12 near wall AB) [see Figures 4a and 5a]. It is also found that active zones of entropy generation due to fluid friction also occur near the middle portion of walls DA (Sψ = 12.30) and BC (Sψ = 1.54). It is interesting to observe that the entropy generation value corresponding to wall BC is much less as compared to that of wall DA. This is due to the presence of multiple circulation near the wall BC. Dense contours of Sψ near wall CD corresponding to Sψ = 2.39 also signify the active zone of entropy generation due to fluid friction near that zone. As φ increases to 45° (see Figure 5b), isotherms are slightly distorted. The lower portion of the wall BC receives larger heat as compared to that along the wall DA similar to φ = 15°. Thus, the thermal boundary layer is more compressed toward the wall BC. Therefore, significant entropy generation due to heat transfer is observed near lower portions of the wall BC as compared to Sθ near the wall DA. It may also be noted that the lower portions of the side walls receive larger heat as compared to the top portion of the side walls. Note that significant Sθ (Sθ,max = 3528) is observed along the lower portion of side walls (DA and BC) in contrast to the top portion of the side walls (DA and BC). However, the local maxima of Sθ near the top portion of the walls DA (Sθ = 1.03) and BC (Sθ = 1.11) increase as φ increases from 15° to 45° [see Figure 5a,b]. It is observed that streamlines at φ = 45° are qualitatively similar to that of φ = 15° except that |ψ|max values are larger (|ψ|max = 5.5) for φ = 45° [see Figure 5a and 5b]. The effects of entropy generation due to fluid friction irreversibility (Sψ) are negligible near the central portion of the cavity due to high intensity fluid circulation cells, which leads to less velocity gradients at the core. It may be noted that significant entropy generation for flow irreversibilities occurs in the middle portion of walls AB and DA due to high velocity gradients between the cavity wall and the flow circulation cell adjacent to the walls. It may be noted that entropy generation due to fluid friction is high as compared to Ra = 103 as Sψ,max = 27.69 occurs for Ra = 104, whereas it was 0.35 in the case of Ra = 103 with the same inclination angle (φ = 45°) [see Figures 4b and 5b]. Note that Sψ = 15.49 occurs near the middle portion of the wall DA, Sψ = 11.76 occurs at the wall BC, and Sψ = 7.58 occurs at the wall CD (see Figure 5b). The thermal boundary layer thickness along the wall DA increases due to less compression of the isotherms as φ increases from 45° to 75° [see Figure 5b,c]. Consequently, the local Sθ (Sθ = 0.89) near the top portion of the wall DA decreases as compared to that of φ = 45° (Sθ = 1.03). On the other hand, there is no considerable change in the boundary layer thickness along the wall BC. However, the local Sθ (Sθ = 0.94) along the wall BC also decreases. Similar to φ = 45°, active zones of entropy generation due to heat transfer are larger near corners A and B of the cavity for φ = 75° due to the direct contact between hot and cold walls [see Figure 5c]. Similar to the previous inclination angles (φ = 15° and 45°), the maximum entropy generation due to fluid friction occurs at the almost middle portion of the wall AB for φ = 75° (Sψ,max = 30.24). The local maxima of Sψ on walls DA, BC, and CD are found as 13.11, 20.59, and 10.62, respectively, for φ = 75° at Ra = 104. Note that local maxima of Sψ occurring at the wall BC (Sψ = 20.59) and wall CD (Sψ = 10.62) are comparatively higher than those for φ = 45°. It is interesting to observe that local maxima on the wall DA for φ = 75° (Sψ = 13.11) are less as compared to that of φ = 45° (Sψ = 15.49) due to lesser 13312



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near the top portion of wall BC occur for φ = 45°, whereas Sθ = 6.63 and 6.96 were observed along the top portion of the walls DA and BC, respectively, for φ = 15° [see Figure 6a,b]. Similar to the previous inclination angle (φ = 15°), insignificant Sθ is found within the core region of the cavity due to a uniform temperature distribution in that region (θ = 0.3−0.5). It may be noted that active zones of entropy generation due to heat transfer (Sθ) for φ = 45° remain identical to the previous inclination angle (Sθ,max = 3528). The intensity of fluid flow is weaker for φ = 45° as indicated by |ψ|max values with |ψ|max = 19 for φ = 45°, whereas |ψ|max = 20 is observed for φ = 15°. However, the entropy generation due to fluid friction is high for φ = 45° (Sψ,max = 855.20) as compared to the φ = 15° case (Sψ,max = 577.46) due to large velocity gradients near the interface between cavity wall and circulation cells [see Figure 6a,b]. It is interesting to note that the local entropy generation due to fluid friction on the cavity walls for φ = 45° is higher as compared to that for φ = 15°. The local entropy generation due to fluid friction, Sψ = 586.03, 548.77, and 311.22, occurs on wall DA, wall BC, and wall CD, respectively for φ = 45°, whereas Sψ = 529.08, 237.16, and 167.95 are observed on wall DA, wall BC, and wall CD, respectively, for φ = 15°. The pattern of isotherms and entropy generation maps due to heat transfer (Sθ) is similar for φ = 75° [see Figure 6b,c]. However, the local maxima of Sθ near the top portion of wall DA (Sθ = 5.38) and wall BC (Sθ = 5.65) for φ = 75° are smaller as compared to those of φ = 45° [see Figure 6b,c]. Maximum entropy generation due to heat transfer still occurs at corners between hot wall AB and cold walls BC and DA. It may be noted that Sθ is almost negligible at the core region and also near the wall CD due to low thermal gradient at those regions. It may be noted that the intensity of fluid circulation gradually becomes weaker as φ increases at a higher Rayleigh number (Ra = 105). Note that |ψ|max = 20 for φ = 15°, |ψ|max = 19 for φ = 45°, and |ψ|max = 15 for φ = 75° [see Figure 6a−c]. The maximum entropy generation due to fluid friction decreases for φ = 75° (Sψ,max = 837.98), whereas Sψ,max = 855.20 was observed for φ = 45°. Dense contours of the Sψ are observed near the cavity walls, signifying the active zones of entropy generation due to fluid friction. It is interesting to observe that local Sψ on the wall DA for φ = 75° is smaller as compared to previous inclination angles (φ = 15° and 45°). Note that Sψ = 529.08, 586.03, and 435.53 occur on the wall DA for the φ = 15°, 45°, and 75°, respectively [see Figure 6a−c]. The active zones of entropy generation due to fluid friction are also observed on wall BC (Sψ = 435.53) and wall CD (Sψ = 370.04) for φ = 75°. Similar to the previous case, Sψ has a negligible value at the core of the cavity due to less velocity gradient for all of the inclination angles. As Pr increases to 0.7, the thermal boundary layer near the top portion of the side walls (DA and BC) is thinner for φ = 15° as compared to that in the previous case with Pr = 0.015 [see Figures 6a and 7a]. The active zones of entropy generation due to heat transfer (Sθ) are found to be more intense than those of Pr = 0.015 near the top portion of the side walls (Sθ = 21.67 near wall DA and Sθ = 20.43 near wall BC), whereas Sθ = 6.63 and 6.96 were observed near the top portion of walls DA and BC, respectively, in the case of lower Pr (Pr = 0.015) [see Figures 6a and 7a]. Isotherms with θ ≥ 0.5, which are highly compressed near wall AB except a small middle region, result in active sites of Sθ in those regions. The magnitude of Sθ,max is observed to be similar to the previous case with Pr = 0.015 as observed near corner regions A and B (Sθ,max = 3528).

Comparative studies on Figures 7a and 6a show that the temperature increases at the core of the cavity as Pr increases from 0.015 to 0.7. However, insignificant Sθ is observed at the core of the cavity for the present case (Pr = 0.7) similar to the previous case with lower Pr (Pr = 0.015) at Ra = 105. The fluid rises along the geometrical symmetric line and flows down along the side walls (DA and BC), forming two symmetric rolls along the geometrical symmetric line at Pr = 0.7 in contrast to lower Pr (Pr = 0.015) where a completely circular fluid pattern is observed for φ = 15° [see Figures 6a and 7a]. Convection dominant flow is clearly illustrated by the high intensity of circulation cells as |ψ|max = 13 with clockwise circulation cells and |ψ|max = 14 with anticlockwise circulation cells occur at Ra = 105. It is interesting to observe that the small multiple circulation cells disappear at the corner regions of the cavity due to enhanced convection motion for Pr = 0.7. Maximum entropy generation due to fluid friction (Sψ,max = 734.94) occurs at almost the middle portion of the wall DA due to a high velocity gradient between the wall DA and flow circulation cell adjacent to the wall DA. Active zones of entropy generation for fluid friction also occur near wall AB (Sψ = 310.94) and wall BC (Sψ = 634.10) due to a significant velocity gradient between the cavity wall and the flow circulation cell adjacent to the walls [see Figure 7a]. The local maximum of Sψ on adiabatic wall CD is found to be 204.71. It is interesting to observe that the entropy generation due to fluid friction occurs at the core of the cavity due to a moderate velocity gradient between the adjacent circulation cells. Note that local maximum of entropy generation due to fluid friction, Sψ = 10, occurs at the core of the cavity, whereas insignificant Sψ (Sψ = 0.01) occurs at the core for Pr = 0.015 and φ = 15° (see Figures 6a and 7a). At φ = 45°, isotherms are dispersed near the bottom portion of wall DA, whereas those are compressed near the upper half of wall DA [see Figure 6b]. On the other hand, isotherms are highly compressed near the bottom portion of wall BC, whereas those are dispersed near the top portion of wall BC in contrast to φ = 15°. As a result, the magnitude of entropy generation due to heat transfer decreases along the bottom to top portion of wall BC (3628 ≤ Sθ ≤ 1.26), whereas 3.71 ≤ Sθ ≤ 1.8 within 0.11 ≤ Y ≤ 0.2 and 1.8 ≤ Sθ ≤ 8.51 within 0.2 ≤ Y ≤ 1 are observed along wall DA [see Figure 6b]. The entropy generation due to heat transfer is almost negligible at the central zone of the cavity due to a low temperature gradient as seen from the nearly uniform temperature (θ = 0.3−0.5) at the core of the cavity. Similar to the previous case with φ = 15°, the active zone of entropy generation due to heat transfer (Sθ) is larger near the corners A and B of the cavity (Sθ = 3528) based on direct contact between hot wall AB and cold walls BC and DA. The trend of streamlines is not circular in contrast to the previous case with lower Pr (Pr = 0.015), and they are skewed diagonally due to enhanced momentum transfer at larger Pr (Pr = 0.7) [see Figures 6b and 7b]. A small tiny multiple circulation is found at the top portion of the cavity unlike the previous angle with φ = 15°. The intensity of fluid flow is stronger for φ = 45° as compared to φ = 15° as seen by |ψ|max values. It may be noted that |ψ|max = 14 for φ = 15°, whereas |ψ|max = 20 is observed for φ = 45°. It may also be noted that maximum entropy generation due to fluid friction is higher for φ = 45° (Sψ,max = 850.25 near wall AB) as compared to φ = 15° (Sψ,max = 734.94 near wall DA) [see Figure 7a,b]. Active zones of entropy generation due to fluid friction are also found near the middle portions of wall DA (Sψ = 402.26) and wall BC (Sψ = 171.23) 13313



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found near wall DA (Sψ = 497.42) and wall AB (Sψ = 448.03) for φ = 15°. The small zones of entropy generation due to fluid friction are found on the wall CD (Sψ = 69.29). As φ increases to 45°, the isotherms are less compressed near the top portion of the side walls (DA and BC) [see Figure 8b]. Therefore, less magnitude of Sθ is observed near the top portion of wall DA (Sθ = 15.06) and wall BC (Sθ = 6.62) as compared to that of φ = 15° (Sθ = 25.75 near wall DA and Sθ = 25.60 near wall BC) [see Figure 8a,b]. A large region of the core of the cavity is maintained at uniform temperature (θ = 0.4−0.6), which leads to a low temperature gradient at the core of the cavity, and hence insignificant Sθ is observed in that region [see Figure 8b]. At φ = 45°, strong primary anticlockwise circulation cells occupy almost the entire part of cavity except top corner. The strength of the flow circulation increases as |ψ|max = 20 for φ = 45°, whereas |ψ|max = 14 is observed for φ = 15° [see Figure 8a,b]. It may be noted that Sψ,max at Pr = 1000 (Sψ,max = 631.10) is lower than that at Pr = 0.7 (Sψ,max = 850.25) due to low velocity gradients near the interface between cavity wall and circulation cells for φ = 45° [see Figures 7b and 8b]. Further, it is observed that the local maxima of Sψ on the wall DA decrease from 402.26 to 308.30 as Pr increases 0.7 to 1000. Significant value of Sψ (Sψ = 20) is also found at the interface between primary and secondary circulation near corner C. At φ = 75°, the distribution of Sθ is almost similar to that of Pr = 0.7. Streamlines form the single circulation cell with |ψ|max = 17. A dense contour of Sψ near walls DA and BC indicates the active zones of entropy generation due to fluid friction. It may be noted that the value of Sψ near walls DA and BC decreases with an increase of inclination angle as Sψ = 497.42, 308.30, and 55.50 occur for φ = 15°, 45°, and 75°, respectively, near wall DA, and Sψ = 601.40, 106.34, and 98.70 occur for φ = 15°, 45°, and 75°, respectively, near wall BC. It is found that Sψ,max = 601.40, 631.10, and 876.54 occur for φ = 15°, 45°, and 75°, respectively, for Pr = 1000 [see Figure 8a−c], whereas Sψ,max = 734.94, 850.25, and 915.30 occur for φ = 15°, 45°, and 75°, respectively, for Pr = 0.7 at Ra = 105 [see Figure 7a−c]. 3.3. Average Nusselt Number, Total Entropy Generation, and Average Bejan Number. The variations of total entropy generation due to heat transfer and fluid friction irreversibilities (Stotal), average Bejan number (Beav), and average Nusselt number on the bottom wall (NuAB) versus logarithmic Rayleigh number are presented in the bottom, middle, and top panels of Figure 9a−c, respectively, for the various Prandtl numbers (Pr = 0.015, 0.7, and 1000). Figure 9a represents distributions of Stotal, Beav, and NuAB for Pr = 0.015. The total entropy generation in the cavity is maintained constant up to 8 × 103 irrespective of inclination angles [see bottom panel of Figure 9a]. This is due to smaller entropy generation with fluid friction (Sψ,total) as compared to Sθ,total as seen from Sθ and Sψ maps in Figures 4a−c and 5a−c for all inclination angles. Note that Sθ,total = 5.70 and Sψ,total = 0.0056 for φ = 15°, Sθ,total = 5.72 and Sψ,total = 0.015 for φ = 45°, Sθ,total = 5.73 and Sψ,total = 0.0223 for φ = 75° at Ra = 103, whereas Sθ,total = 6.16 and Sψ,total = 0.41 for φ = 15°, Sθ,total = 6.32 and Sψ,total = 0.65 for φ = 45°, and Sθ,total = 6.34 and Sψ,total = 0.66 for φ = 75° at Ra = 8 × 103. It is interesting to note that Stotal increases exponentially for Ra ≥ 104 due to a gradual increase of fluid friction irreversibility (Sψ,total) and heat transfer irreversibility (Sθ,total) due to high convection of the fluid as seen from contour plots of ψ and Sψ for all of the inclination angles [see Figures 5a−c and 6a−c]. Note that |ψ|max = 5 for

where the circulation cells are in contact with the walls of the cavity due to a high velocity gradient for φ = 45°. It is interesting to observe that the local Sψ on the adiabatic wall CD for φ = 45° (Sψ = 76.92) is less as compared to φ = 15° (Sψ = 204.71). The entropy generation due to fluid friction is negligible near the central portion of the cavity due to less velocity gradients at the core. The isotherms and entropy generation due to heat transfer for φ = 75° are qualitatively similar to that of φ = 45° [see Figure 6b,c]. The maximum entropy generation due to heat transfer is still observed at the corner regions of wall AB with the same magnitude (Sθ,max = 3528) as seen in the previous case with φ = 45°. It is interesting to observe that the Sθ along the wall BC varies as 3528 ≤ Sθ ≤ 0.64 for φ = 75°, which is almost similar to that of φ = 45°. On the other hand, the local Sθ near the top portion of the wall DA for φ = 75° is smaller as compared to that of φ = 45° and 15°. Note that Sθ = 21.67 for φ = 15°, Sθ = 8.51 for φ = 45°, and Sθ = 4.83 for φ = 75° occur near the top portion of the wall DA [see Figure 7a−c]. Similar to φ = 45°, the large portion of the core of the cavity is maintained at uniform temperature (θ = 0.3−0.5) due to enhanced thermal mixing, which leads to less thermal gradients for φ = 75° [see Figure 7c]. Therefore, entropy generation due to heat transfer is negligible at the core of the cavity. The primary circulation cells span the entire cavity, whereas secondary circulation cells completely disappear for φ = 75° [see Figure 7c]. It is interesting to observe that the active zone of entropy generation due to fluid friction is found near all of the walls of the cavity due to a high velocity gradient near the interface between circular cells and the walls of the cavity. It may be noted that Sψ at the core of the cavity is insignificant due to small velocity gradients in that regime. The maxima of entropy generation due to fluid friction increase with inclination angles. The wall DA of the cavity acts as the strong active sites of Sψ as Sψ,max = 734.94 occurs near the middle portion of wall DA for φ = 15°. In addition, wall AB of the cavity acts as the strong active sites of Sψ for higher inclination angles as Sψ,max = 850.25 and Sψ,max = 915.30 occur near wall AB for φ = 45° and 75°, respectively [see Figure 7a−c]. The active zones of entropy generation due to fluid friction are also observed near walls DA and BC for all of the inclination angles. The value of Sψ near wall DA decreases with inclination angle as Sψ = 734.94 for φ = 15°, Sψ = 402.26 for φ = 45°, and Sψ = 137.53 for φ = 75° occur near the top portion of wall DA. It may also be noted that an increase in Pr to 0.7 also makes an observable increase in Sψ,max (Sψ,max = 734.94, 850.25, and 915.30 for φ = 15°, 15°, and 75°, respectively) as compared to that in Pr = 0.015 (Sψ,max = 577.46, 855.20, and 837.98 for φ = 15°, 45°, and 75°, respectively). The isotherms and entropy generation due to heat transfer at Pr = 1000 are qualitatively similar to that of Pr = 0.7 at Ra = 105 [see Figures 7a−c and 8a−c]. The active zones of the Sθ near the top portion of walls DA (Sθ = 25.75) and BC (Sθ = 25.60) are found to be higher for Pr = 1000 as compared to Pr = 0.7 as Sθ = 21.67 and 20.43 were observed near the top portion of walls DA and BC, respectively, for Pr = 0.7 at φ = 15° [see Figures 7a and 8a]. It may be noted that Sθ = 3528 is still observed at the lower corners of the wall AB similar to the previous case with Pr = 0.7. The distribution of Sψ is different from that of the Pr = 0.7 case. It is observed that Sψ,max near wall AB is shifted toward wall BC for φ = 15°. It may be noted that wall BC acts a strong active region of Sψ (Sψ,max = 601.40). The active zones of entropy generation due to fluid friction are also 13314



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Figure 9. Variation of total entropy generation (Stotal), Bejan number (Beav), and average Nusselt number (Nu AB) with Rayleigh number for a tilted square φ = 15°(- - -), φ = 45° (− − −), and φ = 75° (−) at (a) Pr = 0.015, (b) Pr = 0.7, and (c) Pr = 1000.

φ = 15°, |ψ|max = 5.5 for φ = 45°, and |ψ|max = 5 for φ = 75° at Ra = 104, whereas |ψ|max = 20 for φ = 15°, |ψ|max = 19 for φ = 45°, and |ψ|max = 15 for φ = 75° at Ra = 105. It is worthwhile to note that Stotal for φ = 45° cavity is found to be higher for Ra ≥ 104 as compared to the inclination angles, φ = 15° and 75°. It may be noted that Sθ,total = 6.44, Sψ,total = 0.9 for φ = 45°, Sθ,total = 6.43, Sψ,total = 0.89 for φ = 75°, and Sθ,total = 6.29, Sψ,total = 0.61 for φ = 15° at Ra = 104, whereas Sθ,total = 8.16, Sψ,total = 18.81 for φ = 45°, Sθ,total = 8.06, Sψ,total = 15.96 for φ = 75°, and Sθ,total = 8.04, Sψ,max = 16.75 for φ = 15° at Ra = 105. Average Bejan number (Beav) displays the comparative significance of entropy generation due to heat transfer (Sθ) or fluid friction (Sψ) irreversibilities. As mentioned earlier, Beav > 0.5 indicates that entropy generation is heat transfer dominant, whereas Beav < 0.5 indicates the fluid friction dominant entropy generation. 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 conduction dominant mode. Although Beav is higher for φ = 75° (Beav = 0.34) followed by φ = 45° (Beav = 0.32) and φ = 15° (Beav = 0.30) at higher Ra (Ra = 105) [seen in middle panel plots of Figure 9a]. It may be noted that Beav decreases with Ra for all of the inclination angles as seen in the middle panel plots of Figure 9a. The physical reason is that as Ra increases, fluid friction irreversibility (Sψ,total) increases, and that dominates over heat transfer irreversibility (Sθ,total) due

to enhanced convection heat transfer within the cavity. It may be noted that Sθ,max ≥ Sψ,max is observed for all inclination angles, and the dominant fluid friction irreversibility leads to increase in a Stotal at higher Ra (Ra = 105). Consequently, a large amount of available energy is utilized to overcome the irreversibilities due to fluid friction at high Ra (Ra = 105). Total entropy generation in the cavity is found to be invariant with Ra due to negligible Sψ,total over Sθ,total in Stotal at Ra ≤ 104. Hence, heat transfer rate due to temperature gradient (NuAB) is also maintained constant with Ra. In contrast, total entropy generation increases exponentially with Ra due to significant Sψ,total for Ra ≥ 104, and heat transport due to temperature gradient (NuAB) also increases. The thermal gradient for the hot wall AB is high enough to maintain a high heat transfer rate even after spending some available energy to remove high fluid friction entropy generation (Sψ). The distributions of Stotal, Beav, and NuAB for Pr = 0.7 are shown in Figure 9b. The characteristics of Stotal, Beav, and NuAB are qualitatively similar to those of Pr = 0.015 for Ra ≤ 105. Yet Stotal values are higher for Pr = 0.7 at the convection dominant regimes (Ra ≥ 104) due to higher momentum diffusivity. Note that the value of Stotal is 39.86, 32.95, and 32.84 for 15°, 45°, and 75°, respectively, for Pr = 0.7, whereas they were 24.79, 26.97, and 24.02 for 15°, 45°, and 75°, respectively, for Pr = 0.015. Similar to the Pr = 0.015 case, Stotal increases with Ra for all of the inclination angle at Ra ≥ 104. The distribution of Beav 13315



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is qualitatively similar for all inclination angles. The value of Beav at Ra = 105 is lower for the present case as compared to the previous case (Pr = 0.015) due to high fluid friction irreversibility at Pr = 0.7, which is further due to high momentum diffusivity. It may be noted that the heat transfer rate (NuAB) is comparatively higher for Pr = 0.7 than for Pr = 0.015 for convection dominant regimes (Ra ≥ 104) [see top panel plots of Figure 9b]. Further increase in Pr (Pr = 1000) results in an increase of Sψ,total due to increase in momentum diffusivity, which further results in an increase of Stotal and reduction in Beav for φ = 15°. On the other hand, the reduction in Stotal and increase in Beav are observed for the φ = 45° and 75° as Pr increases from 0.7 to 1000 at Ra = 105. It is also found that Sθ,total = 9.56, Sψ,total = 36.40, and Stotal = 45.96 for φ = 15°, Sθ,total = 8.57, Sψ,total = 22.42, and Stotal = 30.99 for φ = 45°, and Sθ,total = 8.72, Sψ,total = 24.39, and Stotal = 33.11 for φ = 75° for Pr = 1000 at Ra = 105. It is also found that NuAB is larger for the φ = 15° as compared to φ = 45° and 75°. Note that NuAB = 9.72, 8.71, and 8.90 for φ = 15°, 45°, and 75°, respectively, occur at Pr = 1000 at Ra = 105.

4. CONCLUSION In the current study, the analysis of thermal mixing and entropy generation due to heat transfer and fluid friction irreversibilities during natural convection within tilted square cavities with hot wall AB and cooled side walls (DA and BC) in the presence of adiabatic wall CD has been performed. The flow and temperature distributions are obtained for Prandtl number (0.015 ≤ Pr ≤ 1000), Rayleigh number (103 ≤ Ra ≤ 105), and inclination angles (φ = 15°, 45°, and 75°). Element basis sets are used via the Galerkin finite element method to evaluate the entropy generation due to heat transfer (Sθ) and fluid friction (Sψ), which are functions of thermal (θ) and flow fields (ψ). Further analysis of average Nusselt number, total entropy generation, and average Bejan number are presented in detail. Important findings of this study are summarized as follows: Streamlines and Isotherms. • The isotherms are smooth and monotonic, indicating conduction dominant heat transfer at Ra = 103, whereas distorted isotherms are observed for higher Ra = 105 due to convection dominant heat transfer. • At Ra = 105, complete circular pattern of streamlines with multiple circulation cells at corner regions is observed in case of lower Pr (Pr = 0.015) irrespective of φ. On the other hand, asymmetric flow pattern with strong anticlockwise circulation and weak clockwise circulation occurs for φ = 15°, whereas anticlockwise circulation cells span almost the entire cavity for higher inclination angles (φ = 45° and 75°) for higher Pr (Pr = 0.7 and 1000). Entropy Generation versus Heat Transfer Rate. • The entropy generations due to the heat transfer are higher near the lower corners (A and B) of the cavity (Sθ,max = 3528) due to the high temperature gradient at those points based on the direct contact of hot and cold walls, irrespective of φ, Ra, and Pr. The active zones of entropy generation due to heat transfer are also found to occur near the top or left portion of walls DA and BC due to compression of isotherms at those regime, and the value of Sθ decreases as φ increases, irrespective of Pr. • The entropy generation due to fluid friction arises within the cavity at two regions. One is due to high velocity

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gradient near the region where the circulation cells are in contact with the walls of the cavity. Another is due to velocity gradient between the adjacent circulation cells. The maximum value of entropy generation due to fluid friction increases with the inclination angle irrespective of Pr except for the case of φ = 75° at Ra = 105. • Insignificant Sψ is observed at the core of the cavity for the case of lower Pr (Pr = 0.015) due to less velocity gradients, whereas significant Sψ is observed at the core of the cavity for higher Pr (Pr = 0.7 and Pr = 1000) due to friction between the adjacent circulation cells. • The inclination angle (φ) has an almost identical effect on heat transfer rate and Stotal for the lower Pr (Pr = 0.015). As Pr increases to 0.7, no remarkable change in NuAB is observed, but Stotal is lower for φ ≥ 45°. Tilted square cavities with φ = 75° may be the optimal geometrical shape for thermal processing of lower Pr (Pr = 0.015 and 0.7). • At Pr = 1000, both heat transfer rate and Stotal change with the inclination angle. Even though the heat transfer rate is low for φ = 45°, the heating strategy is energy efficient due to its less total entropy generation.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS One of the authors, Abhishek Kumar Singh thanks Council of Scientific and Industrial Research, New Delhi (SRF fellowship) to carry out research at the Department of Mathematics and Department of Chemical Engineering, Indian Institute of Technology, Madras, India.

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NOMENCLATURE Be = Bejan number f = function g = acceleration due to gravity, m s−2 k = thermal conductivity, W m−1 K−1 L = side of the tilted square cavity, m N = total number of nodes n = normal vector to the plane p = pressure, Pa P = dimensionless pressure Nu = local Nusselt number Nu = average Nusselt number 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, 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 dx.doi.org/10.1021/ie3013665 | Ind. Eng. Chem. Res. 2012, 51, 13300−13318


Article

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V = y component of dimensionless velocity X,Y = dimensionless distance along x and y coordinate, respectively Greek Symbols

α = thermal diffusivity, m2 s−1 β = volume expansion coefficient, K−1 γ = penalty parameter Γ = boundary θ = dimensionless temperature φ = inclination angle with the positive direction of X axis ν = kinematic viscosity, m2 s−1 ρ = density, kg m−3 Φ = basis functions ψ = streamfunction Ω = two-dimensional domain Subscripts

i = residual number k = node number av = average

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