Analysis of Entropy Generation during Conjugate Natural Convection

Article pubs.acs.org/IECR

Analysis of Entropy Generation during Conjugate Natural Convection within a Square Cavity with Various Location of Wall Thickness Tanmay Basak,*,† Abhishek Kumar Singh,‡ and R. Anandalakshmi† †

Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai 600036, India Department of Mathematics, Indian Institute of Technology Madras, Chennai 600036, India

‡

ABSTRACT: One of the important objectives in thermal systems engineering is to analyze utilization of thermal energy in an efficient manner. Analysis based on second law of thermodynamics may give an insight about efficient usage of thermal energy in various industrial systems. In this regard, entropy generation analysis during conjugate natural convection within a differentially heated square cavity enclosed by vertical conducting walls of different thicknesses (t1 and t2) has been carried out for various thermal systems. Finite element based numerical simulations are carried out for various location of vertical wall thicknesses (cases 1, 2, and 3) in the range of parameters, Ra (103 ≤ Ra ≤ 105), Pr (Pr = 0.015−1000), wall thicknesses (t = 0.2 and 0.8), and conductivity ratios (K = 0.1,1 10 and ∞). Maximum entropy generation due to heat transfer (Sθ,max) occurs near the solid−fluid interface region due to high temperature gradient whereas maximum magnitude of the entropy generation due to fluid friction (Sψ,max) occurs near the cavity walls due to friction between the circulation cells and cavity walls. Larger heat transfer and high intensity fluid flow lead to larger Sθ,max and Sψ,max for K = 10 compared to that of K = 0.1 and K = 1 irrespective of Pr and location of solid wall thickness. Qualitative features of θ, ψ, Sθ and Sψ for t1 + t2 = 0.8 are identical with t1 + t2 = 0.2. However, the magnitude of Sθ,max and Sψ,max is less for t1 + t2 = 0.8 compared to t1 + t2 = 0.2. Based on detailed discussion of average Nusselt number (Nul), average Bejan number (Beavg) and total entropy generation (Stotal) vs various governing parameters (Ra, Pr and t), it may be concluded that thermal processing is invariant of K in conduction dominant region (103 ≤ Ra ≤ 104) irrespective of Pr and t. On the other hand, K ≤ 1 with t1 + t2 ≈ 0.8 may be optimal for thermal processing within the convection dominant region (104 ≤ Ra ≤ 105) due to less entropy generation and reasonable heat transfer rate, irrespective of Pr.

1. INTRODUCTION Natural convection is a convective heat transfer process which arises due to density difference as a result of the temperature difference in the fluid. The fundamental problems of natural convection heat transfer in a closed cavity have received a considerable attention from researchers due to its numerous engineering applications as solar-collectors, electronic device cooling, heat exchanger, melting of metals and design of thermal insulation and chemical reactions, etc.1−9 There are number of studies in the literature related to natural convection within square/rectangular cavities subjected to various thermal boundary conditions.10−15 However, most of the studies on the natural convection were carried out within enclosures in the absence of wall thickness. In general, enclosures have finite thickness and the heat transport within the fluid of the cavity is significantly influenced by the heat transport through the walls (conjugate heat transfer) subjected to various thermal boundary conditions. A few earlier works were carried out for various applications with conjugate heat transfer.16−21 All the real processes related to thermal convection system are associated with thermal gradient and frictional effects and hence some amount of available energy is destroyed during the process due to irreversibilities. All irreversible processes correspond to energy loss or loss of available energy which is quantified in terms of entropy generation. This is based on the Guoy−Stodola theorem which states that the rate of the available work dissipation is proportional to the rate of internal entropy generation.22 The minimization of entropy generation © 2014 American Chemical Society

results in maximum reduction of irreversibilities associated with the process, and thus, the overall system efficiency is enhanced. Hence, entropy minimization is a major challenge for energy saving processing and optimal design criteria for thermal systems. The entropy generation and its minimization for analyzing various systems were investigated extensively by Bejan.23 Basak et al.24 studied the entropy generation during natural convection within a square cavity subjected to various thermal boundary conditions. The thermal and velocity gradients are evaluated accurately based on the elemental basis set via Galerkin finite element method for the entropy production. Kaluri and Basak25 analyzed the role of distributed heating in order to enhance the thermal mixing and temperature uniformity during natural convection within square cavities filled with fluid-saturated porous media using heatline approach. Anandalakshmi and Basak26 also analyzed the entropy generation within the rhombic enclosures of various angles filled with porous media and bounded by adiabatic top wall, cold side walls, and isothermally and nonisothermally heated bottom wall. Kaluri and Basak27 proposed an alternative approach for maintaining uniform temperature with minimum entropy generation via various distributed/discrete heating strategies. Basak et al.28 also investigated the entropy generation during natural convection in trapezoidal enclosures with various Received: September 13, 2013 Accepted: January 3, 2014 Published: February 24, 2014 3702

dx.doi.org/10.1021/ie403033f | Ind. Eng. Chem. Res. 2014, 53, 3702−3722

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inclination angles for different types of temperature boundary conditions. There are significant amount of works which have been reported based on the entropy generation minimization on various physical systems to analyze the efficiency of the thermal systems.29−32 However, only a few articles on analysis of conjugate convection exist with/without entropy generation minimization as summarized below: Varol et al.33 studied the entropy generation due to conjugate natural convection within square cavities for various wall thicknesses. Varol et al.34 also performed numerical analysis on entropy generation due to buoyancy induced convection and conduction with right angle trapezoidal enclosure filled with fluid saturated porous medium. Shuja et al.35 presented the numerical study of natural convection/conduction heat transfer within a heat generating solid body. They also analyzed the entropy generation of the system to determine the irreversibility ratio for each location of the solid body in the cavity. Chu and Liu36 analyzed the entropy generation due to heat transfer and fluid friction within two-dimensional hightemperature confined jet flow. Antar37 evaluated the energy destruction due to heat transfer across a building block with multiple cavities and conduction heat transfer is considered within solid material whereas both natural convection and radiation occur within air-filled cavities. However, the detailed analysis of effect of location of wall thickness and the influence of thermal conductivity ratio between solid wall and the fluid on the entropy generation due to heat transfer and fluid friction is yet to appear in literature. The prime objective of this study is to analyze the entropy generation due to heat transfer and fluid friction within the square cavity where the right wall is isothermally hot, left wall is cold whereas the horizontal walls are well insulated. It is worthwhile to mention that recent works by Basak and coworkers24−28 involve entropy generation in various cavities. The numerical computation on entropy generation was performed using finite element method, and the detailed discussion on the advantage of the finite element method to compute entropy generation was already discussed in earlier articles.38−44 Entropy generation computation in conjugate heat transfer problems requires careful attention due to the presence of solid and fluid phases. In particular, entropy generation due to heat transfer (Sθ) in solid and fluid phases are not identical as heat transfer within solid and liquid phases are based on thermal conductivity of solid (ks) and liquid phases (kf ), respectively. Subsequently, thermal conductivity ratio (K = (ks/ kf )) plays a major role to obtain entropy balance and entropy generation terms, which have been reported for the first time in this work. In addition, current strategy on entropy generation in composite domain is nontrivial due to presence of solid fluid interference and finite element based technique involving adaptive meshing and in built boundary conditions may be efficient on computation of entropy generation via incorporating thermal conductivity ratio and interface boundary conditions. Three cases are considered based on the location of the wall thickness (see Figure 1). Note that, thickness of the wall (t1 + t2) is assumed as 0.2 and 0.8. In the current study, Galerkin finite element method has been employed to solve the non linear equations of fluid flow, thermal field, and entropy generation. Numerical investigation was carried out for a range of parameters as Rayleigh numbers (103 ≤ Ra ≤ 105), Prandtl numbers (Pr = 0.015 and 1000), and thermal conductivity ratios (K = 0.1 and 10). Simulation results are presented in

Figure 1. Schematic diagrams of the physical system for case 1 [finite wall thickness on left side (t1 ≠ 0, t2 = 0)], case 2 [finite wall thickness on right side (t1 = 0, t2 ≠ 0)], and case 3 [finite wall thickness on both side vertical walls (t1 = t2 ≠ 0)]. Note that, t1 + t2 = constant for all cases. The left wall is maintained cold, and the right wall is maintained hot with top and bottom walls as adiabatic.

terms of streamlines (ψ), isotherms (θ), and entropy generation due to heat transfer (Sθ) and fluid friction (Sψ). The effects of Rayleigh number on average Nusselt number, total entropy generation and average Bejan number are also presented to analyze the efficiency of overall heat transport process.

2. MATHEMATICAL FORMULATION 2.1. Governing Equations, Boundary Conditions, and Solution Strategy. The physical domain of the square cavity with various wall thicknesses is shown in Figure 1. Fluid properties are assumed to be constant except the density in the body force term. The change in density due to temperature variation is calculated using the Boussinesq approximation. Using following dimensionless variables y x uL vL , V= , X= , Y= , U= L L αf αf P= αs =

pL2 ρf αf

2

θ=

,

ks , (ρc p) s

T − Tc , Th − Tc

αf =

kf , (ρc p)

Pr =

ν , αf

Ra =

gβ(Th − Tc)L3Pr

f

ν2 (1)

the governing equations for the flow and temperature characteristics during natural convection are given as ∂U ∂V + =0 ∂X ∂Y U

U

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

(2)

(3)

⎛ ∂ 2V ∂V ∂V ∂P ∂ 2V ⎞ +V =− + Pr ⎜ 2 + ⎟ + RaPrθ ∂X ∂Y ∂Y ⎝ ∂X ∂Y 2 ⎠ (4)

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α ⎛ ∂ 2θ ∂θ ∂θ ∂ 2θ ⎞ +V = i⎜ 2 + ⎟· αf ⎝ ∂X ∂X ∂Y ∂Y 2 ⎠

For large values of γ, the continuity equation (eq 2) is automatically satisfied. Typical value of γ = 107 gives consistent solutions. Applying eq 9 in eqs 3 and 4, we get

(5)

Here i = f and s for the fluid and solid phases, respectively; X and Y are dimensionless coordinates varying along horizontal and vertical directions, respectively; U and V are dimensionless velocity components in the x and y directions; θ is the dimensionless temperature. ν is kinematic viscosity; ρs and ρf are the density of solid and fluid respectively; αs and αf are thermal diffusivity in solid and fluid, respectively; Th and Tc are the temperature at hot and cold walls, respectively; P is the dimensionless pressure; p is the pressure, and L is the height of the cavity. Note that, Ra and Pr are Rayleigh and Prandtl number, respectively. It may be noted that, αi/αf = 1 for fluid phase but at the solid region, αi ≠ αf and U = V = 0. It may also be noted that eq 5 is also applicable for solid phase; however, the pure conduction model of eqs 5 reduces to αs ⎛ ∂ 2θ ∂ 2θ ⎞ ⎟=0 ⎜ 2 + αf ⎝ ∂X ∂Y 2 ⎠

U

U

∂ 2ψ ∂ 2ψ ∂U ∂V + = − 2 2 ∂Y ∂X ∂X ∂Y

U = V = 0, U = V = 0,

θ = 1,

for X = 1 + t1 + t 2 ,

∂θ (in solid region) ∂X ∂θ =− (in liquid region) ∂X

Nu = −K

(13)

Here, K (ks/kf) is the thermal conductivity ratio. The local Nusselt numbers at right wall (Nur) and left wall (Nul) depend on the conductivity ratio of solid and fluid phases, and they are defined for various cases as follows: 1. Case 1: Thick solid wall is placed on the left side of the cavity

∀Y (8)

Note that, t1 and t2 are wall thicknesses of left side and right side of the square cavity, respectively, and they are classified as follows: (1) Case 1: Thick solid wall is placed on the left side of the cavity (t1 ≠ 0 and t2 = 0) (2) Case 2: Thick solid wall is placed on the right side of the cavity (t1 = 0 and t2 ≠ 0) (3) Case 3: Thick solid walls are placed on the both sides of the cavity (t1 = t2 ≠ 0) Equations 3−5 associated with boundary conditions (eq 8) are solved using Galerkin finite element method.45 Note that, the continuity equation (eq 2) has been used as a constraint due to the conservation of mass. In order to solve eqs 3 and 4, a penalty finite element method has been employed to eliminate the pressure (P) with a penalty parameter (γ) and the incompressibility criteria given by (eq 2) via the following relationship: ⎛ ∂U ∂V ⎞⎟ P = −γ ⎜ + ⎝ ∂X ∂Y ⎠

(12)

Positive sign of ψ denotes anticlockwise circulation and clockwise circulation is represented by negative sign of ψ. Equation 12 has been solved by finite element method as discussed in earlier work.46 The heat transfer coefficient in terms of the local Nusselt number (Nu) is defined as

In the above equation, K is the thermal conductivity ratio, K = ks/kf. No slip conditions are assumed at solid boundaries of square cavity (see Figure 1), and the boundary conditions for the velocity components and temperature are as follows:

U = V = 0,

(11)

The system of equations (eqs 5, 10, and 11) with boundary conditions (eq 8) are solved using Galerkin finite element method.45 Since the solution procedure is explained in an earlier work,46 the detailed description is not included in this paper. 2.2. Streamfunction and Nusselt Number. The relationship between streamfunction (ψ) and velocity components for two-dimensional flows yields a single equation

(7)

∂θ = 0, for Y = 0, ∀ X ∂Y ∂θ = 0, for Y = 1, ∀ X ∂Y θ = 0, for X = 0, ∀ Y

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

Heat flux is assumed to be continuous at the interface between the solid and fluid regions, and it is represented as

U = V = 0,

(10)

and

(6)

∂θ ∂θf =K s ∂X ∂X

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

9

Nul = K ∑ θi i=1

9

Nur = −∑ θi i=1

∂Φi (for solid region, X = 0) ∂X

(14)

∂Φi (for fluid region, X = 1 + t1 + t 2) ∂X (15)

2. Case 2: Thick solid wall is placed on the right side of the cavity 9

Nul =

∑ θi i=1

∂Φi (for fluid region, X = 0) ∂X 9

Nur = −K ∑ θi i=1

(9)

(16)

∂Φi (for solid region, X = 1 + t1 + t 2) ∂X (17)

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3. Case 3: Thick solid walls are placed on the both sides of the cavity 9

∂Φ Nul = K ∑ θi i (for solid region, X = 0) ∂X i=1 9

Nur = −K ∑ θi i=1

(18)

∂Φi (for solid region, X = 1 + t1 + t 2) ∂X

1

Nus dY

(20)

Here dY denotes the elemental length along sides of the cavity. 2.3. Entropy Generation: Enclosures Filled with Fluid Medium with Solid Conducting Walls. Entropy generation per unit volume Sgen may be estimated using Second Law of Thermodynamics for an open system by the following principle:

+ entropy convected in and out of the system + time rate of entropy accumulations on the control (21)

In a conjugate natural convection system, the associated irreversibilities are due to (i) heat transfer within thick solid wall, (ii) heat transfer within fluid phase or solid−fluid interface temperature difference, and (iii) fluid friction due to fluid−solid interaction as well as fluid−fluid interaction. By considering a control volume subjected to mass flux and energy interactions, the total local entropy generation in the system based on eq 21 and by assuming local thermodynamic equilibrium of linear transport theory,22 for a two-dimensional heat and fluid flow in Cartesian coordinates in explicit form may be written as s ST,gen

f ST,gen =

S(fu , v),gen

(22)

⎪

(27)

(28)

9

∑ f ke k=1

∂Φek ∂n

(29)

f ek

where, 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, 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), i.e.,

(23)

2⎫ ⎧ ⎡ 2 ⎛ ∂v ⎞2 ⎤ ⎛ ∂u μ ⎪ ⎢⎛ ∂u ⎞ ∂v ⎞ ⎪ ⎥ ⎜ ⎟ = ⎨2 +⎜ ⎟ +⎜ + ⎟⎬ ⎪ ⎢⎝ ∂x ⎠ ⎪ ∂x ⎠ ⎭ T⎩ ⎝ ∂y ⎠ ⎥⎦ ⎝ ∂y ⎣

∂fi

1 = e ∂n N

(24)

SsT,gen

⎪

⎪

μf To ⎛ α ⎞2 ⎜ ⎟ k f ⎝ LΔT ⎠

∂f e = ∂n

2⎤

⎛ ∂T ⎞ k f ⎡⎛ ∂T ⎞ ⎢⎜ ⎟ + ⎜ ⎟ ⎥ ⎝ ∂Y ⎠ ⎦ T 2 ⎣⎝ ∂X ⎠ 2

⎪

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 as discussed in next section. 2.4. Post-Processing for Evaluation of Entropy Generation. It may be noted that no residual equations are required to solve for estimation of entropy generation. However, accurate evaluation of derivatives is the key issue for proper estimation of Ssθ, Sfθ, and Sfψ. Small error in the calculations of temperature and velocity gradients would propagate to a much larger error since the derivatives are powered to 2. As mentioned earlier, the derivatives are evaluated based on finite element method. Current approach offers special advantage over finite difference or finite volume solutions where derivatives are calculated using some interpolation functions which are avoided in the current work and elemental basis set are used to estimate Ssθ, Sfθ, and Sfψ. The derivative of any function f over an element e is written as

= entropy transfer associated with heat transfer

k ⎡⎛ ∂T ⎞2 ⎛ ∂T ⎞2 ⎤ = s2 ⎢⎜ ⎟ + ⎜ ⎟ ⎥ ⎝ ∂Y ⎠ ⎦ T ⎣⎝ ∂X ⎠

(26)

ϕ=

entropy generation

volume

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

Here, Ssθ and Sfθ are local entropy generation due to heat transfer in the solid phase and fluid phase, respectively; whereas, Sfψ is the local entropy generation due to fluid friction in the fluid phase and K is the thermal conductivity ratio between solid (ks) and fluid (kf) phases. In the above equation, ϕ is called irreversibility distribution ratio, defined as

The average Nusselt numbers at the side walls are

∫0

(25)

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

(19)

Nus =

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

SfT,gen

Note that, and are local entropy generation due to heat transfer in the solid and fluid phase, respectively, whereas Sf(u,v),gen is entropy generation due to viscous effects or fluid friction in fluid phase. It is worthwhile to mention that the entropy generation due to heat transfer and fluid friction are characterized by the temperature and velocity gradients, respectively. The dimensionless form of above equation in individual terms of entropy generation due to heat transfer (Sθ) and entropy generation due to heat transfer (Sψ) may be written as

Ne

∑ e=1

∂f ie ∂n

(30)

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

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

(31)

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

(32)



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out for various fluids of Pr (Pr = 0.015 and 1000), wall thickness, t1 + t2 = 0.2 and 0.8, thermal conductivity ratios (K = 0.1 and 10) and 103 ≤ Ra ≤ 105. To validate the code, benchmark studies were carried out for the differentially heated square cavity with hot left wall and cold right wall in presence of adiabatic horizontal walls, similar to the case reported by Ilis et al.47 The results based on the present algorithm are shown in Figure 2, and current simulation results are found to be in good agreement with the earlier work.47 3.2. Numerical Simulations for Total Wall Thickness, t1 + t2 = 0.2. Figures 3−8 show isotherms (θ), streamlines (ψ), entropy generation due to heat transfer (Sθ), and fluid friction

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

(33)

The derivatives, ∂θi/∂X, ∂θi/∂Y, ∂Ui/∂X, ∂Ui/∂Y, ∂Vi/∂X, and ∂Vi/∂Y are evaluated according to the 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. Stotal =

∫Ω,s Sθs dΩ + ∫Ω,f Sθf dΩ + ∫Ω,f Sψf dΩ

= Sθ ,total + Sψ ,total

(34)

where Sθ ,total

⎧⎡ ⎤2 ⎡ ∂ N ⎤2 ⎫ ⎪ ⎪ ∂ N ⎢ ⎥ ⎢ ⎥ ⎨ =K (∑ θk Φk ) + (∑ θk Φk ) ⎬ dX dY ⎪ Ω,s ⎪⎢ ⎥ ⎢ ⎥ ∂ X ∂ Y ⎦ ⎣ ⎦⎭ k=1 k=1 ⎩⎣

∫

⎧ ⎤2 ⎡ ∂ N ⎤2 ⎫ ⎪ ⎪⎡ ∂ N ⎢ ⎥ ⎢ ⎥ ⎨ + (∑ θk Φk ) + (∑ θk Φk ) ⎬ dX dY Ω,f ⎪⎣ ⎥⎦ ⎢⎣ ∂Y k = 1 ⎥⎦ ⎪ ⎭ ⎩⎢ ∂X k = 1

∫

(35)

and Sψ ,total = ϕ

⎧ ⎡ ⎪

N

⎩ ⎣

k=1

⎤2

⎡ ∂ N ⎤2 + 2⎢ (∑ Vk Φk )⎥ ⎢⎣ ∂Y k = 1 ⎥⎦ ⎦

∫Ω,f ⎨⎪2⎢⎢ ∂∂X (∑ Uk Φk)⎥⎥

N ⎡∂ N ⎤2 ⎫ ⎪ ∂ (∑ Vk Φk )⎥ ⎬ dX dY + ⎢ (∑ Uk Φk ) + ⎢⎣ ∂Y k = 1 ⎥⎦ ⎪ ∂X k = 1 ⎭

(36)

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

Sθ ,total Sθ ,total + Sψ ,total

=

Sθ ,total Stotal

(37)

Note that, 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 Tests. The computational domain is defined in three cases based on location of the wall thickness in the cavity. In case 1, wall thickness on the right side (t2) of the cavity is neglected and the left wall (t1) has considerable wall thickness. In case 2, wall thickness on the left wall of the cavity is neglected and considerable wall thickness is maintained on the right side of the cavity whereas finite wall thickness on both vertical sides (t1 and t2) of the cavity are considered in case 3. The wall thickness is maintained identical in all three cases. The domain consists of 24 × 24 biquadratic elements corresponding to 49 × 49 grid system for solid wall thickness, t1 + t2 = 0.2, and 28 × 28 biquadratic elements corresponding to 57 × 57 grid points for solid wall thickness, t1 + t2 = 0.8 with adaptive grid refinement near the solid−fluid interface in all cases. Detailed computations have been carried

Figure 2. Local entropy generation due to heat transfer Sθ and fluid friction Sψ for a square enclosure with hot left wall, cold right wall and adiabatic horizontal walls for Pr = 0:71, (a) Ra = 103 (top figure: present work; lower figure: work reported by Ilis et. al.47) and (b) Ra = 105 (top figure: present work; lower figure: work reported by Ilis et. al.47). Lower images of (a) and (b) have been reprinted from ref 47 with permission from Elsevier. Copyright 2008. Elsevier. 3706



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Figure 3. Isotherms (θ), local entropy generation due to heat transfer (Sθ), streamlines (ψ), and local entropy generation due to fluid friction (Sψ) for (a) case 1 (t1 = 0.2, t2 = 0), (b) case 2 (t1 = 0, t2 = 0.2), and (c) case 3 (t1 = t2 = 0.1) at Pr = 0.015, K = 0.1, and Ra = 103.

within solid phase of the cavity based on presence of Sθ contours with high magnitude in those regions (Sθ = 1.0− 1.17). This is due to maximum thermal gradient (θ = 0.1−0.65) within the solid phase of the cavity. The thick cold wall with lower thermal conductivity ratio (K = 0.1) will not allow the cooling effect to penetrate through the wall. Thus, θ = 0.7−0.9 is maintained within the fluid which leads to a lesser temperature gradient in the fluid phase. As a result, insignificant entropy generation due to heat transfer (Sθ ≈ 0:03 − 0:15) is observed in the fluid phase. It may be observed that the thick solid wall on left portion of the cavity acts as a strong site of Sθ compared to fluid phase due to high temperature gradient in the solid phase. It is interesting to note that isotherms are more compressed within the top portion of solid left wall leading to high temperature gradients in that region. Therefore, distribution of local entropy generation due to heat transfer

(Sψ) for different cases based on the location of wall thickness in the cavity for various fluids (Pr = 0.015 and 1000), conductivity ratios (K = 0.1 and 10), and Rayleigh numbers (103 ≤ Ra ≤ 105) within a square cavity with wall thickness, t1 + t2 = 0.2. Figure 3a−c illustrates the effect of location of wall thickness for Ra = 103, K = 0.1, and t1 + t2 = 0.2 for cases 1−3. At low Rayleigh number (Ra = 103), the isotherms are parallel to the side walls indicating that heat transfer is primarily due to conduction, irrespective of cases (see Figure 3a−c). It may be noted that isotherms with θ ≥ 0.7 occur within the fluid region whereas isotherms with θ ≤ 0.65 was observed within the solid region in case 1 (see Figure 3a, case 1). Presence of isotherms with θ ≤ 0.65 within the solid wall indicates that thermal gradient is high due to lower thermal conductivity ratio (K = 0.1). The distribution of local entropy generation due to heat transfer (Sθ) depicts that the entropy generation is higher 3707



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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) case 1 (t1 = 0.2, t2 = 0), (b) case 2 (t1 = 0, t2 = 0.2), and (c) case 3 (t1 = t2 = 0.1) at Pr = 0.015, K = 0.1 and Ra = 105.

phase of the cavity is maintained quite low temperature (θ = 0.1−0.35) compared to that of case 1; where isotherms with θ = 0.7−0.9 are observed within the fluid region. This is due to low conductivity of right hot solid wall (K = 0.1) which corresponds to high resistance to distribute heat to fluid. Similar to case 1, high temperature gradient is observed within solid wall based on high range of isotherms with θ = 0.35−1 within the right solid wall for the case 2. Therefore, local active zones of entropy generation due to heat transfer occur within solid walls (Sθ = 0.8−1.17). In contrast, Sθ is almost negligible (Sθ ≈ 0.10−0.15) in the fluid phase of cavity due to the lesser temperature gradient (θ ≤ 0.3) in the fluid phase. It may be noted that Sθ,max is observed near lower portion of right wall due to high temperature gradients in that region. It is

(Sθ) for case 1 is higher (Sθ,max = 1.17) near top portion of left wall. The flow intensity is very weak at low Ra (Ra = 103) as seen from the less magnitude of streamfunction (|ψ|max = 0.4). Thus, maximum entropy generation due to fluid friction (Sψ,max = 0.04) which corresponds to the middle portion of right wall, is insignificant relative to Sθ (Sθ,max = 1.17). The active regions of entropy generation due to fluid friction are also observed near middle portion of the horizontal walls (Sψ = 0.03) and left wall (Sψ = 0.02) due to friction between fluid circulation cells and cavity walls. Figure 3b illustrates the temperature distribution, fluid flow, and entropy generation maps for case 2 (t1 = 0 and t2 = 0.2). The distribution of θ, Sθ, ψ, and Sψ for case 2 follows qualitatively similar trend as that of case 1. However, the fluid 3708



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interesting to observe that the magnitudes of Sθ,max (Sθ,max = 1.17) and Sψ,max (Sψ,max = 0.04) for case 2 are similar to case 1. Figure 3c illustrates that temperature of the fluid phase of the cavity varies within θ = 0.4−0.6 as isotherms with θ ≥ 0.7 occurring in right wall and θ ≤ 0.3 is observed in the left wall for case 3 (t1 = 0.1 and t2 = 0.1). Thick side walls of the cavity act as strong site of entropy generation due to heat transfer (Sθ = 1.05−1.21) due to high temperature gradient within solid walls in case 3 (θ ≥ 0.7 corresponds to right wall and θ ≤ 0.3 corresponds to left wall). Insignificant Sθ within fluid region remains the same to previous cases (case 1 and case 2) due to less temperature gradient in the fluid region of the cavity. It is also found that isotherms within fluid phase are compressed along the lower portion of right wall and top portion of left wall and hence maximum entropy generation due to heat transfer (Sθ,max) occurs near those regions for case 3 (Sθ,max = 1.21). It is interesting to observe that the magnitude of Sθ,max is lower for case 1 and case 2 compared to case 3. Similar to previous cases (case 1 and case 2), the maximum magnitude of the entropy generation due to fluid friction (Sψ,max = 0.04) is less due to weak fluid flow (|ψ|max = 0.4) for case 3. The presence of Sθ contours with high magnitude within solid walls of cavity indicates the high entropy production due to heat transfer in the solid walls, irrespective of the location of wall thickness. Figure 4 illustrates the effect of location of the wall thickness on the streamlines, isotherms, and entropy generation due to heat transfer and fluid friction for K = 0.1 at Ra = 105 and Pr = 0.015. Isotherms are highly distorted in the fluid phase due to enhanced convection for cases 1−3 at Ra = 105 (see Figure 4). Figure 4a illustrates the temperature distribution, fluid flow, and corresponding entropy maps for the case 1 (t1 = 0.2 and t2 = 0). Similar to previous test study, the parallel isotherms with θ ≤ 0.8 occur within the solid left wall, indicating conduction dominant heat transfer in solid wall. The high thermal gradient with θ = 0.1−0.8 is found in the left cold solid wall, and hence, this wall acts as a strong active site of the entropy generation due to heat transfer corresponding to Sθ = 1.5−1.84 which are higher than that of Ra = 103 (Sθ = 1.0−1.17) for case 1. It is interesting to observe that the temperature gradient of the central portion of the fluid region is less (θ = 0.85−0.95) compared to that of Ra = 103 (θ = 0.7−0.9) for case 1. Insignificant Sθ (Sθ = 0.01−0.42) occurs within the fluid phase due to less temperature gradient in that region; whereas, Sθ = 0.1−0.12 is observed in the central portion of fluid phase at low Ra (Ra = 103) for case 1. This is further due to uniform temperature distribution in the central portion of fluid region leading to less temperature gradients. Another active region of Sθ is also observed on the lower portion of the right wall (Sθ = 0.42) due to comparatively higher temperature gradient as compression of isotherms is observed in that region. The maximum value of Sθ is found to be 1.84 at the almost top portion of the cold solid wall. The intensity of the fluid circulation is very high as seen from the high magnitude of the streamfunction (|ψ|max = 5.2). The high velocity gradient exists near to cavity walls due to friction between high intense fluid circulation and walls leading to high entropy generation due to fluid friction. The maximum magnitude of Sψ is observed at the middle portion of the right wall (Sψ,max = 43.17). It is observed that the local maxima of Sψ at the bottom wall (Sψ = 37.47) is higher than that at the top wall (Sψ = 35.07). Another local maxima of Sψ (Sψ = 10.06) occurs near the middle portion of the left wall−fluid interface region. Comparative studies of Sθ and Sψ indicate that the value of Sψ is higher than the value of Sθ

exhibiting the convection dominance at high Rayleigh number (Ra = 105). Figure 4b illustrates the isotherms, streamlines, and entropy generation due to heat transfer and fluid friction for case 2 (t1 = 0,t2 = 0.2) with identical parameters. It may be noted that isotherms up to θ ≤ 0.2 are distorted in the fluid region due to convection, but the entire cavity is maintained at very low temperature. Isotherms up to θ ≥ 0.2 are compressed along the hot thick right wall with K = 0.1 and hence high temperature gradient is observed along the right thick wall. Therefore, high magnitudes of Sθ (Sθ = 1.5−1.8) occur within the right solid wall. In contrast, insignificant Sθ (Sθ ≈ 0.015−0.2) is observed in the central portion of fluid phase due to less temperature gradients in the fluid phase (θ ≤ 0.2). Sθ,max (Sθ,max = 1.84) is found to occur within bottom portion of right wall. Due to larger gradient of temperature on top portion of left wall, Sθ = 0.42 is observed at that zone. The patterns of the fluid flow for the case 2 are qualitatively similar as that of case 1 and maximum magnitude of streamfunction (|ψ|max = 5.2) is also same for both the cases (case 1 and case 2) [see Figure 4a and b]. There is no considerable change in the Sψ,max (Sψ,max = 43.17) for case 2 based on the identical intensity of the fluid flow pattern for both cases (case 1 and case 2). Note that, local maxima of Sψ with values as 37.47 and 35.07 occur at the top and bottom walls, respectively. Another local Sψ is observed near middle portion of right wall-solid interface region (Sψ = 10.06) due to significant velocity gradient in that region. This is further due to friction between the fluid circulation cells and solid wall. The distribution of the fluid flow, temperature, and entropy generation due to heat transfer and fluid friction for the case 3 (t1 = 0.1, t2 = 0.1) with identical parameters is shown in Figure 4c. Highly distorted isotherms with θ = 0.4−0.6 occur in the fluid phase, indicating convection mode of heat transfer in the fluid phase. Uniform temperature distribution within fluid phase (θ = 0.4−0.6) leading to low temperature gradient and, hence, insignificant distributions of Sθ (Sθ = 0.01−0.2) are observed within the fluid phase. Parallel isotherms with θ ≤ 0.3 are confined in the left solid wall, and θ ≥ 0.7 are confined in the right solid wall as a high temperature gradient is observed within both the solid walls. Therefore, both the solid walls act as active zones of entropy generation due to heat transfer. High magnitude of Sθ contours (Sθ = 1.35−1.96) occur within the both the solid walls at high Ra (Ra = 105); whereas, Sθ = 1.05− 1.21 are observed within the solid walls at low Ra (Ra = 103) for case 3. The maximum value of Sθ is found to be 1.96 which occurs within the lower portion of the right wall as well as within the upper portion of the left wall. Similar to the previous two cases (cases 1 and 2), streamlines follow a circular pattern in case 3 but the intensity of fluid flow is less for case 3 (|ψ|max = 4.6) compared to cases 1 and 2 (|ψ|max = 5.2) [Figure 4]. Therefore, the maximum magnitude of Sψ,max for case 3 (Sψ,max = 32.76) is also comparatively less than those of cases 1 and 2 (Sψ,max = 43.17). It is also observed that Sψ,max occurs along a central portion of the horizontal walls for case 3 but Sψ,max occurs along one of the vertical walls for cases 1 and 2. It is interesting to observe that Sθ,max for case 3 (Sθ,max = 1.96) is higher than the previous two cases (Sθ,max = 1.84); whereas, Sψ,max for case 3 (Sψ,max = 32.76) is less compared to the previous two cases (Sψ,max = 43.17). The interface region of the solid and fluid phase as well as solid walls acts as a strong site of entropy generation due to heat transfer, irrespective of the location of wall thickness. It is also found that the solid−fluid 3709



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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) case 1 (t1 = 0.2, t2 = 0), (b) case 2 (t1 = 0, t2 = 0.2), and (c) case 3 (t1 = t2 = 0.1) at Pr = 0.015, K = 10, and Ra = 105.

1.1−1.84). As mentioned earlier, fluid region offers high resistance compared to solid wall (ks/kf = K = 10) and that further leads to θ = 0.2−1 within the fluid region with large compression of isotherms near top left portion of the fluid region immediate to solid−fluid interface as seen in Figure 5a. Hence, Sθ offers its maximum (Sθ,max = 81.55), which occurs due to large temperature gradient at the solid−fluid interface at K = 10 whereas Sθ,max = 1.84 occurs within the solid region due to high thermal gradient, which is purely offered by large resistance (ks/kf = K = 0.1) to heat transfer in the solid region at K = 0.1. It may also be noted that local entropy production due to heat transfer (Sθ) near the lower portion of right wall at K = 10 (Sθ = 30.66) is higher than that of K = 0.1 (Sθ = 0.42). This is due to high thermal resistance offered by the fluid (θ =

interface causes significant entropy production due to fluid flow (Sψ ≈ 11). Figure 5 illustrates the isotherms, streamlines, and entropy generation maps due to heat transfer and fluid friction for Pr = 0.015 and Ra = 105 for cases 1−3 at K = 10. As the conductivity ratio increases from K = 0.1 to 10, the resistance offered by solid wall on heat distribution is gradually reduced and the solid wall does not impose any extra resistance to fluid heating and flow characteristics. Therefore isotherms with θ = 0−0.1 are confined within the left thick wall whereas θ = 0−0.8 were confined within the solid wall at K = 0.1 (see Figures 4a and 5a). Due to lesser thermal gradient within the solid wall (θ = 0−0.1), comparatively less distribution of Sθ (Sθ = 0.2−1.6) occurs within the solid region for K = 10 than K = 0.1 (Sθ = 3710



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Figure 6. Isotherms (θ), local entropy generation due to heat transfer (Sθ), streamlines (ψ), and local entropy generation due to fluid friction (Sψ) for (a) case 1 (t1 = 0.2, t2 = 0), (b) case 2 (t1 = 0, t2 = 0.2), and (c) case 3 (t1 = t2 = 0.1) at Pr = 1000, K = 0.1, and Ra = 103.

wall compared to K = 0.1 (Sψ,max = 43.17). The local entropy generation due to fluid friction is observed on the middle portions of bottom wall (Sψ = 367.30) and top wall (Sψ = 365.07); whereas, a significant amount of entropy is also generated near to left wall−fluid interface region (Sψ = 97.55). Insignificant Sψ is observed at core region of the fluid phase due to less velocity gradients at those regions. Figure 5b illustrates the isotherms, streamlines, and entropy generation maps due to heat transfer and fluid friction for case 2 (t1 = 0 and t2 = 0.2) with identical parameters (Pr = 0.015, Ra = 105, and K = 10). Due to high conductivity ratio (K = 10), the thick right wall offered less resistance on the temperature distribution through solid wall. As a result, isotherms with θ ≥ 0.9 are confined along the solid wall whereas isotherms with θ = 0−0.8 are confined the fluid phase in case 2. Similar to case 1,

0.2−1) at high conductivity ratio (K = 10) compared to lower conductivity ratio, K = 0.1 (θ = 0.8−1). A large thermal gradient leads to a loss of available energy or entropy generation, and therefore, resistance offered by the solid wall/ fluid (K) plays a significant role in the entropy production due to heat transfer (Sθ). Due to overall thermal gradient based on θ = 0.2−1 within the fluid region, convection currents are strong at Ra = 105 for case 1. Similar to previous case with low conductivity ratio (K = 0.1), streamlines follow the circular pattern, irrespective of location of the wall thickness. The intensity of fluid flow increases due to increase in conductivity ratio from K = 0.1 (|ψ|max = 5.2) to 10 (|ψ|max = 7.25). As a result, the maximum magnitude of entropy generation due to fluid friction (Sψ,max) is higher for K = 10, Sψ,max = 396.84 at middle portion of right 3711



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left wall (Sθ = 30.66) which is higher than that local maxima of Sθ at low K (K = 0.1). The entropy distribution within the fluid region is high due to higher temperature gradient within the fluid region (θ ≤ 0.8) at K = 10 whereas θ ≤ 0.2 is observed within the fluid region at K = 0.1. It may be noted that fluid flow pattern for the case 2 is qualitatively similar to that of case 1 with same maximum magnitude of streamfunction (|ψ|max = 7.25). High velocity gradients exist near the walls due to no-slip boundary conditions and that leads to significant entropy generation due to fluid friction irreversibility. On the basis of intensity of fluid flow, high entropy generation due to fluid friction for case 2 (Sψ,max = 396.85) is also similar to that of case 1 and that occurs on the middle portion of left wall. As it may be seen from the maps of Sψ, the active sites for Sψ are also

insignificant entropy generation due to heat transfer (Sθ = 0.2− 1.5) is observed within the solid wall due low temperature gradient within solid phase (θ ≥ 0.9) for case 2. It is interesting to observe that isotherms are highly compressed along the top portion of cold left wall and bottom portion of hot right wall. The high magnitudes of Sθ are observed on solid−fluid interface, and hence, these regions act as high entropy production regions due to heat transfer. The magnitude of maximum entropy generation due to heat transfer in case 2 (Sθ,max = 81.55) is similar to case 1 (see Figure 5a and b). It is interesting to observe that maximum magnitude of the entropy generation due to heat transfer (Sθ,max) occurs near the solid− fluid interface due to high conductivity ratio (K = 10). Another local maxima of Sθ is also observed near to upper portion of the 3712



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that the maximum magnitude of entropy generation due to heat transfer is higher for case 3 (Sθ,max = 1.21) compared to cases 1 and 2 (Sθ,max = 1.17). The intensity of fluid flow is less as |ψ|max is observed at 0.4 for all three cases due to low conductivity ratio (K = 0.1) and low Rayleigh number (Ra = 103). The pattern of streamlines is circular at the core; whereas, it takes the shape of the cavity near the walls. The entropy generation due to fluid friction is less due to less intense fluid circulation cells. The maximum magnitude of Sψ (Sψ,max) is found to be 0.03 on the cavity walls due to friction between the walls and fluid circulation cells for all three cases. Figure 7 shows the isotherms, streamlines, and entropy generation due to heat transfer and fluid friction at Pr = 1000 with Ra = 105 and K = 0.1 for all three cases. At high Ra (Ra = 105), the heat transfer in the cavity is due to convection mode based on high intensity of fluid flow and hence isotherms are found to be highly distorted within the fluid region. The presence of parallel isotherms within solid walls indicates the conduction mode of heat transfer within solid walls. It may be noted that isotherms are slightly compressed near the lower portion of the right wall and top portion of the left wall. This further leads to active zones of Sθ near those regions. A high temperature gradient is observed within solid walls at low conductivity ratio, K = 0.1, which does not allow the cooling or heating effect to penetrate through the wall, for all the three cases. Isotherms with θ ≤ 0.8 occur within the left solid wall for case 1 whereas θ ≥ 0.25 occurs in the solid right wall for case 2. On the other hand, isotherms with θ ≤ 0.3 and θ ≥ 0.7 are observed within the left and right solid walls, respectively, for case 3. The entropy generation due to heat transfer is very high within the solid wall region due to high temperature gradient, for all three cases. The local distribution of Sθ within the solid phase of the cavity at Ra = 105 is more significant compared to that of Ra = 103. It may be noted that Sθ = 1.45−1.89 and Sθ = 1.5−1.93 occur within the solid walls for cases 1 and 2, respectively; whereas, Sθ = 1.2−2.35 is observed within the solid walls for case 3 at high Ra (Ra = 105) with K = 0.1 and Pr = 1000 (see Figure 7). On the other hand, the solid wall is confined with the Sθ = 1−1.17 for cases 1 and 2 and Sθ = 1− 1.21 are observed within the solid walls for case 3 at Ra = 103 with K = 0.1 and Pr = 1000 (see Figures 6). It may be noted that Sθ,max occurs within the left solid wall for case 1 (Sθ,max = 1.89) and within right solid wall for case 2 (Sθ,max = 1.93); whereas, Sθ,max = 2.35 is observed within the left and right solid walls for case 3 at Ra = 105 and K = 0.1. Another local maxima of Sθ is observed near lower portion of right wall for case 1 (Sθ = 0.74) and top portion of left wall for case 2 (Sθ = 0.52) [see Figure 7a and b]. It may be noted that temperature gradient of fluid phase is much less as θ = 0.8−1, θ = 0−0.2, and θ = 0.4− 0.55 are observed within the fluid phase of the cavity for cases 1, 2, and 3, respectively. As a result, insignificant entropy generation due to heat transfer is observed within the fluid region for all three cases. On the other hand, entropy generation due to heat transfer within the core region of fluid phase is higher for Ra = 103. Note that, Sθ = 0.09−0.15 is observed within the core region of the fluid phase at Ra = 103, irrespective of the location of wall thickness; whereas, Sθ = 0.005−0.1 occurs within the core region of the fluid phase at Ra = 105, irrespective of the location of wall thickness (see Figures 6 and 7). Similar to lower Rayleigh number case (Ra = 103), fluid flow patterns are almost circular at the core and try to take the shape of the cavity near the walls of the cavity at Ra = 105 with K =

found at the bottom wall (Sψ = 365.06), top wall (Sψ = 367.30), and right wall (Sψ = 97.55). These local active zones of Sψ are due to the friction between primary fluid circulation cells and cavity walls. There exists insignificant entropy generation corresponding to fluid friction at the core of the cavity due to less velocity gradient, in contrast to the wall side region. It may be noted that isotherms with θ = 0.3−0.8 are distributed throughout the fluid region and isotherms with θ ≤ 0.1 are confined in the left wall; whereas, θ ≥ 0.9 are confined in the right wall for case 3 (see Figure 5c). The isotherms are smooth and parallel in the solid wall based on conduction mode of heat transfer whereas it is distorted in the fluid region based on convection mode of heat transfer, irrespective of location of wall thickness. At the high conductivity ratio (K = 10), insignificant entropy generation due to heat transfer (Sθ = 0.1−1.5) is observed within both the solid walls due to less temperature gradients within the solid walls (θ ≤ 0.1 correspond to left wall θ ≥ 0.9 correspond to right wall). The maximum magnitude of entropy generation due to large thermal gradient occurs near the top portion of left wall and lower portion of right wall on the solid wall−fluid interface region for case 3 (Sθ,max = 84.85). It may be noted that Sθ,max in case 3 (Sθ,max = 84.85) is larger compared to case 1 and case 2 (Sθ,max = 81.55). It is interesting to observe that temperature gradient within the fluid region is high at K = 10, and hence, entropy generation within the fluid region is high compared to that of K = 0.1, irrespective of location of wall thickness. The fluid flow patterns are circular for all the three cases, but the intensity of fluid flow is slightly less for case 3 (|ψ|max = 7.22) compared to cases 1 and 2 (|ψ|max = 7.25). The maximum magnitude of the Sψ for case 3 is observed near the middle portion of horizontal walls (Sψ,max = 364.91) which is slightly less compared to those of cases 1 and 2 (Sψ,max = 396.84). Overall, maximum entropy generation due to heat transfer and fluid friction for K = 10 is higher compared to those of K = 0.1, irrespective of location of wall thickness. Thus, higher K may not be found advantageous due to high entropy production. Figure 6 shows the isotherms, streamlines, and entropy generation due to heat transfer and fluid friction at Pr = 1000 with Ra = 103 and K = 0.1 for all three cases. At low Ra (Ra = 103) and low conductivity ratio (K = 0.1), conduction mode of heat transfer is observed. Isotherms are almost smooth and parallel within solid phase as well as fluid region due to conduction dominant heat transfer, irrespective of solid wall location. High temperature gradient within the solid phase is observed due to low conductivity ratio, K = 0.1 with θ = 0.1− 0.6 for case 1 and θ = 0.4−1 for case 2 within solid phase. As a result, entropy generation due to heat transfer is high as Sθ = 1.08−1.17 is observed within the solid phase for cases 1 and 2. Insignificant Sθ (Sθ ≈ 0.1−0.15) occurs due to the lesser temperature gradient within the fluid region. The maximum magnitude of Sθ is found to be within top portion of solid wall at left face for case 1 and within lower portion of solid wall at right face for case 2 with the same magnitude (Sθ,max = 1.17) [see Figure 6a and b]. However, Sθ,max is not large due to low conductivity ratio (K = 0.1). It is interesting to observe that isotherms with θ = 0.1−0.3 and θ = 0.7−1 are confined within left solid wall and right solid wall, respectively; whereas, θ = 0.4−0.6 are confined in the fluid region for case 3 (see Figure 6c). Similar to cases 1 and 2, solid walls act as the active site of for entropy generation due to heat transfer (Sθ = 1.05−1.21) and insignificant entropy generation due to heat transfer is observed within the fluid phase of the cavity for case 3. Note 3713



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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) case 1 (t1 = 0.2, t2 = 0), (b) case 2 (t1 = 0, t2 = 0.2), and (c) case 3 (t1 = t2 = 0.1) at Pr = 1000, K = 10, and Ra = 105.

solid fluid interface region. The dense contours of Sψ near the horizontal walls also indicate the active zone of Sψ near those regions for all three cases. It may be noted that, at top and bottom walls, Sψ is found to be 7.02 and 12.87, respectively, for case 1, Sψ = 10.88 and 6.97, respectively, for case 2, and Sψ = 6.96 and 7.66, respectively, for case 3 at Ra = 105 and Pr = 1000 with K = 0.1 (see Figure 7). Insignificant Sψ is observed at the central region of fluid phase due to less velocity gradient in that region, for all three cases. At K = 10 and Pr = 1000 (figure not shown) with low Rayleigh number (Ra = 103), the distributions of θ, Sθ, ψ, and Sψ remain qualitatively similar to those of K = 0.1 (Figure 6). However, the maximum magnitudes of Sθ and Sψ are high for K = 10 compared to that for K = 0.1, irrespective of the location of wall thickness. In addition, high magnitude of Sψ,max at K = 10 (figure not shown) is also observed due to high intensity of

0.1. But the magnitude of streamfunction is high for the present case with high Ra (Ra = 105) compared to that of lower Ra (Ra = 103) with low conductivity ratio case (K = 0.1) and Pr = 1000. Note that, |ψ|max = 5.7 for case 1, |ψ|max = 5.5 for case 2, and |ψ|max = 4.7 for case 3 at Ra = 105; whereas, |ψ|max = 0.4 for all three cases at Ra = 103 with K = 0.1 and Pr = 1000 (see Figures 6 and 7). Due to high intensity fluid circulation cells, the entropy generation for fluid friction is larger at Ra = 105 compared to that at Ra = 103 with low conductivity ratio (K = 0.1). Note that, Sψ,max = 43.54, 32.27, and 11.64 for cases 1, 2, and 3, respectively at Ra = 105; whereas, Sψ,max is found to be 0.03 for all three cases at Ra = 103 with K = 0.1 (see Figures 6 and 7). The local entropy generation due to fluid friction (Sψ) is observed near the solid−fluid interface region due to friction between the fluid circulation cells and solid walls. Local maxima of Sψ is observed as 10.22 for case 1 and 11.63 for case 2 on the 3714



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fluid flow compared to low conductivity ratio, K = 0.1 (Figure 6), irrespective of location of wall thickness. The solid wall with high conductivity ratio (K = 10) offers very less resistance on the heat transfer in the solid phase and high intensity of fluid flow occurs due to larger buoyancy effect even at Ra = 103 for K = 10. Distributions of isotherms, streamlines, entropy generation due to heat transfer, and fluid friction for all three cases at Pr = 1000 and Ra = 105 with K = 10 are shown in Figure 8. Interesting features on thermal, flow, and entropy generation characteristics are observed at high conductivity ratio (K = 10) and Ra = 105. Isotherms are highly distorted and compressed

along the side walls, signifying the dominance of convection at Ra = 105 and Pr = 1000. Temperature gradients are much less within the solid walls for all three cases as seen by the temperature distribution within the solid walls in contrast of lower conductivity ratio (K = 0.1). Isotherms with θ ≤ 0.1 occur within the solid left wall for case 1, and those with θ ≥ 0.9 occur within the solid right wall for case 2; whereas, θ ≤ 0.1 and θ ≥ 0.9 are observed within the left and right solid walls, respectively for case 3 at K = 10 with Ra = 105 and Pr = 1000 (see Figure 8). On the other hand, isotherms with θ ≤ 0.7 occur in the left solid wall for case 1 and isotherms with θ ≥ 0.2 occur in the right wall for case 2; whereas, isotherms with θ ≤ 3715



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Figure 10. Isotherms (θ), local entropy generation due to heat transfer (Sθ), streamlines (ψ), and local entropy generation due to fluid friction (Sψ) for (a) case 1 (t1 = 0.8, t2 = 0), (b) case 2 (t1 = 0, t2 = 0.8), and (c) case 3 (t1 = t2 = 0.4) at Pr = 1000, K = 10, and Ra = 105.

0.3 and θ ≥ 0.6 are observed within the left and right solid walls, respectively for case 3 for K = 0.1 with Ra = 105 and Pr = 1000 (see Figure 7). It may be noted that the solid−fluid interface acts as a strong site of entropy generation corresponding to heat transfer due to high resistance offered by the fluid region compared to the solid wall (ks/kf = K = 10), irrespective of the location of wall thickness. Significant Sθ values are observed within the solid walls as Sθ = 0.5−4 occurs within the solid walls for cases 1 and 2; whereas, Sθ = 0.5−5 occurs within the both solid walls for case 3. The maximum magnitude of entropy generation due to heat transfer (Sθ,max) for K = 10 within solid−fluid interface is found to be high compared to that of K = 0.1 due to higher thermal gradient in the fluid phase for K = 10. Note that, Sθ,max = 147.05, 157.60, and 172.82 occurs for cases 1, 2, and 3, respectively, at higher K (K = 10); whereas, Sθ,max is found to be 1.89, 1.93, and 2.35 for cases 1, 2, and 3, respectively, for the lower K (K = 0.1) with

identical parameters (Pr = 1000 and Ra = 105) [see Figures 7 and 8]. It is interesting to note that an additional local maxima of entropy generation due to heat transfer (Sθ) is found near the lower portion of the right wall for case 1 (Sθ = 66.64) and the top portion of the left wall for case 2 (Sθ = 60.66); whereas, Sθ,max is observed near both the side walls for case 3 (Sθ,max = 174.82) at K = 10 (see Figure 8). On the other hand, the local maxima of Sθ for cases 1−3 are also found to be less for K = 0.1 as seen from Figure 7. It may be noted that temperature gradient of the fluid phase is very high compared to that of K = 0.1. Note that, θ = 0.2−1, θ = 0−0.8, and θ = 0.2−0.8 occur within the fluid phase of the cavity for cases 1, 2, and 3, respectively at K = 10; whereas, θ = 0.8−1, θ = 0−0.2, and θ = 0.4−0.55 are observed within the fluid phase for cases 1, 2, and 3, respectively, at K = 0.1 with identical parameters (Pr = 1000 and Ra = 105) [see Figures 7 and 8]. A high temperature gradient of fluid phase is observed for K = 10, and hence, local 3716



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distributions of Sθ within the fluid phase are more significant at high K (K = 10) compared to that of lower K (K = 0.1). Note that, Sθ = 0.1−66.64, Sθ = 0.1−60.66, and Sθ = 0.1−20 are observed within the fluid region for cases 1, 2, and 3, respectively, at K = 10; whereas, Sθ = 0.005−0.74, 0.005−0.52, and 0.1−0.15 occur within the fluid phase of cavity for cases 1, 2, and 3, respectively, at K = 0.1 with identical parameters (Pr = 1000 and Ra = 105). The fluid circulation cells in the central core of the cavity are split into two small circulation cells, and they expand and take the shape of the cavity near cavity walls due to intense convection at high Ra (Ra = 105), K (K = 10), and Pr (Pr = 1000), irrespective of wall location. As K increases from 0.1 to 10, the intensity of fluid flow increases as seen by high magnitude of streamfunction, irrespective of location of wall thickness. The intensity of fluid flow is almost identical for all three cases as |ψ|max is found to be 10.6, 10.6, and 10.4 for cases 1, 2, and 3, respectively at K = 10; whereas, |ψ|max = 5.7, 5.5, and 4.7 for cases 1, 2, and 3, respectively, were found at K = 0.1 (see Figures 7 and 8). The magnitude of Sψ at walls are found to be very high and dense contours of Sψ over a large regime near side walls illustrate the zones of high entropy generation due to fluid friction. The maximum entropy generation due to fluid friction (Sψ,max) at K = 10 is high compared to that of K = 0.1 due to enhanced fluid flow at higher K (K = 10). Note that, Sψ,max = 564.44,509.57 and 178.08 for cases 1, 2, and 3, respectively, for high K (K = 10) [see Figure 8]; whereas, Sψ,max = 43.54, 32.27, and 11.64 are observed for cases 1, 2, and 3, respectively for lower K (K = 0.1) with identical parameters (Pr = 1000 and Ra = 105) [see Figure 7]. The dense contours of Sψ occur on the wall of the cavity as well as solid−fluid interface region indicating the local distribution of entropy generation due to fluid friction. The local maxima for Sψ are also found to be larger for K = 10 compared to those at K = 0.1 values. The velocity gradients are found to be small at the core, and therefore, Sψ at core region is small compared to the Sψ at walls, irrespective of wall location. 3.3. Numerical Simulations for Total Wall Thickness, t1 + t2 = 0.8. Numerical simulations have been carried out for different Prandtl numbers (Pr = 0.015 and 1000) and thermal conductivity ratios (K = 0.1 and 10) with the higher wall thickness (t1 + t2 = 0.8) at Ra = 103 and 105. The distributions of the θ, Sθ, ψ, and Sψ for K = 0.1 and 10, and Pr = 0.015 with higher wall thickness, t1 + t2 = 0.8 (figure not shown), are qualitatively similar to that of lower wall thickness, t1 + t2 = 0.2 (see Figure 5) for identical parameters (K = 0.1 and 10 and Pr = 0.015) at Ra = 103 and 105. The magnitudes of Sθ,max and Sψ,max are lower for the higher thickness, t1 + t2 = 0.8, compared to those of lower wall thickness, t1 + t2 = 0.2. Similar to lower wall thickness, t1 + t2 = 0.2, isotherms are highly compressed in lower portion of right wall and top portion of left wall and hence active zones of entropy generation due to heat transfer occur in those regions, irrespective of location of wall thickness for higher wall thickness, t1 + t2 = 0.8 particularly for Ra = 105 (figure not shown). It is interesting to observe that intensity of fluid flow is almost similar to the case with lower wall thickness, t1 + t2 = 0.2 for cases 1 and 2, but |ψ|max is larger compared to lower wall thickness, t1 + t2 = 0.2, for case 3 with identical parameters (Pr = 0.015, K = 10, and Ra = 105). It may be noted that the magnitude of Sψ,max is lower for higher wall thickness, t1 + t2 = 0.8 (figure not shown), compared to that of lower wall thickness, t1 + t2 = 0.2.

Figures 9 and 10 show isotherms (θ), streamlines (ψ), and entropy generation maps due to heat transfer (Sθ) and fluid friction (Sψ) for different conductivity ratios (K = 0.1 and 10) at Pr = 1000 and Ra = 105 within square cavity for three cases which are based on the location of wall thickness, t1 + t2 = 0.8. At low conductivity ratio, K = 0.1, Pr = 1000, and Ra = 105, the isotherms are compressed along the lower portion of right wall and top portion of left wall (see Figure 9). As seen in earlier cases, the parallel isotherms in solid phase indicate conduction mode of heat transfer and distorted isotherms are observed in the fluid phase due to convective heat transfer. It is observed that θ ≥ 0.9 for case 1, θ ≤ 0.1 for case 2, and 0.44 ≤ θ ≤ 0.5 for case 3 are maintained within the fluid phase of cavity. The temperature distribution with lower thermal gradient within fluid phase at higher wall thickness, t1 + t2 = 0.8 occurs, irrespective of location of wall thickness. Therefore, insignificant Sθ is observed within the fluid phase for all the three cases as Sθ = 0.001−0.15, 0.001−0.04, and 0.001−0.18 are observed within the fluid region of the cavity for cases 1, 2, and 3, respectively (see Figure 9). It may be noted that thermal gradient is high within the solid walls as seen from the temperature distribution within the solid phases. The solid walls confined with isotherms with θ ≤ 0.9 for case 1 and θ ≥ 0.1 for case 2; whereas, θ ≤ 0.4 and θ ≥ 0.5 within the left and right solid walls, respectively, for case 3 are observed (see Figures 9). It is also interesting to observe that insignificant Sθ (Sθ = 0.12−0.15) is observed within the solid walls even at high temperature gradients due to low conductivity ratio (K = 0.1). The magnitude of entropy generation due to heat transfer Sθ varies 0.12−0.15 within solid walls for cases 1 and 2; whereas, Sθ varies from 0.11 to 0.18 within the left wall and right wall for case 3. Similar to lower wall thickness case, t1 + t2 = 0.2, isotherms are highly compressed in the lower portion of the right wall and the top portion of the left wall, and hence, active zones of entropy generation due to heat transfer occur in those regions, irrespective of the location of wall thickness in the present case (t1 + t2 = 0.8). It may be noted that Sθ is maximum for case 3 (Sθ,max = 0.18) followed by cases 1 and 2 (Sθ,max = 0.15). The fluid flow follow circular pattern at the core and the flow pattern takes the shape of the cavity near the cavity walls at high Pr (Pr = 1000), irrespective of wall thickness. It is interesting to observe that intensity of fluid flow is high due to enhanced convection at high Ra (Ra = 105). Note that, |ψ|max = 3.98, 3.76, and 3.40 for cases 1, 2, and 3, respectively, for high wall thickness, t1 + t2 = 0.8 with Pr = 1000, K = 0.1, and Ra = 105 (see Figure 9). Due to significant velocity gradients, significant entropy generation due to fluid friction is observed near the cavity walls. Note that Sψ,max = 10.12, 6.57, and 3.02 occurs for cases 1, 2, and 3, respectively (see Figure 9). The another local maxima of Sψ are also observed near the cavity walls as Sψ occurs on the top and bottom walls with magnitude 2.43 and 3.90, respectively for case 1, 3.20 and 2.26, respectively for case 2 and 2.21 and 2.34, respectively for case 3. The dense contours of Sψ occur near the solid−fluid interface region indicating the active site of entropy generation due to fluid friction on those regions. Note that Sψ = 2.47, 2.87, and 1.72 occurs near the solid−fluid interface region for cases 1, 2, and 3, respectively. Figure 10 illustrates the temperature distribution (θ), streamlines (ψ), and entropy generation maps (Sθ and Sψ) at Pr = 1000, K = 10, with Ra = 105 and wall thickness, t1 + t2 = 0.8. Due to higher thermal conductivity of solid wall, temperature gradient is very less within thick solid walls 3717



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Figure 11. Variation of total entropy generation (Stotal), average Bejan number (Beav), and average Nusselt number (Nus) with Rayleigh number (Ra) at Pr = 0.015 for (a) case 1, (b) case 2, and (c) case 3 at K = 0.1 (···), K = 1 (- - -), and K = 10 () with wall thickness, t1 + t2 = 0.2 and t1 + t2 = 0.8. Results are compared with the conductivity ratio, K = ∞ [zero wall thickness] (•) in each panel plots.

Ra = 105) [see Figures 8 and 10]. Thus, the conductivity ratio has strong effect on the evaluation of entropy generation due to heat transfer in solid wall. In contrast of the thermal gradient within solid phase, thermal gradient for the fluid phase of cavity is high at K = 10 compared to that of K = 0.1, irrespective of location of wall thickness. The fluid phase of the cavity is maintained with isotherms with θ ≥ 0.3 for case 1, θ ≤ 0.7 for case 2, and θ = 0.2−0.8 for case 3 at high conductivity ratio (K = 10). Due to high temperature gradient within fluid phase, high entropy generation is observed within fluid phase at K = 10 (Sθ = 0.1−34.07 for cases 1 and 2 and Sθ = 0.1−88.03 for case 3) compared to that of K = 0.1 (Sθ = 0.001−0.06 for case 1, Sθ = 0.001−0.04 for case 2, and Sθ = 0.001−0.18 for case 3). Local distribution of Sθ within fluid phase are found to be 34.07 near the wall with zero thickness (t = 0) for cases 1 and 2 at K = 10. It may be noted that fluid flow pattern is stretched horizontally at the central portion of fluid media and they take the shape of the cavity near cavity walls due to intense convection at high K (K = 10). The intensity of flow circulations increases as seen from the high magnitude of streamfunctions. Note that, |ψ|max = 10, 9.5, and 9.5 for cases 1,

compared to that of low conductivity ratio (K = 0.1) as seen from the isotherms distribution, irrespective of wall thickness. The solid phase of the cavity is confined with isotherms with θ ≤ 0.3 for case 1 and θ ≥ 0.7 for case 2; whereas, θ ≤ 0.2 and θ ≥ 0.8 are observed within the left and right solid walls, respectively, for case 3 at K = 10. At high conductivity ratio (K = 10), the solid−fluid interface acts as strong active site of entropy generation due to heat transfer. This is due to the fact that fluid region offers high resistance compared to solid wall (ks/kf = K = 10). Note that Sθ,max = 75.64, 79.47, and 88.03 for cases 1, 2, and 3, respectively, which corresponds to a higher conductivity ratio (K = 10); whereas, Sθ,max values are observed at 0.15, 0.15, and 0.18 for cases 1, 2 and 3, respectively, which corresponds to a lower conductivity ratio (K = 0.1) with identical parameters (Pr = 1000 and Ra = 105) [see Figures 9 and 10]. Compression of isotherms is more pronounced near the lower portion of the right wall and the top portion of the left wall within the fluid phase as K increases from 0.1 to 10. It is interesting to observe that the maximum magnitude of entropy generation due to heat transfer for the higher wall thickness (t1 + t2 = 0.8) is less compared to lower wall thickness (t1 + t2 = 0.2) with identical parameters (Pr = 1000, K = 10, and 3718



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2, and 3, respectively, at high Pr (Pr = 1000); whereas, |ψ|max are found to be 3.98, 3.76, and 3.40 for cases 1, 2, and 3, respectively, at low K (K = 0.1) with same wall thickness (t1 + t2 = 0.8) [see Figures 9 and 10]. Due to high intense fluid circulation and as a consequence of no-slip boundary conditions along the wall, the high velocity gradients exist near the cavity walls in case of higher K (K = 10). Note that Sψ,max = 420.11, 371.07, and 128.81 occur for cases 1, 2, and 3, respectively, for K = 10; whereas, Sψ,max = 10.12, 6.57, and 3.02 was observed for K = 0.1 with same wall thickness, t1 + t2 = 0.8 and Pr = 1000 at Ra = 105 (see Figures 9 and 10). It is also observed that the intensity of fluid flow decreases with the increase of the wall thickness as seen from the lower magnitude of streamfunction for high wall thickness compared to that of lower wall thickness at identical parameters. Note that |ψ|max = 10, 9.5, and 9.5 for cases 1, 2, and 3, respectively for high wall thickness, t1 + t2 = 0.8; whereas, |ψ|max values are found to be 10.6, 10.6, and 10.4 for cases 1, 2, and 3, respectively for lower wall thickness, t1 + t2 = 0.2 with Pr = 1000, K = 10, and Ra = 105 (see Figures 8 and 10). As a consequence, the maximum magnitude of entropy production due to fluid friction (Sψ,max) is low for the present case with high wall thickness (t1 + t2 = 0.8) compared to that of less wall thickness (t1 + t2 = 0.2) due to less flow strength in high wall thickness (t1 + t2 = 0.8) with identical parameters (Pr = 1000, Ra = 105, and K = 10). Various local maxima of Sψ such as Sψ = 67.93 and 81.63 occur near top and bottom walls, respectively for case 1 and Sψ = 83.50 and 71.59 occur near top and bottom walls, respectively for case 2 whereas Sψ are found to be 30.53 and 75.41 for top and bottom walls, respectively for case 3. Another local maxima of Sψ also occurs near the middle portion of the left wall or solid−fluid interface region for case 1 and case 2 with same magnitude (Sψ = 83.50) and case 3 (Sψ = 23.62). 3.4. Average Nusselt Number, Total Entropy Generation, and Average Bejan Number. The variation of the total entropy generation due to heat transfer and fluid friction (Stotal), average Bejan number (Beav) and average Nusselt number (Nus) vs logarithmic Rayleigh number (Ra) at different conductivity ratios (K = 0.1, 1, and 10) and different wall thicknesses (t1 + t2 = 0.2 and 0.8) are presented in Figure 11. In order to compare the current result with zero wall thickness, the variation of Stotal, Beav, and Nus with Rayleigh number (Ra) for the square cavity with zero wall thickness are also presented with symbols (•). Figure 11 (a−c) correspond to cases 1, 2, and 3, respectively, which are based on the location of wall thickness. In case 1, thick solid wall is placed on the left side of cavity (t1 ≠ 0 and t2 = 0); in case 2, thick solid wall is placed on the right side of cavity (t1 = 0 and t2 ≠ 0); whereas, thick solid walls are placed on the both sides of the cavity (t1 = t2 ≠ 0) in case 3. The lower, middle, and upper panels of the Figure 11 (a−c) represent the variation of Stotal, Beav, and Nus with Rayleigh number, respectively. The total entropy generation (Stotal) in the cavity is the combination of total entropy generation due to heat transfer (Sθ) and total entropy generation due to fluid friction (Sψ). The relative dominance of entropy generation due to heat transfer and fluid friction is given by the average Bejan number (Beav). Note that, Beav > 0.5 indicates that entropy generation is heat transfer dominant, whereas Beav < 0.5 indicates the fluid friction dominant entropy generation. Average Nusselt number represents the overall heat transfer rate along the cavity side walls.

Figure 11a represents distributions of Stotal, Beav, and Nus for Pr = 0.015 for case 1 with different wall thickness, t1 = 0.2 or 0.8, and t2 = 0. The symbols (•) represent the variation of Stotal, Beav, and Nus with Rayleigh number for the square cavity with zero wall thickness (t1 = t2 = 0). It may be noted that Stotal is almost constant up to Ra = 2 × 104 for K = 0.1 with the lower wall thickness, t1 + t2 = 0.2. This is due to small increment in entropy generation due to fluid friction (Sψ,total) compared to entropy generation due to heat transfer (Sθ,total) as the heat transfer is almost conduction dominant. Thereafter, Stotal slowly increases with Ra due to significant Sψ,total of Stotal for Ra ≥ 2 × 104. This is further due to onset of convection for the high Rayleigh number region with significant fluid flow irreversibilities. In the case of high K (K = 1, 10, and ∞), Stotal is almost constant within a narrow Ra region (103 ≤ Ra ≤ 4 × 103); whereas, that increases exponentially for Ra ≥ 5 × 103 due to sharp increase of fluid fiction irreversibility (Sψ,total) compared to heat transfer irreversibility (Sθ,total) of Stotal. This is due to enhanced convection at the higher Rayleigh number region which further leads to significant Sψ. It may be noted that the distribution pattern of Stotal for the t1 + t2 = 0.8 is qualitatively similar to that of t1 + t2 = 0.2. However, the magnitude of Stotal is quantitatively lower for t1 + t2 = 0.8 compared to that of t1 + t2 = 0.2. This is due to low entropy generation due to heat transfer and fluid friction in case of higher wall thickness compared to lower wall thickness, t1 + t2 = 0.2 with identical situation. It may also be noted that the trend of Stotal for K = ∞ (square cavity with zero wall thickness) is qualitatively similar to that for K = 10, irrespective of wall thickness (t1 + t2 = 0.2 and 0.8) with identical boundary conditions. But Stotal is higher for K = ∞ (square cavity with zero wall thickness) due to significant Sψ,total of Stotal followed by K = 10, 1, and 0.1 (significant wall thickness) at Ra = 105. The higher magnitude of Stotal corresponding to K = ∞ (square cavity with zero wall thickness) indicates the large amount of available energy spent to overcome the irreversibilities due to fluid friction. The lower panel of the Figure 11b and c shows the distributions of Stotal for case 2 (t1 ≠ 0 and t2 = 0) and case 3 (t1 = t2 ≠ 0). Similar to case 1, the distribution of Stotal is almost constant for the entire region of Rayleigh number (103 ≤ Ra ≤ 105) at low conductivity ratio (K = 0.1) whereas increasing trends in Stotal are observed within convection dominant region (104 ≤ Ra ≤ 105) at high conductivity ratios (K = 1, 10, and ∞), irrespective of solid wall thickness (t1 + t2 = 0.2 and 0.8). Average Bejan number (Be av ) indicates the relative importance of entropy generation due to heat transfer (Sθ) and fluid friction (Sψ) irreversibilities. The middle panel of Figure 11 (a−c) represents the variation of the Bejan number with Rayleigh number for the wall thickness, t1 + t2 = 0.2. A common trend of decrease in Beav with Ra for all the conductivity ratios (K = 0.1, 1, 10, and ∞) is observed. The maximum value of Beav occurs at low Ra (Ra = 103) which indicates that entropy generation in the cavity is primarily due to heat transfer irreversibility at conduction dominant mode. As Ra increases to 105, fluid friction irreversibility (Sψ,total) increases and that dominates over heat transfer irreversibility (Sθ,total) due to enhanced convection heat transfer in the cavity. Therefore, Beav decreases with Ra as seen in middle panel plots of Figure 11 parts a−c. It may be noted that Beav is higher for K = 0.1 followed by K = 1, 10, and ∞ within conduction dominance region (103 ≤ Ra ≤ 104). For the convection dominance region (104 ≤ Ra ≤ 105), the magnitude of Beav for K = 1, 10, and ∞ closely follows the magnitude of K = 0.1. This 3719



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is due to approximately constant Sθ,total values with various Sψ,total values over the range of Ra = 104−105 for all K values (K = 0.1, 1, 10, and ∞). The insets of the middle panel of Figure 11 (a−c) represent the variation of Beav versus Rayleigh number (Ra) with higher wall thickness, t1 + t2 = 0.8 which follow qualitatively similar trend that of lower wall thickness, t1 + t2 = 0.2. Overall, the plots of the Beav are almost similar for the each K = 0.1, 1, 10, and ∞, irrespective of wall thicknesses. Common to all cases, higher magnitudes of Beav (Beav > 0.5) are observed within the conduction region (103 ≤ Ra ≤ 104) whereas Beav < 0.5 are observed within convection region (104 ≤ Ra ≤ 105), irrespective of conductivity ratio (K) and wall thicknesses (t1 + t2 = 0.2 and 0.8). The upper panels of Figure 11 (a−c) represent the variation of the Nusselt number with Rayleigh number for all the conductivity ratios as K = 0.1, 1, and 10 (cavity with significant wall thicknesses, t1 + t2 = 0.2 and 0.8) and K = ∞ (cavity with zero wall thickness). The heat transfer rate due to thermal gradient (Nus) is found to be constant for entire range of Rayleigh number (103 ≤ Ra ≤ 105) and K = 0.1 with wall thickness, t1 + t2 = 0.2 and K = 0.1 and 1 with wall thickness, t1 + t2 = 0.8. This is due to smaller entropy generation due to fluid friction (Sψ,total) compared to entropy generation due to heat transfer (Sθ,total) for these cases. It is also found that total entropy generation due to K = 0.1 and 1 are almost constant, irrespective of Ra. Increasing trends in Nus are observed for the high conductivity ratio (K = 10) in the convection dominant regions (104 ≤ Ra ≤ 105) due to larger heat transfer rate corresponding to thermal gradient than the exergy loss or entropy generation due to heat transfer and fluid flow, irrespective of wall thicknesses (t1 + t2 = 0.2 or 0.8). It may be noted that heat transfer rate (Nus) with zero wall thickness (K = ∞) closely approaches to the trend of Nus at K = 10 with solid wall thickness (t1 + t2 = 0.2 and 0.8). However, the magnitudes of Nus are higher for zero wall thickness (K = ∞) compared to the cases with significant solid wall thicknesses (t1 + t2 = 0.2 and 0.8). It is interesting to observe that the thermal gradients for the side walls are high enough to maintain a high heat transfer rate even after spending large amount of energy due to high fluid friction irreversibility for K = 10 and ∞, irrespective of wall thickness (t1 + t2 = 0.2 and 0.8). The variations of Stotal, Beav, and Nus with Rayleigh number (Ra) are also studied for the higher Pr (Pr = 1000) for all three cases which are based on location of wall thickness (figure not shown). It is found that the variations of Stotal, Beav, and Nus with Ra at Pr = 1000 are qualitatively similar to that of Stotal, Beav, and Nus with Pr = 0.015. The magnitudes of Beav and Nus for Pr = 1000 are almost similar to these of Pr = 0.015, but the magnitude of Stotal is higher for Pr = 1000 compared to that of Pr = 0.015. The details discussion on the variation of Stotal, Beav, and Nus with Ra at Pr = 1000 may be outlined following the similar manner for Pr = 0.015.

cavity. Finite wall thickness on both vertical sides (t1 and t2) of the cavity is considered for case 3. Current work deals with evaluation of entropy generation using finite element based technique for composite domain including fluid−fluid interface and solid−fluid interface region. Flow (ψ), temperature (θ), and entropy generation zones (Sθ and Sψ) are analyzed for various Rayleigh numbers (103 ≤ Ra ≤ 105), Prandtl numbers (Pr = 0.015, 0.7, and 1000), conductivity ratios (K = 0.1, 1, and 10), and wall thicknesses (t1 + t2 = 0.2 and 0.8). Detailed investigations on the variation of total entropy generation, average Bejan number, and average Nusselt number with Rayleigh number are also presented. Important inferences are summarized as below: 4.1. Temperature Distribution and Heat Flow Irreversibility. • At low Ra (Ra = 103) and low conductivity ratio (K = 0.1), isotherms are smooth and monotonic within both solid and fluid regions, indicating conduction dominant heat transfer. On the other hand, highly distorted isotherms are observed within the fluid phase of cavity due to enhanced convection whereas parallel isotherms occur in the solid region due to conductive heat transfer within solid walls at Ra = 105. • At K = 0.1, the magnitudes of Sθ,max are very low and they occur within the solid wall due to larger thermal resistance in the solid wall whereas higher magnitudes of Sθ,max are observed near the solid−fluid interface at K = 10 due to higher thermal conductivity in the solid region, irrespective of Ra and Pr. The magnitude Sθ,max is higher for case 3 followed by cases 1 and 2, irrespective of K, Ra, and Pr. • The characteristics of θ and Sθ for the higher wall thickness, t1 + t2 = 0.8 is qualitatively similar to that for lower wall thickness, t1 + t2 = 0.2 but the magnitude of Sθ,max is less for higher wall thickness, t1 + t2 = 0.8 compared to lower wall thickness, t1 + t2 = 0.2 4.2. Flow Maps and Fluid Flow Irreversibility. • The intensity of the fluid flow is very less as small magnitudes of streamfunction are observed at low Ra (Ra = 103), irrespective of K and location of solid wall thickness. Circular flow cells are observed near central region of the cavity for Pr = 0.015; whereas, flow circulation cells near the core of the cavity are split into two small circulation cells due to intense convection for Pr = 1000 at Ra = 105, irrespective of K and location of wall thickness. • At Ra = 105, the intensity of the fluid flow is less for case 3 compared to cases 1 and 2 with K = 0.1; whereas, the fluid flow pattern is independent of the location of wall thickness with K = 10, irrespective of Pr. • Dense contours of Sψ occur on the cavity walls and solid fluid interface regions indicating active zones of entropy generation due to fluid friction (Sψ). The magnitude of Sψ,max increases with conductivity ratio (K) and that is minimum for case 3 compared to that of cases 1 and 2, irrespective of Ra and Pr. • Qualitative features of ψ and Sψ for the higher wall thickness, t1 + t2 = 0.8, are identical with those of lower wall thickness, t1 + t2 = 0.2. The magnitude of the |ψ|max and Sψ,max is lower for the higher wall thickness, t1 + t2 = 0.8 compared to the lower wall thickness, t1 + t2 = 0.2, irrespective of Pr and Ra.

4. CONCLUSION Analysis on entropy generation within differentially heated square cavities during conjugate natural convection has been performed for various conductivity ratios (K) and wall thicknesses (t1 and t2). Three cases have been considered based on positions of wall thickness in the cavity. In case 1, wall thickness on the right side (t2) of the cavity is neglected and the left wall (t1) has considerable wall thickness. In case 2, wall thickness on the left wall of the cavity is neglected and considerable wall thickness is maintained on right side of the 3720



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Y = dimensionless distance along y coordinate

4.3. Entropy Generation vs Heat Transfer Rates. • At the conduction dominant region (103 ≤ Ra ≤ 104), the total entropy generation (Stotal) is less while significant heat transfer rate (Nus) is observed for both the wall thicknesses (t1 + t2 = 0.2 and 0.8), irrespective of conductivity ratio (K). Therefore, any K values can be used for efficient processing within low Ra regions (103 ≤ Ra ≤ 104), irrespective of wall thickness. • At the convection dominant region (104 ≤ Ra ≤ 105), high K (K = ∞ and 10) corresponds to high heat transfer rate as well as high entropy generation whereas low K values (K = 0.1 for t1 + t2 = 0.2 or K = 0.1 and 1 for t1 + t2 = 0.8) correspond to almost constant heat transfer rate as well as constant entropy generation for entire range of Ra, irrespective of wall thickness. • The wall thickness, t1 + t2 = 0.8, may be energy efficient due to its less total entropy generation (Stotal) but low heat transfer rate (Nus). Therefore conductivity ratio, K = 1 with wall thickness t1 + t2 = 0.8 is preferable due to its reasonable heat transfer rate and less entropy production at convection dominant region (104 ≤ Ra ≤ 105).

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Greek Symbols

α = thermal diffusivity, m2 s−1 β = volume expansion coefficient, K−1 γ = penalty parameter Γ = boundary of two-dimensional domain θ = dimensionless temperature μf = dynamic viscosity of fluid phase, m2 s−1 ν = kinematic viscosity, m2 s−1 ρ = density, kg m−3 ψ = dimensionless streamfunction Ω = two-dimensional domain Subscripts

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c = cold f = fluid h = hot l = left wall r = right wall s = solid av = average

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AUTHOR INFORMATION

Corresponding Author

*E-mail address: [email protected]. Notes

The authors declare no competing financial interest.

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NOMENCLATURE Be = Bejan number cp = specific heat capacity g = acceleration due to gravity, m s−2 k = thermal conductivity, W m−1 K−1 kf = thermal conductivity of fluid, W m−1 K−1 ks = thermal conductivity of solid, W m−1 K−1 K = thermal conductivity ratio, (ks/kf) L = height of the cavity, m N = total number of nodes Nu = local Nusselt number Nu = average Nusselt number p = pressure, Pa P = dimensionless pressure Pr = Prandtl number Ra = Rayleigh number S = dimensionless entropy generation Ssθ = dimensionless entropy generation due to heat transfer in the solid phase Sfθ = dimensionless entropy generation due to heat transfer in the fluid Sψ = dimensionless entropy generation due to fluid friction Stotal = dimensionless total entropy generation due to heat transfer and fluid friction t = wall thickness, m T = temperature, 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 = distance along x coordinate, m X = dimensionless distance along x coordinate y = distance along y coordinate, m 3721



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Analysis of Entropy Generation during Conjugate Natural Convection

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