Natural convection in rhombic enclosures with isothermally heated side or bottom wall: Entropy generation analysis

European Journal of Mechanics B/Fluids 54 (2015) 27–44

Contents lists available at ScienceDirect

European Journal of Mechanics B/Fluids journal homepage: www.elsevier.com/locate/ejmflu

Natural convection in rhombic enclosures with isothermally heated side or bottom wall: Entropy generation analysis R. Anandalakshmi a , Tanmay Basak b,∗ a

Department of Chemical Engineering, Indian Institute of Technology Guwahati, Guwahati - 781039, India

b

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

highlights • • • • •

Energy efficient convection systems can be designed by reducing irreversibilities. Rhombic enclosures with differential and Rayleigh–Benard heating situations are chosen. Heat transfer (Sθ ) and fluid friction irreversibilities (Sψ ) have been reported. Heating patterns and geometrical orientations have been analyzed based on entropy generation. Appropriate rhombic angles (ϕ ) have been proposed based on irreversibility analysis.

article

info

Article history: Received 23 September 2013 Received in revised form 13 May 2015 Accepted 19 May 2015 Available online 1 June 2015 Keywords: Entropy generation Natural convection Rhombic enclosures Convective heat transfer Irreversibilities

abstract The energy efficient convection systems can be designed by reducing exergy losses. In this context, the analysis on the entropy generation during natural convection in the fluid-filled (Prandtl number, Pr = 0.015 − 1000) rhombic enclosures with various inclination angles (ϕ ) has been carried out for the efficient thermal processing in various applications such as the chemical reactor modeling, underground coal gasification, and nuclear reactors. The enclosure is subjected to the differential heating (case 1) and Rayleigh–Benard convection (case 2). The conduction based static solution occurs only for ϕ = 90◦ and it is observed that the conduction based static solution disappears with a slight perturbation of ϕ at Rayleigh number, Ra ≥ 2 × 103 irrespective of Pr in case 2. The active zones of the heat transfer irreversibility (Sθ ) and fluid friction irreversibility (Sψ ) are found to occur near the isothermal walls for all ϕ s irrespective of Pr in both the cases at Ra = 105 . In addition, the active zones of Sψ are also found to occur near the adiabatic walls of the cavity for all ϕ s irrespective of Pr in both the cases at Ra = 105 . Also, the region between the fluid layers of primary circulation cells acts as the strong active zone of Sψ for all ϕ s in case 2 at lower Pr (Pr = 0.015) and Ra = 105 . The total entropy generation (Stotal ) and maximum heat transfer rates (Nu) are found to be significantly low for ϕ = 30◦ in both the cases at Ra = 105 irrespective of Pr. Analysis of heating patterns and geometrical orientations relates the exergy to irreversibilities which establishes that the rhombic cavity (ϕ = 30◦ ) with the differential heating pattern may be the optimal design based on the energy efficient perspective. © 2015 Elsevier Masson SAS. All rights reserved.

1. Introduction The thermodynamic efficiency of any irreversible process has to be calculated based on the reversible process as a reference scale according to second law of thermodynamics instead of the energy balance of a particular irreversible process according to

∗

Corresponding author. E-mail addresses: [email protected] (R. Anandalakshmi), [email protected] (T. Basak). http://dx.doi.org/10.1016/j.euromechflu.2015.05.004 0997-7546/© 2015 Elsevier Masson SAS. All rights reserved.

the first law of thermodynamics. As the heat transfer processes involving natural convection are inevitable in most of the industrial (i.e. electronic cooling) and natural processes (i.e. building insulation) [1–7], the increase in the thermodynamic efficiency of the heat transfer processes is of vital importance to conserve the heat energy. The increase in the thermodynamic efficiency of the heat transfer processes can be done either by increasing the minimum heat input or by utilizing existing heat energy in more appropriate manner. The former approach is usually not preferred in many engineering and natural applications, because it needs an additional energy requirement or thermal system dimensions.

28

R. Anandalakshmi, T. Basak / European Journal of Mechanics B/Fluids 54 (2015) 27–44

Nomenclature

Be f g k L N n Nu Nu p P Pr R Ra S Sθ Sψ Stotal T To Th Tc u U

v

V X x Y y

Bejan number Function Acceleration due to gravity, m s−2 Thermal conductivity, W m−1 K−1 Side of the rhombic cavity, (m) Total number of nodes Normal vector to the plane Local Nusselt number Average Nusselt number Pressure, Pa Dimensionless pressure Prandtl number Residual of weak form Rayleigh number Dimensionless entropy generation Dimensionless entropy generation due to heat transfer Dimensionless entropy generation due to fluid friction Dimensionless total entropy generation due to heat transfer and fluid friction Temperature of the fluid, K Bulk temperature, K Temperature of hot wall, K Temperature of cold wall, K x Component of velocity, m s−1 x Component of dimensionless velocity y Component of velocity, m s−1 y Component of dimensionless velocity Dimensionless distance along x coordinate Distance along x coordinate, m dimensionless distance along y coordinate Distance along y coordinate, m

Greek symbols

α β γ θ µ ν ρ φ ϕ Φ ψ Ω

Thermal diffusivity, m2 s−1 Volume expansion coefficient, K−1 Penalty parameter Dimensionless temperature Dynamic viscosity, kg m−1 s−1 Kinematic viscosity, m2 s−1 Density, kg m−3 Irreversibility distribution ratio Inclination angle with the positive direction of X axis Basis functions Dimensionless streamfunction Two dimensional domain

Subscripts i k b l r s av total

Global node number Local node number Bottom wall Left wall Right wall Side wall Spatial average Summation over the domain

Superscripts e

Element

Therefore, the second option is more convenient to apply, but it is constrained by the irreversibilities of the conventional heat transfer processes (i.e. heat transfer, fluid friction, etc.). The heat flow visualization on natural convection in rhombic enclosures with the isothermal hot side or bottom wall has been addressed in an earlier article [8] (to be referred as part 1) while current work attempts to quantify the generation of irreversibilities (entropy) during natural convection in rhombic enclosures with the isothermal hot side or bottom wall. The heat transfer and fluid friction irreversibilities are measured in terms of the entropy generation based on second law of thermodynamics. Applying the new methodology termed as Exergy Analysis and its optimization tool via Entropy Generation Minimization (EGM) proposed by Bejan [9–11] in heat transfer applications, the energy destruction can be easily identified to meet the requirement of energy efficient heat transfer processes. ‘Exergy’ quantitatively represents ‘useful energy’. By accounting all the exergy streams of the system, it is possible to determine the extent of exergy destruction. This deviation of the destroyed exergy is proportional to the entropy generation which is further responsible for the poor thermal efficiency of the system. In recent years, there is a large volume of the literature in the concept of the efficient energy transfer systems based on the entropy generation analysis for various applications and the application categories may be based on cavities of different geometrical configurations, cavities filled with different fluids, etc. The prediction of irreversibilities (entropy active zones) during natural convection is important to improve the thermodynamic efficiency of the heat transfer processes. The precise and accurate knowledge about flow and heat transfer characteristics during natural convection is needed to model and mathematically predict active zones of the entropy generation. The accurate evaluation of derivatives is the key issue for the estimation of irreversibilities as the small error in the calculation of temperature and velocity gradients would lead to the irreversibilities with larger errors since the derivatives are powered to 2 in the entropy generation equation. In recent past, the significant amount of works have been reported on the entropy generation analysis for various physical situations [12–18] due to its importance in enhancing the thermal design of the system. Many studies on the analysis of the entropy generation for enclosures with various shapes involving square/non-square enclosures have been reported in the literature. A brief review on the entropy generation for buoyancy-induced flows in the cavity and channels is presented by Oztop and Al-Salem [19]. Numerical simulations conducted by Saleem et al. [20] to study the influence of thermocapillary forces on natural convection of a Newtonian fluid contained in an open cavity revealed the fact that the active spot of the maximum entropy generation depends on the magnitudes of Grashof number, Prandtl number and Marangoni number. Natural convection of the laminar air flow in the Gamma-shaped enclosure with circular corners was investigated by Ziapour and Dehnavi [21] using the finite-volume method and it is reported that Bejan number increases with the decrease in the irreversibility ratio in the cases of the large radius corners. The entropy generation in natural convection through an inclined rectangular cavity was numerically calculated by Bouabid et al. [22] using the Control Volume Finite Element Method (CVFEM). They found that the structure of the flows inside the cavity depends on four dimensionless parameters such as thermal Grashof number, the inclination angle, the irreversibility distribution ratio and the aspect ratio of the cavity. Further, the effect of the magnetic field on the entropy generation with the thermosolutal convection [23] and the discrete heater at the vertical wall of the cavity [24] is also reported. Current work focuses on the analysis of the entropy generation during natural convection in rhombic enclosures based on inclination angles to meet the requirement of various industrial applications involving the electronic cooling, solar heating, etc.

R. Anandalakshmi, T. Basak / European Journal of Mechanics B/Fluids 54 (2015) 27–44

a

29

b

Fig. 1. Schematic diagram of a rhombic cavity for various inclination angles ϕ in (a) Case 1: Differential heating and (b) Case 2: Rayleigh–Benard convection.

The prime objective of this study is to analyze the heat transfer and entropy generation during natural convection in rhombic cavities with various inclination angles (ϕ ) in order to achieve the maximum heat transfer and minimum entropy generation for various applications. Two different cases are considered: (1) the cavity with the hot isothermal left wall and cold isothermal right wall; (2) the cavity with the hot isothermal bottom wall and cold isothermal top wall. Horizontal walls are maintained adiabatic in the former case and vertical side walls are maintained adiabatic in the latter case. In the current study, the Galerkin finite element method has been employed to solve the non-linear equations for the fluid flow, energy and entropy. It may be noted that, the estimation of the entropy generation rate involves the accurate evaluation of the thermal and velocity gradients or derivatives. The finite element approach offers special advantage over the finite difference or finite volume methods as the elemental basis sets are used for the calculation of gradients or derivatives. Simulations are carried out for a range of parameters, Ra = 103 –105 and Pr = 0.015, 0.7 and 1000 and the results are presented in terms of the contours of isotherms (θ ), streamlines (ψ ) and entropy generation maps (Sθ and Sψ ). The effects of Rayleigh number on the total entropy generation (Stotal ), average Bejan number (Beav ) and average Nusselt number (Nu) are also presented to analyze the relative importance of thermal and fluid flow irreversibilities. 2. Mathematical formulation and simulation 2.1. Velocity and temperature distributions The physical domain of the rhombic cavity with side, L and the left wall inclined at an angle ϕ with X axis is shown in Fig. 1(a)–(b) for various boundary conditions (case 1 and case 2). The fluid is considered as Newtonian and the fluid properties are assumed to be constant except the density for the body force term where the density varies linearly with the temperature according to the Boussinesq approximation. The incompressible and laminar fluid flow with the no slip condition is assumed along solid boundaries of rhombic cavities. Under these assumptions, the governing equations for the steady two-dimensional natural convection flow in a rhombic cavity using the conservation of mass, momentum and energy can be written in terms of the following dimensionless variables or numbers X = V =

x L

,

vL , α

Y =

y L

θ=

,

U =

T − Tc Th − Tc

uL

α

,

P =

pL2

ρα

2

,

Pr =

ν , α

Ra =

g β(Th − Tc )L3 Pr

ν2

(1)

as:

∂U ∂V + = 0, ∂X ∂Y  2  ∂U ∂U ∂P ∂ U ∂ 2U U +V =− + Pr + , ∂X ∂Y ∂X ∂X2 ∂Y 2  2  ∂V ∂V ∂P ∂ V ∂ 2V U +V =− + Pr + + Ra Pr θ , ∂X ∂Y ∂Y ∂X2 ∂Y 2 U

∂θ ∂θ ∂ 2θ ∂ 2θ +V = + . ∂X ∂Y ∂X2 ∂Y 2

(2) (3)

(4)

(5)

The boundary conditions for case 1 and case 2 are given as follows: Case 1: Differential heating

∂θ = 0 for Y = 0, 0 ≤ X ≤ 1 on AB, ∂Y U = 0, V = 0, θ =1 for X sin(ϕ) − Y cos(ϕ) = 0, 0 ≤ Y ≤ sin(ϕ) on DA, U = 0, V = 0, θ =0 for X sin(ϕ) − Y cos(ϕ) = sin(ϕ), 0 ≤ Y ≤ sin(ϕ) on BC, ∂θ U = 0, V = 0, = 0 for Y = sin(ϕ), ∂Y cos(ϕ) ≤ X ≤ 1 + cos(ϕ) on CD. (6) U = 0,

V = 0,

Case 2: Rayleigh–Benard convection U = 0,

V = 0,

θ = 1 for Y = 0, 0 ≤ X ≤ 1 on AB

U = 0,

V = 0,

n · ∇θ = 0

for X sin(ϕ) − Y cos(ϕ) = 0, 0 ≤ Y ≤ sin(ϕ) on DA, U = 0,

V = 0,

n · ∇θ = 0

for X sin(ϕ) − Y cos(ϕ) = sin(ϕ), 0 ≤ Y ≤ sin(ϕ) on BC, U = 0,

V = 0,

for Y = sin(ϕ),

θ =0 cos(ϕ) ≤ X ≤ 1 + cos(ϕ) on CD.

(7)

Note that, in Eqs. (1)–(7), 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, respectively; θ is the dimensionless temperature; P is the dimensionless pressure; Ra and Pr are Rayleigh and Prandtl numbers, respectively; Th and Tc are the temperatures at hot and cold walls, respectively; L is each side of the rhombic cavity; ϕ is the inclination angle with the positive direction of X axis.

30

R. Anandalakshmi, T. Basak / European Journal of Mechanics B/Fluids 54 (2015) 27–44

The momentum and energy balance equations (Eqs. (3)–(5)) are solved using the Galerkin finite element method. The continuity equation (Eq. (2)) has been used as a constraint due to the mass conservation. In order to solve Eqs. (3) and (4), we use the penalty finite element method where the pressure (P), penalty parameter (γ ) and incompressibility criteria given by Eq. (2) result in

 P = −γ

∂V ∂U + ∂X ∂Y



.

(8)

The continuity equation (Eq. (2)) is automatically satisfied for large values of γ . The typical value of γ which yields consistent solutions is 107 . Using Eq. (8), the momentum balance equations (Eqs. (3) and (4)) reduce to

∂U ∂ ∂U +V =γ U ∂X ∂Y ∂X



∂U ∂V + ∂X ∂Y



 + Pr

∂ 2U ∂ 2U + ∂X2 ∂Y 2



,

(9)

′′′ The entropy generation per unit volume (Sgen ) may be estimated using second law of thermodynamics for an open system based on the following principle [11]:

Entropy generation

= Entropy transfer associated with heat transfer + Entropy convected in and out of the system + Time rate of the entropy accumulation on the control volume.

∂V ∂V ∂ U +V =γ ∂X ∂Y ∂Y

 

+ Pr

∂U ∂V + ∂X ∂Y

Sgen = −



∂ V ∂ V + ∂X2 ∂Y 2 2

2

Uk Φk (X , Y ),

V ≈

k=1

N  k=1

and θ ≈

N 

T2

+2 + Ra Pr θ .

(10)

The system of equations (Eqs. (5), (9) and (10)) with the boundary conditions (Eqs. (6) or (7)) is solved using the Galerkin finite element method [25]. Expanding the velocity components (U , V ) and temperature (θ ) using the basis set {Φk }Nk=1 as, N 



k

 

Vk Φk (X , Y ),

(14)

In a natural convection system, the associated irreversibilities are due to heat transfer and fluid friction and therefore the entropy generation in natural convection system may be calculated based on Eq. (14) as follows: ′′ ′

and

U ≈

2.3. Entropy generation

∂v ∂x

∂T ∂x

2

 +

2

 +

∂T ∂y

2 

∂ u ∂v + ∂y ∂x

+

µ T

2 

  2

∂u ∂x

2

.

(15)

The dimensionless total local entropy generation terms for the two-dimensional heat and fluid flow in Cartesian coordinates are written as

 Sθ =

∂θ ∂X

2

 +

∂θ ∂Y

2 

,

(16)

and (11)

θk Φk (X , Y ).

Sψ = φ

  2

∂U ∂X

2

 +

∂V ∂Y

2 

 +

∂V ∂U + ∂Y ∂X

2 

,

(17)

k=1

The Galerkin finite element method yields a set of nonlinear residual equations for Eqs. (9), (10) and (5) at the nodes of the internal domain Ω . The detailed solution procedure is given in our earlier work [26]. 2.2. Streamfunction The fluid motion is displayed using the streamfunction (ψ ) obtained from velocity components (U and V ). The relationships between the streamfunction and velocity components for two dimensional flows are U =

∂ψ ∂Y

and

V =−

∂ψ , ∂X

(12)

which yield a single equation

∂ 2ψ ∂U ∂V ∂ 2ψ + = − . ∂X2 ∂Y 2 ∂Y ∂X

(13)

The no slip condition is valid at all boundaries as there is no cross flow. Hence, ψ = 0 is used as residual equations at the nodes for boundaries. Using the above definition of the streamfunction, the positive sign of ψ denotes the anticlockwise circulation and the clockwise circulation is represented by the negative sign of ψ . Expanding the streamfunction (ψ ) using the basis set {Φk }Nk=1

as ψ = k=1 ψk Φk (X , Y ) and the relationships for U , V from Eq. (11), the Galerkin finite element method yields the linear residual equations for Eq. (13) and the detailed solution procedure to obtain ψ s at each node points is given in our earlier work [26].

N

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

φ=

µT o  α  2 . k L1T

(18)

In the current study, φ is taken as 10−4 . A similar value for φ was considered by Ilis et al. [27]. The accurate evaluation of derivatives is the key issue for the proper estimation of Sθ and Sψ as the derivatives of the velocity components (U and V ) and temperature (θ ) are powered to 2 in Eqs. (16) and (17). As mentioned earlier, the derivatives are evaluated based on the finite element method. Current approach offers the special advantage over the finite difference or finite volume solutions where the derivatives are calculated using some interpolation functions which are avoided in the current work and the elemental basis sets, {Φk }Nk=1 are used to estimate Sθ and Sψ . The derivative of any function f over an element e is written as 9  ∂ Φke ∂f e = fke ∂n ∂n k=1

(19)

where, fke is the value of the function at the 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 (N e ), i.e., Ne ∂ fi 1  ∂ fi e = e . ∂n N e=1 ∂ n

(20)

R. Anandalakshmi, T. Basak / European Journal of Mechanics B/Fluids 54 (2015) 27–44

Therefore, at each node, the local entropy generation terms for thermal (Sθ,i ) and flow fields (Sψ,i ) are given by,

 Sθ ,i =

∂θi ∂X

2

 +

∂θi ∂Y

2 

,

(21)

and Sψ,i = φ

  2

∂ Ui ∂X

2

 +

∂ Vi ∂Y ∂θ

2 

∂θ

+

∂ Ui ∂ Vi + ∂Y ∂X

∂U

∂U



∂V

2 

.

(22)

∂V

Note that, the derivatives, ∂ Xi , ∂ Yi , ∂ Xi , ∂ Yi , ∂ Xi , ∂ Yi are evaluated following Eq. (20). The combined total entropy generation (Stotal ) in the cavity is given by the summation of the 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 follows: Stotal = Sθ,total + Sψ,total

(23)

31

Table 1a Comparison of average Nusselt number (case 1: Nul ) for various elements at Ra = 105 and Pr = 0.7 with various inclination angles (ϕ ).

ϕ

Case 1

30° 45° 75° 90°

7×7

14 × 14

20 × 20

28 × 28

32 × 32

34 × 34

2.43 3.62 4.79 4.97

2.47 3.51 4.56 4.73

2.45 3.47 4.47 4.61

2.45 3.46 4.42 4.56

2.44 3.46 4.42 4.55

2.44 3.46 4.41 4.56

Table 1b Comparison of average Bejan number (case 1: Beavg ) for various elements at Ra = 105 and Pr = 0.7 with various inclination angles (ϕ ).

ϕ

Case 1 7×7

14 × 14

20 × 20

28 × 28

32 × 32

34 × 34

30° 45° 75° 90°

0.3053 0.2524 0.2104 0.2043

0.3041 0.2510 0.2030 0.1949

0.3038 0.2509 0.2027 0.1944

0.3036 0.2508 0.2026 0.1944

0.3035 0.2507 0.2026 0.1944

0.3035 0.2508 0.2026 0.1943

where, Sθ ,total

  2   N ∂  θk Φk = Ω  ∂X k=1   2  N  ∂  θk Φk dXdY , +  ∂ Y k=1

Nul =

9  i=1

  ∂Φe ∂Φe θie sin ϕ i − cos ϕ i , ∂X ∂Y

and (24)

9 

Nur = −

i =1

and

 2  2      N N ∂  ∂  Uk Φk +2 Vk Φk Sψ,total = φ 2 ∂ X k=1 ∂ Y k=1 Ω      2  N N  ∂  ∂  + Uk Φk + Vk Φk dXdY . (25)  ∂ Y k =1 ∂ X k=1 The integrals are evaluated using the three-point element-wise Gaussian quadrature integration method. The relative dominance of the entropy generation due to heat transfer and fluid friction is given by Bejan number (Beav ), defined as, Beav =

Sθ,total Sθ,total + Sψ,total

=

Sθ,total Stotal

(26)

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

  ∂Φe ∂Φe θie sin ϕ i − cos ϕ i . ∂X ∂Y

(31)

The average Nusselt numbers at the bottom, top and inclined side walls are given by

1 0

Nub =

Nub dX

X 10 1+cos ϕ

|

 Nut =

cos ϕ

 =

1

Nub dX ,

0

Nut dX ,

1



Nul dS1 ,

Nul = 0

and 1

 .

(30)

Nur dS2 .

Nur =

(32)

0

Here dS1 and dS2 are the small elemental lengths along the left and right walls, respectively. The average Nusselt numbers are also useful to benchmark the overall heat balance within the cavity. Note that, Nul = Nur for case 1 and Nub = Nut for case 2.

2.4. Nusselt number 3. Results and discussion The dimensionless heat transfer rate in terms of the local Nusselt number (Nu) is defined as Nu = −

∂θ , ∂n

(27)

where n denotes the normal direction on a plane. The normal derivative is evaluated by the biquadratic basis set in ξ −η domain. The local Nusselt numbers at the bottom wall (Nub ), top wall (Nut ), left wall (Nul ) and right wall (Nur ) are defined as Nub =

9 

θie

i=1

Nut = −

9  i=1

∂ Φie , ∂Y θie

∂ Φie , ∂Y

(28)

(29)

3.1. Numerical procedure and validation The computational grid within the rhombus is generated via mapping the rhombus into a square domain in the ξ −η coordinate system as mentioned in our earlier work [28]. Current solution scheme produces grid invariant results in terms of the average Nusselt number (Nul (case 1) or Nub (case 2)) and average Bejan number (Beavg ) as shown in Tables 1a–b (case 1: Table 1a for Nul and 1b for Beavg ) and 2a–b (case 2: Table 2a for Nub and 2b for Beavg ). It was found that 32 × 32 biquadratic elements are adequate to obtain grid independent results for the current study. Therefore, the computational grid in the ξ − η coordinate system consists of 32 × 32 biquadratic elements, which correspond to 65 × 65 grid points for the current study.

32

R. Anandalakshmi, T. Basak / European Journal of Mechanics B/Fluids 54 (2015) 27–44

Table 2a Comparison of average Nusselt number (case 2: Nub ) for various elements at Ra = 105 and Pr = 0.7 with various inclination angles (ϕ ).

ϕ 30° 45° 75° 90°

Case 2 7×7

14 × 14

20 × 20

28 × 28

32 × 32

34 × 34

2.97 3.48 4.13 4.23

3.18 3.61 4.03 4.03

3.09 3.52 3.95 3.95

3.03 3.47 3.92 3.93

3.00 3.46 3.91 3.92

3.00 3.45 3.91 3.92

Table 2b Comparison of Bejan number (case 2: Beavg ) for various elements at Ra = 105 and Pr = 0.7 with various inclination angles (ϕ ).

ϕ 30° 45° 75° 90°

Case 2 7×7

14 × 14

20 × 20

28 × 28

32 × 32

34 × 34

0.2134 0.1569 0.1218 0.1179

0.2119 0.1564 0.1222 0.1183

0.2116 0.1563 0.1222 0.1184

0.2114 0.1563 0.1193 0.1184

0.2114 0.1563 0.1223 0.1185

0.2114 0.1563 0.1223 0.1185

In addition, current results based on 32 × 32 elements for the maximum entropy generation values (maximum entropy generation due to heat transfer (Sθ,max ), maximum entropy generation due to fluid friction (Sψ,max ) and maximum entropy generation due to heat transfer and fluid friction (Stotal,max )) are also compared with the earlier work [27] (see Table 3a). Current simulation results are in well agreement with the earlier work [27] at Ra = 103 . However, the results show that there are major differences in values (17.23% error in Sψ,max , 0.88% error in Sθ,max and 16.46% error in Stotal,max ) between the present results and earlier work [27] at Ra = 105 (see Table 3a). The variations in results between the present work (based on elements 32 × 32) and earlier work [27] at Ra = 105 may be due to the different numerical strategy for the estimation of the temperature and velocity gradients in Eqs. (16) and (17) in the earlier work [27]. Further, in order to examine the convergence of the present numerical approach, the maximum values of the streamfunction (ψmax ), isotherm (θmax ), entropy generation due to fluid friction (Sψ,max ), entropy generation due to heat transfer (Sθ ,max ) and entropy generation due to heat transfer and fluid friction (Stotal,max ) with the identical boundary conditions of the earlier work [27] are compared for various elements, 7 × 7, 14 × 14,

20 × 20, 28 × 28, 32 × 32 and 34 × 34 and it was shown that 32 × 32 elements are adequate to achieve the non-linear variations of the field variables (see Table 3b). In the current investigation, the Gaussian quadrature based finite element method provides smooth solutions in the computational domain as the evaluation of residuals depends on the interior Gauss points. The spatial distributions of the entropy generation (Sθ and Sψ ) based on the current numerical scheme are illustrated for the square cavity (ϕ = 90°) with the hot left wall and cold right wall in the presence of the adiabatic top and bottom walls, similar to the case reported by Ilis et al. [27]. Current simulation results are in excellent agreement with the earlier work [27] (see Fig. 2). Also, in order to illustrate the convergence of the spatial distributions of the isotherm (θ ), entropy generation due to heat transfer (Sθ ), streamfunction (ψ ), and entropy generation due to fluid friction (Sψ ), the results are shown for various elements, 7 × 7, 14 × 14, 20 × 20, 28 × 28, 32 × 32 and 34 × 34 at Ra = 103 and Ra = 105 (see Figs. 3 and 4). Based on the numerical evolution with the refinement of elements, 32 × 32 are found to be adequate for converge solutions. Detailed computations and analysis (based on 32×32 elements) have been carried out for rhombic cavities with various values of Pr (Pr = 0.015, 0.7 and 1000), inclination angles (ϕ = 30°, 45°, 75° and 90°) and Ra (Ra = 103 − 105 ). Also, the detailed discussion on the variation of the average total entropy generation due to heat transfer and fluid friction (Stotal ), average Bejan number (Beav ) and average heat transfer rate (Nu) as the function of Ra is presented in the following sections. 3.2. Case 1: differential heating Figs. 5–8 show the 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 in the rhombic cavities of various inclination angles (ϕ = 30°, 45°, 75° and 90°). In this case, the left wall of the cavity is heated isothermally and the right wall is maintained isothermally cold with horizontal adiabatic walls. Slightly distorted isotherms illustrate the conduction dominant heat transfer for all ϕ s at Ra = 103 and Pr = 0.015. The

Table 3a Comparisons of present results with the earlier work [27] for natural convection in air (Pr = 0.71) filled square cavity with 32 × 32 biquadratic elements. Present (32 × 32)

Ra

3

10 105

Ilis et al. [27]

ψmax

θmax

Sψ,max

Sθ ,max

Stotal,max

ψmax

θmax

Sψ,max

Sθ,max

Stotal,max

1.17 9.60

1 1

0.26 660.24

2.27 60.49

2.29 695.42

– –

– –

0.25 563.20

2.27 61.03

2.29 597.11

Table 3b Comparison of maximum values of streamfunction (ψmax ), temperature (θmax ), entropy generation due to heat transfer (Sθ ,max ), entropy generation due to fluid friction (Sψ,max ) and total entropy generation due to heat transfer and fluid friction (Stotal,max ) for various elements during natural convection in air (Pr = 0.71) filled square cavity at Ra = 105 with identical boundary conditions of the earlier work [27]. Ra

ψmax

θmax

Sψ,max

Sθ ,max

Stotal,max

7×7 103 105

1.18 9.57 1.17 9.60

1 1

0.23 282.37

2.29 65.26

2.31 317.65

1.17 9.60

1 1

0.26 616.27

2.27 62.78

2.29 651.36

1.17 9.60

1.17 9.60

Sψ,max

Sθ,max

Stotal,max

1 1

0.25 545.32

2.27 65.98

2.30 579.03

1 1

0.26 651.75

2.27 60.88

2.29 686.41

1 1

0.26 660.24

2.27 60.49

2.29 695.42

1.17 9.60

1 1

0.26 663.57

2.27 60.29

2.29 698.43

28 × 28

32 × 32 103 105

θmax

14 × 14

20 × 20 103 105

ψmax

34 × 34

R. Anandalakshmi, T. Basak / European Journal of Mechanics B/Fluids 54 (2015) 27–44

33

a

b

Fig. 2. Local entropy generation due to heat transfer (Sθ ) and local entropy generation due to fluid friction (Sψ ) for hot left wall, cold right wall and adiabatic horizontal walls for Pr = 0.71, ϕ = 90° and (a) Ra = 103 and (b) Ra = 105 . Top figures in (a) and (b) are reproduced from Ilis et al. [27] with permission from Elsevier. Bottom figures in (a) and (b) are obtained from present simulation strategy.

34

R. Anandalakshmi, T. Basak / European Journal of Mechanics B/Fluids 54 (2015) 27–44

a

b

c

d

e

f

Fig. 3. Temperature (θ ), local entropy generation due to heat transfer (Sθ ), Streamfunction (ψ ) and local entropy generation due to fluid friction (Sψ ) for hot left wall, cold right wall and adiabatic horizontal walls at Pr = 0.71, ϕ = 90° and Ra = 103 for various elements ((a) 7 × 7 (b) 14 × 14(c) 20 × 20(d) 28 × 28 (e) 32 × 32 and (f) 34 × 34) based on current simulation strategy.

isotherms are also found to be distorted much near the center of the cavity for all ϕ s (see Fig. 5(a)–(d)). The boundary layer thickness is high near the lower left portion and top right portion of the

cavity as indicated via isotherms with θ ≥ 0.9 and θ ≤ 0.1, respectively at ϕ = 30° and 45° (see Fig. 5(a)–(b)). Therefore, the large portion of the lower left wall is stagnant with the hot

R. Anandalakshmi, T. Basak / European Journal of Mechanics B/Fluids 54 (2015) 27–44

a

b

c

d

e

f

35

Fig. 4. Temperature (θ ), local entropy generation due to heat transfer (Sθ ), Streamfunction (ψ ) and local entropy generation due to fluid friction (Sψ ) for hot left wall, cold right wall and adiabatic horizontal walls at Pr = 0.71, ϕ = 90° and Ra = 105 for various elements ((a) 7 × 7 (b) 14 × 14(c) 20 × 20(d) 28 × 28 (e) 32 × 32 and (f) 34 × 34) based on current simulation strategy.

fluid whereas the large portion of the top right wall is stagnant with the cold fluid. The lesser heat transfer rates near those regions are due to the inclination of the side wall with the adiabatic wall

which further obstructs the fluid and heat flow (see Fig. 5(a)–(b)). As ϕ increases (see Fig. 5(b)–(d)), the boundary layer thicknesses near the lower left portion and the top right portion of the cavity

36

R. Anandalakshmi, T. Basak / European Journal of Mechanics B/Fluids 54 (2015) 27–44

a a

b b

c

c

d

d

Fig. 5. Isotherms (θ ), local entropy generation due to heat transfer (Sθ ), streamlines (ψ ) and local entropy generation due to fluid friction (Sψ ) in case 1 for Pr = 0.015 at Ra = 103 for (a) ϕ = 30°, (b) ϕ = 45°, (c) ϕ = 75° and (d) ϕ = 90°.

decrease significantly due to the increase in heat transfer near those regions. The distribution of the local entropy generation due to heat transfer (Sθ ) depicts that the entropy generation is higher at the top left junction of the isothermal–adiabatic walls and the bottom right junction of the isothermal–adiabatic walls for the rhombic cavities with ϕ = 30° and 45° (Sθ,max = 254 and 52 for ϕ = 30° and 45°, respectively) due to the high thermal gradient near

Fig. 6. Isotherms (θ ), local entropy generation due to heat transfer (Sθ ), streamlines (ψ ) and local entropy generation due to fluid friction (Sψ ) in case 1 for Pr = 0.015 at Ra = 105 for (a) ϕ = 30°, (b) ϕ = 45°, (c) ϕ = 75° and (d) ϕ = 90°.

R. Anandalakshmi, T. Basak / European Journal of Mechanics B/Fluids 54 (2015) 27–44

a

a

b

b

37

c c

d d

Fig. 7. Isotherms (θ ), local entropy generation due to heat transfer (Sθ ), streamlines (ψ ) and local entropy generation due to fluid friction (Sψ ) in case 1 for Pr = 0.7 at Ra = 105 for (a) ϕ = 30°, (b) ϕ = 45°, (c) ϕ = 75° and (d) ϕ = 90°.

those regions as seen from the large compression of isotherms (see Fig. 5(a)–(b)). It may be noted that the rhombic cavities of ϕ = 30° and 45° show the higher thermal gradient near the lower left and top right portions of the cavity compared to other portions due to its inclination towards the bottom wall and imposed thermal boundary conditions (see Fig. 5(a)–(b)). As ϕ increases, the cavity is maintained with almost constant thermal gradient throughout the

Fig. 8. Isotherms (θ ), local entropy generation due to heat transfer (Sθ ), streamlines (ψ ) and local entropy generation due to fluid friction (Sψ ) in case 1 for Pr = 1000 at Ra = 105 for (a) ϕ = 30°, (b) ϕ = 45°, (c) ϕ = 75° and (d) ϕ = 90°.

domain as in the case of ϕ = 75° and 90° which further correspond to the lower entropy generation due to heat transfer (Sθ ) as seen in Fig. 5(a)–(d). It is observed that, Sθ ,max = 2.6 and 2.1 occur for ϕ = 75° and 90°, respectively. It is also interesting to note that, Sθ is almost negligible at the core of the cavity as there is no variation of the thermal gradient in that regime for all ϕ s (see Fig. 5(a)–(d)).

38

R. Anandalakshmi, T. Basak / European Journal of Mechanics B/Fluids 54 (2015) 27–44

Several interesting features on the entropy generation due to the flow irreversibilities are also observed in Fig. 5(a)–(d). Due to buoyancy and imposed thermal boundary conditions, the hot fluid rises from the left wall and the cold fluid flows down along the right wall with the clockwise unicellular flow pattern within the cavity. It may be noted that, the fluid circulation cell is not circular and elongated diagonally due to the geometrical asymmetry with respect to the central vertical line for ϕ = 30° and 45° (Fig. 5(a)–(b)). As ϕ increases, the non-circular flow gradually approaches to be circular and it is almost circular for ϕ ≥ 75° (Fig. 5(c)–(d)). The strength of the fluid circulation cell tends to become stronger with ϕ . It may also be noted that some significant entropy generation due to the flow irreversibilities (Sψ ) occurs near the top left corner as well as the bottom right corner for ϕ = 30° and 45° (Fig. 5(a)–(b)) and the middle portions of the walls of the cavity for ϕ = 75° and 90° (Fig. 5(c)–(d)). This is due to significant velocity gradients as the flow circulation cells are pushed towards those regions resulting in high velocity gradients compared to other portions of the cavity. However, the overall flow strength within the cavity is very weak due to the conduction dominant heat transfer at Ra = 103 . Therefore, effects of the entropy generation due to the fluid friction irreversibility (Sψ ) are almost negligible throughout the cavity due to the less intense fluid circulation cells. In addition, due to weak fluid flow at the low Ra (Ra = 103 ), the entropy generation due to fluid friction (Sψ ) is smaller than Sθ . Isotherms are largely distorted in the center and compressed along the top portion of the left wall and bottom portion of right wall, signifying the dominance of convection at Ra = 105 and Pr = 0.015 for ϕ = 30° (see Fig. 6(a)). As ϕ increases, isotherms are more distorted at the center and the zone of compression of isotherms near the left hot wall move towards the bottom portion of the same wall whereas that near the right cold wall move towards the top portion of the same wall at Ra = 105 and Pr = 0.015 (see Fig. 6(b)–(d)). As Ra increases from 103 to 105 , the thermal boundary layer thickness near isothermal side walls decreases due to the enhanced heat transfer during the convection dominant heat transfer for all ϕ s (see Fig. 6(a)–(d)). The compression of isotherms results in larger thermal gradients near those regions and hence the significant entropy generation due to heat transfer (Sθ ) is observed near those regions as shown in Fig. 6(a)–(d) at Ra = 105 and Pr = 0.015. It is also interesting to note that, the entropy generation is higher at the top left junction of isothermal–adiabatic walls as well as the bottom right junction of isothermal–adiabatic walls for ϕ = 30° and 45° at Ra = 105 and Pr = 0.015 (Sθ,max = 81 and 22 for ϕ = 30° and 45°, respectively) (see Fig. 6(a)–(b)). As ϕ increases, the zone of Sθ also moves away from the adiabatic wall and progresses along the isothermal side walls of the cavity and Sθ ,max is found along the isothermal side walls where the compressed isotherms are observed for ϕ = 75° and 90° at Ra = 105 and Pr = 0.015 (Sθ,max = 29 and 35 for ϕ = 75° and 90°, respectively) (see Fig. 6(c)–(d)). It is also interesting to note that, Sθ ,max at Ra = 105 is lower compared to Sθ,max at Ra = 103 due to the conduction dominant effect near those regions for ϕ = 30° and 45° (see Figs. 5(a)–(b) and 6(a)–(b)). On the other hand, Sθ ,max at Ra = 105 is higher for ϕ = 75° and 90° compared to that at Ra = 103 due to the convection dominant effect near those regions (see Figs. 5(c)–(d) and 6(c)–(d)). As mentioned earlier, the comparatively higher distortion in isotherms indicates that the onset of convection mode occurs for ϕ = 75° and 90° even at Ra = 103 for Pr = 0.015 and that further leads to increase in Sθ,max as Ra increases from 103 to 105 for ϕ = 75° and 90° at Pr = 0.015 (see Figs. 5(c)–(d) and 6(c)–(d)). At Ra = 105 , the dominant buoyancy forces lead to the enhanced circulation as seen from the larger magnitudes of streamfunctions, |ψ|max compared to Ra = 103 for all ϕ s (Figs. 6(a)–(d) and 5(a–d)). A number of multiple circulation cells are observed

near the top right corner and bottom left corner of the cavity for ϕ = 30° and 45° at Ra = 105 and Pr = 0.015 (Fig. 6(a)–(b)). As ϕ increases, the central primary circulation cell grows bigger and fills the entire part of the cavity at Ra = 105 and Pr = 0.015 (Fig. 6(c)–(d)). It is also interesting to note that, the multiple circulation cells are found to occur near all corners of the cavity at ϕ = 90° (Fig. 6(d)). Due to the intense fluid circulation and as a consequence of the no-slip boundary conditions along the wall, the high velocity gradients exist near the left top portion and bottom right portion of the cavity leading to the larger entropy generation (Sψ,max = 290 and Sψ,max = 401 for ϕ = 30° and 45°, respectively) at Ra = 105 and Pr = 0.015 (see Fig. 6(a)–(b)). In addition, a few more entropy active regions of the fluid friction are found near the top left corner and bottom right corner for ϕ = 30° and 45° (Sψ = 74 and Sψ = 116 for ϕ = 30° and 45°, respectively) (see Fig. 6(a)–(b)). As ϕ increases, all the walls tend to act as the source of the entropy generation due to the fluid friction (Sψ ) (see Fig. 6(c)–(d)) based on the circular fluid circulation cells which are confined within the cavity. The maxima in Sψ is still found along the isothermal side walls of the cavity (Sψ,max = 562 and Sψ,max = 619 for ϕ = 75° and 90°, respectively) where the velocity gradients are higher compared to that near the adiabatic walls (see Fig. 6(c)–(d)). In contrast, the velocity gradients between the circulation cells are very less and thus no significant active zones of Sψ occur near those regions for all ϕ s at Ra = 105 and Pr = 0.015 (see Fig. 6(c)–(d)). Based on the intense convection, Sψ,max at Ra = 105 is higher compared to Sψ,max at Ra = 103 due to the enhanced convection effect for all ϕ s (see Figs. 5(a)–(d) and 6(a)–(d)). Isotherms, streamlines and entropy maps for Ra = 103 and Ra = 104 at Pr = 0.7 are qualitatively similar to Ra = 103 and Ra = 104 at Pr = 0.015. Therefore results for Pr = 0.7 are omitted here for the brevity of the manuscript. Fig. 7(a)–(d) illustrates isotherms, streamlines and entropy maps for Pr = 0.7 and Ra = 105 . In contrast to the previous case with Pr = 0.015, isotherms are compressed along the side wall illustrating the high heat transfer rate from that regime for all ϕ s (Fig. 7(a)–(d)). Thus, the boundary layer thickness is small near that regime compared to the previous case with Pr = 0.015 (Figs. 6(a)–(d) and 7(a)–(d)). Note that, Sθ ,max (Sθ ,max = 68) still occurs near the junction of the adiabatic and isothermal side walls for Pr = 0.7 similar to the previous case of Pr = 0.015 for ϕ = 30° (Fig. 7(a)). As ϕ increases, the zone of Sθ ,max also moves away from the adiabatic wall towards the isothermal side wall of the cavity and further towards the other intersection point of the side wall and the adiabatic wall which corresponds to the highly compressed isotherms for ϕ = 45° − 75° (see Fig. 7(b)–(d)). It is found that, Sθ ,max increases with Pr except at ϕ = 30° due to the largely compressed isotherms at Pr = 0.7 (Figs. 6(a)–(d) and 7(a)–(d)). It is also found that, Sθ active zones are more confined along the isothermal side walls due to the largely compressed isotherms at Pr = 0.7 compared to Pr = 0.015 for all ϕ s. This occur further due to the decrease in the thermal diffusivity which leads to the larger thermal gradients near the isothermal side walls (Figs. 6(a)–(d) and 7(a)–(d)). The intensity of the fluid circulation is found to be stronger for Pr = 0.7 compared to Pr = 0.015 due to the larger momentum diffusivity at Pr = 0.7 as seen from the maximum value of the streamfunction (|ψ|max ) near the core of the cavity for all ϕ s (Figs. 7(a)–(d) and 6(a)–(d)). Further, the streamlines expand and take the shape of the cavity near the cavity walls due to the intense convection which are in contrast with the previous case for Pr = 0.015. In contrast to previous cases with Pr = 0.015, two circulation cells are also formed near the core of the cavity for higher ϕ s (ϕ = 75° − 90°) (Fig. 7(c)–(d)). The multiple fluid circulation cells near the corners of the cavity as observed for Pr = 0.015 are absent for Pr = 0.7 (Fig. 7(a)–(d)). Similar to Pr = 0.015, Sψ,max is found along the isothermal side walls of the cavity for all ϕ s at

R. Anandalakshmi, T. Basak / European Journal of Mechanics B/Fluids 54 (2015) 27–44

Pr = 0.7 (Fig. 7(a)–(d)). However, Sψ,max values for ϕ = 30° and 45° at Pr = 0.7 are comparatively lesser than those observed at Pr = 0.015. This may be due to the increase in the momentum diffusivity in the case of Pr = 0.7 and that further leads to the decrease in the velocity gradients (Figs. 7(a)–(b) and 6(a)–(b)). It is also found that, the significant active zones of Sψ are also found near the top right corner and bottom left corner due to the significant velocity gradients between the circulation cells for all ϕ s at Pr = 0.7 in addition to the regions similar to Pr = 0.015 case (Figs. 7(a)–(b) and 6(a)–(b)). Note that, the maximum value of Sψ (Sψ,max ) is higher for ϕ = 75° and 90° at Pr = 0.7 compared to Pr = 0.015 due to the strong convective cells at the center of the cavity and those convective cells further compress the fluid layers near the walls at ϕ = 75° and 90° leading to the higher velocity gradients near the isothermal side walls at Pr = 0.7 (Figs. 7(c)–(d) and 6(c)–(d)). The distributions for Pr = 1000 and Ra = 105 are shown in Fig. 8(a)–(d). It is observed that, the thermal entropy generation map (Sθ ) and flow entropy generation map (Sψ ) remain qualitatively similar to that of Pr = 0.7 for all ϕ s and the maximum value of Sθ also remains almost the same for all ϕ s. Also, the increase in Sψ,max is not significant at Pr = 1000 compared to that of Pr = 0.7 (Figs. 7(a)–(d) and 8(a)–(d)).

39

a

b

c

3.3. Case 2: Rayleigh–Benard convection In this case, the bottom wall of the cavity is heated isothermally in contrast to the left wall in the previous case. Distributions of θ , Sθ , ψ and Sψ for Pr = 0.015 at Ra = 103 are depicted in Fig. 9(a)–(d). It is interesting to note that the large amount of fluid near the top right portion of the cavity is maintained cold at the lower ϕ (ϕ = 30°) compared to the higher ϕ (ϕ = 90°) (see Fig. 9(a)–(d)). It may also be noted that, the isotherms are parallel to the isothermal horizontal walls signifying the conduction dominant heat transfer at Ra = 103 for ϕ = 90° (see Fig. 9(d)). Less distorted isotherms illustrate comparatively less heat transfer rate in case 2 compared to case 1 for all ϕ s as displayed in Figs. 9(a)–(d) and 5(a)–(d). It is also observed that, identical Sθ active zones, but higher magnitudes of Sθ,max occur due to the high local thermal gradient at the top left and bottom right corners for all ϕ s except ϕ = 90° at the identical Pr, ϕ and Ra of case 1 (see Figs. 9(a)–(d) and 5(a)–(d)). At ϕ = 90°, magnitudes of Sθ ,max are constant throughout the cavity (Sθ,max = 1) due to the constant thermal gradient throughout the domain and that is depicted by parallel isotherms, which denotes the conduction dominant regime (see Fig. 9(d)). Note that, the conduction dominant zone is expected for the Rayleigh–Benard convection at ϕ = 90° as reported in an earlier work by Venturi et al. [29]. Although streamlines exhibit the qualitatively similar trends of the differential heating case for Pr = 0.015 and Ra = 103 as seen in Figs. 5 and 9, both the magnitudes of streamfunctions and intensity of circulations are found to be lower for the Rayleigh–Benard convection (case 2) with all ϕ s except ϕ = 90°. This is due to the fact that the convection is initiated at the critical Ra only in case 2 in contrast to case 1 where the onset of convection occurs at even Ra = 103 (see Figs. 5 and 9). As mentioned earlier, the fluid is stagnant at ϕ = 90° for the boundary conditions applied in this case as there is no circulation (ψ = 0) (see Fig. 9(d)). Due to the weak fluid flow at the low Ra (Ra = 103 ) and Pr (Pr = 0.015), Sψ is almost negligible for all ϕ s (see Fig. 9(a)–(c)). As a result of the static condition, Sψ = 0 for ϕ = 90° at the low Ra (Ra = 103 ) and Pr (Pr = 0.015) (see Fig. 9(d)). The conduction dominant static fluid solution is the characteristics of the square domain for the Rayleigh–Benard convection. However, the pure conduction dominant heat transfer does not occur for other rhombic angles (ϕ ≤ 90°) even at the low Ra (Ra = 103 ) (see Fig. 9(a)–(c)).

d

Fig. 9. Isotherms (θ ), local entropy generation due to heat transfer (Sθ ), streamlines (ψ ) and local entropy generation due to fluid friction (Sψ ) in case 2 for Pr = 0.015 at Ra = 103 for (a) ϕ = 30°, (b) ϕ = 45° (c) ϕ = 75° and (d) ϕ = 90°.

It is interesting to note that, the conduction based static fluid solution is still observed in addition to the convection based dynamic solution for ϕ = 90° irrespective of Pr at Ra ≥ 2 × 103 in case 2 (see Figs. 10(a)–(d) and 11(a)–(d)). The conduction based static solution occurs only for ϕ = 90° (as reported in Fig. 9(d)) and it is observed that the conduction based static solution disappears with a slight perturbation of ϕ at higher Ra (Ra ≥ 2 × 103 ) irrespective of Pr (see Figs. 10(a)–(d)–11(a)–(d)). The multiple

40

R. Anandalakshmi, T. Basak / European Journal of Mechanics B/Fluids 54 (2015) 27–44

a a

b b

c c

d d

Fig. 10. Isotherms (θ ), local entropy generation due to heat transfer (Sθ ), streamlines (ψ ) and local entropy generation due to fluid friction (Sψ ) in case 2 for Pr = 0.015 at Ra = 105 for (a) ϕ = 30°, (b) ϕ = 45° (c) ϕ = 75° and (d) ϕ = 90°.

solutions involving the conduction based static fluid solutions were also reported by Venturi et al. [29] for the square cavity (ϕ = 90°). The present analysis on the entropy generation maps is restricted to only the convection based dynamic solution at ϕ = 90°.

Fig. 11. Isotherms (θ ), local entropy generation due to heat transfer (Sθ ), streamlines (ψ ) and local entropy generation due to fluid friction (Sψ ) in case 2 for Pr = 1000 at Ra = 105 for (a) ϕ = 30°, (b) ϕ = 45° (c) ϕ = 75° and (d) ϕ = 90°.

R. Anandalakshmi, T. Basak / European Journal of Mechanics B/Fluids 54 (2015) 27–44

Fig. 10(a)–(d) represent θ , Sθ , ψ and Sψ distributions for Pr = 0.015 at Ra = 105 . Isotherms are largely distorted at the central portion of the cavity for ϕ = 30° and these further illustrate the high thermal mixing in that region compared to case 1. Also, large portions near the top right and bottom left corners are filled with the stagnant cold and hot fluid, respectively for ϕ = 30° due to the inadequate heat flow compared to case 1 at the identical Pr and Ra (see Figs. 10(a) and Fig. 6(a)). As ϕ increases, the thermal mixing near the central portion of the cavity (θ = 0.4–0.6) as well as the heat flow within the cavity increases as depicted in Figs. 10(b)–(d) compared to case 1 (see 6(b)–(d)). It may also be noted that the boundary layer thickness near the top right or bottom left corner of the cavity also decreases with ϕ as seen in Fig. 10(a)–(d). Overall, the largely distorted isotherms illustrate the high heat transfer rate for case 2 compared to case 1 for identical parameters (Pr , Ra and ϕ ). At Ra = 105 and Pr = 0.015, the compression of isotherms near the left portion of the top wall and the right portion of the bottom wall result in active sites of Sθ near those regions for all ϕ s. It is also interesting to note that, Sθ,max is found to occur near the top left corner and bottom right corner due to the high heat flow as seen from largely compressed isotherms in those regions for ϕ = 30° and 45° (see Fig. 10(a)–(b)). As ϕ increases, the zone of Sθ also grows along the isothermal horizontal walls of the cavity and Sθ,max is found to be shifted along the isothermal horizontal walls (the region near the top left corner and bottom right corner), where the compressed isotherms are observed for ϕ = 75° and 90° at Ra = 105 and Pr = 0.015 (see Fig. 10(a)–(d)). However, Sθ ,max is much larger for the present case at ϕ = 30° and ϕ = 45° than that for case 1 at identical parameters (Pr, Ra and ϕ ) (see Figs. 6(a)–(b) and Fig. 10(a)–(b)). As convection is dominant at Ra = 105 , the fluid flow intensifies and that results in the enhanced convective transport inside the cavity for all ϕ s at Pr = 0.015 (see Fig. 10(a)–(d)). The maximum magnitudes of the streamline cells are comparatively higher than those in case 1 for all ϕ s. As ϕ increases, the size and magnitude of the primary circulation cells increase as seen from Fig. 10(a)–(d). It is also interesting to note that the strength of the circulation cells near the bottom left and top right corners increases with ϕ for ϕ = 30–45° (see Fig. 10(a)–(b)). Further increase in ϕ results in tiny multiple circulation cells near all corners of the cavity at ϕ = 75° and 90° for Pr = 0.015 and Ra = 105 in case 2 (see Fig. 10(c)–(d)). Although active zones of Sψ for ϕ = 30° of case 2 are qualitatively similar to those of case 1, Sψ,max is higher (Sψ,max = 738) for the present case (see Figs. 6(a) and 10(a)). This occurs due to the high flow strength and/or high flow gradient in case 2 based on large magnitudes of streamfunction compared to case 1 (see Figs. 10(a) and 6(a)). As ϕ increases, the strength of primary circulation cell increases and therefore, Sψ active zones are found to occur around primary fluid circulation cells and the interface between primary and secondary fluid circulation cells (Sψ = 50) in addition to Sψ active zones near walls of the cavity as in case 1 (see Figs. 6(b)–(d) and 10(b)–(d)). Note that, Sψ,max in case 2 is larger than that in case 1 for all ϕ s at Ra = 105 and Pr = 0.015 (see Figs. 10(a)–(d) and 6(a)–(d)). Fig. 11(a)–(d) illustrates isotherms, streamlines and entropy generation maps for Pr = 1000 and Ra = 105 in case 2. The results are qualitatively similar for Pr = 0.7 and Pr = 1000 and therefore the results are discussed only for Pr = 1000. It may also be noted that the heat flow is enhanced near the top left and bottom right portions of the cavity as illustrated by highly compressed isotherms near those regions for all ϕ s compared to low Pr values (Pr = 0.015) (see Figs. 10(a)–(d) and 11(a)–(d)). This further results in comparatively less boundary layer thickness near isothermal walls for the present case compared to low Pr case

41

(Pr = 0.015) (see Figs. 10(a)–(d) and 11(a)–(d)). It is also found that the isotherms are largely distorted near the center for all ϕ s and the thermal mixing increases with ϕ as the uniform temperature with θ = 0.4–0.6 is maintained near the central portion of the cavity at ϕ = 90° for the present case compared to case 1 at Pr = 1000 and Ra = 105 (see Figs. 8(a)–(d) and 11(a)–(d)). It is observed that, the heat transfer entropy generation map (Sθ ) remains qualitatively similar to that of Pr = 0.015 for all ϕ s and the maximum value of Sθ increases due to the increase in Pr for all ϕ s. It is also found that, Sθ ,max is quite higher for the present case (case 2) compared to that of the differential heating case (case 1) for all ϕ s except ϕ = 90°. This occurs due to the higher thermal gradient near isothermal horizontal walls based on highly compressed isotherms. Note that, Sθ ,max = 2376, 798, 79 and 28 occur for ϕ = 30°, 45°, 75° and 90°, respectively for case 2 whereas Sθ ,max = 62, 28, 60 and 77 occur for ϕ = 30°, 45°, 75° and 90°, respectively for case 1 at Pr = 1000 and Ra = 105 (see Figs. 8 and 11). The intensity of fluid circulation is found to be stronger for Pr = 1000 compared to Pr = 0.015 as seen from the maximum values of streamfunctions (|ψ|max ) near the core of the cavity for all ϕ s (Figs. 10(a)–(d) and 11(a)–(d)). Similar to case 1, multiple circulation cells are absent for the present case due to the increase in the momentum diffusivity at Pr = 1000 (Figs. 10(a)–(d) and 11(a)–(d)). Although qualitative features of streamfunction contours are similar in both the cases (cases 1 and 2) for all ϕ s, |ψ|max is higher for case 2 compared to case 1 for all ϕ s (see Figs. 8(a)–(d)–11(a)–(d)). It is interesting to observe the reduction in maximum entropy generation values due to the fluid friction irreversibility (Sψ,max ) with Pr signifying that the velocity gradients are smaller for Pr = 1000 (see Figs. 10(a)–(d)–11(a)–(d)). It is also found that, multiple circulation cells do not appear for higher Pr (Pr = 1000). Hence, there are no significant active zones of Sψ at the interface between the circulation cells due to the less velocity gradients at Pr = 1000 in contrast to Pr = 0.015 (see Figs. 10(a)–(d)–11(a)–(d)). It is also interesting to note that, Sψ,max is higher for case 2 compared to case 1 for ϕ ≤ 45° at Pr = 1000 and Ra = 105 due to intense fluid cells which further lead to high velocity gradients. In contrast, Sψ,max is lower for case 2 compared to case 1 for ϕ ≥ 45° at Pr = 1000 and Ra = 105 due to the low velocity gradients in case 2 compared to case 1 as fluid cells at the core split into tiny cells which further push the primary circulation cells near the walls in case 1 (see Figs. 8(a)–(d)–11(a)–(d)). As a result of the Rayleigh–Benard convection (case 2), Sθ ,max and Sψ,max are increased drastically compared to the differential heating situation (case 1) for rhombic shapes (ϕ ≤ 45°) at lower Pr (Pr = 0.015) (see Figs. 6(a)–(b)–10(a)–(b)). Although Sθ ,max values are much higher in case 2, there is no substantial change in Sψ,max values for both the cases at higher Pr (Pr = 1000) for ϕ ≤ 45° (see Figs. 8(a)–(b)–11(a)–(b)). The large spatial distribution of entropy active zones due to heat transfer (Sθ ) and fluid friction irreversibility (Sψ ) in case 2 reveals that, the total entropy production due to heat transfer and fluid friction irreversibilities are much larger than those with the differential heating case (case 1) for the entire Ra and Pr. Further, a detailed analysis on the total entropy generation (Stotal ), average Bejan number (Beav ) and average Nusselt number (Nu) are presented in the following section to study the optimized heat transfer processes based on the differential and Rayleigh–Benard heating cases. 3.4. Characteristics of total entropy generation (Stotal ), average Bejan number (Beav ) and average Nusselt number (Nu) The variations of total entropy generation due to heat transfer and fluid friction irreversibilities (Stotal ), average Bejan number (Beav ) and average Nusselt number at the left or bottom wall (Nul or Nub ) for case 1 and case 2 vs logarithmic Rayleigh number (Ra) are

42

R. Anandalakshmi, T. Basak / European Journal of Mechanics B/Fluids 54 (2015) 27–44

a

b

c

d

Fig. 12. Variations of total entropy generation (Stotal : bottom panel), average Bejan number (Beavg : middle panel) and average Nusselt number (Nul or Nub : top panel) with Ra for various ϕ : 30° (· · · ), 45° (- - -), 75° (– –) and 90° (—) in cases 1 and 2 for Pr = 0.015 (a and b) and Pr = 1000 (c and d).

presented in bottom, middle and top panels, respectively in Fig. 12 for low (Pr = 0.015) and high (Pr = 1000) Pr fluids. Fig. 12(a) represents distributions of Stotal , Beav and Nub , for Pr = 0.015 in case 1. The total entropy generation in the cavity is found to be constant till Ra ≤ 104 for all ϕ s (see bottom panel plots of Fig. 12(a)). This occurs due to the smaller entropy generation with the fluid friction (Sψ,total ) compared to Sθ,total as seen from Sθ and Sψ maps of Fig. 5(a)–(d) for all ϕ s. It may also be noted that, Stotal for ϕ = 90° cavity is found to be higher at Ra ≤ 104 followed by ϕ = 75°, ϕ = 45° and ϕ = 30° cavities. It is interesting to note that Stotal increases exponentially for Ra ≥ 104 due to the gradual increase of the fluid friction irreversibility (Sψ,total ) in addition

to the heat transfer irreversibility (Sθ ,total ) based on the high convective motion of the fluid as seen from contour plots of ψ and Sψ for all ϕ s (see Fig. 6(a)–(d)). Similar to the low Ra regime (Ra ≤ 104 ), it is observed that, Stotal is higher for ϕ = 90° at Ra = 105 due to the significant Sψ,total (see the bottom panel plot of Fig. 12(a)). The average Bejan number (Beav ) indicates the importance of the entropy generation due to heat transfer (Sθ ) or fluid friction (Sψ ) irreversibilities. A common decreasing trend in Beav with Ra is observed for all ϕ s in the middle panel plot of Fig. 12(a). The maximum value for Beav (Beav ≈ 1) occurs at the low Ra (Ra = 103 ), indicating that the entropy generation in the cavity is primarily due to the heat transfer irreversibility (Sθ ,total ) at the conduction

R. Anandalakshmi, T. Basak / European Journal of Mechanics B/Fluids 54 (2015) 27–44

dominant mode. It may be noted that, ϕ = 30° and 45° correspond to the minimum entropy generation at Ra ≤ 104 compared to other ϕ s and consequently, the exergy loss due to the fluid flow irreversibilities is less as Beav ≥ 0.5 occurs due to the maximum contribution of Sθ,total at Ra ≤ 104 (see the middle panel plot of Fig. 12(a)). As Ra increases from 103 , Beav starts decreasing profoundly for ϕ = 75° and ϕ = 90° and that further illustrates the significant contribution of Sψ,total in Stotal at Ra ≤ 104 compared to ϕ = 30° and 45° (see the middle panel plots of Fig. 12(a)). As Ra increases to 105 , the fluid friction irreversibility (Sψ,total ) increases and that dominates over the heat transfer irreversibility (Sθ ,total ) due to the enhanced convection heat transfer in the cavity. Consequently, the large amount of available energy (exergy loss is maximum) is utilized to overcome the irreversibilities due to the fluid friction at the high Ra (Ra = 105 ). It may also be noted that, Beav ≤ 0.5 occurs for all ϕ s due to the Sψ,total dominance at Ra = 105 (the middle panel plot of Fig. 12(a)). The available energy required for heating the fluid decreases with the entropy generation in such a way that a part of the available energy is utilized to overcome irreversibilities. The total entropy generation in the cavity (Stotal ) is almost maintained constant with Ra due to the negligible Sψ,total over Sθ,total in Stotal at Ra ≤ 104 for all ϕ s. The heat transfer rate due to the temperature gradient (Nul ) is also maintained constant with Ra till Ra ≤ 104 as the significant energy is not required to overcome irreversibilities (Sψ,total ) for all ϕ s (the top panel plot of Fig. 12(a)). As Ra increases, the total entropy generation (Sθ,total and Sψ,total ) increases due to the significant Sψ,total for all ϕ s. Simultaneously, the average heat transfer rate at the left wall (Nul ) is also larger at higher Ra. This is due to the fact that the heat transport due to the temperature gradient is much larger than the lost energy due to the fluid friction irreversibilities (Sψ,total ) along the left wall (see bottom and top panel plots of Fig. 12(a)) at the onset of convection (Ra ≥ 104 ). Therefore, there is an increase in the heat transfer rate to the fluid (Nul ) at Ra ≥ 104 as illustrated in the top panel plot of Fig. 12(a) for all ϕ s. It is also interesting to note that, the compression of isotherms or large thermal gradient occurs in a large zone at higher ϕ s for Ra ≥ 104 (see Fig. 6(a)–(d)) and that further illustrates that the temperature gradient is still higher enough to exhibit the increasing trend for the heat transfer rate in the range of Ra = 104 − 105 even after the considerable amount of energy is utilized to overcome significant fluid friction irreversibilities (Sψ,total ) in the convection dominant region (Ra = 104 − 105 ). Hence, there is an increase in the heat transfer rate to the fluid (Nul ) for all ϕ s as shown in the top panel plot of Fig. 12(a). The distributions of Stotal , Beav and Nul for ϕ = 75° and 90° closely follow each other for Pr = 0.015 at all Ra range (see Fig. 12(a)). It is interesting to note that, the maximum heat transfer rate (Nul ) with the higher entropy generation (Stotal ) occurs for ϕ = 90° cavities at Ra = 105 (see the bottom and top panel plots of Fig. 12(a)). It is also found that, the total entropy generation (Stotal ) for ϕ = 90° is drastically increased compared to ϕ = 30° at Ra = 105 whereas the maximum average heat transfer rate (Nul ) for ϕ = 90° is not significantly improved compared to ϕ = 30° at Ra = 105 . Although the maximum heat transfer rate occurs at ϕ = 90°, the entropy generation is drastically increased due to the significant fluid friction irreversibilities and that further illustrates the considerable amount of exergy loss at ϕ = 90° (see bottom and top panel plots of Fig. 12(a)). Fig. 12(b) shows Stotal , Beav and Nub , for the Rayleigh–Benard convection case (case 2). The trend of Stotal , Beav and Nub is qualitatively similar to case 1. However, case 2 shows comparatively lower Stotal , higher Beav and lower Nub than case 1 at Ra ≤ 2 × 103 and higher Stotal , lower Beav and higher or lower Nub than case 1 at Ra ≥ 2 × 103 for all ϕ s (see Fig. 12(b)). This is due to the negligible active zones of the entropy generation with the fluid friction

43

(Sψ,total ) for case 2 compared to case 1 at Ra ≤ 2 × 103 as seen from Sψ maps of Figs. 5(a)–(d) and 9(a)–(d) and that further leads to higher Beav and lower Nub due to the lower thermal gradient at Ra ≤ 2 × 103 (see Fig. 12(b)). On the other hand, significant active zones of the entropy generation with the fluid friction (Sψ,total ) are observed for case 2 compared to case 1 at Ra ≥ 2 × 103 due to the high intense convection as seen from Sψ maps of Figs. 6(a)–(d) and 10(a)–(d) and that further leads to the lower Beav and higher or lower Nub at Ra ≥ 2 × 103 (see Fig. 12(b)). It is also interesting to note that, the very low Beav (0.23 ≤ Beav ≤ 0.12) at Ra = 105 for case 2 corresponds to the larger amount of the available energy (exergy) to overcome comparatively higher fluid friction irreversibilities than case 1 for all ϕ s (see bottom and middle panel plots of Fig. 12(b)). The distributions of Stotal , Beav and Nub for higher Pr fluids (Pr = 1000) in cases 1 and 2 are shown in Fig. 12(c)–(d). The characteristics of Stotal , Beav and Nub are qualitatively similar to those of Pr = 0.015. Due to the higher momentum diffusivity at higher Pr , Sψ,total for Pr = 1000 is comparatively higher than that for Pr = 0.015 and therefore, Stotal values are higher for Pr = 1000 at the convection dominant regime (Ra ≥ 104 ) (see bottom panel plots of Fig. 12(c)–(d) and 12(a)–(b)). Although the distributions of Beav are qualitatively similar for all ϕ s, the values of Beav at Ra = 105 for the present case (Pr = 1000) are lower than those for Pr = 0.015 case due to the high fluid friction irreversibility (Sψ,total ) at Pr = 1000 (see middle panel plots of Figs. 12(c)–(d) and 12(a)–(b)). The heat transfer rate is also comparatively higher for Pr = 1000 than that for Pr = 0.015 at the onset of the convection (Ra ≥ 104 ) and that occurs due to the larger temperature gradient (Nul or Nub ) which results in high heat transport at the onset of the convection (Ra ≥ 104 ) compared to the lost energy due to the fluid friction irreversibilities (Sψ,total ) along the left or bottom wall (see top panel plots of Figs. 12(c)–(d) and 12(a)–(b)). Overall, the rhombic cavities with ϕ = 30° configurations may be recommended for almost all model fluids (Pr = 0.015–1000) due to the minimum entropy generation values (Stotal ) despite the minimum heat transfer rate (Nul or Nub ) compared to square cavities during the convection dominant heat transfer in both the cases. It is interesting to note that, the conduction based static fluid solution is the unique characteristic of the square domain (ϕ = 90°). However, the initiation of convection in the square domain (ϕ = 90°) leads to the convection based dynamic solution. It may also be noted that the Rayleigh–Benard convection (case 2) is not the energy efficient compared to the differential heating case (case 1) at the onset of the convection. 4. Conclusion In the present study, the analysis of the entropy generation due to irreversibilities during natural convection in rhombic cavities has been carried out for various controlling parameters such as ϕ , Pr and Ra with the differential (case 1) and Rayleigh–Benard (case 2) heating situations. The conduction based static fluid solution as well as convection based dynamic solution is observed for ϕ = 90° for all Pr at Ra ≥ 2 × 103 in the Rayleigh–Benard convection. The dimensionless entropy generation due to heat transfer (Sθ ) and fluid friction (Sψ ) irreversibilities, which are the functions of the temperature (θ ) and flow fields (ψ ), respectively are obtained for both the heating situations. Important results of this study are summarized as follows: 4.1. Weak convection regime: 103 ≤ Ra ≤ 104

• The active zones of Sθ and Sψ are found to occur near the junction of the adiabatic–isothermal walls for ϕ = 30° in both

44

R. Anandalakshmi, T. Basak / European Journal of Mechanics B/Fluids 54 (2015) 27–44

the cases irrespective of Pr. At ϕ ≥ 30°, active zones of Sθ and Sψ move along the isothermal walls. In addition, Sψ active zones are also found to occur near adiabatic walls at ϕ ≥ 45° in both the cases irrespective of Pr. • Case 2 shows comparatively lower Stotal , higher Beav and lower Nub than case 1 at Ra ≤ 2 × 103 and higher Stotal , lower Beav and higher or lower Nub than case 1 at 2 × 103 ≤ Ra ≤ 104 for all ϕ s. • The conduction based static solution occurs only for ϕ = 90° and it is observed that the conduction based static solution disappears with a slight perturbation of ϕ at higher Ra (Ra ≥ 2 × 103 ) irrespective of Pr in case 2. 4.2. Dominant convection regime: 104 ≤ Ra ≤ 105

• The active zones of Sθ and Sψ are found to occur along isothermal walls for all ϕ s at Pr = 0.015 in both the cases.

•

•

•

•

In addition, active zones of Sψ are also found to occur near adiabatic walls of the cavity for all ϕ s irrespective of Pr in both the cases. Also, the intermediate regimes between the fluid layers of primary circulation cells also act as strong active zones of Sψ for all ϕ s in case 2 at lower Pr (Pr = 0.015). The comparison of magnitudes indicates that the maximum entropy generation due to heat transfer, Sθ,max decreases with Ra for ϕ ≤ 45° and that increases with Ra for ϕ ≥ 45° in case 1 for all Pr. In contrast to case 1, Sθ,max increases with Ra for all ϕ s in case 2 irrespective of Pr. The maximum entropy generation due to the fluid friction (Sψ,max ) is higher at Ra = 105 due to the enhanced fluid flow irrespective of heating situations, ϕ and Pr. Increase in Stotal with Ra is smaller for lower ϕ (ϕ = 30°) due to the negligible Sψ,total of Stotal and larger for higher ϕ (ϕ = 75°) due to the significant Sψ,total of Stotal at the onset of the convection (Ra ≥ 104 ) irrespective of Pr. The total entropy generation, Stotal , is found to be significantly low for ϕ = 30° in both the cases for all Pr at the high Ra. Analysis of the variation of Beav with Ra for high Pr fluids indicates that, the fluid friction irreversibility contributes significantly for the increase in Stotal . Smaller Beav at Ra ≥ 104 in case 2 illustrates that the maximum amount of the available energy (exergy) is needed to overcome the fluid friction irreversibilities irrespective of Pr and ϕ .

4.3. Overall conclusions

• The conduction dominant static fluid solution is the special characteristics of the square cavity for the Rayleigh–Benard convection. Thus to avoid the conduction based static solution and to get the enhanced convective heat transfer with the minimum loss of available energy, the rhombic cavity (ϕ ≤ 90°) may be an alternative geometrical design for the thermal processing of fluids with the vertical thermal gradient. • Overall, the rhombic cavities with ϕ = 30° configurations may be used for almost all model fluids (Pr = 0.015–1000) due to the minimum entropy generation values (Stotal ) despite the minimum heat transfer rate (Nul or Nub ) compared to square cavities at the convective heat transfer regime in both the cases. Current work attempts to analyze the optimal rhombic geometry design for the enhanced thermal processing of materials using the energy efficient approach. Although average heat transfer rate (Nu) is identical or larger for the Rayleigh–Benard convection (case 2) at specific ϕ s, Stotal is higher or Beav is lower for case 2 compared to the differential heating case (case 1) for any ϕ , especially at larger Ra. Therefore, the Rayleigh–Benard convection (case 2) is not energy efficient compared to the differential heating case (case 1) due to its large fluid friction irreversibilities compared to case 1 at identical parameters.

Acknowledgment The authors would like to thank the anonymous reviewer for critical comments which improved the quality of the manuscript. References [1] A.V. Kuznetsov, Non-oscillatory and oscillatory nanofluid bio-thermal convection in a horizontal layer of finite depth, Eur. J. Mech. B Fluids 30 (2011) 156–165. [2] A. Lazarovici, V. Volpert, J.H. Merkin, Steady states, oscillations and heat explosion in a combustion problem with convection, Eur. J. Mech. B Fluids 24 (2005) 189–203. [3] C. Karcher, Y. Kolesnikov, O. Andreev, A. Thess, Natural convection in a liquid metal heated from above and influenced by a magnetic field, Eur. J. Mech. B Fluids 21 (2002) 75–90. [4] S. Hussain, P.H. Oosthuizen, Numerical investigations of buoyancy-driven natural ventilation in a simple three-storey atrium building and thermal comfort evaluation, Appl. Therm. Eng. 57 (2013) 133–146. [5] K. Ardaneh, S. Zaferanlouei, A lumped parameter core dynamics model for MTR type research reactors under natural convection regime, Ann. Nucl. Energy 56 (2013) 243–250. [6] G. Authie, T. Tagawa, R. Moreau, Buoyant flow in long vertical enclosures in the presence of a strong horizontal magnetic field. Part 2. Finite enclosures, Eur. J. Mech. B Fluids 22 (2003) 203–220. [7] K. Kahveci, S. Oeztuna, MHD natural convection flow and heat transfer in a laterally heated partitioned enclosure, Eur. J. Mech. B Fluids 28 (2009) 744–752. [8] R. Anandalakshmi, T. Basak, Heat flow visualization analysis on natural convection in rhombic enclosures with isothermal hot side or bottom wall, Eur. J. Mech. B Fluids 41 (2013) 29–45. [9] A. Bejan, Study of entropy generation in fundamental convective heat transfer, J. Heat Transfer—Trans. ASME 101 (1979) 718–725. [10] A. Bejan, Entropy Generation Through Heat and Fluid Flow, John Wiley & Sons, New York, 1994. [11] A. Bejan, Entropy Generation Minimization, CRC Press, Boca Raton, 1996. [12] Y. Azoumah, P. Neveu, N. Mazet, Optimal design of thermochemical reactors based on constructal approach, AIChE J. 53 (2007) 1257–1266. [13] S. Lems, H.J. van der Kooi, J.D. Swaan Arons, Thermodynamic optimization of energy transfer in (bio)chemical reaction systems, Chem. Eng. Sci. 58 (2003) 2001–2009. [14] J.C. Slattery, P.G.A. Cizmas, A.N. Karpetis, S.B. Chambers, Role of differential entropy inequality in chemically reacting flows, Chem. Eng. Sci. 66 (2011) 5236–5243. [15] G. de Koeijer, R. Rivero, Entropy production and exergy loss in experimental distillation columns, Chem. Eng. Sci. 58 (2003) 1587–1597. [16] W.R. Dammers, M. Tels, Thermodynamic stability and entropy production in adiabatic stirred flow reactors, Chem. Eng. Sci. 29 (1974) 83–90. [17] H. Khorasanizadeh, M. Nikfar, J. Amani, Entropy generation of Cu–water nanofluid mixed convection in a cavity, Eur. J. Mech. B Fluids 37 (2013) 143–152. [18] Y. Aihara, A thermodynamic approach to the boundary layer flow system, Eur. J. Mech. B Fluids 20 (2001) 47–55. [19] H.F. Oztop, K. Al-Salem, A review on entropy generation in natural and mixed convection heat transfer for energy systems, Renewable Sustainable Energy Rev. 16 (2012) 911–920. [20] M. Saleem, M.A. Hossain, S. Mahmud, I. Pop, Entropy generation in Marangoni convection flow of heated fluid in an open ended cavity, Int. J. Heat Mass Transfer 54 (2011) 4473–4484. [21] B.M. Ziapour, R. Dehnavi, Finite-volume method for solving the entropy generation due to air natural convection in Gamma-shaped enclosure with circular corners, Math. Comput. Modelling 54 (2011) 1286–1299. [22] M. Bouabid, M. Magherbi, N. Hidouri, A. Ben Brahim, Entropy generation at natural convection in an inclined rectangular cavity, Entropy 13 (2011) 1020–1033. [23] M. Bouabid, N. Hidouri, M. Magherbi, A. Ben Brahim, Analysis of the magnetic field effect on entropy generation at thermosolutal convection in a square cavity, Entropy 13 (2011) 1034–1054. [24] M.A. Delavar, M. Farhadi, K. Sedighi, Effect of discrete heater at the vertical wall of the cavity over the heat transfer and entropy generation using Lattice Boltzmann method, Therm. Sci. 15 (2011) 423–435. [25] J.N. Reddy, An Introduction to the Finite Element Method, McGraw-Hill, New York, 1993. [26] S. Roy, T. Basak, Finite element analysis of natural convection flows in a square cavity with non-uniformly heated wall(s), Internat. J. Engrg. Sci. 43 (2005) 668–680. [27] G.G. Ilis, M. Mobedi, B. Sunden, Effect of aspect ratio on entropy generation in a rectangular cavity with differentially heated vertical walls, Int. Commun. Heat Mass Transfer 35 (2008) 696–703. [28] R. Anandalakshmi, T. Basak, Analysis of entropy generation due to natural convection in rhombic enclosures, Ind. Eng. Chem. Res. 50 (2011) 13169–13189. [29] D. Venturi, X.L. Wan, G.E. Karniadakis, Stochastic bifurcation analysis of Rayleigh–Benard convection, J. Fluid Mech. 650 (2010) 391–413.