A computational framework for natural convective hydromagnetic flow via inclined cavity: An analysis subjected to entropy generation

Journal of Molecular Liquids 287 (2019) 110863

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Journal of Molecular Liquids journal homepage: www.elsevier.com/locate/molliq

A computational framework for natural convective hydromagnetic flow via inclined cavity: An analysis subjected to entropy generation Seyyed Masoud Seyyedi a, A.S. Dogonchi a, M. Hashemi-Tilehnoee a, Zeeshan Asghar b, M. Waqas c,⁎, D.D. Ganji d a

Department of Mechanical Engineering, Aliabad Katoul Branch, Islamic Azad University, Aliabad Katoul, Iran Department of Mathematics and Statistics, The University of Lahore Gujrat Campus, Gujrat, 50700, Pakistan NUTECH School of Applied Sciences and Humanities, National University of Technology, Islamabad, 44000, Pakistan d Department of Mechanical Engineering, Babol Noshirvani University of Technology, Babol, Iran b c

a r t i c l e

i n f o

Article history: Received 22 December 2018 Received in revised form 20 April 2019 Accepted 25 April 2019 Available online 3 May 2019 Keywords: Natural convection Entropy generation ECOP CVFEM FVM MHD

a b s t r a c t One of the most interested and essential subjective in mechanical science is natural convection analysis in a cavity. Thus, natural convective flow and entropy generation are scrutinized numerically subjected to magnetic field effect in a semi-annulus enclosure which known as the effect of magneto-hydrodynamic (MHD). Firstly, the governing expressions are rewritten in non-dimensional form utilizing the definition of dimensionless parameters, stream function, and vorticity. Secondly, the entropy generation equation is expressed in non-dimensional form. Then governing non-dimensional expressions are computed by control finite element method (CVFEM). Furthermore, the governing expressions are computed employing finite volume method (FVM) utilizing ANSYS Fluent CFD code. A novel criterion for determination of thermal characteristics of cavity based on thermodynamics second relation is introduced that is called ecological coefficient of performance (ECOP). Flow and heat transport features in addition to entropy generation number are examined for distinct values of the Rayleigh number (Ra = 103,104,105), the orientation of the magnetic field (β = 0°,15°, 30°,45°, 60°,75°, 90°), and Hartmann number (Ha = 0,5,10, 15,20). For validation, the entropy generation number and average Nusselt number are compared with those available in literature and excellent agreement is observed. Isotherms and streamlines are calculated using CVFEM and FVM. Some correlations for entropy generation number are proposed. The results show that for constant Rayleigh number, the entropy generation number decays with increasing Hartmann number. Also, for each Hartmann number, there is an optimum inclination angle of magnetic field that gives a minimum for entropy generation number and a maximum for ECOP at each Rayleigh number. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Analysis of natural convective heat transportation in inclined cavities is a subject that has extensive utilizations in numerous engineering areas such as electronic equipment cooling, solar energy, heating and cooling of buildings and so on. A furthermore useful step in the investigation of these problems is the evaluation of entropy generation. Many researches are focused on cavity and entropy generation analyses in the past [1-32]. A square cavity subjected to top and bottom walls temperature investigated for natural convection and entropy generation by Yilbas et al. [1]. Here entropy generation rises via an increment in wall temperature. In 2000, entropy generation in an inclined cavity was elaborated by Baytas [2]. In 2002, heat transportation and fluid flow features in an enclosure by adiabatic straight and isothermal wavy walls were presented by Mahmud et al. [3]. In 2003, Magherbi et al. [4] scrutinized the impact of irreversibility distribution ratio versus maximal entropy ⁎ Corresponding author. E-mail address: [email protected] (M. Waqas).

https://doi.org/10.1016/j.molliq.2019.04.140 0167-7322/© 2019 Elsevier B.V. All rights reserved.

generation in a closed cavity. In 2005, the natural convection in a cavity considering two walls (horizontal straight, vertical wavy) was examined by Misirliogluin et al. [5]. Their study reveals that for larger estimations of Rayleigh number and moderate estimations of aspect ratio, the heat engendered in porous medium cannot be transported through porous medium from hot wall towards cold wall. In 2007, investigation of heat transfer inside a square cavity was performed numerically for heated thin plate placed vertically/horizontally by Kandaswamy et al. [6]. They determined that heat transport in the vertical situation is more than in a horizontal situation. In 2008, the influence of aspect ratio versus entropy generation inside a rectangular cavity was investigated by Ilis et al. [7]. Their work demonstrated that entropy generation firstly rises via an increment in aspect ratio, gets to a maximal value, and then decays. In 2008, Pirmohammadi et al. [8] formulated the naturalconvection laminar flow considering magnetic field inside a cavity heated from left and cooled from right. Their results indicated that the conduction heat transportation mechanism turns out to be dominant when Hartmann number is increased. Also, the heat transportation mechanisms and flow features inside cavity rely intensely upon the

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magnetic field strength and Rayleigh number. In 2009, the magnetic field effect on convection heat transportation within a tilted square enclosure was investigated by Pirmohammadi and Ghassemi [9]. They concluded that for a specific inclination angle, as the Hartmann number value rises, the convective heat transportation diminishes. In 2010, the effectiveness of an externally oriented magnetic field against entropy generation for a fixed cavity was considered by El Jery et al. [10]. Their results for lower Prandtl number values show that an increment in Hartmann number corresponds to a decay in magnitude of entropy generation. In 2011, Bouabid et al. [11] numerically analyzed entropy generation and natural convection in a rectangular cavity filled with air. In their work, vertical walls associated with cavities were at distinct constant temperatures whereas the cavities horizontal walls were adiabatic. In 2012, Esmaeilpour and Abdollahzadeh [12] modeled free convective nanofluid flow subjected to entropy generation in an enclosure considering distinct patterns of wavy vertical walls. In 2012, Cho et al. [13] numerically scrutinized the natural convective heat transportation characteristics of nanofluid inside an enclosed cavity. Their results show that for Rayleigh numbers in an extensive range of Rayleigh number, the average Nusselt number rises via larger nanoparticles volume fraction. In 2013, entropy generation impact in magneto nanofluid flow by rotating permeable disk was described by Rashidi et al. [14]. The prime intention of their research was to enable the researchers to utilize thermodynamics second relation effectively in computations of rotating fluidic structures. In 2014, the magnetic field impact in viscous material incompressible flow subjected to entropy generation was studied by Sanatan Das and Rabindra Nath Jana [15]. They concluded that entropy generation increases with an increment in magnetic parameter. In 2014, Shavik et al. [16] addressed entropy generation influence in natural convective laminar flow within an inclined square cavity. Their results illustrated that an increment in inclination angle, the entropy generation owing to fluid friction rises however entropy generation owing to heat transport and Bejan number diminishes. In 2015, the aspect ratio impact on natural convection inside a cavity with wavy walls was considered by Arici et al [17]. They modeled the cavity by ANSYS Fluent CFD code and concluded that increasing aspect ratio enhances the heat transport. Also, the inclination angle has also a very significant impact on the flow field and heat transport depending on aspect ratio and Rayleigh number. In 2015, Srinivasacharya and Hima Binduthe [18] calculated entropy generation owing to heat transport and fluid friction for steady and incompressible magneto-micropolar fluid flow inside a rectangular duct considering constant wall temperatures. In 2016, the entropy generation effectiveness in magneto-micropolar material flow between rotating concentric cylinders having infinite length was elaborated by Jangili et al. [19]. They determined that raising the effect of couple stress in micropolar fluid diminishes the velocity and increments micro-rotation and temperature. Influence of entropy generation in unsteady 3D laminar natural convective flow by cubical inclined trapezoidal cavity filled with air was examined by Hussein et al. [20]. Their outcomes witness that when Rayleigh number is increased, the flow structure changes specifically in 3D results and circulation of flow rises. Besides, the inclination angle impact versus entropy generation turn out to be insignificant for lower Rayleigh number. Moreover, by increasing Rayleigh number, the average Nusselt number also rises. In 2017, Shi et al. [21] modeled 2D natural convective flow subjected to air entropy generation considering quadrupole magnetic field. Their outcomes reveal that magneto buoyancy force has prospective utilizations for improving the heat transportation, however this leads to increment of entropy generation. Entropy generation and natural convection impacts in 3D cubical cavity having vertical cooled walls was analyzed by Rashed et al. [22]. They reported that rate of entropy generation rises when Rayleigh number rises however Bejan number diminishes when Rayleigh number is increased. Dogonchi and Ganji [23] modeled magneto-nanofluid flow by nonparallel plates. Their outcomes elaborate that an increment in Schmidt number corresponds to a rise in Nusselt number and thermal field

respectively. In another research, Dogonchi and Ganji [24] scrutinized MHD impact in nanofluid flow employing improved Fourier expression. Go-water nanoliquid flow and heat transportation in a porous channel considering radiation and MHD aspects were inspected by Dogonchi et al. [25]. They demonstrated that Nusselt number and thermal field are proportional to volume fraction and have an opposite relationship with radiation parameter. Dallaire and Gosselin [26] expressed the effect of density change and elastic wall on convection of a cavity through phase change materials. They implement the mathematical model in ANSYS Fluent CFD code. In 2018, Alsabery et al. [27] considered mixed convection in a square double lid-driven cavity subjected to waternanofluid containing a solid internal body. Marangoni convection influence inside a cubical cavity subject to nanoparticles was scrutinized by Sheremet and Pop [28]. The goal of the present research is to evaluate natural convection characteristics and the entropy generation in a semi-annulus enclosure using CVFEM and FVM subjected to magnetic field at different Rayleigh number. Few recent researches related to considered assumptions for formulation of problem in addition to employed simulation scheme are given in Refs. [33-50]. Besides, the advantages of this attempt are: (1) Introducing a new criterion for evaluation of cavity thermal performance that it is called ECOP. (2) Proposing new correlations for entropy generation number. (3) Comparing the capability of the CVFEM and FVM for modeling the natural convection inside a cavity by considering entropy generation. 2. Formulation The physical configuration and mesh of semi-annulus enclosure utilized in the current CVFEM program are illustrated in Fig. 1. The walls (inner, outer) are retained at constant temperatures Th and Tc (Tc b Th), respectively, whereas the other two walls are thermally insulated. ! ! Furthermore, it is supposed that uniform magnetic field ð B ¼ Bx e x þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ! By e y Þ is applied with a constant magnitude B0 ¼ B2x þ B2y , where e x ! and e y signify unit vectors in Cartesian system. The magnetic field orientation forms an angle β = tan−1(By/Bx) with horizontal axis. Electrically conducting fluid having Prandtl number 0.733 [8] is utilized to fill the cavity. The following suppositions are considered to formulate the problem (1) The flow is steady and laminar and fluid is incompressible. (2) The cavity is vertical and therefore gravitational force must be accounted. (3) It is supposed that Boussinesq approximation is valid. 3. Mathematical model 3.1. Basic governing equations ! ! The electric current density ðJ Þ and electromagnetic force ðF EM Þ are expressed as follows, respectively: ! ! ! ! J ¼σ E þV  B

ð1Þ

! ! ! ! ! ! ! F EM ¼ J  B ¼ σ E þ V  B  B

ð2Þ

! ! ! where E represents electrical field, V velocity vector and B uniform magnetic field.

S.M. Seyyedi et al. / Journal of Molecular Liquids 287 (2019) 110863

3

Fig. 1. Physical configuration.

It is supposed that electric field is negligible [10]. Therefore, Eq. (1) and (2) can be simplified as: ! ! ! J ¼σ V  B

ð3Þ

! ! ! ! ! ! F EM ¼ J  B ¼ σ V  B  B

ð4Þ

In our problem, the electromagnetic force must be projected with respect to x and y directions. Therefore, the governing equations, namely continuity, x-momentum, y-momentum and energy equations are presented as follows, respectively: ∂u ∂v þ ¼0 ∂x ∂y

ð5Þ

! 2 2 ∂u ∂u 1 ∂P ∂ u ∂ v þ ¼− þν u þv ρ ∂x ∂x ∂y ∂x2 ∂y2 2  σ B0 v sinβ cosβ−u sin2 β þ ρ

2

2

∂v ∂u ∂ψ ∂ψ − and u ¼ v¼− ð9Þ ∂x ∂y ∂y ∂x Also, non-dimensional variables and variables in this study are described as follows: ω¼

x y T−T c ωL2 ψ ;Ψ ¼ ;Ω ¼ X ¼ ;Y ¼ ;θ ¼ T h −T c α L L α

ð10Þ

The pressure term in the momentum equations (Eqs. 6 and 7) can be removed utilizing vorticity-stream function approach. Thus, using Eqs. (9) and (10), Eqs. (5)–(8) can be converted to non-dimensional forms, as follows: !  2 2 ∂Ψ ∂Ω ∂Ψ ∂Ω ∂ Ω ∂ Ω ∂θ − ¼ Pr þ þ RaPr ∂Y ∂X ∂X ∂Y ∂X ∂X2 ∂Y2 2

þHa2 Pr

ð6Þ

cos2 β

∂ Ψ 2

∂X

2

þ sin2 β

∂ Ψ ∂Y

2

þ 2 sinβ cosβ

! 2 ∂ Ψ ∂X∂Y

ð11Þ

! 2 2 ∂v ∂v 1 ∂P ∂ v ∂ v u þv þ ¼− þν ρ ∂y ∂x ∂y ∂x2 ∂y2  σ B20  u sinβ cosβ−v cos2 β þ gβT ðT−T c Þ þ ρ ∂T ∂T ∂ T ∂ T u þ þv ¼α ∂x ∂y ∂x2 ∂y2

The stream function (ψ) and vorticity (ω) are defined as follows:

 ð7Þ

∂Ψ ∂θ ∂Ψ ∂θ ¼ − ∂Y ∂X ∂X ∂Y

2

∂ θ ∂X2

2

þ

∂ θ

! ð12Þ

∂Y2

! ð8Þ

Table 2 Comparison between the values of average Nusselt number for this work and Refs. [8, 17]. Ra

Table 1 Test of grid for Nuave and Ngen when β = 60°, Ha = 20, Ra = 105 and Φ = 10−4.

10

Ha

3

Number of mesh

51 × 81

61 × 91

71 × 101

81 × 121

91 × 131

104

Nuave Ngen

5.113 125.69

5.184 128.71

5.225 131.34

5.275 137.16

5.290 138.82

105

0 0 10 0 25

Average Nusselt number Pirmohammadi et al. [8]

Arici et al. [17]

Present study

2.29 1.97 4.62 3.51

1.115 2.240 4.528 -

1.110 2.249 1.934 4.571 3.470

4

S.M. Seyyedi et al. / Journal of Molecular Liquids 287 (2019) 110863

Also, the vorticity in non-dimensional form is: 2

∂ Ψ 2

∂ X

Eq. (18), can be simplified as follows:

2

þ

∂ Ψ 2

∂ Y

¼ −Ω

ð13Þ

In Eq. (11), Ha, Pr and Ra are the Hartmann, Prandtl and Rayleigh numbers and these are expressed as follows:

Ha ¼ B0 L

rffiffiffiffiffiffi σ ν gβ ðT −T c ÞL3 ; Pr ¼ ; Ra ¼ T h να ρν α

ð14Þ

! 2 ! !! ! ! !! ! j −Q V E þV  B j V  B σ V ! B ≅ ¼ T T T 2 σ ðuB0 sinβ−vB0 cosβÞ ¼ T

By substitution Eq. (19) into Eq. (18) and using Eqs. (9), (10) and (14) the non- dimensional form of local entropy generation equation can be obtained as follows: " 2  2 # S_ gen ∂θ ∂θ ! #¼ þ ∂X ∂Y k ΔT 2

NL;gen ¼ "

The boundary conditions as presented in Fig. 1 are: 2

θ ¼ 1:0 on the inner circular boundary θ ¼ 0:0 on the outer circular boundary ∂θ ¼ 0 on two other insulation boundaries ∂n ψ ¼ 0 on all solid boundaries

Nulocal ¼ −ð∂θ=∂r Þ r¼r

ð16Þ

in

where r is the radial direction. The average Nusselt number on hot circular wall is evaluated as: Nuave ¼

2 π

Z

π=2 0

Nulocal ðζ Þdζ

ð17Þ

3.2. Entropy generation Exergy is elaborated as the maximum work that one can obtain from a flow of matter or energy. The demolished exergy is proportional to generated entropy. Exergy is usually destroyed in real procedures, moderately or totally, in line with thermodynamics second relation. The demolished exergy, or generated entropy, is accountable for in the less-than-ideal competences of systems or procedures [29]. With considering the magnetic force, as an external force, the entropy generation rate (derived from energy and entropy balances) is introduced by its general formula for a two- dimensional (2D) flow as follows [30]: " 2  2 # "  2  2  2 # k ∂T ∂T μ ∂u ∂v ∂u ∂v þ þ þ2 þ 2 þ S_ gen ¼ 2 T ∂x ∂y ∂x ∂y ∂y ∂x T !    ! ! ! ! j −Q V E þV  B þ T ð18Þ The local entropy generation Eq. (18) involves three terms, the first term is the local entropy generation due to the heat transfer, the second term is local entropy generation due to viscous dissipation, while the third term is local entropy generation due to the effect of the magnetic field (Joule heating or Ohmic heating). In Eq. (18) Q is the electric charge density and it is assumed that J ≫ QV [10], and therefore, the last term in

Ra

103 105

Ilis et al. [7]

Shavik et al. [16]

Present study

Ngen

Beave

Ngen

Beave

Ngen

Beave

1.20 23.5

0.96 0.2

1.150 23.27

0.970 0.194

1.134 23.20

0.973 0.215

2

∂ Ψ 2

∂Y

−

!2 3 2 ∂ Ψ 5 ∂X

2

 þ ΦHa2

∂Ψ ∂Ψ sinβ þ cosβ ∂Y ∂X

2

ð20Þ where NL, gen is the local entropy generation number (which is called the local dimensionless entropy generation, too) and Φ is the irreversibility distribution ratio, that Φ is defined as follows: Φ¼

μT 0  α 2 k LΔT

ð21Þ

where T0 is the mean temperature (T0 = (Th + Tc)/2) and ΔT is temperature difference in the cavity (ΔT = Th − Tc). The Eq. (20) can be rewritten as follows: NL;gen ¼ NL;HT þ NL;FF þ NL;MF

ð22Þ

The first term on the right-hand side of this equation denotes the local entropy generation number due to heat transfer irreversibility (NL, HT), the second term represents the local entropy generation number due to fluid friction irreversibility (NL, FF) and the third term represents the local entropy generation number due to magnetic field (NL, MF). Therefore, we have: "

∂θ ∂X

NL;HT ¼

2

2 # ∂θ þ ∂Y 

2

NL;FF

!2 2 ∂ Ψ ¼ Φ44 þ ∂X∂Y 

NL;MF ¼ ΦHa2

ð23Þ

2

∂ Ψ ∂Y2

−

!2 3 2 ∂ Ψ 5

ð24Þ

∂X2

∂Ψ ∂Ψ sinβ þ cosβ ∂Y ∂X

2 ð25Þ

The total entropy generation is obtained by integrating the local entropy generation over the system volume: Z NT;HT ¼

NL;HT dV

ð26Þ

NL;FF dV

ð27Þ

NL;MF dV

ð28Þ

V

Z NT;FF ¼ NT;MF ¼

Table 3 Comparison of the entropy generation number and the average Bejan number for this work versus Refs. [7, 16] when Pr = 0.71.

L

T 20

!2 2 ∂ Ψ þ þΦ44 ∂X∂Y

ð15Þ

The local Nusselt number along the hot wall can be expressed as:

ð19Þ

ZV

V

Table 4 Comparison between the values of average Nusselt number from CVFEM and FVM when Pr = 0.733, Ha = 0 at Ra = 105.

Nuave

CVFEM (FORTRAN code)

FVM (Fluent code)

5.81

5.87

S.M. Seyyedi et al. / Journal of Molecular Liquids 287 (2019) 110863

The sum of NT, HT, NT, FF and NT, MF is Ngen which is called entropy generation number. Ngen ¼ NT;HT þ NT;FF þ NT;MF

ð29Þ

3.3. Bejan number The local Bejan number indicates the strength of the entropy generation due to heat transfer irreversibility [4]: BeL ¼ NL;HT =NL;gen

ð30Þ

The average Bejan number is used to determine the relative importance of the heat transfer irreversibility for the entire cavity: R Beave ¼

A BeRL ðX; YÞdA A dA

ð31Þ

5

3.4. Ecological coefficient of performance (ECOP) Previously, the ecological coefficient of performance (ECOP) has been defined as a criterion for evaluation of the performance of extended surfaces (fins) by Seyyedi et al. [31]. In this section, ECOP for determination of the thermal performance of cavity is introduced. ECOP is based on the second law of thermodynamics and is defined as the ratio of the Nusselt number to entropy generation number. Bejan number only determines the contribution of heat transfer irreversibility with respect to total irreversibility. It does not give any information about the rate of heat transfer, whereas ECOP indicates indeed the ratio heat transfer rate to total irreversibility rate. The value of Bejan number is always between zero and one but, the ECOP starts from zero and does not have an upper limit. Increasing of the ECOP is desirable (this is similar to COP for a refrigerator, but COP is based on the first law of thermodynamics whereas ECOP is based on the second law of thermodynamics). ECOP can be defined as follows: ECOP ¼

Nuave Ngen

Fig. 2. Streamlines and isotherms contour distribution at Ra = 103 by CVFEM.

ð32Þ

6

S.M. Seyyedi et al. / Journal of Molecular Liquids 287 (2019) 110863

Fig. 3. Streamlines and isotherms contour distribution at Ra = 105 by CVFEM.

4. Numerical procedure

the computational results, the average Nusselt number on heat wall is calculated by the ANSYS Fluent code.

4.1. FVM by ANSYS Fluent 4.2. CVFEM by programing The control-volume-based commercial computational fluid dynamics (CFD) code ANSYS Fluent is utilized to solve the governing equations. The understudy non-dimensional problem is implemented in dimensional ANSYS Fluent CFD code by considering the dimensionless active parameters in post-processing. Rayleigh number is 105 and the Prandtl number is taken to be 0.733 that corresponds to that of air in very cold temperature. The Ra = 105 assures that the convective viscous flow should be laminar. The computational domain is generated by ICEM meshing module. Triangular elements are employed to form the computational grids. The pressure-velocity coupling is implemented by using the SIMPLE (Semi-Implicit Method for Pressure Linked Equations) algorithm. The PRESTO scheme computes the staggered-mesh pressure. The second-order upwind is adopted to deal with the discretization of spatial terms in the governing equations. The MHD model with potential method applied to equations at Ha = 10. In addition, the absolute convergence criteria are set to be 10−3 and 10−6 for velocity components and temperature, respectively. In order to check

The control volume finite element method (CVFEM) combines interesting characteristics from both the FVM and FEM (finite element method). In this study, a FORTRAN code is developed based on the Control Volume Finite Element Method (CVFEM) which introduced by Saabas and Baliga [32]. In this code, firstly Eqs. (11)–(13) are numerically solved for obtaining the vorticity, stream function, and dimensionless temperature. Then, Eq. (29) is used for calculating the entropy generation number. 4.3. Accuracy In order to perform a grid independence study, the average Nusselt number as a function of control volume number has been calculated for β = 60°, Ha = 20, Ra = 105, and Φ = 10−4. In addition, the entropy generation number has been calculated. Table 1 shows the test results of grid number. Regarding Table 1, the number of the grid was selected to be 81 × 121.

S.M. Seyyedi et al. / Journal of Molecular Liquids 287 (2019) 110863

Fig. 4. Streamlines, isotherms, total pressure, velocity vector, and local entropy generation contour at Ra = 105 by FVM.

7

8

S.M. Seyyedi et al. / Journal of Molecular Liquids 287 (2019) 110863

Fig. 5. Entropy generation number versus inclination angle of the magnetic field for different Hartmann number Ra = 103, (b) Ra = 104, (c) Ra = 105.

Fig. 6. ECOP versus inclination angle of the magnetic field for different Hartmann number Ra = 103, (b) Ra = 104, (c) Ra = 105.

4.4. Validation

5. Results and discussion

For validation of CVFEM code (that it has been written in FORTRAN), the average Nusselt number and the entropy generation number with different values of Ha and Ra are calculated and then compared with the results of the literature. As shown in Tables 2 and 3, the excellent agreement between these studies is observed. Table 4. presents the average Nusselt number that has been obtained by CVFEM (FORTRAN code) and FVM (ANSYS Fluent) where shows the satisfied domain grid.

5.1. Streamlines and isotherms contours In this section, streamlines and isotherms of natural convection are presented. Figs. 2 and 3 show contours for stream function and temperature distribution for Ha = 0 and Ha = 10 for β = 45° at Ra = 103, and Ra = 105, respectively. These figures are obtained by CVFEM method (FORTRAN code). One can detect that natural convection's intensity in the enclosure goes up with climbing Ra. So, it can be deduced that

Fig. 7. (a) Entropy generation number and (b) ECOP versus Hartmann number for β = 0° and β = 30° at =103, Ra = 104 and Ra = 105.

S.M. Seyyedi et al. / Journal of Molecular Liquids 287 (2019) 110863

9

Fig. 8. The comparison between the values of (NT, HT), (NT, FF) and (NT, MF) at Ra = 104 (a) for different orientation of the magnetic field at Ha = 5, (b) for different Hartmann number at β = 45°.

there is a positive correlation between Ra and |Ψmax|. Turning on the magnetic field suppresses the power of natural convection so that setting power of magnetic field at Ha = 10 leads to descending natural convection's power by 55.94% and 22.27% at Ra = 103 and Ra = 105, respectively. In fact, turning on the magnetic field causes to ascend the power of conduction mode of heat transfer. So, one can apply the magnetic field to control the natural convection's intensity in the engineering systems. Further, it can be discovered that by ascending the power

of natural convection which is the result of ascending Ra, the main core convection moves toward the vertical adiabatic wall and is also expanded. As also seen in these figures, isotherms have uniform shape at lower Ra. This is because of prevailing conduction mode of heat transfer in the system. However, at higher Ra the power of convection mode ascends in the system which causes to distort the isotherms. These treatments of isotherms are completely vice versa with Ha. That means the conduction mode's power ascends with ascending Ha, so isotherms

Fig. 9. Entropy generation number and ECOP versus Rayleigh number for β = 30° in two different Hartmann number.

Fig. 10. Curve fitting for entropy generation number versus inclination angle of the magnetic field (a) Ra = 103 (b) Ra = 105.

10

S.M. Seyyedi et al. / Journal of Molecular Liquids 287 (2019) 110863

Table 5 The proposed correlations for Ngen that are corresponding to Ha = 5, 10 at Ra = 103, 104, 105 when 0° ≤ β ≤ 90°. Ngen = a4β4 + a3β3 + a2β2 + a1β + a0 for 0° ≤ β ≤ 90°

Ra = 103 Ra = 104 Ra = 105

Ha = 5 Ha = 10 Ha = 5 Ha = 10 Ha = 5 Ha = 10

a4

a3

a2

a1

a0

No. of proposed correlations

-1.23E-09 1.06E-09 -1.99E-08 -9.28E-09 -2.36E-07 -3.22E-07

1.52E-07 -2.19E-07 4.10E-06 2.81E-06 3.77E-05 5.38E-05

1.85E-05 4.56E-05 5.35E-06 4.82E-04 9.44E-04 5.01E-03

-1.66E-03 -2.47E-03 -2.11E-02 -6.20E-02 -1.99E-01 -6.02E-01

1.32E+00 1.20E+00 7.57E+00 7.24E+00 1.26E+02 1.15E+02

Cor. (1) Cor. (2) Cor. (3) Cor. (4) Cor. (5) Cor. (6)

Fig. 11. Curve fitting for entropy generation number versus Hartmann number (a) Ra = 103 (b) Ra = 105.

will have more uniform shape. So, engineering and industrial apparatus can be controlled using this parameter in the viewpoint of heat transfer. Furthermore, the results of ANSYS Fluent CFD code are presented for Ha = 0 and Ha = 10 (for β = 45°) at Ra = 105 in Fig. 4. The figure shows the influences of Ra and Ha on the velocity and temperature distribution, and local entropy generation contour. It is obvious that the natural convection's intensity in the cavity increases with rising Ra. In addition, the magnetic field decreases the velocity of fluid flow in the cavity. Also, by increasing the Hartmann number the local entropy generation decreases. However, due to a vortex flow the entropy generation is distorted in the upper section of the enclosure (Fig. 4(j)).

5.2. Effects of active parameters In this section, the effects of Hartmann number, the orientation of the magnetic field and Rayleigh number are investigated in the cavity. Fig. 5 shows the entropy generation number versus the orientation of the magnetic field for different Hartmann number at Ra = 103, Ra = 104 and, Ra = 105. The figure shows that there is an orientation of the magnetic field that gives a minimum value for entropy generation number in each Hartmann number. β = 30° gives the minimum value for entropy generation number at Ra = 103 and Ra = 104 and β = 45° gives the minimum value for entropy generation number at Ra = 105. As it

Table 6 The proposed correlations for Ngen for different Ha and Ra when 0 ≤ Ha ≤ 20. Ngen = a4Ha4 + a3Ha3 + a2Ha2 + a1Ha + a0 for 0 ≤ Ha ≤ 20

Ra = 103 and β = 0° Ra = 103 and β = 15° Ra = 103 and β = 30° Ra = 103 and β = 45° Ra = 103 and β = 60° Ra = 103 and β = 75° Ra = 103 and β = 90° Ra = 104 and β = 0° Ra = 104 and β = 15° Ra = 104 and β = 30° Ra = 104 and β = 45° Ra = 104 and β = 60° Ra = 104 and β = 75° Ra = 104 and β = 90° Ra = 105 and β = 0° Ra = 105 and β = 15° Ra = 105 and β = 30° Ra = 105 and β = 45° Ra = 105 and β = 60° Ra = 105 and β = 75° Ra = 105 and β = 90°

a4

a3

a2

a1

a0

No. of proposed correlations

-7.13E-06 -8.31E-06 -8.37E-06 -7.96E-06 -7.31E-06 -5.97E-06 -3.72E-06 2.50E-05 -3.20E-06 -2.56E-05 -3.54E-05 -3.26E-04 -8.46E-04 -9.34E-04 -2.53E-04 -5.39E-04 1.21E-03 2.80E-03 2.52E-03 4.96E-03 6.93E-05

3.33E-04 3.71E-04 3.67E-04 3.50E-04 3.29E-04 2.83E-04 1.95E-04 -9.05E-04 5.29E-04 1.59E-03 2.08E-03 1.23E-02 3.08E-02 3.39E-02 1.63E-02 2.90E-02 -2.20E-02 -6.98E-02 -5.49E-02 -1.40E-01 1.35E-03

-4.48E-03 -4.58E-03 -4.32E-03 -4.13E-03 -4.10E-03 -3.87E-03 -3.08E-03 2.06E-03 -1.88E-02 -3.23E-02 -3.82E-02 -1.40E-01 -3.23E-01 -3.50E-01 -3.26E-01 -5.16E-01 -8.28E-02 3.46E-01 1.80E-01 1.07E+00 -1.41E-01

-2.25E-03 -6.61E-03 -9.73E-03 -9.98E-03 -7.24E-03 -3.24E-03 -1.23E-04 1.46E-02 2.61E-02 2.38E-02 1.99E-02 3.14E-01 8.69E-01 9.92E-01 3.12E-01 4.47E-01 -9.86E-01 -2.19E+00 -1.47E+00 -3.62E+00 8.33E-02

1.41E+00 1.41E+00 1.41E+00 1.41E+00 1.41E+00 1.41E+00 1.41E+00 7.54E+00 7.54E+00 7.54E+00 7.54E+00 7.54E+00 7.54E+00 7.54E+00 1.31E+02 1.31E+02 1.31E+02 1.31E+02 1.31E+02 1.31E+02 1.31E+02

Cor. (7) Cor. (8) Cor. (9) Cor. (10) Cor. (11) Cor. (12) Cor. (13) Cor. (14) Cor. (15) Cor. (16) Cor. (17) Cor. (18) Cor. (19) Cor. (20) Cor. (21) Cor. (22) Cor. (23) Cor. (24) Cor. (25) Cor. (26) Cor. (27)

S.M. Seyyedi et al. / Journal of Molecular Liquids 287 (2019) 110863

11

Fig. 12. Curve fitting for entropy generation number versus Rayleigh number for β = 30° (a) Ha = 5 (b) Ha = 10.

can be seen from the figure, the entropy generation number increases with increasing the Rayleigh number. Fig. 6 shows the ECOP versus the orientation of the magnetic field for different Hartmann number at Ra = 103, Ra = 104 and, Ra = 105. The figure shows that there is an orientation of the magnetic field that gives a maximum value for ECOP in each Hartmann number. β = 30° gives the maximum value for ECOP at Ra = 103 and Ra = 104 and β = 45° gives the maximum value for ECOP at Ra = 105. As can be seen from the figure, ECOP decreases when the Rayleigh number increases. Fig. 7 presents the entropy generation number and ECOP versus Hartmann number for β = 0° and β = 30° at Ra = 103, Ra = 104 and, Ra = 105. The figure shows that the entropy generation number decreases with increasing of Hartmann number at Ra = 103, Ra = 10 4 and for β = 0 ° at Ra = 10 5 while it has a minimum for β = 30° at Ra = 105. Also, the figure presents that the ECOP increases with increasing of Hartmann number at Ra = 103, and it gives a maximum for β = 30° at Ra = 105. Fig. 8 shows the contribution of the entropy generation number due to heat transfer irreversibility (NT, HT), the entropy generation number due to fluid friction irreversibility (NT, FF) and the entropy generation number due to magnetic field (NT, MF) in the total entropy generation number (a) for Ha = 5 and β = 15 ° , 30 ° , 45 ° , 60° at Ra = 104 and (b) for β = 45° and Ha = 5,10,15,20 at Ra = 104 (see. Eq. 29). As it can be seen in Fig. 8(a), the effect of orientation of the magnetic field on the entropy generation number can be neglected. Moreover, Fig. 8 (b) shows that the contribution of the entropy generation number due to the magnetic field (NT, MF) increases with increasing of Hartmann number while the contribution of the entropy generation number due to fluid friction irreversibility (NT, FF) decreases with increasing of Hartmann number. Also, Fig. 8(b) demonstrates that with increasing Hartmann number from 5 to 10, the contribution of the entropy generation number due to magnetic field increases from 5% to 18% (i.e. 13%) and with increasing Hartmann number from 15 to 20, the contribution of the entropy generation number due to magnetic field increases from 36% to 49% (i.e. 13%). Fig. 9 shows the entropy generation number and ECOP versus Rayleigh number for β = 30° in two different Hartmann number. The figure shows that with increasing of Rayleigh number the entropy generation number increases while ECOP decreases. As shown in Fig. 9(a), the

difference between the values of entropy generation number corresponding to Ha = 5 and Ha = 10 increases as Rayleigh number increases. 5.3. Proposed correlations for entropy generation number In this section, new correlations are proposed for calculation of entropy generation number. Fig. 10 shows the entropy generation number with respect to the orientation of the magnetic field and the 4th order polynomial that was fitted by using MATLAB software. Figs. 10(a) and (b) are corresponding to Ha = 5 at Ra = 103 and Ra = 105, respectively. Table 5 represents the proposed correlations that are for =103, 104, 5 10 , Ha = 5, 10. In this table 0° ≤ β ≤ 90°. Fig. 11 shows the entropy generation number with respect to Hartmann number and 4th order polynomial that was fitted by using MATLAB software. Figs. 11(a) and 11(b) are corresponding to Ra = 103 and Ra = 105. Table 6 represents the proposed correlations that are for Ra = 103, 4 10 , 105. In this table 0 ≤ Ha ≤ 20. Fig. 12 shows the entropy generation number with respect to Rayleigh number and 6th order polynomial that was fitted by using MATLAB software. Figs. 12(a) and (b) are corresponding to Ha = 5 and Ha = 10. The proposed correlations are summarized in Table 7. The importance of the proposed correlations is when a researcher wants to know the values of entropy generation number for this cavity and one does not have CVFEM code. Therefore, one can use from these correlations and obtains the entropy generation. Table 8 shows some validation (four selected cases) of the proposed correlations with CVFEM (FORTRAN code). 6. Conclusion In this study, a semi-annulus enclosure was investigated. The average Nusselt number and the entropy generation were obtained by CVFEM and FVM in the presence of a magnetic field where its strength is specified by the Hartmann number in the non-dimensional form. A criterion for evaluation of thermal performance of the cavity was introduced that was called ECOP. The effects of Hartmann number, orientation of the magnetic

Table 7 The proposed correlations for Ngen for Ha = 5 and Ha = 10 at β = 30° when 103 ≤ Ra ≤ 105. β = 30°

Ha = 5 Ha = 10

Ngen = a6Ra6 + a5Ra5 + a4Ra4 + a3Ra3 + a2Ra2 + a1Ra + a0 for 103 ≤ Ra ≤ 105 a6

a5

a4

a3

a2

a1

a0

No. of proposed correlations

-9.49E-29 -9.22E-30

3.03E-23 2.70E-24

-3.35E-18 -1.08E-19

9.69E-14 -5.60E-14

9.95E-09 1.10E-08

5.17E-04 4.12E-04

7.68E-01 7.38E-01

Cor. (28) Cor. (29)

12

S.M. Seyyedi et al. / Journal of Molecular Liquids 287 (2019) 110863

Table 8 The values of entropy generation number using proposed correlations and FORTRAN code. Case 1 (Table 5)

Case 2 (Table 6)

Case 3 (Table 6)

Case 4 (Table 7)

Ha = 10 and β = 52.3∘ at Ra = 105

Ha = 8.25 and β = 60∘ at Ra = 103

Ha = 18.2 and β = 15∘ at Ra = 105

Ha = 10 and β = 30∘ at Ra = 6.25 × 104

Cor. (6) 102.31

Fortran code 102.29

Error (%) 0.02

Cor. (11) 1.2202

Fortran code 1.2176

Error (%) 0.21

Cor. (22) 83.667

k L Nu P Pr Q Ra S_ gen

field and Rayleigh number on the average Nusselt number, the entropy generation number, the average Bejan number and the ECOP were investigated. New correlations for the entropy generation number were obtained. The contours of isothermal, streamlines, local entropy generation were plotted for some of the Hartmann numbers at a fixed Rayleigh number. The important results can be summarized as follows: 1. The results of the ANSYS Fluent CFD code are comparable with the results of the CVFEM FORTRAN code. However, ANSYS Fluent CFD code shows more details in the local entropy generation (Figs. 2–4). 2. No entropy is generated in the cavity center (Fig. 4). 3. There is an orientation of magnetic field that it gives a minimum value for the entropy generation number for each Hartmann number (Fig. 5). The entropy generation number increases with increasing the Rayleigh number whereas it decreases as Hartmann number increases. 4. There is an orientation of magnetic field that it gives a maximum value for the ECOP for each Hartmann number (Fig. 6). The ECOP decreases with increasing the Rayleigh number whereas it increases as Hartmann number increases. 5. The minimum value for the entropy generation number and the maximum value for the ECOP occur at Ha = 15 for β = 30° at Ra = 105 (Fig. 7). 6. The contribution of the entropy generation number due to the magnetic field (NT, MF) increases as Hartmann number increases (Fig. 8). 7. The entropy generation number increases with increasing the Rayleigh number whereas the behavior of the ECOP is in contrast with the entropy generation number (Fig. 9). 8. The proposed correlations for entropy generation number are validated with CVFEM FORTRAN code (Table 8).

T u,v ! V x, y X, Y

Fortran code 83.438

Error (%) 0.27

Cor. (29) 56.001

Fortran code 56.002

Error (%) −0.002

thermal conductivity (Wm−1K−1) length of the cavity (m) Nusselt number pressure (Nm−2) Prandtl number electric charge density (C m−3) Rayleigh number rate of entropy generation per unit volume (J s−1 K−1 m−3) temperature (K) dimensional x and y components of velocity (m s−1) velocity vector (ms−1) dimensional coordinates (m) dimensionless coordinates

Greek letter α thermal diffusivity (m2s−1) β Inclination angel (°) thermal expansion coefficient (K−1) βT θ dimensionless temperature μ dynamic viscosity (N s m−2) ν kinematic viscosity (m2s−1) Π lagrange shape function ρ density (Kg m−3) σ electrical conductivity (Ω/m) ΔT temperature difference Φ irreversibility distribution ratio ψ dimensional stream function (m2 s−1) Ψ dimensionless stream function ω dimensional vorticity (s−1) Ω dimensionless vorticity Subscripts ave average c cold EM electromagnetic FF fluid friction gen generation h hot HT heat transfer L local MF magnetic field

Nomenclature magnetic field strength B0 B magnetic field (T) Be Bejan number E electric field (V m−1) ECOP ecological coefficient of performance g gravitational acceleration (ms−2) Ha Hartmann number J current density (A m−2) Appendix A

Tables A.1–A.3 show the values of average Nusselt number, entropy generation number, average Bejan number, and ECOP at Ra = 103, Ra = 104, and Ra = 105, respectively. These values were obtained by CVFEM (FORTRAN code). The value of irreversibility ratio (Φ) is considered to be 10−4 [7,16]. Tables A.1 The values of average Nusselt number, entropy generation number, average Bejan number, and ECOP for different values of Hartmann number and inclination angle at Ra = 103. Ra = 103

Nuave

β

Ha

0 15 30

Ngen Ha

0

5

10

15

20

0

5

10

15

20

1.6289 1.6289 1.6289

1.576 1.566 1.5612

1.508 1.496 1.491

1.474 1.467 1.4647

1.4598 1.456 1.455

1.4082 1.4082 1.4082

1.3221 1.3018 1.2921

1.1995 1.1719 1.1616

1.1299 1.1098 1.1032

1.0958 1.0822 1.0778

S.M. Seyyedi et al. / Journal of Molecular Liquids 287 (2019) 110863

13

Tables A.1 (continued) Ra = 103

Nuave

β

Ha

Ngen Ha

0

5

10

15

20

0

5

10

15

20

45 60 75 90

1.6289 1.6289 1.6289 1.6289

1.5613 1.5666 1.5762 1.5883

1.4923 1.4983 1.5098 1.5268

1.4654 1.4690 1.4762 1.4883

1.4554 1.4574 1.4615 1.4688

1.4082 1.4082 1.4082 1.4082

1.2937 1.3060 1.3269 1.3526

1.1652 1.1817 1.2121 1.2564

1.1059 1.1179 1.1418 1.1818

1.0796 1.0876 1.1045 1.1352

Ra = 103

Beave

β

Ha

0 15 30 45 60 75 90

ECOP Ha

0

5

10

15

20

0

5

10

15

20

0.8899 0.8899 0.8899 0.8899 0.8899 0.8899 0.8899

0.9002 0.9077 0.9121 0.9127 0.9098 0.9038 0.8955

0.9250 0.9387 0.9451 0.9456 0.9407 0.9299 0.9130

0.948 0.9605 0.9657 0.9658 0.9611 0.9505 0.9320

0.9634 0.9734 0.9773 0.9772 0.9732 0.9641 0.9471

1.1567 1.1567 1.1567 1.1567 1.1567 1.1567 1.1567

1.1924 1.2032 1.2082 1.2068 1.1995 1.1878 1.1743

1.2568 1.2765 1.2838 1.2807 1.2680 1.2455 1.2152

1.3045 1.3219 1.3277 1.3251 1.3141 1.2929 1.2593

1.3322 1.3455 1.3498 1.3481 1.3400 1.3232 1.2939

Tables A.2 The values of average Nusselt number, entropy generation number, average Bejan number, and ECOP for different values of Hartmann number and inclination angle at Ra = 104. Ra = 104

Nuave

β

Ha

0 15 30 45 60 75 90

Ha

0

5

10

15

20

0

5

10

15

20

3.3 3.3 3.3 3.3 3.3 3.3 3.3

3.204 3.172 3.148 3.138 3.144 3.1654 3.197

2.94 2.839 2.769 2.741 2.759 2.821 2.922

2.594 2.434 2.333 2.3 2.405 2.697 2.82

2.255 2.078 1.983 1.966 2.031 2.269 2.483

7.543 7.543 7.543 7.543 7.543 7.543 7.543

7.57 7.268 7.037 6.925 6.952 7.1197 7.3997

7.24 6.423 5.884 5.644 5.734 6.195 7.00

6.437 5.333 4.7 4.46 5.76 8.802 10.764

5.42 4.275 3.717 3.602 4.012 6.283 9.096

Ra = 104

Beave

β

Ha

0 15 30 45 60 75 90

Ngen

ECOP Ha

0

5

10

15

20

0

5

10

15

20

0.4707 0.4707 0.4707 0.4707 0.4707 0.4707 0.4707

0.4055 0.419 0.4267 0.4313 0.4306 0.4212 0.4019

0.3746 0.3965 0.4139 0.428 0.4273 0.4083 0.3738

0.3785 0.4031 0.4284 0.4538 0.4423 0.3741 0.3384

0.3997 0.4291 0.4643 0.4998 0.5113 0.425 0.3593

0.4375 0.4375 0.4375 0.4375 0.4375 0.4375 0.4375

0.4232 0.4364 0.4473 0.4532 0.4523 0.4446 0.432

0.4060 0.4420 0.4706 0.4857 0.4812 0.4553 0.4174

0.40298 0.4564 0.4965 0.5154 0.4176 0.3064 0.2620

0.4161 0.4861 0.5335 0.5457 0.5062 0.3611 0.273

Tables A.3 The values of average Nusselt number, entropy generation number, average Bejan number, and ECOP for different values of Hartmann number and inclination angle at Ra = 105. Ra = 105

Nuave

β

Ha

0 15 30 45 60 75 90

Ha

0

5

10

15

20

0

5

10

15

20

5.808 5.808 5.808 5.808 5.808 5.808 5.808

5.759 5.746 5.738 5.736 5.741 5.752 5.766

5.629 5.581 5.549 5.54 5.559 5.6 5.651

5.433 5.327 5.233 5.115 5.075 5.362 5.478

5.182 4.945 5.098 5.204 5.275 5.254 5.257

130.54 130.54 130.54 130.54 130.54 130.54 130.54

125.82 123.17 121.54 121.27 122.40 124.7 127.64

114.79 107.04 102.45 101.44 104.14 110.36 119.3

103.98 91.75 83.95 81.71 91.29 93.72 108.09

96.12 78.81 94.86 114.76 137.16 155.33 97.62

Ra = 105

Beave

β

Ha

0 15 30

Ngen

ECOP Ha

0

5

10

15

20

0

5

10

15

20

0.1354 0.1354 0.1354

8.65e-2 8.82e-2 9.27e-2

7.51e-2 7.87e-2 8.51e-2

8.31e-2 9.11e-2 0.101

9.91e-2 0.1077 0.114

4.45e-2 4.45e-2 4.45e-2

4.58e-2 4.67e-2 4.72e-2

4.90e-2 5.21e-2 5.42e-2

5.22e-2 5.81e-2 6.23e-2

5.39e-2 6.27e-2 5.37e-2

(continued on next page)

14

S.M. Seyyedi et al. / Journal of Molecular Liquids 287 (2019) 110863

Tables A.3 (continued) Ra = 105

Beave

β

Ha

45 60 75 90

ECOP Ha

0

5

10

15

20

0

5

10

15

20

0.1354 0.1354 0.1354 0.1354

0.1 0.1134 0.1275 0.1325

9.45e-2 0.1083 0.1252 0.1343

0.111 0.1117 0.1305 0.1405

0.111 0.1106 0.1202 0.1347

4.45e-2 4.45e-2 4.45e-2 4.45e-2

4.73e-2 4.69e-2 4.61e-2 4.52e-2

5.46e-2 5.34e-2 5.07e-2 4.74e-2

6.26e-2 5.56e-2 5.72e-2 5.07e-2

4.53e-2 3.85e-2 3.38e-2 5.38e-2

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