# Effect of Inclined Magnetic Field on the Entropy Generation in an Annulus Filled with NEPCM Suspension

Authors: Seyyed Masoud Seyyedi; M. Hashemi-Tilehnoee; M. Sharifpur
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

The encapsulation technique of phase change materials in the nanodimension is an innovative approach to improve the heat transfer capability and solve the issues of corrosion during the melting process. This new type of nanoparticle is suspended in base fluids call NEPCMs, nanoencapsulated phase change materials. The goal of this work is to analyze the impacts of pertinent parameters on the free convection and entropy generation in an elliptical-shaped enclosure filled with NEPCMs by considering the effect of an inclined magnetic field. To reach the goal, the governing equations (energy, momentum, and mass conservation) are solved numerically by CVFEM. Currently, to overcome the low heat transfer problem of phase change material, the NEPCM suspension is used for industrial applications. Validation of results shows that they are acceptable. The results reveal that the values ofNuave descend with ascending Ha while Ngen has a maximum at Ha=16. Also, the value of NT,MF increases with ascending Ha. The values of Nuave and Ngen depend on nondimensional fusion temperature where good performance is seen in the range of 0.35<θf<0.6. Also, Nuave increases 19.9% and ECOP increases 28.8% whereas Ngen descends 6.9% when ϕ ascends from 0 to 0.06 at θf=0.5. Nuave decreases 4.95% while Ngen increases by 8.65% when Ste increases from 0.2 to 0.7 at θf=0.35.


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## Body

## 1. Introduction

Nanofluids can be produced by adding the nanosized particles (such as copper and silver) into the base fluids (i.e., oils, water, and so on). Many researchers scrutinized the effects of conventional nanofluids on fluid flow in different cases [1–14]. Recently, a new type of nanofluids has been constructed using phase change materials (PCMs) which are known as NEPCMs. As we know, PCMs have various applications such as energy-efficient buildings [15], cooling of electronic equipment [16], waste heat recovery [17], ventilation systems [18], and solar energy storage [19]. Although PCMs are used in many industrial applications, the low thermal conductivity of PCMs is a disadvantage for systems with charge and discharge cycles. Some techniques were proposed to overcome this problem, such as inserting the fins in the PCM enclosure [20, 21], carbon nanotubes in the PCM [22], and using the multilayer PCMs [23, 24]. Furthermore, microencapsulated phase change materials (MPCMs) are used for thermal energy storage [25].In 2004, the melting process of a PCM was analytically investigated by Hamdan and Al-Hinti [26], where a constant heat flux was imposed on the vertical wall. In 2016, the melting of PCM in a cylindrical medium was studied experimentally and numerically by Azad et al. [27]. In 2017, simulation of melting of a PCM for thermal energy storage was performed by Vikas et al. [28]. They modeled the melting of a rectangular PCM domain using ANSYS (Fluent). In 2017, a simplified model for melting of a PCM in the presence of radiation and natural convection was proposed by Souayfane et al. [29]. They found that natural convection has an important role during the PCM melting process. In 2018, Selimefendigil et al. [30] studied the free convection of CuO–water nanofluid in an enclosure where a PCM and a conductive partition were attached to its vertical wall by the finite element method. Hosseini et al. [31, 32] studied heat transfer in a horizontal shell-and-tube heat exchanger with a PCM. In 2018, characterizations of natural convection in vertical cylindrical shell-and-tube latent heat thermal energy storage (LHTES) systems were numerically and experimentally investigated by Seddegh et al. [33]. In 2018, Jmal and Baccar [34] studied the PCM solidification numerically where vertical fins were installed in a rectangular module. In 2019, Ghalambaz et al. [35] concluded that fusion temperature defines a dynamic behavior for NECPMs. In a similar work, Hajjar et al. [36] reached to the previous result in analyzing the transient fluid flow in a cavity filled by NEPCM. In 2020, Hasehmi-Tilehnoee et al. [37] studied natural convection and entropy generation in a complex medium filled with nanofluid and NEPCM suspension. They used ANSYS-Fluent to solve the nondimensional form of the governing equations. Different methods have been employed to solve the governing equations such as finite difference method (FDM) [38–40], finite element method (FEM) [41, 42], control volume finite element method (CVFEM) [43–48], and homotopy perturbation method (HPM) [49–51].In this work, the entropy generation and natural convection are scrutinized in a medium located between concentric horizontal wavy-circular wall and elliptical enclosure filled by NEPCMs suspension.Preference of this study regarding the earlier studies can be summarized as follows:(1)
A complex porous medium enclosure filled with NEPCM suspension is considered(2)
An inclined magnetic field is applied to the fluid flow(3)
The effects of decision parameters such asRa,Ha,β,Da,A,N,Ste,θf,ϕ,Nc,Nv, and λ (see the Nomenclature) are investigated on heat transfer and entropy generation(4)
The ratio of heat transfer to entropy generation is evaluated


## 2. Problem Definition

### 2.1. Physical Model

The porous medium shown in Figure1 consists of a heated horizontal wavy cylinder at Th that is concentrically placed inside an elliptic enclosure held at Tc. An external uniform magnetic field B0=Bx2+By21/2andβ=tan−1By/Bx has an effect on the fluid flow. The wavy wall of the inner cylinder can be implemented with the following equation:(1)r=rin+AcosNζ−ζ0.Figure 1
Schematic view of the considered problem.


### 2.2. Mathematical Model

#### 2.2.1. Dimensionless Governing Equations

The fluid flow is supposed to be steady, incompressible, two-dimensional, and laminar with no radiation effect. The fluid properties are constant but Boussinesq approximation is applied to the density in the buoyancy term. The dimensionless conservation equations are as follows [14, 35]:(2)∂2Ψ∂X2+∂2Ψ∂Y2=−Ω,∂Ψ∂Y∂Ω∂X−∂Ψ∂X∂Ω∂Y=μb/μfρb/ρfPr∂2Ω∂X2+∂2Ω∂Y2+βT,bβT,fRaPr∂θ∂X−μb/μfρb/ρfPrDaΩ+σb/σfρb/ρfHa2Prfcos2β∂2Ψ∂X2+sin2β∂2Ψ∂Y2+2sinβcosβ∂2Ψ∂X∂Y,∂Ψ∂Y∂θ∂X−∂Ψ∂X∂θ∂Y=kb/kfCr∂2θ∂X2+∂2θ∂Y2,where Pr, Ra, Da, Ha, and Cr are given as follows:(3)Ra=gβfL3Th−Tcνfαf,Pr=νfαf,Ha=B0Lσfρfνf,Da=KL2,Cr=ρCpbρCpf,where(4)kbkf=1+Ncϕ,(5)σbσf=1+Nsϕ,(6)μbμf=1+Nvϕ.The heat capacity ratioCr can be rewritten in the nondimensional form as follows [35]:(7)Cr=ρCpbρCpf=1−ϕ+ϕλ+ϕδStef.Also,f is the nondimensional fusion function defined by the following equation [52]:(8)f=π2sinπδθ−θf+δ2×0,θ<θf−δ2,1,θf−δ2<θ<θf+δ2,0,θf+δ2<θ.The boundary conditions of the system are as follows:(9)Ψ=0andθ=1.0on the inner boundary,Ψ=0andθ=0.0on the outer boundary.The average Nusselt number can be calculated using the local Nusselt number along the hot wall as follows:(10)Nulocal=−kbkf∂θ∂rr=R,Nuave=12π∫02πNulocalζdζ.


#### 2.2.2. Entropy Generation

The performance of energy systems has been evaluated by equation (11) which is named as the rate of entropy generation as follows:(11)S˙gen=kbfT02∂T∂x2+∂T∂y2+μbfT02∂u∂x2+2∂v∂y2+∂u∂y+∂v∂x2+μbu2+v2KT0+σbB02usinβ−vcosβ2T0.The local entropy generation in nondimensional form can be rewritten as follows [14]:(12)NL,gen=S˙genkf/T02ΔT/L2=kbkf∂θ∂X2+∂θ∂Y2︸NL,HT+μbμfΦf4∂2Ψ∂X∂Y2+∂2Ψ∂Y2−∂2Ψ∂X22︸NL,FF+1DaμbμfΦf∂Ψ∂X2+∂Ψ∂Y2︸NL,PM+σbσfHa2Φf∂Ψ∂Ysinβ+∂Ψ∂Xcosβ2︸NL,MF.The total entropy generation and the entropy generation numberNgen can be obtained, respectively, as follows:(13)NT,j=∫VNL,jdV,Ngen=∑jNT,j,where j denotes the HT, FF, PM, and MF. For more details, see Refs. [13, 14].Furthermore, the ecological coefficient of performance (ECOP) is calculated by the following [53]:(14)ECOP=NuaveNgen.


## 2.1. Physical Model

The porous medium shown in Figure1 consists of a heated horizontal wavy cylinder at Th that is concentrically placed inside an elliptic enclosure held at Tc. An external uniform magnetic field B0=Bx2+By21/2andβ=tan−1By/Bx has an effect on the fluid flow. The wavy wall of the inner cylinder can be implemented with the following equation:(1)r=rin+AcosNζ−ζ0.Figure 1
Schematic view of the considered problem.


## 2.2. Mathematical Model

### 2.2.1. Dimensionless Governing Equations

The fluid flow is supposed to be steady, incompressible, two-dimensional, and laminar with no radiation effect. The fluid properties are constant but Boussinesq approximation is applied to the density in the buoyancy term. The dimensionless conservation equations are as follows [14, 35]:(2)∂2Ψ∂X2+∂2Ψ∂Y2=−Ω,∂Ψ∂Y∂Ω∂X−∂Ψ∂X∂Ω∂Y=μb/μfρb/ρfPr∂2Ω∂X2+∂2Ω∂Y2+βT,bβT,fRaPr∂θ∂X−μb/μfρb/ρfPrDaΩ+σb/σfρb/ρfHa2Prfcos2β∂2Ψ∂X2+sin2β∂2Ψ∂Y2+2sinβcosβ∂2Ψ∂X∂Y,∂Ψ∂Y∂θ∂X−∂Ψ∂X∂θ∂Y=kb/kfCr∂2θ∂X2+∂2θ∂Y2,where Pr, Ra, Da, Ha, and Cr are given as follows:(3)Ra=gβfL3Th−Tcνfαf,Pr=νfαf,Ha=B0Lσfρfνf,Da=KL2,Cr=ρCpbρCpf,where(4)kbkf=1+Ncϕ,(5)σbσf=1+Nsϕ,(6)μbμf=1+Nvϕ.The heat capacity ratioCr can be rewritten in the nondimensional form as follows [35]:(7)Cr=ρCpbρCpf=1−ϕ+ϕλ+ϕδStef.Also,f is the nondimensional fusion function defined by the following equation [52]:(8)f=π2sinπδθ−θf+δ2×0,θ<θf−δ2,1,θf−δ2<θ<θf+δ2,0,θf+δ2<θ.The boundary conditions of the system are as follows:(9)Ψ=0andθ=1.0on the inner boundary,Ψ=0andθ=0.0on the outer boundary.The average Nusselt number can be calculated using the local Nusselt number along the hot wall as follows:(10)Nulocal=−kbkf∂θ∂rr=R,Nuave=12π∫02πNulocalζdζ.


### 2.2.2. Entropy Generation

The performance of energy systems has been evaluated by equation (11) which is named as the rate of entropy generation as follows:(11)S˙gen=kbfT02∂T∂x2+∂T∂y2+μbfT02∂u∂x2+2∂v∂y2+∂u∂y+∂v∂x2+μbu2+v2KT0+σbB02usinβ−vcosβ2T0.The local entropy generation in nondimensional form can be rewritten as follows [14]:(12)NL,gen=S˙genkf/T02ΔT/L2=kbkf∂θ∂X2+∂θ∂Y2︸NL,HT+μbμfΦf4∂2Ψ∂X∂Y2+∂2Ψ∂Y2−∂2Ψ∂X22︸NL,FF+1DaμbμfΦf∂Ψ∂X2+∂Ψ∂Y2︸NL,PM+σbσfHa2Φf∂Ψ∂Ysinβ+∂Ψ∂Xcosβ2︸NL,MF.The total entropy generation and the entropy generation numberNgen can be obtained, respectively, as follows:(13)NT,j=∫VNL,jdV,Ngen=∑jNT,j,where j denotes the HT, FF, PM, and MF. For more details, see Refs. [13, 14].Furthermore, the ecological coefficient of performance (ECOP) is calculated by the following [53]:(14)ECOP=NuaveNgen.


## 2.2.1. Dimensionless Governing Equations

The fluid flow is supposed to be steady, incompressible, two-dimensional, and laminar with no radiation effect. The fluid properties are constant but Boussinesq approximation is applied to the density in the buoyancy term. The dimensionless conservation equations are as follows [14, 35]:(2)∂2Ψ∂X2+∂2Ψ∂Y2=−Ω,∂Ψ∂Y∂Ω∂X−∂Ψ∂X∂Ω∂Y=μb/μfρb/ρfPr∂2Ω∂X2+∂2Ω∂Y2+βT,bβT,fRaPr∂θ∂X−μb/μfρb/ρfPrDaΩ+σb/σfρb/ρfHa2Prfcos2β∂2Ψ∂X2+sin2β∂2Ψ∂Y2+2sinβcosβ∂2Ψ∂X∂Y,∂Ψ∂Y∂θ∂X−∂Ψ∂X∂θ∂Y=kb/kfCr∂2θ∂X2+∂2θ∂Y2,where Pr, Ra, Da, Ha, and Cr are given as follows:(3)Ra=gβfL3Th−Tcνfαf,Pr=νfαf,Ha=B0Lσfρfνf,Da=KL2,Cr=ρCpbρCpf,where(4)kbkf=1+Ncϕ,(5)σbσf=1+Nsϕ,(6)μbμf=1+Nvϕ.The heat capacity ratioCr can be rewritten in the nondimensional form as follows [35]:(7)Cr=ρCpbρCpf=1−ϕ+ϕλ+ϕδStef.Also,f is the nondimensional fusion function defined by the following equation [52]:(8)f=π2sinπδθ−θf+δ2×0,θ<θf−δ2,1,θf−δ2<θ<θf+δ2,0,θf+δ2<θ.The boundary conditions of the system are as follows:(9)Ψ=0andθ=1.0on the inner boundary,Ψ=0andθ=0.0on the outer boundary.The average Nusselt number can be calculated using the local Nusselt number along the hot wall as follows:(10)Nulocal=−kbkf∂θ∂rr=R,Nuave=12π∫02πNulocalζdζ.


## 2.2.2. Entropy Generation

The performance of energy systems has been evaluated by equation (11) which is named as the rate of entropy generation as follows:(11)S˙gen=kbfT02∂T∂x2+∂T∂y2+μbfT02∂u∂x2+2∂v∂y2+∂u∂y+∂v∂x2+μbu2+v2KT0+σbB02usinβ−vcosβ2T0.The local entropy generation in nondimensional form can be rewritten as follows [14]:(12)NL,gen=S˙genkf/T02ΔT/L2=kbkf∂θ∂X2+∂θ∂Y2︸NL,HT+μbμfΦf4∂2Ψ∂X∂Y2+∂2Ψ∂Y2−∂2Ψ∂X22︸NL,FF+1DaμbμfΦf∂Ψ∂X2+∂Ψ∂Y2︸NL,PM+σbσfHa2Φf∂Ψ∂Ysinβ+∂Ψ∂Xcosβ2︸NL,MF.The total entropy generation and the entropy generation numberNgen can be obtained, respectively, as follows:(13)NT,j=∫VNL,jdV,Ngen=∑jNT,j,where j denotes the HT, FF, PM, and MF. For more details, see Refs. [13, 14].Furthermore, the ecological coefficient of performance (ECOP) is calculated by the following [53]:(14)ECOP=NuaveNgen.


## 3. Numerical Procedure

A FORTRAN code based on the control volume finite element method (CVFEM) has been extended to solve the governing equations.


### 3.1. Grid Test

The results of the grid independence study are provided in Table1. The calculations have been performed with Ra=105, Pr=6.2, Da=10−2, ϕ=0.05, Nv=Nv=3δ=0.05, Ste=0.313, θf=0.3, λ=0.4, Ha=30, β=60°, A=0.3,N=6, and Φ=10−4. Hence, 81 × 971 can be considered as the grid size.Table 1
The effect of grid size on the maximum absolute value of stream function, the average Nusselt number, and the entropy generation number.
31×37141×49151×61161×73171×85181×97191×1091Ψmax11.5010.9610.8810.3410.2510.1810.16Nuave3.93293.85803.80123.75233.73913.73453.7342Ngen38.1937.2337.0236.9836.7136.6936.68


### 3.2. Validation

In order to validate the results, two validations are performed. Firstly, the isotherms, streamlines, andCr contour for a square cavity are obtained for θf=0.5 at Ra=105. Figure 2 shows the results of the present code and those of Ghalambaz et al. [35]. Secondly, the values of average Nusselt number are calculated for several cases that Table 2 presents the outcomes. Other parameters are constant such as ϕ=0.05, λ=0.4, ρp/ρf=0.9, and δ=0.05. For both validations, the results are acceptable.Figure 2
The isotherms, streamlines, and Cr contour atθf=0.5 and Ra=105 for (a) Ghalambaz et al. [35] and (b) present work Crmax=5.988andCrmin=0.97.
(a)(b)Table 2
The comparison ofNuave between the Ghalambaz et al. [35] and present work.
No.θfSteNcNvNuaveGhalambaz et al. [35]Present workError (%)10.30.313335.19325.22460.620.30.313665.44775.47050.430.10.313334.95504.96850.340.30.2335.35415.36520.2


## 3.1. Grid Test

The results of the grid independence study are provided in Table1. The calculations have been performed with Ra=105, Pr=6.2, Da=10−2, ϕ=0.05, Nv=Nv=3δ=0.05, Ste=0.313, θf=0.3, λ=0.4, Ha=30, β=60°, A=0.3,N=6, and Φ=10−4. Hence, 81 × 971 can be considered as the grid size.Table 1
The effect of grid size on the maximum absolute value of stream function, the average Nusselt number, and the entropy generation number.
31×37141×49151×61161×73171×85181×97191×1091Ψmax11.5010.9610.8810.3410.2510.1810.16Nuave3.93293.85803.80123.75233.73913.73453.7342Ngen38.1937.2337.0236.9836.7136.6936.68


## 3.2. Validation

In order to validate the results, two validations are performed. Firstly, the isotherms, streamlines, andCr contour for a square cavity are obtained for θf=0.5 at Ra=105. Figure 2 shows the results of the present code and those of Ghalambaz et al. [35]. Secondly, the values of average Nusselt number are calculated for several cases that Table 2 presents the outcomes. Other parameters are constant such as ϕ=0.05, λ=0.4, ρp/ρf=0.9, and δ=0.05. For both validations, the results are acceptable.Figure 2
The isotherms, streamlines, and Cr contour atθf=0.5 and Ra=105 for (a) Ghalambaz et al. [35] and (b) present work Crmax=5.988andCrmin=0.97.
(a)(b)Table 2
The comparison ofNuave between the Ghalambaz et al. [35] and present work.
No.θfSteNcNvNuaveGhalambaz et al. [35]Present workError (%)10.30.313335.19325.22460.620.30.313665.44775.47050.430.10.313334.95504.96850.340.30.2335.35415.36520.2


## 4. Results and Discussion

In this paper, the second law of thermodynamics and the free convection heat transfer are numerically studied in a medium between concentric horizontal wavy-circular and elliptical cylinders loaded with a dilute suspension by CVFEM. The effects of nondimensional parameters on the characteristics of the flow and entropy generation number are considered. The effect of an external inclined magnetic field on the fluid flow is specified by Hartmann number (Ha) and angle of the magnetic fieldβ. Darcy number (Da) is used for modeling porous medium. Some decision variables are employed for modeling of NEPCM such as ϕ, λ, δ, Ste, Nv, Nv,Ns, and θf. Also, the geometry of enclosure is specified by AandN. The default values of the nondimensional parameters for calculations are presented in Table 3.Table 3
The default values of the nondimensional parameters.
ANRaHaβDaΦPrθfϕλSteδNcNvNsρp/ρfϕ×βp/βf0.261052030°10−210−46.20.50.050.40.3130.053310.90Figure3 shows the isotherms, the streamlines, and Cr contours at Ra=103,104, and 105. The power of the flow increases with the Rayleigh number as it is obvious from the maximum value of the absolute stream function, Ψmax. The values of Ψmax are 0.44, 3.86, and 16.93 for Ra=103,104, and 105, respectively. The isotherms are concentric circles around the hot wall at Ra=103. The convective heat transfer is predominant in this case, while the role of convection in heat transfer becomes more significant with increasing the Rayleigh number. The red lines in isotherms go up from the upper section of the inner wall to the outer wall at Ra=105. The average Nusselt number ascends with ascending the Rayleigh number where the values of Nuave are 2.5784, 2.9136, and 5.8065, for Ra=103,104, and 105, respectively.Figure 3
The streamlines, the isotherms, and Cr contour at Ra = 103, 104, and 105.The last row in this figure presents theCr contours for each Rayleigh number. According to equations (7) and (8), Cr has a minimum value that is corresponding to f=0 (other parameters are assumed to be constant). For example, for ϕ=0.05, λ=0.4, δ=0.05, and Ste=0.313, the minimum value of Cr is 0.97 (when f=0). Also, Cr has a maximum value that depends on θf in the f correlation. Here, the maximum value of Cr is 5.988 (this value can be obtained by solving the velocity and temperature fields). The values of 0.97 and 5.988 are the same for each value of the Rayleigh number in this figure. Notice that the heat capacity changes since the phase change occurs. Other decision parameters for this figure are Ha=15, Da=10, θf=0.65, β=0°, and A=0.1.Figure4 presents the effects of the magnetic field and its angle on the entropy generation number and the average Nusselt number. Figure 4(a) shows that Nuave decreases with ascending Ha while Ngen has a maximum at Ha=16. For instance, Nuave descends from 5.5496 to 3.6542 (34.15% decreasing) when Ha increases from 0 to 40. Also, Ngen increases from 42.60 to 44.06 (3.43% increasing) when Ha increases from 0 to 16 while it decreases from 44.06 to 35.43 (19.59% decreasing) when Ha increases from 16 to 40. This figure indicates that increasing the Hartmann number is desirable after Ha=16. Figure 4(b) shows that Nuave firstly ascends, attains a maximum value, and then descends with ascending the angle of the magnetic field. The maximum value of Nuave is 4.9214, which occurs at β=60°. On the other hand, Ngen decreases continuously from 44.03 to 33.36 (24.23% decreasing) when β increases from 0° to 90°. Other decision parameters were assumed to be θf=0.6, β=0°, A=0.1,andN=8 for Figure 4(a) while they are Ha=10, Ste=0.5, θf=0.6, ϕ=0.04, A=0.15,andN=4 without porosity for Figure 4(b).Figure 4
The variations ofNuave and Ngen versus (a) the Hartmann number and (b) the angle of magnetic field.
(a)(b)Figure5 presents variations of the entropy generation number and the average Nusselt number versus the fusion temperature of the NEPCM core θf at Ra=105 and ϕ=0.05. It shows approximately a regular symmetrical behavior for θf=0.5 due to the symmetry of both the boundary conditions and geometry. The variations of the average Nusselt number are very small in the range of 0.3<θf<0.65, and there is a local minimum at θf=0.5. The maximum value of Nuave is 3.268 that occurs at θf=0.4 and θf=0.6. Besides, the entropy generation number firstly decreases to a minimum value and then goes up as the fusion temperature increases. The minimum value of Ngen is 39 that occurs at θf=0.5 although it is approximately constant in the range of 0.35<θf<0.6. This figure discovers that the best range of θf is 0.35<θf<0.6 from both second law analysis and heat transfer point of view. Other decision parameters were selected as Ha=30, β=0°, and A=0.3.Figure 5
The variations ofNuave and Ngen versus the fusion temperature.Figure6(a) demonstrates the variations of Nuave and Ngen versus Ste while the values of Ψmax and ECOP are calculated and plotted in Figure 6(b) as a function of Ste. It is obvious from the figure that Nuave decreases with ascending the values of Ste whilst the values of Ngen ascend as the values of Ste go up. Regarding the definition of the Stefan number Ste, which is the ratio of the sensible heat to the latent heat of the PCM, the increasing Ste means descending the PCM core latent heat. The latter leads to a decrease in the heat storage capacity of the NEPCM particles, which concluded a lower rate of heat transfer. For instance, Nuave ascends from 4.6998 to 4.4673 (4.95% decreasing) while Ngen increases from 41.24 to 44.81 (8.65% increasing) when Ste increases from 0.2 to 0.7 at θf=0.35. Figure 6(b) illustrates that the values of Ψmax increase while the ECOP descends with ascending Ste. Therefore, the figure discovers that the heat transfer rate decreases although the fluid velocity enhances with increasing the Stefan number. Also, Ψmax increases 9.13% while the ECOP decreases 12.54% when Ste increases from 0.2 to 0.7. Other decision parameters for this figure are Ha=22, β=0°, Nc=6, and A=0.15.Figure 6
(a) The variations ofNuave and Ngen versus Stefan number and (b) the variations of ψmax and ECOP versus Stefan number.
(a)(b)Figure7 represents the variations of Nuave, Ngen, Ψmax, and ECOP versus ϕ. As shown, increasing ϕ is desirable since Nuave and ECOP raise and Ngen decreases with ascending ϕ. Albeit, the values of Ψmax decrease as ϕ increases. For example, Nuave increases from 5.4591 to 6.5446 (19.9% increasing) and ECOP ascends from 0.1273 to 0.1640 (28.8% amplified) while Ψmax decreases from 17.24 to 16.34 (5.2% decreasing) and Ngen decreases from 42.88 to 39.89 (6.9% decreasing) when ϕ increases from 0 to 0.06. Regarding equation (6), decreasing the values of Ψmax is justifiable since the viscosity of nanofluid increases when the values of ϕ go up. Other decision parameters were assumed to be Ha=18, β=60°, Nc=6, Da=10, Ste=0.2, A=0.1, and N=5.Figure 7
(a) The variations ofNuave and Ngen versus volume fraction of nanoparticles and (b) the variations of ψmax and ECOP versus volume fraction of nanoparticles.
(a)(b)Figure8(a) presents variations of Nuave and Ngen versus the Darcy number while Figure 8(b) shows variations of Ψmax and ECOP against the Darcy number. From the figure, Nuave, Ψmax, and ECOP ascend while the values of Ngen descend with ascending the Darcy number. For instance, Nuave increases from 6.0034 to 6.6509 (10.8% increasing), Ψmax increases from 15.42 to 19.46 (26.2% increasing), and ECOP ascends from 0.1270 to 0.1428 (12.4% amplified) while Ngen decreases from 47.26 to 46.57 (1.5% decreasing) when Darcy number ascends from 10−2 to 102 at θf=0.45. Other decision parameters are Ha=10, β=35°, Nc=6, A=0.1, and N=4.Figure 8
(a) The variations ofNuave and Ngen versus Darcy number and (b) the variations of ψmax and ECOP versus Darcy number.
(a)(b)Figure9 depicts the streamlines, the isotherms, and the Cr contours for different values of amplitude A=0.1,0.2,and0.3. The streamlines follow the shape of the inner wall, especially near the hot wall. The flow velocity decreases with increasing the amplitude parameter as it can be understood from the values of Ψmax. It can be due to this fact that the available space for fluid circulation decreases with increasing the values of A. The rate of heat transfer can be reflected by the shape of isotherms, where Nuave determines the value of it. The values of Nuave are 4.7929, 4.1158, and 3.4985 for A=0.1,0.2,and0.3, respectively. It is obvious from the figure that the values of Nuave descend with ascending the values of A. Here, it can be stated that increasing the A values is similar to increasing the Ra values. Also, it should be noticed that the values of Ngen are 41.01, 40.86, and 40.31 for A=0.1,0.2,and0.3, respectively. Therefore, increasing the values of amplitude is desirable from the second law of thermodynamics viewpoint. The last row of this figure demonstrates the Cr contours where the phase change region is specified by the color region. The value of Cr is 0.964 (outside the fusion region corresponding to ϕ=0.06) while it changes inside the fusion region where it has a maximum value of 10.389 in this figure for each value of the amplitude parameter. The Cr contour is similar to a plume for each value of A. In this figure, a horizontal magnetic field Ha=25,β=0° has been considered.Figure 9
The streamlines, the isotherms, and Cr contour atA = 0.1, 0.2, and 0.3.Figure10 illustrates the streamlines, the isotherms, and the Cr contours for the various numbers of undulation N=4,5,and6. As shown by the figure, the values of Ψmax are 12.83, 12, 65, and 12.21 that indicate the velocity of fluid flow decreases slightly as the values of N increase. The reason could be an increase in the number of obstacles in the path. The red lines in the isotherms show higher temperatures that move from the wavy wall to the enclosure wall. The values of Nuave are 5.0886, 4.6966, and 4.4593 for N=4,5,and6, respectively. So, the value of Nuave decreases 12.4% when ascends from 4 to 6. The values of Ngen are 44.07, 43.46, and 44.80 for N=4,5,and6, respectively. Thus, a minimum value can be seen for Ngen at N=5. As shown by the Cr contours, the width of the ribbon shape is not the same everywhere. The ribbon shape is wider where the temperature gradients are smooth. Here, we have Crmin=0.976and Crmax=7.2592 for ϕ=0.04. Other decision parameters are Ha=18, β=0°, and Ste=2.Figure 10
The streamlines, the isotherms, andCr contour at N=4,5, and 6.Figures11(a)–11(d) present the contribution of the heat transfer NT,HT, the fluid friction NT,FF, and the magnetic field NT,MF in the total entropy generation number for an enclosure without porous medium. Comparison between Figures (a) and (b) discovers that NT,FF increases from 10% to 43% while NT,HT decreases from 74% to 15% as the Rayleigh number attains from 104 to 105. Its reason is increasing fluid flow velocity (ascending the values of Ψmax) with ascending the Rayleigh number. As we know, the friction losses enhance with enhancement of fluid flow velocity. Also, a comparison between Figures (c) and (d) indicates that NT,HT ascends from 15% to 19% as ϕ ascends from 0.01 to 0.06. Regarding the first term in the right-hand side of equation (12), increasing of NT,HT is justifiable. Figures 11(e)–11(h) show the contribution of the heat transfer NT,HT, the fluid friction NT,FF, the magnetic field NT,MF, and the porous medium NT,PM in Ngen. Comparison between Figures (e) and (f) reveals that NT,MF increases from 22% to 49% while other type of irreversibilities decreases when the Hartmann number increases from 12 to 24. The value of NT,MF increases with ascending Ha as shown by the last term in equation (12).Figure 11
Contribution of components inNgen for different values of Ra, ϕ, Ha, and Ste.


## 5. Conclusion

The entropy generation and heat transfer analysis were investigated in a complex cavity filled with NEPCMs suspension. The nondimensional governing equations were solved by CVFEM. Here, significant outcomes are remembered as follows:(i)
The values ofΨmax, Nuave, and Ngen rise with ascending Ra. The values of Ψmax are 0.44, 3.86, and 16.93, the values of Nuave are 2.5784, 2.9136, and 5.8065, and the values of Ngen are 3.03, 4.61, and 45.91, for Ra=104,105, and 106, respectively (Figure 3). These values indicate that fluid velocity and rate of heat transfer increase with increasing the Rayleigh number.(ii)
The values ofNuave descend with ascending Ha while Ngen has a maximum at Ha=16. Also, there is a maximum value for Nuave at β=60° (Figure 4).(iii)
Symmetrical behavior was seen forNuave and Ngen with respect to θf=0.5. The maximum value of Nuave is 3.268 that occurs at θf=0.4 and θf=0.6 while there is a minimum value for Ngen that equals 39 and happens at θf=0.5 (Figure 5).(iv)
Lower values ofSte are desirable because Nuave descends and Ngen ascends with Ste (Figure 6). Indeed, as the latent heat of the PCM cores rises, the Stefan number decreases.(v)
Nuave increases 19.9%, and ECOP increases 28.8% while Ngen decreases by 6.9% when ϕ enhances from 0 to 0.06 (Figure 7).(vi)
Higher values ofDa are suitable since the values of ECOP and Nuave increase while Ngen descends with ascending Da (Figure 8). It must be recalled that the ECOP represents the performance of the cavity.(vii)
The effects ofA and N on the streamlines, the isotherms, and the Cr contours show that lower values of these parameters are proper (Figures 9 and 10).(viii)
The contribution ofNT,MF ascends from 22% to 49% as Ha goes up from 12 to 24. Also, the value of NT,FF increases while the value of NT,HT descends with ascending Ra (Figures 11).



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*Source: 8103300-2021-08-16.xml*