# Impact of Fusion Temperature on Hydrothermal Features of Flow within an Annulus Loaded with Nanoencapsulated Phase Change Materials (NEPCMs) during Natural Convection Process

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

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## Abstract

A new type of nanofluids is nanoencapsulated phase change materials (NEPCMs), where nanoparticles are made of a shell and a core. In the current study, characteristics of free convection flow, entropy generation, and heat transfer of NEPCMs in an enclosure are investigated. The enclosure is an annulus between concentric horizontal circular and square cylinders with a porous medium. The governing equations (i.e., continuity, energy, and momentum) are written in the nondimensional form and then numerically solved by the control volume finite element method (CVFEM). The results of the validation are in good agreement with those of the literature. The effects of decision variables on the entropy generation number and the average Nusselt number are investigated. The outcomes discovered that there is a maximum forNuave and a minimum for Ngen at θf=0.4 for each value of the Stefan number. Also, Nuave and ECOP increase by 8.8% and 24.8%, respectively, while Ngen decreases by 12.8% when ϕ increases from 0 (pure fluid) to 0.05 at θf=0.4.


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## Body

## 1. Introduction

Nowadays, nanofluids are used to enhance heat transfer in many various industries. Simulation of their behavior in enclosures is an interesting subject for researchers because of its many industrial applications.The literature review shows that there are many studies about natural convection analysis in enclosures [1–8]. As we know, a considerable amount of energy can be stored by a small amount of Phase Change Materials (PCMs) during a solid to liquid phase change process. PCMs can be used in many industrial applications such as cooling electronic devices [9, 10], solar heating systems [11, 12], waste heat recovery [13, 14], and thermal energy storage systems [15, 16]. But their disadvantage is low thermal conductivity, and therefore, many researchers have tried to improve their thermal conductivity. Recently, they proposed nanoencapsulated phase change materials (NEPCMs). NEPCMs are a new category of nanofluids that nanoparticle includes a shell and a PCM core [17]. In 2015, NEPCM force convection of slurry was studied by Lu et al. [18]. They considered flow on a backward-facing step and investigated the effect of Reynolds number, wall heat flux, and nanoparticle volume fraction on heat transfer and flow. In 2015 and 2018, two good review papers about NEPCMs were published by Liu et al. [19] and Chai et al. [20], respectively. The former investigated the increment of heat transfer, methods of preparation, and characterization, while the last focused on the study of mini/microchannel heat sinks with nano/microencapsulated phase change slurry as working fluid. In 2019, Ghalambaz et al. [21] investigated the thermal behavior of NEPCMs in a porous medium over a vertical flat plate. They discovered that the Nusselt number increases when the fusion temperature of the NEPCMs decreases. In 2019, Ahmad et al. [22] investigated the transient free convection in a NEPCM. They concluded that the heat transfer is affected by the variation of the PCM core fusion temperature. In 2019, the behavior of NECPMs in a square enclosure was investigated by Ghalambaz et al. [17]. They concluded that the fusion temperature is an important variable that has a major effect on heat transfer. Also, in 2019, they conducted an unsteady free convection flow of NEPCMs in a porous medium [23]. In 2020, the influence of nth order chemical reaction in an unsteady nanofluid bioconvective channel flow was investigated by Basir et al. [24]. In 2020, the natural convection flow of a suspension containing nanoencapsulated phase change particles in an eccentric annulus was investigated by Mehryan et al. [25]. In 2020, Ho et al. [26] studied forced convection heat transfer of NEPCM suspension in a minichannel heatsink. In 2020, thermohydrodynamic and irreversibility analyses for NEPCMs were performed by Hashem Zadeh et al. [27, 28].The literature review shows that the second law analysis has not been performed for an enclosure filled by NEPCMs. In this work, the entropy generation and natural convection analyses are performed in an annulus between square cylinders and concentric horizontal circular filled with NEPCMs.The characteristics of this paper in comparison with the previous works are as follows:(1)
The geometry is a complex enclosure with a porous medium(2)
The nanoparticles are NEPCM that are dispersed in the base fluid(3)
The effects of important parameters on Nusselt number are studied(4)
The entropy generation number is calculated(5)
The thermal performance of the enclosure is evaluated by a criterion known as the ecological coefficient of performance (ECOP)


## 2. Problem Statements

### 2.1. Physical Model

A schematic view of the problem is shown in Figure1. The system consists of a heated horizontal cylinder that is concentrically placed inside a square enclosure with a porous medium. The outer square and inner circle are held at fixed temperatures Tc and ThTc<Th, respectively.Figure 1
Schematic view of the considered problem.


### 2.2. Mathematical Model

#### 2.2.1. Basic Governing Equations

The natural convective flow is assumed to be two-dimensional, steady, laminar, and incompressible. Also, it is assumed that the variation of fluid properties and the radiation effects are negligible. Albeit, the variations of density in the buoyancy term are considered where the Boussinesq approximation is used. The core of NEPCM undergoes a phase change at the fusion temperature ofTf where Tc<Tf<Th. Thus, the governing equations can be written as follows [6, 17]:(1)∂u∂x+∂v∂y=0,(2)ρbu∂u∂x+v∂u∂y=−∂P∂x+μb∂2u∂x2+∂2u∂y2−μbKu,(3)ρbu∂v∂x+v∂v∂y=−∂P∂y+μb∂2v∂x2+∂2v∂y2+gρbβbT−Tc−μbKv,(4)ρCpbu∂T∂x+v∂T∂y=kb∂2T∂x2+∂2T∂y2.


#### 2.2.2. Governing Equations in Nondimensional Form

The vorticityω, stream function ψ, and dimensionless variables are as follows:(5)u=∂ψ∂y,v=−∂ψ∂x,ω=∂v∂x−∂u∂y,(6)X=xL,Y=yL,AR=RL,θ=T−TcTh−Tc,Ω=ωL2αf,Ψ=ψαf.Thus, using equations (5) and (6), the nondimensional form of equations (1)–(4) can be rewritten as(7)∂2Ψ∂X2+∂2Ψ∂Y2=−Ω,(8)∂Ψ∂Y∂Ω∂X−∂Ψ∂X∂Ω∂Y=μb/μfρb/ρfPr∂2Ω∂X2+∂2Ω∂Y2+βbβfRaPr∂θ∂X−μb/μfρb/ρfPrDaΩ,(9)∂Ψ∂Y∂θ∂X−∂Ψ∂X∂θ∂Y=kb/kfCr∂2θ∂X2+∂2θ∂Y2,(10)Ra=gβfL3Th−Tcνfαf,Pr=νfαf,Da=KL2,Cr=ρCpbρCpf,where(11)ρb=1−ϕρf+ϕρp,(12)ρCpb=1−ϕρCpf+ϕρCpp,(13)βb=1−ϕβf+ϕβp,(14)kbkf=1+Ncϕ,(15)μbμf=1+Nvϕ.The heat capacity ratioCr can be written in the nondimensional form as [17](16)Cr=ρCpbρCpf=1−ϕ+ϕλ+ϕδStef.In the previous equation,λ, δ, and Ste are the ratio of the heat capacity of the NEPCM nanoparticles to the base fluid, the nondimensional parameter of fusion range, and Stefan number, respectively. For more details, see [17]. Also, f is the nondimensional fusion function defined by the following equation:(17)f=π2sinπδθ−θf+δ2×0θ<θf−δ21θf−δ2<θ<θf+δ20θf+δ2<θ.The boundary conditions corresponding to Figure1 are(18)θ=1.0on the inner boundary,θ=0.0on the outer boundary,Ψ=0on all solid boundaries.The average Nusselt number can be calculated using the local Nusselt number along the hot wall:(19)Nulocal=−kbkf∂θ∂rr=R,(20)Nuave=12π∫02πNulocalζdζ.


#### 2.2.3. Entropy Generation

The rate of entropy generation can be computed as follows:(21)S˙gen=kbfT02∂T∂x2+∂T∂y2+μbfT02∂u∂x2+2∂v∂y2+∂u∂y+∂v∂x2+μbu2+v2KT0.The local entropy generation in a nondimensional form can be written as follows [6]:(22)NL,gen=S˙genkf/T02ΔT/L2=kbkf∂θ∂X2+∂θ∂Y2+μbμfΦf4∂2Ψ∂X∂Y2+∂2Ψ∂Y2−∂2Ψ∂X22+1DaμbμfΦf∂Ψ∂X2+∂Ψ∂Y2,(23)Φf=μfT0kfαfLΔT2,where T0=Th+Tc/2 is the mean temperature and ΔT is the temperature difference within the enclosure ΔT=Th−Tc.Equation (22) can be rewritten as follows:(24)NL,gen=NL,HT+NL,FF+NL,PM,(25)NL,HT=kbkf∂θ∂X2+∂θ∂Y2,(26)NL,FF=μbμfΦf4∂2Ψ∂X∂Y2+∂2Ψ∂Y2−∂2Ψ∂X22,(27)NL,PM=1DaμbμfΦf∂Ψ∂X2+∂Ψ∂Y2.The total entropy generation can be obtained by(28)NT,HT=∫VNL,HTdV,(29)NT,FF=∫VNL,FFdV,(30)NT,PM=∫VNL,PMdV.The entropy generation number can be calculated by(31)Ngen=NT,FF+NT,HT+NT,PM.


#### 2.2.4. Ecological Coefficient of Performance (ECOP)

It is known as the ecological coefficient of performance (ECOP) proposed by Seyyedi et al. [4] is a criterion for the evaluation of enclosure thermal performance:(32)ECOP=NuaveNgen.


## 2.1. Physical Model

A schematic view of the problem is shown in Figure1. The system consists of a heated horizontal cylinder that is concentrically placed inside a square enclosure with a porous medium. The outer square and inner circle are held at fixed temperatures Tc and ThTc<Th, respectively.Figure 1
Schematic view of the considered problem.


## 2.2. Mathematical Model

### 2.2.1. Basic Governing Equations

The natural convective flow is assumed to be two-dimensional, steady, laminar, and incompressible. Also, it is assumed that the variation of fluid properties and the radiation effects are negligible. Albeit, the variations of density in the buoyancy term are considered where the Boussinesq approximation is used. The core of NEPCM undergoes a phase change at the fusion temperature ofTf where Tc<Tf<Th. Thus, the governing equations can be written as follows [6, 17]:(1)∂u∂x+∂v∂y=0,(2)ρbu∂u∂x+v∂u∂y=−∂P∂x+μb∂2u∂x2+∂2u∂y2−μbKu,(3)ρbu∂v∂x+v∂v∂y=−∂P∂y+μb∂2v∂x2+∂2v∂y2+gρbβbT−Tc−μbKv,(4)ρCpbu∂T∂x+v∂T∂y=kb∂2T∂x2+∂2T∂y2.


### 2.2.2. Governing Equations in Nondimensional Form

The vorticityω, stream function ψ, and dimensionless variables are as follows:(5)u=∂ψ∂y,v=−∂ψ∂x,ω=∂v∂x−∂u∂y,(6)X=xL,Y=yL,AR=RL,θ=T−TcTh−Tc,Ω=ωL2αf,Ψ=ψαf.Thus, using equations (5) and (6), the nondimensional form of equations (1)–(4) can be rewritten as(7)∂2Ψ∂X2+∂2Ψ∂Y2=−Ω,(8)∂Ψ∂Y∂Ω∂X−∂Ψ∂X∂Ω∂Y=μb/μfρb/ρfPr∂2Ω∂X2+∂2Ω∂Y2+βbβfRaPr∂θ∂X−μb/μfρb/ρfPrDaΩ,(9)∂Ψ∂Y∂θ∂X−∂Ψ∂X∂θ∂Y=kb/kfCr∂2θ∂X2+∂2θ∂Y2,(10)Ra=gβfL3Th−Tcνfαf,Pr=νfαf,Da=KL2,Cr=ρCpbρCpf,where(11)ρb=1−ϕρf+ϕρp,(12)ρCpb=1−ϕρCpf+ϕρCpp,(13)βb=1−ϕβf+ϕβp,(14)kbkf=1+Ncϕ,(15)μbμf=1+Nvϕ.The heat capacity ratioCr can be written in the nondimensional form as [17](16)Cr=ρCpbρCpf=1−ϕ+ϕλ+ϕδStef.In the previous equation,λ, δ, and Ste are the ratio of the heat capacity of the NEPCM nanoparticles to the base fluid, the nondimensional parameter of fusion range, and Stefan number, respectively. For more details, see [17]. Also, f is the nondimensional fusion function defined by the following equation:(17)f=π2sinπδθ−θf+δ2×0θ<θf−δ21θf−δ2<θ<θf+δ20θf+δ2<θ.The boundary conditions corresponding to Figure1 are(18)θ=1.0on the inner boundary,θ=0.0on the outer boundary,Ψ=0on all solid boundaries.The average Nusselt number can be calculated using the local Nusselt number along the hot wall:(19)Nulocal=−kbkf∂θ∂rr=R,(20)Nuave=12π∫02πNulocalζdζ.


### 2.2.3. Entropy Generation

The rate of entropy generation can be computed as follows:(21)S˙gen=kbfT02∂T∂x2+∂T∂y2+μbfT02∂u∂x2+2∂v∂y2+∂u∂y+∂v∂x2+μbu2+v2KT0.The local entropy generation in a nondimensional form can be written as follows [6]:(22)NL,gen=S˙genkf/T02ΔT/L2=kbkf∂θ∂X2+∂θ∂Y2+μbμfΦf4∂2Ψ∂X∂Y2+∂2Ψ∂Y2−∂2Ψ∂X22+1DaμbμfΦf∂Ψ∂X2+∂Ψ∂Y2,(23)Φf=μfT0kfαfLΔT2,where T0=Th+Tc/2 is the mean temperature and ΔT is the temperature difference within the enclosure ΔT=Th−Tc.Equation (22) can be rewritten as follows:(24)NL,gen=NL,HT+NL,FF+NL,PM,(25)NL,HT=kbkf∂θ∂X2+∂θ∂Y2,(26)NL,FF=μbμfΦf4∂2Ψ∂X∂Y2+∂2Ψ∂Y2−∂2Ψ∂X22,(27)NL,PM=1DaμbμfΦf∂Ψ∂X2+∂Ψ∂Y2.The total entropy generation can be obtained by(28)NT,HT=∫VNL,HTdV,(29)NT,FF=∫VNL,FFdV,(30)NT,PM=∫VNL,PMdV.The entropy generation number can be calculated by(31)Ngen=NT,FF+NT,HT+NT,PM.


### 2.2.4. Ecological Coefficient of Performance (ECOP)

It is known as the ecological coefficient of performance (ECOP) proposed by Seyyedi et al. [4] is a criterion for the evaluation of enclosure thermal performance:(32)ECOP=NuaveNgen.


## 2.2.1. Basic Governing Equations

The natural convective flow is assumed to be two-dimensional, steady, laminar, and incompressible. Also, it is assumed that the variation of fluid properties and the radiation effects are negligible. Albeit, the variations of density in the buoyancy term are considered where the Boussinesq approximation is used. The core of NEPCM undergoes a phase change at the fusion temperature ofTf where Tc<Tf<Th. Thus, the governing equations can be written as follows [6, 17]:(1)∂u∂x+∂v∂y=0,(2)ρbu∂u∂x+v∂u∂y=−∂P∂x+μb∂2u∂x2+∂2u∂y2−μbKu,(3)ρbu∂v∂x+v∂v∂y=−∂P∂y+μb∂2v∂x2+∂2v∂y2+gρbβbT−Tc−μbKv,(4)ρCpbu∂T∂x+v∂T∂y=kb∂2T∂x2+∂2T∂y2.


## 2.2.2. Governing Equations in Nondimensional Form

The vorticityω, stream function ψ, and dimensionless variables are as follows:(5)u=∂ψ∂y,v=−∂ψ∂x,ω=∂v∂x−∂u∂y,(6)X=xL,Y=yL,AR=RL,θ=T−TcTh−Tc,Ω=ωL2αf,Ψ=ψαf.Thus, using equations (5) and (6), the nondimensional form of equations (1)–(4) can be rewritten as(7)∂2Ψ∂X2+∂2Ψ∂Y2=−Ω,(8)∂Ψ∂Y∂Ω∂X−∂Ψ∂X∂Ω∂Y=μb/μfρb/ρfPr∂2Ω∂X2+∂2Ω∂Y2+βbβfRaPr∂θ∂X−μb/μfρb/ρfPrDaΩ,(9)∂Ψ∂Y∂θ∂X−∂Ψ∂X∂θ∂Y=kb/kfCr∂2θ∂X2+∂2θ∂Y2,(10)Ra=gβfL3Th−Tcνfαf,Pr=νfαf,Da=KL2,Cr=ρCpbρCpf,where(11)ρb=1−ϕρf+ϕρp,(12)ρCpb=1−ϕρCpf+ϕρCpp,(13)βb=1−ϕβf+ϕβp,(14)kbkf=1+Ncϕ,(15)μbμf=1+Nvϕ.The heat capacity ratioCr can be written in the nondimensional form as [17](16)Cr=ρCpbρCpf=1−ϕ+ϕλ+ϕδStef.In the previous equation,λ, δ, and Ste are the ratio of the heat capacity of the NEPCM nanoparticles to the base fluid, the nondimensional parameter of fusion range, and Stefan number, respectively. For more details, see [17]. Also, f is the nondimensional fusion function defined by the following equation:(17)f=π2sinπδθ−θf+δ2×0θ<θf−δ21θf−δ2<θ<θf+δ20θf+δ2<θ.The boundary conditions corresponding to Figure1 are(18)θ=1.0on the inner boundary,θ=0.0on the outer boundary,Ψ=0on all solid boundaries.The average Nusselt number can be calculated using the local Nusselt number along the hot wall:(19)Nulocal=−kbkf∂θ∂rr=R,(20)Nuave=12π∫02πNulocalζdζ.


## 2.2.3. Entropy Generation

The rate of entropy generation can be computed as follows:(21)S˙gen=kbfT02∂T∂x2+∂T∂y2+μbfT02∂u∂x2+2∂v∂y2+∂u∂y+∂v∂x2+μbu2+v2KT0.The local entropy generation in a nondimensional form can be written as follows [6]:(22)NL,gen=S˙genkf/T02ΔT/L2=kbkf∂θ∂X2+∂θ∂Y2+μbμfΦf4∂2Ψ∂X∂Y2+∂2Ψ∂Y2−∂2Ψ∂X22+1DaμbμfΦf∂Ψ∂X2+∂Ψ∂Y2,(23)Φf=μfT0kfαfLΔT2,where T0=Th+Tc/2 is the mean temperature and ΔT is the temperature difference within the enclosure ΔT=Th−Tc.Equation (22) can be rewritten as follows:(24)NL,gen=NL,HT+NL,FF+NL,PM,(25)NL,HT=kbkf∂θ∂X2+∂θ∂Y2,(26)NL,FF=μbμfΦf4∂2Ψ∂X∂Y2+∂2Ψ∂Y2−∂2Ψ∂X22,(27)NL,PM=1DaμbμfΦf∂Ψ∂X2+∂Ψ∂Y2.The total entropy generation can be obtained by(28)NT,HT=∫VNL,HTdV,(29)NT,FF=∫VNL,FFdV,(30)NT,PM=∫VNL,PMdV.The entropy generation number can be calculated by(31)Ngen=NT,FF+NT,HT+NT,PM.


## 2.2.4. Ecological Coefficient of Performance (ECOP)

It is known as the ecological coefficient of performance (ECOP) proposed by Seyyedi et al. [4] is a criterion for the evaluation of enclosure thermal performance:(32)ECOP=NuaveNgen.


## 3. Numerical Procedure

A FORTRAN code has been developed using the control volume finite element method (CVFEM) 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, Da=0.01, Pr=6.2, ϕ=0.05, δ=0.05, Ste=0.313, θf=0.3, λ=0.4, AR=0.2, and Φ=10−4. Hence, 111 × 481 can be considered as the grid size.Table 1
The effect of grid size onΨmax, Nuave, and the Ngen.
31×36151×38171×40191×421111×481121×521Ψmax6.56.336.266.226.216.20Nuave5.00455.17415.18345.18595.18535.1852Ngen50.0251.0151.6351.9452.1352.15


### 3.2. Validation

For validation of the written code, it is validated by comparison of the outcomes with the published results in the literature. Table2 shows the values of the average Nusselt number on the hot surface (inner circular in Figure 1), for Pr=0.71 at Ra=104and105. A good agreement can be seen between the present results and those of previous works. Also, Figure 2 depicts the isothermals and streamlines of the literature and present work at Ra=105 corresponding to Table 2.Table 2
Comparison of the average Nusselt numberNuave on the inner circular for the present study with previous numerical studies for AR = R/L = 0.2 and Pr = 0.71.
RaMoukalled and Acharya [29]Kim et al. [30]Present study1043.3313.4143.3391055.085.13855.1021Figure 2
The isotherms (left) and streamlines (right) at Ra = 105 for (a) Kim et al. [30] and (b) the present work.
(a)(b)Furthermore, another validation has been provided for the isotherms, streamlines, and Cr contour forθf=0.3 at Ra=105. Figure 3 presents a comparison of the results between Ghalambaz et al. [17] and presents work that is acceptable.Figure 3
The isotherms, streamlines, and Cr contour atθf = 0.3 and Ra = 105 for (a) Ghalambaz et al. [17] and (b) the present study.
(a)(b)


## 3.1. Grid Test

The results of the grid independence study are provided in Table1. The calculations have been performed with Ra=105, Da=0.01, Pr=6.2, ϕ=0.05, δ=0.05, Ste=0.313, θf=0.3, λ=0.4, AR=0.2, and Φ=10−4. Hence, 111 × 481 can be considered as the grid size.Table 1
The effect of grid size onΨmax, Nuave, and the Ngen.
31×36151×38171×40191×421111×481121×521Ψmax6.56.336.266.226.216.20Nuave5.00455.17415.18345.18595.18535.1852Ngen50.0251.0151.6351.9452.1352.15


## 3.2. Validation

For validation of the written code, it is validated by comparison of the outcomes with the published results in the literature. Table2 shows the values of the average Nusselt number on the hot surface (inner circular in Figure 1), for Pr=0.71 at Ra=104and105. A good agreement can be seen between the present results and those of previous works. Also, Figure 2 depicts the isothermals and streamlines of the literature and present work at Ra=105 corresponding to Table 2.Table 2
Comparison of the average Nusselt numberNuave on the inner circular for the present study with previous numerical studies for AR = R/L = 0.2 and Pr = 0.71.
RaMoukalled and Acharya [29]Kim et al. [30]Present study1043.3313.4143.3391055.085.13855.1021Figure 2
The isotherms (left) and streamlines (right) at Ra = 105 for (a) Kim et al. [30] and (b) the present work.
(a)(b)Furthermore, another validation has been provided for the isotherms, streamlines, and Cr contour forθf=0.3 at Ra=105. Figure 3 presents a comparison of the results between Ghalambaz et al. [17] and presents work that is acceptable.Figure 3
The isotherms, streamlines, and Cr contour atθf = 0.3 and Ra = 105 for (a) Ghalambaz et al. [17] and (b) the present study.
(a)(b)


## 4. Results and Discussion

In this work, the entropy generation and the free convection heat transfer in an annulus between concentric horizontal circular and square cylinders loaded with a dilute suspension are numerically investigated by CVFEM. The effects of nondimensional parameters on the characteristics of flow and entropy generation number are investigated. Table3 presents the default values of the nondimensional parameters for calculations unless the new values are stated.Table 3
The default values of the nondimensional parameters.
ARRaDaΦPrθfϕλSteδNcNvρp/ρfϕ×βp/βf0.210510–210–46.20.30.050.40.3130.05330.90Figure4 shows the streamlines, the isotherms, and Cr contours at Ra=104,105, and 106. There are two overall rotating symmetric eddies with two inner vortices at Ra=104, but when Rayleigh number goes up to 105 and 106, the recirculating eddies core is placed only at the upper half. In other words, the circulation of the flow increased with increasing the Rayleigh number. The maximum absolute value of the stream function, Ψmax, increases when the Rayleigh number increases. The values of Ψmax are 0.804, 6.81, and 21.64 for Ra=104,105, and 106, respectively. As it can be seen from the isotherms, for Ra=104, the heat transfer in the cavity is mainly dominated by the conduction mode, whereas the heat transfer in the enclosure is mainly dominated by the convection mode at Ra=106. The role of convection in heat transfer becomes more significant with increasing the Rayleigh number, and the thermal boundary layer on the hot wall becomes thinner. The average Nusselt number ascends with ascending the Rayleigh number where the values of Nuave are 4.1636, 6.0981, and 10.962, for Ra=104,105, and 106, respectively. The last column in Figure 4 presents the Cr contours for each Rayleigh number. According to equations (16) and (17), 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=0.04, 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 8.82 (this value can be obtained by solving the velocity and temperature fields). The values of 0.97 and 8.82 are the same for each value of the Rayleigh number in Figure 4. Notice that the heat capacity changes since the phase change has occurred.Figure 4
The streamlines, the isotherms, and Cr contour at Ra = 104, 105, and 106.Figure5 presents the entropy generation number, the average Nusselt number, and ECOP versus the fusion temperature of the NEPCM core θf for three values of the Stefan number at Ra=105 and ϕ=0.05. Figure 5(a) shows that there is a maximum value for Nuave at θf=0.4 for each value of the Stefan number. Albeit, it can be stated that the variations of the average Nusselt number are very small in the range of 0.3<θf<0.7 for each value of the Stefan number. Also, the average Nusselt number decreases with increasing the Stefan number for a constant value of θf. It should be noted that the Stefan number increases as the PCM cores latent heat decreases. Also, the heat storage capacity of the NEPCM particles descends with descending the PCM cores latent heat. Finally, the convective heat transfer rate in the cavity diminishes when the heat storage capacity of the NEPCM particles decreases. Figure 5(b) demonstrates that the entropy generation number firstly decreases, reaches a minimum value, and then goes up as the fusion temperature increases for each value of the Stefan number. Also, decreasing the values of Stefan number is desirable from the second law of thermodynamics point of view since the entropy generation number decreases when the Stefan number decreases for each value of θf. Figure 5(c) illustrates a maximum value for ECOP when it is plotted versus θf for each Stefan number. Therefore, Figure 5 shows that Nuave and ECOP increase whereas Ngen decreases as Ste decreases. For example, Nuave increases from 5.1135 to 5.3484 (4.6% increase) and ECOP increases from 0.0971 to 0.1070 (10.2% increase) while Ngen decreases from 52.67 to 50.00 (5.3% decrease) when Ste decreases from 0.5 to 0.2 at θf=0.4.Figure 5
Nuave, Ngen, and ECOP versus the fusion temperature for different values of the Ste at Ra = 105 and ϕ = 0.05.
(a)(b)(c)Figure6 presents the ECOP, the entropy generation number, and the average Nusselt number versus the volume fraction of the nanoparticle for three values of fusion temperature. As could be seen, raising the volume fraction of nanoparticle is desirable since Nuave and ECOP rise and Ngen descends with raising ϕ for θf. For instance, Nuave increases from 4.7911 to 5.2148 (8.8% increase) and ECOP increases from 0.0812 to 0.1013 (24.8% increase) while Ngen decreases from 58.99 to 51.46 (12.8% decrease) when ϕ increases from 0 (pure fluid) to 0.05 at θf=0.4. On the contrary, Figure 6 shows that the variations of the entropy generation number, the average Nusselt number, and ECOP is small when the nondimensional fusion temperature varies from 0.2 to 0.4. For another example, Nuave increases from 4.9915 to 5.0534 (1.2% increase) and ECOP increases from 0.0906 to 0.0935 (3.2% increase) while Ngen decreases from 55.10 to 54.07 (1.9% decrease) when θf increases from 0.02 to 0.04 at ϕ=0.03.Figure 6
The average Nusselt number, the entropy generation number, and ECOP versus the volume fraction of nanoparticle for three values of fusion temperature.
(a)(b)(c)Figure7 presents Nuave, Ngen, and ECOP versus Da for three values of Ste at θf=0.4 and Ra=105. The result demonstrates that Nuave, Ngen, and ECOP ascend with increasing Da for Ste. For example, Nuave ascends from 5.1135 to 5.3853 (5.3% increase), Ngen enhances from 52.67 to 53.93 (2.4% increase), and ECOP increases from 0.0971 to 0.0999 (2.9% increase) when Darcy number goes up from 0.01 to 100 at Ste=0.5.Figure 7
Nuave, Ngen, and ECOP versus the Da for three values of the Stefan number at θf = 0.4 and Ra = 105.
(a)(b)(c)Figure8 depicts the streamlines, the isotherms, and the Cr contours for different values of aspect ratio 0.1,0.2and0.3 at θf=0.45 and Ra=105. The streamlines show that the flow circulation decreases with increasing the aspect ratio values. In other words, the strength of the flow (indicated by Ψmax) decreases as the values of the aspect ratio go up. Its reason can be the fact that the available space for flow circulation descends when the aspect ratio ascends. On the contrary, Nuave increases with increasing aspect ratio values because of the increased heat transfer surface area associated with higher AR values. Also, Nuave increases since the space between the walls decreases with increasing the values of aspect ratio. The isotherms follow the shapes of inner and outer walls near the hot and cold walls, respectively. At AR=0.1, the boundary layer along the hot wall separates from the surface near the cylinder top and constructs a thermal plume. The last row of Figure 8 presents the Cr contours where the color region (except for the blue color region) illustrates the phase change region. The value of Cr is 0.97 (outside the fusion region corresponding to ϕ=0.05) while it changes inside the fusion region where it has a maximum value of 5.99 in Figure 8 for all aspect ratios. The Cr contour is similar to a plume at AR=0.1, and it is similar to a ring at AR=0.3. It shows that the fusion region is near to the bottom of the inner cylinder, and it moves further upward at the top of the inner cylinder for AR=0.1. On the contrary, the distance of the fusion region from the inner cylinder and outer wall is approximately the same at AR=0.3.Figure 8
The streamlines, the isotherms, and the Cr contours for various values ofAR at θf = 0.45 and Ra = 105.Table4 presents the effects of aspect ratio on Ψmax, Nuave, and Ngen at three values of Ra 0,104and105. The first row at the second column is corresponding to the value of the average Nusselt number for pure conduction (i.e., Ra=0). As seen, the values of Ψmax, Nuave, and Ngen increase with increasing the Rayleigh number for each aspect ratio. The increasing in Ψmax means that the fluid flows more rapidly and increasing Nuave indicates that convection will be the dominant heat transfer mechanism as the Rayleigh number increases. Increasing the value of Ngen is undesirable (from view point of the thermodynamics second law) when Ra increases. At a constant value of Rayleigh number, the values of Ψmax decrease while the values of Nuave and Ngen increase with increasing the value of AR. For example, Ψmax descends from 7.558 to 3.740 (50.52% decrease), Nuave increases from 4.3144 to 7.6795 (78% increase), and Ngen ascends from 51.6553 to 61.9543 (19.94% increase) when AR increases from 0.1 to 0.3 at Ra=105. The last row of Table 4 is corresponding to Figure 8.Table 4
The values ofΨmax, Nuave and Ngen for different values of aspect ratio.
ΨmaxNuaveNgenARARARRa0.10.20.30.10.20.30.10.20.30———2.37894.09226.95568.269711.982423.20931041.0390.64750.3492.49344.13426.96929.399512.593723.52941057.5586.1553.7404.31445.68877.679551.655354.635861.9543Figure9(a) depicts variations of Nulocal along the hot wall of the cavity for different values of Stefan number. It has a maximum value at the top of the inner wall ζ=90° for each value of the Stefan number. Figure 9 demonstrates that the difference between the values of the local Nusselt number is very small when the Stefan number increases from 0.2 to 0.5 at each place of the hot wall. The variations of the local Nusselt number along the inner cylinder for different values of Rayleigh number and various values of aspect ratio have been shown in Figures 9(b) and 9(c), respectively. As expected, Figure 9(b) shows that the curve corresponding to Ra=105 is placed above two other curves. As seen, there is a peak for Ra=105 at ζ=90° whereas the values of the local Nusselt number approximately are equal to each other at ζ=0°,90°,180°,270°,and360°for Ra=103. Figure 9(c) shows that the behavior of the local Nusselt number along the hot wall approximately is the same in the range of 45°<ζ<135° for each aspect ratio. Also, the maximum value of the local Nusselt number decreases as the aspect ratio increases at ζ=90°.Figure 9
Variations of the local Nusselt number versus the rotation angle for three values of (a) the Stefan number, (b) The Rayleigh number, and (c) the aspect ratio.
(a)(b)(c)Figure10 demonstrates the Cr contours for three values of the dimensionless fusion temperature θf=0.1,0.3,and0.5. In Figure 10, the red area presents the phase change region in which a phase change occurs. The region with the phase change approaches the cold wall (inner cylinder) when the fusion temperature increases. The red region is similar to a square (cold wall) at θf=0.1. It is similar to an irregular ring at θf=0.3, while it is similar to a plume at θf=0.5. In Figure 10, the values of Crmin and Crmax are 0.97 and 5.988, respectively. Notice that the value of 0.97 is corresponding to ϕ=0.05 and f=0.Figure 10
The Cr contours for three values of the nondimensional fusion temperature. (a)θf = 0.1, (b) θf = 0.3, and (c) θf = 0.5.
(a)(b)(c)Figure11(a) presents the contribution of NT,HT and NT,FF in the total entropy generation number for an enclosure without porous medium at various values of Rayleigh number. Figure 11 represents the contribution of NT,FF increases as the Ra increases. As evident, the contribution of the NT,HT increases as the Rayleigh number increases. For example, NT,FF ascends from 4% to 67% when the Rayleigh number ascends from 104 to 105. The reason can be the increasing velocity of fluid flow (or in other words increasing the values of Ψmax) as the Rayleigh number increases. As we know, the friction losses enhance with the enhancement of fluid flow velocity. Figure 11(b) depicts the contribution of NT,HT, NT,FF, and NT,PM in the total entropy generation number for two values of volume fraction of nanoparticles ϕ=0.01and0.05 at two values of the fusion temperature θf=0.2and0.4. As it can be seen, NT,HT increases when ϕ increases from 0.01 to 0.05 whereas NT,FF decreases with ascending the nanoparticles volume fraction for each value of fusion temperature. For example, NT,HT goes up from 25% to 30% as ϕ increases from 0.01 to 0.05 at θf=0.4. There is no significant change in NT,PM with the change in the ϕ. Besides, the values of NT,HT, NT,FF, and NT,PM are approximately constant when θf enhances from 0.2 to 0.4 for each value of ϕ.Figure 11
Contribution of components in the total entropy generation number (a) for different values of Ra and (b) for two values ofϕ and θf.
(a)(b)Table5 shows the influence of the effective parameters on Nuave, Ngen, and ECOP. The first row in Table 5 is corresponding to the pure fluid ϕ=0. The comparison between the first and second rows discovers that Nuave and ECOP enhance and Ngen reduces when ϕ increases from 0 to 0.04. The comparison between the second and third rows reveals that Nuave, Ngen, and ECOP increase with increasing Nc (i.e., increasing the thermal conductivity). The comparison between the second and fourth rows displays that Nuave and Ngen decrease, and ECOP increases with increasing Nv (i.e., increment the dynamic viscosity). The comparison between the second and fifth rows discovers that Nuave, Ngen, and ECOP increase with increasing Nc and Nv, simultaneously. The comparison between the second and sixth rows shows that Nuave and ECOP descend and Ngen ascends when λ descends from 0.4 to 0.3. Hence, the higher values of λ are desirable. The comparison between the second and eighth rows demonstrates that Nuave decreases from 5.3568 to 5.3161 (0.8% decrease), Ngen decreases from 54.0578 to 54.9515 (1.7% decrease), and ECOP decreases from 0.0991 to 0.0967 (2.4% decrease) when θf decreases from 0.52 to 0.22 (57.7% decreasing). These values indicate that the changes in Nuave, Ngen, and ECOP are very small compared to the θf changes. The comparison between the second and last rows reveals that Nuave and ECOP ascend and Ngen descends when Ste descends from 0.313 to 0.25. These numbers indicate that lower values of the Stefan number are desirable. For calculations of Table 5, the decision parameters are Ra=105, AR=0.2, Da=0.1, Pr=6.2, and Φ=10−4.Table 5
The effect of active parameters onNuave, Ngen, and ECOP.
No.θfϕλρp/ρfSteδNcNvNuaveNgenECOP10.520.000.40.90.3130.05335.018659.67060.084120.520.040.40.90.3130.05335.356854.05780.099130.520.040.40.90.3130.05635.755057.16480.100740.520.040.40.90.3130.05365.199851.67900.100650.520.040.40.90.3130.05665.588954.56110.102460.520.040.30.90.3130.05335.351054.10490.098970.520.040.40.70.3130.05335.344153.56200.099880.220.040.40.90.3130.05335.316154.95150.096790.520.040.40.90.250.05335.403353.38420.1012


## 5. Conclusion

In this work, entropy generation and free convection were investigated in a complex cavity filled with NEPCMs. The nondimensional governing equations were solved by CVFEM. The CVFEM method could properly determine the behavior of the fluid in the understudy cavity. In future work, the effect of NEPCM suspension can be compared with nanofluids in an exclusive property like non-Newtonian fluid in a complicated geometry by considering the effect of some other fields and parameters. The important highlights can be summarized as follows:(i)
The values ofΨmax and Nuave increase when the Rayleigh number ascends. The values of Ψmax are 0.804, 6.81, and 21.64, and the values of Nuave are 4.1636, 6.0981, and 10.962, for Ra=104,105 and 106, respectively (Figure 4).(ii)
There is a maximum value forNuave and a minimum value for Nuave at θf=0.4 for each value of the Stefan number (Figure 5).(iii)
Nuave increases by 8.8% while Ngen decreases by 12.8% when ϕ goes up 5% at θf=0.4 (Figure 6).(iv)
Nuave, Ngen, and ECOP increase by 5.3%, 2.4%, and 2.9%, respectively, as thr Darcy number goes up from 0.01 to 100 at Ste=0.5 (Figure 7).(v)
The values ofΨmax decrease while the values of Nuave and Ngen increase with increasing the values of aspect ratio. The values of Ψmax descend 50.52%, whereas Nuave and Ngen ascend by 78% and 19.94%, respectively, when AR ascends from 0.1 to 0.3 at Ra=105 (Figure 8 and Table 4).(vi)
There is a maximumNulocal at the top of the hot wall ζ=90° for each value of the Stefan number (Figure 9).(vii)
The red region is similar to a square (cold wall) atθf=0.1. It is similar to an irregular ring at θf=0.3, while it is similar to a plume at θf=0.5 (Figure 10).(viii)
NT,HT goes up from 25% to 30% as ϕ increases from 0.01 to 0.05 at θf=0.4 (Figure 11).(ix)
The values ofNuave, Ngen, and ECOP increase with ascending Nc, but the values of Nuave and Ngen decrease and ECOP increases with increasing Nv (Table 5).



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*Source: 4276894-2021-11-10.xml*