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

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 of Nu ave descend with ascending Ha while N gen has a maximum at Ha � 16. Also, the value of N T, MF increases with ascending Ha. The values of Nu ave and N gen depend on nondimensional fusion temperature where good performance is seen in the range of 0 . 35 < θ f < 0 . 6. Also, Nu ave increases 19.9% and ECOP increases 28.8% whereas N gen descends 6.9% when ϕ ascends from 0 to 0.06 at θ f � 0 . 5. Nu ave decreases 4.95% while N gen increases by 8.65% when Ste increases from 0.2 to 0.7 at θ f �


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][2][3][4][5][6][7][8][9][10][11][12][13][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]. ey 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]. ey 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. ey 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][39][40], finite element method (FEM) [41,42], control volume finite element method (CVFEM) [43][44][45][46][47][48], and homotopy perturbation method (HPM) [49][50][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) e effects of decision parameters such Figure 1 consists of a heated horizontal wavy cylinder at T h that is concentrically placed inside an elliptic enclosure held at T c . An external uniform magnetic field ) has an effect on the fluid flow. e wavy wall of the inner cylinder can be implemented with the following equation: (1)

Dimensionless Governing Equations.
e fluid flow is supposed to be steady, incompressible, two-dimensional, and laminar with no radiation effect. e fluid properties are constant but Boussinesq approximation is applied to the density in the buoyancy term. e dimensionless conservation equations are as follows [14,35]: where Pr, Ra, Da, Ha, and Cr are given as follows: where e heat capacity ratio (Cr) can be rewritten in the nondimensional form as follows [35]: Also, f is the nondimensional fusion function defined by the following equation [52]: e boundary conditions of the system are as follows: Ψ � 0 and θ � 1.0 on the inner boundary, Ψ � 0 and θ � 0.0 on the outer boundary.
e average Nusselt number can be calculated using the local Nusselt number along the hot wall as follows:

Entropy
Generation. e performance of energy systems has been evaluated by equation (11) which is named as the rate of entropy generation as follows:  Mathematical Problems in Engineering e local entropy generation in nondimensional form can be rewritten as follows [14]: e total entropy generation and the entropy generation number (N gen ) can be obtained, respectively, as follows: 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]:

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

Grid Test.
e results of the grid independence study are provided in Table 1. e calculations have been performed with Ra � 10 5 , Pr Hence, 81 × 971 can be considered as the grid size.

Validation.
In order to validate the results, two validations are performed. Firstly, the isotherms, streamlines, and Cr contour for a square cavity are obtained for θ f � 0.5 at Ra � 10 5 . 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, ρ p /ρ f � 0.9, and δ � 0.05. For both validations, the results are acceptable.

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. e effects of nondimensional parameters on the characteristics of the flow and entropy generation number are considered. e effect of an external inclined magnetic field on the fluid flow is specified by Hartmann number (Ha) and angle of the magnetic field (β). 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 θ f . Also, the geometry of enclosure is specified by A and N. e default values of the nondimensional parameters for calculations are presented in Table 3. Figure 3 shows the isotherms, the streamlines, and Cr contours at Ra � 10 3 , 10 4 , and 10 5 . e power of the flow increases with the Rayleigh number as it is obvious from the maximum value of the absolute stream function, |Ψ max |. e values of |Ψ max | are 0.44, 3.86, and 16.93 for Ra � 10 3 , 10 4 , and 10 5 , respectively. e isotherms are concentric circles around the hot wall at Ra � 10 3 . e convective heat transfer is predominant in this case, while the role of convection in heat transfer becomes more significant with increasing the Rayleigh number. e red lines in isotherms go up from the upper section of the inner wall to the outer wall at Ra � 10 5 . e average Nusselt number ascends with ascending the Rayleigh number where the values of Nu ave are 2.5784, 2.9136, and 5.8065, for Ra � 10 3 , 10 4 , and 10 5 , respectively. e last row in this figure presents the Cr 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 θ f 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). e 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, θ f � 0.65, β � 0°, and A � 0.1. Figure 4 presents the effects of the magnetic field and its angle on the entropy generation number and the average Nusselt number. Figure 4(       (19.59% decreasing) when Ha increases from 16 to 40. is figure indicates that increasing the Hartmann number is desirable after Ha � 16. Figure 4(b) shows that Nu ave firstly ascends, attains a maximum value, and then descends with ascending the angle of the magnetic field. e maximum value of Nu ave is 4.9214, which occurs at β � 60°. On the other hand, N gen 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 θ f � 0.6, β � 0°, A � 0.1, and N � 8 for Figure 4(a) while they are Ha � 10, Ste � 0.5, θ f � 0.6, ϕ � 0.04, A � 0.15, and N � 4 without porosity for Figure 4(b). Figure 5 presents variations of the entropy generation number and the average Nusselt number versus the fusion temperature of the NEPCM core (θ f ) at Ra � 10 5 and ϕ � 0.05. It shows approximately a regular symmetrical behavior for θ f � 0.5 due to the symmetry of both the boundary conditions and geometry. e variations of the average Nusselt number are very small in the range of 0.3 < θ f < 0.65, and there is a local minimum at θ f � 0.5. e maximum value of Nu ave is 3.268 that occurs at θ f � 0.4 and θ f � 0.6. Besides, the entropy generation number firstly decreases to a minimum value and then goes up as the fusion temperature increases. e minimum value of N gen is 39 that occurs at θ f � 0.5 although it is approximately constant in the range of 0.35 < θ f < 0.6. is figure discovers that the best range of θ f is 0.35 < θ f < 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 6(a) demonstrates the variations of Nu ave and N gen versus Ste while the values of |Ψ max | and ECOP are calculated and plotted in Figure 6(b) as a function of Ste. It is obvious from the figure that Nu ave decreases with ascending the values of Ste whilst the values of N gen 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. e latter leads to a decrease in the heat storage capacity of the NEPCM particles, which concluded a lower rate of heat transfer. For instance, Nu ave ascends from 4.6998 to 4.4673 (4.95% decreasing) while N gen increases from 41.24 to 44.81 (8.65% increasing) when Ste increases from 0.2 to 0.7 at θ f � 0.35. Figure 6(b) illustrates that the values of |Ψ max | increase while the ECOP descends with ascending Ste. erefore, the figure discovers that the heat transfer rate decreases although the fluid velocity enhances with increasing the Stefan number. Also, |Ψ max | 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 7 represents the variations of Nu ave , N gen , |Ψ max |, and ECOP versus ϕ. As shown, increasing ϕ is desirable since Nu ave and ECOP raise and N gen decreases with ascending ϕ. Albeit, the values of |Ψ max | decrease as ϕ increases. For example, Nu ave increases from 5.4591 to 6     us, a minimum value can be seen for N gen at N � 5. As shown by the Cr contours, the width of the ribbon shape is not the same everywhere. e ribbon shape is wider where the temperature gradients are smooth. Here, we have Cr min � 0.976 and Cr max � 7.2592 for ϕ � 0.04. Other decision parameters are Ha � 18, β � 0°, and Ste � 2.  Figures (a) and (b) discovers that N T,FF increases from 10% to 43% while N T,HT decreases from 74% to 15% as the Rayleigh number attains from 10 4 to 10 5 . Its reason is increasing fluid flow velocity (ascending the values of |Ψ max | ) 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 N T,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 N T,HT is justifiable.

Conclusion
e entropy generation and heat transfer analysis were investigated in a complex cavity filled with NEPCMs suspension. e nondimensional governing equations were solved by CVFEM. Here, significant outcomes are remembered as follows: while N gen has a maximum at Ha � 16. Also, there is a maximum value for Nu ave at β � 60°( Figure 4). (iii) Symmetrical behavior was seen for Nu ave and N gen with respect to θ f � 0.5. e maximum value of Nu ave is 3.268 that occurs at θ f � 0.4 and θ f � 0.6 while there is a minimum value for N gen that equals 39 and happens at θ f � 0.5 ( Figure 5). (iv) Lower values of Ste are desirable because Nu ave descends and N gen ascends with Ste ( Figure 6). Indeed, as the latent heat of the PCM cores rises, the Stefan number decreases. (v) Nu ave increases 19.9%, and ECOP increases 28.8% while N gen decreases by 6.9% when ϕ enhances from 0 to 0.06 ( Figure 7). (vi) Higher values of Da are suitable since the values of ECOP and Nu ave increase while N gen descends with ascending Da (Figure 8). It must be recalled that the ECOP represents the performance of the cavity. (vii) e effects of A 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) e contribution of N T,MF ascends from 22% to 49% as Ha goes up from 12 to 24. Also, the value of N T,FF increases while the value of N T,HT descends with ascending Ra (Figures 11).