Energy and Entropy Production of Nanofluid within an Annulus Partly Saturated by a Porous Region

The flow and heat transfer fields from a nanofluid within a horizontal annulus partly saturated with a porous region are examined by the Galerkin weighted residual finite element technique scheme. The inner and the outer circular boundaries have hot and cold temperatures, respectively. Impacts of the wide ranges of the Darcy number, porosity, dimensionless length of the porous layer, and nanoparticle volume fractions on the streamlines, isotherms, and isentropic distributions are investigated. The primary outcomes revealed that the stream function value is powered by increasing the Darcy parameter and porosity and reduced by growing the porous region’s area. The Bejan number and the average temperature are reduced by the increase in Da, porosity ε, and nanoparticles volume fractions ϕ. The heat transfer through the nanofluid-porous layer was determined to be the best toward high rates of Darcy number, porosity, and volume fraction of nanofluid. Further, the local velocity and local temperature in the interface surface between nanofluid-porous layers obtain high values at the smallest area from the porous region (D=0.4), and in contrast, the local heat transfer takes the lower value.


Introduction
Natural or free convection has vast, important applications in the industry field, such as electronic systems cooling [1], energy collectors solar [2], heat exchangers [3], electric ovens [4], desalination solar systems [5], melting [6,7] and freezing processes [8,9], etc. The heat transfer rate in natural convection is low as a result of the fluid velocity connected with natural convection is minimum. In natural convection, the fluid velocity is relatively low due to the fluid movement caused by the buoyancy force. Therefore, the heat transfer becomes weak also. For these reasons, a technique must be found to enhance the natural convection of heat transfer inside enclosures. One of the methods is to add nanoparticles to the base fluids, or use the porous media inside the enclosures.
The porous medium, for example, is a hard sedimentary rock, wood, bread, and a lung or a substance consisting of a solid structure with an interconnected vacuum. The fluid flow through the interconnected void depends on the porosity and the permeability of the media. Regarding shape and size, when the distribution of pores is unequal, the media is called natural porous media, for example, sandstone and bones [10]. Over the years, numerous researchers have made great progress in the subject of porous media inside enclosures. Porous media are perfect for improving conduction inside enclosures. On the other hand, it causes resistance to the fluid flow and weakens convection. Nanomaterials of infinitesimal sizes play a successful role in the heat transfer, so the nanofluids are used simultaneously with porous media to improve heat transfer [11][12][13]. Toosi and Siavashi [11] investigated the natural convection inside a square cavity partially filled with porous media and of Cu-water nanofluid. They used Corcione's model for a two-phase model and Darcy-Brinkman-Forchheimer to simulate fluid flow in porous media. They found that the convection becomes weak by using nanofluid and porous media, and also the conduction can be strengthened. Baghsaz et al. performed a study on natural convection heat transfer and entropy generation inside a porous cavity filled with Al 2 O 3 /water nanofluid. They found that the natural convection heat transfer, circulation, and irreversibility of the production increased clearly by increasing the Rayleigh number (Ra) and Darcy number (Da). Miroshnichenko et al. [14] simulated the natural convection in an open cavity containing porous layers and filled with alumina/water nanofluid, which penetrated the cavity from the open boundary. They used a single-phase nanofluid approach and the Brinkman-extended Darcy model for porous layers. Their results showed that the heat transfer rate was increased by the expansion of the range between the wall and the porous layer. Cho [15] analyzed the heat transfer of natural convection inside a porous cavity filled with nanofluid and containing a partially-heated vertical wall. He showed that for high values of Da and Ra, convection heat transfer dominates. Thus, the entropy generation and mean Nusselt number increases, whereas the Bejan number decreases. Using the finite volume method, Selimefendigil and Öztop [16] performed a study on magnetohydrodynamics forced convection of carbon nanotube of water nanofluid in a U-shaped cavity containing a porous region under the impact of a wall rippling. They illustrated that the variation of the Reynolds number and permeability of the porous medium affected the flow field and heat transfer. Using a Local Thermal Non-Equilibrium (LTNE) approach, Mehryan et al. [17] reported the fluid flow and heat transfer of natural convection inside a porous enclosure filled with Ag-MgO/water nanofluids. Alsabery et al. [18], Raizah et al. [19], Aly and Raizah [20], Javaherdeh et al. [21], Rao and Barman [22], Wang et al. [23] and Esfe et al. [24] executed numerical imitation studies to analyze the natural convection heat transfer of a nanofluid in porous cavities. They illustrated that one could enhance the heat transfer performance by assuming both porous media and nanofluid.
As noticed from the earlier research surveys, and to the best of the authors' knowledge, there is no outcome with the convection flow within horizontal annulus filled with nanofluid superposed porous layers. Accordingly, this work introduces the understanding of nanofluid superposed horizontal annulus porous layers via the fluid flow and heat transfer distributions.

Mathematical Formulation
The steady natural convection and heat transfer of water-Al 2 O 3 nanofluid within an annulus by radius rr and having an inner hot (T h ) cylinder by radius r is explained in Figure 1. The analyzed composite annulus cavity is divided into two segments, the first one (outer portion) is filled by nanofluid and the second one (inner portion) is loaded by a porous region saturated by a nanofluid. The outer surface is fixed with a cold temperature of T c . The edges of the domain (except for the interface surface between the nanofluid-porous layers) are supposed to remain impermeable. The mixed liquid inside the composite cavity performs as a water-based nanofluid holding Al 2 O 3 nanoparticles. The Forchheimer-Brinkman-extended Darcy approach and the Boussinesq approximation remain appropriate. In contrast, the nanofluid phase's convection and the solid matrix are in the local thermodynamic equilibrium condition. The set of porous media applied in the following output is glass balls (k m = 1.05 W/m· • C). Examining earlier specified hypotheses, the continuity, momentum and energy equations concerning the Newtonian laminar flow survive, formulated as: For the nanofluid region: (4) For the porous region: The subscripts n f and m view the nanofluid layer and porous layer. x and y are the fluid velocity elements, |u| = √ u 2 + v 2 denotes the Darcy velocity, g displays the acceleration due to gravity, ε signifies the porosity of the medium and K is the permeability of the porous medium, which is determined as [25]: Here, d m represents the average particle size of the porous region. The thermo-physical characteristics regarding the adopted nanofluid for 33 nm particle-size will be prepared as [26]: where Re B is defined as The molecular diameter of water (d f ) is given as [26] Presently we present the employed non-dimensional variables: The set scheme leads to the following dimensionless governing equations: In the nanofluid region: In the porous region: 1 1 The dimensionless boundary conditions of Equations (18)- (25) are: In the boundary of inner cylinder: In the boundary of outer cylinder: and the dimensionless boundary forms toward the interface between the nanofluid and porous layers will be obtained from (1) the continuity of tangential and normal velocities, (2) shear and normal stresses, (3) temperature and the heat flux that crossed the central interface and allowing an identical dynamic viscosity (µ n f = µ m ) into both layers. Therefore, the interface dimensionless boundary conditions can be addressed as the following: here D denotes the porous layer's thickness, and the subscripts + and − indicated that the corresponding measures are estimated while addressing the interface of the nanofluid and porous layers, respectively. Ra = signify the Rayleigh number and Prandtl number related to the used base liquid. Local Nusselt numbers (Nu n f and Nu i ) at the inner cylinder and interface wall are, respectively, as follows: here n and D determine the entire length of the inner cylinder heat source and interface wall, respectively. Lastly, the average Nusselt number at the inner heated cylinder is addressed by the following: The entropy production related to the nanofluid layer is provided by [27][28][29]: While the entropy production-related to the porous layer is addressed by [30]: In the dimensionless pattern, Equation (35) can be signified as: where N n f denotes the irreversibility distribution ratio within the nanofluid region and can be represented by: and S GEN,n f explains the dimensionless entropy production rate: The terms of Equation (39) will be divided based on the following scheme: here, S n f ,θ and S n f ,Ψ obtain the entropy generation regarding the heat transfer irreversibility (HTI) and fluid friction irreversibility (FFI) of the nanofluid layer, respectively.
Into the dimensionless structure, Equation (36) remains exposed as: where is the irreversibility distribution ratio in the porous layer and The terms of Equation (43) can be separated into the following form: where S m,θ and S m,Ψ are the entropy generation due to heat transfer irreversibility (HTI) and fluid friction irreversibility (FFI) of the porous layer, respectively.
In the dimensionless form, the local entropy generation can be expressed as: The global entropy generation (GEG) is obtained by integrating Equation (47) over the domain The Bejan number Be is defined as: When Be > 0.5, the HTI is dominant, while when Be < 0.5, the FFI is dominant.

Numerical Method and Validation
The governing dimensionless equations Equations (18)-(25) ruled with the boundary conditions Equations (26)-(31) are solved by the Galerkin weighted residual finite element technique. The computational region does discretize toward small triangular portions, as shown in Figure 2.
These small triangular Lagrange components with various forms are applied to each flow variable within the computational region. Residuals for each conservation equation exist and are accomplished through substituting the approximations within the governing equations. The Newton-Raphson iteration algorithm is adopted for clarifying the nonlinear expressions into the momentum equations. The convergence from the current numerical solution is considered, while the corresponding error toward each of the variables fills the resulting convergence criteria: For confirming the independence regarding the existing numerical solution at the grid size of the numerical region, different grid sizes are used to determine the average Nusselt number (Nu n f ), average temperature (θ avg ) and Bejan number (Be) for the case of Ra = 10 6 , Da = 10 −3 , φ = 0.02, D = 0.5 and ε = 0.5. The outcomes given in Table 1 designate irrelevant variations for the G5 grids and beyond. Hence, for all calculations into this work and similar obstacles to this subsection, the G5 uniform grid is applied.   Regarding the validation for the numerical data, the current results have been compared with earlier experimental and numerical outcomes achieved by Kuehn and Goldstein [31], as shown in Figure 3 concerning the natural convection heat transfer case within a horizontal concentric annulus saturated with pure fluid (water). Furthermore, the results are considered among earlier declared experimental results obtained by Beckermann et al. [32] via a natural convection problem inside a square cavity having both fluid and porous layers, as implemented in Figure 4. As explained by the earlier performed comparisons, the numerical outcomes of the present numerical code are essential to a high degree of reliability.

Results and Discussion
The outcomes described by streamlines, isotherms, and isentropic distributions are addressed within this segment. We have modified the following four parameters; Darcy number (10 −6 ≤ Da ≤ 10 −2 ), nanoparticle volume fraction (0 ≤ φ ≤ 0.04), dimensionless porous layer length (0.4 ≤ D ≤ 0.7) and the porosity of the medium (0.2 ≤ ε ≤ 0.8). The values of the Rayleigh number and Prandtl number are fixed at Ra = 10 6 and Pr = 4.623, respectively. Table 2 displays the thermophysical properties of the base fluid (water) and the solid Al 2 O 3 phases at T = 310 K. The reliance of the streamlines, isotherms, and isentropic lines on the Darcy numbers (Da) when φ = 0.02, D = 0.5 and ε = 0.5 has been shown in Figure 5. It is remarked that as Da increases from 10 −5 to 10 −2 , the absolute value of the streamline's maximum |ψ| max is strongly increasing. The reason is the altitude porous resistance of the nanofluid flow at lower Da. Thus, the streamlines' intensity around a circular cylinder within an annulus raises as Da increases. The isothermal lines filled the annulus at the lower Darcy parameter Da = 10 −5 compared to the higher Darcy parameter Da = 10 −2 . The isotherms' strength is rising across the top area of an annulus according to the influence in Da. Thus, the intensity of the isentropic lines is strongly rising as Da increases. At the lower Darcy parameter, Da ≤ 10 −4 , isentropic lines are distributed around the heater of an inner circular cylinder with fewer lines near the cold outer wall. Furthermore, an increment in Da augments the isentropic lines around the top area of an annulus, and it seems that the distributions of the isentropic lines are following the strength of the isothermal and streamlines contours.  Figure 6 presents the influences of Da toward the local velocity, local temperature, and local interface Nusselt number on the interface wall within nanofluid-porous layers, W at φ = 0.02 and ε = 0.5. In Figure 6a, it is observed that there are three peaks on the local velocity V across W around the locations of W = 0.5, 1, and 1.5, in which the local velocity is (V ≤ 0). Further, the local velocity, V is improving clearly as Da boosts. Figure 6b shows fluctuations in the local temperature across W below the variations on Da. When  Figure 7 depicts the impacts of porosity parameter ε, at the streamlines, isotherms, and isentropic lines at Da = 10 −3 , φ = 0.02 and D = 0.5. It is observed that the value of |ψ| max increases by 32.37%, as ε raises from 0.2 to 0.8. Thus, the streamline contours are affected by the variations on ε. There are minor reductions in the isothermal lines according to an increase in the value of ε. Whilst, the contours of the isentropic lines are clearly improved within an annulus according to an increase in the value of ε. Figure 8 presents the influences of ε on the local velocity, local temperature, and local interface Nusselt number on the interface surface within nanofluid-porous layers, W at Da = 10 −3 , φ = 0.02 and D = 0.5. The local velocity, V, showed a clear increase from W = 0.5 to 1, as ε and in the other parts of the interface wall, there is almost no change on the values of V according to the variations on ε. Further, there is a slight enhancement in the local temperature and local interface Nusselt number according to a reduction in ε. The physical representations of these results are returning to the medium's porosity definition as it is characterized from the ratio of pore volume and the total volume of material, and an increase in ε towards the unity represents the free fluid.  The dependency of the streamlines, isotherms, and isentropic lines on the various dimensionless length of porous layer D at Da = 10 −3 , φ = 0.02 and ε = 0.5 have been shown in Figure 9. The first remark is that |ψ| max reduces by 25% as D raises from 0.4 to 0.7. The reason returns to the nanofluid flow's porous resistance within an annulus at an extra length of a porous layer D. There is a slight enhancement in the isothermal contours according to an increase in D. Thus, the contours of isentropic lines are influenced by the variations on the length of a porous layer D. Figure 10 presents the local velocity, local temperature, and local interface Nusselt number at the interface wall between nanofluidporous layers under the variations on D at Da = 10 −3 , φ = 0.02 and ε = 0.5. It is observed that the lower value of D gives the highest values of the local velocity at the interface wall. Due to the expansion of D, the values of local velocity fluctuate between positive and negative values, as shown previously for an interface wall's location over the streamlines (Figure 9). Further, the local temperature's highest peak across the interface wall appears at the lowest value of D = 0.4. The profiles of Nu i are decreasing according to an increment in D.

Conclusions
This study applied the entropy generation on nanofluid superposed horizontal annulus porous layers via the fluid flow and heat transfer characteristics. Here, the values of local velocity, local temperatures, local Nusselt numbers, average Nusselt number, average temperature, and Bejan number are measured in the interface surface within nanofluid-porous layers. The following points are summarizing the main findings: • Increasing the Darcy parameter from 10 −5 to 10 −2 powers the value of streamlines' maximum and strengthens the intensity of the isotherms and isentropic lines within an annulus due to higher porous resistance at a lower Darcy parameter. Acknowledgments: We thank the respected reviewers for their constructive comments, which clearly enhanced the quality of the manuscript.

Conflicts of Interest:
The authors declare no conflict of interest.

Be
Bejan number C p specific heat capacity (J/kg K) Da Darcy number g gravitational acceleration, (m/s 2 ) GEG dimensionless global entropy generation D dimensionless length of the porous layer, (D = d/rr) k thermal conductivity, (W/m K) K permeability of porous medium, (m 2 ) N Avogadro number N m irreversibility distribution ratio related to the porous layer N n f irreversibility distribution ratio related to the nanofluid layer