Impact of Surface Undulation on Flow and Heat Transfer Characteristics in an Enclosure Filled with Nanoencapsulated Phase Change Materials (NEPCMs)

'e present study investigates the natural convection in a wavy enclosure caused by a thermal difference between a cold wall and a hot undulated wall. 'e enclosure is filled with hybrid nanofluids. 'e hybrid nanofluids are formed of a phase change material (PCM) suspended in the water. 'e PCM utilizes polyurethane as the shell and nonadecane as the core. 'e core absorbs or releases its energy in the shape of latent heat inside the water and contributes to thermal energy storage and heat transfer. 'e governing equations are expressed in PDEs and solved by the finite element method (FEM). Parametric studies were used to analyze the solid concentration, fusion temperature, amplitude of corrugation, number of corrugations, and Rayleigh number. It is found that the heat transfer rate enhances by the rise of the latent heat of the NEPCM cores. 'e global heat transfer can be improved by more than 12% by adding 1% of NEPCM particles volume fraction. However, the heat transfer tends to decrease by applying the wavy surface.


Introduction
Nanotechnology deals with extremely small things that enable us to modify individual atoms. Choi [1] designed nanometersized materials, nanofluids, to increase the thermal conductivity of the fluids. Natural convection in enclosures filled with nanofluid was conducted by Khanafer et al. [2,3], Hwang et al. [4], and Oztop and Abu-Nada [5,6].Öǧüt [7] considered the dispersed argentum in the water, while Rashmi et al. [8] engineered the alumina-water nanofluids. Sheikhzadeh et al. [9] and Motlagh and Soltanipour [10] studied flow field and heat transfer in an enclosure where the transport mechanisms include Brownian and thermophoretic diffusions. Khanafer et al. [2] and Motlagh and Soltanipour [10] concluded that dispersing nanoparticles in host fluid would considerably improve the heat transfer.
Heat and mass transfer efficiency are important in many engineering devices. Surface waviness and nanofluids are two of the techniques with the goal of reducing the amount of energy required. e change of flow field with changes in wavy walls together with conductive fluid is a special feature that is not encountered in the conventional cooling system. Convection heat transfer in enclosures filled with nanofluids having wavy wall was studied by Abu-Nada and Oztop [11]. ey concluded that the addition of alumina nanoparticles to the water increases the heat transfer rate. Esmaeilpour and Abdollahzadeh [12] studied natural convection and entropy generation of copper-water inside an enclosure having different types of vertical wavy walls. Nikfar and Mahmoodi [13] exhibited that considerable differences occur between the rates of heat transfer in the wavy enclosure for the two viscosity models of the nanofluids. Mansour and Bakier [14] indicated that the heat transfer rate can be optimized by adjusting the surface waviness together with the Rayleigh number and nanoparticle concentration. e analysis considering three different nanofluids was investigated by Cho et al. [15] and later Cho [16] considered wavy enclosure with partial heating. Enhancement of the heat transfer rate of a wavy surface by employing hybrid nanofluid was obtained by Takabi and Salehi [17]. Sheikholeslami et al. [18] included the magnetic parameter with different orientations of the wavy enclosure. Sheikholeslami and Ellahi [19] studied the hydrothermal treatment utilizing electrohydrodynamic nanofluid in an enclosure with a corrugated upper wall. e convection enhancement of nanofluid with double wavy walls having different phase deviations was investigated by Tang et al. [20]. Shirvan et al. [21] applied a cosine corrugated surface and Sheremet et al. [22] applied a nonuniform heating. An open wavy cavity filled hybrid nanofluid was considered by Ashorynejad and Shahriari [23]. Hashim et al. [24] observed that heat transfer inside the wavy enclosure is enhanced by introducing alumina and choosing the optimal number of undulations. Dogonchi et al. [25] applied CVFEM to predict magnetohydrodynamic convective flow of copper-water in the wavy enclosure. Dogonchi et al. [26] considered the impact of different shapes of nanoparticles in a locally heated complex wavy enclosure. ey concluded that the particle shape parameter plays a crucial role in the heat performance.
Solidification of nanoenhanced phase change material (PCM) in a wavy enclosure was considered by Kashani et al. [27]. A horizontal heated square cavity filled with nanoenhanced PCM for the flat surface was initially started by Sebti et al. [28].
en, Ho and Gao [29] experimentally studied this configuration. Later, Tasnim et al. [30] studied the convection influence on the melting process of the new type of PCM inside the porous enclosure. Abdollahzadeh and Esmaeilpour [31] proposed phase change material having nanofluids and corrugate surface. ey concluded that the increasing effects of heat transfer on solidification are due to distortion on the surface. Dhaidan [32] considered several types of enclosures such as rectangular, spherical, cylindrical, and annular. ey found that the nanostructures assist the heat transfer rate. Ghalambaz et al. [33] demonstrated that the hybrid nanoparticles enhance the fusion performance of nanoenhanced PCM in a square enclosure. Al-Jethelah et al. [34] compared melting processes between pure PCM and nano-PCM. Selimefendigil et al. [35] studied the conjugate heat transfer characteristic of nanofluid in a square enclosure, where a PCM is attached on the vertical wall. Bondareva et al. [36] reported that nano-PCM melts earlier for the tilted angle greater than 90°at the Rayleigh number.
e latest nanotechnology product is nanoencapsulated phase change materials (NEPCMs).
is new material separates a shell as a capsule and a core inside. e core is PCM with a specific fusion temperature. It enables absorbing or releasing large energy from the latent heat [37]. Fang et al. [38] proposed the NEPCMs composed in situ polymerization techniques. ey utilized the n-tetradecanes as the core and urea and formaldehyde for the cover. Sodium dodecyl sulfate was utilized as the emulsifier and resorcin was utilized as the system modifier. Jamekhorshid et al. [39] reviewed the methods utilized in the microencapsulated phase change materials (MEPCMs). Shchukina et al. [40] summarized some techniques used for the encapsulation phase change materials for the new design of energy storage device. Nazir et al. [41] proposed new ideas in enhancing the thermal energy storage that applies the PCMs through encapsulation. Recently, Ghalambaz et al. [42] and Hajjar et al. [43] investigated the enhancement heat transfer of NEPCMs in an enclosure having a flat surface. e present study aims to study the natural convection in a wavy enclosure filled with water containing nanoencapsulated phase change material. e main parameters in this simulation are Rayleigh number, solid volume fraction, fusion temperature, amplitude, and number of corrugations.
e natural convection problem of NEPCMs in an enclosure with corrugated surface could be applied to the solar heating heat exchangers. Here, the corrugated absorber improves the exergetic performance and the NEPCMs increase the thermal performance.

Model Formulation
A schematic model under consideration is illustrated in Figure 1. e enclosure has a size L. e right wall has a cold temperature T c , while the left corrugated wall has a hot temperature T h . Other walls are assumed to be insulated. e function of the left and the right wavy surfaces is where A and N are amplitude and number of corrugations, respectively. e NEPCMs are made of phase change materials. is advanced material contains an outer shell and a core inside. e outer shell is polyurethane and the core is nonadecane. e thermophysical properties of the applied materials are tabulated in Table 1. e inner subjects to a phase change to fluid in the fusion temperature interval with T h < T f < T c . e phase change temperature and the latent heat of the inner core are 305 (K) and 211 (kJ/kg), respectively. Specifically, the PCM core of the hybrid nanoparticles can absorb/release a notable amount of energy in the form of latent heat. Moreover, the latent heat of the nanoparticle cores can contribute to energy transport inside the base fluid.
e Boussinesq approximation is applied for the density variation. e suspension of NEPCMs and hot fluid is assumed to be uniform and stable. Based on these considerations, the continuity, momentum, and heat transfer equations can be written as follows [42,43]: where subscript b refers to the bulk properties of the hybrid nanofluid, g is the acceleration due to gravity, and ρ b , k b , and β b are the density, thermal conductivity, and thermal expansion coefficient of the base liquid at room temperature, respectively. e density of the mixture is written as a weighted relation of the water and NEPCM dispersed particles [45]: 2 Mathematical Problems in Engineering e symbols p and f denote NEPCM particles and the water, respectively. e density of NEPCM particles is defined as follows: where ρ sh is density of the shell, while ρ co is density of the core. ι is the core-shell mass ratio: ι ∼ 0.447 [44]. Furthermore, the density of the core is the mean of the fluid and solid phases of the phase change material. e specific heat capacitance of the suspension can be evaluated as e specific heat capacitance subject to no phase change, C p,p , is evaluated by the following function: It is noted that the heat capacity of the core is the mean of the heat capacities of the fluid and solid states. is is due to the nanoparticles core subject to solid and fluid phase; then the latent heat was engaged in the heat capacity of the NEPCM. is heat can be formulated by performing rectangular, triangular, or sinusoidal functions as follows [46]: where T Mr is the temperature range. is range prevents the discontinuity in energy balance. e overall heat capacity of the NEPCM core including the sensible and fusion temperature is evaluated based on T Mr : where   [44].
e functions shown below are performed to calculate the thermal conductivity of suspension consisting of nanoencapsulated particles [42]: e dynamic conductivity of suspension is where Nc and Nv in the above equations refer to the numbers of thermal conductivity and viscosity, respectively. e larger the values of numbers of thermal conductivity and viscosity, the larger the increase in the thermal conductivity and dynamic viscosity of the suspension. ese values are constant. Ghalambaz et al. [42] evaluated these values for some nanofluids and hybrid nanofluids. It is noted that these functions are valid only for nanofluids suspension when ϕ < 0.05. Dimensionless quantities were used as follows: e nondimensional mathematical formulations become e dimensionless boundary conditions are zΘ zY � 0 on bottom and top walls, and the Rayleigh number, Ra, and Prandtl number, Pr, are nondimensional quantities defined as It is considered that the thermal expansion of the NEPCMs is equal to the water and, therefore, Cr is the heat capacity of the mixture over the sensible heat capacity of the water: where the sensible heat capacity ratio λ, the melting range ε, and Stefan number Ste can be stated as follows: As extra information, the dimensionless fusion relation, f, is stated as where 4 Mathematical Problems in Engineering Here, Θ fu , the nondimensional fusion temperature, is e local Nusselt number calculated at the hot wall is Finally, the averaged Nusselt number at the hot surface is calculated as where W is the length of corrugated wall. e solution element of the mathematical calculation is generated by a triangular mesh. e triangular mesh arrangements calibrate specifically for fluid dynamics problems.

Computational Methodology
e Galerkin FEM uses the same function for both the test and basis functions. e integral equations can be solved using Galerkin weighted residual technique. e residual integrations of the continuity, momentum, and heat transfer equations in X and Y directions are employed by using Gauss's method. A Newton method is used to solve the system nonlinear equations in the matrix form. e iteration approach is adapted through PDEs solver COMSOL. e iteration is considered to be a convergence solution when the corresponding error of velocities and temperatures is equal to or less than 10 − 6 .
To achieve the optimal grid size, the grid independency test is conducted to make definite the variables independent from the grid calibration. Grid configurations are coarser, coarse, normal, fine, finer, extra fine, and extremely fine and are employed as shown in Table 2. e numerical results are observed to be free from variation by increasing the number of elements. Considering both time calculation and the error, an extra fine was selected for all the computations done in this paper. As a verification, current works (left) of streamlines are validated with that obtained by Adjlout et al. [47] (right) for the sinusoidal wavy wall at Ra � 10 5 , A � 0.05, and N � 1, 3 without NEPCM as shown in Figure 2.
Additional validation is performed by comparison with the published work of Ghalambaz et al. [42], which is natural convection of NEPCMs in an enclosure having flat surface at ϕ � 0.05, θ f � 0.3, Ste � 0.313, and Ra � 10 5 as shown in Figure 3. e comparison reveals good agreement with the reported studies. ese comprehensive verification efforts demonstrated the robustness and accuracy of the present computation.

Results and Discussion
Parametric studies were used to analyze the related nondimensional quantity: the amplitude of undulated surface, 0 ≤ A ≤ 0.3, the number of undulated surface, 3 ≤ N ≤ 6, the fusion temperatures, 0.1 ≤ θ f ≤ 0.5, the Rayleigh number, 10 0 ≤ Ra ≤ 10 6 , and nanoparticles concentration, 0.0 ≤ ϕ ≤ 0.05. e core weight of the nanoparticles is below water, and their shell weight is above water, so that the density ratio is below unity. e thermal expansion coefficient water is greatly above these particles. e dilute solution of NEPCMs has Nc � 23.8 and Nv � 12.5 [44]. e sensible capacity ratio of the NEPCM is 0.32. Figure 4 shows the development of the streamlines, isotherms, and heat capacity ratio distribution inside the wavy enclosure filled with 5% NEPCM for various numbers of corrugations N at θ f � 0.1, Ste � 0.2, A � 0.3, and Ra � 10 6 . e strength of the flow movement enhances by increasing the wavy frequency. e formation of eye vortices is also modified by calibrating the shape of undulated wall. e isotherms are almost unchanging by varying the undulated number. It is noted that the fluid volumes for all enclosures are equal. Figure 5 exhibits the development of the streamlines, isotherms, and heat capacity ratio distribution by increasing the fusion temperatures at Ra � 10 6 , ϕ � 0.05, Ste � 0.2, A � 0.3, and N � 4. Varying the fusion temperatures inconsiderably affects the streamlines and isotherms. e distribution of temperature at the boundary was not changed by varying θ f . At θ f � 0.1, 0.5, the double eyes of vortex appear in the core. e heat capacity ratio exhibits a ribbon-form area near the fixed temperature line of fusion. e blue area displayed in the Cr contours refers to the area of phase change. e ribbon form of the melting area is extremely thin adjacent to the right of the top wall. is observation justifies why the large temperature gradients were revealed at these regions. e blue area gets larger when the fusion temperatures get higher. When the fusion temperature approximates the hot or cold surfaces temperature, the heat transfer improvement because of the phase change is low. e melting zone moves to the right when the fusion temperature increases. is mechanism indicates that a large amount of the NEPCMs releases the latent heat and solidifies. Figure 6 shows the evolution of the streamlines, isotherms, and heat capacity ratio distribution by increasing the Rayleigh number at ϕ � 0.05, θ f � 0.1, A � 0.3, and N � 4. At a small Rayleigh number, the flows movement is weak; meanwhile, at a larger Rayleigh number, the flows movement is strong and denser thermal layer close to the right surface is formed.
is happens because the high Rayleigh numbers extend heating intensity, more particles are spread out inside recirculating zones, and, consequently, more deposition occurs in the lower zone of the enclosure. Phase change of the inner core happens when the temperature reaches the fusion temperature. is conveys that adjusting the heating intensity would modify the area of the phase change in terms of directions, locations, and area. is result is supported by the appearance of the heat capacity ratio, where the blue-ribbon-shaped area refers to the melting and solidification region. e area of phase change gets larger when the Rayleigh number gets stronger. Figure 7 shows a connection between the average Nusselt number, Nu, and the Rayleigh number, Ra, at ϕ � 0.05, θ f � 0.1, Ste � 0.2, and A � 0.3. Heat transfer rate decreases  Figure 2: Validation of present streamlines with that obtained by Adjlout et al. [47] for the sinusoidal wavy wall at Ra � 10 5 , A � 0.05, and N � 1, 3 without NEPCM. with increasing the undulation number. e reduction is more pronounced at a high Rayleigh number, Ra. is fact is due to the fact that the high convective flow covering almost the entire wavy cavity induces a sharply thermal gradient near the wavy boundary but, at the same time, the surface volume area increases so that the Nu ratio is reduced. e effect of waviness frequency on heat transfer becomes negligible for low heating intensity. Details relationship of Nusselt number against undulation amplitude for a small and a high gravitational force will be illustrated as follows. Figure 8 exhibits a connection between the average Nusselt number, Nu, and the Rayleigh number, Ra, at ϕ � 0.05, θ f � 0.1, and N � 4. Heat transfer performance is slightly modified by adjusting the undulation amplitude. e corrugated properties of the flow have insignificant effect because of the plain nature of the flow and the presence of eye convective currents in the center of the wavy cavity. Figure 9 exhibits a relationship between the average Nusselt number, Nu, and the Rayleigh number, Ra, at ϕ � 0.05, A � 0.3, Ste � 0.2, and N � 4. Heat transfer performance increases with increasing the fusion temperature. e fusion temperature is the change in its enthalpy resulting from providing heat to a specific quantity of the NEPCM to alter its properties from a solid to a liquid at constant pressure. e average Nusselt number is unchanged by increasing latent heat from 0.3 to 0.5. Effect of low or high heat of fusion on heat transfer becomes negligible for small Rayleigh number. e results in this figure confirm that the optimum fusion temperature of the NEPCM significantly depends on the heating intensity. temperature where the phase change area moves closer to isothermal surfaces. e highest heat transfers were achieved about θ f � 0.3 and θ f � 0.7. Between these values, there exists the local minimum of the heat transfer, especially at a relative small Stefan number. Figure 11 exhibits the influence of the heating intensity on the average Nusselt number versus the NEPCM particles volume fraction at A � 0.3 and N � 4. e enhancement of heat transfer is regardless of the Rayleigh number. e Nusselt number increases by increasing the NEPCM particle volume fraction. e influence of Rayleigh number on the heat transfer rate is different at small and high NEPCM particles volume fraction. At high NEPCM density and small buoyancy force, the core absorbs more heat from the heated surface and then releases fairly large energy when latent heat passes from one state. Hereby, using only 1% of NEPCM particles and with the high Rayleigh number, the heat transfer can be improved more than 12%.

Conclusions
e present study scrutinized the impact of NEPCM, waviness, magnitude of the temperature difference, and the nanoparticle concentration. e equations governing continuity, momentum, and energy inside the wavy cavity are stated in the dimensionless form. e latent heat released and absorbed throughout the phase change by the NEPCM is given in the formulation. e dimensionless forms of the mathematical model are solved numerically by means of the Galerkin FEM. Graphical result of the local and average Nusselt numbers is given and the plots of isotherms, streamlines, and heat capacity ratio are exhibited. e fluids are agitated by the melting, heater size, heating intensity, and  volume fraction parameters. Some important conclusions from the study are given as follows: (1) e undulation number, fusion temperature, and Rayleigh number are essential aspects in the flow circulation, structure, and its strength (2) e structure of the isotherm is not affected by the different fusion temperature value, but the area of phase change shifts upwards when the fusion temperatures get higher (3) Altering the latent heat from the wavy surface temperatures improves the success of the NEPCM in terms of thermal performance enhancement (4) e global heat transfer can be improved by more than 12% by adding 1% of NEPCM particles volume fraction. However, the heat transfer tends to decrease by applying the wavy surface

A:
Amplitude of undulations C p : Specific heat capacitance g: Accelerating due to gravity k: ermal conductivity L: Width and height of cavity N: Number of undulations Nu: Nusselt number Pr: Prandtl number Ra: Rayleigh number S: Surface area of undulated wall Ste: Stefan number T: Temperature u, v: Velocity components in the x-direction and y-direction x, y and X, Y: Space coordinates and dimensionless space coordinates.

Data Availability
e data used to support the findings of this study are included in this paper and available without any restrictions.