**1. Introduction**

In the last several decades, the heat transport of nanofluids in various shapes with varying outset and boundary conditions has been a popular research point. This is explained by the fact that these geometries have been widely used in real-world applications, such as building thermal management, electronic device cooling, biochemical and food processing, and renewable energy applications [1–7]. Raizah et al. [8] examined nanofluid natural convection (NC) flow inside a V-shaped cavity saturated with porous media. The findings showed that the best-case scenario for porous media is a horizontal heterogeneous porous medium. The buoyancy force is augmented with a Rayleigh number increase,

**Citation:** Abderrahmane, A.; Al-Khaleel, M.; Mourad, A.; Laidoudi, H.; Driss, Z.; Younis, O.; Guedri, K.; Marzouki, R. Natural Convection within Inversed T-Shaped Enclosure Filled by Nano-Enhanced Phase Change Material: Numerical Investigation. *Nanomaterials* **2022**, *12*, 2917. https:// doi.org/10.3390/nano12172917

Academic Editor: S M Sohel Murshed

Received: 3 August 2022 Accepted: 19 August 2022 Published: 24 August 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

which improves convective transport. In a triangular fin-shaped cavity, Khan et al. [9] presented a computational investigation of the convective heat transport of a hybrid nanofluid. Their findings demonstrate that both the nanoparticles' Rayleigh number and solid volume percentage raise the local and average Nusselt numbers. In a uniquely formed cavity, Ghalambaz et al. [10] analyzed the heat transmission and irreversibility of a hybrid nanosuspension. The findings demonstrate that raising the nanoparticle concentration accelerates the rate of entropy formation for all Rayleigh number values. Asmadi et al. [11] investigated how a hybrid nanofluid transfers heat naturally by convection within a U-shaped container with varying heating configurations. The findings indicate that the continuous heating setting delivers the optimum heat removal performance, whereas the oscillating heating setting performed the poorest. The magnetohydrodynamic free convection flow and heat transfer in an angled U-shaped enclosure loaded with Cu-water nanofluid were studied by Nabwey et al. [12]. The findings demonstrate that the mean Nu increases with dimensionless heat source position but decreases with heat source length and Ha number. A numerical study on the free convection flow of a hybrid nanofluid in a reservoir with a trapezoidal form under the impact of partial magnetic fields was conducted by Geridonmez et al. [13]. According to the results, heat transmission and fluid movement are inhibited by the partial magnetic field's vast effect zone.

Conventional heating and cooling systems rely on a working fluid with a limited thermal capacity. Several systems incorporate a greater flow or bigger volume to address this problem, which is an inadequate solution in some applications. In this context, NEPCM is one of the effective approaches for increasing the thermal efficiency of various systems. In this encapsulation technique, PCM is sealed inside nanoshells to prevent leakage. In this structure, the PCM-containing core layer may store and release enormous amounts of energy during melting and solidification at a constant fusion temperature. The use of NEPCM rather than only heat transfer fluid offers several benefits in many applications. NEPCM benefits from both heat transfer fluid and PCM characteristics [14–17]. For instance, it was recently shown that these could improve thermal storage characteristics and also for applications such as triple tube heat exchangers [18] or shell-and-tube heat exchangers [19]. To keep the temperature of lithium-ion batteries (LIBs) stable between 35 and 45 ◦C, Cao et al. [20] utilized water loaded with NEPCM particles. They found that raising the Reynolds number value from 70 to 100 increased the rate of heat transmission of LIBs by 12.1–17.2%. Raising the volume fraction from 0 to 3% also increased the heat transmission rate by 8.2–13.6%. Mohammadpour et al. [21] investigated NEPCM slurry's hydrodynamic and heat transfer characteristics inside a microchannel heat sink featuring two circular synthetic jets. The thermal performance improvement is maximized at 28.5% at 0.2 nanoparticle volume fraction and 180 out-of-phase actuation, according to simulation findings. On the other hand, the figure of merit falls when the concentration of NEPCM rises. In a conical diffuser, Iachachene et al. [22] investigated the turbulent flow of Al2O3, NEPCM, and a mixture of the two. The NEPCM nanofluid had the lowest pressure drop and the largest heat transfer improvements inside the diffuser, according to the findings of this investigation. The Nusselt numbers of NEPCM/Al2O<sup>3</sup> hybrid and Al2O<sup>3</sup> nanofluids were enhanced by 10% and 6%, respectively, whereas the Nusselt number of NEPCM nanofluids were raised by 15%. Analytical research on the laminar flow and heat transmission of water jet impingement augmented with NEPCM slurry was conducted by Mohaghegh et al. [23]. The results indicate that NEPCM slurry may greatly increase the system's cooling performance by increasing the liquid jet's ability to store latent heat. However, an ideal NEPCM concentration results in the system's maximal cooling performance (15%). To improve total heat transmission and lessen pressure drop, Doshi et al. [24] developed a water-based NEPCM nanofluid within a new microchannel with a wavy and irregular shape. The result demonstrates that the presence of NEPCM nanoparticles lowers the fluid domain temperature. NEPCM slurry and more conductive materials combined with heat sinks lessen the influence of thermal and frictional entropy creation as well.

Nano-encapsulated phase change material (NEPCM) is compact, has a high specific surface area, is thermally reliable, and has a wide range of potential applications. However, most of the process conditions in use today are rather complex, making it challenging to create NEPCM with good microscopic morphology and outstanding thermal characteristics [25–29]. Liu et al. [30] created a series of NEPCM using a simple sol-gel technique, using disodium hydrogen phosphate dodecahydrate for the core material and silicon dioxide for the shell material. The encapsulation ratio and melting enthalpy of the produced NEPCM reached maximum values of 70.1% and 165.6 J/g, respectively. According to the data, the adequate component ratio and suitable reaction conditions contribute to NEPCM's superior microscopic morphology and thermal characteristics. Stearic acid (SA)/Ag nanocapsules were synthesized by Huanmei Yuan et al. [31] utilizing a Pickering emulsifier and a chemical reduction process. The results demonstrated that the thermal dependability of the nanocapsules was assessed after 2000 thermal cycles, during which time their latent heat marginally decreased by 0.55%. The fabrication of N-Hexacosane-encapsulated Titania phase change composite using a sol-gel approach was achieved by Khanna et al. [32]. The findings showed that the NEPCM was solidified and liquefied at 52.08 ◦C and 54.02 ◦C, respectively, with latent heats of 127.37 J/g and 142.09 J/g. The thermogravimetric curves showed that the composite's overall thermal stability increased with the increasing titanium concentration. In another study [33], by using a chemical technique, they established how to manufacture silica NEPCM layered between exfoliated-graphite nanosheets. The results indicated that there were no chemical processes that occurring in the phase transition material, which had a diameter of 120–220 nm. Furthermore, at 57.9 ◦C and 48.1 ◦C, respectively, with latent heats of 126.7 J/g and 117.6 J/g, the solid–liquid phase transition of the NEPCM nanocomposite was observed. After 300 heat cycles, the NEPCM composites showed very high durability against thermal deterioration and 15.74 W/m K thermal conductivity.

Lately, the suspension of NEPCMs as a novel type of nanofluid was examined in various enclosures. Cao et al. [34] investigated the free convective of NEPCM nanofluid within an insulated chamber with two pipes acting as cooler and heater sources with a constant temperature boundary condition. They demonstrated how the NEPCM phase transition happens at low Rayleigh numbers but that it has no bearing on the heat transmission rate. Instead, it is heavily linked to the thermal conductivity of the nanofluid. The free convective heat transport in NEPCM nanofluid within a square enclosure that has been differentially heated and rotated with a constant, uniform counterclockwise rotational velocity was studied by Alhashash et al. [35]. The rotation parameter primarily influences the number of cell circulations, the number of inner vortexes, and the strength of those vortexes. Higher rotational speed results in less NEPCM phase transition and slower heat transfer. In an angled L-shaped chamber, Sadeghi et al. [36] studied the free convection and entropy formation of NEPCM. The findings show that the micro-rotation parameter, Stefan number, and nondimensional fusion temperature all negatively affected the NC heat transfer of NEPCMs and decrease the Nuavg by up to 42%, 25%, and 15%, respectively. On the other hand, the Nuavg was increased by up to 36% when more nanoparticles were present. Zidan et al. [37] investigated the NEPCM–water mixture NC flow in a reversed T-shaped porous cavity with two heated corrugated baffles. This research shows that increasing the Raleigh number causes escalation velocity fields and phase change zone structural changes, whereas decreasing the Darcy number has the opposite impact. Hussain et al. [38] looked into the free convection of NEPCM in a grooved enclosure with an oval form and saturated with a porous medium. According to the study's findings, raising the Darcy parameter reduces the porous flow's resistance, which in turn enhances the streamline strength and nanofluid movement. Furthermore, the enlarged radius of the inner oval form creates a barrier in the grooved cavity, which slows the passage of the nanofluid within.

In the literature, there are no studies on the natural convection of NEPCM in an inversed T-shaped cavity, including a trapezoidal fin subjected to a magnetic field. A better understanding of the impact of the studied parameters on heat transfer rates would be beneficial for design engineers, as cavities saturated with porous media are widely founded

in engineering applications, including (but not limited to) heat exchangers and electrical components. In this work, the natural convection of NEPCM in an enclosure saturated with porous media is handled by the high-order GFEM. The run simulations look at how viral parameters affect the contours of temperature, heat capacity ratio, and nanofluid velocity within an inverse T-shaped cavity. These parameters include the Darcy number, the concentration of NEPCM nanoparticles in the base fluid, the Rayleigh number, and the Hartmann number representing the intensity of the magnetic field. nanofluid velocity within an inverse T-shaped cavity. These parameters include the Darcy number, the concentration of NEPCM nanoparticles in the base fluid, the Rayleigh number, and the Hartmann number representing the intensity of the magnetic field. **2. Problem Formulation** As shown in Figure 1, we consider an inverted T-shaped porous cavity with a

the enlarged radius of the inner oval form creates a barrier in the grooved cavity, which

In the literature, there are no studies on the natural convection of NEPCM in an inversed T-shaped cavity, including a trapezoidal fin subjected to a magnetic field. A better understanding of the impact of the studied parameters on heat transfer rates would be beneficial for design engineers, as cavities saturated with porous media are widely founded in engineering applications, including (but not limited to) heat exchangers and electrical components. In this work, the natural convection of NEPCM in an enclosure saturated with porous media is handled by the high-order GFEM. The run simulations look at how viral parameters affect the contours of temperature, heat capacity ratio, and

*Nanomaterials* **2022**, *12*, x FOR PEER REVIEW 4 of 17

slows the passage of the nanofluid within.

### **2. Problem Formulation** trapezoidal fin at the bottom. The porous cavity is loaded with a nanofluid consisting of water as a base fluid and nano-encapsulated PCM (NEPCM) as nanoparticles. This novel

As shown in Figure 1, we consider an inverted T-shaped porous cavity with a trapezoidal fin at the bottom. The porous cavity is loaded with a nanofluid consisting of water as a base fluid and nano-encapsulated PCM (NEPCM) as nanoparticles. This novel substance's nanoparticles are made up of an outer shell and an inner core. The core is made of nonadecane, while the exterior is mostly made of polyurethane. Their thermal properties are shown in Table 1. The fusion temperature, which is restricted to this range *T<sup>h</sup>* < *T<sup>f</sup>* < *T<sup>c</sup>* is what characterizes the single-particle core. The latent heat of the core and the phase transition temperatures are estimated to be 211 (kJ/kg) and 305 (K), respectively. Overall, the PCM cores' capacity to absorb, store, and release heat energy distinguishes them. When combined with a base liquid such as water, they also effectively transmit heat energy. substance's nanoparticles are made up of an outer shell and an inner core. The core is made of nonadecane, while the exterior is mostly made of polyurethane. Their thermal properties are shown in Table 1. The fusion temperature, which is restricted to this range <sup>ℎ</sup> < < is what characterizes the single-particle core. The latent heat of the core and the phase transition temperatures are estimated to be 211 (kJ/kg) and 305 (K), respectively. Overall, the PCM cores' capacity to absorb, store, and release heat energy distinguishes them. When combined with a base liquid such as water, they also effectively transmit heat energy.

**Figure 1.** Physical problem. **Figure 1.** Physical problem.

**Table 1.** Thermophysical properties of the shell and core of the NEPCMs and the base fluid.


)

Water: base fluid 21 × 10−5 4.179 0.613 997.1 The governing equations are as follows [39,40]:

The governing equations are as follows [39,40]: The side walls are maintained at low temperature, *Tc*, while the trapezoidal fin walls are heated and kept at high temperature, Th with (*T<sup>h</sup>* > *Tc*), and the other walls are thermally isolated. For this problem, the flow is assumed to be steady and laminar. The driving force in the geometry under study is the buoyancy force due to the temperature difference between the trapezoidal fin and the sidewalls. A uniform magnetic field is applied. It is believed that the effects of Joule heating, radiation, displacement currents, and viscous dissipation are insignificant. Natural convection is approximated using the Boussinesq approximation in the buoyancy element of the momentum equation. Pressure adjustments do not affect the density of nanoliquids. Temperature gradients, on the other hand, alter the density. The particles are distributed uniformly throughout the host fluid, and dynamic and thermal equilibrium between the nano-additives and the base fluid is established.

$$\nabla \cdot \mathbf{v} = \mathbf{0} \tag{1}$$

$$
\rho\_b \cdot \nabla \mathbf{v} = -\nabla p + \nabla \cdot (\mu\_b \nabla \mathbf{v}) + (\rho \mathcal{J})\_b \mathbf{g} (T - T\_c) - \frac{\sigma\_{nf} B\_0^2 v}{\rho\_{nf}} \tag{2}
$$

$$(\rho \mathbb{C}\_p)\_b \mathbf{v} \cdot \nabla T = \nabla \cdot (k\_b \nabla T) \tag{3}$$

The current suspension is distinguished by its global density, which is expressed as [41]

$$
\rho\_b = (1 - \phi)\rho\_f + \phi\rho\_p \tag{4}
$$

where the symbols *f* , and *p* denote the base fluid and the added nanoparticles, respectively. The nanoparticle density of NePCM is provided below:

$$\rho\_p = \frac{(1+t)\rho\_{\rm co}\rho\_{\rm sh}}{\rho\_{\rm sh} + i\rho\_{\rm co}}\tag{5}$$

where the symbols *ρ*sh , *ρ*co and *ι* denote the shell's density, the core's density and the mass ratio of the core-shell (*ι* ∼ 0.447) [41], respectively.

Additionally, the core density of a PCM is the average of its solid and liquid phases. When using NEPCM, the water's specific heat capacitance value may be calculated as

$$\mathbb{C}\_{p,b} = \frac{(1-\phi)\left(\rho \mathbb{C}\_p\right)\_f + \phi \left(\rho \mathbb{C}\_p\right)\_p}{\rho\_p} \tag{6}$$

Heat capacitance is considered for a single-phase state, *ip*, *p*, is defined according to the following expression:

$$\mathcal{C}\_{p,p} = \frac{\left(\mathcal{C}\_{p,co} + \iota \mathcal{C}\_{p,sh}\right)\rho\_{co}\rho\_{sh}}{(\rho\_{sh} + \iota \rho\_{co})\rho\_{p}} \tag{7}$$

The heat capacity of the inner substance (core) is considered the average of the heat capacities of both states, solid and fluid. This is because whether the nanoparticle's core is in a solid or fluid state, the latent heat is changed in the form of the heat capacity of the NEPCM. This new form of heat can be defined as follows [42]:

$$\mathcal{C}\_{P,p} = \mathcal{C}\_{P,co} + \frac{h\_{sf}}{T\_{Mr}} \tag{8}$$

$$\mathcal{C}\_{P,P} = \mathcal{C}\_{p,\it 0} + \left\{ \frac{\pi}{2} \cdot \left( \frac{h\_{\rm sf}}{T\_{Mr}} - \mathcal{C}\_{P,\it 0} \right) \cdot \sin \left( \pi \frac{T - T\_{fu} + (T\_{Mr}/2)}{T\_{Mr}} \right) \right\} \tag{9}$$

$$\mathbb{C}\_{P,P} = \mathbb{C}\_{p,\text{c0}} + 2\left(\frac{h\_{fs}}{T\_{Mr}^2} - \frac{\mathbb{C}\_{p,\text{C0}}}{T\_{Mr}}\right)\left(T - T\_{fu} + \frac{T\_{Mr}}{2}\right). \tag{10}$$

Most of the researchers employ linear interpolation to deal with the phase change due to it is simplicity. However, in this work, we chose to characterize phase change by employing the sine function to assure function continuity in the whole domain, where *TMr* is the range of the temperature. This interval circumvents the discontinuity in the stability of energy. The total heat capacity of the NEPCM core incorporating fusion temperature and the sensible is determined based on *TMr*<sup>∗</sup>

$$\mathcal{C}\_{p,p} = \mathcal{C}\_{p,co} + \left\{ \frac{\pi}{2} \cdot \left( \frac{h\_{sf}}{T\_{Mr}} - \mathcal{C}\_{p,c0} \right) \cdot \sin \left( \pi \frac{T - T\_{fu} + \left( T\_{Mr}/2 \right)}{T\_{Mr}} \right) \right\} \gamma \tag{11}$$

where

$$\gamma = \begin{cases} 0, T < T\_{fu} - \frac{T\_{\text{Mr}}}{2} \\ 1, T\_{fu} - \frac{T\_{\text{Mr}}}{2} < T < T\_{fu} + \frac{T\_{\text{Mr}}}{2} \\ 0, T > T\_{fu} + \frac{T\_{\text{Mr}}}{2} \end{cases} \tag{12}$$

The suspension's thermal volume expansion rate is calculated as

$$
\beta\_b = (1 - \phi)\beta\_f + \phi\beta\_{p^-} \tag{13}
$$

To estimate the thermal conductivity of a combination including nano-encapsulated particles, the following definitions are stated [43]:

$$\frac{k\_b}{k\_f} = 1 + \mathcal{N}c\phi.\tag{14}$$

The dynamic conductivity of suspension is

$$\frac{\mu\_b}{\mu\_f} = 1 + Nv\phi \tag{15}$$

where *Nc* and *Nv* in the above expressions define the numbers of thermal conductivity and viscosity, respectively.

The higher the thermal conductivity and viscosity values, the higher the increment in the mixture's thermal conductivity and dynamic viscosity (water and PCM particles). These constant values were established by Ghalambaz et al. [44] for several hybrid nanofluids and nanofluids. It is determined that these expressions are acceptable only for nanofluids if *φ* < 5%. The used quantities were considered in dimensionless form as follows:

$$X = \frac{x}{L'}$$

$$Y = \frac{y}{L'}$$

$$\delta = \frac{\delta^\*}{L}$$

$$U = \frac{uL}{a\_f}$$

$$V = \frac{vL}{a\_f}$$

$$P = \frac{p\ell^2}{\rho\_f a\_f^2}$$

$$\Theta = \frac{T - T\_c}{T\_b - T\_c}$$

The nondimensional mathematical formulations become

$$\frac{\partial \mathcal{U}}{\partial X} + \frac{\partial V}{\partial Y} = 0 \tag{16}$$

$$
\left(\frac{p\_b}{\rho\_f}\right)\left(U\frac{\partial U}{\partial X} + V\frac{\partial U}{\partial Y}\right) = \frac{\partial P}{\partial X} + \text{Pr}\left(\frac{\mu\_b}{\mu\_f}\right)\left(\frac{\partial^2 U}{\partial X^2} + \frac{\partial^2 U}{\partial Y^2}\right) \tag{17}
$$

$$
\left(\frac{p\_t}{\rho\_f}\right)\left(\mathcal{U}\frac{\partial V}{\partial X} + V\frac{\partial V}{\partial Y}\right) = \frac{\partial P}{\partial Y} + \text{Pr}\left(\frac{\mu\_b}{\mu\_f}\right)\left(\frac{\partial^2}{\partial X^2} + \text{RaPr}\frac{(\rho\beta)\_b}{(\rho\beta)\_f}\right)\Theta \tag{18}
$$

$$\mathrm{Cr}\left(U\frac{\partial\Theta}{\partial X} + V\frac{\partial\Theta}{\partial Y}\right) = \frac{k\_b}{k\_f}\left(\frac{\partial^2\Theta}{\partial X^2} + \frac{\partial^2\Theta}{\partial Y^2}\right) - \frac{\sigma\_{\mathrm{hnf}}}{\sigma}Ha^2V\tag{19}$$

The dimensionless boundary conditions are

*U* = *V* = 0, Θ = 1 *on the trapezoidal fin at the bottom U* = *V* = 0, Θ = 0 *on the side walls U* = *V* = 0, *<sup>∂</sup>*<sup>Θ</sup> *<sup>∂</sup><sup>Y</sup>* = 0 *on the rest adiabatic walls*,

and Ra and Pr are no dimensional quantities of Rayleigh and Prandtl numbers, respectively:

$$\mathbf{Ra} = \frac{g\rho\_f\beta\_f\Delta T e^3}{\mathfrak{a}\_f\mu\_f} \tag{20}$$

$$\text{Pr} = \frac{\mu\_f}{\rho\_f \mathfrak{a}\_f} \tag{21}$$

also,

$$
\left(\frac{\rho\_b}{\rho\_f}\right) = (1 - \phi) + \phi \left(\frac{\rho\_p}{\rho\_f}\right) \tag{22}
$$

$$
\left(\frac{\beta\_b}{\beta\_f}\right) = (1 - \phi) + \phi \left(\frac{\beta\_p}{\beta\_f}\right) \tag{23}
$$

Given that it is presumed that the thermal expansion of water is equivalent to that of NEPCMs, *βb*/*β <sup>f</sup>* ∼ 1, Cr describes the ratio of heat capacity of the suspension over the water heat capacity: *Cr* = (*ρCp*)*<sup>b</sup>* (*ρCp*)*<sup>f</sup>* = (1 − *φ*) + *φλ* + *φ <sup>δ</sup>* Ste *f* . (Ste) is the Stefan number, and it is defined as follows:

$$\lambda = \frac{\left(\mathbb{C}\_{p,co} + \imath \mathbb{C}\_{p,\text{sh}}\right) \rho\_{cof} \rho\_{\text{sh}}}{\left(\rho \mathbb{C}\_{p}\right)\_{f} \left(\rho\_{\text{sh}} + \rho\_{\text{co}}\right)} \tag{24}$$

$$
\varepsilon = \frac{T\_{Mr}}{\Delta T} \tag{25}
$$

$$\text{Ste} = \frac{\left(\rho \mathcal{C}\_p\right)\_f \Delta T (\rho\_{\text{sh}} + \rho\_{\text{co}})}{\alpha\_f \left(h\_{\text{sf}} \rho\_{\text{co}} \rho\_{\text{sh}}\right)} \tag{26}$$

Additionally, the nondimensional fusion expression, *f* , is given as

$$f = \frac{\pi}{2} \sin \left( \frac{\pi}{\varepsilon} \left( \Theta - \Theta\_{f\mu} + \frac{\varepsilon}{2} \right) \right) \sigma \tag{27}$$

where

$$\sigma = \begin{cases} 0, & \Theta < \Theta\_{\rm fu} - \frac{\varepsilon}{2}, \\ 1, & \Theta\_{\rm fu} - \frac{\varepsilon}{2} < \Theta < \Theta\_{\rm fu} + \frac{\varepsilon}{2} \\ 0, & \Theta > \Theta\_{\rm fu} + \frac{\varepsilon}{2}. \end{cases} \tag{28}$$

Here, Θ*f u*, the nondimensional fusion temperature is

$$
\Theta\_{fu} = \frac{T\_{fu} - T\_c}{\varepsilon T}.\tag{29}
$$

The local Nusselt number of the heated side is obtained:

$$Nu = -(1 + Nc\phi)\frac{\partial\Theta}{\partial Y} \tag{30}$$

Additionally, the averaged form of the Nusselt number of the heated side is given as

$$
\overline{Nu} = \int\_{-0.5}^{0.5} Nu \mathbf{d}Y \tag{31}
$$
