**2. Methodology and Problem Definition**

Figure 1 depicts an illustration of the current computational models. Al2O3/noctadecane paraffin was used as the working fluid. Fins were modeled in four different configurations. Figure 2 depicts the specifics of four examples. At the start, all fluids had a solidus temperature. The fins and inner cylinder are maintained at hot temperature (312 K), and the outer cylinder is adiabatic.

**Figure 1.** Computational setup of LTESS: 3D model 2D cross‐section. **Figure 1.** Computational setup of LTESS: (**a**) 3D model (**b**) 2D cross-section. **Figure 1.** Computational setup of LTESS: 3D model 2D cross‐section.

ଷ

1

1

**Figure 2.** Different cases considered in this study. **Figure 2.** Different cases considered in this study. **Figure 2.** Different cases considered in this study.

∂ ሬ⃗ ⋅ ∇൰ ൌ ሺ െ <sup>1</sup>ሻଶ

#### *2.1. Problem Formulation 2.1. Problem Formulation 2.1. Problem Formulation*

൬ ∂

൬ ∂

To model this transitory process, we assumed that the flow was Newtonian and lam‐ inar. To account for the gravity force effect, Boussinesq estimation was used. The calcula‐ tion formulas are as follows [46–49]: ∇⋅ሬ⃗ ൌ 0 (1) To model this transitory process, we assumed that the flow was Newtonian and lam‐ inar. To account for the gravity force effect, Boussinesq estimation was used. The calcula‐ tion formulas are as follows [46–49]: To model this transitory process, we assumed that the flow was Newtonian and laminar. To account for the gravity force effect, Boussinesq estimation was used. The calculation formulas are as follows [46–49]:

$$
\nabla \cdot \vec{V} = 0 \tag{1}
$$

ିଵ

(5)

(5)

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

$$\left(\frac{\partial v}{\partial t} + \vec{V} \cdot \nabla v\right) = v\mathbf{C} \frac{\left(\lambda - 1\right)^2}{\varepsilon + \lambda^3} + \frac{1}{\rho\_{nf}} \left(-\nabla P + \mu\_{nf} \nabla^2 v\right) + \frac{1}{\rho\_{nf}} (\rho \mathfrak{F})\_{nf} \mathbf{g} \left(T - T\_{ref}\right) \tag{2}$$

$$\frac{\partial u}{\partial t} + \stackrel{\rightarrow}{V} \cdot \nabla u = u \mathcal{C} \frac{\left(\lambda - 1\right)^2}{\varepsilon + \lambda^3} + \frac{1}{\rho\_{nf}} \left(-\nabla P + \mu\_{nf} \nabla^2 u\right) \tag{3}$$

$$\left(\rho \mathbb{C}\_{p}\right)\_{nf} \frac{\partial \left(\rho L \lambda\right)\_{nf}}{\partial t} + \left(\rho \mathbb{C}\_{p}\right)\_{nf} \frac{\partial T}{\partial t} - k\_{nf} \nabla^{2} T = -\left(\rho \mathbb{C}\_{p}\right)\_{nf} \stackrel{\rightarrow}{V} \cdot \nabla T \tag{4}$$

൫൯ ିଵ൫൯ ൌ <sup>ሺ</sup><sup>1</sup> െ ሻ ൫൯ ൫൯ ିଵ൫൯ We considered *ε* = 10−<sup>3</sup> and *C* = 10<sup>5</sup> .

 ൌ ௦ ሺ1 െ ሻ (6) ൌ <sup>ሺ</sup><sup>1</sup> െ ሻ ൫൯ ௦ ൫൯ To forecast the NEPCM attributes, a single-phase model was used:

$$\left(\rho \mathbb{C}\_{\mathcal{P}}\right)^{-1}\_{f} \left(\rho \mathbb{C}\_{\mathcal{P}}\right)\_{nf} = \left(1 - \phi\right) + \phi \left(\rho \mathbb{C}\_{\mathcal{P}}\right)\_{s} \left(\rho \mathbb{C}\_{\mathcal{P}}\right)^{-1}\_{f} \tag{5}$$

௦ ൫൯ 

$$
\rho\_{nf} = \phi \rho\_s + \rho\_f (1 - \phi) \tag{6}
$$

$$
\dots \quad \dots \quad \dots \quad \dots \quad (6)
$$

$$(\rho \mathfrak{G})\_{nf} = \phi (\rho \mathfrak{G})\_s + (1 - \phi)(\rho \mathfrak{G})\_f \tag{7}$$

$$(\rho L)\_f = \frac{(\rho L)\_{nf}}{(1 - \phi)} = \frac{(\rho L)\_{nf}}{(1 - \phi)} \tag{8}$$

$$k\_{nf} = \frac{2k\_f + 2\phi\left(k\_s - k\_f\right) + k\_p}{k\_p - \phi\left(k\_s - k\_f\right) + 2k\_f}k\_f\tag{9}$$

$$
\mu\_{nf} = \frac{\mu\_f}{(1-\phi)^{2.5}}\tag{10}
$$

Table 1 lists the characteristics of both nanoparticles and the PCM.

**Table 1.** Properties of the PCM and alumina [33].


The enthalpy is formulated as [50]:

$$h = h\_{ref} + \int\_{T\_{\rm ret}}^{T} \left(\mathbb{C}\_p\right)\_{nf} dT \tag{11}$$

$$\lambda = \begin{cases} 1 & T < T\_l \\ \frac{T - T\_s}{T\_l - T\_s} & T\_s < T < T\_{l\prime} H\_\varepsilon = h + \lambda L \\ 0 & T < T\_s \end{cases} \tag{12}$$

The formulas of *S*gen, total , *S*gen, th , and *S*gen, f are:

$$\begin{array}{lcl} \mathcal{S}\_{\text{gen,total}} &= \mathcal{S}\_{\text{gen,th}} + \mathcal{S}\_{\text{gen,f}} \\ &= \frac{k\_{nf}}{T^2} \left[ \left(\frac{\partial T}{\partial \mathbf{x}}\right)^2 + \left(\frac{\partial T}{\partial \mathbf{y}}\right)^2 \right] \\ &+ \frac{\mu\_{nf}}{T} \left\{ 2\left[ \left(\frac{\partial u\_{\text{x}}}{\partial \mathbf{x}}\right)^2 + \left(\frac{\partial u\_{\text{y}}}{\partial \mathbf{y}}\right)^2 \right] + \left(\frac{\partial u\_{\text{x}}}{\partial \mathbf{y}} + \frac{\partial u\_{\text{y}}}{\partial \mathbf{x}}\right)^2 \right\} \end{array} \tag{13}$$

The no-slip boundary conditions are subjected to the previous system (*u* = *v* = *T* = 0) on the outer boundaries, while on the inner fins *u* = *v* = 0, *T* = *T<sup>h</sup>* .
