Justification of Pore Configuration of Metal-Foam-Filled Thermal Energy Storage Tank: Optimization of Energy Performance

Abstract

Thermal energy storage (TES) is a crucial technology for mitigating energy supply–demand mismatches and facilitating the integration of renewable energy. This study proposes a novel horizontal phase change TES unit integrated with partially filled metal foam (MF) and fins, divided into six sub-regions (ε₁–ε₆) with graded pore parameters. A comprehensive numerical model is developed to investigate the synergistic heat exchange mechanism and energy storage performance. The results demonstrate that porosity in Porosity-1 (ε₁) and Porosity-2 (ε₂) regions dominates melting dynamics. Through multi-objective optimization, targeting both minimal energy storage time and maximal energy storage rate, an optimal configuration (Case TD) is derived after technical design. Case TD features porosity values ε₁ = ε₂ = ε₃ = ε₅ = ε₆ = 0.97 and ε₄ = 0.98, where the graded porosity distribution balances heat conduction efficiency and energy storage capacity. Compared to the uniform MF case (Case 1) and the fin-only case (Case 6), Case TD reduces TES time by 51.75% and 17.39%, respectively, while increasing the mean TES rate by 102.55% and 19.12%, respectively. This design minimizes the TES capacity loss (only decreasing by 2.14% compared to Case 1) while maximizing the energy storage density and improving the efficiency–cost trade-off of the phase-change material-based system. It provides a scalable solution for rapid-response TES applications in solar thermal power plants and industrial waste heat recovery.

1. Introduction

In terms of mitigating climate change and reducing the reliance on fossil fuels, the global transition to a sustainable energy system has become a crucial task [1]. Renewable energy sources play a key role in this transformation by providing clean, abundant, and environmentally friendly alternative energy sources [2]. However, the inherent intermittency and variability of these energy sources pose significant challenges to energy stability [3]. Energy storage technology has become a crucial driving force in addressing these challenges, facilitating the effective integration of renewable energy into the power system [4,5]. Energy storage solutions not only improve the economic feasibility but also accelerate the broader energy transition.

Thermal energy storage (TES) has garnered significant attention due to its efficiency and cost effectiveness [6], in particular latent heat storage, which absorbs and releases thermal energy in PCMs [7]. These characteristics make phase change heat storage (PCHS) particularly suitable for application in solar energy and building heat control [8]. However, the properties of the PCHS system are significantly affected by the thermal conductivity of phase change materials (PCMs). Most commercial PCMs inherently have low thermal conductivity [9], slow heat transfer rates, prolonged heat storage/release cycles, and uneven phase transitions. These factors severely limit the system efficiency and power density [10].

To alleviate these challenges, researchers have explored various strategies [11], including the addition of high-thermal-conductivity additives, packaging technologies, and optimization of heat exchanger designs [12,13]. These improvements aimed to enhance thermal diffusion while maintaining high energy storage capacity. Adding high thermal conductivity through metal foams or fins can effectively improve the overall heat exchange performance [14]. Zhuang et al. [15] proposed a metal foam (MF) structure with multi-layer gradient porosity for optimizing the TES characteristics and modeled the melting process. The influence of porosity gradient at non-equal layer heights and PPI on energy storage time and heat transfer process was researched. The TES of the negative gradient was greater than that of the positive gradient, but the mean TES rate was slightly lower. Regardless of the porosity gradient structure, an increase in pores per inch (PPI) could enhance the average TES rate and the heat conduction process. Righetti et al. [16] experimentally studied the improvement of the TES and discharging processes of PCMs in a central tube heat exchanger with three kinds of metal foam. Pore densities of 10, 20, and 40 pores per inch, water flow velocity of 2–81 L/min⁻¹, and a temperature of the heat transfer fluid of 45–55 °C were selected as the research variables. The MF with 20 holes per inch minimized the TES and discharging time, and the inclination angle had no effect on the charge/discharge times of the foam-saturated PCMs. Mirshekar et al. [17] designed four energy storage structures with/without PCM and with/without copper MF and studied the experimental processes of the heating and cooling processes of the radiator. By adjusting various thermal power and inlet air velocity parameters during the TES process, the TES time of sample 1 (paraffin in copper MF) was reduced by 20.1% compared to sample 4 (without MF and paraffin), and the overall temperature was significantly reduced during the cooling process. For the triple tube heat exchanger, NematpourKeshteli et al. [18] conducted numerical research on the TES process of PCMs using three enhanced heat exchange methods (MFs, nanoparticles, and fins). The melting time of the PCM/MF structure with a porosity ranging from 0.98 to 0.92 was greatly reduced. After adding nanoparticles with a volume fraction of 5% to it, the maximum TES time was reduced by 83.48%. The melting time of the nanoparticles coated with foam was shortened by 53.17%. Shirbani et al. [19] investigated the influence of pore arrangement in uniform closed-cell aluminum MF on the TES performance of PCMs. The simulation of the heat flow problem was carried out through the lattice Boltzmann solver. The results showed that the alternating arrangement of layers/holes with an average pore diameter of 10 mm had the best effect, which was sufficient to promote the overall heat transfer. Compared to the other two arrangements, the TES time was reduced by 20%. However, large average apertures (10 mm and 15 mm) require more energy storage time. Li et al. [20] experimentally researched the influence of MF (20 PPI, 40 PPI) and parallel fins (5, 10) of shell-and-tube TES devices. The results showed that, with an energy density loss of only 7%, ten metal fins reduced the energy storage time of PCM by 47%. Moreover, the power density of the entire device increased significantly by 70%. The use of MF suppresses natural convection and increases thermal conductivity, but an overly dense structure can lead to a deterioration in heat transfer.

However, MFs still have disadvantages, including high cost, increased weight, reduced effective capacity of PCM, long-term reliability issues (such as thermal cycling stability and corrosion), and possible suppression of natural convection [21,22]. It is still worthy of further exploration to further optimize the matching between the geometric parameters of metal addition (porosity and fins’ spacing) and working conditions (such as temperature field and flow rate) through means such as gradienting the porosity of MF or fin layout and optimizing the synergy of heat transfer and energy storage. Therefore, in this paper, for the TES process of a horizontal PCHS unit, we propose a novel horizontal PCHS unit divided into six sub-regions with graded MF porosity and establish a comprehensive numerical model to analyze its melting dynamics, thermal behavior, and energy storage characteristics. We identify the influence of porosity distribution (especially in key sub-regions) on the melting rate, storage efficiency, and uniformity of PCMs. We derive an optimal porosity configuration (via multi-objective optimization targeting minimal TES duration and maximal mean TES rate) that balances heat conduction efficiency and PCM storage capacity. We validate the superiority of the optimized design by comparing its performance with uniform MF configurations and fin-only structures, thereby providing a scalable solution for rapid-response TES applications in solar thermal systems and industrial waste heat recovery. Through an in-depth performance comparison, the enhancement mechanism of the synergistic heat transfer in this PCHS unit is revealed, highlighting its impact on energy storage characteristics.

2. Problem Description

2.1. Physical Model

As illustrated in Figure 1, this diagram depicts the application process of phase change heat storage (PCHS) technology in the mobile energy storage and power supply system for renewable energy. By integrating phase change materials (PCMs), the system converts intermittently generated renewable energy into thermal energy and stores it efficiently, thereby addressing issues related to energy supply fluctuations and delayed energy demand. Compared with traditional energy storage methods, PCHS can ensure energy continuity while improving energy utilization rate and system flexibility. Figure 2a presents a structural schematic of the PCHS unit. Six copper fins are attached to the inner tube, through which the heat transfer fluid (HTF) flows inside. The heat exchanger adopts a symmetric structure, where the right half is identical to the left half in terms of porosity distribution. The inner and outer tubes are filled with PCMs or MF composite PCMs. As shown in Figure 2b, it is a two-dimensional structural schematic diagram of the PCHS unit. The inner (R₃) and outer diameter (R₁) of the outer tube are 50 mm and 20 mm, and the thickness of the wall is 1 mm. The internal PCM area is divided into six sub-areas through fins, and the inner diameter R₂ = 35 mm. The length (l) and width (w) of the copper fins are 25 mm and 1 mm, and the six sub-regions are ε₁–ε₆, respectively. They are used to fill aluminum metal foam structures with different porosities. Due to the relatively small influence of pore density, the pore density of all structures is 30. The properties of the PCMs and metals are given in

Table 1. Thermophysical parameters [24,25].

Property	Cu	Lauric Acid	Paraffin RT82	Unit
Isobaric specific heat (cp)	381	2275	2000	J/kg·K
Density (ρ)	8978	915	860	kg/m3
Volumetric coefficient of thermal expansion (β)	-	0.0008	0.001	K−1
Melting temperature (Tm)	-	319.15	353.15	K
Thermal conductivity (k)	387.6	0.14	0.2	W/m·K
Latent heat of fusion (λ)	-	187,210	176,000	J/kg
Solidus temperature (Ts)	-	316.65	351.15	K
Dynamic viscosity (μ)	-	0.008	0.03499	Pa·s
Liquidus temperature (Tl)	-	321.35	355.15	K

. The different combinations of porosity filled in the internal sub-regions of different structures are shown in Table 2.

2.2. Mathematical Model

First, we put forward the following hypotheses [26,27]:

(1)

The copper foam is isotropic and uniform;

(2)

The flow in the pores is laminar flow;

(3)

During the phase transformation process, the volume change of molten PCM is ignored;

(4)

The thermal performance parameters of PCM are constant during the phase transition process.

The equations are as follows [28,29]:

$${\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }x}}+{\displaystyle \frac{\mathsf{\partial }v}{\mathsf{\partial }y}}=0$$

(1)

$${\displaystyle \frac{\mathsf{\partial }\left(\rho u\right)}{\mathsf{\partial }t}}+u{\displaystyle \frac{\mathsf{\partial }\left(\rho u\right)}{\mathsf{\partial }x}}+v{\displaystyle \frac{\mathsf{\partial }\left(\rho u\right)}{\mathsf{\partial }y}}={\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\left(\mu {\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }x}}\right)+{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }y}}\left(\mu {\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }y}}\right)-{\displaystyle \frac{\mathsf{\partial }p}{\mathsf{\partial }x}}+Su$$

(2)

$$\begin{array}{l}{\displaystyle \frac{\mathsf{\partial }\left(\rho v\right)}{\mathsf{\partial }t}}+u{\displaystyle \frac{\mathsf{\partial }\left(\rho uv\right)}{\mathsf{\partial }x}}+v{\displaystyle \frac{\mathsf{\partial }\left(\rho vv\right)}{\mathsf{\partial }y}}=\\ {\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\left(\mu {\displaystyle \frac{\mathsf{\partial }v}{\mathsf{\partial }x}}\right)+{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }y}}\left(\mu {\displaystyle \frac{\mathsf{\partial }v}{\mathsf{\partial }y}}\right)-{\displaystyle \frac{\mathsf{\partial }p}{\mathsf{\partial }y}}+\rho g\beta \left(T-T_0\right)+Sv\end{array}$$

(3)

The S of Darcy’s law is defined as

$$S={\displaystyle \frac{{\left(1-f_m\right)}^2}{\left(f_m^3+\epsilon \right)}}{A}_{mush}$$

(4)

The Boussinesq hypothesis in this study is as follows:

$$\left(\rho -{\rho }_0\right)g=-{\rho }_0\beta \left(T-T_0\right)$$

(5)

The energy equation is as follows:

• where in the paste region, A_mush is 10⁶ kg/m³, which refers to the velocity momentum [30] in order to prevent the denominator from being 0 [31], and $\epsilon ={10}^{-4}$. $f$ is the liquid fraction:

  $$f=\left\{\begin{array}{cc}0\hfill & T_{\mathrm{PCM}}<T_{solidus}\hfill \\ {\displaystyle \frac{T_{\mathrm{PCM}}-T_{solidus}}{T_{liquidus}-T_{solidus}}}\hfill & T_{solidus}\le T_{\mathrm{PCM}}\le T_{liquidus}\hfill \\ 1\hfill & T_{\mathrm{PCM}}<T_{liquidus}\hfill \end{array}\right.$$

  (6)

($\stackrel{·}{f}$) is the melting rate [32]:

$$\stackrel{·}{f}={\displaystyle \frac{\mathsf{\partial }f}{\mathsf{\partial }t}}$$

(7)

The total enthalpy is h

$$h=h_{sen}+h_{lat}$$

(8)

$$h_{sen}=h_{ref}+{\displaystyle {\int }_{T_{ref}}^T{c}_{p}dT}$$

(9)

$$h_{lat}=f\lambda$$

(10)

The amounts of heat absorbed are

$$Q_s=m{c}_p\left(T_c-T_i\right)$$

(11)

$$Q_l=m\lambda f$$

(12)

$$Q=Q_s+Q_l$$

(13)

The instantaneous and mean TES rates are [33]

$$\stackrel{·}{q}={\displaystyle \frac{\mathsf{\partial }Q}{\mathsf{\partial }t}}$$

(14)

$$\overline{q}={\displaystyle \frac{\mathsf{\partial }{Q}_m}{\mathsf{\partial }{t}_m}}$$

(15)

The calculation process of MF is shown in Table 2.

Table 2. MF parameter calculation.

Parameter	Refs.	Correlation
Specific area (${\alpha }_{sf}$)	[34]	${\alpha }_{sf}={\displaystyle \frac{1.18\omega }{0.0224}}\sqrt{3\pi \left(1-\epsilon \right)}$
Effective thermal conductivity ($k_e$)	[35]	$k_e={\displaystyle \frac{k_s\left(1-\epsilon \right)}{\left(1-e+\frac{3e}{2a}\right)\left[3\left(1-e\right)+\frac{3}{2}ae\right]}}+k_f\epsilon , a=1.5, e=0.3$
Inertial coefficient (F)	[36]	$F=0.095{\displaystyle \frac{c_d}{12}}\sqrt{{\displaystyle \frac{\epsilon }{3\left(\tau -1\right)}}}{\left({\displaystyle \frac{1.18}{G}}\sqrt{{\displaystyle \frac{1-\epsilon }{3\pi }}}\right)}^{-1}, c_d=1.56$
Local heat transfer coefficient ($h_{sf}$)	[37,38]	$h_{sf}=\left\{\begin{array}{c}\left(0.35+0.5R{e}^{0.5}\right)k_f/d_f, 0<Re ⩽ 1\\ 0.76R{e}^{0.4}P{r}^{0.37}{k}_f/d_f, 0<Re ⩽ 40\\ 0.52R{e}^{0.5}P{r}^{0.37}{k}_f/d_f, 40<Re ⩽1000\\ 0.26R{e}^{0.6}P{r}^{0.37}{k}_f/d_f, 1000<Re ⩽ 20000\end{array}\right.$
Permeability (K)	[38]	$K={\displaystyle \frac{\epsilon \left[1-{\left(1-\epsilon \right)}^{1/3}\right]}{108\left[{\left(1-\epsilon \right)}^{1/3}-\left(1-\epsilon \right)\right]}}{d}_p^2$
Thermal dispersion conductivity ($k_{td}$)	[39]	$k_{td}={\displaystyle \frac{C_{td}}{1-\epsilon }}{\rho }_f{C}_{p,f}{d}_f\sqrt{u^2+v^2+w^2}, C_{td}=0.36$ $d_p=0.0224/\omega$ $\tau ={\displaystyle \frac{\epsilon }{1-{\left(1-\epsilon \right)}^{1/3}}}$, $G=1-e^{-\left(1-\epsilon \right)/0.04}$

Table 2. MF parameter calculation.

Parameter	Refs.	Correlation
Specific area (${\alpha }_{sf}$)	[34]	${\alpha }_{sf}={\displaystyle \frac{1.18\omega }{0.0224}}\sqrt{3\pi \left(1-\epsilon \right)}$
Effective thermal conductivity ($k_e$)	[35]	$k_e={\displaystyle \frac{k_s\left(1-\epsilon \right)}{\left(1-e+\frac{3e}{2a}\right)\left[3\left(1-e\right)+\frac{3}{2}ae\right]}}+k_f\epsilon , a=1.5, e=0.3$
Inertial coefficient (F)	[36]	$F=0.095{\displaystyle \frac{c_d}{12}}\sqrt{{\displaystyle \frac{\epsilon }{3\left(\tau -1\right)}}}{\left({\displaystyle \frac{1.18}{G}}\sqrt{{\displaystyle \frac{1-\epsilon }{3\pi }}}\right)}^{-1}, c_d=1.56$
Local heat transfer coefficient ($h_{sf}$)	[37,38]	$h_{sf}=\left\{\begin{array}{c}\left(0.35+0.5R{e}^{0.5}\right)k_f/d_f, 0<Re ⩽ 1\\ 0.76R{e}^{0.4}P{r}^{0.37}{k}_f/d_f, 0<Re ⩽ 40\\ 0.52R{e}^{0.5}P{r}^{0.37}{k}_f/d_f, 40<Re ⩽1000\\ 0.26R{e}^{0.6}P{r}^{0.37}{k}_f/d_f, 1000<Re ⩽ 20000\end{array}\right.$
Permeability (K)	[38]	$K={\displaystyle \frac{\epsilon \left[1-{\left(1-\epsilon \right)}^{1/3}\right]}{108\left[{\left(1-\epsilon \right)}^{1/3}-\left(1-\epsilon \right)\right]}}{d}_p^2$
Thermal dispersion conductivity ($k_{td}$)	[39]	$k_{td}={\displaystyle \frac{C_{td}}{1-\epsilon }}{\rho }_f{C}_{p,f}{d}_f\sqrt{u^2+v^2+w^2}, C_{td}=0.36$ $d_p=0.0224/\omega$ $\tau ={\displaystyle \frac{\epsilon }{1-{\left(1-\epsilon \right)}^{1/3}}}$, $G=1-e^{-\left(1-\epsilon \right)/0.04}$

The instantaneous and average melting uniformities are calculated, respectively, by calculating the liquid phase rate in different regions [13]:

$$\sigma \left(f\right)=\sqrt{{\displaystyle \frac{1}{6}}{{\displaystyle {\sum }_{i=1}^6\left[f({\epsilon }_i,t)-f(t)\right]}}^2}$$

(16)

$$\overline{\sigma }\left(f\right)={\displaystyle \frac{{\displaystyle {\int }_o^{t_m}\sigma \left(f\right)dt}}{t_m}}$$

(17)

where $f(t)$, $f({\epsilon }_i,t)$, respectively, represent the liquid fraction of the entire region and the melting fraction within sub-regions.

The uniformity of instantaneous and average temperatures in different regions is shown as follows:

$$\sigma \left(T\right)=\sqrt{{\displaystyle \frac{1}{6}}{{\displaystyle {\sum }_{i=1}^6\left[{\overline{T}}_{\mathrm{PCM}}({\epsilon }_i,t)-{\overline{T}}_{\mathrm{PCM}}(t)\right]}}^2}$$

(18)

$$\overline{\sigma }\left(T\right)={\displaystyle \frac{{\displaystyle {\int }_o^{t_m}\sigma \left(T\right)dt}}{t_m}}$$

(19)

where ${\overline{T}}_{\mathrm{PCM}}(t)$ and ${\overline{T}}_{\mathrm{PCM}}({\epsilon }_i,t)$, respectively, represent the temperature of the entire region and the temperature within sub-regions.

2.3. Initial and Boundary Conditions

Relevant usage conditions

$$t=0, T_i=300.15 \mathrm{K}$$

(20)

$$r=R_1=20 \mathrm{mm}, T=T_w=348.15 \mathrm{K}$$

(21)

Heat exchange boundary

$$-{\lambda }_{fin}{\left.{\displaystyle \frac{\mathsf{\partial }{T}_{fin}}{\mathsf{\partial }\overrightarrow{n}}}\right|}_{\Omega }=-{\lambda }_{pcm}{\left.{\displaystyle \frac{\mathsf{\partial }{T}_{pcm}}{\mathsf{\partial }\overrightarrow{n}}}\right|}_{\Omega }$$

(22)

$$T_{fin}{|}_{\Omega }=T_{pcm}{|}_{\Omega }$$

(23)

where $\Omega$ represents the contact surface between the fin and the PCMs.

3. Numerical Method and Model Validation

3.1. Numerical Processes

This study conducted numerical calculations using the CFD software Fluent 2023. Firstly, a structured triangular mesh was generated using ICEM, and local mesh refinement was performed on the heat source, fins, and metal foam boundaries (see Figure 2b for a typical structure). In terms of numerical settings, for the solution method, the first-order implicit format was used to handle the transient equations; the SIMPLE algorithm was employed to achieve pressure–velocity coupling; and spatial discretization was set up for energy, pressure, gradient, momentum, etc. The convergence criterion was that the residuals of the continuity equation in the X/Y direction reached momentum at 10⁻⁶, and the residual of the energy equation reached momentum at 10⁻⁸. At the same time, corresponding numerical values were set for the control of pressure, density, liquid phase fraction, momentum, and energy, with pressure control set at 0.3, density at 1, liquid phase fraction at 0.9, energy and momentum control at 0.8 and 0.7, respectively. The relevant settings are shown in

Table 3. Relevant numerical settings.

Solution Methods	Object
First-Order Implicit	Transient Formulation
SIMPLE Scheme	Pressure–Velocity Coupling
Spatial Discretization	Spatial Discretization	Energy
		Pressure
		Gradient
		Momentum
Absolute Convergence Criteria		Control of Solution
Continuity Equation	10−6	Pressure	0.3
Momentum Equation in X,Y Direction	10−6	Density	1
		Liquid Fraction	0.9
Energy Equation	10−8	Momentum	0.7
		Energy	0.8

, thereby establishing the numerical simulation process for the study.

3.2. Grid Independence and Time-Step Study

In Figure 3a, the temperature curves corresponding to different numbers of grid nodes (25,827, 50,167, 103,634, 157,331) are presented. With the advancement of the heat storage process, the values eventually tend to be similar. When the number of grid nodes is relatively small (25,827), the curve discreteness in the initial stage is slightly larger. When the number of nodes increases (such as 103,634, 157,331), the curve tends to concentrate earlier. It is indicated that when the number of grid nodes reaches a certain scale (103,634), the temperature calculation result is less affected by the number of grids, reflecting grid independence. The liquid phase rate curves in different grid numbers are given in Figure 3b. The curves with more nodes (103,634, 157,331) converge more quickly, while those with fewer nodes (25,827, 50,167) initially exhibit a finer discretization. This indicates that when the grid number is sufficient (103,634), the calculation result of the liquid phase rate is stable and meets the grid-independent requirement.

As shown in Figure 4, with different time steps (0.50, 0.25, 0.20, 0.10 s), the overall variation trend of the temperature curve is consistent. When the step size is reduced to a certain extent (0.20, 0.10 s), the curves almost overlap, indicating that the results converge under small step sizes. When the time step size is 0.20 s or smaller, the temperature calculation results are weakly affected by the step size, satisfying the validity of time-step size. Moreover, when the time step is lower than 0.20 s, the calculation result of the rate of liquid phase is stable, verifying the validity of the time step. For subsequent calculations, the number of grid nodes at 0.20 s and 103,634 is selected to ensure accuracy.

3.3. Model Validation

Initially, the phase transition process of the finned PCHS unit in this investigation is contrasted and evaluated against the experimental research presented by Abidi et al. [40]. RT-82 paraffin is selected as the PCM, and the constant surface temperature is maintained at 348.15 K. The heat charging results are presented in Figure 5a. The comparison reveals that the discrepancies in the liquid phase rates over time are below 4.37%, thereby validating the dependability of the melting/solidification model in characterizing the PCHS process.

Subsequently, to further validate the numerical solution accuracy of the MF model, it is compared with the experimental data of Tian et al. [41]. As shown in Figure 5b, the phase change energy storage structure used in this experiment is depicted. The RT58 was filled into a copper foam with dimensions of 200 × 200 × 25 mm to form an energy storage structure. The bottom surface heat flux of this structure was 1600 watts per square meter. The other three side walls were at a room temperature of 20 °C. The simulation process adopted the aforementioned boundary conditions and initial conditions and carried out corresponding numerical calculations. The natural convection heat transfer coefficient was assumed to be 3 W/m² K. Figure 5c illustrates the numerical calculation results under the local thermal non-equilibrium model employed in this study, which are in strong agreement with the experimental measurement results from Tian et al. Nevertheless, a minor deviation was observed during the phase transition completion stage, attributed to a degree of heat loss in the actual experiment. The comparison confirms the good reliability of the MF non-thermal equilibrium model and the numerical model of phase transition melting and solidification established in this study.

4. Results and Discussion

4.1. Study of Temperature and Liquid Fraction

This section selects different pore combinations to conduct comparative research of the energy storage process. After technical design, Case TD refers to the optimal porosity configuration obtained through multi-objective optimization, targeting a minimal TES time and a maximal mean TES rate. In this configuration, the six sub-regions of the PCHS unit are filled with aluminum metal foam with graded porosities: ε₁ = 0.97, ε₂ = 0.97, ε₃ = 0.97, ε₄ = 0.98, ε₅ = 0.97, and ε₆ = 0.97 (as shown in

Table 4. Internal porosity of different structures.

Case	${\mathit{\epsilon }}_1$	${\mathit{\epsilon }}_2$	${\mathit{\epsilon }}_3$	${\mathit{\epsilon }}_4$	${\mathit{\epsilon }}_5$	${\mathit{\epsilon }}_6$
1	0.99	0.99	0.98	0.97	0.99	0.98
2	0.99	0.97	0.99	0.98	0.99	0.98
3	0.98	0.98	0.99	0.97	0.98	0.99
4	0.97	0.99	0.99	0.99	0.99	0.99
5	0.97	0.98	0.98	0.98	0.98	0.98
6	0.97	0.97	0.97	0.97	0.98	0.98
TD	0.97	0.97	0.97	0.98	0.97	0.97

). This design strikes a balance between heat conduction efficiency and energy storage capacity, resulting in significant improvements in TES performance compared to uniform metal foam (Case 1) and fin-only (Case 6) configurations. As shown in Figure 6 and Figure 7, these figures represent, respectively, the liquid phase and temperature evolution processes of different cases, as well as the optimal structures at various moments during the PCHS process.

Through liquid phase comparison, the fin–MF structure significantly enhances the heat exchange efficiency of the phase change material to varying degrees. At the initial stage, t = 20 s, since the MF is not yet obvious, heat exchange is mainly carried out through the fins. Therefore, the liquid phase rate distributions of the seven structures are similar. During the mid-melting stage (t = 160–420 s), the porosity of 0.97 is relatively rapid for the expansion of the liquid phase. Therefore, the melting process of Case 6 is faster than Cases 1–5, while the liquid phase rate of Case TD increases uniformly, and there is no obvious lag area, indicating that its pore distribution balances the heat conduction and TES capabilities (t = 780–1500 s). Case TD is nearly completely melted at 1200 s. However, Cases 5 and 6 have melting delays due to different combinations of porosity, and there is still unmelted solid phase at 1500 s. However, compared with Cases 1–4, a significant improvement is achieved.

By comparing the temperature gradient distribution, the temperature uniformity field at each moment of Case TD is higher than that of the other structures, especially in the middle/later stages of TES. However, Cases 1–2 are affected by the high porosity area, resulting in local high temperatures in the middle of melting. However, the overall temperature of PCMs close to the outer tube is relatively low; the heat conduction efficiency is limited; and more time is needed to reach the phase transition temperature. At t = 1500 s, Case TD has almost completed the melting process, while the phase transformation of the remaining structures has not been completed.

As shown in Figure 8, the internal melting properties of the seven structures are compared. Through comparison of the mean temperature of PCM, the average temperature of Case TD is always higher than that of other structures. However, due to its shortest heat storage time, the temperature at the end of TES is lower than that of other structures. Through comparison of the liquid phase rate, the liquid phase rate of Case TD approaches 1 at t = 1500 s. The instantaneous melting rate of Case TD is highest in the early/middle stages of melting, and Cases 1–6 are all affected by the low melting rate for a relatively long time in the later stage of PCHS. Case TD is affected by the low melting rate for the shortest time; therefore, its melting is completed earliest.

Through comparison of the TES performance of the seven structures, as shown in Figure 9, the heat absorption of the TES unit is the combination of latent and sensible TES absorption. Since latent TES is related to the liquid phase rate, the latent TES change rate of Case TD is the fastest. Because the PCM temperature at the end of its melting is the lowest, the sensible TES absorption is lower than that in Cases 1–7. By comparing the instantaneous TES rates, it is evident that the instantaneous TES rate of Case TD is consistently the highest. However, in the middle and later stages, due to the weakened heat exchange strengthening effect of MF and metal fins, the heat storage rates of each structure continuously decrease.

Figure 10 shows the quantitative comparison at the end of TES. To quantitatively study the changes in melting time, a rate of change was proposed for comparison. Based on Case 1, the rate of change in melting time for the remaining structures is (tm, Case i − tm, Case 1)/tm, Case 1 × 100%. The total energy storage of Case TD decreased by 2.14% and 1.37%, respectively, compared with Case 1 and Case 6, while TES time decreased by 51.75% and 17.39%, respectively. Further comparison is made through the average heat storage rate. The mean TES rate of Case TD is increased by 102.55% and 19.12%, respectively. Meanwhile, the time required for heat storage is reduced.

This section compares the differences in uniformity characteristics of the internal melting process between Case 1 and Case TD, focusing on melting and temperature uniformity. Firstly, the liquid phase changes and temperature changes in different sub-regions are presented in Figure 11. In Case 1, the liquid phase rates of the ε₁, ε₃, and ε₅ regions completed the melting earlier than those of the ε₂, ε₄, and ε₆ regions. Moreover, due to the relatively weak enhanced heat transfer effect of MFs and fins on the PCM near the outer tube, the overall TES process is delayed. Compared with the internal TES process of Case TD, although the liquid phase rates in the ε₁, ε₃, and ε₅ regions also reach 1 earlier, the time required to complete the TES in the ε₂, ε₄, and ε₆ regions is also shorter. Combined with the temperature changes in different regions, in Case 1, due to the longer heating time, the temperature at the end of melting of the PCM is higher than in Case TD. Notably, the PCM temperatures in the ε₃ and ε₄ regions are almost equal to the heat source temperature at the end of melting. The PCM temperature in Case TD increased rapidly and maintained a high temperature for a short time.

4.2. Comparative Analysis of Melting and Temperature Uniformity

Figure 12 shows a comparison of instantaneous melting and temperature uniformity in the sub-regions of Case 1 and Case TD. After the melting begins, the uniformity of melting in different sub-regions rapidly increases to its maximum value, then decreases, and ultimately reaches 0. The instantaneous melting uniformity of Case TD is higher than that of the internal area of Case 1, and the action time is shorter. Especially when comparing the melting uniformity of the two ε₃ regions, the overall uniformity of Case 1 is significantly worse than that of Case TD. Combining the temperature uniformity in different regions, the internal uniformity of temperature of Case TD is higher, proving that the combination of its internal porosity greatly improves overall heat exchange uniformity and makes the melting process more balanced.

5. Conclusions

This paper focuses on the heat storage process of the horizontal PCHS unit. Heat exchange is carried out through the cooperative heat exchange between the MF and fins, and the heat storage unit is divided into six sub-regions to fill the metal foam with different pore parameters. A comprehensive numerical model is constructed, and the influences on the melting rate, energy storage rate, and uniformity are investigated. The main conclusions are as follows:

(1)

The use of low-porosity metal foam in the near-inner tube is more conducive to heat diffusion. The synergistic effect of fins and the metal foam is worth considering;

(2)

Compared with Cases 1 and 6, the total energy storage of Case TD decreased by 2.14% and 1.37%, respectively, while the PCHS time decreased by 51.75% and 17.39%, respectively;

(3)

Further comparison is made through the mean PCHS rate. The mean PCHS rate of Case TD was enhanced by 102.55% and 19.12%, respectively;

(4)

Through the comparison of uniformity of melting and temperature, the combination of pore structure of Case TD had a good improvement effect on the liquid phase evolution, temperature distribution, and TES efficiency of the PCHS process;

(5)

On the premise of only reducing a small amount of heat storage capacity, the TES rate of this unit was significantly increased, and the time required for heat storage was reduced accordingly.

(6)

The practical application of mobile thermal energy storage technology should undergo experiments and tests under actual working conditions, and it is worthy of further research.