Numerical Modeling of Charging and Discharging of Shell-and-Tube PCM Thermal Energy Storage Unit

Abstract

This paper presents the results of a numerical study on transient temperature distributions and phase fractions in a thermal energy storage unit containing phase change material (PCM). The latent heat storage unit (LHSU) is a compact shell-and-tube exchanger featuring seven tubes arranged in a staggered layout. Three organic phase change materials are investigated: paraffin LTP 56, fatty acid RT54HC, and fatty acid P1801. OpenFOAM software is utilized to solve the governing equations using the Boussinesq approximation. The discretization of the equations is performed with second-order accuracy in both space and time. The three-dimensional (3D) computational domain corresponds to the inner diameter of the LHSU. Calculations are conducted assuming constant thermal properties of the fluids. The experimental and numerical results indicate that for paraffin LTP56, the charging time is approximately 8% longer than the discharging time. In contrast, the discharging times for fatty acids RT54HC and P1801 exceed their charging times, with time delays of about 14% and 49% for RT54HC and 25% and 30% for P1801, according to experimental and numerical calculations, respectively.

1. Introduction

The widespread use of renewable energy sources (RESs) necessitates addressing the challenges of energy storage, as highlighted in previous studies [1,2]. One effective method for retaining thermal energy involves the use of phase change materials (PCMs), as discussed in works such as [3,4]. However, one of the main drawbacks of latent heat storage units (LHSUs) is their suboptimal performance, which often leads to extended operational times. The performance of LHSUs can be improved by incorporating enhanced heat transfer methods [5,6].

For brevity, this article focuses solely on numerical studies regarding the application of PCMs in shell-and-tube LHSUs. Hendra et al. [7] numerically modeled the melting process in a horizontal shell-and-tube LHSU with a staggered tube bundle arrangement, positioning the organic PCM on the shell side. In their study, conduction was assumed to be the controlling heat transfer mechanism, employing a 2D model with constant thermophysical properties of the PCM. Luo et al. [8] investigated the influence of a tube layout on the melting process using a Lattice Boltzmann simulation. Esapour et al. [9] examined how the number and configuration of tubes affect the melting process of paraffin RT35. Their 3D calculations indicated that an increase in tube number results in shorter melting times. In another study [10], Esapour et al. demonstrated that raising the inlet temperature of the heat transfer fluid (HTF) leads to shorter melting times.

Al-Mudhafar et al. [11] conducted a numerical study on the melting process of RT82 in a tube bundle consisting of four tubes arranged in-line. Their findings revealed that additional plates welded to the tubes significantly accelerated the melting process. Sodhi et al. [12] established that increasing the tube number while simultaneously reducing their diameter shortens both melting and solidification times. Abreha et al. [13] numerically examined melting and solidification processes in an LHSU with nineteen finned tubes arranged in a staggered configuration, using erythritol as the PCM. They found that the solidification time was shorter than the melting time.

Park [14] studied the impact of the tube number and arrangement on the melting time and energy density in tube bundles with centrosymmetric, in-line, and staggered layouts. They concluded that although the tube number has little effect on the melting time, it significantly influences the energy density. Kudachi et al. [15] employed Fluent software to investigate charging and discharging processes in a shell-and-tube LHSU using OM65 as the PCM, discovering that the melting process took twice as long as solidification. Mahdi et al. [16] examined the effect of four-tube arrangements on lauric acid melting time, finding that an arc arrangement near the shell bottom was the most promising configuration for a three-tube bundle.

Shaikh et al. [17] demonstrated that increasing the number of tubes and raising the HTF inlet temperature substantially reduce the melting time of solar salt in a small tube bundle, considering a maximum of four tubes. Zaglanmis et al. [18] highlighted the significant impact of HTF inlet temperature on melting processes in a vertical LHSU, conducting 3D numerical simulations using an enthalpy porosity approach. Song et al. [19] researched the melting process of lauric acid across four in-line tubes, determining that finned tube bundles significantly outperform smooth tube bundles. Qaiser [20] investigated how tube geometry affects the melting process of stearic acid in a three-tube bundle by testing in-line and staggered layouts with various external cross-sections, concluding that triangular tubes with downward vertex orientation yield better performance. Vikas et al. [21] analyzed the effect of a five-tube bundle on paraffin wax melting using a 2D model based on an enthalpy porosity approach and Ansys Fluent 19.0 code, comparing ten layouts and finding that the staggered arrangement generally outperformed the in-line configuration, with a notable emphasis on the role of thermal stratification. In [22], Vikas et al. established that higher eccentricity leads to improved performance of a four-tube bundle.

This paper numerically and experimentally investigates the melting and solidification processes in a shell-and-multi-tube heat exchanger serving as an LHSU. Charging and discharging times are determined for three organic PCMs. The novelty of this study lies in the developed calculation procedure implemented within the OpenFOAM environment. Additionally, an original method for determining the liquid fraction has been proposed in the experimental studies.

2. Materials and Methods

The experimental setup consists of a latent heat storage unit (LHSU), cold and hot water loops, and a data acquisition (DAQ) system [23].

2.1. Experimental LHSU

The LHSU is constructed from acrylic glass and features a tube bundle. All fundamental dimensions are depicted in Figure 1a. The tube bundle is composed of seven stainless steel 1.4301 tubes with an outer diameter (OD) of 6 mm. Temperature measurements were taken using Class A Pt100 thermometers with an uncertainty of ±0.2 K. Figure 1b illustrates the positions of 18 thermometers used to measure the temperature distribution of the PCM. Figure 1c presents the numbering of the tubes in a staggered arrangement, with a pitch ratio s/OD = 4.5.

2.2. PCM Tested

The tested PCMs, namely, paraffin LTP 56, fatty acid RT54HC, and fatty acid P1801, were sourced from Polwax S.A. in Jaslo, Poland; Rubitherm Technologies GmbH in Berlin, Germany; and Konimpex Chemicals in Konin, Poland, respectively. The detailed chemical composition of these materials was determined using a GC-MS-QP2010 PLUS Shimadzu gas chromatograph [23], while their thermophysical properties were assessed using a TA Instruments Q1000 differential scanning calorimeter.

Table 1. Thermophysical properties [24,25,26].

Property	LTP56		RT54HC		P1801
	Solid	Liquid	Solid	Liquid	Solid	Liquid
λ [W/(mK)]	0.237	0.152	0.210	0.151	0.198	0.157
cp [kJ/(kgK)]	2.48	2.34	1.83	2.07	2.01	2.12
μ [mPas]	-	9.83	-	10.95	-	13.13
ht [kJ/kg]	189.6		195.4		181.9
hs [kJ/kg]	182.2		200.1		181.4
$T_t^{\prime }$ [K]	323.8		326.3		326.5
$T_t^{\prime \prime }$ [K]	335.6		336.3		338.1
$T_s^{\prime }$ [K]	325		324.7		325.8
$T_s^{\prime \prime }$ [K]	318.4		316.8		318.7
ρ [kg/m3]	850	800	850	800	920	850
β [1/K]	-	0.00068	-	0.00075	-	0.000815

summarizes the thermophysical properties of the tested PCMs, which were assumed during numerical simulations. For numerical calculations, the onset melting temperature $T_t^{\prime }$ for the charging process and the onset solidification temperature $T_s^{\prime }$ for the discharging process were regarded as the phase transition temperatures.

2.3. Experimental Procedure

Each experimental run commenced by ensuring that the temperature of the entire setup was maintained at 20 °C ± 0.5 K. Molten PCM at a temperature of 80 °C was poured into the prepared LHSU to initiate the solidification process. The end of solidification was defined as the moment when the readings from the 18 thermometers placed within the PCM stabilized at 20 °C ± 0.5 K. The charging process began by heating the feed tubes to 80 °C ± 0.5 K while maintaining the LHSU temperature at 20 °C ± 0.5 K. The end of the charging process was determined when the average readings from the 18 thermometers measuring the PCM temperature did not vary by more than 0.5 K over a period of 15 min.

3. Numerical Modeling

3.1. Governing Equations

The velocity, pressure, and temperature fields for multiphase, incompressible Newtonian fluid flow, accounting for mass and heat transfer processes between phases, were modeled using the Volume of Fluid (VoF) method. Considering the individual phases—namely, the liquid phase and stationary solid—the transport equation is formulated as follows [27,28]:

$${\displaystyle \frac{\partial }{\partial \tau }}\left({\rho }_i{\alpha }_i\right)+𝛻·\left({\rho }_i{\alpha }_{i}u\right)=S_{mi}$$

(1)

where ${\mathsf{\alpha }}_{\mathrm{i}}$ is the volume fraction of the i-th phase, $u$represents the mixture velocity vector, $S_{mi}$ denotes the mass source of the i-th phase, and $\tau$is time. The source of the mass $S_{mi}$ is directly related to the mass transport between the phases:

$$S_{mi}={\dot{m}}_{sc}-{\dot{m}}_{cs}$$

(2)

where ${\dot{m}}_{sc}$ is the mass flow rate of the melting solid phase, and ${\dot{m}}_{cs}$ is the mass flow rate of the solidifying liquid phase.

Interphase mass transport is described by the following equations [29]:

For melting ${\dot{m}}_{cs}=0$,

$${\dot{m}}_{sc}=C{\rho }_{sc}{\alpha }_{sc}{\displaystyle \frac{T-T_t}{T_t}}$$

(3)

For solidification ${\dot{m}}_{sc}=0$,

$${\dot{m}}_{cs}=C{\rho }_{cs}{\alpha }_{cs}{\displaystyle \frac{T_k-T}{T_k}}$$

(4)

where $C$ is a model constant ($C=1$), $\mathrm{T}$ is temperature, $T_t$ is the melting temperature, and $T_k$ is the solidification temperature.

To satisfy the conservation of mass, the sum of the mass sources $S_{mi}$ must equal zero:

$${\sum }_i{S}_{mi}=0$$

(5)

The sum of the volume fractions of the phases must also equal one:

$${\sum }_i{\alpha }_i=1$$

(6)

The mass conservation equation for the mixture can be expressed as

$$\nabla ·\mathrm{u}=0$$

(7)

The Navier–Stokes equation, which governs the flow dynamics, is given by [27]

$${\displaystyle \frac{\partial }{\partial \tau }}\left(\rho u\right)+𝛻·\left(\rho uu\right)=\rho g+f_{\sigma }-𝛻p+𝛻·\left(2\mu D\right)+S_u$$

(8)

where $g$ is the gravitational acceleration vector, $p$ is the pressure, $D$ is the rate of the deformation tensor, $f_{\sigma }$ is the term related to surface tension (considered negligible), and $\mu$ is the dynamic viscosity of the mixture:

$$\mu ={\displaystyle \frac{1}{\rho }}{\sum }_i{\alpha }_i{\rho }_i{\mu }_i$$

(9)

and $\rho$ is the mixture density:

$$\rho ={\sum }_i{\alpha }_i{\rho }_i$$

(10)

For melt-related issues, the Boussinesq approximation is employed:

$$\rho g={\rho }_{0}g-{\rho }_0\beta \left(T-T_0\right)g$$

(11)

where $\beta$ is the thermal expansion coefficient, ${\rho }_o$ is the reference density, and $T_o$ is the reference temperature of the liquid phase.

The source term for the interface $S_u$ is expressed as

$$S_u=-Au$$

(12)

where $A$ is defined by the Carman–Kozeny equation for flows in porous media:

$$A=C_u{\displaystyle \frac{{\alpha }_s^2}{{\alpha }_c^3+{10}^{-3}}}$$

(13)

Here, $C_u$ is a constant that needs adjustment. The literature indicates that the values of $C_u$ fluctuate within the range Cu = 10³–10⁸ [30] or Cu = 10⁵–10⁸ [31]. Our preliminary calculations found that the optimal experimental agreement occurs at the upper limit of these ranges, specifically ${\mathrm{C}}_{\mathrm{u}}={10}^8$.

For the solid phase ${\alpha }_s=1$, the source term $S_u$ dominates the other terms in the Equation (8). For the liquid phase ${\alpha }_s=0$, the source term $S_u$ in Equation (8) does not contribute to Equation (8). This allows Equation (8) to effectively map semi-liquid regions throughout the melting process [32,33].

The energy transport equation for the problem under consideration has the form [27]

$${\displaystyle \frac{\partial }{\partial \tau }}\left(\rho c_{p}T\right)+𝛻·\left(\rho c_{p}Tu\right)=𝛻·\left(\lambda 𝛻T\right)+S_h$$

(14)

where $c_p$ is the specific heat of the mixture:

$$c_p={\displaystyle \frac{1}{\rho }}{\sum }_i{\alpha }_i{\rho }_i{c}_{pi}$$

(15)

$\lambda$ is the thermal conductivity of the mixture:

$$\lambda ={\displaystyle \frac{1}{\rho }}{\sum }_i{\alpha }_i{\rho }_i{\lambda }_i$$

(16)

The source term $S_h$ represents heat transfer due to mass transfer between phases [32]:

$$S_h=-h\left({\displaystyle \frac{\partial }{\partial \tau }}\left({\rho }_i{\alpha }_i\right)+𝛻·\left({\rho }_i{\alpha }_{i}u\right)\right)\approx -\rho h{\displaystyle \frac{\partial {\alpha }_i}{\partial \tau }}$$

(17)

where $h$ represents the heat of phase transition. It is assumed that the convective terms associated with the liquid phase in Equation (17) are negligible compared to the unsteady terms.

3.2. Spatial Discretization of Equations

The Finite Volume Method (FVM) was employed to discretize Equations (1), (7), (8), and (14). The resulting algebraic systems of equations were solved using OpenFOAM software [34]. Since the mass conservation equation, momentum conservation equation, and energy conservation equation share a similar form, it suffices to discretize only one transport equation for the quantity ϕ [27]:

$${\displaystyle \frac{\partial \varphi }{\partial \tau }}+𝛻·\left(\varphi u\right)=𝛻·\left(\Gamma 𝛻\varphi \right)+S_{\varphi }$$

(18)

Source terms $S_{\varphi }$ should be linear functions:

$$S_{\varphi }=S_u+S_p\varphi$$

(19)

The diffusivity coefficient for the quantity $\varphi$ is denoted as $\Gamma$. The corresponding discrete version of transport Equation (18) for the control volume $V_p$ takes the form [27]

$${\displaystyle \frac{d{\varphi }_p}{d\tau }}\left|V_p\right|+{\sum }_f{\varphi }_f{u}_f{·S}_f={\sum }_f{\Gamma }_f{\left(𝛻\varphi \right)}_f{·S}_f+S_u\left|V_p\right|+S_p\left|V_p\right|{\varphi }_p$$

(20)

where $S_f$ denotes the outward-facing surface normal.

The control volume $V_p$ around the centroid p comprises $f$ flat surfaces $S_f$. The terms associated with divergence, such as convective terms $𝛻·\left(\varphi u\right)$ and diffusion terms $𝛻·\left(\Gamma 𝛻\varphi \right)$, utilize the Gaussian formula. Discretized convective terms are interpolated using values from the finite volume centroids, as the values of ${\varphi }_f$ are situated at the edge centroids of this volume. Linear interpolation with second-order accuracy is employed. Van Leer limiters were used for terms related to the volume fractions ${\alpha }_i$. Second-order orthogonal discretization schemes were also employed for the discretization of diffusion terms containing gradients of the form ${\left(𝛻\varphi \right)}_f{·S}_f$.

3.3. Time Discretization

Time discretization is carried out using a backward differencing scheme, which requires computed values for three different time steps:

$${\displaystyle \frac{d{\varphi }_p}{d\tau }}={\displaystyle \frac{{3\varphi }_p^{n+1}-{4\varphi }_p^n-{\varphi }_p^{n-1}}{2∆\tau }}$$

(21)

The set of equations is solved using the PISO algorithm [35]. The modified pressure equation is handled with a GAMG (generalized geometric–algebraic multi-grid) solver accompanied by a DIC (diagonal-based incomplete Cholesky) preconditioner and additional Gauss–Seidel (GS) smoothing. Symmetric solvers using GS smoothing are applied for the volume fraction, while the temperature field is solved using the PBiCG (preconditioned bi-conjugate gradient) solver with DILU (diagonal-based incomplete LU) preconditioning. The sub-relaxation coefficients are set to 1.

The time step is variable and calculated based on maximum values for the Courant number Co = 1 and the Fourier number Fo = 200. For Co = 1 and Fo < 200, the time step is computed using

$$∆\tau ={\displaystyle \frac{Co\delta \cdot S_f}{u_f\cdot S_f}}$$

(22)

where $\delta$ is the vector connecting two adjacent finite volume centroids. For Fo = 200 and Co < 1, the time step is calculated from the relation:

$$∆\tau ={\displaystyle \frac{Fo\rho c_p{L}^2}{\lambda }}$$

(23)

where L is the length over which conduction occurs. This length is generally defined as

$$L={\displaystyle \frac{{‖\delta ‖}^2}{\widehat{n}·\delta }}$$

(24)

where $\widehat{n}$ is a vector normal to the surface. For an orthogonal mesh, Equation (24) simplifies to

$$L=‖\delta ‖$$

(25)

3.4. Boundary Conditions

The velocities on all solid surfaces, including the shell and tubes, are set to zero $u=0$. The surfaces of the resistance thermometers satisfy the adiabatic condition $\partial T/\partial n=0$. This condition also applies to the shell walls during the charging process. For discharging on the walls, the following Neumann condition is applied:

$$\dot{q}=\alpha \left(T-T_o\right)$$

(26)

where $T_o=298 \mathrm{K}$, and the heat transfer coefficient was set at $\alpha =7 \mathrm{W}/\left({\mathrm{m}}^2\mathrm{K}\right)$; however for LTP56, better agreement with experimental data was achieved with $\alpha =2 \mathrm{W}/\left({\mathrm{m}}^2\mathrm{K}\right)$.

For the discharging process, a Neumann condition of T = 293 K is applied to the surfaces of the pipes, while during the charging process, the temperature is set between 349 K and 353 K, depending on the outlet HTF temperatures measured during experimentation. On surfaces perpendicular to the axis of symmetry (Figure 2), all variables satisfy the condition $\partial /\partial n=0$. The model of the LHSU, along with boundary conditions, is illustrated in Figure 2.

3.5. Mesh Parameters

The cross-sectional area (Figure 3a) corresponds to the inner diameter of the LHSU shell D_w = 70 mm. A thin layer with a thickness of 10 mm was assumed for the calculations. This approach simultaneously accounts for the presence of resistance thermometers (Figure 3b).

To ensure mesh quality in proximity to physical walls (e.g., tubes), three additional boundary layers were introduced, with the first layer being 0.1 mm thick.

Figure 4 presents the calculated temperature distributions at three positions of resistance thermometers TA, TB, and TC (Figure 2a) for three different mesh configurations—see

Table 2. Mesh parameters.

	Mesh 1	Mesh 2	Mesh 3
Elements	28,119	68,333	139,791
Nodes	25,455	60,216	129,208
Calculation time [s]	7521	7858	18,279

. The results obtained from the three tested meshes exhibit minimal differences; however, the use of a mesh with a larger number of nodes (Mesh 3) resulted in significantly longer computation times compared to the other meshes.

Using a mesh with a small number of nodes (Mesh 1) may lead to inaccurate flow mapping due to overly large finite volumes, which—as demonstrated in Figure 4—results in lower temperatures compared to the results obtained with Mesh 2 and Mesh 3. Considering these factors, Mesh 2 was chosen for the calculations, balancing the accuracy of calculations with computational duration.

4. Results and Discussion

Figure 5 shows a comparison of locally measured and numerically calculated temperatures for the charge and discharge processes of the S3 LHSU cross-section (Figure 1). Thermometer T9 was placed along a 29.4 mm diameter, between tubes 1 and 2. Thermometer T10 was positioned on the right side of the shell, below tube 5, at a diameter of 43.6 mm. Thermometer T11 was installed directly below the centrally located tube 4, between tubes 6 and 7, at a diameter of 21.3 mm. A satisfactory agreement was obtained, regardless of the PCM used and the location of the resistance thermometers (T9, T10, and T11). In the case of LTP56 (Figure 5a), during the charging process, the maximum discrepancy between the measured (solid lines) and calculated (dashed lines) values was observed for the resistance thermometer located closest to the bottom of the LHSU (T11). After 1150 s into the charging process, the measured temperature was found to be about 23% lower than the calculated temperature. For the discharging process (Figure 5b), the maximum difference between the experimentally measured temperature and the numerical calculations occurred at 2550 s for the T9 resistance thermometer located at the highest point of the LHSU, resulting in a difference of approximately 22%.

For RT54HC, it was observed that, regardless of whether it was the charging (Figure 5c) or discharging (Figure 5d) process, the maximum differences between the measured and calculated values were recorded for the T9 resistance thermometer during the charging process. The maximum difference, approximately 20%, occurred after 330 s, while for discharging, a maximum difference of about 30% was observed at 3140 s. In the case of P1801 (Figure 5e), the maximum discrepancy between the measured (solid lines) and calculated (dashed lines) temperatures was found for the T10 resistance thermometer, located just below the LHSU axis during the charging process. The maximum difference amounted to about 16% at 2020 s. During the LHSU discharging process (Figure 5f), the maximum difference between the measured values and those calculated numerically was again noted for the T9 resistance thermometer. This difference, about 23%, occurred at 3410 s.

In Figure 6, the measured and numerically determined temperature distributions during the charging and discharging processes are compared. The measured temperature was calculated in two ways: as the arithmetic mean of the readings from 18 thermometers mounted in the LHSU and as the arithmetic mean of the readings from thermometers (T9–T11) placed in the S3 cross-section (Figure 1). The numerically calculated temperature was also the arithmetic mean of the three values determined for the S3 cross-section (Figure 1). Regardless of the PCM used, satisfactory agreement between the measured and calculated temperature distributions was achieved.

For LTP56 during the charging process (Figure 6a), the maximum difference between the measured and numerically calculated values, approximately 23%, was observed at 1200 s. During the discharging process (Figure 6b), the maximum difference was noted at 2350 s, amounting to about 10%. For RT54HC during the charging process (Figure 6c), the maximum difference between the measured and numerically calculated values (approximately 3%) occurred at 1250 s, while during the discharging process (Figure 6d), a difference of around 2% was observed at 3550 s. As shown in Figure 6e and 6f, for P1801 during the charging process, the maximum difference between the measured values and those calculated numerically was observed at 1980 s, with a difference of about 5%. For the discharging process, the maximum difference at 4100 s was about 14%.

The negligible difference in the measured temperature, calculated as the average of the 18 thermometer readings and as the arithmetic mean of the readings from the three thermometers located in the S3 section (Figure 1), justifies the adoption of the numerical model.

Figure 7 illustrates examples of the temperature fields and phase distributions resulting from the numerical calculations performed for P1801 during the charging and discharging processes. As seen in Figure 7a, in the early stages of the charging process (τ < 240 s), melting of P1801 occurs symmetrically around the tubes. This is visible in the form of black fringes for the phase distribution and red fringes for the calculated temperature field. At this stage, conduction is the prevailing mode of heat transfer. By τ = 600 s, the areas of molten material around the tubes cease to be symmetrical due to greater density variations in the PCM, leading to free convection becoming the dominant heat transfer regime. After τ < 1800 s, liquid is present in almost all spaces; unmelted P1801 is observed between the tubes—in the middle row and in the lowest row. After τ = 2400 s, only a small amount of P1801 remains solid below the bottom row of the tubes. It is not until τ = 3600 s that all P1801 is fully in the liquid phase.

In the discharging process (Figure 7b), after τ = 240 s, a layer of solidified PCM forms on the surface of the tubes, with thickness increasing over time. After τ < 600 s, solid PCM bridges form towards the mantle, indicating insufficient thermal insulation of the LHSU. By τ = 1800 s, the contour of the interface resembles a six-arm star that decreases in size over time (from 2400 s to 3600 s), and after τ = 4200 s, all PCM is solidified.

Liquid Fraction

The liquid fraction was determined from the temperature measurements in three zones of the LHSU, as shown in Figure 8. In each zone, there were six thermometers located along the three radii: r1—zone 1; r2—zone 2; and r3—zone 3. It was assumed that if the average temperature from the readings of the six thermometers located on a given radius reaches the phase transition temperature ($T_t$), then the PCM in that zone is in the liquid phase.

Figure 9 shows measured (red lines) and calculated (blue lines) liquid fraction distributions over time.

As seen in Figure 9a for the charging process of LTP56, the maximum discrepancy between the measured and calculated liquid fraction occurred after 1480 s, amounting to about 114%. During the discharging process (Figure 9b), at 280 s, the difference was about 42%. The difference between the measured and calculated liquid fractions decreases over time. For LTP56 at the end of the charging process—specifically at 2530 s—the difference was about 3%, while for the discharging process at 2310 s, it was about 4%.

For RT54HC during the charging process (Figure 9c), the maximum difference between the liquid fractions, as determined from temperature distributions and numerical calculations, was about 67% at 1050 s. In the discharging process (Figure 9d), a difference of approximately 19% was observed after 670 s. At the end of the charging process—namely at 2070 s—the difference was about 5%. At the conclusion of the discharging process (τ = 3060 s), the difference was about 3%.

For the charging process of P1801 (Figure 9e), the maximum difference (about 121%) was noted at 1370 s, while for the discharging process (Figure 9f), the difference was about 34% at 1360 s. At the end of the charging process, specifically at 2310 s, the difference between the measured and calculated liquid fractions was about 4%, and at the end of the discharging process (τ = 2990 s), it was about 8%.

5. Conclusions

The computational model presented successfully reproduces the experimental results with satisfactory agreement. The high length-to-diameter ratio of the tested latent heat storage Unit (LHSU) indicates that the thin cutout (10 mm) covering the resistance thermometers is adequate for accurately capturing both the temperature distribution and the liquid fraction.

Numerical calculations reveal that heat conduction is the predominant heat transfer mechanism during LHSU discharging. In contrast, during the initial phase of LHSU charging, heat conduction predominates initially, followed by the emergence of free convection. Both experimental results and numerical calculations indicate that, regardless of the phase change material (PCM) used, the discharging times of the LHSU are approximately 40% longer than the charging times.

The findings demonstrate that the use of paraffin (LTP56) leads to increased charging times compared to discharging times, showing an increase of about 8% in experimental tests and approximately 24% in numerical calculations. The range of melting and solidification temperatures is broad, covering nearly the entire range of tested temperatures, and the times for solidification and melting are similar. It is important to note that below the final solidification temperature (T_k^’’), the paraffin attains a plastic-like consistency.

Conversely, the discharging times of fatty acids (RT54HC and P1801) exceed the charging times, with experimental studies showing a difference of about 49% for RT54HC, and numerical calculations indicating a 14% difference. For P1801, both experimental and numerical results show a difference in charge/discharge times of about 30% and 25%, respectively. This disadvantage in the operation of the LHSU is attributed to the dominant heat transfer mechanism during discharging, primarily heat conduction. The low intensity of heat conduction is a result of the low thermal conductivity values of the organic PCMs employed in the tests. A potential method to enhance the thermal conductivity of PCMs is the fabrication of nanoPCMs, which entail mixtures of PCM with high thermal conductivity nanoparticles [36].