
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)12136-6
10.1016/j.heliyon.2024.e36105
e36105
Research Article
Optimization of the thermal performance of a lobed triplex-tube solar thermal storage system equipped with a phase change material
NematpourKeshteli Abolfazl
Iasiello Marcello marcello.iasiello@unina.it
⁎
Langella Giuseppe
Bianco Nicola
Dipartimento di Ingegneria Industriale, Università degli Studi di Napoli Federico II, P.le Tecchio 80, 80125, Napoli, Italy
⁎ Corresponding author. marcello.iasiello@unina.it
13 8 2024
30 8 2024
13 8 2024
10 16 e3610514 6 2024
5 8 2024
9 8 2024
© 2024 The Authors
2024
https://creativecommons.org/licenses/by-nc/4.0/ This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
The thermal performance of a PCM-based triple-tube lobed heat exchanger storage system is here simulated and optimized, including performance improvements via lobed surfaces, Y-shaped fins, dispersed multi-walled carbon nanotubes, and metal foams, to be used in combination, or singly. Such computations are done with the finite volume method under different operating conditions. The reason behind this study is to look for solutions to improve the poor thermal performance of phase change materials (PCMs) as thermal energy storage materials, that limits their compactness and instantaneous heat stored/released. This is the first time that a throughout analysis of this aspect is presented. The result showed that higher modified Stefan number allow to improve melting time of a 50.88 %. The inclusion of lobes and fins resulted in a reduction of roughly 30.54 % in time needed for melting completion, compared to straight tubes. This reduction increases to 74.26 % when lobes are combined with both nanoparticles and metal foam, and to 73.60 % with just foam. The best solution also provides a 228.34 W mean heat rate. This study becomes an option to design tube-in-tube energy storage systems, where the best improvement is achieved by considering a lobed surface together with nano/PCM and foam, whereas the highest enhancement comes from using a metal foam.

Highlights

• A PCM-based lobed triple-tube heat exchanger is modeled.

• Performances improvement via y-shaped fin, nanopowder, and metal foams.

• Four lobes with a Y-shaped fin allow a reduction in process time of up to 30.54 %.

• Melting time savings of up to of 74.26 % are achieved with some of the solutions.

• These results are useful to improve heat transfer and compactness of these systems.

Keywords

Phase change materials
Metal foams
Thermal energy storage
Triplex-tube heat exchanger
Nanofluid
==== Body
pmc Nomenclature				
Am	Constant of mushy zone (kg/m3 s)	V	Volume (m3)	
Asf	Interfacial surface area (1/m)	x, y, z	Cartesian coordinates (m)	
c	Specific heat capacity (J/kg∙K)		Greek letters	
dl	Ligament diameter (m)	α	Thermal diffusivity (m2/s)	
dp	Pore size (m)	β	Coefficient of thermal expansion (1/K)	
D	Pipe Diameter (m)	λ	Liquid fraction	
Fo	Fourier number	δ	Enthalpy-porosity constant	
g	Gravity acceleration (m/s2)	ε	Porosity	
hsf	Interfacial heat transfer coefficient (W/m2∙K)	φ	Volume fraction of nanoparticles	
k	Thermal conductivity (W/m∙K)	μ	Dynamic viscosity (kg/m∙s)	
K	Permeability of porous foam (m2)	v	Kinematic viscosity (m2/s)	
L	Pipe length (m)	ρ	Fluid density (kg/m3)	
Lf	Latent heat (J/kg)	σ	Constant for thermal conductivity	
mpcm	PCM mass (kg)		Subscripts	
m˙	Mass flow rate (kg/s)	e	Endpoint	
M	Constant for thermal conductivity	s	Solid/Porous foam	
p	Pressure (Pa)	se	Solid foam effective value	
Pr	Prandtl number	f	Liquid PCM/nanoPCM	
pL,m	Average melting heat rate (W)	fe	Liquid PCM/nanoPCM effective value	
pL,m*	Dimensionless average melting heat rate	ini	Initial	
Q	Energy (J)	l	Liquid	
Ra	Rayleigh number	m	Melting	
Re	Reynolds number	out	Outlet	
Ste∗	Modified Stefan number			
t	Time (s)			
T	Temperature (K)			
u, v, w	Velocity components (m/s)			

1 Introduction

1.1 State-of-the-art

Thermal energy storage (TES) offers a viable solution to the challenges posed by the intermittent and uneven distribution of solar energy resources [1]. TES encompasses various thermal storage principles, with three prominent categories: sensible thermal energy storage [2], thermochemical energy storage [3], and latent thermal energy storage (LTES) [4]. Among the various techniques used in LTES, latent heat thermal energy storage (LHTES) is characterized by its ease of installation, significant energy storage capacity, constant storage temperature and controllable operation [5]. Phase change materials (PCMs) are an interesting option for TES due to their high latent heat of fusion. However, PCMs also have weak thermal properties, generally reflected in low values of thermal conductivity, often below 0.3 W/m K. This property limits their usefulness during both the melting and solidification processes [6,7]. Previous research has shown that highly thermally conductive materials in PCMs can enhance their heat transfer and conduction capabilities. To overcome this barrier, a variety of approaches have been developed to improve the heat transfer capability of PCMs. Several potential options are available, such as the incorporation of solid nanoparticles with high thermal conductivity [8,9], the use of expanded fins [10], and the use of metal foams or porous matrices [[11], [12], [13]]. Furthermore, different thermal energy storage systems can be designed depending on different constraints such as the available source, working fluid, etc. In other words, each geometric configuration is unique [14], and requires dedicated enhancement solutions. Some of these are PCM tanks with coils, shell-and-tube heat exchangers, or, for example, tube-in-tube heat exchangers, the latter of which can be subdivided into different sub-categories, such as double-tube heat exchangers (DPHX) or triplex-tube heat exchangers (TTHX).

The simplest and mostly used way to improve the thermal performance of PCMs is based on extended surfaces embedded in the PCM, in particular conventional fins or other solutions such as corrugated walls or lobed configurations to increase the heat transfer area per unit volume, or on other similar structures like twisted-way tubes [15]. Abdulateef et al. [16,17] performed an experimental study to examine the heat storage capacity of a triplex-tube heat exchanger storage system with a focus on different fins arrangements. The results shows that the outer triangular fin design showed the most significant improvement in solidification characteristics, resulting in a 18 % reduction in solidification time. Al-Abidi et al. [18,19] investigated the effect of various design features such as fin length, thickness and number, and unit shape on solidification performance, using a two-dimensional (2D) numerical model of a TTHX unit. The results showed that the most optimal solidification effect was achieved by the model with eight longitudinal fins, which resulted in a significant reduction of solidification time by 35 %. Najafabadi et al. [20] presented a new approach to heat storage in solar collectors by investigating the use of PCM in a twisted double-lobed tube heat exchanger. The results showed that the three-lobed DPHX design had the largest surface area for heat transfer and achieved a faster melting rate (by 6 %) compared to the five-lobed DPHX. Dhaidan et al. [21] conducted a study on the PCM melting process in heat exchangers, comparing those with and without fins. Their research showed that the fin with perforations has superior thermal properties compared to the fin without perforations. In a study of PCM melting and convention in different-shaped containers, Dhaidan et al. [22] focused on geometrical characteristics including aspect ratio and eccentricity in annular cavities. The authors found that conduction dominates heat transfer initially, but convection dominates fusion as the molten liquid layer grows.

Another solution is to directly improve the thermal properties of the storage material, say the PCM, with additional materials. To achieve this, the incorporation of nanoparticles into PCMs increases both the thermal conductivity and the supercooling degree of the base PCMs. In their investigation of shell-and-tube PCM heat exchangers, Nie et al. [23] found that the use of top-injected heat transfer fluid (HTF) rather than bottom-injected HTF resulted in a more positive enhancement, referred to as vertical shell-and-tube latent heat storage (LHS) unit performance. Abdulateef et al. [24] investigated the effect of incorporating nano-sized alumina powders on the thermal conductivity of paraffin rubitherm 82 (RT-82). They found that the combination of these two techniques resulted in a significant reduction in solidification time of approximately 17 %. Moghaddam and Ganji [25] carried out a comprensive evaluation of a PCM-triplex tube heat exchanger equipped with nanoparticles at different concentrations, showing that a 18.5 % improvement in performance can be achieved. Mozafari et al. [26] used different combinations of PCMs in terms of their location in the cross section and found that nanoparticles can achieve a 23.43 % improvement in terms of stored energy with reference to a standard reference case. Dhaidan [27] analyzed nanoparticles phase change materials (nanoPCM) analytical, computational, and experimental studies and examined how cavity geometry affects nanostructures-assisted phase-change material melting, recommending also to use nanoPCMs reported thermophysical characteristics in numerical studies instead of mixed models. Chibani et al. [28] also emphasized by means of a predictive model that the bulk materials for nanoparticles in nanoPCM would have an important role in improving the phase change characteristics of the thermal storage material, remarking that this parameter is also a relevant one in storage design.

Currently, numerous simulations and experiments have been carried out to enhance the phase change through porous media [29]. Among the several methods, metal foam, has shown significant promise in enhancing the thermal conductivity of PCMs. This is because a metal foam has a large surface area per unit volume, which provides more space for the PCM to spread out and facilitate heat transfer, thereby increasing the overall conduction too. When foams are used, the phase change is generally improved because of the higher overall thermal conductivity due to the metallic parts to be incorporated in the PCM, causing a slight reduction in terms of natural convection motion, since the foams are an obstacle. The result is that more conduction makes the melting front to be more parallel to the heated surface than that of a pure PCM, and this effect is amplified at lower porosity, where the thermal performances are generally higher, with a slight reduction in terms of available PCM volume for storage. Air-based PCM storage systems have been shown to have interesting performance if they use metal foams to improve the convective heat transfer within the air as the heat transfer fluid [30]. Mancin et al. [31] conducted an experimental study on the melting of PCM in a copper metal foam with varying densities of pores per inch (PPIs). They determined that the presence of a metal foam enhances the rate at which the PCM melts. Moreover, it was observed that variations in the density of pores per inch (PPI) within the foam do not exert a noteworthy influence on the melting rate of the PCM. Guo et al. [32] investigated the potential of metal foam to save energy and showed that compressing the foam results in an increase in charging time of approximately 129.4 %. The research carried out by NematpourKeshteli et al. [33] focused on improving the thermal conductivity of PCMs within TTHX systems. By incorporating nanoparticles and metal foam into the multilayer PCM-TTHX configuration (Case B), a significant 83.48 % reduction in melting time was achieved compared to the simple PCM-TTHX configuration (Case A). Yang et al. [34] investigated the contribution of metal foam in improving heat transfer. They suggested that replacing the original PCM light pipe with metal foam significantly improved internal heat transfer, resulting in an 88.6 % reduction in total melting time. Based on the literature survey, it is evident that both nanoparticles and metal foam are promising approaches for enhancing the thermal performance of PCM, and that this enhancement might depend on the single storage system design. Combining metal foams and nanoparticles has been shown also to be something that could be applied to larger scale storage devices [35]. In conclusions, these solutions have the potential to significantly improve melting time, solidification time and other related factors, referred to different available storage designs.

1.2 Objectives and novelty of the current study

The studies presented in this introduction show that there are several ways to improve the thermal performance of PCMs. The role of each thermal performance technique is related to the device – i. e., the type of heat exchanger – used, and there is room for improvement, especially when it comes to storage systems based on tube-in-tube heat exchangers, such as the triplex-tube heat exchanger. As per the previously described state-of-the-art, no solutions for this type of heat exchanger for latent heat thermal storage have been proposed by simultaneously consider different thermal enhancement solutions, and no studies assess the weighted importance of each of these compared to other potential ones. The focus of this research is on a novel optimized energy storage system for renewable energy applications like solar collectors, which is a TTHX with a unique lobed cross section, with the aim of improving its thermal performance as much as possible. In particular, combining all the proposed solutions will be done to assess the extent of improvement in thermal performance. These solutions include the incorporation of metal foams, the addition of nanoparticles, and the use of extended surfaces such as Y-shaped fins by considering a lobed configuration. This analysis marks the first time that such techniques have been brought together in the LHTES-TTHX system. The paraffin Rubitherm 54HC (RT-54HC) is used as the PCM that stores heat for applications such as domestic hot water systems; aluminum foams with a porosity range of 0.88–0.95 and 20 PPI, together with 0.04 % of multi-walled carbon nanotubes (MWCNTs) nanoparticles, are used to improve its thermal characteristics. The main objective is to identify the most effective solution among various alternatives to improve factors such as melting time and velocity, as well as stored energy or melting heat rate. The present thermal storage system, say a triplex-tube heat exchanger that might be coupled with a solar concentration system, generally includes three innovative ways to expedite the melting process, thus enhance the heat transfer rate of energy storage. The optimization procedure outline starting from the baseline heat exchanger equipped with the PCM is reported as follow.1) Enhancing the heat exchanger design by using a lobed pipe with Y-shaped fins; 2) dispersing nano-sized MWCNT powders into RT54HC for dispersion; 3) using aluminum foam with different porosities.

2 Triplex-tube heat exchanger modeling

2.1 Geometry

It is widely known that solar collector systems generally require an appropriate storage system. In this study, even if a thermal energy storage system could be applied everywhere, references are done to a solar system as an example. A generic sketch is shown in Fig. 1a and described in the following. Throughout the day, solar collectors absorb solar energy and transfer it to the working fluid, water. Since hot water is not available during the night, a storage system is included in the overall system. It is remarked that the present storage device might be applied also in other cases if one thinks about power-to-x applications or other thermal energy recovery situations. In Fig. 1a, such a system is a triplex-tube heat exchanger, with a PCM to be employed as the storage medium, placed between inner and outer water tubes. Hot water coming from the solar collector heats up the PCM during the day, then cold water from the user is heated during the night thanks to the previously charged PCM. Therefore, the role of boundary condition from the solar system, or other thermal sources, is of primary importance in modeling, and this aspect will be clarified in a detailed section. In the same scheme, a hot water tank is included too as a backup solution, even if the core of the present analysis is the main storage system. A sketch of a cross-section of the storage system is shown in Fig. 1a, where it is shown that three different cases, say Cases A, B, and C, are here analyzed to consider a simpler reference case (Case A), a case with four lobes (Case B), and a case with four lobes including Y-shaped fins (Case C). These cases are also summarized in Table 1. A more detailed sketch, together with sizes, external conditions to be described later, and materials used with their own characteristics, is shown in Fig. 1b-1d. Some details about the mesh used, to be described later, are also included here. In this study, the same amount of PCM volume is always considered, so the middle diameter of the PCM region, middle diameter (Dmiddle), is assumed to vary freely. Y-shaped fins are used here because of their high heat transfer surface area and the fact that they promote vortex formation, which could enhance PCM liquid convection. This effect is particularly valuable in low velocity flow scenarios where laminar flow could hinder efficient heat transfer [[36], [37], [38]]. Geometrical values reported in Fig. 1 for length and so on are justified in the following. Values here used are of the order of magnitude of typical experimental studies in this field. For instance, Al-Abidi et al. [18] proposed a half-meter length triplex-tube heat exchanger, while Abdulateef et al. [16] designed a 3 m length device. With references to diameters, in the former diameters where 25.4 mm, 75 mm, and 100 mm; in the latter, such diameters where 76.2 mm, 381 mm, and 500 mm. This means that Abdulateef et al. [16] essentially studied a scaled device for higher storage medium quantities. In this study, it is assumed that the heat exchanger is 1 m long, with diameters of 25 mm, 54 mm, and 90 mm, meaning that sizes are really similar to the ones from Al-Abidi et al. [18], typical of thermal storage applications for domestic hot water production.Fig. 1 Configuration of the solar thermal collector system equipped with the LHTES system (a) Scheme with the three-cases geometries, (b) Simple TTHX, (c) Lobed TTHX, and (d) Lobed-Fin TTHX and generated mesh.

Fig. 1

Table 1 A resume of the cases here investigated, say Cases A, B, and C, in terms of tube geometries.

Table 1Cases	Systems	
Case A	Simple TTHX	
Case B	Lobed TTHX	
Case C	Lobed-Fin TTHX	

2.2 Assumptions and mathematical formulation

The following is a list of assumptions made after the geometry has been determined and before the governing equations are written. The flowing water is assumed to be laminar thanks to the non-high Reynolds numbers simulated. The copper tube is assumed to present a negligible thermal resistance. With references to the PCM, this is assumed to have negligible volume expansion and viscous dissipation. The open-cell foam to be used is assumed to be isotropic, in local thermal non equilibrium with either the PCM or the nano-based PCM, and with negligible micro-inertial forces and thermal dispersion [39]. Finally, the nanoparticles are assumed to be homogeneously distributed within the PCM, and the former are assumed to be in local thermal equilibrium with the latter, with no relative motion. A summary of continuity, momentum, and energy equations used here is given in Table 2. These governing equations are presented in a generic form, referring to the case of nanoPCM with foam, to be applied to all cases and domains investigated. For example, if porosity and permeability are unitary and infinite, respectively, and also if the interfacial heat transfer coefficient hsf is infinite (Tf = Ts = T), such governing equations refer to the cases with no porous media and nanoparticles. Such equations become valid for pure PCM if, for example, a zero percentage of nanoparticles is assumed too. It is also emphasized here that such equations are used in the well-known single-phase form to simulate the flowing water as the heat transfer fluid.Table 2 Continuity, momentum, and energy equations for the current system to be simulated [29,33].

Table 2Equation	Expression	
Continuity	∇⋅V→=0	
x-Momentum equation	ρ‾ε2(ε∂u∂t+V→⋅∇u)=−∂p∂x+(Am(1−λ)2λ3+δ−μ‾K)u+μ‾ε∇2u	
y-Momentum equation	ρ‾ε2(ε∂v∂t+V→⋅∇v)=−∂p∂y+(ρβ‾)g(Tf−Tini)+(Am(1−λ)2λ3+δ−μ‾K)v+μ‾ε∇2v	
z-Momentum equation	ρ‾ε2(ε∂w∂t+V→⋅∇w)=−∂p∂z+(Am(1−λ)2λ3+δ−μ‾K)w+μ‾ε∇2w	
PCM/NanoPCM energy equation	ρc‾(ε∂Tf∂t+V→⋅∇Tf)=hsfAsf(Ts−Tf)+kfe∇2Tf−ερLf‾∂λ∂t	
Solid foam energy equation	(1−ε)(ρc)s∂Ts∂t=kse∇2Ts−hsfAsf(Ts−Tf)	
Parameter	Expression	
Carman-Kozeny damping term [40]	Am=106,δ=0.001	
Liquid fraction, λ	λ={0→ifTf≤TsolidusTf−TsolidusTliquidus−Tsolidus→ifTsolidus<Tf<Tliquidus1→ifTf≥Tliquidus	
Modified Stefan number, Ste*[11]	Ste*=c(Tm−Tini)+c(Tinlet−Tm)Lf	
Fourier number, Fo [11]	Fo=αtL2	
Rayleigh number, Ra [11]	Ra=gβ(Tinlet−Tm)L3αν	

It is necessary here to employ porous media closure coefficients to describe a variety of characteristics, including permeability or effective thermal conductivities, since metal foams are used in this research. Correlations expressing these values as a function of morphological characteristics such as PPIs and porosity are summarized in Table 3 for a comprehensive understanding.Table 3 Correlations for determining the closure coefficients of metallic foam.

Table 3Parameter	Correlations	Reference	
Permeability(K)	Kdp2=0.00073(1−ε)−0.224(dldp)−1.11dldp=1.181−ε3π(11−e−(1−ε)/0.04)wheredp=0.0254(m)/PPIε=0.88,0.90,0.92,0.95;PPI=20	[41]	
Effective thermal conductivity of PCM or nanoPCM (kfe)
Effective thermal conductivity of the porous medium (kse)	kfe=22(MA+MB+MC+MD)|ks=0
kse=22(MA+MB+MC+MD)|kf=0
MA=4σ(2e2+πσ(1−e))ks+(4−2e2+πσ(1−e))kf
MB=(e−2σ)2(e−2σ)e2ks+(2e−4σ−(e−2σ)e2)kf
MC=(2−2e)22πσ2(1−2e2)ks+2(2−2e−πσ2(1−2e2)))kf
MD=2ee2ks+(4−e2)kf
σ=2(2−6/8e32−2ε)π(3−4e2−e),ande=0.339	[42]	
Interfacial heat transfer coefficient (hsf), Specific surface area, (Asf)	hsf={0.76Rel0.4Pr0.37(kfdl),1≤Rel≤400.52Rel0.5Pr0.37(kfdl),40≤Rel≤10000.26Rel0.6Pr0.37(kfdl),1000≤Rel≤2×105
Asf=3πdl(1−e−(1−ε)/0.04)(0.59dp)2	[41,43]	

MWCNT were considered as the nanoparticles dispersed within the PCM for the current model. In all cases, single-phase equivalent formulations were used to compute velocity, pressure, and temperature. As the nanomaterial volumetric concentrations were assumed to be less than 0.04 %, a methodology using a homogeneous mixture approach was proposed to estimate the properties [44,45]. The thermal conductivity of nanoPCM was determined by using the Maxwell model [46]. A summary of the equations used to describe the nanoPCM (NEPCM, MWCNT + RT54HC) properties is given in in Table 4. In the absence of nanoparticles, these properties refer only to the RT54HC PCM. Furthermore, when nanoparticles are taken into consideration in the study of nanoPCM, a homogeneous single-phase method is used. Specifically, this approach assumes that the governing equations for the PCM are averaged by considering the dispersion of nanoparticles with weighted averages.Table 4 Properties for the nanoPCM simulated; for φ = 0 these are referring to pure PCM [47,48].

Table 4Parameter	Equation	
ρ‾	ρ‾=(1−φ)ρRT54HC+φρMWCNT	
ρc‾	ρc‾=(1−φ)(ρc)RT54HC+φ(ρc)MWCNT	
ρβ‾	ρβ‾=(1−φ)(ρβ)RT54HC+φ(ρβ)MWCNT	
ρLf‾	ρLf‾=(1−φ)(ρLf)RT54HC	
μ‾	μ‾=μRT54HC(1−φ)2.5	
kf	kfkRT54HC=kMWCNT+2kRT54HC−2φ(kRT54HC−kMWCNT)kMWCNT+2kRT54HC−φ(kRT54HC−kMWCNT)	

When modelling PCMs, variables such as stored energy or melting fraction are usually taken into account. In order to get an idea of what the potential might be to use latent heat continuously to obtain available thermal energy, the melting heat transfer rate pL,m is used here. Its definition, which is useful in determining the rate at which melting occurs, is given below.(1) pL,m=Qtm=mPCM(∫TiniTsoliduscdT+Lf+∫TliquidusTecdT)tm=mPCM[cPCM(Tsolidus−Tini)+Lf+cPCM(Te−Tliquidus)]tm

This parameter emphasized how the system is able to store heat per unit time. It can be interpreted as the mean melting velocity as it represents an average heat rate over the melting time [47,48]. This parameter is also presented in a scaled form here via pL,m* to give an idea of how efficient heat transfer is compared to the sensible enthalpy variation rate. Variables that are necessary to finally close the model are resumed in Table 5.(2) pL,m*=pL,mm˙cwater(Tinlet−Tini)=AverageheatrateSensibleenthalpyvariationrate

Table 5 Variables to close equations from Table 1 also referred to Eqs. (1), (2)

Table 5Parameter	cPCM (J/kg K)	cwater (J/kg K)	Tsolidus (K)	Tliquidus (K)	Tini (K)	Tinlet (K)	Lf (kJ/kg)	
Value	2000	4182	327	330	303	333, 336, 339	200	

2.3 Boundary conditions and foam/nanoparticles characteristics

The boundary and initial conditions for the problem are shown below. In all cases, the whole system has an initial temperature (t = 0 s) of 303 K. This temperature would be referred to a typical quasi-steady condition, where all the storage system must start to accumulate heat during peak temperatures periods during the day. To simulate the phase change process, a three-dimensional (3-D) model was used with HTF water inlet temperatures of 333 K, 336 K, and 339 K, chosen as a different boundary conditions for the inner and outer tubes; these temperatures refer to modified Stefan numbers (Ste*) of 0.317, 0.349, and 0.381, respectively. Obviously, water incoming temperature, that might come from the solar collector, has to be higher than the system initial temperature in order to start accumulating heat. The simulated inlet temperature might be typical of water that comes out from a solar collector in typical conditions.

In the present study, a PCM with a melting temperature always lower than the assumed inlet temperatures was used to assess what happens during PCM loading. Indeed, in latent thermal storage design, the PCM choice has a primary role, and this depends on the working temperature desired. In fact, as already anticipated about typical order of magnitudes, a typical working temperature in industrial and domestic hot water applications is 333 K. Based on this, RT54HC is chosen as the PCM as it starts to melt at the liquidus temperature Tl = 330 K. It is assumed here that 0.0945 kg/s of water flows in, at which value the flow regime is laminar; this value is typical of this applications, as for instance done in the experimental studies on triplex-tube heat exchanger from Ref. [18]. This value might allow to have reasonable instantaneous heat rate for the storage device. In this study, the outer tube of the TTHX is assumed to be adiabatic, so no heat losses are simulated in order to focus on the storage medium. The boundaries between adjacent regions are assumed to have temperature and heat flux continuity, while the HTF inlet/outlet sections are assumed to have no slip and outflow conditions with respect to the PCM domain. Highly porous foams with porosities between 0.88 and 0.95 and 20 PPIs were used for this study. In addition, nanoparticles with low MWCNT volume fractions of approximately 0.04 % were incorporated along with copper fins and tubes. Previous studies have shown that the foam porosity has a greater influence on the overall PCM/foam heat transfer than PPI [[49], [50], [51], [52], [53], [54]]. Such nanoparticle volume fraction values have been chosen because higher nanoparticles concentrations can cause PCM segregation and, in addition to an increase in thermal conductivity, a significant increase in melt viscosity, causing some changes within the melt flow hydrodynamics. A comprehensive list of thermophysical properties used in the current study can be found in Table 6 [[54], [55], [56]]. In terms of the numerical method employed – which will be analyzed in more detail later in terms of grid convergence, etc. - the finite volume method (FVM) is used here; input parameters for the solver are resumed in Fig. 2, which also provides more details on the numerical methods employed.Table 6 Properties of the materials used in this work.

Table 6Property	Foam [54]	RT54HC [55]	MWCNT [56]	Cu	
Tsolidus/Tliquids (K)	–	327/330	–	–	
ρsolid/ρliquid (kg/m3)	2700	850/800	2100	8920	
csolid/cliquid (J/kg K)	900	2000	711	380	
k (W/m K)	220	0.2	2000	400	
β (1/K)	–	0.00075	–	–	
μ (kg/m s)	–	0.002508	–	–	
Lf (kJ/kg)	–	200	–	–	

Fig. 2 Input parameters and solution approaches in finite volume simulations.

Fig. 2

2.4 Effective management of mesh size and time step

The FVM commercial code ANSYS FLUENT was used to model the problem. To enhance computational performance, it is necessary to optimize two crucial parameters: grid size and time step. By achieving the right balance between solution precision and computational effort, the overall processing time can be reduced. As illustrated in Fig. 3, optimized values for the simulation were obtained using a grid with 1 950 762 nodes and a time step of Δt = 0.5 s. Interestingly, further refinement of the grid and reduction of the time step did not significantly improve the results, as the liquid fraction values obtained were nearly identical. Even if not represented here, grid with higher nodes provide similar results to the ones referred as the very fine mesh with references to the average melting percentage. This suggests that using a coarser grid and larger time step can achieve results of comparable quality while reducing the computational cost required for the simulation.Fig. 3 (a) Mesh (b) and time independence check on Case A with pure PCM liquid fraction evolution.

Fig. 3

3 Validation and comparisons

Before proceeding with the modeling approach, it is crucial to check the current code, and numerical details, by comparing the simulation results to previously published research. Firstly, a validation between the present results and the findings from Mat et al. [57] was carried out by comparing the evolution of the liquid fraction of the PCM melting in a TTHX without internal fins. As shown in Fig. 4a, results obtained through the current study exhibit a reasonable level of agreement with those reported by Mat et al. [57]. In Mat et al. [57], the authors simulate a very similar problem with internal, external, and external/internal fins. For the sake of comparisons, models for triplex-tube heat exchangers with no fins, are compared, by assuming the same geometrical and boundary conditions, including also materials employed, reported in the paper. It is also mentioned that computations done in Mat et al. [57] have been validated with experiments. Li et al. [58] investigated the melting process in a spherical tank with a surface temperature of 333 K, and the liquid fraction values are compared with previous data to assess the accuracy of the numerical code (Fig. 4b). Comparisons are done with that paper to appreciate if the model performs well also with a slightly different geometry. In particular, same conditions of Li et al. [58] have been tested too, with particular references to a sphere with a 4 cm radius; in that paper, a good agreement has been shown with experiments too. Comparisons with results from Li et al. [58] indicate that there is a satisfactory agreement between the current numerical predictions and their outcomes. Comparisons with the numerical findings of Liu et al. [59], who modeled the melting of RT58 and copper foam inside a circular annulus, are also included. In Liu et al. [59], the authors used a 3D model to analyze a PCM annulus embedded in a metal foam, with adiabatic condition at the exterior and a uniform temperature of 350 K at the interior. Comparisons with Liu et al. [59] are helpful to appreciate if the model performs well. It is worthwhile mentioning that the authors in Ref. [59] verified their model with experimental data. Liquid fractions derived from the simulations (Fig. 4c) demonstrate a strong correspondence with the prior studies. The little difference between these results may be attributed to the fact that in Liu et al. [59] the authors assumed a uniform and constant boundary condition, differently to what has been done here, where axial conduction is considered too. Moreover, Table 7 provides a summary of the average and maximum discrepancies between different studies, demonstrating that deviations between current results and those from Mat et al. [57], Li et al. [58], and Liu et al. [59] are relatively small, with the average differences being especially small. Besides, with references to the using of nanoparticles and metal foam, no experimental detailed study for the triplex-tube configuration are available. Therefore, the double pipe, which is similar to the one here shown, has been already compared in Ref. [60] with predictions from Ref. [61], that have been in turn validated with experiments from Refs. [62,63].Fig. 4 Comparison of liquid fraction and average PCM temperature between the present work and (a) Mat et al. [57], (b) Li et al. [58] for pure PCM, and (c) Liu et al. [59] for metal foam/PCM composite, showing a good agreement between computations and available literature data.

Fig. 4

Table 7 Deviation between the liquid fraction and average temperature from this study compared to data from Mat et al. [57], Li et al. [58], and Liu et al. [59], showing a good agreement between computations and available literature data; LF stands for liquid fraction, and K stands for kelvin degrees.

Table 7	Average difference	Maximum difference	
Present study vs. Mat et al. [57]	LF: 0.0107	LF: 0.0270	
Present study vs. Li et al. [58]	LF: 0.0006	LF: 0.0351	
Present study vs. Liu et al. [59]	K: 0.381	K: 4.969	

4 Results and discussion

This section provides an in-depth investigation of the melting process of RT54HC/MWCNT saturated aluminum foam. The investigation focuses on different design configurations implemented in a TTHX, using water as the working fluid on the opposite side. Three approaches have been here combined to improve the melting rate in the present work: 1) improving the heat exchanger geometry by using a lobed heat exchanger configuration; 2) increasing the heat transfer surface area by using Y-shaped fins; 3) improving the heat transfer performance by incorporating metal foam and/or nanomaterials. The investigation focuses on exploring the influence of metal foam placement within the storage unit on fluid flow and heat transfer performance. Porosities ranging from 0.88 to 0.95 and a 20 PPI configuration are considered in the study. The type of nanoparticles is MWCNT, with a concentration of 0.04 %. In this study, all the numerical simulations conducted were based on four unique compositions of PCMs, applied on each of three geometrical configurations previously resumed in Table 1: 1) Pure PCM (PP), 2) Nanoparticle-PCM (NP), 3) Metal Foam-PCM (PP-MF), and 4) Nanoparticle-Metal Foam-PCM (NP-MF).

4.1 Geometry analysis

4.1.1 Simple TTHX-Case A-PP

Fig. 5 shows melting time, average PCM temperature and stored energy for Case A using pure PCM (Case A-PP). The average temperature refers to the average temperature of the PCM during the melting process, while the melting time refers to the time taken for the PCM to change from a solid to a liquid phase. Energy storage describes how much energy the PCM absorbs during the melting process. The results show that the curves representing the liquid fraction and the increase in energy storage decrease over time until they reach a plateau. This occurs when complete melting is achieved and no further latent heat effects occur. The results show that the slope of the PCM average temperature profile changes at around 900 s. This is because a significant amount of the PCM begins to undergo a phase change, resulting in an overall increase in resistance that causes the average temperature not to vary as much over time. If the simulation ends at the point where all the PCM has melted, the temperature can rise significantly due to sensible heat. The data presented in the figures shows that the melting process is complete after exactly 5811 s.Fig. 5 Case A - PP during melting process: (a) liquid fraction, (b) PCM-averaged temperature, (c) and stored energy, remarking in all cases an asymptotic trend due to the end of melting.

Fig. 5

Fig. 6 shows liquid fraction, temperature field, and liquid PCM streamlines after 1800, 3600, and 5400 s, for the Case A-PP. The range of the liquid fraction is between 0 and 1, referring to pure solid and liquid PCM, respectively. In addition, the liquid fraction is plotted with the isotherms contours and the average temperature contour in the x = 0 and z = L/2 section, to show both longitudinal and transverse sections. Contours have also been used to show velocity distributions and streamlines. The hot wall (Tinlet = 333 K) is located on the inner and outer walls along the flow direction, while the initial temperature (Tinitial = 303 K) of the whole system is lower than the melting temperature. During the early stage of the transient (Fig. 6a), there is only conduction from the hot walls since the PCM is still solid, causing the RT54HC to gradually increase its temperature until it reaches its melting point. As the process progresses into the liquid phase, the importance of natural convection becomes more relevant (Fig. 6b). When a significant percentage of the PCM has melted, it causes an increase in temperature. As the melting zone expands, natural convection intensifies, leading to increased heat transfer; the heat transfer process in advanced stages is primarily influenced by the motion of the liquid PCM. This aspect rises from the fact that the molten PCM presents a rolling shape thanks to free convection that arise in the liquid (Fig. 6b and 6c). The streamlines also highlight characteristics of natural convection motions, which exhibit similar behavior and intensity in Case A-PP (Tinlet = 333 K) during the early stages (t = 1800 s) before gradually expanding after some time due to the higher temperature gradients inside the annulus region. It is also noted that the presence of natural convection causes the temperature fields to be asymmetric within the cross section.Fig. 6 Liquid fraction (LF) and temperatures for Case A-PP after (a) 1800 s, (b) 3600 s, and (c) 5400 s, remarking liquid PCM recirculation and also comparisons in terms of achieved liquid fraction and peak temperatures.

Fig. 6

Liquid fractions evolution with time and combination among Fourier, Stefan, and Rayleigh numbers, namely Ste∗2FoRa0.25, is shown in Fig. 7. A modified Stefan number is here used, to consider the sensible heating effects of solid and liquid states of the PCM, as presented in Table 2. Here, modified Stefan numbers of 0.317, 0.349, and 0.381, that correspond to wall temperatures of 333, 336, and 339 K, respectively, are used. Increasing modified Stefan number means dealing with higher temperature differences, so more potential for thermal energy; it is then relevant to understand how the melting time, and related variables, change depending on a modified Stefan number increase. Results presented in Fig. 7a demonstrate that the melting time decreases as the modified Stefan number increases. The liquid fractions are then correlated using a combination of dimensionless numbers, namely Fourier, Stefan and Rayleigh, in accordance with the dimensional analysis performed by Kamkari and Shokouhmand [11]. The terms Ra0.25 and Ste∗2Fo examine different effects on the melting process, where Ra0.25 considers the effect of natural convection and Ste∗2Fo includes transient heat conduction and phase change. The increase in convective flow is associated with an increase in heat transfer; this also causes a reduction in melting time in enclosures. Fig. 7b clearly shows that the melting time for Ste* = 0.349 and Ste* = 0.381 is approximately 34.91 % and 50.88 % shorter, respectively, compared to Ste* = 0.317. It is obviously noticed that the scaling analysis makes curves far closer in Fig. 7b than in Fig. 7a. This reduction in melting time in TTHX is mainly due to the intensification of the inlet temperature, which generally improves heat transfer, even with respect to convective flows. Thus, the observed higher liquid fraction with higher Ste* in Fig. 7b indicates that the latent heat contribution might increase as the HTF inlet temperature grows. These results underscore the significant role of these factors in reducing the melting time in TTHX compared to conventional melting methods.Fig. 7 A comparison between different Ste* number for the liquid fraction as a function of (a) melting time, and (b) Ste*2FoRa0.25, remarking the role of the modified Stefan number.

Fig. 7

4.1.2 Lobed TTHX-Case B-PP

Fig. 8 shows melting time, average temperature and stored energy for Case A and Case B with pure PCM. Compared to the previous case, the lobed heat exchanger is included here. The figures show that the change in liquid fraction is slower for the simple TTHX (Case A-PP) compared to the lobed configuration. This discrepancy could be attributed to the larger heat transfer area offered by the finned configuration. According to the available data, the melting time for the Case B-PP case was 14.90 % shorter than for the Case A-PP case. Similar conclusions can be drawn for both the average PCM temperature and the stored energy; in all cases, Case B-PP appears to have better performance, so the average temperature rises earlier, and the stored energy appears to be higher than in Case A-PP. However, it is noted that differences between cases A and B here investigated are not that relevant because this part of the geometry optimization is referred to minor changes to the structure. Fig. 9 presents the simulated TTHX results for Case A and Case B with pure PCM as a liquid fraction and isotherm contours at t = 3600 s and Ste* = 0.317, for a cross-section (z = L/2) of the domain investigated. Fig. 9 shows that the use of a lobed heat exchanger speeds up the melting process. This is due to the fact that the heat transfer area becomes larger when switching from Case A to Case B inner tube geometry, allowing deeper heat penetration. On the other hand, it is also emphasized that natural convection can be hindered by the lobed shape; even if this happens, the lobed configuration still allows a faster melting evolution. It is remarked that the convection role is not that relevant since lobes are not a huge constraint to liquid motion. Besides, this effect is dumped by the fact that lobes present some smaller cross sections for the liquid PCM.Fig. 8 Case A-PP, and Case B-PP, during melting: (a) liquid fraction, (b) PCM-averaged temperature, (c) and stored energy, showing better performances for the Case B configuration.

Fig. 8

Fig. 9 Liquid fraction and temperature fields for (a) Case A-PP, and (b) Case B-PP at 3600 s, and z = L/2.

Fig. 9

4.1.3 Lobed TTHX with Y-shaped fins-case C-PP

Fig. 10 illustrates the melting time evolution, average temperature, and energy stored over time for all systems, namely a basic TTHX design (Case A), a TTHX design with lobes (Case B), and a TTHX design with lobes and a Y-shaped fins (Case C), including just pure PCM. he average temperature of the PCM is higher when both lobes and Y shaped-fins are considered, resulting in a faster charging process and faster completion of the entire liquid area. This gain can be attributed to the greater surface area of the highly conductive fin, coupled with the greater surface area of the lobed structure, which allows for faster charging times and more effective heat transfer. When comparing the Case C-PP case to the Case B-PP and Case A-PP cases, the phenomenon completion time was reduced by approximately 14.90 % and 30.54 % respectively. It is also remarked that, even if fins might be an obstacle to liquid PCM convection, the advantage in terms of heat transfer area is far higher than this obstacle. Temperatures observed are also generally slightly higher because the heat mainly becomes sensible.Fig. 10 All the Cases simulated with Pure PCM (PP) during charging (melting) process: (a) liquid fraction, (b) average of PCM temperature, and (c) energy stored, showing better performances for the Case C configuration.

Fig. 10

Fig. 11 displays a comprehensive comparison of the liquid fraction fields and temperature contours among all cases involving pure PCM. The volume of PCM was kept constant in all cases studied, ensuring that an equal amount of paraffin was used, except for a negligible amount coming from the fins. Figures are presented after 3600 s, to underline the impact of natural convection previously analyzed in Fig. 6; the molten paraffin starts to move upwards because of buoyancy, with liquid recirculation that slightly enhance heat transfer. The higher the velocity of the fluid containing the liquid paraffin, the faster the melting process. However, as shown in Fig. 9, natural convection is dumped by the fin too; even if this happens, melting is much faster for Case C-PP because the fin provides a further increase in terms of heat transfer area, in addition to the lobed shape that promotes heat penetration. It is also evident that at equal time the case with fins presents more or less all liquid within the volume. This underlines the role of fins also compared to lobes.Fig. 11 Liquid fraction and temperatures for (a) Case A-PP, (b) Case B-PP, and (c) Case C-PP at 3600 s, and z = L/2, with some emphasis on natural convection outcomes.

Fig. 11

4.2 Thermal enhancement techniques

Having demonstrated the effectiveness of improving thermal performances through geometric modifications in the previous sections, the focus now shifts to exploring advanced techniques aimed at directly improving the performance of PCMs through the incorporation of nanoparticles and/or foams. These innovative approaches offer promising ways to improve the thermal properties and overall functionality of PCMs, making them more suitable for a wide range of technical applications. This section of the current paper will delve into the details of these techniques to provide a comprehensive understanding of how nanoparticles and foams can be used to improve phase change in terms of thermal performance.

4.2.1 Effects of nanoparticles

In this section, thermal performances of the TTHX are evaluated when nanoparticles are dispersed within the PCM, i. e. when 0.04 % MWCNT nanoparticles are included. Fig. 12 shows the liquid fraction, average PCM temperature and melting time for all the scenarios presented so far in this study involving the dispersion of MWCNT nanopowders in pure PCM. Among the investigated scenarios, the longest complete melting takes 5811 s and corresponds to cases without lobed tube and Y-shaped fins, i.e. the baseline case introduced earlier. Conversely, the fastest process occurs when the nanomaterial is placed in a lobed Y fin container, which takes 3590 s to complete.Fig. 12 All Cases with PP and NP during charging (melting): (a) evolution of liquid fraction and (b) average of PCM temperature, and (c) melting time showing better performances for the Case C configuration with nanoparticles.

Fig. 12

The inclusion of MWCNT nanoparticles appears to speed up the process and this phenomenon can be attributed to the increased driving force, i.e. thermal conduction, provided by the dispersed nanoparticles. In addition, the use of a lobed tube with Y-shaped fins provides a

Further improvement, as the melting becomes more uniform due to the improved heat transfer surface. At the same time, the average temperature appears higher because the faster melting allows more parts of the domain to reach the liquid phase at the same time. With reference to the present simulations, when nanomaterials are included in the PCM for cases B and C, reductions of 24.14 % and 38.23 % respectively in terms of melting time have been found compared to case A with only PCM. It is also remarked that all this is valid under the assumption of a uniform nanoparticles distribution with the PCM, for which the present model can be employed. On the other hand, if one considers a different reference parameter to just make references to nanoparticles inclusion, in other words if one assumes the nanoparticle inclusions with reference to each Case A, B, and C, then the improvement coming from Case A-NP, Case B-NP, and Case C-NP, would be 20.49 %, 10.86 % and 11.07 %, respectively.

Table 8 shows melting time (t), time-saving (%), energy stored (kJ), and heat transfer rate (pL,m and pL,m*) for all cases with pure PCM and MWCNT nanopowder dispersed within the pure PCM. When comparing PP with NP in all cases, it is shown that dispersing MWCNT generally reduces melting times; in particular, when taking into account lobed and Y-shaped fins (Case C-NP), a strong reduction on melting time has been found. However, the energy stored after 600 s appears relatively similar across all investigated cases. This outcome can be attributed to the lower energy storage capacity of the NP compared to pure PCM, resulting in a smaller overall heat capacity; because of this, the cases that have somewhat larger stored energy are those with pure PCM. The heat storage rate pL,m, for Case B-NP and Case C-NP, is higher than for Case A-PP. This is because, even though the stored energy is similar, the melting time is shorter. As a result, the average heat rate decreases; the same conclusions can be applied for the dimensionless heat storage rate. This means that NEPCMs represent a cause of thermal performances improvement thanks to a higher energy transfer rate; certainly, the significance of this particular parameter in this context is notable as it assesses the rate of melting by considering the fusion time and the stored energy for each scenario, providing insight into the melting process velocity.Table 8 A comparison between various configuration LHTES-TTHXs in terms of melting time, time saving, energy stored, and heat transfer rate.

Table 8Cases	Melting time (s)	Time-saving (%)	Energy stored (kJ) after 600s	pL,m(W) at completed melting time	pL,m* at completed melting time	
Case A-PP	5811	0 %	89.76	60.22	0.0253	
Case B-PP	4945	14.90 %	92.345	70.77	0.0298	
Case C-PP	4036.5	30.54 %	105.20	86.71	0.0365	
Case A-NP	4620	20.49 %	86.23	73.92	0.0311	
Case B-NP	4408	24.14 %	88.75	77.47	0.0326	
Case C-NP	3589.5	38.23 %	101.01	95.14	0.0401	

4.2.2 Effects of porous metal foam

This section focuses on the impact of metal foams with references to Case B, i. e. a case that to be exhaustively analyzed which has been already shown to perform better than the reference Case A. In this study, three different porosities ranging from 0.88 to 0.95, with equal PPIs (i.e., 20), are compared. Fig. 13a and Fig. 13b illustrate comparisons by means of liquid fraction and average temperature evolution among Case B-PP, Case B-NP, Case B-PP-MF, and Case B-NP-MF, with a porosity of 0.88. The total melting time of metal foam/PCM, with ε = 0.88, is 73.60 %, 68.98 %, and 61.99 %, less than that of Case A-PP, Case B-PP, and Case C-PP, respectively. Based on the findings presented in Fig. 13, it is evident that the incorporation of nanomaterials and foam in Case B-NP-MF offers the most optimal solution. Fig. 13 indicates that combining nanomaterial dispersion and foam, Case B-NP-MF, is the most effective approach for reducing melting time. Specifically, this approach results in a melting time reduction by 74.26 % compared to Case A-PP, and 69.76 %, when compared to Case B– PP. These results clearly demonstrate the significant impact of coupling nanoparticles and foam on the performance of the system, emphasizing the importance of considering both factors when optimizing thermal behavior. From the figure, it is also evident that foams provide a huge advantage for this application in terms of melting time and related variables, thanks to an overall conductivity improvement, and also thanks to a higher contact area between the inserted metal and the PCM. This would witness the fact that in this context, say triplex-tube heat exchangers, using foams would boost the thermal performances more than the other geometries adjustments done. Besides, if one looks at Fig. 13a, melting time for liquid fraction are about 5 times lower if foams are included, making melting time on a different stage. This outcome is also balanced from the temperature evolution, where, in Fig. 13b, it is clear that when foams are used then temperature increases very fast. All the heat becomes sensible after a very short time, while for the case with no foam the temperature increases very slow, making the melting very long. From a first point of view, one might think that the latter is the best solution since the system works in phase change; however, on the other hand, when metal foams are used, the instantaneous heat rate is by far higher, so the thermal storage system would be able to fulfill requirements from the user, that is the working fluid that requires thermal energy. A resume of the melting times is shown in Fig. 13b, underlining the drop that happens when comparing melting times if foams are not used. Fig. 13c provides a comprehensive summary of the melting time saves when the Case B geometry – lobed heat exchanger – is considered; in this figure, results obtained with different porosities are shown too. This graph shows a considerable relationship between porosity and melting time, with porosity levels that affect the system behavior; in particular, lower porosity levels leading to better thermal conduction since there is more metal. It is worth noting that using foams has the highest impact on reducing melting time, even if it is assumed here that there is less PCM available for melting because a volume fraction is occupied by the solid foam. When considering PP and NP melting times without foams (4945 and 4408 s, respectively), introducing these leads to a considerable drop in melting time, reducing it to between 1495.5 and 1767 s. This reduction is primarily due to the increased surface area of the foam structure, which enhances heat transfer between the phase-change material and the heat transfer fluid. Furthermore, the use of foams also helps to prevent the formation of hot spots and promotes more uniform melting because of the generally higher thermal conductivity; the latter is further improved by the nanoparticles dispersion. Therefore, the incorporation of foams is a highly effective approach for improving the performance of phase-change TES systems. From Fig. 13, it is clear that, among the solutions presented, the best case is the lobed configuration with a 0.88 porosity foam including nanoparticles too, i. e. say Case B-NP-MF with ε = 0.88.Fig. 13 Case B with PP, NP, PP-MF, NP-MF during charging (melting): (a) liquid fraction, (b) average of PCM temperature, and (c) time-saving comparisons, showing better performances when generally foams are employed.

Fig. 13

Fig. 14 presents a comprehensive analysis comparing Case B-PP and Case B-NP-MF with a porosity value of ε = 0.88. The comparison focuses on the liquid fraction and temperature fields after a duration of 1200 s. The figure also includes the cross-sections at x = 0 and z = L/2. Looking at the melt fraction at t = 1200 s, there are notable discrepancies in the results obtained. Compared to pure PCM, the metal foam/NEPCM composite shows a significantly higher temperature, mainly due to the improved melting process. This also because at equal time the PCM is completely melted, so the heat is sensible and temperatures are higher. Even if one could not store much heat when sensible, it is also important to remark that heat rates are not that high during phase change if melting is considered, as previously underlined. Therefore, having higher temperature with foams is better because this would mean that melting is faster. This accelerated melting mechanism drives the composite material towards a state of almost complete melting during this phase. Besides, the fact that foams are conductive and almost a regular porous material, reduces thermal stratification, making temperature more uniform across the domain. In other words, it looks like an internal heat generation for the phase change material. By introducing the metal foam, the composite experiences a reduction in temperature fluctuations, resulting in a more homogeneous temperature distribution throughout the metal foam/NEPCM composite. Even if the heat is sensible, the temperature still becomes uniform because of the foam conductivity, so another advantage related to foam is that it is able to maintain temperatures as uniform as possible even during the PCM liquid phase. Therefore, this feature promotes improved heat transfer efficiency and a more balanced thermal response within the composite. Therefore, one could conclude that the advantage when considering all the material improvements, say nanoparticles and foam, for the investigated case, is really huge, making melting times totally different. It is also remarked that temperatures are generally uniform along the radius in Fig. 14b, while this not happens in Fig. 14a. This means that for the heat is really difficult to go from a radial position to another one if thermal improvement techniques are not used.Fig. 14 Liquid fraction (LF) and temperatures for (a) Case B-PP, and (b) Case B-NP-MF, after 1200s, where it is evident that using thermal enhancers makes melting far faster.

Fig. 14

A final summary of melting time (t), time-saving (%), energy stored (kJ), and heat transfer rate (pL,m and pL,m*), for all the cases here investigated is shown in Table 9. In this table, results for Case C are presented too. Again, the energy stored after 600 s is used to compute the average melting heat rates based on formulas from equations. The denominator in the calculation is the melting time, which varies depending on the specific scenario being analyzed. Compared to the pure PCM case, the NEPCM/foam cases generally show a much higher heat storage rate, especially for conditions with lower porosities. The heat storage rate for NEPCM is slightly higher than that for PCM, for the reasons given above about the reduced heat capacity of nanoparticles. This suggests that while foams are the main contributor to the improved thermal behavior, NEPCM also has the potential to provide a further increase. The use of metal foam results in a significant improvement in both pL,m and pL,m* compared to the use of pure PCMs. This means that combined solutions have the potential to achieve a significant increase in the rate of stored energy, primarily due to the incorporation of metal foams. The inclusion of nanoparticles and metal foam for the lobed-TTHX (Case B), reduces charging times by 74.26 % compared to simple TTHX (Case A) with pure PCM. The best case simulated in this study appears to be Case B, which involves the use of both nanoparticles and a foam with 0.88 porosity. This particular case demonstrates slightly better performance compared to Case C, primarily due to the pronounced impact of natural convection, which makes Case C slightly inferior to Case B. On the other hand, when comparing Cases B and C with nanoparticles and foams possessing a 0.95 porosity, Case C seems to perform slightly better than Case B. This is because the convection impact is reduced in Case C, resulting in improved overall performance.Table 9 A comparison between various configuration on instantaneous: melting time, time-saving, energy stored (after 10 min), and heat transfer rates.

Table 9Cases	Melting time (s)	Time saving (%)	Energy stored (kJ) at 600s	pL,m (W) at completed melting time	pL,m* at completed melting time	
Case A-PP	5811	0 %	89.76	60.22	0.0253	
Case B-PP	4945	14.90 %	92.345	70.77	0.0298	
Case C-PP	4036.5	30.54 %	105.20	86.71	0.0365	
Case A-NP	4620	20.49 %	86.23	73.92	0.0311	
Case B-NP	4408	24.14 %	88.75	77.47	0.0326	
Case C-NP	3589.5	38.23 %	101.01	95.14	0.0401	
Case B– PP-MF-0.95	1767	69.60 %	149.81	186.38	0.0786	
Case B– PP-MF-0.92	1614.5	72.22 %	156.95	203.99	0.0860	
Case B– PP-MF-0.90	1560	73.15 %	159.70	211.12	0.0890	
Case B– PP-MF-0.88	1534	73.60 %	161.39	214.69	0.0905	
Case A- NP-MF-0.95	1829.5	68.52 %	138.24	186.66	0.0787	
Case B– NP-MF-0.95	1714.5	70.49 %	142.37	199.18	0.0839	
Case C– NP-MF-0.95	1651.5	71.58 %	145.03	206.78	0.0872	
Case B– NP-MF-0.88	1495.5	74.26 %	153.24	228.34	0.0962	
Case C– NP-MF-0.88	1523.5	73.78 %	151.48	224.15	0.0945	

5 Conclusions and future recommendations

The present study presents an optimized design idea for a latent thermal energy storage system (LHESS), based on a triplex-tube heat exchanger (TTHX), where the focus is on the optimization of thermal performances. The optimization is performed considering different heat transfer enhancement solutions, to be employed alone or together. Since a low-medium temperature solar application is analyzed here, the paraffin Rubitherm 54HC (RT-54HC) has been chosen as the latent thermal storage material. Techniques used here to improve the thermal performances are based on lobed surfaces, Y-shaped fins, metal foams (porosity in between 0.88 and 0.95, with 20 pores per inches) and nanoparticles (with a volume fraction of 0.04 %) in the storage material. The predictive model is setup by using the enthalpy-porosity method to predict the phase change; on the other hand, the local thermal non-equilibrium model and the homogeneous single-phase approach have been used to model the cases that include foams and nanoparticles, respectively. The main conclusions from the present work are listed as follows.• Modified Stefan numbers of 0.317, 0.349, and 0.381, that correspond to wall temperatures of 333, 336, and 339 K, respectively, are simulated. It is showed that the melting time for Ste* = 0.349 and Ste* = 0.381 is approximately 34.91 % and 50.88 % shorter, respectively, compared to Ste* = 0.317 for Case A with pure PCM (Case A-PP). This generally means that increasing wall temperature allows to have faster melting here, with small temperature increase that have a big impact on melting times.

• The current research suggests, as a first step, that the use of lobed heat exchangers, alone or together with Y-shaped fins, makes melting faster. When comparing the four lobes with Y-shaped fins (Case C-PP) to the four lobes (Case B-PP) and reference case (Case A-PP), the phenomenon completion time was reduced by approximately 14.90 % and 30.54 %, respectively.

• When nanoparticles with a volume fraction of 0.04 % are included, for Case A-NP, Case B-NP and Case C-NP, the melting time is reduced by 20.49 %, 24.14 %, and 38.23 %, respectively, compared to the Case A-PP.

• On the other hand, the addition of metal foams, e. g. with porosities in between 0.88 and 0.95, has been found to have a significant effect on the melting time, even more so than for nanoparticles. For example, in the TTHX (Case B-PP-MF), using foam porosities of 0.88, 0.90, 0.92, and 0.95, the melting times were reduced by 73.60 %, 73.15 %, 72.22 %, and 69.60 %, respectively, compared to the Case A-PP.

• Finally, when all the improvement methods are combined, NP-MF with porosities of 0.88, for Case B, i. e. with only the lobed configuration, the time saving and the rate of thermal energy storage become 74.26 % and 228.34 W, respectively.

In the future, a thorough analysis of the present device when integrated into a day/night solar collector could be carried out, including economic and environmental aspects. This analysis might also refer to larger systems, in order to remark the role of scalability in such devices. This would also remark how relevant is to give emphasis to a single component of the overall energy system, say the storage system. Indeed, assessing how this device is integrated in an energy system, to quantify sustainability impact and so on, is something that must be done to appreciate the advantage of the device in terms of a polygeneration system. With references to the single component, a throughout CFD analysis with advanced artificial intelligence tools might be done including also other parameters for the numerical optimization. Besides, other porous media, say for instance triply-periodic minimal surfaces, lattices, and so on, might be included in this throughout optimization analysis to appreciate if there is further room to improve the storage performances with other novel materials. Furthermore, thermal cycling analysis will be also carried out to appreciate the behavior of the storage materials under long-term conditions. In this way, one might consider also different phase change materials for a further optimization, also referring to the whole energy system. The present study will be then a starting point to remark which solutions improve thermal performances, with an eye on future improvements starting from the solutions shown in this paper.

CRediT authorship contribution statement

Abolfazl NematpourKeshteli: Writing – original draft, Methodology, Conceptualization. Marcello Iasiello: Writing – review & editing, Methodology, Conceptualization. Giuseppe Langella: Writing – review & editing, Methodology, Conceptualization. Nicola Bianco: Writing – review & editing, Methodology, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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