© The Author(s), published by EDP Sciences, 2025

Abbreviations

A_c : Area of collector

Q_u : Overall heat transfer in flat plate

F_r : Fractional efficiency

G_t : Solar constant

α_t : Transmissivity of the glass

U_L : Overall heat loss

T_in : Input temperature in flat plate

T_out : Output temperature

T_i : Inlet temperature in PCM

m : Mass of the working fluid

c_p : Specific heat capacity of heat transfer fluid

U_t : Top heat loss in collector

U_b : Bottom heat loss collector

U_e : Side heat loss in collector

L_b : Insulation thickness

K_b : Insulation thermal conductivity

h_b,a : Convection heat loss of collector at bottom

L_e : Side insulations loss

h_e,a : Heat coefficient of collector in side

R_pc : Overall heat transfer coefficient

R_ca : Resistance of glass cover and surrounding

h_pc : Convection heat coefficient of absorber

σ : Boltzmann constant

T_pm : Mean temperature of collector

T_c : Glass cover temperature

ϵ_p : Absorber plate emissivity

ϵ_c : Glass cover emissivity

h_w : Wind heat transfer coefficient

T_s : Sky temperature effects

T_a : Ambient temperature

ŋ_th : Thermal efficiency of collector

dT : _p,avg Average temperature of the phase change

d_t : Diameter of the tubes

Q_ch : Charging of the PCMs

ŋ₀ : Output efficiency of collector

I : Irradiance

Q_dis : Discharging of the PCMs

T_pcm : Temperature of PCM

R_tot : Total resistance loss

A_pcm : Area of PCM occupy

ŋ_ch : Efficiency of PCMs charging

ŋ_dis : Efficiency of PCMs during discharging

E_x,stored : Total exergy stored

E_x,in : Inlet exergy

E_x,out : Output exergy during discharging

$\mathrm{In}\left(\frac{T_{\mathrm{in}}}{T_{\mathrm{out}}}\right)$ : PCM total input and output temperature

PCM: Phase Change Material

1 Introduction

Enhancing the efficiency of renewable energy systems has become a prevailing trend in the current era, aiming to diminish reliance on fossil fuels and promote the generation of clean energy. Renewable energy resources stand out as environmentally friendly alternatives, contributing to the mitigation of global warming. Among these, Flat Plate Collectors (FPCs) emerge as a robust source in the field of solar thermal energy. Operating effectively at both high and moderate temperatures, FPCs adeptly absorb both beam and diffused sunlight. FPC has versatile applications in renewable energy, serving purposes such as heating and cooling. Despite their advantages, a significant challenge lies in the intermittent availability of renewable energy resources like solar power, which is contingent upon specific times of the day. Thermal energy storage emerges as a pivotal solution to address this temporal limitation, ensuring a more consistent and reliable supply of energy [1–3].

In [4], a latent heat thermal energy storage system is employed to store thermal energy within Phase Change Materials (PCMs). This thermal energy storage method plays a crucial role in shifting load demand and finds application in diverse areas such as air-conditioning and heat recovery. Notably, latent heat storage is recognized for its capacity to store high levels of thermal energy. In comparison to sensible heat storage, latent heat storage exhibits a remarkable advantage, storing thermal energy at a rate 3–4 times higher per unit volume. Addressing thermal heat storage is a primary consideration when utilizing a system for storing heat for future use. Initially, water is employed as a medium to store thermal heat, and subsequently, phase-change materials are introduced to enhance the storage capacity. In the charging process, the PCM undergoes a transition from a solid to a liquid state, effectively storing thermal heat. During the discharging process, the stored heat is released and harnessed to power various applications. This cyclical process facilitates the efficient utilization of stored thermal energy for practical purposes [5].

In [6], it is noted that PCMs exhibit a sustained high thermal energy density even at low temperatures. This stored thermal energy proves valuable for cooling applications. The system utilizes salt-based PCMs with a melting temperature set at 142 °C to store the thermal energy. Thermal oil serves as the working fluid for the collection of thermal heat, which is then stored in sodium-based salts. The simulation of this system is carried out using TRNSYS, providing a comprehensive analysis of its performance and efficiency. Nanoparticles are added with the PCMs to produce a nanocomposite to improve thermo-physical properties of the PCMs [7]. In the system described, triple-PCMs featuring distinct melting point temperatures (63 °C, 54 °C, and 43 °C) are employed to effectively store thermal heat energy. This innovative approach involves storing PCM within triple concentric tubes. The thermal heat is stored through a latent heat process, wherein both the transfer of heat and the storage of heat in the PCM contribute to the overall efficiency of the system. This design optimizes the utilization of latent heat for thermal energy storage [8].

In [9, 10], the investigation focuses on enhancing the performance of FPCs through the incorporation of PCMs and helically corrugated heat-collecting tubes. The results reveal a significant improvement in the efficiency of the FPC, with a notable decrease in heat loss by 39.8%, as indicated by the output working fluid temperature difference. Specifically, the use of PCMs contributes to a remarkable efficiency boost, ranging from 41.5% to 48.9% for the FPC. This underscores the positive impact of integrating PCMs and innovative tube designs on overall collector performance. In [11], a nano-based PCM is employed for thermal energy storage, utilizing three distinct nano PCMs. The study reveals that the charging and discharging efficiency of the PCM system is significantly improved, specifically by 24%–28%, through the use of amino-functional PCMs. This signifies a notable advancement in the effectiveness of the thermal energy storage process, highlighting the positive impact of incorporating amino-functional PCMs in enhancing both charging and discharging efficiency.

A novel heat storage tank, outlined in [12, 13], has been developed to capture and store heat from PCMs, alongside water collection. The tank can store this heat within a span of 2 h. The performance of the tank loaded with PCMs shows a 10% improvement over the non-loaded PCM tank. The enhanced heat transfer efficiency of the collector can be attributed to the presence of PCMs. In [14], a novel supercooled PCM is introduced, designed to enhance the efficiency of flat-type solar collectors to 15%. The supercooled PCM has a melting point of 323.15 K, and it operates with a cooling temperature of 30 K, resulting in a 15% increase in the cooling rate. In [15], various types of PCMs are tested to evaluate their compatibility with highly efficient energy storage applications. A small-scale prototype system is designed to assess the suitability of PCM for such energy storage materials.

Abu-Hamdeh et al. [16] describes a method where thermal energy is stored in PCM to meet the energy demands for both heating and cooling loads in a building. The thermal heat is collected through the use of an FPC. The PCM employed in this context consists of paraffin wax combined with graphene or graphene oxide-based materials. This incorporation enhances the efficiency of the FPC and extends the storage time for thermal energy. In [17], PCMs designed for low-temperature thermal heat energy storage are introduced. These materials are engineered to absorb energy at the lowest temperatures and release it when needed at similarly low values. The document delves into the details of thermal energy storage and management associated with these low-temperature-based PCMs. In [18], the enhancement of thermal properties in PCMs is explored by altering the size and shape of the PCM storage container. The thermal energy storage capacity is improved by increasing the storage area, as discussed in the study.

In [19], thermal heat is stored using paraffin wax, and an FPC is employed to gather this heat. This stored heat becomes valuable during periods of insufficient solar radiation intensity. The incorporation of PCMs enhances the output of heat storage by 15%. In [20], the presence of PCMs results in a temperature difference that is 2 °C–9 °C higher compared to a basic FPC without such materials. The collector’s efficiency is notably improved with the incorporation of PCMs. In [21], the thermal stability of heat storage is enhanced by incorporating nanoparticles of inorganic materials along with PCMs. The efficiency is further elevated by utilizing rotating tubes within the FPC. Specifically, 3% iron-oxide nanoparticles are introduced into the PCM.

In [22], the collector’s efficiency experiences a notable 15% improvement through the introduction of alumina nanoparticles combined with PCMs. These PCMs not only facilitate the cooling of the collector but also serve the additional function of acting as a super cooler. In [23], sodium dodecyl sulfate serves as a PCM for the storage of thermal energy in conjunction with multiwall carbon nanotubes. The study employs a nanocomposite material consisting of polyethylene glycol 8,000 with varying mass fractions (0%, 0.5%, 1%, 2%, and 5%) to augment its performance. In [24], a graphene-based aerogel is employed to mitigate the leakage behavior of tubes containing PCMs. The approach involves the use of graphene oxide-based nanosheets to coat carbon-based nanotubes, resulting in an enhancement of the heating and cooling efficiency by up to 93.44%. In [25], tetradecane is utilized as the PCM, and the efficiency is augmented by incorporating porous nano carbon tubes. Figure 1 shows the FPC integrated with PCMs.

Fig. 1 PCMs integrated with FPC.

Paraffin wax PCMs combined with three types of nanoparticles, namely metal foams and wavy fin-based FPCs, aim to improve the performance of FPCs. Various weight percentages of nanoparticles are incorporated into the PCM to enhance its efficiency. Specifically, aluminum foam nanoparticles are used at weight percentages of 0.92%, 0.95%, and 5% in conjunction with the collector. During sunlight exposure, this PCM stores energy, which can then be utilized when solar radiation is unavailable [26].

The double-pipe heat exchanger is integrated into the FPC to improve its performance. The geometry of the collector pipes plays a crucial role in enhancing efficiency. Utilizing a lobes-type geometry reduces melting time by 18.32%, thereby boosting collector performance. Additionally, various foam materials are incorporated with the PCM to further enhance the collector’s efficiency [27].

A triple-tube heat exchanger pipe is employed in the FPC to enhance its performance significantly. This design reduces the melting rate of PCM and NePCM by 11%–12%, facilitated by the use of internal and external fins. Furthermore, the addition of alumina increases the thermal conductivity of the PCM by 0.2 W/m K, further improving overall efficiency [28].

In this research, the efficacy of the FPC is enhanced through the incorporation of PCMs. Carbon-based nanotubes are employed, serving as vessels for the PCMs, allowing them to efficiently gather thermal heat and store thermal energy. The integration of inorganic-based PCMs, particularly sodium sulfate decahydrate, is a noteworthy aspect of the design. This specific salt is chosen for its superior thermal conductivity and substantial thermal energy storage capacity. To address potential corrosion effects on the tubes, copper-oxide-based nanocomposite materials are introduced alongside the inorganic salts. This addition aims to mitigate corrosion and enhance the durability of the system. The efficiency of the FPCs is assessed by testing the temperature of the working fluid at various Reynolds (Re) numbers, providing insights into the overall performance of the design.

2 Methodology

2.1 FPC integrated with PCM

An FPC serves as a mechanism for converting solar thermal energy into a practical and usable form. This involves the utilization of PCMs integrated within the collector to store the thermal energy derived from the sun. The stored heat becomes instrumental during periods when sunlight is unavailable, such as during nighttime or on rainy days. In these instances, the stored heat is employed to power the connected load, compensating for the decreased performance of the FPC.

The applications of this stored heat are diverse and include driving cooling systems, meeting hot water requirements, and facilitating drying processes [29]. The FPC consists of four key components: Absorber plates, glass, insulation, and working fluids. Absorber plates play a crucial role in absorbing the sun’s intense radiation and transforming it into a usable and efficient form of energy.

Figure 2 illustrates the integration of PCMs with an FPC using nanotubes, with the PCM housed within these carbon-based tubes. To ensure leak-free containment of the PCM, graphene gel is applied. This setup allows the storage of solar thermal energy in the tubes when sunlight is available, and the accumulated energy serves the purpose of providing heat during nighttime periods. As an additional protective measure against corrosion of the tubes, nanocomposite materials based on copper oxide are introduced into the PCM. This integration enhances the durability and longevity of the system, contributing to its overall efficiency and performance.

Fig. 2 FPC with integration of the PCMs in nanotubes.

2.2 Nanotubes construction and performance in FPC

In this design, carbon-based nanotubes play a crucial role in extending the thermal heat storage duration and enhancing the overall efficiency of the FPC. The thermal energy of the collector is stored using a salt-based PCM. To mitigate corrosion and further optimize thermal heat storage, nanocomposite materials with a copper oxide base are introduced into the PCM. This addition not only prevents corrosion but also contributes to prolonging the storage time of thermal heat.

Figure 3 illustrates the configuration of carbon-based nanotubes integrating phase-change materials within the FPC design. The thermal conductivity of the multiwall nanotubes, being carbon-based, is notably efficient [30]. Within these tubes, the PCM is housed, featuring a hollow middle section. The heat transfer fluid circulates through this central space on both sides of the multiwall nanotubes.

Fig. 3 Nanotubes construction and process of PCMs energy storage.

To prevent corrosion, a copper insulation layer is strategically positioned between the passage of heat transfer fluid and the tubes containing PCM. When sunlight is present, the absorber plate captures thermal heat and transfers it to the working fluids. These fluids accumulate the heat, with a portion directed towards the PCM for storage. The remaining thermal energy is then transferred to the load. To enhance thermal stability, nanocomposite materials are added in limited amounts, specifically up to 0.05%. This addition improves the PCM’s thermal stability by up to 30%. The stored heat serves as a valuable resource for subsequent use, particularly during periods of inconsistent sunlight, such as rainy days, stabilizing the load requirements by utilizing the stored energy.

2.3 Classification of PCMs

Utilizing thermal energy storage emerges as the optimal solution to address energy crises and meet the growing demand for energy in buildings and various industrial applications. Among the available options, PCMs stand out as the most efficient choice for storing thermal energy. These materials excel in absorbing and retaining substantial amounts of energy over extended periods [31].

Figure 4 illustrates the categorization of PCMs according to their material properties and availability. These materials are divided into four sub-categories based on their phase transition: Solid–solid, solid–liquid, liquid–gas, and solid–gas. The liquid–gas and solid–gas forms, however, are deemed impractical due to the significant space they occupy. The prevalent choice among PCMs is the solid-liquid category, with organic and inorganic types being the primary classifications [32]. In the proposed design, the utilization of inorganic-based PCMs is advocated for storing the thermal heat generated by FPCs.

Fig. 4 Classification of the PCMs [32].

2.4 Inorganic-based PCMs

Inorganic PCMs typically comprise metallic salts and salt hydrates for the storage of thermal heat. In this particular configuration, PCMs based on salt hydrates are employed to capture the thermal energy within the FPC. Within salt hydrates, ion-dipole forces in the PCM create connections between the two hydrogen bonds [33]. Weak forces of attraction exist among water molecules, and when latent heat energy is absorbed, these forces break. This results in the dissolution of the water molecule, and sodium sulfate begins to store thermal energy. The latent heat is generated as the salt initiates the hydration/dehydration process.

Figure 5 illustrates the phase transition of the PCM, depicting the absorption and release of latent heat of fusion. In this configuration, sodium sulfate decahydrate salt is employed for energy storage. Initially, during the first stage, the salt is in a solid form. When solar radiation is present, the absorber plate collects thermal heat and channels it to the PCM, initiating the phase transition. The salt absorbs and stores this heat in the form of latent heat. In the subsequent stage, when sunlight is unavailable, the stored heat begins to release, facilitating the operation of the load. As the entire energy is discharged, the PCM solidifies completely.

Fig. 5 PCMs transition from solid to liquid and liquid to solid.

Sodium sulfate decahydrate exhibits commendable thermal conductivity, making it effective for heat transfer, and it possesses a high temperature release capability. The introduction of nanocomposite materials serves to augment the thermal stability of these salts, thereby enhancing their capacity for bearing and transmitting heat [34].

Figure 6 provides a comprehensive overview of the melting points and heat-bearing capacities of various common PCMs. Fatty acid-based PCMs exhibit a melting point of 65 °C, while paraffin wax surpasses this with a melting point of 125 °C. Salts, on the other hand, boast the highest melting point at 225 °C. The PCMs based on salts are particularly advantageous for thermal heat storage, given their elevated melting points and superior thermal conductivity [35].

Fig. 6 Melting points of different PCMs [34, 35].

2.5 Copper oxide-based nanocomposite material added with PCM

Nanocomposite materials undergo supplementation with PCMs to improve their thermal conductivity. Typically, two categories of materials, organic and inorganic composites, are employed for this purpose [35]. In terms of thermal conductivity, inorganic-based composite materials outperform their organic counterparts. Among inorganic composite materials, metals and metal alloys are frequently utilized due to their high heat-bearing capacity [36]. In the proposed system, copper oxide composite material is introduced to augment the thermal stability of sodium sulfate decahydrate salt.

In Figure 7a, the illustration depicts the preparation of nanocomposite materials incorporating PCMs, specifically sodium sulfate decahydrate salts. To enhance the thermal conductivity of the PCMs, copper oxide is introduced along with the sodium sulfate salt. Furthermore, the inclusion of copper serves the dual purpose of enhancing thermal conductivity and providing corrosion protection for the microtubes in the FPCs.

Fig.7 (a) PCM preparation with copper oxide nanocomposite (b) Nanocomposite material types.

In Figure 7b, the illustration showcases the varieties of nanocomposite materials suitable for incorporation with phase change. Two prevalent types are organic and inorganic. In the scope of inorganic materials, metals and metalloids find common application, whereas organic materials typically involve the use of graphite, carbon nanotubes, and carbon nanofibers to augment the thermal stability of thermal heat storage. The elevated melting points and heat-bearing capacity of inorganic materials, in comparison to organic counterparts, result from the robust bonding between molecules in metals. Ionic, covalent, or coordinate covalent bonding exists, necessitating more energy for molecule breaking.

2.6 Thermal heat storage categories for PCMs

There are two main categories of thermal energy storage: Sensible heat storage and latent heat storage. Latent heat storage exhibits a higher thermal energy storage capacity per unit volume when compared to sensible heat storage [37].

Figure 8 illustrates the thermal energy storage categories specific to PCMs. The process of adding or removing heat in PCMs is indicative of sensible heat, while the energy needed to alter the shape or state of materials is termed the latent heat stage. Sensible heat storage materials include water and rocks, whereas organic and inorganic materials serve as latent heat storage materials. In this particular design, latent heat storage materials are employed for the storage of thermal energy.

Fig. 8 Thermal energy storage categories in PCM.

PCMs undergo a transition between solid and liquid or solid and gas states through processes such as melting or freezing. When the temperature rises, these materials, known as PCMs, store energy. During this phase, the molecules of the PCM begin to break, initiating an endothermic process in which the PCM absorbs energy. In the absence of heat, the PCM starts discharging, releasing stored energy and transitioning from its liquid state to a solid form in an exothermic reaction. This charging and discharging cycle of phase-change materials occurs with a minimal change in volume fraction, typically less than 10% [38].

3 Mathematical modeling

3.1 Assumptions

This study investigates the integration of NePCM into an FPC to enhance its thermal performance. The analysis encompasses heat-loss comparisons (with and without NePCM), output temperature variations during NePCM charging and discharging, heat transfer rates, and melting fraction rates. Energy balance equations and boundary conditions are formulated for both heat transfer fluids. Key assumptions for this study include:

• The Reynolds number for the heat transfer fluid is maintained between 1,500 and 4,000.

• The use of highly conductive copper and aluminum for the heat transfer fluid tubes results in negligible thermal resistance between the NePCM and the heat transfer fluid.

• The full insulation of the PCM storage tank’s outer layer results in negligible heat loss from the outer surface.

• Unlike the heat transfer fluid, whose properties remain constant regardless of temperature, the NePCM used in this study exhibits properties that vary with the temperature of the FPC.

• The collector experiences inlet and outlet temperatures ranging from 26 °C to 70 °C during peak hours.

• During charging, the NePCM completely transitions from solid to liquid, achieving a melting fraction of 1, which signifies maximum energy storage efficiency.

• The liquid NePCM is characterized by laminar flow, incompressibility, unsteady state, and Newtonian flow behavior.

• Nanomaterials like copper oxide are evenly distributed within the PCM, where ∇·U = 0.

• The study assumes a no-slip boundary condition, meaning the fluid velocity at the walls is zero.

3.2 Thermo-physical properties and boundary conditions

Table 1 presents the thermo-physical properties of the PCM and metal oxide used in this study. It includes specific heat capacity, melting and solidification temperatures, density, thermal conductivity, latent heat, dynamic viscosity, and thermal expansion volume for both the PCM and metal oxide.

Table 2 outlines the boundary conditions for the NePCM in this study, detailing the melting and solidification processes. The velocity of the NePCM is 4.6 during melting and 1.4 during solidification, with a turbulence intensity of 5 for both processes. The melting temperature of the NePCM is 65 °C, solidification occurs at 15 °C, and the pressure outlet is set at 0.3. Fluid flow rates are specified as 15 and 9 for different conditions.

3.3 Analytical model of heat transfer in nanoparticles of NePCM

The application of NePCM involves multiple cycles of charging and discharging over an extended period. Heat transfer within the NePCM occurs through sedimentation, Brownian diffusion, and thermophoresis diffusion of the nanoparticles, which are non-homogeneously distributed throughout the PCM. In this study, an aluminum copper pipe is used to transfer heat through diffusion. The analytical modeling of the system is outlined as follows. First, the density and specific heat capacity of the NePCM used in this work are determined. Then, using the provided equations, the latent heat value, thermal expansion, dynamic viscosity, thermal conductivity, and melting fraction of the NePCM are calculated. The equations in Table 3 are used to determine the energy balance and energy transfer in the nanoparticles of the NePCM. These equations illustrate the continuity value, momentum, energy, liquid fraction, buoyancy term, and momentum sources of the NePCM. Here, u, t, T, P, and g depict the velocity vector, time, temperature, pressure, and gravity, respectively. Where μ_m is dynamic viscosity, ρm is density, C_pm is specific heat capacity, km is thermal conductivity, and β_m is volumetric expansion coefficient. S_m is the momentum source term, Sh is the enthalpy source term, T_l is liquid temperature, T_S is solid temperature, T_me is melting temperature, and C and b are constants with a value of 1.6 × 10⁶. The term g_k represents the Boussinesq gravity term.

Equation (1) determines the density of the NePCM, while equation (2) determines the specific heat capacity of the NePCM, where ρ represents the density of the materials, φ is the nano-additive volume fraction, and ρ_NM is the density of nanomaterials [41–43]:$${ \rho }_{\mathrm{NePCM}}=\left(1-\phi \right){\mathrm{\rho }}_{\mathrm{PCM}}\phi {\mathrm{\rho }}_{\mathrm{NM}},$$(1) $${\left(\rho C_{\mathrm{p}}\right)}_{\mathrm{NePCM}}=\left(1-\phi \right){\left(\rho C_p\right)}_{\mathrm{PCM}}+\phi {\left(\rho C_{\mathrm{p}}\right)}_{\mathrm{NM}}.$$(2)

Equation (3) is used to determine the latent heat of the NePCM, where L represents the latent heat of the materials used in the NePCM.$${\left(\rho L\right)}_{\mathrm{NePCM}}=\left(1-\phi \right){\left(\rho L\right)}_{\mathrm{PCM}}.$$(3)

Equation (4) is used to determine the thermal expansion of the NePCM, where β represents the coefficient of thermal expansion:$${\left(\rho \beta \right)}_{\mathrm{NePCM}}=\left(1-\phi \right){\left(\rho \beta \right)}_{\mathrm{PCM}}.$$(4)

Equation (5) is used to determine the dynamic viscosity of the NePCM, where μ represents the dynamic viscosity:$${\mathrm{\mu }}_{\mathrm{NePCM} }=0.9197{\mathrm{e}}^{22.8539\phi }{\mu }_{\mathrm{PCM}}.$$(5)

Equation (6) is used to determine the thermal conductivity of the NePCM, where k_NP is the thermal conductivity of the nanomaterial and k_PCM is the thermal conductivity of the PCM.$$k_{\mathrm{NePCM}}=k_{\mathrm{PCM}}\left[\frac{k_{\mathrm{NP}}+{2k}_{\mathrm{pcm}}-2\phi (k_{\mathrm{PCM}}-k_{\mathrm{NP}})}{k_{\mathrm{NP}}+{2k}_{\mathrm{PCM}}+\phi (k_{\mathrm{pcm}}-k_{\mathrm{NP}})}\right]$$(6)

Equation (7) is used to determine the melting point of the NePCM, where T_m,l and T_m,u are the lower and upper temperature point values, respectively.$$\phi =\left\{\begin{array}{c}\frac{T-T_{\mathrm{m},1}}{T_{\mathrm{m},\mathrm{u}}-T_{\mathrm{m}1}}, 0T<T_{\mathrm{m},1}\\ T_{\mathrm{m},1}\le T\le T_{\mathrm{m},\mathrm{u}}\\ 1T>T_{\mathrm{m},\mathrm{u}}\end{array}\right\}$$(7)

The FPC undergoes mathematical modeling by incorporating PCMs, with the addition of nanocomposite materials to augment the thermal storage capacity of these PCMs. Several steps have been implemented to assess the efficiency of the FPC featuring PCMs shown in Figure 9. Initially, the total heat transfer of the FPC is quantified, followed by the determination of its total heat loss. Subsequently, the overall efficiency of the FPC is calculated. Finally, the charging and discharging processes of the PCMs are investigated [11, 35, 44].

Fig. 9 Mathematical modeling flow chart for simulation.

Equation (8) is used to determine the total heat transfer of the FPCs, where Q_u represents the overall heat transfer of the FPC, A_c is the area of the collector, F_r is the fractional efficiency of the collector, G_t is the solar constant value, α_t is the transmissivity of the glass, U_L is the overall heat loss coefficient of the FPC, T_in is the inlet temperature, T_i is the inlet temperature during phase change, m is the mass of the fluid, c_p is the specific heat capacity of the heat transfer fluid, and T_out is the outlet temperature of the collector.$$Q_{\mathrm{u}}={A_{\mathrm{c}}F}_{\mathrm{r}}\left[G_{\mathrm{t}}{\alpha }_{\mathrm{t}}-U_{\mathrm{L}}\left(T_{\mathrm{in}}-T_{\mathrm{i}}\right)\right]=m{c}_{\mathrm{p}}\left(T_{\mathrm{in}}-T_{\mathrm{out}}\right).$$(8)

Equation (9) is used to determine the overall heat loss of the FPCs. U_L, the overall heat loss coefficient, is separated into: U_t, representing the top heat loss; U_b, representing the bottom heat loss; and U_e, also representing the side loss of the FPC.$$U_{\mathrm{L}}=U_{\mathrm{t}}+U_{\mathrm{b}}+U_{\mathrm{e}}.$$(9)

The bottom heat loss coefficient (U_b) of an FPC, as shown in equation (10), is determined by the combined effects of conduction through the insulation and convection to the surrounding air. The conduction component is governed by the insulation thickness (L_b) and its thermal conductivity (K_b), while the convective heat loss is represented by h_b,a.$$U_{\mathrm{b}}=\left(\frac{L_{\mathrm{b}}}{K_{\mathrm{b}}}+\frac{1}{h_{\mathrm{b},\mathrm{a}}}\right).$$(10)

The side heat loss coefficient (U_e) of the FPC, as shown in equation (11), is determined by the combined effects of conduction through the side insulation and convection to the surrounding air. The conduction component is governed by the insulation thickness (L_e), while the convective heat loss is represented by h_e,a.$$U_{\mathrm{e}}=\left(\frac{L_{\mathrm{e}}}{K_{\mathrm{b}}}+\frac{1}{h_{\mathrm{e},\mathrm{a}}}\right)$$(11)

The top heat loss coefficient (U_t) of the collector, as shown in equation (12), is determined by the combined effects of the overall heat transfer coefficient between the absorber plate and the glass cover (R_pc) and the thermal resistance between the glass cover and the surrounding environment (R_ca).$$U_{\mathrm{t}}=\frac{1}{R_{\mathrm{pc}}}+\frac{1}{R_{\mathrm{ca}}}.$$(12)

In equation (13), R_pc represents the overall heat transfer coefficient, which combines the radiative and convective heat transfer between the absorber plate and the glass cover. The convective component, hpc, depends on several factors, including the Stefan-Boltzmann constant (σ), the mean temperature of the collector (T_pm), the glass cover temperature (T_c), and the emissivity of the absorber plate (ϵ_p) and glass cover (ϵ_c).$$R_{\mathrm{pc}}={\left[h_{\mathrm{pc}}+\frac{\frac{\sigma \left(T_{\mathrm{pm}}+T_{\mathrm{c}}\right)\left(T_{\mathrm{pm}}^2+T_{\mathrm{c}}^2\right)}{1}}{{ϵ}_{\mathrm{p}}}+\frac{1}{{ϵ}_{\mathrm{c}}}-1\right].}^{-1}$$(13)

In equation (14), R_ca denotes the thermal resistance between the glass cover and the surrounding environment. This resistance is influenced by the wind heat transfer coefficient (h_w), the emissivity of the glass cover (ϵ_c), the Boltzmann constant (σ), the temperature of the glass cover (T_c), the sky temperature (T_s), and the ambient temperature (T_a).$$R_{\mathrm{ca}}={\left[h_{\mathrm{w}}+\frac{{ϵ}_{\mathrm{c}}\sigma \left(T_{\mathrm{c}}^4-T_{\mathrm{s}}^4\right)}{T_{\mathrm{c}}}-T_{\mathrm{a}}\right]}^{-1}.$$(14)

Equation (15) is used to determine the radiation heat flux, where E_e is the flux density difference of the irradiation, J_e is the radiosity (the sum of the emitted radiation), and ∂_γ is the wavelength of the radiation.$$Q_{\mathrm{r}}=E_{\mathrm{e}}-J_{\mathrm{e}}={\int }_0^{\infty }{E}_{\mathrm{e},\mathrm{\gamma }}, {\partial }_{\mathrm{\gamma }}-{\int }_0^{\infty }{J}_{\mathrm{e},\mathrm{\gamma }}, {\partial }_{\mathrm{\gamma }}.$$(15)

Equation (16) is used to determine the overall thermal efficiency (η_th) of the flat-plate collector with PCMs. It considers the collector’s thermal efficiency (η_th), the overall heat transfer (Q_u) of the collector, the solar constant (G_t), and the collector area (A_c).$${\mathrm{ŋ}}_{\mathrm{th}}=\frac{\int Q_{\mathrm{u}}\mathrm{d}t}{\int G_{\mathrm{t}}{A}_{\mathrm{c}}\mathrm{d}t}.$$(16)

Equation (17) is used to determine the charging of the PCMs, where mp is the mass of the PCMs, c_p is their specific heat capacity, T is the average temperature of the PCMs, d_t represents the diameter of the tubes, m is the mass of the working fluid, T_out is the output temperature, and T_in is the input temperature. Q_ch represents the charging of the PCMs, and the output efficiency of the collector is I, while A_c is the area of the collector and U_L is the overall heat loss.$$m_{\mathrm{p}}{c}_{\mathrm{p}}\left(\frac{{\mathrm{dT}}_{\mathrm{p},\mathrm{avg}}}{d_{\mathrm{t}}}\right)+m{c}_{\mathrm{p}}\left(T_{\mathrm{out}}-T_{\mathrm{in}}\right)+Q_{\mathrm{ch}}={ŋ}_{0}I{A}_{\mathrm{c}}-U_{\mathrm{L}}\left(T_{\mathrm{p},\mathrm{avg}}-T_{\mathrm{c}}\right)T_{\mathrm{c}}.$$(17)

Equation (18) is used to determine the discharging process of the PCMs, where Q_dis is the discharging of the PCMs.$$m_{\mathrm{p}}{c}_{\mathrm{p}}\left(\frac{{\mathrm{dT}}_{\mathrm{p},\mathrm{avg}}}{d_{\mathrm{t}}}\right)+m{c}_{\mathrm{p}}\left(T_{\mathrm{out}}-T_{\mathrm{in}}\right)=Q_{\mathrm{dis}}-U_{\mathrm{L}}\left(T_{\mathrm{p},\mathrm{avg}}-T_{\mathrm{c}}\right)T_{\mathrm{c}}.$$(18)

Equation (19) is used to determine the total heat stored by the PCMs with the addition of nanocomposite materials. A_c is the area of the collector, F_r is the fractional efficiency of the collector, G_t is the solar constant value, and U_L is the overall heat loss of the FPC. Where T_in is the inlet temperature, T_i is the inlet temperature in the phase change, T_out is the output temperature of the collector, T_p is the temperature of the plate, T_pcm is the temperature of the PCM, R_tot is the total resistance loss, and A_pcm depicts the area where the PCM is placed.$$Q_{\mathrm{u}}={A_{\mathrm{c}}F}_{\mathrm{r}}\left[G_{\mathrm{t}}{\alpha }_{\mathrm{t}}-U_{\mathrm{L}}\left(T_{\mathrm{in}}-T_{\mathrm{i}}\right)\right]-T_{\mathrm{p}}-\frac{T_{\mathrm{pcm}}}{R_{\mathrm{tot} }}\times A_{\mathrm{pcm}}.$$(19)

Equation (20) is used to find the overall efficiency of the PCMs during the charging and discharging processes. The efficiency of the PCM during charging is represented by η_ch, and the efficiency of the PCM during the discharging mode is represented by η_dis.$${ŋ}_0={ŋ}_{\mathrm{ch}}\times {ŋ}_{\mathrm{dis}}$$(20)

In equation (21), E_x,stored denotes the total exergy stored within the PCM, while E_x,in represents the total exergy entering the PCM system.$${ŋ}_{\mathrm{ch}}=\frac{E_{\mathrm{x},\mathrm{stored}}}{.E_{\mathrm{x},\mathrm{in} }}$$(21)

In equation (22), E_x,out represents the total exergy released or discharged by the PCMs, while E_x,stored denotes the total exergy stored within the PCMs.$${ŋ}_{\mathrm{dis}}=\frac{E_{\mathrm{x},\mathrm{out}}}{E_{\mathrm{x}, \mathrm{stored}}}.$$(22)

Equation (23) determines the inlet exergy of the PCMs (E_x,in) based on their mass (m), specific heat capacity (c_p), inlet temperature (T_in), and outlet temperature (T_out), as well as the ambient temperature (T_a).$$E_{\mathrm{x},\mathrm{in}}=m{c}_{\mathrm{p}}\left[\left(T_{\mathrm{in}}-T_{\mathrm{out}}\right)-T_{\mathrm{a} }\mathrm{In}\left(\frac{T_{\mathrm{in}}}{T_{\mathrm{out}}}\right)\right].$$(23)

Equation (24) is used to determine the exergy out during the discharging process$$E_{\mathrm{x},\mathrm{out}}=m{c}_{\mathrm{p}}\left[\left(T_{\mathrm{out}}-T_{\mathrm{in}}\right)-T_{\mathrm{a} }\mathrm{In}\left(\frac{T_{\mathrm{out}}}{T_{\mathrm{in}}}\right)\right].$$(24)

Equation (25) is used to determine the total exergy of the PCMs$$E_{\mathrm{x}}= m{c}_{\mathrm{p}}\left[\left(T_{\mathrm{in}}-T_{\mathrm{out}}\right)\left[1-\frac{T_{\mathrm{a}}}{T_{\mathrm{m}}}\right]\right].$$(25)

Equation (26) determines the exergy efficiency of the PCMs, both for simple PCMs and nanocomposite PCMs. This efficiency (η) relates the output exergy from the PCMs (E_x,out) to the input exergy (E_x,in).$$ŋ=E_{x,\mathrm{out}}/E_{x,\mathrm{in}}.$$(26)

4 Results

PCMs are incorporated into FPCs alongside nanocomposite materials to improve the thermal stability of energy storage. The numerical analysis includes the design aspects of heat transfer, heat loss, fluid fractions, PCM charging and discharging energy, exergy, and overall efficiency of the FPC. The collector is positioned at a latitude of 5.1679 and a longitude of 100.4783. A comparison is made between the presence of nanotubes in the proposed FPC and a conventional plain tube collector. Sodium sulfate decahydrates serve as PCM, and copper oxide is introduced to enhance the thermal stability of the PCMs.

Figure 10 illustrates the temperature variations observed in FPCs. Inlet, outlet, and ambient temperatures exhibit fluctuations throughout the day. The consistent weather conditions in Malaysia persist throughout the year. At 8:00 am, the FPC’s inlet temperature initiates at 28 °C, gradually rising throughout the day. During peak hours (1–2 pm), the inlet temperature reaches its maximum at 35 °C. Simultaneously, the outlet temperature begins at 26 °C in the morning, increasing as heat is stored and peaking at 33 °C during the same peak hours. The ambient temperature starts at 28.3 °C in the morning, reaching 36 °C at the day’s zenith. Weather data is sourced from real-time meteorological data. The irradiance value commences at 200 W/m², peaks at 1,000 W/m² during midday, subsequently decreases, and eventually returns to 200 W/m², reaching zero at sunset.

Fig. 10 Temperature and irradiance values at the inlet, outlet, and ambient conditions for the FPC.

Figure 11 depicts the heat loss characteristics of the FPC, encompassing top heat loss, bottom heat loss, side heat loss, and overall heat loss. Equations (10)–(14) are used to determine the overall heat loss. The top heat loss accounts for losses in the glazing and absorber of the FPC. In the early morning, when the ambient temperature is low at 28 °C, the top heat loss is 90 W, the side loss is 50 W, the bottom loss is 70 W, and the overall heat loss is 240 W. As the day progresses and temperatures rise, the losses in the FPC escalate. At the peak temperature of 36 °C, the losses reach their maximum, with top loss at 420 W, side loss at 300 W, bottom loss at 310 W, and overall heat loss at 840 W. At sunset, the heat loss begins to decrease, paralleled by a reduction in the output of the FPC.

Fig. 11 Heat losses in the FPC.

Figure 12 illustrates the output temperature of the working fluid released after expelling absorbed energy. Without the integration of PCMs, the output temperature is high. However, when PCMs are incorporated into the FPC, the output temperature decreases because the PCM absorbs more energy compared to a simple working fluid.

Fig. 12 Output temperature of FPC with and without integration of PCM.

The output temperature is 28 °C without PCM, while with PCM, it is 26 °C at a working fluid flow rate of 1,500 Re. As the temperature rises, the working fluid rate is increased. At a flow rate of 3,700 Re, the output temperature without PCMs is 36 °C, and with PCMs, it is 34 °C. With a further increase in the working fluid rate, the output temperature begins to decline due to a reduction in the absorber rate. During the simulation, a high upper limit for the temperature value was set, leading to a peak value at a Reynolds number of 3,700. This local peak in the simulation results can be attributed to the change in heat transfer fluid viscosity as the flow transitions from laminar to turbulent.

Figure 13 illustrates the charging and discharging processes of PCM under solar radiation, comparing cases with and without the incorporation of nanocomposite materials containing sodium sulfate decahydrate. Equations (17) and (18) is used to determine the charging and discharging process of PCM. The performance of PCM during these processes is notably enhanced with the addition of nanocomposite material, particularly due to the presence of copper oxide, which improves thermal conductivity and storage duration.

Fig. 13 Charging and discharging of PCM with and without nanocomposite materials.

At the beginning of the day at 8:00 am, when solar radiation intensity is 100 W/m², the charging capacity of PCM without nanocomposite materials is 50 kJ, whereas with the addition of nanocomposite materials, it reaches 70 kJ. As the day progresses and solar radiation intensity increases, the storage capacity experiences a corresponding rise. During the peak hours, PCM reaches its maximum charging capacity.

By 5:00 pm, at the day’s conclusion, the charging capacity of PCM without nanocomposite materials stands at 350 kJ, while PCM with nanocomposite materials reaches 400 kJ. In the absence of solar radiation, the stored energy is utilized to power the thermal load. Notably, during the discharging process, PCM without nanocomposite materials discharges earlier.

The highest intensity of sunlight is reached between 1 and 2 pm, allowing the FPC to operate at peak efficiency and the NePCM to charge at its maximum rate. Although the sunlight intensity decreases after this peak, the FPC continues to operate and store energy in the NePCM until the sun sets, extending the charging process until 6 PM. After sunset, the stored energy is used to power the load.

Figure 14 depicts the heat transfer rates of both simple PCM and PCM enhanced with nanocomposite materials. The heat transfer rate is significantly higher in PCM with nanocomposite materials compared to simple PCM.

Fig. 14 Heat transfer rate of simple and nanocomposite PCM.

At the beginning of the day, at 8:00 am, the heat transfer rate of the FPC in simple PCM is 100 W, whereas with the inclusion of nanocomposite materials in the PCMs, it reaches 160 W. Throughout the day, the heat transfer rate experiences a continuous increase. By the day’s conclusion, at 5:00 pm, the heat transfer rates reach their maximum values, with simple PCM at 550 W and nanocomposite PCM at 650 W.

Figure 15 illustrates the melting fraction over time for both simple PCM and PCM enhanced with nanocomposite materials in the context of FPCs. The inclusion of sodium sulfate decahydrate allows for the absorption of heat, breaking down water molecules to store thermal energy. The introduction of copper oxide nanocomposite materials further enhances the thermal stability of the PCM.

Fig. 15 Melting fraction of simple and nanocomposite PCM.

Starting at 8:00 am, the melting fraction of simple PCM is 0.1, while nanocomposite PCM registers a higher value of 0.2. As the day unfolds, the melting fraction rates for both materials increase. By 5:00 pm, at the day’s conclusion, the melting fraction of simple PCM reaches 0.9, whereas nanocomposite PCM shows a higher value of 1.2, indicating an advanced level of thermal transformation and stability.

Figure 16 presents the liquid fraction over time for both simple PCM and PCM enhanced with nanocomposite materials. Commencing at 8:00 am, the liquid fraction for simple PCM is 0.12, while nanocomposite PCM starts at a slightly higher value of 0.2. As the day advances, the liquid fraction values for both materials increase, corresponding to an augmentation in thermal energy storage. Equation (7) is used to determine.

Fig. 16 Liquid fraction of simple and nanocomposite PCM.

By 5:00 pm, at the day’s conclusion, the liquid fraction for simple PCM reaches 0.90, indicating complete phase change. In contrast, the liquid fraction for nanocomposite materials is elevated to 0.99, attributed to the inclusion of metal oxide materials, signifying enhanced thermal characteristics and increased thermal energy storage capability.

Figure 17 depicts the temporal variations in exergy and mass flow rates for the FPC incorporating PCMs. Commencing at 8:00 am, the exergy rate for the FPC, devoid of PCM integration, stands at 1.2 W. However, with the inclusion of phase change and nanocomposite materials, the exergy rate increases to 1.6 W, accompanied by a mass flow rate of 4. Over the course of time, both the exergy rate and mass flow rate exhibit an upward trend. At the conclusion of the observation period, the maximum exergy transfer rate is attained, reaching 10 W with the integration of PCMs and 9.2 W without such integration. Simultaneously, the mass flow rate reaches 8.5. Equations (23) and (24) is used to determine the exergy rate in and out in PCMs.

Fig. 17 Exergy rate of the FPC.

Figure 18 illustrates the exergy efficiency of the FPC with both simple PCM and nanocomposite PCMs in relation to irradiation intensity. At the commencement of the day at 8:00 am, the exergy efficiency for simple PCM stands at 20%, while for nanocomposite PCM, it is 35%, with an irradiance value of 200 W/m².

Fig. 18 Exergy efficiency of the PCMs and nanocomposite.

As the day progresses, the exergy efficiency experiences a corresponding increase. During the peak hours, specifically between 1:00 pm and 2:00 pm, the exergy efficiency reaches 78% for simple PCM and 85% for nanocomposite PCMs, coinciding with an irradiance intensity value of 1000 W/m². Following this peak period, as the irradiance value starts to decrease, the exergy efficiency of the FPC also exhibits a decline. Equation (26) is used to determine the overall exergy efficiency of the FPCs.

Figure 19 depicts the overall efficiency of the FPC utilizing nanotubes and nanocomposite materials in relation to sun intensity values. At sunrise, 8:00 am, the sun intensity is 200 W/m², and the collector’s efficiency stands at 30%. As the day unfolds, the sun’s intensity increases, leading to a corresponding rise in the collector’s efficiency. During the peak hours, specifically between 1:00 pm and 2:00 pm, when the irradiance intensity reaches its zenith at 1,000 W/m², the collector achieves its highest efficiency at 85%. Post this peak period, as the sun’s intensity diminishes, the efficiency of the FPC also experiences a decline.

Fig. 19 Overall efficiency of the FPC concerning radiation intensity of the sun.

Figure 20 illustrates the enhancement in overall efficiency in FPCs achieved through the incorporation of nanocomposite materials and nanotubes. The utilization of nanocomposite materials leads to a noteworthy 12% improvement in the FPC’s efficiency. This improvement is attributed to the enhanced thermal conductivity of PCMs facilitated by the integration of nanocomposite materials, resulting in improved efficiency in thermal energy collection and storage within the PCMs.

Fig. 20 FPC efficiency improvement.

Simultaneously, the use of nanotubes contributes to a 6% increase in the efficiency of the FPC. When both nanocomposite materials and nanotubes are employed, the combined effect results in an overall efficiency improvement of 15% in the FPC.

4.1 Comparison of results

In Figure 21, the outcomes of Harish Kumar Sharma and Shuai Zhang are depicted, showcasing the integration of PCMs with FPCs [11, 45]. Figure 21a reveals Harish Kumar’s utilization of Nano-enhanced PCM and finned PCM to enhance FPC efficiency with respect to mass fraction. NePCM, consisting of nano paraffin wax and graphene, demonstrates an efficiency of 72.5%, while efficiency improves to 74% with finned NePCM, and simple PCM achieves 70%.

Fig. 21 Results obtained from references [11, 45] for comparison of proposed design.

Figure 21b illustrates the thermal energy storage characteristics of these phase-change materials. Simple PCM exhibits thermal energy storage of 200 kJ, whereas NePCM enhances it to 250 kJ, and finned NePCM further increases the storage to 270 kJ.

In Figure 21c, the results from Shuai Zhang’s study are presented, featuring a design incorporating pure salt and the addition of ceramic foam to augment collector efficiency and extend thermal energy storage duration [45]. The use of pure molten salt yields an energy storage capacity of 140 kJ, while the incorporation of ceramic foam enhances this capacity to 170 kJ.

In Figure 22, the outcomes of the suggested design are illustrated, encompassing the overall efficiency of the FPC integrated with Nano-Enhanced PCM and the corresponding charging and discharging times of the PCMs.

Fig. 22 Proposed design results for comparison.

Figure 22a depicts the overall efficiency of the FPC with the inclusion of sodium sulfate decahydrate and copper oxide. The integration of the FPC with nanotubes for PCM storage leads to an impressive 85% enhancement in overall efficiency.

Figure 22b showcases the charging and discharging processes of the PCMs, demonstrating superior performance compared to the approaches of Harish Kumar and Shuai Zhang [11, 45].

4.2 Practical comparison of FPC with NePCM

Figure 23a demonstrates the practicality of integrating an FPC with NePCM to enhance its thermal performance. According to reference [46], the thermal efficiency of a collector using paraffin wax NePCM reaches 82% at 2 pm. In contrast, the collector in this study, which utilizes sodium sulfate decahydrate and copper oxide, achieves an efficiency of 85% at the same time. This shows that the thermal efficiency in this work surpasses that of the compared study.

Fig. 23 (a) FPC efficiency comparison with the work from [46] (b) FPC with HnPCM and without PCM [47].

Figure 23b depicts the experimental comparison of the thermal efficiency of the FPC with HnPCM and without PCM. The efficiency of the FPC without HnPCM is 63.8%, and with the addition of HnPCM, the efficiency improves to 71.93%.

4.3 Limitation

NePCM improves energy storage performance and enhances the efficiency of FPCs when integrated with them. Incorporating NePCM into an FPC reduces load fluctuations and conserves energy. Salt hydrates combined with metal oxide-based phase change materials enhance storage duration and performance, although NePCM have certain limitations. The thermal conductivity of salt hydrate PCM is low, and the preparation of NePCM involves chemical and physical processes that are harmful to the environment. Additionally, the inclusion of nanomaterials in PCM is expensive, driving up the cost of NePCM. Selecting appropriate materials for NePCM preparation is also challenging; if suitable materials are unavailable, it can lead to instability in the NePCM. Moreover, the high leakage rate of inorganic phase change materials negatively impacts the performance of the FPC.

5 Conclusion

Thermal energy storage not only meets energy demands but also reduces the burden on renewable energy resources. PCM are particularly effective for storing thermal energy because of their high heat of fusion and excellent thermal conductivity. These materials absorb thermal energy, causing their molecules to break apart and transition from a solid to a liquid state, thus storing the energy efficiently. In this design, FPCs are integrated with PCMs to store thermal energy. Sodium sulfate decahydrate is used as the PCM for energy storage in this setup. To enhance thermal conductivity and improve storage efficiency, a copper oxide-based nanocomposite material is added to the salt. The PCMs and the energy they store are contained within nanotubes in the FPC. The key findings of the work include:

• The incorporation of nanocomposites with PCMs results in a 12% increase in efficiency, while the use of nanotubes contributes an additional 6% improvement, leading to an overall improvement of 15%. Consequently, the overall efficiency of the FPC in this design reaches 85%.

• With the use of sodium sulfate decahydrate, copper oxide, and nanotubes as a NePCM, the exergy efficiency of the FPC also improves by 85%.

• The charging and discharging of the NePCM are more efficient compared to simple PCM. The NePCM stores 50 kJ more energy than the simple PCM.

• The liquid fraction of the simple PCM is 0.90, whereas in this work, the NePCM achieves a liquid fraction of 0.99, demonstrating a significant improvement over the simple PCM.

• The flow rate is maintained within the range of 1,500–4,000 Reynolds numbers.

System simulation is conducted using Anaconda Jupyter Notebook with Python language. For future work, exploring various types of nanocomposite materials to further enhance thermal conductivity and extend thermal energy storage duration in PCMs is recommended. Additionally, implementing a rotating absorber pipe could be considered to enhance the efficiency of the FPC even further.

Funding

The research work is financially supported by the Universiti Sains Malaysia Short-Term Grant 304/PELECT/6315747.

Conflicts of 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.

References

All Tables

All Figures

	Fig. 1 PCMs integrated with FPC.
In the text

	Fig. 2 FPC with integration of the PCMs in nanotubes.
In the text

	Fig. 3 Nanotubes construction and process of PCMs energy storage.
In the text

	Fig. 4 Classification of the PCMs [32].
In the text

	Fig. 5 PCMs transition from solid to liquid and liquid to solid.
In the text

	Fig. 6 Melting points of different PCMs [34, 35].
In the text

	Fig.7 (a) PCM preparation with copper oxide nanocomposite (b) Nanocomposite material types.
In the text

	Fig. 8 Thermal energy storage categories in PCM.
In the text

	Fig. 9 Mathematical modeling flow chart for simulation.
In the text

	Fig. 10 Temperature and irradiance values at the inlet, outlet, and ambient conditions for the FPC.
In the text

	Fig. 11 Heat losses in the FPC.
In the text

	Fig. 12 Output temperature of FPC with and without integration of PCM.
In the text

	Fig. 13 Charging and discharging of PCM with and without nanocomposite materials.
In the text

	Fig. 14 Heat transfer rate of simple and nanocomposite PCM.
In the text

	Fig. 15 Melting fraction of simple and nanocomposite PCM.
In the text

	Fig. 16 Liquid fraction of simple and nanocomposite PCM.
In the text

	Fig. 17 Exergy rate of the FPC.
In the text

	Fig. 18 Exergy efficiency of the PCMs and nanocomposite.
In the text

	Fig. 19 Overall efficiency of the FPC concerning radiation intensity of the sun.
In the text

	Fig. 20 FPC efficiency improvement.
In the text

	Fig. 21 Results obtained from references [11, 45] for comparison of proposed design.
In the text

	Fig. 22 Proposed design results for comparison.
In the text

	Fig. 23 (a) FPC efficiency comparison with the work from [46] (b) FPC with HnPCM and without PCM [47].
In the text