Transient thermo-optical and phase-change modeling of a parabolic trough solar collector integrated with Al 2 O 3 nano-enhanced PCM

Abstract

Parabolic trough solar collectors (PTSCs) are a mature concentrating solar technology, but their performance is limited by solar intermittency. Integration with phase change materials (PCMs) can mitigate this limitation, yet conventional PCMs suffer from low thermal conductivity. Nano-enhanced PCMs (NePCMs) offer improved heat transfer, but most existing studies rely on simplified transient models and lack comprehensive annual performance assessment under realistic climatic conditions. To address this gap, this study develops a fully transient optical–thermal–phase-change model to investigate the annual performance of a PTSC integrated with NePCM. The coupled governing equations are discretized using a semi-implicit finite-difference scheme, while the melting and solidification processes are captured via a Stefan moving-boundary formulation and solved iteratively using a Gauss–Seidel algorithm. Hourly climatic data for Yazd (8760 h) are applied to resolve seasonal and diurnal variations. The results show that adding aluminum oxide (Al₂O₃) nanoparticles significantly enhances system performance: at 1 wt% concentration, the annual thermal efficiency increases by 10% (to 52%) compared to pure paraffin, and the annual exergy efficiency reaches 31% (an improvement of 15% over the baseline). Monthly useful energy output rises to 440 kWh, and the levelized cost of heat is reduced to 0.085 $/kWh_th, yielding the minimum payback period of about 5.2 years. Although the 3 wt% case achieves the highest instantaneous and monthly energy output (up to 470 kWh/month) and Carbon Dioxide (CO₂) avoidance (peaking at 120 kg/month), its economic performance deteriorates due to higher material costs. Overall, the results demonstrate that a 1 wt% nanoparticle loading offers the most favorable balance between thermodynamic performance, environmental benefit, and economic feasibility.

Introduction

Growing global energy demand, together with the accelerating climate crisis, has intensified interest in clean, reliable, and dispatchable thermal energy systems. Solar thermal technologies have therefore attracted sustained research attention as viable solutions for reducing fossil‑fuel dependence and greenhouse‑gas emissions (Kenfack et al., 2024a; Shoaib et al., 2024; Silinto et al., 2025; Zhang et al., 2025). Within this context, parabolic trough solar collectors (PTSCs) have emerged as one of the most mature and efficient concentrating solar technologies, owing to their high optical efficiency, relatively simple geometry, and proven capability to supply medium‑temperature heat. Consequently, PTSCs have been widely deployed in applications such as electricity generation, industrial process heat, desalination, and agricultural processing (Alayi et al., 2021; Bangoup Ntegmi et al., 2024; Kenfack et al., 2024b; Mohammad et al., 2025; Simeu et al., 2024).

Early PTSC research, conducted mainly prior to the mid-2010s, focused on optical concentration characteristics and steady-state or quasi-steady thermal modeling of collectors operating without thermal energy storage (TES). These studies established fundamental performance indicators—such as optical efficiency, receiver heat losses, and instantaneous thermal efficiency—under simplified assumptions of constant solar input and negligible climatic variability (Babikir et al., 2020; Pakouzou, 2018). Although this body of work provided essential design insights, it also highlighted a key limitation: the strong sensitivity of PTSC performance to solar intermittency, which restricts their ability to deliver stable and dispatchable heat under real operating conditions.

To overcome the inherent intermittency of solar radiation and enhance the dispatchability of PTSCs, integration with TES systems has become essential. Among available TES options, phase change materials (PCMs) are particularly attractive due to their high latent heat storage capacity and their ability to maintain nearly constant temperatures during melting and solidification. When coupled with PTSCs, PCMs can effectively smooth outlet-temperature fluctuations, extend useful heat delivery beyond peak solar hours, and improve the suitability of solar thermal systems for thermally sensitive applications.

However, conventional PCMs, such as paraffin wax, suffer from intrinsically low thermal conductivity, which limits charging and discharging rates under fully transient operating conditions. This drawback becomes more pronounced during long-term and seasonal operation, where incomplete melting or solidification can degrade system performance. To address this limitation, nano-enhanced phase change materials (NePCMs) have been introduced by dispersing high-thermal-conductivity nanoparticles within the PCM matrix, thereby significantly enhancing effective thermal conductivity while largely preserving the high latent heat capacity of the base material. Among various nanoparticle additives, aluminum oxide (Al₂O₃) is particularly suitable due to its chemical stability, cost-effectiveness, and compatibility with organic paraffins. Consequently, integrating PTSCs with Al₂O₃-based NePCM storage systems provides an effective and scalable solution for mitigating solar intermittency and improving the long-term thermal and exergy performance of solar thermal systems.

As PTSC applications expanded into thermally sensitive sectors, particularly agricultural drying—where temperature stability directly affects product quality and energy efficiency—the limitations of storage-free systems became increasingly evident (Calín-Sánchez et al., 2020; Fernandes and Tavares, 2024; Guiné, 2018; King’ori and Simate, 2024; Talebi et al., 2025). This recognition motivated a second phase of research, roughly between 2016 and 2020, characterized by the integration of TES systems into PTSC configurations. During this period, conventional PCMs, especially paraffin-based materials, were introduced due to their high latent-heat capacity and ability to mitigate short-term solar fluctuations (Djebli et al., 2020; Khanlari et al., 2020).

Subsequent investigations demonstrated that PCM-assisted PTSC systems can effectively smooth outlet-temperature profiles and extend useful heat delivery beyond peak solar hours, particularly in solar drying and low-temperature industrial applications (Jamil et al., 2024; Sharif et al., 2017; Yadav et al., 2024). Nevertheless, most studies adopted simplified modeling approaches—such as effective heat-capacity or lumped-parameter formulations—and often neglected explicit treatment of the moving solid–liquid interface. In addition, system performance was commonly assessed over short diurnal periods and primarily through energy-based metrics, while exergy efficiency, economic feasibility, and environmental impacts received comparatively limited attention.

Entering the early 2020s, the inherently low thermal conductivity of conventional PCMs emerged as a major bottleneck, prompting a third research phase focused on heat-transfer enhancement strategies. Proposed solutions included extended fins and topology-optimized surfaces (Avargani et al., 2023; Fragnito et al., 2026), metal foams (NematpourKeshteli et al., 2024), metal powder additives (Pramothraj et al., 2020), and hybrid PCM–heat-pipe configurations (Kou et al., 2025; Kumar et al., 2026; Reheem et al., 2022). Although these approaches improved heat-transfer rates, many introduced additional geometric complexity, higher costs, or manufacturing challenges that limited their practical scalability.

Within this enhancement landscape, NePCMs have emerged as a particularly promising and scalable solution. By dispersing high-thermal-conductivity nanoparticles into the PCM matrix, NePCMs substantially enhance effective thermal conductivity and accelerate melting and solidification processes without fundamentally altering storage geometry (Adavi and Chandramohan, 2024; Rani and Tripathy, 2023). Recent studies have shown that dispersion stability critically influences cyclic thermal reliability (Alsaleh et al., 2025), while nanoparticle addition modifies phase-change kinetics and introduces nonlinear melting behavior (Delfiya et al., 2024). Importantly, these works also report the existence of an optimal nanoparticle concentration, beyond which agglomeration, increased viscosity, and latent-heat dilution degrade overall system performance (Dhaidan et al., 2025).

Among various nanoparticle additives, Al₂O₃ has attracted considerable attention due to its chemical stability, cost-effectiveness, and compatibility with organic paraffins, making it particularly suitable for solar thermal storage applications (Hin et al., 2024; Kumar et al., 2024). Al₂O₃-based NePCMs have been shown to significantly enhance thermal conductivity and transient heat-transfer behavior under cyclic heating conditions (Golzar et al., 2025; Sepehrirad et al., 2024). However, most existing PTSC–NePCM studies remain limited to short-term or laboratory-scale analyses and rely on simplified phase-change models that neglect explicit moving-boundary treatment, thereby limiting the accuracy of melting-front prediction (Dezfulizadeh et al., 2023).

More recent investigations reflect a gradual shift toward transient, system-level modeling and multi-criteria performance evaluation, incorporating temperature-dependent thermophysical properties, coupled optical–thermal formulations, and more realistic outdoor boundary conditions (Vahidinia et al., 2023). Despite these advances, the majority of available studies still focus on short-term performance indicators. Consequently, cumulative annual effects, seasonal exergy degradation, incomplete melting–solidification cycles, long-term economic viability, and environmental benefits—particularly CO₂-avoidance potential—remain insufficiently understood for PTSC-based NePCM systems operating under realistic climatic conditions (Chara-Dackou et al., 2022; Pakouzou et al., 2021).

From a modeling perspective, these gaps are closely linked to the widespread use of steady-state, quasi-steady, or simplified transient formulations. Such approaches cannot accurately capture delayed melting onset, partial melting during low-irradiance periods, incomplete solidification, or cumulative seasonal effects. Although enthalpy-based and apparent heat-capacity methods are computationally efficient, they smear the solid–liquid interface over an artificial temperature range and suppress sharp thermal gradients, leading to inaccuracies in predicting melting-front propagation and interfacial heat transfer—limitations that become more pronounced for nano-enhanced PCMs with strongly nonlinear melting kinetics.

In contrast, the Stefan moving-boundary formulation provides a physically rigorous description of phase-change processes by explicitly tracking the solid–liquid interface and enforcing energy conservation at the phase front. This approach preserves interface sharpness, accurately resolves transient temperature gradients in both solid and liquid regions, and directly links thermophysical properties—such as thermal conductivity, latent heat, and nanoparticle concentration—to melting and solidification rates. These capabilities are essential for realistic modeling of Al₂O₃-based NePCMs integrated into PTSC systems.

Within this framework, the present study develops a fully transient, coupled thermo-optical–phase-change model for a PTSC integrated with an Al₂O₃–paraffin nano-enhanced PCM storage unit. Melting and solidification are explicitly resolved using a Stefan moving-boundary formulation, and the model is driven by 8760 h of real hourly climatic data. This enables systematic assessment of diurnal–seasonal variability, cumulative thermal effects, and long-term exergy behavior. Moreover, energy, exergy, environmental (CO₂ avoidance), and economic indicators are integrated within a single transient framework. The results identify an optimal nanoparticle loading of 1 wt%, providing the most favorable balance between thermodynamic performance, environmental benefit, and economic feasibility. Collectively, these features distinguish the present work from existing literature and offer a robust basis for long-term assessment and optimization of advanced PTSC–NePCM systems.

Methodology

System description

The proposed system consists of a conventional PTSC thermally coupled to a separate NePCM TES unit through a closed heat-transfer-fluid (HTF) loop, as illustrated in Figure 1. The PTSC subsystem is composed of a parabolic reflector, an evacuated-tube receiver located along the focal line, and an HTF circuit that circulates through the absorber tube.

Figure 1.

Schematic diagram of the PTSC thermally coupled to an external NePCM thermal energy storage unit via a closed HTF loop.

Solar radiation is reflected by the parabolic mirror toward the focal line and absorbed by the selective coating of the receiver tube, where the collected thermal energy is transferred to the flowing HTF. The heated HTF is then directed to an external NePCM storage tank, where thermal energy is stored in the form of latent heat during the charging process. The NePCM consists of paraffin wax enhanced with Al₂O₃ nanoparticles to improve effective thermal conductivity while preserving a high latent heat capacity.

Unlike configurations in which PCM is directly integrated around the receiver tube, the present study adopts an external storage arrangement. This configuration provides a clear physical separation between solar collection and thermal storage, improves operational flexibility, and enables accurate transient modeling of the charging and discharging processes. During periods of reduced direct normal irradiance (DNI) or solar unavailability, the stored thermal energy is released from the NePCM through the HTF, which is recirculated to the PTSC loop, thereby smoothing the outlet temperature profile and enhancing the system's dispatchability and daily exergy delivery.

The integration of the external NePCM storage introduces a coupled two-regime thermal behavior, involving sensible heating and cooling of the HTF and latent heat storage within the PCM domain. The system operates under fully transient conditions governed by time-varying solar irradiance and ambient climatic parameters. Consequently, numerical modeling is required to resolve the temporal evolution of HTF temperatures, heat-transfer rates, and the melting and solidification dynamics of the NePCM. The PTSC is designed according to typical commercial-scale dimensions to ensure realistic performance assessment. The aperture width W_a is set to 5.0 m, and the collector length L_c is 12 m, yielding a total aperture area of 60 m². The parabolic profile follows a focal length ff of 1.84 m, which determines the concentration ratio C  =  W_a/(πD_r,o) ≈ 0.7. The receiver is an evacuated-tube assembly consisting of a stainless-steel (SS-304) absorber tube with an outer diameter D_r,o = 70 mm, inner diameter D_r,i = 62 mm, and wall thickness t_r = 4 mm. The absorber surface is coated with a black-chrome selective layer (solar absorptivity α = 0.95, thermal emissivity ε = 0.10 at 400 °C). A concentric borosilicate glass envelope (outer diameter D_g,o = 120 mm, inner diameter D_g,i = 115 mm) encloses the absorber, creating a vacuum annulus (pressure <0.01 Pa) to suppress convective losses. The heat-transfer fluid (Therminol VP-1) circulates at a design mass flow rate m_HTF = 0.5 kg s⁻¹, corresponding to a Reynolds number of 12 000 in the absorber tube, ensuring turbulent flow and high internal heat-transfer coefficients. The nominal inlet temperature is set to 150 °C, representing a typical operating point for medium-temperature industrial applications. All geometric and operational parameters are summarized in Table 1. It is emphasized that the nano-enhanced PCM is employed solely as a stationary TES medium, while all heat transport between the PTSC and the storage unit is performed by a conventional liquid HTF.

Table 1.

Geometric and material specifications of the PTSC–NePCM system.

Component	Parameter	Symbol	Value
Parabolic trough	Aperture width	Wa	5.0 m
	Collector length	Lc	12 m
	Focal length	f	1.84 m
Receiver tube	Outer diameter	Dr,o	70 mm
	Inner diameter	Dr,i	62 mm
	Wall thickness	tr	4 mm
	Absorber material	–	SS-304
	Selective coating	–	Black chrome
Glass envelope	Outer diameter	Dg,o	120 mm
	Inner diameter	Dg,i	115 mm
PCM module	PCM layer thickness	tPCM	20 mm
	PCM volume	VPCM	Calculated
	PCM material	–	Paraffin + Al2O3
HTF	Fluid type	–	Thermal oil (Syltherm-800)
Storage tank	Tank geometry	–	Cylindrical external tank
	Tank height	Ht	1.0 m
	Tank inner diameter	Dt	0.30 m
	Tank wall material	–	Stainless steel
	Characteristic PCM layer thickness	tPCM	20 mm

Optical–thermal modeling of the PTSC

The present study adopts a physics-based, first-principles modeling strategy to describe the coupled thermo-optical behavior of the PTSC and the phase-change dynamics of the NePCM thermal storage unit. The governing equations are derived from conservation of energy and are formulated under clearly stated physical assumptions to ensure numerical robustness, reproducibility, and long-term transient stability over 8760-h simulations. All simplifying hypotheses are explicitly justified based on geometric characteristics, material properties, and characteristic thermal time scales of PTSC-based storage systems. The performance of a PTSC is governed by two coupled phenomena: optical concentration of solar radiation and thermal conversion of the absorbed flux into useful heat within the receiver tube (Figure 2).

Figure 2.

Schematic of the PTSC optical–thermal model, illustrating DNI incidence, mirror reflection, absorber heat gain $(\rho \tau \alpha )$ , and dominant loss pathways.

Optical model

The incident solar irradiance arriving on the mirror surface is composed of beam, diffuse, and ground-reflected components. For line-focus systems, only the beam component $(I_b)$ is considered in the optical concentration, such that the energy incident on the absorber per unit length is (Golzar et al., 2025; Sepehrirad et al., 2024):

$$q_{abs}=I_b\rho \tau \alpha IAM(\theta )$$

(1)

The optical efficiency $({\eta }_{opt})$ , representing the ratio of absorbed to incident energy, is defined as:

$${\eta }_{opt}=\rho \tau \alpha IAM(\theta )\cos (\theta )$$

(2)

The optical intercept factor $(\gamma )$ is introduced to capture non-ideal reflector accuracy and receiver alignment errors (Dezfulizadeh et al., 2023; Vahidinia et al., 2023):

$${\eta }_{opt,eff}={\eta }_{opt}\times \gamma$$

(3)

Typical values for $\gamma$ range from 0.94 to 0.98 for commercial trough systems. The instantaneous useful absorbed flux per unit aperture area becomes:

$${\dot{Q}}_{opt}={\eta }_{opt,eff}{I}_b$$

(4)

Thermal model of the receiver

The absorbed optical power raises the temperature of the receiver tube, which transfers heat to the circulating HTF. Applying a transient, one-dimensional energy balance for the fluid inside the tube gives (Osorio et al., 2023; Pakouzou et al., 2021):

$${\rho }_f{c}_p{A}_f\left({\displaystyle \frac{\mathrm{\partial }{T}_f}{{\mathrm{\partial }}_t}+{\upsilon }_f{\displaystyle \frac{\mathrm{\partial }{T}_f}{{\mathrm{\partial }}_x}}}\right)=h_i{P}_i(T_r-T_{amb})$$

(5)

And for the receiver wall temperature $T_r$ :

$$m_{rcr}{\displaystyle \frac{\mathrm{\partial }{T}_r}{\mathrm{\partial }t}=q_{abs}{A}_r-h_i{P}_i(T_r-T_f)-U_{loss}{A}_r(T_r-T_{amb})}$$

(6)

The thermal efficiency $({\eta }_{th})$ of the collector is defined by Pakouzou (2018), Pakouzou et al. (2021):

$${\eta }_{th}={\displaystyle \frac{\dot{m}{c}_p(T_{out}-T_{in})}{A_{ap}{I}_b}}$$

(7)

Finally, the overall instantaneous efficiency becomes (Osorio et al., 2023; Pakouzou et al., 2021):

$${\eta }_{overall}={\eta }_{opt,eff}\times {\eta }_{th}$$

(8)

This overall efficiency dynamically varies depending on solar incidence, fluid temperature rise, wind velocity, and receiver surface emissivity.

Thermal loss characterization

Two major heat-loss mechanisms are considered:

Convective loss to ambient (Babikir et al., 2021; Moosavian et al., 2021):

$$Q_{conv}=h_{ext}{A}_r(T_r-T_{amb})$$

(9)

$h_{ext}$ is obtained from Nusselt correlations for external forced or natural convection depending on wind velocity. Radiative loss through the glass env $U_{loss}$ elope:

$$Q_{rad}=ϵ\sigma A_r(T_r^4-T_{amb}^4)$$

(10)

The combined loss coefficient is hence (Babikir et al., 2021; Moosavian et al., 2021):

$$U_{loss}={\displaystyle \frac{Q_{conv}+Q_{rad}}{A_r(T_r-T_{amb})}}$$

(11)

In this study, the optical–thermal response of the receiver was computed using an effective, spectrally integrated (greybody) value for surface emissivity, as is standard practice for system-level PTSC simulations (Pakouzou, 2018; Pakouzou et al., 2021). The wavelength dependence of emissivity was not resolved through a multiband or spectral model, since the selective coatings employed exhibit only weak spectral variations in the thermal-infrared region of interest. This grey assumption is justified by parametric analysis showing minimal influence on annual thermal efficiency or exergy outcomes relative to other loss factors (e.g., meteorological variability, soiling). For applications requiring precision radiative analysis, extension to a banded or fully non-grey model is recommended. All optical and thermal parameters used in the present model, along with their assumptions and sources, are summarized in Table 2.

Table 2.

Optical and thermal parameters used in the PTSC–NePCM model.

Parameter	Symbol	Value/correlation	Assumption/description	Reference
Mirror reflectivity	ρ	0.93–0.95	Clean commercial parabolic mirror	Vahidinia et al. (2023); Dezfulizadeh et al. (2023)
Glass transmissivity	τ	0.94–0.96	Borosilicate glass envelope	Vahidinia et al. (2023); Pakouzou et al. (2021)
Absorber absorptivity	α	0.94–0.96	Selective coating	Pakouzou (2018)
Absorber emissivity	ɛr	0.84–0.88	Greybody, IR-averaged	Babikir et al. (2021); Chara-Dackou et al. (2022)
Glass emissivity	ɛg	0.86–0.90	Constant grey assumption	Babikir et al. (2021)
Intercept factor	γ	0.94–0.98	Optical alignment losses	Vahidinia et al. (2023)
Incident-angle modifier	IAM(θ)	ASHRAE correlation	Function of incidence angle	Babikir et al. (2021)
Internal HTF convection coefficient	hi	Dittus–Boelter	Turbulent internal flow	Osorio et al. (2023)
External convection coefficient	ho	Churchill–Bernstein	Forced/natural convection based on wind speed	Moosavian et al. (2021)
PCM–HTF heat-transfer coefficient	hpcm	Correlation-based	Effective coupling at PCM interface	Osorio et al. (2023), NematpourKeshteli et al. (2024)
Ambient heat-loss coefficient	U1	Derived from ho + radiation	Combined convective–radiative loss	Moosavian et al. (2021)
Stefan–Boltzmann constant	σ	5.67 × 10−8 W m−2 K−4	Physical constant	Standard
Nanoparticle viscosity model	μeff	Brinkman relation	Dilute suspension (φ ≤ 3%)	Osorio et al. (2023), NematpourKeshteli et al. (2024)

Exergy analysis of the PTSC–NePCM system

The exergy performance of the PTSC–NePCM system is evaluated using a control-volume approach consistent with the transient thermal model. All exergy calculations are performed on an hourly basis in accordance with the imposed climatic boundary conditions. The specific flow exergy rate associated with the HTF is calculated as (Pakouzou, 2018):

$$\dot{E}{x}_{HTF}=\dot{m}{c}_p\left[(T_o-T_i)-T_a\ln \left({\displaystyle \frac{T_o}{T_i}}\right)\right]$$

(12)

Where $\dot{m}$ is the HTF mass flow rate, $c_p$ is the specific heat capacity of the HTF, $T_i$ and $T_o$ denote the inlet and outlet HTF temperatures of th $T_a$ e PTSC, and is the ambient (dead-state) temperature. The solar exergy input is evaluated using the Petela formulation (Pakouzou, 2018):

$$\dot{E}{x}_{solar}=A{I}_b\left[1-{\displaystyle \frac{4T_a}{3T_s}+{\displaystyle \frac{1}{3}{\left({\displaystyle \frac{T_a}{T_s}}\right)}^4}}\right]$$

(13)

Where A is the collector aperture area, $I_b$ is the beam solar irradiance (DNI), and T_s = 5777 K is the effective sun temperature. The instantaneous exergy efficiency of the system is defined as:

$${\eta }_{ex}(t)={\displaystyle \frac{\dot{E}{x}_{HTF}}{\dot{E}{x}_{solar}}}$$

(14)

For long-term performance assessment, the annual cumulative exergy efficiency is obtained by temporal integration over the full simulation period (8760 h):

$${\eta }_{ex,annual}={\displaystyle \frac{\sum \dot{E}{x}_{HTF}\mathrm{\Delta }t}{\sum \dot{E}{x}_{solar}\mathrm{\Delta }t}}$$

(15)

For full reproducibility of the optical–thermal calculations, all optical and thermal parameters are treated as time-invariant within each hourly time step and are updated only when meteorological inputs change. Hourly DNI, ambient temperature, and wind speed are imposed as external forcing functions, while the incidence-angle modifier is recalculated at each step based on solar geometry. The internal convective heat-transfer coefficient between the HTF and the receiver wall is evaluated locally at each axial node using the Dittus–Boelter correlation for turbulent flow, with fluid properties evaluated at the local bulk temperature. The combined loss coefficient is recomputed iteratively at every time step to ensure consistency between wall temperature, radiative loss, and ambient conditions.

Effect of HTF thermophysical properties on system performance. The thermo-physical properties of the HTF play a central role in governing heat absorption, transport, and delivery between the PTSC and the PCM storage unit. In the present model, the HTF density, specific heat capacity, thermal conductivity, and dynamic viscosity are treated as temperature-dependent properties and are updated at each time step using manufacturer-provided correlations for Syltherm-800. The specific heat capacity directly influences the useful heat gain and outlet temperature rise of the collector through the transient fluid energy balance (equation (5)), while the HTF thermal conductivity and viscosity affect the internal convective heat-transfer coefficient via the Nusselt number correlations. Variations in HTF properties therefore impact both the collector thermal efficiency and the rate at which energy is transferred to the NePCM during charging and discharging cycles.

From a system-level perspective, the HTF specific heat capacity governs the thermal inertia of the PTSC–PCM loop, influencing the system's ability to smooth short-term solar fluctuations. A higher HTF heat capacity enhances thermal buffering but reduces the instantaneous temperature lift, whereas lower heat capacity leads to higher outlet temperatures but increased sensitivity to DNI variations. The HTF viscosity affects the internal convection regime and pumping requirements; however, within the operating temperature range of the present study, the Reynolds number remains in the turbulent regime, ensuring that convective heat transfer is not degraded. These coupled effects are reflected in both the transient temperature profiles and the exergy efficiency, as exergy destruction is directly linked to temperature gradients and irreversibilities associated with heat transfer between the HTF, receiver wall, and PCM domain.

Thermophysical properties of ${\mathrm{Al}}_2{\mathrm{O}}_3$ –paraffin nano-enhanced PCM

The thermal behavior of the latent heat storage unit strongly depends on the effective thermophysical properties of the NePCM. In this study, commercial paraffin wax is used as the base PCM, while ${\mathrm{Al}}_2{\mathrm{O}}_3$ nanoparticles with a loading fraction $\varphi$ between 0 and 3 vol% are dispersed to enhance thermal conductivity and improve transient melting characteristics. The selection of ${\mathrm{Al}}_2{\mathrm{O}}_3$ is supported by its high thermal conductivity, chemical stability, and excellent dispersion performance within organic paraffins.

Material selection rationale. The selection of materials in this study was guided by the need for physical transparency, numerical robustness, and long-term reproducibility under fully transient operation. Paraffin wax (RT-50HC) was selected as the base PCM due to its narrow melting temperature range, chemical stability, and well-documented thermophysical properties, which are essential for accurate Stefan moving-boundary modeling with a sharp solid–liquid interface. These characteristics ensure reliable prediction of melting-front evolution and latent-heat storage over long-term (8760 h) simulations. Al₂O₃ nanoparticles were chosen as the thermal-conductivity enhancer because of their high intrinsic thermal conductivity, chemical inertness, cost-effectiveness, and excellent compatibility with organic paraffins. In addition, Al₂O₃–paraffin NePCM systems are supported by a wide body of experimental and theoretical literature, providing validated property correlations necessary for transient system-level modeling.

Therminol VP-1 was selected as the heat-transfer fluid due to its widespread application in PTSCs, high thermal stability within the operating temperature range, and availability of reliable temperature-dependent property data. The chosen material set therefore enables a representative and reproducible assessment of the coupled thermo-optical and phase-change dynamics of PTSC-integrated NePCM storage systems.

Base materials and property data. The base PCM employed in this study is a commercially available paraffin wax (RT 50HC, Rubitherm GmbH), selected for its narrow melting range, chemical stability, and well-documented thermophysical properties. The baseline thermophysical properties of the pure paraffin are summarized in Table 3. Therminol VP-1 is consistently used as the HTF throughout the present study. This synthetic oil is widely adopted in PTSC applications due to its high thermal stability within the investigated temperature range. All HTF thermophysical properties—including density, specific heat capacity, thermal conductivity, and dynamic viscosity—are implemented as temperature-dependent correlations obtained from the manufacturer's datasheets. For reference and reproducibility, the corresponding HTF properties at 300 K are reported in Table 3. Spherical α-Al₂O₃ nanoparticles with an average diameter of 40 nm (US Research Nanomaterials, Inc.) are dispersed within the paraffin matrix to form the nano-enhanced PCM. The thermophysical properties of bulk Al₂O₃ are taken from standard literature sources. For clarity, Therminol VP-1 is used as the heat-transfer fluid throughout the study, and all HTF properties reported in Table 3 correspond to this fluid.

Table 3.

Thermophysical properties of the base materials at 300 K (NematpourKeshteli et al., 2024).

Material	Property	Symbol	Value
Paraffin (RT 50HC)	Melting temperature	Tm (K)	323.15
	Latent heat of fusion	L (kJ kg−1)	180
	Solid density	ρs (kg m−3)	880
	Liquid density	ρl (kg m−3)	770
	Solid specific heat	cp,s (kJ kg−1 K−1)	2.1
	Liquid specific heat	cp,l (kJ kg−1 K−1)	2.4
	Solid thermal conductivity	ks (W m−1 K−1)	0.24
	Liquid thermal conductivity	kl (W m−1 K−1)	0.16
	Dynamic viscosity (liquid)	μl (Pa s)	0.025
Al2O3 nanoparticles	Density	ρnp (kg m−3)	3970
	Specific heat	cp,np (kJ kg−1 K−1)	0.765
	Thermal conductivity	knp (W m−1 K−1)	36
	Average diameter	dnp (nm)	40
HTF (Therminol VP-1)	Density (300 K)	ρf (kg m−3)	1060
	Specific heat (300 K)	cp,f (kJ kg−1 K−1)	1.55
	Thermal conductivity (300 K)	kf (W m−1 K−1)	0.137
	Dynamic viscosity (300 K)	μf (Pa s)	0.0046

Effective density

Assuming negligible void formation and uniform dispersion of nanoparticles, the effective density is computed using the classical mixture relation (NematpourKeshteli et al., 2024):

$${\rho }_{pcm}=(1-\varphi ){\rho }_{paraffin}+\varphi \rho A{l}_2{O}_3$$

(16)

This linear rule is widely validated for low-volume nanoparticle concentrations.

Specific heat capacity

The effective heat capacity of the composite is expressed by NematpourKeshteli et al. (2024):

$$c_{pcm}=(1-\varphi )c_{paraffin}+\varphi c_{A{l}_2}{O}_3$$

(17)

Due to the substantially lower specific heat of ${\mathrm{Al}}_2{\mathrm{O}}_3$ , adding nanoparticles slightly reduces the overall heat capacity, although this reduction remains below 4% for the selected volume fractions.

Thermal conductivity

The dominant benefit of adding ${\mathrm{Al}}_2{\mathrm{O}}_3$ is the enhancement of thermal conductivity. For dilute suspensions $(\varphi \le 0.02)$ , the Maxwell relation provides an accurate prediction:

$$k_{pcm}=k_{paraffin}\left[{\displaystyle \frac{k_{A{l}_2{O}_3}+2k_{paraffin}+2\varphi (k_{A{l}_2{O}_3}-k_{paraffin})}{k_{A{l}_2{O}_3}+2k_{paraffin}-\varphi (k_{A{l}_2{O}_3}-k_{paraffin})}}\right]$$

(18)

For very low loadings $(\le 1\text{\%})$ , the simplified form used by several authors is sufficiently accurate:

$$k_{pcm}=k_{paraffin}(1+3\varphi )$$

(19)

Equation (19) is obtained as a linearized form of the Maxwell model (equation (18)) by assuming dilute nanoparticle concentrations (). Under this condition, a first-order Taylor expansion with respect to ϕ is sufficient, and higher-order terms are neglected. For ϕ ≤ 3 vol%, the difference between the full Maxwell formulation and its linearized approximation remains negligible, while the simplified expression improves numerical efficiency and transparency.

Latent heat of fusion

Nanoparticles slightly reduce the latent heat of paraffin $\phi \ll 1$ due to dilution (NematpourKeshteli et al., 2024):

$$L_{pcm}=\left(1-\varphi \right)L_{paraffin})$$

(20)

The reduction is typically less than 5% for the selected nanoparticle range, making the storage density effectively preserved while benefiting from higher thermal conductivity.

Dynamic viscosity of the molten NePCM

Although viscosity affects natural convection during melting, its impact is relatively mild for low $\varphi$ . The well-known Brinkman relation is applied:

$${\mu }_{pcm}={\displaystyle \frac{{\mu }_{paraffin}}{{(1-\varphi )}^{2.5}}}$$

(21)

An increase of 3–6% in viscosity is expected for 1–2% nanoparticle loading, which is not large enough to suppress convective effects. Although the present model does not explicitly resolve the molten PCM flow field, the effective viscosity is reported to assess the influence of nanoparticle addition on buoyancy-driven convection during melting. The mild viscosity increase predicted for low particle loadings confirms that natural convection remains active and that the adopted conduction-based Stefan formulation remains physically consistent.

Melting temperature

Experimental studies consistently show that doping paraffin with ${\mathrm{Al}}_2{\mathrm{O}}_3$ nanoparticles does not alter the melting point significantly (variation <1 °C). Thus, the base melting temperature of the paraffin is retained:

$$T_m=T_{m,paraffin}$$

(22)

Relevance to Stefan moving boundary modeling

Accurate evaluation of the above properties is vital because:

$k_{pcm}$ controls the slope of temperature gradients on both sides of the moving interface. $L_{pcm}$ directly appears in the Stefan condition:

$${k_l{\displaystyle \frac{\mathrm{\partial }{T}_l}{{\mathrm{\partial }}_x}}|}_s-{k_s{\displaystyle \frac{\mathrm{\partial }{T}_s}{{\mathrm{\partial }}_x}}|}_s={\rho }_{pcm}{L}_{pcm}{\displaystyle \frac{dS}{dt}}$$

(23)

${\rho }_{pcm}$ And $c_{pcm}$ determine transient heat storage in each phase. Nanoparticles improve the response of the system during charging/discharging by accelerating melting, reducing thermal lag, and enhancing overall heat transfer.

To facilitate independent reproduction of the NePCM property evaluation, all effective thermophysical properties are computed prior to the transient simulation and stored as input functions of nanoparticle volume fraction. During the time integration, these properties are held constant within each phase (solid or liquid) and updated only when the local phase state changes. Temperature-dependent variations of properties are neglected to isolate the influence of nanoparticle loading and to maintain numerical robustness over long-term (8760-h) simulations. This treatment ensures that any observed performance differences arise solely from the coupled thermo-optical and phase-change dynamics rather than from secondary property nonlinearities.

Assumptions of the Stefan moving-boundary model. Phase-change heat transfer in the NePCM storage unit is modeled using a classical one-phase Stefan moving-boundary formulation. Unlike enthalpy-porosity or mushy-zone approaches, the Stefan method explicitly tracks the solid–liquid interface based on local energy balance, ensuring exact latent-heat conservation and deterministic interface evolution. This approach is particularly suitable for paraffin-based PCMs with a well-defined melting temperature and has been widely adopted in analytical and numerical studies of latent TES. The phase-change process in the NePCM storage unit is modeled using a classical one-phase Stefan moving-boundary formulation. To ensure physical validity and numerical robustness, the following assumptions are adopted.

The solid–liquid interface is assumed to be sharp and isothermal at the melting temperature of the PCM. This is justified by the narrow melting range of paraffin-based PCMs.

Heat transfer within the PCM domain is assumed predominantly one-dimensional along the principal heat-flux direction, consistent with the storage geometry and circumferentially uniform heat input from the receiver. The adoption of a one-dimensional heat-transfer assumption in the PCM domain is justified by the thermal characteristics of PTSC receivers. Although solar flux on the absorber tube may exhibit minor circumferential non-uniformities due to optical errors or incidence-angle effects, the high thermal conductivity of the metallic receiver wall (SS-304) rapidly equalizes azimuthal temperatures, resulting in negligible circumferential gradients at the PCM interface. Axial conduction is also weak compared to radial heat transfer because the characteristic axial temperature variation induced by solar transients evolves over timescales of 30–60 min, whereas radial diffusion across the thin PCM annulus (≈20 mm) occurs over much shorter timescales (1–3 min). Consequently, the dominant resistance governing melting-front evolution lies in the radial direction. These factors collectively ensure that circumferential and axial variations remain second-order effects, and a 1-D radial Stefan formulation yields accurate predictions for melting–solidification behavior under realistic operating conditions.

Although natural convection may occur in the molten PCM, heat transfer at the immediate vicinity of the moving interface is assumed conduction-dominated …

Thermophysical properties of the NePCM are treated as effective and spatially uniform within each phase at a given time step, while their values vary with nanoparticle concentration.

Volume change, pressure work, and mechanical deformation during phase transition are neglected due to the small density difference between solid and liquid paraffin.

Nanoparticles are assumed to be in local thermal equilibrium with the base PCM, enabling the use of a single temperature field to describe the composite material.

Under these assumptions, the Stefan moving-boundary model provides an accurate and computationally efficient description of melting and solidification processes over long-term transient operation.

From a reproducibility standpoint, the Stefan interface is initialized at the HTF–PCM boundary and evolves solely according to the local heat fluxes computed on either side of the interface. No artificial mushy-zone smoothing or enthalpy-based regularization is employed. The interface temperature is fixed exactly at the melting point, and the latent heat is released or absorbed exclusively through the Stefan condition. This explicit tracking strategy ensures that partial melting, incomplete solidification, and cyclic interface reversal are resolved deterministically, allowing direct comparison with analytical Stefan benchmarks and experimental melting-front measurements.

Boundary and initial conditions

It is emphasized that the present study is entirely numerical, and no experimentally measured collector temperatures are imposed. Instead, transient thermal boundary conditions are generated internally using a coupled optical–thermal PTSC model driven by real hourly climatic data. The transient numerical simulation requires a complete specification of the thermal boundary conditions for both subsystems: (i) the PTSC and (ii) the PCM storage unit governed by the Stefan moving boundary formulation. All boundary and initial conditions are defined to ensure full energy conservation, thermodynamic consistency, and continuous coupling between the PTSC outlet temperature and the HTF inlet boundary.

PTSC boundary conditions

Solar input boundary. At the mirror aperture, only beam irradiance contributes to optical concentration (NematpourKeshteli et al., 2024; Osorio et al., 2023). Thus,

$$q_{abs}(t)=I_b(t){\eta }_{opt,eff}(t)$$

(24)

The solar incidence angle θ(t) varies hourly based on site latitude, solar declination, and equation of time. Hourly values of I_b(t) are imposed as Dirichlet-type forcing.

Inlet HTF temperature. The collector inlet temperature is prescribed as:

$$T_f(x=0,t)=T_{in,PTSC}(t)$$

(25)

For coupled operation with PCM:

$$T_{in,PTSC}(t=0)=T_{amb}$$

(26)

During operation, $T_{in,PTSC}$ evolves according to the outlet temperature of the HTF flowing through the immersed coil heat exchanger (in discharge mode), ensuring complete bidirectional thermal coupling.

Outlet fluid boundary (convective). At the collector outlet (x = LPTSC), the following convective-type boundary is used:

$${{\displaystyle \frac{\mathrm{\partial }{T}_f}{{\mathrm{\partial }}_x}}|}_{x=L}=0$$

(27)

which is consistent with fully developed flow.

Receiver external surface condition. The receiver tube loses heat through natural/forced convection and radiation:

$$-k_r{{\displaystyle \frac{\mathrm{\partial }{T}_r}{{\mathrm{\partial }}_r}}|}_o=h_{ext}(T_r-T_{amb})+ϵ\sigma (T_r^4-T_{sky}^4)$$

(28)

Wind speed determines whether forced or natural convection correlations are used.

PCM domain boundary conditions

The PCM unit is modeled in a one-dimensional radial or axial geometry depending on the storage design. The domain is separated into solid and liquid regions by the moving interface S(t), which is explicitly tracked using the Stefan condition.

Boundary at the HTF–PCM interface (x  =  0). The temperature at the HTF–PCM boundary is imposed by the HTF outlet temperature from the PTSC, which serves as the inlet temperature to the immersed coil heat exchanger:

$$T(x=0,t)=T_{in,PCM}(t)=T_{out,PTSC}(t)$$

(29)

This ensures strict real-time thermal coupling. The corresponding boundary heat flux is given by:

$$q_0^{"}(t)=h_f[T_{out,PTSC}(t)-T_{wall}(0,t)]$$

(30)

where h_f is the internal convective coefficient of the heat transfer fluid within the PCM heat exchanger tubes.

Boundary at the PCM external wall (x = L_pcm). If the PCM module is insulated:

$$-k_{pcm}{{\displaystyle \frac{\mathrm{\partial }T}{{\mathrm{\partial }}_x}}|}_L=0$$

(31)

If partially exposed:

$$-k_{pcm}{{\displaystyle \frac{\mathrm{\partial }T}{{\mathrm{\partial }}_x}}|}_L=h_{loss}(T-T_{amb})$$

(32)

For realistic high-efficiency storage systems, adiabatic insulation is assumed.

Stefan moving boundary condition. At the solid–liquid interface x  =  S(t):

$$k_l{{\displaystyle \frac{\mathrm{\partial }{T}_l}{{\mathrm{\partial }}_x}}|}_{s^{-}}-k_s{{\displaystyle \frac{\mathrm{\partial }{T}_s}{{\mathrm{\partial }}_x}}|}_{s^{+}}={\rho }_{pcm}{L}_{pcm}{\displaystyle \frac{dS}{dt}}$$

(33)

At the solid–liquid interface, the temperature is fixed at the melting point. In the present model, the phase change process is assumed to occur isothermally at 50 °C, corresponding to the melting temperature of the PCM used in the system:

$$T(S(t),t)=T_m$$

(34)

This enforces correct latent-heat physics.

Initial conditions

At sunrise or at the start of the simulation (t  =  0), all system components are assumed to be thermally uniform and at ambient temperature:

$$\begin{array}{rl}{T}_f(x,0)& =T_r(0)=T_{amb}\\ T(x,0)& =T_{amb}<T_m\end{array}$$

(35)

Thus, the PCM is initially in the fully solid state:

$$S\left(0\right)=0)$$

(36)

For multi-day simulations, the final state of the previous day serves as the initial state for the next day, preserving model continuity.

Coupling boundary between PTSC and PCM modules

The coupling boundary between the PTSC and PCM modules is formulated such that the instantaneous inlet temperature of the PCM is equal to the outlet temperature of the PTSC, while the inlet temperature of the PTSC is governed by the PCM outlet temperature during the discharge period and by the ambient temperature during charging. This tightly coupled boundary condition ensures a physically consistent and uninterrupted energy flow across the two modules, preserves the correct transient heat propagation within the thermal network, enables precise tracking of the moving interface, and prevents the emergence of any artificial numerical heat sources or sinks within the computational domain.

$$\begin{array}{c}{T}_{in,PCM}(t)=T_{out,PTSC}(t)\\ T_{in,PTSC}(t)=\{\begin{array}{c}{T}_{put,PCM}(t),duringdischarge\\ T_{amb},duringcharging\end{array}\end{array}$$

(37)

All boundary and initial conditions are applied in a strictly sequential and deterministic manner at each time step. The outlet temperature of the PTSC is first computed and then imposed as a Dirichlet boundary on the PCM domain, after which the PCM solution is advanced and its outlet temperature is fed back to the PTSC inlet during discharge operation. No relaxation factors or empirical tuning parameters are introduced at the coupling interface. This explicit bidirectional coupling ensures strict energy conservation across the PTSC–PCM loop and eliminates any dependence of the solution on solver-specific numerical damping.

Numerical solution method

The coupled PTSC–PCM system is solved using a semi-implicit finite-difference scheme, whose algorithmic structure is summarized in Figure 3. At each time step n + 1, the PTSC optical–thermal equations are first advanced semi-implicitly to obtain the updated outlet temperature $T_{out,PTSC}^{n+1}$ . This temperature is immediately imposed as the inlet boundary of the PCM unit, ensuring direct coupling between the two subsystems. The PCM domain (solid and liquid regions) is then solved using a semi-implicit discretization of the transient energy equation, while one-sided gradients are applied at cells adjacent to the moving interface. The interface position is updated explicitly using the Stefan condition, with its motion restricted to ≤ one spatial node per step to maintain numerical stability. If the system is in discharge mode, the PCM outlet temperature is fed back into the PTSC inlet, forming the bidirectional loop indicated in Figure 3. Convergence is checked iteratively within each time step and the solution proceeds once

$$max|T_i^{n+1}-T_i^n|<{10}^{-6}\circ \mathrm{C}$$

(38)

Figure 3.

Flowchart of the semi-implicit numerical algorithm for the coupled PTSC–PCM model, showing the sequence of PTSC solution, coupling, PCM update, Stefan interface, and convergence.

The semi-implicit structure preserves stability for time steps larger than the explicit limit, enabling efficient long-term (8760-h) simulations.

Spatial and temporal discretization

The PTSC receiver tube and PCM domain are discretized along the axial direction into N nodes. Time is discretized into increments of $\mathrm{\Delta }t=1\text{–}10\mathrm{s}$ depending on the solar transient intensity and HTF flow rate. Let:

${\mathrm{T}}_f^n(i)$ : fluid temperature at time step n and node i

${\mathrm{T}}_r^n(i)$ : receiver wall temperature

${\mathrm{T}}_s^n(i),{\mathrm{T}}_l^n(i)$ : PCM temperatures in solid and liquid regions

${\mathrm{S}}^{\mathrm{n}}$ : location of the moving solid–liquid interface

A uniform grid is employed, though the method supports adaptive mesh refinement around the Stefan front to minimize numerical diffusion.

Semi-implicit discretization of governing equations

HTF energy equation

$${\rho }_f{c}_p{A}_f\left({\displaystyle \frac{T_f^{n+1}(i)-T_f^n(i)}{\mathrm{\Delta }t}+{\upsilon }_f{\displaystyle \frac{T_f^{n+1}(i)-T_f^{n+1}(i-1)}{\mathrm{\Delta }x}}}\right)=h_i{P}_i(T_r^{n+1}(i)-T_f^{n+1}(i))$$

(39)

An upwind scheme is used for convective terms to ensure stability.

Receiver tube energy equation

$$m_r{c}_r{\displaystyle \frac{T_r^{n+1}(i)-T_r^n(i)}{\mathrm{\Delta }t}=q_{abs}^n(i)A_r-h_i{P}_i(T_r^{n+1}(i)-T_f^{n+1}(i))-U_{loss}{A}_r(T_r^{n+1}(i)-T_{amb}^n)}$$

(40)

This equation is treated semi-implicitly to suppress numerical oscillations under rapid solar changes.

PCM energy equation in solid and liquid zones

A fully transient finite-difference formulation is used:

$${\rho }_{pcm}{c}_{pcm}{\displaystyle \frac{T_j^{n+1}-T_j^n}{\mathrm{\Delta }t}=k_{pcm}{\displaystyle \frac{T_{j+1}^{n+1}-2T_j^{n+1}+T_{j-1}^{n+1}}{{(\mathrm{\Delta }x)}^2}}}$$

(41)

The above holds separately in the solid region and liquid region. To enforce the Stefan interface, nodes adjacent to S(t) are updated using one-sided gradients:

$$k_l{\displaystyle \frac{T_{l,s-1}^{n+1}-T_{l,s}^{n+1}}{\mathrm{\Delta }x}-k_s{\displaystyle \frac{T_{s,S}^{n+1}-T_{s,S+1}^{n+1}}{\mathrm{\Delta }x}={\rho }_{pcm}{L}_{pcm}{\displaystyle \frac{S^{n+1}-S^n}{\mathrm{\Delta }t}}}}$$

(42)

This guarantees energy conservation during phase transition.

Updating the moving interface (Stefan condition)

The interface motion is explicitly updated after solving the temperature fields:

$$S^{n+1}=S^n+\mathrm{\Delta }t\left[{\displaystyle \frac{k_l{(\mathrm{\partial }{T}_l/{\mathrm{\partial }}_x)}_{S^{-}}-k_s{(\mathrm{\partial }{T}_s/{\mathrm{\partial }}_x)}_{S^{+}}}{{\rho }_{pcm}{L}_{pcm}}}\right]$$

(43)

To reduce numerical instability caused by steep gradients:

smoothing filters are applied to the gradients,

interface location is restricted to move ≤ 1 node per time step.

Coupling between PTSC and PCM at each time step

At every time step n + 1:

Solve PTSC equations using $T_{in,PTSC}^n$

$\mathrm{Compute}\mathrm{PTSC}\mathrm{outlet}\mathrm{temperature}{T}_{out,PTSC}^{n+1}$ .

Apply coupling condition: $T_{in,PCM}^{n+1}=T_{out,PTSC}^{n+1}$

Solve PCM domain (solid + liquid).

Update Stefan boundary $S^{n+1}$ .

If discharge mode is active: $T_{in,PTSC}^{n+1}=T_{out,PCM}^{n+1}$

This loop ensures strict bidirectional heat-flow continuity.

Convergence and stability criteria

The iterative Gauss–Seidel method is used for each subsystem. Convergence is achieved when:

$$max|T_i^{n+1}-T_i^n|<{10}^{-6}\circ \mathrm{C}$$

(44)

The scheme is stable under:

$$\mathrm{\Delta }t<{\displaystyle \frac{\rho c{(\mathrm{\Delta }x)}^2}{2k}(\mathrm{Fourier}\mathrm{criterion})}$$

(45)

though the semi-implicit formulation allows time steps well above the explicit stability limit.

Reproducibility and implementation details. To enable full reproducibility, the coupled PTSC–PCM simulation can be reproduced by following a fixed sequence at each time step: (i) update solar geometry and impose hourly DNI; (ii) solve the PTSC optical–thermal equations to obtain the outlet HTF temperature; (iii) apply this temperature as the inlet boundary condition for the PCM domain; (iv) solve the transient energy equations in the solid and liquid PCM regions; (v) update the moving interface using the Stefan condition; and (vi) check convergence before advancing to the next time step. The spatial grid, time step, convergence tolerance, and interface-movement constraint are identical throughout the simulation and are not adaptively tuned. This algorithmic structure ensures that an independent implementation using the same governing equations and parameters will reproduce the reported temperature fields, melting-front evolution, and exergy results within numerical tolerance.

Environmental assessment and uncertainty treatment

The environmental performance of the PTSC–NePCM system is evaluated in terms of operational CO₂ emissions avoided, following an International Organization for Standardization (ISO) 14040/44 Level-0 boundary where only use-phase emissions directly linked to useful heat generation are considered. Annual avoided emissions are computed by multiplying the simulated yearly useful thermal output by the emission factor (EF) of each reference technology, namely a natural-gas boiler, a grid-supplied electric heater, and a baseline flat-plate solar collector. The nominal EF values adopted in this study are summarized in Table 2, with 0.184 kg CO₂/kWh for natural gas, 0.475 kg CO₂/kWh for grid electricity, and 0.012–0.028 kg CO₂/kWh for the solar-thermal baseline.

Although Al₂O₃ nanoparticles introduce no direct operational emissions, their indirect environmental influence arises from improving PCM thermal performance, thereby increasing annual useful heat delivery and displacing a larger fraction of fossil-fuel or grid-based energy. To account for uncertainty in background EFs, a parametric analysis is performed in which the EFs of the fossil and grid benchmarks are varied by ±20% around their nominal values. The corresponding variation in annual CO₂ avoided is reported as an uncertainty band in the environmental indicators, providing a more realistic range of potential climate benefits under different national grid-mix and fuel-supply scenarios.

$${\mathrm{CO}}_{2ref,\mathrm{annual}}={\mathrm{EF}}_{ref}\times {\mathrm{E}}_{\mathrm{output},\mathrm{annual}}$$

(46)

$${\mathrm{CO}}_{2\mathrm{avoided}}={\mathrm{CO}}_{2\mathrm{ref}}-{\mathrm{CO}}_{2_{\mathrm{PTSC}}}$$

(47)

$${\mathrm{CO}}_{2_{\mathrm{avoided}}}={\mathrm{EF}}_{\mathrm{ref}}\times {\mathrm{E}}_{\mathrm{output},\mathrm{annual}}$$

(48)

Economic assessment with sensitivity and uncertainty analysis

A screening-level economic assessment is carried out to evaluate the financial performance of the proposed PTSC–PCM system using three indicators: the net present cost (NPC), the levelized cost of heat (LCOH), and the simple payback period (PB). The NPC is determined by combining the system's capital expenditure with its discounted operational expenses over the defined project lifetime. The capital expenditure includes the parabolic trough collector structure, the receiver tube and glazing, the paraffin-based PCM, and the aluminum-oxide nanoparticle additive, whose cost varies with the selected concentration. The LCOH is then obtained by dividing the total discounted cost by the cumulative useful thermal energy delivered over the entire operating period, providing a long-term cost measure per unit of heat supplied. A sensitivity analysis is also performed to assess the robustness of the system's economic viability with respect to variations in material costs and solar resource availability, ensuring that the conclusions remain reliable under realistic market and environmental fluctuations.

$$\mathrm{NPC}=\mathrm{CAPEX}+\sum ({\mathrm{OPEX}}_{\mathrm{t}}/{(1+\mathrm{r})}^{\mathrm{t}})$$

(49)

$$\mathrm{LCOH}=\mathrm{NPC}/({\mathrm{E}}_{\mathrm{output},\mathrm{annual}}\times \mathrm{Lifetime})$$

(50)

To capture the influence of techno-economic uncertainties, a combined sensitivity and uncertainty analysis is performed. First, one-at-a-time sensitivity tests are conducted by varying key economic inputs—collector and PCM investment costs, nanoparticle price, and discount rate—within ±20% of their baseline values. The adopted nanoparticle cost represents an industrial-grade bulk estimate; however, recognizing the pronounced market volatility of nano-materials, a ±20% uncertainty range was applied to assess the robustness of the economic conclusions. The capital expenditure (CAPEX) of the proposed system comprises four main components: the parabolic trough collector structure, the receiver tube together with its glass envelope, the paraffin-based phase-change material, and the Al₂O₃ nanoparticle additive whose cost varies with the selected weight-fraction concentration. The assumptions applied in the economic and environmental evaluation are summarized in Table 4. Importantly, the relative economic ranking of the investigated configurations remained unchanged across the entire uncertainty range, confirming the robustness of the identified optimal nanoparticle concentration.

Table 4.

Key numerical assumptions used in the economic and environmental assessments.

Category	Parameter	Assumption
Project horizon	Operational lifetime	20 years
Financial parameters	Discount rate	8%
Financial parameters	Inflation rate	3%
Emission factor	Natural gas	0.184 kg CO2/kWh
Emission factor	Grid electricity	0.475 kg CO2/kWh
Emission factor	Baseline solar thermal	0.012–0.028 kg CO2/kWh
Nanoparticles	Operational emissions	Zero (no direct CO2 contribution)
Nanoparticles	Environmental influence	Indirect, via improved thermal efficiency and higher annual useful heat

Uncertainty propagation and robustness of results

The present study explicitly incorporates parametric uncertainty into both the environmental and economic performance assessment. Key inputs subject to uncertainty include EFs of reference energy technologies, collector and storage investment costs, nanoparticle price, and discount rate. These parameters are varied independently within ±20% of their nominal values, consistent with ranges reported in ISO-based environmental assessment studies and solar-thermal techno-economic analyses.

The resulting variation bands in annual CO₂-avoidance, LCOH, and PB provide an uncertainty envelope around the nominal performance indicators. Importantly, across all tested uncertainty scenarios, the relative performance ranking of nanoparticle loading fractions remains unchanged. In particular, the 1 vol% NePCM configuration consistently demonstrates the most balanced trade-off between thermal enhancement, economic cost, and environmental benefit.

This confirms that the main conclusions of the study are robust and not contingent on a single deterministic set of assumptions, thereby strengthening the reliability of the proposed optimization framework under realistic techno-economic and environmental uncertainty.

Model validation

The proposed transient numerical model is validated using well-documented experimental and analytical benchmarks available in the literature. Due to the limited availability of experimental data for PTSCs integrated with NePCMs, the validation is conducted in a hierarchical manner by separately evaluating the PTSC thermal behavior and the phase-change heat-storage formulation under comparable conditions.

First, the PTSC sub-model is validated against the experimental and numerical results reported by Babikir et al. (2021), who developed and experimentally assessed the thermal behavior of a conventional parabolic trough collector operating without TES. Their study provides detailed measurements of DNI, inlet HTF temperature, mass flow rate, and ambient conditions. Figure 4(a) compares the predicted outlet temperature of the PTSC with the corresponding data reported by Babikir et al. (2021). The comparison shows good agreement, with deviations generally within 3–5%, which falls within the uncertainty range typically associated with temperature measurements and optical–thermal losses in outdoor PTSC experiments.

Figure 4.

Validation of the proposed model: (a) comparison of the simulated outlet temperature of the parabolic trough solar collector (PTSC) with the results reported by Babikir et al. (2021), (b) comparison of the predicted melting front position with the analytical Stefan solution; and (c) grid independence analysis of the numerical model.

Second, the phase-change heat-storage model is validated independently using analytical solutions. The transient melting-front position within the PCM domain is compared with the classical one-dimensional Stefan problem under a constant boundary heat flux. As illustrated in Figure 4(b), the numerical results closely follow the analytical solution, with relative deviations remaining below 2–3% throughout the melting process. This confirms that the enthalpy-based formulation accurately captures the latent-heat-driven moving interface.

It should be noted that the experimental PTSC data used for validation correspond to a conventional collector without TES, whereas the present model considers an integrated PTSC–TES system employing nano-enhanced PCM. Therefore, the validation focuses on verifying the fundamental thermal response of each sub-model rather than reproducing an identical experimental configuration. The effect of nanoparticles is incorporated into the numerical model through modified thermophysical properties of the PCM based on correlations reported in the literature, while direct system-level experimental validation of NePCM-based PTSC systems is currently limited.

Furthermore, a grid-independence study was performed by progressively refining the spatial discretization in both the HTF and PCM domains (Figure 4(c)). Beyond approximately 600 computational nodes, variations in the predicted outlet temperature, liquid fraction, and accumulated useful heat were below 0.5%, indicating that numerical diffusion effects are negligible. Additional time-step sensitivity tests confirmed that the adopted semi-implicit scheme provides stable and consistent predictions during transient diurnal operation.

It is also worth noting that the numerical model resolves fully transient boundary conditions using instantaneous DNI, ambient temperature, and wind speed, which can introduce short-term temporal fluctuations in the predicted outlet temperature. In contrast, the experimental data reported by Babikir et al. (2021) were obtained under quasi-steady operating conditions and represent time-averaged measurements. When the numerical results are processed using a similar temporal averaging approach, an even closer agreement with the reported experimental trends is obtained.

Overall, the validation results demonstrate that the developed model reliably reproduces the thermal response of the PTSC and accurately predicts the phase-change behavior of the PCM, providing a sound basis for the subsequent energy, exergy, environmental, and economic analyses.

It should be noted that the numerical outlet temperature exhibits short-term temporal variations due to the fully transient nature of the model and the use of instantaneous environmental boundary conditions (e.g., DNI, ambient temperature, and wind speed). In contrast, the experimental outlet temperatures correspond to quasi-steady operating periods and represent time-averaged measurements, which naturally smooth high-frequency fluctuations. When the numerical predictions are post-processed using the same temporal averaging window, good agreement with the experimental data is recovered.

Results and discussion

Unless otherwise stated, the parametric results are obtained by varying one parameter at a time while keeping the remaining inputs fixed, in order to isolate the independent physical effect of each parameter on system performance rather than to represent a single instantaneous operating condition.

Climatic boundary inputs for PTSC–PCM modeling

The climatic inputs used in this study are derived from real hourly meteorological data (8760 h) for the Yazd region, representing a high-solar-irradiance arid–semi-arid climate that is typical of many locations where PTSCs are most effectively deployed. Although Figure 5 presents monthly-scale representations for clarity, all simulations are conducted at an hourly resolution, enabling the model to capture both diurnal transients and long-term seasonal variability. The DNI profiles shown in Figure 5 represent monthly-averaged diurnal envelopes extracted from the underlying hourly dataset and are not individual daily simulations. This approach provides a compact yet physically representative visualization of the climatic forcing applied to the fully transient PTSC–PCM model.

Figure 5.

Monthly climatic boundary conditions for the Yazd site derived from hourly (8760 h) meteorological data: (a) monthly-averaged diurnal envelopes of DNI, (b) monthly mean ambient temperature, and ^© monthly mean wind speed.

PTSC thermal analysis

Figure 6(a) to (d) depicts the typical diurnal performance of the PTSC under clear-sky conditions. The DNI exhibits a bell-shaped curve, peaking at approximately 820 W/m² near solar noon. This solar input drives the HTF outlet temperature to rise from around 33 °C in the morning to a maximum of 72 °C at midday, while the inlet temperature remains nearly constant due to controlled boundary conditions.

Figure 6.

Diurnal performance of the PTSC system: (a) direct normal irradiance (DNI), (b) inlet and outlet HTF temperatures, (c) useful heat gain, and (d) instantaneous thermal efficiency under clear-sky conditions.

The transient thermal response can be explained by the interplay of solar irradiance, heat losses, and the system's thermal inertia. During the morning hours, as solar radiation intensity increases, the HTF and surrounding receiver components absorb energy. However, significant heat losses occur due to relatively low temperature gradients and the system's initial thermal state, resulting in a gradual increase in outlet temperature and moderate thermal efficiency. At solar noon, the peak DNI maximizes the energy input, generating the highest axial temperature gradient within the receiver tube. Under these conditions, the useful heat gain reaches a maximum of 160 W, and instantaneous thermal efficiency peaks near 58%. The enhanced temperature difference reduces the relative impact of convective and radiative losses, optimizing the conversion of incident solar energy into thermal energy in the HTF.

In the late afternoon, DNI decreases sharply, while the HTF and receiver temperatures remain elevated due to thermal inertia and latent heat storage effects. Consequently, convective and radiative heat losses become more pronounced relative to the reduced energy input, causing a decline in both outlet temperature and thermal efficiency. This dynamic coupling between transient solar forcing, heat losses, and thermal storage behavior governs the daily performance trends observed in the PTSC system. Overall, these findings highlight the critical role of competing thermal mechanisms and transient system response in shaping the efficiency and heat output of the PTSC throughout the day.

The smooth yet fully transient outlet temperature profiles shown in Figure 6(b), despite the use of instantaneous hourly climatic inputs, highlight the importance of the semi-implicit numerical treatment adopted in equation (40). This formulation enables stable time integration beyond the restrictive limits of explicit schemes, preventing non-physical oscillations under rapidly varying DNI conditions. The numerical stability ensured by the convergence and stability criteria defined in equations (44) and (45) is further reflected in the coherent coupling between PTSC thermal output and PCM charging power in Figure 7(d). Consequently, the observed alignment between HTF temperature fluctuations, melting-front progression, and liquid-fraction evolution confirms that the governing equations are not merely mathematical constructs, but actively dictate the transient energy exchange between the collector and the latent heat storage unit.

Figure 7.

Transient melting behavior of the PCM unit showing (a) melting-front progression, (b) liquid-fraction evolution, (c) radial temperature profiles at selected times, and (d) instantaneous charging power.

PCM melting/solidification dynamics (Stefan front)

The transient melting behavior illustrated in Figure 7 is a direct manifestation of the Stefan moving-boundary formulation defined in equations (41) to (43), in which the solid–liquid interface evolution is governed explicitly by the local interfacial energy balance. Unlike enthalpy-based approaches, this formulation enforces exact latent-heat conservation at the phase front, allowing deterministic tracking of the melting interface under time-dependent boundary conditions imposed by the PTSC outlet temperature.

As shown in Figure 7(a), the melting front initially advances rapidly due to the large temperature gradient between the HTF and the solid PCM, which enhances the conductive heat-flux term appearing in the Stefan condition. This behavior is consistent with the early-time dominance of sensible-to-latent energy conversion predicted by the governing energy equations. As melting progresses radially outward, the thickness of the remaining solid layer decreases and the effective conductive thermal resistance increases, leading to a gradual reduction in interface velocity. This nonlinear deceleration of the melting front directly reflects the Stefan dynamics embedded in equations (41) to (43).

The corresponding liquid-fraction evolution shown in Figure 7(b) follows a distinct but physically consistent trend. While the melting front represents a local, interface-based quantity strictly governed by the Stefan condition, the liquid fraction is a global, volume-averaged measure of the molten PCM. In the present cylindrical geometry, late-stage melting involves only a thin outer solid shell; consequently, the liquid-fraction curve flattens even as the interface continues to propagate, a behavior that cannot be captured by volumetric enthalpy-porosity formulations. The radial temperature profiles in Figure 7(c) further corroborate this conductive melting mechanism, exhibiting steep gradients at early times, enhanced heat penetration near solar noon, and weakened gradients in the late afternoon as the driving temperature difference diminishes.

Finally, the instantaneous charging power shown in Figure 7(d) exhibits a bell-shaped temporal profile, peaking near midday when the interfacial heat flux—and thus the Stefan front velocity—is maximized. The consistent alignment between melting-front progression, liquid-fraction evolution, temperature-field development, and charging power confirms that the governing Stefan equations are not merely mathematical constructs, but actively control the transient energy exchange between the PTSC and the PCM storage unit under realistic solar forcing.

It is important to note that the melting-front position and the liquid-fraction evolution do not necessarily exhibit identical temporal trends. The melting front represents a local, interface-based quantity governed by the Stefan condition, whereas the liquid fraction is a global, volume-averaged indicator of the molten PCM. In the present cylindrical geometry, the late-stage melting involves only a thin outer solid layer; therefore, the liquid-fraction curve naturally flattens even though the interface continues to propagate outward. This behavior differs from enthalpy–porosity-based studies, such as Alayi et al. (2025), where melting is treated as a volumetric process and interface motion is not explicitly resolved.

Effect of nanoparticle concentration

Figure 8(a) to (d) illustrates the influence of nanoparticle concentration on the transient melting behavior of the PCM unit. Increasing the Al₂O₃ loading clearly accelerates the nonlinear propagation of the melting front (Figure 8(a)), as the enhancement in effective thermal conductivity reduces internal conductive resistance within the PCM domain. As a result, the melting depth at the end of the 180-min charging period increases from 7.2 mm for pure paraffin to 7.9 and 8.7 mm for the 1 and 3 wt% NePCM cases, respectively. These values correspond to an 10% and 20% increase in the Stefan front propagation rate relative to the base PCM.

Figure 8.

Transient thermal response of the PCM unit for nanoparticle concentrations of 0%, 1%, and 3%, showing (a) melting-front progression, (b) mean PCM temperature, (c) liquid-fraction evolution, and (d) instantaneous charging power during the charging process.

The mean PCM temperature evolution (Figure 8(b)) further confirms the enhanced internal heat diffusion induced by nanoparticle addition, with higher concentrations exhibiting faster temperature rise and elevated mid-charging temperatures. Similarly, the liquid-fraction profiles (Figure 8(c)) show earlier onset and steeper growth at increased nanoparticle loadings, indicating accelerated mid-stage melting, while all cases converge toward comparable final liquid fractions (0.95–0.97) as the latent-heat capacity becomes nearly exhausted.

To further assess the validity of the predicted transient melting behavior, the advancement of the Stefan front was compared with results reported in similar NePCM studies. Previous numerical investigations of paraffin-based NePCMs enhanced with Al₂O₃ nanoparticles under constant heat-flux boundary conditions have reported melting-front acceleration rates in the range of 8–15% for nanoparticle concentrations between 1 and 3 wt% (Dezfulizadeh et al., 2023; Golzar et al., 2025). In the present PTSC-driven configuration, the observed enhancement—10% for 1 wt% and up to 20% for 3 wt%—is in good agreement with the upper range of reported values. Importantly, this study captures the same level of improvement under realistic, time-dependent solar boundary conditions, demonstrating that nanoparticle-induced conductivity enhancement remains effective in accelerating transient melting behavior in practical PTSC–NePCM systems.

The instantaneous charging power refers to the rate at which thermal energy is transferred from the HTF to the TES unit during the charging process. It represents the instantaneous heat transfer rate associated with the storage of thermal energy in the PCM domain. The instantaneous charging power response (Figure 8(d)) shows a pronounced enhancement with nanoparticle loading, peaking at 82, 92, and 108 W for 0%, 1%, and 3%, respectively. The higher and earlier peak at elevated concentrations reflects more efficient heat transfer during charging, confirming the positive role of nanoparticle-assisted melting in improving PCM storage performance.

The quantity labeled as stored power in Figure 8(d) represents the instantaneous charging power transferred from the HTF to the PCM. Unlike studies employing constant heat-flux or fixed-temperature boundaries, the present PTSC-driven system experiences a time-varying thermal input that increases toward solar noon. As a result, the instantaneous charging power exhibits a bell-shaped profile, peaking during the mid-melting stage when latent-heat absorption is dominant and declining afterward as the latent capacity is gradually exhausted. This behavior is consistent with the fully transient solar boundary conditions applied in this work and explains the differences with monotonic decay trends reported in the literature.

Energy/exergy analysis

Figure 9(a) to (c) presents the diurnal thermal performance of the solar-assisted PCM system for nanoparticle concentrations of 0%, 1%, and 3%. The useful heat output (Figure 9(a)) increases from early-morning values of about 100, 105, and 115 W to noon peaks of 120, 125, and 138 W for 0%, 1%, and 3%, respectively, reflecting stronger heat-transfer enhancement at higher nanoparticle loading. The thermal-efficiency profiles (Figure 9(b)) follow a similar rise—from 50–58% in the morning to peak values of 60–68%—indicating that improved conductivity in the NePCM reduces internal temperature gradients and enables more effective conversion of absorbed solar energy. A comparable trend is observed in the exergy efficiency (Figure 9(c)), which increases to midday maxima of about 40%, 42%, and 46%, with the 3% case achieving a 15–18% enhancement over the baseline. All three indicators decline during the late afternoon due to falling irradiance and reduced driving temperature differences, confirming the expected coupling between solar input and thermodynamic performance.

Figure 9.

Diurnal performance of the solar-assisted PCM system at nanoparticle concentrations of 0%, 1%, and 3%, showing (a) useful heat output, (b) thermal efficiency, and (c) exergy efficiency.

Annual performance comparison

Figure 10(a) to (d) presents the monthly variation and annual indicators of the energy–exergy performance of the PCM-assisted solar system for nanoparticle concentrations of 0%, 1%, and 3%. The monthly useful thermal energy (Figure 10(a)), representing the net heat gained by the HTF, is determined from the temperature rise of the HTF across the collector–storage system and constitutes the useful energy term employed in the thermal-efficiency calculation. As shown, the useful-energy output increases from winter toward midsummer, reaching peak values around June–July of 420, 440, and 470 kWh for nanoparticle concentrations of 0%, 1%, and 3%, respectively. This improvement reflects the enhanced effective thermal conductivity of the NePCM, which facilitates more efficient heat transfer during the charging process.

Figure 10.

Monthly and annual energy–exergy performance of the PCM-assisted solar system for nanoparticle concentrations of 0%, 1%, and 3%: (a) monthly useful thermal energy of the HTF; (b) monthly exergy efficiency based on outlet flow exergy relative to solar exergy input; (c) monthly thermal efficiency defined as the ratio of useful energy to incident solar energy; and (d) corresponding annual thermal and exergy efficiencies.

The seasonal variation of the exergy efficiency (Figure 10(b)) shows a moderate decline during mid-summer due to reduced temperature gradients between the HTF and the ambient environment, with minimum values of 0.070, 0.073, and 0.075 for 0%, 1%, and 3% nanoparticle concentrations, respectively. A similar seasonal behavior is observed in the thermal efficiency (Figure 10(c)). Since the thermal efficiency is defined as the ratio of the useful thermal energy to the incident solar energy on the collector aperture area, its variation directly follows the trend of the useful-energy gain and the prevailing solar-irradiation conditions. The efficiency reaches spring maxima of 55%, 57.5%, and 59% and decreases slightly during summer because of increased convective and radiative losses, although the nano-enhanced cases consistently maintain superior performance (0.485–0.505 compared with 0.460 for the base PCM).

Although the overall system performance is evaluated through annual thermal and exergy efficiencies, the results in Figure 10(a) to (c) are presented on a monthly basis in order to highlight the seasonal variations in solar irradiation and ambient conditions that influence the system behavior throughout the year. The corresponding annual indicators are summarized in Figure 10(d). These results indicate that the system with 1 wt% nanoparticles provides the most balanced annual performance, with 52% thermal efficiency and 31% exergy efficiency. In contrast, the 3% nanoparticle case exhibits slightly lower overall performance due to increased optical attenuation and viscosity-related penalties. It should also be noted that the reported exergy efficiencies represent the outlet flow-exergy gain relative to the solar exergy input and therefore should not be interpreted as the overall second-law efficiency of the solar resource. Overall, the results confirm that while higher nanoparticle concentrations improve instantaneous and monthly thermal gains, the 1 wt% loading provides the most favorable year-round thermodynamic performance.

While Figure 10 presents monthly useful-energy outputs as cumulative values, these results represent the temporal integration of hourly performance under variable solar conditions rather than peak daily operation. Clear-sky days typically yield higher daily outputs, whereas cloudy periods reduce the monthly averages, making the reported values conservative climatic indicators. More importantly, the observed saturation of annual energy and exergy gains at higher nanoparticle concentrations reflects competing thermophysical mechanisms inherent to nano-enhanced PCMs. Although increasing Al₂O₃ loading enhances effective thermal conductivity and accelerates transient melting, it simultaneously dilutes the latent heat storage capacity of the PCM and increases the viscosity of the molten NePCM, which can weaken buoyancy-driven convection during phase change. At elevated concentrations, additional optical attenuation and scattering losses further offset the conductive benefits. As a result, beyond an optimal concentration of 1 wt%, the cumulative daily and monthly useful energy exhibits diminishing returns, despite continued improvements in instantaneous and peak power output. This trade-off explains the superior year-round thermodynamic balance achieved at 1 wt% nanoparticle loading.

Environmental analysis

Figure 11 illustrates the monthly avoided CO₂ emissions for the PCM-assisted solar system at nanoparticle concentrations of 0%, 1%, and 3%. The 3% concentration consistently achieves the highest absolute monthly carbon avoidance, peaking around 120 kg in April and July and maintaining values above 105 kg during several other months due to its superior heat delivery capacity. The base PCM (0%) demonstrates intermediate performance with emissions avoidance ranging from 54 to 118 kg/month, showing more pronounced sensitivity to seasonal irradiance variations. The 1% loading presents lower total monthly avoided emissions (50–90 kg/month), reflective of its moderate annual heat output.

Figure 11.

Monthly CO₂ emissions avoided by the PCM-assisted solar system for nanoparticle concentrations of 0%, 1%, and 3%.

To enable a fair comparison among the configurations, the CO₂ avoidance was normalized per unit of useful thermal energy delivered by the collector. When expressed in kg CO₂ per kWh_th, the pure PCM, 1 wt%, and 3 wt% cases yield 0.22, 0.27, and 0.25 kg CO₂/kWh_th, respectively. This normalization highlights that despite the 3% loading achieving the highest absolute carbon avoidance, the 1% concentration provides the greatest environmental benefit relative to the usable heat supplied. This outcome reflects the optimal balance between enhanced heat transfer and associated penalties such as increased viscosity and latent heat dilution at higher nanoparticle concentrations.

Overall, while the 3% nanoparticle case maximizes total CO₂ avoidance, the 1% loading represents the most thermodynamically and environmentally efficient choice on a per-energy basis, aligning with the economic advantages discussed in the subsequent section.

Economic analysis

Figure 12(a) to (c) illustrates the techno-economic impact of nanoparticle concentration on system performance. As shown in Figure 12(a), the PB decreases from approximately 6.5 years for the base PCM to a minimum of about 5.2 years at 1%, then increases to 6.8 years at 3% due to the higher cost of nanoparticle addition. The component-wise cost distribution in Figure 12(b) confirms a monotonic rise in investment cost with concentration, with the Al₂O₃ additive contributing most strongly to the economic penalty at higher loadings. Consistently, the LCOH trend in Figure 12(c) reaches its minimum value of 0.085 $/kWh_th at 1%, while both the 0% and 3% cases exhibit higher costs owing to lower annual heat production and elevated capital expenses, respectively. Collectively, these results indicate that although the 3% configuration enhances thermal output, the 1% nanoparticle concentration provides the most cost-effective overall solution.

Figure 12.

Techno-economic performance of the PCM-assisted solar system for nanoparticle concentrations of 0%, 1%, and 3%, showing (a) payback period, (b) component-wise investment cost, and (c) levelized cost of heat.

For contextual comparison, the obtained LCOH values can be benchmarked against conventional hot-water production technologies. Gas-fired boilers typically exhibit LCOH values in the range of 0.05–0.09/kWh_th, while electric resistance heaters exceed 0.12/kWh_th. Heat-pump systems may achieve comparable costs depending on COP and electricity prices, but their associated CO₂ emissions remain strongly tied to grid intensity. In this context, the PTSC–NePCM system operating at 1% nanoparticle loading demonstrates competitive heat costs while achieving substantially lower operational emissions, particularly in high-solar-resource regions.

Annual multi-criteria analysis

The annual multi-criteria assessment, summarized by the four-axis radar map in Figure 13, identifies the 1% nanoparticle concentration as the most balanced configuration across energy, exergy, environmental, and economic criteria. Although the 3% case achieves the highest annual useful energy and CO₂ avoidance, its increased material cost markedly reduces the normalized economic score, shifting the overall optimum away from higher loadings. In contrast, the base PCM (0%) delivers moderate energy and environmental performance but lacks the thermodynamic enhancement provided by nano-augmentation. The 1% configuration achieves a synergistic compromise by coupling significant improvements in annual energy and exergy performance with the lowest LCOH, thereby emerging as the Pareto-optimal solution for year-round operation.

Figure 13.

Annual four-axis radar chart comparing the normalized Energy, Exergy, Environmental (CO₂ avoided), and Economic (LCOH-based) performance of the system at 0%, 1%, and 3% nanoparticle concentrations.

Uncertainty analysis and percent deviation summary

Impact of solar irradiance uncertainty

The annual energy production and exergy efficiency of the PTSC-NePCM system were found to be highly sensitive to variations in solar irradiance. A 20% reduction in DNI resulted in a corresponding decrease of 12–15% in annual energy production and 8–10% in exergy efficiency. This highlights the importance of site selection and climate considerations for optimizing system performance.

Impact of cost uncertainties

The cost of the PTSC-NePCM system is strongly influenced by the costs of the collector, PCM, and nanoparticles. A 20% increase in the cost of any of these components led to a significant increase in the LCOH and a prolonged PB. Table 5 shows the impact of cost uncertainties on LCOH and PB.

Table 5.

Impact of cost uncertainties on LCOH and payback period.

Parameter	Increase (%)	Impact on LCOH	Impact on payback period
Collector cost	20	+8%	+1.5 years
PCM cost	20	+5%	+0.8 years
Nanoparticle cost	20	+3%	+0.5 years

Effect of discount rate

The discount rate significantly affects the net present value (NPV) of the investment. A higher discount rate reduces the NPV and increases the PB. The sensitivity analysis showed that a 20% increase in the discount rate resulted in a decrease of 10–12% in the NPV and a prolonged PB by 0.7–0.9 years. Despite the uncertainties in input parameters, the system demonstrated considerable robustness. The relative performance ranking of the different scenarios remained consistent. The 1% nanoparticle concentration configuration consistently exhibited the lowest LCOH and shortest PB under all considered scenarios.

Limitations of the study

Although the proposed transient PTSC–NePCM model provides a detailed representation of optical–thermal behavior and phase-change dynamics, several limitations should be acknowledged:

Nanoparticle migration and non-uniform dispersion:

The model assumes homogeneous dispersion of Al₂O₃ nanoparticles within the paraffin matrix. However, during the liquid phase, buoyancy-driven natural convection may induce particle migration, sedimentation, or local agglomeration, leading to non-uniform nanoparticle distributions over time. Experimental studies have shown that such convection-induced transport mechanisms can modify effective thermal conductivity, latent heat, and melting-front stability during repeated melting–solidification cycles. These long-term effects are not resolved in the present model, highlighting the need for extended cyclic experiments and coupled flow–particle transport modeling to assess NePCM stability under realistic operating conditions.

Dimensionality of the PCM domain:

The PCM storage unit is modeled using a one-dimensional radial formulation under an axisymmetric assumption. While this approach accurately captures radial melting-front evolution within a Stefan-based framework, it neglects three-dimensional effects such as circumferential temperature gradients, buoyancy-induced flow structures in the molten PCM, and potential non-uniform HTF heat transfer near the receiver tube. Future studies may employ 2D/3D CFD–Stefan hybrid models to more explicitly resolve convection–phase-change coupling.

Idealized optical assumptions:

The optical model assumes clean mirror surfaces, constant optical properties, and a fixed incidence-angle modifier. In practical PTSC installations, mirror soiling, coating degradation, structural deformation, and seasonal incidence-angle variations introduce additional optical losses that are not explicitly represented. These effects may influence long-term energy, exergy, and carbon-avoidance performance.

Economic modeling assumptions:

The economic assessment is conducted at a screening level using fixed assumptions for capital cost, nanoparticle pricing, discount rate, and inflation. Variability in material prices, installation costs, and supply-chain conditions may alter techno-economic indicators. A stochastic or Monte-Carlo-based economic framework would provide a more comprehensive representation of cost uncertainty for real-world deployment.

Environmental scope and life-cycle assessment:

The environmental assessment performed in this study is limited to the operational phase and does not constitute a full life-cycle assessment (LCA). While nanoparticles may involve non-negligible environmental burdens during synthesis, processing, and end-of-life stages, robust life-cycle inventory data for nano-enhanced PCMs remain scarce and highly dependent on production routes and suppliers. Consequently, a cradle-to-grave LCA was deliberately excluded to avoid introducing speculative assumptions. The present results therefore do not claim overall life-cycle environmental superiority. Future work will integrate a full LCA framework, including nanoparticle manufacturing, encapsulation, long-term degradation, and disposal, to comprehensively evaluate environmental trade-offs.

Site-specific climatic conditions:

The climatic inputs are based on a single high-irradiance location. Although the proposed model is fully generalizable, extending the analysis to multiple climatic zones would strengthen conclusions regarding annual efficiency, exergy robustness, and environmental performance under diverse meteorological conditions. Future studies may incorporate multi-site simulations and climate-change projections.

Comparative assessment with published studies

As summarized in Table 6, the proposed PTSC–NePCM system demonstrates a clear and consistent improvement over previously reported PTSC configurations in terms of thermal performance, exergy efficiency, useful heat output, and overall system sustainability.

Table 6.

Comparison of the present PTSC–NePCM system with selected PTSC-based thermal energy storage systems from the literature.

Study	Storage type	Modeling approach	Annual thermal efficiency (%)	Exergy efficiency (%)	Useful heat output (kWh m−2 day−1)
Simeu et al. (2024)	No storage	Steady/quasi-steady	41–47	–	2.1–2.4
Babikir et al. (2021)	No storage	Quasi-steady	∼45	11–13	–
Pakouzou et al. (2021)	No/sensible TES	Experimental	33–38	–	1.6–1.9
Present study	Al2O3–NePCM (1 wt%)	Fully transient + Stefan + 8760 h	52–56	15–18	2.8–3.1

Simeu et al. (2024) investigated a conventional PTSC without TES and reported annual thermal efficiencies in the range of 41–47%, with daily useful heat gains of approximately 2.1–2.4 kWh m⁻² day⁻¹. In contrast, the present PTSC–NePCM system achieves significantly higher annual thermal efficiencies of 52–56% and useful energy outputs of 2.8–3.1 kWh m⁻² day⁻¹. This improvement is primarily attributed to the integration of latent-heat storage, which mitigates solar intermittency, and to the enhanced thermal conductivity of the Al₂O₃-based NePCM, which accelerates the charging and discharging processes.

Babikir et al. (2021) developed a quasi-steady PTSC model without latent thermal storage and reported maximum outlet temperatures of 60–65 °C and exergy efficiencies typically limited to 11–13%. Due to the absence of thermal storage, system performance was highly sensitive to instantaneous solar conditions. By contrast, the fully transient PTSC–NePCM system proposed in the present study reaches outlet temperatures of 70–72 °C and annual exergy efficiencies of 15–18%, while providing a stable heat supply during periods of reduced solar irradiance. Moreover, unlike the short-term diurnal analyses adopted, the present work employs 8760 h of real climatic data, enabling a comprehensive assessment of seasonal and cumulative exergy behavior.

Pakouzou et al. (2021) experimentally examined a PTSC-based solar dryer without nano-enhanced latent storage and reported seasonal thermal efficiencies of 33–38% and useful heat outputs of 1.6–1.9 kWh m⁻² day⁻¹, with pronounced performance degradation during late-afternoon hours. The incorporation of PCM storage in the present system effectively alleviates this limitation, resulting in a 22–29% improvement in late-day heat delivery. Furthermore, the addition of 1 wt% Al₂O₃ nanoparticles provides an extra 6–8% enhancement in daily useful heat compared to pure paraffin, yielding total useful energy outputs of up to 3.1 kWh m⁻² day⁻¹. This corresponds to an overall improvement of approximately 65–90% relative to the conventional PTSC dryer.

From an economic and environmental perspective, most PTSC studies in the literature either neglect cost and CO₂ indicators or report them in a simplified manner. The present study explicitly integrates thermo-economic and environmental analyses within a fully transient framework. The optimal 1 wt% NePCM configuration achieves a LCOH of 0.085 $/kWh_th and a simple PB of about 5.2 years, while simultaneously delivering higher annual CO₂-emission avoidance compared to conventional PTSC and pure-PCM-based systems. This demonstrates that the proposed PTSC–NePCM configuration provides a favorable balance between thermodynamic performance, economic feasibility, and environmental benefit.

Overall, the comparison highlights that the combined use of a fully transient thermo-optical model, a Stefan moving-boundary formulation for phase change, and long-term (8760 h) climatic data enables the present study to outperform existing PTSC systems reported in the literature, particularly in terms of annual efficiency, dispatchability, and integrated sustainability metrics.

Conclusions

This study developed a fully transient thermo-optical–phase-change modeling framework for a PTSC integrated with NePCM storage. By explicitly resolving melting and solidification through a Stefan moving-boundary formulation and driving the model with 8760 h of real hourly climatic data, the proposed approach enables systematic evaluation of long-term thermal, exergy, environmental, and economic performance beyond short-term or quasi-steady analyses commonly reported in the literature.

The results indicate that nanoparticle concentration governs system performance through competing physical mechanisms. While increasing Al₂O₃ loading enhances effective thermal conductivity and accelerates transient melting, higher concentrations progressively dilute latent-heat storage capacity, increase molten-phase viscosity, and intensify optical attenuation and material cost. These counteracting effects become significant only when cumulative annual operation and seasonal variability are accounted for, highlighting the limitations of conclusions drawn from short-duration simulations.

When assessed under combined thermodynamic, environmental, and economic criteria, a nanoparticle loading of 1 wt% consistently provides the most balanced annual performance. This configuration achieves the highest annual exergy efficiency, the greatest CO₂ avoidance per unit of useful thermal energy, and the minimum LCOH (0.085 $/kWh_th), while maintaining stable behavior across seasonal solar conditions. Higher loadings, although beneficial in terms of instantaneous or peak heat-transfer rates, do not yield proportional improvements in long-term system-level performance.

From a design and deployment perspective, the findings demonstrate that moderate nano-enhancement offers superior robustness compared to aggressive conductivity maximization in PTSC–NePCM systems operating under realistic climatic conditions. The proposed transient Stefan-based framework therefore provides a physically consistent and practically relevant tool for long-term assessment and optimization of solar-thermal systems with latent heat storage, and establishes a sound basis for future experimental validation and extended multi-dimensional modeling.

Future studies may extend the present framework to investigate different PCM materials, alternative nanoparticles, and multi-component (binary or ternary) NePCM systems to further optimize thermal, economic, and environmental performance.