Solar power is the most widespread source of energy in the world. The energy of sunlight that strikes the earth in an hour is more than the energy consumption of the earth’s population in a year. Despite its extensiveness, solar energy has an extremely low density (Kim et al., 2022; Mustafa et al., 2022; Hassan et al., 2023). The Sun is not only an enormous source of energy but also the origin of life and the source of all other energies (Izadi et al., 2021; Roper et al., 2022). Fossil energy also originates from solar energy because fossil fuels were once plants with sun-dependent life (Ascione et al., 2021; Hamzat et al., 2021; Li et al., 2021). To harness this source, we need to find a way for the high-productive and low-cost conversion of its dispersed energy into consumable electrical energy (Ejaz et al., 2021; Jamil et al., 2021; Wang et al., 2021; Xiong et al., 2021). Therefore, many researchers have focused their studies on solar energy. Bansal et al. (1994) shown that the wind tower-connected solar chimney can create an airflow rate of 1.4 kg/s at an incoming solar radiation of 700 W/m2 using a solar chimney

linked with a wind tower-assisted ventilation system. The findings also revealed that an air exchange rate of 35–73 air changes per hour may be reached utilising a mass flow rate of 0.75–1.3 kg/s in an environment with a wind speed of 1 m/s. Using a spreadsheet computer software, Aboulnaga (1998) provided a parametric analytical evaluation of a rooftop solar chimney paired with a cooled hollow. According to the findings, the suggested rooftop solar chimney may produce up to 1.6 kg/s of airflow at an incident solar radiation level of 850 w/m2 on average. Moreover, the chimney’s highest air velocity was close to 1.1 m/s. The findings also suggest that with a wind speed of 0.4 m/s, the cooled cavity may provide a mass flow rate of 0.35 kg/s at an average incoming solar energy of 575 w/m2 and an air velocity of 0.9 m/s in the chimney. Nowadays, the use of photovoltaic (PV) systems to convert solar into electrical energy is regarded as a solution to meet energy needs (Chandrasekar and Senthilkumar, 2021; Salari et al., 2021; Siah Chehreh Ghadikolaei, 2021). A small fraction of absorbed solar-radiated energy is converted into electrical energy by PV cells, while a large part of it is converted into thermal energy and raises the cells’ temperature (Chan et al., 2010; Kumar et al., 2015; Said et al., 2018; Ali, 2020). PV cells are the most crucial part of PV systems, and the increased temperature of PV cells reduces the efficiency of PV systems. Hence, the PV cells’ temperature should be reduced as possible to improve the efficiency and productivity of such systems. Various researchers have proposed different cooling systems with different conclusions (Katekar and Deshmukh, 2020; Vaithilingam et al., 2020; Aghakhani et al., 2022). The photovoltaic phenomenon is the one that generates electricity through light radiation without any driving mechanism. A PV system is the PV phenomenon and any system that employs it. A solar PV system or solar energy system is a renewable energy system that uses PV modules to convert sunlight into electricity (Hernández-Callejo et al., 2019; Shittu et al., 2019; Wahab et al., 2019). The generated electricity can be either stored or harnessed directly. PV systems are widely-used systems concerning new energies. Kazemian et al. (2019) presented a 3D simulation of a comprehensive model of a PCM-integrated PV thermal system. The findings show that as compared to the PVT system only, the PCM-integrated PV thermal system has a lower surface temperature and coolant output temperature. The findings suggest that raising the melting point of PCMs from 40°C to 65 °C raises the surface temperature from 51.53°C to 58.78°C while lowering the percentage of melted PCMs from 82.7% to 9.6%. The electrical and thermal energy efficiency of the PCM-integrated PV thermal system is also shown to improve when the thermal conductivity of the PCM rises.

Given the significance of energy restoration in solar devices by PCMs due to the limited time of solar radiation, the present article performs a numerical study on the effect of using PCMs in a solar panel (Mondal, 2008; Sharma et al., 2009; Agyenim et al., 2010; Rostami et al., 2020). To this end, some PCM is used under a solar panel along with two pipes with a NFD flow. Several PFs are embedded over the pipes, and the effect of PFs shapes, i.e., square, rectangular, triangular, and circular, is investigated. The transient study continues until the variations are stabilized, and the values of outlet NFD‘s temperature, panel temperature, HC, and PCM’s temperature and melting contours are studied.

The problem’s geometry includes a 100 × 85 cm solar panel, under which a type- Paraffin wax PCM compartment is employed. Two pipes with an internal diameter (ID) of 5 and 4 cm are embedded inside the compartment with a center-to-center spacing of 46 cm. There is a flow of NFD inside the pipe with an inlet temperature of 298.15 K and a constant Reynolds number. Several circular (Cir), square (Sqr), rectangular (Rec), and triangular (Tri)-shaped PFs are placed on the pipes with a height of 2.5 cm. Figure 1 depicts a view of the problem’s geometry.

Table 1 shows the values for the properties of the nanoparticle, water, and PCM (Li, 2013).

The continuity, momentum, and energy equations for mixture as well as the volume fraction equation for the secondary phase are therefore solved using the mixture model. (Ishii, 1975). ∇ ⋅ ρ m v → m = 0 (1) ∇ ⋅ ρ m v → m v → m = − ∇ p + ∇ . μ m ∇ v → m + ∇ . ( ∑ k = 1 n φ k ρ k v → d r , k v → d r , k ) (2) ∇ . ( ∑ k = 1 n φ k v → k ρ k H k + P = ∇ . k m ∇ T (3) ∇ . φ p ρ p   v → m = ∇ . φ p ρ p   v → d r , p (4) where v → _m is the mass average velocity: v → m = ∑ k = 1 2 ϕ k ρ k v → k ρ m (5)

The drift velocity for nanoparticles, v → _dr;k, is defined as v → d r , k = v → k − v → m (6)

The nanoparticles’ velocity in relation to the base fluid’s velocity is used to compute the slip velocity (relative velocity). v → p f = v → p − v → f (7)

This is the relationship between slip velocity and drift velocity: v → d r , p = v → p f − ∑ k = 1 n φ k ρ k   ρ m v → f k (8)

The mixture model requires a constitutive equation to be solved. As a result, the slip velocity may be calculated using the Manninen et al. (Mikko et al., 1996) proposed equation (Schiller, 1933). v → p f = ρ p   d p 2 18 μ c f d   ρ P − ρ m ρ P g → − v → m . ∇ v → m (9) f drag = 1 + 0.15   Re p 0.687   Re p ≤ 1000   0.0183 Re P   Re p > 1000   (10)

Enthalpy-Porosity is a well-considered approach for mimicking a porous material in a domain.

The value of the volum fraction linked with each cell in this sort of media indicates the porosity of the cells (Daneh-Dezfuli and Pordanjani, 2022). ∂ ρ ∂ t + ∇ . ρ u → = 0 (11) ∂ ∂ t ρ u → + ∇ . ρ u → u → = μ ∇ 2 u → − ∇ P + ρ g → + S → . (12) ∂ ∂ t ρ H + ∇ . ρ u → H = ∇ . k ∇ T (13)

Moreover, the source word expressed by the letter S was thought to be as follows. (Kalbasi et al., 2019; Daneh-Dezfuli and Pordanjani, 2022): S = A m u s h y 1 − γ 2 γ 3 + ε u → . (14)

The term “enthalpy” was used to refer to the variation of “H" in Eq. 13, which contains the latent and sensible enthalpies (Kalbasi, 2021; Daneh-Dezfuli and Pordanjani, 2022). H = h + ∆ H (15)

According to the following mathematical formulation, the variant of “h” reflects the sensible enthalpy: h r e f + ∫ T r e f T C P ∆ T (16)

Moreover, the latent heat was represented by the symbol ∆ H and was described numerically as follows (Kalbasi, 2021; Daneh-Dezfuli and Pordanjani, 2022): ∆ H = γ L h (17) where the γ is determined as follows: γ = 0   i f   T < T s ; T − T s T l − T s   i f T s ≤ T < T l ;   1   i f   T l ≤ T ; (18)

The density was predicted and applied using this simulation’s boussinesq model. The aforementioned supposition is formally expressed in the following equation. (Kalbasi et al., 2019; Kalbasi, 2021). ρ = ρ l β T − T l + 1 (19)

The finite element approach is used to conduct the simulation (FEM). The equations of the NFD are solved using the two-phase mixing technique. With COMSOL 6, all equations are resolved.

Two articles on solar energy and solar thermal panels are employed to validate the present article. In Figure 2, the values of panel temperature in different heat fluxes are compared among two articles, i.e., Nasrin et al. (2018) and Rahman et al. (2017). The given articles investigate solar panel temperature in different heat fluxes. The result comparison indicates that the results of the present simulation are close to those of the two previous articles.

Figure 3 shows the vectors that denote the direction of the melted PCM current with temperature coloration for different circular, rectangular, triangular, and square-shaped PFs on the NFD flow pipe. As with other fluids, the melted PCM contains convective flow due to the temperature difference within, which gives rise to the buoyant force in the melted PCM; as a result, the hotter parts of PCM moves upward against the gravity force, and the colder and heavier parts move downward along the direction of the gravity force. Of course, the high viscosity of the melted PCM leads to weak buoyant force and extremely slow PCM movement. However, the vectors depict the direction of the melted PCM’s movement in the Solar System, showing that lower-temperature parts of PCM move downward, and higher-temperature parts of melted PCM move upward. The highest upward velocity is seen in the middle of the two pipes, where PCM melts more quickly and has a higher temperature. In contrast, in lower-temperature parts, the melted PCM moves downward to be replaced by higher-temperature PCM.

Figure 4 shows the temperature of the Solar System for different circular, rectangular, triangular, and square-shaped PFs on the NFD -containing pipes at four different times. The analysis of the PCM’s melting process over time shows that at the beginning, PCM starts to melt sooner in the upper and middle parts of the two pipes, especially in the outlet section. On the contrary, there is less melted and more solid PCM in the lower part, especially around the pipes. Since the NFD enters these areas, there is a high heat transfer from PCM, which leaves PCM solid in the given area at all times. On the contrary, the melting process gradually progresses in other parts over time, covering the side parts of PCM in the Solar System. In the upper layer, it can be seen that there is less melted PCM in the area where the two pipes pass, which is due to the colder NFD -containing pipes compared to other parts. Especially, a part of solid PCM is seen in the inlet part of the NFD for all pin din shapes at all times, which is due to the lower temperature of the NFD s in this particular section. The PF shape variation has no significant effect on the amount of melted PCM at different times, and slow NFD is one of the reasons behind limited variations.

Figure 5 shows the temperature of NFD -containing pipes for different circular, rectangular, triangular, and square-shaped PFs on the NFD flow pipe at the final time. In all PF shapes, there is low-to-high temperature variation from the inlet toward the outlet of the NFD flow. On the inlet side, the pipe’s temperature is lower, while on the outlet side, the pipe has the highest temperature. The maximum temperatures on the pipe are close for different PF shapes, and the minimum temperature on the pipe is different according to PF shape variations. The addition of PFs on pipes results in the heat transferring better from PCM to the pipe, which leads to increased time of PCM melting and higher temperature of NFD in the outlet. There is a lower minimum temperature on the pipe in the circular and rectangular PFs compared to the triangular and square PF shapes. The outlet NFD flow with a high temperature can be employed for domestic use and hot water generation. As a consequence, the enhanced heat transfer to the NFD raises the temperature of the outlet NFD and lowers the temperature of the panel, both of which improve the overall efficiency of the Solar System.

Figure 6 gives the temperature on the middle layer of the Solar System for different circular, rectangular, triangular, and square-shaped PFs on the NFD flow pipe at the final time. The PF shape variation changes the heat transfer area between the pipe and PCM. Low-temperature NFD flow enters the pipe and takes in the PCM’s heat. The extent of contact between the pipe and PCM affects the amount of heat transfer from PCM to the NFD and the required time to reach the steady state conditions. The maximum temperature in this section of PCM shows no significant change in the PF shape variations. The considerable energy restoration by PCM mitigates the effect of PF shape. In a circular PF shape, the NFD‘s temperature is lower to some extent in the middle of the pipes and to some lower extent in the inlet compared to the three PF shapes. The low NFD velocity is also a reason behind the limited effect of PF shapes. However, the minimum temperature in this section of PCM significantly changes with PF shape variations. The minimum temperature is seen in the rectangular PF shape, followed by two triangular and square cases. The highest temperature minimum occurs in the circular PF. The close the minimum and maximum temperatures in a Solar System, the higher the temperature uniformity, which is desirable for solar systems and panels.

Figure 7 shows the transient study of the NFD temperature in the outlet of the Solar System for different circular, rectangular, triangular, and square-shaped PFs on the NFD -containing pipes. The NFD temperature in the outlet depends on the temperature of the panel and PCM and is seen to be increased up to 1800 s; but, after that, the NFD temperature in the outlet remains constant as most conditions of the problem, and the panel temperature become constant. After 1800 s, a steady state is established in the system, and the transient problem becomes a steady-state one with quite limited variations in the results. The PF shape variation has a low effect on the outlet NFD temperature, which is primarily due to the use of PCM, which allows a considerable amount of thermal energy in the Solar System to be used for melting PCM; as a result, PF shape variation, which leads to the heat transfer variation between the NFD and PCM, causes limited changes in the results. The use of circular PFs lowers the NFD temperature in the outlet, while the use of triangular PFs increases the outlet NFD temperature.

Figure 8 shows the transient study of the HC between the NFD and Solar System for different circular, rectangular, triangular, and square-shaped PFs on the NFD -containing pipes. The HC between the NFD flow and its surrounding is also increasing up to 1800 s, after which it reaches the steady state and takes a constant value. The increase in the panel temperature enhances the temperature difference in PCM, leading to a rise in the temperature gradient in the NFD and an increase in its HC. When the NFD and panel temperatures are fixed, the HC also has very limited variations after 18000 s. The values of the HC are exceedingly close in square, rectangular, and triangular PF shapes, and the only significantly different value of the HC belongs to the circular PF shape, in which the value of the HC is lower than that of other PF shapes, which is evidently clear at most times. Among various PF shapes, the triangular PF shape has the highest HC. Given the averaging process on the local HCs to obtain the average value of the HC, a higher surface on the PFs leads to a reduction in the average HC on circular PFs.

Figure 9 shows the transient study of the volume fraction of PCM for different circular, rectangular, triangular, and square-shaped PFs on the NFD -containing pipes. The melting process of PCM starts from the early times, and the amount of melted PCM increases over time until after approximately 1800 s, after which the variations in the amount of melted PCM become considerably limited. From this time forward, the conditions of the problem get close to the steady state, and the variations in the problem significantly decrease. From this time forward, the amount of melted PCM remains near 73% with small variations. Melting PCM during the daytime helps supply the energy in the solar water heater system at night. The changes in the shape of PFs on the pipes have limited effects on the amount of melted PCM at various times, and the highest level of variations in melted PCM upon PF shape variation is observed when the problem reaches the steady state. At this time, the use of circular and triangular PFs is seen to cause the minimum and maximum amount of melted PCM, respectively. The amount of melted PCM is higher in square PFs compared to rectangular PFs. The higher the aspect ratio of the PF, the greater the contact between the PFs and PCM, which results in more heat transferring from PCM to the NFD and, finally decreased amount of melted PCM. Hence, the use of circular PFs leads to the minimum amount of melted PCM.

Figure 10 shows the average temperature of the solar panel for different circular, rectangular, triangular, and square-shaped PFs on the NFD -containing pipes. Given the inclusion of PCM under the panel, the high amount of restored energy in it, and the limited velocity of the NFD in the pipe, PF shape variation has a low effect on the panel’s average temperature. The panel’s temperature has an increasing trend upon receiving solar energy, rising from the initial temperature of 293 K to more than 332 K. The temperature trend in the panel also implies a significant decrease in its variation after 1800 s. Given the constant conditions of the problem, the panel temperature also remains constant, and the problem becomes a steady-state one. Under the transient conditions, the shape variations of PFs on the pipes have a lower effect on the panel temperature, and the greatest variations on the average panel temperature are seen upon PF shape variations at the steady-state time after 1800 s when the use of triangular PFs is seen to give rise to higher temperatures in the panel. On the contrary, the use of circular PFs generates the lowest temperatures in the panel. The increased cross-section of the PFs results in improved heat transfer from the PCM to the pipe and the NFD flow, which, in turn, leads to a greater drop in the panel temperature.

The present article performs a simulation on the effect of the use of differently-shaped PFs on the pipes in a solar thermal panel system. The corresponding PFs include circular, triangular, square, and rectangular PFs placed on two pipes under the solar panel. The two pipes are embedded in the middle of the PCM, and the study is continued transiently until the problem reaches a steady state. The studies on the effect of PF shapes show:

1. The minimum and maximum panel temperatures are obtained upon the use of circular and triangular PFs, respectively, and the corresponding variations in the panel temperature are mostly observed after 1800 s.