Impact of the infrared properties of the backsheet on the efficiency of an encapsulated photovoltaic silicon cell

The efficiency of an encapsulated photovoltaic (PV) silicon cell is studied by modeling the radiative exchanges between its layers, focusing on the influence of the radiative properties of the backsheet on the cell temperature. The model uses the Monte-Carlo method to solve the radiative transfer equation. Two-dimensional simulations were performed, showing how the radiative properties of the backsheet affected the silicon cell temperature, which was then used to calculate the current-voltage characteristic and the efficiency of two PV modules. The results revealed a significant sensitivity of the model to the emissivity of the top-side backsheet, with a 19.4°C temperature difference for the silicon cell and, a 9.5% and 13.3% increase in the maximum power between a fully reflective and a fully absorptive backsheet for the two PV modules. Overall, the study highlights the importance of considering radiative properties in the design and optimization of PV panels, as they can have a substantial impact on their performances.


Introduction
In order to foster the use of renewable energy sources, the implementation of photovoltaic (PV) generation systems has widely been endorsed, notably by integrating these systems into buildings.The PV technology has greatly been diversified, with the emergence of devices exhibiting tailored aspects and functionalities [1].Colored cell modules, photovoltaic tiles and facade elements represent notable examples of solar energy systems offering conventional building functionalities coupled with electricity generation.Solar modules commonly consist of solar cells, a glass layer on the front, an encapsulant and a protective backsheet on the rear side.The backsheet, usually composed of a polymer material, serves as a barrier against moisture and ultraviolet radiation, while also acting as an electrical insulator.Moreover, a reflective foil backsheet could be utilized to improve the PV module efficiency by diffusely or non-diffusely reflecting the incident light with possible gains in the range of 1 to 3% [2].White is the most widely used color for backsheets, whereas black backsheets are frequently encountered for aesthetic purposes [3].According to research [4], optimizing the gap between cells and using white backsheets lead to 1 to 2% of improvement in the maximal power compared with the usage of black ones.Luxembourg et al. demonstrated that the application of black powder coatings with low IR absorption on standard films used in the photovoltaic industry can significantly reduce cell-to-module (CtM) losses for all the black modules [5].The infrared radiation properties of the underlying substrate are largely preserved when such films are applied.Compared with conventional absorbent black foils, an improvement of the short-circuit current of 1.4 to 1.8% at the PV module level was achieved.McIntosh et al. reported that around 25% of the photons that fall on the backsheet between the PV cells are internally reflected back to the cells, resulting in a 2.4% increase in the module power [6].They also found that black foil modules have 2.0 to 3.0% less power than white foil modules.Burgers and al. showed that adding mirrors to the front and back surfaces reduces the amount of light emitted from these surfaces [7].However, some of the emitted light is absorbed by the mirrors due to their imperfect reflectivity.Additionally, the temperature is one of the fundamental parameters that affects the efficiency of the photovoltaic module and hence, the thermal properties of the backsheets can have a significant impact on the efficiency of the PV module.The temperature of a PV module is influenced by the ambient temperature (directly proportional), the cell irradiance (directly proportional), and the wind speed (inversely proportional) [8].Some possible modifications in the structure of a PV module to maximize the efficiency are discussed in [9].This can be realized by augmenting the reflection of light for PV conversion and dissipating heat for diminishing the operating cell temperatures with conductive and radiative-adapted materials.The use of encapsulating materials with high thermal conductivity enables to lower the operating temperature of a PV module [10][11][12].Vaishak et al. conducted a study where they examined the efficiency of a refrigerant-based PV/T system using three different backsheet materials: glass, TPT, and copper [13].They employed a validated numerical model for their analysis, and the results indicated that the PV/T collector with the TPT backsheet exhibited lower cell temperature, while the collector with the copper backsheet demonstrated the highest coefficient of performance.
In this study, we address the issue of heat flux knowledge within a PV module by examining the impact of the radiative properties of the backsheet on the output parameters of an encapsulated PV cell.For this purpose, a model was developed to understand the radiative transfers inside an encapsulated silicon cell by accounting for the infrared radiations and their influence on the temperature and efficiency of the PV cell.The radiative exchanges between the semi-transparent layers of an encapsulated silicon cell are determined by CFD simulations taking into account the spectral properties of the surfaces for wavelengths ranging from 0.1 to 100 µm.The obtained results are analyzed.

Photovoltaic system
The subject of this study is an encapsulated silicon photovoltaic (PV) cell composed of the following layers: a backsheet, an ethylene vinyl acetate (EVA) encapsulant, a silicon (Si) cell, and a cover glass, as depicted in Figure 1, which also indicates their sizes.The case of exposure under the lamp of a solar simulator with controlled temperature conditions is considered.

Geometry and meshing
In view of performing the numerical simulations of the photovoltaic system, we used the commercial code CFD-Ace.The geometry was drawn according to the dimensions specified in Figure 1, and the PV cell domain was meshed using "quadrilateral" elements with a size of 0.2×0.2mm 2 .The choice of quadrilateral elements was made knowingly, as these elements can accurately capture the shape of the PV cell domain without introducing excessive computational cost.By carefully selecting the mesh resolution, we were able to achieve reliable and accurate results in our simulations.The resulting mesh is shown in Figure 2. The subdomain of the PV cell is surrounded by two others, one above having a height of 17 cm corresponding to the distance between the emitting lamp of the solar simulator and the PV cell, and the second, down, having the same dimensions.Both allow to model the incident radiations and the heat exchanges with the ambient air and the environment.Special attention was given to the subdomain of the PV cell, where high accuracy was required, the mesh being refined with shrinkage in this area.The presence of the electrodes of the PV cells is not taken into consideration in the present model.

Governing equations
To analyze the thermal behavior of the PV cell, two governing equations were employed here [14].First, the heat conservation equation considering the different heat transfer processes, such as conduction in solids or convection in gases, is used to determine the temperature distribution: where the total mass enthalpy is: is the internal mass energy.It is a variable that depends on the density  and the temperature  ;  and  are the components of the fluid velocity  according to the directions  and  ;  is the thermal conductivity;  is the static pressure;   is the viscosity tensor;   represents a variable containing terms from additional heat sources such as a radiative source or chemical reaction.
The second governing equation used was the radiative heat transfer equation so as to study the behavior of a medium that emits, absorbs, and scatters energy through radiation.

𝛺. 𝑔𝑟𝑎𝑑(𝐼(𝑟, 𝛺
where:  is the propagation direction of the beam; (, ) is the intensity of the radiation.It is a function of the position  and the direction  ;  et  are respectively the absorption and diffusion coefficient;   () is the intensity of the blackbody at medium temperature; () is a phase function of the energy transfer from the direction of arrival  to the direction of exit  ′ ; .((, )) is the intensity gradient in the direction of propagation  ; −( + )(, ) represents respectively the losses due to the absorption  and the diffusion  ;   () corresponds to the radiative emission of the body; is the gain due to diffusion.
On the surface, the intensity is: with (, ′) : intensity of the energy of the radiation leaving the surface;  : surface emissivity;  : surface reflectivity;  : unit vector normal to the surface.
The Monte-Carlo method was used to solve the radiative transfer equation by simulating the heat exchanges by radiation.This method traces the rays emitted by different surfaces until they are absorbed by another surface, allowing for the calculation of the radiative heat flux.The method is described in [15] and the details of the photon packet tracing are explained in [16].The surfaces were discretized into patches, and the radiative heat flux for each patch () was determined by the incident radiation from all the other patches () as well as its own emission: where   is the heat transfer rate;   is the heat flux density;   is the area of the surface under consideration;   is the exchange matrix;   is the symbol of Kronecker;   is the emissivity of the patch  ;  is Stephan Boltzmann's constant of 5.669.10 −8 . −2 . −4 ;   is the average temperature of patch .

Photovoltaic conversion
Upon computing the values of the cell temperature from the model of the photovoltaic module, the diode equivalent circuit of a photovoltaic panel was applied to simulate the changes in the current-voltage characteristics and to assess the variations in the output power as a function of the cell temperature [17].
The basic equation that mathematically describes the current-voltage () characteristic of an ideal photovoltaic cell in the semiconductor theory is expressed by the following equation: where   is the current generated by the incident light (it is directly proportional to the irradiance of the Sun) [],  0, is the saturation or inverse leakage current of the diode [],  is the electronic charge The current generated by the incident light on the photovoltaic cell is indeed directly proportional to the solar irradiance and is also influenced by the temperature according to equation (7): where  , is the current generated by the light in its nominal state (usually for 25°C and 1000 / 2 ) with ∆ =  −   ,  and   being the actual and nominal temperatures [],  is the irradiance at the surface of the device, and   is the nominal irradiance [/ 2 ].The diode saturation current  0 and its temperature dependence can be expressed as: where   is the gap energy of the semiconductor (  ≈ 1,12 eV for crystalline Si at 25°), and  0, is the nominal saturation current: To account for the electrical losses, equation ( 6) can be written with the series and parallel resistances   and   : where   and  0 are the photovoltaic and saturation current of the PV cell, respectively; is the thermal voltage [] for   cells connected in series.

Volume properties
Table 1 presents the thermal properties applied to the different subdomains.A Tedlar-based backsheet was considered, although other stack variants are possible with varying thermal properties [20,21].The initial temperature is 20°C.The air above and below the encapsulated solar cell is initially considered with a velocity of  =  = 0.

Boundary conditions
To model the photovoltaic system when being exposed to irradiance, the upper and lower parts were subjected to Neuman-type boundary conditions (Figure 3.a).Applied heat flux values were found to achieve an incident irradiance of 1000 W/m² on the top surface of the encapsulated PV cell while obtaining the silicon cell at a temperature of 25°C as per the STC conditions.An emissivity of 1 was considered for the upper limit representing the emission area of the lamp to obtain a blackbody emission so as to approximate an AM1.5 spectral distribution.A blackbody property was also taken for the bottom surface of the backsheet and the bottom boundary of the subdomain.The sides of the system were assumed to be fully reflective and were subjected to a convective flux condition to account for convective heat transfer.Spectral properties for a wavelength range from 0.1 μm to 100 μm were introduced for the interfaces involving the glass, EVA, and silicon layers (Figure 3.b and c).The backsheet was considered as an opaque material with a graybody property for its upper surface.A fully absorptive condition (emissivity   = 1, reflectivity   = 0) was initially taken for the latter.(c) Radiative properties of the silicon [18,22,23].

Calculations
Equations ( 1) to (5) were solved using the finite volume method with the CFD-Ace code considering the Algebraic Multi Grid solver for faster convergence, and the solutions of equations ( 6) to (10) were determined utilizing a Matlab program.Parametric calculations of the temperature field were carried out varying the emissivity   of the top side of the backsheet from 1 to 0 with a step of 0.1.At the same time, the reflectivity   = 1 −   was scaled from 0 to 1.The average temperature of the silicon cell was calculated for each step and the value found was introduced as the cell temperature in the single diode model for finding the output parameters of a photovoltaic module based on the radiative properties of the backsheet.The input parameters of two commercial PV modules having silicon PV cells of size 156 mm × 156 mm close to the one in the present model were entered (Table 2).

Photovoltaic cell temperature
The temperature field obtained for an emissivity of the top side of the backsheet of 1 is shown in Figure 4.The temperature within the silicon solar cell was found to be homogeneous and its mean value was calculated and introduced into the single diode model to calculate the output parameters for the two PV modules as previously mentioned.The variation of the mean PV cell temperature according to the emissivity of the upper surface of the backsheet is plotted in Figure 5.The temperature of the silicon cell is increased from 26.2°C to 45.6°C when the emissivity is reduced from 1 to 0. This temperature difference of 19.4°C between a fully reflective and a fully absorptive backsheet shows that the PV cell temperature is sensitive to the change in the radiative properties in the infrared domain.When the emissivity is reduced, the absorptivity is also decreased and the reflectivity is augmented, so the radiative heat flux from the upper surface of the backsheet towards the silicon cell becomes more significant.A second-degree polynomial is found as a fitting curve with a determination coefficient R 2 close to 99%.This means that the derivative of the temperature with respect to the emissivity is an affine function of equation 21,192.  − 28,262.The variation appears to be more significant for lower emissivities.When backsheets are exposed to UV radiation [24] or to various weather conditions [25], they can be degraded by chemical mechanisms or cracked, changing their reflectance and possibly inducing variations in the cell temperature.The ideal surface property for the backsheet would be to be fully absorptive in the IR domain and fully reflective in the visible range.The aim is to maintain the operating temperature of the cell as low as possible in order to improve its performance and to reflect the maximum of light towards the PV cell.

Variations of the electrical performance
The I-V curves of both the PV modules are modified when the emissivity of the top-side backsheet is varied.The example of the Q6LM PV module is displayed in Figure 6.As the emissivity increases, the cell temperature is lowered and hence both the open-circuit voltage and the maximum power are raised.We observe a noticeable improvement in the open-circuit voltage and no significant effect on the shortcircuit current.The fill factor is slightly increased when the temperature becomes more important.(11) where  is the illuminated surface.When the emissivity increases, the efficiency is improved.Its relative variation for the two cases is 9.5% for the KC200GT module and 13.3% for the Q6LM one (Figure 7).This discrepancy is due to the temperature coefficients which are different for both the PV modules.This result highlights the importance of absorbing infrared radiations at the bottom of the PV module to reduce the cell temperature at its best.

Conclusion and perspectives
The purpose of this study was to investigate the radiative transfers within the layers of an encapsulated photovoltaic cell to gain a better understanding of its properties and to improve its performance.To achieve this, a two-dimensional model of the encapsulated silicon cell was constructed, and the conservation equations accounting for the heat exchanges were specified.The radiative transfer equation was then solved using the Monte-Carlo method.The simulation results showed that the thermal emissivity of the backsheet had a significant impact on the temperature of the solar cell, with a 13% variation in the efficiency observed when the emissivity was changed from 0 to 1.The sensitivity of the cell temperature to the IR properties of the backsheet was also ascertained.A backsheet that is fully reflective in the wavelength range corresponding to the spectral response of the PV cell and strongly absorbent in the IR range was identified as the ideal setting.A performance calculation was carried out on two PV modules, showing that a temperature difference of 19.4°C between a totally reflective and a totally absorptive backsheet could lead to a significant gain of 13% on the maximum power.This highlights the importance of considering the radiative properties of the backsheet when designing and manufacturing encapsulated PV cells.Further research could include a comparison between simulation results and experimental tests by manufacturing an encapsulated cell prototype.Additionally, a coupling of the PV conversion could be considered by modeling the PN junction while considering the spectral and scattering properties of the carriers with the finite volume code.A parametric study varying the optical indices of the backsheet could also be handled to minimize the cell temperature and maximize the power gain.Future calculations will consider the spectral properties of the backsheet.Overall, this study provides valuable insights into the radiative properties of encapsulated photovoltaic cells and highlights the importance of considering the radiative properties of the backsheet in improving their performance.

Figure 1 .
Figure 1.Encapsulated photovoltaic PV cell with its dimensions.

Figure 2 .
Figure 2. Meshing of the system with tight mesh for the encapsulated PV cell.

Figure 3 .
(a) Boundary conditions.(b) Radiative properties of the glass and of the encapsulant.

Figure 6 .
Figure 6.I-V curves obtained for several emissivities of the backsheet for the Q6LM model.

Figure 7 .
Figure 7. Efficiency of the PV versus top-side backsheet emissivity for the two PV modules.

Table 1 .
Materials and air properties.

Table 2 .
Input parameters of the two commercial PV modules considered.