# Implicit Equation for Photovoltaic Module Temperature and Efficiency via Heat Transfer Computational Model

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Photovoltaic (PV) Panel Operating Temperature

^{2}, ambient temperature is 20 °C, average wind speed is 1 m/s with zero electrical loads (i.e., open circuit), and the cell surface is perpendicular to the solar noon direction [23,24]. The NOCT model assumes that both sides of the PV module contact the same ambient temperature, and therefore the temperature difference (Tc − Ta) is not a function of ambient temperature. The NOCT equation for PV module operating temperature includes PV electric efficiency, which is a function of the PV module operating temperature. Therefore, the NOCT equation is an implicit approach for PV module operating temperatures, and it is only suitable for PV modules mounted in a free-standing manner. Hence, this is largely unrealistic [16].

#### 1.2. Heat Transfer Model of Thermal Collectors and Solar Panels

#### 1.3. Aim of This Study

## 2. Methods

#### 2.1. PV Module Structure and Setup

#### 2.2. Environment Surrounding the PV Module

#### 2.3. Theory

#### 2.3.1. Solar Flux on the PV Module

#### 2.3.2. Radiation to/from the PV Module

#### 2.3.3. PV Module Output

^{2}, and $\mathsf{\eta}$ is the PV module efficiency.

#### 2.3.4. Wind Speed Effect on the PV Module

_{w}is the forced convective coefficient of heat transfer in W/m

^{2}·°C and V

_{w}is the free stream wind velocity parallel to the module near the surface in m/s.

#### 2.3.5. Heat Transfer Convection in the PV Module

_{w}> 5 m/s), the force convection is also regarded. On the other hand, an inclined plate (e.g., the PV module) has free convection heat transfer, which has an empirical equation that is a function of gravity, tilt angle, airflow properties, and temperature difference between the plate and the environment. The free convection equations have categories based on the Rayleigh number, which specifies the thermal boundary layer laminar and turbulence states. The equation is considered to be specific when the PV module is horizontal with a tilt angle of zero. There are also equations for the inclined plate at other tilt angles. In this study, we employ these equations to define the numerical model.

_{s}is a panel surface and P

_{s}is the panel perimeter [33].

^{9}and Ra < 10

^{11}, respectively, for inclined and flat PV modules [32,33,34,35]. In Equation (8), g must convert to $\mathrm{gsin}\mathsf{\theta}$ [33], and the Nusselt number for inclined and horizontal PV modules are defined with Equations (9) and (10), respectively [33]

^{2}), $\mathsf{\beta}$ is the volumetric thermal expansion coefficient, $\mathsf{\alpha}$ is thermal diffusivity, $\mathsf{\upsilon}$ is kinematic viscosity, $\mathsf{\theta}$ is tilt angle, and Pr is the Prandtl number.

#### 2.3.6. PV Module Heat Transfer Model

- -
- Solar flux.
- -
- Solar beam orientation.
- -
- PV module tilt angle.
- -
- Airflow properties, most of which function as ambient temperature.
- -
- PV module operating temperature related to ambient temperature.
- -
- PV module temperature is a function of the PV module area.
- -
- There is a functionality between the PV module operating temperature and the PV module efficiency.

#### 2.3.7. Numerical Solution

## 3. Results

#### 3.1. Rayleigh Number Evaluation

^{11}, validating Equation (10) for the application of a flat PV module.

^{9}, and Equation (9) are suggested for use in engineering applications. Ra < 10

^{9}[33] covers our study domain. It is well known that a higher Ra reflects turbulence conditions in the boundary layer and is expected to have more desirable convective heat transfer than a smaller Ra [32,33,34,35]. Figure 4 indicates a tilt angle of 30° with a Rayleigh number higher than that of the other tilt angles. This reflects the larger tilt angle, causing a lower Ra and less convective heat transfer.

#### 3.2. Convection Heat Transfer Coefficient on the PV Module

#### 3.3. PV Module Efficiency and Temperature

_{t}have better efficiency. In the literature, efficiency is a function of the PV module operating temperature and solar flux [3,31]. Nevertheless, there is no significant investigation of efficiency and solar flux correlations [31]. Figure 6 dictates an analogous flat PV panel with different solar flux, and each curve has a unique efficiency compared to the corresponding inclined PV module with similar solar flux.

_{t}= 1000 W/m

^{2}at $\mathsf{\gamma}=90\xb0$. This shows that an increasing tilt angle leads to a colder PV module, resulting in less energy gain. For all efficiency curves, a tilt angle of $\mathsf{\theta}=30\xb0$ has the best efficiency at this setup. This figure illustrates that a hotter operating temperature causes less efficiency.

#### 3.4. PV Module Tilt Angle and Temperature

#### 3.5. PV Module Temperature and Cooling Systems

#### 3.6. Solar Orientation and PV Module Temperature

^{2}, $\mathsf{\eta}=12\%$, an inclined PV module with a tilt angle $\mathsf{\theta}=30\xb0$, and a flat PV panel. This flat PV module assumes a solar angle of incidence changing from 30° to 150° with respect to ground level. The PV module temperature increases to the maximum temperature at noon and then drops. The curve is symmetric to the noon assumed angle, $\mathsf{\gamma}=90\xb0$, because, in this assessment, the ambient temperature is constant, and cloudy skies, pollutants, and humidity are not considered.

- -
- The ambient temperature is not constant, and it changes during the daytime.
- -
- The sky can sometimes be cloudy, causing changes to the solar flux.
- -
- Humidity changes cause differences in temperature.
- -
- For wind speeds > 5 m/s, forced convection can occur, affecting PV module temperature.
- -
- Air pollution can affect heat transfer.

## 4. Discussion

## 5. Conclusions

- -
- The proposed equation has an implicit scheme that determines the environmental and operational characterizations of PV modules.
- -
- PV module efficiency is a linear function of PV module temperature, and it depends on the solar flux.
- -
- The PV panel temperature changes with tilt angle.
- -
- The PV module temperature depends on the solar angle of incidence.
- -
- The inclined PV module becomes hotter than a flat PV panel owing to convection heat transfer, assuming no forced convection and no conduction vis-a-vis a cooling system with a non-isolated backside PV module.
- -
- The optimum operating condition is available with an inclined PV module, and the flat panel has minimum energy generation.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Rayleigh number variation at horizontal $\mathsf{\theta}=0\xb0$ against PV module operating temperature with various solar flux at $\mathsf{\gamma}=90\xb0$.

**Figure 4.**Rayleigh number variation on inclined PV module operating temperature, with various solar fluxes at $\mathsf{\gamma}=90\xb0$ and $30\xb0\le \mathsf{\theta}\le 80\xb0$.

**Figure 6.**PV module efficiency vs. PV module operating temperature with specified solar flux at $\mathsf{\gamma}=90\xb0$ on the horizontal module and inclined PV module with $\mathsf{\theta}=30\xb0$.

**Figure 8.**PV module operating temperature vs. tilt angle with specified solar flux at $\mathsf{\gamma}=90\xb0$ on an inclined PV module. The tilt angle is $30\xb0\le \mathsf{\theta}\le 80\xb0$ and the solar flux is Gt = 1000 W/m

^{2}.

**Figure 9.**PV module operating temperature vs. solar flux with $\mathsf{\gamma}=90\xb0$, horizontal PV module and an inclined PV module at $\mathsf{\eta}=12\%$, without a cooling system.

**Figure 10.**PV module operating temperature vs. solar flux with $\mathsf{\gamma}=90\xb0$, a horizontal PV module and an inclined PV module with a tilt angle of $\mathsf{\theta}=30\xb0$, $\mathsf{\eta}=12\%$ for various values of R

_{cool}Km

^{2}/W. The cooling system results from [31] are presented in this figure (which has an ellipse mark on the curve) to compare the proposed model output with the literature.

**Figure 11.**PV module operating temperature vs. solar ray orientation $\mathsf{\gamma}$, horizontal PV module, and inclined PV module, $\mathsf{\eta}=12\%$; solar flux is constant at G

_{t}= 1000 W/m

^{2}.

**Figure 12.**PV module operating temperature vs. daytime hours from 7 a.m. to 7 p.m. from experimentally recorded data over 4 days, 15 April, 15 May, 11 June, and 17 July, at Tehran’s meteorological station in 2012.

**Figure 13.**PV module operating temperature vs. daytime hours from 8:10 a.m. to 3:15 p.m. The filled red color circle curve is experimentally recorded data from [67]. The filled black color square curve has been calculated based on the input from [67] to compare the proposed model and empirical result.

**Table 1.**Variables and constants taken from [1] used in the numerical solution.

Description | Character | Unit | Value |
---|---|---|---|

PV solar absorptivity | ${\mathsf{\alpha}}_{\mathrm{pv}}$ | - | 1 |

PV module surface | $\mathrm{A}$ | ${\mathrm{m}}^{2}$ | 1 |

Stefan–Boltzmann constant | $\mathsf{\sigma}$ | $\mathrm{W}/{\mathrm{m}}^{2}{\mathrm{K}}^{-4}$ | $5.67\times {10}^{-8}$ |

PV surface emissivity | $\mathsf{\epsilon}$ | - | 0.855 |

Ambient temperature | ${\mathrm{T}}_{\mathrm{a}}$ | °C | 25 |

Air effective thermal conductivity | $\mathrm{k}$ | $\mathrm{W}/\mathrm{mK}$ | $0.026$ |

PV module length | $\mathrm{L}$ | $\mathrm{m}$ | 1 |

Gravity acceleration | $\mathrm{g}$ | $\mathrm{m}/{\mathrm{s}}^{2}$ | 9.81 |

Volumetric thermal expansion coefficient | $\mathsf{\beta}$ | ${\mathrm{K}}^{-1}$ | $3.35\times {10}^{-3}$ |

Air thermal diffusivity | $\mathsf{\alpha}$ | ${\mathrm{m}}^{2}/\mathrm{s}$ | $22.4\times {10}^{-6}$ |

Air kinematic viscosity | $\upsilon $ | ${\mathrm{m}}^{2}/\mathrm{s}$ | $15.7\times {10}^{-6}$ |

Air Prandtl number | $\mathrm{Pr}$ | - | 0.7 |

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**MDPI and ACS Style**

Hassanian, R.; Riedel, M.; Helgadottir, A.; Yeganeh, N.; Unnthorsson, R. Implicit Equation for Photovoltaic Module Temperature and Efficiency via Heat Transfer Computational Model. *Thermo* **2022**, *2*, 39-55.
https://doi.org/10.3390/thermo2010004

**AMA Style**

Hassanian R, Riedel M, Helgadottir A, Yeganeh N, Unnthorsson R. Implicit Equation for Photovoltaic Module Temperature and Efficiency via Heat Transfer Computational Model. *Thermo*. 2022; 2(1):39-55.
https://doi.org/10.3390/thermo2010004

**Chicago/Turabian Style**

Hassanian, Reza, Morris Riedel, Asdis Helgadottir, Nashmin Yeganeh, and Runar Unnthorsson. 2022. "Implicit Equation for Photovoltaic Module Temperature and Efficiency via Heat Transfer Computational Model" *Thermo* 2, no. 1: 39-55.
https://doi.org/10.3390/thermo2010004