# In Situ Solar Panel Convective Heat Transfer Study: Methodology, Limitations, and Seasonal Analysis

Liliane Bou Nassif, Stéphanie Giroux-Julien, Hervé Pabiou

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## To cite this version:

Liliane Bou Nassif, Stéphanie Giroux-Julien, Hervé Pabiou. In Situ Solar Panel Convective Heat Transfer Study: Methodology, Limitations, and Seasonal Analysis. Journal of Thermal Analysis and Calorimetry, 2025, 10.1007/s10973-024-13983-y. hal-04985650

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Distributed under a Creative Commons Attribution 4.0 International License
In Situ Solar Panel Convective Heat Transfer
Study: Methodology, Limitations, and Seasonal
Analysis

Liliane Bou Nassif¹*, Stéphanie Giroux-Julien²† and Hervé Pabiou³†

¹*INSA Lyon, CNRS, CETHIL, UMR5008, Villeurbanne, 69621, France.

²Université Claude Bernard Lyon 1, CNRS, LAGEPP, UMR5007,
Villeurbanne, 69100, France.

³CNRS, INSA Lyon, CETHIL, UMR5008, Villeurbanne, 69621, France.

*Corresponding author(s). E-mail(s): liliane.bou-nassif@insa-lyon.fr;

Contributing authors: stephanie.giroux@univ-lyon1.fr;

herve.pabiou@insa-lyon.fr;

†These authors contributed equally to this work.

Abstract

This paper addresses the challenges associated with predicting the temperature of solar cells during operational phases - a critical consideration for improving efficiency and preventing potential system failures in photovoltaic (PV) systems. The traditional approach to determining solar cell temperature, using thermal models based on energy balance, introduces significant uncertainties - particularly in estimating the convective heat transfer coefficient - an intricate parameter influenced by various factors. To address this issue, the paper outlines a measurement protocol for the convective heat transfer coefficient (HTC), enhanced by data pre-processing, and examines its dependence on wind characteristics across seasons. This work is novel as a result of its year-long direct measurements and validation of the protocol through energy balance methods using an instrumented plate. The pre-processing used to clean the dataset is carefully described and compared to the classical selection at solar noon. The analysis of the linear correlation for convective HTC against wind speed highlights the importance of the dataset used to build the correlation. The analysis covering the largest dataset at solar noon leads to the linear relationship $h_{conv} = 2.8U + 5.0$ with a correlation coefficient of $R^2 = 0.58$ which is in line with relationships found in the literature. Indeed, the different correlations found in the seasonal analysis are
consistent with the discrepancy observed between correlations from the litera-
ture, which highlights the importance of contextualizing findings within outdoor
conditions.

**Keywords:** Solar Energy, Convective Heat Transfer Coefficient, Measurement, Outdoor Conditions, PV Panels

# 1 Introduction

Technology based on mono or polycrystalline silicon dominates the global photovoltaic (PV) panel market, representing 95% of the industry [1]. Despite its widespread use, this technology suffers from a temperature-dependent efficiency loss, where increased cell temperatures result in decreased power output [2, 3]. For instance, under extreme conditions, solar panel temperatures can reach 70 °C [4], leading to a 4-point efficiency drop compared to standard test conditions. As a consequence, accurately predicting solar cell temperature under operating conditions is crucial in order to optimize PV system performance.

Estimating solar cell temperature is challenging due to its dependence on multiple fac-
tors, including ambient temperature, solar irradiance, and wind speed. To address this
complexity, various explicit and implicit correlations have been developed to link the
operating temperature to standard weather variables and material/system-dependent
properties [5]. However, caution is necessary when applying these correlations, as they
are often derived from experimental data for specific PV systems without any phys-
ical consideration which may not be suitable for all scenarios. Indeed, recent studies
reveal substantial variations in temperature predictions among different implicit cor-
relations [6].

At the other end of the spectrum, Notton et al. [7] present a detailed thermal model including models for both radiative and convective heat exchanges, and additional parameters such as ground and sky temperatures in addition to the panel’s radiative properties in the infrared region. A similar approach is proposed by Masson et al. [8] who integrated solar panels into the mesoscale model, addressing their impact on the urban microclimate through radiative and convective fluxes. Building on this, Heusinger et al. [9] developed a refined energy balance model for photovoltaic panels using standard meteorological inputs and key parameters such as ground and sky temperatures, showing improved temperature representation compared to that of Masson et al. [8], as well as the estimates provided by the commercial software PVsyst and the Python package PVLIB.

In these approaches, the radiative heat exchange models are based on accurate knowl-
edge of radiative parameters such as incoming radiations and emissivity of the panel
in both short and long wavelength ranges. Moreover, considering the emissivity, it also
varies with the angle of incidence [10] and soiling which increases the uncertainty with
regard to the input of these thermal models. However, as highlighted by Heusinger
et al. [9], the uncertainty in predicting PV module temperature is primarily due to estimation of the convective heat transfer coefficients, indicating a need for better modeling of convective heat transfer. Indeed, in numerical codes used to predict the output power of photovoltaic systems, convective HTCs are computed from empirical equations that are dependent on meteorological parameters such as wind speed. Hence, in 2008, Faiman [11] proposed a simple model for heat transfer based on physical principles where the solar cell temperature is estimated by identifying the energy balance of the module. In this case, heat transfer from the panel to the environment is modeled using a total heat transfer coefficient (HTC). Faiman [11] derived a total HTC through a steady-state energy balance of a PV panel, presenting a simplified model where the HTC is a linear function of wind speed. Following the studies based on a detailed physical approach ([7], [9]), the global heat transfer coefficient can be decomposed as the sum of both a convective and a radiative component [12]. However, if radiative models are physically well established, modeling convective heat transfer is a tricky endeavor, especially in outdoor conditions.

Convective heat transfer over a PV panel can be modeled by using the boundary layer theory and more specifically resembles the classical configuration of forced convection over a heated flat plate [13]. In terms of dimensionless number, the dimensionless convective HTC is the Nusselt number ($Nu$) which can be expressed as a function of the Prandtl ($Pr$) and Reynolds numbers ($Re$): $Nu = \alpha Re^{\beta}$. This theory implies that the convective HTC is a function of the mean flow velocity ($U$). Moreover, in photovoltaic applications, the Prandtl number is constant (air flow) and the Reynolds number varies over one decade. As a consequence, the power law can be linearized without loss of accuracy. Therefore, in terms of physical parameter, the Nusselt-Reynolds relationship leads to the classical form for the convective HTC: $h_{conv} = aU + b$ [14, 15].

Building on these theoretical foundations, Kumar and Mullick [16] conducted an empirical study to estimate the wind heat transfer coefficient for flat plate solar collectors in outdoor conditions, correlating the convective HTC with wind speed using linear and power regression. Their results, normalized by plate length, show significant dispersion compared to previous studies, indicating a significant scatter in the values predicted by the different laws. Interestingly, the correlations reported in this article differ primarily in their constant coefficients, which the authors attribute to variations in turbulence levels. It is worthy of note that the correlations commonly used in the literature on PV models relate the convective HTC only to the mean wind speed, which is in line with boundary layer theory. However, heat transfer over a PV panel in outdoor conditions may depart significantly from the academic configuration of forced convection over flat plates due to tilt angle, wind direction, fluctuations in solar radiation and wind speed.... Ghabuzyan et al. [17] explored the impact of wind direction and ambient temperature using CFD simulation and experimental data. They find that wind direction has little effect on PV performance and they also give a linear relationship between convective HTC and mean wind speed. More recently, Keddouda et al. [18] developed a simulation algorithm utilizing experimental data yielding an accurate prediction of PV module temperature, as well as convective and radiative heat transfers showing that only 16% of energy is converted to electrical power, while the rest is lost by radiation and convection at 50 and 34%, respectively. Their study
demonstrated significant variability in convective heat transfer coefficients across dif-
ferent methods, emphasizing the need for accurate heat transfer mechanisms.
To further refine the understanding of HTC, direct measurements using heat flux sen-
sors have been employed for high flux applications [19], buildings [20, 21] and more
recently for floating photovoltaic modules [14]. In these cases, the primary source of
uncertainty is the determination of the convective heat transfer coefficient.

The objective of this study is therefore twofold. The first objective is to improve
the measurement method of the convective heat transfer coefficient (HTC) presented
in [14] by incorporating a comprehensive pre-processing of the dataset using environ-
mental variables and validating flux measurement methods with traditional energy
balance approaches using an instrumented plate. The second objective is to evaluate
the robustness of classical linear laws h<sub>conv</sub> = aU + b by using different subsets of data
across seasons. This work is novel as a result of its year-long direct measurements,
including *in situ* comparisons with an instrumented plate, allowing for a comprehen-
sive analysis of heat transfer dynamics. The conclusions include our proposal of a
measurement protocol for real systems without the need for an instrumented plate,
supported by reference data. The article is organized as follows: the experimental
setup, including data pre-processing, is described in Section 2, then the results are pre-
sented in a seasonal and yearly analysis in Section 3. Finally, the authors conclusions
and perspectives for future research are provided.

2 Experimental Setup

## 2.1 Platform Presentation

The solar platform in the CETHIL laboratory - *Centre for Energy and Thermal Sciences of Lyon* - is a 33 m² area on the roof of a building on the Campus Lyon Tech la Doua in Lyon, France, as shown in Fig.1. It is dedicated to the analysis of solar components in outdoor conditions. For this study, 12 PV modules are installed

<figure><img src="image_3.png" /><figcaption>**Fig. 1:** Experimental setup on the roof of the building. (a) PV panels viewed from the east, (b) sun tracker (top) and measuring devices (bottom), (c) PV panels viewed from the west, (d) 6 mpole with four weather stations, (e) heat fluxmeter at the center of the module under study.</figcaption></figure>

on the platform in a bi-slope power plant configuration. The panels are installed
in an east-west dual-sided format with a tilt of 12° (Fig.2). D₁ and D₂ are the
orientation of the panels, and D₁ : S → N indicates a wind blowing from south to
<figure><img src="image_4.png" /><figcaption>**Fig. 2:** (left) Top view of the experimental setup on the roof of the building at 16 m above ground. Panel in light blue is the panel under study. (right) Side views of 12 solar panels with an inclination angle of 12°. $D_1$ and $D_2$ are the orientation of the panels, N indicates a northerly direction.</figcaption></figure>

north. The campaign focuses on the thermal and electrical monitoring of a module in the center of the group. This module under study operates at the maximum power point thanks to a programmable electronic load using the incremental conductance method. Thermal monitoring is carried out by direct temperature measurement using 120 µm diameter K-type thermocouples glued to the back of the panel at multiple points. All thermocouples are calibrated using a calibration well (Fluke 9142), resulting in an uncertainty of ±0.1 K. A custom-made heat fluxmeter (Captec) is used to locally measure the total heat flux at the center of the front surface of the module. The dimension of the sensor (10×100 mm²) is adapted to fit the size of the busbar to avoid shading effects. It is painted white, and the normal emissivity is measured in the short wave range [0.27 – 1.1] µm at the Institute of Nanotechnology of Lyon and in the long wave range [2.5 – 20] µm at CETHIL. Assuming that emissivities are independent of the wavelength (gray hypothesis), one can calculate the average values of the emissivities in the shortwave ($\varepsilon_{sw}$) and longwave ($\varepsilon_{lw}$) ranges (Table 1). However, it is important to note that these values are measured in the direction

<table><caption>**Table 1:** Normal emissivities measured in the short wave and long wave ranges.</caption><thead><tr><td>$\varepsilon_{sw}$</td><td>$\varepsilon_{lw}$</td></tr></thead><tbody><tr><td>0.40</td><td>0.54</td></tr></tbody></table>

normal to the surface and the white painted surface is also assumed to be isotropic.

Regarding the monitoring of weather data, the solar platform is equipped with a Kipp-Zonen CMP22 pyranometer and a Kipp-Zonen CGR4 pyrgeometer, both
installed in the same plane as the Captec fluxmeter. This plane is oriented 12 degrees from the horizontal plane (Fig.2). These instruments are used to measure the global tilted irradiation in the short wave (GTI<sub>sw</sub>) and long wave (GTI<sub>lw</sub>), respectively. All these sensors are connected to a digital multimeter (Keithley 3706), and the data is collected at a sampling rate of 1 Hz. Wind is measured by four weather stations located at an altitude of 2 m, 3.25 m, 4.75 m, and 6 m above the roof on a mast located 30 m from the solar platform (Fig.2). Two components of the wind speed and temperature are collected at a sampling rate of 1 Hz. A 3D sonic anemometer is located very close to the solar panel (<1m from the heat fluxmeter). It records the instantaneous values of the three velocity components at a sampling rate of 1 Hz.

Data is pre-processed to obtain a homogeneous set. The data is initially averaged over a one minute interval to synchronize the measurement at a frequency of 1/60 Hz. Subsequently, a 10 minute rolling average is applied, resulting in a 10 minute-averaged sample every minute. In this study, data collection occurred on specific days chosen from three seasons during 2023 (spring, summer, and autumn), each encompassing a 24-hour period. These days were intentionally selected to coincide with the sunniest and warmest conditions, creating an optimal environmental setting for evaluating the measurement methodology.

## 2.2 Measurement Protocol for Convective HTC and Validation Technique

The convective HTC is measured at the module's center, assuming that this local measurement is close to the value averaged over the entire PV module. The measurement method relies on a fluxmeter sensor already described in [14], utilizing the tangential gradient flux measurement technique [22], and positioned on a module at the center of the platform.

The total heat flux measured by the sensor ($q_{fluxm}$) on the front surface is the sum of both a radiative ($q_{rad}$) and a convective part ($q_{conv}$):

$$q_{fluxm} = q_{rad} + q_{conv} \qquad (1)$$

The convective HTC ($h_{conv}$) is defined by Eq.(2):

$$q_{conv} = h_{conv}(T_{fluxm} - T_{air}) \qquad (2)$$

where $T_{fluxm}$ is the temperature of the sensor and $T_{air}$ is the air temperature measured 2 meters above the roof and 30 meters from the solar panels by weather station A. To estimate the radiative part of the total heat flux, it is assumed that the surface of the sensor is diffuse and gray over the both the short wave and the long wave range. Under these assumptions, the net radiative heat flux is calculated by Eq.(3).

$$q_{rad} = \varepsilon \sigma T_{fluxm}^4 - (\alpha_{sw}GTI_{sw} + \alpha_{lw}GTI_{lw}) \qquad (3)$$
where $\sigma = 5.67 \times 10^{-8} \text{W} \cdot \text{m}^{-2} \cdot \text{K}^{-4}$ is the Stephan-Boltzmann constant, and $\varepsilon$ is the total emissivity of the heat fluxmeter. As the emission in the short wave range is negligible, one can assume that $\varepsilon = \alpha_{lw} = 0.54$. Therefore, $q_{rad}$ is estimated from measurements, and it is subtracted from the total heat flux to obtain the convective heat flux and finally the convective HTC.

This convective HTC measuring method is validated by an experimental approach using an instrumented plate. It consists of a 200×1.900 mm instrumented plate made up of a 50 mm-thick insulating board covered with 50 µm black aluminum tape (Fig.3a). A heat fluxmeter is positioned on the reverse of the tape to measure the heat rate through the insulating board ($q_{HF,back}$). A thermocouple is placed between the heat fluxmeter and the aluminum tape. The test plate is positioned between two PV modules (Fig.3b).

<figure><img src="image_5.png" /><figcaption>Fig. 3: Experimental device to validate the measurement of the convective HTC. (a) measuring principle; (b) location of the instrumented plate in the setup.</figcaption></figure>

The convective HTCs on both the instrumented plate and the solar panel are supposed to be equal. For the system depicted in Fig.3, the temperature of the aluminum tape is supposed to be homogeneous, with the energy equation described in Eq.(4):

$$ \rho e C_p \frac{dT_p}{dt} = -h_{conv}(T_p - T_{air}) - \varepsilon \sigma T_p^4 + \alpha_{lw} GTI_{lw} + \alpha_{sw} GTI_{sw} - q_{HF,back} \quad (4) $$

where $\rho$, $e$, $C_p$, $T_p$ are the density, thickness, thermal capacity, and temperature of the aluminum tape, respectively. Therefore, the coefficient $h_{conv}$ can be obtained from the measurements at each time.

By installing the instrumented plate shown in Fig.4, the convective heat transfer coefficient (HTC) obtained from the fluxmeter method is compared with the reference value provided by the instrumented plate for a 1-day sample at solar noon in Fig.5. The measurements obtained by the heat flux meter ($h_{HFM,raw}$) are close to the values obtained by the instrumented plate ($h_{valid}$) but the difference may amount to as much as 30%. Therefore, corrections are applied to $h_{HFM,raw}$.
<figure><img src="image_6.png" /><figcaption>Fig. 4: Photo showing the instrumented plate used for the validation method, positioned next to the module undergoing heat fluxmeter measurement.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 5: Time evolution of the convective coefficients: The raw values obtained by the fluxmeter method (gray) and the reference values measured by the instrumented plate (green) are displayed, along with the corrected values from the fluxmeter method (black and red), for a 1-day dataset at solar noon.</figcaption></figure>

Upon visual inspection, it is evident that the sensor's paint has deteriorated with an accumulation of dust. Therefore, the error is first attributed to the absorptivity value in the short wave range.

Let $\delta\alpha_{sw}$ be the gap between the real value of the absorptivity ($\alpha_{sw, True}$) and the value measured on the freshly painted sensor ($\alpha_{sw}$):

$$ \alpha_{sw} = \alpha_{sw, True} + \delta\alpha_{sw} \qquad (5) $$

This error leads to a variation in the net radiative heat flux:

$$ \begin{aligned} q_{rad} &= q_{rad, True} + \frac{\partial q_{rad}}{\partial \alpha_{sw}} \delta\alpha_{sw} \\ &= q_{rad, True} - GTI_{sw} \delta\alpha_{sw} \end{aligned} \qquad (6) $$

where Eq. (3) has been used. Then, considering Eqs. (1) and (2), Eq. (6) leads to:

$$ h_{HFM,raw} = \frac{q_{fluxm} - q_{rad, True}}{T_{fluxm} - T_{air}} + \frac{GTI_{sw} \delta\alpha_{sw}}{T_{fluxm} - T_{air}} \qquad (7) $$

where $h_{HFM,raw}$ is the convective HTC without correction. As a result, the corrected value of the convective HTC is obtained ($h_{HFM,\delta\epsilon_{sw}}$):

$$ h_{HFM,\delta\epsilon_{sw}} = h_{conv,raw} - \frac{GTI_{sw} \delta\alpha_{sw}}{T_{fluxm} - T_{air}} \qquad (8) $$

To find the error $\delta\alpha_{sw}$, the values of the convective HTC $h_{HFM,raw}$ shown in Fig. 5 are computed with different values of absorptivity to match the measurements obtained
<figure><img src="image_8.png" /></figure>

**Fig. 6:** Convective HTC measured using both the heat fluxmeter and the instrumented plate method as a function of wind speed, measured by the 3D anemometer, highlighting the effect of the correction for the paint absorptivity.

<figure><img src="image_9.png" /></figure>

**Fig. 7:** Comparison of the HTC correlation with wind speed (U) for the instrumented plate and the two corrected values from the fluxmeter method.

by the instrumented plate. An error $\delta\alpha_{sw}$ of 0.06 is obtained. Therefore, the raw value of the convective coefficient measured by the heat flux sensor method can be corrected by adjusting the value of the absorptivity $\alpha_{sw}$ from 0.4 to 0.34. Fig. 5 clearly shows that correcting the shortwave absorptivity adds an offset to the raw measurement, significantly reducing bias. As the objective of this work is to analyze the relationship between the convective heat transfer coefficient (HTC) and wind speed, Fig. 6 presents the reference value, raw value, and corrected value of the convective HTC as a function of wind speed. As expected, the corrected value is closer to the reference values. However, the dispersion remains significant compared to the values obtained with the instrumented plate.

By comparing the reference value $h_{valid}$ with the corrected value $h_{HFM,\delta\alpha_{sw}=0.06}$ on Fig. 5, a phase shift between the green and black curves can be observed. A 9-minute time delay in convective HTC measurements between the heat fluxmeter method and the instrumented plate is observed, most likely due to the differences in thermal mass between the PV panel and the instrumented plate. The convective heat transfer coefficient (HTC) is calculated from five measurements: $T_{fluxm}$, $T_{air}$, $q_{fluxm}$, $G_{sw}$ and $G_{lw}$ (Eqs. (1)-(3)). The temperature sensor closely tracks the module temperature as a result of excellent thermal contact, but this leads to a significant response time caused by the module's thermal inertia. In contrast, the heat fluxmeter, which has a lower thermal capacity, responds more quickly. The 50 µm thick aluminum tape of the instrumented plate provides a reference measurement $h_{valid}$ without such bias. Consequently, under unsteady conditions, a phase shift occurs between the convective HTC measured by the heat fluxmeter and the reference value from the instrumented plate, potentially increasing dispersion. To mitigate this effect, the convective HTC $h_{HFM,\delta\alpha_{sw}=0.06}$ is advanced by 9 min and the resulting convective HTC $h_{HFM,\delta\alpha_{sw}=0.06,\Delta t=9 \text{ min}}$ is
shown in Fig. 5. Concerning the effect of this correction on the scatter plot $h_{conv}(U)$, Fig. 7 shows the reference value $h_{valid}$, and the two corrected values ($h_{HFM,\delta\alpha_{sw}=0.06}$, $h_{HFM,\delta\alpha_{sw}=0.06,\Delta t=9 \text{ min}}$) of the convective HTC as a function of wind speed. The green line corresponds to the correlation $h_{valid} = 2.5U + 6$ obtained on the reference values. The second correction clearly reduces the dispersion that can be quantified by calculating the standard deviation of the difference between experimental data and the correlation. Results are shown in Table 2. Clearly, the second correction enables a

<table><caption>**Table 2:** Standard deviation of the difference between the convective HTC measurements and the correlation $h_{valid} = 2.5U + 6$. Results are expressed in W m$^{-2}$ K$^{-1}$</caption><thead><tr><th>$h_{valid}$</th><th>$h_{HFM,\delta\alpha_{sw}=0.06}$</th><th>$h_{HFM,\delta\alpha_{sw}=0.06,\Delta t=9 \text{ min}}$</th></tr></thead><tbody><tr><td>0.18</td><td>1.08</td><td>0.47</td></tr></tbody></table>

reduction in the dispersion. To further mitigate the effect of inertia, data was selected during periods of maximum direct solar radiation received by the panel, corresponding to times with minimal time variations on sunny days. For simplicity's sake, this time is referred to hereafter as *solar noon*, although it does not correspond exactly to the maximum azimuth of the sun, since the panel is not horizontal. Despite the effect of wind, which continues to induce temperature variations, this condition tends to limit the bias on the measured convective HTC. In the rest of the article, the value of the convective HTC is corrected by applying a correction on the absorptivity in the short wave range and a phase shift on the measured convective HTC denoted $h_{conv}$.

## 2.3 Dataset Presentation

The analysis of the HTC's evolution is based on data collected on eleven specific days for each season: spring (March & May 2023), summer (June 2023), and autumn (September 2023). These days provide a favorable environmental context for evaluating the measurement methodology, characterized by sunny conditions with minimal cloud passage, as observed from visualizations at 2-second intervals. This is especially relevant during solar noon, where the maximum short wave solar irradiation $G_{sw}$ reached 758 W/m, with a standard deviation of 109 W/m. This approach helps reduce the effect of measurement inertia discussed in the previous section. A comprehensive dataset is then compiled for the entire year 2023, incorporating the data from these three seasons. The acquisition loop is configured to record data every minute (Section 2.2). All measurements undergo filtering using a Gaussian low-pass filter with a typical time of 10 min.

However, as with any continuously collected data, the datasets need to be checked before processing in order to eliminate data biased by spurious effects such as shading and condensation. The pre-processing applied to the dataset is shown below for a subset of 4 days collected in September 2023 illustrated in Fig. 8.
<figure><img src="image_10.png" /><figcaption>**Fig. 8:** Temporal evolution of dataset point of September 2023. The horizontal line in subplot θ corresponds to the condition given by Eq. (9b). The highlighted plots corresponds to the filter applied to the data given by Eqs. (9). Orange bands indicate solar noon (Eqs. (10a)). (The legends are detailed in the text).</figcaption></figure>

In Fig. 8, $h_{conv}$ is the convective HTC, $U$ is the magnitude of the wind speed measured by the 3D sonic anemometer, $G_{sw}$ is the global solar radiation in the plane of the panel, $\theta$ is the incidence angle on the panel, $q_{HFM}$ is the total heat flux measured by the sensor, $q_{conv}$ is the convective heat flux, $q_{rad}$ is the radiative heat flux, $T_{air}$ is
the temperature of the air measured by the weather station, $T_{DP}$ is the dew point, $T_{HFM}$ is the temperature of the sensor and RH is the relative humidity of the air.

To analyze the shading of the panel under study, a simulation of the test bench using SketchUp software is carried out shown in Fig. 9. This figure shows the time at which the sensor is illuminated in the morning on a given day. A similar analysis is carried out at sunset, and the range of the angle of incidence $\theta$ of direct solar radiation corresponding to an illuminated sensor can be identified. Considering that the incidence angle is defined with respect to the normal of the panel, the range is defined by $\theta \le \theta_s$ where $\theta_s$ evolves during one year. For example, the sensor is illuminated at 8 a.m. on September 1, 2023. At this time, the elevation of the sun is 9°11' and the azimuth measured clockwise from the north is 87°49' while the incident angle of the direct radiation on the panel is 69°. For the sake of simplicity, $\theta_s$ is set at 73° for all seasons and shown in Fig. 8a. Similarly, data corresponding to a solar azimuth angle $\psi_s$ between 230° and 260° are suppressed to account for anemometer shading (not shown in Fig. 8).

<figure><img src="image_11.png" /><figcaption>Fig. 9: Example of shading analysis on September 1, 2023: the sensor is illuminated at 8 a.m. N indicates the north. Scheme created with SketchUp</figcaption></figure>

Another parasitic effect comes from condensation, which can appear early in the morning. When condensation occurs, the heat flow measured by the sensor is mainly due to the evaporation of the drops, which act as discretely distributed heat sinks, disrupting the measurement. This effect can be shown on Fig. 8 where the temperature of the sensor becomes equal to the dew point. However, the measurement is biased by condensation until the sensor is completely dry, a process whose duration cannot be predicted. Consequently, these data are filtered out based on relative humidity. Indeed, condensation is found to occur for high levels of relative humidity, and data with a relative humidity greater than 75% are filtered out. Finally, a threshold for $\Delta T = T_{HFM} - T_{air}$ of 5°C is set, as the convective HTC is inversely proportional to $\Delta T$.
The different filters applied to the dataset are given in Eqs. (9):

$$RH < 75\%, \qquad (9a)$$

$$\theta < 73^{\circ}, \qquad (9b)$$

$$\psi_s < 230^{\circ} \quad \& \quad \psi_s > 260^{\circ}, \qquad (9c)$$

$$\Delta T > 5^{\circ}\text{C}. \qquad (9d)$$

To limit the bias in convective HTC measurement due to inertia problems in an unsteady environment (see end of Section 2.2), the correlations given in this article are limited to solar noon where the short wave radiation are almost constant during sunny days. As the sun's maximum azimuth angle changes over the course of the year, the condition used to define solar noon is different for the three seasons (Eqs. (10)).

$$\theta < 55^{\circ}, \text{Autumn September 2023} \qquad (10a)$$

$$\theta < 40^{\circ}, \text{Summer June 2023} \qquad (10b)$$

$$\theta < 55^{\circ}, \text{Spring March 2023} \qquad (10c)$$

$$\theta < 40^{\circ}, \text{Spring May 2023} \qquad (10d)$$

Therefore, for results obtained at solar noon, Eqs. (10) is applied after the pre-processing techniques defined in Eqs. (9). After analyzing the data for the entire year 2023 and applying appropriate filters (Eqs. (9)) to eliminate errors from condensation and shadows, a total of 13,142 data points were collected during this period (approximately 1 data point per minute for 6.6 hours on 33 days), and a total of 8,514 at solar noon.

# 3 Results

## 3.1 Seasonal Analysis

In this section, the results for the data collected in spring, summer, and autumn 2023 are presented. To describe the environmental conditions, Figs. 10 show the histograms of the mean wind speed and those of wind direction (measured clockwise from a southerly direction), highlighting two main directions: in orange the directions parallel to the surface of the panels ($D_1: S \to N =160^{\circ}$ and $D_1: N \to S =340^{\circ}$), and in green the directions perpendicular to $D_1$ ($D_2: E \to W =70^{\circ}$, $D_2: W \to E =250^{\circ}$).

At first glance, the prevailing wind is southerly. Fig. 10a reveals variations in the range and amplitude distribution of wind speeds. In Spring and Autumn, the wind speed distribution is concentrated between 1 and 2 m s⁻¹, whereas in summer, the distribution tends to be flat between 1 and 2.8 m s⁻¹. In terms of direction, the autumn distribution is tightly packed around $D_1: S \to N$, while the spring and summer distributions show wind occurrences in the transverse direction $D_2$.
<figure><img src="image_12.png" /><figcaption>Fig. 10: Wind characteristics measured for 3 seasons. (a) Histogram of the magnitude. (b) Histogram of the direction measured clockwise from the south. The colored area corresponds to specific directions shown in Fig. 2 top-left.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 11: HTC versus wind speed for the three seasons. The filtered dataset is represented in black, with the linear correlation in blue. The selected set at solar noon is depicted in orange, and its corresponding correlation in red. a and b are the coefficients defined by Eq. (11), and R² indicates the goodness of fit of the regression analysis.</figcaption></figure>

To establish a classical linear correlation, the convective HTC is plotted against wind intensity measured by the 3D sonic anemometer in Fig. 11. The linear correlation is written:

$$h_{\text{conv}} = aU + b \qquad (11)$$

where, a and b are defined for velocity and convective HTC, expressed in m s⁻¹ and W m⁻² K⁻¹, respectively. These coefficients are calculated based on pre-processed
data, as defined by Eqs. (9), and data collected specifically at solar noon, as defined by Eqs. (10).

This observation indicates that the solar noon data has fewer outliers compared to the filtered data, as the effect of inertia is reduced. The coefficient of determination, $R^2$, increases from 0.599 to 0.739 in spring, indicating a better model fit at solar noon. Although this coefficient may appear small, such values are typical in outdoor conditions. Indeed, for data at solar noon, the widely used Faiman's correlation yields a coefficient of determination of 0.63 [11].

However, in summer and autumn, the coefficients of determination are smaller, particularly for the filtered data, which exhibits outliers. In 2023, these seasons were particularly cloudy in France, making it difficult to find fully sunny days, which may explain the presence of outliers. Moreover, the presence of outliers may significantly change the coefficients of the linear correlation, thus lowering the degree of confidence with regard to the correlation. Therefore, the analysis of these different subsets, which all have the same length, highlights the difficulty of building a robust correlation under varying conditions. The calculation of the correlation at solar noon minimizes this effect, which can be attributed to the small variation in solar radiation at that time, which tends to reduce the bias due to thermal inertia despite the variations due to wind fluctuations that remain present. However, at solar noon, the slope of the correlation varies by approximately 18% between seasons, whereas the coefficient *b* shows an even more significant variation of about 40%. These differences may be attributed to the effects of turbulence intensity in the vicinity of the module, which must be explored in future works.

## 3.2 Yearly Analysis

Seasonal datasets are combined into a single dataset representing the whole year.

Fig. 12a presents the linear correlation for both the filtered dataset and the solar noon dataset. The results show that both correlations yield similar coefficients: (a = 2.4, b = 5.7, R = 0.405) for the filtered dataset, and (a = 2.8, b = 5, R = 0.575) for the solar noon dataset. The determination coefficient shows that measurements around solar noon provide more reliable correlation since they correspond to more stable environmental conditions, as was already observed in the seasonal analysis. All correlations at solar noon conditions are reported in Table 3. The differences in correlations is a mean-

<table><caption>**Table 3:** Linear laws at solar noon, determination coefficient and wind speed range for the 3 seasons and the whole dataset.</caption><thead><tr><th>Period</th><th>Correlation</th><th>R<sup>2</sup></th><th>Wind speed range</th></tr></thead><tbody><tr><td>Spring</td><td>h<sub>conv</sub> = 3.0U + 4.1</td><td>0.74</td><td>[0.7 - 5.3]m.s<sup>-1</sup></td></tr><tr><td>Summer</td><td>h<sub>conv</sub> = 2.8U + 5.1</td><td>0.42</td><td>[0.6 - 3.4]m.s<sup>-1</sup></td></tr><tr><td>Autumn</td><td>h<sub>conv</sub> = 2.5U + 6.1</td><td>0.48</td><td>[0.7 - 4.3]m.s<sup>-1</sup></td></tr><tr><td>Year</td><td>h<sub>conv</sub> = 2.8U + 5.0</td><td>0.58</td><td>[0.6 - 5.3]m.s<sup>-1</sup></td></tr></tbody></table>
<figure><img src="image_14.png" /><figcaption>**Fig. 12:** Convective HTC against wind speed for data in the year 2023. (a) The entire dataset is represented in black, the selected set at solar noon is highlighted in orange. (b) Main directions are highlighted (see Fig. 2).</figcaption></figure>

sure of the uncertainties associated with these simple models $h_{conv}(U)$. Table 3 clearly shows that the choice of dataset significantly influences the correlation coefficients. As explained in Section 3.1, the presence of outliers greatly affects the reliability of the correlation. However, other factors can also account for discrepancies between correlations. For instance, each dataset is subject to different meteorological conditions. In the case of the summer dataset, the correlation is calculated over a narrow wind speed range, which may explain the correspondingly low determination coefficient.

To go further in the analysis, Fig. 12b highlights data corresponding to the specific directions $D_1$ and $D_2$ during solar noon for the whole dataset. As $D_1: S \rightarrow N$ is close to the direction of the prevailing wind (Fig. 10b), most of the points shown in Fig. 12b correspond to this direction. In the transverse direction $D_2$, the data is analyzed depending on the wind direction, either $E \rightarrow W$ or $W \rightarrow E$. The wind from $D_2: W \rightarrow E$ more or less follows the same trend, while points corresponding to the direction $D_2: E \rightarrow W$ deviate significantly. Indeed, unlike in direction $D_1$, the wind orientation could have an impact in direction $D_2$. When the wind blows from $D_2: W \rightarrow E$ (Fig. 2), the sensor is positioned in a leeward location, whereas when it blows from $D_2: E \rightarrow W$, the sensor is located in the wake of the other row of panels. Data for these two orientations are depicted in Fig. 12b.

The convective heat transfer coefficient (HTC) in the $D_2: W \rightarrow E$ orientation appears to be slightly higher than in the opposite orientation. In the $D_2: W \rightarrow E$ case, the wind speed remains moderate, and the panel's inclination angle is small, most likely enabling the dynamic boundary layer to remain attached to the panel. As a result, the air velocity above the sensor remains close to that measured just above the panel. Conversely, when the wind blows from $D_2: E \rightarrow W$, the sensor is in the wake of the other row of panels, causing the mean wind speed above the sensor to be lower than that measured by the sonic anemometer, which leads to a lower convective
HTC. However, the number of data points for direction $D_2$ is limited, and more data
is needed to validate these observations.

<figure><img src="image_15.png" /><figcaption>**Fig. 13:** Comparison with other correlations in
the literature.</figcaption></figure>

Fig. 13 compares the correlation obtained in the present study (represented by the blue line) with several existing correlations from the literature on convective heat transfer coefficients presented in [16]. The correlation from the present study aligns well with the results of Wattmuff et al. [23], derived from wind tunnel experiments, though there is a slight difference in the slope. The correlation derived in this study compares well with the correlation in Test et al. [24] which is also obtained in outdoor conditions. It differs significantly from the correlations given by Sparrow et al. [25], which is derived from boundary layer theory, as well as from the simplified exponential theoretical relation of Sartotri [15]. The correlations in Kumar et al. [26], Test et al. [24], Sharples et al. [27], and Mc Adams [28] are all linear, based on various experimental setups, including indoor, outdoor, in situ, and wind tunnel measurements.

However, comparing several studies is a tricky endeavor. Indeed, as shown in the
seasonal analysis (Sec. 3.1), the results depend on the range covered by the environ-
mental parameters within the dataset. For example, the autumn data subset (Fig. 10b)
gives little detail as to when the wind blows in the transverse direction $D_2$, conse-
quently, the range of validity of the correlation calculated from this subset of data
should be limited to the main direction $D_1$. Therefore, to qualify the accuracy of a
given correlation, a description of the dataset is required. The number of points and
the wind amplitude and direction histograms help to define the validity range of the
correlation. Concerning the correlation for the whole dataset (Table 3), even though
the transverse direction $D_2$ is explored in summer, most of the points correspond to
a wind blowing in direction $D_1$. Consequently, the validity of the correlation is prob-
ably limited to wind parallel to the panel surface. Finally, the comparison with other
results (Fig. 13) shows that the discrepancy observed between the subsets (Fig. 11) is similar to the dispersion between the correlations found in the literature. Indeed, the correlation found in autumn (Table 3) is similar to that given by Test et al. [24]: $h_{conv} = 2.56U + 8.55$. In the same way, the correlation found in Spring compares well with that of Wattmuf et al. [23]: $h_{conv} = 3.0U + 2.8$. As a consequence, the measurement method derived from the use of a heat fluxmeter enables the retrieval of a correlation $h_{conv}(U)$ that compares well with the results found in the literature and therefore validates the measurement method. However, such simple correlations fail to take into account the dispersion observed in all experiments and more research is needed in order to identify the parameters that drive this dispersion.

# 4 Conclusions and Perspectives

This study analyzes the methodology for *in situ* measurement of the convective heat transfer coefficient on a photovoltaic module under outdoor conditions using a heat flux sensor. The analysis reveals that the measurements are affected by time drift due to dust accumulation on the surface of the sensor and that the thermal inertia of the module induces a phase shift. These biases have been corrected and the results are validated through comparison with an instrumented plate.

For the comprehensive dataset at solar noon, the HFM method effectively estimates the following correlation for the convective heat transfer coefficients for specific sunny day conditions ($h_{conv} = 2.8U + 5.0$) with a determination coefficient ($R^2 = 0.575$). However, distinct correlations are observed in three subsets (corresponding to three seasons) and these differences are consistent with the dispersion observed in the literature. This suggests that the dataset used to construct the correlations are of major importance, meaning further investigations are required to determine the influence of other parameters. Hence, an initial insight with regard to the influence of the wind direction is given, but in the experiment used in this study, the prevailing wind direction significantly shapes heat transfer dynamics, meaning it is difficult to draw conclusions as to the influence of wind direction.

Looking ahead, data-driven methods, such as clustering, hold promise for identifying configurations where specific correlations may apply, and this approach will be further explored. In particular, clustering methods could help to identify a specific parameter or a group of parameters that drive heat transfer in order to develop more accurate correlations.

# Acknowledgements

The authors would like to thank Fabien Mandorlo of the Institut des Nanotechnologies de Lyon and Agnès Delmas of CETHIL for their spectrometry measurements.

## Open Access

For the purpose of Open Access, a CC-BY public copyright license has been applied by the authors to the present document and will be applied to all subsequent versions up to the Author Accepted Manuscript arising from this submission.
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