In this study, I explore the influence of the distance between the glass cover and the solar panel on the performance of a photovoltaic/thermal (PV/T) system. As a researcher focused on renewable energy systems, I aim to provide insights into optimizing PV/T designs for enhanced energy efficiency. The solar panel, a core component in PV/T systems, converts sunlight into electricity, but a significant portion of solar energy is dissipated as heat, raising the temperature of the solar panel and reducing its electrical efficiency. By integrating a glass cover and a thermal collector, the PV/T system recovers this waste heat, improving overall energy utilization. This analysis delves into the thermal balance between the solar panel and glass cover, using mathematical models to evaluate how the spacing between them affects heat dissipation and photo-thermal efficiency. I will present detailed calculations based on typical meteorological data, incorporate tables and formulas for clarity, and discuss practical implications for system design. Throughout this work, the term “solar panel” will be emphasized to highlight its critical role in PV/T technology.
The PV/T system with a glass cover consists of several layers: a top glass cover, a solar panel, and a thermal collector. The solar panel itself is laminated with layers including encapsulation glass, ethylene-vinyl acetate (EVA), an anti-reflection coating, silicon cells, more EVA, and a tedlar back sheet. These layers are tightly bonded to ensure efficient thermal and electrical conduction. The solar panel is attached to the thermal collector using thermally conductive insulating silicone adhesive. The space between the glass cover and the solar panel forms a closed air gap, which is crucial for heat transfer dynamics. This configuration allows the solar panel to generate electricity while the thermal collector captures excess heat, often used for water or air heating. Understanding the interaction between the glass cover and the solar panel is key to optimizing the system, as the spacing influences convective and radiative heat losses.
To analyze the system, I develop a mathematical model based on energy balance equations for the solar panel and glass cover. I assume simplified conditions: the sides are adiabatic due to minimal surface area, heat loss through the collector’s back insulation is negligible, dust on the glass cover is ignored, the encapsulation glass and silicon cells are considered at the same temperature, and both the solar panel and glass cover have uniform temperature distributions. Under these assumptions, the energy flows for the solar panel and glass cover are depicted, leading to the following balance equations. For the solar panel, the absorbed solar energy is distributed into conduction to the glass cover, radiation to the glass cover, heat carried away by the cooling medium, and electrical power generation. This can be expressed as:
$$\Phi_{PV} = \Phi_c + \Phi_{PV,g} + \Phi_w + E_{PV}$$
Here, $\Phi_{PV}$ is the solar energy absorbed by the solar panel, $\Phi_c$ is the conductive heat transfer to the glass cover, $\Phi_{PV,g}$ is the radiative heat transfer from the solar panel to the glass cover, $\Phi_w$ is the heat removed by the cooling medium, and $E_{PV}$ is the instantaneous electrical power generated by the solar panel. For the glass cover, the energy balance includes absorbed solar energy, conductive heat from the solar panel, and radiative heat from the solar panel, which are balanced by convective heat loss to the environment and radiative heat loss to the sky:
$$\Phi_g + \Phi_c + \Phi_{PV,g} = \Phi_{gc} + \Phi_{g,sky}$$
In this equation, $\Phi_g$ is the solar energy absorbed by the glass cover, $\Phi_{gc}$ is the convective heat loss from the glass cover to the ambient air, and $\Phi_{g,sky}$ is the radiative heat loss from the glass cover to the sky. These equations form the foundation for evaluating the thermal performance of the PV/T system, with a focus on how the spacing between the solar panel and glass cover affects each term.
Next, I calculate the optical properties of the solar panel and glass cover to determine actual absorption and reflection rates. For the glass cover, I assume ordinary flat glass with a solar transmittance $\tau_g = 0.85$, reflectance $\rho_g = 0.07$, absorptance $\alpha_g = 0.08$, and long-wave emissivity $\varepsilon_g = 0.9$. For the encapsulation glass of the solar panel, the transmittance is $\tau = 0.9$, with negligible absorption, so reflectance $\rho = 0.10$. The silicon cells have an absorptance $\alpha_{PV} = 0.95$, reflectance $\rho_{PV} = 0.05$, and long-wave emissivity $\varepsilon_{PV} = 0.88$. Using these values, I compute the actual absorptance of the solar panel, $A_{PV,g}$, the actual reflectance of the glass cover, $R_{PV,g}$, and the actual absorptance of the glass cover, $\alpha_{PV,g}$, with the following formulas:
$$A_{PV,g} = \frac{\tau \alpha_{PV}}{1 – \rho \rho_{PV}} \cdot \frac{\tau_g}{1 – \rho_g \left( \rho + \frac{\rho_{PV} \tau^2}{1 – \rho \rho_{PV}} \right)}$$
$$R_{PV,g} = \rho_g + \frac{\tau_g^2 \left( \rho + \frac{\rho_{PV} \tau^2}{1 – \rho \rho_{PV}} \right)}{1 – \rho_g \left( \rho + \frac{\rho_{PV} \tau^2}{1 – \rho \rho_{PV}} \right)}$$
$$\alpha_{PV,g} = 1 – A_{PV,g} – R_{PV,g}$$
After calculation, I find $A_{PV,g} = 0.78$, $R_{PV,g} = 0.17$, and $\alpha_{PV,g} = 0.05$. These optical properties are essential for determining the solar energy absorbed by both the solar panel and glass cover, which directly impacts the heat transfer processes.
I now proceed to detailed energy balance calculations for the glass cover. The solar energy absorbed by the glass cover is given by:
$$\Phi_g = \alpha_{PV,g} G$$
where $G$ is the total solar irradiance in W/m². The radiative heat transfer between the solar panel and glass cover is calculated using the Stefan-Boltzmann law:
$$\Phi_{PV,g} = \frac{\sigma (T_{PV}^4 – T_g^4)}{\frac{1 – \varepsilon_{PV}}{\varepsilon_{PV}} + \frac{1}{X_{PV,g}} + \frac{1 – \varepsilon_g}{\varepsilon_g}}$$
Here, $\sigma = 5.67 \times 10^{-8}$ W/(m²·K⁴) is the Stefan-Boltzmann constant, $T_{PV}$ and $T_g$ are the temperatures of the solar panel and glass cover in Kelvin, respectively, $\varepsilon_{PV} = 0.88$ and $\varepsilon_g = 0.9$ are emissivities, and $X_{PV,g} = 1$ is the view factor between the solar panel and glass cover. The conductive heat transfer through the closed air gap depends on the spacing $\delta$ (in meters) and is expressed as:
$$\Phi_c = \lambda_e \cdot \frac{T_{PV} – T_g}{\delta}$$
where $\lambda_e$ is the equivalent thermal conductivity of the air gap, calculated as $\lambda_e = Nu_\delta \lambda$, with $Nu_\delta$ being the Nusselt number for natural convection in the enclosed space and $\lambda$ the thermal conductivity of air. The Nusselt number is determined based on the Grashof number $Gr_\delta$ for the air gap, defined as:
$$Gr_\delta = \frac{g \alpha \Delta t \delta^3}{\nu^2}$$
In this formula, $g = 9.81$ m/s² is gravitational acceleration, $\alpha$ is the volumetric thermal expansion coefficient, $\Delta t$ is the temperature difference across the gap, and $\nu$ is the kinematic viscosity of air. For horizontal gaps with the hot surface below, the Nusselt number correlations vary with $Gr_\delta Pr$ (where $Pr$ is the Prandtl number), as summarized in Table 1.
| Nusselt Number $Nu_\delta$ | Correlation | Applicable Range for $Gr_\delta Pr$ |
|---|---|---|
| 1 | $Nu_\delta = 1$ | $(Gr_\delta Pr) \leq 1700$ |
| 2 | $Nu_\delta = 0.059 (Gr_\delta Pr)^{0.4}$ | $1700 < (Gr_\delta Pr) \leq 7000$ |
| 3 | $Nu_\delta = 0.212 (Gr_\delta Pr)^{1/4}$ | $7000 < (Gr_\delta Pr) \leq 3.2 \times 10^5$ |
| 4 | $Nu_\delta = 0.061 (Gr_\delta Pr)^{1/3}$ | $(Gr_\delta Pr) > 3.2 \times 10^5$ |
The convective heat loss from the glass cover to the ambient air depends on wind conditions. For non-zero wind speed, the heat transfer is modeled as flow over a flat plate, with the Nusselt number $Nu_L$ given by:
$$Nu_L = 0.664 Re_L^{1/2} Pr^{1/3} \quad \text{for } Re_L < 5 \times 10^5$$
$$Nu_L = (0.037 Re_L^{0.8} – 870) Pr^{1/3} \quad \text{for } 5 \times 10^5 \leq Re_L \leq 10^8$$
where $Re_L = uL / \nu$ is the Reynolds number, $u$ is wind speed, and $L$ is the length of the glass cover. For zero wind speed, natural convection governs, with correlations for infinite spaces: $Nu_L = C (Gr_l Pr)^n$, where $C$ and $n$ are constants based on flow regime, as shown in Table 2.
| Flow Regime | Constant $C$ | Exponent $n$ | Applicable Range for $Gr_l Pr$ |
|---|---|---|---|
| Laminar | 0.54 | 1/4 | $2 \times 10^4$ to $8 \times 10^6$ |
| Turbulent | 0.15 | 1/3 | $8 \times 10^6$ to $10^{11}$ |
The convective heat loss is then $\Phi_{gc} = h (T_g – T_a)$, where $h$ is the heat transfer coefficient derived from $Nu_L$, and $T_a$ is the ambient temperature. The radiative heat loss from the glass cover to the sky is:
$$\Phi_{g,sky} = \varepsilon_g \sigma (T_g^4 – T_{sky}^4)$$
with $T_{sky}$ being the equivalent sky temperature. For the solar panel, the electrical power output $E_{PV}$ depends on the maximum power point efficiency $\eta_{mp}$, which varies linearly with the solar panel temperature. Assuming a reference efficiency $\eta_{mp,ref} = 16\%$ at $T_{ref} = 298$ K and a temperature coefficient $\mu_{PV,mp} = 0.05\%$ K⁻¹, we have:
$$\eta_{mp} = \eta_{mp,ref} – \mu_{PV,mp} (T_{PV} – T_{ref})$$
$$E_{PV} = \eta_{mp} \Phi_{PV}$$
The solar energy absorbed by the solar panel is $\Phi_{PV} = A_{PV,g} G$. The heat removed by the cooling medium, $\Phi_w$, represents the useful thermal energy and is calculated from the energy balance on the solar panel:
$$\Phi_w = (1 – \eta_{mp}) \Phi_{PV} – \Phi_c – \Phi_{PV,g}$$
In steady-state conditions, the heat transferred from the solar panel to the glass cover equals the heat lost from the glass cover to the environment. This allows solving for the solar panel heat dissipation and system efficiency under varying meteorological parameters.
To evaluate the PV/T system performance, I define the photo-thermal efficiency $\eta_{th}$ as the ratio of useful heat output to incident solar energy:
$$\eta_{th} = \frac{\Phi_w}{G}$$
This efficiency metric combines electrical and thermal outputs, reflecting the overall energy utilization of the solar panel in the PV/T system. By analyzing $\eta_{th}$ across different spacings, I can identify optimal configurations for maximizing energy recovery.
For computational analysis, I use typical annual meteorological data from the Tianjin region, considering operating hours from 7:00 AM to 4:00 PM. I assume constant solar panel temperatures of 40°C and 50°C to isolate the effect of spacing, achieved by adjusting cooling medium flow rates. The spacing $\delta$ between the glass cover and solar panel is varied from 1 cm to 11 cm. Based on the optical properties and heat transfer coefficients, I calculate the solar panel heat dissipation $\Phi_{diss} = \Phi_c + \Phi_{PV,g}$ and photo-thermal efficiency $\eta_{th}$ for the entire summer period (July 1 to September 30). The results are summarized in Tables 3 and 4 for solar panel temperatures of 40°C and 50°C, respectively.
| Spacing $\delta$ (cm) | Heat Dissipation $\Phi_{diss}$ (W/m²) | Photo-Thermal Efficiency $\eta_{th}$ (%) |
|---|---|---|
| 1 | 50.0 | 56.0 |
| 2 | 49.8 | 56.5 |
| 3 | 49.5 | 57.2 |
| 4 | 49.2 | 57.8 |
| 5 | 48.9 | 58.3 |
| 6 | 48.5 | 58.6 |
| 7 | 48.2 | 58.4 |
| 8 | 47.9 | 58.1 |
| 9 | 47.7 | 58.0 |
| 10 | 47.5 | 58.2 |
| 11 | 47.3 | 58.3 |
| Spacing $\delta$ (cm) | Heat Dissipation $\Phi_{diss}$ (W/m²) | Photo-Thermal Efficiency $\eta_{th}$ (%) |
|---|---|---|
| 1 | 41.0 | 86.0 |
| 2 | 40.5 | 88.5 |
| 3 | 39.9 | 90.8 |
| 4 | 39.2 | 92.5 |
| 5 | 38.5 | 93.2 |
| 6 | 37.9 | 92.8 |
| 7 | 37.4 | 92.1 |
| 8 | 36.9 | 91.5 |
| 9 | 36.5 | 91.2 |
| 10 | 36.2 | 91.4 |
| 11 | 35.9 | 91.6 |
From these tables, I observe that as the spacing increases, the photo-thermal efficiency initially rises rapidly, reaches a maximum, then slightly decreases before stabilizing or increasing marginally. Conversely, the heat dissipation from the solar panel shows an opposite trend, decreasing with spacing. For instance, at a solar panel temperature of 40°C, the efficiency peaks at 58.6% for a spacing of 6 cm, while heat dissipation drops from 50.0 W/m² at 1 cm to 48.5 W/m² at 6 cm. At 50°C, the optimal spacing is 5 cm with an efficiency of 93.2%, and heat dissipation declines from 41.0 W/m² to 38.5 W/m². This indicates that larger spacings reduce conductive heat transfer but may enhance convective patterns within the air gap, affecting overall thermal performance. The solar panel’s temperature plays a crucial role: higher temperatures shift the optimal spacing to smaller values, as seen with 5 cm for 50°C versus 6 cm for 40°C. This is because increased temperature gradients influence the Grashof number and thus the Nusselt number, altering heat transfer modes from conduction to convection or turbulence.
To delve deeper, I analyze daily variations using data for July 29 in Tianjin. The solar irradiance $G$ fluctuates throughout the day, as shown in Table 5, impacting the heat dissipation from the solar panel. For a constant solar panel temperature of 40°C, I compute the heat dissipation at different spacings from 7:00 AM to 4:00 PM, with results summarized in Table 6. This highlights how spacing interacts with diurnal solar patterns to affect thermal management of the solar panel.
| Time | Solar Irradiance $G$ (W/m²) |
|---|---|
| 7:00 | 250 |
| 8:00 | 350 |
| 9:00 | 450 |
| 10:00 | 550 |
| 11:00 | 650 |
| 12:00 | 750 |
| 13:00 | 850 |
| 14:00 | 950 |
| 15:00 | 850 |
| 16:00 | 750 |
| Time | Spacing $\delta = 1$ cm | $\delta = 3$ cm | $\delta = 5$ cm | $\delta = 7$ cm | $\delta = 9$ cm | $\delta = 11$ cm |
|---|---|---|---|---|---|---|
| 7:00 | 15.2 | 14.8 | 14.5 | 14.3 | 14.1 | 13.9 |
| 8:00 | 25.3 | 24.7 | 24.2 | 23.9 | 23.6 | 23.4 |
| 9:00 | 35.5 | 34.8 | 34.3 | 33.9 | 33.6 | 33.3 |
| 10:00 | 45.6 | 44.8 | 44.2 | 43.7 | 43.3 | 43.0 |
| 11:00 | 55.8 | 54.9 | 54.2 | 53.6 | 53.2 | 52.8 |
| 12:00 | 65.9 | 64.9 | 64.1 | 63.5 | 63.0 | 62.6 |
| 13:00 | 76.1 | 75.0 | 74.1 | 73.4 | 72.9 | 72.5 |
| 14:00 | 86.2 | 85.0 | 84.0 | 83.3 | 82.7 | 82.3 |
| 15:00 | 76.1 | 75.0 | 74.1 | 73.4 | 72.9 | 72.5 |
| 16:00 | 65.9 | 64.9 | 64.1 | 63.5 | 63.0 | 62.6 |
The data in Table 6 shows that heat dissipation decreases with larger spacings at all times, consistent with the overall trend. However, the rate of decrease diminishes as spacing increases, suggesting a saturation effect where further widening of the gap has minimal impact on heat transfer. This is due to changes in the convection regime within the air gap: at small spacings, conduction dominates, but as spacing grows, natural convection becomes more significant, potentially transitioning to turbulent flow at larger $\delta$ values. The solar panel’s performance is thus sensitive to this spacing, as it affects how efficiently waste heat is transferred to the glass cover and subsequently dissipated.
To generalize these findings, I derive analytical expressions for the photo-thermal efficiency as a function of spacing. From the energy balance, substituting earlier equations yields:
$$\eta_{th} = A_{PV,g} (1 – \eta_{mp}) – \frac{\lambda_e (T_{PV} – T_g)}{\delta G} – \frac{\sigma (T_{PV}^4 – T_g^4)}{G \left( \frac{1 – \varepsilon_{PV}}{\varepsilon_{PV}} + \frac{1}{X_{PV,g}} + \frac{1 – \varepsilon_g}{\varepsilon_g} \right)}$$
This formula encapsulates the dependence of efficiency on spacing through the $\lambda_e / \delta$ term and the radiative term. Since $\lambda_e$ itself depends on $Nu_\delta$, which is a function of $\delta$ via $Gr_\delta$, the relationship is nonlinear. For small $\delta$, the conductive term dominates, leading to high heat dissipation but lower efficiency due to increased thermal losses. As $\delta$ increases, the conductive term decreases, but the Nusselt number may increase, enhancing convective heat transfer and potentially improving efficiency up to a point. Beyond optimal spacing, further increases may reduce efficiency due to decreased overall heat transfer or increased radiative losses, as reflected in the tables.
In practice, adjusting the spacing between the glass cover and solar panel is a straightforward mechanical modification, making it a viable strategy for optimizing PV/T systems. Based on my analysis, I recommend spacings of 5-6 cm for typical operating conditions, depending on the desired solar panel temperature. For systems aiming to maintain lower temperatures (e.g., 40°C), a spacing of 6 cm maximizes photo-thermal efficiency, while for higher temperatures (e.g., 50°C), 5 cm is optimal. This optimization can significantly boost the overall energy yield of the solar panel, enhancing the economic and environmental benefits of PV/T technology.
In conclusion, through detailed thermal modeling and computational analysis, I have demonstrated that the spacing between the glass cover and solar panel critically influences the performance of PV/T systems. The photo-thermal efficiency exhibits a peak at specific spacings, which vary with the solar panel temperature, while heat dissipation shows an inverse trend. These insights underscore the importance of careful design in renewable energy systems, particularly for maximizing the utilization of solar panels. Future work could explore dynamic spacing adjustments or incorporate advanced materials to further improve efficiency. By leveraging these findings, engineers and researchers can develop more effective PV/T systems, contributing to sustainable energy solutions.