The quest for sustainable building solutions has intensified, driven by the substantial global energy consumption attributed to the built environment. Harnessing solar energy through building-integrated systems presents a compelling pathway to reduce operational energy demands while improving indoor environmental quality. Among various passive and active solar architectural features, the Trombe wall concept has been extensively studied for its ability to provide space heating and natural ventilation. This article delves into the advanced integration of photovoltaic (PV) and photocatalytic functionalities within a Trombe wall structure, proposing a holistic, multi-parameter optimization framework. The primary aim is to maximize the synergistic outputs of electrical power generation, indoor air purification, and thermal energy delivery from a single, building-integrated solar system.
The proposed solar system, termed a Photovoltaic Photocatalytic Trombe Wall (PV-PTrombe), is a composite structure designed for multifunctional performance. It typically consists of four primary layers: an outer glazing, a photocatalytic coating applied to the inner surface of the glass, an air channel, and an absorber wall integrated with photovoltaic cells. This configuration allows for the simultaneous conversion of solar energy. The photocatalytic layer utilizes ultraviolet (UV) spectrum radiation to degrade volatile organic compounds (VOCs) like formaldehyde. The PV panel converts a portion of the broader solar spectrum into electricity, while the absorbed but unconverted energy heats the panel. This heat is transferred to the air in the adjacent channel, inducing a buoyancy-driven flow (natural convection) that draws polluted indoor air through the photocatalytic zone, delivers purified air back into the room, and provides space heating. The physical arrangement of this integrated solar system is crucial for its overall efficacy.
To analyze and optimize this solar system, a comprehensive mathematical model encompassing thermodynamic and kinetic analyses is established. The model is based on several simplifying assumptions: all surfaces are gray bodies and diffuse; material properties are constant; the photocatalytic coating only absorbs UV light; and the air in the channel is an ideal gas participating solely in convective heat transfer. The energy balance equations for the key components form the core of the thermodynamic model.
For the outer glazing, the energy balance accounts for convection with the outdoor environment, radiative exchange with the sky, convection with the channel air, radiation exchange with the PV panel, and absorption of incident solar radiation:
$$
q_g = h_{amb,g}(T_{amb} – T_g) + h_{sky,g}(T_{sky} – T_g) + h_{a,g}(T_a – T_g) + h_{pv,g}(T_{pv} – T_g) + \alpha_g G
$$
where $h$ represents heat transfer coefficients, $T$ is temperature, $\alpha_g$ is the glass absorptivity, and $G$ is the solar irradiance. The convective coefficient with ambient air ($h_{amb,g}$) considers wind speed, while the radiative sky coefficient ($h_{sky,g}$) uses the Stefan-Boltzmann constant $\sigma$ and emissivity $\varepsilon$. The convection within the channel ($h_{a,g}$) is calculated using the Nusselt number ($Nu$) correlation for natural convection in a vertical cavity, which depends on the Rayleigh number ($Ra$) and Prandtl number ($Pr$).
The energy balance for the air flowing in the channel includes convective heat gain from the glass and the PV wall, and the energy carried by the airflow itself:
$$
q_a = h_{g,a}(T_g – T_a) + h_{pv,a}(T_{pv} – T_a) – \rho_a u_a c_a \delta_a \frac{dT_a}{dH}
$$
Here, $u_a$ is the air velocity induced by buoyancy, $\rho_a$ is density, $c_a$ is specific heat, and $\delta_a$ is the channel width. The induced air velocity is a critical performance parameter for this solar system and is derived from the balance between buoyancy and flow resistance:
$$
u_a = \sqrt{ \frac{2 g \beta (T_{out} – T_{in}) H}{f_{in}(\frac{A_a}{A_{in}})^2 + f_{out}(\frac{A_a}{A_{out}})^2 + f(\frac{H}{d_h})} }
$$
where $g$ is gravity, $\beta$ is the thermal expansion coefficient, $H$ is the channel height, $f$ terms are friction loss coefficients, $A$ represents cross-sectional areas, and $d_h$ is the hydraulic diameter. The friction factor $f$ is calculated differently for laminar and turbulent flows.
The energy balance for the PV panel incorporates heat exchange with the glass and channel air, absorption of solar radiation (transmitted through the glass), and the removal of energy via electrical conversion:
$$
q_{pv} = h_{g,pv}(T_g – T_{pv}) + h_{a,pv}(T_a – T_{pv}) – G \tau_g \eta_e [1 – 0.0045(T_{pv} – 298.15)] + \tau_g \alpha_{pv} G
$$
In this equation, $\tau_g$ is glass transmittance, $\alpha_{pv}$ is PV panel absorptivity, and $\eta_e$ is the PV cell’s reference efficiency. The term $0.0045(T_{pv} – 298.15)$ captures the temperature-dependent efficiency loss characteristic of silicon-based PV cells in this solar system.
Concurrently, the kinetic analysis models the purification performance of the solar system. The steady-state mass balance for a pollutant like formaldehyde (HCHO) in the air stream is given by:
$$
\delta_a \frac{dC}{dH} = -u_a \frac{dC}{dH} + h_m (C_s – C)
$$
where $C$ is the formaldehyde concentration in the bulk air, $C_s$ is the concentration at the photocatalytic surface, and $h_m$ is the convective mass transfer coefficient, related to the Sherwood number ($Sh$). The surface reaction rate is governed by a Langmuir-Hinshelwood type kinetics, where the apparent reaction rate constant $k_{app}$ depends on UV irradiance ($G_{UV}$) and surface temperature ($T_s$):
$$
k_{app} = \frac{k’_{HCHO} G_{UV}^n e^{(-\frac{E_{HCHO}}{RT_s})} K’_{HCHO} e^{(-\frac{H_{HCHO}}{RT_s})}}{1 + K’_{HCHO} e^{(-\frac{H_{HCHO}}{RT_s})} C_s}
$$
The performance of this multifunctional solar system is evaluated using three key metrics: Useful Thermal Energy Gain ($Q$), PV Electrical Efficiency ($\eta$), and Clean Air Delivery Rate ($CADR$). These are defined as:
$$
Q = \rho_a u_a c_a \delta_a \frac{dT_a}{dH}, \quad \eta = \tau_g \eta_e [1 – 0.0045(T_{pv} – 298.15)], \quad CADR = 3600 \times u_a A_a \frac{C_{in} – C_{out}}{C_{in}}
$$
$CADR$ represents the volumetric flow rate of air that has been 100% purified of the target pollutant, indicating the purification capacity of the solar system.
The geometric design parameters, specifically the channel height ($H$) and width ($\delta_a$ or $W$), profoundly influence the performance of this solar system. A single-parameter analysis reveals their distinct effects, summarized in the table below.
| Performance Metric | Effect of Increasing Channel Height ($H$) | Effect of Increasing Channel Width ($W$) |
|---|---|---|
| Thermal Gain per unit area ($Q$) | Decreases. Outlet temp. increases but area increases more. | Exhibits a maximum. Initially increases with flow area, then decreases as temp. drop dominates. |
| PV Efficiency ($\eta$) | Decreases. Higher $H$ leads to larger PV area and higher operating temperature. | Exhibits a maximum. Optimal convection cooling exists at intermediate widths. |
| Purification Rate ($CADR$) | Increases. Higher $H$ increases buoyancy pressure and airflow rate. | Increases then plateaus. Larger flow area increases flow rate, but residence time and conversion may decrease. |
Given the competing trends, a multi-parameter, multi-objective optimization is essential to find the best compromise for the solar system. The objective is to maximize $Q$, $\eta$, and $CADR$ simultaneously according to a weighted priority. A coupled evaluation index $J$ is formulated as the objective function for minimization:
$$
J = \frac{1}{ f_1 \left( \frac{Q}{Q_m} \right) + f_2 \left( \frac{\eta}{\eta_m} \right) + f_3 \left( \frac{CADR}{CADR_m} \right) }
$$
Here, $Q_m$, $\eta_m$, and $CADR_m$ are the maximum achievable values for each metric under given conditions (found via single-objective optimization), and $f_1$, $f_2$, $f_3$ are weighting factors summing to 1, reflecting the desired importance of each function in the overall solar system (e.g., $f_1=0.25$, $f_2=0.35$, $f_3=0.4$ to emphasize air purification).
A simplified conjugate gradient method is employed for the optimization, with $H$ and $W$ as the design variables. The algorithm iteratively adjusts these parameters to find the minimum of $J$. The search direction $p_n^{(k)}$ for variable $n$ at iteration $k$ is updated using its gradient and a conjugate coefficient $\gamma_n^{(k)}$:
$$
p_n^{(k)} = \left( \frac{\partial J}{\partial x_n} \right)^{(k)} + \gamma_n^{(k)} p_n^{(k-1)}, \quad \text{where} \quad \gamma_n^{(k)} = \left[ \left( \frac{\partial J}{\partial x_n} \right)^{(k)} \right]^2 / \left[ \left( \frac{\partial J}{\partial x_n} \right)^{(k-1)} \right]^2
$$
The parameters are then updated as $x_n^{(k+1)} = x_n^{(k)} – \beta_n p_n^{(k)}$, where $\beta_n$ is a fixed step length. The process continues until $J$ converges within a specified tolerance.
The optimization was conducted for different indoor temperatures ($T_{room}$ = 18, 20, 22°C) under a constant solar irradiance ($G$ = 800 W/m²). First, single-objective optimizations were performed to determine the baseline maxima ($Q_m$, $\eta_m$, $CADR_m$). The results are shown below.
| Target Metric ($T_{room}$) | Optimal Value | Optimal Geometry (H, W) |
|---|---|---|
| $Q_m$ at 18°C | 148.20 W/m² | (1.12 m, 0.073 m) |
| $Q_m$ at 20°C | 138.82 W/m² | (1.00 m, 0.071 m) |
| $\eta_m$ at 20°C | 9.86 % | (1.41 m, 0.033 m) |
| $CADR_m$ at 20°C | 22.46 m³/h | (3.00 m, 0.078 m) |
Subsequently, the multi-objective optimization was run with the aforementioned weights. The algorithm consistently converged to similar optimal geometries regardless of the starting point, validating the robustness of the model for this solar system. The final optimized results for the multi-performance solar system are presented in the following table.
| Indoor Temp. $T_{room}$ (°C) | Optimal Geometry (H, W) | Optimized Performance (Q, η, CADR) | Coupled Index $J$ (vs. Initial) |
|---|---|---|---|
| 18 | (3.00 m, 0.073 m) | 138.94 W/m², 9.87%, 22.03 m³/h | 6.78% lower (improved) |
| 20 | (3.00 m, 0.075 m) | 130.75 W/m², 9.80%, 22.47 m³/h | 6.85% lower (improved) |
| 22 | (3.00 m, 0.077 m) | 122.44 W/m², 9.73%, 22.92 m³/h | 6.90% lower (improved) |
The results demonstrate clear trends. The optimal channel height consistently converges to the maximum allowed value (3 m). This is because $CADR$, which has the highest weighting factor ($f_3=0.4$), benefits significantly from increased height due to stronger buoyancy-driven flow. While increased height slightly reduces specific thermal gain ($Q$) and efficiency ($η$), the substantial boost in purification capacity dominates the coupled objective $J$, leading to this choice for the overall solar system performance.
The optimal channel width settles around 0.075 m. At this width, the solar system achieves a balance. It is wide enough to allow sufficient airflow to boost $CADR$ and provide adequate convective cooling to the PV panel (benefiting $η$), yet not so wide that the air temperature rise and thermal gain ($Q$) become excessively diminished or that photocatalytic contact time is severely reduced. This width represents a “sweet spot” for the multi-functional solar system.
In conclusion, the integrated PV-PTrombe solar system represents a promising technology for concurrent energy harvesting and indoor environmental remediation. Through systematic mathematical modeling and the application of a simplified conjugate gradient multi-objective optimization, the complex trade-offs between electrical efficiency, air purification rate, and thermal energy delivery can be effectively managed. The analysis reveals that the purification performance ($CADR$) is the most sensitive to geometric design and exerts the strongest influence on the overall coupled performance under the chosen weights. The optimization framework successfully identifies Pareto-optimal geometries that maximize the holistic output of the solar system, providing a valuable theoretical and practical foundation for the design and implementation of high-performance, building-integrated, multi-functional solar systems. This approach enables the tailoring of solar system parameters to specific climatic conditions and performance priorities, advancing the development of sustainable built environments.