## 3 Results and Discussion
### SEM Characterisation and Projections of Biomass Yields Obtainable from the System
When analysing the long-term projections (Supplementary Materials), the models consistently identified Exp3 as the most favourable operating condition, with higher biomass accumulation observed in wastewater (W) compared to the standard culture medium (A), as illustrated in Figure 9. This finding aligns with previous literature [13; 14]. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Treatment** & **Model** & **AICc** & **BIC** & **SSE** & **MSE** & **RMSE** & \(\mathbf{R}^{2}\) \\ \hline \multirow{4}{*}{A} & Second-degree polynomial & 678.35 & 736.57 & 142.71 & 0.47 & 0.69 & 0.96 \\ \cline{2-7}  & 3-parameter logistic & 773.64 & 831.85 & 193.12 & 0.64 & 0.80 & 0.95 \\ \cline{2-7}  & 2-parameter exponential & 787.01 & 827.41 & 208.62 & 0.68 & 0.83 & 0.95 \\ \hline \multirow{4}{*}{W} & Second-degree polynomial & 716.85 & 775.07 & 161.26 & 0.54 & 0.73 & 0.97 \\ \cline{2-7}  & 3-parameter logistic & 718.05 & 776.26 & 161.87 & 0.54 & 0.73 & 0.97 \\ \cline{1-1} \cline{2-7}  & 2-parameter exponential & 763.52 & 803.93 & 193.63 & 0.63 & 0.80 & 0.96 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of the three mathematical regression curves fitted to the growth data for conditions A and W, respectively. For Exp3, the extrapolated growth trends in Figure 9 suggest potentially significant long-term biomass yields. Among the models tested, the exponential regression provided the most optimistic projection, indicating that a surface biomass density of 1 kgDw/m2 could be achieved within approximately two months (54 days for W, 64 days for A). It is important to emphasise that these projections are purely theoretical. Both polynomial and exponential models lack saturation constraints, mathematically suggesting continuous biomass accumulation. However, in real-world conditions, multiple limiting factors--including nutrient depletion, light attenuation, and physical constraints of the growth substrate--would inevitably halt further biomass accumulation [50]. The attached-biomass nature of the twin-layer system adds an additional layer of complexity, as vertical biofilm thickening would eventually reduce light penetration and slow down photosynthetic efficiency. Moreover, from a physical point of view, there must be a maximum surface density threshold above which the gravitational force exerted on the algal biofilm exceeds the adhesive forces between the polycarbonate substrate and the biofilm, which behaves as a high-density fluid matrix [51]. This would cause the biofilm to 'drip' off the vertical panel, necessitating biomass collection before this occurs. Figure 8: SEM observations on biomass samples grown with W and A and collected at days 2 and 14. As suggested, intrinsic phenomena related to biofilm stratification must be considered. Indeed, the dense biomass layer can be viewed as a stratification of cellular layers, with some layers more in contact with the internal interface, receiving more moisture and nutrients through the polycarbonate membrane, while other layers are more exposed to the atmospheric environment, promoting better gas exchange and more efficient light exposure [50]. As the biofilm thickens due to cell reproduction, the inner layers become progressively less efficient in terms of light, gas exchange and nutrient supply [51]. This progressive decrease in efficiency for the inner layers might correlate with a slowdown in growth rates after a certain period, thus necessitating the use of more complex and comprehensive mathematical models [50; 52]. For instance, [PERSON] et al. [53] employed an automata model to investigate the behaviour of _Chlorella vulgaris_ (Chlorophyta) algal biofilms under varying environmental and nutrient conditions. Their study highlighted two key findings: first, the existence of a critical depth (30-70 \(\upmu\)m in their case) beyond which light penetration is insufficient, resulting in an active growth zone where cells actively engage in photosynthesis and contribute to biofilm thickening.