## 4 Conclusions  
This paper presents a novel mathematical model for the geometric optimization of a new solar collector type--a single-glazed vacuum tube collector (SGVTC) with a SnAl2O3 selective flat absorber plate and gravity-driven single-stage direct water flow. The proposed optimization approach integrates operative, geometric, economic and environmental indicators to determine two key design parameters: the flat absorber plate width and glass tube thickness.  
The development of a comprehensive iterative calculation algorithm that enables the precise thermal behavior modeling of the new solar collector is undertaken. The model accounts for thermal efficiency, useful heat power, heat losses, incident solar irradiance and optical efficiency.  
A novel two-step multi-criteria analysis (MCA)-based optimization strategy is proposed, which simultaneously evaluates heat power and thermal efficiency through logarithmic performance curves.  
The identification of the optimal operational conditions, where useful heat power and thermal efficiency peak at an absorber width of 87-88 mm, confirm the strong correlation between geometric parameters and thermal performance.  
Compared to traditional parametric studies and single-objective optimization approaches, this multi-criteria optimization method allows for a more balanced trade-off between thermal efficiency and useful heat power, rather than optimizing one parameter at the expense of another. While conventional gradient-based algorithms or evolutionary techniques (e.g., genetic algorithms) can optimize collector geometry, they often require extensive computational resources and may overlook practical constraints such as material limitations and economic feasibility. In contrast, the proposed iterative MCA approach ensures computational efficiency and integrates economic and ecological considerations, making it more applicable to real-world solar thermal systems.  
Further research should explore the integration of the proposed optimization model with machine learning algorithms to enhance its predictive accuracy and allow it to adapt dynamically to changing meteorological conditions. Additionally, coupling the model with finite element analysis (FEA) and computational fluid dynamics (CFD) simulations could provide deeper insights into heat transfer mechanisms. From an application perspective, implementing the SGVTC concept in hybrid solar thermal energy systems--particularly in residential heating applications--could significantly improve energy efficiency and sustainability.  
As climate change progresses, the European residential sector remains particularly vulnerable to energy supply fluctuations. Advancing renewable energy-based heating solutions, such as the proposed solar collector design, offers a dual benefit: increased energy self-sufficiency for end users and reduced environmental impact.  
Conceptualization, A.N. and R.K.; methodology, A.N.; software, R.K.; validation, A.N. and R.K.; formal analysis, R.K.; investigation, A.N.; resources, A.N.; data curation, R.K.; writing--original draft preparation, A.N. and R.K.; visualization, A.N. and R.K. All authors have read and agreed to the published version of the manuscript.  
* This research received no external funding.