## 1 Introduction
The resulting equations demonstrated a reliable capability to estimate optimal insulation thickness values with accuracy. [PERSON] [26] analyzed the environmental impact of insulation materials while determining the optimal insulation thickness for two different wall types through a combined thermo-economic and environmental approach. This method incorporated the entransy concept, which characterizes an object's ability to transfer heat. The study assessed insulation thickness across various wall models and insulation materials in different Turkish cities using a thermo-economic method integrated with entransy analysis. The results showed that insulation thickness ranged from 6.15 cm to 0.01 cm, depending on the wall model and insulation material. The study further evaluated entransy-based cost savings, combustion parameters, and environmental impacts, revealing that insulation thickness determined through the entransy-based environmental analysis was higher compared to thermo-economic calculations. It was concluded that while the thermo-economic approach offered greater energy and cost savings, the environmental perspective prioritized sustainability by reducing fuel consumption and CO2 emissions. [PERSON] and [PERSON] [27] employed three-dimensional numerical analysis to determine the optimal phase change material (PCM) doping content (i.e., thickness) for a double-glazed window designed for the summer climate in Baghdad, Iraq. Using a FORTRAN (F90) program, the study combined the Finite Volume method with the Enthalpy method to analyze conduction-related phase transition issues within the wax. The transient heat transfer characteristics of the insulation were meticulously examined using the developed numerical model. The findings revealed that to maintain indoor temperatures within the optimal range, the ideal doping content of paraffin wax (melting point 40 \({}^{\circ}\)C) varied as follows: 2 cm in May, 3 cm in June, 3.5 cm in July and August, 3 cm in September, and 1.5 cm in October. [PERSON] [28] conducted a climate-geographical assessment of Jaipur, India, which features a Mid-Latitude Steppe and Desert Climate. The study employed Cooling Degree Days (CDDs) to calculate energy demand, optimal insulation thickness (\(X_{opt}\)), payback periods, and heat losses in exterior walls for two insulation materials, extruded polystyrene (XPS) and expanded polystyrene (EPS). The results indicated that \(X_{opt}\) ranged from 3.83 cm to 7.31 cm, with payback periods varying between 2.35 and 1.79 years. [PERSON] et al. [29] investigated the applicability of artificial neural networks (ANNs) for modeling _Xopt_, annual total net savings (ATS), and CO2 emission reductions resulting from building insulation. The study utilized insulation market data, economic parameters, fuel prices, and heating degree days (HDDs) as input variables for the ANN model. Three learning algorithms and five statistical indices were employed to assess the model's accuracy, while visualization techniques were used to compare the predicted and calculated values for \(X_{opt}\), ATS, and CO2 reduction. [PERSON] et al. [30] assessed the energy, environmental, and economic implications of thermal insulation in different climates, highlighting that optimal insulation thickness varies significantly with climatic conditions and building usage patterns. Their findings underscore that insulation requirements should not be static but rather responsive to both climate change projections and urbanization-driven temperature variations. In Saudi Arabia, [PERSON] et al. [31] used seven insulation materials and seven wall configurations to investigate the impact of wall configuration on \(X_{opt}\). Under ideal conditions, the type of insulation material used had the greatest impact on yearly transmission load and R-value, while the wall configuration had the least impact. The most economical material was polystyrene, which had a \(X_{opt}\) average for all walls of 7.8 cm.