## 1 Introduction  
Global warming due to climate change has emerged as a critical issue worldwide. According to reports from the Intergovernmental Panel on Climate Change (IPCC), greenhouse gas emissions have been identified as a primary driver of global warming [1], with the refrigerants used in air-conditioning systems being highlighted as one of the major contributors [2]. The HCFC and CFC refrigerants that were commonly used in the past were found to cause ozone depletion and were subsequently restricted by the Montreal Protocol, which came into effect in 1987. They have since been replaced by refrigerants with zero ozone depletion potential (ODP), such as HFCs. However, these zero ODP refrigerants have high global warming potential (GWP) and contribute to global warming when released into the atmosphere. Through various climate agreements, the use of refrigerants with high GWP is being increasingly restricted [3], and research is being conducted on low-GWP refrigerants as alternatives to these [4, 5]. As refrigerants are replaced, changes in their thermophysical properties affect the heat transfer performance of heat exchangers [6]. In response, studies have been conducted to determine how heat transfer performance can be improved by adopting smaller-diameter tubes in the evaporators of air conditioners that use low-GWP refrigerants, thereby increasing the heat transfer area of the heat exchangers [7]. Additionally, since the air-side resistance accounts for over 90% of the total thermal resistance in fin-tube heat exchangers [8], various studies have investigated how air-side heat transfer can be enhanced by optimizing fin designs [9]. Several fin designs have been employed to enhance air-side heat transfer in the indoor units of air conditioners; common types include louvered fins, slit fins, and wavy fins. Dimensionless numbers, such as the j-factor and f-factor, are commonly used to evaluate the performance of different fin types. The j-factor, derived from the Stanton and Prandtl numbers, represents heat transfer performance, while the f-factor quantifies frictional losses and pressure drop across the fin [10]. [PERSON] et al. [11] experimentally compared the heat transfer and pressure drop characteristics of plain fins, slit fins, and louvered fins to find that the slit fin exhibited the highest j-factor while the louvered fin had the highest f-factor. Despite the louvered fin's higher f-factor, the slit fin demonstrated superior performance in terms of heat transfer and pressure drop due to its higher j-factor. [PERSON] et al. [12] conducted experiments comparing the heat transfer coefficient and friction factor of sinusoidal-wave fins and herringbone-wave fins. They found that, at the same waffle height, the sinusoidal-wave fin demonstrated higher heat transfer and friction coefficients as well as a faster transition to turbulence. Research has also been conducted on wavy-slit fins, which combine the characteristics of wave fins and slit fins. [PERSON] et al. [13] demonstrated that wavy-slit fins exhibit a higher heat-transfer-enhancement effect compared to wave fins. [PERSON] et al. [14] used computational fluid dynamics (CFD) to compare the thermohydraulic performance of wavy fins and wavy-slit fins, revealing that the j-factor of wavy-slit fins improved by 20-39% compared to wavy fins, although the f-factor also increased by 20-38%. Therefore, it is necessary to optimize fin geometry based on an objective function that considers both heat transfer and pressure drop to achieve optimal performance. Studies on shape optimization typically involve fabricating multiple models with different geometric parameters and then conducting experiments to derive the optimal point statistically based on the results. This process can be time-consuming and expensive, but these challenges can be mitigated using CFD. Additionally, using CFD allows for the analysis of localized flow and temperature distributions, which helps identify the mechanisms responsible for improved performance in the optimal geometry.