## 2 Method
### Calculation Conditions  
In this study, numerical simulations were conducted using the proposed model to analyze PM deposition and oxidation behavior inside a DPF with catalysts. The conditions applied in the calculations are summarized in Tables 1-3. Table 1 presents the thermal properties of solid materials used in this study, including the oxidation heat of carbon, the specific heat capacity of cordierite, and the density of cordierite. These parameters were incorporated into the numerical model to evaluate heat transfer and temperature changes during PM oxidation. The oxidation heat of carbon determines the energy released during combustion, while the specific heat capacity and density of cordierite influence the thermal response of the DPF structure. Table 2 provides the structural and flow-related parameters of the DPF, including the total length and diameter of the filter, wall thickness, porosity, and exhaust flow rate. These values were implemented in the numerical calculations to determine the exhaust flow distribution, pressure drops across the filter, and the impact of PM accumulation on engine performance. The wall thickness and porosity are particularly important in defining the permeability of the filter, which affects PM-trapping efficiency and pressure loss. Table 3 lists the activation energy values used for the solid-state reaction (SSR) and gas-phase reaction (GPR) in the calculations. The activation energy values, ranging from 90 to 200 kJ/mol, were selected based on the previous literature and experimental data. Platinum-based and silver-based catalysts typically exhibit lower activation energies in a range of 90-140 kJ/mol, while cerium oxide-based and manganese-based catalysts fall within a range of 100-180 kJ/mol. Non-catalytic conditions generally require activation energies exceeding 180 kJ/mol. By evaluating different activation energies, this study aims to determine the optimal catalyst type for efficient PM oxidation under various operating conditions [10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32].  
PM deposition amount of 1 g represents a recently regenerated DPF or a low-emission condition, typically observed under continuous operation with efficient regeneration. A PM deposition amount of 10 g corresponds to a typical level of PM accumulation before regeneration, which is commonly encountered in standard engine cycles. A PM deposition amount of 20 g represents a high PM accumulation scenario, such as frequent stop-and-go urban driving, where incomplete regeneration leads to a higher PM load. A PM deposition amount of 30 g represents a worst-case scenario where excessive PM accumulation leads to significant back pressure, reduced engine efficiency, and the potential clogging of the DPF. These PM deposition levels were selected based on realistic PM accumulation estimates. For example, a 1.5 L diesel engine operating at a space velocity of 50,000 h\({}^{-1}\) with a PM accumulation rate of 10 g/L results in an estimated PM accumulation of 30 g in a DPF volume of 3 L. This estimation aligns with real-world PM emission rates, where an engine emitting 0.1 g/kWh at 50 kW produces approximately 5 g/h of PM. Under normal regeneration conditions, PM is oxidized before reaching 30 g, but in cases of regeneration failure, excessive accumulation can occur, leading to increased pressure drop and decreased fuel efficiency. By incorporating these selected PM deposition amounts into the numerical model, this study aims to capture a wide range of practical operating conditions. This approach ensures that the results of the study apply to real-world DPF performance evaluation, allowing for the better optimization of catalyst selection and exhaust system design.