## 1 Introduction  
Effective fluid flow control is a fundamental requirement in a wide array of industrial processes, where precision and efficiency are predominant. Within this context, discharge valves play a pivotal role in regulating fluid flow within pipelines and systems. A comprehensive understanding of airflow dynamics within these valves is essential for their optimal design and operation. The stability of safety relief valves, essential for their reliable performance is influenced by various factors, such as the design of the valve, spring pre-deformation, the length of the inlet pipeline, valve opening length, the position of regulator rings and the volume of the pressure vessel [1]. [PERSON] and [PERSON] [2] conducted research on the dynamics of a pilot-operated pressure relief valve using the Bondgraph simulation technique. They derived the governing equations of the system from their model and, in the process of numerically solving these equations, they considered the different pressure-flow characteristics of the valve ports. The simulation study identified specific critical design parameters that exerted a substantial impact on the transient response of the system. Similarly, [PERSON] and [PERSON] [3] created a straightforward highly precise model for analyzing the dynamic behavior of a pressure reducing valve. The primary contributions of their work included the formulation of multiple dynamic models, encompassing two phenomenological models, one behavioral model, and one linear model, all designed to effectively describe the functioning of pressure reducing valves. Advancements in computational techniques have made it increasingly practical to analyze the intricate flow patterns within valves [4; 5; 6]. As a result, there has been a significant increase in research related to the functionality and design of valves [4; 5; 6; 7; 8; 9; 10; 11]. The study [7] develops a numerical model using advanced computational fluid dynamics techniques to analyze a direct-operated safety relief valve. The model accurately simulates the entire process from valve opening to closure, providing insights into compressible fluid flow and the effects of design parameters. A three-dimensional numerical analysis with CFD was performed in [1] to examine a safety relief valve's performance under various conditions, including different openings and inlet pressures. The study compared mass flow and discharge coefficients from simulations with ASME 2011a Section 1 standards and the model proved consistent for openings below 12 mm and closely matched the ASME standard. Additionally, CFD simulations revealed that the average normal disc force was approximately 19% lower than the theoretical ASME force, which could help prevent valve oversizing. A simplified dynamic model (based on the equation of motion for a single degree of freedom system, with the assumption that damping effects are negligible due to their minimal impact on reclosing time) was developed in [12] to predict the dynamic behavior during both the reclosing and blowdown phases of a conventional pressure relief valve. The study introduces a novel approach by integrating static computational fluid dynamics (CFD) analysis results, calculating the lift force coefficient of the valve at various fixed lift positions using static CFD analysis and then incorporates these values into the simplified dynamic model as the inherent characteristics of the valve. CFD simulation was also used by [PERSON] and [PERSON] [13] to explore the dynamic instability (flutter/chatter) of gas-service direct spring operated pressure relief valves, specifically focusing on the acoustic coupling between the valve body dynamics and the upstream piping. To enhance accuracy and enable analytical computations, the paper introduces two CFD-based methods. The first involves steady-state CFD runs to validate and improve the reduced-order models.