## 1 Introduction  
The Ruhrstahl-Heraeus (RH) vacuum degassing process is a kind of secondary steel-making process between the Basic Oxygen Furnace (BOF) and the Continuous Casting (CC) process [1; 2; 3; 4]. It aims to improve the cleanliness of steel by reducing impurities such as hydrogen, nitrogen and oxygen. It also controls the temperature of the molten steel to achieve the requirement of the CC process. In particular, due to the excellent decbarurization capability of the RH process, it is the most popular vacuum steel refining unit for the production of Ultra-Low Carbon (ULC) steel and Interstitial Free (IF) steel for automobiles [5; 6; 7; 8]. In order to produce high-quality ULC steel, it is important to estimate the decbarurization rate of the molten steel during the operation and to determine the decbarurization end time exactly. If the operation is terminated too early due to the incorrect prediction of the decbarurization rate, the mechanical and chemical properties of steel products may deteriorate due to the high carbon content and additional work may be required to remove the residual carbon in the melt. In contrast, if the operation time is delayed longer than necessary, the consumption of utilities such as circulation gas, ejection steam and electricity increases and the erosion of refractory materials in the vacuum chamber and submerged snorkels accelerates. Since the temperature of the molten steel may become too low due to the excessively long operating time, it may take additional energy and time to reheat the molten steel. In addition, some ULC steels, such as Bake Hardened (BH) steel [9], have very narrow upper and lower limits of target carbon content, so it is necessary to determine the decarburization end time through more accurate estimation of the decarburization rate. Because it is difficult to measure the dissolved carbon content of the molten steel in real time, process operators typically rely on intermittent sampling measurements or their long experience to estimate the dissolved carbon content during the RH process. So there have been various studies to predict the decarburization rate accurately and to understand the degassing reaction in the RH process. Based on mass and thermal balance, the decarburization reaction can be expressed as a first-order equation with the apparent rate constant of decarburization [10; 11; 12; 13]. Many studies have attempted to determine the influence of various factors affecting the apparent rate constant of decarburization, such as circulation flow rate, initial carbon and oxygen concentration, internal pressure of vacuum chamber and flow rate of Top Oxygen Blowing (TOB) [11; 14; 15; 16; 17; 18]. In addition, they have tried to obtain sophisticated prediction models by identifying decarburization mechanisms according to the reaction zone where the decarburization reactions take place and by analyzing changes in decarburization rates over operating time [19; 14; 17; 18; 19; 20; 21]. However, the RH process involves multidimensional physicochemical phase changes at high temperature and the model parameters may change depending on the capacity and conditions of the equipment. Therefore, it is difficult to obtain a generalized model through experiments, especially in large-scale commercial facilities where experiments are time-consuming and expensive. Some researchers have used Computational Fluid Dynamics (CFD) to calculate nonlinear reactions in the RH processes, taking into account various operating conditions that are difficult to test directly. CFD is particularly effective in identifying decarburization rates according to reaction area or operating time and in finding optimal process conditions such as circulation flow rate, oxygen blowing and snorkel dimensions [22; 23; 24]. However, the CFD models have limitations to account for all important responses due to computational power, lack of prior knowledge of the system, or modeling complexity.