## 1 Introduction  
In oxygen steelmaking, carbon, silicon, and manganese dissolved in the hot metal are removed by an oxidation process to produce liquid steel. The BOF is charged with molten pig iron and steel scrap, and high-purity oxygen gas is introduced at high flow rates via a lance to interact with the metal bath. This interaction triggers chemical reactions that remove elements and produce crude steel. A supersonic copper tip is typically used as a part of an oxygen lance to deliver high-purity oxygen to perform required process functions in modern steelmaking. However, it has been found that oxygen jets also contribute to some undesirable phenomena in the BOF. The upward splashing caused by the supersonic oxygen lance during the steelmaking process can reduce productivity and metal yield [1]. This splashing can also lead to significant loss of MgO-C refractory lining and damage to the oxygen lance and vessel mouth due to skull formation [2, 3]. Moreover, the refractory lining of the BOF wearsunequally due to different chemical, mechanical, and temperature loads. For this reason, a zonal lining is implemented that integrates several types of refractory materials, which are designed based on their specific application in each zone of the BOF lining [4]. The careful design of a lance plays a pivotal role in regulating the dynamics of the jets, thereby facilitating the achievement of higher reactor efficiencies, and ensuring the stable operation of processes in the BOF environment [5]. A well-designed lance serves as a critical component in optimizing operating parameters and improving overall system performance in the BOF environment. As BOF vessel-specific volume continues to increase, metallurgists have refined and improved the oxygen lance to meet the demands of efficient and stable process [6]. Metallurgical experts have optimized the structure of the supersonic nozzle, increased the number of nozzle holes, changed the manufacturing method of the nozzle, increased cooling in the nozzle area, and more [7]. The oxygen lance consists of three centrally aligned steel tubes (Figure 1a). The inner tube carries high-purity oxygen for metallurgical processes and the other concentric tubes carry cooling water [8]. The steel body of the lance is topped by the lance tip, which is made of high thermal conductivity cast copper with precisely machined de Laval nozzles to achieve the desired flow rate and flow parameters (Figure 1b). The copper tips are most commonly available in four, five, or six nozzle configurations [9]. The angle of divergence of the nozzle from the central axis of the lance is a function of the reaction area that the oxygen jet from the nozzle strikes during the steelmaking process [10,11]. The research conducted on the conventional lance has resulted in significant process optimization and advancement in technology. By studying the intricate dynamics of lances, insights have been gained that have enabled the improvement of industrial applications. The insights gained from experimentation and analysis have enabled stakeholders to refine operations, enhance efficiency, and develop innovative solutions [12; 13; 14; 15; 16; 17; 18; 19; 20]. In order to reduce costs and increase efficiency in steel production, the traditional design of oxygen nozzles has technological limitations that only allow working with the nozzle dimensions and their inclination angle from the central axis. There are many research papers that study and propose new design solutions for shaping and arranging nozzles on the copper tip. A number of researchers have investigated the topic of coherent jet oxygen lance [9; 21; 22], which allows for greater depth of impact and better stirring of the molten bath [22; 23; 24; 25; 26; 27]. Currently, a popular and innovative solution and modification of the oxygen nozzle is the double-parameter lance [28; 29], which divides the nozzles into two groups characterized by different angles of inclination, diameter and speed of the oxygen jet.