## 1 Introduction  
Basic oxygen furnace (BOF) steelmaking is a complex physical and chemical process used in the metallurgy and smelting fields (Figure 1). It involves various phenomena such as multiphase flow, heat and mass transfer, dissolution, and chemical reactions. In the BOF process, high-pressure oxygen is blown into the molten bath at supersonic speed via an oxygen lance, where it promotes the oxidation of C, Mn, Si, P, and other impurities in the bath; the removal of these impurities determines the steel quality. In industrial production, the oxygen top-blowing process is generally limited to approximately 15 min. However, less than 2% of the energy is transferred from the oxygen lance to the molten bath [1], with most losses occurring at the gas-liquid interface [2]. Therefore, acceleratingthe slagging reaction and improving energy efficiency in BOF steelmaking are the key objectives for improving steel production capacity [3]. Researchers worldwide have been studying the BOF process to address the aforementioned issues. [PERSON] et al. [4] researched the oxygen-slag interaction using numerical simulations and cold model experiments and discussed the relationship between the blowing number and cavity depth. [PERSON] et al. [5] predicted the oxygen top-blowing process conducting standard \(k\)-\(\varepsilon\), realizable \(k\)-\(\varepsilon\), standard \(k\)-\(\omega\), and SST \(k\)-\(\omega\) models and suggested that, among these models, the standard \(k\)-\(\omega\) model is most accurate for describing BOF processes. [PERSON] et al. [6] evaluated the transfer characteristics from the oxygen lance to the molten bath and discussed the evolution of momentum, energy, and turbulence; their results indicated that the efficiency of momentum transfer can be improved by reducing the lance height or increasing the operating pressure but at the expense of the efficiency of kinetic energy transfer. [PERSON] et al. [7] confirmed the optimum angle of the oxygen lance to be 16\({}^{\circ}\) by comparing the slagging time under different nozzle inclination angles. They used a modified lance in PANSTEEL and achieved significant improvements, in which the average time of early slag formed decreased from 5.25 to 4.1 min, the rate of dephosphorization increased from 80.02% to 82.49%, the life of oxygen lance was raised from 428 heats to 800 heats. The aforementioned literature indicates that the jet process is an important dynamic behavior that triggers complex physical and chemical phenomena. Elucidating the evolution mechanism of the multiphase interface and optimizing the oxygen top-blown process would be highly beneficial in accelerating the slagging reaction. In this context, a modified multi-nozzle oxygen lance with a twist angle (also known as a nozzle-twisted lance) has been proposed and has attracted extensive attention in the industry because it can induce a considerable tangential velocity component [8] and intensify stirring effects [9; 10]. [PERSON] et al. studied the blow behavior of nozzle twisted lance and verified that a nozzle twist angle (NTA) of 11.4\({}^{\circ}\) was most effective for reducing spillage in BOF steelmaking [11]. [PERSON] et al. investigated the effects of the NTA on the swirling flow intensity, slagging characteristics, and blow dynamic parameter distributions [12]. [PERSON] et al. [13] analyzed the stirring effects and flow field characteristics of different lances with NTAs of 4\({}^{\circ}\), 8\({}^{\circ}\), and 12\({}^{\circ}\) and confirmed that the 8\({}^{\circ}\) oxygen lance resulted in a better dephosphorization effect in a 120 t steel converter. The aforementioned literature implies that the nozzle-twisted lance has substantial advantages for the slagging reaction but that the optimum NTA is affected by the flow field [14]. Current research is mainly based on numerical simulations and water-model experiments to provide theoretical support; a method for determining the optimum NTA  
Figure 1: Basic oxygen furnace steelmaking and slagging reaction. has not been developed.