The article presents the results of numerical modeling of the interaction of a supersonic non-calculated gas-powder jet with the surrounding strongly heated environment of the converter cavity, including slag particles ejected by the jet. The method of determining the depth of penetration of the jet into the melt is the subject of many years of discussion, but it has not been solved analytically. A new approach to solving this problem is proposed, which consists in the fact that in this work the structure is considered not according to the laws of subsonic turbulent flow, but as supersonic, not calculated. In this paper, there is a task - to increase the stability of the slag crust, nitrogen supply of MgO powder is provided, which, in turn, enters the jet and slows it down. To solve this problem, the solution of many parametric equations is proposed to establish the effect of heating on the qualitative characteristics of the slag crust on the surface linings. A physical model has been created that predicts the outflow of a gas-powder jet (N2 + powder) from a Laval nozzle. The formation outside the jet nozzle is established, which becomes a supersonic non-computational structure with a shock wave. The most important hydro-gas-dynamic characteristic of the gas jet flow is its momentum, which depends on a number of factors and physical effects. The scheme for solving a fairly high-tech problem involves the following: calculation of the parameters of the gas-powder flow at the exit from the nozzle; calculation of the amount of added gas mass from the environment; temperature: pulse speed; jet power, etc. It was established that in some cases, when a chemical connection between the slag and the lining is ensured (when there is an optimal value of MgO and CaO in the slag), it is possible to stop the supply of refractory powder and then the slag will be inflated only with a nitrogen jet. Using the law of constancy of momentum in different cross-sections of the jet (for example, in the outlet cross-section of the nozzle and in the fixed cross-section of the XX stream), it is possible to calculate: total momentum, average mass velocity of the nitrogen-powder jet, average mass temperature of the jet. It is shown that the power of the gas-powder jet flowing into the melt is most significantly affected by the heating of the gas-dispersed flow to the temperature before the nozzles. It was established that an increase in temperature from 50°C to 600°C leads to an increase in momentum by almost 2.75 times. The joint solution of almost 50 equations (messages 1 and 2) makes it possible to visually present the picture of the interaction of the supersonic jet with shock waves and melt, as well as to develop recommendations for improving the energy efficiency of slag-blowing technology. Since conducting an experiment in the converter cavity is difficult, the results of numerical calculations are verified by the method of special cases
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