The stability and positivity of linear positive Markovian jump systems with respect to part of the variables is considered. The methodologies of stability of positive systems with known transition probabilities based on common linear copositive Lyapunov function and stability of linear systems with respect to part of the variables are combined to find sufficient conditions of the stochastic stability and positivity of Markovian jump systems with respect to part of the variables. The results are extended for a class of nonlinear positive Markovian jump systems with respect to part of the variables. An example is given to illustrate the obtained results.
Purging the liquid steel with inert gases is a commonly used treatment in secondary metallurgy. The main purposes for which this method is used are: homogenization of liquid steel in the entire volume of the ladle, improvement of mixing conditions, acceleration of the absorption process of alloy additives and refining of liquid steel from non-metallic inclusions. The basic processing parameters of this treatment are: gas flow rate and the level of gas dispersion in liquid steel. The level of gas dispersion depends on the design and location of the porous plug in the ladle. Therefore, these parameters have a significant impact on the phenomena occurring in the contact zone of liquid steel with slag. Their improper selection may cause secondary contamination of the bath with exogenous inclusions from the slag, or air atmosphere due to discontinuity of the slag and exposure of the excessive surface of the liquid steel free surface. The article presents the results of modelling research of the effect of liquid steel purging with inert gases on phenomena occurring in this zone. The research was carried out using the physical (water) model of steel ladle. As a modelling liquid representing slag, paraffin oil was used, taking into account the conditions of similarity with particular reference to the kinematic viscosity. The results of the conducted research were presented in the form of visualization of phenomena occurring on the surface of the model liquid free surface in the form of photographs. The work is a part of a bigger study concerning modelling of ladle processes.
This paper deals with the possibilities of using physical modelling to study the degassing of metal melt during its treatment in the refining ladle. The method of inert gas blowing, so-called refining gas, presents the most common operational technology for the elimination of impurities from molten metal, e.g. for decreasing or removing the hydrogen content from liquid aluminium. This refining process presents the system of gas-liquid and its efficiency depends on the creation of fine bubbles with a high interphase surface, uniform distribution, long period of its effect in the melt, and mostly on the uniform arrangement of bubbles into the whole volume of the refining ladle. Physical modelling represents the basic method of modelling and it makes it possible to obtain information about the course of refining processes. On the basis of obtained results, it is possible to predict the behaviour of the real system during different changes in the process. The experimental part focuses on the evaluation of methodical laboratory experiments aimed at the proposal and testing of the developed methods of degassing during physical modelling. The results obtained on the basis of laboratory experiments realized on the specific physical model were discussed.