## 1 Introduction
Subsequently, many studies in the literature have been devoted to clarifying this empirical relationship: [PERSON] and [PERSON] [18] took into account the influence of the excavation inclination angle, [PERSON] and [PERSON] [19] proposed determining the dimensionless HRR of a fire through the equivalent diameter of the excavation, [PERSON] and [PERSON] [20] studied the propagation of smoke in tunnels at different ratios of their transverse dimensions, and [PERSON] [21] studied the effect of channel width on the critical air velocity. The maximum temperature of the gas-air mixture at the roof of the tunnel [22; 23] and the length of the return flows of the gas-air mixture [24; 25] were also studied. [PERSON] and [PERSON] [26] introduced a new term: confinement velocity. This is the minimum longitudinal velocity induced by extraction that is necessary to prevent smoke layer development after the last exhaust vent has been activated. [PERSON] et al. [27] studied the influence of the tunnel aspect ratio on the temperature distribution in tunnels. [PERSON] et al. [28] investigated the thermal effect of the latent heat of vaporization in concrete linings. [PERSON] et al. [29] utilized computer-vision-based deep learning methods to analyze smoke diffusion characteristics and achieved real-time predictions of smoke temperature fields. Several researchers examined the dynamics of the temperature field and the harmful effects of fire when implementing natural ventilation, which is often used in tunnels [30; 31; 32]. Most modern work is aimed at studying the characteristics of the spread of heat and smoke during fires in subway and road tunnels, which have their own characteristics in terms of organizing ventilation [32; 33]. At the same time, mines have their own specifics of ventilation, inclination angles of mine workings, and branched systems of mine workings ventilated by one or more fans [34; 35]. When calculating the mine ventilation processes in the event of fires, it is important to take into account not only the local effects of thermal convection at the site of the fire but also their subsequent impact on the entire ventilation network as a whole. For example, fires can lead to a significant redistribution of air flows in mine workings due to the effect of thermal draught [36; 37]. Today, several approaches are being developed to account for the complex processes of return air flows in mine ventilation systems: one-dimensional models of ventilation networks [38], modified one-dimensional models representing individual workings as a network of branches in a ventilation graph [39], as well as CFD simulations [40]. All these approaches have not been sufficiently developed to date and have certain problems (difficulty in parameterizing models, computational complexity, failure to take into account some effects, such as 'thermal slug' of the air flow, etc.). One of the promising and currently undeveloped areas of research is the improvement of operational methods for calculating air distribution during the development of underground fires in mines by introducing effective empirical dependencies into one-dimensional models of ventilation networks to account for the thermal effect of a fire on air flow parameters. Moreover, such empirical dependencies can be obtained based on multiparameter modeling of heat and mass transfer in a relatively small area of the mine near the fire source in a three-dimensional setting. Obtaining such dependencies is the global goal of this study. At the same time, in this study, we solved the primary issue related to the validation of the mathematical model by comparing the data of numerical calculations with the data of experimental measurements of the parameters of the air stream on a laboratory bench that we developed, simulating an inclined mine working.