## 2 Materials and Methods
### Problem Statement and Boundary Conditions  
The basis stage of the tests conducted was to develop a numerical model for the real-world (actual) region of the longwall under analysis. This area covers the goaves with caving, the longwall and the longwall headings. The actual tensile strength of the roof rocks forming the caving in this longwall amounted to 3.06 MPa. Additionally, tests were also conducted for the following strength values of these rocks: 2, 4, 5, 6 and 7 MPa. The geometric parameters of the entire region under analysis were taken into consideration during the development of the model. The geometric model of the longwall under analysis, ventilated with the Y-type system along with the equivalence conditions adopted, is presented in Figure 2.  
The geometric model of the longwall area investigated takes into account the following:  
* a section of the longwall gallery with a length of 20.0 m and a cross sectional area of A = 15.0 m\({}^{2}\). * a tailgate maintained along the goaves with caving, with a length 525.0 m.
* a longwall with a height of 3.0 m and a length of 220.0 m, and an inclination of 0\({}^{\circ}\). * a section of goaves with caving with a length 500.0 m (constituting \({}^{2}\)/3 of the longwall panel length). In order to examine the air flow through the caving goaves of longwalls, it was also necessary to determine the vertical extent of this flow. Generally, it is assumed that the zone of air flow through the goaves with caving is equal to three to four times the thickness of the exploited seam (layer) [36]. The height of this flow is determined by the extent of the full caving and is equal to three times the thickness of the exploited layer. Taking into consideration the fact that the air also flows in the space (void) of the mined longwall, we obtain a flow zone measuring four times the thickness of the seam. On the other hand, taking into account the settlement of the basic roof by a value of 0.5 \(\div\) 0.6 of the seam thickness, it was assumed that the height of the air flow in the goaves amounts to 3.5 times the thickness of the exploited seam (layer). Such geometrical models were subjected to the process of discretization. The selection of the right size of the numerical mesh elements was preceded by an analysis of its sensitivity to the calculation results obtained. Based on the analysis, it was concluded that, for model-based tests of air flow through goaves with caving, one may adopt a structural numerical mesh with the size of cubic elements equal to 0.05 m \(\times\) 0.05 m \(\times\) 0.05 m for a longwall and longwall headings, as well as a structural numerical mesh consisting of cuboidal elements (type of mesh: hexahedron) measuring 0.025 m \(\times\) 0.025 m \(\times\) 0.025 m for goaves with caving. Smaller elements significantly extend the time of calculations, without making any changes in the results obtained. The \"inlet\" and \"outlet\" boundary conditions were defined in longwall galleries. The \"inlet\" boundary condition was set in the distance of 20.0 m from the longwall, in the maingate as well as in the tailgate. It was assumed that the length of longwall galleries amounting to 20.0 m would allow for full development of the speed profile for the air stream supplied to the longwall. The volumetric flow rate of the air supplied to the longwall under analysis was equal to 14,250 m\({}^{3}\)/min. The volumetric flow rate of the air supplied through the tailgate amounted to 410 m\({}^{3}\)/min. The \"outlet\" boundary condition (pressure-outlet) was defined in the tailgate (this reflects the actual condition in the longwall region).