## 2 Materials and Methods
### General Description of the CFD Method  
Numerical fluid flow simulation involves solving systems of differential equations describing the analysed phenomena. Basic equations solved for all tasks are the equations describing the velocity and pressure field (continuity equation) and Navier-Stokes movement equations. For calculations of the smoke and heat spread, additional equations of energy and radiation conservation and smoke transport are solved. The description of the CFD method used in fire safety engineering was presented by [PERSON] [27] and [PERSON] [28]. Theoretical foundations of the CFD method are described [29; 30; 31]. The set of equations presented by [PERSON] is the same, as the mathematical model of ANSYS Fluent software [18] used by the Authors in the experiments presented in the paper. In the context of this paper, the mass conservation Equation (1) of the CFD model means that the fire-extinguishing gas can be removed from the model only when it flows out through the domain boundary. The density change in any volume equals the mass flow through its boundaries (leakages) or the mass introduced through the source in the volume (\(\dot{m}_{i}^{{}^{\prime\prime\prime}}\)), which in this case is equal to 0. \[\frac{\partial\rho}{\partial t}+\
abla\cdotp\dot{\vec{u}}=\dot{m}_{i}^{{}^{ \prime\prime\prime}} \tag{1}\]  
where: \(\rho\)--density, \(t\)--time and \(u\)--velocity. The extinguishing gas within the air-gas mixture is modelled as a single mixture ingredient (\(Y_{\text{gas}}\)), and its transport can describe by the Equation (2). \[\frac{\partial(\rho Y_{i})}{\partial t}+\
abla\cdotp(\rho Y_{i}\dot{\vec{u}} )=\
abla\cdotp(\rho D_{i}\
abla Y_{i})+\dot{m}_{i}^{{}^{\prime\prime\prime}} \tag{2}\]  
where: \(D_{i}\) is the dispersion coefficient of \(i\)-th species. The momentum conservation Equation (3) is expressing the preservation of the [PERSON]'s Second Law of Motion. The forces causing the fluid flow are composed of the pressure field \(\
abla p\), tension (tensor \(\overline{\overline{\pi}}\)), buoyant force \(\rho\overrightarrow{g}\) and external forces \(\overset{\rightarrow}{F}\), which in this case are equal to 0, (3). \[\frac{\partial(\rho\overset{\rightarrow}{u})}{\partial t}+\
abla\cdot(\rho \overset{\rightarrow}{u}\overset{\rightarrow}{u})=-\
abla p+\
abla\cdot( \overline{\overline{\pi}})+\rho\overrightarrow{g}+\overset{\rightarrow}{F} \tag{3}\]  
Since the tests performed took into account the difference in the temperature inside and outside the test chamber, as well as the temperature gradient outside the chamber, the heat flow in the system was modelled using the energy conservation Equation (4), which identifies that the enthalpy (5) in any point changes according to the stream of energy flowing into the control volume. The equation also considers the possibility of heat generation directly in the finite element (\(\dot{q}^{\prime\prime\prime}\)), which in this case amounts to 0. Heat can also be delivered as a result of the fluid kinetic energy dissipation as a result of friction (\(\varepsilon\)), the impact of pressure (\(\frac{Dp}{Df}\)) or heat radiation. In fire-safety related applications, the term responsible for the pressure field impact or kinematic energy dissipation is usually neglected. \[\frac{\partial(\rho h)}{\partial t}+\
abla\cdot(\rho h\overset{\rightarrow} {u})=\frac{Dp}{Dt}+\dot{q}^{\prime\prime\prime}-\
abla\cdot\overset{ \rightarrow}{q}+\varepsilon \tag{4}\]  
\[h=\int\limits_{T_{0}}^{T}c_{p}dT \tag{5}\]  
A change in the density can be sufficiently described by a perfect gas Equation (6). \(M_{avg}\) (7) in Equation (6) stands for the averaged molecular weight of the gas mixture ingredients, whose concentration can be identified in any volume using (2). \[p=\frac{\rho\Re T}{M_{avg}} \tag{6}\]  
\[M_{avg}=\frac{1}{\sum\frac{Y_{i}}{M_{i}}} \tag{7}\]  
where: \(p\)--pressure, \(\Re\)--gas constant, \(T\)--temperature, \(Y_{i}\)--vol. concentration of \(i\)-th species, \(M_{i}\)--molar mass of the \(i\)-th species.