## 2 Materials and Methods
### Mixing Process Tracking
#### 2.2.2 Diffusional Model  
Two effects needed to be studied to determine the variation of concentration over time: (a) molecular diffusion, the transfer of individual molecules due to a concentration, pressure, or temperature gradient, among others; and (b) convection transport, caused by the overall movement of the fluid [51]. As the gases used in these experiments have high molecular weight, it was considered that the gas mixing process happened not only due to the concentration gradient, but also to the differences in the density of the components or natural convection. Considering the described two effects, a modified Fick's generalized expression was obtained:  
\[\frac{dC}{dt}=D\ \frac{\partial^{2}C_{i}}{\partial z^{2}}+\frac{\partial(V_{i} \ C_{i})}{\partial z} \tag{3}\]  
where \(C_{i}\) is the concentration, \(D\) (m\({}^{2}\) s\({}^{-1}\)) is the diffusion coefficient, \(t\) represents the time, and \(z\) the height. \(V_{i}\) is the velocity caused by the differences between densities:  
\[V_{i}=K\ (\rho_{m}-\rho_{i})\Big{[}\text{m s}^{-1}\Big{]} \tag{4}\]  
where \(K\) (m\({}^{4}\) kg\({}^{-1}\) s\({}^{-1}\)) is the convection constant. The density of the heavy component in the binary mixture (\(\rho_{m}\)) and the density of the mixture at each point in time (\(\rho_{i}\)) were calculated by:  
\[\rho_{m} =\frac{P\ MW_{m}}{R\ T}\Big{[}\text{kg m}^{-3}\Big{]} \tag{5}\] \[\rho_{i} =\frac{P\ MW_{i}}{R\ T}\Big{[}\text{kg m}^{-3}\Big{]} \tag{6}\]  
\(T\) and \(P\) are the temperature and pressure at which the experiments were carried out, 298 K and 1 atm (101,325 Pa), and \(R\) is the ideal gas constant. \(MW_{m}\) is the molecular weight of the heavy component of the binary mixture, and \(MW_{i}\) indicates the molecular weight of the mixture at each point in time, which was calculated by the weighted average of the \(MW_{m}\) of each component of the mixture:  
\[MW_{i}=\sum_{c}^{n}y_{m}\ MW_{m}\Big{[}\text{kg mol}^{-1}\Big{]} \tag{7}\]  
The equations were based on the fact that for a binary mixture consisting of components A and B, the diffusivity of A in B (\(D_{AB}\)), and vice versa (\(D_{BA}\)), are the same [51]. Both molecular diffusion and natural convection were considered to determine the molar flux (\(J\)):  
\[J=-D\ \frac{dC_{i}}{dz}+V_{i}\ C_{i}\Big{[}\text{mol m}^{-2}\ \text{s}^{-1} \Big{]} \tag{8}\]  
Theoretical studies of multicomponent diffusion are based on the Stefan-Maxwell equation, which, in turn, was derived from the solution of the Boltzmann equation [52]. For isothermal and isobaric conditions, this equation is a strong approximation that satisfies the practical requirement. The variation of concentration over time for multicomponent mixtures is represented in the following equation:  
\[\left(\frac{dC}{dt}\right)=[D]\ \left(\frac{\partial^{2}C_{i}}{\partial z^{2}} \right)+\left(\frac{\partial(V_{i}\ C_{i})}{\partial z}\right) \tag{9}\]  
The element \([D]\) is a size \(n-1\) squared matrix that represents the diffusion coefficients, \(n\) being the number of components in the mixture. The elements of the square matrix \([D]\)are called practical diffusion coefficients [52]. In this study, a gas mixture composed of three elements was made; therefore, the matrix was composed of 2 \(\times\) 2 elements. As well as in binary gas mixtures, the convection effect was also considered in multicomponent mixing:  
\[\left( \begin{array}{c}\frac{dC_{1}}{dt}\\ \frac{dC_{2}}{dt}\end{array} \right)=\left[ \begin{array}{cc}D_{11}&D_{12}\\ D_{21}&D_{22}\end{array} \right]\left( \begin{array}{c}\frac{\partial^{2}C_{1} }{\partial x^{2}}\\ \frac{\partial^{2}C_{2}}{\partial z^{2}}\end{array} \right)+\left( \begin{array} []{c}\frac{\partial(V_{1}\ C_{1})}{\partial z}\\ \frac{\partial(V_{2}\ C_{2})}{\partial z}\end{array}