## 2 Materials and Methods
### Mixing Process Tracking
#### 2.2.2 Diffusional Model
\right) \tag{10}\]  
Hence, the concentration differential of each component over time depends on the practical diffusion coefficients of both components and their concentration differential over the position in the gas chamber [52]:  
\[\frac{dC_{1}}{dt}=D_{11}\frac{\partial^{2}C_{1}}{\partial z^{2}}+D_{12}\frac{ \partial^{2}C_{2}}{\partial z^{2}}+\frac{\partial(V_{1}\ C_{1})}{\partial z} \tag{11}\]  
\[\frac{dC_{2}}{dt}=D_{21}\frac{\partial^{2}C_{1}}{\partial z^{2}}+D_{22}\frac{ \partial^{2}C_{2}}{\partial z^{2}}+\frac{\partial(V_{2}\ C_{2})}{\partial z} \tag{12}\]  
Furthermore, the effective diffusivity coefficients (\(D_{eff}\)) were calculated using the practical diffusion coefficients [52]. [PERSON] et al. [53] stated that the molar fraction gradients could be replaced by molar fraction differences to simplify the calculations:  
\[D_{1 eff}=\sum_{k=1}^{n-1}D_{1k}\frac{\
abla y_{k}}{\
abla y_{1}}\Big{[}\text{m }^{2}\text{ s}^{-1}\Big{]} \tag{13}\]  
Furthermore, according to [PERSON] and [PERSON] [52], the diffusivity eigenvalues of the components (\(\hat{D}_{1}\) and \(\hat{D}_{2}\)) must be positive:  
\[\hat{D}_{1}=\frac{1}{2}\Big{(}tr[D]+\sqrt{disc[D]}\Big{)} \tag{14}\]  
\[\hat{D}_{2}=\frac{1}{2}\Big{(}tr[D]-\sqrt{disc[D]}\Big{)} \tag{15}\]  
where \(tr[D]\) is the trace and \(disc[D]\) is the discriminant of the matrix. Because of Equations (14) and (15), the conditions presented in the following equations must be met:  
\[D_{11}+D_{22}>0 \tag{16}\]  
\[D_{11}\ D_{22}-D_{12}\ D_{21}>0 \tag{17}\]  
\[\left(D_{11}-D_{22}\right)^{2}+4\ D_{12}\ D_{21}>0 \tag{18}\]  
The proposed mathematical model was reproduced in an algorithm created in Scilab. The purpose of the program is the numerical calculation of the differential Equations (3), (11) and (12) to obtain the optimized diffusion and convection coefficients by minimizing an error objective function (OF):  
\[\text{OF}=\frac{\sum_{1}^{4N}\big{(}C_{exp}-C_{model}\big{)}^{2}}{4N} \tag{19}\]  
Such OF was defined as the sum of the squared differences between the experimental values of the concentration (\(C_{exp}\)) and the ones obtained in the model (\(C_{model}\)) divided by the whole amount of measured concentrations in the lines (\(N\)). The relative standard error (\(RSE\)) [54] between the experimental concentration values and the model was also calculated by referencing them to the final concentration value (\(C_{final}\)):  
\[RSE=\sqrt{\frac{\sum_{1}^{4N}\Big{(}\frac{C_{exp}-C_{model}}{C_{final}}\Big{)} ^{2}}{4N-1}}\ 100\ [\%] \tag{20}\]The optimization algorithm looked for the values of \(D\) and \(K\) in the differential equations so that the theoretical model fitted the experimental values and also calculated the relative standard error. Additionally, the binary diffusion coefficients (\(D_{E}\)) were also calculated using the correlation proposed by [PERSON], [PERSON], and [PERSON] [52,55]:  
\[\frac{dC_{2}}{dt}=D_{21}\frac{\partial^{2}C_{1}}{\partial z^{2}}+D_{22}\frac{ \partial^{2}C_{2}}{\partial z^{2}}+\frac{\partial(V_{2}\ C_{2})}{\partial z} \tag{12}\]0  
The results were compared with the numerically obtained optimized diffusion constants of all binary mixtures. The binary diffusion coefficient between components A and B (\(D_{AB}\)) depends on the temperature (\(T\), K), pressure (\(P\), Pa), molecular weight (\(M\), g mol\({}^{-1}\)), and atomic diffusion volumes (\(v\)). The values of \(v\) for the atoms C, H, O, and F are 15.9, 2.31, 6.11, and 14.7, respectively [56]. The values of \(D_{E}\) of the ternary mixture were calculated by dividing the mixture into two pseudo-binary mixtures between one heavy component against the other two components as a whole. The estimated value for each heavy component was compared with the numerically calculated \(D_{\mathit{eff}}\) values.