## 3 Results and Discussion
Simulation results obtained from the MOT application based on [PERSON]'s model are presented on \(\alpha_{\text{O2/N2}}=\text{f(PO_{2})}\) dependency graphs (Figures 5-8). The above charts also take into account [PERSON]'s line. Figure 4: Menu window: More options. Figure 3: [PERSON]’s model window. This Robeson's upper bound line represents the limits on the selectivity-permeability trade-off behavior achievable with polymer membranes. Nowadays, the main research objective is to find polymeric structures that exceed the empirical upper bound limits. The main parameters for gas separation are the permeability \(P\) of a specific component of the gas mixture and the selectivity coefficient \(\alpha\). They are trade-off parameters, because usually, the selectivity coefficient \(\alpha\) decreases with increasing permeability \(P\) of the more permeable gas component. This trade-off relationship was  
Figure 5: Comparison of experimental results (navy blue point) for the zeolite 4A-2, 2\({}^{\prime}\)-BAPB + BPADA mixed-matrix membranes with 3 different filler additions (M1, M2 and M3) with [PERSON] theoretical results (red point) and their position relative to the [PERSON] upper bound line (green line). shown to be related to an upper bound relationship where the \(log\)\(\alpha\) versus the \(\log\)\(P\) determined a limit for achieving the desired result of a high separation factor combined with a large permeability for polymeric membranes. The upper bound relationship was shown to be valid for many of the gas pairs, including O\({}_{2}\)/N\({}_{2}\), CO\({}_{2}\)/CH\({}_{4}\), H\({}_{2}\)/N\({}_{2}\), He/N\({}_{2}\), H\({}_{2}\)/CH\({}_{4}\), He/CH\({}_{4}\), He/H\({}_{2}\), H\({}_{2}\)/CO\({}_{2}\) and He/CO\({}_{2}\)[50]. The upper bound relationship could be expressed by the following equation [50]:  
\[Pi=k\alpha_{ij}^{n}, \tag{7}\]  
where:  
_Pi_--the permeability of the more permeable gas,  
\(\alpha\)--the selectivity coefficient (\(Pi\)/\(P\)), and  
\(n\)--the slope of the log-log limit. Figure 6: Comparison of experimental results (navy blue point) for the zeolite 4A-Ultem mixed-matrix membranes with 2 different filler additions (M1 and M2) with [PERSON] theoretical results (red point) and their position relative to the Robeson upper bound line (green line). The presented simulation and experimental results concerned hybrid membranes based on polymer matrices, such as 2, 2\({}^{\prime}\)-BAPB + BPADA, Ultem and poly (vinyl acetate), and an inorganic additive in the form of zeolite 4A. The experimental results (Tables 1 and 2) concerning mixed-matrix membranes based on rigid polyimide matrices have shown improvements in the transport parameters [39]. It was stated that the selectivity values were similar to the predicted ones. In contrast, the permeabilities decreased significantly, especially for larger filler addition (Figures 5 and 6). This could be caused by polymer adsorption onto the surface of the sieve and its chain inhibition. In order to explain this phenomenon, a theoretical model of a hybrid membrane was created, which included the existence of three phases: polymeric, inorganic and interphase. In the last one, the permeability is lower than in the bulk polymeric phase [28; 29; 39]. In this case could be used the modified Maxwell model, where gas-penetrant permeability can be predicted using a two-step approach.