## 4 Conclusions  
In this paper, the authors proposed a modern computer application MOT (Membrane Optimization Tool) to model the gas transport processes through mixed-matrix membranes MMMs.  
The current version of application was based on the Maxwell model, which, can be successfully used to model the gas transport through the simplest types of hybrid membranes without any defects, consisting of two phases, organic and inorganic. The operation of the computer application has been verified on the basis of experimental results of O\({}_{2}\)/N\({}_{2}\) separation for four types of hybrid membranes, consisting of various types of the polymer matrix, such as poly (vinyl acetate), 2, 2\({}^{\prime}\)-BAPB + BPADA, Ultem, hyperbranched (ODPA-MTA) and zeolite 4A.  
Two errors (the average absolute relative error and the root-mean-square error) were calculated in order to compare the theoretical MOT-predicted results with the experimental data obtained from the literature and the authors' own research.  
It was found that the mean absolute relative percentage ranges from 29% to 36%, while the root-mean-square error was in the range of 10% to 29%. The article presents also the preliminary comparison of MOT predicted gas transport data through zeolite 4A-HBPI membranes, obtained with two models, namely Maxwell and Bruggeman. It was found that the errors calculated from the Bruggeman model were comparable with errors obtained based on the Maxwell model proposed earlier, namely AARE was 33% and RMSE was 18%.  
To obtain more accurate reproduction of experimental results, further versions of the proposed application will be extended with next-generation permeation models, allowing for the description of transport in more complex systems with the possibility of taking into account possible membrane defects.  
Figure 10: (**a**) [PERSON]’s model MOT window and (**b**) comparison of the theoretical MOT-predicted data for two different models: Maxwell (red points) and Bruggeman (blue points) on HBPI mixed-matrix membranes with various volume fractions of zeolite 4A particles relative to the Robeson upper bound line (green line).  
The created application will allow us to verify the behavior and effectiveness of innovative membranes before they are made. In addition, it will allow us to determine their performance, verify the properties of alternative membranes and reduce capital-intensive research and time.  
It will be an excellent tool for scientists studying the separation of various gas mixtures using the inorganic-organic hybrid membranes.  
Conceptualization, A.R. ([PERSON]) and A.R. ([PERSON]); Methodology, A.R. ([PERSON]) and A.R. ([PERSON]); Software, A.R. ([PERSON]); Investigation, A.R. ([PERSON]) and A.R. ([PERSON]); Resources, P.S.; Data Curation, A.R. ([PERSON]) and A.R. ([PERSON]); Writing-Original Draft Preparation, A.R. ([PERSON]) and A.R. ([PERSON]); Writing-Review & Editing, A.R. ([PERSON]) and A.R. ([PERSON]); Supervision, A.R. ([PERSON]).; Project Administration, A.R. ([PERSON]); Funding Acquisition, A.R. ([PERSON]).  
This research was funded by Narodowe Centrum Nauki (NCN; The National Science Centre Poland), grant number DEC-2017/01/X/ST10/00878 and The Silesian University of Technology, grants number 04/040/RGJ17/0058 and 06/010/RGJ18/0026 (Rector's Grants in Research and Development).  
The authors would like to thank The National Science Centre for providing financial support under the project no. DEC-2017/01/X/ST10/00878. This publication was also supported under the Rector's Grants in Research and Development. The Silesian University of Technology, Grant no.04/040/RGJ17/0058 and 06/010/RGJ18/0026.  
The authors declare no conflict of interest.