## 1 Introduction
### Study Progress  
The membrane gas-capture process has been examined for several separation purposes, such as air separation, hydrogen purification, and carbon-capture methods. For the membrane CO\({}_{2}\) removal process, several membrane materials have been studied and enhanced to meet carbon dioxide separation standards, like molecular sieves, fixed-site carriers (FSC), inorganic, and polymeric membranes [22]. The materials must be prepared and designed to be harnessed for a particular CO\({}_{2}\) removal process under any conditions. The essential properties of any membrane material are defined in terms of its mechanical durability, as well as its chemical and thermal stability, in addition to its resistance against impurities such as SO\({}_{2}\), NOx, fly ash, etc. Consequently, for a specific CO\({}_{2}\)-capture process, the selection of the membrane materials depends on these properties and operating conditions [20; 23]. Membranes should have an effective selectivity and high CO\({}_{2}\) permeability for the membrane CO\({}_{2}\)-capture process to be compatible with the chemical absorption process. For most polymeric membranes, \"the Robeson upper-bound\", is a trade-off between permeability and selectivity [24]. Due to significant advancements in membrane performance since the original upper limit was established in the 1950s, the updated upper bound has an improved bound [25]. Up to the present, permeability and selectivity for different materials have been studied and considered the main features for selecting membrane materials for carbon dioxide recovery; different articles have studied membrane materials with a high permeability, as can be found in references [26; 27]. [PERSON] et al. (2010) investigated various membrane layers at different operating conditions to examine their influence on membrane performance to recover CO\({}_{2}\) from different streams in terms of permeability and selectivity. The author found that improving the membrane material with a higher CO\({}_{2}\) permeable layer helped to increase the CO\({}_{2}\) recovery rate with a higher purity. Enhancing the membrane permeability and selectivity are the main challenges to study for a more efficient CO\({}_{2}\) separation process. However, polymeric materials are the most mature materials harnessed for CO\({}_{2}\) removal [28; 29]. The pressure difference across the membrane module is the mass transport driving force of a carbon-capture process. In the case of flue gas flows at atmospheric pressure, improving the pressure difference involves a compressor before the membrane unit, a vacuum pump to extract flue gas stream on the permeate side, or both for the membrane system to compensate for the low CO\({}_{2}\) composition in the flue gas generated from a CFPP [30; 31; 32]. To compete with chemical absorption utilization, various membrane systems in post-combustion with different stages were studied to reduce the total cost, which is fundamentally caused by electrical energy demands and the membrane size [33; 34; 35; 36; 37]. [PERSON] et al. (2022) researched the effect of the compression ratio across the membrane on energy consumption with a flue gas flow of 18.7 kmol/s. The cost and capture rate of the process were estimated by changing several configurations with different indicators, such as the pressure difference across the membrane unit and membrane surface. The authors declared that an increasing pressure ratio helped to raise the energy consumption and the investment and operating costs for the process. Despite the fact that the energy requirements could be decreased by a suitable membrane configuration, an extensive economic assessment of the considered process was not accomplished. [PERSON] et al. (2013) discussed a membrane configuration of two units integrated into a conventional coal-fired power plant (CFPP) for carbon dioxide emissions reduction.