Carbon capture is a promising option to mitigate CO2 emissions from existing coal-fired power plants, cement and steel industries, and petrochemical complexes. Among the available technologies, membrane-based carbon capture presents the lowest energy consumption, operating costs, and carbon footprint. In addition, membrane processes have important operational flexibility and response times. On the other hand, the major challenges to widespread application of this technology are related to reducing capital costs and improving membrane stability and durability. To upscale the technology into stacked flat sheet configurations, high-fidelity computational fluid dynamics (CFD) that describes the separation process accurately are required. High-fidelity simulations are effective in studying the complex transport phenomena in membrane systems. In addition, obtaining high CO2 recovery percentages and product purity requires a multi-stage membrane process, where the optimal network configuration of the membrane modules must be studied in a systematic way. In order to address the design problem at process scale, we formulate a superstructure for the membrane-based carbon capture, including up to three separation stages. In the formulation of the optimization problem, we include reduced models, based on rigorous CFD simulations of the membrane modules. Numerical results indicate that the optimal design includes three membrane stages, and the capture cost is 45.4 $/t-CO2.