The design of adsorption systems for separation of CO2/N2 in carbon capture applications is notoriously challenging because it requires constrained multiobjective optimization to determine appropriate combinations of a moderately large number of system operating parameters. The status quo in the literature is to use the nondominated sorting genetic algorithm II (NSGA-II) to solve the design problem. This approach requires 1000s of time-consuming process simulations to find the Pareto front of the problem, meaning it can take days of computational time to obtain a solution. As an alternative approach, we have employed a Bayesian optimization algorithm, the Thompson sampling efficient multiobjective optimization (TSEMO). For constrained productivity/energy usage optimization, we find that the TSEMO algorithm is able to find an essentially identical solution to the design problem as that found using NSGA-II, while requiring 14 times less computational time. We have used the TSEMO algorithm to design a postcombustion carbon capture system for a 1000 MW coal fired power plant using two adsorbent materials, zeolite 13X and ZIF-36-FRL. Although ZIF-36-FRL showed promising process-scale performance in previous studies, we find that the industrial-scale performance is inferior to the benchmark zeolite 13X, requiring a 21% greater cost per tonne of CO2 captured. Finally, we have also tested the performance of the Bayesian design framework when coupled with a data-driven machine learning process modeling framework. In this instance, we find that the incumbent NSGA-II offers better computational performance than the Bayesian approach by a factor of 3.