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7.4 PSA Case Study - Post Combustion CO2 Capture (7.14) hold. Then one has that lim ∥∇ψ(xk)∥ = 0 (7.16) k→∞ In other words, all limit points for Algorithm I converge to x∗, a first-order optimal point for Problem (7.1), when all the assumptions (AF1)–(AF3), and (A1)–(A4) hold, and a sufficient decrease (7.14) can be ensured. As mentioned before, (AF1)–(AF3), (A1) and (A4) are assumed to be true in this work. With Zero-order Correction, we can satisfy (A2), while First-order Correction can ensure both (A2) and (A3) are satisfied. With assumptions (A2) and (A3) being true, it can be shown that as the trust-region gets small enough, the linear part of ψkR(xk + s) dominates and we can always compute a step sk which lies in the steepest descent direction of ψkR(xk + s) (Cauchy step) and satisfies the fraction of Cauchy decrease condition (7.14). Moreover, with FOC, the Cauchy step obtained for Problem (7.11) coincides with the exact Cauchy step of Problem (7.11) with ψkR(xk +s) replaced with ψ(xk +s). Hence, Algorithm I is globally convergent. Assumptions (A2), and (A3) should necessarily be satisfied at the optimum to ensure that the Algorithm I converges to a solution that corresponds to the optimum of the original optimization problem (7.1) [29, 77]. Since we satisfy (A2) and (A3) with FOC for all the trust-region iterations, it is ensured that the algorithm will converge to the true optimum. We follow the approach presented in section 6.5.4.1 for our case studies, in order to verify if Algorithm I converges to the true optimum. 7.4 PSA Case Study - Post Combustion CO2 Capture 7.4.1 PSA Process and Model Equations We consider a 2-bed 4-step isothermal PSA process with an 85%-15% N2-CO2 feed mixture which is a typical composition of a post-combustion flue gas stream. In particular, the process is a Skarstrom cycle shown in Figure 2.3, and described in detail in section 2.4.1. Zeolite 13X is chosen as the adsorbent to separate CO2. We make following assumptions to develop a Chapter 7. Trust-region Framework for ROM-based Optimization 146PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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