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7.4 PSA Case Study - Post Combustion CO2 Capture Table 7.6: Comparison of rigorous model and ROM based on the performance variables ROM 92.51% 80.71% 38.29% 67.44% Performance variables N2 purity N2 recovery CO2 purity CO2 recovery Rigorous model 92.57% 80.21% 37.76% 66.27% Since the comparison is reasonable, we solve (7.31) with the POD-based ROM in Table 7.4 constructed and updated with every trust-region iteration with λ∗ = 0.05. 7.4.3.2 Optimization with Zero-order Correction We first solve (7.31) with the Zero-order Correction (7.8) applied for the objective function and the purity constraint. As mentioned before, ZOC can be computed cheaply as it doesn’t involve evaluation of the original objective and the constraint gradients. However, it doesn’t satisfy assumption (A3), although it ensures (A2) holds. Therefore, we cannot ensure that ZOC satisfies the fraction of Cauchy decrease (7.14). Our focus in this section is to observe if the POD-based ROM, besides accurately predicting the dynamics, can also predict the de- scent direction accurately without involving the actual gradients from the original optimization problem. In other words, since POD-based ROMs are reasonably accurate at any xk, we are inquisitive about whether they can also implicitly satisfy (A3) by their very construction, or whether a first-order correction (7.9) with accurate gradients is vital for ROM-based optimiza- tion for convergence to an optimum. As mentioned before, we use a box trust-region in this work. For problem (7.31), with a current trust-region radius ∆k, the shape of the box is defined as below |Ph − Ph,k| ≤ ∆k, |tp − tp,k| ≤ ∆k/2, |ua − ua,k| ≤ ∆k/4 |Pl − Pl,k| ≤ ∆k/4, |ta − ta,k| ≤ ∆k/2, (in kPa) (in sec) (7.32) (in cm/sec) With penalty parameter μ = 1000, we apply the exact penalty trust-region algorithm. Table B.1 in Appendix B lists all the trust-region iterations together with the values of the decision variables after the iteration ends (i.e., xk + sk ), CO2 purity pC O2 , and recovery rC O2 at xk + sk , Chapter 7. Trust-region Framework for ROM-based Optimization 156PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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