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4.4 Case Studies and Computational Results is chosen for the lower bound on CO2 purity. Besides Pads and Pdes, we also consider bed length BLen, and cycle time Tc as decision variables. Since a moving finite element strategy is adopted, the length of each finite element is also considered as an optimization variable. Because none of the decision variables are functions of time, the optimal control problem (3.1) becomes a dynamic optimization problem, which becomes the following NLP after discretizing PDAEs in both space and time. max CO2 recovery (from Equation (3.14d)) s.t. c(w) = 0 (fully discretized Equations (3.2)-(3.13)) CO2 purity ≥ 0.4 (from Equation (3.14b)) Pads ≥ Pd Pdes ≤ Pfeed Pa ≥ Pfeed 1 m ≤ BLen ≤ 6 m 20 sec ≤ Tc ≤ 2400 sec 101.32 kPa ≤ Pads ≤ 1000 kPa Pdes ≥ 10 kPa (4.1a) (4.1b) (4.1c) (4.1d) (4.1e) (4.1f) (4.1g) (4.1h) (4.1i) Here w and c(w) = 0 represent the set of completely discretized variables and model equations, respectively. Constraint (4.1c) ensures that the pressure always drops around the pressure reducing valve in the superstructure. Similarly, constraints (4.1d) and (4.1e) ensure that the gas is never expanded by the heavy gas and the feed compressors, respectively. The rest of the inequalities are bounds on the decision variables. With 24 temporal finite elements and 20 spatial finite volumes, we solved the NLP in AMPL [78] using IPOPT. Table 4.3 includes a summary of the optimization results. With 35,022 variables and 29 degrees of freedom, we were able to solve it to optimality in around 3 CPU hours on an Intel Quad core 2.4 GHz machine with 8 GB RAM. Optimal moving finite element lengths and cycle time of 2140 sec. yield an optimal step time of 760 sec. for the pressurization (and depressurization) step, and 410 sec. for the feed (and light reflux) step. Such a long pressurization step is due to a small amount of feed during that step, which requires longer time for bed to get pressurized and CO2 to adsorb. At the optimum, the cycle Chapter 4. Superstructure Case Study: Post-combustion CO2 Capture 55PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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