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product. If required, it can be directly included in the following equations. γRTd Wtotal=γ−1 i (φ(t)vfeedCfeed,i + α(t)(−vd)Cd,i) η P −1 feed γ−1 Pa γ 3.4 Solution Strategy c i α(t)(−vd)Cd,i + η minP ,P −1 h atm des γ−1 i (−vd )Cd,i Patm γ Pfeed Pfeed γ−1 γ γ−1 γ +ηmax0,P −1 v γ φ(t)vf eed Pf eed des γ−1 Pf eed γ +γ−1η P −1dt f g inlet (3.15a) (3.15b) Power = Wtotal (1 − α(t))vd(t)Cd,H (t) dt Here, the max function ensures that the work done by the vacuum generator is zero when Pdes is more than the atmospheric pressure Patm. Similarly, since the vacuum generator discharges heavy reflux at Patm, the min function ensures a proper upstream pressure for the heavy product compressor. Since min and max functions introduce non-differentiability, the following smoothing approximations are adopted [20]. A value of 0.01 is used for ε in the following equations. min(f1(x), f2(x)) = f1(x) − max(0, f1(x) − f2(x)) (3.16a) max(0, f(x)) = 0.5 f(x) + f(x)2 + ε2 (3.16b) 3.4 Solution Strategy We adopt a complete discretization approach to solve the system of PDAEs in Table 3.1. The PDAEs are converted into a set of algebraic equations by discretizing the state and the control variables both in space and time. As a result, the PDAE-constrained optimal control problem (3.1) gets converted into a large-scale nonlinear programming (NLP) problem. One of the advantages of this approach is that it directly couples the solution of the PDAE system with the optimization problem. The model equations are solved only once at the optimum and the Chapter 3. PSA Superstructure 44PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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