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β(t), and φ(t) shown in Figure 3.3 translate into the classical 2-bed 4-step Skarstrom cycle (FP,F,CnD,LR) [176]. CoB generates pressurization (FP) and feed (F) steps, while CnB si- multaneously generates depressurization (CnD) and light reflux (LR) steps. Thus, the overall cycle includes these four steps (FP,F,CnD,LR). In an actual 2-bed PSA unit, after performing its steps, CoB will follow the steps of CnB and vice-versa. However, in the mathematical framework, this is realized by giving final conditions of CoB as the initial conditions for CnB and vice-versa, thus modeling the true 2-bed behavior. In other words, we utilize Multibed approach and simulate both beds simultaneously (cf. section 2.4.3). 3.2.3 Optimal Control Problem It is possible to obtain a plethora of distinct PSA configurations with this superstructure because of an endless number of shapes that the profiles of α(t), β(t), φ(t), Pads(t) and Pdes(t) can take. As a consequence of this, we obtain an optimal sequence of operating steps, along with other decision variables such as cycle time, step times and bed dimensions, by solving the following optimization problem. min Φ(z(x, tf ), y(x, tf ), α(tf ), β(tf ), φ(tf ), Pads(tf ), Pdes(tf ), z0, p) ∂z ∂z = 0 s.t. f ∂t , ∂x, z(x, t), y(x, t), α(t), β(t), φ(t), Pads(t), Pdes(t), z0, p zCoB(x, 0) = zCnB(x, tf ) zCnB(x, 0) = zCoB(x, tf ) s(z(x, t), y(x, t), α(t), β(t), φ(t), Pads(t), Pdes(t), p) = 0 g(z(x, t), y(x, t), α(t), β(t), φ(t), Pads(t), Pdes(t), p) ≤ 0 0 ≤ (α(t), β(t), φ(t)) ≤ 1 bL ≤ (Pads(t), Pdes(t), p) ≤ bU (3.1) Here Φ is the objective function related to overall power consumption, component purity or recovery. It can depend upon differential variables z(x,t), algebraic variables y(x,t), control variables α(t), β(t), φ(t), Pads(t), and Pdes(t), initial conditions z0 and other decision variables Chapter 3. PSA Superstructure 40 3.2 MethodologyPDF Image | Design and Operation of Pressure Swing Adsorption Processes
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