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M yiR(x,t) = y0i(x)+ j=1 M T R (x, t) = T0 (x) + j=1 ayij(t)φyij(x) aT j (t)φT j (x) qiR(x,t) = q0i(x)+ P R (x, t) = P0 (x) + M j=1 M j=1 aqij(t)φqij(x) aP j (t)φP j (x) Here y0i(x), q0i(x), T0(x), and P0(x) are snapshot averages for mole fraction, solid loading, temperature, and pressure, respectively. Gas density, specific heat and equilibrium solid con- centrations are calculated explicitly in terms of yiR, TR, and PR. Table 6.6 shows DAEs of the reduced-order model obtained after Galerkin projection of the model in Table 6.2 on to these POD basis functions. Since the snapshots are obtained after the rigorous model achieves cyclic steady state, snapshots for both beds for the corresponding steps are identical. For instance, snapshots collected during pressurization step for bed 1 are identical to the snapshots of the pressurization step for bed 2. Therefore, we construct reduced-order model for only one bed and the other bed is ignored. Coupling for the adsorption/desorption steps of the two beds is ensured by the adsorption and desorption steps of the same bed. Hence, a greater model reduction is achieved 6.5 Case Study - Hydrogen PSA The snapshots are generated only after the 2-bed system attains cyclic steady state. To achieve CSS, we use a successive substitution method in which first the PSA cycle is simulated with random initial conditions, and then a series of simulations are performed with initial conditions of each new cycle taken from the final condition of the previous cycle. This is executed successively until the bed conditions do not change from cycle to cycle. Since a bench-scale PSA model is considered in this case, we achieve cyclic steady state after 100-120 cycles. After obtaining snapshots, separate POD basis functions are generated for pressurization, adsorption, depressurization and desorption steps. Moreover, we derive separate POD basis functions for gas phase mole fractions, solid phase loadings, temperature, and pressure. Similar to Equation (6.17), these state variables are then expressed in terms of the correspoding POD basis as below Chapter 6. Reduced-order Modeling for Optimization 117PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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