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Figure 6.3: Singular values of gas-phase mole fraction of methane and temperature operating steps with only 5-6 spatial POD basis functions, instead of 35 spatial volumes. With 35 spatial volumes, the rigorous model comprises a total of 2800 DAEs (including both beds and all four operating steps), while with 5 basis functions the ROM obtained after Galerkin projection comprises a mere 200 DAEs which is 1/14th of the rigorous system. Hence, we observe a significant model reduction. We discretize DAEs of the reduced-order model for all the operating steps in time with 30 finite elements, and the resulting algebraic equations are solved simultaneously in AMPL using IPOPT. Instead of solving the discretized ROM repeatedly for CSS, we consider initial conditions as decision variables and reduced CSS conditions (shown in Table 6.6) are solved simultaneously with the model equations in AMPL. Figure 6.4 compares profiles of gas-phase methane mole fraction obtained for the rigorous model and ROM with 5 POD basis functions. We observe that 5-rank approximation captures the dynamics of the problem very well and profiles are nearly identical. Temperature profiles in Figure 6.5 also depict such similarity. We note that all the profiles plotted are at CSS. We also observe that the mole fraction profiles 6.5 Case Study - Hydrogen PSA Pressurization Adsorption Depressurization Desorption 1121 10 10 10 10 00 10 10 00 10 −1 10 −1 10 −2 10 −2 10 −4 10 10 −1 10 −2 −3 −6 −2 10 10 10 10 0 5 10 0 5 10 0 5 10 0 5 10 (a) Singular values for gas−phase mole fraction of methane (on log scale) Pressurization Adsorption Depressurization Desorption 2354 10 10 10 10 12 10 10 010 10 10 10 −1 0 10 10 2 10 0 10 −2 −1 −5 −2 10 10 10 10 0 5 10 0 5 10 0 5 10 0 5 10 (b) Singular values for temperature (on log scale) Chapter 6. Reduced-order Modeling for Optimization 120PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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