Design and Operation of Pressure Swing Adsorption Processes

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Design and Operation of Pressure Swing Adsorption Processes ( design-and-operation-pressure-swing-adsorption-processes )

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αf ∇f􏳧R sCf= k τf= f 7.5 Hybrid Filter Trust-region Algorithm in this section, we rely on the gradients of the objective and the constraints obtained from the reduced-order model to promote such descent. Formally, we evaluate Cauchy steps (steepest descent direction) for both objective function and infeasibility as below   0 if sC = 0 ∥∇f􏳧R∥ k f􏳧R(x )−f􏳧R(x +sC) kkkkf otherwise ifsCθ =0 otherwise (7.41) (7.42)  αf ∥∇f􏳧R∥ k 0 sC= k τ= R R C αθ∇θ􏳧R θ θ θ􏳧 ( x ) − θ􏳧 ( x + s ) ∥∇θ􏳧R∥ kkkkθ k  αθ ∥∇θ􏳧R ∥ k for some αf , αθ ∈ (0, 1). Here f􏳧R is constructed with ZOC in equation (7.8), and θ􏳧R is given kk by equation (7.35) with 􏳧cR(x), and 􏳧cR(x) given by equation (7.8). We note that since f􏳧R and EIk θ􏳧R are based on the ROM for kth iteration, their gradients, and thus sC and sC can be cheaply kfθ evaluated for each trust-region iteration. Hence, with a little computational expense, we can determine if the reduced-order model can predict descent for f􏳧R or θ􏳧R or both. If τf > 0 or τθ > 0, it can guarantee descent for the reduced objective function f􏳧R or reduced infeasibility k θ􏳧R, respectively. Therefore, the normal subproblem with ZOC is solved only if τθ > 0, and k similarly, the tangential subproblem with ZOC is solved only when τf > 0. If both cannot be ensured, algorithm moves to Section II where FOC with exact gradients is used. In Section I, we also incorporate POD subspace augmentation which involves adding more POD basis functions to improve accuracy of the existing ROM for the kth iteration, and thus enhancing its ability to accurately predict the descent direction for f􏳧R or θ􏳧R or both. To kk construct a reduced-order model, we choose the number of basis functions M by deciding on an error tolerance level λ∗. During the course of the algorithm, if we encounter a situation when both τf ≤ 0 and τθ ≤ 0, tolerance level λ∗ is reduced to increase the POD subspace dimension M and more basis functions are added to the ROM. This is repeated until either one of the τf or τθ or both become positive, or we hit the maximum allowable limit for POD subspace dimension Mmax. Once we reach Mmax, algorithm switches to Section II. kk Chapter 7. Trust-region Framework for ROM-based Optimization 169

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