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excessive computational effort of getting intermediate solutions is avoided [28]. However, the performance of this approach substantially depends upon the optimization solver, and therefore it is crucial to choose an efficient NLP solver. Hence, we use the state-of-the-art NLP solver IPOPT 3.4 for our case studies. This interior point solver uses a barrier method to handle inequalities and exact second derivative information for faster convergence to the optimum [195]. To capture steep adsorption fronts, avoid oscillations in the solution, and model conser- vative properties of the system, we apply a first-order finite volume method for spatial dis- cretization. For the temporal domain, we apply orthogonal collocation on finite elements with a Radau collocation scheme. Radau collocation allows us to set constraints at the ends of the finite elements [103]. A 3-point collocation scheme is used for state variables while con- trol variables are considered to be piecewise constant. While control variables are allowed to be discontinuous, we ensure state variables demonstrate continuity in their profiles. We also consider a moving finite element strategy in which the size of each temporal finite element is considered a decision variable. With moving finite elements, it is possible to locate optimal breakpoints of the control variables with variable element lengths. Appropriate bounds are imposed on the variable element lengths of each finite element to guarantee accuracy of the discretization. Because spatial discretization together with a pre-determined temporal discretization with- out any error checking mechanism for temporal integration can cause inaccuracies to creep in the NLP solution obtained from IPOPT, verification of the solution with an accurate dynamic simulation is essential. Therefore, we perform dynamic simulations in MATLAB [1] at the optimal values of the decision variables obtained from IPOPT. The DAE system obtained af- ter applying method of lines is integrated in MATLAB at the optimal values of the decision variables. The profiles and performance variables obtained from MATLAB are then compared with those obtained from IPOPT. For the method of lines approach in MATLAB, we use a first-order finite volume method for spatial discretization and ode15s for temporal integration. 3.4 Solution Strategy Chapter 3. PSA Superstructure 45PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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