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dy dt i,j = dy (ti,j ) with ti,j = ti−1 + hiτj , and Ωj (τ ) is a polynomial of order nC , satisfying dt τ 0 lj(τ)dτ, τ ∈ [0,1], n t ∈ [ti−1,ti] j=1,...,nC i = 1,...,nE Ωj(τ) = wherelj(τ)= τ −τ , C τ−τk k=1,̸=j j k (2.25) 2.6 Concluding Remarks Ωj(0) = 0, dΩj = δj,k, j,k = 1,...,nC dτk Here we use Radau collocation points with τj < τj+1, j = 1,...,nC −1, and τnC = 1 for every element. Since the last collocation point lies at the end of the finite element i, continuity of the state profiles is easily ensured by setting yi,0 = y(ti−1,nC ) = y(ti−1). C yi,0=yi−1,0+hi−1 (2.26) n j =1 dy Ωj(1) dt While state variables are continuous across the finite elements, control variables can present discontinuities at the boundaries of the elements. We prefer Radau collocation points because they allow us to set constraints easily at the end of each element in an optimization problem, and to stabilize the system more efficiently if high index DAEs are present [33, 28]. To de- termine polynomial coefficients dy , we substitute Equation (2.24) into Equation (2.23) and dt i,j enforce the resulting algebraic equations at the collocation points τj, which leads to dy dy = dt(ti,j) = f(y(tij),tij), j = 1,...,nC, i = 1,...,nE (2.27) where y(tij) is obtained from Equation (2.24). dt i,j 2.6 Concluding Remarks Beginning with a review of the fundamentals of adsorption phenomena, we described how the physics of pressure swing adsorption processes can be modeled mathematically in the form of PDAEs. It is clear from the description that the fundamentals at the molecular level are relatively well understood and characterized, and it is possible to construct fairly i−1,j Chapter 2. Pressure Swing Adsorption 33PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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