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they get translated as where f ̄ , f ̄ 0 −1 , f ̄ N+1 , and f ̄ N+2 are average values for ghost cells. 2.5.2 Temporal Discretization f ̄ 0 f ̄ −1 f ̄ N+1 f ̄ N+2 = = = = 2f − f ̄ inlet 1 4f − 3f ̄ inlet 1 2f − f ̄ outlet N 4f − 3f ̄ outlet N (2.22a) (2.22b) (2.22c) (2.22d) For temporal discretization of the DAE system obtained after spatial discretization, we employ orthogonal collocation on finite elements (OCFE) technique for our work [40, 69]. OCFE is a discretization scheme which combines the method of weighted residuals with the finite element methods. The state temporal profiles are approximated at a finite number of points - the collocation points - by a family of polynomials. To illustrate the concept, we consider the following set of ordinary differential equation (ODE) dy = f(y(t), t), y(t0) = y0 (2.23) dt For discretization, the time domain is partitioned into nE finite elements of length hi, i ∈ [1,...,nE] such that nE hi = tf −t0, where t0 and tf are initial and final time, respectively. 2.5 Simulation Methodologies i=1 Thus, time at the end of each element i is defined as ti = t0 + im=1 hm. Next, we represent the time derivative of the state as a Lagrange polynomial of order nC, where nC is the number of collocation points, for each element i [17]. This leads to t = ti−1 + hiτ, y(t)=yi,0+hi τ ∈ [0,1], t ∈ [ti−1,ti], i = 1,...,nE n C dy (2.24) Ωj(τ) dt i,j Here τj , j = 1, . . . , nC are the collocation points which are usually the roots of an orthogonal polynomial of degree nC, yi,0 is the value of the state at the beginning of the element i, j=1 Chapter 2. Pressure Swing Adsorption 32PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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