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-1 0 1 2 j-1 j j+1 N N+1 N+2 Ghost cells for boundary flux 2.5 Simulation Methodologies f1 f2 f j−1 fj f j+1 fN Ghost cells for boundary flux j-1/2 j+1/2 Figure 2.7: Finite volume discretization scheme In a finite volume method, the spatial domain is divided into discrete volume elements (or cells) and we define average values for state variables over each element. For instance, for one-dimensional finite volume method, spatial division is done as shown in Figure 2.7, and average density over each element j is defined as xj+1/2 xj −1/2 Here j ± 1/2 are walls of volume j, ∆ is the length of the volume j, and f ̄ is the volume jj average of f(x) which for example can represent bulk phase mass concentration or enthalpy. PDAEs are then integrated in the spatial domain and the state variables are replaced by their cell average values. For instance, Equation (2.14) after applying finite volume discretization becomes (for j-th cell and i-th component) f(x)dx = ∆ f ̄ (2.18) jj dC ̄dq ̄1 ε i,j + (1 − ε )ρ i,j + v C − v C = 0 (2.19) b dt b s dt ∆j j+1/2 i,j+1/2 j−1/2 i,j−1/2 Here vj+1/2Ci,j+1/2 and vj−1/2Ci,j−1/2 are mass fluxes across the walls of volume j, resulting from the approximation of the integral in Equation (2.18). Since Equation (2.17) (or any other equation) always evaluates velocity at cell walls, only Ci,j+1/2,Ci,j−1/2 (or in general, wall values of the densities) need to be approximated in terms of cell average values by interpolation. For upwind finite volume methods, such an interpolation depends on the direction of the flux. In this work, we use the following flux direction-based formulation to approximate wall values of densities, such as bulk phase mass concentration or enthalpy [59, 91, 125] Chapter 2. Pressure Swing Adsorption 30PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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