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(2.20a) (2.20b) (2.20c) (2.20d) Here f ̄ is the cell average value for state variable such as gas-phase concentration C, solid j loading q, or temperature T, θ(r) is the flux limiter and rj+1/2 is a ratio which measures the smoothness of the profile. If rj+1/2 is close to 1, the profile is presumably smooth. If rj+1/2 is far from 1, there must be a sharp discontinuity at xj. Depending on the value of rj+1/2, θ(r) applies proper correction. If the profile is smooth, θ(r) preserves second or higher-order nature of the discretization, otherwise near steep regions θ(r) reduces the order of the discetization to eliminate oscillations. Flux limiters take various forms to perform aforementioned functions. Darwish et al. [59] and Hirsch [91] give a detailed description of several flux limiters such as Minmod, Superbee, and Van Leer limiters. While Minmod limiter is too diffusive and Superbee doesn’t perform adequately for smooth regions, Van Leer has properties between the two and thus is more desirable. Hence, we use the Van Leer flux limiter for our case studies, which has the following form θ(r) = r + |r| (2.21) 1 + |r| Boundary conditions are incorporated in the finite volume scheme with the help of “ghost cells”, as illustrated in Figure 2.7. Ghost cells are required because Dirichlet or Neumann boundary conditions specified for the problem apply to the walls of the first or Nth finite volume and they need to be translated to corresponding cell average value to get accounted for in the discretization scheme. Thus, boundary conditions at the walls are usually translated to the average values of fictitious ghost cells using some form of extrapolation. For instance, if finlet and foutlet are given as boundary conditions and a linear extrapolation scheme is chosen, rj+1/2 = Forv<0f =f ̄+1θ(r)f ̄−f ̄ j+1/2 j+1/2 j+1 2 j+1/2 j j+1 f ̄ −f ̄ rj+1/2 = j+1 j+2 f ̄ −f ̄ j+1 j 2.5 Simulation Methodologies For v j+1/2 ≥ 0 f = j+1/2 f ̄ + 1 θ(r j 2 f ̄ − f ̄ j j−1 ) f ̄ j+1 − f ̄ j j+1/2 f ̄ − f ̄ j j+1 Chapter 2. Pressure Swing Adsorption 31PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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