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and axial directions. We let ξ= r Rbed u (ξ) = ε⃗v (ξ) ⟨v⟩ za= z Rbed′ c∗ = c − c c′′ − c′ t1=⟨v⟩t Rbed ′ n∗ = n − n n′′ − n′ (4.2) Per = 2⟨v⟩Rbed εDr Pea = 2⟨v⟩Rbed εDz where ⟨v⟩ is the mean superficial velocity, c′ and n′ are presaturated concentrations (zero for an initially clean bed), c′′ is the feed concentration, and n′′ is the corre- sponding adsorbed-phase concentration. Equation 4.1 becomes n′′ − n′ ∂n∗ ′ ∂c∗ ∂c∗ 2 1 ∂ ∂c∗ 2 ∂2c∗ ρbc′′−c′ ∂t +ε∂t +u(ξ)∂z =Pe ξ∂ξ ξ∂ξ +Pe ∂z2 (4.3) 11araa A new coordinate is defined to follow the stoichiometric front of the adsorption wave. We let where Λ, the partition ratio, is given by Λ=ρb c′′ −c′ ζ=za− t1 Λ+ε′ (4.4) (4.5) n′′ − n′ Assuming the bed length L is large, a constant pattern profile will develop. Further- more, for Λ ≫ 1, Coppola and LeVan4 show that the fluid-phase accumulation term in eq 4.3 is negligible, transforming it to ∂n∗ ∂c∗ 2 1 ∂ ∂c∗ 2 ∂2c∗ −∂ζ +u(ξ)∂ζ =Perξ∂ξ ξ∂ξ +Pea ∂ζ2 (4.6) with boundary conditions in the axial direction of c∗→1 as ζ→−∞ c∗→0 as ζ→+∞ (4.7) We describe the local velocity by a small perturbation from plug flow using u (ξ) = 1 + γf (ξ) (4.8) 53PDF Image | TEMPERATURE SWING ADSORPTION COMPRESSION AND MEMBRANE SEPARATIONS
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