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This result stems from a detailed numerical analysis of the advection-diffusion problem for channel possessing aspect ratio α>5, by solving the complete advection-diffusion equation Eq., enforcing both the infinite-column approximation and the Danckwerts condition. For fixed Peeff, the accuracy of the approximation in which axial diffusion is neglected improves as the aspect ratio α increases.
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This result stems from a detailed numerical analysis of the advection-diffusion problem for channel possessing aspect ratio α>5, by solving the complete advection-diffusion equation Eq., enforcing both the infinite-column approximation and the Danckwerts condition. For fixed Peeff, the accuracy of the approximation in which axial diffusion is neglected improves as the aspect ratio α increases.
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23
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72449172604
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The integrals Eq. converge for any finite value of Peeff. For 2D straight channels, and 3D circular channels in the presence of no-slip boundary conditions (Poiseuille flow), it can be proved that, for n>2, m out (n) ∼ Pe eff (n-1) /3, while m out (1) ∼log Peeff, as discussed in Sec. . In the presence of slip boundaries, the nth moment for any n>1 converges for Peeff →∞ towards a constant value corresponding to the nth moment of the purely kinematic residence-time distribution.
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The integrals Eq. converge for any finite value of Peeff. For 2D straight channels, and 3D circular channels in the presence of no-slip boundary conditions (Poiseuille flow), it can be proved that, for n>2, m out (n) ∼ Pe eff (n-1) /3, while m out (1) ∼log Peeff, as discussed in Sec.. In the presence of slip boundaries, the nth moment for any n>1 converges for Peeff →∞ towards a constant value corresponding to the nth moment of the purely kinematic residence-time distribution.
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24
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72449184262
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The eigenvalues/eigenfunctions have been obtained numerically by expanding ψ (y) in cosines, ψ (y) = σ k=0 N xk cos (kπy), and by solving the generalized eigenvalue problem -λBx=Ax, where x= (x0, x1,..., xN), A= (Ah,k =- (πh) 2 sh δh,k), s0 =1, sh =1/2, h>1, B= (Bh,k = i;y) dy).
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The eigenvalues/eigenfunctions have been obtained numerically by expanding ψ (y) in cosines, ψ (y) = σ k=0 N xk cos (kπy), and by solving the generalized eigenvalue problem -λBx=Ax, where x= (x0, x1,..., xN), A= (Ah,k =- (πh) 2 sh δh,k), s0 =1, sh =1/2, h>1, B= (Bh,k = ∫01 u (y) cos (hπy) cos (kπy) dy).
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25
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72449209891
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The fact that limε→0 (ε) = r0 approaches a constant value derives from the most critical situation for the c spectrum expressed by Eq. The observation r0 ∼O (10-1) for no-slip Poiseuille flow is numerical.
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The fact that limε→0 (ε) = r0 approaches a constant value derives from the most critical situation for the c spectrum expressed by Eq.. The observation r0 ∼O (10-1) for no-slip Poiseuille flow is numerical.
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