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THE USE OF FLUX-AVERAGED DIFFUSIVITIES TO MODEL MULTICOMPONENT DIFFUSION

THE USE OF FLUX-AVERAGED DIFFUSIVITIES TO MODEL MULTICOMPONENT DIFFUSION. Claudio Olivera-Fuentes TADiP Group Thermodynamics and Transport Phenomena Dept. Simón Bolívar University Caracas, Venezuela. Contents. Constitutive Equations Exact and Approximate Solutions

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THE USE OF FLUX-AVERAGED DIFFUSIVITIES TO MODEL MULTICOMPONENT DIFFUSION

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  1. THE USE OF FLUX-AVERAGED DIFFUSIVITIES TO MODEL MULTICOMPONENT DIFFUSION Claudio Olivera-Fuentes TADiP Group Thermodynamics and Transport Phenomena Dept. Simón Bolívar University Caracas, Venezuela

  2. Contents • Constitutive Equations • Exact and Approximate Solutions • Definition of Flux-Averaged Diffusivities (FADs) • Examples • Comments and Conclusions

  3. Constitutive Equations

  4. Maxwell-Stefan equations [1] • Molecular diffusion at constant T, P • Fik friction coefficients (resistances) = Fki ? • Dik MS diffusivities (conductances) = Dki ?

  5. Maxwell-Stefan equations [2] • Ji, diffusive or relative fluxes • Ni, net or absolute fluxes

  6. Diffusive interactions • Principle of equipresence • Duncan & Toor (1962) • Diffusion barrier: Potential gradient exists, yet component does not diffuse • Reverse diffusion: Component diffuses “uphill” from smaller to larger potential • Osmotic diffusion: Component diffuses although no potential gradient exists

  7. Binary diffusion • MS for 2-component systems • Fick’s “law” • DAB binary or “ordinary” diffusivity  DBA

  8. Exact v. Approximate Solutions

  9. Ideal gas mixtures • Thermodynamic & diffusive ideality • Simplified MS equations

  10. Continuity equations • Steady-state diffusion • One-dimensional transport • No homogeneous reaction

  11. Unified treatment

  12. Transformed equations • Continuity • Constitutive • Determinacy: boundary values, flux ratios, etc.

  13. Matrix solution, exact • Solve as vector/matrix ODE (Amundson, 1966)

  14. Matrix solution, linearized • Stewart & Prober (1964), Toor (1964) • Uncouple by diagonalization, solve as Fick’s law in pseudo-composition space

  15. Equivalent diffusivity methods • Assume pseudo-binary behavior, solve as Fick’s law in true composition space • Dim “equivalent” or “effective” diffusivity • must depend on fluxes and/or gradients, otherwise

  16. Typical assessment • Dim’s have the advantage of simplicity, but… • They are not system properties • They fail to account for diffusive interaction effects • They depend on transport fluxes which are not always known in advance • They vary with position or with composition in a manner that may require iteration • They may give large errors when used in situations they were not designed to handle • They are no longer justified, in view of the availability of more rigorous analytical and computational tools

  17. Definition of Flux-Averaged Diffusivities

  18. A rigorous definition • Average diffusive interactions for each component • Define flux-average diffusivity (FAD) implicitly as

  19. General FAD solution • Define mean film equivalent diffusivity

  20. Practical application • If correct average FAD can be identified a priori, method is exact • If approximate FAD must be used, this is the only source of error in the method • Arithmetic mean (AM) value used in the literature is most likely in error, because exact solution shows exponential dependence of (y) on h.

  21. Example 1 Evaporation of a pure liquid into a gas mixture in a Stefan tube

  22. Slattery (1999) • Pure liquid A evaporates into a gas mixture of A, B and C in a Stefan tube, arranged in such a manner that the liquid-gas interface remains fixed in space (h = 0) as the evaporation takes place. Species B and C are insoluble in liquid A, nB = nC = 0. At the prevailing conditions, the gas-phase composition of A in equilibrium with the liquid is yA0. The gas mixture blowing past the top of the tube (h = 1) has constant composition yB1, yC1. It is desired to find the rate of evaporation of A.

  23. Problem specifications Pure liquid A Gas mixture nB= nC = 0 yA0 yA1, yB1 interface top of tube h = 0 h = 1

  24. Exact solution: B & C (stagnant ) • From MS equations • Solve for nA, yB0, yC0

  25. FAD solution: B & C (stagnant) • From general FAD solution • From general FAD definition • Hence FAD solution is exact

  26. Exact solution: C (mobile) • From MS equations • Mean film compositions • From solution for B & C

  27. The correct average • Exact • FAD • i.e. mean FAD at log-mean (LM) compositions

  28. What not to do? • Do not use AM compositions to compute FAD • and if you do, don’t blame FAD method for errors! • Do not solve equation for C alone • Use of FAD approach does not overrule system determinacy • Use A & B, A & C or B & C. All combinations give exact solution

  29. AM v LM averages [1] • Parametric analysis • Exact (LM): • Approximate (AM):

  30. AM v LM averages [2] yA2 << yA1 DAB << DAC

  31. Generalization • n– 1 stagnant components • 1 mobile component

  32. Example 2 Diffusion of one component in a stagnant mixture of known average composition

  33. Treybal (1980) • Ammonia (A) is diffusing through a stagnant gas mixture consisting of 1/3 nitrogen (B) and 2/3 hydrogen (C) by volume at 206.8 kPa and 54 °C. Calculate the rate of diffusion of A through a film of gas 0.5 mm thick when the concentration change across the film is 10 to 5 % A by volume

  34. Problem specifications Gas mixture Gas mixture nB= nC = 0 Average A-free composition y´av yA1 yA0 h = 0 h = 1

  35. What average? [1] • This work

  36. What average? [2] • Hsu and Bird (1960) • Does not give

  37. Not all averages are equal!

  38. Example 3 Condensation of mixed vapors in presence of a noncondensable gas

  39. Sherwood et al. (1975) • A condenser operates with a mixture of vapors of ammonia (A), water (B) and hydrogen (C) at 3.36 atm and 200 °F. Mole fractions are yA0 = 0.30, yB0 = 0.40 in the gas bulk (h = 0), and yA1 = 0.455, yB1 = 0.195 at the interface (h = 1), where equilibrium with the local liquid is assumed. Binary diffusivities are DAB = 0.294, DAC = 1.14, DBC = 1.30 cm2.s-1. It is desired to estimate the ratio of condensation fluxes of water and ammonia.

  40. Problem specifications Gas mixture Condensate nC = 0 yA0, yB0 yA1, yB1 bulk gas interface h = 0 h = 1

  41. Exact solution • Gilliland (in Sherwood, 1937) • cf. Toor (1957), Sherwood et al. (1975), Taylor (1981)

  42. Exact, yet false [1] • The special value … • … reduces both equations to • Discarded on physical grounds

  43. Exact, yet false [2] • System is always satisfied by … • … but then this problem becomes overspecified (nmix = 0 in addition to original data!)

  44. FAD solution • From general FAD solution • For any stagnant component (C in this case) … • … but no obvious suggestion for the others

  45. FAD-1 • Literature: Use AM compositions … • … but AM is inconsistent with exact solution

  46. FAD-2 • Use LM compositions for all components … • … but LM is exact only for stagnant components

  47. FAD-3 Not yi, but xi yi – bi! • Recall general FAD solution … • … use flux-averaged compositions

  48. Exact v FAD solutions

  49. Exact v FAD-3

  50. Example 4 Diffusion with rapid heterogeneous catalytic reaction

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