394:
53:
17:
389:{\displaystyle a_{i}{\frac {\nabla \mu _{i}}{R\,T}}=\nabla a_{i}=\sum _{j=1 \atop j\neq i}^{n}{{\frac {\chi _{j}}{{\mathfrak {D}}_{ij}}}({\vec {v}}_{j}-{\vec {v}}_{i})}=\sum _{j=1 \atop j\neq i}^{n}{{\frac {c_{j}}{c{\mathfrak {D}}_{ij}}}\left({\frac {{\vec {J}}_{j}}{c_{j}}}-{\frac {{\vec {J}}_{i}}{c_{i}}}\right)}}
589:
The basic assumption of the theory is that a deviation from equilibrium between the molecular friction and thermodynamic interactions leads to the diffusion flux. The molecular friction between two components is proportional to their difference in speed and their mole fractions. In the simplest case,
609:
and are therefore not tabulated. Only the diffusion coefficients for the binary and ternary case can be determined with reasonable effort. In a multicomponent system, a set of approximate formulas exist to predict the
Maxwell–Stefan-diffusion coefficient.
613:
The
Maxwell–Stefan theory is more comprehensive than the "classical" Fick's diffusion theory, as the former does not exclude the possibility of negative diffusion coefficients. It is possible to derive Fick's theory from the Maxwell–Stefan theory.
460:
572:
498:
527:
598:
solutions, and other drivers, such as a pressure gradient, the equation must be expanded to include additional terms for interactions.
40:
in multicomponent systems. The equations that describe these transport processes have been developed independently and in parallel by
763:
431:
623:
606:
684:
541:
467:
668:, Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften Wien, 2te Abteilung a, 1871,
602:
505:
418:
8:
530:
41:
758:
412:
33:
594:
of chemical potential is the driving force of diffusion. For complex systems, such as
605:, with the exception of the diffusion of dilute gases, do not correspond to the
736:
Measurement and calculation of multicomponent diffusion coefficients in liquids
752:
666:Über das Gleichgewicht und Bewegung, insbesondere die Diffusion von Gemischen
628:
406:
595:
583:
45:
20:
Thermal diffusion coefficients vs. temperature, for air at normal pressure
37:
591:
601:
A major disadvantage of the
Maxwell–Stefan theory is that the
575:
586:, i.e., the neglect of time derivatives in the velocity.
400:
16:
544:
508:
470:
434:
56:
681:
682:Bird, R.B.; Stewart, W.E.; Lightfoot, E.N. (2007).
566:
521:
492:
454:
388:
750:
651:, The Scientific Papers of J. C. Maxwell, 1965,
697:
695:
701:
692:
48:for liquids. The Maxwell–Stefan equation is
730:
728:
719:Diffusion – Mass Transfer in Fluid Systems
721:(2 ed.). Cambridge University Press.
88:
641:
15:
725:
716:
658:
751:
710:
462:: Maxwell–Stefan-diffusion coefficient
455:{\displaystyle {\mathfrak {D}}_{ij}}
500:: Diffusion velocity of component i
438:
423:i, j: Indexes for component i and j
289:
166:
13:
675:
238:
119:
98:
70:
14:
775:
738:, Fluid Phase Equilibria, 2007,
702:Taylor, R.; Krishna, R. (1993).
649:On the dynamical theory of gases
567:{\displaystyle {\vec {J}}_{i}}
552:
493:{\displaystyle {\vec {v}}_{i}}
478:
356:
320:
226:
214:
192:
182:
1:
634:
607:Fick's diffusion coefficients
734:S. Rehfeldt, J. Stichlmair:
704:Multicomponent Mass Transfer
536:c: Total molar concentration
401:vector differential operator
7:
624:Advanced Simulation Library
617:
10:
780:
30:Stefan–Maxwell diffusion
26:Maxwell–Stefan diffusion
426:n: Number of components
717:Cussler, E.L. (1997).
603:diffusion coefficients
568:
523:
494:
456:
390:
268:
149:
21:
582:The equation assumes
569:
524:
522:{\displaystyle c_{i}}
495:
457:
391:
233:
114:
44:for dilute gases and
19:
688:(2 ed.). Wiley.
542:
506:
468:
432:
54:
764:James Clerk Maxwell
685:Transport Phenomena
531:Molar concentration
42:James Clerk Maxwell
564:
519:
490:
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413:Chemical potential
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647:J. C. Maxwell:
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36:for describing
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578:of component i
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533:of component i
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3:
2:
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629:Pervaporation
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407:Mole fraction
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27:
18:
739:
735:
718:
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612:
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596:electrolytic
588:
584:steady state
581:
50:
46:Josef Stefan
29:
25:
23:
664:J. Stefan:
753:Categories
635:References
759:Diffusion
672:, 63–124.
553:→
479:→
357:→
345:−
321:→
255:≠
235:∑
215:→
205:−
193:→
155:χ
136:≠
116:∑
99:∇
75:μ
71:∇
38:diffusion
742:, 99–104
706:. Wiley.
655:, 26–78.
618:See also
592:gradient
419:Activity
32:) is a
34:model
590:the
576:Flux
28:(or
24:The
740:256
417:a:
411:μ:
405:χ:
399:∇:
755::
727:^
694:^
670:63
574::
529::
653:2
560:i
550:J
515:i
511:c
486:i
476:v
448:j
445:i
439:D
382:)
374:i
370:c
364:i
354:J
338:j
334:c
328:j
318:J
308:(
299:j
296:i
290:D
284:c
278:j
274:c
265:n
258:i
252:j
247:1
244:=
241:j
231:=
227:)
222:i
212:v
200:j
190:v
183:(
176:j
173:i
167:D
159:j
146:n
139:i
133:j
128:1
125:=
122:j
112:=
107:i
103:a
96:=
90:T
86:R
79:i
63:i
59:a
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