Page 558 - Bird R.B. Transport phenomena

538 Chapter 17 Diffusivity and the Mechanisms of Mass Transport
§17.9 THE MAXWELL-STEFAN EQUATIONS FOR
MULTICOMPONENT DIFFUSION IN GASES
AT LOW DENSITY
For multicomponent diffusion in gases at low density it has been shown ' 1 2 that to a very good
approximation

(
N
V* = - 2 ^ (v - V = - 2 -or- ^ - ~ *° *> ct = 1,2,3 N (17.9-1)
N
e e
/8 = 1 ^a$ /3 = 1 C^afi
The Q) here are the binary diffusivities calculated from Eq. 17.3-11 or Eq. 17.3-12. There-
aj3
fore, for an N-component system, \N(N — 1) binary diffusivities are required.
Equations 17.9-1 are referred to as the Maxwell-Stefan equations, since Maxwell 3
suggested them for binary mixtures on the basis of kinetic theory, and Stefan 4 gener-
alized them to describe the diffusion in a gas mixture with N species. Later Curtiss
and Hirschfelder obtained Eqs. 17.9-1 from the multicomponent extension of the
Chapman-Enskog theory.
For dense gases, liquids, and polymers it has been shown that the Maxwell-Stefan
equations are still valid, but that the strongly concentration-dependent diffusivities ap-
pearing therein are not the binary diffusivities. 5
There is an important difference between binary diffusion and multicomponent dif-
fusion. 6 In binary diffusion the movement of species A is always proportional to the neg-
ative of the concentration gradient of species A. In multicomponent diffusion, however,
other interesting situations can arise: (i) reverse diffusion, in which a species moves
against its own concentration gradient; (ii) osmotic diffusion, in which a species diffuses
even though its concentration gradient is zero; (iii) diffusion barrier, when a species does
not diffuse even though its concentration gradient is nonzero. In addition, the flux of a
species is not necessarily collinear with the concentration gradient of that species.

QUESTIONS FOR DISCUSSION

1. How is the binary diffusivity defined? How is self-diffusion defined? Give typical orders of
magnitude of diffusivities for gases, liquids, and solids.
2. Summarize the notation for the molecular, convective, and total fluxes for the three transport
processes. How does one calculate the flux of mass, momentum, and energy across a surface
with orientation n?
3. Define the Prandtl, Schmidt, and Lewis numbers. What ranges of Pr and Sc can one expect to
encounter for gases and liquids?
4. How can you estimate the Lennard-Jones potential for a binary mixture, if you know the pa-
rameters for the two components of the mixture?
5. Of what value are the hydrodynamic theories of diffusion?
6. What is the Langevin equation? Why is it called a "stochastic differential equation"? What in-
formation can be obtained from it?

С F. Curtiss and J. O. Hirschfelder, /. Chem. Phys., 17, 550-555 (1949).
1
For applications to engineering, see E. L. Cussler, Diffusion: Mass Transfer in Fluid Systems, 2nd
2
edition, Cambridge University Press (1997); R. Taylor and R. Krishna, Multicomponent Mass Transfer,
Wiley, New York (1993).
3
J. C. Maxwell, Phil. Mag., XIX, 19-32 (1860); XX, 21-32, 33-36 (1868).
4
J. Stefan, Sitzungsber. Kais. Akad. Wiss. Wien, LXIIK2), 63-124 (1871); LXV(2), 323-363 (1872).
5
С F. Curtiss and R. B. Bird, Ind. Eng. Chem. Res., 38, 2515-2522 (1999); /. Chem. Phys., I l l ,
10362-10370(1999).
6
H. L. Toor, AIChE Journal, 3,198-207 (1959).