Page 546 - Bird R.B. Transport phenomena

526 Chapter 17 Diffusivity and the Mechanisms of Mass Transport

It is assumed that the concentration profile co (y) is very nearly linear over distances
A
of several mean free paths. Then we may write
<»A\y±a = 0) \ ± |A —j^ (17.3-6)
A y
Combination of the last two equations then gives for the combined mass flux at plane y:

(17.3-7)
dy
This is the convective mass flux plus the molecular mass flux, the latter being given by Eq.
17.1-1. Therefore we get the following expression for the self-diffusivity:
ЯЬ * = \пк (17.3-8)
АА
Finally, making use of Eqs. 17.3-1 and 3, we get
2 УкТ/тгт А i 2
^ = =
which can be compared with Eq. 1.4-9 for the viscosity and Eq. 9.3-12 for the thermal
conductivity.
The development of a formula for % AB for rigid spheres of unequal masses and di-
1
ameters is considerably more difficult. We simply quote the result here:

3 - 1 0 )
That is, \/m A is replaced by the arithmetic average of \/m A and l/m , and d A by the
B
arithmetic average of d A and d .
B
The preceding discussion shows how the diffusivity can be obtained by mean free
path arguments. For accurate results the Chapman-Enskog kinetic theory should be
used. The Chapman-Enskog results for viscosity and thermal conductivity were given in
§§1.4 and 9.3, respectively. The corresponding formula for сЯЬ is: ' 2 3
АВ

M

= 2.2646 ХНГ 5 IlTF + ^ - h - ^ (17.3-11)
i
V \M A M J Одвпд^в
B
Or, if we approximate с by the ideal gas law p = cRT, we get for Э л в

3
= 0.0018583 /T [ 4 - + T V I — r ^ (17.3-12)
2
In the second line of Eqs. 17.3-11 and 12, % [=] cm /s, a [=] А, Г [=] К, and p [=] atm.
AB AB

1
A similar result is given by R. D. Present, Kinetic Theory of Gases, McGraw-Hill, New York (1958), p. 55.
2
S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, 3rd edition,
Cambridge University Press (1970), Chapters 10 and 14.
3
J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids, 2nd corrected
printing, Wiley, New York (1964), p. 539.