Page 541 - Bird R.B. Transport phenomena

§17.2 Temperature and Pressure Dependence of Diffusivities 521

§17.2 TEMPERATURE AND PRESSURE
DEPENDENCE OF DIFFUSIVITIES
In this section we discuss the prediction of the diffusivity 2) for binary systems by cor-
л в
responding-states methods. These methods are also useful for extrapolating existing
data. Comparisons of many alternative methods are available in the literature. 12
For binary gas mixtures at low pressure, ЯЬ is inversely proportional to the pressure,
АВ
increases with increasing temperature, and is almost independent of the composition for a
given gas pair. The following equation for estimating ЯЬ at low pressures has been devel-
АВ
3
oped from a combination of kinetic theory and corresponding-states arguments:
I
рР + /M ) 1 / 2
B
2
Here ЯЬ [=] cm /s, p [=] atm, and T [=] K. Analysis of experimental data gives the di-
АВ
4
mensionless constants a = 2.745 X 1СГ and b = 1.823 for nonpolar gas pairs, excluding
4
helium and hydrogen, and a = 3.640 X 1СГ and b = 2.334 for pairs consisting of H O
2
and a nonpolar gas. Equation 17.2-1 fits the experimental data at atmospheric pressure
within an average deviation of 6 to 8%. If the gases A and В are nonpolar and their
Lennard-Jones parameters are known, the kinetic-theory method described in the next
section usually gives somewhat better accuracy.
At high pressures, and in the liquid state, the behavior of 2) лв is more complicated.
The simplest and best understood situation is that of self-diffusion (interdiffusion of la-
beled molecules of the same chemical species). We discuss this case first and then extend
the results approximately to binary mixtures.
A corresponding-states plot of the self-diffusivity % * for nonpolar substances is
AA
4
given in Fig. 17.2-1. This plot is based on self-diffusion measurements, supplemented by
molecular dynamics simulations and by kinetic theory for the low-pressure limit. The or-
dinate is сЯЬ * at pressure p and temperature T, divided by c% * at the critical point.
АА AA
This quantity is plotted as a function of the reduced pressure p y = p/p c and the reduced
temperature T = T/T . Because of the similarity of species A and the labeled species A*,
r c
the critical properties are all taken as those of species A.
From Fig. 17.2-1 we see that сЯЬ * increases strongly with temperature, especially
АА
for liquids. At each temperature сЯЬ * decreases toward zero with increasing pressure.
АА
With decreasing pressure, с2)дл* increases toward a low-pressure limit, as predicted by
kinetic theory (see §17.3). The reader is warned that this chart is tentative, and that the
lines, except for the low-density limit, are based on data for a very few substances: Ar,
Kr, Xe, and CH .
4
The quantity {c% *) may be estimated by one of the following three methods:
c
AA
(!) Given сЯЬ * at a known temperature and pressure, one can read (с2) *),. from
АА Л/4
the chart and get {(fdb *) c = с^ А
AA
1
R. C. Reid, J. M. Prausnitz, and В. Е. Poling, The Properties of Gases and Liquids, 4th edition,
McGraw-Hill, New York (1987), Chapter 11.
2
E. N. Fuller, P. D. Shettler, and J. С Giddings, Ind. Eng. Chem., 58, No. 5,19-27 (1966); Erratum:
ibid. 58, No. 8,81 (1966). This paper gives a useful method for predicting binary gas diffusivities from the
molecular formulas of the two species.
J. С Slattery and R. B. Bird, AIChE Journal 4,137-142 (1958).
3
Other correlations for self-diffusivity at elevated pressures have appeared in Ref. 3 and in
4
L. S. Tee, G. F. Kuether, R. C. Robinson, and W. E. Stewart, API Proceedings, Division of Refining, 235-243
(1966); R. C. Robinson and W. E. Stewart, 1EC Fundamentals, 7, 90-95 (1968); J. L. Bueno, J. Dizy,
R. Alvarez, and J. Coca, Trans. Inst. Chem. Eng., 68, Part A, 392-397 (1990).