Page 42 - Bird R.B. Transport phenomena

§1.4 Molecular Theory of the Viscosity of Gases at Low Density 27

In the second form of this equation, if Г [ = ] К and a [ = ] A, then /x [ = ] g/cm • s. The di-
mensionless quantity П is a slowly varying function of the dimensionless temperature
м
кТ/e, of the order of magnitude of unity, given in Table E.2. It is called the "collision in-
tegral for viscosity/' because it accounts for the details of the paths that the molecules
take during a binary collision. If the gas were made up of rigid spheres of diameter a (in-
stead of real molecules with attractive and repulsive forces), then H^ would be exactly
unity. Hence the function П may be interpreted as describing the deviation from rigid-
м
sphere behavior.
Although Eq. 1.4-14 is a result of the kinetic theory of monatomic gases, it has been
found to be remarkably good for polyatomic gases as well. The reason for this is that, in
the equation of conservation of momentum for a collision between polyatomic mole-
cules, the center of mass coordinates are more important than the internal coordinates
[see §0.3(fr)]. The temperature dependence predicted by Eq. 1.4-14 is in good agreement
with that found from the low-density line in the empirical correlation of Fig. 1.3-1. The
viscosity of gases at low density increases with temperature, roughly as the 0.6 to 1.0
power of the absolute temperature, and is independent of the pressure.
To calculate the viscosity of a gas mixture, the multicomponent extension of the
Chapman-Enskog theory can be used. ' 4 5 Alternatively, one can use the following very
satisfactory semiempirical formula: 7

fi mix = 2 ^ (1.4-15)

in which the dimensionless quantities Ф are
а/3
/2
1 / M У^Г fa V /M«V/4~|2

Here N is the number of chemical species in the mixture, x a is the mole fraction of species
a, fi is the viscosity of pure species a at the system temperature and pressure, and M is
a a
the molecular weight of species a. Equation 1.4-16 has been shown to reproduce mea-
sured values of the viscosities of mixtures within an average deviation of about 2%. The
dependence of mixture viscosity on composition is extremely nonlinear for some mix-
tures, particularly mixtures of light and heavy gases (see Problem 1 A.2).
To summarize, Eqs. 1.4-14,15, and 16 are useful formulas for computing viscosities
of nonpolar gases and gas mixtures at low density from tabulated values of the intermol-
ecular force parameters a and s/к. They will not give reliable results for gases consisting
of polar or highly elongated molecules because of the angle-dependent force fields that
exist between such molecules. For polar vapors, such as H O, NH , CHOH, and NOC1,
2
3
an angle-dependent modification of Eq. 1.4-10 has given good results. 8 For the light
gases H and He below about 100K, quantum effects have to be taken into account. 9
2
Many additional empiricisms are available for estimating viscosities of gases and
gas mixtures. A standard reference is that of Reid, Prausnitz, and Poling. 10

7
С. R. Wilke, /. Chem. Phys., 18, 517-519 (1950); see also J. W. Buddenberg and C. R. Wilke, hid. Eng.
Chem., 41,1345-1347 (1949).
8
E. A. Mason and L. Monchick, /. Chem. Phys., 35,1676-1697 (1961) and 36,1622-1639, 2746-2757
(1962).
9
J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, op. cit., Chapter 10; H. T. Wood and C. F. Curtiss, /.
Chem. Phys., 41,1167-1173 (1964); R. J. Munn, F. J. Smith, and E. A. Mason, /. Chem. Phys., 42, 537-539
(1965); S. Imam-Rahajoe, С F. Curtiss, and R. B. Bernstein, /. Chem. Phys., 42, 530-536 (1965).
10
R. C. Reid, J. M. Prausnitz, and В. Е. Poling, The Propeties of Gases and Liquids, McGraw-Hill, New
York, 4th edition (1987).