Page 45 - Bird R.B. Transport phenomena

30 Chapter 1 Viscosity and the Mechanisms of Momentum Transport

coordinate directions in jumps of length a at a frequency v per molecule. The frequency
is given by the rate equation

^ (1.5-1)

In which к and h are the Boltzmann and Planck constants, N is the Avogadro number,
and R = NK is the gas constant (see Appendix F).
In a fluid that is flowing in the x direction with a velocity gradient dv /dy, the fre-
x
quency of molecular rearrangements is increased. The effect can be explained by consid-
ering the potential energy barrier as distorted under the applied stress r yx (see Fig. 1.5-1),
so that

(f)(^) 0.5-2)

where Vis the volume of a mole of liquid, and ±(a/8)(r V/2) is an approximation to the
yx
work done on the molecules as they move to the top of the energy barrier, moving with
the applied shear stress (plus sign) or against the applied shear stress (minus sign). We
now define v + as the frequency of forward jumps and v_ as the frequency of backward
jumps. Then from Eqs. 1.5-1 and 1.5-2 we find that
+
^ = ^exp(-AG /KT) exp(±ar V/28RT) (1.5-3)
0
yx
The net velocity with which molecules in layer A slip ahead of those in layer В (Fig.
1.5-1) is just the distance traveled per jump (a) times the net frequency of forward jumps
(v+ - *O; this gives

V
a
v xA ~ XB = (v+ ~ v-) (1.5-4)
The velocity profile can be considered to be linear over the very small distance 8 between
the layers A and B, so that

- | - ( ! > ' • - ' • '
By combining Eqs. 1.5-3 and 5, we obtain finally

+
= (f ( f «Р(-АС„ /1Ш)(2 sinh § g ) (1.5-6)
)
This predicts a nonlinear relation between the shear stress (momentum flux) and the ve-
locity gradient—that is, non-Newtonian flow. Such nonlinear behavior is discussed further
in Chapter 8.
The usual situation, however, is that ar V/28RT « 1. Then we can use the Taylor
yx
3
series (see §C2) sinh x = x + О/З!)* + (l/5!)r s + • • • and retain only one term. Equation
1.5-6 is then of the form of Eq. 1.1-2, with the viscosity being given by
+
|
= I Y Щ exp(AG /KT) (1.5-7)
0
The factor 8 /a can be taken to be unity; this simplification involves no loss of accuracy,
since AGj is usually determined empirically to make the equation agree with experimen-
tal viscosity data.
It has been found that free energies of activation, AGj, determined by fitting Eq. 1.5-7
to experimental data on viscosity versus temperature, are almost constant for a given