Page 48 - Bird R.B. Transport phenomena

§1.6 Viscosity of Suspensions and Emulsions 33

Another approach for concentrated suspensions of spheres is the "cell theory/' in
which one examines the dissipation energy in the "squeezing flow" between the spheres.
As an example of this kind of theory we cite the Graham equation 8
Meff 1 , 5 . , 9 ( 1 \ n , ~,
-ГГ- = 1 + - ф + - (1.6-3)
in which ф = 2[1 - ^Ф/ф )/^Ф/Ф \, where ф т а х is the volume fraction corre-
тах
тах
sponding to the experimentally determined closest packing of the spheres. This expres-
9
sion simplifies to Einstein's equation for ф —> 0 and the Frankel-Acrivos equation when
Ф ""* Фтах-
For concentrated suspensions of nonspherical particles, the Krieger-Dougherty equation 10
can be used:

The parameters A and ф тах to be used in this equation are tabulated 11 in Table 1.6-1 for
suspensions of several materials.
Non-Newtonian behavior is observed for concentrated suspensions, even when the
suspended particles are spherical. 11 This means that the viscosity depends on the veloc-
ity gradient and may be different in a shear than it is in an elongational flow. Therefore,
equations such as Eq. 1.6-2 must be used with some caution.

Table 1.6-1 Dimensionless Constants for Use in Eq. 1.6-4
System A Фтах Reference

Spheres (submicron) 2.7 0.71 a
Spheres (40 /xm) 3.28 0.61 b
Ground gypsum 3.25 0.69 с
Titanium dioxide 5.0 0.55 с
Laterite 9.0 0.35 с
Glass rods (30 X 700 /im) 9.25 0.268 d
Glass plates (100 X 400 /xm) 9.87 0.382 d
Quartz grains (53-76 /xm) 5.8 0.371 d
Glass fibers (axial ratio 7) 3.8 0.374 b
Glass fibers (axial ratio 14) 5.03 0.26 b
Glass fibers (axial ratio 21) 6.0 0.233 b
0 С G. de Kruif, E. M. F. van Ievsel, A. Vrij, and W. B. Russel, in
Viscoelasticity and Rheology (A. S. Lodge, M. Renardy, J. A. Nohel,
eds.), Academic Press, New York (1985).
'' H. Giesekus, in Physical Properties of Foods (J. Jowitt et al., eds.),
Applied Science Publishers (1983), Chapter 13.
R. M. Turian and T.-F. Yuan, AIChE Journal, 23, 232-243 (1977).
r
y
' B. Clarke, Trans. Inst. Chem. Eng., 45, 251-256 (1966).

s
A. L. Graham, Appl. Sci. Res., 37, 275-286 (1981).
4
N. A. Frankel and A. Acrivos, Chem. Engr. Sci., 22, 847-853 (1967).
1 0 1. M. Krieger and T. J. Dougherty, Trans. Soc. Rheoi, 3,137-152 (1959).
11 H. A. Barnes, J. F. Hutton, and K. Walters, An Introduction to Rheology, Elsevier, Amsterdam
(1989), p. 125.