Page 258 - Bird R.B. Transport phenomena

242 Chapter 8 Polymeric Liquids

Table 8.3-2 Parameters in the Carreau Model for Some
Solutions of Linear Polystyrene in 1-Chloronaphthalene rt

Properties of Parameters in Eq. 8.3-4
solution (r] is taken to be zero)
x
M w с Vo Л n
(g/mol) (g/ml) (Pa • s) (s) (...)
3.9 X 10 5 0.45 8080 1.109 0.304
3.9 X 10 5 0.30 135 3.61 X 10" 2 0.305
1.1 X 10 5 0.52 1180 9.24 X 10" 2 0.441
1.1 X 10 5 0.45 166 1.73 X 10~ 2 0.538
3.7 X 10 4 0.62 3930 1 X 1 0 1 0.217
a Values of the parameters are taken from K. Yasuda, R. C.
Armstrong, and R. E. Cohen, Rheol. Ada, 20,163-178 (1981).

(b) A better curve fit for most data can be obtained by using the four-parameter Car-
3
reau equation, which is
(8.3-4)

in which T] is the zero shear rate viscosity, 77*, is the infinite shear rate viscosity, A is a pa-
0
rameter with units of time, and n is a dimensionless parameter. Some sample parameters
for the Carreau model are given in Table 8.3-2.
We now give some examples of how to use the power law model. These are exten-
sions of problems discussed in Chapters 2 and 3 for Newtonian fluids. 4

EXAMPLE 8.3-1 Derive the expression for the mass flow rate of a polymer liquid, described by the power law
model. The fluid is flowing in a long circular tube of radius R and length L, as a result of a
Laminar Flow of an pressure difference, gravity, or both.
Incompressible Power
Law Fluid in a Circular SOLUTION
Tube ' 4 5
Equation 2.3-13 gives the shear stress distribution for any fluid in developing steady flow in a
circular tube. Into this expression we have to insert the shear stress for the power law fluid
(instead of using Eq. 2.3-14). This expression may be obtained from Eqs. 8.3-2 and 3 above.

(8.3-5)
Tr
Since v z is postulated to be a function of r alone, from Eq. B.I-13 we find that у = (7:7) =
^
2
\Z(dv /dr) . We have to choose the sign for the square root so that у will be positive. Since
z
dvjdr is negative in tube flow, we have to choose the minus sign, so that
'~ x dv 7
=m
dr Tr VTr (8.3-6)
3
P. J. Carreau, Ph.D. thesis, University of Wisconsin, Madison (1968). See also K. Yasuda,
R. C. Armstrong, and R. E. Cohen, Rheol Ada, 20,163-178 (1981).
4 For additional examples, including nonisothermal flows, see R. B. Bird, R. C. Armstrong, and
O. Hassager, Dynamics of Polymeric Liquids, Vol. 1. Fluid Mechanics, Wiley-Interscience, New York, 2nd
edition (1998), Chapter 4.
5 M. Reiner, Deformation, Strain and Flow, Interscience, New York, 2nd edition (I960), pp. 243-245.