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Macosko, Christopher W - Rheology - Principles, Measurements and Applications-John Wiley & Sons (1994)

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SHEAR RHEOMETRY: DRAG FLOWS I 235 
6 
The water in the tube, when 
it encounters the resistance 
of the wall, is not able to move 
as a solid cylinder. The 
middle thread mustjlow much 
faster than the outer one. 
G. H. L. Hagen (1839) 
Figure i.l.1. 
Hagen’s capillary tube (A) 
was supplied with water from 
tank (B). The pressure head 
was recorded by a pointer 
(C) attached to the float (D) 
and flow rate was checked by 
weighing the effluent. 
SHEAR 
RHEOMETRY: 
PRESSURE 
DRIVEN FLOWS 
6.1 Introduction 
The first measurements of viscosity were done using a small straight 
tube or capillary (Figure 6.1.1). Hagen (1 839) in Germany and in- 
dependently Poiseuille (1 840) in France used small diameter cap- 
illaries to measure the viscosity of water. A key development that 
made their work possible was the advent of precision diameter, 
small bore tubing. For water, use of larger diameters usually results 
in turbulent flow. Precision is required, as we shall see, because 
the tube radius enters the viscosity equation to the fourth power. 
237 
A capillary rheometer is a pressure-driven flow, the theme 
of this chapter, in contrast to the drag flows of Chapter 5. As Ha- 
gen first observed, when pressure drives a fluid through a channel, 
velocity is maximum at the center. The velocity gradient or shear 
rate and also the shear strain will be maximum at the wall and 
zero in the center of the flow. Thus all pressure-driven flows are 
nonhomogeneous. This means that they are only used to measure 
steady shear functions: the viscosity and normal stress coefficients 
q ( P ) , $I(P), and 1,b2(9). Equations 5.1.1-5.1.3 define these func- 
tions, and Figure 11.3 indicated how they are related to the other 
material functions. 
If pressure-driven rheometers can measure only the steady 
shear functions, why are they so widely used? The first reason, of 
course, is that they are relatively inexpensive to build and simple 
to operate. Despite their simplicity, long capillaries can give the 
most accurate viscosity data available. A second major advantage 
is that closed-channel flows have no free surface in the test region. 
In Chapter 5 , for example, we saw how edge effects in the cone and 
plate geometry seriously limit the maximum shear rate in rotational 
instruments. In fact, for viscous polymer melts, capillary or slit 
rheometers appear to be the only satisfactory means of obtaining 
data at shear rates greater than 10 s-’ . Capillary rheometers can 
also eliminate solvent evaporation and other problems that plague 
rotational devices with free surfaces. Because the sample flows 
through a capillary or slit, these rheometers can be readily adapted 
for on-line measurements (see Chapter 8). 
Another reason capillary rheometers are so,widely used is 
that they are very similar to process flows like pipes and extrusion 
dies. A capillary run is an excellent first test of processibility for a 
small amount of a new polymer or coating formulation. 
Just as a rotational rheometer designed to produce cone and 
plate flow can typically be used for concentric cylinder or parallel 
plate geometries, a capillary rheometer usually can be adapted for 
slit or annular flows. This chapter focuses mainly on capillary 
flow but also treats these other channel geometries. Flow over a 
narrow channel or “pressure hole” gives data. Such flow can be 
generated by both drag and pressure, but since it is usually measured 
in a pressure-driven slit geometry, we discuss it here. Extrudak 
swell and exit pressure can also give information on normal stresses 
and are discussed. We also look at two important pressure-driven 
indexers: the melt index and squeezing flow. At the end of the 
chapter we compare all the shear rheometers, summarizing their 
advantages and limitations. Chapter 8 has a section addressing 
capillary rheometer design. 
6.2 Capillary Rheometer 
A capillary was the first rheometer, and this device remains the 
most common method for measuring viscosity.