Page 62 - Instrumentation Reference Book 3E

Measurement of the shear viscosity 47

The popular book on viscometry by Van The non-trivial extensions to (2.5) when the fluid
Wazer etal. (1963) and that of Wilkinson (1960) is non-Newtonian may be found in Walters
on non-Newtonian flow are now out of date in (1975), Whorlow (1980), and Coleman et al.
some limited respects, but they have stood the test (1966). For example. in the case of the power-
of time remarkably well and are recommended to law fluid (2.3), the formula is given by
readers, provided the dates of publication of the
books are appreciated. More modern treatments,
developed from different but complementary
viewpoints, are given in the books by Lodge One of the major advantages of the capillary vis-
(1974), Walters (1975); and Whorlow (1980). The cometer is that relatively high shearrates can be
text by Dealy (1982) already referred to is limited attained.
to polymer-melt rheometry, but much of the Often, it is not possible to determine the pressure
book is of general interest to those concerned gradient over a restricted section ofthe capillary and
with the measurement of viscosity. it is then necessary, especially in the case of non-
Newtonian liquids, to study carefully the pressure
losses in the entry and exit regions before the results
2.3 Measuremlent of the shear can be interpreted correctly (see, for example.
viscosity Dealy (1982) and Whorlow (1980)). Other possible
sources of error include viscous heating and flow
It is clearly impracticable to construct visco- instabilities. These and other potential problems
meters with the infinite planar geometry asso- are discussed in detail by Dealy (1982), Walters
ciated with Newton’s postulate (Figure 2.1): (1975), and Whorlow (1980).
especially in the case of mobile liquid systems, The so-called “kinetic-energy correction” is
and this has led to the search for convenient important when it is not possible to limit the
geometries and flows which have the same basic pressure drop measurement to the steady simple
steady simple shear flow structure. This problem shear flow region and when this is taken over the
has now been resolved and a number of the complete length L of the capillary. For a Neiv-
so-called “viscometric flows” have been used as tonian fluid, the kinetic energy correction is given
the basis for viscometer design. (The basic mathe- (approximately) by
matics is non-trivial and may be found in the
texts by Coleman et al. (1966), Lodge (1974); and P=Po-- l.lpQ2
Walters (1 979.) Most popular have been (i) capil-
lary (or Poiseuille) flow, (ii) circular Couette flow, where P is the pressure drop required in (2.5), Po
and (iii) cone-and-plate flow. For convenience, is the measured pressure drop and p is the density
we shall briefly describe each of these flows and of the fluid.
give the simpie operating formulae for Newtonian Since a gas is highly compressible; it is more
liquids: rererring the reader to detailed texts for the convenient to measure the IEUSS rate of flow, rii.
extensions to non-Newtonian liquids. We also Equation (2.5) has then to be replaced by (see, for
include in Section 2.3.4 a discussion of the paral- example, Massey (1968))
lel-plate rheometer, which approximates closely
the flow associated with Newton’s postulate. v=- 7ru4pMP
8mRTL
2.3.1 Capillary viscometer where p is the mean pressure in the pipe, M is the
Consider a long capillary with a circular cross- molecular weight of the gas, R is the gas constant
section of radius a. Fluid is forced through the per mole and T is the Kelvin temperature. The
capillary by the application of an axial pressure kinetic-energy correction (2.7) is still valid and must
drop. This pressure drop Pis measured over a length be borne in mind, but in the case of a gas, this
L of the capillary, far enough away from both correction is usually very small. A “slip correction”
entrance and exit for the flow to be regarded as is also potentially important in the case of gases, but
‘-fullydeveioped” steady simple shear flow. The only at low pressures.
volume rate of flow Q through the capillary is In commercial capillary viscometers for non-
measured for each pressure gradient PIL and the gaseous materials, the liquids usually flow
viscosity 7 for a Newtonian liquid can then be deter- through the capillaries under gravity. A good
mined from the so-called Hagen-Poiseuille law: example is the Ostwald viscometer (Figure 2.5).
In this b. c, and d are fixed marks and there are
reservoirs at D and E. The amount of liquid must
be such that at equilibrium one meniscus is at d.
To operate, the liquid is sucked or blown so lhat