Page 63 - Instrumentation Reference Book 3E

48 Measurement of viscosity
If the outer cylinder of radius ro rotates with
angular velocity 00 and the inner cylinder of radius
rl is stationary, the torque C per unit length of
cylinder on the inner cylinder for a Newtonian
liquid is given by

(2.10)

so that measurement of C at each rotational speed
00 can be used to determine the viscosity q. The
extensions to (2.10) when the fluid is non-New-
tonian are again non-trivial (unless the annular
gap is very small) but the relevant analysis is con-
tained in many texts (see, for example, Walters
(1975) and Whorlow (1980)). With reference to
possible sources of error, end effects are obvious
candidates as are flow instabilities, misalignment
of axes, and viscous heating. A detailed discussion
of possible sources of error is to be found in Dealy
Figure 2.5 Schematicdiagramofan (1982), Walters (1975), and Whorlow (1980).
Ostwald viscometer

the other meniscus is now a few milimeters above 2.3.3 Cone-and-plate viscometer“
b. The time t for the level to fall from b to c is
measured. The operating formula is of the form Consider the cone-and-plate arrangement shown
schematically in Figure 2.6. The cone rotates with
v = At - B/t (2.9) angular velocity 00 and the torque C required
where v is the kinematic viscosity (E VIP). The to keep the plate stationary is measured. The
second term on the right-hand side of equation gap angle 80 is usually very small (<4’) and, in
(2.9) is a correction factor for end effects. For any the interpretation of results, edge effects are
particular viscometer, A and B are given as cali- neglected. It is then easy to show that for a New-
bration constants. Viscometers with pipes of dif- tonian liquid, the operating formula is
ferent radii are supplied according to British
Standards specifications and a “recommended (2.1 1)
procedure” is also given in B.S. Publication 188:
1957. where a is the radius of the cone.
Relying on gravity flow alone limits the range In contrast to the capillary-flow and Couette-
of measurable stress to between 1 and 15Nm-2. flow situations, the operating formula for non-
The upper limit can be increased to 50Nm-2 by
applying a known steady pressure of inert gas
over the left-hand side of the U-tube during
operation.
Fluid Stationary
\ Plate
2.3.2 Couette viscometer
The most popular rotational viscometer is the
Couette concentric-cylinder viscometer. Fluid is
placed in the annulus between two concentric
cylinders (regarded as infinite in the interpret-
ation of data) which are in relative rotation about
their common axis. It is usual for the outer cylin-
der to rotate and for the torque required to keep
the inner cylinder stationary to be measured, but Figure 2.6 Basic cone-and-plate geometry.
there are variants, as in the Brookfield vis- *The torsional-flow rheometer in which the test fluid is
cometer, for example, where a cylindrical bob (or contained between parallel plates is similar in operation
sometimes a disc) is rotated in an expanse of test to the cone-and-plate rheometer, but the data interpret-
liquid and the torque on this same bob is ation is less straightforward, except in the Newtonian
recorded; see Section 2.4. case (see, for example, Walters (1975)).