Figure 1: Horizontal section of a concentric cylinder viscometer and deformation of a fluid element.
| 1: Introduction |
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2: Theory of
viscosity measurement
The two main types of viscometer are the tube and rotational instruments [ref. 1], The former observe the rate of flow through tubes due to a known pressure difference. These types are unsuitable for this application, because a suitable viscometer must allow the nest of the experiment to proceed normally in the intervals between measurements, The lubrication fluid is contained in a small lest tube or beaker and therefore any tube-like measurements would be impractical and hard to automate.
The vast majority of rotational viscometers fall into two categories: those where two concentric cylinders rotate relative to one another around a common axis; and those consisting of a cone of large vertical angle (approaching 180 degrees), and plate whose plane is through the apex of the cone. Many variations on this theme are possible, but in all types the test fluid is sheared between the rotating parts. The cone on plate type is again rejected for this application, as it would not be possible to perform oxidation experiments inside the viscosity measurement apparatus.
A concentric cylinder viscometer can easily be formed by regarding a
beaker in which the experitnents are penformed as the outer cylinder, and
placing a rotating inner cylinder centrally within it. The suitability
and simplicity of this arrangement makes it the ideal choice here. Hence
the foliowaig theoretical derivations are only concerned with instruments
of this type.
Prior to detailed mathematical consideration, it is necessary to define two variables used in the description of fluid flow: shear stress and shear strain. Stress is measured in units of Pascals (1 Pa = I Nm-2). Consider a point P in a body, surrounded by a plane of area A. The material above, below and to the sides of P exert a resultant force F on the element. As the area is varied the force changes, and the ratio F/A approaches a limit as A tends to zero, imown as the traction across the area. This traction has a perpendicular component, the 'normal stress', and a parallel component the 'shear stress' s. Shear strain y is defined as the relative displacememt of two layers in the fluid, divided by their separation.
A Newtonian fluid is one in which the ratio of shear stress to the rate
of shear strain is constant [ref. l].This
parameter is the viscosity n. That is,
n = s / y.
The unit of viscosity is the poise. Kinematic viscosity v is often
used and is defined as
v= n / p,
where p is the density of the fluid. The unit of kinematic viscosity
is the Stokes; lubricants are usually specified, by convention, in terms
of their kinematic viscosity in centistokes (cst).
A non-Newtonian fluid is one in which the viscosity is not a constant
parameter, it depends in some way on the shear rate. The Newtonian model
is accurate over a large range for most low molecular weight fluids, including
water and many aqueous solutions, liquid metals, organic compounds, and
silicones. Other fluids such as suspensions obey various more complex models.
A large number of such models have been proposed, for example the Bingham
[ref. 2], power law fluid [ref.
3], and Casson [ref. 1] models.
This report is only concerned with the measurement of viscosity for Newtonian
fluids, although modifications to allow for Non-Newtonian behaviour would
not be difficult (See discussion, Section 5).
2.3 Concentric cylinder viscometer.
The formulae derived apply to the measurement of Newtonian fluids, confined
between concentric cylinders of infinite length, and neglecting any inertial
effects [ref. 1]. The inner and outer
cylinders are of radius R1 and R2 respectively, and rotate with a relative
angular velocity O. Considering the fluid between the inner cylinder and
a tadius r; each particle moves with a constant angular velocity, such
that the net torque on the fluid is zero. The torque G per unit length
on a cylindrical surface at radius r is
G = 2 pi R1^2 s1
where s1 is the shear stress on the inner cylinder, The shear stress
at any radius r is
s = G / 2 pi r^2
and in particular, at the outer cylinder is
s2=G / 2 pi R2^2.
An expression for the strain rate may be derived using figure 1, which
shows sectors of two cylindrical surfaces separated by a small distance
dr. In a time dt the radial line AB moves to AB', as opposed to A'C, had
the fluid been a rigid body. Now,
BB' = (r + dr) (w + dw) dt
and
BC= (r + dr) w dt
so the shear strain is
y = B'C / CA' = (r + dr) dw dt / dr,
which in the limit as dr tends to zero, gives
dy / dt = r dw / dr
Substituting these results into the Newtonian fluid equation leads
to
r dw / dr = G / 2 pi v r^2
Applying the boundary conditions to w = 0 at r = R1, w = O at r = R2,
and integrating gives
O = G ( 1 / R1^2 - 1 / R2^2 ) / 4 pi n
For a Newtonian fluid a graph of angular velocity against torque per
unit length will be linear through the origin, and have gradient
(1 / R1^2 - 1 / R2^2) / 4 pi n
This formula will be used to obtain the viscosity. If the instrument
is calibrated with a liquid of known viscosity, or used to measure relative
viscosity, then all subsequent measurements can be referenced to this and
it is not necessary to know R1 or R2 explicitly, provided they remain constant.
2.4 Deviations
from the ideal model
A practical concentric cylinder viscometer must be of finite size, and
therefore the top and base of the inner cylinder will also exert a torque.
Furthermore, near to the ends, the torque per unit length will be reduced
since the velocity gradient is no longer radial. This necessitates complex
corections to the formula derived above; however, the torque is still proportional
to angular velocity. Hence provided calibration is performed, the end effects
will not cause error.
Viscosity is highly dependent on temperature. The relation is often
found to approximate
v = A exp ( Ev / k T)
over a large temperature range, where v is the kinematic viscosity,
k Boltzmann's constant, and T the temperature. The constants A and Ev (known
as the activation energy for viscous flow) exhibit a large variation between
different fluids. This relation was tested, see section
4.2.
The fluid must he kept at a known and constant temperature throughout
the measurement. If the concentric cylirider viscometer is used with very
viscous fluids at high shear rates, temperature rise due to shear heating
can be troublesome. This effect is neglected here, but is considered further
in the discussion, section 5.
2.4.3. Departure from circular flow
In concentric cylinder arrangements, fast moving fluids near to the
inner cylinder try to move outwards due to the centripetal force. Such
a movement is impossible for the liquid as a whole, so local circulation
occurs [ref 4]. These 'Taylor vortices'
are only formed above a certain rate of rotation, as in figure 2. This
secondary flow is still regular but complex, and the relations derived
above no longer apply. At still higher speeds the flow becomes turbulent
For Newtonian fluids, the 'Reynolds number' is delined as
Re = O R (R2 - R2) / v
where R is the radius of the moving cylinder, and the other variables
are as before.
For inner cylinder rotation, Taylor [ref.
5] found that vortices occurred for
Re > 41.3 (R2 / (R2 - R1))
^ l/2
At a rotation rate of 300 rpm, and with inner and outer cylinders of
radius 1 and 2 cm respectively, the
corresponding minimum kinematic viscosity which may measured is 0.005
cst. The current viscometer will not handle such low viscosities.
Figure 2: Secondary flow patterns at high rotation rates, known as
Taylor vortices.
| 1: Introduction |
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