Rheology Measurement
3.2 Rotational visco meters
CHAPTER 3
Power Law index, n 0.2 0.4 1.0
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RHEOLOGY MEASUREMENT
Figure 3.3 The velocity profile for the laminar flow of power-law liquids, calculated for the same volumetric throughput [Barnes et al. (1989»)
Equation 3.9 shows how the viscosity can be calculated in a capillary viscometer if the flowrate Q, the pressure difference across the capillary, and the dimensions of the capillary are known.
The pressure gradient can be measured using various devices. The most common method is by connecting holes in wall of the capillary to pressure transducers. To avoid end effects the holes on the capillary wall must be far away from the ends [Barnes et al. (1989)). Usually, end effects (entrance effects and exit effects) are avoided by ensuring that the aspect ratio of the capillary is greater than one hundred.
CHAPTER 3 RHEOLOGY MEASUREMENT
controlled shear stress test, a torque is applied to the measuring system and a result the deflection angle or speed which the measuring system carries out towards the fluid is measured [Barnes et al. (1989) and Slatter et al. (2000)].
The pseudo-fluids used in dense medium separations are seldom Newtonian in behaviour. They can either be dilatant or plastic depending on the shear rate and the solids volume concentration.
The biggest difficulty in choosing the right type of viscometer to use in measuring the viscosity of dense medium suspensions is the rapid settling rate of the solids in the suspension. The suspension needs to be continuously agitated in order to minimise this settling effect. As a result many viscometers have been modified to test suspensions of different solids concentration and composition.
For rotational viscometers there are two types of measuring principles, as shown in Figure 3.4 [Slatter et al (2000), Harris (1977)]:
(a) (b)
Figure 3.4 Cylinder measuring system according to (a) Searle's principle (b) Couette's principle rSlatter et al. (2000»)
CHAPTER 3 RHEOLOGY MEASUREMENT
a) Searle's principle: Rotating measuring bob, stationary measuring cup (or still standing lower plate respectively). Most rheometers work according to this method. The rheometer used in this project was also based on this principle.
b) Couette's principle: Rotating measuring cup (or rotating lower plate respectively), still standing measuring bob. Only a few rheometers work according to this method.
Measuring systems according to Couette's principle cause a problem concerning temperature control with liquids because of the friction of the sealings, since a rotating surface has to be sealed against the temperating medium (e.g. water). For this reason they are seldom used unless in a temperated room.
Most of the work presented in this section will be based on rheometers working according to Searle's principle.
3.2.1 Concentric-cylinder viscometer
A typical concentric-cylinder viscometer operating under Searle's principle is shown in Figure 3.4 (a). The accuracy of the results from a concentric-cylinder viscometer depends to great extent on the measuring gap between the concentric cylinders. Ideally, the shear rate and shear stress should be linearly distributed between the measuring gap.
(a) (b)
Figure 3.5 Cross sections of measuring systems with different gap sizes. The function of the circumferential speed v(r) is presented [Slatter et aI. (2000»)
CHAPTER 3 RHEOLOGY MEASUREMENT
A narrow measuring gap (Figure 3.5 (a)) gives the desired linear course of speed function quite accurately. When using a wide measuring gap (Figure 3.5 (b)) the speed function has a curve-like course which is difficult to calculate and which is rheologically undesired, particularly for pseudoplastic fluids and those with an apparent yield. This leads to poor results which are often irreproducible when measuring'data are analyzed.
For solid-liquid suspensions great care must be taken in selecting the fight type of measuring gap.
The gap should be big enough to accommodate the solid particles in suspension, and the same time the gap should be able to give reproducible results when measured data are an·alyzed.
3.2.2 Double gap concentric-cylinder viscometer
The double gap viscometer is primarily used to measure the viscosity of fluids with low viscosity. The viscometer has a large shearing area which is used to detect sufficiently high torque values to enable meaningful evaluation of the results. A diagram of a double gap viscometer is shown in Figure 3.6.
R4
D
R3
"'1 I '"'"l r--
, i
~.
! I
i !
L
R2 RJ
6;:-.. I 1:::::::=
, i
,...""" - ,
Figure 3.6 Double gap viscometer [Slatter et aI. (2000»)
CHAPTER 3 RHEOLOGY MEASUREMENT
The viscometer also works under Searle's principle, with a rotating bob and stationary cup. The measuring bob is shaped like a tube. The measuring cup has a cylindrical core part and therefore shows a ring gap [Slatter et al. (2000)]. This type of viscometer is unlikely to be used to measure the viscosity of dense medium suspensions since the viscosity of these suspensions are moderately high. However, because of the high shearing area, this type of viscometer can be used to measure the flow curves at low shear rates. This can be used to estimate the relative yield stresses of dense medium suspensions, which is often difficult to obtain with other viscometers.
According to DIN 54453 standards, the radii and immersion length of the viscometer (figure 3.5) should be constrained under the following relation [Slatter et al. (2000)]:
(3.10)
(3.11 )
The relatively large shearing area, which detects the torque or shear stress of the fluid, is composed of the inner and outer surface of the measuring bob. This makes the viscometer more sensitive to torque measurement than concentric-cylinder viscometers.
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3.2.3 High-shear viscometer
This type of viscometer is used to obtain high shear rates. Sometimes, high shear rates are required to measure the viscos,ity of samples with high viscosities. The high shear rates are obtained by using a very small relation of radii between the cylinders. The radii are shown in Figure 3.7 [Slatter et al. (2000)].
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Figure 3.7 High-shear viscometer [Slatter et al. (2000)]
The shear rate is calculated from the following equation:
• v
r = h
(Where h = Ra - Ri) (3.12)From Equation 3.12, the shear rate increases as the cylinder gap width is reduced if the viscometer is running at the same speed. High-shear viscometers are run at shear rates not in excess of 10 000 S-l because of the thermal energy encountered as the fluid is sampled at these rates, which would influence the apparent viscosity by heating up the fluid. For the measurement of suspensions the value of the measuring gap 'h' should be large enough to accommodate the solid particles.
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3.2.4 Cone-plate viscometer
In the cone-plate viscometer the shear rate can be considered to be the same everywhere in the fluid provided the gap angle
13
is small. The value of13
is usually less than 4°. Above this value, the requirements for the calculation of the viscosity of the fluid are no longer given [Barnes et al.(1989) and Slatter et al. (2000)]. Figure 3.8 shows a diagram of a cone-plate viscometer with a truncated cone tip.
R
a
Figure 3.8 Cone-plate viscometer [Barnes et al. (1989) and Slatter et al. (2000»)
The shear rate in the fluid is given by the following equation:
r=-
OJfJ
(3.13)The shear stress, measured from the torque at the surface of the cone, is given by the following equation:
(3.14)
The viscosity ofthe fluid is calculated by dividing Equation 3.14 by Equation 3.13:
CHAPTER 3 RHEOLOGY MEASUREMENT
(3.15)
Cone-plate viscometers are usually truncated because this reduces the abrasion of the cone and the bottom plate since there is no friction between them. Friction between the cone and plate would also result in an additional torque being required to drive the viscometer. This would result in falsified high viscosities. However, because of the small values of
13,
cone-plate viscometers cannot be used to measure the viscosity of solid suspensions whose particle sizes are greater than 10 microns.3.2.5 Plate-plate viscometer
In the plate-plate viscometer shown in Figure 3.9, the fluid is placed between the plates. In order to get meaningful results, the value of the plate distance 'h' should be small compared to the plate radius'R'.
~D .,.
. I j , j j'
R j
-
t Figure 3.9 Plate-plate viscometer [Barnes et at. (1989) and Slatter (2000))The shearing area is equal to the area of the top plate. Unlike cone-plate viscometers, the shear rate in plate-plate viscometers is a function of the radius of the plate [Slatter et al. (2000)]. The shear rate increases linearly from zero at the axis to its maximum at the rim of the plate. The shear rate at the rim of the plate is given by the following equation:
Y= -evR
h (3.16)
CHAPTER 3 RHEOLOGY MEASUREMENT
The shear stress at the rim of the plate is equal to Equation 3.14. Thus, the viscosity at the rim of the plate is given by the following equation:
(3.17)
When testing suspensions, in order to get meaningful results, the value of the measuring gap 'h' should be at least ten times greater than the size of the aggregate particles.