2.3 C OAXIAL - CYLINDERS RHEOMETER – T HE BML V ISCOMETER
2.3.1 Principles of measurement in BML viscometer
During the test, the outer cylinder rotates at different angular velocities ωo and the torque due to shear resistance of the test material that is necessary to keep the inner cylinder stationary is measured (Figs.2.9 and 2.11). As shown in Fig.2.10, the inner cylinder unit consists of three parts. This includes the upper unit that measures torque, the lower unit, and the top ring. The function of the lower unit is to eliminate the influence of shearing from the bottom plate of the outer cylinder. In this way, height independence can be assumed and this will simplified the process for mathematical deduction of effective shear rate in the test material. The function of the top ring is to keep a constant height H (Fig.2.10) where torque is measured.
Furthermore, the inner and outer cylinder consists of ribs aligned as a cylinder
(Fig.2.11, left). This leads to a greater cohesion between the two cylinders and the test material in-between. This configuration also reduces the danger for slippage.
In the evaluation of the flow properties of fresh concrete using the coaxial cylinders viscometer, the following general equation (Tattersall, 1991) is adapted:
T = (g-value) + (h-value).N (2.8)
where T is the measured torque at rotational speed N, g-value is the intercept, and h- value is the gradient of the relationship. The measured torque is a function of the viscosity (or the extent of flowability) of the test material and the rotational speed of the outer cylinder (N). The outer cylinder rotates at the angular velocity ωo = 2π.N.
Also, the g-value is known as the flow resistance of the concrete and is a measure of the force necessary to start the movement of the concrete. The h-value is the relative viscosity and is a measure of the resistance of the concrete against an increased speed of the movement. Furthermore, it has been shown that the g-value is a measure of the yield stress while the h-value is a measure of the plastic viscosity, these being the two parameters that can be used to describe the workability of a material whose behaviour fits the Bingham model given by Equation (2.4). Although the g-value and h-value do provide measures of the yield value and plastic viscosity, respectively, their actual numerical values also depend on characteristics of the apparatus used, specifically on the design and dimensions. Therefore, the rheological parameters of the yield stress and plastic viscosity are used in favour to the g-value and h-value in the current study so as to eliminate this dependency. When measurement of rheological parameters is conducted on a Bingham fluid by measuring the torque on the inner cylinder at different angular velocities, the resulting relationship is a linear one. Figure 2.13 shows a typical screen shot of the chart in the BML viscometer software. The chart is plotted on T-N axis. Substituting the slope (h-value) and the intercept (g-value) in the
Reiner-Riwlin Equation (2.9) with some mathematical manipulation, the plastic viscosity and yield stress of the tested material can be calculated.
Fig.2.13 – A typical chart of torque-rotational speed in BML viscometer software.
For a sample conforming to the Bingham model, the flow can be described by the well-known Reiner-Riwlin equation (Reiner, 1949; Bird et al., 1987) for coaxial cylinders viscometer:
⎥⎥
⎥⎥
⎥
⎦
⎤
⎢⎢
⎢⎢
⎢
⎣
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −
η ω
⎟⎟+
⎠
⎜⎜ ⎞
⎝ τ ⎛ π
=
2 2 i
p o i o 0
Ro
1 R
1 R ln R H 4
T (2.9)
where ωo is the angular velocity of the outer cylinder (rad/s), H is the effective height of the inner cylinder (m) as shown in Fig.2.11, Ri the radius of the inner cylinder (m), and Ro the radius of the outer cylinder (m), according to Fig.2.9. By substituting ωo
with 2π.N in Equation (2.9), the total torque can be determined as follows:
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −
η π
⎟⎟+
⎠
⎜⎜ ⎞
⎝ τ ⎛ π
=
2 2 i
p 2 i
o 0
Ro
1 R
1
HN R 8
ln R H 4
T (2.10)
By comparing Equations (2.8) and (2.10), one has
0
2 2 i
i o
Ro
1 R
1 R ln R H 4
g τ
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −
⎟⎟⎠
⎜⎜ ⎞
⎝ π ⎛
= (2.11)
p
2 2 i
2
Ro
1 R
1 H
h 8 η
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −
= π (2.12)
Therefore, the units of g-value and h-value can be transferred into fundamental units of yield stress (Pa) and plastic viscosity (Pa s), respectively, expressed as follows:
g R ln R H 4
R 1 R
1
i o 2 2 i 0
o
⎟⎟⎠
⎜⎜ ⎞
⎝ π ⎛
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −
=
τ (2.13)
H h 8
R 1 R
1
2 2 2 i p
o
π
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −
=
η (2.14)
Wallevik J.E. (1998) has given a detailed mathematical derivation of the above rheological equations for the coaxial cylinders viscometer.
The effective shear rate γ varies with position r (measured from centre of cylinders) in the annulus, and may be calculated from τ
&
and η :
0 p
H ) r 2 ( T 1
2 0 p
τ π −
= η (2.15)
γ&
Assuming no plug flow occurs, the Equations (2.10) and (2.15) may be combined to give:
p 0 i
o p 0 o 1 2 o 2 i
2 )]
R ln(R [
R ) 1 R ( 1 r
2
η
− τ η
+ τ ω
−
=
γ −
& (2.16)
From Equation (2.16), the shear rate is not a constant within the test sample as it varies with position r, measured from the centre of the cylinders. The shear rate is the greatest at the region nearest to the inner cylinder, i.e. at Ri, and decreases outward towards the outer cylinder. In terms of possible suspended particle migration within the test sample, it is always from the region of high shear rate to the region of low shear rate (Barnes et al. 1989). As such the gradient of the shear rate
dr
dγ& is an
important factor that influences the rate of particle migration. The gradient of the shear rate is given by:
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎟⎟⎠
⎜⎜ ⎞
⎝
⎛ η + τ
⎟⎟ ω
⎠
⎜⎜ ⎞
⎝
⎛ −
−
γ = −
i o p
0 o 1 2 o 2 i
3 R
ln R R
1 R
1 r
4 dr
d& (2.17)
With increasing angular velocity ωo, the difference in the shear rate will increase, which results in a higher likelihood of particle migration leading to segregation of the test sample during the measurement. Hence, the shear rate during testing of concrete should be kept low to minimise segregation of the test sample. The range of shear rate usually encountered in practice is shown in Fig.1.2.
Based on Equation (2.16), the calculated shear rate experienced by the concrete sample at the interface of the inner cylinder (r = 0.1 m) ranged from about 2 to 22 s-1 in the current study, depending on the rheological parameters of the tested concrete mixtures. According to Fig.1.2, this range corresponds to the typical processing operations encounter in the handling of concrete, such as pouring, mixing and pumping.