2.4 Modelling Slurry Rheology
2.4.1 The viscosity of spherical-particle suspensions
2.4.1.2 Suspensions with a high solid-phase concentration
Suspensions with a high solid-phase concentration are considered to be those in which the solid phase exceeds 10% by volume i.e. 0.1 < 0
o
Dmax. Omax is the maximum attainable concentration and depends on the void fraction corresponding to a more or less compact arrangement of the spheres [Jinescu (1974)]:<1>max =~=l-& V
V (2.12)
CHAPTER 2 LITERATURE REVIEW
Where 0= V
JV
is the porosity, and Vo the void volume. The porosity depends on the particle- size distribution and uniformity.For suspenSIOns used in DMS the spheres are not equal In size, and 0 can be determined experimentally from the following equation:
&=l_ Pv =1-<1> max (2.14)
P
Where Q, is the bulk density of the spheres, and Dis the skeletal density.
The relative viscosity can be found from the Arrhenius relationship:
f.ir
=
exp(k<l» (2.15)Mooney (1951) also extended Einstein's equation for an infinitely dilute suspension of spheres to apply to a suspension of finite spheres. For a monodisperse system (i.e. a system having only one disperse phase) he obtained the following equation:
f.ir
=
exp ( 2.5<1>J
1-k<l>
(2.16)
Mooney's analysis was limited to rigid, spherical particles. Also, his approach was partly empirical in that the interaction parameters were left for experimental determination, no effort being made to obtain their values from hydrodynamic theory. Mooney introduces the space- crowding effect of suspended spheres on each other. That is, in a two component system, for example, spheres of size r\ and partial volume concentration 0\ crowd spheres of size r2 into the remaining free volume 1-=l2~\' where
[12
is the crowding factor. The Einstein viscosity equation postulates a suspension so dilute that there is no appreciable interaction between spheres. Mooney postulates that in extending the equations to higher concentrations, first order interactions must be taken into account. This interaction is described as essentially a crowding effect.The modelling of sluny rheology can also be extended to include different types and size range of contaminants. Ferrara et al. (1990, 1992) considered the solid phase of the medium as made up of
CHAPTER 2 LITERA nJRE REVIEW
by a binary blend of a magnetic component and a non-magnetic component. Two series of experiments were carried out using FeSi as the magnetic component, and two types of limestone as the non-magnetic contaminant. Under steady-state. conditions, they supposed that the characteristics of the medium (chemical composition, size and particle shape) were constant.
These assumptions lead to a two component model.
Using a semi-empirical model (Quemada model) as their basic model, the following expression was written for a two component system, comprised of a non-magnetic component (1) and magnetic component (2):
(2.l7)
Where:
(2.18)
where 2 = 110; q = 1/01; Q = 1102; q = Otic), the volume fraction of the non-magnetic component in the total solid; and Q
=
(I-CD, the volume fraction of the magnetic component in the total solid.The above two equations are concerned with the rheological properties made up of a non- magnetic component (1) and a non-magnetic component (2). A major constraint for the above equations is the maintenance of a constant medium density. To maintain a constant medium density, q a variation in the content of one component should be compensated by an appropriate variation in the total solid phase composition. This is expressed by the following constraint relation:
0= G{1-0) + (Cl[j+ QG)O (2.l9)
where Q, q, and G are the fluid density, density of the non-magnetic solid, and density of the magnetic component, respectively.
CHAPTER 2 LITERATURE REVIEW
The conclusions drawn from the experimental work, using ferro silicon and two types of limestone as the non-magnetic contaminant (fine limestone 0.2 mm and coarse limestone in the size class - 0.5mm+0.2mm) were:
• The results showed that the apparent viscosity of the tested suspensions depended mainly on the ratio of the volume occupied by the solid phase to the total volume of the medium and to a lower extent on the volume fraction of the non-magnetic component in the solid.
The apparent viscosity decreased slowly as the shear rate was increased, indicating a pseudo-plastic behaviour of the tested samples.
• The results of the correlation were better for the fine limestone, compared with the coarser limestone. This was attributed to experimental errors which would have arisen from the lower stability of the coarser limestone.
• It was noted that the prediction capability of the model did not vary appreciably when using less experimental values for calculating the model parameters. This fact is very important for the practical use of the model in plant control.
Napier-Munn et al. (1996) modelled the rheology of slurries based on the output reading of a Debex on-line viscometer [Reeves (1985); Napier-Munn et al (1996)]
The Debex on-line slurry viscometer is rotated at a constant speed by a DC micrometer or stepper motor in a baffled cup. The retarding torque on the bobbin caused by the viscous resistance of the slurry in the cup is reflected in an increase in the current draw of the motor which is measured and displayed in units of mill i-volt (mV). The mV reading and associated bobbin rotational speed are then converted to a turbulence-corrected shear stress flow curve using a special calibration algorithm [Napier-Munn et al. (1996)].
To predict the viscometer raw readings a polynomial equation describing the Debex readings (m V) as a function of bobbin angular velocity (0) was employed. The equation is of the general form:
(2.20)
CHAPTER 2 LITERATURE REVIEW
The following phenomenological assumptions were made in developing the model:
• Coefficient
ao
is only determined by a variable which is related to the yield stress of the slurry, and therefore reflects the effect of particle hydrodynamics.• Coefficient al is also a factor of the hydrodynamic interaction of the particles. Solids volume fraction, particle size, chemical conditions and slurry 'temperature may affect the intensity of these interactions.
• Coefficient a2 may also be affected by particle interactions. However, the effect of these interactions will become weaker with the bobbin action becoming stronger.
• With the bobbin rotation a strong function in the cubic term, particle interactions may be ignored and coefficient a3 mainly reflects the mechanical action of the bobbin.
This lead to the following expression:
Where:
and,
o
= volume solids concentration (proportion) T=
temperature (0C)o
= bobbin angular velocity (rad/s) P20=
20% passing size solids (mm) Pao=
80% passing size solids (mm)C1-C6
=
model parameters to be determined experimentally for particular slurries or slurry types(2.21)
(2.22)
CHAPTER 2 LITERATURE REVIEW
The flow curve for the slurry, and local viscosity, can be obtained by applying the calibration algorithm discussed by Napier-Munn et al. (1996). The model is a semi-empirical model, and does not account explicitly for other variables such as particle shapes, extreme values of particle size (particularly fine), electrochemical conditions or complex interaction of these variables.
It was observed that the set of parameters obtained from the model are not universal, but differ for different slurries. This is due to the complexity of the mineralogy, chemistry and particle shape of the slurries. Under such circumstances part, or all, of the parameters may need to be re-fitted to measurement data from particular slurry.
From their model, Napier-Munn et al. (1996) showed that a simple type of slurry can exhibit a wide range of rheological behaviour, from pseudo-plastic to dilatant, and with varying yield stress, depending on the prevailing particle size and solids concentration:
• When a suspension contains relatively low solids concentration and a high content of fines, its rheological character is likely to be dilatant.
• When the suspension density increases and fines content reduce, the suspension is likely to exhibit a pseudo-plastic character.
• Increasing either fines content or suspension density leads to increased yield stress for both dilatant and pseudo-plastic flows.