III.1. Rheology of cohesive powders in a pilot scale planetary blender
2. Materials and methods
Table 2: Physical characteristics of the powders studied: mean particle size as obtained by LASER diffraction, bulk densities measured by a volumenometer Erweka®, true density measured by a He pycnometer, cohesion
measured by a FT4 rheometer. Available properties are mentioned for the mixtures as well as their compositions
Powder d50 (µm) ρp (kg.m-3) ρb (kg.m-3) ρt (kg.m-3) Carr Index (%)
Cohesion (kPa) Properties of Pure Powders
Semolina 312 1463 679 720 5.8 0.244
±0.098
Lactose 61 1533 661 795 16.9 0.745
±0.116 Fine
lactose
26 1539 495 638 22.5 1.115
±0.133
Talc 16 2772 501 848 40.9 0.535
±0.070
Sand 33 2643 887 1222 27.4 0.593
±0.039 Properties of Powder Mixtures by wt%:
M1 (Lactose : Semolina) and M2 (Fine lactose : Semolina)
M1 (10:90) N.A N.A 724 792 8.6 0.17
M1 (20:80) N.A N.A 761 840 9.4 0.17
M1 (30:70) N.A N.A 793 901 12.0 0.24
M1 (40:60) N.A N.A 817 935 12.6 0.36
M1 (50:50) N.A N.A 781 946 17.4 0.50
M1 (60:40) N.A N.A 761 929 18.1 0.65
M1 (70:30) N.A N.A 744 900 17.3 0.70
M1 (80:20) N.A N.A 695 864 19.6 0.78
M1 (90:10) N.A N.A 685 824 16.9 0.75
M2 (10:90) N.A N.A 704 766 8.1 0.16
M2 (20:80) N.A N.A 706 816 13.5 0.17
M2 (30:70) N.A N.A 698 854 18.2 0.34
M2 (40:60) N.A N.A 681 859 20.7 0.57
M2 (50:50) N.A N.A 655 833 21.4 0.94
M2 (60:40) N.A N.A 599 785 23.6 1.23
M2 (70:30) N.A N.A 582 741 21.5 1.29
M2 (80:20) N.A N.A 517 697 25.9 1.16
M2 (90:10) N.A N.A 482 660 26.9 1.20
Figure 1: SEM pictures of semolina (top left), lactose (top right), fine lactose (mid left), sand (mid right) and talc (bottom left)
The particle’s median diameter d50 was measured with a LASER diffraction particle sizer Mastersizer3000 (Malvern) under an air pressure of 3.5 bar, the particle density ρp was measured using an Accumulator Pyc 1330 (Micromeritics) with the 10 cc cell. The bulk and tapped densities, ρb and ρt, were measured with a volumenometer (Erweka) with 110 g of powder, the tapped density being measured after 1000 taps, which is enough to be sure that the powder cannot be more packed by the apparatus since the volume stabilizes around 300 taps. Each characterization was made at least in duplicate, the mean values being given in Table 2 and Figure 2. Mixtures were prepared by adding the ingredients to a half liter transparent container, and further blend it with a spatula for at least one minute until the mixture appears homogenous. The mixture quality was good enough to have no
Fine lactose Sand
Semolina Lactose
Talc
influence on the density measurement, which is verified by the low standard deviation on the repeated tests.
Figure 2: Apparent densities measured for mixture 1 (left) and mixture 2 (right) as a function of composition
Bulk and tapped densities allow calculation of the Carr Index (eq. 1), which is a characteristic of powder flowability [19].
� = .��− �
�� eq. 1
Table 2 and Figure 3 represent Carr Index values for single powders and mixtures. As reported by Leturia et al. [20], the typical scale of flowability used to classify powder flow behavior shows that powders of Carr Index between 5 and 15% are easy flowing, between 15 and 22% the behavior is intermediate, while when higher than 22%, it reveals a bad flowing powder. To ease the interpretations in this study, we will consider that for Carr Index below 15% the powder is free flowing and above 15% the powder is cohesive. Considering the data in Table 2, semolina is free flowing and lactose, fine lactose, talc and sand can be considered as cohesive. For mixture 1, if the lactose content is smaller than 50% the powder is free flowing, and when it is bigger the powder is cohesive (Figure 3). For mixture 2, powders are cohesive when the fine lactose content is bigger than 30%.
The FT4® rheometer by Freeman Technology is used as shear device, with its rotational shear cell (Figure 4). This apparatus imposes a normal stress to the powder bed while the shear stress is recorded. Cohesion is measured according to the standard protocol for the FT4 shear testing. The powder is poured into the cylindrical vessel, and its volume is fixed using the rotating part on the top of the vessel that can be seen in Figure 4. The powder is conditioned by a blade getting downward and upward through the testing cell.
Figure 3: Carr Indexes calculated for both mixtures
Yield locus are built as follows: (1) powder is pre-sheared at a normal pre-consolidation load (2) the sample is sheared under smaller consolidation values, increasing up to the pre-consolidation one, in order to measure the shear stress peaks characteristics of the powder bed rupture.
Figure 4: FT4 Freeman rheometer (left) and shear cell (right)
For a given pre-consolidation, the major principal stress1 and the unconfined yield strength
c [21] are determined graphically from the Yield Locus (Figure 5 (a)) and lead to build flow functions (Figure 5 (b)) that are needed for silo design, for example. Flow functions allow the description of the flowing behavior of powders whatever their consolidation state. They give the possibility to gather results obtained under different procedures [22]. The flow function’s representative numerical value is the slope but as far as flow functions are often not straight-lines passing by the origin like in Figure 5 (b) (see [23]), the ratio ffc of 1 and c for given pre- consolidation states is widely used. That limits the interest of the method.
(a)
(b)
Figure 5: Examples of lactose Yield locus (a) and flow function (b)
As far as cohesion quantifies the interaction forces between particles, it should be considered as a key property that influences the way particles are flowing and insofar the flow pattern during agitation in a vessel. Cohesions derived from the Yield locus correspond to 2, 4, 8 and 16 kPa. Each measurement was repeated twice for three powders. As far as levels of consolidation expected for powders during a mixing operation are small, we chose to consider cohesion measurements obtained at 4 kPa (Table 2). It must be noted that experiments at 2 kPa are not reproducible for finer powders like talc and fine lactose.
2.2. Experimental set up and procedure
The Triaxe® is a four bladed mixer that operates thanks to a dual motion of rotation and gyration (Figure 6). It is designed so that the impeller system covers the whole volume of the blender. The four rectangular blades, made of stainless steel, are inclined at about 45°, the angle between each blade being 90°. The angle between the horizontal and the gyration axis is about 15°.
The stainless steel spherical tank has a volume of 48 l and the distance between the vessel and the blades is about 1 mm. The powder is loaded by the top and drained from the bottom. More detailed information about this kind of mixer can be found in Demeyre’s PhD thesis [24].
Figure 6: Triaxe blender (left) and schematic diagram (right)
Two torque-meters record the rotational and the gyrational torque. If ω is the angular speed (rad.s-1) and T is the torque (N.m), the power P (W) needed to stir the powder load can be calculated as follows (eq. 2):
� = � . (��− �0) + �� . (��� − ��0) eq. 2
With subscripts "g" for gyration, "r" for rotation, "f" for filled tank, "0" for empty tank and "m"
for motor.
Angular speeds are expressed as the speed of motors, using reduction ratios of the apparatus;
the angular speeds of axis ωga and ωra can be calculated thanks to equations 3 and 4. Speeds of the axis reach about 20 revolutions per minute (rpm) and 100 rpm for gyration and rotation respectively.
� = �
.79
eq. 3
�� =��
+ . 9. � eq. 4
Each pilot scale experiment is performed with 30 kg of powder, following the protocol: torque stabilization during 2 hours with ωgm and ωrm about 2000 rpm, measurement of mean torques with empty mixer (Tg0 and Tr0), loading of the powder, stirring during 10 min with ωgm and ωrm about 2000 rpm to mix the powder enough in order to stabilize the torque, recording of mean torques with filled mixer (Tgf and Trf), and finally tank emptying and cleaning.
Torque measurements correspond to combinations of speeds ωgm and ωrm ranging from 0 to 3000 rpm (0, 75, 150, 300, 600, 900, 1500, 2100 and 3000 rpm). This means that for each powder studied, 81 couples (ωgm, ωrm) have been carried out. Figure 7 describes the sequence of gyrational
speeds and rotational speeds used during experiments. First, the gyration is fixed and the rotation increases. When the rotation reaches its maximum value, the gyration speed is increased to the following value and the rotation starts again at its smaller value. It is important to notice that because of the dependence of the rotational angular speed of blades ωra on the gyrational angular speed of blades ωga (equation 4). Agitation angular speeds are expressed as angular speeds given by the motor ωgm and ωrm for a better understanding.
Figure 7 : Diagram showing the procedure followed for rotation and gyration changes during the experiments
The filling ratio f may be calculated using the bulk density ρb, the tank volume Vtank and the mass of powder poured mp (eq. 5). The filling weight of 30 kg of each powder corresponds to f values in the range 0.70 - 1.25. For f values greater than 1, the powder was forcibly compacted into the blender.
� = ��
� . �� ��
eq. 5
Experiments with varying filling weights of a same powder were conducted in order to complete these runs. For this, three additional filling ratios were investigated for semolina and lactose: 0.42, 0.63 and 0.83 (Table 3).
Table. 3: Lactose and semolina weights, volumes and filling ratios studied
Semolina Lactose
Mass (kg) 13.7 20.5 27.1 30.0 13.3 20.0 26.3 30
Volume (L) 20 30 40 44 20 30 40 45
f 0.42 0.63 0.83 0.92 0.42 0.63 0.83 0.94
ωgm = 0 rpm ωrm = 0 rpm
ωgm = 0 rpm ωrm = 75 rpm
ωgm = 0 rpm ωrm =3000 rpm
ωgm = 75 rpm ωrm = 0 rpm
ωgm = 75 rpm ωrm = 75 rpm
ωgm = 75 rpm ωrm =3000 rpm
ωgm = 3000 rpm ωrm = 0 rpm
ωgm = 3000 rpm ωrm = 75 rpm
ωgm = 3000 rpm ωrm = 3000 rpm
A characteristic tip speed uch has been set up by Delaplace et al. [25] to combine gyrational and rotational speeds into a single value. It corresponds to the maximum linear tip blade velocity divided by π. Equations 6 and 7 can be used to calculate uch with: ds=0.112 m the diameter of the spherical reducer of the blender and D=0.448 m the distance between two opposite blade tips. In both equations, angular speeds are written as ga and ra for gyration and rotation. The units should therefore be in revolutions per second.
Ω� . �
Ω . � < → �ℎ = √(Ω + Ω� ). � + � eq. 6
Ω� . �
Ω . � > → � ℎ = Ω� . � + Ω . � eq. 7
The impact of both rotational and gyrational speeds on uch can be appreciated on the 3-D graph in figure 8. It can be stated that rotational speed has a greater influence on the characteristic speed than the gyrational speed. The surface represented is almost planar except for the small values of the rotational speed and the highest ones of the gyrational speed, which are not in the expected range of operation of a planetary mixer. In addition, if a projection of these values on the (uch, ra) plane is made, the whole values are collapsing in a single line, mainly because ds is much smaller than D.
Figure 8: Influence of both rotational and gyrational speeds on uch
Through the use of uch, a dimensionless correlation (eq. 10) [16] links a modified Froude number FrM (eq. 8) and a modified power number NpM (eq. 9) [25], where g=9.81 m.s-2.
���= �ℎ . �
eq. 8
��� = �
� . � ℎ. �
eq. 9
��� = �. ��� eq.10
To get a better understanding of the meanings of this correlation, the power can be expressed directly as a function of uch and the two coefficients a and b (eq. 11) from equations 8, 9 and 10.
� = � � − − � � ℎ+ eq.11
It has been previously reported that for a free flowing powder b = -1 [17]. In equation 11, it means that the power is a linear function of uch. If the powder is cohesive, b will be closer to -3/2, meaning that the power is not dependent on uch [17].