Fenómenos de recuperación

Wall friction tests were performed under ambient conditions (20–24oC, 35–55% relative humidity) with the same annular shear cell (Brookfield Engineering Laboratories Inc., USA). The hopper wall material was the PFT-507 lid, which had a smooth bottom surface made of 304 stainless steel with 2B finish. All operations were computer-controlled and the default test option was Even Spacing of Displacement Levels. The powder was first consolidated up to ~4.8 kPa and sheared to steady state conditions. The powder was then consolidated and sheared at 10 decreasing w that ranged from ~4.8 kPa to ~0.48 kPa; the steady state w at each normal stress was measured. Ten pair values of (w, w) were used to construct the wall yield locus.

3.4.4 Analysis

3.4.4.1 Powder flow function and Jenike’s arbitrary powder flow divisions

Unconfined yield stress, y, was plotted against major consolidation stress, c. The Jenike’s limiting flow function values and arbitrary powder flow divisions, namely very cohesive or non- flowing, cohesive, easy flowing, and free flowing, were superimposed on the c:y plot; powder flow behaviour was read directly from the plot as a function of pre.

3.4.4.2 Cohesion

A primary aim of this work was to correlate C with d*32, and pre; this was done following the Coulomb Yield Criterion, =μ+C (Equation 3.3). According to the Coulomb criterion, the shear

stress required to fracture a consolidated powder bed is the sum of the frictional contact stresses involved in the sliding between particles, term μ, and C. The processes that take place when a powder bed undergoes shear deformation are complex; they have been described and reviewed by Schulze (2008). According to Schulze (2008), shear failure occurs in a zone and not a simple plane, and the thickness of the shear zone is apparently dependent on mean particle size, ~5–20 particle diameters for particle larger than ~100 μm and ~200 particle diameters for very fine powder. When a bed of consolidated particles is sheared, the particles in the shear zone react against the applied normal stress to free themselves sufficiently to force themselves past one another. This relative movement results in bed dilation, which affects the maximum shear stress at incipient flow.

Following Molerus (1993) who derived a theoretical expression for C in an unconsolidated powder, C in a polydisperse bulk powder under consolidation is expected to relate to the number of interparticle contact points, and hence co-ordination number. The number of particle-particle contacts is not directly measurable, but is expected to depend on the particle surface area per unit volume. For a shear zone of constant cross-sectional area, the number of particle contacts will depend on the zone thickness, which can be deduced if the following are known: i) B as a function of pre, and ii) bed dilation.

For (i), it is assumed that the B measured in static tests is predictive of the B when the bed is sheared at the same pre. For (ii), direct measurement of bed dilation is not available; however as C relates to the forces that must be overcome before flow commences, the dilation will produce a normal reaction stress that is assumed to be equal to the pre. Thus, C is postulated to be a function of particle surface area per unit volume of the powder bed and the dilation force per unit area across the shear zone, per Equation 3.19.

Cf

(

)

By definition, the d32 of a particle of density p enables the direct calculation of the surface area, Ap, of a volume of material, Vp, also of density p, when it is divided into spheres of

d32; Ap/Vp = 6/d32. However in this work, the parameter of interest is the particle surface area divided by the volume of bulk powder, VB. For a powder of bulk density B comprising particles of density p, it follows from simple algebra that Ap/VB = 6B/pd32. The p of milled and spray- dried lactose powders is ~1,540 kg m–3 _{(G. Niro, 2012). The following are the}

p of the other powders in kg m–3 measured with a specific gravity bottle and distilled water: ~2,470 for S3, ~2,120 for S1, ~2,130 for S2, ~3,200 for RD3, ~3,010 for RD1, and ~2,750 for RD2.

The dilation force per unit area term was written non-dimensionally and relative to the minimum preconsolidation stress, pre,min, applied to the bed, and became pre/pre,min. Equation 3.19 was written algebraically per Equation 3.20, with the assumption of a power law

relationship; the integer 6 in the term 6B/pd32 was lumped with the prefactor m; m, a, and b are experimental fitting parameters and their values were determined by regression analysis.

C=m B pd*32 a pre pre,min b (3.20)

3.4.4.3 Bulk density under consolidation

The profiles of B at pre of 0.31 kPa, 0.61 kPa, 1.20 kPa, 2.41 kPa, and 4.85 kPa were modelled with Equations 3.6, 3.8, 3.10, 3.12, and 3.14. By linear regression, Equations 3.6, 3.9, 3.11, 3.13, and 3.15 were used to estimate the values of fitting parameters ks,M1,ks,M2, as,bs, kN1,kN2, kJ1, kJ2,

kG1, and kG2. Each fitting parameter was then plotted against 1/d*32 and the trend displayed was examined.

3.4.4.4 Hopper outlet B

Hopper outlet B was calculated with Equations 3.16 and 3.17 following Jenike’s procedure outlined in Section 3.2.5; B was then plotted against d*32 and the trend was examined and discussed with regards to powder flowability.

3.5 Results