Procesos de conformado por deformación plástica

Jenike (1975) proposed the determination of B, which is the minimum width of hopper outlet required to assure powder mass flow under gravity from hoppers, as a measure of powder flowability.

Powder mass flow is a uniform and steady state flow through the hopper outlet under gravity, in which the bed surface of the powder remains level until it reaches the sloping section of the hopper. Mass flow is influenced by hopper half angle, p, which is the angle between the sloping hopper wall and the vertical. Hopper half angle is primarily dependent on the angle of internal friction and the angle of wall friction; the angle of wall friction represents the friction between the powder and the wall of the hopper (Jenike, 1964).

Following Jenike (1975), B is given by Equation 3.16, where g is gravity acceleration [m s–2_{] and}

crit is the critical stress developed in an arch surface [Pa]. H(p) is a factor determined by the hopper half angle, and Equation 3.17 gives an approximate expression for conical hoppers. To calculate B, a list of steps following the Jenike’s hopper design procedure (Jenike, 1964) is required; they are summarized below.

B= H(p)crit Bg (3.16) H(p)=2.0+ p 60 (3.17)

The first few steps involve the measurement of a family of yield loci of a powder with a shear cell under different loads, followed by Mohr circles analysis to obtain the values of y and

c, and hence powder flow functions, and the values of e; these have been reviewed in Sections 3.2.1 and 3.2.2.

Next is the measurement of kinematic angle of wall friction, w; it is determined from the wall yield locus measured by shear testing in which a powder is sheared against a sample of the hopper wall material, usually at decreasing applied normal consolidation stresses (Jenike, 1964; Schulze, 2008). The consolidation stress acting between the powder and wall material is the wall normal stress, w. The powder is first sheared at a selected w until the wall shear stress,

w, becomes constant; shearing at steady state conditions is reached. Subsequently, w is reduced and the powder is sheared until another constant value of w is obtained. This procedure is repeated to obtain several pair values of (w, w) which are used to construct the wall yield locus; the locus is typically a straight line that passes through the origin on the w:w plot. The wall yield locus is a yield limit that describes the w necessary to shift the powder continuously across the wall surface under certain w at steady state conditions, and Equation 3.18 gives the value of w.

w=tan1 w w (3.18)

To obtain ff and p, the flow factor charts developed by Jenike (1964) are required. A flow factor chart is the plot of w versus p at a specified value of e. On the chart is a series of lines that represent different values of ff that range from 1.1 to 4.0, and a line that gives the limiting value of p to ensure mass flow as a function of w. The line divides into core and mass flow; mass flow is achieved below this line and above it core flow takes place. The ff values in the Jenike’s flow factor charts were measured for a conical hopper and a wedge-shaped hopper with a slot outlet for e values of 30o, 40o, 50o, 60o, and 70o. An example on how to use the chart to estimate the values of p and ff is available in Rhodes (1998). In this thesis, all calculations have been done using the flow factor charts in Rhodes (1998).

Fitzpatrick, Barringer and Iqbal (2004) reported the flow properties of 13 food powders which included salt, sugar, starch, flour, and cellulose powder. FF and e were measured with an annular shear cell at consolidation stresses below 8 kPa, and w was measured with a Jenike shear cell at consolidation stresses below 6 kPa; the cylindrical base of the cell was replaced with a flat plate made of 304 stainless steel. It was found that e ranged from 40o to 65o, w ranged from ~12o to ~27o, and p ranged from 15o to 35o. The authors noted that as a rule of thumb, p of 20o _{is often used for achieving mass flow, but further demonstrated that}

p could vary by up to 15o for some food powders.

The final steps in calculating B involve the determination of crit, the critical stress developed in an arch surface, which requires information on FF and ff, and the value of bulk density, B,crit, that is associated with crit. An illustration on how crit is obtained is given in Figure 3.4; the intercept of the flow function with the D=(1/ff)c line gives the value of crit. Note that no flow occurs when y is greater than c/ff, and flow happens when y is lower than c/ff. The value of B,crit can be obtained if its relationship with (c, crit) is known.

In spite of the proposal by Jenike (1975) on the use of B as a measure of flowability, there is no traceable work in the literature that emphasizes on the utility of B as a powder flow indicator, and on how B relates to the output of other flow characterization methods.

Figure 3.4 Determination of critical stress developed in an arch surface with powder flow function and hopper flow factor

3.2.6 Summary of literature review

The literature review of this chapter covers powder shear testing and the protocol developed by Jenike (1964); it includes yield locus, cohesion, powder flow function, B under consolidation, and B, the minimum width of hopper outlet required for mass flow. Parameters C, C0, and B have been singled out for further investigation.

The shear testing by Jenike provides a scientific basis for the qualitative and quantitative characterization of powder flowability; caution should be practised because shear tests that are restricted to a particular stress level is not capable of providing information on powder flow behaviour at other stress regions. Therefore, measurement over a range of consolidation stresses should be carried out to generate several sets of yield locus and powder flow function for comprehensive and accurate flow information.

Cohesion is the shear stress of a consolidated powder in the absence of consolidation stress; it relates to the interparticle forces that must be overcome before failure or flow commences. It has been demonstrated by Orband and Geldart (1997) that C relates directly to d32 and a critical d32 or diameter range exists; C increases almost linearly with decreasing d32 below this critical value. However, the combined influence of d32 and pre on C has not yet been fully explored and quantified.

It has also been shown that C, which is a measure of interparticle forces in consolidated powders, can relate directly to powder avalanche activity in the GDR (Vasilenko, et al., 2011; Vasilenko, et al., 2013); powder avalanche reflects powder flowability under unconfined conditions. The reason for such observation is still largely unknown, and there has not been further assessment of the relationship between powder avalanching and C0, the cohesion of

unconsolidated powder that is more likely to directly influence powder avalanching; this will be addressed later in Chapter 6.

Powders are consolidated in shear testing; B changes as a function of pre and empirical correlations with two fitting parameters each can be used to model B and pre, recall Kawakita and Lüdde (1971), Malave et al. (1985), and Gu et al. (1992). Little is known about the relationships between the fitting parameters and powder properties such as d32, though early evidence suggests connections between the fitting parameters, C, and particle diameter, see for example Kawakita and Lüdde (1971), Peleg (1978), and Yamashiro et al. (1983).

Hopper outlet B can be used as a measure of powder flowability (Jenike, 1975); however the relationships between B and particle size distribution and the output of other flow characterization methods remain unknown.

3.3 Aims

1. To measure the yield locus, c, y, e, B, and w of samples of milled and spray-dried lactose powders, fine sand, and refractory dust with an annular shear cell at pre below 5 kPa.

2. To characterize the flowability of the selected powders according to the Jenike’s arbitrary powder flow divisions.

3. To determine the C and C0 of the selected powders, and correlate them with d*32 and pre. 4. To model the B of the selected powders with Equations 3.6, 3.8, 3.10, 3.12, and 3.14,

and correlate the fitting parameters of the equations with d*32.

5. To determine the B for the selected powders, investigate its relationship with d*32, and discuss its utility as a measure of powder flowability.