Introduction & Context

In process engineering, the flowability of bulk solids is a critical parameter for the design of storage vessels, hoppers, and silos. The construction of a yield locus from shear cell data allows engineers to quantify the internal friction and cohesive strength of a powder. By determining the relationship between consolidation stress and unconfined yield strength, engineers can predict whether a material will arch or rathole within a hopper, ensuring reliable discharge and preventing operational downtime.

Methodology & Formulas

The analysis relies on two primary metrics: the Flow Function Coefficient (ffc) and the Wall Friction Angle (φ_w). These parameters are derived from experimental shear cell measurements.

The Flow Function Coefficient is defined as the ratio of the major consolidation stress to the unconfined yield strength:

\[ ffc = \frac{\sigma_{c}}{\sigma_{y}} \]

The wall friction angle represents the frictional resistance between the bulk solid and the hopper wall material. It is calculated using the arctangent of the ratio between the wall shear stress and the normal wall stress:

\[ \phi_{w} = \arctan\left(\frac{\tau_{w}}{\sigma_{w}}\right) \]

The following table outlines the classification criteria used to categorize the flow behavior of the bulk solid based on the calculated ffc value:

Flow Function Coefficient (ffc) Range	Flowability Classification
ffc < 2.0	Poor flowability (cohesive)
2.0 ≤ ffc < 4.0	Moderate flowability
4.0 ≤ ffc < 10.0	Good flowability
ffc ≥ 10.0	Excellent flowability (free flowing)

To ensure statistical significance and accurate extrapolation of the major consolidation stress, we recommend the following:

Perform a minimum of three to four shear points at different consolidation stresses.
Ensure the points cover the range of pressures expected in your specific industrial application.
Verify that the resulting yield locus curve exhibits a consistent, concave-downward shape typical of cohesive powders.

The unconfined yield strength is a critical parameter derived from the yield locus that represents the stress required to fail a free-standing arch of powder. It is essential because:

It determines the minimum outlet dimension required to prevent cohesive arching.
It allows engineers to calculate the flow factor of the material.
It helps predict whether a material will exhibit ratholing or bridging behavior in a silo.

Moisture significantly alters the inter-particle forces, which directly shifts the yield locus. When testing samples with varying moisture levels, consider these factors:

Increased moisture typically shifts the yield locus upward, indicating higher cohesive strength.
Samples must be conditioned to the exact moisture content expected in the process environment to avoid underestimating flow resistance.
Small fluctuations in humidity can lead to non-linear changes in the yield locus, requiring multiple test sets for sensitivity analysis.

Worked Example: Determining Powder Flowability

A process engineer is evaluating a pharmaceutical excipient to determine if it will discharge reliably from a conical hopper. Using a Jenike-style shear cell, the engineer performs a consolidation test to determine the flow function coefficient (ffc) and the wall friction angle.

Knowns:

Consolidation stress (σ_w): 5.0 kPa
Wall shear stress (τ_w): 2.1 kPa
Unconfined yield strength (σ_unconfined): 2.5 kPa
Major consolidation stress (σ_{consolidation}): 10.0 kPa

Step-by-Step Calculation:

Calculate the wall friction angle (φ_w) in degrees. The coefficient of wall friction is defined as the ratio of wall shear stress to consolidation stress: \[ \tan(\phi_w) = \frac{\tau_w}{\sigma_w} = \frac{2.1}{5.0} = 0.420 \] \[ \phi_w = \arctan(0.420) \approx 22.782^\circ \]
Calculate the flow function coefficient (ffc). The ffc is the ratio of the major consolidation stress to the unconfined yield strength: \[ ffc = \frac{\sigma_{consolidation}}{\sigma_{unconfined}} = \frac{10.0}{2.5} = 4.000 \]
Classify the flowability based on the calculated ffc. Given the range thresholds (Poor: < 2.0, Moderate: 2.0–4.0, Good: 4.0–10.0), an ffc of 4.000 indicates the transition between moderate and good flow.

Final Answer:

The calculated wall friction angle is 22.782 degrees, and the flow function coefficient is 4.000. Based on these results, the material is classified as having Good flowability.

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