Reference ID: MET-C588 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
The Flow Function (symbol \( ff_c \)) is a dimensionless index used to characterize the flowability of bulk solids. It is defined as the ratio of the major consolidation stress (\( \sigma_1 \)) to the unconfined yield strength (\( \sigma_c \)).
In accordance with ASTM D6128 (the Jenike classification), \( ff_c \) is the critical parameter for determining the minimum outlet dimensions required to prevent flow obstructions such as arching (bridging) or ratholing. Misclassifying a powder—specifically labeling cohesive materials as easy-flowing—presents a significant engineering risk, often resulting in undersized hopper outlets and total system failure.
Methodology & Formulas
Major Consolidation Stress (\( \sigma_1 \))
The principal stress acting on the material during steady-state flow or compaction:
\[ \sigma_1 \quad [\text{kPa}] \]
Unconfined Yield Strength (\( \sigma_c \))
The stress required to cause a consolidated bulk solid to fail (break) in an unconfined state:
\[ \sigma_c \quad [\text{kPa}] \]
Flow Function Coefficient (\( ff_c \))
The dimensionless ratio:
\[ ff_c = \frac{\sigma_1}{\sigma_c} \]
Flowability Classification
\( ff_c \) Range
Engineering Implication
Non-flowing / Hardened
\( ff_c < 1 \)
Material gains significant strength; gravity flow unlikely.
Very Cohesive
\( 1 \le ff_c < 2 \)
High risk of arching; requires flow aids or very large outlets.
Cohesive
\( 2 \le ff_c < 4 \)
Standard industrial powders; requires precise outlet calculation.
Easy-flowing
\( 4 \le ff_c < 10 \)
Flows well; standard hopper geometries are typically sufficient.
Free-flowing
\( ff_c \ge 10 \)
Non-cohesive; flow is limited only by orifice effects or air resistance.
Note: Testing must be conducted at the specific temperature and moisture content expected in the process, as these variables significantly alter the yield locus and resulting \( ff_c \).
The Flow Function (\( ff_c \)) classifies powders based on the ratio of consolidation stress to yield strength:
\( ff_c < 1 \): Non-flowing (hardened)
\( 1 \le ff_c < 2 \): Very cohesive
\( 2 \le ff_c < 4 \): Cohesive
\( 4 \le ff_c < 10 \): Easy-flowing
\( ff_c \ge 10 \): Free-flowing
A lower \( ff_c \) indicates a more difficult-to-handle material.
A Shear Cell Test (such as a Jenike Shear Cell, Schulze Ring Shear Tester, or Brookfield PFT) is required. The test measures the shear stress at failure under varying normal loads to generate a Yield Locus. From this locus, the Mohr’s circle analysis determines the unconfined yield strength (\( \sigma_c \)) and the major consolidation stress (\( \sigma_1 \)).
An \( ff_c \) of 3 is Cohesive. Labeling it "Easy Flow" (which requires \( ff_c > 4 \)) leads to arching (where a stable bridge forms over the outlet) or ratholing, causing complete cessation of discharge.
Increased moisture typically increases liquid bridging and van der Waals forces between particles, significantly increasing the unconfined yield strength (\( \sigma_c \)). Since \( \sigma_c \) is in the denominator, a higher moisture content usually results in a lower \( ff_c \), moving the material from a free-flowing regime into a cohesive or non-flowing regime.
Yes. \( ff_c \) is used in conjunction with the Flow Factor (\( ff \)) of the hopper. Mass flow occurs when the Flow Function of the material lies above the Flow Factor of the hopper (\( ff_c > ff \)). This analysis determines the minimum hopper steepness and outlet size required to ensure all material is in motion during discharge.
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Worked Example: Flowability Assessment of Skim Milk Powder
A process engineer is evaluating a skim milk powder transfer line. The powder must pass through a 150 mm diameter drop chute. To determine if the powder will arch, the engineer performs a shear cell test at the maximum expected moisture level.
Test Parameters
Temperature: 25 °C
Moisture content (wet basis): 3.2 %
Major consolidation stress (\( \sigma_1 \)): 6.0 kPa
The calculated \( ff_c \) is 2.86. According to the Jenike classification (ASTM D6128), this material is classified as Cohesive (\( 2 \le ff_c < 4 \)).
Risk Assessment: Because the powder is cohesive, the engineer cannot assume a 150 mm chute is sufficient. A critical arching diameter calculation must be performed. If the critical diameter exceeds 150 mm, the chute will bridge, requiring a larger diameter or mechanical vibration.
"Un projet n'est jamais trop grand s'il est bien conçu."— André Citroën
"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise."— Charles de Gaulle
