How flow function and flow factor influence hopper design
Q: What's the difference between flow function and flow factor when designing a hopper?
People commonly confuse these two similarlooking technical terms. Andrew Jenike (1961) was the first person to use flow function and flow factor in hopper design. Jenike related soil mechanics and bulk solids yield strength to create a design procedure for a massflow bin or hopper. This procedure is still commonly used by many industries around the world to design massflow bins.
Flow function
Flow function (FF) is the relationship between a material's cohesive strength and a consolidation stress applied to the material and provides information about the material's ability to shear on itself at various consolidation pressures. The material will flow if its yield strength limit is overcome by the force acting on the material.
A material's flow function depends on the material's properties and can be determined experimentally using a shear tester. The tester applies several consolidating stresses to a material sample and determines the material's unconfined yield strength at each consolidating stress using the principle of Mohr's stress circle, as shown in Figure 1. A Mohr's circle is a graphical representation of the relationship between a normal stress and a shear stress acting on a bulk solid. The flow function value and its relation to flowability are shown in Table 1. A higher flowfunction value indicates easy material flow, which means material has low strength. A lower flowfunction value indicates a more difficult material flow, which means material has high strength. Time consolidation can also influence a material's flowfunction, so storage time must also be considered when testing a material's flow properties.
Table 1
The flow function and its relation to flowability
Flow Function Range 
Flowability 
FF > 10 
Free flowing 
4 > FF > 10 
Easy flowing 
2 > FF > 4 
Cohesive 
1 > FF > 2 
Very cohesive 
FF < 1 
No flowability 
Flow factor
Flow factor (ff) is the ratio of the major consolidation stress to the major principal stress in a stable arch. For a bulk solid in any position in a massflow hopper, this ratio is constant and can be described using the following equation (described in Powders and Bulk Solids: Behavior, Characterization, Storage and Flow by D. Schulze, 2008):
As the equation shows, the flow factor depends on the hopper geometry (wedge or cone), the wall surface characteristics, and the material properties such as angle of internal friction. Jenike used this equation to develop flow factor graphs showing wall friction angle versus hopper angle for each effective angle of internal friction for both wedgeshaped and conical hoppers. The flow factor values relating wall friction angle to hopper angle for a 50degree effective angle of internal friction in a conical hopper is shown in Table 2.
Table 2
Flow factor values for conical hopper and
50degree effective angle of internal friction
Wall Friction Angle 
Hopper Angle
(from vertical) 
Reduced Hopper Angle
(3° deducted as factor of safety) 
Flow Factor 
10° 
38° 
35° 
1.40 
15° 
33° 
30° 
1.38 
20° 
27° 
24° 
1.33 
25° 
20° 
17° 
1.30 
How flow function and flow factor are used in hopper design
The relationship between a material’s flow function and flow factor is shown in Figure 2. In the graphs, the intersection between the flow function and flow factor lines represents the critical stress acting on the material at the hopper outlet. In the graph, if flow factor lies above flow function, the material will always flow because the material's yield strength is less than the stress required to form the arch. Similarly, if flow factor lies below flow function, the material won't flow because the material's strength is greater than the force acting on the material.
In Scenario I, material won't flow at consolidation stresses to the left of the critical stress intersection because the material strength is greater than the force acting on the material. This will cause material arching above the hopper outlet. At consolidation stresses to the right of the critical stress intersection, however, the material won't have enough strength to overcome the force acting on it, so the material will flow.
In Scenario II the opposite is true. Material will flow at consolidation stresses to the left of the critical stress intersection but not to the right. In this scenario, the material gained strength over the force acting on it, blocking flow. Hopper designers use this relationship between flow function and flow factor to ensure that the hopper outlet dimension is large enough to prevent arching.
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