Arching in silos: critical outlet diameter
An arch of bulk solid is a stable, self-supporting vault above the discharge opening. It forms when the unconfined compressive strength of the consolidated bulk solid is greater than the stress acting in the arch – discharge then comes to a standstill.
From the measured flow function we calculate the critical outlet diameter above which no stable arch can hold.
How an arch of bulk solid forms
Arching occurs mainly with fine-grained, cohesive bulk solids. Adhesive forces between the particles give the consolidated bulk solid an unconfined compressive strength. As the hopper width decreases, the stress in the bulk solid becomes smaller – below the critical outlet diameter it is no longer sufficient to make an arch collapse.
Contributing factors
- Discharge opening too small
- High cohesion and unconfined compressive strength of the bulk solid, low flowability ffc
- Flat hopper inclination or high wall friction or adhesion on the wall material
- Higher moisture content
- Time consolidation during storage at rest
Consequences in operation
- Discharge stops or is greatly reduced
- Irregular, intermittent material flow
- Dynamic loads on hopper and feeder when the arch collapses
- Air entrainment as material slides down, uncontrolled flooding of fine bulk solids
Calculating the critical outlet diameter
Flow function and stress in the arch
The flow function describes the unconfined compressive strength σc of the bulk solid as a function of the consolidation stress σ1. The stress acting in the arch above the outlet is linked to σ1 through the flow factor ff: σ1′ = σ1 / ff. The flow factor depends on the hopper inclination, the effective angle of internal friction φe and the wall friction angle φw.
The critical unconfined compressive strength σc,crit lies at the intersection of both curves. If the stress in the arch is greater, no stable arch can hold.
From the critical point to the outlet
The critical outlet diameter follows from σc,crit, the bulk density ρb in the outlet region and the hopper shape:
bmin = H(Θ) · σc,crit / (ρb · g)
H(Θ) accounts for the hopper shape (conical or wedge-shaped) and the hopper inclination Θ. The calculation tool evaluates several recognised methods side by side – after Jenike (standard and critical point method) and after Arnold, McLean, Roberts & Enstad – each for your actual hopper geometry.
Measurements for assessing arching
The flow function and the related parameters are measured at the stresses that occur in the outlet region of your silo.
| Measurement | Parameters | Use |
|---|---|---|
| Yield locus | flow function σc(σ1), effective angle of internal friction φe, flowability ffc | critical point σc,crit, flow factor ff |
| Compression density | density function ρb(σ) | bulk density in the outlet region |
| Wall friction | wall friction angle φw on the hopper material | flow factor ff, mass flow |
| Time consolidation | time flow function σc,t(σ1) (optional) | critical outlet after storage at rest |
Measures against arching
Which measure works is shown by comparing the existing and the critical outlet – geometry and wall material can be varied in the calculation tool.
- GeometryDischarge opening larger than the critical outlet diameter; steeper hopper inclination or a different hopper shape.
- Wall materialWall material or coating with lower wall friction.
- Bulk solidAdjusting particle size, moisture or flow aids – the effect can be proven by comparative measurements.
- DischargeA feeder that activates the whole outlet cross-section; discharge aids such as vibration or air blasts.
- OperationLimit the storage time at rest or activate discharge regularly if the bulk solid tends to time consolidation.
Frequently asked questions about arching
- Which measurements does the calculation of the critical outlet diameter need?
- Yield loci at a minimum of three consolidation stresses for the flow function σc(σ1) and the effective angle of internal friction φe, compression density for the bulk density in the outlet region, and wall friction on the hopper material. For longer storage times, time consolidation is added.
- Does the storage time change the critical outlet diameter?
- Yes. Through time consolidation the unconfined compressive strength of the bulk solid increases at rest. The time flow function gives a larger critical outlet diameter than for discharge directly after filling.
- Does mass flow prevent arching?
- No. A stable arch can also form above an outlet that is too small in mass flow. Mass flow and arching are two separate criteria; both are calculated from the same measured parameters.
- Does the critical value apply equally to circular and slot outlets?
- No. Conical hoppers with a circular outlet give a critical diameter, wedge-shaped hoppers with a slot outlet a critical slot width. The hopper shape enters the calculation through the function H(Θ).
Have the critical outlet diameter determined
Describe your bulk solid and your hopper – we put together the right measurements and prepare a non-binding quote for you.
- Flow function at the stresses in your silo
- Critical outlet diameter by several methods
- With time consolidation for longer storage times on request