Reference ID: MET-1B4F | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
The angle of repose is the steepest angle (measured from horizontal) at which a granular material can be piled without slumping. In Process Engineering it is a key indicator of:
Flowability of powders and bulk solids
Design slopes for hoppers, chutes and stockpiles
Segregation and stability risks during storage and transport
Typical applications include silo discharge design, conveyor transfer chutes, pharmaceutical tablet press feed frames, and mine tailings dam geometry.
Methodology & Formulas
Form a conical heap of the material under gravity; measure its base radius \(R\) and vertical height \(H\).
The tangent of the angle of repose \(\alpha\) is the ratio of height to radius:
\[
\tan\alpha = \frac{H}{R}
\]
Solve for \(\alpha\) in radians:
\[
\alpha_{\text{rad}} = \arctan\left(\frac{H}{R}\right)
\]
Convert to degrees when required:
\[
\alpha_{\text{deg}} = \alpha_{\text{rad}}\times\frac{180}{\pi}
\]
Flow Regime
Angle of Repose Range
Typical Handling Strategy
Free-flowing
25–30°
Standard chutes and belt conveyors
Cohesive
30–38°
Steep hopper walls, vibration assistance
Very cohesive / fibrous
>38°
Mechanical agitation, mass-flow hoppers
The angle of repose is the steepest angle (measured from horizontal) at which a bulk solid will remain stable without sliding. It matters because it predicts:
Maximum safe stockpile heights
Chute and hopper angles needed to prevent arching and rat-holing
Conveyor discharge trajectories
Feeder loads and downstream equipment sizing
A low angle (<30°) signals free-flowing material; a high angle (>45°) warns of cohesive, difficult-to-handle solids.
Use the fixed funnel method for repeatable design numbers:
Place a closed funnel 10 cm above a level plate
Quickly open the valve and let the material form a conical pile
Measure the angle between the plate and the pile slope with a protractor or laser scan
Run triplicates and average; discard any test where the pile collapses
This mimics the static conditions in bins and stockpiles. Rotating drum or tilting-table tests are useful for dynamic situations but give lower values.
Moisture increases cohesion and raises the angle roughly linearly until surface moisture appears. A practical rule:
Below critical moisture Mc (typically 3–6 % for most ores), add 1.2° per 1 % moisture
Above Mc, the angle can jump 5–8° within 1 % moisture change
Always run lab tests at the highest expected moisture for hopper and chute design
If plant moisture varies seasonally, design for the worst-case angle plus a 3° safety factor.
Switch when:
Angle of repose exceeds 40°—flow is likely to be erratic
You need to calculate minimum hopper angles and opening sizes to guarantee mass flow
You are scaling beyond 50 t h⁻¹ or using tall, slender silos (>5 m height/diameter)
Shear cell testing gives effective angle of internal friction and wall friction, letting you use Jenike or Eurocode methods for reliable, obstruction-free discharge.
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Worked Example – Estimating the Angle of Repose for a New Catalyst Powder
A process engineer at a petrochemical plant needs to characterise the flow behaviour of a fresh alumina-based catalyst before it is loaded into the regenerator hopper. A small heap is gently formed on a flat bench and the resulting cone is measured to obtain the angle of repose.
Knowns
Cone height, H = 4.2 cm
Cone base radius, R = 7.0 cm
Step-by-step calculation
Compute the tangent of the angle of repose: \[ \tan\alpha = \frac{H}{R} = \frac{4.2}{7.0} = 0.600 \]
Convert the tangent to the angle in radians: \[ \alpha_{\text{rad}} = \arctan(0.600) = 0.540 \text{ rad} \]
Convert radians to degrees for reporting: \[ \alpha_{\text{deg}} = 0.540 \times 57.296 = 30.964^\circ \]
Final Answer
Angle of repose = 30.96°
"Un projet n'est jamais trop grand s'il est bien conçu."— André Citroën
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