There are many ways to test a bulk solid. Through testing we can poke and prod a given material, generating reams of data by which we might better know the material. One of the first properties of a particulate solid differentiating it from its liquid counterparts is that solids form piles.
Pour a glass of liquid onto the ground and what results is a relatively flat surface (along with a mess), whereas pour out the same glass now full of a powder onto a flat surface and as seen in figure 1, a conical shape will result.
Now that we have formed one pile, we could pour out a different powder in the same way and generate a pile whose cone has a different slope. The angle formed by this heap is referred to as a bulk solid’s angle of repose.
The angle of repose is the slope a free surface of a bulk solid retains, typically when the material is at rest. Standard nomenclature defines angle of repose in degrees from horizontal. For anyone with a flat surface, a bulk solid, and some means of determining the angle that has been formed, they too can measure angle of repose.
Our evaluation here becomes more nuanced as the term serves as a catch-all for a family of shapes retained by the free surface of a bulk solid. The three most common as shown in figure 2, are poured angle of repose, drained angle of repose, and dynamic angle of repose. [1]
The poured angle of repose is the predominant of the three and requires a conical pile be formed in some manner, typically by pouring material through a funnel a fixed distance above a flat surface. Other standards stipulate that the flat service should have a vessel (cup/cylinder) on the flat surface to collect the material [2].
For the latter approach, the base of the pile rests on other material, reducing the role of friction between the material and another surface. The drained angle of repose is the angle formed when a pile of material is emptied (drained) through an orifice in the surface on which material sits.
The dynamic angle of response is the angle formed by product inside a slowly rotating drum. For a given material its angle of repose will vary for the different styles; poured, drained, and dynamic. The focus of this discussion will be on poured angle of repose, as it is the most ubiquitous.
When considering a product’s angle of repose both the type and method must be specified. As well it is necessary to employ a consistent means of measuring what the given slope is with each test. A pile will take on a specific shape such as concave up, concave down, straight, or some combination of the three.
As the angle of repose may vary throughout the pile a sort of overall or average can be employed. This is achieved by forming the pile on an elevated platform (or in a cylinder), thus fixing the outer diameter of the base, and then measuring the height of the heap from base to tip.
From this the angle of repose is calculated as the arctangent of the ratio of the pile height to the radius of the pile base. At any given point of the heap, the angle of repose may deviate from this calculated value.
Alternate styles of angle of repose interact with the pile in some way after its formation to see how the shape may change, for instance by striking the surface on which the pile rests in a specific manner the angle of fall (or slump) is determined [3].
A great challenge in angle of repose is that once a type and method has been prescribed, it is important to have a test that can be carried out for a breadth of materials. If the method passes product through a funnel the inherent problem arises that many materials will not readily pass through the funnel outlet.
This has led to alternate approaches to the procedure ranging from agitation and stirring product within the funnel, passing material through a vibrating screen prior to the funnel, or using a cylinder in place of a funnel. Other method variations focus on the size of the pile base and rate of pile formation. All of these permutations can influence the resulting angle.
Beyond the method, changes in a bulk solid’s attributes can impact the angle of repose. These include particle shape, particle size, particle surface, and moisture content. One study on a sample of coal found that the angle of repose could increase from 40° to 65° with a 5% moisture content increase [4].
At this point the discussion must turn a corner because despite the ease at which we may form and measure angle of repose a more important question should be asked.
Once we know angle of repose, what can we do with it?
The foremost use of angle of repose is to make capacity estimates for bins and stockpiles. Bins are sized based on capacity requirements for a bulk solid. A silos liquid volume indicates the total internal space, but bulk solids are not liquids, so the available storage space depends on the angle of repose.
Capacity loss in the cylinder section because of this upper pile is shown in figure 3. Maximising capacity can often be achieved by reducing the angle of repose, typically by changing how the pile of material is formed in the silo.
The second use of a materials’ angle of repose is to design for the containment of material. Examples of this include applications where belt conveyors are transporting bulk solids. The amount of product that can be moved without spilling material over the sides of a traditional belt will be limited by the angle of repose. The angle a material maintains during belt conveying is referred to as the surcharge angle.
Historically this value has been estimated to be within 10-25° less than the poured angle of repose [5]. A more precise prediction of angle of surcharge involves modelling to simulate the affect from the motion of the belt passing over the idlers. Other types of feeders and conveyors rely on knowing the angle of repose or some variant of it to allow for better control and containment of the bulk solid.
For both items above, it is important to note the state of the material forming the pile. For example, fine powder delivered pneumatically to a silo will form a much shallower pile than if the material was discharged via a bucket elevator.
Certain materials in dynamic states become difficult to control in a feeding device as the angle of repose becomes significantly less than might have been anticipated. To make a meaningful measurement of angle of repose we must first have knowledge of the state of the material and how the pile will be formed.
Many have gone further with angle of repose, employing it to understand flowability. Most notably is the Carr Indices, named after Ralph L Carr. From his work with bulk solids, angle of repose became incorporated into a set of 7 other measured properties to rank the flowability of bulk solids [3].
Although there is a commercially available apparatus to run all 8 of these measurements, many employ a shorthand version of Carr’s work based solely on the poured angle of repose [1].
This metric postulates that the steeper the angle, the worse the flowability becomes. Those materials with an angle of repose less than 30° are deemed “very free flowing.” Meanwhile, those attempting to handle a material whose angle of repose exceeds 55° should be warned that their material is, “very cohesive.” The complete flowability ranking from angle of repose is found in figure 4.
The benefit of this assessment is it offers a quick and cheap means of comparing different materials. For instance, 10 variations on the same powder can be readily formed into piles and the angle of repose of each can be measured. The pile that formed the steepest pile is typically assumed to the be worst flowing.
Unfortunately, there are materials which defy this trend, as can be seen with certain fine powders. A fine and low moisture material may have high levels of cohesive strength in a silo yet can exhibit a very low angle of repose.
The ranking from angle of repose may not align with how the products flow, Titanium Dioxide can be one such offender where angle of repose may not properly identify which variations of the material possess the best and worst flow with regard to hopper discharge [6].
Certain fibrous or irregular shaped particles may not readily form a steep angle in a pile, yet present daunting and unique problems during handling. A particle size limitation also exists for the standard Carr Indices procedure with a cutoff of 2mm particles (no. 10 mesh sieve) at the largest.
As well, the procedure designated that the ranking is only valid for those materials which are free flowing and moderately cohesive powders. Those handling materials with cohesive levels above “moderate” must look elsewhere to rank their material’s flowability.
Is angle of repose worth measuring?
It depends on why we want this value. As a design tool, angle of repose is limited in its use. For estimates of bin capacity and designing material containment for feeding and mechanical conveying equipment the angle of repose can be a helpful if not overly precise tool.
Beyond this we should tread with light footsteps and open eyes. Many powders with a steeper angle of repose may possess worse flowability, but this is not a universal rule. Certain bulk solids do not lend themselves readily to characterisation via angle of repose. Finally, such flow rankings are qualitative not quantitative in nature.
A ranking implies either comparison or reference. Angle of repose by itself for a single material does not allow for the design of a silo or a piece of feeding equipment. The results of this type of flow testing are qualitative with regards to flowability. For a facility or designer who typically deals with fluffy carbon black, “poor flowability” may mean something radically different than another who typically handles grades of dry sand. Ranking without reference as a point of anchor is difficult at best.
There are far more vital properties intrinsically linked to flowability than angle of repose, most notably a bulk solid’s cohesive properties, wall friction properties, compressibility, and permeability [7].
Furthermore, a pass/fail or general assessment of flowability is not of prime importance. Testing for design must go beyond a personality quiz (ex. does powder X have good flowability?) and should rather determine what is required to design a bin and handling equipment to result in the desired performance.
For any given case, good or bad flowability for a bulk solid is better defined as the material’s ability to meet a certain performance criterion with respect to the environment, state of the bulk solid, equipment employed, and its operation.
Testing to identify good flowability at that point becomes far more predictive and actionable. Such testing asks if the material will behave in a specific manner for a specific situation. Only when handling multiple similar materials or variations on the same product, general flowability rankings then might offer a useful purpose.
For those looking for a test of a bulk solid to use as a fundamental design basis for storage vessels, angle of repose is not an appropriate tool. Angle of repose consists of loose and unconfined material forming a pile. Conversely product in a bin is typically not loose but under some pressure level from the weight of other material.
Testing which neglects to study the result of compacting stress; almost universally present within storage vessels, cannot serve as a design basis. As well, product within a silo is confined within a flow channel. Although angle of repose may at times be mildly informative on flow, it is vital to remember that it is typically not the same thing as flowability.
Flowability is not defined by one parameter nor is it a standardised value. To make an engineered decision based on material properties we must understand what we are trying to accomplish and then act on appropriate data [8].
Because bulk solids vary in their properties, behaviour, and flowability it is important to understand that every silo or piece of handling equipment should not be evaluated simply on its own merit but rather on how it engages with a specific material under specific conditions.
Any silo should readily be deemed wrong if used for something other than its intended use and material. In a similar manner, any type of material characterisation, including angle of repose, should be assessed on its helpfulness to a specific end. Measuring a materials’ angle of repose may serve as a useful testing tool, yet those viewing the results must understand the limits of what this simple test tells us.
So where do we go from here then? Not very far and with the utmost caution.
- D. McGlinchey, Characterisation of Bulk Solids. Blackwell Publishing Ltd, 2005.
- ISO International, “Surface active agents – Powders and granules – Measurement of angle of repose,” ISO 4324:1977
- ASTM International, “Standard Test Method for Bulk Solids Characterisation by Carr Indices,” ASTM D6393/D6393M – 21, West Conshohocken, PA (2021)
- R.L. Brown and J.C. Richards, Principles of Powder Mechanics. Pergamon Press Lt., Headington Hill Hall, Oxford, 1970
- H. Colijn, Mechanical Conveyors for Bulk Solids. Elsevier Science Publishers B.V., 1985
- Žídek, Martin & Zádrapa, J & Jezerská, Lucie & Hlosta, Jakub & Necas,, “Comparison TiO2 powders flowability tests,” January 2018
- J. Marinelli and J.W. Carson, “Characterise bulk solids to ensure smooth flow,” Chemical Engineering, April 1994
- E. Maynard, “Ten Steps to an Effective Bin Design. Chemical Engineering Progress,” 109. 25-32. 2013
