Feeders are widely used for metering bulk solids and discharging the contents of hoppers and silos. At the smaller scale power requirements may not be economically significant, but for larger units become very important, as an underassessment can have expensive consequences. It can also be difficult to determine what power is necessary, on what to allow a safety factor and gross over-assessments incur excess capital costs and long term inefficiencies. The topic has commanded considerable interest in recent years, as attached references, but the subject is complex and general conclusions and formula have been derived from results that may only be valid in similar circumstances to the experimental conditions.
The first task in an investigation is to define the conditions of interest. In the case of feeder loads the main objective is usually to determine the maximum drive power that has to be provided. The worst case scenario is that of starting with a full bin after a period of time consolidation, taking into account any local vibrations, input loading during filling and any time-related effects on the product condition. The result may indicate values many ties that needed for normal running. However, practical steps can be taken to mitigate initial start and re-start conditions by both design and operating practice. Also, as running periods normally occupies most of the operating time, it may be practical to accept a temporary drive overload on starting, provided this is within industrial tolerance.
The main operating conditions of interest that may apply are:
1. Initial start with full bin following maximum static storage time.
2. Initial start with full bin immediately after filling.
3. Re-start with full bin.
4. Continuous running with full bin. (In practice, running with lower contents tends to take similar power until nearly empty).
The geometry off the bin and feeder must then be considered. These can have different characteristics as below:
1. Mass flow with square or round outlet.
2. Mass flow with slot outlet.(Feeder should have progressive extraction).
3. Funnel flow with square or round outlet.
4. Funnel flow with slot outlet and progressive extraction feeder.
5. Funnel flow with slot outlet and local extraction feeder.
The size of outlet that is required to ensure reliable flow from a hopper depends on whether the flow regime generated during discharge is of mass flow or non-mass flow design, so the mode of flow prevailing will affect the size of opening, the shape of the stressed arch that forms over the outlet when flow has occurred and the degree of support given to this arch.
The nature of the bulk material handled has an important influence on the power required by a feeder, particularly: -
1. Wall friction on the hopper contact surface. (Incipient value).
2. Wall friction on the hopper contact surface. (Sustained value).
3. Bulk density in differing conditions of compaction.
4. Internal shear strength. (Incipient value). In specific state of compaction.
5. Internal shear strength. (Sustained value). In specific state of compaction.
Note that moisture, chemical, thermal and time effects may bear significantly on these values, in which case historical and ambient conditions may be relevant.
It will be seen that combinations of the above factors results in many permutations that makes research on this topic an extensive subject to investigate and involve considerable overlapping to devise a formula to suit each case. The way forward chosen here is to identify individual components that may be assembled to reflect a given total situation.
A correctly designed hopper will have outlet dimensions that exceeds the ‘critical arching size’ for the product in the given bin geometry by a comfortable, but not excessive, margin that is sufficient to satisfy the required discharge rate and accommodate foreseeable adverse operating conditions. A hopper outlet of ‘critical arch’ dimensions would offer no additional load on the feeder from material above the arch, but flow reliability would be very suspect unless the ‘worst’ operating circumstances on which a design allowance was made was very unlikely to occur. Clearly, the amount by which the size of opening exceeds the critical arching size has a great influence on the superimposed loads passing through the outlet.
A calculated ‘safe’ hopper opening dimension for reliable flow itself is likely to incorporate a degree of safety, but an additional factor is also usually applied as a form of insurance and to accommodate and uncertainty. The resulting dimension must relate to the minimum size of opening specified, so a taper slot outlet that is incorporated to provide an incremental extraction by a belt feeder in order to generate live flow along the whole length of a slot outlet, will create a further increase in span that allows additional over-pressure to develop. Also, if the condition to be stored has a variable condition due to source, process, age or time consolidation, then the design must be based on the ‘worst’ flow conditions. When then handling the ‘easiest’ flow conditions that would pass through a smaller opening, an even greater over-pressure will be generated. In practice, therefore, a practical hopper outlet size is likely to be significantly larger than the critical arching span dimension.
A key concept is that material in a hopper with outlet larger than the ‘critical arch span’ will collapse and flow out, as the principle stress in the arch exceeds the failure strength of the bulk material. The magnitude of the principle stress that is in excess of the unconfined failure strength, times the principle stress ratio of the bulk solid, produces a minimum principle stress normal force to the surface of the arch that acts down onto the feeder, whether flow is taking place or not.
To this superimposed load on the feeder must be added the weight of the mass under the stressed arch that is totally unsupported by the hopper walls. i.e. from the material in the space between the stressed arch and the shear plane that is developed when extraction takes place.
An important consideration is that, even if flow has not taken place, there is usually some degree of settlement in the lower region of the contents as the bin fills due to the increasing load on the prior contents. The degree to which settlement takes place will be much less for hard, granular material that settle quickly to a stable structure than it is for powders, which tend to land in a dilated condition and consolidate to settle to a higher density as excess air escapes by percolating through the bed. The compacting load on the contents will be limited by Janssen effects, but any settlement will cause a reaction at the bins walls as microscopic slip takes place and develops stressed arches across the bin walls. Through these arches are not sufficiently robust to prevent flow taking place when the feeder allows extraction, they do offer some support for the material above the stressed arches when flow is not occurring.
The arches themselves follow a network of continuous load path between contacting particles and roughly adopt a catenary shape, small deviations being possible because contact friction between particles allows slight angular deviations of particle-to-particle contact forces from straight lines of action. Surface tension, cohesive, tensile, electrostatic, molecular and other inter-particle forces may allow other particles to stick on the underside of this arch at the outlet and be subjected to this stress relieving action of the unstable arch, but these effects can normally be neglected for practical purposes.
The line of action at the toes of the arch is dictated by the slope of the hopper wall and angle of wall friction in the case of mass flow hoppers, or by the internal angle of friction of the bulk solid for a funnel flow regime. The arch shape generally follows the form of a catenary, but this can be closely approximated by a parabola, allowing simple calculations to establish the height of the arch and the mass of material between the stressed arch and the feeder that is solely borne by the feeder. Added to this downward value acting on the feeder is the force acting through the stressed arch in either the static or discharge condition. Stresses in the static arch can be calculated by the method used to establish the ‘critical arching size’. It follows that spans exceeding this size are not capable of bearing the compressive stress applied and that, to prevent failure, the arch must be supported by a force equivalent to the value by which the principle stress in the arch exceeds the compressive failure stress multiplied by the principle stress ratio of the bulk material.
The combination of the unsupported load and force required to prevent the arch failing act directly on the feeder and the stress required to shear this normal load times the area over which this stress applies is the ‘drag-out force necessary for the feeder to initiate discharge. Once flow has started the material dilates to a weaker bulk state and lower density. Pressures decrease towards the outlet, by an amount depending on the degree of flow restraint offered by the rate at which the feeder extracts material. However, the shear value also decreases due to the extra freedom of movement allowed by the dilatation, so the ‘drag-out’ resistance offered to the feeder also diminishes.
The rub is that the recognised Jenike type method of measuring the initial shear force to ‘drag out’ material from under a hopper outlet is applicable to fine powders, but is not appropriate for coarse, firm granular material, because the confining force does not reflect passive stresses that oppose the expansion of the shear plane in totally confined conditions.
For initial shear to take place the mass must dilate locally so that particles that were overlapping in a close-nested, settled structure can move past each other/ This dilatation is significantly for large particles. If expansion is resisted by confinement, passive stresses can be exceptionally high, in some cases requiring failure of the particles to allow shear to take place, even though the material in a loose condition may be very free-flowing, as with granular sugar or salt. This condition may be avoided operational by extracting a small amount of material at an early stage of initial loading and retaining a small heel of material when re-filling. It also may be avoided by design by providing a local region of voidage into which the shear plane expansion my move, by way of inserts in the hopper outlet or offsetting the outlet to remove the compacting stresses on the shear plane or providing a local void region.
A second practical point is that the development of the shear plane between the gravity flow of the hopper contents and the translationary motion of material on the feeder belt is highly sensitive to the exit geometry from the hopper. All to often the exit profile from the hopper does not align with the stressed arch formed in the hopper, losing the benefit of the expansion in the shear plane.
Detail design matters hugely and is not a task for amateurs. Many project engineers consider detail design is the responsibility of draughtsmen, who pay close attention to engineering aspects but are not specialists in powder technology.
Subject to the proviso that wall friction has been mobilised, the approach below may be used to assess starting and re-starting forces. It can be adopted for extended slot length feeders that do not provide progressive extraction, where the feeder must drag out product from under a static bed, provided flow has taken place or accommodation is make for shear expansion. The force required to extract product through a flowing bed is lower than the force needed to initiate discharge because mass dilated by a flowing condition is easier to shear; the higher the rate of flow, the lower the transverse shear strength. It may be difficult to assess the shear strength of a bulk material in dynamic flow condition because of its sensitivity to flow rate, but any feeder capable of initiating flow can usually sustain discharge continuously.
Unsupported load under parabola that follows the form of y = Kx2
The slope at any point = dy/dx = 2K = Tan(α +ф) when x = w/2
Where α = angle of hopper wall from vertical
Ф = angle of wall friction
W = width of hopper outlet (generally greater than 1.1 x ‘critical span’)
K = Tan(α +ф) /2 when x = W/2 so K = Tan(α +ф) /W
Therefore y = Tan(α +ф)x2 /W
Area, A, under a parabola = 2/3 WH, where H = height of centre of stressed arch
A = Tan(α +ф)x2 /W
So area = 2Tan(α +ф)x2 /3 when x = W/2
So area under stressed arch at hopper outlet = W2.Tan(α +ф) /6
Hence weight of unsupported load under stressed arch = L.ʎ.W2.Tan(α +ф) /6
Where ʎ = bulk density
The overpressure from the stressed arch = Op = (∫p - ∫f). Rp.W.L
Where ∫p = principle stress in arch
∫f = failure stress of the bulk material
Rp = principle stress ratio
L = Length of hopper outlet
Clearance between hopper outlet and shear plane of extraction = d
Load acting from material in clearance space = d.W.L. ʎ
Total load on feeder = (∫p - ∫f). Rp.W.L + L.ʎ.W2.Tan(α +ф) /6 + d.W.L. ʎ
To minimise the feeder load, the width of the exit from the hopper should be based on the critical arching span of the bulk material, with a suitable safety factor and allowance for the increased span incorporated over the hopper outlet length to secure progressive extraction.
Reason why outlets are larger than critical span
- Predictive theories are undoubtedly conservative.
- To provide margin to guarantee flow.
- To guarantee the flow rate required.
- To secure progressive extraction from slot outlets.
- Accommodate possible change or uncertainty of product condition.
- Provide extra storage capacity.
- Save storage headroom.
- Poor design confidence.
The feeder must accommodate both initial start-up and running loads. The forces required to initiate discharge can be many times that to sustain running, but they only occur for a short time, are usually infrequent and, most importantly. It is therefore good practice to take steps to mitigate these. By contrast, running loads continue through the total operating life of the equipment and determine the essential energy consumption. This paper therefore concentrates on the dynamic condition of operation.
The top profile of the exit from the hopper to the belt should follow the parabolic shape of the shear plane from the outlet end of the increasing taper of the transition at the bottom edge of the ‘V’ section of the hopper to the translation of movement along the direction of travel. Good practice would make the sides of the outlet inclined at the angle of repose of the product, and carry this shape back as a decreasing height along the hopper outlet length to ease the flow transition to the conveying section, avoid trapping under the diverging skirt and minimise leakage.
The maximum transfer capacity that can be handled with a flat belt is a triangular cross section with a slope at the angle of repose, θ, of the material, so the width and height of the sides of the exit and the product repose angle determines the belt width required. For a firm, granular product that is likely to be free flowing, the hopper outlet width should be small in relation to the height, to secure the largest slope possible for the shear plane to expand along the hopper length.
The hopper outlet width is likely to be dictated by the span necessary to avoid the formation of a structural arch of large granular materials, whereas a wide span needed for a cohesive product may have lower heights because the expansion along the shear plane is less for fine products and a shallower shear plane may be utilised.
This review highlights the great benefits of generating ‘live flow’ over the total hopper outlet area, paying attention to design detail for the feeder interface, making special design arrangements for initiating the discharge of either hard, granular bulk materials to avoid unacceptable load conditions or cohesive material that would otherwise demand an unacceptably wide span when handling such products. Brute force is a very poor solution. Project engineers tend to consider detail equipment design to be the responsibility of the draughtsman, who pays attention to engineering aspects, but is not a specialist in powder technology. The interface between hoppers and feeders merits close attention as it is a major factor in capital and running costs.
References
1. Roberts, A. University of Newcastle. ‘Design and application of feeders for the
controlled loading of bulk solids onto conveyor belts’.
2 . Brian Moore, Thesis 1998, University of Wollongong
‘Flow properties and design procedures for coal storage bins’
3. Ooi.J.Y. Rotter.J.M. , BCURA Project B54. 2005
‘Arching propensity in coal bunkers with non-symmetric geometries
4. Corin Holmes, et al., 10th ICBMH. Brisbane. 2009
‘Start-up and Running Loads Exerted by Bulk Solid Materials on Extractive Belt Feeders: Experimental findings compared with available models’
5. Leo Allen, CEEP Seminar. 2013. ‘Belt feeder head load investigation’
6. Jie Guo, Thesis 2014, University of Wollongong
‘Investigation of arching behaviour under surcharge pressure in mass flow bins and stress states at hopper/feeder interface
7. Karol Ariza-Zafra, Thesis 2015, The Wolfson Centre. ‘Improving performance of discharge equipment for coals with poor handling characteristics’.
8. Bates.L., Ajax publication. The blockage of a hopper outlet or other flow channel by Lumps’
