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• Muhd Fahmi

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CHAPTER 5 HOPPER DESIGN 5.1 Bulk solid handling • Measuring the flow properties of bulk solids and how to use this information for the design of storage vessel. • Definitions: Bin: Any upright container for storing bulk solids. Silo: A tall bin, where H > 1.5D Bunker: A shallow bin, where H < 1.5D Hopper: A converging sloping wall section attached to the bottom of a silo. 5.2 Solid flow pattern • As solid flow from a bin, the boundaries between flowing and non-flowing regions define the flow pattern. • Three types of common pattern: i) Funnel flow / core flow ii) Mass flow iii) Expanded flow 5.2.1Flow obstruction • Interruption of solid flow in a bin can be caused by 2 types of obstructions: i) An arch • Sometimes called as a bridge • Formed across a flow channel

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ii)

Bin opening / rathole • Formed when the flow channel empties, leaving the surrounding stagnant material in place.

• Important in understanding the forces acting on the wall of the bin and to the material. 5.3 Types of flow pattern 5.3.1Funnel flow / core flow: • Occurs in bin with flat bottom or hopper having slopes too shallow or too rough to allow solid to slide along the wall during the flow. • Funnel flow through an entire bin - Rathole, formed when stagnant materials gains sufficient strength to remain in place as flow channel empties (refer figure 10.1, page 266) • Material near to the bin wall becomes stagnant. • First in, last out or do not come out at all. • Rathole / pipe could form. • In severe cases, the material can form a bridge or arch over the discharged opening. • The flow channel may not well defined o particle segregation might occur. o material surrounding the channel may be unstable o this will cause stop and start flowing, pulsating or “jelly” flow. o could lead to the damage of material structure. 68

• As bin emptied; solid continually slough off the top surface into the channel. • Storage bin having a funnel flow pattern is most common in industry. o the design do not consider the stagnant materials. o thus, resulting in less discharged capacity. • Funnel flow is usually the least costly design. • It has several disadvantages when handling certain materials: i)

Flow rate from the discharged opening can be erratic: • Arches tends to form and break. • Flow channel becomes unstable. • Upset volumetric feeder installed at the silo discharged • Powder density at discharged vary widely due to varying stresses in flow.

ii)

Fine powders : • Flush/aerated uncontrollably • Sudden collapse of rathole/arch

iii)

Caking/degrading of solid: • Left under consolidating stresses in the stagnant areas.

iv)

A stable rathole/ pipe formed • Stagnant material gain sufficient strength to remain stagnant.

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v)

Level indicator • Would not give correct signal on materials level. • Submerged in stagnant area

•

Despite all of the above, funnel flow is still adequate for (advantage) : i) Non-caking or non-degradable ii) Discharge opening adequately sized to prevent bridging/ratholing.

•

However, many mechanical devices could be used to promote flow.

5.3.2

Mass flow:

• Occurs in bin having steep and smooth hoppers. • Material discharges are fully active. • Flow channel coincides with the bin and hopper walls i.e all materials is in motion and sliding against the wall of bin and hopper. Advantages: i)

Erratic flow, channeling and flooding of powders are avoided.

ii)

Stagnant regions in the silo are eliminated.

iii)

First in, first out flow occurs. Resulting in minimizing caking, degrading and segregation during process.

iv)

Little particle segregation or eliminated 70

v)

Uniform flow at the hopper outlet • flow is easily controlled • pressures are well predictable.

Disadvantages: i)

Friction between moving solids and the silo. • resulting in erosion of the wall • could give rise to contamination of the solids by the material of the hopper wall. • Serious erosion of the wall material.

ii)

For conical hoppers, the slope angle required to ensure mass flow depends on the powder-powder friction and the powder-wall friction.

• There is no such thing as mass flow hopper - a hopper that gives mass flow with one powder may give core flow with another. Mass Flow

Figure 10.2 pg 266

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Funnel / core flow

Figure 10.3 pg 267

Expanded flow: • Term used to describe flow in a vessel that combines a core flow converging hopper with a mass flow attached below it. • The mass flow hopper section ensures a uniform, controlled flow from the outlet. Its upper diameter is sized such that no stable pipe can form in the core flow hopper portion above it. • Expanded flow is used where a uniform discharged in desired, but where space or cost restrictions rule out a fully mass flow bin. • This arrangement can be used to modify existing funnel flow bins to correct flow problems. 72

• Multiple mass flow hoppers are sometimes mounted under a large core flow silo. 5.4 The Design Philosophy

Powder

Arch of powder with sufficient strength to prevent flow

Blockage or obstruction to flow = arching.

Figure 10.5 page 268

• From above diagram:
powders develop strength under the action of compacting stresses 73




the greater the compacting stress, the greater the strength developed

(Free-flowing solids such as coarse sand will never develop compacting stress) 5.5 Flow- no flow criterion Flow to occur: • strength developed by the solids under the action of consolidating pressure to support obstruction to flow is less than gravity flow of the solids. An arch occurs: • when the strength developed by the solid greater than the stresses acting within the surface of the arch. The hopper flow factor (ff) • the ff relates the stress developed in a particulate solid within the compacting stress acting in a particular hopper.

ff =

σ C compacting ⋅ stress ⋅ in ⋅ the ⋅ hopper = σ D stress ⋅ developed ⋅ in ⋅ the ⋅ powder

• high value of ff means low flowability. • High σC means greater compaction. • Low σD means more chance of an arch forming. The hopper depends on: • The nature of the solid • The nature of the wall material • The slope of the hopper wall 74

5.5.1

Unconfined yield stress, σy

• σy = yield stress of the powder in the exposed surface of the arch. • For flow to occur: - stresses developed in the powder forming the arch are greater than the unconfined yield stress of the powder in the arch, flow will occur. - For flow, σD> σy or 5.5.2

σC ff

>σy

Powder flow function

• the unconfined yield stress, σY of the solid varies with compacting stress, σC ;i.e:

σ y = fn (σ C ) • the relationship is determined experimentally • the relationship is called powder flow function 5.5.3

Critical condition for flow

 The limiting condition for flow:

σc ff

=σy

 to reveal conditions under which the flow will occur.

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σD/ σy (b)

Powder flow function

(c)

σc

(a) – powder has a yield stress greater than σc/ff ⇒ no flow occurs. (b) – if actual stress developed < σcrit : ⇒ no flow. If actual stress developed > σcrit : ⇒ flow occurs. (c) – the powder has a yield stress less flow occurs.

than σc/ff ⇒

5.6 Critical outlet dimension  For a given hopper geometry, the stress developed in the arch is related to the size of the hopper outlet, B, and the bulk solid, ρs, of the material.  Minimum outlet dimension, B

B=

H (θ )σ crit ρb g

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where

H(θ) – factor determined by the slope of the wall. g – acceleration due to gravity.

 For conical hoppers:

H (θ ) = 2.0 +

θ 60

Summary  In designing to ensure mass flow form a conical hopper, all the things below are required: • relationship between σc and σy. • the variation of ff by knowing: 1. the nature of powder- from knowing the effective angle of the internal friction,δ. 2. The nature of the hopper wall- by knowing the angle of wall friction, Φw. 3. The slope of the hopper wall – by knowing θ, the semi-included angle of the conical section, i.e. the angle between the sloping hopper wall and the vertical.  Knowing δ, Φw and θ, the hopper flow factor, ff can be fixed.  ff ( geometry, material of construction of the wall).  knowing ff and the powder flow function (σy vs. σc), then the critical stress in the arch can be determined.  Thus; the minimum size of the outlet found corresponds to this stress. 77

5.7 Shear cell test  The Jenike shear cell test allows powder to be compacted to any degree and sheared under controlled load conditions. At the same time, shear and stress can be measured.  Powders change bulk solid under shear. Under the action of shear: - a loosely packed powder would contract (↓ρB) - a very tightly packed powder would expand (↑ρB) - a critically packed powder would not change in volume.

Figure 10.8 page 272

 5 or 6 samples of powder are prepared.  Normal force and shear force are recorded.  The pair of values (normal and shear force) are plotted to give a yield locus. 78

 The end point of yield locus corresponds to critical condition. Where initiative of flow is not accompanied by a change in the bulk density.

Figure 10.9 pg 273

5.8 Mohr`s circle  Represents the possible combination of normal and shear stresses acting on any plane in a powder (or a body) under stress.  The entire process is repeated 2 or 3 times with samples prepared with different ρB.  In this way, a family of yield loci is generated.  These yield loci characterize the flow properties of the unaerated powder.

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Figure 10.10 pg 273

Figure 10.12 pg 275

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 Each point on a yield locus represents that point on a particular Mohr`s circle for which failure or yield of powders occurs.  A yield locus is tangent to all the Mohr`s circle representing stress system when the powder fail to flow.  a and b represents stress system under which the powder would fail.  c - stresses are insufficient to cause flow.  d – not relevant since the system cannot support stress combinations above the yield locus.  a and b – interest us in analyzing the flow. 5.9 Determination of σy and σc

Figure 10.13 pg 275

 Circle A- represents condition of the free surface of the arch. - at the free surface zero shear and zero normal stress. 81

- circle A must pass the origin. - gives the value of unconfined yield stress, σy  Circle B- the Mohr`s circle is tangent to the yield locus at its critical condition of failure.  Major principle stress = compacting stress, σc 5.10

Determination of δ from Shear Cell Test.

δ- effective angle of internal friction of the solid. - tangent of the ratio of shear stress to normal stress. YL – yield locus  For a free-flowing solid, there is only one yield locus and coincides with the effective yield locus.  The relationship between normal stress and shear stress is known as friction.

Figure 10.14 pg 276

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5.11

Kinematic angle of friction between powder and wall, Φw

 Also known as the angle of wall friction.  Gives relationship between the normal stress acting between powder and wall and the shear stress under conditions.  Wall yield locus is determined by shearing powder against a sample of the wall material under various normal load.

Figure 10.15 pg 277

 Kinematic angle of wall friction is the gradient of the wall yield locus:

tan Φ w =

shear .stress.at.the.wall normal .stress.at.the.wall

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5.12 Determination of Hopper Flow Factor, ff. Determination of the hopper flow factor, ff Eg: δ= 30o Φw= 19o From the graph; θ = 30.5 (X) o allow 3 margin for safety Thus; the semi-included angle of conical hopper = 27.5o (Y)  Thus; the hopper flow factor, ff = 1.8    

Figure 10.16 pg 277

Figure 10.17 pg 278

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5.13 Summary of design procedure  Shear cell test on powder give a family yield loci.  Mohr`s circle stress analysis gives pairs of values of σy and σc and the value of the effective of internal friction, δ.  Pairs of values of σc and σy give the powder flow function.  Shear cell tests on the powder and the material of the hopper wall give the kinematic angle of wall friction, Φw.  Φw and δ are used to obtain hopper flow factor, ff and semi- included angle of conical hopper wall slope,θ.  Powder flow function and hopper flow factor are combined to give the stress corresponding to the critical flow- no flow condition, σcrit.  σcrit , H(θ) and bulk density, ρB are used to calculate the minimum diameter of the conical hopper outlet, B.

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