Flow Behavior of Granular Materials and Powders

Part II

Asst. Prof. Dr. Muanmai Apintanapong

Mechanical properties : Flow Properties

To compare and optimize powders regarding flowability.

To design powder handling equipment: silos, feeders, flow

promoting devices and other bulk solid handling equipment (so that no

flow problems occur e.g. flow obstructions, segregation, irregular

flow, flooding,..)

Flow properties depend on: Particle size distribution Particle shape Chemical composition of particles Moisture Temperature

It is necessary to determine the flow properties in appropriate testing devices.

Stresses in bulk solids Bulk solid element in

container. Assumptions: infinite

filling height, frictionless internal walls

No shear stresses are ac ting at the lateral walls, s ince the lateral walls wer

e assumed as frictionless.

v = positive normal stress in vertical direction

h = horizontal normal stress (result of vertical stress)

= 0

In bulk solid, h < v

K or = ratio of horizontal stress to vertical stress (h/v)

Typical values of K are between 0.3 and 0.6

I n analogy to solids – in a bulk solid different stresses can be found in di

fferent cutting planes. Stresses in cutting planes other tha

n the vertical and the horizontal ca n be analyzed using a simple equili

brium of forces

Thus only the normal stresses shown are acting on the b ulk solid from outside. Using a simple equilibrium of force

- s at a volume element with triangular cross section cut fr om the bulk solid element shown in

T he normal stress (σα ) and the shear stress (τα) acting

on a plane inclined by an arbitrary angle α , can be calculated.

The pair of values (σα , τα ), which are to b e calculated for all possible angles α , can

be plotted in a σ,τ- diagram (normal stres - s, shear stress diagram).

If one joins all plotted pairs of values, a ci rcle emerges; i.e., all calculated pairs of

values form a circle in the σ,τ-diagram. This circle is called "the Mohr stress circl

e".

Mohr stress circle The Mohr stress circle represents the

stresses in all cutting planes at arbitr ary inclination angles α , i.e., in all pos

sible cutting planes within a bulk soli d element.

Its centre is located at σm = (σv+σh 2)/ and τm = 0.

The radius of the circle is σm = (σ v -σh) 2/ .

Since the centre of the Mohr stress circle is always located on the σ- axis, each Mohr stress circle has t wo points of intersection with the σ- axis.

The normal stresses defined through these points of intersection are called the principal stresses, wh

ereby the larger principal stress – the major princi pal stress – is designated as σ

1 and the smaller pri

ncipal stress – the minor principal stress – is desig nated as σ

2 . If both principal stresses are given, th

e Mohr stress circle is well defined.

Summary A bulk solid can transmit shear stres

ses even if it is at rest. In different cutting planes different s

tresses are acting. Stress conditions can be represente

d with Mohr stress circles. A stress c ircle is defined clearly only if at least two numerical values are given, i.e.,

σ1

and σ2

.

Uniaxial compression test A - hollow cylinder filled with a fine grained bulk

solid (crosssectional area A; internal wall of th e hollow cylinder assumed as frictionless).

The bulk solid is loaded by the stress σ1

– the consolidation stress – in the vertical direction.

The more the volume of the bulk solid specimen is reduced, the more compressible the bulk solid is.

In addition to the increase in bulk density from co nsolidation stress, one will observe also an increa

se in strength of the bulk solid specimen. Hence, t he bulk solid is both consolidated and compressed

through the effect of the consolidation stress. After consolidation, the bulk solid specimen is reli

eved of the consolidation stress, σ1

, and the hollo w cylinder is removed. If subsequently the consoli

dated cylindrical bulk solid specimen is loaded wit h an increasing vertical compressive stress, the s

pecimen will break (fail) at a certain stress. The st ress causing failure is called compressive strength

or unconfined yield strength, σc (another common designation is fc).

I ncipient Flow A t failure the consolidated bulk solid speci

men starts to flow. I ncipient flow is plastic deformation with d

ecrease of bulk density. Since the bulk soli d fails only at a sufficiently large vertical s

tress, which is equal to the compressive st -rength, there must exist a material specifi

c yield limit for the bulk solid. Only when t his yield limit is reached does the bulk soli

d start to flow. Yield limits depend on materials and also

its stress history.

Yield limit and Mohr stress circles

During the increasing ve rtical load in the second

part of the test, the stres s states at different load

steps are represented by stress circles with increa

sing diameter (stress cir cles B1, B2, B3). The less

er principal stress, which is equal to the horizontal

stress, is equal to zero at all stress circles.

At failure of the specimen the Mohr stres s circle B3 represents the stresses in the

bulk solid sample. Since the load corresp onding to this Mohr stress circle causes i

ncipient flow of the specimen, the yield li mit of the bulk solid must have been atta

inedi n one cut t i ng pl ane of t he speci men. Thus, Mohrst r ess ci r cl e B3must r each t he yi el

d l i mi t i n t he σ,τ- diagram. The real cou rse of the yield limit can not be determin

ed with only the uniaxial compression test.

The Mohr stress circles B1 and B2 , whi ch are completely below the yield limit, c

ause only an elastic deformation of the b ulk solid specimen, but nof ai l ur e and/or

fl ow . Stress circles larger than stress cir cle B3, and thus partly above the yield li

mit, arenotpossible: Thespecimenwoul d al r eady be fl owi ng when t he Mohr stress circle reaches the yield limit

(failure), so that no larger load could be exerted on the specimen.

If, during the second part of the experim ent(measurementofcompr essi ve st r engt h), one wer e t o

apply also a constant horizontal stress σh >0onthespeci men (i n addi t i on t o t he ver t

icalst r ess, σv ), one would likewise find str esscirclesthatindicatefailureoft he speci men and r each t he yi el d l

imit (e.g. stress circle C in Figure). Thus t he yield limit is the envelope of all stress

circles that indicate failure of a bulk solidsample.

Jenike shear tester

This tester was the first one design ed for the purposes of powder tech

nology (e.g. measurement at small stresses), and still today shear test

ers are compared to the Jenike She ar Tester.

Jenike shear tester

Shear test procedure (yield locus)

Shear test data analysis

Cohesion 1

Free flow

No flowi

i = angle of internal friction

= C + tan i

envelope

Cohesion 1

Free flow

No flowi

envelope

Effective angle of internal friction

Part II

Flow function (FF)

It is the relationship between the unconfined yield pressure of a solid and the consolidation pressure

c = (1)

FF = d1 /dc

Flow function (FF)

To obtain FF curve of a solid, shear tests are conduct for several consolidating loads and the resulting 1 and c are plotted.

Jenike (1964) classified solids according to their limiting flow function (FF) FF < 2 very cohesive and non-

flowing 4 > FF > 2 cohesive 10 > FF > 4 easy-flowing FF> 10 free flowing

Jenike (1964) showed that for a given hopper there exists a critical line such that as long as the flow function curve below this line, the strength of the solid is insufficient to support an arch c and 1 there would be no obstruction to flow. This critical line was referred to as the “critical

flow factor” or “ff” of the hopper. It is the ratio of critical major consolidating pressure (1) to the critical

unconfined yield pressure (c) determined experimentally.

ff = (1 /c)critical = (F1 /Fc)critical This is the condition at which the bulk material is just on

the point of forming an obstruction to flow. In other words, in a free-flowing solid.

c = f(1)<ccritical = (1/ff) 1

Or 1 /c > ff

Example Tests were conducted using a shear cell

on a granular material. The consolidating load was 8 lb and the cell diameter was 3.75 in. Estimate the following:

Yield locus Angle of internal friction Effective yield locus Effective angle of internal friction Minor and major consolidating stress Unconfined yield stress Cohesion Value of flow function of the material

Normal load (lb)

Shear load (lb)

8.06.04.53.01.5

5.04.43.52.81.7

y = 0.5082x + 1.1424

R2 = 0.9777

0

2

4

6

8

10

12

0 5 10 15 20 25 30

normal load (lb)

shear load (lb)

Consolidating load = 8 lb

Minor consolidating stress

Unconfined yield stress

Major consolidating stress

cohesion Effectiv

e yie

ld lo

cus

Yield locus

FF = 1 /c

Adhesive forces The flowability of a bulk solid depends on the

adhesive forces between individual particles . Different mechanisms create adhesive forces. - fine grained, dry bulk solids : van der Waals inter

actions. moist bulk solids: liquid bridges between the part

icles

(Both types are dependent on the distance betwe en particles and on particle size.)

T ime Consolidation

The effect is occurred when s ome bulk solids continue to gain strengt

h if stored at rest under compressiv e stress for a longer time interval

(from the effects of adhesive forces).

Possible Mechanismsof T ime Consolidation Solids bridges due to solid crystallizing when drying moist

bulk solids, where the moisture is a solution of a solid and a solvent (e.g. sand and salt water).

Solid bridges from the particle material itself, e.g. after so me material at the contact points has been dissolved by m

oisture (e.g. crystal sugars with slight dampness). Bridges due to sintering during storage of the bulk solid at

temperatures not much lower than the melting temperatur e. This can appear e.g. at ambient temperature during the

storage of plastics with low melting points. Plastic deformation at the particle contacts, which leads to

an increase in the adhesive forces through approach of the particles and enlargement of the contact areas.

Chemical processes (chemical reactions at the particle contacts).

Biological processes (e.g. due to fungal growth).

Particle size

A dhesive forces increase with decrea sing in particle size . Thus, as a rule, a bulk solid flows more poorly with decr

easing particle size. - Fi ne gr ai ned bul k sol i ds wi t h mod

erate or poor flow behaviour due to a dhesive forces are called cohesive bu

lk solids.

Flow function (FF)

It is the relationship between the unconfined yield pressure of a solid and the consolidation pressure

c = (1)

FF = d1 /dc

Flow function (FF)

To obtain FF curve of a solid, shear tests are conduct for several consolidating loads and the resulting 1 and c are plotted.

Flow factor (ff) Jenike (1964) showed that for a given hopper there exists

a critical line such that as long as the flow function curve below this line, the strength of the solid is insufficient to support an arch c and 1 there would be no obstruction to flow. This critical line was referred to as the “critical

flow factor” or “ff” of the hopper. It is the ratio of critical major consolidating pressure (1) to the critical

unconfined yield pressure (c) determined experimentally.

ff = (1 /c)critical = (F1 /Fc)critical This is the condition at which the bulk material is just on

the point of forming an obstruction to flow. In other words, in a free-flowing solid.

c = f(1)<ccritical = (1/ff) 1

Or 1 /c > ff

ffc = 3.5

ffc = 1

For flow function : curve A If ff = 1 : there is no obstruction to flow If ff = 3.5 : there is obsrtruction to flow (no flow)

Flow function FF depends on the material only, the flow factor ff depends on both the material and the hopper geometry, surface wall characteristics, etc.

If a solid with certain flow properties, represented by its FF curve, is placed in a hopper with a certain critical flow function ff, represented by a straight on the c versus 1 coordinates, then the critical values of c and 1 are given by the intersection of of the curve and the line. For different size hopper outlets, this point of intersection will determine the minimun size of the outlet required to avoid arching or piping.

Jenike (1964) classified solids according to their limiting flow function (FF) FF < 2 very cohesive and non-

flowing 4 > FF > 2 cohesive 10 > FF > 4 easy-flowing FF> 10 free flowing

Time consolidation (caking)

Some bulk solids increase in strength if t hey ar e st or ed f or a l onger t i me at res

t under a compressive stress (e.g. in a sil o or an intermediate bulk container). Thi s eff ect i s cal l ed time consolidation o

r caking. Time consolidation can be determined wi

t h uniaxial compression test (to simulate long-term storage in a silo).

Flow function and time flow fun ctions for two different storage times t1

and t2

> t1

.

T he unconfined yield strength,σc , increases with increasing

S torage time. This result is tru e for many bulk solids, but not

for all (e.g. dry quartz sand). These differences are due to t

he different physical, chemical , or biological effects that are t

he causes of consolidation ove r time, e.g. chemical processe s, crystallizations between the

particles, enlargement of the contact areas through plastic

deformation, capillary conden sation, or biological processes

such as fungal growth.

Influence of storage time o n flowability