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Peripheral Pressure Distribution and Characteristic Angles

The energy absorption is determined by the width and the shape of the pressure diagram, means by the peripheral pressure distribution. The width of the distribution is equal to the distance between compression and relaxation angle. The compression angle is depending on the inner and outer friction of the bulk material, on the relation of maximum feed size to roller diameter and on the roller profile. The relaxation angle depending on the elastic deformation behaviour of the material and on rearrangement processes of particles in the relaxation zone. The shape of the pressure distribution is influenced by the resistance of the material bed against deformation which again is depending on the material itself but also is depending on the particle size distribution, the moisture and the velocity of stress. In the following the influence of different parameters on the shape of the pressure distribution, the location of the pressure maximum and the size of the characteristic angles will be discussed, such as the influence of material, speed, axial position of the measurement sensor.

a) Material and shape of the pressure diagram

The influence of the material on the shape of the pressure diagram shall be discussed based on the following examples (pressure over rotation angle):

Comminution
grinding results for different materials

Throughput
characteristic of different rollers

Pressure Distribution
literature pressure distribution, pressure diagrams, peripheral pressure distribution, axial pressuredistrib., effective pressure and specific grinding force, compression angle and specific throughput

Press Tests
tests with a molding press

Modelling
mathematic model of the process

Signs and Symbols
nomenclature

Quartz 1.0/2.0 mm, limestone 1.2/1.8 mm, u = 0.3 m/s, l = 0

Quartz 4.0/6.0 mm, limestone 4.0/6.0 mm, u = 1.1 m/s, l = 0.19

In the first diagram no significant differences can be seen. This is also valid for the finer fraction and the other circumfential speeds. The coarser fraction creates  a more unregular curve for quartz than for limestone. The reason for this is the different material introduction into the gap. While the limestone is introduced in form of a material bed the quartz is introduced in direct contact: It means the gap width of 2.8 to 3.7 mm is significantly smaller than the quartz particles itself. Tests in a press cylinder have shown that the compression curve will not be influenced by the differences in breakage behaviour of quartz and limestone if a material bed situation is given. In the direct contact situation the breakage of the brittle quartz particles creates a sudden decrease of the pressure as can be seen in the diagrams for the coarser quartz fraction. The averaged pressure diagrams are similarly unregular. Only if many single measurements would be used to calculate an average curve smooth pressure diagrams with usable average values could be expected.

b) Axial location of the measurement layer

In the following diagram the pressure diagrams in the edge zone of the roller (l = 0.38) and in the middle layer (l = 0) are compared, again for the fractions 1.0/2.0 mm (quartz) and 1.2/1.8 mm (limestone) at a specific grinding force of 2.7 respectively 2.9 N/mm2 and u = 0.3 m/s. The pressure maximum in the edge zone is much smaller what means it exists a clearly visible axial pressure profile as well. This will be discussed more detailed in the chapter about the axial pressure distribution. Surprisingly the compression angle and the relaxation angle in the edge zone are only insignificantly smaller than in the middle of the roller. 

Average pressure diagrams at l = 0 and l = 0.38, Quartz 1.0/2.0 mm, limestone 1.2/1.8 mm, u = 0.3 m/s, Fsp = 2.7 N/mm²

c) Location of the pressure maximum

From the metal rolling it is known known that the position of the pressure maximum is above the shaft level. Therefore the same assumption is made quite often for briquetting rollers and roller presses. But all averaged pressure diagrams of the fine fractions at circumferential speeds of u = 0.3 and 1.1 m/s show a pressure maximum between a = 0° and 1°, see for example the diagrams above.

From this the conclusion can be drawn that at these operational conditions which are characterized by material bed conditions and a compression of dS < 0.9 the pressure maximum is located in the shaft layer or slightly above. The pressure diagrams of the coarse fractions are not taken into account for this analysis as they show unsystematical effects due to statistical uncertainties.

From some additional tests with a high speed of 3.2 m/s a tendency for a shifting of the maximum to a = 2° can be seen, see diagram below.

A possible explanation could be as follows: At higher speeds the material introduction into the gap between the rollers and into the compression zone is reduced and locally an inhomogeneous material flow may occur. This leads to local overpressing of material to dS > 0.9. In such areas the material will start to move faster than the rollers as seen during the metal rolling which will shift the pressure maximum to a higher position respectively bigger angle. This effect still needs more deeper investigations.

 

d) Compression angle, force attack angle and relaxation angle

The compression angle a0 and the relaxation angle gamma describe the width of the peripheral pressure distribution. Their value can be determined with an accuracy of +/- 0.5° in the average pressure diagrams - even looking at the fact that there is only a slight pressure increase at the very beginning of the curve. The single values of the fine and the medium fractions for both materials are in a +\- 1° trust interval with a probability of 95 %. The same can be said about the compression angle for the fractions 4.0/6.0 mm, the corresponding relaxation angle shows higher variations in this case.

The force attack angle results from the specific grinding force and specific torque:

b = Tsp / Fsp

The angle b can be determined with a much higher accuracy than a0 and g because it is based on average values of Tsp and Fsp which were calculated from the trends during the whole test duration.

The following diagrams show a0b and g In dependence of the specific grinding force. For the limestone fractions 0.5/0.8, 1.2/1.8, 4.0/6.0 mm and the quartz fraction 1.0/2.0 mm the measured values in all three layers are shown at a speed of u = 0.3 m/ s. For the other quartz fractions and the speed u = 1.1 m/s measurements were only done for the layer l = 0.19. Looking at the diagrams the results can be summarized as follows:

Quartz 1,1 m/s

The particle size has a certain influence which will be discussed later. The first conclusions are:

  •  for limestone fractions and a low speed of 0.3 m/s all three angles are independent of the particle size. At 1.1 m/s lower values for compression and relaxation angles are observed for the two finer fractions. The force attack angle is almost not affected.
  • for quartz the compression and relaxation angles of the coarse fractions are always bigger than those of the fine fraction. The force attack angle is only slightly dependent on the particle size.

The grinding force has no impact on the compression angle but the relaxation angle increases with increasing grinding force while the force attack angle decreases.

 

Compression angle

A comprehensive explanation of the results without contradictions regarding the compression angle has to take into account the two border cases "material bed" and "direct contact". One has also to consider the fact that the dynamic compression angle will be smaller than the static one because the material introduction into the compression zone decreases with increasing circumferential speed (see chapters about material transport).

As limit values for the material bed situation and the direct contact situation (s/xquer) > 2 respectively (s/xquer) < 0.5 can be assumed. Between these limit values a transition zone exists. In the following table these characteristic relations were calculated based on the average of the measured gap width:

Table: Gap width s in mm of different compression tests: Fsp = 1 to 5 N/mm2, MB = Material bed situation, Tr = Transition between material bed and direct contact situation

Materials:                         Quartz

0,5/1,0

1,0/2,0

4,0/6,0

mm

Limestone

0,5/0,8

1,0/2,0

4,0/6,0

mm

Quartz, u = 0,3 m/s

1,6....2,1

2,3...3,3

3,1...4,0

mm

 

MB

Tr

Tr

 

Quartz, u = 1,1 m/s

1,5...2,2

2,1...2,8

2,8...3,7

mm

 

MB

Tr

Tr

 

Limestone, u = 0,3 m/s

3,0...3,8

3,1....3,9

4,1....5,3

mm

 

MB

MB

Tr

 

Limestone, u = 1,1 m/s

2,4....3,2

3,0....3,8

3,7....4,0

mm

 

MB

MB

Tr

 

For the “material bed” situation only the inner and outer friction determine the compression angle. As long as material transport and friction angle are independent of the particle size a0 is not changing. The corresponding test results in the previous chapters show that this is the case for the tested material fractions.

For limestone at a speed of u = 0.3 m/s no influence of the particle size can be seen. This is in line with the finding that all the limestone fractions are introduced as material bed or almost as material bed. A reduction of the material transport can not be expected as this low roller speed, so more or less the static compression angle can be found which is not dependent of the particle size as can be seen in the spreadsheet with the friction values in the previous chapters.

The finest quartz fraction 0.5/1.0 mm is also introduced as a material bed but the compression angle a0 is smaller than for limestone. This can explained with the already discussed effect of the limestone-coated rollers (soft material is creating a coating layer on the rollers). Therefore the introduction of quartz is dependent on the outer friction and not on the inner friction as for limestone which is always bigger than the outer friction, see corresponding spreadsheet.

The significantly bigger compression angle for the coarsest quartz fraction 4.0/6.0 mm is a result of the transition from material bed situation to direct contact. In the direct contact situation the roller speed has an indirect influence on the compression angle only when the gap width is significantly reduced. For the coarse quartz this is not the case, so a0 stays constant.

The decrease of the the compression angle for the fine fractions with increasing speed results from the reduction of the material flow. This effect can be seen more or less significantly for both materials.

 

Relaxation angle

With the relaxation angle g and the relative gap width s the relaxation e can be calculated based on a simple geometric deduction:

 

e = (d-s) / d      d = flake thickness

d = s + D (1 - cos g) ~   s [1 + (g2 D / 2 s)]

e = g2 / (2 s + g2)

 

With the corner values of the e curves from the previous diagrams and the corresponding gap width values from the table above relaxation values of e = 15 to 45 % for quartz and e = 9 to 41 % for limestone are resulting. This big relaxation exceeds by far the elastic relaxation and can only be explained by rearrangement processes in the particle bed [20].

As expected the relaxation increases with increasing grinding force, means with bigger tensions in the loaded material bed.

 

Force attack angle

The force attack angle is almost independent of material and roller speed. Only in transition to direct contact slightly higher b-values can be found.

With increasing grinding force the force attack angle is slightly decreasing because the pressure maximum is increased and with a constant a0 the inclination of the pressure diagram gets steeper towards the narrowest gap.In general the relation of force attack angle and compression angle is approximately 1/4.

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