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Observed Phenomenons in the Throughput Behavior

The findings described on the page Throughput Characteristic show that there is a complex relationship between the infuencing variables. In the following it is tried to discover those relationships by phenomenological description. For this purpose it is reasonable to look at the following border cases:

  • Circumferential speed against null: Acceleration effects and streaming effects disappear.
  • Material bed: The particles are significant smaller than the gap width, cohesionless material can flow through the gap with idle rollers, see drawings material bed.
  • Direct contact: The particles are much bigger than the gap width and are clamped in the entrance cross section of the compression zone, see drawings material bed.

Industrial mills are usually operated in a transition range between direct contact and material bed situation.

Material bed, smooth rollers

D/xmax = 200, s = s/D = 0,015, a0 = 0,14 = 8°

Material bed, moulded rollers

D/xmax = 200, s = s/D = 0,025, a0 = 0,12 = 7°

Comminution
grinding results for different materials

Exper. Setup
lab-scale mill, techn. data mill, intrumentation, feed materials, testing program, roller surface data, test execution

Throughput
literature, decoupling of throughput and comminution, energy absorption, characteristic of different rollers, phenomenons, throughput results, elasticity, friction angle, specific throughput

Pressure Distribution
pressure diagrams

Press Tests
tests with a molding press

Modelling
mathematic model of the process

Signs and Symbols
nomenclature

Direct contact, smooth rollers

D/xmax = 33, s = s/D = 0,015, a0 = 0,21 = 12°

Direct contact, moulded rollers

D/xmax = 33, s = s/D = 0,035, a0 = 0,16 = 9°

The throughput is determined by the material flow through the entrance cross section of the compression zone whose position is characterized by the compression angle a0. Using the distance between the rollers h and the solid volume portion d0 at a0 the following relationships can be deducted:

h = s+D (1-cos a0) » s+ (a02 /2) D           (4)

Mt = d0rhLv0

n = d0 (v0 /u) [s + (a02 /2)]                          (5)

The convex bending of the throughput characteristic and - following out of this - the decrease of the specific throughput n with increasing speed can be evoked by a decrease of d0, (v0 /u), s , and a0. These four influencing parameters are discussed below.

The relative gap width s can be expressed as a function of d0, a0 and the standardized compaction qm:

qm = (dS - d0)/(1 - d0)                                (6)

s = [d0 (1 - cos a0)] / [qm (1 - d0)]             (7)

According to this equation s is a parameter which is depending on d0 and a0 and which can be adjusted to any value within a certain range by variation of the grinding force.

On the page Decoupling it has been discussed and proved that throughput and comminution are almost decoupled. Therefore the relative gap width can be adjusted principally independent of the throughput and must not be considered in the following.

There is nothing known about the slip at the beginning of the compression zone. When the Coulomb friction law is valid the slip should be independent of the speed. This means that the quotient (v0 /u) remains constant. For the following considerations this will be assumed.

The compression angle a0 is determined on the one hand by the inner and outer friction of the material and on the other hand by the material inflow from the acceleration zone. The latter is probably dependent on the circumferential speed and will decrease as the speed increases. The maximum value for the compression angle is achieved for the border case u ® 0 and shall be named static compression angle a0,st .

Equation 5 implicates that the solid volume portion d0 has to be seen as an average value over the whole entrance cross section. For the border cases “material bed” and “direct contact” different values will result. In the material bed situation the forces are transmitted by particle contacts; therefore d0 must be in a range equivalent to the bulk density of poured and shaked material; for different speeds it can only vary within these borders. For direct contact d0 is smaller and its maximum value is given by the densest package of a single particle layer. When the material flow decreases with higher speed than d0 must also decrease.

The static compression angle a0,st .for the direct contact between smooth rollers can be deducted by geometric considerations. For single particles it is approximately:

cos a0 = (D + s) / (D + x) tan a0 < m

In a filled feeding funnel also two particles can be clamped between the rollers. Video records show this effect.The statistically occurring clamping can lead to a bigger value of the over time and roller length measured average of the compression angle. To consider this the upper equation is modified as follows:

cos a0 = (D + s) / (D + xx)x >1, tan a0 < m                  (8)

For the material bed situation no explicit equation for the calculation of the static compression angle exists. Nevertheless it can be approximately estimated with the Johanson theory [8] which uses the inner and outer friction. Therefore the influence of roller speed, material and roller profile on the inner and outer friction has to be discussed.

With validity of the Coulomb friction law the inner friction as well as the outer friction is independent of the speed. This shall be assumed for the following.

According earlier [2] and own measurements the inner and outer friction of quartz and limestone are almost the same. The observed influence of the material cannot be explained with it.

With formation of a coating layer on the roller the inner friction which is much higher than the outer friction has to be used to characterize the friction between roller and material. The coating layer formation is dependent on the hardness of the material. Limestone forms a coating on the rollers however quartz does not. This effect explains the bigger throughput for limestone. The influence of the material is then principally a consequence of the different formation of such coating layers.

The moulding of the roller surface causes a better entrainment of the outer particle layers by form closure. For the material entrainmanet in the material bed situation it means that again the inner friction and not the outer friction is decisive.When moulded rollers are used the throughput increase will be bigger for hard and brittle materials that don’t form coating layers compared to soft materials.For direct contact the form closure facilitates an easier clamping of particles.

For a general approach the introduction of the throughput number n0 is reasonable. The throughput number results from the first border case u ® 0 assuming a non-slip entrainment with (v0 /u) = 1. In equation 5 then a0 has to be replaced by a0,st . Considering equations 6 and 7 it follows:

n0 = (a0,st2 / 2) d0 [1 + d0 / qm (1 - d0)]           (9)

With a solid volume portion of 0,5 equation 9 can be simplified to:

n0 = (a0,st2 / 4) [1 + (1 / qm)]

The deducted relationship between n0 and qm shows that throughput and comminution are still coupled to a certain extent. For the usual range of the standardized compression of qm = 0,6 to 0,8 (@ dS = 0,8 to 0,9) the result for a static compression angle of a0,st = 0,18 are n0 values of 2,16 to 1,82 % what is equal to a 16 % decrease.

The observed decrease out of the measured Mt(Fsp)-functions was 10 % in maximum which is less than the deducted percentage of 16 %. Therefore the simplified decoupled treatment of throughput and comminution is still acceptable and reasonable.

At higher speeds the material introduction is determined more and more by dynamic processes in the acceleration zone. That means the theoretically highest throughput can only be achieved if always the maximum grindable amount of material is transported into the compression zone. The continuity equation postulates the acceleration of the material in the acceleration zone.

This happens by normal force and by friction between the material and the roller surface.The latter results from the multiplication of friction coefficient and normal force which is again determined by the vertical and horizontal pressure of the material column and by the centrifugal force on the particles as a result of the particle movement along the roller surface. When fine particles below approximately 0,100 mm are fed to the mill the back-streaming of air pressed out of the compression zone has to be considered which also interferes with the material flow.

The centrifugal force grows proportional with the square of the circumferential speed. As long as only this force is decisive for the reduction of the normal force and back-streaming of air can be neglected it should be possible to present the throughput as follows:

Mt = C1u - C2u2   with C1, C2 = constant          (10)

For the specific throughput  this means:

n = n0 - (1/k) z  with n0, k = constant              (11)

with z = u / uc and uc = (gD/2)0,5

The parameter k shall be named throughput elasticity. The linear euqation formed with it offers the possibility for an easy approximation of the throughput characteristic.

That the centrifugal force actually reduces the normal force and that particles even leave the roller surface is made clear by the sketches of video pictures shown below. The dotted particles lift off the roller surface, the reason can only be the centrifugal force.

Centrifugal effect by particle acceleration, particle peels away from the roller surface, limestone 6,0/8,0 mm, upper picture: smooth rollers, u = 2,0 m/s; lower picture: moulded rollers with profile B (1-8-3/4), u = 1,0 m/s

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