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Charge motion model for vibration mills with high excitation
Powder Technology 105 Ž1999. 311–320 www.elsevier.comrlocaterpowtec
Charge motion model for vibration mills with high excitation Uwe Bock, Klaus Schonert ¨
)
Institute of Mechanical Engineering and Institute of Mineral Processing, UniÕersity Clausthal, Waether-Nerust-Str. 38678, Clausthal-Zellerfeld, Germany
Abstract The charge motion determines the power draft, the stressing intensity and probability, the mixing kinetic, the internal material transport, and by all that the performance of a vibration mill. Excitations higher than usual Žacceleration number - 10, relative amplitude - 3%. change the motion pattern because the charge impacts against the upper shell part. The power draft does not longer obey the known relations. A simple model is established for describing approximately the charge motion in such a way that the power draft can be predicted. The key point in the model is the consolidation phase of the charge after impacting, which delays the charge take-off. This effect was quantified with experimental observations. The model does not need any additional parameter for adjusting it to the reality. Resume ´ ´ Le mouvement du chargement de boulets determine la puissance absorbee, la ´ ´ l’intensite´ et la probabilite´ de la charge mecanique, ´ cinetique du procede le transport de la matiere ´ ´ ´ de melange, ´ ` a` moudre et ainsi l’efficacite´ d’un broyeur vibrant. Habituellement l’etat ´ de marche des broyeurs vibrants se caracterise par un chiffre d’acceleration inferieur a` 10 et une amplitude relative inferieure a` 3% et le ´ ´´ ´ ´ chargement ne se souleve ` que peu du fond de l’appareil. En cas de stimulation plus elevee ´ ´ le chargement bute contre la partie superieure ´ de l’appareil et la forme du mouvement change. La puissance absorbee etablies. Un simple modele ´ ne se calcule plus selon les theories ´ ´ ` permet de decrire approximativement le mouvement du chargement de boulets et de calculer a` l’avance la puissance absorbee. ´ ´ Le point fort de ce modele de la phase de consolidation du chargement de boulets qui apres ` est la prise en consideration ´ ` chaque impact repousse le moment du rejet. Cet effet est quantifie´ conformement aux resultats experimentaux. Le modele aucun parametrage ´ ´ ´ ` ne necessite ´ ´ supplementaire pour l’adapter aux resultats reels. ´ ´ ´ q 1999 Elsevier Science S.A. All rights reserved. Keywords: Charge motion; Vibration mill; High excitation
1. Introduction The charge motion determines the power draft, the stressing intensity and stressing probability, the mixing kinetic inside the charge, the material transport and by all that the performance of a vibration mill. The motion depends on many parameters, most important are the oscillating mode, the mill speed, the amplitudes, the ball and material filling ratio. The ball size and the feed material has an influence, only if the ratio of the millrball-diameter or the flowability and agglomeration behaviour of the material differs much from usual operation. The first theoretical approach was published by Bachmann w1x in 1940. He considers the charge as a fully plastic body resting on a horizontal plane which oscillates circularly. The charge does not impact other walls than the )
Corresponding author
bottom plane. Although this model was helpful for initiating the theoretical treatment, it does not fit the real situation of vibration mills with a circular cylindrical mill body and usual ball filling ratios of 50 to 80%. High speed movies Že.g., Ref. w2x. have shown that the charge is lifted somewhat and rotates slowly oppositely to the mill rotation. In a centrifugal mill which can be considered as a vibration mill with a large relative amplitude above 10% and high acceleration numbers, the charge is shaped almost like a circular cylinder w3x. A similar observation is reported in Ref. w4x for a vibration mill filled with rods and excited highly with a relative amplitude of 17% and an acceleration number of 16. In recent publications it was shown that the charge motion and the power draft can be modelled with the discrete elements method w5,6x. However, some model parameters have to be adjusted to experimental results of standard operations. A more fundamental approach for
0032-5910r99r$ - see front matter q 1999 Elsevier Science S.A. All rights reserved. PII: S 0 0 3 2 - 5 9 1 0 Ž 9 9 . 0 0 1 5 3 - 9
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U. Bock, K. Schonertr Powder Technology 105 (1999) 311–320 ¨
predicting the power draft is developed in Ref. w7x, which considers the momentum and energy transfer from the shell to the charge and inside the charge. A power density field is derived showing strong gradients and indicating only small regions, which are effective in grinding. This paper considers the situations in a vibration mill running with acceleration numbers Ž A v 2rg . s z up to 60 and relative amplitudes Ž2 ArD . s a up to 13% which exceeds the usual limits of z s 6 to 10 and a s 3%. These operation parameters have been chosen because this mill was developed in a study dealing with a gas–solid reaction in the mill simultaneously to the comminution w8x, which demands to loosen widely the charge and to mix well. However, it was found that the model discussed in this paper can also be used for usual operation conditions.
2. Experimental device The mill body is mounted in a strong frame, which rests on 12 springs, see Fig. 1. Two shafts with eccentric weights drive the frame. The adjustment can be varied to operate with different amplitudes and oscillating modes Že.g., linear, circular or elliptical oscillation.. For taking high speed videos Ž1000 framesrs. an additional mill body was mounted at the outer side. Mill and operation data are as follows. Mill body D = L s 100 mm Ø = 50 mm, ball diameter d s 3 mm, ball filling ratio f B s 50 to 80%, material filling ratio f M s 0 and 100%, number of revolutions n s 1360 to 2500 rpm, amplitude A s 1 to 6.5 mm, acceleration number Ž A v 2rg . s z s 6 to 63, relative amplitude Ž2 Ard . s a s 2 to 13%. The device is equipped with a tachometer for determining the number of revolutions and a triaxial acceleration gauge, which signals were integrated to determine the
velocity and the motion. A special insulated mill body was made to measure calorimetrically the power draft of the mill charge. 3. Outline of the model The model approach is similar to the one used by Bachmann w1x, however, it takes into account the circular cylindrical mill body, different oscillating modes, the impacts of the charge against the upper part of the mill shell, the ball filling ratio, and the consolidation phase of the charge after an impact. The impact energy of the charge can be calculated to derive the power draft. 3.1. Take-off scheme of the charge The characteristic feature of a vibration mill is that the charge is periodically thrown off the shell and moves freely in the mill. The power draft and the comminution depend on the charge free motion. The model, therefore, is based on the calculation of the charge trajectories which needs to know the initial conditions of the free motion. The first step is to formulate a scheme showing at which moment the free motion is beginning. This moment depends on the mill vibration and the preceding free motion. The scheme, therefore, has to relate the take-off with the moment at which the preceding free motion is terminated by the collision of the charge with the shell. The take-off scheme is developed in the following three steps: Ž1. defining the throw conditions, Ž2. formulating the take-off scheme for an incompressible charge, Ž3. modification due to experimental observations. The classical condition for an upward throw w1x states that a charge leaves the lower part of the shell if the vertical component of the mill acceleration is equal to the gravity acceleration. For expressing this condition, a harmonic vertical mill oscillation y Ž t . s A y sin v t s A y sin c Ž t . is considered, the acceleration number z s A y v 2rg introduced and the throw angle denoted with cthU , it follows:
cthU s arcsin Ž 1rz .
Ž 1.
This equation has two solutions within the range 0 F c F 2p . Therefore, an additional condition is needed which is: the mill moves upwards at cthU , given by y Ž cthU . ) 0. If a vibration mill is operated with a high excitation, then the charge can hit also the upper part of the shell and stay in contact with it. In this case, a particular angle exists at which the charge leaves the shell and is thrown downwards. For evaluating that angle the same consideration can be applied. Therefore, the general throw conditions are given as:
cthU s arcsin Ž 1rz . and y Ž cthU . ) 0 throw upwards Fig. 1. Systematical sketch of the test mill; mill body Ž1., frame Ž2., twelve springs Ž3., two driving shafts Ž4., four excentric double weights Ž5..
cthU s arcsin
Ž 1rz . and y Ž
cthU
Ž 2.
. - 0 throw downwards. Ž 3.
U. Bock, K. Schonertr Powder Technology 105 (1999) 311–320 ¨
Fig. 3. Measurements of the throw-off angle over the acceleration number and the theoretical curve after Bachmann w1x. w B r%; `, 50; v, 60; I, 70; B, 80.
Fig. 2. Sketch for explaining the throw-off conditions.
The second condition can easily to evaluated by y s A y v cos v t s A y v cos c Ž t . .
Ž 4.
The vertical component of the contact force FC is defined as the sum of the charge weight and its vertical acceleration if it moves with the shell: FC s y Ž aosc q g . M s Ž A y v 2 sin c Ž t . y g . M
313
Ž 5.
M: charge mass, g: gravity acceleration. FC can be calculated with Eq. Ž5. as a function of c . If a fully plastical collision is assumed as it is done in the model ŽSection 3.2., then by definition the velocities of the charge and the shell are equal at the moment of collision. The modelling has to consider the general situation that the charge collides at time t with either the lower or the upper part of the shell and leaves again the shell at time tX , which defines the inial conditions of the next free motion. Obviously the take-off can happen only within restricted time intervals which are defined firstly by the vertical component of the contact force FC between the charge and the shell and secondly by the direction of the shell motion y. ˙ Therefore, FC expresses the contact situation in that particular moment. If FC is directed downwards, FC - 0, then the charge can stay in contact with the lower shell part, and for FC ) 0 a contact with the upper shell is possible. The question whether a charge is thrown upwards or downwards if it collides with the lower or the upper shell is to answer, that in the first case FC and y˙ has to be positive and in the second case negative. Both of these ranges are indicated in Fig. 2 and defined as cthU F c - pr2
and 180 y cthU F c - 3pr2, respectively. However, what happens at a collision with the lower shell in the range pr2 F c - 180 y cthU where FC ) 0 and y˙ - 0 and at a collision with the upper shell in y3pr2 F c - cthU where FC - 0 and y˙ ) 0? In the first case the charge moves further downwards following the shell, however, no contact can be built up. In the second case the charge begins a free fall motion because no contact can be established. From all this considerations, the take-off scheme of an incompressible charge can be established as shown in Table 1. The theoretical throw angle controls the scheme. Therefore, the throw angle was determined from high speed videos of stationary mill operations to prove if the theory predicts properly. The results are shown in Fig. 3. They differ very much from the theory to which cthU decreases strongly with increasing z as it can seen from the curve in this figure. The experimental readings are poor in accuracy, therefore, the results scatter widely, especially for small amplitudes. Relating the c U-values on cthU and plotting this ratio over z gives Fig. 4. The ratios are assembled around a straight line and the scattering in the lower range is suppressed in this kind of diagram because of their low values. For simplification the ratio can be approximated with Ž c U rcthU . s z. These experimental observation demands a modification of the throw-up condition Eq. Ž1., which now is to write as: c U s z arcsin Ž 1rz . and y˙ Ž c U . ) 0 Ž 6. Eq. Ž6. expresses a delay of the throw and two reasons could be thought for this effect: Ž1. The charge is not a rigid body but is loosen during the flight. After falling back upon the shell, the charge has
Table 1 Take-off scheme of an incompressible charge. c : phase angle of colliX sion, c : phase angle of take-off Lower shell 0 F c - cthU cthU F c -908 908 F c - 3608
c X s cthU cX sc c X s 3608q cthU
Žthrow upwards. Žthrow upwards. Žthrow upwards.
Upper shell 0 F c - cthU U cth F c -1808y cthU 1808y cthU F c - 3608
cX sc c X s1808y cthU cX sc
Žfall. Žthrow downwards. Žfall.
Fig. 4. Ratio Ž c U rcthU . over the acceleration number. w B r%; `, 50; v, 60; I, 70; B, 80.
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U. Bock, K. Schonertr Powder Technology 105 (1999) 311–320 ¨
Table 2 Take-off scheme for z G 2 under consideration of the consolidation effect Lower shell 08 908 y D c C 3608 q cthU y D c C
F c - 908 y D c C F c - 3608 q cthU y D c C F c - 3608
c X s c q D cC c X s cthU c X s c q D cC
Žthrow upwards. Žthrow upwards. Žthrow upwards.
Upper shell 08 1808 y cthU y D c C 2708 y D c C 3608 q cthU y D c C
F c - 1808 y cthU y D c C F c - 2708 y D cC F c - 3608 q cthU y D c C F c - 3608
c X s 1808 y cthU c X s c q D cC c X s c q D cC c X s 540 y cthU
Žthrow downwards. Žthrow downwards. Žfall. Žthrow downwards.
to consolidate before it can thrown upwards again. The consolidation angle D c C and the consolidation time D tC are given with: D c C s c U y cthU s Ž z y 1 . arcsin Ž 1rz . D tC s Ž pr1808. Ž 1rv . D c C
Ž 7.
Ž2. The circular cylindrical shape of the mill tube hinders the throw. Eq. Ž6. can be expanded as c U s z Ž1808rp .Ž1rz . q Ž1r6 z 3 . q . . . 4 and converges rapidly against 57.308 f 578. The acceleration number z influences c U only in the lower range; this applies also to the consolidation angle. The consolidation time, however, shortens as the oscillation frequency is raised. The operators of vibration mills know that the charge should ‘breathe’ for a proper operation. This means nothing else than loosening during the free motion and consolidating during the collision with the shell. The videos show the consolidation after each impact. Therefore, the considerations on the take-off have to take into account this effect as given in Table 2. For simplification the scheme is given for z G 2, because then D c C G cthU . 3.2. Assumptions Nine assumptions are introduced in the model. Ž1. Circular cylindrical mill tube. Ž2. Circular cylindrical shape of the ball charge with a cylinder volume equal to the bulk volume. This assumption differs much from the situation at usual operation conditions, however, the videos show that at a higher excitation the charge shape looks almost like a cylinder. This is already known from observations with centrifugal mills w3x and vibration mills with high excitation w4x. Ž3. No relative motion between charge and shell as long as they are in contact to each other. This means that sliding or rolling down are neglected. The high video tapes show, that these effects are at least marginal. Ž4. Harmonical oscillation according to x s A x cos v t and y s A y sin v t; A x s A y s A gives a circular oscillation and A x s 0 a vertical linear oscillation. Ž5. Two-dimensional problem. Ž6. The charge leaves the shell according the take-off scheme of Table 2.
Ž7. The ball charge shape remains constant. This is not the case in reality as the charge loosens during its flight and consolidates after each impact. This effect is taken into account with the consolidation phase as discussed in Section 3.1. Ž8. The charge flight trajectories are calculated according to Newtonian mechanics. Ž9. Fully plastical impact. Assumptions 2, 3 and 7 appear to be rather strange, however, they can be justified in some extent with the observations of the video tapes. The essential point of the model is whether the introduced consolidation phase can bridge the gap between the model and the physical reality. This can only be judged by comparing the results with the experimental measurements. 3.3. Equations Based on the assumptions the equations can be written down straight forwardly. The notations are, given in Fig. 5, D: mill diameter, DC : charge diameter, e: distance between mill axis and charge center as the charge is in contact with the shell, x and y: horizontal and vertical position of the mill axis, j and h : horizontal and vertical position of the charge, f B : ball filling ratio, A x and A y : horizontal and vertical amplitude, v : circular frequency, t: time, z: acceleration number, c : phase angle of the oscillation, cthU and tUth : angle and time referring to the theoretical throw-off condition, c U : modified throw-off condition, D c C : consolidation angle. All parameters referring to charge take-off are denoted with a dash, they are c X , tX , xX , yX , j X , h X .
( /
e s Ž 1r2 . Ž D y DC . s Ž Dr2 . 1 y f B
ž
x s A x cos v t s A x cos c Ž t . y s A y sin v t s A y sin c Ž t .
cthU s arcsin Ž 1rz . tUth s Ž pr1808. Ž cthUrv . c U s z arcsin Ž 1rz . tU s Ž pr1808. Ž c Urv . D c C s Ž z y 1 . arcsin Ž 1rz . D tC s Ž pr1808. Ž D c C rv . . The take-off angle c X is given with the scheme in Table 2. xX s A x cos c X yX s A y sin c X tX s Ž pr1808. Ž c Xrv . x˙X s A x v cos c X
yX s A y v sin c X .
U. Bock, K. Schonertr Powder Technology 105 (1999) 311–320 ¨
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The impact happens as the distance between the mill axis and charge center becomes again equal to e. Therefore, the time tY of impacting and the corresponding coordinates xY , yY follows from
(j
2
Ž tY . q h 2 Ž tY . y (x 2 Ž tY . q y 2 Ž tY . s e.
Ž 8. U
Fig. 5. Sketch for explaining the coordinates.
The take-off coordinates j X and hX of the charge center is to calculate from its coordinates at the preceding collision under consideration of assumption 3. The trajectory follows from:
j Ž t . s j X q x˙X Ž t y tX . 2
h Ž t . s hX q y˙X Ž t y tX . y Ž gr2 . Ž t y tX . .
The model calculation starts at the angle c , then the first flight trajectory is calculated and by that the first collision time tY and by that the coordinates xY , yY , j Y , hY . The take-off scheme gives the start of the second flight, then the procedure is repeated. For a vertical linear oscillation the periodicity of the motion is achieved as the take-off angle of a succeeding flight is equal to c U . This condition can always be satisfied. In the case of an elliptical oscillation Ža circular oscillation is a special mode of the elliptical family. a periodical motion can only be achieved at particular conditions; one of them is an operation with low excitation, in
Fig. 6. Modelled charge motion and video frames, f B s 50%, a s 12.2%, z s 27.
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U. Bock, K. Schonertr Powder Technology 105 (1999) 311–320 ¨
which the charge does not impact the upper part of the shell. Otherwise a periodicity in the modelled charge motion is almost unlikely because it demands, that one of the succeeding flights starts at c kY s Ž k P 3608 q c U . and at the same contact point xU s A x cos c U , yU s A y sin c U . However, it happens always that the starting point at c kX is in the neighbourhood of Ž xU , yU ., then an approximative periodicity appears. In reality a periodicity arises because the charge contacts the shell over a certain range instead of in one point. This effect can also be included in the model with an additional assumption. The above equations can be rearranged in a dimensionless form by relating the coordinates x, y, j , h and the amplitudes A x , A y to the mill radius Ž Dr2. and the time t to the oscillation duration T s 2prv . Then the time tY and the angle c Y referring to the collision depends only on the relative amplitudes a y s 2 A yrD, a x s 2 A xrD, the acceleration number z and the flight start angle c X which
depends also only on z. Therefore, the charge motion is describable in terms of the three dimensionless parameters a y , a x and z and is independent on the mill size.
4. Results 4.1. Modelled charge motion and comparison with obserÕation About 40 experiments were performed for comparing the modelled charge motion of a circular oscillation with high speed videos. The operation parameters have been varied as follows: f B s 50–90%, n s 1370–2500 miny1 , A s 1.1–6.2 mm, a s 2.2–12.4%, z s 2.3–43. Four examples are shown in the following to demonstrate typical appearances of agreement and deviation ŽFigs. 6–9..
Fig. 7. Modelled charge motion and video frames, f B s 70%, a s 5.6%, z s 12.5.
U. Bock, K. Schonertr Powder Technology 105 (1999) 311–320 ¨
Fig. 6 Fig. 7 Fig. 8 Fig. 9
f B s 50% f B s 70% f B s 70% f B s 80%
n s 1980 miny1 n s 1980 miny1 n s 1980 miny1 n s 1980 miny1
A s 6.1 mm A s 2.8 mm A s 6.1 mm A s 6.1 mm
a s 12.2% a s 5.6% a s 12.2% a s 12.2%
The expansion of the charge can be seen clearly by comparing the charge area in the frames and drawings. This effect depends on the filling ratio and is affected only a little by the amplitude and the frequency. In the examples with a s 12.2% and z s 27 the expansions are 46, 19 and 7% at f B s 50, 70, 80%. It is obviously, that the time between the first contact and the succeeding take-off is delayed by the consolidation of the charge. The charge shape is not simple and changes continuously. Already an approximative description of this com-
317
z s 27 z s 12 z s 27 z s 27
plex appearance would be extremely tedious and demand additional assumptions taken from experimental investigations. If one want to neglect this appearance, then it seems reasonable to try a radical simplification as introduced with the assumption 2 and to see what happens. The real charge shape differs from the model drawings. However, it could be proven that in almost all tests for comparing the model with the video the consolidation phases agree well over some periods. The model, therefore, seems to be reasonable, or with other words, the idea with the consoli-
Fig. 8. Modelled charge motion and video frames, f B s 70%, a s 12.2%, z s 27.
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U. Bock, K. Schonertr Powder Technology 105 (1999) 311–320 ¨
Fig. 9. Modelled charge motion and video frames, f B s 80%, a s 12.2%, z s 27.
dation and its quantifying according to Eq. Ž7. are sufficient enough to compensate the simplification. Because the model describes reasonably the reality of the charge motion, it can be used to study the influence of the operation parameters; Figs. 10 and 11 demonstrate this in respect to the filling ratio and the relative amplitude. The free space in the mill becomes smaller with increasing filling, therefore, the trajectories are shorten and the oscillation angle at collision shifts to smaller values, e.g., from 2708 to 1758 for the first collision. The measured power draft of the charge Pexp depends on f B much more strongly than it could be expected with the idea that the power is proportional to the charge mass w9x. This would cause an increase by only a factor of 80r50s 1.6 instead of the measured factor of 299r51s 5.9. The reason could be the higher number of collisions per oscillation period which will be discussed in a succeeding paper.
An amplitude increase raises the take-off velocity and shortens the flight time, therefore the collision angle drops down, here from 2958 down to 2108. The classic approach to the power draft relates P with the square of the amplitude, this gives a factor of 4.7, however, the power ratio is 13.6. Again the higher number of collisions is the reason for that. 4.2. Power draft on the charge The model enables to calculate the dissipated energy Ep,i of each impact of the charge against the shell at time t i . The power draft follows from: P s lim k™`
½
k
Ž 1rt k .
Ý Ep ,i is1
5
Ž 9.
In a special test programm, the dissipated energy in the mill was measured calorimetrically for operation condi-
U. Bock, K. Schonertr Powder Technology 105 (1999) 311–320 ¨
319
Fig. 12. Comparison of the measured Ž Pexp . and the calculated Ž Pmod . power draft of the charge, f B s 70%. Ž`. z s 5.8; Žv . z s15; ŽI. z s 37; ŽB. z s 59; Že. n s1350 miny1 ; Žq. n s1640 miny1 ; Žl. n s1980 miny1 ; Ž=. n s 2460 miny1 .
Fig. 10. Influence of the ball filling ratio f B on the charge motion.
tions as discussed here. The power ranges in the wide span from 1 to 550 W. The comparison shows that 80% of the model calculation with f B s 60, 70 and 80%, and 60% of those with f B s 50% deviate less than "20% with the measurements; Fig. 12 shows the results with f B s 70%. The power issue will be dealt in an additional publication. In this paper, it is only mentioned to support the statement that the here discussed motion model is useful. 5. Conclusions The charge motion in vibration mills with high excitation can be modelled well with the simple idea to calculate the charge flight trajectories. After introducing nine assumptions the equation can be written down straight forwardly. The key point of the model is the consolidation of the charge after each collision, which was observed with high speed videos and quantified experimentally in this way. Obviously the consolidation angle helps to bridge the gap between the simplifying assumptions and the physical reality. The model enables to predict the power consumption and helps to understand the complex situation in such a mill. Acknowledgements This project was financed by the Deutsche Forschungsgemeinschaft ŽDFG.. The authors are grateful for this help. References
Fig. 11. Influence of the relative amplitude a on the charge motion.
w1x D. Bachmann, Bewegungsvorgange in Schwingmuhlen mit trockner ¨ ¨ Mahlkorperfullung, VDI Beiheft Verfahrenstechnik 2 Ž1940. 43–55. ¨ ¨ w2x K.-E. Kurrer, J. Jeng, E. Gock, Analyse von Rohrschwingmuhlen, ¨ VDI, Fortschr.-Ber., Reihe 3, no. 282 Ž1992..
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w3x A.A. Bradley, A.L. Hinde, P.J.D. Lloyd, K. Schymurak, Development in centrifugal milling, Prepr. Eur. Symp. Particle Technol. A Ž1980. 153–170. w4x S. Bernotat, Leistungsaufnahme von Schwingmuhlen mit großen ¨ Schwingkreisdurchmessern, Freiberg. Forschungsh. A 798 Ž1989. 33–39. w5x T. Yokoyama, K. Tamura, H. Usui, G. Jimbo, Simulation of ball behaviour in a vibration mill, in: K.S.E. Forssberg, K. Schonert ¨ ŽEds.., Comminution, Elsevier, Amsterdam, 1994, pp. 413–424. w6x T. Inoue, K. Okaya, Grinding mechanism of centrifugal mills, in:
ŽEds.., Comminution, Elsevier, AmsK.S.E. Forssberg, K. Schonert ¨ terdam, 1994, pp. 425–435. w7 x K.-E. Kurrer, Zur inneren Kinematik und Kinetik von Rohschwingmuhlen, PhD thesis, Techn. Univ. Berlin, 1986. ¨ w8x S. Bade, Einsatz einer Rohrschwingmuhle zur simultanen ¨ Zerkleinerung und chemischen Reaktion von Ferrosilicium mit Chlorwasserstoff, PhD thesis, Techn. Univ. Clausthal, 1995. w9x H.E. Rose, R.M.E. Sullivan, Vibration Mills and Vibration Milling, Constable, London, 1961.
Charge motion model for vibration mills with high excitation Uwe Bock, Klaus Schonert ¨
)
Institute of Mechanical Engineering and Institute of Mineral Processing, UniÕersity Clausthal, Waether-Nerust-Str. 38678, Clausthal-Zellerfeld, Germany
Abstract The charge motion determines the power draft, the stressing intensity and probability, the mixing kinetic, the internal material transport, and by all that the performance of a vibration mill. Excitations higher than usual Žacceleration number - 10, relative amplitude - 3%. change the motion pattern because the charge impacts against the upper shell part. The power draft does not longer obey the known relations. A simple model is established for describing approximately the charge motion in such a way that the power draft can be predicted. The key point in the model is the consolidation phase of the charge after impacting, which delays the charge take-off. This effect was quantified with experimental observations. The model does not need any additional parameter for adjusting it to the reality. Resume ´ ´ Le mouvement du chargement de boulets determine la puissance absorbee, la ´ ´ l’intensite´ et la probabilite´ de la charge mecanique, ´ cinetique du procede le transport de la matiere ´ ´ ´ de melange, ´ ` a` moudre et ainsi l’efficacite´ d’un broyeur vibrant. Habituellement l’etat ´ de marche des broyeurs vibrants se caracterise par un chiffre d’acceleration inferieur a` 10 et une amplitude relative inferieure a` 3% et le ´ ´´ ´ ´ chargement ne se souleve ` que peu du fond de l’appareil. En cas de stimulation plus elevee ´ ´ le chargement bute contre la partie superieure ´ de l’appareil et la forme du mouvement change. La puissance absorbee etablies. Un simple modele ´ ne se calcule plus selon les theories ´ ´ ` permet de decrire approximativement le mouvement du chargement de boulets et de calculer a` l’avance la puissance absorbee. ´ ´ Le point fort de ce modele de la phase de consolidation du chargement de boulets qui apres ` est la prise en consideration ´ ` chaque impact repousse le moment du rejet. Cet effet est quantifie´ conformement aux resultats experimentaux. Le modele aucun parametrage ´ ´ ´ ` ne necessite ´ ´ supplementaire pour l’adapter aux resultats reels. ´ ´ ´ q 1999 Elsevier Science S.A. All rights reserved. Keywords: Charge motion; Vibration mill; High excitation
1. Introduction The charge motion determines the power draft, the stressing intensity and stressing probability, the mixing kinetic inside the charge, the material transport and by all that the performance of a vibration mill. The motion depends on many parameters, most important are the oscillating mode, the mill speed, the amplitudes, the ball and material filling ratio. The ball size and the feed material has an influence, only if the ratio of the millrball-diameter or the flowability and agglomeration behaviour of the material differs much from usual operation. The first theoretical approach was published by Bachmann w1x in 1940. He considers the charge as a fully plastic body resting on a horizontal plane which oscillates circularly. The charge does not impact other walls than the )
Corresponding author
bottom plane. Although this model was helpful for initiating the theoretical treatment, it does not fit the real situation of vibration mills with a circular cylindrical mill body and usual ball filling ratios of 50 to 80%. High speed movies Že.g., Ref. w2x. have shown that the charge is lifted somewhat and rotates slowly oppositely to the mill rotation. In a centrifugal mill which can be considered as a vibration mill with a large relative amplitude above 10% and high acceleration numbers, the charge is shaped almost like a circular cylinder w3x. A similar observation is reported in Ref. w4x for a vibration mill filled with rods and excited highly with a relative amplitude of 17% and an acceleration number of 16. In recent publications it was shown that the charge motion and the power draft can be modelled with the discrete elements method w5,6x. However, some model parameters have to be adjusted to experimental results of standard operations. A more fundamental approach for
0032-5910r99r$ - see front matter q 1999 Elsevier Science S.A. All rights reserved. PII: S 0 0 3 2 - 5 9 1 0 Ž 9 9 . 0 0 1 5 3 - 9
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predicting the power draft is developed in Ref. w7x, which considers the momentum and energy transfer from the shell to the charge and inside the charge. A power density field is derived showing strong gradients and indicating only small regions, which are effective in grinding. This paper considers the situations in a vibration mill running with acceleration numbers Ž A v 2rg . s z up to 60 and relative amplitudes Ž2 ArD . s a up to 13% which exceeds the usual limits of z s 6 to 10 and a s 3%. These operation parameters have been chosen because this mill was developed in a study dealing with a gas–solid reaction in the mill simultaneously to the comminution w8x, which demands to loosen widely the charge and to mix well. However, it was found that the model discussed in this paper can also be used for usual operation conditions.
2. Experimental device The mill body is mounted in a strong frame, which rests on 12 springs, see Fig. 1. Two shafts with eccentric weights drive the frame. The adjustment can be varied to operate with different amplitudes and oscillating modes Že.g., linear, circular or elliptical oscillation.. For taking high speed videos Ž1000 framesrs. an additional mill body was mounted at the outer side. Mill and operation data are as follows. Mill body D = L s 100 mm Ø = 50 mm, ball diameter d s 3 mm, ball filling ratio f B s 50 to 80%, material filling ratio f M s 0 and 100%, number of revolutions n s 1360 to 2500 rpm, amplitude A s 1 to 6.5 mm, acceleration number Ž A v 2rg . s z s 6 to 63, relative amplitude Ž2 Ard . s a s 2 to 13%. The device is equipped with a tachometer for determining the number of revolutions and a triaxial acceleration gauge, which signals were integrated to determine the
velocity and the motion. A special insulated mill body was made to measure calorimetrically the power draft of the mill charge. 3. Outline of the model The model approach is similar to the one used by Bachmann w1x, however, it takes into account the circular cylindrical mill body, different oscillating modes, the impacts of the charge against the upper part of the mill shell, the ball filling ratio, and the consolidation phase of the charge after an impact. The impact energy of the charge can be calculated to derive the power draft. 3.1. Take-off scheme of the charge The characteristic feature of a vibration mill is that the charge is periodically thrown off the shell and moves freely in the mill. The power draft and the comminution depend on the charge free motion. The model, therefore, is based on the calculation of the charge trajectories which needs to know the initial conditions of the free motion. The first step is to formulate a scheme showing at which moment the free motion is beginning. This moment depends on the mill vibration and the preceding free motion. The scheme, therefore, has to relate the take-off with the moment at which the preceding free motion is terminated by the collision of the charge with the shell. The take-off scheme is developed in the following three steps: Ž1. defining the throw conditions, Ž2. formulating the take-off scheme for an incompressible charge, Ž3. modification due to experimental observations. The classical condition for an upward throw w1x states that a charge leaves the lower part of the shell if the vertical component of the mill acceleration is equal to the gravity acceleration. For expressing this condition, a harmonic vertical mill oscillation y Ž t . s A y sin v t s A y sin c Ž t . is considered, the acceleration number z s A y v 2rg introduced and the throw angle denoted with cthU , it follows:
cthU s arcsin Ž 1rz .
Ž 1.
This equation has two solutions within the range 0 F c F 2p . Therefore, an additional condition is needed which is: the mill moves upwards at cthU , given by y Ž cthU . ) 0. If a vibration mill is operated with a high excitation, then the charge can hit also the upper part of the shell and stay in contact with it. In this case, a particular angle exists at which the charge leaves the shell and is thrown downwards. For evaluating that angle the same consideration can be applied. Therefore, the general throw conditions are given as:
cthU s arcsin Ž 1rz . and y Ž cthU . ) 0 throw upwards Fig. 1. Systematical sketch of the test mill; mill body Ž1., frame Ž2., twelve springs Ž3., two driving shafts Ž4., four excentric double weights Ž5..
cthU s arcsin
Ž 1rz . and y Ž
cthU
Ž 2.
. - 0 throw downwards. Ž 3.
U. Bock, K. Schonertr Powder Technology 105 (1999) 311–320 ¨
Fig. 3. Measurements of the throw-off angle over the acceleration number and the theoretical curve after Bachmann w1x. w B r%; `, 50; v, 60; I, 70; B, 80.
Fig. 2. Sketch for explaining the throw-off conditions.
The second condition can easily to evaluated by y s A y v cos v t s A y v cos c Ž t . .
Ž 4.
The vertical component of the contact force FC is defined as the sum of the charge weight and its vertical acceleration if it moves with the shell: FC s y Ž aosc q g . M s Ž A y v 2 sin c Ž t . y g . M
313
Ž 5.
M: charge mass, g: gravity acceleration. FC can be calculated with Eq. Ž5. as a function of c . If a fully plastical collision is assumed as it is done in the model ŽSection 3.2., then by definition the velocities of the charge and the shell are equal at the moment of collision. The modelling has to consider the general situation that the charge collides at time t with either the lower or the upper part of the shell and leaves again the shell at time tX , which defines the inial conditions of the next free motion. Obviously the take-off can happen only within restricted time intervals which are defined firstly by the vertical component of the contact force FC between the charge and the shell and secondly by the direction of the shell motion y. ˙ Therefore, FC expresses the contact situation in that particular moment. If FC is directed downwards, FC - 0, then the charge can stay in contact with the lower shell part, and for FC ) 0 a contact with the upper shell is possible. The question whether a charge is thrown upwards or downwards if it collides with the lower or the upper shell is to answer, that in the first case FC and y˙ has to be positive and in the second case negative. Both of these ranges are indicated in Fig. 2 and defined as cthU F c - pr2
and 180 y cthU F c - 3pr2, respectively. However, what happens at a collision with the lower shell in the range pr2 F c - 180 y cthU where FC ) 0 and y˙ - 0 and at a collision with the upper shell in y3pr2 F c - cthU where FC - 0 and y˙ ) 0? In the first case the charge moves further downwards following the shell, however, no contact can be built up. In the second case the charge begins a free fall motion because no contact can be established. From all this considerations, the take-off scheme of an incompressible charge can be established as shown in Table 1. The theoretical throw angle controls the scheme. Therefore, the throw angle was determined from high speed videos of stationary mill operations to prove if the theory predicts properly. The results are shown in Fig. 3. They differ very much from the theory to which cthU decreases strongly with increasing z as it can seen from the curve in this figure. The experimental readings are poor in accuracy, therefore, the results scatter widely, especially for small amplitudes. Relating the c U-values on cthU and plotting this ratio over z gives Fig. 4. The ratios are assembled around a straight line and the scattering in the lower range is suppressed in this kind of diagram because of their low values. For simplification the ratio can be approximated with Ž c U rcthU . s z. These experimental observation demands a modification of the throw-up condition Eq. Ž1., which now is to write as: c U s z arcsin Ž 1rz . and y˙ Ž c U . ) 0 Ž 6. Eq. Ž6. expresses a delay of the throw and two reasons could be thought for this effect: Ž1. The charge is not a rigid body but is loosen during the flight. After falling back upon the shell, the charge has
Table 1 Take-off scheme of an incompressible charge. c : phase angle of colliX sion, c : phase angle of take-off Lower shell 0 F c - cthU cthU F c -908 908 F c - 3608
c X s cthU cX sc c X s 3608q cthU
Žthrow upwards. Žthrow upwards. Žthrow upwards.
Upper shell 0 F c - cthU U cth F c -1808y cthU 1808y cthU F c - 3608
cX sc c X s1808y cthU cX sc
Žfall. Žthrow downwards. Žfall.
Fig. 4. Ratio Ž c U rcthU . over the acceleration number. w B r%; `, 50; v, 60; I, 70; B, 80.
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Table 2 Take-off scheme for z G 2 under consideration of the consolidation effect Lower shell 08 908 y D c C 3608 q cthU y D c C
F c - 908 y D c C F c - 3608 q cthU y D c C F c - 3608
c X s c q D cC c X s cthU c X s c q D cC
Žthrow upwards. Žthrow upwards. Žthrow upwards.
Upper shell 08 1808 y cthU y D c C 2708 y D c C 3608 q cthU y D c C
F c - 1808 y cthU y D c C F c - 2708 y D cC F c - 3608 q cthU y D c C F c - 3608
c X s 1808 y cthU c X s c q D cC c X s c q D cC c X s 540 y cthU
Žthrow downwards. Žthrow downwards. Žfall. Žthrow downwards.
to consolidate before it can thrown upwards again. The consolidation angle D c C and the consolidation time D tC are given with: D c C s c U y cthU s Ž z y 1 . arcsin Ž 1rz . D tC s Ž pr1808. Ž 1rv . D c C
Ž 7.
Ž2. The circular cylindrical shape of the mill tube hinders the throw. Eq. Ž6. can be expanded as c U s z Ž1808rp .Ž1rz . q Ž1r6 z 3 . q . . . 4 and converges rapidly against 57.308 f 578. The acceleration number z influences c U only in the lower range; this applies also to the consolidation angle. The consolidation time, however, shortens as the oscillation frequency is raised. The operators of vibration mills know that the charge should ‘breathe’ for a proper operation. This means nothing else than loosening during the free motion and consolidating during the collision with the shell. The videos show the consolidation after each impact. Therefore, the considerations on the take-off have to take into account this effect as given in Table 2. For simplification the scheme is given for z G 2, because then D c C G cthU . 3.2. Assumptions Nine assumptions are introduced in the model. Ž1. Circular cylindrical mill tube. Ž2. Circular cylindrical shape of the ball charge with a cylinder volume equal to the bulk volume. This assumption differs much from the situation at usual operation conditions, however, the videos show that at a higher excitation the charge shape looks almost like a cylinder. This is already known from observations with centrifugal mills w3x and vibration mills with high excitation w4x. Ž3. No relative motion between charge and shell as long as they are in contact to each other. This means that sliding or rolling down are neglected. The high video tapes show, that these effects are at least marginal. Ž4. Harmonical oscillation according to x s A x cos v t and y s A y sin v t; A x s A y s A gives a circular oscillation and A x s 0 a vertical linear oscillation. Ž5. Two-dimensional problem. Ž6. The charge leaves the shell according the take-off scheme of Table 2.
Ž7. The ball charge shape remains constant. This is not the case in reality as the charge loosens during its flight and consolidates after each impact. This effect is taken into account with the consolidation phase as discussed in Section 3.1. Ž8. The charge flight trajectories are calculated according to Newtonian mechanics. Ž9. Fully plastical impact. Assumptions 2, 3 and 7 appear to be rather strange, however, they can be justified in some extent with the observations of the video tapes. The essential point of the model is whether the introduced consolidation phase can bridge the gap between the model and the physical reality. This can only be judged by comparing the results with the experimental measurements. 3.3. Equations Based on the assumptions the equations can be written down straight forwardly. The notations are, given in Fig. 5, D: mill diameter, DC : charge diameter, e: distance between mill axis and charge center as the charge is in contact with the shell, x and y: horizontal and vertical position of the mill axis, j and h : horizontal and vertical position of the charge, f B : ball filling ratio, A x and A y : horizontal and vertical amplitude, v : circular frequency, t: time, z: acceleration number, c : phase angle of the oscillation, cthU and tUth : angle and time referring to the theoretical throw-off condition, c U : modified throw-off condition, D c C : consolidation angle. All parameters referring to charge take-off are denoted with a dash, they are c X , tX , xX , yX , j X , h X .
( /
e s Ž 1r2 . Ž D y DC . s Ž Dr2 . 1 y f B
ž
x s A x cos v t s A x cos c Ž t . y s A y sin v t s A y sin c Ž t .
cthU s arcsin Ž 1rz . tUth s Ž pr1808. Ž cthUrv . c U s z arcsin Ž 1rz . tU s Ž pr1808. Ž c Urv . D c C s Ž z y 1 . arcsin Ž 1rz . D tC s Ž pr1808. Ž D c C rv . . The take-off angle c X is given with the scheme in Table 2. xX s A x cos c X yX s A y sin c X tX s Ž pr1808. Ž c Xrv . x˙X s A x v cos c X
yX s A y v sin c X .
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The impact happens as the distance between the mill axis and charge center becomes again equal to e. Therefore, the time tY of impacting and the corresponding coordinates xY , yY follows from
(j
2
Ž tY . q h 2 Ž tY . y (x 2 Ž tY . q y 2 Ž tY . s e.
Ž 8. U
Fig. 5. Sketch for explaining the coordinates.
The take-off coordinates j X and hX of the charge center is to calculate from its coordinates at the preceding collision under consideration of assumption 3. The trajectory follows from:
j Ž t . s j X q x˙X Ž t y tX . 2
h Ž t . s hX q y˙X Ž t y tX . y Ž gr2 . Ž t y tX . .
The model calculation starts at the angle c , then the first flight trajectory is calculated and by that the first collision time tY and by that the coordinates xY , yY , j Y , hY . The take-off scheme gives the start of the second flight, then the procedure is repeated. For a vertical linear oscillation the periodicity of the motion is achieved as the take-off angle of a succeeding flight is equal to c U . This condition can always be satisfied. In the case of an elliptical oscillation Ža circular oscillation is a special mode of the elliptical family. a periodical motion can only be achieved at particular conditions; one of them is an operation with low excitation, in
Fig. 6. Modelled charge motion and video frames, f B s 50%, a s 12.2%, z s 27.
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which the charge does not impact the upper part of the shell. Otherwise a periodicity in the modelled charge motion is almost unlikely because it demands, that one of the succeeding flights starts at c kY s Ž k P 3608 q c U . and at the same contact point xU s A x cos c U , yU s A y sin c U . However, it happens always that the starting point at c kX is in the neighbourhood of Ž xU , yU ., then an approximative periodicity appears. In reality a periodicity arises because the charge contacts the shell over a certain range instead of in one point. This effect can also be included in the model with an additional assumption. The above equations can be rearranged in a dimensionless form by relating the coordinates x, y, j , h and the amplitudes A x , A y to the mill radius Ž Dr2. and the time t to the oscillation duration T s 2prv . Then the time tY and the angle c Y referring to the collision depends only on the relative amplitudes a y s 2 A yrD, a x s 2 A xrD, the acceleration number z and the flight start angle c X which
depends also only on z. Therefore, the charge motion is describable in terms of the three dimensionless parameters a y , a x and z and is independent on the mill size.
4. Results 4.1. Modelled charge motion and comparison with obserÕation About 40 experiments were performed for comparing the modelled charge motion of a circular oscillation with high speed videos. The operation parameters have been varied as follows: f B s 50–90%, n s 1370–2500 miny1 , A s 1.1–6.2 mm, a s 2.2–12.4%, z s 2.3–43. Four examples are shown in the following to demonstrate typical appearances of agreement and deviation ŽFigs. 6–9..
Fig. 7. Modelled charge motion and video frames, f B s 70%, a s 5.6%, z s 12.5.
U. Bock, K. Schonertr Powder Technology 105 (1999) 311–320 ¨
Fig. 6 Fig. 7 Fig. 8 Fig. 9
f B s 50% f B s 70% f B s 70% f B s 80%
n s 1980 miny1 n s 1980 miny1 n s 1980 miny1 n s 1980 miny1
A s 6.1 mm A s 2.8 mm A s 6.1 mm A s 6.1 mm
a s 12.2% a s 5.6% a s 12.2% a s 12.2%
The expansion of the charge can be seen clearly by comparing the charge area in the frames and drawings. This effect depends on the filling ratio and is affected only a little by the amplitude and the frequency. In the examples with a s 12.2% and z s 27 the expansions are 46, 19 and 7% at f B s 50, 70, 80%. It is obviously, that the time between the first contact and the succeeding take-off is delayed by the consolidation of the charge. The charge shape is not simple and changes continuously. Already an approximative description of this com-
317
z s 27 z s 12 z s 27 z s 27
plex appearance would be extremely tedious and demand additional assumptions taken from experimental investigations. If one want to neglect this appearance, then it seems reasonable to try a radical simplification as introduced with the assumption 2 and to see what happens. The real charge shape differs from the model drawings. However, it could be proven that in almost all tests for comparing the model with the video the consolidation phases agree well over some periods. The model, therefore, seems to be reasonable, or with other words, the idea with the consoli-
Fig. 8. Modelled charge motion and video frames, f B s 70%, a s 12.2%, z s 27.
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Fig. 9. Modelled charge motion and video frames, f B s 80%, a s 12.2%, z s 27.
dation and its quantifying according to Eq. Ž7. are sufficient enough to compensate the simplification. Because the model describes reasonably the reality of the charge motion, it can be used to study the influence of the operation parameters; Figs. 10 and 11 demonstrate this in respect to the filling ratio and the relative amplitude. The free space in the mill becomes smaller with increasing filling, therefore, the trajectories are shorten and the oscillation angle at collision shifts to smaller values, e.g., from 2708 to 1758 for the first collision. The measured power draft of the charge Pexp depends on f B much more strongly than it could be expected with the idea that the power is proportional to the charge mass w9x. This would cause an increase by only a factor of 80r50s 1.6 instead of the measured factor of 299r51s 5.9. The reason could be the higher number of collisions per oscillation period which will be discussed in a succeeding paper.
An amplitude increase raises the take-off velocity and shortens the flight time, therefore the collision angle drops down, here from 2958 down to 2108. The classic approach to the power draft relates P with the square of the amplitude, this gives a factor of 4.7, however, the power ratio is 13.6. Again the higher number of collisions is the reason for that. 4.2. Power draft on the charge The model enables to calculate the dissipated energy Ep,i of each impact of the charge against the shell at time t i . The power draft follows from: P s lim k™`
½
k
Ž 1rt k .
Ý Ep ,i is1
5
Ž 9.
In a special test programm, the dissipated energy in the mill was measured calorimetrically for operation condi-
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319
Fig. 12. Comparison of the measured Ž Pexp . and the calculated Ž Pmod . power draft of the charge, f B s 70%. Ž`. z s 5.8; Žv . z s15; ŽI. z s 37; ŽB. z s 59; Že. n s1350 miny1 ; Žq. n s1640 miny1 ; Žl. n s1980 miny1 ; Ž=. n s 2460 miny1 .
Fig. 10. Influence of the ball filling ratio f B on the charge motion.
tions as discussed here. The power ranges in the wide span from 1 to 550 W. The comparison shows that 80% of the model calculation with f B s 60, 70 and 80%, and 60% of those with f B s 50% deviate less than "20% with the measurements; Fig. 12 shows the results with f B s 70%. The power issue will be dealt in an additional publication. In this paper, it is only mentioned to support the statement that the here discussed motion model is useful. 5. Conclusions The charge motion in vibration mills with high excitation can be modelled well with the simple idea to calculate the charge flight trajectories. After introducing nine assumptions the equation can be written down straight forwardly. The key point of the model is the consolidation of the charge after each collision, which was observed with high speed videos and quantified experimentally in this way. Obviously the consolidation angle helps to bridge the gap between the simplifying assumptions and the physical reality. The model enables to predict the power consumption and helps to understand the complex situation in such a mill. Acknowledgements This project was financed by the Deutsche Forschungsgemeinschaft ŽDFG.. The authors are grateful for this help. References
Fig. 11. Influence of the relative amplitude a on the charge motion.
w1x D. Bachmann, Bewegungsvorgange in Schwingmuhlen mit trockner ¨ ¨ Mahlkorperfullung, VDI Beiheft Verfahrenstechnik 2 Ž1940. 43–55. ¨ ¨ w2x K.-E. Kurrer, J. Jeng, E. Gock, Analyse von Rohrschwingmuhlen, ¨ VDI, Fortschr.-Ber., Reihe 3, no. 282 Ž1992..
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w3x A.A. Bradley, A.L. Hinde, P.J.D. Lloyd, K. Schymurak, Development in centrifugal milling, Prepr. Eur. Symp. Particle Technol. A Ž1980. 153–170. w4x S. Bernotat, Leistungsaufnahme von Schwingmuhlen mit großen ¨ Schwingkreisdurchmessern, Freiberg. Forschungsh. A 798 Ž1989. 33–39. w5x T. Yokoyama, K. Tamura, H. Usui, G. Jimbo, Simulation of ball behaviour in a vibration mill, in: K.S.E. Forssberg, K. Schonert ¨ ŽEds.., Comminution, Elsevier, Amsterdam, 1994, pp. 413–424. w6x T. Inoue, K. Okaya, Grinding mechanism of centrifugal mills, in:
ŽEds.., Comminution, Elsevier, AmsK.S.E. Forssberg, K. Schonert ¨ terdam, 1994, pp. 425–435. w7 x K.-E. Kurrer, Zur inneren Kinematik und Kinetik von Rohschwingmuhlen, PhD thesis, Techn. Univ. Berlin, 1986. ¨ w8x S. Bade, Einsatz einer Rohrschwingmuhle zur simultanen ¨ Zerkleinerung und chemischen Reaktion von Ferrosilicium mit Chlorwasserstoff, PhD thesis, Techn. Univ. Clausthal, 1995. w9x H.E. Rose, R.M.E. Sullivan, Vibration Mills and Vibration Milling, Constable, London, 1961.