Validating the instrumented ball outputs with simple trajectories

Validating the instrumented ball outputs with simple trajectories
Sudarshan Martins a,*,Wei Li a ,Peter Radziszewski a ,Sylvain Caron b ,Marc Aguanno a ,Michael Bakhos a ,Emma Lee Petch a
a Department of Mechanical Engineering,McGill University,817Sherbrooke Street West,Montreal,Québec,Canada H3A 2K6b
COREM,1180rue de la Minéralogie,Québec,Canada H3A 2K6
a r t i c l e i n f o Article history:
Received 20December 2007Accepted 30May 2008
Available online 23July 2008Keywords:
Process instrumentation Modeling
a b s t r a c t
An instrumented ball capable of measuring a number of physical quantities within a highly dynamic environment,such as a tumbling mill,has been designed,built and tested.To ensure that the instru-mented ball is operating as designed,it is made to follow a number of known trajectories.The physical quantities measured by the instrumented ball are consistent with the expected results along the trajec-tories.An example of the use of a properly functioning instrument is also shown.
Ó2008Elsevier Ltd.All rights reserved.
1.Introduction
Milling is not an efficient process.According to Mishra (2003),only 20%of the supplied energy is directed towards comminution processes.The remaining energy is wasted.This inefficiency has driven the effort to understand the dynamics of a mill;a greater understanding of milling processes could lead to an increased effi-ciency.Measurements are the basis of any improved understand-ing –they underpin new models and theories.A review by Caron and Roy (2004)found only a limited number of instruments having stemmed from all the recent research effort –the development of measurement instruments for a tumbling mill is a challenging problem,particularly if internal physical quantities are to be mea-sured.An instrument dedicated to the measurement of an internal physical quantity usually implies that,while in operation,the instrument must withstand and survive the action of the mill,without affecting the dynamics of the mill.
One possible internal measurement system is the instrumented ball .Once deployed within an operating mill,the instrumented ball is subject to the same environment as the charge (Martins et al.,2006;Rolf,1999;Dunn and Martin,1978).Any measure-ments thus obtained provide some insight into the internal dynamics of a mill.
The objective of the present work is to demonstrate that (i)the instrumented ball is an accurate measurement system and (ii)to demonstrate a possible application of the instrumented ball.
2.Instrumented ball
The development of the McGill University instrumented ball (iBall)was motivated by major advancement in electronics.During the development of the prior instrumented balls,very high density computer memory and MEMs (micro-e lectro m echanical s ystems)sensors were not commercially available.In addition,the current consumer electronics market has driven the production of small,low-power electronic components.These new advances in elec-tronics are leveraged by the McGill University iBall.The iBall,pic-tured in Fig.1,consists of an electronic data acquisition system embedded within a protective shell.
Several subsystems form the electronics:1.Power supply.
2.Microcontroller and clock.
3.Storage (data and instructions).
4.Input/output subsystem (analog and digital).
5.
Measurement sensors.
In the current configuration,the measurement sensors are com-posed of one 3-axis accelerometer,three angular rate sensors and a temperature sensor.The adjustable sample frequency is set at 1kHz.From the measured data,a number of physical quantities,such as the rotational kinetic energy,E rotational KE ,the net applied mo-ment,~M Applied ,the net applied force,~F Applied and the angular momentum,L *
,can be found (Goldstein,1980).
E rotational KE ¼
1x *T I $x *ð1ÞL *
¼I $
~x
ð2Þ
0892-6875/$-see front matter Ó2008Elsevier Ltd.All rights reserved.doi:10.1016/j.mineng.2008.05.016
*Corresponding author.
E-mail address:sudarshan.martins@mcgill.ca (S.Martins).
Minerals Engineering 21(2008)
782–788
Contents lists available at ScienceDirect
Minerals Engineering
j o u r n a l h o m e p a g e :w w w.e l s e vier.c om/loc
ate/mineng
~M Applied ¼_
L *þ~x Â~L
ð3Þ~F Applied ¼m ða *þx *Âðx *Âr *Þþ_x *Âr *Þ
ð4Þ
where ~x is the rotation rate,I $
is the moment of inertia tensor,~r is the accelerometer displacement from the instrument center of
mass,~a is the accelerometer acceleration and m is the instrument mass.The components of the moment of inertia tensor are sums over the mass elements,m a ,that form the iBall (Thornton and Mar-ion,2004):
I ij
X
a
m a d ij
X 3k ¼1
x 2a ;k Àx a ;i x a ;j
!
ð5Þ
where all mass elements m a of the instrument electronics and shell
are located at a position (x a ,1,x a ,2,x a ,3)in the x 0,y 0,z 0(body
fixed
Fig.3.Cubic shells –aluminum (left)and Delrin (right).These shells are used to change the geometry of the instrumented
ball.
Fig.4.Delrin instrumented ball.This shell has the same geometric properties as the aluminum ball.
Table 1
Calculated and measured physical parameters of the aluminum instrumented ball Property
Value
Mass (kg)[calculated] 1.2049±0.0005Mass (kg)[measured]
1.1708±0.0001Moment of inertia tensor (kg m 2)[calculated]0:0013770:0000030:0000010:0000030:001377À0:0000010:000001À0:0000010:001387
2
43
5All values ±0.000084Diameter (cm)
10.2±0.2
Table 2
Calculated and measured physical parameters of the Delrin instrumented box Property
Value
Mass (kg)[calculated]0.7031±0.0005Mass (kg)[measured]
0.6935±0.0001Moment of Inertia Tensor (kg m 2
)[calculated]0:0013910:000001À0:0000010:0000010:0014000:0000010:000001À0:0000010:001457
2
43
5All values ±0.000099Length (cm)
10.2±
0.2
Fig.5.The screen analysis graph –an example of CDF in mineral
processing.
Fig.1.Instrumented ball opened to display the stacked sensor,microcontroller,communication and power
electronics.
Fig.2.Instrumentation coordinate system.A body fixed coordinate system (x 0,y 0,z 0)is used.This can be related to a fixed laboratory frame (x ,y ,z )when possible.
S.Martins et al./Minerals Engineering 21(2008)782–788
783
with the origin at the center of mass)coordinate system (Fig.2).The Kronecker delta,d ij is defined as
d if ¼
0i ¼j
1i ¼j
ð6Þ
All measurements are taken in the x 0,y 0,z 0coordinate system.The sensors are fixed in this coordinate system.2.1.Instrument shells
Several protective shells have been built:an aluminum ball (Fig.1),a Delrin box (Fig.3),an aluminum box (Fig.3)and a Delrin
ball (Fig.4).By changing shell,a number of physical properties of the instrumented ball can be modified.
Since the properties of the electronics and the aluminum ball shell are well known,some physical properties of the alu-minum instrumented ball can be determined (Table 1).Simi-larly,the properties of the Delrin instrumented box can be determined (Table 2).The moment of inertia tensor is a calcu-lated value,obtained from a CAD program.Since the percent difference between the calculated mass,also obtained from a CAD program,and the measured mass is less than 5%,the error in the moment of inertia is assumed to be small.The calculated moment of inertia is used in place of the measured moment of inertia.
By changing the properties of the instrumented ball,such as changing the shell,a change in its behavior is expected (Martins et al.,2007)as it evolves within a specific
charge.
Fig.6.Drawing of the lab mill,including lifter
details.
b mill at rest.The wooden charge,the instrumented ball and the lifters are
visible.
b mill in operation.
784S.Martins et al./Minerals Engineering 21(2008)782–788
2.2.Standard mill measurement procedure
The procedure to collect mill data,using either the instrumental ball or the instrumented box,begins with the instrumented ball (or box)being placed within the mill,with the wood balls.Before the instrument starts to any collect data,the mill is allowed to reach steady state.The instrumented ball collects data from within the mill,which is set at a rotational speed of 30%,60%,70%or 80%of the critical speed.In each case,the ball remains in the mill for at least 5min,allowing the ball to collect over 300,000measurement sets.Once the mill is stopped,the instrument is retrieved and opened.The I/O port is connected to a computer for the purpose of transferring and saving the data.
For the experimental tests,the aluminum ball and the Delrin box are used since they form a set of very dissimilar shells –signif-icant differences in the measurements should be observed.
2.3.Cumulative distribution of the results
Due to the stochastic nature of the mill,a statistical analysis of the results is performed.From the analysis,a number of cumula-tive distribution functions are obtained.The probability that a ran-dom variable v takes on a value less than or equal to x is given by the function CDF(x ),the cumulative distribution function (Freund and Walpole,1987).
CDF ðx Þ¼P ðv 6x Þ
ð7Þ
The cumulative distribution function represents a statistical description of a physical quantity.In mineral processing,one appli-cation of the CDF is the recording of the results of a sieving test (Wills,2006).When plotting the cumulative undersize with respect to particle size,a CDF of the particle size is constructed (Fig.5
).
Fig.9.Rotation rate measurement –instrumented ball measurement vs.direct
measurement.Fig.10.Measured applied net force (N)at 32%critical speed,30%fill in spherical
coordinates.
Fig.11.Average rotational kinetic energy of the instrumented ball as a function of the mill speed (30%fill by volume,90°lifter angle).
S.Martins et al./Minerals Engineering 21(2008)782–788785
3.Experimental platform
The instrumented ball has been used to measure the charge dynamics of the Lab Mill (Martins et al.,2006;Radziszewski et al.,2006).The Lab Mill is a 5ft diameter,1ft-long cam-driven glass-faced mill (Fig.6).
The mill charge is composed of 5cm diameter wooden balls,with a mass of 44g (Fig.7).The selection of a wooden charge,as opposed to a ceramic or steel charge,was motivated by the low cost of wood.The fill level is 30%by volume.The speed of the mill is adjusted prior to its operation and is held constant throughout each test.Between 100and 140rotations of the mill occur over the duration of each test (Fig.8).Over 300,000data sets are col-lected within this period of time.
To ensure that any observations and trends are due to the experimental configuration and not any sensor errors or faults,a calibration exercise is undertaken.
4.Calibration
An improperly functioning sensor may skew some of the results observed.Therefore,the instrumented ball requires a test exercise.The instrumented ball is fixed to the mill shell,and follows a circu-lar trajectory at the set speed.The rotation rate of the mill is then compared to the measurements of the instrumented ball (Fig.9
).
Fig.12.Rotational kinetic energy cumulative distribution of the Delrin box and aluminum ball shells at three different mill speeds (30%fill by volume,90°lifter
angle).
Fig.13.Angular momentum cumulative distribution of the Delrin box and aluminum ball shells at three different mill speeds (30%fill by volume,90°lifter angle).
Table 3
Rotating kinetic energy of the instrumented Al Ball and Delrin box when they rotate at the same rate as the mill Mill speed (rad/s)E rotational KE
Al Ball (J)E rotational KE
Delrin box (J)
2.15(60%critical)0.00320.00322.51(70%critical)0.00430.00442.87(80%critical)
0.0057
0.0058
786S.Martins et al./Minerals Engineering 21(2008)782–788
The line represents the1:1relation.Within error,the rotation rate measured by the instrumented ball is accurate–the sensors prop-erly measure the rotation rate.
5.An example of usage–instrument application to a laboratory tumbling mill
Instrumented ball measurements of the McGill Lab Mill have produced a number of interesting results.For instance,the mea-sured net applied force is plotted in spherical coordinates(h,/ coordinatesfixed to the body as in Fig.2,with the origin at the instrument center of mass)for the direction–with a color bar to show the magnitude(Fig.10).The distribution of the force is uni-form when the forces are small.This is not the case for the larger forces.
Additionally,the rotational kinetic energy and the angular momentum were determined and analysed.The cumulative distri-bution functions of the rotational kinetic energy(Fig.12)and the angular momentum(Fig.13)are presented.The CDF of the rota-tional kinetic energy gives some information of the behavior of the instrument within the mill.80%of the time(a cumulative prob-ability value of0.8),the rotational kinetic energy of the aluminum instrumented ball has a value less than0.068J.In this case,the mill is set at a speed of60%critical.In the case of the instrumented box, at a speed of60%critical,the rotational kinetic energy is less than 0.060J,80%of the time.Moreover,in this configuration,at any gi-ven speed,the aluminum ball has,on average,more rotational ki-netic energy than the Delrin box(Fig.11).This suggests that the instrumented cube has a more difficult time rotating within a pop-ulation of wood spheres than the aluminum sphere.Sometimes, the rotation of a cube will require it to push wooden balls away –this action hinders the rotation of the cube.Since this does not occur with the instrumented ball,then the instrumented cube is expected to have a higher probability of being in lower rotational kinetic energy states than the instrumented sphere(Fig.12).This situation is reflected by the angular momentum distribution (Fig.13).
An increase in the mill speed provides more energy to the charge.Since a fraction of the imparted energy promotes rotation, an increase in rotational kinetic energy should occur as the mill speed is increased,regardless of the shell type.Fig.11shows that as the mill speed is increased,the average of the rotational kinetic energy also increases.This trend breaks down when the mill speed is much greater than the critical speed,since the charge becomes centrifugally locked(Gupta and Yan,2006).
At different mill speeds or when using different shells(the alu-minum ball or the Delrin box),there are noticeable differences in the rotational kinetic energy and the angular momentum cumula-tive distributions(Figs.12and13).Different CDFs have different moments(mean,standard deviation,skewness,etc.).From each test configuration,a set of characterizing numbers are obtained. Two CDFs are similar if their moments are similar.Any differences can be used as discriminants.One such discriminant is thefirst moment,also known as the average or the mean(Fig.11).
If the instrumented ball(or box)were to rotate at the same rate as the mill,its rotational kinetic energy,as defined in Eq.(1),would have the values given in Table3.
The results show that,often,the instrumented ball and box have higher rotational kinetic energy values–they often rotate fas-ter than the mill(Fig.12).
Furthermore,it is assumed that the Lab Mill operates in steady state.Therefore,the time rate of change of the rotational kinetic energy should average to zero.A non-zero value would indicate either the accumulation or the loss of rotational energy over the lifetime of the test,which is not a steady state situation.In Fig.14,the probability density of the time rate of change of the rotational kinetic energy is plotted.
At60%critical,the mean value of the rotational kinetic energy time rate of change is4Â10À4W,with a standard deviation of 1.2W.The probability densities are similar for the other ball and box cases,and at all other speeds.Only the standard deviations dif-fer.In all cases,the mill operates about steady state.
6.Repeatability
The cumulative distributions of the rotational kinetic energy do not change significantly when the experiment is repeated.In Fig.15,the60%critical distributions,obtained six months apart (2006and2007)are not significantly different.A greater difference is observed when these60%critical distributions are compared
to Fig.14.Probability density of the time rate of change of the rotational kinetic energy(Al Ball).
S.Martins et al./Minerals Engineering21(2008)782–788787
the 70%critical distributions.Therefore,the change in the mill speed has a stronger effect on the results than any instrument or configuration error.Similar results are observed for changes in other physical quantities.7.Other measurements
Though tested by the manufacturer,the accelerometer and the temperature sensor will undergo similar laboratory calibration and repeatability tests,validating the accelerometer and temperature measurements.A shaker table test is planned for the accelerome-ter.A thermal bath test is planned for the temperature sensor.8.Conclusion
By following a number of circular trajectories at constant speed,it was found that the instrumented ball was capable of accurately measuring the rotation rate.Therefore,the measurements and the physical quantities derived thereof are valid.
A possible application of the instrumented ball is the measure-ment of a number of physical quantities from within the mill.A number of derived quantities are determined and statistically ana-lyzed.From the analysis,the assumption of steady state operation is confirmed.When two different shells are used,differences in the measurements are observed.For example,the aluminum ball ro-tates faster than a Delrin box.The moments describing the statis-tics are used to quantitatively describe the mill dynamics.They can be used in conjunction with the toe and shoulder position to characterize the global dynamics of the mill.
The repeatability of the instrument shows that the results ob-tained are valid.They are due to the dynamics of the mill and not any systematic sensor or configuration error.Therefore,a DEM simulation –which is constructed to accurately model the
mill –should be able to reproduce the results obtained with the instrumented ball.At present,the Comminution Dynamics Labora-tory is working on comparing DEM simulation and instrumented ball results.Concurrently,work continues in the validation of the other sensors.References
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Fig.15.Repeatability of the results.
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