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Acoustic emissions simulation of tumbling mills using charge dynamics
Minerals Engineering 24 (2011) 1440–1447
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Minerals Engineering journal homepage: www.elsevier.com/locate/mineng
Acoustic emissions simulation of tumbling mills using charge dynamics Poorya Hosseini a,⇑, Sudarshan Martins a, Tristan Martin b, Peter Radziszewski a, Francois-Raymond Boyer b a b
Department of Mechanical Engineering, McGill University, Montreal, Quebec, Canada Département de génie informatique et génie logiciel, École Polytechnique de Montréal, Canada
a r t i c l e
i n f o
Article history: Received 26 January 2011 Accepted 4 July 2011 Available online 4 August 2011 Keywords: Discreet element modelling Simulation Process instrumentation Mineral processing SAG milling
a b s t r a c t Knowledge of the internal variables of a mill is of importance in design and performance optimization of the mill, notwithstanding the difficulty in measuring these variables within the harsh mill environment. To overcome this problem, the research has focused on measuring the internal parameters through noninvasive measurement methods such as the use of the vibration/acoustic signal obtained from the mill. Alternatively, virtual instruments, such as discrete element methods (DEM), are employed. Here, a methodology is developed to simulate on-the-shell acoustic signal emitted from tumbling mills using the information extracted from a DEM simulator. The transfer function which links the forces exerted on the internal surface of the mill and the acoustic signal measured on the outer surface is measured experimentally. Given this transfer function and the force distribution obtained from the DEM simulation, and assuming a linear time-invariant response, the on-the-shell acoustic of a laboratory scale ball mill has been simulated. Comparison of this simulated signal with the signal measured experimentally can be used as a criterion to judge the validity of the DEM simulations, and as a tool for enhancing our understanding of both DEM simulations and the use of acoustics within the context of mineral processing. The results derived from preliminary experiments on a laboratory scale mill shows satisfactory agreement between the actual measurement and the simulated acoustic signal. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Tumbling mills are a class of comminution devices, and are frequently used for the size reduction of the ore in the mineral processing industry. Better understanding of the mode and mechanism of energy utilization, and the dynamics of the charge can potentially lead to significant energy savings. Due to the harsh environment inside the mills, as well as severe charge–charge and charge-liner impacts, the use of on-line sensors presents some practical problems (Martins et al., 2008). An alternative solution is the use of discrete element models (DEM) to simulate internal mill dynamics and the charge motion (Cleary, 2001; Cleary et al., 2003; Mishra, 2003; Mishra and Rajamani, 1992; Powell and Nurick, 1996). Significant advances in computer technology have had a role in the growing interest in using DEM to simulate dynamics of the mill; however, DEM simulations still lack in accuracy. In addition, there are ongoing challenges for DEM simulations of tumbling mills, such as shortcomings in simulating the fine progeny and behaviour of the slurry (Morrison and Cleary, 2008). It is known that ball mills undergo strong mechanical vibrations, caused by the impacts and collisions. As a result, they generate a loud noise. Though noise and vibration may be harmful, from the
⇑ Corresponding author. E-mail address: [email protected] (P. Hosseini). 0892-6875/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.mineng.2011.07.002
viewpoint of a human operator, and are a waste of the energy, they can serve as a useful tool in studying the operation of the mills. The acoustic/vibration signal contains information directly related to the operating state of the mill and the mill charge dynamics. Anecdotal evidence has long suggested that a skilled grinding mill operator can evaluate the operating state of the mill by listening to the generated sound. The measurement of the sound of the mill by means of instrumentation has the benefit of full-time on-line operation, increased precision, while having a greater tolerance to perilous or harsh working environments (Zeng and Forssberg, 1993). Over recent decades, different studies have been conducted on laboratory and industrial scale mills to correlate the acoustic/vibration signal with the operating parameters of the mill such as power draw, feed rate, mill load, pulp density, ore type and particles size distribution (Aldrich and Theron, 2000; Das et al., 2010; Kolacz, 1997; Spencer et al., 1999; Tang et al., 2010; Watson, 1985; Zeng and Forssberg, 1993). Moreover, there are some dynamic values which play an important role in optimizing the mill performance and mill design. The shoulder and toe angles are two such examples. Correlating the acoustic/vibration signal with these dynamic features has been much less studied than the relation between operating parameter and the mill sound (Huang et al., 2009; Martins et al., 2006). Considering that the use of acoustic/vibration signal is a non-invasive, low cost tool of studying comminution machines, there remains room for more studies, specifically for industrial applications.
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Nomenclature p i h ft fn K dn ~ v m f
l ~ n R
acoustic pressure impact force impulse response tangential component of contact force normal component of contact force spring coefficient in contact model relative normal displacement at contact translational velocity at contact mass of particle viscous damping ratio in contact model sliding friction coefficient unit normal vector at the contact point radius of particle
Through the simulation of the charge motion inside tumbling mills, DEM models calculate distribution and magnitude of forces and impacts. These forces and impacts engender the vibrations of the structure, and are the main cause of the sound signal generated by the mill. If the relation between these impacts and the acoustic/ vibration signal emitted from the mill is known, the vibration/ acoustic signal can be simulated using the distribution of impacts extracted from the DEM simulator; such a simulation of acoustic/ vibration signal was held to be unworkable in the past (McElroy et al., 2009). Replacement of surface vibration with DEM modelling allows the implementation of DEMs for soft-sensors design approaches, with the objective of measuring the internal variables of the mill (McElroy et al., 2009). Furthermore, the comparison between the simulated signal and the signal measured experimentally can be used as a criterion for evaluating validity of DEM simulations, and as tool for enhancing our understanding of the dynamics of the mill. If the inverse approach is taken, it may be possible to determine dynamic values currently obtained from other methods, such as impacts inside the mill, using only the acoustic/vibration signal. Implementing such an approach for a similar application – a vibratory ball mill containing a single ball – produced promising result in the prediction of impact force using the vibration signal (Huang et al., 1997). These so-called inverse techniques have been extensively used to predict features of mechanical systems which are difficult or impossible to measure directly. A categorization of these techniques for force-prediction models, various applied examples and the required theoretical background has been presented by Wang (2002). Acoustic signal and vibration signal of
f F d G A r rm h hm
u t
impact on mill shell total force acting on mill shell dirac delta function green’s Function displacement of mill shell position vector position vector of microphone angular position of impact angular position of microphone angular difference between impact position and microphone position time
the mill are highly correlated; however, the acoustic signal is more of interest, since its measurement is more practical and has the potential of being captured through sensors which are not necessarily attached to the structure. In this paper, as mentioned earlier, it is demonstrated that an acoustic signal can be calculated from DEM models. This simulated acoustic signal will be shown to be comparable to the measured acoustic signal. 2. Experimental setup A brief description of the laboratory-scale ball mill used in the experiments, the experimental setup used to capture the acoustic signal and to measure impact forces as well as the methodology implemented to process the primary measurements are presented in this section. 2.1. The laboratory-scale ball mill For the experiment, a laboratory-scale ball mill featuring a cam drive is used, as illustrated in Fig. 1. A large diameter aluminium disc is fixed to a shaft mounted on a bearing. The aluminium disc has two functions. Firstly, the followers for the cam drive are fixed to its face. Secondly, the mill drum (or shell) is bolted to the disc. A transparent Plexiglas face closes the mill at the free end of the drum, allowing for the observation of the charge. The drum consists of a steel cylinder, with a diameter of 1.524 m and a length of 0.3048 m. A set of twelve plates are fixed to the inner surface
Fig. 1. Schematic of the laboratory scale ball mill.
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of the shell. These plates are called lifters. Their role is to further promote the tumbling action of the charge, as induced by the rotation of the mill. The birch balls, which form the charge, have an average diameter of 5.1 cm and an average mass of 43 g. 2.2. Impulse response measurement This section describes the method by which the relationship between a single impact, inside the mill shell, at a specific position and the corresponding acoustic signal captured on the mill shell is established. Impact force and the resulting acoustic signal are measured using an impact hammer with embedded force transducer and a pressure-field microphone, respectively. A multi-channel data acquisition system is used to concurrently amplify and digitize both signals and to stream the resulting signals to a computer. Fig. 2 shows a sample of the impact force signal along with the resulting acoustic pressure signal. Assuming the vibrations of the shell is linear and time invariant, meaning different impact at the same position at different times produce the same response, the shell response to the impact force can be characterized through the system impulse response. Consequently, the relationship between acoustic pressure p(t) and impact force i(t) can be expressed using the convolution integral (Phillips et al., 2008):
pðtÞ ¼ iðtÞ hðtÞ ¼
Z
1
iðsÞhðt sÞ ds
ð2Þ
where P(jx), I(jx) and H(jx) are Fourier Transforms of p(t), i(t) and h(t) respectively. To obtain the impulse response in time domain, the Inverse Fourier Transform, then, is applied to the Eq. (2),
PðjxÞ IðjxÞ
The same microphone and recording system described in Section 2.2 were used to capture the acoustic signal of the mill while rotating with the charge. The microphone is mounted on the mill shell and rotates with the mill. The position of the microphone is tracked through adding a fingerprint to the acoustic signal every time the microphone passes by a certain point. To detect the position of the microphone, a fibber unit proximity sensor has been used. At any time the sensor detects the microphone, a voltage is generated; this voltage becomes amplified and finally activates a buzzer. Since the frequency of the sound generated by this buzzer is unique, it can be distinguished from other sound sources in the signal. The sensor has been used during the whole measurement; however, once the location of the microphone is known at a specific time, its position can be found thereafter using the mill rotation speed. 3. DEM simulation
where h(t) is the system impulse response to the impacts at a specific position. By applying the Fourier transform to both sides of Eq. (1), the time domain convolution integral becomes a multiplication in the frequency domain (Phillips et al., 2008):
hðtÞ ¼ F 1
2.3. Acoustic signal of the mill
ð1Þ
1
PðjxÞ ¼ IðjxÞ HðjxÞ
a constant amplitude in the frequency domain – the denominator of Eq. (3) becomes constant and the impulse response, in this case, is equal to the acoustic pressure divided by a constant (the magnitude of the impact). However, in this research deconvolution of the actual excitation exerted by hammer is used for calculations rather than the ideal impact assumption. Fig. 3 shows a sample of the calculated impulse response in the time domain.
ð3Þ
If the force exerted by impact hammer is assumed to be an ideal impact – a perfect impulse that has an infinitely small duration causing
The DEM simulator used in this work has been originally developed for broader research purposes in comminution, including breakage efficiency, and mill equipment design. DEM, in general, is a numerical iterative method which calculates the dynamics of a discontinuous system of particles (Cundall and Strack, 1979). In the case of tumbling mills, it is the charge of the mill that is represented by a collection of particles of defined properties. As shown in Fig. 4, at each cycle or time step, the calculator initially resolves collisions and calculates the corresponding forces generated by these collisions. The collision forces along with external forces, such as gravity or electrostatic forces are subsequently applied to the appropriate particles. By repeating this calculation cycle, the simulator generates trajectories and forces of particles as a
Fig. 2. Excitation exerted by impact hammer inside the mill shell (top), and response to the excitation captured by a pressure microphone on the mill shell (bottom).
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Fig. 3. A typical impulse response.
Table 1 Parameters of simulation and contact model. Parameter
Value
Drum, d l (cm) Normal viscous damping ratio Tangential viscous damping ratio Normal stiffness (N/m) Tangential stiffness (N/m) Particle diameter, D (mm) Particle density (g cm3) Sliding friction coefficient Filling percentage of the mill (%) Rotation speed of the mill (rpm)
152.4 30.5 0.3 0.4 10,000 20,000 50.6 0.49 0.5 30 24
pffiffiffiffiffiffiffiffiffiffi fn ¼ K n dn~ v ~nÞ~n n 2 K n m fn ð~ R pffiffiffiffiffiffiffiffiffi ft ¼ minflfn ; K t ~ v dt 2 K t m ft ð~ v ~nÞ ~ng
Fig. 4. Calculations steps of the DEM.
function of time. The time step of the simulation, varying from 10 to 100 microseconds, is dynamically adjusted to achieve the best possible compromise between precision and performance. While different contact models may be used to describe the contacts in DEM simulations (Zhu et al., 2007), here the software adopts the linear spring-dashpot contact model (Cundall and Strack, 1979; Martins, 2011; Xiang et al., 2009). The tangential component (ft) and normal component (fn) of interparticle contact force in this model are calculated as below,
ð4Þ
where dn is the relative normal displacement at contact, ~ v is the translational velocity, m is the mass of the particle, f is the viscous damping ratio, l is the sliding friction coefficient, ~ n is the unit normal vector at the contact point, and finally Kn and Kt are normaland tangential spring coefficients, respectively. The torque resulted from the tangential force (mt) can be easily calculated using equation below,
mt ¼ R ft
ð5Þ
where R is the particle radius. The numerical value of the parameters used in the contact model, obtained through experimental measurement and by comparing the simulation results to experimental results, are all given in Table 1. Fig. 5 demonstrates a sample of the
Fig. 5. Demonstration of charge motion obtained by DEM simulations and experiment.
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charge motion profile obtained from the DEM simulator and compares it with the ones obtained from the laboratory-scale ball mill. 4. Transfer function In Section 2.2, the method by which the impulse response of a single impact at a specific position is recorded is explained. The acoustic signal, captured by the microphone, is a superposition of the acoustic signals caused by all impacts at a specific time. For this reason, a methodology or procedure is needed in order to relate all the impacts at a specific time to the acoustic signal measured by the microphone while the mill is running. An equation is derived using basic mathematics and fundamental concepts of mechanical vibrations, in conjunction with some experimental measurements. 4.1. Formulation Acoustic signal emitted from the mill is generated by the direct impacts on the shell as well as collision of the balls with one another. In the model and methodology adopted here, only direct impacts on the shell are taken into account; acoustic signal resulted from ball-on-ball impacts are neglected. This assumption is based on the fact that shell vibrations induced by an acoustic wave resulting from a ball-on-ball impact (indirect stimulation of the shell) is of a much lower strength than those caused by a direct impacts (or stimulation) on the shell. Forces and displacements are assumed to be perpendicular to the shell surface; in other words, the effect of the tangential component of the impact forces in vibration of the mill shell is neglected. For this reason, despite the vector nature of forces and displacements, they are shown as scalars and their direction is assumed to be collinear with the outward radial direction throughout the paper. Fig. 6 depicts the geometry, coordinate system and the normal vector which corresponds to the direction of forces and displacements. The total excitation force inside the mill is a summation of all the impacts, each having a certain magnitude (fi) and being exerted at a specific location (ri) and at a specific time (ti),
Fðr; tÞ ¼
n X
fi dðr ri Þ dðt ti Þ
ð6Þ
i¼1
where n is the total number of impacts on the shell, throughout the duration of the simulation. These impacts induce vibrations all over
Fig. 6. Geometry and coordinate system.
the shell. Assuming these vibrations have a linear response, they have the same frequency and pattern as the near-field radiated sound (Williams, 1999). Therefore, vibration response of the system to the excitation expressed by Eq. (6) at the position of the microphone has the same frequency as the acoustic pressure captured by the on-the-shell microphone. Fig. 7 depicts position of the impacts (ri) and that of microphone (rm) relative to reference point. For the excitation force expressed by Eq. (6), the displacement response (A) at the point where the microphone is mounted (rm) is calculated using Green’s Function (G) of the structure and is given by,
Aðr m ; sÞ ¼
Z Z Z t
Fðr; tÞGðr m ; r; s; tÞdsdt
ð7Þ
s
where the integration is over the area of the source, in this case the outer surface of the mill (Hansen and Snyder, 1997). Assuming axial symmetry, there is no dependence on z, the surface element of the integration is simplified to a linear element as,
Aðr m ; sÞ ¼
Z I t
Fðr; tÞGðr m ; r; s; tÞdrdt
ð8Þ
r
Based on the physics of this problem, there is a rotational symmetry for the Green’s Function of the structure. According to this rotational symmetry, the impacts which are symmetric about the position of the microphone are assumed to have the same acoustic response,
GðuÞ ¼ Gð2p uÞ
ð9Þ
where u is the angular difference between the position vector and the position of the microphone (see Fig. 7) expressed in the equation below,
u ¼ hm h
ð10Þ
Moreover, assuming the response of the system is time-invariant; the Green’s Function can be taken to be a convolution operator, that is,
Gðs; tÞ ¼ Gðs tÞ
ð11Þ
Considering two above-mentioned assumptions, Eq. (8) can be rewritten as,
Aðr m ; sÞ ¼
Z I t
Fðr; tÞGðu; s tÞdrdt
r
Fig. 7. Impacts and microphone position.
ð12Þ
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The forcing function, as defined by Eq. (6), is substitute into Eq. (12),
Aðr m ; sÞ ¼
n X
fi
Z I t
i¼1
Lifters Gðu; s tÞdðt t i Þdðr r i Þdrdt
ð13Þ
r
The magnitude of impact forces are pulled out of the integral since it is not dependent on the integral variable. The time integration can be readily done using the property of the delta function. For the radial integration to be done, u can be written in terms of the position vectors using equation below,
u ¼ cos1
r rm ¼ cos1 ð^r ^r m Þ jrjjrm j
ð14Þ
where ^r and ^r m are the unit vectors of r and rm, respectively. Putting Eq. (14) into Eq. (13) while conducting the time integration, we get,
Aðr m ; sÞ ¼
n X
fi
I
Gðcos1 ð^r ^r m Þ; s ti Þdðr r i Þdr
ð15Þ
Fig. 8. Stationary mill and its lifters.
i¼1
The properties of the delta function allows for the evaluation of the radial integral,
Aðr m ; sÞ ¼
n X
fi Gðui ; s t i Þ
ð16Þ
i¼1
where ui is the angular difference between the impact position and microphone position. It is necessary to note that the excitation of the mill shell resulting from the impacts travels at the very same speed as the microphone. Thus, the relative position of the excitations and microphone (ui) remains constant over time. Therefore, the amplitude is a function of s only,
AðsÞ ¼
n X
fi T e ðui ; s t i Þ
ð17Þ
i¼1
The modified Green’s Function in the Eq. (16) is regarded as the Transfer Function between forces (impacts) and the acoustic signal measured experimentally using the pressure microphone. In conformance with this interpretation, the letter G is replaced by the letter Te standing for the experimental Transfer Function. To simulate the acoustic signal using Eq. (17), impact forces (fi) are obtained from DEM simulations (Section 3) and the Transfer Function, a response property of the mill, is measured experimentally (Section 4.2). Together, these two quantities allow for the calculation of the simulated acoustic signal using Eq. (17). To facilitate the comparison of the simulated and experimental acoustic signal, the power of the acoustic signal will be plotted instead of the acoustic signal itself. The acoustic signal is directly proportional to the amplitude of the vibration (A) while the power of the acoustic signal is proportional to the amplitude square (Phillips et al., 2008). These relations are shown in equation below,
Acoustic signal ðpressureÞ / A Power of the acoustic signal / A2
ð18Þ
As illustrated in Fig. 8, the mill has 12 adjustable lifters. The angles of the lifters are variable, with a range of settings between 0 and 180 °. For reasons of experimental ease, as shown in Fig. 9, the impacts were applied in the space between every two lifters. Thus, the impact response resolution is 24, for the spatial angle. Moreover, due to the symmetry of the structure, results from 180 to 360 ° should mirror those of 0 to 180 °; consequently, number of impacts could be reduced to 12. It should be noted that the experimental procedure and precision of the measurements may be customized according to the specific feature or phenomenon which is of the utmost interest. For instance, depending on whether events at the low- or high-frequency range are objectives of the analysis, different types of microphones can be implemented or specific type of algorithms and filters may be chosen for the mathematical analysis. 5. Results and discussion To assess the validity of the model presented in this paper, the simulated acoustic signal and the one measured experimentally are compared for the operating conditions that follow. The mill was run at 70% of its critical speed (the speed in which the charge begins to centrifuge). For the laboratory-scale mill used in the experiments, the critical speed is equal to 34 rpm which is
θ = 15
4.2. Transfer function measurement To measure the experimental transfer function, a certain number of impacts are exerted to the inner surface of the mill and the corresponding acoustic signal is recorded using the procedure explained in Section 2.2. To determine the continuous transfer function, as described by the Eq. (17), a discrete number of impacts are initially applied to the inner surface of the mill. Finally, the method of interpolation (built-in function in MATLAB) is used to calculate the response in between two measured points, wherever data is not available experimentally.
Fig. 9. Impacts positions.
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equivalent to a frequency of 0.56 Hz. The mill charge or percentage of the mill volume filled with wooden balls is approximately 30%. The lifters are held constant at a 90 degrees position. As explained in Section 3, initially, DEM simulator calculates dynamic features of the mill including the distribution of impact forces on the shell. Using this impact distribution and implementing Eq. (17), a computer code was developed using MATLAB software. The code inputs are the magnitude of forces (fi) and their corresponding positions relative to the microphone (ui), obtained from DEM simulations, as well as the experimental transfer function (Te) obtained by the methodology explained in Section 4.2. The code output is unsurprisingly the simulated acoustic signal of a certain time length. For the same operating conditions described above, acoustic signal was measured experimentally using the methodology described in Section 2.3. Using Eq. (18), the power of both acoustic signals is calculated by squaring the acoustic signal. It should be noted that, in this step of the research, both the simulated and measured acoustic signals are not calibrated; therefore, the absolute value of the acoustic signal is not a subject of comparison. Consequently, for the sake of better comparison, the amplitude of the power of both acoustic signals is normalized with respect to the maximum value of each signal. Fig. 10 shows the power of experimental and simulated acoustic signals in time domain. At first glance, the simulated acoustic signal and the one measured experimentally have more or less a general harmonic behaviour. As the microphone follows the mill, the following behaviours are observed in proximity to the shell and dominate the microphone signal. As the mill turns, the charge is lifted and is subject to small forces, producing little sound. As the charge reaches the top, it is thrown. There is no longer any contact with the shell, and the sound is at its lowest. Finally, the charge falls to the bottom, producing large impact forces. These forces stimulate vibration modes of the shell, producing a strong microphone response. Thus, the microphone response is expected to increase and decrease as the microphone rotates about the mill. Therefore, the observed periodic behaviour of the acoustic signals is ascribed to the periodic nature of the mill; however, because of a degree of randomness in the balls movement and probable limitations in the model/methodology/assumptions, periodic behaviour could be fully achieved neither in simulation nor in the experiment. Fourier Transform of the experimental and the simulated signals provides a powerful tool for a more quantitative comparison of the signals and identifying the fingerprints linked to the operating parameters. Fig. 11 shows the Fourier Transform of the measured and
Fig. 10. Power of the measured (blue) and simulated (red) acoustic signals. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 11. Power Spectrum of the acoustic signal.
simulated acoustic signals in the low-frequency domain which is also called the power spectrum of the acoustic signals. This frequency range is chosen because of the specific phenomena which are of the most interest to us is expected to be observed in this range. The frequency axis is normalized with respect to the mill frequency or the frequency that corresponds to the operating speed of the mill (fmill = 0.4 Hz). The Fourier Transform of the acoustic signal contains features associated to the operating parameters of the mill (rotation speed) and mill configuration (number of lifters). Plumbing the physical interpretation of all these features is not the objective of this paper, nor could it be described without detailed signal processing and extensive correlation of the acoustic/vibration signal to the internal variables of the mill. Nonetheless, there are at least two distinguishing features which can be explained. The first major peak in both measured and simulated signals corresponds to the rotation speed of the mill. According to the location of these peaks, the periodicities of the simulated and measured signals are 2.44 s and 2.50 s, respectively. Equivalently, the periods of the simulated and experimental signal correspond to frequencies of 0.41 and 0.40 Hz, in that order. The mill speed or periodicity of the signal can be additionally obtained through measuring the average time interval between two major peaks in the time domain (Fig. 10). The harmonics of the mill frequency are observed on the diagram, in the simulated and experimental results; their amplitude must normally decrease as the frequency increase, until finally, they become lost in the noise. Unexpectedly, the amplitude of the fourth harmonic in the simulated signal is higher than other harmonics. The reason for this is not fully clear to the authors; it may be due to the shortcomings in the DEM simulator, or the transfer function window length. If it were due to the simulation, identifying such deficiencies in the DEM with the aid of the simulated acoustic signal was one of the main objectives of this research work. Another interesting signal feature is the peaks seen in both signals approximately at 12; this frequency corresponds to lifter effects as there are 12 lifters installed in the mill. Interestingly, it has been postulated that variations in the amplitude of the observed peaks in the frequency domain is related to the variations of the operating parameters (Das et al., 2010; Tang et al., 2010). It has been reported that parameters such as type of the load (balls), filling percentage or even grinding condition (dry or wet) has a effect on the amplitude of the observed peaks such as the one corresponding to the mill speed (Tang et al., 2010). Consequently, excellent agreement between the periodicity of the experimental and simulated signal (about 2% discrepancy) shows the validity and reliability of this specific simulation for such a study.
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6. Conclusion
References
In this paper, a methodology is proposed to simulate the acoustic/vibration signal emitted from the tumbling mills. To this end, the dynamic features of the mill, force distribution within the mill, have been calculated using a DEM simulator. These forces and impacts are the major cause for the acoustic signal emitted from the mill surface. The relation between a single force and the resulted acoustic signal is measured experimentally. Thereafter, the acoustic signal captured by the microphone has been simulated as the superposition of all those single phenomena. Furthermore, the on-the-shell acoustic signal of the mill has been experimentally measured using pressure microphones mounted on the mill shell. The simulated and measured acoustic signals have been plotted in both time and frequency domain and discussed. The peaks in the frequency domain of acoustic signals (simulated and experimental) have been correlated to the operating parameters of the mill (rotation speed) and mill configuration (number of lifters). It has been postulated that the variations in the amplitude of these peaks are linked to the variations of the operating parameters. The simulated signal is in good agreement with the actual signal in terms of sensitivity to the mill speed and also number of lifters. In a general sense, the discrepancy between simulated and measured acoustic signal may be used as a criterion to judge the validity of DEM simulator and thereby mend its possible deficiencies. Several questions remain to address, such as the signal power and the relative amplitude discrepancies. While using of the online sensors faces some practical difficulties due to the harsh environment inside the mill, comparison of the simulated and measured acoustic/vibration signal can be a criterion in assessing the validity of DEM simulations. Moreover, as the potential extension of this research, through applying the inverse approach, it is theoretically possible to obtain the force distribution on the inner surface of the mill using vibration/acoustic signals on the outer surface of the shell in conjunction with the DEM simulation. Though this work discusses acoustic signal simulation of tumbling mills only, acoustic/vibration signals of other mill types may be simulated through identification of the phenomena causing the acoustic emissions in those mills and generalizing this methodology accordingly.
Aldrich, C., Theron, D., 2000. Acoustic estimation of the particle size distributions of sulphide ores in a laboratory ball mill. Journal of the South African Institute of Mining and Metallurgy (South Africa) 100 (4), 243–248. Cleary, P., 2001. Modelling comminution devices using DEM. International Journal for Numerical and Analytical Methods in Geomechanics 25 (1), 83–105. Cleary, P., Morrisson, R., Morrell, S., 2003. Comparison of DEM and experiment for a scale model SAG mill. International Journal of Mineral Processing 68 (1–4), 129–165. Cundall, P.A., Strack, O., 1979. A discrete numerical model for granular assemblies. Geotechnique 29 (1), 47–65. Das, S., Das, D., Behera, S., Mishra, B., 2010. Interpretation of mill vibration signal via wireless sensing. Minerals Engineering. Hansen, C., Snyder, S., 1997. Active control of noise and vibration, Taylor & Francis. Huang, H., Pan, J., McCormick, P., 1997. Prediction of impact forces in a vibratory ball mill using an inverse technique. International Journal of Impact Engineering 19 (2), 117–126. Huang, P., Jia, M., Zhong, B., 2009. Investigation on measuring the fill level of an industrial ball mill based on the vibration characteristics of the mill shell. Minerals Engineering 22 (14), 1200–1208. Kolacz, J., 1997. Measurement system of the mill charge in grinding ball mill circuits. Minerals Engineering 10 (12), 1329–1338. Martins, S., 2011. Exploring tumbling mill dynamics through sensor development. In: Mechanical Engineering, McGill University, Montreal. Martins, S., Zepeda, J., Picard, B., Radziszewski, P., Roy D., 2006. Investigating onthe-Shell Acoustics. In SAG, Vancouver, vol. 3 (September), pp. 300–310. Martins, S., Li, W., Radziszewski, P., Caron, S., Aguanno, M., Bakhos, M., Petch, E., 2008. Validating the instrumented ball outputs with simple trajectories. Minerals Engineering 21 (11), 782–788. McElroy, L., Bao, J., Yang, R., Yu, A., 2009. Soft-sensors for prediction of impact energy in horizontal rotating drums. Powder Technology. Mishra, B., 2003. A review of computer simulation of tumbling mills by the discrete element method: Part I-contact mechanics. International Journal of Mineral Processing 71 (1–4), 73–93. Mishra, B., Rajamani, R., 1992. The discrete element method for the simulation of ball mills. Applied Mathematical Modelling 16 (11), 598–604. Morrison, R., Cleary, P., 2008. Towards a virtual comminution machine. Minerals Engineering 21 (11), 770–781. Phillips, C.L., Parr, J.M., Riskin, E.A., 2008. Signals, systems, and transforms, Prentice Hall. Powell, M., Nurick, G., 1996. A study of charge motion in rotary mills part 2experimental work⁄ 1. Minerals Engineering 9 (3), 343–350. Spencer, S., Campbell, J., Weller, K., Liu, Y., Miner, C., Kenmore, Q., 1999. Acoustic emissions monitoring of SAG mill performance. Tang, J., Zhao, L., Zhou, J., Yue, H., Chai, T., 2010. Experimental analysis of wet mill load based on vibration signals of laboratory-scale ball mill shell. Minerals Engineering. Wang, B., 2002. Prediction of impact and harmonic forces acting on arbitrary structures: theoretical formulation. Mechanical Systems and Signal Processing 16 (6), 935–953. Watson, J., 1985. An analysis of mill grinding noise. Powder Technology 41 (1), 83– 89. Williams, E., 1999. Fourier Acoustics: Sound Radiation and Nearfield Acoustical Holography. Academic Press. Xiang, J., Munjiza, A., Latham, J.P., Guises, R., 2009. On the validation of DEM and FEM/DEM models in 2D and 3D. Engineering Computations 26 (6), 673–687. Zeng, Y., Forssberg, E., 1993. Monitoring grinding parameters by signal measurements for an industrial ball mill. International Journal of Mineral Processing 40 (1–2), 1–16. Zhu, H.P., Zhou, Z.Y., Yang, R.Y., Yu, A.B., 2007. Discrete particle simulation of particulate systems: theoretical developments. Chemical Engineering Science 62, 3378–3396.
Acknowledgments Authors are grateful to Prof. Luc Mongeau and Prof. Annie Ross for their advices and helpful discussion during the course of this research project and Dr. Amar Sabih for his help during the primary stages of the work. We also would like to thank Mr. Arnaud Faucher and Dr. Sami Makni from COREM for their invaluable assistance regarding the DEM simulations.
Contents lists available at ScienceDirect
Minerals Engineering journal homepage: www.elsevier.com/locate/mineng
Acoustic emissions simulation of tumbling mills using charge dynamics Poorya Hosseini a,⇑, Sudarshan Martins a, Tristan Martin b, Peter Radziszewski a, Francois-Raymond Boyer b a b
Department of Mechanical Engineering, McGill University, Montreal, Quebec, Canada Département de génie informatique et génie logiciel, École Polytechnique de Montréal, Canada
a r t i c l e
i n f o
Article history: Received 26 January 2011 Accepted 4 July 2011 Available online 4 August 2011 Keywords: Discreet element modelling Simulation Process instrumentation Mineral processing SAG milling
a b s t r a c t Knowledge of the internal variables of a mill is of importance in design and performance optimization of the mill, notwithstanding the difficulty in measuring these variables within the harsh mill environment. To overcome this problem, the research has focused on measuring the internal parameters through noninvasive measurement methods such as the use of the vibration/acoustic signal obtained from the mill. Alternatively, virtual instruments, such as discrete element methods (DEM), are employed. Here, a methodology is developed to simulate on-the-shell acoustic signal emitted from tumbling mills using the information extracted from a DEM simulator. The transfer function which links the forces exerted on the internal surface of the mill and the acoustic signal measured on the outer surface is measured experimentally. Given this transfer function and the force distribution obtained from the DEM simulation, and assuming a linear time-invariant response, the on-the-shell acoustic of a laboratory scale ball mill has been simulated. Comparison of this simulated signal with the signal measured experimentally can be used as a criterion to judge the validity of the DEM simulations, and as a tool for enhancing our understanding of both DEM simulations and the use of acoustics within the context of mineral processing. The results derived from preliminary experiments on a laboratory scale mill shows satisfactory agreement between the actual measurement and the simulated acoustic signal. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Tumbling mills are a class of comminution devices, and are frequently used for the size reduction of the ore in the mineral processing industry. Better understanding of the mode and mechanism of energy utilization, and the dynamics of the charge can potentially lead to significant energy savings. Due to the harsh environment inside the mills, as well as severe charge–charge and charge-liner impacts, the use of on-line sensors presents some practical problems (Martins et al., 2008). An alternative solution is the use of discrete element models (DEM) to simulate internal mill dynamics and the charge motion (Cleary, 2001; Cleary et al., 2003; Mishra, 2003; Mishra and Rajamani, 1992; Powell and Nurick, 1996). Significant advances in computer technology have had a role in the growing interest in using DEM to simulate dynamics of the mill; however, DEM simulations still lack in accuracy. In addition, there are ongoing challenges for DEM simulations of tumbling mills, such as shortcomings in simulating the fine progeny and behaviour of the slurry (Morrison and Cleary, 2008). It is known that ball mills undergo strong mechanical vibrations, caused by the impacts and collisions. As a result, they generate a loud noise. Though noise and vibration may be harmful, from the
⇑ Corresponding author. E-mail address: [email protected] (P. Hosseini). 0892-6875/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.mineng.2011.07.002
viewpoint of a human operator, and are a waste of the energy, they can serve as a useful tool in studying the operation of the mills. The acoustic/vibration signal contains information directly related to the operating state of the mill and the mill charge dynamics. Anecdotal evidence has long suggested that a skilled grinding mill operator can evaluate the operating state of the mill by listening to the generated sound. The measurement of the sound of the mill by means of instrumentation has the benefit of full-time on-line operation, increased precision, while having a greater tolerance to perilous or harsh working environments (Zeng and Forssberg, 1993). Over recent decades, different studies have been conducted on laboratory and industrial scale mills to correlate the acoustic/vibration signal with the operating parameters of the mill such as power draw, feed rate, mill load, pulp density, ore type and particles size distribution (Aldrich and Theron, 2000; Das et al., 2010; Kolacz, 1997; Spencer et al., 1999; Tang et al., 2010; Watson, 1985; Zeng and Forssberg, 1993). Moreover, there are some dynamic values which play an important role in optimizing the mill performance and mill design. The shoulder and toe angles are two such examples. Correlating the acoustic/vibration signal with these dynamic features has been much less studied than the relation between operating parameter and the mill sound (Huang et al., 2009; Martins et al., 2006). Considering that the use of acoustic/vibration signal is a non-invasive, low cost tool of studying comminution machines, there remains room for more studies, specifically for industrial applications.
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Nomenclature p i h ft fn K dn ~ v m f
l ~ n R
acoustic pressure impact force impulse response tangential component of contact force normal component of contact force spring coefficient in contact model relative normal displacement at contact translational velocity at contact mass of particle viscous damping ratio in contact model sliding friction coefficient unit normal vector at the contact point radius of particle
Through the simulation of the charge motion inside tumbling mills, DEM models calculate distribution and magnitude of forces and impacts. These forces and impacts engender the vibrations of the structure, and are the main cause of the sound signal generated by the mill. If the relation between these impacts and the acoustic/ vibration signal emitted from the mill is known, the vibration/ acoustic signal can be simulated using the distribution of impacts extracted from the DEM simulator; such a simulation of acoustic/ vibration signal was held to be unworkable in the past (McElroy et al., 2009). Replacement of surface vibration with DEM modelling allows the implementation of DEMs for soft-sensors design approaches, with the objective of measuring the internal variables of the mill (McElroy et al., 2009). Furthermore, the comparison between the simulated signal and the signal measured experimentally can be used as a criterion for evaluating validity of DEM simulations, and as tool for enhancing our understanding of the dynamics of the mill. If the inverse approach is taken, it may be possible to determine dynamic values currently obtained from other methods, such as impacts inside the mill, using only the acoustic/vibration signal. Implementing such an approach for a similar application – a vibratory ball mill containing a single ball – produced promising result in the prediction of impact force using the vibration signal (Huang et al., 1997). These so-called inverse techniques have been extensively used to predict features of mechanical systems which are difficult or impossible to measure directly. A categorization of these techniques for force-prediction models, various applied examples and the required theoretical background has been presented by Wang (2002). Acoustic signal and vibration signal of
f F d G A r rm h hm
u t
impact on mill shell total force acting on mill shell dirac delta function green’s Function displacement of mill shell position vector position vector of microphone angular position of impact angular position of microphone angular difference between impact position and microphone position time
the mill are highly correlated; however, the acoustic signal is more of interest, since its measurement is more practical and has the potential of being captured through sensors which are not necessarily attached to the structure. In this paper, as mentioned earlier, it is demonstrated that an acoustic signal can be calculated from DEM models. This simulated acoustic signal will be shown to be comparable to the measured acoustic signal. 2. Experimental setup A brief description of the laboratory-scale ball mill used in the experiments, the experimental setup used to capture the acoustic signal and to measure impact forces as well as the methodology implemented to process the primary measurements are presented in this section. 2.1. The laboratory-scale ball mill For the experiment, a laboratory-scale ball mill featuring a cam drive is used, as illustrated in Fig. 1. A large diameter aluminium disc is fixed to a shaft mounted on a bearing. The aluminium disc has two functions. Firstly, the followers for the cam drive are fixed to its face. Secondly, the mill drum (or shell) is bolted to the disc. A transparent Plexiglas face closes the mill at the free end of the drum, allowing for the observation of the charge. The drum consists of a steel cylinder, with a diameter of 1.524 m and a length of 0.3048 m. A set of twelve plates are fixed to the inner surface
Fig. 1. Schematic of the laboratory scale ball mill.
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of the shell. These plates are called lifters. Their role is to further promote the tumbling action of the charge, as induced by the rotation of the mill. The birch balls, which form the charge, have an average diameter of 5.1 cm and an average mass of 43 g. 2.2. Impulse response measurement This section describes the method by which the relationship between a single impact, inside the mill shell, at a specific position and the corresponding acoustic signal captured on the mill shell is established. Impact force and the resulting acoustic signal are measured using an impact hammer with embedded force transducer and a pressure-field microphone, respectively. A multi-channel data acquisition system is used to concurrently amplify and digitize both signals and to stream the resulting signals to a computer. Fig. 2 shows a sample of the impact force signal along with the resulting acoustic pressure signal. Assuming the vibrations of the shell is linear and time invariant, meaning different impact at the same position at different times produce the same response, the shell response to the impact force can be characterized through the system impulse response. Consequently, the relationship between acoustic pressure p(t) and impact force i(t) can be expressed using the convolution integral (Phillips et al., 2008):
pðtÞ ¼ iðtÞ hðtÞ ¼
Z
1
iðsÞhðt sÞ ds
ð2Þ
where P(jx), I(jx) and H(jx) are Fourier Transforms of p(t), i(t) and h(t) respectively. To obtain the impulse response in time domain, the Inverse Fourier Transform, then, is applied to the Eq. (2),
PðjxÞ IðjxÞ
The same microphone and recording system described in Section 2.2 were used to capture the acoustic signal of the mill while rotating with the charge. The microphone is mounted on the mill shell and rotates with the mill. The position of the microphone is tracked through adding a fingerprint to the acoustic signal every time the microphone passes by a certain point. To detect the position of the microphone, a fibber unit proximity sensor has been used. At any time the sensor detects the microphone, a voltage is generated; this voltage becomes amplified and finally activates a buzzer. Since the frequency of the sound generated by this buzzer is unique, it can be distinguished from other sound sources in the signal. The sensor has been used during the whole measurement; however, once the location of the microphone is known at a specific time, its position can be found thereafter using the mill rotation speed. 3. DEM simulation
where h(t) is the system impulse response to the impacts at a specific position. By applying the Fourier transform to both sides of Eq. (1), the time domain convolution integral becomes a multiplication in the frequency domain (Phillips et al., 2008):
hðtÞ ¼ F 1
2.3. Acoustic signal of the mill
ð1Þ
1
PðjxÞ ¼ IðjxÞ HðjxÞ
a constant amplitude in the frequency domain – the denominator of Eq. (3) becomes constant and the impulse response, in this case, is equal to the acoustic pressure divided by a constant (the magnitude of the impact). However, in this research deconvolution of the actual excitation exerted by hammer is used for calculations rather than the ideal impact assumption. Fig. 3 shows a sample of the calculated impulse response in the time domain.
ð3Þ
If the force exerted by impact hammer is assumed to be an ideal impact – a perfect impulse that has an infinitely small duration causing
The DEM simulator used in this work has been originally developed for broader research purposes in comminution, including breakage efficiency, and mill equipment design. DEM, in general, is a numerical iterative method which calculates the dynamics of a discontinuous system of particles (Cundall and Strack, 1979). In the case of tumbling mills, it is the charge of the mill that is represented by a collection of particles of defined properties. As shown in Fig. 4, at each cycle or time step, the calculator initially resolves collisions and calculates the corresponding forces generated by these collisions. The collision forces along with external forces, such as gravity or electrostatic forces are subsequently applied to the appropriate particles. By repeating this calculation cycle, the simulator generates trajectories and forces of particles as a
Fig. 2. Excitation exerted by impact hammer inside the mill shell (top), and response to the excitation captured by a pressure microphone on the mill shell (bottom).
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Fig. 3. A typical impulse response.
Table 1 Parameters of simulation and contact model. Parameter
Value
Drum, d l (cm) Normal viscous damping ratio Tangential viscous damping ratio Normal stiffness (N/m) Tangential stiffness (N/m) Particle diameter, D (mm) Particle density (g cm3) Sliding friction coefficient Filling percentage of the mill (%) Rotation speed of the mill (rpm)
152.4 30.5 0.3 0.4 10,000 20,000 50.6 0.49 0.5 30 24
pffiffiffiffiffiffiffiffiffiffi fn ¼ K n dn~ v ~nÞ~n n 2 K n m fn ð~ R pffiffiffiffiffiffiffiffiffi ft ¼ minflfn ; K t ~ v dt 2 K t m ft ð~ v ~nÞ ~ng
Fig. 4. Calculations steps of the DEM.
function of time. The time step of the simulation, varying from 10 to 100 microseconds, is dynamically adjusted to achieve the best possible compromise between precision and performance. While different contact models may be used to describe the contacts in DEM simulations (Zhu et al., 2007), here the software adopts the linear spring-dashpot contact model (Cundall and Strack, 1979; Martins, 2011; Xiang et al., 2009). The tangential component (ft) and normal component (fn) of interparticle contact force in this model are calculated as below,
ð4Þ
where dn is the relative normal displacement at contact, ~ v is the translational velocity, m is the mass of the particle, f is the viscous damping ratio, l is the sliding friction coefficient, ~ n is the unit normal vector at the contact point, and finally Kn and Kt are normaland tangential spring coefficients, respectively. The torque resulted from the tangential force (mt) can be easily calculated using equation below,
mt ¼ R ft
ð5Þ
where R is the particle radius. The numerical value of the parameters used in the contact model, obtained through experimental measurement and by comparing the simulation results to experimental results, are all given in Table 1. Fig. 5 demonstrates a sample of the
Fig. 5. Demonstration of charge motion obtained by DEM simulations and experiment.
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charge motion profile obtained from the DEM simulator and compares it with the ones obtained from the laboratory-scale ball mill. 4. Transfer function In Section 2.2, the method by which the impulse response of a single impact at a specific position is recorded is explained. The acoustic signal, captured by the microphone, is a superposition of the acoustic signals caused by all impacts at a specific time. For this reason, a methodology or procedure is needed in order to relate all the impacts at a specific time to the acoustic signal measured by the microphone while the mill is running. An equation is derived using basic mathematics and fundamental concepts of mechanical vibrations, in conjunction with some experimental measurements. 4.1. Formulation Acoustic signal emitted from the mill is generated by the direct impacts on the shell as well as collision of the balls with one another. In the model and methodology adopted here, only direct impacts on the shell are taken into account; acoustic signal resulted from ball-on-ball impacts are neglected. This assumption is based on the fact that shell vibrations induced by an acoustic wave resulting from a ball-on-ball impact (indirect stimulation of the shell) is of a much lower strength than those caused by a direct impacts (or stimulation) on the shell. Forces and displacements are assumed to be perpendicular to the shell surface; in other words, the effect of the tangential component of the impact forces in vibration of the mill shell is neglected. For this reason, despite the vector nature of forces and displacements, they are shown as scalars and their direction is assumed to be collinear with the outward radial direction throughout the paper. Fig. 6 depicts the geometry, coordinate system and the normal vector which corresponds to the direction of forces and displacements. The total excitation force inside the mill is a summation of all the impacts, each having a certain magnitude (fi) and being exerted at a specific location (ri) and at a specific time (ti),
Fðr; tÞ ¼
n X
fi dðr ri Þ dðt ti Þ
ð6Þ
i¼1
where n is the total number of impacts on the shell, throughout the duration of the simulation. These impacts induce vibrations all over
Fig. 6. Geometry and coordinate system.
the shell. Assuming these vibrations have a linear response, they have the same frequency and pattern as the near-field radiated sound (Williams, 1999). Therefore, vibration response of the system to the excitation expressed by Eq. (6) at the position of the microphone has the same frequency as the acoustic pressure captured by the on-the-shell microphone. Fig. 7 depicts position of the impacts (ri) and that of microphone (rm) relative to reference point. For the excitation force expressed by Eq. (6), the displacement response (A) at the point where the microphone is mounted (rm) is calculated using Green’s Function (G) of the structure and is given by,
Aðr m ; sÞ ¼
Z Z Z t
Fðr; tÞGðr m ; r; s; tÞdsdt
ð7Þ
s
where the integration is over the area of the source, in this case the outer surface of the mill (Hansen and Snyder, 1997). Assuming axial symmetry, there is no dependence on z, the surface element of the integration is simplified to a linear element as,
Aðr m ; sÞ ¼
Z I t
Fðr; tÞGðr m ; r; s; tÞdrdt
ð8Þ
r
Based on the physics of this problem, there is a rotational symmetry for the Green’s Function of the structure. According to this rotational symmetry, the impacts which are symmetric about the position of the microphone are assumed to have the same acoustic response,
GðuÞ ¼ Gð2p uÞ
ð9Þ
where u is the angular difference between the position vector and the position of the microphone (see Fig. 7) expressed in the equation below,
u ¼ hm h
ð10Þ
Moreover, assuming the response of the system is time-invariant; the Green’s Function can be taken to be a convolution operator, that is,
Gðs; tÞ ¼ Gðs tÞ
ð11Þ
Considering two above-mentioned assumptions, Eq. (8) can be rewritten as,
Aðr m ; sÞ ¼
Z I t
Fðr; tÞGðu; s tÞdrdt
r
Fig. 7. Impacts and microphone position.
ð12Þ
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The forcing function, as defined by Eq. (6), is substitute into Eq. (12),
Aðr m ; sÞ ¼
n X
fi
Z I t
i¼1
Lifters Gðu; s tÞdðt t i Þdðr r i Þdrdt
ð13Þ
r
The magnitude of impact forces are pulled out of the integral since it is not dependent on the integral variable. The time integration can be readily done using the property of the delta function. For the radial integration to be done, u can be written in terms of the position vectors using equation below,
u ¼ cos1
r rm ¼ cos1 ð^r ^r m Þ jrjjrm j
ð14Þ
where ^r and ^r m are the unit vectors of r and rm, respectively. Putting Eq. (14) into Eq. (13) while conducting the time integration, we get,
Aðr m ; sÞ ¼
n X
fi
I
Gðcos1 ð^r ^r m Þ; s ti Þdðr r i Þdr
ð15Þ
Fig. 8. Stationary mill and its lifters.
i¼1
The properties of the delta function allows for the evaluation of the radial integral,
Aðr m ; sÞ ¼
n X
fi Gðui ; s t i Þ
ð16Þ
i¼1
where ui is the angular difference between the impact position and microphone position. It is necessary to note that the excitation of the mill shell resulting from the impacts travels at the very same speed as the microphone. Thus, the relative position of the excitations and microphone (ui) remains constant over time. Therefore, the amplitude is a function of s only,
AðsÞ ¼
n X
fi T e ðui ; s t i Þ
ð17Þ
i¼1
The modified Green’s Function in the Eq. (16) is regarded as the Transfer Function between forces (impacts) and the acoustic signal measured experimentally using the pressure microphone. In conformance with this interpretation, the letter G is replaced by the letter Te standing for the experimental Transfer Function. To simulate the acoustic signal using Eq. (17), impact forces (fi) are obtained from DEM simulations (Section 3) and the Transfer Function, a response property of the mill, is measured experimentally (Section 4.2). Together, these two quantities allow for the calculation of the simulated acoustic signal using Eq. (17). To facilitate the comparison of the simulated and experimental acoustic signal, the power of the acoustic signal will be plotted instead of the acoustic signal itself. The acoustic signal is directly proportional to the amplitude of the vibration (A) while the power of the acoustic signal is proportional to the amplitude square (Phillips et al., 2008). These relations are shown in equation below,
Acoustic signal ðpressureÞ / A Power of the acoustic signal / A2
ð18Þ
As illustrated in Fig. 8, the mill has 12 adjustable lifters. The angles of the lifters are variable, with a range of settings between 0 and 180 °. For reasons of experimental ease, as shown in Fig. 9, the impacts were applied in the space between every two lifters. Thus, the impact response resolution is 24, for the spatial angle. Moreover, due to the symmetry of the structure, results from 180 to 360 ° should mirror those of 0 to 180 °; consequently, number of impacts could be reduced to 12. It should be noted that the experimental procedure and precision of the measurements may be customized according to the specific feature or phenomenon which is of the utmost interest. For instance, depending on whether events at the low- or high-frequency range are objectives of the analysis, different types of microphones can be implemented or specific type of algorithms and filters may be chosen for the mathematical analysis. 5. Results and discussion To assess the validity of the model presented in this paper, the simulated acoustic signal and the one measured experimentally are compared for the operating conditions that follow. The mill was run at 70% of its critical speed (the speed in which the charge begins to centrifuge). For the laboratory-scale mill used in the experiments, the critical speed is equal to 34 rpm which is
θ = 15
4.2. Transfer function measurement To measure the experimental transfer function, a certain number of impacts are exerted to the inner surface of the mill and the corresponding acoustic signal is recorded using the procedure explained in Section 2.2. To determine the continuous transfer function, as described by the Eq. (17), a discrete number of impacts are initially applied to the inner surface of the mill. Finally, the method of interpolation (built-in function in MATLAB) is used to calculate the response in between two measured points, wherever data is not available experimentally.
Fig. 9. Impacts positions.
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equivalent to a frequency of 0.56 Hz. The mill charge or percentage of the mill volume filled with wooden balls is approximately 30%. The lifters are held constant at a 90 degrees position. As explained in Section 3, initially, DEM simulator calculates dynamic features of the mill including the distribution of impact forces on the shell. Using this impact distribution and implementing Eq. (17), a computer code was developed using MATLAB software. The code inputs are the magnitude of forces (fi) and their corresponding positions relative to the microphone (ui), obtained from DEM simulations, as well as the experimental transfer function (Te) obtained by the methodology explained in Section 4.2. The code output is unsurprisingly the simulated acoustic signal of a certain time length. For the same operating conditions described above, acoustic signal was measured experimentally using the methodology described in Section 2.3. Using Eq. (18), the power of both acoustic signals is calculated by squaring the acoustic signal. It should be noted that, in this step of the research, both the simulated and measured acoustic signals are not calibrated; therefore, the absolute value of the acoustic signal is not a subject of comparison. Consequently, for the sake of better comparison, the amplitude of the power of both acoustic signals is normalized with respect to the maximum value of each signal. Fig. 10 shows the power of experimental and simulated acoustic signals in time domain. At first glance, the simulated acoustic signal and the one measured experimentally have more or less a general harmonic behaviour. As the microphone follows the mill, the following behaviours are observed in proximity to the shell and dominate the microphone signal. As the mill turns, the charge is lifted and is subject to small forces, producing little sound. As the charge reaches the top, it is thrown. There is no longer any contact with the shell, and the sound is at its lowest. Finally, the charge falls to the bottom, producing large impact forces. These forces stimulate vibration modes of the shell, producing a strong microphone response. Thus, the microphone response is expected to increase and decrease as the microphone rotates about the mill. Therefore, the observed periodic behaviour of the acoustic signals is ascribed to the periodic nature of the mill; however, because of a degree of randomness in the balls movement and probable limitations in the model/methodology/assumptions, periodic behaviour could be fully achieved neither in simulation nor in the experiment. Fourier Transform of the experimental and the simulated signals provides a powerful tool for a more quantitative comparison of the signals and identifying the fingerprints linked to the operating parameters. Fig. 11 shows the Fourier Transform of the measured and
Fig. 10. Power of the measured (blue) and simulated (red) acoustic signals. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 11. Power Spectrum of the acoustic signal.
simulated acoustic signals in the low-frequency domain which is also called the power spectrum of the acoustic signals. This frequency range is chosen because of the specific phenomena which are of the most interest to us is expected to be observed in this range. The frequency axis is normalized with respect to the mill frequency or the frequency that corresponds to the operating speed of the mill (fmill = 0.4 Hz). The Fourier Transform of the acoustic signal contains features associated to the operating parameters of the mill (rotation speed) and mill configuration (number of lifters). Plumbing the physical interpretation of all these features is not the objective of this paper, nor could it be described without detailed signal processing and extensive correlation of the acoustic/vibration signal to the internal variables of the mill. Nonetheless, there are at least two distinguishing features which can be explained. The first major peak in both measured and simulated signals corresponds to the rotation speed of the mill. According to the location of these peaks, the periodicities of the simulated and measured signals are 2.44 s and 2.50 s, respectively. Equivalently, the periods of the simulated and experimental signal correspond to frequencies of 0.41 and 0.40 Hz, in that order. The mill speed or periodicity of the signal can be additionally obtained through measuring the average time interval between two major peaks in the time domain (Fig. 10). The harmonics of the mill frequency are observed on the diagram, in the simulated and experimental results; their amplitude must normally decrease as the frequency increase, until finally, they become lost in the noise. Unexpectedly, the amplitude of the fourth harmonic in the simulated signal is higher than other harmonics. The reason for this is not fully clear to the authors; it may be due to the shortcomings in the DEM simulator, or the transfer function window length. If it were due to the simulation, identifying such deficiencies in the DEM with the aid of the simulated acoustic signal was one of the main objectives of this research work. Another interesting signal feature is the peaks seen in both signals approximately at 12; this frequency corresponds to lifter effects as there are 12 lifters installed in the mill. Interestingly, it has been postulated that variations in the amplitude of the observed peaks in the frequency domain is related to the variations of the operating parameters (Das et al., 2010; Tang et al., 2010). It has been reported that parameters such as type of the load (balls), filling percentage or even grinding condition (dry or wet) has a effect on the amplitude of the observed peaks such as the one corresponding to the mill speed (Tang et al., 2010). Consequently, excellent agreement between the periodicity of the experimental and simulated signal (about 2% discrepancy) shows the validity and reliability of this specific simulation for such a study.
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6. Conclusion
References
In this paper, a methodology is proposed to simulate the acoustic/vibration signal emitted from the tumbling mills. To this end, the dynamic features of the mill, force distribution within the mill, have been calculated using a DEM simulator. These forces and impacts are the major cause for the acoustic signal emitted from the mill surface. The relation between a single force and the resulted acoustic signal is measured experimentally. Thereafter, the acoustic signal captured by the microphone has been simulated as the superposition of all those single phenomena. Furthermore, the on-the-shell acoustic signal of the mill has been experimentally measured using pressure microphones mounted on the mill shell. The simulated and measured acoustic signals have been plotted in both time and frequency domain and discussed. The peaks in the frequency domain of acoustic signals (simulated and experimental) have been correlated to the operating parameters of the mill (rotation speed) and mill configuration (number of lifters). It has been postulated that the variations in the amplitude of these peaks are linked to the variations of the operating parameters. The simulated signal is in good agreement with the actual signal in terms of sensitivity to the mill speed and also number of lifters. In a general sense, the discrepancy between simulated and measured acoustic signal may be used as a criterion to judge the validity of DEM simulator and thereby mend its possible deficiencies. Several questions remain to address, such as the signal power and the relative amplitude discrepancies. While using of the online sensors faces some practical difficulties due to the harsh environment inside the mill, comparison of the simulated and measured acoustic/vibration signal can be a criterion in assessing the validity of DEM simulations. Moreover, as the potential extension of this research, through applying the inverse approach, it is theoretically possible to obtain the force distribution on the inner surface of the mill using vibration/acoustic signals on the outer surface of the shell in conjunction with the DEM simulation. Though this work discusses acoustic signal simulation of tumbling mills only, acoustic/vibration signals of other mill types may be simulated through identification of the phenomena causing the acoustic emissions in those mills and generalizing this methodology accordingly.
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Acknowledgments Authors are grateful to Prof. Luc Mongeau and Prof. Annie Ross for their advices and helpful discussion during the course of this research project and Dr. Amar Sabih for his help during the primary stages of the work. We also would like to thank Mr. Arnaud Faucher and Dr. Sami Makni from COREM for their invaluable assistance regarding the DEM simulations.