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Ultrasonic velocity measurements in powders and their relationship to strength in particles formed by agglomeration
Powder Technology 208 (2011) 694–701
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Powder Technology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / p ow t e c
Ultrasonic velocity measurements in powders and their relationship to strength in particles formed by agglomeration P.J. Coghill ⁎, P. Giang CSIRO Process Science and Engineering, Lucas Heights, Australia
a r t i c l e
i n f o
Article history: Received 10 June 2010 Received in revised form 10 October 2010 Accepted 26 November 2010 Available online 27 January 2011 Keywords: Particle strength Powder Ultrasonic velocity Alumina
a b s t r a c t Many industries produce pellets, powders, and granules as finished or intermediate products. The strength of these products is important to their marketability, as dusting and breakdown from weak product is undesirable and results in poor handling properties. Measurement of “strength” usually takes the form of some empirical test involving crushing or attrition. The possibility of using sound velocity measurements to replace these techniques (especially for composite materials such as pellets or agglomerates) is investigated by performing measurements on glass sphere powders and test industrial materials. The ultrasonic velocity measurements in glass spheres were shown to be described by effective medium theories of sound propagation and can be used to probe the packing structure of the powder. For coarser powders sensible values of the co-ordination number Z could be calculated during compaction of the material. For powders finer than 0.2 mm the sound velocity measurements are no longer repeatable and rapid changes of the configuration may occur though the packing density only changes slowly. Two industrially important materials, iron ore pellets and calcined alumina powders were studied. It was found that reproducible measurements can be made in these materials that correlate to standard industrial “strength” measurements. In the case of iron ore pellets the correlation coefficient was 0.94 and the variability of the sound velocity based “strength” measurement was a factor of five smaller than the particle breakage method. For alumina powders the standard attrition index measurements could be correlated with the ultrasonic velocity and bulk density for both hydrated and calcined alumina powders with correlation coefficients of up to 0.95. The sound velocity strength measurement technique has significant advantages in repeatability, sample size and provides direct measurement of a physical parameter related to strength. Crown Copyright © 2011 Published by Elsevier B.V. All rights reserved.
1. Introduction Ultrasonic velocity measurements are relatively simple to make in bulk solids and can be related to the various elastic moduli, especially for isotropic solids. For these bulk solids the sound speed may be weakly related to the crush or abrasion strength of the material. In agglomerates, as apart from single crystals, where the material is held together through contacts there is reason to relate “strength” to the sound speed. A well described example is in sinters where the interparticle contact areas increase over time as the material is heated leading to an increase in sound velocity and strength during sintering [1]. Many industries produce pellets, powders, and granules as finished or intermediate products that are either grown directly by processes like crystallisation and spray drying, or agglomerated by pelletisation. As a rule the strength of these particles is important to their marketability, as dusting and breakdown from weak product is
⁎ Corresponding author. E-mail address: [email protected] (P.J. Coghill).
undesirable and results in poor handling properties. Often particle size is closely monitored but particle “strength”, which is harder to define and measure, is less closely monitored. The “strength” of these products is generally defined by the type of wear mechanism expected during the particle's life. For some products impact and crushing are the most important comminution mechanism, for others attrition may be more important. Where the wear due to impact and crushing is the important mechanism, strength is measured on a pellet by pellet basis from a small sub-sample; for example, pellets of iron ore are generally destroyed by crushing to test their strength [2]. An example where attrition wear is the important comminution mechanism is in the aluminium industry where vast quantities of alumina powder are produced by crystallisation and calcining to meet size as well as purity standards. The strength of these powders is consistently rated as one of the most important product quality parameters [3]. The current method used for measuring alumina and hydrate strength is a modification of the Forsythe-Hertwig attrition index (AI) method [4]. In this technique the −45 μm mass fraction is measured before and after treatment in a fluidised bed unit. The limitations of this method are: high sensitivity of the measured AI to the operating parameters of the fluid bed unit, and size dependence of
0032-5910/$ – see front matter. Crown Copyright © 2011 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2010.11.040
P.J. Coghill, P. Giang / Powder Technology 208 (2011) 694–701
AI. Significantly different products can have the same or similar measured AI values and in some cases a product with less desirable breakage and attrition properties can even have a lower AI. For both crushing and attrition wear the current measurement techniques are unsatisfactory in some aspects [4]. It is proposed in this work that the mean “strength” of the particles or pellets, in some cases, can be estimated ultrasonically by measuring the sound speed of a sample of powder or pellets placed under pressure. These measurements then can be compared to the standard “strength” measurements in use to determine whether in those particular cases the sound speed measurement is a feasible alternative. The sound measurement, if demonstrated to be feasible, may well have extensive practical advantages over the old techniques in terms of repeatability, ease of use, sample size, and having a direct relationship with a well known physical parameter. The two example materials chosen, iron ore pellets and alumina powder, are agglomerates. In this case the Young's Modulus and hence the sound speed of individual particles may be expected to be related to the strength, as is the case in sinters [1]. In other particles, especially those made of single crystals, the Young's Modulus and hence the sound speed in the particles is unlikely to be related to particle strength. For some larger particles, such as iron ore pellets, measurements of a single layer of particles may be sufficient. This is not the case for finer particles when measurements of sound velocity must take place over a significant distance and hence many layers of particles. In this case the sound velocity in powders is related to the packing of the particles as well as their intrinsic material properties. The theoretical framework of an effective medium theory (EMT) [5] is used to model the behaviour of sound transmission through compressed powders in this paper. In these theories the lumpy nature of the physical properties are smoothed to calculated values for an equivalent homogenous medium. The experiments are then set up so that the conditions match those required for an EMT to work. These include that the sound wavelength is greater than ten times the particle size to prevent scattering from becoming important, and that
695
the pressure is in a moderate range to prevent either crushing at the contacts or insufficient sound transmission [6]. As the sound transmission depends on both the properties of the particles and their configuration, care must be taken to understand the preparation of the sample for measurement. This includes shaking steps to consolidate the powder and prepare as uniform as possible configuration of the powders at measurement. Many investigations of the densification of glass balls have been carried out [7,8], principally to study the glassy or jammed dynamics. In these experiments the packing density ρ, defined by the ratio of grain volume to the total volume occupied by the packing, is measured as the system evolves under mechanical excitation. Following the notation of Richard [9], the minimal value of ρ for a mechanically stable random loose packing of spheres is ρRLP = 0.55, the maximal random packing fraction of spheres is ρRCP = 0.64, which is still significantly less than the maximal packing fraction obtained for a face centred cubic packing of ρMAX = 0.74. In tapping experiments the excitation level is defined by Γ the ratio of the maximum acceleration of the tapping to gravitational acceleration. In all cases the evolution of density proceeds slowly, in some cases a final state not being reached after 105 taps. Density does not constitute a full description of the state of the granular material and the evolution of the packing. Other experiments have observed the behaviour of the packing in more detail for example X-Ray tomography of particle arrays at various stages of compaction has also been undertaken [9,10]. These studies focus on structural properties such as pore size and pair distribution function. One study claims that densification under gentle agitation is mostly evident in the decrease of pore size rather than any change in the average difference between neighbouring particles. The approach taken in this paper is to use the EMT framework to examine experimental results from our apparatus for in the first instance uniformly sized glass spheres, then glass spheres with broader size distributions and finally the alumina powders. This will allow us to assess the importance of the various factors such as size
Fig. 1. Photograph and diagram of the experimental set up.
P.J. Coghill, P. Giang / Powder Technology 208 (2011) 694–701
distribution and shape to the behaviour of the industrial particles of interest and help devise a repeatable measurement procedure. Conversely the measurements of sound velocity will also provide insight to the compaction behaviour of the standard particles. 2. Experimental apparatus and procedure The apparatus for the sound velocity measurement is shown in Fig. 1. The cylindrical sample holder contains the receive transducer in its base. During measurement the sample holder is fixed in position on the apparatus. The powder to be measured is placed in the sample holder and a fixed pressure is applied to the sample by lowering the ram. Between the top of the sample and the ram is the transmit transducer which is pressed onto the sample when the ram is lowered. A sound pulse is then transmitted through the sample from top to bottom and the time of flight is measured. Fig. 1a shows schematics of transmit and receive transducer pairs used for longitudinal and shear measurements. The internal diameter of the sample holder is 60 mm. The external diameter of the top transducer piece (for both longitudinal and shear measurements) is 56 mm to avoid powder becoming stuck between the top transducer piece and the wall of the sample holder. If the powder becomes stuck in this way, then the full pressure is not applied to the powder being measured. For each measurement the weight of the powder is known and the distance between the two ultrasonic transducers measured so the bulk density can be calculated. 2.1. Ultrasonic transducers and pulsing equipment Longitudinal ultrasonic measurements were made with Krautkramer Steinkamp transducers of nominal 50 kHz resonance frequency. The top transducer was attached to the ram which was lowered into the sample holder to apply pressure to the sample being measured. A toneburst pulse of 1 cycle was used to excite the transducer at 48 kHz. The amplitude was adjusted to a maximum of 20 V PTP depending on the attenuation of the material. Transducer width for the longitudinal transducers was 40 mm, which equated to 0.44 of the sample's area having sound applied to it. The received ultrasonic signal was digitised by an A/D card at stored in the computer. A typical received pulse is shown in Fig. 2 for 30 g of 180 μm glass spheres. At this frequency the expected wavelength is between 15 and 20 mm, sufficient for the long wavelength assumption to apply to all samples except the iron ore pellets. 2.2. Measurement procedure The measurement procedure for a single powder sample was standardised on a range of materials to enable comparison of results. Pressures of 0.15, 0.3 or 0.6 bar are applied to the sample. These values were measured directly by a calibrated load cell. Two types of measurement were then made, the first being consolidation experiments where the various materials were compressed at a given pressure with the sound velocity and transducer separation measured periodically. The second measurement procedure was to obtain single measurements of sound velocity and density for a particular sample. In the first consolidation measurement scheme a program triggers horizontal tapping at 100 Hz with a maximum peak acceleration recorded by a calibrated accelerometer of 16.16 times gravitational acceleration. The tapping is applied horizontally unlike the typical compaction experiments where the tapping is vertical. After 20 s of vibration a measurement of sound transmission and compaction are recorded. Each of the 5 measurements is an average of 10 pulses. After each measurement period the 20 s of continuous vibration resumes automatically. The total time to complete the procedure for 1 sample is approximately 1 hour.
0.5 0.4 0.3
Amplitude (V)
696
0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 0.00
50.00 100.00 150.00 200.00 250.00 300.00 350.00 400.00 450.00
Time (microseconds) Fig. 2. Recorded signal of an ultrasonic wave passing through 30 g of SAC powder at 0.6 bar pressure.
The second type of experiment is used for single measurements from a sample. As will be demonstrated later simple compression is insufficient to create a reproducible state and cycling of the pressure to remove hysteresis is also required [6]. The sample is horizontally tapped for 15 min at 0.15 bar, then the pressure is manually cycled 10 times and a measurement of the velocity and density made at 0.3 bar. The velocity and compaction measurement is repeated 100 times and averaged. The measurement system determines time of flight (TOF) as the time elapsed to the first positive pressure peak above half the maximum amplitude of the signal (see Fig. 2). This technique requires consistently shaped waveforms to be accurate and has a finite value for zero delay. This offset was determined by measuring samples of the different materials at different weights and determining the delay projected at zero height from these measurements. Standard measurements were performed with 30 g of powder unless otherwise stated. 2.3. Materials tested Glass ballotini and glass spheres were used as the standard materials to perform tests of the EMT in this experimental apparatus. The glass ballotini has different size grades, with relatively broad size distributions for each grade. Some of the AC glass ballotini was sieved to produce a narrow distribution and called SAC glass. The values of the physical parameters used in the rest of the paper for the glass spheres and ballotini are shear modulus μg = 31 GPa, Poissons Ratio νg = 0.24 and density ρg = 2.5 g/cm3. All of the materials used are shown in Table 1. 3. Sound transmission through granular solids The transmission of an ultrasonic burst through granular material produces a ballistic pulse with a scattering induced coda following in its aftermath [11] as may be seen in Fig. 2. Sound propagation of this sort has been well studied due to its relevance to seismic wave propagation, sonar, and sound transmission in sinters [5,6,12]. Hostler and Brennen Table 1 Physical properties of materials tested. Material
D50 (μm)
Size distribution
Shape
AH glass AE glass AC glass SAC glass 0.5 mm glass 1 mm glass Iron ore pellets Alumina — hydrate Alumina — calcined
70 115 191 181 500 1000 10,000 100 100
Broad Broad Broad Narrow Narrow Narrow Broad Broad Broad
Spheres Spheres Spheres Spheres Spheres Spheres Rough Spheres blocky blocky
P.J. Coghill, P. Giang / Powder Technology 208 (2011) 694–701
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K + 4 = 3μ V= : ρ
ð1Þ
The two elastic moduli K the bulk modulus and μ the shear modulus can be calculated with an EMT theory when uniform pressure is applied. Using the formulation of Makse et al. [6] in an assembly of spheres under pressure P, the sound velocity can be calculated by substituting the values of the elastic constants for frictionless grains into Eq. 1 as
V =Z
1=3
φ
−1 = 6
6πP kn
1 = 6 sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3kn 20πρg
ð2Þ
where Z is the average number of contacts per particle, ϕ is the packing fraction of solids, kn = 4μg/(1 − νg), and νg is the Poisson's ratio of glass. The g subscript refers to the values of the physical constants for the glass beads used in the subsequent experiments. Corresponding to the maximal random packing density and maximal density for beds of spheres are the values of 6 and 14 for Z [6]. In the case of our experiment however the force is applied uniaxially in the direction that the sound is propagating. Under these conditions the powder is not isotropic so the bulk and shear moduli cannot be defined. The EMT framework can still be used to calculate the longitudinal sound velocity parallel to the applied force using the formulation of Walton [14]. Once again assuming frictionless grains V =Z
1=3
φ
−1 = 6 1 = 6
P
sffiffiffiffiffiffiffiffiffiffiffiffiffi ε 4π2 ρg
ð3Þ
where ε is a function of the glass elastic moduli given by ε=
!1 = 3 3π2 Bð2B + CÞ 1 B =3 + C = 10 B
Sample
Mean crush strength (N)
Std dev (% of mean)
Mean velocity (m/s)
Std dev (% of mean)
1 2 3
333.1 257.1 238.5
27.3 29.5 29.9
1864 1792 1747
3.5 4.0 6.2
λg is the second Lame parameter of the glass. Eq. 2 or 3 provides a list of all the factors that influence sound speed in the EMT framework. These factors include the intrinsic physical properties of the particles, the external pressure, the packing fraction of the particles and its configuration. To measure the intrinsic properties of the particles, and then relate them to their strength, the packing configuration must be controlled, not just the packing density. To examine the packing behaviour measurements can be made of standard glass particles with known physical properties where Z, which represents the influence of the packing geometry of the glass, is the sole variable in the range of applicability of the EMT theory. Measurements of sound velocity in these conditions, while the granular material is compacting, can lead to knowledge of how geometry of the packing is evolving. If the theory is accurate the measurements can be used to calculate Z directly, if quantitative measurements are not possible, the velocity measurements can still be used to observe that the structure of the packing changes. Sound velocity measurements therefore have the potential to further elucidate the mechanisms of packing and the glassy behaviour of granular packing. 4. Experimental measurements The experimental program had three phases. In the first the ultrasonic velocity of iron ore pellets were measured. These pellets are approximately 1 cm diameter agglomerates and as such their strength could be tested individually in the standard fashion and compared to the sound velocity. In the second phase standard glass particles of various sizes are measured by the two procedures described in section 2. This is to confirm the predictions of the EMT model in our apparatus, observe the packing behaviour, and confirm that the sound velocity measurements are size independent below 200 μm, the size range of interest in alumina powders. In the third phase alumina powders are measured in the apparatus and the results compared to the standard industrial strength tests based on attrition methods. 4.1. Iron ore pellets The first industrial material measured was iron ore pellets. Three sets of iron ore pellets were produced so they would have different 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 0.00
!
where B =
Table 2 Iron ore pellet measurements.
Amplitude (V)
[13] have also observed sound propagation in vibrated granular beds. These measurements were made without measurement of the state of compaction and under no external pressure. In our experiments the compaction procedure is modified so a low pressure is continuously applied and sound transmission is possible through the granular assembly during its compaction. In these conditions higher values of Γ have to be used to attain compactions similar to the no pressure case. For long wavelength conditions Effective Medium Theories (EMTs) [14] have been developed to describe sound propagation through a medium of spheres where the wavelength is long enough for scattering from grains to have a negligible effect (usually greater than a factor of 10). In these theories the elastic constants of the medium are derived from a suitable average of the behaviour of the grain to grain contacts. They depend upon assuming Hertz-Mindlin [15] contact behaviour between the spheres and result in a scaling of the sound velocity to the 1/6th power of the applied pressure. Measurements have been made demonstrating the P1/6dependence velocity for assemblages of suitably small glass spheres where the applied pressure is below about 10 MPa [6], a much higher pressure than those used in our experiments. At very low pressures some authors report higher dependence of pressure approximately P1/4 attributing this result to the pressure change modifying the configuration of the packed state thus resulting in more contacts on average at higher pressure. The longitudinal sound speed (P-wave speed) V in an isotropic solid is given by
697
1 1 1 1 1 1 + − and C = 4π μg μg + λg 4π μg μg + λg
!
50.00 100.00 150.00 200.00 250.00 300.00 350.00 400.00 450.00
Time (microseconds) Fig. 3. A log/log plot of ultrasonic velocity against pressure for four types of glass spheres.
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P.J. Coghill, P. Giang / Powder Technology 208 (2011) 694–701
a
b
0.64
0.61
0.63 0.60 0.62 0.59
1mm
0.61
0.5mm
φ
φ
0.60
0.58
0.18mm
0.59 0.57 0.58
1mm 0.5mm
0.57 0.56 1000
0.56
0.18mm 10000
100000
0.55 1000
10000
100000
N
N
Fig. 4. Relative density as a function of tapping number N for confining pressures of (a) 0.15 bar and (b) 0.6 bar.
strengths. These strengths were then measured by standard crush tests [16] where many individual pellets are crushed one at a time. The pellets were approximately 10 mm in diameter and of slightly variable spherical shape. The pellet size was too large for measurement in bulk in our apparatus. Instead 2 thin rubber mats were placed on each transducer and the transmission measured at a pressure of 0.6 bar for one layer of 9 pellets using 500 kHz Krautkramer transducers. This measurement was repeated 5 times. This is a direct test of the relationship between sound speed and material strength in a composite material without requiring any intervening theory dealing with assemblies of particles. The sound speed measurements could be related directly to the “strength” as there was no appreciable change in density between the pellets. To obtain sound speed the transducer separation was measured, and the measured TOF was the difference between TOF with and without pellets at 0.6 bar. The results of the measurements are shown in Table 2. The standard deviation of the mean crush strength refers to the variation between individual measurements, not the expected error in the mean value. The sound speed results follow the mean crush strength results with a correlation coefficient of 0.94. These measurements provide evidence that for composite particles sound speed can be related to strength. The principal advantage of the sound measurements for real time evaluation of mean particle strength is the reduced variability (approximately a factor of 5 as shown in the last two columns of Table 2) and speed of the test measurement. Additionally, the ultrasonic measurement is non-destructive. 4.2. Glass spheres The first experiment conducted on the standard particles was to measure the variation of velocity with pressure using the second experimental method described in Section 2.2. The results for 1 mm, 0.5 mm, 0.181 mm narrow size range glass spheres and broadly distributed AH glass ballotini are shown in Fig. 3. In all cases the results show a P1/6 power law, within experimental error, as predicted by the EMT over the range of pressure used in our measurements. The EMT assumes uniformly sized spheres nevertheless the value of power index for the broadly distributed AH ballotini was compatible with a value of 0.167. 30 g of the 3 narrowly sized glass spheres were then subjected to tapping and measurement according to scheme 1. Fig. 4a) and b) show the evolution of density under tapping for the glass spheres at
0.6 and 0.15 bar. The 1 mm glass beads had an initial density of 0.56, the 0.5 mm spheres 0.57 and the 0.18 mm spheres 0.53 at 0.6 bar. The individual curves are an average of 3 measurements. For all the samples 100,000 taps was insufficient to produce a steady density at any pressure. In all cases the maximum density attained was below 0.64, the maximum random packing. The largest densities were achieved with the lightest constraining pressure. For the two coarsest materials the compaction scheme produced no significant change in the measured velocity, excluding the measurement made with no tapping at all. Table 3 shows the sound velocity and the equivalent co-ordination number Z for 1 mm and 0.5 mm spheres. It also shows the standard deviation of the Z measurement throughout the 100,000 tap procedure. Z is calculated by re-arranging Eq. 3 and substituting in the glass material properties and pressure values giving, for 0.6 bar of pressure and velocity in m/s, Z = ϕ1/2 (V/321)3. In the 0.181 mm glass spheres significant changes in velocity occurred during the tapping procedure. Instead of presenting the average velocity of several measurements three series of Z values are presented at two pressures in Fig. 5. The velocities behave differently from run to run and change by up to a third of their magnitude during the run. This occurs despite repeatable behaviour of the density. The relatively low average values of Z are most likely the result of mismatched physical property measurements in new ballotini material. All the grades of ballotini also showed the variability in velocity along with the steady densification and the P1/6 dependence of velocity, despite the EMT being derived for uniformly sized spheres. To test the size independence of velocity for fine powders the velocity was measured by the second scheme five times in each of the three ballotini grades. For AC, AE and AH the measured velocities were 405, 400, and Table 3 The measured velocities and calculated Z values for 1 mm and 0.55 mm glass spheres as they were densified by tapping in the first measurement scheme. Material type
Pressure (Bar)
Average velocity (m/s)
Average Z
St dev Of Z
1 mm
0.15 0.3 0.6 0.15 0.3 0.6
539 574 636 482 510 582
7.04 6.56 6.08 4.96 4.4 4.6
0.48 0.57 0.33 0.26 0.31 0.26
0.5 mm
P.J. Coghill, P. Giang / Powder Technology 208 (2011) 694–701
a
b
9 8
699
7
6
7 5 6 4
5
Z
Z 4
3
3 2 2 1 1 0 1000
100000
10000
0 1000
10000
N
100000
N
Fig. 5. Evolution of Z calculated using Eq. 3 as a function of tap number N for 0.18 mm glass spheres at pressures for (a) of 0.15 bars and for (b) of 0.6 bars.
424 m/s with an error of the mean of 15 m/s, consistent with the velocity being unaffected by particle size in this range.
The results are shown in Figs. 6 and 7 and summarised in Table 5 where the sign of the correlation co-efficients a and b is recorded.
4.3. Alumina powders 5. Discussion and summary
ð4Þ
AI = aV + bD
where a and b are unknown co-efficients determined by the correlation procedure. Table 4 The repeatability of ultrasonic velocity measurements in various powders shown as the % value of the measurement standard deviation divided by the average value. Material S.D./Av. % Description
SAC 3.0% Glass/narrow size range
AH 4.8% Glass/wide size range
Hydrate 7.9% Alumina/wide
Alumina 10.7% Alumina/wide/rough particles
The goal of the experimental work described in this paper was to provide an initial assessment of the feasibility of sound measurement in aggregates, evaluate the relevance of published theory, and develop criteria by which to judge the feasibility of any potential application, especially in alumina powders. Sound measurement was achieved in aggregates and powders for ratios of pressure to Young's modulus of 10− 6, small enough not to cause permanent deformation or damage to the particles. To avoid unacceptable scattering losses the ratio of the sound wavelength to the diameter of the particle had to be greater than 10. At this ratio sound was transmitted over 500 particle diameters without significant attenuation. This is a basic requirement in designing a feasible measurement system in any given material.
15
Malvern Coulter
AI by correlation
A set of uncalcined (hydrate) and calcined alumina samples was obtained for sound velocity measurements. The crystallisation process that forms the powders involves a great deal of agglomeration of individual crystal seeds as well as direct crystal growth. The calcined alumina samples were produced by calcining (heating) a portion of each of the uncalcined samples. This process drives off water from the particles and leads to more porous and friable particles. Each of these samples had the attrition index (AI) measured in another laboratory with the fraction less than 45 μm measured both by particle counting (Coulter) and optical diffraction (Malvern) instruments. As a preliminary step one alumina sample had the velocity measured as a function of pressure. This gave a value for the pressure index of 0.161 with a standard error of 0.016, again consistent with the value of 1/6th. The repeatability of velocity measurement by the second scheme was also investigated. Several of the test materials were measured 5 times and the results reported in Table 4 below. The measurement repeatability decreased progressively from narrow size range, smooth particles to wide size range, rough particles. Measurements of ultrasonic velocity V, and bulk density D were made according to scheme 2 for the total set of 12 alumina samples. These two values were then correlated with the A.I. measurements to determine whether there was any relationship in the simple linear relation shown in Eq. 4.
10
5
0 0
5
10
15
AI value Fig. 6. Comparison between the measured A.I. value for hydrated alumina and the value derived from ultrasonic velocity and bulk density measurements.
700
P.J. Coghill, P. Giang / Powder Technology 208 (2011) 694–701
35 Malvern
30
Coulter
AI by correlation
25 20 15 10 5 0 0
5
10
15
20
25
30
35
AI value Fig. 7. Comparison between the measured A.I. value for calcined alumina and the value derived from ultrasonic velocity and bulk density measurements.
The simplest theory that can be developed to describe sound propagation in powders under pressure is the EMT. It applies in principle to uniformly sized spheres where the sound wavelength is much greater than the particle size and the particles have a Hertzian contact force. In both Eqs. 2 and 3 it is clear that to measure the physical properties of the particles not only the packing density must be measured but the configuration of the packing must be controlled. This is expressed by the co-ordination number Z in the equations. The validity of the EMT formulation was tested first by checking materials for the P1/6 pressure dependence of the velocity. This was found to occur not only in uniformly sized spheres but spheres with broader size distributions and blocky material with broad distributions. This is not unexpected as the P1/6 dependence is calculated from the elastic deformation of the contact points between particles. Next the values of Z can be calculated from ultrasonic measurements in packings of glass spheres with known physical properties. For 1 mm and 0.5 mm spheres as the pack was densified by tapping there was no significant change in the velocity or calculated Z values. The lack of any significant change in velocity suggests that rather than re-organising the structure of the material significantly the tapping procedure shrank the pore size within the material. This corresponds to the results of Richards [9] in their tomographic study where the volume reduction was principally caused by reduction of pore volume. The calculated values of Z closely correspond with the value 6 expected for the densest random packing. This lends support to the validity of the EMT for describing these systems even before they reach a steady state of density. For smaller particles, both glass spheres of uniform size, broadly size distributed glass spheres, and broadly distributed blocky material, the velocity did not remain uniform under tapping. For the 0.181 mm ballotini spheres though the densification was uniform, the
Table 5 The comparison between the AI values and the ultrasonic data. Correlation
Correl'. co'eff
Velocity sign
Density sign
Velocity only CC
Density only CC
Malvern hydrate Coulter hydrate Malvern calcined Coulter calcined
0.99 0.77 0.77 0.95
+ + – –
– – – –
0.405 0.388 0.432 0.414
0.147 0.017 0.053 0.067
ultrasonic velocity varied by up to a third within experiments and changed from run to run. The amount of variation increased as the confining pressure decreased. As the dynamics of the point to point contacts seem unlikely to change the most reasonable assumption is that the configuration of the packing is changing more rapidly than for the larger spheres. The authors have no clear explanation why this should be so. What is clear is that simply tapping the powder for a fixed period of time and measuring the packing density is unable to produce stable velocity measurements and hence measurements of particle properties. For this reason the second measurement scheme was developed that involved tapping then cycling of the pressure ten times. This produced much improved repeatabilities in the velocity measurement of 3.0% for the 0.181 mm glass compared to variations of 30% with tapping alone. Having established a measurement procedure with acceptable repeatability the next step is to relate the velocity and bulk density measurements to the properties of the particles as measured by standard strength tests. The first demonstration in iron ore pellets was with a single layer of particles. This demonstrated that the measurement of ultrasonic velocity could be related to strength, as measured in a crush test, at least for this class of agglomerated particles. For powders where many layers of particles the bulk density and the ultrasonic velocity were used to correlate with the attrition index strength measurement. At least for alumina powders, which are irregular in shape unlike the glass spheres, the direct correction by theory for bulk density was insufficient to produce satisfactory results and correlation had to be used. The most interesting feature of these correlations was that the sign of the velocity term changes from before to after calcining for both particle size measurement techniques. The more cracked and porous calcined alumina being stronger with faster ultrasonic velocity, matching the iron ore pellets. This suggests that the uncalcined alumina is not able to be modelled as an agglomerate, but the calcined alumina, where the structure has been weakened maybe able to be modelled in this way. In all cases the bulk density only acts as a minor correction to the velocity term in the correlation. Ultrasonic velocity measurement is shown to be a feasible technique for assessing particle the strength in a class of industrially useful materials. These include iron ore pellets and alumina powders. To perform these measurements in powders careful attention must be paid to the packing state of the particles, especially for powders where the diameter was less than 0.2 mm. Even in packing of uniformly sized glass spheres the configuration of the packing could change relatively rapidly under tapping consolidation. The ultrasonic velocity technique has several advantages over standard techniques where it can be applied, these include ease of use, sample size and independence of the size of the tested particles. It does depend on an initial calibration to a direct strength measurement technique, though in cases, such as the example materials, where the composition of the measured powders effectively does not vary this is not a significant drawback. List of symbols V ultrasonic longitudinal velocity K bulk modulus P pressure Z average number of contacts per particle B,C elastic constants of convenience D bulk density kn elastic constant defined by kn = 4μg/(1 − νg) a,b calibration coefficients μ shear modulus ν Poisson's ratio ρ density ϕ packing fraction λ second Lame parameter
P.J. Coghill, P. Giang / Powder Technology 208 (2011) 694–701
ε Γ
elastic constant of convenience maximum acceleration caused by tapping
Acknowledgements The authors wish to acknowledge the assistance of M Schibeci for preparing the alumina samples and measuring their attrition indexes. References [1] V.J. Garcia, C.H. Schilling, Ultrasonic velocity and reduction of surface area during solid-state sintering, Mater. Sci. Eng. A265 (1998) 42–49. [2] K. Meyer, Pelletizing of Iron Ore, Springer Verlag, Berlin, 1980. [3] S. Chandrashekar, D. Jackson and J. Kisler, Alumina Fines' Journey from Cradle to Grave, 7th Alumina Quality Workshop, 2005. [4] C.B. Benrose, J. Bridgewater, A review of attrition and attrition test methods, Powder Technol. 49 (1987) 97–126. [5] P.J. Digby, The effective elastic moduli of porous granular rocks, J. Appl. Mech. 48 (1981) 803–808.
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[6] H.A. Makse, N. Gland, D.L. Johnson, L. Schwartz, Granular packings: nonlinear elasticity, sound propagation, and collective relaxation dynamics, Phys Rev. E. 70 (061302) (2004) 1–19. [7] J.B. Knight, C. G. Fandrich, C.N. Lau, H.M Jaeger and S.R. Nagel,Density relaxation in a vibrated granular material Phys. Rev. E 51 (1995) 3957–63. [8] P. Phillippe and Bideau Europhys. Lett. Compaction dynamics of a granular medium under vertical tapping 60 (2002) 677–83. [9] P. Richard, M Nicodemi, R. Delannay, P. Ribiere and D. Bideau Slow relaxation and compaction of granular systems Nat. Mat. 4 (2005)121–8. [10] T. Aste, M.Saadatfar, A. Sakellariou and T. Senden, Investigating the geometrical structure of disordered sphere packings Physica A. 339 (2004)16–23. [11] X. Jia, C. Caroli, B. Velicky, Ultrasound propagation in externally stressed granular media, Phys. Rev. Lett. 82 (9) (1999) 1863–1866. [12] P.J. Digby, The effective elastic moduli of porous granular rocks, J. App. Mech. 48 (1981) 803–808. [13] S.R. Hostler, C.E. Brennen, Pressure wave propagation in a granular bed, Phys. Rev. E. 72 (031303) (2005) 1–13. [14] K. Walton, The effective elastic moduli of a random packing of spheres, J. Mech. Phys. Solids 35 (2) (1987) 213–226. [15] R.D. Mindlin, Compliance of elastic bodies in contact, J. Appl. Mech. 71 (1949) 259–268. [16] ISO Standard 4700: Iron ore pellets for direct reduction feedstocks — determination of the crushing strength, 2007.
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Powder Technology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / p ow t e c
Ultrasonic velocity measurements in powders and their relationship to strength in particles formed by agglomeration P.J. Coghill ⁎, P. Giang CSIRO Process Science and Engineering, Lucas Heights, Australia
a r t i c l e
i n f o
Article history: Received 10 June 2010 Received in revised form 10 October 2010 Accepted 26 November 2010 Available online 27 January 2011 Keywords: Particle strength Powder Ultrasonic velocity Alumina
a b s t r a c t Many industries produce pellets, powders, and granules as finished or intermediate products. The strength of these products is important to their marketability, as dusting and breakdown from weak product is undesirable and results in poor handling properties. Measurement of “strength” usually takes the form of some empirical test involving crushing or attrition. The possibility of using sound velocity measurements to replace these techniques (especially for composite materials such as pellets or agglomerates) is investigated by performing measurements on glass sphere powders and test industrial materials. The ultrasonic velocity measurements in glass spheres were shown to be described by effective medium theories of sound propagation and can be used to probe the packing structure of the powder. For coarser powders sensible values of the co-ordination number Z could be calculated during compaction of the material. For powders finer than 0.2 mm the sound velocity measurements are no longer repeatable and rapid changes of the configuration may occur though the packing density only changes slowly. Two industrially important materials, iron ore pellets and calcined alumina powders were studied. It was found that reproducible measurements can be made in these materials that correlate to standard industrial “strength” measurements. In the case of iron ore pellets the correlation coefficient was 0.94 and the variability of the sound velocity based “strength” measurement was a factor of five smaller than the particle breakage method. For alumina powders the standard attrition index measurements could be correlated with the ultrasonic velocity and bulk density for both hydrated and calcined alumina powders with correlation coefficients of up to 0.95. The sound velocity strength measurement technique has significant advantages in repeatability, sample size and provides direct measurement of a physical parameter related to strength. Crown Copyright © 2011 Published by Elsevier B.V. All rights reserved.
1. Introduction Ultrasonic velocity measurements are relatively simple to make in bulk solids and can be related to the various elastic moduli, especially for isotropic solids. For these bulk solids the sound speed may be weakly related to the crush or abrasion strength of the material. In agglomerates, as apart from single crystals, where the material is held together through contacts there is reason to relate “strength” to the sound speed. A well described example is in sinters where the interparticle contact areas increase over time as the material is heated leading to an increase in sound velocity and strength during sintering [1]. Many industries produce pellets, powders, and granules as finished or intermediate products that are either grown directly by processes like crystallisation and spray drying, or agglomerated by pelletisation. As a rule the strength of these particles is important to their marketability, as dusting and breakdown from weak product is
⁎ Corresponding author. E-mail address: [email protected] (P.J. Coghill).
undesirable and results in poor handling properties. Often particle size is closely monitored but particle “strength”, which is harder to define and measure, is less closely monitored. The “strength” of these products is generally defined by the type of wear mechanism expected during the particle's life. For some products impact and crushing are the most important comminution mechanism, for others attrition may be more important. Where the wear due to impact and crushing is the important mechanism, strength is measured on a pellet by pellet basis from a small sub-sample; for example, pellets of iron ore are generally destroyed by crushing to test their strength [2]. An example where attrition wear is the important comminution mechanism is in the aluminium industry where vast quantities of alumina powder are produced by crystallisation and calcining to meet size as well as purity standards. The strength of these powders is consistently rated as one of the most important product quality parameters [3]. The current method used for measuring alumina and hydrate strength is a modification of the Forsythe-Hertwig attrition index (AI) method [4]. In this technique the −45 μm mass fraction is measured before and after treatment in a fluidised bed unit. The limitations of this method are: high sensitivity of the measured AI to the operating parameters of the fluid bed unit, and size dependence of
0032-5910/$ – see front matter. Crown Copyright © 2011 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2010.11.040
P.J. Coghill, P. Giang / Powder Technology 208 (2011) 694–701
AI. Significantly different products can have the same or similar measured AI values and in some cases a product with less desirable breakage and attrition properties can even have a lower AI. For both crushing and attrition wear the current measurement techniques are unsatisfactory in some aspects [4]. It is proposed in this work that the mean “strength” of the particles or pellets, in some cases, can be estimated ultrasonically by measuring the sound speed of a sample of powder or pellets placed under pressure. These measurements then can be compared to the standard “strength” measurements in use to determine whether in those particular cases the sound speed measurement is a feasible alternative. The sound measurement, if demonstrated to be feasible, may well have extensive practical advantages over the old techniques in terms of repeatability, ease of use, sample size, and having a direct relationship with a well known physical parameter. The two example materials chosen, iron ore pellets and alumina powder, are agglomerates. In this case the Young's Modulus and hence the sound speed of individual particles may be expected to be related to the strength, as is the case in sinters [1]. In other particles, especially those made of single crystals, the Young's Modulus and hence the sound speed in the particles is unlikely to be related to particle strength. For some larger particles, such as iron ore pellets, measurements of a single layer of particles may be sufficient. This is not the case for finer particles when measurements of sound velocity must take place over a significant distance and hence many layers of particles. In this case the sound velocity in powders is related to the packing of the particles as well as their intrinsic material properties. The theoretical framework of an effective medium theory (EMT) [5] is used to model the behaviour of sound transmission through compressed powders in this paper. In these theories the lumpy nature of the physical properties are smoothed to calculated values for an equivalent homogenous medium. The experiments are then set up so that the conditions match those required for an EMT to work. These include that the sound wavelength is greater than ten times the particle size to prevent scattering from becoming important, and that
695
the pressure is in a moderate range to prevent either crushing at the contacts or insufficient sound transmission [6]. As the sound transmission depends on both the properties of the particles and their configuration, care must be taken to understand the preparation of the sample for measurement. This includes shaking steps to consolidate the powder and prepare as uniform as possible configuration of the powders at measurement. Many investigations of the densification of glass balls have been carried out [7,8], principally to study the glassy or jammed dynamics. In these experiments the packing density ρ, defined by the ratio of grain volume to the total volume occupied by the packing, is measured as the system evolves under mechanical excitation. Following the notation of Richard [9], the minimal value of ρ for a mechanically stable random loose packing of spheres is ρRLP = 0.55, the maximal random packing fraction of spheres is ρRCP = 0.64, which is still significantly less than the maximal packing fraction obtained for a face centred cubic packing of ρMAX = 0.74. In tapping experiments the excitation level is defined by Γ the ratio of the maximum acceleration of the tapping to gravitational acceleration. In all cases the evolution of density proceeds slowly, in some cases a final state not being reached after 105 taps. Density does not constitute a full description of the state of the granular material and the evolution of the packing. Other experiments have observed the behaviour of the packing in more detail for example X-Ray tomography of particle arrays at various stages of compaction has also been undertaken [9,10]. These studies focus on structural properties such as pore size and pair distribution function. One study claims that densification under gentle agitation is mostly evident in the decrease of pore size rather than any change in the average difference between neighbouring particles. The approach taken in this paper is to use the EMT framework to examine experimental results from our apparatus for in the first instance uniformly sized glass spheres, then glass spheres with broader size distributions and finally the alumina powders. This will allow us to assess the importance of the various factors such as size
Fig. 1. Photograph and diagram of the experimental set up.
P.J. Coghill, P. Giang / Powder Technology 208 (2011) 694–701
distribution and shape to the behaviour of the industrial particles of interest and help devise a repeatable measurement procedure. Conversely the measurements of sound velocity will also provide insight to the compaction behaviour of the standard particles. 2. Experimental apparatus and procedure The apparatus for the sound velocity measurement is shown in Fig. 1. The cylindrical sample holder contains the receive transducer in its base. During measurement the sample holder is fixed in position on the apparatus. The powder to be measured is placed in the sample holder and a fixed pressure is applied to the sample by lowering the ram. Between the top of the sample and the ram is the transmit transducer which is pressed onto the sample when the ram is lowered. A sound pulse is then transmitted through the sample from top to bottom and the time of flight is measured. Fig. 1a shows schematics of transmit and receive transducer pairs used for longitudinal and shear measurements. The internal diameter of the sample holder is 60 mm. The external diameter of the top transducer piece (for both longitudinal and shear measurements) is 56 mm to avoid powder becoming stuck between the top transducer piece and the wall of the sample holder. If the powder becomes stuck in this way, then the full pressure is not applied to the powder being measured. For each measurement the weight of the powder is known and the distance between the two ultrasonic transducers measured so the bulk density can be calculated. 2.1. Ultrasonic transducers and pulsing equipment Longitudinal ultrasonic measurements were made with Krautkramer Steinkamp transducers of nominal 50 kHz resonance frequency. The top transducer was attached to the ram which was lowered into the sample holder to apply pressure to the sample being measured. A toneburst pulse of 1 cycle was used to excite the transducer at 48 kHz. The amplitude was adjusted to a maximum of 20 V PTP depending on the attenuation of the material. Transducer width for the longitudinal transducers was 40 mm, which equated to 0.44 of the sample's area having sound applied to it. The received ultrasonic signal was digitised by an A/D card at stored in the computer. A typical received pulse is shown in Fig. 2 for 30 g of 180 μm glass spheres. At this frequency the expected wavelength is between 15 and 20 mm, sufficient for the long wavelength assumption to apply to all samples except the iron ore pellets. 2.2. Measurement procedure The measurement procedure for a single powder sample was standardised on a range of materials to enable comparison of results. Pressures of 0.15, 0.3 or 0.6 bar are applied to the sample. These values were measured directly by a calibrated load cell. Two types of measurement were then made, the first being consolidation experiments where the various materials were compressed at a given pressure with the sound velocity and transducer separation measured periodically. The second measurement procedure was to obtain single measurements of sound velocity and density for a particular sample. In the first consolidation measurement scheme a program triggers horizontal tapping at 100 Hz with a maximum peak acceleration recorded by a calibrated accelerometer of 16.16 times gravitational acceleration. The tapping is applied horizontally unlike the typical compaction experiments where the tapping is vertical. After 20 s of vibration a measurement of sound transmission and compaction are recorded. Each of the 5 measurements is an average of 10 pulses. After each measurement period the 20 s of continuous vibration resumes automatically. The total time to complete the procedure for 1 sample is approximately 1 hour.
0.5 0.4 0.3
Amplitude (V)
696
0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 0.00
50.00 100.00 150.00 200.00 250.00 300.00 350.00 400.00 450.00
Time (microseconds) Fig. 2. Recorded signal of an ultrasonic wave passing through 30 g of SAC powder at 0.6 bar pressure.
The second type of experiment is used for single measurements from a sample. As will be demonstrated later simple compression is insufficient to create a reproducible state and cycling of the pressure to remove hysteresis is also required [6]. The sample is horizontally tapped for 15 min at 0.15 bar, then the pressure is manually cycled 10 times and a measurement of the velocity and density made at 0.3 bar. The velocity and compaction measurement is repeated 100 times and averaged. The measurement system determines time of flight (TOF) as the time elapsed to the first positive pressure peak above half the maximum amplitude of the signal (see Fig. 2). This technique requires consistently shaped waveforms to be accurate and has a finite value for zero delay. This offset was determined by measuring samples of the different materials at different weights and determining the delay projected at zero height from these measurements. Standard measurements were performed with 30 g of powder unless otherwise stated. 2.3. Materials tested Glass ballotini and glass spheres were used as the standard materials to perform tests of the EMT in this experimental apparatus. The glass ballotini has different size grades, with relatively broad size distributions for each grade. Some of the AC glass ballotini was sieved to produce a narrow distribution and called SAC glass. The values of the physical parameters used in the rest of the paper for the glass spheres and ballotini are shear modulus μg = 31 GPa, Poissons Ratio νg = 0.24 and density ρg = 2.5 g/cm3. All of the materials used are shown in Table 1. 3. Sound transmission through granular solids The transmission of an ultrasonic burst through granular material produces a ballistic pulse with a scattering induced coda following in its aftermath [11] as may be seen in Fig. 2. Sound propagation of this sort has been well studied due to its relevance to seismic wave propagation, sonar, and sound transmission in sinters [5,6,12]. Hostler and Brennen Table 1 Physical properties of materials tested. Material
D50 (μm)
Size distribution
Shape
AH glass AE glass AC glass SAC glass 0.5 mm glass 1 mm glass Iron ore pellets Alumina — hydrate Alumina — calcined
70 115 191 181 500 1000 10,000 100 100
Broad Broad Broad Narrow Narrow Narrow Broad Broad Broad
Spheres Spheres Spheres Spheres Spheres Spheres Rough Spheres blocky blocky
P.J. Coghill, P. Giang / Powder Technology 208 (2011) 694–701
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K + 4 = 3μ V= : ρ
ð1Þ
The two elastic moduli K the bulk modulus and μ the shear modulus can be calculated with an EMT theory when uniform pressure is applied. Using the formulation of Makse et al. [6] in an assembly of spheres under pressure P, the sound velocity can be calculated by substituting the values of the elastic constants for frictionless grains into Eq. 1 as
V =Z
1=3
φ
−1 = 6
6πP kn
1 = 6 sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3kn 20πρg
ð2Þ
where Z is the average number of contacts per particle, ϕ is the packing fraction of solids, kn = 4μg/(1 − νg), and νg is the Poisson's ratio of glass. The g subscript refers to the values of the physical constants for the glass beads used in the subsequent experiments. Corresponding to the maximal random packing density and maximal density for beds of spheres are the values of 6 and 14 for Z [6]. In the case of our experiment however the force is applied uniaxially in the direction that the sound is propagating. Under these conditions the powder is not isotropic so the bulk and shear moduli cannot be defined. The EMT framework can still be used to calculate the longitudinal sound velocity parallel to the applied force using the formulation of Walton [14]. Once again assuming frictionless grains V =Z
1=3
φ
−1 = 6 1 = 6
P
sffiffiffiffiffiffiffiffiffiffiffiffiffi ε 4π2 ρg
ð3Þ
where ε is a function of the glass elastic moduli given by ε=
!1 = 3 3π2 Bð2B + CÞ 1 B =3 + C = 10 B
Sample
Mean crush strength (N)
Std dev (% of mean)
Mean velocity (m/s)
Std dev (% of mean)
1 2 3
333.1 257.1 238.5
27.3 29.5 29.9
1864 1792 1747
3.5 4.0 6.2
λg is the second Lame parameter of the glass. Eq. 2 or 3 provides a list of all the factors that influence sound speed in the EMT framework. These factors include the intrinsic physical properties of the particles, the external pressure, the packing fraction of the particles and its configuration. To measure the intrinsic properties of the particles, and then relate them to their strength, the packing configuration must be controlled, not just the packing density. To examine the packing behaviour measurements can be made of standard glass particles with known physical properties where Z, which represents the influence of the packing geometry of the glass, is the sole variable in the range of applicability of the EMT theory. Measurements of sound velocity in these conditions, while the granular material is compacting, can lead to knowledge of how geometry of the packing is evolving. If the theory is accurate the measurements can be used to calculate Z directly, if quantitative measurements are not possible, the velocity measurements can still be used to observe that the structure of the packing changes. Sound velocity measurements therefore have the potential to further elucidate the mechanisms of packing and the glassy behaviour of granular packing. 4. Experimental measurements The experimental program had three phases. In the first the ultrasonic velocity of iron ore pellets were measured. These pellets are approximately 1 cm diameter agglomerates and as such their strength could be tested individually in the standard fashion and compared to the sound velocity. In the second phase standard glass particles of various sizes are measured by the two procedures described in section 2. This is to confirm the predictions of the EMT model in our apparatus, observe the packing behaviour, and confirm that the sound velocity measurements are size independent below 200 μm, the size range of interest in alumina powders. In the third phase alumina powders are measured in the apparatus and the results compared to the standard industrial strength tests based on attrition methods. 4.1. Iron ore pellets The first industrial material measured was iron ore pellets. Three sets of iron ore pellets were produced so they would have different 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 0.00
!
where B =
Table 2 Iron ore pellet measurements.
Amplitude (V)
[13] have also observed sound propagation in vibrated granular beds. These measurements were made without measurement of the state of compaction and under no external pressure. In our experiments the compaction procedure is modified so a low pressure is continuously applied and sound transmission is possible through the granular assembly during its compaction. In these conditions higher values of Γ have to be used to attain compactions similar to the no pressure case. For long wavelength conditions Effective Medium Theories (EMTs) [14] have been developed to describe sound propagation through a medium of spheres where the wavelength is long enough for scattering from grains to have a negligible effect (usually greater than a factor of 10). In these theories the elastic constants of the medium are derived from a suitable average of the behaviour of the grain to grain contacts. They depend upon assuming Hertz-Mindlin [15] contact behaviour between the spheres and result in a scaling of the sound velocity to the 1/6th power of the applied pressure. Measurements have been made demonstrating the P1/6dependence velocity for assemblages of suitably small glass spheres where the applied pressure is below about 10 MPa [6], a much higher pressure than those used in our experiments. At very low pressures some authors report higher dependence of pressure approximately P1/4 attributing this result to the pressure change modifying the configuration of the packed state thus resulting in more contacts on average at higher pressure. The longitudinal sound speed (P-wave speed) V in an isotropic solid is given by
697
1 1 1 1 1 1 + − and C = 4π μg μg + λg 4π μg μg + λg
!
50.00 100.00 150.00 200.00 250.00 300.00 350.00 400.00 450.00
Time (microseconds) Fig. 3. A log/log plot of ultrasonic velocity against pressure for four types of glass spheres.
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a
b
0.64
0.61
0.63 0.60 0.62 0.59
1mm
0.61
0.5mm
φ
φ
0.60
0.58
0.18mm
0.59 0.57 0.58
1mm 0.5mm
0.57 0.56 1000
0.56
0.18mm 10000
100000
0.55 1000
10000
100000
N
N
Fig. 4. Relative density as a function of tapping number N for confining pressures of (a) 0.15 bar and (b) 0.6 bar.
strengths. These strengths were then measured by standard crush tests [16] where many individual pellets are crushed one at a time. The pellets were approximately 10 mm in diameter and of slightly variable spherical shape. The pellet size was too large for measurement in bulk in our apparatus. Instead 2 thin rubber mats were placed on each transducer and the transmission measured at a pressure of 0.6 bar for one layer of 9 pellets using 500 kHz Krautkramer transducers. This measurement was repeated 5 times. This is a direct test of the relationship between sound speed and material strength in a composite material without requiring any intervening theory dealing with assemblies of particles. The sound speed measurements could be related directly to the “strength” as there was no appreciable change in density between the pellets. To obtain sound speed the transducer separation was measured, and the measured TOF was the difference between TOF with and without pellets at 0.6 bar. The results of the measurements are shown in Table 2. The standard deviation of the mean crush strength refers to the variation between individual measurements, not the expected error in the mean value. The sound speed results follow the mean crush strength results with a correlation coefficient of 0.94. These measurements provide evidence that for composite particles sound speed can be related to strength. The principal advantage of the sound measurements for real time evaluation of mean particle strength is the reduced variability (approximately a factor of 5 as shown in the last two columns of Table 2) and speed of the test measurement. Additionally, the ultrasonic measurement is non-destructive. 4.2. Glass spheres The first experiment conducted on the standard particles was to measure the variation of velocity with pressure using the second experimental method described in Section 2.2. The results for 1 mm, 0.5 mm, 0.181 mm narrow size range glass spheres and broadly distributed AH glass ballotini are shown in Fig. 3. In all cases the results show a P1/6 power law, within experimental error, as predicted by the EMT over the range of pressure used in our measurements. The EMT assumes uniformly sized spheres nevertheless the value of power index for the broadly distributed AH ballotini was compatible with a value of 0.167. 30 g of the 3 narrowly sized glass spheres were then subjected to tapping and measurement according to scheme 1. Fig. 4a) and b) show the evolution of density under tapping for the glass spheres at
0.6 and 0.15 bar. The 1 mm glass beads had an initial density of 0.56, the 0.5 mm spheres 0.57 and the 0.18 mm spheres 0.53 at 0.6 bar. The individual curves are an average of 3 measurements. For all the samples 100,000 taps was insufficient to produce a steady density at any pressure. In all cases the maximum density attained was below 0.64, the maximum random packing. The largest densities were achieved with the lightest constraining pressure. For the two coarsest materials the compaction scheme produced no significant change in the measured velocity, excluding the measurement made with no tapping at all. Table 3 shows the sound velocity and the equivalent co-ordination number Z for 1 mm and 0.5 mm spheres. It also shows the standard deviation of the Z measurement throughout the 100,000 tap procedure. Z is calculated by re-arranging Eq. 3 and substituting in the glass material properties and pressure values giving, for 0.6 bar of pressure and velocity in m/s, Z = ϕ1/2 (V/321)3. In the 0.181 mm glass spheres significant changes in velocity occurred during the tapping procedure. Instead of presenting the average velocity of several measurements three series of Z values are presented at two pressures in Fig. 5. The velocities behave differently from run to run and change by up to a third of their magnitude during the run. This occurs despite repeatable behaviour of the density. The relatively low average values of Z are most likely the result of mismatched physical property measurements in new ballotini material. All the grades of ballotini also showed the variability in velocity along with the steady densification and the P1/6 dependence of velocity, despite the EMT being derived for uniformly sized spheres. To test the size independence of velocity for fine powders the velocity was measured by the second scheme five times in each of the three ballotini grades. For AC, AE and AH the measured velocities were 405, 400, and Table 3 The measured velocities and calculated Z values for 1 mm and 0.55 mm glass spheres as they were densified by tapping in the first measurement scheme. Material type
Pressure (Bar)
Average velocity (m/s)
Average Z
St dev Of Z
1 mm
0.15 0.3 0.6 0.15 0.3 0.6
539 574 636 482 510 582
7.04 6.56 6.08 4.96 4.4 4.6
0.48 0.57 0.33 0.26 0.31 0.26
0.5 mm
P.J. Coghill, P. Giang / Powder Technology 208 (2011) 694–701
a
b
9 8
699
7
6
7 5 6 4
5
Z
Z 4
3
3 2 2 1 1 0 1000
100000
10000
0 1000
10000
N
100000
N
Fig. 5. Evolution of Z calculated using Eq. 3 as a function of tap number N for 0.18 mm glass spheres at pressures for (a) of 0.15 bars and for (b) of 0.6 bars.
424 m/s with an error of the mean of 15 m/s, consistent with the velocity being unaffected by particle size in this range.
The results are shown in Figs. 6 and 7 and summarised in Table 5 where the sign of the correlation co-efficients a and b is recorded.
4.3. Alumina powders 5. Discussion and summary
ð4Þ
AI = aV + bD
where a and b are unknown co-efficients determined by the correlation procedure. Table 4 The repeatability of ultrasonic velocity measurements in various powders shown as the % value of the measurement standard deviation divided by the average value. Material S.D./Av. % Description
SAC 3.0% Glass/narrow size range
AH 4.8% Glass/wide size range
Hydrate 7.9% Alumina/wide
Alumina 10.7% Alumina/wide/rough particles
The goal of the experimental work described in this paper was to provide an initial assessment of the feasibility of sound measurement in aggregates, evaluate the relevance of published theory, and develop criteria by which to judge the feasibility of any potential application, especially in alumina powders. Sound measurement was achieved in aggregates and powders for ratios of pressure to Young's modulus of 10− 6, small enough not to cause permanent deformation or damage to the particles. To avoid unacceptable scattering losses the ratio of the sound wavelength to the diameter of the particle had to be greater than 10. At this ratio sound was transmitted over 500 particle diameters without significant attenuation. This is a basic requirement in designing a feasible measurement system in any given material.
15
Malvern Coulter
AI by correlation
A set of uncalcined (hydrate) and calcined alumina samples was obtained for sound velocity measurements. The crystallisation process that forms the powders involves a great deal of agglomeration of individual crystal seeds as well as direct crystal growth. The calcined alumina samples were produced by calcining (heating) a portion of each of the uncalcined samples. This process drives off water from the particles and leads to more porous and friable particles. Each of these samples had the attrition index (AI) measured in another laboratory with the fraction less than 45 μm measured both by particle counting (Coulter) and optical diffraction (Malvern) instruments. As a preliminary step one alumina sample had the velocity measured as a function of pressure. This gave a value for the pressure index of 0.161 with a standard error of 0.016, again consistent with the value of 1/6th. The repeatability of velocity measurement by the second scheme was also investigated. Several of the test materials were measured 5 times and the results reported in Table 4 below. The measurement repeatability decreased progressively from narrow size range, smooth particles to wide size range, rough particles. Measurements of ultrasonic velocity V, and bulk density D were made according to scheme 2 for the total set of 12 alumina samples. These two values were then correlated with the A.I. measurements to determine whether there was any relationship in the simple linear relation shown in Eq. 4.
10
5
0 0
5
10
15
AI value Fig. 6. Comparison between the measured A.I. value for hydrated alumina and the value derived from ultrasonic velocity and bulk density measurements.
700
P.J. Coghill, P. Giang / Powder Technology 208 (2011) 694–701
35 Malvern
30
Coulter
AI by correlation
25 20 15 10 5 0 0
5
10
15
20
25
30
35
AI value Fig. 7. Comparison between the measured A.I. value for calcined alumina and the value derived from ultrasonic velocity and bulk density measurements.
The simplest theory that can be developed to describe sound propagation in powders under pressure is the EMT. It applies in principle to uniformly sized spheres where the sound wavelength is much greater than the particle size and the particles have a Hertzian contact force. In both Eqs. 2 and 3 it is clear that to measure the physical properties of the particles not only the packing density must be measured but the configuration of the packing must be controlled. This is expressed by the co-ordination number Z in the equations. The validity of the EMT formulation was tested first by checking materials for the P1/6 pressure dependence of the velocity. This was found to occur not only in uniformly sized spheres but spheres with broader size distributions and blocky material with broad distributions. This is not unexpected as the P1/6 dependence is calculated from the elastic deformation of the contact points between particles. Next the values of Z can be calculated from ultrasonic measurements in packings of glass spheres with known physical properties. For 1 mm and 0.5 mm spheres as the pack was densified by tapping there was no significant change in the velocity or calculated Z values. The lack of any significant change in velocity suggests that rather than re-organising the structure of the material significantly the tapping procedure shrank the pore size within the material. This corresponds to the results of Richards [9] in their tomographic study where the volume reduction was principally caused by reduction of pore volume. The calculated values of Z closely correspond with the value 6 expected for the densest random packing. This lends support to the validity of the EMT for describing these systems even before they reach a steady state of density. For smaller particles, both glass spheres of uniform size, broadly size distributed glass spheres, and broadly distributed blocky material, the velocity did not remain uniform under tapping. For the 0.181 mm ballotini spheres though the densification was uniform, the
Table 5 The comparison between the AI values and the ultrasonic data. Correlation
Correl'. co'eff
Velocity sign
Density sign
Velocity only CC
Density only CC
Malvern hydrate Coulter hydrate Malvern calcined Coulter calcined
0.99 0.77 0.77 0.95
+ + – –
– – – –
0.405 0.388 0.432 0.414
0.147 0.017 0.053 0.067
ultrasonic velocity varied by up to a third within experiments and changed from run to run. The amount of variation increased as the confining pressure decreased. As the dynamics of the point to point contacts seem unlikely to change the most reasonable assumption is that the configuration of the packing is changing more rapidly than for the larger spheres. The authors have no clear explanation why this should be so. What is clear is that simply tapping the powder for a fixed period of time and measuring the packing density is unable to produce stable velocity measurements and hence measurements of particle properties. For this reason the second measurement scheme was developed that involved tapping then cycling of the pressure ten times. This produced much improved repeatabilities in the velocity measurement of 3.0% for the 0.181 mm glass compared to variations of 30% with tapping alone. Having established a measurement procedure with acceptable repeatability the next step is to relate the velocity and bulk density measurements to the properties of the particles as measured by standard strength tests. The first demonstration in iron ore pellets was with a single layer of particles. This demonstrated that the measurement of ultrasonic velocity could be related to strength, as measured in a crush test, at least for this class of agglomerated particles. For powders where many layers of particles the bulk density and the ultrasonic velocity were used to correlate with the attrition index strength measurement. At least for alumina powders, which are irregular in shape unlike the glass spheres, the direct correction by theory for bulk density was insufficient to produce satisfactory results and correlation had to be used. The most interesting feature of these correlations was that the sign of the velocity term changes from before to after calcining for both particle size measurement techniques. The more cracked and porous calcined alumina being stronger with faster ultrasonic velocity, matching the iron ore pellets. This suggests that the uncalcined alumina is not able to be modelled as an agglomerate, but the calcined alumina, where the structure has been weakened maybe able to be modelled in this way. In all cases the bulk density only acts as a minor correction to the velocity term in the correlation. Ultrasonic velocity measurement is shown to be a feasible technique for assessing particle the strength in a class of industrially useful materials. These include iron ore pellets and alumina powders. To perform these measurements in powders careful attention must be paid to the packing state of the particles, especially for powders where the diameter was less than 0.2 mm. Even in packing of uniformly sized glass spheres the configuration of the packing could change relatively rapidly under tapping consolidation. The ultrasonic velocity technique has several advantages over standard techniques where it can be applied, these include ease of use, sample size and independence of the size of the tested particles. It does depend on an initial calibration to a direct strength measurement technique, though in cases, such as the example materials, where the composition of the measured powders effectively does not vary this is not a significant drawback. List of symbols V ultrasonic longitudinal velocity K bulk modulus P pressure Z average number of contacts per particle B,C elastic constants of convenience D bulk density kn elastic constant defined by kn = 4μg/(1 − νg) a,b calibration coefficients μ shear modulus ν Poisson's ratio ρ density ϕ packing fraction λ second Lame parameter
P.J. Coghill, P. Giang / Powder Technology 208 (2011) 694–701
ε Γ
elastic constant of convenience maximum acceleration caused by tapping
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