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Particles settling studies using ultrasonic techniques
Powder Technology 177 (2007) 102 – 111 www.elsevier.com/locate/powtec
Particles settling studies using ultrasonic techniques A. Shukla, A. Prakash ⁎, S. Rohani Department of Chemical and Biochemical Engineering, The University of Western Ontario, London, Ontario, Canada N6A 5B9 Received 14 August 2006; received in revised form 12 December 2006; accepted 7 February 2007 Available online 15 February 2007
Abstract Effects of acoustic velocity and attenuation measurements during settling of 43-, 110- and 168-μm glass beads in water are reported. Ultrasonic waves were generated at a frequency of 3.2 MHz. An abrupt increase in acoustic velocity and a sharp peak in the attenuation characterized the onset of settled bed. The observed attenuation peak at the transition between suspended and settled bed was attributed to dissipation caused by viscous absorption losses. The critical concentration at which increase in acoustic velocity and attenuation peak occurred was estimated for these particle sizes. © 2007 Elsevier B.V. All rights reserved. Keywords: Ultrasonic techniques; Acoustic velocity; Attenuation; Settling
1. Introduction Sedimentation and settling of particles in liquid is important in various chemical engineering operations such as fluidized beds, hydrosizers, thickeners, and hydraulic conveying. Fluidized beds are commonly used in physical, chemical and biochemical processes. These systems provide good heat transfer, mass transfer and mixing characteristics. An important requirement for such systems is to keep the particles in suspensions for achieving the desirable properties. Similarly, efficient suspension of particles is also desired in hydraulic conveying. On the other hand, hydrosizers rely on controlled settling for separating particles of different sizes and/or densities by utilizing the difference between their settling velocities. Thickeners rely on maximization of sediment rate for efficient operation. Hence, a real time measurement technique is required for the online detection and control of settling in industrial processes. Experimental techniques available for monitoring sedimentation can be broadly summarized in the following categories based on the underlying measurement principle [1]: 1. External radiation e.g. Neutrons, X-ray, γ-ray, microwaves
⁎ Corresponding author. Tel.: +1 519 661 2111x88528. E-mail address: [email protected] (A. Prakash). 0032-5910/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2007.02.003
2. Emitted radiation e.g. Radioactive tracers, NMR, magnetic tracers 3. Electrical properties e.g. Sedimentation potential, capacitance, conductance 4. Physical properties e.g. Sedimentation balance, pressure measurements 5. Direct methods e.g. Sampling, physical interruptions A comprehensive review of these measurement principles have been carried out by Williams et al. [1]. However, most of these techniques can have limited industrial application due to either stringent safety requirements (NMR, radioactive tracers, neutron, X-ray, γ-ray) or need for specific properties of solids under investigation (magnetic, electrical) or are highly intrusive. Direct measurement techniques require sampling and hence are not suitable for real-time online measurement. Measurement of physical properties such as pressure and viscosity can only be used to obtain macroscopic properties such as net solid flux. Measurement principles of techniques, which are non/less intrusive and capable of online application in dense suspensions, are discussed in brief. These include techniques based on the measurement of external radiation, electrical and physical properties. Measurement of electrical properties (capacitance/ permittivity, conductance) of the system has been extensively used for online monitoring of sedimentation in optically opaque and dense systems. The relationship between effective permittivity
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and particles concentration (ϕ) for spherical particles and negligible effect of system conductance is given by Eq. (1) [2]. 3 1=3 ee ¼ ð1−/Þef þ /e1=3 ð1Þ p
dense suspensions is demonstrated and the effects of concentration on measured parameters are evaluated. Furthermore, this technique has the ability to clearly detect the transition from suspension to settled bed.
where, εf, εp are the permittivity of liquid and solid phase and the effective permittivity εe can be obtained from the measured capacitance. In most systems, conductance affects the capacitance and hence has to be measured for accurate determination of effective capacitance. Measurement of total hydrostatic pressure is also a useful tool for analysis of sedimentation. The hydrostatic pressure at a given height is the sum of pressure due to the fluid and particles above the measurement location and is given by Eq. (2) [2].
2. Experimental details
Pm ¼ qmd gd z
ð2Þ
where ρm is the effective mass density given by Eq. (3) and z is the height of suspension above the measurement location. qm ¼ ð1−/Þqf þ /qp
ð3Þ
Particle concentration can be derived from the measured hydrostatic pressure at a given height and time by combining Eqs. (2) and (3). 1 Pm ðz; tÞ /ð z; t Þ ¼ −qf ð4Þ ðqp −qf Þ gz External radiation methods are based on the monitoring of a beam of radiation, which has passed through the suspension and can be used in dense and optically opaque suspensions. The energy removed from the incident beam is proportional to the particle size and their concentrations. Hence, the energy loss can be used to determine the volume fraction and size of the constituent particles. For a polydisperse suspension this relationship can be expressed using Eq. (5) [1]. I ¼ I0 e
−x
n P i¼1
a i /i
ð5Þ
Ultrasonic technique can be categorized under external radiation measurement principle but is free from radiation safety requirements. Like other radiation based techniques, it can operate in dense and optically opaque suspensions and can measure particle size and concentration. This technique provides an added advantage over other X-ray and γ-ray techniques as both energy loss and acoustic velocity of the pulse can be used to characterize settling. These parameters are dependent on the propagation regime of the acoustic pulse. The propagation regimes are a function of the wave number (kr), which is a ratio of particle size and the wavelength of the acoustic pulse. While describing the ultrasonic propagation mechanisms during polydisperse sedimentation, the dynamic changes in particle size and concentration have to be accounted for. In this study the response of acoustic velocity and attenuation during settling in different propagation regimes are studied for various practical applications. Its applicability in
Measurements of acoustic velocity and attenuation were conducted to study the effects of particle settling and sedimentation on acoustic properties in liquid–solid systems. A schematic diagram of the experimental setup and data acquisition system is shown in Fig. 1. The solid phase used in the experiments was glass beads of 43-, 110- and 168-μm mean diameter and the suspension medium was distilled water. Particles were suspended in plexiglass column of 0.102 m inside diameter and 1.02 m height by introducing air through a porous polyethylene distributor plate with 35 μm pore size (Fig. 2). Settling experiments were also conducted for mixed particle system with equal proportions of 43- and 168-μm particles. The suspended particles were allowed to settle by shutting off the gas flow and measurement of acoustic properties was obtained at a height of 0.17 m above the distributor. The total suspension height was 0.528 m and the concentration of solids in the suspended slurry was 28 vol.%. Measurements of acoustic velocity and attenuation during the settling of 43- and 110-μm particles were also conducted in a stirred tank to investigate the effect of slurry height on settling characteristics of the suspension. The initial slurry concentration of particles was 36 vol. % and distilled water was the suspension medium. A schematic representation of the stirred tank used to carry out the ultrasonic measurements is shown in Fig. 3. The stirred tank is made of plexiglass with diameter 0.1 m (4ʺ) and height 0.252 m (10ʺ). It is equipped with radial ports for mounting the transducers and a vertical stirrer for suspending the particles. The operating temperature for experiments in this study varied between 21 and 25 °C. The change of acoustic velocity in water for this temperature range is 2.58 m/s/°C [3]. The variations in measured acoustic velocity are reported as (Δv = Vsuspension − Vwater) at the operating temperature. Hence, the effect of temperature on the measurements is normalized and the reported change in acoustic velocities is comparable for different experiments. The change due to temperature during settling for a given particle sample was negligibly small (≤0.78 m/s) and the error introduced in measured Δv was less than 0.64%. Acoustic parameters were measured using an ultrasonic pulser/receiver unit (Fallon Ultrasonic Inc.) at 3.2 MHz
Fig. 1. Block diagram of experimental setup.
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Fig. 2. Schematic diagram of measurement column.
frequency in through transmission mode. Transducer separation used during this study was 25 mm and 50 mm for the column and stirred tank respectively. This arrangement of transducers is
intrusive in nature but was preferred to maximize signal strength. Due to the intrusive nature of measurements used in this study the transducers were enclosed in a cylindrical housing
Fig. 3. Schematic diagram of stirred tank.
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to minimize resistance to the flow of slurry and bubbles. Small ratios of the particle size to the distance between transducers also reduced distortions to particles motion in the column. A TDS 210 (Tektronix) digital oscilloscope was used to visualize and select the ultrasonic pulse. The pulser/receiver unit activates the receiving transducer at a time fixed by the delay and for duration of the gate width to measure the flight time of the signal. The pulses repetition rate was 1 kHz and the sampling interval was 1 s. Hence each sampled value of transit time and amplitude represent the average of 1000 acoustic pulses. 3. Results and discussions Fig. 4 shows changes in measured acoustic velocity (Δv = Vsuspension − Vwater) during settling of 43-μm particles in the column and the stirred tank. Acoustic velocity measurements in the column resemble an ‘S’ shape and can be divided in four sections based on the settling characteristics of the suspension. In section-I, acoustic velocity measurements do not change with time. Measurements showed a gradual increase in section-II and are followed by an abrupt increase of 110 m/s in section-III before attaining a stable value in section-IV. Acoustic velocity in a two-phase suspension is governed by the particle size, concentration, physical parameters of the constituent phases (density and compressibility) and the frequency of the acoustic pulse [4–6]. Experimental measurements in liquid–solid systems have shown that the acoustic velocity increases with increase in concentration and decrease in particle size [4–9]. Hence, constant acoustic velocity in section-I indicates no change in the average particle size and concentration at the measurement height. The gradual increase in the measured acoustic velocity in section-II could be attributed to the change in either one or both of the above parameters. The average particle size at the measurement location could decrease due to particle segregation or/and the concentration could increase due to formation of a dense layer. Attenuation measurements during this phase of settling can be used to identify the dominant effect and is discussed in the next section. The abrupt increase of acoustic velocity in section-III was observed when the bed reached the transducer height and can be attributed to the high
Fig. 4. Change in acoustic velocity during settling of 43-μm particles in column and stirred tank.
Fig. 5. Particle size distribution of 43-, 110- and 168-μm particles using Malvern Mastersizer®.
concentration of particles in the settled bed. This is followed by a gradual increase in acoustic velocity over time in section-IV and can be attributed to bed compaction (hence further increase in concentration) due to the weight of particles and fluid above the measurement location. Measured acoustic velocity in the stirred tank was higher as the initial particle concentration was 8 vol.% more than that in the column. However, the trends in acoustic velocity change were similar for both systems. The measurements show an initial region of constant concentration followed by a gradual increase in acoustic velocity. The onset of settled bed was also characterized by an abrupt increase in acoustic velocity. However, the settling duration in the stirred tank was much shorter than the column as the transducers were located only 0.07 m above the bottom. Furthermore, gradual increase in acoustic velocity was also much smaller in magnitude than the measurements made in the column. This indicates that the extent of particle segregation in the stirred tank is much smaller and is attributable to the shorter settling height (0.115 m) compared to the column (0.526 m). The acoustic velocity measurement in the settled bed of the stirred tank was higher than that of the column. This could be attributed to lower concentration of particles in the settled bed of the column due to segregation, if the average particle size is less than 30-μm [10]. Fig. 5 shows that 30% of particles are smaller than 34-μm in the size distribution of the 43-μm particle sample. Since the transducer location is about 33% below that settled bed height, it is likely that the average particle size at this location was less than 30-μm and hence the concentration was lower. Fig. 6 shows the acoustic velocity measurements during the settling of 110-μm particles in the column and in the stirred tank. The relative trends during settling in the column and in the stirred tank for 110-μm particles were similar to those observed for 43-μm particles. The magnitude of increase in acoustic velocity during settling in the column before the settled bed reached the transducer was more than the stirred tank. This is because the settling duration and the extent of segregation in the stirred tank was small compared to the column and can be attributed to the small settling height in the stirred tank (Fig. 3). The acoustic velocity measurements in the settled bed of the stirred tank were higher than that of the column. Unlike 43-μm
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Fig. 6. Change in acoustic velocity during settling of 110-μm particles in column and stirred tank.
particle suspension, this deviation cannot be attributed to the presence of particles smaller than 30-μm, as this size range is not present in the 110-μm samples. An alternative explanation for this discrepancy could be the lack of particle segregation in the stirred tank. The measured size distribution (Malvern, Mastersizer®) of 110-μm particles used in this study was lognormal in nature and has a standard deviation of 0.498. Lognormally distributed particles show higher concentrations in settled beds for standard deviations greater than 0.15 [11–13]. These studies show that the concentration increases by 6 vol.% for increase in particle size standard deviation from 0.15 to 0.5. The size distribution of particles in the column is expected to be more uniform due to segregation. The transducers location in the settled bed is 33% below the maximum bed height and hence should contain particle in the lower 30% (70–90 μm) of the size distribution of 110-μm (Fig. 5). However, the increase in magnitude of acoustic velocity due to the decrease in average particles size is less than 2 m/s for particles of similar sizes [8]. Hence, higher acoustic velocity measurements in settled bed of the stirred tank may be attributed to increase in concentration due to higher particle size standard deviation for 110-μm particles. The standard deviation of 43-μm particles was 0.533 and could be contributing to high acoustic velocity measurements in the stirred tank along with reason already discussed. Fig. 7 compares the change in measured acoustic velocity during the settling of 110 and 168-μm particles in the column. Unlike 110- and 43-μm particles, 168-μm particles show an intermediate step in the S-shaped curve. It should also be observed that the region of constant concentration for 168-μm particles was small. This can be attributed to higher settling velocities of larger particles. The abrupt increase of 37 m/s in acoustic velocity during the intermediate step cannot be attributed to a decrease in the average particle size, as this effect is relatively small [9]. During the experiments it was observed that an interface between the slurry and clear water was formed almost immediately after the gas was shut off. The location of this interface was calculated to be 0.39 m after 20 s of settling and would result in a 10 vol.% increase in the concentration. Acoustic velocity measurements at 38 vol.% for similar particle sizes are comparable to the measurements made after 20 s during the settling of 168-μm particles. Hence, the
first abrupt increase in acoustic velocity measurements appears to be due to the formation of a dense layer in the suspension. The oscillation of measured acoustic velocity during this increase could be attributed to the transient disturbances of the dense layer interface moving through the measurement volume. This was followed by a gradual increase in acoustic velocity due to hindered settling effects till the settled bed reached the transducer. Fig. 7 shows that the measured acoustic velocities in the settled beds of 110- and 168-μm particles were similar. However, acoustic velocity decreases with increase in particle size and has been reported in literature for suspensions of varying concentrations [5,7,9]. Higher acoustic velocity in the settled beds of 43-μm particles (Fig. 4) as compared to 110and 168-μm (Fig. 7) is in agreement with the trends observed for particles in suspension. Similar acoustic velocity measurement for 110- and 168-μm particles indicates that the concentration in the settled bed of 168-μm particles is higher. As discussed earlier, increase in standard deviation of lognormally distributed particles from 0.15 to 0.5 results in a 6 vol. % increase in concentration [11–13]. The initial standard deviation of 110- and 168-μm particles was 0.498 and 0.464 respectively. The size distribution of 110-μm particles in the column is expected to be more uniform as its settling time is twice that of 168-μm particles. Hence, similar acoustic velocity measurements observed in Fig. 7 can be attributed to higher concentration in the settled bed of 168-μm particles due to smaller settling time. The time required by the settled bed to reach the transducer height can be theoretically calculated using the settling velocities and compared with the time at which the abrupt increase in acoustic velocity was observed using Eqs. (6) and (7). Ubed ¼
Hbed UTorH Hsus −Hbed
b;Tr TTorH ¼
Hbed Ubed
ð6Þ
ð7Þ
The maximum settling velocity of a single particle in the fluid is called the terminal settling velocity. The settling
Fig. 7. Comparison of change in acoustic velocity during settling of 110- and 168-μm particle suspension in column.
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criterion K (Eq. (8)) determines whether the settling conditions are within or outside the Stoke's limit [14]. ! gqf ðqp −qf Þ 1 3 K¼d ð8Þ l2f Depending on the settling criterion appropriate equations were used to calculate the settling velocities of 43-, 110- and 168-μm particles. The drag coefficients for particles with Reynolds number greater than one were estimated using the relationship given by Clift et al. [15]. The terminal settling velocities are however reduced due to the fluid up flow generated by neighboring particles [16]. The reduced velocity is called the hindered settling velocity and is a function of particle concentration. The Richardson and Zaki [17] relationship was used to obtain the hindered settling velocity for the average particle size in the suspensions (Eq. (9)). UH ¼ UT ð1−/Þn
ð9Þ
Here, UT is the terminal settling velocity and n is the empirical settling exponent, which is a function of particle size, column diameter and Reynolds number of the particles. Table 1 shows the bed rise time calculated using the terminal and hindered settling velocities. Hindered settling velocity based calculations for bed rise time showed good agreement with the measurements. The small deviations between measured and calculated bed rise times can be attributed to polydispersity of the particles (Fig. 5). Experimental measurement of acoustic velocity was also conducted in a mixed suspension of 43- and 168-μm particles containing equal volume fractions of each particle size (Fig. 8). The acoustic velocity shows three abrupt increases during settling. The first and second increases in the acoustic velocity indicate the settling of two large particle fractions below the measurement volume. The first increase occurs due to dense layer formation by 168-μm particles. This dense layer is a result of very high settling velocities of these particles compared to 43-μm particles (Table 1). This was followed by the second increase when all the particles in the 168-μm sample have settled. However, the settling of 168-μm particles in the mixed particle system was slow compared to the single particle species suspension and could be attributed to hindered settling effect caused by the presence of 43-μm particles. The third increase was observed when the settled bed reached the transducer height. Schneider et al. [18] developed a model to determine the interface velocities during the sediment of a bi-disperse suspension. They use the drift flux concept [19] to obtain the
Table 1 Comparison of measured and calculated bed rise times for 43-, 110- and 168-μm particles using terminal and hindered settling velocities d (μm) UT (m s− 1) Rep 43 110 168
0.0017 0.0105 0.0197
n
b,Tr UH (m s− 1) TTb,Tr (s) THb,Tr (s) TUS (s)
0.08 4.66 0.00037 1.3 4.35 0.00252 3.71 3.92 0.00544
110 17 9
525 76.8 35.5
534 87 38
Fig. 8. Change in acoustic velocity and attenuation during settling of 43- and 168-μm mixed particle suspension in column.
volume flux densities of the two components to obtain a relationship between sediment and suspension composition (Eq. (10)). In this equation ‘ϕL’ and ‘ϕS’ are the suspension concentration for the smaller and larger particle size just above the sediment–suspension interface. Composition of sediment constituting of two different particle sizes (Eq. (11)) is a function of their diameters ‘dS/dL’ and was obtained by a theoretical investigation of the limiting case ‘dS/dL→0’ by Jeschar [20]. Eqs. (10) and (11) can be iteratively solved to obtain the sediment composition for a given suspension composition. ½1−/L ð1 þ F Þ/SV−½ F−/S ð1 þ F Þ/LV ¼ /S −/L F where;
F¼
ð10Þ
/S dS 2 /L dL
/SV ¼ ð1−eb Þð1−/LVÞ−eb /LVðdS =dL Þm ;
0V/LVV1−eb
ð11Þ
where;
p/SV m ¼ 0:6sin /SVþ /LV
The suspension and sediment concentrations of the two particle species can be used to obtain the propagation velocities of different interfaces. The sediment–suspension interface velocity for large particle species is given by Eq. (12). mL;sed ¼ where;
jL /LV−/L
ð12Þ
jL ¼ /L ð1−/L ÞUH;L −/L /S UH;S
In the above equation ‘jL’ denotes the volumetric flux densities of the constituent particle sizes. Similar equations can be used to obtain the other interfacial propagation velocities. It should be noted that this technique is based on the assumption of absence of continuous transition between concentrations at the various interfaces and is limited to bi-disperse suspensions. Numerically intensive models such as that of Bürger et al. [21] maybe used to overcome these drawbacks. However, location of
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these interfaces calculated using Schneider et al. [18] model shows good agreement with experimental observations and numerical simulations of Bürger et al. [21]. For this study the interface locations predicted by the Schneider et al. [18] model for a bidisperse suspension of 168- and 43-μm particles are compared with the ultrasonic measurements. The interface position can be plotted on a space–time diagram to obtain the sediment and suspension concentration as a function of sediment height and time. Fig. 9 shows the concentration variation along the bed height with settling time for a twoparticle suspension containing 43- and 168-μm particles. The y-axis is the settling height ‘z’ and the x-axis is settling height divided by the terminal settling velocity of the smallest particle. The acoustic velocity measurements during the settling of mixed particle suspension and the measurement height are shown on the secondary y-axis of the figure. Section A of the plot shows that a constant concentration region exists between suspended and fixed bed. This is in agreement with the experimental measurements, as the acoustic velocity does not show any change in this section. Section B consists of intermediate concentration of small particles between fixed and settled bed. Acoustic velocity in this section is initially constant and then shows a gradual increase before the suspension–sediment interface (between section B and D) is reached. Section C consists of the settled bed with particles of both size fractions and does not affect acoustic velocity as this section lies below the measurement location. Section D consists of settled bed with only small particles as the larger fraction has already settled. The model predicted interface between sections A–B and B–D is in good agreement with the time at which abrupt increase in the acoustic velocity were measured. The first interface (between 43- and 168-μm particles) predicted by the model lies between the first and second abrupt increases in the acoustic velocity. The second interface (between 43-μm particle suspension and sediment) occurs about 30 s after the third abrupt increase in acoustic velocity. Section E is the clear liquid region after all the particles have settled and does not affect the measurements as this region lies above the transducer location. Deviation between measured and predicted interfaces could be attributed
Fig. 9. Simulation of bi-disperse suspension (43- and 168-μm) using the Schneider et al. [18] model and comparison with change in acoustic velocity measurements.
Fig. 10. Change in attenuation during settling of 43-μm particles in column and stirred tank.
to the polydispersity of the particles. The gradual change in acoustic velocity before the transition from suspended to settled bed of small particles could be attributed to concentration gradient caused due to hindered settling [21]. 3.1. Attenuation measurements during settling Fig. 10 shows the measured attenuation of the ultrasonic pulse in the column and stirred tank during settling of 43-μm particles. Attenuation measurements in the column show a slight increase (3 Np/m) during the initial phase of settling (Section-Ia). The increase in attenuation can be attributed to removal of a few small bubbles entrapped in the suspension. Once the small bubbles escape, the attenuation measurements remain constant up to 225 s (Section-Ib). This is followed by a decrease in attenuation up to 500 s (Section-II) and can be attributed to the decrease in average particle size due to segregation. The transition from suspended to settled bed is characterized by a peak in the attenuation (Section-III). The secondary attenuation peak observed just before the settled bed is reached could be attributed to a non-uniform settling due to the presence of transducers. After the bed has settled, measurements in the column gradually increase before stabilizing (section-IV). This increase can be attributed to the effect of bed compaction due to the height of settled bed above the measurement location (80 mm). Compaction causes a decrease in the porosity of the settled bed and hence an increase in concentration. Attenuation measurements in the stirred tank do not show an initial increase as the mode of suspension was mechanical and no fine bubbles where entrapped (section-Ia and Ib). No decrease in the attenuation measurements in the stirred tank, before the settled bed reaches the transducer, indicates the absence of particle segregation (section-II). This can be attributed to the small settling height in the stirred tank. Similar to the measurements made in the column, the transition between suspended and settled bed was characterized by a peak in the attenuation (section-III). The reason for this attenuation peak is discussed in the section on viscous losses in concentrated systems. In section-IV the attenuation measurements in the column gradually increase by 18 Np/m to achieve values similar to that of the column. A simultaneous increase in acoustic
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velocity (by 6 m/s) during this phase of settling (Fig. 4) in the column indicates an increase in concentration of the settled bed and can be attributed to compaction. Absence of compaction in the stirred tank is due to the small height of settled bed over the measurement location. Fig. 11 shows the acoustic velocity and attenuation measurements during the settling of 110-μm particles in the column and in the stirred tank. A peak in attenuation measurements accompanied the abrupt increase in acoustic velocity. Similar observations were also made during the settling of 168-μm particles (Fig. 12). However, acoustic velocity measurement during the settling of 168-μm particles also shows an intermediate abrupt increase due to an increase in concentration by 10 vol.% (Fig. 7). This increase was not captured by attenuation measurements (Fig. 12) and can be attributed to relatively small effect of concentration change on attenuation in dense suspensions of this particle size [9]. The effect of slurry height on suspension composition during settling can be observed in the attenuation measurements for 110-μm particles in column and in the stirred tank. Absence of particle segregation in the stirred tank is indicated by the constant attenuation measurements during settling. Furthermore attenuation measurements in settled bed of the stirred tank were higher than that of the column and can be attributed to higher concentration. As discussed in the velocity section this increase in concentration is a result of higher particle size standard deviation in the absence of segregation. However, unlike 43-μm particles, there was no increase in attenuation measurements in the column after the settled bed has reached the transducer. This indicates that there is no significant compaction in beds of larger particle. These results show that ultrasonic techniques can be used to monitor suspension composition during settling and can be used to control the performance of hydrosizers. 3.2. Viscous dissipation in highly concentrated suspensions The transition from suspended to settled bed shows a peak in the attenuation measurements. The increase in attenuation could be attributed to increase in energy loss due to high particle concentration in the bed. However, the subsequent drop in attenuation requires a more in-depth analysis of the various
Fig. 11. Change in attenuation and acoustic velocity during settling of 110-μm particles in column and stirred tank.
109
Fig. 12. Change in attenuation and acoustic velocity during settling of 168-μm particles in column.
energy loss mechanisms and their effects. Energy loss in a suspension is a function of the particle size, particle concentration and frequency of the acoustic pulse. The various energy loss mechanisms during wave propagation can be attributed to viscous, thermal, scattering, intrinsic and structural dissipation effects. For a glass beads–water suspension the thermal and intrinsic effects are negligible. Thermal effects are only dominant for soft particles with size smaller than 1-μm and low-density difference with the suspending medium. Glass beads are rigid, have high-density contrast compared to water and the size used in this study was much larger than 1-μm. Intrinsic attenuation for water has been shown as negligible and independent of frequency in the range of 3 to 100 MHz [22]. Significant role of structural losses in settled beds of large particles is unlikely. Its use even for particles in the colloidal range is only justified when, other mechanisms fail to provide a suitable explanation for experimental measurements [22]. Viscous and scattering effects will play a dominant role during the settling of suspensions and in the settled bed of the particle studied. Measurements made during this study showed that the significant increase in attenuation occurs before the sharp increase in acoustic velocity (Figs. 11 and 12). The increase in acoustic velocity coincided with a decrease in attenuation. This was followed by nearly constant measurements of both these parameters for 110- and 168-μm particles, indicating that the bed has settled. For 43-μm particles the abrupt changes in the measured attenuation was followed by a gradual increase and can be attributed to increase in concentration due to compaction as discussed in the previous section. These observations indicate that an attenuation mechanism comes in to existence near the settled bed concentrations. It causes maximum dissipation just before the bed settles and the mechanism ceases to exist in the settled as indicated by the drop in attenuation. Scattering can account for the increase in attenuation just before sediment formation. This mode of attenuation is caused by the deflection of a part of the energy in the incident beam and is proportional to the particle size and concentration. However, it cannot explain attenuation drop in the settled bed, which is more concentrated than the suspension. The attenuation peak in the vicinity of settled bed concentration could be explained by viscous dissipation. Viscous losses are caused by the oscillation
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of particles due to the pressure gradient generated by the acoustic pulse. The oscillations have a shearing effect on the fluid surrounding the particles hence leading to the formation of shear waves. Propagation of shear waves leads to energy loss, as these waves can only exist over short distances before being completely dissipated. The distance of shear wave propagation is known as the viscous layer thickness and can be calculated using the equation given below [22]. sffiffiffiffiffiffiffiffi 2lf dv ¼ ð13Þ xqf For dilute suspensions the energy dissipation due to this mechanism is maximum when the viscous layer thickness becomes comparable to the particle radius. However, in concentrated suspensions, the inter-particle separation overtakes particle radius as the deciding parameter for maximum viscous dissipation [22]. Hence, viscous dissipation due to concentration comes into existence when the inter-particle distances are comparable to the viscous layer thickness and is independent of particle size (Eq. (14)). sffiffiffiffiffiffiffiffi 2lf lcr cdv ¼ ð14Þ xqf With further increase in concentration the inter-particle distances become less than the viscous layer depth and begins to inhibit the relative oscillatory motion between the particles and fluid. This leads to a decrease in attenuation till the settled bed is reached where greater numbers of adjacent particles are in contact and no significant relative oscillation is possible. For 3.2 MHz frequency the critical inter-particle distance at which viscous energy dissipation will be maximum is 0.3-μm and should be achieved near the settled bed concentration. Eq. (15) can be used to calculate the settled bed concentration assuming mono-sized particles. The measured fixed bed height for all particle sizes was 0.252 m and corresponds to a concentration of 57.5 vol.%. /V ¼
M Hbed pr2 qp
ð15Þ
The actual concentration of the settled bed in the current study is expected to be higher than 57.5 vol.% due to polydispersity of the particles. Random close packing of spherical particles has a maximum concentration of 63.5 vol.% and can be safely assumed as the packing structure of the sediment. Using this packing structure the critical inter-particle distance is achieved at 60.9, 62.5 and 62.8 vol.% for 43-, 110- and 168-μm respectively. 4. Conclusions Ultrasonic measurements captured the change in average particle size due to segregation during settling. The ultrasonic parameters could clearly distinguish between suspended and settled bed. Acoustic velocity showed a sharp increase at the transition between settled and suspended bed. Attenuation
measurements showed a peak just before the settled bed reached the transducer location and this was attributed to the increase in viscous dissipation effects. The critical concentration at which these losses became prominent was calculated to be above 60 vol.% for the particles investigated. In the settled bed this technique was able to detect the effect of particle size standard deviations and the existence of bed compaction. These results show that ultrasonic techniques can detect transition from suspended to settled bed and has potential for online monitoring of suspension composition. Nomenclature D Column diameter (m) d Particle diameter (m, μm) dL Large particle diameter (m) dS Small particle diameter (m) g Acceleration due to gravity (m s− 2) Hsus Height of suspension (m) Hbed Height of bed (m) Δh Distance between suspension height and fixed bed height (m) I Incident intensity (W m− 2) I0 Received intensity (W m− 2) i = 1…n Particle size index jL Volume flux density of larger particles (m s− 1) K Settling criterion kr Non-dimensional wavenumber lcr Inter-particle distance for maximum viscous dissipation (m) M Mass of solids (kg) n Hindered settling exponent Pm Total hydrostatic pressure (Pa) Rep Reynolds number r Particle radius (m) rcr Particle radius for maximum viscous dissipation at a given circular frequency (m) b,Tr TUS Measured time required for settled bed to reach measurement location (s) THb,Tr Calculated time using hindered settling velocity for settled bed to reach measurement location (s) TTb,Tr Calculated time using terminal settling velocity for settled bed to reach measurement location (s) t Settling time (s) Ubed Settled bed rise velocity (m s− 1) UH Hindered settling velocity (m s− 1) UH,L Hindered settling velocity of large particles (m s− 1) UH,S Hindered settling velocity of small particles (m s− 1) UT Terminal settling velocity (m s− 1) V Acoustic Velocity (m s− 1) Δv Difference in acoustic velocity between suspension and water (m s− 1) x Path length of radiation in slurry (m) z Height of slurry above measurement location (m) αi Attenuation due to ith size fraction (Np m− 1) δv Viscous layer thickness (m) εb Settled bed voidage εe Effective permittivity (F m− 1)
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εf εp ϕ ϕL ϕS ϕ′ ϕL′ ϕS′ μf νL,sed ω ωcr ρm ρp ρf
Fluid permittivity (F m− 1) Solids permittivity (F m− 1) Particle volume fraction in suspension Large particle volume fraction in suspension Small particle volume fraction in suspension Particle volume fraction in sediment Large particle volume fraction in sediment Small particle volume fraction in sediment Fluid viscosity (Pa s) Sediment–suspension interface velocity for large particles (m s− 1) Circular frequency (rad s−1) Circular frequency for maximum viscous dissipation for a given particle radius (rad s− 1) Effective density (kg m− 3) Particles density (kg m− 3) Fluid density (kg m− 3)
References [1] R.A. Williams, C.G. Xie, R. Bragg, W.P.K. Amarasinghe, Experimental techniques for monitoring sedimentation in optically opaque suspensions, Colloids and Surfaces 43 (1) (1990) 1–32. [2] C.G. Xie, R.A. Williams, S.J.R. Simons, M.S. Beck, R. Bragg, A novel sedimentation analyser, Measurement Science & Technology 1 (1990) 1216–1227. [3] T.Y. William, Brandon R. Macias, Peihong Cao, Alan R. Hargens, T. Ueno, System for determination of ultrasonic wave speeds and their temperature dependence in liquids and in vitro tissues, Journal of the Acoustical Society of America 117 (2) (2005) 646–652. [4] W.S. Ament, Sound propagation in gross mixtures, Journal of Acoustical Society of America 25 (1953) 638–641. [5] C.M. Atkinson, H.K. Kyotomaa, Acoustic wave speed and attenuation in suspensions, International Journal of Multiphase Flow 18 (4) (1992) 577–592. [6] R.J. Urick, A sound velocity method for determining the compressibility of finely divided substance, Journal of Applied Physics 18 (1947) 983–987.
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[7] S. Alizadeh-Khaivi, Ultrasonic Measurement Techniques in Slurries, Department of Chemical and Biochemical Engineering, University of Western Ontario, 2002. [8] A. Shukla, Size and Concentration Measurement in Slurries by Ultrasonic Methods, Department of Chemical and Biochemical Engineering, University of Western Ontario, 2003. [9] V. Stolojanu, A. Prakash, Characterization of slurry systems by ultrasonic techniques, Chemical Engineering Journal 84 (2001) 215–222. [10] J.G. Dueck, D. Purevjav, D.Y. Kilimnik, A contribution to the theory of porosity of fine grade sediments, Journal of Engineering Physics and Thermophysics 77 (1) (2004) 93–102. [11] D. He, N.N. Ekere, Computer simulation of powder compaction of spherical particles, Journal of Material Science Letters 17 (20) (1998) 1723–1725. [12] D. He, N.N. Ekere, L. Cai, Computer simulation of random packing of unequal particles, Physical Review E 60 (6) (1999) 7098–7104. [13] R.J. Wakeman, Packing densities of particles with log-normal size distributions, Powder Technology 11 (1975) 297–299. [14] W.L. McCabe, J.C. Smith, P. Harriott, Unit Operations of Chemical Engineering, McGraw-Hill, Boston, 2001. [15] R. Clift, J.R. Grace, M.E. Weber, Bubbles, Drops, and Particles, Academic Press, New York, 1978. [16] G.J. Kynch, A theory of sedimentation, Transactions of Faraday Society 48 (1952) 166–176. [17] J.F. Richardson, W.N. Zaki, Sedimentation and fluidization: part I, Transactions of the Institution of Chemical Engineers 32 (1954) 35–53. [18] W. Schneider, G. Anestis, U. Schaflinger, Sediment composition due to settling of particles of different sizes, International Journal of Multiphase Flow 11 (3) (1985) 419–423. [19] G.B. Wallis, One Dimensional Two Phase Flow, McGraw-Hill, New York, 1969. [20] R. Jeschar, Druckverlust in Mehrkornschüttungen aus Kugeln, Archiv für das Eisenhüttenwesen 2 (1964) 91–108. [21] R. Bürger, K.K. Fjelde, K. Hofler, K.H. Karlsen, Central difference solutions of the kinematic model of settling of polydisperse suspensions and three-dimensional particle-scale simulations, Journal of Engineering Mathematics 41 (2001) 167–187. [22] A.S. Dukhin, P.J. Goetz, Ultrasound for characterizing colloids — particle sizing, zeta potential, rheology, in: D. Möbius, R. Miller (Eds.), Studies in Interface Science, vol. 15, Elsevier, Boston, 2002.
Particles settling studies using ultrasonic techniques A. Shukla, A. Prakash ⁎, S. Rohani Department of Chemical and Biochemical Engineering, The University of Western Ontario, London, Ontario, Canada N6A 5B9 Received 14 August 2006; received in revised form 12 December 2006; accepted 7 February 2007 Available online 15 February 2007
Abstract Effects of acoustic velocity and attenuation measurements during settling of 43-, 110- and 168-μm glass beads in water are reported. Ultrasonic waves were generated at a frequency of 3.2 MHz. An abrupt increase in acoustic velocity and a sharp peak in the attenuation characterized the onset of settled bed. The observed attenuation peak at the transition between suspended and settled bed was attributed to dissipation caused by viscous absorption losses. The critical concentration at which increase in acoustic velocity and attenuation peak occurred was estimated for these particle sizes. © 2007 Elsevier B.V. All rights reserved. Keywords: Ultrasonic techniques; Acoustic velocity; Attenuation; Settling
1. Introduction Sedimentation and settling of particles in liquid is important in various chemical engineering operations such as fluidized beds, hydrosizers, thickeners, and hydraulic conveying. Fluidized beds are commonly used in physical, chemical and biochemical processes. These systems provide good heat transfer, mass transfer and mixing characteristics. An important requirement for such systems is to keep the particles in suspensions for achieving the desirable properties. Similarly, efficient suspension of particles is also desired in hydraulic conveying. On the other hand, hydrosizers rely on controlled settling for separating particles of different sizes and/or densities by utilizing the difference between their settling velocities. Thickeners rely on maximization of sediment rate for efficient operation. Hence, a real time measurement technique is required for the online detection and control of settling in industrial processes. Experimental techniques available for monitoring sedimentation can be broadly summarized in the following categories based on the underlying measurement principle [1]: 1. External radiation e.g. Neutrons, X-ray, γ-ray, microwaves
⁎ Corresponding author. Tel.: +1 519 661 2111x88528. E-mail address: [email protected] (A. Prakash). 0032-5910/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2007.02.003
2. Emitted radiation e.g. Radioactive tracers, NMR, magnetic tracers 3. Electrical properties e.g. Sedimentation potential, capacitance, conductance 4. Physical properties e.g. Sedimentation balance, pressure measurements 5. Direct methods e.g. Sampling, physical interruptions A comprehensive review of these measurement principles have been carried out by Williams et al. [1]. However, most of these techniques can have limited industrial application due to either stringent safety requirements (NMR, radioactive tracers, neutron, X-ray, γ-ray) or need for specific properties of solids under investigation (magnetic, electrical) or are highly intrusive. Direct measurement techniques require sampling and hence are not suitable for real-time online measurement. Measurement of physical properties such as pressure and viscosity can only be used to obtain macroscopic properties such as net solid flux. Measurement principles of techniques, which are non/less intrusive and capable of online application in dense suspensions, are discussed in brief. These include techniques based on the measurement of external radiation, electrical and physical properties. Measurement of electrical properties (capacitance/ permittivity, conductance) of the system has been extensively used for online monitoring of sedimentation in optically opaque and dense systems. The relationship between effective permittivity
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and particles concentration (ϕ) for spherical particles and negligible effect of system conductance is given by Eq. (1) [2]. 3 1=3 ee ¼ ð1−/Þef þ /e1=3 ð1Þ p
dense suspensions is demonstrated and the effects of concentration on measured parameters are evaluated. Furthermore, this technique has the ability to clearly detect the transition from suspension to settled bed.
where, εf, εp are the permittivity of liquid and solid phase and the effective permittivity εe can be obtained from the measured capacitance. In most systems, conductance affects the capacitance and hence has to be measured for accurate determination of effective capacitance. Measurement of total hydrostatic pressure is also a useful tool for analysis of sedimentation. The hydrostatic pressure at a given height is the sum of pressure due to the fluid and particles above the measurement location and is given by Eq. (2) [2].
2. Experimental details
Pm ¼ qmd gd z
ð2Þ
where ρm is the effective mass density given by Eq. (3) and z is the height of suspension above the measurement location. qm ¼ ð1−/Þqf þ /qp
ð3Þ
Particle concentration can be derived from the measured hydrostatic pressure at a given height and time by combining Eqs. (2) and (3). 1 Pm ðz; tÞ /ð z; t Þ ¼ −qf ð4Þ ðqp −qf Þ gz External radiation methods are based on the monitoring of a beam of radiation, which has passed through the suspension and can be used in dense and optically opaque suspensions. The energy removed from the incident beam is proportional to the particle size and their concentrations. Hence, the energy loss can be used to determine the volume fraction and size of the constituent particles. For a polydisperse suspension this relationship can be expressed using Eq. (5) [1]. I ¼ I0 e
−x
n P i¼1
a i /i
ð5Þ
Ultrasonic technique can be categorized under external radiation measurement principle but is free from radiation safety requirements. Like other radiation based techniques, it can operate in dense and optically opaque suspensions and can measure particle size and concentration. This technique provides an added advantage over other X-ray and γ-ray techniques as both energy loss and acoustic velocity of the pulse can be used to characterize settling. These parameters are dependent on the propagation regime of the acoustic pulse. The propagation regimes are a function of the wave number (kr), which is a ratio of particle size and the wavelength of the acoustic pulse. While describing the ultrasonic propagation mechanisms during polydisperse sedimentation, the dynamic changes in particle size and concentration have to be accounted for. In this study the response of acoustic velocity and attenuation during settling in different propagation regimes are studied for various practical applications. Its applicability in
Measurements of acoustic velocity and attenuation were conducted to study the effects of particle settling and sedimentation on acoustic properties in liquid–solid systems. A schematic diagram of the experimental setup and data acquisition system is shown in Fig. 1. The solid phase used in the experiments was glass beads of 43-, 110- and 168-μm mean diameter and the suspension medium was distilled water. Particles were suspended in plexiglass column of 0.102 m inside diameter and 1.02 m height by introducing air through a porous polyethylene distributor plate with 35 μm pore size (Fig. 2). Settling experiments were also conducted for mixed particle system with equal proportions of 43- and 168-μm particles. The suspended particles were allowed to settle by shutting off the gas flow and measurement of acoustic properties was obtained at a height of 0.17 m above the distributor. The total suspension height was 0.528 m and the concentration of solids in the suspended slurry was 28 vol.%. Measurements of acoustic velocity and attenuation during the settling of 43- and 110-μm particles were also conducted in a stirred tank to investigate the effect of slurry height on settling characteristics of the suspension. The initial slurry concentration of particles was 36 vol. % and distilled water was the suspension medium. A schematic representation of the stirred tank used to carry out the ultrasonic measurements is shown in Fig. 3. The stirred tank is made of plexiglass with diameter 0.1 m (4ʺ) and height 0.252 m (10ʺ). It is equipped with radial ports for mounting the transducers and a vertical stirrer for suspending the particles. The operating temperature for experiments in this study varied between 21 and 25 °C. The change of acoustic velocity in water for this temperature range is 2.58 m/s/°C [3]. The variations in measured acoustic velocity are reported as (Δv = Vsuspension − Vwater) at the operating temperature. Hence, the effect of temperature on the measurements is normalized and the reported change in acoustic velocities is comparable for different experiments. The change due to temperature during settling for a given particle sample was negligibly small (≤0.78 m/s) and the error introduced in measured Δv was less than 0.64%. Acoustic parameters were measured using an ultrasonic pulser/receiver unit (Fallon Ultrasonic Inc.) at 3.2 MHz
Fig. 1. Block diagram of experimental setup.
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Fig. 2. Schematic diagram of measurement column.
frequency in through transmission mode. Transducer separation used during this study was 25 mm and 50 mm for the column and stirred tank respectively. This arrangement of transducers is
intrusive in nature but was preferred to maximize signal strength. Due to the intrusive nature of measurements used in this study the transducers were enclosed in a cylindrical housing
Fig. 3. Schematic diagram of stirred tank.
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to minimize resistance to the flow of slurry and bubbles. Small ratios of the particle size to the distance between transducers also reduced distortions to particles motion in the column. A TDS 210 (Tektronix) digital oscilloscope was used to visualize and select the ultrasonic pulse. The pulser/receiver unit activates the receiving transducer at a time fixed by the delay and for duration of the gate width to measure the flight time of the signal. The pulses repetition rate was 1 kHz and the sampling interval was 1 s. Hence each sampled value of transit time and amplitude represent the average of 1000 acoustic pulses. 3. Results and discussions Fig. 4 shows changes in measured acoustic velocity (Δv = Vsuspension − Vwater) during settling of 43-μm particles in the column and the stirred tank. Acoustic velocity measurements in the column resemble an ‘S’ shape and can be divided in four sections based on the settling characteristics of the suspension. In section-I, acoustic velocity measurements do not change with time. Measurements showed a gradual increase in section-II and are followed by an abrupt increase of 110 m/s in section-III before attaining a stable value in section-IV. Acoustic velocity in a two-phase suspension is governed by the particle size, concentration, physical parameters of the constituent phases (density and compressibility) and the frequency of the acoustic pulse [4–6]. Experimental measurements in liquid–solid systems have shown that the acoustic velocity increases with increase in concentration and decrease in particle size [4–9]. Hence, constant acoustic velocity in section-I indicates no change in the average particle size and concentration at the measurement height. The gradual increase in the measured acoustic velocity in section-II could be attributed to the change in either one or both of the above parameters. The average particle size at the measurement location could decrease due to particle segregation or/and the concentration could increase due to formation of a dense layer. Attenuation measurements during this phase of settling can be used to identify the dominant effect and is discussed in the next section. The abrupt increase of acoustic velocity in section-III was observed when the bed reached the transducer height and can be attributed to the high
Fig. 4. Change in acoustic velocity during settling of 43-μm particles in column and stirred tank.
Fig. 5. Particle size distribution of 43-, 110- and 168-μm particles using Malvern Mastersizer®.
concentration of particles in the settled bed. This is followed by a gradual increase in acoustic velocity over time in section-IV and can be attributed to bed compaction (hence further increase in concentration) due to the weight of particles and fluid above the measurement location. Measured acoustic velocity in the stirred tank was higher as the initial particle concentration was 8 vol.% more than that in the column. However, the trends in acoustic velocity change were similar for both systems. The measurements show an initial region of constant concentration followed by a gradual increase in acoustic velocity. The onset of settled bed was also characterized by an abrupt increase in acoustic velocity. However, the settling duration in the stirred tank was much shorter than the column as the transducers were located only 0.07 m above the bottom. Furthermore, gradual increase in acoustic velocity was also much smaller in magnitude than the measurements made in the column. This indicates that the extent of particle segregation in the stirred tank is much smaller and is attributable to the shorter settling height (0.115 m) compared to the column (0.526 m). The acoustic velocity measurement in the settled bed of the stirred tank was higher than that of the column. This could be attributed to lower concentration of particles in the settled bed of the column due to segregation, if the average particle size is less than 30-μm [10]. Fig. 5 shows that 30% of particles are smaller than 34-μm in the size distribution of the 43-μm particle sample. Since the transducer location is about 33% below that settled bed height, it is likely that the average particle size at this location was less than 30-μm and hence the concentration was lower. Fig. 6 shows the acoustic velocity measurements during the settling of 110-μm particles in the column and in the stirred tank. The relative trends during settling in the column and in the stirred tank for 110-μm particles were similar to those observed for 43-μm particles. The magnitude of increase in acoustic velocity during settling in the column before the settled bed reached the transducer was more than the stirred tank. This is because the settling duration and the extent of segregation in the stirred tank was small compared to the column and can be attributed to the small settling height in the stirred tank (Fig. 3). The acoustic velocity measurements in the settled bed of the stirred tank were higher than that of the column. Unlike 43-μm
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Fig. 6. Change in acoustic velocity during settling of 110-μm particles in column and stirred tank.
particle suspension, this deviation cannot be attributed to the presence of particles smaller than 30-μm, as this size range is not present in the 110-μm samples. An alternative explanation for this discrepancy could be the lack of particle segregation in the stirred tank. The measured size distribution (Malvern, Mastersizer®) of 110-μm particles used in this study was lognormal in nature and has a standard deviation of 0.498. Lognormally distributed particles show higher concentrations in settled beds for standard deviations greater than 0.15 [11–13]. These studies show that the concentration increases by 6 vol.% for increase in particle size standard deviation from 0.15 to 0.5. The size distribution of particles in the column is expected to be more uniform due to segregation. The transducers location in the settled bed is 33% below the maximum bed height and hence should contain particle in the lower 30% (70–90 μm) of the size distribution of 110-μm (Fig. 5). However, the increase in magnitude of acoustic velocity due to the decrease in average particles size is less than 2 m/s for particles of similar sizes [8]. Hence, higher acoustic velocity measurements in settled bed of the stirred tank may be attributed to increase in concentration due to higher particle size standard deviation for 110-μm particles. The standard deviation of 43-μm particles was 0.533 and could be contributing to high acoustic velocity measurements in the stirred tank along with reason already discussed. Fig. 7 compares the change in measured acoustic velocity during the settling of 110 and 168-μm particles in the column. Unlike 110- and 43-μm particles, 168-μm particles show an intermediate step in the S-shaped curve. It should also be observed that the region of constant concentration for 168-μm particles was small. This can be attributed to higher settling velocities of larger particles. The abrupt increase of 37 m/s in acoustic velocity during the intermediate step cannot be attributed to a decrease in the average particle size, as this effect is relatively small [9]. During the experiments it was observed that an interface between the slurry and clear water was formed almost immediately after the gas was shut off. The location of this interface was calculated to be 0.39 m after 20 s of settling and would result in a 10 vol.% increase in the concentration. Acoustic velocity measurements at 38 vol.% for similar particle sizes are comparable to the measurements made after 20 s during the settling of 168-μm particles. Hence, the
first abrupt increase in acoustic velocity measurements appears to be due to the formation of a dense layer in the suspension. The oscillation of measured acoustic velocity during this increase could be attributed to the transient disturbances of the dense layer interface moving through the measurement volume. This was followed by a gradual increase in acoustic velocity due to hindered settling effects till the settled bed reached the transducer. Fig. 7 shows that the measured acoustic velocities in the settled beds of 110- and 168-μm particles were similar. However, acoustic velocity decreases with increase in particle size and has been reported in literature for suspensions of varying concentrations [5,7,9]. Higher acoustic velocity in the settled beds of 43-μm particles (Fig. 4) as compared to 110and 168-μm (Fig. 7) is in agreement with the trends observed for particles in suspension. Similar acoustic velocity measurement for 110- and 168-μm particles indicates that the concentration in the settled bed of 168-μm particles is higher. As discussed earlier, increase in standard deviation of lognormally distributed particles from 0.15 to 0.5 results in a 6 vol. % increase in concentration [11–13]. The initial standard deviation of 110- and 168-μm particles was 0.498 and 0.464 respectively. The size distribution of 110-μm particles in the column is expected to be more uniform as its settling time is twice that of 168-μm particles. Hence, similar acoustic velocity measurements observed in Fig. 7 can be attributed to higher concentration in the settled bed of 168-μm particles due to smaller settling time. The time required by the settled bed to reach the transducer height can be theoretically calculated using the settling velocities and compared with the time at which the abrupt increase in acoustic velocity was observed using Eqs. (6) and (7). Ubed ¼
Hbed UTorH Hsus −Hbed
b;Tr TTorH ¼
Hbed Ubed
ð6Þ
ð7Þ
The maximum settling velocity of a single particle in the fluid is called the terminal settling velocity. The settling
Fig. 7. Comparison of change in acoustic velocity during settling of 110- and 168-μm particle suspension in column.
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criterion K (Eq. (8)) determines whether the settling conditions are within or outside the Stoke's limit [14]. ! gqf ðqp −qf Þ 1 3 K¼d ð8Þ l2f Depending on the settling criterion appropriate equations were used to calculate the settling velocities of 43-, 110- and 168-μm particles. The drag coefficients for particles with Reynolds number greater than one were estimated using the relationship given by Clift et al. [15]. The terminal settling velocities are however reduced due to the fluid up flow generated by neighboring particles [16]. The reduced velocity is called the hindered settling velocity and is a function of particle concentration. The Richardson and Zaki [17] relationship was used to obtain the hindered settling velocity for the average particle size in the suspensions (Eq. (9)). UH ¼ UT ð1−/Þn
ð9Þ
Here, UT is the terminal settling velocity and n is the empirical settling exponent, which is a function of particle size, column diameter and Reynolds number of the particles. Table 1 shows the bed rise time calculated using the terminal and hindered settling velocities. Hindered settling velocity based calculations for bed rise time showed good agreement with the measurements. The small deviations between measured and calculated bed rise times can be attributed to polydispersity of the particles (Fig. 5). Experimental measurement of acoustic velocity was also conducted in a mixed suspension of 43- and 168-μm particles containing equal volume fractions of each particle size (Fig. 8). The acoustic velocity shows three abrupt increases during settling. The first and second increases in the acoustic velocity indicate the settling of two large particle fractions below the measurement volume. The first increase occurs due to dense layer formation by 168-μm particles. This dense layer is a result of very high settling velocities of these particles compared to 43-μm particles (Table 1). This was followed by the second increase when all the particles in the 168-μm sample have settled. However, the settling of 168-μm particles in the mixed particle system was slow compared to the single particle species suspension and could be attributed to hindered settling effect caused by the presence of 43-μm particles. The third increase was observed when the settled bed reached the transducer height. Schneider et al. [18] developed a model to determine the interface velocities during the sediment of a bi-disperse suspension. They use the drift flux concept [19] to obtain the
Table 1 Comparison of measured and calculated bed rise times for 43-, 110- and 168-μm particles using terminal and hindered settling velocities d (μm) UT (m s− 1) Rep 43 110 168
0.0017 0.0105 0.0197
n
b,Tr UH (m s− 1) TTb,Tr (s) THb,Tr (s) TUS (s)
0.08 4.66 0.00037 1.3 4.35 0.00252 3.71 3.92 0.00544
110 17 9
525 76.8 35.5
534 87 38
Fig. 8. Change in acoustic velocity and attenuation during settling of 43- and 168-μm mixed particle suspension in column.
volume flux densities of the two components to obtain a relationship between sediment and suspension composition (Eq. (10)). In this equation ‘ϕL’ and ‘ϕS’ are the suspension concentration for the smaller and larger particle size just above the sediment–suspension interface. Composition of sediment constituting of two different particle sizes (Eq. (11)) is a function of their diameters ‘dS/dL’ and was obtained by a theoretical investigation of the limiting case ‘dS/dL→0’ by Jeschar [20]. Eqs. (10) and (11) can be iteratively solved to obtain the sediment composition for a given suspension composition. ½1−/L ð1 þ F Þ/SV−½ F−/S ð1 þ F Þ/LV ¼ /S −/L F where;
F¼
ð10Þ
/S dS 2 /L dL
/SV ¼ ð1−eb Þð1−/LVÞ−eb /LVðdS =dL Þm ;
0V/LVV1−eb
ð11Þ
where;
p/SV m ¼ 0:6sin /SVþ /LV
The suspension and sediment concentrations of the two particle species can be used to obtain the propagation velocities of different interfaces. The sediment–suspension interface velocity for large particle species is given by Eq. (12). mL;sed ¼ where;
jL /LV−/L
ð12Þ
jL ¼ /L ð1−/L ÞUH;L −/L /S UH;S
In the above equation ‘jL’ denotes the volumetric flux densities of the constituent particle sizes. Similar equations can be used to obtain the other interfacial propagation velocities. It should be noted that this technique is based on the assumption of absence of continuous transition between concentrations at the various interfaces and is limited to bi-disperse suspensions. Numerically intensive models such as that of Bürger et al. [21] maybe used to overcome these drawbacks. However, location of
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these interfaces calculated using Schneider et al. [18] model shows good agreement with experimental observations and numerical simulations of Bürger et al. [21]. For this study the interface locations predicted by the Schneider et al. [18] model for a bidisperse suspension of 168- and 43-μm particles are compared with the ultrasonic measurements. The interface position can be plotted on a space–time diagram to obtain the sediment and suspension concentration as a function of sediment height and time. Fig. 9 shows the concentration variation along the bed height with settling time for a twoparticle suspension containing 43- and 168-μm particles. The y-axis is the settling height ‘z’ and the x-axis is settling height divided by the terminal settling velocity of the smallest particle. The acoustic velocity measurements during the settling of mixed particle suspension and the measurement height are shown on the secondary y-axis of the figure. Section A of the plot shows that a constant concentration region exists between suspended and fixed bed. This is in agreement with the experimental measurements, as the acoustic velocity does not show any change in this section. Section B consists of intermediate concentration of small particles between fixed and settled bed. Acoustic velocity in this section is initially constant and then shows a gradual increase before the suspension–sediment interface (between section B and D) is reached. Section C consists of the settled bed with particles of both size fractions and does not affect acoustic velocity as this section lies below the measurement location. Section D consists of settled bed with only small particles as the larger fraction has already settled. The model predicted interface between sections A–B and B–D is in good agreement with the time at which abrupt increase in the acoustic velocity were measured. The first interface (between 43- and 168-μm particles) predicted by the model lies between the first and second abrupt increases in the acoustic velocity. The second interface (between 43-μm particle suspension and sediment) occurs about 30 s after the third abrupt increase in acoustic velocity. Section E is the clear liquid region after all the particles have settled and does not affect the measurements as this region lies above the transducer location. Deviation between measured and predicted interfaces could be attributed
Fig. 9. Simulation of bi-disperse suspension (43- and 168-μm) using the Schneider et al. [18] model and comparison with change in acoustic velocity measurements.
Fig. 10. Change in attenuation during settling of 43-μm particles in column and stirred tank.
to the polydispersity of the particles. The gradual change in acoustic velocity before the transition from suspended to settled bed of small particles could be attributed to concentration gradient caused due to hindered settling [21]. 3.1. Attenuation measurements during settling Fig. 10 shows the measured attenuation of the ultrasonic pulse in the column and stirred tank during settling of 43-μm particles. Attenuation measurements in the column show a slight increase (3 Np/m) during the initial phase of settling (Section-Ia). The increase in attenuation can be attributed to removal of a few small bubbles entrapped in the suspension. Once the small bubbles escape, the attenuation measurements remain constant up to 225 s (Section-Ib). This is followed by a decrease in attenuation up to 500 s (Section-II) and can be attributed to the decrease in average particle size due to segregation. The transition from suspended to settled bed is characterized by a peak in the attenuation (Section-III). The secondary attenuation peak observed just before the settled bed is reached could be attributed to a non-uniform settling due to the presence of transducers. After the bed has settled, measurements in the column gradually increase before stabilizing (section-IV). This increase can be attributed to the effect of bed compaction due to the height of settled bed above the measurement location (80 mm). Compaction causes a decrease in the porosity of the settled bed and hence an increase in concentration. Attenuation measurements in the stirred tank do not show an initial increase as the mode of suspension was mechanical and no fine bubbles where entrapped (section-Ia and Ib). No decrease in the attenuation measurements in the stirred tank, before the settled bed reaches the transducer, indicates the absence of particle segregation (section-II). This can be attributed to the small settling height in the stirred tank. Similar to the measurements made in the column, the transition between suspended and settled bed was characterized by a peak in the attenuation (section-III). The reason for this attenuation peak is discussed in the section on viscous losses in concentrated systems. In section-IV the attenuation measurements in the column gradually increase by 18 Np/m to achieve values similar to that of the column. A simultaneous increase in acoustic
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velocity (by 6 m/s) during this phase of settling (Fig. 4) in the column indicates an increase in concentration of the settled bed and can be attributed to compaction. Absence of compaction in the stirred tank is due to the small height of settled bed over the measurement location. Fig. 11 shows the acoustic velocity and attenuation measurements during the settling of 110-μm particles in the column and in the stirred tank. A peak in attenuation measurements accompanied the abrupt increase in acoustic velocity. Similar observations were also made during the settling of 168-μm particles (Fig. 12). However, acoustic velocity measurement during the settling of 168-μm particles also shows an intermediate abrupt increase due to an increase in concentration by 10 vol.% (Fig. 7). This increase was not captured by attenuation measurements (Fig. 12) and can be attributed to relatively small effect of concentration change on attenuation in dense suspensions of this particle size [9]. The effect of slurry height on suspension composition during settling can be observed in the attenuation measurements for 110-μm particles in column and in the stirred tank. Absence of particle segregation in the stirred tank is indicated by the constant attenuation measurements during settling. Furthermore attenuation measurements in settled bed of the stirred tank were higher than that of the column and can be attributed to higher concentration. As discussed in the velocity section this increase in concentration is a result of higher particle size standard deviation in the absence of segregation. However, unlike 43-μm particles, there was no increase in attenuation measurements in the column after the settled bed has reached the transducer. This indicates that there is no significant compaction in beds of larger particle. These results show that ultrasonic techniques can be used to monitor suspension composition during settling and can be used to control the performance of hydrosizers. 3.2. Viscous dissipation in highly concentrated suspensions The transition from suspended to settled bed shows a peak in the attenuation measurements. The increase in attenuation could be attributed to increase in energy loss due to high particle concentration in the bed. However, the subsequent drop in attenuation requires a more in-depth analysis of the various
Fig. 11. Change in attenuation and acoustic velocity during settling of 110-μm particles in column and stirred tank.
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Fig. 12. Change in attenuation and acoustic velocity during settling of 168-μm particles in column.
energy loss mechanisms and their effects. Energy loss in a suspension is a function of the particle size, particle concentration and frequency of the acoustic pulse. The various energy loss mechanisms during wave propagation can be attributed to viscous, thermal, scattering, intrinsic and structural dissipation effects. For a glass beads–water suspension the thermal and intrinsic effects are negligible. Thermal effects are only dominant for soft particles with size smaller than 1-μm and low-density difference with the suspending medium. Glass beads are rigid, have high-density contrast compared to water and the size used in this study was much larger than 1-μm. Intrinsic attenuation for water has been shown as negligible and independent of frequency in the range of 3 to 100 MHz [22]. Significant role of structural losses in settled beds of large particles is unlikely. Its use even for particles in the colloidal range is only justified when, other mechanisms fail to provide a suitable explanation for experimental measurements [22]. Viscous and scattering effects will play a dominant role during the settling of suspensions and in the settled bed of the particle studied. Measurements made during this study showed that the significant increase in attenuation occurs before the sharp increase in acoustic velocity (Figs. 11 and 12). The increase in acoustic velocity coincided with a decrease in attenuation. This was followed by nearly constant measurements of both these parameters for 110- and 168-μm particles, indicating that the bed has settled. For 43-μm particles the abrupt changes in the measured attenuation was followed by a gradual increase and can be attributed to increase in concentration due to compaction as discussed in the previous section. These observations indicate that an attenuation mechanism comes in to existence near the settled bed concentrations. It causes maximum dissipation just before the bed settles and the mechanism ceases to exist in the settled as indicated by the drop in attenuation. Scattering can account for the increase in attenuation just before sediment formation. This mode of attenuation is caused by the deflection of a part of the energy in the incident beam and is proportional to the particle size and concentration. However, it cannot explain attenuation drop in the settled bed, which is more concentrated than the suspension. The attenuation peak in the vicinity of settled bed concentration could be explained by viscous dissipation. Viscous losses are caused by the oscillation
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of particles due to the pressure gradient generated by the acoustic pulse. The oscillations have a shearing effect on the fluid surrounding the particles hence leading to the formation of shear waves. Propagation of shear waves leads to energy loss, as these waves can only exist over short distances before being completely dissipated. The distance of shear wave propagation is known as the viscous layer thickness and can be calculated using the equation given below [22]. sffiffiffiffiffiffiffiffi 2lf dv ¼ ð13Þ xqf For dilute suspensions the energy dissipation due to this mechanism is maximum when the viscous layer thickness becomes comparable to the particle radius. However, in concentrated suspensions, the inter-particle separation overtakes particle radius as the deciding parameter for maximum viscous dissipation [22]. Hence, viscous dissipation due to concentration comes into existence when the inter-particle distances are comparable to the viscous layer thickness and is independent of particle size (Eq. (14)). sffiffiffiffiffiffiffiffi 2lf lcr cdv ¼ ð14Þ xqf With further increase in concentration the inter-particle distances become less than the viscous layer depth and begins to inhibit the relative oscillatory motion between the particles and fluid. This leads to a decrease in attenuation till the settled bed is reached where greater numbers of adjacent particles are in contact and no significant relative oscillation is possible. For 3.2 MHz frequency the critical inter-particle distance at which viscous energy dissipation will be maximum is 0.3-μm and should be achieved near the settled bed concentration. Eq. (15) can be used to calculate the settled bed concentration assuming mono-sized particles. The measured fixed bed height for all particle sizes was 0.252 m and corresponds to a concentration of 57.5 vol.%. /V ¼
M Hbed pr2 qp
ð15Þ
The actual concentration of the settled bed in the current study is expected to be higher than 57.5 vol.% due to polydispersity of the particles. Random close packing of spherical particles has a maximum concentration of 63.5 vol.% and can be safely assumed as the packing structure of the sediment. Using this packing structure the critical inter-particle distance is achieved at 60.9, 62.5 and 62.8 vol.% for 43-, 110- and 168-μm respectively. 4. Conclusions Ultrasonic measurements captured the change in average particle size due to segregation during settling. The ultrasonic parameters could clearly distinguish between suspended and settled bed. Acoustic velocity showed a sharp increase at the transition between settled and suspended bed. Attenuation
measurements showed a peak just before the settled bed reached the transducer location and this was attributed to the increase in viscous dissipation effects. The critical concentration at which these losses became prominent was calculated to be above 60 vol.% for the particles investigated. In the settled bed this technique was able to detect the effect of particle size standard deviations and the existence of bed compaction. These results show that ultrasonic techniques can detect transition from suspended to settled bed and has potential for online monitoring of suspension composition. Nomenclature D Column diameter (m) d Particle diameter (m, μm) dL Large particle diameter (m) dS Small particle diameter (m) g Acceleration due to gravity (m s− 2) Hsus Height of suspension (m) Hbed Height of bed (m) Δh Distance between suspension height and fixed bed height (m) I Incident intensity (W m− 2) I0 Received intensity (W m− 2) i = 1…n Particle size index jL Volume flux density of larger particles (m s− 1) K Settling criterion kr Non-dimensional wavenumber lcr Inter-particle distance for maximum viscous dissipation (m) M Mass of solids (kg) n Hindered settling exponent Pm Total hydrostatic pressure (Pa) Rep Reynolds number r Particle radius (m) rcr Particle radius for maximum viscous dissipation at a given circular frequency (m) b,Tr TUS Measured time required for settled bed to reach measurement location (s) THb,Tr Calculated time using hindered settling velocity for settled bed to reach measurement location (s) TTb,Tr Calculated time using terminal settling velocity for settled bed to reach measurement location (s) t Settling time (s) Ubed Settled bed rise velocity (m s− 1) UH Hindered settling velocity (m s− 1) UH,L Hindered settling velocity of large particles (m s− 1) UH,S Hindered settling velocity of small particles (m s− 1) UT Terminal settling velocity (m s− 1) V Acoustic Velocity (m s− 1) Δv Difference in acoustic velocity between suspension and water (m s− 1) x Path length of radiation in slurry (m) z Height of slurry above measurement location (m) αi Attenuation due to ith size fraction (Np m− 1) δv Viscous layer thickness (m) εb Settled bed voidage εe Effective permittivity (F m− 1)
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εf εp ϕ ϕL ϕS ϕ′ ϕL′ ϕS′ μf νL,sed ω ωcr ρm ρp ρf
Fluid permittivity (F m− 1) Solids permittivity (F m− 1) Particle volume fraction in suspension Large particle volume fraction in suspension Small particle volume fraction in suspension Particle volume fraction in sediment Large particle volume fraction in sediment Small particle volume fraction in sediment Fluid viscosity (Pa s) Sediment–suspension interface velocity for large particles (m s− 1) Circular frequency (rad s−1) Circular frequency for maximum viscous dissipation for a given particle radius (rad s− 1) Effective density (kg m− 3) Particles density (kg m− 3) Fluid density (kg m− 3)
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