- Email: [email protected]
Enhanced synchronized ultrasonic and flow-field fractionation of suspensions
Enhanced synchronized ultrasonic and flowfield fractionation of suspensions Z. Mandralis*, W. Bolek% W. Burger% E. Benest and D.L. Feke* *Department of Chemical Engineering, Case Western Reserve University, Cleveland, OH 44106, USA tlnstitute of General Physics, Technical University of Vienna, Vienna, Austria
Received 30 June 1993; revised 15 September 1993 A fractionation method for fine-particle suspensions based on variations in the speed of response to ultrasonic standing wave fields has been developed. The method is sensitive to a combination of the particle size as well as the particle and suspending fluid density and compressibility. The fractionation is accomplished within a narrow-gap rectangular channel having a solid barrier positioned along its midplane. Ultrasonic standing wave fields of two alternating frequencies are applied across the gap to induce a partial separation at short irradiation times. A co-ordinated bidirectional laminar flow field is used to transform these partial separations into useful separations along the chamber. In comparison with similar fraction strategies that use other types of fields to accomplish the separation, the acoustic forces acting on particles can be quite strong, thereby enabling fast, continuous, and controllable fractionation of micrometre-sized particles. An analytical model based on the trajectories of particles in response to the acoustic and flow cycles was developed. Model predictions indicate how the fractionation can be controlled through the choice of cycle parameters. Experimental results using a 325 mesh polystyrene particle suspension demonstrate that sharp fractionations of nearly neutrally buoyant micrometre-sized particles can be achieved.
Keywords: suspensions; fractionation; standing wave; acoustic force
Nomenclature 2b c Ea¢ f F g j2 m
Q R t uf Uo x y
channel height (m) speed of sound (m s- i) time averaged acoustic energy density (J m-3) frequency (s- l) acoustic contrast (Equation (5)) (dimensionless) gravity acceleration (9.81 m s-2) -1 4~(pp _ pf)R 3 = buoyant particle mass (kg) flow rate per unit width ( m 2 s - 1 ) particle radius (m) time (s) local flow velocity (m s- l) velocity amplitude of incident acoustic wave (m s- i) particle position from acoustic velocity node (m) particle displacement in y-direction (m)
Greek symbols ct
Q2t2/Qltl
fl
Eac,2/Eac,1
F 7 6
f2/fl l/pc 2 = compressibility (m 2 N - l) QE/Q1
0041 -624X/94/020113-09 O 1994 Butterworth-HeinemannLtd
K 2 u P
27z/2 = wave-number (m-l) wavelength of sound (m) viscosity (Pa s) density (kg m - 3)
Superscripts and Subscripts f if p zc
x 2 prod *
fluid infinitely fast particle particle zero contrast particle step 1 step 2 product stream dimensionless variable
The ability to fractionate fine particles sharply may be useful in a variety of applications. For example, nearly monodisperse particles may have advantages for diagnostic and treatment purposes in the life sciences. More effective separation of mammalian cells, bacteria, or yeasts is desirable in biotechnology. Fabrication of ceramics using narrow-sized ceramic particles may result in superior physical properties of the resulting materials. Also, the efficiency of a chromatographic packing is known to correlate with its degree of monodispersity.
Ultrasonics 1994 Vol 32 No 2
113
Ultrasonic and flow-field fractionation of suspensions. Z. Mandralis et al. Many conventional fine-particle fractionation methods require excessive time, high pressure drops, or intense fields to accomplish the separation of micrometre-sized, neutrally buoyant, or neutrally charged particles. Also, the resolution of these methods is often quite limited. In this paper, we report on a fractionation method that uses ultrasonic standing wave fields to accomplish a fast, sharp, efficient and controllable fractionation of fineparticles in liquid suspension. During the past decade, several filtration methods based on the response of fine particles to ultrasonic standing wave fields have been developed 1 9. A method based on the combined transient response of particles to ultrasonic standing wave fields and bidirectional laminar flow fields (denoted SUFFF, an acronym for Synchronized Ultrasonic and Flow Field Fractionation) has also been reported 1°. This method was shown to enable fast, sharp and controllable fractionation of fine particle suspensions but had a limited processing capacity per unit volume of the separator chamber. Here, we report on an enhanced version of this method (denoted E S U F F F for Enhanced Synchronized Ultrasonic and Flow Field Fractionation) that is a hybrid of S U F F F and Periodic Flow Slurry Fractionation 11 (denoted PFSF). The P F S F method fractionates particles on the basis of differences in the gravitational sedimentation rates of the various particles. In cycles combining steps of sedimentation, re-suspension and bidirectional axial flow, the P F S F method transports faster sedimenting particles to one end of a long tube while slower sedimenting particles move to the other end. Mineral particles as small as 10/~m or polymeric particles as small as 100/tm, differing by only few percent in their sedimentation rates due to size and/or density differences, can be effectively separated. The advantage of PFSF over classical processes (screening, flotation, etc) is that it is continuously adjustable as to the cut point and sharpness of separation without hardware changes. However, for small particle sizes or low density contrast between the particles and suspending fluid, production rates are extremely low due to the slow sedimentation and re-suspension steps of the cycle. Operation of the method in centrifugal fields enhances sedimentation rates but re-suspension and secondary fluid flows become problematic 12. The E S U F F F method adopts the basic fractionation philosophy of P F S F with the exception that, instead of using gravitational fields, acoustic fields are employed to drive particle motion. Similar to SUFFF, the fractionation takes place in a relatively long and wide rectangular chamber wherein bidirectional, laminar flows along the chamber are co-ordinated with a cycle of alternating low and high frequency ultrasonic standing wave fields propagated across the chamber. The low and high frequency steps are analogues of the sedimentation and re-suspension steps in PFSF. Unlike SUFFF, the chamber is divided into two equal subvolumes by a thin barrier parallel to the transducer, located at the midplane of the chamber. As will become evident in the next section, the presence of this barrier and the resulting laminar flow profile enhances the processing rate per unit volume of the separator by about an order of magnitude, in comparison with SUFFF. Like PFSF, E S U F F F retains the advantages of the potential for narrow and precise cuts, continuous adjustability as to the cut point and sharpness of fractionation without hardware changes. However, for
114
Ultrasonics 1994 Vol 32 No 2
micrometre-sized particles, acoustic forces can be ordersof-magnitude higher than gravitational forces, and hence efficient fractionation of particles in this size range are anticipated. Also, unlike PFSF, E S U F F F can be used for neutrally buoyant particles as long as they exhibit a different compressibility than the suspending medium. The next section details the process concept. The theoretical background on the acoustic force that acts on the particles, the resonant frequencies, the energy density, and the location of particle collection positions in the standing wave field in an acoustic chamber are given next. Model predictions of the fractionation efficiency are presented and the results are compared with equivalent predictions for SUFFF. Finally, proof-ofprinciple experimental results of the fractionation of 325 mesh polystyrene particle suspensions are presented and compared with model predictions.
Process concept Figure 1 shows the geometry of the separation chamber. The lateral walls comprise an acoustic transducer at the bottom and a reflector at the top. A thin barrier placed at the midplane divides the chamber into two identical subvolumes. To perform the fractionation, a two-step acoustic cycle is applied. One of the two frequencies for the acoustic field is chosen so that a velocity antinode is positioned on the midplane barrier; the second frequency is selected so that velocity antinodes are positioned midway between the barrier and the lateral walls. As will be discussed in the Theoretical background section, these conditions can readily be achieved. The primary acoustic force urges the particles to move in the x-direction towards the velocity antinodes of whatever standing wave field is being propagated. Thus, the effect of the acoustic cycle is to drive particles alternatively towards the barrier and then towards the midway point between the barrier and a wall. In practice, the duration of each step of the cycle is chosen to be short compared with the time it takes for particles to arrive at the velocity antinodes. Synchronized with the two-step acoustic cycle is a bidirectional flow cycle. The particles are assumed to track fluid motion perfectly in the y-direction. During the low frequency step of the cycle, suppose that the laminar flow is directed toward the right. Conversely, during the higher frequency portion of the cycle, the flow is assumed to be leftward. To demonstrate and quantify the operation principle more precisely, the trajectories of three types of particle are shown in Figure 2. These particles are distinguished on the basis of their speed of response to the acoustic field: (1) an infinitely fast particle which responds
~//////////J////S~ REFLECTOR
Acoustic
Acoustic
Force
vet~ity
O/2
Separet~
Q/2
y
2/////////////////////////////..¢'////////////////////~, TRANSDUCER
Figure 1
Geometry of the acoustic chamber
Ultrasonic and flow-field fractionation of suspensions: Z. Mandralis et al. O Zero Contrast
• Infinitely East
• Real
FT////////////////////////A
Step 1
where Acoustic Acoustic Velocity Force
Qltx Qzt2
Flow
Acoustic Acoustic Veloclt
Force__
Step 2 b
Figure 2 Illustration of the fractionation concept. The trajectories of an infinitely fast particle and a zero-contrast particle, as well as a real particle are s h o w n for (a) Step 1; (b) Step 2
instantaneously to the sound field; (2) a zero-contrast particle that is insensitive to acoustic forces (and thus behaves equivalently to a fluid element); (3) a real particle with intermediate response to the sound field. Due to symmetry of the acoustic and flow fields about the separator, only the upper half of the acoustic chamber is shown in Figure 2. Also, for simplicity, the effect of gravity is neglected in this discussion. During the first step of the cycle, an acoustic standing wave field of frequency f l is established in the chamber so that a velocity antinode is located inside the barrier. The acoustic force drives particles towards the central barrier. Simultaneously, a laminar flow of magnitude QI transports the particles to the right for time tl. Figure 2a shows that the infinitely fast particle moves immediately to the separator and maintains its original y-position, the real particle moves in the same direction at a finite rate, and the zero contrast particle maintains its original x-position but moves rightward with the flow. Owing to the parabolic flow profile, the zero contrast particle moves farther to the right than the real particle, which in turn moves farther to the right than the infinitely fast particle. For the second step, the sound frequency is switched to f2 so that a velocity antinode is located midway between the barrier and the wall. Particles are urged by the sound field towards a plane located midway between the wall and the barrier. They are also transported to the left for time t2 by a laminar flow field of flow rate of magnitude Qz. Figure 2b shows the particle trajectories during the second step. As a consequence of the parabolic flow profile, the infinitely fast particle moves farther to the left than the real particle. The flow velocity profile for the narrow slit geometry is
uf = ~ 3 x ( b - x)
(1)
Over the course of one cycle the net displacement of the zero constant and of the infinitely fast particle can be readily calculated from Equation (1) since the x-positions of those particles are constant during individual steps of the cycle. Hence, these y-direction displacements are Ayz¢ _ 3x(b - x)Q2t2(~ - 1) b3 Ayif --
3
~'~ Q2t2
(2a)
(2b)
represents the relative volumes of fluids displaced in the two steps of the cycle. Real particles will be displayed some intermediate distance between the limits given by Equations (2a) and (2b). According to Equations (2a) and (2b), for ~ = 1, zero contrast particles return to their original position while infinitely fast particles move to the left over one cycle. Real particles would also tend to move leftward. Repetition of this two-step cycle results in a type of chromatographic separation where particles of different acoustic response speeds arrive at the left end of the chamber at different times. Operation with ~ # 1 results in a net displacement of suspension through the chamber. The net amount of suspension obtained at the right end of the chamber, averaged over the time of the cycle, Qprod, is Oprod --
Q,tl -- Q2t2 tl + t2
--
Q2(a - 1)
(Q2/Qt o~+ 1)
(3)
For continuous operation of the separation method, feed can be introduced at the left end of the acoustic chamber at an equivalent rate. When the process is operated with ~ > 1, there will be a net displacement of zero contrast particles to the right (Ayzc > 0) and infinitely fast particles to the left (Ayif < 0) over the course of one cycle. Consequently, some types of the real particles would move to the left and others to the right. For each combination of flow rates and sound intensities there will be one type of real particle (denoted as the cut particle) that will have zero displacement over one cycle. Particles faster than the cut particle will concentrate at the left end of the chamber while particles slower than the cut particle will emerge at the product stream at the right end of the chamber.
Theoretical background The basic theory and simplifying assumptions that are used to model the particle response in a standing wave field maintained in the acoustic chamber are described here. First, a brief summary of the primary acoustic force acting on particles is presented. This is followed by a discussion about the resonant frequencies and the location of the velocity antinodes in a composite acoustic resonator composed, in general, of piezoelectric (acoustic transducer) and non-piezoelectric (reflector, separator, suspension) layers. Forces acting on particles suspended in acoustic standing wave fields The primary acoustic force acting on an individual spherical particle with radius much smaller than the wavelength of sound, suspended in a planar standing. wave acoustic field, created by oppositely travelling, single frequency, sinusoidal sound waves is 13 Fac = 4rncR3Eac F sin (2xx)
(4)
where F is the acoustic contrast defined as l ( 5 p p - 2pf V = 3 \ 2pp + pf
7p~ ~/
Ultrasonics 1994 Vol 32 No 2
(5)
115
Ultrasonic and flow-field fractionation of suspensions: Z. Mandralis If F is positive, as is the case in the experiments presented later, particles are urged to the velocity antinodes of the standing wave pattern; if F is negative, the particles are collected at the velocity nodes. In addition to the primary acoustic force, other forces might arise due to diffraction effects and/or non-uniform generation of the acoustic field v'~3. Secondary acoustic forces 14 caused by the interaction of the secondary acoustic fields emitted from each particle to other particles may cause agglomeration of particles. Bernoullitype attraction forces also arise from the rapid oscillation of fluid between neighbouring particles as. Scaling arguments ~6 show that the primary acoustic force is at least two orders of magnitude larger than all other acoustic forces and hence these are to be neglected in the analytical model. For the micrometre-sized particles used in this study, inertia and Brownian forces can also be neglected. In order to judge the effectiveness of the E S U F F F for fractionation of fine particles, we compare the magnitude of acoustic forces to the gravitational force used in P F S F to move the particles within the separation chamber. The ratio of the maximum primary acoustic force to gravitational force is
Fac f g,
3xE~¢F (pp -- pf)g
Resonant frequencies and acoustic velocity profiles in acoustic resonators The acoustic force that acts on a particle suspended in a standing wave field depends on the acoustic energy density of the sound field and on the position of the particle relative to the velocity node. The general one-dimensional treatment of layered piezoelectric resonators iv was applied to determine the resonant frequencies and the acoustic velocity profiles in the acoustic chamber for each of the two steps of the E S U F F F cycle. This theory gives an exact solution of the fundamental equations of linear piezoelectricity subject to ascribed boundary conditions at the terminating surfaces and at the electrical ports of the resonator. The properties of the various layers, chosen to reflect those of the materials used in the experiments to be described later, are given in Table 1. Since only the longitudinal mode is excited electrically, only the elastic (e .E. . . ), dielectric (~s=) and piezoelectric (e==) constants affecting this mode are used. The constants of the transducer were determined by fitting the formula for the admittance for stiffened modes 18 to the measured admittance of the transducer in air 19 using many data points in the vicinity of resonance. In the simulation, the amplifier was replaced by an ideal voltage source with a 50 fl resistance in series. Figures 3 and 4 show the acoustic velocity profiles for the predicted resonant frequencies of 0.486 MHz and 1.385 MHz respectively. The positions of each member of this resonator were chosen to mimic the actual dimensions of the experimental chambers used in this study. Driving voltages of 1 V peak-to-peak were assumed for both cases. At the lower resonant frequency (Figure 3), a velocity antinode is located at the centre of the acoustic chamber (inside the midplane barrier) and the velocity profile exhibits some asymmetry. The velocity profile for the higher resonant frequency (Figure 4) shows greater symmetry with respect to the midplane, and two velocity antinodes which are positioned midway between the barrier and the walls. In our application, the separation efficiency is enhanced if the velocity antinodes are positioned exactly midway between the midplane barrier and the nearest wall during the high-frequency step. The lack of symmetry in the acoustic velocity profile during the low frequency step is not as critical to separation efficiency as long as the antinode is positioned within the midplane. Although we did not attempt to optimize the geometry to provide symmetric acoustic fields in both steps of the cycle, it should be possible to achieve this condition. The acoustic energy density used in Equation (4) for the acoustic force acting on the particle for each step of
(6)
For a typical polystyrene-water suspension (R = 3 ktm, pp -- pf = 50 kg m-3, F = 0.23), irradiated by 1 MHz, 1 0 0 J m -3 sound (~c=4234m-1), Equation (6) shows that the acoustic force is three orders of magnitude larger than the gravitational force. The speed at which particles are urged towards the collection plane, which can be found by equating the Stokes law drag on the particle with the primary acoustic force, is a sinusoidal function of the particles' distance from the velocity node (see Equation (4))
2tCEa~RZFsin (2Kx) up =
et al.
(7)
3/~
The maximum velocity at which a particle traverses the acoustic chamber under the influence of the acoustic field is about 0.58 mm s- 1 according to Equation (7). This is equivalent to sedimentation rates of equivalent polystyrene particles larger than 400 #m. Thus, utilizing acoustic fields in E S U F F F has a strong potential to extend the processing philosophy of P F S F to submicrometre sized particles or neutrally buoyant particles.
Table 1 Values of the material constants and thicknesses for the transducer, water, the steel separator, and the glass reflector used by the general one-dimensional treatment of layered piezoelectric resonators Transducer
c~zzz,(GPa)
123.3
e.... ( N V
16.7
1m-r)
+ j4.11
Water 2.2
+ j0.0011
Separator 286.3
-
-
+ j0.0573
Reflector 74.3
~sz
770 + j3.85
80 + j100
-
Thickness (mm)
2.03
0.54
0.076
0.9
4.7 + j0.018
Density (kg m -3)
7700
998
8087
2500
Frequency parameter (m s 1)
2310
742
2975
2725
30
2000
5000
1000
Quality factor
116
Ultrasonics
1994 Vol 32 No 2
+ j0.0743
Ultrasonic
and flow-field
U o (cbservatic~ chamber)
0.15 0.10
Transd ....
/
u wor in
~
/
r
~
Reflector
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Figure 3 Prediction of the acoustic velocity profile for the observation chamber at the lower frequency (0.486 MHz) assuming 1 V peak-to-peak driving voltage. The dotted line indicates the magnitude of the maximum acoustic velocity in the actual (clamped) working chamber if it was driven at 1 V peak-to-peak
the cycle can be calculated by integrating the kinetic and potential energy contributions of the sound field over the internal volume of the chamber. For a simple harmonic acoustic field, this yields pfU 2
(8)
4
y*
(1 lb)
(2x,/n)y
=
(12)
3XZQlt/n 2
=
c3x*
t~t*
Position (mm)
-
(1 la)
The height of the chamber is approximately equal to one half wavelength of the lower frequency used. In dimensionless form, the particle trajectories for the first step of the cycle are
~\
0.00
E.c
where Uo is the maximum acoustic velocity generated. Even though the acoustic velocity profiles given in Figures 3 and 4 are not perfectly harmonic, Equation (8) is believed to give a reasonable estimate of the acoustic energy density for the chambers used in this study. The results shown in Figures 3 and 4 can be extended to determine the acoustic velocity profiles under operating conditions other than for a 1 V peak-to-peak driving voltage by assuming proportionality of the acoustic velocity profile to the driving voltage. In this case, the acoustic velocity profiles can be found by multiplying the results in Figures 3 and 4 by the ratio of the actual driving voltage to 1 V peak-to-peak.
Process m o d e l The process model contains a description of particle trajectories due to the combined response of acoustic, gravitational and laminar flow fields. At steady-state, the force balance in the x-direction includes the drag force on the particle (described by the Stokes drag law) and the acoustic force given by Equation (4)
-
A sin (nx*)
-
4x*(1
(13a)
and 0y* 0t*
-
x*)
+
(13b)
B
For the second step, a sound field of higher frequency and possibly different intensity is applied. The trajectory equations in this case are 0x*
0t*
-
FflA sin (Fnx*)
(14a)
and ay* 0t*
- -46x*(1 - x*) + B
(14b)
Five dimensionless groups arise. Here, A is the ratio between the transverse speed of the particles due to acoustic forces and the lateral speed of fluid flow
A
-
41zEae'lR2F
(15)
9#Q1 while B is the ratio between the gravitational sedimentation speed of the particles and the lateral speed of fluid flow
B = 4nRE(pp
(16)
pf)g 27#xtQ1 - -
Also, F is the ratio of the higher to the lower frequency, fl is the ratio of the intensity of the acoustic field at the high frequency relative to that at the low frequency, and 6 is the ratio of the flow rate of the second step to the flow rate of the first step.
0.20
-
et al.
x* = (2xl/n)x t*
0.05
<
o f s u s p e n s i o n s . Z. M a n d r a l i s
frequency step
0.20
o ~
fractionation
6rc#R~tt + 4~zR3xEacFsin (2xx) = 0
The motion of the particle in the y-direction, in each of the subvolumes, is governed by the drag force due to the fluid flow and the gravitational force (particle-wall and particle-particle interactions are neglected)
O.15
+ mg=O
(10)
where uf is the local flow velocity given by Equation (1) and gravity is arbitrarily defined to be in the positive y-direction. For convenience, these trajectory equations are cast in a dimensionless form. The characteristic length is based on the wave-number at the lower frequency and the characteristic time on the flow speed during the low
~3 U o (observation chamber)
0o 0.10 o
8
6z#R u f -
Reflector
Transducer
(9)
0.05 U o (w~king chamber) -'T'-~-'~
0.00
0.0
0.5
,~.---'~-~,,,, , , , 10
1.5
2.O
Position
2.5
2
B.O
3.5
4.0
(mm)
Figure 4 Prediction of the acoustic velocity profile for the observation chamber at the higher frequency (1.387 M Hz) assuming 1 V peak-to-peak driving voltage. The dotted line indicates the magnitude of the maximum acoustic velocity in the actual (clamped) working chamber if it was driven at 1 V peak-to-peak
Ultrasonics 1994 Vol 32 No 2
117
Ultrasonic and flow-field fractionation of suspensions." Z. Mandralis et al. 0.40 t, \ 0.35 '
t~ =82.86
~x 1.25
030 '\ \,
~.75
a cut 0.°25 150.20 I \' :'\'~ ~ ~ _ ~ ,
i - - ~
~-~-h~ ~ ~
2.752.25
0.10
3.25
0.05
X~g
0.00 O. 10
Figure 5
= 0.30
~ 0.50
= 0.90
'
0.70
1.10
Sample model predictions forNoot as a function of//and e
0.40 0.35 \\ \ 030 \ A cut 0 2 5
020
t =82.86 c~
\ k\
\
\\
015 010
\
125 3.75 ' " ~-_ -_
125 '~SLJFFF] -'-''--- _ _ -_
005 000 0.10
0.30
0.50
070
090
110
Figure 6 Comparison of the fractionation potential for ESUFFF and SUFFF
Model predictions A numerical integration method 2° was used to solve the system of Equations (13) and (14). The trajectories of a large number of particles, initially distributed uniformly over the cross-section of the channel at y * = 0, were traced. The average displacement of the group of particles at the end of the two-step cycle is calculated as the arithmetic mean of the individual displacements. The case of negligible gravity (B = 0) will be presented here. The effect of gravity on the fractionation performance has been discussed in relation to the S U F F F method 1°. After simulating about five cycles, for fixed values of ~, /3, F and 6, a unique A (denoted Acut) can be identified, by a trial-and-error procedure, for which the average displacement of particles over one cycle is zero. Particles with A > Aeut have displacements of opposite direction from particles with A < Ac,t. As an example, consider Figure 5 in which A~ut is shown as a function of/3 for various values of the parameter and for t* = 82.86. As ~ decreases, Aeut decreases. The production rate, given by Equation (3), also decreases. As/3 increases, Aeut decreases because particles respond faster to higher energy density acoustic fields. F r o m Figure 5 it is apparent that A¢,t is most sensitive to fl in the range 0.1 < fl < 0.5. Next, we compare the efficiency of E S U F F F relative to S U F F E in Fioure 6. At equivalent production rates (~ = 1.25 in this example) the value of Acre is much lower in E S U F F F than in S U F F F indicating its improved capability for handling smaller particles. Also, it is
118
Ultrasonics 1 9 9 4 Vol 32 No 2
possible to compare production rates achievable by the two methods in accomplishing a given separation task. F r o m Figure 6 it can be seen that approximately the same Aout is obtained for ~ = 3.75 in E S U F F F as for = 1.25 in SUFFF. For a cycle operated with 6 = 1, Equation (3) leads to the result that the production rate in E S U F F F would be 5.2 times as high as in SUFFF.
Experimental procedure Proof-of-principle experiments were performed in the acoustic chamber shown in Figure 7. A rectangular lead zirconate titanate transducer (APC 880, 38.1 × 25.4 × 2.03 mm) supplied by the APC company was supported on a rubber gasket (Neoprene 0.81 mm) in an aluminium frame. A rectangular glass reflector (38.1 × 25.4 x 0.9 mm) was similarly placed in another aluminium frame. Two Teflon spacers (0.533 m m each) were placed between the two frames. Their central sections were cut out so that when the whole assembly was clamped together a sealed rectangular channel was defined. A thin stainless steel sheet (0.076mm) was placed between the Teflon spacers so that it separated the rectangular channel into two identical subvolumes. At the two ends of the channel, ports allowed feed introduction and product removal with the help of a computerized peristaltic pump (Masterflex 7550-90). The electrical excitation for the transducer was generated by a K R O H N - H I T E 2100A generator and amplified by an E N I 240L power amplifier. The electrical energy input was measured by a B I R D 4410A wattmeter. The experiment was controlled by an AT-type computer. The low and high resonant frequencies were experimentally determined to be 0.481 M H z and 1.328 MHz. To avoid the complications of gravity, the chamber was oriented such that fluid flow was in the vertical direction with feed introduced at the bottom of the chamber and the product stream withdrawn from the top. Particles smaller than the cut size, emerging at the product stream, are collected and analysed. Particles larger than the cut size concentrate at the bottom of the acoustic chamber and are removed at the end of the experimental run. A purge stream for removal of particles from the bottom of the chamber would be required for truly continuous operation. However, the experimental procedure followed closely resembles continuous operation for low feed concentrations and short operation times. The particle size distribution and particle number density of the product stream was determined by placing a small sample of the stream between two microscope
FEED
,J, v
2ASKET
TEFLON
~V////////////////////A~ | /
<
Gravity
TRANSDUCEIR ~
I1<
38.1 mm
l
>1 /~z PRODUCT
Figure 7 Schematic of the fractionation chamber used in the experiments
Ultrasonic and flow-field fractionation of suspensions. Z. Mandralis et al.
E x p e r i m e n t a l results - c o m p a r i s o n s w i t h theory
(a)
(b)
(c) Figure 8 Photographs illustrating the acoustic response of particles: (a) suspension before the application of ultrasound; (b) collection of particles on the separator during the low frequency step; (c) collection into planes located midway between the separator and the walls during the high frequency step
slides that were spaced 0.076 mm apart. The CUE-2 image analysis software installed on an AT computer and an optical microscope were used to analyse the particle size distribution. For each measurement, a large number of particles was analysed in order to get statistically meaningful results. The same procedure was followed for determining the characteristics of the feed stream. Using this method, particles larger than 2 #m were accurately detected and analysed.
In order to confirm that resonant acoustic fields could be used to drive particles toward the midplane barrier and to positions midway between the barrier and a wall, a second cell was constructed for visualization purposes. It incorporated an acoustic transducer and a reflector similar to those used by the chamber described above. The reflector was attached on a micrometer controlled stage so that precise positioning ( _+5/~m) was achieved. The side walls were made of Plexiglas that permitted direct observation through the chamber. The first of the photographs of Figure 8 shows a uniform suspension of 325 Mesh polystyrene particles between the transducer and the reflector. The thin stainless steel separator can also be seen at the midplane. The second photograph demonstrates the collection of the particles on the separator, after irradiation with 0.486 MHz sound. The third photograph illustrates the collection of particles midway between the centre and the walls of the acoustic chamber, after irradiation with 1.387 MHz sound. For the working separation cell in which the fractionation was attempted, the experimentally determined resonant frequencies (0.481 and 1.328 MHz) for the two-step cycle are slightly lower than the resonant frequencies predicted by the general one-dimensional treatment of piezoelectric resonators. This is attributed to the fact that the constants for the transducer were obtained by measuring the admittance of the transducer in the observation cell and not in the working chamber. Unlike the transducer of the observation cell, the transducer of the working chamber is tightly clamped. Also, there is some uncertainty in the dimensions of the various layers within the composite resonator. The use of a conventional hydrophone to measure acoustic energy density was not possible due to the small dimensions of the acoustic chamber. One way to calculate the acoustic energy density is by using Equation (8) in conjunction with the acoustic velocity profiles from Figures 3 and 4 and a measurement of the driving voltage as discussed previously. However, the results shown in Figures 3 and 4 were obtained for the observation chamber, which had an unclamped transducer, while the transducer of the working chamber was clamped. This clamping is expected to reduce the intensity of the sound field. An alternative method, developed in our laboratories, was used to measure the acoustic energy density in the actual chamber 2°. According to this method, a micrometre-sized sphere is suspended in the standing wave field of the actual chamber used for the fractionation experiments, and its equilibrium position is precisely determined using a microscope. By balancing the acoustic force predicted by Equation (4) with the buoyant weight of the particle, the acoustic energy density can be determined to within 20%. The values of acoustic energy density determined this way were used for comparisons of the experimental results with theoretical predictions. To visualize the effects of clamping on the energy density in the chamber, Equation (8) was used to calculate U o in the working chamber from the value of Eac determined from the levitated sphere measurements described above. Again, assuming proportionality of the acoustic velocity profile to the driving voltage, it is possible to calculate the magnitude of U o expected if a
Ultrasonics 1994 Vol 32 No 2
119
Ultrasonic and flow-field fractionation of suspensions: Z. Mandralis T a b l e 2 Particle number density and size distribution in the product stream as a function of electric energy input, c~-1.25, -0.16. The table entry for 0 W indicates the statistics of the process feed Power input (W)
Particle concentration (cm 3)
Mean radius (#m)
Standard deviation (/zm)
0 0.25 1.05 2.40 3.60
2 244,094 980,314 318,897 297,900 212,598
10.0 6.76 4.54 3.82 3.23
5.50 3.82 2.40 2.11 1.49
O
et al.
stream for the same experimental conditions used in Table 2. The vertical dashed line is the theoretical prediction from the model described in the previous section. The agreement between the experiments and the model are generally good; however, Figure 9 shows that the experimentally achieved fractionation is not as sharp as the theoretical prediction. In all cases, few particles larger than the theoretically predicted cut size emerge at the product stream. On the other hand, many of the smaller particles that should, theoretically, emerge at the product stream, were trapped in the acoustic chamber. This is a result of the secondary forces that cause agglomeration of smaller particles into clusters, which then behave as large particles. For the higher energy densities used, there are some particles in the 5-10 #m range that emerge at the product stream. This can be explained by the fact that smaller particles are only slightly affected by the combined effects of the alternating sound field and the bidirectional flow field. Thus, small flow disturbances due to wall and edge effects or deviations from the uniform, planar standing wave field configuration might change the final destination of these particles.
Conclusions
o
6
1o
16
20
Particle
2~
~
Radius
e
~
1o
15
2o
2~
m
(¢~,m)
Figure 9 Product stream particle size distributions obtained for several values of Eat,1 with ~ = 1.25 and /~ = 0.16. Also shown in each graph are the particle size distributions for the polystyrene suspension fed to the chamber. The vertical dashed lines represent the theoretical cut particle size predicted from the model
1 V peak-to-peak driving voltage was used. These values are shown on Figures 3 and 4 as dotted lines. Note that values of U o for the working (clamped) chamber are roughly one-third of the predicted values for an unclamped chamber. Several experiments were performed to demonstrate the effectiveness of the fractionation technique and obtain data to complement the theoretical development. The ratio of the acoustic energy density of the second step to that of the first step was set arbitrarily to fi ~ 0.16 for all the experimental runs. The flow rate was fixed at Q1 = Q2 = 7.36 x 1 0 - 6 m 2 s -1. Suspensions containing 2% w/w of 325 mesh polystyrene particles in distilled water, which contained about 0.01% w/w sodium dodecyl sulfate (SDC) to prevent particle agglomeration, were fractionated. Experiments were performed at a = 1.25 (tl = 4.13 s, t 2 ---- 5.16 s) which resulted in Q p r o d : 0.81 X 10- 6 m 2 s- 1. The electric energy input varied in the range 0.2-3.7 W. Table 2 shows the particle number density, mean particle radius and standard deviation for the feed and product streams, at steady-state, for various energy density levels. The number of particles in the product stream as well as the mean particle radius and standard deviation decrease as the acoustic energy density increases, in agreement with theoretical expectations. A more detailed picture of the effect of the energy density on the particle cut size is given in Figure 9, which shows the particle size distribution of the feed and the product
120
Ultrasonics
1994 Vol 32 No 2
A fractionation method for fine particles in liquid suspensions has been developed. The fractionation is based on the transient response of fine particles to alternating low and high frequency ultrasonic fields. The separation strategy is similar to that of PFSF. However, substitution of the gravitational feld with more powerful and controllable acoustic fields enables sharp fractionation of micrometre-sized and neutrally buoyant particles. Sharp fractionation of micrometre-sized particles was demonstrated for a 325 mesh polystyrene suspension. The sharpness of fractionation can be further increased by increasing the number of separation stages. For a given device, this can be achieved without hardware changes by reducing the volume of liquid displaced in each step of the cycle. For a given separation task, higher production rates than S U F F F can be achieved by ESUFFF. The method can be scaled up by constructing chambers in which several transducers are combined into a mosaic structure. In addition, the distance between the transducer and the reflector can be several wavelengths, thus increasing the active volume of the acoustic chamber. Thin or acoustically transparent layers can subdivide the chamber into many parallel channels serviced by a single generator, amplifier and pumping mechanism. For general computer-aided design of separation chambers useful for ESUFFF, the general one-dimensional treatment of piezoelectric resonators can be used. However, the complicating effects of clamping on the cell make the acoustic energy density generated within a working chamber more difficult to predict. Instead, a simple method for determining the acoustic energy density in the operating chamber was used. This was based on a balance ~)f the acoustic and gravitational forces acting on a microsphere suspended in the sound field. The predictions were generally in agreement with the experimental results.
Acknowledgements Nestle has provided partial support for this research through a Westreco Fellowship. The authors also wish
Ultrasonic and flow-field fractionation of suspensions. Z. Mandralis et al.
to thank the Austrian Science Foundation (project P7198-PHY) and the Erwin Schroedinger Society for financial support.
10 11
12
References 1 2 3 4 5 6 7 8 9
Tolt, T.L. and Feke, D.L. Analysis and application of acoustics to suspension processing ASME. 23rd lntersociety Energy Conversion Engineering Conference (1988) 4 327 Tolt, T.L. and Feke, D.L. Separation of dispersed phases from liquids in acoustically driven chambers Chem En O Sci (1993) 48 527-540 Tolt, T.L. and Feke, D.L. Separation devices based on forced coincidence response of fluid-filled pipes J Acoust Soc Am (1992) 91(6) 3152-3156 Hager, F., Benes, E., Bolek, W. and Gr6sehl, M. Investigation of a new ultrasonic drifting resonance field cell for the refinement of aerosols Proc 9th FASE Symposium Hungary (1991) 261-266 Whitworth, G., Grundy, M.A. and Coakley, W.T. Transport and harvesting of suspended particles using modulated ultrasound Ultrasonics (1991) 29 439-444 Schram, C.J. US Patent No. 4,743,361 (1988) Schram, C.J. and Rendell, M. Manipulation of particles in megahertz standing waves Proceedings Ultrasonics International Conference (1989) 262-267 Peterson, S.C. US Patent No. 4,759,775 (1988) Mandralis, Z.I. and Feke, D.L. Continuous suspension fractionation
13 14 15 16 17 18 19 20
using acoustics and divided flow fields Chem Eng Sci (1993) 48 3897-3905 Mandralis, Z.I. and Feke, D.L. Fractionation of fine particle suspensions by synchronized ultrasonic and flow fields AIChE J (1993) 39 197-206 Adler, R.J., Palepu, P.T. and Wu, C.K. Periodic flow slurry fractionation Chemeca '88 (1988) 892-897
Papann, J.S., Adler, R.J., Gorensek, M.B. and Menon, M.M. Separation of fine particle dispersions using periodic flows in a spinning coiled tube AIChE J (1986) 32 798-808 Yosioka, K. and Kawasima, Y. Acoustic radiation pressure on a compressible sphere Acustica (1955) 5 167-173 Apfel, R.E. Acoustically induced square law forces and some speculations about gravitation Am J Phys (1988) 56 726-729 Weiser, M.H., Apfel, R.E. and Neppiras, E.A. Interparticle forces on red cells in a standing wave field Acustica (1984) 56 114-119 Mandralis, Z.I., Feke, D.L. and Adler, R.J. Transient response of fine-particle suspensions to mild, planar, ultrasonic fields Fluid/ Particle Sepn J (1990) 3 115-121 Nowotny, H., Benes, E. and Sehmid, M. Layered piezoelectric resonators with an arbitrary number of electrodes (general one-dimensional treatment) J Acoust Soc Am (1991 ) 90 1238-1245 IEC Standard 483 Guide to Dynamic Measurements of Piezoelectric Ceramics with High Electromechanical Coupling (1976) Geneve, Switzerland Schmid, M., Benes, E. and Sedlaszek, R. A computer controlled system for the measurement of complete admittance spectra of piezoelectric resonators Meas Sci Technol (1990) 1 970-975 Mandrafis, Z.I. Fractionation of fine-particle suspensions by combined ultrasonic and flow fields. PhD Dissertation (1993) Case Western Reserve University, Cleveland; Ohio, USA
Ultrasonics 1994 Vol 32 No 2
121
Received 30 June 1993; revised 15 September 1993 A fractionation method for fine-particle suspensions based on variations in the speed of response to ultrasonic standing wave fields has been developed. The method is sensitive to a combination of the particle size as well as the particle and suspending fluid density and compressibility. The fractionation is accomplished within a narrow-gap rectangular channel having a solid barrier positioned along its midplane. Ultrasonic standing wave fields of two alternating frequencies are applied across the gap to induce a partial separation at short irradiation times. A co-ordinated bidirectional laminar flow field is used to transform these partial separations into useful separations along the chamber. In comparison with similar fraction strategies that use other types of fields to accomplish the separation, the acoustic forces acting on particles can be quite strong, thereby enabling fast, continuous, and controllable fractionation of micrometre-sized particles. An analytical model based on the trajectories of particles in response to the acoustic and flow cycles was developed. Model predictions indicate how the fractionation can be controlled through the choice of cycle parameters. Experimental results using a 325 mesh polystyrene particle suspension demonstrate that sharp fractionations of nearly neutrally buoyant micrometre-sized particles can be achieved.
Keywords: suspensions; fractionation; standing wave; acoustic force
Nomenclature 2b c Ea¢ f F g j2 m
Q R t uf Uo x y
channel height (m) speed of sound (m s- i) time averaged acoustic energy density (J m-3) frequency (s- l) acoustic contrast (Equation (5)) (dimensionless) gravity acceleration (9.81 m s-2) -1 4~(pp _ pf)R 3 = buoyant particle mass (kg) flow rate per unit width ( m 2 s - 1 ) particle radius (m) time (s) local flow velocity (m s- l) velocity amplitude of incident acoustic wave (m s- i) particle position from acoustic velocity node (m) particle displacement in y-direction (m)
Greek symbols ct
Q2t2/Qltl
fl
Eac,2/Eac,1
F 7 6
f2/fl l/pc 2 = compressibility (m 2 N - l) QE/Q1
0041 -624X/94/020113-09 O 1994 Butterworth-HeinemannLtd
K 2 u P
27z/2 = wave-number (m-l) wavelength of sound (m) viscosity (Pa s) density (kg m - 3)
Superscripts and Subscripts f if p zc
x 2 prod *
fluid infinitely fast particle particle zero contrast particle step 1 step 2 product stream dimensionless variable
The ability to fractionate fine particles sharply may be useful in a variety of applications. For example, nearly monodisperse particles may have advantages for diagnostic and treatment purposes in the life sciences. More effective separation of mammalian cells, bacteria, or yeasts is desirable in biotechnology. Fabrication of ceramics using narrow-sized ceramic particles may result in superior physical properties of the resulting materials. Also, the efficiency of a chromatographic packing is known to correlate with its degree of monodispersity.
Ultrasonics 1994 Vol 32 No 2
113
Ultrasonic and flow-field fractionation of suspensions. Z. Mandralis et al. Many conventional fine-particle fractionation methods require excessive time, high pressure drops, or intense fields to accomplish the separation of micrometre-sized, neutrally buoyant, or neutrally charged particles. Also, the resolution of these methods is often quite limited. In this paper, we report on a fractionation method that uses ultrasonic standing wave fields to accomplish a fast, sharp, efficient and controllable fractionation of fineparticles in liquid suspension. During the past decade, several filtration methods based on the response of fine particles to ultrasonic standing wave fields have been developed 1 9. A method based on the combined transient response of particles to ultrasonic standing wave fields and bidirectional laminar flow fields (denoted SUFFF, an acronym for Synchronized Ultrasonic and Flow Field Fractionation) has also been reported 1°. This method was shown to enable fast, sharp and controllable fractionation of fine particle suspensions but had a limited processing capacity per unit volume of the separator chamber. Here, we report on an enhanced version of this method (denoted E S U F F F for Enhanced Synchronized Ultrasonic and Flow Field Fractionation) that is a hybrid of S U F F F and Periodic Flow Slurry Fractionation 11 (denoted PFSF). The P F S F method fractionates particles on the basis of differences in the gravitational sedimentation rates of the various particles. In cycles combining steps of sedimentation, re-suspension and bidirectional axial flow, the P F S F method transports faster sedimenting particles to one end of a long tube while slower sedimenting particles move to the other end. Mineral particles as small as 10/~m or polymeric particles as small as 100/tm, differing by only few percent in their sedimentation rates due to size and/or density differences, can be effectively separated. The advantage of PFSF over classical processes (screening, flotation, etc) is that it is continuously adjustable as to the cut point and sharpness of separation without hardware changes. However, for small particle sizes or low density contrast between the particles and suspending fluid, production rates are extremely low due to the slow sedimentation and re-suspension steps of the cycle. Operation of the method in centrifugal fields enhances sedimentation rates but re-suspension and secondary fluid flows become problematic 12. The E S U F F F method adopts the basic fractionation philosophy of P F S F with the exception that, instead of using gravitational fields, acoustic fields are employed to drive particle motion. Similar to SUFFF, the fractionation takes place in a relatively long and wide rectangular chamber wherein bidirectional, laminar flows along the chamber are co-ordinated with a cycle of alternating low and high frequency ultrasonic standing wave fields propagated across the chamber. The low and high frequency steps are analogues of the sedimentation and re-suspension steps in PFSF. Unlike SUFFF, the chamber is divided into two equal subvolumes by a thin barrier parallel to the transducer, located at the midplane of the chamber. As will become evident in the next section, the presence of this barrier and the resulting laminar flow profile enhances the processing rate per unit volume of the separator by about an order of magnitude, in comparison with SUFFF. Like PFSF, E S U F F F retains the advantages of the potential for narrow and precise cuts, continuous adjustability as to the cut point and sharpness of fractionation without hardware changes. However, for
114
Ultrasonics 1994 Vol 32 No 2
micrometre-sized particles, acoustic forces can be ordersof-magnitude higher than gravitational forces, and hence efficient fractionation of particles in this size range are anticipated. Also, unlike PFSF, E S U F F F can be used for neutrally buoyant particles as long as they exhibit a different compressibility than the suspending medium. The next section details the process concept. The theoretical background on the acoustic force that acts on the particles, the resonant frequencies, the energy density, and the location of particle collection positions in the standing wave field in an acoustic chamber are given next. Model predictions of the fractionation efficiency are presented and the results are compared with equivalent predictions for SUFFF. Finally, proof-ofprinciple experimental results of the fractionation of 325 mesh polystyrene particle suspensions are presented and compared with model predictions.
Process concept Figure 1 shows the geometry of the separation chamber. The lateral walls comprise an acoustic transducer at the bottom and a reflector at the top. A thin barrier placed at the midplane divides the chamber into two identical subvolumes. To perform the fractionation, a two-step acoustic cycle is applied. One of the two frequencies for the acoustic field is chosen so that a velocity antinode is positioned on the midplane barrier; the second frequency is selected so that velocity antinodes are positioned midway between the barrier and the lateral walls. As will be discussed in the Theoretical background section, these conditions can readily be achieved. The primary acoustic force urges the particles to move in the x-direction towards the velocity antinodes of whatever standing wave field is being propagated. Thus, the effect of the acoustic cycle is to drive particles alternatively towards the barrier and then towards the midway point between the barrier and a wall. In practice, the duration of each step of the cycle is chosen to be short compared with the time it takes for particles to arrive at the velocity antinodes. Synchronized with the two-step acoustic cycle is a bidirectional flow cycle. The particles are assumed to track fluid motion perfectly in the y-direction. During the low frequency step of the cycle, suppose that the laminar flow is directed toward the right. Conversely, during the higher frequency portion of the cycle, the flow is assumed to be leftward. To demonstrate and quantify the operation principle more precisely, the trajectories of three types of particle are shown in Figure 2. These particles are distinguished on the basis of their speed of response to the acoustic field: (1) an infinitely fast particle which responds
~//////////J////S~ REFLECTOR
Acoustic
Acoustic
Force
vet~ity
O/2
Separet~
Q/2
y
2/////////////////////////////..¢'////////////////////~, TRANSDUCER
Figure 1
Geometry of the acoustic chamber
Ultrasonic and flow-field fractionation of suspensions: Z. Mandralis et al. O Zero Contrast
• Infinitely East
• Real
FT////////////////////////A
Step 1
where Acoustic Acoustic Velocity Force
Qltx Qzt2
Flow
Acoustic Acoustic Veloclt
Force__
Step 2 b
Figure 2 Illustration of the fractionation concept. The trajectories of an infinitely fast particle and a zero-contrast particle, as well as a real particle are s h o w n for (a) Step 1; (b) Step 2
instantaneously to the sound field; (2) a zero-contrast particle that is insensitive to acoustic forces (and thus behaves equivalently to a fluid element); (3) a real particle with intermediate response to the sound field. Due to symmetry of the acoustic and flow fields about the separator, only the upper half of the acoustic chamber is shown in Figure 2. Also, for simplicity, the effect of gravity is neglected in this discussion. During the first step of the cycle, an acoustic standing wave field of frequency f l is established in the chamber so that a velocity antinode is located inside the barrier. The acoustic force drives particles towards the central barrier. Simultaneously, a laminar flow of magnitude QI transports the particles to the right for time tl. Figure 2a shows that the infinitely fast particle moves immediately to the separator and maintains its original y-position, the real particle moves in the same direction at a finite rate, and the zero contrast particle maintains its original x-position but moves rightward with the flow. Owing to the parabolic flow profile, the zero contrast particle moves farther to the right than the real particle, which in turn moves farther to the right than the infinitely fast particle. For the second step, the sound frequency is switched to f2 so that a velocity antinode is located midway between the barrier and the wall. Particles are urged by the sound field towards a plane located midway between the wall and the barrier. They are also transported to the left for time t2 by a laminar flow field of flow rate of magnitude Qz. Figure 2b shows the particle trajectories during the second step. As a consequence of the parabolic flow profile, the infinitely fast particle moves farther to the left than the real particle. The flow velocity profile for the narrow slit geometry is
uf = ~ 3 x ( b - x)
(1)
Over the course of one cycle the net displacement of the zero constant and of the infinitely fast particle can be readily calculated from Equation (1) since the x-positions of those particles are constant during individual steps of the cycle. Hence, these y-direction displacements are Ayz¢ _ 3x(b - x)Q2t2(~ - 1) b3 Ayif --
3
~'~ Q2t2
(2a)
(2b)
represents the relative volumes of fluids displaced in the two steps of the cycle. Real particles will be displayed some intermediate distance between the limits given by Equations (2a) and (2b). According to Equations (2a) and (2b), for ~ = 1, zero contrast particles return to their original position while infinitely fast particles move to the left over one cycle. Real particles would also tend to move leftward. Repetition of this two-step cycle results in a type of chromatographic separation where particles of different acoustic response speeds arrive at the left end of the chamber at different times. Operation with ~ # 1 results in a net displacement of suspension through the chamber. The net amount of suspension obtained at the right end of the chamber, averaged over the time of the cycle, Qprod, is Oprod --
Q,tl -- Q2t2 tl + t2
--
Q2(a - 1)
(Q2/Qt o~+ 1)
(3)
For continuous operation of the separation method, feed can be introduced at the left end of the acoustic chamber at an equivalent rate. When the process is operated with ~ > 1, there will be a net displacement of zero contrast particles to the right (Ayzc > 0) and infinitely fast particles to the left (Ayif < 0) over the course of one cycle. Consequently, some types of the real particles would move to the left and others to the right. For each combination of flow rates and sound intensities there will be one type of real particle (denoted as the cut particle) that will have zero displacement over one cycle. Particles faster than the cut particle will concentrate at the left end of the chamber while particles slower than the cut particle will emerge at the product stream at the right end of the chamber.
Theoretical background The basic theory and simplifying assumptions that are used to model the particle response in a standing wave field maintained in the acoustic chamber are described here. First, a brief summary of the primary acoustic force acting on particles is presented. This is followed by a discussion about the resonant frequencies and the location of the velocity antinodes in a composite acoustic resonator composed, in general, of piezoelectric (acoustic transducer) and non-piezoelectric (reflector, separator, suspension) layers. Forces acting on particles suspended in acoustic standing wave fields The primary acoustic force acting on an individual spherical particle with radius much smaller than the wavelength of sound, suspended in a planar standing. wave acoustic field, created by oppositely travelling, single frequency, sinusoidal sound waves is 13 Fac = 4rncR3Eac F sin (2xx)
(4)
where F is the acoustic contrast defined as l ( 5 p p - 2pf V = 3 \ 2pp + pf
7p~ ~/
Ultrasonics 1994 Vol 32 No 2
(5)
115
Ultrasonic and flow-field fractionation of suspensions: Z. Mandralis If F is positive, as is the case in the experiments presented later, particles are urged to the velocity antinodes of the standing wave pattern; if F is negative, the particles are collected at the velocity nodes. In addition to the primary acoustic force, other forces might arise due to diffraction effects and/or non-uniform generation of the acoustic field v'~3. Secondary acoustic forces 14 caused by the interaction of the secondary acoustic fields emitted from each particle to other particles may cause agglomeration of particles. Bernoullitype attraction forces also arise from the rapid oscillation of fluid between neighbouring particles as. Scaling arguments ~6 show that the primary acoustic force is at least two orders of magnitude larger than all other acoustic forces and hence these are to be neglected in the analytical model. For the micrometre-sized particles used in this study, inertia and Brownian forces can also be neglected. In order to judge the effectiveness of the E S U F F F for fractionation of fine particles, we compare the magnitude of acoustic forces to the gravitational force used in P F S F to move the particles within the separation chamber. The ratio of the maximum primary acoustic force to gravitational force is
Fac f g,
3xE~¢F (pp -- pf)g
Resonant frequencies and acoustic velocity profiles in acoustic resonators The acoustic force that acts on a particle suspended in a standing wave field depends on the acoustic energy density of the sound field and on the position of the particle relative to the velocity node. The general one-dimensional treatment of layered piezoelectric resonators iv was applied to determine the resonant frequencies and the acoustic velocity profiles in the acoustic chamber for each of the two steps of the E S U F F F cycle. This theory gives an exact solution of the fundamental equations of linear piezoelectricity subject to ascribed boundary conditions at the terminating surfaces and at the electrical ports of the resonator. The properties of the various layers, chosen to reflect those of the materials used in the experiments to be described later, are given in Table 1. Since only the longitudinal mode is excited electrically, only the elastic (e .E. . . ), dielectric (~s=) and piezoelectric (e==) constants affecting this mode are used. The constants of the transducer were determined by fitting the formula for the admittance for stiffened modes 18 to the measured admittance of the transducer in air 19 using many data points in the vicinity of resonance. In the simulation, the amplifier was replaced by an ideal voltage source with a 50 fl resistance in series. Figures 3 and 4 show the acoustic velocity profiles for the predicted resonant frequencies of 0.486 MHz and 1.385 MHz respectively. The positions of each member of this resonator were chosen to mimic the actual dimensions of the experimental chambers used in this study. Driving voltages of 1 V peak-to-peak were assumed for both cases. At the lower resonant frequency (Figure 3), a velocity antinode is located at the centre of the acoustic chamber (inside the midplane barrier) and the velocity profile exhibits some asymmetry. The velocity profile for the higher resonant frequency (Figure 4) shows greater symmetry with respect to the midplane, and two velocity antinodes which are positioned midway between the barrier and the walls. In our application, the separation efficiency is enhanced if the velocity antinodes are positioned exactly midway between the midplane barrier and the nearest wall during the high-frequency step. The lack of symmetry in the acoustic velocity profile during the low frequency step is not as critical to separation efficiency as long as the antinode is positioned within the midplane. Although we did not attempt to optimize the geometry to provide symmetric acoustic fields in both steps of the cycle, it should be possible to achieve this condition. The acoustic energy density used in Equation (4) for the acoustic force acting on the particle for each step of
(6)
For a typical polystyrene-water suspension (R = 3 ktm, pp -- pf = 50 kg m-3, F = 0.23), irradiated by 1 MHz, 1 0 0 J m -3 sound (~c=4234m-1), Equation (6) shows that the acoustic force is three orders of magnitude larger than the gravitational force. The speed at which particles are urged towards the collection plane, which can be found by equating the Stokes law drag on the particle with the primary acoustic force, is a sinusoidal function of the particles' distance from the velocity node (see Equation (4))
2tCEa~RZFsin (2Kx) up =
et al.
(7)
3/~
The maximum velocity at which a particle traverses the acoustic chamber under the influence of the acoustic field is about 0.58 mm s- 1 according to Equation (7). This is equivalent to sedimentation rates of equivalent polystyrene particles larger than 400 #m. Thus, utilizing acoustic fields in E S U F F F has a strong potential to extend the processing philosophy of P F S F to submicrometre sized particles or neutrally buoyant particles.
Table 1 Values of the material constants and thicknesses for the transducer, water, the steel separator, and the glass reflector used by the general one-dimensional treatment of layered piezoelectric resonators Transducer
c~zzz,(GPa)
123.3
e.... ( N V
16.7
1m-r)
+ j4.11
Water 2.2
+ j0.0011
Separator 286.3
-
-
+ j0.0573
Reflector 74.3
~sz
770 + j3.85
80 + j100
-
Thickness (mm)
2.03
0.54
0.076
0.9
4.7 + j0.018
Density (kg m -3)
7700
998
8087
2500
Frequency parameter (m s 1)
2310
742
2975
2725
30
2000
5000
1000
Quality factor
116
Ultrasonics
1994 Vol 32 No 2
+ j0.0743
Ultrasonic
and flow-field
U o (cbservatic~ chamber)
0.15 0.10
Transd ....
/
u wor in
~
/
r
~
Reflector
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Figure 3 Prediction of the acoustic velocity profile for the observation chamber at the lower frequency (0.486 MHz) assuming 1 V peak-to-peak driving voltage. The dotted line indicates the magnitude of the maximum acoustic velocity in the actual (clamped) working chamber if it was driven at 1 V peak-to-peak
the cycle can be calculated by integrating the kinetic and potential energy contributions of the sound field over the internal volume of the chamber. For a simple harmonic acoustic field, this yields pfU 2
(8)
4
y*
(1 lb)
(2x,/n)y
=
(12)
3XZQlt/n 2
=
c3x*
t~t*
Position (mm)
-
(1 la)
The height of the chamber is approximately equal to one half wavelength of the lower frequency used. In dimensionless form, the particle trajectories for the first step of the cycle are
~\
0.00
E.c
where Uo is the maximum acoustic velocity generated. Even though the acoustic velocity profiles given in Figures 3 and 4 are not perfectly harmonic, Equation (8) is believed to give a reasonable estimate of the acoustic energy density for the chambers used in this study. The results shown in Figures 3 and 4 can be extended to determine the acoustic velocity profiles under operating conditions other than for a 1 V peak-to-peak driving voltage by assuming proportionality of the acoustic velocity profile to the driving voltage. In this case, the acoustic velocity profiles can be found by multiplying the results in Figures 3 and 4 by the ratio of the actual driving voltage to 1 V peak-to-peak.
Process m o d e l The process model contains a description of particle trajectories due to the combined response of acoustic, gravitational and laminar flow fields. At steady-state, the force balance in the x-direction includes the drag force on the particle (described by the Stokes drag law) and the acoustic force given by Equation (4)
-
A sin (nx*)
-
4x*(1
(13a)
and 0y* 0t*
-
x*)
+
(13b)
B
For the second step, a sound field of higher frequency and possibly different intensity is applied. The trajectory equations in this case are 0x*
0t*
-
FflA sin (Fnx*)
(14a)
and ay* 0t*
- -46x*(1 - x*) + B
(14b)
Five dimensionless groups arise. Here, A is the ratio between the transverse speed of the particles due to acoustic forces and the lateral speed of fluid flow
A
-
41zEae'lR2F
(15)
9#Q1 while B is the ratio between the gravitational sedimentation speed of the particles and the lateral speed of fluid flow
B = 4nRE(pp
(16)
pf)g 27#xtQ1 - -
Also, F is the ratio of the higher to the lower frequency, fl is the ratio of the intensity of the acoustic field at the high frequency relative to that at the low frequency, and 6 is the ratio of the flow rate of the second step to the flow rate of the first step.
0.20
-
et al.
x* = (2xl/n)x t*
0.05
<
o f s u s p e n s i o n s . Z. M a n d r a l i s
frequency step
0.20
o ~
fractionation
6rc#R~tt + 4~zR3xEacFsin (2xx) = 0
The motion of the particle in the y-direction, in each of the subvolumes, is governed by the drag force due to the fluid flow and the gravitational force (particle-wall and particle-particle interactions are neglected)
O.15
+ mg=O
(10)
where uf is the local flow velocity given by Equation (1) and gravity is arbitrarily defined to be in the positive y-direction. For convenience, these trajectory equations are cast in a dimensionless form. The characteristic length is based on the wave-number at the lower frequency and the characteristic time on the flow speed during the low
~3 U o (observation chamber)
0o 0.10 o
8
6z#R u f -
Reflector
Transducer
(9)
0.05 U o (w~king chamber) -'T'-~-'~
0.00
0.0
0.5
,~.---'~-~,,,, , , , 10
1.5
2.O
Position
2.5
2
B.O
3.5
4.0
(mm)
Figure 4 Prediction of the acoustic velocity profile for the observation chamber at the higher frequency (1.387 M Hz) assuming 1 V peak-to-peak driving voltage. The dotted line indicates the magnitude of the maximum acoustic velocity in the actual (clamped) working chamber if it was driven at 1 V peak-to-peak
Ultrasonics 1994 Vol 32 No 2
117
Ultrasonic and flow-field fractionation of suspensions." Z. Mandralis et al. 0.40 t, \ 0.35 '
t~ =82.86
~x 1.25
030 '\ \,
~.75
a cut 0.°25 150.20 I \' :'\'~ ~ ~ _ ~ ,
i - - ~
~-~-h~ ~ ~
2.752.25
0.10
3.25
0.05
X~g
0.00 O. 10
Figure 5
= 0.30
~ 0.50
= 0.90
'
0.70
1.10
Sample model predictions forNoot as a function of//and e
0.40 0.35 \\ \ 030 \ A cut 0 2 5
020
t =82.86 c~
\ k\
\
\\
015 010
\
125 3.75 ' " ~-_ -_
125 '~SLJFFF] -'-''--- _ _ -_
005 000 0.10
0.30
0.50
070
090
110
Figure 6 Comparison of the fractionation potential for ESUFFF and SUFFF
Model predictions A numerical integration method 2° was used to solve the system of Equations (13) and (14). The trajectories of a large number of particles, initially distributed uniformly over the cross-section of the channel at y * = 0, were traced. The average displacement of the group of particles at the end of the two-step cycle is calculated as the arithmetic mean of the individual displacements. The case of negligible gravity (B = 0) will be presented here. The effect of gravity on the fractionation performance has been discussed in relation to the S U F F F method 1°. After simulating about five cycles, for fixed values of ~, /3, F and 6, a unique A (denoted Acut) can be identified, by a trial-and-error procedure, for which the average displacement of particles over one cycle is zero. Particles with A > Aeut have displacements of opposite direction from particles with A < Ac,t. As an example, consider Figure 5 in which A~ut is shown as a function of/3 for various values of the parameter and for t* = 82.86. As ~ decreases, Aeut decreases. The production rate, given by Equation (3), also decreases. As/3 increases, Aeut decreases because particles respond faster to higher energy density acoustic fields. F r o m Figure 5 it is apparent that A¢,t is most sensitive to fl in the range 0.1 < fl < 0.5. Next, we compare the efficiency of E S U F F F relative to S U F F E in Fioure 6. At equivalent production rates (~ = 1.25 in this example) the value of Acre is much lower in E S U F F F than in S U F F F indicating its improved capability for handling smaller particles. Also, it is
118
Ultrasonics 1 9 9 4 Vol 32 No 2
possible to compare production rates achievable by the two methods in accomplishing a given separation task. F r o m Figure 6 it can be seen that approximately the same Aout is obtained for ~ = 3.75 in E S U F F F as for = 1.25 in SUFFF. For a cycle operated with 6 = 1, Equation (3) leads to the result that the production rate in E S U F F F would be 5.2 times as high as in SUFFF.
Experimental procedure Proof-of-principle experiments were performed in the acoustic chamber shown in Figure 7. A rectangular lead zirconate titanate transducer (APC 880, 38.1 × 25.4 × 2.03 mm) supplied by the APC company was supported on a rubber gasket (Neoprene 0.81 mm) in an aluminium frame. A rectangular glass reflector (38.1 × 25.4 x 0.9 mm) was similarly placed in another aluminium frame. Two Teflon spacers (0.533 m m each) were placed between the two frames. Their central sections were cut out so that when the whole assembly was clamped together a sealed rectangular channel was defined. A thin stainless steel sheet (0.076mm) was placed between the Teflon spacers so that it separated the rectangular channel into two identical subvolumes. At the two ends of the channel, ports allowed feed introduction and product removal with the help of a computerized peristaltic pump (Masterflex 7550-90). The electrical excitation for the transducer was generated by a K R O H N - H I T E 2100A generator and amplified by an E N I 240L power amplifier. The electrical energy input was measured by a B I R D 4410A wattmeter. The experiment was controlled by an AT-type computer. The low and high resonant frequencies were experimentally determined to be 0.481 M H z and 1.328 MHz. To avoid the complications of gravity, the chamber was oriented such that fluid flow was in the vertical direction with feed introduced at the bottom of the chamber and the product stream withdrawn from the top. Particles smaller than the cut size, emerging at the product stream, are collected and analysed. Particles larger than the cut size concentrate at the bottom of the acoustic chamber and are removed at the end of the experimental run. A purge stream for removal of particles from the bottom of the chamber would be required for truly continuous operation. However, the experimental procedure followed closely resembles continuous operation for low feed concentrations and short operation times. The particle size distribution and particle number density of the product stream was determined by placing a small sample of the stream between two microscope
FEED
,J, v
2ASKET
TEFLON
~V////////////////////A~ | /
<
Gravity
TRANSDUCEIR ~
I1<
38.1 mm
l
>1 /~z PRODUCT
Figure 7 Schematic of the fractionation chamber used in the experiments
Ultrasonic and flow-field fractionation of suspensions. Z. Mandralis et al.
E x p e r i m e n t a l results - c o m p a r i s o n s w i t h theory
(a)
(b)
(c) Figure 8 Photographs illustrating the acoustic response of particles: (a) suspension before the application of ultrasound; (b) collection of particles on the separator during the low frequency step; (c) collection into planes located midway between the separator and the walls during the high frequency step
slides that were spaced 0.076 mm apart. The CUE-2 image analysis software installed on an AT computer and an optical microscope were used to analyse the particle size distribution. For each measurement, a large number of particles was analysed in order to get statistically meaningful results. The same procedure was followed for determining the characteristics of the feed stream. Using this method, particles larger than 2 #m were accurately detected and analysed.
In order to confirm that resonant acoustic fields could be used to drive particles toward the midplane barrier and to positions midway between the barrier and a wall, a second cell was constructed for visualization purposes. It incorporated an acoustic transducer and a reflector similar to those used by the chamber described above. The reflector was attached on a micrometer controlled stage so that precise positioning ( _+5/~m) was achieved. The side walls were made of Plexiglas that permitted direct observation through the chamber. The first of the photographs of Figure 8 shows a uniform suspension of 325 Mesh polystyrene particles between the transducer and the reflector. The thin stainless steel separator can also be seen at the midplane. The second photograph demonstrates the collection of the particles on the separator, after irradiation with 0.486 MHz sound. The third photograph illustrates the collection of particles midway between the centre and the walls of the acoustic chamber, after irradiation with 1.387 MHz sound. For the working separation cell in which the fractionation was attempted, the experimentally determined resonant frequencies (0.481 and 1.328 MHz) for the two-step cycle are slightly lower than the resonant frequencies predicted by the general one-dimensional treatment of piezoelectric resonators. This is attributed to the fact that the constants for the transducer were obtained by measuring the admittance of the transducer in the observation cell and not in the working chamber. Unlike the transducer of the observation cell, the transducer of the working chamber is tightly clamped. Also, there is some uncertainty in the dimensions of the various layers within the composite resonator. The use of a conventional hydrophone to measure acoustic energy density was not possible due to the small dimensions of the acoustic chamber. One way to calculate the acoustic energy density is by using Equation (8) in conjunction with the acoustic velocity profiles from Figures 3 and 4 and a measurement of the driving voltage as discussed previously. However, the results shown in Figures 3 and 4 were obtained for the observation chamber, which had an unclamped transducer, while the transducer of the working chamber was clamped. This clamping is expected to reduce the intensity of the sound field. An alternative method, developed in our laboratories, was used to measure the acoustic energy density in the actual chamber 2°. According to this method, a micrometre-sized sphere is suspended in the standing wave field of the actual chamber used for the fractionation experiments, and its equilibrium position is precisely determined using a microscope. By balancing the acoustic force predicted by Equation (4) with the buoyant weight of the particle, the acoustic energy density can be determined to within 20%. The values of acoustic energy density determined this way were used for comparisons of the experimental results with theoretical predictions. To visualize the effects of clamping on the energy density in the chamber, Equation (8) was used to calculate U o in the working chamber from the value of Eac determined from the levitated sphere measurements described above. Again, assuming proportionality of the acoustic velocity profile to the driving voltage, it is possible to calculate the magnitude of U o expected if a
Ultrasonics 1994 Vol 32 No 2
119
Ultrasonic and flow-field fractionation of suspensions: Z. Mandralis T a b l e 2 Particle number density and size distribution in the product stream as a function of electric energy input, c~-1.25, -0.16. The table entry for 0 W indicates the statistics of the process feed Power input (W)
Particle concentration (cm 3)
Mean radius (#m)
Standard deviation (/zm)
0 0.25 1.05 2.40 3.60
2 244,094 980,314 318,897 297,900 212,598
10.0 6.76 4.54 3.82 3.23
5.50 3.82 2.40 2.11 1.49
O
et al.
stream for the same experimental conditions used in Table 2. The vertical dashed line is the theoretical prediction from the model described in the previous section. The agreement between the experiments and the model are generally good; however, Figure 9 shows that the experimentally achieved fractionation is not as sharp as the theoretical prediction. In all cases, few particles larger than the theoretically predicted cut size emerge at the product stream. On the other hand, many of the smaller particles that should, theoretically, emerge at the product stream, were trapped in the acoustic chamber. This is a result of the secondary forces that cause agglomeration of smaller particles into clusters, which then behave as large particles. For the higher energy densities used, there are some particles in the 5-10 #m range that emerge at the product stream. This can be explained by the fact that smaller particles are only slightly affected by the combined effects of the alternating sound field and the bidirectional flow field. Thus, small flow disturbances due to wall and edge effects or deviations from the uniform, planar standing wave field configuration might change the final destination of these particles.
Conclusions
o
6
1o
16
20
Particle
2~
~
Radius
e
~
1o
15
2o
2~
m
(¢~,m)
Figure 9 Product stream particle size distributions obtained for several values of Eat,1 with ~ = 1.25 and /~ = 0.16. Also shown in each graph are the particle size distributions for the polystyrene suspension fed to the chamber. The vertical dashed lines represent the theoretical cut particle size predicted from the model
1 V peak-to-peak driving voltage was used. These values are shown on Figures 3 and 4 as dotted lines. Note that values of U o for the working (clamped) chamber are roughly one-third of the predicted values for an unclamped chamber. Several experiments were performed to demonstrate the effectiveness of the fractionation technique and obtain data to complement the theoretical development. The ratio of the acoustic energy density of the second step to that of the first step was set arbitrarily to fi ~ 0.16 for all the experimental runs. The flow rate was fixed at Q1 = Q2 = 7.36 x 1 0 - 6 m 2 s -1. Suspensions containing 2% w/w of 325 mesh polystyrene particles in distilled water, which contained about 0.01% w/w sodium dodecyl sulfate (SDC) to prevent particle agglomeration, were fractionated. Experiments were performed at a = 1.25 (tl = 4.13 s, t 2 ---- 5.16 s) which resulted in Q p r o d : 0.81 X 10- 6 m 2 s- 1. The electric energy input varied in the range 0.2-3.7 W. Table 2 shows the particle number density, mean particle radius and standard deviation for the feed and product streams, at steady-state, for various energy density levels. The number of particles in the product stream as well as the mean particle radius and standard deviation decrease as the acoustic energy density increases, in agreement with theoretical expectations. A more detailed picture of the effect of the energy density on the particle cut size is given in Figure 9, which shows the particle size distribution of the feed and the product
120
Ultrasonics
1994 Vol 32 No 2
A fractionation method for fine particles in liquid suspensions has been developed. The fractionation is based on the transient response of fine particles to alternating low and high frequency ultrasonic fields. The separation strategy is similar to that of PFSF. However, substitution of the gravitational feld with more powerful and controllable acoustic fields enables sharp fractionation of micrometre-sized and neutrally buoyant particles. Sharp fractionation of micrometre-sized particles was demonstrated for a 325 mesh polystyrene suspension. The sharpness of fractionation can be further increased by increasing the number of separation stages. For a given device, this can be achieved without hardware changes by reducing the volume of liquid displaced in each step of the cycle. For a given separation task, higher production rates than S U F F F can be achieved by ESUFFF. The method can be scaled up by constructing chambers in which several transducers are combined into a mosaic structure. In addition, the distance between the transducer and the reflector can be several wavelengths, thus increasing the active volume of the acoustic chamber. Thin or acoustically transparent layers can subdivide the chamber into many parallel channels serviced by a single generator, amplifier and pumping mechanism. For general computer-aided design of separation chambers useful for ESUFFF, the general one-dimensional treatment of piezoelectric resonators can be used. However, the complicating effects of clamping on the cell make the acoustic energy density generated within a working chamber more difficult to predict. Instead, a simple method for determining the acoustic energy density in the operating chamber was used. This was based on a balance ~)f the acoustic and gravitational forces acting on a microsphere suspended in the sound field. The predictions were generally in agreement with the experimental results.
Acknowledgements Nestle has provided partial support for this research through a Westreco Fellowship. The authors also wish
Ultrasonic and flow-field fractionation of suspensions. Z. Mandralis et al.
to thank the Austrian Science Foundation (project P7198-PHY) and the Erwin Schroedinger Society for financial support.
10 11
12
References 1 2 3 4 5 6 7 8 9
Tolt, T.L. and Feke, D.L. Analysis and application of acoustics to suspension processing ASME. 23rd lntersociety Energy Conversion Engineering Conference (1988) 4 327 Tolt, T.L. and Feke, D.L. Separation of dispersed phases from liquids in acoustically driven chambers Chem En O Sci (1993) 48 527-540 Tolt, T.L. and Feke, D.L. Separation devices based on forced coincidence response of fluid-filled pipes J Acoust Soc Am (1992) 91(6) 3152-3156 Hager, F., Benes, E., Bolek, W. and Gr6sehl, M. Investigation of a new ultrasonic drifting resonance field cell for the refinement of aerosols Proc 9th FASE Symposium Hungary (1991) 261-266 Whitworth, G., Grundy, M.A. and Coakley, W.T. Transport and harvesting of suspended particles using modulated ultrasound Ultrasonics (1991) 29 439-444 Schram, C.J. US Patent No. 4,743,361 (1988) Schram, C.J. and Rendell, M. Manipulation of particles in megahertz standing waves Proceedings Ultrasonics International Conference (1989) 262-267 Peterson, S.C. US Patent No. 4,759,775 (1988) Mandralis, Z.I. and Feke, D.L. Continuous suspension fractionation
13 14 15 16 17 18 19 20
using acoustics and divided flow fields Chem Eng Sci (1993) 48 3897-3905 Mandralis, Z.I. and Feke, D.L. Fractionation of fine particle suspensions by synchronized ultrasonic and flow fields AIChE J (1993) 39 197-206 Adler, R.J., Palepu, P.T. and Wu, C.K. Periodic flow slurry fractionation Chemeca '88 (1988) 892-897
Papann, J.S., Adler, R.J., Gorensek, M.B. and Menon, M.M. Separation of fine particle dispersions using periodic flows in a spinning coiled tube AIChE J (1986) 32 798-808 Yosioka, K. and Kawasima, Y. Acoustic radiation pressure on a compressible sphere Acustica (1955) 5 167-173 Apfel, R.E. Acoustically induced square law forces and some speculations about gravitation Am J Phys (1988) 56 726-729 Weiser, M.H., Apfel, R.E. and Neppiras, E.A. Interparticle forces on red cells in a standing wave field Acustica (1984) 56 114-119 Mandralis, Z.I., Feke, D.L. and Adler, R.J. Transient response of fine-particle suspensions to mild, planar, ultrasonic fields Fluid/ Particle Sepn J (1990) 3 115-121 Nowotny, H., Benes, E. and Sehmid, M. Layered piezoelectric resonators with an arbitrary number of electrodes (general one-dimensional treatment) J Acoust Soc Am (1991 ) 90 1238-1245 IEC Standard 483 Guide to Dynamic Measurements of Piezoelectric Ceramics with High Electromechanical Coupling (1976) Geneve, Switzerland Schmid, M., Benes, E. and Sedlaszek, R. A computer controlled system for the measurement of complete admittance spectra of piezoelectric resonators Meas Sci Technol (1990) 1 970-975 Mandrafis, Z.I. Fractionation of fine-particle suspensions by combined ultrasonic and flow fields. PhD Dissertation (1993) Case Western Reserve University, Cleveland; Ohio, USA
Ultrasonics 1994 Vol 32 No 2
121